The J & P
Transformer Book
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The J & P
Transformer Book
Thirteenth edition
A PRACTICAL TECHNOLOGY OF THE
POWER TRANSFORMER
Martin J. Heathcote, CEng, FIEE
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Newnes is an imprint of Elsevier
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Newnes
An imprint of Elsevier
Linacre House, Jordan Hill, Oxford OX2 8DP
30 Corporate Drive, Burlington, MA 01803
First published 1925 by Johnson & Phillips Ltd
Ninth edition 1961
Reprinted by Illiffe Books Ltd 1965
Tenth edition 1973
Reprinted 1967 (twice), 1981
Eleventh edition 1983
Reprinted 1985, 1988, 1990, 1993, 1995
Twelfth edition 1998
Reprinted 2003, 2005
Copyright © 2007, Elsevier Ltd. All rights reserved
No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means electronic, mechanical, photocopying,

recording or otherwise without the prior written permission of the publisher
Permissions may be sought directly from Elsevier’s Science & Technology Rights
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e-mail: Alternatively you can submit your request online
by visiting the Elsevier web site at and selecting
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Notice
No responsibility is assumed by the publisher for any injury and/or damage to persons
or property as a matter of products liability, negligence or otherwise, or from any use or
operation of any methods, products, instructions or ideas contained in the material herein.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
A catalog record for this book is available from the Library of Congress.
ISBN-13: 978-0-7506-8164-3
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Printed in Great Britain
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visit our website at www.newnespress.com
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Contents
Foreword ix
Preface xi
Acknowledgements xiii
1 Transformer theory 1
1.1 Introduction 1
1.2 The ideal transformer: voltage ratio 2
1.3 Leakage reactance: transformer impedance 4
1.4 Losses in core and windings 5
1.5 Rated quantities 9

1.6 Regulation 11
2 Design fundamentals 14
2.1 Types of transformers 14
2.2 Phase relationships: phasor groups 17
2.3 Volts per turn and fl ux density 22
2.4 Tappings 24
2.5 Impedance 25
2.6 Multi-winding transformers including tertiary windings 27
2.7 Zero-sequence impedance 33
2.8 Double secondary transformers 34
2.9 General case of three-winding transformer 36
3 Basic materials 41
3.1 Dielectrics 41
3.2 Core steel 42
3.3 Winding conductors 54
3.4 Insulation 60
3.5 Transformer oil 75
4 Transformer construction 105
4.1 Core construction 106
4.2 Transformer windings 119
4.3 Dispositions of windings 143
4.4 Impulse strength 149
4.5 Thermal considerations 158
4.6 Tappings and tapchangers 169
4.7 Winding forces and performance under short-circuit 230
4.8 Tanks and ancillary equipment 250
4.9 Processing and dry out 285
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5 Testing of transformers 319
5.1 Testing and quality assurance during manufacture 319

5.2 Final testing 321
5.3 Possible additional testing for important transformers 384
5.4 Transport, installation and commissioning 392
6 Operation and maintenance 406
6.1 Design and layout of transformer installations 406
6.2 Neutral earthing 415
6.3 Transformer noise 430
6.4 Parallel operation 454
6.5 Transient phenomena occurring in transformers 495
6.6 Transformer protection 530
6.7 Maintenance in service 588
6.8 Operation under abnormal conditions 622
6.9 The infl uence of transformer connections upon third-harmonic
voltages and currents 646
7 Special features of transformers
for particular purposes 670
7.1 Generator transformers 670
7.2 Other power station transformers 682
7.3 Transmission transformers and autotransformers 688
7.4 Transformers for HVDC converters 689
7.5 Phase shifting transformers and quadrature boosters 701
7.6 System transformers 710
7.7 Interconnected star earthing transformers 714
7.8 Distribution transformers 718
7.9 Scott- and Le Blanc-connected transformers 743
7.10 Rectifi er transformers 750
7.11 AC arc furnace transformers 752
7.12 Traction transformers 757
7.13 Generator neutral earthing transformers 764
7.14 Transformers for electrostatic precipitators 769

7.15 Reactors 771
8 Transformer enquiries and tenders 782
8.1 Transformer enquiries 782
8.2 Assessment of tenders 807
8.3 Economics of ownership and operation 812
vi Contents
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APPENDICES
1 Transformer equivalent circuit 822
2 Geometry of the transformer phasor diagram 833
3 The transformer circle diagram 839
4 Transformer regulation 844
5 Symmetrical components in unbalanced three-phase systems 848
6 A symmetrical component study of earth faults in transformers
in parallel 871
7 The use of fi nite-element analysis in the calculation of
leakage fl ux and dielectric stress distributions 921
8 List of national and international standards relating to
power transformers 950
9 List of principal CIGRE reports and papers relating to
transformers 961
10 List of reports available from ERA Technology Ltd 964
Index 969
Contents vii
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Foreword
The J & P Transformer Book continues to withstand the test of time as a key
body of reference material for students, teachers, and all whose careers are
involved in the engineering processes associated with power delivery, and par-

ticularly with transformer design, manufacture, testing, procurement, applica-
tion, operation, maintenance, condition assessment and life extension. As a
measure of its popular and wide-scale appeal an internet search on the title
results in over 15,000 reference hits from a large number of countries and in
multiple languages. Now in print for over 80 years since initial publication
in 1925 by Johnson & Phillips Ltd., the text has grown and changed over time
from its original intent as a guide for transformer theory, design, and construc-
tion from a manufacturing perspective, and continues to steadily expand on
incorporation of the information needed for a user to apply transformer equip-
ment knowledgeably and effectively. Evolutions in transformer experience,
knowledge, design calculation capability, manufacturing process, MVA and
voltage ratings, and application needs, together with the understanding of new
problems and development of new solutions have all been refl ected as this
book has been revised and updated.
Current experience and knowledge have been brought into this thirteenth
edition with discussions on moisture equilibrium in the insulation system, veg-
etable based natural ester insulating fl uids, industry concerns with corrosive
sulfur in oil, geomagnetic induced current (GIC) impacts, transportation issues,
new emphasis on measurement of load related noise, and enhanced treatment
of dielectric testing (including Frequency Response Analysis), Dissolved Gas
Analysis (DGA) techniques and tools, vacuum LTC’s, shunt and series reactors,
and HVDC converter transformers. The historical basis of the book continues to
highlight British experience, but these changes in the thirteenth edition together
with updates of IEC reference Standards documentation and inclusion for the fi rst
time of IEEE reference Standards, provide a more universal fl avor to the volume
and a recognition that the transformer industry and market is truly global in scale.
The book was last updated in 1998 by Martin J. Heathcote, and in prepar-
ing this edition, Mr. Heathcote has continued to draw on his contacts and
experience in both the manufacturing and user arenas to provide relevance
and value to the industry. Upon graduating from the University of Sheffi eld

in Electrical Power Engineering, his career began immediately with the
Transformer Division of Ferranti Limited, where his training led to a position
as a transformer designer. After successfully working on designs up to 400 kV,
Mr. Heathcote moved from manufacturing to the other side of the aisle and
worked for over 20 years on various power engineering projects for the
Central Electricity Generating Board (CEGB) in the UK, culminating in a role
as CEGB Generating Division’s Transformer Engineer. Subsequently, since
1992 he has provided consulting services specializing in power transform-
ers, primarily working for utilities. In this context he has established working
relationships with transformer manufacturers on several continents.
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Transformer talent is at a premium today, and all aspects of the power
industry are suffering a diminishing of the supply of knowledgeable and
experienced engineers. Manufacturers still retain highly capable transformer
expertise, but their depth of engineering manpower resources is sometimes
reduced – perhaps mistakenly - by market economics. Utilities have diffi culty
retaining young engineers in the fi eld long enough to develop expert status,
and in many cases transformer engineering is outsourced. From my perspec-
tive, maintaining appropriate power systems and equipment expertise is neces-
sary for a utility to support the reliability, availability, and quality of service
goals demanded by energy consumers now and into the future. As a result,
there is a drive in the industry to capture knowledge and experience before it is
lost. While familiar with the stature of the book when Mr. Heathcote asked me
to consider this Foreword, it was not until I reviewed through this edition that
I learned the enduring value of the J&P Transformer Book as a thorough and
vital collection of transformer learning to facilitate that continuing education.
In a single volume it brings back to mind much of the experience and knowl-
edge I had been exposed to in over 25 years working for a major US utility
with technical responsibility for the health and well-being of a fl eet of over
1000 power transformers, and in participating during that same period in the

standards development and maintenance work of the IEEE/PES Transformers
Committee, a group for which I am privileged to presently serve as Chair.
The J&P Transformer Book has served well to educate prior and present
generations, and this thirteenth edition continues to be a steady reference to
those in the industry and a source of new knowledge to future students and
engineers. I will enjoy reviewing it often as my career in this interesting fi eld
continues, and I trust you will fi nd it a useful addition to your technical library.
Donald J. Fallon
Newark, NJ
July 5, 2007
x Foreword
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Preface to the thirteenth edition
The design and manufacture of transformers is not cutting edge technology,
and compared with many fi elds of engineering, progress is slow and change
is gradual. The question might therefore be asked as to what is the need for a
new and revised edition of The J & P Transformer Book. It is also true that in
many branches of the industry the book has become well known and widely
respected, and many owners will not consider replacing their trusted old edi-
tion. Indeed, many transformer engineers swear by copies handed down to
them from older colleagues. After all, transformers are expected to have long
lives so that many that are currently in service could have been designed 20 or
30 years ago using practices that have long been considered out of date.
True, but the industry does change and the Twelfth edition certainly did
appear at a time of great change. Electricity supply privatisation in the UK
was beginning to have a signifi cant impact on procurement strategy; the single
European market was becoming well established, and its effects strongly felt.
There have been many mergers in the industry; many well known and well
respected manufacturers in Europe and the USA have disappeared. The result
is that the industry has become truly an international one, national standards

are disappearing and international standards, in particular those of the IEC, are
becoming dominant.
The Twelfth edition of J & P, as it has tended to become known, certainly
introduced change; it was the fi rst to have been written by an author who was
not primarily a manufacturer. The impression is that the change has been
appreciated by readers. The Twelfth edition was, however, written from a UK
viewpoint, and with UK experience and practices in mind. This Thirteenth edi-
tion has aimed to be less parochial and recognize that, practices may differ in
different parts of the world; that there is often more than one way of resolv-
ing a problem; and the way that is familiar might not necessarily be the only
way. That view has been acknowledged primarily by reducing the references
to British Standards with which a high proportion of readers will not be famil-
iar, and replacing these with European and International Standards.
It is clear that these industry changes can and will go further, so that there
will undoubtedly be a need for further revision to this work in the not so dis-
tant future, but it is hoped that the changes introduced in this edition will
prove helpful, will not disappoint readers, and neither will they detract from
the appeal of The J & P Transformer Book to a wide international readership.
M J H
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Acknowledgements
The author wishes to express grateful thanks to many friends and colleagues
who have provided assistance in this revision of the J & P Transformer Book.
Though less extensive than the work involved in the production of the previ-
ous, Twelfth, edition, there is still a large amount of effort involved, which
would certainly not have been possible without their help. In particular to
Jeremy Price, National Grid Company, for much constructive comment and
advice on the sections relating to insulation coordination and the many spe-
cialised transformers including arc furnace transformers, HVDC converter

transformers, traction transformers and rectifi er transformers; to Alan Darwin
of Areva T & D for much helpful advice, specialist input on transformer noise,
as well as fi nding the very impressive cover photograph; to Rajinder Pal (Raj)
of Areva T & D for a very comprehensive review of the Twelfth edition high-
lighting the many areas of revision necessary, and to Greg Anderson of the
IEEE PES Transformers Committee for enthusiastic support and assistance in
giving this Thirteenth edition a slightly more transatlantic outlook.
Because the main substance of the Twelfth edition remains and because, with-
out that the Thirteenth edition would have come to nothing, a great debt of grati-
tude is still owed to all the friends who assisted in its preparation; these include
W.J. (Jim) Stevens who read every word of the Twelfth edition and provided
invaluable criticism and comment; to Professor Denis Allan, FEng, from whom
much help and guidance was received; to Dr. Colin Tindall of the Department of
Electrical and Electronic Engineering, the Queen’s University, Belfast, who read
my fi rst chapter and helped me to brush up on my somewhat rusty theory; to other
friends who provided written contributions; Aziz Ahmad-Marican, University of
Wales, Cardiff, on Petersen Coil earthing, Mike Newman, Whiteley Limited, on
transformer insulation; Cyril Smith, Bowthorpe EMP Limited, on surge arresters.
Grateful thanks are also offered to many organisations who freely provided assist-
ance, as well as data, diagrams and photographs which enabled the chapters to be
so generously illustrated.
These include:
ABB Power T & D Limited
Accurate Controls Limited
Allenwest-Brentford Limited
Areva T&D
Associated Tapchangers Limited
Bowthorpe EMP Limited
British Standards
Brüel & Kjær Division of Spectris (UK) Limited

Brush Transformers Limited
Carless Refi ning & Marketing Limited (Electrical Oil Services)
CIGRÉ
Copper Development Association
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Emform Limited
ERA Technology Limited
GEA Spiro-Gills Limited
GEC Alsthom Engineering Research Centre
GEC Alsthom T&D Protection and Control Limited
Hawker Siddeley Transformers Limited
Maschinenfabrik Reinhausen
Peebles Transformers
Schneider Electric
South Wales Transformers Limited
Strategy and Solutions
TCM Tamini
Whiteley’s Limited (H. Weidmann AG)
In the relatively short time since the Twelfth edition was produced a sig-
nifi cant number of the above have disappeared or been absorbed into larger
groupings.
Finally, despite the extensive revision involved in the production of the
Twelfth and Thirteenth editions, some of the work of the original authors,
H. Morgan Lacey, the late S.A. Stigant, the late A.C. Franklin and D.P. Franklin,
remains; notably much of the sections on transformer testing, transformer pro-
tection, magnetising inrush, parallel operation, and third harmonic voltages
and currents, and for this due acknowledgement must be given.
xiv Acknowledgements
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1

1
Transformer theory
1.1 INTRODUCTION
The invention of the power transformer towards the end of the nineteenth cen-
tury made possible the development of the modern constant voltage AC supply
system, with power stations often located many miles from centres of elec-
trical load. Before that, in the early days of public electricity supplies, these
were DC systems with the source of generation, of necessity, close to the point
of loading.
Pioneers of the electricity supply industry were quick to recognise the ben-
efi ts of a device which could take the high current relatively low voltage out-
put of an electrical generator and transform this to a voltage level which would
enable it to be transmitted in a cable of practical dimensions to consumers
who, at that time, might be a mile or more away and could do this with an effi -
ciency which, by the standards of the time, was nothing less than phenomenal.
Todays transmission and distribution systems are, of course, vastly more
extensive and greatly dependent on transformers which themselves are very
much more effi cient than those of a century ago; from the enormous gener-
ator transformers such as the one illustrated in Fig. 7.5, stepping up the output
of up to 19 000 A at 23.5 kV, of a large generating unit in the UK, to 400 kV,
thereby reducing the current to a more manageable 1200 A or so, to the thou-
sands of small distribution units which operate almost continuously day in day
out, with little or no attention, to provide supplies to industrial and domestic
consumers.
The main purpose of this book is to examine the current state of transformer
technology, inevitably from a UK viewpoint, but in the rapidly shrinking and
ever more competitive world of technology it is not possible to retain one’s
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2 Transformer theory
place in it without a knowledge of all that is going on the other side of the

globe, so the viewpoint will, hopefully, not be an entirely parochial one.
For a reasonable understanding of the subject it is necessary to make a brief
review of transformer theory together with the basic formulae and simple
phasor diagrams.
1.2 THE IDEAL TRANSFORMER: VOLTAGE RATIO
A power transformer normally consists of a pair of windings, primary and sec-
ondary, linked by a magnetic circuit or core. When an alternating voltage is
applied to one of these windings, generally by defi nition the primary, a current
will fl ow which sets up an alternating m.m.f. and hence an alternating fl ux in the
core. This alternating fl ux in linking both windings induces an e.m.f. in each of
them. In the primary winding this is the ‘back-e.m.f’ and, if the transformer were
perfect, it would oppose the primary applied voltage to the extent that no current
would fl ow. In reality, the current which fl ows is the transformer magnetising
current. In the secondary winding the induced e.m.f. is the secondary open-circuit
voltage. If a load is connected to the secondary winding which permits the fl ow of
secondary current, then this current creates a demagnetising m.m.f. thus destroying
the balance between primary applied voltage and back-e.m.f. To restore the balance
an increased primary current must be drawn from the supply to provide an exactly
equivalent m.m.f. so that equilibrium is once more established when this add-
itional primary current creates ampere-turns balance with those of the secondary.
Since there is no difference between the voltage induced in a single turn whether
it is part of either the primary or the secondary winding, then the total voltage
induced in each of the windings by the common fl ux must be proportional to the
number of turns. Thus the well-known relationship is established that:
E
1
/E
2
ϭ N
1

/N
2
(1.1)
and, in view of the need for ampere-turns balance:
I
1
/N
2
ϭ I
2
/N
2
(1.2)
where E, I and N are the induced voltages, the currents and number of turns
respectively in the windings identifi ed by the appropriate subscripts. Hence, the
voltage is transformed in proportion to the number of turns in the respective
windings and the currents are in inverse proportion (and the relationship holds
true for both instantaneous and r.m.s. quantities).
The relationship between the induced voltage and the fl ux is given by refer-
ence to Faraday’s law which states that its magnitude is proportional to the rate
of change of fl ux linkage and Lenz’s law which states that its polarity such as
to oppose that fl ux linkage change if current were allowed to fl ow. This is nor-
mally expressed in the form
e ϭ ϪN(dφ/dt)
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Transformer theory 3
but, for the practical transformer, it can be shown that the voltage induced per
turn is
E/N ϭ KΦ
m

f (1.3)
where K is a constant, Φ
m
is the maximum value of total fl ux in Webers link-
ing that turn and f is the supply frequency in Hertz.
The above expression holds good for the voltage induced in either primary or
secondary windings, and it is only a matter of inserting the correct value of N for
the winding under consideration. Figure 1.1 shows the simple phasor diagram cor-
responding to a transformer on no-load (neglecting for the moment the fact that
the transformer has reactance) and the symbols have the signifi cance shown on the
diagram. Usually in the practical design of transformer, the small drop in voltage
due to the fl ow of the no-load current in the primary winding is neglected.
If the voltage is sinusoidal, which, of course, is always assumed, K is 4.44
and Eq. (1.3) becomes
E ϭ 4.44f ΦN
Figure 1.1 Phasor diagram for a single-phase transformer on open
circuit. Assumed turns ratio 1:1
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4 Transformer theory
For design calculations the designer is more interested in volts per turn and
fl ux density in the core rather than total fl ux, so the expression can be rewrit-
ten in terms of these quantities thus:
E/N ϭ 4.44B
m
Af ϫ 10
Ϫ6
(1.4)
where E/N ϭ volts per turn, which is the same in both windings
B
m

ϭ maximum value of fl ux density in the core, Tesla
A ϭ net cross-sectional area of the core, mm
2
f ϭ frequency of supply, Hz.
For practical designs B
m
will be set by the core material which the designer
selects and the operating conditions for the transformer, A will be selected
from a range of cross-sections relating to the standard range of core sizes pro-
duced by the manufacturer, whilst f is dictated by the customer’s system, so
that the volts per turn are simply derived. It is then an easy matter to determine
the number of turns in each winding from the specifi ed voltage of the winding.
1.3 LEAKAGE REACTANCE: TRANSFORMER IMPEDANCE
Mention has already been made in the introduction of the fact that the trans-
formation between primary and secondary is not perfect. Firstly, not all of the
fl ux produced by the primary winding links the secondary so the transformer
can be said to possess leakage reactance. Early transformer designers saw
leakage reactance as a shortcoming of their transformers to be minimised to
as great an extent as possible subject to the normal economic constraints. With
the growth in size and complexity of power stations and transmission and dis-
tribution systems, leakage reactance – or in practical terms since transformer
windings also have resistance – impedance, gradually came to be recog-
nised as a valuable aid in the limitation of fault currents. The normal method
of expressing transformer impedance is as a percentage voltage drop in the
transformer at full-load current and this refl ects the way in which it is seen by
system designers. For example, an impedance of 10 per cent means that the
voltage drop at full-load current is 10 per cent of the open-circuit voltage, or,
alternatively, neglecting any other impedance in the system, at 10 times full-
load current, the voltage drop in the transformer is equal to the total system
voltage. Expressed in symbols this is:

VZ
IZ
E
z
FL
%ϭϭ ϫ100

where Z is ͙(R
2
ϩ X
2
), R and X being the transformer resistance and leak-
age reactance respectively and I
FL
and E are the full-load current and open-
circuit voltage of either primary or secondary windings. Of course, R and X
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Transformer theory 5
may themselves be expressed as percentage voltage drops, as explained below.
The ‘natural’ value for percentage impedance tends to increase as the rating
of the transformer increases with a typical value for a medium sized power
transformer being about 9 or 10 per cent. Occasionally some transformers are
deliberately designed to have impedances as high as 22.5 per cent. More will
be said about transformer impedance in the following chapter.
1.4 LOSSES IN CORE AND WINDINGS
The transformer also experiences losses. The magnetising current is required
to take the core through the alternating cycles of fl ux at a rate determined by
system frequency. In doing so energy is dissipated. This is known variously
as the core loss, no-load loss or iron loss. The core loss is present whenever
the transformer is energised. On open circuit the transformer acts as a single

winding of high self-inductance, and the open-circuit power factor averages
about 0.15 lagging. The fl ow of load current in the secondary of the trans-
former and the m.m.f. which this produces is balanced by an equivalent
primary load current and its m.m.f., which explains why the iron loss is inde-
pendent of the load.
The fl ow of a current in any electrical system, however, also generates loss
dependent upon the magnitude of that current and the resistance of the system.
Transformer windings are no exception and these give rise to the load loss or
copper loss of the transformer. Load loss is present only when the transformer
is loaded, since the magnitude of the no-load current is so small as to pro-
duce negligible resistive loss in the windings. Load loss is proportional to the
square of the load current.
Reactive and resistive voltage drops and phasor diagrams
The total current in the primary circuit is the phasor sum of the primary load
current and the no-load current. Ignoring for the moment the question of resist-
ance and leakage reactance voltage drops, the condition for a transformer sup-
plying a non-inductive load is shown in phasor form in Fig. 1.2. Considering
now the voltage drops due to resistance and leakage reactance of the trans-
former windings it should fi rst be pointed out that, however the individual
voltage drops are allocated, the sum total effect is apparent at the secondary
terminals. The resistance drops in the primary and secondary windings are eas-
ily separated and determinable for the respective windings. The reactive volt-
age drop, which is due to the total fl ux leakage between the two windings, is
strictly not separable into two components, as the line of demarcation between
the primary and secondary leakage fl uxes cannot be defi ned. It has therefore
become a convention to allocate half the leakage fl ux to each winding, and
similarly to dispose of the reactive voltage drops. Figure 1.3 shows the phasor
relationship in a single-phase transformer supplying an inductive load having a
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6 Transformer theory

lagging power factor of 0.80, the resistance and leakage reactance drops being
allocated to their respective windings. In fact the sum total effect is a reduc-
tion in the secondary terminal voltage. The resistance and reactance voltage
drops allocated to the primary winding appear on the diagram as additions to
the e.m.f. induced in the primary windings.
Figure 1.4 shows phasor conditions identical to those in Fig. 1.3, except
that the resistance and reactance drops are all shown as occurring on the sec-
ondary side.
Of course, the drops due to primary resistance and leakage reactance are
converted to terms of the secondary voltage, that is, the primary voltage drops
are divided by the ratio of transformation n, in the case of both step-up and
step-down transformers. In other words the percentage voltage drops con-
sidered as occurring in either winding remain the same.
To transfer primary resistance values R
1
or leakage reactance values X
1
to
the secondary side, R
1
and X
1
are divided by the square of the ratio of trans-
formation n in the case of both step-up and step-down transformers.
Figure 1.2 Phasor diagram for a single-phase transformer supplying
a unity power factor load. Assumed turns ratio 1:1
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Transformer theory 7
The transference of impedance from one side to another is made as follows.
Let

Z
s
ϭ total impedance of the secondary circuit including leakage and
load characteristics
Z
s
Ј ϭ equivalent value of Z
s
when referred to the primary winding.
Then

′
I
N
N
I
N
N
E
Z
E
N
N
E
2
2
1
2
2
1

2
2
2
1
1
ϭϭ ϭ
s
and

so

′
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
I
N
N
E
Z

2
2
1
2
1
ϭ
s
(1.5)
Figure 1.3 Phasor diagram for a single-phase transformer supplying
an inductive load of lagging power factor cos φ
2
. Assumed turns ratio
1:1. Voltage drops divided between primary and secondary sides
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8 Transformer theory
Also,
V
1
ϭ E
1
ϩ I
Ј
2
Z
1
where
E
1
ϭ I
Ј

2
Z
s
Ј
Therefore
I
Ј
2
ϭ E
1
/Z
s
Ј (1.6)
Comparing Eqs (1.5) and (1.6) it will be seen that Z
s
Јϭ Z
s
(N
1
/N
2
)
2
.
The equivalent impedance is thus obtained by multiplying the actual imped-
ance of the secondary winding by the square of the ratio of transformation n,
that is, (N
1
/N
2

)
2
. This, of course, holds good for secondary winding leakage
reactance and secondary winding resistance in addition to the reactance and
resistance of the external load.
Figure 1.4 Phasor diagram for a single-phase transformer supplying
an inductive load of lagging power factor cos φ
2
. Assumed turns
ratio 1:1. Voltage drops transferred to secondary side
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Transformer theory 9
Figure 1.5 is included as a matter of interest to show that when the load
has a suffi cient leading power factor, the secondary terminal voltage increases
instead of decreasing. This happens when a leading power factor current
passes through an inductive reactance.
Preceding diagrams have been drawn for single-phase transformers, but they
are strictly applicable to polyphase transformers also so long as the conditions
for all the phases are shown. For instance Fig. 1.6 shows the complete phasor
diagram for a three-phase star/star-connected transformer, and it will be seen
that this diagram is only a threefold repetition of Fig. 1.4, in which primary
and secondary phasors correspond exactly to those in Fig. 1.4, but the three
sets representing the three different phases are spaced 120º apart.
1.5 RATED QUANTITIES
The output of a power transformer is generally expressed in megavolt-amperes
(MVA), although for distribution transformers kilovolt-amperes (kVA) is gen-
erally more appropriate, and the fundamental expressions for determining
these, assuming sine wave functions, are as follows.
Single-phase transformers
Output ϭ 4.44 f Φ

m
NI with the multiplier 10
Ϫ3
for kVA and 10
Ϫ6
for MVA.
Figure 1.5 Phasor diagram for a single-phase transformer supplying
a capacitive load of leading power factor cos φ
2
. Assumed turns
ratio 1:1. Voltage drops transferred to secondary side
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10 Transformer theory
Three-phase transformers
Output ϭ 4.44 f Φ
m
NI ϫ ͙3 with the multiplier 10
Ϫ3
for kVA and 10
Ϫ6

for MVA.
In the expression for single-phase transformers, I is the full-load current in
the transformer windings and also in the line; for three-phase transformers, I is
the full-load current in each line connected to the transformer. That part of the
expression representing the voltage refers to the voltage between line termi-
nals of the transformer. The constant ͙3 is a multiplier for the phase voltage
in the case of star-connected windings, and for the phase current in the case of
Figure 1.6 Phasor diagram for a three-phase transformer supplying
an inductive load of lagging power factor cos φ

2
. Assumed turns
ratio 1:1. Voltage drops transferred to secondary side. Symbols
have the same signifi cance as in Fig. 1.4 with the addition of A, B
and C subscripts to indicate primary phase phasors, and a, b and c
subscripts to indicate secondary phase phasors
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