Operational Amplifiers G. B. CLAYTON BSc FInstP Principal Lecturer in Physics, Liverpool Polytechnic ELBS English Language Book Society/Butterworths Butterworth & Co. (Publishers) Ltd 88 Kingsway, London WC2B 6AB ) Butterworth & Co. (Publishers) Ltd, 1979 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, including photocopying and recording, without the written permission of the copyright holder, application for which should be addressed to the Publishers. Such written permission must also be obtained before any part of this publication is stored in a retrieval system of any nature. First published 1971 Reprinted 1974, 1975, 1977 Second edition 1979 Reprinted 1980 ELBS edition first published 1986 Reprinted 1987 ISBN 0 408 00985 3 Photoset by Butterworths Litho Preparation Department Printed in England by Anchor Brendon Ltd, Tiptree, Essex CONTENTS Chapter 1 Fundamentals 1 1.1 1.2 1.3 Introduction The ideal operational amplifier. Operational feedback More examples of the ideal amplifier at w o r k 1.4 Integrated circuit operational amplifiers Exercises Chapter 2 Understanding operational amplifier performance parameters 16 2.1 Amplifier output and input limitations 2.2 Gain Terminology. Feedback principles 2.3 Summary of some of the effects of negative feedback 2.4 Frequency response characteristics 2.5 Small signal closed loop frequency response 2.6 2.7 Closed loop stability considerations Frequency compensation (phase compensation) 2.8 Transient response characteristics 2.9 Full power response 2.10 Offsets, bias current and d r i f t 2.11 Common mode rejection 2.12 Amplifier impedances 2.13 Noise in operational amplifier circuits Exercises Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 Amplifier testing. Measurement of parameters 100 Measurement of bias current, input difference current, input offset voltage and their d r i f t coefficients Amplifier noise Measurement of input impedances Maximum common mode voltage Measurement of open loop voltage gain and o u t p u t dynamic range Anomalies in d.c. open loop gain values revealed by open loop transfer curves 3.7 Input/output transfer curves allow the measurement of amplifier d.c. parameters 3.8 3.9 Dynamic response measurements Large signal response measurements Chapter 4 Applications. Baste scaling circuits 132 4.1 Introduction 4.2 Voltage sealers and impedance conversion 4.3 Voltage summation 4.4 Differential input amplifier configurations (voltage subtractors) 4.5 Current scaling 4.6 Voltage to current conversion 4.7 A.C. amplifiers Exercises Chapter 5 Nonlinear circuits 163 5.1 Amplifiers w i t h defined nonlinearity 5.2 Synthesised nonlinear response 5.3 Logarithmic conversion w i t h an inherently logarithmic device 5.4 Log amplifiers; practical design considerations 5.5 Some practical log and antilog circuit configurations 5.6 Log antilog circuits for computation 5.7 A variable transconductance four quadrant multiplier Exercises Chapter 6 Integrators and differentiators 223 6.1 The basic integrator 6.2 Integrator run, set and hold modes 6.3 Integrator errors 6.4 Extensions to a basic integrator 6.5 Integrator reset 6.6 A.C. integrators 6.7 Differentiators 6.8 Practical considerations in differentiator design 6.9 Modifications to the basic differentiator 6.10 Analogue computation 6.11 A simple analogue computer Exercises Chapter 7 7.1 Switching and positive feedback circuits Comparators 7.2 Multivibrators 7.3 Sine wave oscillators 263 7.4 Waveform generators Exercises Further measurement and processing applications Chapter 8 8.1 Transducer amplifiers 8.2 Resistance measurement 8.3 Capacitance measurement 8.4 Hot wire anemometer with constant temperature operation 8.5 Chemical measurements 8.6 Active filters 8.7 Phase shifting circuit (all pass filter) 8.8 Capacitance multipliers 8.9 Averaging 8.10 Precise diode circuits 8.11 Sample hold circuits 8.12 Circuit with switched gain polarity 8.13 Voltage to frequency conversion 8.14 Frequency to voltage conversion 8.15 Modulation Exercises Practical considerations Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 Amplifier selection, design specification Selection processes Attention to external circuit details Avoiding unwanted signals Ensure closed loop stability Offset nulling techniques Importance of external passive components Avoiding fault conditions Modifying an amplifier's output capability Speeding up a low drift amplifier Single power supply operation for operational amplifiers Quad operational amplifiers Exercises Appendices Answers to Exercises Index 342 407 290 PREFACE TO SECOND EDITION Since the publication of the first edition of this book many more operational amplifier types have become available and the performance characteristics of integrated circuit devices have been greatly improved. The decreased cost of integrated circuit amplifiers has resulted in their wider usage and many new circuit ideas and areas of application for operational amplifiers have been devised. The first edition has been extensively rewritten to provide a more comprehensive coverage of known modes of operational amplifer action. A greater emphasis has been given to the factors influencing the performance limitations of practical circuits to make the book immediately useful to the ever increasing number of operational amplifier users. Chapter 1 gives a preliminary introduction to the capabilities of operational amplifiers; it is intended mainly for the newcomer to the subject. Chapter 2, which explains the significance of the performance parameters of practical amplifiers, has been rewritten to make it more applicable to present day amplifiers. In a sense, it serves as a reference chapter, to which the applications sections of the book (chapters 4 to 8) constantly refer when accuracy limitations are under consideration. The applications sections themselves have been extended and modified to take account of recent developments. Chapter 9 provides a resume and an overview of the practical considerations which the designer must take into account if he is to exploit fully the operational amplifier approach to electronic instrumentation. The chapter on amplifier testing (Chapter 3) has been completely revised to include test procedures suitable for integrated circuit amplifiers. Many new numerical exercises have been included in the second edition; these are not simply numerical 'hoops' through which the student is expected to 'jump', but are for the most part directly related to practical design considerations. This text is intended for both the user and the potential user of operational amplifiers and as such it should prove equally valuable to both the undergraduate student and the practising engineer in the measurement sciences. I wish to express my gratitude to Mrs J. Davies for typing the manuscript, Mr D. McLuskey for help in the preparation of sketches and diagrams, and Mr J. Anderson for help in connection with the experimental work and the construction of circuits described in the text. CHAPTER ONE FUNDAMENTALS 1.1 Introduction The term 'operational amplifier' was originally introduced by workers in the analogue computer field to denote an amplifier circuit which performed various mathematical operations such as integration, differentiation, summation and subtraction. In such circuits the required response is obtained by the application of negative feedback to a high gain d.c. amplifier by means of components connected between amplifier input and output terminals in a particular manner referred to as 'operational feedback'. The term operational amplifier is now more loosely used to designate any high performance d.c. amplifier suitable for use with this type of feedback. Operational amplifiers are still widely used for analogue computation and, although analogue techniques are being superseded by digital methods in computer engineering, the range of applications for operational amplifiers in instrumentation and control system engineering is rapidly expanding. The increase in the use of operational amplifiers in analogue systems is to a large extent attributable to the availability of high performance ready built amplifiers in discrete component modular form and more recently in the form of inexpensive integrated circuits. The use of such ready built amplifiers makes it possible to adopt a new approach to many electronic design problems. The approach involves the selection of a suitable amplifier and the connection of a few discrete components to it to form a complete sub-system. In many cases this is more economical than designing with individual components and it frees the engineer from the tedious and often time consuming task of d.c. amplifier design; it seems likely that it will displace discrete component designs in a wide variety of fields. Amplification is probably the most important basic capability of electronic circuits, for with an amplifier it is possible to synthesise most other important circuits. The range of possible applications in which the operational amplifier approach may be usefully adopted is enormous and is only really limited by the ingenuity of the user. It is not essential that the user of operational amplifiers be familiar with the intricacies of their internal circuit details, but he must under1 stand the function of the external terminals provided by the manufacturer and the terms used to specify the amplifier's performance if he is t o be able t o select the best amplifier for a particular application. 1.2 The ideal operational amplifier. Operational feedback As far as input and o u t p u t terminals are concerned operational amplifiers w i t h three varieties of single ended and differential connections are available. Amplifier input and o u t p u t terminals are said to be single ended if one of the pair of terminals is earthed (or common), otherwise they are said to be earth free or floating. In the case of floating amplifier input or output terminals, the signal of importance is the difference in the voltages at the t w o terminals irrespective of any common voltage they may both have w i t h respect to earth. The conventional circuit symbols for the three basic varieties of amplifier are illustrated in Figure 1.1. Negative and positive signs are used t o distinguish between the phase-inverting and non-phase-inverting input terminals of the differential amplifier. If the input terminal marked w i t h a positive sign is earthed and a signal is applied to the input Single ended input single ended output (Earth r e t u r n may be o m i t t e d ) Differential input single ended output ( E a r t h r e t u r n may be o m i t t e d ) Î Input Output l t 1 Input i Output Differential differential input output + t Input i > Figure 1.1 Circuit symbols for operational amplifiers 2 4 Output - t ° terminal marked with a negative sign it will appear phase inverted at the output of the amplifier. The converse holds if the terminal marked with a negative sign is earthed. In a first consideration of an operational feedback it is found convenient to assume that the amplifier has certain ideal characteristics. The effects of departures from the ideal, exhibited by real operational amplifiers, may then be expressed later in terms of the errors to which they give rise. Differential input single ended output operational amplifiers are the ones most commonly used and we will refer specifically to this type of amplifier in the treatment which follows. A differential input amplifier allows a greater flexibility in the choice of feedback configuration than a single ended input amplifier. The output of the ideal differential input amplifier depends only on the difference between the voltages applied to the two input terminals. Any common voltage that the two input terminals may have with respect to earth is called a common mode input voltage. The output of the ideal differential amplifier is unaffected by any common mode input voltage. The ideal amplifier is further assumed to have the following characteristics. Infinite Gain as we shall show makes the performance entirely dependent on input and feedback networks. Infinite Input Impedance ensures that no current flows into the amplifier input terminals. Infinite Bandwidth is a bandwidth extending from zero to infinity, ensuring a response to d.c. signals, zero response time and no phase change with frequency. Zero Output Impedance ensures that the amplifier is unaffected by the load. Zero Voltage and Current Offset ensures that when the input signal voltage is zero the output signal will also be zero regardless of the input source resistance. There are two basic ways of applying operational feedback to a differential amplifier: Figure 1.2a shows the inverting configuration, the non-inverting configuration being illustrated in Figure 1.2b. In both circuits the signal fed back is proportional to the output voltage, feedback takes place via the resistor /?2 connected between the output and the phase-inverting input terminal of the amplifier. Phase inversion through the amplifier ensures that the feedback is negative. In the in3 verting circuit the feedback signal is effectively applied in shunt w i t h the external input signal, in the non-inverter input signal and feedback are effectively in series. The action of both circuits may be understood if a small voltage e€ is assumed to exist between the differential input terminals of the amplifier. The signal fed back w i l l be in opposition t o e e and w i t h the amplifier gain tending to infinity the voltage ee will be forced towards zero. This is an extremely important point and is w o r t h restating in an alternative f o r m . Thus, when the amplifier output is fed back t o the inverting input terminal the output voltage will always take on that value required to drive the signal between the amplifier input terminals t o zero. In the case of an amplifier having large but finite gain the error voltage ee is small but non-zero, and the effect will be discussed in the next chapter. Summing point J" R ,— (b) Figure ^.2 The two basic operational feedback circuits A second basic aspect of the ideal circuit follows f r o m the assumed infinite input impedance of the amplifier. No current can f l o w into the amplifier so that any current arriving at the point X, as a result of an applied input signal, must of necessity flow through the feedback path R2. If instead of the single resistor /?·, connected to the inverting input terminal there are several alternative signal paths the sum of these several currents arriving at X must f l o w through the feedback path. It is for this reason that the phase-inverting input terminal of an operational amplifier (point X) is sometimes referred t o as the amplifier summing point. The t w o basic aspects of ideal performance are called the summing point restraints; they are so important that they are repeated again. 4 1. When negative feedback is applied to the ideal amplifier the differential input voltage approaches zero. 2. No current flows into either input terminal of the ideal amplifier. The t w o statements form the basis of all simplified analyses of operational feedback circuits; we use them t o derive closed loop gain expressions for the circuits of Figure 1.2. Inverter closed loop gain. (Refer to Figure 1.2a) The first summing point restraint, gives e \ l\ = — #1 e and / f = o R 2 The second restraint gives l\ = /f Thus, = #2 — R, and the closed loop gain, ^ Non-inverter V C L B i£ = _ *2_ ( 1 1 ) closed loop gain. (Refer to Figure 1.2b) Since no current flows into either input of the amplifier the voltage at the inverting input terminal, e R. = Ri + R2 But e€ = 0 Therefore e-{ = eB or e-t = R, +R2 tfi and the closed loop gain ^VCL R eQ 2 = — = 1 + — e\ R<i (1.2) 5 Apart from the difference in the sign of the closed loop gain the main difference between the two circuits lies in the effective input resistance which they present to the signal source ej. The effective input resistance of the ideal inverter measured at the amplifier summing point is zero. Feedback prevents the voltage at this point from changing; the point acts as a virtual earth. Note that any current supplied to this point does not actually flow to earth but flows through the feedback path R2. The input current /j in Figure 1.2a is thus determined by the resistor R^ and the input resistance presented to the signal source is /?-j. The ideal non-inverting circuit Figure 1.2b takes no current from the signal source and thus presents an essentially infinite input impedance. It is worth noting that even if the differential input impedance of the amplifier itself is not infinite, as assumed, an infinite amplifier gain is a sufficient condition to ensure that the effective input impedance of the non-inverting circuit is infinite. Infinite amplifier gain causes the voltage fed back to the inverting input terminal to be exactly equal in magnitude to the applied input voltage. The feedback voltage is effectively in series with the input voltage, it opposes the input voltage and so prevents any current from being driven into the amplifier. The inverter and non-inverter illustrate the difference between negative feedback applied in shunt and in series with the external signal. In general shunt negative feedback lowers input impedance, series negative feedback increases input impedance. The comparative simplicity of the closed loop gain expressions should not make the reader forgetful of their significance. Closed loop gains in the ideal case depend only on the values of series and feedback components, not on the amplifier itself. Real amplifiers introduce departures from the ideal, and these are conveniently treated as errors. Using currently available operational amplifiers, errors can be made very small and one of the main features of the operational amplifier approach to instrumentation is the accuracy with which it is possible to set gain and impedance values. 1.3 More examples of the ideal operational amplifier at work The ideal operational amplifier serves as a valuable starting point for a preliminary analysis of operational amplifier applications and in this section we present a few more examples illustrating the usefulness of 6 the ideal operational amplifier concept. Once the significance of the summing point restraints are firmly understood, ideal circuit analysis involves little more than the intelligent use of Ohm's law. Remember that the ideal differential input operational amplifier with negative feedback applied to it has an output voltage which takes on just that value required to force the differential input voltage to zero and to cause all currents arriving at the inverting input terminal to flow through the feedback path. 1.3.1 T H E I D E A L O P E R A T I O N A L A M P L I F I E R ACTS AS A C U R R E N T TO V O L T A G E CONVERTER An ideal operational amplifier can act as a current to voltage converter. In the circuit of Figure 1.3, the ideal amplifier maintains its inverting input terminal at earth potential and forces any input current to flow Feedback forces the potential of this point to zero e o = -'mRf Figure 1.3 An ideal operational amplifier acts as a current to voltage converter through the feedback resistance. Thus /j n = /f and eQ = — /\nRf. Notice that the circuit provides the basis for an ideal current measurement; it introduces zero voltage drop into the measurement circuit, and the effective input impedance of the circuit measured directly at the inverting input terminal is zero. 1.3.2 T H E IDEAL O P E R A T I O N A L A M P L I F I E R ADDS VOLTAGES OR CURRENTS I N D E P E N D E N T L Y The principle involved in the current to voltage converter circuit of Figure 1.3 may be extended. In the ideal operational amplifier circuit of Figure 1.4(a) the operational amplifier forces the sum of the several currents arriving at the inverting input to flow through the feedback 7 path (there is no where else for them to go). The inverting input terminal is forced to be at earth potential (a virtual earth) and the output voltage is thus: eQ = - t'i + / 2 + '3] tff In Figure 1.4(b) a number of input voltages are connected to resistors which meet at the inverting input terminal. The ideal operational amplifier maintains the inverting input at earth potential, thus each input e O=-[V ] 2 + I 3] R Figure 1.4 An ideal operational amplifier adds voltages and currents independently current is independently determined by each applied input voltage and series input resistor. The sum of the input currents is forced to flow through /?f and the output voltage must take on a value which is equal to the sum of the input currents multiplied by /?f. Ri f ii Figure 1.5 An ideal operational amplifier can act as a voltage to current converter 8 + Î*. + ill Figure 1.6 An ideal operational amplifier can act as an ideal unity gain buffer stage 1.3.3 THE IDEAL OPERATIONAL AMPLIFIER CAN ACT AS A VOLTAGE TO CURRENT CONVERTER In maintaining its differential input voltage at zero the amplifier shown in the circuit of Figure 1.5 forces a current / = e-m/R to flow through the load in the feedback path. The value of this current is independent of the nature or size of the load. 1.3.4 THE IDEAL OPERATIONAL AMPLIFIER CAN ACT AS A PERFECT BUFFER In the circuit of Figure 1.6 the amplifier output voltage must take on a value equal to the input voltage in order to force the differential input signal to zero. The ideal circuit has infinite input impedance, zero output impedance and unity gain, and acts as an ideal buffer stage. ?"R T [v e 2 ] Figure 1.7 An ideal operational amplifier can act as a sub tractor 1.3.5 THE IDEAL OPERATIONAL AMPLIFIER CAN ACT DIFFERENTIALLY ASASUBTRACTOR The circuit shown in Figure 1.7 illustrates the way in which an operational amplifier can act differentially as a subtracter. The voltage at the inverting input terminal rs (by superposition) e_ = e2 R2 n i + r?2 + e0 /?i n2 r?i + The voltage at the non-inverting input is e+ = ΘΛ R2 /?I1 +/? ^n2 The operational amplifier forcese_ = e+ 9 Thus or e2 R2 — /?! + / ? 2 R "l + eQ = <?i #1 + #2 ^2 r e0 = — 2 #1 +/? 2 i [e-i - e 2 J 1.3.6 THE IDEAL OPERATIONAL AMPLIFIER CAN ACT AS AN INTEGRATOR In the circuit of Figure 1.8 negative feedback is applied by the capacitor C connected between the output and the inverting input terminal. The amplifier output voltage acting via this capacitor maintains the inverting input terminal at earth potential and forces any current arriving at the inverting input terminal t o flow as capacitor charging current. Thus: - 'ιη d Vc *?in u - R df The output voltage is equal in magnitude but opposite in sign to the capacitor voltage. Therefore: ^in_ R and e0 C - = - — àe0 CRJ df /6? i n d f The output is proportional to the integral w i t h respect to time of the input voltage. 1.3.7 LIMITATIONS OF THE IDEAL OPERATIONAL AMPLIFIER CONCEPT Real operational amplifiers have characteristics which approach those of an ideal operational amplifier but do not quite attain them. They have an open loop gain which is very large (in the region 10 5 to 10 6 ) but the gain is not infinite. They have a large but finite input impedance. They draw small currents at their input terminals (bias currents), they require a small differential input voltage to give zero output voltage (the input offset voltage), and they do not completely reject common mode signals (finite c.m.r.r.). In our discussion of ideal operational amplifier circuits no mention has been made of frequency response characteristics—real amplifiers have a frequency dependent gain which can have a marked effect on the performance of operational amplifier applications. 10 Figure 1.8 An ideal operational amplifier can act as an integrator The above features of real operational amplifiers cause the performance of a real operational amplifier circuit to differ from that predicted by an analysis based upon the assumption of ideal amplifier performance. In many respects the differences between real and ideal behaviour are quite small but in some aspects of performance, particularly those involving frequency dependent performance parameters, the differences are marked. The next chapter presents detailed discussions about the parameters which are normally given on the data sheets of practical operational amplifiers, and shows how a knowledge of these parameter values can be used to predict the behaviour of practical circuits. In the next section a brief practical introduction to real amplifiers is given to prepare the reader for his study of Chapter 2. 1.4 Integrated circuit operational amplifiers Inexpensive integrated circuit operational amplifiers are now freely available, they are easy to use and they permit the comparatively inexperienced electronicist to build working circuits rapidly. The newcomer to operational amplifiers is strongly advised1 to connect up a few of the basic operational amplifier applications and practically evaluate their performance: this forms a useful learning and familiarisation exercise which is worth performing before delving more deeply into the finer aspects of operational amplifier performance. A preliminary practical evaluation of operational amplifier applications is most conveniently carried out using a general purpose or internally compensated operational amplifier type. The significance of this terminology will be discussed in the next chapter. There are several general purpose amplifier types to choose from; at 11 the present time the most widely used is the type 741 amplifier. Most manufacturers produce their version of this amplifier type, for example Analog Devices, A.D.741 J,K,L and S series, National Semiconductors L.M.741, Motorola's MC1741, SigneticsMA 741. It seems likely that the 741 will eventually be superseded by the newer improved general purpose designs that are now appearing on the market. These new amplifier designs incorporate f.e.t. input stages formed on the same integrated circuit chip with standard bipolar transistors, for example, National Semiconductors Bi Fet amplifiers, type L.F. 155, 156 and 157 series, and R.C.A.'s Bi-Mos amplifiers type C A 3140 series. As a user of operational amplifiers one does not need a detailed knowledge of their internal circuitry, and fortunately most general purpose operational amplifiers are pin compatible. It is with the function of the external pin connections that the operational amplifier user is primarily concerned. The eight pin dual-in-line plastic package and the metal can are the most commonly used operational amplifier circuit packages. These are shown in Figure 1.9. Operational amplifiers are normally used with dual power supplies; input and output voltages are measured with respect to the potential of the power supply common terminal which acts as the zero signal reference point. The use of dual supplies allows input and output voltages to swing both positive and negative with respect to the zero reference point. Figure 1.10 shows the circuit connections which are required to make a practical form of the inverting amplifier circuit previously encountered in Section 1.2. Amplifier pin connections which are not shown in Figure 1.10 should be left with no connections made to them—their function will be described later. In his preliminary experimentation with operational amplifiers the newcomer will inevitably make wrong connections; fortunately most general purpose operational amplifiers will tolerate many wrong connections. Particular care however should be taken to ensure that the power supplies are connected to the correct pins, as incorrect power supply connections can permanently damage an amplifier. Input signals should not be applied to an amplifier before power supplies are switched on, as application of input signals with no power supplies connected can damage an amplifier. Practical forms of the ideal amplifier circuits given in the previous sections should be tried out. The ideal circuits require simply the addition of appropriate power supply connections in order to make working circuits. A preliminary acquaintance with real amplifiers 12 rT° Offset null TiVg»-ve Inverting inputo'""]' > - 0 0 u t f Non-inverting i n p u t ^ ^ c^ O fOffset f s e t null V ve s O f f s e t null . \-q 0 NC h-/ Inverting J ^ L w, -Put-^>>V s + ve fr-^Output Non-inverting input Offset null V-ve-^ (Top v i ew ) M e t a l package Figure 1.9 General purpose Op A m p e.g. 741 o r l _ F 3 5 6 N ( Top v i e w ) Plastic package I.C. operational amplifier packages Dual Power s u p p l y + - + - Power s u p p l y c o m m o n line Figure 1.10 Connections for practical inverting amplifier circuit obtained in this w a y w i l l help the reader t o understand m o r e readily t h e discussion of amplifier characteristics w h i c h is t o be presented in t h e next chapter. REFERENCE 1. C L A Y T O N , G. B., Experiments with Operational Amplifiers—Learning by Doing, Macmillan (1975). 88 Practical Op Amp Circuits you can Build, Tab Books (1977) 13 50kO 20kn -ŒD— 100kO 2V ί ■+ 10kO (b) 2V 20kQ ^ 10kQ r -Z.V ■2V (c) 20kQ (</) 50kQ 100kQ 10kO -CZZ310kO 3V 20kO 2V (f) 100kO IjuF -IIHOV/sec -t Figure 1.11 14 to) Circuits for exercise 1.2 Exercises 1.1 Give component values and sketch diagrams of operational amplifier circuits for the following applications. Assume ideal operational amplifier performance. (a) An amplifier voltage gain - 5 and input resistance 100 kf2. (b) An amplifier voltage gain —20 and input resistance 2 k£2. (c) An amplifier voltage gain +100 with ideally infinite input resistance. (d) An integrator with the circuit performance equation eQ = — 100 fe\ndt and an input resistance 100 k£2. (e) A circuit which when supplied by an input signal of 2 V will drive a constant current of 5 mA through a variable load resistor. 1.2 Find the value of the amplifier output voltage for each of the circuits given in Figure 1.11. In all cases assume that the operational amplifier behaves ideally. 15 CHAPTER TWO UNDERSTANDING OPERATIONAL AMPLIFIER PERFORMANCE PARAMETERS A wide variety of operational amplifier types are now commercially available. The selection of the best amplifier type for a particular application is becoming an increasingly difficult problem. This is particularly true for the new user of operational amplifiers, for when first delving into manufacturers' catalogues, he is faced w i t h what seems t o be at first sight a bewildering variety of specifications and different amplifier types. It will usually be found that the more demanding the performance requirements the more costly w i l l be the amplifier necessary t o f u l f i l these requirements; so the choice of amplifier then w i l l probably ultimately be governed by economic considerations. The least expensive one which w i l l meet the design specifications w i l l be chosen. In order to make this choice the design objectives must be completely defined and the relationship between published amplifier parameters and their effects on overall circuit performance in any intended applicat i o n must be well understood. This chapter will introduce the reader to the various amplifier specifications normally included in a manufacturer's data sheet; the significance of the parameters w i l l be discussed. A n added difficulty exists here in that there is not as yet any standardisation of specificat i o n definitions amongst the various manufacturers so that, when consulting data sheets, it is important t o understand under exactly what conditions a particular parameter is defined. Differences are not great, but they do exist. The important question of amplifier selection will be | 1" Positive saturation Negative saturation Figure 2.1 Idealised transfer curve for an operational amplifier 16 returned to in a later chapter after the reader has been introduced to amplifier applications; he should then more clearly appreciate design objectives and the way in which amplifier parameters limit their achievement. 2.1 Amplifier output and input limitations The output of a differential amplifier is related to its inputs by the somewhat idealised open loop transfer curve illustrated in Figure 2.1; with open loop gains greater than 10 4 being quite typical only a very small voltage between the two input terminals is needed to cause saturation of the amplifier output voltage. The maximum output voltage swing V0 m a x is the maximum change in output voltage, (positive and negative) measured with respect to earth, that can be achieved without clipping of the signal waveform caused by amplifier saturation. Values of V0 m a x are quoted for the amplifier working into a specified load (sometimes at full rated output current) and with specified values for amplifier power supplies. For applications requiring a d.c. response and an output voltage capable of swinging above and below earth potential (as shown by the transfer curve) operational amplifiers are generally designed to operate from symmetrical positive and negative voltage supplies (V+, V~). Maximum values for supply voltages are normally specified and should not be exceeded. Values for VQ m a x will be found to be dependent on the magnitude of the supply voltages actually used. Maximum voltage between inputs. The voltage between the input terminals of an operational amplifier is maintained at a very small value under most operating conditions by the feedback circuit in which the amplifier is used. If the application is such that the voltage between the terminals may be appreciable, care must be taken to ensure that it does not exceed the maximum allowable value for the particular amplifier as otherwise permanent damage to the amplifier may be caused. Some amplifiers are internally protected against input overload conditions; where such internal protection is not provided parallel back to back diodes may be connected externally to the amplifier input terminals to provide the necessary protection. Maximum common mode voltage. The voltage at both inputs of a differential amplifier may be raised above earth potential. The input common mode voltage is the voltage above earth at each input when 17 both inputs are at the same voltage. The maximum common mode voltage, E c m , is the maximum value of this voltage which can be applied w i t h o u t producing clipping or nonlinearity at the o u t p u t . If an amplifier is to be used under conditions in which excessive common mode voltage may cause damage, protection can be obtained by the use of a suitable pair of zener diodes. The circuit of Figure 2.2 illustrates a method of protecting both against excessive voltage between inputs and excessive common mode voltage; current limiting by resistance in series w i t h the input terminals is assumed. 2.2 Gain terminology. Feedback principles 2.2.1 OPEN LOOP VOLTAGE GAIN The open loop voltage g a i n , / 4 0 L / of a differential input operational amplifier may be defined as the ratio, change of output voltage change of input voltage the input voltage being that measured directly between the inverting and non-inverting input terminals of the amplifier. AOL is normally specified for very slowly varying signals and can in principle be determined f r o m the slope of the non-saturated portion of the i n p u t / o u t p u t transfer curve (Figure 2.1). The magnitude of AOL for a particular amplifier depends on the amplifier load and on the value of the power supplies. Values of AOL are normally quoted for specified supply voltages and load. Operational amplifiers are seldom used open loop, being normally connected in negative feedback circuits in order t o define precise operat i o n ; the significance of open loop gain is that it determines the accuracy limits in such applications. A n assessment of the quantitative effects of the open loop gain magnitude requires a study of the principles underlying feedback amplifier operation. 2.2.2 NEGATIVE FEEDBACK AMPLIFIERS In a negative feedback amplifier circuit a signal is fed back f r o m the output t o the input terminal, and this signal is in opposition t o the externally applied input signal. The signal which actually drives the input terminals of the amplifier is that which results from a subtraction 18 Silicon diodes ^>Zeners Figure 2.2 Circuit protection with diodes process. The larger the gain of the amplifier w i t h o u t feedback (the open loop gain) the smaller is the signal voltage applied between the amplifier input terminals. If the open loop gain of the amplifier is infinite (as assumed for an ideal operational amplifier) negative feedback forces the amplifier input signal to zero; however, w i t h a large, but finite amplifier open loop gain, a small input signal must exist between the amplifier input terminals. It is convenient to think of this as an input error voltage, which arises because the real amplifier has a finite open loop gain. A differential input amplifier allows the application of negative feedback w i t h a variety of external circuit configurations; this together w i t h the power of the feedback techniques themselves accounts for the great versatility of the differential input operational amplifier. Certain general terminology is commonly used to distinguish between different types of feedback configuration. A negative feedback circuit which causes the feedback signal t o be proportional to the o u t p u t voltage produced by an amplifier is called a voltage feedback circuit. A circuit in which the signal fed back is proportional to o u t p u t current is called a current feedback circuit. It is important to note that the terminology relates t o the way in which the feedback signal is derived, and not to what is actually being fed back. Negative voltage feedback acts on the o u t p u t voltage; it works towards making the amplifier output behave like an ideal voltage generator, i.e. as a source of e.m.f. linearly related to the externally applied input signal (no distortion) and having zero output impedance (output voltage unaffected by loading). Current feedback, on the other hand, works towards making the amplifier o u t p u t behave like an ideal current source, i.e. a source of current linearly related to the input signal and unaffected by changes in the external load to which the current is supplied. The extent to which negative feedback achieves these performance ideals w i l l be examined in the analyses which follow shortly. 19 T w o other terms are used to distinguish feedback circuits: series feedback and shunt feedback. These terms relate to the way in which the feedback signal is introduced at the amplifier input terminals. In a series feedback circuit the feedback signal is applied in series w i t h the external input signal to the amplifier input terminals; in a shunt feedback circuit the feedback signal and the external input signal are applied in parallel. 2.2.3 SERIES VOLTAGE FEEDBACK (THE FOLLOWER CIRCUIT) A differential input amplifier w i t h series negative voltage feedback applied to it is shown in Figure 2.3. The amplifier is represented in terms of its Thevenin equivalent. The output behaves like a source of e.m.f. — A O L e€ in series w i t h the amplifier output impedance Z 0 . (Note the minus sign simply comes f r o m the assumed positive direction of the differential input signal ee). A signal e f which is directly proportional to the output voltage eQ is fed back t o the inverting input terminal of the amplifier (negative feedback) ef = ßeQ The constant of proportionality ß is called the voltage feedback fraction; it is an important quantity when analysing the quantitative effects of feedback. If, in Figure 2.3, we assume Z i n > R^ and neglect the shunting effect of Z i n on /?-| we may write β = - ^ - (2.1) The output voltage of the amplifier may be written as *o = - iQZo ^OL^e - (2.2) It is simply a use of the general equation for the output voltage produced by a loaded source of e.m.f., Output voltage = Open circuit voltage — Internal volts drop. ee is the difference between the externally applied input signal e} and the feedback signal ef. Note that e f and β] are effectively applied in series to the differential input terminals of the amplifier. - ee = * j - ef (2.3) Substitution for e€ in equation 2.2 gives ^o = ^ O L tel 20 of) - i0Z0 Figure 2.3 Series voltage feedback S u b s t i t u t i n g ^ = ßeQ and rearrangement give *OL 1+ßA O L e\ - (2.4) in 1 + M O L According to equation 2.4 the circuit behaves like an amplifier w i t h open circuit gain 4 0 ι _ / Π + ßA0[_) and output impedance Z 0 / ( 1 + J M 0 L ) · These are the closed loop parameters for the circuit. Thus: <CL 1 1 *OL 1 + L and ßAOLJ (2.6) -oCL 1 Note that if ßAOL (2.5) ß 1+iMoL + M O L is very large the quantity 1 1 + 1 /MOL is as near unity as makes no difference and the closed loop gain is determined almost entirely by the value of the feedback fraction. The 21 closed loop output impedance is made very small (i.e. output voltage little affected by loading). The product of the feedback fraction and the open loop gain is the gain around the feedback loop and it is called the loop gain. Loop gain, ]3>4OL is a most important parameter in determining the quantitative effects of feedback. Let us derive an expression for the input impedance of the circuit. Note that e f is applied in opposition to e} and tends to prevent Θ] from driving a current into the circuit. Series negative feedback may thus be expected to increase effective input impedance. We write e, = é?f - e€ = ßeQ - ee But eQ = - AOL e€ - — (assuming R2 > Z L ) ZQ+ZL Substitution gives e, = - ee Γΐ+jMoL L Now But Thus Z t 1 *o+*J U+ßAOL-*±-] ZQ+ ZLJ ^inCL = ^ = - ^ 'in 'in L e - V {- = Z in Z i n C L = Zj ^in ZL n+^OL 1 (27) Series voltage feedback increases input impedance to an extent determined by the loop gain ßAOL. 2.2.4 SHUNT VOLTAGE FEEDBACK (THE INVERTER) In Figure 2.4 the externally applied input signal voltagees and the output voltage eQ are effectively applied in parallel to the differential input terminals of the amplifier. The signal e€ which drives the differential input terminals is a superposition of the effects of es and eQ. R2 /?2 +/?1 +/? s + eQ # 1 +/? 8 + R2 It is assumed that Z j n > R^ + Rs and Z 0 < R2 22 #1 + Rs -1 1— (2.8) fli The feedback fraction + #s The output voltage may be written as = -A OL Î0Z0 Substitution for e€ and rearrangement give en = - R2 AOL R2 +/?i M + " sA?e T e« - 1 +]3>4OL ir 1+MoL The closed loop signal gain of the circuit is thus /?2 »CL = ^2 >4QL /?2 +/? 1 +/? s 1 + j M O L /?1 + #* (2.9) 1 +WOL and the closed loop output impedance is L oCL = 1+jMoL (2.10) Compare equations 2.9 and 2.10 with equations 2.5 and 2.6. Again, notice the importance of the loop gain ßAOL. If the loop gain is sufficiently large the closed loop performance is determined by the value of the components used to fix the feedback fraction j3. If R^ < /?s and the loop gain is large the closed loop signal gain approximates to Figure 2.4 Shurn voltage feedback 23 The main difference between shunt and series negative feedback lies in the effect on the input impedance of the circuit. In Figure 2.4 'in = ' ' + /f ee-en Z|n If*. >Zn e o - ~ ^oL^e and L *in R? J f e f Figure 2.5 Series current feedback The effect of the shunt feedback is to reduce the effective input impedance measured at the differential input terminals of the amplifier. A n additional impedance /? 2 /(1 + ^ Q L ) is effectively put in shunt w i t h Z i n and if A0L is very large the impedance measured at the differential input terminals is very small. The input impedance of the circuit is then effectively equal to the value of the resistor βΛ. 2.2.5 NEGATIVE CURRENT FEEDBACK In the circuit of Figure 2.5 a feedback signal is developed by the output current / 0 flowing through a current sensing resistor Rf. The object of 24 applying negative current feedback is to achieve a stable output current. The circuit is analysed in order to find an expression for this output current. We may write >4oLé?e Ό = ZQ+ZL+Rf But ee = e f — e\ = / 0 /?f — e{. (Assume Z i n > Rf). Substitution for e and rearrangement give € /"o = - ^OL » Z 0 + Z L + (1 + AOL) <2-11> Rf Note that the effect of the negative current feedback is to increase the effective output impedance of the circuit (the term AOLRf is added). Equation 2.11 may be written in an alternative f o r m as 1 Rf 1 + L 1 (2.12) MOLJ where in this case the voltage feedback fraction ß = Rf/(Rf + Z L + Z 0) has a value which depends upon the load impedance. You should again notice the importance of the loop gain and the factor [1/(1 + V[ßAOL] )] which is sometimes referred to as the gain error factor. In the current feedback circuit if ßAOL is very large, the output current is almost entirely determined by the input voltage and the value of the resistor Rf which is used t o set the feedback. 2.3 Summary of some of the effects of negative feedback It is instructive to summarise the effects of negative feedback as shown by the above analysis. If y o u analyse any negative feedback amplifier circuit you will find that the following general principles hold. 1. Series negative feedback increases input impedance. 2. Shunt negative feedback decreases input impedance. 3. Negative voltage feedback makes for a stable distortion-free output voltage. 4. Negative current feedback makes for a stable distortion-free output current. 25 2.3.1 GAIN TERMINOLOGY It is important to distinguish between the several 'gain' terms which are used when discussing operational amplifier feedback circuits. Open loop gain Aoi_ is defined as the ratio of a change of o u t p u t voltage to the change in the input voltage which is applied directly t o the amplifier input terminals. The other gains are dependent upon both the amplifier and the circuit in which it is used and are controlled by the feedback fraction ß. The feedback fraction ß, is the fraction of the amplifier output voltage which is returned to the input. It is a function of the entire circuit from output back to input, including both designed and stray circuit elements and the input impedance of the amplifier. Loop gain ) 3 / l O L / is the total gain in the closed loop signal path through the amplifier and back to the amplifier input via the feedback network. The magnitude of the loop gain is of prime importance in determining how closely circuit performance approaches the ideal. Closed loop gain Is the gain for signal voltages connected directly to the input terminals of the amplifier. The closed loop gain for an ideal amplifier circuit is 1/0 and for a practical circuit is 1 ß 1 + L The quantity [1/(1 + M[ßAOL] 1 /MOL J )] is called the gain error factor. The amount by which this factor differs from unity represents the gain error (usually expressed as a percentage). Signal gain Is the closed loop transfer relationship between the output and any signal input to an operational amplifier circuit. Care should be taken to avoid confusion between closed loop gain and signal gain. In some circuits (the follower for example) the t w o 26 gains are identical. Reference to the circuit shown in Figure 2.6 should clarify the distinction between the two types of gain. The signals e 4 and e3 are applied directly to the amplifier input terminals; they appear at the output multiplied by the closed loop gain 1/0. The signals βΛ and e2 are applied to the inverting input terminal through resistors R^ and R2 respectively; there are thus two different signal gains: — Rf/R^ for signal e<[ and —Rf/R2 for signal e2. R<i R 2 ß= + Rf Rf Rf 1 1 e, R > e 2 " R ~ 2 - ^ J 3 - e 3 ^ 3 f e- Signal gain r* I 1 +. £A 0 L L Gain error factor Closed loop 1 e ? Signal gain a „. ,, Closed loop ^gain gain Signal source impedances are assumed negligibly small and the effects of amplifier offsets and input and output impedances are neglected Figure 2.6 Differences between signal gain and closed loop gain In evaluating an expression for ß due account must always be taken for all paths from the inverting input terminal to earth. A general expression for the feedback fraction ß in any circuit may be written as ß = Zn+Zf (2.13) where Z p = parallel sum of all impedances between the inverting input terminal and earth and Z f = feedback impedance: the impedance between the output terminal and the inverting input terminal. 27 The performance equation for an operational amplifier feedback circuit can always be put in the f o r m Γ Actual Closed loop 1 _ Γ Lperformance equationj Ideal Closed loop "I |_performance equationj ßAOL The ideal closed loop performance equation is that obtained assuming that the operational amplifier behaves ideally (Chapter 1); ideal performance depends entirely upon the values of components external to the amplifier. If the loop gain ßAOL is very large, actual closed loop performance is very close to the ideal. Note that it does not matter if the open loop gain of an amplifier is non-linear, for provided that ßAOL is very large the closed loop performance will be linear if linear components are used to f i x feedback conditions. 2.4 Frequency response characteristics In our treatment of feedback amplifiers presented thus far no specific reference has yet been made to dynamic response characteristics. Gain has been defined as the ratio change of o u t p u t voltage t o change of input voltage, but we have not yet discussed the way in which amplifier response is governed by the rate of change of the input signals applied to it. It is usual to distinguish between sinusoidal and transient response characteristics: sinusoidal response parameters describe the way in which an amplifier responds to sinusoidal signals, in particular they show how response depends upon signal frequency; transient response parameters characterise the way in which an amplifier reacts to a step or squarewave input signal. A n added complication arises when dealing w i t h dynamic response parameters, in that it is necessary to distinguish between small signal and large signal response parameters; differences arise because of dynamic saturation effects which occur w i t h large signals. This section is concerned w i t h small signal sinusoidal response characteristics. A n ideal operational amplifier is assumed t o have an open loop gain which is independent of signal frequency but the gain of real amplifiers exhibits a marked frequency dependence. Both the magnitude and the phase of the open loop gain are frequency dependent; this frequency dependence has a marked effect on closed loop performance. 28 2.4.1 BODE PLOTS Gain/frequency characteristics are often presented graphically. It is usual to plot gain magnitude in dB against frequency on a log scale. Gain in dB is determined f r o m the relationship Voltage gain in dB = 20 log Va 10 Vin (2.14) The reader who is unfamiliar w i t h the use of dB should get practice in working out the dB equivalents of some voltage ratios (try Exercises 2.5 and 2.6). \VQ/V]n\= \V0/V]n\= ^o/^in \V0/V]n 10 represents a voltage gain of 20 log 1 0 10 = 20 d B ; 100 represents 40 d B ; \V0/V]n l = I = 1000 represents 60 d B ; ^ represents-20 d B ; \VQ/V]n I = y/2 represents 3 d B ; I = l / v / 2 represents - 3 dB. Note, since power is proportional to (Voltage) 2 , a fall in gain of 3 dB represents a halving of the output power. Gain/frequency plots are often given as a series of straight line approximations rather than as continuous curves. The straight lines are called Bode approximations and the graphs are called Bode diagrams. The significance of Bode plots should emerge f r o m the study of specific examples which will now be given. The open loop frequency response of many operational amplifiers is designed to f o l l o w an equation of the form ^OL(jf) = A L ° (2.15) 1 +/ — U ^ O L ( j f ) is a complex quantity representing the magnitude and phase characteristics of the gain at frequency f. ^ O L represents the d.c. value of the gain. fc is a constant, sometimes called the break frequency. Equation 2.15 describes what is sometimes called a first order lag response; its magnitude and phase characteristics are shown plotted in Figure 2.7. The magnitude of the response is U *OL(jf) oLiifil = _ A°L 12 + . ,- (ΘΊ (2.16) 29 A t low frequencies for which f < fc, \Α^ AOL and the straight line is t n e , o w (jf) I - ^ O L frequency asymptote. A t high frequencies for which f> fCl the response is asymptotic to the line \A^) <OL fjf which has a slope of —20 dB/decade change in frequency. Approximation Magnitude errors Phase Bode approximation Actual response i 0°- 10fc f 0.1fc c 1 log f X \ X \ I / RO . 4 D \ \X ^x 90°Figure 2.7 1 First order low pass magnitude and phase response and Bode approximations For each ten times increase in frequency the magnitude decreases by 1/10, or - 2 0 dB. (Note that a slope of - 2 0 dB/decade is sometimes expressed as —6 dB per octave; it goes down by 6 dB for each doubling of the frequency.) Gain attenuation w i t h increase in frequency is referred to as the roll off in the frequency response. The t w o straight lines intersect at the frequency f = fc and at this 30 frequency l ^ ( j f c ) I = 4 O L f = fc. The frequency fc A / 2 ; the response is thus 3 dB down when is sometimes referred t o as the 3 dB band- w i d t h limit. The phase/frequency characteristic associated w i t h equation 2.15 is determined by Θ = - tan~1 — fc (2.17) Forf<fc6^ 0, for f = fc Θ = - 4 5 ° and for f > fc Θ -> 9 0 ° . The Bode phase approximation approximates the phase shift by the asymptotic limits of 0° and —90° for frequencies a decade below and above fc respectively. The asymptotes are connected by a line whose slope is —45° per decade of frequency as shown in Figure 2.7. The errors involved in using the straight line approximation for the magnitude and phase behaviour of equation 2.15 are tabulated in Figure 2.7. Operational amplifier data sheets normally give values of AOL and the unity gain frequency ίΛ, which is the frequency at which the open loop gain has fallen to 0 dB because of open loop roll off. In the case of amplifiers which exhibit a first order frequency response w i t h a 20 dB per decade roll off down to unity gain the frequency f^ is related to the 3 dB bandwidth frequency fc by the expression fc = — (2.18) Frequency response characteristics are readily plotted f r o m a knowledge of AOL and /"·,. The Bode magnitude approximations are obtained by simply drawing t w o straight lines, one horizontal line at the value of AOL and the second through ίΛ w i t h a slope of —20 dB/decade. The t w o intersect at the frequency fc. Bode diagrams are useful in evaluating the frequency response characteristics of cascaded gain stages. The gain of a multistage amplifier is obtained as the product of the gains of the individual stages, but since gain is represented logarithmically in Bode plots the overall response may be determined by linearly adding the Bode plots for the separate stages as shown in Figure 2.8. Note that the final roll off and limiting phase shift depend upon the number of gain attenuating stages. T w o stages give a final roll off of —40 dB/decade and a limiting phase shift of 180°; three stages give - 6 0 dB/decade and 270° phase shift. 31 A /decade log f c1 Figure 2.8 'c2 N 'c3' Frequency response of cascaded gain stages Bode diagrams, as will be shown later, are particularly useful in assessing the stability and frequency response of feedback circuits and for this reason we give more examples of Bode diagrams for commonly encountered frequency response functions. The Bode magnitude and phase approximations for the function f fc2 (jf) (2.19) 1 +/■ ' d are given in Figure 2.9. A response of this kind is produced by a socalled lag, lead network (see Exercise 2.8). Note that the response is (a obtained by adding the Bode approximation for 1/(1 +j[flfc<\]) response which exhibits lagging phase shift), to the Bode approximation of 1 + j(f/fC2) ( a response which exhibits leading phase shift). 32 2.5 Small signal closed loop frequency response The desirable characteristics of operational amplifier circuits stem from the application of negative feedback. The quantitative effects of negative feedback, as was shown in Section 2.3, are related to the loop gain log f 20 d B / d e c a d e 4"(a y wrÏÏ ' ί^ 21 1 + ('c2 ) 1 + Λ Ν >2" — ta,/ J I Θ. + 90 + 45 0 -45 -90 Figure 2.9 / θ= + tan~1 J _ fc2 0.1f cl1 fr1 / cl/ f 9 c2 • Log f Overall phase shift ■tan-1 fcl Bode magnitude and phase approximations for a lag lead type of response (J/4 OL . Real operational amplifiers exhibit a frequency dependent AOL and in some applications the feedback fraction β is also frequency 33 dependent. Practical operational amplifier circuits therefore exhibit a frequency dependent loop gain and this has a marked effect on closed loop performance. It should always be remembered that a frequency dependence implies both a magnitude change and a phase change w i t h frequency and in a circuit designed to apply negative feedback it only needs an excess phase shift of 180° in the feedback loop t o make the circuit apply positive feedback and this can have most undesirable consequences. A n operational amplifier feedback circuit will produce self sustained oscillations if the phase shift in the loop gain reaches 180° at frequencies at which the magnitude of the loop gain is greater than u n i t y ; the amplifier and the circuit in which it is used must not allow this to happen. Loop gain phase shifts w i t h frequency of greater than 90° but less than 180°, whilst not resulting in sustained oscillations, can cause a feedback circuit to have a closed loop frequency response which peaks up at the bandwidth limit before it rolls off. Associated w i t h this closed loop gain peaking the circuit will have a transient response which exhibits overshoot and ringing. Transient response refers to the output changes produced in response to a step or squarewave input signal (see Section 3.8.2). A way of expressing the relative stability of a closed loop amplifier circuit is in terms of the so called phase margin. The phase margin is the amount by which the excess phase shift (phase shift over and above the inherent 180 required for negative feedback and obtained by returning the feedback signal t o the inverting input terminal) is less than 180° at that frequency at which the magnitude of the loop gain is unity. A closed loop circuit w i t h 90 phase margin shows no gain peaking; as phase margin is reduced gain peaking becomes noticeable for phase margins of approximately 60 (about 1 dB peaking) and becomes more marked w i t h further reduction in phase margin (20 phase margin gives approximately 9 dB of gain peaking). In order that they should be unconditionally stable under any value of resistive feedback most general purpose operational amplifiers are designed to have an open loop frequency response which follows a first order characteristic down to unity gain. This type of response was discussed in the previous section; it has a 20 dB/decade roll off down to unity gain and the phase shift associated w i t h gain attenuation never exceeds 90°. The phase margin for any value of resistive feedback is therefore never less than 9 0 ° . The effect of open loop gain frequency dependence on closed loop gain frequency dependence is most conveniently demonstrated in 34 graphical form by sketching the appropriate Bode plots. We look for the effect of AOL on loop gain and then to the effect of loop gain on the gain error factor. We may write ßAOL(jf) *OL(Jf) 1 which when expressed in dB form gives |loop gain (in dB| = |open loop gain (in dB) | (in dB) (2.20) The magnitude of the loop gain in dB at any frequency is equal to the difference between the open loop gain magnitude in dB and 1/ß in dB. R2 d B 4 99kQ ^OLljf)! Figure 2.10 Bode plots show frequency dependence of loop gain As an example of the graphical approach consider an operational amplifier with a first order frequency response used with resistive feedback in the follower configuration. The circuit and its Bode plots are illustrated in Figure 2.10. In order to display the frequency dependence of the loop gain we merely superimpose the plot of Mß (in dB) on the open loop frequency response plot of the amplifier. If feedback is purely resistive, as it is in the example under consideration, ß is independent of 35

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