heathcote, m. j. (1998). the j & p transformer book (12th ed.)

TheJ&P
Transformer Book
J & P Books
The J&P Transformer Book and The J&P Switchgear Book were published originally by Johnson
& Phillips Ltd, and have for many years been accepted as standard works of reference by
electrical engineers concerned with transformers and switchgear. They now appear under the
Newnes imprint.
TheJ&P
Transformer Book
Twelfth edition
A PRACTICAL TECHNOLOGY OF THE
POWER TRANSFORMER
Martin J. Heathcote, CEng, FIEE
Newnes
OXFORD BOSTON JOHANNESBURG MELBOURNE NEW DELHI SINGAPORE
Newnes
An imprint of Butterworth-Heinemann
Linacre House, Jordan Hill, Oxford OX2 8DP
225 Wildwood Avenue, Woburn, MA 01801-2041
A division of Reed Educational and Professional Publishing Ltd
A member of the Reed Elsevier plc group
First published 1925 by Johnson & Phillips Ltd
Ninth edition 1961
Reprinted by Iliffe Books Ltd 1965
Tenth edition 1973
Reprinted 1967 (twice), 1981
Eleventh edition 1983
Reprinted 1985, 1988, 1990, 1993, 1995
Twelfth edition 1998
© Reed Educational and Professional Publishing Ltd 1998
All rights reserved. No part of this publication may be reproduced in any material form (including

photocopying or storing in any medium by electronic means and whether or not transiently or
incidentally to some other use of this publication) without the written permission of the copyright
holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988
or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham
Court Rd, London, England W1P 9HE. Applications for the copyright holder’s written permission
to reproduce any part of this publication should be addressed to the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library.
ISBN 07506 1158 8
Library of Congress Cataloguing in Publication Data
A catalogue record for this book is available from the Library of congress.
Typeset by Laser Words, Madras, India
Printed in Great Britain
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 10
1.6 Regulation 11
2 Design Fundamentals 13
2.1 Types of transformers 13
2.2 Phase relationships
phasor groups 17

2.3 Volts per turn and flux density 22
2.4 Tappings 23
2.5 Impedance 24
2.6 Multi-winding transformers including tertiary windings 27
2.7 Zero-sequence impedance 32
2.8 Double secondary transformers 33
2.9 General case of three-winding transformers 35
3 Basic Materials 40
3.1 Dielectrics 40
3.2 Core steel 41
3.3 Winding conductors 53
3.4 Insulation 59
3.5 Transformer oil 74
4 Transformer construction 103
4.1 Core construction 104
4.2 Transformer windings 118
4.3 Disposition of windings 143
4.4 Impulse strength 148
4.5 Thermal considerations 156
4.6 Tappings and tapchangers 167
4.7 Winding forces and performance under short-circuit 226
4.8 Tanks and ancillary equipment 245
4.9 Processing and drying out 280
vi Contents
5 Testing of transformers 313
5.1 Testing and quality assurance during manufacture 313
5.2 Final testing 315
5.3 Possible additional testing for important transformers 377
5.4 Transport, installation and commissioning 384
6 Operation and maintenance 398

6.1 Design and layout of transformer installations 398
6.2 Neutral earthing 408
6.3 Transformer noise 422
6.4 Parallel operation 445
6.5 Transient phenomena occurring in transformers 485
6.6 Transformer protection 519
6.7 Maintenance in service 560
6.8 Operation under abnormal conditions 612
6.9 The influence of transformer connections upon third-harmonic
voltages and currents 636
7 Special features of transformers for particular purposes 661
7.1 Generator transformers 661
7.2 Other power station transformers 673
7.3 Transmission transformers and autotransformers 679
7.4 Transformers for HVDC converters 681
7.5 Phase shifting transformers and quadrature boosters 690
7.6 System transformers 697
7.7 Interconnected-star earthing transformers 703
7.8 Distribution transformers 707
7.9 Scott and Le Blanc connected transformers 729
7.10 Rectifier transformers 736
7.11 AC arc furnace transformers 739
7.12 Traction transformers 745
7.13 Generator neutral earthing transformers 750
7.14 Transformers for electrostatic precipitators 756
7.15 Series reactors 758
8 Transformer enquiries and tenders 764
8.1 Transformer enquiries 764
8.2 Assessment of tenders 789
8.3 Economics of ownership and operation, cost of losses 793

APPENDICES
1 Transformer equivalent circuit 803
2 Geometry of the transformer phasor diagram 814
3 The transformer circle diagram 820
Contents vii
4 Transformer regulation 825
5 Symmetrical components in unbalanced three-phase systems 829
6 A symmetrical component study of earth faults in
transformers in parallel 851
7 The use of finite element analysis in the calculation of
leakage flux and dielectric stress distributions 904
8 List of National and International Standards relating to
power transformers 931
9 List of principal CIGRE reports and papers relating
to transformers 934
10 List of reports issued by ERA Technology Limited relating to
transformers and surge phenomena therein 937
Index 941
Foreword
The J & P Transformer Book has been in print for 75 years and during that
time it has been a rewarding work of reference for students, young engineers,
older engineers who have changed the direction of their careers to become
involed with transformers, practising designers and for generations of applica-
tions engineers. In the previous eleven editions the publishers endeavoured to
revise the work, extend it and to bring it up to date. The fact that The J & P
Transformer Book is still in demand is a tribute to the publishers and to the
authors who have carried the torch to light our way for 75 years. The first
edition was prepared by Mr H. Morgan Lacey in 1925, based on a series of
pamphlets entitled Transformer Abstracts that were first printed in 1922. The
book was welcomed as a key reference, giving a guide to British experience at

a time of great change in transformer technology. It was reprinted and revised
many times during the next three decades.
The ninth edition was produced in 1958 by Mr A. C. Franklin together
with his co-author Mr S. A. Stignant. The tenth edition was produced in
1961 by the same authors, and was revised in 1965. Mr Stignant later retired
leaving Mr Franklin, as the main author of the eleventh edition, to carry on
the work. This edition was published in 1983 with some assistance from
Mr D. P. Franklin, who had been appointed as his co-author.
The current twelfth edition has been prepared by Martin J. Heathcote.
Unlike the previous authors, Mr Heathcote has experience as both a
manufacturer and a purchaser. His most recent appointment was with
PowerGen, a successor company to CEGB, where he gained a wide experience
in the design and manufacturing techniques adopted by many different
transformer manufacturers both in Britain and overseas. His strong relationship
with manufacturers and users has allowed him access to a wide range of
information that has been included in this edition. In particular he has
completely rewritten many sections of the book to bring it up to date and
reflect current experience. The latest information on transformer materials has
been included, the modern trend to design transformers with the lowest lifetime
costs has been addressed, and interface problems with other equipment has
been considered in each section. Mr Heathcote’s extensive experience in the
operation and maintenance phases of transformer life has been included in this
edition, together with a more complete analysis of the many specialist types
of transformer that are installed on supply systems and in industrial networks.
This edition contains a wealth of new technical information that has been
freely made available by transformer manufacturers, the electrical supply
x Foreword
industry, learned institutions and industrial associations such as CIGRE. It
is intended that the information contained in this twelfth edition of The J & P
Transformer Book will update the knowledge of the current generation of

engineers and will be of as much use to new generations of engineers as the
previous editions have been to their predecessors.
Professor Dennis J. Allan FEng
Stafford, 16 March 1998
Preface to the twelfth edition
A brief history of the J & P Transformer Book and of its many distinguished
previous authors appears elsewhere in this volume. From this it will be seen
that most were chief transformer engineers or chief designers for major manu-
facturers. The effect of this has been twofold. One, all have tended to write
from a manufacturer’s point of view, and two, all have held very demanding
‘day jobs’ whilst attempting to bring the benefit of their particular knowl-
edge and experience to the task of revising and updating the efforts of their
predecessors. This is a task of great magnitude, and as a result of the many
conflicting demands for their time, even the many ‘complete revisions’ of the
J & P Transformer Book have not greatly changed the unique character that
can be traced back to 1925.
The production of the twelfth edition has been taken as an opportunity to
carry out an almost total rewrite, and, as well as making significant changes
to the structure, to change the viewpoint significantly towards that of the
transformer user.
It is hoped that the book will, nevertheless, still be of value to the young
graduate engineer embarking upon a design carreer, as well as to the student
and those involved in transformer manufacture in other than a design capacity.
To provide more specialist design information than this would require a very
much larger volume and would probably have had the effect of discouraging
a significant proportion of the prospective readership. For the more advanced
designer, there are other sources, the work of CIGRE, many learned society
papers, and some textbooks.
Primarily the objective has been to provide a description of the principles
of transformer design and construction, testing operation and maintenance, as

well as specification and procurement, in sufficient depth to enable those engi-
neers who have involvement with transformers in a system design, installation
or maintenance capacity to become ‘informed users,’ and it is hoped that, in
addition, all of that valuable operational guidance contained in earlier editions
has been retained and made more relevant by being brought fully into line
with current thinking.
Above all, the hope is that the successful formula which has led to the
enormous popularity of earlier editions has not been lost and it is hoped that
the information contained in this edition will prove even more useful to today’s
engineers than those editions which have gone before.
MJH
Acknowledgements
The author wishes to express grateful thanks to many friends and colleagues
who have provided assistance in this major revision of the J&P
Transformer Book. In particular to my good friend W. J. (Jim) Stevens
who has read every word and provided invaluable criticism and comment;
to Professor Dennis 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 first
chapter and helped me to brush up on my somewhat rusty theory; to other
friends who have read and commented on specific sections, and to those
who have provided written contributions; Aziz Ahmad-Marican, University
of Wales, Cardiff, on Petersen coil earthing; Alan Darwin, GEC Alsthom,
on transformer noise; Mike Newman, Whiteley Limited, on transformer
insulation; Cyril Smith, Bowthorpe EMP Limited, on surge arresters; to
Jeremy Price, National Grid Company, for much constructive comment and
advice on the sections relating to many specialised transformers including arc
furnace transformers, HVDC converter transformers, traction transformers and
rectifier transformers. Grateful thanks are also offered to many organisations
who freely provided assistance, 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
Associated Tapchangers Limited
Bowthorpe EMP Limited
British Standards
Br
¨
uel & Kjær Division of Spectris (UK) Limited
Brush Transformers Limited
Carless Refining & Marketing Limited
CIGR
´
E
Copper Development Association
Emform Limited
ERA Technology Limited
GEA Spiro-Gills Limited
GEC Alsthom Engineering Research Centre
GEC Alsthom T & D Transformers Limited
GEC Alsthom T & D Protection and Control Limited
Hawker Siddeley Transformers Limited
Merlin Gerin Lindley Thompson Transformers
Merlin Gerin Switchgear
Peebles Transformers
South Wales Transformers Limited
xiv Acknowledgements
Strategy and Solutions

TCM Tamini
Whiteley limited
In addition to these, special thanks must be expressed to National Power Plc
for the loan of the original artwork for over 50 illustrations which originally
appeared in my chapter on transformers in Volume D of the Third Edition of
Modern Power Station Practice published by Pergamon Press.
Finally, despite the extensive revision involved in the production of the
Twelfth Edition, 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 protection,
magnetising inrush, parallel operation, and third harmonic voltages and
currents, and for this due acknowledgement must be given.
1 Transformer theory
1.1 INTRODUCTION
The invention of the power transformer towards the end of the nineteenth
century made possible the development of the modern constant voltage AC
supply system, with power stations often located many miles from centres of
electrical 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 bene-
fits of a device which could take the high-current, relatively low-voltage output
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 efficiency
which, by the standards of the time, was nothing less than phenomenal.
Today’s transmission and distribution systems are, of course, vastly more
extensive and greatly dependent on transformers which themselves are very
much more efficient than those of a century ago; from the enormous gener-
ator transformers such as the one illustrated in Figure 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 1200A or so, to
the thousands 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, primarily from a UK viewpoint, but in the rapidly shrinking and
ever more competitive world of technology it is not possible to retain one’s
1
2 Transformer theory
place in it without a knowledge of all that is going on 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
secondary, linked by a magnetic circuit or core. When an alternating voltage
is applied to one of these windings, generally by definition the primary, a
current will flow which sets up an alternating m.m.f. and hence an alternating
flux in the core. This alternating flux 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 flow. In reality, the current which flows 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 flow 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 additional 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 flux must be proportional to the number
of turns. Thus the well-known relationship is established that:
E
1
/E
2
D N
1
/N
2
1.1
and, in view of the need for ampere-turns balance:
I
1
N
1
D 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 identified 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 flux is given by refer-
ence to Faraday’s law which states that its magnitude is proportional to the
rate of change of flux linkage, and Lenz’s law which states that its polarity
is such as to oppose that flux linkage change if current were allowed to flow.
This is normally expressed in the form
e DNd/dt
Transformer theory 3
but, for the practical transformer, it can be shown that the voltage induced per
turn is
E/N D K8
m
f1.3
where K is a constant, 8
m
is the maximum value of total flux in Webers
linking 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 corresponding 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 a transformer,
the small drop in voltage due to the flow of the no-load current in the primary
winding is neglected.
Figure 1.1 Phasor diagram for a single-phase transformer on open
circuit. Assumed turns ratio 1:1
If the voltage is sinusoidal, which, of course, is always assumed, K is 4.44
and equation (1.3) becomes
E D 4.44f8N
4 Transformer theory

For design calculations the designer is more interested in volts per turn and
flux density in the core rather than total flux, so the expression can be rewritten
in terms of these quantities thus:
E/N D 4.44B
m
Af ð 10
6
1.4
where E/N D volts per turn, which is the same in both windings
B
m
D maximum value of flux density in the core, tesla
A D nett cross-sectional area of the core, mm
2
f D 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
produced 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 specified 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
flux 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
distribution systems, leakage reactance
or, in practical terms, impedance,
since transformer windings also have resistance
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 reflects the way in which it is seen by
system designers. For example, an impedance of 10% means that the voltage
drop at full-load current is 10% 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:
V
z
D %Z D
I
FL
Z
E
ð 100
where Z is

R
2
C X
2
, R and X being the transformer resistance and leakage
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 may
themselves be expressed as percentage voltage drops, as explained below.
The ‘natural’ value for percentage impedance tends to increase as the rating
Transformer theory 5
of the transformer increases with a typical value for a medium-sized power
transformer being about 9 or 10%. Occasionally some transformers are delib-
erately designed to have impedances as high as 22.5%. 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 flux 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 flow of load current in the secondary of the transformer
and the m.m.f. which this produces are balanced by an equivalent primary
load current and its m.m.f., which explains why the iron loss is independent
of the load.
The flow of a current in any electrical system, however, also generates loss
dependent upon the magnitude of that current and the resistance of the system.
Figure 1.2 Phasor diagram for a single-phase transformer
supplying a unity power factor load. Assumed turns ratio 1:1
6 Transformer theory
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 produce
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 resistance and leakage reactance voltage drops, the condition for a trans-
former supplying a non-inductive load is shown in phasor form in Figure 1.2.
Considering now the voltage drops due to resistance and leakage reactance
of the transformer windings it should first 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 wind-
ings are easily separated and determinable for the respective windings. The
Figure 1.3 Phasor diagram for a single-phase transformer
supplying an inductive load of lagging power factor cos 
2
.
Assumed turns ratio 1:1. Voltage drops divided between primary
and secondary sides
Transformer theory 7
reactive voltage drop, which is due to the total flux leakage between the two
windings, is strictly not separable into two components, as the line of demar-
cation between the primary and secondary leakage fluxes cannot be defined.
It has therefore become a convention to allocate half the leakage flux 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 lagging power factor of cos
2
, the resistance and
leakage reactance drops being allocated to their respective windings. In fact
the sum total effect is a reduction in the secondary terminal voltage. The resis-
tance 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 Figure 1.3, except
that the resistance and reactance drops are all shown as occurring on the
secondary side.
Figure 1.4 Phasor diagram for a single-phase transformer
supplying an inductive load of lagging power factor cos 
2
.
Assumed turns ratio 1:1. Voltage drops transferred to secondary
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
8 Transformer theory
step-down transformers. In other words the percentage voltage drops consid-
ered 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
transformation n in the case of both step-up and step-down transformers.
The transference of impedance from one side to another is made as follows:
Let Z
s
D total impedance of the secondary circuit

including leakage and load characteristics
Z
0
s
D equivalent value of Z
s
when referred to
the primary winding
Then I
0
2
D
N
2
N
1
I
2
D
N
2
N
1
E
2
Z
s
and E
2
D

N
2
N
1
E
1
so I
0
2
D

N
2
N
1
2
E
1
Z
s
1.5
Also, V
1
D E
1
C I
0
2
Z
1

where E
1
D I
0
2
Z
0
s
Therefore I
0
2
D E
1
/Z
0
s
1.6
Comparing equations (1.5) and (1.6) it will be seen that Z
0
s
D Z
s
N
1
/N
2

2
.
Figure 1.5 Phasor diagram for a single-phase transformer

supplying a capacitive load of leading power factor cos 
2
.
Assumed turns ratio 1:1. Voltage drops transferred to secondary
side
Transformer theory 9
The equivalent impedance is thus obtained by multiplying the actual
impedance of the secondary winding by the square of the ratio of
transformation n,i.e.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 1.5 is included as a matter of interest to show that when the load
has a sufficient leading power factor, the secondary terminal voltage increases
instead of decreasing. This happens when a leading current passes through an
inductive reactance.
Preceding diagrams have been drawn for single-phase transformers, but they
are strictly applicable to polyphase transformers, so long as the conditions for
Figure 1.6 Phasor diagram for a three-phase transformer supplying
an inductive load of lagging power factor cos 
2
. Assumed turns
ratio 1:1. Voltage drops transferred to secondary side. Symbols
have the same significance as in Figure 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
10 Transformer theory
all the phases are shown. For instance Figure 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 Figure 1.4, in which primary
and secondary phasors correspond exactly to those in Figure 1.4, but the three
sets representing the three different phases are spaced 120
°
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 generally more appropriate, and the fundamental expressions for determining
these, assuming sine wave functions, are as follows:
Single-phase transformers
Output D 4.44f8
m
NI with the multiplier 10
3
for kVA
and 10
6
for MVA
Three-phase transformers
Output D 4.44f8
m
NI ð
p
3 with the multiplier 10
3

for kVA
and 10
6
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 terminals
of the transformer. The constant
p
3 is a multiplier for the phase voltage in
the case of star-connected windings, and for the phase current in the case of
delta-connected windings, and takes account of the angular displacement of
the phases.
Alternatively expressed, the rated output is the product of the rated
secondary (no-load) voltage E
2
and the rated full-load output current I
2
although these do not, in fact, occur simultaneously and, in the case of
polyphase transformers, by multiplying by the appropriate phase factor and
the appropriate constant depending on the magnitude of the units employed.
It should be noted that rated primary and secondary voltages do occur
simultaneously at no-load.
Single-phase transformers
Output D E
2
I
2
with the multiplier 10

3
for kVA
and 10
6
for MVA
Transformer theory 11
Three-phase transformers
Output D E
2
I
2
ð
p
3 with the multiplier 10
3
for kVA
and 10
6
for MVA
The relationships between phase and line currents and voltages for star- and
for delta-connected three-phase windings are as follows:
Three-phase star connection
phase current D line current I D VA /E ð
p
3
phase voltage D E/
p
3
Three-phase delta connection
phase current D I/

p
3 D VA/E ð
p
3
phase voltage D line voltage D E
E and I D line voltage and current respectively
1.6 REGULATION
The regulation that occurs at the secondary terminals of a transformer when
a load is supplied consists, as previously mentioned, of voltage drops due
to the resistance of the windings and voltage drops due to the leakage reac-
tance between the windings. These two voltage drops are in quadrature with
one another, the resistance drop being in phase with the load current. The
percentage regulation at unity power factor load may be calculated by means
of the following expression:
copper loss ð100
output
C
percentage reactance
2
200
This value is always positive and indicates a voltage drop with load.
The approximate percentage regulation for a current loading of a times
rated full-load current and a power factor of cos 
2
is given by the following
expression:
percentage regulation D aV
R
cos 
2

C V
X
sin 
2

C
a
2
200
V
X
cos 
2
 V
R
sin 
2

2
1.7
where V
R
D percentage resistance voltage at full load
D
copper loss ð100
rated kVA
12 Transformer theory
V
X
D percentage reactance voltage D

I
2
X
00
e
V
2
ð 100
Equation (1.7) is sufficiently accurate for most practical transformers; however,
for transformers having reactance values up to about 4% a further simplifica-
tion may be made by using the expression:
percentage regulation D aV
R
cos 
2
C V
X
sin 
2
1.8
and for transformers having high reactance values, say 20% or over, it is some-
times necessary to include an additional term as in the following expression:
percentage regulation D aV
R
cos 
2
C V
X
sin 
2


C
a
2
2 ð 10
2
V
X
cos 
2
 V
R
sin 
2
2

C
a
4
8 ð 10
6
V
X
cos 
2
 V
R
sin 
2


4
1.9
At loads of low power factor the regulation becomes of serious consequence
if the reactance is at all high on account of its quadrature phase relationship.
This question is dealt with more fully in Appendix 4.
Copper loss in the above expressions is measured in kilowatts. The expres-
sion for regulation is derived for a simplified equivalent circuit as shown
in Figure 1.7, that is, a single leakage reactance and a single resistance in
series between the input and the output terminals. The values have been repre-
sented in the above expressions as secondary winding quantities but they could
equally have been expressed in primary winding terms. Since the second term
is small it is often sufficiently accurate to take the regulation as equal to the
value of the first term only, particularly for values of impedance up to about
4% or power factors of about 0.9 or better.
XR
Input
terminals
Output
terminals
Figure 1.7 Simplified equivalent circuit of leakage impedance of
two-winding transformer
V
X
may be obtained theoretically by calculation (see Chapter 2) or actually
from the tested impedance and losses of the transformer. It should be noted that
the per cent resistance used is that value obtained from the transformer losses,
since this takes into account eddy-current losses and stray losses within the
transformer. This is sometimes termed the AC resistance, as distinct from the
value which would be measured by passing direct current through the windings
and measuring the voltage drop (see Chapter 5, Testing of transformers).

2 Design fundamentals
2.1 TYPES OF TRANSFORMERS
There are two basic types of transformers categorised by their winding/core
configuration: (a) shell type and (b) core type. The difference is best under-
stood by reference to Figure 2.1.
Figure 2.1 Transformer types
In a shell-type transformer the flux-return paths of the core are external to
and enclose the windings. Figure 2.1(a) shows an example of a three-phase
shell-type transformer.
While one large power transformer manufacturer in North America was
noted for his use of shell-type designs, core-type designs predominate in the
UK and throughout most of the world, so that this book will be restricted to
the description of core-type transformers except where specifically identified
otherwise.
13