Table Of ContentMAGNETIC CIRCUITS AND TRANSFORMERS
PRINCIPLES OF
ELECTRICAL ENGINEERING
SERIES
Magnetic Circuits
and Transforlners
A FIRST COURSE
FOR POWER AND COMMUNICATION
ENGINEERS
By Members of the Staff of the
Department of Electrical Engineering
Massachusetts Institute of Technology
1111111
THE M.LT. PRESS
MASS ..\CIIUSETTS I�STITUTE OF TECII1\"OLOGY
CA:-'1BRIDGE, i\IASSACIICSETTS
COPYRIGHT, 1943
BY
THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AU Right8 Re8erved
This book 01' any part thereof must not
be reproduced in any form ,t'ilhout the
mUm permission of lhe publisher.
FOURTEENTH
F1ITF.ENTli
PRINTIID IN THB
PRISTISG,
PRISTISG,
UNITB'>
1962
1965 JUI.Y,
j.�SUARY.
STATIIS or "'UIIRleA
Foreword
The statT of the Department of Electrical Engineering at the :\Iassa
chusetts Institute of Technology has for some years been engaged in an
extensin: program of revising as a unit substantially its entire presenta
tion of the basic technological principles of electrical engineering. TIns
yolume is one of a series coyering tIns revised presentation.
The decision to undertake so comprehensive a plan rather than to
add here and patch there came from the belief that the Department's
large staff, with its varied interests in teaching and related research, could
effect a new synthesis of educational material in the field of electrical
engineering and evol\·e a set of textbooks ,vith a breadth of view not
easily approached by an author working indhidually.
Such a comprehensive series, it was felt, should be free from the dupli
cations, repetitions, and unbahnces so often present in unintegrated
series of textbooks. It should possess
the organization of a
a unity and a breadth arising from
subject as a whole. It should appeal to the student
of ordinary preparation and also prmide a depth and rigor challenging to
the exceptional student and acceptable to the advanced scholar. It should
comprise a basic course adequate for all students of electrical engineering
regardless of their ultimate specialty. Restricted to material which is of
fundamental importance to all branches of electrical enI,,>ineering, the
course should lead naturally into anyone branch.
This book and the reorganized prob'Tam of teaching out of which it has
grown arc thus products of a major research project to improve educa
tional methods. The fact that this book and its companions arc the result
of collaborative endeavor im·oh·ing substantially the entire staff of the
Institute's Department of Electrical Enb>ineering and that this collabo
ration has been carried out ,,,ith sustained enthusiasm is striking testi
mony to each contributor's devotion to this broad objective of improved
instruction. To those of us who have obsernxl the proI,,'Tess of the work
from the outside, this demonstration of unity and
part of a
esprit de corps on the
of the entire
large staff is perhaps the most impressive aspect
eiTort.
KARL
v
T. COMPTON
Preface
This book is the second volume in the Principles of Electrical Engineer
ing series described in the Foreword, and extends into another field the
circuit theory begun in the first volume. Computation of magnetic-circuit
performance, principles associated with the concepts of interlinked electric
and magnetic circuits , and their application to the analysis of transformers
are the essential subject matter.
Like the other volumes in this series, this book is
basic course.
intended for a first
Emphasis, hence, again is placed on fundamental principles
important to students of electrical engineering regardless of what their
special fields ultimately may be; both po\\"er and communication problems
are considered. Rigor of thought and analysis rather than extensiveness
of scope is likewise the intended feature of this book.
The treatment assumes that the reader has knowledge of electric-circuit
theory as given in the first volume of the series or in other textbooks on a
similar level. l\Iathematics through differen tial equations is
freely
used i
not, however, as a substitute for the explanation of physical phenomena
but rather as a means of describing quantitatively their consequences.
Parallel to the mathematical analysis, explanations are given in terms of
physical interpretations.
The text is divided into two parts. Part I, " .:\Iagnetic Circuits," starts
with a discussion of the current theory of ferromagnetism, derives the
magnetic-circuit concept, and continues \\"ith development of the funda
mental principles for computation of the behavior of magnetic circuits.
;\ rha!1ter is devoted to discussion of iron-core reactors by means of model
theory - a method of analysis of great power whenever nonlinear phenomena are involved. A brief summary of thermal-circuit problems is
also included. P,ut II, " Transformers," begins with a short discussion
pointing out that the theories from "'hich the ch arac teristics of all electric
apparatus can be computed are developed through the combination of the
fundamental concepts of the electric, dielectric, magnetic, thermal, and
mechanical circuits contained in them, and that costs and other practical
matters are of paramount importance in their influence on the development of electric apparatus. Part II then continues with a study of the
applications of these general principles to transformers. The theory of
transformers is developed both from the resolution of their magnetic
fields into Jeakage and resultant mutual components and from the classical
VH
PREFACE
theory of coupled circuits, the interrelations between the two methods of
analysis being emphasized.
Although the text is by no means exhaustive in anyone field, nor is it
intended to be a design book, an a ttempt has been made to discuss a
variety of representative practical problems in both the heavy-current
power and the light-current control, measurement, and communication
applications of magnetic materials and transformers. As in the other
volumes of the series, more material than is usually covered in a first
course is presen ted, to provide for additional study by particularly apt or
advanced students, and to increase the usefulness of the text as a reference
book.
CAMBRIDGE, MASSACHUSETTS
January, 1913
Contents
PAGE
FOREWORD .
PREFACE . •
TABLE OF SnmOLS
PART I.
V
vii
xix
MAGNETIC CIRCUITS
CHAPTER I
PROPERTIES OF FERRmlAGNETIC l\IATERIALS .
Art.
.
1. Importance of Ferromagnetic Material;
2. Theory of Magnetism . . .
2a. Electrons and Kuclei .
2b. The Atom .
2c. The Domain
. . . .
2d. The Crystal
2e. Polycrystalline Substances
3. General Properties of Magnetic Materials.
4. Characteristics and Magnetization Curves of
5. Magnetic Testing . . . . . . . . . . . .
3
3
4
5
5
7
8
10
15
�Iagnetic l\laterials
. . .
6. Engineering Utilization of Fer romagnetic Materials
18
29
34
40
CHAPTER II
TIlE l\lAGKETIc-CIRClJIT COXCEPT
Art.
. . . . . . . • . . . •
1. The General Problem . . . . . . . . . . . . .
2. Analogies between Electric and Magnetic Circuits
41
41
44
CHAPTER III
PRIXCIPLES OF MAGNETIC CIRCUIT CO�Il'UTATIO::\'S
Art. 1. Units for Practical Magnetic Calculations . . .
2. Calculations for Magnetic Paths Wholly in Iron
3. Illustrative Example of Calculations for Magnetic Structu re of Uniform Material and Cross-Sectional Area . . . . . . . . . . . .
4. Illustrative Examples of Calculations for Magnet i c Structure of Uni-
form Material and Nonuniform Cross-Sectional Area
5. Occurrence of Air Gaps in Ferromagnetic Circuits
6. Fringing and Leakage Fluxes . . . . . . . . . . .
7. Methods of Calculation for Short Air Gaps . . . . .
8. Illustrative Example of Calculations for a Simple Ferromagnetic Cir-
cuit with a Short Air Gap
zx
. . . . . . .
•
. . . . . . .
54
54
55
59
61
65
67
68
74
CONTENTS
PAGE
9. Illustrative Examples of Calculations for Series-Parallel Iron and
Short Air-Gap Paths . . . . . . . . . . . . . . . . . . . .
10. Calculations for Long Air Gaps and for Leakage Flux Paths . . .
11. Illustrative Example of Calculation of Leakage Flux for the l\Iagnetic
Circuit of an Oscillograph . . . . . . . . . . . . . . . . . .
12. Calculation of the Magnetization Curve of a Direct-Current Dynamo
Problems. . . . . • . . . . . . . . . . . . . . . . . . . .
78
81
82
84
94
CHAPTER IV
PERMANENT MAGNETS
Art.
A}."D ENERGY IN THE l\IAGNETlC FIELD
1. Magnetic Circuits Involving Permanent l\Iagnets .
2. Design for Minimum Amount of Magnetic Material
3. Fringing and Leakage . . . . . . .
4. Stabilization of Permanent Magnets .
5. Energy in the Magnetic Field.
6. Magnetic Pull
Problems . . . . . .
98
99
103
105
111
114
117
121
CHAPTER V
LOSSES
Art.
IN MAGNETIC CORES CONTAIXIKG TDIE-VARYING FLUXES
1. Hysteresis Loss . . . . . . . . .
2. Eddy-Current Loss and Skin EfIect .
124
125
. . .
. . . .
. . .
3. Total Core Loss. . . . . . . . . . . . . .
4. Reduction of To tal Core Loss; Design Considerations
5. l\leasurement of Total Core Loss . . . . . . . . .
6. Experimental Loss Curves, Experimental Coefficients
7. Separation of Hysteresis and Eddy-Current Losses
Problems. . . . . . . . . . . . . . . . . .
132
138
142
144
148
151
154
CHAPTER VI
ALTERNATING-CURRENT EXCITATION CHARACTERISTICS OF IRON-CORE REACTORS AND TRANSFORMERS . • . • • • .
Art. 1. General Properties of Iron-Core Reactors
2. Iron Cores in Transformers.
3. Assumptions . . .
3a. Capacitance
3b. Resistance .
156
156
157
158
158
158
159
160
164
166
169
9. General Properties of Fourier Series
9a. The Constant Component.
173
174
175
176
9d. Power . . . . .
ge. Effective Values. . . . .
180
CONTENTS
X2
PAGE
10. Fourier Analysis of the Exciting Current for Sine-Wave Flux
181
184
lOa. EfIect of an Air Gap
. . . . . . . .
11. Vector Representation of the Exciting Current . . . . . 186
12. ,\lternating-Current Excitation Characteristics . . . . . 189
12a. Illustrative Example of Iron-Core Reactor Design
193
14. Characteristics of Iron Subjected to Superposed Direct and Alternat-
ing ::\Iagnctizing Forces
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
197
15. Iron-Core Reactors with Superposed Direct and Alternating Excita-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
tions
1530. Illustrative Example of Calculation of Apparent Inductance
Problems.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
202
203
207
CHAPTER VII
::\10DEL THEORY AXD DESIGN OF IROX-CORE REACTORS
212
212
2. EfIects of Changes in the Winding
3. EfIects of Changes in Linear Dimensions .
216
218
228
Sa. Reduction of N"onlinear EfTects by Means of an Air Gap
5b . EfTects of an Air Gap on
Quality Factor
6. Model Theory Applied to Entire Ketworks .
6a. Relations among the Variables
.
.
.
. . . .
. . .
.
.
.
.
.
.
.
6b. Relations between Corresponding Linear Parameters .
6c. Relation between the Derived and Prototype Reactors
Problems
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
228
230
232
233
235
236
238
CHAPTER Vln
TIIER�IAL PROPERTIES AXD HEAT FLOW
Art.
1. EfTects of Heat on Materials
2. Conduction of Heat
242
242
244
3. Radiation of Heat. . . . . 2-16
4. Convection of Heat . . . . 248
5. Cooling of Electrical Apparatus
Sa. Transformers
.
.
.
5b. Rotating Machines
6. Transient lIeating
PART II.
250
251
252
253
TRANSFORMERS
CHAPTER IX
GENERAL DESIGN AXD COST CONSIDERATIONS
Art.
1. Electrical Machinery
2. Power Systems
.
.
.
.
.
.
.
3. Electronic Devices
. . . . .
4. Electronic and Other Circuits.
259
259
261
262
263
XII
CHAPTER X
PAGE
Art.
TRAXSFOR�IERS; GEXERAL PRIXCIl'LES 265
1. Elementary Transformer Theory 265
2. The Ideal Transformer 268
3. Uses of Transformers . .
4. Evolution of Transformers 272
278
5. Transformer Problems
5a. Losses . .
5b. Cooling
279
280
2g0
5c. ;,Iagnetic Leakage 280
5lI. Exciting Currcn t 281
5e. Thc Electric Field 282
5f. �ranufacturing Problems
5g. Othcr Considerations
. .
282
282
283
CIL\PTER XI
Problems . . .
PHYSICAL FEATCRES OF TIL\XSFOIL\IERS .
.\ rt. 1. Cores . . . . . . . .
2. Windings. . . . . . .
3. Cooling and Insulation
3a. c\ir Cooling . . .
3b. Air-blast Cooling
3c. Transformer Oil .
3d. Oil-immersed, Sell-Cooled
3e. Oil-Immersed, Forced-Air-Cooled
3f. Oil-Immersed, Water-Cooled . . .
3g. Forced-Oil Cooling . . . . . .
3h. Pyranol, Inerteen, or Chlorextol.
4. Tanks . . . . . .
S. Bushings. . . . .
6. Protective Devices
7. Shielding. . . . .
286
21\6
296
297
2lJlJ
29lJ
2lJlJ
2lJ9
300
300
300
300
300
302
30-l
310
CHAPTER XII
MAGNETIC LEAKAGE
Art.
IN TRANSFORMERS . . . . . . . . . . .
1. Magnetic Leakage at No Load . . . . . . . . . .
2. Voltage Equations and Flux Distribution under Load
2a. Component Fluxes. . . . .
. . . . .
2b. Resultant Flux Distribution ' "
3. Leakage Inductances . . . . . . . . . . . . . .
4. Exciting and Load Components of the Primary Current
S. An Equivalent Circuit . . . . . . . . . . . . . . .
6. Effects of the Iron Core and Simplifying Approximations
Problems . . . . . . . . . . . . . . . . . . . . .
3 13
313
317
31g
320
32-t
325
326
328
329
CHAPTER XIII
LEAKAGE IXDUCTAXCES
Art.
. . .
1. Vector Diagrams and
.
an Equivalent Circuit
332
332
