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Vout
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Figure 1221 Active low-pass lter
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Note the similarity between ZF and the low-pass characteristic of the passive RC circuit! The closed-loop gain ALP (j ) is then computed to be ALP (j ) = ZF RF /RS = ZS 1 + j CF RF (1250)
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RF + VS ~ CF + Vout
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Figure 1222 Passive low-pass lter
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This expression can be factored into two terms The rst is an ampli cation factor analogous to the ampli cation that would be obtained with a simple inverting ampli er (ie, the same circuit as that of Figure 1221 with the capacitor removed); the second is a low-pass lter, with a cutoff frequency dictated by the parallel combination of RF and CF in the feedback loop The ltering effect is completely analogous to that which would be attained by the passive circuit shown in Figure 1222 However, the op-amp lter also provides ampli cation by a factor of RF /RS It should be apparent that the response of this op-amp lter is just an ampli ed version of that of the passive lter Figure 1223 depicts the amplitude response of the active low-pass lter (in the gure, RF /RS = 10 and 1/RF CF = 1) in two different graphs; the rst plots the amplitude ratio Vout (j ) versus radian frequency, , on a logarithmic scale, while the second plots the amplitude ratio 20 log10 VS (j ) (in units of dB), also versus on a logarithmic scale You will recall from 6 that dB frequency response plots are encountered very frequently Note that in the dB plot, the slope of the lter response for frequencies signi cantly higher than the cutoff frequency, 0 = 1 RF CF (1251)
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is 20 dB/decade, while the slope for frequencies signi cantly lower than this cutoff frequency is equal to zero The value of the response at the cutoff frequency is found to be, in units of dB, |ALP (j 0 )|dB = 20 log10 where 20 log10 2 = 3 dB RF 20 log10 2 RS (1252)
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(1253)
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Thus, 0 is also called the 3-dB frequency
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Electronics
10 Amplitude ratio 8 6 4 2 0 10 1
Amplitude response of low-pass active filter 20 0 dB 20 40 60 80 10 1
dB amplitude response of low-pass active filter
100 101 102 103 104 Radian frequency (logarithmic scale)
100 101 102 103 104 Radian frequency (logarithmic scale)
Figure 1223 Normalized response of active low-pass lter
Among the advantages of such low-pass active lters is the ease with which the gain and the bandwidth can be adjusted by controlling the ratios RF /RS and 1/RF CF , respectively It is also possible to construct other types of lters by suitably connecting resistors and energy-storage elements to an op-amp For example, a high-pass active lter can easily be obtained by using the circuit shown in Figure 1224 Observe that the impedance of the input circuit is ZS = RS + 1 j CS (1254)
ZS RS CS + VS
+ Vout
and that of the feedback circuit is ZF = RF Then, the following gain function for the op-amp circuit can be derived: AHP (j ) = ZF RF j CS RF = = ZS RS + 1/j CS 1 + j RS CS (1256) (1255)
Figure 1224 Active high-pass lter
As approaches zero, so does the response of the lter, whereas as approaches in nity, according to the gain expression of equation 1256, the gain of the ampli er approaches a constant:
lim AHP (j ) =
RF RS
(1257)
That is, above a certain frequency range, the circuit acts as a linear ampli er This is exactly the behavior one would expect of a high-pass lter The highpass response is depicted in Figure 1225, in both linear and dB plots (in the
Amplitude response of high-pass active filter 20 0 dB 20 40 10 3
10 Amplitude ratio 8 6 4 2 0 10 3
dB amplitude response of high-pass active filter
10 2 10 1 100 101 102 Radian frequency (logarithmic scale)
10 2 10 1 100 101 102 Radian frequency (logarithmic scale)
Figure 1225 Normalized response of active high-pass lter
12
Operational Ampli ers
RF CF RS CS +
gure, RF /RS = 10, 1/RS C = 1) Note that in the dB plot, the slope of the lter response for frequencies signi cantly lower than = 1/RS CS = 1 is +20 dB/decade, while the slope for frequencies signi cantly higher than this cutoff (or 3-dB) frequency is equal to zero As a nal example of active lters, let us look at a simple active band-pass lter con guration This type of response may be realized simply by combining the high- and low-pass lters we examined earlier The circuit is shown in Figure 1226 The analysis of the band-pass circuit follows the same structure used in previous examples First, we evaluate the feedback and input impedances: ZF = RF 1 RF = j CF 1 + j CF RF 1 1 + j CS RS = j CS j CS (1258)
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