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(636)
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and this is plotted as a function of frequency on a log10 scale Note that the use of decibels implies that one is measuring a ratio Decibel plots are usually displayed on semilogarithmic paper, with decibels on the linear axis and frequency on the logarithmic axis Bode plots are named after Hendrik W Bode, a research mathematician who is among the pioneers in modern electrical network analysis and feedback ampli er design Let us examine the appearance of dB plots for typical low-pass and high-pass lter circuits From Figure 628, we can see that both plots have a very simple appearance: either the low-frequency part of the plot (for a low-pass lter) or the
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dB plot of low-pass filter amplitude response
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Amplitude ratio, dB
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10 3 100 103 106 Radian frequency (logarithmic scale)
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dB plot of high-pass filter amplitude response
20 0 20 40 60 80
100
Amplitude ratio, dB
100 10 3
100 103 106 Radian frequency (logarithmic scale)
Figure 628 dB magnitude plots of low- and high-pass lters
high-frequency part (for the high-pass lter) is well approximated by a at line, indicating that for some range of frequencies, the lter has a constant amplitude response, equal to 1 Further, the lter cutoff frequency, 0 , appears quite clearly as the approximate frequency where the lter response starts to fall The response of the circuit decreases (or increases) with a constant slope with respect to (on a logarithmic scale) For the high-pass and low-pass lters described earlier, this slope is equal to 20 dB/decade ( for the low-pass lter, + for the high-pass), where a decade is a range of frequencies f1 to f2 such that f2 = 10 f1 What kind of decrease in gain is 20 dB/decade The expression |H (j )|dB = 20 dB means that 20 = 20 log10 |H (j )| or |H (j )| = 01 That is, the gain decreases by a factor of 10 for every increase in frequency by a factor of 10 You see how natural these units are Further, if 0 is known, (639) (638) (637)
Part I
Circuits
a plot of |H (j )|dB versus (on a logarithmic scale) may be readily sketched using the asymptotic approximations of two straight lines, one of slope zero and the other with slope equal to 20 dB/decade, and with intersection at 0 The homework problems and exercises provide a good number of practical examples of this technique
Check Your Understanding
61 Derive an expression for HI (j ) =
IL (j ) IS
for the circuit of Figure 61
62 Use the method of node voltages to derive HY (j ) for the circuit of Figure 61 63 Use the method of mesh currents to derive HV (j ) for the circuit of Figure 61 64 Connect the lter of Example 63 to a 1-V sinusoidal source with internal resistance
of 50 to form a circuit similar to that of Figure 612 Determine the circuit cutoff frequency, 0
65 Determine the cutoff frequency for each of the four prototype lters shown in Figure 629 Which are high-pass and which are low-pass
R C L
(a) C L
66 Show that it is possible to obtain a high-pass lter response simply by substituting an inductor for the capacitor in the circuit of Figure 69 Derive the frequency response for the circuit 67 Determine the cutoff frequency for the high-pass RC lter shown in Figure 630 [Hint: First nd the frequency response in the form j a/(1 + j b), where a and b are constants related to R1 , R2 , and C1 , and then solve numerically] Sketch the amplitude and frequency responses 68 A simple RC low-pass lter is constructed using a 10- F capacitor and a 22-k
resistor Over what range of frequencies will the output of the lter be within 1 percent of the input signal amplitude (ie, when will VL 099VS )
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