《电子工程师手册》学习资料（英文版）chapter 29 Active Filters

29.1 Synthesis of Low-Pass Forms Passive and Active Filters Active Filter Classification and J W. Steadman Sensitivity. Cascaded Second-Order Sections. Passive Ladder University of wyoming

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29 Active Filters Robert e. massar 29.1 Synthesis of Low-Pass Forms Passive and Active Filters Active Filter Classification and J W. Steadman Sensitivity. Cascaded Second-Order Sections. Passive Ladder University of wyoming Simulation. Active Filters for Ics B M. Wilamowski 29.2 Realization University of Wyoming Transformation from Low-Pass to Other Filter Types.Circuit Realizations James A Svoboda 29.3 Generalized Impedance Convertors and Simulated 29.1 Synthesis of Low-Pass Forms Robert e massara Passive and Active filters There are formal definitions of activity and passivity in electronics, but it is sufficient to observe that passive filters are built from passive components; resistors, capacitors, and inductors are the commonly encountered building blocks although distributed RC components, quartz crystals, and surface acoustic wave devices are sed in filters working in the high-megahertz regions. Active filters also use resistors and capacitors, but the ductors are replaced by active devices capable of producing power gain. These devices can range from single transistors to integrated circuit(IC)-controlled sources such as the operational amplifier(op amp), and more exotic devices, such as the operational transconductance amplifier(OTA), the generalized impedance converter (GIC), and the frequency-dependent negative resistor(FDNR) The theory of filter synthesis, whether active or passive, involves the determination of a suitable circuit topology and the computation of the circuit component values within the topology, such that a required network response is obtained. This response is most commonly a voltage transfer function(VTF)specified in the frequency domain Circuit analysis will allow the performance of a filter to be evaluated, and this can be don by obtaining the VTE, H(s), which is, in general, a rational function of s, the complex frequency variable. The poles of a VTF correspond to the roots of its denominator polynomial. It was established early in the histor of filter theory that a network capable of yielding complex-conjugate transfer function(TF)pole-pairs is quired to achieve high selectivity. a highly selective network is one that gives a rapid transition between passband and stopband regions of the frequency response Figure 29.1(a) gives an example of a passive low- pass LCR ladder network capable of producing a VTF with the necessary pole pattern. The network of Fig. 29.1(a) yields a VTF of the form +as+ c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 29 Active Filters 29.1 Synthesis of Low-Pass Forms Passive and Active Filters • Active Filter Classification and Sensitivity • Cascaded Second-Order Sections • Passive Ladder Simulation • Active Filters for ICs 29.2 Realization Transformation from Low-Pass to Other Filter Types • Circuit Realizations 29.3 Generalized Impedance Convertors and Simulated Impedances 29.1 Synthesis of Low-Pass Forms Robert E. Massara Passive and Active Filters There are formal definitions of activity and passivity in electronics, but it is sufficient to observe that passive filters are built from passive components; resistors, capacitors, and inductors are the commonly encountered building blocks although distributed RC components, quartz crystals, and surface acoustic wave devices are used in filters working in the high-megahertz regions. Active filters also use resistors and capacitors, but the inductors are replaced by active devices capable of producing power gain. These devices can range from single transistors to integrated circuit (IC) -controlled sources such as the operational amplifier (op amp), and more exotic devices, such as the operational transconductance amplifier (OTA), the generalized impedance converter (GIC), and the frequency-dependent negative resistor (FDNR). The theory of filter synthesis, whether active or passive, involves the determination of a suitable circuit topology and the computation of the circuit component values within the topology, such that a required network response is obtained. This response is most commonly a voltage transfer function (VTF) specified in the frequency domain. Circuit analysis will allow the performance of a filter to be evaluated, and this can be done by obtaining the VTF, H(s), which is, in general, a rational function of s, the complex frequency variable. The poles of a VTF correspond to the roots of its denominator polynomial. It was established early in the history of filter theory that a network capable of yielding complex-conjugate transfer function (TF) pole-pairs is required to achieve high selectivity. A highly selective network is one that gives a rapid transition between passband and stopband regions of the frequency response. Figure 29.1(a) gives an example of a passive lowpass LCR ladder network capable of producing a VTF with the necessary pole pattern. The network of Fig. 29.1(a) yields a VTF of the form H s (29.1) V s V s a s a s a s a s a s a ( ) ( ) ( ) = = + + + + + out in 1 5 5 4 4 3 3 2 2 1 0 Robert E. Massara University of Essex J. W. Steadman University of Wyoming B. M. Wilamowski University of Wyoming James A. Svoboda Clarkson University

rYL (a) s=0+Jo (dB)passband FIGURE 29.1 (a)Passive LCR filter;(b)typical pole plot;(c) typical frequency response FIGURE 29.2 RC-active filter equivalent to circuit of Fig. 29.1(a) Figure 29. 1(b) shows a typical pole plot for the fifth-order VtF produced by this circuit Figure 29.1(c) gives a sample sinusoidal steady-state frequency response plot. The frequency response is found by setting s jo in q(29.1)and taking H(io). The LCR low-pass ladder structure of Fig. 29.1(a)can be altered to higher or lower order simply by adding or subtracting reactances, preserving the series-inductor/shunt-capacitor pattern. In general terms, the higher the filter order, the greater the selectivity. This simple circuit structure is associated with a well-established design theory and might appear the perfect solution to the filter synthesis problem. Unfortunately, the problems introduced by the use of the inductor as a circuit component proved a serious difficulty from the outset. Inductors are intrinsically nonideal componen and the lower the frequency range of operation, the greater these problems become Problems include significant ries resistance associated with the physical structure of the inductor as a coil of wire, its ability to couple by electromagnetic induction into fields emanating from external components and sources and from other indue tors within the filter, its physical size, and potential mechanical instability. Added to these problems is the fact that the inductor tends not to be an off-the-shelf component but has instead to be fabricated to the required value as a custom device. These serious practical difficulties created an early pressure to develop alternative approaches to electrical filtering. After the emergence of the electronic amplifier based on vacuum rules, it wa discovered that networks involving resistors, capacitors, and amplifiers-RC-active filterswere capable of producing TFs exactly equivalent to those of LCR ladders. Figure 29. 2 shows a single-amplifier multiloop ladder tructure that can produce a fifth-order response identical to that of the circuit of Fig. 29.1(a) The early active filters, based as they were on tube amplifiers, did not constitute any significant advance over their passive counterparts. It required the advent of solid-state active devices to make the Rc-active filter a viable Iternative. Over the subsequent three decades, active filter theory has developed to an advanced state, and this development continues as new IC technologies create opportunities for novel network structures and applications Active Filter Classification and Sensitivity There are two major approaches to the synthesis of RC-active filters. In the first approach, a TF specification is factored into a product of second-order terms. Each of these terms is realized by a separate RC-active c 2000 by CRC Press LLC

© 2000 by CRC Press LLC Figure 29.1(b) shows a typical pole plot for the fifth-order VTF produced by this circuit. Figure 29.1(c) gives a sample sinusoidal steady-state frequency response plot. The frequency response is found by setting s = jw in Eq. (29.1) and taking *H(jw)*. The LCR low-pass ladder structure of Fig. 29.1(a) can be altered to higher or lower order simply by adding or subtracting reactances, preserving the series-inductor/shunt-capacitor pattern. In general terms, the higher the filter order, the greater the selectivity. This simple circuit structure is associated with a well-established design theory and might appear the perfect solution to the filter synthesis problem. Unfortunately, the problems introduced by the use of the inductor as a circuit component proved a serious difficulty from the outset. Inductors are intrinsically nonideal components, and the lower the frequency range of operation, the greater these problems become. Problems include significant series resistance associated with the physical structure of the inductor as a coil of wire, its ability to couple by electromagnetic induction into fields emanating from external components and sources and from other inductors within the filter, its physical size, and potential mechanical instability. Added to these problems is the fact that the inductor tends not to be an off-the-shelf component but has instead to be fabricated to the required value as a custom device. These serious practical difficulties created an early pressure to develop alternative approaches to electrical filtering. After the emergence of the electronic amplifier based on vacuum rules, it was discovered that networks involving resistors, capacitors, and amplifiers—RC-active filters—were capable of producing TFs exactly equivalent to those of LCR ladders. Figure 29.2 shows a single-amplifier multiloop ladder structure that can produce a fifth-order response identical to that of the circuit of Fig. 29.1(a). The early active filters, based as they were on tube amplifiers, did not constitute any significant advance over their passive counterparts. It required the advent of solid-state active devices to make the RC-active filter a viable alternative. Over the subsequent three decades, active filter theory has developed to an advanced state, and this development continues as new IC technologies create opportunities for novel network structures and applications. Active Filter Classification and Sensitivity There are two major approaches to the synthesis of RC-active filters. In the first approach, a TF specification is factored into a product of second-order terms. Each of these terms is realized by a separate RC-active FIGURE 29.1 (a) Passive LCR filter; (b) typical pole plot; (c) typical frequency response. FIGURE 29.2 RC-active filter equivalent to circuit of Fig. 29.1(a)

FIGURE 29.3 Biquad cascade realizing high-order filter. subnetwork designed to allow for non-interactive interconnection. The subnetworks are then connected in cascade to realize the required overall TF, as shown in Fig. 29.3. A first-order section is also required to realize dd-order TF specifications. These second-order sections may, depending on the exact form of the overall TF specification, be required to realize numerator terms of up to second order. An RC-active network capable of alizing a biquadratic TF(that is, one whose numerator and denominator polynomials are second-order)is alled a biquad This scheme has the advantage of design ease since simple equations can be derived relating the co of each section to the coefficients of each factor in the VTE. Also, each biquad can be independently adjusted relatively easily to give the correct performance. Because of these important practical merits, a large number of alternative biquad structures have been proposed, and the newcomer may easily find the choice overwhelming The second approach to active filter synthesis involves the use of RC-active circuits to simulate passive LCR ladders. This has two important advantages. First, the design process can be very straightforward: the wealth of design data published for passive ladder filters(see Further Information) can be used directly so that the sometimes difficult process of mponent value synthesis fror eliminated. Second, the LCR ladder offers optimal sensitivity properties [Orchard, 1966], and RC-active filters designed by ladder simulation share the same low sensitivity features. Chapter 4 of Bowron and Stephenson [1979] gives an excellent intro- duction to the formal treatment of circuit sensitivity to which a change in the value of any given component affects the response of the filter. High sensitivity in an RC-active filter should also alert the designer to the possibility of oscillation. a nominally stable design will be unstable in practical realization if sensitivities are such that component value errors cause one or more pairs of poles to migrate into the right half plane. Because any practical filter will be built with components that are not exactly nominal in value, sensitivity information provides a practical and useful indication of how different filter structures will react and provides a basis for comparison. Cascaded Second-Order Sections This section will introduce the cascade approach to active filter design. As noted earlier, there are a great many second-order RC-active sections to choose from, and the present treatment aims only to convey some of the main ideas involved in this strategy. The references provided at the end of this section point the reader to several comprehensive treatments of the subject Sallen and Key Section aple example of a second-order section building block [Sallen and Key, 1955]. It remain commonly used filter despite its age, and it will serve to illustrate some key stages in the design of all suc RC-active sections. The circuit is shown in Fig. 29.4. A straightforward analysis of this circuit yields a VTF K H(s) CIC2RR (29.2 C2R2C2R1CR1」CC2RR2 This is an all-pole low-pass form since the numerator involves only a constant term. c 2000 by CRC Press LLC

© 2000 by CRC Press LLC subnetwork designed to allow for non-interactive interconnection. The subnetworks are then connected in cascade to realize the required overall TF, as shown in Fig. 29.3. A first-order section is also required to realize odd-order TF specifications. These second-order sections may, depending on the exact form of the overall TF specification, be required to realize numerator terms of up to second order. An RC-active network capable of realizing a biquadratic TF (that is, one whose numerator and denominator polynomials are second-order) is called a biquad. This scheme has the advantage of design ease since simple equations can be derived relating the components of each section to the coefficients of each factor in the VTF. Also, each biquad can be independently adjusted relatively easily to give the correct performance. Because of these important practical merits, a large number of alternative biquad structures have been proposed, and the newcomer may easily find the choice overwhelming. The second approach to active filter synthesis involves the use of RC-active circuits to simulate passive LCR ladders. This has two important advantages. First, the design process can be very straightforward: the wealth of design data published for passive ladder filters (see Further Information) can be used directly so that the sometimes difficult process of component value synthesis from specification is eliminated. Second, the LCR ladder offers optimal sensitivity properties [Orchard, 1966], and RC-active filters designed by ladder simulation share the same low sensitivity features. Chapter 4 of Bowron and Stephenson [1979] gives an excellent introduction to the formal treatment of circuit sensitivity. Sensitivity plays a vital role in the characterization of RC-active filters. It provides a measure of the extent to which a change in the value of any given component affects the response of the filter. High sensitivity in an RC-active filter should also alert the designer to the possibility of oscillation. A nominally stable design will be unstable in practical realization if sensitivities are such that component value errors cause one or more pairs of poles to migrate into the right half plane. Because any practical filter will be built with components that are not exactly nominal in value, sensitivity information provides a practical and useful indication of how different filter structures will react and provides a basis for comparison. Cascaded Second-Order Sections This section will introduce the cascade approach to active filter design. As noted earlier, there are a great many second-order RC-active sections to choose from, and the present treatment aims only to convey some of the main ideas involved in this strategy. The references provided at the end of this section point the reader to several comprehensive treatments of the subject. Sallen and Key Section This is an early and simple example of a second-order section building block [Sallen and Key, 1955]. It remains a commonly used filter despite its age, and it will serve to illustrate some key stages in the design of all such RC-active sections. The circuit is shown in Fig. 29.4. A straightforward analysis of this circuit yields a VTF (29.2) This is an all-pole low-pass form since the numerator involves only a constant term. FIGURE 29.3 Biquad cascade realizing high-order filter. H s K s s C C R R C R C R K C R C C R R ( ) = + È Î Í ˘ ˚ + + ˙ + - 1 1 1 1 1 1 212 2 2 2 1 1 1 1 212 2

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

H a) FIGURE 29.11 FDNR active filter K Z(s) Z(s (29.10) where K is a constant the final filter. sind impedance by the designer and which provides the capacity to scale component values in ce transformations do not affect voltage and current transfer ratios, the VTF remains unaltered by this change. The Bruton transformation is applied directly to the elements in the prototype network, and it follows from Eq (29.10)that a resistance R transforms into a capacitance C=K/R, while an inductance L transforms into a resistance R= KL. The elimination of inductors in favor of resistors is the key purpose of the Bruton transform method. Applying the Bruton transform to a prototype circuit capacitance C gives s SC $2C S2D where d=C/K is the parameter value of a new component produced by the transformation, which is usually referred to as a frequency-dependent negative resistance(FDNR). This name results from the fact that the sinusoidal steady-state impedance Z Gjo)=-(1/o?D)is frequency-dependent, negative, and real, hence, resis tive. In practice, the FDNR elements are realized by RC-active subnetworks using op amps, normally two per FDNR. Figure 29 11(a) and (b) shows the sequence of circuit changes involved in transforming from a third- bd t lcr prototype ladder to an FDNR circuit. Figure 2911(c) gives an RC-active realization for the FDNR based on the use of a GIC, introduced in the previous subsection Active filters for Ics It was noted earlier that the advent of the IC op amp made the RC-active filter a practical reality. a typical ate-of-the-art 1960-70s active filter would involve a printed circuit board-mounted circuit comprising discrete passive components together with IC op amps. Also appearing at this time were hybrid implementations, which involve special-purpose discrete components and op amp ICs interconnected on a ceramic or glass substrate It was recognized, however, that there were considerable benefits to be had from producing an all- IC active filter. Production of a quality on-chip capacitor involves substantial chip area, so the scaling techniques referred to earlier must be used to keep capacitance values down to the low picofarad range. The consequence of this is that, unfortunately, the circuit resistance values become proportionately large so that, again, there is a chip- area problem. The solution to this dilemma emerged in the late 1970s/early 1980s with the advent of the switched-capacitor (SC) active filter. This device, a development of the active-RC filter that is specifically intended for use in IC form, replaces prototype circuit resistors with arrangements of switches and capacitors that can be shown to simulate resistances, under certain circumstances. The great merit of the scheme is that the values of the capacitors involved in this process of resistor simulation are inversely proportional to the values of the prototype resistors; thus, the final IC structure involves principal and switched capacitors that are c 2000 by CRC Press LLC

© 2000 by CRC Press LLC (29.10) where K is a constant chosen by the designer and which provides the capacity to scale component values in the final filter. Since impedance transformations do not affect voltage and current transfer ratios, the VTF remains unaltered by this change. The Bruton transformation is applied directly to the elements in the prototype network, and it follows from Eq. (29.10) that a resistance R transforms into a capacitance C = K/R, while an inductance L transforms into a resistance R = KL. The elimination of inductors in favor of resistors is the key purpose of the Bruton transform method. Applying the Bruton transform to a prototype circuit capacitance C gives (29.11) where D = C/K is the parameter value of a new component produced by the transformation, which is usually referred to as a frequency-dependent negative resistance (FDNR). This name results from the fact that the sinusoidal steady-state impedance ZT(jw) = –(1/w2 D) is frequency-dependent, negative, and real, hence, resistive. In practice, the FDNR elements are realized by RC-active subnetworks using op amps, normally two per FDNR. Figure 29.11(a) and (b) shows the sequence of circuit changes involved in transforming from a thirdorder LCR prototype ladder to an FDNR circuit. Figure 29.11(c) gives an RC-active realization for the FDNR based on the use of a GIC, introduced in the previous subsection. Active Filters for ICs It was noted earlier that the advent of the IC op amp made the RC-active filter a practical reality. A typical state-of-the-art 1960–70s active filter would involve a printed circuit board-mounted circuit comprising discrete passive components together with IC op amps. Also appearing at this time were hybrid implementations, which involve special-purpose discrete components and op amp ICs interconnected on a ceramic or glass substrate. It was recognized, however, that there were considerable benefits to be had from producing an all-IC active filter. Production of a quality on-chip capacitor involves substantial chip area, so the scaling techniques referred to earlier must be used to keep capacitance values down to the low picofarad range. The consequence of this is that, unfortunately, the circuit resistance values become proportionately large so that, again, there is a chiparea problem. The solution to this dilemma emerged in the late 1970s/early 1980s with the advent of the switched-capacitor (SC) active filter. This device, a development of the active-RC filter that is specifically intended for use in IC form, replaces prototype circuit resistors with arrangements of switches and capacitors that can be shown to simulate resistances, under certain circumstances. The great merit of the scheme is that the values of the capacitors involved in this process of resistor simulation are inversely proportional to the values of the prototype resistors; thus, the final IC structure involves principal and switched capacitors that are FIGURE 29.11 FDNR active filter. Z s K s Z s T ( ) = × ( ) Z s K s sC K s C s D T ( ) = × = = 1 1 2 2

small in magnitude and hence ideal for IC realization. a good account of SC filters is given, for example, in Schaumann et al. [1990] and in Taylor and Huang [1997]. Commonly encountered techniques for SC filter design are based on the two major design styles(biquads and ladder simulation)that have been introduced in Many commercial IC active filters are based on SC techniques, and it is also becoming usual to find custom and semicustom IC design systems that include active filter modules as components within a macrocell library that the system-level design can simply invoke where analog filtering is required within an all-analog or mixed Defining Terms Active filter: An electronic filter whose design includes one or more active devices el nator polynomials in the frequency variable Comprises a ratio of second-order numerator and denomi- Biquad: An active filter whose transfer function c tronic filter: An electronic circuit designed to transmit some range of signal frequencies while rejecting others. Phase and time-domain specifications may also occur of the extent to which a giver ffected by a component within the circuit. Related Topic erences A. Antoniou, " Realization of gyrators using operational amplifiers and their use in RC-active network synthesis, ProC.IEE,vol.116,p.1838-1850,1969 P. Bowron and F.w. Stephenson, Active Filters for Communications and Instrumentation, New York: McGraw Hill. 1979 L.T. Bruton,Network transfer functions using the concept of frequency dependent negative resistance, IEEE Trans., voL. CT-18,pp.406-408,1969 W.J. Kerwin, L.P. Huelsman, and R W. Newcomb, State-variable synthesis for insensitive integrated circuit transfer functions, IEEE J., vol. SC-2, PP. 87-92, 1967. H ]. Orchard,"Inductorless filters, " Electron. Letters, vol. 2, Pp. 224-225, 1966. PR Sallen and EL. Key, " A practical method of designing RC active filters, "IRE Trans., vol CT-2, Pp. 74-85, R Schaumann, M.S. Ghausi, and K.R. Laker, Design of Analog Filters, Englewood Cliffs, N J: Prentice-Hall, 1990 J.T. Taylor and Q. Huang, CRC Handbook of Electrical Filters, Boca Raton, Fla. CRC Press, 1997. A. Waters, Active Filter Design, New York: Macmillan, 1991 Further information Tabulations of representative standard filter specification functions appear in the sources in the References by Schaumann et al. [1990] and Bowron and Stephenson [1979), but more extensive tabulations, including pro- totype passive filter component values, are given in A I. Zverev, Handbook of Filter Synthesis(New York: John wiley, 1967). More generally, the Schaumann text provides an admirable, up-to-date coverage of filter design with an extensive list of references as does Taylor and Huang [ 1997] The field of active filter design remains active, and new developments appear in IEEE Transactions on Circuits and Systems and IEE Proceedings Part G(Circuits and Systems). The IEE publication Electronic Letters provides for short contributions. A number of international conferences(whose proceedings can be borrowed through technical libraries)feature active filter and related sessions, notably the IEEE International Symposium on Circuits and Systems(ISCAS)and the European Conference on Circuit Theory and Design(ECCtD) c 2000 by CRC Press LLC

© 2000 by CRC Press LLC small in magnitude and hence ideal for IC realization. A good account of SC filters is given, for example, in Schaumann et al. [1990] and in Taylor and Huang [1997]. Commonly encountered techniques for SC filter design are based on the two major design styles (biquads and ladder simulation) that have been introduced in this section. Many commercial IC active filters are based on SC techniques, and it is also becoming usual to find custom and semicustom IC design systems that include active filter modules as components within a macrocell library that the system-level design can simply invoke where analog filtering is required within an all-analog or mixedsignal analog/digital system. Defining Terms Active filter: An electronic filter whose design includes one or more active devices. Biquad: An active filter whose transfer function comprises a ratio of second-order numerator and denominator polynomials in the frequency variable. Electronic filter: An electronic circuit designed to transmit some range of signal frequencies while rejecting others. Phase and time-domain specifications may also occur. Sensitivity: A measure of the extent to which a given circuit performance measure is affected by a given component within the circuit. Related Topic 27.2 Applications References A. Antoniou, “Realization of gyrators using operational amplifiers and their use in RC-active network synthesis,” Proc. IEE, vol. 116, pp. 1838–1850, 1969. P. Bowron and F.W. Stephenson, Active Filters for Communications and Instrumentation, New York: McGrawHill, 1979. L.T. Bruton, “Network transfer functions using the concept of frequency dependent negative resistance,” IEEE Trans., vol. CT-18, pp. 406–408, 1969. W.J. Kerwin, L.P. Huelsman, and R.W. Newcomb, “State-variable synthesis for insensitive integrated circuit transfer functions,” IEEE J., vol. SC-2, pp. 87–92, 1967. H.J. Orchard, “Inductorless filters,” Electron. Letters, vol. 2, pp. 224–225, 1966. P.R. Sallen and E.L. Key, “A practical method of designing RC active filters,” IRE Trans., vol. CT-2, pp. 74–85, 1955. R. Schaumann, M.S. Ghausi, and K.R. Laker, Design of Analog Filters, Englewood Cliffs, N.J: Prentice-Hall, 1990. J.T. Taylor and Q. Huang, CRC Handbook of Electrical Filters, Boca Raton, Fla.: CRC Press, 1997. A. Waters, Active Filter Design, New York: Macmillan, 1991. Further Information Tabulations of representative standard filter specification functions appear in the sources in the References by Schaumann et al. [1990] and Bowron and Stephenson [1979], but more extensive tabulations, including prototype passive filter component values, are given in A. I. Zverev, Handbook of Filter Synthesis (New York: John Wiley, 1967). More generally, the Schaumann text provides an admirable, up-to-date coverage of filter design with an extensive list of references as does Taylor and Huang [1997]. The field of active filter design remains active, and new developments appear in IEEE Transactions on Circuits and Systems and IEE Proceedings Part G (Circuits and Systems). The IEE publication Electronic Letters provides for short contributions. A number of international conferences (whose proceedings can be borrowed through technical libraries) feature active filter and related sessions, notably the IEEE International Symposium on Circuits and Systems (ISCAS) and the European Conference on Circuit Theory and Design (ECCTD)

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