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* See note 21 in Appendix.
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Vi R jXC
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CHAPTER 9 Impedance Transformation
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thus, again by Ohm s law, V jXC " " Vo I jXC i R jXC
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j 1 , you should nd Now, dividing both sides by Vi and noting that jXC !C j!C that " G 1 1 j!RC 316
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" where G is the SINUSOIDAL STEADY-STATE VOLTAGE GAIN of the basic RC lowpass lter of Fig. 190, where ! is any frequency in radians per second (! 2f ). In eq. (316), let us regard the product RC as having a constant value in any given case, with the variable being the frequency !. We now wish to develop certain important " relationships that exist between G and !. This can be done in an interesting way algebraically, as follows. Let us begin by creating, by de nition, the equation RC 1=!1 317
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where, since RC is constant, the frequency !1 is also constant. It is permissible to write such an equation, because the unit of measurement for both RC and 1=! is seconds (eq. (91) in Chap. 5, and note 14 in the Appendix). Thus, replacing RC with the right-hand side of eq. (317), eq. (316) becomes " G 1 1 j !=!1 318
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The advantage of eq. (318) over (316) is that we no longer have to work with actual absolute values of ! (such as ! 508 rad/sec or ! 10,750 rad/sec, and so on), but only with the simple ratio of ! to !1 ; the frequency is now said to be normalized with respect to the reference frequency !1 . Eq. (318) contains all the information, concerning both the amplitude and phase response, of the network of Fig. 190. Let us rst investigate the amplitude response, as follows. The AMPLITUDE response is determined by the MAGNITUDE of eq. (318); thus the amplitude response of Fig. 190 is equal to 1 " jGj q 1 !=!1 2 1=2 1 !=!1 2 319
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" Now, as noted in note 19 in the Appendix, because G is a voltage ratio, the decibel relationship would be written " dB 20 log jGj
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320
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" and thus, using the value of jGj from eq. (319) and remembering that log A n log A, eq. (320) gives the value dB 10 log 1 !=!1 2 321
The minus sign appears because the voltage gain in Fig. 190 is less than 1 for all values of ! greater than zero, and the logarithm of a number less than 1 is negative. The procedure now is to plot eq. (321) on semilog paper, putting decibel gain on the linear vertical axis and the frequency ratio !=!1 on the logarithmic horizontal axis.
CHAPTER 9 Impedance Transformation
To draw the curve, we rst calculate a table of values of decibel gain for various values of the independent variable !=!1 . Thus, the following table was obtained by using eq. (321) to calculate dB gain for each of the following given values of !=!1 .
!=!1 0.2 0.4 0.6 0.8 1.0 !=!1 2.0 4.0 6.0 8.0 !=!1 10 20 30 40 50
dB gain 0.17 0.65 1.34 2.15 3.01
dB gain 6.99 12.31 15.68 18.13
dB gain 20.04 26.03 29.55 32.04 33.98
Plotting decibel gain versus frequency ratio, we get the curve shown in Fig. 191.
Fig. 191
Remember that negative decibel gain means signal attenuation; that is, the output voltage is less than the input voltage. Figure 191 shows that Fig. 190 is a low-pass network, because the higher the frequency !, the greater is the attenuation of the output signal relative to the constant amplitude input signal. This is an appropriate time to introduce another term, called the HALF-POWER frequency, that is widely used in the evaluation of the frequency response of networks and systems. The de nition is as follows. A half-power frequency is a frequency at which the OUTPUT POWER of a network is reduced to ONE HALF its maximum value under the condition of constant amplitude of input signal. Thus, setting P 1=2 0:5 in eq. (315), we have that dB 10 log 0:5 3 decibels; very nearly showing that a half-power frequency is a frequency at which the power gain of a network is down 3 decibels from its maximum or reference power. Note that, for the PARTICULAR
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