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

15-6 Passive, Active, and Digital FiltersWe obtain readily from Equations 15.11 through 15.13 the following (rounded) component values:for T 1 : R ¼ 1:5 kV QR ¼ 2:7 kV R=k ¼ 2:7 kVfor T 2 : R ¼ 0:9 kV QR ¼ 3:6 kV R=k ¼ 1:7 kVfor T 3 : R ¼ 2:6 kV QR ¼ 10 kV R=k ¼ 1kVThe three sections are then interconnected in cascade in the order T 1 T 2 T 3 .15.3 Multiple-Loop Feedback RealizationsThese topologies are also based on biquad building blocks, which are then embedded, as the nameimplies, into multiple-loop resistive feedback configurations. The resulting coupling between sections isselected such that transfer function sensitivities are reduced below those of cascade circuits. It has beenshown that the sensitivity behavior of the different available configurations is comparable. We shall,therefore, concentrate our discussion only on the FLF and, as part of the ladder simulation techniques, onthe leapfrog (LF) topologies, which have the advantage of being relatively easy to derive without anysacrifice in performance. Our derivation will reflect the fact that both configurations* are particularlyconvenient for geometrically symmetrical bandpass functions and that the LF topology is obtained from adirect simulation of an LC lowpass ladder.15.3.1 Follow-the-Leader Feedback TopologyThe FLF topology consists of a cascade of biquads whose outputs are fed back into a summer at thefilter’s input. At the same time, the biquad outputs may be fed forward into a second summer atthe filter’s output to permit an easy realization of arbitrary transmission zeros. The actual implementationof the summers and the feedback factors is shown in Figure 15.3; if there are n noninteractingbiquads, the order of the realized transfer function H(s) is2n. Assuming that the two summer op-ampsare ideal, routine analysis yieldsV 0 ¼ R F0R inV in þ Xni¼1R F0R FiV i ¼ aV in þ Xni¼1F i V i (15:14)R FnR FiR F 2V inR F1 T 1–V 1 V 2 V i Vn+ A T 1 T 2 T i T nV 0R inR F0R o0R o1R o2R oi RonR A–+ AV outFIGURE 15.3 FLF circuit built from second-order sections T i (s) and a feedback network consisting of an op-ampsummer with resistors R Fi . Also shown is an output summer with resistors R oi to facilitate the realization of arbitrarytransmission zeros.* As are all other multiple-loop feedback circuits.