Electronic Devices for Analog Signal Processing ppt

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/>Yu. K. Rybin
Electronic Devices for
Analog Signal Processing
123
Yu.K. Rybin
Tomsk Polytechnic University
Electro Physical Department
Lenin street 30
634050 Tomsk
Russia

ISSN 1437-0387
ISBN 978-94-007-2204-0 e-ISBN 978-94-007-2205-7
DOI 10.1007/978-94-007-2205-7
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011940132
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Abstract
This book deals with modern devices for analog signal processing. A particular
attention is paid to the main element of such devices: integral operational amplifiers
(op-amps) and electronic devices based on them, including scaling, summing,
integrating, and filtering linear devices. The principles of construction of nonlinear

devices in op-amps are presented along with various circuit solutions for limiting,
rectification, and piecewise linear conversion of input signals. Sine wave and pulse
oscillators are analyzed. Some examples of applying these devices to processing of
signals from resistance, inductive, optical, and temperature sensors are presented.
This book is intended for engineers and post graduated students, learning the
course “Instrument Making” and for advanced learning of the courses “Electronics
part III” and “Electronics and Microprocessor Hardware,” but is can be also used
by other students and engineers dealing with the design of electronic devices and
systems.
This book has been prepared at the Chair “Computer Measuring Systems and
Metrology” of the Tomsk Polytechnic University.
v

Introduction
This book considers electronic devices applied to process analog signals in in-
strument making, automation, measurements, and other branches of technology.
They perform various transformations of electrical signals: scaling, integration,
logarithming, etc. Such devices are considered in tutorials on electronics. The need
in their deeper study is caused, on the one hand, by the great demands of extending
the range of input signals, as well as increasing the accuracy and speed of such
devices, which usually receive insufficient attention. On the other hand, new devices
arise permanently, which are not considered in electronic tutorials yet, but already
widely applied in practice.
Chapter 1 concerns the principles of design of modern operational amplifiers
(op-amps). This choice is caused by the fact that an op-amp is now one of the
most popular and versatile semiconductor components of almost any electronic
device. Since the advent of operational amplifiers, their circuits and fabrication
technology have been permanently improved. The efforts of developers were aimed
at the design and fabrication of different op-amp types with various characteristics.
As a result, the parameters of amplifiers with the traditional structure (voltage-

controlled amplifiers) have been improved and new current-controlled op-amps,
rail-to-rail amplifiers, clamping amplifiers, and specialized amplifiers of sensor
signals appeared. The information about these amplifiers is mostly concentrated
in scientific journals and manufacturers’ materials, but is almost lacking in the
educational literature.
Chapter 2 is devoted to the consideration of features of linear and nonlinear
operations with signals. The experience in teaching the electronics shows that reader
not always are able to determine correctly the function performed by an electronic
device, fail to select the method for its analysis, and, as a consequence, obtain
mistaken results. Therefore, this chapter considers the principal differences of linear
and nonlinear transformations by invoking the concepts of the spectrum of input and
converted signals.
Chapter 3 presents linear functional devices based on op-amps: inverting,
noninverting,summing, and instrumental amplifiers with the normalized gain. These
devices are now widely used for the primary processing of measuring, acoustic,
vii
viii Introduction
and video information, where they execute the functions of matching, precision
amplification, coupling with information transmission lines, etc.
Chapter 4 is devoted to nonlinear devices. It concerns the general issues of the
theory of nonlinear devices in op-amps and the practical circuits of such devices:
comparators, logarithmators, rectifiers, limiters, functional signal converters.
Chapters 5 and 6 consider sine wave and pulse oscillators. The range of
applicability of such oscillators is extremely wide. They are used in devices for
exciting sensors of physical parameters, in meters of frequency characteristics of
amplifiers and filters, in devices for transformation of signal spectra, in clocking
and synchronization devices, etc. As was mentioned in book (Horowitz P., Hill W.
The Art of Electronics. Second Edition. Cambridge University Press, England,
1998), a device without generator either is capable of nothing or is designed to
be connected to other device (which, most probably, includes a generator). Despite

this, such devices receive insufficient attention in the educational literature. Their
considerationis often fragmentary and does not favor the understandingof processes
occurring in them. Chapter 5 considers sine wave oscillators and the main known
approaches to the analysis of the processes of self-oscillation excitation and settling
in them. In particular, the analysis by the method of complex amplitudes, the method
of differential equations, the method of phase plane, and the two port method is
discussed. The preferable areas of application of these methods are demonstrated.
The well-known amplitude and phase balance conditions are criticized. Chapter 6 is
devoted to pulse oscillators. It is well-known that pulsed signals and their derivatives
have some features: parts with fast and slow change, wide spectrum. Pulsed signals
are generated by specific oscillating systems, for which the general conditions of
self-oscillation excitation are obtained.
Chapter 7 is devoted to the consideration of practical circuits for processing of
signals from sensors of physical parameters: resistance, inductive, semiconductor
sensors and coupling of sensors with electronic devices.
This book is organized nontraditionally. Its main goal is not only to give some
knowledge on modern electronic devices, but also to inspire students to the more
detailed study of these devices, understanding of their operation, ability to analyze
circuits, synthesize new devices, and assess the possibilities of their application for
solution of particular practical problems.
As was already mentioned, the course is divided into seven chapters. Each
chapter includes the theoretical material, questions, and tests to check how the
students have learned the theoretical material in the process of independent cognitive
work, as well as how ready he or she is to practical and laboratory works. The most
difficult questions are marked by asterisk

and can be given to advanced readers.
Paragraphs way of writing by italics are very important for the understanding of
the studied material and together they can serve a brief summary of a section. The
text marked by italic indicates new or non-traditional concepts. Calculated examples

are indicated by .
Contents
1 Modern Operational Amplifiers 1
1.1 Introduction 1
1.2 Application of Operational Amplifiers 2
1.3 Amplifiers with Potential Input 3
1.4 Electrical Models of Operational Amplifiers 8
1.5 Analysis of the Effect of Signal Source and Load 13
1.6 Amplifiers with Current Input 14
1.7 Amplifiers with Current Output 19
1.8 Current-Differencing Amplifiers 24
1.9 Rail-to-Rail Amplifiers 26
1.10 Instrumental Amplifiers 27
1.11 Clamping Amplifiers 27
1.12 Isolation Amplifiers 28
1.13 Conclusions 29
References 33
2 Functional Transformations of Signals 35
2.1 Introduction 35
2.2 Linear Transformations of Signals 36
2.3 Nonlinear Transformations of Signals 40
2.4 Conclusions 42
References 44
3 Linear Functional Units in Operational Amplifiers 45
3.1 Introduction 45
3.2 General Circuit Designs of Linear Devices 45
3.3 Scalers 49
3.3.1 Inverting Amplifiers 49
3.3.2 Noninverting Amplifier 53
3.3.3 Amplifiers Based on Inverting and Noninverting Amplifiers 54

3.4 Integrating Amplifiers 60
3.4.1 Inverting Integrating Amplifiers 60
ix
x Contents
3.4.2 Noninverting Integrating Amplifier 64
3.4.3 Integrating Amplifier with Two Inputs 65
3.4.4 Double Integrating Amplifier 66
3.5 Differentiating Amplifier 67
3.6 Active Filters Constructed in Op-amps 69
3.7 Conclusions 77
References 80
4 Nonlinear Devices in Op-amps 81
4.1 Introduction 81
4.2 Voltage Comparator 83
4.3 Logarithmic Amplifier 84
4.4 Operational Rectifiers 90
4.5 Full-Wave Operational Rectifiers 92
4.6 Voltage Limiters and Overload Protection Circuits 99
4.7 Op-amp Function Generators 103
4.8 Conclusions 108
References 110
5 Sine Wave Oscillators 111
5.1 Introduction 111
5.2 Oscillatory Processes 118
5.2.1 Analysis by the Method of Phase Plane 118
5.2.2 Analysis by the Method of Complex Amplitudes 123
5.2.3 Analysis by the Method of Differential Equations 126
5.2.4 Analysis by the Two-Port Network Method 130
5.3 Features of Oscillating Systems 133
5.4 RC Sine-Wave Oscillators 134

5.4.1 Principles of the Theory of RC Oscillators 134
5.4.2 The Oscillation Amplitude Stabilization and
Nonlinear Distortions 141
5.5 LC Sine Wave Oscillators 148
5.5.1 Transformer-Coupled LC Oscillators 148
5.5.2 Three-Point Oscillators 152
5.6 Quartz Oscillators 154
5.7 Negative Resistance Oscillators 155
5.8 Synthesis of Oscillating Systems of RC Oscillators 160
5.9 Conclusions 167
References 172
6 Pulse Oscillators 173
6.1 Introduction 173
6.2 Selected Issues of Theory of Pulse Oscillators 174
6.2.1 The Conditions for Excitation of Pulsed Oscillations 176
6.3 Op-amp Pulse Oscillators 184
6.4 Possible Circuits of Op-amp Oscillators 194
Contents xi
6.5 Logic-Gate Oscillator 197
6.6 Integrated Timer Oscillator 198
6.7 Oscillators in Elements with Negative Resistance 202
6.8 Conclusions 207
References 210
7 Signal Conditioners 211
7.1 Introduction 211
7.2 Resistive Sensor Signal Conditioners 212
7.3 Inductive Sensor Signal Conditioner 219
7.4 Optical Sensor Signal Conditioners 223
7.5 Thermocouple Signal Conditioners 225
7.6 Voltage and Current Sensor Signal Conditioners 227

7.7 Conclusions 228
References 229
Appendix 231
Abbreviations 243
Parameters 245
Conclusions 247
Glossary 249
Index 255

Chapter 1
Modern Operational Amplifiers
Abstract The purpose of this chapter is to introduce specific features of circuit
design of modern op-amps, their parameters, characteristics and macromodels to
ensure effective use and proper design of electronic devices based on these op-
amps. The necessary prerequisite is the knowledge of theory of amplifiers within
the course “Electronics” or “Electronics in Instrument Making.”
Having studied this Chapter, one will be able to determine the structure of an
operational amplifier, analyze circuits, basic parameters and characteristics, and
know their structural differences.
1.1 Introduction
An operational amplifier is a direct current (DC) amplifier intended for executing (together
with external elements) various operations on (above) input signals and capable of working
with the large feedback.
This term arose in the 1930s [1],
1
and initially it applied to DC amplifiers used
in telephony and analog computers.
First operational amplifiers (op-amps) were based on electronic tubes; they
executed linear mathematical operations with input voltages: multiplication by
a constant, differentiation, and integration, and allowed electronic modeling of

differential equations. These op-amps had large size and several supply voltages
and consumed power up to several watts.
With further developmentof semiconductor industry,hybrid op-amps (assembled
of separate elements: transistors and resistors) were designed, and later on op-amps
were manufactured on a single piece of silicon crystal (chip). Specifications and
characteristics of these op-amps are persistently improved. Now such op-amps are
1
Appearance of operational amplifiers is associated with Harold S. Black, who, working in Bell
Labs, proposed op-amps for telephony in 1934 [2].
Y.K. Rybin, Electronic Devices for Analog Signal Processing, Springer Series
in Advanced Microelectronics 33, DOI 10.1007/978-94-007-2205-7
1,
© Springer ScienceCBusiness Media B.V. 2012
1
2 1 Modern Operational Amplifiers
called integral operational amplifiers. Though their basic application has changed
since appearance of digital computers, these amplifiers are still referred to as
operational and widely used in various electronic devices. This term no longer
carries the meaning that had at the beginning. The word “operational” assumes some
operation on signal, but an operational amplifier itself performs no other operations
without external elements, but only signal amplification, which is its main and,
perhaps, sole function. Modern op-amps perfectly carried out this function.
Op-amps are characterized by the high gain (1,000,000 and more), low input offset voltage
(from 0,1 V), wide frequency band (up to 2,000 MHz), and high slew rate (up to 3,000
V/s) [3]. These op-amp parameters are continuously improving.
Nowadays the industry produces a large number(several hundreds) of various op-
amps; therefore, even simple enumeration of their parameters and characteristics,
in particular, those that earlier believed atypical for op-amps (for example, low
input or output resistance) is a certain problem. It is difficult to orient oneself in
this abundance of types and parameters without the necessary structured knowledge

about them.
Thus, consideration of op-amps starts with their electrical models, rather than
parameters and features of circuitry and production technologies (these issues are
sufficiently addressed in the literature). It is assumed that the students already have
the basic knowledge about the input and output parameters of op-amps (parameters
and characteristics of some of them are presented in Appendix 1).
1.2 Application of Operational Amplifiers
Op-amps are now used in the systems for data acquisition and signal processing of
measurement information, entering of the analog signals into the computer, in audio and
medical systems, etc. [4–7]
They are characterized by small size, wide range of power supply voltages, low
consumed power, and others. Besides, they are suitable for any operating conditions.
However, the main reason for wide application of op-amps is that the parameters and
characteristics of a device are independent of the parameters and characteristics of
the op-amp itself, because, as known, the op-amp parameters are usually instable
in time and vary with temperature and frequency, and so developers of electron
devices try all ways to minimize their effect. A large feedback allows reaching it.
The needed functions of a device are rather readily achieved in this case using of
external elements.
The relative easiness of designing various circuits with op-amps caused a
simplified attitude to them. Now the knowledge of parameters and characteristics of
operational amplifiers sometimes substitutes for the recognition of their structure.
The common opinion is that for application of op-amps it is not needed (rather,
not necessarily needed) to know their circuit, but it is sufficient to be aware of the
1.3 Amplifiers with Potential Input 3
input and output (interface) characteristics and to consider the op-amp itself as a
black box. This statement applies not only to amplifiers. Thus, most of computer
manufacturers scarcely know the principle circuit of the Pentium microprocessor,
and this does not hinder them to create excellent computers. Op-amps are considered
as such circuit elements, for example, resistor or capacitor, with only somewhat

more complex internal structure. Moreover, the wide usage of software for mod-
eling electronic devices on personal computers (Multisim, Electronics Workbench,
DesignLab, Orcad, Protel, and others) approves this approach, because op-amps
in such a case are selected from a library, as any other element. Nevertheless,
the system modeling assuming the knowledge of the structure, structural relations,
and principles of construction of various operational amplifiers allows one to more
competently design and operate electron devices based on them. This concept can
be supported by the following.
First, any op-amp model is certainly more simple than the principle circuit and,
even more so, its physical prototype.
Second, from the system point of view, the amplifier scheme corresponds to a
higher level of modeling, including any model of a black box with all its parameters.
Third, the knowledge of the internal structure allows one to more efficiently apply
op-amps and to use methods for correction of their characteristics, in particular,
those not documented by the manufacturer.
Finally, alphanumeric indexes of operational amplifiers give no information
about their structure (for example, 140UD1 and 1401UD1 (Russian) amplifiers have
absolutely different structures and different applications).
Op-amps have widely different designs, parameters, and characteristics, and the
main problem for developer is to find the best op-amp for some device or another
one, because the correct and reasonable choice of an op-amp determines the cost,
reliability, and quality of the device under development.
All amplifiers can be divided into two groups: amplifiers with potential (high
resistance) input and amplifiers with current (low resistance) input. Let us consider
these two types.
1.3 Amplifiers with Potential Input
The circuit of the K157UD4 op-amp with potential input made using the bipolar technology
is shown on Fig. 1.1 . The circuit [3] includes three amplifier stages.
The first (input) stage is a symmetric differential one; it is constructed in VT1—
VT4 transistors. The input signal is given to one of the bases of the VT1andVT2

transistors or to the both bases simultaneously.The signal amplified by the first stage
comes to the second (intermediate) stage constructed in VT5andVT6 transistors,
and after amplification by the second stage it comes to the third (output) stage
designed in VT7—VT10 transistors. The output stage is connected in the circuit
of a push-pull compound emitter follower constructed in VT8, VT9andVT7, VT10
complementary transistors, respectively. Note that each arm of the stage includes
4 1 Modern Operational Amplifiers
Fig. 1.1 Circuit of K157UD4-type op-amp with the VCVS structure
Fig. 1.2 Drawing of
K157UD4-type op-amps in
figures (a) and simplified
representation (b)
the current sources I3andI4. Current sources are usually represented by transistors,
and for their normal operation the voltage drop no less than 1–1.5 V is needed.
Consequently, the output voltage of the amplifier is always lower than the supply
voltage by 1.5–2 V.
The basic amplifier parameters are determined by the parameters of the stages. Thus, the
input resistance, current, and offset voltage are determined by the input stage, while the
output resistance and the maximal values of the output voltage and current are determined
by the output stage.
The op-amp gain is equal to the product of the stage gains. But, as known, the
emitter follower does not amplify the signal voltage. Therefore, the whole gain of
the amplifier is determined by the product of the gains of the input and intermediate
stages only.
The circuit symbols for it are shown on Fig. 1.2.
One of the basic characteristics of op-amps is the frequency dependence of the gain, which
is called the gain-frequency characteristic (GFC) or the open-loop-gain characteristic.
1.3 Amplifiers with Potential Input 5
Fig. 1.3 GFC of op-amp without (0) and with feedback (1–3)
The GFC shape in the general case depends on the number of amplifier stages,

type of the transistors, circuit of their connection, operating mode, etc.
It is a specific of the op-amp GFC that frequency of the input signal increases, and
the gain varies widely: from several tens or even hundreds thousands to 1 and even
smaller. In addition, in many circuits the op-amp is to operate with a large feedback,
and the gain-frequency and the phase-response (PRC) characteristics should have a
certain form, providing for some marginal stability. Therefore, the op-amp GFC
is corrected. For example, for correction of the K157UD2 amplifier, the circuit
includes the capacitor C
fc
connected to frequency correction (FC) terminals. In this
case, the gain of the intermediate stage and consequently, of the op-amp as a whole
depends on the signal frequency. With accordance C
fc
, the overall op-amp gain is
P
K D
P
k
1
P
k
2
P
k
3
D
k
1
1 C jf =f
1


k
2
1 Cjf =f
2

k
3
1 C jf =f
3
'
K
0
1 C jf =f
cut
;
where
P
k
1
,
P
k
2
,and
P
k
3
are the complex gains of the input, intermediate, and output
stages; K

0
is the gain at f D0; f
cut
Df
2
is the cutoff frequency of the op-amp GFC.
The cutoff frequency depends on many factors, first of all, on the collector
currents of the transistors: the higher the currents, the higher the cutoff frequency.
But the input resistance in this case decreases, because the emitter current increases.
A way to increase the input resistance is to decrease the emitter currents of
the input transistors. The typical values of the input resistance are from 4 k for the
140UD1 op-amp to 1.5 M for the A725 op-amp. However, this decrease in the
currents of the input transistors results in the impossibility of quick recharge of
the correcting capacitor. Therefore, these amplifiers are characterized by the low
frequency properties and the low slew rate. The GFC cutoff frequency for these op-
amps usually ranges within 10–100 Hz, and the slew rate does not exceed 10 V/s.
Figure 1.3 shown GFC of the K157UD4 op-amp in the log scale. At the low
frequencies, the gain is constant, independent of the signal frequency, and equal
6 1 Modern Operational Amplifiers
Fig. 1.4 Circuit of inverter
amplifier with parallel
feedback based on op-amp
with VCVS structure
Fig. 1.5 PRC of amplifier without (0) and with feedback (1–3)
to K
0
. Starting from the cutoff frequency f
cut
D20 Hz, the gain monotonically
decreases with the rate of 20 dB/dec because of the decrease in the gain of the

intermediate stage caused by the presence of the correcting capacitor with the
capacity C
fc
D30 pF. At the frequency f
T
D1,000 kHz the gain becomes equal to
1, and there is no amplification. This frequency is called the threshold amplification
frequency of op-amp.
The gain decreases with the negative feedback (see Fig. 1.4). The op-amp gain
with a large feedback is K
fb
R
2
/R
1.
It is independent of the op-amp parameters,
but determined by external elements. The characteristics at K
fb
equal to 1,000, –
100, and 10 are shown by lines 1, 2,and3 on Fig. 1.3. As the gain decreases, the
feedback increases and the frequency band becomes wider. Amplifiers of this kind
are characterized by the roughly constant amplification area, that is the product of
the gain by the upper threshold frequency (cutoff frequency).
The phase-response characteristic is connected with the GFC and dependent on
this. PRC of the K157UD4 without and with feedback is shown on Fig. 1.5. It can be
seen that at the frequencies higher than the cutoff frequency the op-amp phase is al-
most equal to –/2.
2
If op-amp is enveloped by the negative frequency-independent
feedback, the total phase shift in the feedback loop only slightly exceeds –3/2, and

the amplifier has the stability margin about 60–70
ı
at the threshold frequency. The
amplified frequency band extends where phase shift is zero.
Another important parameter of an amplifier is the gain characteristic (GC), which is the
dependence of instant output voltage vs. the instant input voltage.
2
It is PRC for the noninverting input. For the inverting input, – should be added at any frequency.
1.3 Amplifiers with Potential Input 7
Fig. 1.6 Typical gain
characteristic of op-amp
without (0) and with (1)
feedback
This characteristic is measured as slow variation of the input voltage and has
wider variety as compared to GFC. However, all GCs are features with limited
output values. Typical GC is shown on Fig. 1.6.
In the general case, it does not pass through the origin, because almost any
amplifier has the input offset voltage V
off
. As can be seen from Fig. 1.6, GC becomes
more linear with feedback; it is a smoothly increasing (for the noninverting input) or
decreasing (for the inverting input) curve limited by the maximum allowable levels
of the output voltage, which, naturally, cannot exceed the supply voltage.
For practical calculations accordingly the op-amp nonlinear properties, its GC
without feedback can be described through the hyperbolic tangent function
V
out
D
8
ˆ

ˆ
<
ˆ
ˆ
:
V
m
; if F.V
in
/  V
m
=k
2
;
k
2
F.V
in
/; if  V
m
=k
2
<F.V
in
/<V
m
=k
2
;
V

m
; if F.V
in
/ ÄV
m
=k
2
; where F.V
in
/ D V
m
tanh Œ.V
in
C V
off
/='
T
;
(1.1)
V
off
is the input offset voltage reduced to the op-amp input; ®
T
is the temperature
voltage (25.6 mV at T D20
ı
C); V
m
is the maximum allowable voltage at op-amp
output, k

2
is the gain of the intermediate stage. The gain at a small signal in this case
is K D k
2
V
m
='
T
.
When amplifying pulsed signals and operating in the switching mode, the transient response
characteristic (TC) is important.
Remind that TC is the time dependence of the output voltage at a stepwise change
of the input voltage. TC for small and large input signals are usually distinguished.
Small signals are the signals, at which the output voltage remains within the linear
range of the gain characteristic and does not achieve the maximum allowable value,
or the signals, variation of whose amplitude does not result in a change of the
amplifier parameters. Large signals are the signals, at which the output voltage can
8 1 Modern Operational Amplifiers
Fig. 1.7 Typical transient
response characteristics of
A741 amplifier at small (1)
and large (2) signals
take limit values. In this case, transistors operate in the cutoff or saturation ranges,
that is, high signals force the op-amp into the significantly nonlinear operation
mode.
TC of the A741 amplifier are shown on Fig. 1.7. It can be seen that at small
signals (curve 1) the transient response process is long enough (see the lower scale
of the axis t). At large signals, the op-amp quickly enters the nonlinear mode with
the rate limited only by the rate of increase of the op-amp output voltage.
1.4 Electrical Models of Operational Amplifiers

Modern op-amps are made by the integral technology, and so they are chips with very-large-
scale integration (VLSI).
The exact analysis of circuits with such op-amps is almost impossible without
computer. Even in this case, the circuit including dozens of transistors, resistors, and
capacitors do not analyzed. Frequently equivalent circuit is used, whose input and
output voltages and currents are equal to the input and output voltages and currents
of the op-amp.
The modern analysis uses various equivalent circuits of op-amps, from simplest to very
complicated.
In this case, the choice is usually caused by the demanded accuracy and the ac-
ceptable time of analysis. Simple equivalent circuits do not guarantee high accuracy
or they even fail to determine some needed parameters and characteristics, but allow
fast tentative analysis. Complex circuits (so-called macromodels), to the contrary,
give rather accurate results, but they are labor consuming and time expensive.
We are considered the known equivalent circuits by the principle “from simple
to complex.”
1.4 Electrical Models of Operational Amplifiers 9
Fig. 1.8 The op-amp linear models in a kind of two-port form
From electrical circuits theory it is well known that any two-port in the linear
approximation can be represented by one of equivalent circuits on Fig. 1.8.
3
Perhaps, it is the simplest electrical models of op-amps. They have different input
and output parts depending on the chosen independent input and output electrical
characteristics. Parameters of equivalent circuits are denoted as Z, Y, F and H with
the corresponding indexes. The meaning and values of these parameters are well
known.
4
The output circuit is represented by a voltage source in Figs. 1.8a, c and by
a current source in Figs. 1.8b, d, and the both sources are dependent. In the first
circuit (1.8a) voltage depends on the input current: E

i
DZ
21
I
in
, and in the second
one (1.8c) it depends on the input voltage: E
V
DF
21
V
in
. Similarly, the currents of
the controlled current sources depend on the input voltage in the circuit shown in
Fig. 1.8b(I
V
DY
21
V
in
) and on the input current in the circuit shown on Fig. 1.8d
(I
i
DH
21
I
in
). As applied to amplifiers, the parameter Z
21
DZ

tr
is transresistance,
F
21
DK
V
is voltage gain, Y
21
DS is transconductance, and H
21
DK
i
, is current
gain, that characterized the op-amp amplifying properties.
It should be noted that the amplifying parameters are measured in different units:
K
V
and K
i
are dimensionless parameters, while Z
tr
is measured in the units of
resistance, and S is measured in the units of conductance.
Depending on the type of the output source and the controlling electrical characteristic, the
simple equivalent circuits present, respectively: 1.8a – Current controlled voltage source
(CCVS), 1.8b – voltage controlled current source (VCCS), 1.8c – VCVS, and 1.8d – CCCS.
Each of these circuits can be described by a system of equations.
3
For simplicity, reverse transfer elements are excluded in Fig. 1.8.
4

See, for example, A.F.Beletskii, Principles of Theory of Linear Electrical Circuits (Svyaz,
Moscow, 1967).
10 1 Modern Operational Amplifiers
For example, the two-port on Fig. 1.8c (VCVS) can be described by the system
of two equations:
I
in
D F
11
V
in
I
V
out
D K
V
V
in
C F
22
I
out
:
)
(1.2)
The independent variables here are the input voltage (V
in
) and the output current
(I
out

). The first and second equations describe, respectively, the input and output
op-amp circuits. In the first equation, the input circuit is represented by the input
conductance F
11
D1/Z
in
. The input resistance serves a load for the signal source
and consumes the corresponding power from it. The higher the input resistance, the
lower the input current I
in
, so the greater voltage part of the signal source comes to
the op-amp input, and the lower is the power needed from the signal source.
Most of modern op-amps are characterized by high input resistance (1–10 M).
Due to this fact, the necessary current from the signal source is low. The output
circuit includes the voltage source E
V
depended on the input voltage V
in
and the
output resistance F
22
DZ
out
. The relation between Z
out
and Z
load
determines what
part of voltage E
V

will be separated at the load resistance.
The described equivalent circuits can be used only for approximate calculations of such
device parameters as the gain and input and output resistance, because they ignore the
following op-amp disadvantages:
– input offset voltage, and input currents;
– limited output voltage;
– rising of the input voltage, etc.
Some of these disadvantages are eliminated in more complex equivalent circuits.
The linear one-port equivalent op-amp circuit (macromodel) used in the Electronics
Workbench software is shown on Fig. 1.9.
It more accurately models the op-amp frequency properties, the input currents
of transistors and the input offset voltage. The frequency properties are presented
by two frequency-dependent RC-circuits: (R
i
, C
i
, C
fc
, R
in2
)and(R
out
and C
out
), and
one of the capacitors (the frequency correcting capacitor (C
fc
)) is connected to the
external terminals and can be changed. The transistor input currents are determined
by the sources of input currents (I

b1
, I
b2
), and the input offset voltage (V
off
)issetby
the voltage source.
The circuit includes two (rather than one) depended sources, which are enclosed by
the dashed rectangle. The disadvantages of this circuit are the impossibility to consider
common-mode parameters and the limited output voltage.
In circuit on Fig. 1.10 these disadvantages are removed. Here the input resistance
Z
in
is represented more specifically by the resistors R
in
and input capacitors (C1and
C2) for the symmetric input. The elements R
cm
and C
cm
account for the common-
mode input resistance and common-mode input capacitance. The elements R
i
, C
i
and R
out
, C
out
model the op-amp frequency properties, while the elements VD1,

1.4 Electrical Models of Operational Amplifiers 11
Fig. 1.9 One-port linear equivalent circuit of op-amp
Fig. 1.10 Nonlinear equivalent circuit (macromodel) of op-amp
VD2, V1andV2 accounts for the effect of the limited output voltage at the level
of V
1
and V
2
voltages. Diodes in this circuit makes it nonlinear, unlike the previous
circuits.
Certainly, now the use of the macromodel is more complicated, and the calcu-
lations become more complexes. Therefore, it suits for computations as a PSpice
macromodel in the Electronics Workbench and DesignLab software. Such a model
can be easily constructed not only for the VCCS structure, but also for any other.
The further improvement of the equivalent circuit allows us to take into account
the input currents and the input offset voltage, to determine more accurately the
frequency properties, limitedness of not only output voltage, but also the output
current, etc.
One of the most perfect op-amp macromodels, namely, the Boyle-Cohn-Pederson model [8]
on Fig. 1.11 is also used in Electronics Workbench and DesignLab.
12 1 Modern Operational Amplifiers
Fig. 1.11 Boyle–Cohn–Pederson macromodel
This model includes a differential stage consisting of NPN transistors VT1and
VT2
5
one uncontrolled (I1) and four controlled (I2–I5) current sources, output
current limiters assembled in diodes VD1andVD2, and output voltage limiters
assembled in diodes VD3andVD4. The built-in current sources in their structure
are similar to VCCS. The effect of the op-amp input parameters is modeled by the
differential stage, the frequency properties are determined by the capacitors C1and

C
fc
, and the output resistance is modeled by the resistors R7andR9.
As would be expected, the more exactly is a model, the more complicated one,
and it is the nearer to the op-amp circuit. But the analyses in this case bring much
time. This is explained by the ancient contradiction between the accuracy and the
simplicity of a model. However there are no miracles. Hence, it can be concluded
that the op-amp principle circuit serves the most accurate op-amp macromodel, just
which was stated in the beginning of this Chapter.
Thus, for approximate calculations of the gain and the input and output DC resistance, we
can use the op-amp models shown on Fig. 1.8. If it is necessary to take into account the
op-amp frequency properties, the descriptions of these models should be supplemented with
the frequency dependence of their parameters. The one-port linear equivalent circuit on
Fig. 1.9 is better suited, when it is needed to more accurately take into consideration the
frequency properties in the form of two time constants, as well as the effect of the input
currents and the input offset voltage. The nonlinear equivalent circuit (macromodel) shown
on Fig. 1.10 represents better the common-mode parameters and the level of restriction of
the output voltage. Finally, the Boyle-Cohn-Pederson macromodel on Fig. 1.11 accounts for
all the listed dependences.
5
There are similar models constructed in bipolar (PNP) and field-effect (FET) transistors.