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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
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