springer series in advanced microelectronics 33 · chapter 1 concerns the principles of design of...

Springer Series in Advanced Microelectronics 33

The Springer Series in Advanced Microelectronics provides systematic informationon all the topics relevant for the design, processing, and manufacturing of micro-electronic devices. The books, each prepared by leading researchers or engineersin their fields, cover the basic and advanced aspects of topics such as waferprocessing, materials, device design, device technologies, circuit design, VLSIimplementation, and subsystem technology. The series forms a bridge betweenphysics and engineering and the volumes will appeal to practicing engineers as wellas research scientists

Series Editors:Dr. Kiyoo ItohHitachi Ltd., Central Research Laboratory, 1-280 Higashi-KoigakuboKokubunji-shi, Tokyo 185-8601, Japan

Professor Thomas LeeDepartment of Electrical Engineering, Stanford University, 420 Via Palou Mall,CIS-205 Stanford, CA 94305-4070, USA

Professor Takayasu SakuraiCenter for Collaborative Research, University of Tokyo, 7-22-1 RoppongiMinato-ku, Tokyo 106-8558, Japan

Professor Willy M.C. SansenESAT-MICAS, Katholieke Universiteit Leuven, Kasteelpark Arenberg 103001 Leuven, Belgium

Professor Doris Schmitt-LandsiedelLehrstuhl fur Technische Elektronik, Technische Universitat MunchenTheresienstrasse 90, Gebaude N3, 80290 Muanchen, Germany

For further volumes:http://www.springer.com/series/4076

Yu. K. Rybin

Electronic Devices forAnalog Signal Processing

123

Yu.K. RybinTomsk Polytechnic UniversityElectro Physical DepartmentLenin street 30634050 [email protected]

ISSN 1437-0387ISBN 978-94-007-2204-0 e-ISBN 978-94-007-2205-7DOI 10.1007/978-94-007-2205-7Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2011940132

© Springer Science+Business Media B.V. 2012No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or byany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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Springer is part of Springer Science+Business Media (www.springer.com)

Abstract

This book deals with modern devices for analog signal processing. A particularattention 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 nonlineardevices in op-amps are presented along with various circuit solutions for limiting,rectification, and piecewise linear conversion of input signals. Sine wave and pulseoscillators are analyzed. Some examples of applying these devices to processing ofsignals from resistance, inductive, optical, and temperature sensors are presented.

This book is intended for engineers and post graduated students, learning thecourse “Instrument Making” and for advanced learning of the courses “Electronicspart III” and “Electronics and Microprocessor Hardware,” but is can be also usedby other students and engineers dealing with the design of electronic devices andsystems.

This book has been prepared at the Chair “Computer Measuring Systems andMetrology” 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 needin their deeper study is caused, on the one hand, by the great demands of extendingthe range of input signals, as well as increasing the accuracy and speed of suchdevices, which usually receive insufficient attention. On the other hand, new devicesarise permanently, which are not considered in electronic tutorials yet, but alreadywidely 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 themost popular and versatile semiconductor components of almost any electronicdevice. Since the advent of operational amplifiers, their circuits and fabricationtechnology have been permanently improved. The efforts of developers were aimedat 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 sensorsignals appeared. The information about these amplifiers is mostly concentratedin scientific journals and manufacturers’ materials, but is almost lacking in theeducational literature.

Chapter 2 is devoted to the consideration of features of linear and nonlinearoperations with signals. The experience in teaching the electronics shows that readernot always are able to determine correctly the function performed by an electronicdevice, fail to select the method for its analysis, and, as a consequence, obtainmistaken results. Therefore, this chapter considers the principal differences of linearand nonlinear transformations by invoking the concepts of the spectrum of input andconverted signals.

Chapter 3 presents linear functional devices based on op-amps: inverting,noninverting, summing, and instrumental amplifiers with the normalized gain. Thesedevices are now widely used for the primary processing of measuring, acoustic,

vii

viii Introduction

and video information, where they execute the functions of matching, precisionamplification, coupling with information transmission lines, etc.

Chapter 4 is devoted to nonlinear devices. It concerns the general issues of thetheory 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 ofapplicability of such oscillators is extremely wide. They are used in devices forexciting sensors of physical parameters, in meters of frequency characteristics ofamplifiers and filters, in devices for transformation of signal spectra, in clockingand 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 tobe connected to other device (which, most probably, includes a generator). Despitethis, such devices receive insufficient attention in the educational literature. Theirconsideration is often fragmentary and does not favor the understanding of processesoccurring in them. Chapter 5 considers sine wave oscillators and the main knownapproaches to the analysis of the processes of self-oscillation excitation and settlingin them. In particular, the analysis by the method of complex amplitudes, the methodof differential equations, the method of phase plane, and the two port method isdiscussed. The preferable areas of application of these methods are demonstrated.The well-known amplitude and phase balance conditions are criticized. Chapter 6 isdevoted to pulse oscillators. It is well-known that pulsed signals and their derivativeshave some features: parts with fast and slow change, wide spectrum. Pulsed signalsare generated by specific oscillating systems, for which the general conditions ofself-oscillation excitation are obtained.

Chapter 7 is devoted to the consideration of practical circuits for processing ofsignals from sensors of physical parameters: resistance, inductive, semiconductorsensors and coupling of sensors with electronic devices.

This book is organized nontraditionally. Its main goal is not only to give someknowledge on modern electronic devices, but also to inspire students to the moredetailed study of these devices, understanding of their operation, ability to analyzecircuits, synthesize new devices, and assess the possibilities of their application forsolution of particular practical problems.

As was already mentioned, the course is divided into seven chapters. Eachchapter includes the theoretical material, questions, and tests to check how thestudents have learned the theoretical material in the process of independent cognitivework, as well as how ready he or she is to practical and laboratory works. The mostdifficult questions are marked by asterisk � and can be given to advanced readers.

Paragraphs way of writing by italics are very important for the understanding ofthe studied material and together they can serve a brief summary of a section. Thetext marked by italic indicates new or non-traditional concepts. Calculated examplesare indicated by�.

Contents

1 Modern Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Application of Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Amplifiers with Potential Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Electrical Models of Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Analysis of the Effect of Signal Source and Load . . . . . . . . . . . . . . . . . . . 131.6 Amplifiers with Current Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Amplifiers with Current Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Current-Differencing Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.9 Rail-to-Rail Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.10 Instrumental Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.11 Clamping Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.12 Isolation Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Functional Transformations of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Linear Transformations of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Nonlinear Transformations of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Linear Functional Units in Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 General Circuit Designs of Linear Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Scalers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.1 Inverting Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.2 Noninverting Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.3 Amplifiers Based on Inverting and Noninverting Amplifiers 54

3.4 Integrating Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.1 Inverting Integrating Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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3.4.2 Noninverting Integrating Amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.3 Integrating Amplifier with Two Inputs . . . . . . . . . . . . . . . . . . . . . . . 653.4.4 Double Integrating Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Differentiating Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.6 Active Filters Constructed in Op-amps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Nonlinear Devices in Op-amps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Voltage Comparator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Logarithmic Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Operational Rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5 Full-Wave Operational Rectifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.6 Voltage Limiters and Overload Protection Circuits . . . . . . . . . . . . . . . . . . 994.7 Op-amp Function Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Sine Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Oscillatory Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 Analysis by the Method of Phase Plane . . . . . . . . . . . . . . . . . . . . . . 1185.2.2 Analysis by the Method of Complex Amplitudes . . . . . . . . . . . 1235.2.3 Analysis by the Method of Differential Equations.. . . . . . . . . . 1265.2.4 Analysis by the Two-Port Network Method . . . . . . . . . . . . . . . . . 130

5.3 Features of Oscillating Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.4 RC Sine-Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.1 Principles of the Theory of RC Oscillators. . . . . . . . . . . . . . . . . . . 1345.4.2 The Oscillation Amplitude Stabilization and

Nonlinear Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.5 LC Sine Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.5.1 Transformer-Coupled LC Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . 1485.5.2 Three-Point Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.6 Quartz Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545.7 Negative Resistance Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.8 Synthesis of Oscillating Systems of RC Oscillators . . . . . . . . . . . . . . . . . 1605.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6 Pulse Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.2 Selected Issues of Theory of Pulse Oscillators . . . . . . . . . . . . . . . . . . . . . . . 174

6.2.1 The Conditions for Excitation of Pulsed Oscillations . . . . . . . 1766.3 Op-amp Pulse Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.4 Possible Circuits of Op-amp Oscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Contents xi

6.5 Logic-Gate Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.6 Integrated Timer Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.7 Oscillators in Elements with Negative Resistance . . . . . . . . . . . . . . . . . . . 2026.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7 Signal Conditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2117.2 Resistive Sensor Signal Conditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2127.3 Inductive Sensor Signal Conditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.4 Optical Sensor Signal Conditioners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.5 Thermocouple Signal Conditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257.6 Voltage and Current Sensor Signal Conditioners . . . . . . . . . . . . . . . . . . . . 2277.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Chapter 1Modern Operational Amplifiers

Abstract The purpose of this chapter is to introduce specific features of circuitdesign of modern op-amps, their parameters, characteristics and macromodels toensure effective use and proper design of electronic devices based on these op-amps. The necessary prerequisite is the knowledge of theory of amplifiers withinthe course “Electronics” or “Electronics in Instrument Making.”

Having studied this Chapter, one will be able to determine the structure of anoperational amplifier, analyze circuits, basic parameters and characteristics, andknow their structural differences.

1.1 Introduction

An operational amplifier is a direct current (DC) amplifier intended for executing (togetherwith external elements) various operations on (above) input signals and capable of workingwith the large feedback.

This term arose in the 1930s [1],1 and initially it applied to DC amplifiers usedin telephony and analog computers.

First operational amplifiers (op-amps) were based on electronic tubes; theyexecuted linear mathematical operations with input voltages: multiplication bya constant, differentiation, and integration, and allowed electronic modeling ofdifferential equations. These op-amps had large size and several supply voltagesand consumed power up to several watts.

With further development of semiconductor industry, hybrid op-amps (assembledof separate elements: transistors and resistors) were designed, and later on op-ampswere manufactured on a single piece of silicon crystal (chip). Specifications andcharacteristics of these op-amps are persistently improved. Now such op-amps are

1Appearance of operational amplifiers is associated with Harold S. Black, who, working in BellLabs, proposed op-amps for telephony in 1934 [2].

Y.K. Rybin, Electronic Devices for Analog Signal Processing, Springer Seriesin 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 changedsince appearance of digital computers, these amplifiers are still referred to asoperational and widely used in various electronic devices. This term no longercarries the meaning that had at the beginning. The word “operational” assumes someoperation on signal, but an operational amplifier itself performs no other operationswithout 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 3,000V/�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, lowinput or output resistance) is a certain problem. It is difficult to orient oneself inthis abundance of types and parameters without the necessary structured knowledgeabout them.

Thus, consideration of op-amps starts with their electrical models, rather thanparameters and features of circuitry and production technologies (these issues aresufficiently addressed in the literature). It is assumed that the students already havethe basic knowledge about the input and output parameters of op-amps (parametersand 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 ofmeasurement information, entering of the analog signals into the computer, in audio andmedical systems, etc. [4–7]

They are characterized by small size, wide range of power supply voltages, lowconsumed 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 andcharacteristics of a device are independent of the parameters and characteristics ofthe op-amp itself, because, as known, the op-amp parameters are usually instablein time and vary with temperature and frequency, and so developers of electrondevices 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 ofexternal elements.

The relative easiness of designing various circuits with op-amps caused asimplified attitude to them. Now the knowledge of parameters and characteristics ofoperational 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 ablack box. This statement applies not only to amplifiers. Thus, most of computermanufacturers scarcely know the principle circuit of the Pentium microprocessor,and this does not hinder them to create excellent computers. Op-amps are consideredas such circuit elements, for example, resistor or capacitor, with only somewhatmore 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-ampsin 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 morecompetently design and operate electron devices based on them. This concept canbe 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 ahigher 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 applyop-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 informationabout their structure (for example, 140UD1 and 1401UD1 (Russian) amplifiers haveabsolutely different structures and different applications).

Op-amps have widely different designs, parameters, and characteristics, and themain problem for developer is to find the best op-amp for some device or anotherone, 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 (highresistance) input and amplifiers with current (low resistance) input. Let us considerthese two types.

1.3 Amplifiers with Potential Input

The circuit of the K157UD4 op-amp with potential input made using the bipolar technologyis 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 VT1 and VT2transistors or to the both bases simultaneously. The signal amplified by the first stagecomes to the second (intermediate) stage constructed in VT5 and VT6 transistors,and after amplification by the second stage it comes to the third (output) stagedesigned in VT7 — VT10 transistors. The output stage is connected in the circuitof a push-pull compound emitter follower constructed in VT8, VT9 and VT7, VT10complementary transistors, respectively. Note that each arm of the stage includes


Fig. 1.1 Circuit of K157UD4-type op-amp with the VCVS structure

Fig. 1.2 Drawing ofK157UD4-type op-amps infigures (a) and simplifiedrepresentation (b)

the current sources I3 and I4. 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 supplyvoltage by 1.5–2 V.

The basic amplifier parameters are determined by the parameters of the stages. Thus, theinput resistance, current, and offset voltage are determined by the input stage, while theoutput resistance and the maximal values of the output voltage and current are determinedby the output stage.

The op-amp gain is equal to the product of the stage gains. But, as known, theemitter follower does not amplify the signal voltage. Therefore, the whole gain ofthe amplifier is determined by the product of the gains of the input and intermediatestages 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, whichis called the gain-frequency characteristic (GFC) or the open-loop-gain characteristic.


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, andthe gain varies widely: from several tens or even hundreds thousands to 1 and evensmaller. 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 acertain form, providing for some marginal stability. Therefore, the op-amp GFCis corrected. For example, for correction of the K157UD2 amplifier, the circuitincludes the capacitor Cfc connected to frequency correction (FC) terminals. In thiscase, the gain of the intermediate stage and consequently, of the op-amp as a wholedepends on the signal frequency. With accordance Cfc, the overall op-amp gain is

PK D Pk1 Pk2 Pk3 D k1

1C jf =f1� k2

1C jf =f2� k3

1C jf =f3' K0

1C jf =fcut;

where Pk1, Pk2, and Pk3 are the complex gains of the input, intermediate, and outputstages; K0 is the gain at f D 0; fcut D f2 is the cutoff frequency of the op-amp GFC.

The cutoff frequency depends on many factors, first of all, on the collectorcurrents 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 ofthe input transistors. The typical values of the input resistance are from 4 k� for the140UD1 op-amp to 1.5 M� for the �A725 op-amp. However, this decrease in thecurrents of the input transistors results in the impossibility of quick recharge ofthe correcting capacitor. Therefore, these amplifiers are characterized by the lowfrequency 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 lowfrequencies, the gain is constant, independent of the signal frequency, and equal


Fig. 1.4 Circuit of inverteramplifier with parallelfeedback based on op-ampwith VCVS structure

Fig. 1.5 PRC of amplifier without (0) and with feedback (1–3)

to K0. Starting from the cutoff frequency fcut D 20 Hz, the gain monotonicallydecreases with the rate of 20 dB/dec because of the decrease in the gain of theintermediate stage caused by the presence of the correcting capacitor with thecapacity Cfc D 30 pF. At the frequency fT D 1,000 kHz the gain becomes equal to1, and there is no amplification. This frequency is called the threshold amplificationfrequency of op-amp.

The gain decreases with the negative feedback (see Fig. 1.4). The op-amp gainwith a large feedback is Kfb � �R2/R1. It is independent of the op-amp parameters,but determined by external elements. The characteristics at Kfb equal to �1,000, –100, and �10 are shown by lines 1, 2, and 3 on Fig. 1.3. As the gain decreases, thefeedback increases and the frequency band becomes wider. Amplifiers of this kindare characterized by the roughly constant amplification area, that is the product ofthe gain by the upper threshold frequency (cutoff frequency).

The phase-response characteristic is connected with the GFC and dependent onthis. PRC of the K157UD4 without and with feedback is shown on Fig. 1.5. It can beseen 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-independentfeedback, the total phase shift in the feedback loop only slightly exceeds –3 /2, andthe amplifier has the stability margin about 60–70ı at the threshold frequency. Theamplified frequency band extends where phase shift is zero.

Another important parameter of an amplifier is the gain characteristic (GC), which is thedependence of instant output voltage vs. the instant input voltage.

2It is PRC for the noninverting input. For the inverting input, – should be added at any frequency.


Fig. 1.6 Typical gaincharacteristic of op-ampwithout (0) and with (1)feedback

This characteristic is measured as slow variation of the input voltage and haswider variety as compared to GFC. However, all GCs are features with limitedoutput values. Typical GC is shown on Fig. 1.6.

In the general case, it does not pass through the origin, because almost anyamplifier has the input offset voltage Voff. As can be seen from Fig. 1.6, GC becomesmore linear with feedback; it is a smoothly increasing (for the noninverting input) ordecreasing (for the inverting input) curve limited by the maximum allowable levelsof the output voltage, which, naturally, cannot exceed the supply voltage.

For practical calculations accordingly the op-amp nonlinear properties, its GCwithout feedback can be described through the hyperbolic tangent function

Vout D

8ˆ<

ˆ:

V m; if F.Vin/ � Vm =k2 ;

k2F.Vin/; if � Vm =k2 < F.Vin/ < Vm =k2;

�Vm; if F.Vin/ � �Vm =k2 ; where F.Vin/ D Vm tanh Œ.Vin C Voff/ ='T � ;

(1.1)

Voff is the input offset voltage reduced to the op-amp input; ®T is the temperaturevoltage (25.6 mV at T D 20 ıC); Vm is the maximum allowable voltage at op-ampoutput, k2 is the gain of the intermediate stage. The gain at a small signal in this caseis K D k2Vm ='T .

When amplifying pulsed signals and operating in the switching mode, the transient responsecharacteristic (TC) is important.

Remind that TC is the time dependence of the output voltage at a stepwise changeof 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 linearrange 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 theamplifier parameters. Large signals are the signals, at which the output voltage can


Fig. 1.7 Typical transientresponse 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 operationmode.

TC of the �A741 amplifier are shown on Fig. 1.7. It can be seen that at smallsignals (curve 1) the transient response process is long enough (see the lower scaleof the axis t). At large signals, the op-amp quickly enters the nonlinear mode withthe 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 withoutcomputer. Even in this case, the circuit including dozens of transistors, resistors, andcapacitors do not analyzed. Frequently equivalent circuit is used, whose input andoutput voltages and currents are equal to the input and output voltages and currentsof the op-amp.

The modern analysis uses various equivalent circuits of op-amps, from simplest to verycomplicated.

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 accuracyor they even fail to determine some needed parameters and characteristics, but allowfast 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 simpleto 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 linearapproximation 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 inputand output parts depending on the chosen independent input and output electricalcharacteristics. Parameters of equivalent circuits are denoted as Z, Y, F and H withthe corresponding indexes. The meaning and values of these parameters are wellknown.4

The output circuit is represented by a voltage source in Figs. 1.8a, c and bya current source in Figs. 1.8b, d, and the both sources are dependent. In the firstcircuit (1.8a) voltage depends on the input current: E i D Z21Iin, and in the secondone (1.8c) it depends on the input voltage: E V D F21Vin. Similarly, the currents ofthe controlled current sources depend on the input voltage in the circuit shown inFig. 1.8b (I V D Y21Vin) and on the input current in the circuit shown on Fig. 1.8d(I i D H21Iin). As applied to amplifiers, the parameter Z21 D Ztr is transresistance,F21 D K V is voltage gain, Y21 D S is transconductance, and H21 D K i , is currentgain, 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 Ztr is measured in the units ofresistance, and S is measured in the units of conductance.

Depending on the type of the output source and the controlling electrical characteristic, thesimple 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.

3For simplicity, reverse transfer elements are excluded in Fig. 1.8.4See, for example, A.F. Beletskii, Principles of Theory of Linear Electrical Circuits (Svyaz,Moscow, 1967).


For example, the two-port on Fig. 1.8c (VCVS) can be described by the systemof two equations:

Iin D F11VinIVout D KV Vin C F22Iout:

)

(1.2)

The independent variables here are the input voltage (Vin) and the output current(Iout). The first and second equations describe, respectively, the input and outputop-amp circuits. In the first equation, the input circuit is represented by the inputconductance F11 D 1/Zin. The input resistance serves a load for the signal sourceand consumes the corresponding power from it. The higher the input resistance, thelower the input current Iin, so the greater voltage part of the signal source comes tothe 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 outputcircuit includes the voltage source E V depended on the input voltage Vin and theoutput resistance F22 D Zout. The relation between Zout and Zload determines whatpart of voltage E V will be separated at the load resistance.

The described equivalent circuits can be used only for approximate calculations of suchdevice parameters as the gain and input and output resistance, because they ignore thefollowing 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 ElectronicsWorkbench software is shown on Fig. 1.9.

It more accurately models the op-amp frequency properties, the input currentsof transistors and the input offset voltage. The frequency properties are presentedby two frequency-dependent RC-circuits: (Ri, Ci, Cfc, Rin2) and (Rout and Cout), andone of the capacitors (the frequency correcting capacitor (Cfc)) is connected to theexternal terminals and can be changed. The transistor input currents are determinedby the sources of input currents (Ib1, Ib2), and the input offset voltage (Voff) is set bythe voltage source.

The circuit includes two (rather than one) depended sources, which are enclosed bythe dashed rectangle. The disadvantages of this circuit are the impossibility to considercommon-mode parameters and the limited output voltage.

In circuit on Fig. 1.10 these disadvantages are removed. Here the input resistanceZin is represented more specifically by the resistors Rin and input capacitors (C1 andC2) for the symmetric input. The elements Rcm and Ccm account for the common-mode input resistance and common-mode input capacitance. The elements Ri, Ci

and Rout, Cout 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, V1 and V2 accounts for the effect of the limited output voltage at the levelof V1 and V2 voltages. Diodes in this circuit makes it nonlinear, unlike the previouscircuits.

Certainly, now the use of the macromodel is more complicated, and the calcu-lations become more complexes. Therefore, it suits for computations as a PSpicemacromodel in the Electronics Workbench and DesignLab software. Such a modelcan 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 accountthe input currents and the input offset voltage, to determine more accurately thefrequency properties, limitedness of not only output voltage, but also the outputcurrent, 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.


Fig. 1.11 Boyle–Cohn–Pederson macromodel

This model includes a differential stage consisting of NPN transistors VT1 andVT25 one uncontrolled (I1) and four controlled (I2–I5) current sources, outputcurrent limiters assembled in diodes VD1 and VD2, and output voltage limitersassembled in diodes VD3 and VD4. The built-in current sources in their structureare similar to VCCS. The effect of the op-amp input parameters is modeled by thedifferential stage, the frequency properties are determined by the capacitors C1 andCfc, and the output resistance is modeled by the resistors R7 and R9.

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 muchtime. This is explained by the ancient contradiction between the accuracy and thesimplicity of a model. However there are no miracles. Hence, it can be concludedthat the op-amp principle circuit serves the most accurate op-amp macromodel, justwhich was stated in the beginning of this Chapter.

Thus, for approximate calculations of the gain and the input and output DC resistance, wecan use the op-amp models shown on Fig. 1.8. If it is necessary to take into account theop-amp frequency properties, the descriptions of these models should be supplemented withthe frequency dependence of their parameters. The one-port linear equivalent circuit onFig. 1.9 is better suited, when it is needed to more accurately take into consideration thefrequency properties in the form of two time constants, as well as the effect of the inputcurrents and the input offset voltage. The nonlinear equivalent circuit (macromodel) shownon Fig. 1.10 represents better the common-mode parameters and the level of restriction ofthe output voltage. Finally, the Boyle-Cohn-Pederson macromodel on Fig. 1.11 accounts forall the listed dependences.

5There are similar models constructed in bipolar (PNP) and field-effect (FET) transistors.

1.5 Analysis of the Effect of Signal Source and Load 13

1.5 Analysis of the Effect of Signal Source and Load

The signal source and the load influence a significant effect on the amplifier properties as awhole. From the circuit of op-amp connection on Fig. 1.12 , it can be seen that the resistorsZs, Zin, Zout, and Zload form voltage dividers.

� In this case, the overall gain with regard for all resistances is

Ke D Vout

EsD KV

Zin

Zs CZin

Zload

Zout CZload: (1.3)

Designers of circuits with amplifiers usually aim to increase the gain K e , but atthe given Zs and Zload this can be achieved only by selecting optimal Zin and Zout.From Eq. 1.3 it is clear that, to increase the gain K e (given K V ), it is necessaryto increase Zin and decrease Zout. The limit value of the gain K e at Zin D 1 andZout D 0 is equal to K V . On the other hand, at the high resistance of the signalsource (Zs >>Zin) or any low input resistance of the amplifier, the gain K e ! 0according to Eq. 1.3. This effect can be explained by following. As the internalresistance of the signal source increases, the input voltage and output voltage alsointend to zero. However it is not a case. In a real amplifier, the input current (base oremitter current) continues to pass through the input circuit. Due to the properties ofsemiconductor devices, this current will induce the output voltage. Therefore, at thehigh resistance Zs it is better using another equivalent circuit with other independentvariables.

� If we describe the same two-port in Z-parameters (at the independent variablesIin and Iout), then we obtain the following system of equations:

Vin D Z11IinIVout D ZtrIin CZ22Iout:

�

(1.4)

In this system, the op-amp output voltage is followed by amplification of theinput current and its conversion into the output voltage. The op-amp in this case isconsidered as a Current controlled voltage source (CCVS), the relation betweenwhose input current and output voltage is the following: Vout D IinZtr D E i at

Fig. 1.12 Equivalent circuit of op-amp connection: SS is signal source; Amp is amplifier; L isload; Es, E V are voltage of the signal source and amplifier; Zs, Zout D F22 are output resistancesof the signal source and amplifier; Zload is the load resistance; Zin D 1/F11 is the input amplifierresistance


Iout D 0, where Ztr is the transresistance. In this case, just transresistance Ztr, ratherthan the gain K V , reflects the op-amp amplifying properties. Describing the overallgain with Ztr, we obtain:

Ke D Vout

EsD Ztr

1

Zs CZ11

Zload

Z22 CZload: (1.5)

Now to increase the gain, it is necessary to decrease the input resistance, but notincrease. The maximum value of K e at Z11 D 0 and Z22 D 0 is equal to Ztr/Zs, that isa finite nonzero value. The minimum value of K e is now achieved at Z11 ! 1, thatis, the situation is quite opposite to that given by Eq. 1.3. What a paradox!

However there is no paradox here. It was not by accidentally that we considereddifferent approaches to determination of the amplifying properties of op-amps.These approaches correspond to different physical realizations of op-amps. Thefirst of them is characteristic of ordinary (traditional) op-amps with the high inputresistance.

Op-amps with the high input resistance are referred to as amplifiers with potential input.This term describes the fact that the output voltage in them is controlled at low inputcurrents, by the input potential. In the electrical circuits theory such two-ports are knownas voltage controlled voltage sources (VCVS).

Another approach is needed when considering op-amps with low inputresistance.

Op-amps with the low input resistance are referred to as amplifiers with current input,because the output voltage in them is depended on the input current, rather than voltage. Byanalogy, they can be classified as Current controlled voltage sources (CCVS).

These amplifiers are known for a long time, but as elements of integratedcircuits they appeared only recently and did not receive wide acceptance yet. Theyare created as a result of the progress in the complementary bipolar technology.However, because of their numerous advantages, they can find sufficient placein electronics. To understand the advantages of these new op-amps, considerdifferences in their circuits.

1.6 Amplifiers with Current Input

Consider now op-amps with the CCVS structure. A simplified circuit of AD844 op-ampis shown on Fig. 1.13 [9]. The op-amp has a symmetrical circuit design based oncomplementary transistors and includes three stages: offset voltage compensation stage,amplification stage and a voltage follower.

Offset voltage compensation stage is assembled using transistors VT1 and VT2and sources of current IA and IB (see the first dashed rectangle on Fig. 1.13).A voltage equal to the offset voltage, necessary for transistors VT5 and VT6, isgenerated on bases of both transistors. The first stage does not amplify the signal.

1.6 Amplifiers with Current Input 15

Fig. 1.13 AD844 op-amp with the CCVS structure

Main amplification is performed by the next stage, assembled using transistorsVT3 – VT12 in accordance with a “current mirror” circuit. Output current Iout

on terminal 5 is equal to input current on inverting input –Vin. Thus the stagedoes not amplify the signal current but provides significant amplification of signalvoltage. Finally, the signal amplified by the intermediate stage follows to the inputof the third (output) stage in VT 13–VT 18 transistors. As in the previous circuit(Fig. 1.1) it is designed as the emitter follower circuit and does no provide voltageamplification.

Only the intermediate stage has the voltage gain higher than 1 among all thestages, so the overall gain is not high – about 60,000 for the noninverting input.Consider the input stage in more detail. At the “CVin” input the transistors areconnected in the circuit with common collector-base (VT1) and common emitter(CE)(VT5), and at the “–Vin” input the transistors are connected in the circuit withcommon base (CB), so they have different input resistance. It is quite natural thatat the “CVin” input the resistance is much higher than at the “–Vin” input, becausethe input resistance of the CE circuit is “ times higher (“ are the current transferratios of the transistor base) than the input resistance of the CB circuit, and thedifference between the resistances may be significant. Thus, for the AD844-typeop-amp the resistances are 10 M� and 50 �, [9]. The difference between the inputresistances allows using op-amps with both potential (noninverting) and current(inverting) inputs. The maximal output voltage connected with the terminal 6, as


Fig. 1.14 Nonlinear equivalent circuit of AD844-type op-amp

in the circuit shown on Fig. 1.1, is 1.5–2 V lower than the supply voltage, and at thecurrent terminal 5 it is almost equal to the supply voltage.

Nonlinear equivalent circuit of AD844-type op-amp is shown on Fig. 1.14. Thecircuit includes two voltage followers OA1 and OA2, a current-controlled currentsource assembled in accordance with a “current mirror” circuit – CM, RC – a circuit,which simulates inertial properties, and a double output voltage limiter.

One of the inputs of the op-amp (CVin) is a noninverting current input withhigh input resistance (voltage input), and another one (�Vin) inverting current inputwith low input resistance (Rin2). There is a voltage follower OA1 fur the purpose ofreflection of various input resistances. It is known that this op-amp has two outputs:a “current” output (Iout) and “voltage” output (Vout). Conventional voltage output(Vout) is made on the OA2 follower output after output voltage limiter (VD1 and VD2with voltage sources E1 and E2). Current output (Iout) has a high output resistanceRi. The main specific feature of this circuit is that it includes a current limitershown in CM (current mirror) block. Output current Iout is equal to input current oninverting input due to use of “current mirror”, but is limited by maximum allowablevalue I D fIin2 if jIin2j � ImaxI Imax if Iin2 > ImaxI �Imax if Iin2 < �Imaxg. It is thisfeature that distinguishes the circuit design from other known models.

Such op-amps in books and articles are often referred to as current-feedbackoperational amplifiers – CFOA.6

Figure 1.15 shows schematic symbols of an op-amp with voltage (3) and current(2) input as well as voltage (6) and current (5) output.

The fact that the op-amp has two separate inputs and outputs makes it possibleto use it as follows:

– VCVS (input y, output w);– VCCS (input y, output z);

6Abbreviation CFOA is an improper one because there is no feedback in the amplifier. Moreoverthe op-amp can be used without the feedback.

1.6 Amplifiers with Current Input 17

Fig. 1.15 AD844 type op-amp on electric circuits: equivalent circuit – (a) and simplified – (b) withnumerical and corresponding letter identification of outputs (Inverting and non-inverting inputs aremarked with letters x and y or numbers 2 and 3 for AD844CH op-amp, and current and voltageoutputs are marked with letters z and w or numbers 5 and 6, correspondingly)

Fig. 1.16 Circuit of invertingamplifier with the CCVSstructure

– CCVS (input x, output w);– CCCS (input x, output z).

As this op-amp is not a traditional one, let us consider some circuits which use it.Determine the gain in the circuit with parallel negative feedback for the output

voltage (the feedback is added to the current input). The circuit of op-ampconnection is shown on Fig. 1.16. To demonstrate the current control, the op-ampcircuit shows the direction to the inverting current input.

� The gain can be calculated by the Fig. 1.17. The input (Rin2) and output (Rout)op-amp resistors on Fig. 1.17 have low resistance. The gain with feedback atRin2 D Rout D 0 can be determined from the following:

Vout D �Iin2ZtrI Iin2 D IR1 � IR2 � Vin=R1 � Vout=R2

Substituting Iin2 from the second equation into the first one, we obtain Kfb:

Kfb D Vout

VinD �R2

R1

Ztr=R2

1CZtr=R2Š �R2

R1: (1.6)


Fig. 1.17 Equivalent circuit of inverting amplifier with op-amp the CCVS structure

It can be seen that change of the potential input to the current one does notchange the gain of the op-amp with feedback. As in the previous case (with the largefeedback F D 1 C Ztr/R2), it is independent of the op-amp amplifying properties, butdetermined only by the resistances of the external resistors R1 and R2.

Consider now how the amplifier GFC is changed. For this purpose, represent thetransresistance Ztr in the complex form:

Ztr.jf / D Ztr0

1C jf =fcut:

Then

Kfb.jf / Š �R2R1

1

1C jf =Œfcut.1CZtr0=R2/�: (1.7)

It is seen from the last equation that the GFC cutoff frequency of the amplifierwith feedback, as in the circuit shown on Fig. 1.4, depends on feedback. Thefeedback is larger the cutoff frequency is higher. It is agreement with general theoryof feedback. However, the difference is following. At the same gain, this circuitcan provide for different feedback and different frequency, which depend on the R2resistance. GFC for different values of the transresistance Ztr and the resistance R2 isshown on Fig. 1.18. Curve 0 corresponds to the amplifier without feedback. Curves1, 2, and 3 are for amplifier’s GFC with different feedback.

The analysis of the plots shows that at the same gain K D 100 the GFC cutofffrequency varies upon variation of R2. Thus, the amplifier of this structure ischaracterized by the dependence of the amplification area on the feedback.

Note that GFC here is plotted in a wider frequency band comparing to Fig. 1.3,and it is not accidentally. From Fig. 1.13 follows that in this structure the frequencyband is determined by the VT5 and VT6 transistors, and the correcting capacitor(not shown on the circuit) is recharged by other transistors: VT10 and VT11, whose

1.7 Amplifiers with Current Output 19

Fig. 1.18 GFC of amplifier with the CCVS structure without (0) and with (1, 2, and 3) feedback

collector currents do not affect the input resistance and can be taken high to speedup recharging. In addition, the current input is the input of transistors connectedin the CB circuit. As well known, it is characterized by the higher frequency bandof the amplified signals as compared to the CE circuit. The low input resistancefor the inverting input neutralizes the effect of the input capacitance, so amplifierswith the CCVS structure also have the wider frequency band. That is why thethreshold frequency of the AD844 op-amp is 80 MHz, and the slew rate achieves2,000 V/�s [9].

1.7 Amplifiers with Current Output

Amplifiers designed in the VS (voltage source) structures have significant disadvantages:low load-carrying capacity, sensitivity to the output short circuit, etc.

The limited allowable load resistance characterizes it, because most of them areintended for operation only with high-resistance load (no less than 2–5 k�).

They are unsuitable for operation in matched high-frequency amplification channels withthe resistance of 50 and 75�. It is better using op-amps with the CS (current source)structure for this purpose.

Consider the circuits of such amplifiers. Amplifiers with VCVS structure oftenhave a current output also, therefore the op-amp with the VCVS structure can beeasily transformed into the op-amp with the VCCS structure. For this operation thesignal from the intermediate amplification stage should be used. On Fig. 1.1 thisoutput is connected to the terminal 5.

Figure 1.19 shown the amplifier with feedback based on this op-amp. The circlewith two arrows near the output terminal indicates the current output.

Determine the gain and the frequency properties of this amplifier. Since theoutput current is controlled potentially, the op-amp input current can be ignoredby taking the input resistance equal to infinity.


Fig. 1.19 Inverting amplifierconstructed in op-amp withVCCS structure

Fig. 1.20 Equivalent circuitof the amplifier with theVCCS structure

The equivalent VCCS circuit on Fig. 1.20 corresponds to the system of equationsin Y-parameters:

I0 D GinV0IIout D SV0 CGoutVout:

�

(1.8)

� The gain will be calculated for the ideal source of output current, in whichthe output resistance is equal to infinity and, consequently, Gout D 0. The system ofequations is:

V0 D Vin � SV0R1IVout D Vin � SV0.R1 CR2/:

(1.9)

Having determined V0 from the first equation, substitute it in the second equationand obtain the equation for the gain

Kfb D 1 � SR21C SR1

Š �SR21C SR 1

Š �R2R1: (1.10)

From denominator of Eq. 1.10 follows that only external resistors R1 and R2 alsodetermine the gain for this circuit at the large feedback. It is interesting that feedbackdepends here only on the transconductance S and the R1 resistance. At R1 D 0 thedenominator is equal to 1 and the feedback is not present. Gain is

K0 D 1 � SR2: (1.11)


Fig. 1.21 GFC for the VCCS amplifier with (lines 1, 2, and 3) and without (line 0) feedback

The unity in Eq. 1.11 determines the direct transmission of the input signalto the output through the resistors R1 and R2, by-passing the inverting amplifier.This becomes possible just owing to the output current source. In the equation forcircuits with output voltage sources this 1 is not present. Certainly, it does not affectsignificantly the gain, since it is much smaller than SR2.

Consider now the gain-frequency characteristics. For this purpose, represent thetransconductance in the complex form:

S.jf / D S0

1C jf =fcut;

then

Kfb Š �R2R1

1

1C if =fc; (1.12)

where fc D fcutS0R1 is the GFC cutoff frequency of the amplifier with feedback.From GFC on Fig. 1.21 for the amplifiers with feedback and without it follows

that the gain-frequency characteristics of the amplifiers with the CCVS and VCCSstructures coincide at equal Kfb and fcut.

Examples of amplifiers with the VCCS structure are NE5517 op-amps fabricatedby Philips, and others.

Finally, consider the last op-amp structure based on the current controlled currentsource (CCCS). The circuit of this amplifier can be described, if in the circuit onFig. 1.13 the signal from the intermediate stage (terminal 5) will be used as outputsignal, and the input signal is the input (�Vin) (Fig. 1.22).

� To calculate the gain and analyze the frequency properties of the amplifier, letus take the same inverting amplifier (Fig. 1.20). The op-amp symbol here representsthe current input and output.


Fig. 1.22 Inverting amplifierconstructed in op-amp withCCCS structure

Fig. 1.23 Equivalent circuit of the amplifier with the CCCS structure

The equivalent circuit (Fig. 1.23) is described by the system of H-parameters forcalculation of the gain.

The gain can be found from the following system of equations:

Vout D V0 C VR2 D V0 �KiIin2R2I

Iin2 D V0=H11I

Vin D VR1 C V0 D V0 C .1CKi/ R1Iin;

if H22 D 0. Having excluded Iin and V0 and described Vout through Vin, we receive

Kfb D Vout

VinD Rin2 �KiR2

Rin2 CR1 CKiR1: (1.13)

At the small input resistance Rin2 (the op-amp with the current input just has lowinput resistance), it can be ignored. Then Eq. 1.13 if Ki � 1 can be transformed into

Kfb Š � Ki

1CKi

R2

R1Š �R2

R1: (1.14)


Fig. 1.24 Non-invertingamplifier in op-amp withCCCS structure

It is notable that in case if Ki D1, as for AD844 op-amp, the gain is equal to

Kfb Š �12

R2

R1Š � R2

2R1:

From Eq. 1.14 follows that amplifier with feedback designed in the op-amp withthe CCCS structure, the gain is determined by the resistances of R2 and R1, as in theprevious cases only for Ki � 1.

Consider the amplifier with the series feedback for the current input shown onFig. 1.24. From the equivalent circuit, for Ki � 1 we have

Kfb D Vout

VinD R1

R1 CH11

C .R2 CH11/Ki

.1CKi/.R1 CH11/Š 1C R2

R1: (1.15)

It is notable that in case if Ki D 1 (AD844)Kfb D VoutVin

Š 1C R22R1

.Analyzing the equation, we can separate the passive part of the gain (first term)

and the active one (second term including the current gain Ki). The passive part isindependent of the op-amp amplifying properties and at the higher input resistanceH11 it is roughly equal to 0, while the active one depends on the current gain Ki andunder the same conditions it is equal to the ratio R2/R1. With no feedback and theresistor R2 as a load, the gain is K0 D R2Ki/H11 � SR2.

� Now let us analyze the frequency properties of the amplifier. If Ki depends onthe frequency as

Ki.if / D Ki

1C if=fcut;

then

Kfb .if / Š�

1C R2

R1

�1C if=fcutKfbKi

1C if=fcut.1CKi/(1.16)

where fcut is the GFC cutoff frequency of the amplifier without feedback.


Fig. 1.25 GFC of amplifier with CCCS structure

Analysis of the equation for Kfb(jf ) and the GFC of this amplifier at differentgain on Fig. 1.25 show that GFC decreases monotonically with the cutoff frequency(1 C Ki) times higher than the op-amp cutoff frequency, and the cutoff frequencyis a constant independent of the gain. This is a characteristic feature of the op-ampwith the CCCS structure.

1.8 Current-Differencing Amplifiers

The distinctive feature of all the amplifiers considered above is the input differential stage.

Owing to this stage, it becomes possible to obtain the minimal input offsetvoltage and to perform various operations with input signals regardless of the op-amp parameters. But if the amplifier is supplied from one source, bias circuitsbecome more complex, because input voltage dividers are needed to apply the biasvoltage. In practice, op-amps are rarely operated from one power supply (except forthe use with bridge circuits).

Recently, amplifiers with the so-called current mirror in place of the bipolar circuit at theinput (current-differencing amplifiers) have come into being, and these amplifiers are justintended for operation from one power supply. They turned out to have some advantages,in particular, the minimal number of external elements to provide for bias and others.

But these advantages turn out to be disadvantages at the same time, becausethese amplifiers can amplify only unipolar or variable signals in the presence of ablocking capacitor. Examples of current-differencing amplifiers are LM2900/3900op-amp fabricated by National Semiconductor and Russian 1401UD1 and 1435UD1amplifiers fabricated by Foton, Kvazar, and KMT.

The circuit of the 1401UD1 amplifier on Fig. 1.26 includes three stages: the inputstage in the VT1 and VT2 transistors, the intermediate stage in the VT3 transistor,and the output stage in the VT4 —VT6 transistors. The current sources I1 — I3serve as loads in all stages. The VT1 and VT2 input transistors form the current

1.8 Current-Differencing Amplifiers 25

Fig. 1.26 Circuit of 1401UD1 current-differencing op-amp

mirror. The intermediate stage is the ordinary inverting stage connected in the CEcircuit with frequency correction realized with the aid of the capacitor Cfc. Theoutput stage is a complex voltage follower. The amplifier has two inputs: invertingand noninverting, and the input resistance for the inverting input is about 1 M�.

As the input current is follows to the noninverting input owing to the currentmirror, the current at the inverting input tends to become equal to the inputcurrent. Therefore, the collector currents of the VT1 and VT2 transistors are alwaysmaintained equal in the case of feedback. This results in appearance of the outputvoltage proportional to the feedback resistance. So it is clear that the output voltageVout D �IinRfb is independent of the amplifier parameters and determined only bythe parameters of the feedback elements, as in the circuit with the differential stageat the input.

Figure 1.27 shows the circuit of the inverting alternating-voltage amplifier in thecurrent-differencing op-amp.

The direct current (DC) mode in the circuit is set by the R2 and R3 resistors.If R3 D 2R2, then the direct voltage at the output is equal to the halved supplyvoltage C Vcc, since when the input direct currents are equal, the voltage drop atthe resistor R2 and, consequently, the output direct voltage are twice as low asthe supply voltage. The input alternating voltage, transmitting through the resistorR1 to the inverting input, tends to violate the balance of the currents. Thus, thecurrent difference appears, which is reflected in the title of this op-amp. However,the feedback causes the compensating current from the amplifier output, and itagain balances the currents. So the circuit maintains the balance of the alternating


Fig. 1.27 Inverting amplifierin the current-differencingop-amp

currents through the resistors R1 and R2, that is, the equality Vin/R1 D �Vout/R2 istrue. Therefore, the gain is Kfb D �R2/R1. As in the previous cases, it can be seenthat the gain is independent of the op-amp parameters and determined only by theresistance of the external resistors.

It is likely most appropriate to use current-differencing op-amps in cheapalternating-voltage amplifiers for mobile systems with battery power supply.

1.9 Rail-to-Rail Amplifiers

In ordinary op-amps, the amplifying properties keep within some range of the outputvoltage. The limits of this range are equal to the maximum and minimum allowablevalues. Usually they are 1.5–3 V lower than the corresponding supply voltage. Thatis, the op-amp output voltage is lower than the supply voltages by the residualvoltage at the output transistors, that is, just by 1.5–3 V.

The limits of variation of the output voltage are shown on Fig. 1.6. It can beseen that there is a gap, equal to the residual voltage, between C Vcc1 and Vout m, aswell as between –Vcc2 and –Vout m. Hence, it follows that the supply voltage of theordinary op-amp cannot be lower than the doubled residual voltage equal to 3–6 V.At the same time, on the one hand, now it is necessary to have an amplifier capableof operating at lower supply voltages, for example, in micro-power medical devices,cell phone tools, portable CD players. On the other hand, the residual voltagereduces the efficiency of powerful amplifiers. So the decrease of the residual voltageis an urgent problem of op-amp improvement.

Figure 1.28 shows the possible versions of output voltages for op-amps of various types. Thedashed line indicates the supply voltage level. The ideal amplifier is the amplifier, whoseoutput voltage (Fig. 1.28 c) is equal to the supply voltage. Such amplifiers are called rail-to-rail amplifier7 amplifiers.

The residual voltage of bipolar transistors cannot be lower than the voltage dropacross the open diode, that is, 1–1.2 V. Consequently, the amplifiers in bipolar

7The term “rail-to-rail” is registered trademark of Nippon Motorola Ltd.

1.11 Clamping Amplifiers 27

Fig. 1.28 Output voltages of ordinary amplifiers with potential and current control (a), current-differencing amplifiers (b), and amplifier amplifiers (c)

transistors cannot operate at voltages lower than 2–2.4 V. Therefore, rail-to-railamplifier op-amps are most often constructed in field effect transistors (FETs).Examples of such amplifiers are ICL761 fabricated by Intersil, TS912 fabricated bySTMicroelectronics, 1423UD1, KR1446UD1-5, and 1447UD1 fabricated by Foton,Angstrem, and Pulsar companies, and some others. They are capable of operatingat the supply voltage from 1 to 8 V. Another distinctive feature of these amplifiers isthe possibility of programming the supply current.

1.10 Instrumental Amplifiers

Recently a new type of integral amplifiers has appeared: instrumental amplifiers.

These op-amps are intended for operation in input stages of measuring in-struments, for amplification of signals from high-resistance sensors of physicalparameters, bridge circuits, thermocouples, etc. As a rule, they have the normalizedgain multiple of 10. For example, the gains of LM163 and LM363 op-amps(National Semiconductor), INA258 op-amp (Burr-Brown), or 140UD27 (Kvazar)are equal to 10, 100, and 1,000. Some or other value is selected by closing chipterminals with jumpers. The gain is determined by the resistances of switchedinternal resistors, and the error is 0.1–1%. Adding external resistors, it is possibleto increase the number of the fixed gain values. These amplifiers find the utility inprecision electron devices.

1.11 Clamping Amplifiers

The so-called clamping amplifiers have arisen quite recently. They are amplifierswith switched (clamping) inputs, for example, AD8036 and AD8037 amplifiersfabricated by Analog Devices. Such amplifiers are unique devices, which allow


Fig. 1.29 Clamping amplifier with switched inputs

not only the ordinary amplification function in a wide frequency band, but alsoswitching, rectification, and modulation of signals. Figure 1.29 shows schematicallythe circuit of a clamping amplifier and its operational table.

The circuit includes the amplifier made as two stages A1 and A2, among whichthe op-amp A1 is responsible for the basic amplification, while A2 is a voltagefollower; three extra followers A3 — A5, comparators CH and CL, a logic circuitLC, and a commutator (multiplexer) S. At the –Vin input, the signal is amplifiedin the ordinary way, while amplification at the CVin, VH, and VL inputs occursdepending on the ratio between signals at these inputs in accordance with the table.Analysis of the table shows that at CVin >VH the signal from VH is amplified,while at CVin<VL the signal from VL is amplified, and at VL � CVin � VH thesignal from C Vin is amplified. This opens the possibility for realization of variousfunctions. Examples of devices based on this amplifier are considered in Chap. 4.

1.12 Isolation Amplifiers

Isolation amplifiers are used for galvanic isolation of input and output circuits ofthe amplifier. Unlike conventional amplifiers they don’t have connection betweenthe common wire of the input circuit and the common wire of the output circuit(Fig. 1.30). The absence of galvanic coupling is a requirement when signal sourceand op-amp load have different voltage and when it is necessary to eliminatepotential errors caused by ground circuits. Isolation of input and output circuits ofthe op-amp is made using optical or inductive coupling. E.g. AD215 op-amp usesinductive coupling, and ASPL785J op-amp – optical coupling.

1.13 Conclusions 29

Fig. 1.30 Conventional and isolation amplifier

Isolation amplifiers are used in power measurement circuits, electric motordrive control and monitoring circuits, signal isolation circuits of switching powersupplies etc.

1.13 Conclusions

After consideration of various op-amp structures, a question naturally arises: whatof these structures is the best? Most probably, there is no one answer to this question.The point is that correct comparison is impossible, since all the considered op-ampstructures have different circuits, each structure has a different number of transistorsmade by different technologies and, consequently, having different parameters andcharacteristics.

The more different amplifiers with various properties are likely the better,because this provides for a wider range of choice. In the course of studying variousdevices based on op-amps with some or other structure, their most appropriateapplications will be determined and the corresponding recommendations will begiven. What’s more, Russian and international industries produce now a greatnumber of op-amps with the structures considered above and specialized op-amp models: micropower (140UD28), high-power (LM12 fabricated by NationalSemiconductor), precision (140UD17), high-speed (1412UD6, 544UD10), low-noise (OP-27/37 fabricated by Analog Devices), and high-voltage (1443UD1 andLM163 fabricated by National Semiconductor) op-amps. The information aboutthese and many other amplifiers can be found in the handbook [3] or in Appendix 1.

Thus, the above-said suggests:

1. All operational amplifiers can be divided into amplifiers with potential andcurrent inputs and potential and current outputs.

2. Amplifiers with the VCVS structure (potential control), which have the highinput and low output resistance and whose band depends on the gain withfeedback (constant amplification area), are most widely used nowadays.

3. Op-amps with the CCCS structure (current control), to the contrary, have the lowinput and high output resistance and the constant band regardless of the gain with


feedback. They are more broadband, and the amplification area in them increaseswith the increasing gain.

4. Amplifiers with the CCVS and VCCS structures have, respectively, the low(high) input and output resistances and their bands depend on the resistance ofthe feedback loop at the fixed gain.

5. In recent years, amplifiers of new types with the fixed gain and low supplyvoltage have arisen.

6. Op-amps have various circuit designs, and to represent them correctly, it isnecessary to choose the adequate mathematical model.

Questions

1. What is VCVS (CCVS, VCCS, CCCS)?2. Draw the simplest equivalent circuit of the op-amp.3. Give a definition of the term “operational amplifier.”4. Enumerate the basic electrical parameters of op-amp.5. Enumerate the basic characteristics of op-amp.6. What are the approximate values of the gain, input and output resistance for

VCVS (CCVS, VCCS, CCCS) structure?7. What parameters characterize the op-amp amplifying properties?8. Compare the basic parameters of op-amps with the VCCS and CCVS structure.9. Explain what is the result of variation of the op-amp input current.

10. Explain how variation of the current in the first stage affects the op-amp inputresistance.

11. Explain what are the causes for decrease of the gain with the increasingfrequency.

12. Explain why the correcting capacitor affects the op-amp frequency properties.13. Tell in your own words how to determine the stability of an amplifier with the

resistive feedback from its GFC and PRC.�

14. What is the cause for the drop of the op-amp GFC at high frequencies with therate of 20 dB/dec?�

15. What op-amp parameters characterize the error in amplification of directvoltage?

16. Explain the meaning of the input offset voltage.17. Show the relation between the capacity of the correcting capacitor in the op-

amp circuit and its GFC.18. Prove that the voltage at the op-amp output cannot be more supply voltage.19. What op-amp parameters characterize the amplification error of the square-

wave voltage?20. Give a definition of the reduction coefficient of the in-phase signal.21. How do you understand the term “average input current of the op-amp”?22. Draw the circuit of an amplifier with potential (current) control.23. Threshold frequency is fT D 200 MHz. Determine the possible op-amp

structure.24. Propose a circuit for measurement: (a) bias voltage, (b) input current, (c) input

resistance, (d) GFC, (e) PRC, (f) TC using Electronics Workbench.�

1.13 Conclusions 31

25. What is the difference between op-amps with potential and current control?26. What is the advantage of op-amp with potential (current) control?27. What op-amps have better frequency properties: with potential or current

control and why?�

28. What is the difference between rail-to-rail amplifier amplifiers and ordinaryamplifiers?

29. What are the features of instrumental amplifiers as compared to ordinaryamplifiers?

30. Explain the purpose of using instrumental amplifiers.

Test Yourself

1. What kind of the signal source is characterized by the high input and low outputresistance:

(a) CCCS;(b) CCVS;(c) VCVS;(d) VCCS.

2. What kind of op-amp has low input resistance:

(a) with current control;(b) with force control;(c) with potential control;(d) with power control.

3. What kind of op-amp has high output resistance:

(a) with current output;(b) with force output;(c) with potential output;(d) with power output.

4. Enumerate the op-amp parameters determining the error in amplification ofdirect voltage:

(a) slew rate;(b) input offset voltage;(c) threshold amplification frequency;(d) output resistance.

5. What op-amp parameters determine the frequency component of the error inamplification of small alternating voltages:

(a) input offset voltage;(b) slew rate;(c) threshold amplification frequency;(d) input current.


6. What op-amp parameters determine the error in amplification of small pulsedvoltages:

(a) input offset voltage;(b) input resistance;(c) threshold amplification frequency;(d) input current.

7. What op-amp parameters affect the temperature component of the error inamplification of direct voltages:

(a) drift of input offset voltage;(b) slew rate;(c) threshold amplification frequency;(d) input current.

8. Maximal op-amp output voltage is restricted by:

(a) input offset voltage;(b) power supply voltage;(c) threshold amplification frequency;(d) input current.

9. What are differences between the circuits of the op-amp with current controland the ordinary op-amp:

(a) output stage;(b) intermediate stage;(c) input stage;(d) power supply voltage.

10. The slope of op-amp GFC is:

(a) 10 dB/oct;(b) 20 dB/dec;(c) 15 dB/oct;(d) 6 dB/dec.

11. What parameter characterizes amplifying properties of the amplifier with theCCVS structure:

(a) voltage gain;(b) current gain;(c) transresistance;(d) reverse-transfer impedance.

12. What are differences between the circuits of the current-differencing amplifierand the ordinary op-amp:

(a) output stage;(b) intermediate stage;

References 33

(c) input stage;(d) power supply voltage.

13. What type of the input stage is used in the current-differencing amplifier:

(a) differential;(b) emitter follower;(c) resistive;(d) current mirror.

14. What are differences between the instrumental amplifier and the ordinaryone:

(a) gain;(b) input resistance;(c) error in the gain;(d) accuracy of output resistance.

References

1. Ragazzini J.R., Randall R.N., Russell F.A.: Analysis of problems in dynamics by electroniccircuits. Proceeding of the IRE, vol. 35 (1947)

2. Black H.S.: Stabilized feedback amplifiers. Bell Syst. Tech. J. 13 (1934)3. Operational Amplifiers and Comparators. Handbook. Dodeka-XXI, Moscow (2001) (in Rus-

sian)4. Dostal, J.: Operational Amplifier (Edn Series for Design Engineers). Butterworth-Heinemann,

Boston (1993)5. Graeme J.G.: Applications of Operational Amplifiers: Third Generation Technique. McGraw-

Hill, New York (1973)6. Graeme, J.G., Tobey, G.E., Huelsmann, L.P. (eds.): Operational Amplifiers Design and

Applications. McGraw-Hill Book Company, New York (1971)7. Irvin, R.G.: Operational Amplifier Characteristics and Applications, 3rd edn. Prentice-Hall,

Englewood Cliffs (1994)8. Boyle, G.R., Cohn, B.M., Pederson, D.O., Solomon, J.E.: Macromodeling of integrated circuit

operational amplifiers. IEEE J. Solid State Circ. SC-9(6), 353–364 (1974)9. Analog Deices: 60 MHz, 2000 V/�s Monolitic Op Amp AD844, URL: http://www.analog.

com/static/imported-files/data sheet/AD844.pdf. Assessed 3 Mar 2011

Chapter 2Functional Transformations of Signals

Abstract The main objective of this Chapter is to understand the differencebetween linear and nonlinear transformations of signals and between linear andnonlinear electron devices.

The necessary prerequisite for studying this material is theory of linear spaces –the course of high mathematics.

After studying this Chapter, the readers will be able to determining the form oftransformation performed by any electron device.

2.1 Introduction

Operational amplifiers serve basic elements for various electron devices, whichperform different transformations of input signals.

All transformations of signals: scaling, integration, clipping, synchronous detection, deter-mination of the absolute value of the input signal, etc. can be divided into two large classes:linear and nonlinear. Linear transformations assume fulfillment of the basic principles ofsuperposition and scaling.

In this case the analysis of devices naturally becomes simpler. Nowadays thereexist many accurate and powerful methods for analysis and investigation, namely,the symbolic method, the operator method, etc. All these methods are usuallyanalytical. In the case of nonlinear transformations, the principles mentioned aboveare not fulfilled in the most cases. Methods for investigation become less generaland, as a rule, numerical or qualitative. The basic difference between the linearand nonlinear transformations in the temporal and spectral representations areconsidered below.


35

36 2 Functional Transformations of Signals

2.2 Linear Transformations of Signals

Clarify the difference between linear and nonlinear operations. In mathematics[1], the transformation § of the space V into the space W is called the lineartransformation of V into W, y D§(x), (x 2 V, y 2 W), if some conditions are fulfilled,in particular:

.x1 C x2/ D .x1/C .x2/I .�x/ D � .x/; (2.1)

where œ is a constant.The simplest linear transformation is multiplication by a constant:

y D kx or y.t/ D kx.t/; (2.2)

where k is the constant representing the transformation §.Let x D x1 C x2, then y D k(x1 C x2) D kx1 C kx2 and y D k(œx) D kœx.This transformation is called the scaling transformation, and k, x, and y can be

both real and complex parameters. In the last case, the scaling transformation turnsinto the operation of spectrum conversion, that is, signal filtering.

PY .!/ D PK.!/ PX.!/

where

PX.!/ D1Z

�1x.t/e�j!tdt

Particular cases of linear operations are the operations of integration anddifferentiation. It is known that

y DZ

Œ�1f1.x/C �2f2.x/�dx D �1

Z

f1.x/dx C �2

Z

f2.x/dx;

y D d.x1 C x2/

dtD dx1

dtC dx2

dt

or

PY .!/ D 1

j!

� PX1.!/C PX2.!/� D

PX1.!/j!

CPX2.!/j!

and

PY .!/ D j!� PX1.!/C PX2.!/

� D j!� PX1.!/C PX2.!/

�:

2.2 Linear Transformations of Signals 37

Fig. 2.1 Input and output voltages of a linear device

If we apply the conditions (2.1) to electron devices, then x1 and x2 are thedevice input voltages, which can vary within the range V; y are the output voltagesvarying within the range W, and œ is the gain (conversion gain). If any of these twoconditions is not fulfilled, the transformation is not linear. So, taking into accountthis explanation, we can draw the following definition.

Linear operations are such operations, in which if the amplitude of the input signal changes� times, then the amplitude of the output signal also changes � times, and the output signalcorresponding to the sum of input signals is equal to the sum of output signals from eachinput signal separately.

As follows from the conditions (2.1), the operations of multiplication by a con-stant, addition and subtraction of signals, integration and differentiation, filtering,delay of signals, and some others can be classified as linear.

Figure 2.1 shows the plots representing typical linear operations [2, 3]: multipli-cation by a constant (scaling) and shift in time. It can be seen from Fig. 2.1 that, inthe case of scaling, doubling of the amplitude at the input corresponds to doubling ofthe amplitude at the output. The linear device does not change the sine-wave shapeof the input signal, but can shift it in time by tshift (phase shift).

The conditions (2.1) can be applied to the spectrum conversion as well. It alsoobeys the properties of linearity, that is, the sum of signals corresponds to the sumof their spectrum, and multiplication of a signal by the constant œ corresponds tothe multiplication of the amplitude spectrum of the signal by the same constant.Other properties of linearity (differentiation, integration, delay, etc.) are fulfilled inthe spectral space as well.

As is well-known, the spectrum of the sine-wave signal has only one component.Consequently, there is only one component at the output of a linear circuit. This


Fig. 2.2 Amplitude (a) and phase (b) spectrum of the input and output signals of a linear device

is the basic property of a linear electrical circuit.1 At the other shape of the inputsignal, whose spectrum consists of many components, the shape of the output signalmay differ from that of the input signal because different spectral components mayby transmitted with the different gain. On Fig. 2.1b, at the rectangular shape ofthe input signal, the output signal already has not the same shape, but the directproportionality is still present between the signal amplitudes at the input and outputof a linear device.

The spectral components of the output voltage can change their amplitude and phase andeven vanish (in the case of filtering), but new spectral components, absent in the inputsignal, never arise in linear circuits with constant parameters.

The frequencies of the corresponding harmonic components of the input andoutput signals remain equal to each other. Some components decrease, while othersincrease in the amplitude and shift in phase. For example, in the case of the scalingtransformation, all the components change the amplitude in the same (proportional)way and, in the ideal case, do not shift in phase. To the contrary, if the signal isshifted in time by a delay line, the amplitudes of the spectral components do notchange, while the phase increases in the direct proportion to the frequency of thecomponent. Figure 2.2 shows the examples of the amplitude and phase spectra ofthe input and output signals of a linear device.

The upper panels in Fig. 2.2 show four harmonic components with the frequen-cies f1, f2, f3, and f4 in the spectrum of the output signal, while the lower panelsdepict the spectral components of the output signal. Analyzing the plots, we can

1Certainly, there exist no linear devices in the mathematical sense as discussed above. All elementsand, first of all, active ones are nonlinear, at least, to the lowest degree, but the presence of lownonlinearity does not violate the basic conditions (2.1) within the given error of transformation.

2.2 Linear Transformations of Signals 39

see that the amplitude of the component with the frequency f2 increased, whilethe amplitudes of the other components decreased. The phases of the componentswith the frequencies lower than f2 have gained the positive increment, while forthose with the higher frequencies the increment was negative. Consequently, we candraw the conclusion about separation of the component with the frequency f2 andsuppression of all other components, that is, about the filtering transformation. Itis important to note here that no one new component appeared in the spectrum ofthe output signal. Analyzing the spectra of the input and output signals in the entirefrequency region, it is always possible to determine the type of the transformation,that is, to solve the inverse problem, namely, the problem of synthesis of theoperational transformation.

A particular, but very important case of the linear transformation is the casethat some parameters of the operator of signal transformation depend on time. Anexample of such transformation is the scaling transformation with the coefficientvariable in time. In such case, Eq. 2.2 can be written in the kind

y.t/ D k.t/x.t/; (2.3)

where x(t), k(t), and y(t) are the functions of time. Equation 2.3 differs fromEq. 2.2 by the fact that in the former k depends on time, while in the latter it isa constant. This insignificant, at first sight, difference considerably supplementsour conclusions concerning the linear transformations. First, the transformationsperformed by Eq. 2.3 are called parametric linear transformations. Second, suchtransformations become non-stationary in time.

� But what are their features? Demonstrate them by the following example. Letx(t) and k(t) vary by the sine law independently of each other (in this example, withdifferent frequencies), then

y.t/ D km sin.!1t/xm sin.!2t/ D

D kmxm

2cosŒ.!1 � !2/t� � kmxm

2cosŒ.!1 C !2/t�: (2.4)

The coefficient before the braces in Eq. 2.4 is the amplitude of y m . For comparison,let us present the result of the linear transformation with the constant coefficients byEq. 2.2:

y.t/ D kx.t/ D kxm sin.!2t/ D ym sin.!2t/:

It can be seen that in the linear transformations with the coefficients constantand variable in time, the amplitudes of the transformed signals are proportionalto the amplitudes of the initial signals x m . Just this is the similarity betweenthem. However, there is a difference as well. At the parametric transformation, thecomponents with new frequencies arisen, those are not in the initial signal. And thisis the principal difference between these types of linear transformations.


2.3 Nonlinear Transformations of Signals

In nonlinear transformations, the principle of proportionality of the amplitudes of theinput and the output signals is violated, and the sum of the input signals not necessarilycorresponds to the sum of the output signals.

For nonlinear transformations, Eq. 2.2 can be presented in the following:

y.t/ D kŒx.t/�x.t/ D F Œx.t/�: (2.5)

Here k depends on x, that is a function of the signal and, what’s more, its nonlinearfunction.

The typical nonlinear functions are shown on Fig. 2.3.Consider the plots of the voltage at the input and output of the device performing,

for example, one-side limitation (Fig. 2.3a) of the sine-wave signal. The diagramsof the input and output voltage and gain character-ristic are presented on Fig. 2.4.

It can be seen from Fig. 2.4 that until the input voltage equals Vin1, the gaincharacteristic is almost linear and the input and output voltages are almost directlyproportional. The further increase of the input voltage amplitude no longer leads tothe proportional increase in the amplitude of the output voltage: it is restricted at thelevel of Vmax. As this takes place, the shape of the output voltage is distorted, and itbecomes non-sine-wave in the top part of the positive halfwave. The spectrum of the

Fig. 2.3 Typical nonlinear functions: limitation of signal: one-side (a and b), two-side withoutshift (c), two-side with a shift (d); sign function (e), function returning the absolute value (f),exponential function (g), logarithmic function (h), and parabolic function (i)

2.3 Nonlinear Transformations of Signals 41

Fig. 2.4 Input and output voltages for restriction (a) and gain characteristic (b)

Fig. 2.5 Amplitude spectrumof the output restrictionvoltages at differentamplitudes of the input signal

output signal enriches, and higher harmonics, which were absent in the spectrum ofthe input signal, arise in addition to the first harmonic. The amplitude spectrum ofthe output signal at different values of the input voltage is shown on Fig. 2.5.

The dashed curves in Figs. 2.4 and 2.5 are for the signals and their spectrumbefore limitation, and the solid curves are those after limitation. For more convenientcomparison, the spectral components having equal frequencies are plotted near eachother. It can be seen from Fig 2.4 that until the voltage reaches the threshold valueVin1 (before limitation), the output signal has the nearly sine-wave shape, and itsspectrum in Fig. 2.5 contains only one harmonic.

As the amplitude of the input signal doubles (up to Vin2), the output signal islimited at the level Vmax, due to which its amplitude increases less than twice.


Fig. 2.6 Voltages of a frequency converter (a) and its spectrum (b)

Now the spectrum of the output signal includes the direct voltage and the higherharmonics: second, third, etc, which are not present in the input signal

With the other form of the gain characteristic shown on Fig. 2.6, not only theamplitude, but also frequencies of signal can be transformed.

The gain characteristic here is described by the equation:

Uout D Um

�

2U 2

in

U 2m

� 1

�

:

If we substitute the sine-wave input voltage with the amplitude V m and thefrequency f0 into this equation, the output voltage also has the sine-wave form andthe same amplitude, but its frequency becomes equal to 2f0, thus confirming thefact that nonlinear devices are capable of transforming (multiplying or dividing) thesignal frequency.

2.4 Conclusions

Thus, considered the functional transformations, we can make the followingconclusions.

In linear operational transformations with constant coefficients, the principle ofsuperposition of signals is fulfilled. The frequencies of signals do not change aftertransformation.

In nonlinear transformations, the principle of superposition is not fulfilled inthe general case, and the signals with some frequencies are transformed into

2.4 Conclusions 43

signals with other frequencies. The spectrum of periodic signal enriches with newhigher-harmonic components with the amplitudes and phases depending on thetransformation characteristics.

Under certain conditions, subharmonics (harmonic components having the fre-quency two, three, and more times lower than the signal frequency) and signals withcombination frequencies mf1 C nf2, where m and n are integer numbers can appear.In addition, a direct voltage often arises, depending on the amplitude of the inputsignal, but absent in it. In some cases, continuous sine-wave or other self-oscillationsor even chaotic oscillations, close to random ones in their shape are arisen.

Questions

1. Give examples of linear mathematical operations.2. Signal integration is a linear operation, is not it?3. Squaring is a linear operation, is not it?4. Whether the principle of superposition is fulfilled in a linear circuit and how can

this be proved?5. Can new spectral components arise in a linear electron device?6. Can a signal component with a new frequency arise in a liner electrical circuit?

Test Yourself

1. Which of the following mathematical operations are linear?

(a) addition;(b) square-rooting;(c) taking the logarithm;(d) squaring;(e) integration;(f) limitation;(g) rectifying;(h) exponentiation.

2. A direct voltage arises in a linear electronic device at the input signal. Is thistrue?

(a) Yes;(b) No.

3. New spectral components, which are not present in the input signal, appear in alinear electron device. Is it true?

(a) Yes;(b) No.

4. Spectral components disappear

(a) at subtraction;(b) at addition;(c) at integration;(d) at rejection.


5. Can a linear device distort the shape of a sine-wave signal?

(a) Yes, it can;(b) No, it cannot.

6. Can a linear device distort the shape of a rectangular signal?

(a) Yes, it can;(b) No, it cannot.

References

1. Guillemin, E.A.: The mathematics of circuit analysis: Extensions to the mathematical trainingof electrical engineers. In: Introductory Circuit Theory. Wiley, New York (1950)

2. Guillemin, E.A.: Introductory Circuit Theory. Wiley, New York (1953)3. Paul, C.R.: Analysis of Linear Circuits. McGraw-Hill Book Company, New York (1989)

Chapter 3Linear Functional Units in OperationalAmplifiers

Abstract The objective of this Chapter is to study the design features of functionalelectronic units executing linear operations with input signals: scaling, summing,subtracting, integrating, and filtering.

To become familiar with the material presented, the readers should have a clearidea about these units within the course “Electronics.”

After studying this Chapter, the readers will be able to determine the form ofthe functional transformation performed by some or other device and to explain itsoperation.

3.1 Introduction

Today we have a variety of linear devices, in particular, those made in op-amps.They are thoroughly described in the training and research literature. Certainly, allthese devices cannot be considered within this Chapter, so our consideration will befocused at some of them, which present good examples of the basic principles usedin design of linear electron devices.

3.2 General Circuit Designs of Linear Devices

Operational amplifiers can be used as elements of various devices performing linearoperations with analog input signals: multiplication of signal by a constant (scaling),summation or subtraction, differentiation or integration of signals, transformation of signalspectrum (filtering), and others.

Let us consider and analyze the most widespread general circuit design of thelinear operational device constructed in op-amps.

In this generalized circuit design, A and B are linear electrical circuits includingpassive elements: resistors, capacitors, and inductances. The signal applied to the


45

46 3 Linear Functional Units in Operational Amplifiers

Fig. 3.1 Generalized circuit design of an operational device constructed in op-amps with twoexternal circuits

input of the circuit A is then amplified by op-amp and returns back to the op-ampinput through the circuit B in the kind of the feedback signal.

Many linear operational devices constructed in op-amps can be reduced to thisgeneralized circuit design. For example, if both A and B are the simplest circuitswith the only one resistor between the input and the output, the resulting circuit isan inverting amplifier. If the resistor in the circuit A or B is replaced by a capacitor,then the resulting circuit becomes an integrator or differentiator, etc. Finally, usingmore complex circuits, it is possible to realize various filters of electrical signals.The op-amp in the generalized circuit can have any structure of those considered inthe previous chapters.

� Let us find the basic functions of this circuit: the gain and the input resistance.To determine the transfer function, apply the generalized matrix method [4]. Forthis let us enumerate the elements as shown on Fig. 3.1 and write the conductancematrix. Let the input of the circuit A be left and connected to the input of thegeneralized device Vin, which is common-use. The input of the circuit B is the rightand connected to the op-amp output. The op-amp has the VCCS structure. On theseassumptions, the conductance has the form

Y D

2

664

yA11 C yA12 �yA12 0

�yA12 yA12 C yA22 C yB22 C yB23 C yOAin �yB23

0 SOA � yB23 yB23 C yB33 C yOAout

3

775 ; (3.1)

where y are self-conductances and transfer conductances of the circuits A, B and op-amp as indicated by the superscripts;yX

ij , S, yOAin ,yOA

out – self- and transfer conductanceof nodes, mutual conductance, input and output conductance of OA.

Using matrix Y it is easy to form the equation for determining all secondaryfunctions of the circuit – current or voltage gain, input and output resistance –according to the known rules [5]:

3.2 General Circuit Designs of Linear Devices 47

KV D Vout

VinD .�1/iCj �ij

�ii– voltage gain;

KI D Iout

IinD .�1/iCj �ij

�jj– current gain;

Ztr D Vout


�– transfer resistance;

G D Iout

VinD .�1/iCj �ij

�ii;jj

– transfer conductance;

Zin D Vin

IinD �ii

�– input resistance;

Zout D V Idleout

I Shortout

D �ii;jj

�ii– output resistance,

where�ij – complementary minors of matrix Y, i j – input and output node numbers.For instance, the voltage gain for the circuit in Fig. 3.1 is equal to

KV D .�1/1C3�13

�11

D �yA12.S � yB23/

y2y3 C yB23.S � yB23/:

where y2 D yA12 C yA22 C yB22 C yB23 C yOAin , y3 D yB33 C yB23 C yOA

out – cumulativeconductance of the second and the third nodes.

Assuming that the op-amp transconductance is much higher than any of theconductances (S � y), the first term in the denominator can be ignored. As a result,we obtain the simplified equation

K � �yA12yB23

: (3.2)

It can be seen that the gain of the generalized circuit is determined only by the transferconductances of the circuit’s A and B.

For the ordinary circuit of the inverting amplifier

yA12 D � 1

R1; while yB23 D � 1

R2:

then

K D �R2R1; (3.3)

which coincides with the known equation (1.6). For the input conductance Yin, we obtain:

Yin D yA11 C yA12: (3.4)


Fig. 3.2 Generalized circuit design of an operational device constructed in op-amps with fourexternal circuits

As could be expected, the input conductance Yin is equal to the input conductanceof the circuit A. Thus, using Eqs. 3.3 and 3.4, we can find the most commonly usedfunctions of a particular operational circuit.

A more complex operational circuit is shown on Fig. 3.2. It includes fourelectrical circuits A, B, C, and D, and the both inputs of the operational amplifierare involved. The circuit D here has two ports (sometimes its input 6 is connected tothe op-amp output 3, as shown by the dashed line, then the positive feedback arisesin the amplifier).

Obviously, without circuits C and D, this operational circuit coincides with theprevious one. For the circuit in Fig. 3.2 the conductance matrix takes the followingform:

Y D

2

66666666664

y1 �yA12 0 0 0 0

�yA12 y2 �yB23 �yOAin 0 0

0 S � yB23 y3 �S 0 0

0 �S � yOAin 0 y4 �yC45 �yD46

0 0 0 �yC45 y5 0

0 0 0 �yD46 0 y6

3

77777777775

where y1 D yA11 C yA12, y2 D yA12 C yA22 C yB22 C yB23 C yOAin , y3 D yB33 C yB23 C yOA

out ,y4 D yC44CyC45CyD44CyD46CyOA

in CS , y5 D yC45CyC55, y6 D yC46CyC66– cumulativeconductance of nodes 1–6.

To determine gain factors, input and output resistance for one of inputs it isnecessary to connect the remainder inputs by the common wire (node 0). At thesame time rows and columns with numbers of unused nodes are to be deleted fromthe matrix. For example, to determine the gain factor for node 1 it is necessary to

3.3 Scalers 49

connect nodes 5 and 6 by the common wire, delete rows and columns 5 and 6 frommatrix Y, determine, as earlier, complementary minors �13 i �11 of the remaindermatrix and put them to the formula forKV ; as a result we get

K1 D .�1/1C3�13

�11

� �yA12

yB23: (3.5)

To determine the gain factor for the second input (node 5) we are to connectnodes 1 and 6 by the common wire, then the gain factor will be equal to

K2 D .�1/5C3�53

�55

:

Under signal injection for the both inputs simultaneously output resistance canbe calculated by the following formula:

Vout D Vin1.�1/1C3�13

�11

C Vin2.�1/5C3�53

�55

:

Introducing positive feedback (node 6 is connected to node 3) to conductancematrix Y it is necessary to add elements 6–3 of the rows and columns beforehand,then the gain factors can be determined with the use of the formulas given above.

The equation for the input conductance at the input 1 almost coincides withEq. 3.4. Joining the inputs 1 and 2, we can obtain a new generalized circuit designof an operational device. To describe it one needs to add elements 1–2 of the rowsand columns. Certainly, the whole variety of modern operational devices cannotbe classified to the circuits considered above. Nevertheless, as will be seen below,the most of them can be reduced to one of these circuits and thus described by theobtained equations.

3.3 Scalers

3.3.1 Inverting Amplifiers

The corresponding operational circuit (Fig. 3.3) is a special case of the formulain Fig. 3.1. Recall that the op-amp gain is K D �R2/R1 regardless of the op-ampstructure (3.3). However, this value varies at the varying frequency of the signal.Therefore, here we will consider peculiarities in amplification of signals havingdifferent shape and frequency.

The characteristic diagrams of the output voltage at the sine wave input signalwith different effect factors shown on Fig. 3.4. For the best, the curves are given indifferent time scales.


Fig. 3.3 Inverting amplifier

Fig. 3.4 Diagrams of outputvoltage of the invertingamplifier

Curve 1 shows the output voltage at the low frequency of the input signal (withinthe flat part of GFC). It should be noted that the output voltage has a phase shift of180ı with respect to the input voltage and can be biased upward or downward bythe value of KVoff. Curve 2 is the voltage diagram at the higher frequency, when thesignal frequency corresponds to the GFC drop.

It is seen that the signal amplitude is lower in this case and the output voltage isshifted in time by tsh corresponding to the phase shift ®D¨tsh. Curve 3 shows theoutput voltage at the even higher signal frequency. The output is no longer a sinewave, but have a triangular shape, that is, it is strongly distorted. These distortionsdepend not only on the frequency, but also on the slew rate (W) of the output voltageVout and begin to show themselves at

dVout

dtD V out m! � W: (3.6)

The diagrams of the output voltage at rectangular input signals are shown onFig. 3.5.

3.3 Scalers 51

Fig. 3.5 Diagrams of outputvoltage at a rectangular inputsignal

Curve 1 is the diagram of the output voltage at the low frequency of the inputsignal, when the almost whole signal spectrum fits within the flat part of GFC. Itcan be seen that the voltage has the rectangular shape, but is shifted upward ordownward due to the offset voltage. Curve 2 corresponds to the output voltage atthe higher frequency, when a part of the spectrum lies at the GFC drop. In this case,the high-frequency components of the spectrum decrease in the amplitude, and thefront and the drop of the output voltage become more flat. Curve 3 represents thehigh-frequency signal, when the whole signal spectrum falls on the GFC drop. Sincethe slope is 20 dB/dec, the output signal becomes proportional to the integral of theinput signal and variations of the output voltage are nearly triangular. The curves inFig. 3.5 are given in different time scales. The signal distortions are mostly causedby GFC flatness. With the allowance made for this, we can give recommendationson selection of op-amps for an inverting amplifier.

In the amplifier with the op-amp having the VCVS structure, the GFC cutoff frequencydepends practically only on the ratio of resistances R2/R1 and is independent of theirindividual values. Therefore, at the increase of Kfb, the cutoff frequency decreases andthe frequency properties worsen. If the CCVS structure is used, the cutoff frequency, tothe contrary, is independent of the resistance ratio and remains practically unchanged atvariation of Kfb, but depends on the resistance R2.

Therefore, if we want to control or change the gain with the frequency propertieskept unchanged, the better way is to use the op-amp with the CCVS or CCCSstructure. For amplification of sine wave signals, one has to select op-amps withthe VCVS structure and the upper threshold frequency so that the requirements onthe maximal frequency error to be met: fT>Kfbfmax/•Kfb, where fT is the op-ampthreshold frequency, fmax is the maximal frequency in the operating frequency rangeof the amplifier, •Kfb is the acceptable frequency error.

When amplifying mostly rectangular signals, the principal attention should be paid to theop-amp time parameters, such as pulse rise and decay times.


Fig. 3.6 Inverting amplifierwith additional divider

3.3.1.1 Inverting Amplifier with Additional Divider

Consider different versions of the circuit design for inverting amplifiers. In thiscircuit design shown on Fig. 3.6, the feedback circuit includes an additional voltagedivider constructed in resistors R3 and R4 [1].

� Taking them into consideration and forming the conductance matrix we candetermine the gain factor by the following formula:

K D .�1/1C3�13

�11

� �y1 y2 C y3 C y4

y2y4D �R2R3 CR2R4 CR3R4

R1R3: (3.7)

The advantage of this circuit is if for an amplifier with the gain K D �100 in theordinary circuit (Fig. 3.3) the resistance ratio R2/R1 should be also equal to 100, thenin this circuit the difference between the resistances can be much smaller and thesame gain can be obtained at the resistance ratio R2/R1 � R4/R3 D 10. In turn, thisallows to improve the accuracy of the gain and its stability, because the differencebetween resistances is smaller, the resistance temperature coefficient is smaller also,the difference in the error is smaller also and, consequently, the accuracy is higher.

Note that the industry produces instrumental op-amps, for example, of LM163type made by National Semiconductor, which permit the construction of a scalerhaving the gain of 10, 100, and 1,000 without external resistors. The necessaryresistors are already built in the op-amp chip, and the accuracy in the gain is veryhigh (the error of 0.05%). Just these amplifiers will be considered below.

3.3.1.2 Inverting Current Amplifier

As well known, current amplifiers are used to measure a low current through its conversioninto voltage with the following measurement of the voltage.

The simplest way to convert the current into voltage is to transmit it througha resistor by the Ohm law. However, this simple method has some disadvantages.To improve the sensitivity when measuring low currents, one has to increase theresistance, and this results in the adverse effect of the resistor on the circuit,which measures the current. In addition, it becomes necessary to increase the input

3.3 Scalers 53

Fig. 3.7 Current amplifiers: simplest circuit (a) and photodiode current amplifiers (b)

Fig. 3.8 Noninvertingamplifier

resistance of the following stages, which is accompanied by the increase of the lagof the measuring circuit because of the effect of the input and parasitic capacitances.The current amplifier eliminates these disadvantages to a significant degree. One ofthe simplest current amplifiers is shown on Fig. 3.7.

Figure 3.7 is an inverting amplifier without input resistor. The coefficient ofcurrent-to-voltage conversion (transresistance) is equal to

Ztr D Vout


�� RC Rout

K� �R (3.8)

3.3.2 Noninverting Amplifier

The circuit design of a noninverting amplifier is shown on Fig. 3.8. This circuit canbe obtained from that of Fig. 3.2 in the absence of quadripoles A, C and D, i.e.only under conductance y23 and y22 and signal injection to node 5. The gain of thenoninverting amplifier is K D 1 C R2/R1. This amplifier is distinguished for the highinput and low output resistances. It is characterized by the same signal distortionsas in the inverting amplifier.


Fig. 3.9 Amplifier withcontrollable gain

Fig. 3.10 Simplestdifference amplifier

3.3.3 Amplifiers Based on Inverting and Noninverting Amplifiers

3.3.3.1 Amplifier with Variable Scale Parameter

It is easy to understand that, in the amplifier on Fig. 3.9, the gain varies as a slider ofthe potentiometer R1 moves within – ’<K<C ’, at ’> 1. Both mechanical andelectronic digital potentiometers can be used.

3.3.3.2 Differential-Mode Amplifier (Scaling Subtractor)

The difference amplifier on Fig. 3.10 provides the difference between the voltage inthe input 1 and the voltage in the input 2. As can be seen from Eq. 3.9, the voltage issubtracted with a certain weight specified by the resistors. At R1 D R3 and R2 D R4,these weighting coefficients are equal to R2/R1.

� The output voltage of the amplifier can be determined by the following formulawhen nodes 1 and 5 of the circuit in Fig. 3.2 are connected:

Vout D �Vin1R2

R1C Vin2

R4

R3 CR4

R1 CR2

R1D R2

R1.Vin2 � Vin1/: (3.9)

The disadvantages of the simplest difference amplifier are the low outputresistances and the difficulty of regulating the gain.

3.3 Scalers 55

Fig. 3.11 Difference amplifiers constructed in two op-amps

One of these disadvantages is removed in the circuit on Fig. 3.11, which isthe difference amplifier with the high input resistances. It has two inputs andtwo outputs. Whenever necessary, it is possible to use only one input and onlyone output. The input resistances of this amplifier are increased due to the seriesfeedback.

� To find the output voltages, let us use the principle of superposition, that is,determine each output voltage separately as a sum of input voltages. This calculationis represented by Eqs. 3.10 and 3.11.

The output voltages are equal to

Vout1 D Vin1

�

1C R2

R1

�

� Vin2R2

R1(3.10)

Vout2 D Vin2

�

1C R3

R1

�

� Vin1R3

R1: (3.11)

From these equations, the main feature of the circuit is the difference of theweighting coefficients of the subtracted voltages, which does not permit the exactsubtraction to be realized. Nevertheless, some integrated circuits, for example,MAX4147 fabricated by MAXIM, are designed as shown on Fig. 3.11.

The accuracy of subtraction can be improved in the modernized this circuit onFigs. 3.11b and 3.12.

The first circuit includes the inverting and noninverting amplifiers and has noneed in additional explanations. In the second circuit, the resistors R1, R2, R3 are


Fig. 3.12 Differenceamplifier on two op-ampswith additional balancingresistors

Fig. 3.13 Difference amplifier constructed in three op-amps

supplemented with the resistors R4 and R5, which balance the input signals. As aresult, the gains become equal, which is seen from Eq. 3.12.

Vout1 D .Vin2 � Vin1/

�R2 CR5

R1C R2

R4C 1

�

: (3.12)

This equation assumes the following equality: R2/R4 D R3/R5.The combination of the two circuits considered above (Figs. 3.10 and 3.11) gives

a new version of the difference amplifier (Fig. 3.13).With fulfillment of the conditions from analysis of the circuit on Fig. 3.12, the

equation for the output voltage has the form

Vout1 D .Vin2 � Vin1/R7

R4

�R2 CR3

R1C 1

�

: (3.13)

One of the advantages of this difference amplifier is the high common mode rejections,that significantly decrease noise and the constant component of signals, if they aresimultaneously present at the both inputs.

3.3 Scalers 57

Fig. 3.14 Difference amplifier constructed in inverting op-amps

Owing this, the amplifier is widely used in the Hi-Fi sounders, in communication,etc. Nowadays several integrated circuits are produced, whose structure fullycorresponds to the structure of this amplifier, for example, INA163 fabricated byTexas Instruments (for it Fig. 3.13 gives the resistance values), MAX4144 fabricatedby MAXIM, AD830 fabricated by Analog Devices, and others. These amplifiers fallin the class of instrumental amplifiers. The fact that all elements are made on a singlesilicon crystal and the laser alignment of resistors guarantee the high accuracy of thegain and its temperature stability.

More in-phase rejection provides inverting amplifier as on Fig. 3.14.The both op-amps are connected in the circuit of inverting scaling amplifiers.

Voltages with the high in-phase component can be applied to the inputs Vin1 andVin2; it is only important for the op-amp output to be not overloaded. The outputvoltage can be determined as

Vout D �Vin2R5

R4C Vin1

R2

R1

R5

R3:

At R2 D R3 and R4 D R1, the equation becomes simpler

Vout D .Vin1 � Vin2/R5

R1: (3.14)

All differential amplifiers have a common disadvantage, namely, the difficultyof controlling the gain. To change the gain, one has to change simultaneously theresistances of no less than two resistors. This is inconvenient, because the controlledresistors must be tuned synchronously. For example, in the circuit on Fig. 3.10, it isneeded to tune the resistances R2 and R4, ensuring their equality.

In the circuit on Fig. 3.15 the gain can be changed by turning only theresistance R5. At the condition R1 D R3, R2 D R4, R6 D R7 the output voltage is

Vout D .Vin2 � Vin1/

�R2 CR6

R1C 2

R2R6

R1R5

�

: (3.15)

Differential-mode amplifiers are used where a large external common mode noiseis expected to exit which must be rejected from the input signal.


Fig. 3.15 Difference amplifier with controlled gain

Fig. 3.16 Differential communication line

Fig. 3.17 The differential mode amplifier with the measuring bridge

On Figs. 3.16 and 3.17 shows the simplified schemes of a differential communi-cation line and the sensing circuit for a bridge measuring instrument. Differential-mode amplifiers in these schemes are called a line driver, a line receiver or adifferential receiver.

The input signal Vc comes to the line driver (LD). There it is amplified andconverted into two symmetric antiphasal output signals CVc and –Vc, which arefollows into a long line made as a balanced pair. At the end of the line section,the signal is attenuated somewhat, and then the attenuated signal C’Vc and –’Vc

come to the trunk amplifier (TA). The trunk amplifier compensates for attenuationof the signal by its amplification and again followed it into the line. At the lineoutput, the differential receiver (DR) amplifies the signal and converts it into theasymmetric output signal of the line. Since noise identically disturbs the wires ofthe twisted-pair cable and induces the same noise voltage in them, they come tothe TA and DR inputs in phase and are subtracted, while the valid signals comein anti-phase and are summed. Thus, noise is suppressed, while the valid signal isamplified. To increase the transmission rate, all amplifiers should be broadband.Any of the difference amplifiers, in which one of the inputs is connected withthe common cable, can be used as a line driver. To provide this, for example, in

3.3 Scalers 59

the circuit on Fig. 3.11, we have to connect the input 2 to the common cable. Toensure the equal amplitudes of the output signals, the following relation betweenthe resistances should be met:

1C R2

R1D R3

R1:

However, the best results are achieved when using specialized circuits of linedrivers of the AD815 type fabricated by Analog Devices. This chip is constructed asshown on Fig. 3.11 and capable of supplying up to 0.5 A at the double amplitudeof the output signal up to 40 V. The trunk amplifier can be also represented bythis circuit or the circuits of the MAX4147 type fabricated by MAXIM or AD8132produced by Analog Devices. The bandwidth of these circuits is 350 MHz, while theslew rate is 3,600 V/�s and 1,200 V/�s, respectively. As a differential receiver, it isbetter using the circuit on Fig. 3.13 or the mentioned circuits MAX4144 fabricatedby MAXIM or AD830 produced by Analog Devices, which integrate all the threeamplifiers with resistors.

The reference voltage Vref on Fig. 3.17 comes to the measuring bridge (Br). Thereit is converted into two output signals Vmode C Vc and Vmode – Vc, which are followsinto a line communication. Vmode it is common mode voltage. At the end of the linesection, the signals come to the differential receiver (DR), where Vmode is rejected.

3.3.3.3 Summing Amplifier

This amplifier is intended for adding the voltages applied to the inputs. The voltagesare summed on Fig. 3.18 with certain weighting coefficients specified by resistors.The result of summation will obviously have the opposite sign.

Vout D �Vin1R0

R1� Vin2

R0

R2� Vin3

R0

R3(3.16)

The main disadvantage of this circuit is the different input resistance, because itis determined by the resistors R1, R2, R3. Using an inverting adder and a differentialamplifier, we can create a multi-input adder–subtracter of signals (Fig. 3.19). The

Fig. 3.18 Summingamplifier


Fig. 3.19 Multiinputsumming–differenceamplifier

output voltage in this circuit is equal to the sum of the weighted input voltagesapplied to the inputs 2 with the corresponding weighting coefficient minus the sumof the voltages applied to the inputs 1. It is desirable to satisfy the condition: Y1 D Y2,where Y1 D y1

0 C y100 C y1

000 C y0, Y2 D y20 C y2

00 C y2000 are the sums of

conductances of the corresponding resistors. If this condition is satisfied, then theweighting coefficients for the inverting inputs are determined by the resistance ratio-R0/ R0

1; and those for the noninverting inputs are governed by the ratio R0/ R02:

Otherwise, balancing resistors Ra or Rb are needed [1, 2].In the first approximation all the considered amplifying, summing, and subtract-

ing devices have plane GFC and PRC, characteristic of amplifiers with resistivefeedback without reactive elements. The general shape of GFC for such amplifiersis shown on Figs. 1.8 and 1.10. If capacitors and inductance are introduced into thefeedback, GFC and PRC become frequency-dependent.

3.4 Integrating Amplifiers

Consider operational devices with reactive elements. Integrating amplifier is oneof such devices. As follows from its name, it is designed for integration of theinput signal and is often used in generators of electrical signals, automatic controlsystem, etc.

3.4.1 Inverting Integrating Amplifiers

The circuit of the simplest inverting integrating amplifier is shown on Fig. 3.20.The effect of integration can be seen using the complex transfer function, which(accordingly Eq. 3.2) is determined as

3.4 Integrating Amplifiers 61

Fig. 3.20 Integrating amplifier (a) and diagrams of the output voltage at the different frequencyof the sine wave input voltage (b)

K.j!/ D Vout.j!/

V in.j!/Š �yA12

yB23Š � 1

j!RCD 1

!RCe�j 3�=2:

As well known, division by jw in the complex frequency domain corresponds to integrationof function in the real domain of the variable t, i.e. time. It can be seen from the equationthat the complex transfer function is purely imaginary with jw in the denominator, whichindicates the signal integration.

The absolute value of the transfer function decreases with the increasingfrequency, and the phase at any frequency is equal to �270ı (or C90ı). Figure 3.19bshows the plots 1 and 2 of the output voltage at two different frequencies of thesine wave input voltage. The both output voltages are shifted by �270ı aboutthe input voltage, but have different amplitudes, which is in agreement with theequation for the transfer function. The output voltage with the lower amplitudecorresponds to the input voltage with the higher frequency. The shift by 270ı isexplained by the inversion in the op-amp (shift by 180ı) and integration, which addsthe shift of 90ı. Figure 3.21 shows the gain GFC and PRC plots for the integratingamplifier.

In the complex form, the transfer function of the integrator reflects the ratio of the spectrumof the output and input signals.

� The spectrum of the output signal can be found by multiplying the spectrum ofthe input signal by the complex gain

Vout.j!/ D K.j!/Vin.j!/ D jK.j!/jjVin.j!/jej.'in�3�=2/

Vout.j!/ D 1

!RCjVinjej.'in�3�=2/: (3.17)


Fig. 3.21 GFC and PRC for op-amp (1) and integrator (2)

Fig. 3.22 Diagrams of theoutput voltage of an integratorat a jump of the input voltage

In order to find the amplitude spectrum by Eq. 3.17, all the spectral componentsof the input signal should be multiplied by the absolute value of the gain, each at itsown frequency. To determine the phase spectrum, we should subtract 270ı from thephase of the input signal.

Equation 3.17 is convenient, if the input signal is a sine wave including only oneharmonic component. To determine the output voltage as a function of time, it isbetter using the following equation

Vout.t/ D � 1

RC

tZ

0

Vin.t/dt C V.0/: (3.18)

Consider transient processes proceeding in the integrator as a single pulse isapplied to the input (Fig. 3.22). Ideally, the output voltage varies as curve 1. It starts


Fig. 3.23 Diagrams ofoutput voltage for theintegrator at the rectangularinput voltage

from the initial voltage V(0). In a particular case V(0) D Voff. The voltage continuesto change until the minimal op-amp voltage limited by the supply voltage – Vout m.

Strictly, the variation of the voltage in a real integrator (taking into account thefinite value of the op-amp gain) is the exponential law (curve 2). This process ischaracterized for op-amps with the VCVS structure. For the VCCS op-amp thevoltage varies in a jump at the initial time (curve 3) because the passive signaltransfers from the input to the output, accordingly Eq. 1.10. This transfer occursin the op-amp with output resistance.

Figure 3.23 shows the plots of the voltage at the integrator output for the periodicrectangular input voltage.

Curve 1 is the output voltage at the ideal integration of the rectangular inputsignal. The output voltage here can be shifted upward or downward by the offsetvoltage Voff. Curve 2 corresponds to real integration. Curve 3 is the voltage diagramin the case of the op-amp with the VCCS structure and the higher frequency.

Important factors affecting the error of integration are the input offset voltage and inputcurrents of the op-amp, the quality of the capacitor and resistor in the integrator, and thetime and temperature stability of their parameters.

The effect of the op-amp parameters shows itself in the slow drift of the outputvoltage with time. The effect of the parasitic parameters of the capacitor results inthe spontaneous discharge of the capacitor, which worsens the storage propertiesof the integrator. The best time stability and low leakage current are inherent incapacitors with teflon dielectric. In crucial cases, the integrator elements are placedin an active thermostat to exclude the temperature variations of the capacitanceand resistance. However, even these measures fail to ensure the equivalent constantgreater than 106 M�� F.


Fig. 3.24 Noninvertingintegrating amplifier

Fig. 3.25 Noninvertingintegrator with positivefeedback

3.4.2 Noninverting Integrating Amplifier

The noninverting integrator includes two RC circuits on Fig. 3.24. The circuitR2, C2 is the basic one. With this circuit (but without R1, C1), the transferfunction is K(jw) D 1 C 1/jwR2C2, that is, it includes unity along with the integratingcomponent. This amplifier is called the proportional and integral (PI) amplifier. Thecircuit R1, C1 compensates for the unity in the equation for the transfer function.With this circuit, the amplifier becomes purely integrating. With the equal timeconstants, that is at R1 D R2 D R, C1 D C2 D C, the transfer function is

K.j!/ D Vout.j!/

Vin.j!/D 1

j!RCD 1

!RCe�j�=2: (3.19)

It differs from obtained earlier only by the phase shift. For the noninvertingintegrator, the phase shift is equal to �90ı in place of �270ı, so the maindependences and distortions on Figs. 3.22 and 3.23 are characteristic of thisintegrator as well. Certainly, Eq. 3.19 is valid at the equal rated capacitances andresistances, which is not always possible in practice. Besides, tuning the timeconstant requires the parameters of two elements (two resistors or two capacitors) tobe changed. This disadvantage can be eliminated, if we use the circuit on Fig. 3.25.


Fig. 3.26 Integrators with two inputs

It is the first circuit involving both the negative and positive feedbacks. Thepositive feedback is rarely used in linear devices because of the risk of their self-excitation. It can be easily seen that the op-amp with the resistors R2 and R4 formsa noninverting amplifier constructed as shown on Fig. 3.8. Taking this into account,we can find the transfer function as

K.j!/ D R2 CR4

R2

R1

R1 CR3

1

1 � R2CR4R2

R1R1CR3 C j!CR1jjR3

:

If the real part of the denominator is equal to zero, this transfer function is similarto that determined by Eq. 3.19

K.j!/ D 1

j!.R1jjR3/C :

3.4.3 Integrating Amplifier with Two Inputs

Combining the considered circuits, we can obtain integrator versions with twoand many inputs [1]. Figure 3.26 shows the circuits of the integrators with twoinputs, which have no need in additional explanations, because they are obtainedas a combination of the inverting and noninverting integrators (Fig. 3.26a) and the


Fig. 3.27 Circuits for double integration

difference amplifier (Fig. 3.26b). It can be only noted that the circuit on Fig. 3.26bis characterized by the extended integration precision [1].

3.4.4 Double Integrating Amplifier

The operation of double integration is used in modeling of differential and integralequations, and in construction of sine wave generators. It consists in the consecutiveintegration of the signal itself and then its integral. It is not difficult to imaginethe circuit realizing double integration. It obviously should include two integratorsconnected in series. In the simplest version, for example, they are two invertingintegrators on Fig. 3.19. However, for this circuit two op-amps can be used,which may be economically unjustified. The operation of double integration can berealized with one op-amp in the circuit on Fig. 3.27a (with one input) or Fig. 3.27b(two inputs) [1, 2].� In the first design according to the generalized circuit on Fig. 3.1, the circuits Aand B includes T-shaped RC chains. For the transfer conductance of such chains,we can easily obtain the equations:

yA12 D yR1.yC1 C yR2/

yR1 C yC1 C yR2; yB23 D yC3.yR3 C yC2/

yC2 C yR3 C yC2:

3.5 Differentiating Amplifier 67

Then at

.C2 C C3/R3 D C1R1R2

R1 CR2

we have

K.j!/ D � 1

.j!/2.R1 CR2/R3C2C3: (3.20)

The conditions C1 D C, C2 D C3 D C/2, R3 D R, R1 D R2 D 2R are usually ful-filled. In this case

K.j!/ D � 1

.j!/2R2C 2: (3.21)

The j¨ in the denominator indicates double integration.The circuit on Fig. 3.27b is one of the generalized circuit shown on Fig. 3.2. At

the conditions specified for it, it performs double integration with inversion for theinput 1 and without inversion for the input 2. Recall that it is necessary the equalparameters of the two-ports A and C; B and D, respectively.

3.5 Differentiating Amplifier

The differentiating amplifier performs differentiation of the input signal, that isinverse to integration. The circuit of the simplest differentiating amplifier is shownon Fig. 3.28.

The transfer function of the differentiating amplifier is 3.2

K.j!/ D Vout.j!/

Vin.j!/D �yA12

yB23D �j!RC D !RCe�j�=2:

Unlike the integrating amplifier, the absolute value of K(j!) in this case increasesproportionally to the frequency as it grows, while the phase is constant and equal to �90ı.This means that the amplitudes of the spectral components increase with the growingfrequency, and their initial phases get the increment of �90ı.

Fig. 3.28 Differentiatingamplifier


Fig. 3.29 GFC and PRC of op-amp (1) and differentiator (2); fcut is the op-amp cutoff frequency

Fig. 3.30 Differentiatingamplifier with correction

Taking into account that the high-frequency part of the spectrum increasesat any changes of the input signal, we can say that the differentiating amplifier“emphasizes” these changes. This is also clearly seen from the GFC and PRCof the amplifier on Fig. 3.29. However, this dependence takes place only withinthe plane part of the op-amp GFC up to the frequency fg, and then the feedbackdecreases, because the capacitor resistance becomes negligibly small and thefrequency characteristics of the op-amp and the differentiator almost coincide upto the frequency fT.

Unfortunately, it is not the only disadvantage of the simplest differentiator circuit.The most important disadvantage is the decrease of its stability. If the frequencyincreases, the RC chain connected in the feedback loop adds the phase shift of �90ıto the own op-amp shift of �270ı, and this reduces the stability margin in termsof phase to zero, so the amplifier becomes capable of self-exciting. That is why thesimplest circuit is not used in practice.

More complex circuits, one of which is shown on Fig. 3.30, are used to improvethe stability. In this circuit, the resistor Rfc and the capacitor Cfc serve for correctionof GFC. They improve stability due to reduction of the frequency band, in whichdifferentiation is efficient.

3.6 Active Filters Constructed in Op-amps 69

Fig. 3.31 Inertial (a) and differentiating (b) circuits

Fig. 3.32 GFC and PRC fordifferent versions ofdifferentiator with correction

It is appropriate mention here that the last circuit can be easily transformedinto two other circuits, which find wide utility and are called the inertial anddifferentiating circuits (Fig. 3.31).

Curve 1 on Fig. 3.32 shows, as usually, op-amp GFC, curve 2 is for thedifferential amplifier with correction; curve 3 shows GFC of the inertial circuit,and curve 4 shows GFC of the differential circuit.

3.6 Active Filters Constructed in Op-amps

Electrical filters are devices intended for separation of a signal of some frequencies andsuppression of other frequencies. Another definition is also possible. Electrical filter is anelectrical circuit with the known response to a given excitation.

Filters can be passive and active, depending on whether amplifying elements areused in them or not. Integrators and differentiators, besides their base function, canalso perform filtering, and so they are simplest filters. The former ones performsfiltering of low frequencies, while the latter ones filter high frequencies. However,there are specialized linear devices constructed in op-amps and filtering inputsignals, namely, active filters. The specific feature of active filters is normalization


Fig. 3.33 Gain-frequency characteristics of different filters: LPF (a), HPF (b), bandpass filter (c),and bandstop filter (d)

of their gain-frequency characteristics, which must satisfy certain requirements inthe passband and stopband regions (usually this is not a case for integrators anddifferentiators).

The active filter differently transmits signals of different frequency to the output: somecomponents pass practically unchanged, while others are significantly attenuated. By thisreason, filters have different GFCs, depending on which they are classified into low passfilters (LPF) and high pass filters (HPF), bandpass filters and bandstop filters.

Figure 3.33 shows the GFCs of these filters. The GFC of LPF indicates, forexample, that the filter transmits the signals, whose frequency is lower than ¨low1,and stops the signals, whose frequency is higher than ¨low2, that is, it separates andtransmits only the signals with the frequencies lower than some cutoff frequency.The term transmits means that the signals with the frequency lower than ¨low1 arefollowed to the filter output with the gain K1 nearly equal to 1, while the term stopsmeans that the signals with the frequencies higher than ¨low2 are followed to thefilter output significantly attenuated (with the gain K2 � 1).

Therefore, the frequency band from 0 to ¨low1 is called the passband, while theband from ¨low2 to infinity is called the stopband. The frequency ¨low1 is the LPFcutoff frequency. The signals, whose frequencies fall within the band from ¨low1


to ¨low2,, are followed to the output with the gain ranging from K1 to K2. Thefrequencies ¨low1, ¨low2 and the gains K1 and K2 are the LPF parameters. Theparameters for other filters can be defined in the similar manner. Let us determinethe general properties of the transfer function, whose absolute value depends on thefrequency as shown on Fig. 3.33. In the general case, the equation for this function is

K.s/ D a0sn C a1s

n�1 C a2sn�2 C � � � C an�1s C an

b0sm C b1sm�1 C b2sm�2 C � � � C bm�1s C bm;

where a and b are the real coefficients, p is the Laplace operator.This equation describes the operator transfer function of the filter of m-th order,

since the filter order is determined by the highest power of the polynomial in thedenominator. It is assumed that the transfer function corresponds to a stable filter,that is, all coefficients of the parameter b are positive, nonzero, and satisfying theRouth–Hurwitz criterion. Depending on the parameters a and b, the operator transferfunction describes the corresponding filter.

Determine the requirements to the coefficients for LPF. Since this filters transmitsall low frequencies, including the zero frequency, at ¨D 0 the equations becomessimpler K.j0/ D an =bm :

It is obvious that for LPF the coefficients a n and b m must be nonzero. Toensure transmission of signals without attenuation, it is necessary for K(j0) D 1,which takes place provided that a n D b m . Usually, a n D b m D 1. As the frequencyincreases, the gain must decrease. The rate of this decrease depends on othercoefficients and is maximal at all a i D 0 at i 2 (0, 1, 2, : : : , n – 1), that is, exceptfor a n . Accordingly this, the transfer function of LPF assumes the form:

K.s/ D an

b0sm C b1sm�1 C b2sm�2 C � � � C bm�1s C bm:

The high pass filter transmits, without attenuation, the signals with higherfrequencies (theoretically, up to infinity) and attenuates the signals with lowerfrequencies, therefore the gain at higher frequencies is equal to 1, while at thezero frequency it must be zero. Here from it is clear that the highest powers ofthe polynomials in the numerators and denominators and their coefficients must beequal, that is, m D n and a0 D b0, as well as a n D 0. To provide for the steep descentin the lower frequencies, it is necessary for all a i D 0 at i 2 (1,2, : : : , n), except fora0. Then the equation for the HPF transfer function takes the form:

K.s/ D a0sm


Analogical we can easily obtain for the bandpass filter:

K.s/ D a1sn�1 C a2s

n�2 C � � � C an�1sb0sm C b1sm�1 C b2sm�2 C � � � C bm�1s C bm


Fig. 3.34 Gain-frequency (a) and transfer (b) characteristics of LPF: critical (1), Butterworth (2),and Chebyshev (3) filters (Besides these filters, other, for example, Zolotarev, Bessel, etc. filtersare known)

and for the bandstop filter:

K.s/ D a0sm C a1s

m�1 C a2sm�2 C � � � C am�1s C am


The order of the operator transfer function determines the complexity and thedamping characteristics of a filter. From the above equations, it can be seen thatthe first-order operator transfer functions K.s/ D a1

b0sCb1 ; K.s/ D a0sb0sCb1 allow

realization of only LPF and HPF, respectively. For the bandpass and bandstop filters,the circuit should be of at least second order, for example,

K.s/ D a1s

b0s2 C b1s1 C b2; K.s/ D a0s

2 C a2

b0s2 C b1s C b2:

What’s more, depending on the ratio between a and b, the filter GFCs may havedifferent forms.

Study a particular GFC for LPF. Passing from operator to complex transferfunctions, we can obtain GFCs for LPF at different coefficients of the polynomials.Figure 3.34 shows GFCs in the log scale and the corresponding transfer character-istics (TCs). Curves 1 represent the characteristics of the filter with the aperiodictransfer function and with the shortest time of establishment of the output voltage.However, GFC of this filter has the most gently sloping drop, which indicates thelower suppression of the higher frequencies. Curves 3 correspond to the filter withthe steepest GFC drop, which is called the Chebyshev filter. However this filter hasthe periodic transfer characteristic. Finally, curve 2 corresponds to the filter withintermediate GFC and TC, which is called the Butterworth filter. The more detailedinformation about the properties of these and other filters can be found in [6].


Consider the peculiarities of approximation of GFC for these filters. In theory,it is accepted to bring the characteristics of all filters (LPF, HPF, bandpass andbandstop filters) to the common form corresponding to the so-called prototype filter.Usually, the role of the prototype is played by LPF, whose GFC is normalized asfollows: the gain is equal to unity at the frequency ¨D 0 and the cutoff frequencyis taken equal to the relative frequency �D¨/¨cut D 1. Then, transforming thesecharacteristics, we obtain the characteristic of the needed filter: HPF, bandpass orbandstop. The aim of these manipulations is to reduce the variety of GFCs to a singlecharacteristic and maximally facilitate the design of a filter. As characteristics ofthe prototype filter, the mentioned characteristics described by the Butterworth andChebyshev functions [3]

k2.�/ D 1

1C "2�2n; k2.�/ D 1

1C "2Œcosn.arccos.�/�2;

are used. Here � is the normalized frequency, © is the coefficient (©� 1) char-acterizing the nonuniformity in the passband, n D 1, 2, : : : is the function order,T n (x) D cos(n arccos(x)) is the Chebyshev polynomial of the n-th order.

When designing a filter, the absolute value of the complex transfer characteristic is taken ask(˝). Consequently, the Butterworth and Chebyshev functions are squared absolute valuesof the transfer functions or, what is even simpler, the squared GFCs. The Butterworth andChebyshev functions and their order can be found from analysis of requirements to GFCof the designed filter (GFC uniformity in the passband and suppression of signals in thestopband).

� Assume that the designed filter must be described by the third-order Butterworthfunction k2.�/ D 1

1C�6 : Determine the form of the complex or operator functionof the designed filter. It is known that k2(�) D K(j�) K(�j�), then at �D s/j wehave

k2.�/D 1

1C .s=j /6D 1

1 � s6D 1

.s3 C 2s2 C 2s C 1/� 1

.�s3 C 2s2 � 2s C 1/:

The obtained equation is the product of two terms; the second term is the complexconjugate to the first one and has poles in the rights half-plane. To find the equationfor the normalized operator function of the filter, we should exclude this term.Rejecting the complex-conjugate part, we obtain

K.s/ D 1

s3 C 2s2 C 2s C 1D 1

.s C 1/.s2 C s C 1/D 1

s C 1� 1

s2 C s C 1:

This equation is the product of the first-order and second-order operator transferfunctions, and it demonstrates one of the possible versions of realization of the third-order filter through series connection of the first-order and second-order LPFs.


Finally, upon the substitution s D j�D j¨/¨cut D j¨£, we get the equation for thesought complex transfer function of the third-order Butterworth filter:

K.j!/ D 1

1C j2!� � 2!2�2 � j!3�3;

where £ is the time constant of the filter.Creating the Chebyshev filter, it is take into account the representation of the

Chebyshev function through well-known polynomials [3]:

Tn.x/ D cos.n arccos.x/ D anxn C an�1xn�1 C � � � C a1x C 1;

where a are the coefficients of the polynomial, n is the polynomial order. The firstpolynomials have the form:

T0.x/ D 1; T1.x/ D x; T2.x/ D 2x2 � 1; T3.x/ D 4x3 � 3x:

The obvious relation between them can be described by the following recursionequation:

TnC1.x/ D 2xTn.x/ � Tn�1.x/:

As x changes from �1 to C1, the Chebyshev function varies within �1 to C1,passing through zero point n times and taking the extreme values n C 1 times in turn.Outside this interval of x, that is, at jxj> 1, the functions are undefined, and they arereplaced by the equation T n (x) D ch(n arch(x)), which can be easily derived fromthe basic one, using the relation between the trigonometric and hyperbolic functions.

� As applied to the problems of approximating GFC of a filter of, for example,third order, the Chebyshev function has the form

k2.�/ D 1

1C "2.4�3 � 3�/2 D 1

1 � 9"2s2 � 24"2s4 � 16"2s6 :

Find the form of the normalized transfer function of the Chebyshev filter at©D 1/4:

k2.�/ D 1

s3 C 1; 512s2 C 1; 89s C 1� 1

�s3 C 1; 512s2 � 1; 89s C 1:

Rejecting the second factor, we obtain the normalized operator and complextransfer function of the third-order Chebyshev filter

KCh.s/ D 1

s3 C 1:512s2 C 1:89s C 1D 1

s C 0:756� 1

s2 C 0:756s C 1:32I


Fig. 3.35 Generalized circuits of active RC filters with negative (a) and positive (b) feedback

KCh.j!/ D 1

1C j1:89!� � 1:52!2�2 � j!3�3 :

Comparing the equations for the transfer functions of the Butterworth and Cheby-shev low pass filters, we can see that they have similar forms and close coefficients,but the characteristics of the filters are significantly different (Fig. 3.34).

The LPF characteristic can be transformed into GFC of any other filter through conversionof frequencies. The easiest case is transformation of the LPF characteristic into the HPFcharacteristic. This transformation is accomplished through substitution of 1/˝ for thefrequency ˝ or 1/p for p. As applied to the considered third-order Butterworth LPF, thistransformation brings the following equation:

KHPF.s/ D 1

.1=s/3 C 2.1=s/2 C 2.1=s/ C 1D s3

s3 C 2s2 C 2s C 1:

The transformation of LPF into the bandpass and bandstop filters is performed in thesimilar manner, but with more complicated substitutions:

s0 D 1

ƒ�

�

s C 1

s

�

for the bandpass filter;

s00 D ƒ�

s C 1s

for the bandstop filter,

where �˝D˝max –˝min is the bandwidth.

Consider now the filter realization. Depending on the elements, all filters can beclassified into passive and active ones. Passive filters are created only in passiveelements: resistors, inductances, and capacitors, that is, R, L, and C elements. Theycan be divided into RC, RL, and LC filters. Active filters are created in additionto RLC elements, active amplifying elements, such as transistors and op-amps. Inthe literature, the term active filter is most often applied to RC filters with op-amps. Therefore, in what follows, we will consider just active RC filters constructedinop-amps.

The generalized circuit of an active filter in op-amps often coincides with thegeneralized circuit of the linear device in op-amps on Fig. 3.1 or 3.2 with variousRC chains used in place of the two-ports A and B. However, other circuits havegained the widest utility (Fig. 3.35). The first of these circuits is made in op-amps


Fig. 3.36 Second-order active RC low pass filters

Fig. 3.37 Second-order active RC filters: HPF (a), bandpass (b), and bandstop (c) filters

with negative feedback, since the RC chain is connected between the output and theinverting input of the op-amp. In the second circuit, the op-amp is encompassed bytwo feedback loops: negative resistive feedback and positive frequency-dependentfeedback.

The filter with negative feedback includes an RC chain with two inputs: oneconnected to the op-amp input and another connected to the op-amp and filteroutput. The output of the RC chain is connected to the inverting op-amp input.

3.7 Conclusions 77

In the circuit on Fig. 3.35b, the output of the RC circuit is connected to thenoninverting input of the op-amp. As a result, the RC chain turns to be a part of thepositive feedback loop.

Examples of second-order low pass filters constructed in these circuits [5] areshown on Fig. 3.36. These filters were taken as examples, because realization ofhigher-order filters assumes cascade connection of just first-order and second-orderfilters. Thus, low pass filters of any even order can be assembled of these circuits.In the case of an odd order, a first-order integrating RC chain should be added.Figure 3.37 demonstrates the circuits of HPF, bandpass and bandstop filters.

3.7 Conclusions

In the linear op-amp devices the necessary characteristics are received duefeedbacks. The characteristics of linear devices are independent of the op-ampparameters.

Almost all the variety of linear devices created on op-amps can be reduced toseveral generalized circuit designs, some of which are shown on Figs. 3.1, 3.2,and 3.34.

Questions

1. What are the main properties of a linear electronic circuit?2. How to find the gain of an inverting scaling amplifier?3. How to find the gain of a noninverting scaling amplifier?4. What parameters determine the error in the gain of a scaling amplifier?5. What is the highest rate of variation of the output voltage of a scaling amplifier?6. What circuit can you propose for a summator with high input resistance?7. What op-amp is better to provide for the variable (controllable) gain of a scaling

amplifier with the frequency properties kept unchanged?8. What parameters determine the rate of variation of the output voltage for an

integrating amplifier?9. What can be the highest rate of variation of the output voltage for an integrating

amplifier?10. Using the Electronic Workbench, determine the voltage at the output of an

inverting (noninverting) scaling amplifier, if the input voltage is zero, andcomment the obtained result.

11. Propose a circuit for an integrating (proportional-integrating, differentiating, orany other) device constructed in op-amp. Assemble the circuit using ElectronicWorkbench. Using computer simulation, prove its integrating (proportional-integrating, differentiating) properties.

12. What is the difference between the Butterworth and Chebyshev filters?13. How to determine the operator function of HPF from the operator function of

LPF?


14. Propose a circuit of the Butterworth active RC low pass filter (HPF, bandpassfilter, bandstop filter). Assemble the circuit using Electronic Workbench. Usingcomputer simulation, prove its filtering properties.�

15. How to obtain the operator function of a bandpass or bandstop filter from theoperator function of LPF?

16. An active RC low pass filter is a linear device, is not it?

Test Yourself

1. What is the gain of the inverting amplifier?

(a) K D �R1/R2;(b) K D �R2/R1;(c) K D �(R1 C R2)/R1;(d) K D (R1 C R2)/R2.

2. Determine the output voltage in the circuit of the current amplifier.

(a) Vout D IinR;(b) Vout D �IinR;(c) Vout D �Iin/R.

3. Determine the gain of the noninverting amplifier.

(a) K D �3;(b) K D 2;(c) K D 3;(d) K D 4.

3.7 Conclusions 79

4. Determine the output voltage of the amplifier.

(a) Vout D �2 V;(b) Vout D C3 V;(c) Vout D C2 V;(d) Vout D �4 V.

5. Determine the voltage at the first output of the amplifier (Vout1).

(a) Vout1 D C10 V;(b) Vout1 D �10 V;(c) Vout1 D C7 V;(d) Vout1 D C8 V.

6. Calculate the highest positive output voltage of the summing amplifier.

(a) Vout m D C2 V;(b) Vout m D �3 V;(c) Vout m D C4 V;(d) Vout m D �6 V.


7. Meander is applied at integrator the input. What is the form of the output voltage?

8. Determine the type of the filter and justify your choice.

(a) LPF;(b) HPF;(c) bandpass filter;(d) bandstop filter.

References

1. Tietze, U., Schenk, Ch: Halbleiter-Schaltungstechnik. Springer, Berlin/Heidelberg/New York(1980)

2. Wangenheim, Lutz v: Active RC-Filters and Oscillators. Technosphera, Moscow (2010)3. Horovitz, P., Hill, W.: The Art of Electronics. Cambridge University Press, New York (1989)4. Sigorskii, V.P., Petrenko, A.N.: Principles of Theory of Electronic Circuits. Tekhnika, Kiev

(1967) (in Russian)5. Wangenheim, L.: Aktive Filter und Oszillatoren: Entwurf und Schaltungstechnik mit integrierten

Bausteinen. Springer, Berlin (2007)

Chapter 4Nonlinear Devices in Op-amps

Abstract The main objective of the Chap. 4 theory is to acquaint the readers withthe peculiarities of functional devices construction performing nonlinear operationswith input signals: limiting, logarithmation, rectification of alternating voltages, etc.

To become familiar with the material below, the readers should have a clearknowledge about these units within the course “Electronics” or “Electronics inInstrument Making.”

After learning this Chapter, the readers will know the operation of variousnonlinear devices and can determine the form of the functional transformationperformed by them.

4.1 Introduction

Nonlinear devices are the electronic devices performing such operations with input signals,at which the dependence of the output voltage (or current) on the input voltage (orcurrent) is described by the function other than linear. This class of devices includesa voltage comparator, a voltage limiter, a logarithmic device, a device determining theabsolute value of the input signal, etc.

If linear devices used a limited set of elements: op-amps, resistors, capacitors,and inductances, then nonlinear devices additionally use semiconductor diodes,transistors, nonlinear resistors, etc. This strongly complicates analysis of thenonlinear devices.

As was already noted, when designing and analyzing nonlinear devices, the linear methodsof analysis such as superposition principle, operator method, etc. are inapplicable with rareexception.

So the design of nonlinear devices is a real complicated. Nevertheless, nonlineardevices in op-amps are widely used for limiting and rectification of alternatingvoltages, transformation of the signal shape and spectrum, and for other purposes.The wide application of such devices necessitates the study of the general principlesof their construction.


81

82 4 Nonlinear Devices in Op-amps

Fig. 4.1 Common circuit of the nonlinear operational device

Consider one of the generalized circuits of nonlinear devices on Fig. 4.1. Inthis circuit, A and B are nonlinear elements. Determine the gain characteristic ofthe generalized circuit. The nonlinear elements can be described by the nonlineardependences:

Iin D h .Vin/ ; Ifb D g .Vout/

where h and g are known functions. Then, taking into account that Iin D �Ifb andapplying the inverse functions, we obtain

Vout D g�1 Œ�h.Vin/� : (4.1)

Recall that the inverse functions are the functions satisfying the followingconditions:

V D g�1Œg.V /� or V D gŒg�1.V /�:

As applied to the generalized circuit, these conditions mean that if the elementswith the identical nonlinear dependences are used as the nonlinear elements Aand B, then the output voltage is equal to the input voltage with the oppositesign. Examples of the mutually inverse functions are sin and arcsin, tan andarctan, ln and exp, etc. For example, for them it is known, that V D sin(arcsinV)or V D arcsin(sinV).

Unfortunately, not every function has the inverse one. Nevertheless, this equationis one of the few equations allowing derivation of the analytical equation for theoutput voltage, if the input voltage and the functions g and h are known. This willbe demonstrated below with the examples of particular nonlinear devices. Howeverthe consideration of nonlinear devices starts from the simplest one, without elementB and with ordinary resistor as the element A. It is a voltage comparator.

4.2 Voltage Comparator 83

4.2 Voltage Comparator

The voltage comparator is intended for comparison of two input voltages V1 and V2. In theideal case, the output voltage of the comparator takes two values. The comparator outputvoltage is maximal and positive when the voltage V1 is higher than V2; otherwise, the outputvoltage is minimal and negative.

This relation between the input and output voltages is described by the signfunction Vout D Vout m sign (V1 – V2), that was shown on Fig. 2.3e in Chap. 2.Certainly, this equation and the plot represent the ideal voltage comparator. Theoperational amplifier realizes this function not exactly. Equation 1.1 becomes thesign function only in the limit at infinitely large k and no input offset voltage.

Vout D limk!1; Voff!0

V out m th

�

kV1 � V2 C Voff

'T

�

D Vout m sign.V1 � V2/

Nevertheless, op-amps are widely used as voltage comparators. The ways of op-amp improvement for this application are obvious: increase of the gain, decrease ofthe input offset voltage, decrease of the input current, etc.

Figure 4.2 shows the circuit of op-amp connection in the mode of voltagecomparator.

The input voltages are applied to the inverting and noninverting op-amp inputs.At the difference of the input voltages larger than V00 the output voltage takes themaximal positive value (CVout m), and at the difference smaller than V 0 it takes theminimal negative value (�Vout m). If the voltage difference is between these twovalues, then the output voltage depends on this difference. As the gain increases,the difference between V00 and V 0 tends to zero, for example, at K D 1,000,000 itis 10–15 �V. Thus, the op-amp comparator not only compares two voltage signals

Fig. 4.2 Typical comparator circuit (a) and its gain characteristic (b)


and determines which one is greater, but also (in a narrow range) outputs the signalproportional to the difference of the input voltages. This is both its advantage anddisadvantage, because within this range it is sensitive to various noises causingfalse response. But this disadvantage can be easily eliminated, if we introducea positive feedback into the op-amp. In such case we have the comparator withhysteresis. This comparator will be considered in the more detail in Chap. 6, whereone of its applications in the circuit of a generator’s threshold element will bediscussed.

Recently in connection with development data acquisition and digital processingsystems, the specialized circuits for connection with digital devices have arisen.They are comparators, whose output signal takes the voltages corresponding to theTTL (transistor-transistor logic) or ECL (emitter-coupled logic) logical levels. Infact, now the term “comparator” is associated just with these devices.

The widely used comparator circuits include the LM111, LM211, LM311 circuitsproduced by National Semiconductor. In the circuit design, they differ only slightlyfrom ordinary op-amps. However, there are some differences in the input and outputstages. For example, the output stage allows operation with the input voltages, equalto or even lower than the negative voltage of the power supply. The output stageoften includes an “open” collector, which ensures the matching with all types ofdigital integrated circuits. The latest models of integral comparators include built-indigital logic for processing of the output signals and triggers for operation in theselection–storage mode. An example is a fast-response precision comparator withTTL output and a trigger of the RC4805 or 1165CA1 type. The parameters andcharacteristics of the comparators (mostly corresponding to the op-amp parametersand characteristics) are now continuously improved: the accuracy and the speedincrease, and the functional capabilities are extended.

4.3 Logarithmic Amplifier

Logarithmic amplifiers are intended for obtaining the output voltage proportional to thelog input voltage. They are used in companders and expanders of signals (devices forcompression and expansion of the dynamic range of input signals at magnetic recording),in noise suppression systems, in voltage multiplication systems, etc.

Theoretically, the logarithmic dependence (Fig. 4.3) is determined by thefollowing equation: y D log a x. At a D e we have y D lnx.

In the log scale, the plot is a straight line. This peculiarity is often used toestimate the errors of real logarithmic devices through comparison of experimentalcurves with a straight line. The error of logarithmation is estimated by the deviationfrom the straight line. In the equation for the logarithmic function, y and x aredimensionless parameters. However, the input and output voltages in electrondevices are measured in volts, so some constants, in volts, should necessarily beincluded in this equation. Accordingly, the equation for the theoretical logarithmic

4.3 Logarithmic Amplifier 85

Fig. 4.3 The logarithmic dependence in the linear (a) and log (b) scales

Fig. 4.4 Logarithmic amplifier (a) and its gain characteristic (b)

dependence of a device should have the form Vout D V0 ln(Vin/V0). The function lncan be realized by Eq. 4.1 in two kind:

g D .lnx/�1 D ex; h D a;

g D b; h D ln.x/:

The first version is preferable, because semiconductor diodes have the nonlineardependence close to the exponential one.

I D I0�eV=m 'T � 1

� � I0eV=m 'T ;

where I0 is the thermal current, ® T is the temperature voltage, V is the diodevoltage; m is a factor depending on the semiconductor material.

Shottky diode is characterized by the practically better exponential volt-amperecharacteristic in a wide current range (within five to six decades). According toEq. 4.1, the output voltage for the circuit of the logarithmic amplifier with asemiconductor diode on Fig. 4.4 is:

Vout D �m 'T ln

�Vin

I0R1

�

:


Fig. 4.5 Logarithmic amplifiers with transistor

This equation is obviously meaningful only at the positive values of the inputvoltage, because the argument of the log function can be only positive. Therefore,the effective range in the gain characteristic is the fourth quadrant (Fig. 4.4b).

The disadvantage of this amplifier is the limited input voltage range, in which thelogarithmic dependence keeps true. It is caused by the ohmic resistance of the diodeand the current dependence of the coefficient m (dashed line on Fig. 4.4b). The muchwider voltage range can be achieved using transistors. In transistors, the currentdependence of the coefficient m is compensated for by the inverse dependence of’ (transfer coefficient). As a result, the exponential dependence keeps true in thecurrent range from picoamperes to milliamperes. The output voltage in this case hasthe form:

Vout D �'T ln

�Vin

I0ceR1

�

:

The circuits of transistor amplifiers are shown on Fig. 4.5. However, thecapabilities of these circuits are difficult to implement because of the probability ofunstable operation and self-excitation due to the increase of the loop gain becauseof the high amplifying properties of the transistor.

When used, this circuit is corrected. Figure 4.5b shows one version of thelogarithmic amplifier with correction [1].

The correction by capacitor C c consists in creation of feedback at the highfrequencies of the parallel arm in the circuit, by-passing the transistor, whichsignificantly increases the amplifier stability. The additional resistor R3 improves theop-amp operation, because the op-amp load on Fig. 4.5a is the low input resistanceof the transistor, which is unacceptable for most op-amps. In addition to these


Fig. 4.6 Logarithmic amplifier with temperature compensation for error

elements, the circuit includes the diodes VD1, VD2 and the resistor R2 for overloadprotection of the transistor and the op-amp.

The common disadvantage of the considered logarithmic amplifiers is theirtemperature instability caused by the effect of the temperature voltage ® T , which,as known, is directly proportional to the temperature. The temperature effect canbe weakened through the use of thermostating or temperature compensation of thetransistors.

Thermostating is carried out by placing transistors in a passive or activethermostat. Temperature compensation is performed by applying different elementswith the analogous temperature dependence. The first way is rather expensive, so itis applied only in important cases, while the second is more practically feasible,because analogous transistors are suitable for temperature compensation. Veryoften temperature compensation is performed with integral pairs of transistors, forexample 1NT591, which are made almost identical, that is, with the same parametersand temperature characteristics, due to the use of the corresponding technology [2].Figure 4.6 shows the circuit with thermal compensation including the integral pairof transistors.

At the high op-amp gain, the collector currents of the transistor are

Ic1 D Vin

R1; Ic2 D Vref

R5;

and their ratio is

Ic1

Ic2D e

V1'T ;

where

V1 D VoutR4

R2 CR4:


Fig. 4.7 Voltage multiplier on logarithmic amplifiers

Then the output voltage is determined as

Vout D �'T R2 CR4

R4ln

�R5Vin

R1Vref

�

D �a ln

�Ic1

Ic2

�

:

This logarithmic amplifier without the resistors R1 and R5, but with twoadditional op-amps was realized in the integrated-circuit form [2, 3] in the Log102circuit of Texas Instruments. The logarithmic dependence in this circuit keeps truein the current range from 1 nA to 1 mA at the error of 0.3%. The scale parameter inthis case is equal to -1 V for the tenfold change of the current ratio.

It can be seen from the last equation that, as before, the scale parameter includesthe temperature potential ® T, which increases with the growing temperature (withall disadvantages of the previous circuits). For the temperature compensation ofits changes, the factor following it should be changed in the opposite direction.This can be done, if we take, as resistor R4, the resistor with the temperaturecoefficient having the opposite sign, that is, the so-called posistor. Selecting thecorresponding posistor R4 and, whenever necessary, shunting it by an ordinaryresistor, it is possible to decrease the temperature error to acceptable values.

The considered circuit creates the basis of the voltage multiplier [2], whosemodernized version is shown on Fig. 4.7.

The operating of the multiplier is based on the known method of performing themultiplication and division operations through logarithmation:


Fig. 4.8 Antilogarithmic amplifier (a) and its gain characteristic (b)

if xy D w; then ln.xy/ D lnx C lny D ln w; eln w D w;

ifw

zD v; then ln

�w

z

�

D ln

�xy

z

�

D lnx C ln y � ln z D ln v; eln v D v:

The logarithmic amplifiers are assembled in OA1–OA3 op-amps, while theantilogarithmic (potentiating) amplifier is assembled in OA4 op-amps. The OA1 op-amp together with the transistor VT1 provides the voltage algorithm Vx (ln x), whileOA2 with the transistor VT2 provides the voltage logarithm Vy (ln y), and OA3 withthe transistor VT3 provides the voltage logarithm Vz (ln z). In its turn, the OA4 op-amp with the transistor VT4 determines the exponential function of the sum of x andy logarithms minus z logarithm, that is, potentiates the value of ln v. The multiplieroutput voltage at the output of the OA4 op-amp is

Vout D Ic4R6; but Ic4 D I0eVeb4'T :

Taking into account that Veb4 D Veb1 C Veb3 � Veb2, we find:

Veb4 D 'T ln

�R8

I0R1R2

VxVy

Vz

�

:

Upon substitution of Veb4 into the equation for the current, we finally obtain:

Vout D VxVy

Vz

R6R8

R1R2:

The multiplier involves the antilogarithmic amplifier including the transistorVT4. Consider its operating principle in more detail. Figure 4.8 shows the circuitof the antilogarithmic amplifier.

The gain characteristic of this device can be easily found from the equation:

Vout D �I0R1�

eVin='T � 1

�� I0R1eVin=

'T : (4.2)


Here the effective range of the characteristic is also the fourth quadrant, becausethe exponential function is always positive and does not alternates the sign, as itsargument (input voltage) alternates the sign. The theoretical dependence is shownby the dashed curve in the figure.

4.4 Operational Rectifiers

In the instrument making, instrumentation, and electronics, it is often needed toconvert the alternating voltage into the direct one, for example, in power supplies.For this purpose, passive rectifiers in semiconductor rectifier diodes are used.Operational rectifiers are used for construction of high-accuracy sensors.

Mathematically, the rectification operation is formulated quite simply: Vout.t / D jVin.t /j,where the symbol j j denotes the modulo operation and, as in terms of electronics, theconversion of the bipolar voltage into the positive unipolar voltage.

The operation of rectification often arises in connection with determination ofthe half-period average value of the alternating voltage by the equation

Vout.t/ D 1

T

TZ

0

jVin.t/j dt;

where T is the interval of determination of the half-period average voltage or theperiod of the alternating voltage.

The converters of the half-period voltage are used in voltmeter of alternatingvoltage, in particular, digital voltmeters.

Let us analyze rectifiers starting from the simplest circuit of the half-waverectifier (Fig. 4.9) to consider the rectifier features and possible errors with thisexample.

Figure 4.10 shows the time plots and the spectrum of the input and outputvoltages. At the positive input voltage (curve 1 on Fig. 4.10a), the rectifier diodeopens and the input voltage comes to the output. According to the Kirchhoff law

Fig. 4.9 Simplest rectifier (a) and its gain characteristic (b)

4.4 Operational Rectifiers 91

Fig. 4.10 Diagrams of input and output voltages of the simplest rectifier (a) and their spectra (b)

Vin D V VD C Vout, the output voltage (curve 2) is lower than the input voltage(curve 3) by the diode voltage drop (in the figure, it is denoted as V VD ). The negativevoltage applied to the input blocks the diode. However, the reverse current passingthrough the diode creates a small voltage drop�V at the resistor.

The spectrum on Fig. 4.10b confirms again that a nonlinear circuit enriches thespectrum of the output signal. Now the spectrum includes higher harmonics and theconstant component, which are not present in the input signal. The solid curve onFig. 4.10a is the plot of the output voltage of the real rectifier, while the dashedcurve is for the ideal one. It can be seen that they differ by the voltage drop V VD atthe positive voltages and by �V D I0R at the negative voltages. Just these voltagedrops determine the error of rectification. So it can be concluded that to decreasethe error, it is necessary to decrease the effect of the diode voltage drop at theforward bias and to decrease the reverse current through the diode at the backwardbias.

The spectrum of the input signal includes only one component with the frequencyf0, while in the spectrum of the output signal there are the constant component andthe higher harmonics. The constant component has the same value as the constantcomponent on Fig. 4.10a. Just it is the useful component of the transformation, allother components are not needed. Filtering can decrease them, but this slows downthe rectifier.

Thus, two problems follow from analysis of the simplest rectifier operation: theincrease of the accuracy of transformation of the alternating voltage into the directone and the decrease of the spurious spectral components. The solution of the formerwill increase the accuracy of AC voltmeters, while the solution of the later willincrease their speed.


4.5 Full-Wave Operational Rectifiers

Connecting the diode into the op-amp feedback circuit on Fig. 4.11, it can reducethe error by the diode voltage drop. In this case, we obtain the circuit of an activeoperational rectifier.

Its operating principle can be understood from the corresponding plots of theinput and output voltages on Fig. 4.11c. After amplification, the positive half-waveof the input voltage becomes negative at the output, opens the diode VD2 and blocksVD1. The negative half-wave (shown by the dashed curve in the plots) with theamplitude Vout2 m D �Vin mR3/R1 is formed at output 2. The diode VD1 in this half-period is blocked, and it transmits a low reverse current, which creates a smallvoltage drop at the resistor R2 comparable with that in the passive rectifier. In thenext half-period, the diode VD1 is open, and the positive voltage half-wave (shownby the solid curve in the plots) with the amplitude Vout1 m D �Vin mR2/R1 is createdat output 1. At the equal resistances R2 D R3, the amplitudes of the half-waves areequal. If the output voltage is the voltage between outputs 1 and 2 (symmetricoutput), then it is double-wave, whose both half-waves have the same polarity,because Vout3 D Vout1– Vout2.

Figure 4.11b shows the spectrum of the input voltage and the output voltagesVout2 and Vout3. The spectrum of the output voltages include the constant componentand the harmonics, and at output 3 the constant component is twice as high and the

Fig. 4.11 Circuit of operational rectifier (a), voltage plots (c) and spectrum (b)

4.5 Full-Wave Operational Rectifiers 93

first harmonic is compensated for, because the voltage period is halved. The absenceof the first harmonic in the spectrum allows the time constant of the low-pass filterto be decreased and, consequently, the rectifier speed to be increased.

If the op-amp with the VCVS structure is used, the currents through the diodesVD1, VD2 and the resistors R2 and R3 at the high op-amp gain are determined onlyby the input voltage Vin and the resistance of R1. They are independent of the diodeparameters, identical, and equal to the input current in the corresponding periods.These currents induce the voltage drop at the feedback resistors, whose profilealmost coincides with that of the input voltage and, naturally, which is independent,in the first approximation, on the diode threshold voltage. The more detailed analysisreveals that the effect of the voltage drop decreases 1 C K“D F times, where F isthe amount of the feedback. Certainly, at the large coefficients this effect can beneglected. Thus, the feedback efficiently reduces the error by the voltage drop at thediodes.

Unfortunately, the situation with the reverse diode current is not so simple. Whenone of the diodes is open, another transmits the reverse current. A low voltage nearthe abscissa on Fig. 4.11 shows the effect of the reverse current. The reverse currentof one diode not only induces the parasitic voltage drop �V, but also decreases theamount of the feedback for another (open) diode, so the effect of the reverse currentis not eliminated in this case.

The same disadvantage is also inherent in the circuits of full-wave rectifierswith the asymmetric output. The full-wave rectification mode in the first of them(Fig. 4.12a) is achieved by applying the input voltage through the resistor R3 to theoutput, and then at R3 D R2 D 2R1 the both half-waves have the same amplitudes,and the amplitude gain is

K D jVout m =Vin mj D R2 =R1 D R2 =.R2 CR3/ D 1=2:

In the second circuit (Fig. 4.12b), the full-wave mode is achieved at R3 D R1 D 2R2.The amplitude gain for this circuit is

K D jVout m =Vin mj D R2 =R1 D R3 =.R1 CR3/ D 1=2:

If we take R3 � R1 D R2 in the first circuit and R3 � R1 D R2 in the secondone, then K D 1 for the both ones. In the third (bridge) circuit, the output is thecurrent passing through the load resistor R n . Often a pointer-type instrument,microammeter – is used as a load. Then its current with the sine profile is equalto I D 2 =� jVin =R1j ; that is, almost strictly proportional to the input voltage. Thisequation accounts for the fact that the pointer-type instrument integrates the currentpassing through it and separates only the constant component of the current. In thelast circuit on Fig. 4.12d, the full-wave mode is satisfied in every output.

The error component caused by the reverse current can be reduced by decreasingthe reverse voltage (the reverse current in such a case decreases as well) ordecreasing the reverse current itself. To reduce the reverse voltage, a possible wayis to apply the rectifier circuit with two diodes connected in series on Fig. 4.13.


Fig. 4.12 Operational full-wave rectifiers with asymmetric output

Fig. 4.13 Operational rectifier with extra diodes (a) and the plots of variation of the reversecurrents (b)


Fig. 4.14 Transistoroperational rectifier

In this circuit, the extra diodes VD3 and VD4 are connected in series with therectifier diodes VD1 and VD2, respectively. The rectifier operates similarly to thatshown on Fig. 4.11, but the voltage applied to the blocked main diodes is lower thanin the previous circuit, because it is equal to the voltage drop at the resistors R4 andR5 from the reverse current through the extra diodes VD3 and VD4. Figure 4.13bshows the corresponding plots of the reverse voltages and reverse currents. Thedashed curve shows the voltage and current plots in the previous circuit, andthe solid curve are for the circuit under consideration. One can see that the reversecurrents are significantly lower. Thus, the application of extra diodes can reduce theeffect of the reverse currents on the ten times.

More significant reduction of the reverse currents can be achieved, using, forexample, complementary transistors connected in the circuit of emitter followers inplace of the diodes [3]. The circuit on Fig. 4.14 operates as an ordinary operationalrectifier and is characterized by the same plots. At the positive half-wave of theinput voltage, the negative half-wave is formed at the op-amp input. This half-waveis followed by the emitter follower constructed in the transistor VT2 and the resistorR5. At the same time the transistor VT1 is blocked. It is known that in the blockedstate the emitter reverse current is much lower than the collector inverse current;therefore, its effect on the error is insignificant. Ie0 D Ic0/“. In the other half-period,the follower in the transistor VT1 and the resistor R4 is in operation.


In addition to the reduced effect of the reverse current, the advantages of thetransistor rectifier are the low output resistance and the increased output current,which are ensured by the emitter follower.

Figure 4.15 shows some versions of the circuits of full-wave operational rectifiers[4], which also have low output resistance, because in them the output signal of therectifier is formed at the op-amp output.

All circuits on Fig. 4.15 have an important common advantage, namely, the lowoutput resistance. It permits a relatively low-resistance load allowable for op-ampsof this type to be connected to them.

In the circuit 4.15a designed of two half-wave rectifiers with connected outputs,the output resistance is determined by the output resistance of some or other rectifierdepending on the half-wave of the output voltage. The gain at R1 D R2 is equal to 1.

The circuit on Fig. 4.15b is designed in the half-wave rectifier with the op-amp OA1 and the follower with the op-amp OA2. The output voltage is formedin two half-periods. At the positive input voltage, it comes to the output throughthe resistor R2 and the voltage follower without amplification. The negative inputvoltage opens the diode VD2, connecting the input of the op-amp OA2 to the outputof OA1. Thus, the both op-amps are encompassed by the common feedback throughthe resistor R3. The output voltage again has the positive polarity with the gain –R3/R1. To provide for the equal output voltages regardless of the polarity, the conditionR1 D R3 must be met. The gain here is equal to unity as well.

In the circuits 4.15c, d, and f designed by the same basic principle (rectificationand summation–subtraction of signals), the rectifier is realized in the OA1 op-amp,and the adder is realized in the OA2 op-amp. For the half-waves to be equal onFig. 4.15c, the conditions R2 D R3, R4 D R5, R6 D R7 should be met, then the gainis K D jVout/Vinj D (R2/R1) (R7/R4). In the circuit on Fig. 4.15d under the conditionR1 D R2 D R3 D R4/2, the gain is K D 1, and in the circuit on Fig. 4.15f K D R5/R2

at 2R1R4 D R2R3. In the circuit on Fig. 4.15e the positive voltage comes to the inputof the OA2 op-amp through the OA1 op-amp and the diode VD2 and then to therectifier output. At the same time, the output voltage comes back to the input ofthe OA1 op-amp by the feedback circuit through the resistors R4, R2, R1. Thus, thefeedback loop includes two op-amps. As a result, the gain is K D 1 C (R2 C R4)/R1.The negative input voltage opens the diode VD1 and blocks the diode VD2, andthe voltage from the inverting input of the OA1 op-amp (equal to the input voltage)comes through the inverting amplifier in the OA2 op-amp to the output with thegain –R4/R2. From here we can easily find the condition of the equal amplitudes ofthe output voltages 1 C (R2 C R4)/R1 D R4/R2, which, for example, transforms intoR2 D R4 at R1 ! 1.

The circuit on Fig. 4.15g occupies a particular place, because it combines thepassive and active rectifiers. The passive one is designed in the VD2 diode and theresistors R2 and R5, while the active one is designed in the op-amp, the diodes VD1,VD3, and the resistors R1, R3, and R4. At the positive voltage, the diode VD2 isopen, and the input voltage after amplification opens the diode VD3. At the negativeinput voltage, the diodes VD1 and VD3 are open, while the diode VD2 is blocked. Inthis case, the feedback loop of the op-amp includes the serially connected nonlinear


Fig. 4.15 Full-wave operational rectifiers with low output resistance


Fig. 4.15 (continued)

chains of the resistor R1, VD1, as well as R4, VD3. At the equal resistances of theresistors and identical characteristics of the diodes, the output voltage copies theprofile of the input one with inversion.

It is appropriate giving some explanations here. For this purpose, the circuitcorresponding to the negative input voltage is shown separately on Fig. 4.16. Thenonlinear circuits are circled by the dashed line.

At the equal resistors and the identical volt-ampere characteristics of the diodes,the volt-ampere characteristics of the circuit are described by the same nonlinearequations. According to Eq. 4.1, the output voltage is equal to the input one, butwith the opposite sign. Actually,

Vout D g�1 Œ�f .Vin/� D g�1 Œ�g.Vin/� D �Vin at f D g:

4.6 Voltage Limiters and Overload Protection Circuits 99

Fig. 4.16 Operationalrectifier (Fig. 4.15g) atnegative polarity of the inputvoltage

Under the conditions above, the feature of the circuit on Fig. 4.16 is thatthe nonlinear electrical circuit has a linear relation between the input and outputvoltages. If the input signal of any shape is fed to this circuit, then the output signalcopies it with the opposite sign, as in the ordinary linear inverting amplifier. Forexample, if the input voltage is a sine wave, then the output voltage is a sine wavetoo. Certainly, this is valid at the strict equality of the resistances and the identicalvolt-ampere characteristics of the diodes. Naturally, if these conditions are violated,the nonlinear character of the circuit manifests itself, and the larger is the deviationfrom these conditions, the more significant are the manifestations.

Thus, to ensure the equality of the gains for the positive and negative voltages, itis sufficient to satisfy the condition R1 D R2 D R3 D R4 D R. Then Vout D �Vinj.

4.6 Voltage Limiters and Overload Protection Circuits

Voltage limiters, as follows from their title, are intended for formation of voltages withinsome preset limits. The need in voltage limiting arises in many cases: in protection ofelectron devices against over voltage and incorrect connection of power supplies, innonlinear correction of automatic regulators, in reproduction of voltages of a certain shape,in generation of voltages with a preset spectrum or harmonic content, etc.

No one instrument or household electrical device is produced without voltageand current limiters. Limiters play the secondary, but important role. Very oftenengineers understate their significance, but application of simplest limiters asprotectors improves the reliability of devices and economizes considerably thematerial and financial expenses on repair. It is appropriate mention that now manyintegrated circuits, in particular, op-amps, are fabricated with limiters of inputvoltages and output currents (protectors).

The term “clipper” is very closely related to limiters. Clipping circuits keep avoltage from exceeding some preset value, while limiters restrict the voltage to aspecific range. Figure 4.17 shows the gain characteristics of clippers and limiters.All of the curves are made of straight lines.


Fig. 4.17 Amplitude characteristics of voltage limiters: positive-peak clipper (a), negative-peakclipper (b), limiter (c), clippers with a bias (d, e), and “dead zone” (f)

Fig. 4.18 Voltage limiters (a) and clippers (b, c) with diodes connected in parallel

Semiconductor diodes, stabilitrons (zener diodes), and, sometimes, transistorsare used as nonlinear elements. Figure 4.18 shows some circuits of diode clippersand limiters.

The principle of voltage limiting is very simple. For example, if the input voltageV1 in the circuit on Fig. 4.18a is less than – Vs1, then the diode VD1 opens andsets the output voltage V2 practically equal to – Vs1, regardless of the input voltagevalue. If the input voltage becomes greater than C Vs2, then the diode VD2 opensand sets the output voltage equal to C Vs2. At the intermediate values of the inputvoltage, the diodes are blocked and the input voltage is fed to the output without

4.6 Voltage Limiters and Overload Protection Circuits 101

Fig. 4.19 Limiter (a) and clipping circuits (b, c) with diodes connected in series

limiting. Changing EMF of the sources Vs, it is possible to change the limitinglevels. Interchanging the diodes and resistors, we can obtain limiter or clippingcircuits with diodes connected in series, as shown on Fig. 4.19.

Stabilitrons are widely used as nonlinear elements in limiters, since they have twoconducting parts in the direct and reverse branches of the volt-ampere characteristic,which opens wider possibilities for construction of various protection circuits andlimiters. Figure 4.20 demonstrates some stabilitron limiters, which are used forprotection of the input circuits of op-amps with field-effect transistors, digitalintegrated circuits, etc.

Certainly, the characteristics above are idealized. For example, Fig. 4.19b showsthe ideal gain characteristic of the clipping circuit. At Vs D 0 the clipper is,essentially, half-wave rectifier, whose real characteristic is shown on Fig. 4.9b.It can be seen that the characteristics are different, especially, at low voltages (lowerthan 1 V).

The circuits of practical applications of limiters/clippers will be considered here startingfrom the circuit for protection of the op-amp input (Fig. 4.21).

To protect the op-amp inputs means to provide for the voltage at the op-ampinputs within some safe range at any polarity of the input signals. In this circuitat any value and any polarity, the voltage across the diodes VD1 and VD2 doesnot exceed the direct voltage drop, which is within 0.7–0.8 V for silicon diodes.Therefore, the voltage between the inputs is also limited at the same level. The gaincharacteristic of the protection circuit is that of the voltage limiter (Fig. 4.21b).


Fig. 4.20 Stabilitron voltage limiters (clippers)

Fig. 4.21 Limiter of input differential voltages of op-amp

Some op-amp protection circuits include no resistors, since they are usuallypresent in the external circuit, or they have two diodes connected in series in placeof one diode. The similar protection circuits are built in the integrated op-amps ofthe following types: 105UD1, 154UD2, 140UD14, LM108 produced by NationalSemiconductor, MC1539 produced by Motorola, and some others. However, they donot protect the op-amp from inphase overvoltage. It is necessary for this to protectall of the op-amp inputs from the voltage, exceeding the supply voltage, which oftendisables the op-amp. The most efficient protection from the inphase overvoltage isprovided by the circuit on Fig. 4.22.

In this circuit, the diodes VD1, VD2 and the resistor R1, as well as the diodesVD3, VD4 and the resistor R2, form the voltage limiting at the level of the op-ampsupply voltage. If the positive voltage at the terminal 1 exceeds the supply voltageVs1, then the diode VD1 opens, and if the negative voltage is less than – Vs2, then the

4.7 Op-amp Function Generators 103

Fig. 4.22 Limiter of op-amp input inphase voltage (a) and its gain characteristic (b)

diode VD2 opens. As a result, the voltage at any input terminal cannot be higher thanVs1 C V VD1 or lower than – Vs2– V VD2. Such a circuit is employed for protection ofthe op-amp inputs in, for example, 1423UD1-3.

The considered limiter is a particular circuit of a voltage limiter, whose is shownon Fig. 4.18a. Passive limiters have all the disadvantages inherent in the rectifier onFig. 4.9. Therefore, they are used in the cases, when the high accuracy is not crucial.If such accuracy is necessary, operational function generators are employed.

4.7 Op-amp Function Generators

Function generators are nonlinear devices intended for realization of complex nonlineardependences between input and output voltages.

Generally, all nonlinear devices considered above can be classified as functiongenerators, because the logarithmic amplifier, rectifier, and limiter convert the inputvoltage into the output one in accordance with some nonlinear dependence. How-ever, this dependence is fixed, unchanged, nontunable, and has a strongly definedfunction. For example, the logarithmic amplifier does not rectify the AC voltage,while the rectifier does not take the signal logarithm, etc. Therefore, the functiongenerator is understood in the scientific and training literature as a such device,which can provide for different nonlinear dependences without any changes in thedevice structure, but changing only the parameters of some elements. Theoretically,the function generator can be a rectifier, a logarithmic amplifier, and other device.Besides, function generators are usually characterized by the high accuracy.

Realization of an arbitrary nonlinear dependence is based on the principle of ap-proximate representation of the needed function by other functions, more convenientfor reproduction and ensuring the needed accuracy. The stepwise (piecewise con-stant) and piecewise linear functions have gained the widest utility in interpolation


Fig. 4.23 Initial function andits piecewise linearapproximation

or approximation of arbitrary functions. The former are widely used in the analog-digital technology, while the latter are applied in analog function generators.

Figure 4.23 shows the plots of the initial dependence and the approximatingpiecewise linear one.

The initial nonlinear function y D f (x) is shown by the solid curve in the figure,and the piecewise linear function is shown by the dashed curve. The latter consistsof segments of straight lines 1, 2, and 3 between the nodes O, A, B, and C. Thenumber and the positions of the nodes are determined from the condition of theminimal approximation discrepancy.

Approximation by the piecewise linear functions is simplicity, basing the piece-wise linear dependence can be formed using simple linear operations: summation,subtraction, multiplication by a constant. Consider possible versions of reproductionof the piecewise linear function.

One of these versions on Fig. 4.24 is based on representation of the piecewiselinear function by a finite sum of segments of straight lines. It is easy to check(Fig. 4.24a) that the piecewise linear function y is formed by summation of thesegments of straight lines y D y1 C y2 C y3. Figure 4.24b demonstrates the diagramof computational operations. The output characteristic y in this structure is formedby summation and subtraction of signals of the output units 1, 2, and 3. Thus, theunit 1 generates the output signal corresponding to line 1 on Fig. 4.23 with the angle’1 and zero bias. The unit 2 generates the signal corresponding to the difference oflines 1 and 2 with the angle ’1– ’2 and the bias x A . The unit 3 reconstructs thecharacteristic with the angle ’1– ’2– ’3 and the bias x B . Then the signals from theunits 2 and 3 are subtracted from the signal of the unit 1 in the adder-subtracter 4.

An advantage of this structure is the possibility of independent change ofthe angles and bias, which makes it suitable for formation of different nonlineardependences with any preset accuracy, because the anyone number of units. It suitswell for realization of functions with limiters and adders. Another words, it can beassembled of typical basic units. Figure 4.25 shows the basic unit [1], which realizesone linear segment.

As to the circuit design, it is a half-wave rectifier with the biased gain charac-teristic. The angle of the straight line is determined by the ratio R2/R1, and the bias


Fig. 4.24 Diagram of the piecewise linear function realization

Fig. 4.25 Basic unit of the function generator (a), its graphical (b), and the gain characteristic (c)

is determined by the polarity and the value of the voltage V, the position of thepotentiometer slider R, and the ratio R2/R3. In practice, V and R3 are constant, whilethe position of the potentiometer slider R is changed. So, regulating R, it is possibleto change the bias from 0 to CV and from 0 to –V (see Fig. 4.25c).

Explain how the nonlinear dependence can be obtained with such basic unitsusing the function generator on Fig. 4.26. This circuit involves two basic units inOA1 and OA2, which form the parts 2 and 3 of the gain characteristic, and an addercircuit in OA3, while the resistor R10 forms the part 1.

Changing the voltages V1, V2 and resistances R1–R3, it is possible to receivemany various nonlinear functions with this generator. Some of them are shown onFig. 4.26b and in Table 4.1. The first column of Table 4.1 gives the recommendedresistance and voltage values for generation of the dependence specified in the


Fig. 4.26 Function generator (a) and its gain characteristic (b)

Table 4.1 Variants of functional converters

Voltages and resistances Functional dependenceDevice realizing thisdependence

V1 D � 5 V, V2 D C5 V;R3 D R4 D R5 D R6 D R7 D R8

D R9 D R10 D R11 D R12

See Fig. 4.18a Noninverting limiter

V1 D C 5 V, V2 D �5 V;R3 D R4 D R5 D R6 D R7 D R8

D R9 D R11 D R12

Dependence inverse to thatshown on Fig. 4.18a

Inverting limiter

V1 D C5 V, V2 D �5 V;R4 D R5 D R6 D R7 D R8

D R9 D R11 D R12; R10 D 1See Fig. 4.17f Limiter with a dead zone

V1 D �5 V, V2 D C5 V;R3 D R4 D R5 D R6 D R7 D R8

D R9 D R11 D R12

Piecewise linear sine wavedependence from �90ı

to C90ı

Sine wave voltage generator

V1 D C3.33 B, V2 D �3.33 V;R3 D R4 D R5 D R6 D R7 D R8

See Fig. 4.26b Frequency tripler of triangularvoltage with the amplitudeof 10 VR10 D 2R9 D 2R11, R12 D 3.33R10


Fig. 4.27 Full-wave operational rectifier on the AD8036 chip (a) and its gain characteristic (b)

Fig. 4.28 Full-wave rectifier on the AD8036 chip (a) and its gain characteristic (b)

second column. The last but one row shows how the piecewise linear approximationof the sin function can be obtained. The last row presents the conditions forformation of the piecewise linear dependence, which triples the frequency of thetriangular input signal having the amplitude of 10 V with the pulse shape kepttriangular. The accuracy of formation of the piecewise linear dependences can beincreased by enlarging the basic units number.

Various function generators can be constructed based on the clamping amplifiersconsidered in Chap. 1, for example, AD8036 amplifier. Figures 4.27 and 4.28 showsome versions of the function generators and their gain characteristics.


Such nonlinear function generators can be used in various devices, for example,function generators with the tunable shape of the output signal, compensators ofnonlinear dependences of sensors, as well as for correction of automatic controlsystems, in laboratory mockups for studying nonlinear devices, etc.

4.8 Conclusions

Passive nonlinear devices (including only semiconductor diodes and resistors)are not characterized by high accuracy of reproduction of nonlinear dependencesbecause of the influence of the diode voltage drop at the forward bias and the reversecurrents at the backward bias.

Introduction of the op-amp and the feedback transforms a nonlinear device intoan active one and allows the major components of the error to be decreased.

Questions

1. What are the main differences of the nonlinear mathematical transformationfrom the linear one?

2. Is the function y D sin x (y D tan x, y D arcos x, y D x2) nonlinear?3. Is it possible to change the signal frequency in nonlinear transformation?4. Can the constant component of signal arise at nonlinear transformation?5. Periodic self-oscillations arise in nonlinear devices, is not it?6. If two signals with the frequencies¨1 and¨2 are fed to a nonlinear device, what

will be the frequency (frequencies) of the output signal?7. Explain purposes of the logarithmic amplifier.8. Draw the circuit of the op-amp logarithmic amplifier.9. Is it possible to construct a voltage multiplier (divider) in logarithmic ampli-

fiers?10. Is the mathematical operation of multiplication y D x � z a nonlinear operation?11. What are the applications of operational rectifiers?12. What are the major components of the error of the simplest half-wave rectifier?13. Draw the circuit of the op-amp operational rectifier.14. What are the advantages of the half-wave operational rectifier over the simplest

one constructed in a semiconductor diode and a resistor?15. What are the results from application of the full-wave rectifier?16. Draw the circuit of the op-amp full-wave rectifier.17. What are the applications of voltage limiters?18. How can the op-amp inputs be protected from high voltage?

Test Yourself

1. What of these mathematical operations are nonlinear?

(a) summation;(b) subtraction;

4.8 Conclusions 109

(c) differentiation;(d) squaring.

2. Determine whether the square-rooting operation is a nonlinear one:

(a) Yes;(b) No.

3. Determine whether a constant component arises in the nonlinear electronicdevice at application of the input signal:

(a) Yes;(b) No.

4. Determine whether new spectral component, absent in the input signal, can arisein a nonlinear electronic device:

(a) Yes;(b) No.

5. What of the following operations can result in appearance of new spectralcomponents?

(a) subtraction;(b) summation;(c) squaring;(d) rejection.

6. Find the op-amp output voltage at K D 10,000, Vcc D ˙15 V.

(a) Vout D C1,000 V;(b) Vout D �1,000 V;(c) Vout � C15 V;(d) Vout � �15 V.

7. At what input voltages the op-amp can be considered as a nonlinear device:

(a) at the input voltages lower than Vout m/K;(b) at the input voltages higher than Vout m/K.

8. What is the gain characteristic of this nonlinear device?

(a) Vout D K [exp (Vin/® T )];(b) Vout D �K [exp (Vin/® T )];(c) Vout D �® T ln [Vin/IoR];(d) Vout D �ln [Vin/IoR].


9. Determine the gain characteristics of the following rectifier for the output 1.

10. Choose the transformation with the following amplitude spectrum of the inputand output signals?

(a) rectification;(b) logarithmation;(c) squaring.

References

1. Tietze, U., Schenk, Ch: Halbleiter-Schaltungstechnik. Springer, Berlin/Heidelberg/New York(2002)

2. Gutnikov, V.S.: Integral Electronics in Measuring Devices. Energoatomizdat, Leningrad (1988)(in Russian)

3. USSR Inventor’s Certificate No. 809219, International Catalog of Inventions No. G 06 G 7/12/Rybin Yu.K./ Device for determination of an absolute value. Bulletin of Inventions No. 8 (1981)

4. Volgin, L.I.: Measuring AC-to-DC Voltage Converters. Sov. Radio, Moscow (1977) (in Russian)

Chapter 5Sine Wave Oscillators

Abstract The major objective of this Chapter are to give an introduction intomodern theory of sine wave oscillators, to explain the main problems in constructionof self-oscillating systems, to determine the conditions for excitation and estab-lishment of stable periodic oscillation in such systems, and to acquaint with theprinciples of practical oscillator circuit design.

To become familiar with the presented material, the reader should have the initialknowledge of operational amplifiers and the methods of solution for differential andoperator equations within the course of high mathematics.

After studying this Chapter, the readers will know and be able to explain theoperation of an oscillator, determine the conditions of excitation of periodic self-oscillations in it, and to know the methods for decrease of nonlinear distortions.

5.1 Introduction

An oscillator of electric signals is a device, through which the energy from the power supplyis converted into electric oscillations of a certain waveform with the some amplitude andfrequency. Oscillations arising due to feedback circuit, therefore oscillators do not requirean external applied of input signal.

Oscillators are used as the sources of measuring, stimulating, synchronizing, andcontrol signals. They often are independent measurement devices or componentparts of other instruments and systems.

The Russian domestic and foreign industries fabricate now a variety of signaloscillators. They can be classified by different parameters: waveform and frequencyof oscillations, output power, purpose, kind of the used active element, form ofthe frequency-sensitive feedback loop, etc. However any classification reflects thesubjective point of view and has its own advantages and disadvantages. For example,from the customer’s point of view, it is unimportant which type of feedback or activeelement is used in the oscillator. These parameters are of secondary importance


111

112 5 Sine Wave Oscillators

for customer, while the parameters of primary importance are the waveform of theoutput signal, its level (amplitude), error, power, oscillator overall dimensions andweight, etc. So it is the better using the classification, which is widely applied inpractice and has become legally recognized.

According to the Russian state classification, oscillators are divided into severalgroups, which are designated as follows: G1 – complex oscillators, G2 – noise-signal oscillators, G3 – low-frequency signal oscillators, G4 – high-frequency signaloscillators; G5 – pulsed signal oscillators; G6 – arbitrary waveform oscillators.

Oscillators of the G3 and G4 groups occupy a particular place. Their wideapplication is caused by useful properties of sine-wave signals. In the first turn, itis a constant waveform when passing through a linear electrical circuit. Second,the relation between the amplitude, root-mean-square, and half-period values ofsuch signals is known exactly, and this allows verifying sensors and AC voltmeters.Third, these signals can be used to reveal slight deviations from linearity in electricalelements through measurements of the higher harmonic components, for example,to determine nonlinearity of resistors and capacitors in the production process andthus to control their quality. Fourth, the sine wave signals are very well appropriateto the widely applied methods for theoretical analysis of circuits and devices, inparticular, with symbolic and operator methods, which permits the experimentalcheck of correctness of theoretical calculations.

Oscillators of pulsed signals (group G5) are common in use. They are used tostart, synchronize, and clock the pulsed and digital devices.

As was already mentioned, an oscillator converts the energy from a constant-voltage source (power supply) to the energy of stable periodic oscillations of thespecified waveform, frequency, and amplitude.

One of the main conditions of this conversion is an oscillating system, which determines thewaveform, frequency, and other parameters of generating oscillations. The oscillating sys-tem is constructed of linear frequency-dependent circuits (LFDCs) specifying the frequencyof oscillations and nonlinear elements (NEs) restricting the oscillation amplitude.1

It follows here from that any oscillator is a nonlinear functional device, whichconverts the constant voltage into the alternating one. In sine wave oscillators, justthe oscillating system is a converter.

However, a real system assembled of resistors, capacitors, and inductances always involvesenergy loss. So to keep the energy constant, the consumed part of energy should becompensated for by an external source. This compensation becomes possible by an activeelement (AE) in the oscillating system (transistor, operational amplifier, etc.), whichperiodically resupplies the energy under the effect of the positive feedback, or an elementcompensating for the energy loss owing to the internal feedback inherent in it (tunneldiode, thyristor, etc.). Thus, all the three elements (LFDC, NE, and AE) are necessary inthe oscillator.

1Certainly, this sharing of functions is quite conditional and suitable only for free-runningoscillating systems generating sine wave oscillations. For other waveforms of oscillations, thefrequency can be determined by linear and nonlinear elements.

5.1 Introduction 113

Fig. 5.1 Graph of oscillator

The electrical energy is usually converted without external action, so theoscillator structure may not have inputs in the general case, that is, it must be a ringone. According to this, the generalized oscillator circuit must include the active,nonlinear, and linear frequency-dependent elements, forming one or several closedloops. Taking directed sections having inputs and outputs as element models, wecan represent the oscillator structure as a graph.

It is obvious that based on the enumeration theorem, only two topologicallyindependent closed graphs corresponding to the structures of the oscillating systemscan be constructed of three elements (Fig. 5.1a, b). On Fig. 5.1a, b, the linear,nonlinear, and active elements are connected in series or parallel. Reasoning fromtopology (rather than electrical engineering), it is unimportant in what order theelements are connected or what symbols are used for some or other structureelements. Often the functions of the active and the nonlinear or the active andthe linear elements are conjointed. In this case, the both structures are reduced toa single one on Fig. 5.1c. The functions of the linear frequency-dependent andthe nonlinear elements are not usually combined, because the frequency and theamplitude of oscillations may be interrelated.

These graph-schemes correspond to the block diagrams of the oscillating systemson Fig. 5.2, where LFDC is the linear frequency-dependent element, NE is thenonlinear element, AE is the active element, ANE is the active nonlinear element,ALFDC is the active linear frequency-dependent element. Figure 5.2 letters x1,x2and x3 denote the electric voltages or currents.

Two block diagrams Fig. 5.2a, b are topologically isomorphic to the graph onFig. 5.1a, because they include the same elements connected in the same way(forming a ring). However, electrically, these block diagrams are not equivalent,because the elements in them are connected in different order. In this view, theinput voltages of the nonlinear elements are different in the general case, andconsequently, their output voltages are different as well.

The block diagram Fig. 5.2c represents the graph Fig. 5.1b, while Fig. 5.2dcorresponds to the structure on Fig. 5.1c, which incorporates the active and non-linear elements, and Fig. 5.2e corresponds to the same structure, but incorporatingthe active and linear frequency-dependent elements. Each of them is a free-running (not needing in the external effect) oscillating system, which under certainconditions generates periodic self-oscillations of the given waveform, amplitude,and frequency.


Fig. 5.2 Generalized oscillator block diagrams

Consider the functions of all the elements in this system. LFDC is the only element, whosecomplex transfer characteristic depends on frequency. It can be RC, LC, LR, or RLC – alumped or distributed electrical circuit.

The characteristics of such circuits depend on frequency, at some value of whichthe oscillations are generated, therefore just this element is predominantly respon-sible for the frequency of oscillations. Without it and frequency-independence ofanother elements, the frequency of oscillations is undetermined, and consequently,no periodic oscillations are generated.

AE controls the transfer of the electrical energy from the power supply to the oscillatingsystem. It continuously or periodically resupplies the electrical energy in the oscillatingsystem. NE sees that the amplitude of oscillations is constant , because any its decrease (forexample, if load is connected) results in the stop of oscillations, while the increase leads tothe growth of the oscillation amplitude.

Certainly, it is idealized case. In practice, there are no frequency-independentactive elements, as well as no instantaneous nonlinear elements. What’s more, allresistors and capacitors are nonlinear, though to a small degree. However, nonlin-earity of resistors and capacitors can be ignored since it is small, and the nonlinearelements, whose lag shows itself at high frequencies, can be thought frequency-independent. Thus, idealizing the parameters and characteristics of real elements,we can consider their major properties and facilitate the study of oscillating systems.

The oscillator operation can be divided into several stages: first, excitation and growth ofoscillations; second, stabilization of oscillations, their waveform, amplitude, and frequency,that is their conversion into periodic oscillations; and, third, frequency and amplitudetuning.


The excitation stage begins as the power supply is turned on, and the oscillationamplitude is risen. This stage is usually of short duration and corresponds to thetransient mode of oscillator operation. Then periodic oscillations corresponding tothe main mode, namely, steady-state self-oscillations, are established. In this mode,the oscillator can be operated for a long time. Just this mode is characterized by themain parameters of the generated oscillations. The stage of amplitude and frequencytuning (also unsteady) occurs, when it becomes necessary to change the parametersof the generated oscillations.

It is necessary to say a few words about the excitation way. From here on theoscillations excitation is understood as the continuous increase of the oscillationamplitude from the low values (�V or even nV level) to the high ones (few volts).Some questions arise in this respect. What processes initiate the growth of theoscillation amplitude? What stimulates the growth? Why are oscillations generatedjust at the given frequency? To answer these and other questions, it is necessaryto explain the possible ways of oscillation excitation. The first of them, the so-called “noise” way, is based on the fact that in real oscillators the active and passiveelements (transistors, resistors, diodes) induce low output voltages varying by therandom law, that is, noise. The nature of this noise is attributed to the disorderedmotion of electrons and electron holes in conductors and semiconductors.

For example, the root-mean-square voltage induced at the output of ordinaryresistor is [1]

Vrms Dp4kTBR;

where k D 1:372 � 10�23 /deg is the Boltzmann constant; T is temperature, B is thecircuit pass band, in Hz; R is the resistance, in Ohm.

Thus, for example, at the band of 1 MHz, the resistance of 1 MOhm and thetemperature of 293 K (the room temperature), the rms noise voltage is approxi-mately equal to 127 �V. Its spectrum is uniform in the pass band, that is, it includesoscillations from the constant current up to the frequency of 1 MHz. This noisesignal affects the oscillating system and is amplified by it.

However, different spectral components are amplified nonuniformly. The oscil-lating system amplifies some of them and attenuates others. If we would succeedin creating a system amplifying most strongly only one sine-wave component withthe frequency ¨0, then only this component increases, while the other componentsremain at the low level. Therefore, we can assert that this oscillator excites sine-wave oscillations at the given frequency. Then, when the oscillation amplitudeachieves high values, the nonlinear element restricts it at the needed level, and theoscillations transit onto the steady-state mode. If the oscillating system amplifiessignals in a wide frequency band, then more than one component are increased andthe oscillations may be pulsed.

However, the considered above ignores that the power of one spectral componentof noise with the frequency ¨0 is negligibly small and, theoretically, close to zero.Simulation of the effect of the noise signal on a quasilinear oscillating system usingthe Multisim 2001 software shows that at the noise signal variance 10 times (!)


exceeding the amplitude of stable self-oscillations, the oscillations increase veryslowly. This rate does not correspond to the real rate of oscillation excitation.

The noise explain fails to answer another natural question: why do oscillationsarise at only one frequency equal to the frequency of stable self-oscillations, but donot arise at neighboring frequencies? In addition, it cannot explain the excitation ofself-oscillations in mathematical models of oscillators at analytical description ofprocesses, since models principally contain no noise.

Another way can be invoked to explain the excitation of self-oscillations. It isbased on the effect of voltage jumps as the supply voltage turns on and can be calledthe “impact” way. It starts as the supply voltage is followed to the active elements ofthe oscillator. As a rule, the supply voltage is applied by plugging the oscillator tothe AC line or by connecting the oscillator leads to a battery (accumulator). As thistakes place, the voltages across the active elements change almost immediately, ina jump. These voltage jumps form single pulsed signals with a wide frequencyspectrum, which necessarily include the spectral component with the frequency¨0. Just this component is amplified by the oscillating system, which results inexcitation of sine-wave oscillations. We can see that the impact way is also basedon the spectral representation of the acting pulse and has the same disadvantage asthe noise mechanism. Thus, the spectral approach to explanation of excitation ofself-oscillations proves to be inconsistent.

And, finally, consider the most correct, in our opinion, explanation of excitationof self-oscillations, which is equally suitable for explanation of processes in bothphysical systems and mathematical models of oscillators. When designing botha physical self-oscillating system and its mathematical model, the conditions foroscillation increasing are created by introducing a positive feedback or an activeelement with negative resistance at a certain frequency. That is the way of instabilityof the equilibrium position and, as a consequence, the growth of the amplitude is laidin the structure of the self-oscillating system. Therefore, even minimal any initialvoltage across power-consuming elements of a physical self-oscillating system (aswell as the initial value of a variable or its derivative in the mathematical model)then increases just owing to the positive feedback or the negative resistance of theactive element. Here we do not speak about the amplification of noise, but aboutthe structure instability of the oscillating system, a small initial stimulus in which issufficient for the growth of oscillations. The voltage jump at the instance of voltagesupply can serve as such stimulus. Then the voltage may change, but the oscillationswill still increase.

Recently in connection with the requirement to minimize the time for establish-ment of the steady state, the common tendency is to shorten the duration of theexcitation stage or even remove it, especially, in the sine wave oscillators. Therefore,the way of creation of optimal initial conditions is increasingly often used to exciteoscillations. This way involves creation of optimal initial conditions at the power-consuming elements of the oscillator prior to oscillation generation, namely, thecapacitors are charged up to a certain voltage and are generated in the inductancesinitial currents corresponding to the voltages and currents at some time of the future


Fig. 5.3 Contradictions inthe oscillation circuit of anoscillator

periodic self-oscillations. Therefore, once the power supply turns on, the oscillationsstart just from these initial voltages and currents and immediately become stable,without the excitation stage.

The steady-state mode is characterized by the constant parameters of thegenerated oscillations: waveform, amplitude, and frequency. This is achievedthrough the corresponding circuit design including a nonlinear element or amplitudeand frequency stabilization systems. An important requirement imposed on thesesystems is the minimal effect to the waveform of oscillations, which is almost notaffected by any regulating or stabilizing influence.

Thus, we can see that both the excitation of oscillations and the provision forthe steady state in sine wave oscillators have some features. That is why theseoscillations are considered separately.

The design of sine wave oscillators is distinguished by that sine-wave oscillations can begenerated by the linear oscillating system without nonlinear elements, for example, in an LCoscillation circuit. However, these oscillations are unstable, and their amplitude decreaseseven upon a slight change of the system parameters (for example, when a load resistanceappears). The active element introduced into the circuit compensates for the loss of theelectric energy, and the oscillation amplitude does not decrease. However, if this elementis linear, the exact compensation cannot be achieved for any reason that load resistancecan change. The nonlinear element is principally necessary here. It corrects disturbanceand stabilizes the amplitude of self-oscillations, but introduces nonlinear distortions, thuscausing deviation of the waveform from the sine wave.

And, the better stabilized is the amplitude, the more nonlinear distortions areintroduced. To decrease these distortions, the nonlinearity of the element should below [2]. This is the first contradiction inherent in sine wave oscillators (Fig. 5.3,where Kthd is the total harmonic distortion; test is the amplitude rise time, and •V isthe amplitude instability), namely, the contradiction between the level of nonlineardistortions and the amplitude stability. The second contradiction is the weak of thenonlinear element effect, the slower transient processes are occurring when excitingthe oscillations and tuning the oscillation frequency and amplitude. That means,the duration of the transient processes is proportional to the level of nonlineardistortions: the transient processes are faster, the oscillation waveform is worse.The third contradiction arises between the time needed for establishment of theoscillations and the instability of their amplitude.


Removal of these contradictions is the main problem in construction of sine waveoscillators. This problem is especially important for RC oscillators2. It should bementioned that recently the efficient methods have been proposed for solution of thisproblem. To understand thoroughly this problem and the methods for its solution,analyze the processes occurring in the oscillator.

5.2 Oscillatory Processes

The oscillatory processes are analyzed in order to find the conditions for excitation ofoscillations and to determine the oscillation waveform, amplitude, and frequency. In thiscase, it is very important to properly choose the mathematical methods for analysis. Theresearch and training papers often use the mathematical methods based on the complexamplitudes, phase plane, ordinary differential equations, etc.

Each of the mentioned methods has its advantages and disadvantages. Forexample, if the processes in the sine wave oscillator are described by a system ofdifferential equations, then integration of this system gives all possible solutions.If some of these solutions are periodic, then just they describe the waveform of theoutput voltages in the steady-state mode of the oscillator. All other variables of theseequations are, naturally, periodic too. However, the study of solutions by consideringtheir time variations is inefficient, because there are infinitely many solutionsdepending on the initial conditions. The method of complex amplitudes allowsthe steady-state solutions to be easily revealed, but fails to describe adequately theprocess of their establishment. At the same time, the study can be made illustrativeand even beautiful, if we apply the method of phase plane [3].

5.2.1 Analysis by the Method of Phase Plane

This method (phase plane) essentially consists in the fact that two variables can berepresented as variations of the coordinates of some point on a plane, whose abscissacorresponds to one of these variables (U), while the ordinate corresponds to the another(V). Such a plane is called the phase plane.

At any time ti , the values U(ti ) and V(ti ) determine a point A on this plane; theposition of this point is described by the radius vector. The radius vector connectingthis point with the origin of coordinate forms the phase angle ® i with the abscissa.A different instant corresponds to the different point and the different angle.

2It should be noted that in oscillators of other waveform of signals these contradictions are not sopronounced.

5.2 Oscillatory Processes 119

Fig. 5.4 (a) Phase plane and (b) a limit cycle on it

With time, we can draw the continuous series of such points, and the variation of the phaseangle describes its continuous motion. As a result, a continuous line called a hodograph(time plot) is drawn on the plane. The moving point itself is called an image point.

Not only U and V, but any two variables can be taken as phase variables.The dimensions of the phase plane are determined by physical restrictions, forexample, by the values of the oscillator supply voltages. If the variables U(t) andV(t) are periodic functions, then the image point periodically (with the interval T)passes through a certain point of the phase plane. In this case the hodograph is aclosed curve. If, with time, all trajectories on the plane fall on this curve, then thishodograph is called a limiting cycle (Fig. 5.4b). Studying the behavior of the phasetrajectories on the plane, we can judge the behavior of the variables in time.

We know closed trajectories on a plane from the school physics as Lissajous figures. It isalso known from the physics course that a circle or an ellipse on a plane corresponds totwo periodic sine-wave oscillations shifted in time. Therefore, the voltage variables of thesine wave oscillator, when plotted on the phase plane, form either an ellipse or a circle.Figure 5.5 shows the trajectories formed by two oscillator voltages on the plane. The outputvoltage is taken as V, and the voltage or current in one element of the oscillating system istaken as U3.

At the time when the supply voltage turns on, the variables U and V in theoscillator can take the values different from the values on the limit cycle, thatis, the initial position of the image point can be different, and in the process ofestablishment of the stable oscillations the image point must move to the trajectoryof the limit cycle. In other words, all the trajectories must tend to the limit cyclefrom any position on the plane inside or outside the limit cycle. The shape of the

3It should be noted that the phase plane represents signals ambiguously. For example, the sameform of the limit cycle (see Fig. 5.5) can correspond not only to the sine wave, but also to the othersignal waveform. The matter is that the phase plane does not represent the velocity of the imagepoint.


Fig. 5.5 Directions of phasetrajectories near the limitcycle at stable sine-waveself-oscillations

Fig. 5.6 Singular points of phase trajectories: (a) saddle, (b) node, (c) center

trajectories depends on the form of the system or, more precisely, on the so-calledsingular points of this system. Explain what is singular point.

Let all the trajectories inside the limit cycle go away from the origin of coordinate, tendingto the limit cycle along a spiral line. Then it is said that the origin is a singular point ofthe type of unstable focus. On the other hand, the motion from the outside toward the limitcycle can also follow a spiral line, but the winding one. This motion corresponds to thestable focus at the origin of coordinate. Besides singular points of the focus type, there aresome others, which are shown on Fig. 5.6 .

The concept of singular points is useful, because such points specify thedirections of the trajectories on the entire phase plane or some its part. Each typeof singular points corresponds to a certain combination of roots of the characteristicequation of the system. For example, the singular point of the focus type correspondsto two complex-conjugate roots. Therefore, knowing the positions of the roots, wecan easily determine the conditions for appearance of self-oscillations and theircharacter.

It will be shown below that the equation of the oscillating system of a sine waveoscillator often has complex-conjugate roots, corresponding to spiral trajectories onthe phase plane.

Consequently, the origin of coordinate in the oscillating system of a sine waveoscillator is just the unstable focus, if the image point is inside the limit cycle – acircle, and the stable focus, if the image point lies outside the limit cycle; that is whyall trajectories tend to the limit cycle.


However, the knowledge of the type of singular points still does not guarantee thelimit cycle. At the same time, periodic oscillations of a particular waveform arisein the oscillating system only when the image point moves along the limit cycle.Therefore, one of the critical conditions for appearance of periodic oscillations isthe presence of a system generating them. In sine wave oscillators, a conservativelinear oscillating system (usually, of the second order) serves such a generatingsystem. From Fig. 5.6c it is seen that in the phase plane the sine-wave oscillationsare represented by the limit cycle in the form of closed curve. Analysis of the curveshape suggests that it is described by an ambiguous function in the coordinatesy D V, x D U. For example, the limit cycle of the sine wave oscillator (a circle) isdescribed by the function y2 Cx2 D r2. The ambiguity is that any coordinate eitherx or is determined through another ambiguously in view of y D ˙p

r2 � x2 andevery value of x corresponds to two values of y and vice versa. Therefore, if timeis excluded, the equations of the oscillating system are described by an ambiguousnonlinear function.

Let the system of equations of the generating oscillating system consist, in thegeneral case, two equations

dy

dtD P.x; y/

dx

dtD F.x; y/

9>>=

>>;

: (5.1)

Then, excluding time, we obtain dydx D P.x;y/

F .x;y/, that is the differential equation

describing the curves on the plane (x, y).

�Using the sine wave oscillator as an example, let us demonstrate what is the formof the functions P(x, y) and F(x, y). Determine the derivative dy/dx for the equationof a circle y2 C x2 D r2. Upon differentiation, we obtain dy

dx D �2x˙2pr2�x2 D �x

y,

when P(x,y) D ˙ x, F(x,y) D �x. The derived system of equations of the generatingoscillating system of the sine wave oscillator is linear:

dy

dtD x

dx

dtD �y

9>>=

>>;

or

dy

dtD �x

dx

dtD y

9>>=

>>;

:

Its singular point is a center, corresponding to concentric circles on the phaseplane, and the sine-wave functions are its solution in time. However, the diameterof the circle and, consequently, the amplitude of oscillations depend on the initialconditions and change at their even minor variations. For one circle to be a limitcycle and all trajectories to tend it, the obtained system should be complementedwith a stabilizing operator, leading, with time, all curves on the plane from anyinitial positions to this circle. These curves may be shaped as spiral lines winding


on the circle of the limit cycle from both inside and outside (Fig. 5.5). Accordinglyfor this, the system of equations takes the form

dy

dtD P.x/

dx

dtD y C "x'.x; y/: (5.2)

Here ©x®(x, y) is the stabilizing operator, at which every point converts fromthe center into the focus and trajectories transform into spiral lines. It is clear thatthe stabilizing operator in the second equation must be equal to zero on the circleof the limit cycle, that is, be a function of the circle equation. Let us find its form inthe general case from the equation of the conservative part. As known, this systemhas the first integral

C D y2

2CG.x/;

where G.x/ D R x0P.s/ds; C is the integration constant.

It is also known that the first integral remains constant along each of systemsolutions. On the phase plane, the first integral of the equation of the conservativesystem corresponds to a family (continuum) of closed lines (trajectories), embeddedinto each other. For this, the function G(x) should have a global minimum. In thiscase, a singular point of the center type corresponds to the closed lines. In thecontinuum of trajectories, one of them corresponds to the given steady-state periodicoscillation xst(t). A deflection from the trajectory (at certain C) or, in other words, thetransition to a different trajectory changes the value of the first integral. In fact, theconstant C determines the parameters of the oscillation: its amplitude and frequency.At a particular value of C, the transition to a different trajectory leads to distortionof the equality. A difference arises between the left – and right-hand sides. The signand the value of this difference depend on the deflection from the given trajectory,that is, on the steady-state oscillation x(t). This property of the first integral is used tocontrol oscillations. It is clear that the parameters of the corresponding oscillationsof the conservative system depend on the initial conditions and change significantlyupon even a little variation of these conditions. For the oscillation parameters tobe independent on the initial conditions or on the effect of perturbations and inorder to convert them into self-oscillations, it is needed to convert the singularpoint of the center type into focus, so that at the steady-state trajectory Eq. 5.8transforms into an identity. To convert center into focus, introduce an incrementterm. Thus, the differential equation of the method of steady-state self-oscillationscan be represented in the form

dy

dtD P.x/;

dx

dtD y C "x

�C � y2=2�G.x/� :

9>>=

>>;


In the simplest case then P(x) D � x, it can be the equation of circle: '.x; y/ Dr2 � y2 � x2. Then the system of equations takes the final form

dy

dtD �x;

dx

dtD y C "x.r2 � y2 � x2/:

9>>=

>>;

The solution of this system of equations at ©> 0 and t ! 1 is the strictly sine-wave oscillation y.t/ D r sin. t C t0/.

In Eq. 5.2 the oscillating system describes the active linear frequency-dependentelement (ALFDE), while the stabilizing operator including the squared variables xand y represents the nonlinear element (NE).

It is interesting to note that in the well-known Van der Pol equation

dy

dtD �x;

dx

dtD y C "x.1� y2/�

9>>=

>>;

the stabilizing operator is not zero on the circle (on the trajectory of the limit cycle);therefore, the solutions of the system are not strictly sine waves. They can approachsine waves at the decrease of the small parameter ©, but, theoretically, cannot be sinwaves.

The method of the phase plane allows us to consider the character of thetrajectories, determine the form of the limit cycle, find the amplitude and frequencyof self-oscillations, etc. For second-order systems, it is characterized by simplegeometric structures on a plane, but as the order of the oscillating system increases,it becomes necessary to represent the trajectories in the multidimensional space, andthe method becomes impracticable.

5.2.2 Analysis by the Method of Complex Amplitudes

Let us perform the analysis of the oscillating systems on Fig. 5.2 by the method lesssensitive to the system order and more appropriate for an engineer, namely, by themethod of complex amplitudes.

This method (complex amplitudes) operates with complex transfer functions and complexgains, which can be easily found even for high-order systems.

� For example, for the oscillating system on Fig. 5.2a, we can write the followingequations:

PV3 D PV2 PK; PV2 D PV1 P � PV1�; PV1 D PV3 P�: (5.3)


Here: PV1, PV2 and PV3 are the complex amplitudes of voltages at the outputs ofthe blocks in the diagram. Excluding PV2 and PV3 from these equations, we obtainone nonlinear equation PV1 D PV1 P. PV1/ PK P� . In the general form, this equation hasno analytical solution. However, at the weak nonlinearity of P. PV1/, that is, at thealmost linear function, it can be linearized and replaced by the linear functionPV1 P. PV1/ D PV1ˇ. Then the nonlinear equation transforms into the linear one,

which is called the complex characteristic equation of the oscillating system or theBarkhausen criterion:

1 D PK P� P: (5.4)

In this equation, the right-hand side contains only complex variables. It can betransformed into two equations for the real and imaginary parts:

Re. PK P� P/ D 1;

Im. PK P� P/ D 0

)

(5.5)

or for the products of the absolute values and arguments:

j PKjj P� jj Pj D 1;

arg.K�ˇ/ D 'K C '� C 'ˇ D 0; 2n�:

)

(5.5a)

The first equation in the systems (5.5) and ( 5.5a) states that in the steady-state mode theproduct j PKjj P� jj Pj at the oscillation frequency must be equal to 1. This equation is calledthe amplitude balance . It shows that the sine wave, passing in the circuit through LFDE,AE, and NE, keeps its amplitude unchanged. Given the dependence of ˇ on the voltage V1,this condition permits us to find the amplitude of oscillations. The second equation of thesystem – phase balance – indicates that the shift of the initial phases of the oscillation,having passed through the circuit, must be equal to 0 or 2n� rad (n is any integer number).This condition allows us to determine the oscillation frequency. If the active and nonlinearelements are frequency-independent, then Eq. 5.5a take the simpler form:

Kj P�jˇ D 1;

arg.K P�ˇ/ D '� D 0:

)

(5.5b)

The similar linearized systems of equations can be written for the diagramFig. 5.2b as well.

The diagram on Fig. 5.2c can be described by the following equations:

PV2 D . PV1 C PV 001 /

PK; PV 01 D P� PV2; PV 00

1 D PV2 P. PV2/;

which can be reduced to a single nonlinear equation PV2 D Œ PV2 P. PV2/C P� PV2� PK:After linearization it takes the form:

1 D . P C P�/ PK D Pı PK; (5.6)


where Pı D P C P� . The similar characteristic equation can be also written for thediagram on Fig. 5.2d:

PU2 D PKfb P� PU2 or 1 � PKfb P� D 0: (5.7)

Here Kfb is the complex transfer function of the feedback amplifier, P� is thecomplex transfer function of the linear frequency-dependent circuit.

Equation 5.7 is sometimes called the equation for oscillations appearance, though itdetermines no conditions for appearance or increase of oscillations. What’s more, it evendoes not determine the waveform of these oscillations.

No waveform follows from Eq. 5.7, because it does not include the oscillation V2.It is quite clear, because the transfer functions are determined by the voltage ratio.The same is also true for Eq. 5.4. Both these equations allow us to determine theparameters of only the steady-state mode: amplitude and frequency. In turn, Eq. 5.7can be divided into two other equations, which are also called the amplitude andphase balance equations or the Barkhausen criterion [4]:

j PKfbj j P� j D 1;

arg. PKfb P�/ D 'K C '� D 0:

)

(5.8)

According to the first equation, the absolute value of the amplifier gain must beequal to the absolute value of the inverse gain of the frequency-dependent feedbackcircuit:

ˇˇ PKfb

ˇˇ D j1= P� j. According to the second equation, the sum of phase shifts at

the oscillation frequency must be zero.These conditions allow us to determine the amplitude of self-oscillations (known

the nonlinear dependence of the gain on the output voltage). The conditions make itpossible to determine the frequency of self-oscillations (known the phase-responsecharacteristic).

Sometimes incorrect conclusions about excitation of self-oscillations are drawnfrom these conditions. It is commonly supposed that the conditions for excitation ofself-oscillations are

j PKfbj j P� j > 1;arg. PKfb P�/ D 'K C '� D 0

)

:

However, it is unclear how these conditions can be understood, because they arenonsense for a closed self-oscillating system. Actually, if we consider Fig. 5.2d,where, say, LFDE is a well-known Wien bridge with the gain j P� j D 1=3 at thefrequency of self-oscillations !0 D 1=� , and ANE is an amplifier with negativefeedback and the gainK 0

fb D 3:3; then these conditions would take the more specificform

K 0fb j P�.!0/j D 3:3 � 1

3D 1:1 > 1;

'� .!0/ D 0:

9=

;


Check is it possible to satisfy these conditions in a closed-loop system. Let atsome time the voltage at the amplifier input is 1 V. Pass on the feedback loop throughthe amplifier and the Wien bridge, this voltage becomes equal to 1.1 V at the sametime. But 1.1 V is not equal to 1 V. What is the matter? The matter is that in theclosed self-oscillating system the loop gain (if this concept can be applied here)is equal to 1 in both the steady-state mode and the excitation mode. Because theleft-hand side of Eq. 5.7 it is always equal to the right-hand side. Then what isthe amplifier gain or what is the gain of the Wien bridge during excitation? Theoscillations increase in spite of these questions, but at what frequency? Obviously,it is not the frequency !0, since the gain at this frequency must be 3.3. Also it isabsolutely unclear at what gain the oscillations are damped, when the loop gain, onone hand, must be less than 1, but on the other hand, it is equal to 1.

Thus the Barkhausen criterion and the amplitude balance and phase balance conditionshave limited applicability only to the steady-state mode, but even in this case it is notalways possible to determine Kfb , for example, in the positive feedback in the amplifier. Itis appropriate noting here that it is unclear what these conditions are at other frequencies.And, finally, these conditions are inapplicable to the oscillating systems generating non-sine-wave oscillations.

Nevertheless, in many cases Eqs. 5.5a and 5.8 are quite fruitful, illustrative, andgive grounds for practical calculation of the steady-state parameters: amplitude andfrequency of oscillations. But it should be noted that they give no idea about thewaveform of oscillations and the character of their variation at excitation, as wellas about the oscillations at the frequencies other than the oscillation frequency. Inaddition, as will be shown below, they are necessary but insufficient and valid onlyin the case that K, ”, and “ are determined.

5.2.3 Analysis by the Method of Differential Equations

The mathematical method of ordinary differential equations proves to be much moreinformative for analysis of the oscillations. Represent the models of the linear and activeelements in the form of differential equations:

�2d2V1

d t2C �

�

dV1

dtC V1 D �

dV2

dt; �K

dV2

dtC V2 D KfbV1; (5.9)

where � is the time constant and � is the LFDE gain, while �K is the time constant and Kfb isthe amplifier gain.

Here the linear frequency-dependent circuit is described by the second-orderequations of the selecting circuit, while the active element (amplifier) is describedby the first-order equations. The first equation describes the LC oscillation circuit,RC Wien bridge, integro-differentiating RC circuit, etc. (see Appendix 2), while


Fig. 5.7 Oscillations in the oscillator at different coefficients ’

the second one describes the typical op-amp (VCVS) considered in Chap. 1. Theconstant £K is determined by the cutoff frequency as �K D 1=2�fcut. At the smalltime constant £K , the op-amp can be considered as inert-free, and it is describedby the equation V2 D KfbV1. Let us accept this assumption at the first stage of theanalysis. Then, upon substitution of the second Eq. 5.9 into the first one, we obtainthe differential equation of the self-oscillating system:

�2d2V1

d t2C �

�1

��Kfb

�dV1

dtC V1 D 0: (5.10)

The solution of this equation is the following:

V1.t/ D Vme�˛1t sin.!1t C '/;

where Vm is the oscillation amplitude, ’1 D C(1/” – Kfb)/2£ is the increment(at ’1 < 0) or decrement4 (at ’1 > 0) of oscillations, !1 D p

1 =�2 � ˛12 DD

q

!20 � ˛12 is the frequency of oscillations, and ® is the initial phase.We can see that the solution of the differential equation represents the output

oscillation of V1 as a sine wave, whose amplitude increases at ’< 0, decreases at’> 0, and remains constant at ’D 0. At 1/” – Kfb D 0 the oscillations have thesinusoidal waveform, constant amplitude Vm, and the frequency ¨0 D 1/£, that isthey are periodic. The conditions in this case are just the same as the amplitude andphase balance conditions: ’D C (1/” – Kfb)/2£D 0 and ¨0 D 1/£.

Figure 5.7 shows the plots of voltage at different coefficients ’. Alternation of thesign of ’ results in the qualitative change of the character of oscillations: oscillationsdecrease at positive ’ and increase at negative ’.

The value of ’ affects the rate of this change. At ’D 0 the amplitude doesnot change. The property of the amplitude dependence on ’ is used in oscillators

4The terms increment and decrement mean, respectively, increase or decrease (damping).


Fig. 5.8 Steady-state mode in the oscillating system of an oscillator

to control the amplitude of self-oscillations. The increment results in the increaseof the amplitude, while the decrement leads to its decrease. For example, as thesupply voltage turns on and the oscillation amplitude is still low, with the amplifiergain Kfb > 1/” we can obtain the oscillation build-up. As the oscillation amplitudeexceeds the steady-state value, the gain should be decreased and fixed at the levelKfb < 1/”. In the steady-state mode, the equality Kfb D 1/” should be fulfilled.The stabilizing operator in Eq. 5.2, whose functions are executed by the nonlinearelement, operates just in this way.

With the allowance for nonlinearity, the equation of the oscillating system takes the form:

�2d 2V1

d t2C �

�1

��Kfb.V1/

�dV1

dtC V1 D 0: (5.11)

Unlike Eq. 5.10 , this equation includes the dependence of the gain vs. V1.

In Fig. 5.8a this dependence is shown by the solid curve, while the dashed lineshows the dependence of the inverse gain of the frequency-dependent feedbackcircuit 1/”. Figure 5.8b shows the gain characteristic of the amplifier. It is thedependence of the first-harmonic amplitude of the output voltage on the first-harmonic amplitude of the input voltage for the amplifier with the nonlinear element.It can be seen from the figure that at the low voltage V1, the gain Kfb> 1/”, and theoscillations grow.

But as the oscillation amplitude increases, the gain gradually decreases. Atthe point A the equality Kfb(Vm1) D 1/” is achieved, and the amplitude no longerincreases. The steady-state mode is established. A.A. Andronov (prominent Russianphysicist and mathematician) called such oscillations self-oscillations.

If by some reasons (for example, at increase of load, influence of noise pulse)the amplitude of the input voltage exceeds V1m, then the gain becomes smallerthan the inverse gain of the frequency-dependent feedback circuit: Kfb < 1/”, andthe amplitude begins to decrease down to the steady-state value. From analysisof Fig. 5.8a we can deduce the condition for ensuring the stable amplitude in thesteady-state mode: dKfb/dV1< 0 or, in other words, the derivative of the amplifiergain with respect to the voltage must be negative.


Now take into account the frequency properties of the amplifier and obtain the solution ofthe differential Eq. 5.9 as they are without any simplifications. In this case, the system oftwo equations reduces to a single third-order equation

�K�2 d

3V1

d t3C

�

�2 C ��K1

�

d2V1

d t2C

�

�K C �

�1

��Kfb

�dV1

dtC V1 D 0;

whose solution is

V1.t/ � Vme�˛t sin.!1t C '/C V�e

��t :

Unlike the solution of Eq. 5.10, this one includes the component rapidly dampingby the exponential law (at œ> 0), which does not change principally the characterof oscillations.

Consider another example, when the allowance for the frequency properties ofthe amplifier changes principally the character of oscillations. Take the second-orderrejector circuit (see Appendix 2, circuit 7) as the linear element, then the differentialequations take the form

�2d2V1

d t2C �

�

dV1

dtC V1 D �2

d2V2

d t2C �

ˇ

dV2

dtC V2;

�KdV2

dtC V2 D KfbV1:

Having substituted, as before, the simplified second equation into the first one,we obtain the equation for the self-oscillating system:

�2.1 �Kfb/d2V1

d t2C �

�1

�� Kfb

ˇ

�dV1

dtC V1.1�Kfb/ D 0:

The solution of this equation has the following form:

V1.t/ D Vme�˛2 t sin.!2t C '/;

where Vm is the oscillation amplitude, ’2 D (1/” – Kfb/“)/2(1 – Kfb)£ is the damping

decrement or increment, !2 Dq

1=�2 � ˛22 Dq

!20 � ˛22 is the oscillationfrequency, and ® is the initial phase.

Thus, the solution of this differential equation also represents the output voltagein the form of the sinusoidal voltage V1, whose amplitude grows at ’< 0 anddecreases at ’> 0. At ’D 1/” – Kfb/ “D 0 the oscillation is a sine wave withthe constant amplitude Vm and the frequency ¨2 D 1/£, that is periodic. In thisoscillating system, as well as in the previous one, the amplitude and phase balance


Fig. 5.9 Circuits on active and passive two-ports: series connection (a), parallel connection(b); series–parallel connection (c), parallel-series connection (d), and cascade connection (e)

conditions are met. However, with the allowance for the real frequency propertiesof the amplifier, the character of oscillations changes. Analogously to the abovecase, the equation of oscillations with account the frequency properties of theamplifier becomes a third-order equation, but it solution includes, in addition tothe sine-wave component, the growing exponential component, which, with time,brings the amplifier into the nonlinear limiting mode, when generation of sine-wave oscillations terminates. This oscillating system is inoperative. By the way,the amplitude and phase balance conditions are met in it as well. So we candraw the following conclusion: the amplitude balance and the phase balance arenecessary, but insufficient conditions for establishment of sine-wave oscillations inthe oscillator.

5.2.4 Analysis by the Two-Port Network Method

Represent the structure on the graph scheme Fig. 5.1 c as connected of 2 two-port networks.It is well-known that using 2 two-port networks it is possible to assemble five differentcircuits, which are shown schematically on Fig. 5.9 .

The two-port networks here are denoted by different letters indicating thepreferable system of parameters used for this type of the circuit [5]. The directionsof the input and output currents are shown according to the commonly accepteddesignations for different systems of parameters. Each of the circuits presented


can be described by two systems of equations in the corresponding system ofparameters. For example, the circuit Fig. 5.9b can be represented as follows:

i11 D Y 111 � u11 C Y 112 � u12

i12 D Y 121 � u11 C Y 122 � u12and

i 111 D Y 1111 � u111 C Y 1112 � u112

i 112 D Y 1121 � u111 C Y 1122 � u112 :(5.12)

Taking into account that i 11 D �i 111 D i1; i112 D �i 12 D i2; u11 D u111 D u1;

u12 D u112 D u2; after transformation we obtain

.Y 111 C Y 1111 /u1 D �.Y 112 C Y 1112 /u2 and .Y 121 C Y 1121 /u1 D �.Y 122 C Y 1122 /u2

Excluding the variables u1 and u2, we get the single equation:

Y 121 C Y 1121Y 122 C Y 1122

D Y 111 C Y 1111Y 112 C Y 1112

;

whence it is possible to derive the general equation

Y21

Y22D Y11

Y12or Y21Y12 D Y11Y22; that is; jY j D 0: (5.13)

This condition, namely, the zero in the determinant is necessary for self-oscillations in the system. It can be easily reduced to the condition (5.7):

1 � Y21Y12

Y22Y11D 1 � PKfb P� D 0;

where

PKfb D Y21

Y11; P� D Y12

Y22:

The results of this analysis can be extended to all circuits, and the equations [19]analogous to Eq. 5.13 can be obtained: jZj D 0 for the circuit Fig. 5.9a, jHj D 0for the circuit Fig. 5.9, jGj D 0 for the circuit Fig. 5.9d, and jAj D 0 for the circuitFig. 5.9e.

As can be seen from Fig. 5.9, all the oscillator circuits can be reduced to a singlecascade circuit. Figure 5.10 demonstrates a version of such reduction.

For transition from the system of equations with Y parameters to the system withA parameters, it is needed to alternate the signs of i 112 and i 12 .

In practice, the active element is represented by a transistor or op-amp, whichcan be represented by a three-terminal circuit rather than a two-port network. Thefrequency-dependent two-port network also usually has the joined input and outputleads, and so it can be represented by a three-terminal circuit as well. Regardingthis, all the circuits on Fig. 5.9 can be represented as assembled of three-terminalcircuit. Figure 5.11 shows the possible versions of transformation of the two-port


Fig. 5.10 Reduction of theparallel Y circuit to thecascade A circuit

Fig. 5.11 Transformation of two-port circuits into three-terminal circuits

circuits into the three-terminal circuits. A line inside a two-port circuits indicatesthe common line connecting the input and output terminals.

The two-port circuit on Fig. 5.11a transforms into the three-terminal circuitFig. 5.11c, while the circuit Fig. 5.11b transforms into the circuit Fig. 5.11d. Forconvenience, the two-port terminals are numbered. It can be seen that the circuitFig. 5.11c is reduced to the ordinary connection coinciding with the circuit Fig. 5.2d,all the conditions for appearance of self-oscillations obtained earlier, for example,Eqs. 5.7 or 5.13 are applicable to it.

The circuit on Fig. 5.11d is a connection of one-ports (two-terminal elements):active and passive ones. For this circuit, the system of Eq. 5.12 takes the form

i 11 D Y 112u12

i 12 D �Y 122u12

)

andi 111 D Y 1112 u112

i 112 D Y 1122 i112

)

; (5.14)

5.3 Features of Oscillating Systems 133

since the variables u, i are connected by the following: u11 D u111 D 0; u12 D u112 Du2 ¤ 0; and i 12 D �i 112 D i2:

Equating the second equations of the system (5.14), we obtain the necessary condition forself-oscillations in the circuit consisting of two one-ports:

Y 1122 D �Y 122: (5.15)

This condition means the equality of the complex conductivities of the active and linearfrequency-dependent one-port networks, and it can be presented as two equalities (ampli-tude balance and phase balance)

jYLFDE j D j�YAE j and 'LFDE D 'AE: (5.15a)

The equality (5.15) states that the conductivity of the active one-port must benegative, because the conductivity of the frequency-dependent one-port is positive.The negative conductivity is inherent in a tunnel diode, thyristor, unijunctiontransistor, and other elements. The equalities (5.15a) are indicative of the equalabsolute values of the conductivity and the phase at the oscillation frequencybetween the current and voltage in one-ports networks.

Thus, the analysis of self-oscillating structures based on the two-port networkmethod allows us to justify the possibility of self-oscillations in the two-port, three-terminal, and one-port circuits from the general position.

5.3 Features of Oscillating Systems

The further analysis of the oscillating systems is impossible without considerationof the frequency band of the generated oscillations and the accuracy of setting oftheir amplitude and frequency, that is, the parameters and the circuit features ofoscillators. The frequency range is conditionally divided into the low-frequency(up to 10 MHz) and high-frequency (above 10 MHz) parts. The frequency exertsthe decisive influence on the choice of the frequency-dependent circuit and thenonlinear and active elements. In the low-frequency range the oscillating systemsbased on the RC circuits and op-amps as active elements are mostly used, while inthe high-frequency range the LC circuits and transistors or tunnel diodes are applied.That is why RC and LC oscillators are analyzed separately in this Chapter.

At first, let us answer the question: why are RC and LC oscillators used,respectively, in the low- and high-frequency bands? For this purpose, determine theparameters of the elements of the frequency-dependent circuit needed for generationof low-frequency oscillations.

Let our task be to construct a sine wave oscillator operating at the frequencyof 1 kHz (low frequency). Calculate the elements of the frequency-dependentLC circuit. In the most known circuit of this type – the resonant LC circuit,the frequency of the possible generation is f D 1/2�

pLC , whence the product


LC D p1 =2�f � 12:6 �10�3 s. Taking, for example, L D 2 Hn and C D 0.0063 F,

we can obtain the needed oscillation frequency. However, to ensure the inductanceof 2 Hn, a very large and massive inductance coil with a ferrite or electric steelcore is needed. Capacitors with the capacitance of 0.0063 F D 6,300 �F also havelarge size and are characterized by the wide spread and instability of capacitance.Naturally, the frequency-dependent LC circuit will have large size and heavy mass,and the frequency of the generated oscillations will be inaccurate, unstable, anddifficult to tune.

For comparison, determine the parameters of the RC circuit. Thus, for the Wienbridge f D 1/2 RC, whence the time constant is RC D 1/2 f D 16.6 10�6 s. Thistime constant can be obtained by taking a 10 kOhm resistor and a 1.66 nF capacitor.These elements have small size and mass, accurate and stable parameters, and evencan be implemented in an integral circuit.

Naturally, RC circuits have disadvantages as well, such as low Q-factor, lowtime stability, and others, but the technology of their manufacture is continuouslyimproved and disadvantages are corrected.

In the high-frequency band, the parameters of the RC circuit elements becomelow, comparable with the parasitic ones, therefore the error and instability of theoscillation frequency increase. At the same time, the parameters of the LC elementsare easily realizable and have acceptable values.

5.4 RC Sine-Wave Oscillators

5.4.1 Principles of the Theory of RC Oscillators

Consider RC oscillators constructed in op-amps as shown on Fig. 5.2c, d. Fig-ure 5.12a describes the circuit diagram of the self-oscillating system with the VCVS

Fig. 5.12 Block diagrams of op-amp oscillators

5.4 RC Sine-Wave Oscillators 135

op-amp as an active element. It comprises the op-amp (OA), a linear frequency-dependent circuit (LFDC) of the positive feedback circuit with the gain P�and thenegative feedback circuit P.V2/ including the nonlinear element (NE). The circuiton Fig. 5.12b differs by LFDC is included in the negative feedback circuit, whileNE is included in the positive feedback circuit. In the circuit on Fig. 5.12c NE isincluded in the op-amp amplifier (FBOA).

Determine the conditions, under which stable self-oscillations occur in thesecircuits. Write the linearized equations, assuming, as before, the weak NE nonlin-earity, at which the effect of its nonlinear dependence can be denied in the firstapproximation:

PV2 D . PV 01 � PV 00

1/ PK and PV 01 D PV2 P�; PV 00

1 D PV2 P:

Excluding voltages, we obtain

1 D . P� � P/ PK: (5.16)

For the circuit on Fig. 5.12b the similar condition can be derived:

1 D . P � P�/ PK: (5.17)

From these equalities at PK � 1, we obtain

P� D P;

whence two conditions follow:

j P� j D j Pj; '� D 'ˇ: (5.18)

The conditions (5.18) establish the equality of the gains and phase shifts of the circuitsP� and P at the oscillation frequency. They can be called the feedback balance equations.These equations indicate that at the oscillation frequency the absolute values of the LFDCand NE gains and the phase shifts must be equal.

The second equality does not restrict the phase angle to any value. For example,if '� D 0, then 'ˇ D 0 as well, and if '� D � , then 'ˇ D �: If we ignore the NEinertness, the following equations can be derived:

j P� j D ˇ; '� D 0; �; 2�: (5.19)

Conditions (5.18) and (5.19) are more general than the earlier known amplitudeand phase balance conditions (5.8), because they can be applied to both theoscillating systems Fig. 5.12a and the system Fig. 5.12b. The matter is that theapplication of the conditions (5.8) requires the knowledge of Kfb. But, for example,in the system Fig. 5.12b the feedback through the circuit “ is positive, and beyond


Fig. 5.13 Wien bridge (a) and its gain-frequency and phase-response characteristics (b)

the oscillating system the feedback op-amp transforms into a threshold device likea trigger, that is, it is no longer an amplifier, and its gain cannot be determinedexperimentally. The amplifier may also be unstable when determining Kfb with thecircuit ”. Consequently, the conditions (5.8) cannot be applied to this system.

Appendix 2 presents the frequency-dependent circuits, among which there arecircuits with the phase shift of 0 and radian at the frequency of possible oscillation.Which (negative or positive) feedback loop should include these circuits? Equa-tion 5.17 not always gives an unambiguous answer to this question. For example,for the Wien bridge (see Appendix 2) the unambiguous answer can be given basedon Fig. 5.13, which shows its circuit, GFC, and PRC.

At the frequency ¨0, the circuit has the zero phase shift and the maximum gainof 1/3. Consequently, it should be included in the positive feedback loop as shownon Fig. 5.12a. The negative feedback should also have the zero phase shift at theoscillation frequency and the gain of 1/3.

� On the classic Wien-bridge oscillator on Fig. 5.14a, which is considered inalmost students’ books on electronics, the chains of resistor R and capacitor C(Z1) connected in series and resistor R and capacitor C (Z2) connected in paralleltogether with resistors R1 and R2 form the arms of the bridge with the op-amp(ANE) connected in its diagonal. As the oscillator operates in the steady-state mode,the almost exact equality of the voltages V 0

1 D V 001 is established in it due to the

feedbacks and the high op-amp gain; this condition corresponds to the well-knowncondition of the bridge balance

Z2R2 D Z1R1; (5.20)

which can be easily derived from the feedback balance condition (5.17) as appliedto this circuit:

P� D P D Z2

Z1 CZ2D R1

R1 CR2:


Fig. 5.14 Circuits of oscillating systems of ordinary Wien-bridge oscillators (a) and invertedWien-bridge oscillators (b)

Fig. 5.15 Circuits of the oscillating systems obtained based on the reciprocity theorem

Thus, the feedback balance equation is confirmed for the bridge circuit.Based on the reciprocity theorem, the bridge balance condition (5.20) will be also

fulfilled, if we interchange the input (a, b) or output (c, d) op-amp leads or reversethe op-amp inputs and outputs. In this case, we obtain new circuits of the oscillatingsystems, which are shown on Figs. 5.14b and 5.15a, b.

In the circuits on Fig.5.15a, b, the Wien bridge is divided into two chains. Itshould be noticed that, besides these circuits, their versions with the interchangedinverting and non-inverting inputs are also possible. In this case, for example, in thecircuit on Fig. 5.14a, the Wien bridge is a part of the negative feedback loop, while aresistor chain is a part of the positive feedback loop. Thus, based on the Wien bridgeand one VCVS op-amp, it is possible to create various oscillating systems. Similarcircuits can be also constructed based on other-type op-amps.


Fig. 5.16 Bridged T- RC circuit (a) and its gain-frequency and phase-response characteristics (b)

Simulation of all these oscillating systems with an ideal op-amp (that is, second-order systems) and the nonlinear element connected in parallel with the resistor R2(omitted in the figures) using the MathCAD and

Electronics Workbench software demonstrates the stable sine-wave self-oscillat-ions in them.

In all these circuits, the amplitude balance and phase balance conditions are fulfilled at theoscillation frequency. These conditions are also met when taking into account the op-ampfrequency properties in the form of its single time constant. However, stable self-oscillationsin the system with regard for the op-amp frequency properties can be obtained only in thecircuits, which are shown on Figs. 5.14 and 5.15. In the circuits with the interchangedinverting and non-inverting inputs (not shown on the figures), real amplifiers with theinherent frequency dependence of the gain operate in the limited mode and self-oscillationsare lacking.

What’s more, these circuits allow the trigger mode, when one of two steady statesis established at the op-amp output or pulsed periodic oscillations arise with thefrequency determined not only by the elements of the RC circuit.

Thus, fulfillment the only amplitude and phase balance conditions does not guaranteethe stable sine-wave self-oscillations in the steady-state mode. The amplitude and phasebalance are only necessary, but not sufficient conditions.

The question concerning the construction of the oscillator in other circuits, forexample, the bridged T – RC circuit (see Appendix 2, circuit 7), which, as is seenfrom Fig. 5.16b, also has the zero phase shift at the frequency ¨o, is even moreintricate. It is logical to assume that this circuit should be included in the positivefeedback loop. However, it is not so. This circuit is included in the negative feedbackloop.

Consider another interesting example of construction of the oscillating systembased on the electrical circuit called the 2 T-RC twin-tee bridge (Fig. 5.17). Thiscircuit is also called the rejector, since with the equal time constants R1C1 D R2C2 DR3C3 D RC at the frequency ¨o it maximally suppresses (rejects) the signal and itsgain j P� j is equal to zero (Fig. 5.17b). The phase-response characteristic has a breakfrom C /2 to – /2 rad and the amplitude and phase balance conditions cannot


Fig. 5.17 2T -RC bridge (a) and its gain-frequency and phase-response characteristics (b)

be satisfied at this frequency. To satisfy them, we would have an amplifier withthe infinitely gain and the phase shift of C /2 or – /2 at the oscillation frequency.Therefore, theoretically, at the equal time constants the oscillating system cannotbe created in this circuit. However, in practice the time constants are never exactlyequal, but always different. If R3C3 >RC, then the rejection frequency shifts towardsmaller values, and if R3C3 <RC, then it shifts toward greater values. As this takesplace, the gains j P� j can be equal, but the phase shifts are different. In the first caseat the frequency ¨0 the phase characteristics undergoes a break from C to – ,and in the second case at the frequency ¨00 the phase is zero. According to thephase balance condition (5.18), the circuit “ must be, correspondingly, invertingand noninverting. However, it is unclear to which op-amp input (inverting or non-inverting) the 2 T-RC bridge should be connected. The same is true for other rejectorcircuits as well (see Appendix 2, circuits 9–12). These examples are indicative ofcertain difficulties in construction of the self-oscillating systems.

Thus, based only on the amplitude and phase balance conditions it is difficultto determine the proper connection of some or other RC bridge. It can be foundfrom the analysis of the positions of zeros for the oscillating system. From theoryof electrical circuits, it is known that for the sine-wave oscillations to take place ina system, its characteristic equation must have two complex-conjugate roots. If thereal part of the roots is zero, the oscillations have a strictly harmonic waveform withthe constant amplitude (as on Fig. 5.7). If the real part of the roots is positive, thenthe oscillations grow (Fig. 5.7), but if it is negative, then the amplitude of oscillationsdecreases.

In fact, for appearance and growth of oscillations, it is sufficient that theoscillating system has at least two complex-conjugate roots, that means, it must havethe second order. Such system can be constructed based on Wien bridges and othercircuits (see Appendix 2, circuits 1–7). These circuits include only two resistors andtwo capacitors.


Consider the conditions, the roots of the characteristic equation of the second-order oscillating system must satisfy to, for existence periodic sine-wave oscillationin the system. For this purpose, describe the linearized operator equation of ordinaryoscillating system based on Eq. 5.16, for example, with a Wien bridge:

T .s/ D 1 �K Œ�.s/� ˇ� D 1 �K

�s�

s2�2 C 3s� C 1� ˇ

�

: (5.21)

From here we can obtain the characteristic equation

Q1.s/ D s2�2.1CKˇ/C s� Œ3 � .K � 3Kˇ/�C 1CKˇ;

which at K � 1 has two complex-conjugate roots s1;2 D �˛1 ˙ j!1:

The solution of this equation has the form:

V1.t/ D Vme�˛

1t sin.!1t C '/;

where

!1 Dq

!20 � ˛21; ˛1 D .3ˇ � 1/=2ˇ�:

In the steady-state mode at 3“ – 1 D 0, the roots have only the imaginary part,and the oscillations are strictly sine-wave with the constant amplitude and thefrequency !1 Š 1 =� . This case corresponds to an unstable oscillating system, butthis instability is useful. It is inherent in all oscillating systems, because it gives riseto sine-wave oscillations.

Now let us analyze the oscillating system with a Wien bridge, included in thenegative feedback loop. In this case, the characteristic equation

Q2.s/ D s2�2.1 �Kˇ/C s� Œ3C .K � 3Kˇ/�C 1 �Kˇ

has the practically close roots s1;2 D �˛1 ˙ j!1:

Theoretically, the characteristic equation in this case at 3“ – 1 D 0 also cor-responds to an unstable system and has a sine-wave solution, and with the idealfrequency-independent op-amp the both systems are practically identical. However,with a real op-amp, there are no sine-wave oscillations in such a system. Thesame analysis performed for other oscillating systems with the ideal frequency-independent op-amp shows the same. Consequently, from the roots of the equationaccounting only for the frequency dependence of second- and higher-order linearcircuit, it is impossible to judge the potential efficiency of some or other circuit.

Obviously that in analysis it is necessary to take into account the small inertness of otherelements and, in the first turn, the op-amp time constant. This is just the case, which iscalled the effect of a small parameter (small time constant) in theory. Therefore, to find theproper connection of the frequency-dependent RC-circuit, it is necessary to take the op-ampinertness into account. Without this, the problem cannot be solved.


Taking into account the op-amp inertness determined only by the time constant£1, the characteristic equation Q1(s) of the system on Fig. 5.14a with the Wien bridgecan be written in the form

Q.s/D s3�2�1 C s2.�2.1CKˇ/C 3��1/C s.� Œ3 � .K � 3Kˇ/�C �1/C 1CKˇ:

In addition to two complex-conjugate roots giving the sine-wave oscillation, thisequation has one more root – real negative one. The oscillation process in thissystem has two components: sine-wave and exponential

V1.t/ D Vme�˛t sin.!1t C '/C V�e

��t :

The exponential oscillations caused by the third root damp with time. At thesame time, with the op-amp inertness, the oscillating system with the frequency-dependent Wien bridge included in the negative feedback loop is described bythe other characteristic equation, whose third root is real and positive. Becauseof this root, the oscillation process, to the contrary, grows with time, thus puttingthe amplifier into the saturation mode. Based on this, the Wien bridge should beincluded into the positive feedback loop. Certainly, this analysis can be extended toother, less known frequency-dependent circuits as well.

Thus, for correct connection of some or other frequency-dependent RC circuit and creationof the conditions for excitation of sine-wave oscillations, it is necessary to analyze thepositions of zeros of the characteristic equation taking into account the op-amp inertness.The characteristic equation in this case should have two complex-conjugate roots with thepositive real part, while all other roots should have the negative real part.

5.4.2 The Oscillation Amplitude Stabilization and NonlinearDistortions

Oscillations increase after excitation, tending to the steady state. At this stage, the nonlinearelement of the oscillating system exerts the increasing influence, restricting the rate ofincrease. As this takes place, the positive real part of the roots decreases, while theimaginary part remains practically unchanged. On the complex plane p, the roots of theequation begin to move toward the imaginary axis. As the oscillations achieve the steadystate, their further growth terminates. The roots of the equations move to the imaginaryaxis. Just this task – control of the root positions – is executed by the nonlinear element. Itchanges the decrement ˛ of the oscillating system (the real part of the roots) and determinesthe oscillation amplitude in the steady-state mode. To change the damping decrement, thenonlinear element is usually introduced into the circuit ˇ, that is the resistive feedbackcircuit.

In the linear oscillating system the characteristic equation also can have twocomplex-conjugate roots with the zero real part, and the oscillations will be sine-wave. However, the amplitude of oscillations in it depends only on the initialvalues of voltages across reactive elements and changes at the minor changes of


V1 +

−

V2

R2

R3

VDNE

AE

R1

Fig. 5.18 Circuit of active nonlinear op-amp element (a) and its gain characteristic (b)

the parameters of the oscillating system (shift of the roots from the imaginaryaxis). Stabilization of oscillations in the linear system with distortions is principallyimpossible. Just this contradiction was noted above. Therefore, for stabilization ofthe amplitude in the steady state, the nonlinear element is necessary. It changes ’toward negative values, if the amplitude is damped and should be increased, andtoward positive values, if the amplitude is increased and should be decreased (seeFig. 5.7), that is, makes the oscillations stable to disturbances.

Stabilizing self-oscillations, the nonlinear element usually distorts the signal waveform andenriches the spectrum. This circumstance was also mentioned above as a contradiction,forming the basis of other problem in construction of oscillators, namely, minimization ofnonlinear distortions. Therefore, the following analysis of nonlinear elements is performedkeeping in mind minimization of distortions in the output signal.

The amplitude stabilization can be achieved due to the natural ANE nonlinearity(Fig. 5.2d), which is inherent in all amplifiers, when the output amplitude increasesslowly with the increase of the oscillation amplitude at the ANE input. However,this nonlinearity is difficult to control and cannot be normalized; besides, theoscillation amplitude is not known before. Much better are the circuits, in whichnonlinearity is set by specialized elements with the normalized nonlinear current–voltage characteristic, for example, Zener diode (Fig. 5.18).

The active nonlinear element is a nonlinear voltage former on op-amp and double-sidelimiter VD. At the input voltage V1 <Vm1 (Fig. 5.18 b), the diodes are closed, and the gainis determined only by the resistors R1 and R2, as in the non-inverting op-amp amplifier. Atthe voltage higher than Vm1 , when the difference Vm2 – Vm1 D Vm1 R2 /R1 becomes equal tothe stabilization voltage, the VD diodes open, connecting the resistor R3 in parallel to theresistor R2 , due to which the gain decreases, which can be seen from the smaller slop ofthe gain characteristic. However, the peak of the sine wave becomes blunt, the waveformof the oscillations differs more strongly from the sine wave, and higher harmonics and,consequently, nonlinear distortions arise.

This method of oscillation control is used when the requirements to nonlineardistortions are not strict or at the high selectivity (Q-factor) of the linear frequency-dependent circuit, which can be used to minimize the nonlinear distortions. If the


Fig. 5.19 Circuit of the active inertial nonlinear element (b) and the resistance of the filamentlamp as a function of current (a)

requirements of the nonlinear distortion coefficient are strict enough, inertialnonlinear elements (INEs), such as thermal resistors and filament lamps are used.INEs have the nonlinear resistance depending on the root-mean-square, rather thaninstantaneous current, therefore they do not distort the waveform and, consequently,introduce no distortions. The circuit of such element with the filament lamp is shownon Fig. 5.19. The lamp here plays the same stabilizing role, as the resistors R2, R3

and the VD diode do in the circuit on Fig. 5.18a.It is known that for a lamp with the metal filament, it resistance depends on

the temperature, which depends on the electrical power scattered in the filament.The scattered power is determined by the current value. At the low current, thelamp resistance is low and the gain is Kfb D (1 C R2/Rlamp)> 1/”, so the oscillationsincrease, the current passing through the lamp increases too, and the resistancegrows. At the time when the resistance takes the value R1, at which Kfb D 1/”, theoscillations no longer increase, and the steady state is established. An advantage ofthis nonlinear element is its simplicity.

The main disadvantage of the circuits with INE is that their resistance dependsnot only on the current, but also on the environment temperature, because it alsoaffects the filament heating and the resistance variation. Finally, the amplitude ofthe oscillator output voltage changes. Because of that inertial nonlinear elementsare not practically used in modern oscillators.

Recently, role of nonlinear elements is played by the rather complex systems for automaticstabilization of the oscillation amplitude. The problem for the decision is the same as that ofthe simplest nonlinear element, namely, to restrict the oscillation build-up after excitationand to stabilize the amplitude. Figure 5.20 shows the circuit of the oscillating systemwith automatic stabilization of the oscillation amplitude for the Wien bridge oscillator forexample.

In the rectifier (Rec), the alternating voltage is converted into the direct one,whose value is proportional to the amplitude of the alternating voltage. Then in thecomparison circuit (C) it is compared with the reference voltage from the source


Fig. 5.20 Oscillator withautomatic amplitudestabilization

Fig. 5.21 FET control element (a), its current–voltage characteristics (b), and the regulationcurve (c)

(Ref), and after amplification of the error signal in the error amplifier (EA) itchanges the resistance of the control element (CE). The resistance of the controlelement determines the gain “, so its change affects the damping decrement. Theoscillation amplitude either increases or decreases until reaching the equal voltagesat the inputs of the comparison circuit. The joint action of all elements of this circuitis equivalent to the action of the nonlinear element.

As was shown in Chap. 4, the conversion of the alternating voltage into thedirect one is possible only in nonlinear elements. The control element can be alinear element with the variable resistance that is the parametric one. Therefore, thestabilization system is not simply nonlinear, it is nonlinearly parametric, but this factdoes not change its purpose.

To realize an oscillator, it is possible to take an operational rectifier as a converter.The source of reference voltage can be easily designed in the circuit of the Zenerdiode voltage reference. The error signal can be amplified by an integrating op-ampamplifier. A field-effect transistor (FET) is usually used as a control element.

This should be considered in more details. It is known that if the drain-to-sourcevoltage (Vds) is low, then at variation of the gate-source voltage (Vgs) FET behavesas a controlled resistor, whose resistance depends on Vgs. Just this property allowsFET to be used as a control element. Figure 5.21 shows such a FET element withthe control p–n junction and the p-type channel with linearizing resistors along withits regulation curve.


Fig. 5.22 Oscillator with automatic stabilization circuit with amplitude detector

Fig. 5.23 Operation diagrams of oscillator with the amplitude stabilization circuit

The circuit on Fig. 5.20 has some disadvantages. First, the alternating-voltagerectifier (Rec) performs the conversion not exactly. In addition to the usefulcomponent, its output voltage includes the parasitic variable component, which isamplified by EA after comparison and affects CE, thus giving rise to nonlineardistortions of the output voltage. Second, the stabilization circuit involves severalnonlinear elements and so it can be unstable, which make itself the possibleintermittent generation of oscillations. The circuit on Fig. 5.22 is much better [6].

The oscillation amplitude in this circuit is stabilized by a pulsed automaticregulation system based on an amplitude detector (AD). In the limiter (L), theamplitude is compared with the direct voltage from the reference (Ref). If thisvoltage is exceeded, the cutoff part of the peak is amplified by the error amplifier(EA) and detected by AD. The direct voltage from the AD output controls the CEresistance, setting the needed gain Kfb.

Figure 5.23 shows the diagrams of the voltages at the outputs of the basicelements: oscillator, EA, and AD.


Fig. 5.24 Conservative op-amp oscillation circuit

In the top diagram, the output voltage Vout is shown with the varying amplitudefor illustration. The straight line in this diagram indicates the voltage of the referencedirect voltage source Vref, at whose level the amplitude is limited. The amplifiedcutoff vertices are shown below (Vea). The bottom diagram demonstrates the outputvoltage of the amplitude detector Vad.

Stable oscillations can be easily obtained in this structure. The error amplifierprovides for the main amplification in the stabilization circuit, and detection iscarried out after amplification, so the saw-tooth pulses of the detector voltage arenot amplified and cause only small extra nonlinear distortions.

However, if oscillator with ultra-low nonlinear distortions (about 0.001%) isnecessary, this design is impracticable as well. To obtain very low distortions, itis necessary to apply a set of measures aimed at their reduction.

The sources of distortions in oscillator are active elements, such as amplifiers and thestabilization system. The amplifier distortions can be reduced by linearizing the gaincharacteristic of the amplifier through introduction of large-amount negative feedbacksand the use of the best frequency-dependent circuits. Therefore, the best results are obtainedwith the use of active frequency-dependent circuits [17], in particular, the so-called linearconservative oscillation circuit (Fig. 5.24).

In accordance with the above definitions, it is the active frequency-dependentcircuit. It has three outputs, whose voltages are shifted by 90ı and, consequently,the voltages V1 and V3 are shifted by 180ı.

The system of equations for the conservative oscillating circuit is following:

PV1 D PV2 PK3; PV2 D PV3 PK2; PV3 D PV1 PK1;

where PK are the complex gains of the integrators in OA2 and OA3 and the inverterin OA1.


Excluding the voltages from this system, we find

1 � PK1PK2

PK3 D 1 �K1

�

� 1

j!�1

� �

� 1

j!�2

�

D K1 C !2�1�2 D 0;

where K1 D �R2/R1, £1 D R3C1, £2 D R4C2.As can be seen, the complex characteristic equation involves only the real part.

In the operator form, it has two complex-conjugate imaginary roots with zero realpart. Its solution is a sine-wave oscillation with the arbitrary amplitude determinedby the initial conditions.

The oscillating system constructed by this equation is commonly called con-servative. The oscillations in this system, having appeared when the system isswitched on, can continue as long as desired. Certainly, in practice they decreasegradually, because the op-amp gains are though high but still finite, the capacitorshave losses, and resistors have parasitic parameters. However, if we ignore them,the characteristic equation allows us to determine the frequency of oscillations:

!0 Dp

jK1j =.�1�2/ ;

which depends on all the elements here. The resistors and capacitors are usuallytaken with the equal parameters, then jK1j D 1, £1 D £2 D £, and ¨0 D 1/£.

To control the oscillation amplitude, the oscillating system is complemented withthe resistors R6 and R5, R7. With the equality R6/(R1jjR2) D R7/R5 there is a balancebetween the exciting and damping parts. Any deviation from this equality results ineither increase or decrease of oscillations. Thus, they can be controlled by changingthe resistance of any of these resistors, for example, R5 (using FET instead it).

The nonlinear distortions of the stabilization system can be decreased usingthe oscillator circuit on Fig. 5.25, based on the conservative oscillating system.It employs the automatic stabilization system with a strobe amplitude detector,similar to the circuit on Fig. 5.22. The amplitude detector is constructed in thetransistor switch (Sw) and the capacitor C3, and the switch opens for a short timeby pulses from the pulse shaper (PSh) at the time, when the output voltage achievesits maximum (amplitude) value. The limiter and the comparison circuit here aredesigned in OA5, the diode VD, and the resistors R7–R9, while the error amplifier ismade in OA4 in the kind of proportional integral amplifier (PIA).

The oscillator operation is similar to the operation of the circuit on Fig. 5.22 andcharacterized by the upper diagrams on Fig. 5.23: the output voltage Vout and thevoltage at the EA output Vea. However, the voltage at the output of the amplitudedetector Vad does not decrease between pulses (Fig. 5.23), but remains constant,and in the established steady-state mode it includes no pulsations and introduces nodistortions into the output signal. The oscillator has the frequency band from 1 Hzto 200 kHz (with switching of the capacitors C1, C2 and resistors R3 and R4), andat the output voltage of 2 V it allows obtaining the sine-wave oscillations with thetotal harmonic distortion less than 0.0001% in the audio frequency band [7].


Fig. 5.25 Oscillator with the automatic stabilization circuit with the strobe amplitude detector

In this oscillator circuit, all the contradictions arising in the oscillating system, first of all,between the level of nonlinear distortions and the stability of the oscillation amplitude areremoved, since the distortions of the output voltage in it theoretically can be equal to zero.Practically, if the elements are manufactured perfectly, then the distortions can be decreaseddown to arbitrarily small values. Moreover, short transient processes, whose duration doesnot exceed one to two periods of the output voltage, can be achieved in this oscillator, thatis, the contradiction between nonlinear distortions, stability, and duration of the transientprocesses is removed as well [7].

5.5 LC Sine Wave Oscillators

5.5.1 Transformer-Coupled LC Oscillators

As was already mentioned, the oscillation frequency in LC oscillators is inverselyproportional to the square root of LC, therefore the nominal values of the capac-itance and inductance decrease more slowly with the increasing frequency thanin the RC oscillating systems. At the high frequencies they are higher than theparasitic capacitances and inductances. Therefore, the oscillation frequency of theLC oscillator more weakly depends on the parasitic parameters than in the RCoscillator. This explains why the LC selective circuit is used as the linear frequency-dependent circuit at high frequencies of the generated oscillations, despite that aninductance is less technological and more expensive element than a resistor. At the

5.5 LC Sine Wave Oscillators 149

Fig. 5.26 FET LC oscillator

same time, op-amps serving active elements at low frequencies lose the amplifyingproperties with the increase of the frequency and are not used in LC oscillators.

LC sine wave oscillators are applied most widely, in which the positive feedback isassembled in LC resonance circuits and the active element is constructed in a bipolar orfield-effect transistor. Such a FET oscillator with the resonance circuit in the drain circuitis shown on Fig. 5.26 .

The oscillating system in this circuit is formed by inductance L1 of the trans-former primary winding and the capacitor C. The transformer secondary winding isconnected to the FET gate VT through the chain of the resistor Rg and the capacitorCg. The FET drain is connected to the primary winding tap of the transformer of theoscillation circuit. This connection is called partial and used to decrease the effectof the oscillator load onto the Q-factor of the oscillating system.

The FET and the oscillation circuit form the resonance amplifier. To providefor the positive feedback, the oscillator terminals of the transformer (marked by anasterisk) are connected so that the transformer generates the additional phase shiftof 180ı to the phase shift of the resonance amplifier, which is also equal to 180ı atthe resonant frequency.

Figure 5.27 shows the most illustrative simplified circuit of the oscillator withoutthe power supply and bias circuits and extra elements along with the equivalentcircuit.

The oscillator starts to operate as the power supply voltage Vcc is applied throughthe smoothing filter Rf, Cf to diminish the effect of its possible pulsations. Since thecapacitor Cg was discharged before power-on, the FET gate-to-source voltage Vgs

(Fig. 5.28) was zero. As known, the FET transconductance and the gain of the res-onance amplifier are maximal in this case, so the circuit generates self-oscillationsat the resonant frequency. The positive halfwave of voltage from the transformersecondary winding opens the control p – n junction, and the gate begins to traversethe current Ig, which charges the capacitor Cg. At the negative halfwave, the FETcontrol p – n junction shifts in the backward direction, and this current terminates.Thus, the current through the capacitor is pulsed, and it charges the capacitor,


Fig. 5.27 Simplified (a) and equivalent (b) circuits of LC oscillator

5m

Vol

tage

(A

) Voltage (B

)

−20

−13

−7

0

7

3

20

7m 9m 11m 14m 16m

genlc.ewb

18m 20m

−6

−4

−2

0

2

4

6

Fig. 5.28 LC oscillator voltage diagrams calculated in Electronics Workbench (a) and theoreti-cal (b)

generating the negative voltage at the gate (dashed curve in the V diagram). Atthe negative gate-to-source voltage, the FET transconductance decreases, the gaindecreases too, and at some amplitude the amplitude balance condition is fulfilled.The further growth of oscillations terminates, and the steady-state self-oscillationmode is established. In this mode, the capacitor Cg is periodically recharged bycurrent pulses at the time, when the gate-to-source voltage exceeds the zero level,and most of the time it slowly discharges through the resistor Rg. Thus, just theRg, Cg chain controls the FET transconductance and, consequently, ensures thefulfillment of the amplitude balance condition. Figure 5.28 shows the oscillatoroperation diagrams from excitation of self-oscillations to the establishment of thesteady-state mode.

� Using the linearized equivalent circuit (Fig. 5.27), write the basic equations ofthe oscillator:

id D SVgs; Vgs D nVL D nLdiLdt;

where id is the drain current, S is the FET transconductance, Vgs is the gate-to-sourcevoltage, n is the transformer ratio.


According to the first Kirchhoff law, the drain current of FET VT is equal to thesum of the currents in the FET inductance, capacitor, and internal resistor

id D iL C iC C iRi ;

where

iC D CduCdt:

In its turn,

vc D riL C LdiLdt;

where r is the resistance of the coil wire. Excluding Vc , we obtain

iC D rCdiLdt

C LCd2iL

dt2:

Substituting the value of iC , find the drain current

id D iL C rCdiLdt

CLCd2iL

dt2C r

RiiL C L

Ri

diLdt

D nSLdiLdt;

whence it follows that

d2iL

dt2C

�r

LC 1

RiC� nS

C

diLdt

C 1

LC

�

1C r

Ri

iL D 0: (5.22)

Introduce the following designations:

!0 D 1pLC

is the resonant frequency of the oscillation circuit,

˛ D 1

2

�r

LC 1

RiC� nS

C

is the oscillation increment or decrement, having the same meaning as in Eq. 5.10.Then Eq. 5.22 at Ri � r takes the form

d2iL

dt2C 2˛

diLdt

C !20 iL D 0: (5.22a)


The solution of Eq. 5.22a has the form

iL D IL me�˛t sin.! t C '/; (5.23)

whereIL m is the current amplitude; ! Dq

!20 � ˛2 is the frequency of oscillationsin the circuit.

Equations 5.22a and 5.10 are similar, so at ’< 0 the oscillations in the circuitincrease, while at ’> 0 the oscillations damp. At ’D 0 the steady-state oscillationmode is established in the circuit. It is useful to note that ’ becomes negative becauseof the negative sign of nS

C. This is connected with the sign of the transformer ratio n.

Taking the different sign of n, that is, connecting the transformer secondary windingin a different way (connecting the lead marked by the asterisk on Fig. 5.26 to thecommon wire rather than the resistor Rg), it is impossible to obtain the negativevalue of ’. Consequently, the increase of the oscillations is impossible. The signof n determines the form of the feedback in the circuit. On Fig. 5.26 the positivefeedback is chosen. It should be noted that earlier, when analyzing the processes bythe methods of phase plane and complex amplitudes, we also assumed the presenceof just positive feedback in all the considerations. Without the positive feedback theoscillations do not increase.

From the analysis performed, we can obtain the amplitude and phase balanceconditions in the LC oscillator. The amplitude balance occurs at ’D 0, that is, atrL

C 1RiC

� nSC

�D 0, while the phase balance takes place at !0 D 1p

LC. The

plots of the current in the inductance coil coincide with the plots on Fig. 5.7. Thesteady-state self-oscillation mode is established due to the decrease of the FETtransconductance with the increasing oscillation amplitude.

5.5.2 Three-Point Oscillators

In the transformer-coupled oscillators, to obtain the additional phase shift by 180ı, atransformer with strong inductive coupling is needed, which is a significant disadvantage.Such a transformer usually has not only the inductive, but also the capacitive couplingbetween the windings, which changes the above equations and worsens the accuracy offrequency setting. Therefore, the oscillators employing the oscillation circuit with three LCelements to fulfill the condition of antiphase gate and drain voltages have gained the wideutility. The circuits of these oscillators are shown on Fig. 5.29 .

In the oscillator on Fig. 5.29a, the oscillation circuit consists of the capacitor Cand the two inductanceL1 andL2 connected in series in the autotransformer circuit.In the oscillator on Fig. 5.29b, the inductive divider is replaced by the capacitivedivider consisting of the two capacitors 1 and C2 and the inductance . Another coilLb here plays the blocking role, eliminating the effect of the power supply on thecircuit operations.


Fig. 5.29 Oscillators with three LC elements

Fig. 5.30 Simplified circuits of oscillators with three LC elements and vector diagrams of voltagesacross the circuit elements

To fulfill the phase balance condition, the opposing terminals of the circuit areconnected between the drain and the gate through the capacitor Cg of the self-oscillation control chain Cg, Rg. The simplified circuits and vector diagrams ofoscillators of this type are shown on Fig. 5.30.

It can be seen from the vector diagrams that the voltages VL1, VL2 of the firstcircuit (Fig. 5.30a) and VC1, VC2 of the second circuit (Fig. 5.30b) are in antiphase, asneeded for introduction of the positive feedback and fulfillment of the phase balance.The ratio of the vector lengths is equal to the feedback coefficient, which determinesthe amplitude balance in the oscillator. Hartley and Colpitts proposed other versionsof the oscillators of this kind, in which there is no need in phase inversion,because the transistors in them are connected in the common-base (common-gate)or common-collector (common-drain) circuits. As known, in this case an amplifierdoes not invert the signal phase. The feedback here also should not introduce a phaseshift, therefore it is made with the inductive or capacitive divider. The circuits of theHartley and Colpitts oscillators are shown on Fig.5.31a, b.


Fig. 5.31 Hartley and Colpitts LC oscillators

Both the LC and RC oscillators have a common significant disadvantage, namely,the unstable frequency of the generated oscillations. This instability is about 0.1%in the RC oscillators and about 0.01% in the LC oscillators. Quartz oscillators arecharacterized by the much higher stability (frequency instability from 10�4% to10�9%).

5.6 Quartz Oscillators

Quartz 5 oscillators have gained the wide application owing to the unique electromechan-ical properties of quartz. Mechanic vibrations of quartz plates generate surface electricvoltages with the same frequency, and vice versa: the applied alternating voltage generatesmechanic vibrations, and the frequency of these vibrations is determined by almost only theplate dimensions and is independent of both the temperature and the applied voltage.

The electromechanical oscillating system of the quartz oscillator, i.e. resonator,is made as a quartz plate with electrodes placed in an evacuated case.

Figure 5.32a shows the equivalent circuit of the quartz resonator as an oscillationcircuit, where L is the quartz equivalent inductance, r is the loss resistance, C1 andC2 are the resonator capacitances. This circuit can have the voltage resonance at thefrequency!u � 1=

pLC1 and the current resonance at the frequency!i � 1=

pLC2,

and !u < !i . Due to the low loss resistance, the frequency characteristic of thecircuit has a very sharp peak and high Q. The circuit of the quartz oscillator isshown on Fig. 5.32b, where QR is the quartz resonator.

5Quartz is a natural or artificial mineral having the piezoelectric properties.

5.7 Negative Resistance Oscillators 155

Fig. 5.32 Equivalent circuit of quartz resonator (a) and circuit of quartz oscillator (b)

5.7 Negative Resistance Oscillators

Besides the considered oscillators based on the two-port and three-terminal circuits, one-port oscillators are widely used. In Sect. 5.2 it was shown that the condition for excitationof self-oscillations in such structures is the equality Y 1122 D �Y 122. It means the equalityof the complex conductivities of the linear frequency-dependent element and the activeelement. The conductivity of the latter should be negative, that is, a one-port with negativeconductivity is needed for appearance of oscillations.

Active elements, whose negative resistance is determined by their physical structure,are known. They include some types of semiconductor diodes (dinistors, thyristors, tunneldiodes, impact avalanche transit-time [IMPATT] diodes), unijunction and avalanchetransistors, as well as secondary-emission electron tubes and gas-discharge tubes. Circuitsmodeling one-port with negative resistance are also known; they are negative resistanceconverters.

The current–voltage characteristics for some of these elements are shown onFig. 5.33. Dots mark the boundaries of the negative-resistance field. According tothe shape resembling the letters N and S, all the current–voltage characteristics canbe divided into two types. Thus, the tunnel diode and the tetrode tube have thecharacteristics of the N type, while the characteristics of the dinistor, the unijunctionand avalanche transistors, and the gas-discharge tube can be classified to the Stype. This classification is connected with the different dependence of the elementconductivity.

Figure 5.34 depicts the voltage dependence of the differential and averaged (inthe vicinity of point A) conductivity of the element with the N-type characteristic,while Fig. 5.35 shows the current dependence of the differential and averagedresistance of the element with the S-type characteristic. The selection of the voltagein the first case and the current in the second case as an independent variable iscaused by the unique dependence of, respectively, the conductivity and resistanceon them and the ambiguous dependence on other parameters. Few words shouldbe said about the dependences. The differential dependences are obtained through


Fig. 5.33 Voltage-current characteristics of tunnel diode (a), dinistor (b), unijunction transis-tor (c), avalanche transistor (d), tetrode tube (e), gas-discharge tube (f)

Fig. 5.34 Voltage dependence of the differential (b) and averaged (c) conductivity for the N-typecharacteristic (a)

Fig. 5.35 Current dependence of the differential (b ) and averaged (c) resistance for the S-typecharacteristic (a)


Fig. 5.36 Dependences of the averaged conductivity for the N-type (a) and S-type (b) character-istics

Fig. 5.37 Frequency (a) and amplitude (b) dependence of the absolute value of AE and LFDEconductivity

calculation of the derivatives at any point, while the averaged dependences are foundthrough determination of the first harmonic of Fourier series expansion of current(voltage) at different values of the amplitude of sine-wave voltage (current) in thevicinity of the working point A.

To describe the differences in the character of the conductivity dependence, onFig. 5.36 the plots of the averaged conductivity for the N – and S-type characteristicsis shown. From their comparison, we can see that the negative conductivity Ytd of theelement with the N-type characteristic (tunnel diode) decreases with the increasingvoltage, while the conductivity Yd of the element with the S-type characteristic(dinistor), increases to the contrary.

As was already mentioned, the negative conductivity of the active element is used tocompensate for the loss in the passive elements of the frequency-dependent circuit. If ata certain frequency it is equal to the positive conductivity of the frequency-dependentelement, most often, the oscillation circuit, then, including an active element in its structure,it is possible to compensate for the loss in it and, thus, generate continuous waves. Atother frequencies, the negative conductivity of the active element should be lower than theconductivity of the frequency-dependent one-port to exclude the possibility of appearanceof self-oscillations. It follows herefrom that the frequency-dependent circuit should have theconductivity dependence shown on Fig. 5.37.


Fig. 5.38 Tunnel-diode oscillator

The dependence of the absolute value and the phase of LFDE conductivity onits frequency (its GFC and PRC) are shown on Fig. 5.37a. The AE conductivityis taken frequency-independent, and its phase is taken zero. The equality betweenthe phases of the positive and negative conductivity is achieved at the frequency¨0, and self-oscillations can arise just at this frequency. However, “can” does notmean “must,” because the conditions for the increase of the amplitude should takeplace. So it is needed for the absolute value of the negative conductivity of theactive element to be greater than that of the frequency-dependent element at lowamplitudes. Thus, the amplitude of self-oscillations should be determined from theother plot on Fig. 5.37b. In this plot the averaged AE conductivity becomes equalto the LFDE conductivity, which, by the way, is voltage-independent due to LFDElinearity, at the amplitude V1m. It is just the amplitude of self-oscillations. In fact,the phase and amplitude balance conditions are fulfilled at these frequency andamplitude. It is obvious that such dependences are inherent in the active elementswith the N-type characteristic (for example, the tunnel diode) and in the parallelLC oscillation circuit, whose dependence of the absolute value of the conductivity(GFC) resembles the letter V. It is also clear that the parallel oscillation circuitcannot be used together with the elements having the S-type characteristic, forexample, a dinistor, since the conditions for the increase of the amplitude are notmet, and the tunnel diode cannot be applied together with the serial oscillationcircuit, the dependence of the absolute value of whose conductivity resemblesƒ.

Thus, self-oscillations are possible in the following cases: if the active elementhas the current–voltage characteristic of the N-type and the frequency-dependentelement has the frequency characteristic of conductivity of the V-type or if AE hasthe current–voltage characteristic of the S-type and LFDE has the conductivity ofthe ƒ-type.

Figure 5.38 shows the tunnel-diode oscillator VD along with the current–voltage character-istic of the tunnel diode and the equivalent circuit of the oscillator. The oscillator includesthe tunnel diode, the power supply E, the inductance coil L with the resistance r, and thecapacitor C.


The tunnel diode with current–voltage characteristic on Fig. 5.38b has thenegative differential resistance rdif D � (20 : : : 100) Ohm in the b–d region. Thesupply voltage E and the resistance r are chosen so that the initial position of theworking point A falls within the negative resistance region (b–d region).

� In the equivalent circuit, one can see two one-ports: the linear frequency-dependent parallel oscillation circuit, consisting of the capacitor C and the coil Lwith the resistance r, and the active element represented by the resistor with negativeresistance �R. Write the basic equation of the oscillator, assuming that the sum ofcurrents at the point A is zero according to the Kirchhoff law: ic C il C i�R D 0.After simple transformations, similar to those performed when deriving Eq. 5.22,we obtain

d2iL

dt2C

�r

L� 1

CR

diLdt

C 1

LC

1 � r

R

�iL D 0: (5.24)

Equation 5.24 is similar to Eq. 5.22 of FET LC oscillator on Fig. 5.26 with thedifference that the FET transconductance is replaced by the negative resistance �Rand the last term has the factor 1 – r/R in place of 1 C r/Ri . This does not changeprincipally the processes in the oscillator, because the coil resistance r is usuallymuch lower than the negative resistance –R. If we assume that the resistance of theactive one-port is positive, rather than negative, then the factor in the second term ofEq. 5.24 is positive and, consequently, the oscillations are only damping.

The negative resistance of the active one-port, as well as the positive feed-back, creates the conditions for the increase of oscillations in the oscillator at. rL

� 1RC/<0. With . r

L� 1

RC/ D 0 the increase of the amplitude terminates, and at

. rL

� 1RC/>0 it decreases. To stabilize the amplitude, this factor is to be changed.

But the values of L, C, r are constant and independent of the oscillation amplitude,therefore the negative resistance (�R) changes. This occurs automatically, becauseat the increase of the self-oscillation amplitude the variations of the AE currentgo beyond the negative resistance region b–d. As this takes place, the negativeresistance increases, while the conductivity decreases (see the plot on Fig. 5.37b).As the amplitude decreases, to the contrary, the negative conductivity increases upto the maximum value at the point A (Fig. 5.38b). Thus, in the circuit with the one-port active element, the self-oscillation amplitude stabilizes due to its nonlinearity.However, as was already mentioned, nonlinear distortions of the output signal ariseat such stabilization. In this circuit, as well as in other circuits of LC oscillators, theyare low because of the high Q-factor of the oscillation circuit.

The analysis confirms that the processes in oscillators with two-port, three-terminal, andone-port active elements are identical at the negative resistance of the last ones and withthe positive feedback in the first elements.


5.8 Synthesis of Oscillating Systems of RC Oscillators

It is shown above that an oscillating system (OS) is a basis of any generatorof electric signals. Periodic oscillations are generated and established in it. Itdetermines their form and main parameters. That is why numerous papers (forexample, [8–10]) are devoted to the analysis and synthesis of oscillating systems.These sections consider methods of analysis and synthesis of oscillating systemsat active elements, such as amplifiers, negative impedance converters, gyrators,and so on. In particular, a technique for the synthesis of OS based on the matrixmethod for calculation of electric circuits is proposed [8]. In this technique, the OSsynthesis process is maximally formalized. A disadvantage of its application is theuse of different matrices of types A, Z, Y, and H for the description of elementsof the same OS. The graph method is also used for the synthesis. Often synthesistechniques applicable to OS at some active elements are difficult to apply to OS atother elements. This paper undertakes an attempt to develop a versatile techniquefor the synthesis of OS for sine-wave oscillators.

As was mentioned above, OS should include a linear frequency-dependent circuit (LFDC)and an active element (AE) (for example, amplifier).

Figure 5.39 shows schematically a generalized oscillating system with LFDC andone AE. This circuit connects two two-port circuits (Fig. 5.11).

Letters x and y in the schematic are for electric parameters. Depending on thetype of used AE, they can be either voltages or currents6.

First, we analyze the oscillating circuit. For this purpose, some simplifications areneeded. The four-pole (two-port) LFDC can be easily transformed into a three-polecircuit, if we connect directly two of four external terminals, as in Fig. 5.11c. Here,the four-pole circuit Y1 is represented by the linear frequency-dependent circuit, thefour-pole circuit Y2 is represented by the active element. Just three-pole LFDC aremost often used in practice. It is preferable to represent AE as a three-pole circuitas well.

Fig. 5.39 Generalized schematic of an oscillating system

6Hereinafter, AE is assumed to be an idealized controlled source with the frequency-independenttransfer factor.

5.8 Synthesis of Oscillating Systems of RC Oscillators 161

Fig. 5.40 Schematics of three-pole LFDC (a), AE (b), and their “basic” coupling (c)

Figure 5.40 shows three-pole LFDC, active three-pole element, and their “basic”7

coupling.Three-pole LFDC and AE can be coupled in different ways depending on

the chosen common, input, and output terminals, that is, different topologicaltransformations can be applied to the basic connection in order to obtain newcircuits. Figure 5.40c shows a version of OS, in which the zero-potential point(common terminal 0) is coupled with LFDC terminal 3 and AE terminal c. Couplingother LFDC terminals with the common terminal or changing the circuit input andoutput (LFDC rotation), we can obtain five more OS circuits.

Structures with similar properties can also be obtained through the rotation ofan active element. The rotation of LFDC and AE terminals gives identical OSstructures, if we neglect the method of supply voltage application to AE. It isnatural to expect that they have identical oscillation parameters. Therefore, theAE rotation gives no advantages to a linear OS, but only complicates its practicalimplementation. That is why it is not considered below.

In a particular LFDC, for example, Wien bridge and amplifier, we can see (Fig. 5.14) howthe configuration of the principal OS circuit changes at the alternation of only external

7Here, the basic coupling is any known coupling of LFDC and AE, in which the condition ofamplitude and phase balance can be fulfilled.


Fig. 5.41 Structures of three-pole LFDC with three (a), two (b), and one admittance parame-ters (c)

terminals of the circuit or amplifier. It is natural that the transfer functions of the circuitsand the OS characteristic equations change at these transformations. The amplitude andphase balance conditions can change as well.

As an active element with the unidirectional energy transfer, we can use voltagecontrolled voltage source (VCVS), current controlled current source (CCCS),voltage controlled current source (VCCS), or current controlled voltage source(CCVS). Thus, with one LFDC, using different AEs and changing the connectionof LFDC terminals, we can create 24 structures with possible new structures amongthem. Then we have to determine in which of them the amplitude and phase balancecondition can be fulfilled and periodic oscillations can be obtained. Naturally, wecannot do that without knowledge of the internal circuit of LFDC and AE.

Any LFDC circuit includes R, L, and C elements coupled in a certain way. Whateverdifficult is the connection of elements, mutual admittance between external terminals isalways formed in them. That is why LFDC is represented by three admittance parametersin Fig. 5.41a.

To determine the admittance parameters G1, G2, and G3, we can apply the matrixmethod [6] to any circuit with the following lowering of the matrix order.

Figure 5.41a depicts the structure of the circuit, in which all the mutualadmittance parameters PG1, PG2, and PG3 exist between circuit terminals. An exampleof this structure is the structure of the LC circuit of capacitance (C) or inductance(L) three-point circuits.

Another circuit structure is shown in Fig. 5.41b. The admittance PG1 is absentin this circuit. The structure of the RC circuit of the Wien bridge can serve as anexample here. Topologically, it is isomorphic to other structures, in which PG2 orPG3 is absent. It follows here from that two more isomorphic structures exist for the

structure shown in Fig. 5.41b.Two admittance parameters PG1 and PG3 are absent in the structure shown in

Fig. 5.41c. It also has two isomorphic structures. However, all structures with oneadmittance parameter transform a three-pole circuit into a two-pole one.

Thus, to determine the transfer function of the circuit in Fig. 5.41a, we write itssingular admittance matrix


PY D2

4

PG1 C PG2 � PG2 � PG1� PG2 PG2 C PG3 � PG3� PG1 � PG3 PG1 C PG3

3

5 :

If one of the circuit terminals is connected to the common node, we obtain threenonsingular matrices, which are used below

PY1 D� PG2 C PG3 � PG3

� PG3 PG1 C PG3�

; PY2 D� PG1 C PG2 � PG1


; PY3 D� PG1 C PG2 � PG2


where PY1; PY2 i PY3 are, respectively, circuit matrices obtained when the LFDCterminals 1, 2, or 3 are connected to the common terminal 0.

With allowance for the admittance matrices of the circuit and active element,we can determine LFDC and AE secondary parameters (transfer functions) in theform of ratio of matrix determinants [8]. In this case, characteristic equations canbe easily written through algebraic complements of the matrices. For example, ifLFDC terminal 3 is connected to the common wire and different active elements areused, we obtain the following equations:

�11 �

�12 �Ku D 0;

22 �

12 �Ki D 0;

� ��12 �Kg D 0;

�11;22 �

�12 �Kz D 0;

or

�11 �Ku

aa ��12 �Ku

ab D 0;

22 �Ki

bb �12 �Ki

ab D 0;

� �Kgaa;bb ��

12 �Kgab D 0;

�11;22 �Kz �

�12 �Kz

ab D 0;

where �11,

�12,

22,

12,

v, �12,

�11;22,

�12 are determinants of the algebraic

complements of the matrices PY1; PY2 i PY3 of LFDC voltage and current transferfactors, transfer impedance, and transfer admittance of the circuit, respectively, withsuperscripts �; ; � i ; Ku

aa , Kuab , Ki

bb , Kiab ,

Kgaa;bb ,

Kgab , Kz ,Kz

ab are algebraiccomplements of the matrices of AE transfer factors.

The equations obtained serve as OS matrix characteristic equations. For ex-ample, having determined the algebraic complements of the canonic matrix Y3

corresponding to the connection of terminal 3 to the common wire, we can writeOS characteristic equations with allowance for the admittance parameters PG1, PG2,and PG3and AE, respectively, of the VCVS, CCCS, VCCS, and CCVS type

.1 �Ku/ PG2 C PG3 D 0;

.1 �Ki/ PG2 C PG1 D 0;

. PG1 C PG3 �Kg/ � PG2 C PG1 � PG3 D 0;

1 �KZ � PG2 D 0: (5.25)


Equalities (5.25) allow one to determine possible strategies for the formation ofrequirements to LFDC and AE in the synthesis of a particular circuit. First of all,it follows from the above equalities that if VCVS is used, then the characteristicequation does not include the admittance PG1. Consequently, it does not affectthe conditions of fulfillment of the equality and may be absent from LFDC. Thesame can be said about the admittance PG3 absent in the second equality for OSwith CCCS. An oscillating system with AE of the VCCS type can be true in thepresence of all the three admittance parameters in LFDC. However, OS can alsobe true in the absence of either PG1 ( PG1 D 0) or PG3 ( PG3 D 0). In this case, thethird equality becomes simpler and transforms into either

� PG3 �Kg

� � PG2 D 0 or� PG1 �Kg

� � PG2 D 0. Moreover, the admittance PG2 in them can be formally chosenequal to any nonzero value, in particular, infinity. Then the equalities take the formPG3�Kg D 0 and PG1 �Kg D 0. In the latter case, OS can be implemented in a two-

pole circuit. Analyzing the last equality in (5.25), we can see that the two admittanceparameters PG1 and PG3 can be absent simultaneously in OS with AE of the CCVStype in LFDC, and then the three-pole circuit transforms into a two-pole one. It isknown that the amplitude and phase balance conditions in OS with delayless AE canbe fulfilled only with LFDC of no lower than second order. Therefore, a two-polewith one admittance parameter can be only inductance-capacitance (LC).

The above analysis allows us to draw the following recommendations for thesynthesis of OS:

� the admittance PG2 in LFDC with unidirectional AE should be present for theformation of the feedback loop;

in the absence of admittance PG2 in LFDC, an oscillating system should be based onAE with bidirectional signal transfer;

OS with only inductance-capacitance (LC) LFDC can be based on all types ofunidirectional AE;

OS with resistance-capacitance LFDC can be obtained only with VCVS, CCCS, andVCCS (in the presence of all the admittance parameters PG1; PG2 i PG3);Connecting LFDC terminals 2 or 1 to the common wire, we can write similar

characteristic equations for other OS. They all are summarized in Table 5.1, whereletters and digits correspond to the circuit terminals: input, output, and commonwire.

Analyzing the transfer functions of frequency-dependent circuits in Table 5.1, wecan easily see that they are related by the following equations:

P� au C P� c

u D 1;

P�bu C P�g

u D 1;

P� eu C P�d

u D 1I

9>>=

>>;

Pai C Pe

i D 1;

Pbi C Pd

i D 1;

Pci C Pg

i D 1I

9>>=

>>;

P� au D Pb

i ;

P� cu D Pd

i ;

P�gu D Pe

i I

9>>=

>>;

P�bu D Pa

i ;

P�du D Pc

i ;

P� eu D Pg

i I

9>>=

>>;

P�az D P�b

z ;

P�cz D P�d

z ;

P�gz D P�e

z I

9>>=

>>;

P�ag D P�b

g;

P�cg D P�d

g;

P�gg D P�e

g:

9>=

>;

The relations between the transfer functions are independent of the scheme of elementcoupling in an electric circuit and invariant to it. Therefore, they can be referred to as asystem of invariants of circuit transformations.


Tab

le5.

1M

ain

func

tion

sof

OS

wit

hdi

ffer

ent

AE

and

diff

eren

tL

FDC

conn

ecti

onˇ ˇ

P Yˇ ˇD

P G 1�P G

2C

P G 1�P G

3C

P G 2�P G

3

AE

type

VC

VS

CC

CS

VC

CS

CC

VS

Type

ofL

FDC

conn

ecti

onA

Etr

ansf

erfa

ctor

Ku

Ki

Kg

Kz

�a

�(1

23)

P�a uD

P G2

P G2C

P G3

Pa iD

P G2

P G1C

P G2

P�a zD

P G2 jP Y j

P �a gD

P G 2.1

�K

a u/

�P G2

CP G 3

D0

.1�K

a i/

�P G2

CP G 1

D0

.P G 1

CP G 3

�K

a g/

�P G2

CP G 1

�P G3

D0

1�K

a z�P G

2D0

�b

�(2

13)

P�b uD

P G2

P G1C

P G2

Pb iD

P G2

P G2C

P G3

P�b zD

P G2 jP Y j

P �b gD

P G 2.1

�K

b u/

�P G2

CP G 1

D0

.1�K

b i/

�P G2

CP G 3

D0

.P G 1

CP G 3

�K

b g/

�P G2

CP G 1

�P G3

D0

1�K

b z�P G

2D0

�c

�(3

21)

Equ

atio

nfo

rL

FDC

tran

sfer

func

tion

and

OS

char

acte

rist

iceq

uati

ons

P�c uD

P G3

P G2C

P G3

Pc iD

P G3

P G1C

P G3

P�c zD

P G3 jP Y j

P �c gD

P G 3.1

�K

c u/

�P G3

CP G 2

D0

.1�K

c i/

�P G3

CP G 1

D0

.P G 1

CP G 2

�K

c g/

�P G3

CP G 1

�P G2

D0

1�K

c z�P G

3D0

�d

�(2

31)

P�d uD

P G3

P G1C

P G3

Pd iD

P G3

P G2C

P G3

P�d zD

P G3 jP Y j

P �d gD

P G 3.1

�K

d u/

�P G3

CP G 1

D0

.1�K

d i/

�P G3

CP G 2

D0

.P G 1

CP G 2

�K

d g/

�P G3

CP G 1

�P G2

D0

1�K

d z�P G

3D0

�g

�(3

12)

P�g uD

P G1

P G1C

P G2

Pg iD

P G1

P G1C

P G3

P�g zD

P G1 jP Y j

P �g gD

P G 1.1

�K

g u/

�P G1

CP G 2

D0

.1�K

g i/

�P G1

CP G 3

D0

.P G 2

CP G 3

�K

g g/

�P G1

CP G 2

�P G3

D0

1�K

g z�P G

1D0

�e

�(1

32)

P�e uD

P G1

P G1C

P G3

Pe iD

P G1

P G1C

P G2

P�e zD

P G1 jP Y j

P �e gD

P G 1.1

�K

e u/

�P G1

CP G 3

D0

.1�K

e i/

�P G1

CP G 2

D0

.P G 2

CP G 3

�K

e g/

�P G1

CP G 2

�P G3

D0

1�K

e z�P G

1D0


It should be kept in mind that the transfer functions P�u i Pi are dimensionless,while P�z i P�g have the dimension of resistance and admittance.

The first two systems of invariants show that if the voltage or current transferfunction is determined in one of circuits (basic), then there is another coupling ofthe same circuit, whose transfer function can be found as one’s complement ofthe transfer function of the basic circuit. In the first of these systems, the sumsof the transfer functions of the circuits, which have nodes with the same numbersas an output node (terminal 2 in the first row of the table) and whose input andcommon nodes interchange, are equal to unity. In the second system, to the contrary,the circuits have nodes with the same number as an input node, and the outputand common nodes interchange. In [9], these circuits were called complementary,just because the sum of their transfer functions is equal to unity. We can alsonotice other peculiarities of these transfer functions. For example, the sum of alltransfer functions of the same LFDC at all possible terminal couplings is constant,frequency-independent, and equal to three.

The third and fourth systems of invariants show the relation between the voltageand current transfer functions of circuits. Based on this system, it can be stated thatif we know the voltage transfer function of the circuit, then this circuit has the samecurrent transfer function. If follows here from that if there exists a basic LFDC,which with AE of the VCVS type forms OS, then there exists LFDC with otherterminal coupling, which forms the same OS but with AE of the CCCS type.8

The fifth and sixth systems of invariants relate transfer impedances and transferadmittances of the circuit at different connections.

Now let us find how the properties of LFDC invariants reflect in the transferfactor of AE (amplifier) in the case of an oscillator. To write the equation forthe oscillating system of an oscillator with LFDC with the type a structure (123)Ka

u � PU23 D PU13; we divide the left-hand and right-hand sides of the equation byPU13 and obtain Ka

u � P� au D 1: Similarly, for the oscillating system with the type c

structure, we have Kcu � PU21 D PU31; and dividing the left-hand and right-hand sides

by PU31; we obtain the equation Kcu � P� c

u D Kcu � �

1 � P� au

� D 1: From here we canfind the ratio of the gain coefficients of amplifiers in the oscillating systems withthe basic and complementary circuits Kc

u D �Kau=

�1 �Ka

u

�: The amplifier with

this gain coefficient for OS with the complementary circuit is referred to as thecomplementary amplifier.

Based on Table 5.1, we can also find the relation between the AE transfercoefficients

Kau D �Kc

u=.1 �Kcu/;

Kbu D �Kg

u=.1�Kgu/;

Keu D �Kd

u=.1�Kdu/I

9>=

>;

Kcu D �Ka

u=.1�Kau/;

Kgu D �Kb

u=.1 �Kbu /;

Kdu D �Ke

u=.1�Keu/I

9>=

>;

Kau D Kb

i ; Kbu D Ka

i IKc

u D Kdi ; Kd

u D Kci I

Kgu D Ke

i ; Keu D K

gi :

9>=

>;

8From the viewpoint of the linear theory, such oscillating systems are identical. However, from theviewpoint of practical implementation of active elements, they can differ by the level of nonlineardistortions.

5.9 Conclusions 167

Table 5.2 Relation of amplifier gain coefficients

OS based on LFDC

Basic Complementary

Phase shift of circuitat the frequency ¨0

AE gain coefficient Phase shift of circuitat the frequency ¨0

AE gain coefficient

'a� D 0 Ka

u D Koa

1CKoa � ˇ 'c� D 0 Kc

u D �Koa

1�Koa � .1� ˇ/

'a� D � Ka

u D �Koa

1CKoa � ˇ 'c� D 0 Kc

u D Koa

1CKoa � .1C ˇ/

A remark should be drawn concerning the gain coefficient for the betterunderstanding of the conditions of fulfillment of the amplitude balance and phasebalance in OS. For the fulfillment of the amplitude balance, the high gain coefficientis not usually required. Therefore, the given value ofKa

u is obtained through the useof an op-amp with the high gain coefficient Koa with feedback ˇ for its lowering.Depending on the value of the LFDC phase shift, the gain coefficient Ka

u can beboth positive and negative. Table 5.2 presents the relations for the selection of gaincoefficients of amplifiers at different LFDC parameters.

The characteristic equations summarized in Table 5.1 describe nearly all knownOS schemes of LC and RC oscillators with one AE. For example, the equation ofLFDC circuit with connection a and AE of the VCCS type is described by OSof inductance or capacitance tree-point circuits (Fig. 5.30). The LFDC equation ofcircuit a and AE of the VCVS type describes OS of an oscillator with the Wienbridge, the equation of LFDC of circuit c and AE of the VCVS type describes OSof the type of turned capacitance three-point circuit, and so on.

5.9 Conclusions

1. Generation of stable periodic oscillations is possible only in the nonlinearfrequency-dependent oscillating system including three necessary elements:linear frequency-dependent element, nonlinear element, and active element.

2. In the self-oscillating system of a sine-wave oscillator there are contradictionsbetween the level of nonlinear distortions, the amplitude stability, and theduration of transient processes.

3. The nonlinear distortions of sine waves can be diminished in two ways: throughapplication of a high-Q frequency-dependent circuit (resonance circuit in LCoscillators) at a simple active element or active oscillating systems modeling theresonance circuit and through the use of complex systems for stabilization of theoscillation amplitude (in RC oscillators).

4. In the low-frequency band (up to 1 MHz), it is better using the frequency-dependent RC oscillators, while LC oscillators are better suited for generationof high-frequency oscillations.


5. Based on the equations of invariance of the frequency-determining circuit andcomplementarity of the transfer coefficients of active elements (amplifiers), wehave proposed the method of synthesis, which allows the development of newoscillating systems.

Questions

1. What are the main elements (units, modules) of a sine wave oscillator?2. Draw block diagrams of oscillators.3. Justify the application of every oscillator element.4. What is the main purpose of the frequency-dependent circuit (nonlinear ele-

ment, active element)?5. Write and comment the amplitude and phase balance conditions.6. How can we control the oscillation amplitude in an oscillator?7. How can we control the oscillation frequency in an oscillator?8. What are the main conditions for stability of steady-state self-oscillations?

Justify them.9. What groups can oscillators be classified into depending on the frequency of

generated oscillations?10. What frequencies are RC oscillators used at? Justify this.11. What are the conditions for excitation of self-oscillations in an oscillator?12. Describe examples of frequency-determining circuits of RC oscillators.13. An amplifier has gain of 29 and a phase shift of 180ı. What conditions should

the oscillator frequency-determining circuit satisfy to?14. In which feedback circuit the Wien bridge should be included to assemble an

RC oscillator? What are requirements on an oscillator amplifier?15. Formulate the conditions for the steady-state mode of self-oscillations.16. What factors determine the oscillator self-oscillation frequency?17. What elements are needed to set the self-oscillation amplitude?18. Under which conditions self-oscillations are stable to amplitude deviations?19. Formulate the main contradictions arising in sine-wave oscillators.20. What are the interrelated parameters of self-oscillations of the simplest oscilla-

tor? Explain the relation between them.21. What is the essence of the contradiction between amplitude stability and

nonlinear distortions in an RC oscillator?22. Why the total harmonic distortion is related to the duration of the transient

process in an oscillator?23. What requirements are imposed on the elements and systems for stabilization

of the self-oscillation amplitude?24. What are disadvantages of application of diode limiters to stabilization of the

self-oscillation amplitude?25. What is the essence of the advantage of inertial nonlinear elements for

stabilization of the self-oscillation amplitude?26. Why do RC oscillators use complex systems for stabilization of the self-

oscillation amplitude? Formulate the requirements to such stabilization system.

5.9 Conclusions 169

27. Give an example of the circuit for automatic stabilization of the oscillationamplitude with the use of an amplitude detector. What are its merits anddemerits?

28. What is the advantage of the amplitude detector with the storage circuit over theordinary diode detector in the amplitude stabilization circuit?

29. What is the difference between LC and RC oscillators?30. What are the features of LC oscillators as compared to RC oscillators?31. What are the amplitude and phase balance conditions in LC oscillators?32. In what way is the self-oscillation amplitude stabilized in the FET oscillator?33. Describe the three-point circuits of oscillator.34. Explain how is the phase balance fulfilled in the oscillator with three LC

elements.35. Why the self-oscillation frequency accuracy and stability is higher in quartz

oscillators than in LC or RC oscillators?36. Assemble the LC oscillator circuit in the Electronics Workbench software and

obtain stable periodic self-oscillations.�

37. What are possible applications of quartz oscillators?

Test Yourself

1. To create a sine-wave oscillator, the set of the following functional elements isneeded:

(a) linear frequency-dependent circuit and linear amplifier;(b) linear amplifier, linear frequency-dependent circuit, and nonlinear element;(c) passive nonlinear element and passive linear frequency-dependent circuit.

2. The set of elements, a sine-wave oscillator can be assembled of, includes:

(a) resistor, rectifier diode, and linear RC circuit;(b) Wien bridge, op-amp, resistor, and thermistor;(c) linear RC circuit, stabilitron, inductance coil;(d) inductance, transistor amplifier.

3. The minimal order of the equation of an oscillating system for excitation ofsine-wave self-oscillations is:

(a) first;(b) second;(c) third;(d) fourth.

4. The set of the roots of the linearized characteristic equation of oscillator,corresponding to the condition of excitation of sine-wave oscillations, is

(a) s1 D � 0.1, s2 D C 0.2;(b) s1 D C0.1 C j 1,000, s2 D C 0.1 – j 1,000;(c) s1 D C j 100, s2 D � j 100;(d) s1 D C 0.1, s2 D C 0.2.


5. The set of the roots of the linearized characteristic equation of oscillator,corresponding to the condition of damping of sine-wave oscillations, is

(a) s1 D � 0.2, s2 D � 0.2;(b) s1 D C j 100, s2 D � j 100;(c) s1 D � 0,1 C j 1,000, s2 D � 0.1 – j 1,000;(d) s1 D C 0.1, s2 D C 0.2.

6. The set of the roots of the linearized characteristic equation of oscillator,corresponding to the stable sine-wave oscillations, is

(a) s1 D � 0.2, s2 D � 0.2;(b) s1 D � 0.1 C j 1,000, s2 D � 0.1 – j 1,000;(c) s1 D Cj 100, s2 D � j 100.

7. Conditions for excitation of self-oscillations in an oscillator are:

(a) two roots of the linearized characteristic equation of oscillator are positiveand real;

(b) three roots of the linearized characteristic equation of oscillator are imagi-nary;

(c) two roots of the linearized characteristic equation of oscillator are complex-conjugate with the positive real part, while the others have the negative realpart.

8. Conditions for existence of stable steady-state self-oscillations in an oscilla-tor:

(a) dK/dV > 0;(b) dK/dV < 0;(c) dK/dV D 0.

Note. Here K is the amplifier gain, V is the output voltage.9. Determine whether self-oscillations arise in this circuit:

(a) Yes;(b) No.

5.9 Conclusions 171

10. The frequency of self-oscillations in this circuit is

(a) f � 1,000 Hz;(b) f � 120 Hz;(c) f � 1,500 Hz;(d) f � 10,000 Hz.

11. Determine whether self-oscillations arise in this circuit:

(a) Yes;(b) No.

12. Determine whether self-oscillations arise in this circuit:

(a) Yes;(b) No.


References

1. Robinson, F.N.H.: Noise and Fluctuations in Electronic Devices and Circuits. Clarendon,Oxford (1974)

2. Roitman, M.S.: Amplitude-Stable Generators and Adjustable AC Voltage Gages. Tomsk,Tomsk Polytechnic Institute (1977) (in Russian)

3. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillations. Pergamon Press, Oxford(1966)

4. Barkhausen, H.: Elektronen-Roehen. 3. band: Ruckkopplung. In: Hirzel, S. (ed.) 3. Auflage.Verlag, Leipzig (in German) (1934)

5. Bondarenko, V.G.: RC Sine-Wave Oscillators. Svyaz, Moscow (1976) (in Russian)6. Rybin Yu.K., Budeikin V.P., Gertsiger L.N.: Low-frequency RC sine wave oscillators with

small of total harmonic distortion. Meas. Control Automation. 2(54), 25–37 (in Russian) (1985)7. Rybin Yu.K.: Synthesis of sine-wave oscillators with pulsed amplitude stabilization. Ra-

diotekhnika i Elektronika. 29 (9), 1764–1771 (in Russian) (1984)8. Rybin Yu.K.: The analysis and synthesis of oscillations system of electric signal generators.

Proc. Tomsk Polytechnic Univ. 317(4), 134–139 (in Russian) (2010)9. Volgin, L.I.: Methods of Topological Transformation of Electrical Circuits. Saratov UnivPress,

Saratov (1982) (in Russian)10. Wangenheim, von L.: Aktive Filter und Oszillatoren: Entwurf und Schaltungstechnik mit

integrierten Bausteinen von Springer, Berlin (2007)

Chapter 6Pulse Oscillators

Abstract The aim of this Chapter is provide an integral conception about up-to-date theory of pulse oscillators, the conditions for excitation of periodic pulsedoscillations in them, and practical principles of oscillator designing.

The material presented is not difficult for reader’s familiar with the basic princi-ples of electronics within the course “Electronics” or “Electronics in InstrumentMaking,” as well as with the methods of solution of differential and operatorequations within the course of Higher Mathematics.

Upon learning this chapter, the readers will be familiar with the oscillatoroperation and able of determining the conditions for excitation of periodic pulsedself-oscillations.

6.1 Introduction

Besides sine-wave signals, pulsed signals are widely used in researches andtechnology.

The pulsed signals have some features, namely,

– they include parts of fast and slow change;– they have a sharp transition from slow change to fast change;– they are characterized by a wide spectrum.

It should be noted that sine-wave signals also have parts of fast and slow change,because the rate of change of the sine-wave function depends on the amplitude andfrequency. However, they do not include sharp transitions between these parts, andin addition, they have only one spectral component.

The features mentioned above, together with some others, determine the partic-ular conditions of amplification and generation of pulsed signals. Thus, the widespectrum causes the use of broadband amplifiers for amplification of such signals.A particular attention is paid to matching of a signal source and a load. Generationof pulsed signals also has some features.


173

174 6 Pulse Oscillators

Unfortunately, the electronic literature still fails to give the correct explanation toeither the conditions for appearance of self-oscillations or the conditions ensuringthe steady state. Sad to say, but this is also true for student’s books in radioengineering and electronics. Usual explanations of oscillator operation begin,roughly speaking, as “Let oscillations exist, then the operational oscillograms havethe form : : : ” etc. The author of this manual has nothing against this approach,especially, because it is one of the scientific research methods, but using it, itis hard or even impossible to explain the existence of other operational modesdifferent from that under consideration. For this it would be necessary to supposethe existence of such modes beforehand, which is quite problematic. Therefore, itis impossible with such approach to choose the analysis and design strategy. Thismakes this approach vulnerable, and the design process in this case fails to guaranteethe absence of “supernumerary” operational modes. Here we need a method, whichwould allow us to study all possible modes of operation, in particular, the conditionsfor appearance and establishment of periodic pulsed self-oscillations, their characterand waveform.

6.2 Selected Issues of Theory of Pulse Oscillators

The processes in pulse oscillators, as well as in sine wave oscillators, can be describedby system of differential equations. However, investigation of these processes throughintegration of the differential equations can be performed only numerically because of thenonlinear character of the equations.

Numerical integration allows us to find the solutions, including periodic ones,which satisfy the condition V(t) D V(t C T) and describe the form of the outputvoltage in the steady mode of the pulse oscillator. However, it is a labor – and time-consuming method, which do not guarantee obtaining all possible solutions.

That is why qualitative methods, in particular, the method of phase plane are used to studythe processes in pulse oscillators. Unlike the numerical methods, this method permits us toanalyze all solutions, rather than only one of them.

The voltage variables of the sine wave oscillator form a limit cycle: ellipse orcircle, in the phase plane. What is the shape of the limit cycle of self-oscillations inthe pulse oscillator? Consider Fig. 6.1 to answer this question.

Assume that pulsed signal varies stepwise as on Fig. 6.1. Then the image pointfollows straight lines in the phase plane on Fig. 6.1. The straight line ab correspondsthe signal section ab in the phase plane, while the section bc is described by astepwise transition to the line cd and the following smooth motion of the imagepoint along the line cd to the point d, whereupon the image point returns to itsinitial position, and the cycle repeats. Strictly speaking, this image is incorrect forone signal V(t) (one time variable), because the value of U and its time variationis unknown. The output oscillator voltage can be taken as the variable V, and thevoltage or current in one of the elements of the oscillating system can be taken as U.

6.2 Selected Issues of Theory of Pulse Oscillators 175

Fig. 6.1 Pulsed oscillation of the oscillator (a) and its image in the phase plane (b)

Fig. 6.2 Directions of phasetrajectories near the limitcycle at the rectangle steadyself-oscillations

Therefore, depending on the choice of the variable U, the rectangle of the limitcycle can transform into a parallelogram, and the motion along the line ab can havedifferent speed.1

As in the sine wave oscillator, the variables U and V at the time of switching thesupply voltage on can have the values different from the values on the limit cycle;therefore, when the power supply is turned on, the initial position of the image pointcan be different. In the process of establishment of stationary oscillations, the imagepoint must come to the trajectory of the limit cycle. Therefore, all the trajectoriesfrom any position in the phase plane both inside and outside the limit cycle must bedirected toward the limit cycle.

Figure 6.2 shows the phase space with possible trajectories, whose form fully depends on thesingular points of the oscillating system. Knowing the positions of zeros of its characteristicequation, it is easy to determine the conditions for appearance of self-oscillations and theircharacter.

One of the main conditions for appearance of pulsed oscillations is the presenceof the oscillating system giving rise to them. As was shown in Chap. 5, sine

1Note once again that the phase plane ambiguously represents signals. The same rectangular limitcycle can correspond not only to a square-wave, but also to a trapezoid signal, because the phaseplane does not represent the speed of motion of the image point.


wave oscillators usually employ a conservative linear oscillating system. Pulsedoscillations can be generated only by nonlinear system, whose dynamic behaviorin the phase plane is represented by a closed curve.

A real oscillating system consisting of passive elements always has some losses, thereforethe oscillations decay with time and the limit cycle is not closed. To obtain periodicoscillations described by a closed curve in the plane, the consumed part of the energyshould be compensated for by the active element, whose role in the nonlinear system canbe played by a transistor, an operational amplifier , or an element with negative resistance(tunnel diode, thyristor, etc.).

6.2.1 The Conditions for Excitation of Pulsed Oscillations

In the theory of electric signals generators the problem of conditions for sinusoidoscillations simulation was solved long ago. The establishment criterion for suchoscillations is Barkhausen criterion known as conditions for amplitude and phasebalance (see Chap. 5). However, the problem of conditions for excitation and estab-lishment of periodic non-sinusoid oscillations, e.g. pulsed oscillations, specificallysquare, triangular or of any other form, still needs to be solved. To simulate electricsignals of such forms transistor and operational amplifiers generators are known,described and widely used; among them are multivibrators, ramp generators andothers. The analysis shows that almost all of them are invented, as the theoreticalaspects are not developed enough.

The point is that though there are plenty of works on the theory of nonlinearoscillations, there is no any perspicuous theory of excitation and establishment ofperiodic non-sinusoid oscillations, i.e. it is not known which characteristics shouldbe inherent in oscillatory system and its elements. So, the development of non-sinusoid signals generators theory is topical for practical application, as well as forthe theory of nonlinear oscillation as a whole.

Let us consider block diagram of the elementary oscillatory system (OS) of agenerator, Fig. 6.1, where LFDC stands for linear frequency-dependent circuit andANE – for active nonlinear element. According to this block diagram OS for manygenerators of different oscillations have been realized. In respect to this system it isappropriate to clarify the conditions for:

• excitation of divergent oscillations;• periodic oscillations;• disturbance-resistant oscillations, i.e. self-excited oscillations.

All these problems concerning establishment of sinusoid self-excited oscillationsare more or less considered and solved. Specifically, the condition for excitationof divergent oscillations is a couple of conjugate complex roots of a characteristicequation of a linearized equation system. The criteria and conditions for periodicoscillations establishment for this system are also known: specifically, Barkhausencriterion or conditions for amplitude and phase balance [1, 2]. The conditions for


these oscillations stability are identified as well [2]. Unfortunately, the situation isnot so good when we speak about other self-oscillations establishment study. Asnoted in [1], it is connected with the difficulty of analytical solution of the problemdue to strong nonlinearity of OS equations.

This section deals with the conditions for simulation of periodic non-sinusoidoscillations in OS in Fig. 6.1, i.e. the second of the stipulated questions is examinedthoroughly.

So, let us pose the problem of the conditions under which in OS shown inFig. 6.1 periodic oscillations described by some periodic non-sinusoid function x(t)can exist. We should pay our attention to the fact that this diagram does not differfrom the diagram in Fig. 5.2d, Chap. 5. Thereby, it is assumed that the problem ofthe conditions for simulation of sinusoid and pulsed oscillations is common.

First of all, we need to define the requirements for simulated oscillations, LFDCtransfer function and ANE amplitude characteristics.

To fulfill the assigned task it is necessary to define LFDC transfer functionand ANE amplitude characteristics as well. Let us introduce symbols: P�.!/ DPy.!/= Px.!/ – LFDC complex transfer function and PK.!; x/ D Px.!/= Py.!/ – ANEcomplex characteristic. Multiplying these functions we obtain the known criterion ofexistence of periodic sinusoid oscillations with frequency!0 in this OS: Barkhausencriterion PK.!0; x/ � P�.!0/ D 1. Let us find out whether one could extend thiscriterion to the periodic non-sinusoid oscillations.

First of all, it is necessary to choose and take mathematical models of signals inOS. Analytical descriptions of signals in closed forms are not always feasible. Themost acceptable description of signals is their description in the form of a series bybasic functions system. So, we present output oscillation x.t/ in the form of Fouriertrigonometric series

x.t/ D1X

nD1xmn � sin.n!0t C 'xn/: (6.1)

Let the oscillation described by the function come to the input of LFDC. Then,coming through LFDC all harmonic components of x.t/ change their amplitude andshift in phase according to its complex transfer function P�.!/. As a result, LFDCoutput signal will take the form of y.t/

y.t/ D1X

nD1ymn � sin.n!0t C 'yn/

D1X

nD1xmn j P�.n!0/j � sin.n!0t C 'xn C arg. P�.n!0///: (6.2)

This oscillation does not differ from x.t/ in spectral composition, but differs invalues of amplitudes and initial phases of harmonic components in compliance withamplitude-frequency (AFC) and phase-frequency (PFC) characteristic of LFDC, i.e.


in compliance with expressions ymn D j P�.n!0/jxmn and 'yn D 'xn Carg . P�.n!0//,where !0 is oscillation frequency. Then signal y.t/ comes to ANE input and, goingthrough it becomes oscillation z.t/. Now ANE at its output associates each harmonicof input signal y.t/ with the harmonic of signal z(t)

z.t/ D1X

nD1xmn j P�.n!0/j � ˇ

ˇ PK.n!0; x/ˇˇ

� sin.n!0t C 'xn C arg. P�.n!0//C arg. PK.n!0; x///: (6.3)

Here there is a need to explain some values of components being members of(6.3). Specifically,

ˇˇ PK.n!0; x/

ˇˇ is a modulus of relation between the amplitude of

n-th harmonic of ANE output signal and the same harmonic of the input signal,and arg. PK.n!0; x/ is a difference between phases of n-th harmonic at ANE inputand output. As one can see, these values depend not only on oscillations frequencyand harmonic number, but on output signal amplitude as for any nonlinear element.Naturally, they depend on input signal form as well.

In the case of closed OS z(t) becomes equal to x.t/. Equating right parts ofEqs. 6.1 and 6.3, we get the conditions for simulation of periodic oscillations

j P�.n � !0/j � ˇˇ PK.n � !0; x/

ˇˇ D 1; arg. P�.n!0/C arg. PK.n!0; x/ D 0: (6.4)

The first equation determines the value of transfer ratio for each harmonic atcircuit LFDC-ANE, and the second – their phase shift. When the equations arerealized oscillation y.t/ in ANE transforms back into x.t/. Actually ANE serves asan “inverse” converter of y.t/ into x.t/. It changes amplitudes and shifts phasesof each harmonic so that oscillation x.t/ can be obtained at its output. Suchtransformation occurs only when simulating periodic oscillations.

This simple analysis can be the proving of the following theorem.

Theorem 1. To get periodic oscillations in an oscillatory system containing series-connected LFDC and ANE, complex transfer ratio on feedback loop should be equalto one at the frequency of each harmonic of output signal decomposition into Fouriertrigonometric series.

Corollary 1. The modulus of transfer ratio on feedback loop at the frequency ofeach harmonic should be equal to one.

Corollary 2. Phase shift on feedback loop the frequency of each harmonic shouldbe equal to zero or multiple of 2� radian.

Theorem 2. To let oscillations with given form exist in OS, ANE amplitudecharacteristic should be inverse to LFDC amplitude characteristic (without criticalpoints and self-intersections at amplitude characteristic).

Corollaries 1 and 2 follow from Eq. 6.4 and proved by the given above analysis.As a practical proving of the theorem and its corollaries validity we give an

example further.


Fig. 6.3 Generalized block diagram of pulse generator oscillatory system

Fig. 6.4 Input x(t) andoutput y(t) signals of LFDC

Let OS, Fig. 6.3, be necessary to be formed for simulating periodic squareoscillations shown in Fig. 6.4 with a solid line. It should be immediately stipulatedthat to realize this signal is practically impossible, as its first derivative is equal toinfinity in the moments of polarity change, but this signal was chosen because it isquite thoroughly studied and its mathematical description is known.

The mathematical model of this signal can be the following expression

x.t/ D 4

�

1X

nD1

sin Œ.2n � 1/!0t �2n � 1

� (6.5)

Let us define the requirements that should be set up for OS blocks.As LFDC we use, say, a simple differencing RC-circuit2 and take its time

constant � D RC. In this case the differential equation for determining LFDC outputsignal takes the following form

du.t/

dtD �u.t/

�Cx.t/

�

�du.t/

dtD y.t/; (6.6)

where x.t/, y.t/and u.t/ are input, output voltage and the voltage at RC-circuitcapacitance, correspondingly.

2One could choose any other circuit in the capacity of RC-circuit.


Fig. 6.5 Phase trajectories of limiting cycle: (a) LFDC; (b) ANE

We would remind you that complex transfer function of the circuit at harmonicfrequencies has the following form

P�.n!0/ D jn!0�

1C jn!0�D n!0�

q

1C .n!0�/2

� ej .�=2�arctg.n!0�//: (6.7)

With a glance to transfer function (6.7) one could put down the expression ofLFDC output signal

y.t/ D 4

�

1X

nD1

1

2n � 1 � .2n� 1/!0�q

1C ..2n � 1/!0�/2

� sin Œ.2n � 1/ � !0t C .�=2 � arctg..2n � 1/!0�//� : (6.8)

Input x(t) signal is shown in Fig. 6.4 with a solid line, and the output y(t) one –with a dashed line.

The trajectories of signals of RC-circuits3 on a phase plane are shown inFig. 6.5a in coordinates y.t/ and x.t/. These trajectories represent the amplitudecharacteristic of RC-circuits under different amplitudes as the relation betweenoutput and input signals in the circuit.4 One of them corresponds to the signalsin Fig. 6.4. Comparing Figs. 6.4 and 6.5a it is not difficult to observe that in the

3About phase plane method and its application in nonlinear systems study in detail see [1].4It is clear that amplitude characteristic is individual both for the circuit itself and for its inputsignal, and its form is invariant only with respect to input signal amplitude.


course of time the representation point continuously moves by a closed trajectoryclockwise, and the phase portrait itself under different amplitudes of signal x.t/represents multitude of such trajectories embedded in each other (there are only twoof them shown in the figure).

If ANE makes inverse transformation of signal y.t/ into signal x.t/, thenthe trajectory of its limiting cycle will be analogous, only coordinate axes willchange their places, naturally, with the changing of the representation point movingdirection (see Fig. 6.5). Now it should move counterclockwise retaining traversespeed at each segment.

Actually, the mathematical part of the problem of conditions for simulation ofoscillations of the type (6.5) in OS can be considered completed. The result ofthe solving can be stipulated in the following statement. To let the oscillationsof the type (6.5) exist in OS with the chosen LFDC, it is necessary that ANEphase trajectory be inverse to one of LFDC cycles. Meeting this condition becomespossible with the help of corresponding ANE amplitude characteristic.

Let us consider this question as well.Amplitude characteristic of RC-circuit after time exclusion can be described by

the following equation system:

y1 D g1.x/ Dˇˇˇˇˇ

k.1C�/.Xm C x/ � 0; 5Xm; if �Xm > x � �Xm.1C�/Ix C 0; 5Xm; if �Xm < x � Xm.1C�/I

y2 D g2.x/ Dˇˇˇˇˇ

k.1C�/.x �Xm/C 0; 5Xm; if Xm < x � Xm.1C�/Ix � 0; 5Xm; if �Xm.1C�/ � x � Xm;

(6.9)

where XmD1 – output oscillation amplitude.Here for the unique determination of vertical subcircuits we took an assumption

about their minor vertical deviation specified by the coefficients k � 1 and �� 1,and k D Xm=�.

Amplitude characteristic of ANE making inverse transformation should bedescribed by the function inverse to the amplitude characteristic of RC-circuit. Thisfunction is not difficult to obtain when finding functions inverse to (6.9) at eachsegment of function g(x), notably:

f1.y/ Dˇˇˇˇˇˇ

y C 0; 5Xm

k.1C�/�Xm; if � 1; 5Xm.1C�/ < y � �0; 5XmI

y � 0; 5Xm; if � 0; 5Xm < y � 1; 5Xm.1C�/I

f2.y/ Dˇˇˇˇˇˇ

y � 0; 5Xm

k.1C�/CXm; if 0; 5Xm < y � 1; 5Xm.1C�/I

y C 0; 5Xm; if � 1; 5Xm.1C�/ < y � 0; 5Xm:

(6.10)


However, the amplitude characteristic of ANE itself does not guarantee theexistence of periodic self-oscillations. The required phase trajectory is only thenecessary condition. To let periodic self-oscillations establish it is necessary tofulfill the sufficient conditions for its stability under possible deviations of therepresentation point from this trajectory, including its possible initial positions bothinside and outside the limiting cycle. To make it possible all the trajectories fromany inside point should get to the limiting cycle. Similarly, the outside trajectoriesshould also be directed to the limiting cycle. And these conditions can be met onlywhen OS equation system is studied.

To make OS equation system it is necessary to redefine ANE amplitudecharacteristic outside the limiting cycle. One of variants of the extension of thedefinition is shown in Fig. 6.5b with a dotted line.

f1.y/ Dˇˇˇˇˇ

�Xm; if y � �0; 5XmIy � 0; 5Xm; if � 0; 5Xm < y � 1; 5XmI

f2.y/ Dˇˇˇˇˇ

Xm; if 0; 5Xm < yIy C 0; 5Xm; if � 1; 5Xm � y < 0; 5Xm:

(6.11)

In formulae (6.11) coefficients � and k from (6.10) are taken equal to zero andinfinity, respectively. So, combining LFDC Eq. 6.6 and amplitude characteristicEq. 6.11 with regard to its inertia one could present OS equation system in thefollowing way:

8ˆˆˆˆ<

ˆˆˆˆ:

�du.t/

dtD � u.t/ C x.t/I

�du.t/

dtDy.t/I

�kdx.t/

dtD � x.t/C

"f1 Œy.t/� ; if y.t/ < u.t/

f2 Œy.t/� ; if y.t/ � u.t/

#

:

(6.12)

Equation system (6.12) includes the first two LFDC equations and the third equa-tion of the inertial5 ANE with a small time constant � k . The introduction of ANEsmall “inertia” is explained by the fact that, as shown in [1], periodic oscillationscannot appear in the system described by the first-order equation. To let periodicoscillations appear it would be necessary to introduce the term “discontinuity”,which is theoretically acceptable but not feasible. Thus, the introduction of ANEinertia increases the order of OS and transforms it into the system of the secondorder where periodic oscillations are feasible.

5ANE inertia is also introduced to provide finite values of output signal derivatives.


The solving of system (6.12) under � ' 0; 91 and £k D 0.001 in steady-stateconditions is oscillations close to those shown in Fig. 6.4 with the finite length offront and section.

It is interesting to observe the following fact resulting from the form of signalx.t/. The movement of the representation point on the limiting cycle trajectoryunder the given form of signal (6.5) is realized with different speeds: at the slopingparts of the phase trajectory it is rapid, as they correspond to the pulse front andsection. At the horizontal and vertical parts the movement is relatively slow. Thatgives reasons to believe that the sloping parts of the trajectory in Fig. 6.4 define onlypulse front and section, and under � � �k influence pulse duration and signal cycleto a very little degree.

Theoretically, equation system (6.12) satisfies the posed problem and formssignals shown in Fig. 6.4, but it is difficult for practical realization, as ANEamplitude characteristic includes six linear parts demanding conjugation in definitepoints. Therefore the simplified realization of the amplitude characteristic providesample opportunities for application of inventive wit.

For instance, an unexpected type of the characteristic close to the requiredhysteresis one can be found in amplitude characteristic of a non-inverting amplifierwith OA. The closeness can be explained by the fact that the amplitude characteristicof a non-inverting amplifier, single-valued under slow changing of input signal,becomes multiple-valued (hysteresis) under rapid changing of input signal becauseof inertia inherent in any amplifier. This characteristic and amplifier circuit realizingit are given in [3]. There one can find also a generator circuit. With regard toamplifier characteristic the equation system takes more simple form

8ˆˆˆ<

ˆˆˆ:

�du.t/

dtD � u.t/ C x.t/I

�du.t/

dtD y.t/I

�kdx.t/

dtD � x.t/CXm � tanh ŒK .y.t/ � ˇx.t//� :

(6.13)

where tanh ( ) – hyperbolic tangent, “ – inverse feedback transfer ratio, K – OAamplification factor.

In [4] there is a detailed study of the equation system solving behavior (6.13)at the phase plane, and it is shown that under any deviations of the representationpoint from stationary phase trajectory in the course of time the representation pointreturns to the limiting cycle trajectory and continues to move on it. The solutions ofequation systems (6.12) and (6.13) under “D 0.5 are close to each other. A slightdifference is observed only at the front and the section of square pulse just owing tothe difference in theoretical and practical hysteresis characteristics of ANE. Thus,the analysis of equation system (6.13) solutions can be applied to equation system(6.12) as well.


6.3 Op-amp Pulse Oscillators

Pulse oscillators can be divided into pulse oscillators in op-amps, digital logic gates,timers, elements with negative resistance, and transistors. Transistor pulse oscilla-tors are most widely used in pulsed voltage regulators, where the requirements tothe oscillation waveform are not decisive. That is why transistor oscillators are notconsidered below.

Op-amp pulse oscillators are commonly known and described in the training andresearch literature. The first publications about op-amp pulse oscillators are datedto the 1960s, after the advent of integrated op-amps. Later on the descriptions ofsuch oscillators were included in all op-amp books in the sections devoted op-ampapplications [5, 6]. We need not to prove the advantages of op-amp pulse oscillators;it is sufficient only to note their simplicity and facility of tuning and control of theoscillation parameters.

Figure 6.6 shows the most widely known pulse oscillator. In this circuit, the op-amp isenclosed by the positive feedback through the resistors R1 and R2 and by the negativeone through the resistance–capacitance RC circuit. Owing to the feedback s, this circuit iscapable of generating continuous self-oscillations, whose waveform is show on Fig. 6.7.

Consider the structure of the oscillator in more detail.

The circuit of the resistor R and the capacitor C, as well as the resistors R1 and R2form a resistance–capacitance bridge with the op-amp inputs and output connected in itsdiagonals. On the other hand, the op-amp together with the positive-feedback circuit of theresistors R1 and R2 forms the so-called threshold hysteretic switch (named also the Schmitttrigger), whose circuit and the gain characteristic (GC) are shown on Fig. 6.8.

GC of the threshold element can be easily drawn analyzing the operation of thefeedback op-amp in the whole variation range of the input voltage U from negativeto positive values and vice versa. Obviously the negative voltage, applying to theinverting terminal, carried out the op-amp into the limiter mode, at which the outputvoltage is positive and equal to the maximum value Vm. This voltage applied throughthe feedback circuit to the non-inverting input maintains the op-amp in the limitermode. This state keeps during the increase of the input voltage from the negative

Fig. 6.6 Simplest op-amppulse oscillator

6.3 Op-amp Pulse Oscillators 185

Fig. 6.7 Voltage plots of the simplest op-amp pulse oscillator

Fig. 6.8 Op-amp hysteretic element (Schmitt trigger)

values to U D “Vm (“D R1/(R1 C R2)). As soon as the voltages at the op-amp inputsbecome equal, the regenerative process begins due to the positive feedback, andfinally of this process the op-amp transits into the other state, in which its outputvoltage becomes equal to the minimum value �Vm. The further increase of the inputvoltage does not change the op-amp state.

The branch of the GC at the input voltage variation from negative to positivevalues is shown on Fig. 6.8b by the right arrows. If the input voltage varies in theopposite direction, the output voltage follows another line, which is shown by theleft arrows. From this plot we can see the ambiguity of the GC of the thresholdswitch. This ambiguity is also inherent in magnetic materials during magnetizationby the magnetic field of different direction and in ferroelectric materials exposedto electric fields of different direction. This phenomenon is called hysteresis. It iswidely used in triggers, memory devices, etc. In the considered oscillator, hysteresisserves for generation of periodic rectangular self-oscillations. The future rectangle-shaped limit cycle can be easily seen in the GC.

For analytical description of the hysteretic dependence, it is needed to take intoaccount the direction of motion. As U varies in the positive and negative directions,the sign of the derivative of U alternates, and this dependence with regard for theop-amp GC (1.5) has the form


V D Vmk2tanh Œ.ˇV � U /='T�; atdU

dt> 0; jV j < Vm;

V D �Vmk2tanh Œ.U � ˇV /='T�; atdU

dt< 0; jV j < Vm;

where V is the op-amp output voltage; U is the input voltage of the hystereticelement; k is the coefficient of proportionality; “ is the gain of the positive-feedbackresistive circuit; ®T is the temperature voltage. Regardless of the derivative sign,these two equations can be joined in one:

V D �Vmk2tanhŒ.U � ˇV /='T�; at jV j < Vm:

Now describe the equations allowing quantitative analysis of the oscillatoroperation. For correct describing of the hysteretic element, its mathematical modelshould account for both nonlinear and inertial properties.6 The op-amp inertialproperties can be submitted by the first-order differential equation with the timeconstant �k determined by the op-amp cutoff frequency.

With equation of the external RC circuit, the oscillator can be described by the followingsystem of equations:

8ˆ<

ˆ:

�kdV

dtD �V � Vmk2tanh Œ.U � ˇV /='T�;

�dU

dtD V � U;

(6.14)

where V is the op-amp output voltage, U is the voltage on the capacitor C (at the input of thehysteretic element), £k is the op-amp time constant, £k D 1/2 f cut, £ is the time constantof the RC circuit.

We can see that the result is the system of two differential equations, one of which(nonlinear) describes the threshold switch with the hysteretic characteristic, whileanother (linear) represents the processes in the RC circuit.

For comparison of different oscillator circuits, represent the system of Eq. 6.14 in the gen-eralized form by introducing the new designations: V D x; U D y; Vm k2tanhŒ.ˇV � U/=

'T� D �.x; y/ . Then the system of Eq. 6.14 takes the general form

8ˆ<

ˆ:

�kdx

dtD �x C �.x; y/;

�dy

dtD x � y:

(6.15)

6In fact, in the qualitative description of the threshold switch the op-amp inertial properties arealready taken into account, since without them the GC would not have hysteresis and would varyalong the dashed Z – shaped curve (see Fig. 6.8).


Fig. 6.9 Phase portrait of self-oscillations

Here ¥(x, y) is a nonlinear hysteresis function. It is well-known that in the generalcase the equation with nonlinear hysteresis functions has no analytical solution.Therefore, the processes in nonlinear systems are often studied by numerical orqualitative methods, for example, by the method of phase plane [1]. Let us use thismethod as well.

� Assume that the initial voltages on the capacitor U0 and the op-amp output V0

are zero. Such initial conditions are called zero conditions. In this case, the initialpoint in the phase plane lies at the origin of coordinate. It is a singular point of thesystem (6.14), because the time derivatives in it are equal to zero. It can be easilychecked by the substitution of the zero values for the variables U D 0 and V D 0in Eq. 6.15. The equations transform into identities. It is quite natural that all thefollowing processes begin in the vicinity of this point.

Remind some advantage of the phase plane method. Knowing the type of asingular point, it is possible to determine how the position of the image point willchange further for a certain region of the initial positions of the image point in thevicinity of the singular point.

Determine the type of the singular point and one of solutions of the system (6.14).At the low voltage, the argument of the function tanh is low as well, and thereforein Eq. 6.7 it can be replaced by the first term of expansion into the Taylor series.Obviously, the small argument of the function tanh corresponds to the position ofthe image point in the vicinity of the straight line “V D U, that is, near the diagonalof the limit cycle (lines 1–3 on Fig. 6.9). In this case, the system of Eq. 6.14 takesthe form

8ˆ<

ˆ:

�kdV

dtD �V C Vmk2

ˇV � U

'TI

�dU

dtD V � U:

(6.16)


After such substitution, the equations become linear. Solving them, we can seehow the processes evolve in the vicinity of the coordinate origin and this line.The best way for this is to use the differential equations instead the operatorones. Applying the known rules, we obtain, in place of the system of differentialequations, the system of algebraic equations:

8

<

:

s�kV D �V C Vmk2ˇV � U

'TI

s�U D V � U;

(6.17)

which can be easily reduced to a single second-order algebraic operator equation:

s2�k� C s

�

�k C � � �ˇk2Vm

'T

�

C 1C .1 � ˇ/k2Vm

'TD 0 (6.18)

According to the theory of operator equations, Eq. 6.18 has two roots. Studyingthem, we can determine the behavior of the solution of the initial system and thecharacter of trajectories in the phase plane.

At

�k > 0I � > 0I 1 > ˇ > 0I 'T > 0I Vm > 0

the both roots are real and positive and have different values. The positive rootscorrespond to a singular point of the type of unstable node. The solution of thesystem of equations has the form:

U.t/ DC1es1t C C2es2t

V .t/ DC1.1C s1�/es1t C C2.1C s2�/e

s2t ;(6.19)

where C are constants of integration, s1 K“/2£k , s2 (1 – “)/“£, K D k2Vm/®T isthe op-amp gain.

In the phase plane the solutions of the equations look like the curves shown on Fig. 6.9.

From figure the voltages vary according to the dependence of increasingexponents, and the rate of increase of these exponents is different. The greaterthe pole value, the higher the rate of increase. Since the roots are different, ands1 is much greater than s2 (K � 1), the first terms of the equations increase withtime faster than the second ones. What’s more, the voltage V (across the op-ampoutput) increases much faster than the voltage across the capacitor U does, becauseit has the factor 1 C s1£. Therefore, for a short time V changes significantly, whileU practically has no time to change. This change continues until the values of Vclose to 0.7–0.8 Vm, since the function tanh can be considered as almost linear andEq. 6.16 can be considered valid.


Thus, the oscillations from the coordinate origin and from the straight line“V D U (and, in the general case, from all the points of the rectangle 1–2–3–4 in thephase plane on Fig. 6.9) increase very quickly and go either upward or downwardalmost vertically.

As the image point goes beyond the limits mentioned above, the system ofEq. 6.16 becomes incorrect, because it was derived through replacement of thenonlinear function tanh with the first term of its expansion into the Taylor series.This replacement is valid only if the argument of function tanh varies from 0.8 to 1.To study the character of motion at large values of V, take into account that thefunction tanh becomes close to C1 already at the positive value of the argumentabout 2–3 and about �1 at the negative values of the argument. Then, in the furtherdetermination of the trajectories, the function tanh can be replaced by either C1 or�1 depending on the position of the image point: above or below the mentionedline. In this case, the differential equations take a new form:

8ˆ<

ˆ:

�kdV

dtD �V C Vm;

�dU

dtD V � U:

(6.20)

Find the singular point of the new system of equations by putting the derivativesto zero. It is the point with the coordinates (Vm, Vm). To reveal the character of phasetrajectories, determine the stability of this point. As before, describe the operatorequation and calculate its roots

s2�k� U C s .�k C �/ U C U D Vm: (6.21)

At £k> 0 and £> 0, the roots of the equation are real and negative. The negativevalues of the roots indicate that the singular point is a stable node, and, consequently,all the trajectories tend to the singular point with time.

Analysis of Eq. 6.20 shows that the first of them does not include the variable U,and the value of V is equal to Vm. Therefore, the variable is equal to zero, and thevoltage V does not change. At the same time, the voltage U(t) varies with the timeconstant by the following,

U.t/ Š Vm � ŒVm � U.0/� e� 1� ; (6.22)

where U(0) is the initial value of the voltage U.According to this equation, the image point moves along the line V D Vm

relatively slowly tending to the singular point in (CVm, CVm) at U varying in thepositive direction to the point 1 (Fig. 6.9). It is obvious that at the negative values ofU the derivative and, consequently, the rate of change of the voltage are greater thanat the positive ones (since the sum of the voltage U and its derivative is positive).At this stage the first half-period of oscillations if formed, and the duration of this


half-period can be found from Eq. 6.22 under the conditions:

U.T=2/ D ˇVm and U.0/ D �ˇVm W

ˇVm Š Vm � ŒVm � .�ˇVm/� e� T=2

� :

According to this equation, the duration of the half-period is

T=2 D � ln1C ˇ

1 � ˇ : (6.23)

When the voltage U at the point 1 achieves the value U D “Vm, the system ofequations again takes the form (6.16), the positive feedback arises in the oscillator,and the voltages change in accordance with these equations. The image point transitsto point 2 along the line 1–2 very quickly (almost as a jump).

The following motion is again described by the system (6.20), but already withthe negative value �Vm, and the image point moves along the line V D �Vm toanother singular point (�Vm, �Vm) from point 2 to point 3. At this stage, the secondnegative half-period of the output voltage is formed. Naturally, its duration can bedetermined by Eq. 6.23.

Then the function tanh becomes equal to C1, and the equations of the systemtake the already known form (6.16). The image point jumps upward along the lineU D �“Vm from point 3 to the intersection with the line V D Vm at point 4. Then themotion repeats, following the limit cycle 4–1–2–3.

Thus, we have the followed character of trajectories at the motion of the image point frominside the limit cycle and demonstrated that all the trajectories from any initial positionstend to the limit cycle, fall in it, and remain there. The parameters of this motion are thefollowing: amplitude Vm and period

T D 2� ln1C ˇ

1 � ˇ : (6.24)

Now let us follow the motion from outside the limit cycle. Here we can separatefour areas:

– above the line V D CVm;– to the right of the line U D “Vm;– below the line V D �Vm;– to the left of the line U D �“Vm.

The first and third areas correspond to the op-amp output voltages higher thanthe maximum allowable value C Vm and lower than the minimum allowable value�Vm, and therefore, practically, the image point cannot fall itself in these areas.


Fig. 6.10 Phase plane andthe field of directions ofoscillator trajectories

Nevertheless, theoretically, this is possible. Consider the character of motion of theimage point from these areas.

In the first area, the function tanh up to the line “V D U has the value C1.Therefore, the system of Eq. 6.20, in which the singular point with the coordinates(V D Vm, U D Vm) is a stable node , is valid here, and all the trajectories tend to thispoint. The motion occurs vertically downward to the line V D Vm and then alongthis line in accordance with the second Eq. 6.20 to point 1, where the transition tothe limit-cycle trajectory takes place. All the trajectories of the third area behaveanalogously, but they are directed upward.

In the second area of the phase plane, “V <U outside the limit cycle and thefunction tanh takes the value �1. The system of equations describing the characterof motion from this area is the same as Eq. 6.21 with the only difference that CVm

is substituted by �Vm. The singular point transits to the point with the coordinates(V D �Vm, U D �Vm). As in the previous case, it is a stable node, so all trajectoriestend to it vertically downward and then along the line V D �Vm to the left. Thetrajectories from the third area behave quite analogously, but are directed upward.

Figure 6.10 shows the fields of directions of the trajectories in the phase plane.The light region corresponds to the downward trajectories, while the dark regioncorresponds to the upward trajectories. In the plane there are only two lines, atwhich the trajectories are directed horizontally: the lines V D CVm and V D �Vm.Naturally, this idealization is valid at K ! 1.

The processes considered above are valid for an idealized oscillator with the infinitelylarge gain op-amp and the infinitely broad frequency band. But the real op-amps have alimited frequency band. The boundary frequency (fT ) of the gain area of a general op-amplies within 1–2 MHz (see Chap. 1), and the cutoff frequency (fcut) of the gain-frequencycharacteristic is only tens Hz.

The mathematical model of the op-amp in the system of Eq. 6.14 accounts just forthis frequency. However, experimental investigations show that the real frequency


Fig. 6.11 Voltage curves of op-amp oscillator with regard for its inertial properties

of self-oscillations differs widely from the theoretical one determined from thecalculated period of oscillations (6.24).

Therefore, to study the oscillator, let us apply a more complex op-amp modelwith two time constants and two nonlinear elements. In this case, the mathematicalmodel of the oscillator takes the following form:

8ˆˆˆ<

ˆˆˆ:

�kdY

dtD �Y C YmtanhŒ.U � ˇVm/='T �;

�1dV

dtD �V � Vmk2tanhŒY='T�;

�dU

dtD V � U:

(6.25)

In the system of Eq. 6.25, the op-amp is represented by the macromodel inthe form of a three-stage amplifier (Fig. 1.4). The first equation (a nonlineardifferential equation with the time constant ££K ) describes the first and secondstages. Nonlinearity in this equation is represented by the hyperbolic tangentfunction and corresponds to the input stage, while the time constant £k reflects thefrequency dependence of the second-stage gain and determines the op-amp cutofffrequency (£K D 1/2fcut). The second equation (also nonlinear) describes the third –output – high-frequency stage with the gain k2Vm/®T and the small time constant££1. This stage establishes the maximal levels, as well as the rate of change of theoutput voltage, that is, its amplitude, front, and cutoff. The stage gains and timeconstants depend on a particular op-amp. The third equation, as before, describes theRC circuit of the oscillator. The solution of this system of equations in the generalform is shown on Fig. 6.11.

Along with the voltage V at the op-amp output and U across the capacitor of theRC circuit, this figure shows the plots of the voltage H at the output of the first stage


Fig. 6.12 Dependence of the frequency of the oscillator output voltage fout on the ratio of the timeconstants of the op-amp second stage £k and the RC circuit £ and on the gain “ of the positivefeedback circuit

and Y at the output of the second stage. In contrast to Fig. 6.7, once the voltageacross the capacitor of the RC circuit achieves the threshold value “Vm at the timet1, the hysteretic threshold switch does not respond immediately, but is delayed tothe time t2, when the voltage Y at the output of the second stage becomes equal tozero. As a result, the half-period duration of the output voltage increases and thefrequency decreases. The frequency of the output voltage is mostly affected by thetime constant £ and £k , as shown on Fig. 6.12.

It is seen from Fig. 6.12 that as the ratio of the time constant of the second stage tothe time constant of the frequency-determining RC circuit increases, the frequencyof the output voltage decreases quickly.

� For example, at the estimated oscillator frequency equal to 1 kHz with the idealop-amp, the frequency of the generated oscillations with the real op-amp is about0.9 kHz (at the ratio of the time constants equal to 0,1 and “D 0.5). This ratiocorresponds to the op-amp cutoff frequency of about 4 kHz. Ordinary op-ampshave the cutoff frequency about 20–50 Hz, that is, much lower. Frequency of thegenerated oscillations is even lower using it. For example, in the A709 op-amposcillator (cutoff frequency of 22 Hz) at the estimated frequency of 1 kHz, thefrequency of real oscillations is equal to 480 Hz, that is more than twice as lowas the estimated value.

Thus, to achieve the frequency error less than 1% at “D 0.5, it is necessary tohave the high-frequency op-amp with the cutoff frequency higher than 40–50 kHz,which is characteristic of only few op-amps, for example, AD844.

Hence it follows that we demand much of the frequency properties from theoscillator op-amp. In this case, the dependence of the oscillation frequency on thegain is insignificant and the gain can be low (several thousands). Figure 6.12 allowsthe well-founded selection of op-amps to be carried out.


6.4 Possible Circuits of Op-amp Oscillators

A circuit on Fig. 6.13 of the op-amp oscillator with the controllable duty ratio is known.The oscillogram of its operation is described on Fig. 6.14 .

This circuit generates periodic square waves with the regulable duty ratio underthe same conditions as in the simplest oscillator. The oscillator operates in thefollowing way. Let at the time t0 the maximal positive voltage Vm be applied atthe op-amp output. From this time the capacitor C begins to charge. The chargingcurrent passes from the op-amp output through the open diode VD2, a part of theresistor R2, and the resistor R1 to the capacitor C. The voltage across the capacitorbegins to increase. At the time t1 this voltage becomes equal to “Vm, the regenerativeprocess is developed in the op-amp due to the positive feedback through the resistorsR3 and R4, and the op-amp output voltage takes the value �Vm stepwise. From thetime t1 the capacitor discharges, and the discharge current passes through the diode

Fig. 6.13 Oscillator withregulable duty ratio

Fig. 6.14 Operational diagrams of the oscillator with regulable duty ratio

6.4 Possible Circuits of Op-amp Oscillators 195

Fig. 6.15 Pulse oscillator in the differentiating RC circuit (a) and oscillograms of its operation(b) same parameters

VD1 and the upper part of the potentiometer R2. The discharge continues till t2,when the regenerative process is developed again, the output voltage returns thevalue CVm stepwise, and then all the processes repeat. Since the capacitor chargesand discharges through different parts of the resistor R2, the half-period durations ofthe oscillations can be different. Changing the position of the potentiometer slider,we can regulate the oscillation duty ratio.

Figure 6.15 demonstrates the circuit of the oscillator with the differentiationcircuit instead integration. This circuit was created on the reciprocity theoremapplying to the circuit on Fig. 6.6. The processes in this circuit are described bythe following system of equations:

8ˆ<

ˆ:

�dUC

dtD � UCCV D URI

�kdV

dtD � V C Vm � tanh

�K

�ˇ0 V � UR

��:

(6.26)

where £D RC, £k is the op-amp time constant, “0 D R2/(R1 C R2).The first equation represents the differentiating circuit, and the second one

corresponds to the op-amp amplifier with the negative feedback loop. Comparingthe systems of Eqs. 6.14 and 6.26, we can see that they are identical at the sameparameters. Consequently, the variations of the voltages at the op-amp output and atthe capacitor C in these circuits coincide with the analogous parameters on Fig. 6.11,and there is no need for additional analysis.

Equation system (6.26) coincides with equation system (6.13), so the circuitin Fig. 6.15 can serve as an illustration of the oscillatory system described inSect. 6.2.1. Concerning this circuit it is necessary to make the following observation.Voltage UR at some part of pulse period can exceed supply voltage, so amplifier


Fig. 6.16 Pulse oscillator in four op-amps with regulable frequency and duty ratio

input stage can break into saturation or cutoff mode, and input voltage will notcorrespond to the oscillogram in Fig. 6.15. To prevent it, amplifier output voltageneeds to be limited at the level of not less than Ecc>Vm(1 C “0). Limiter circuit isnot shown in the figure.

There is also a more complex square-wave oscillator in four op-amps [6]. This oscillatorallows separate regulation of the pulse frequency and the duty ratio. It is characterized bythe increased accuracy in setting the frequency and the duty ratio owing to application ofan integrator in place of the integrating RC circuit. The circuit of this oscillator is shownon Fig. 6.16. The oscillograms of its operation are similar to those on Fig. 6.7.

Different-polarity constant voltages depending on the position of the potentiome-ter slider R1 are formed at the outputs of the OA1 and OA2 op-amps. At the centralposition R1(R2 D R3 D R4 D R5 D R6 D R7) the voltages have equal absolute values.Through switches controlled from the OA4 output, the voltages are periodically fedto the input of the OA2 integrator, where they are integrated in turn and applied tothe input of the OA4 threshold element.

The equations describing this circuit largely coincide with Eq. 6.14, and theplots of voltages at the OA4 output are practically identical to the oscillograms onFig. 6.11. The difference is only that the voltage at the OA3 output varies fromone threshold level to another by the linear dependence, instead exponential. As theposition of the potentiometer slider is changed, the ratio of the positive and negativehalf-wave durations changes, while the change of the input voltage Vf changes thepulse repetition frequency. This oscillator can be used in precision signal sourcesbecause of its complexity.

6.5 Logic-Gate Oscillator 197

6.5 Logic-Gate Oscillator

The simplest logic-gate oscillator [5, 6] differs from the considered op-amposcillators by the logic gates creation and supplies from the one power supply, ratherthan two. This is its advantage.

The pulse oscillator on Fig. 6.17 is designed of logic gates LG1 and LG2, resistor R,and capacitor C in the circuit of unstable multivibrator. Periodic oscillations arise in itimmediately after switching the power on. Figure 6.14 shows the voltage oscillograms.

Consider the operation of this oscillator. Let at the time t0 the voltage V2 at theLG2 output changes stepwise from the low level to the high one, that is from 0to 1. The current begins to pass from the LG2 output through the capacitor C andthe resistor R through the LG1 output. This current charges the capacitor C. As itis charged, the current strength decreases, and the voltage V0 reduces to the timet1, when it achieves the threshold level Vthr.7 At that time the transistors of LG1and LG2 begin to operate in the active mode, and the positive feedback comes inforce in the circuit. It initiates the regenerative process, as a result of which thelogic gates transit into the opposite state: with the high level of voltage at the LG1output and the low level at the LG2 output. The change of the voltage at the LG2output is transmitted through the capacitor to the LG1 input and sets the low level ofvoltage in it. This voltage generates the high-level voltage at the LG1 output, whichdischarges the capacitor, because the direction of the current through the capacitoralternates: the current passes through the LG1 output through the resistor and thecapacitor to the LG2 output in the direction opposite to the direction of the chargingcurrent.

The alternation of the discharge current direction is important for understandingof the operation of a real oscillator. It can be said that the capacitor recharges fromthe time t1 to t2, and its voltage tends to the high-level voltage at the LG1 output.At the time t2, the voltage across the capacitor and, consequently at the LG1 inputagain achieves the threshold value Vthr, at which the LG1 and LG2 transistors againoperate in the active mode. The positive feedback is again formed in the circuit,

Fig. 6.17 Simplified circuitof logic-gate oscillator

7For transistor-transistor-logic circuits this voltage is roughly equal to 1.3 V, while for CMOS it isVcc/2.


Fig. 6.18 Oscillograms of operation of a logic-gate oscillator

and the new regenerative process starts, resulting in the alternation of the state ofthe logic gates: high-level voltage at the LG1 output and low-level voltage at theLG2 output. The circuit returns into the state, it was in at the time t0. Then all theprocesses repeat with the period T D t2 – t0.

Analysis of the oscillator operation describes it in the mode of periodic self-oscillations. When the supply voltage switching on, the pattern is somewhatdifferent. At that time the capacitor is fully discharged and the voltage across itis zero. Let the low-level voltage be at the LG2 output and, correspondingly, thehigh-level voltage be at its input, when the supply power is turned on. The processof charging of the capacitor C begins from LG1 through the resistor R. The voltageacross the capacitor increases with charging and, consequently, the voltage at theLG1 input grows as well. As soon as this voltage becomes equal to Vthr, LG1 andLG2 transit into the opposite state, analogous to the state at the time t2, as shown onFig. 6.18.

Then all processes carried out in accordance with this figure. As a result,the period from the power-on time to the beginning of periodic oscillations canbe longer. One can easily notice the similarity between the oscillograms of thelogic-gate oscillator and the op-amp oscillator with the differentiating circuit (seeFig. 6.15), because they both employ the differentiating circuit and the amplifier,which is constructed in logic gates in the latter.

6.6 Integrated Timer Oscillator

Recently specialized integrated microcircuits (chips) called timers have been developed.They allow designing the simple square-wave oscillator s. The most widely known amongsuch chips is the NE555 microcircuit schematically shown on Fig. 6.19.

6.6 Integrated Timer Oscillator 199

Fig. 6.19 Circuit of NE555 timer

0

2

4

6

8

1012

−100mTime (seconds)

555VAR.EWB

Vol

tage

Fig. 6.20 GC (a) and operational oscillograms of timer oscillator (b)

This chip includes two voltage comparators K1 and K2, a trigger T, an outputswitch in transistors VT1 and VT2, a voltage divider in resistors R1, R2, and R3, andan extra switch in open-collector transistor VT3 to obtain output voltages higherthan Vcc.

The circuit of the voltage comparators, trigger, and output switch with the joinedinputs 2 and 6 is similar, in its function, to the op-amp threshold switch (Fig. 6.8)and has the similar GC (Fig. 6.20). Consequently, using the timer as a thresholdswitch, it is possible to generate different square-wave oscillators.

Figure 6.21 shows the circuit of one of such oscillators. The timer DA1 togetherwith the potentiometer R1, resistor R2, and capacitor C1 form a square-waveoscillator. Regulating the resistance of the potentiometer R1, we can change thefrequency of the output voltage. The oscillograms of the output voltage and thevoltage across the capacitor of this oscillator are shown on Fig. 6.22 as V1 andV2 curves. It can be seen that all the processes in this oscillator are similar to theprocesses in the op-amp oscillator described in Sect. 6.3.


Fig. 6.21 Square-wave timer oscillator with controllable frequency and duty ratio

Fig. 6.22 Oscillograms of operation of timer oscillator

Figure 6.20b shows the results of simulation of these processes in the oscillatorbased on the NE555 chip in the Electronics Workbench software. The output voltageis a square wave and varies from zero to C12 V equal to the supply voltage.The voltage across the capacitor C1 changes by the exponential dependence fromone threshold voltage (C4 V) to another (C8 V), because the voltage divider dividesthe supply voltage into three equal parts.

6.6 Integrated Timer Oscillator 201

Fig. 6.23 Oscillator with square-wave, triangular, and sine-wave output voltages

There are some features in the operation of the timer oscillator.First, the output voltage in it is unipolar and varies from zero to C Vcc (rather than from

�Vcc2 to C Vcc1 ).Second, the application of the fast-response comparators allows us to increase the

accuracy in setting the oscillation frequency because of the decreased effect of the dwell(Fig. 6.11).

Third, the precision voltage divider also favors the increase of the accuracy in theoscillation frequency.

The exponential voltage V2 from the capacitor C1 is fed through the voltagefollower to DA3 and the resistor R6 and to the input 6 of DA2 along with thereference direct voltage Vref from the potentiometer R3 for formation of the voltagewith regulable duty ratio.

The comparator of the total voltage V4 and the internal reference voltage 2Vcc/3is made in DA2. If the total voltage is lower than the reference one, then the high-level voltage is generated at the DA2 output; otherwise, the DA2 output voltage isthe low-level voltage. Regulating the reference voltage V4, it is possible to changethe duty ratio of the output voltage, since the ratio of t3 � t1 to t2 � t1 is equal tothe duty ratio of the output voltage. The process of the output voltage formation isexplained by Fig. 6.18, which shown the operational oscillograms.

The advantage of this circuit is that it permits separate regulation of the frequencyand the duty ratio of the output voltage. The results of simulation show that thiscircuit operates with the frequency band from 10 Hz to 100 kHz and the duty ratiofrom 1.1 to 100.

The timer with the analogous design is used in the ISL8038 chips manufacturedby Intersil and MAX8038 manufactured by Maxim. Figure 6.23 shows schemati-


cally the timer oscillator, which generates not only square-wave, but also triangularand sine-wave pulses. This circuit also includes two comparators, a trigger, aresistive divider, and a switch.

In addition, it involves two current sources I0 and 2I0. This circuit operatessimilarly to the circuit in DA1 on Fig. 6.21 with the only difference that thecapacitor C is charged and discharged from the current sources I0 and 2I0, ratherthan through the resistor. Due this, the capacitor is charged and discharged by thelinear dependence. This linear varying voltage is fed through the buffer amplifier(BA) to the output 3, and in the sine-wave shaper (SWS) it is converted into thesine-wave voltage applied to the output 2.

6.7 Oscillators in Elements with Negative Resistance

Besides the considered oscillators with external feedback , there are also oscillators withinternal positive feedback , which is caused by the physical stricture of the active elementused. As was already mentioned in Chap. 5 , such active elements include semiconductordiodes having negative-resistance parts: dinistors, thyristors, tunnel diodes, and well assecondary-emission electronic tubes and gas-discharge lamps. These elements have the N-and S-shaped current-voltage characteristics.

Let us find under what conditions the voltage can change stepwise. Figure 6.24shows the generalized characteristics of these elements.

At the N-shaped characteristic, significant stepwise changes of the voltage occuronly when the working point transits from the ob branch of the characteristic to thedc branch, for example, from the point b to the point c and from the point d to thepoint a. As this takes place, the voltage can change from Vb to Vc and from Vd toVa . The voltage jumps are possible at the unchanged currents I2 and I1. Obviously,such jumps can be achieved, if we connect in series with the element the inductance,whose current does not change stepwise.

Fig. 6.24 Current-voltage characteristics of the N (a) and S types (b)

6.7 Oscillators in Elements with Negative Resistance 203

Fig. 6.25 Tunnel-diode pulse oscillator

Contrary, at the S-shaped characteristic stepwise changes of the current arepossible, for example, at transition from the point b to the point d and from the pointc to the point a at the almost constant voltage. The constant voltage can be obtainedby using a capacitor connected in parallel with the element having the S-shapedcharacteristic.

Consequently, tunnel-diode oscillators should be constructed with an inductanceconnected in series, while the dinistor oscillators should be made with a capacitorconnected in parallel.

Figure 6.25 shows a pulse oscillator in the tunnel diode VD. This oscillatorincludes also a power supply Vcc, an inductance L, and a resistor r. At the sectionb–d, the tunnel diode, whose current-voltage characteristic is on Fig. 6.25, has thenegative differential resistance with the value ranging from �20 to �100. Thepower supply voltage Vcc and the resistor r are chosen so that the initial position ofthe working point A falls within the section b–d. Figure 6.25 shows the load line 1passing through the point A at this section.

When the power is switch on, the current in the inductance is zero and the diodevoltage is zero too, therefore the working point is initially at the position O. Underthe effect of the supply voltage, the current begins to pass through the circuit Vcc, r,L, VD, and the working point moves along the branch 0–b. Upon reaching the pointb, the current can no longer increase, and due to the presence of the inductance in thecircuit, the working point moves to the point c in a jump. Since the supply voltageis lower than Vc , the current begins to decrease, and the working point moves fromc to d, wherefrom it returns to the point a in a jump. Then the process repeats.

It is obvious that supply voltage should be chosen from the condition Vb<Vcc<Vd ,while the resistance r< jrdifj. Since the jumps from the point b to the point c andfrom the point d to the point a occur rather fast, they correspond to the stepwisechanges of the output voltage. At the sections a–b and c–d, the speed depends onthe time constant of the rL-circuit and the diode characteristics. The phase portraitof the oscillator and the plots of the output voltage are shown on Fig. 6.26.


Fig. 6.26 Time diagrams (a) of the tunnel-diode oscillator and its limit cycle (b)

� With allowance for low parasitic capacitance C, the processes in the oscillatorcan be described by the following system of equations:

Vcc D rIL C VL C Vtd; IL D IC C Itd; VC D Vtd; Itd D '.Vtd/;

where ®(Vtd) is the single-valued function describing the current-voltage character-istic of the tunnel diode.

The system of equations obtained can be easily reduced to the system of twofirst-order differential equations:

8ˆ<

ˆ:

LdILdt

D Vcc � rIL � VtdI

CdV td

dtD IL � '.Vtd/:

Introduce new variables: L/r D £, rC D £k , IL D y, Vtd/r D x, Vcc D E, 1/r D g,then the system of equations takes the normalized form:

8ˆ<

ˆ:

�dy

dtD Eg � y � x;

�kdx

dtD y � '.x=g/:

(6.27)

�Another example of an oscillator in elements with negative resistance is a dinistoroscillator on Fig. 6.27.

Besides the dinistor, it includes a power supply Vcc, a resistor R, and a capac-itor C. The current-voltage characteristic of the dinistor has a negative-resistancesection b–d. When the power is switch on, the capacitor C begins to charge, and

6.7 Oscillators in Elements with Negative Resistance 205

Fig. 6.27 Dinistor oscillator (a) and its current-voltage characteristic (b)

Fig. 6.28 Time diagrams of dinistor oscillator and its phase portrait

the working point moves from the origin of coordinates along the line 0–a–b tothe point b. The dinistor VD turns on as soon as the voltage across the capacitorC achieves the value Von. The capacitor discharges quickly, and the output voltagechanges from Von to Voff stepwise. The working point quickly moves along the lineb–c–d to the point d.

If the condition I2<Vcc/R is fulfilled, then the dinistor working point moves tothe point a, after which the dinistor turns off. Then the new cycle of charging of thecapacitor C starts, and the process repeats.

Since the capacitor is charged from the power supply Vcc through the resistorR, the process obeys the exponential dependence with the rate determined by thesupply voltage, the resistance R, and the capacitance C. The shape of the outputvoltage and the phase portrait of the oscillator are shown on Fig. 6.28. The output


Fig. 6.29 Plot illustratingselection of the working pointof dinistor oscillator

voltage on Fig. 6.28a looks like pulsed periodic self-oscillations approximatelydescribed by the function V(t) D Vcc(1–exp(�t/RC)) at the time interval from t1 to t2.

The processes in such oscillator were earlier considered theoretically in [1]. Byanalogy, write the equations describing this circuit

Vcc D R.IC C Idin/C Vdin; IC D CdUC

dtD C

dVdin

dt; Idin D .Vdin/;

where §(Vdin) is the ambiguous function describing the voltage dependence of thedinistor current.

Excluding the current IC , we obtain the first-order nonlinear differential equa-tion:

dUdin

dtD 1

RC.Vcc � Vdin �R .Vdin// : (6.28)

The steady state (singular point) is determined at dUdindt

D 0; that is, from theequation

Vcc � Vdin � R .Vdin/ D Vcc � Vdin � RI.Vdin/ D 0:

The last expression is the equation of a straight line in the coordinates Idin andVdin, and it can be solved graphically. Figure 6.29 shows the solution of this equationat three values of the resistor R. At R D R1 the steady position is at the point A1,and at R D R2 it is at the point A2.

The stability of the steady state is determined by the sign of the function§(Vdin) near the steady position. It is obvious that the points A1 and A3 arestable, while the point A2 is unstable. If the steady position is stable, no oscillationarises. Consequently, the resistor R must be equal to R2 (higher than the value ofthe negative dinistor resistance). However, based on the first-order equation, it isdifficult to justify the periodic oscillations in the system. For this it is necessary tointroduce the hypothesis of “jump” [14]. This can be achieved much more easily,if we assume the presence of one more low reactance, for example, inductance L

6.8 Conclusions 207

of dinistor leads. Then the system is described by two, rather than one, first-orderequations:

8ˆ<

ˆ:

RCdUC

dtD Vcc � VC �RIdin;

LdI

dtD VC � �1 .Idin/;

where §�1(Idin) is the function inverse to §(Vdin), that is already single-valuedfunction.

It is easy to see the analogy of the obtained system of equations with the system(6.14). Toward this end, change the variables: L/R D £k ; RC D £; IdinR D x; VC D y;1/R D g, Vcc D E, then the system takes the form

8ˆ<

ˆ:

�dy

dtD E � y � x;

�kdx

dtD y � .xg/:

(6.29)

Comparing the systems of Eqs. 6.29 and 6.27, we can see that they have the identicalform and differ only in the scaling factors. Thus, the processes in the dinistor and tunnel-diode oscillators are described by the identical systems of equations and the pulsedsignals generated have the similar form. Consequently, the conditions for appearanceand generation of periodic self-oscillations are identical for these oscillators. Unlike theequations of op-amp oscillators (6.15), they include the single-valued nonlinear function ®or , while Eq. 6.15 include the ambiguous hysteresis function ®. It easy to find the relationbetween these functions.

6.8 Conclusions

1. All considered oscillators generate stable periodic pulsed self-oscillations.2. The processes in these oscillators are described by the similar systems of

equations (6.15), (6.27), and (6.29).3. Under any initial conditions, all trajectories of the image point on the phase plane

in the considered oscillating systems are directed toward the single limit cycle(for example, that shown on Fig. 6.10 by a rectangle with the vortices 1–2–3–4or Figs. 6.25 and 6.27 by the lines a–b–c–d). Therefore, the limit cycle trajectoryis stable to hindrances.

4. To achieve the uniqueness and stability of the limit cycle, the system of equationsmust have singular points of the stable node type outside and inside the limitcycle, to which trajectories tend from inside and from outside.

5. There are no other limit cycles and, consequently other forms of output oscilla-tions.


Questions

1. What are the features of pulsed signals?2. The method of phase plane is used to study the processes in pulse oscillators.

Explain what is the essence of this method.3. What is the phase plane (image point, hodograph curve, limit cycle)?4. What is a singular point of the phase plane?5. What are the possible types of singular points? Show them.6. Draw the circuit of an op-amp pulse oscillator (tunnel-diode, dinistor, logic-gate

oscillator).7. Explain the similarity and difference of oscillators in elements with negative

resistance.8. What is the similarity between op-amp oscillators and oscillators in devices with

negative resistance?9. Self-oscillations are possible in the circuit with a tunnel diode, is not it? Give a

detailed answer.

Test Yourself

1. Features of pulsed signals are:

(a) parts with fast and slow change of signal, sharp transition from slowchanges to the fast ones, broad spectrum of signal;

(b) parts with fast and slow change of signal, smooth transition from slowchanges to the fast ones, narrow spectrum of signal.

2. It is desirable to study processes in pulse oscillators in the following way:

(a) through analytical solution of a system of differential equations;(b) through numerical solution of differential equations;(c) by the method of phase plane.

3. Phase trajectory is:

(a) a line in the phase plane, representing one of the solutions of the differentialequation;

6.8 Conclusions 209

(b) a line in the phase plane, connecting two given points;(c) a line in the phase plane, passing through a given point.

4. Singular point is:

(a) a point in the phase plane, at which the tangent to the phase trajectories isdirected horizontally;

(b) a point in the phase plane, at which the direction of the tangent to the phasetrajectories is uncertain;

(c) a point in the phase plane, at which the tangent to the phase trajectories isdirected vertically.

5. Singular point, no one enveloping closed integral curve passes through, is:

(a) saddle;(b) node;(c) center;(d) focus.

6. The amplitude and phase balance conditions in the oscillating system of a pulseoscillator are:

(a) fulfilled;(b) not fulfilled.

7. The set of zeros of the linearized characteristic equation of an oscillator,corresponding to the condition of excitation of pulsed oscillations, is:

(a) s1 D C0.1, s2 D C2,000;(b) s1 D C0.1 C j1,000, s2 D C 0.1 � j 1,000;(c) s1 D Cj100, s2 D � j 100.

8. The output voltage V of the oscillator is shown by the plot:

(a) V 1;(b) V 2;(c) V 3.

9. The voltage across the capacitor C of the oscillator is shown by the plot:

(a) V 1;(b) V 2;(c) V 3.


10. The tunnel-diode oscillator must include:

(a) an inductance;(b) a capacitor;(c) a resistor.

11. The dinistor oscillator must include:

(a) an inductance;(b) a capacitor;(c) a resistor.

References

1. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillations. Pergamon Press, Oxford(1966)

2. Barkhausen, H.: Elektronen-Rochen. 3. band: Ruckkopplung, 3. Auflage. S. Hirzel Verlang,Leipzig (1934) (in German)

3. Rybin, Yu.K.: The condition of reproducing the prescribed shape periodic oscillations ingenerators. Proc. Tomsk Polytechnic Univ. 316(4), 136–140 (in Russian) (2010)

4. Rybin, Yu.K.: Electronic Devices for Analog Signal Processing. Students’ Book. Print Manu-facture Publishers, Tomsk (2005)

5. Horovitz, P., Hill, W.: The Art of Electronics. Cambridge University Press, New York (1998)6. Tietze, U., Schenk, Ch: Halbleiter-Schaltungstechnik. Springer, Berlin/Heidelberg/New York

(1980)

Chapter 7Signal Conditioners

Abstract The object of this Chapter is acquaintance with the construction ofmodern analog electronic signal conditioners, that is, devices placed between asignal source and a readout instrument to change the signal. Examples of readoutinstruments are various sensors, such as mechanic pressure, displacement, andtemperature sensors. In fact, in all previous chapters we already considered devicesintended for signal conditioning. In this Chapter, these devices are considered notseparately, but in the interaction with each other, and the problems of their mutualinfluence and optimal interfacing are investigated.

For in-depth understanding of the material presented, the readers should firststudy the previous chapters of this book and be familiar with, in particular, lineardevices, detectors, and signal generators.

Upon learning this Chapter, the readers will understand the operation of signalconditioners, be able to properly apply them, and determine the parameters andcharacteristics for application with a particular sensor.

7.1 Introduction

Analog electronic devices for processing of signals from sensors of physical quantities arefinding expanding applications in modern measuring instruments.

Such devices serve for interfacing with sensors pressure and temperature, aswell as, sensors of optical characteristics, for example, resistive-strain sensors,thermocouples, optical sensors, ionization detection sensors, etc.

The devices of such kind are usually called signal conditioners, and the operations neededto transform the sensor output into the form necessary to interface with other elements arecalled signal conditioning.

The advent of these devices is the result of efforts to facilitate the developmentof measuring instruments, initiate unification of sensor outputs, improve the sensorreliability, etc.


211

212 7 Signal Conditioners

Fabrication of these devices in the form of integrated chips almost spares theuser the need in development of devices, application of discrete units, and solutionof problems of their interfacing. The integrated technology opens the possibility torealize the sensor itself and electronic devices for interfacing and converting sensoroutputs in a single crystal. With the integrated technology, it is easy to provide forthe measures for linearization and temperature stabilization of the parameters andcharacteristics, overload and misplug protection, operation in a wide range of supplyvoltages, etc. In essence, such microcircuits operate as smart sensors.

Today we know specialized chips for processing of signals from resistance temperature de-vices (RDT), strain gauges, linear variable differential transformer (LVDT) and rotationalvariable differential transformer (RVDT) inductive displacement sensors, thermocouples,photodiodes, etc.

Below we will consider some typical devices, technologies, and solutions, whichhave already de facto become the standards in signal conditioning.

Most of physical transducers transform the measured physical variable into some electricalvariable. Electrical variables are convenient to scale, transform, filter, and transmit to somedistance. All electrical variables can be divided into two classes: passive and active. Passivevariables include resistance, capacitance, inductance, and conductance, while the activeones involve electrical current and electromotive force (EMF).

The variables of the first class are called passive, because they do not showthemselves without the action of the electrical energy, while those of the secondclass, to the contrary, generate the electrical energy. Therefore, the circuits forprocessing of signals from passive sensors should necessarily include, apart fromthe processing circuit itself, the sources for excitation of these sensors.

7.2 Resistive Sensor Signal Conditioners

Resistive sensors are widely used for conversion of changes in temperature, pressure,irradiance, and position into the proportional change in resistance. This class of sensors,naturally, includes thermal resistor, strain gauge, photoresistor, potentiometric resistor, etc.These sensors are distinguished by stable characteristics, are manufacturable, and havemany other advantages. However, resistance is a passive electrical quantity. It cannot betransmitted to a distance from the place of application of an action. That is why the sensorresistance is prior converted into the electrical voltage or current, usually, by transmittingcurrent through the resistor or by applying voltage on the resistor. According to the Ohmlaw, the voltage drop in the resistor is proportional to the resistance and the current.

However, the voltage dependence not only on the resistance, but also on thecurrent necessitates stabilization of the current. In addition, almost all resistivesensors are sensitive to temperature, humidity, vibrations, etc. Therefore, the sensorresistance usually depends on several, rather than one, physical variables. In bridgecircuits, it is possible to realize the sensitivity of the sensor to only one variable andmitigate the influence of other, not measured variables.

7.2 Resistive Sensor Signal Conditioners 213

Fig. 7.1 Bridge resistive circuits supplied from an EMF (voltage) source (a) and a current source(b) and half-bridge circuit (c)

Figure 7.1 shows the typical circuits of bridge resistive transformers.The bridge resistive-capacitive circuit was used in Chap. 5 when considering

the processes in the Wien bridge oscillator or in RC oscillator and op-amp pulseoscillator in Chap. 6. In this case, the bridge consisting of only resistors is applied.

The bridge is used as a measuring converter of the change in the resistance of one or severalresistors into the voltage. It is an electric circuit consisting of four resistors 1 and a powersupply. The bridge circuit includes two diagonals ab and cd, two braches acb and adb, andfour bridge arms ac, cb, ad and bd.

The power supply (EMF (voltage) or current source) is connected to one of thediagonals, and the output voltage or current is read out from the other diagonal. Thebridge can be supplied by both dc and ac current with the sinewave or non-sinewavewaveform. Though the four-resistor bridge circuit was proposed, for the first time,by Christie, it is called the Wheatstone bridge, because Wheatstone set the precedentfor using it to measure resistances.

The resistive sensor is connected in one of the bridge arms. If there are two, three, or fourresistive sensors, then they are connected in the corresponding bridge arms. In such case,the bridge is referred to as one-, two, three-, or four-element bridge.

The change of any bridge resistance results in the change of the current throughthe corresponding branch and, as a consequence, in redistribution of voltage dropsacross the branch elements and variation of the output voltage�V or current in thediagonal cd. Complete theory of bridge circuits can be found in [1].

If the equality

R1 �R4 D R2 �R3 (7.1)

1In the general case, bridge elements may be not only resistors, but also capacitors, inductances,or their combinations.


is fulfilled, the output voltage or current are zero, and therefore the bridge is calledbalanced.

It is obvious that the bridge remains balanced, if the resistances R1 and R2, or R3and R4, or R1 and R3, or R2 and R4 change simultaneously by the same relativevalue. Therefore, the sensors with the same sign of resistance variation shouldbe connected in the opposite bridge arms, while those with the opposing sign ofresistance variation should be included in the neighboring arms. The rated bridgeresistances are usually taken equal. It is worth noting that the condition of bridgebalance does not change, if we connect the power supply in the diagonal cd andread out the output voltage from the diagonal ab.

For the half-bridge circuit, the condition of bridge balance depends on the supply voltages

R2E1 D R1E2: (7.2)

If in the bridge circuit supplied from the EMF source (Fig. 7.1a) the resistancesof one, two, or four2 resistors change simultaneously (in opposing directions), theoutput voltage is determined by the equations:

�V 0 D E�R

4RC 2�R; �V 00 D E

�R

2R; �V 0000 D E

�R

R: (7.3)

At the similar changes of resistances in the circuit supplied from the currentsource, the output voltages are, correspondingly, equal to

�V 0 D IR�R

4RC�R; �V 00 D I

�R

2; �V 0000 D I�R: (7.4)

For the half-bridge circuit, the output voltages are [2]

�V 0 D E�R

2RC�R; �V 00 D E

�R

R: (7.5)

It is obvious that the highest sensitivity and linearity to resistance variations areinherent in the four-element bridge and two-element half-bridge. It is interestingto note that the input parameters in the bridge circuit are the supply voltage E orcurrent I and the change �R. With respect to variations of the voltage E or currentI the bridge behaves as a linear circuit, while with respect to the change of R itbehaves as a parametric circuit (because the parameters of the elements change).That is why it is classified as a linear parametric circuit.

2Bridge circuits with three elements are rarely used.


Fig. 7.2 Circuit of interfacing the measuring resistive bridge to the difference amplifier (a) and itsequivalent circuit (b)

To transmit the voltage from the bridge output to some distance, the bridge is complementedwith signal conditioners (interfacing devices). Figure 7.2 shows the simplest version ofinterfacing the Wheatstone bridge to the difference amplifier.

The circuit on Fig. 7.2a includes a measuring bridge in R1–R4 resistors (arrowsnear resistors indicate the direction of change of their resistances) and the op-ampdifference amplifier in R5–R8 resistors (Fig. 3.10). An instrumental amplifier, suchas INA132 manufactured by Texas Instruments, can be used as the op-amp withresistors. As can be seen from the figure, this circuit employs the four-elementbridge, but one-or two-element bridges can be used as well.

The output voltages of the bridge are the voltages V1 and V2. If the equalityR1R4 D R2R3 is fulfilled, these voltages are equal. In this case, the voltage in thebridge diagonal is zero. For this condition to keep true upon connection of theamplifier, the equalities of the resistances: R5 D R6 and R7 D R8, should be fulfilledas well.

If the bridge balance is disturbed, for example, by changing the resistanceof at least one resistor, the voltage in the bridge diagonal and at the amplifieroutput becomes nonzero. It can be determined from the circuit on Fig. 7.2b,where the dashed rectangle encloses the equivalent circuit of the bridge, whoseparameters (voltages C E1 and C E2 and resistances R00 D R1jjR2 and R0 D R3jjR4)are determined by the Thevenin’s theorem.


Fig. 7.3 Versions of interfacing circuits for the measuring resistive bridge with linearization oftransformation characteristics

For the four-element circuit, the output voltage is

Vout D �E2 R8

R5 CR3jjR4 C E1R7

R6 CR7 CR1jjR2R8 CR5 CR3jjR4R5 CR3jjR4 :

Under the condition R5 D R6, R7 D R8, R1 D R2 D R3 D R4 D R � R6 C R7, weobtain

Vout D �V 0000 R8

R5 CR3jjR4 � E�R

R

R8

R5:

As expected, the output voltage is equal to the product of the bridge outputvoltage and the gain of the difference amplifier. For interfacing with the resistivebridge, it is possible to use any other of the amplifiers in Chap. 3 (Figs. 3.11–3.15).

The considered circuit has one significant disadvantage. With the most widely used one-element bridge, the output voltage depends nonlinearly on the change in the resistancebecause of the term 2�R present in the denominator of Eq. 7.3. To linearize the outputvoltage, other versions of the bridge connection are employed, and some of them are shownon Fig. 7.3 [2].

In the first circuit, the linearizing effect is achieved due to the resistor with thevariable resistance connected as a feedback resistor. Owing to the feedback, thevoltages across the op-amp inputs becomes practically equal, but the voltage atthe noninverting input, at the equal resistances, is equal to the half bridge supplyvoltage. Consequently, the voltage drops at the resistors R1, R2, and R3 are alsoequal to E/2. So we can easily find the currents through the resistors. They are equalto E/2R.


The same current is transmitted by the resistor R4 too; consequently, the outputvoltage is

Vout D E

2� E

2

RC�R

RD �E�R

2R:

In the circuit on Fig. 7.3b, the linearization is also performed due to the feedback.But in this case the op-amp serves to maintain the zero potential at the point c. Theoutput voltage can be calculated by the equation

Vout D �ER2 C�R

R1

R3

R3 CR4C E

R4

R3 CR4D �E�R

2R:

All circuits considered implement the interface with the bridges supplied by the EMF source.Usually, this source and amplifier s are located far away from the sensor. Therefore, theinterfacing circuit is connected with the sensor by long cables. At the low sensor resistance(200–400 Ohm) the resistance of the connecting cables begins to play a significant role.A specific error arises due to the decrease of the bridge supply voltage directly at the placeof the sensor location and the bridge.

Figure 7.4 shows the equivalent circuits explaining the causes for appearance ofthe error due to remoteness of the power supply or the sensor. In this figure theresistances of connecting cables are shown in dashed rectangles as resistors r.

Two connecting cables in the first circuit (a) reduces the bridge supply voltageat the place of the bridge location (the voltage Vab decreases), while in the secondcircuit (b) the cable resistances disturb the bridge balance. If we take into accountthat the connecting cables are often used in a wide range of the environmenttemperatures, then the change of their resistances under the effect of temperaturecan considerably influence the measurement error.

The effect of the resistances of the connecting cables can be partly mitigated using the three-wire connection as shown on Fig. 7.4d. In this case, the resistances of the cables r 1 andr 2 are included in the neighboring bridge arms and no longer affect the bridge balance.In the six-wire connection, the cables with the resistances r 3 and r 4 are added to fed thesignal equal to the bridge supply voltage directly from the place of the bridge location tothe power supply and thus correct its voltage with regard for the resistances of the powercables r 1 and r 2.

However, these disadvantages can be eliminated radically, if the bridge is supplied fromthe current source. Figure 7.5 shows one of the versions of interfacing with the bridgesupplied from the current source.

In this circuit, the measuring bridge is connected in the negative feedback loop,in which the constant current is maintained due to the constant voltage drop at thestandard resistor R0, which is constant to the reference voltage Vref and independentof the resistances r1 and r2 of the connecting cables. The effect of the resistances r3

and r4 of the cables connecting the bridge output with the following devices can beeasily eliminated by applying difference amplifiers with high input resistance (forexample, amplifiers on Figs. 3.11–3.15).


Fig. 7.4 Equivalent circuits of bridge (a) with two-wire (b), three-wire (d), and six-wire (c) con-nection

Fig. 7.5 Circuit of remote measuring bridge supplied from current source

7.3 Inductive Sensor Signal Conditioner 219

Fig. 7.6 Simplified circuit of transformer (a) and autotransformer (b) inductive displacementsensors

Currently fabricated integrated microcircuits combine the functions of amplifica-tion of the bridge disbalance signal and the functions of its rectification (for the acsupplied bridge). Many integrated circuits perform transformation of the rectifiedvoltage into the digital form, etc. Examples of such microcircuits are AD598 andAD698 fabricated by Analog Devices and MAX1467 fabricated by MAXIM.

7.3 Inductive Sensor Signal Conditioner

Inductive sensors are used to transform linear displacements into the proportional voltage.They are characterized by the high sensitivity and stability of parameters with time, resistantto temperature, and capable of operating under difficult production operating conditions.

We know the great number of inductive sensors different in the design and inthe principle of transformation of the displacement into an electrical parameter.For better understanding of the operation of inductive sensor signal conditioners,consider the operating principle of LVDT (Linear Variable Differential Transformer)sensors, which have gained the widest acceptance.

In such sensor, the linear displacement is transformed into the proportional (in theamplitude and phase) alternating voltage. Figure 7.6 shows schematically the displacementsensors and the circuits of their connection.

In the first circuit on Fig. 7.6a, the alternating sine-wave supply voltage E3

induces the current in the transformer primary winding, which induces the magneticfield inside the coil. In turn the magnetic field induces EMF V 0

out and V 00out in the

secondary windings. These EMF values are equal, if the moving core is at the centerof the windings, and different, if the core is displaced. Determining the amplitudedifference of these voltages and the phase, we can find the direction of motion andthe value of the core displacement.

3Here it is worth referring to the sensor supply voltage as the sensor excitation voltage.


The magnetic field in the autotransformer sensor on Fig. 7.6b is excited bytwo alternating voltages with the same amplitude and frequency, but shifted by180ı with respect to each other. The excitation sources and the sensor form thehalf-bridge circuit with the output voltage read out in one of the diagonals. Ifthe coils are symmetric, the amplitudes of the exciting voltages are equal, andthe core is positioned at the center, then the output voltage is zero. As the coredisplaces, the nonzero output voltage is induced with the amplitude proportional tothe displacement and the phase proportional to the direction of motion. Such sensorsare characterized by high accuracy and stability of the coefficient of displacementtransformation into voltage.

Many companies now produce such sensors. They differ constructively in themounting dimensions and settings, but have the same operating principle. Amongthe well-known and widely applied LVDT sensors are Models 75511 producedby Izmeron, E100 fabricated by Schaeviz, 210–220 fabricated by Trans-Tek Inc.,and others. Specialized chips for signal conditioning are also produced for thesesensors. Examples are AD598, AD698 fabricated by Analog Devices, NE5521fabricated by Phillips Semiconductors, MAX1457 fabricated by MAXIM, and others.Some companies produce tools for displacement measurement based on thesemicrocircuits in the form of completed devices with digital indication and interfacefor information transmission into a computer.

Consider the operating principle of signal conditioning circuits with AD6984

taken as an example. Figure 7.7 demonstrates the simplified functional diagram ofthis circuit [3].

The chip AD698 is intended for the use with both transformer and autotransformer sensors.

It includes two independent synchronous demodulation channels. The B-channelserves to detect the amplitude of the excitation voltage applied to LVDT. It consistsof the amplifier A1, comparator C1, voltage multiplier M1, and the low pass filterLPF1. The input sine-wave voltage is converted in C1 into the pulsed rectangularvoltage and is fed, along with the voltage amplified by the amplifier A1, to M1,where they are multiplied. If the voltages to be multiplied are in phase, M1, inessence, performs full-wave rectification of the alternating voltage. The M1 outputvoltage is filtered in LPF1 with an external capacitor C1 connected to Bfil leads,and the direct voltage proportional to the half-period average excitation voltage isapplied to the PWMVD computing circuit.

The A-channel is identical to the B-channel except for the fact that the compara-tor C1 has an individual input. In the A-channel, the input voltage is multipliedwith the Ain voltage in M2 and filtered by LPF2. Then the ratio of the outputvoltages of the channels VA/VB is computed in the pulse-width divider. Introductionof the voltage divider decreases the error in the transformation coefficient due tothe change in the amplitude of the excitation voltage and improves the temperature

4Universal LVDT signal conditioner AD698, http://www.analog.com

7.3 Inductive Sensor Signal Conditioner 221

Fig. 7.7 Functional diagram of AD698 circuit fabricated by Analog Devices: op-amps A, com-parators C, multipliers M, low pass filter LPF, VCCS differential amplifying stage, pulse-widthmodulation voltage divider PWMVD, sawtooth generator STG, shaper of antiphase sine-wavevoltages S

stability. Then the voltage equal to the ratio of the channel voltages is amplified bythe differential stage and the amplifier A3. The differential stage provides for thepossibility of compensation for the offset voltage, for which the trimming resistor isconnected between the corresponding lead Off and the supply voltage – Vs and theneeded gain is set in the amplifier A3 through connection of the resistor betweenthe leads Fb and Vout. If we take into account that the synchronous detector outputsthe voltage proportional to the amplitude and the cosine of the phase angle, then tocompensate for the possible phase shift at the input Ain, it may be needed to connectthe RC chain for compensation for the phase difference in the primary or secondarycircuits of LVDT. Only in this case it is possible to achieve the zero output voltageat the zero LVDT position.

AD698 contains also a low-distortion sine wave oscillator to drive the LVDT. Thesine-wave voltage is formed in the shaper S from the triangular STG voltage. Theprinciple of STG construction is analogous to that of the generator shown on Fig.6.10. The shaper reproduces two voltages of the same amplitude and frequency, butshifted by 180ı with respect to each other, at the leads Exc1 and Exc2. The amplitudeof these voltages is set by the external resistor R1 connected to the leads Lev1 and


Fig. 7.8 Circuit of interfacing the autotransformer LVDT sensor and AD698

Lev2, while the frequency is set by the capacitor C1 connected to the leads Freq1and Freq2. Figure 7.8 shows the circuit of connection of the autotransformer LVDTto AD6985.

Calculation of the circuit elements. The output voltage is determined from therelation [3]

Vout D 500 � 10�6.A/Va

VbR2.Om/;

where Va is the voltage at the input CAin, Vb is the voltage across the inputs Bin.The capacitance of the capacitor C1 setting the generator frequency is deter-

mined as

C1 D 35 � 10�6.F �H z/=fexc :

The capacitances of the LPF capacitors are determined by the filter cutofffrequency fcut. If this frequency is taken equal to 250 Hz, then the capacitancesare equal to 68 nF.

The supply voltage ˙Vs is selected in the range of ˙(3–15) V.At these parameters, the conversion function looks like on Fig. 7.9. Connecting

the analog-to-digital converter to the AD698 output, we can obtain the result in

5This circuit employs the version of LVDT connection other than that recommended in “UniversalLVDT signal conditioner AD698”, http://www.analog.com

7.4 Optical Sensor Signal Conditioners 223

Fig. 7.9 Displacement-to-voltage conversion function

the digital form. The range of displacements varies from 1 to 1,000 mm and isdetermined by the sensor. The total error of the displacement transducer does notexceed 0.3–0.5%.

The AD698 microcircuit can be used not only with inductive, but also withresistive and capacitive bridge circuits.

7.4 Optical Sensor Signal Conditioners

Various types of sensors are used to convert optical parameters into electrical ones. Exam-ples are photoresistors, photodiodes, phototransistors, photoelectric cells, photoelectronicamplifiers, etc. Among them, photodiodes have gained the wide utility. The circuits ofphotodiode signal conditioning differ significantly from the circuits considered above.Photodiode features dictate these differences. First of all, photodiodes, in contrast toresistive and inductive sensors, fall in the class of active sensors. They generate currentor EMF depending on irradiance. Within five to six decades, the photodiode current isproportional to irradiance.

As to EMF, it weakly depends on irradiance, but strongly changes with tem-perature. That is why photodiodes are usually used as current sources. Anotherfeature is that the photodiode current is very low: from 1 pA to 1 �A. Such a lowcurrent practically cannot be transmitted to a distance from the place of the sensorlocation to the signal conditioner because of the influence of interferences andparasitic leakages. Therefore, the photodiode currents are pre-amplified or convertedinto voltage. Op-amp with the CCCS structure (see Chap. 1) would be the idealcandidate for amplification of the current, but there are no op-amps of this typewith low input currents. Therefore, often the photodiode current is converted intovoltage, for example, with the circuits on Figs. 3.6 and 3.7. It is obvious that theincreased requirements are imposed upon the op-amp input currents and the designof the interfacing circuits. The practical photodiode interfacing circuits are shownon Figs. 7.10 and 7.11 [2, 4, 5].

The circuits on Fig. 7.10 are carried out based on current converters (Figs. 3.6and 3.7). They include additional resistors R1 and capacitors C1 and C2 to decreasethe shift of the output voltage due to the op-amp input currents and to reduce


Fig. 7.10 Versions of the photodiode interfacing circuits in one op-amp with resistor in thefeedback loop (a) (Walt Kester. Practical design techniques for sensor signal conditioning,amplifiers for signal conditioning, Analog Devices, http://www.alalog.com) and with T-circuit(b) (Bonnie C. Baker. Comparison of noise performance between a FET transimpedance amplifierand a switched integrator, sboa 034, Texas Instruments, http://www.ti.com/)

Fig. 7.11 Photodiode interfacing circuit in two (output from OA1) or three op-amps

noise. In addition, the circuits include the so-called “guard rings,” which serve todecrease the leakage currents. Generally speaking, serious consideration should bepaid to the correct construction of the input part of low, nano- and micro-currentamplifiers, because the input currents of modern op-amps are negligibly small (theinput current of OPA124 fabricated by Texas Instruments does not exceed 1 pA,and for AD549 fabricated by Analog Devices it is as low as 100 fA) and the parasiticleakage currents through elements and the surface currents of printed circuit begin tohave the pronounced effect. The guard ring neutralizes the effect of these currents.

7.5 Thermocouple Signal Conditioners 225

It is made as an arc-shaped or ring-shaped conductor of the metallized layer ofprinted circuit, which encloses the op-amp input leads. Besides the guard ring, theop-amp input leads are placed on insulating Teflon supports. The ring and the Teflonsupports reduce effect of the leakage currents from the power supplies and otherelements on the op-amp input.

The circuit on Fig. 7.11 is constructed in three op-amps using OPA2111 andINA132 instrumental amplifier fabricated by Texas Instruments.

Upon small development, these circuits of current-to-voltage converters can bealso applied for interfacing with other high-resistance sensors: temperature sensors,pH – sensors, chemical sensors, smoke detectors, etc.

7.5 Thermocouple Signal Conditioners

In 1823 the German physicist Seebeck discovered the phenomenon of thermoelectricity,which appears in two dissimilar conductors or semiconductors A and B joined andmaintained at different temperatures T 1 and T 2. The thermoelectromotive force (E T ) arisesin such circuit.

This circuit is called the thermoelectric converter or thermocouple; componentconductors are called thermoelectrodes, and the place of their joining is called ajunction point. The thermocouple consists of two conductors, most often, alloys,for example, iron–constantan, chromel–alumel, platinum–platinum-rhodium, etc.Thermo EMF arising between them is a nonlinear function of temperature. To de-crease this nonlinearity effect, two junctions are used as shown on Fig. 7.12a. In thiscase, the output voltage is equal to the difference of thermo EMF values. This is thedifference of the functions depending on the component temperatures

ET D f .T1/ � f .T2/ D a .T1 � T2/C b.T1 � T2/2 C c.T1 � T2/3 C : : :

The coefficients a, b, c and others depend on the materials used. They are knownand tabulated for various thermocouples. At the small temperature difference, E T

can be believed directly proportional to the temperature difference

ET Š a .T1 � T2/ :

Fig. 7.12 Thermocouple connection: A and B are thermocouple materials with different junctiontemperatures (a) and with one junction temperature of 0ıC (b)


Fig. 7.13 Circuit ofelectronic compensation forcold-junction temperature

Known and widely used thermocouples consist of both noble (platinorhodium-platinum ofTPP type) and base metals (chromel-alumel of TCA type and chromel-copel of TCC type).Thermocouples can be employed under different temperatures and operating conditions.

Thermocouples, as photodiodes, fall in the class of active sensors, because they generateEMF under the effect of temperature, but this EMF is as low as 10–50 �V/K. Naturally,this EMF is difficult to measure. First, the values of thermo EMF are low and, to amplifythem, amplifiers with low offset voltage are needed. Second, transmission of such lowvoltage to a distance also presents a problem, because the connecting cables needed fortransmission also form new (parasitic) thermocouples at the places of their contact withthe thermocouple. And, finally, thermo EMF is proportional to the temperature difference,rather than the absolute value of the temperature.

Figure 7.12 shows the thermocouple connection.

V 0out D f .T1 � T2/ V

00out D aT1

As can be seen, it is rather difficult to satisfy all the conditions, at least because it isnecessary to ensure, in some way, T 2 D 0ıC, for example, by placing the thermocouple T2in the medium, where water is simultaneously in three its phases: solid, liquid, and gaseous(Fig. 7.12b). As known, the temperature of such medium is equal to 0ıC. This is practicallyinconvenient, and therefore the temperature of cold junction is provided electronically.Figure 7.13 shows the simplified circuit of electronic compensation for the cold-junctiontemperature.

In this case, specialized wires are used to connect the thermocouple to theelectronic circuit. The parasitic junction’s T3 and T4 are naturally formed. ThermoEMFs of these junctions affect the measurement error. To exclude this error,the junction temperatures are balanced by introducing the isothermal unit witha temperature sensor and using the electronic compensation circuit in place ofthe thermocouple T2 (Fig. 7.13) for compensation for the cold-junction temper-ature. The compensation circuit generates the output voltage equal to the sumE T2 C E T3 C E T4, that is, the cold-junction EMF. All these circuits can be easilyimplemented with the integrated-circuit form of the interfacing elements. Forexample, AD594 – AD595 chips fabricated by Analog Devices have the built-in cold-junction compensation circuit and instrumental amplifier to obtain the normalized

7.6 Voltage and Current Sensor Signal Conditioners 227

Table 7.1 Thermocouple options

Junction materialType ofthermocouple

Temperaturerange (ıC)

Rated sensitivitya (�V/K)

ANSI letterdesignation

Chromel–copel TCC �200 : : : C900 61 EIron–copel 0 : : : C800 52 JCopper– copel �200 : : : C300 41 TChromel–Alumela TCA �200 : : :

C1,30039 K

Tungsten (5%)Rhenium–Tungsten(20%)Rhenium

TTR C800 : : :C1,950

16 C

Platinum(87%) C Rhodium(13%) – Platinum

TPR 0 : : : C1,600 11.7 R

Platinum(90%) C Rhodium(10%) – Platinum

TPP 0 : : : C1,300 10.4 S

a Chromel: 90% Ni, 10% Cr. Alumel: 94.83% Ni, 3% Mn, 2% Al, 1% Si, and 0.17% Fe.Constantan: 54% Cu, 45% Ni, 1% Mn. Copel: 56% Cu, 44% Ni

output voltage of 10 mV/ıC. The AD594 chip is operated in conjunction withthe J-type thermocouple, while AD595 is intended for operation with the K-typethermocouple. Microcircuits are used to measure temperature in the range from �55to C125ıC with the error no higher than 1ıC.

The Table 7.1 below summarizes the averaged parameters of thermocouples [2].

7.6 Voltage and Current Sensor Signal Conditioners

Electrical current sensors are represented by shunts, current transformers, field-effecttransistors, Hall-effect sensors, and other devices. Shunts and Hall-effect sensors are usedin transformation of both ac and dc current. Current transformers are used exclusively fortransformation of ac current, while the field-effect transistors are used only for dc current.

The wide variety of sensors has given rise to the variety of interfacing circuits.Consider some of them.

Figure 7.14 shows one of the versions of shunt interfacing with a differenceamplifier. A feature of this circuit is that the load is grounded, while the shunt isnot connected to the earth. At such interfacing, the common-mode component ofthe signal has the high level, because the shunt can be under the voltage up to200 V. With the voltage divider assembled of the resistors R1–R5, this common-mode voltage is decreased down to the level of 10 V, which is safe for the op-amp.This device employs the specialized instrumental amplifier INA117 fabricated byTexas Instruments.


Fig. 7.14 Circuit of shuntinterfacing with differenceamplifier

7.7 Conclusions

The concept of “signal processing” includes the generation, conversion, and trans-mission of signals. The first stage – generation of electric signals – usuallyinvolves the conversion of a non-electrical physical parameter into an electricalone, for example, conversion of temperature, displacement, force, illuminance intoelectrical resistance, capacitance, inductance, voltage, or current. The conversionof one electrical parameter into another is carried out by changing its value, units,temporal form or waveform. Finally, the transmission of a signal to some distance isperformed with the aid of electrical current or electric field. At all the stages of signalprocessing, it is important to minimize the losses and distortion of information.

Therefore, in the practical realization of electronic devices and signal processing,of great significance are the problems of matching with sources of signals (sensorsof physical parameters) and matching with each other. In this aspect, not only circuitproblems, but also design problems become important. In this Chapter, particularexamples are used to demonstrate features of design of signal processors, includingthe coupling with sensors, the conversion of signals to reduce the resistance ofconnecting wires, to decrease the leakage current, etc.

Questions

1. What are the features of bridge signal processor circuits?2. What circuit of a passive resistive bridge has higher sensitivity?3. What circuits of resistive bridges have better linearity?4. How to increase the linearity of the dependence of the output bridge voltage on

the resistance of one of its elements using an op-amp?5. What for are three-, four-, and six-wire connection circuits of a resistive bridge

used?6. Why the sinusoidal voltage is used to excite inductive sensors?7. To convert the output ac voltage from an inductive sensor into the dc voltage,

a synchronous detector, rather than a detector of half-period average values, isused. Why?

References 229

8. What for does AD698 circuit use a duty cycle divider?9. What is the function of a guard ring in circuits for processing of sensor signals?

10. What is the function of a cold-junction temperature compensator in thermome-ters?

Test Yourself

1. A complete resistive bridge circuit of signal processing includes at least:

(a) one resistor;(b) two resistors;(c) three resistors;(d) four resistors.

2. To excite a resistive bridge circuit, it is possible to use:

(a) dc voltage;(b) ac voltage.

3. A resistive bridge circuit is connected with other signal processors by:

(a) one-wire line;(b) two-wire line;(c) three-wire line;(d) four-wire line;(e) six-wire line.

4. To excite an inductive transformer sensor, it is possible to use:

(a) dc voltage;(b) ac voltage.

5. The use of a guard ring leads to:

(a) removal of self-excitation;(b) decrease of the consumed current from a power supply;(c) decrease of the influence of leakage currents.

6. The effect of thermoelectricity used in thermocouples was discovered by:

(a) Gauss;(b) Seebeck;(c) Edison.

References

1. Karandeev, K.B.: Bridge Measuring Methods. Kiev, Gostekhizdat (1953) (in Russian)2. Kester W.: Practical design techniques for signal conditioning, Analog Devices (1999) http://

www.analog.com3. Universal LVDT signal conditioner AD698. http://www.analog.com/static/imported-files/data

sheets/AD698.pdf4. Horovitz, P., Hill, W.: The Art of Electronics. Cambridge University Press, New York (1998)5. Tietze, U., Schenk, Ch: Halbleiter-Schaltungstechnik. Springer, Berlin/Heidelberg/New York

(1980)

App

endi

x1

Tab

le1

Com

mon

oper

atio

nala

mpl

ifier

s

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

140U

D1

140U

D8

140U

D20

153U

D6

157U

D2

157U

D6

Supp

lyvo

ltag

e,V

C12,

�12

C15,

�15

C5:::C

18,�

5:::�

18C1

5,�1

5C3

:::C

18,�

3:::C

18C2

:::C

18,�

2:::�

18Su

pply

curr

ent,

mA

125

2.8

38

6G

ain

2,00

050

,000

50,0

0050

,000

50,0

0050

,000

Inpu

toff

set

volt

age,

mV

720

32

53

Off

setv

olta

gedr

ift,�

V/d

eg2

50–

––

–In

putr

esis

tanc

e,k�

41,

000

400

––

–In

putc

urre

nt,n

A8,

000

0.2

8075

500

500

Inpu

toff

set

curr

ent,

nA–

––

––

–O

utpu

tcur

rent

,mA

––

––

45–

Com

mon

-mod

ere

ject

ion

rati

o,dB

6064

8080

7070

Gai

nba

ndw

idth

,MH

z20

0.5

–1

1Sl

ewra

te,V

/�s

0.5

50.

3–

0.5

0.5

(con

tinu

ed)

Y.K. Rybin, Electronic Devices for Analog Signal Processing, Springer Series 231in Advanced Microelectronics 33, DOI 10.1007/978-94-007-2205-7,© Springer ScienceCBusiness Media B.V. 2012

232 Appendix 1

Tab

le1

(con

tinu

ed)

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

544U

D1

544U

D7

551U

D1

�A

702

Fair

chil

d�

A70

9Fa

irch

ild

�A

725

Fair

chil

d

Supp

lyvo

ltag

e,V

C13.

5:::1

6.5,

�13.

5:::1

6.5,

C1.5:::1

6.5,

�1.5:::1

6.5

C13.

5:::1

6.5,

�13.

5:::1

6.5,

C12,

�6C9

:::C

18,

�9:::�

18C5

:::C

22,

�5:::�

22Su

pply

curr

ent,

mA

3.5

35

6.7

52

Gai

n20

0,00

050

,000

500,

000

2,50

070

,000

3,00

0In

puto

ffse

tvo

ltag

e,m

V20

51.

52

0.6

0.5

Off

setv

olta

gedr

ift,�

V/d

eg50

––

104.

80.

6In

putr

esis

tanc

e,k�

108

––

4015

01,

500

Inpu

tcur

rent

,nA

0.2

100

100

5,00

010

075

Inpu

toff

set

curr

ent,

nA–

––

500

105

Out

putc

urre

nt,m

A–

––

50–

–C

omm

on-m

ode

reje

ctio

nra

tio,

dB13

011

080

100

7080

Gai

nba

ndw

idth

,MH

z0.

51

20–

0.7

1Sl

ewra

te,V

/�s

(con

tinu

ed)

Appendix 1 233

Tab

le1

(con

tinu

ed)

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

�A

741

Fair

chil

dC

A30

33In

ters

ilC

A31

40In

ters

ilL

F147

Nat

iona

lse

mic

ondu

ctor

LF4

11N

atio

nal

sem

icon

duct

orL

M10

1N

atio

nal

sem

icon

duct

or

Supp

lyvo

ltag

e,V

C5:::C

22,

�5:::�

22C3

:::C

18,

�3:::�

18C5

:::C

18,

�5:::�

18C5

:::C

16.5

,�5

:::�

16.5

C5:::C

22,

�5:::�

22C5

:::C

22,

�5:::�

22Su

pply

curr

ent,

mA

2.8

86

111.

81.

8G

ain

50,0

0090

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200,

000

100,

000

200,

000

25,0

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puto

ffse

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ltag

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V6.

02.

620

10,

50.

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73.

0In

putr

esis

tanc

e,M�

0.3

1.5

109

109

109

4�103

Inpu

tcur

rent

,nA

500

700.

040.

050.

0530

Inpu

toff

set

curr

ent,

nA20

035

0.02

0.03

0.02

51.

5O

utpu

tcur

rent

,mA

1044

––

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Com

mon

-mod

ere

ject

ion

rati

o,dB

7010

070

100

100

96

Gai

nba

ndw

idth

,MH

z1

2.7

4.5

44

1Sl

ewra

te,V

/�s

0.5

2.7

913

1510

234 Appendix 1

Tab

le2

Hig

h-sp

eed

oper

atio

nala

mpl

ifier

s

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

6402

UV

114

32U

D1

1432

UD

615

4UD

414

20U

D1

Supp

lyvo

ltag

e,V

C6,�

5C1

5,�1

5C1

5,�1

5C1

5:::�

15C9

,�6

Supp

lycu

rren

t,m

A10

020

127

8G

ain,

K30

020

0,00

012

0,00

08,

000

350

Inpu

toff

set

volt

age,

mV

1015

306.

05

Off

setv

olta

gedr

ift,�

V/d

eg25

0–

7050

18In

putr

esis

tanc

e,k�

–5

0.01

––

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tcur

rent

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1,00

050

,000

10,0

001,

200

10,0

00In

puto

ffse

tcu

rren

t,nA

500

––

300

1,50

0O

utpu

tcur

rent

,mA

50–

4010

5C

omm

on-m

ode

reje

ctio

nra

tio,

dB60

–55

7460

Gai

nba

ndw

idth

,MH

z–

100

300

201,

550

Slew

rate

,V/�

s2,

000

2,00

01,

200

400

280

(con

tinu

ed)

Appendix 1 235

Tab

le2

(con

tinu

ed)

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

154U

D2

574U

D1

140U

D30

AD

844

Supp

lyvo

ltag

e,V

C15,

�15

C15,

�15

C15,

�15

C4,5:::C

18,�

4,5:::�

18Su

pply

curr

ent,

mA

68

65

Gai

n10

,000

50,0

0020

0,00

03,

000

Inpu

toff

set

volt

age,

mV

250

150

Off

setv

olta

gedr

ift,�

V/d

eg20

50–

1In

putr

esis

tanc

e,k�

––

–24

Inpu

tbia

scu

rren

t,nA

100

0.5

401,

200

Inpu

toff

set

curr

ent,

nA20

0.2

–10

0O

utpu

tcur

rent

,mA

25–

–80

Com

mon

-mod

ere

ject

ion

rati

o,dB

8670

94–

Gai

nba

ndw

idth

,MH

z–

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ewra

te,V

/�s

150

100

502,

000

236 Appendix 1

Tab

le3

Prec

isio

nop

erat

iona

lam

plifi

ers

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

140U

D17

140U

D24

140U

D26

544U

D12

OP2

7

Supp

lyvo

ltag

e,V

C15,

�15

C5,�

5C1

5,�1

5C5

.5:::C

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:::C

18,�

3:::�

18Su

pply

curr

ent,

mA

44

53

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ain

300,

000

1,00

0,00

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02,

000,

000

1,50

0,00

0In

puto

ffse

tvo

ltag

e,m

V0.

025

0.00

50.

050.

010.

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ffse

tvol

tage

drif

t,�

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eg0.

60.

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0.2

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tres

ista

nce,

k�–

––

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000

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tbia

scu

rren

t,nA

20.

0160

120

Inpu

toff

set

curr

ent,

nA–

0.00

550

–15

Out

putc

urre

nt,m

A–

––

––

Com

mon

-mod

ere

ject

ion

rati

o,dB

110

120

114

130

126

Gai

nba

ndw

idth

,MH

z0.

250.

820

0.4

8Sl

ewra

te,V

/�s

0.1

211

0.1

2.8

(con

tinu

ed)

Appendix 1 237

Tab

le3

(con

tinu

ed)

Am

plifi

er(m

anuf

actu

rer)

para

met

ers

OP3

7Te

xas

inst

rum

ents

OP4

2A

nalo

gde

vice

sO

P77

Bur

rBro

wn

AD

797

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log

devi

ces

Supp

lyvo

ltag

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18,�

2:::�

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:::C

20,�

8:::�

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:::C

18,�

3:::�

18C5

:::C

15,�

5:::�

15Su

pply

curr

ent,

mA

65,

11.

55

Gai

n50

,000

500,

000

12,0

00,0

0020

,000

,000

Inpu

toff

set

volt

age,

mV

0.01

0.3

0.01

0.02

5O

ffse

tvol

tage

drif

t,�

V/d

eg0.

24

0.1

0.2

Inpu

tres

ista

nce,

k�10

310

645

�103

–In

putb

ias

curr

ent,

nA50

0.08

2.4

250

Inpu

toff

set

curr

ent,

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0.00

40.

5–

Out

putc

urre

nt,m

A–

––

50C

omm

on-m

ode

reje

ctio

nra

tio,

dB12

688

7010

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ain

band

wid

th,M

Hz

6310

0.7

–Sl

ewra

te,V

/�s

1758

0.3

20

Appendix 2

Table 1 RC circuit

# RC circuit Operator transfer function

1 2 3

1 R1

R2

C1

C2

At R1 D R2 D R, C1 D C2 D C, £D RC

K.p/ D p

p22 C 3p C 1; !0 D 1

,

K.!0/ D 1

3

2 R1

R2C1

C2 At R1 D R2 D R, C1 D C2 D C, £D RC

K.p/ D p

p22 C 3p C 1; !0 D 1

,

K.!0/ D 1

3.

3

R1

R2C1

C2


K.p/ D p

p22 C 3p C 1; !0 D 1

,

K.!0/ D 1

3.

4

R1 R2

C1 C2


K.p/ D p

p22 C 3p C 1; !0 D 1

,

K.!0/ D 1

3.

(continued)

239

240 Appendix 2

Table 1 (continued)


1 2 3

5

R1 R2

C1 C2At R1 D R2 D R, C1 D C2 D C, £D RC

K.p/ D p

p22 C 3p C 1; !0 D 1

,

K.!0/ D 1

3.

6 C1

C2

R1 R2

At R1 D R2 D R, C1 D C2 D C, £D RC,

K.p/ D p22 C 2p C 1

p22 C 3p C 1; !0 D 1

,

K.!0/ D 2

3

7 R1

C1 C2

R2

At R1 D R2 D R, C1 D C2 D C, £D RC,

K.p/ D p22 C 2p C 1

p22 C 3p C 1; !0 D 1

,

K.!0/ D 2

3

8

C1 C2

C3

R1 R2

R3

At R1 D R2 D R, R3 D R/2,

C1 D C2 D C, C3 D 2C, £D RC

K.p/ D p22 C 1

p22 C 4p C 1; !0 D 1

,

K.!0/ D 0

9 C1 C2

R1 R2 R3

C3

At R1 D R2 D R3 D R, C1 D C2 D C,

C3 D 12C, £D RC,

K.p/ D 12p33 C 3p22 C 4p C 1

12P 33 C 39p22 C 16p C 1;

!0 D 1

p3; K.!0/ D 0

(continued)

Appendix 2 241

Table 1 (continued)


1 2 3

10 R1 R2

R3

C1 C2 C3

At R1 D R2 D R, R3 D R/12,

C1 D C2 D C3 D C, £D RC

K.p/ D p33 C 4p22 C 3p C 12

P 33 C 16p22 C 39p C 12;

!0 Dp3

; K.!0/ D 0

11 R1

R2 R3

C1 C2 C3

At R1 D R2 D R3 D 12R,

C1 D C2 D C3 D C, £D RC

K.p/ D 12p33 C 3p22 C 4p C 1

12P 33 C 39p22 C 16p C 1;

!0 D 1

p3; K.!0/ D 0

12 C1

R1 R2 R3

C2 C3

At R1 D R2 D R3 D R,

C2 D C3 D C, C1 D C/12, £D RC,

K.p/ D p33 C 4p22 C 3p C 12

P 33 C 16p22 C 39p C 12;

!0 Dp3

; K.!0/ D 0

13 R1 R2 R3

C1 C2 C3

At R1 D R2 D R3 D R,

C1 D C2 D C3 D C, £D RC,

K.p/ D 1

p33 C 5p22 C 6p C 1;

!0 Dp6

; K.!0/ D � 1

29

14

R1 R2 R3

C1 C2 C3 At R1 D R2 D R3 D R,

C1 D C2 D C3 D C, £D RC,

K.p/ D p33

p33 C 6p22 C 5p C 1;

!0 D 1

p6; K.!0/ D � 1

29

(continued)

242 Appendix 2

Table 1 (continued)


1 2 3

15 C1

R1R2

R3

C2

C3

At R1 D R2 D R, R3 D 2R,

C2 D C3 D C, C1 D C/2, £D RC,

K.p/ D p22 C 1

p22 C 4p C 1; !0 D 1=;

K.!0/ D 0.

16 R1

C2 C1 R3

R2 C3

At R1 D R2 D R, R3 D 2R,

C1 D C2 D C, C3 D C/2, £D RC,

K.p/ D p22 C 1

p22 C 4p C 1; !0 D 1= ;

K.!0/ D 0:

Abbreviations

AC Alternative currentAD Amplitude detectorAE Active elementALFDE Active linear frequency-dependent elementANE Active nonlinear elementBA Buffer amplifierCB Common-base circuitsCC-CE Common collector and common emitterCCCS Current controlled current sourceCCVS Current controlled voltage sourceCFOA Current feedback operational amplifierCE Common-emitter circuitsCE Control elementCS Current sourceDC Direct currentE, J, T, K, C, R, S Types of thermocouplesEA Error amplifierECL Emitter-coupled logicEMF Voltage or current sourceFBOA Perational amplifier with nonlinear feedbackFET Field-effect transistorsGC Gain characteristicGFC Gain-frequency characteristicHPF High-pass filterINE Inertial nonlinear elementsLFDC Linear frequency-dependent circuitLFDE Linear frequency-dependent elementsLG Logic gateLPF Low-pass filterLVDT Linear variable differential transformer

243

244 Abbreviations

NE Nonlinear elementNPN, PNP Types of bipolar transistorsOA Op-ampOp-amp Operational amplifierOS Oscillation systemPIA Proportional integral amplifierPRC Phase-response characteristicPSpice Program for analysis of circuitsPWMVD Pulse-width modulation voltage dividerRC, RL, and LC Circuits or filters consisting of resistors, capacitors, and induc-

tancesRTD Resistance temperature deviceRVDT Rotational variable differential transformerSTG Saw-tooth generatorSWS Sine-wave shaper2T-RC, T-RC T-shaped resistance-capacitance networkTC Transfer characteristicTCA-type Chromel-alumel (type of thermocouples)TCC-type Chromel-copel (type of thermocouples)TPP-type Platinum (90%) C Rhodium (10%) – Platinum (type of ther-

mocouples)TPR-type Platinum (87%) C Rhodium (13%) – Platinum (type of ther-

mocouples)TTL Transistor–transistor logicTTR-type Tungsten (5%) Rhenium–Tungsten (20%) Rhenium (type of

thermocouples)VCCS Voltage controlled current sourceVCVS Voltage controlled voltage sourceVFOA Voltage feedback operational amplifierVD1 Diode position numberVS Voltage sourceY, Z, F, S, H, K Two-port network parameters

Parameters

Ccm The common-mode capacitanceCfc Capacitor for GFC correctionCi, Cout Output capacitanceF Amount of the feedbackfT Threshold frequency (unit gain frequency)Iin Input currentIout Output currentG Transfer conductance (transconductance)Kfb Gain with a large feedbackKV Voltage gainKI Current gainKg Coefficient harmonicKthd Total harmonic distortionRi, Rout, Zout Output resistanceRin, Zin Input resistanceVout m Maximum or minimum allowable voltage at op-amp outputVgs Gate-to-source voltageVin Input voltageVoff Input offset voltageVout Output voltageVst Stabilitron voltageZi Complex output resistanceZtr TransresistanceRfc Resistor for GFC correction® T Temperature voltage¨cut, fcut Cutoff frequency

245

Conclusions

Electronic methods are widely used for processing of analog signals because allphysical phenomena around us and the human activity itself are continuous in timeand space. Even natural disasters occur though fast, but still in the finite time.Therefore, it is always necessary to convert changes in physical parameters intoelectrical signals and to process them. An analog sensor is usually in a “directcontact” with a physical parameter, and, due to the continuity of the physicalparameter, the output electrical signal of the sensor is continuous too. The masspractical implementation of digital devices for signal processing not only did notreduce the significance of analog methods and devices of signal processing, buteven considerably increased the requirements to them in respect of the accuracyand stability. Analog methods should be as good as digital ones in the accuracy.Moreover, devices for analog processing of signals should be in advance of thedigital ones in the conversion error, resolution, noise level, etc., because the error ofanalog devices cannot be always compensated in the following digital processing.

Summarizing the material presented in this book, it is possible to concludewith confidence that the key point in the investigation, design, and use of variouselectronic devices is the understanding of their structure and operating principles,as well as analytical and modeling skills.

The minimal requirements to knowledge, a reader acquires after the study ofthis book, can be formulated as follows. The reader should be able to analyzethe principal circuit of a signal processing device and to determine the form ofconversions, to draw up a macromodel of conversions adequate to the problemto be solved, to draw up a design circuit of the device, and to determine its mainparameters and characteristics. The reader should know the principal features ofdesigning amplifiers, converters, and generators of sine-wave and pulsed signals,understand the general regularities of their mathematical description and modeling.The learning of the information presented here will allow the student to bettercomprehend the operation of such devices, to make the best use of them, and toextend the areas of their practical application.

Y.K. Rybin, Electronic Devices for Analog Signal Processing, Springer Seriesin Advanced Microelectronics 33, DOI 10.1007/978-94-007-2205-7,© Springer ScienceCBusiness Media B.V. 2012

247

248 Conclusions

The author hopes that the information presented in this book will help studentsand engineers to take a closer look at the problems of design of electronic devicesand initiate the interest to their further enhancement in accordance with moderntendencies in development of electronics.

Glossary

Amplifier Electronic device, intended for amplification of electric signalsAmplifier of rail-to-rail type Amplifier, whose maximal output voltage is practi-

cally equal to the supply voltageAmplitude balance The case that the loop gain in the feedback circuit is equal to

unityAmplitude of self-oscillations Maximal or minimal value of an oscillationBias current Input current at the base of a BJT differential amplifierCenter Isolated point on the phase plane, near which trajectories look like enclosed

closed curves, for example, ellipsesCoefficient of suppression of the common-mode signal Ratio of the increment

in the output voltage, related to the op-amp input, to the corresponding incrementin the common-mode voltage

Consumption current Current consumed by an amplifier with no loadCurrent input Input, whose input resistance is much lower than the resistance of

the signal sourceCurrent source The source of electric energy, whose output current is independent

of the voltage appliedCurrent-controlled current source (CCCS) Controlled source of electric energy,

whose output current depends on the control current (input current) and does notdepend on the load voltage

Current-controlled voltage source (CCVS) Controlled source of electric energy,whose output voltage depends on the control current (input current) and does notdepend on the load current

Current-difference amplifier Amplifier, whose output signal is proportional tothe difference between the input signals of a current mirror

Current feedback operational amplifier (CFOA) A type of electronic amplifierwhose inverting input is sensitive to current, rather than to voltage as ina conventional voltage-feedback operational amplifier (VFOA). The currentfeedback operational amplifier is a type of current controlled voltage source(CCVS)

249

250 Glossary

Cutoff frequency Frequency, at which the absolute value of the gain becomesequal to 0.707 of the value at the direct current

Decay time Time of decay of the output voltage, when it changes in the negativedirection

Difference amplifier Amplifier, whose output voltage is proportional to the differ-ence of input voltages

Differentiating amplifier (differentiator) Amplifier, whose output voltage is pro-portional to the derivative of the input signal

Fast op-amp Amplifier with high rate of rise of the output voltage (by thebeginning of 2002, the rate of rise higher than 10 V/�s was considered as high)

Feedback Transmission of a part of the output signal power to the amplifier inputFocus Trajectory on the phase plane, looking like an unwinding (unstable focus)

or winding (stable focus) spiralGain bandwidth The product of the open-loop voltage gain and the frequency at

which it is measuredGain characteristic Dependence of the amplitude of the output voltage on the

amplitude of the input voltageGain Dimensionless coefficient, showing how many times an output parameter

(voltage, current, power) exceeds the corresponding input parameterGain-frequency characteristic Dependence of the absolute value of the gain on

the frequency of the input signalHarmonics Sinusoidal component of a signal with the frequency multiple of the

signal frequency, obtained through expansion into the Fourier seriesHigh-power op-amp Amplifier with the output power higher than 100 mWInput capacitance of op-amp Capacitance between the op-amp inputsInput common-mode voltage Half-sum of voltages between the op-amp inputsInput differential voltage Voltage between the noninverting and inverting op-amp

inputsInput offset current Difference between currents in the op-amp input terminalsInstrumental amplifier Amplifier with the built-in feedback circuit intended for

performing linear operations with input signalsIntegrating amplifier Amplifier, whose output signal is proportional to the inte-

gral of the input signalLC oscillator Oscillator, whose frequency-determining circuit is realized in induc-

tance coils and capacitorsLimit cycle Closed trajectory on the phase planeLimiter (clipper) Functional device, performing the nonlinear operation of limit-

ing of the input signal from above or from below (clipper) or from above andbelow simultaneously (limiter)

Linear functional devices Devices performing linear operations with input signals(multiplication by a constant, addition, subtraction and so on)

Logarithmic amplifier Amplifier, whose output signal is proportional to logarithmof the input signal

Micro power op-amp Amplifier supplying the power no higher than 100 �W

Glossary 251

Multiplier of signals Functional device, performing the operation of multiplica-tion of the input signals

Negative feedback Feedback resulting in decrease of the op-amp gainNode Isolated point on the phase plane, all trajectories near which are directed

toward (stable node) or from which (unstable node)Nonlinear functional device Device, performing nonlinear operations with input

signals (limiting, rectification, taking the logarithm, etc.)Offset current Difference between the two bias currents entering the bases of a

differential amplifier in an op-ampOffset voltage drift Time derivative of the output voltage (time drift) upon tem-

perature variation (temperature drift) and so onOffset voltage DC error voltage that appears at the output of an op-amp with a

feedback referred back to the input terminals through the noninverting gainOffset voltage Voltage, which should be applied to one of the amplifier inputs for

the output voltage to be equal to zeroOp-amp input current Current, passing through the op-amp input terminalOp-amp input resistance Differential resistance between the op-amp inputs. The

resistance between the input and the output of the power supply is called theinput common-mode resistance

Op-amp input voltage Voltage between any input and the common wireOp-amp inverting input Op-amp input, whose voltage is amplified and transmit-

ted to the output with the initial phase equal to the initial phase of the inputvoltage plus 180ı

Op-amp noninverting input Input, whose voltage is amplified and transmitted tothe output with the same initial phase as the input voltage

Operational amplifier (op-amp) Amplifier with the large gain, intended for per-forming, in combination with external elements, different operations with inputsignals

Operational amplifier with current input Amplifier, whose output electric pa-rameter (current or voltage) is proportional to the current in one of the inputs

Operational rectifier Functional device, performing the nonlinear operation ofdetermining the absolute value of the input signal

Oscillator The electronic device intended for generation of electric signals with acertain waveform, amplitude, and frequency

Output voltage range Values of the output voltage, within which the op-ampparameters have the rated values

Q-factor Quality factor of the resonans circuitPhase balance The case that the shift of initial phases in the feedback circuit is

equal to zero or 2 n, where n is an integer numberPhase portrait Phase plane with phase trajectoriesPhase-response characteristic (PRC) Dependence of the change in the difference

of initial phases between the input and output voltages on the signal frequencyPositive feedback Feedback resulting in increase of the op-amp gainPotential input Input, whose input resistance is much higher than the resistance of

the signal source

252 Glossary

Potentiating amplifier Amplifier, whose output signal is proportional to anti-logarithm of the input signal

Precision rectifier Circuit that half-wave rectifies a dual-polarity input signal asine wave using an amplifier so the output voltage always has one polarity withrespect to zero volts

Pulse oscillator The electronic device intended for generation of electric signalswith alternating fast and slow changes of the output voltage

RC oscillator Oscillator, whose frequency-determining circuit is realized in resis-tors and capacitors

Rise time Time, during which the op-amp output voltage, changes from 0.1 to0.9 of its established value or from 0.1 to 0.9 of the difference between theestablished and the initial values (if the initial value is nonzero)

Saddle Isolated point on the phase plane, near which trajectories are generalizedhyperbolas (a saddle is always unstable)

Self-oscillations Oscillations, whose amplitude and frequency are constant andindependent of the initial conditions

Settling time Time needed for an amplifier with a feedback to settle within a set ofspecified limits near the steady-state output voltage once a stepwise function hasbeen applied to the input

Sine-wave oscillator Electronic device intended for generation of electric signalswith sinusoidal waveform

Slew rate Most rapid rate that an amplifier can go from one voltage level toanother when a step or square wave input voltage is applied

Slew rate Rate of rise of the output voltage upon the stepwise change of the inputvoltage

Small signal Signal, the change of whose amplitude does not lead to the change inthe amplifier parameters

S-plane Hypothetical plane having a real axis of ˙ ¢ and an imaginary axisof ˙ j¨. All poles and zeros lie somewhere on the s-plane

Square-wave oscillator Electronic device intended for generation of electric sig-nals with square waveform of the output voltage

Stability Property of an amplifier or a functional device based on it, consisting inthe fact that all free oscillations in it decay with time

Stationary mode Mode of oscillations with constant parametersSumming amplifier Amplifier, whose output signal is proportional to the sum of

the input signalsTemperature drift of input offset current Increment to the difference of the in-

put currents as the temperature changes by 1 KTemperatures drift of offset voltage Change of the offset voltage with changes in

temperatureThreshold voltage Voltage value that a circuit must cross in order to change a

conditionTime for establishment of the output voltage Time, during which the amplitude

of the output voltage of an oscillator increases up to 0.9 of the established value

Glossary 253

Transconductance Coefficient, equal to the ratio of the increment in the op-ampoutput current to the corresponding increment in the input voltage, measured inmilliamperes per volt

Transfer characteristic (TC) Change of the output voltage upon the stepwisechange of the input voltage

Transient resistance Coefficient, equal to the ratio of the increment in the outputvoltage to the corresponding increment in the input current, measured in ohms

Transimpedance (the same as transient resistance) Ratio of the increment in theoutput voltage to the corresponding increment in the input current

Transresistance Output voltage in response to an input currentTwo-port network Circuit having two input terminals and two output terminals.

The inputs and outputs are related by parametric equationsUnit gain frequency Frequency, at which the gain, decreasing with the increase of

the frequency, becomes equal to unity; or frequency, above which an amplifierdoes not amplify a signal

Voltage source The source of electric energy, whose output voltage is independentof current

Voltage-controlled current source (VCCS) Controlled source of electric energy,whose output current depends on the control voltage (input voltage) and does notdepend on the load voltage

Voltage-controlled voltage source (VCVS) Controlled source of electric energy,whose output voltage depends on the control voltage (input voltage) and doesnot depend on the load current

Index

AAmplifier, 1–29, 45–77, 83–90, 96, 99, 103,

107, 112, 125, 126, 128–130, 135, 136,138, 139, 141, 142, 144–147, 149, 153,160–162, 166–168, 173, 176, 183, 192,196, 197, 199, 203, 217–219, 222, 223,225–230

Amplitude balance, 124, 126, 130, 133, 138,150, 152, 153, 158, 167

Amplitude of self-oscillations, 117, 125, 127,158

BBias current, 24, 91, 104, 108, 149

CCCCS. See Current-controlled current sourceCCVS. See Current controlled voltage sourceCenter, 120–122, 221, 222CFOA. See Current-feedback operational

amplifierCommon-mode input, 10Common-mode voltage, 59, 229Current-controlled current source (CCCS), 9,

16, 17, 21–24, 29, 51, 162–166, 225Current controlled voltage source (CCVS), 9,

13–15, 17–19, 21, 30, 51, 162–165Current-differencing amplifier, 24–26Current-feedback operational amplifier

(CFOA), 16Current (low resistance) input, 3Current output, 16, 19–24, 29Current source, 4, 9, 12, 16, 19, 21, 24, 162,

203, 215, 216, 219, 220, 225

Cutoff frequency, 5, 6, 18, 23, 24, 51, 68, 70,73, 127, 186, 191, 192, 194, 224

DDecay time, 51, 176Difference amplifier, 54–58, 60, 66, 217–219,

229, 230Differentiating amplifier, 67–69Differentiator, 46, 68–70

FFeedback, 1, 2, 5–7, 16–21, 23, 25, 29, 30, 46,

48, 49, 52, 55, 60, 64, 65, 68, 75–77, 84,86, 92, 93, 96, 108, 111, 112, 116, 125,126, 128, 135–138, 140, 141, 146, 149,152, 153, 159, 164, 167, 178, 183–186,190, 194–196, 198, 203, 218, 219, 226

Focus, 45, 120, 122

GGain, 2, 4–7, 9, 10, 12–15, 17–24, 26, 27,

29, 30, 37, 38, 40–42, 46–49, 51–54,56–58, 61–63, 70–73, 82, 83, 85–87,89, 90, 93, 96, 99, 101, 103–107, 123,125, 126, 128, 135, 136, 138, 139,142–147, 149, 150, 152, 154, 166, 167,184, 186, 188, 191, 192, 194, 218, 223

Gain bandwidth,Gain characteristic, 4, 6, 7, 40–42, 82, 83, 85,

86, 89, 90, 99, 101, 103–107, 109, 110,128, 142, 146, 184

Gain-frequency, 4, 5, 21, 70, 72, 136, 138, 139,191

Y.K. Rybin, Electronic Devices for Analog Signal Processing, Springer Seriesin Advanced Microelectronics 33, DOI 10.1007/978-94-007-2205-7,© Springer ScienceCBusiness Media B.V. 2012

255

256 Index

Gain-frequency and the phase-response (PRC)characteristics, 5, 136, 138, 139

Gain-frequency characteristic, 4, 21, 70, 191

HHarmonics, 41, 42, 91, 92, 142

IInput capacitance, 10, 19Input current, 14–19, 21–23, 29Input differential voltage, 102Input offset, 2, 7, 10–12, 24, 63, 83Input voltage, 1, 6, 7, 9, 10, 13, 24, 37, 40–42,

50, 55, 60–63, 81–84, 86, 90–93, 95, 96,98–100, 103, 113, 128, 142, 184–186,197, 222

Instrumental amplifier, 27, 57, 217, 227–229Integrating amplifier, 60–67Invariant, 164, 166, 180

LLC oscillator, 133, 148–152, 154, 159, 167Limit cycle, 119–123, 174–176, 185, 187, 190,

191, 205, 208Limiter, 12, 16, 81, 99–104, 106, 142, 145,

147, 184, 197Limiter (clipper), 99–102Logarithmic amplifier, 84–90, 103

MMatrix, 46–49, 52, 160, 162, 163Multiplier, 88, 89, 222, 223

NNegative feedback, 6, 17, 76, 125, 135–138,

140, 141, 146, 196, 219Node, 46–49, 53, 54, 104, 120, 163, 166, 188,

189, 191, 208Non-inverting op-amp amplifier, 17, 23, 137,

138, 142, 183, 184Nonlinear functional device, 112

OOffset voltage, 2, 4, 7, 10–12, 14, 24, 51, 63,

83, 223, 228Op-amp input, 7, 10, 12, 46, 76, 83, 95,

101–103, 137, 139, 184, 185, 218, 227

Op-amp input (inverting/non-inverting), 6, 7,15, 17–23, 25, 26, 46, 49–60, 76, 77,83, 96, 137–139, 142, 183, 184, 218

Op-amp input current, 19, 225Op-amp input resistance, 5, 10, 14–16, 19, 22,

23, 86Operational amplifiers (op-amps), 1–30, 35,

45–77, 81–108, 111, 112, 126, 127,131, 133–142, 144, 146, 147, 149, 167,176, 184–200, 208, 215, 217–219, 223,225–227, 229

Operational rectifier, 90–99, 107, 144Oscillator, 111–168, 173–208, 215, 223Output voltage, 4, 6–8, 10–17, 21, 25–27, 37,

38, 40–42, 49–51, 54–57, 60–63, 72,81–86, 88–92, 96, 98–100, 103, 113,115, 118, 119, 125, 128, 129, 143,145–148, 170, 174, 179, 184–186, 190,192, 194–197, 200–202, 204, 206,215–219, 222–225, 227–229

PPhase balance, 124–127, 129, 130, 133, 135,

138, 139, 152, 153, 161, 162, 164, 167,176

Phase portrait, 181, 187, 204, 206Phase-response (PRC), 5, 6, 60–62, 68, 69,

125, 136, 138, 139, 158Positive feedback, 48, 49, 64, 65, 77, 84, 112,

116, 126, 135–138, 141, 149, 152, 153,159, 184–186, 190, 194, 195, 198, 203

Potential input, 3–8, 14, 18Potentiating amplifier, 89Pulse oscillator, 173–208, 215

RRc oscillator, 118, 134–141, 148, 154,

160–167, 215Rectifier, 90, 93, 94, 96Rise time, 117

SSaddle, 120Self-oscillations, 43, 108, 113, 115–117, 120,

122, 123, 125, 127, 128, 132, 133, 135,138, 142, 149, 150, 152, 155, 157–159,174, 175, 177, 182, 184, 185, 187, 192,207, 208

Sine-wave oscillator, 111–167, 174, 175, 223Slew rate, 2, 5, 19, 50, 59Small signal, 7, 8

Index 257

Square-wave oscillator, 197, 199, 200Stability, 5, 6, 52, 57, 63, 68, 86, 117, 134,

148, 154, 167, 177, 182, 189, 207, 208,221–223

Summing amplifier, 59–60

TThreshold voltage, 93, 201Total harmonic distortion, 117, 147Transconductance, 9, 20, 21, 47, 149, 150, 152,

159

Transresistance, 9, 14, 18, 53Two-port network, 130–133

VVoltage controlled current source (VCCS), 9,

11, 12, 16, 19–21, 30, 46, 63, 162–165,167, 223

Voltage controlled voltage source (VCVS), 4,6, 9, 10, 14, 16, 19, 29, 51, 63, 93, 126,134, 137, 162–167

Voltage source, 9, 10, 13, 16, 19, 21, 112, 146

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