Kennedy, E.J., Wait, J.V. “Operational Amplifiers” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 27 Operational Amplifiers 27.1Ideal and Practical Models The Ideal Op Amp?Practical Op Amps?SPICE Computer Models 27.2Applications Noninverting Circuits 27.1 Ideal and Practical Models E.J. Kennedy The concept of the operational amplifier (usually referred to as an op amp) originated at the beginning of the Second World War with the use of vacuum tubes in dc amplifier designs developed by the George A. Philbrick Co. [some of the early history of operational amplifiers is found in Williams, 1991]. The op amp was the basic building block for early electronic servomechanisms, for synthesizers, and in particular for analog computers used to solve differential equations. With the advent of the first monolithic integrated-circuit (IC) op amp in 1965 (the mA709, designed by the late Bob Widlar, then with Fairchild Semiconductor), the availability of op amps was no longer a factor, while within a few years the cost of these devices (which had been as high as $200 each) rapidly plummeted to close to that of individual discrete transistors. Although the digital computer has now largely supplanted the analog computer in mathematically intensive applications, the use of inexpensive operational amplifiers in instrumentation applications, in pulse shaping, in filtering, and in signal processing applications in general has continued to grow. There are currently many commercial manufacturers whose main products are high-quality op amps. This competitiveness has ensured a marketplace featuring a wide range of relatively inexpensive devices suitable for use by electronic engineers, physicists, chemists, biologists, and almost any discipline that requires obtaining quantitative analog data from instrumented experiments. Most operational amplifier circuits can be analyzed, at least for first-order calculations, by considering the op amp to be an “ideal” device. For more quantitative information, however, and particularly when frequency response and dc offsets are important, one must refer to a more “practical” model that includes the internal limitations of the device. If the op amp is characterized by a really complete model, the resulting circuit may be quite complex, leading to rather laborious calculations. Fortunately, however, computer analysis using the program SPICE significantly reduces the problem to one of a simple input specification to the computer. Today, nearly all the op amp manufacturers provide SPICE models for their line of devices, with excellent correlation obtained between the computer simulation and the actual measured results. The Ideal Op Amp An ideal operational amplifier is a dc-coupled amplifier having two inputs and normally one output (although in a few infrequent cases there may be a differential output). The inputs are designated as noninverting (designated + or NI) and inverting (designated – or Inv.). The amplified signal is the differential signal, v e , between the two inputs, so that the output voltage as indicated in Fig. 27.1 is E.J. Kennedy University of Tennessee John V. Wait University of Arizona (Retired) ? 2000 by CRC Press LLC (27.1) The general characteristics of an ideal op amp can be summarized as follows: 1. The open-loop gain A OL is infinite. Or, since the output signal v out is finite, then the differential input signal v e must approach zero. 2.The input resistance R IN is infinite, while the output resistance R O is zero. 3.The amplifier has zero current at the input (i A and i B in Fig. 27.1 are zero), but the op amp can either sink or source an infinite current at the output. 4.The op amp is not sensitive to a common signal on both inputs (i.e., v A = v B ); thus, the output voltage change due to a common input signal will be zero. This common signal is referred to as a common- mode signal, and manufacturers specify this effect by an op amp’s common-mode rejection ratio (CMRR), which relates the ratio of the open-loop gain (A OL ) of the op amp to the common-mode gain (A CM ). Hence, for an ideal op amp CMRR = ￥. 5.A somewhat analogous specification to the CMRR is the power-supply rejection ratio (PSRR), which relates the ratio of a power supply voltage change to an equivalent input voltage change produced by the change in the power supply. Because an ideal op amp can operate with any power supply, without restriction, then for the ideal device PSRR = ￥. 6.The gain of the op amp is not a function of frequency. This implies an infinite bandwidth. Although the foregoing requirements for an ideal op amp appear to be impossible to achieve practically, modern devices can quite closely approximate many of these conditions. An op amp with a field-effect transistor (FET) on the input would certainly not have zero input current and infinite input resistance, but a current of <10 pA and an R IN = 10 12 W is obtainable and is a reasonable approximation to the ideal conditions. Further, although a CMRR and PSRR of infinity are not possible, there are several commercial op amps available with values of 140 dB (i.e., a ratio of 10 7 ). Open-loop gains of several precision op amps now have reached values of >10 7 , although certainly not infinity. The two most difficult ideal conditions to approach are the ability to handle large output currents and the requirement of a gain independence with frequency. Using the ideal model conditions it is quite simple to evaluate the two basic op amp circuit configurations, (1) the inverting amplifier and (2) the noninverting amplifier, as designated in Fig. 27.2. For the ideal inverting amplifier, since the open-loop gain is infinite and since the output voltage v o is finite, then the input differential voltage (often referred to as the error signal) v e must approach zero, or the input current is (27.2) The feedback current i F must equal i I , and the output voltage must then be due to the voltage drop across R F , or (27.3) FIGURE 27.1 Configuration for an ideal op amp. vAvv out OL B A =-() i vv R v R I I I = - = - e 11 0 viRviR R R v oFF IF F I =- + =- =- ? è ? ? ? ÷e 1 ? 2000 by CRC Press LLC The inverting connection thus has a voltage gain v o /v I of – R F /R 1 , an input resistance seen by v I of R 1 ohms [from Eq. (27.2)], and an output resistance of 0 W. By a similar analysis for the noninverting circuit of Fig. 27.2(b), since v e is zero, then signal v I must appear across resistor R 1 , producing a current of v I /R 1 , which must flow through resistor R F . Hence the output voltage is the sum of the voltage drops across R F and R 1 , or (27.4) As opposed to the inverting connection, the input resistance seen by the source v I is now equal to an infinite resistance, since R IN for the ideal op amp is infinite. Practical Op Amps A nonideal op amp is characterized not only by finite open-loop gain, input and output resistance, finite currents, and frequency bandwidths, but also by various nonidealities due to the construction of the op amp circuit or external connections. A complete model for a practical op amp is illustrated in Fig. 27.3. The nonideal FIGURE 27.2 Illustration of (a) the inverting amplifier and (b) the noninverting amplifier. (Source: E.J. Kennedy, Opera- tional Amplifier Circuits, Theory and Applications, New York: Holt, Rinehart and Winston, 1988, pp. 4, 6. With permission.) vR v R v R R v oF I I F I = ? è ? ? ? ÷ +=+ ? è ? ? ? ÷ 11 1 ? 2000 by CRC Press LLC effects of the PSRR and CMRR are represented by the input series voltage sources of DV supply /PSRR and V CM /CMRR, where DV supply would be any total change of the two power supply voltages, V + dc and V – dc , from their nominal values, while V CM is the voltage common to both inputs of the op amp. The open-loop gain of the op amp is no longer infinite but is modeled by a network of the output impedance Z out (which may be merely a resistor but could also be a series R-L network) in series with a source A(s), which includes all the open-loop poles and zeroes of the op amp as (27.5) where A OL is the finite dc open-loop gain, while poles are at frequencies w p1 , w p2 , . . . and zeroes are at w Z 1 , etc. The differential input resistance is Z IN , which is typically a resistance R IN in parallel with a capacitor C IN . Similarly, the common-mode input impedance Z CM is established by placing an impedance 2Z CM in parallel FIGURE 27.3 A model for a practical op amp illustrating nonideal effects. (Source: E.J. Kennedy, Operational Amplifier Circuits, Theory and Applications, New York: Holt, Rinehart and Winston, 1988, pp. 53, 126. With permission.) As A s ss OL Z pp () () () = + ? è ? ? ? ÷ + ××× + ? è ? ? ? ÷ + ? è ? ? ? ÷ + ××× 11 111 1 12 w ww ? 2000 by CRC Press LLC with each input terminal. Normally, Z CM is best represented by a parallel resistance and capacitance of 2R CM (which is >> R IN ) and C CM /2. The dc bias currents at the input are represented by I B + and I B – current sources that would equal the input base currents if a differential bipolar transistor were used as the input stage of the op amp, or the input gate currents if FETs were used. The fact that the two transistors of the input stage of the op amp may not be perfectly balanced is represented by an equivalent input offset voltage source, V OS , in series with the input. The smallest signal that can be amplified is always limited by the inherent random noise internal to the op amp itself. In Fig. 27.3 the noise effects are represented by an equivalent input voltage source (ENV), which when multiplied by the gain of the op amp would equal the total output noise present if the inputs to the op amp were shorted. In a similar fashion, if the inputs to the op amp were open circuited, the total output noise would equal the sum of the noise due to the equivalent input current sources (ENI + and ENI – ), each multiplied by their respective current gain to the output. Because noise is a random variable, this summation must be accomplished in a squared fashion, i.e., (27.6) Typically, the correlation (C) between the ENV and ENI sources is low, so the assumption of C ? 0 can be made. For the basic circuits of Fig. 27.2(a) or (b), if the signal source v I is shorted then the output voltage due to the nonideal effects would be (using the model of Fig. 27.3) (27.7) provided that the loop gain (also called loop transmission in many texts) is related by the inequality (27.8) Inherent in Eq. (27.8) is the usual condition that R 1 << Z IN and Z CM . If a resistor R 2 were in series with the noninverting input terminal, then a corresponding term must be added to the right hand side of Eq. (27.7) of value –I B + R 2 (R 1 + R F )/R 1 . On manufacturers’ data sheets the individual values of I B + and I B – are not stated; instead the average input bias current and offset current are specified as (27.9) The output noise effects can be obtained using the model of Fig. 27.3 along with the circuits of Fig. 27.2 as (27.10) where it is assumed that a resistor R 2 is also in series with the noninverting input of either Fig. 27.2(a) or (b). The thermal noise (often called Johnson or Nyquist noise) due to the resistors R 1 , R 2 , and R F is given by (in rms volt 2 /Hz) EAAA Ov 2222 1 22 12 2 rms volt /Hz ENV ENI ENI 2 ( ) =+ + +- () () () vV V V R R IR oOS CM F BF =+ + ? è ? ? ? ÷ + ? è ? ? ? ÷ + - CMRR PSRR supply D 1 1 R RR As F 1 1 1 + ? è ? ? ? ÷ >>() I II III B BB BB = + =- +- +- 2 ; offset ** EE R R EE R R RR R R F F F F F out rms volts /Hz ENV ENI ENI 22 1 2 1 2 22 2 2 1 2 22 2 2 2 1 2 11 () () () () = ? è ? ? ? ÷ ++ + ′ + ? è ? ? ? ÷ ++ + ? è ? ? ? ÷ -+ ? 2000 by CRC Press LLC (27.11) where k is Boltzmann’s constant and T is absolute temperature (°Kelvin). To obtain the total output noise, one must multiply the E 2 out expression of Eq. (27.10) by the noise bandwidth of the circuit, which typically is equal to p/2 times the –3 dB signal bandwidth, for a single-pole response system [Kennedy, 1988]. SPICE Computer Models The use of op amps can be considerably simplified by computer-aided analysis using the program SPICE. SPICE originated with the University of California, Berkeley, in 1975 [Nagel, 1975], although more recent user-friendly commercial versions are now available such as HSPICE, HPSPICE, IS-SPICE, PSPICE, and ZSPICE, to mention a few of those most widely used. A simple macromodel for a near-ideal op amp could be simply stated with the SPICE subcircuit file (* indicates a comment that is not processed by the file) .SUBCKT IDEALOA 1 2 3 *A near-ideal op amp: (1) is noninv, (2) is inv, and (3) is output. RIN 1 2 1E12 E1 (3, 0) (1, 2) 1E8 .ENDS IDEALOA (27.12) The circuit model for IDEALOA would appear as in Fig. 27.4(a). A more complete model, but not including nonideal offset effects, could be constructed for the 741 op amp as the subcircuit file OA741, shown in Fig. 27.4(b). .SUBCKT OA741 1 2 6 *A linear model for the 741 op amp: (1) is noninv, (2) is inv, and *(6) is output. RIN = 2MEG, AOL = 200,000, ROUT = 75 ohm, *Dominant open - loop pole at 5 Hz, gain - bandwidth product *is 1 MHz. RIN 1 2 2MEG E1 (3, 0) (1, 2) 2E5 R1 3 4 100K C1 4 0 0.318UF ; R1 2 C1 = 5HZPOLE E2 (5, 0) (4, 0) 1.0 ROUT 5 6 75 .ENDS OA741 (27.13) The most widely used op amp macromodel that includes dc offset effects is the Boyle model [Boyle et al., 1974]. Most op amp manufacturers use this model, usually with additions to add more poles (and perhaps zeroes). The various resistor and capacitor values, as well as transistor, and current and voltage generator, values are intimately related to the specifications of the op amp, as shown earlier in the nonideal model of Fig. 27.3. The appropriate equations are too involved to list here; instead, the interested reader is referred to the article by Boyle in the listed references. The Boyle model does not accurately model noise effects, nor does it fully model PSRR and CMRR effects. A more circuits-oriented approach to modeling op amps can be obtained if the input transistors are removed and a model formed by using passive components along with both fixed and dependent voltage and current sources. Such a model is shown in Fig. 27.5. This model not only includes all the basic nonideal effects of the op amp, allowing for multiple poles and zeroes, but can also accurately include ENV and ENI noise effects. EkTR EkTR EkTR FF 1 2 1 2 2 2 2 4 4 4 = = = ? 2000 by CRC Press LLC The circuits-approach macromodel can also be easily adapted to current-feedback op amp designs, whose input impedance at the noninverting input is much greater than that at the inverting input [see Williams, 1991]. The interested reader is referred to the text edited by J. Williams, listed in the references, as well as the SPICE modeling book by Connelly and Choi [1992]. FIGURE 27.4Some simple SPICE macromodels. (a) A near ideal op amp. (b) A linear model for a 741 op amp. (c) The Boyle macromodel. ? 2000 by CRC Press LLC A comparison of the SPICE macromodels with actual manufacturer’s data for the case of an LM318 op amp is demonstrated in Fig. 27.6, for the open-loop gain versus frequency specification. Defining Terms Boyle macromodel:A SPICE computer model for an op amp. Developed by G.R. Boyle in 1974. Equivalent noise current (ENI):A noise current source that is effectively in parallel with either the nonin- verting input terminal (ENI + ) or the inverting input terminal (ENI – ) and represents the total noise contributed by the op amp if either input terminal is open circuited. Equivalent noise voltage (ENV): A noise voltage source that is effectively in series with either the inverting or noninverting input terminal of the op amp and represents the total noise contributed by the op amp if the inputs were shorted. Ideal operational amplifier: An op amp having infinite gain from input to output, with infinite input resistance and zero output resistance and insensitive to the frequency of the signal. An ideal op amp is useful in first-order analysis of circuits. Operational amplifier (op amp):A dc amplifier having both an inverting and noninverting input and normally one output, with a very large gain from input to output. SPICE:A computer simulation program developed by the University of California, Berkeley, in 1975. Versions are available from several companies. The program is particularly advantageous for electronic circuit analysis, since dc, ac, transient, noise, and statistical analysis is possible. Related Topic 13.1 Analog Circuit Simulation FIGURE 27.5A SPICE circuits-approach macromodel. V2D 6 D 5 I sc + I sc – R 01 R 02 G 0 = I/R 02 D 4 D 3 E 2 V OUT L 0 V1 (7) (18)(17)(14)(13) (3) (+) (–) (Input) (2) (1) (4) (7) (10) (3) (9) (11) (12) (5) (15) (23) (16) (6)(19) (20) (6) (21) (22) RpsIps V CC Cp4 Rp4 Rp3 Rp2 VN VP DN2 DP2 DN1 ECMRR EPSRR CCMI RCMI CCM2 RCM2 DP1 RN1 CN2 RN2>>RN1 Rp1 Rslew E1=1xV(15) E 2 =1xV(6) Cp3 Cp2 Cp1 Rz1 G 3 G 2 G 1 G 4 D2 D 1 V EE V OS I s – I s + V CC C IN R IN -V EE (4) +– +–+–+– +– +– –++ – + – I slew + I slew – ? 2000 by CRC Press LLC References G.R. Boyle et al., “Macromodeling of integrated circuit operational amplifiers,” IEEE J. S. S. Circuits, pp. 353–363, 1974. J.A. Connelly and P. Choi, Macromodeling with SPICE, Englewood Cliffs, N.J.: Prentice-Hall, 1992. FIGURE 27.6Comparison between manufacturer’s data and the SPICE macromodels. ? 2000 by CRC Press LLC E.J. Kennedy, Operational Amplifier Circuits, Theory and Applications, New York: Holt, Rinehart and Winston, 1988. L.W. Nagel, SPICE 2: A Computer Program to Simulate Semiconductor Circuits, ERL-M520, University of Cali- fornia, Berkeley, 1975. J. Williams (ed.), Analog Circuit Design, Boston: Butterworth-Heinemann, 1991. 27.2 Applications John V. Wait In microminiature form (epoxy or metal packages or as part of a VLSI mask layout) the operational amplifier (op amp) is usually fabricated in integrated circuit (IC) form. The general environment is shown in Fig. 27.7. A pair of + and – regulated power supplies (or batteries) may supply all of the op amp in a system, typically with ±10 – ±15 V. The ground and power supply buses are usually assumed, and an individual op-amp symbol is shown in Fig. 27.8. Such amplifiers feature: 1.A high voltage gain, down to and including dc, and a dc open loop gain of perhaps 10 5 (100 dB) or more 2.An inverting (–) and noninverting (+) symbol 3.Minimized dc offsets, a high input impedance, and a low output impedance 4.An output stage able to deliver or absorb currents over a dynamic range approaching the power supply voltages It is important never to use the op amp without feedback between the output and inverting terminals at all frequencies. A simple inverting amplifier is shown in Fig. 27.9. Here the voltage gain is V out /V in = –K = –R F /R 1 The circuit gain is determined essentially by the external resistances, within the bandwidth and output-driving capabilities of the op amp (more later). If R F = R 1 = R, we have the simple unity gain inverter of Fig. 27.10. Figure 27.11 shows a more flexible summer-inverter circuit with v 0 = –(K 1 v 1 + K 2 v 2 + . . . + K n v n ) where K i = R F /R i . FIGURE 27.7Typical operational amplifier environment. ? 2000 by CRC Press LLC FIGURE 27.8 Conventional operational amplifier symbol. Only active signal lines are shown, and all sig- nals are referenced to ground. FIGURE 27.9 Simple resistive inverter-amplifier. FIGURE 27.10 A simple unity gain inverter, showing (a) detailed circuit; (b) block-diagram symbol. FIGURE 27.11 The summer-inverter circuit, showing (a) complete circuit; (b) block-diagram symbol. ? 2000 by CRC Press LLC The summer-inverter is generally useful for precisely combining or mixing signals, e.g., summing and inverting. The signal levels must be appropriately limited but may generally be bipolar (+/–). The resistance values should be in a proper range since (a) too low resistance values draw excessive current from the signal source, and (b) too high resistance values make the circuit performance too sensitive to stray capacitances and dc offset effects. Typical values are from 1 MW and 10 kW. The circuit of Fig. 27.12 shows a circuit to implement v 0 = –4v 1 – 2v 2 Noninverting Circuits Figure 27.13(a) shows the useful noninverting amplifier circuit. It has a voltage gain V 0 /V 1 = (R 2 + R 1 )/R 1 = 1 + (R 2 /R 1 ) Figure 27.13(b) shows the important unity gain follower circuit, which has a very high input impedance, which lightly loads the signal source but which can provide a reasonable amount of output current milliamps. It is fairly easy to show that the inverting first-order low-pass filter of Fig. 27.14 has a dc gain or –R 2 /R 1 and a –3-dB frequency = 1/(2pR 2 C). Figure 27.15 shows a two-amplifier differentiator and high-pass filter circuit with a resistive input impedance and a low-frequency cutoff determined by R 1 and C. FIGURE 27.12Simple summer-inverter. FIGURE 27.13 Noninverting amplifier circuit with resistive elements. (a) General circuit; (b) simple unity gain follower. ? 2000 by CRC Press LLC Op amps provide good differential amplifier circuits. Figure 27.16 is a single amplifier circuit with a differ- ential gain A d = R 0 /R 1 Good resistance matching is required to have good common-mode rejection of unwanted common-mode signals (static, 60-Hz hum, etc.). The one-amplifier circuit of Fig. 27.16 has a differential input impedance of 2R 1 . R 1 may be chosen to provide a good load for a microphone, phono-pickup, etc. The improved three-amplifier instrumentation amplifier circuit of Fig. 27.17, which several manufacturers provide in a single module, provides 1.Very high voltage gain 2.Good common-mode rejection 3.A differential gain 4.High input impedance FIGURE 27.14First-order low-pass filter circuit. FIGURE 27.15A two-amplifier high-pass circuit. ? 2000 by CRC Press LLC Operational amplifier circuits form the heart of many precision circuits, e.g., regulated power supplies, precision comparators, peak-detection circuits, and waveform generators [Wait et al., 1992]. Another important area of application is active RC filters [Huelsman and Allen, 1980]. Microminiature electronic circuits seldom use inductors. Through the use of op amps, resistors, and capacitors, one can implement precise filter circuits (low-pass, high-pass, and bandpass). Figures 27.18 and 27.19 show second-order low-pass and bandpass filter circuits that feature relatively low sensitivity of filter performance to component values. Details are provided in Wait et al. [1992] and Huelsman and Allen [1980]. FIGURE 27.16Single-output differential-input amplifier circuit. FIGURE 27.17 A three-amplifier differential-input instrumentation amplifier featuring high input impedance and easily adjustable gain. A R R R R VV d =- + ? è ? ? ? ÷ - ( ) 0 2 1 21 1 2 ? 2000 by CRC Press LLC Of course, the op amp does not have infinite bandwidth and gain. An important op-amp parameter is the unity-gain frequency, f u . For example, it is fairly easy to show the actual bandwidth of a constant gain amplifier of nominal gain G is approximately f –3 dB = f u /G Thus, an op amp with f u = 1 MHz will provide an amplifier gain of 20 up to about 50 kHz. FIGURE 27.18Sallen and Key low-pass filter. FIGURE 27.19State-variable filter. ? 2000 by CRC Press LLC When a circuit designer needs to accurately explore the performance of an op-amp circuit design, modern circuit simulation programs (SPICE, PSPICE, and MICRO-CAP) permit a thorough study of circuit design, as related to op-amp performance parameters. We have not here treated nonlinear op-amp performance limitations such as slew rate, full-power bandwidth, and rated output. Surely, the op-amp circuit designer must be careful not to exceed the output rating of the op amp, as related to maximum output voltage and current and output rate-of-change. Nevertheless, op-amp circuits provide the circuit designer with a handy and straightforward way to complete electronic system designs with the use of only a few basic circuit components plus, of course, the operational amplifier. Defining Terms Active RC filter: An electronic circuit made up of resistors, capacitors, and operational amplifiers that provide well-controlled linear frequency-dependent functions, e.g., low-, high-, and bandpass filters. Analog-to-digital converter (ADC): An electronic circuit that receives a magnitude-scaled analog voltage and generates a binary-coded number proportional to the analog input, which is delivered to an interface subsystem to a digital computer. Digital-to-analog converter (DAC): An electronic circuit that receives an n-bit digital word from an interface circuit and generates an analog voltage proportional to it. Electronic switch: An electronic circuit that controls analog signals with digital (binary) signals. Interface: A collection of electronic modules that provide data transfer between analog and digital systems. Operational amplifier: A small (usually integrated circuit) electronic module with a bipolar (+/–) output terminal and a pair of differential input terminals. It is provided with power and external components, e.g., resistors, capacitors, and semiconductors, to make amplifiers, filters, and wave-shaping circuits with well-controlled performance characteristics, relatively immune to environmental effects. Related Topic 29.1 Synthesis of Low-Pass Forms References Electronic Design, Hasbrook Heights, N.J.: Hayden Publishing Co.; a biweekly journal for electronics engineers. (In particular, see the articles in the Technology section.) Electronics, New York: McGraw-Hill; a biweekly journal for electronic engineers. (In particular, see the circuit design features.) J.G. Graeme, Applications of Operational Amplifiers, New York: McGraw-Hill, 1973. L.P. Huelsman, and P.E. Allen, Introduction to the Theory and Design of Active Filters. New York: McGraw-Hill, 1980. J. Till, “Flexible Op-Amp Model Improves SPICE,” Electronic Design, June 22, 1989. G.E. Tobey, J.G. Graeme, and L.P. Huelsman, Operational Amplifiers, New York: McGraw-Hill, 1971. J.V. Wait, L.P. Huelsman, and G.A. Korn, Introduction to Operational Amplifier Theory and Applications, 2nd ed., New York: McGraw-Hill, 1992. Further Information For further information see J.V. Wait, L.P. Huelsman, and G.A. Korn, Introduction to OperationalAmplifier Theory and Applications, 2nd ed., New York: McGraw-Hill, 1992, a general textbook on the design of operational amplifier circuits, including the SPICE model of operational amplifiers; and L.P. Huelsman and P.E. Allen, Introduction to the Theory and Design of Active Filters, New York: McGraw-Hill, 1980, a general textbook of design considerations and configurations of active RC filters. ? 2000 by CRC Press LLC