tag:blogger.com,1999:blog-80495708156258963032024-03-14T05:52:38.727-04:0017 Frequency Response of Amplifiers - conocimientos.com.veFrequency Response of Amplifiers. Concept of frequency response, Human ear response to audio frequencies, significance of Octaves and Decades. The decibel unit. Square wave testing of amplifiers, Miller's theorem. Effect of coupling, bypass, junction and stray capacitances on frequency response for BJT and FET amplifiers. Concept of dominant pole.Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.comBlogger28125tag:blogger.com,1999:blog-8049570815625896303.post-67341363409793938212010-03-21T10:50:00.002-04:302010-03-23T17:49:47.772-04:30Frequency compensation<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In electrical engineering, frequency compensation is a technique used in amplifiers, and especially in amplifiers employing negative feedback. It usually has two primary goals: To avoid the unintentional creation of positive feedback, which will cause the amplifier to oscillate, and to control overshoot and ringing in the amplifier's step response.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Explanation</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Most amplifiers use negative feedback to trade gain for other desirable properties, such as decreased distortion or improved noise reduction. Ideally, the phase characteristic of an amplifier's frequency response would be constant; however, device limitations make this goal physically unattainable. More particularly, capacitances within the amplifier's gain stages cause the output signal to lag behind the input signal by 90° for each pole they create.[1] If the sum of these phase lags reaches 360°, the output signal will be in phase with the input signal. Feeding back any portion of this output signal to the input when the gain of the amplifier is sufficient will cause the amplifier to oscillate. This is because the feedback signal will reinforce the input signal. That is, the feedback is then positive rather than negative.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency compensation is implemented to avoid this result.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Another goal of frequency compensation is to control the step response of an amplifier circuit as shown in Figure 1. For example, if a step in voltage is input to a voltage amplifier, ideally a step in output voltage would occur. However, the output is not ideal because of the frequency response of the amplifier, and ringing occurs. Several figures of merit to describe the adequacy of step response are in common use. One is the rise time of the output, which ideally would be short. A second is the time for the output to lock into its final value, which again should be short. The success in reaching this lock-in at final value is described by overshoot (how far the response exceeds final value) and settling time (how long the output swings back and forth about its final value). These various measures of the step response usually conflict with one another, requiring optimization methods.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency compensation is implemented to optimize step response, one method being pole splitting.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkUDYYp0ykrZi79lyw_h8ErNs8QU0z4RbJYQk2xJToQ7ORNCWswH-REbjEnjP-PF_MVBdKez5pXA-TfK1XhjMmEj2-oAyL18KKnNrh4eC4zl5pOhLzpA7Sh_b8YN2wxI7XZKtkPpUdvdY/s1600-h/w1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451079878098644578" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkUDYYp0ykrZi79lyw_h8ErNs8QU0z4RbJYQk2xJToQ7ORNCWswH-REbjEnjP-PF_MVBdKez5pXA-TfK1XhjMmEj2-oAyL18KKnNrh4eC4zl5pOhLzpA7Sh_b8YN2wxI7XZKtkPpUdvdY/s320/w1.bmp" style="cursor: pointer; height: 145px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Use in operational amplifiers</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Because operational amplifiers are so ubiquitous and are designed to be used with feedback, the following discussion will be limited to frequency compensation of these devices.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">It should be expected that the outputs of even the simplest operational amplifiers will have at least two poles. An unfortunate consequence of this is that at some critical frequency, the phase of the amplifier's output = -180° compared to the phase of its input signal. The amplifier will oscillate if it has a gain of one or more at this critical frequency. This is because (a) the feedback is implemented through the use of an inverting input that adds an additional -180° to the output phase making the total phase shift -360° and (b) the gain is sufficient to induce oscillation.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A more precise statement of this is the following: An operational amplifier will oscillate at the frequency at which its open loop gain equals its closed loop gain if, at that frequency,</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">1. The open loop gain of the amplifier is ≥ 1 and</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">2. The difference between the phase of the open loop signal and phase response of the network creating the closed loop output = -180°. Mathematically,</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">ΦOL – ΦCLnet = -180°</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Practice</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency compensation is implemented by modifying the gain and phase characteristics of the amplifier's open loop output or of its feedback network, or both, in such a way as to avoid the conditions leading to oscillation. This is usually done by the internal or external use of resistance-capacitance networks.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">[edit]Dominant-pole compensation</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The method most commonly used is called dominant-pole compensation, which is a form of lag compensation. A pole placed at an appropriate low frequency in the open-loop response reduces the gain of the amplifier to one (0 dB) for a frequency at or just below the location of the next highest frequency pole. The lowest frequency pole is called the dominant pole because it dominates the effect of all of the higher frequency poles. The result is that the difference between the open loop output phase and the phase response of a feedback network having no reactive elements never falls below −180° while the amplifier has a gain of one or more, ensuring stability.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Dominant-pole compensation can be implemented for general purpose operational amplifiers by adding an integrating capacitance to the stage that provides the bulk of the amplifier's gain. This capacitor creates a pole that is set at a frequency low enough to reduce the gain to one (0 dB) at or just below the frequency where the pole next highest in frequency is located. The result is a phase margin of ≈ 45°, depending on the proximity of still higher poles.[2] This margin is sufficient to prevent oscillation in the most commonly used feedback configurations. In addition, dominant-pole compensation allows control of overshoot and ringing in the amplifier step response, which can be a more demanding requirement than the simple need for stability.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Though simple and effective, this kind of conservative dominant pole compensation has two drawbacks:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">1. It reduces the bandwidth of the amplifier, thereby reducing available open loop gain at higher frequencies. This, in turn, reduces the amount of feedback available for distortion correction, etc. at higher frequencies.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">2. It reduces the amplifier's slew rate. This reduction results from the time it takes the finite current driving the compensated stage to charge the compensating capacitor. The result is the inability of the amplifier to reproduce high amplitude, rapidly changing signals accurately.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Often, the implementation of dominant-pole compensation results in the phenomenon of Pole splitting. This results in the lowest frequency pole of the uncompensated amplifier "moving" to an even lower frequency to become the dominant pole, and the higher-frequency pole of the uncompensated amplifier "moving" to a higher frequency.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Other methods</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Some other compensation methods are: lead compensation, lead–lag compensation and feed-forward compensation.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Lead compensation. Whereas dominant pole compensation places or moves poles in the open loop response, lead compensation places a zero[3] in the open loop response to cancel one of the existing poles.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Lead–lag compensation places both a zero and a pole in the open loop response, with the pole usually being at an open loop gain of less than one.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Feed-forward compensation uses a capacitor to bypass a stage in the amplifier at high frequencies, thereby eliminating the pole that stage creates.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The purpose of these three methods is to allow greater open loop bandwidth while still maintaining amplifier closed loop stability. They are often used to compensate high gain, wide bandwidth amplifiers.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Dominant Pole approximation</span></span></b></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Reduction of a second order system to first order</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Consider a second order system with a transfer function that is reduced to first order.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3HPKPstCdXvLocJOgynLUxkDfcBmky1k14loS-SjV6msogAxT97f_8Jje_aiym-aHPjEo9uHI_4bP8ETNESMAjnjO1hVLJo2hXtyJcp5pMeQ3pFzqG55XXe98Ql6Kxtw9vzAitKzi7Sw/s1600-h/w2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451082588727445986" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3HPKPstCdXvLocJOgynLUxkDfcBmky1k14loS-SjV6msogAxT97f_8Jje_aiym-aHPjEo9uHI_4bP8ETNESMAjnjO1hVLJo2hXtyJcp5pMeQ3pFzqG55XXe98Ql6Kxtw9vzAitKzi7Sw/s320/w2.bmp" style="cursor: pointer; height: 46px; width: 231px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">This assumes that a>>b, or that the pole at b is dominant. The coefficient "a" remains in the denominator so that the DC gain (which is also the final value of the output with a unit step input) remains unchanged. Recall that the DC gain is G(0).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The graph below shows the exact response (red) and the dominant pole approximation (green) for a=8 and b=1. Following the graph is Matlab code in which you can set a with b=1 to see how accurate the dominant pole approximation is.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><i>Example</i></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs9QdkaQb6iGjBeMbu4B6iTbT58eqJCcCFrA2bYVCzig2Q6lVzZVTdcg4CuFIV6PuEIxRxH4dJHTlY-yFnj3vbvAaTqePKY31HekqXb6WdX0obYH4ZgP7iymXNMv2wd0HKtNT0QNL-XA4/s1600-h/w3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451082846243644482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs9QdkaQb6iGjBeMbu4B6iTbT58eqJCcCFrA2bYVCzig2Q6lVzZVTdcg4CuFIV6PuEIxRxH4dJHTlY-yFnj3vbvAaTqePKY31HekqXb6WdX0obYH4ZgP7iymXNMv2wd0HKtNT0QNL-XA4/s400/w3.bmp" style="cursor: pointer; height: 305px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Higher Order</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The dominant pole approximation can also be applied to higher order systems. Here we consider a third order system with one real root, and a pair of complex conjugate roots.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrX0SgTNhzuhIuHF3NWi05FaPjAbByzu25bwUIlQVbDMQBodxpX8IqZUIRnC-P37R5-ib9cON4YBsyXKLgp8KIvDC601gAFNPaGJ66qPxW9v4KRlilaPdVzRvSy7foTwrnGLqV6lSK77c/s1600-h/wn.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451083151233463458" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrX0SgTNhzuhIuHF3NWi05FaPjAbByzu25bwUIlQVbDMQBodxpX8IqZUIRnC-P37R5-ib9cON4YBsyXKLgp8KIvDC601gAFNPaGJ66qPxW9v4KRlilaPdVzRvSy7foTwrnGLqV6lSK77c/s400/wn.bmp" style="cursor: pointer; height: 86px; width: 349px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In this case the test for the dominant pole compare "a" against "zwn". This is because "zwn" is the real part of the complex conjugate root (we only compare the real parts of the roots when determining dominance because it is the real part that determines how fast the response decreases). Note that the DC gain of the exact system and the two approximate systems are equal.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In the examples and Matlab code below, the second order pole has zeta=0.4 and wn=1 (which yields roots with a real part of 0.4 and an imaginary part of +/-0.92j). There are three graphs. In the first graph a=0.1 (the real pole dominates), in the second graph a=4 (the complex conjugate poles dominate) and in the third graph a=0.4 (neither dominates and the response is obviously more complicated than a simple second order response). In all three graphs the exact response is in red, the approximate response in which the first order pole dominates is in green, and the approximate response in which the second order pole dominates is in blue.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><i>Examples:</i></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsNUbfEcvhaAoUkubbe0ISV2GawZx-twMC_zvN3GVRMPPOiQl2WpLcR98nFLg6izPyf3ExVE0s44cAthLylQnCwS1Xavg8UK30qVIbWteaGYgLglrFVix7QUw4f_5DDo0Xx08xbzDIaL4/s1600-h/w4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451083427421890434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsNUbfEcvhaAoUkubbe0ISV2GawZx-twMC_zvN3GVRMPPOiQl2WpLcR98nFLg6izPyf3ExVE0s44cAthLylQnCwS1Xavg8UK30qVIbWteaGYgLglrFVix7QUw4f_5DDo0Xx08xbzDIaL4/s400/w4.bmp" style="cursor: pointer; height: 400px; width: 329px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></u></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ-0k5zRdrbFY0mMlcuW6So7pEp3n6uXKKHfnpeMXQ8fnZVJ8eW3M6TtuSqlrbpzzG58VcyYSiUABm2rT-lQZ_1hnyYoLMJ-XuRCDsK-Y45ca-6ZhF5t8gT6n6B5Ob80EY-u_-RFTyoiQ/s1600-h/w5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451083667556435890" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ-0k5zRdrbFY0mMlcuW6So7pEp3n6uXKKHfnpeMXQ8fnZVJ8eW3M6TtuSqlrbpzzG58VcyYSiUABm2rT-lQZ_1hnyYoLMJ-XuRCDsK-Y45ca-6ZhF5t8gT6n6B5Ob80EY-u_-RFTyoiQ/s400/w5.bmp" style="cursor: pointer; height: 213px; width: 363px;" /></a></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span" style="font-size: small;"><b>Lenny Z Perez M</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span" style="font-size: small;"><b>EES</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/frequency-compensation.html"><span class="Apple-style-span" style="font-size: small;"><b>http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/frequency-compensation.html</b></span></a></span></div></div></span> <br />
<hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href="http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us" target="_new">Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-1256277275696106092010-03-21T10:48:00.002-04:302010-03-23T17:49:30.935-04:30Understanding Speaker Frequency Response<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;">The Secret Behind The Industry's Most-Cited Spec.</span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Here's a quick quiz: which of these two speakers sounds better: Speaker A with a frequency response range of 45Hz to 18kHz or, Speaker B with a range of 20Hz to 25kHz? The truth is there's simply not enough data in these numbers to know anything of value. Taken out of context and without other data, a simple set of numbers don't tell you much about real world sound quality. But people make audio buying decisions based on published specifications, such as the frequency response spec, everyday. I'd like to demystify the process for you; let you in on a little industry secret about "The Frequency Response Spec."</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>My Frequency Response</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Frequency Response specification attempts to describe the range of frequencies or musical tones a speaker can reproduce, measured in Hertz (known to old-timers as "Cycles per Second"). The range of human hearing is generally regarded as being from 20Hz, very low bass tones, through 20kHz (20,000Hz), the very highest treble. Presumably a speaker that could reproduce that range would sound lifelike. Alas, it is no guarantee. The most important determinant of a speaker's frequency performance is not its width or range, but whether it's capable of reproducing all the audible frequencies at the same volume at which they were recorded.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">You don't want the speaker to change the "mix" of tones; that would ruin the timbre of voices and instruments, making them sound unnatural. Ideally, you want the sounds that are on the recording to be reproduced as they were recorded, without the speaker changing the sound. To say it another way: if you made a recording of all the audible tones at the same volume and played that recording through a speaker, you'd want all the audible tones to come out at the same volume. In fact, that's one way of measuring speakers. A signal that's comprised of all frequencies at equal volume is fed into a speaker that sits in a room with no reflective surfaces. A calibrated microphone is placed in front of the speaker and feeds the speaker's output into a machine that plots the frequency vs. amplitude as shown in Figure A.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBVhWeDqrB4ptaYZ4O-uWxMP_A34GpFo_wGNxl9NgtBz5MQsPvla_n5IWi_00uM4flaDUSw-eMu9vyFYHiBrWEjQjtxLlEpiVzJS_XOfgN3KOlxghPpJBkPgxgXVPiDnmxSaMZtPxtTas/s1600-h/dos1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450904851576565938" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBVhWeDqrB4ptaYZ4O-uWxMP_A34GpFo_wGNxl9NgtBz5MQsPvla_n5IWi_00uM4flaDUSw-eMu9vyFYHiBrWEjQjtxLlEpiVzJS_XOfgN3KOlxghPpJBkPgxgXVPiDnmxSaMZtPxtTas/s320/dos1.jpg" style="cursor: pointer; display: block; height: 224px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Now take a look at the graph in Figure B. That's the frequency response of the Erehwon Model 10, with drivers and tweeters made of pure Unobtainium ("Half the carbs, all the sound!"). The flat line on the graph indicates that the speaker is "flat"; it reproduces all the musically relevant tones at the same volume. That doesn't mean that a "flat" speaker will play all recorded sounds at the same volume -- bear with me here -- it means that it will treat all sounds equally; it won't impose its will on the music but will allow you to hear the music as it was recorded. Flat is good. Flat response means that the speaker reproduces sound accurately.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqK3rtTim_y0z0DXcIyk6NH_ByTZs2qp7UQSPkiaHElvDM2_BG7l-o3MFJPzxP9W07xpnM50nFt9uKnoiHX0hhyqDQZNox8VKb7KpteNEpRjUv01x43Nn3vPDg62JXi43zZAkvppyk7JM/s1600-h/dos2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450905054932368162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqK3rtTim_y0z0DXcIyk6NH_ByTZs2qp7UQSPkiaHElvDM2_BG7l-o3MFJPzxP9W07xpnM50nFt9uKnoiHX0hhyqDQZNox8VKb7KpteNEpRjUv01x43Nn3vPDg62JXi43zZAkvppyk7JM/s320/dos2.jpg" style="cursor: pointer; display: block; height: 224px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
<div><div style="text-align: center;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Too bad that the Erehwon Model 10 doesn't really exist, and neither does Unobtainium. Today's technologies allow speaker designers to get closer to the "flat" ideal than ever before, but they still fall far short of "perfection." So if a frequency range spec is not adequate, what is?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency Response In Context</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A big improvement would be a frequency response number that also includes the amplitude tolerance, expressed as "XHz-YkHz +/- 3dB." This tells you that the amplitude of the speaker's response relative to frequency does not deviate more than 3 Decibels from the center line. The "plus or minus 3dB" spec is regarded as a standard of sorts. The theory is that 3dB differences are "just perceptible," so a speaker whose response curve lies within that tolerance window is a reasonably accurate speaker. Let's see if that idea holds water.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><u><br />
</u></span></div></div></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGXKvVnfoAj1fSHwcv_IkMzf0aErbb6vjomV7UDm3G-nzcwO5VSnqagLGoXM6h_INo90xyaiYS-QMvrYgYf6EG-kyqiYvsXductdK7gReGr7Am4Lz2xM9M0QxUWF0_Ksa1tkCsvOrnLEI/s1600-h/dos3.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450905558833372114" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGXKvVnfoAj1fSHwcv_IkMzf0aErbb6vjomV7UDm3G-nzcwO5VSnqagLGoXM6h_INo90xyaiYS-QMvrYgYf6EG-kyqiYvsXductdK7gReGr7Am4Lz2xM9M0QxUWF0_Ksa1tkCsvOrnLEI/s320/dos3.jpg" style="cursor: pointer; display: block; height: 224px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Take a look at Figure C. This speaker has response that can be specified as 20Hz-20kHz +/- 3dB. Take a look at Figure D; it, too, can have the exact same specification as Speaker C! Do you think they will sound similar? NOT! They won't sound even remotely like one another. Speaker C will have "one note" bass and will make voices and other instruments sound unnatural, but Speaker D will sound smooth and more natural.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge4U3m5eAknlp-1fP_HRVabrSPr78Xm0yWdPmFlXeEC3RX-rmZ6tynoeCQ5xMBv8-Uf7hx0p-tV5NZ7zCJ_xfJEuQF3uSelTlrLkGBt_cpyI5vhE_qaMwAawqbgGifczymlzKsVpc5rW4/s1600-h/dos4.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450906325418711666" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge4U3m5eAknlp-1fP_HRVabrSPr78Xm0yWdPmFlXeEC3RX-rmZ6tynoeCQ5xMBv8-Uf7hx0p-tV5NZ7zCJ_xfJEuQF3uSelTlrLkGBt_cpyI5vhE_qaMwAawqbgGifczymlzKsVpc5rW4/s320/dos4.jpg" style="cursor: pointer; display: block; height: 224px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
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</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If I had to choose strictly by the response curves, I'd choose speaker D because its amplitude variations are smoother and gentler. In contrast, speaker C's amplitude variations are more extreme and "spikey." Experience has shown speaker designers that those rapid changes in response produce a sound that is more fatiguing, less pleasing and subjectively less accurate.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><br />
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<div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrtojOt7vkAzpZdA-SeFp-SgE-SGM1m25ZUXcF85oQ-d440WgXFW9odDneSkBjTF0sWs6XBlEi7u3k_6OBE05x68Yozi24rFEL_dqBBfWrn8lHDoTGyg48_TSjt46EEKZgfsyhYV76LOE/s1600-h/dos5.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450906501443837874" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrtojOt7vkAzpZdA-SeFp-SgE-SGM1m25ZUXcF85oQ-d440WgXFW9odDneSkBjTF0sWs6XBlEi7u3k_6OBE05x68Yozi24rFEL_dqBBfWrn8lHDoTGyg48_TSjt46EEKZgfsyhYV76LOE/s320/dos5.jpg" style="cursor: pointer; display: block; height: 224px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Now look at the response of the speaker in Figure E. This speaker exhibits a smooth response curve with low amplitude variations so you'd expect a fairly natural sound; however, the bandwidth of these errors is very broad, and experience has shown us that even low volume variations are audible if they cover a broad range of frequencies. In this case, Speaker E would have rich bass, prominent treble and be somewhat recessed or "laid back" in the midrange. Audiophiles call this "The Smile Curve." It's not the desirable trait it sounds like but it's a very "sellable" trait to naïve buyers.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>My Response To Frequency</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Now that you know the importance (and limitations) of amplitude variations in frequency response graphs, you might ask: "does the frequency range tell us anything at all?" Yes, it does. As long as you know the amplitude tolerance (+/- 3dB), the frequency response range or width tells you how high or low the speaker goes. A speaker rated as 20Hz - 25kHz +/- 3dB will play lower bass and higher treble sounds than a speaker that measures 40Hz - 20kHz +/- 3dB. I wouldn't bet money that it would be the better, more enjoyable speaker, but at least I'd know something of value.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">And now that you know how to interpret these numbers, you're ready to run right out and buy a speaker just by looking at the response curve, right? I wouldn't recommend it. Despite many advances in technology over the past 20 years, frequency response measurement is an imperfect science. The same speaker measured by two different labs may yield different response graphs. And some companies just plain cheat when they publish response curves. If it looks hand drawn, it probably was. ( Yes, the graphs were hand drawn for illustration purposes.)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The Third Dimension</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">So far we've talked about frequency (the X axis of the graph) and amplitude (Y axis) but we left out an important third dimension: time. When a speaker responds to an impulse, for example a rim shot -- "THWACK!" -- it should start instantly and stop the instant the instrument stops making sound. If the speaker keeps vibrating or resonating and making sound after the source sound stops it's changing, or "coloring," the sound of the original recording. And that's bad.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhloEZKCmQrxSl9MUREfB2hfj8mjr7R6zf4l7ozaV0g-KPVE360kLkldsyN29L0iaZiRlpmEY8KRUpBHZcqCzeCAn4fxzEojJ-JasnWBJUwVGM8HXKMwOOq8eXpQWdaSgCjnU4S8W71vDE/s1600-h/dos6.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450958853300036386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhloEZKCmQrxSl9MUREfB2hfj8mjr7R6zf4l7ozaV0g-KPVE360kLkldsyN29L0iaZiRlpmEY8KRUpBHZcqCzeCAn4fxzEojJ-JasnWBJUwVGM8HXKMwOOq8eXpQWdaSgCjnU4S8W71vDE/s400/dos6.gif" style="cursor: pointer; height: 115px; width: 154px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Figure F shows a bandwidth limited impulse signal. You can see that it starts and stops abruptly. Figure G shows that same impulse coming out of a speaker. You can see that the sound persists after the impulse input has stopped -- it resonates or "rings." The speaker is changing the timbre or character of the original recording. In order to see to what extent and at which frequencies the "ringing" is happening, we use a sophisticated computer algorithm called MLLSA (affectionately called "Melissa" by engineers who don't date much) to measure the response of a speaker in frequency, amplitude and time. Figure H is a MLLSA spectral decay graph of a prototype speaker. The third axis of this graph is time, so graph lines closest to you are measurements taken later than the ones in the back. Think of it as a series of slices with each slice being a frequency response graph taken at a different point in time.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If we were to measure the perfect speaker the MLLSA graph would look like a straight line in back with no lines in front. Real speakers fall far short of this ideal and continue to resonate after an impulse has stopped, such as in Figure H. Figure I is a Polk LSi9, and we can see that the speaker stops responding sooner in the midrange than the speaker pictured in Figure H, indicating that the LSi9 is a better sounding speaker.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguW8T6HMZrykIyEbyOrnsYJLq-VNliUyFz2kmmy1ndgtLHIwAez0IgZpbV5weD7WWHew9bR3xCBsao0ocB27_dyeXltJiYpuCZEqSVr-jcMeac32BI4Kj4LKOFH0Zw_U0TZnaCeDw3n54/s1600-h/dos7.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450959056805641650" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguW8T6HMZrykIyEbyOrnsYJLq-VNliUyFz2kmmy1ndgtLHIwAez0IgZpbV5weD7WWHew9bR3xCBsao0ocB27_dyeXltJiYpuCZEqSVr-jcMeac32BI4Kj4LKOFH0Zw_U0TZnaCeDw3n54/s400/dos7.gif" style="cursor: pointer; height: 115px; width: 154px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">While no measurement technique can fully describe the subjective sound of a louds peaker, MLLSA and other frequency response measurements are of great help to Polk engineers in developing better sounding speakers. Only a fool would design a speaker based on measurements alone and only a total fool would design a speaker based solely on subjective listening. A speaker that might sound good on a particular recording may in fact be flawed - it may have what is commonly called a "euphonic coloration." It may be pleasing to the ear under certain conditions, but it sure ain't right.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">We use both measurements and subjective listening to design and evaluate speakers. The measurements save us time and are a great help in pointing us in the right design direction, avoiding mistakes that may come back to bite us later. The measurements give us a means of selecting which experimental designs are worth listening to. But we have to be satisfied with the total subjective experience before a new design becomes a Polk Audio speaker. We spend countless hours listening to music and movies. Several experienced listeners have to listen to a proposed design and sign off on the sound before a model can even go into production. The Project Manager, Systems Engineer, VP of Engineering, Product Line Manager, and especially Matthew Polk, all have to agree that the prototype delivers the kind of rewarding listening experience that you expect from Polk Audio.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>What's Your Frequency?</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">You now know the secret: a frequency response specification is a very weak predictor of the actual performance of a loudspeaker. A frequency response chart can be more helpful, but it's missing the important time measurement. You now know to look for overall curve smoothness and to avoid rapid swings in amplitude. Some magazines and review sites publish MLSSA graphs of reviewed speakers, and now you'll understand how to interpret them. More power to you!</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">No matter how adept you might be at interpreting frequency response data, it should only be one data point among many in choosing a speaker. There is so much more to a speaker's performance than just its response - like its dispersion and imaging, dynamic range and detail resolution as well as size, cosmetics and price. Looking at good frequency response data can help you eliminate speakers with obvious and obnoxious errors. Once you've eliminated the boom & tizz pseudo-fi speakers, you can settle down to careful listening and making a more informed choice.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>How Polk Specifies Frequency Response</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Polk Audio publishes two frequency response specifications: "Overall" and "-3dB." "Overall" describes the frequency range limits of the speaker within an amplitude drop off of 9dB. Any frequency re p roduced more than 9dB down from the rest of the frequencies will contribute little to the sound. The "-3dB" spec describes the frequency range limits of the speaker within an amplitude drop off of 3dB.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">I just wrote this big article making the case that these kinds of numbers are not terribly useful in making buying decisions. So why does Polk use them? For better or for worse, these numbers are the norm in the audio industry. To not publish them would leave an impression that our products were not competitive. A better question would be: why don't we publish frequency response and MLSSA graphs in addition to the simple numbers? We feel that these graphs would not be meaningful to the vast majority of consumers. It takes years of working with measurements and loudspeakers before you get a good sense of how the graphs correlate to subjective sound quality. Incorrect interpretation of graphs can easily lead to misinformation and bad choices. Finally, the variation in measurement techniques can make comparing graphs from two different labs or manufacturers unreliable and misleading.</span></span></div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;"><span class="Apple-style-span" style="font-size: small;"><b>Lenny Z Perez M</b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-size: small;"><b>EES</b></span></div><div style="text-align: justify;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/understanding-speaker-frequency_20.html"><span class="Apple-style-span" style="font-size: small;"><b>http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/understanding-speaker-frequency_20.html</b></span></a></div></div></span> <br />
<hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href="http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us" target="_new">Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-70727785319689355932010-03-21T10:46:00.002-04:302010-03-23T17:49:15.041-04:30Basic Circuits - Bypass Capacitors<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">The Function</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The definition of a bypass capacitor can be found in the dictionary of electronics.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Bypass capacitor: A capacitor employed to conduct an alternating current around a component or group of components. Often the AC is removed from an AC/DC mixture, the DC being free to pass through the bypassed component.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In practice, most digital circuits such as microcontroller circuits are designed as direct current (DC) circuits. It turns out that variations in the voltages of these circuits can cause problems. If the voltages swing too much, the circuit may operate incorrectly. For most practical purposes, a voltage that fluctuates is considered an AC component. The function of the bypass capacitor is to dampen the AC, or the noise. Another term used for the bypass capacitor is a filter cap.</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhexqKd-ESijFisqZrXeQkxnBpQNZi6yGKmm1Va8kEc5BNDpExUKKED1wmuqUCqLfWrqN9G8VyIYbzyINxsJ7ahTKuBdDdrEeUwoUwWKESWjok6RTxT49KdYGZtA4ZB7G4eeHGeVWn2YHM/s1600-h/b1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451096561733870994" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhexqKd-ESijFisqZrXeQkxnBpQNZi6yGKmm1Va8kEc5BNDpExUKKED1wmuqUCqLfWrqN9G8VyIYbzyINxsJ7ahTKuBdDdrEeUwoUwWKESWjok6RTxT49KdYGZtA4ZB7G4eeHGeVWn2YHM/s320/b1.bmp" style="cursor: pointer; height: 243px; width: 320px;" /></span></span></a></span></div></span><br />
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</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">n the chart on the left, you can see the what happens to a noisy voltage when a by-pass capacitor is installed. Notice that the differences in voltage are pretty small (between 5 and 10 millivolts). This graph represents a small range of 4.95 volts to 5.05 volts. Random electrical noise causes the voltage to fluctuate, as you can see in graph. This is often called 'noise' or 'ripple'. The blue line, represents the voltage of a circuit that doesn't have a bypass. The pink line is a circuit that has a bypass. Ripple voltages are present in almost any DC circuit. You can see even with the bypass, the voltage does fluctuate, even though it is to a smaller degree. The key function of the bypass capacitor is to reduce the amount of ripple in a circuit. Too much ripple is bad, and can lead to failure of the circuit. Ripple is often random, but sometimes other components in the circuit can cause this noise to occur. For example, a relay or motor switching can often times cause a sudden fluctuation in the voltage. Much like disturbing the water level in a pond. The more current the other component uses, the bigger the ripple effect.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A fair question to ask is why does this small fluctuation matter? Gee, isn't the voltage close enough? The answer depends on the type of circuit you are designing. If you are just running a motor connected to a battery, or perhaps an LED, then chances are the ripple doesn't matter much to you. However, if you are using digital logic gates, things get slightly more complex, and this ripple can cause problems in the circuit.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Lets consider for just a moment what the effect of the ripple voltage is. Basic electrical theory tells us that a voltage is a difference in potential. It tells us that a current will flow across this difference in potential. We know that the larger the voltage, the larger the current. We also know the direction of the voltage determines the direction of the current.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Consider the graphs on the right. The top graph shows a pair of ripple voltages that I enlarged to make them easier to see. Just like the previous graph, the blue line represents the circuit without the bypass cap, and the other line is with the bypass cap. By looking along the bottom axis of the graph, you can see that starting at point 2 that the voltage is increasing. By looking in the Ripple Current chart, point 2 shows that the current is a relatively large magnitude in one direction. In contrast, point 5 shows the voltage and current going the other direction.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Notice the difference between the values with and without the bypass cap. By dampening the ripple voltage, the bypass cap also dampens the ripple current. I would like to point out that the Ripple Voltage chart and the Ripple Current charts clearly show an alternating current. You can see how the voltage swings, and how the current changes directions. Even though this is is a DC circuit, the ripple is causing an AC component. The bypass capacitor is helping to reduce this AC component.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The ripple current acts like an eddie or backflow in the circuit. As the fluctuating voltages and currents propogate through the circuit, differences in voltages and currents can occur that cause the circuit to fail. For example, assume that a AND gate is holding its state because the semiconductors that make up the gate are in a stable state. Transistors work by currents flowing one direction through the gate. If the current stops flowing, the transistor shuts down. If a ripple current comes through where the current momentarily flows the wrong direction, the gate will shutdown, and you will see a change it its output. This can cause a cascading failure, because one gate may be connected to many other gates.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">To summarize, the bypass capacitor is used to dampen the AC component of your DC circuits. By installing bypass capacitors, your DC circuit will not be as susceptable to ripple currents and voltages.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Using Bypass capacitors</span></span></b></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmN-CUB-SY_vrCnm4ZetZbRoYdOVsBAL0RrjQsEXPGtvzdDX8DbSqpWVJEaKIPRDT1x_XPjVF7QKsuQVNC2oBtR-P0ToizKErVGRi2KbpqWusxUI-YBWu7xkrH7uIxZTJ9PxBJ976erqI/s1600-h/b3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451097159891797794" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmN-CUB-SY_vrCnm4ZetZbRoYdOVsBAL0RrjQsEXPGtvzdDX8DbSqpWVJEaKIPRDT1x_XPjVF7QKsuQVNC2oBtR-P0ToizKErVGRi2KbpqWusxUI-YBWu7xkrH7uIxZTJ9PxBJ976erqI/s320/b3.bmp" style="cursor: pointer; height: 164px; width: 106px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Many schematics that you find published in magazines and books leave the bypass capacitors out. They assume you know to put them in. Other times you will find a little row of capacitors (caps) stuck off in the corner of the schematic with no apparent function. These are usually the bypass (or filter) caps. If you pickup almost any digital circuit, you will find a bypass capacitor on it.</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW6irA6OHwvQmT0dDqQA86N-Dr6Eh16lt2dRgdYMOZ94xNEVMUBGP4ZUXO_sbCDnkxPoRrl2fCkMv_y7hOcjIEi23d3JURQZ8CoRk-HlOwLQ7kMAX3OnZL3TPA7WunXbMGn-PiXSv0pp4/s1600-h/b4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451097958007833986" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiW6irA6OHwvQmT0dDqQA86N-Dr6Eh16lt2dRgdYMOZ94xNEVMUBGP4ZUXO_sbCDnkxPoRrl2fCkMv_y7hOcjIEi23d3JURQZ8CoRk-HlOwLQ7kMAX3OnZL3TPA7WunXbMGn-PiXSv0pp4/s320/b4.bmp" style="cursor: pointer; float: left; height: 88px; margin-bottom: 10px; margin-left: 0px; margin-right: 10px; margin-top: 0px; text-align: justify; width: 172px;" /></a><span class="Apple-style-span" style="font-family: arial;"></span><br />
<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;"><span class="Apple-style-span">The most simple incarnation of the bypass capacitor is a cap connected directly to the power source and to ground, as shown in the diagram to the left. This simple connection will allow the AC component of VCC to pass through to ground. The cap acts like a reserve of current. The charged capacitor helps to fill in any 'dips' in the voltage VCC by releasing its charge when the voltage drops. The size of the capacitor determines how big of a 'dip' it can fill. The larger the capacitor, the larger the 'dip' it can handle. A common size to use is a .1uF capacitor. You will also see .01uF as a common value. The precise value of a bypass cap isn't very important.</span></div></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: left;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">So, how many bypass capacitors do you really need? A good rule of thumb I like to use is each IC on my board gets its own bypass capacitor. In fact, I try to place the bypass cap so it is directly connected to the Vcc and Gnd pins. This is probably overkill, but it has always served me well in the past, so I will recommend it to you. It turns out you can even by DIP sockets that have the bypass caps built in. I suppose once you reach more than a few capacitors per square inch, you might be able to let up a bit!</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Another great place for a bypass cap is on power connectors. Anytime you have a power line heading off to another board or long wire, I would recommend putting in a bypass cap. Any long length of wire is going to act like a little antenna. It will pick up electrical noise from any magnetic field. I always put a bypass cap on both ends of such lengths of wire.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The frequency of the ripple can have a role in choosing the capacitor value. Rule of thumb is the higher the frequency, the smaller the bypass capacitor you need. If you have very high frequency components in your circuit, you might consider a pair of capacitors in parallel. One with a large value, one with a small value. If you have very complex ripple, you may need to add several bypass capacitors. Each cap is targeting a slightly different frequency. You may even need to add a larger electrolytic cap in case the amplitude of the lower frequencys is too great. For example, the circuit on the right is using three different capacitor values in parallel. Each will respond better to different frequencies. The 4.7uF cap (C4) is used to catch larger voltage dips which are at relatively low frequencies. The cap C2 should be able to handle the midrange frequencies, and C3 will handle the higher frequencies. The frequency response of the capacitors is determined by their internal resistance and inductance.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Summary</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Bypass capacitors help filter the electrical noise out of your circuits. They do this by removing the alternating currents caused by ripple voltage. Most digital circuits have at least a couple of bypass capacitors. A good rule of thumb is to add one bypass capacitor for every integrated circuit on your board. A good default value for a bypass cap is 0.1uF. Higher frequencies require lower valued capacitors.</span></span></div></div></div><div style="text-align: justify;"><span class="Apple-style-span"><br />
</span></div><div style="text-align: justify;"><span class="Apple-style-span">Lenny Z Perez M</span></div><div style="text-align: justify;"><span class="Apple-style-span">EES</span></div><div><span class="Apple-style-span" style="color: #aabbcc; font-family: arial;"><span class="Apple-style-span" style="line-height: 17px;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/basic-circuits-bypass-capacitors_21.html">http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/basic-circuits-bypass-capacitors_21.html</a></span></span></div></div></span> <br />
<hr />Connect to the next generation of MSN Messenger <a href="http://imagine-msn.com/messenger/launch80/default.aspx?locale=en-us&source=wlmailtagline" target="_new">Get it now! </a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-24681156868998364582010-03-21T10:44:00.002-04:302010-03-23T17:48:57.582-04:30The Miller’s theorem<div><br />
</div><div><span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Miller's theorem establishes that in a linear circuit, if there exists a branch with impedance Z, connecting two nodes with nodal voltages V1and V2, we can replace this branch by two branches connecting the corresponding nodes to ground by impedances respectively Z / (1-K) and KZ / (K-1), where K = V2 / V1.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK4L-ExPcNjX7BSEVSUYdiV1-1yAuI_U-aqlEERgwtWUeVCTexB8OmtMXsu-_2X1z86HGHkW6SlYZwjfwkyX1tqZaLRgwX0uhOgBgFw19bcxYxwJ-Qz6iVMgi9cZfLLPnK_pT-ysPT5aM/s1600-h/q1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451070601890981202" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK4L-ExPcNjX7BSEVSUYdiV1-1yAuI_U-aqlEERgwtWUeVCTexB8OmtMXsu-_2X1z86HGHkW6SlYZwjfwkyX1tqZaLRgwX0uhOgBgFw19bcxYxwJ-Qz6iVMgi9cZfLLPnK_pT-ysPT5aM/s320/q1.bmp" style="cursor: pointer; height: 91px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div><div style="text-align: justify;"><span class="Apple-style-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span></div><span class="Apple-tab-span" style="white-space: pre;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinwvsz9hfZ1zLTDQ1cDkCc_EuiD-qajVVfomFke24TlFnrmnufEQxaCKGJ39Yf3-Fi4-9YwIcVWtLjryD7Z8W3XlTtgtX-UWr_X_K26OrKzoWA1MZmuAKWXekYzdASkjUr0rgeKyTMbt4/s1600-h/q2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451070796197500258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinwvsz9hfZ1zLTDQ1cDkCc_EuiD-qajVVfomFke24TlFnrmnufEQxaCKGJ39Yf3-Fi4-9YwIcVWtLjryD7Z8W3XlTtgtX-UWr_X_K26OrKzoWA1MZmuAKWXekYzdASkjUr0rgeKyTMbt4/s200/q2.bmp" style="cursor: pointer; float: left; height: 112px; margin-bottom: 10px; margin-left: 0px; margin-right: 10px; margin-top: 0px; text-align: justify; width: 176px;" /></a></span><br />
<span class="Apple-tab-span" style="white-space: pre;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In fact, if we use the equivalent two-port network technique to replace the two-port represented on the right to its equivalent, it results successively:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJWMvDcF3mAEWndnLUjVQHjMo_dB2kNRnjcpXVJ1JBbZY1DvV3AzL1FvlswtxWy3ftJKXm7VvqhUXt-mcAkTyOLvP1pNmYYr7eKnEXrN2pvvdZjuhnPwunk624NFrXXFlsQlH1BXLzPUg/s1600-h/q3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451071347627340274" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJWMvDcF3mAEWndnLUjVQHjMo_dB2kNRnjcpXVJ1JBbZY1DvV3AzL1FvlswtxWy3ftJKXm7VvqhUXt-mcAkTyOLvP1pNmYYr7eKnEXrN2pvvdZjuhnPwunk624NFrXXFlsQlH1BXLzPUg/s400/q3.bmp" style="cursor: pointer; height: 120px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">And, according to the source absorption theorem, we get the following:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTSpMSoZdlJeE16nHNzDYOCbAbqp5vRDZZ1K9jWzlSTkCf9UxChNgzE6-zshPWJGYKQmRqMRdNsJD3C16higIRtq55NJ_1ocw_AfOWFkUyn-_7zzoCtheZUADnqMkY552o0920eWQ1ZzI/s1600-h/q4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451071663850480658" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTSpMSoZdlJeE16nHNzDYOCbAbqp5vRDZZ1K9jWzlSTkCf9UxChNgzE6-zshPWJGYKQmRqMRdNsJD3C16higIRtq55NJ_1ocw_AfOWFkUyn-_7zzoCtheZUADnqMkY552o0920eWQ1ZzI/s400/q4.bmp" style="cursor: pointer; height: 108px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTSpMSoZdlJeE16nHNzDYOCbAbqp5vRDZZ1K9jWzlSTkCf9UxChNgzE6-zshPWJGYKQmRqMRdNsJD3C16higIRtq55NJ_1ocw_AfOWFkUyn-_7zzoCtheZUADnqMkY552o0920eWQ1ZzI/s1600-h/q4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As all the linear circuit theorems, the Miller's theorem also has a dual form:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Miller's dual theorem</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If there is a branch in a circuit with impedance Z connecting a node, where two currents I1 and I2 converge, to ground, we can replace this branch by two conducting the referred currents, with impedances respectively equal to (1+ a) Z and (1+ a) Z / a, where a = I2 / I1.</span></span></div></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy93hA03dv30U2gifiLgLCYqQMxQfVHlX0hKiTABtu1pHNSmD0HywLeloHwcT4FJ19yIJw6vhuJuUvlxDrgO_XSX1pbnMZmhFhVjgniWAEcOj1M0TBMdbX9wsk2d8kK61ZnM2zdPCyE0o/s1600-h/q5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451072019886733986" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy93hA03dv30U2gifiLgLCYqQMxQfVHlX0hKiTABtu1pHNSmD0HywLeloHwcT4FJ19yIJw6vhuJuUvlxDrgO_XSX1pbnMZmhFhVjgniWAEcOj1M0TBMdbX9wsk2d8kK61ZnM2zdPCyE0o/s400/q5.bmp" style="cursor: pointer; height: 104px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; line-height: 20px;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; line-height: normal;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEneDYKEJttgeDa2xu13iyn6aIj0Gva8kmYfxZc-Ukcwy4JuwUHVO6JaGFc2sh3dU3X7cQsMvQdGmHI-_k1SVcNlk4XMOkCeOWRTsU1I5JICXkKQY4ASGy1sjqDssCzINb_2RkcSQhapA/s1600-h/q6.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451072491027985762" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEneDYKEJttgeDa2xu13iyn6aIj0Gva8kmYfxZc-Ukcwy4JuwUHVO6JaGFc2sh3dU3X7cQsMvQdGmHI-_k1SVcNlk4XMOkCeOWRTsU1I5JICXkKQY4ASGy1sjqDssCzINb_2RkcSQhapA/s200/q6.bmp" style="cursor: pointer; float: right; height: 74px; margin-bottom: 10px; margin-left: 10px; margin-right: 0px; margin-top: 0px; text-align: justify; width: 200px;" /></a><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="color: #0000ee;"><u><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></u></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In fact, replacing the two-port networkby its equivalent, as in the figure,</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">it results the circuit on the left in the next figure and then, applying the source absorption theorem, the circuit on the right.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGGhmMoGCOgpZPWVwTx8LYHFR-sdLdRYPdY-ilrIqcmtMU-Gy2Ajf7Go1iIT9O7Q56-3fGIygWLaCemlO_6XFTMHN5jWoErqYoYFdFEEkqyoNbuAOrzyXZmNzmEAMmaZV_VY_n-mvErNk/s1600-h/q7.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451072790115242162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGGhmMoGCOgpZPWVwTx8LYHFR-sdLdRYPdY-ilrIqcmtMU-Gy2Ajf7Go1iIT9O7Q56-3fGIygWLaCemlO_6XFTMHN5jWoErqYoYFdFEEkqyoNbuAOrzyXZmNzmEAMmaZV_VY_n-mvErNk/s400/q7.bmp" style="cursor: pointer; height: 111px; width: 400px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Miller's theorem applies to the process of creating equivalent circuits. This general circuit</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">theorem is particularly useful in the high-frequency analysis of certain transistor amplifiers at</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">high frequencies.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE4zHRXwOtewNqtI1F_ExT6UcGzI_iakIcQ-LUR46T9vxsyGYKV-aP_sCAjWfpNJOmGbQH4UMAMzrXc8wD6mc7L40VvcsHm2JIZJn-9C11EjyRttfFLwo6WalBSfDzBFOkUye8En5SkYs/s1600-h/q10.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451073633487755026" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE4zHRXwOtewNqtI1F_ExT6UcGzI_iakIcQ-LUR46T9vxsyGYKV-aP_sCAjWfpNJOmGbQH4UMAMzrXc8wD6mc7L40VvcsHm2JIZJn-9C11EjyRttfFLwo6WalBSfDzBFOkUye8En5SkYs/s400/q10.bmp" style="cursor: pointer; height: 192px; width: 400px;" /></span></span></a></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></b></div><div style="text-align: center;"><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Miller Theorem (and "Effect")</span></span></b></div><div style="text-align: center;"><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></b></div><div style="text-align: left;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Suppose that we have two networks separated by a bridging element Y. The equivalent circuits shown above represent particular important examples of such a situation</span></span></div><div style="text-align: left;"><b><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></b></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguPWohOEwfdIHz9GL01_x95PVyc-PqjANmSBUSyDo9eYXXEmZdMHC2ZDWpzgBjTtUbmy23JfTbSbq86Y-NL4qBJoEAKpRTmyw5kR-GTGswBYNysb1_Xa36IR7y5vU4eIl2C7D1iFDsX_g/s1600-h/a1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451087247144987650" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguPWohOEwfdIHz9GL01_x95PVyc-PqjANmSBUSyDo9eYXXEmZdMHC2ZDWpzgBjTtUbmy23JfTbSbq86Y-NL4qBJoEAKpRTmyw5kR-GTGswBYNysb1_Xa36IR7y5vU4eIl2C7D1iFDsX_g/s400/a1.bmp" style="cursor: pointer; height: 138px; width: 388px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Further, suppose that we can establish the following "gain relationship" by independent means:</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQH9HUhiHRr2XkuxdLt-11r7bGZmyTokhIBPwagy2P5Pc2QHRr919YLSXKw4qdvud0PUZGv0IXwVo-WKii1IQCOQoWuvSGOX9h07VKGd3bWhjFt377oqAsmcReyCMYBDlvr7Z_XSuAhBA/s1600-h/a2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451087440130658930" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQH9HUhiHRr2XkuxdLt-11r7bGZmyTokhIBPwagy2P5Pc2QHRr919YLSXKw4qdvud0PUZGv0IXwVo-WKii1IQCOQoWuvSGOX9h07VKGd3bWhjFt377oqAsmcReyCMYBDlvr7Z_XSuAhBA/s400/a2.bmp" style="cursor: pointer; height: 33px; width: 100px;" /></span></span></a></span></div></span></span><br />
<div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">and, thus, we may write</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglseaA75IhVPwAy07LpAN4PQEeKbhV8nPgth3zW67WQ-xm34wnDaxS4BvwWV_U9KwQe-oDyhNK6cccYZqpU_5IS9r2lnOIhHGm0UJyiPqKvQLi_Oes5quuBjRxovTmoUDh06mkBKXcK2Q/s1600-h/a3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451087613867757218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglseaA75IhVPwAy07LpAN4PQEeKbhV8nPgth3zW67WQ-xm34wnDaxS4BvwWV_U9KwQe-oDyhNK6cccYZqpU_5IS9r2lnOIhHGm0UJyiPqKvQLi_Oes5quuBjRxovTmoUDh06mkBKXcK2Q/s400/a3.bmp" style="cursor: pointer; height: 93px; width: 270px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If everything else remains unchanged, this bridged configuration can be replaced by a configuration of "decoupled" networks as follows:</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-GNajPgevihkSo8XqDVtgcJyfst7NNLkbGQz5l25NnL45SUndJSbvN09Bw-3uOqcWfm2L1AV-BqWR4jO9m6wcMVdvKANVjd07GuFo3ox45XHSxEP5W96PucsnV4SNO1Ng5Ron3-oMYjw/s1600-h/a5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451087846160013922" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-GNajPgevihkSo8XqDVtgcJyfst7NNLkbGQz5l25NnL45SUndJSbvN09Bw-3uOqcWfm2L1AV-BqWR4jO9m6wcMVdvKANVjd07GuFo3ox45XHSxEP5W96PucsnV4SNO1Ng5Ron3-oMYjw/s400/a5.bmp" style="cursor: pointer; height: 138px; width: 387px;" /></span></span></a></span></div></span></span><br />
<div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">where by equivalence we must have</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw6NdR4TvnhzWsuzBMqUTZrOIAXcJvSoAvRlzJs887RfxiMX6Upzq-8dJZb80sRtHZvdQydte8IZPFP989LMMj6UOUG48Nt9aa9rehG0d2FtNKaGQhYLchQnPwA-3L-5Chz5XZgv0Sn68/s1600-h/a6.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451088042309235426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw6NdR4TvnhzWsuzBMqUTZrOIAXcJvSoAvRlzJs887RfxiMX6Upzq-8dJZb80sRtHZvdQydte8IZPFP989LMMj6UOUG48Nt9aa9rehG0d2FtNKaGQhYLchQnPwA-3L-5Chz5XZgv0Sn68/s400/a6.bmp" style="cursor: pointer; height: 113px; width: 191px;" /></a></span></span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div></span></span><br />
<div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The Classic Solution to the "Miller Effect"</b></span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The Cascode Amplifier</b></span></span></div><div><br />
</div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUNCFZPEUAIRUGmjQLEFG1F3vxPCJzfefSW8LLpboFIyaj7dOtB9FJ9J6gEz6_Rks2-AHezSD9ujg4YWrc3E_-baMnxG_x6HzlL1b7ZbA8G3dh9s9KOnpTP6J6STk7G3_dabq53vPc9Io/s1600-h/a7.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451088245030797330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUNCFZPEUAIRUGmjQLEFG1F3vxPCJzfefSW8LLpboFIyaj7dOtB9FJ9J6gEz6_Rks2-AHezSD9ujg4YWrc3E_-baMnxG_x6HzlL1b7ZbA8G3dh9s9KOnpTP6J6STk7G3_dabq53vPc9Io/s400/a7.bmp" style="cursor: pointer; height: 235px; width: 180px;" /></a></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh10RziaZet06TMK1nFbySJgn0tjZg9RubuPQv-fqU0uU81L-oh2jix-JE5RhFd7lVWSS3dXd9cdxkTwddGYqHxOFRZ5rrghcIcXJHnp6IpYPHjPLydWUGmPXnRRfLGDOXlZmmlZpTtAYM/s1600-h/a8.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: none; color: black;"></span></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh10RziaZet06TMK1nFbySJgn0tjZg9RubuPQv-fqU0uU81L-oh2jix-JE5RhFd7lVWSS3dXd9cdxkTwddGYqHxOFRZ5rrghcIcXJHnp6IpYPHjPLydWUGmPXnRRfLGDOXlZmmlZpTtAYM/s1600-h/a8.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451088383314286178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh10RziaZet06TMK1nFbySJgn0tjZg9RubuPQv-fqU0uU81L-oh2jix-JE5RhFd7lVWSS3dXd9cdxkTwddGYqHxOFRZ5rrghcIcXJHnp6IpYPHjPLydWUGmPXnRRfLGDOXlZmmlZpTtAYM/s400/a8.bmp" style="cursor: pointer; height: 286px; width: 400px;" /></a></div><div style="text-align: center;"><br />
</div><div style="text-align: justify;">Lenny Z Perez M</div><div style="text-align: justify;">EES</div><div style="text-align: justify;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/millers-theorem.html">http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/millers-theorem.html</a></div></span></div><br />
<hr />Explore the seven wonders of the world <a href="http://search.msn.com/results.aspx?q=7+wonders+world&mkt=en-US&form=QBRE" target="_new">Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-50616208626111634122010-03-21T10:42:00.002-04:302010-03-23T17:48:38.887-04:30Square Wave Testing.<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;">It is perhaps unfortunate that the most common test for stability is to look for 'ringing' on a square-wave test signal. It is instructive to look at some examples, here using a 2kHz square wave input.</span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDyatkAoZtR_oDmPAXLwlFse7j2z5h8vJqxPU5327gRsFoTkCl0jZwoPnmT_ab9PhtUzUOp3jYX4_56HbUllnpKFwU-XFEiXBvsJjM5daBWwyPZciKDWQZ2pFuuR-Czs1yFzPm84i0gvw/s1600-h/6.1.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450943679992748978" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDyatkAoZtR_oDmPAXLwlFse7j2z5h8vJqxPU5327gRsFoTkCl0jZwoPnmT_ab9PhtUzUOp3jYX4_56HbUllnpKFwU-XFEiXBvsJjM5daBWwyPZciKDWQZ2pFuuR-Czs1yFzPm84i0gvw/s200/6.1.gif" style="cursor: pointer; height: 133px; width: 200px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHHtoPOBDfjEZnFHuO-rrcUcB8xiODDrYbGpfm-rNLYsFmjYGxRIbyGd1ZjewmcIvjmHK55OUW1GAijc1ts4GOkKf-D9sG-aSg28EEXIgVvdjshIkVoripuSxi79keyKOW_a3aVtLhJyE/s1600-h/6.2.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450943780048598802" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHHtoPOBDfjEZnFHuO-rrcUcB8xiODDrYbGpfm-rNLYsFmjYGxRIbyGd1ZjewmcIvjmHK55OUW1GAijc1ts4GOkKf-D9sG-aSg28EEXIgVvdjshIkVoripuSxi79keyKOW_a3aVtLhJyE/s200/6.2.gif" style="cursor: pointer; height: 133px; width: 200px;" /></span></span></a></div><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDyatkAoZtR_oDmPAXLwlFse7j2z5h8vJqxPU5327gRsFoTkCl0jZwoPnmT_ab9PhtUzUOp3jYX4_56HbUllnpKFwU-XFEiXBvsJjM5daBWwyPZciKDWQZ2pFuuR-Czs1yFzPm84i0gvw/s1600-h/6.1.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The first looks like sustained low level oscillation around 30kHz, while the second looks like damped oscillation at the same frequency. Actually the first diagram has nothing at all added to the square wave, the only thing done was to remove everything above the 15th harmonic. Everything up to and including 30kHz is being reproduced with no distortion, no phase error and flat frequency response. (If possible see 'A check on Fourier' by M.G.Scroggie, Wireless World, Nov 1977. p79-82. His Fig.5 is a better drawn version showing the harmonics and how they add.) The lack of higher frequency components however gives the impression of a serious problem, when in fact the audio frequency reproduction is perfect, and there is nothing at all added or removed in this range. The symmetrical variation of the 'oscillation' amplitude gives a clue to the origin of the effect, but practical low pass filters give a less sharp cut off of high harmonics together with frequency dependant phase shift which will give a different appearance. The suggestion that 'ringing' needs to be minimised is not entirely convincing when even an ideal low-pass filter gives the above result. Using an audio signal with no frequency components above 30kHz instead of the square wave there would be no effect at all from this filter.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The second diagram can also be the result of low-pass filtering, and something similar is often produced by the interaction of output inductors with capacitive loads, which is not related in any direct way to stability. Checking the signal ahead of the inductor may reveal a smooth signal without the 'ringing' effect, though some amplifiers have an output impedance with a small internal inductive component which will add some small effect. The square-wave response shown in the MJR-6 test results shows low level 'ringing' which is estimated at 120kHz. This is close to the expected resonance frequency of the 0.4uH output inductor with the 4uF load capacitance used in that test. Increasing loop gain to the point where the amplifier becomes unstable caused oscillation around 6MHz, as expected from the feedback loop unity gain frequency. This demonstrates that output 'ringing' is generally not related to instability, which can occur in an entirely different frequency range, and unless the input signal includes components close to the LC resonance frequency, or the inductance used is too high, there will be little effect. Leaving out the output inductor to eliminate 'ringing' caused by this LC resonance may seriously reduce the phase margin at higher frequencies with some capacitive loads, dangerously increasing the risk of instability.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">A square wave test to investigate stability into capacitive loads is therefore of limited usefulness, and may be seriously misleading. My experience is that amplifiers sometimes have a stable state and an unstable state, and triggering them into instability may need a precise choice of load and input signal, in one case driving the amplifier heavily into clipping and then removing the input signal caused a dramatic latch-up and oscillation effect. Failure to oscillate with just any square-wave input and the usual 2uF test load may be necessary, but is no guarantee of unconditional stability. I also use high level sinewave signals at various frequencies, and look for signs of instability close to clipping as the signal level is adjusted to give different levels of clipping. Going into or out of clipping the loop gain is changing, and so the feedback loop unity gain frequency is in effect shifted over a wide range, revealing potential stability problems over a similar range. To limit dissipation it is convenient to use a toneburst signal for these clipping tests.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The next two photos are oscilloscope traces showing examples of clipping behaviour:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTLs0OCEULP4F8nr4qa9ENfbu3D6YD2Ut-Ab_0cTOHIjySWXh6BkNkkX9LP2WMGjFYdO0RR3toVwSDqpnl2eQgc-YQWZ6TcV89jNHilxxwjz8wvrWqIeIwvmCCCPygkrfJ0Ggv8K9AMzs/s1600-h/6.3.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450943978301113714" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTLs0OCEULP4F8nr4qa9ENfbu3D6YD2Ut-Ab_0cTOHIjySWXh6BkNkkX9LP2WMGjFYdO0RR3toVwSDqpnl2eQgc-YQWZ6TcV89jNHilxxwjz8wvrWqIeIwvmCCCPygkrfJ0Ggv8K9AMzs/s200/6.3.jpg" style="cursor: pointer; height: 99px; width: 200px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUUH1L27jxj9Hn2wMPQgLGDt8nNx8YGjA9ROxIpW7UXfz5pRRxp1Ngkem9mXCF4Q0zf37QRv79hhQIKUjzJ-cAgXS9as71rQO1W_04djLhMo8gGENMoPDuWKYfEaYd7PgUo0DBjRCwRk4/s1600-h/6.4.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450944486393177874" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUUH1L27jxj9Hn2wMPQgLGDt8nNx8YGjA9ROxIpW7UXfz5pRRxp1Ngkem9mXCF4Q0zf37QRv79hhQIKUjzJ-cAgXS9as71rQO1W_04djLhMo8gGENMoPDuWKYfEaYd7PgUo0DBjRCwRk4/s200/6.4.jpg" style="cursor: pointer; height: 172px; width: 200px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The first of these is just a single notch when coming out of clipping, and this is typical of latch-up effects rather than instability. In this case it was caused by a bad choice of frequency compensation circuit such that the compensation capacitor charged up during clipping and had to discharge before normal linear operation could return. A change to the compensation arrangement was needed to cure this.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Stability problems generally have a different appearance of the type shown in the second photo. Here a short burst of oscillation occurs when coming out of clipping, but in this case the effect continues long after this as seen from a slight ripple on the trace. A change in the value of the compensation capacitor was needed to remove this effect. The positive and negative clipping look different, which is not uncommon, here the positive clipping appears to include a latch-up effect in addition to the stability problem.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Had I relied only on observations of square-wave ringing with a 2uF load below clipping I would have said there were no stability problems to worry about, and stopped there without doing the necessary modifications.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">It is known that the choice of test signal rise-time can often have a great effect on observed 'ringing', and it is possible to claim 'excellent transient response' just by careful choice of the rise-time of the test signal. This was mentioned in one of the Douglas Self articles, "The Audio Power Interface", Electronics World Sept.1997 p717-722.</span></span></div><div style="text-align: left;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The low-pass filter used at the input of my own amplifiers helps give a smooth square wave output with little ringing, but it was not included for this purpose. Anyone who still wants to reduce ringing further in the mosfet amplifiers could try reducing the damping resistor in parallel with the inductor, maybe to one ohm.</span></span></div><div style="text-align: center;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>Waveform and Spectrum Analysis</b></span></span></div><div style="text-align: center;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>by Lloyd Butler VK5BR</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The article is divided into two sections. Section A deals with typical CRO waveforms which might indicate certain characteristics or fault conditions in the electronic equipment being tested. The section shows various waveforms associated with square wave testing, sine wave testing, measurement of rise time and overshoot and measurement of phase shift. This section is part of the article "Measurement of Distortion" published in Amateur Radio, June 1989 (ref.1).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Section B displays Spectrum Analyser waveforms for sine wave, square wave, triangular wave and modulated signals. Also displayed are typical spectrograms made to measure frequency response or the characteristics of filters. This section was originally published in Amateur Radio, September 1987 (Ref 2). )</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Section A</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Waveforms using the Cathode Ray Oscilloscope (CRO)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>SQUARE WAVE TESTING</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">One method of assessing frequency response (and sometimes other characteristics) is to feed a square wave to the input of the device under test and examine its output on a CRO. The square wave is made up of a fundamental frequency and all odd harmonies, theoretically to infinity. A deficiency within the frequency spectrum, from the fundamental upwards, will show a change in the shape of waveform. The test is subjective rather than precise but gives a good indication of the response.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0PaqCPwFTnr2OwyUKbFvSk-LUJnuIs52NZPRywe5oQ1FIxJ6hSprhwQy2f3-mn-s1BSXkuUbfkua6hoXketcP8GOEVdY03dv_YdNLShIUxwY2j1bgva6OrIydOJgB8TeZwlmKU_GHaUQ/s1600-h/6.5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450946264371563026" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0PaqCPwFTnr2OwyUKbFvSk-LUJnuIs52NZPRywe5oQ1FIxJ6hSprhwQy2f3-mn-s1BSXkuUbfkua6hoXketcP8GOEVdY03dv_YdNLShIUxwY2j1bgva6OrIydOJgB8TeZwlmKU_GHaUQ/s400/6.5.bmp" style="cursor: pointer; height: 240px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Typical response patterns taken from a reference source are shown in figure 1. The captions under the patterns decribe the various operational conditions and the effect of loss of low or high frequency response is illustrated. Further patterns shown in figure 2 also illustrate the effect on the waveforms when relative phase delay is changed over part of the frequency spectrum. Also observe in figure 1(J) how the ringing from oscillation in the circuit under test is initiated by the steep edge of the square wave. This is a test result on how the circuit might handle a transient which might not have been detected in carrying out a sine wave frequency response check.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Related to frequency response, there is a specification called "transient response' which is the ability of a device to respond to a stop function. "Rise time" is one measure of transient response and is the time taken for the signal, initiated from a stop function, to rise from 10 percent to 90 percent of its stable maximum value. Another measure is the percentage of the stable maximum value that the signal over-shoots in responding to the step. Figure 3 shows how the square wave, in conjunction with a calibrated CRO, can be used to measure rise time and overshoot.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirEpwxPS7GQFu_9KZmjMGGGfmQ0-zEwLgpAW4o9l5qyB5nIORmuKXHO-nE43nPKgemfxzNzLGhQa8raninqUxSK1A8pOrPjv5s2IzwXNfHzC6SmGP6hd7qmn5yZrnBeROlyvzYlxC00qc/s1600-h/6.7.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450946721533347250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirEpwxPS7GQFu_9KZmjMGGGfmQ0-zEwLgpAW4o9l5qyB5nIORmuKXHO-nE43nPKgemfxzNzLGhQa8raninqUxSK1A8pOrPjv5s2IzwXNfHzC6SmGP6hd7qmn5yZrnBeROlyvzYlxC00qc/s400/6.7.bmp" style="cursor: pointer; height: 179px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Rise time is also measure of the maximum slope of any sine wave component and hence is directly related to the limits in high frequency response. Together, rise time and overshoot define the ability of a device to reproduce transient type signals. Another specification commonly used in operational amplifiers is the 'slew rate" given in volts per microsecond. Such amplifiers have limitations in the rate of change that the output can follow and this is defined by the slew rate. The greater the output voltage, the greater is the rise time and hence the greater the output voltage, the lower is the effective bandwidth. Slew rate is equal to the output voltage step divided by the rise time as measured over the 10 percent to 90 percent points, discussed previously. It is an interesting observation that, in specifying frequency response, output voltage should also be part of the specification.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>HARMONIC DISTORTION</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Harmonic distortion in any signal transmission device results from non-linearity in the device transfer characteristic. Additional frequency components, harmonically related to frequencies fed into the input, appear at the output in addition to the reproduction of the original input components.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Measurement of harmonic distortion can be carried out by feeding a sine wave into the input of the device and separating the sine wave from its harmonics at the output. Distortion is measured as the ratio of harmonic level to the level of the fundamental frequency. This is usually expressed as a percentage but sometimes also expressed as a decibel. Distortion Meters and types of distortion are described in the original article (Ref 1). Intermodulation distortion is described in ref 3.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>SINE WAVE TESTING</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Subjective testing for harmonic distorton can be carried out by feeding a good sine wave signal into the device under test and examining the device output on a CRO. Quite low values of distortion can be detected in this way.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsog_w2docLJsSyd8dZvSDQzsv-pN9NLU9ekosU-aW-A5GfAE1vBo5US4MqRTbZ8zo1sPEpiFLu1s7FOGeZ7tOwnRWVCpbvQX_HXloV77pFpnHCbbTGhQNIMtGIqv0RudiwtvLb-x_w2A/s1600-h/6.8.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450946938921611762" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsog_w2docLJsSyd8dZvSDQzsv-pN9NLU9ekosU-aW-A5GfAE1vBo5US4MqRTbZ8zo1sPEpiFLu1s7FOGeZ7tOwnRWVCpbvQX_HXloV77pFpnHCbbTGhQNIMtGIqv0RudiwtvLb-x_w2A/s400/6.8.bmp" style="cursor: pointer; height: 152px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Some idea of the order of the harmonic can often be determined from the shape of the waveform. Figure 4 illustrates the formation of a composite waveform from a fundamental frequency and its second harmonic at one quarter of the fundamental Amplitude. Figure 5 illustrates similar formation from a fundamental frequency and its third harmonic, also a quarter of the fundamental amplitude. In Figure 5(b), the phase of the harmonic is shifted 180 degrees to that in Figure 5(a), and in Figure 5(c), the phase is shifted 90 degrees to that in (5a). The figures show that the composite wave forms can be quite different for different phase conditions making resolution sometimes tricky.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCneDE9pdhq3NCo_TClOi-YAfZ9j9atvyRMXPhHo0tKA-dlDksVw3ahuC7BFy98JSmhJaYTO_3UWatK48AvNQRIxUUs3sKPLIhnnoK9G0DstJxnWmL7YAasc4UtGASvEd9OEGkVAVtcoA/s1600-h/6.9.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450947186428628386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCneDE9pdhq3NCo_TClOi-YAfZ9j9atvyRMXPhHo0tKA-dlDksVw3ahuC7BFy98JSmhJaYTO_3UWatK48AvNQRIxUUs3sKPLIhnnoK9G0DstJxnWmL7YAasc4UtGASvEd9OEGkVAVtcoA/s400/6.9.bmp" style="cursor: pointer; height: 257px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Some distorted waveforms directly indicate an out of adjustment or incorrect operating condition. The clipped waveform of Figure 6(a) shows the output of an amplifier driven to an overload or saturated condition. Figure 6(b) is clipped in one direction indicating an off-centre setting of an amplifier operating point. Figure 6(c) shows crossover distortion in a Class B amplifier.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6h19TBZoQsXtapB-qoao-d-3rg-x32RfsiTvO3yjPkAhOQFsJksPYupxO57PQHvBNHizXOAa_WYZAKEG3qle72A4RVxmtaPQN172wxWRvl3C6NKgR-yZtu8Bx3s5mWQk-giq2h5dRs8E/s1600-h/6.10.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450947446808094706" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6h19TBZoQsXtapB-qoao-d-3rg-x32RfsiTvO3yjPkAhOQFsJksPYupxO57PQHvBNHizXOAa_WYZAKEG3qle72A4RVxmtaPQN172wxWRvl3C6NKgR-yZtu8Bx3s5mWQk-giq2h5dRs8E/s400/6.10.bmp" style="cursor: pointer; height: 143px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Another method of testing, using sine waves, is to feed the monitored device input signal to the X plates input of the CRO and the device output signal to the Y plates input of the CRO. This plots the transfer characteristic of the device, that is, instantaneous output voltage as a function of instantaneous input voltage. X and Y gain is adjusted for equal vertical and horizontal scan. A perfect response is indicated by a diagonal line on the screen, or with phase shift, an ellipse or circle. Figure 10 shows various fault wave forms taken from one reference source. The different effects are explained in the diagram captions</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The same connection can be used to measure phase shift between two sine wave signals of the same frequency such as measuring the phase shift between the output and input of an amplifier. Typical measurements are shown in Figure 8. A straight forward sloped diagonal line indicates no phase shift. A straight reverse sloped diagonal line indicates 180°. A circle indicates 90° and an elipse 45° or 135°.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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<div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">If a dual trace CRO is available, the two signals can be displayed together, one on each vertical or Y trace with normal X sweep. In this case, it is simply a matter of scaling off the phase difference along the Y axis graticle.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>Section B - Spectrum Analyser Waveforms</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Over the years, the cathode ray oscilloscope (CRO) has been a universal instrument for examining analogue signals. Rapid advances in technology have led to a era of microcomputer controlled, digitally controlled test equipment, not the least of which is the modern spectrum analyser which enables greater precision analysis of analogue signals than is possible with the CRO. A spectrum analyser plots signal amplitude (or signal power) as a function of frequency compared to the CRO which plots signal amplitude as a function of time.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">The spectrum analyser is not the type of equipment normally within the reach of the radio amateur and because of this, it was thought that it would be of interest to illustrate a few spectrum plots of well-known waveforms.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>BASIC WAVEFORMS</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 1 shows the spectrum of a sine wave oscillator with fundamental at 1000 Hz and harmonics up to 20 kHz. The highest level harmonic at 7 kHz is 70 dB below the fundamental, representing a harmonic distortion of 0.03 percent. This is a very good oscillator which would not be matched by many laboratory instruments. It can also be seen that the noise floor is about 95 dB below the fundamental and this is also very good. The oscillator noise level might be even better than this as much of the noise is due to the spectrum analyser itself.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 2 shows a 1000 Hz square wave. A perfect square wave generates odd harmonics to infinity with an amplitude 1/n relative to that of the fundamental or (20 log n) dB below the fundamental. ('n' is the order of harmonic). For n = 3, 5, 7 and 9 this calculates to -9.5, -14, -16.9 and -19.1 dB respectively, very close to the readings shown in Figure 2.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbBHszFxGSlxTY3HvLddPdaWIQBo5iVZZbsMQ46v2jB0jjyk06l7xzMI4cdAjGPjGPSJc3G7fYF6EXyMEgbWqtJHjBBKFBJcY3P5_JwajfgvlNngVennMmp8bc_R1UsConTePq7wGZq6A/s1600-h/6.13.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450948196695791218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbBHszFxGSlxTY3HvLddPdaWIQBo5iVZZbsMQ46v2jB0jjyk06l7xzMI4cdAjGPjGPSJc3G7fYF6EXyMEgbWqtJHjBBKFBJcY3P5_JwajfgvlNngVennMmp8bc_R1UsConTePq7wGZq6A/s400/6.13.bmp" style="cursor: pointer; height: 205px; width: 400px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 3 is the same square wave plotted out to 200 kHz and showing the apparently unlimited spread of harmonics. From this, it is easy to see why a low frequency square wave oscillator can be used as a marker generator over a wide frequency range.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 4 shows a 1000 Hz triangular wave. A perfect triangular wave also generates odd harmonics to infinity, but each amplitude is (l/n) squared relative to the fundamental or (40 log n) dB below the fundamental. For n = 3, 5, 7, and 9, the calculation is -19, -28, -33.8, and -38.2 dB respectfully, again very close to the readings shown.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGNFg6rATYOi3dstk9BAmbbse-QmPSFoyXtI2NnqK8MuaCXcogj1CuDuSzEZfMDgva_K4UUqCYM7yb0T0NZTlEhTV46kumkJSQhJme1lE7rjJweyPwBVotFoorNAsEyczNtsghtvdiaJQ/s1600-h/6...bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450948906856761154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGNFg6rATYOi3dstk9BAmbbse-QmPSFoyXtI2NnqK8MuaCXcogj1CuDuSzEZfMDgva_K4UUqCYM7yb0T0NZTlEhTV46kumkJSQhJme1lE7rjJweyPwBVotFoorNAsEyczNtsghtvdiaJQ/s400/6...bmp" style="cursor: pointer; height: 196px; width: 400px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>MODULATION</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 5 shows a 1 MHz carrier frequency, amplitude modulated by a frequency of 1 kHz to a modulation depth of 50 percent. For this case, the two side frequencies, 1 kHz either side of the carrier, are 12 dB below the carrier level, or a quarter of its amplitude. Other side frequencies at 2 kHz and 3 kHz, either side of the carrier, are the result of harmonics either in the original modulating tone or distortion caused by the modulation process. The 2 kHz side frequencies are about 30 dB below the 1 kHz side frequencies representing about three percent distortion in the system.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">In Figure 6, the modulation level has been increased to 100 percent and the side frequencies, 1 kHz either side of the carrier, are now 6 dB below carrier level, or half its amplitude. The spectrum has been expanded to show many more harmonically related sideband components which now appear. Except for those close to the carrier, most of the components are more than 50 dB down and not of any great concern.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX-D39f23fVTSmZJPJQW4UBIq5U5fzd4ASkmVcQ6j8xh9_rzES_dX-a5UhK1y-HddV6KqLgWEGZjuJF429x4gvitxWPCFlEqtdGP1_irQL2P_n-AngHtOcKT9njuqqQaN7XO7zp-sHl34/s1600-h/6.14.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450949166776978578" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX-D39f23fVTSmZJPJQW4UBIq5U5fzd4ASkmVcQ6j8xh9_rzES_dX-a5UhK1y-HddV6KqLgWEGZjuJF429x4gvitxWPCFlEqtdGP1_irQL2P_n-AngHtOcKT9njuqqQaN7XO7zp-sHl34/s400/6.14.bmp" style="cursor: pointer; height: 205px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX-D39f23fVTSmZJPJQW4UBIq5U5fzd4ASkmVcQ6j8xh9_rzES_dX-a5UhK1y-HddV6KqLgWEGZjuJF429x4gvitxWPCFlEqtdGP1_irQL2P_n-AngHtOcKT9njuqqQaN7XO7zp-sHl34/s1600-h/6.14.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span><br />
<div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">In Figure 7, the carrier is over-modulated and there is now a spread of sideband components about 30 dB down. If this were an amateur radio transmitter, other amateur stations in nearby suburbs would be complaining about sideband splatter.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Figure 8 Shows a 1 MHz carrier, frequency modulated by a 1 kHz tone with a deviation of 8.650 kHz, representing a modulation index of 8.650. It can be seen that there are many side frequencies all spaced by an amount equal to the modulating frequency (1 kHz). For this signal, a significant bandwidth of about 20 to 30 kHz is being utilised.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivSpsD7aQqr8duJG2_gSSPy1Dt6Y6m5kyTXZ8nr99vB7Xfui6cB1MJ3Y9XvjN0TvApzbb2D7JPyVHX6q63aSNAut-fAn6si0PPhEQZsofZKgY18eta4pRtctzCir0JFGcz6jS_PWK8EUA/s1600-h/6.15.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450949590311571122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivSpsD7aQqr8duJG2_gSSPy1Dt6Y6m5kyTXZ8nr99vB7Xfui6cB1MJ3Y9XvjN0TvApzbb2D7JPyVHX6q63aSNAut-fAn6si0PPhEQZsofZKgY18eta4pRtctzCir0JFGcz6jS_PWK8EUA/s400/6.15.bmp" style="cursor: pointer; height: 215px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">If we now examine Figure 9, which plots the amplitude of the carrier and side frequencies against the value of modulation index, we can see that there are a number of values of modulation index where the carrier level becomes zero. These are very convenient references to calibrate the amount of deviation. In Figure 8, the deviation has been set to produce the third carrier null at a modulation index of 8.650, so we know precisely that with our modulating frequency of 1000 Hz, our deviation is 8.650 x 1000 = 8850 Hz.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
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</span></span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXKvkmGA9OHu2GiG5eP9nJX5JZmBVlDfdGjOGVZDKzm41GoWdC6oXXsvix3BbvAWINRXtPifGHhJV3hYwojo4b9DOylgBaF8xSRloX7dMBArgIl35IHD5RUHYc1TncCUTJ7hlXV1BR-vw/s1600-h/6.16.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450949837970732194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXKvkmGA9OHu2GiG5eP9nJX5JZmBVlDfdGjOGVZDKzm41GoWdC6oXXsvix3BbvAWINRXtPifGHhJV3hYwojo4b9DOylgBaF8xSRloX7dMBArgIl35IHD5RUHYc1TncCUTJ7hlXV1BR-vw/s400/6.16.bmp" style="cursor: pointer; height: 248px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><b>FREQUENCY RESPONSE</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;">Another useful function of the spectrum analyser is to plot the frequency response of a four terminal device such as an amplifier or a filter. In this case, the analyser frequency sweep generator is fed to the input of the device and the output of the device is fed to the input of the analyser. Typical plots of a low pass filter and a bandpass filter are shown in Figures 10 and 11 respectively</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></span></div></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: none; color: black;"></span><span class="Apple-style-span"><span class="Apple-style-span" style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEickNSlX014sOCYNCLszYCOZ83uOFq_iIXDLgy8ZSjluIlbktXRas9WXaElvdOsjjX7q3jK5IR8_qPQsWQZWKVnXH2YZT_gLUDBtXUdVTLt-gJtybw05mAphHHXIZZhtz6Ofi3Iz5mruU8/s1600-h/6.17.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450950077473991906" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEickNSlX014sOCYNCLszYCOZ83uOFq_iIXDLgy8ZSjluIlbktXRas9WXaElvdOsjjX7q3jK5IR8_qPQsWQZWKVnXH2YZT_gLUDBtXUdVTLt-gJtybw05mAphHHXIZZhtz6Ofi3Iz5mruU8/s400/6.17.bmp" style="cursor: pointer; height: 194px; width: 400px;" /></a></span></span></div><div style="text-align: center;"><span class="Apple-style-span"><br />
</span></div><div style="text-align: justify;"><span class="Apple-style-span"><b>Lenny Z Perez M</b></span></div><div style="text-align: justify;"><span class="Apple-style-span"><b>EES</b></span></div><div style="text-align: justify;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/square-wave-testing.html"><span class="Apple-style-span"><b>http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/square-wave-testing.html</b></span></a></div></span> <br />
<hr />Get news, entertainment and everything you care about at Live.com. <a href="http://www.live.com/getstarted.aspx" target="_new">Check it out!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-43098657841199805562010-03-21T10:39:00.002-04:302010-03-23T17:48:19.673-04:30The decibel unit<div><br />
</div><div><span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Decibel</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering (particularly acoustics, electronics, and control theory) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.775 volts RMS.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi10bhLwiFlrWGCsnvlVdKVG21xuiMHkTnTvV6I9ptfY42VFEyQfhHYDmWN13_EodVs5Di2a6dHju4wA9UarXEMZMxzMvVuC4H3F3o1oYMG7QeGMyezjrhXFlNy7QDHaMfaBsGW7-QgKus/s1600-h/z1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451075160909868114" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi10bhLwiFlrWGCsnvlVdKVG21xuiMHkTnTvV6I9ptfY42VFEyQfhHYDmWN13_EodVs5Di2a6dHju4wA9UarXEMZMxzMvVuC4H3F3o1oYMG7QeGMyezjrhXFlNy7QDHaMfaBsGW7-QgKus/s200/z1.bmp" style="cursor: pointer; height: 200px; width: 160px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span"><br />
</span></b></span></div><div style="text-align: center;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">History</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz) and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).[citation needed]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The transmission unit or TU was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to reference power.[2] The definitions were conveniently chosen such that 1 TU approximately equalled 1 MSC (specifically, 1.056 TU = 1 MSC).[3] Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, which was named the "bel" in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was a factor of ten larger than the TU, such that 1 TU equalled 1 decibel.[4] In many situations, the bel proved inconveniently large, so the decibel has become more common.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.[5] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC).[6] The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Merits</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><i><span class="Apple-style-span">The use of the decibel has a number of merits:</span></i></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) + ...</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Acoustics</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The human ear has a large dynamic range in audio perception. The ratio of the sound pressure that causes permanent damage during short exposure to the quietest sound that the ear can hear is above a trillion. Such large measurement ranges are conveniently expressed in logarithmic units: the base-10 logarithm of one trillion (1012) is 12, which is expressed as an audio level of 120 dB. Since the human ear is not equally sensitive to all sound frequencies, noise levels at maximum human sensitivity — for example, the higher harmonics of middle A (between 2 and 4 kHz) — are factored more heavily into some measurements using frequency weighting.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Further information: Examples of sound pressure and sound pressure levels</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbvABUYQzZ8lS95aUK7yD1-Ny2Othkf-U3ZFOq4OvkTb2kbXA85i-BKiG54WuY3h1SNxOHpI8CcsiP9xMLu31RF76y4HJ9rqG21Maz1l9R9AUB1F1LxXgcqf0DKw3qJMSanF4NU-NCAI8/s1600-h/z3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451076515987662226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbvABUYQzZ8lS95aUK7yD1-Ny2Othkf-U3ZFOq4OvkTb2kbXA85i-BKiG54WuY3h1SNxOHpI8CcsiP9xMLu31RF76y4HJ9rqG21Maz1l9R9AUB1F1LxXgcqf0DKw3qJMSanF4NU-NCAI8/s320/z3.bmp" style="cursor: pointer; height: 289px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Electronics</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a link budget.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in telephone audio circuits.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPoS5pSRJ4ZEgn3PLPhaPmwtOvufVOJtRHAepdg5A8ZGi2rVZEwhGnDI-PkTIxTNawQl3WtFMPFLu8XxreXuzeDgfQkbUDSj8XIWnaffX3NLuR-cf7MB2VXJaSBMocPbI8Cfge8WgTsP8/s1600-h/z2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451076864605578962" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPoS5pSRJ4ZEgn3PLPhaPmwtOvufVOJtRHAepdg5A8ZGi2rVZEwhGnDI-PkTIxTNawQl3WtFMPFLu8XxreXuzeDgfQkbUDSj8XIWnaffX3NLuR-cf7MB2VXJaSBMocPbI8Cfge8WgTsP8/s200/z2.bmp" style="cursor: pointer; height: 149px; width: 200px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Optics</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In spectrometry and optics, the blocking unit used to measure optical density is equivalent to −1 B. In astronomy, the apparent magnitude measures the brightness of a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">]Video and digital imaging</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In connection with digital and video image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square. Thus, a camera signal-to-noise ratio of 60 dB represents a power ratio of 1000:1 between signal power and noise power, not 1,000,000:1</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><i><b><span class="Apple-style-span"><br />
</span></b></i></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><i><b><span class="Apple-style-span">Common reference levels and corresponding units</span></b></i></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Absolute and relative decibel measurement</span></b><span class="Apple-style-span">s</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">−6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 μW (0.25 mW).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative. The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Absolute measurements</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span" style="font-weight: normal;"><b><span class="Apple-style-span">Electric power</span></b></span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBm or dBmW</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBW</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Voltage</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.</span></span></div></div></div></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzot5izM4_oasMlL4aQx5W4OQ9NAi1vsYZWTn7WOygk5XySlJDUZACQ0rVslYpKFhdJyOCvAxBuB_NNOD4V57CKZY1PdktCpT_kNO7F6cI7nocFThTAYTCn7o2GzibN_uTMkZuR2RwoDM/s1600-h/z4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451077818207998642" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzot5izM4_oasMlL4aQx5W4OQ9NAi1vsYZWTn7WOygk5XySlJDUZACQ0rVslYpKFhdJyOCvAxBuB_NNOD4V57CKZY1PdktCpT_kNO7F6cI7nocFThTAYTCn7o2GzibN_uTMkZuR2RwoDM/s400/z4.bmp" style="cursor: pointer; height: 93px; width: 283px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBmV</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 Ω coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm) or ~13 nW.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBμV or dBuV</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Acoustics</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Probably the most common usage of "decibels" in reference to sound loudness is dB SPL, referenced to the nominal threshold of human hearing:[11]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(SPL)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB (sound pressure level) — for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10−5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a reference pressure of 1 μPa is used.[12]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(PA)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB — relative to 1 Pa, often used in telecommunications.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB SIL</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB sound intensity level — relative to 10−12 W/m2, which is roughly the threshold of human hearing in air.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB SWL</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB sound power level — relative to 10−12 W.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(A), dB(B), and dB(C)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">These symbols are often used to denote the use of different weighting filters, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB HL or dB hearing level is used in audiograms as a measure of hearing loss. The reference level varies with frequency according to a minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.[citation needed]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting[citation needed]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Radar</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBZ</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBsm</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBsm - decibel (referenced to one) square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of dBsm, large flat plates or non-stealthy aircraft have positive values.[14]</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Radio power, energy, and field strength</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBc</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBc — relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, used in acoustics.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBJ</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBm</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. -120 dBm = 0.224 microvolts.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBμV/m or dBuV/m</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(μV/m) — electric field strength relative to 1 microvolt per meter.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBf</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(fW) — power relative to 1 femtowatt.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBW</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(W) — power relative to 1 watt.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBk</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(kW) — power relative to 1 kilowatt.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Antenna measurements</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBi</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(isotropic) — the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBd</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(dipole) — the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBiC</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(isotropic circular) — the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBq</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(quarterwave) — the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0 dBq = -0.85 dBi</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Other measurements</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">dBFS or dBfs</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(full scale) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS (peak) would equal the highest level (number) the processor is capable of representing. Measured values are always negative or zero, since they are less than the maximum or full-scale. Full-scale is typically defined as the power level of a full-scale sinusoid, though some systems will have extra headroom for peaks above the nominal full scale.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB-Hz</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations. Also used in carrier-to-noise-density ratio (not to be confused with carrier-to-noise ratio, in dB).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBov or dBO</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(overload) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBr</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB(relative) — simply a relative difference from something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dBrn</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">dB above reference noise. See also dBrnC.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span"><b>Lenny Z Perez M</b></span></div><div style="text-align: justify;"><span class="Apple-style-span"><b>EES</b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/decibel-decibel-db-is-logarithmic-unit.html"><span class="Apple-style-span"><b>http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/decibel-decibel-db-is-logarithmic-unit.html</b></span></a></span></div></div></span></div><br />
<hr />Get news, entertainment and everything you care about at Live.com. <a href="http://www.live.com/getstarted.aspx" target="_new">Check it out!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-1237461586309996732010-03-21T10:37:00.002-04:302010-03-23T17:47:58.726-04:30Significance of Octaves and Decades<div><br />
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<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>OCTAVE AND OTHER FREQUENCY INTERVALS</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The octave in music</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In music, the octave is the interval between two frequencies which are in the ratio of 2-to-1 (i.e. the higher frequency is exactly twice the lower frequency). Check out the following examples.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI6c1XM2kC5e0pgwqr2-CO7biuL4TZKBwihqk4gd5bc0MjtqIVFsAt6eDET0ICULsohyphenhyphentT1bvHUpNbPONTVsWqXL11RXv9ovyBsGhJFWyTOhUoxOg9WSlwGdA5yhM1sJVrsXN36MLMHbY/s1600-h/5.1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450952533103466482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI6c1XM2kC5e0pgwqr2-CO7biuL4TZKBwihqk4gd5bc0MjtqIVFsAt6eDET0ICULsohyphenhyphentT1bvHUpNbPONTVsWqXL11RXv9ovyBsGhJFWyTOhUoxOg9WSlwGdA5yhM1sJVrsXN36MLMHbY/s400/5.1.bmp" style="cursor: pointer; height: 89px; width: 378px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The complete musical scale is generated by defining one frequency (the note A which is 440 Hz) and working out all of the other musical notes, with sharps and flats etc, from that defined frequency.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The octave in acoustics and audio</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In acoustics and audio, the octave is the interval between two frequencies which are in the ratio of 10 0.3 to 1 10 0.3 = 1.995 (Calculator key strokes: [shift] [log] [0] [.] [3] [=] )</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The standard octave intervals in acoustics are worked out starting from the Reference Frequency of 1 kHz which is 10 3.0 Hz. The full sequence of frequencies is:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitXh5NE4pcuvNjAGUCxLVopkHiW-LK52PEsOmrG2TJULnmtYb0RHq6DEFNts4wLtTXM6M9Gd7mCot98-zTK1TqmdN2F4nBpUCmY4DNuE8oNdmrCaxqg0tKwi2R-QZIVBnLc2mP0vW6IZE/s1600-h/5.2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450952699433005346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitXh5NE4pcuvNjAGUCxLVopkHiW-LK52PEsOmrG2TJULnmtYb0RHq6DEFNts4wLtTXM6M9Gd7mCot98-zTK1TqmdN2F4nBpUCmY4DNuE8oNdmrCaxqg0tKwi2R-QZIVBnLc2mP0vW6IZE/s400/5.2.bmp" style="cursor: pointer; height: 44px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">On page 2 there is an image of an octave band graphic equalizer. Look below the sliders for the nominal frequencies listed on the bottom row of the table above.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Fractions of an octave</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">On the Klark-Teknik graphic equalizer, on page 3, there are two sliders between each of the standard octave sliders. For example, between the sliders at 250 Hz and 500 Hz there is one at 315 Hz and one at 400 Hz. The sliders on this graphic equaliser are arranged at one-third-octave intervals.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZvtsT2nOtUySGBzzje-zUQLrZ0FImKNvtAhV-d_axcWqy1WjkgE_Xt0k47OAwJTrEPMG2N9iqevOimIsupP8ehQ_7G8HSd0jPw00ESU9Ucv5d4T69QvDQ-HSd9gXRarXYCNKk5geiBJ4/s1600-h/5.3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450952976856157586" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZvtsT2nOtUySGBzzje-zUQLrZ0FImKNvtAhV-d_axcWqy1WjkgE_Xt0k47OAwJTrEPMG2N9iqevOimIsupP8ehQ_7G8HSd0jPw00ESU9Ucv5d4T69QvDQ-HSd9gXRarXYCNKk5geiBJ4/s400/5.3.bmp" style="cursor: pointer; height: 50px; width: 400px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">For an octave, the ratio is 10 0.3 . The number 0.3 is the exponent (i.e. the power of ten).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">We work out an interval which is a given fraction of an octave by taking the same fraction of 0.3</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">So, for one-third-octave, the exponent is a third of 0.3, i.e. 1/3 × 0.3 = 0.1 and 10 0.1 = 1.2589 .</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">This table explains how a number of fractions of an octave can be calculated.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5Kmfg3CZXlyclo688rKjBUuX6jGu2CL7EG4PwltrQB88Uxmj0mEbw9ZAcr0LnubZq3yFXhluBvS08hTvNsrJ5GrtlTgPrmsZ0Qfg055U8OELZA3-uEqM0SutLZODmcpoc_tZSh_ItCko/s1600-h/5.4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450953376051393618" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5Kmfg3CZXlyclo688rKjBUuX6jGu2CL7EG4PwltrQB88Uxmj0mEbw9ZAcr0LnubZq3yFXhluBvS08hTvNsrJ5GrtlTgPrmsZ0Qfg055U8OELZA3-uEqM0SutLZODmcpoc_tZSh_ItCko/s400/5.4.bmp" style="cursor: pointer; height: 67px; width: 400px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Octave bands</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The image below is a graphics user interface (GUI) from some audio-processing computer software and it shows an octave band equaliser. There are 10 octave bands in the audio range; hence 10 sliders.</span></span></div></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDBgt7n11ckrnvhDqnlNcuYhkf2a6yCNF8stLMsZvDBKhIBZ6whRYVdQyErAE7aELfi4xKPUnGazUHGOPHVBHN2aia3LCXzk0Ek_PnzleoICRkeyY3aMZkTT4xLh4cZ_km3jNS8hyphenhyphenMLTg/s1600-h/5.5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450954052233824114" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDBgt7n11ckrnvhDqnlNcuYhkf2a6yCNF8stLMsZvDBKhIBZ6whRYVdQyErAE7aELfi4xKPUnGazUHGOPHVBHN2aia3LCXzk0Ek_PnzleoICRkeyY3aMZkTT4xLh4cZ_km3jNS8hyphenhyphenMLTg/s320/5.5.bmp" style="cursor: pointer; height: 194px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Below the sliders, you can see the octave interval nominal frequencies *. Each slider controls the gain for a one-octave-wide band of frequencies. The gain, in dB, is shown in the window above each slider.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Each slider=s quoted frequency is at the centre of its respective octave band. The quoted frequency is called the octave band centre frequency. We identify which octave band we are controlling by quoting the octave band centre frequency. Each of the ten octave bands reaches half an octave above the band centre frequency and half an octave below the band centre frequency. So:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">the Band Upper Limit is half an octave above (i.e. × 10 0.15) the band centre frequency; and</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">the Band Lower Limit is half an octave below (i.e. ÷ 10 0.15) the band centre frequency.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">For the 500 Hz octave band (shown with a gain of + 2.9 dB in the image):</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">the exact band centre frequency is 10 2.7 (see page 1 for how this is worked out);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">the band upper limit is 10 2.7 × 10 0.15 = 10 2.7 + 0.15 = 10 2.85 = 707.9 Hz</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">the band lower limit is 10 2.7 ÷ 10 0.15 = 10 2.7 - 0.15 = 10 2.55 = 354.8 Hz</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The slider labelled 500 Hz does not adjust the gain only for the frequency of 500 Hz. It adjusts the gain for an octave band of frequencies from 354.8 Hz up to 707.9 Hz. 500 Hz is at the centre of that octave band. The slider adjusts the gain for the whole of the 500 Hz octave band.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The lowest frequency being processed by the software above, is not 31.5 Hz *, but is the frequency at the band lower limit of the 31.5 Hz octave band. (See question 2(e) on page 4.)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The highest frequency being processed by the software is not 16 kHz, but is the frequency at the band upper limit of the 16 kHz octave band. (See question 2(a) on page 4.)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The above image is of an octave band equaliser. The following page shows an example of a one-third-octave band equaliser. The frequency bands are narrower and the EQ control much finer.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">(* 31 Hz should be labelled 31.5 Hz; and 62 Hz should be labelled 63 Hz. Many manufacturers get these two wrong.)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">One-third-octave Bands</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The image below shows a one-third-octave band graphic equaliser. There are 10 octaves in the audio range; so there are 30 one-third-octaves in the audio range.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Each one of the 30 sliders adjusts the gain of the signal for a band of frequencies which is one-third-of-an-octave wide.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">These professional equalisers offer fine detailed adjustment to the equalisation (EQ) of the audio signal.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The frequency quoted under each slider is the nominal Band Centre Frequency. So each third-octave band extends from one-sixth of an octave below the band centre frequency to one-sixth of an octave above the band centre frequency. The two extremes are called the Band Lower Limit and the Band Upper Limit.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Notice that the vertical position of the sliders is calibrated in dB with markings at +12, +6, +3, 0, -3, -6, and -12 dB, showing how the power level of each band is increased or decreased relative to the flat response , 0 dB, position.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic64OJdrGxIh0C9p4v1hFFwMkuMyan_Vo3XzEp2Z2axam8z1Nb8m6geFWFqr5KJmG9e9vLU-ext4l2mDuuL2khV1BoHjGvIKp-ZVefor7Lsx8b2dCGQs8csJP0L6GMqmwtgdkAIBNdqAI/s1600-h/5.6.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450954414371292706" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic64OJdrGxIh0C9p4v1hFFwMkuMyan_Vo3XzEp2Z2axam8z1Nb8m6geFWFqr5KJmG9e9vLU-ext4l2mDuuL2khV1BoHjGvIKp-ZVefor7Lsx8b2dCGQs8csJP0L6GMqmwtgdkAIBNdqAI/s320/5.6.bmp" style="cursor: pointer; height: 105px; width: 320px;" /></span></span></a></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Decade (log scale)</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">One decade is a factor of 10 difference between two numbers (an order of magnitude difference) measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronic systems, such as audio amplifiers and filters.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Calculations</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The factor-of-ten in a decade can be in either direction: so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, so 3.14 rad/s is one decade down from 31.4 rad/s.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">To determine the number of decades between two frequencies, use the logarithm of the ratio of the two values:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">How many decades is it from 15 rad/s to 150,000 rad/s?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">log10(150000 / 15) = 4 decades</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">How many decades is it from 3.2 GHz to 4.7 MHz?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">log10(4.7x10´6 / 3.2x10´9) = -2.86 decades</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">How many decades is one octave?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">One octave is a factor of 2, so log10(2) = 0.301 decades per octave</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">To find out what frequency is a certain number of decades from the original frequency, multiply by appropriate powers of 10:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">What is 3 decades down from 220 Hz?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">220x10´3=0.22 Hz</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">What is 1.5 decades up from 10?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">10x10´1.5=316.23 Hz</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">To find out the size of a step for a certain number of frequencies per decade, raise 10 to the power of the inverse of the number of steps:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">What is the step size for 30 steps per decade?</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">101 / 30 = 1.079775 - or each step is 7.9775% larger than the last.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgexQB41bbYRbidTQBXwL-fj-QJmZtFF3gf75KVyfaKsYfGoMdIqdSMl6pxYdRB9KnvVnxrwMM0_6NtSQaji5olOSqtKA_hxJnfIaxjr_OKFpOR6MMbr4V-1ZPRssDTJQ0SpyxNX_WYOTc/s1600-h/5.8.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450954703645013426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgexQB41bbYRbidTQBXwL-fj-QJmZtFF3gf75KVyfaKsYfGoMdIqdSMl6pxYdRB9KnvVnxrwMM0_6NtSQaji5olOSqtKA_hxJnfIaxjr_OKFpOR6MMbr4V-1ZPRssDTJQ0SpyxNX_WYOTc/s320/5.8.bmp" style="cursor: pointer; height: 227px; width: 320px;" /></span></span></a></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;"><span class="Apple-style-span"><b>Graphical representation and analysis</b></span></div><div style="text-align: justify;"><span class="Apple-style-span">Decades on a logarithmic scale, rather than unit steps (steps of 1) or other linear scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in Bode plots, since depicting large frequency ranges on a linear scale is often not practical. For example, an audio amplifier will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (100) and go up to perhaps 100 kHz (105), to comfortably include the full audio band in a standard-sized graph paper, as shown below. Whereas in the same distance on a linear scale, with 10 as the major step-size, you might only get from 0 to 50.</span></div></span><br />
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<div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: none; color: black;"></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhClcNVKmegNA0ymIOyc6Z8Bc83_mL8_6AivICIbbNYsljj9DU832i95IwYmLxhjx4VX_7H-EAPI_soa-IFsspKnN4rXLXJMzdcqMNFpTD39jE-db1WD_VLdu_YbJ1EDl7E6gF4Sah_e_A/s1600-h/5.9.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450954923823721938" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhClcNVKmegNA0ymIOyc6Z8Bc83_mL8_6AivICIbbNYsljj9DU832i95IwYmLxhjx4VX_7H-EAPI_soa-IFsspKnN4rXLXJMzdcqMNFpTD39jE-db1WD_VLdu_YbJ1EDl7E6gF4Sah_e_A/s320/5.9.bmp" style="cursor: pointer; height: 126px; width: 320px;" /></a></span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><br />
</span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Lenny Z Perez M.</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>EES</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/octave-and-other-frequency-intervals_20.html"><span class="Apple-style-span"><b>http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/octave-and-other-frequency-intervals_20.html</b></span></a></span></div></span></div><br />
<hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href="http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us" target="_new">Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-52408175334665574242010-03-21T10:33:00.002-04:302010-03-23T17:47:37.953-04:30Human ear response to audio frequencies<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Music and The Human Ear</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">This section contains information on the softest and loudest sounds we can hear, the range of frequencies we can hear, subjective vs. objective loudness, how we locate the source of a sound, and sound distortion. This section focuses mainly on the ear itself, but the brain is an integral part of the human hearing system. A separate section considers the function of the brain in more detail.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The human ear is a truly remarkable instrument. At one point in my life I designed Electronic Counter Measures (ECM) systems for the U. S. military. The primary function of an ECM system is to detect an enemy before he (it's rarely a she) detects you, for self-defense. It is interesting to compare the characteristics of a good ECM system and human hearing:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><i><span class="Apple-style-span">Comparison of characteristics</span></i></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="color: #0e0010;"><b><u><span class="Apple-style-span" style="color: black;"><span class="Apple-style-span" style="font-weight: normal;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></span></span></u></b></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTTWi8EieOqRWBqtI2OC2EwGEnJk83ffiWQF7Xbf_K-zWQh-nUB_7yIJYCQm7TqYWWcYBHfWxmHwBeROFS65h14yc1CAjjgUJQKhbp-qVdYVwHxQBJo-KCB022KLMx1FcXAnPyPAW-Bok/s1600-h/2.1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450894175130588162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTTWi8EieOqRWBqtI2OC2EwGEnJk83ffiWQF7Xbf_K-zWQh-nUB_7yIJYCQm7TqYWWcYBHfWxmHwBeROFS65h14yc1CAjjgUJQKhbp-qVdYVwHxQBJo-KCB022KLMx1FcXAnPyPAW-Bok/s320/2.1.bmp" style="cursor: pointer; display: block; height: 80px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Human hearing is a superior defensive system in every respect except source location accuracy. Note: Jourdain (page 23) states that human accuracy is 1-2 degrees in azimuth.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In contrast, a military system designed for communications (rather than detection) would typically have a much smaller ratio of highest-to-lowest frequency, no source location capability, and often a narrow directional coverage. For human communication a frequency ratio of 10:1 and a ratio of strongest to weakest signal of 10,000:1 would suffice. The far larger actual ratios strongly imply a purpose other than communication.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">All of this tells me that the ear evolved primarily for self-defense (or perhaps hunting, as one reader pointed out), and language and enjoyment of music are delightful evolutionary by-products. A defensive purpose also suggests some direct hard-wiring between the ears and primitive parts of the brain, which may account for the powerful emotional impact of music - and its virtual universality among human cultures. A few years after writing this paragraph I found the very interesting book This is Your Brain on Music which confirms the speculation on wiring to primitive parts of the brain, but argues that music has a definite evolutionary function.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Soft Sounds and Loud Sounds</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span"><br />
</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Acknowledgment: a good part of the material in the remainder of this section is derived from an excellent book The Master Handbook of Acoustics by F. Alton Everest, and from the chapter he contributed to the Handbook for Sound Engineers. Seereferences. These sources also contain much additional interesting material. David Worrall has posted his course notes of Physics and Psychophysics of Music on the web, which includes an informative section on the physiology of hearing. A series of tutorial papers on hearing and other related topics has also been posted by HeadWize.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Sound pressure level (SPL) is given in dB SPL. This is a scale that is defined such that the threshold of hearing is close to 0 dB. The threshold of pain is about 135 dB. This is a logarithmic scale where power doubles for each 3 dB increase; the 135 dB difference between the thresholds of hearing and pain means the power doubles about 45 times - an increase of 32 trillion (32x1012) in the power level. This is an incredible dynamic range, and totally blows away anything human engineers are capable of creating. (Actually in a Dec 99 Newsgroup post Dick Pierce states that B&K 4138 microphones have a dynamic range of 140 dB, so I was underrating human engineers). At the low end of the range the ears lose function due to background noise. At 0 dB SPL noise created by blood flow in the ear is one source. It is shown elsewhere that the noise of molecules colliding with the eardrum is not far below this level. At the threshold sound level of 0 dB SPL Everest states that the eardrum moves a distance smaller than the diameter of a hydrogen molecule! At first I was incredulous when I read this, but it is consistent with the change in diameter of the balloon example used in the previous section. For a 0 dB SPL the change in balloon diameter is 6x10-10 inches, which is about 1/10 of the diameter of a hydrogen atom. The sensitivity of the ear is truly mind-boggling.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Pressure is an objective physical parameter. The relationship of SPL to the subjective sensitivity to sound is discussed below. The human ear is most sensitive in a band from about 2,000-5,000 Hz. This is an important region for understanding speech, and could be construed to imply that hearing evolved to match speech. However, did the ear evolve to be sensitive to the speech frequency band, or did human speech evolve to match the band where the ear is most sensitive? (I read somewhere that babies cry in the frequency band where the ear is most sensitive). As measured by Voss and Allen, a typical eardrum absorbs about 75% of the incident sound energy at 5 kHz. The sensitivity vs. frequency behavior has a fair resemblance to the response of a piston load matched to the impedance of air, as shown in the physics section. Music levels vary from about 50 dB for quiet background music to maybe 120 dB for a very loud rock band. Subjectively, a 2-3 dB change in sound level is barely perceptible; if someone asks you to "turn up the volume a little," you will probably increase the sound by at least 3 dB. (Note that if you have a 100-Watt amplifier and it doesn't play loud enough, you need a 200-Watt amplifier to turn up the volume 3 dB. This can get very expensive very quickly). Interestingly there were some ABX test results on the web which indicate that a 0.3 dB difference in level can be detected (link no longer exists). However the test procedure allows switching between the two levels as much as you want before making a decision, and the test used pink noise for the sound. You can hear what a 3 dB difference sounds like yourself with sound files in the sound demo section.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A full orchestra can also hit a sound level of 110 dB and more, and then play a quiet passage at 20-30 dB. To reproduce this faithfully requires a recorded sound source capable of covering this 80+ dB dynamic range. (Everest quotes one researcher who claims a 118 dB range is required). A vinyl record is good for about 50-70 dB; a standard compact disc with 16-bit encoding can cover a 96 dB range, and the 24-bit DVD disk format a 144 dB range - in theory. Real D/A converters tend to be noise limited to a somewhat lower range.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A problematic aspect of music for a sound system designer is that there are brief transients ("spikes") in sound level that far exceed average power levels. Usually people talk about average, or root-mean-square (RMS) power. RMS power is really only important with respect to the generation of heat. In my opinion, peak power is far more important, since this is when a speaker could be driven into a non-linear region, and when an amplifier would clip. These two effects are major causes of distortion. Using Cool Edit 96, I recorded 10-20 second segments from Talking Heads "Burning Down the House," Diana Krall "All or Nothing at All," and Shostakovich Symphony #5. I then processed the cuts in Matlab, to generate the outputs of a 3-way crossover. The crossover frequencies are 300 and 3000 Hz. Both 1st order Butterworth and 4th order Linkwitz-Riley filters were modeled. Finally I calculated the average and peak power in each driver band, with results as shown in the tables below.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></div></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgftst2ofG1BTEOwqC1YEeEythBHw78DqbkzzyLiOBghis7noAZHIhvSi4MPVp0eNXWIFBJ0fHE-VR03bQl5dv5yJir4VUHJxK_72nlBydHbRYtUkMhwZcJoOUOb6q-_hHwqy5yeDkYOzA/s1600-h/2.2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450894573855669026" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgftst2ofG1BTEOwqC1YEeEythBHw78DqbkzzyLiOBghis7noAZHIhvSi4MPVp0eNXWIFBJ0fHE-VR03bQl5dv5yJir4VUHJxK_72nlBydHbRYtUkMhwZcJoOUOb6q-_hHwqy5yeDkYOzA/s320/2.2.bmp" style="cursor: pointer; display: block; height: 164px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: justify; width: 320px;" /></a><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">All powers are shown as a percentage of the same quantity in the unfiltered music. Note that the average power for the Butterworth adds to 100%, but the Linkwitz-Riley adds to less than 100%. The voltage output of a Linkwitz-Riley coherently adds to unity, but the power addition is less than unity. The peak power is obtained by computing the time-domain waveform of the signal output by the crossover. Then the peak value is found. Typically the peaks occur at different times for the tweeter, midrange, and woofer, so there is no physical significance to the sum of the three powers in this case. The startling result is that by far the greatest demands on peak power are in the midrange for the Krall and Shostakovich. The 4th order reduces the demands in the high and low bands, but there is little difference in the mid-band. Only the Talking Heads cut has a greater demand in the bass. It is also quite significant that even though the average tweeter power is low, the peak tweeter power is not all that much lower than other bands, and in fact is greater than the woofer in some cases!</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">When I play the Talking Heads cut, my CLIO sound measurement system shows a peak sound level of 100 dB SPL in the room, and an average of around 95 dB. Judging from the oscilloscope connected to the amp outputs, the average amplifier output power appears to be about 17 watts. The ratio of peak power to RMS power was 40:1, 40:1 and 30:1 for the Talking Heads, Diana Krall, and Shostakovich cuts respectively. Therefore, for 17 watt RMS, the peak power demands are on the order of 700 Watts. This indicates that either my amps can put out peaks much higher than their rated power (possible, but I'm not sure), or they are clipping. There are demo files In the sound demo section which simulate clipping by tube and solid-state amplifiers. For more on this subject see the section on amplifier distortion.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Jourdain (page 41) states that an orchestra produces 67 watts of acoustic power at full blast. Loudspeakers have efficiencies on the order of 0.5 to 2% converting electrical power to acoustic power. Even at 2% efficiency this implies that well over 3,000 watts of electrical power would be required to duplicate this sound level. Of course an orchestra plays in a large auditorium, and no doubt less power is needed for a small room. This still indicates that power requirements should not be underestimated.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">The Audio Spectrum</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span"><br />
</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A major criterion of a good sound system is its frequency response. The usual frequency range considered "hi-fi" is 20-20,000 Hz. These sample tones are audible with good loudspeakers or headphones, but many computer speakers will not reproduce them at all: a 100 Hz tone, (12 kb wav file) and a 10,000 Hz tone (44 kb wav file). Yesterday I did a test using the very accurate signal generator built into my CLIO system. I can clearly hear, and certainly can feel, a 10 Hz tone. My sound system totally poops out below 10 Hz, so I can't test any lower than that. The lowest notes on organs and pianos are 16.4 and 24.5 Hz respectively. Testing at the other extreme, as a 61 year-old male (when I originally wrote this) I can hear a 13,500 Hz tone, but no higher. (It is generally agreed that women are more sensitive to high frequencies). However, good high frequency response is required to produce sharp transients, such as a snap of the fingers. I performed a test using a Ry Cooder CD, "Talking Timbuktu." Track 10 on this disk has some very sharp transients that just leap out at you from a good sound system. My pre-amp has a filter that cuts off frequencies above 12,000 Hz. With this filter in, the transients limp out rather than leap out. This shows that even though I cannot hear a pure tone in most of the range of frequencies cut out by the filter, I can clearly hear the difference in the sound quality of the transients. I repeated this test recently (at age 67) with a segment of this cut recorded as a .wav file, and digitally processed with a 12kHz filter. This time the test was a double-blind ABX test, and I can't reliably detect any difference (I can still hear a 13,000 Hz tone). I now doubt the validity of the earlier test. See the discussion on high frequency tests in the section on sound demos.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">James Boyk at Caltech has posted an interesting paper on the frequencies generated by musical instruments between 20kHz and 102 kHz! He also cites a paper that states that people react to sounds above 26 kHz even when they cannot consciously hear the sound. Jourdain (page 42) states that sound can be heard up to 40 kHz if sufficiently loud (A knowledgeable reviewer of the book is skeptical about this claim. Unfortunately the link to the review no longer works).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The ear tends to combine the sound within critical bandwidths, which are about 1/6 octave wide (historically thought to be 1/3 octave). This has led to the practice of averaging frequency response over 1/3 octave bands to produce beautiful-looking frequency response curves. In my opinion this is misleading. Suppose a loudspeaker has a bad dropout (very weak response) over a narrow frequency range; the dropout will be totally obscured by averaging. But when a musical instrument plays a note that just happens to fall in the dropout notch, you will not be able to hear the note. See the example of a warts-and-all response(28.2 kb) vs. a 1/3 octave smoothed response (24.5 kb) from my final system measurements section. Since we can barely hear a 2-dB difference in sound level, it is reasonable to accept ±2 dB as an excellent level of performance for frequency response. In fact this is impossible to achieve in the real world, due to room acoustics. (see the section on room acoustics). Personally I would say a more-or-less practical goal for a sound system installed in a room is a frequency response ±5 dB from 200-20,000 Hz, and maybe ±10 dB from 10-200 Hz. It is also worth noting that the ear itself has a quite variable frequency response, as shown by measured data on head-related transfer functions, and as discussed in the next section.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">What is the minimum audible change in frequency? I created two .wav files: case #1 was a series of 1/2 second tone bursts, all at a frequency of 800 Hz; for case #2 the bursts alternated between 800 and 805 Hz. I can reliably distinguish between these two cases in a double-blind test. This difference in frequency is less than 1/100 of an octave. I could also distinguish between 400 and 402 Hz. According to Jourdain (page 18) this is about normal for a young person; at age 61 I'm not supposed to be able to detect a difference of less than about 8 Hz at 400 Hz. But I can. (I repeated this test at age 67, and I still can do it). Sample files are described in the sound demo section. An interesting detail is that tone bursts that start and stop abruptly are easier to discriminate than bursts with a fade-in fade-out. I don't know if this is simply a timing issue, or if the brain is making use of the higher Fourier transform sidelobes that occur for a square window (the spectrum for a tapered burst is extremely narrow, the square burst spectrum has extensive sidelobes about 40 dB below the peak).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">For music the audio spectrum is divided into discrete notes. A brief discussion of the interesting subject of musical scales is given in a separate section.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Subjective vs. Objective Sound Levels</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">SPL is an objective measurement of sound pressure, or power in watts, and is independent of frequency. In 1933 Fletcher and Munson of Bell Labs did a study that showed that subjective sound levels varied significantly from the SPL level. That is, when two tones were played at the exactly the same SPL level, one sounded louder than the other. And the results were very dependent on how loud the tones were to begin with. This is illustrated by the set of Fletcher-Munson curves [102 Kb]. The vertical axis is the objective SPL sound level. Each of the curves in the graph represents a constant subjective sound level, which are in units called "phones." The lowest curve is the minimum audible level of sound. As noted above, the ear is most sensitive around 2-5 kHz. To be audible at this minimum level, a sound at 20Hz must be 80 dB (100 million times!) more powerful than a sound at 3 kHz.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Near the top, the curve at 100 phones is a fairly loud level. To sound equally loud at this level the sound at 20 Hz must be about 40 dB more powerful. This change in subjective level for different loudness levels means that music played softly will seem to be lacking in bass. For years pre-amps have come equipped with "loudness" controls to compensate for this. For me, part of "Hi-fidelity" means playing music at the same level it was originally played, so this is all academic - but interesting none the less.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Source Location</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span"><br />
</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">An important characteristic of a sound system is the "sound image." An ideal system would create a vivid illusion of the location of each musical instrument. In designing a system it is important to understand, as well as current knowledge permits, how we locate the source of a sound. One thing that is clear is that the brain processes several different types of data to extract directional information. The data include:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">•</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">shape of the sound spectrum at the eardrum</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">•</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">difference in sound intensity between the left and right ears</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">•</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">difference in time-of-arrival between the left and right ears</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">•</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">difference in time-of-arrival between reflections from the ear itself</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A remarkable fact is that the pinna, the cartilage-filled structure surrounding the ear canal (commonly simply called the "ear"), is a vital part of direction sensing. Test subjects can be trained to locate sound using only one ear. But when the ridges of the pinna are gradually filled in, the ability is lost, in proportion to the filled in area. Apparently the brain uses reflections from the ridges of the pinna (19.4 kb) to determine direction. The head and pinna have a major effect on the sound that arrives at the ear. This effect is mathematically represented by a head-related transfer function (HRTF). There are files in the sound demo section where a monophonic sound source is processed with HRTFs to synthesize sound arriving from various directions. The full HRTFs contain both the difference in sound intensity, and difference in time-of-arrival. There are two other demo files where only one of these two differences are retained. When I listen to these files I perceive the apparent direction almost equally well with all three files, indicating that the brain has a remarkable capability of making good use of whatever information it gets.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The significance of the pinna reflection experiments for a sound system designer is that time delays on the order of 0.1 millisecond can effect sound imaging. Time delays between the left and right ear are on the order of 0.5 milliseconds, and are quite important. On the other hand, researchers have found that echoes in the range of 1 to 50 milliseconds are lumped together by the brain with the direct sound, so they are not actually heard as distinct echoes. Delays greater than 50 milliseconds are heard as echoes. My own echo research is described in the sound demo section, and you can listen to the results yourself. Echoes in the range of 25 to 100 milliseconds give a "cavernous" quality to the sound. What is commonly called an "echo," a distinct repetition of the original sound, only occurs for echoes of 400 milliseconds or longer. Echoes in the range of 0.1 to 2 milliseconds do cause changes in the apparent direction of the source.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A regular CD sampled at 44.1 kHz is theoretically capable of reproducing frequencies up to 22 kHz, which corresponds to a transient duration of .05 milliseconds. However, as discussed in a recent paper by Mike Story (e-mail mstory@dcsltd.co.uk to request a copy) the anti-aliasing filters required to record within this band cause the transients to be blurred, in effect smearing the ability of our ears to distinguish direction. Mike reports that in listening tests 96 kHz recordings provide notably better spatial resolution. In the Handbook for Sound Engineers Steve Dove says anti-aliasing filters "....exhibit serious frequency dependent delay and convoluted frequency/phase characteristics... leaving mangled audio in their wake". He also advocates sampling around 100 kHz, and says the result is a more open and spacious sound. Humans perceive left-right direction more accurately than up-down direction. Presumably this is due to the fact that we generally move in two dimensions along a more-or-less level surface. All of this information is important for the sound system designer, particularly regarding the control of sound diffractionand reflection, both of which can muddle the sound image.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Distortion</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Distortion is a commonly accepted criterion for evaluating high-fidelity sound equipment. It is usually understood to mean the tones in the reproduced sound that were not present in the original sound. An ideal sound system component has a perfectly linear response. This means that the ratio of the output and the input signal magnitude is always exactly the same, and the relative phase is constant, regardless of the strength of the signal. For a non-linear response (anything other than a linear response), distortion will occur. It is commonly categorized as total harmonic distortion (THD) and intermodulation distortion. Harmonic distortion means that a pure 1000 Hz input tone results in spurious outputs at 2000 Hz, 3000 Hz, and other integer multiples of the input frequency. Intermodulation distortion means two input tones at 1000 Hz and 100 Hz result in spurious outputs at 900 Hz, and 1100 Hz, among others.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The audibility of phase distortion is controversial. Some loudspeaker manufacturers, such as Dunlavy (apparently now out of business), cite flat phase response as a significant feature of their products. There is no question that under some artificial circumstances phase distortion is audible. Further discussion on the interesting topic of phase audibility can be found here.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">So called "Doppler" distortion is produced by the motion of the loudspeaker cone itself. This creates some harmonic distortion, but the most significant effect is intermodulation distortion. This class of distortion can only be reduced by reducing the cone motion. A large surface, such as the membrane of an electrostatic speaker, will produce very little Doppler distortion. See theanalysis for a piston in a tube for technical details.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Also see the discussion above on "clipping."</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Everest quotes research indicating that amplitude distortion has to reach a level of 3% to be audible. However this varies greatly depending on the distortion harmonic products, and on the sound source. More on this below. Good CD players, amplifiers and pre-amplifiers typically have distortion levels of 0.1% or less. (Tube amps typically have higher distortion). Loudspeakers are the weak link regarding distortion. It is hard to even get information on loudspeaker distortion since it looks embarrassing compared to the values advertised for electronics. I measured 2nd and 3rd harmonic distortion of my sound system end-to-end using myCLIO sound measuring system. Since speaker distortion dominates, this is essentially a measurement of speaker distortion. The measurement was made using one speaker; with two speakers the distortion would be the same, but the SPL levels would increase 6 dB for the two lower frequency bands, and 3 dB for the upper bands. The entire measured distortion curve at the higher power level is shown in the section on final system measurements.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Distortion is universally considered to be bad, and it is perhaps not generally realized that musical instruments introduce overtones that have similarities to distortion. I imagine most music lovers are aware that all musical instruments produce a fundamental tone (the "note"), and a series of overtones. The overtones are at frequencies higher than the fundamental tone, and give the sound a rich quality not possessed by a pure tone. Overtones are generally harmonics (integer multiples) of the fundamental frequency. The relative strength of the various harmonics gives the instrument its characteristic sound. You can hear a comparison of a real piano note [42kb] and a tone {42kb] with the same fundamental frequency, but lacking in overtones. There is also additional description of the spectrum of this note.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The ear is not perfectly linear and produces distortion. A short discussion of the non-linear behavior of the ear can be found in aseparate section. Finally, air itself is non-linear, and harmonic distortion grows steadily as a wave propagates (see plane waves in the physics section). This is usually a very small effect, but can be significant in the throat of a horn speaker.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The subject of sound quality is not at all clear-cut. Even though tube amplifiers have higher measured distortion, a lot of knowledgeable people swear that they sound better. I finally dove into this subject in August 2006. I can clearly hear THD at 0.5% for a pure 440 Hz tone and the type of harmonics produced by a typical solid-state amp; for the type of harmonics produced by a single-ended triode amp I could not detect distortion until it reached a level of 10%. This amazing difference is covered in detail in the section on amplifier distortion. For music samples the difference is not quite as big, but is still quite significant. Many people have come to the conclusion that THD is a terrible way to judge amplifier quality, and I totally agree. Norman Koren, an advocate of tube amplifiers, has posted a very interesting commentary on the subject of distortion and the effect of feedback.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Human Hearing</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span"><br />
</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The human ear is an exceedingly complex organ. To make matters even more difficult, the information from two ears is combined in a perplexing neural network, the human brain. Keep in mind that the following is only a brief overview; there are many subtle effects and poorly understood phenomena related to human hearing.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Figure 22-1 illustrates the major structures and processes that comprise the human ear. The outer ear is composed of two parts, the visible flap of skin and cartilage attached to the side of the head, and the ear canal, a tube about 0.5 cm in diameter extending about 3 cm into the head. These structures direct environmental sounds to the sensitive middle and inner ear organs located safely inside of the skull bones. Stretched across the end of the ear canal is a thin sheet of tissue called the tympanic membrane or ear drum. Sound waves striking the tympanic membrane cause it to vibrate. The middle ear is a set of small bones that transfer this vibration to the cochlea (inner ear) where it is converted to neural impulses. The cochlea is a liquid filled tube roughly 2 mm in diameter and 3 cm in length. Although shown straight in Fig. 22-1, the cochlea is curled up and looks like a small snail shell. In fact, cochlea is derived from the Greek word for snail.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">When a sound wave tries to pass from air into liquid, only a small fraction of the sound is transmitted through the interface, while the remainder of the energy is reflected. This is because air has a low mechanical impedance (low acoustic pressure and high particle velocity resulting from low density and high compressibility), while liquid has a high mechanical impedance. In less technical terms, it requires more effort to wave your hand in water than it does to wave it in air. This difference in mechanical impedance results in most of the sound being reflected at an air/liquid interface.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The middle ear is an impedance matching network that increases the fraction of sound energy entering the liquid of the inner ear. For example, fish do not have an ear drum or middle ear, because they have no need to hear in air. Most of the impedance conversion results from the difference in area between the ear drum (receiving sound from the air) and the oval window (transmitting sound into the liquid, see Fig. 22-1). The ear drum has an area of about 60 (mm)2, while the oval window has an area of roughly 4 (mm)2. Since pressure is equal to force divided by area, this difference in area increases the sound wave pressure by about 15 times.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Contained within the cochlea is the basilar membrane, the supporting structure for about 12,000 sensory cells forming thecochlear nerve. The basilar membrane is stiffest near the oval window, and becomes more flexible toward the opposite end, allowing it to act as a frequency spectrum analyzer. When exposed to a high frequency signal, the basilar membrane resonates where it is stiff, resulting in the excitation of nerve cells close to the oval window. Likewise, low frequency sounds excite nerve cells at the far end of the basilar membrane. This makes specific fibers in the cochlear nerve respond to specific frequencies. This organization is called the place principle, and is preserved throughout the auditory pathway into the brain.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Another information encoding scheme is also used in human hearing, called the volley principle. Nerve cells transmit information by generating brief electrical pulses called action potentials. A nerve cell on the basilar membrane can encode audio information by producing an action potential in response to each cycle of the vibration. For example, a 200 hertz sound wave can be represented by a neuron producing 200 action potentials per second. However, this only works at frequencies below about 500 hertz, the maximum rate that neurons can produce action potentials. The human ear overcomes this problem by allowing several nerve cells to take turns performing this single task. For example, a 3000 hertz tone might be represented by ten nerve cells alternately firing at 300 times per second. This extends the range of the volley principle to about 4 kHz, above which the place principle is exclusively used.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Table 22-1 shows the relationship between sound intensity and perceived loudness. It is common to express sound intensity on a logarithmic scale, called decibel SPL (Sound Power Level). On this scale, 0 dB SPL is a sound wave power of 10-16watts/cm2, about the weakest sound detectable by the human ear. Normal speech is at about 60 dB SPL, while painful damage to the ear occurs at about 140 dB SPL.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The difference between the loudest and faintest sounds that humans can hear is about 120 dB, a range of one-million in amplitude. Listeners can detect a change in loudness when the signal is altered by about 1 dB (a 12% change in amplitude). In other words, there are only about 120 levels of loudness that can be perceived from the faintest whisper to the loudest thunder. The sensitivity of the ear is amazing; when listening to very weak sounds, the ear drum vibrates less than the diameter of a single molecule!</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The perception of loudness relates roughly to the sound power to an exponent of 1/3. For example, if you increase the sound power by a factor of ten, listeners will report that the loudness has increased by a factor of about two (101/3 ≈ 2). This is a major problem for eliminating undesirable environmental sounds, for instance, the beefed-up stereo in the next door apartment. Suppose you diligently cover 99% of your wall with a perfect soundproof material, missing only 1% of the surface area due to doors, corners, vents, etc. Even though the sound power has been reduced to only 1% of its former value, the perceived loudness has only dropped to about 0.011/3 ≈ 0.2, or 20%.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The range of human hearing is generally considered to be 20 Hz to 20 kHz, but it is far more sensitive to sounds between 1 kHz and 4 kHz. For example, listeners can detect sounds as low as 0 dB SPL at 3 kHz, but require 40 dB SPL at 100 hertz (an amplitude increase of 100). Listeners can tell that two tones are different if their frequencies differ by more than about 0.3% at 3 kHz. This increases to 3% at 100 hertz. For comparison, adjacent keys on a piano differ by about 6% in frequency.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJQZx1xptAtCHjnehUWbphuXhVFilcN4tNnkWxgrddwnr0HsRBkltYjM8gb2PpH2JvGTeeYpWD8yPqecawf6tWub2yETU55zIzG5qyXoFwg1Kz6usu0LQcDiVRHsXKgvWDCU7RSyqrT2g/s1600-h/2.6.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450896652099376658" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJQZx1xptAtCHjnehUWbphuXhVFilcN4tNnkWxgrddwnr0HsRBkltYjM8gb2PpH2JvGTeeYpWD8yPqecawf6tWub2yETU55zIzG5qyXoFwg1Kz6usu0LQcDiVRHsXKgvWDCU7RSyqrT2g/s400/2.6.gif" style="cursor: pointer; display: block; height: 214px; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center; width: 400px;" /></a><span class="Apple-style-span" style="font-family: arial;"></span><br />
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</span></div><div style="text-align: justify;"><span class="Apple-style-span">The primary advantage of having two ears is the ability to identify the direction of the sound. Human listeners can detect the difference between two sound sources that are placed as little as three degrees apart, about the width of a person at 10 meters. This directional information is obtained in two separate ways. First, frequencies above about 1 kHz are stronglyshadowed by the head. In other words, the ear nearest the sound receives a stronger signal than the ear on the opposite side of the head. The second clue to directionality is that the ear on the far side of the head hears the sound slightly later than the near ear, due to its greater distance from the source. Based on a typical head size (about 22 cm) and the speed of sound (about 340 meters per second), an angular discrimination of three degrees requires a timing precision of about 30 microseconds. Since this timing requires the volley principle, this clue to directionality is predominately used for sounds less than about 1 kHz.</span></div><div style="text-align: justify;"><span class="Apple-style-span">Both these sources of directional information are greatly aided by the ability to turn the head and observe the change in the signals. An interesting sensation occurs when a listener is presented with exactly the same sounds to both ears, such as listening to monaural sound through headphones. The brain concludes that the sound is coming from the center of the listener's head!</span></div><div style="text-align: justify;"><span class="Apple-style-span">While human hearing can determine the direction a sound is from, it does poorly in identifying the distance to the sound source. This is because there are few clues available in a sound wave that can provide this information. Human hearing weakly perceives that high frequency sounds are nearby, while low frequency sounds are distant. This is because sound waves dissipate their higher frequencies as they propagate long distances. Echo content is another weak clue to distance, providing a perception of the room size. For example, sounds in a large auditorium will contain echoes at about 100 millisecond intervals, while 10 milliseconds is typical for a small office. Some species have solved this ranging problem by using active sonar. For example, bats and dolphins produce clicks and squeaks that reflect from nearby objects. By measuring the interval between transmission and echo, these animals can locate objects with about 1 cm resolution. Experiments have shown that some humans, particularly the blind, can also use active echo localization to a small extent.</span></div></span></span><br />
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</div><div style="text-align: justify;"><span class="Apple-style-span"><b>Lenny Z Perez M.</b></span></div><div style="text-align: justify;"><span class="Apple-style-span"><b>EES</b></span></div><div style="text-align: justify;"><b><a href="http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/music-and-human-ear-this-section.html">http://eesfrequencyresponseofamplifiers.blogspot.com/2010/03/music-and-human-ear-this-section.html</a></b></div></span> <br />
<hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href="http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us" target="_new">Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-68745660926846442892010-03-21T10:29:00.002-04:302010-03-23T17:47:19.098-04:30Frequency Response of BJT Amplifiers<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"><div><span class="Apple-style-span" style="font-family: arial;">All amplifiers typically exhibit a band-pass frequency response as in Figure 1. The cut-off frequency on the low end is usually determined by the coupling and bypass capacitors (if there are no such capacitors the low end extends all of the way to DC). The high frequency limit is typically determined by internal capacitances in the transistor itself.</span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnZSq2mbgVC3vhkKLP95C9evHkzIB2UuZnYLnBG8Oz8eMYD4SF_-Zb_ThCgS4Q6d8gEzxKz_NnHWO0fko_PCCkDklhIs7NH5m-jWkKQ1GT9N1FtRjXQrrHUgjSYgcdgF-HbpDIuM47aq0/s1600-h/p1.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450960321581118466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnZSq2mbgVC3vhkKLP95C9evHkzIB2UuZnYLnBG8Oz8eMYD4SF_-Zb_ThCgS4Q6d8gEzxKz_NnHWO0fko_PCCkDklhIs7NH5m-jWkKQ1GT9N1FtRjXQrrHUgjSYgcdgF-HbpDIuM47aq0/s400/p1.bmp" style="cursor: pointer; height: 149px; width: 400px;" /></span></span></a></span></div></span></span><br />
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</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Low frequency response</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If an amplifier does not have coupling or bypass capacitors, then in general the low frequency response goes all of the way down to DC. However, as we discussed in class, it is desirable to have these capacitors in the circuit to isolate the amplifiers DC bias point from the outside world.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In the most general case (Figure 2), the input and output coupling capacitors lead to a high-pass filter response determined by the resistances they see:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD9xTYFCwyuvfXgMxubO_C_ZZyAkSjMRiNj2E0lsn-X5zTECMPRkzyXO_jSuKpZTkDs_WJw5VQUSKJY6-SY_kIBB5Y_YlScTyVB3URvP-i61Del8QaqJxrLrlgEm2YDz5eiGt1QxrI8JM/s1600-h/p2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451064702952570834" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD9xTYFCwyuvfXgMxubO_C_ZZyAkSjMRiNj2E0lsn-X5zTECMPRkzyXO_jSuKpZTkDs_WJw5VQUSKJY6-SY_kIBB5Y_YlScTyVB3URvP-i61Del8QaqJxrLrlgEm2YDz5eiGt1QxrI8JM/s400/p2.bmp" style="cursor: pointer; height: 119px; width: 400px;" /></span></span></a><br />
<div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD9xTYFCwyuvfXgMxubO_C_ZZyAkSjMRiNj2E0lsn-X5zTECMPRkzyXO_jSuKpZTkDs_WJw5VQUSKJY6-SY_kIBB5Y_YlScTyVB3URvP-i61Del8QaqJxrLrlgEm2YDz5eiGt1QxrI8JM/s1600-h/p2.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a>In general, we can calculate their cutoff frequencies using the following formulas:</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijjbFwNeqv7xU4Vnt3HJL8DbOJbvzWjTJNk-n_Zy-mhhuHHBOTL9Y1yFvAg1lyEOym8CurNoN3Ko9Zl6XxJRgckxwKr8GOOlvtlWT8gu22MPTMTDBxMavFxg7QgpvijZMVX8xaKfoG9qg/s1600-h/p3.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451065876784026802" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijjbFwNeqv7xU4Vnt3HJL8DbOJbvzWjTJNk-n_Zy-mhhuHHBOTL9Y1yFvAg1lyEOym8CurNoN3Ko9Zl6XxJRgckxwKr8GOOlvtlWT8gu22MPTMTDBxMavFxg7QgpvijZMVX8xaKfoG9qg/s320/p3.bmp" style="cursor: pointer; height: 52px; width: 248px;" /></span></span></a><br />
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</span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExOufhR7gjMyrNIoDDjTwn1Klh_hNdIdgZaghaCWNMevDUEZGtHQBG90_HA3Gud_VogZ4RsUmnEs9kHiglJjDk8uWwj6qzp7q2w6ssQ8LFRom-ijlkQDrbZ6gADEEzpNdgA9jfClDrO4/s1600-h/p4.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451066350352057714" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExOufhR7gjMyrNIoDDjTwn1Klh_hNdIdgZaghaCWNMevDUEZGtHQBG90_HA3Gud_VogZ4RsUmnEs9kHiglJjDk8uWwj6qzp7q2w6ssQ8LFRom-ijlkQDrbZ6gADEEzpNdgA9jfClDrO4/s320/p4.bmp" style="cursor: pointer; height: 50px; width: 250px;" /></span></span></a><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Where Ri is the source resistance, RL is the load resistance, Rin is the input resistance for your amplifier and Ro is the output resistance for your amplifier. These last two are calculated based on the type of amplifier you are working with (See the handout on small signal amplifier calculations).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Once you have calculated the frequencies due to C1 and C2, the cutoff is determined by the following rules:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">1)</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If the two frequencies are more than a decade apart then flow2 in Figure 1 (the 3db point of the amp) is simply the higher of the two values.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">2)</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If the two frequencies are closer than one decade, then the actual cutoff frequency of the amp is somewhat larger than either of the two calculated frequencies.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">3)</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If the amplifier has a bypass capacitor, then it can also influence the cutoff frequency. Typical, emitter bypass capacitors are chosen large enough so that their effects are negligible.</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>High Frequency Response</b></span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As previously stated, the high frequency response of a discrete transistor amp is determined by the internal capacitances of the transistor itself (Figure 3).</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><div style="text-align: center;"><span class="Apple-style-span" style="font-family: Georgia, serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSejRLBHyqnc4OIykj8lhEldinySZjAIt_xrdh48FUnzFR4Iz4BxyjS22pILNXvKAtVpJbgA3zMMrSx62wdf6iWulKnEBDfI__rSug7yP8pleNivOCLDqBAT3_z5VTpPurf8X4XL00oUI/s1600-h/p5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451066622825778914" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSejRLBHyqnc4OIykj8lhEldinySZjAIt_xrdh48FUnzFR4Iz4BxyjS22pILNXvKAtVpJbgA3zMMrSx62wdf6iWulKnEBDfI__rSug7yP8pleNivOCLDqBAT3_z5VTpPurf8X4XL00oUI/s320/p5.bmp" style="cursor: pointer; height: 110px; width: 320px;" /></span></span></a></span></div></span></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSejRLBHyqnc4OIykj8lhEldinySZjAIt_xrdh48FUnzFR4Iz4BxyjS22pILNXvKAtVpJbgA3zMMrSx62wdf6iWulKnEBDfI__rSug7yP8pleNivOCLDqBAT3_z5VTpPurf8X4XL00oUI/s1600-h/p5.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a>If either Cbe or CoB short out at high frequencies, then the transistor stops acting as an amplifier and so the response is cut off. The values of Cbe and CoB can be found or calculated from the transistor spec sheet. Typically, CoB is on the spec sheet and Cbe is calculated from fT (the gain-bandwidth product) also found on the spec sheet using:</span></span></div><div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicziCFLOkBakgFEg3HqisxSffhH3ytLOP0WpilvYx2SgUbu0QbjR3BwFl2SUfGDOemyh7CUzGRiZ9Qr-zS3EtYWv2EAfwGPg9ujb3bjUpK7cZDU9j98wSwhS8VoifL8rZDNJp0d5eM48c/s1600-h/p10.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451067594475696018" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicziCFLOkBakgFEg3HqisxSffhH3ytLOP0WpilvYx2SgUbu0QbjR3BwFl2SUfGDOemyh7CUzGRiZ9Qr-zS3EtYWv2EAfwGPg9ujb3bjUpK7cZDU9j98wSwhS8VoifL8rZDNJp0d5eM48c/s320/p10.bmp" style="cursor: pointer; height: 54px; width: 187px;" /></span></span></a><br />
<div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Once the capacitance values are known, the high frequency cutoff value can be calculated from the following formulas:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><i><b>Common Emitter Amp</b></i></span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpapH5PmmQl2daIvoApxb5Ozpu_VpjvSyT2P8xtUn7UGPmsAZ5AVoGahKKicAqygKE2RaDnnMmFV078rMk6mdXXFVgm2i2Xw-c8skzMzV22zSbIpVcAJrdpItaxeo4uCoT_Qe36L6ds0U/s1600-h/p6.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451067947646360034" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpapH5PmmQl2daIvoApxb5Ozpu_VpjvSyT2P8xtUn7UGPmsAZ5AVoGahKKicAqygKE2RaDnnMmFV078rMk6mdXXFVgm2i2Xw-c8skzMzV22zSbIpVcAJrdpItaxeo4uCoT_Qe36L6ds0U/s320/p6.bmp" style="cursor: pointer; height: 53px; width: 275px;" /></span></span></a><br />
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</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><i>Common Collector Amp</i></b></span></span></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4GeaxjMVo6AwB7Hvf2zx9CMbgpcDEc3qLcGZFq_h_PAsy65Po_tqy1C1gaCiy-prLu2VoDY03LgKPDfA_5y-smurfNjvRqypqe_zo7sNLAj_HBlRVXZJn5ySUiTjhTeGxi-DkH-EzRCk/s1600-h/p7.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451068478157454866" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4GeaxjMVo6AwB7Hvf2zx9CMbgpcDEc3qLcGZFq_h_PAsy65Po_tqy1C1gaCiy-prLu2VoDY03LgKPDfA_5y-smurfNjvRqypqe_zo7sNLAj_HBlRVXZJn5ySUiTjhTeGxi-DkH-EzRCk/s320/p7.bmp" style="cursor: pointer; height: 59px; width: 235px;" /></span></span></a><br />
<div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><i>Common Base</i></b></span></span></div></div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUMRLsWf9zn8pdTAorM9AH6S3UfPQ6K_q0rYwFKpMU2vaKBwokJ1-Vhd7l6BGjrARJgOb1g9eLId34cWHtetg224q1AQ9erV1qpPn5EmW7uUhec8HoQeAXmwk7UjfwczxfVaZrg_lZs_o/s1600-h/p8.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5451068760941502482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUMRLsWf9zn8pdTAorM9AH6S3UfPQ6K_q0rYwFKpMU2vaKBwokJ1-Vhd7l6BGjrARJgOb1g9eLId34cWHtetg224q1AQ9erV1qpPn5EmW7uUhec8HoQeAXmwk7UjfwczxfVaZrg_lZs_o/s320/p8.bmp" style="cursor: pointer; height: 61px; width: 181px;" /></a></span></span><br />
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<hr />Discover the new Windows Vista <a href="http://search.msn.com/results.aspx?q=windows+vista&mkt=en-US&form=QBRE" target="_new">Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-518918637821932182010-03-21T10:26:00.002-04:302010-03-23T17:47:00.429-04:30Frequency Response<span class="Apple-style-span" style="font-family: Georgia, serif; font-size: 16px;"></span><br />
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<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;">Frequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables, category cables, video switchers and wireless communications devices. Subsonic frequency response measurements can include earthquakes and electroencephalography (brain waves).</div><div><span class="Apple-style-span"><br />
</span></div></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency response requirements differ depending on the application. In high fidelity audio, an amplifier requires a frequency response of at least 20–20,000 Hz, with a tolerance as tight as ±0.1 dB in the mid-range frequencies around 1000 Hz, however, in telephony, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency response curves are often used to indicate the accuracy of electronic components or systems. When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency response of a low pass filter with 6 dB per octave or 20 dB per decade</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The frequency response is typically characterized by the magnitude of the system's response, measured in decibels (dB), and the phase, measured in radians, versus frequency. The frequency response of a system can be measured by applying a test signal, for example:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">· applying an impulse to the system and measuring its response (see impulse response)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">· sweeping a constant-amplitude pure tone through the bandwidth of interest and measuring the output level and phase shift relative to the input</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">· applying a signal with a wide frequency spectrum (for example digitally-generated maximum length sequence noise, or analog filtered white noise equivalent, like pink noise), and calculating the impulse response by deconvolution of this input signal and the output signal of the system.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">These typical response measurements can be plotted in two ways: by plotting the magnitude and phase measurements to obtain a Bode plot or by plotting the imaginary part of the frequency response against the real part of the frequency response to obtain a Nyquist plot.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Once a frequency response has been measured (e.g., as an impulse response), providing the system is linear and time-invariant, its characteristic can be approximated with arbitrary accuracy by a digital filter. Similarly, if a system is demonstrated to have a poor frequency response, a digital or analog filter can be applied to the signals prior to their reproduction to compensate for these deficiencies.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Frequency response measurements can be used directly to quantify system performance and design control systems. However, frequency response analysis is not suggested if the system has slow dynamics.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Frecuency Response of system</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Systems are analyzed in the time domain by using convolution. A similar analysis can be done in the frequency domain. Using the Fourier transform, every input signal can be represented as a group of cosine waves, each with a specified amplitude and phase shift. Likewise, the DFT can be used to represent every output signal in a similar form. This means that any linear system can be completely described by how it changes the amplitude and phase of cosine waves passing through it. This information is called the system's frequency response. Since both the impulse response and the frequency response contain complete information about the system, there must be a one-to-one correspondence between the two. Given one, you can calculate the other. The relationship between the impulse response and the frequency response is one of the foundations of signal processing: A system's frequency response is the Fourier Transform of its impulse response. Figure 9-6 illustrates these relationships.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Keeping with standard DSP notation, impulse responses use lower case variables, while the corresponding frequency responses are upper case. Since h[ ] is the common symbol for the impulse response, H[ ] is used for the frequency response. Systems are described in the time domain by convolution, that is: x[n] ∗ h[n] = y[n]. In the frequency domain, the input spectrum is multiplied by the frequency response, resulting in the output spectrum. As an equation: X[f] × H[f] = Y[f]. In other words, convolution in the time domain corresponds to multiplication in the frequency domain.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Figure 9-7 shows an example of using the DFT to convert a system's impulse response into its frequency response. Figure (a) is the impulse response of the system. Looking at this curve isn't going to give you the slightest idea what the system does. Taking a 64 point DFT of this impulse response produces the frequency response of the system, shown in (b). Now the function of this system becomes obvious, it passes frequencies between 0.2 and 0.3, and rejects all others. It is a band-pass filter. The phase of the frequency response could also be examined; however, it is more difficult to interpret and lessinteresting. It will be discussed in upcoming chapters.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Figure (b) is very jagged due to the low number of samples defining the curve. This situation can be improved by padding the impulse response with zeros before taking the DFT. For example, adding zeros to make the impulse response 512 samples long, as shown in (c), results in the higher resolution frequency response shown in (d).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">How much resolution can you obtain in the frequency response? The answer is: infinitely high, if you are willing to pad the impulse response with an infinite number of zeros. In other words, there is nothing limiting the frequency resolution except the length of the DFT. This leads to a very important concept. Even though the impulse response is a discrete signal, the corresponding frequency response is continuous. An N point DFT of the impulse response provides N/2 + 1 samples of this continuous curve. If you make the DFT longer, the resolution improves, and you obtain a better idea of what the continuous curve looks like. Remember what the frequency response represents: amplitude and phase changes experienced by cosine waves as they pass through the system. Since the input signal can contain any frequency between 0 and 0.5, the system's frequency response must be a continuous curve over this range.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGs_eKncNdA4t4rqFzMS04iAgZXQk0SpIMF-xfQPiUY1Isk9lfYL0XXuZNfjvtkcxiA4GdhdLKkJbsYqDLG5FZdv74u2F2mIRQvAP3rntfrdVolmVBuRvLJlkzLlVvCSW_c8gyw_nPSxc/s1600-h/publicacion+1.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450926594141109746" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGs_eKncNdA4t4rqFzMS04iAgZXQk0SpIMF-xfQPiUY1Isk9lfYL0XXuZNfjvtkcxiA4GdhdLKkJbsYqDLG5FZdv74u2F2mIRQvAP3rntfrdVolmVBuRvLJlkzLlVvCSW_c8gyw_nPSxc/s400/publicacion+1.gif" style="cursor: pointer; height: 366px; width: 400px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">This can be better understood by bringing in another member of the Fourier transform family, the Discrete Time Fourier Transform (DTFT). Consider an N sample signal being run through an N point DFT, producing an N/2 + 1 sample frequency domain. Remember from the last chapter that the DFT considers the time domain signal to be infinitely long and periodic. That is, the N points are repeated over and over from negative to positive infinity. Now consider what happens when we start to pad the time domain signal with an ever increasing number of zeros, to obtain a finer and finer sampling in the frequency domain. Adding zeros makes the period of the time domain longer, while simultaneously making the frequency domain samples closer together.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Now we will take this to the extreme, by adding an infinite number of zeros to the time domain signal. This produces a different situation in two respects. First, the time domain signal now has an infinitely long period. In other words, it has turned into an aperiodic signal. Second, the frequency domain has achieved an infinitesimally small spacing between samples. That is, it has become a continuous signal. This is the DTFT, the procedure that changes a discrete aperiodic signal into a frequency domain that is a continuous curve. In mathematical terms, a system's frequency response is found by taking the DTFT of its impulse response. Since this cannot be done in a computer, the DFT is used to calculate a sampling of the true frequency response. This is the difference between what you do in a computer (the DFT) and what you do with mathematical equations (the DTFT).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The frequency response of a system can be viewed two different ways:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Via the Bode plot or via the Nyquist diagram. Both methods display the same information; the difference lies in the way the information is presented. We will study both methods in this tutorial.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The frequency response is a representation of the system's response to sinusoidal inputs at varying frequencies. The output of a linear system to a sinusoidal input is a sinusoid of the same frequency but with a different magnitude and phase. The frequency response is defined as the magnitude and phase differences between the input and output sinusoids. In this tutorial, we will see how we can use the open-loop frequency response of a system to predict its behavior in closed-loop.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">To plot the frequency response, we create a vector of frequencies (varying between zero or "DC" and infinity) and compute the value of the plant transfer function at those frequencies. If G(s) is the open loop transfer function of a system and w is the frequency vector, we then plot G(j*w) vs. w. Since G(j*w) is a complex number, we can plot both its magnitude and phase (the Bode plot) or its position in the complex plane (the Nyquist plot).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Bode Plots</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As noted above, a Bode plot is the representation of the magnitude and phase of G(j*w) (where the frequency vector w contains only positive frequencies). To see the Bode plot of a transfer function, you can use the Matlab bode command. For example,</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode(50,[1 9 30 40])</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">displays the Bode plots for the transfer function:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">50 / (s^3 + 9 s^2 + 30 s + 40)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Please note the axes of the figure. The frequency is on a logarithmic scale, the phase is given in degrees, and the magnitude is given as the gain in decibels.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Note: a decibel is defined as 20*log10 ( |G(j*w| )</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Gain and Phase Margin</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Let's say that we have the following system:</span></span></div></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNQaz9xTPRgVo2pTFG8F21cJNMDquL5nszLbnaIuCMXDNwL4e_aBhi5xnAjBsTl0Nkc_HtWJfpLe8dyvIhZB7D0Sc3LakBacBSO3Uh1vqAQxshS8U_OngSOv81hBBrMET9Nh6SxnJQYSc/s1600-h/uno2.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450927400915032722" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNQaz9xTPRgVo2pTFG8F21cJNMDquL5nszLbnaIuCMXDNwL4e_aBhi5xnAjBsTl0Nkc_HtWJfpLe8dyvIhZB7D0Sc3LakBacBSO3Uh1vqAQxshS8U_OngSOv81hBBrMET9Nh6SxnJQYSc/s320/uno2.gif" style="cursor: pointer; height: 80px; width: 320px;" /></span></span></a></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;"><span class="Apple-style-span">Where K is a variable (constant) gain and G(s) is the plant under consideration. The gain margin is defined as the change in open loop gain required to make the system unstable. Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop.</span></div><span class="Apple-style-span"></span><br />
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</div><div style="text-align: justify;">The unity gain in magnitude is equal to a gain of zero in dB.</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable.</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">The phase margin also measures the system's tolerance to time delay. If there is a time delay greater than 180/Wpc in the loop (where Wpc is the frequency where the phase shift is 180 deg), the system will become unstable in closed loop. The time delay can be thought of as an extra block in the forward path of the block diagram that adds phase to the system but has no effect the gain. That is, a time delay can be represented as a block with magnitude of 1 and phase w*time_delay (in radians/second).</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">For now, we won't worry about where all this comes from and will concentrate on identifying the gain and phase margins on a Bode plot:</div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;">The phase margin is the difference in phase between the phase curve and -180 deg at the point corresponding to the frequency that gives us a gain of 0dB (the gain cross over frequency, Wgc). Likewise, the gain margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase cross over frequency, Wpc).</div></span></span><span class="Apple-style-span" style="font-family: arial;"></span><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMbIRYfDtni324TNp4mknsfueQfG_08jCw47qoXhyLskFLuaDYCLQN2g8HkymtHewZN68CIsg8qQNYyxqUc55pMTz1F_QzxGjQjcH0CKryxwNaKrY9FwlHJySV8Z0pwImQNkIxRnPqc6w/s1600-h/uno3.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450927639401846514" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMbIRYfDtni324TNp4mknsfueQfG_08jCw47qoXhyLskFLuaDYCLQN2g8HkymtHewZN68CIsg8qQNYyxqUc55pMTz1F_QzxGjQjcH0CKryxwNaKrY9FwlHJySV8Z0pwImQNkIxRnPqc6w/s320/uno3.gif" style="cursor: pointer; height: 261px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">One nice thing about the phase margin is that you don't need to replot the Bode in order to find the new phase margin when changing the gains. If you recall, adding gain only shifts the magnitude plot up. This is the equivalent of changing the y-axis on the magnitude plot. Finding the phase margin is simply the matter of finding the new cross-over frequency and reading off the phase margin. For example, suppose you entered the command bode(50,[1 9 30 40]). You will get the following bode plot:</span></span></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLV7_IuJvslAZ7B_nP1LK-4Zp-7nKdgUZr8eK0-ANJUunEmtTaFuWC_dC15dL9Wv9211nxzK5GlnyfosGccqWiwLXn5ruaIe1gl4GBKZhcZmsXZVwD0vEVH2DM7j4k1rrmdyyndk72Z3c/s1600-h/uno4.GIF" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450928020332668306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLV7_IuJvslAZ7B_nP1LK-4Zp-7nKdgUZr8eK0-ANJUunEmtTaFuWC_dC15dL9Wv9211nxzK5GlnyfosGccqWiwLXn5ruaIe1gl4GBKZhcZmsXZVwD0vEVH2DM7j4k1rrmdyyndk72Z3c/s320/uno4.GIF" style="cursor: pointer; height: 229px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">You should see that the phase margin is about 100 degrees. Now suppose you added a gain of 100, by entering the command bode(100*50,[1 9 30 40]). You should get the following plot (note I changed the axis so the scale would be the same as the plot above, your bode plot may not be exactly the same shape, depending on the scale used):</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjl7R3P7ZOMDZ7q1r74jXFwntwqT2C1hBAOZsgSch-FWKe739kcGShMIkmAJguooQpDf7CxuRBZP_T97x7QXIXmT1G1m6JmIA1bxBgkKhs3LzwbrHIYusxlqQrRSChtMvxHjlTbufaPorI/s1600-h/uno5.GIF" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450928504415834146" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjl7R3P7ZOMDZ7q1r74jXFwntwqT2C1hBAOZsgSch-FWKe739kcGShMIkmAJguooQpDf7CxuRBZP_T97x7QXIXmT1G1m6JmIA1bxBgkKhs3LzwbrHIYusxlqQrRSChtMvxHjlTbufaPorI/s320/uno5.GIF" style="cursor: pointer; height: 232px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As you can see the phase plot is exactly the same as before, and the magnitude plot is shifted up by 40dB (gain of 100). The phase margin is now about -60 degrees. This same result could be achieved if the y-axis of the magnitude plot was shifted down 40dB. Try this, look at the first Bode plot, find where the curve crosses the -40dB line, and read off the phase margin. It should be about -60 degrees, the same as the second Bode plot.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">We can find the gain and phase margins for a system directly, by using Matlab. Just enter the margin command. This command returns the gain and phase margins, the gain and phase cross over frequencies, and a graphical representation of these on the Bode plot. Let's check it out:</span></span></div><div></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">margin(50,[1 9 30 40])</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhg_CPaJr3kmH57L5VJIv4D6_GjV39qBwNC-l8OMZGWXYMmIA6y6hj0kQRFRDHH8JWukcmcULeQk_hjR16qU6ofAu_miXZEPDsIzuAS9TgN-Y2Zix7AzZdADAvFztO4xTqE9vXSjbIa_jc/s1600-h/uno6.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450928739862349474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhg_CPaJr3kmH57L5VJIv4D6_GjV39qBwNC-l8OMZGWXYMmIA6y6hj0kQRFRDHH8JWukcmcULeQk_hjR16qU6ofAu_miXZEPDsIzuAS9TgN-Y2Zix7AzZdADAvFztO4xTqE9vXSjbIa_jc/s320/uno6.gif" style="cursor: pointer; height: 258px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
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</span></span></div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhg_CPaJr3kmH57L5VJIv4D6_GjV39qBwNC-l8OMZGWXYMmIA6y6hj0kQRFRDHH8JWukcmcULeQk_hjR16qU6ofAu_miXZEPDsIzuAS9TgN-Y2Zix7AzZdADAvFztO4xTqE9vXSjbIa_jc/s1600-h/uno6.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span><br />
<div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>Bandwidth Frequency</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The bandwidth frequency is defined as the frequency at which the closed-loop magnitude response is equal to -3 dB. However, when we design via frequency response, we are interested in predicting the closed-loop behavior from the open-loop response. Therefore, we will use a second-order system approximation and say that the bandwidth frequency equals the frequency at which the open-loop magnitude response is between -6 and - 7.5dB, assuming the open loop phase response is between -135 deg and -225 deg. For a complete derivation of this approximation, consult your textbook.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If you would like to see how the bandwidth of a system can be found mathematically from the closed-loop damping ratio and natural frequency, the relevant equations as well as some plots and Matlab code are given on our Bandwidth Frequency page.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In order to illustrate the importance of the bandwidth frequency, we will show how the output changes with different input frequencies. We will find that sinusoidal inputs with frequency less than Wbw (the bandwidth frequency) are tracked "reasonably well" by the system. Sinusoidal inputs with frequency greater than Wbw are attenuated (in magnitude) by a factor of 0.707 or greater (and are also shifted in phase).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Let's say that we have the following closed-loop transfer function representing a system:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">1 / (s^2 + 0.5 s + 1)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">First of all, let's find the bandwidth frequency by looking at the Bode plot:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode (1, [1 0.5 1 ])</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil4bSi4AKswUmM-5JtAxqR2mTAm30bG0HH0hJ_VJpMpW0poVkUyGQcZ8L4f7KLPeZF8P7H8JVTFLsGqKphvmjsGRW_2_Xe8Y3TSxcsyrjBoFhffDjeYZy2Re6zXfR70FBHgTN3CMa2XCk/s1600-h/uno7.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450929137125919154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil4bSi4AKswUmM-5JtAxqR2mTAm30bG0HH0hJ_VJpMpW0poVkUyGQcZ8L4f7KLPeZF8P7H8JVTFLsGqKphvmjsGRW_2_Xe8Y3TSxcsyrjBoFhffDjeYZy2Re6zXfR70FBHgTN3CMa2XCk/s320/uno7.gif" style="cursor: pointer; height: 261px; width: 320px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil4bSi4AKswUmM-5JtAxqR2mTAm30bG0HH0hJ_VJpMpW0poVkUyGQcZ8L4f7KLPeZF8P7H8JVTFLsGqKphvmjsGRW_2_Xe8Y3TSxcsyrjBoFhffDjeYZy2Re6zXfR70FBHgTN3CMa2XCk/s1600-h/uno7.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"></span></a></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Since this is the closed-loop transfer function, our bandwidth frequency will be the frequency corresponding to a gain of -3 dB. looking at the plot, we find that it is approximately 1.4 rad/s. We can also read off the plot that for an input frequency of 0.3 radians, the output sinusoid should have a magnitude about one and the phase should be shifted by perhaps a few degrees (behind the input). For an input frequency of 3 rad/sec, the output magnitude should be about -20dB (or 1/10 as large as the input) and the phase should be nearly -180 (almost exactly out-of-phase). We can use the lsim command to simulate the response of the system to sinusoidal inputs.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">First, consider a sinusoidal input with a frequency lower than Wbw. We must also keep in mind that we want to view the steady state response. Therefore, we will modify the axes in order to see the steady state response clearly (ignoring the transient response).</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">w= 0.3;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = 1;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1 0.5 1 ];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">t=0:0.1:100;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">u = sin(w*t);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">[y,x] = lsim(num,den,u,t);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">plot(t,y,t,u)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">axis([50,100,-2,2])</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHQbCBi9reJexUy5dRVUBn42lCcMZPasVn1DbA5lMI2dURH239GsWH9M1bf1U7h4-PQ56Z4I8d-8e-k-5fabD_cN8Lr-3JasMxrTBpVGg38ZHN8DiSCMfjnBP2k-F-cy4m00yVyC9yDIA/s1600-h/uno8.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450929319009850306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHQbCBi9reJexUy5dRVUBn42lCcMZPasVn1DbA5lMI2dURH239GsWH9M1bf1U7h4-PQ56Z4I8d-8e-k-5fabD_cN8Lr-3JasMxrTBpVGg38ZHN8DiSCMfjnBP2k-F-cy4m00yVyC9yDIA/s320/uno8.gif" style="cursor: pointer; height: 135px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Note that the output (blue) tracks the input (purple) fairly well; it is perhaps a few degrees behind the input as expected.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">However, if we set the frequency of the input higher than the bandwidth frequency for the system, we get a very distorted response (with respect to the input):</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">w = 3;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = 1;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1 0.5 1 ];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">t=0:0.1:100;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">u = sin(w*t);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">[y,x] = lsim(num,den,u,t);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">plot(t,y,t,u)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">axis([90, 100, -1, 1])</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9QaAgaoDUh9lV0Agh3GVFZs6rpVvwxTgSh05K0hzgsmk5MXMEaMBMnTIKC9TmOEtO2bXcqYpXd8h40X3O3WrA5AxSsz8PzFZcFuAfbNfsHEgt1Lw_fgnkd7__Z9y2FPk7nE1AKF-pFiM/s1600-h/uno9.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450929810342390482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9QaAgaoDUh9lV0Agh3GVFZs6rpVvwxTgSh05K0hzgsmk5MXMEaMBMnTIKC9TmOEtO2bXcqYpXd8h40X3O3WrA5AxSsz8PzFZcFuAfbNfsHEgt1Lw_fgnkd7__Z9y2FPk7nE1AKF-pFiM/s320/uno9.gif" style="cursor: pointer; height: 135px; width: 320px;" /></span></span></a></div><div><span class="Apple-style-span" style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9QaAgaoDUh9lV0Agh3GVFZs6rpVvwxTgSh05K0hzgsmk5MXMEaMBMnTIKC9TmOEtO2bXcqYpXd8h40X3O3WrA5AxSsz8PzFZcFuAfbNfsHEgt1Lw_fgnkd7__Z9y2FPk7nE1AKF-pFiM/s1600-h/uno9.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span"></span></a></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Again, note that the magnitude is about 1/10 that of the input, as predicted, and that it is almost exactly out of phase (180 degrees behind) the input. Feel free to experiment and view the response for several different frequencies w, and see if they match the Bode plot.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Closed-loop performance</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In order to predict closed-loop performance from open-loop frequency response, we need to have several concepts clear:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The system must be stable in open loop if we are going to design via Bode plots.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">If the gain cross over frequency is less than the phase cross over frequency(i.e. Wgc <></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">For second-order systems, the closed-loop damping ratio is approximately equal to the phase margin divided by 100 if the phase margin is between 0 and 60 deg. We can use this concept with caution if the phase margin is greater than 60 deg.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">For second-order systems, a relationship between damping ratio, bandwidth frequency and settling time is given by an equation described on the bandwidth page.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">A very rough estimate that you can use is that the bandwidth is approximately equal to the natural frequency.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Let's use these concepts to design a controller for the following system:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm9T8RdzCfzX5JGVlBTl4O48pUjJYxMqKSyOblq3dUe75aBMk89gTiK5IuFUzAh0IQIruCkWixzXVYyZOuw1_wgdf_KPJptRDLsW8ceLcS4jKf4wVgkzInPs1-eddMAQdhdQUZUm4c7N8/s1600-h/uno10.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450930017021818914" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm9T8RdzCfzX5JGVlBTl4O48pUjJYxMqKSyOblq3dUe75aBMk89gTiK5IuFUzAh0IQIruCkWixzXVYyZOuw1_wgdf_KPJptRDLsW8ceLcS4jKf4wVgkzInPs1-eddMAQdhdQUZUm4c7N8/s320/uno10.gif" style="cursor: pointer; height: 87px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm9T8RdzCfzX5JGVlBTl4O48pUjJYxMqKSyOblq3dUe75aBMk89gTiK5IuFUzAh0IQIruCkWixzXVYyZOuw1_wgdf_KPJptRDLsW8ceLcS4jKf4wVgkzInPs1-eddMAQdhdQUZUm4c7N8/s1600-h/uno10.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Where Gc(s) is the controller and G(s) is:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">10 / (1.25s + 1)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The design must meet the following specifications:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">• Zero steady state error.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">• Maximum overshoot must be less than 40%.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">• Settling time must be less than 2 secs.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">There are two ways of solving this problem: one is graphical and the other is numerical. Within Matlab, the graphical approach is best, so that is the approach we will use. First, let's look at the Bode plot. Create an m-file with the following code:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = 10;</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1.25,1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode(num, den)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0qTfkKIoXYEDGNN2ZR-RKzSCzZWNcpFu1NT0rxmzaRczt63RSzc6RiKIRd-WOiNJC9L6ppQ_QVjaxhcktnnkr-cn4sIvlnMKGlrGs5pjWxfHgmwul6KM2xIyQQzIHHVIsfKMmvJtM2ng/s1600-h/uno11.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450930229284918674" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0qTfkKIoXYEDGNN2ZR-RKzSCzZWNcpFu1NT0rxmzaRczt63RSzc6RiKIRd-WOiNJC9L6ppQ_QVjaxhcktnnkr-cn4sIvlnMKGlrGs5pjWxfHgmwul6KM2xIyQQzIHHVIsfKMmvJtM2ng/s320/uno11.gif" style="cursor: pointer; height: 260px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">There are several several characteristics of the system that can be read directly from this Bode plot. First of all, we can see that the bandwidth frequency is around 10 rad/sec. Since the bandwidth frequency is roughly the same as the natural frequency (for a second order system of this type), the rise time is 1.8/BW=1.8/10=1.8 seconds. This is a rough estimate, so we will say the rise time is about 2 seconds.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The phase margin for this system is approximately 95 degrees. This corresponds to a damping of PM/100=95/100=0.95. Plugging in this value into the equation relating overshoot and the damping ratio (or consulting a plot of this relation), we find that the damping ratio corresponding to this overshoot is approximately 1%. The system will be close to being overdamped.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The last major point of interest is steady-state error. The steady-state error can be read directly off the Bode plot as well. The constant (Kp, Kv, or Ka) are located at the intersection of the low frequency asymptote with the w=1 line. Just extend the low frequency line to the w=1 line. The magnitude at this point is the constant. Since the Bode plot of this system is a horizontal line at low frequencies (slope = 0), we know this system is of type zero. Therefore, the intersection is easy to find. The gain is 20dB (magnitude 10). What this means is that the constant for the error function it 10. Click here to see the table of system types and error functions. The steady-state error is 1/(1+Kp)=1/(1+10)=0.091. If our system was type one instead of type zero, the constant for the steady-state error would be found in a manner similar to the following</span></span></div></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;"><span class="Apple-style-span"><br />
</span></div></span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2aderRy0VTuLsMOqnhZFamH9uySE6EHuy_CxTnw2p9SQHrlO1m0yVGKpcJwkPwoMkvTIkR9ILxkivTYyhx5AsaDavxH8k3HQycxm4IyPGmL4Ct0QDhLWCpQ0coYhm2NPgr6FY-69i38/s1600-h/uno12.GIF" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450930441487336994" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY2aderRy0VTuLsMOqnhZFamH9uySE6EHuy_CxTnw2p9SQHrlO1m0yVGKpcJwkPwoMkvTIkR9ILxkivTYyhx5AsaDavxH8k3HQycxm4IyPGmL4Ct0QDhLWCpQ0coYhm2NPgr6FY-69i38/s320/uno12.GIF" style="cursor: pointer; height: 212px; width: 254px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Let's check our predictions by looking at a step response plot. This can be done by adding the following two lines of code into the Matlab command window.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">[numc,denc] = cloop(num,den,-1);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">step(numc,denc)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2LPeCkvCrpJc0OaDjiUSSZdy847fvR4Ua_TfOYHepUF8Dyl7Ij3NalhLSQeBcmYIfziyLimbUUjrKr0mOZcQ22GDUZolizSBFBsVOz_YC9Bs_TcPV7UTbcYpyFu8anMPDnqDpq_YYKDs/s1600-h/uno14.GIF" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450932506556125442" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2LPeCkvCrpJc0OaDjiUSSZdy847fvR4Ua_TfOYHepUF8Dyl7Ij3NalhLSQeBcmYIfziyLimbUUjrKr0mOZcQ22GDUZolizSBFBsVOz_YC9Bs_TcPV7UTbcYpyFu8anMPDnqDpq_YYKDs/s320/uno14.GIF" style="cursor: pointer; height: 258px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As you can see, our predictions were very good. The system has a rise time of about 2 seconds, is overdamped, and has a steady-state error of about 9%. Now we need to choose a controller that will allow us to meet the design criteria. We choose a PI controller because it will yield zero steady state error for a step input. Also, the PI controller has a zero, which we can place. This gives us additional design flexibility to help us meet our criteria. Recall that a PI controller is given by:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Gc(s) = K*(s+a)/ s</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The first thing we need to find is the damping ratio corresponding to a percent overshoot of 40%. Plugging in this value into the equation relating overshoot and damping ratio (or consulting a plot of this relation), we find that the damping ratio corresponding to this overshoot is approximately 0.28. Therefore, our phase margin should be approximately 30 degrees. From our Ts*Wbw vs damping ratio plot, we find that Ts*Wbw ~ 21. We must have a bandwidth frequency greater than or equal to 12 if we want our settling time to be less than 1.75 seconds which meets the design specs.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Now that we know our desired phase margin and bandwidth frequency, we can start our design. Remember that we are looking at the open-loop Bode plots. Therefore, our bandwidth frequency will be the frequency corresponding to a gain of approximately -7 dB.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Let's see how the integrator portion of the PI or affects our response. Change your m-file to look like the following (this adds an integral term but no proportional term):</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = [10];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1.25, 1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">numPI = [1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">denPI = [1 0];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newnum = conv(num,numPI);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newden = conv(den,denPI);</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode(newnum, newden, logspace(0,2))</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBY6D1xcu9h3cYayzK8F89ofiVuDjpDX4NwYIb3YrQ7zVlhE2fTzuC7Qlu1Es95b78EZ3rsc2hwMqqgdU0ZWFd4Dynx4nWrPnb8N9D_5nYXj_qtI173WJxQqFUXNgj83BN6oBlUkimze8/s1600-h/uno13.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450933101481315554" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBY6D1xcu9h3cYayzK8F89ofiVuDjpDX4NwYIb3YrQ7zVlhE2fTzuC7Qlu1Es95b78EZ3rsc2hwMqqgdU0ZWFd4Dynx4nWrPnb8N9D_5nYXj_qtI173WJxQqFUXNgj83BN6oBlUkimze8/s320/uno13.gif" style="cursor: pointer; height: 261px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">Our phase margin and bandwidth frequency are too small. We will add gain and phase with a zero. Let's place the zero at 1 for now and see what happens. Change your m-file to look like the following:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = [10];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1.25, 1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">numPI = [1 1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">denPI = [1 0];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newnum = conv(num,numPI);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newden = conv(den,denPI);</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode(newnum, newden, logspace(0,2))</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7kvQG4LIlthqz9MhGFcXvG2o5U9xd-meLqBWYnW1uLwEws8FJC5oPifkYXZn1GMEZM7MJr5_yM3ggWwjhq3hTkA70tNCu90d84zsT4FXjco-g422E6IpASwz6fQttiQpB-PxWZYaf9Hg/s1600-h/uno15.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450933397960741698" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7kvQG4LIlthqz9MhGFcXvG2o5U9xd-meLqBWYnW1uLwEws8FJC5oPifkYXZn1GMEZM7MJr5_yM3ggWwjhq3hTkA70tNCu90d84zsT4FXjco-g422E6IpASwz6fQttiQpB-PxWZYaf9Hg/s320/uno15.gif" style="cursor: pointer; height: 261px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">It turns out that the zero at 1 with a unit gain gives us a satisfactory answer. Our phase margin is greater than 60 degrees (even less overshoot than expected) and our bandwidth frequency is approximately 11 rad/s, which will give us a satisfactory response. Although satisfactory, the response is not quite as good as we would like. Therefore, let's try to get a higher bandwidth frequency without changing the phase margin too much. Let's try to increase the gain to 5 and see what happens .This will make the gain shift and the phase will remain the same.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">num = [10];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">den = [1.25, 1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">numPI = 5*[1 1];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">denPI = [1 0];</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newnum = conv(num,numPI);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">newden = conv(den,denPI);</span></span><span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"> </span></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">bode(newnum, newden, logspace(0,2))</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><span class="Apple-style-span" style="font-family: arial;"></span><br />
<span class="Apple-style-span" style="font-family: arial;"><div style="text-align: justify;"><span class="Apple-style-span"><br />
</span></div></span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge6xsIeEruHEdHJBpAGScO9aYwL-9hfXn2AQk8xxFSYWVzilDj-lJWR9wJvEFTxztBwAGgqehomAfzIYEPapKz7t-FvvrqH2V1y1u9eCiJxSHimJsIQFi_nn-V01yYb6dMtSN0BR2zFQs/s1600-h/uno16.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450933589847878098" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge6xsIeEruHEdHJBpAGScO9aYwL-9hfXn2AQk8xxFSYWVzilDj-lJWR9wJvEFTxztBwAGgqehomAfzIYEPapKz7t-FvvrqH2V1y1u9eCiJxSHimJsIQFi_nn-V01yYb6dMtSN0BR2zFQs/s320/uno16.gif" style="cursor: pointer; height: 261px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">That looks really good. Let's look at our step response and verify our results. Add the following two lines to your m-file:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">[clnum,clden] =cloop(newnum,newden,-1);</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">step(clnum,clden)</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5VYwUtxpFQVQkWc6jHYITRKbhu23E4YXMBlBjydkVnW9G9A4-giXQxkZ3G6p68EmEr_Ryq4JSJ6s7ZXEimDYvQV5lKnpfxYjkzKYHSPh-21bmCVd9fbEjGvy1i_WPj-xLpxJecMdgSR4/s1600-h/uno17.gif" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450934042920256770" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5VYwUtxpFQVQkWc6jHYITRKbhu23E4YXMBlBjydkVnW9G9A4-giXQxkZ3G6p68EmEr_Ryq4JSJ6s7ZXEimDYvQV5lKnpfxYjkzKYHSPh-21bmCVd9fbEjGvy1i_WPj-xLpxJecMdgSR4/s320/uno17.gif" style="cursor: pointer; height: 135px; width: 320px;" /></span></span></a></div><div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">As you can see, our response is better than we had hoped for. However, we are not always quite as lucky and usually have to play around with the gain and the position of the poles and/or zeros in order to achieve our design requirements.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b>The Nyquist Diagram</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Nyquist plot allows us to predict the stability and performance of a closed-loop system by observing its open-loop behavior. The Nyquist criterion can be used for design purposes regardless of open-loop stability (remember that the Bode design methods assume that the system is stable in open loop). Therefore, we use this criterion to determine closed-loop stability when the Bode plots display confusing information. The following movie will help you visualize the relationship between the Bode plot and the Nyquist diagram.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">The Nyquist diagram is basically a plot of G(j* w) where G(s) is the open-loop transfer function and w is a vector of frequencies which encloses the entire right-half plane. In drawing the Nyquist diagram, both positive and negative frequencies (from zero to infinity) are taken into account. We will represent positive frequencies in red and negative frequencies in green. The frequency vector used in plotting the Nyquist diagram usually looks like this (if you can imagine the plot stretching out to infinity):</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ79mynBEfsinPng-cfc5j1_7a7S8zVqBV2X7cSZyxuRjYdEdfAvQA6bjV3Y95VeEbRG8DVhLTs-p2qK19qywAbxrYvy7P95nOWXBd73GDH63c9vHLerKM_PHC555z5ztjZOb6F60SbzY/s1600-h/uno18.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450934536196582930" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ79mynBEfsinPng-cfc5j1_7a7S8zVqBV2X7cSZyxuRjYdEdfAvQA6bjV3Y95VeEbRG8DVhLTs-p2qK19qywAbxrYvy7P95nOWXBd73GDH63c9vHLerKM_PHC555z5ztjZOb6F60SbzY/s320/uno18.bmp" style="cursor: pointer; height: 130px; width: 320px;" /></span></span></a></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div><div><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ79mynBEfsinPng-cfc5j1_7a7S8zVqBV2X7cSZyxuRjYdEdfAvQA6bjV3Y95VeEbRG8DVhLTs-p2qK19qywAbxrYvy7P95nOWXBd73GDH63c9vHLerKM_PHC555z5ztjZOb6F60SbzY/s1600-h/uno18.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"></a></span></span><br />
<div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">In order to see how the frequency vector contributes to the Nyquist diagram more clearly, you can view our movie.</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span">However, if we have open-loop poles or zeros on the jw axis, G(s) will not be defined at those points, and we must loop around them when we are plotting the contour. Such a contour would look as follows:</span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><br />
</span></span></div></div><div style="text-align: center;"><span class="Apple-style-span" style="-webkit-text-decorations-in-effect: none; color: black;"></span><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFrGexLqD2WCipBpIZCjYAuh3YQwoQS-SoUSVkz7QMO3e18pOfIlumLYdJ3m0HLVPszl8uAuVxCvUk-XOM0JWceIBQMFThxn0uNb60rizSZRa2QEE-8S2Y13zdeGoDSysbSRivPrZ4KqA/s1600-h/uno19.bmp" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5450934815669138850" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFrGexLqD2WCipBpIZCjYAuh3YQwoQS-SoUSVkz7QMO3e18pOfIlumLYdJ3m0HLVPszl8uAuVxCvUk-XOM0JWceIBQMFThxn0uNb60rizSZRa2QEE-8S2Y13zdeGoDSysbSRivPrZ4KqA/s320/uno19.bmp" style="cursor: pointer; height: 131px; width: 320px;" /></a></span></span></div><div style="text-align: center;"><span class="Apple-style-span" style="font-family: arial;"><span class="Apple-style-span"><b><br />
</b></span></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">Lenny Z Perez M</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><b><span class="Apple-style-span">EES</span></b></span></div><div style="text-align: justify;"><span class="Apple-style-span" style="font-family: arial;"><a href="http://www.blogger.com/post-edit.g?blogID=6409606414427111243&postID=4324343711946070628"><b><span class="Apple-style-span">http://www.blogger.com/post-edit.g?blogID=6409606414427111243&postID=4324343711946070628</span></b></a></span></div></span> <br />
<hr />Connect to the next generation of MSN Messenger <a href="http://imagine-msn.com/messenger/launch80/default.aspx?locale=en-us&source=wlmailtagline" target="_new">Get it now! </a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-38165476963673737092010-02-15T22:44:00.001-04:302010-02-16T10:46:30.962-04:30Respuesta en frecuencia del oído humano<span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><u><span style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-bidi-font-family: 'Times New Roman'; mso-bidi-font-size: 12.0pt"><span style="COLOR: windowtext"><font color=#bfbfbf>Sistema auditivo periférico<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></SPAN></SPAN></U></B></P></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El sistema auditivo humano podemos dividirlo en cuatro etapas:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*la fisiológica, de la que se encarga el sistema auditivo períferico. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*la psicológica (percepción) de la que se encarga el sistema auditivo central. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*El Sistema auditivo periférico es el responsable de los procesos fisiológicos de la audición. Estos procesos que permiten captar el sonido y transformarlo en impulsos eléctricos suceptibles de ser enviados al cerebro a través de los nervios auditivos.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*El sistema auditivo periférico lo constituye el oído.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El oído humano se divide en tres partes:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*el oído externo, que canaliza la energía acústica. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*el oído medio, que transforma la energía acústica en energía mecánica, transmitiéndola - y amplificándola- hasta el oído interno. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*el oído interno, donde se realiza la definitiva transformación de la energía mecánica en impulsos eléctricos. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Cuando el sonido llega al oído, las ondas sonoras son recogidas por el pabellón auricular (o aurícula). El pabellón auricular, por su forma helicoidal, funciona como una especie de "embudo" que ayuda a dirigir el sonido hacia el interior del oído.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Sin la existencia del pabellón auricular, los frentes de onda llegarían de forma perpendicularmente y el proceso de audición resultaría ineficaz (gran parte del sonido se perdería):<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Parte de la vibración penetraría en el oído. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Parte de la vibración rebotaría sobre la cabeza y volvería en la dirección de la que procedía. (reflexión). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Parte de la vibración lograría rodear la cabeza y continuar su camino. (difracción). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El pabellón auricular humano es mucho menos direccional que el de otros animales (como los perros) que poseen un control voluntario de su orientación. (Los perros pueden mover las orejas a voluntad, los humanos no).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Una vez que ha sido recogido el sonido, las vibraciones provocadas por la variación de presión del aire cruzan el canal auditivo externo y llegan a la membrana del tímpano, ya en el oído medio.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El conducto auditivo actúa como una etapa de potencia natural que amplifica automáticamente los sonidos más bajos que proceden del exterior. Al mismo tiempo, en el caso contrario, si se produce un sonido muy intenso que puede dañar el oído, el conducto auditivo segrega cerumen (cera), con lo que cierra parcialmente el conducto, protegiéndolo.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>En el oído medio, se produce la transducción, es decir, la transformación la energía acústica en energía mecánica. En este sentido, el oído medio es un transductor mecánico-acústico.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Además de transformar la señal, antes de que ésta llege al oído interno, el oído medio la habrá amplificado.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La presión de las ondas sonoras hace que el tímpano vibre empujando a los osículos, que, a su vez, transmiten el movimiento del tímpano al oído interno. Cada osículo empuja a su adyacente y, finalmente a través de la ventana oval. Es un proceso mecánico, el pie del estribo empuja a la ventana oval, ya en el oído interno.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Esta fuerza empuja a la venta oval es unas 20 veces mayor que la que empujaba a la membrana del tímpano, lo que se debe a la diferencia de tamaño entre ambas.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Esta presión ejercida sobre la ventana oval, gracias a la helicotrema penetra en el interior de la cóclea (caracol) y pone en movimiento el líquido linfático (endolinfa o linfa) que ésta contiene.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El líquido linfático se mueve como una especie de ola y, transmite las vibraciones a las dos membranas que conforman la cóclea (membrana tectorial, la superior, y la membrana basilar, la inferior).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Entre ambas membranas se encuentra el órgano de Corti, que es el transductor propiamente dicho. En el órgano de Corti se encuentran las células receptoras. Existen aproximadamente 24 000 de estas fibras pilosas, dispuestas en 4 largas filas que son las que recogen la vibración de la membrana basilar.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Como la membrana basilar varía en masa y rigidez a lo largo de su longitud su frecuencia de resonancia no es la misma en todos los puntos:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>En el extremo más próximo a la ventana oval y al tímpano, la membrana es rígida y ligera, por lo que su frecuencia de resonancia es alta. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Por el contrario, en el extremo más distante, la membrana basilar es pesada y suave, con lo que su resonancia es baja frecuencia. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El margen de frecuencias de resonancia disponible en la membrana basilar determina la respuesta en frecuencia del oído humano, las audiofrecuencias que van desde los 20 Hz hasta los 20 kHz. Dentro de este margen de audiofrecuencias, la zona de mayor sensibilidad del oído humano se encuentra en los 1000 y los 4000 Hz.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Esta respuesta en frecuencia del oído humano</SPAN></B><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">, permite que seamos capaces de tolerar un rango dinámico que va desde los 0 db (umbral de audición) a los 120 dB (umbral de dolor)<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El movimiento de la membrana basilar afecta las células del órgano de Corti de forma diferencial, en función de su frecuencia de resonancia. Al ser empujadas contra la membrana tectorial, las células pilosas generan patrones característicos de cada tono o (frecuencia), que al llegar aquí, al final del proceso fisiológico, son idénticas a la sonido original.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>En función de estos patrones, al ser estimuladas las células pilosas producen un componente químico que genera los impulsos eléctricos que son trasmitidos a los tejidos nerviosos adyacentes (nervio auditivo y, de ahí, al cerebro), donde se producirá la percepción del sonido gracias al sistema auditivo central.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Las células del órgano de Corti, (células ciliares, capilares o pilosas), no tienen capacidad regeneradora, es decir, cuando se lesionan se pierde audición de forma irremediable. Además, con la edad, desciende la agudeza auditiva de los seres humanos.</FONT></SPAN></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>La audición Humana<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El umbral de audición, para la media de los humanos, se fija en 20 µPa (20 micro pascales = 0.000002 pascales), para frecuencias entre 2KHz y 4KHz. Por encima y por debajo de estas frecuencias, la presión requerida para excitar el oído es mayor. Esto significa que nuestro oído no responde igual a todas las frecuencias (tiene una respuesta en frecuencia desigual). Un tono puro, a la frecuencia de 125 Hz y con 15 dB de nivel, sería prácticamente inaudible, mientras que si aumentamos la frecuencia, hasta 500 Hz, sin variar el nivel de presión, se obtendría un tono claramente audible.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p><font color=#bfbfbf><img height=302 src="http://www.estudioveracruz.com/imagenes/grafica.gif" width=324></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Las líneas discontinuas marcan los niveles de presión necesarios a cada frecuencia, para que el oído detecte (subjetivamente) la misma sonoridad en todas. Esto quiere decir que si reproducimos un tono de 31.5 Hz a 100 dB (NPS), luego otro de 63 Hz a 90 dB y otro de 125 Hz a 80 dB, el oyente dirá que todos sonaban al mismo volumen. En 2 KHz el umbral de audición se fija en 0 dB y a 4 KHz es incluso menor de 0 dB, ya que a 3600 Hz se encuentra la frecuencia de resonancia del oído humano.Por debajo de 2000 Hz y según se va bajando en frecuencia, el oído se vuelve menos sensible.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Los umbrales de audición para frecuencias menores de 2 KHz son: 5 dB a 1 KHz, 7 dB a 500 Hz, 11 dB a 250 Hz, 21 dB a 125 Hz, 35 dB a 63 Hz, 55 dB a 31 Hz. Recuerda que estos dB's son de nivel de presión sonora (NPS o SPL).Por encima de los 4 KHz, el oído es menos sensible, pero no tanto como en bajas frecuencias. Sin embargo, se producen fluctuaciones a frecuencias cercanas, debido a las perturbaciones que produce la cabeza del oyente en el campo sonoro.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Los umbrales de audición son: 15 dB a 8 KHz y 20 dB a 16 KHz Todos los receptores de sonido, tienen un comportamiento que varía con la frecuencia. En el caso del oído humano, sucede lo mismo, ya que se trata el receptor más complicado y (aunque parezca extraño) más eficiente que existe.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El umbral de audición define la mínima presión requerida para excitar el oído. El límite del nivel de presión sonora se sitúa generalmente alrededor de 130 dB, coincidiendo con el umbral del dolor (molestias en el oído).La pérdida de audición de manera súbita, por daños mecánicos (en el oído medio) se produce a niveles mucho mayores. La exposición suficientemente prolongada a niveles superiores a 130 dB produce pérdida de audición permanente y otros daños graves. En acústica, las frecuencias siempre se tratan de manera logarítmica: representaciones, gráficas y demás. El motivo principal es que el oído humano interpreta las frecuencias de manera casi logarítmica.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>En el eje de frecuencias de cualquier gráfica de las vistas hasta ahora, las marcas pasan de una frecuencia (p. ej. 1000 Hz) al doble (2000 Hz). La apreciación subjetiva de un oyente será que hay la misma distancia entre un tono de 200 Hz y otro de 400, que entre uno de 1000 Hz y otro de 2000 Hz. Sin embargo la "distancia" en frecuencia en el primer caso es de 200 Hz y en el segundo de 1000 Hz.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>SONIDO, FRECUENCIA Y EL OIDO HUMANO<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>SONIDO<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Sonido es cualquier fenómeno que involucre la propagación en forma de ondas elásticas, audibles o casi audibles, generalmente a través de un fluido (u otro medio elástico) que esté generado por el movimiento vibratorio de un cuerpo.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>FRECUENCIA<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Frecuencia es una medida para indicar el número de repeticiones de cualquier fenómeno o suceso periódico en la unidad de tiempo. Para calcular la frecuencia de un evento, se contabilizan un número de ocurrencias de este teniendo en cuenta un intervalo temporal, luego estas repeticiones se dividen por el tiempo transcurrido.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p><font color=#bfbfbf><img src="http://go2.wordpress.com/?id=725X1342&site=nottheremin.wordpress.com&url=http%3A%2F%2Fnottheremin.files.wordpress.com%2F2008%2F11%2Ffreqond.jpg"></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>OIDO HUMANO<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El oído humano es capaz de percibir frecuencias entre 20 y 20.000 Hz, aunque va disminuyendo por la edad. Esta respuesta en frecuencia se conoce como audiofrecuencia, pero el espectro sonoro es más amplio.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p><font color=#bfbfbf><img src="http://go2.wordpress.com/?id=725X1342&site=nottheremin.wordpress.com&url=http%3A%2F%2Fnottheremin.files.wordpress.com%2F2008%2F11%2Fsiaues21.gif"></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>OSCILADORES <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La frecuencia del oscilador fijo es de 170 KHz y la del variable oscila entre 168 y 170 KHz.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Frecuencias audibles de 0 a 2 KHz para ejecutar el instrumento.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><img style="WIDTH: 597px; HEIGHT: 195px" height=416 src="http://go2.wordpress.com/?id=725X1342&site=nottheremin.wordpress.com&url=http%3A%2F%2Fnottheremin.files.wordpress.com%2F2008%2F11%2Fnotasyfrecsbg.gif" width=984></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN> </P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Gráfica de frecuencia.<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>A la vista de la gráfica se diría que este altavoz tiene una respuesta en frecuencia de 450 Hz. a 4 KHz. con una variación de +/- 3dB.... <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Caídas de más de10 dB en la respuesta en frecuencia equivale a decir que el aparato no trabaja en esa frecuencia. De este altavoz conocemos a través de la gráfica de respuesta en frecuencia que si se le alimenta con dos señales de igual nivel, una por ejemplo de 800 Hz. y otra de 4000 Hz, la segunda tendrá un nivel de presión sonora (NPS) 6 dB menor que la señal de 800 Hz. Esto significa que reproduciendo música o cualquier otra señal, las frecuencias cercanas a 800 Hz. se escucharán más que las cercanas a 3 KHz.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El caso más favorable (e imposible) de respuesta en frecuencia sería una línea recta que cubra todo el espectro. En este caso hablaríamos de respuesta en frecuencia plana. Como esto es imposible, se suele hablar de la "zona de respuesta plana", aunque realmente se trata de una aproximación. En el caso anterior diríamos que la zona de respuesta plana es la definida entre 800 y 3000 Hz, ya que en esta zona es donde es útil el altavoz.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Zona de respuesta idealmente plana entre 200 Hz y 10 KHz<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El oído humano tiene dificultad para detectar variaciones de nivel de presión de menos de 0.3 dB. Esto significa que si exponemos a una persona a un ruido (sonido continuo) y vamos variando el nivel de presión sonora (dando más volumen o menos al ruido), el sujeto notará variación cuando la diferencia de NPS (nivel de presión sonora) antes y después se aproxime a los 0.3 dB. Esto da una idea, de cuanta variación de respuesta en frecuencia es aceptable, por ejemplo en unos altavoces. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Este apartado ha tenido como ejemplo un altavoz; sin embargo todos los aparatos de audio tienen su respuesta en frecuencia característica. En una cadena se sonido, donde la señal pasa por varios equipos uno tras de otro, las respuestas en frecuencia de cada aparato sen van sumando para conformar la respuesta en frecuencia total del equipo completo.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Respuesta en frecuencia de tres sistemas. Representación superpuesta.<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La respuesta en frecuencia del conjunto de aparatos será la suma en dB de todas (azul). El amplificador del ejemplo provoca una caída en la respuesta de 6 dB a 6600 Hz y el sistema de altavoces provoca 6 dB de caída a esa misma frecuencia, la respuesta total tendrá una caída de 12 dB en esa frecuencia.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Respuesta en frecuencia total de los tres sistemas en cadena<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Como se ha dicho todos los elementos por los que pasa la señal de sonido en una cadena de audio (o una cadena de música) van dejando su huella en el espectro de la señal, recortándola y limitándola. Es por esto que es importante que todos los equipos por los que atraviesa la señal de audio tengan la máxima calidad posible. En cualquier caso todos han de ser de calidad similar, ya que el elemento de peor calidad será el que pondrá el límite a la calidad del conjunto.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Actualmente, gracias al desarrollo de la electrónica, los equipos electrónicos suelen tener una respuesta en frecuencia bastante buena. El punto crítico suele estar en los altavoces, que son elementos mecánicos que no han evolucionado tanto como la electrónica por lo que sigue siendo muy costoso fabricar buenos altavoces. Suelen ser los altavoces los que más limitan la respuesta en frecuencia del conjunto y por lo tanto la calidad del conjunto. Por este motivo en las cadenas domésticas, un parámetro de calidad a tener en cuenta son los altavoces, ya que la electrónica es muy similar en todos los casos.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Otra filosofía muy acertada a la hora de buscar el mejor sonido a base de no modificar el espectro, es desechar todas las etapas en la cadena de sonido que no sean necesarias; ya que por muy buena calidad que tengan, siempre alterarán la señal. Así se explica que los equipos HI-FI más caros y de mayor calidad no tengan funciones como distintos tipos de ecualización, controles de tono, efecto de cine... El motivo es que en estos aparatos de alta gama, la electrónica está cuidada al máximo y ese tipo de funciones "ensucian" la señal, alejándola de la original.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Otro aspecto que no se ha tratado es el tema de la fase. Todos los dispositivos electrónicos modifican la fase y mucho, por esto los equipos HI-FI de alta gama reducen al máximo la electrónica.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Lenny Z. Perez M.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>EES</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Referencias:</FONT></SPAN></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://es.wikipedia.org/wiki/Sistema_auditivo_perif%C3%A9rico"><font face=Calibri color=#bfbfbf>http://es.wikipedia.org/wiki/Sistema_auditivo_perif%C3%A9rico</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://www.estudioveracruz.com/audicionhumana.html"><font face=Calibri color=#bfbfbf>http://www.estudioveracruz.com/audicionhumana.html</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://nottheremin.wordpress.com/2008/11/09/sonido-frecuencia-y-el-oido-humano/"><font face=Calibri color=#bfbfbf>http://nottheremin.wordpress.com/2008/11/09/sonido-frecuencia-y-el-oido-humano/</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://www.sonido-zero.com/biblioteca-de-sonido/grafica-de-frecuencia..html"><font face=Calibri color=#bfbfbf>http://www.sonido-zero.com/biblioteca-de-sonido/grafica-de-frecuencia..html</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"></SPAN></o:p></SPAN> </P><hr />Get news, entertainment and everything you care about at Live.com. <a href='http://www.live.com/getstarted.aspx ' target='_new'>Check it out!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-17358991543132102762010-02-15T22:13:00.002-04:302010-02-16T10:46:10.880-04:30Spectrum Analysis<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-bidi-font-family: Arial"><span style="COLOR: windowtext"><font color=#bfbfbf>Waveform and Spectrum Analysis<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></SPAN></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: Arial"><font color=#bfbfbf>Waveforms using the Cathode Ray Oscilloscope (CRO)<o:p></o:p></FONT></SPAN></B></P><p style="TEXT-ALIGN: justify"><font size=3><font face="Times New Roman"><font color=#bfbfbf><b><span lang=EN-US style="mso-ansi-language: EN-US">SQUARE WAVE TESTING</SPAN></B><span lang=EN-US style="mso-ansi-language: EN-US"> <o:p></o:p></SPAN></FONT></FONT></FONT></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman" color=#bfbfbf>One method of assessing frequency response (and sometimes other characteristics) is to feed a square wave to the input of the device under test and examine its output on a CRO. The square wave is made up of a fundamental frequency and all odd harmonies, theoretically to infinity. A deficiency within the frequency spectrum, from the fundamental upwards, will show a change in the shape of waveform. The test is subjective rather than precise but gives a good indication of the response</FONT></FONT></SPAN></P><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><o:p> <p style="TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig1.jpg" align=middle></FONT></CENTER></TD></TR>
</TBODY></TABLE></P><p style="TEXT-ALIGN: justify"><font color=#bfbfbf><strong>Figure 1: Typical Square-Wave Response Patterns<br />
<br />
</STRONG>(A) Excellent Response<br />
(B) Poor Low Frequency Response<br />
(C) Fair Low Frequency Response<br />
(D) Poor High Frequency Response<br />
(E) Improved High Frequency Response<br />
(F) Emphasised Low Frequency Amplification<br />
(G) Reduced Low Frequency Amplification<br />
(H) Frequency or Band of Frequencies Emphasised<br />
(I ) Frequency or Band of Frequencies Attenuated<br />
(J) Oscillation or "Ringing"<br />
</FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Typical response patterns taken from a reference source are shown in figure 1. The captions under the patterns decribe the various operational conditions and the effect of loss of low or high frequency response is illustrated. Further patterns shown in figure 2 also illustrate the effect on the waveforms when relative phase delay is changed over part of the frequency spectrum. Also observe in figure 1(J) how the ringing from oscillation in the circuit under test is initiated by the steep edge of the square wave. This is a test result on how the circuit might handle a transient which might not have been detected in carrying out a sine wave frequency response check.</FONT></SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><span style="COLOR: gray; text-effect: emboss; mso-color-alt: windowtext"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig2.jpg" align=middle></FONT></TD></TR>
</TBODY></TABLE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><strong>Figure 2 - Typical Square-Wave Response Patterns<br />
<br />
</STRONG>(A) Deficient in Low Frequency, No Phase Error<br />
(B) Deficient in Low Frequency, Phase Advance<br />
(C) Excess Low Frequency, No Phase Error<br />
(D) Excess Low Frequency, Phase Delay<br />
(E) Deficient in High Frequency, No Phase Error<br />
(F) Deficient in High Frequency, Phase Delay<br />
(G) Sharp Cut-off above Square Wave Frequency<br />
(H) Gradual Cut-off above Square Wave Frequency <br />
(I) Gradual Cut-off at higher frequency than (h)<br />
Related to frequency response, there is a specification called "transient response' which is the ability of a device to respond to a stop function. "Rise time" is one measure of transient response and is the time taken for the signal, initiated from a stop function, to rise from 10 percent to 90 percent of its stable maximum value. Another measure is the percentage of the stable maximum value that the signal over-shoots in responding to the step. Figure 3 shows how the square wave, in conjunction with a calibrated CRO, can be used to measure rise time and overshoot. </FONT></P><center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig3.jpg" align=middle></FONT></CENTER> <center><font color=#bfbfbf></FONT> </CENTER> <center><strong><font color=#bfbfbf>Figure 3 - Transient Response - Measurement of Rise Time and Overshoot</FONT></STRONG></CENTER> <center><strong><font color=#bfbfbf></FONT></STRONG> </CENTER> <center> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Rise time is also measure of the maximum slope of any sine wave component and hence is directly related to the limits in high frequency response. Together, rise time and overshoot define the ability of a device to reproduce transient type signals. Another specification commonly used in operational amplifiers is the 'slew rate" given in volts per microsecond. Such amplifiers have limitations in the rate of change that the output can follow and this is defined by the slew rate. The greater the output voltage, the greater is the rise time and hence the greater the output voltage, the lower is the effective bandwidth. Slew rate is equal to the output voltage step divided by the rise time as measured over the 10 percent to 90 percent points, discussed previously. It is an interesting observation that, in specifying frequency response, output voltage should also be part of the specification. <o:p></o:p></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="mso-ansi-language: EN-US">HARMONIC DISTORTION</SPAN></B><span lang=EN-US style="mso-ansi-language: EN-US"><o:p></o:p></SPAN></FONT></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Harmonic distortion in any signal transmission device results from non-linearity in the device transfer characteristic. Additional frequency components, harmonically related to frequencies fed into the input, appear at the output in addition to the reproduction of the original input components. <o:p></o:p></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Measurement of harmonic distortion can be carried out by feeding a sine wave into the input of the device and separating the sine wave from its harmonics at the output. Distortion is measured as the ratio of harmonic level to the level of the fundamental frequency. This is usually expressed as a percentage but sometimes also expressed as a decibel. <o:p></o:p></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="mso-ansi-language: EN-US">SINE WAVE TESTING</SPAN></B><span lang=EN-US style="mso-ansi-language: EN-US"><o:p></o:p></SPAN></FONT></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Subjective testing for harmonic distorton can be carried out by feeding a good sine wave signal into the device under test and examining the device output on a CRO. Quite low values of distortion can be detected in this way. </FONT></SPAN></P><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#000000><o:p> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig4.jpg" align=middle></FONT></CENTER> <center><b><font color=#bfbfbf>Figure 4 - Formulation of Waveform from Fundamental Frequency (a1)<br />
and Second Harmonic Frequency (a2). </FONT></B></CENTER> <center><font color=#bfbfbf></FONT> </CENTER> <div align=left><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Some idea of the order of the harmonic can often be determined from the shape of the waveform. Figure 4 illustrates the formation of a composite waveform from a fundamental frequency and its second harmonic at one quarter of the fundamental Amplitude. Figure 5 illustrates similar formation from a fundamental frequency and its third harmonic, also a quarter of the fundamental amplitude. In Figure 5(b), the phase of the harmonic is shifted 180 degrees to that in Figure 5(a), and in Figure 5(c), the phase is shifted 90 degrees to that in (5a). The figures show that the composite wave forms can be quite different for different phase conditions making resolution sometimes tricky.</FONT></SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><span style="COLOR: gray; text-effect: emboss; mso-color-alt: windowtext"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig5ab.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>A B</B> </FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig5c.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><b><font color=#bfbfbf>C</B?< center> </FONT></B></CENTER><br />
<tr> <td> <center><b><font color=#bfbfbf>Figure 5 - Formation of Waveforms from Fundamental Frequency and Third Harmonic.<br />
Diagrams (a), (b) & (c) show different phase relationships between harmonic and fundamental.</FONT></B></CENTER></TD></TR><br />
</TBODY></TABLE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf></FONT> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Some distorted waveforms directly indicate an out of adjustment or incorrect operating condition. The clipped waveform of Figure 6(a) shows the output of an amplifier driven to an overload or saturated condition. Figure 6(b) is clipped in one direction indicating an off-centre setting of an amplifier operating point. </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Figure 6(c) shows crossover distortion in a Class B amplifier</SPAN></FONT></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig6.jpg" align=middle></FONT></CENTER></TD> <td><font color=#bfbfbf><b>Figure 6 - Distortion Waveforms.</B><br />
<br />
(a) Amplifier Overdriven.<br />
(b) Amplifier Operating Point not centered.<br />
(c) Crossover Distortion in Class B Stage.<br />
</FONT></TD></TR>
</TBODY></TABLE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Another method of testing, using sine waves, is to feed the monitored device input signal to the X plates input of the CRO and the device output signal to the Y plates input of the CRO. This plots the transfer characteristic of the device, that is, instantaneous output voltage as a function of instantaneous input voltage. X and Y gain is adjusted for equal vertical and horizontal scan. A perfect response is indicated by a diagonal line on the screen, or with phase shift, an ellipse or circle. Figure 10 shows various fault wave forms taken from one reference source. </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">The different effects are explained in the diagram captions.</SPAN></FONT></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig7.jpg" align=middle></FONT></CENTER></TD> <td><font color=#bfbfbf><b>Figure 7: Sine Wave Testing with Amplifier Input<br />
and Output fed to X and Y Plates of CRO, respectively<br />
<br />
</B>(a) Amplifier Overdriven<br />
(b) Anode Bend Distortion in Valve Amplifier<br />
(c) Curvature Distortion<br />
(d) Crossover Distortion in a Class B Output Stage<br />
(e) Magnetising Current Distortion<br />
(f) As (e) with Phase Distortion later in the Chain<br />
(g) As (d) with Phase Distortion earlier in the Chain<br />
(h) As (c) with Phase Distortion earlier in the Chain<br />
(i) As (a) with Phase Distortion later in the Chain<br />
</FONT></TD></TR>
</TBODY></TABLE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>The same connection can be used to measure phase shift between two sine wave signals of the same frequency such as measuring the phase shift between the output and input of an amplifier. Typical measurements are shown in Figure 8. A straight forward sloped diagonal line indicates no phase shift. A straight reverse sloped diagonal line indicates 180°. A circle indicates 90° and an elipse 45° or 135°.</FONT></SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#000000><span style="COLOR: gray; text-effect: emboss; mso-color-alt: windowtext"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/WaveformsFig8.jpg" align=middle> </FONT></CENTER></TD></TR>
<tr> <td><font color=#bfbfbf><b>Figure 8 - Measuring Phase Difference between two voltages of the same frequency.</B> </FONT> <center></CENTER></TD>
<tr><font color=#bfbfbf></FONT></TR>
</TBODY></TABLE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf>If a dual trace CRO is available, the two signals can be displayed together, one on each vertical or Y trace with normal X sweep. In this case, it is simply a matter of scaling off the phase difference along the Y axis graticle. </FONT></P><font face=Arial size=5><span style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><strong><font color=#bfbfbf><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><strong><span lang=EN-US style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: #BFBFBF; mso-ansi-language: EN-US"><font color=#bfbfbf>Spectrum Analyser Waveforms</FONT></SPAN></STRONG><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><o:p></o:p></SPAN></P></o:p></FONT></STRONG></SPAN> <span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>Over the years, the cathode ray oscilloscope (CRO) has been a universal instrument for examining analogue signals. Rapid advances in technology have led to a era of microcomputer controlled, digitally controlled test equipment, not the least of which is the modern spectrum analyser which enables greater precision analysis of analogue signals than is possible with the CRO. A spectrum analyser plots signal amplitude (or signal power) as a function of frequency compared to the CRO which plots signal amplitude as a function of time. <o:p></o:p></FONT></FONT></FONT></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>The spectrum analyser is not the type of equipment normally within the reach of the radio amateur and because of this, it was thought that it would be of interest to illustrate a few spectrum plots of well-known waveforms. <o:p></o:p></FONT></FONT></FONT></SPAN><br />
<span><strong><font size=3><font face="Times New Roman"><font color=#bfbfbf>BASIC WAVEFORMS<o:p></o:p></FONT></FONT></FONT></STRONG></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>Figure 1 shows the spectrum of a sine wave oscillator with fundamental at 1000 Hz and harmonics up to 20 kHz. The highest level harmonic at 7 kHz is 70 dB below the fundamental, representing a harmonic distortion of 0.03 percent. This is a very good oscillator which would not be matched by many laboratory instruments. It can also be seen that the noise floor is about 95 dB below the fundamental and this is also very good. The oscillator noise level might be even better than this as much of the noise is due to the spectrum analyser itself. <o:p></o:p></FONT></FONT></FONT></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>Figure 2 shows a 1000 Hz square wave. A perfect square wave generates odd harmonics to infinity with an amplitude 1/n relative to that of the fundamental or (20 log n) dB below the fundamental. ('n' is the order of harmonic). For n = 3, 5, 7 and 9 this calculates to -9.5, -14, -16.9 and -19.1 dB respectively, very close to the readings shown in Figure 2. <o:p></o:p></FONT></FONT></FONT></SPAN><br />
<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig1.jpg" align=middle></FONT></CENTER></TD> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig2.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>Figure 1:<br />
1000 Hz Sine Wave and Harmonics.</B> </FONT></CENTER></TD> <td> <center><b><font color=#bfbfbf>Figure 2:<br />
1000 Hz Square Wave Showing Harmonics to 20 kHz.</FONT></B></CENTER></TD></TR>
</TBODY></TABLE></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>Figure 2 shows a 1000 Hz square wave. A perfect square wave generates odd harmonics to infinity with an amplitude 1/n relative to that of the fundamental or (20 log n) dB below the fundamental. ('n' is the order of harmonic). For n = 3, 5, 7 and 9 this calculates to -9.5, -14, -16.9 and -19.1 dB respectively, very close to the readings shown in Figure 2. <o:p></o:p></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>Figure 3 is the same square wave plotted out to 200 kHz and showing the apparently unlimited spread of harmonics. From this, it is easy to see why a low frequency square wave oscillator can be used as a marker generator over a wide frequency range. <o:p></o:p></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>Figure 4 shows a 1000 Hz triangular wave. A perfect triangular wave also generates odd harmonics to infinity, but each amplitude is (l/n) squared relative to the fundamental or (40 log n) dB below the fundamental. For n = 3, 5, 7, and 9, the calculation is -19, -28, -33.8, and -38.2 dB respectfully, again very close to the readings shown. <o:p></o:p></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig3.jpg" align=middle></FONT></CENTER></TD> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig4.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>Figure 3: 1000 Hz Square Wave<br />
showing harmonics to 200 kHz</B> </FONT></CENTER></TD> <td> <center><b><font color=#bfbfbf>Figure 4: 1000 Hz Triangular Wave</FONT></B></CENTER></TD></TR>
</TBODY></TABLE></P><p style="TEXT-ALIGN: justify"><b><font color=#bfbfbf size=3>MODULATION</FONT></B></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>Figure 5 shows a 1 MHz carrier frequency, amplitude modulated by a frequency of 1 kHz to a modulation depth of 50 percent. For this case, the two side frequencies, 1 kHz either side of the carrier, are 12 dB below the carrier level, or a quarter of its amplitude. Other side frequencies at 2 kHz and 3 kHz, either side of the carrier, are the result of harmonics either in the original modulating tone or distortion caused by the modulation process. The 2 kHz side frequencies are about 30 dB below the 1 kHz side frequencies representing about three percent distortion in the system. <o:p></o:p></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>In Figure 6, the modulation level has been increased to 100 percent and the side frequencies, 1 kHz either side of the carrier, are now 6 dB below carrier level, or half its amplitude. The spectrum has been expanded to show many more harmonically related sideband components which now appear. Except for those close to the carrier, most of the components are more than 50 dB down and not of any great concern. <o:p></o:p></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig5.jpg" align=middle></FONT></CENTER></TD> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig6.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>Figure 5: 50 percent Amplitude Modulated Signal<br />
- Modulating Frequency 1000 Hz.</B> </FONT></CENTER></TD> <td> <center><b><font color=#bfbfbf>Figure 6: 100 percent Amplitude Modulated Signal<br />
- Modulating Frequency 1000 Hz.</FONT></B></CENTER></TD></TR>
</TBODY></TABLE></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>In Figure 7, the carrier is over-modulated and there is now a spread of sideband components about 30 dB down. If this were an amateur radio transmitter, other amateur stations in nearby suburbs would be complaining about sideband splatter. <o:p></o:p></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#bfbfbf>Figure 8 Shows a 1 MHz carrier, frequency modulated by a 1 kHz tone with a deviation of 8.650 kHz, representing a modulation index of 8.650. It can be seen that there are many side frequencies all spaced by an amount equal to the modulating frequency (1 kHz). For this signal, a significant bandwidth of about 20 to 30 kHz is being utilised. <o:p></o:p></FONT></FONT></SPAN></P><table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig7.jpg" align=middle></FONT></CENTER></TD> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig8.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>Figure 7: Over modulated AM Signal<br />
- Modulating Frequency 1000 Hz</B>.</FONT></CENTER></TD> <td> <center><b><font color=#bfbfbf>Figure 8. Frequency Modulated Signal<br />
- Modulating Frequency 1000 Hz.<br />
- Modulation Index 8.650<br />
and showing Third Carrier Null.</FONT></B></CENTER></TD></TR>
</TBODY></TABLE><p><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf size=3>If we now examine Figure 9, which plots the amplitude of the carrier and side frequencies against the value of modulation index, we can see that there are a number of values of modulation index where the carrier level becomes zero. These are very convenient references to calibrate the amount of deviation. In Figure 8, the deviation has been set to produce the third carrier null at a modulation index of 8.650, so we know precisely that with our modulating frequency of 1000 Hz, our deviation is 8.650 x 1000 = 8850 Hz.</FONT></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font color=#000000><o:p> <table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig9.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><b><font color=#bfbfbf>Figure 9: Plot of Bessel Functions<br />
(Third carrier null at modulation index = 8.650)</FONT></B></CENTER></TD></TR>
</TBODY></TABLE><font color=#bfbfbf><b><span lang=EN-US style="mso-ansi-language: EN-US">FREQUENCY RESPONSE</SPAN></B><span lang=EN-US style="mso-ansi-language: EN-US"><o:p></o:p></SPAN></FONT><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Another useful function of the spectrum analyser is to plot the frequency response of a four terminal device such as an amplifier or a filter. In this case, the analyser frequency sweep generator is fed to the input of the device and the output of the device is fed to the input of the analyser. Typical plots of a low pass filter and a bandpass filter are shown in Figures 10 and 11 respectively <o:p></o:p></FONT></SPAN><br />
<table borderColor=red border=1><tbody>
<tr> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig10.jpg" align=middle></FONT></CENTER></TD> <td> <center><font color=#bfbfbf><img src="http://users.tpg.com.au/users/ldbutler/SpectrumFig11.jpg" align=middle></FONT></CENTER></TD></TR>
<tr> <td> <center><font color=#bfbfbf><b>Figure 10: Low Pass Filter Response (fco = 20 kHz).</B> </FONT></CENTER></TD> <td> <center><b><font color=#bfbfbf>Figure 11: Bandpass Fitter Response (fc = 10 kHz, B = 8 kHz</FONT></B></CENTER></TD></TR>
</TBODY></TABLE><font color=#bfbfbf></FONT> <br />
<p style="TEXT-ALIGN: justify"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><u><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><span style="COLOR: windowtext">Microphone Frequency Response</SPAN></SPAN></U></B><b style="mso-bidi-font-weight: normal"><u><span lang=EN-US style="FONT-SIZE: 14pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><o:p></o:p></SPAN></U></B></FONT></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><font color=#bfbfbf><i><span lang=EN style="mso-ansi-language: EN">Frequency response</SPAN></I><span lang=EN style="mso-ansi-language: EN"> refers to the way a microphone responds to different frequencies. It is a characteristic of all microphones that some frequencies are exaggerated and others are attenuated (reduced). For example, a frequency response which favours high frequencies means that the resulting audio output will sound more trebly than the original sound.<o:p></o:p></SPAN></FONT></P><h3 style="BACKGROUND: white; MARGIN: 10pt 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: windowtext; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency Response Charts<o:p></o:p></FONT></SPAN></H3><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>A microphone's frequency response pattern is shown using a chart like the one below and referred to as a frequency response curve. The <i>x</I> axis shows frequency in Hertz, the <i>y</I> axis shows response in decibels. A higher value means that frequency will be exaggerated, a lower value means the frequency is attenuated. In this example, frequencies around 5 - kHz are boosted while frequencies above 10kHz and below 100Hz are attenuated. This is a typical response curve for a vocal microphone.</FONT></SPAN></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><o:p><font color=#bfbfbf><img height=286 alt="Frequency Response Chart" src="http://www.mediacollege.com/audio/images/mic-frequency-response1.gif" width=430></FONT></o:p></SPAN></P><span lang=EN style="mso-ansi-language: EN"><o:p> <h3 style="BACKGROUND: white; MARGIN: 10pt 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: windowtext; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Which Response Curve is Best?<o:p></o:p></FONT></SPAN></H3><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>An ideal "flat" frequency response means that the microphone is equally sensitive to all frequencies. In this case, no frequencies would be exaggerated or reduced (the chart above would show a flat line), resulting in a more accurate representation of the original sound. We therefore say that a flat frequency response produces the purest audio.<o:p></o:p></FONT></SPAN></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>In the real world a perfectly flat response is not possible and even the best "flat response" microphones have some deviation.<o:p></o:p></FONT></SPAN></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>More importantly, it should be noted that a flat frequency response is not always the most desirable option. In many cases a tailored frequency response is more useful. For example, a response pattern designed to emphasise the frequencies in a human voice would be well suited to picking up speech in an environment with lots of low-frequency background noise.<o:p></o:p></FONT></SPAN></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>The main thing is to avoid response patterns which emphasise the wrong frequencies. For example, a vocal mic is a poor choice for picking up the low frequencies of a bass drum.<o:p></o:p></FONT></SPAN></P><h3 style="BACKGROUND: white; MARGIN: 10pt 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: windowtext; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency Response Ranges<o:p></o:p></FONT></SPAN></H3><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>You will often see frequency response quoted as a range between two figures. This is a simple (or perhaps "simplistic") way to see which frequencies a microphone is capable of capturing effectively. For example, a microphone which is said to have a frequency response of 20 Hz to 20 kHz can reproduce all frequencies within this range. Frequencies outside this range will be reproduced to a much lesser extent or not at all.<o:p></o:p></FONT></SPAN></P><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>This specification makes no mention of the response curve, or how successfully the various frequencies will be reproduced. Like many specifications, it should be taken as a guide only.<o:p></o:p></FONT></SPAN></P><h3 style="BACKGROUND: white; MARGIN: 10pt 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: windowtext; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Condenser vs Dynamic<o:p></o:p></FONT></SPAN></H3><p style="BACKGROUND: white; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>Condenser microphones generally have flatter frequency responses than dynamic. All other things being equal, this would usually mean that a condenser is more desirable if accurate sound is a prime consideration.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p> </o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><font color=#bfbfbf>Lenny Z. Perez M</FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><font color=#bfbfbf>EES</FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><font color=#bfbfbf>Referencias:</FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><a href="http://www.mediacollege.com/audio/microphones/frequency-response.html"><font color=#bfbfbf>http://www.mediacollege.com/audio/microphones/frequency-response.html</FONT></A></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><a href="http://users.tpg.com.au/users/ldbutler/Waveforms.htm"><font color=#bfbfbf>http://users.tpg.com.au/users/ldbutler/Waveforms.htm</FONT></A></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p></o:p></SPAN> </P><p style="BACKGROUND: white; TEXT-ALIGN: justify"></o:p></SPAN> </P></o:p></FONT></FONT></SPAN></o:p></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"></FONT></o:p></SPAN></FONT></SPAN></o:p></SPAN></o:p></SPAN></o:p></SPAN></SPAN></o:p></FONT></SPAN> </P></DIV></CENTER></o:p></SPAN></SPAN></o:p></FONT></FONT></SPAN> <br />
<hr />Connect to the next generation of MSN Messenger <a href='http://imagine-msn.com/messenger/launch80/default.aspx?locale=en-us&source=wlmailtagline' target='_new'>Get it now! </a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-54038753583780873062010-02-15T21:01:00.002-04:302010-02-16T10:45:41.641-04:30Respuesta en Frecuencia.<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><u><span style="FONT-SIZE: 14pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-fareast-language: ES-VE; mso-bidi-font-size: 12.0pt"><span style="COLOR: windowtext"><font color=#bfbfbf>Respuesta en frecuencia<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></SPAN></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La respuesta en frecuencia es un parámetro que describe las frecuencias que puede grabar o reproducir un dispositivo.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Respuesta en Frecuencia en audio. En audio, para que sea un equipo de calidad debe cubrir al menos el margen de las audiofrecuencias (20-20.000 Hz).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Por el mismo motivo, cuanto mayor sea la respuesta en frecuencia de un equipo, más calidad tendrá el sonido final. Así, a los nuevos formatos de audio digital que sobrepasan sobradamente este margen (SACD, 20-100 KHz. y DVD-Audio, 20-80 kHz) se los cataloga como formatos HI-FI (High Fidelity) "Alta Fidelidad".<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La respuesta en frecuencia de cualquier sistema debería ser plana, lo que significa que el sistema trata igual a todo el sonido entrante, con lo que nos lo devuelve igual.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>No obstante, en la práctica, la respuesta en graves y agudos, normalmente no es la misma. Hecho que se nota más en unos equipos que en otros. (En los altavoces, por ejemplo, esta diferencia entre la respuesta a graves o agudos es muy acusada, pudiendo estar por encima de los 10 dB de más o de menos, entre una y otra).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Un equipo con una respuesta inapropiada afectará al sonido final:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Si un equipo enfatiza los agudos, el sonido resultante será "vibrante y chillón", mientras que si, por el contrario, pierde agudos, todo lo que reproduzca tendrá un "matiz oscuro". <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Si un equipo enfatiza los graves, el sonido resultante resulta "atronador", mientras que si, por el contrario, pierde graves, todo lo que reproduzca tendrá un "matiz metálico". <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Si se acentúan las frecuencias medias se produce un sonido "nasal". <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>En la mayoría de equipos, en las especificaciones técnicas, además de indicar cuál es la respuesta en frecuencia típica, se indica también la variación en dB entre una y otra.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Para ello, lo habitual es eligir -como nivel de referencia para indicar la respuesta en frecuencia- 1 kHz y a esta frecuencia se le da el valor de 0 dB. Luego, los fabricantes analizan todo el margen de frecuencias y establecen la diferencia en dBs entre la frecuencia más baja y la más alta.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Con esto, en las especificaciones técnicas nos dicen, por ejemplo, tal lector de CD tiene una respuesta en frecuencia de 20-20 kHz (+/-5 dB).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Salvo en los transductores (micrófonos, altavoces, etc), este margen, para asegurarnos "calidad", debe ser:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Inferior a +/- 1 dB, si hablamos de formatos digitales. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Inferior a +/- 3 dB si son equipos analógicos. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Como mucho +/- 6 dB, si son micros o altavoces. En la práctica, los muchos transductores: altavoces y micrófonos (salvo los más "profesionales") llegan a una variación de +/- 10. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Una mala respuesta en frecuencia no es lo peor que puede suceder, lo peor, es una respuesta desigual. Es decir, como a ciertas frecuencias sube, en otras baja, por lo que el sonido resultante sale distorsionado</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf></FONT></SPAN> </P><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Décadas y Octavas <o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Cuando dos frecuencias están separadas por una OCTAVA significa que una frecuencia es el doble<span style="mso-spacerun: yes"> </SPAN>que la otra. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>*Cuando dos frecuencias están separadas por una DÉCADA significa que una frecuencia es 10 veces la otra. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Dos frecuencias f1 y f2 están separadas n décadas cuando: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN>log10 (f2/f1) = n <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Dos frecuencias f1 y f2 están separadas n octavas cuando: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN>log2 (f2/f1) = n<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>El Decibel<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Equivale a la décima parte de un bel. Una unidad de referencia para medir la potencia de una señal o la intensidad de un sonido. El nombre bel viene del físico norteamericano Alexander Graham Bell (1847-1922). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>El decibel es una unidad relativa de una señal, tal como la potencia, voltaje, etc. Los logaritmos son muy usados debido a que la señal en decibeles (dB) puede ser fácilmente sumada o restada y también por la razón de que el oído humano responde naturalmente a niveles de señal en una forma aproximadamente logarítmica. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>GANANCIA DE POTENCIA EN DECIBELES <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La ganancia de Potencia G de un amplificador es la razón entre la potencia de salida a la potencia de salida a la potencia de entrada. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = P2 / P 1<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si la potencia de salida (P2) es de 15 W y la de entrada (P1) de 0.5 W, <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = 15 W / 0.5 W = 30<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Lo que significa que la potencia de salida es 30 veces mayor que la de entrada. por lo tanto la ganancia de potencia en decibeles se define como: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = ganancia de potencia (sin unidades) <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>donde G' = ganancia de potencia en decibeles <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G'(dB) = 10*log10(G) <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si un circuito determinado tiene una ganancia de potencia de 100, su ganancia en decibeles es: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10*log10(100) = 20 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La ganancia G' es adimensional, pero para estar seguros de no confundirla con la ganancia normal de potencia G, se añade la palabra decibel (dB). Cada vez que una respuesta se expresa en decibeles automáticamente se sabrá que se trata de la ganancia en decibeles de potencia y no de la ganancia normal de potencia. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Para transformar de decibeles a unidades absolutas: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>P= 10 x/10donde x esta dado en decibeles<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>3 dB por cada factor de 2 <o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Supóngase que la ganancia de potencia es 2, la ganancia en decibeles de potencia es: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 2 = 3.01 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si G = 4 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 4 = 6.02 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Si G= 8 <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 8 = 9.01 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Por lo general, se redondean estos valores tomando 3 dB, 6 dB y 9 dB. Se observa que cada vez que la potencia se aumenta al doble, la ganancia expresada en decibeles se incrementa 3 dB. (ver siguiente tabla)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G G´<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1<span style="mso-spacerun: yes"> </SPAN>0 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>2<span style="mso-spacerun: yes"> </SPAN>3 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>4<span style="mso-spacerun: yes"> </SPAN>6 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>8<span style="mso-spacerun: yes"> </SPAN>9 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>16<span style="mso-spacerun: yes"> </SPAN>12 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Decibeles negativos <o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si la ganancia de potencia es menor que la unidad, existe una pérdida de potencia (atenuación) y la ganancia de potencia en decibeles es negativa. Por ejemplo, si la potencia de salida es 1.5 W para una potencia de entrada de 3 W, se tiene: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = 1.5 W / 3 W = 0.5<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>y la ganancia de potencia en decibeles será: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 0.5 = -3.01 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Cuando la ganancia de potencia es de 0.25 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 0.25 = -6.02 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Y la ganancia de potencia es de 0.125, entonces <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 0.125 = -9.03 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>También en este caso se redondean estas cantidades a -3 dB, -6 dB y 9 dB. Cada vez que la ganancia disminuye en un factor de 2, la ganancia de potencia en decibeles disminuye en aproximadamente 3 dB. (ver siguiente tabla)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">G<span style="mso-spacerun: yes"> </SPAN>G'<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1<span style="mso-spacerun: yes"> </SPAN>0 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.5<span style="mso-spacerun: yes"> </SPAN>-3 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.25<span style="mso-spacerun: yes"> </SPAN>-6 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.125<span style="mso-spacerun: yes"> </SPAN>-9 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.0625<span style="mso-spacerun: yes"> </SPAN>-12 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10 dB corresponden a un factor de 10 <o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Supóngase que la ganancia de potencia es 10, la ganancia de potencia en decibeles será <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 10 = 10 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si la ganancia de potencia fuera 100, entonces <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 100 = 20 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Si la ganancia de potencia fuera de 1000 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 10 log 1000 = 30 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>En este caso el patrón que se observa es que la potencia en decibeles aumenta en 10 dB cada vez que la ganancia de potencia se incrementa por un factor de 10. (ver siguiente tabla). Un resultado similar se obtiene cuando las ganancias de potencia son inferiores a la unidad.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">G<span style="mso-spacerun: yes"> </SPAN>G'<span style="mso-spacerun: yes"> </SPAN>G<span style="mso-spacerun: yes"> </SPAN>G'<span style="mso-spacerun: yes"> </SPAN></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1<span style="mso-spacerun: yes"> </SPAN>0 dB<span style="mso-spacerun: yes"> </SPAN>1<span style="mso-spacerun: yes"> </SPAN>0 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10<span style="mso-spacerun: yes"> </SPAN>10 dB<span style="mso-spacerun: yes"> </SPAN>0.1<span style="mso-spacerun: yes"> </SPAN>-10 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>100<span style="mso-spacerun: yes"> </SPAN>20 dB<span style="mso-spacerun: yes"> </SPAN>0.01<span style="mso-spacerun: yes"> </SPAN>-20 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1000<span style="mso-spacerun: yes"> </SPAN>30 dB<span style="mso-spacerun: yes"> </SPAN>0.001<span style="mso-spacerun: yes"> </SPAN>-30 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10000<span style="mso-spacerun: yes"> </SPAN>40 dB<span style="mso-spacerun: yes"> </SPAN>0.0001<span style="mso-spacerun: yes"> </SPAN>-40 dB<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Las ganancias normales se multiplican entre sí <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>En la siguiente figura (a) se muestran dos etapas de un amplificador. A la primera etapa se le aplica una potencia de entrada de P1 y sale de ella una potencia P2, lo que significa una ganancia de potencia. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G 1 = P 2 / P 1<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La segunda etapa tiene una entrada de potencia P2 y sale una potencia P3, lo que equivale a una ganancia de <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G 2 = P 3 / P 2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La segunda total de potencia de ambas etapas es <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = (P 2 /P 1 )*(P 3 /P 2 )= P 3 /P 1<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Es decir, que <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = G1 G2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Esto demuestra que la ganancia total de potencia de etapas amplificadas en cascada es igual al producto de las ganancias de las etapas. No importa cuantas etapas sean, siempre puede determinarse la ganancia total de potencia multiplicando todas las ganancias individuales entre sí. En la figura del inciso (b), por ejemplo, indica una ganancia de potencia de 100 para la primera etapa y una ganancia de potencia de 200 para la segunda. La ganancia de potencia total será: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = 100 x 200 = 20,000 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 212.4pt; TEXT-INDENT: -212.4pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf><img height=250 src="http://www.eveliux.com/imagenes/db01.gif" width=350 border=0><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">figura.- etapas en cascada <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Las ganancias en decibeles se suman <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Puesto que la ganancia total de potencia de dos etapas en cascada es de <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = G1G2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>pueden tomarse logaritmos en ambos lados para obtener <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>log G = log G1G2 = logG1 + logG2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>y, al multiplicar ambos miembros por 10, se tiene <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10 logG = 10 logG1 + 10 logG2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>lo que también puede escribirse como <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = G'1 + G'2 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>donde G' = ganancia de potencia total en decibeles <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G'1 = ganancia de potencia en decibeles de la primera etapa <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G'2 = ganancia de potencia en decibeles de la segunda etapa <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La ecuación nos dice que la ganancia de potencia total en decibeles de dos etapas en cascada es igual a la suma de las ganancias en decibeles de cada etapa. La misma idea es valida para n etapas. La figura del inciso ©, por ejemplo nos muestra las mismas dos etapas de la figura (b) con la salvedad de que las ganancias están representadas en este caso en decibeles. La ganancia de potencia total en decibeles es <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G' = 20 dB + 23 dB = 43 dB <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La respuesta puede expresarse así o pasarla de nuevo a la forma normal de ganancia de potencia como sigue: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>G = 10 G'/10 = antilog( 43/10) = 20,000 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La respuesta en dB tiene la ventaja de ser más compacta y fácil de escribir.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Facilidad de medida <o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La ventaja de usar dBm es que simplifica la medición de la potencia. Algunos instrumentos, por ejemplo, tienen dos escalas para indicar el nivel de potencia, como se muestra en la siguiente figura inciso (a). La escala superior está graduada en miliwatts. Supóngase que se mide la potencia de entrada y la potencia de salida de la etapa de la figura (b). En la escala superior se lee 0.25 mW (aguja del trazo continuo) para la potencia de entrada y 1 mW (aguja de línea punteada para la de salida) <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La escala inferior, en la figura (a), es la escala de dBm. Como se indica en la figura, 0 dBm equivale a 1mW, -3 dBm equivale a 0.5 mW, -6 dBm equivalen a 0.25 mW, etc. Si se usa esta escala para medir las potencias indicadas en la figura (b), se leerá -6 dBm para la potencia de entrada y 0 dBm para la potencia de salida, como se muestra en la figura ©. Puesto que la aguja se mueve de -6 dBm significa que el amplificador tiene una ganancia de potencia de 6 dB. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf><strong><img height=220 src="http://www.eveliux.com/imagenes/db02.gif" width=340 border=0></STRONG> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Significado de dBm </SPAN></B><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>A continuación se da una tabla de conversión de Watts y miliwatts a dBW y a dBm.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Watts<span style="mso-spacerun: yes"> </SPAN>mW<span style="mso-spacerun: yes"> </SPAN>dBW<span style="mso-spacerun: yes"> </SPAN>dBm<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.01<span style="mso-spacerun: yes"> </SPAN>10<span style="mso-spacerun: yes"> </SPAN>-20<span style="mso-spacerun: yes"> </SPAN>10<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.10<span style="mso-spacerun: yes"> </SPAN>100<span style="mso-spacerun: yes"> </SPAN>-10<span style="mso-spacerun: yes"> </SPAN>20<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.63<span style="mso-spacerun: yes"> </SPAN>630<span style="mso-spacerun: yes"> </SPAN>-2<span style="mso-spacerun: yes"> </SPAN>28<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>0.79<span style="mso-spacerun: yes"> </SPAN>790<span style="mso-spacerun: yes"> </SPAN>-1<span style="mso-spacerun: yes"> </SPAN>29<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1<span style="mso-spacerun: yes"> </SPAN>1000<span style="mso-spacerun: yes"> </SPAN>0<span style="mso-spacerun: yes"> </SPAN>30<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1.12<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>0.5<span style="mso-spacerun: yes"> </SPAN>30.5<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1.26<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>1<span style="mso-spacerun: yes"> </SPAN>31<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1.58<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>2<span style="mso-spacerun: yes"> </SPAN>32<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>2<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>3<span style="mso-spacerun: yes"> </SPAN>33<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>3.16<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>5<span style="mso-spacerun: yes"> </SPAN>35<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>4<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>6<span style="mso-spacerun: yes"> </SPAN>36<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>5.01<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>7<span style="mso-spacerun: yes"> </SPAN>37<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>10<span style="mso-spacerun: yes"> </SPAN>40<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>100<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>20<span style="mso-spacerun: yes"> </SPAN>50<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1,000<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>30<span style="mso-spacerun: yes"> </SPAN>60<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>10,000<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>40<span style="mso-spacerun: yes"> </SPAN>70<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>100,000<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>50<span style="mso-spacerun: yes"> </SPAN>80<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>1'000,000<span style="mso-spacerun: yes"> </SPAN>-<span style="mso-spacerun: yes"> </SPAN>60<span style="mso-spacerun: yes"> </SPAN>90<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Conclusión</SPAN></B><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">: <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>La mayoría de los amplificadores usados en electrónica son especificados en decibeles. Por ejemplo: si adquirimos un amplificador con Ganancia de 20 dB, significa que éste amplificará la señal de entrada 100 veces. En cambio un amplificador de 30 dB (10 dB más que el anterior) amplificara 1,000 veces la señal de entrada. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Por ultimo para recalcar, el término dbm se emplea más comúnmente cuando nos estamos refiriendo a potencias entre 0 y 1 Watt. (en este caso es más fácil hablar en términos de miliwatts o dBm).</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf></FONT></SPAN> </P><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b style="mso-bidi-font-weight: normal"><u><span style="FONT-SIZE: 14pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: Arial"><a href="http://circuitos-de-electronica.blogspot.com/2007/11/regla-de-sustitucin-teorema-de-miller.html"><span style="BACKGROUND: white; COLOR: windowtext"><font color=#bfbfbf>Regla de Sustitución. Teorema de Miller</FONT></SPAN></A><o:p></o:p></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-bidi-font-weight: bold">Si se conoce la relación u = Z(D)·i ó i = Y(D)·u entre los terminales de un elemento pasivo o de una rama de un circuito, estos elementos pueden sustituirse por una fuente de tensión, cuya forma de onda sea Z(D)·i, o por una fuente de intensidad dada por Y(D)·u.</SPAN><u><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p></o:p></SPAN></U></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><img style="WIDTH: 427px; HEIGHT: 117px" height=117 alt=[Figura+15.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_-cRpc7WEuwQcI9ny5XIuDTlTHz_2Rj4xMIe4jJAQIHR92UzxENQ1G5DPgGGe7hAcz82jQkIi5s8TcJ_FWm2EOqloIFnp1XgPLiRXYWQa28gWypUym7xPdmvZotNjE2EtRmzg8BMNktA/s1600/Figura+15.jpg" width=380 border=0> </FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Esta regla está fundada en que la sustitución indicada no altera las ecuaciones que se deducen a partir de las Leyes de Kirchhoff.<br />
Estas fuentes de sustitución son dependientes y ha de tenerse en cuenta que se comportan de forma distinta que las fuentes ideales. En particular se puede explicar esta regla a un circuito abierto y a un circuito en corto.<br />
Si la tensión entre dos terminales A y B de un circuito activo es Uo, no se altera en nada el estado del circuito al conectar esos terminales mediante una fuente de tensión e0 = Uo de la misma polaridad que la existente entre A y B (Fig. 16).</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p><font color=#bfbfbf><img style="WIDTH: 375px; HEIGHT: 125px" height=125 alt=[Figura+16.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_bfxh9Dok6Nr2BuqX-RoA9RCzFxGaW1_E2qGGMRbzHS_oFCGe6364j7KtZMb3H8sBRVa-ri1A_tRjB-_SWAshCFqUcSGSdkcYmb_tbXLvnYFEmZKmvRvYJ0Vm5xEBDYnyF-51WIoaVTg/s1600/Figura+16.jpg" width=431 border=0> </FONT></o:p></SPAN></P></o:p></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">Su configuración cambia aparentemente, pues aumenta en una unidad el número de mallas, pero ha de tenerse en cuenta que en esa malla no circula intensidad alguna, luego no aumenta el número de incógnitas.<br />
Análogamente, en un conductor de resistencia nula, por el que circula una intensidad i0, puede intercalarse una fuente ideal de intensidad igual a i0 sin que se altere el estado del circuito.<br />
Un condensador con carga eléctrica inicial o una bobina con flujo inicial admiten una representación sencilla haciendo uso de la regla de sustitución. En el caso de un condensador cargado inicialmente a una tensión Uo, la ecuación de definición es:</SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><img alt=[Ecuacion+14.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLRbB821INbWn016B93pnTKMu1sro6r2Y3Zjhyphenhyphenwk_y39nnbJAvjM8zINxYpuHKvTb8zRD8wKoqjY8zQxrj2XrrBQQ8F_-Ir75-ihwYq-wcdukiXn1kExcYAjZWCHPtD1pWp-S0ewUW5wk/s1600/Ecuacion+14.jpg" border=0> </FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf>que corresponde al circuito </FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><font color=#bfbfbf><img style="WIDTH: 388px; HEIGHT: 191px" height=266 alt=[Figura+17.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidK2KVQBsjRz7upC9rF3uoRowwV_gSMmz-i1UAn9VzsfM4oDZf60Z-Gt3sb5zCq21FkD4_BERy6uw4lWe8N8LlI4tFqPacY84k2adp-VnJ-Pdf0JU6QRm4KZPHnj4DJQd1x05O5Re5Uf8/s1600/Figura+17.jpg" width=499 border=0> </FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es decir, se sustituye por una fuente de tensión en serie con un condensador inicialmente descargado.<br />
Para una bobina por la que circula inicialmente una intensidad I0, la ecuación de definición es:</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p><font color=#bfbfbf><img height=76 alt=[Ecuacion+15.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6SrxAVs6iUXOfNCOhbi4UX5eFxYRCN8LyyQR8jW4CA2WZqI-tcFshHtzz1fDEDQxCINZ8GnmzHVDM3UxI860xfhWBTLKVgnm2OWLEoUtDmFx_dJuZB5hyphenhyphenqaP9FWqJF_f7sQdgaGU77E4/s1600/Ecuacion+15.jpg" width=173 border=0> </FONT></o:p></SPAN></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>que corresponde al circuito </FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La regla de sustitución es una herramienta muy útil en la demostración de teoremas. Por ejemplo el Teorema de Miller (en Electrónica):<br />
Si en un circuito se sustituye la impedancia Z por un circuito abierto y se une el nudo 1 a otro 0, este último tomado como referencia, mediante una impedancia Z/(1-k) y el 2 se une también al mismo nudo O por medio de otra impedancia de valor ZK/(k-1), en donde k es la relación U<sub>2</SUB>/U<sub>1</SUB> entre las tensiones de los nudos 2 y 1 respecto a 0, no se altera la intensidad que circula entre los nudos 1 y 2.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img alt=[Figura+18.jpg] src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6fVBh8Ohjw_ZRNs_hnZJ7QRSFvHk_N6ZYn3Vo6Op7y8wYT1MYpM0n3-9LOgcUgg50xSLvaV1IdhcFPw9AHTDqhFAdXpcrNUi9JEfnOMf4eoAHdkK8ZMEs7-sGP1JRBmXkoux-N2Fy-yQ/s1600/Figura+18.jpg" border=0> </FONT></SPAN></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>APROXIMACIÓN DEL POLO DOMINANTE.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Una vez que se tiene el equivalente en pequeña señal del circuito, este método de análisis consiste en lo siguiente: <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Análisis en BAJA frecuencia.<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Se analizarán los individualmente los condensadores que afectan en baja frecuencia (acoplo y desacoplo). Para ello los condensadores de alta frecuencia se consideran circuitos abiertos y los de baja frecuencia que no sean el condensador que se analiza encada momento se considerarán cortocircuitos. Cada condensador de baja frecuencia aportará una frecuencia candidata a ser considerada la frecuencia de corte inferior. La MAYOR de todas ellas será considerada la frecuencia de corte inferior del circuito, si dista de las demás como una distancia igual o superior a 1 década. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Análisis en ALTA frecuencia.<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Se analizarán los individualmente los condensadores que afectan en alta frecuencia (paralelo). Para ello los condensadores de baja frecuencia (acoplo y desacoplo) se consideran cortocircuitos y los de alta frecuencia que no sean el condensador que se analiza en cada momento se considerarán circuitos abiertos. Cada condensador de alta frecuencia aportará una frecuencia candidata a ser considerada la frecuencia de corte superior. La MENOR de todas ellas será considerada la frecuencia de corte superior del circuito, si dista de las demás como una distancia igual o superior la década.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf></FONT> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf>Lenny. Z Perez M</FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf>EES</FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf>Referencias:</FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><o:p><font face=Calibri color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://www.eveliux.com/mx/el-decibel.php"><font face=Calibri color=#bfbfbf>http://www.eveliux.com/mx/el-decibel.php</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://circuitos-de-electronica.blogspot.com/2007/11/regla-de-sustitucin-teorema-de-miller.html"><font face=Calibri color=#bfbfbf>http://circuitos-de-electronica.blogspot.com/2007/11/regla-de-sustitucin-teorema-de-miller.html</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><cite><span style="FONT-FAMILY: 'Arial','sans-serif'"><a href="http://www.dte.upct.es/personal/jsuardiaz/docencia/.../CIALTema4.pdf"><span lang=EN style="mso-ansi-language: EN"><font color=#bfbfbf>www.dte.upct.es/personal/jsuardiaz/docencia/.../CIALTema4.pdf</FONT></SPAN></A></SPAN></CITE><cite><span lang=EN style="FONT-FAMILY: 'Arial','sans-serif'; mso-ansi-language: EN"><o:p></o:p></SPAN></CITE></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p></o:p></SPAN> </P></P></o:p></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><o:p></o:p></SPAN> </P></o:p></SPAN> <br />
<hr />Discover the new Windows Vista <a href='http://search.msn.com/results.aspx?q=windows+vista&mkt=en-US&form=QBRE' target='_new'>Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-54397638331744202412010-02-15T19:57:00.002-04:302010-02-16T10:45:16.369-04:30Dominant Pole...<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><u><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency compensation<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In </FONT><a title="Electrical engineering" href="http://en.wikipedia.org/wiki/Electrical_engineering"><span style="COLOR: windowtext"><font color=#bfbfbf>electrical engineering</FONT></SPAN></A><font color=#bfbfbf>, <b>frequency compensation</B> is a technique used in </FONT><a title=Amplifiers href="http://en.wikipedia.org/wiki/Amplifiers"><span style="COLOR: windowtext"><font color=#bfbfbf>amplifiers</FONT></SPAN></A><font color=#bfbfbf>, and especially in amplifiers employing negative feedback. It usually has two primary goals: To avoid the unintentional creation of </FONT><a title="Positive feedback" href="http://en.wikipedia.org/wiki/Positive_feedback"><span style="COLOR: windowtext"><font color=#bfbfbf>positive feedback</FONT></SPAN></A><font color=#bfbfbf>, which will cause the amplifier to </FONT><a title="Electronic oscillation" href="http://en.wikipedia.org/wiki/Electronic_oscillation"><span style="COLOR: windowtext"><font color=#bfbfbf>oscillate</FONT></SPAN></A><font color=#bfbfbf>, and to control </FONT><a title="Overshoot (signal)" href="http://en.wikipedia.org/wiki/Overshoot_(signal)"><span style="COLOR: windowtext"><font color=#bfbfbf>overshoot</FONT></SPAN></A><font color=#bfbfbf> and </FONT><a title="Ringing (signal)" href="http://en.wikipedia.org/wiki/Ringing_(signal)"><span style="COLOR: windowtext"><font color=#bfbfbf>ringing</FONT></SPAN></A><font color=#bfbfbf> in the amplifier's </FONT><a title="Step response" href="http://en.wikipedia.org/wiki/Step_response"><span style="COLOR: windowtext"><font color=#bfbfbf>step response</FONT></SPAN></A><font color=#bfbfbf>.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=mw-headline><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN"><font color=#bfbfbf size=3>Explanation</FONT></SPAN></SPAN></B></SPAN><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Most amplifiers use negative feedback to trade gain for other desirable properties, such as decreased distortion or improved noise reduction. Ideally, the phase characteristic of an amplifier's frequency response would be constant; however, device limitations make this goal physically unattainable. More particularly, capacitances within the amplifier's gain stages cause the output signal to lag behind the input signal by 90° for each pole they create.[1] If the sum of these phase lags reaches 360°, the output signal will be in phase with the input signal. Feeding back any portion of this output signal to the input when the gain of the amplifier is sufficient will cause the amplifier to oscillate. This is because the feedback signal will reinforce the input signal. That is, the feedback is then positive rather than negative.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency compensation is implemented to avoid this result.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Another goal of frequency compensation is to control the step response of an amplifier circuit as shown in Figure 1. For example, if a step in voltage is input to a voltage amplifier, ideally a step in output voltage would occur. However, the output is not ideal because of the frequency response of the amplifier, and ringing occurs. Several figures of merit to describe the adequacy of step response are in common use. One is the rise time of the output, which ideally would be short. A second is the time for the output to lock into its final value, which again should be short. The success in reaching this lock-in at final value is described by overshoot (how far the response exceeds final value) and settling time (how long the output swings back and forth about its final value). These various measures of the step response usually conflict with one another, requiring optimization methods.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency compensation is implemented to optimize step response, one method being pole splitting<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Use in operational amplifiers<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Because operational amplifiers are so ubiquitous and are designed to be used with feedback, the following discussion will be limited to frequency compensation of these devices.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>It should be expected that the outputs of even the simplest operational amplifiers will have at least two poles. An unfortunate consequence of this is that at some critical frequency, the phase of the amplifier's output = -180° compared to the phase of its input signal. The amplifier will oscillate if it has a gain of one or more at this critical frequency. This is because (a) the feedback is implemented through the use of an inverting input that adds an additional -180° to the output phase making the total phase shift -360° and (b) the gain is sufficient to induce oscillation.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>A more precise statement of this is the following: An operational amplifier will oscillate at the frequency at which its open loop gain equals its closed loop gain if, at that frequency,<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>1. The open loop gain of the amplifier is ≥ 1 and <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>2. The difference between the phase of the open loop signal and phase response of the network creating the closed loop output = -180°. Mathematically, <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>ΦOL – ΦCLnet = -180° <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Practice<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Frequency compensation is implemented by modifying the gain and phase characteristics of the amplifier's open loop output or of its feedback network, or both, in such a way as to avoid the conditions leading to oscillation. This is usually done by the internal or external use of resistance-capacitance networks.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Dominant-pole compensation<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The method most commonly used is called dominant-pole compensation, which is a form of lag compensation. A pole placed at an appropriate low frequency in the open-loop response reduces the gain of the amplifier to one (0 dB) for a frequency at or just below the location of the next highest frequency pole. The lowest frequency pole is called the dominant pole because it dominates the effect of all of the higher frequency poles. The result is that the difference between the open loop output phase and the phase response of a feedback network having no reactive elements never falls below −180° while the amplifier has a gain of one or more, ensuring stability.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">Dominant-pole</SPAN></B><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"> compensation can be implemented for general purpose operational amplifiers by adding an integrating capacitance to the stage that provides the bulk of the amplifier's gain. This capacitor creates a pole that is set at a frequency low enough to reduce the gain to one (0 dB) at or just below the frequency where the pole next highest in frequency is located. The result is a phase margin of ≈ 45°, depending on the proximity of still higher poles.This margin is sufficient to prevent oscillation in the most commonly used feedback configurations. In addition, dominant-pole compensation allows control of overshoot and ringing in the amplifier step response, which can be a more demanding requirement than the simple need for stability.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Though simple and effective, this kind of conservative dominant pole compensation has two drawbacks:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>1. It reduces the bandwidth of the amplifier, thereby reducing available open loop gain at higher frequencies. This, in turn, reduces the amount of feedback available for distortion correction, etc. at higher frequencies. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>2. It reduces the amplifier's slew rate. This reduction results from the time it takes the finite current driving the compensated stage to charge the compensating capacitor. The result is the inability of the amplifier to reproduce high amplitude, rapidly changing signals accurately. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Often, the implementation of dominant-pole compensation results in the phenomenon of Pole splitting. This results in the lowest frequency pole of the uncompensated amplifier "moving" to an even lower frequency to become the dominant pole, and the higher-frequency pole of the uncompensated amplifier "moving" to a higher frequency.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Other methods<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Some other compensation methods are: lead compensation, lead–lag compensation and feed-forward compensation.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Lead compensation. Whereas dominant pole compensation places or moves poles in the open loop response, lead compensation places a zero[3] in the open loop response to cancel one of the existing poles. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Lead–lag compensation places both a zero and a pole in the open loop response, with the pole usually being at an open loop gain of less than one. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Feed-forward compensation uses a capacitor to bypass a stage in the amplifier at high frequencies, thereby eliminating the pole that stage creates. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The purpose of these three methods is to allow greater open loop bandwidth while still maintaining amplifier closed loop stability. They are often used to compensate high gain, wide bandwidth amplifiers.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font color=#bfbfbf>Footnotes<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>^ In this context, a pole is the point in a frequency response curve where the amplitude decreases by 3db due to an integrating resistance and capacitive reactance. Ultimately, each pole will result in a phase lag of 90°, i.e., the output signal will lag behind the input signal by 90° at this point. For the mathematical concept of a pole, see, Pole (complex analysis). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>^ The dominant pole produces a phase shift approximating -90° from approx. 10 times the pole frequency to a frequency a factor of ten below the next higher pole position. The next higher pole, in turn, adds another -45° for a frequency at its location for a total of -135° (neglecting the still higher poles). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>^ In this context, a zero is the point in a frequency response curve where the amplitude increases by 3db due to a differentiating resistance and capacitive reactance. Ultimately, each zero will result in a phase lead of 90°, i.e., the phase of the output signal will be 90° ahead of the phase of the input signal at this point. For the mathematical concept of a zero, see, Zero (complex analysis).</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf></FONT></SPAN> </P><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p> <h2 style="MARGIN: auto 0cm"><u><span style="FONT-SIZE: 14pt; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>The Dominant Pole approximation<o:p></o:p></FONT></SPAN></U></H2><h3 style="MARGIN: 10pt 0cm 0pt"><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font size=3><font face=Cambria><font color=#bfbfbf>Reduction of a second order system to first order<o:p></o:p></FONT></FONT></FONT></SPAN></H3><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Consider a second order system with a transfer function that is reduced to first order</FONT></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><o:p><font color=#bfbfbf><img height=30 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole1.gif" width=212 align=bottom></FONT></o:p></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><o:p> <p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>This assumes that a>>b, or that the pole at b is dominant. The coefficient "a" remains in the denominator so that the DC gain (which is also the final value of the output with a unit step input) remains unchanged. Recall that the DC gain is G(0).<o:p></o:p></FONT></SPAN></P><span lang=EN-US style="FONT-SIZE: 11pt; FONT-FAMILY: 'Calibri','sans-serif'; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-fareast-language: EN-US; mso-bidi-language: AR-SA"><font face="Times New Roman" color=#bfbfbf>The graph below shows the exact response (red) and the dominant pole approximation (green) for a=8 and b=1. Following the graph is Matlab code in which you can set a with b=1 to see how accurate the dominant pole approximation is</FONT></SPAN><br />
<span lang=EN-US style="FONT-SIZE: 11pt; FONT-FAMILY: 'Calibri','sans-serif'; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-fareast-language: EN-US; mso-bidi-language: AR-SA"><font color=#bfbfbf><img style="WIDTH: 409px; HEIGHT: 223px" height=345 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole2.gif" width=532 align=bottom></FONT></SPAN><br />
<span lang=EN-US style="FONT-SIZE: 11pt; FONT-FAMILY: 'Calibri','sans-serif'; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-fareast-language: EN-US; mso-bidi-language: AR-SA"> <h3 style="MARGIN: 10pt 0cm 0pt"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: windowtext; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US"><font color=#bfbfbf>Higher Order<o:p></o:p></FONT></SPAN></H3><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman" color=#bfbfbf>The dominant pole approximation can also be applied to higher order systems. Here we consider a third order system with one real root, and a pair of complex conjugate roots.</FONT></FONT></SPAN><br />
<span lang=EN-US style="mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3></FONT></SPAN> <br />
<span lang=EN-US style="mso-ansi-language: EN-US"></SPAN><span style="mso-no-proof: yes"><font color=#bfbfbf><img height=70 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole3.gif" width=333 align=bottom></FONT></SPAN><br />
<span style="mso-no-proof: yes"> <p style="TEXT-ALIGN: justify"><font size=3><font face="Times New Roman"><font color=#bfbfbf><span lang=EN-US style="mso-ansi-language: EN-US">In this case the test for the dominant pole compare "a" against "</SPAN>zw<sub><span lang=EN-US style="mso-ansi-language: EN-US">n</SPAN></SUB><span lang=EN-US style="mso-ansi-language: EN-US">". This is because "</SPAN>zw<sub><span lang=EN-US style="mso-ansi-language: EN-US">n</SPAN></SUB><span lang=EN-US style="mso-ansi-language: EN-US">" is the real part of the complex conjugate root (we only compare the real parts of the roots when determining dominance because it is the real part that determines how fast the response decreases). Note that the DC gain of the exact system and the two approximate systems are equal.<o:p></o:p></SPAN></FONT></FONT></FONT></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>In the examples and Matlab code below, the second order pole has zeta=0.4 and wn=1 (which yields roots with a real part of 0.4 and an imaginary part of +/-0.92j). There are three graphs. In the first graph a=0.1 (the real pole dominates), in the second graph a=4 (the complex conjugate poles dominate) and in the third graph a=0.4 (neither dominates and the response is obviously more complicated than a simple second order response). In all three graphs the exact response is in red, the approximate response in which the first order pole dominates is in green, and the approximate response in which the second order pole dominates is in blue.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></P><p align=center><font color=#bfbfbf><img height=205 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole4.gif" width=456 align=bottom></FONT></P><p align=center><font color=#bfbfbf><img height=205 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole5.gif" width=456 align=bottom></FONT></P><p align=center><font color=#bfbfbf><img height=205 src="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole6.gif" width=456 align=bottom></FONT></P><p align=center><font color=#bfbfbf></FONT> </P><h2 style="MARGIN: auto 0cm"><u><span lang=EN-US style="FONT-SIZE: 14pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: Arial"><font color=#bfbfbf>Dominant-Pole Compensation<o:p></o:p></FONT></SPAN></U></H2><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font color=#bfbfbf>Dominant-pole compensation is the simplest kind, though its action is subtle. Simply take the lowest pole to hand (P1), and make it dominant, i.e., so much lower in frequency than the next pole P2 that the total loop-gain (i.e., the open-loop gain as reduced by the attenuation in the feedback network) falls below unity before enough phase-shift accumulates to cause HF oscillation. With a single pole, the gain must fall at 6 dB/octave, corresponding to a constant 90 phase shift. Thus the phase margin will be 90 , giving good stability.<o:p></o:p></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="mso-ansi-language: EN-US">Figure 7.1 a shows the traditional Miller method of creating a dominant pole. The collector pole of TR4 is lowered by adding the external Miller-capacitance Cdom to that which unavoidably exists as the internal Cbc of the VAS transistor. However, there are some other beneficial effects; Cdom causes <i>pole-splitting</I>, in which the pole at TR2 collector is pushed up in frequency as P1 is moved down most desirable for stability. </SPAN>Simultaneously the local NFB through Cdom linearises the VAS.</FONT></P><p align=center><font color=#bfbfbf></FONT> </P><p align=center><font color=#bfbfbf><span><img height=373 src="http://images.books24x7.com/bookimages/id_25410/fig209_01.jpg" width=295></SPAN> </FONT></P><p align=center><font color=#bfbfbf></FONT> </P><p style="TEXT-ALIGN: justify"><o:p><font color=#bfbfbf> </FONT></o:p></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Figure 7.1: (a) The traditional Miller method of making a dominant pole. (b) Shunt compensation shows a much less satisfactory method the addition of capacitance to ground from the VAS collector. (c) Inclusive Miller compensation. (d) Two-pole compensation<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Assuming that input-stage transconductance is set to a plausible 5 mA/V, and stability considerations set the maximal 20kHz open-loop gain to 50 dB, then from Equations 3.1 3.3 on pages 63 and 64, Cdom must be 125 pF.<o:p></o:p></FONT></SPAN></P><p align=center><font color=#bfbfbf></FONT> </P><p align=left><font color=#bfbfbf>Lenny Z. Perez M</FONT></P><p align=left><font color=#bfbfbf>EES</FONT></P><p align=left><font color=#bfbfbf>Referencias</FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><o:p><font face=Calibri color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://en.wikipedia.org/wiki/Frequency_compensation"><font face=Calibri color=#bfbfbf>http://en.wikipedia.org/wiki/Frequency_compensation</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole.html"><font face=Calibri color=#bfbfbf>http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole.html</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN style="mso-ansi-language: EN"><a href="http://www.globalspec.com/reference/22133/203279/Dominant-Pole-Compensation"><font face=Calibri color=#bfbfbf>http://www.globalspec.com/reference/22133/203279/Dominant-Pole-Compensation</FONT></A><o:p></o:p></SPAN></P><p align=left> </P></o:p></SPAN> </SPAN> <br />
<span style="mso-no-proof: yes"> <br />
<p style="TEXT-ALIGN: center" align=center><?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" /><v:shape id=Imagen_x0020_85 style="VISIBILITY: visible; WIDTH: 249.75pt; HEIGHT: 52.5pt; mso-wrap-style: square" type="#_x0000_t75" alt="http://www.swarthmore.edu/NatSci/echeeve1/Class/e58/SpecialTopics/DominantPole3.gif" o:spid="_x0000_i1025"><v:imagedata o:title="DominantPole3" src="file:///C:\DOCUME~1\Usuario\CONFIG~1\Temp\msohtmlclip1\01\clip_image001.gif"></v:imagedata></v:shape></P></SPAN> </SPAN> <br />
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<hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href='http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us' target='_new'>Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-49399398326385700922010-02-15T19:21:00.002-04:302010-02-16T10:44:44.401-04:30Frequency Response of Amplifiers. Cascode<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN; mso-bidi-font-size: 12.0pt"><span style="COLOR: windowtext">Cascode<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></SPAN></SPAN></I></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode is a two-stage amplifier composed of a transconductance amplifier followed by a current buffer. Compared to a single amplifier stage, this combination may have one or more of the following advantages: higher input-output isolation, higher input impedance, higher output impedance, higher gain or higher bandwidth. In modern circuits, the cascode is often constructed from two transistors (BJTs or FETs), with one operating as a common emitter or common source and the other as a common base or common gate. The cascode improves input-output isolation (or reverse transmission) as there is no direct coupling from the output to input. This eliminates the Miller effect and thus contributes to a much higher bandwidth.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">History<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode (sometimes verbified to cascoding) is a universal technique for improving analog circuit performance, applicable to both vacuum tubes and transistors. The word "cascode" is a contraction of the phrase "cascade to cathode". It was first used in an article by F.V. Hunt and R.W. Hickman in 1939, in a discussion for application in low-voltage stabilizers.They proposed a cascode of two triodes (first one with common cathode, the second one with common grid) as a replacement of a pentode.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=mw-headline><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">Operation</SPAN><o:p></o:p></SPAN></B></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Figure 1 shows an example of cascode amplifier with a common source amplifier as input stage driven by signal source Vin. This input stage drives a common gate amplifier as output stage, with output signal Vout.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The major advantage of this circuit arrangement stems from the placement of the upper Field Effect Transistor (FET) as the load of the input (lower) FET's output terminal (drain). Because at operating frequencies the upper FET's gate is effectively grounded, the upper FET's source voltage (and therefore the input transistor's drain) is held at nearly constant voltage during operation. In other words, the upper FET exhibits a low input resistance to the lower FET, making the voltage gain of the lower FET very small, which dramatically reduces the Miller feedback capacitance from the lower FET's drain to gate. This loss of voltage gain is recovered by the upper FET. Thus, the upper transistor permits the lower FET to operate with minimum negative (Miller) feedback, improving its bandwidth.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The upper FET gate is electrically grounded, so charge and discharge of stray capacitance Cdg between drain and gate is simply through RD and the output load (say Rout), and the frequency response is affected only for frequencies above the associated RC time constant: τ = Cdg RD//Rout, namely f = 1/(2πτ), a rather high frequency because Cdg is small. That is, the upper FET gate does not suffer from Miller amplification of Cdg.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p> </o:p></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">If the upper FET stage were operated alone using its source as input node (i.e. common-gate (CG) configuration), it would have good voltage gain and wide bandwidth. However, its low input impedance would limit its usefulness to very low impedance voltage drivers. Adding the lower FET results in a high input impedance, allowing the cascode stage to be driven by a high impedance source.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">If one were to replace the upper FET with a typical inductive/resistive load, and take the output from the input transistor's drain (i.e. a common-emitter (CE) configuration), the CE configuration would offer the same input impedance as the cascode, but the cascode configuration would offer a potentially greater gain and much greater bandwidth.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Stability<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode arrangement is also very stable. Its output is effectively isolated from the input both electrically and physically. The lower transistor has nearly constant voltage at both drain and source and thus there is essentially "nothing" to feed back into its gate. The upper transistor has nearly constant voltage at its gate and source. Thus, the only nodes with significant voltage on them are the input and output, and these are separated by the central connection of nearly constant voltage and by the physical distance of two transistors. Thus in practice there is little feedback from the output to the input. Metal shielding is both effective and easy to provide between the two transistors for even greater isolation when required. This would be difficult in one-transistor amplifier circuits, which at high frequencies would require neutralization.<b style="mso-bidi-font-weight: normal"><o:p></o:p></B></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Biasing<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">As shown, the cascode circuit using two "stacked" FET's imposes some restrictions on the two FET's―namely, the upper FET must be biased so its source voltage is high enough (the lower FET drain voltage may swing too low, causing it to leave saturation). Insurance of this condition for FET's requires careful selection for the pair, or special biasing of the upper FET gate, increasing cost.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode circuit can also be built using bipolar transistors, or MOSFETs, or even one FET (or MOSFET) and one BJT. In the latter case, the BJT must be the upper transistor; otherwise, the (lower) BJT will always saturate (unless extraordinary steps are taken to bias it).<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Advantages<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode arrangement offers high gain, high slew rate, high stability, and high input impedance. The parts count is very low for a two-transistor circuit.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p> </o:p></SPAN><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Disadvantages<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The cascode circuit requires two transistors and requires a relatively high supply voltage. For the two-FET cascode, both transistors must be biased with ample VDS in operation, imposing a lower limit on the supply voltage.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Dual-gate version<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">A dual-gate MOSFET often functions as a "one-transistor" cascode. Common in the front ends of sensitive VHF receivers, a dual-gate MOSFET is operated as a common-source amplifier with the primary gate (usually designated "gate 1" by MOSFET manufacturers) connected to the input and the 2nd gate grounded (bypassed). Internally, there is one channel covered by the two adjacent gates; therefore, the resulting circuit is electrically a cascode composed of two FETs, the common lower-drain-to-upper-source connection merely being that portion of the single channel that lies physically adjacent to the border between the two gates.</SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><img src="http://upload.wikimedia.org/wikipedia/commons/f/f3/NMOS-cascode.png"></SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'">Other applications<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">With the rise of integrated circuits, transistors have become cheap in terms of silicon die area. In MOSFET technology especially, cascoding can be used in current mirrors to increase the output impedance of the output current source.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">A modified version of the cascode can also be used as a modulator, particularly for amplitude modulation. The upper device supplies the audio signal, and the lower is the RF amplifier device.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'">Two-port parameters<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">The cascode configuration can be represented as a simple voltage amplifier (or more accurately as a g-parameter two-port network) by using its input impedance, output impedance, and voltage gain. These parameters are related to the corresponding g-parameters below.[2] Other useful properties not considered here are circuit bandwidth and dynamic range.</SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-fareast-language: EN-US; mso-bidi-language: AR-SA"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN">BJT Cascode: low-frequency small-signal parameters<o:p></o:p></SPAN></B></P><p style="BACKGROUND: #f8fcff"><span lang=EN style="mso-ansi-language: EN">The idealized <a title=Small-signal href="http://en.wikipedia.org/wiki/Small-signal"><span style="COLOR: windowtext">small-signal</SPAN></A> equivalent circuit can be constructed for the circuit in figure 2 by replacing the current sources with open-circuits and the capacitors with short circuits, assuming they are large enough to act as short-circuits at the frequencies of interest. The BJTs can be represented in the small-signal circuit by the <a title="Hybrid-pi model" href="http://en.wikipedia.org/wiki/Hybrid-pi_model"><span style="COLOR: windowtext">hybrid-pi model</SPAN></A>.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><img style="WIDTH: 368px; HEIGHT: 204px" height=383 src="http://upload.wikimedia.org/wikipedia/commons/d/da/BJT_Cascode.png" width=415></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify">Figure 2: BJT Cascode using ideal current sources for DC bias and large coupling capacitors to ground and to the AC signal source; capacitors are short circuits for AC</P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'">MOSFET Cascode: low-frequency small-signal parameters<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">Similarly the small-signal parameters can be derived for the MOSFET version, also replacing the MOSFET by its hybrid-pi model equivalent. This derivation can be simplified by noting that the MOSFET gate current is zero, so the small-signal model for the BJT becomes that of the MOSFET in the limit of zero base current:<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=tex style="DISPLAY: inline-block; FONT-SIZE: 0px; BORDER-LEFT-COLOR: black; BACKGROUND-IMAGE: none; BORDER-BOTTOM-COLOR: black; VERTICAL-ALIGN: middle; BORDER-TOP-COLOR: black; BORDER-RIGHT-COLOR: black"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/5/8/5/5857e5b99e1ce3f04057376dc3e8967d.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN> <span class=tex style="DISPLAY: inline-block; FONT-SIZE: 0px; BORDER-LEFT-COLOR: black; BACKGROUND-IMAGE: none; BORDER-BOTTOM-COLOR: black; VERTICAL-ALIGN: middle; BORDER-TOP-COLOR: black; BORDER-RIGHT-COLOR: black"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/d/f/0/df09aea884019cb88a2957126faba316.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN><span class=tex style="DISPLAY: inline-block; FONT-SIZE: 0px; BORDER-LEFT-COLOR: black; BACKGROUND-IMAGE: none; BORDER-BOTTOM-COLOR: black; VERTICAL-ALIGN: middle; BORDER-TOP-COLOR: black; BORDER-RIGHT-COLOR: black"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/d/7/1/d711b8bca72cc4b4fa77358cf4b3d52e.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN>, </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face=Calibri>where <i>V<sub>T</SUB></I> is the </FONT><a title="Boltzmann constant" href="http://en.wikipedia.org/wiki/Boltzmann_constant#Role_in_semiconductor_physics:_the_thermal_voltage"><span style="COLOR: windowtext"><font face=Calibri>thermal voltage</FONT></SPAN></A><font face=Calibri>.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"></SPAN><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><img style="WIDTH: 390px; HEIGHT: 248px" height=290 src="http://upload.wikimedia.org/wikipedia/en/2/27/MOSFET_Cascode.png" width=429></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify">Figure 3: MOSFET Cascode using ideal voltage sources for DC gate bias and a DC current source as active load</P>The combination of factors <i>g<sub>m</SUB>r<sub>O</SUB></I> occurs often in the above formulas, inviting further examination. For the bipolar transistor this product is (see <a title="Hybrid-pi model" href="http://en.wikipedia.org/wiki/Hybrid-pi_model"><font color=#002bb8>hybrid-pi model</FONT></A>):<br />
<dl><dd>
<dl><dd><span class=tex style="DISPLAY: inline-block; FONT-SIZE: 0px; BORDER-LEFT-COLOR: black; BACKGROUND-IMAGE: none; BORDER-BOTTOM-COLOR: black; VERTICAL-ALIGN: middle; BORDER-TOP-COLOR: black; BORDER-RIGHT-COLOR: black"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/1/e/e/1ee88919f9b58143332a17a102ae79a0.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN>. </DD></DL></DD></DL>In a typical discrete bipolar device the Early voltage <i>V<sub>A</SUB></I> ≈ 100 V and the thermal voltage near room temperature is <i>V<sub>T</SUB></I> ≈ 25 mV, making <i>g<sub>m</SUB>r<sub>O</SUB></I> ≈ 4000, a rather large number. From the article on <a title="Hybrid-pi model" href="http://en.wikipedia.org/wiki/Hybrid-pi_model"><font color=#002bb8>hybrid-pi model</FONT></A>, we find for the MOSFET in the active mode:<br />
<dl><dd>
<dl><dd><span class=tex style="DISPLAY: inline-block; FONT-SIZE: 0px; BORDER-LEFT-COLOR: black; BACKGROUND-IMAGE: none; BORDER-BOTTOM-COLOR: black; VERTICAL-ALIGN: middle; BORDER-TOP-COLOR: black; BORDER-RIGHT-COLOR: black"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/4/8/d/48d8f6891ecb0ae91e4579c8219435fc.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN></DD></DL></DD></DL>At the <a title="65 nanometer" href="http://en.wikipedia.org/wiki/65_nanometer"><font color=#002bb8>65 nanometer</FONT></A> technology node, <i>I<sub>D</SUB></I> ≈ 1.2 mA/μ of width, supply voltage is <i>V<sub>DD</SUB></I> = 1.1 V; <i>V<sub>th</SUB></I> ≈ 165 mV, and <i>V<sub>ov</SUB> = V<sub>GS</SUB>-V<sub>th</SUB> ≈ 5%V<sub>DD</SUB></I> ≈ 55 mV. Taking a typical length as twice the minimum, <i>L</I> = 2 <i>L<sub>min</SUB></I> = 0.130 μm and a typical value of λ ≈ 1/(4 V/μm <i>L</I>), we find 1/λ ≈ 2 V, and <i>g<sub>m</SUB>r<sub>O</SUB></I> ≈ 110, still a large value.<sup class=reference id=cite_ref-Baker_4-0><a href="http://en.wikipedia.org/wiki/Cascode#cite_note-Baker-4"><font color=#5a3696><span>[</SPAN>5<span>]</SPAN></FONT></A></SUP> <sup class=reference id=cite_ref-Sansen_5-0><a href="http://en.wikipedia.org/wiki/Cascode#cite_note-Sansen-5"><font color=#5a3696><span>[</SPAN>6<span>]</SPAN></FONT></A></SUP> The point is that because <i>g<sub>m</SUB>r<sub>O</SUB></I> is large almost regardless of the technology, the tabulated gain and the output resistance for both the MOSFET and the bipolar cascode are very large. That fact has implications in the discussion that follows.<br />
<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"> </P></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=mw-headline><b style="mso-bidi-font-weight: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'">Low frequency design</SPAN></SPAN></U></B></SPAN><b style="mso-bidi-font-weight: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><o:p></o:p></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN">The g-parameters found in the above formulas can be used to construct a small-signal voltage amplifier with the same gain, input and output resistance as the original cascode (an <a title="Equivalent circuit" href="http://en.wikipedia.org/wiki/Equivalent_circuit"><span style="COLOR: windowtext">equivalent circuit</SPAN></A>). This circuit applies only at frequencies low enough that the transistor parasitic capacitances do not matter. The figure shows the original cascode (top panel) and the equivalent voltage amplifier or g-equivalent two-port (bottom panel). The equivalent circuit allows easier calculations of the behavior of the circuit for different drivers and loads. In the figure a <a title="Thévenin equivalent" href="http://en.wikipedia.org/wiki/Th%C3%A9venin_equivalent"><span style="COLOR: windowtext">Thévenin equivalent</SPAN></A> voltage source with Thévenin resistance <i>R<sub>S</SUB></I> drives the amplifier, and at the output a simple load resistor <i>R<sub>L</SUB></I> is attached. Using the equivalent circuit, the input voltage to the amplifier is (see article on <a title="Voltage divider" href="http://en.wikipedia.org/wiki/Voltage_divider"><span style="COLOR: windowtext">voltage division</SPAN></A>):</SPAN><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><font face="Times New Roman"> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"> </P></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/8/3/c/83c3f10055ca5e4bf92f028d970eb322.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN>, </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">which shows the importance of using a driver with resistance RS << Rin to avoid attenuation of the signal entering the amplifier. From the above amplifier characteristics, we see that Rin is infinite for the MOSFET cascode, so no attenuation of input signal occurs in that case. The BJT cascode is more restrictive because Rin = rπ2.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">In a similar fashion, the output signal from the equivalent circuit is<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/0/1/d/01db6e184426cd402db723072c0d7464.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN>, </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify">In low frequency circuits, a high voltage gain typically is desired, hence the importance of using a load with resistance RL >> Rout to avoid attenuation of the signal reaching the load. The formulas for Rout can be used either to design an amplifier with a sufficiently small output resistance compared to the load or, if that cannot be done, to decide upon a modified circuit, for example, to add a voltage follower that matches the load better.</P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify">The earlier estimate showed that the cascode output resistance is very large. The implication is that many load resistances will not satisfy the condition RL >> Rout. (An important exception is driving a MOSFET as load, which has infinite low frequency input impedance.) However, the failure to satisfy the condition RL >> Rout is not catastrophic because the cascode gain also is very large. If the designer is willing, the large gain can be sacrificed to allow a low load resistance; for RL << Rout the gain simplifies as follows:</P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/math/c/b/2/cb20617abfd4bb507b94bf1a1b2a0726.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN>. </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Because the amplifiers are wide bandwidth, the same approach can determine the bandwidth of the circuit when a load capacitor is attached (with or without a load resistor). The assumption needed is that the load capacitance is large enough that it controls the frequency dependence, and bandwidth is not controlled by the neglected parasitic capacitances of the transistors themselves.</SPAN></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'"><img style="WIDTH: 396px; HEIGHT: 271px" height=537 src="http://upload.wikimedia.org/wikipedia/commons/4/4f/BJT_Cascode_Small-signal_Circuit.png" width=586></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN; mso-bidi-font-family: 'Times New Roman'">High frequency design<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">At high frequencies, the parasitic capacitances of the transistors (gate-to-drain, gate-to-source, drain-to body, and bipolar equivalents) must be included in the hybrid pi models to obtain an accurate frequency response. The design goals also differ from the emphasis on overall high gain as described above for low-frequency design. In high frequency circuits, impedance matching at the input and output of the amplifier is typically desired in order to eliminate signal reflections and maximize power gain. In the cascode, the isolation between the input and output ports still is characterized by a small reverse transmission term g12, making it easier to design matching networks because the amplifier is approximately unilateral.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"></o:p></SPAN> </P></SPAN></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p>Lenny Z. Perez M.</o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p>EES</o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><a href="http://en.wikipedia.org/wiki/Cascode#Two-port_parameters">http://en.wikipedia.org/wiki/Cascode#Two-port_parameters</A></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"> </P></P> <br />
<hr />Explore the seven wonders of the world <a href='http://search.msn.com/results.aspx?q=7+wonders+world&mkt=en-US&form=QBRE' target='_new'>Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-64580273960850672662010-02-15T18:49:00.002-04:302010-02-16T10:43:51.085-04:30Basic Circuits<span lang=EN-US style="COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss"><font size=5><font color=#bfbfbf>Basic Circuits - <strong><span style="FONT-FAMILY: 'Tempus Sans ITC'">Bypass Capacitors</SPAN></STRONG></FONT></FONT></SPAN><br />
<p style="TEXT-ALIGN: justify"><span lang=EN-US><font size=3><font face="Times New Roman"><font color=#bfbfbf>This time in Basic Circuits, I would like to discuss bypass capacitors. This article will explain the function of a bypass capacitor, when its appropriate to use them, and what values you should consider using. </FONT></FONT></FONT></SPAN></P><h2 style="TEXT-ALIGN: justify"><u><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>The Function</FONT></SPAN></U></H2><p style="TEXT-ALIGN: justify"><span lang=EN-US><font size=3><font face="Times New Roman"><font color=#bfbfbf>The definition of a bypass capacitor can be found in the dictionary of electronics.</FONT></FONT></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font size=3><font color=#bfbfbf><strong><span lang=EN-US><font face=Calibri>Bypass capacitor</FONT></SPAN></STRONG><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'">: A capacitor employed to conduct an alternating current around a component or group of components. Often the AC is removed from an AC/DC mixture, the DC being free to pass through the bypassed component. </SPAN></FONT></FONT></P><p style="TEXT-ALIGN: justify"><span lang=EN-US><font size=3><font face="Times New Roman"><font color=#bfbfbf>In practice, most digital circuits such as microcontroller circuits are designed as direct current (DC) circuits. It turns out that variations in the voltages of these circuits can cause problems. If the voltages swing too much, the circuit may operate incorrectly. For most practical purposes, a voltage that fluctuates is considered an AC component. The function of the bypass capacitor is to dampen the AC, or the noise. Another term used for the bypass capacitor is a filter cap. </FONT></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><font size=3><font face="Times New Roman" color=#bfbfbf><span><strong><img style="WIDTH: 375px; HEIGHT: 286px" height=342 hspace=10 src="http://www.seattlerobotics.org/Encoder/jun97/basics1.gif" width=455 align=left vspace=10 border=1></STRONG></SPAN></FONT></FONT></P><p style="TEXT-ALIGN: justify"><font size=3><font face="Times New Roman" color=#bfbfbf><span><strong></STRONG></SPAN></FONT></FONT> <span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In the chart, you can see the what happens to a noisy voltage when a by-pass capacitor is installed. Notice that the differences in voltage are pretty small (between 5 and 10 millivolts). This graph represents a small range of 4.95 volts to 5.05 volts. Random electrical noise causes the voltage to fluctuate, as you can see in graph. This is often called 'noise' or 'ripple'. The blue line, represents the voltage of a circuit that doesn't have a bypass. The pink line is a circuit that has a bypass. Ripple voltages are present in almost any DC circuit. You can see even with the bypass, the voltage does fluctuate, even though it is to a smaller degree. The key function of the bypass capacitor is to reduce the amount of ripple in a circuit. Too much ripple is bad, and can lead to failure of the circuit. Ripple is often random, but sometimes other components in the circuit can cause this noise to occur. For example, a relay or motor switching can often times cause a sudden fluctuation in the voltage. Much like disturbing the water level in a pond. The more current the other component uses, the bigger the ripple effect. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>A fair question to ask is why does this small fluctuation matter? Gee, isn't the voltage close enough? The answer depends on the type of circuit you are designing. If you are just running a motor connected to a battery, or perhaps an LED, then chances are the ripple doesn't matter much to you. However, if you are using digital logic gates, things get slightly more complex, and this ripple can cause problems in the circuit. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Lets consider for just a moment what the effect of the ripple voltage is. Basic electrical theory tells us that a voltage is a difference in potential. It tells us that a current will flow across this difference in potential. We know that the larger the voltage, the larger the current. We also know the direction of the voltage determines the direction of the current. </FONT></SPAN></P><table class=ecxMsoNormalTable cellPadding=0 border=0><tbody>
<tr> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 0.75pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 0.75pt; PADDING-BOTTOM: 0.75pt; BORDER-LEFT: #ece9d8; PADDING-TOP: 0.75pt; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"> <p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Consider the graphs. The top graph shows a pair of ripple voltages that I enlarged to make them easier to see. Just like the previous graph, the blue line represents the circuit without the bypass cap, and the other line is with the bypass cap. By looking along the bottom axis of the graph, you can see that starting at point 2 that the voltage is increasing. By looking in the Ripple Current chart, point 2 shows that the current is a relatively large magnitude in one direction. In contrast, point 5 shows the voltage and current going the other direction. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Notice the difference between the values with and without the bypass cap. By dampening the ripple voltage, the bypass cap also dampens the ripple current. I would like to point out that the Ripple Voltage chart and the Ripple Current charts clearly show an alternating current. You can see how the voltage swings, and how the current changes directions. Even though this is is a DC circuit, the ripple is causing an AC component. The bypass capacitor is helping to reduce this AC component. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The ripple current acts like an eddie or backflow in the circuit. As the fluctuating voltages and currents propogate through the circuit, differences in voltages and currents can occur that cause the circuit to fail. For example, assume that a AND gate is holding its state because the semiconductors that make up the gate are in a stable state. Transistors work by currents flowing one direction through the gate. If the current stops flowing, the transistor shuts down. If a ripple current comes through where the current momentarily flows the wrong direction, the gate will shutdown, and you will see a change it its output. This can cause a cascading failure, because one gate may be connected to many other gates. </FONT></SPAN></P></TD></TR>
</TBODY></TABLE><p style="TEXT-ALIGN: justify"><span lang=EN-US><font size=3><font face="Times New Roman"><font color=#bfbfbf><strong><img height=240 src="http://www.seattlerobotics.org/Encoder/jun97/basics2.gif" width=352></STRONG> </FONT></P><strong><font color=#bfbfbf><img height=241 src="http://www.seattlerobotics.org/Encoder/jun97/basics3.gif" width=353></FONT></STRONG><br />
</FONT></FONT></SPAN><span lang=EN-US><font size=3><font face="Times New Roman"><font color=#bfbfbf>To summarize, the bypass capacitor is used to dampen the AC component of your DC circuits. By installing bypass capacitors, your DC circuit will not be as susceptable to ripple currents and voltages.</FONT></FONT></FONT></SPAN><br />
<p style="TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3><font color=#bfbfbf>Using Bypass capacitors</FONT></FONT></SPAN></U></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Many schematics that you find published in magazines and books leave the bypass capacitors out. They assume you know to put them in. Other times you will find a little row of capacitors (caps) stuck off in the corner of the schematic with no apparent function. These are usually the bypass (or filter) caps. If you pickup almost any digital circuit, you will find a bypass capacitor on it. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img height=163 src="http://www.seattlerobotics.org/Encoder/jun97/basics5.gif" width=104></FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The most simple incarnation of the bypass capacitor is a cap connected directly to the power source and to ground, as shown in the diagram to the left. This simple connection will allow the AC component of VCC to pass through to ground. The cap acts like a reserve of current. The charged capacitor helps to fill in any 'dips' in the voltage VCC by releasing its charge when the voltage drops. The size of the capacitor determines how big of a 'dip' it can fill. The larger the capacitor, the larger the 'dip' it can handle. A common size to use is a .1uF capacitor. You will also see .01uF as a common value. The precise value of a bypass cap isn't very important. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>So, how many bypass capacitors do you really need? A good rule of thumb I like to use is each IC on my board gets its own bypass capacitor. In fact, I try to place the bypass cap so it is directly connected to the Vcc and Gnd pins. This is probably overkill, but it has always served me well in the past, so I will recommend it to you. It turns out you can even by DIP sockets that have the bypass caps built in. I suppose once you reach more than a few capacitors per square inch, you might be able to let up a bit!</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Another great place for a bypass cap is on power connectors. Anytime you have a power line heading off to another board or long wire, I would recommend putting in a bypass cap. Any long length of wire is going to act like a little antenna. It will pick up electrical noise from any magnetic field. I always put a bypass cap on both ends of such lengths of wire. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The type of capacitor you use can be important. I would recommend you use a monolithic ceramic capacitor. They are small, cheap, and readily available. I usually use a .1uF 50Volt +-20% with .1" or .2" spacing. Again, .01uF is also acceptable. I would avoid larger voltage capacitors as they are physically too large. Electrolytic capacitors are not well suited to the role of bypass capacitors as they typically have larger capacitance values and don't respond as well to high frequency changes.</FONT></SPAN></P><table class=ecxMsoNormalTable cellPadding=0 border=0><tbody>
<tr> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 0.75pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 0.75pt; PADDING-BOTTOM: 0.75pt; BORDER-LEFT: #ece9d8; PADDING-TOP: 0.75pt; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"> <p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The frequency of the ripple can have a role in choosing the capacitor value. Rule of thumb is the higher the frequency, the smaller the bypass capacitor you need. If you have very high frequency components in your circuit, you might consider a pair of capacitors in parallel. One with a large value, one with a small value. If you have very complex ripple, you may need to add several bypass capacitors. Each cap is targeting a slightly different frequency. You may even need to add a larger electrolytic cap in case the amplitude of the lower frequencys is too great. For example, the circuit on the right is using three different capacitor values in parallel. Each will respond better to different frequencies. The 4.7uF cap (C4) is used to catch larger voltage dips which are at relatively low frequencies. The cap C2 should be able to handle the midrange frequencies, and C3 will handle the higher frequencies. The frequency response of the capacitors is determined by their internal resistance and inductance. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img height=135 src="http://www.seattlerobotics.org/Encoder/jun97/basics6.gif" width=304></FONT></SPAN></B></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 0.75pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 0.75pt; PADDING-BOTTOM: 0.75pt; BORDER-LEFT: #ece9d8; PADDING-TOP: 0.75pt; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><font color=#bfbfbf></FONT></TD></TR>
</TBODY></TABLE><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><b><u><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>Summary</FONT></SPAN></U></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Bypass capacitors help filter the electrical noise out of your circuits. They do this by removing the alternating currents caused by ripple voltage. Most digital circuits have at least a couple of bypass capacitors. A good rule of thumb is to add one bypass capacitor for every integrated circuit on your board. A good default value for a bypass cap is 0.1uF. Higher frequencies require lower valued capacitors. </FONT></SPAN></P><p style="TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><span style="TEXT-DECORATION: none"><font color=#bfbfbf size=3> </FONT></SPAN></SPAN></U></B></P><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font style="FONT-SIZE: 16pt" size=4> <p style="TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-SIZE: 14pt; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>Coupling (electronics)</FONT></SPAN></U></B></P></FONT></SPAN></U></B> <p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In electronics and telecommunication, coupling is the desirable or undesirable transfer of energy from one medium, such as a metallic wire or an optical fiber, to another medium, including fortuitous transfer.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Coupling is also the transfer of power from one circuit segment to another, e.g., an alternating voltage may be transferred to a segment at a different direct voltage by use of a capacitor or transformer; power may be efficiently transferred to a segment with different impedance by use of a transformer.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Types of coupling</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*Electromagnetic coupling:inductive coupling, most commonly transformer coupling, also called magnetic coupling </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*capacitive coupling, capacitor coupling, also called electrostatic coupling </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*RF coupling </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*electromagnetic interference (EMI), sometimes called radio frequency interference (RFI), is unwanted coupling. Electromagnetic compatibility (EMC) requires techniques to avoid such unwanted coupling, such as electromagnetic shielding. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*wireless energy transfer </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>*Other kinds of energy coupling:acoustic coupling with an acoustic coupler, evanescent wave coupling</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><u><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'">Capacitive coupling</SPAN></U></B><b><u><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></U></B></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In </FONT><a title=Electronics href="http://en.wikipedia.org/wiki/Electronics"><span style="COLOR: windowtext"><font color=#bfbfbf>electronics</FONT></SPAN></A><font color=#bfbfbf>, <span>capacitive <a title="Coupling (electronics)" href="http://en.wikipedia.org/wiki/Coupling_(electronics)"><span style="COLOR: windowtext">coupling</SPAN></A></SPAN> is the transfer of energy within an </FONT><a title="Electrical network" href="http://en.wikipedia.org/wiki/Electrical_network"><span style="COLOR: windowtext"><font color=#bfbfbf>electrical network</FONT></SPAN></A><font color=#bfbfbf> by means of the </FONT><a title=Capacitance href="http://en.wikipedia.org/wiki/Capacitance"><span style="COLOR: windowtext"><font color=#bfbfbf>capacitance</FONT></SPAN></A><font color=#bfbfbf> between circuit nodes. This coupling can have an intentional or accidental effect. Capacitive coupling is typically achieved by placing a </FONT><a title=Capacitor href="http://en.wikipedia.org/wiki/Capacitor"><span style="COLOR: windowtext"><font color=#bfbfbf>capacitor</FONT></SPAN></A><font color=#bfbfbf> in </FONT><a title="Series and parallel circuits" href="http://en.wikipedia.org/wiki/Series_and_parallel_circuits#Series_Circuits"><span style="COLOR: windowtext"><font color=#bfbfbf>series</FONT></SPAN></A><font color=#bfbfbf> with the signal to be coupled.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Use in analog circuits</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In analog circuits, a coupling capacitor is used to connect two circuits such that only the AC signal from the first circuit can pass through to the next while DC is blocked. This technique helps to isolate the DC bias settings of the two coupled circuits. Capacitive coupling is also known as AC coupling and the capacitor used for the purpose is known as a coupling or DC blocking capacitor. Capacitive coupling has the disadvantage of degrading the low frequency performance of a system containing capacitively coupled units. Each coupling capacitor along with the input electrical impedance of the next stage forms a high-pass filter and each successive filter results in a cumulative filter with a -3dB frequency that may be higher than each individual filter. So for adequate low frequency response the capacitors used must have high capacitance ratings. They should be high enough that the reactance of each is at least ten times the input impedance of each stage, at the lowest frequency of interest. This disadvantage of capacitively coupling DC biased, transistor amplifier circuits is largely minimized in directly coupled designs.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Use in digital circuits</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>AC coupling is also widely used in digital circuits to transmit digital signal with a zero DC component, known as DC-balanced signals. DC-balanced waveforms are useful in communications systems, since they can be used over AC-coupled electrical connections to avoid voltage imbalance problems and charge accumulation between connected systems or components.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>For this reason, most modern line codes are designed to produce DC-balanced waveforms. The most common classes of DC-balanced line codes are constant-weight codes and paired-disparity codes.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Gimmick</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>A "gimmick" is a very simple kind of capacitive coupling: a piece of wire that is placed in proximity to another one, providing a capacitive coupling between two nodes of a few picofarads in value. Sometimes the wires are twisted together for physical stability.[1][2]</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Parasitic capacitive coupling</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Capacitive coupling is often unintended, such as the capacitance between two wires or PCB traces that are next to each other. Often one signal can capacitively couple with another and cause what appears to be noise. To reduce coupling, wires or traces are often separated as much as possible, or ground lines or ground planes are run in between signals that might affect each other. Breadboards are particularly prone to these issues due to the long pieces of metal that line every row creating a several-picofarad capacitor between lines. To prototype high-frequency (10s of MHz) or high-gain analog circuits, often the circuits are built over a ground plane so that the signals couple to ground more than to each other. If a high-gain amplifier's output capacitively couples to its input it often becomes an oscillator.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3> </FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3> </FONT></SPAN></B><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3>BJT NPN</FONT></SPAN></I></B></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3></FONT></SPAN></I></B> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3><img style="WIDTH: 338px; HEIGHT: 290px" height=290 src="http://macao.communications.museum/images/exhibits/small/2_16_3_1_eng.png" width=369></FONT></SPAN></I></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3><font color=#bfbfbf>Coupling Capacitor (C<sub>1</SUB>)<sub> </SUB></FONT></FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The cutoff frequency due to C<sub>1</SUB> can be calculated by </FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">f<sub>L1</SUB>= 1/(2∏(R1+Ri)*C1)</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Where</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>R<sub>i</SUB>=R2</FONT></SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span style="TOP: 5pt"><font color=#bfbfbf> </FONT></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>R3</FONT></SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span style="TOP: 5pt"><font color=#bfbfbf> </FONT></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>ßre</FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>Coupling Capacitor (C<sub>2</SUB>)<sub> </SUB></FONT></SPAN></B></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The cutoff frequency due to C<sub>2</SUB> can be calculated with</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f<sub>L2</SUB>= 1/(2∏(Ro+R<sub>6</SUB>)*C2)</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Where</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>R<sub>o</SUB>=R2</FONT></SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span style="TOP: 5pt"><font color=#bfbfbf> </FONT></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>r<sub>0</SUB></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3><img style="WIDTH: 286px; HEIGHT: 282px" height=266 src="http://macao.communications.museum/images/exhibits/small/2_16_6_1_eng.png" width=278></FONT></SPAN></I></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3></FONT></SPAN></I></B> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3>J</FONT></SPAN></I></B><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3>FET</FONT></SPAN></I></B></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3><font color=#bfbfbf>Coupling Capacitor (C<sub>1</SUB>)<sub> </SUB></FONT></FONT></SPAN></I></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The cutoff frequency due to C<sub>1</SUB> can be calculated with</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>F<sub>L1</SUB>= 1/(2∏(R1+R<sub>i</SUB>)*C1)</FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Where</FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Ri=Rg</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font size=3><font color=#bfbfbf>Coupling Capacitor (C<sub>2</SUB>)<sub> </SUB></FONT></FONT></SPAN></I></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The cutoff frequency due to C<sub>1</SUB> can be calculated with</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>F<sub>L1</SUB>= 1/(2∏(Ro+R<sub>5</SUB>)*C2)</FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Where</FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">R<sub>0</SUB>=</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> R3</SPAN></FONT><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span style="TOP: 5pt"><font color=#bfbfbf> </FONT></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>r<sub>d</SUB></FONT></SPAN></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><sub><font color=#bfbfbf></FONT></SUB></SPAN> </P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span> <p class=ecxMsoNormal style="tab-stops: list 35.45pt"><b><u><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"><a title="BIPOLAR JUNCTION TRANSISTORS" href="http://www.allaboutcircuits.com/vol_3/chpt_4/index.html"><font color=#bfbfbf><span lang=EN-US>BIPOLAR JUNTION</SPAN><span lang=EN-US style="COLOR: windowtext"> TRANSISTORS</SPAN></FONT></A></SPAN></U></B><b><u><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></U></B></P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'">The cascode amplifier</SPAN></B><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></B></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">While the C-B (common-base) amplifier is known <span class=ecxhilite51><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN> wider bandwidth than the C-E (common-emitter) configuration, the low input impedance (10s of </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Ω</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">) of C-B is a limitation <span class=ecxhilite51><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN> many applications. The solution is to precede the C-B stage by a low gain C-E stage which has moderately high input impedance (k</SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Ω</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">s). See Figure </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html#03500.png"><span lang=EN-US style="COLOR: windowtext">below</SPAN></A></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">. The stages are in a <i>cascode</I> configuration, stacked in series, as opposed to cascaded <span class=ecxhilite51><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN> a standard amplifier chain. See "Capacitor coupled three stage common-emitter amplifier" </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html#03142L"><span lang=EN-US style="COLOR: windowtext">Capacitor coupled</SPAN></A></SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> <span class=ecxhilite51><span lang=EN-US><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN></SPAN><span lang=EN-US> a cascade example. The cascode amplifier configuration has both wide bandwidth <span class=ecxhilite7>and</SPAN> a moderately high input impedance.</SPAN></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf></FONT></SPAN></SPAN> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf><img style="WIDTH: 457px; HEIGHT: 193px" height=205 src="http://sub.allaboutcircuits.com/images/03500.png" width=501> </FONT></SPAN></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf></FONT></SPAN></SPAN> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><em><font color=#bfbfbf>The cascode amplifier is combined common-emitter <span class=ecxhilite7>and</SPAN> common-base. This is an AC circuit equivalent with batteries <span class=ecxhilite7>and</SPAN> capacitors replaced by short circuits.</FONT></EM></SPAN></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><em><font color=#bfbfbf></FONT></EM></SPAN></SPAN> </P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US> <p class=ecxMsoNormal style="BACKGROUND: white; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">The key to understanding the wide bandwidth of the cascode configuration is the Miller effect Miller effect. It is the multiplication of the bandwidth robbing collector-base capacitance by beta. This C-B capacitance is smaller than the E-B capacitance. Thus, one would think that the C-B capacitance would have little effect. However, in the C-E configuration, the collector output signal is out of phase with the input at the base. The collector signal capacitively coupled back opposes the base signal. Moreover, the collector feedback is beta times larger than the base signal. Thus, the small C-B capacitance appears beta times larger than its actual value. This capacitive gain reducing feedback increases with frequency, reducing the high frequency response of a C-E amplifier. <br />
A common-base configuration is not subject to the Miller effect because the grounded base shields the collector signal from being fed back to the emitter input. Thus, a C-B amplifier has better high frequency response. To have a moderately high input impedance, the C-E stage is still desirable. The key is to reduce the gain (to about 1) of the C-E stage to reduce the Miller effect C-B feedback to 1•CCB. The total C-B feedback is the Miller capacitance 1•CCB plus the actual capacitance CCB for a total of 2•CCB. This is a considerable reduction from β•CCB. <br />
The way to reduce the common-emitter gain is to reduce the load resistance. The gain of a C-E amplifier is approximately RC/RE. The internal emitter resistance REE at 1mA emitter current is 26Ω. For details on the 26Ω, see "Derivation of REE", see REE. The collector load RC is the resistance of the emitter of the C-B stage loading the C-E stage, 26Ω again. CE gain amplifier gain is approximately RC/RE=26/26=1. We now have a moderately high input impedance C-E stage without suffering the Miller effect, but no dB voltage gain. The C-B stage provides a high voltage gain. Thus, the cascode has moderately high input impedance of the CE, good gain, and good bandwidth of the C-B. <br />
</SPAN></FONT></P><p class=ecxMsoNormal style="BACKGROUND: white; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">. </SPAN></FONT></P><font color=#bfbfbf><img src="http://sub.allaboutcircuits.com/images/03502.png"> </FONT><br />
<i><font color=#bfbfbf>SPICE: Cascode <span class=ecxhilite7>and</SPAN> common-emitter <span class=ecxhilite5>for</SPAN> comparison.</FONT></I><br />
<p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">The SPICE version of both a cascode amplifier, <span class=ecxhilite7>and</SPAN> <span class=ecxhilite51><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN> comparison, a common-emitter amplifier is shown in Figure </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html#03502.png"><span lang=EN-US style="COLOR: windowtext"><font color=#bfbfbf>above</FONT></SPAN></A></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>.</FONT> The netlist is in Table </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html#cascode.tbl"><span lang=EN-US style="COLOR: windowtext"><font color=#bfbfbf>below</FONT></SPAN></A></SPAN></FONT><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">. The AC source V3 drives both <span class=ecxhilite9>amplifiers</SPAN> via node 4. The bias resistors <span class=ecxhilite51><font style="BACKGROUND-COLOR: #ffccdd">for</FONT></SPAN> this circuit are calculated in an example problem </SPAN><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html#CASCODEB"><span lang=EN-US style="COLOR: windowtext"><font color=#bfbfbf>cascode</FONT></SPAN></A></SPAN></FONT><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">.</SPAN></FONT></P><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img src="http://sub.allaboutcircuits.com/images/23044.png"> </FONT><br />
<i><font color=#bfbfbf>SPICE waveforms. Note that Input is multiplied by 10 <span class=ecxhilite5>for</SPAN> visibility.</FONT></I><br />
<a name=cascode.tbl><i><font color=#bfbfbf>SPICE netlist <span class=ecxhilite5>for</SPAN> printing AC input <span class=ecxhilite7>and</SPAN> output voltages</FONT></I><br />
</A></SPAN></SPAN></SPAN> <p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf></FONT></SPAN></SPAN> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf>Lenny Z. Perez M</FONT></SPAN></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf>EES</FONT></SPAN></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US><font color=#bfbfbf>Referencias:</FONT></SPAN></SPAN></P><span style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span lang=EN-US> <p class=ecxMsoNormal><span lang=EN><a href="http://www.seattlerobotics.org/Encoder/jun97/basics.html"><font face=Calibri color=#bfbfbf>http://www.seattlerobotics.org/Encoder/jun97/basics.html</FONT></A></SPAN></P><p class=ecxMsoNormal><span lang=EN><a href="http://en.wikipedia.org/wiki/Coupling_(electronics)"><font face=Calibri color=#bfbfbf>http://en.wikipedia.org/wiki/Coupling_(electronics)</FONT></A></SPAN></P><p class=ecxMsoNormal><span lang=EN><a href="http://macao.communications.museum/images/exhibits/small/2_16_6_1_eng.png"><font face=Calibri color=#bfbfbf>http://macao.communications.museum/images/exhibits/small/2_16_6_1_eng.png</FONT></A></SPAN></P><p class=ecxMsoNormal><span lang=EN><a href="http://macao.communications.museum/images/exhibits/small/2_16_3_1_eng.png"><font face=Calibri color=#bfbfbf>http://macao.communications.museum/images/exhibits/small/2_16_3_1_eng.png</FONT></A></SPAN></P><p class=ecxMsoNormal><cite><span lang=EN style="FONT-FAMILY: 'Arial','sans-serif'"><a href="http://www.dartec.com/MIC4120/CHAP9a.ppt"><font color=#bfbfbf>www.dartec.com/MIC4120/CHAP9a.ppt</FONT></A></SPAN></CITE></P><p class=ecxMsoNormal><span lang=EN><a href="http://www.allaboutcircuits.com/vol_3/chpt_4/8.html"><font face=Calibri color=#bfbfbf>http://www.allaboutcircuits.com/vol_3/chpt_4/8.html</FONT></A></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify; tab-stops: list 35.45pt"></SPAN></SPAN> </P><p class=ecxMsoNormal style="tab-stops: list 35.45pt"></SPAN></SPAN> </P><hr>Explore the seven wonders of the world <a href="http://search.msn.com/results.aspx?q=7+wonders+world&mkt=en-US&form=QBRE">Learn more!</A> <br />
<hr />Explore the seven wonders of the world <a href='http://search.msn.com/results.aspx?q=7+wonders+world&mkt=en-US&form=QBRE' target='_new'>Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-78772410072046587532010-02-15T17:48:00.001-04:302010-02-16T10:42:27.260-04:30FW: General Frequency Considerations<style> .ExternalClass .ecxhmmessage P {padding:0px;} .ExternalClass body.ecxhmmessage {font-size:10pt;font-family:Verdana;} </STYLE> <span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"> <p class=ecxMsoNormal><font color=#bfbfbf><b><i><u><span lang=EN-US style="FONT-SIZE: 13pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss"><span style="COLOR: windowtext">General Frequency Considerations</SPAN></SPAN></U></I></B><i><u><span lang=EN-US style="FONT-SIZE: 13pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss"></SPAN></U></I></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The <b>frequency response</B> of an amplifier refers to the frequency range in which the amplifier will operate with negligible effects from capacitors and device internal capacitance. This range of frequencies can be called the <b>mid-range</B>.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-INDENT: -14.15pt; TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span>•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">At frequencies above and below the midrange, capacitance and any inductance will affect <span> </SPAN>the gain of the amplifier.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-INDENT: 0cm; TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span>•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">At low frequencies the coupling and bypass capacitors lower the gain.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-INDENT: 0cm; TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span>•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">At high frequencies stray capacitances associated with the active device lower the gain.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-INDENT: 0cm; TEXT-ALIGN: justify; tab-stops: list 35.45pt"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><span>•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">Also, cascading amplifiers limits the gain at high and low frequencies.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3> </FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3>Bode Plot</FONT></SPAN></U></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>A <span>Bode plot</SPAN> is a </FONT><a title="Plot (graphics)" href="http://en.wikipedia.org/wiki/Plot_(graphics)"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>graph</FONT></SPAN></A><font color=#bfbfbf> of the </FONT><a title=Logarithm href="http://en.wikipedia.org/wiki/Logarithm"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>logarithm</FONT></SPAN></A><font color=#bfbfbf> of the </FONT><a title="Transfer function" href="http://en.wikipedia.org/wiki/Transfer_function"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>transfer function</FONT></SPAN></A><font color=#bfbfbf> of a </FONT><a title="LTI system theory" href="http://en.wikipedia.org/wiki/LTI_system_theory"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>linear, time-invariant</FONT></SPAN></A><font color=#bfbfbf> system versus </FONT><a title=Frequency href="http://en.wikipedia.org/wiki/Frequency"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>frequency</FONT></SPAN></A><font color=#bfbfbf>, plotted with a log-frequency axis, to show the system's </FONT><a title="Frequency response" href="http://en.wikipedia.org/wiki/Frequency_response"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>frequency response</FONT></SPAN></A><font color=#bfbfbf>. It is usually a combination of a <span>Bode magnitude plot</SPAN> (usually expressed as </FONT><a title=Decibel href="http://en.wikipedia.org/wiki/Decibel"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>dB</FONT></SPAN></A><font color=#bfbfbf> of </FONT><a title=Gain href="http://en.wikipedia.org/wiki/Gain"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>gain</FONT></SPAN></A><font color=#bfbfbf>) and a <span>Bode phase plot</SPAN> (the </FONT><a title="Phase (waves)" href="http://en.wikipedia.org/wiki/Phase_(waves)"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>phase</FONT></SPAN></A><font color=#bfbfbf> is the </FONT><a title="Imaginary part" href="http://en.wikipedia.org/wiki/Imaginary_part"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>imaginary part</FONT></SPAN></A><font color=#bfbfbf> of the </FONT><a title="Complex logarithm" href="http://en.wikipedia.org/wiki/Complex_logarithm"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none"><font color=#bfbfbf>complex logarithm</FONT></SPAN></A><font color=#bfbfbf> of the complex transfer function).</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img style="WIDTH: 372px; HEIGHT: 327px" height=424 src="http://upload.wikimedia.org/wikipedia/commons/6/63/Bode_High-Pass.PNG" width=345></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>Figure 1(a): The Bode plot for a first-order (one-pole) highpass filter; the straight-line approximations are labeled "Bode pole"; phase varies from 90° at low frequencies (due to the contribution of the numerator, which is 90° at all frequencies) to 0° at high frequencies (where the phase contribution of the denominator is −90° and cancels the contribution of the numerator).</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN><font face=Calibri color=#bfbfbf size=3></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf><img style="WIDTH: 381px; HEIGHT: 355px" height=469 src="http://upload.wikimedia.org/wikipedia/commons/c/cc/Bode_Low-Pass.PNG" width=418></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Figure 1(b): The Bode plot for a first-order (one-pole) lowpass filter; the straight-line approximations are labeled "Bode pole"; phase is 90° lower than for Figure 1(a) because the phase contribution of the numerator is 0° at all frequencies.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>Cutoff frequency</FONT></SPAN></U></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is the minimum frequency or energy of an incident wavelength required by an electron to overcome its binding energy.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example as defined by a 3 dB corner, a frequency for which the output of the circuit is -3 dB of the nominal passband value. Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Cutoff frequency can also refer to the plasma frequency.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Electronics</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>In electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a line, amplifier, or electronic filter is the power of the passband.[1] In electronics, decibel numbers refer to power. When directly comparing two powers P1 and P2, this is simply 10 * log(P1 / P2). However, one frequently compares two different voltages V1 and V2. In this case, the decibel is defined as 20 * log(V1 / V2). (In many circuits, power is proportional to the square of the voltage; this square results in multiplying the logarithm by 20 instead of 10.) The 3dB point refers to the point at which the power is -3dB of the unattenuated signal. -3dB corresponds to 1/2 the unattenuated power; when comparing voltages, this corresponds to of the unattenuated voltage.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>A bandpass circuit has two corner frequencies; their geometric mean is called the center frequency</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'">Roll-Off of Gain in the Bode Plot</SPAN></B><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></B></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">The Bode plot not only indicates the cutoff frequencies of the various capacitors it also indicates the amount of attenuation (loss in gain) at these frequencies.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">The amount of attenuation is sometimes referred to as roll-off.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">The roll-off is described as dB loss-per-octave or dB loss-per-decade.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf>Roll-off Rate (-dB/Decade)</FONT></SPAN></B></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">-dB/decade</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> refers to the attenuation for every 10-fold change in frequency.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>For attenuations at the low-frequency end, it refers to the loss in gain from the lower cutoff frequency to a frequency that is one-tenth the cutoff value.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">In this example:</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f<sub>LS</SUB> = 9kHz gain is 0dB</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f<sub>LS</SUB>/10 = .9kHz gain is –20dB</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Thus the roll-off is 20dB/decade</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The gain decreases by –20dB/decade </FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">-dB/octave</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"> refers to the attenuation for every 2-fold change in frequency.</SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>For attenuations at the low-frequency end, it refers to the loss in gain from the lower cutoff frequency to a frequency one-half the cutoff value.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'">In this example:</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f<sub>LS</SUB> = 9kHz gain is 0dB</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f<sub>LS </SUB>/ 2 = 4.5kHz gain is –6dB</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Therefore the roll-off is 6dB/octave. </FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>This is a little difficult to see on this graph because the horizontal scale is a logarithmic scale. </FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"><font color=#bfbfbf size=3>Miller Capacitance</FONT></SPAN></U></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'">General</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"> </SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The term "Miller capacitance" is often seen when reading about guitar amplifier circuit design. It refers to the effective multiplication of the plate-to-grid capacitance in a triode tube (or transistor) by the gain of the amplifying stage. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>When a tube is amplifying a signal, it has to work against the plate-to-grid capacitance, charging and discharging it as the signal changes. Because the grid is a high impedance, and doesn't sink or source any current, this charging current must be sourced or sinked through the driving source resistance of the previous stage. This forms a lowpass filter, with a corner frequency determined by the source resistance of the previous stage and the input capacitance. </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The Miller capacitance in a triode tube is equal to the plate-to-grid capacitance multiplied by a factor equal to the stage gain plus one. Pentode and tetrode tubes don't suffer as much from the effects of Miller capacitance because of the shielding effect of the screen grid, which drastically lowers the plate-to-grid capacitance?</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'">Why is it important?</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"> </SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The majority of the input capacitance of a triode stage is made up of the combination of the grid-to-cathode capacitance, plus the Miller capacitance formed by the grid-to-plate capacitance multiplied by the stage gain plus one. The formula for determining the total input capacitance of a triode stage is as follows: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Cin = Cgk + Cgp*(A+1)</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Where: Cin = input capacitance <br />
Cgk = grid-to-cathode capacitance, composed of the internal tube capacitance plus the stray capacitance <br />
Cgp = grid-to-plate capacitance, composed of the internal tube capacitance plus the stray capacitance <br />
A = stage gain </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The typical interelement capacitances are very small, but, as can be seen from the above equation, the grid-to-plate capacitance is multiplied by the gain of the tube stage plus one, so if the gain is large, the capacitance can very easily become significant, resulting in audible rolloff in frequency response.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'">Example</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"> </SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>For example, a typical 12AX7 stage has the following capacitances and gain: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Cgk = 1.6pF + 0.7pF stray = 2.3pF <br />
Cgp = 1.7pF + 0.7pF stray = 2.4pF <br />
A = 61</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Therefore, the total input capacitance would be: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Cin = 2.3pF + (61+1)* 2.4pF = 151.1pF</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'">Effect on frequency response</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"> </SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The input capacitance of the tube, in conjunction with the source impedance of the previous stage, forms a simple, single-pole RC lowpass filter with a -6dB/octave (-20dB/decade) slope, and an upper -3dB cutoff frequency equal to: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f = 1/(2*pi*R*C)</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>The -3dB point with a 68K resistor (such as at the input stage of an amplifier) would be: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f = 1/(2*pi*68K*151.1pF) = 15.5kHz</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>This is not too bad, considering the frequencies involved in guitar amplification. However, if the source resistance were increased to 470K, the cutoff frequency would be: </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>f = 1/(2*pi*470K*151.1pF) = 2.2kHz</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>This would result in considerable rolloff in the upper part of the passband at guitar frequencies. Note that the Miller effect applies to any triode, not just small signal triodes. Pentode output tubes operated in triode mode will exhibit less high frequency response because of the higher input capacitance of the tube in triode mode. This, in combination with the normally lower level of higher order harmonics in triode mode, will cause the overall tone to be less bright than pentode mode.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'">What to do about it</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"> </SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Miller capacitance can kill the frequency response rather easily, so there are a few things to consider when designing guitar amplifiers. If you desire to minimize the effect of the Miller capacitance on frequency response, you can do the following things: </FONT></SPAN></P><ul type=disc><li class=ecxMsoNormal style="COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Reduce the output impedance of the previous stage. This can be accomplished by lowering the value of the plate load resistor, using a tube with a lower internal plate resistance, or lowering the value of any series or shunt attenuation resistors. Obviously, all of these things will affect the gain of the stages, so this must be taken into account as well. </FONT></SPAN></LI></UL><ul type=disc><li class=ecxMsoNormal style="COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Reduce the gain of the stage. The Miller capacitance is proportional to the gain of the amplifying stage, so using a lower stage gain will reduce the Miller capacitance, thereby increasing the frequency response. </FONT></SPAN></LI></UL><ul type=disc><li class=ecxMsoNormal style="COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Use a pentode or cascode stage instead of a triode stage. Both the pentode and cascode configuration suffer very little from the effects of Miller capacitance because of the AC grounding of the screen grid in the pentode, and the AC grounding of the upper tube grid in the cascode, which drastically lowers the plate-to-grid capacitance in both configurations. The cascode configuration has the added advantage of lower noise, when compared to the pentode, because it does not have the "division" noise created by the screen grid of the pentode. </FONT></SPAN></LI></UL><ul type=disc><li class=ecxMsoNormal style="COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Use the Miller capacitance to your advantage. Since the Miller capacitance forms a lowpass filter in conjunction with the output resistance of the previous stage, it can be used as a "free" lowpass filter, thus saving the cost and trouble of adding a capacitor or RC network in the amplifier when a tailored frequency response is desired. This is commonly used in a not-so-apparent manner in the typical input stage of most guitar amplifiers. The input resistor, in conjunction with the input capacitance, forms a lowpass filter that reduces the susceptibility of the tube to parasitic oscillations, such as those that can occur when the input stage is being driven by a long guitar cable. As shown in the example above, the typical cutoff point of this lowpass filter, when using a 68K resistor, is around 15.5kHz. Also, some high-gain guitar amplifiers have a distortion channel that uses large value series resistors in front of the grids of the preamp tubes. These resistors not only aid in minimizing blocking distortion, they also act as lowpass filters to reduce some of the high frequency content of the distortion signal, in order to reduce some of the "buzziness" in the tone. As shown in the above example, the typical cutoff frequency using a series 470K resistor would be 2.2kHz, which would roll off most of the upper harmonics, producing a subjectively smoother distortion tone. </FONT></SPAN></LI></UL><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font size=3><font color=#bfbfbf><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'">Multistage Frequency Effects</SPAN></U></B><u><span style="FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></U></FONT></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>Each stage will have its own frequency response, but the output of one stage will be affected by capacitances in the subsequent stage. This is especially so when determining the high frequency response. For example, the output capacitance (C<sub>o</SUB>) will be affected by the input Miller Capacitance (C<sub>Mi</SUB>) of the next stage.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Summary of frequency response of single-stages:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>CE/CS: suffers from Miller effect</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>CC/CD: "wideband" -- see Section 10.5</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>CB/CG: "wideband" -- see Section 10.6</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>(Wideband means that the stage operates to near the frequency</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Limit of the device ...<i>fT</I>)</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>How to fine the Bode plot for a general multistage amplifier?</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Can´t handle <i>n</I> poles and <i>m</I> zeroes analytically --> SPICE </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Develop analytical tool for an important special case:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>* No zeroes</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><font color=#bfbfbf><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">* Exactly one "dominant" pole </SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">(w1<<w2, w3, ... ,w<i>n</I>)<b></B></SPAN></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>(The example shows a voltage gain ... it could be<i>Iout</I>/<i>Vin</I>or<i>Vout</I>/<i>Iin</I>)</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3> </FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font size=3><font color=#bfbfbf><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'">Square Wave Testing</SPAN></U></B><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></U></FONT></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>In order to determine the frequency response of an amplifier by experimentation, you must apply a wide range of frequencies to the amplifier.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>One way to accomplish this is to apply a square wave. A square wave consists of multiple frequencies (by Fourier analysis: it consists of odd harmonics).</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><font color=#bfbfbf><b><u><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'">Square Wave Response Waveforms</SPAN></U></B><u><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'"></SPAN></U></FONT></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>If the output of the amplifier is not a perfect square wave then the amplifier is 'cutting' off certain frequency components of the square wave.</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3></FONT></SPAN> </P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>Lenny Z. Perez M</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf size=3>EES</FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'"><font face=Verdana color=#bfbfbf><u>en.wikipedia.org/wiki/Bode_plot</U></FONT></SPAN></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"></SPAN><a href="http://www.aikenamps.com/MillerCapacitance.html"><font color=#bfbfbf>http://www.aikenamps.com/MillerCapacitance.html</FONT></A></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><a href="http://www.prenhall.com/howe3/microelectronics/pdf_folder/lectures/mwf/lecture31.fm5.pdf"><font color=#bfbfbf>http://www.prenhall.com/howe3/microelectronics/pdf_folder/lectures/mwf/lecture31.fm5.pdf</FONT></A></P><p class=ecxMsoNormal style="TEXT-ALIGN: justify"><u><font color=#bfbfbf>www.dartec.com/MIC4120/CHAP9a.ppt</FONT></U></P> <br />
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It's easy! <a href='http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us' target='_new'>Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-81703390746412879942010-02-15T17:42:00.003-04:302010-02-16T10:42:55.101-04:30General Frequency Considerations<span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><b><i style="mso-bidi-font-style: normal"><u><span lang=EN-US style="FONT-SIZE: 13pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><span style="COLOR: windowtext">General Frequency Considerations</SPAN></SPAN></U></I></B><i style="mso-bidi-font-style: normal"><u><span lang=EN-US style="FONT-SIZE: 13pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN-US"><?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">The <b>frequency response</B> of an amplifier refers to the frequency range in which the amplifier will operate with negligible effects from capacitors and device internal capacitance. This range of frequencies can be called the <b>mid-range</B>.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 35.45pt; TEXT-INDENT: -14.15pt; TEXT-ALIGN: justify; mso-list: l4 level2 lfo1; tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'"><span style="mso-list: Ignore">•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">At frequencies above and below the midrange, capacitance and any inductance will affect <span style="mso-spacerun: yes"> </SPAN>the gain of the amplifier.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 21.3pt; TEXT-INDENT: 0cm; TEXT-ALIGN: justify; mso-list: l4 level2 lfo1; tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'"><span style="mso-list: Ignore">•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">At low frequencies the coupling and bypass capacitors lower the gain.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 21.3pt; TEXT-INDENT: 0cm; TEXT-ALIGN: justify; mso-list: l4 level2 lfo1; tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'"><span style="mso-list: Ignore">•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">At high frequencies stray capacitances associated with the active device lower the gain.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 21.3pt; TEXT-INDENT: 0cm; TEXT-ALIGN: justify; mso-list: l4 level2 lfo1; tab-stops: list 35.45pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'"><span style="mso-list: Ignore">•<span style="FONT: 7pt 'Times New Roman'"> </SPAN></SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Also, cascading amplifiers limits the gain at high and low frequencies.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US"><o:p><font size=3> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US"><font size=3>Bode Plot<o:p></o:p></FONT></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 7.1pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">A <span style="mso-bidi-font-weight: bold">Bode plot</SPAN> is a <a title="Plot (graphics)" href="http://en.wikipedia.org/wiki/Plot_(graphics)"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">graph</SPAN></A> of the <a title=Logarithm href="http://en.wikipedia.org/wiki/Logarithm"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">logarithm</SPAN></A> of the <a title="Transfer function" href="http://en.wikipedia.org/wiki/Transfer_function"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">transfer function</SPAN></A> of a <a title="LTI system theory" href="http://en.wikipedia.org/wiki/LTI_system_theory"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">linear, time-invariant</SPAN></A> system versus <a title=Frequency href="http://en.wikipedia.org/wiki/Frequency"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">frequency</SPAN></A>, plotted with a log-frequency axis, to show the system's <a title="Frequency response" href="http://en.wikipedia.org/wiki/Frequency_response"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">frequency response</SPAN></A>. It is usually a combination of a <span style="mso-bidi-font-weight: bold">Bode magnitude plot</SPAN> (usually expressed as <a title=Decibel href="http://en.wikipedia.org/wiki/Decibel"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">dB</SPAN></A> of <a title=Gain href="http://en.wikipedia.org/wiki/Gain"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">gain</SPAN></A>) and a <span style="mso-bidi-font-weight: bold">Bode phase plot</SPAN> (the <a title="Phase (waves)" href="http://en.wikipedia.org/wiki/Phase_(waves)"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">phase</SPAN></A> is the <a title="Imaginary part" href="http://en.wikipedia.org/wiki/Imaginary_part"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">imaginary part</SPAN></A> of the <a title="Complex logarithm" href="http://en.wikipedia.org/wiki/Complex_logarithm"><span style="COLOR: windowtext; TEXT-DECORATION: none; text-underline: none">complex logarithm</SPAN></A> of the complex transfer function).</SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 7.1pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><img style="WIDTH: 372px; HEIGHT: 327px" height=424 src="http://upload.wikimedia.org/wikipedia/commons/6/63/Bode_High-Pass.PNG" width=345></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 7.1pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font size=3>Figure 1(a): The Bode plot for a first-order (one-pole) highpass filter; the straight-line approximations are labeled "Bode pole"; phase varies from 90° at low frequencies (due to the contribution of the numerator, which is 90° at all frequencies) to 0° at high frequencies (where the phase contribution of the denominator is −90° and cancels the contribution of the numerator).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><o:p><font face=Calibri size=3></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><img style="WIDTH: 381px; HEIGHT: 355px" height=469 src="http://upload.wikimedia.org/wikipedia/commons/c/cc/Bode_Low-Pass.PNG" width=418></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">Figure 1(b): The Bode plot for a first-order (one-pole) lowpass filter; the straight-line approximations are labeled "Bode pole"; phase is 90° lower than for Figure 1(a) because the phase contribution of the numerator is 0° at all frequencies.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN">Cutoff frequency<o:p></o:p></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is the minimum frequency or energy of an incident wavelength required by an electron to overcome its binding energy.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example as defined by a 3 dB corner, a frequency for which the output of the circuit is -3 dB of the nominal passband value. Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">Cutoff frequency can also refer to the plasma frequency.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Electronics<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">In electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a line, amplifier, or electronic filter is the power of the passband.[1] In electronics, decibel numbers refer to power. When directly comparing two powers P1 and P2, this is simply 10 * log(P1 / P2). However, one frequently compares two different voltages V1 and V2. In this case, the decibel is defined as 20 * log(V1 / V2). (In many circuits, power is proportional to the square of the voltage; this square results in multiplying the logarithm by 20 instead of 10.) The 3dB point refers to the point at which the power is -3dB of the unattenuated signal. -3dB corresponds to 1/2 the unattenuated power; when comparing voltages, this corresponds to of the unattenuated voltage.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">A bandpass circuit has two corner frequencies; their geometric mean is called the center frequency<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US">Roll-Off of Gain in the Bode Plot</SPAN></B><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US"><o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">The Bode plot not only indicates the cutoff frequencies of the various capacitors it also indicates the amount of attenuation (loss in gain) at these frequencies.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">The amount of attenuation is sometimes referred to as roll-off.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">The roll-off is described as dB loss-per-octave or dB loss-per-decade.</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US">Roll-off Rate (-dB/Decade)<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">-dB/decade</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"> refers to the attenuation for every 10-fold change in frequency.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">For attenuations at the low-frequency end, it refers to the loss in gain from the lower cutoff frequency to a frequency that is one-tenth the cutoff value.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">In this example:</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">f<sub>LS</SUB> = 9kHz gain is 0dB<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">f<sub>LS</SUB>/10 = .9kHz gain is –20dB<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">Thus the roll-off is 20dB/decade<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">The gain decreases by –20dB/decade <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">-dB/octave</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"> refers to the attenuation for every 2-fold change in frequency.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">For attenuations at the low-frequency end, it refers to the loss in gain from the lower cutoff frequency to a frequency one-half the cutoff value.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">In this example:</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">f<sub>LS</SUB> = 9kHz gain is 0dB<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">f<sub>LS </SUB>/ 2 = 4.5kHz gain is –6dB<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">Therefore the roll-off is 6dB/octave. <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold">This is a little difficult to see on this graph because the horizontal scale is a logarithmic scale. <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'"><font size=3>Miller Capacitance<o:p></o:p></FONT></SPAN></U></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">General</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"> <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The term "Miller capacitance" is often seen when reading about guitar amplifier circuit design. It refers to the effective multiplication of the plate-to-grid capacitance in a triode tube (or transistor) by the gain of the amplifying stage. <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">When a tube is amplifying a signal, it has to work against the plate-to-grid capacitance, charging and discharging it as the signal changes. Because the grid is a high impedance, and doesn't sink or source any current, this charging current must be sourced or sinked through the driving source resistance of the previous stage. This forms a lowpass filter, with a corner frequency determined by the source resistance of the previous stage and the input capacitance. <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The Miller capacitance in a triode tube is equal to the plate-to-grid capacitance multiplied by a factor equal to the stage gain plus one. Pentode and tetrode tubes don't suffer as much from the effects of Miller capacitance because of the shielding effect of the screen grid, which drastically lowers the plate-to-grid capacitance?<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Why is it important?</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"> <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The majority of the input capacitance of a triode stage is made up of the combination of the grid-to-cathode capacitance, plus the Miller capacitance formed by the grid-to-plate capacitance multiplied by the stage gain plus one. The formula for determining the total input capacitance of a triode stage is as follows: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Cin = Cgk + Cgp*(A+1)<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Where: Cin = input capacitance <br />
Cgk = grid-to-cathode capacitance, composed of the internal tube capacitance plus the stray capacitance <br />
Cgp = grid-to-plate capacitance, composed of the internal tube capacitance plus the stray capacitance <br />
A = stage gain <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The typical interelement capacitances are very small, but, as can be seen from the above equation, the grid-to-plate capacitance is multiplied by the gain of the tube stage plus one, so if the gain is large, the capacitance can very easily become significant, resulting in audible rolloff in frequency response.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Example</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"> <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">For example, a typical 12AX7 stage has the following capacitances and gain: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Cgk = 1.6pF + 0.7pF stray = 2.3pF <br />
Cgp = 1.7pF + 0.7pF stray = 2.4pF <br />
A = 61<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Therefore, the total input capacitance would be: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Cin = 2.3pF + (61+1)* 2.4pF = 151.1pF<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Effect on frequency response</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"> <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The input capacitance of the tube, in conjunction with the source impedance of the previous stage, forms a simple, single-pole RC lowpass filter with a -6dB/octave (-20dB/decade) slope, and an upper -3dB cutoff frequency equal to: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">f = 1/(2*pi*R*C)<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">The -3dB point with a 68K resistor (such as at the input stage of an amplifier) would be: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">f = 1/(2*pi*68K*151.1pF) = 15.5kHz<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">This is not too bad, considering the frequencies involved in guitar amplification. However, if the source resistance were increased to 470K, the cutoff frequency would be: <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">f = 1/(2*pi*470K*151.1pF) = 2.2kHz<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">This would result in considerable rolloff in the upper part of the passband at guitar frequencies. Note that the Miller effect applies to any triode, not just small signal triodes. Pentode output tubes operated in triode mode will exhibit less high frequency response because of the higher input capacitance of the tube in triode mode. This, in combination with the normally lower level of higher order harmonics in triode mode, will cause the overall tone to be less bright than pentode mode.<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><o:p> </o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: #333300; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">What to do about it</SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"> <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; COLOR: black; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Miller capacitance can kill the frequency response rather easily, so there are a few things to consider when designing guitar amplifiers. If you desire to minimize the effect of the Miller capacitance on frequency response, you can do the following things: <o:p></o:p></SPAN></P><ul type=disc><li class=MsoNormal style="MARGIN: 0cm 0cm 10pt; COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l3 level1 lfo2; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Reduce the output impedance of the previous stage. This can be accomplished by lowering the value of the plate load resistor, using a tube with a lower internal plate resistance, or lowering the value of any series or shunt attenuation resistors. Obviously, all of these things will affect the gain of the stages, so this must be taken into account as well. <o:p></o:p></SPAN></LI></UL><ul type=disc><li class=MsoNormal style="MARGIN: 0cm 0cm 10pt; COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l0 level1 lfo3; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Reduce the gain of the stage. The Miller capacitance is proportional to the gain of the amplifying stage, so using a lower stage gain will reduce the Miller capacitance, thereby increasing the frequency response. <o:p></o:p></SPAN></LI></UL><ul type=disc><li class=MsoNormal style="MARGIN: 0cm 0cm 10pt; COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l1 level1 lfo4; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Use a pentode or cascode stage instead of a triode stage. Both the pentode and cascode configuration suffer very little from the effects of Miller capacitance because of the AC grounding of the screen grid in the pentode, and the AC grounding of the upper tube grid in the cascode, which drastically lowers the plate-to-grid capacitance in both configurations. The cascode configuration has the added advantage of lower noise, when compared to the pentode, because it does not have the "division" noise created by the screen grid of the pentode. <o:p></o:p></SPAN></LI></UL><ul type=disc><li class=MsoNormal style="MARGIN: 0cm 0cm 10pt; COLOR: black; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-list: l2 level1 lfo5; tab-stops: list 36.0pt"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE">Use the Miller capacitance to your advantage. Since the Miller capacitance forms a lowpass filter in conjunction with the output resistance of the previous stage, it can be used as a "free" lowpass filter, thus saving the cost and trouble of adding a capacitor or RC network in the amplifier when a tailored frequency response is desired. This is commonly used in a not-so-apparent manner in the typical input stage of most guitar amplifiers. The input resistor, in conjunction with the input capacitance, forms a lowpass filter that reduces the susceptibility of the tube to parasitic oscillations, such as those that can occur when the input stage is being driven by a long guitar cable. As shown in the example above, the typical cutoff point of this lowpass filter, when using a 68K resistor, is around 15.5kHz. Also, some high-gain guitar amplifiers have a distortion channel that uses large value series resistors in front of the grids of the preamp tubes. These resistors not only aid in minimizing blocking distortion, they also act as lowpass filters to reduce some of the high frequency content of the distortion signal, in order to reduce some of the "buzziness" in the tone. As shown in the above example, the typical cutoff frequency using a series 470K resistor would be 2.2kHz, which would roll off most of the upper harmonics, producing a subjectively smoother distortion tone. <o:p></o:p></SPAN></LI></UL><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font size=3><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'">Multistage Frequency Effects</SPAN></U></B><u><span style="FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-bidi-font-weight: bold"><o:p></o:p></SPAN></U></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><font size=3>Each stage will have its own frequency response, but the output of one stage will be affected by capacitances in the subsequent stage. This is especially so when determining the high frequency response. For example, the output capacitance (C<sub>o</SUB>) will be affected by the input Miller Capacitance (C<sub>Mi</SUB>) of the next stage.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Summary of frequency response of single-stages:<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p> </o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">CE/CS: suffers from Miller effect<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">CC/CD: "wideband" -- see Section 10.5<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">CB/CG: "wideband" -- see Section 10.6<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">(Wideband means that the stage operates to near the frequency<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Limit of the device ...<i>fT</I>)<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">How to fine the Bode plot for a general multistage amplifier?<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Can´t handle <i>n</I> poles and <i>m</I> zeroes analytically --> SPICE <o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Develop analytical tool for an important special case:<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">* No zeroes<o:p></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">* Exactly one "dominant" pole </SPAN></B><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">(w1<<w2, w3, ... ,w<i>n</I>)<b><o:p></o:p></B></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">(The example shows a voltage gain ... it could be<i>Iout</I>/<i>Vin</I>or<i>Vout</I>/<i>Iin</I>)<o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><o:p><font size=3> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font size=3><b><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'">Square Wave Testing</SPAN></U></B><u><span lang=EN-US style="FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-font-weight: bold"><o:p></o:p></SPAN></U></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><font size=3>In order to determine the frequency response of an amplifier by experimentation, you must apply a wide range of frequencies to the amplifier.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><font size=3>One way to accomplish this is to apply a square wave. A square wave consists of multiple frequencies (by Fourier analysis: it consists of odd harmonics).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b><u><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'">Square Wave Response Waveforms</SPAN></U></B><u><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN-US; mso-bidi-font-family: 'Times New Roman'; mso-bidi-font-weight: bold"><o:p></o:p></SPAN></U></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-bidi-font-weight: bold"><font size=3>If the output of the amplifier is not a perfect square wave then the amplifier is 'cutting' off certain frequency components of the square wave.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"></SPAN> </P><hr />Connect to the next generation of MSN Messenger <a href='http://imagine-msn.com/messenger/launch80/default.aspx?locale=en-us&source=wlmailtagline' target='_new'>Get it now! </a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-61570078127076768722010-02-15T00:41:00.002-04:302010-02-16T10:41:07.812-04:30Frequency response<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span class="Apple-style-span" style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: large;"><span class="Apple-style-span" style="font-size: 18px;"><span class="Apple-style-span" style="color: black; font-family: 'Times New Roman'; font-size: medium;"></span></span></span></div><span class="Apple-style-span" style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: large;"></span><br />
<span class="Apple-style-span" style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: large;"><h2><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="color: black; font-family: sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;"><br />
<h1 class="firstHeading" id="firstHeading" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; border-bottom-color: rgb(170, 170, 170); border-bottom-style: solid; border-bottom-width: 1px; font-size: 24px; font-weight: normal; line-height: 1.2em; margin-bottom: 0.1em; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-top: 0.5em;"><span class="Apple-style-span" style="color: silver;">Frequency response</span></h1></span></span></h2><div><span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"></span><br />
<span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><b><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Frequency response</span></span></b><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> is the measure of any system's output </span></span><a href="http://en.wikipedia.org/wiki/Frequency_spectrum" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Frequency spectrum"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">spectrum</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> in response to an input signal.</span></span><sup class="reference" id="cite_ref-Stark51_0-0" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">[</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">1</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]</span></span></a></sup><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> In the audible range it is usually referred to in connection with </span></span><a href="http://en.wikipedia.org/wiki/Electronic_amplifier" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Electronic amplifier"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">electronic amplifiers</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, </span></span><a href="http://en.wikipedia.org/wiki/Microphone" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Microphone"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">microphones</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> and </span></span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Loudspeakers" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Loudspeakers"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">loudspeakers</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">. Radio spectrum frequency response can refer to measurements of </span></span><a href="http://en.wikipedia.org/wiki/Coaxial_cable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Coaxial cable"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">coaxial cables</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, </span></span><a href="http://en.wikipedia.org/wiki/Category_6_cable" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Category 6 cable"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">category cables</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, </span></span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Video_switcher" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Video switcher"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">video switchers</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> and </span></span><a href="http://en.wikipedia.org/wiki/Wireless" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Wireless"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">wireless</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> communications devices. Subsonic frequency response measurements can include </span></span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Earthquakes" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Earthquakes"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">earthquakes</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> and </span></span><a href="http://en.wikipedia.org/wiki/Electroencephalography" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Electroencephalography"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">electroencephalography</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> (brain waves).</span></span></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Frequency response requirements differ depending on the application.</span></span><sup class="reference" id="cite_ref-Luther141_1-0" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Luther141-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">[</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">2</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]</span></span></a></sup><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> In </span></span><a href="http://en.wikipedia.org/wiki/High_fidelity" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="High fidelity"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">high fidelity</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> audio, an amplifier requires a frequency response of at least 20–20,000 </span></span><a href="http://en.wikipedia.org/wiki/Hertz" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Hertz"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Hz</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, with a tolerance as tight as ±0.1 </span></span><a href="http://en.wikipedia.org/wiki/Decibel" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Decibel"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">dB</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> in the mid-range frequencies around 1000 Hz, however, in </span></span><a href="http://en.wikipedia.org/wiki/Telephony" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Telephony"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">telephony</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.</span></span><sup class="reference" id="cite_ref-Luther141_1-1" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Luther141-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">[</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">2</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]</span></span></a></sup></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Frequency response curves are often used to indicate the accuracy of electronic components or systems.</span></span><sup class="reference" id="cite_ref-Stark51_0-1" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">[</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">1</span></span><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]</span></span></a></sup><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.</span></span><sup class="reference" id="cite_ref-Stark51_0-2" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">[</span></span></a><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">1</span></span></a><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none; white-space: nowrap;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]</span></span></a></sup></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><sup class="reference" id="cite_ref-Stark51_0-2" style="font-style: normal; font-weight: normal; line-height: 1em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/60/Butterworth_response.svg/300px-Butterworth_response.svg.png" /></span></span></sup></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><sup class="reference" id="cite_ref-Stark51_0-2" style="font-style: normal; font-weight: normal; line-height: 1em;"><span class="Apple-style-span" style="line-height: 19px;"></span></sup></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">The frequency response is typically characterized by the </span></span><i><a href="http://en.wikipedia.org/wiki/Amplitude" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Amplitude"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">magnitude</span></span></a></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of the system's response, measured in decibels (dB), and the </span></span><i><a href="http://en.wikipedia.org/wiki/Phase_(waves)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Phase (waves)"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">phase</span></span></a></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, measured in </span></span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Radians" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Radians"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">radians</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, versus frequency. The frequency response of a system can be measured by applying a </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">test signal</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, for example:</span></span></div><ul style="line-height: 1.5em; list-style-image: url(http://bits.wikimedia.org/skins-1.5/monobook/bullet.gif); list-style-type: square; margin-bottom: 0.5em; margin-left: 1.5em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">applying an impulse to the system and measuring its response (see </span></span><a href="http://en.wikipedia.org/wiki/Impulse_response" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Impulse response"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">impulse response</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">)</span></span></li>
<li style="margin-bottom: 0.1em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">sweeping a constant-amplitude pure tone through the </span></span><a href="http://en.wikipedia.org/wiki/Bandwidth_(signal_processing)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Bandwidth (signal processing)"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">bandwidth</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of interest and measuring the output level and phase shift relative to the input</span></span></li>
<li style="margin-bottom: 0.1em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">applying a signal with a wide frequency spectrum (for example digitally-generated</span></span><a href="http://en.wikipedia.org/wiki/Maximum_length_sequence" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Maximum length sequence"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">maximum length sequence</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> noise, or analog filtered </span></span><a href="http://en.wikipedia.org/wiki/White_noise" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="White noise"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">white noise</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> equivalent, like </span></span><a href="http://en.wikipedia.org/wiki/Pink_noise" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Pink noise"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">pink noise</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">), and calculating the impulse response by </span></span><a href="http://en.wikipedia.org/wiki/Deconvolution" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Deconvolution"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">deconvolution</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of this input signal and the output signal of the system.</span></span></li>
</ul><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">These typical response measurements can be plotted in two ways: by plotting the magnitude and phase measurements to obtain a </span></span><a href="http://en.wikipedia.org/wiki/Bode_plot" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Bode plot"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Bode plot</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> or by plotting the imaginary part of the frequency response against the real part of the frequency response to obtain a </span></span><a href="http://en.wikipedia.org/wiki/Nyquist_plot" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Nyquist plot"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Nyquist plot</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">.</span></span></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Once a frequency response has been measured (e.g., as an impulse response), providing the system is </span></span><a href="http://en.wikipedia.org/wiki/LTI_system_theory" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="LTI system theory"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">linear and time-invariant</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, its characteristic can be approximated with arbitrary accuracy by a </span></span><a href="http://en.wikipedia.org/wiki/Digital_filter" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Digital filter"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">digital filter</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">. Similarly, if a system is demonstrated to have a poor frequency response, a digital or</span></span><a class="mw-redirect" href="http://en.wikipedia.org/wiki/Analog_filter" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; text-decoration: none;" title="Analog filter"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">analog filter</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> can be applied to the signals prior to their reproduction to compensate for these deficiencies.</span></span></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Frequency response measurements can be used directly to quantify system performance and design control systems. However, frequency response analysis is not suggested if the system has slow dynamics</span></span></div></span></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">The </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">frequency response</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of an </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html" onmouseover="return escape('The topic of <i>linear systems theory</i> is primarily about <i>linear, time-invariant (LTI) filters</i>. A <i>linear</i> filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A <i>time-invariant</i> filter, or \'constant-coefficient\' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their <i>impulse response</i>, or their <i>frequency response</i>, or their <i>poles and zeros</i> plus a gain, or the like. Also, only LTI filters have the <i>superposition property</i> for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no <i>intermodulation distortion</i> for audio signals. — Click for http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">LTI filter</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> may be defined as the </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html" onmouseover="return escape('Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">spectrum</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of the output </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Definition_Signal.html" onmouseover="return escape('A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://www.dsprelated.com/dspbooks/filters/Definition_Signal.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">signal</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> divided by the </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html" onmouseover="return escape('The <i>spectrum</i> of a signal gives the distribution of signal energy as a function of frequency. — Click for http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">spectrum</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> of the input signal. In this section, we show that the frequency response of any </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html" onmouseover="return escape('The topic of <i>linear systems theory</i> is primarily about <i>linear, time-invariant (LTI) filters</i>. A <i>linear</i> filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A <i>time-invariant</i> filter, or \'constant-coefficient\' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their <i>impulse response</i>, or their <i>frequency response</i>, or their <i>poles and zeros</i> plus a gain, or the like. Also, only LTI filters have the <i>superposition property</i> for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no <i>intermodulation distortion</i> for audio signals. — Click for http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">LTI</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> </span></span><a href="http://www.dsprelated.com/dspbooks/filters/What_Filter.html" onmouseover="return escape('Click for http://www.dsprelated.com/dspbooks/filters/What_Filter.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">filter</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> is given by its </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Transfer_Function_Analysis.html" onmouseover="return escape('The <i>transfer function</i> is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for http://www.dsprelated.com/dspbooks/filters/Transfer_Function_Analysis.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">transfer function</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> <img align="MIDDLE" alt="$ H(z)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img308.png" width="39" /> evaluated on the unit circle, </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">i.e.</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, <img align="MIDDLE" alt="$ H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img340.png" width="63" />. We then show that this is the same result we got using </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Sinusoids.html" onmouseover="return escape('A <i>sinusoid</i> is any function of the form <i>A</i> sin(<i>ω t+φ</i>), where <i>t</i> is the independent variable, and <i>A, ω, φ</i> are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any \'pure\' vibration, such as that of an ideal <i>tuning fork</i> or <i>mass-spring</i> system. — Click for http://www.dsprelated.com/dspbooks/mdft/Sinusoids.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">sine-wave</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> analysis in Chapter </span></span><a href="https://ccrma.stanford.edu/~jos/fp/Simplest_Lowpass_Filter.html#chap:partOne" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">1</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">.</span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Beginning with Eq.<img align="BOTTOM" alt="$ \,$" border="0" height="15" src="https://ccrma.stanford.edu/~jos/fp/img94.png" width="7" />(</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Z_Transform_Convolution.html#eq:tpsixteen" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">6.4</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">), we have</span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\displaystyle Y(z) = H(z)X(z) $" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img822.png" width="128" /></span></span></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">where X(z) is the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">z</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> transform of the filter input signal <img align="MIDDLE" alt="$ x(n)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img88.png" width="35" />, <img align="MIDDLE" alt="$ Y(z)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img307.png" width="37" /> is the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">z</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> transform of the output signal <img align="MIDDLE" alt="$ y(n)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img90.png" width="34" />, and <img align="MIDDLE" alt="$ H(z)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img308.png" width="39" /> is the filter transfer function.</span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">A basic property of the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">z</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> transform is that, over the unit circle <img align="BOTTOM" alt="$ z=e^{j\omega T}$" border="0" height="17" src="https://ccrma.stanford.edu/~jos/fp/img277.png" width="65" />, we find the </span></span><i><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html" onmouseover="return escape('The spectrum of a signal gives the distribution of signal energy as a function of frequency. — Click for http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">spectrum</span></span></a></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> [</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#MDFT" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">84</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">].</span></span><a href="https://ccrma.stanford.edu/~jos/fp/footnode.html#foot13843" name="tex2html123" style="text-decoration: none;"><sup><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">8.1</span></span></sup></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">To show this, we set <img align="BOTTOM" alt="$ z=e^{j\omega T}$" border="0" height="17" src="https://ccrma.stanford.edu/~jos/fp/img277.png" width="65" /> in the definition of the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">z</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">transform, Eq.<img align="BOTTOM" alt="$ \,$" border="0" height="15" src="https://ccrma.stanford.edu/~jos/fp/img94.png" width="7" />(</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Z_Transform.html#eq:zt" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">6.1</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">), to obtain</span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\displaystyle X(e^{j\omega T}) = \sum_{n=-\infty}^\infty x(n) e^{-j\omega T n} $" border="0" height="60" src="https://ccrma.stanford.edu/~jos/fp/img823.png" width="207" /></span></span></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">which may be recognized as the definition of the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">bilateral</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> </span></span><i><a href="http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html" onmouseover="return escape('The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">discrete time Fourier transform</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> (</span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html" onmouseover="return escape('The Discrete Time Fourier Transform (DTFT) is the appropriate Fourier transform for discrete-time signals of arbitrary length. It can be obtained as the limit of a Discrete Fourier Transform (DFT) as its length goes to infinity. — Click for http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">DTFT</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">)</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><a href="" name="13924"></a> when <img align="BOTTOM" alt="$ T$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img96.png" width="16" /> is normalized to 1 [</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#Oppenheim89" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">59</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">,</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#MDFT" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">84</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">]. When <img align="BOTTOM" alt="$ x$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img101.png" width="13" /> is </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Causal_Recursive_Filters.html" onmouseover="return escape('A <i>causal filter</i> is any filter whose impulse response is zero prior to time zero. — Click for http://www.dsprelated.com/dspbooks/filters/Causal_Recursive_Filters.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">causal</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, this definition reduces to the usual (unilateral) DTFT definition:<a href="" name="13925"></a></span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><a href="" name="eq:dtft"></a></span></span><br />
<table align="CENTER" cellpadding="0"><tbody>
<tr valign="MIDDLE"> <td align="CENTER" nowrap=""><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">DTFT<img align="MIDDLE" alt="$\displaystyle _\omega(x) \isdefs \sum_{n=0}^\infty x(n) e^{-j\omega n} \protect$" border="0" height="60" src="https://ccrma.stanford.edu/~jos/fp/img824.png" width="163" /></span></span></td> <td align="RIGHT" nowrap="" width="10"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">(8.1)</span></span></td></tr>
</tbody></table></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><br clear="ALL" /> </span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Applying this relation to <img align="MIDDLE" alt="$ Y(z)=H(z)X(z)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img825.png" width="128" /> gives</span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><a href="" name="eq:fr"></a></span></span><br />
<table align="CENTER" cellpadding="0"><tbody>
<tr valign="MIDDLE"> <td align="CENTER" nowrap=""><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\displaystyle Y(e^{j\omega T}) = H(e^{j\omega T})X(e^{j\omega T}). \protect$" border="0" height="36" src="https://ccrma.stanford.edu/~jos/fp/img826.png" width="204" /></span></span></td> <td align="RIGHT" nowrap="" width="10"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">(8.2)</span></span></td></tr>
</tbody></table></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><br clear="ALL" /> </span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Thus, the spectrum of the filter output is just the input spectrum times the spectrum of the </span></span><a href="http://www.dsprelated.com/dspbooks/filters/Impulse_Response_Representation.html" onmouseover="return escape('The <i>impulse response</i> of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for http://www.dsprelated.com/dspbooks/filters/Impulse_Response_Representation.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">impulse response</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> <img align="MIDDLE" alt="$ H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img340.png" width="63" />. We have therefore shown the following:</span></span><br />
<blockquote><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\textstyle \parbox{0.8\textwidth}{\emph{The frequency response of a linear tim... ...uated on the unit circle in the $z$ plane, \textit{i.e.}, $H(e^{j\omega T})$.}}$" border="0" height="49" src="https://ccrma.stanford.edu/~jos/fp/img827.png" width="600" /></span></span></blockquote><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">This immediately implies the following:</span></span><br />
<blockquote><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\textstyle \parbox{0.8\textwidth}{\emph{The frequency response of an LTI filter equals the discrete-time Fourier transform of the impulse response.}}$" border="0" height="49" src="https://ccrma.stanford.edu/~jos/fp/img828.png" width="600" /></span></span></blockquote><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">We can express this mathematically by writing</span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\displaystyle \zbox {H(e^{j\omega T}) = \mbox{{\sc DTFT}}_{\omega T}(h).} $" border="0" height="47" src="https://ccrma.stanford.edu/~jos/fp/img829.png" width="189" /></span></span></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">By Eq.<img align="BOTTOM" alt="$ \,$" border="0" height="15" src="https://ccrma.stanford.edu/~jos/fp/img94.png" width="7" />(</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html#eq:fr" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">7.2</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">), the frequency response specifies the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">gain</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> and </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">phase shift</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> applied by the filter at each frequency. Since <img align="BOTTOM" alt="$ e$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img182.png" width="12" />, <img align="MIDDLE" alt="$ j$" border="0" height="28" src="https://ccrma.stanford.edu/~jos/fp/img202.png" width="12" />, and <img align="BOTTOM" alt="$ T$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img96.png" width="16" /> are constants, the frequency response <img align="MIDDLE" alt="$ H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img340.png" width="63" /> is only a function of radian frequency <img align="BOTTOM" alt="$ \omega$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img228.png" width="15" />. Since <img align="BOTTOM" alt="$ \omega$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img228.png" width="15" /> is real, the frequency response may be considered a </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">complex-valued function of a real variable</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">. The response at frequency <img align="MIDDLE" alt="$ f$" border="0" height="29" src="https://ccrma.stanford.edu/~jos/fp/img143.png" width="14" /> Hz, for example, is <img align="MIDDLE" alt="$ H(e^{j2\pi f T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img830.png" width="76" />, where <img align="BOTTOM" alt="$ T$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img96.png" width="16" /> is the </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theorem.html" onmouseover="return escape('The sampling interval (or sampling period) of a discrete-time signal is defined as the reciprocal of the sampling rate. Since the sampling rate is normally in units of samples per second (Hz), the sampling interval is normally in units of seconds. See also: sampling rate, sampling theorem — Click for http://www.dsprelated.com/dspbooks/mdft/Sampling_Theorem.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">sampling period</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> in seconds. It might be more convenient to define new functions such as <img align="MIDDLE" alt="$ H^\prime(\omega) \isdeftext H(e^{j\omega T})$" border="0" height="36" src="https://ccrma.stanford.edu/~jos/fp/img831.png" width="123" /> and write simply <img align="MIDDLE" alt="$ Y^\prime(\omega) = H^\prime(\omega) X^\prime(\omega)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img832.png" width="149" /> instead of having to write <img align="BOTTOM" alt="$ e^{j\omega T}$" border="0" height="17" src="https://ccrma.stanford.edu/~jos/fp/img833.png" width="36" /> so often, but doing so would add a lot of new functions to an already notation-rich scenario. Furthermore, writing <img align="MIDDLE" alt="$ H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img340.png" width="63" /> makes explicit the connection between the transfer function and the frequency response.</span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Notice that defining the frequency response as a function of <img align="BOTTOM" alt="$ e^{j\omega T}$" border="0" height="17" src="https://ccrma.stanford.edu/~jos/fp/img833.png" width="36" /> places the frequency ``axis'' on the </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">unit circle</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> in the complex <img align="BOTTOM" alt="$ z$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img45.png" width="12" /> plane, since <img align="MIDDLE" alt="$ \left\vert e^{j\omega T}\right\vert=1$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img834.png" width="76" />. As a result, adding multiples of the </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html" onmouseover="return escape('Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">sampling</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> frequency to <img align="BOTTOM" alt="$ \omega$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img228.png" width="15" /> corresponds to traversing whole cycles around the unit circle, since</span></span><br />
<div align="CENTER"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"><img align="MIDDLE" alt="$\displaystyle e^{j(\omega + k 2\pi f_s)T} = e^{j(\omega T + k 2\pi)} = e^{j\omega T}, $" border="0" height="37" src="https://ccrma.stanford.edu/~jos/fp/img835.png" width="240" /></span></span></div><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">whenever <img align="BOTTOM" alt="$ k$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img79.png" width="13" /> is an integer. Since every discrete-time spectrum repeats in frequency with a ``</span></span><a href="http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals" onmouseover="return escape('A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">period</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">'' equal to the </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html" onmouseover="return escape('The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon\'s Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">sampling rate</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, we may restrict <img align="BOTTOM" alt="$ \omega T$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img238.png" width="26" /> to one traversal of the unit circle; a typical choice is <img align="MIDDLE" alt="$ -\pi \leq \omega T < \pi$" border="0" height="29" src="https://ccrma.stanford.edu/~jos/fp/img836.png" width="100" /> [ <img align="MIDDLE" alt="$ \omega T\in[-\pi,\pi)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img318.png" width="95" />]. For convenience, <img align="MIDDLE" alt="$ \omega T \in[-\pi,\pi]$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img334.png" width="93" /> is often allowed.</span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">We have seen that the spectrum is a particular slice through the transfer function. It is also possible to go the other way and generalize the spectrum (defined only over the unit circle) to the entire <img align="BOTTOM" alt="$ z$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img45.png" width="12" /> plane by means of </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">analytic continuation</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> (§</span></span><a href="https://ccrma.stanford.edu/~jos/fp/Analytic_Continuation.html#sec:analcont" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">D.2</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">). Since analytic continuation is unique (for all filters encountered in practice), we get the same result going either direction.</span></span><br />
<span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Because every </span></span><a href="http://www.dsprelated.com/dspbooks/mdft/Complex_Numbers.html" onmouseover="return escape('Click for http://www.dsprelated.com/dspbooks/mdft/Complex_Numbers.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">complex number</span></span></a><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> <img align="BOTTOM" alt="$ z$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img45.png" width="12" /> can be represented as a magnitude <img align="MIDDLE" alt="$ r=\vert z\vert$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img837.png" width="50" /> and angle <img align="BOTTOM" alt="$ \theta=\angle z$" border="0" height="14" src="https://ccrma.stanford.edu/~jos/fp/img838.png" width="53" />, </span></span><i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">viz.</span></span></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">, <img align="MIDDLE" alt="$ z=r\exp(j\theta)$" border="0" height="31" src="https://ccrma.stanford.edu/~jos/fp/img839.png" width="96" />, the frequency response <img align="MIDDLE" alt="$ H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img340.png" width="63" /> may be decomposed into two real-valued functions, the </span></span><i><a href="http://www.dsprelated.com/dspbooks/filters/Amplitude_Response_I_I.html" onmouseover="return escape('The <i>amplitude response</i> of an LTI filter is simply the magnitude of the (complex) frequency response — Click for http://www.dsprelated.com/dspbooks/filters/Amplitude_Response_I_I.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">amplitude response</span></span></a></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> <img align="MIDDLE" alt="$ \vert H(e^{j\omega T})\vert$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img840.png" width="72" /> and the </span></span><i><a href="http://www.dsprelated.com/dspbooks/filters/Phase_Response_I_I.html" onmouseover="return escape('The <i>phase response</i> of an LTI filter is the angle of the (complex) frequency response — Click for http://www.dsprelated.com/dspbooks/filters/Phase_Response_I_I.html')" style="text-decoration: none;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">phase response</span></span></a></i><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;"> <img align="MIDDLE" alt="$ \angle H(e^{j\omega T})$" border="0" height="34" src="https://ccrma.stanford.edu/~jos/fp/img841.png" width="74" />. Formally, we may define them as follows:</span></span><br />
<div style="color: black;"><span class="Apple-style-span" style="font-family: Arial, Helvetica, Verdana, Sans; font-size: 13px;"></span></div><h3 style="font-weight: bold; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 6px; padding-left: 0px; padding-right: 0px; padding-top: 10px;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">Frequency Response Ranges</span></span></h3><div style="margin-bottom: 14px; margin-left: 0px; margin-right: 0px; margin-top: 14px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: justify;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">You will often see frequency response quoted as a range between two figures. This is a simple (or perhaps "simplistic") way to see which frequencies a microphone is capable of capturing effectively. For example, a microphone which is said to have a frequency response of 20 Hz to 20 kHz can reproduce all frequencies within this range. Frequencies outside this range will be reproduced to a much lesser extent or not at all.</span></span></div><div style="margin-bottom: 14px; margin-left: 0px; margin-right: 0px; margin-top: 14px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px; text-align: justify;"><span class="Apple-style-span" style="color: silver;"><span class="Apple-style-span" style="font-size: small;">This specification makes no mention of the response curve, or how successfully the various frequencies will be reproduced. Like many specifications, it should be taken as a guide only.</span></span></div><div style="color: black;"><br />
</div><div style="color: black;"><br />
</div></span><br />
HENDERSON PARADA<br />
<div><a href="http://en.wikipedia.org/wiki/Frequency_response">http://en.wikipedia.org/wiki/Frequency_response</a></div><div><a href="https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html">https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html</a></div><div><a href="http://www.mediacollege.com/audio/microphones/frequency-response.html">http://www.mediacollege.com/audio/microphones/frequency-response.html</a></div>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-83507270275557211522010-02-15T00:33:00.002-04:302010-02-16T10:40:46.567-04:30Frequency response<div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><b><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">Frequency response</span></b><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">is the measure of any system's output</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Frequency_spectrum" title="Frequency spectrum"><span style="color: #bfbfbf;">spectrum</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">in response to an input signal.<sup><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0"><span style="color: #bfbfbf;">[1]</span></a></sup></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">In the audible range it is usually referred to in connection with</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Electronic_amplifier" title="Electronic amplifier"><span style="color: #bfbfbf;">electronic amplifiers</span></a>,</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Microphone" title="Microphone"><span style="color: #bfbfbf;">microphones</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">and</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Loudspeakers" title="Loudspeakers"><span style="color: #bfbfbf;">loudspeakers</span></a>. Radio spectrum frequency response can refer to measurements of</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Coaxial_cable" title="Coaxial cable"><span style="color: #bfbfbf;">coaxial cables</span></a>,</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Category_6_cable" title="Category 6 cable"><span style="color: #bfbfbf;">category cables</span></a>,</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Video_switcher" title="Video switcher"><span style="color: #bfbfbf;">video switchers</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">and</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Wireless" title="Wireless"><span style="color: #bfbfbf;">wireless</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">communications devices. Subsonic frequency response measurements can include</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Earthquakes" title="Earthquakes"><span style="color: #bfbfbf;">earthquakes</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">and</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Electroencephalography" title="Electroencephalography"><span style="color: #bfbfbf;">electroencephalography</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">(brain waves).</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">Frequency response requirements differ depending on the application.<sup><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Luther141-1"><span style="color: #bfbfbf;">[2]</span></a></sup></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">In</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/High_fidelity" title="High fidelity"><span style="color: #bfbfbf;">high fidelity</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">audio, an amplifier requires a frequency response of at least 20–20,000 <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz"><span style="color: #bfbfbf;">Hz</span></a>, with a tolerance as tight as ±0.1 <a href="http://en.wikipedia.org/wiki/Decibel" title="Decibel"><span style="color: #bfbfbf;">dB</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">in the mid-range frequencies around 1000 Hz, however, in</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Telephony" title="Telephony"><span style="color: #bfbfbf;">telephony</span></a>, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.<sup><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Luther141-1"><span style="color: #bfbfbf;">[2]</span></a></sup></span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">Frequency response curves are often used to indicate the accuracy of electronic components or systems.<sup><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0"><span style="color: #bfbfbf;">[1]</span></a></sup></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.<sup><span style="color: #bfbfbf;"><a href="http://en.wikipedia.org/wiki/Frequency_response#cite_note-Stark51-0">[1]</a></span></sup></span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><sup><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/6/60/Butterworth_response.svg/300px-Butterworth_response.svg.png" /></sup></span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><sup></sup></span></div><div class="MsoNormal" style="background: #F9F9F9; line-height: 16.8pt; margin-bottom: 9.6pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 8.5pt;">Frequency response of a low pass filter with 6 dB per octave or 20 dB per decade</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">The frequency response is typically characterized by the</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Amplitude" title="Amplitude"><span style="color: #bfbfbf;">magnitude</span></a></span></i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">of the system's response, measured in decibels (dB), and the</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Phase_(waves)" title="Phase (waves)"><span style="color: #bfbfbf;">phase</span></a></span></i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">, measured in</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Radians" title="Radians"><span style="color: #bfbfbf;">radians</span></a>, versus frequency. The frequency response of a system can be measured by applying a</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">test signal</span></i><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">, for example:</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 1.2pt; margin-left: 18.0pt; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span style="color: #bfbfbf; font-family: Wingdings; font-size: 10pt;">§<span style="font: normal normal normal 7pt/normal 'Times New Roman';"> </span></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">applying an impulse to the system and measuring its response (see</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response"><span style="color: #bfbfbf;">impulse response</span></a>)</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 1.2pt; margin-left: 18.0pt; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span style="color: #bfbfbf; font-family: Wingdings; font-size: 10pt;">§<span style="font: normal normal normal 7pt/normal 'Times New Roman';"> </span></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">sweeping a constant-amplitude pure tone through the</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)"><span style="color: #bfbfbf;">bandwidth</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">of interest and measuring the output level and phase shift relative to the input</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 1.2pt; margin-left: 18.0pt; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; tab-stops: list 36.0pt; text-indent: -18.0pt;"><span style="color: #bfbfbf; font-family: Wingdings; font-size: 10pt;">§<span style="font: normal normal normal 7pt/normal 'Times New Roman';"> </span></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">applying a signal with a wide frequency spectrum (for example digitally-generated<a href="http://en.wikipedia.org/wiki/Maximum_length_sequence" title="Maximum length sequence"><span style="color: #bfbfbf;">maximum length sequence</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">noise, or analog filtered</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/White_noise" title="White noise"><span style="color: #bfbfbf;">white noise</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">equivalent, like</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Pink_noise" title="Pink noise"><span style="color: #bfbfbf;">pink noise</span></a>), and calculating the impulse response by</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Deconvolution" title="Deconvolution"><span style="color: #bfbfbf;">deconvolution</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">of this input signal and the output signal of the system.</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">These typical response measurements can be plotted in two ways: by plotting the magnitude and phase measurements to obtain a</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Bode_plot" title="Bode plot"><span style="color: #bfbfbf;">Bode plot</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">or by plotting the imaginary part of the frequency response against the real part of the frequency response to obtain a</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Nyquist_plot" title="Nyquist plot"><span style="color: #bfbfbf;">Nyquist plot</span></a>.</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">Once a frequency response has been measured (e.g., as an impulse response), providing the system is</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/LTI_system_theory" title="LTI system theory"><span style="color: #bfbfbf;">linear and time-invariant</span></a>, its characteristic can be approximated with arbitrary accuracy by a</span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"><a href="http://en.wikipedia.org/wiki/Digital_filter" title="Digital filter"><span style="color: #bfbfbf;">digital filter</span></a>. Similarly, if a system is demonstrated to have a poor frequency response, a digital or<a href="http://en.wikipedia.org/wiki/Analog_filter" title="Analog filter"><span style="color: #bfbfbf;">analog filter</span></a></span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">can be applied to the signals prior to their reproduction to compensate for these deficiencies.</span></div><div class="MsoNormal" style="line-height: 18.0pt; margin-bottom: 6.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 4.8pt;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 10pt;">Frequency response measurements can be used directly to quantify system performance and design control systems. However, frequency response analysis is not suggested if the system has slow dynamics.<sup>[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span style="color: #bfbfbf;">citation needed</span></a></i>]</sup></span></div><div class="MsoNormal"><span style="color: #bfbfbf;"> <span class="Apple-style-span" style="line-height: normal;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">The</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">frequency response</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">of an</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html"><span style="color: #bfbfbf; text-decoration: none;">LTI filter</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">may be defined as the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html"><span style="color: #bfbfbf; text-decoration: none;">spectrum</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">of the output</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Definition_Signal.html"><span style="color: #bfbfbf; text-decoration: none;">signal</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">divided by the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html"><span style="color: #bfbfbf; text-decoration: none;">spectrum</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">of the input signal. In this section, we show that the frequency response of any</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Linear_Time_Invariant_Digital_Filters.html"><span style="color: #bfbfbf; text-decoration: none;">LTI</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/What_Filter.html"><span style="color: #bfbfbf; text-decoration: none;">filter</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">is given by its</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Transfer_Function_Analysis.html"><span style="color: #bfbfbf; text-decoration: none;">transfer function</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(z)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image001.gif" width="39" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">evaluated on the unit circle,</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">i.e.</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">,</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image002.gif" width="63" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. We then show that this is the same result we got using</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Sinusoids.html"><span style="color: #bfbfbf; text-decoration: none;">sine-wave</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">analysis in Chapter</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="https://ccrma.stanford.edu/~jos/fp/Simplest_Lowpass_Filter.html#chap:partOne"><span style="color: #bfbfbf; text-decoration: none;">1</span></a>.</span></span></span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Beginning with Eq.<img alt="$ \,$" border="0" height="15" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image003.gif" width="7" />(<a href="https://ccrma.stanford.edu/~jos/fp/Z_Transform_Convolution.html#eq:tpsixteen"><span style="color: #bfbfbf; text-decoration: none;">6.4</span></a>), we have</span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\displaystyle Y(z) = H(z)X(z) $" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image004.gif" width="128" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">where X(z) is the <i>z</i> transform of the filter input signal </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ x(n)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image005.gif" width="35" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ Y(z)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image006.gif" width="37" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is the <i>z</i> transform of the output signal </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ y(n)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image007.gif" width="34" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, and </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(z)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image001.gif" width="39" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is the filter transfer function.</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">A basic property of the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">z</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">transform is that, over the unit circle</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z=e^{j\omega T}$" border="0" height="17" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image008.gif" width="65" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, we find the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Example_Applications_DFT.html"><span style="color: #bfbfbf; text-decoration: none;">spectrum</span></a></span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">[<a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#MDFT"><span style="color: #bfbfbf; text-decoration: none;">84</span></a>].<a href="" name="tex2html123"></a><a href="https://ccrma.stanford.edu/~jos/fp/footnode.html#foot13843"><sup><span style="color: #bfbfbf; text-decoration: none;">8.1</span></sup></a>To show this, we set</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z=e^{j\omega T}$" border="0" height="17" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image008.gif" width="65" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">in the definition of the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">z</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">transform, Eq.<img alt="$ \,$" border="0" height="15" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image003.gif" width="7" />(<a href="https://ccrma.stanford.edu/~jos/fp/Z_Transform.html#eq:zt"><span style="color: #bfbfbf; text-decoration: none;">6.1</span></a>), to obtain</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\displaystyle X(e^{j\omega T}) = \sum_{n=-\infty}^\infty x(n) e^{-j\omega T n} $" border="0" height="60" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image009.gif" width="207" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">which may be recognized as the definition of the <i>bilateral</i> <i><a href="http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html"><span style="color: #bfbfbf; text-decoration: none;">discrete time Fourier transform</span></a> (<a href="http://www.dsprelated.com/dspbooks/mdft/Discrete_Time_Fourier_Transform.html"><span style="color: #bfbfbf; text-decoration: none;">DTFT</span></a>)</i><a href="" name="13924"></a> when </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ T$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image010.gif" width="16" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is normalized to 1 [<a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#Oppenheim89"><span style="color: #bfbfbf; text-decoration: none;">59</span></a>,<a href="https://ccrma.stanford.edu/~jos/fp/Bibliography.html#MDFT"><span style="color: #bfbfbf; text-decoration: none;">84</span></a>]. When </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ x$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image011.gif" width="13" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is <a href="http://www.dsprelated.com/dspbooks/filters/Causal_Recursive_Filters.html"><span style="color: #bfbfbf; text-decoration: none;">causal</span></a>, this definition reduces to the usual (unilateral) DTFT definition:</span><a href="" name="13925"></a><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div align="center"><table border="0" cellpadding="0" class="MsoNormalTable" style="mso-cellspacing: 1.5pt; mso-padding-alt: 0cm 0cm 0cm 0cm; mso-yfti-tbllook: 1184; width: 100.0%;"><tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"> <td nowrap="" style="padding: 0cm 0cm 0cm 0cm;"><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><a href="" name="eq:dtft"></a><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;">DTFT<img alt="$\displaystyle _\omega(x) \isdefs \sum_{n=0}^\infty x(n) e^{-j\omega n} \protect$" border="0" height="60" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image012.gif" width="163" /></span></div></td> <td nowrap="" style="padding: 0cm 0cm 0cm 0cm; width: 7.5pt;" width="10"><div align="right" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;">(8.1)</span></div></td> </tr>
</tbody></table></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><br clear="all" style="mso-special-character: line-break;" /> </span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Applying this relation to </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ Y(z)=H(z)X(z)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image013.gif" width="128" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> gives</span></div><div align="center"><table border="0" cellpadding="0" class="MsoNormalTable" style="mso-cellspacing: 1.5pt; mso-padding-alt: 0cm 0cm 0cm 0cm; mso-yfti-tbllook: 1184; width: 100.0%;"><tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"> <td nowrap="" style="padding: 0cm 0cm 0cm 0cm;"><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><a href="" name="eq:fr"></a><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"><img alt="$\displaystyle Y(e^{j\omega T}) = H(e^{j\omega T})X(e^{j\omega T}). \protect$" border="0" height="36" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image014.gif" width="204" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div></td> <td nowrap="" style="padding: 0cm 0cm 0cm 0cm; width: 7.5pt;" width="10"><div align="right" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: right;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;">(8.2)</span></div></td> </tr>
</tbody></table></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><br clear="all" style="mso-special-character: line-break;" /> </span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Thus, the spectrum of the filter output is just the input spectrum times the spectrum of the <a href="http://www.dsprelated.com/dspbooks/filters/Impulse_Response_Representation.html"><span style="color: #bfbfbf; text-decoration: none;">impulse response</span></a> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image002.gif" width="63" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. We have therefore shown the following:</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\textstyle \parbox{0.8\textwidth}{\emph{The frequency response of a linear tim... ...uated on the unit circle in the $z$ plane, \textit{i.e.}, $H(e^{j\omega T})$.}}$" border="0" height="49" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image015.gif" width="600" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">This immediately implies the following:</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\textstyle \parbox{0.8\textwidth}{\emph{The frequency response of an LTI filter equals the discrete-time Fourier transform of the impulse response.}}$" border="0" height="49" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image016.gif" width="600" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">We can express this mathematically by writing</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\displaystyle \zbox {H(e^{j\omega T}) = \mbox{{\sc DTFT}}_{\omega T}(h).} $" border="0" height="47" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image017.gif" width="189" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">By Eq.<img alt="$ \,$" border="0" height="15" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image003.gif" width="7" />(<a href="https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html#eq:fr"><span style="color: #bfbfbf; text-decoration: none;">7.2</span></a>), the frequency response specifies the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">gain</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">and</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">phase shift</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">applied by the filter at each frequency. Since</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ e$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image018.gif" width="12" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">,</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ j$" border="0" height="28" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image019.gif" width="12" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, and</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ T$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image010.gif" width="16" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">are constants, the frequency response</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image002.gif" width="63" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">is only a function of radian frequency</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image020.gif" width="15" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. Since</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image020.gif" width="15" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">is real, the frequency response may be considered a</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">complex-valued function of a real variable</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. The response at frequency</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ f$" border="0" height="29" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image021.gif" width="14" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Hz, for example, is</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j2\pi f T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image022.gif" width="76" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, where</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ T$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image010.gif" width="16" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">is the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theorem.html"><span style="color: #bfbfbf; text-decoration: none;">sampling period</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">in seconds. It might be more convenient to define new functions such as</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H^\prime(\omega) \isdeftext H(e^{j\omega T})$" border="0" height="36" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image023.gif" width="123" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">and write simply</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ Y^\prime(\omega) = H^\prime(\omega) X^\prime(\omega)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image024.gif" width="149" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">instead of having to write</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ e^{j\omega T}$" border="0" height="17" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image025.gif" width="36" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">so often, but doing so would add a lot of new functions to an already notation-rich scenario. Furthermore, writing</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image002.gif" width="63" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">makes explicit the connection between the transfer function and the frequency response.</span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Notice that defining the frequency response as a function of</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ e^{j\omega T}$" border="0" height="17" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image025.gif" width="36" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">places the frequency ``axis'' on the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">unit circle</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">in the complex</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image026.gif" width="12" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">plane, since</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \left\vert e^{j\omega T}\right\vert=1$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image027.gif" width="76" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. As a result, adding multiples of the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html"><span style="color: #bfbfbf; text-decoration: none;">sampling</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">frequency to</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image020.gif" width="15" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">corresponds to traversing whole cycles around the unit circle, since</span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$\displaystyle e^{j(\omega + k 2\pi f_s)T} = e^{j(\omega T + k 2\pi)} = e^{j\omega T}, $" border="0" height="37" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image028.gif" width="240" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">whenever </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ k$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image029.gif" width="13" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is an integer. Since every discrete-time spectrum repeats in frequency with a ``<a href="http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals"><span style="color: #bfbfbf; text-decoration: none;">period</span></a>'' equal to the <a href="http://www.dsprelated.com/dspbooks/mdft/Sampling_Theory.html"><span style="color: #bfbfbf; text-decoration: none;">sampling rate</span></a>, we may restrict </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega T$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image030.gif" width="26" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> to one traversal of the unit circle; a typical choice is </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ -\pi \leq \omega T < \pi$" border="0" height="29" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image031.gif" width="100" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> [ </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega T\in[-\pi,\pi)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image032.gif" width="95" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">]. For convenience, </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \omega T \in[-\pi,\pi]$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image033.gif" width="93" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> is often allowed.</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">We have seen that the spectrum is a particular slice through the transfer function. It is also possible to go the other way and generalize the spectrum (defined only over the unit circle) to the entire</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image026.gif" width="12" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">plane by means of</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">analytic continuation</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">(§<a href="https://ccrma.stanford.edu/~jos/fp/Analytic_Continuation.html#sec:analcont"><span style="color: #bfbfbf; text-decoration: none;">D.2</span></a>). Since analytic continuation is unique (for all filters encountered in practice), we get the same result going either direction.</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 12pt;"></span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">Because every</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/mdft/Complex_Numbers.html"><span style="color: #bfbfbf; text-decoration: none;">complex number</span></a></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image026.gif" width="12" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">can be represented as a magnitude</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ r=\vert z\vert$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image034.gif" width="50" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">and angle</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \theta=\angle z$" border="0" height="14" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image035.gif" width="53" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">,</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">viz.</span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">,</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ z=r\exp(j\theta)$" border="0" height="31" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image036.gif" width="96" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">, the frequency response</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image002.gif" width="63" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">may be decomposed into two real-valued functions, the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Amplitude_Response_I_I.html"><span style="color: #bfbfbf; text-decoration: none;">amplitude response</span></a></span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \vert H(e^{j\omega T})\vert$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image037.gif" width="72" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">and the</span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><a href="http://www.dsprelated.com/dspbooks/filters/Phase_Response_I_I.html"><span style="color: #bfbfbf; text-decoration: none;">phase response</span></a></span></i><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"> </span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"><img alt="$ \angle H(e^{j\omega T})$" border="0" height="34" src="file:///C:\DOCUME~1\ADMINI~1\CONFIG~1\Temp\msohtmlclip1\01\clip_image038.gif" width="74" /></span><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;">. Formally, we may define them as follows:</span></div><div class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #bfbfbf; font-family: 'Times New Roman', serif; font-size: 13.5pt;"></span></div><h3 style="margin-top: 0cm;"><span style="color: #bfbfbf; font-family: Arial, sans-serif;">Frequency Response Ranges</span></h3><div style="margin-bottom: 10.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 10.5pt; text-align: justify;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 11pt;">You will often see frequency response quoted as a range between two figures. This is a simple (or perhaps "simplistic") way to see which frequencies a microphone is capable of capturing effectively. For example, a microphone which is said to have a frequency response of 20 Hz to 20 kHz can reproduce all frequencies within this range. Frequencies outside this range will be reproduced to a much lesser extent or not at all.</span></div><div style="margin-bottom: 10.5pt; margin-left: 0cm; margin-right: 0cm; margin-top: 10.5pt; text-align: justify;"><span style="color: #bfbfbf; font-family: Arial, sans-serif; font-size: 11pt;">This specification makes no mention of the response curve, or how successfully the various frequencies will be reproduced. Like many specifications, it should be taken as a guide only.</span></div></div>-- <br />
HENDERSON PARADA<br />
<div><br />
</div><div><a href="http://en.wikipedia.org/wiki/Frequency_response">http://en.wikipedia.org/wiki/Frequency_response</a></div><div><a href="https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html">https://ccrma.stanford.edu/~jos/fp/Frequency_Response_I.html</a></div><div><a href="http://www.mediacollege.com/audio/microphones/frequency-response.html">http://www.mediacollege.com/audio/microphones/frequency-response.html</a></div>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com2tag:blogger.com,1999:blog-8049570815625896303.post-81678017769291948312010-02-13T14:51:00.002-04:302010-02-13T20:04:55.336-04:30Teorema De Miller<span class="Apple-style-span" style="color: silver;"><br clear="all" /></span><br />
<div class="MsoNormal" style="line-height: 100%; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 5.8pt; margin-right: 49.45pt; margin-top: 0cm; mso-layout-grid-align: none; mso-pagination: none; text-align: justify; text-autospace: none; text-indent: -.6pt;"><b><span style="font-size: 11.5pt; line-height: 100%;"><span class="Apple-style-span" style="color: silver;">. </span></span></b><span style="font-size: 11.5pt; letter-spacing: -0.05pt; line-height: 100%;"><span class="Apple-style-span" style="color: silver;">E</span></span><span style="font-size: 11.5pt; line-height: 100%;"><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">ál</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.75pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">de</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span class="Apple-style-span" style="color: silver;">pu</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">al</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">f</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">cu</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">nc</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.7pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">f</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.95pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">de</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">f</span></span><span class="Apple-style-span" style="color: silver;">u</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">e c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">ú</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">y </span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">is</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">ún</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span class="Apple-style-span" style="color: silver;">e </span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">le</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">a </span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">é</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">p</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">za</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">a </span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">it</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">nc</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">pun</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">o </span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">(</span></span><span class="Apple-style-span" style="color: silver;">C</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">C</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">µ</span></span><span class="Apple-style-span" style="color: silver;">)</span><span style="letter-spacing: 1.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 1.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">na</span><span style="letter-spacing: 1.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">p</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">nc</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 1.9pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 1.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">da</span><span style="letter-spacing: 1.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">qu</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">v</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">en</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">.</span><span style="letter-spacing: 2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">E</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 1.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">é</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 1.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">y</span><span style="letter-spacing: 1.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">ú</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">ti</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 1.6pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">y</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">f</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">ti</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">v</span></span><span class="Apple-style-span" style="color: silver;">a </span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">á</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">b</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">un</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">id</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">M</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">il</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">r</span></span><span style="font-size: 13.5pt; line-height: 100%;"><span class="Apple-style-span" style="color: silver;">.</span></span><span style="font-size: 13.5pt; line-height: 100%;"><span class="Apple-style-span" style="color: silver;"></span></span></div><div class="MsoNormal" style="line-height: 101%; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 5.8pt; margin-right: 49.85pt; margin-top: .4pt; mso-layout-grid-align: none; mso-pagination: none; text-align: justify; text-autospace: none; text-indent: -.6pt;"><span style="height: 147px; left: 0px; margin-left: 112px; margin-top: 133px; position: absolute; width: 160px; z-index: -1;"> </span></div><table cellpadding="0" cellspacing="0"><tbody>
<tr> <td height="147" style="vertical-align: top;" width="160"><span style="left: 0pt; position: absolute; z-index: -1;"> </span><br />
<span style="left: 0pt; position: absolute; z-index: -1;"><table cellpadding="0" cellspacing="0"><tbody>
<tr> <td><div class="shape" style="padding: 0pt 0pt 0pt 0pt;"><div class="MsoNormal" style="line-height: 107.0pt; margin-bottom: .0001pt; margin-bottom: 0cm;"><span style="font-family: 'Times New Roman', serif; font-size: 12pt;"><span class="Apple-style-span" style="color: silver;"><img height="142" src="file:///C:/DOCUME~1/ADMINI~1/CONFIG~1/Temp/msohtmlclip1/01/clip_image002.jpg" width="156" /></span></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;"><br />
</div></div></td> </tr>
</tbody></table></span></td> </tr>
</tbody></table><span style="font-size: 11.5pt; line-height: 101%;"><span class="Apple-style-span" style="color: silver;">C</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">ns</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.7pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">u</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">ió</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span class="Apple-style-span" style="color: silver;">u</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">f</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span class="Apple-style-span" style="color: silver;">ura</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">p</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.7pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">á</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.6pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.7pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">q</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">no se</span><span style="letter-spacing: 2.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">u</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">,</span><span style="letter-spacing: 2.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 2.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">h</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 2.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">do</span><span style="letter-spacing: 2.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 2.1pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 2.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">de</span><span style="letter-spacing: 2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">rc</span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s,</span><span style="letter-spacing: 2.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">am</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 2.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">1</span><span style="letter-spacing: 1.85pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">y</span><span style="letter-spacing: 2.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">2</span></span><span class="Apple-style-span" style="color: silver;">,</span><span style="letter-spacing: 2.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 2.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 2.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 2.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span class="Apple-style-span" style="color: silver;">e c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">p</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">Z</span></span><span class="Apple-style-span" style="color: silver;">.</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">L</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">1 y</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">2 </span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">tam</span></span><span class="Apple-style-span" style="color: silver;">b</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">é</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">es</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">tá</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">ta</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.65pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;"> o</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">r</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">p</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">te</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span class="Apple-style-span" style="color: silver;">rcu</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">,</span><span style="letter-spacing: 1.5pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 1.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">s</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.95pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">ic</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 1.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 1pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">lí</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">ne</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.35pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">pun</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span style="letter-spacing: -0.2pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.6pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">q</span></span><span class="Apple-style-span" style="color: silver;">ue</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 1.4pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 1.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">s.</span><span style="letter-spacing: 1.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">M</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">á</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 1.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">ún,</span><span style="letter-spacing: 1.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">se ha s</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">pu</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.45pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">q</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">de</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">al</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">g</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">u</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span class="Apple-style-span" style="color: silver;">a</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">se </span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">h</span></span><span class="Apple-style-span" style="color: silver;">a d</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;">r</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">m</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">do</span><span style="letter-spacing: 0.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">que</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> v</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">l</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span style="letter-spacing: -0.1pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">j</span></span><span class="Apple-style-span" style="color: silver;">e</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.15pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">2 </span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">s</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">á r</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">el</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span class="Apple-style-span" style="color: silver;">c</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">d</span></span><span class="Apple-style-span" style="color: silver;">o</span><span style="letter-spacing: 0.55pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">c</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: 0.3pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">l</span><span style="letter-spacing: 0.2pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">n</span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">o</span></span><span class="Apple-style-span" style="color: silver;">do</span><span style="letter-spacing: 0.25pt;"><span class="Apple-style-span" style="color: silver;"> </span></span><span class="Apple-style-span" style="color: silver;">1</span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;"> m</span></span><span style="letter-spacing: -0.05pt;"><span class="Apple-style-span" style="color: silver;">e</span></span><span class="Apple-style-span" style="color: silver;">d</span><span style="letter-spacing: 0.1pt;"><span class="Apple-style-span" style="color: silver;">i</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">a</span></span><span style="letter-spacing: -0.15pt;"><span class="Apple-style-span" style="color: silver;">n</span></span><span style="letter-spacing: 0.05pt;"><span class="Apple-style-span" style="color: silver;">t</span></span><span class="Apple-style-span" style="color: silver;">e</span></span><br />
<div class="MsoNormal" style="line-height: 101%; margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 5.8pt; margin-right: 49.85pt; margin-top: .4pt; mso-layout-grid-align: none; mso-pagination: none; text-align: justify; text-autospace: none; text-indent: -.6pt;"><span style="font-size: 11.5pt; line-height: 101%;"><span class="Apple-style-span" style="color: grey; font-family: Arial, sans-serif; font-size: 12px; line-height: 16px;"></span></span></div><div style="background: inherit; margin-bottom: 5px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="font-weight: bold;"><span class="Apple-style-span" style="color: silver;">Si se conoce la relación u = Z(D)·i ó i = Y(D)·u entre los terminales de un elemento pasivo o de una rama de un circuito, estos elementos pueden sustituirse por una fuente de tensión, cuya forma de onda sea Z(D)·i, o por una fuente de intensidad dada por Y(D)·u.</span></span><span class="Apple-style-span" style="color: silver;"><br />
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</span></div><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_-cRpc7WEuwQcI9ny5XIuDTlTHz_2Rj4xMIe4jJAQIHR92UzxENQ1G5DPgGGe7hAcz82jQkIi5s8TcJ_FWm2EOqloIFnp1XgPLiRXYWQa28gWypUym7xPdmvZotNjE2EtRmzg8BMNktA/s1600-h/Figura+15.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129376781220854050" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_-cRpc7WEuwQcI9ny5XIuDTlTHz_2Rj4xMIe4jJAQIHR92UzxENQ1G5DPgGGe7hAcz82jQkIi5s8TcJ_FWm2EOqloIFnp1XgPLiRXYWQa28gWypUym7xPdmvZotNjE2EtRmzg8BMNktA/s400/Figura+15.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Figura 15 (Click para ampliar)</span></span><span class="Apple-style-span" style="color: silver;"><br />
</span></div><span class="Apple-style-span" style="color: silver;"><br />
Esta regla está fundada en que la sustitución indicada no altera las ecuaciones que se deducen a partir de las Leyes de Kirchhoff.<br />
Estas fuentes de sustitución son dependientes y ha de tenerse en cuenta que se comportan de forma distinta que las fuentes ideales. En particular se puede explicar esta regla a un circuito abierto y a un circuito en corto.<br />
Si la tensión entre dos terminales A y B de un circuito activo es Uo, no se altera en nada el estado del circuito al conectar esos terminales mediante una fuente de tensión e0 = Uo de la misma polaridad que la existente entre A y B (Fig. 16).<br />
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</span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_bfxh9Dok6Nr2BuqX-RoA9RCzFxGaW1_E2qGGMRbzHS_oFCGe6364j7KtZMb3H8sBRVa-ri1A_tRjB-_SWAshCFqUcSGSdkcYmb_tbXLvnYFEmZKmvRvYJ0Vm5xEBDYnyF-51WIoaVTg/s1600-h/Figura+16.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129376785515821362" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_bfxh9Dok6Nr2BuqX-RoA9RCzFxGaW1_E2qGGMRbzHS_oFCGe6364j7KtZMb3H8sBRVa-ri1A_tRjB-_SWAshCFqUcSGSdkcYmb_tbXLvnYFEmZKmvRvYJ0Vm5xEBDYnyF-51WIoaVTg/s400/Figura+16.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Figura 16 (Click para ampliar)</span></span><span class="Apple-style-span" style="color: silver;"><br />
</span></div><span class="Apple-style-span" style="color: silver;"><br />
Su configuración cambia aparentemente, pues aumenta en una unidad el número de mallas, pero ha de tenerse en cuenta que en esa malla no circula intensidad alguna, luego no aumenta el número de incógnitas.<br />
Análogamente, en un conductor de resistencia nula, por el que circula una intensidad i0, puede intercalarse una fuente ideal de intensidad igual a i0 sin que se altere el estado del circuito.<br />
Un condensador con carga eléctrica inicial o una bobina con flujo inicial admiten una representación sencilla haciendo uso de la regla de sustitución. En el caso de un condensador cargado inicialmente a una tensión Uo, la ecuación de definición es:<br />
<br />
</span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLRbB821INbWn016B93pnTKMu1sro6r2Y3Zjhyphenhyphenwk_y39nnbJAvjM8zINxYpuHKvTb8zRD8wKoqjY8zQxrj2XrrBQQ8F_-Ir75-ihwYq-wcdukiXn1kExcYAjZWCHPtD1pWp-S0ewUW5wk/s1600-h/Ecuacion+14.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129376995969218882" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLRbB821INbWn016B93pnTKMu1sro6r2Y3Zjhyphenhyphenwk_y39nnbJAvjM8zINxYpuHKvTb8zRD8wKoqjY8zQxrj2XrrBQQ8F_-Ir75-ihwYq-wcdukiXn1kExcYAjZWCHPtD1pWp-S0ewUW5wk/s400/Ecuacion+14.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Ecuacion 14</span></span><span class="Apple-style-span" style="color: silver;"><br />
</span></div><span class="Apple-style-span" style="color: silver;"><br />
que corresponde al circuito de la Fig. 17a.<br />
<br />
</span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidK2KVQBsjRz7upC9rF3uoRowwV_gSMmz-i1UAn9VzsfM4oDZf60Z-Gt3sb5zCq21FkD4_BERy6uw4lWe8N8LlI4tFqPacY84k2adp-VnJ-Pdf0JU6QRm4KZPHnj4DJQd1x05O5Re5Uf8/s1600-h/Figura+17.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129377000264186194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidK2KVQBsjRz7upC9rF3uoRowwV_gSMmz-i1UAn9VzsfM4oDZf60Z-Gt3sb5zCq21FkD4_BERy6uw4lWe8N8LlI4tFqPacY84k2adp-VnJ-Pdf0JU6QRm4KZPHnj4DJQd1x05O5Re5Uf8/s400/Figura+17.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Figura 17 (Click para ampliar)</span></span><span class="Apple-style-span" style="color: silver;"><br />
</span></div><span class="Apple-style-span" style="color: silver;"><br />
Es decir, se sustituye por una fuente de tensión en serie con un condensador inicialmente descargado.<br />
Para una bobina por la que circula inicialmente una intensidad I0, la ecuación de definición es:<br />
<br />
</span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6SrxAVs6iUXOfNCOhbi4UX5eFxYRCN8LyyQR8jW4CA2WZqI-tcFshHtzz1fDEDQxCINZ8GnmzHVDM3UxI860xfhWBTLKVgnm2OWLEoUtDmFx_dJuZB5hyphenhyphenqaP9FWqJF_f7sQdgaGU77E4/s1600-h/Ecuacion+15.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129377004559153506" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6SrxAVs6iUXOfNCOhbi4UX5eFxYRCN8LyyQR8jW4CA2WZqI-tcFshHtzz1fDEDQxCINZ8GnmzHVDM3UxI860xfhWBTLKVgnm2OWLEoUtDmFx_dJuZB5hyphenhyphenqaP9FWqJF_f7sQdgaGU77E4/s400/Ecuacion+15.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Ecuacion 15</span></span><span class="Apple-style-span" style="color: silver;"><br />
</span></div><span class="Apple-style-span" style="color: silver;"><br />
que corresponde al circuito de la Fig. 17b.<br />
La regla de sustitución es una herramienta muy útil en la demostración de teoremas. Por ejemplo el Teorema de Miller (en Electrónica):<br />
<br />
Si en un circuito como el de la Fig.18a se sustituye la impedancia Z por un circuito abierto y se une el nudo 1 a otro 0, este último tomado como referencia, mediante una impedancia Z/(1-k) y el 2 se une también al mismo nudo O por medio de otra impedancia de valor ZK/(k-1), en donde k es la relación U</span><sub><span class="Apple-style-span" style="color: silver;">2</span></sub><span class="Apple-style-span" style="color: silver;">/U</span><sub><span class="Apple-style-span" style="color: silver;">1</span></sub><span class="Apple-style-span" style="color: silver;"> entre las tensiones de los nudos 2 y 1 respecto a 0, no se altera la intensidad que circula entre los nudos 1 y 2. (Fig. 18b).<br />
<br />
</span><br />
<div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6fVBh8Ohjw_ZRNs_hnZJ7QRSFvHk_N6ZYn3Vo6Op7y8wYT1MYpM0n3-9LOgcUgg50xSLvaV1IdhcFPw9AHTDqhFAdXpcrNUi9JEfnOMf4eoAHdkK8ZMEs7-sGP1JRBmXkoux-N2Fy-yQ/s1600-h/Figura+18.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="background: inherit; text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5129377262257191282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6fVBh8Ohjw_ZRNs_hnZJ7QRSFvHk_N6ZYn3Vo6Op7y8wYT1MYpM0n3-9LOgcUgg50xSLvaV1IdhcFPw9AHTDqhFAdXpcrNUi9JEfnOMf4eoAHdkK8ZMEs7-sGP1JRBmXkoux-N2Fy-yQ/s400/Figura+18.jpg" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; cursor: pointer; display: block; margin-bottom: 10px; margin-left: auto; margin-right: auto; margin-top: 0px; text-align: center;" /></a><span style="font-size: 10px;"><span class="Apple-style-span" style="color: silver;">Figura 18 (Click para ampliar)</span></span></div>-- <br />
HENDERSON PARADA<br />
<div><br />
</div><div><a href="http://circuitos-de-electronica.blogspot.com/2007/11/regla-de-sustitucin-teorema-de-miller.html">http://circuitos-de-electronica.blogspot.com/2007/11/regla-de-sustitucin-teorema-de-miller.html</a></div>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-26860353764366958362010-02-10T19:30:00.002-04:302010-02-10T20:36:29.188-04:30La unidad de decibelios<div><span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"></span><br />
<span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><b>Decibelio</b> es la unidad relativa empleada en <a href="http://es.wikipedia.org/wiki/Ac%C3%BAstica" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Acústica">acústica</a> y <a href="http://es.wikipedia.org/wiki/Telecomunicaci%C3%B3n" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Telecomunicación">telecomunicaciones</a> para expresar la relación entre dos magnitudes, acústicas o eléctricas, o entre la magnitud que se estudia y una magnitud de referencia.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">El decibelio, cuyo símbolo es <i>dB</i>, es una unidad <a href="http://es.wikipedia.org/wiki/Logaritmo" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Logaritmo">logarítmica</a>. Es un submúltiplo del<b>belio</b>, de símbolo <i>B</i>, que es el logaritmo de la relación entre la magnitud de interés y la de referencia, pero no se utiliza por ser demasiado grande en la práctica, y por eso se utiliza el decibelio, la décima parte de un belio. El belio recibió este nombre en honor de<a href="http://es.wikipedia.org/wiki/Alexander_Graham_Bell" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Alexander Graham Bell">Alexander Graham Bell</a>.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Un belio equivale a 10 decibelios y representa un aumento de potencia de 10 veces sobre la magnitud de referencia. Cero belios es el valor de la magnitud de referencia. Así, dos belios representan un aumento de cien veces en la potencia, 3 belios equivalen a un aumento de mil veces y así sucesivamente.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">El oído humano no percibe igual las distintas frecuencias y alcanza el máximo de percepción en las medias, de ahí que para aproximar más la unidad a la realidad auditiva, se ponderen las unidades (para ello se utilizan las llamadas <a href="http://es.wikipedia.org/wiki/Curva_isof%C3%B3nica" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Curva isofónica">curvas isofónicas</a>).</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Por este motivo se definió el decibelio A (dBA), una unidad de nivel sonoro medido con un filtro previo que quita parte de las bajas y las muy altas frecuencias. De esta manera, después de la medición se filtra el sonido para conservar solamente las frecuencias más dañinas para el oído, razón por la cual la exposición medida en dBA es un buen indicador del riesgo auditivo.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Hay además otras unidades ponderadas, como dBC, dBD, adecuadas para medir la reacción del oído ante distintos niveles de sonoridad.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Como el decibelio es adimensional y relativo, para medir valores absolutos se necesita especificar a qué unidades está referida la medida:</div><ul style="line-height: 1.5em; list-style-image: url(http://bits.wikimedia.org/skins-1.5/monobook/bullet.gif); list-style-type: square; margin-bottom: 0px; margin-left: 1.5em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><b>dB<sub style="line-height: 1em;">SPL</sub></b>: Hace referencia al <a href="http://es.wikipedia.org/wiki/Nivel_de_presi%C3%B3n_sonora" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Nivel de presión sonora">nivel de presión sonora</a>. Es la medida, por ejemplo, usada para referirse a <a class="mw-redirect" href="http://es.wikipedia.org/wiki/Ganancia_(audio)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Ganancia (audio)">ganancia</a> o <a href="http://es.wikipedia.org/wiki/Atenuaci%C3%B3n" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Atenuación">atenuación</a> de <a href="http://es.wikipedia.org/wiki/Volumen_(sonido)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Volumen (sonido)">volumen</a>. Toma como unidad de referencia 20 <a class="mw-redirect" href="http://es.wikipedia.org/wiki/Pascal_(unidad_de_presi%C3%B3n)" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Pascal (unidad de presión)">micropascal</a>.</li>
<li style="margin-bottom: 0.1em;"><b>dBW</b>: La W indica que el decibelio hace referencia a <a href="http://es.wikipedia.org/wiki/Vatio" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Vatio">vatios</a>. Es decir, se toma como referencia 1 W (vatio). Así, a un vatio le corresponden 0 dBW.</li>
</ul><div class="thumb tright" style="border-bottom-color: white; border-bottom-style: solid; border-bottom-width: 0.8em; border-left-color: white; border-left-style: solid; border-left-width: 1.4em; border-right-color: white; border-right-style: solid; border-right-width: 0px; border-top-color: white; border-top-style: solid; border-top-width: 0.5em; clear: right; float: right; margin-bottom: 0.5em; width: auto;"><div class="thumbinner" style="background-color: #f9f9f9; border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; font-size: 12px; overflow-x: hidden; overflow-y: hidden; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: center; width: 202px;"><a class="image" href="http://es.wikipedia.org/wiki/Archivo:Relationship_between_dBu_and_dBm.png" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;"><img alt="" class="thumbimage" height="57" src="http://upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Relationship_between_dBu_and_dBm.png/200px-Relationship_between_dBu_and_dBm.png" style="border-bottom-color: rgb(204, 204, 204); border-bottom-style: solid; border-bottom-width: 1px; border-color: initial; border-left-color: rgb(204, 204, 204); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(204, 204, 204); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(204, 204, 204); border-top-style: solid; border-top-width: 1px; border-width: initial; vertical-align: middle;" width="200" /></a><br />
<div class="thumbcaption" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; font-size: 11px; line-height: 1.4em; padding-bottom: 3px !important; padding-left: 3px !important; padding-right: 3px !important; padding-top: 3px !important; text-align: left;"><div class="magnify" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; float: right;"><a class="internal" href="http://es.wikipedia.org/wiki/Archivo:Relationship_between_dBu_and_dBm.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; color: #002bb8; display: block; text-decoration: none;" title="Aumentar"><img alt="" height="11" src="http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png" style="background-attachment: initial !important; background-clip: initial !important; background-color: initial !important; background-image: none !important; background-origin: initial !important; background-position: initial initial !important; background-repeat: initial initial !important; border-bottom-style: none !important; border-color: initial !important; border-color: initial; border-left-style: none !important; border-right-style: none !important; border-top-style: none !important; border-width: initial !important; border-width: initial; display: block; vertical-align: middle;" width="15" /></a></div>Relación entre dBu y dBm</div></div></div><ul style="line-height: 1.5em; list-style-image: url(http://bits.wikimedia.org/skins-1.5/monobook/bullet.gif); list-style-type: square; margin-bottom: 0px; margin-left: 1.5em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><b>dBm</b>: Cuando el valor expresado en vatios es muy pequeño, se usa el milivatio (mW). Así, a un mW le corresponden 0 <a href="http://es.wikipedia.org/wiki/DBm" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="DBm">dBm</a>.</li>
<li style="margin-bottom: 0.1em;"><b>dBu</b>: El dBu expresa el nivel de señal en decibelios y referido a 0,7746 <a href="http://es.wikipedia.org/wiki/Voltio" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Voltio">voltios</a> <img alt="\left ( \sqrt { \frac{3}{5}} \right ) \,\!" class="tex" src="http://upload.wikimedia.org/math/3/3/3/333f0f04a594b2b5dccb0d6c3e40ce9b.png" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; vertical-align: middle;" />. 0,7746 V es la tensión que aplicada a una <a href="http://es.wikipedia.org/wiki/Impedancia" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Impedancia">impedancia</a> de 600 <a href="http://es.wikipedia.org/wiki/Ohmio" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Ohmio">Ω</a>, desarrolla una potencia de 1 mW. Se emplea la referencia de una impedancia de 600 Ω por razones históricas.<sup class="reference" id="cite_ref-1" style="font-style: normal; font-weight: normal; line-height: 1em;"><a href="http://es.wikipedia.org/wiki/Decibelio#cite_note-1" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;">2</a></sup></li>
</ul><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">En algunos casos (especialmente en telecomunicaciones), al medir niveles relativos en decibelios, se da un nombre específico a la unidad, dependiendo del tipo de medida.</div><ul style="line-height: 1.5em; list-style-image: url(http://bits.wikimedia.org/skins-1.5/monobook/bullet.gif); list-style-type: square; margin-bottom: 0px; margin-left: 1.5em; margin-right: 0px; margin-top: 0.3em; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><li style="margin-bottom: 0.1em;"><b>dBc</b>: Nivel relativo entre una señal <a class="mw-redirect" href="http://es.wikipedia.org/wiki/Portadora" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Portadora">portadora</a> (<i>carrier</i>) y alguno de sus <a href="http://es.wikipedia.org/wiki/Arm%C3%B3nico" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Armónico">armónicos</a>.</li>
<li style="margin-bottom: 0.1em;"><b>dBi</b>: Decibel medido con respecto a una antena isotropica</li>
<li style="margin-bottom: 0.1em;"><b>dBd</b>: Decibel medido con respecto a una antena dipolo</li>
<li style="margin-bottom: 0.1em;"> <br />
</li>
</ul></span></div>-- <span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;">El <b>decibelio</b> es quizá la unidad más utilizada en el campo de las Telecomunicaciones por la simplificación que su naturaleza logarítmica posibilita a la hora de efectuar cálculos con valores de potencia de la <a href="http://es.wikipedia.org/wiki/Se%C3%B1al" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Señal">señal</a> muy pequeños.</span><br />
<span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Como relación de <a href="http://es.wikipedia.org/wiki/Potencia" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Potencia">potencias</a> que es, la cifra en decibelios no indica nunca el valor absoluto de las dos potencias comparadas, sino la relación entre ellas. A diferencia de lo que ocurre en el sonido, donde siempre se refiere al mismo nivel de referencia, en telecomunicación, el nivel de referencia es cambiante.<br />
Esto permite, por ejemplo, expresar en decibelios la ganancia de un <a href="http://es.wikipedia.org/wiki/Amplificador" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Amplificador">amplificador</a> o la pérdida de un <a href="http://es.wikipedia.org/wiki/Atenuador" style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: none; background-origin: initial; background-position: initial initial; background-repeat: initial initial; color: #002bb8; text-decoration: none;" title="Atenuador">atenuador</a> sin necesidad de referirse a la potencia de entrada que, en cada momento, se les esté aplicando.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">La pérdida o ganancia de un dispositivo, expresada en decibelios viene dada por la fórmula:</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><br />
<img alt=" {dB}= 10\times log \frac{P_S}{P_E}" class="tex" src="http://upload.wikimedia.org/math/9/7/f/97f1038f86bb4a1dc73b799a3797ed4c.png" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><br />
en donde <b>P<sub style="line-height: 1em;">E</sub></b> es la potencia de la señal en la entrada del dispositivo, y <b>P<sub style="line-height: 1em;">S</sub></b> la potencia a la salida del mismo.<br />
Si hay ganancia de señal (amplificación) la cifra en decibelios será negativa, mientras que si hay pérdida (atenuación) será positiva.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">Para sumar ruidos, o señales en general, es muy importante considerar que no es correcto sumar directamente valores de las fuentes de ruido expresados en decibelios. Así, dos fuentes de ruido de 21 dB no dan 42 dB sino 24 dB.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;">En este caso se emplea la fórmula:</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><br />
<img alt="10\cdot \log_{10}(10^{\frac{X_1}{10}}+10^{\frac{X_2}{10}}+ ... ) =dB totales" class="tex" src="http://upload.wikimedia.org/math/3/8/f/38ffff40b838524a68124c4091202572.png" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><br />
Donde <span class="texhtml" style="font-family: serif;"><i>X</i><sub style="line-height: 1em;"><i>n</i></sub></span> son los valores de ruido o señal, expresados en decibelios, a sumar.</div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><br />
<img alt="10 \cdot \log_{10} \left( antilog\left( \frac{X_1}{10} \right )+ antilog \left( \frac{X_2}{10} \right )+ ... \right) = dB totales" class="tex" src="http://upload.wikimedia.org/math/a/4/5/a455c2f135a6d7e457ececfb795f694a.png" style="border-bottom-style: none; border-color: initial; border-left-style: none; border-right-style: none; border-top-style: none; border-width: initial; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; vertical-align: middle;" /></div><div style="line-height: 1.5em; margin-bottom: 0.5em; margin-left: 0px; margin-right: 0px; margin-top: 0.4em;"><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: Verdana, Arial, sans-serif; font-size: 12px; line-height: 15px;"></span></div><h4 style="color: #224455; font-weight: bold; text-align: left;">Belios y decibelios</h4>Un bueno tocadiscos (de alta fidelidad) o un buen magnetófono tienen una potencia de salida de sonido constante, desde 20 c/s a 15.000 c/s, esto es, que una representación gráfica de la potencia de salida, sobre todo el intervalo de frecuencias audibles, es casi una línea recta horizontal, desde el extremo de baja frecuencia (20 c/s) al de frecuencia alta (15.000 c/s). Esto quiere decir que el instrumento amplifica todos los sonidos en la proporción correcta. Las notas altas no son aumentadas a expensas de las bajas. La salida de los amplificadores, en el eje vertical de la representación gráfica, se mide en una unidad algo compleja, llamada decibelio (decibel), y, a veces, en unidades mayores: belios (beles) -10 decibeles (dB) = 1 belio. Es difícil comprender estas unidades por dos razones. Primero, porque no son unidades como el gramo o el centímetro, que tienen un valor definido y fijo. Son medidas de una potencia de salida comparada con otro nivel de potencia, que se usa como referencia. Hay mucha confusión acerca de los niveles de referencia, y por esta causa, los patrones no se aceptan de modo general. La segunda dificultad es que el decibelio son unidades logarítmicas. Cuando la potencia de salida es diez veces mayor que la de referencia se expresa con un belio (o 10 decibelios); 1 es el logaritmo de 10. Sin embargo, si la potencia de salida es cien veces mayor, son solamente 2 belios o 20 decibelios; el logaritmo de 100 es 2. Del mismo modo, una amplificación de potencia de 3 belios (30 decibelios) significa que la potencia aumento mil veces; el logaritmo de 1.000 es 3 (el número de ceros de la primera cifra). El oído humano puede percibir notas dentro de un amplio intervalo de intensidad de sonido. Un nivel de referencia que se toma casi siempre para establecer la escala de decibelios es la potencia sonora mas baja que puede detectar el oído. El sonido mas fuerte que se puede detectar es de unos 13 belios. Quizá parezca que no es mucha potencia, pero tal cifra quiere decir que la potencia asociada con el sonido es superior en 10.000.000.000.000 veces a la del sonido mas bajo. Se pueden usar decibelios para comparar dos corrientes, dos tensiones, dos potencias, dos intensidades de sonido o dos presiones de sonido. De hecho, son una medida de la ganancia en cualquier sistema físico. La ganancia de un amplificador se expresa, a veces, en decibelios. La mayoría de los aparatos "Hi-Fi" (alta fidelidad) tiene varios amplificadores. La ganancia total es el resultado de multiplicar las ganancias individuales. Cuando éstas se expresan en forma logarítmica es muy fácil averiguar la ganancia total, sumando los logaritmos. Sumar logaritmos es equivalente a multiplicar los números ordinarios. Normalmente, a la parte lisa de la curva de respuesta de un tocadiscos se le da el valor de cero decibelio. Cualquier pico o hendidura en la curva se convierte en ganancia de decibelios o en pérdida de decibelios.<br />
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</div>HENDERSON PARADA TOVAR<br />
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</div><div><span class="Apple-style-span" style="font-family: sans-serif; font-size: 13px; line-height: 19px;"><a href="http://es.wikipedia.org/wiki/Decibelio#Unidades_basadas_en_el_decibelio">http://es.wikipedia.org/wiki/Decibelio#Unidades_basadas_en_el_decibelio</a></span></div><div><a href="http://tecnologia.idoneos.com/index.php/La_medicion_del_sonido:_belios_y_decibelios">http://tecnologia.idoneos.com/index.php/La_medicion_del_sonido:_belios_y_decibelios</a></div>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-21612472903836797592010-02-08T13:25:00.001-04:302010-02-09T10:56:44.313-04:30Square wave testing of amplifiers, Miller's theorem.<p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><b><i style="mso-bidi-font-style: normal"><span lang=EN-US style="FONT-SIZE: 16pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-bidi-font-size: 14.0pt; mso-bidi-font-family: 'Times New Roman'; mso-color-alt: windowtext; mso-ansi-language: EN-US"><span style="COLOR: windowtext"><font color=#bfbfbf>Square Wave Testing for Frequency Response of Amplifiers<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p></o:p></FONT></SPAN></SPAN></I></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Square waves are rich in odd numbered harmonics and have a very simple shape that makes it easy to observe frequency response limitations in amplifiers. This note discusses how to use square waves to measure the approximate low and high cutoff frequencies of an amplifier. Amplifiers generally have AC coupled sections that limit the low frequency response and have shunt capacitances either parasitic or intentional that limit high frequency response. This not is divided into two sections </SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US"><font face=Calibri>–</FONT></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">one section discusses the measurement of low frequency response and the other discusses the measurement of high frequency response. In each case we are looking for the low and high frequencies where the power transfer has dropped to one-half the mid-band value.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>These are known as the -3dB frequencies or cutoff frequencies.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN-US"><font color=#bfbfbf>Low Frequency Response<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>An AC coupled stage acts as a derivative network for frequencies below the series RC cutoff frequency. This will significantly modify a square wave passing through it.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>A useful relation that gives the approximate low cutoff frequency based on the slope of the AC coupled square wave is as follows<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN>Fcl = <span style="mso-spacerun: yes"> </SPAN>((Vmax_pp </SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US"><font face=Calibri>–</FONT></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Vmin_pp) * F) / (Vmax_pp * </SPAN><font face=Calibri><span style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman">π</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US">)<o:p></o:p></SPAN></FONT></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Where:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Vmax_pp is the overall peak-peak value of the waveform<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Vmin_pp is the peak-peak value of the low-points of the decaying slopes<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>F is the frequency in Hz of the square wave<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman"><font face=Calibri>Π </FONT></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">is 3.14159<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Equation 1 is reasonably accurate providing the decaying slopes are fairly linear.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN-US style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN-US"><font color=#bfbfbf>High Frequency Response<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>An RC roll-off acts as an integrator for signals above the RC cutoff frequency and this reduces the amplitude of signals roughly be the frequency is above the cutoff frequency.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">A useful relation that gives the approximate high frequency bandwidth is derived from the time it takes the square wave to go from one state to the other. Since it is usually difficult to pick the starting point and a fully settled point, an alternative is to measure the time, T</SPAN><span lang=EN-US style="FONT-SIZE: 8pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">10-90</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">, required to move from the ten percent point to the 90 percent point.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN>For a first order section the high frequency bandwidth, Fc, is:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN>Fc = 0.35 / T</SPAN><span lang=EN-US style="FONT-SIZE: 8pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">10-90 <span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN><span style="mso-spacerun: yes"> </SPAN></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Eq. 2<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Equation 2 also gives good results when there are multiple low-pass sections. For<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">Equation 2 to be valid the square wave must fully settle as in Figures 8 and 9. As an example the 10 to 90 percent rise time in Figure 8 is approximately 113 </SPAN><font face=Calibri><span style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman">μ</SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US">s</SPAN></FONT><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">which infers a bandwidth of 3.1 kHz </SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US"><font face=Calibri>–</FONT></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">fairly close to the actual bandwidth of 3 kHz.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>In application using an oscilloscope you adjust the frequency of the square wave to be low enough so that complete settling exists. The exact frequency does not matter. Then you set the amplitude on the scope display to make it easy to pick out the 10 and 90 percent time points. A good choice is five major vertical divisions peak-peak. Once the amplitudes are set then you should expand the horizontal display such that the rise portion is generally centered on the screen and occupies most of the horizontal width.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>Then visually note (probably using a cursor on a digital scope) the point at which the wave has risen one-half division. Visually note again (probably using a second cursor on a digital scope) the point at which the waveform passes though the upper last half division. Then measures the time between these points (if the cursors are in relative time mode that will be easy) and apply Equation 2.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">It is left as an exercise for the student to derive Equation 2. It is not hard </SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: TimesNewRoman; mso-bidi-font-family: TimesNewRoman; mso-ansi-language: EN-US"><font face=Calibri>–</FONT></SPAN><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US">just derive an exponential equation for the 10 and 90 percent points and relate the time difference to the cutoff frequency of the exponential which is one divided by two pi times the exponential time constant.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-outline-level: 2"><b><i style="mso-bidi-font-style: normal"><span lang=EN-US style="FONT-SIZE: 14pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>Square Wave Testing.<o:p></o:p></FONT></SPAN></I></B></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>It is perhaps unfortunate that the most common test for stability is to look for 'ringing' on a square-wave test signal. It is instructive to look at some examples, here using a 2kHz square wave input.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span style="mso-fareast-language: ES-VE; mso-no-proof: yes"><?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" /><v:shapetype id=_x0000_t75 stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600"><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path o:connecttype="rect" gradientshapeok="t" o:extrusionok="f"></v:path><o:lock aspectratio="t" v:ext="edit"></o:lock></v:shapetype><font color=#bfbfbf> </FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>The first looks like sustained low level oscillation around 30kHz, while the second looks like damped oscillation at the same frequency. Actually the first diagram has nothing at all added to the square wave, the only thing done was to remove everything above the 15th harmonic. Everything up to and including 30kHz is being reproduced with no distortion, no phase error and flat frequency response. (If possible see 'A check on Fourier' by M.G.Scroggie, Wireless World, Nov 1977. p79-82. His Fig.5 is a better drawn version showing the harmonics and how they add.) The lack of higher frequency components however gives the impression of a serious problem, when in fact the audio frequency reproduction is perfect, and there is nothing at all added or removed in this range. The symmetrical variation of the 'oscillation' amplitude gives a clue to the origin of the effect, but practical low pass filters give a less sharp cut off of high harmonics together with frequency dependant phase shift which will give a different appearance. The suggestion that 'ringing' needs to be minimised is not entirely convincing when even an ideal low-pass filter gives the above result. Using an audio signal with no frequency components above 30kHz instead of the square wave there would be no effect at all from this filter.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>The second diagram can also be the result of low-pass filtering, and something similar is often produced by the interaction of output inductors with capacitive loads, which is not related in any direct way to stability. Checking the signal ahead of the inductor may reveal a smooth signal without the 'ringing' effect, though some amplifiers have an output impedance with a small internal inductive component which will add some small effect. The square-wave response shown in the MJR-6 test results shows low level 'ringing' which is estimated at 120kHz. This is close to the expected resonance frequency of the 0.4uH output inductor with the 4uF load capacitance used in that test. Increasing loop gain to the point where the amplifier becomes unstable caused oscillation around 6MHz, as expected from the feedback loop unity gain frequency. This demonstrates that output 'ringing' is generally not related to instability, which can occur in an entirely different frequency range, and unless the input signal includes components close to the LC resonance frequency, or the inductance used is too high, there will be little effect. Leaving out the output inductor to eliminate 'ringing' caused by this LC resonance may seriously reduce the phase margin at higher frequencies with some capacitive loads, dangerously increasing the risk of instability.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>A square wave test to investigate stability into capacitive loads is therefore of limited usefulness, and may be seriously misleading. My experience is that amplifiers sometimes have a stable state and an unstable state, and triggering them into instability may need a precise choice of load and input signal, in one case driving the amplifier heavily into clipping and then removing the input signal caused a dramatic latch-up and oscillation effect. Failure to oscillate with just any square-wave input and the usual 2uF test load may be necessary, but is no guarantee of unconditional stability. I also use high level sinewave signals at various frequencies, and look for signs of instability close to clipping as the signal level is adjusted to give different levels of clipping. Going into or out of clipping the loop gain is changing, and so the feedback loop unity gain frequency is in effect shifted over a wide range, revealing potential stability problems over a similar range. To limit dissipation it is convenient to use a toneburst signal for these clipping tests.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>In two photos are oscilloscope traces showing examples of clipping behaviour</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>The first of these is just a single notch when coming out of clipping, and this is typical of latch-up effects rather than instability. In this case it was caused by a bad choice of frequency compensation circuit such that the compensation capacitor charged up during clipping and had to discharge before normal linear operation could return. A change to the compensation arrangement was needed to cure this.<br />
Stability problems generally have a different appearance of the type shown in the second photo. Here a short burst of oscillation occurs when coming out of clipping, but in this case the effect continues long after this as seen from a slight ripple on the trace. A change in the value of the compensation capacitor was needed to remove this effect. The positive and negative clipping look different, which is not uncommon, here the positive clipping appears to include a latch-up effect in addition to the stability problem. <br />
Had I relied only on observations of square-wave ringing with a 2uF load below clipping I would have said there were no stability problems to worry about, and stopped there without doing the necessary modifications.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>It is known that the choice of test signal rise-time can often have a great effect on observed 'ringing', and it is possible to claim 'excellent transient response' just by careful choice of the rise-time of the test signal. This was mentioned in one of the Douglas Self articles, "The Audio Power Interface", Electronics World Sept.1997 p717-722.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: ES-VE"><font color=#bfbfbf>The low-pass filter used at the input of my own amplifiers helps give a smooth square wave output with little ringing, but it was not included for this purpose. Anyone who still wants to reduce ringing further in the mosfet amplifiers could try reducing the damping resistor in parallel with the inductor, maybe to one ohm.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> <img style="WIDTH: 188px; HEIGHT: 499px" height=537 src="http://www.joeltunnah.com/images/6L6GCamp_sqwaves_1uF.JPG" width=234></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span style="FONT-SIZE: 16pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-bidi-font-size: 10.0pt; mso-color-alt: windowtext"><span style="COLOR: windowtext"><font color=#bfbfbf>Miller's theorem<o:p></o:p></FONT></SPAN></SPAN></I></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: CMTI12; mso-bidi-font-family: CMTI12; mso-ansi-language: EN-US"><o:p><font face=Calibri color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><i style="mso-bidi-font-style: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: CMTI12; mso-bidi-font-family: CMTI12; mso-ansi-language: EN-US"><font face=Calibri><font color=#bfbfbf>"Thus the apparent input capacity can become a number of times greater than the actual capacities between the tube electrodes . . ."<o:p></o:p></FONT></FONT></SPAN></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><i style="mso-bidi-font-style: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: CMTI12; mso-bidi-font-family: CMTI12; mso-ansi-language: EN-US"><o:p><font face=Calibri color=#bfbfbf> </FONT></o:p></SPAN></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In electronics, the Miller effect accounts for an increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of capacitance between the input and output terminals. Although Miller effect normally refers to capacitance, any impedance connected between the input and another node exhibiting high gain can modify the amplifier input impedance via the Miller effect.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>This increase in input capacitance is given by<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>C<sub>M</SUB>= C (1 - A<sub>v</SUB>)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Where Av is the gain of the amplifier and C is the feedback capacitance.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The Miller effect is a special case of Miller's theorem.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><font color=#bfbfbf><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p> </o:p></SPAN><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN">Notes<o:p></o:p></SPAN></B></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>As most amplifiers are inverting amplifiers (i.e. Av < 0) the effective capacitance at the input is larger. For non-inverting amplifiers, the Miller effect results in a negative capacitor at the input of the amplifier (compare Negative impedance converter).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Naturally, this increased capacitance can wreak havoc with high frequency response. For example, the tiny junction and stray capacitances in a Darlington transistor drastically reduce the high frequency response through the Miller effect and the Darlington's high gain.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The Miller effect applies to any impedance, not just a capacitance. A pure resistance or pure inductance will be divided by 1 − Av. In addition if the amplifier is non-inverting then a negative resistance or inductance can be created using the Miller effect.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>It is also important to note that the Miller capacitance is the capacitance seen looking into the input. If looking for all of the RC time constants (poles) it is important to include as well the capacitance seen by the output. The capacitance on the output is often neglected since it sees C(1 − 1 / Av) and amplifier outputs are typically low impedance. However if the amplifier has a high impedance output, such as if a gain stage is also the output stage, then this RC can have a significant impact on the performance of the amplifier. This is when pole splitting techniques are used.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The impact of the Miller effect is often reduced by using a cascode or cascade amplifier rather than a common emitter. For feedback amplifiers the Miller effect can actually be very beneficial since stabilizing the amplifier may require a capacitor too large to practically include in the circuit, typically a concern for an integrated circuit where capacitors consume significant area.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><o:p><font color=#bfbfbf></FONT></o:p></SPAN></B></P><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Impact on frequency response<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><font color=#bfbfbf><span style="mso-fareast-language: ES-VE; mso-no-proof: yes"></SPAN><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf><img height=235 src="http://upload.wikimedia.org/wikipedia/commons/c/cf/Miller_before_transform.PNG" width=391>Figure 2<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><font color=#bfbfbf><span style="mso-fareast-language: ES-VE; mso-no-proof: yes"></SPAN><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: center; mso-layout-grid-align: none" align=center><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf><img style="WIDTH: 413px; HEIGHT: 214px" height=197 src="http://upload.wikimedia.org/wikipedia/commons/f/f1/Miller_after_transform.PNG" width=357>Figure 3<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Figure 2: Operational amplifier with feedback capacitor CC. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Figure 3: Circuit of Figure 2 transformed using Miller's theorem, introducing the Miller capacitance on the input side of the circuit.Figure 2 shows an example of Figure 1 where the impedance coupling the input to the output is the coupling capacitor CC. A Thévenin voltage source VA drives the circuit with Thévenin resistance RA. At the output a parallel RC-circuit serves as load. (The load is irrelevant to this discussion: it just provides a path for the current to leave the circuit.) In Figure 2, the coupling capacitor delivers a current jωCC( vi - vo ) to the output circuit.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p></o:p></SPAN><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Figure 3 shows a circuit electrically identical to Figure 2 using Miller's theorem. The coupling capacitor is replaced on the input side of the circuit by the Miller capacitance CM, which draws the same current from the driver as the coupling capacitor in Figure 2. Therefore, the driver sees exactly the same loading in both circuits. On the output side, a dependent current source in Figure 3 delivers the same current to the output as does the coupling capacitor in Figure 2. That is, the R-C-load sees the same current in Figure 3 that it does in Figure 2.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In order that the Miller capacitance draw the same current in Figure 3 as the coupling capacitor in Figure 2, the Miller transformation is used to relate CM to CC. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><font color=#bfbfbf>This result is the same as <i>C<sub>M</SUB></I> of the <i>Derivation Section</I>.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p></o:p></SPAN><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>It is important to notice that the effect of CM upon the amplifier bandwidth is greatly reduced for low impedance drivers (CM RA is small if RA is small). Consequently, one way to minimize the Miller effect upon bandwidth is to use a low-impedance driver, for example, by interposing a voltage follower stage between the driver and the amplifier, which reduces the apparent driver impedance seen by the amplifier.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>The present example with <i>A<sub>v</SUB></I> frequency independent shows the implications of the Miller effect, and therefore of <i>C<sub>C</SUB></I>, upon the frequency response of this circuit, and is typical of the impact of the Miller effect (see, for example, </FONT></SPAN><a href="http://www.answers.com/topic/common-source" target=_top><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>common source</FONT></SPAN></A><font size=3><font face="Times New Roman"><font color=#bfbfbf><span lang=EN-US style="mso-ansi-language: EN-US">). If <i>C<sub>C</SUB></I> = 0 F, the output voltage of the circuit is simply <i>A<sub>v</SUB> v<sub>A</SUB></I>, independent of frequency. However, when <i>C<sub>C</SUB></I> is not zero, Figure 3 shows the large Miller capacitance appears at the input of the circuit. </SPAN>The voltage output of the circuit now becomes</FONT></FONT></FONT></P><p style="TEXT-ALIGN: justify"><font face="Times New Roman" color=#bfbfbf size=3></FONT> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt 36pt; TEXT-ALIGN: center" align=center><span style="mso-fareast-language: ES-VE; mso-no-proof: yes"><font color=#bfbfbf></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><font size=3><font face="Times New Roman"><font color=#bfbfbf><span lang=EN-US style="mso-ansi-language: EN-US">and rolls off with frequency once frequency is high enough that </SPAN>ω<span style="mso-ansi-language: EN-US"> <i><span lang=EN-US>C<sub>M</SUB>R<sub>A</SUB></SPAN></I><span lang=EN-US> ≥ 1. It is a </SPAN></SPAN></FONT></FONT></FONT><a href="http://www.answers.com/topic/low-pass-filter" target=_top><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>low-pass filter</FONT></SPAN></A><font size=3><font face="Times New Roman"><font color=#bfbfbf><span lang=EN-US style="mso-ansi-language: EN-US">. In analog amplifiers this curtailment of frequency response is a major implication of the Miller effect. In this example, the frequency </SPAN>ω<i><sub><span lang=EN-US style="mso-ansi-language: EN-US">3dB</SPAN></SUB></I><span lang=EN-US style="mso-ansi-language: EN-US"> such that </SPAN>ω<i><sub><span lang=EN-US style="mso-ansi-language: EN-US">3dB</SPAN></SUB></I><span lang=EN-US style="mso-ansi-language: EN-US"> <i>C<sub>M</SUB>R<sub>A</SUB></I> = 1 marks the end of the low-frequency response region and sets the </SPAN></FONT></FONT></FONT><a href="http://www.answers.com/topic/bandwidth-signal-processing" target=_top><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>bandwidth</FONT></SPAN></A><span lang=EN-US style="mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3> or </FONT></SPAN><a href="http://www.answers.com/topic/cutoff-frequency" target=_top><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>cutoff frequency</FONT></SPAN></A><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf> of the amplifier.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p style="TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>It is important to notice that the effect of <i>C</I><sub>M</SUB> upon the amplifier bandwidth is greatly reduced for low impedance drivers (<i>C</I><sub>M</SUB> <i>R</I><sub>A</SUB> is small if <i>R</I><sub>A</SUB> is small). Consequently, one way to minimize the Miller effect upon bandwidth is to use a low-impedance driver, for example, by interposing a </FONT></SPAN><a href="http://www.answers.com/topic/buffer-amplifier" target=_top><span lang=EN-US style="COLOR: windowtext; mso-ansi-language: EN-US"><font face="Times New Roman" color=#bfbfbf size=3>voltage follower</FONT></SPAN></A><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf> stage between the driver and the amplifier, which reduces the apparent driver impedance seen by the amplifier.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The output voltage of this simple circuit is always Av vi. However, real amplifiers have output resistance. If the amplifier output resistance is included in the analysis, the output voltage exhibits a more complex frequency response and the impact of the frequency-dependent current source on the output side must be taken into account.[3] Ordinarily these effects show up only at frequencies much higher than the roll-off due to the Miller capacitance, so the analysis presented here is adequate to determine the useful frequency range of an amplifier dominated by the Miller effect.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Miller approximation<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>This example also assumes Av is frequency independent, but more generally there is frequency dependence of the amplifier contained implicitly in Av. Such frequency dependence of Av also makes the Miller capacitance frequency dependent, so interpretation of CM as a capacitance becomes a stretch of imagination. However, ordinarily any frequency dependence of Av arises only at frequencies much higher than the roll-off with frequency caused by the Miller effect, so for frequencies up to the Miller-effect roll-off of the gain, Av is accurately approximated by its low-frequency value. Determination of CM using Av at low frequencies is the so-called Miller approximation.[2] With the Miller approximation, CM becomes frequency independent, and its interpretation as a capacitance at low frequencies is secure.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><span lang=EN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p><font color=#bfbfbf>Lenny. Z Perez</FONT></o:p></SPAN></B></P><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN-US style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><a href="http://www.angelfire.com/ab3/mjramp/sw.html"><font color=#bfbfbf size=2>http://www.angelfire.com/ab3/mjramp/sw.html</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN-US style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><a href="http://www.kennethkuhn.com/students/ee351/text/square_wave_testing.pdf"><font color=#bfbfbf size=2>http://www.kennethkuhn.com/students/ee351/text/square_wave_testing.pdf</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><span lang=EN-US style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN-US"><a href="http://en.wikipedia.org/wiki/Miller_effect"><font color=#bfbfbf size=2>http://en.wikipedia.org/wiki/Miller_effect</FONT></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; TEXT-ALIGN: justify; mso-layout-grid-align: none"></o:p></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: normal; mso-layout-grid-align: none"><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN-US style="FONT-SIZE: 16pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-bidi-font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman'; mso-color-alt: windowtext; mso-ansi-language: EN-US"><o:p> </o:p></SPAN></I></B></P><hr />Invite your mail contacts to join your friends list with Windows Live Spaces. It's easy! <a href='http://spaces.live.com/spacesapi.aspx?wx_action=create&wx_url=/friends.aspx&mkt=en-us' target='_new'>Try it!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com1tag:blogger.com,1999:blog-8049570815625896303.post-24930723741872838742010-02-06T14:32:00.000-04:302010-02-09T10:56:16.090-04:30Significance of Octaves and Decades. The decibel unit<b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN style="FONT-SIZE: 12pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-bidi-font-size: 11.0pt; mso-color-alt: windowtext; mso-ansi-language: EN"><span style="COLOR: windowtext"><font face="Times New Roman"><?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><span style="COLOR: windowtext"></SPAN></SPAN></I></B><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-ansi-language: EN"><o:p> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: #BFBFBF; mso-ansi-language: EN; mso-themecolor: background1; mso-themeshade: 191"><font color=#bfbfbf>Decade (log scale)</FONT></SPAN></I></B><span lang=EN-GB style="COLOR: #bfbfbf; mso-ansi-language: EN-GB; mso-themecolor: background1; mso-themeshade: 191"><o:p></o:p></SPAN></P></o:p></SPAN></FONT></o:p></FONT></SPAN></SPAN></I></B> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3>One <b>decade</B> is a factor of 10 difference between two numbers (an </FONT><a title="Order of magnitude" href="http://en.wikipedia.org/wiki/Order_of_magnitude"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>order of magnitude</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3> difference) measured on a </FONT><a title="Logarithmic scale" href="http://en.wikipedia.org/wiki/Logarithmic_scale"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>logarithmic scale</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3>. It is especially useful when referring to frequencies and when describing </FONT><a title="Frequency response" href="http://en.wikipedia.org/wiki/Frequency_response"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>frequency response</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3> of </FONT><a title=Electronics href="http://en.wikipedia.org/wiki/Electronics"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>electronic systems</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3>, such as </FONT><a title="Audio amplifier" href="http://en.wikipedia.org/wiki/Audio_amplifier"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>audio amplifiers</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3> and </FONT><a title="Electronic filter" href="http://en.wikipedia.org/wiki/Electronic_filter"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>filters</FONT></SPAN></A><o:p></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=mw-headline><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><font face="Times New Roman"><font color=#bfbfbf><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; mso-ansi-language: EN">Calculations</SPAN><o:p></o:p></FONT></FONT></SPAN></I></B></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>The factor-of-ten in a decade can be in either direction: so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, so 3.14 rad/s is one decade down from 31.4 rad/s.<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>To determine the number of decades between two frequencies, use the logarithm of the ratio of the two values:<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-How many decades is it from 15 rad/s to 150,000 rad/s? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>log10 (150000 / 15) = 4 decades <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-How many decades is it from 3.2 GHz to 4.7 MHz? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>Decades <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-How many decades is one octave? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>One octave is a factor of 2, so log10(2) = 0.301 decades per octave <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>To find out what frequency is a certain number of decades from the original frequency, multiply by appropriate powers of 10:<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-What is 3 decades down from 220 Hz? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>220x10<sup>-3</SUP> =0.22Hz <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-What is 1.5 decades up from 10? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>10x10<sup>1.5 </SUP>=316.23<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN>To find out the size of a step for a certain number of frequencies per decade, raise 10 to the power of the inverse of the number of steps:<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>-What is the step size for 30 steps per decade? <o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-US style="mso-ansi-language: EN-US"><font size=3><font face="Times New Roman"><font color=#bfbfbf>10<sup> 1/ 30 </SUP>= 1.079775 - or each step is 7.9775% larger than the last<o:p></o:p></FONT></FONT></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=editsection><span lang=EN style="mso-ansi-language: EN"><o:p><font face="Times New Roman" color=#bfbfbf size=3> </FONT></o:p></SPAN></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=editsection><span lang=EN style="mso-ansi-language: EN"><o:p><font face="Times New Roman" color=#bfbfbf size=3> </FONT></o:p></SPAN></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span class=mw-headline><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><font face="Times New Roman"><font color=#bfbfbf><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; mso-ansi-language: EN">Graphical representation and analysis</SPAN><o:p></o:p></FONT></FONT></SPAN></I></B></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3>Decades on a </FONT><a title="Logarithmic scale" href="http://en.wikipedia.org/wiki/Logarithmic_scale"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>logarithmic scale</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3>, rather than unit steps (steps of 1) or other </FONT><a title=Linear href="http://en.wikipedia.org/wiki/Linear"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>linear</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3> scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in </FONT><a title="Bode plot" href="http://en.wikipedia.org/wiki/Bode_plot"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>Bode plots</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3>, since depicting large frequency ranges on a linear scale is often not practical. For example, an </FONT><a title="Audio amplifier" href="http://en.wikipedia.org/wiki/Audio_amplifier"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>audio amplifier</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3> will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (10<sup>0</SUP>) and go up to perhaps 100 kHz (10<sup>5</SUP>), to comfortably include the full audio band in a standard-sized graph paper, as shown below. Where as in the same distance on a linear scale, with 10 as the major step-size, you might only get from 0 to 50.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3></FONT></SPAN> </P><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3><o:p> <a class=image title="1,10,100,1k,10k,100k using decades vs. 0,10,20,30,40,50 using linear scale" href="http://en.wikipedia.org/wiki/File:Decade_vs_Linear.svg"><span style="BORDER-TOP-WIDTH: 2px; DISPLAY: inline-block; BORDER-LEFT-WIDTH: 2px; FONT-SIZE: 0px; BORDER-LEFT-COLOR: #800080; BACKGROUND-IMAGE: none; BORDER-BOTTOM-WIDTH: 2px; BORDER-BOTTOM-COLOR: #800080; VERTICAL-ALIGN: middle; CURSOR: hand; BORDER-TOP-COLOR: #800080; BORDER-RIGHT-WIDTH: 2px; BORDER-RIGHT-COLOR: #800080"><span style="DISPLAY: inline-block; FILTER: progid:DXImageTransform.Microsoft.AlphaImageLoader(src='http://upload.wikimedia.org/wikipedia/en/thumb/8/86/Decade_vs_Linear.svg/400px-Decade_vs_Linear.svg.png'); WIDTH: 1px; HEIGHT: 1px"></SPAN></SPAN></A><br />
</o:p></FONT></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3>Electronic frequency responses are often described in terms of "per decade". The example Bode plot shows a slope of -20 </FONT><a title=Decibel href="http://en.wikipedia.org/wiki/Decibel"><span style="COLOR: windowtext"><font face="Times New Roman" color=#bfbfbf size=3>dB</FONT></SPAN></A><font face="Times New Roman" color=#bfbfbf size=3>/decade in the stopband, which means that for every factor-of-ten increase in frequency (going from 10 rad/s to 100 rad/s in the figure), the gain decreases by 20 dB.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" color=#bfbfbf size=3><o:p><img style="WIDTH: 441px; HEIGHT: 271px" height=537 src="http://upload.wikimedia.org/wikipedia/commons/6/66/Butterworth_filter_bode_plot.png" width=767></o:p></FONT></SPAN></P><span lang=EN style="mso-ansi-language: EN"><font face="Times New Roman" size=3><o:p> <div class=Section1><span> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss; mso-color-alt: windowtext; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt"><strong><span><em><font color=#bfbfbf><span style="COLOR: windowtext"></SPAN><o:p><span lang=EN style="FONT-SIZE: 12pt; COLOR: #bfbfbf; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: EN-US; mso-ansi-language: EN; mso-themecolor: background1; mso-bidi-language: AR-SA; mso-themeshade: 191">OCTAVE AND OTHER FREQUENCY INTERVALS</SPAN></o:p></FONT></EM></SPAN></STRONG></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><o:p><font color=#bfbfbf></FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-GB style="mso-ansi-language: EN-GB">The octave in music</SPAN></B><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><font color=#bfbfbf><i><span lang=EN-GB style="mso-ansi-language: EN-GB">In music</SPAN></I><span lang=EN-GB style="mso-ansi-language: EN-GB">, the octave is the interval between two frequencies which are in the ratio of 2-to-1 (i.e. the higher frequency is exactly twice the lower frequency).<span style="mso-spacerun: yes"> </SPAN>Check out the following examples.<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><table class=MsoNormalTable style="BORDER-RIGHT: medium none; BORDER-TOP: medium none; MARGIN: auto auto auto 88.2pt; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-table-layout-alt: fixed; mso-border-alt: solid black .25pt; mso-padding-alt: 0cm 6.0pt 0cm 6.0pt; mso-border-insideh: .25pt solid black" cellSpacing=0 cellPadding=0 border=1><tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 124.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=166> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>a one octave <span style="mso-spacerun: yes"> </SPAN><i style="mso-bidi-font-style: normal">rise</I> <span style="mso-spacerun: yes"> </SPAN>from<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 62.3pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=83> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>125 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 25.5pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=34> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>to<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt; mso-border-right-alt: solid black .25pt" vAlign=top width=94> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>250 Hz<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 1"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 124.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=166> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>a one octave <span style="mso-spacerun: yes"> </SPAN><i style="mso-bidi-font-style: normal">fall</I> <span style="mso-spacerun: yes"> </SPAN>from<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 62.3pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=83> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>800 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 25.5pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=34> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>to<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt; mso-border-right-alt: solid black .25pt" vAlign=top width=94> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>400 Hz<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 2"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 124.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=166> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>a two octave <span style="mso-spacerun: yes"> </SPAN><i style="mso-bidi-font-style: normal">rise</I> <span style="mso-spacerun: yes"> </SPAN>from<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 62.3pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=83> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>4 kHz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 25.5pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=34> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>to<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt; mso-border-right-alt: solid black .25pt" vAlign=top width=94> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>16 kHz<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 3; mso-yfti-lastrow: yes"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 124.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=166> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>a three octave <span style="mso-spacerun: yes"> </SPAN><i style="mso-bidi-font-style: normal">rise</I> <span style="mso-spacerun: yes"> </SPAN>from<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 62.3pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=83> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>500 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 25.5pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt" vAlign=top width=34> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>to<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-top-alt: solid black .25pt; mso-border-bottom-alt: solid black .25pt; mso-border-right-alt: solid black .25pt" vAlign=top width=94> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>4 kHz<o:p></o:p></FONT></SPAN></P></TD></TR>
</TBODY></TABLE><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The complete musical scale is generated by defining one frequency (the note <i>A</I> which is 440 Hz) and working out all of the other musical notes, with sharps and flats etc, from that defined frequency.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-GB style="mso-ansi-language: EN-GB">The octave in acoustics and audio</SPAN></B><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>In acoustics and audio, the octave is the interval between two frequencies which are in the ratio of<span style="mso-spacerun: yes"> </SPAN>10<sup> 0.3</SUP> to 1<span style="mso-spacerun: yes"> </SPAN>10<sup> 0.3</SUP> = 1.995<span style="mso-spacerun: yes"> </SPAN>(Calculator key strokes: <span style="mso-spacerun: yes"> </SPAN>[shift] <span style="mso-spacerun: yes"> </SPAN>[log] <span style="mso-spacerun: yes"> </SPAN>[0] <span style="mso-spacerun: yes"> </SPAN>[.] <span style="mso-spacerun: yes"> </SPAN>[3] <span style="mso-spacerun: yes"> </SPAN>[=] )<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The standard octave intervals in acoustics are worked out starting from the <i>Reference Frequency</I> of 1 kHz which is 10<sup> 3.0</SUP> Hz.<span style="mso-spacerun: yes"> </SPAN>The full sequence of frequencies is:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><div align=center><table class=MsoNormalTable style="BORDER-RIGHT: medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-table-layout-alt: fixed; mso-border-alt: solid black .25pt; mso-padding-alt: 0cm 6.0pt 0cm 6.0pt; mso-border-insideh: .25pt solid black; mso-border-insidev: .25pt solid black" cellSpacing=0 cellPadding=0 border=1><tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 107.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt" vAlign=top width=144> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Calculated from<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 1.5</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 1.8</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 2.1</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 2.4</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 2.7</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 3.0</SUP><o:p></o:p></FONT></SPAN></B></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 3.3</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 3.6</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 3.9</SUP><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.9pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 4.2</SUP><o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 1"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 107.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=144> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Exact frequency (Hz)<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>31.62<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>63.10<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>125.9<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>251.2<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>501.2<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>1 k<o:p></o:p></FONT></SPAN></B></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>1.995k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>3.981k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>7.943k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.9pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>15.85k<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 2; mso-yfti-lastrow: yes"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 107.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=144> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Nominal frequency (Hz)<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>31.5<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>63<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>125 <o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>250<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>500<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>1 k<o:p></o:p></FONT></SPAN></B></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>2 k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>4 k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>8 k<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 35.9pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=48> <p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 2.8pt; TEXT-ALIGN: justify; mso-line-height-rule: exactly"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 2.8pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>16 k<o:p></o:p></FONT></SPAN></P></TD></TR>
</TBODY></TABLE></DIV><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>On page 2 there is an image of an octave band graphic equalizer.<span style="mso-spacerun: yes"> </SPAN>Look below the sliders for the nominal frequencies listed on the bottom row of the table above.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-GB style="mso-ansi-language: EN-GB">Fractions of an octave</SPAN></B><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>On the Klark-Teknik graphic equalizer, on page 3, there are two sliders between each of the standard octave sliders.<span style="mso-spacerun: yes"> </SPAN>For example, between the sliders at 250 Hz and 500 Hz there is one at 315 Hz and one at 400 Hz.<span style="mso-spacerun: yes"> </SPAN>The sliders on this graphic equaliser are arranged at one-third-octave intervals.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><table class=MsoTableGrid style="BORDER-RIGHT: medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-border-insideh: none; mso-border-insidev: none; mso-yfti-tbllook: 480" cellSpacing=0 cellPadding=0 border=0><tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top width=70> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Slider:<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>250 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>315 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.75pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>400 Hz<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.8pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>500 Hz<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 1"> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70 rowSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Interval:<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: windowtext 1.5pt double; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top width=70> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P></TD> <td style="BORDER-RIGHT: windowtext 1pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent; mso-border-left-alt: double windowtext 1.5pt; mso-border-right-alt: solid windowtext .5pt" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>< one-third-octave ><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: windowtext 1pt solid; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.7pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid windowtext .5pt; mso-border-right-alt: solid windowtext .5pt" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>< one-third-octave ><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: windowtext 1.5pt double; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 105.8pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent; mso-border-left-alt: solid windowtext .5pt" width=141 colSpan=2> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>< one-third-octave ><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.9pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent; mso-border-left-alt: double windowtext 1.5pt" vAlign=top width=71> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 2; mso-yfti-lastrow: yes"> <td style="BORDER-RIGHT: windowtext 1.5pt double; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.85pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" vAlign=top width=70> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P></TD> <td style="BORDER-RIGHT: windowtext 1.5pt double; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; BACKGROUND: #e6e6e6; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 317.2pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; mso-border-left-alt: double windowtext 1.5pt" width=423 colSpan=6> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf><<span style="mso-spacerun: yes"> </SPAN>one octave<span style="mso-spacerun: yes"> </SPAN>><o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: #ece9d8; PADDING-RIGHT: 5.4pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 5.4pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 52.9pt; PADDING-TOP: 0cm; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent; mso-border-left-alt: double windowtext 1.5pt" vAlign=top width=71> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P></TD></TR>
<tr height=0> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=70><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=71><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=71><font color=#bfbfbf></FONT></TD> <td style="BORDER-RIGHT: #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent" width=71><font color=#bfbfbf></FONT></TD></TR>
</TBODY></TABLE><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>For an octave, the ratio is<span style="mso-spacerun: yes"> </SPAN>10 <sup>0.3</SUP> .<span style="mso-spacerun: yes"> </SPAN>The number 0.3 is the exponent (i.e. the power of ten).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>We work out an interval which is a given fraction of an octave by taking the same fraction of 0.3<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>So, for one-third-octave, the exponent is a third of 0.3, i.e.<span style="mso-spacerun: yes"> </SPAN><sup>1</SUP>/<sub>3</SUB> × 0.3<span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>0.1<span style="mso-spacerun: yes"> </SPAN>and<span style="mso-spacerun: yes"> </SPAN>10<sup> 0.1</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>1.2589 .<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>This table explains how a number of fractions of an octave can be calculated.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><div align=center><table class=MsoNormalTable style="BORDER-RIGHT: medium none; BORDER-TOP: medium none; BORDER-LEFT: medium none; BORDER-BOTTOM: medium none; BORDER-COLLAPSE: collapse; mso-table-layout-alt: fixed; mso-border-alt: solid black .25pt; mso-padding-alt: 0cm 6.0pt 0cm 6.0pt; mso-border-insideh: .25pt solid black; mso-border-insidev: .25pt solid black" cellSpacing=0 cellPadding=0 border=1><tbody>
<tr style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 4cm; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt" width=151> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>1-octave<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>2</SUB>-octave<o:p></o:p></SPAN></FONT></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>3</SUB>-octave<o:p></o:p></SPAN></FONT></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>6</SUB>-octave<o:p></o:p></SPAN></FONT></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: black 1pt solid; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>12</SUB>-octave<o:p></o:p></SPAN></FONT></P></TD></TR>
<tr style="mso-yfti-irow: 1"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 4cm; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=151> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Value of the exponent<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>1 × 0.3 <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 0.3<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>2</SUB> × 0.3 <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 0.15<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>3</SUB> × 0.3 <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 0.10<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>6</SUB> × 0.3<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN>= 0.05<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><sup><span lang=EN-GB style="mso-ansi-language: EN-GB">1</SPAN></SUP><span lang=EN-GB style="mso-ansi-language: EN-GB">/<sub>12</SUB> × 0.3 <o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 0.025<o:p></o:p></FONT></SPAN></P></TD></TR>
<tr style="mso-yfti-irow: 2; mso-yfti-lastrow: yes"> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: black 1pt solid; WIDTH: 4cm; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" vAlign=top width=151> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Which gives a ratio of<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 0.3</SUP> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 1.995<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 0.15</SUP> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 1.413<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 0.10</SUP> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 1.259<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 0.05</SUP> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 1.122<o:p></o:p></FONT></SPAN></P></TD> <td style="BORDER-RIGHT: black 1pt solid; PADDING-RIGHT: 6pt; BORDER-TOP: #ece9d8; PADDING-LEFT: 6pt; PADDING-BOTTOM: 0cm; BORDER-LEFT: #ece9d8; WIDTH: 70.05pt; PADDING-TOP: 0cm; BORDER-BOTTOM: black 1pt solid; BACKGROUND-COLOR: transparent; mso-border-alt: solid black .25pt; mso-border-left-alt: solid black .25pt; mso-border-top-alt: solid black .25pt" width=93> <p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>10<sup> 0.025</SUP> <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 2pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>= 1.059<o:p></o:p></FONT></SPAN></P></TD></TR>
</TBODY></TABLE></DIV></DIV><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-GB style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: EN-US; mso-ansi-language: EN-GB; mso-bidi-language: AR-SA"><br style="PAGE-BREAK-BEFORE: auto; mso-break-type: section-break" clear=all><font color=#bfbfbf></FONT></SPAN> </P><div class=Section2><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><b><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Octave bands<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The image below is a graphics user interface (GUI) from some audio-processing computer software and it shows an octave band equaliser.<span style="mso-spacerun: yes"> </SPAN>There are 10 octave bands in the audio range; hence 10 sliders.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Below the sliders, you can see the octave interval nominal frequencies *.<span style="mso-spacerun: yes"> </SPAN>Each slider controls the gain for a one-octave-wide band of frequencies.<span style="mso-spacerun: yes"> </SPAN>The gain, in dB, is shown in the window above each slider.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-ALIGN: justify"><font color=#bfbfbf><span lang=EN-GB style="mso-ansi-language: EN-GB">Each slider</SPAN><span lang=EN-GB style="FONT-FAMILY: 'WP TypographicSymbols'; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-ansi-language: EN-GB; mso-char-type: symbol; mso-symbol-font-family: 'WP TypographicSymbols'"><span style="mso-char-type: symbol; mso-symbol-font-family: 'WP TypographicSymbols'">=</SPAN></SPAN><span lang=EN-GB style="mso-ansi-language: EN-GB">s quoted frequency is at the centre of its respective octave band.<span style="mso-spacerun: yes"> </SPAN>The quoted frequency is called the <i>octave band centre frequency</I>.<span style="mso-spacerun: yes"> </SPAN>We identify which octave band we are controlling by quoting the octave band centre frequency.<span style="mso-spacerun: yes"> </SPAN>Each of the ten octave bands reaches half an octave above the band centre frequency and half an octave below the band centre frequency.<span style="mso-spacerun: yes"> </SPAN>So:<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-INDENT: 36pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>the <i>Band Upper Limit</I> is half an octave above (i.e. × 10<sup> 0.15</SUP>) the <i>band centre frequency</I>; and<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-INDENT: 36pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>the <i>Band Lower Limit</I> is half an octave below (i.e. ÷ 10<sup> 0.15</SUP>) the <i>band centre frequency</I>.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>For the 500 Hz octave band (shown with a gain of + 2.9 dB in the image):<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-INDENT: 36pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>the exact band centre frequency is 10<sup> 2.7</SUP><span style="mso-spacerun: yes"> </SPAN>(see page 1 for how this is worked out);<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-INDENT: 36pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>the band upper limit is 10<sup> 2.7</SUP> × 10<sup> 0.15</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>10<sup> 2.7 + 0.15</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>10<sup> 2.85</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>707.9 Hz<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-INDENT: 36pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>the band lower limit is 10<sup> 2.7</SUP> ÷ 10<sup> 0.15</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>10<sup> 2.7 - 0.15</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>10<sup> 2.55</SUP><span style="mso-spacerun: yes"> </SPAN>=<span style="mso-spacerun: yes"> </SPAN>354.8 Hz<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 3.3pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The slider labelled 500 Hz does not adjust the gain only for the frequency of 500 Hz.<span style="mso-spacerun: yes"> </SPAN>It adjusts the gain for an octave band of frequencies from 354.8 Hz up to 707.9 Hz.<span style="mso-spacerun: yes"> </SPAN>500 Hz is at the centre of that octave band.<span style="mso-spacerun: yes"> </SPAN>The slider adjusts the gain for the whole of the <i>500 Hz octave band</I>.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The lowest frequency being processed by the software above, is not 31.5 Hz *, but is the frequency at the band lower limit of the 31.5 Hz octave band.<span style="mso-spacerun: yes"> </SPAN>(See question 2(e) on page 4.)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The highest frequency being processed by the software is not 16 kHz, but is the frequency at the band upper limit of the 16 kHz octave band.<span style="mso-spacerun: yes"> </SPAN>(See question 2(a) on page 4.)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The above image is of an octave band equaliser.<span style="mso-spacerun: yes"> </SPAN>The following page shows an example of a one-third-octave band equaliser.<span style="mso-spacerun: yes"> </SPAN>The frequency bands are narrower and the EQ control much finer.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm -2.8pt 0pt 0cm; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>(*<span style="mso-spacerun: yes"> </SPAN>31 Hz should be labelled 31.5 Hz; and 62 Hz should be labelled 63 Hz.<span style="mso-spacerun: yes"> </SPAN>Many manufacturers get these two wrong.)<o:p></o:p></FONT></SPAN></P></DIV><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN-GB style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-language: EN-US; mso-ansi-language: EN-GB; mso-bidi-language: AR-SA"><br style="PAGE-BREAK-BEFORE: always; mso-break-type: section-break" clear=all><font color=#bfbfbf></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span lang=EN-GB style="mso-ansi-language: EN-GB">One-third-octave Bands</SPAN></B><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The image below shows a one-third-octave band graphic equaliser.<span style="mso-spacerun: yes"> </SPAN>There are 10 octaves in the audio range; so there are 30 one-third-octaves in the audio range. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Each one of the 30 sliders adjusts the gain of the signal for a band of frequencies which is one-third-of-an-octave wide.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>These professional equalisers offer fine detailed adjustment to the equalisation (EQ) of the audio signal.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>The frequency quoted under each slider is the nominal <i>Band Centre Frequency</I>.<span style="mso-spacerun: yes"> </SPAN>So each third-octave band extends from one-sixth of an octave below the band centre frequency to one-sixth of an octave above the band centre frequency.<span style="mso-spacerun: yes"> </SPAN>The two extremes are called the <i>Band Lower Limit</I><span style="mso-spacerun: yes"> </SPAN>and the <i>Band Upper Limit</I>.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><font color=#bfbfbf>Notice that the vertical position of the sliders is calibrated in dB with markings at<span style="mso-spacerun: yes"> </SPAN>+12, +6, +3, 0, -3, -6, and -12 dB, showing how the power level of each band is increased or decreased relative to the <i>flat response</I> , 0 dB, position.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN><font color=#bfbfbf><span lang=EN-GB style="mso-ansi-language: EN-GB"><o:p> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><i style="mso-bidi-font-style: normal"><span lang=EN style="FONT-SIZE: 14pt; COLOR: gray; FONT-FAMILY: 'Calibri','sans-serif'; text-effect: emboss; mso-bidi-font-size: 11.0pt; mso-color-alt: #BFBFBF; mso-ansi-language: EN; mso-themecolor: background1; mso-themeshade: 191"><font color=#bfbfbf>The decibel</FONT></SPAN></I></B><span lang=EN-GB style="COLOR: #bfbfbf; mso-ansi-language: EN-GB; mso-themecolor: background1; mso-themeshade: 191"><o:p></o:p></SPAN></P></o:p></SPAN></FONT> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering (particularly acoustics, electronics, and control theory) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.775 volts RMS.[1]<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>History<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz) and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).[citation needed]<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The transmission unit or TU was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to reference power.[2] The definitions were conveniently chosen such that 1 TU approximately equalled 1 MSC (specifically, 1.056 TU = 1 MSC).[3] Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, which was named the "bel" in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was a factor of ten larger than the TU, such that 1 TU equalled 1 decibel.[4] In many situations, the bel proved inconveniently large, so the decibel has become more common.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><font color=#bfbfbf> </FONT></o:p></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.[5] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC).[6] The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf><img style="WIDTH: 178px; HEIGHT: 290px" height=537 src="http://www.easypedia.gr/el/images/shared/archive/1/10/20050304161331!Alexander_Graham_Bell.jpg" width=358></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><o:p><span lang=EN style="COLOR: black; FONT-FAMILY: 'Verdana','sans-serif'; mso-ansi-language: EN"><font size=2><font color=#bfbfbf>Alexander Graham Bell<o:p></o:p></FONT></FONT></SPAN></P></o:p></SPAN> <p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Merits<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The use of the decibel has a number of merits:<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.) <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) + ... <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Uses<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Acoustics<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Main article: Sound pressure<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>A reason for using the decibel is that the ear is capable of detecting a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage during short exposure to quietest sound that (undamaged) ears can hear is above a million. To deal with such a range, logarithmic units are useful: the base-10 logarithm of a trillion is 12, so a level difference of 120 dB represents a power ratio of this amount. Since the human ear is not equally sensitive to all the frequencies of sound, noise levels at maximum human sensitivity — for example, the higher harmonics of middle A (between 2 and 4 kHz) — are factored more heavily into sound descriptions using a process called frequency weighting.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Further information: Examples of sound pressure and sound pressure levels<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Electronics<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a link budget.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in telephone audio circuits.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Optics<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In spectrometry and optics, the blocking unit used to measure optical density is equivalent to −1 B. In astronomy, the apparent magnitude measures the brightness of a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Video and digital imaging<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>In connection with digital and video image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square. Thus, a camera signal-to-noise ratio of 60 dB represents a power ratio of 1000:1 between signal power and noise power, not 1,000,000:1.[8]<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Common reference levels and corresponding units<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf><span style="mso-spacerun: yes"> </SPAN><i style="mso-bidi-font-style: normal"><u>Absolute and relative decibel measurements</U></I><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>*−6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 μW (0.25 mW). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative. The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI.[9] However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 14pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Absolute measurements<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Electric power<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBm or dBmW<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30. dBW<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Voltage<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>A schematic showing the relationship between dBu (the voltage source) and dBm (the power dissipated as heat by the 600 Ω resistor)<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBV<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance.<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBu or dBv<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(0.775 VRMS) — voltage relative to 0.775 volts.[1] Originally dBv, it was changed to dBu to avoid confusion with dBV.[10] The "v" comes from "volt", while "u" comes from "unloaded". dBu can be used regardless of impedance, but is derived from a 600 Ω load dissipating 0 dBm (1 mW). Reference voltage<span style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBmV<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 Ω coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm) or ~13 nW. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBμV or dBuV<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Acoustics<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Probably the most common usage of "decibels" in reference to sound loudness is dB SPL, referenced to the nominal threshold of human hearing:[11]<o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(SPL)<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB (sound pressure level) — for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10−5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a reference pressure of 1 μPa is used.[12] <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB SIL<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB sound intensity level — relative to 10−12 W/m2, which is roughly the threshold of human hearing in air. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB SWL<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB sound power level — relative to 10−12 W. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(A), dB(B), and dB(C)<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>These symbols are often used to denote the use of different weighting filters, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">dB HL</SPAN></B><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"> or dB hearing level is used in audiograms as a measure of hearing loss. The reference level varies with frequency according to a minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.[citation needed]<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><font color=#bfbfbf><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">dB </SPAN></B><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN">Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting[citation needed]<o:p></o:p></SPAN></FONT></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Radar<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBZ<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.[13] <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBsm<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBsm - decibel (referenced to one) square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of dBsm, large flat plates or non-stealthy aircraft have positive values.[14] <b style="mso-bidi-font-weight: normal"><o:p></o:p></B></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Radio power, energy, and field strength<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBc<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBc — relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, used in acoustics. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBJ<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBm<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. -120 dBm = 0.224 microvolts. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBμV/m or dBuV/m<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(μV/m) — electric field strength relative to 1 microvolt per meter. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBf<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(fW) — power relative to 1 femtowatt. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBW<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(W) — power relative to 1 watt. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBk<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(kW) — power relative to 1 kilowatt. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Antenna measurements<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBi<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(isotropic) — the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise. <b style="mso-bidi-font-weight: normal"><o:p></o:p></B></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBd<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(dipole) — the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBiC<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(isotropic circular) — the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBq<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(quarterwave) — the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0 dBq = -0.85 dBi <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><i style="mso-bidi-font-style: normal"><u><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Tempus Sans ITC'; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN"><font color=#bfbfbf>Other measurements<o:p></o:p></FONT></SPAN></U></I></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBFS or dBfs<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(full scale) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS (peak) would equal the highest level (number) the processor is capable of representing. Measured values are always negative or zero, since they are less than the maximum or full-scale. Full-scale is typically defined as the power level of a full-scale sinusoid, though some systems will have extra headroom for peaks above the nominal full scale. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB-Hz<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations. Also used in carrier-to-noise-density ratio (not to be confused with carrier-to-noise ratio, in dB). <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBov or dBO<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(overload) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBr<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB(relative) — simply a relative difference from something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance. <o:p></o:p></FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><b style="mso-bidi-font-weight: normal"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dBrn<o:p></o:p></FONT></SPAN></B></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>dB above reference noise. See also dBrnC.</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Lenny Z. Perez M</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>EES</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>Referencias:</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><a href="http://en.wikipedia.org/wiki/Octave"><font color=#bfbfbf>http://en.wikipedia.org/wiki/Octave</FONT></A></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><font color=#bfbfbf>www.acoustics.salford.ac.uk/.../09%20<b>Octave</B>%20and%20<b>other</B>%20<b>intervals</B>.<wbr>doc -</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><a href="http://en.wikipedia.org/wiki/Decade_(log_scale"><font color=#bfbfbf>http://en.wikipedia.org/wiki/Decade_(log_scale</FONT></A><font color=#bfbfbf>)</FONT></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><a href="http://en.wikipedia.org/wiki/Decibel"><font color=#bfbfbf>http://en.wikipedia.org/wiki/Decibel</FONT></A></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="FONT-SIZE: 12pt; LINE-HEIGHT: 115%; FONT-FAMILY: 'Times New Roman','serif'; mso-ansi-language: EN"><a href="http://www.sizes.com/units/decibel.htm"><font color=#bfbfbf>http://www.sizes.com/units/decibel.htm</FONT></A></SPAN></P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"></o:p></FONT></SPAN> </P><p class=MsoNormal style="MARGIN: 0cm 0cm 10pt; TEXT-ALIGN: justify"><span lang=EN style="mso-ansi-language: EN"><font size=3><font face=Calibri><o:p></o:p></FONT></FONT></SPAN> </P> <br />
<hr />Explore the seven wonders of the world <a href='http://search.msn.com/results.aspx?q=7+wonders+world&mkt=en-US&form=QBRE' target='_new'>Learn more!</a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com0tag:blogger.com,1999:blog-8049570815625896303.post-80711280996602772012010-02-06T12:37:00.001-04:302010-02-06T16:46:15.379-04:30Medidas en oído real mediante sonda microfónica.<b><i><span style="FONT-SIZE: 16pt; COLOR: gray; FONT-FAMILY: 'Tempus Sans ITC'; text-effect: emboss"><span style="COLOR: windowtext"><font color=#bfbfbf>Medidas en oído real mediante sonda microfónica.<span> </SPAN><span> </SPAN>Definición y aplicaciones.</FONT></SPAN></SPAN></I></B><br />
<p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Las medidas de oído real ha permitido al audioprotesista disponer de un criterio para la valoración de la adaptación de audífonos fiable y válido. El uso de estas medidas en la estimación de la bondad de la adaptación nos permite, entre otras ventajas, tener en cuenta las diferencias individuales al facilitarnos parámetros referidos al rendimiento del audífono en oído real. En el presente artículo llevaremos a cabo una revisión de las principales medidas que pueden registrarse con audioanalizador y sonda microfónica en oído real y sus principales aplicaciones.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Medidas con sonda microfónica en oído real</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: 439.45pt"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La terminología al uso en este tipo de medidas puede resultar un tanto confusa toda vez que son muchos los registros que podemos llevar acabo. En los siguientes párrafos presentamos las diferentes medidas en oído real y acoplador al uso y una definición de cada una de ellas siguiendo las normas ANSI S3.46-1997 (23).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: 439.45pt"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Para facilitar la comprensión de las siguientes definiciones es conveniente que el lector distinga entre aquellas medidas que hacen referencia a la "respuesta" y aquellas que se refieren a la "ganancia". La respuesta indica una medida absoluta de salida en dB SPL, mientras que la ganancia indica la diferencia entre dos medidas relativas no absolutas.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify; tab-stops: 439.45pt"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REUR - Respuesta en oído real no amplificado.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">Real-Ear Unaided Response</SPAN></I></B><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">.</SPAN></B></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la respuesta obtenida en dB SPL, en función de la frecuencia, medida en un punto determinado del CAEa la presentación de un estímulo sonoro específico a campo abierto sin amplificación. (ANSI S3.46-1997) (23).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La principal aplicación del REUR es medir las características individuales de resonancia en el oído que vienen determinadas por las características anatómicas de la pabellón auditivo, concha y el CAE.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: center" align=center><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REUG - Ganancia en oído real no amplificado.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">Real-Ear Unaided Gain</SPAN></I></B><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">.</SPAN></B></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en dB SPL, en función de la frecuencia, medida en un punto determinado del CAE y la señal de entrada, para un estímulo sonoro específico a campo abierto sin amplificación (ANSI S3.46-1997) (23). Es decir, la ganancia dada por el pabellón auricular y el conducto auditivo con el consecuente efecto de difracción de la cabeza al medir con la sonda microfónica en el conducto auditivo y sustraer este valor al obtenido en campo abierto. Para su cálculo deberemos restar la intensidad de la señal de entrada a la del REUR en todas las frecuencias. Esta medida también es conocida como Opean Ear Gain.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: center" align=center><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REUG = E – REUR</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Esta medida es requerida por algunos fabricantes para él calculo de la Ganancia de Inserción (REIG:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear Insertion Gain). El REUG también es utilizado para ajustar la ganancia del audífono en el acoplador de 2 cc tomando como referencia el objetivo a alcanzar por el REIG. En los métodos que utilizan</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>SPL-O-Gramas para las adaptaciones protésicas (p.e.:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Desired Sensation Level [DSL]), el REUR se utiliza para la conversión de los dB HL obtenidos en las audiometrías a campo abierto en valores SPL. Una última aplicación del REUG es alertarnos ante registros inusuales de posibles anormalidades en el CAE o en el oído medio tal como ocurre con las perforaciones de la membrana timpánica (22).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: center" align=center><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REAR - Respuesta en oído real amplificado.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear Aided Response.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la respuesta obtenida en dB SPL, en función de la frecuencia, medida en un punto determinado del CAE a la presentación de un estímulo sonoro específico a campo abierto con los audífonos en funcionamiento y el molde auditivo en el oído. (ANSI S3.46-1997)</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>(23).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REAG - Ganancia en oído real amplificado.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear Aided Gain.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en dB SPL, en función de la frecuencia, entre las medidas obtenidas en un punto determinado del CAE y la señal de entrada, a la presentación de un estímulo sonoro específico a campo abierto con los audífonos en funcionamiento y el molde auditivo en el oído (ANSI S3.46-1997) (23). Es decir, la sustracción entre el estímulo presentado y el REAR obtenido a través de todas las frecuencias estudiadas.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: center" align=center><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REAG = E – REAR</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El REAR y el REAG se suele llevar acabo para calcular posteriormente el REIG o Ganancia de Inserción.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Hay algunos métodos de prescripción de la ganancia, por ejemplo el DSL, que necesitan el REAR y el</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REAG para poder calcular el REIG y por lo tanto requieren de estas medidas durante el proceso de adaptación. La principal ventaja al disponer de este parámetro es que podemos obtener un SPL-O-Grama para cada paciente en dB SPL en vez de dB HL. De esta manera podremos de un solo vistazo (empleando mel REAR) determinar a que intensidad un estímulo sonoro en particular es audible, confortable o inconfortable para un paciente en concreto.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REIG - Ganancia de inserción. <i>Real-Ear</I></FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Insertion Gain.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en decibelios en función de la frecuencia entre el REAG y el REUG obtenido en el mismo punto de medida del CAE y en las mismas condiciones a campo abierto (ANSI S3.46-1997)</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>(23). Es decir, la ganancia dada por el audífono sustrayendo el REUG del REAG o el REUR del REAR para todas las frecuencias.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La principal aplicación del REIG es determinar en que punto, el ajuste del audífono a alcanzado un valor particular establecido previamente por algún método de prescripción de la ganancia (ver Figura. 3). Tal como cita Muller (22) sin no tenemos este objetivo establecido previamente por algún método de prescripción de la ganancia, la medición del REIG carece de sentido.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: center" align=center><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></I></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REOR - Respuesta en oído real ocluido.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear Occluded Response.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la respuesta obtenida en dB SPL, en función de la frecuencia, medida en un punto determinado del CAE a la presentación de un estímulo sonoro específico a campo abierto con los audífonos apagados y el molde auditivo en el oído. (ANSI S3.46-1997) (23). Es decir un REAR pero con el audífono apagado.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>REOG - Ganancia en oído real ocluido.</FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear Occlude Gain.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en dB SPL, en función de la frecuencia, entre las medidas obtenidas en un punto determinado del CAE y la señal de entrada, a la presentación de un estímulo sonoro específico a campo abierto con los audífonos apagados y el molde auditivo en el oído (ANSI S3.46-1997) (23). Es decir, la sustracción entre la señal de entrada y la del REOR en todas las frecuencias.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Como podemos esperar, ya que el audífono está apagado y ocluyendo el oído, el REOR suele estar por encima del REUR. Hay casos en los que esto no ocurre y son en los que un molde no llega a ocluir él CAE este posee un ventig grande capaz de producir un efecto de resonancia.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>La razón por la que obtenemos el REUR y REOG es disponer de una medida del efecto del venting en el resultado final de la adaptación (22). Esto nos permitirá valorar sí el ventig se está comportando de la forma esperada. El REOR y el REOG también puede emplearse para averiguar en que medida el venting está introduciendo efectos acústicos no deseados (resonancias asociadas al venting) que puedan modificar la respuesta final de la amplificación (22).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf> </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>RECD – Diferencia entre oído real y acoplador. </FONT></SPAN></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><b><i><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Real-Ear-To-Coupler Difference.</FONT></SPAN></I></B></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en dB SPL, en función de la frecuencia, entre las medidas obtenidas en un punto determinado del CAE y los obtenidos en un acoplador de 2 cc, a la presentación de un estímulo sonoro específico con los audífonos en funcionamiento y el molde auditivo en el oído (ANSI S3.46-1997) (23). Es decir, la diferencia entre las medidas obtenidas en oído real y las obtenidas en acoplador.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Dadas las diferencias en volumen e impedancia entre el oído y el acoplador, los valores del RECD son generalmente mayores o iguales a 0. Los valores de RECD pueden variar sustancialmente a través de los grupos de edad (los niños suelen tener RECD más largos que los adultos) y entre grupos (24). Un valor negativo del RECD indica un sellamiento inadecuado del trasductor en el oído, un oído mayor que la media, un tímpano perforado o un tubo de drenaje por miringotomía (25).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El RECD es una herramienta muy valiosa toda vez que nos puede ayudar en los diferentes estadios del proceso de la adaptación. El RECD nos permite convertir de una forma precisa la información obtenida en dB HL durante el diagnóstico en dB SPL (26). El RECD pueden usarse para ayudarnos en la selección del audífono sobre la base de las especificaciones técnicas dada por el fabricante en sus fichas técnicas, al permitirnos convertir los valores esperados en oído real en valores objetivo en acoplador 2cc. Seewal (27) resumió las principales ventajas de poder predecir la salida de un audífono en los siguientes cuatro puntos:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>1. El audioprotesista puede conocer de antemano los valores esperados de amplificación en él CAE del paciente.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>2. Se tiene en cuenta las propiedades del molde una vez adaptado en el CAE. Esto nos ayuda a evitar errores cuando usamos métodos de prescripción de la ganancia basados en baremos de referencia durante el proceso de adaptación.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>3. Todos los ajustes sobre el aparato pueden llevarse en cámaras anecoicas bajo condiciones acústicas controladas.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>4. La necesidad de colaboración y el tiempo requerido por el paciente en el proceso de adaptación se reduce considerablemente.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">REDD – Diferencia entre oído real y el dial. </SPAN></B><b><i><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">Real-Ear-To Dial Difference.</SPAN></I></B><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></B></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la diferencia en dB SPL, en función de la frecuencia, entre las medidas obtenidas en un punto determinado del CAE y el valor dado por el dial del audiómetro a la presentación de un estímulo sonoro específico a través de los auriculares del audiómetro (ANSI S3.46-1997) (23). Es decir, la diferencia entre las medidas obtenidas en oído real y la intensidad marcada por el dial del audiómetro.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Generalmente obtenemos un registro superior a los 0 dB. Como cabe esperar los valores del REDD pueden variar sustancialmente según los individuos (28).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Un REDD negativo puede indicar un sellamiento inapropiado del transductor en el oído, obstrucción de la sonda o una colocación inapropiada de la sonda.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El REDD es empleado para convertir información audiométrica (p.e.: umbrales o niveles de incomfort) de dB HL a dB SPL. El REDD nos permite expresar los valores audiométricos en un SPL-O -Grama. Clínicamente el REDD suele usarse cuando la audiometría ha sido obtenida en dB HL.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><font color=#bfbfbf><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">RESR – Respuesta de saturación en oído real. </SPAN></B><b><i><span lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'">Real-Ear Saturation Response.</SPAN></I></B><b><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"></SPAN></B></FONT></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es la respuesta obtenida en dB SPL, en función de la frecuencia, medida en un punto determinado del</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>CAE a la presentación de un estímulo sonoro específico a campo abierto capaz de inducir al audífono al punto de máxima presión de salida con la ganancia del audífono al máximo o en punto previo a la retroalimentación del mismo y el molde auditivo en el oído.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Es decir, la respuesta en frecuencia del audífono medida en el oído con una señal de entrada lo suficientemente intensa para llevar al instrumento a la máxima presión de salida.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Dada la alta intensidad a la que se debe llevar acabo esta prueba es conveniente hacer este examen en acoplador (utilizando el RECD para predecir la salida en oído real).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El valor del RESR puede variar sustancialmente dependiendo de la señal de entrada utilizada. Las señales de banda estrecha (p.e.: tonos puros o modulados) suelen dar mejor respuesta que los de banda ancha (p.e.: ruido blanco, ruido shaped speech) (29).</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>El RESR suele obtenerse para determinar el valor máximo de presión de salida que el audífono es capaz de dar en el oído del paciente. Esta información sirve para ajustar la salida máxima del audífono y para asegurar que las señales amplificadas no exceden los valores de inconfort del paciente. Así mismo permite al audioprotesista comprobar en que medida los ajustes de máxima presión de salida del audífono son correctos.</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf></FONT></SPAN> </P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Lenny Z Perez M</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>EES</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><font color=#bfbfbf>Referencia:</FONT></SPAN></P><p class=ecxMsoNormal style="LINE-HEIGHT: normal; TEXT-ALIGN: justify"><span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman','serif'"><a href="http://www.auditio.com/revista/vol1/1/040101.pdf"><font color=#bfbfbf>http://www.auditio.com/revista/vol1/1/040101.pdf</FONT></A></SPAN></P><hr>Discover the new Windows Vista <a href="http://search.msn.com/results.aspx?q=windows+vista&mkt=en-US&form=QBRE">Learn more!</A> <br />
<hr />Connect to the next generation of MSN Messenger <a href='http://imagine-msn.com/messenger/launch80/default.aspx?locale=en-us&source=wlmailtagline' target='_new'>Get it now! </a>Tecnología en Telecomunicaciones - conocimientos.com.vehttp://www.blogger.com/profile/13517798918797491823noreply@blogger.com1