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

15-14 Passive, Active, and Digital Filtersi ig i–i 2V 3+i 4 –i 6+ +–z 1 y 2 –z 3 y 4 –z 5g oV i+ + +–V 1 –V 5 –V 0FIGURE 15.7 Signal-flow graph block diagram realizing Equation 15.42.implementing Equation 15.42 is shown in Figure 15.7. As is customary, all voltage signals are drawn atthe bottom of the diagram, and those derived from currents at the top. We observe that the circuitconsists of a number of interconnected loops of two transmittances each and that all loop-gains arenegative as required for stability. Notice that redrawing this figure in the form of Figure 15.5 results in anidentical configuration, i.e., as mentioned earlier, the leapfrog method is derived from a ladder simulationtechnique.To determine how the transmittances are to be implemented, we need to know which elements are inthe ladder arms. Consider first the simple case of an all-pole lowpass ladder where Z i ¼ 1=(sC i ) andY j ¼ 1=(sL j ), i.e., z i ¼ 1=(sC i R) and y i ¼ 1=(sL i =R) (see Figure 15.11). Evidently then, for this case alltransmittances are integrators. Suitable circuits are shown in Figure 15.8 where for each integrator wehave used two inputs in anticipation of the final realization, which has to sum two signals as indicated inFigure 15.7. The circuits realizeV o ¼ G 1V 1 þ G 2 V 2sC þ G 3(15:43)where the plus sign is valid for the series transmittances y i (s) in Figure 15.8b and the minus sign for theshunt transmittances z i (s)* in Figure 15.8a. G 3 is zero if the integration is lossless as required in aninternal branch of the ladder; in the two end branches, G 3 is used to implement the source and loadresistors of the ladder.R 3R 3R 2R 2V 2 V 2CCV 1–VV 1+R 1 +oR 1 –r–+rV o(a)(b)FIGURE 15.8(a) Inverting lossy Miller integrator; (b) noninverting lossy phase-lead integrator.* These two circuits are good candidates for building two-integrator loops as required in Figure 15.7 because the op-amp inthe Miller integrator causes a phase lag, whereas the op-amps in the noninverting integrator cause a phase lead of the samemagnitude. In the loop, these two phase shifts just cancel and cause no errors in circuit performance. Notice that theÅckerberg–Mossberg biquad in Figure 15.2 is also constructed as a loop of these two integrators.