Mathematics for electronics : with applications

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Mathematics for electronics : with applications

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HENRY M. NODELMAN Head of Mathematics and Physics Departments RCA Institutes, and Instructor in Mathematics, Polytechnic Institute of Brooklyn

FREDERICK W. SMITH Television Engineer f National Broadcasting Company


19 5 6



MATHEMATICS TOR ELECTRONICS with Applications Copyright © 1056 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. AH rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number 55-9549 IX









In Mathematics for Electronics with Applications, the emphasis has been placed on application rather than on mathematical theory. This approach, which represents a new departure for a mathematics text in this field, is deliberate. It is a direct result of the authors7 many years of experience in the teaching of mathematics to students of electronic engineering and in engineering practice. It is our conviction that the wide barrier which exists between mathematics and its applications can best be surmounted by demonstrating the practical uses which industry makes of mathematics. In this way, student motivation is increased considerably, and the student is not confronted with the problem of interrelating two discrete sets of information, one mathematical, the other electronic, as unfortunately has too often been the case. In order to accomplish this objective, the text offers a complete set of up-to-date problems based on current engineering practice, the result of more than five years of research in the technical literature. All of the leading American and foreign technical periodicals devoted to communi¬ cations and electronics were carefully examined to determine which mathematical topics were to be included. The mathematical applica¬ tions are accompanied by worked-out examples and are extensively supported with references so that the reader can supplement his knowledge of the problem by consulting the original article from which the problem was obtained. Part I of the book describes the methods by which mathematics has been used in electronics and presents a series of electronic engineering problems which involve the use of the calculus. Part II introduces the reader to dimensional analysis as it is applied in electronics and shows the methods which can be used to check the validity of engineering equations and formulas independent of the engineering considerations used to establish such equations. Part III deals with the algebra of determi¬ nants. The theory is then applied to the solution of steady-state elec¬ tronic circuits. Matrix algebra is discussed, and the solution of elec¬ tronic networks, including those which contain transistors, is undertaken by the matrix method. In Part IV the fundamental properties of series approximations are presented. The use of series to predict the behavior of nonlinear electronic devices is treated; the processes of harmonic gen-



eration and intermod illation testing are described.

Part V is devoted

to solutions of differential equations employing both classical and Laplace transform methods. Part. VI gives the elements of Boolean algebra and includes material on switching circuits. In addition, it presents an analysis of the more advanced mathematical topics which logically follow those incorporated in the text. The various parts of the book are independent and may be studied and read as separate units if desired. The book is intended for those readers who have a background of elementary calculus, physics, and elementary electrical network theory. It is designed for use in industry as well as for use in undergraduate tech¬ nical institute and engineering college courses in applied mathematics, networks, transients, and nonlinear systems. Because the technical references incorporated in this book are such an integral part of t, it is obvious that the book could not have become a reality without the generous cooperation of the editors of the various technical publications identified in the footnote references, for which the authors are grateful. We also wish to extend thanks to the industrial organizations elsewhere identified who made illustrations and valuable technical data available. Hen by M. Non elman Fandeuick W. Smith

Con fen fs



PART 1* The Uses of Mathematics in Electronics


1- ('use Histories of Practical Applications


2. Electronic Applications of Calculus


PART 2* Equation Testing prior to Mathematical Operations 3. Dimensional Formulas and Systems 4* Checking Equations and Predicting Solutions

37 '

39 54

PART 3* The Algebra of Circuit Analysis


5. Theory of Determinants


6. Network Solutions by Determinants


7. Matrix Algebra


8. Network Solutions by Matrices


PART 4_ Theory and Applications of Series


9. Basic Properties and Applications


10. Nonlinear Electronic Devices

PART 5. Differentia! Equations: Theory and Applications



11. Classical Solutions and Non-network Applications vii




12. Transient Solutions and Elementary Laplace Transforms


PART 6. The Direction of Mathematics in Electronics


13. Elements of Boolean Algebra


14. A Study Plan in Mathematics for Specialists in Electronics




Table A-4. Common Logarithms


Table A-2. Natural Logarithms


Table A-3. Values of



Table A-4. Trigonometric Functions


Table A-5. Integrals


Author Index


Subject Index



Resistance network analog computer provides solutions to problems of temperature distribution in solids, lamellar flow in fluids, and field patterns of electrostatic and magnetic fields, {National Bureau of Standards.)

PART 1 The Uses of Mathematics in Electronics

Chapter 1


A crucial question often proposed concerning mathematics is: Of what practical use is it? In the chapters which follow, it is hoped that the reader will discover sufficient evidence of its practical use to agree with J. J* Thomson that ^mathematical analysis is the greatest mental labor saving device ever invented/*1 Actually, the practicing engineer now¬ adays is enormously indebted to the mathematician* Once he is able to express in symbols and equations the physical relations of a problem, lie is in a position to forget the original problem for a moment and to solve the equations by the methods of mathematical analysis, which have usually been worked out previously on an abstract basis by the mathematician. How the engineer thus came to be dependent on the mathematician has been described by Van der Pol:2 Consider the relation between mathematics on the one hand and physics and technology on the other*

History shows us that mathematics has, to a great

extent, been developed by pure mathematicians wTho were under the spell of the beauty of it; who were entranced by its mysterious generality and who spent their life discovering new facts and relations in this very wide and wonderful domain. Whether their results could be applied to astronomy, physics, chemistry, or say, technology, as a rule did not interest them primarily*

Their main concern was

to make their theorems cover cases of a more and more abstract nature, and generality in then work was their main purpose. Some mathematicians even went so far as to he proud of the fact that none of their results could be applied in practice.

It is said that one great mathema-

1 Jh R. Carson.. Mathematics and Electrical Communication, Bell Labs. Record, 14 (8): 397-399, August, 1936. 2 B. Van der Pol, Radio Technology and the Theory of Numbers. J. Franklin Inet., 266 (G): 475-495, June, 1953. 5



tician (Cayley) remarked that “Bessel functions are beautiful functions hi spite of their many applications/’ Now, physics and technology were developed to a certain extent independently of pure mathematics. It is, therefore, natural that at a given moment the physicists or technologists discovered, to their surprise, that precisely the mathe¬ matical tools and methods they needed to solve their problems had already been fully developed by pure mathematicians who, at the time, did not have the slightest notion of, or even interest in, a possible practicable application of their results. An adaptation of mathematics to the purposes of the engineer is aptly portrayed in the history of the complex number.1 In 1572, Raphael Bombelli pointed out that in order to obtain a solution for an equation of the general form x2 + a = 0, one must either accept the concept of the imag¬ inary number \f—a or conclude that the equation is meaningless. Although a dim view was taken of this sugges¬ tion by Bombelli*s mathematical con¬ temporaries, the concept of the im¬ Fig. 1-1

The Argand diagram.

aginary number flourished. Later, the mathematician Gauss in extending the

work of a Norwegian surveyor, Weasel, and a Parisian bookkeeper, Argand, developed the idea of a complex plane wherein points represented by pure imaginaries jy or complex numbers2 of the form x A jy could be graphically displayed3 as in Fig. 1-1. These developments held little significance for the electrical engi¬ neer until the late nineties, when C. P. Stcinmctz of the General Electric Company and others observed that the multiplication of a directed line in the complex plane by C01’ f) represents a rotation of 90°, and thereby established quite ingeniously a connection between the complex number, which had been up to that time a mathematical curio, and altemating-current theory. This result, amazing at the time, was then further applied by showing that if the inductances and capacitances in electric networks were represented by impedances jwL and 1 fju>C. respec¬ tively, the voltages and currents in the networks could then be calcu1 See E. Kasner and .1, Newman, "Mathematics and the Imagination,” Simon and Schuster, Trie., New York, 1941, for stimulating essays on this and other mathematical topics. 2 Euler originally represented the imaginary \/ — i by the symbol i, and mathe¬ maticians still use this notation. However, electrical engineers prefer the symbol \/“l — j to prevent confusion with the standardized symbol for electric current i, * Such a graph is still known as an Argand diagram because of its origin. See A. T. Starr, "Electric Circuits and Wave Filters," p. 19, Sir Isaac Pitman tfcSons, Ltd., London.



lat-ed by the simple algebraic methods previously devised for d-c circuits. Thus, through the keen insight of an engineer, an abstract mathematical idea was converted to a practical laborsaving method for the solution of a-c circuit problems which had heretofore required continual recourse to tile drawing board for the graphical construction of phasor diagrams.1 Sometimes, of course, the mathematical theory evolves from physical discovery and is then confirmed by further experiment, as in the development of electromagnetic wave theory, as described by Van der Pol.2 One will remember how Faraday, working entirely alone in the Royal Institute in Albemarle Street in London, developed the fundamentals of the science of electricity, his researches being crowned with the famous law of induction.

It is

most remarkable that he set to work absolutely experimentally and that all his results, visualized in three-dimensional space, were represented without any formulae.

Then Maxwell arrived on the scene and translated Faraday's ideas

into the language of mathematics; at first, however, without employing the vector notation, which was not introduced into the field of electricity until some¬ time later by Heaviside a.nd Lorentz. Thus Maxwell (by solving the famous wave equation) was able to arrive at the conclusion that electromagnetic waves exist; to achieve this result without mathe¬ matical symbols would have been beyond human possibility*

It was not until

1886 that Heinrich Hertz, stimulated partly by a problem set by Yon Helmholtz, based on the Maxwell theory, demonstrated the existence of electromagnetic waves in the laboratory.

In the modern research laboratory, the application of the mathematical method is no less important. reported by Gilman.3

Consider the engineering


To evaluate a proposed development [the engineer] usually starts with an exploratory study.

Here, whenever possible, he takes advantage of mathematics

and physical theory which have frequently demonstrated their capacity to save him time and money.

There are notable instances in which mathematical

studies have effectively disclosed in quantitative terms basic controlling factors of complex problems without any practical experimentation at all.


matical analysis has proved a powerful tool, for example, in clarifying the phi¬ losophy of the transmission of information and in the application of statistical theory to telephone trunking, 1 So-called "vector” diagrams of the phase and amplitude relationships of voltages and currents are properly described as phasor diagrams; and their usage has been so standardized by the Institute of Radio Engineers. See N* R* LcPnge, Symbolic Nomenclature in Sinusoids, Elec. Enq.} 68 (7), July, 1949. Although complex quan¬ tities add vectorially in the complex plane, they are not vectors, because they do not possess the vector and scalar products, 2 B. Van der Pol, Mathematics and Radio Problems, Philips Research Repis., 3: 17*1-100, 194S. J G. W. Gilman, Systems Engineering in Roll Telephone Laboratories, Bell Labs, Recordt 31 (1), January, 1953.



The level of the mathematics employed need not necessarily be high, either.

As Fry1 says,

There need be no apology associated with the statement that such simple processes as algebra, trigonometry and the elements of calculus are the most common and the most productive in modern industrial research. quently lead to results of the greatest practical importance.

They fre¬

The single sideband

system of carrier transmission, for example, was a mathematical invention.


virtually doubled the number of long distance calls that could be handled simul¬ taneously over a given line.

Yet the only mathematics involved in its develop¬

ment was a single trigonometric equation, the formula for the sine of the sum of two angles.

Of course, where blind reliance is placed on it, the application of mathe¬ matics can sometimes produce negative results, too. Occasionally, a theoretical result obtained mathematically will prove to be in conflict with the physical facts as determined by experiment because the analyst made some incorrect assumptions, ignored certain factors which in prac¬ tice may not be neglected, or overgeneralized for all cases from the particular case under study* Thus, on the basis of an insufficiently general theoretical analysis, J. It* Carson2 concluded in 1928 that fC . , , static, like the poor, wall always be with usY Posterity, through the benefits of hindsight, now recognizes that the riddle of static elimination can be solved by means of many of the recently developed modulation schemes, but it is inter¬ esting to speculate on the extent to wdiich research into modulation systems other than amplitude was delayed by such a discouraging prophecy so early in the development of radio. It is often quite surprising to discover how essential the use of mathe¬ matics is, particularly for the solution of problems which are apparently very simple. Problems, for instance, which involve solutions in terms of integers only can prove to be quite complicated, and the reader is invited to test his skill on the four which follow.3 Problems Prob. 1-1

One of 27 precision resistors has been improperly marked

and is known to have a value which differs considerably from the remain¬ der of the lot. A Wheatstone bridge is available; consequently the

1 T. C. Fry, Industrial Mathematics, Bell System Tech. J.} 20 {S): 267, July, 1941. a J, R. Carson, The Reduction of Atmospheric Disturbances, Ptqc* I.R.E,, 16 (7): 966, July, 1928. a For information on the solution of these sec, for example, P. S. Henvitz, The Theory of Numbers, Sd. Amer£canr 186 (1): 52-55, July, 1051; O. Ore, f< Number Theory and Its History," McGraw-Hill Rook Company, Inc., New York, 1948.



resistors may be checked against one another by placing them in series in opposite arms of the bridge- What is the minimum number of checks which must be made with the bridge in order to locate the culprit? Prob* 1-2

Ans. 3 if unknown is known to be high or low; otherwise, 4 A radar observer errs in reporting the distance to a target

as read on a radar range indicator which can be read to an accuracy of Y\ oo mile. He accidently reverses his readings and reports in miles what should have been hundredths of a mile and vice versa.

The radar

plotter notes that if the erroneous range is decreased by 0.62 mile, it is about twice what it should be. What is the correct range of the target? Arts. 12.25 miles Prob* 1-3 The over-all height above ground of a television superturnstile antenna and the steel tower on which it is mounted is 364 ft exclud¬ ing the code beacon at the top. Since such a structure constitutes a hazard to air navigation, it must be painted in alternate uniform bands of international orange and white, the orange bands to be exactly twice the height of the white bands. If each white section must be from 6 to 9 per cent of the over-all tower height, and the base and the top of the tower must be painted orange, lay out a tower painting plan, Prob. 1-4 In the central stockroom of a television broadcasting net¬ work, a shipment of monitor kinescopes is received. The standard pro¬ cedure is for such a shipment to be divided into three equal parts which are dispatched to the eastern, central, and western divisions of the net¬ work, respectively. Before the division occurs, however, one of the tubes is always placed in reserve stock. On this particular occasion the clerk responsible for shipments to the eastern division is alone when the ship¬ ment arrives* After consignment of one tube to reserve stock, he forth¬ with divides the remainder into three equal parts and ships one part to the eastern division. Subsequently the second clerk, who is responsible for the central division, arrives to find the stockroom unattended. He divides the remaining kinescopes into three groups, after first dispatching one to reserve stock* He then forwards one group to the central division. Next the clerk for the western division arrives and, in the absence of the other clerks, repeats the previous process of dividing the remaining tubes into three lots, after earmarking and sending one tube to reserve stock. He then sends one lot to the western division. Finally, during lunch relief, the stockroom supervisor appears and, assuming that the kinescope shipment has not yet been processed, sends one to reserve stock and ships out the three groups remaining to the vari¬ ous divisions of the network* Before the invoice is checked for payment, the minimum number of kinescopes in the original shipment must be known* Ans. 79

Chapter 2



Applications of Differentiation

Example 1.*

For a thermistor used as a detector in microwave power

measurements, R = where R is the thermistor resistance; B, J, and C are constants; P is the applied power; and K is the absolute ambient temperature. The detector sensitivity of the thermistor is defined as the change in resistance with respect to the change in applied power, and an expression for this quantity must be derived. In the limiting case, Detector sensitivity = ^ dr and

In R = In J + v rix -h L


By difTerentiation, 1 dR _ R dP

BC {K + CP)2



BCR c k + cpy

But recall from above that



K + CP ~ 111 J so that

i Y Jt + cp



* C, G. Montgomery, "Technique of Microwave Measurements," Radiation Laboratory Series, vol. 11, p* 104, MeGraw-IIill Book Company, Inc,, New York, 1947. 10



Substitution of this result in the expression for dR/dP yields the detector sensitivity ns

which is the desired result. Example 2. In the design of nonlinear precision potentiometers, non¬ linearity of resistance with rotation is obtained by suitably shaping the card on which the resistance wire is wound. It can be shown that the card shape is proportional to the derivative of the resistance as a function of angular rotation. For example, a potentiometer card which

Direction of movement of potentiometer arm along card

Fig. 2-2 network.

Fro, 2-1 Potentiometer card shape used for approximating logarithmic variation of resistance with rotation of potentiometer.


consists of a true triangular section will give a variation of resistance with shaft rotation conforming to a true square-law function. Suppose it is desired to obtain a logarithmic variation of resistance with potentiometer shaft rotation such that R — k In 0, where k is a constant. According to this method for designing a potentiometer card shape, Potentiometer card shape

dR = * dO 6

The potentiometer card shape is therefore hyperbolic, as shown in Fig,

2-1. Example 3.*

The ratio of the output to


input voltage of the

90°-phase-shift network, shown in Fig. 2-2, is C*






R2 R%If jwCz

* I. S. Greenwood, J\ V. Holdain, and Dr MacRao, “Electronic Instruments/’ Radiation Laboratory Series, voL 21, pp. 149-150, McGraw-Hill Book Company, Inc,, New York, 1948.



€2_ or

1 ~h


Ti ~ 1 - u-RiRvCiCi + MlhG'i + R2Ci)

and the phase shift is 90° if


= RiRzCiC,

Under this condition the expression for e2fex may bo written as cv = _1 + (cd/tup)2_


1 — {6j/cl)o)2 ~T~


If the fraction is multiplied in both numerator and denomiuator by the complex conjugate of the denominator, it becomes

e2 _ [1 + (WmY}[ 1 - {ufmY - MR1C1 + RzC*)]


[1 - (w/wo)2P + «■(RiCi + RzCzY


— Crj(i?lUl

^ = tan-

R2C 2)

1 - (*/uo)2 where