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

12-2 Passive, Active, and Digital FiltersI 33I 1 I1 Passive 2 2networkV 1 V 2–+V3FIGURE 12.1General infinite-gain single-amplifier structure.For the amplifier, ideal except for finite gain AV 3 ¼ AV 2 (12:4)Noting that I 2 ¼ 0, the above equations reduce to the following expression for the voltage transferfunction:V 3 y 31¼V 1 y 32 þ y 33A(12:5)As A !1, which we can expect at low frequencies, the above expression reduces to the more familiarV 3V 1¼ y 31y 32(12:6)Theoretically, a wide range of transfer characteristics can be realized by appropriate synthesis of thepassive network [1]. However, it is not advisable to extend synthesis beyond second-order functions forstructures containing only one operational amplifier due to the ensuing problems of sensitivity andtuning. Furthermore, notch functions require double-element replacements [2] or parallel ladderarrangements [3], which are nontrivial to design, and whose performance is inferior to that resultingfrom other topologies such as those discussed in Chapters 13 and 14.While formal synthesis techniques could be used to meet particular requirements, the most commonapproach is to use a double-ladder realization of the passive network, as shown in Figure 12.2. Thisarrangement, commonly referred to as the multiple-loop feedback (MFB) structure, is described by thefollowing voltage transfer ratio:V 3Y 1 Y 3¼(12:7)V 1 Y 5 (Y 1 þ Y 2 þ Y 3 þ Y 4 ) þ Y 3 Y 4Y 4 Y 5Y 1 Y 3–+Y V1 2V 3FIGURE 12.2General double ladder multiple-feedback network.