FIGURE 29.6 Form of fifth-order Sallen and Key cascade. Clearly, the designer has some degrees of freedom here since there are two equations in five unknowns Choosing to set both(normalized) capacitor values to unity, and fixing the dc stage gain K= 5, gives C1=C2=1F;R1=1.8134gR2=137059;R=4g;R,=1g Note that Eq.(29.5)is a normalized specification giving a filter cut-off frequency of 1 rad s-l. These normalized component values can now be denormalized to give a required cut-off frequency and practical omponent values. Suppose that the filter is, in fact, required to give a cut-off frequency f= 1 kHz. The necessary shift is produced by multiplying all the capacitors (leaving the resistors fixed)by the factor o/op where ON is the normalized cut-off frequency(1 rad s- here)and @p is the required denormalized cut-off frequency(2T X 1000 rad s-). Applying this results in denormalized capacitor values of 159.2 uE. A useful rule of thumb [Waters, 1991] advises that capacitor values should be on the order of magnitude of (10/f)uF, which suggests that the capacitors should be further scaled to around 10 nF. This can be achieved without altering of the filters fo by means of the impedance scaling property of electrical circuits. Providing all circuit impedances are scaled by the same amount, current and voltage TFs are preserved. In an RC-active circuit, this requires that all resistances are multiplied by some factor while all capacitances are divided by it(since capacitive impedance is proportional to 1/0). Applying this process yields final values as follows C,G2=10nF;R1=29.86kg2;R2=21.81kgRx=6366kR=15.92kg Note also that the de gain of each stage, H(o), is given by K[see Eq (29.2)and Fig 29.4] and, when several stages are cascaded, the overall dc gain of the filter will be the product of these individual stage gains. This feature of the Sallen and Key structure gives the designer the ability to combine easy-to-manage amplification with prescribed filtering Realization of the complete fifth-order Chebyshev VTF requires the design of another second-order section to deal with the second quadratic term in Eq.(29.5), together with a simple circuit to realize the first-order term arising because this is an odd-order VTE. Figure 29. 6 shows the form of the overall cascade. Note that the op amps at the output of each stage provide the necessary interstage isolation. It is finally worth noting that an extended single-amplifier form of the Sallen and Key network exists-the circuit shown in Fig. 29.2 is an cample of this-but that the saving in op amps is paid for by higher component spreads, sensitivities, and design complexity. State-Variable Biquad The simple Sallen and Key filter provides only an all-pole TE; many commonly encountered filter specifications are of this form-the Butterworth and Chebyshev approximations are notable examples--so this is not a serious limitation. In general, however, it will be necessary to produce sections capable of realizing a second-order denominator together with a numerator polynomial of up to second-orde (29.7) The other major filter approximation in common use-the elliptic (or Cauer)function filter--involves quadratic numerator terms in whic b, coefficient in Eq.(29.7)is missing. The resulting numerator c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Clearly, the designer has some degrees of freedom here since there are two equations in five unknowns. Choosing to set both (normalized) capacitor values to unity, and fixing the dc stage gain K = 5, gives C1 = C2 = 1F; R1 = 1.8134 W; R2 = 1.3705 W; Rx = 4 W; Ry = 1 W Note that Eq. (29.5) is a normalized specification giving a filter cut-off frequency of 1 rad s–1. These normalized component values can now be denormalized to give a required cut-off frequency and practical component values. Suppose that the filter is, in fact,required to give a cut-off frequency fc= 1 kHz. The necessary shift is produced by multiplying all the capacitors (leaving the resistors fixed) by the factor wN/wD where wN is the normalized cut-off frequency (1 rad s–1 here) and wD is the required denormalized cut-off frequency (2p ¥ 1000 rad s–1). Applying this results in denormalized capacitor values of 159.2 mF. A useful rule of thumb [Waters, 1991] advises that capacitor values should be on the order of magnitude of (10/fc ) mF, which suggests that the capacitors should be further scaled to around 10 nF. This can be achieved without altering of the filter’s fc , by means of the impedance scaling property of electrical circuits. Providing all circuit impedances are scaled by the same amount, current and voltage TFs are preserved. In an RC-active circuit, this requires that all resistances are multiplied by some factor while all capacitances are divided by it (since capacitive impedance is proportional to 1/C). Applying this process yields final values as follows: C1, C2 = 10 nF; R1 = 29.86 kW; R2 = 21.81 kW; Rx = 63.66 kW; Ry = 15.92 kW Note also that the dc gain of each stage, *H(0)*, is given by K [see Eq. (29.2) and Fig. 29.4] and, when several stages are cascaded, the overall dc gain of the filter will be the product of these individual stage gains. This feature of the Sallen and Key structure gives the designer the ability to combine easy-to-manage amplification with prescribed filtering. Realization of the complete fifth-order Chebyshev VTF requires the design of another second-order section to deal with the second quadratic term in Eq. (29.5), together with a simple circuit to realize the first-order term arising because this is an odd-order VTF. Figure 29.6 shows the form of the overall cascade. Note that the op amps at the output of each stage provide the necessary interstage isolation. It is finally worth noting that an extended single-amplifier form of the Sallen and Key network exists—the circuit shown in Fig. 29.2 is an example of this—but that the saving in op amps is paid for by higher component spreads, sensitivities, and design complexity. State-Variable Biquad The simple Sallen and Key filter provides only an all-pole TF; many commonly encountered filter specifications are of this form—the Butterworth and Chebyshev approximations are notable examples—so this is not a serious limitation. In general, however, it will be necessary to produce sections capable of realizing a second-order denominator together with a numerator polynomial of up to second-order: (29.7) The other major filter approximation in common use—the elliptic (or Cauer) function filter—involves quadratic numerator terms in which the b1 coefficient in Eq. (29.7) is missing. The resulting numerator FIGURE 29.6 Form of fifth-order Sallen and Key cascade. H s b s b s b s a s a ( ) = + + + + 2 2 1 0 2 1 0
polynomial, of the form b2 3+ bo gives rise to s-plane zeros on the jo axis corresponding to points in the stopband of the sinusoidal frequency response where the filter's transmission goes to zero. These notches or mission zeros account for the elliptic's very rapid transition from passband to stopband and, hence, its optimal selectivity A filter structure capable of producing a VTF of the form of Eq (29.7)was introduced as a state-variable real- ization by its originators [Kerwin et al, 1967 The struc- ture comprises two op amp integrators and an op amp summer connected in a loop and was based on the inte- grator-summer analog computer used in control/analo AC stems analysis, where system state is characterized by some set of so-called state variables. It is also often referred to as a ring-of-three structure Many subsequent FIGURE 29.7 Circuit schematic for state-variable refinements of this design have appeared(Schaumann biquad et al.,[ 1990 gives a useful treatment of some of these evelopments)and the state-variable biquad has achieved considerable popularity as the basis of many com- mercial universal packaged active filter building blocks. By selecting appropriate chip/package output terminal and with the use of external trimming components, a very wide range of filter responses can be obtained. Figure 29.7 shows a circuit developed from this basic state-variable network and described in Schaumann et al. [1990]. The circuit yields a VTF H(s) Vout (s) As +Oo(B-D)s Eoo /RC (29.8) +0 By an appropriate choice of the circuit component values, a desired VTF of the form of Eq (29.8)can be realized Consider, for example, a specification requirement for a second-order elliptic filter cutting off at Assume that a suitable normalized (1 rad/s)specification for the VTF is 056772+7464) (299 s2+0.9989s+1.1701 From Eq.(29.8)and Eq.(29.9), and referring to Fig. 29.7, normalized values for the components are mputed as follows. As the s term in the numerator is to be zero, set B=D=0(which obtains if resistors R/B and R/d are simply removed from the circuit). Setting C= l F gives the following results AC=0.15677F;R=1/CO0=0.924462;QR=1.082909;R/E=0.92446 Removing the normalization and setting C=(10/10 k)uF=1 nF requires capacitors to be multiplied by 10-9 and resistors to be multiplied by 15.9155 x 10. Final denormalized component values for the 10-kHz filter are C=1nF;AC=0.15677nF;R=R/E=14713kg;QR=17.235k9 Passive ladder simulation As for the biquad approach, numerous different ladder-based design methods have been proposed. Two representative schemes will be considered here: inductance simulation and ladder transformation. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC polynomial, of the form b2 s 2 + b0, gives rise to s-plane zeros on the jw axis corresponding to points in the stopband of the sinusoidal frequency response where the filter’s transmission goes to zero. These notches or transmission zeros account for the elliptic’s very rapid transition from passband to stopband and, hence, its optimal selectivity. A filter structure capable of producing a VTF of the form of Eq. (29.7) was introduced as a state-variable realization by its originators [Kerwin et al., 1967]. The structure comprises two op amp integrators and an op amp summer connected in a loop and was based on the integrator-summer analog computer used in control/analog systems analysis, where system state is characterized by some set of so-called state variables. It is also often referred to as a ring-of-three structure. Many subsequent refinements of this design have appeared (Schaumann et al., [1990] gives a useful treatment of some of these developments) and the state-variable biquad has achieved considerable popularity as the basis of many commercial universal packaged active filter building blocks. By selecting appropriate chip/package output terminals, and with the use of external trimming components, a very wide range of filter responses can be obtained. Figure 29.7 shows a circuit developed from this basic state-variable network and described in Schaumann et al. [1990]. The circuit yields a VTF (29.8) By an appropriate choice of the circuit component values, a desired VTF of the form of Eq. (29.8) can be realized. Consider, for example, a specification requirement for a second-order elliptic filter cutting off at 10 kHz. Assume that a suitable normalized (1 rad/s) specification for the VTF is (29.9) From Eq. (29.8) and Eq. (29.9), and referring to Fig. 29.7, normalized values for the components are computed as follows. As the s term in the numerator is to be zero, set B = D = 0 (which obtains if resistors R/B and R/D are simply removed from the circuit). Setting C = 1 F gives the following results: AC = 0.15677F ; R = 1/Cw0 = 0.92446 W; QR = 1.08290 W; R/E = 0.92446 W Removing the normalization and setting C = (10/10 k) mF = 1 nF requires capacitors to be multiplied by 10–9 and resistors to be multiplied by 15.9155 ¥ 103 . Final denormalized component values for the 10-kHz filter are thus: C = 1 nF ; AC = 0.15677 nF ; R = R/E = 14.713 kW; QR = 17.235 kW Passive Ladder Simulation As for the biquad approach, numerous different ladder-based design methods have been proposed. Two representative schemes will be considered here: inductance simulation and ladder transformation. FIGURE 29.7 Circuit schematic for state-variable biquad. H s V s V s As B D s E s Q s ( ) RC ( ) ( ) ( ) = = - , + - + + + out in with / 2 0 0 2 2 0 0 2 0 1 w w w w w D H s s s s ( ) . ( . ) . . = - + + + 0 15677 7 464 0 9989 1 1701 2 2
converter/inverter K(s) real Impedance Converter(NIC) complex: Generalized Impedance Converter(GIC) converter:Zin=(s).Zt inverter classes. K(o/ real, positive: Positive Impedance Inverter(Pll or Gyrator) real, negative: Negative Impedance Inverter(NII FIGURE 29.8 Generic impedance converter/inverter networks. cm「zu-"2=a S CR2 indici hean ied uts k of effect th gyration resistance of the gyrator FIGuRE 29.9 gyrator simulation of an inductor. (a) 1 p p l FIGURE 29.10(a)Practical gyrator and( b)simulation of floating inductor. ( Source: A Antoniou, Proc. IEE, vol. 116, Pp 1838-1850, 1969. With permission. Inductance simulation In the inductance simulation approach, use is made of impedance converter/inverter networks. Figure 29.8 gives a classification of the various generic forms of device. The NIC enjoyed prominence in the early days of active filters but was found to be prone to instability. Two classes of device that have proved more useful in the longer term are the GiC and the gyrator Figure 29.9 introduces the symbolic representation of a gyrator and shows its use in simulating an inductor The gyrator can conveniently be realized by the circuit of Fig. 29.10(a), but note that the simulated inductor is grounded at one end. This presents no problem in the case of high-pass filters and other forms requiring a grounded shunt inductor but is not suitable for the low-pass filter. Figure 29.10(b)shows how a pair of back to-back gyrators can be configured to produce a floating inductance, but this involves four op amps per inductor The next section will introduce an alternative approach that avoids the op amp count difficulty associated with simulating the floating inductors directly. Ladder Transformation The other main approach to the RC-active simulation of passive ladders involves the transformation of a prototype ladder into a form suitable for active realization. A most effective method of this class is based on the use of the Bruton transformation [Bruton, 1969], which involves the complex impedance scaling of prototype passive LCR ladder network. All prototype circuit impedances Z(s)are transformed to Z,(s) with c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Inductance Simulation In the inductance simulation approach, use is made of impedance converter/inverter networks. Figure 29.8 gives a classification of the various generic forms of device. The NIC enjoyed prominence in the early days of active filters but was found to be prone to instability. Two classes of device that have proved more useful in the longer term are the GIC and the gyrator. Figure 29.9 introduces the symbolic representation of a gyrator and shows its use in simulating an inductor. The gyrator can conveniently be realized by the circuit of Fig. 29.10(a), but note that the simulated inductor is grounded at one end. This presents no problem in the case of high-pass filters and other forms requiring a grounded shunt inductor but is not suitable for the low-pass filter. Figure 29.10(b) shows how a pair of backto-back gyrators can be configured to produce a floating inductance, but this involves four op amps per inductor. The next section will introduce an alternative approach that avoids the op amp count difficulty associated with simulating the floating inductors directly. Ladder Transformation The other main approach to the RC-active simulation of passive ladders involves the transformation of a prototype ladder into a form suitable for active realization. A most effective method of this class is based on the use of the Bruton transformation [Bruton, 1969], which involves the complex impedance scaling of a prototype passive LCR ladder network. All prototype circuit impedances Z(s) are transformed to ZT(s) with FIGURE 29.8 Generic impedance converter/inverter networks. FIGURE 29.9 Gyrator simulation of an inductor. FIGURE 29.10 (a) Practical gyrator and (b) simulation of floating inductor. (Source: A. Antoniou, Proc. IEE, vol. 116, pp. 1838–1850, 1969. With permission.)