Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

13-22 Passive, Active, and Digital FiltersR 1R 2–+R 0 C CR–R+R3–+R 6V iV 1 V 2 V 3V oR 7–+(a)R 4R 5RV 11 R 10R 11 R 101VV 1oV 3– R 12–R 13+V +3R 12 R R 14 13V 2 V 2V o(b)(c)FIGURE 13.26 (a) State-variable filter with output amplifier, (b) output amplifier with inverting and noninvertinginputs, and (c) output amplifier with three inverting inputs.If we solve Equations 13.68 through 13.70 for the three output voltages and substitute them in Equation13.79 with normalized variables s we obtainH(s) ¼ V o ja 1 js 2 þ ja 2 js þ ja 3 j¼ K 2V i D(s)(13:80)All numerator coefficients have the same sign. Therefore, the zeros of the transfer function are in the lefts-halfplane. If we set a 2 ¼ 0, i.e., if we delete the voltage divider R 12 , R 14 and ground the noninvertinginput terminal of the op-amp, we obtain zeros on the jv axis.When designing the biquad, R 14 and R 10 may be used to scale the impedance level of the two resistivesubnetworks. Then from the three numerator coefficients or from the overall gain constant of the transferfunction, the zero frequency v z , and the zero Q-factor Q z we can easily determine the remaining resistorsR 11 , R 12 , and R 13 .The output amplifier in Figure 13.26c has three inverting inputs. Summing the voltages V 1 , V 2 , and V 3leads to a numerator polynomial where the sign of the middle coefficient is different from the sign of theother two. Thus, the zeros are in the right-s-half plane. Again, we can delete the resistor R 12 to realize zeroon the jv axis.A second state-variable biquad circuit proposed by Tow and Thomas [10–12] yields similar performanceto that of the Kerwin–Huelsman–Newcomb circuit. It uses a feedback loop with one dampedintegrator, one integrator, and one inverting amplifier; see Figure 13.27a. Figure 13.27b shows the Tow–Thomas circuit with three op-amps.The damped integrator has a transfer functionV 1V 3¼1sT þ aVi¼0(13:81)