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

13-24 Passive, Active, and Digital FiltersR QRCRCR 1R 0–– + – + +R 2 R 3 RV 4iV oFIGURE 13.28Generalized Tow–Thomas biquad.with K 0 , a 2 , and K Q as defined in Equation 13.84 anda 3 ¼ R R 3, a 4 ¼ R 0R 4(13:86)Thus, arbitrary numerator coefficients can be predescribed. In particular, if we choose a 3 ¼ a 4 ¼ 0weobtain the low-pass filter circuit in Figure 13.27b and the transfer function in Equation 13.83.When the state-variable filters described above are used to realize high-Q filter functions, the Qpractically obtained is usually higher than that desired in the design. This effect is called Q enhancementand is caused by the phase lag introduced by the nonideal op-amps. One way to solve this problem is touse integrators with phase compensation.Figure 13.29 shows a noninverting integrator with an additional op-amp for phase lag compensation.A detailed description of this circuit can be found in Ref. [13]. Putting this noninverting integratortogether with an inverting integrator in a feedback loop results in a resonator with a Q-factor that isalmost independent of the gain-bandwidth product of the op-amps. Thus, nearly no Q enhancementoccurs.Exactly this feedback loop is used in the Åkerberg–Mossberg biquad [14]; see Figure 13.30. In thiscircuit, a noninverting integrator with phase lag compensation together with an inverting dampedintegrator is connected as a feedback loop. More details about this filter section can be found in Refs.[4,13,14].Finally, let us consider the general biquad proposed by Berka and Herpy [15]; see Figure 13.31. Thisbiquad is also based on a state-variable representation and requires a second-order differentiator and adamped integrator. One of the main advantages of this circuit is extremely low sensitivities. A detaileddescription of the filter circuit and its design can be found in Ref. [4].R 1R 1R+–C–+FIGURE 13.29Noninverting integrator using an additional op-amp to compensate for phase lag.