have a lower cutoff frequency equal to the larger of INC) and figure 10-12 shows the frequency response for the case i(CI) = 10 Hz and (C) = lollop The mid-band gain is assumed to he 20 dB. It is clear that = 100 Hz, because the gain at that frequency is 20 3 = 17 3rd. The gain at 10 Hz is much lower (approximately 3rd) because at frequencies below 100 Hz both CI and C1 cause gain redact n. Note that Resell-12(a) the gain asymptotic has slope 20 dB/decade (6 dBI octave) between 10 Hz and 100 Hz but breaks downward at 10 Hz with a slope of 40 Del/decade (12 Del/octave). There arc, therefore, two break frequencies in this example. i.c., two frequencies where the asymptotic changes slope 10Hz and 100 J-Iz. The phase shift is approximately 510 at = 100 Hz and 1290 at 10Hz and chess as frequency approaches O. The response characteristics shown in Figure 10-12 arc valid for two high-pass RC net works connected in series, provided the networks are isolated from each other, in the sense that one docs not load the other. In our illustration, the two networks arc assumed to be isolated by an amplifier having gain 20 Del. If the frequencies fl( CI) and fl( C1) are closer than one decade to each other, then the overall lower cutoff frequency is somewhat higher than the larger of the two. In such cases it is usually adequate to assume that /i equals the larger of the two. The exact value of II can be found using the following rather cumbersome equation (derived in Appendix C):
Crucify) = equation 10-23 can be used to show the overall lower cutoff frequency is 1.55 times the value of either. For example. if N CI) = fie = H)O Hz, j, = 155 Hz. Figure 10-13 shows the normalized gain and phase response for the special case fl( CI) = fl( C1). Note that the asymptotic breaks downward at / = II(CI) = /1(C2) and has a slope of 40 dB/decade, or 12 dB/octave. The actual response is 6 tin below the asymptotic at that frequency. The cutoff frequency is II = 1.55r where the gain is down J dB. Note that the break frequency (f) is not the same as the cutoff frequency (II) in this case. The phase shift is 900 at / and approaches WHO as frequency approaches O. Once again. the plots shown in Figure 10-13 are valid only when the two high-pass RC networks whose response they represent arc isolated from each other. In our case, we assume that the isolation is provided by an amplifier (having unity gain, since the gain in Figure IO-U(a) approaches () dB at high frequencies).The amplifier shown in Figure 10-14 has midland gain = 140. Find
1. the approximate lower cutoff frequency;
2. the gain Vol
3. the lower cutoff frequency when Cz is changed to 50 ,uF; 4. the approximate gain in dB, at 2.9 Hz with C1 = 50 ,uF: and 5. the value that C2 would have to be in order to obtain a lower cutoff frequency of approximately 100 Hz.