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

15-24 Passive, Active, and Digital FiltersA+ –1I 2I 1iRi II oI 3 I C 21 4 I osk : 1I 2VR 1RiV oV i– +V oAV o = V i I o = sk I i(a)(b)k = CR 1FIGURE 15.17General impedance converter: (a) circuit; (b) symbolic representation.transconductors (voltage-to-current converters) and are widely used in transconductance-C filters; seeSection 16.3. However, no high-quality gyrators with good performance beyond the audio range havebeen designed to date with op-amps. If op-amps are to be used, a different kind of impedance converter isemployed, one that converts a load resistor R L into an inductive impedance, such thatZ in (s) ¼ (sk)R L (15:60)A good circuit that performs this function is Antoniou’s general impedance converter (GIC) shown inFigure 15.17a. The circuit, with elements slightly rearranged, was encountered in Figure 15.4, where weused the GIC to realize a second-order bandpass function. The circuit is readily analyzed if we recall thatthe voltage measured between the op-amp input terminals and the currents flowing into the op-ampinput terminals are zero. Thus, we obtain from Figure 15.17a the set of equationsV o ¼ V iI 4sC ¼ I 3R I 2 R ¼ I 1 R 1 I 2 ¼ I 3 I i ¼ I 1 I 4 ¼ I o (15:61)These equations indicate that the terminal behavior of the general impedance converter is described byV o ¼ V i I o ¼ sCR 1 L i ¼ skI i (15:62)that isV iI i¼ Z in (s) ¼ sk V ol o¼ sk Z L (s) (15:63)Notice that the input impedance is inductive as prescribed by Equation 15.60 if the load is a resistor.*Figure 15.17b also shows the circuit symbol we will use for the GIC in the following to keep the circuitdiagrams simple. This impedance converter and its function of converting a resistive load into aninductive input impedance is the basis for Gorski-Popiel’s embedding technique [5, Chapter 6], [9,Chapter 14.4], which permits replacing the inductors in an LC filter by resistors.* To optimize the performance of the GIC, i.e., to make it optimally independent of the finite gain-bandwidth product of theop-amps, the GIC elements should be chosen as follows: For an arbitrary load Z L (s) one chooses v c C ¼ 1=*Z L (jv c )*.v c is some critical frequency, normally chosen at the upper passband corner. If the load is resistive, Z L ¼ R L , selectC ¼ 1=(v c R L ) [6].