Magnetoelectric Devices SLEMON, Notas de estudo de Engenharia Mecânica

Baixe Magnetoelectric Devices SLEMON e outras Notas de estudo em PDF para Engenharia Mecânica, somente na Docsity! ,,'n'ill 'j "r "II' Magnetoelectric Devices TRANSDUCERS, TRANSFORMERS, AND MACHINES Gordon R. Slemon Professor and Head, Department of ElectricaI Engineering University of Toronto \ t. '), 1 L John Wiley and Sons, Ine. Io)Mf='~---'O~ L?JLkJ:::-- ,-:,Inlr~é':'---O',,',~L:J':.~=-', ~ New York . London . Sydney Copyright © 1966 by John Wiley & Sons, Inc, Ali rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 66-21039 Printed in the United States of America I ! \ ,~ PREFACE ... This book is concerned with understanding, modeling, analyzing, and designing those devices that are used to convert, transfonn, and control electrical energy. Included are transducers, actuators, transforrners, mag- netic arnplifiers, anel rotating machines. In introducing the generic term "magnetoelectric" to denote this group of devices, 1 have sought to emphasize their predorninantly magnetic nature, while distinguishing them from other well-established groupings such as electronic and microwave devices. Chapter 1 is devoted to a study of some of the basic energy conversion processes anel to an examination 01' the properties 01' lhe importam materiais used in exploiting thcsc processes. Particular emphasis is placcd ou a study ofIerromagnctism, inclueling thc mechanism offorce production betwcen sections 01' ferrornagneric material. Chapter 2 introduces some 01' the concepts that are useful in elcvising approximate analytical models for nonlinear magnetic elements. Thc derivation of equivalent electric circuirs to represent rnultilirnbed magnetic systems is discussed. ln Chapter 3, the modcling concepts of Chapter 2 are applicd in analyzing the performance of transforrners, saturable reactors, and other static magnetic devices. Chapters 4 anel 5 are devoted to rotaling electric machines. General equivalent circuit rnodels are developed for two broad classes ofmachines, narnely cornrnutator and polyphase rnachines. Specific machine lypes- such as shunt, series, cornpound, induction, synchronous, reluctancc, polyphase commutator, single-phase, anel synchro machines-are analyzcd by adaptation of the two basic equivalem circuit models. A featurc 01' lhe analytical approach is that the effects 01' magnetic n on linearitv can be includcd approxirnately in any 01' the modcIs. It is my conviction that lhe essence 01' cngineering is designo Thc ultimare objective of each phase 01' preparatory study should therefore be to incrcase lhe studenls' capabil ity to design uscful components ând syslems. The routc to this ultimate objcctive has four distinct but v ( ( ( ( ( ( ( ( ( ( ( ( ( ( (- ( ( ( ( ( ( ( ( { ( ( ( ( ( ( { ( -< c ,( l j ), ( ) u I( J ) ti ) 11 ( ) ,) ) ) ) ) ) ) ) ) ) ) ) ) ! ) ) í I) ), ) I ) I ) I ) x 5 Polyphase Machines 5.1 General Magnetic and Winding Arrangements 5.2 Magnetic Field Analysis 5.3 Equivalent Circuits 5.4 Polyphase lnduction Machines 5.5 Polyphase Synchronous Machines 5.6 Polyphase Synchronous-Reluctance Machines 5.7 Polyphase Commutator Machines 5.8 Unbalanced Operation of Polyphase Machines 5.9 Synchronous ControI Devices Problems lndex CONTENTS 334 334 340 361 371 392 439 457 470 497 503 Magnetoelectric Devices TRANSDUCERS, TRANSFORMERS, AND MACHrnNES 539 Chapter 1 ELECTRIC ENERGY CONVERSION PROCESSES 1.1 THE PURPOSE AND SCOPE OE THIS BOOK This book is concerned with electrical devices that are used for the con- version of electric energy to or from mechanical energy, or to electric energy in a different formo Let us look first at sÇ>meof the needs for e1ectrical devices or machines since it is only by appreciating the functions which they must perform and lhe environment in which they operare that we can assess the properties which will be significant. Electrical devices are required for the following. 1. The proeluction 01' largc amounts 01' c1ectric power from hydraulic or steam turbincs. (Need for thcsc rnachines may be partially superscded in the futurc by the devclopmcnt 01" dircct methods for converting thcrrnal energy to electrical Iorrn.) 2. The transformation of electric energy to a voltage suitable for bulk transmission anel its retransforrnation to voltages suitable for distribution and use. 3. The conversion of available electric power to other desired forrns involving a change of frequency, source characteristics, etc. 4. The production of mcchanical energy with relatively constant mechanical speed for a majority of the applications of electric motors, large anel srnall. 5. The control, with varying degrees of accuracy, of the speed and position of mechanical systems. 6. The measurcment of mcchanical quantities by the production of proportional electric signals. If electrical machines predominare in all these roles, it is because they allow economical centralized production of power, relatively simple distribution, economical reconversion to useful form, and a high degrce of control and automation. To contrivc a useful device, various materiais must be assembled in 1 ? ~ ( ;, ( ( / ( ) ) ) ) ) I í) ) I ) I ) JI . ) f ) I ) I ) I ) t ) i ) I ) I ,) ), SOME ENERGY CONVERSION PHENOMENA 3 2 ELECTRIC ENERGY CONVERSlON PROCESSES of that pragmatic approach by which the engineer seeks to obtain the maximum utility in his design and analysis with the minimum expenditure of effort. such a way as to exploit some physical phenomenon. Chapter I of this book is devoted to a study of some of the basic energy conversion processes on which electrieal devices are based and to an examination of lhe prop- erties of the more important materiaIs that are used. The concepts of electric and magnetic fields are very useful in reaching an understanding of these basic energy conversion processes. But when these processes are exploited in complex machines, the wealth 01' infor- mation contained in the fieId approach generally becomes unwieldy. It is usually possible to extract only a fcw descriptive parameters through whieh the important operational properties of the machine may be adequateJy described. In many instances this extraction results in an equivalent electric circuit; in others, a set of equations is produeed. When making this step from the fieId to the cireuit point of view for purposes of easy analysis, we should keep in mind that the maehine designer must reverse this process, and, starting with the required terminal properties of the maehine, must specify its dimensions. Chapter 2 covers methods by which simple models such as equivalent circuits may be produced for magnetic systems. These models are then used in Chapter 3 to analyze the properties of transformers and some other stationary electrical-to- eIectricaI conversion devices. The remainder of the book is devoted to rotating machines and to systems of these machines. Although these machines are, for convenience, divided into a nurnber of categories, they are ali based on a small and essentially common set of energy conversion phenornena. The appropriate pararnetcrs of these machines are deveIoped frorn a knowIedge of the dimensions and materials, These pararnctcrs are then incorporated into analytieal models, usually in the form of equivalent eircuits. From an analysis of these models, the behavior of these machines may be predicted and their applications studied. Thus the purposes of this book are to introduce the reader to those physical principIes that have been most widely exploited by engineers to produce electrical devices and maehines, to show how materiaIs are arranged to produce the device and how the performance is limited by the physical limits of the material properties, to develop means by which the behavior of the machine may be readily predicted in various applications, and to provide some background for the understanding and even for the invention of new devices. Throughout the book, numerous approximations are made to provide the maximum simplicity and yet to retain enough information for the problem under study. Naturally, these approximate analyses neglect various secondary factors which may be significant in certain circumstances. A further purpose of the book is the development 1.2 SOME ENERGY CONVERSION PI-IENOMENA [11 this section let us examine, in a qualitative manner, some of the elcctrical phenomena that result in the production of mechanicaI force. Figure 1.1 shows lhe familiar forces of repulsion between similarly r~ (a) ...;"";'-!- "f. ~+lr~- t1-/t- ,r' ")<. i- ~f (6) Fig. J.1 Force betwecn "electrically charged bodies. (a) Opposite charges. (b) Like charges. charged bodies. lf in Fig. l.la the two bodies are allowed to move in response to these forces, mechanical work is done. During this motion some of the energy which was originally used to separate the positive charge from the negative is converted to mechanical energy. To pull apart the bodies in Fig. I.la requires an input of mechanical energy that is converted into an electrical formo By generalizing from this simple example, it follows that any system exhibiting mechanical forces of eIectrical origin is capable of hoth electrical-to-rnechanical and mechanical- to-electrical energy conversion. Figure 1.2a shows the forces that exist on parallelcurrent-carrying 8 ELECTRIC ENERGY CONVERSION PROCESSES ENERGY CONVERS.ION USING ELECTROSTATIC FORCES 9 Area ilA Charge ilQ The rcason that this method of analysis is alrnost never used for manual calculations may be appreciated Irorn this brief discussion. The method is, however, suitable for solution on a digital cornputer for which the repctition of the sim pIe steps in lhe method presents no difficulty. To determine the work lha! is done as the right-hand plate in Fig. 1.8 moves between two positions Xl and X2, it is necessary to evaluate the force J for a number of values of X between Xl and X2. When the force vector J for each vectorial incrernent of motion d i: is known with sufficient accuracy the work W is given by* w=Jx.J'. X1 joules (1.3) surface could be divided into a large number N of incremental areas D.A, .and the charge !lQ on each area !lA would be known. Suppose we regard each charge increment !lQ as an approximate point charge. Each charge increment !lQ on the positively charged pIa te experiences an incremental force cornponent !lJ directed toward each of the N charge increments on Although the work is expressed here as an integral, a surnmation over finitc increments ofz is normally made in numerical calculations, The intention in introducing this method Iirst is no! so much to develop an ability to calculare forces as 10 examine lhe thought process involved in qualitatively assessing such questions as; How is the charge distributed? In what direction does the net force act ? How does the force change as the spacing is changed '? J= I!lJ 1.3.1 Elcctrk Ficld Approach In a systcm such as that 01' Fig. 1.8, the use of the sornewhat more abstract conccpts of an electric Iield and a few judjcious assumptions leads to a much simpler method of calculating forces and energies. This approach depencls on the determination of the electrie field intensity 6; which is defined by the property that an increment of concentrated charge !l Q placed into the electric field at the point where the field, intensity is rF experiences a force !l/of ,,. Fig. 1.8 Two parallcl charged conducting plates. the negative plate; the magnitude of each component is found by use of eq. 1.1. The total force J may then be deterrnined by vectorial addition of the N2 incremental force vectors. * newtons ( 1.2) sr = L\Q;'5'~ newtons "--,, (I.4) Since the distribution of the charge on the conductor surfaces is generally not known, it must be found by using a surnmation 01' force incrernents for each incremental surface charge. The force component parallcl to lhe concluctor surface acting on each incremental eharge !l Q is zero, since otherwise the eharge would move until this was so. Ifthis force cornponent is evaluated in terms of the N unknown charge incrernents I'1Q and equatecl to zero for each area, the result is N sirnultaneous equations, which lllay be solvecl to obtain the charge distribution. The elec~ric field intensity is relateel to another vector, the clectric flux density D by ", _..'- -_'I. D = E/i" coulorn bsjmeter i--'/ (1.5) for vacuurn anel, for ali practical purposes, for air or other gases. The electric flux density may be considered as emanating from a chargeel body. By Gauss's theorem, lhe total electric flux emanating from the area A 01' a closed surface is numerically equal to the charge Q enclosed * Througho~t this book veetor quantities are denoted by an arrow over the symbol (for example.j"). The magnitude of a veetor quantity is dcnoted by the symbol with no arrow (for example,f). * The scalar or dot product of these two vectors is a sealar quantity equal to the product of lhe magnitudefof lhe force, the magnitude d.c of the direeted length, anel the eosine of the angle hetwcen these two vcctors. ) 10 within that surface. Thus ELECTRIC ENERGY CONVERSION PROCESSES 11 ) ;' ) ) '. ) ) i I I ;1 ~ ): i ~ .~ ~ 1! ENERGY CONVERSION USING ELECTROSTATIC FORCES f closcd 15· dA = Q"ncloscd coulombssur í ace (1.6) To the extent that the assumptions made are valid, eq. 1.9 shows that the force/is independent of the plate spacing x. The work W done by this force in moving the plate from a spacing Xl to a spacing X2 is obtained by evaluation of eq. 1.3. J""w= I d:GXl T z If the spacing X in Fig. 1.8 is much less than either of the dimensions y or z, it is reasonably accurate to assume that the charges are distributed uniformly over the inside surfaces of the plates. Actually, there will be a denser charge distribution near the edges, but the region of highly nonuniform distribution is assumed to be only a small part of the total area. Figure 1.9 shows the assumed uniform sheet of charge from the left- hand plate of Fig. 1.8. A field of electric flux density 15 of constant magnitude emerges uniformly and perpendicular1y from the surfaces of this sheet. This field is assumed to remain constant in magnitude and direction for some distance outward from the sheet. These assumptions of uniformity allow us to evaluate the integral in eq. 1.6. For a closed surface around thc plate in Fig. 1.9, we find that Si) . JA ~ D 2yz = Q, from which 2yz,,-o It is now assumed that the whole of the negative charge Q on the right-hand plate of Fig. 1.8 is placed in this uni- form electric field intensity of magnitude 8. The magnitude of the force on this right-hand plate is then given by y ~ Fig. 1.9 Electric flux dcnsity field assumcd to cxist around a uniform shect of chargc. This force is directed to the left. - ~(XI - x2)- 2yuo joules (1.10) D =.2.... 2yz ~rom eq. 1.5, the electric ficld intensity 6' arising from the charge on the left- hand plate has a magnitude (J' of If Xl > x2, the work W is positive, and mechanical energy is supplied to the mechanical system attached to the movable plate. The assumption of uniformity of charge distribution could have been made in the method of analysis based on Coulomb's law in Section 1.3. Although this would have eliminated one step in the calculation, the calculation and surnmation of force incrernents in eq. '1.2 would still have been required. Thus, the field approach allows a fuller exploitation of the assurnption of uniformity than does the nurnerical summation approach. On the othcr hand, unlcss there is some symmetry in the physical charged object that pcrrnits assurnptions leading to an easy evaluation of the integral in eq. 1.6, lhe numerical complexity in determining the fíux density [j at ali significant points may be as great as that of the earlier rnethod described in Section 1.3. Since the elcctric field approach gives solutions which are functional rather than numerical, it provides a better appreciation of the relations between forces or energies and lhe dimensions of the object. To exploit this approach, it is usual to replace the physical object under study by one or more idealized charge distributions (such as that of Fig. 1.9) for which the elcctric field is readily calculable. 1.3.2 Approach through the Conservation of Energy Consider any device having provision for the input of energy and the output of useful energy, as is shown in Fig. 1.10. Within the device there are generally several ways in which energy can be stored and by which energy can be lost or rendered useless. If we suppose that ali the energy must be in one of these categories, it follows from the law of conservation of energy that coulornbsjmeter" (1.7) 6'=!!.. EO =-ª- newtons{coulomb ( 1.8) J = Q(J' Q2 =-- 2Y"Eo (1.11) (1.9) Energy = Energy + Jncrease in + Energy input lost energy stored output Suppose now that the device is electromechanical with an input of electric energy and an 'output of mechanical energy. If we allow an infinitesimal increment dx in mechanical displacement, the corresponding increments 12 ELECTRIC ENEI~GY CONVERSrON PROCESSES ENERGY CONVERSION USING ELECTROSTATIC FORCES 13 of energy input, Ioss, storage, and output are related by The stored energy of a two-conductor electrostatic system may beexpressed in terrns 01' its capacitance C and íts stored charge Q as dWelectrical = d~VI(I", + dWst.orc'l + (HVII11'('II;IIlÍ<",1I Input outpu t ( 1.12) Q2 . ~'KI(lr"d = -- ioules 2C ( 1.16) The force Ix exerted by the mechanical mernber ín the direction 01' the displacement dx is dWmcch.olll.\lUI.Ix = dx ( 1.13) Substítuting into eC[. 1.15 gives . Q2 ec I -j = - - newtons . x 2C2 à« Q"" ('O!lsl. (1.17) This reduces the problern to one of finding lhe system capacitance C as a function of :/;. One method of accomplishing this would be the experi- mental measurernent of the capacitance at a sufficient nurnber of values 01' x. Alternatively, the capacitance can be calculated as 'r If the input, loss; and stored energy terms in eC[o 1.12 can be evaluated, the mechanical encrgy output is then known and may be used in eq. 1.13 to determine the force. D"'''-It.- fj'; I C=ª farads ( 1.18)e Input 11 1 Loss Output . where e is lhe electric potential difTerence between the plates and is given by f'~ e = â'· d » volts (joulcs/coulomb) (1.19) • u Fig. 1.10 Energy flow in a device. For the parallel-plate arrangement 01' Fig. 1.8, the assurnption of uniform charge distribution on the inside platc surfaces resu1ts in a uniforrn electric flux density betwecn the plates of magnitudeA1though this method on first examination may appear to be more involved than a direct deterrnination 01' force, ir often leac\s to sim pIe and general expressions for torce. Consider, for cxamplc, the system 01' two parallel charged conductors in Fig. 1.8. An clectrical source may once have been connected to the plates in order to place on them the separated charge of Q coulornbs ; but since this source is not now connected, there can be no e1ectricaI input. Disp1acement of one plate relative to the other may cause some redistribution of charge, anel, as this charge Ilows through the imperfectly conclucting material 01' lhe platcs, some energy may be lost. But, since the charge distribution remains reasonably constant, this loss may safely be neglected, particularly ir the motion is reasonably slow. Let us further assume that the energy loss due to friction is part of the mechanical output. Tt therefore Iollows, for this system, that the mechanical energy output is equa1 to the arncunt of the decrease in stored energy. Thus we have D=2. coulorn bsjmeter2 ( 1.20) !JZ The vector quantity !5 is direcied to the right. lt is noted that the quantity D in eq. 1.20 has twice the value or eq. 1.7, because the fields of both charged plates are considered simultaneously'. From eqs. 1.19 anel 1.20, we have Q:l: voltse =-- !FXo C = !JZEo .r farads (1.21 ) and the force in lhe dircction 01' positive :/: is dWmcl'!1. = -dW,lor"d out.put ( 1.14) t.= Q~ J!... (YZEO) 2(Y'~~Or ox :r and, from eq. 1.13, we have fx = _ a Wslored I OX Q~consl. newtons (1.1 5) - Q~ 2YZEu newtons (1.22) )~ 18 ELECTRIC ENERGY CONVERSION PROCESSES )" It is significant to note that if the fringing field had been entirely neglected in developing the capacitance expression of eq. 1.33, the resulting force expression of eq. 1.34 would have been fortuitously correct, in spite of our previous reasoning that the force ariscs from this fringing charge! This fact emphasizes some of the strengths and weaknesses of lhe methods we have used. The direct approaches to the determination of force using either Coulomb's law or the e1ectric field are most useful in obtaining a physical appreciation of a situation, but a quantitative analysis using these approaches is often difficult or tedious. The preceding example shows the ease with which functional expressions for force often may be obtained by using the energy conservation approach. But, by its nature, this approach can give no physical picture ofthe origin and area of action ofthe force~ The methods are therefore often complementary rather than competi tive. ) ) ) ) ) ~ 1 f 1 f ( ( ; I) ( ( j J; { \ 1.3.4 A Rotating Electrostatic Machine For a device in which the permitted motion is rotary rather than translational, the mechanical work is expressed in terms of the torque vector T anel the angle O of rotation. The torque component To in the elirection of positive angie O is given by dWmecll. ou(pu( newton-meters ( 1.35) To = dO By analogy with eq. 1.30, the torque may be expressed as To = !e2 dC 2 dO newton-meters (1.36) That is, there is a torque component in the direction of positive angle () if the capacitance C increases with increase of the angle O. Figure 1.l3a shows a simple electrostatic machine with stationary or stator plates and rotating or rotor plates. lf the rotor and stator axes are coincident at () = O, the capacitance C between them may be approxi- mated by regarding the stator and rotor sectors as complete quarter- circular disks for which ,2 to7T1 ,Co~o = li farads (1.37) where d is the spacing between the plates. The variation of capacitance with angle is shown in Fig. 1.13b to be essentially a set of straight-line , ENERGY CONVERSION USING ELECTROSTATIC FORCES 19 I. () P======i~ (a) c 'Ir (b) Fig. 1.13 (a) A simplc rotating elcctrostatic machine. (b) The capacitance of the machine as a function of angular position, segments with rounding at the intersections because of fringing fields. ln the interval of O < O < 7T/2, the capacitance C can be approximated by C = t07Tr2(1 _ 2(J) 2d 7T If the potential difference e applied to the plates is constant at a value E, the torque, from eq. 1.36, is farads (1.38) To = _ E2tor2 2d newton-meters (1.39) for O< ()< 7T/2. As shown in Fig. 1.14a, this constant negative torque is followed by a constant positive torque in the interval7T/2 < (J < 7Tover which the capacitance again increases. Since the average torque over a 20 I}: ELECTRIC ENERGY CONVERSION PROCESSES complete revolution is zero, it is notcd that this modc 01' operation cannot provide a continuous meehanical output. A useful unidirectional output torque can be obtained by restricting the motion to an are not greater than 'iT12 radians, as in an electrostatic voltrneter. The shape of the torque curve in Fig. J. 14(/ suggests that ir we wish to have torque in the positive direction only, wc can eonncct the machine to w To \ ---;r 71" 3,,-"2 "2 \ -:>-e (a) e 271" 8. L~ (b) T~e __n.n , 7 71" ªf 27l" e (c) Fig.1.[4 (a) Torquc of clectrostatic machine with constam vollage. (h) Interrniucnt alternating voltage switched on to machinc terminais. (c) Torque rcsulting Irem applicd voltage of (b). a source which applies a potential difTcrcncc c = E over the intervals 01' rotation nl2 < O < n and 3n/2 < O < 2n, anel e = ° over the remaining intervals. Beeause the torque is proportional to the square 01' the voltage, the polarity of the potential difference e is immaterial. Thus, application 01' the voltage e shown in Fig, J. 14b wil! result in the unidirectional torque of Fig. 1.14c. Examination of the voltage wave 01' Fig. J .14b suggests that this machine could produce useful torque ir supplied with an alternating sinusoidal voltage e, the period of which is equal to the time required for one complete rotation. Let e = Ê cos 0)( volts (1.40) ENERGY CONVERSION USING ELECTROSTATIC FORCES 21 As we are dealing with sinusoidal functions, lct us approximate the periodically varying capaeitance by the first two terms of its Fourier series expansion. c = Cu + C2 eos 20 farads (1.41) If the angular velocity of the rotor is v radiansjsecond and the angular position 01' the rotor at time t = ° is O = -o, 0= vt - (j radians (J .42) Substitution Irorn cq. 1.40,1.41, and 1.42 into eq. 1.36 and manipulating trigonometrically give T.o =~(E cos (I)f)~ ~ [Co + c., cos 2(vt - o)]- dO - _ ~Ê2C2{ sin 2(vt - 15) +~ sin [2(0) + v)t - 20] + .~sin [2((1) - J')I - 2ô]} (1.43) Examination 01' eq. 1.43 shows that ali terms vary sinusoidally with time and havc zero averagc value unless the angular velocity 01' the rotor is v = ±(f). This is consistent with our previous conclusion that the period of rotation must be eq ual to the period of the applied voltage, If, in eq. 1.43, l' is set equal to (I) for counterclockwise or positive rotation, the average iorque Tis T = lC2Ê2 sin 215 (1.44)newton-rneters This machine, operating on alternating voltage, can produce useful torque only at the forward or backward rotational velocities that cor- respond to the angular frequency of its supply voltage. It is therefore known as a synehronous machine. If a rnechanical load is applied to this machinc, the rotor morncntarily slows down, thereby increasing the anglc O of its lag behind the maximum capacitance position O = ° at the instant of maxirnum voltagc t = O. Equation 1.44 shows that this increase in angle ô results in an increase of lhe average torque T, thus providing the required load torque as long as (S is in the range ° < O < n14. Figure 1.15 shows lhe variation of average torque with the angle o. The maximurn average torque which the machine can produce is Tm:,x = lC2Ê2 (1.45)newton-rneters If the load torque exceeds this value, the lag angle O continues to increase, and the machine eventually stops. This machine may also be used as a generator to convert mechanicaI energy to clectrical forrn. 1I' torque is applied to the shaft by a prime mover in the direction of rotation, the machine momentarily speeds up. The )' ) ) ) ELECTRIC ENERGY CONVERSION PROCESSES22 ang1e b and the average torque both become negative, indicating that mechanical power is being absorbed by thc machine and is being Ied back into the electrical supply. If lhe prime-mover lorque exceeds thc value given in eq. 1.45 the machine accelerates in a runaway condition. The mechanical power output of the machine has an average value of P = Tv watts lt is generally impractical to use air as a dielectric medium between the stator and rotor plates, becausc the output per unit volume is much smaller than can be obtained frorn other types of machinc. To date no significant (1.46) T ·1 1 1 0,..-8 Gener~tor I Motor ~)o( Fig. 1.15 Variatian of average tarque f with angle af lag O. arca of practical application has been found for clectrostatic synchronous machin~s. The use of vacuurn, which permits a much higher electric field intensity in the machine may, howevcr, make this type of machine com- petitive in the future for some applications. 1.4 PROPERTIES AND USE OF DIELECTRIC MATERIALS Various nonconducting materiais are used to provide e1ectrical in- sulation, mechanical support, and electric energy storage in electrical apparatus. ln this section some of the material properties that are most per~inent to energy conversion devices are examined briefty. In Section 1.3 the discussion of electrostatic forces was restricted to conductors immersed in a vacuum. If the vacuum is replaced by a gas at normal pressure and temperature, the electric field and force relationships remain essentially unchanged. The only significant electrical difference in the gaseous media is in the electric field intensity which each medium can withstand before breakdown. Air, at normal temperature and pressure, can withstand an electric field intensity of 2.5 to 8.5 megavolts/meter without appreciable conduction, the value of breakdown stress increasing as the electrode spacing is decreased. With excessive electric stress, stray ·1, 1 ~ PROPERTIES ANO USE OF OIELECTRIC MATERIALS 23 electrons in the air can achieve sufficient kinetic energy to dislodge other electrons as a result of collisions with atoms. If this ionization process is continued, a stream of conducting electrons and ions is produced between the electrodes, and breakdown occurs. At reduced pressure, free electrons can travel longer distances in the electric field and acquire greater kinetic energy between collisions. This leads to ionization and breakdown at lower values of electric field intensity. Conversely, an increase in pressure results in an increased breakdown stress. Theoretically, a vacuum cannot conduct because it contains no charged partic1es. Practically, electrical breakdown between a pair of electrodes immersed in high vacuum is almost entireIy dependent on the electrode material. At sufficiently high electrical stress, gas and metal atoms are torn from the surface of the electrode material to initiate the breakdown processo Solid and liquid insulating materiais are characterized by their ability to withstand clectrical stress, their electrical losses, their ability to store e1ectrostatic energy, thcir thcrrnal propertics, and their stability with time. Conduction in these materiais may be due to the flow of electrons, holes, or ions. Although a small amount of conduction may occur safe1y under normal stress, breakdown in the insulation usually occurs when the conduction becomes thermally unstable aIong some concentrated path between the electrodes. As the number of free charged partic1es increases with temperature in dielectric materiais, the conductivity of these materiais increases with temperature. The breakdown stress generally drops rapidly as temperature is increased. Excessive temperature may also cause struc- turaI changes, particularly in organic insuIating materials, which may lead to a deterioration of both electricaI and mechanical strength. Breakdown stresses in insulating materiais range from 10 to 200 megavolts/meter depending on the material and its thickness. Let us now examine the use of solid or liquid dieIectric materiaIs as electrostatic energy storage media. When a material is subjected to an electric field intensity, the electrons within each atom are displaced somewhat with respect to the positively charged nucleus (eIectronic polarization), ions are strained from their normal positions in the material structure (ionic polarization), and asymetrically charged molecules are rotated toward alignment with the applied fie1d (molecular polarization). All of these charged bodies return to their normal positions on removal of the fie1d. Thus, energy may be stored in these various strain mechanisms and recovered as the field is removed. In most dielectric media, the electronic, ionic, and molecular strains are directly proportional to the applied e1ectric stress. If the whole of the '·i': I~'~ 28 ELECT!UC ENERGY CONVERSrON PROCESSES vector B, the magnetostatic force J is" J= Q(iJ X B) newtons (1.49) ',f li, I· ,! If the moving charge is in a conductor, it is generally described by either the electric current i or by the electric current dcnsity 1. If the conductor material contains 11 Irce charges per unit volume, each onc of value Q., and alI moving with common velocity v perpendicular to the eonductor cross section of area A, the eurrent i in lhe conducror is i = nQeAv amperes (1.50) For the current element of length d7 in Fig. 1.18b, the force 40n its eharge nQ A di is rJ! --> e' d = i(d X B) newtons (J .51) In this expression the element length dl is reprcsented by a vector directed along the conductor axis in the direction of positive current i. This ex- pression is useful when the coneluctor area A can be considered as negligibly smal!. When the area is large, it is preferable to consider lhe force 4 on the incremental conductor volume dV in Fig. l.l8e in which the current density J is specified. From eqs. 1.49 and 1.50 we, have 4 = dv(1 X B) newtons (1.52) I! li. t·;~ p. ;~ l ~ Ht,.. One further elefining expression for flux density is of particular use for small coils anel spinning clcctrons in matcrials. Figure l.ISd shows a small closeel loop enclosing an arca A and carrying a current i. The torque T on this loop can be shown to be equal to the proeluct 01' lhe current i, the área A, the flux elensity B in lhe region of lhe loop, and lhe sine of the angle between the normal to the area A anel the flux density. It is convenient to describe the area, current, andorientution of lhe loop by its magnetie rnornent F", a vector having a magnitude Pm = iA arnpere-metcr" (1.53) a direction normal to the area, anel a sense following the right-hand screw convention, as shown in Fig. 1.l8e1. The torque on the loop may then be expressed as t = !'m X li newton-meters (1.54) The rotational direction of the torque i in Fig. 1.18<1 is the sarne as would be foIlowed by the turning of a right-hanel serew proceeding along the direction of the torque vector t: * The vector or cross product (u x B) is a vector quantity equal in magnitude to the product ofthe magnitudes of the two vcctors and the sine of lhe anglc o: betwccn thcrn. It is dire~ted perpendicularly to both u and n, and its direction is the same as would be followed by a right-hand screw rotating from the first vcctor v to lhe second vector B. ENERGY CONVERSION USING MAGNETOSTATIC FORCES 29 The magnetic flux elensity Ê is reI ateei to another vector, the magnetic field intensity H, by the relation B = PoFi newtonsfampere-meter (1.55) in vacuurn. For engineering purposes, the relation is also valid for essentialIy alI materiaIs except the ferromagnetic materiaIs to be diseussed in Section 1.6. The dimensional magnetie constant Po, often called the permeability of free space, is Po = 47T X 10-7 webersj arn pere- meter (1.56) The magnetic Iield intensity Fi is eonsielered to be prod uced by thc moving charges or currents of the system. The ineremental eurrent /~" / ~ I A dIl T'ig. 1.19 Magnetic fielcJ intcnsity of a current clcrnent. element of length d! anel cross-sectional area A shown ín Fig. 1.19 produees in the region around it an ineremental magnetic field intensity dH given by sn = _i-o (dI x fi) 47T/'- arnperes/meter ( 1.57) where fi is a vector of magnitude equal to one directed from the element towarel the point of interest, and r is the distance from the element to the point at which dFi is specified. The field intensity c/H rnay also be written in terms of the incremental volume C/V of the element, its current density J, anel the uni t vector fi as t - dV ~d H = -,,(1 X fi) 47Tr- am peresjmeter (1.58) Combining eqs. 1.51, 1.55, and 1.57 allows the development of an expression for the force of interaction between two current elements of incremental lengths dI] anel d12, anel currents i1 anel i2, respectively. The " ( )1 / ) ) ) 30 ELECTRIC ENERGY CONVERSION PROCESSES rcsulting expression for the i,ncremental forceon elernent c/lL is d/L = 110" iLi2[cill X (dl2 X [1)] 417"" newtons ( 1.59) I ) where fi is the unit vector directed from element 2 toward elernent I. This expression may be used to determine the forces on any system of con- ductors by' summation of the incremental force vectors caused by ali pairs of current elements. As has been already mentioned, the expression of eq. 1.59 is not very useful as a basis for understanding lhe nature of lhe forces. The magnetic field intensity also has the useful property that its line integral around a closed path is equal to the current enclosed by that path, that is I J ) IR· dS' = iCllctosC(jJ ampcr es (1.60) ) This circuital law is very useful as a mcans of obtaining the magnetic field intensity in situations that have symmetry. Let us now apply these field concepts in determining the force between the two Jong parallel conductors shown in Fig. J .20. Consider an element di of conductor 2. From eq. 1.57 it can bc secn that nonc of lhe other length elcmcnts of conductor 2 can producc a magnetic field intensity in the clement di, because the angle bctween c/l and the unit vector i1 to these elements is either O or 17. Thus the magnetic field in which the element di is situated is produced only by the current in conductor 1. The intensity RI of this magnetic field could be found by summation of eq. 1.57 over the length of conductor l. But because 01' the circular symmetry of the field which is produced by conductor 1 acting alone, the circuital law can be used to evaluate the field much more simply. From eq. 1.60, we have ) ) ) ) ) I ( ) §Rl' ds = HI 217X = il and i I Hl = - amperejrneter 217X By using eqs. 1.51 and 1.55, the force dj on the element di is dj = i2(dl X RI) = flOi2(c/t x RI) (1.6l) ) ) ) ( ( The magnitude of the force is df = 110ili2 dL 217X newtons (1.62) ) r' ENERGY CONVERSION lJSING MAGNETOSTATIC FORCES 31 Provided the conductors are long and straight for a distance much greater than x from the point of interest, the symmetry condition obtains, and the force per unit Icngth on each conductor is constant in magnitude and direction. Let us investigate thc energy conversion properties of the system of Fig. 1.20 by pulling the conductors apart against the force of attraction •....---~, // -, Conduclor 1 (Iixed) /Path 01 rntegratior, \ ~ C ductor 2 (movable)I to nnd H 1 \ '1 0n..I \ , \ - --I ~-- x i2 _--_)) , \ - _/ »> ' \ - ", »> »> \ \" • ", ':-",--»> ~\ ar" _--' _:ç;", I \ -- --~í\ _--\ ~-\ / \ / \ I -, /BI.HI' /'- ./'----- Fig. 1.20 Force between parallel conductors. from x = XI to X = xz. Frorn eqs. 1.3 anel 1.62, the mechanical energy input for ~ length I of conductor is 1"'2110ili21Wm"rh. = --- dxinput x, 217X = l1oili21 In ~ 217 Xl joules (1.63) This mechanicaI-input energy can also be expressed in terms ofthe amount of magnetic flux 1> crossed by conductor 2 during its displacement. The magnetic ftux through an area A is defined as 1> =J B' dA Arca webers or joulesjampere (1.64) The ftux crossed by conductor 2 in the displacement from Xl to X 2 is Ix,1> = Bll dxx, = 110il1n x2 217 Xl joulesjampere (1.65) 32 ELECTRIC ENERGY CONVERSION PROCESSES ENERGY CONVERSION USING MAGNErOSTATIC FORCES of energy reguires that Energy = Energy + Increase in + Energy input lost energy stored output 33 't. . From this equation we can see that when a conductor carrying a current i sweeps across a magnetic field, the amount of energy converted is equal to the product of the current i and the magnetic flux cP crossed by the con- ductor. (1. 72)i (I'.:,, , 1(, j W = icPcrossed hy conductor joules (1.66) If we allow an incremental displacement dx of the movable pia te in Fig. 1.21, the increments of electric input energy, loss energy, storage energy, and mechanical output energy are related by, "1'.!'i i i I"':\1 If mechanical energy is supplied to conductor 2 in Fig. 1.20, it might be expected to appear as electric energy supplied to the source of the current i2• As the conductor moves from Xl to X2 at velocity V, any electric charge Q within the conductor wiU experience a force J of " J = Q(v X Bl) newtons (1.67) In Fig. 1.20, this force is directed in such a way as to assist the current i2• The electromotive force e2 in a length / of conductor 2 is the energy acquired by each coulomb of charge in passing along the length. e2 = L I Q dWe1ect. = d~08S + d~Vstored + dWmceh. íuput , output (1.73) Following the same pattern as we used in Section 1.3.2 for electrostatic forces, let us attempt to find the mechanical output by evaluating all r.!! de2 = - (cP crossed by conductor 2) volts dt (1.69) --1--1- Movable plate [1 ri.i" ;( í" "I [J 'li ~t if illf{r, (j: ti fi'~.i: = vEIl joules/coulornb or volts ( 1.68) //'/. ,/.' /',/.'//(( 11 1i "1i11 1i 1i 1i II " --?"- f Since V = dxldt, it can be seen, by comparison of eqs. 1.65 and 1.68, that the electromotive force may aiso be expressed as ~ ti' P2 = i2e2 waUs (1.70) /,// / / / / / / / / / The power converted to e1ectrical form in conductor 2 is íj' ~ " li' and the electric energy produced in conductor 2 is Welec. =f P2 dt =f ;2 ~~ dt =f ;2 dcP = i2 (cP crossed by conductor 2) joules Source (1.71) Fig. 1.21 Parallcl-plate conductors. terms in eg. 1.73, except dW mechanical output, and then by determining the force in the direction 01' displacement from ti H~neeh. OUl",,[I; = newtons (1.74) Thus, alI the mechanical input energy is converted to electric output energy in the moving conductor. Jf the conductor moves in the opposite direction, energy is converted from electrical to mechanical formo 1.5.2 Approach through the Conservatíon of Energy Consider the sirnple system of conductors 'shown in Fig. 1.21. For any displacernent of one plate relative to that other, the law of conservation dx There will be resistance to the flow of the current i in the plates. Suppose we considcr an eguivalcnt systcm with resistanceless plates, the aetua! ) ) ) ) ~ ) ~ j ( ) I ) ) ) ) ) ) ) ) I I 1 ~ ) I ) 1 ) ) ) !; ( ,. ) ! ) i ( ) ! ) ! 38 ELECTRIC ENERGY CONVERSION PROCESSES The flux linkages of the coils may be expressed in terrns 01' their sclf- inductances Lll and L22 and their mutual inductance Lu· }'l = Lnil + Ll2iz À2 = Ll2il + L22iz webers (1.95) (1.96)webers Sincc the self-inductances of the rigid coils are constant, substitution from eqs. 1.95 and 1.96 into eq. 1.94 gives dWc1cc.·= i1Lll di, + i1Ljz di2 + iji2 dLl2 inpu t + izLjz dil + izi1 dLl2 + izL22 di2 (1.97) Thc encrgy storcd in a Iwo-coil systcm may be statcd in tcrrns or lhe sclf- and mutual inductanccs as WsLorC(!= ~Llli12 +~L2Zi22 + Ll2i1i2 (1.98) The incrcase in stored energy resulting from an incremental displaccment, which may change the currents and the mutual inductance, is dW,torcd = Lllil dil + L22i2di2 + Ll2il di2 + i1i2 dLl2 + i2Ll2 di, (1.99) Thus, with no internallosses in the system, we have dWmcch. = dWc1c<". - dWstorcd output input = i1iz dL12 (1.100) The component of force on coil2 in the positive r direction is then given by Ix = d~nCCh. output dx u.; = iliz dx newtons (1.101) Hence, the force of interaction between two current-carrying coils is always such as to increase their mutual inductance, provided that the currents in the assigned direction of measurement in both coils are both positive or both negative. Generally, the force is directed in a way that tends to increase the magnitude of the flux linkage of each of the coils. Instead of a lateral motion of coil 2 in Fig. 1.23, a rotational motion through an incremental angle de about some arbitrary axis may be made. The component of torque on the coil in the direction of positive eis then, PROPERTIES or MAGNETlC MATERfALS 39 from eq. 1.100, d~ncch. out.put To = de .. dLl2= 11/2-- de lf there are more than two rigid coils, the force on any coil designated as nurnber 1 may be expressed as newton-meters (1.102) "\ f .. dLI2 .. dLI3 •• dL14= I 10 -- + I I -- + I 1 -- + ...x 1 - dx 1 3 dx 1 4 dx Thc forces and torques that depend on change of mutual inductance are inherently more uscful than those that depend on a ehange in self- inductance as a result 01' lhe ability to control the direction of the foree or torque. Examination of eq. 1.84 shows that the force in a single coi1 always aets to inerease its self-inductance, regardless of the direction of its current. The direction or lhe force in eq. 1.101 or the torque in eq. Ll02 may, however, be reversed by reversing the sign of one of the cur- rents. Application of the force and torque expressions of eqs. 1.84, 1.101, and 1.102 requires that the self- and mutual inductances of the system be evaulated as functions of coil position. In a physical system, these in- ductances can be measured. ln the predietion of the performance of a proposed system, they may be calculated using magnetic field concepts to evaluate individual flux linkages per unit of coil current. newtons (1.103) 1.6 PROPERTIES OF MAGNETIC MATERIALS Very few practical eIectromechanical devices are built using only current-carrying conductors because of the relatively small forces or torques that can be obtained per unit of machine volume. While there is no lirnit on the intensity of the magnetic field that can exist in air or space, the value of the fieId intensity is limited practically by the current density permissible in the conductors producing the field. With copper or aluminum conductors at normal operating ternperatures, current densities must general1y be limited to about 106_107 arnperes/rneter''. With the introduction of superconducting materiais, like niobium tin, which have zero resistance at low temperatures in the region of the boiling point of helium (4.2°K), much higher current densities are possible, and practical machines operating at these temperatures may be developed. Supercooled coils can produce flux densities of 10webers/meter" or higher. In comparison, it is only with difficulty that a flux density greater than 0.1 weberjrneter" can be produced using normal coils ai room temperature. 40 ELECTRIC ENERGY CONVERSION PROCESSES PROPERTIES OF MAGNETIC MATERIALS 41 ' , At present, the easiest means of producing flux densities up to about 1.5-2.0 weber/rneter'' is by the use of ferromagnetic rnaterials. The structure of most electromechanical devices is therefore mainly composed of ferromagnetic material and conductors, with the magnetic material often predominating in volume. An understanding of the properties and limitations of magnetic materiaIs is therefore necessary for the analysis or design of these devices. 1.6.1 Magnetic Moments of Atoms Magnetic effects in materiaIs arise from the orbital motions of the electrons about the uuclei and the spin motion ofeach elcctron, as sug- 1.6.2 Alignrnent of Magnetic Moments In a solid material, the magnetic mornents of adjacent atoms may be aIigned as shown in Fig. 1.25a, producing a ferromagnetic material. Alternatively, the magnetic moments may be opposed, as shown in Fig. 1.25b, producing an antiferromagnetic material. In a ferrdmagnetic material, the magnetic tlux is continuous along the alignment direction and is potentially useful. Tn an antiferromagnetic material, however, the flux paths merely link adjacent magnetic mornents and cannot be ma de to link external current-carrying coils. 'Pm Pm ,,-, /- .... ~ . '-=-'----+--~-'--;:'--- - -.------~-- ---- Pm -- -------~---- - ___ -:::-----r-- ~ __ \..._/ H \,_./ /-, /-, ~ /', ~/" -- •..•.---- ., - "/ ~ H ;t /" - ,,', // \ I ~\ If \. 1\\ f f ~- \ \ ( I -' \ \+~ I I I I t~ \~\, (-" /' / \ \~(-') /' / " ' :.:. " // \, ' :::'_/ /1............,.... .........._/ - Pm- -, I I t I \ J \ I•.... _/ ,_/ . e Electron ~ (~~ __ ~_~/~~ -- -- --- --~._- -- --::- --------~----- -- -- -::-:: --~._- --;::: -- ',_ ..../ "_/'(a) (b) (a) (b) Fig. 1.24 (a) Orbital and (li) spin magnctic momcnts of elcctron. Fig. 1.25 (11) Fcrrornagnctic and (li) anlifcrromagnctic alignrnent of atorns togcther with thc magnctic ficlds that each would produce. gested by Fig. 1.24. Each electron produces a magnetic morncnt that is an integral multiple of the Bohr magneton, whose value is P130hr = 9.27 X 10-24 arnpere-meter" (1.104) While an application of the torque expression of eq. 1.102 might lead us to cxpect the two types of alignment shown in Fig. J .25, the magnetic alignment torques between adjacent magnetic moments are found to be much smaller than the torques that actually occur. The type of alignment is detcrmined by lhe exchange forces which operare only over short distances and are electrostatic rather than magnetic in nature. It is only in a srnall group of the pure elements-iron, cobalt, and nickel-that the alignment torques cause ferromagnetism at normal temperatures. At somewhat below normal ternperatures, some other elements, like gadolinium and dysprosiurn, are also ferromagnetic. Fortunately, this does not restrict ferromagnetic behavior to only these few materiaIs. The exchange forces are critically dependent on the spacing between those electrons producing the magnetic moment in adjacent atoms. Antiferromagnetism occurs when this spacing is toa small. The spacing betwcen the atorns of an antiferromagnetic material may be increased by alloying it with a nonmagnetic material, for exarnple, rnan- ganese with copper or tino With the increased spacing, ferromagnetic alignment can occur. Alternatively, two groups of atoms or molecular Because of the symrnetry of the electron arrangcment, the net magnetic moment of most atorns is zero. It is only in atorns having incompleto inner shells of electrons that the atom has a significant magnetic moment. This net moment arises principally frorn the spins of electrons that are not paired with electrons of opposite spin direction. Exarnination of a periodic table of the clernents shows that an in- complete inner shell, and therefore a net magnetic mornent, may be present in any element within the ranges of atornic nurnbers 21 to 28, 39 to 45, 57 to 78 and 89 and above. Thc most important range is the first, which inc1udes vanadium, chrorniurn, manganese, iron, cobalt, and nickel. When visualizing the magnetic effects within materials, each atom may be regarded as equivalent to a small constant-current loop. The orientation of this loop may be altered by changing only lhe dircctions of the spin axes of the electrons. '. 42 ELECTRIC ENERGY CONVERSION PROCESSES PROPERTIES OF MAGNETIC MATERIALS 43 J)l groups having differcnt magnetic morncnts may bc arranged in anti- alignment, but, because 01' the differcncc in thc rnoments, lhe rcsult- can be magnetic. This structure is called ferrimagnetic, and is characteristic of the group of magnetic materiais known as ferrites. Suppose a material has the magnetic mornents of its atoms aligned as illustrated in Fig. 1.25a. To determine thc net magnetic cffcct of such an alignment, let us consider each atorn as equivalent to a current i circulating around the square perimeter of the volume (j3 occupied by the To appreciate the magnitude of the field produced by this internal alignment of atornic morncnts, consider iron ; on the average, it has a magnetic mornent of,2.21 Bohr magnetons (eq. 1.104) per atom, and an atornic density 01' about 0.85 X 1029 atoms/rneter''. Thus if aU the atoms have their magnetic moments aligned, vII = np",n = 0.85 X 1029 X 2.21 X 9.27 X 10-24 1.74 X 106 amperes/meter (1.108) / Path j I ) ! ~ r f ( ( ( ~ ( I) ( ( ( / ( / //'m~--'-g:r~- -, r~ -, " '--l-+--', OU", ,[g J :. .... '.'. .AI ~ j ) ~, ( ,~', d 1/ i~ Fig. 1.26 Modcling the magnelic morncnts of atoms as circulaling currcnts. Te T Fig. 1.27 Thc rclaiion bctwcen the magnelic momcnt rer unit volume J( and the absolute tempcrature T (OK). atom, as shown in Fig. 1.26. The current i is then relatedto the magnetic moment of the atorn Pma by Pma = id2 ampere-rneter" (1.105) Because the magnetic moment jl per unit volume is equivalent to a field intensity, the magnetic flux density which the alignment can produce in the material isIf we consider a path along the direction of alignment, we see that the currents i form an equivalent solenoid around this path and produce a magnetic field intensity Hm of n =-'- m d B = /10'.4 = 47TX 10-7 X 1.74 X 106 = 2.18 webers/ meter" (1.109) = Pma. d3 Since da is the volume occupied by one atorn, the number /1 of atorns per meter" is l/d3• The internally produced magnetic field intensity Hm is norrnally designated by the symbol jl, and is equal to the net aligned magnetic moment per unit volume. amperesjrneter (1.106) For cobalt and nickel, the numbers of Bohr magnetons per atom are 1.72 and 0.6, respectively. Their values of magnetic moment vii per unit volume are correspondingly lower than that for iron. The exchange forces that tend to hold the atomic magnetic moments in alignment are always in competition with the thermal forces in the material, which tend to rnake the electron axes=-and therefore the mag- netic moments-point in random directions. As a result the net magnetic moment pcr unit volume of a magnetic material decreases as the tem- perature of the material is increased, as shown in Fig. 1.27. At a tem- perature Te' known as the Curie temperature, the randomization of jl=Hm = npma ,ampere-meter2fmeter or ampercs/meter (1.107) 48 ELECTRIC ENERGY CONVERSION PROCESSES ~ ~ ~ ~~--?~~t ',~ =r:« --'? -;>- /í tt" --....,...-?o- ~ / t t t/>--.;;~~--:=-<,:~ t t //~ ~ ~ ~ "" + Easy direction01magnetization Fig. 1.33 A crystal with closure domains. in Fig. 1.33 is similar to that of Fig. 1.30b, except that there are two .smaller domains at the ends of the crystal that are aligned in directions perpendicular to those of the main domains. The magnetic flux doses on itself through these two closure domains and is entirely confined to the crystal. Although this arrangement has somewhat greater domain wall area than tliat of Fig. 1.30b, its magnetostatic energy is zero. We might expect that any crystal having rnutually perpendicular axes of easy 'magnetization, regardless of its size, would have the sim pie four-domain arrangement of Fig. 1.33, since this arrangement has a smaller walI area than would obtain with a larger nurnber of domains. There is, however, one additional property that generally tends to limit the size of domains. Most magnetic materiais are likely to change their dimensions along the direction of the magnetization. An iron crystal usually expands along its magnetization direction, while nickel and cobalt contract. The long horizontally directed domains in Fig. 1.33 are free to expand or contract along most of their length. But the vertically directed closure domains are constrained frorn motion in the direction of magnetization. If they expand, they will be in compression; if they con- tract, they will be in tension. These mechanical strains in the c1osure- domain region involve stored mechanical energy throughout this region. The overall energy stored in the crystal is therefore often lower in a domain arrangement such as the one shown in Fig, 1.34. ln this figure, the volume of the strained closure domains at the ends of the crystal is reduced below that of Fig. 1.33 at the expense of a greater total domain wall area. From the foregoing discussion, a mental picture of a normal ferro- magnetic crystal emerges. Most of the magnetic moments of the crystal f>..~--~--~ -~-_=--?_--/,} /. ,~ +- +- +- '~ t>-~+-v!!.~ --~--~---/t v ~ ~ _ ~ " " ~.At ~ - ~ /1 +Easy directiont>-----------------<:.} 01. . / +- +- +- +- '. magnetization Fig. 1.34 Effect of strain energy on the number of dornains in a crystal. .I :[ PROPERTlES OF MAGNETlC MATERJALS 49 are aligned with their nearest neighbors along one of the easy directions of magnetization, the exceptions being the small number in the region 01' domain walls. For materiais with mutually perpendicular easy directions of magnetization, there are generally closure domains providing closed paths for magnetic flux which are collinear with the magnetic moment vector di within the crystal. Application of the circuital law of eq. 1.60 around a closed path of fíux, as in Figs. 1.33 or 1.34, results in the con .. c1usion that ali of the crystal, except a small portion in the region of- domain walls, has a large and constant flux density B, equal to B, = flo.ÁI webersjmeter? (1.111) This is generally called the saturation flux density. lts value is approxi .. mately 2.2 webersjrneter- for iron, 1.8 for cobalt, and 0.64 for nickel. 1.6.4 Domain Wall Motion and Coercive Force in Crystals The discussion of Section 1.6.3 shows that the normal state 01' most ferromagnetic crystals is characterizcd by internally closed flux paths, ali parts of which are at the saturation flux density of the material. Most of the uses of such magnetic materiais depend on our ability to change their magnetic state by application of external forces. If we consider the relatively high values of saturation flux density in these materiais, it is evident that a very high external magnetic field intensity H would have to be applied to produce any significant change in this flux density. Thus control of the magnetic state is affected, not by changing the magnitude of the flux density within the dornains of a crystal, but rather by shifting the domain walls so that the orientation of this saturation flux density is changéd, Figure 1.35 shows a single ferromagnetic crystal, which has a window cut in its center. The sides of the window are parallel to the easy directions of magnetization. Domain walls are shown between regions that have different directions of the magnetic moment per unit volume di. In the steady state shown in Fig. 1.35, there are two oppositeJy directcd streams of magnetic flux within the crystal. Each of these streams closes on itself. The net magnetic flux cp through a cross section ofthe crystal in a c\ockwise direction is .~ cp = Bs[zx - z(d - x)] = Bsz(2x - d) webers (1.112) :1 ~I I I :1 where B, is the saturation flux density of the material. Let us now enc\ose the crystal in a uniformly distributed coil of N turns carrying a current i. This coil produces a clockwise-directed magnetic field intensity H within the crystal. In the region of the main 1800 dornain wall, the magnetic ) ) ) ) 50 ELECTRIC ENERGY CONVERSION PROCESSES PROPERTIES OF MAGNETIC MATERIALS 51 field intensity can be assumed to be uniform over the whole wall length at the value H = Ni 2(1 + 2b + 8:c If this magnetic field intensity is sufficient to cause the doinain wall in Fig. 1.35 to move outward, the magnetic flux cp of eq. 1.112 will be in- creased. Let us therefore examine the effect of this externally applied field umpcrcs/meter (1.113) fi magnetic mornents, cach of value ]7"" through 1800 is, using eq. IV = nI" TdO = 2nPmfloH joulesjmeter" From eqs. 1.108 and 1.111, this energy may ais o be expressed as w = 2J1floH 1.114, (1.115) on the wall. = 2BsH joules/meter" (1.116) I {;, ) .~ J ")'ti) ~ The energy of eq. 1.116 is used in moving the domain wall past micro- scopic impurities, irregularities, or strain regions in the crystal. It ali Domain walls H t Pm j !!li !JJI/II/II"~\\ 1111 Pm t b _~l Wall motion " ~ ~'~"-+-dJ~ (a) x H Fig. 1.35 Control of magnctization in a sing1c-crystal core. Figure 1.36a shows thc magnctic momcnt vectors i: of atoms in the region of the domain wall along lhe left-hand side of lhe crystal. In an applied magnetic field of intensity Fi, the torque 1acting on a magnetic moment is given, from eqs. 1.54 and 1.55, by ,/ /' ,/ /' /' /' /' -: Pm 1= PIII X Fi newton-Illeters (1.114) (b) \ This torque is zero on the magnctic morncnts on lhe right of Fig. ) .36a because these are already aligned with Fi; it is also zero on those on the left, which are directed 1800 away from the vector fi. It is only in the region of the domain wall that the torque acts, the direction being that shown in Fig. 1.36b. This torque causes the domain wall to move to the left in Fig. 1.36a in the samc way a twist can move along a paper strip. Suppose a domain wall moves across a unit volume of material causing ali the magnetic moments to turn from antiparallel to parallel with the applied field intensity Fi. The work done per unit volume in turning the Fig. 1.36 Wal\ motion due to applied magnetic field of intensity H. ( J ) f / \ eventualJy appears as heat within the crystal, and is called hysteresis loss. The value of the magnetic field intensity H, required to move the domain wall across a material is known as the coercive force. ln a very pure iron crystal this coercive force is of thc order of 1-10 amperesjmeter. Let us now return to our discussion of the control of the magnetic state in the crystal of Fig. 1.35. When the current i in .the coil is raised to a value that is sufficient to create a magnetic field intensity H in eq. 1.113 52 ELECTRIC ENERGY CONVERSION PROCESSES equal to the coercive force H, at the dornain wall, the wall moves outward, increasing x and increasing the net flux <p of eg. 1.112. The flux !inkage ít of the coi!, given by ít = N<p webers (1.117) is plotted as a function of the coil current i in Fig. 1.37. Before the current .i is applied, the state of the core in Fig. 1.35 can be represented by the point O in Fig. 1.37. As the current i is increased, the flux linkage À remains i, amp 3 8 9 Fig. 1.37 Flux Iinkage .l.versus current i characteristic for lhe single-crysral core of Fig.1.35. essentially constant until the point 1, where coercive force H; is produced (eq. 1.113) for the position x of the dornain wall shown in Fig. 1.35. Although this process starts the outward displacernent of the dornain wall, an increase in current i is required to produce coercive force for larger values of x in order to keep the wall moving. At point 2, the coil current is ;2 = (2a + 2b + 4d)Hc N amperes . (1.118) and the wall has moved to midpositíon, resulting in zero flux linkage. At point 3, the wall has moved to the outside of the crystal, that is, x = d, and further increase of the coil current produces an increase in flux linkage of only the small amount which the magnetic field intensity }{ would create in free space, As the current is decreased to zero, the domain wall position rernains fixed and the coil flux linkage remains essentially constant along the PROPERTIES OF MAGNETIC MATERIALS 53 locus 3-4. Application of a nega tive current i of a value indicated by the state point 5 causes a new set of domain walls to start from the inner periphery of the crystal. These domain walls move outward as the negative value of the current i is increased. Ir this current is removed at a point 6 in Fig. 1.37 before it rises to a magnitude sufficient to move the wall across the crystal, the wall is left in some interrnediate position, and the flux Iinkage À of the coil remains constant at point 7. Ir the coil current is cycled, first in a positive direction then in a negative direction so that the À-i characteristic of Fig. 1.37 moves once around the path 9-3-5-8, the wall will have swept twice across the complete volume V of the crystal. The energy W dissipated in this operation is found by inserting the value of coercive force H; in eq. 1.116 to give rtíoss = 2 Vw joules (1.119) By substituting from eqs. 1.118, 1.112, and 1.117 into 1.119, it can be shown that this dissipated energy can be expressed as ltíoss = 4(NBsZd>[:C (2a + 2b + 4d)] = 4À4i2 joules (1.120) From Fig. 1.37, it can be seen that this energy is equal to the area enc\osed by the closcd path 9-3-5-8 followcd by À and i during this operation. This conc1usion may also be reached by determining the nct e\ectrical energy input to the coil, exclusive of its resistance loss, during one cycle of ít and i, and assuming that ali this encrgy is 105t within the material. Then the energy loss for one cycle of operation following the locus 4-5-8-9-3-4 is Wloss =J i dít cotu p le t e cycle' = 4ít4i2 joules For a material having a saturation flux density of 2 webers/rneter" and a coercive force of 10 amperesjrneter, this hysteresis cnergy loss would be 80 joulesjrneter" for one cycle. The slope of the sides of the flux-linkage-current characteristic of Fig. 1.37 is determined by the shape of the crystal core in Fig. 1.35. If thc width d of the material is made small in comparison with the length of the inner periphery, this slope is correspondingly large. Although lhe discussion in this section has been limited to the rather impractical case of a single crystal, the properties described are the basis for understanding the behavior of practical polycrystalline magnetic materiaIs. J ) ) ) ~ -1 ( t ) -. ) ) ) ) ) ) ) ) ) ) ) ) PROPEUTIES 011 MAGNETIC MATElHAI.S 59 58 ELECTIUC ENERGY CONVlmSION I'IWCESSES as great. The residual Ilux dcnsity is of lhe same order of magnitude as soft metallic magnelic matcrials. Perrnanent-magnet materials may also be made of ferrites such as barium ferrite. While the residual flux density of ferrite magnets is only about 0.2 to 0.4 weberjmeter-, exceptionally high coercive forces of the order of 200,000 amperesjmeter can be obtained. When residual flux density has been established in such a material, an energy of the order of 60,000 joulesjrneter" of material is required to remove it. Magnets of these materials are therefore particularly stable. 1.6.7 M agnetostrictíon Magnetostriction is the elastic deformation of a magnetic material when its magnetic state is changed. Some rnaterials, like nickel and cobalt, 1.6.6 Permancnt-Magnct Materials The materials we have discussed are those whose magnetic state can be easily controlled by an externaliy applied magnetic field intensity. Per- manent-magnet materiaIs are those which can produce a magnetic field outside the material and maintain it in spite of a large externaJly applied field intensity. These materiaIs are therefore characterized by very high values of coercive force. Section 1.6.3 discussed the fact that vcry small crystals would not have domain walls and, therefore, Ilux closurc, within themsclvcs. It was also B, Wbjm2 ~xl06 I 10 Fe Co ____ ~L----+_------t-~J"'v.!-',v:::v=-v"----~N, ampjm 60,000 H, ampjm I ~ -10 -20 ,-- Fig. 1.41 Flux density B vcrsus magnetic ficld intensity H for pcrmancnt-magnct material (Alnico, V). -30 ,- -40 ---------Ni noted that mechanical strain has a marked effect on the preferred direction of magnetic orientation. Most permanent-magnet materials achieve high coercive force by a combination of the following features: (I) high crystal anísotropy energy, as in cobalt; (2) very small crystals, often in the form of long slivers to increase the directional effect; and (3) high internal strain. The metallic permanent magnets are genera][y alloys of several of the elements Fe, Ni, Co, Cu, Cr, AI, V, and Mo. The srnall particles and the high strain are produced by heating and quenching the material, and also by cold rolling for some materiaIs. It is possible to produce some preferred orientation by cooling the materiais from the Curie temperature in a strong magnetic fieJd or by directional cooling. A typical B-H characteristic for a common metallic permanent magnet (Alnico V: 51 % Fe; 24% Co, 14% Ni, 8% AI, 3% Cu) is shown in Fig. 1.41. Although the general shape of this characteristic is similar to that of soft magnetic materiaIs, the coercive force is of the order of 1000 times Fig. J .42 Magnetostriction in iron, cobalt, and nickel as a function of magnetic field intensity H. contract in the direction of magnetization, while others expando If a ferromagnetic bar is subjected to an applied magnetic field (Fig. 1.5), the per unit change in dimension along the field direction is as shown in . Fig. 1.42, for iron, cobalt, and nickel. Note that large values of magnetic field intensity H are required to produce the largest magnetostriction effect. The reason for this is that most of the change in dimension results from domain rotation near the saturation condition rather than from domain wall motion; the eífect of wall motion is usually to reverse the magnetic moments, leaving them similarly oriented with respect to the length axis of the material. 60 . ELECTRIC ENERGY CONVERSION PROCESSES i' .j~ The magnetostrictive effect is also reversible. 1f tension is applied to a material such as nickel, which normally contracts under magnetization, a much larger magnetic field intensity is required to produce a given flux density in the material, as shown in Fig. 1.43. Since magnetostriction is an elastic effect, the additional energy that must be applied to magnetize the material is recoverable when the magnetic field is removed. Magnetostriction is important in many a-c devices because of the noise which is produced by the small but rapid changes in the dimensions , I, I~ B, Wb/m2 l i ~. ,I 'j' ~lilili 1:I, ri ~ ~, 0.5 o -~~---~~---~:;;;--------;15SroiOoH. arnp/rn500 1000 Fig. 1.43 Influence of tension on magnctic flux density B versus magnetic field intensity H for nickel. of the device. The dimensional change shown in Fig. 1.42 is unchanged if the direction of the magnetic field is reverseel. With an alternating field, the same dimensional change occurs during each half cyc1e. Thus the basic frequency of the noise produced by magnetostriction is normal1y twice that of the frequency of the electrical supply, The applications of magnetostrictive materiais as electromechanical energy converters are similar to those of piezoelectric materiais, because both combine small motion anel large force. The major uses of these material are in the production of ultrasonic waves, in electrical filters, and as strain gauges. 1.7 USE OF MAGNETIC MATERIALS TO PRODUCE FIELDS AND FORCES In the preceding section we saw that the magnetic flux in a closed path of soft magnetic material can be controlled by the application of a rel- atively small external magnetic field. While closed magnetic paths are possible in transformers and other stationary magnetic devices, energy USE OF MAGNETIC MATERIALS 61 conversion elevices normally require an air gap between the stationary anel moving parts of the system. One of the major uses of magnetic materiais is to produce intense concentrated magnetic fields in air gaps. In this section we eliscuss the magnetization properties of a magnetic path with an air gap. We also examine anel determine the forces that are exerted on the magnetic materiais on each side of an air gap. 1.7.1 Magnetic Field in an Air Gap Figure 1.44 shows a core of soft magnetic material which has an air gap of length x anel area yz. The core is encirc1ed by a concentrated coil of N turns. Typical magnetization characteristics of the core material are =>» í/Ássigned direction 't of positive flux <p Fig. J.44 Magnetic system with air gap. shown in Fig. 1.45a. As the air gap length x is small in comparison to the dimensions y and z, let us assume that the magnetic flux <p is confined to the core and to the air gap volume. The average ftux density B in the core and in the air gap is therefore given by B = 1. webers/rneter! (1.123)yz To support a magnetic f1ux density B in the air gap, a magnetic field intensity Ha is required, where B Ha =- fio amperes/meter (1.124) The same flux density can be produced in the magnetic material with the magnetic field intensity H", given in the B-H characteristic of Fig. 1.45a. According to the circuital law of eq. 1.60, the summation of the magnetic field intensity around the closed path of the magnetic flux equals the current encircling this path. Thus we have Ni = Hn/ + Hax amperes (1.125) / " I j J, --1 ) I 1 1 ) ) ) ) ) ) i ( I ) i ? ! ) 1 ( ( ( ? ;.I ( ( ? I ) i 62 ELECTIUC ENERGY CONVERSION PHOCESSES For a particular value of flux density B, the values of H; and fI", (found from eq. 1.124 and Fig. l.45a) may be inserted into eq. 1.125 to find the required coil current. Fig. 1.45b shows the relatiol1ship between the magnetic flux rp and the coil current i. For most soft magnetic materiais, the magnctic ficld intensity fI", in lhe core is several orders of magnitude Icss than lfa" For example, for B = 1.0 weber/meter", a field intensity of H" = 800,000 ampcresjmetcr is requircd in the air, while in an iron core the field intensity H", may be B <1>, Wb n, i, ampHm (a) (b) Fig. 1.45 (a) Flux density B versus field intensity H for the core material. (b) Rclation betwccn flux é and coil current i for core of Fig. 1.44. of the order of 20 amperesjmeter. Unless the gap length x in Fig. 1.44 is very srnall, the term HlI,l in eq. 1.125 may often be neglected in analysis, provided the flux dcnsity in the core is we\l below its saturation value. Thus by combining eqs. 1.123, 1.124, and 1.125, the air gap Ilux can be approximated by ~= yzP,oN . I x webers (1.126) The effect of the magnetic material is therefore to concentrate the magneto- motive force Ni of the coil on the reluctance of the air gap. An alternative approach to the analysis of this system follows from a consideration of the energy input to the coil to produce a magnetic field in the air gap. The magnetic energy stored in the air gap at a tlux density Bis 82W = - Xl;Z a 2P,o' The energy required to change the flux density in the core from zero to a joules (1.127) I' USE OF MAGNETIC MATEHIALS value B is 63 J 13 W'" = /yz o I-I", dB For B = 1 weber/rncter-, the energy density in the gap wiII be about 400,000 joules/mctcr', while thc energy density required by an iron core may be only about 20 joulcsjmeter". Thus W", may often be negleeted, and the whole of the electric energy input to the coi! may be assumed to be stored in the concentrated magnetic field in the air gap. The means by which the magnetic material concentrates the effect of the coil to produce an air gap field may be appreciated by considering the behavior of the microscopic streams of magnetic flux that exist in the dornains of the material. When the air gap flux density is zero, essentially alI these streams of flux may be considered to dose on themselves within the material. Figure 1.46a suggests, in very simplified form, the condition joules (1.128) ~-------J-----~ ~ ~C -------------- i ~------_!_----; ~ ----~-------- ~-----:--!_---- • ~--))E--------:-----? -----~-------D --------~----) ~-----~------D~ ~ D------,,------) \ ~ DcmainbOundary:J-'--- (a) -------;---- ) c=-==---=?- (------~ ê ~ E _.:. ) --- - c-=~=-..::: s;~--~--g:1E1----é.~-~----- ------ __ .-:--) C ------::- (b) Fig. 1.46 Sirnplified model of the magnetic dornains in the region of the air gap of Fig. 1.44. (a) With no air-gap flux, (b) Shifted domain boundaries, oriented domain strcams terminating on air-gap surface, and air-gap flux. 68 ELECTRIC ENERGY CONVERSION PROCESSES ÀÀ Locus during displacement Locus during À2~---~i~~~~) 11 I I I I I I I I I I I I Àl c i[ oo (a) (b) Locus during displ'acement 2 ÀÀ 2 oo (d)(c) Fig. 1.48 Deterrnination of force in a nonlinear magnetic system. (li) Displaccment with constant current, (b) Displacement with constant flux linkage. (c) General A-i locus during displacement. (li) lnfinitesimal displacement. 1.130 must be stated as !Y..We1ec. = !Y..W10ss + !Y..~Vstored + !Y..Wmrch. input ou t.pu t (1.143) The electrical energy input to the system during the displacement !Y..x is !Y..Wclcc. =f).,;[ dÀ = Arca (A + B) . ínput ).1 in Fig. 1.48a ( 1.144) The loss energy is assumed to be zero because hysteresis has been neglected, and the resistance and friction losses have been moved to the external electric and mechanical systems. The increase in stored energy is equal to the stored energy W2 at state 2, minus the stored energy W[ at state I. The energy W[ is equal to the electric energy input to the system to increase the current from zero to ;1 at a constant gap spacing Xo· Thus J, l' USE OF MAGNETIC MATERIALS 69 we have J )., W1 = o ; dI, = Arca (C + D) in Fig. 1.48a (1.145) The stored energy W2 is equal to the energy input to increase the current from zero to i1 ut gap spacing x() - !Y..x along the ?-i locus 0-2. Thus, we have (À" W2 =Jo ; dI. = Area (A + C) The increase in stored energy during the elisplacement is (1.146)in Fig. 1.48a L'lW.torcll = W2 - 11"1 = Area (A - D) Sub:tituting these energy coniponents into eq. 1.143 gives L'lW",cch. = Arca (A + B) - Arca (A - D) OlltPllt (1.147) = Area(B + D) (1.148)joules Note that the mechanical output energy is represented in Fig. 1.48a by the area bounded by the locus 0-1-2-0. An alternative method of determining the mechanical output energy of eq. 1.148 involves cornparison of two ways of arriving at state 2 in Fig. 1.48a. Suppose the gap length is set at x = Xo while the current is increased from zero to ij, and is then reduced to x = Xo - !Y..x with i = i . For this 5equence, the total electric input energy is the area (A + B + C + D) to lhe left of thc locus 0-1-2 in Fig. 1.48a. Now considcr a second sequence of operations in which the gap is initially reduced to x = Xo - Is» with the current equal to zero, anel the current is then increased to ;1' The electric input for this sequence is the area (A + C) in Fig. 1.48a to the left of the 0-2 locus. It is assumed that neithcr sequence involves any loss of energy. The mechanical energyin the second sequence is zero, because the displacement is made at zero current. Both sequences arrive at the same final magnetic state for the system. Because the electric input energy for the first sequence exceeds that for the second sequence by an arnount represented by the area (B + D), it is concluded that this must be the mechanical output energy. Let us now consider a rcduction of gap length from x = Xo to x = Xo - L'lx, which is made 50 rapidly that the flux linkage ). of the coil remains substantially constant during the displacement. While the flux linkagc may not change appreciably, its rate of change is largc. The resultant induccd voltage e causes the current ; to be reduced during the displacement, as shown in the 10CllS 1-2 of Fig. 1.48b. After the displace- ment is completcd, the coil current may rise along the locus 2-3 toward its . ! ), )1 )! I J! ) ~ I -l I ) ~ i ~ ; i • i -( i I. I ~ ( ( 1 ( ( ( ( ( ( ( ( ) 1 70 ELECTRIC ENERGY CONVERSION PROCESSES 71USE OF MAGNETlC MATERIALS original value i.. all of the input energy going into storage in the magnetic field. As the flux linkage },remains constant at )'1 during the displacement, the electric input energy associated with thc locus 1-2 is equal to zero. The losses in the system are assumed to be zero. Thus (frorn eq. 1.143) the mechanical output energy must be obtained from the stored energy of the The expression of eq. 1.150 is convenient for analytical determination of the force when the current i is expressed as a function of the flux linkage À and the distance x. . An alternative expression for force can be derived by considering the current to be constant during the infinitesirnal displacement. From eq. 1.148 and Fig. 1.48a, the force is equal to the rate of increase with displacement of the area under the À-i curve. Let W' =IÀdi system Ê!. P/mech. = - Ê!. W,torl"l out.puf • fÁ' ]_ _ i dJ,. - i c/À - • Il o a.loll~ :.Llong . IOCIIK tocu s 0'- 2 0-1 Note that the mechanical output energy is represented in Fig. 1.48b by the area bounded by the locus 0-1-2-0. Figure 1.48c shows a general locus 1-2 of the Ilux linkage-cllrrent relation, which might be followed during a displaccment 6x. The shape of this locus dcpcnds on thc dynamic charactcristics ar both the elcctrical sourcc system anel the external mcchunical systet11. Thc rncchunical output energy is cqual to the clectric input energy required to arrivc at state 2 along the locus 0-1-2, minus the electrical input required to arrive at the sarne state 2 along the locus 0-2. This mechanical output energy is represented in Fig. 1.48c as thc area bounded by the static },-i relations for the initial and fina! states and the locus fol1owed in transition from the initial to the fina! state. The average force during the displacement can be determined by dividing the mechanical output encrgy in eqs. 1.148 or 1.149 by Ê!.x. To Iind lhe force at a particular valuc of spacing ;1:, thc displaccment 6x may bc contractcd to an infinitesimal displaccmcnt (h. Figure 1.48c/ shows that as the locus 0-2 for x = :00 - do becomes coincidcnt with locus O-I for x = xo, the shapc of lhe locus 1-2 ccases to be significant in dclermining the arca boundcd by lhe locus 0-1-2-0. The force com- ponent Ix acting in the positive x dircction is equal to lhe dilferential mechanical energy (represented by the area within locus 0-1-2-0) divided by the differential displacement dx. If thc flux linkage is considered constant during the infinitesimal displacement, a general expression for the force cornponent j', can be deduced from eq. 1.149. (1.149) joules (1.152) where W' is generally called the coenergy. Then dIx = - Wmcch. d.x 0111,,111 =~w'l():~; i". co Jl:-l I. • (1.153) This expression is convenient when the flux linkage À is expressed as a function of the current i and the distance x. The expressions for force in eqs. 1.148, Ll49, 1.150, and 1.153 in general do not apply ir losses occur in the magnetic material as a result of a displacement. Suppose the material has significant hysteresis. The flux linkage-currcnt relation for a given spacing is then no Ionger single valued but depends on the previous magnetic history of the material. The methods for determining force developed earlier in this section may still be applied in some special circumstances. Suppose the magnetic mate- rial of Fig. 1.47 has a B-H relation of the form shown in Fig. L45a. If the air gap is held constant at x = xo, while the coil current is increased from O to i1, the },-i relation for the coil follows a locus such as 0-1 in Fig. L49. The energy input during the operation is represented by the are a (C + D). Part of this input energy is stored and part is hysteresis loss. Now let us suppose the air gap length is decreased to Xo - Ê!.x, while the coil current is held constant. The energy input during this operation is represented by the arca (A + B). Part of this energy is stored, part is lost as hysteresis, and part is rnechanical output energy. Because the initial magnetization at x = Xo along locus O-I incrcased the flux density in the material, and the displacement increased the flux density still further, the net effect on the material should be essentially the same as would be obtained if the current were increased from Oto i1 as spacing Xo - Ê!.x. This would not be true if the flux density decreased as a result of the displacement. In that case, the material would be operating along some minor hystercsis loop. The effect also would not bc the sarne ir the distribution of the flux in the dIx = - Wrncch. dx nutput o I- - WSlorc(\ 0;1: i.=colIsI:t1l1 newtons (USO) where 1)·W~lorcd = i c/ J,.) ( 1.151)joules 72 ELECTRIC ENERGY CONVERSION PROCESSES material weré significantly different at the two values of gap spacing. Let us assume that the magnetization characteristic for spacing Xo - t..x with increasing 'current is the locus 0-2 in Fig. 1.49. The energy input to 'reach state 2 with spacing Xo - t..x is represented by the area (A + C), part being stored and part being hysteresis loss. The total input energy required to reach state 2 along the locus 0-1-2 is represented by the area (C + D + A + B). Assuming that the energy stored in state 2 is the same for either sequence of operations, and assuming that the sarne hysteresis loss occurs for either sequence of operations, the mechanical x 2 c I I I I 1 I 1 I I I, x = xo - t.x o 'l t Fig. 1.49 Determination of mechanical energy in a system with magnetic hysteresis. output energy is represented by thc diffcrence arca (B + D), that is, the area enclosed by the locus 0-1-2-0. Analysis is more complicated for displacements that cause a decrease in the f1ux linkage. The average force during an increase in gap length is not equal to that obtained for an equivalent decrease in gap length when the material has hysteresis. The rnechanical energy input required to increase the gap length must exceed the mechanical energy output obtained during the decrease of the gap length by the total hysteresis loss involved in the operation. Another situation in which the force can be found in spite of the presence of hysteresis loss occurs when the displaccment is made at essentiaIly constant f1ux. If we suppose that the f1ux density does not change at any point in the material as a result of the displacement, it foIlows thatthe hysteresis loss is zero and that the energy stored in the material is unchanged, For this condition, the mechanicaloutput energy must be obtained from a reduction in the energy stored in the air gap. USE OF MAGNETIC MATEIUALS 73 Equation 1.150 may therefore be used, the energy storeel being equal to the gap energy plus a constant stored energy in the material. While the hysteresis loss may be zero for the situation just elescribeel, the eddy current Iosses in the material may be significant. These losses depend on the rate of change of the flux in the material anel on the resis- tivity of the material (see Section 2.2). Because of eddy current losses, the force produced by a magnetic actuator at a given gap length x and current i is not lhe sarne if x and i are varying rapidly with time as it is for the sarne set of vaIues of constant length x and current i. ., 1.7.3 Physical Origin of Force 00 Magnetic MateriaIs The energy conservation approach ofSection 1.7.2leads to a number of very useful expressions by which the force in a magnetic system may be deterrnined. But, by its nature, this approach is incapable of explaining the physical origin of the force. A similar situation was met in the analysis t ~ --- a"i 8Prní ---------------'\;- Pm2f~-- ,,~-- -------r--ccc----;;,.,. ~;--- =f ~- x '---------?- Fig. 1.50 Force between two aligned magnetic moments. of electrostatic systems (section 1.3.3), where it was found that the most useful expressions arose from the energy conservation approach, while Coulornb's law was most useful in obtaining a physical understanding. Therefore, it woulel be helpfuI if we could develop an approach to forces between magnetic materiaIs which would be analogous to the use of Coulornb's law in assessing forces between charged coneluctors. The significant properties of ferromagnetic materiaIs arise from the magnetic moments of the inelividual atoms. Thus, it should be possible to determine the force between two pieces of ferromagnetic material as the summation of the forces produceel by the magnetic moments of one on the magnetic moments of the other.. Let us first determine the force between two small circular coils carrying currents t, and i2 that are aligned as shown in Fig. 1.50. From Section 1.5.3 we know that the force between the coils may be found from the expression ix = i1i2 dL12 dx newtons (1.154) ) ) ) ) j : i l 1 1 ~ i ~r ,{ ( .) ) ) { ~ I 1 ) 1 1 ( t ( ) I ( ( ( ( ( ( ( ( ) I I) 78 ELECTRIC ENERGY CONVERSION PROCESSES magnitude and radial\y directed. There are, however, fringing fie1ds beyond the regions of ovcrlap, as ShOWIl in Fig. 1.53b. As thc air path lengths become longer, the average magnetic field intensity in these fringing areas becomes proportionately smaller. As was already discussed in Section J .7.3, the magnetic f1ux cntcrs the air gap from the ends of elementary solcnoids of the magnetic mornents of the fcrromagnctic atorns. Thcsc solcnoid cnds Illay be considcrcd as a elislribulcd ílux source over, lhe rcgion where lhe Ilux emerges Irem the material. Where thc ílux rccntcrs thc material thcrc is a distributcd f1ux sink. The elensity of thc f1ux source on a surfucc is numcrically equal to the magnetic flux density B out of the surface, Let us now use eq. 1. J 62 in a qualitative manner to examine the forces and torques in this system. There will be raelially directeel forces of attraction bctween the uniforrnly distributcd flux sourccs on one sielc of the gap, and tbe flux sinks on the other side in the regions of overlap. Since the motion of the rotor has bccn restricted to rotation about its axis, these radial forces are not useful in producing mechanical output. Thc useful forces in this system arise Irom ihe attraction bctwccn flux sources and sinks bounding the fringing portion of thc air gap Iicld. These forces are in such a direction as to rotate the rotor increasing the angle of overlap O between rotor anel stator. To afirst approximation, the pattem and the intensity of the field in the fringing arcas are independent of the position O of the rotor if lhe coil current is hcld constant. lncreasing the angle O increascs lhe f1ux in the overlap region, but leaves the total fringing flux substantially constant. This ceases to be true as the angle of ovcrlap approachcs its maximum value. We could therefore cxpcct thal with constanl coil current i, the rotor torque teneling to increase the ovcrlap angle () would be independent of () except in the range where () approaches its maxirnum valuc, In the analysis of a machine such as lhe one illustrated in Fig. 1.53a the physical concept of flux sources anel sinks indicates the naturc of the forces, the region ôf their action, and some 01' their relationships. But as we have already noted, this approach is scldom useful in numerical ca1culation. For this type of calculation we revert to the energy conscr- vation approach of Section 1.7.2. If the ferromagnetic material is assumed to be lossless and to require negligible magnetic field intensity, there will be a linear relation between the magnetic flux in the magnctic structure and the coil current. For a linear system, the torque may be expressed as a variant of eq. J .84. To=!i2dL2 dO newton-rnetcrs (1.165) I",·,.. USE OF MAGNETIC MATERIALS 79 where L is the inductance ofthe coi!. This inductance may be approximated by assuming a uniform ficld in each of the air gaps of Jength g, as given in eq. 1.164 and considering the total flux 1> to be the sum of this uniformly distributed flux plus a fringing flux 1>1 proportional tothe current i. N(p L=-Thell where , N( N' )= -:- flo -.-!; zrO + 1>1 I 2g = N2 (fl;Z;'O+ LI) LI = N~I I. (1.167) henrys (1.166) henrys Substitution from this inductance expression into eq. 1.165 gives N3IJ,n::.ri~ To = -'-'-- 4g newton-meters (1.168) Note that this expression shows the torque to be independent of the angle O, to be in such a elirection as to increase the overlap angle 0, and to be proportiona! to the square of the current i. Ali these properties are as expected from the previous qualitative analysis. If the coi! current in Fig. 1.53a is held constant, the torque always acts to increasc the angle of overlap O. To obtain continuous rotation with a uscful average iorquc, it is necessary to control the current i or the f1ux 1> in such a way that the torque will be greater while the overlap angle O is increasing than while il is decreasing. This can be done by using an a-c supply iil a manner similar to that discussed in reIation to the electrostatic machine in Section J .3.4. Supposc we apply to the coil terminais a source voltage of e = Ê cos cot volts (1.169) If the coil resistance is negligibJy small, the magnetic flux in the system must change at a rate sufficient to produce an induced voItage equal to e, anel is therefore given by 1 J'1> = N e di Ê . = - sm ou Nw webers (1.170) 80 ELECTRlC ENERGY CONVERSION PROCESSES The torque on the rotor can be expressed as a vari~nt of eq. 1.87. To == _ ~ ~2 ~~ (1.171)newton-meters The variation of the reluctance fJf of the magnetic system with angle () is of the forrn shown in Fig. l.54, where O is the angle of the rotor axis. This reluctance may be approximated by the first two terms of its Fourier series expansion. . fJf = fJfo + fJf2 cos 20 amperesjweber (1.172) .'1l - Actual reluctance ('APprOXimation \ \ I \, 1 \ / \ / / :t \ .óJl2 / l_L l_ \ I \ / \ / - ---·'1l0 ~------~~--------~--------,~-----~~ 2Jr oo Jr Fig. 1.54 Variation of reluctance 8i with angle O for lhe machinc of Fig. 1.53a. If the angular velocity of the rotor is 1) radiansjsecond, and the angular position of the rotor at t = O is O = -15, 0= 1'/ - o radians ( 1.173) Ifwe now substitute from eqs. 1.170, 1.172, anel 1.173 into eq. 1.171, thc instantaneous torque is obtained as Ê2 . 9 d,,, ( ~1T.o = - -- SII1-(I!!- ['Wo + .'ft •• cos 21'/ - 2u)2N2W~ dO - Ê2fJf2 . .= -- [SII1 (21'/ - 20) - ~, sm (21// - 2(1)/ - 215) 2N2w2 - - t sin (2111 + 2w/ - 2ô)] (1.174) All terms in this expression will be of sine Iorm with zero average value unless l' = ±w and o ~ O. Setting the rotational velocity l' equal to the angular frequency w of the supply gives an average torque of _ 1 Ê2fJf.. . T = - --- SIl1 20 4 N2ü/ newton-meters (1.175) PROBLEMS 81 This torque characteristic is of the same form as that developed in eq. 1.44 and shown in Fig. 1.15 for an electrostatic machine. The rnaximum torque occurs when the angle of lag 15 is equal to 7T/4 or 45°. The maximum torque obtainabJe from this type of synchronous reluctance machine is limited by the maxirnum permissible flux ~ and the value of the reiuctance variation fJf2. These quantities are interrelated, because with the maximum value of reluctance r,the overlap between rotor and stator is zero, and ali the flux is in a highly concentrated fringing field nearthe tips ofthe rotor and stator poles. Ifthis leads to saturation in the ferromagnetic material, the maximum value of the reluctance fJf of the system is reduced . . \' Problems 1.1 As an exercise in the application of Coulomb's law, consider the system of four small spheres shown in Fig. P 1.1. The charges on the spheres are: Ql = Q,! = 2 X 10-11 Q2 = Q4 = -IO-H couJomb ooulornb The charge Q4 is free to move along the x axis. The other charges are fixed in position. (a) Find the force on the charge Q4 as a function of the distance x. (b) At what value of x will the force on the charge Q4 be zero? Ans.: 0.81 cm. T lcm t oQ, Q40OQ~ f--- x lcm J_ ~-, x = 0.2 m ._-~o (i:! Fig. PI.! Fig. P1.2 I,. I 1.2 In Section 1.3 three methods are described for determining the e1ectrostatic forces between oppositely charged bodies. To compare these methods, let us apply each of them to the pair 01' long, parallel, cylindrical conductors shown in Fig. P 1.2. The conductors are oppositely charged t ) ) ) J J 1 ~ t i 1 ( ) I ) ) ) I ~ ( ( ,. ( ( I) ( ( ( ,) ( I) ( I) ( I) I) ( 82 ELECTRIC ENERGY CONVERSION PROCESSES and each carries a charge of 1.5 microcoulornbs per meter 01' length. The problcm .is to find the force acting on each meter lcngth 01' con- d uctor. (a) First, let us use Coulomb's law in solving lhe problem. Since lhe conductor radius r is small in comparison with the spacing :l:, the conductor may be considered as two filaments of charge, essentially infinito in length. Write a general expression for the force hetween the charges on increments of length on each of the conductors. Satisfy yourself that the components of force parallel to the conductor on any length incrernent will sum to zero because of symmetry. Determine the force produced by one con- ductor on an incrementallength of the other. AI/s.: 0.202 newton/m. (b) Next, let us use the electric field approach on the same problem, Using Gauss's theorern, find the c!ectric field intensity produced by onc conductor in the space occupied by lhe other conductor, and from this find the force per unit length on the conductor. (c) Let us now approach the problem using the principie of conservation of energy. Derive an expression for lhe capacitance per unit length of thc conductors as a function of the spacing x, anel use this to Iind the force. (eI) Which of the preceding methods appears to give the solution most simply ? (e) Determine the force between the conductors in Fig. P1.2 when the potential difference between them is 250,000 V. 1.3 Two parallel plates (as in Fig. 1.11) are maintained at a potential difference e of 104 V. Each plate has an area of 0.02 rn", (a) Find the force between the plates as a function of their spacing x. (b) Find the energy converted to mechanical form as the plate spacing xis reduccd Irorn I em to 0.5 cm. Ans.: 8.85 X 10-1 joule. 1.4 The force between charged plates is exploited in the electrostatic loudspeaker shown in Fig. P 1.4. Two circular metallic plates are separated by a compressible ring of insulating material. When a voltage et is applied between the plates, the resulting force causes a change in the plate spacing z, which in turn results in an acoustic wave from the plate surfaces. For accurate sound reproduction, the variation in spacing x should be proportional to the output voltage e of the audio amplifier. Unfortunately, the electrostatic force is proportional to the square of the terminal voltage ec. To overcome this difficulty, a terminal voltage et = E + e is applied, where E is a constant voltage much larger than the audio frequency signal voltage e. (a) Suppose the voltage Eis 1000 V. Assume the spacing x between the .plates does not vary appreciably from a value of 0.5 mm. If the relative permittivity of the insulating ring is assumed to be 1.0, show that the force PROBLEMS 83 between the platcs is given approximately by .l = 0.139 + 2.78 x 10-4 e newtons (b) Suppose lhe rillg provides a spring constant of 300 newtonsjmeter. Ir e = 100 sin (I)/, determine lhe pcak-to-pcak oscillation in thc platc spacing. Ans.: 0.185 mm. í 0.1 m 1 -1xr- t !i I I. r, ;w.ç?-~~ Fig. Pl.4 Fig. Pl.5 1.5 The electrostatic voltmeter movement shown in Fig. P1.5 consists of three semicircular metal plates. The middle plate is attached to a taut suspension wire wh ich acts as a torsional spring. The rest position of the middle plate is at O = O, where it is just about to enter the space between the ou ter plates. Suppose the plates have a radius of 4 em and an air spacing of 1 'Ylm between lhe center plate and the ou ter plates. (a) Determine the capacitance of the system as a function of the angle O. (b) Determine the torque 011 the center plate as a function of the applied voltage e. (c) Suppose lhe voltmeter is to have a full-scale deflection of O = 3 rad with an applied voltage of 1000 V. What should be the spring constant of the suspension system? Ans.: 2.36 X 10-6 newton-mjrad. (d) When used on alternating voltage, does this instrument measure average, rms or peak values? (e) Supposc an alternating voHage of 600 Vrms at a frequency of 2000 c/s is applied to the meter. Determine the deflection and also deter- mine the input impedance of the instrument. Ans.: 5.2 X 106 n. (f) How could the plates be redesigned to make the scale of the instru- ment linear over a significant part of the voltage range? I !L,. 88 ELECTRIC ENERGY CONVERSION PROCESSES (a) Detei:ínine the inductance of the system. (b) Determine the total force acting on the plasma cloud. (c) Determine the pressure being applied to the plasma cloud. Ans.: 62,8 newtonsjm", 1.15 A fuse is made of wire having a radius r of 0.25 mm bent into a circular loop with its ends connected to adjacent terminais. The radius R . of the loop .is 1 cm. The problem is to determine' the stress in the wire when a current of 5 amp is passed through the fuse. The inductance of a circular Ioop can be expressed approximately by . L= floR In (1.083 ~) henrys . (If you have difficulty relating the force on the wire to the stress in it, consider finding the force on each of six equal sectors of the Ioop and solving the force system for the tension on one of the sectors.) Ans.: 60.6 newtons/m-, 1.16 The long solenoidal coil shown in Fig. PI.16 hus 200 turns. Since its Iength is much greater than its diameter, the Iield inside the coil may be considered uniform. . Fig. PI.16 (a) Derive an approxirnate expression for the inductance of this coi!. (b) If the current ;1 is 100 amp, determine the force tending to change the length of this soIenoid. Ans.: 2.84 newtons. (c) A smaller coil of 500 turns and diameter 3 cm is placed inside the solenoid. The axes of the two coils are aligned. Derive an expression for the mutual inductance between the two coils. (d) If i2 = 0.5 amp and ;1 = 100 amp, what will be the force tending to move the smaller coil along the axis of the solcnoid ? PROBLEMS 89 (e) Suppose the smaller coil is free to rotate so that its axis is at an angle O to that of the solenoid. With ;1 = 100 amp and ;2 = 0.5 amp, what w.ill be the :aximunj torque on the srnaller coil? Ans .. 8.9 x 10 newtoh-rn. (f) Suppose that the coils are initially aligned and carry currents ;1 = 100 amp and ;2 = 0.5 amp. How much mechanical energy is required to withdraw the smaller coil from the solenoid " " Ans.: 8.9 x 10--3 joule . 1.17- A dynamometer ammeter has one stationary coil and one moving coi!. The sclf and mutual inductances of the two coils are Lll = 0.01 henry, L22 = 0.004 henry, and L12 = 0.003 cos O henry, where () is the angle between the axes of the two coils. The coils are connected in series and carry a current of i = f sin cot . (a) Derive an expression for the instantaneous torque on the moving coil as a function of the angle O and the current i. (b) Determine the average torque as a function of O and f. (c) The helical restraining spring on the moving coil is adjusted to give a rest position of O = 90°. What should the spring constant be if the scalc are is to bc 600 and full-scalc dcflcction is to occur at 0.5 amp rrns ? Ans.: 3.58 X 104 newton-mfrad. (d) What is the current required to produce half-scale deflcction ? Ans.: 0.269 amp. (e) Suppose coil 2 is short circuited and an alternating current is passed through coil I. What rms value of current will be required to produce full-scalc deflection ? The resistances of the coils may be assurncd neg- ligible. Ans.: 0.621 amp. 1.18 Nickel has a magnetic moment of 0.6 Bohr magneton per atom. The density of nickel is 8850 kg/m:l and its atornic weight is 58.7. Ir all the magnetic mornents are aligned, what will be the flux density in the material? ' Ans.: 0.63 Wb/m2. 1.19 Equation 1.102 statcs thal there is a rorque tending to align the axes of two current-carrying coils. It might be thought that this would be . e 8 ;&pm2_ -r--------- . 8Pml ----- ,..-- -(-.--------. x ,... figo 1'1.19 the mechanism that causes alignment of magnetic moments in a ferro- magnctic material. Let us examine. this rnechanisrn by considering the two magnetic rnoments of Fig. P 1.19. )1 J 90 ELECTlUC ENERGY CONVERSrON PROCESSES (a) Using the currcnt clcmcnt law (eq. 1.57), determine thc flux density produccd by mornent PU/I at the center of mornent Pm2' In doing this, the moment P",l may be eonsidered as a circular loop of any radius rI for which "i «x. Ans.: flOPml/27TX3 Wb/m2. (b) Show that the magnitude of the torque acting on each of the moments ); n )'i\ !). ~ is T = IlOP1IltPm2 sin O 27TX3 ncwton-rn (c) For iron, each atorn has a magnetic mornent of 2.2 Bohr magnetons. The interatomic spacing can be derived from the atem density ofO.85 X 1029 atomsjrn". Determine the torque tending to align two adjacent mornents. (d) Determine the energy required to turn one moment frorn its aligned position of O = O to the unaligned condition of O = 7T/2. Ans.: 7.09 x 10-24 joule. (e) Iron remains ferromagnetic up 'to the Curie temperature of Te = 1043°K. The average thcrmal energy per atorn tending to randomizc the alignments of the momcnts is kT, where k = 1.38 X 10-23 joule/deg (Boltzrnann's constant). Compare this thermal energy with the energy of magnetic origin in part (d). J ) ) ) ) ) ) ), I ~ )1 J J ~ 1.20 A crystal of iron has a magnetic mornent per unit volume of Jlt = 1.74 x 106 amp/rn. If a magnetie field intensity of 8 arnp/rn is required to move a domain wall across the crystal, determine the hysteresis loss per unit volume when the crystal is magnetized to saturation flux density at a frequeney of 100 e/s. Ans.: 6970 Wjm3• 1.21 To a first approximation the fractional change in length 1::..111, due to magnetostriction in iron is given by the expression /::",1 = 1.5 x IO-GB2 I Consider an iron bar, 0.1 m in length and 10-4 m2 in cross-sectional area. Suppose the bar is surrounded by a uniformly distributed winding having 100 turns and negligible resistance. (a) If a supply of 200 V rms at a frequeney of 10 Kc/s is connected to the coil, determine the peak-to-peak variation in the length of the bar. Ans.: 3.04 X 10-8 m. (b) What is the frequency of the length oscillation in part (a)? (c) Suppose a transducer is required in which the change in length will be proportional to an applied voltage. By analogy with Probo 1.4, suggest a means by which this could be accomplished. I,. w~:· ,'-~...", PROBLEMS 91 1.22 A toroidal magnctic core lias a magnctization characteristic that can be approximatcd by the idealized B·-H relation of Fig. PL22. The core has a cross-sectional area of I em" and a mean diameter.of 6 em, The f1ux density may be assumed uniform across the core. B, Wb/m2 1.5 -20 20 -1.5 li, arnp/m Fig. P1.22 (a) Ir the core has an initial flux density of B = 1.5 Wbjm2, how much energy must be dclivered to the core to create a flux density B of -1.5 Wb/m2. Ans.: 1.13 X 10-3 joule. (b) Suppose a coil of 400 tUI'l1S is wound around the core. What current is required in this coil to change the flux density from B = + 1.5 to B = -1.5 Wbjm2? (c) Suppose a source of 1 Vis connected to the coil, If the coi1 resistance is 25 Q, what time will be required for the flux density to change from + l.5 to -1.5 Wbjm2?" Ans.: 0.157 seco 1.23 An air gap of 0.05 rnm is cut in the toroidal core described in Probo 1.22. Fringing of magnetic flux around this air gap may be neglected. (a) What current is required in the coil to produce a flux density of 1.5 Wb/m2 in the air gap? Ans.: 0.159 amp. (b) When the current of part (a) is reduced to zero, what will be the flux density in the air gap? Ans.: 0.0945 Wbfm2. (c) Draw a flux,linkage-current characteristic for the coil, What is the approxirnate value of the inductance of the coil? Ans.: 0.377 henry. 1.24 A magnetic field of'-L? Wbjm2 is required in a space 4 em by 10 em by 0.6 cm. An electromagnet of the form shown in Fig. P1.24 is proposed. Our task is to design a suitable coil and core arrangement. 92 ELECTRlC ENERGY CONVERSrON PROCESSES (a) Assuming that up to a ftux density of 1.2 Wb/n12, the magnetic field intensity required by the core is negligible, determine the magneto- motive force required of the coi!. (b) The temperature of the coi! probably will be reasonably low if the current density in the copper conductors does not exceed 2 amp/mrnê, About 0.6 of the area of the window in the core can be occupied by the copper conductors, the rest being requireel for insulation and mechanical cIearance. Determine appropriate values for the height anel width of the core window. t e I lOcm -~4cm~ Fig. Pl.24 (c) Suppose the coi! is to opera te from a 100 V d-e supply, The resis- tivity of the corper at the operating temperature is about 2 X 10.-8 D-m. Determinethe required number of turns and the cross-sectional area of the wire in the coi!. . (c) Determine the power that must be dissipated as heat per unit of outside súrface area of the coi!. With organic insulating materiais this should not exceed about 0.2 W/cm2• If it does, the current density should be appropriately reduced. 1.25 A phonograph pickup shown in Fig. P1.25 consists of a 5-turn cyIindrical coi! of l-em diameter that is attached to the sty!us. The coil is situated in the air gap of a cylindrical permanent magnet. The flux density is 0.3 Wbfm2• The stylus-coil assembly is attached to the magnet by a flexible mounting, and the horizontal motion 01' the coil is approximately equal to that of the stylus. 11' the tip of the stylus oscillates sinusoidalIy with a peak-to-peak amplitude of O. I rnm at 600 cfs, determine the rms output voltage of the pickup coi!. . Ans.: 6.26 mV. pnOBLEMS ) )93 Magnet ) ) ) ) ) Rubber mounting Stylus \í\r'V\/VV\r~ Record Fig. P1.2S 1.26 Figure P1.26 shows the essential components of a loudspeaker. The permanent magnet produccs a uniform radial magnetic flux density of 0.8 Wbjrn" across a cylindrical air gap. The coi! of 30 turns is wound on a fiber cylinder of dia meter d = 2 em. When assembled, the coi! 1S inserted into the air gap of lhe magnet. (a) Determine the force on the cone as a function of the current i. Ans.: 1.51 newtons/amp. (b) Determine the induced voltage in the coil per unit 01' coil velocity. (c) Neglecting coil resistance, show that lhe electrical power input to the coil is equal to lhe mechanical power delivered to the cone. (d) Over most ofthe audio frequency range, the force is absorbed by the damping action of the air being driven by motion 01' the cone. Suppose the damping coefficient of the cone is 0.3 newton per mfsec of velocity. Ir the current in the coil is f sin cot, determine the velocity of the coi! and the voltage induced in the coil. Show that the impedance !ooking into the coil terminaIs is resistive and has a value or 7.6 D. ~ ) ) ) ) ) ) j ) ) ) ) ) ) ) ) T d _L Magnet Fig. Pl.26 ) li (I ~ li 1/ 98 ELECTRIC ENERGY CONVERSION PROCESSES to P7IIV2 newtonsjm-, where Pm is the density of the material and v is the peripheral velocity. Suppose we use a copper alloy having a density of '8800 kg/m3 and a maxirnurn working stress of 2.8 x 108 newtons/rn''. Determine the maximum peripheral velocity õ. Write an expression for the induced voltage betwcen the ends 01' the cylinder. AI/s. ú = 178 m/seco (b) Current is passed through lhe cylinder by means of stationary brushes, which are distributed around both the inside and the outside rims. In high-current machines, the electrical contacts may take lhe form 01' a liquid metal channel, The current i that can be carried by the cylinder is limited by the heat that can be extracted from its surfaces. Let h denote the heat in W 1m2 that can be removed by air flow from the curved cylin- drical surfaces (inside and outside) of the cylinder. If the thickness of the material is c and the resistivity is p, derive expressions for the current density J and the current i in the material. (c) Show that the power P converted in the machine can be expressed as ( ? h(')~GP = 27T1':dJv ':'--;. W ~ I I,,, l l (d) With forced air cooling, h might be about 5000 W/m2• The copper alloy would have a resistivity of about 2 x 10-8 D.-m at the operating temperature. The flux density B in the air gap would probably not exceed 1.0 Wbfm2• The iron core cannot support a flux density in excess of about 1.6 Wb/m2. Determine the required relation between the dimensions x and r to provide adequare magnetic materialin the central cylindrical core of the machine. If the thickness c of the metal cylinder is made 1 em, determine the radius r and the length x for a l-MW machine. Ans.: r = x = 0.125 m. (e) Determine the resistance of the metal cylinder. Find the fraction of the converted power that wil! be dissipated as heat in the cylinder. Ans.: 0.0008. (f) If the current i were brought out from the inside brushes through holes in the magnetic structure, as shown in Fig. P1.31, what effect would this current have on the magnetic material? Show that this effect can be substantially eliminated by bringing out the current i along a stationary metal cylinder placed outside the revolving cylinder in the air gap. (g) If the air gap length is 2.5 em, estimate the magnetomotive force required in the coil. Ans.: > 20,000 amp. (h) Determine approximately the required dimensions of the magnetic path around the coil and air gap. lf"""" l' ::, I~ t- ~ . ~ I, I ! PROBLEMS 99 1.32 The magnetic core shown in Fig. Pl.32 has a square cross section 3 em by 3 em. When lhe two sections of the core are fitted together, air gaps, each of length x = 1 mm, separate them. The coil has 250 turns and a resistancc of 7.5.0. Thc magnetic field intensity required by.the magnetic material is negligible. . --±. tx Fig. P1.32 (a) Supposc a d-e sourcc of 40 V is connectecl to the coi!. Determine the total force holding the two scctions 01' lhe core together. Ans.: 503 newtons. (b) Suppose an a-c source of 100 V rrns at 60 c/s is connected to the coil. Determine the force holding the sections together. What is its average value? Ans.: 757 newtons. (c) The magnetic field is not entirely confined to the air gap volume. It actuaJJy extends out into the space around the gap with diminishing density. This effect may be incJuded in approximate calculations by assuming that the magnetic flux density is uniform for a distance equal to half the gap length x out frorn each side of the air gap. Consider the effect of this correction for fringing on the results of parts (a) and (b). 1.33 A toroidal core has a cross-sectional are a of 10-3 m2 and a mean length of 0.4 m. The material of the core has a maxirnum magnetic mornent per unit volume of ..;tI = 1.5 X 106 amp/rn. For vii> J!( > -.it, the required field intensity is negligible. The core is uniformly wound with a coil of 400 turns. (a) Draw a B-H characteristic for the core material. (b) Determine the flux density in the core if the coil current is 50 amp. Ans.: 1.95 Wbjrn". (c) Now suppose an air gap of I-rnrn length is cut across the core. What minirnum value of coil current is required to make J!( = ..;tIin the core? Ans.: 3.75 amp. I 100 ":-.,0o...~._--y ELECTRIC ENERGY CONVERSION PROCESSES (d) What is the air gap flux density with a coil current of 50 amp? .Ans.: 1.94 Wbfm2. (e) What is the force acting on the core material to dose the air gap for the condition of part (d)? Ans.: 1415 newtons. (f) Ir the coil of 400 turns had been concentrated around the side of the torus opposité to the air gapped side, would the air gap flux dcnsity reach the value giyen in part (d)? Would the force found in part (e) have changed? . 1.34 The rriá'gnetic actuator shown in Fig. P 1.34 is to be used to raise a mass m through a distance y. The coil has 5000 turns and can carry a -'24cm---,»>: L ===:::;:::'" d part-e, L. :;i' ~ ~ '---v ~v~ -: - 1 . Fixe 16cm I /Y.//f M p I ~ I - 'I 1_ 8cm-- J4cmt 4cm Fig. PI.34 current of 2 amp without overheating. The magnetic material can support a flux density of 1.5 Wbfm2 with negligibJe field intensity. Fringing of flux at the air gaps may be neglected. (a) Determine the maximurn air gap y for which a flux density of 1.5 Wbfm2 can be established with a current of 2.0 amp. Ans.: 4.18 rnm, (b) With the air gap determined in part (a), what is the force exerted by the actuator? (c) The mass density of the material is 7800 kgfm:1. Determine the approximate value of the net mass m of the load which can be lifted against the force of gravity (at sea levei) by lhe actuator. Ans.: 578 kg. (d) What current is required in the coil to lift lhe unloaded actuator? (e) What is the initial acceleration 01' lhe unloadcd actuator ir it is released when the coil current i is 0.3 amp? Ans.: 11.7 mfsec2• 1.35 Figure P1.35 shows a cross scction 01' a cylindrical magnetic actuator, The plunger, of cross-sectional arca 15 em", is Iree to slide ::s- '(j (fJ PRORLEMS 101 vertically through a circular hole in the outer magnetic casing, the air gap between the two being negligible. The coil has 3000 ,turns and a resistance of 8 ohms. I t is con- nectqd to a 12 V li-c source. The magnetic material may be assumed perfect up to its saturation flux density of 1.6 Wbjrn", (a) Determine the static force on the plunger as a function 01' the gap length y. (b) Over what range of gap length y will the force on the plunger be essentially con- stant because saluration flux density has Fig. PI.35 beerrreached ? Ans.: y ~ 3.53 mm. (c) Suppose the plunger is constrained to move slowly from a gap of 1 em to the fully closed position. What will be the mechanical energy produced? Ans.: 8.9 joules. (d) Suppose that the plunger is allowed to close 50 quickly from an initial gap of I em that the flux linkage of the coil does not change ap- preciably . during the motion. How much rnechanical energy will be produced? Ans.: 1.9 joules. 1.36 The magnetic actuators discussed in the two preceding problems producc a large force over a relatively short displacement. Figure P1.36 shows a type 01' actuator that is capable 01' exerting a smaller force over a larger displaccmcnt. The central cylindrical plunger is separated from the outer cylindrical casing by a uniforrn air gap of 0.5 mm. The coil has 500 turns. The magnetic material of the plunger and casing can be con- sidered perfect up to 1.6 Wbfm2. (a) One of the lirnitations of this actuator is the magnetic flux that can be supported by the plunger. Ir the flux density in this member is not to exceed 1.6 Wbfm2, what is the maximum permissible current in the coil ? Ans.: 0.255 amp. ~ Casing """''''\ ~!t -.:re:: ~ ~ O.5mm I .2c_m ~r ~~-~5cm~ Fig. P1.36 J -fi ) dl ' ( l- f, I, -J ~ 1 l,, ( -i í ,. ! \ 102 ELECTIUC ENERGY CONVERSION PROCESSES (b) With the current held conslanl al the valuc Iound in part (a), derive an exprcssion for the force as a function of lhe displaeement ~;. Determine the maxirnum and minimurn valucs of this force. Ans.: 1.28 newtons (max). (c) Suppose the coil current is increased to five times the value given in (a). This will cause the plunger to be magnetical!y saturated at B = 1.6 Wb/m2 for most of the displacement .e. Determine the force as a function of x for this condition. ls the force at .1: = 5 em increased above thc value found in (b)? 1.37 Figure P1.37 shows a moving-iron ammeter in which a curved ferromagnetic rod is drawn into a curved solenoidal coil against the Scale ç--'> <;--" ;::::~ Stationary coil ~<, Fig.PI.37 torque of a restraining spring. The inductance of lhe coil is L = 5 + 200 f.1H, where O is the deflection angle in radians. The spring constant is 7 X 10-4 newton-m/rad. (a) Show that the inslrument measures the root-rnean-squarc value of the current. (b) What will be the full-seale deflection if the rated current is lO amp? Ans.: 1.43 rad. (c) What will be the potential difference across the coil when the current is 5 amp rms at a frequency 01' 180 c/s? The coil resistance is 0.6 I n. Ans.: 85 mv. 1.38 A rotating actuator of the form shown in Fig. 1.53a has the following dimensions: g = 1 mrn, r = 2 em, Z = 4 em, The coil has 400 turns. The magnetic material may be considered perfect up to 1.2 Wb/m2. '••••• ""1 'i PROBLEMS 103 (a) Determine the maximum coil eurrent if the flux density is to be limited to 1.2 Wb/m2 in lhe material bounding the air gap. (b) With the eurrent found in part (a), determine the torque produced. What work will be done as the shaft moves from e = 71"/2 to e = O? Ans.: 1.44 joules. 1.39 In some eontrol applications a rotating actuator with reversible torque is requircd. Figure PI.39 shows a deviee that has this feature. i1 , i I I I I!. :I I y ~ I ~' ~ i, rr ,\ '{/\ \ -, \ 'o. Fig. P1.39 Tn addition, it provides a torque that is lineariy proportional to the product of its two coil currents. Each of the four stator poles has an angular width of 45° and a length z. The rotor faces eaeh have an angular width of 90° and a radius r. The air gap length is g, and the flux outside the air gaps may be neglccted. Each stator pole has two N-turn coils, one carrying a current t, and the other carrying a current i2. The coils are connected so that the magnetomotive force on caeh of the horizontal poles is N(il + i2), while that on eaeh of the vertical poles is N(i1 - i2) •• (a) Note that because'of the symmetry of the magnetic system, magneto- motive forces on the horizontal poles do not produce flux in the vertical poles. Derive an expression for the torque due to the magnetomotive forces on the horizontal poles. Derive a similar expression for' the torque due to the magnetomotive forces on lhe vertical poles. Combine these two expressions and show that T= 4N~f.1ozrili2 newton-m g over the range O < O < 71"/4. 108 B ANALYSIS OF MAGNETIC SYSTEMS III /VV/f )o li Hr 1\ 1\ fi' f\ ( "VV\[VV Fig. 2.2 A loeus of the B-H eharacteristic for an alternating intcnsity H of variable magnitude. B I1I1A7/IJ I ) H Fig. 2.3 A set of hysteresis loops and the normal magnetization curve .for a ferro- magnetie material. APPROXIMATE MODELS FOR B-H CHARACTERISTICS I } ) ) ) ) ) . ) c ) ~ .J ) ) ) ) ) ) ) j ) ) ) ) ) ) ) I .I / ) 109 a number of cycles with variable peak magnitude. Note that at a given value of magnetic field intensity, lhe flux density may have any value within a wicle range. To determine the appropriate value, it is necessary to know the past history of the B-H locus. It is obviously impossible to record the loci for ali possible past histories; thus actual E-H loci are seldom used in analysis. Simplified approximations general1y give results 01' «de-vuate accuracy. \. . For alternating current of constant peak magnitude, the set of closed, symmetrical B-H loops shown in Fig. 2.3 represent the behavior 01' the material. The analytical difficulty in using such loops is that it is necessary to know, in advance, the pcak amplitude of either B or H to decide which loop is applicable. 2.1.1 Normal Magnetization Curve Severa I simple and useful approxirnations can be obtained ir the hys- teresis effect in the material is neglected. Without its memory, the E-H re\ationship becomes single valuecl. The most commonly used approxi- mation, known as the normal magnetization curve, is shown in Fig. 2.3. This curve is lhe locus of the tips of a set of syrnmetrical hysteresis loops. This is lhe curve that is most reaclily available in lhe descriptive literature 011 most types of soft magnelic material. Il is obtained by use of a flux meter, which measures the total change in the Ilux of a sample when its exciting current is reversed. Numerical techniques are applicable to analyses which involve the normal magnetization curve. As a sim pIe example, consider the circuit of Fig. 2.4a in which a constant voltage E is applied to a coil at time I = O by closing the switch. The coil is represented by its resistance R and by the Ilux linkage }, versus currcnt i curve of Fig. 2.4/1. This curve is cleriveel Irorn lhe normal magnetization curve 01' Fig. 2.3 by rescaling lhe B-H curve using eqs. 2.2 and 2.3. At 1 = O, thc current i is zero. We want to finei the current i as a function of time I. Frorn eq. 2.1, we have 'liÀ)-- - E(dI I~O- (2.5) Assuming that this rate of change of flux linkage remains approximately constant for a short interval of lime .6.1, lhe flux linkage ÀJ at time 6.1 can be approxirnated by ÀJ = E 6.1 (2.6) Referring to the },-i curve, the corresponeling current is ir. The slope of the À-I curve at 1 = 6.1 can now be deterrnined as (dÀ) = E - tu, dt I=At (2.7) /J;I f' I. ) i 1- 110 ANALYSIS or MAGNETlC SYSTEMS (a) À j, ) ) - ) S ) ) ) ) ) ) I"( ( ( ) I ) I··) ) ) \) ) !. j ) ) ) :... ) y ~~G1' À ij o t:.t 2t:.t (c) Fig. 2.4 (a) Circuit. (b) Flux linkagc À-i charactcr istic. (c) Flux linkagc À and currcnt i as functions of time f. and the flux linkage at time 2 61 can be approximated by • A (di)1'2 = ÀI + D.I - di 1="1 (2.8) Continued repetition of this calculation results in the data for the curves of flux linkage and current as functions of time, plotted in Fig. 2.4c. This sirnple numerical method of solution is adequare for many non- linear systems. Reference should be rnade to books on numerical analysis for more elaborate techniques that provi de either a greater accuracy or fewer steps in the computation, or both. APPROXIMATE MODELS FOR B-H CHARACTERISTICS 111 2.1.2 Piecewisc Lincarization of thc Normal Magnetization Curve In Illany analyscs lhe normal magnetization curve (Fig. 2.3) may be adcqualely rcprcscntcd hy lhe lJ-H charactcristic of Fig. 2.5. This consists of a linear portion in lhe range - B" < B < B" having an unsaturated relative permeability 11n and two linear portions for the ranges lEI> E", each having a slope ,U,IIO where fls is terrned the saturated relative perrne- ability. Within the unsaturated range, the flux linkage-current relationship .. H Fig. 2.5 Pieccwisc linearization of B-H curve. of a coil (such as that 01' Fig. 2.1) may be expressed as the unsaturated value of inductance L; where, from eq. 2.4, À N2A L" = i = ~i- fl"flo henrys (2.9) In this expression, A is the core area, i the mean length of the flux path, and N the number of turns. If the magnitude of the coil current i is less than i" = H"if N, the voltage-current relationship of the coil may be expressed by the lineardifferential equation . di e = Ri + L" - volts (2.10) dt Ir the magnitude of lhe current excecds i", the voltage-current relationship becomes . di e = RI + Ls- dI volts (2.11) 112 ANALYSIS OF MAGNETIC SYSTEMS where N2Ai; = ,fisfio Figure 2.6 shows the current versus time curve for the circuit of Fig. 2.4a for two values of applied voltage E. Tn the lower cUQ'e the current does not reach the value ik; in the upper curve this value is exceeded. Since eqs. 2.10 and 2.11 are of the first order and have constant coefficients, alI terms of the solutions are simple exponentials. To arrive at the solution for case 2 in Fig. 2.6, eq. 2.10 is used until i = ik. The final conditions henrys (2.12) 2 ."" ,\ 1 o I" I Fig.2.6 Current-time curves for circuil of Fig. 2.4a. (I) E/R < ik• Currcnt insufficicnt to reach saturated region. (2) With increased applicd voltage. E/R> ik• (ik, fk) of eq. 2.10 are then used as the initial conditions in the solution of eq. 2.11, which applies for i> i". The process of simplifying the B-H relation ll1ay be carried further for those situations where the magnetic field intensity H is negligible as long as the flux density B does not approach its saturation value, The unsaturated relative permeability fi" of Fig. 2.5 ll1ay then be set at infinity. This approximation, shown in Fig. 2.7a, proves to be very useful in the analysis of devices, such as saturable reactors, which opera te far into the saturated region of the B-H curve. " . Sometimes the further sirnplification of making the saturated relative permeabilitye, equal to zero is justified. Physically, we know that even a perfectly grain-oriented material cannot have a value of /1,,, less than unity. This approximation, shown in Fig. 2.7b, is applicable in those situations where the saturated inductance L" of eq. 2.12 is negligible in rclation to the other pararneters of the systern under analysis. To dernonstrate the use of these simplc lincarizcd models, consider the circuit of Fig. 2.8a in which a voltage e = I? COS (01 is applied to a coil at t = O. Suppose the À-i charaeteristic of the coil is represented by the APPROXIMATE MODELS FOR B-H CHARACTERISTICS 113 B B BkL--C----B" Slope= O HH -Bk ----~-Bk (a) (b) Fig. 2.7 (a) Linearization of E-H curve, similar to Fig. 2.6, with fi••= 00. (b) fi. = 00 and 11, = O. X Xk R 7:' ..•. ~y .J~E:.~ -->-i di ) -Xk (a) (1)) e wt I z~ z - 2 ~ "~ ~ " ~ wt ~--+- (c) Fig.2.8 (li) Circuit with e = t. COS WI. (b) Idealizcd relation between flux linkage iI. and current i. (c) Waveforms of e, dil./dl, and i. r;~ )] J, ," ANALYSIS OF MAGNETIC SYSTEMS118 2.1.4 Impcdancc Model Most a-c apparatus operates with a voltage that is approximately sinusoidal. It would therefore bc useful to have a simple reprcscntation for a nonlinear inductor operating under this condition. Suppose a sinusoidal voltage of E" . 1 (221)e =, SII1 o)! vo ts . )-, rI! is applied to a coil whose resistance is ncgligible and whosc core has the Il (i) }.. ), ) ) (a) ) j .) .:. (b)) U :\ ),..... II niX'UE< El t R" s, R" I jXo,) t .. J \\) (J , ) (c) (d) (e) Fig. 2.12 (a) E-H or Â-i loop. (b) Wavcforms of applicd voltagc e, currcnt i, funda- mental componcnt i" in-phase component i" and quadrature componcnt ix• (c) Funda- mental-frcquency equivalent circuit. (ti) Variation of equivalent circuit parameters with applicd voltage El' (e) Third harmonic cquivalcnt circuit. ,... .... ) :.../ ,} I' I ) APPROXIMATE MODELS FOR E-lI CHARACTERISTICS 119 fiux linkage-current characteristic of Fig. 2.12a. This characteristic is obtained by rescaling a B-·H loop of the material using eqs. 2.2 and 2.3. The flux linkage of the coil is },=J e di É . ( 7T)= - SIIl cot - - w . 2 webers (2.22) The current i in the coil has the periodic but nonsinusoidal form shown in Fig. 2.l2b. This current may be expressed as a Fourier series. i = 11 sin (WI - e) + odd harmonic terms = 1,. sin oit + l; sin (wt - ~) + odd harmonic terms (2.23) In many anaJyses the harmonic components of the current can be ignored and only the fundamentaJ-frequency component preserved. One reason is that if the volt age is sinusoidal, the harmonic currents deliver no net power. If necessary, the behavior of the harmonics may be studied separately after a first approximation to the solution has been obtained using fundamental-frequency quantities only. ln eq. 2.23 the fundamental cornponent t. of the current i has been separated into a componcnt ir in phase with the voltage e and a component ix lagging the voltagc e by 71'/2radians. The relation between the fundamen- tal-frequency components of voltage and current can be represented by the equivalent ~ircuit of Fig. 2.l2c. In this figure EI, lI, Ir' and Ix are the phasors corresponding to e, i17 ir' and t; respectively. The hysteresis losses in the core are equal to the 105s in the resistance RI. where R _ El Ê h--- Ir - Jr ohms (2.24) The magnetizing reactance Xo is given by E "Xo = ~ _ EI --,:;- z . Ix ohms (2.25) The hysteresisloop and waveforms of Fig. 2.12a and b apply for only one value of applied voltage. As the voltage is varied in magnitude, the equivalent circuit parameters R" and Xo also change. The reactance Xo drops rapidly in value as Lhecore enters its saturated region. The resistance 120 ANALYSIS OF MAGNETIC SYSTEMS Rh generálly tends to rise with increasing applied voltage, indicating that the hysteresis losses are proportional to a power of lhe applied voltage which is less than 2. It may be shown that in a core having the idealized B-H loop of Fig. 2.10, the hysteresis losses at a given frequency are directly proportional to the applied voltage until saturation is reached. For this condition, R" is directly proportional to El' The values of the parameters R" anel Xo could be derived from the B-H loops of the material as elescribed in relation to Fig. 2.12. This processis very tedious. The parameters can usually be derived directly from published data of the power loss anel reactive volt-amperes per unit 'of volume of the material when tested at constant frequency and variable sinusoidal flux density. When using the equivalent circuit of Fig. 2.12c to represent a nonlinear magnetic element, the parameters R" and Xo should be adjusted to lhe values appropriate for the applied voltage El' 1f this voltage E, varies over only a small range of magnitude, it is often possible to assume Rh and Xo to be constant at appropriate average values. The equivalent circuit of Fig. 2,12c represents only the fundamental- frequenéy behavior of the element. Examination of Fig. 2.12b shows that ifthe applied voltage e is sinusoidal, the current iconsists of a fundamental frequency term plus a series of odd harmonic terrns. The most important of these is the third harmonic, the magnitude 01' which may be as high as 70 % of the fundamental component. Since 'the third harmonic is the most important one, it is useful to have some circuit model which permits at least a qualitative analysis of the third-harmonic behavior under steady-state a-c operation. Supposewe regard the nonlinear magnetic element as a source of third harmonics. When the voltage applied to the nonlinear clerncnt is sinusoidal, this third-harmonic source can be considercd as short circuited. When the current in the nonlinear element is sinusoidal, the third-harmonic source is open circuited. Fig. 2.12e shows a simple third-harmonic equivalent circuit that can be used to represent approximately lhe third-harmonic behavior of the nonlinear element. It consists 'of a source voltage E:IO equal tothe third-harrnonic voltage wjth sinusoidal current in the element, in series with an inductive impedance jX:l> where X:1 is lhe ratio of E30 to the third-harmonic current with sinusoidal voltage applied to the element.. ' This third-harmonic equivalent cireuit is very useful in qualitative analysisandalso may be used to a limiteel extent for quantitative analysis. The value of E30 is normally in the range of 0.3-0.7 El> depending on the degree of saturation in the magnetic element. The value of X3 is generally of the sameorder as Xo in Fig. 2.12c. EDDY CURRENTS 121 ) ) ) ) ) ) 2.2 EDDY CURRENTS When a magnetic flux changes with time, an induced electric field is proeluced around the region of changing f1ux. Normally, we are most interesteel in the electromotive force which this electric field establishes in the winelings encircling the magnetic paths. But this electric field is also produced within the magnetic material, anel, if the material is a conductor, currents known as eddy currents are establisheel. Consieler the long cylinelrical solenoiel of Fig. 2.13. Suppose the coil current i is positive anel increasing. We would expect a positive and ) ) ) L/ / / / //: --- It I~ /":"'B I" \ \ 1 \ ~ I \ \ ~ 'J<.:: / ft path~ /~ f---;:--H I------,----~r, / Fig.2.13 Thc cddy currents in a magnctic core. (The currcnt i is incrcasing with time.) increasing flux density B in lhe material elirected along its axis. The circular path shcwn encircles a magnetic f1ux (P of ./ 4)c,,('1.= JB . dA webers (2.26) By Faraday's law, the integral of the electric field intensity {/ in a counter- clockwise direction around this path is equal to the rate of change of this flux. Because of the circular symmetry, " {;'= _1__ d 1>,,"<'1. volts/meter (2.27) 27fr di ) 122 ANALYSIS OF MAGNETIC SYSTEMS ) If the material has a resistivity collinear current density -, 6' J=- p (2.28) p, this electric field intensity sets up a arnperes/meter" ) This eddy current density is in a direction that causes it to oppose the change in the enc!osed magnetic fluxo Thc opposition to the change in fíux is greatest along the axis of the solenoid (r = O), since ali the eddy current encircles this axis. The effect becomes zero at the periphery of the solcnoid where r = rc' Therefore one effect of eddy currents is to cause the time-varying magnetic flux density to be nonuniform within the material. An alternating magnetic flux tends to be concentrated toward the outside surface of the material, since the effect of eddy currents in preventing variation of magnetic fiux is greatest near the central axis. This is known as the magnetic skin ejJect. If a magnetic material is to be used to best advantage, the magnetic flux density should be reasonably uniform over its cross-sectional area. Thus, there is a practicaI limit to the thickness of a solid conducting magnetic material which should be uscd at any given frequency of oper- ation. A second cffect of the cddy currents is to producc power loss in the material. The eddy currenl loss per unit 01' volume of material is p = pJ2 wattsjrneter" (2.29) One means of controllíng eddy current cffects is the use of high-resis- tivity materiaIs. Pure iron has a resistivity of about 1O-70hm-meter. The addition of about 4 % silicon to the ircn increases its resistivity to about 6 X 10-7 ohm-rneter. The Icrrite materials elescribed in Section 1.6.5 are oxides rather than metaIs and have very high resistivities. For example, nickel-zinc ferrite has a resistivity of about 1O-40hm-meter. With rnetallic magnetic rnaterials, the principal means of controlling eddy currents is the use of thin sheets of laminations. Figure 2.14 shows ) ) " ) ) ) ) ) ) T ) ) Fig. 2.14 Toroidal core made frorn a long, thin strip of material. i :1 r i li "li ~ ~ . II, I' i !, i EDDY CURRENTS how a toroidal core can be made from a long, thin strip of magnetic material. The surfaces of lhe material are covered with a thin insulating coating. When the mag- netic flux in lhe core changes, an electric field is set up in the material, as in Fig. 2.13. But the resultant eddy currents cannot flow from layer to Iayer anel are restricted to paths within the cross section of the strip. We now derive an approxirnatc cxpres- sion for thc eddy current losses within a laminated magnetic material. Figure 2.15 shows an enlargeel cross section of the strip used in the toroidal core of Fig. 2.14. We assume that the eddy currents are not Iarge enough to infIuence significantly the magnetic field within thc lamination. The flux density is considered to bc uuiforrn. Consider the closed path shown within the lamiuation in Fig. 2.15. Thc sidcs of this path are at distance :l: Irorn lhe ccntcrlinc of the Iarnination. This path encloses a magnetic flux of 4>x = 2xyB webers (2.30) 123 .> Fig. 2.15 The determination of eddy-current loss in a lamination. Since y » x, the change of this magnetic fIux may be assumed to produce an electric íield of constant magnitude down one side of the path and up the other. By Faraday's law, 6',,)y = d4>x volts (2.31) dt Combining eqs. 2.30 and 2.31 gives tff' = xç!,!! x dt The current density at a distance x from the center plane of the lamination is therefore ' J _ 6"x x- - P :!!dB P dt amperes/meter" voltsjmeter (2.32) (2.33) 128 ANA LYSIS OF MAGNETIC SYSTEMS '\ shows that the instantaneous power Pl entering winding 1 is equal to the instantaneous power P2 leaving winding 2. Pl = e1i1 = (N1 e2) (N2 i2) N2 N1 = e2i2 = ])2 watts (2.48) This power invariance of the idealized transformer follows from the assumption that it has no power loss and no energy storage. 10 approximate analyses, transformers can often be considered ideal. Where the idealizing assumptions are not valid, a transformer can be considered as an ideal transformer in i1 i2 combination with other pararneters rep- DC resenting its imperfections. lt is there-N e fore convenient to have a symbolic circuit2 2. I elernent to represent an ideal transformer,as shown in Fig. 2.18. Its terminal vari- F· ..,18 E'" 1 t "1 ables are related by ecs. 2.44 and 2.47.. Ig. k. qUlva en ClrCUI sym- '1 • . boI for an ideal transformer. Where necessary, for clarity, the starting . . ends of boI h windings are identified by a doto The polarity of the potential difference between the dotted end and the undotted end is the same for each winding. Consíderthe circuit shown in Fig. 2.19(1 in which a resistance RL is connected to winding 2 and a source to winding I of an ideal transformer. The voltage-current relationship at the source is ~1 = (N1/N2)e2 i. (N2/N1)i2 or , (N1)2RRL = - L N2 ohms (2.49) Thus the source sees an equivalent resistance R/, as shown in Fig. 2.19b. A shunt resistance on one side of an ideal transformer may therefore be moved across the transformer, the value of the equivalent shunt resistance being equal'.to the original resistance, multiplied by the square of the turns ratio. Similarly, a resistance that is in series on one side of an ideal transformei .may be moved to a scrics position on thc other sidc, using the same multiplier. This ability 10 transfcr elcments across ideal trans- formers is nót restricted to resistances but applies equally to inductances, capacítances.isources, and irnpedances. For example, Fig. 2.19d shows a circuit that is ~quivalent to the circuit of Fig, 2.19('. lt is noted that "'" j EQUIVALENT CIRCUITS: COMPLEX MAGNETIC SYSTEMS 129 i1 i2(]\)2 (n) me\ R1_' \ (I) ) i1 L fi \c e2) (c) L' () (li) " ) / -' / Fig. 2.19 «(I) and (b) transfer of a resistance element across an ideal transformer. (c) and (d) the transfer of a nctwork of elcrncnts across an ideal transformer. (N )' (N )' (N )'R' = N: R; L' = N: L; C' = N: C. inductances and resistances are multiplied by the square of the turns ratio, whereas capacitances are dividcd by lhe sqllarc of lhe turns ratio. J 2.4 EQUIV ALENT cmcurrs FOR COMPLEX MAGNETIC SYSTEMS Whcn we encounter a cornplex system of clcclrical elements and wish to analyze its performance, our normal approach is to develop an electric equivalent circuit for lhe system. Wc put into the cquivalent circuit only ) 130 ANALYSrS OF MAGNETIC SYSTEMS EQUIVALENT CIRCUITS: COMPLEX MAGNETIC SYSTEMS 131 ;. ) ) ) those parameters that are considered to be significant in intluencing the performance. The equivalent circuit is thus a simplified mathernatical model of the real system. Having developed an adequate equivalent circuito we ernploy the well-known techniques of e1ectric circuit analysis to deter- mine its behavior. The accuracy with which the solution of the electric circuit behavior represcnts the perforrnancc of the real systcm is lirnited onJy by the adequacy of the cquivalcnt circuit modcl. In electricaJ machines, transforrncrs and other electrical devices, ferro- magnetic material is used in a wide variety of shapes. Various parts of muJtili~bed magnetic struetures are encircIed by coils. In this section we show how such complex magnetic systems may be represented approxi- mately by equivalent circuits. The system is first represented by a magnetic equivalent circuit, whieh is then transforrned into un equivalent electric circuit. The methods of electric circuit analysis may then be used. force .'17 and the rnagnetic fíux 1> may be expressed symbolically by the equivalent magnetic circuit of Fig. 2.20b. The .magnetic properties of the material and the dirnensions of the core determine its reluctance !?li. Under the idealized condition where the relative permeability can be regarded as constant, that is, B = f1r!loH the reluetance can be cxpresscd as :7 .0/1 =- 1> (2.51) i Pr!loA arnpcrcs/weber (2.52) 2.4.1 Derivation of Magnetic Equivalent Circuits Let us first derive a magnetic equivalent circuit for a simplc magnetic system. Figure 2.20a shows a torus of magnetic material with a winding In general, the rcluctance of a ferromagnetic core is nonlinear. The reIuctance syrnbol is then used in a magnetic equivalent circuit merely to denote a magnetic elernent for which a ,,? -1> relation exists. This relation may be obtained by rescaling the B-H characteristic for the material using the expressions <I> and 1> = BA ,'17 = ut webers amperes (2.53) (2.54) ) ) ) ) ) t c I ff = Ni amperes (2.50) For purposes of analysis, any of the approximate models for the B-H characteristic dcvcloped in Section 2.1 may be used. Let us now considcr, as an cxarnple, the magnetic system of Fig. 2.2Ia. This system consists of a three-legged magnetic core, two of the legs having windings arid the third leg having an air gap. Basically, this is a complex three-dimensional magnetic field problem. But by the use of simplifying assumptions, the magnetic field can be reduced to a magnetic circuit of lumped reluctances. Let us assume that except in the air gap, ali magnetic flux is confined to the rnagnetic material. The leakage flux in theair paths around the windings is considered negligible. . The magnetic system may now be divided into four sections, each of which has a uniform flux over its length. Three of these sections represent magnetic paths in the material, and the fourth represents the air-gap path. Each section may be represented by a reluctance which relates the fíux to the magnetornotive force required to establish that fíux along the length of the section. Figure 2.21b shows the magnetic equivalent circuit that results from the foregoing assurnptions. Reluctances !?lil, fJ1!z, and fJ1!3 represent the three paths in the magnetic material carrying magnetic ftuxes 1>1' 1>2' and 1>3' respectively. The air gap is represented by the linear reluctance !?li4, and its magnetic flux is 1>3' The circuital law of eq. 1.60 applies to any closed (b) (a) t e I (c) ) ) ) ) ) Fig. 2.20 (a) A simplc rnagnctic clemcnt. (/;) A magnctic cquivalent circuit for the clement. (c) An electric equivalcnt circuit for lhe elcment. of N turns, The magnetomotive force around the magnetic path is ) This magnetomotive force establishes a magnetic field intensity H, which in turn produces a magnetic flux density B in the material. Integration of this fiux density over the cross-sectional area A of the core gives the magnetic flux 4>. The cause-effect relationship between the magnetomotive ) 132 ANALYSIS OF MAGNETIC SYSTEMS ia ib 't eu F I I, :1 I: !! I' I' cJ>1 4>2 m'l Fig.2.21 yI (b) (a) A magnetic systcm. (b) A magnctic equivalem circuit for lhe systcm. -- - path in a magnetic field system. In a magnetic equivalent circuit this law is represented by the following relation: Around any closed path, the total magnetomotive force of thc windings is cqual to the surn of the products of reluctance and fluxo I,ff= I,.%/> around closed path (2.55) The continuity of magnetic flux in the magnetic field is represented by equating the sum of the fluxes entering any junction of magnetic paths in the equivalent circuit to zero L <Pinto = O (2.56) [unctlon Normal methods of circuit analysis may now be ernployed to determine the f1uxes in Fig. 2.21b for a given set of magnetomotive forces. If the relative permeability of the magnetic material can be considereel constam, the reluctance of the sections may be deterrnined by use of eq. 2.52, in which 1 is the mean length of the flux path in each section anel A is the cross-sectional area, If the permeability cannot be considered constant, each reluctance element may be represented by a graph of the relation EQVIVALENT cmCVITs: COMPLEX MAGNETlC SYSTEMS 133 between its flux anel its magnetomotive force. Graphical or trial-and-error methoels may then be used for analysis of lhe equivalent circuit. lt should be noted that all the assumptions are introduced in the process of deriving the magnetic equivalem circuit from the magnetic system. With elifferent assumptions, a difTerent equivalent circuit is obtaineel. For exarnple, if the leakage fluxes in the air paths around the windings in Fig.' 2.21 a had not been neglectcel, the reluctances of thesc air leakage paths wculd havc been connecteel across the respective magnetomotive forces in Fig. 2.21 b. There is therefore no uniq ue eq uivalent magnetic circuit for a magnetic system. The chosen circuit shoulel contain just the information required for the problem to be solveel. 2.4.2 Derivation of Electrie Equivalent Cireuits A magnetic equivalent circuit, such as that shown in Fig. 2.2Ib, is most useful in the analysis anel design of a elevice. However, ir lhe device is connected to other electric elements, it is elesirable to have an equivalent circuit for the device frorn which the relationships between the terminal voltages and currcnts can be obtained directly, In this section it is shown that the electric equivalent circuit can be elerived directly anel uniquely frorn the magnetic equivalent circuit. First, let us consieler the simple magnetic circuit of Fig. 2.20b, which relates two variables--the coil magnetomotive force ff anel the flux c/>-hy the rcluctancc para meter /.;1. .'F = .'.;1</, amperes (2.57) Tn the equivalent electric eireuit, the variables are the voltage e between the coil terminais and the current i in the coil. Let us assume that the core reluciancc is constant and that the coil resistance is negligible. The electric circuit variables are related to lhe magnetic circuit variables by the two \ relations .':1--:- i =- arnpercs (2.58) N and d(p volts (2.59)e= N- ~. dr I. By substituting from eqs. 2.57 anel 2.58 into eq. 2.59, the relation between the two electric circuit variables is e = N !:!..(ff) dI .<JIl N d.'F N~ di !.!4 dt ;;1 dt = LqJ dr volts (2.60) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ó ) . ) ) ) (" ) 138 ANALYSIS OF PERMANENT-MAGNET SYSTEMS 139ANALYSIS OF MAGNETIC SYSTEMS residual value Br. Suppose a reversed magnetic field intensity of magnitude H, is now applied to the path. On its removal and reapplication, the B-H locus follows a minor loop, as shown. Jn most analyses this minor loop may be considered as a single straight line. Application of a reversed magnetic fic\d intensity of magnitude less than fi s causes an excursion along this Jine. I f the reversed magnetic field intensity has a magnitude greater than H" the locus of operation moves down to a lower and more or less parallc\ Jine. The incremental slope of these lines representing minor loops is known as the recai! pcrmeability, For Alnico magnets it is in the range of 3-5 !-lo, whereas for ferrite magnets it may be as low as 1.1 !-lo. After a magnet has been initially magnetized, usual practice is to stabilize it by subjecting it to a demagnetizing magnetic field intensity H, that is somewhat larger than the magnet is expected to encounter in service. As long as this demagnetizing intensity is not subsequently exceeded, the magnet should operate along an essentially straight-line minor loop. around the system gives HII/'" + Hgl" = O, or '~, I"rr", = - H" --'- . I", amperes/meter (2.69) The continuity of flux around the path requires that BmAm = BuA", or Bm = B A" . " Am webersjmeter 2 (2.70) By substituting from eq. 2.69 into 2.70 and noting that B, = !-loHg, we arrive at the following relation between the flux density and the field intensity in the material. A" i;Bm = =u« - - H m Am I" webers/meter" (2.71) The second relation between Bm and H", is the B-H curve of Fig. 2.24b. As shown, the opcrating point of the material is at the intersection of the B-H curve and the straight line representing eq. 2.71. As the magnet keeper is removed, the operating point of the material moves along the IOCllS a-b. If the keeper is reinserted, the operating point of the material moves along the essentially straight-line recoil locus b-c. The foregoing analysis shows that the operating point of an air-gap magnet is determined by the demagnetization portion of its B-H loop and by the dimensions of the magnet. The corresponding design problem is to choose the operating point and dimensions of the material so that a given air-gap field may be produced with a minimum of magnet material. Suppose we wish to obtain a flux density Bg in an air gap of length lu and cross-sectional arca Ag. By using eqs. 2.69 and 2.70, the required volume of magnet material is 2.5.1 Pcrmancnt-Magnct Systcms with Air Gaps Figure 2.24a shows a permanent magnet with an air gap. Supposc this magnet has bcen magnetized with a soft iron keeper in its air gap lcaving Em = (BuAg) (-HgIg) e; u; ~ T lg A,lm - JJ.ü A--;;:r; Vm = A"J", -L----~ .Hm (a) (b) Fig. 2.24 (a) A permanent magnet with an air gap. (b) Graphical analysis of air-gap magnet. B/Vg !-lo IB",Hml (2.72) meter" it in the residually magnetized state denoted as a in Fig. 2.24b. Wbat will be the flux density in the magnet and in the air gap if the keeper is now withdrawn from the air gap? Let us assume that the magnetic fíux is confined to the area Am of the magnet and to an effective arca Ag of the air gap (making allowance for some fringing of flux around the gap). Application of the circuital law Thus, to produce a fíuxdensity B. in an air gap ofvolume V g , a minimum volume of magnet material is required if the material is operated where the magnitude of the product BmH m is greatest. This product is a measure of the energy that can be supplied per unit volume ofmaterial to an air-gap field. It is known as the energy product of the material and its value is in the range of 5000-50,000 joules/rneter" in the better permanent-magnet materiaIs. 140 ANALYSIS OF MAGNETIC SYSTEMS When an operating point such as b in Fig. 2.24b has been chosen, the length 1m and area Am of the magnet material may be chosen to make the.Tntersection of lines occur at that point. Jn the simple magnet of Fig. 2:24a the areas Ag of the air gap anel A", of lhe material are nearly equal. The air-gap fíux density must then be essentially the sarne as the flúx density in the material. lf these llux e1ensities are to be difTerent, the magnet may be fitted with pole shoes of soít magnetic material to increase or decrease the gap area as required. 2.5.2 Linear Models for Permanent Magnets The fact that the operating locus 01' a permanent magnet is an es- sentially straight line, as shown in locus b-c of Fig. 2.24b, suggests that the magnet might be represented by a linear moelel. Such a model woulel facilitate calculations, particularly in cornplex systems that incluele permanent magnets. Figure 2.25a shows a permanent magnet of area A", and length '-m' which forms part of a closeel magnetic path encircled by a coi!. Let us assume that the soft magnetic material req uires negligible magnetic field intensity. Figure 2.25b shows the relation between the magnetic Ilux 1>manel the magnetomotive force .'Y", of the block 01' permanent-magnet material. This curve may be obtained by rcscaling the e1emagnetizing portion of the B-H curve of the material using 4>", = BmAm webers (2.73) anel ffm = Hu/", amperes (2.74) Suppose that the magnet is initially magnetizeel using a positive current i anel then stabilizeel by application of a negative currenl sufficient to make ff m = Ni = -#. This brings the operating point on the'1>m~ffm locus to point a, where the magnet fíux is 'P<L" lf, in lhe subsequent operation of this magnet, the magnitude of thAemagnelomotive force applied in the negative direction does not exceed ff, the magnet operates along the locus a-b-c. This locus may be closeIy approximated by a straight line of slope lj9t o denoted by the expression amperes (for .'F", >-.#)ffm = .:«, + 9104>", (2.75) This equation describes the equivalent magnetic circuit of Fig. 2.25c. The magnet is represented as a source of magnetomotive force .'Yo in series with a reluctance ~o· The part of the system externa! to the magnet is simply a magnetomotive force ff", = Ni in this case. Equation 2.75 may be elivided through by ,q,fo anel rewritten in the form ff", 1>m= 1>0+ 91 o webers (for 1>",> 1>,,). (2.76) ANALYSIS OF PERMANENT-MAGNET SYSTEMS 141 ) ) ) ) ) ) ) Soft magnetic material c ) ) ) ..- ..-/...- .--/...- ...-/.--...- --L 1 -Slõ .) j ) ~ [ilto ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) I ) () <Pm x .Y .Y (c) (d) Fig. 2.25 (a) A closed systcm containing a permancnt magnet and a coil. (b) Graphical analysis, (c) Equivalcnt circuit with magnctomotive force sourcc. (ti) Equivalem circuit with rnagnctic flux source. This eq uation e1escribes the alternative form of equivalent magnetic circuit of Fig. 2.25d. In this circuit the magnet is representeel as a fiux source 10 in parallel with a reluctance :#.0' This form of equivalent circuit is prefcr- able to that of Fig. 2.25c for rnagnets that approach the ideal behavior of constant flux and very high equivalem reluctance 810, The two equivalent circuits of Fig. 2.25c and d may be considereel as' analogous to the Thévenin anel Norton forms of electric equivalent circuit. When the magnet is closeel by a zero reluctance path, its "short- circuit" Ilux is 4>0' The incrernental reluctance encountered by a magneto- motive force applied to the magnet is Mo. The magnet cannot, of course, ') ) 142 ANALYSIS OF MAGNETIC SYSTEMS ) ) \ be "open circuited," since the air path between the ends of lhe magnet a!ways has, a finite reluctanee. ln addition, the mode1 applics only for s;> -»: As an cxarnplc of the use ar thcsc linear ruodcls, let us determine the magnetic flux in the air-gapped magnet 01' Fig. 2.24(/. Suppose that the magnet has been stabilized 50 as to operare along thc locus h-c in Fig. ) ) <P ) ~.flg Fig. 2.26 Equivalcnt magnctic circuit for thc air-gappcd magnct of Fig. 2.24a. ) ) 2.24b. The magnet may be represented by a fiux souree 4>0 in parallel .with a reluctance Sio. lf B() is the flux density at point c ill Fig. 2.24h, and ir the s!ope of the line l;-c is the recoil permeability f.lrP'o, 4>0 = BoA '" webers (2.77) ) and .;,b I .71 n == - '''- I"r,u.()11 ampcrcsjwcbcr (2.78) ) ) ) ) ) ) ) ) ) ) ) ) the Illagnet consists of lhe rcluctanceThe magnetic systcrn externa I to .0f. of the air gap, where ~=lo l"oAg arnperesjweber (2.79) The system may therefore be represented by the equivalent eircuit of Fig. 2.26. The magneticfiux 4> in the air gap is given by SioJ. J. - '1'0 'I' - []i{o + Si. webers (2.80) Problems 2.1 A toroida! magnetic core (as shown in Fig. 2.1) has a cross-sectional arca 01' 1 em" and an average radius of 2.5 em. Thc material has the set of cyclic hysteresis loops shown in Fig. P2.1. The coi! on the core has 200 turns and neg!igib!e resistance. ) ) PROBLEMS 143 1.61-1-r-==-:l~===1======j===::~~1 ~-~--~-~--~~ N E?08 cQ 04~1-vt+ ~1-II4J11j I 80 100 120 OO~ 20 40 so-20 o r- I I I \ ! I \, H, arnp/rn Fig. P2.J (a) Suppose a sqllare-wave voltage having a peak-to-peak value of 12 V is applied to the coi!. What value of the period of this voltage wave will produce a peak flux density of 1.6 Wb/m2? Ans.: 21.4 msec. (b) Sketch the current waveforrn for the condition of part (a). Determine the peak value of the current in the coi!. Ans.: 0.0942 arnp. (c) What rms value of a sine-wave voltage at 50 c/s would produce a peak flux density of 1.2 Wb/m2? Ans.: 5.34 V. (d) Sketch the current waveforrn for the condition of part (c). Determine the value of the eurrent at thc instant of maximum voltage. Ans.: 8.64 X \0-3 amp. 2.2 (a) Using the B-H loops of Fig. P2.1, plot a normal magnetization curve for the magnetic material. 148 ANALYSlS 01' MAGNETIC SYSTEMS the supply voltage. The relay wil! elose when its half-cycle average current (disregarding polarity) reaches 10 m amp. The inductor is to have a toroidal core rnade of 50% nickel, 50% iron. The B-H characteristic of this material may be idealized as shown in Fig. 2.7h wilh E" = 1.5 Wb/m2. To insure rnechanical reliability, the cross-sectional arca 01" the wire should not be less than about 0.1 mm", Design a suiiablc inductor, specifying the core dimensions and the nurnber 01" turns on the coil, 2.12 Figure P2.12 shows a proposed system for measuring the frequency of an alternating-voltage signa!. A nonlinear inductor in series with a R d-e voltmeter Variable frequency signal Fig. P2.12 resistor R is connected to the signal source. The voltage across the inductor is applied to a full-wave bridge rcctifier and rncasurcd using a high- resistance d-e volt meter. (a) Suppose the signal voltage is of sufficient amplitude to saturate the inductor core before the end of each half cycle of the signal. The inductor may be assumed to have an idealized B-H characteristic, as in Fig. 2.7b, with a saturation flux density Bk' Let the cross-sectional area of the core be A, and let N be the number of turns. Assuming the coil resistance to be negligible in comparison with R, show that the voltmeter reading wil! be E = 4NABd V, wherefis the signal frequency. (b) The meter is to measure frequency over the range of 25-500 c/s. Let Bk = 0.7 Wb/m2, A = 0.2 cm-, and N = 500. What minimum rms values of sinusoidal voltage at 25 c/s and at 500 c/s must be applied to give correct measurement of frequency? An.\". 15.6 V for 500 c/s. (c) Show that the volt meter reading will be 0.028 V per eIs for any waveform of adequate amplitude, provided that the volt age changes sign only once in each half cycle. pnOBLEMS 149 2.13 An iron-cored inductor, comrnonly known as a choke, is often useel to smooth out the ripples on the current Irorn a rectifier. To be effective, the choke must have an appreciable induetance when carrying the elirect current of the rcciifier. Saturation 01' the magnetic path is preventcd by lhe use of an appropriate air gap. Dcsign a choke to have a n inductancc of 10 H whcn carrying a dircct current 01' 0.5 amp. The magnetic material ma} be assurned to require no appreciable magnetic field intensity up to a flux density 01' 1.4 Wb/m2• A current density 01' 2 arnpjmrn" may be used in the coi!. To start the design, the volume 01' the air gap required to store the magnetic field energy may be found. A reasonably shaped air gap may then be chosen. Next, the number of turns may be found; allowing adequate space for these turns and their insulation, the dimensions of the magnetic core may be chosen. 2.14 A magnetic core 01' the Iorrn shown in Fig. P2.14 is made 01' a material for which lhe B-H loops can be represented by the lincarizcd 4C;(Z /~ 7'--- --f-- f . ~[ ~.6cm--~J ~--lOcm-- t c I N=300 Fig. P2.J4 model shown in Fig. 2.9, with H, = 10 ampjrn, Bk = 1.2 Wb/m2, fi" = 105, and 11, = 10.1. (a) Finei the peak value of sinusoidal voltage e at 50 c/s that will drive the core arounel a B-H loop passing through Bk anel Hc' The coil resistance is negligible. (b) Sketch the magnetizing current for lhe applicd voltage eleterrilined in part (a). What is its peak value ? An.:.: 0.026 amp. (c) Determine thc hysteresis loss in the core for the operating conelition 01' part (a). Alls.: 0.76 W. 2.15 The toroidal magnetic core shown in Fig. P2.15a has a uniformly distributed winding of 200 turns. The magnetization characteristie of the ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . ) 150 ANALYSIS 01' MAGNETIC SYSTEMS magnetic material can be approximated by the idealized Iorrn shown in Fig. pi15b. Since the outside diameter of the core is considerably greater than lhe inside diameter, the magnetic ficld intensity cannot be considered uniform over the cross-sectional area of lhe core. b/ -r- IT 10[L JJ,VV 1.5 -15 15 fi, . -- arnp/rn / G 1.5 (a) (/J) Fig. P2.15 (a) Sketch the relationship between lhe flux linkage and the current of the coil. (b) Suppose a sinusoidal voltage at a frcquency of 400 c/s is applied to the coil and increased in amplitude until core is operating between its saturation limits, The coil resistance may be ignored. Sketch the waveform of the magnetizing current, and determine its peak value. Alls.: 0.0236 amp. , (c) Determine the in-phase and quadrature fundamental-frequency components of the magnetizing current, (d) Compare the magnetizing current of this core with that of another core having the same cross-sectional area and volume, but having a near-unity ratio of outside to inside diarneter. Show that the quadrature component of fundamental-frequency magnetizing current arises from the shape of the core and not frorn its E-H characteristic. (e) Show that the hysteresis loss obtained from the in-phase component of fundamental-frequency magnetizing current and the applied voltage is equal to the area of the B-H loop multip1ied by the core volume and the frequency. 2.16 A relay is to be energized 0.25 sec after a switch is closed. The proposed scheme, shown in Fig. P2.16a, consists of a nonlinear inductor having two identical windings, one of which is connected in series with the relay coi!. This winding is intended to absorb most of tbe battery '. I I t PROBLEMS 151 voltage for the 0.25-scc interval after the switch is set to position a. At the end of this interval the core saturates, and most of the battery voltage is applied to the relay coil. The second winding is required to reset the core flux density to negative saturation when the switch is operated to positi.on b. The relay will operate with a currcnt of 0.01 amp. This current may be increased to 0.02 amp without overheating. The relay coil may be con- Relay coil ------7----- "'~ (a) .-i t O.2dB ""'---- . .<1---- -, -'-1---- ., k-O.2d (b) Fig. P2.16 sidered to have negligible inductance and to have a resistance of 200 O. Toroidal cores having the shape shown in Fig. P2.l6b are available with inside diameters of 2.5 em, 4 em, 5 em, and 6 em. Figure P2.l5b is a reasonable idealization of the E-H charaeteristics of the magnetic material. The current density in the inductor coils should not exceed 2 ampjmm-. The same wire size may be used for both coils. Design a suitable inductor for this application. 2.17 Rectangular voltage pulses of the form shown in Fig. 2.llb are produced each time a pàckage 011 a conveyor belt passes a detector. The pulses have a height of 3 V and a duration of 1 msec. A magnetic counter, which will give a 3-V output pulse across a 20-0 load for each tenth package is required. An available toroidal core has a eross-seetional 152 ANALYSIS OF MAGNETIC SYSTEMS area of 0.5 em" and a mean dia meter of 3 cm. Its B-H characteristic is shown in Fig. P2.15b. Determine the approximate number of turns required on the core. Ans.: 172. 2.18 Figure P2.l8 shows typical ferrite cores in the memory system of a digital computer. Each core has a rectangular hysteresis loop of the form shown in Fig. 2.10, with BT = 0.3 Wb/m2 and H, = 25 arnp/rn. Each core has a mean diameter of 2 mrn and a cross-sectional area of 10-7 m". -. iy z Fig. P2.18 (a) Determine the magnitude of a current pulse i" that will provide a magnetie field intensity equal to 0.7 11" in the core. Ans.: 0.11 amp. Note that this pulse will not ehange the core fluxo (b) To change the magnetic state 01' a core frorn negative to positive saturation, simultaneous current pulses i", and ', are applied, each having the amplitude found in part (a). Simultaneous pulses of reverscd polarity return the core to the state of negative saturation. A winding, denoted as z, passes through ali the cores. The appearance of a voltage pulse e; on this winding indicares that flux switching is taking place in the core linked by the simultaneously applied eurrents ix and iy. If the core flux switches at a uniform rate between saturation lirnits in l.0 usec, determine the amplitude of the pulse ez• Ans.: 0.06 V. (c) If the core is already at positive saturation, will any voltage ez be produced when the core is pulsed with currents ix and i,,? 2.19 When a 115-V rrns, 60-cjs source is connected to a nonlinear inductor, the current is 0.23 amp rrns and thc input power is 14.1 W. The coil resistance is negligible. Observation of the eurrent waveforrn , r PROBLEMS 153 on an oscilloscope indicates that the third-harmonie component of the magnetizing current is about 0.3 of the fundamental component and that higher harmonics are negligible. Determine the values 01' the two parameters in an equivalent circuit of the form shown in Fig. 2.12c. Ans.: 940 i2; 630 n. 2.20 The magnetizing current 01' an iron-cored induetor can be expressed approximately as i = 0.28 sin (0)/ + 30°) - 0.12 sin 3(0/ + 0.05 sin 50)/ The applied voltage is 160 cos O)! V. (a) Sketch the waveform of the magnetizing current. (b) Determine the rms value of the magnetizing eurrent. Ans.: 0.218 amp. (e) Determine the average power entering the inductor. Ans.: 11.2 W. 2.21 A sinusoidal voltage of 200 V rms at 500 c/s is applied to an 80- turn coil on a toroidal core of eross-sectional area 10-3 m2 anel mean diameter 0.1 m. Thc core material can be rcprescnted by rhc idealizcd model of Fig. 2.10 with B, = 1.6 Wbjm2 and H, = 20 ampjm. The coil resistance may be ignored. (a) Determine the amplitude and waveiorrn of the magnetizing current. (b) Derive a Fourier series representation for the magnetizing current, anel evaluate the rms value of the third-harmonic component. AI/s.: 0.0235 <lmp. (e) Determine thc ratio of the rms valuc of the fundamental-frequency component to the rrns value of the total magnetizing current. Ans.: 0.9. 2.22 A manufacturer lists a grain-oriented magnetie material in sheets which are 0.3 mrn thick. The resistivity of the material is given as 5 x J 0-7 l2-Ill. The published static B-H loop for the material is essentially rec- tangular in forrn, with a coercive force of 12 ampjrn for ali values of peak ílux density in lhe range 0.8 to 1.6 Wb/rn2. The quoted value for the total loss in lhe material is 1.2 W/kg, with a sinusoidal f1ux density of 1.0 Wbjm2 (peak) at 100 cjs. The mass dcnsiry of the material is 7650 kgjm3• Calculare the loss in lhe material from its properties, and compare the value obtaincd with the quoted valuc. Suggest possible reasons for the errors in predicting the loss. 2.23 The magnetic core shown in Fig. P2.14 is made of laminations of thiekness 0.35 mrn and resistivity 6 X 10-7.o-m. The statie B-II-char- acteristic of lhe material may bc approximuted by the idealized Iorrn shown in Fig. 2.7, where Bk = 1.2 Wbjm2• The coil resistance is negligible. (a) Show that the inductor can bc represented as a resistance connected ) ) 158 ) ) ) ~... " ) Air gap ) ANALYSIS OF MAGNETIC SYSTEMS ) ) ) Fig. P2.33 (a) Develop an equivalent eleetrie eireuit for the system deseribing each inductive element either by its inductanee value or by its flux linkage- eurrent relationship. (b) Neglecting the resistancc of the 120-turn coil and assuming the 35-turn coil to be open eireuited, determine the peak amplitude and the width of the output pulse. Ans.: 11.56 V; 0.196 msee. (c) Using the equivalent circuit of part (a), devise a system having similar terminal properties to those of the system or Fig. P2.33 and using a saturable toroidal core with 120-turn and 35-turn windings. Ir the toroidal core is made of the same material as that used in the system of Fig. P2.33, speeify the required cross-seetional arca. 2.34 A rectangular block of permanent-magnet material is to be mag- netized using an arrangement similar to the one shown in Fig. 2.25a. ) ) ) ) ) ;. ) ) ) ) ) ) ) ) ) ) ) ) ) ) , ) o. 1.6 1.2 ",. E:o- 3: 0.8 A:1' -60 -40 -20 o 20 H, karnp/rn Fig. P2.34 40 60 80 r I I \ ! \ , ~ ~ i li i H /; I l' I : ~. I ~~ I j, ,1 I ~ ~~ PROBLEMS 159 The block has a 1ength of I", = 5 em and a cross-sectional are a of 10 em". The soft magnetic material may bc assurned to have infinite permeability. The coil lias 100 turns. Figure P2.34 shows the E-H characteristic pub- lished by the manufacturer of the material. (a) Suppose that the material has an initialflux density of zero. Deter- mine the peak value of the coil current i required to magnetize the mag- net so it will have a residual flux density of 1.35 Wbjm2• Ans.: About 40 amp. (b) Determine approximately the energy required to magnetize the magnet under the conditions or part (a). Coil resistance may be neglected. (c) One simple method of magnetizing a magnet eonsists of connecting a charged capacitor to the winding in the system shown in Fig. 2.25a. If the capacitance is 100 {tf, to what voltage should the capacito r be eharged to provide the necessary magnetization cunent? Coil resistance may be neglected. Ans.: 290 V. (d) The method of magnetization discussed in part (c) produces an oscillatory eurrcnt, which might tend to dcmagnetize the magnet. Show that this demagnetization can be prevented by placing a rectifier in series with the capacitor. Soft magnetic material Fig. P2.35 2.35 In the system shown in Fig. P2.35, the permanent-magnet material has the E-H characteristics shown in Fig. P2.34. The soft magnetic material may be assumcd to have infinite permeability. Initially, a keeper made of soft magnetic material is inserted into the air gap, and the per- manent magnet is magnetized to a residual flux density of 1.35 Wbjm2 by means of a coil. (a) Determine the flux density in the air gap when the keeper is removed. Fringing ftux around the air gap may bc ignorcd. AlIs.: 0.745 Wbjm", (b) The permancnt-magnet material has a recoil permeability of about 2.0 {to. Suppose the keeper is reinserted into the air gap. What is the new value of residual f1ux density in the permanent-magnet material? Ans.: About 1.25 Wbjm2• 160 ANALYSrS OF MAGNETIC SYSTEMS 2.36 In Probo 2.35 the air flux was assumed to be confined within the air gap. There is, however, some fringing flux around the gap edges. This may be included by the method discussed in relation to Probo 2.32. In addition, there is a leakage ftux between the upper and lower horizontal sections of soft magnetic material. Let us assume that this leakage flux density is uniform throughout the volume of the air space bctween lhe two horizontal members and is zero outside this volume. (a) lnitially, the permanent magnet has a residual flux density of 1.35 Wb/m2 with the keeper inserted in the air gap. Determine the flux density in the permanent magnet when the keeper is removed. Ans.: 1.23 WbJm2. (b) Determine the air-gap flux density and compare it with the value found in Probo 2.35(a). (c) The leakage flux of the arrangement in Fig. P2.35 may be reduced considerably by placing the permanent-magnet material closer to the air gap. Sketeh an improved rearrangement of the system using the same volume ofpermanent-magnet material and the same air-gap dimensions. 2.37 A magnetic flux density of 0.8 Wbjrn" is required in an air gap having a length of 2 em and a circular cross section of radius 10 em. Permanent-rnagnet material having the E-H characteristic of Fig. P2.34 is available in cylinders of any length and radius, Soft magnetic material is also available in any shape ; it can be regarded as pcrfect up to a Ilux density of 1.4 Wbfm2• Since the permanent-magnet material is relatively costly, its volume should be minimized. (a) At what value 01' l1ux dcnsity should thc pcrrnnncnt-mugnct material be operated? (b) Design a permanent-magnet assembly that wiJl have reasonably low leakage. (e) What volume of permanent magnet material is required? Ans.: 5670 em", approximately. Permanent magnet air gap 1.5 em-1 r" Fig. P2.38 PROBLEMS 161 2.38 Figure P2.38 shows a half-sectional view of the permanent-magnet assernbly of a loudspeaker (see Probo 1.26). A magnetic fíux density of 1.2 Wbjrn" is required in the cylindrical air gap. Perrnanent-rnagnetic material having lhe characteristics shown in Fig. P2.34 may be used. (a) Neglecting leakage and fringing tluxes, determine the appropriate dimensions for the various parts of the magnet assernbly. The flux density in the soft iron should not exceed 1.2 Wb/m2. (b) Determine the mass 01' the perrnanent magnet if the material has a mass density of 7330 kg/m3• Ans.: 0.31 kg. 2.39 Suppose lhe pcrrnanent magnct in Fig. P2.35 has becn stabilized so that its residual ílux density is 1.2 Wbim~ (with the keepcr in the air gap). The recoil permeability 01' lhe material is 2.0 lho. (a) Represent the systern by a linearized magnetie eircuit. Include a reluctance elernent to represent, approxirnately, the leakage f1ux path between the urper and lower horizontal mernbers. For sirnplicity, the field may be assumed to be un iform within the space between those mern- bers. Make the usual allowance for fringing flux in calculating the reluctance of the air gap. (b) Use lhe equivalent circuit to determine the air-gap fluxo Ans.: 2.46 x 10-:1 Wb, including the fringing fluxo 2.40 Figure P2.40a shows a srnall magnet assembly intended ror use as a dom holder. The kccpcr is to be attached to the door, whilc the rernaindcr of the asscmbly is to bc attnchcd to thc door frame. Thc perrnanent magnet is made 01' a Ierrite ceramic material. The E-H characteristic for this material is shown, in sornewhat idealized form, B. Wb/m2. J04 Permanent magnet / ~Oft rron I~I Keeper 2cm ~" 1/ ?rt2emj~~~Ç= J !~ I, 2 em I 2 em 1,1 I.I I rOSem -173.000 ---------10.15 I ~ H. arnpjrn (a) (b) Fig. 1'2.40 ) ) ) ) ) ) ) 162 ANAL YSIS OF MAGNETlC SYSTEMS PROBLEMS in Fig. P2.40b. The soft iron may be assumed to have infinito perme- ability. (a) The recoil permeability of the permanent-rnagnet material is 1.1511-0. Show ihat the sloped part of the demagnetization characteristic may be uscd as the recoil characteristic for this material. (b) Derive a linearized magnetic circuit for the magnet asscrnbly, neglecting leakage and fringing f1ux components. Determine the max imum value of air-gap 1ength x for which this magnetic circuit is valido AI/S.: 1.81 mrn. (c) In one of its possible forms, the equivalent magnetic circuit consists of a constant magnetomotive force in series with three reluctance elements, two of which represent the air gaps. Use the analogy suggested by this cquivalcnt circuit to derive an expression for the force aeting on the keeper at any value of gap spaeing x. Evaluate the cxprcssion forr = 1 mrn. Ans.: 55 newtons. (d) The linear model derived in part (b) prediets that the magnet will be demagnetized if the air gap exceedsa length of 1.81 rnm. Show that the leakage reluctance between the upper and lower soft-iron mernbers is low enough to prevent demagnctization frorn occurring. ) ) 163 Soft iron pole piece ~ :: Pivot " , Rubber mounting (c) Show that lhe induccd vollage c in lhe coil may be expressed as e=N<Po(. !JtIo )dÕ t :Jfo + _1_ di 11-0 A (d) Design a reluctance pickup assigning any reasonable set of dimen- sions to lhe various parts. A ferrite magnet having the demagnetization characteristic of Fig. P2.40b may be used. 2.43 A magnetic field of density B is required in the rectangular channel of width x, height y, and Iength z of a magnetohydrodynamic machine (see Probo 1.28). The proposed design, shown in Fig. P2.43, consists of a ) ) ) ) ) ) ) ) ) ) ,) ) ) ) ) ) ) ) ) ) ) j. ) Fig. P2.42 2.41 A meter movernent of the type shown in Fig. P 1.27 has a magnetic flux clensity of 1.1 Wbjrn" in the spaces between the pole faces and the central core. The total volume of this magnetic ficld is 1.5 em". (a) If Fig. P2.40b represents the B-H characteristic of the material, determine the minirnurn volume of permanent-magnet material required to establish the magnetic field. Ans.: 52.1 em", (b) lf the material whose characteristics are shown in Fig. P2.34 were used, what would be the required volume of the magnet? 2.42 Figure P2.42 shows a magnetic reluctance type of phonograph pickup. As the stylus moves from side to side in the groove of the recording, the central magnetic mernber rotates about its pivot. A displacement of the stylus causes a proportional increase () in the lengths of two of the air gaps while the lengths of lhe other two air gaps are decreased by o. The rubber mountings restrain the deflections o to small values so that Õ is much less than the average gap spacing t. Each gap has a cross-sectional area A. (a) Develop a magnetic equivalent circuit for this device. The magnet may be represented as a flux source <Po in parallel with a reluctance ·!J1!o. Denote the flux <P linking each turn of lhe pickup coi!. (b) Show that, if o « I, the reluctance seen by the magnet is essentially constant. z// -L/ x/2 Iron m+v::;7~/? l1IDJ Copper.:>ttllt? 00 coil I .. Fig. P2.43 y t 168 TRANSFORMATION OF ELECTRIC ENERGY EQUIVALENT CrRCUlTS FOR T1~ANSFORMERS 169 electronic or communication apparatus, transformers are required to operate over a wide range of audio and radio frequencies. Further applications of transformers arise frorn their ability to couple two electric systems without requiring a conductive connection between them. This isolation feature is of particular importance in the transforrners designed for measurement purposes. In analyzing many transformer applications, eqs. 3.1 and 3.2 are obeyed to within a few per cent, and ideal behavior can bc assurned. There is probably no otherelectrical device that approximates its ideal behavior so closely, but there are many situations in which even smal! imperfections are important. For example, an efficiency of 98 ~{, may be considercd as essentially ideal. The heat arising from the remaining 2 % loss, however, must be withdrawn from the transformer; it is thus of major concern to the designer. Even a 2 % loss in a large transformer is economically important. In some transforrners, significant imperfections are inten- tionally introduced to achieve a desired operating characteristic. In the following sections, the imperfections of transforrners are discussed, and means of analyzing their effects are developed. (a) Ia 2a 3.1 EQUlVALENT CIRCUITS FOR TRANSFORMERS In this section we develop some equivalem circuits that are uscful in predicting the behavior 01' transformcrs. In cach instancc lhe cquivalcnt eircuit eonsists of one or more ideal transforrners and elernents to rep- resent the physical properties that cause a departure frorn ideal behavior. Some of the imperfections in the transforrner are readily appreciated. There is resistance in each of the windings, and this prcduces both a loss of power and a drop in voltage. The Icrromugnctic core requires Iinite magnetic field intensity to produce its magncric fluxo Thc ideal currcnt relation of eq. 3.2 does not apply exactly, bccausc lhe surn of the magneto- motive forces of the coils must be sufficicnt to supply this field intensity. Perhaps the most important imperfection in a transforrner arises from magnetic flux that fails to link aI! the windings. As this leakage flux varies with time it produces an induced voltage in some 01' the windings, but not in others; it thus causes a departure from the ideal voltage relation of eq. 3.1. To minimize leakage flux, the windings are arrunged close to each other. Figure 3.1 shows severa I common arrangements of cores and coils. In the shell-type construction of Fig. 3.1 a the wi ndings are concentric, winding 2 being placed outside winding I. The core consists of laminations that are stacked so that the windings encirc1c the centrallcg. The core-type construction of Fig. 3.1 b has two identical sets of concentric coils placed on the two legs of a single-path laminated core. Figures 3.1 c and ti show similar shell- and core-type arrangements using cores that are wound 2 (c) 16 26 Fig. 3.1 Winoing ano core arrangemcnls. (a) Shell-type with core of larninated shccts. (b) Corc-typc with core of laminatcd shccts. (e) Shcll-typc with wound core. (d) Core- lype with wound core. Irorn a long strip of magnclic material. This wound-corc arrangcrncnt is particularly advantageous whcn grain-oricnled magnctic material is uscd, 3.1.1 Equivalent Circuit for Two-Winding Shell-Type Transformcr Methods for developing an cquivalent electric circuit to represent a magnetic system approximately were discussed in Section 2.4. Let us now apply these methods to the two-winding shell-type transformer, a cross section of which is shown in Fig. 3.2a. The first step in proclucing an equivalent electric circuit is to reduce the significant features of the magnetic systern to a magnetic equivalent circuit. The pertinent assump- tions are introduced at t his stuge. The flux pattern in Fig. 3.2a occurs whcn the magnetomotive force 5"1 of lhe inner winding I is sligh!ly grcaler than the rnagncromotivc force ,'Y2 of the outer winding 2. A magnetic ílux </)1 passes upward through thc central leg and divides into two equal parts that proceed outward along lhe yokes. A part ,h of this Ilux takes the path down / ) ) ) ) 170 TRANSFORMATrON OF ELECTRIC ENERGY (a) ) ) ) ) ) ) Ii ~ ) I ) I I i ,I . ) I ) i 1 I )- I ) I )\ ) ) ) '<PI. b ~~- (~ /-1/ -, - -- " \:'11\ <; \ -, I :n,~I L- ~ ~ / / .--/// - - -:.. -=- -=-."'7."--=~...::.::::-~::::_./ " (u) R~jwL/~ lz (c) Fig. 3.2 (a) A two-wincling, shcll-typc transformcr ..(h) Magnctic circuit. (c) Equivalent clcctric circuit. ) ) through the air space betwcen the windings to avoid the oppositely directed magnetomotive force of winding 2. The rernainder cp2 passes down the outer Jcgs and thereby links winding 2. This flux pattern suggests that the rerromagnetic core consists of two significant paths, each of which carries a' distinetive magnetic flux. Path I, carrying the flux CPl> includes the centralleg and about half of each of the yoke sections, Path 2, carrying the total flux cp2' consists of the outer legs and the remainder of the yoke sections. This second path consists of two physically separate parts, but, because of the symmetry, they may be regarded as being in parallel magnetically. I ~ EQUIVALENT CIRCUITS FOR TRANSFORMERS 171!' Figure 3.2b shows a magnetic equivalent circuit in which the assumed paths of flux are reprcsented by their reluctances. In this circuit, winding 1 . (represented by its magnetomotive force :FI) is linked by the flux CPJ, which exists in thc rcluctance ,911 of path l. This reluctance may be depcndcnt on lhe valuc of the flux cp, and is therefore indicated asnonlinear in lhe cquivalcnt magnctic circuit. The magnetic flux cpL is established in the air spacc bctwccn the windings. Mcthods for determining the rcluctance :?f/, of this leakage path are discussed later in Section 3.1.4. As this path is assurned to be entirely in the air space, its reluctance is constant. The flux 1>2 = cpI - cpL is established in the reluctance ~2 or the combined parts of path 2. This flux CP2 links winding 2 which is represented by its magnetomotive force :F2. Since path 2 isferromagnetic, its reluctance is regarded as nonlinear. In developing an eleetric circuit that is equivalent to the magnetic circuit of Fig. 3.2b, let us use the number N1 of the turns of winding 1 as a reference nurnbcr. The Iorrn of the electric circuit may be derived by use of lhe simpie topological technique described in Section 2.4.2. The result is lhe set of dashed lines joining lhe nodes a, b, and o in Fig. 3.2b. Suppose the transforrner is to operate at an angular frequency of w radians per second. The equivalent eJcctric circuit of Fig. 3.2c includes an inductivc impedancc.,jwLL connected between nodes a and b. The value of the leakage inductance LL is related to the leakage reluctance by NI2 LL = ~L henrys (3.3) as the reference winding has N] turns. The equivalent circuit inc1udes two magnetizing branches representing the reluctances' ~I and [Jf2 of the magnetic circuit. lf these reluetances could be considered constant, the corresponding magnetizing branches would consist of constant induetive impedances. To include the effects of nonlinearity, hysteresis, and eddy current losses in paths 1 and 2, each branch of the eleetric circuit cor- responding to a nonlinear re1uctance is represented by parallel nonlinear resistive and inductive impedances. This mode1 is similar to that developed for simple nonlinear inductors in Figs. 2.l2c and 2.16. For convenience in notation, these nonlinear branches are identified by their admittances Y1and Y2. The induced voltage phasor E, in winding 1 is given by E, = j(ONt(!)1 volts (3.4) where <DI is the phasor representing the magnetic flux CPl' The terminal voltage Eu' differs from the indueed voltage by the drop across the winding 172 TRANSFORMAnON OF ELECTRIC ENERGY EQUJVALENT CWCUJTS FOR T1~A!'\SFORMEHS jwL" , Nc1.. = --, I" - N I - The voltage L:II, in Fig. 3,2(' is cqual to lhe induced voltage in a winding of N[ turns causcd by lhe rate of change of lhe lcakage flux rpL' Note that lhe equivalem circuit of Fig. 3,2/ is not lhe only one which could huvc been developeel for this transforrner. In Iact, the leakage paths are not as simple as inelicated in Fig, 3,2([, Some of lhe leakage Ilux links only a part of each 01' lhe windings anel passes through parts of lhe core lcgs. To rcpresent lhe increaseel complexity in flux paths, more cornplex equivalent magnelic anel electric circuits are required. For- tunately, such complcx rnodels are selelom needed. The circuit may, in fact, be sirnplified by assurning that lhe difference between the fluxes 1>1 anel 1>2 is so small that single reluctance 24\ + f}f2 may be used to represent lhe whole magnetic core, In lhe electric equivalent circuit, this has lhe effect of paraIleling lhe aelmittances YI and Y2, This resultant admittance may be connected bctween nodes a and o if the whole core is (;-=-===-====:=-~~~.:~:- r- l t l (r--, r--i: J I ( 'I [;I' :'", r, "'I ""'.I • I I I 11 x I • • I I I I I I • Ix • I I I x I • • I x I i I 1I I x I • + I tÜ x t' . I x I I I I I : Ix A x trt e tt e I x I I I I I !l.r: ,',"-ur- • J' Ittrt . jI I I I I »: • • " • Ix J • I I 1I I I I x I I I I • I ~ 1 : 1(1)'1(1,1 ::: ::. : (b_l~,: : I I I , I I I x I • • I" I I I I I I~,:~ .."'"'''~rêfi ~I,-_)~~~~~-::-::-=~~~_:'J Winding Ia Winding2a 'xy-Winding lb~IWinding 2b " x " ,'PLb [Ia RI nO I <l>La h'hR't I1,1> ~ L,~ Leg a Leg I> (a) g;a <~-;b / ........•/ I ) i/<I>la / / /I _ I mia I / I / 1I \18T (t' " I« / I , , \ ( -- ----'',-----------~-- (b) Fig. 33 (a) A core-type transformer. (b) Magnctic circuit. (c) Equivalem elcctric circuit as four-winding transforrner, (ri) Equivalcnt elcctric circuit with windings Ia and 1h in parallel, windings 2a and 2/) in séries. resistance RI' The voltage Ez' is given by Ez' = jwNICI)z volts (3,5) I This is the iriduced voltage which would apply for wineling 2 if Nz = NI. The actual induced voltage in winding 2 is Ez, which appears on the other side of the ideal transformer of ratio NI: Nz. Th us E - N2 E'2 - 2 N1 = jwN2Q>2 volts (3.6) Winding 2 has aresistance Rz and a current phasor [2' Ir winding 2 were replaced by a winding of NI turns, its current would have the value noted on the other side of the ideal transformer, that is, (c) 111/2 jwL/./2 r u, I I'---- (eI) Fig, LI. (COIII<I,) a m peres 173 O R~l~aT I " I R·) ! ~'r ':"~ __ .. I~r:, 2f1~ I:~ 1 10'" (3,7) 180 TRANSFORMATION OF ELECTRIC ENERGY It is noted that the departure of lhe voltage ratio form its ideal value N1:N2 not only depends on the resistance and leakage reactance of the windings but ais o on the magnitude and phase angle of the load im- pedance. In large transformers the leakage reactance X1L is considerably greater than the resistance R1e- For a resistive load, the voltage j Xll}2' across the leakage reactance adds in quadrature to the voltage E2' pro- ducing a phase shift between EI and E2' but little difference in magnitude. For an inductive load in which 12/ lags E2' by 90°, EI is equal to the sum of the magnitudes of E2' and X1L/2'. It is this inductive type of load which gives the lowest voltage ratio E2/El' On the other hand, E/ may well be greater than EI in magnitude for capacitivc loads. Suppose we are given the resistance RIe and reactance Xl]; in ohms for the transformer of Fig. 3.6a. To assess the effects of these parameters on the voltage ratio, we might multiply each by the rated current of winding 1 and compare it with the rated voltage of winding 1. The resulting ratios are known as the per-unit resistanee and leakage reactance, and are given by lu R1e = - R1,(ohm,)e; per unit (3.24) and / X1L = Elb XIL(ohms) per unit (3.25) lb where Elb = IEllrated and 11&= 1/1Iratc,!' These per-unit values are rnuch easier to interpret than their ohrnic counterparts. ln large high-voltage power transformers the leakage reactance is generally between 0.\ and 0.15 per unit, whereas the resistance is usually less than 0.005 per unit. As the size of the transforrner is reduced, the pcr-unit rcsistance incrcases, . while the reactance remains in the range 0.03-0.15 in ali but a few special transformers. The departure of the current ratio Irorn its ideal value is causcd by the magnetizing current lhn, as shown in Fig. 3.6b. It is again convenient to use a per-unit notation to describe the magnitude of this current relative to the rated current. Thus Ill1l(IlIIlPcres) 11m= / lb per unit (3.26) Magnetizing currents in transformers are normally Iess than 0.05 per unit. Using wound-core construction (Fig. 3.1) and grain-oriented mate- riaIs, the magnetizing current can be reduced to less than 0.005 per unit. Essentially ali electric load devices are designed to operate well only over a narrow range of supply voltagc. Ir the voltagc is 100 high, heating appliances may overheat, insulation may be ovcrstressed, and lighting I I \ \ ~J OPERATr;-.!G CHARACTERISTlCS OF TR..••:-;SFORIYffiRS 181 ( devices may burn out prernaturelv. If the voltage is too low, motors ma)' overheat or fail to start, and illumination levels are lowered. Even small transient changes in supply voltage result in an undesirable fticker in the illumination from lighting devices. The contribution of a transformer to this change of supply voltage with changing load may be expressed in terms of its voltage regulation. This is defined as lhe per-unit increase in magnitude of lhe output voltage when a specified load is rernoved, the input voltage being held constanl. Using syrnbols frorn Fig. 3.6a, we have ( ( ( ( ( ( ( IE21(oPCH :- IE21(on !oad) Rcgulation = _....:,.:.::'il.:.:·'·'c::1il:.ó.) _ 1E21(on 1o,,,!) The normally specified load is one requmng lhe rated output current and the rated output volt age. To make the regulation meaningful, the power factor of the load must be specified. per unit (3.27) 3.2.1 Effícíency The efliciency '1) 01' an energy transformation device is a per-unit quantity indicating lhe ratio of lhe averagc output power to the average input power. (OUtpllt power 1) = . - 111pu t power In general, transformers are highly efficient devices. In large units the efficiency is 50 close to unity that any attempt to measure this srnall difference from unity by measuring output and input power is frustrated by the small inaccuracies of instruments. An alternative approach is to express the efficiency as per unit (3.28) ( ( output power 1) = output power + losscs The power losses in the transformer may be calculated readily by reference to the equivalent circuit of Fig. 3.6a. The eddy current and hysteresis losses in the core are represented by the real part G\1II of the admittance Y1m and are given by (3.29)per unit ( ( ( r, = G[,u IE,12 (3.30)watts Alternatively, P, may be measured as lhe input power on open circuit as a function of El' The power losses in the windings are approximated by P", = R", 1[2/12 ( J= R". 1/,12 (3.31)watts since the magnctizing current is generally small and partially in quadrature with lhe load current. . c ( ( 178 TRANSFORMATION OF ELECTIUC ENERGY Let I bc lhe mean circurnferential lcngt h 01' lhe windings. Then the energy stored in lhe volume V of lhe magnetic field is w =J ~f10H2 dV .. J "I+d2+d31 o = -f-lol1"!J1 d x o 2 J N2i2/(d11 d2)=2f-l°-Y 3+(3+3 Since W = }Li2, lhe leakage inductancc of the pair of N-turn windings is joules (3.17) L N21(dl . I d2')=/1---j-(.+-o 3 .1 3Y , henrys (3.18) The actual Icakage inductancc is somcwhat lcss than lhe valuc calculated in cq. 3.1 í), bccausc lhe rcl urn path outsidc lhe windings has some re- luctance, and bccause some of the lcakagc flux lcavcs thc sidcs 01' the windings. The expression of eq. 3.18 indicates how the transformer designer can vary lhe leakage inductance by varying lhe dimensions and turns of the windings. To achieve a low valuc of lcakage inductancc, thc winding height y should be large, lhe widths of lhe windings and the spacing between thern should be small, and a minimurn number of turns should be used. A further reduction of inductance may be achieved by inter- leavinglayers of the two windings. In general, high-voltage transformers have rclatively high values of lcakagc inductance because the insulation requirements prevent dose proximity of lhe windings. 3.2 OPERATING CHARACTERISnCS OF TRANSFORMERS We have now rcpresented a transforrncr by an equivalent circuit, which includes an ideal transformer elcment and other elements to modcl the imperfections of the transformer. In this section we use this information to predict some of the operating characteristics of transformers. Let us first examine how much the voltage and current ratios depart from ideal when the transformer is loaded. Figure 3.6a shows an ap- proximate equivalent circuit that is applicable for most transformer studies. The winding resistances RI and R2 have been combined into one equivalent resistance RI. on the winding "1" side of the ideal transforrner, where ( NI)2 RI. = RI + N 2 R2 ohms (3.19) OPERATfNG CHARACTERfSTICS OF TRANSFORMERS 179 RI' jX1D Supply Load f E1 I (a) jXlIh' El 11m 11 (b) Fig. 3.6 (a) Equivalent circuit of transformer with supply and load. (b) Phasor diagram 011 load. The leakage reactance XIL and the magnetizing admittance Ylm are the values as seen from winding 1, which has N1 turns. Let us sllppose that the phasor voltage E2 and current 12 of a load are known and that we wish to determine the corresponding quantities EI and lI' The voltage E2' and current 12' on the winding 1 side of the ideal transforme r are and , NI E 1E2 = - 2 vo ts N2 (3.20) I' - N2 I2 - 2N1 The supply voltage E1 is then given by E1 = E2' + (Rle + jX1L)I2' amperes (3.21) (3.22) and the supply current is 11 = 12' + Y1mE1 (3.23) The reIations of eqs. 3.22 anel 3.23 are shown in exaggerated form in the phasor diagram of Fig. 3.6b. l J ) 182 TRANSFORMATION OF ELECTIUC ENERGY ) ) ) ) ) If a transformer is opcrating at its rated voltage and eurrent, its losses are essentially constant. But its power output at ratcd load depends on the power factor of thc load. Thus the efficiency is greatest at unity power factor in the load and is, of coursc, zero for a zero power Iactor load. A quantity that is more meaningful than cfficicncy is thc pcr-unit loss at rated loud. Ir E 1/, and '1/, are thc r.ucd vollagc and currcnt of winding I of a transformer, its voltampere rating is voltamperes (3.32) ) ) U; = EI/,11b The per-unit loss is then.givcn by P _ 1\ + P",l- u" G1",EI/,2 + R1)1/,2 per unit (3.33)) ) ) E 1/,11/, In large power transforrners the pcr-unit loss is usually in the range of 0.01-0.04. An important feature in the application of some transformers is the way in which the efficicncy varies with load. The output power, using voltage and current magnitudes, is P2 = EZ/2 cos O ) ) ) = E2' /2' cos O watts where cos e is the load power faetor. Let us neglect any ehange in the voltages EI and E; as the lcad current 12is varied at constant power facto r. The core loss P; is then eonstant. The efficiency may be stated, frorn eq. 3.29, as ) ) ) Ez'I2' cos O per unit (3.34) 1) = E2' I 2' ~os O + P" + Rlc(I2,)2 ) ) ) ) ) To determine the conelition for maximum cfficiency, eq. 3.34 may be differentiated with respeet to the variable 12', ' [E2' 12' cos O + r, '+ R1c(I2')2} E2 cos O dn - Ez' 12' cos O(E2' cos () + 212' RI.) d1 2 ' = [E2' 12' cos O + r, + R1P2')2}2 =0 (3.35) from which the maxirnum effieieney eondition is founel to be P; = RIcU2')2 = P w (3.36) Thus the maxirnum efficiency at any loael power factor occurs for that load at which the constant core losses are equal to the winding losses. ) ~ j :1 ~ ~ I I I OPERATlNG CHARACTEIUSTICS OF TRANSFORMERS 183 The transformer designer may vary the reIative volumes of conductor material and magnetic material in the transformer to achieve maximum efficiency at some average value of load. 3.2.2 Varlable-Frequency Opcration In electron ic and COI11Jllll n ica Iion systcrns, tra nsf ormers are oftcn used over a widc range of Ircqucncics. An examplc is the transformer useel in an auelio amplificr to eouple the output stage to the speaker. Ideally, this audio transformer should operate well over the audible frequency range of about 50--15000 cycles per second. Another example is a pulse transformer. A train of pulses may be represented as a Fourier series of frequency components. The relationship between the voltage ratio ofa transformer and frequency can be found by using a general equivalent cireuit such as that of Fig. 3.4a. In certain frequeney ranges several of the parameters are negligible and the analysis may be simplified. At high frequencies, the effects of interwinding capacitanees may have to be included to obtain an adequate analysis. Figure 3.7a shows an equivalent circuit that is applicable over the middle of lhe frequency range for which the transforrner is designed. ln this range, the leakage reactance and the magnetizing current generally can be neglected. l n smal] transformers, the winding resistances may be significant. In Fig. 3.7, the souree (for examp1e, e1ectronic amplifier) is represented as a variable-frequency source voltage E, in series with an internal resistance R; The load (for example, speaker) is represented by its resistance RL- In the midfrequeney range, the ratio of the load voltage to the source voltage is given approximately by EL = nRL e, Rll + R22 where Rll = RI + Rs, R22 = n2(R2 + RL), and n is the turns ratio N1!N2. To obtain maximurrr power input to the load, the transformer param- eters should be chosen so that the load resistanee RL is equal to the magnitude of the equivalent Thévenin impedance as seen from the load terminais. In the midfrequency range, the load resistance for maximum load power is given by (3.37) Rs + RI + R 2R = -" 2I, n (3.38) Under this condition Er,=_ Es 2n (3.39)