GIFT OF
MICHAEL REE&E
TRANSFORMERS
WOBKS BY GISBEBT KAPP,
M. INST. C. E., M. INST. E. E.
ELECTRICAL ENGINEERING DESIGNS. A Series of
Dynamos and other Electrical Machines, with Descriptive
Articles. Containing examples of English, American,
German, and French practice. (In the press.}
ELECTRIC TRANSMISSION OF ENERGY, AND ITS
Transformation, Subdivision, and Distribution. A Practical
Handbook. Fourth Edition, mostly rewritten, 455 pp. xii.
pp. With 166 Illustrations. Crown 8vo, 10s. 6d.
* # * The work has been brought up to date, both as regards
theory and practice.
'This book is one which must of necessity be found in the hands of
every one who desires to become acquainted with the best and latest in-
formation on the subject.' Electrical Engineer.
'The book is an excellent one in every way, and will, we imagine, long
be regarded as the standard treatise on the electrical transmission of
energy. ' Mechanical World.
' Although, therefore, the book will be of greater interest to the trained
specialist, it has an intrinsic value for the average manufacturer who is
willing to give a little study to the subject.' Textile Recorder.
'Is one of the most generally useful books to the electrical engineer
which has been published.' Industries and Iron.
ALTERNATING CURRENTS OF ELECTRICITY,
their Generation, Measurement, Distribution, and Applica-
tion. With 37 Illustrations and 2 Plates. 4*. Qd.
DYNAMOS, ALTERNATORS and TRANSFORMERS.
With 138 Illustrations. Crown 8yo, 10s. Qd.
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LONDON, WHITTAKER & CO., PATERNOSTER SQUARE.
TBANSFORMEKS
SINGLE AND MULTIPHASE CURRENTS
A TREATISE ON
THEIR THEORY, CONSTRUCTION, AND USE
GISBERT KAPP
, /
MEMB. 1NST. C. E., MEMB. 1NST. E. E.
WITH ONE HUNDRED AND THIRTY-THREE ILLUSTRATIONS
WHITTAKER AND CO.
WHITE HART STREET, PATERNOSTER SQUARE, LONDON
AND 66, FIFTH AVENUE, NEW YORK
1896
[All rights reserved']
Ct.
OF THE
PREFACE
IN what may be termed the literature of heavy electrical
engineering there is no lack of books on dynamos, motors,
cables, and a host of auxiliary apparatus, but transformers
have been somewhat neglected. The reason may possibly
be that scientists have not considered it worth their
trouble to investigate so seemingly simple a piece of
apparatus as a transformer, whilst so much more interest-
ing problems connected with machinery in motion
remained to be solved. Practical engineers, on the other
hand, who, in carrying on their profession, must investigate
whatever they design, cannot be expected to publish the
result of their researches for the benefit of competitors.
Notwithstanding the apparent simplicity of the trans-
former, a study of this apparatus leads to a number of
questions which are not only highly interesting from a
purely scientific point of view, but are also very important
for practical or commercial reasons. On most alternating-
current lighting installations the transformers are at work
night and day all the year round, and cause a waste of
power which is always going on. Under these circum-
stances, even a small improvement in the efficiency of the
apparatus has a large monetary value, but such improve-
ments can only be made as the result of "careful study
combined with practical experience,
vi PREFACE.
The object of the present book is to enable the reader to
judge the design of transformers and to design such appara-
tus for himself. The mathematical treatment of the
subject has been kept as short and as simple as possible.
I am well aware that through the omission of many theore-
tical considerations which were not immediately pertinent
to the practical object I had in view, my book, considered
as a scientific treatise on transformers, is incomplete.
I have omitted to include detailed researches on the
influence of the shape of current and E.M.F. waves on
hysteresis loss ; I also have omitted the analysis of such
waves by Fourier's series, and generally the analytical
treatment of problems connected with transformers.
These are sins of omission for which I plead the indulg-
ence of the reader, on the ground that my book is not
intended for mathematicians or physicists, but for engineer-
ing students and practical engineers, whose object must
always be to obtain a maximum of practical success with
a minimum expenditure of mental labour.
GISBERT KAPP.
Berlin, September 1896.
\
OF THE
IVERSITTJ
J
CONTENTS
CHAPTER I
PRINCIPLE OF ACTION MAGNETIC LEAKAGE ARRANGEMENT OF COILS
FUNDAMENTAL EQUATION ... ... ...page 1
CHAPTER II
LOSSES IN TRANSFORMERS INFLUENCE OF THE E.M.F. CURVE ON
HYSTERESIS LOSS INFLUENCE OF THE SHAPE OF CORE AND
COILS UPON THE LOSSES CORE AND SHELL TRANSFORMERS 19
CHAPTER III
USUAL TYPES CONSTRUCTION OF THE IRON PART PROPORTIONS OF
THE IRON PART HEATING OF TRANSFORMERS RESULTS OF
TESTS INFLUENCE OF THE LINEAR DIMENSIONS ... 39
CHAPTER IV
POWER OF ALTERNATING CURRENT COMBINING CURRENTS OR PRES-
SURES DETERMINATION OF NO LOAD CURRENT INFLUENCE OF
JOINTS ... ... ... ... ... ... G3
CHAPTER V
DESIGN OF A TRANSFORMER BEST DISTRIBUTION OF COPPER COST
OF ACTIVE MATERIAL BEST DISTRIBUTION OF LOSSES ECONOMY
JN WORKING CONSTRUCTION DETAILS ... 83
viii CONTENTS.
CHAPTER VI
CLOCK DIAGRAMS WORKING ON OPEN CIRCUIT WORKING UNDER
LOAD MAGNETIC LEAKAGE WORKING DIAGRAM OF TRANS-
FORMER HAVING LEAKAGE VOLTAGE DROP GRAPHIC DETER-
MINATION OF DROP 103
CHAPTER VII
THE DYNAMOMETER THE WATTMETER MEASUREMENT OF IRREGULAR
CURRENTS OTHER METHODS OF POWER-MEASUREMENTTESTING
TRANSFORMERS TESTING SHEET-IRON 141
CHAPTER VIII
SAFETY APPLIANCES FOR TRANSFORMERS SUB-STATIONS AND HOUSE-
TRANSFORMERS BOOSTERS CONNECTION IN SERIES CHOKING
COILS COMPENSATING COILS THREE-WIRE SYSTEM BALANCING
TRANSFORMER SCOTT'S SYSTEM 175
CHAPTER IX
ILLUSTRATED DESCRIPTION OF SOME TRANSFORMERS 204
LI
OF THE
XTNIVERSIT
LIST OF ILLUSTRATIONS
1. Faraday's ring 2
2. Magnetic leakage 5
3. Faraday's ring improved 6
4. Magnetic pressure ... ... ... ... ... ... 7
5. Subdivision of coils 8
6 Eectangular magnetic link 9
7. Rotating loop 11
8. Clock diagram 15
9. Curve of losses for different brands of iron 23
10. Graphic representation of E.M.F. curve ... ... ... 26
11 Relation of E.M.F. curve to induction curve 28
12. Types of shell transformer 34
13. Hedgehog transformers ... ... ... ... ... 37
14. Typical shell transformer 39
15. Kapp's core transformer ... ... ... ... ... 40
16,17. Core transformer for polyphase work 40
18. Core of Westinghouse transformer 42
19. Core of Ferranti's transformer 42
20. Dovetail joint of Kapp's core 43
21. Crompton's core 43
22. Mordey core 44
23. Mordey's additional stampings ... ... ... ... 44
24. Mode of inserting insulation in cores 46
25. Ditto, another way 47
26. Design for proportion of iron parts 47
27. Design for shell transformer with short cores 48
* LIST OF ILLUSTRATIONS.
FIG. PAOE
28. Table of heat curves ... ... ... ... ... 51
29. Table of tests of heat 53
30. Curves of E.M.F. and current 64
31. Clock diagram to show periodic variation ... ... 66
32. Curves of instantaneous value 67
33. Diagram of effective power 70
34. Diagram of alternators coupled ... ... ... ... 71
85. Clock diagram of coupled alternators ... ... ... 72
36. Diagram of four currents 73
37. Two alternators coupled to voltmeters ... ... ... 74
38. Determination of phase ... ... ... ... ... 74
39. Determination of no load ... 77
40. Design for core 84
41 to 44. Plans for transformer 98-101
45. Clock of transformer on open circuit ... ... ... 104
46. Ditto, under load 106
47. Diagram of magnetic leakage - 109
48. Subdivisions of core to lessen magnetic leakage ... 110
49,50. Ditto, ditto Ill
51. Choking coils to avoid leakage ... ... ... ... 112
52. Clock diagram of transformer having leakage 114
53. Clock diagram of short-circuited transformer 116
54. Diagrams of voltage " drop " 118
55. Ditto ... 119
56. Graphic determination of drop 120
57. Diagram of self-induction 122
58. Ditto, with lamps in circuit 122
59. Effect of phase difference 1 24
60. Effect of constant load 125
61. E.M.F. vector 127
62. Effect of capacity %. 129
63. Diagram of inductive resistance ... ... ... ... 130
64. Diagram of current passing through liquid resistance ... 131
65. Diagram of current leads on pressure 132
66. Diagram of pressures for different currents 133
67. Diagram of pressure with different phases 136
68. Scale for power factor 139
69. Dynamometer 145
70. Measuring main current ... ... ... ... ... 148
71. Measuring shunt current 148
LIST OF ILLUSTRATIONS. xi
FIG. PACK
72. Measuring power 151
73. Connections to be avoided 1 52
74. Graphic representation of measurement ... ... ... 153
75. Three voltmeter method ... ... ... ... ... 158
76. Clock diagram of three- volt method ... ... ... 159
77. Three amperemeter method 161
78. Clock diagram of ditto 161
79. Transformer testing 164
80. Dobrowolsky's sheet-iron tester 170
81. Kapp's sheet-iron tester 171
82. Ewing's ditto 173
83. Thomson- Houston safety device 177
84. Cardew's safety device 178
85. Ferranti's ditto 179
86. Sub-station transformer 181
87. Booster 187
88. Booster connected to primary circuit ... ... ... 188
89. Booster with no switch ... ... ... ... ... 189
90. Transformer connected in series 191
91. Transformer for lamps in series ... ... ... ... 193
92. Diagram of regulation ... ... ... ... ... 194
93. Choking-coil arrangement ... ... ... ... ... 195
94. Clock diagram of ditto 195
95. Diagram of compensating coils ... ... ... ... 196
96. Ditto, with three lamps and three coils 198
97. Three wire system 199
98. Ditto, with balancing transmitter 200
99. Scott's system 201
100. Graphic representation of Scott's system 202
101. Complete plant on Scott's system 203
102. Siemens and Halske three-phase transformer, with and
without case 205
103. Siemens and Halske transformer with perforated case ... 206
104. Dimensioned plans and elevations of ditto 208
105. Ditto 209
106. Schukert's single-phase transformer ... ... ... 210
107. Ditto i 211
108. Schukert's three-phase transformer 212
109. Schwartykopff single-phase transformer ... ... ... 213
110. Ditto, with perforated casing 213
xii LItil 1 OF ILLUSTRATIONS.
FIG. 1'AG K
111. Siemen ,< Bros. & Co.'s transformers ... ... ... 214
112. Parts of Brown, Bo veri & Co.'s transformer 216
113. Brown, Boveri & Co.'s transformer complete 216
114. W. Lahmeyer & Co.'s single-phase transformer 217
115. Ditto polyphase dityo 218
116. Mordey-Brush Electrical Engineering Co.'s transformer 219
117. Ditto, incase 220
118. Johnson & Phillips' transformers 221
119. Ditto 222
120. Ganz & Co.'s transformers 223
121. Maschinenfabrik Oerlikon single-phase transformer ... 224
122. Ditto, with perforated cover 225
123. Ditto, without side-cover 225
124 Ditto, three-phase transformer 226
125. Ditto, with circular yokes 227
126. Ditto, with straight yokes 228
127. Electric Construction Co.'s transformer 229
128. Ditto 230
129. Ditto ... 231
130. Allgemeine Elektricitatsgesellshaft's transformer ... 232
131. Westinghouse Electric Manufacturing Co.'s transformer 234
132. Ditto 235
133. Ditto .. 236
OF THE
UNIVERSIT
RANSFORMERS
CHAPTER I
rillNCII'LE OF ACTION MAGNETIC LEAKAGE ARRANGEMENT OF COILS
FUNDAMENTAL EQUATION
Principle of action, If the magnetic flux N passing
through a coil changes, an E.M.F. is induced in the coil
which is proportional to the time-rate of change/ - ) and
the number of turns n. Conversely, if a current be sent
through the coil it produces a magnetic flux, threading
the coil, which is within certain limits proportional to the
current. If this current changes, a corresponding change
takes place in the magnetic flux. Let now two coils be so
arranged that the flux produced by the current in one
passes wholly or partially through the other, then any
change in the current strength in the former coil will
produce an E.M.F. in the latter coil. Such an arrangement
is shown in Fig. 1, where a ring of iron is threaded through
the two coils I, II. A current passing through coil I
produces a magnetic field which passes partly through the
iron ring, and partly through the air surrounding this
coil. The flux will therefore be strongest in the centre
of the coil, at a, and weakest at b, in the centre of coil II.
The iron ring acts as a vehicle for carrying the flux
produced by coil I through coil II, though as an imperfect
vehicle, since part of the flux is lost on the way. In a
sense, the iron ring may be regarded as a magnetic link
between the two coils. Even without iron the coils can
be linked together by the magnetic flux passing through
air. Thus in the position shown, the field produced by
I would in part pass through II, though its strength
would be much diminished. The same holds good if the
two coils are laid upon each other, in which position a
somewhat stronger field would pass through II, though
not so strong as with an iron ring. If, however, whilst
Fig. 1.
still omitting the iron ring, the coils are relatively so
placed that the axis of I lies in the plane of II, or vice
versa, then none of the lines produced by I can pass
through II, and a change in the current passing through
I cannot produce any E.M.F. in II. By suitably
placing the coils, an inductive effect of one upon the other
can therefore be produced, even without the use of an iron
link, but the employment of such a link has the advantage
that not only is the inductive action increased, but it
becomes to a greater extent independent of the mutual
position of the coils. An apparatus consisting of two coils,
interlinked with an iron coil common to both, is called a
transformer.
PRINCIPLE OF ACTION. 8
It has already been mentioned that the E.M.F.
produced in II, which we may call the secondary coil, is
proportional to the time-rate of change of the current
in the primary coil I. Since the current in this coil
cannot alter indefinitely in the same sense without
becoming infinite, it follows that periods of growing
current strength must alternate with periods of declining
current strength. If, then, with a growing current
in the primary coil the E.M.F. induced in the secondary
coil acts in one way, it must act in the opposite way if the
current diminishes, and it is thus clear that changes in the
current strength in the primary coil, even if not accompanied
by changes in direction, must produce an alternating
E.M.F. in the secondary coil. This alternating E.M.F.
produces, in an external circuit connected to the terminals
of the secondary coil, an alternating current. We are thus
able to convert a unidirected pulsatory current into an
alternating current, but never into a continuous current.
Instead of using a pulsating current in the primary coil,
we may with advantage use an alternating current, and
thus obtain from the secondary coil another alternating
current, the E.M.F. of which is dependent on that of the
primary current, and on the ratio between the number of
turns in the two windings.
Magnetic leakage. Since the lines of force not only pass
through iron, but in a lesser degree also through air, it
follows that only part of the magnetic flux in a actually
threads through the secondary coil at &, the rest closing
round the primary coil in air. The difference between
the flux in a and & will be the greater the farther the
coils are from each other, and the greater the resistance
which the iron offers to the passage of the lines of force.
In consequence of this resistance (sometimes also called
4 T
magnetic reluctance) lines of force are caused to leak out
laterally, and form thus a leakage field which does not
contribute in any way to the production of E.M.F. in the
secondary coil. The more leakage there is, the smaller is
consequently the E.M.F. induced in the secondary coil
through an alternating current in the primary coil.
In order to understand the conditions which influence
leakage, we assume for the present that the primary coil
carries a continuous current, whilst through the secondary
coil there passes, either no current at all, or also a
continuous current of such direction as will tend to weaken
the field produced by the primary current.
The coil I drives, then, a magnetic flux in a certain
direction through the iron ring. If no current flows
through coil II, then the lines of force have only to over-
come the magnetic resistance of the iron path, which may
'be so small that comparatively few lines are crowded out.
If, however, the coil II also carries a current, it will tend
to produce a magnetic flux in the opposite direction, which,
colliding with the original flux, must can sea strong leakage
field, thus weakening considerably the flux actually passing
through the secondary coil.
This condition of things may easily be explained by
hydraulic analogy. Let, in Fig. 2, a ring-shaped tube of
porous material be filled with and immersed in water, and
let the water in the tube, as shown by the arrow, be kept
in motion by a propelling fan I. This fan produces a
difference of pressure between its inlet and outlet side,
which pressure is absorbed by the frictional resistance of
the tube. Since the pressure above the fan is greater
than that below, water will, as indicated by the dotted
lines, pass out through the pores of the tube in its upper
half, and enter the tube through the pores in its lower half.
MAGNETIC LEAKAGE.
The velocity of the water must consequently be greater at a
than at b. If the tube is wide and the propelling power of
the fan small, little head will suffice to overcome the friction ;
and the quantity of water leaking out and in, as well as the
difference of velocity in a and b } will be small. Let now a
second fan (II) be inserted at b, which for the present we
will imagine to be frictionless ; then this fan will be set in
rotation by the stream of water, but it will not increase the
Fig. 2.
leakage nor diminish the velocity of the water. If, however,
we retard the motion of fan II by letting its spindle
transmit mechanical energy, the free flow of the water will
be impeded, and the difference of pressure between the
upper and lower halves of the tube will be increased. As a
result, the leakage will be augmented, and the quantity of
water passing the point a in unit time will be appreciably
more than that passing the point b. At the same time the
speed of fan II will be reduced ; and this for two reasons.
In the first place because the load on the spindle of II
must retard its motion, and in the second place because
the velocity of the water is smaller than before. If we
wish to limit the loss of speed due to the latter cause,
we can do so by placing the fan I as near as possible to
fan II.
Now let us substitute for the porous tube the iron ring,
and for the two fans the driving and the driven coil ; then
we see that the magnetic flux through the driven coil
(which corresponds to the velocity of the water at 1>) will
be the smaller the stronger the current is in the driven
coil.
The arrangement of coils shown in Fig. 1 is bad, on
account of their great distance. It does not give a strong
magnetic flux through the driven coil if a large current is
permitted to flow through this coil. We can improve the
design by spreading the coils each over half the circum-
ference of the ring, as shown in Fig. 3. In this case the
Fig. 3.
magnetic pressure tending to force the lines through the
air is no longer constant over each half of the ring, but it
attains its previous value only in the points c and cl. It
diminishes on either side of the vertical diameter, and
becomes zero in a and I. The leakage field is therefore
not only quantitatively smaller, but, owing to its distribu-
tion and the distribution of the two windings, its quali-
tative influence is also lessened, as compared with the
arrangement shown in Fig. 1.
The distribution of the leakage field may be approxi-
mately determined if we remember that the magnetic
MAGNETIC LEAKAGE. 7
pressure, which forces the lines to leave the ring at any
point, is proportional to the ampere turns counted up to
that point. Imagine now the windings evenly distributed,
and the direction of the currents thus, that the magnetic
pressure is from iron to air in the upper left quadrant, and
from air to iron in the lower left quadrant. Corresponding
pressures must of course exist in the right quadrants.
Let now the ring be cut at a and straightened out, then
the zig-zag line in Fig. 4 gives a graphic representation
Fig. 4.
of the magnetic pressure producing leakage. Positive
ordinates represent a pressure from iron to air, i. c. north
polarity, and negative ordinates the opposite pressure, or
south polarity. The leakage lines are shown dotted in
Fig. 3, but only inside the ring. There are, of course,
also leakage lines in the whole of the air space Surrounding
the ring. If we assume, as a very rough approximation,
that the magnetic resistance along any path through air is
the same, then the number of lines passing through unit
surface in any point of the ring will be proportional to
the magnetic pressure at that point, and the shaded areas
in Fig. 4 may be taken to represent roughly the leakage
field. The assumption that all the paths through air
have the same magnetic resistance is of course not strictly
correct; as we are, however, at present only concerned
with a general investigation of the leakage field, it is not;
8
TRANSFORMERS.
necessary to enter minutely into the question of how the
magnetic resistance of any particular path through the
air varies, and we may as a rough approximation assume
that Fig. 4 represents the leakage field.
We have up to the present assumed that the two coils
carry continuous currents, but it is obvious that our
reasoning applies equally to the case of alternating currents,
provided that the change in direction occurs in both coils
nearly simultaneously, a condition which is always fulfilled
in transformers when working under a load.
Arrangement of coils. We have seen that in point of
Fig. 5.
leakage Fig. 3 is an improvement on Fig. 1. We may,
however, carry the improvement still further by sub-
dividing each unit into several parts. In Fig. 5 we
have six separate coils, arranged to uniformly cover the
ring, and connected alternately with the primary and
secondary circuit. The greatest magnetic pressure is also
in this case at the junction of two coils ; since, however,
the number of turns in each coil is reduced to one-third,
this pressure is also reduced to one-third of its previous
value. The surface through which lines can leak is at
the same time also reduced to one-third, so that the total
ARRANGEMENT OF COILS. 9
leakage field now only amounts to * X * = J of its previous
value. If, instead of subdividing each coil into three parts,
we subdivide it into four, the leakage field would be reduced
to y F of its previous value, and so on. It will thus be
seen that, by carrying the principle of subdivision suffici-
ently far, we can reduce the leakage field to any desired
extent. It could even be reduced to zero if we were to
interweave the primary and secondary coils. This would,
however, lead to difficulties as regards insulation, and is
not necessary, since experience has shown that it suffices
for all practical purposes to subdivide the windings so far
as to limit the effective ampere turns in each individual
coil to 500 or GOO.
We have hitherto assumed that the magnetic link
between the two windings is a circular ring, but it will
Fte. 6.
be obvious that its geometric form is immaterial, and
that any shape of ring may be used. We might, for
instance, employ a magnetic link in the shape of a
rectangular frame, and place the coils over the two longer
sides of the rectangle (Fig. 6). The arrangement shown
on the left corresponds to Fig. 3. In this case there is
only one secondary and one primary coil, and the magnetic
leakage must therefore be very great. In the arrangement
shown on the right, the primary winding is subdivided
into five coils, which alternate in position with five coils
10 TRANSFORMERS.
of the secondary winding. The leakage is thereby re-
duced to about o^ part. Another, and with regard to
the reduction of leakage equally effective, arrangement
consists in placing the coils axially within each other.
This arrangement has the advantage of a reduction in
the number of separate coils to be wound and handled,
whilst at the same time the insulation between primary
and secondary coils is of simple shape (plain cylinders),
and can therefore easily be made perfect. v
Fundamental equation. The E.M.F. induced in a coil
is, by a well-known law of electro-dynamics, proportional
to the number of turns of wire, and the time-rate of
change of the magnetic flux. In symbols
dt
In order to be able to calculate the E.M.F. occurring at
any given moment of time, we must know the relation
between N and t. The magnetic flux N is produced by
the current passing through the primary coil, and if the
magnetization of the iron core remains within such limits
that the permeability may be considered to remain
constant, then N may be considered to be proportional to
the primary current. We assume for the present that
the secondary coil is open, so that no current can flow
through it which would mask or disturb the magnetizing
effect of the primary current. There are, as a matter of
fact, certain secondary actions which interfere with the
strict proportionality of primary current and magnetic
flux, but the consideration of these we must postpone.
We also assume that the primary current is obtained from
an alternator, the E.M.F. of which follows a true sine law.
This is not always, and indeed very seldom, the case ; but
FUNDAMENTAL EQUATION.
11
it will be shown later on that the equations obtained
under these assumptions remain applicable in all cases
occurring in practice.
Imagine, then, a wire coil of one turn, including an area of
s. square centimetres, traversed at right angles to its plane
by a magnetic flux, which varies according to a periodic
sine function between the limits + N and N. The
maximum value of the induction is obviously B = N : s.
Call the time required for the performance of a complete
/
^
^
x
X
X
"^.
\
i
^
^
x
\
\
XN
X
/
s
^
^
s
cycle from +N to N, and back to +A T , T, and the
number of cycles occurring per second ~ ; then
Since the E.M.F. is dependent on the change in the
flux passing through the wire loop, but not on the angle
at which the lines of force thread through the loop, we
may replace the rectilinear and oscillating field by a con-
stant and homogeneous field, provided we revolve the loop
round an axis in its plane with a speed of ~ revolutions
per second. Let, in Fig. 7, the field be represented by the
vertical lines, and be the axis around which the wire
loop is rotated. If the rotation takes place in the direc-
12 TRANSFORMERS.
tion shown by the arrow, and if we count the time from
the moment the loop is horizontal, then let, at time t, the
coil occupy the angular position a. Through the cutting
of the lines of force there will be induced in the upper
half of the loop an E.M.F. directed towards the observer,
and in the lower half from the observer. In this and all
the following diagrams we mark these directions respect-
ively by a dot and a cross inscribed into the little circle
representing the cross-section of the wire ; these signs
meaning the point and the feathers of an arrow which
indicates the direction of E.M.F. or current.
Let co be the angular velocity of the loop, then a = co t,
and co = 2 IT ~ ; from which it follows that
a = 2 TT ~ t.
The magnetic flux threading through the loop is ob-
viously N cos a, and its rate of change
d NCOS a , r . da
-~3T -**-&
Q. da
mce eft = M ~ ^ "*"' we h ave f r the instantaneous
value of the E.M.F. the expression
E = 2 TT ~ Nsin a.
The loop has only one turn of wire. If there are n turns
the same E.M.F. is induced in each, and the total E.M.F.,
measured at the terminals of a coil of n turns, is therefore
in absolute measurement
E 2 77 ~ N n sin a.
To obtain it in volts we must multiply by 10~ 8 . If the
coil is horizontal, the flux threading through it is a maxi-
mum, and the E ? M.F, is zero (a = v). If the coi] is
FUNDAMENTAL EQUATION. 13
vertical, i. c. parallel to the direction of tlie field, the flux
passing through the coil is zero, and the E.M F. has its
maximum value
.... (1)
The instantaneous value of the E.M.F. is therefore
E t E sin a,
and the instantaneous value of the magnetic flux is
N t = N cos a ;
if by N we denote its maximum value. ^
These equations show that between E.M.F. and magnetic
flux there is a difference in phase of a quarter period.
Imagine now the terminals of the coil connected with an
incandescent lamp '.of resistance E. The current I t passing
through the lamp must vary as the E.M.F., E t . Calling /
a
the maximum value of the current, we should have / =-77,
H
and l t = / sin a. Although ordinarily the resistance of a
lamp depends on and varies with the current, we are
justified in assuming the resistance in our case to be con-
stant, since the changes in current strength occur so rapidly
that the filament has no time to grow hotter or cooler as
the current grows stronger or weaker, but assumes a mean
temperature, and has consequently a constant resistance.
Let, then, the lamp be fed, first by our alternating current
derived from the coil, and secondly by a continuous current,
but let in both cases the filament be raised to the same
temperature, so as to get the same amount of light. The
lamp will then in both cases require the same supply of
electric power, and we may consider the strength of the
continuous current as a measure of the effective strength
of the alternating current. Since I t = I sin a, we may
1 4< TRANSFORMERS.
represent the instantaneous value of an alternating current
by the projection of a vector of length /, which revolves
with an angular speed of a = 2 TT ~. In the same
manner other quantities of a periodic character may be
represented, and the diagrams used for this purpose are
called clock diagrams. Let then, in our case, the maximum
value of the current be graphically represented to a certain
scale by the length of a line /. Let the line revolve round
one of its ends, and take the projection of the other end
at stated times. The length of the projection, measured
with the same scale, gives the instantaneous value of the
current.
To find the total work which was supplied to the lamp
by the current during the time of, one period, we should
divide the circle described by the current vector into a
sufficiently large number of equal parts, distant by the time
interval A t from each other, project these points to get
T t , and form the expression 2 If R A t. This would be a
laborious process, but it can be simplified if we imagine
the additions made twice over by counting together
such positions of the vector / as are 90 apart. The
members of the series we have to sum up would then
be of the form
R (/ 2 sin 2 a + 7 2 cos\} A t : = R I' 2 A t,
as will easily be seen by reference to Fig. 8, in which the
vector is shown in two positions differing by 90. The
projections are I and OI 2 , and as the sum of their
squares is obviously equal / 2 , we find that each member
of our series has the same value, namely H I 2 A t. Let m
T
be the number of members, so that m -r- ; then we find
the total work done by the current during one period, by
FUNDAMENTAL EQUATION.
15
multiplying the value of one member of the series by m
and dividing by 2, the latter because we have by taking
the vectors in pairs counted them twice
done during one period is
. _RI' 2 T
~2~
and the power ; that is, the rate at which work is done is
If the lamp is fed by a continuous current i the power is
P = i -R. Let the power be the same in both cases, then
i may be considered the effective value of the alternating
current, and will be given by the expression
This relation is of course only valid if the current is a
sine' function of the time. If it follows any other law the
ratio between its maximum and effective value will be
given, not by *J 2, but by some other co-efficient. If by
1 TRANSFORMERS.
I we deiiote the instantaneous, and by i the effective,
value of the current, we have for any form of current
curve
.... (3)
In words : The effective current is the square root of the
mean squares of the instantaneous values. j/
The same reasoning applies, of course, to the pressure at
the terminals of our coil, and in fact to any alternating
E.M.F. Since in all instruments intended for the measure-
ment of alternating pressure (hot wire, electrostatic and
electro-dynamic) the action is dependent on the square of
the E.M.F. applied to the instrument, the quantity actually
indicated is the square root of the mean squares ; or in
symbols
/r r T
1 E-dt .... (4)
i r
!/
If E changes according to a sine law so that the instant-
aneous value E t = E sin %TT ~ t, where E is the maximum
value of the E.M.F., then is the effective value
It has been previously shown that E = ZTT ~ Nn 1O S
represents the maximum value of the E.M.F. in volts in-
duced in a coil of n turns through which the magnetic
flux N passes, if the frequency is ~ complete cycles per
FUNDAMENTAL EQUATION. 17
second. Combining equation 5 with this expression, we
obtain
2 7T ^ , r IAS
e = ^ Nn 10~ 8
V2
e = 4-44 JVw, 10- 8 (6)
This is the fundamental equation for the determination
of the effective E.M.F. in the coils of a transformer, pro-
vided the E.M.F. curve is sinusoidal. For other shapes of
the E.M.F. curve the equation will be given in the next
chapter. Equation 6 applies, of course, to both the primary
and secondary coils. If we distinguish the corresponding
quantities by the indices 1 and 2, and assume for the pre-
sent that the flax is the same in both coils, we have
! = 4-44 ~ Nn^ 10- 8
e. 2 = 4-44 ~ N n 2 lO' 8
These equations are therefore only applicable if there be
no magnetic leakage. If there be leakage, a correction
must be applied, as will be shown later on.
Since the current does work which is absorbed by the
primary coil, the direction of e L must, on the whole, be op-
posed to the direction of the current. The secondary coil
gives work to its external circuit, and the secondary cur-
rent must therefore, on the whole, be of the same direction
as e 2 . The power supplied to or given off by the trans-
former can, however, not generally be considered to be
correctly represented by the product effective current x
effective pressure, since the phases of these two quantities
are, as a rule, not coincident ; that is to say, the current
attains its maximum value at a different time from the
E.M.F., and the times at which they pass through zero are
also different. In the primary coil the product of instant-
1 8 TRANSFORMERS
aneous current and instantaneous E.M.F. is therefore not
always negative, and in the secondary coil this product is
not always positive ; the work impressed on or given off
by the transformer during the time of a period is therefore
smaller than T ei\ and Te 2 i 2 respectively. The deter-
mination of the true work and true power will be given in
Chapter IV.
CHAPTER II
LOSSES IN TRANSFORMERS INFLUENCE OF THE E.M.F. CURVE ON
HYSTERESIS LOSS INFLUENCE OF THE SHAPE OF CORE AND COILS
UFON THE LOSSES CORE AND SHELL TRANSFORMERS.
Losses in transformers. The losses occurring in trans-
formers are of various kinds. There are first of all losses
due to the ohmic resistance of the coil, causing so-called
" current heat." These may be easily determined by
Ohm's law, and need no further consideration here. Then
there may be losses caused by eddy currents in the con-
ductors, or other metallic parts of the apparatus. To
calculate these is exceedingly difficult, and often im-
possible ; on the other hand, it is always possible, by
suitably placing or subdividing the metallic parts of a
transformer, to reduce the eddy current losses so far as to
make them a negligible quantity. Finally, we have to
consider the losses occurring in the iron core, which are
due to two causes : hysteresis and eddy currents.
If the induction in the core passes through a complete
cycle from + B through zero to B, and back through
zero to + B, a certain amount of electrical work is trans-
formed into heat. This amount depends on the quantity
and quality of the iron affected, and the maximum value
of the induction. The corresponding power is proportional
to the frequency, but is independent of the shape of the
20 TRANSFORMERS.
wave representing the induction, provided there is in each
cycle only one maximum and one minimum. According
to Steinmetz, the work lost per unit weight of iron per
cycle is represented by an expression of the form
A = 1i rG ,
where h is a co-efficient depending on the quality of the
iron, and on the magnitude of the unit of weight we have
chosen.
The magnetic flux surging to and fro within the mass
of the iron produces in it E.M. forces, which in turn give
rise to eddy currents, and thus cause loss. Let us assume
an iron core of rectangular cross-section with sides a and
6; and let us compare different sections all of the same
width a, but of different thickness 6, this dimension being,
however, always small in comparison with a. The E.M.F.
producing eddy currents must obviously be a maximum
in the contour of the rectangle, that is, the skin of the
core, and must be proportional to the whole of the flux
passing through the cross-section ; namely B a 8. For a
given value of B the E.M.F. varies therefore as a 8, and
the same holds for all the smaller E.M. forces which
occur in the lower layers throughout the mass of the
metal. The currents are inversely proportional to the
resistances of the different layers, that is to say, the larger
8 the smaller are all the resistances, and the stronger are
all the eddy currents. With increasing thickness 8 we
have therefore a proportional increase in the E.M.F., pro-
ducing eddy currents, and an increase in quadratic ratio
of these currents themselves. The power lost increases
therefore as 8 3 .
For a core of circular cross-section the E.M.F. in each
layer must obviously be proportional to the square of its
LOSSES IN TRANSFORMERS. 21
diameter, whilst for layers of similar proportions (radial
thickness of the layer a definite fraction of its diameter)
the resistance is constant and independent of the diameter.
The currents are therefore proportional to the square of
the diameter, and the losses increase as the fourth power
of the diameter.
In order to reduce these losses, it is therefore only
necessary to reduce the diameter or the thickness b, that
is to say, the core must not be solid, but made up of
wires or plates. In a wire core the loss is proportional to
the fourth power of the diameter of the wire, and in a
plate core to the third power of the thickness of the
plates. By reducing the thickness to one-half or a third,
the loss may thus be brought down to ^ or ^ T of its
previous value respectively. It is thus possible to make
the loss negligibly small by using plates thin enough, but
in practice the subdivision is not earned to the extreme
limit, because the expense involved and the loss of space
through insulation would outweigh the possible gain. It
is sufficient to carry the subdivision to such a point that
the eddy current losses become reasonably small, and this
point has been found in practice to lie between a thickness
of plates of 0-35 to 0'5 mm. (14 to 20 mils.). The
stouter plates are used for frequencies of about 50, and
the thinner plates ^or higher frequencies up to about 100.
For a very low frequency and a low induction, plates
thicker than J mm. may be used. How far it is per-
missible to increase the thickness of plates can best be y
shown by an example. Let us assume that we have
found in practice plates of '5 mm. suitable for ~ =50 and
JJ = 4,000, and that we have to design a transformer for
~ = 20 and B = 3,000. How thick may the plates be
made in order to have the same eddy current lass per
22 TRANSFORMERS.
kilogram of iron ? In the transformer for 50 frequency the
E.M.F. which produces eddy currents is proportional to
B ~ 200,000. If at the lower frequency and lower
induction we use the same thickness of plates, the E.M.F.
would be proportional to 20 X 3000 = 60,000, or only
*3 of the former value. Since the loss is proportional to
the square of the E.M.F., it would amount to only '09 of
its former value. As, however, the same loss is per-
missible in the second case, we may increase the thickness
of the plates in the ratio .
3 / ^
9-99
0~9
We may thus use plates of 1*1 mm. (48 mils.) thickness,
and yet only lose as much power per kilogram of iron as
in the former case.
If, as is always the case in practice, the thickness of the
plates is such as to make the eddy current loss unim-
portant, then its exact and separate determination becomes
unnecessary. It forms only a small fraction of the
hysteresis loss, and may therefore be determined jointly
with the latter. It suffices for practical purposes to
measure the joint losses in sample plates of about the
usual thickness, and with inductions and periodicities
such as are generally employed ; the results may then be
used to predetermine the joint loss for thicknesses and
periodicities, not differing too widely from the respective
values obtaining in the experiments. I have made such
experiments with finished transformers and with samples
of plates, using for the latter purpose the special measuring
instrument which is described in Chapter VII., and have
thus determined the relation which exists between the
loss of work per cycle at different inductions. The thick-
LOSSES IN TMANSFO&MERS. 23
ness of the plates varied between '4 and '5 mm. (16 and
20 mils.), and the periodicity between 50 and 75. Since
eddy current losses form only a very small fraction of the
hysteresis losses, the variation of the periodicity between
the narrow limits of 50 and 75 does not materially
influence the magnitude of the joint loss. The results of
my measurements are graphically represented in Fig. 9.
For convenience this figure gives, not the loss of work
per cycle, but the loss of power calculated for a frequency
of 100 cycles per second. For a different frequency the
losses must be proportionately altered. Two curves are
Fig. 9.
3000
2000
5000 SOW 7000
given ; the upper represents ordinary wrought-iron plates
of fair quality, the lower plates of extra good quality,
specially rolled for use in transformers. The dotted
curve gives the permeability, which has been found by
measuring the primary current and loss of energy in
transformers when working an open circuit, as will be
more fully explained later on. The permeability has
been found for both qualities of iron to be approximately
the same.
Influence of the E.M.F. curve on hysteresis loss. The
24 TRANSFORMERS.
shape of the E.M.F. curve has not only an influence on
the relation between the effective and maximum E.M.F.,
but also on the hysteresis loss which necessarily ac-
companies the production of any given effective E.M.F.
If we assume that the flux has only one maximum in
each half period, then the hysteresis loss is dependent on
this maximum and the frequency, but is independent of
the shape of the E.M.F. curve, and of the shape of the
curve which represents N as a function of the time. We
may imagine a series of N, t curves which all have the
same maximum, but are otherwise of different shape.
The hysteresis loss with all these will be the same, but
not the effective E.M.F. Of these curves we would
naturally prefer that which gives the greatest E.M.F. for
any given maximum value of the flux. Since the flux is
dependent on the shape of the E.M.F. curve, we may
state the problem also in the following terms. Let there
be at our disposal different alternators, all of which
produce the same effective E.M.F., but with different
forms of E.M.F. curve. That form is to be selected in
which the flux, and therefore the hysteresis loss is least.
To solve this problem it is of course necessary to
assume certain forms of E.M.F. curve. As a starting-
point we may conveniently select the sine curve, and then
determine what influence an alteration in its shape has
on the relation between maximum and effective E.M.F.,
and on the hysteresis loss. The shape may be altered in
two ways ; we may either flatten the curve, or make it
steeper. If we carry the flattening process to its theo-
retically (though not practically) possible limit, we obtain
a broken line consisting of vertical and horizontal parts.
The vertical parts represent the instantaneous change
from E to + E, and the length of each horizontal
INFLUENCE OF THE E.M.F. CURVE. 25
part represents the time of half a period. Such a curve
might be obtained by the commutation of a continuous
E.M.F., but only as long as no appreciable current is
allowed to flow. With an alternator of special design,
the rectangular shape of E.M.F. curve might also approxi-
mately be obtained, though never completely. We are
therefore justified in regarding this shape as the extreme
limit of the flattening out of the sine curve with which
we started. The effective E.M.F., c, is then equal to the
maximum E.M.F., E, or in symbols
e = E.
d N
Since E is a constant for each half period, -= must
d t
during that time be also a constant, and the flux must be
represented by a zig-zag line, with joints having the same
abscissa3 as the vertical parts of the E.M.F. line. From
Fig. 10 it will be seen that
d N _ 4 N
dt ~~T
Since for one turn E = c , and since e = E, we have
d t
for n turns
e = 4 ~ n N 10~ 8 (7)
For a sine curve we have by (6)
e = 4-44 ~nN 10- 8 (G)
If, then, the effective E.M.F., e, is to be the same in
both cases, then the induction N must, with a rectangular
E.M.F. curve, be greater than with a sine curve in the
ratio of 4 '44 : 4 ; that is to say, the maximum value of
the flux N must be by 11 per cent, greater, or with the
same induction B we must put 11 per cent, more iron
26 TRANSFORMERS.
into the core. The hysteresis loss is therefore increased
by at least 11 per cent. As already pointed out, the
rectangular form is an extreme case, scarcely attainable
in practice. In reality, the curve will assume a shape
roughly represented by the dotted line where the sharp
corners are rounded, and thus the peaks of the N, t curve
will also be rounded and lowered. The hysteresis loss
will therefore be somewhat smaller than corresponds to
the extreme case, but certainly greater than with a sine
curve. Our investigation thus far has therefore shown
that an alternator giving a flat E.M.F. curve is dis-
advantageous for working transformers.
Let us now see what the conditions of working are
with a peaky E.M.F. curve. In this case the limit is not
definable. We may imagine the E.M.F. curve composed
of a series of inclined straight lines, thus forming a suc-
cession of triangles; but we may also imagine this form
exaggerated, i. e. the lines convex towards each other, like
the sides of a tent, whereby the peaks would be raised
higher than with the triangle form. As a matter of fact,
INFLUENCE OF THE E.M.F. CURVE. 27
certain alternators, in which the armature is provided
with projecting teeth, give an E.M.F. curve having very
steep and high peaks, so that E is very large in com-
parison with e. We shall not attempt to investigate a
curve of this kind, since the investigation is extremely
complicated, and for our present purpose not required.
All we care to learn is whether a more pointed curve
than the sine curve is better or worse as regards hysteresis
loss, and for this purpose we may take the simple case of
a triangular shape. If we find that this gives us a
smaller hysteresis loss than a sine curve, we may conclude
that the exaggerated triangle must, in this respect at
least, be still better.
Let, then, line E in Fig. 11 represent the E.M.F. given
by the alternator ; what will be the curve of induction ?
Since for one turn
in absolute units, the curve in question must be such that
its trigonometric tangent in any point a, whose abscissa
is t, is equal to the ordinate of the E.M.F. line having the
same abscissa. In symbols
E t = tto.no. = -"';
dt *
JV r f It tan a d t +
constant ;
N t = constant - t' 2 tan a.
z
The constant can be found from the condition that for
t = 0, N t = N. We thus obtain
N t = N - I t 2 tan a,
28
TRANSFORMERS.
rn
the equation of a parabola. Since E Ian a, we have
4
also
T
For t = , N t = ; from which we find
4
* 7 1* /T\' 2 4 E
N = - I - )
IM r
= __ . in absolute measure.
8
Since = ~ t we have
E = 8 -
volts.
We see thus that with a triangular E.M.F. line the
Fig. 11.
maximum E.M.F. is, for the same induction, exactly
doubled as compared with the rectangular form, and the
stress on the insulation is therefore also doubled. The
point of interest is, however, not the maximum, but the
effective value of the E.M.F.
INFLUENCE OF THE E.M.F. CURVE. 23
This is for a quarter period given by the following
equation
e = / I ? Ef A t
'/*
= i tan a
e= /4te* /I.;. /!*?_ 1**
-t o o4
T 1
c = ~7 tan a -7=
By inserting this value in the above equation for E we
find
e = 4-62 N 10~ s for one turn, and
e = 4-62 - n N 1Q- 8 ....... (8)
for a coil of n turns.
Since the co-efficient is greater than with a sine curve, it
follows that a smaller induction N with the same quantity
of iron, or a smaller quantity of iron with the same induc-
tion, will now suffice to produce the desired effective E.M.F.
The equations (6), (7), (8) have all the form
e = c ~ n N 10~ 8 ,
and differ only in the co-efficient c, which has the following
values
(1) With a rectangular E.M.F. curve . . c '= 4'00
(2) sine . . . c = 4'44
(3) triangular . . . c = 4*62
Imagine now that we have three alternators with
E.M.F. curves of these shapes, and let us supply the same
30 TRANSFORMERS.
transformer successively with currents from these machines,
and at the same E.M.F. The maximum induction ^will
be different in each case. It is greatest in case 1, and
smallest in case 3. If now we take the iron loss in case
2, where the E.M.F. is represented by a sine curve as our
standard of comparison, and call it unity, then the loss
will be
With a rectangular E.M.F. curve Ill
With a triangular E.M.F. curve O96
The last figure shows that there is, as regards hysteresis
loss, some, though not an overwhelming advantage in em-
ploying a machine giving a peaky E.M.F. curve ; whilst, on
the other hand, machines with a flat E.M.F. curve are
disadvantageous.
Influence of the shape of core and coils upon the losses.
Since the hysteresis loss is proportional to the weight of
iron, we must aim at making this as small as possible. In
designing the iron core of a transformer, we are limited by
two conditions. First, the iron of the core must suffice to
pass the total flux N with a moderate induction ; and
secondly, its length must suffice for housing the coils. On
the other hand, it is desirable to make the length of each
turn of wire as short as possible, in order to reduce the
current heat. These conditions are in part contradictory,
and cannot each and all be fully met. The best design is
consequently merely a compromise, and can only be
obtained by a method of trial and error carried to a point
at which any farther change in dimensions or winding does
not reduce the sum of all the losses.
The shape of the cross-section of the core is of great
importance for the length of wire required in the coils and
the resistance. Thus, a rectangular shape is worse than a
INFLUENCE OF SHAPE OF COPE AND COILS. 31
square, since it requires more length of wire for the enclosure
of an equal area. For the same reason, a circle is better than
a square of equal area, though if for constructive reasons
the diameter of the circle cannot be made greater than the
side of the square (i.e. the condition of equal area cannot be
fulfilled), then a slight advantage lies with the square core,
4
which contains - times the amount of iron as compared
7T
with a circular core. Let r be the radius of the circle, and
2 r the side of the square, 8 the thickness of the insulating
covering on the core, and d the depth of the winding. For
the same induction B and the same number of turns the
E.M. forces will be in the ratio of TT r 2 : 4 r 2 for the circular
and square core respectively. The mean length of wind-
ing is, for the circular core, TT (2 (r-\-b)-\-d), and for the
square core 8 (r+fy + ir d. The E.M.F. induced per unit
length of winding is therefore proportional to TT r 2 :
TT (2(r + 8) + d), and 4 r 2 : (8(r + 5) + 7r d), and the ratio
between these values is
from which it will be seen that with a square core the
E.M.F. induced per unit length of winding is greater than
with a round core, and that the difference becomes the
more marked the greater the depth of winding. The
explanation for this paradoxical result (that a square
core requires less wire than a circular) lies in this, that the
two cores have not the same area. The square core con-
tains more iron, and the electrical output is greater. The
larger apparatus has, of course, an advantage over the
32 TRANSFORMER!*.
smaller which, more than compensates for the less advan-
tageous shape of the core.
It has already been mentioned that the core of a trans-
former must be built up of wire or plates, in order that
eddy currents may be avoided. If wire be used for this
purpose, no special insulation is required, and the space
actually filled by iron is from 78 to 80 per cent, of the
total. When plates are used, they must be insulated
against each other, though no specially good insulating
medium is required. The insulation may consist of a
layer of oxide on the plate itself, or a coat of varnish, or
paper insertion. The latter method of insulating is the
most reliable, and occasions a loss of space of from 12 to 15
per cent., so that about 87 \ per cent, of the total space is
actually occupied by iron. The available space is, therefore,
better utilized with plates than with wire, and for this
reason, as well as on account of the more mechanical con-
struction which is possible with plates, the latter are
nearly always used in the building up of transformer cores.
Core and shell transformers. As already explained, the
action of a transformer is due to the linking together of
two coils by means of an iron core. This principle of inter-
linking can be carried out in a variety of ways ; one of the
simplest is represented in Fig. 6. The link is a rectangular
frame, and the coils are placed upon the two longer limbs
of the rectangle. Such an arrangement is called a core
transformer, and is characterized by the fact that most of
the iron is within the coils, whilst the external surface of
the coils is everywhere exposed to the cooling action of
the air.
We may, however, also change the relative position of
iron and copper. We may assume that the rectangular
frame is composed of copper wire, forming the two coils,
CORE AND SHELL TRANSFORMERS. 38
which are laid close upon each other, and the external
cylindrical parts of Fig. 6 of iron discs, or a winding of iron
wire which forms a kind of iron shell in which parts of the
coils are imbedded. Such a transformer is called a shell
transformer.
Whether the core or the shell type is better cannot be
generally decided, but depends on a variety of circumstances.
In the core type, the weight of iron is small and the turns
of wire short. On the other hand, the number of turns
is great (because N is small), and thus the total weight of
copper is, notwithstanding the small perimeter (length of
one turn), fairly large. The length of the path of the flux
is also great, and the ampere turns required to produce the
flux are large. On the other hand, we have the advantage
that the coils are accessible, and exposed to the cooling
effect of the air.
The shell type has the advantage of a short magnetic
path, which requires but few ampere turns to produce the
flux ; the coils have fewer turns, and the total weight of
copper is, notwithstanding the larger perimeter, small.
It has,_however, the disadvantage of requiring considerably
more iron, whilst the coils are not accessible, nor so well
exposed to the cooling effect of the air.
In order to compare in a general way the two types of
transformer, we may show what effect alterations in the
arrangement of iron and copper have in a particular case.
For this purpose, we may start with any concrete design
and alter it in various ways, but always under the condition
that the induction in the iron, current density in the
copper, and total output shall be the same. For the sake of
simplicity we may assume that the same gauge of wire shall
be used in all cases, so that the number of turns in each
coil will be proportional to the winding space. The current
34
TRANSFORMERS,
and pressure being the same for all designs, it follows that
the flux must be inversely proportional to the winding
space ; and as the induction is to be the same, it further
follows that the area of the core and that of the winding
Fig. 12.
space are also inversely proportional. Under these con-
ditions we may roughly judge the merit of each design by
the weight of iron and the length of wire.
As a starting-point, we may take the design shown in
Fig. 12a, in which the area of the core is 400 sq. c.m.
CORE AND SHELL TRANSFORMERS, 35
(inclusive of the space wasted, in insulation), and that of
the winding space GO sq. c.m. The iron weighs 200 kgr.,
and the mean perimeter of the winding is 119 c.m. With
100 turns of wire we have thus a total length of 119 m.
Let us now alter this design by reducing the core area to
one-quarter of the previous value and quadruple the
winding space. The iron part will then be 40 c.m., but
only 10 c.m. long, in a direction at right angles to the
plates, and the winding space must now be 12 x 20 c.rn.
We arrive thus at the design Fig. 12&. The mean
perimeter of the coils is now only 78 c.m., but as we must
have four times as many turns as before, the total length
of wire has been increased to 312 in., that is nearly
three times the former length. On the other hand, the
quantity of iron has been much reduced ; it is only 73
kgr. ,If the iron is cheap but of good quality and
copper is dear, then type a is preferable. With cheap
copper and bad iron type 1} will be the better design.
Both of these designs may, however, be still improved.
We may, for instance, so alter type a as to embed both
sides of the coil in iron and thus arrive at a true shell
transformer, Fig. 12c. This does not alter the length of
winding, which is still 119 m., but it reduces the weight of
iron considerably. The flux through the coil divides now
to both sides, and only half the previous area is required
in the outside shell. The iron weight is thus reduced to
112 kgr. We thus obtain the design of shell transformer
now largely used in England and America.
By making similar alterations in Fig. 12& we arrive at
the type Fig. 12d, which is also a shell transformer, but
lacks the advantage of a short length of winding. This
length is as before 312 m. ; on the other hand, the weight
of iron has been reduced to 59 kgr. This design is only
36 TRANSFORMERS.
justifiable if no iron of good quality can be obtained, and
there is no need to be economical in copper. In former
times, before rolling mills were capable of turning out such
first-rate transformer plates as now, there was some reason
for building transformers of this type. Now-a-days, how-
ever, excellent iron can be obtained from a variety of mills,
and there is no reason to practise economy in iron at the
cost of an increased weight of copper. It is therefore
preferable to alter the design Fig. 12& in such way as to
save copper, and this can be done by dividing the winding
and placing it upon both limbs. We arrive thus at the
type, Fig. 120, which is a true core transformer. The
perimeter of the winding is on account of its smaller depth
considerably reduced, and the iron weight is not excessive.
It is 73 kgr., and the length of wire is 236 m. This
type of transformer is much used in England and
Germany.
For convenience of comparison the above results are
summarized in the following table :
Type. Weight of Iron. Length of wire.
Kgr. Meters.
a 210 119
I ... ... 73 312
c 112 119
d 59 312
e 73 236
In all these types the magnetic path lies completely in
iron. There exists, however, also another type of trans-
former in which the lines of force pass only partly through
iron and close themselves through air. This is the so-
called hedgehog transformer, Fig. 13$, introduced by
Swinburne with the intention of reducing the iron loss.
CORE AND SHELL TRANSFORMERS.
37
For this purpose Swinburne winds the coils upon a core
consisting of a bundle of iron wires with their ends spread
out like the back of a hedgehog. The lines of force pass
then from one end of the core to the other, through air, as
shown by the dotted lines, and hysteresis loss only takes
place in the small quantity of iron which forms the core
proper ; in the shell of air surrounding the coils there is,
of course, no hysteresis loss. This type of transformer
has not been successful in practice. If we imagine two
such transformers placed side by side and the ends of the
iron wires bent so as to meet (Fig. 136), we obtain an
ordinary core transformer, the hysteresis loss in which can
Fig. 13.
only be very slightly greater than in two single hedgehogs,
the small increase being due to the extra length of iron
wire required to make a perfect junction between the two
cores. On the other hand, there must be an increase in the
hysteresis loss, in each core of the two single hedgehogs,
because the induction in the middle of the core is
considerably greater than at the ends. The E.M.F. is of
course proportional to the mean induction and the
hysteresis loss to the 1 'V of the mean values of B rG . It is
therefore clear that any variation in the value of the
induction along the core must result in a greater hysteresis
loss than in the case where the induction is constant
38 TRANSFORMERS.
throughout the length of the core. The hedgehog trans-
former has also another drawback, namely, that it requires
an exceedingly large primary current at no load. Whilst
in the types shown in Fig. 12 the no load current is but
a small fraction (a few per cent.) of the full load current,
the hedgehog takes with an open secondary up to GO per
cent, of the full load primary current. This is a property
which renders the hedgehog transformer unfit for use in
central station work where a small day current is of the
utmost importance. There is however one purpose for
which this type of transformer is very suitable, namely, as
a choking coil, whereby its capability of letting large
currents pass under moderate E.M.F. is the very thing
desired. For transformer work proper the types shown in
Fig. 12 are however far preferable, especially the designs
12c and 12c.
CHAPTER III
USUAL TYPES CONSTRUCTION OF THE IEON PART PROPORTIONS OF
THE IRON PART HEATING OF TRANSFORMERS RESULTS OF TESTS
INFLUENCE OF THE LINEAR DIMENSIONS.
Usual types. The designs commonly used belong all to
the two great groups of shell and core transformers. The
former are of the kind shown in Fig. 14, where P and $
Fig. 14.
are the primary and secondary coils somewhat oblong in
shape and placed either within or upon each other, whilst
the iron part consists of rectangular plates each with two
openings in which the winding is embedded so that only
the rounded ends of the coils remain accessible. In a
variety of this design the coils are circular, and the iron
part is arranged in sections placed all round the circle.
Beyond a slight increase in the cooling surface, this form
has no advantage over the more usual form shown in Fig.
40
TEAN3FOHMEUS.
14. The coils are first wound and the iron plates are
afterwards threaded over them in a manner to be presently
described. The centre part of the plates K forms the core
proper which carries the flux through the coils, whilst the
external part M forms the shell.
Fie. 15.
In the core transformer the iron part has the shape of a
rectangular frame (Fig. 15), the longer limbs forming the
cores K, and the shorter limbs the yokes J. The primary
and secondary coils (P and S) are cylindrical, and may be
placed within each other as shown in the figure, or as a
Fig. 16.
Fii;. 17.
pile of narrow discs upon each other. For polyphase
work the core type of transformer is almost exclusively used,
and Figs. 16 and 17 show two arrangements of the iron
part for this purpose. In the former the yokes J are formed
of circular iron discs (armature discs may conveniently be
USUAL TYPES. 41
used), and the three cores K are placed at an angular
distance of 120 from each other and pressed against the
yoke discs by top and bottom plates of cast-iron with slant-
ing brackets engaging the tapered ends of the core pieces.
The coils are placed over the cores as in Fig. 15. In Fig.
17 the true symmetrical arrangement is abandoned; the
three cores K being in line and joined by top and bottom
yokes J precisely as in Fig. 15. Each core is surrounded
by the primary and secondary coil belonging to one phase.
The advantage of this design lies in the fact that the yoke
and core plates are parallel and not at right angles to each
other, which renders the construction mechanically easier
and magnetically more perfect.
Construction of the iron part. In Figs. 14, 15 and 17 the
plates forming the iron part are shown as complete surfaces
without any joint between that part which forms the core
and that which forms the yoke or shell. The use of such
plates is possible, and would have the advantage that no
interruption of any kind is offered to the flow of the lines.
On the other hand, there would be the great disadvantage
that the coils could only be wound after the iron part is
built up. The wire would have to be threaded through
the openings, and could therefore not be wound on a lathe.
Moreover if the wire is stout it could scarcely be wound
properly by hand, and to obtain a proper and reliable
insulation would be next to impossible, whilst a fault in
the insulation could only be detected after the transformer
is completely finished. These are such serious drawbacks
that the use of closed plates is not commercially possible.
It is better to arrange the iron part in such way that the
plates may be inserted after the coils are wound and tested,
and for this purpose some sort of joint in each plate is
necessary. It is true that the joint interrupts the con-
42 TRANSFORMERS.
tinuity of the magnetic path and is in this respect im-
perfect, but the imperfection can with a suitable arrange-
ment of plates be so far reduced as to be without appreciable
influence on the magnetic conductivity of the iron part
taken as a whole. For this purpose the plates are so
arranged and laid upon each other that the joint in one
plate is covered by the solid part of the next. The lines
of force instead of leaping across the joint can then in
part pass through the solid metal of the neighbouring
plates, and since the surface through which they pass from
Fig. 18. Fig. 19.
one plate to the other is immensely greater than the
sectional area of each plate, the magnetic resistance of this
bye-pass becomes negligible, and we may regard the iron
part so built up as practically equivalent to one having no
joints whatever.
The principle here explained may be illustrated by the
way the iron parts of some of the usual types of trans-
former are built up. In the WcstingJwuse transformer the
plates are stampings of the form shown in Fig. 18. In-
clined slits are made to both sides of the central bridge
piece forming the core, and the two parts of the shell are
bent up as shown. In this condition the plate is inserted
into the coils and the bent-up pieces are bent back so as
CONSTRUCTION OF THE IRON PART.
43
to lie flat ; the next plate is then inserted from the
opposite side and its bent-up pieces laid flat. In this
manner the joints in each plate are covered by the solid
parts of its two neighbours, and the continuity of the
magnetic path is thereby secured.
The core in Fermnti's transformer (Fig. 19) consists of a
bundle of straight iron plates which is inserted into the
coils. The plates are then one by one doubled back to
form the shell. The length of the plates is such that
Fig. 20.
Fi. 21.
at the joint they overlap slightly, and thus form an easy
path for the magnetic flux.
In the Authors transformer (Fig. 15) the plates for the
core and the yoke are straight, and are dove-tailed into
each other at their junction as shown in Fig. 20. In order
to show the construction more clearly the thickness of the
plates is much exaggerated in this figure. After the iron
part is built up it is held together at the corners by
insulated bolts. Since all the plates are rectangular there
is no waste of material in cutting out.
In Cromptoris transformer (Fig. 21) the plates are cut
TRANSFORMERS.
out in L form and inserted into the coils alternately from
one side and the other so as to cover the joints in one
layer by the solid parts of the next. In cutting or punch-
ing the plates there is on account of the special shape
some loss of material.
In stamping the plates for the Westinghouse transformer
there is also some loss of material, namely, that punched
out to form the two windows. To obviate this loss Mordey
has altered the design in such way as to utilize the material
punched out as the bridge-piece or core, Fig. 22. Each
punching yields two pieces, the shell which is a rectangular
frame and the core which is laid across it. The shell
Fi. 22.
plates are placed over and the core plates through the coils.
With the plates made in this way the winding space is of
course dependent on the thickness d of the core, and can-
not be chosen arbitrarily as in Fig. 18. The height of each
window is d and its width , as will be seen by Fig. 22.
2i
The external dimensions of the shell are 3rf and 2d. Con-
tact between core and shell plates only takes place over
the shaded areas ; in all other parts there is a clearance
between neighbouring plates equal to their thickness. In
consequence there is only half the space within the coils
actually filled by iron, and to carry the same flux the peri-
meter of the coils must be greater than in Fig. 18, where
CONSTRUCTION OF THE IRON PART. 45
the whole space is filled by iron. To remedy this defect,
Mordey uses additional stampings of the form shown in
Fig. 23, which can also be done without any waste of
material. The internal square of side d is used to fill up
the interstices in the core of Fig. 22, and the two side
pieces to fill up the interstices in the shell. There are
thus five pieces required for every two layers, and all the
material cut up is also utilized. Although the proportions
resulting from this method of building up the iron part are
suitable, it is sometimes desirable to vary slightly the
dimensions of the winding space. This may be done
whilst still retaining the Mordey system of punching, but
a slight waste of material is then unavoidable.
In all the methods of building up described above, the
principle of breaking joint is maintained and the iron part
is magnetically equivalent with one having no joints. It
is however also feasible to abandon this principle and
build up the iron part with joints between core and yoke
or core and shell. In Fig. 16 such joints are unavoidable ;
in transformers of the type shown in Figs. 15 and 17 they
are sometimes used to facilitate the fitting up. It is then
possible to completely finish each part of the apparatus
and to piece it together with very little labour. In case
of repair a coil may be taken out and replaced without
having to take the plates out one by one as is the case
with transformers constructed as described above. There
are thus from a practical point of view many advantages
in the use of joints ; there is however the disadvantage
that the no load or idle current becomes greater, as will
be shown in the next chapter, and that unless the joints
are very carefully made there is loss of power through eddy
currents at the joint. To make the magnetic path as
perfect as possible, it is of course desirable to reduce the
46 TRANSFORMERS.
distance between the edges of the plates at the joint as far
as practicable. It is however not practicable to reduce it
to zero, partly for obvious mechanical reasons, and partly
because direct contact between the edges would give rise
to eddy currents. With the design shown in Fig. 16 this
is at once obvious. The plates cross at right angles, and if
there were contact between them there would be formed
closed circuits through which considerable eddy currents
would flow and produce heat whilst wasting power. To
avoid this defect a sheet of insulating material must be
inserted at the joint, and the latter can therefore not be
magnetically perfect.
Even where the plates do not cross, but are parallel to
Fig. 24.
I
each other, the insertion of a sheet of insulating material
is desirable, as will be seen on inspection of Fig. 24, which
represents a joint, metal to metal. The thickness of the
plates is shown exaggerated, and the stout black lines
between the plates represent the insulation. Although
the thickness of the plates is supposed to be the same, we
cannot expect that at the joint insulation and metal will
register perfectly. There must be portions on the surface
where there is a slight displacement, so that the insulation
in part A touches not the insulation but the plate in part
B, as shown in the figure. It is obvious that in this case
there is metallic connection between plates a and I,
and that eddy currents as indicated by the wavy line must
CONSTRUCTION OF THE
flow. To interrupt the path of such currents we must
separate parts A and B by inserting a sheet of insulation
as shown in Fig. 25. Thus the magnetic resistance of the
joint must be increased in order to avoid heating and loss
of power.
Proportions of the iron part. The value of a trans-
former design is largely dependent on the proportions of
the iron part. If, for instance, the cores in Fig. 15 are
made very short, the yoke pieces must be made correspond-
ingly long in order to obtain sufficient winding space.
At the same time, the depth of winding, and consequently
the weight of copper is increased, and we thus make the
Fig. 25.
I
A
B
Fi
K
g. 26
-A
H
f
* a. -
i
-
design worse by shortening the cores. On the other
hand, if to get a small perimeter for the winding we make
the cores very long and thin, we get a long magnetic
path, and consequently a large no load current. We shall
do better to increase the core section, and reduce the
number of turns, but not too much, because of hysteresis
loss. No hard and fast rules can be given for the best
proportion, and the only way to determine them is by a
method of trial and error in a series of tentative designs,
paying due regard to such questions as the desired
efficiency, the average and maximum loads, the cost of
48
TRANSFORMERS.
iron and copper, etc. As a starting-point for the design,
the proportions shown in Fig. 26 may be taken. The
section of the cores and yokes is supposed to be a square,
and if the junctions are made by dove-tailing, the edges
of the cores may be chamfered so as to reduce the mean
perimeter of the winding. It is convenient to bring all
dimensions into relation with the thickness d of the core
given in millimeters. We thus have as a first approxima-
tion to a good design
a = 10 + l-2d
I =100 + 2'Qd
A = 10 + 3'2d
B = 100 + 4-Gd
Fig. 27,
-
-J
i
-^
-i-
^
i
For shell transformers with short cores, the plates of
which are to be punched without any loss of material, we
have (Fig. 27)
-i
b = d
If however a slight waste of material is to be permitted,
the dimension a may be somewhat increased so as to
HEATING- OF TRANSFORMERS. 49
get more room for winding and generally better pro-
portions.
a = O'Qd b = d
A = 3'2d B = 2'2d
or a = Q-*7d I = d
The depth of the core and shell measured at right angles
to the plane of the plates may be 2d to 4td.
Heating of Transformers, If a transformer is at work,
a part of the energy supplied to it is transformed into
heat which must be dissipated by radiation or convection.
A necessary condition for this dissipation is an increase
of temperature of the transformer over its surrounding
medium, and this increase is naturally the greater the
smaller the surface as compared with the power lost.
S
The rise in temperature is thus a function of the cr = -
Jr v
if by S we denote the total extemal surface of the trans-
former, and by P v the power transformed into heat, that is
lost between the primary and secondary. The character
of this function can only be determined experimentally,
and varies of course with the type of the transformer
tested. Is the arrangement such that the external air
has free access to the iron part and the coils, the cooling
effect is greater than in transformers placed in a case.
The rise of temperature T = f () respectively.
The instantaneous value of the power is
P = ET sin a sin (a 0).
If the vectors perform ~ revolutions per second so that
T = is the periodic time, and if by to we denote the
angular speed, the following equations obtain
a>T= 2 77
a = w t
da = (a d t
da = 2 TT ~ dt
Since work is the product of power and time, we have
Fig. 32.
for the work performed by the current in the time dt the
expression
d A = Pdt
In Fig. 32 are drawn the curves E and / to represent
respectively E.M.F. and current. By multiplying their
ordinates we obtain the ordinates of a third curve marked
P, which represents the instantaneous value of the power
whilst the area enclosed between P and the horizontal
68 TRANSFORMERS.
represents work. For ordinates above the horizontal,
the power is positive, or given to the circuit ; for those
below the horizontal it is negative, or taken from the
circuit. To obtain the work given to the circuit during a
complete cycle, we must measure the area of P between
the ordinates for t = o and t = T, counting the small
shaded parts below the horizontal as negative. The
work corresponding to a complete cycle is
A = /Pdt
The instantaneous power varies, as will be seen from
Fig. 32, between a small negative and a larger positive
maximum. Let us now suppose that we substitute for
this varying power the constant power of a continuous
current, so that the work taken over the time T is the
same in both cases, then the constant power (which in
future we call effective power) is the quotient of work and
time, or in symbols
p -
Substituting for P, dt, and T the values given above,
we obtain also
/*
P . - / E I sin a sin (a - $) d a,
2 77 "".yo
or after integration
p _ I E I cos q>
The same result can be obtained graphically, as was
first shown by Blakesley, by using the diagram Fig. 31.
POWER OF ALTERNATING CURRENT. 69
Let E and 1 represent the instantaneous positions of the
vectors, then E sin a I sin /3 represents the instantaneous
value of the power. To find the effective power we
would have to draw the vectors in a number of positions,
determine the sum of all the expressions E sin a / sin /3,
and divide by the number of positions. Instead of
performing this operation for neighbouring positions of
the vectors in their right sequence, we may do so for a
series of positions ^ which are 90 apart. In this way we
would count the power twice over, and to get the true
value of the effective power we must divide the result by
2. The effective power will therefore be given by the
expression
P 2 2 (E 1 sin a sin /3 + E I cos a cos /3)
m being the number of positions counted in their natural
sequence. The expression in brackets is E I cos (a - /3),
or ET cos $, and is independent of the particular position
or value of a chosen. All members of the series have
therefore the same value, and the sum is simply m E I
cos (f). We thus obtain
p m E I cos (f)
2 m
P = - E I cos (f>
2i
the same expression as before.
E and / are the maxima of E.M.F. and current respect-
ively. Between these and their effective values we have,
E ^
as already shown, the relations e =='> i = ~^
v2 v^
The effective power may therefore be expressed by the
equation
P = e i cos <
70 TEANSFOEMEES.
In this expression e and i denote the effective values of
E.M.F. and current respectively, and < the lag between
current and E.M.F. For a given E.M.F. and current we
have maximum power, when = 90 the
power is zero.
Since i cos $ represents the projection of the current
vector on the E.M.F. vector, we may represent the power
graphically by the area of a rectangle, one side of which
is the E.M.F. and the other the projection of the current
upon the E.M.F. line. Or, we may project the E.M.F. on
Fig. 33.
the current line, and thus form the rectangle. In
measuring the surface it is of course necessary to bear in
mind the scale to which E.M.F. and current have been
plotted. If, for instance, the scale is 1 m.m. per volt and
1 m.m. per ampere, then the number of square m.m.
contained in the rectangle is the number of watts. If,
however, we plot the current to a scale of 1 m.m. per
ampere, and the E.M.F. to a scale of 1 m.m. per 100
volts, then each square m.m. of surface represents 100
watts. In Fig. 33, Oi is the effective current and Oe the
effective E.M.F. which leads by the angle (/>. The shaded
rectangle gives the effective power which is equivalent
with that represented by a current i h flowing under a
COMBINATION OF CURRENTS AND PRESSURES. 71
potential difference e, but having no lag. We may thus
assume the real current i replaced by two currents i h and
i^ between which there is a phase difference of 90.
These can be regarded as the components of the real
current, the component i n conveying all the power, and
being therefore called the watt-component of the current,
whilst ip conveys no power, and is called the wattless
component, or briefly the idle current.
Combination of currents and pressures. We have in the
above an adaptation of the parallelogram of forces to the
combination of currents of different phase; and this leads
Fig. 34.
Cf
/wwwv
to the question whether such combinations are generally
admissible, provided the currents are sine functions of the
time and have the same frequency. Let us assume that
the currents are produced by the alternators M 1 and J/ 2
(Fig. 34), which are mechanically geared so as to insure
equality of phase and a constant lag. The machine
currents are measured on the amperemeters I and II,
whilst the resulting current is measured on the ampere-
meter O. The problem is to predetermine the current
passing through the conductor C, if the machine currents
and their phase difference are known. Let, in the clock
diagram (Fig. 35), /' and /" represent the maximum of the
UNIVERSITY
72 TRANSFORMERS.
machine currents as regards strength and relative phase.
At the moment to which the diagram refers, the alternator
M^ gives the current Oi' and the alternator Af 2 the current
Oi'', so that the total current passing through C is Oi' -f
Oi". If we draw the parallelogram OF II" we see at a
glance that the vertical distance between the points / and
T is equal to the height of point /" over the horizontal.
In other words the distance Oi is equal to the sum of Oi"
and Oi' ; so that Oi represents correctly the strength of
the resultant current at the moment for which the clock
diagram has been drawn. The length Oi is nothing else
Fig. 35.
than the projection of 01 upon the vertical, and since the
above reasoning holds good for any position of the vectors,
it is clear that the projection of 01 will at all times give
the instantaneous value of the resultant current. We may
therefore imagine that the conductor G is traversed by a
single current the maximum value of which is graphically
represented by the resultant of the two current vectors 1'
and /", and the phase of which lies between the phases
of these two currents. If we further imagine all lengths
in the diagram reduced in the ratio ^2 : 1 there will be
no change in the angles, nor in the ratios of vectors, but
the resultant vector will then represent not the maximum,
but the effective value of the resultant current. It will be
COMBINATION OF CURRENTS AND PRESSURED 73
clear that the above method of combining currents can be
applied to more than two currents. We first find the
resultant of two currents, then combine this resultant
with the third current, and obtain a new resultant, and so
on. It is not necessary in this operation to draw out the
various parallelograms of currents ; all we need do is to
add the currents graphically after the manner of the
polygon of forces. The last line which closes the polygon
Fig. 36.
represents in magnitude and phase the resultant current.
Let, in Fig. 36, the lines % to i represent four currents
as regards phasal position, direction, and magnitude, then
by drawing the polygon of 5 sides, of which 4 correspond
to the current vectors, we obtain in the fifth side the
vector of the resultant current as regards phase, direction,
and magnitude.
Electromotive forces of the same frequency, but differing
in phase and magnitude, may be combined in the same
manner.
TRANSFORMERS.
Let, in Fig. 37, M l and M 2 represent two alternators of
the same type, and mechanically coupled together so as to
insure equal frequency. Let three voltmeters 0, 1, and II,
be connected as shown, then / will indicate the terminal
pressure of Jf 1} // that of M 2 , and will show the resultant
pressure. The latter is not necessarily equal to the alge-
braic sum of the two other readings, but will as a rule be
smaller. It depends on the magnitude of the component
pressures and their phasal difference. After what has
already been said with regard to the combination of currents,
we need not explain the combination of electromotive
Fig. 37.
Fig. 38.
forces at length. If the three voltmeter readings are
available we can use them to determine the difference of
phase in the E.M. Forces of the two machines, as will
easily be understood from Fig. 38. Let Oc^ be the
voltage indicated on /, and draw round ^ as centre a
circle with radius equal to the voltage indicated by 77.
The resultant pressure must be a line joining with
some as yet unknown point on the circle. To find this
point we need only describe a circle round as centre with
a radius equal to the reading on the voltmeter 0. The two
circles have two points of intersection, either of which may
DETERMINATION OF THE NO LOAD CURRENT. 75
be the end of the vector of resultant E.M.F. If the
phase of J/ : is in advance over that of M 2 then the vector
of M. 2 will lie behind and therefore above that of M ly the
rotation of vectors being clock-wise. The resultant E.M.F.
will then have the position shown in the diagram, and the
angle of lag is e Oe z .
Determination of the no load current. In the beginning
of this chapter we have considered the general principles
for the determination of the power of an alternating
current. The practical methods for measuring power will
be given later on, as also the extension of these methods
for the measurement of currents of iion-sinnsoidal form of
wave. The investigation as far as it has been carried up
to the present is, however, sufficient to enable us to de-
termine the currents which the primary of a transformer
takes when the secondary is open, and we now proceed
to study this subject, which is of great practical importance.
The importance lies in this, that transformers used for
lighting are generally only loaded during very few hours
daily, and if each were to take a large no load current,
the ampere output of the central station might remain
considerable even during those hours when very little or
no light is required. To make this matter clear, let us
assume a central station for the supply of 100,000 50-
watt lamps installed. We shall then require a number
of transformers with a total output of 5000 Kwt., since
the case may arise that one or other of the users lights
all the lamps installed in his house at the same time.
The total output of the alternators, however, need not be
as large as 5000 Kwt. Experience has shown that of all
the lamps connected to a central station only some 30 to
70 per cent, are ever lit simultaneously, the exact per-
centage depending on the character of the locality served
76 TRANSFORMERS.
from the station. Assuming 60 per cent, as an example,
we would have to supply alternators for a total supply of
3000 Kwt. to the lamps. The day supply is of course
very small, and may be taken as from 3 to 4 per cent.
Assuming 3J per cent, as an average, we should have to
provide a little over 100 Kwt. during the day-time. To
this output must be added the loss in hysteresis, for which
we may take 2 per cent, of the total transformer output,
or say another 100 Kwt., so that an alternator of 200 Kwt.
output ought to suffice. In reality, however, a larger
alternator will be required, because more current than
corresponds to the power is taken by the transformers,
the discrepancy being the greater the larger the no load
current of each transformer. With a no load current of
10 per cent., the current output of the central station,
apart from the true power output of 200 Kwt., would
amount to an apparent power of 500 Kwt., and with a
no load current of 5 per cent, the current output would
correspond to an apparent power of 250 Kwt. It is thus
clear that in order to reduce as far as possible the amount
of machinery which must be kept going during the day-
time, the transformers must be so constructed as to
require a minimum of no load current.
The no load current is required for magnetizing the
iron to that value of B which corresponds to the primary
E.M.F., and for supplying the power lost through hysteresis.
The magnetizing current can be calculated according to
the well-known laws of electro-magnets, when the dimen-
sions of the iron part, the quality of the iron, and the
number of turns of primary wire are known.
Let I in Fig. 39 represent the mean length of the path
of the magnetic flux as determined from the drawing, and
ft the permeability of the particular quality of iron used
DETERMINATION OF THE NO LOAD CURRENT. 77
at the induction B, then the magnetizing force of the
current 1^ passing through n turns is 4 TT n 1^ : I, where
Fig. 39.
. J
IP is taken in absolute measure. If 1^ be taken in
amperes the magnetizing force becomes 0*4 TT n / M : I
and the induction produced is
I
1^ being the maximum value of the current wave.
i^ represents the effective value, we have / M = ^
and
If
B =
V =
1-78 n \
I
Bl
1-78 ju n
The change in the magnetic flux produces in the coil an
E.M.F., the phase of which is 90 behind that of the
current * M , as is easily seen from the following. If the
current has attained its maximum, B is also a maximum,
and the E.M.F. is therefore zero. If the current , and
therefore the flux pass through zero, the E.M.F. is a
maximum. To the maximum of current corresponds
the E.M.F. zero and vice versa ; a relation which can only
exist if the angle in Fig. 31 is 90. Then e i cos = ;
there is no power given out by the current. We thus
find that the magnetizing current i^ carries no power;
its vector in the clock diagram (Fig. 33) stands at right
78 TRANSFORMERS.
angles to the vector of the E.M.F. as shown in the
diagram.
Power is, however, required to cover the hysteresis
loss ; and the component of the no load current which
carries this power must coincide in phase with the E.M.F.
In Fig. 33 this component is i h . The total no load
current is therefore the resultant of two currents, namely
The wattless magnetizing current i^
The watt current required to cover the hysteresis loss ih.
Since these components are at right angles to each other,
we can calculate the resultant no load current i o from
the equation
Up to the present we have assumed that the magnetizing
force is solely required to drive the flux through iron ; in
other words, that the path of lines is either not inter-
rupted by joints, or that if joints there be, they have no
magnetic resistance. This condition is fulfilled with over-
lapping joints, such as are shown in Figs. 18 22 ; it is,
however, not fulfilled in but-joints, as shown in Figs. 16
and 25. If for the convenient fitting together but-joints
are used, it is obvious that at every joint a certain
magnetic resistance is introduced, since, for reasons already
stated in Chapter III., no perfect junction face to face can
be obtained nor permitted, even if it were mechanically
possible. We must therefore take account of the fact
that the faces are separated by a certain distance, and
that the lines of force have to pass through a non-magnetic
medium at every joint. Let 8 be the combined thickness
5s, 5\
of these non-magnetic layers (that is - or - the thick-
4f
ness of each layer with two and four joints respectively),
DETERMINATION OF THE NO LOAD CURRENT. 79
and B represent the induction at the joint, then, since
JJL = 1, the magnetizing force required for overcoming
the resistance of the joints is
0'4 77 n I = 5 B
and the effective ampere-turns required are
:
The current i must be added to the magnetizing current
previously determined, and we thus obtain the magnetizing
current for a transformer with but-joints of total thickness
6" by the equation
B II
q - . _ _ I _ -4-0
1-78 n V
I and 5 being given in centimeter. If there are over-
O O
lapping joints or no joints at all b = o, and we obtain the
expression previously given.
If for the moment we confine ourselves to transformers
having no but-joints, we are able to utilize the formula
here given for the determination of the permeability by
experimenting with finished transformers. For this
purpose the transformer is worked with open secondary,
whilst the primary current and power supplied are measured.
It is convenient to use for this experiment the low-pressure
winding as the primary. The E.M.F. and frequency are
also observed, and from these data can be calculated the
induction B and the component i h required to cover the
loss of power through hysteresis. If P v is the loss and c
the supply pressure, we have
i h = P v : e
With good, and even with middling good, transformers
with closed magnetic circuit the no load current is so small
that loss of power by ohmic resistance may be neglected ;
80 TRANSFORMERS.
hence P v is simply the power measured on open circuit.
Let i be the no load current observed, then the mag-
netizing current can be calculated from the formula
V = V V 2 ^ 2
and the permeability from the formula
Bl
~ li"78 n v
If the experiment be repeated for different values of
primary E.M.F., we are thus able to find a series of corre-
sponding values for B and ju, and the results obtained may
then be used in designing new transformers.
The following table gives corresponding values of B and
fj, which were found experimentally in the manner above
explained. The iron contained in the transformers under
test was of different quality, but lay, as regards hysteresis,
between the limits indicated by the two curves in Fig. 9.
For the rest the experiments showed that the permeability
cannot be taken as a measure for the value of the iron
as regards low hysteresis loss, and that the difference in
permeability for different brands of transformer-plates are
generally small. The values here given are averages.
B = 2000 3000 4000 5000 6000 7000
M= 1300 1720 2070 2330 2570 2780
The dotted curve in Fig. 9 represents these values
graphically.
Influence of but-joints. It remains yet to investigate
what effect but-joints produce as regards the no load
current. On that component of it which is required for
covering the hysteresis and eddy current losses, they have,
of course, no influence, since these losses do not depend
on the magnetic resistance. The wattless component of
INFLUENCE OF BUT-JOINTS 81
the no load current, being directly proportional to the
magnetic resistance, is, however, strongly influenced by
the presence of but-joints. To show this we may apply
the formula
B 1+1
l x 78 n
to a practical example. In core transformers of the types
Figs. 15 or 16 we should have four but-joints. The gap
in each joint can, even with the most careful workmanship
and the use of specially tough material as an insertion, not
be made smaller than 0'5 millimeter. We thus obtain
8 = 0'2 centimeter. The permeability is from Fig. 9 of
the order 2000. The mean length of magnetic path
varies of course with the size of the transformer. With
small transformers of 1 to 10 Kwt. it may be taken as
between 70 and 160 centimeter; with large transformers
of 100 Kwt. or thereabouts, I would be of the order 300
centimeter. Assuming as a fair average for I 100 c.m. in
small and 250 c.m. in large transformers, then the fraction
I : ^ will lie between the limits 0*05 and 015. The term
in brackets in the equation for i^ will therefore be
increased from O'Oo to 0'25 in small, and from 0*15 to
0'35 in large transformers. This is an increase of 400
and 133 per cent, respectively.
If in a small transformer without but-joints the
magnetizing current is 4 per cent, and the hysteresis
current 3 per cent, of the full load current, the no load
current will be
i = \/ 3 2 + 4 2 = 5 per cent.
Now let us build the same transformer with but-joints.
This will not alter the hysteresis current, which remains
82 TRANSFORMERS.
3 per cent, as before, but the magnetizing current will be
increased to 20 per cent., giving a no load current of
i = ^/ 3 2 + 20 2 = 20'2 per cent.
Let in a large transformer without but-joints
i h = 1-5 % and v =2%, then i = ^ 1'5 2 + 2 2 = 2-5 %.
The same transformer built with but-joints would have
= 4-6 and
By using but-joints we have thus in the small trans-
former quadrupled, and in the large transformer nearly
doubled, the no load current.
Our examples referred to core transformers. With shell
transformers the use of but-joints would be still more
objectionable, since the length of the magnetic path is
only about one-third as compared with core transformers,
and the influence of an increase in magnetic resistance
correspondingly greater. For this reason shell trans-
formers are never built with but-joints, but always with
over-lapping joints according to one or other of the
methods described in Chapter III. For large core trans-
formers but-joints may be permitted, partly because their
influence is less felt than in small transformers, but chiefly
because large transformers are nearly always working with
some load, and then the influence of a larger or smaller
magnetizing current is hardly felt. Small transformers,
especially those for private house lighting from a central
station, should never be made with but-joints, as they
have long periods of no load during which they burden
the station with a large current output out of all pro-
portion with the useful power output.
CHAPTER Y
DESIGN OF A TRANSFORMER BEST DISTRIBUTION OF COPPER COST OF
ACTIVE MATERIAL BEST DISTRIBUTION OF LOSSES ECONOMY IN
WORKING CONSTRUCTION DETAILS.
Construction of a transformer. In order to demonstrate
the practical application of the rules and formulae given
in the previous chapters, we now proceed to work out
a design for a particular transformer, with a transform-
ing ratio of 2000 : 100. For this purpose we select a
core transformer of the type Fig. 15, and assume the
thickness of core to be 125 m.m. The winding space
thus becomes by our formulas a = 160 m.m ; I = 450 m.m.
Let the frequency be ~ = 5 0. To start with, let us
assume B = 5000, with the reservation, that should it be
found advantageous we will alter the induction. The
core and yoke-plates are held together by insulated bolts,
as already explained, and the coils are wound on separate
cylinders of paper, or preferably micanite, so that the
windings may be independently prepared and slipped on.
It is immaterial which winding is inside and which out-
side ; we shall place the low pressure or secondary wind-
ing next to the core, and the high-pressure winding
outside.
To save copper, we chamfer the edges of the core by
say 20 m.m., as shown in Fig. 40. This is done by
84 TRANSFORMERS.
stepping the width of the outer plates. After all the
plates are assembled the core is bound up by strong tape
to keep the plates from bulging. The thickness of this
tape serving is about 2 m.m. In order to enable the low-
pressure coil to be easily slipped on, we require a clearance
of 2 m.m. over the corners, so that the inner diameter of
the supporting cylinder becomes 160 m.m. Its thickness
need not exceed 5 m.m., bringing the inner diameter of
the coil itself to 170 m.m. The depth of winding and the
Fig. 40.
mean perimeter can at present only be fixed approxi-
mately from the following consideration. The dimensions
of the iron part being given, we find the distance of centres
of cores to be 125 -h 160 = 285 m.m. The external
diameter of the primary winding can therefore not exceed
285 m.m., and must in reality be somewhat smaller, since
a certain clearance is necessary not only to avoid touching
of the two coils, but also to make up for possible irregu-
larities in the manufacture, and leave a margin in case the
same size of transformer should have at any future time to
BEST DISTRIBUTION OF COPPER. 85
be made for a higher primary pressure, when finer wire re-
quiring more space would have to be used. A certain
clearance is also necessary to allow for the circulation of
air or oil, whereby the coils are kept from over-heating.
If we fix this clearance at 20 m.m., we obtain for the
external diameter of each primary coil 265 m.m., so that
between the inner surface of the secondary and the outer
surface of the primary winding, there remains a distance
of J (265 - 170) = 47'5 m.m. This distance is made up of
a, the depth of winding in the secondary coil ; 5, the
clearance between the latter and the inner side of the
primary cylinder ; c, the thickness of the latter ; and d, the
depth of the primary winding.
For clearance we may reckon 4 m.m., and for the thick-
ness of primary cylinder 5 m.m., leaving 47*5-9 = 38%5
m.m., to be apportioned between the two windings. It is
obvious that the depth of primary winding will be more
than half this amount, because not only is more space
required for the insulation of the thinner wire, but the
perimeter being larger the gauge of the wire must be
correspondingly increased in order to avoid getting too
much resistance in the primary. As a first attempt in
apportioning the space we may assume that 60 per cent,
of it will be filled by the primary, and 40 per cent, by the
secondary winding.
Best distribution of copper. The correct distribution of
the available space between the two windings is in so far
important as the total loss by ohmic resistance becomes a
minimum if it is equally divided between the two windings.
This may be shown as follows. Let the total winding
space available for both coils be I (in our case 5 = 38'5
m.m.), and the depth of the secondary winding a, then 5 - a
is the depth of the primary winding. Let D be the inner
86 TRANSFORMERS.
diameter of the secondary coil. Its resistance is propor-
tional to the number of turns n 2 and to the mean perimeter
of one turn T: (D + a) ; it is inversely proportional to the
cross section of wire ; or to put it in another way, the
resistance is with a given length of wire inversely propor-
tional to the depth of winding, a. By comprising all
constants into one co-efficient K, we have for the loss in
the secondary coil
f^i$*
a
For the primary coil we obtain a similar expression,
except that the co-efficient K must be multiplied by the
ratio of windings % n 2> and we thus obtain
p-Kn, D + a + bn
*- Vl -M-/C/1 = -1 tj
fc- n 2
Since n? if = n/ i 2 2 , we can also write
P - Kv -* + a + & 2
L V \ ^v. n 2 ^ 2
u-a
The total loss by ohmic resistance is therefore
p = **
a U a
In order to make this loss a minimum, a must be so
chosen that the term in brackets becomes a minimum.
By differentiating and equating to zero, we obtain after
some reduction
a 2 + aD- =
Zi
Since a negative depth of winding is physically im-
possible, we take the upper sign
, 4. _
2 ' Y 4 2
BEST DISTRIBUTION OF COPPER. 87
By equalling the two terms in brackets we also get
9 , r\ bD r\
a 2 + a D - =0
and we see from this that there will be minimum loss by
ohmic resistance if the loss is equally divided between the
two circuits. If therefore our first assumption regarding
the depth of winding does not produce equality of losses,
we must redistribute the available winding space accord-
ingly so as to satisfy the above equation for a. We design
for B = 5000 and ~ = 50 the windings, retaining for the
present the ratio of 40 to 60 per cent, for secondary and
primary winding respectively. On calculating the ohmic
losses, we find that the primary loses too much and the
secondary too little power. We therefore reduce the
depth of winding in the secondary, and increase it in
the primary. It is superfluous to give the calculation in
detail ; the result is
Best depth of winding for the secondary 14 m.m.
primary 24 m.m.
We may now draw the coils, and determine from
the drawing the exact mean perimeter of each. This
gives
7r 2 = 0,575 m. TTj. = 0,755 m.
The next step in the design is the determination of the
hysteresis loss from the weight of iron, the induction, and the
frequency. Assuming that we use best transformer plates,
the lower of the two curves in Fig. 9 is to be taken.
According to this curve the loss of power with B = 5000,
and ~~ = 100 is 1*8 watt per kilogram, and since our
frequency is ~ = 50, we have a loss of 0*9 watt per
kilogram. The net section of iron in the cores is 130
88 TRANSFORMERS.
square c.m., and in the yokes which are not chamfered 136
square c.m. The weights can be calculated from the dimen-
sions, and we thus obtain for the total hysteresis loss
2 Cores 116'8 kg. B = 5000 at 0'9 watt ... 105 watt
2 Yokes 61-2 B = 4770 at 0'S5 ... 52
Total weight 179 kg. Total hysteresis loss 157 watt
We next determine from the drawing the total cooling
surface with 12,000 square centimeters. Assuming for the
present that the transformer is to be worked in a case with-
out oil, and that its temperature rise is to be 54 C., we find
from the curve for air Fig. 29 a- == 37, and can now deter-
mine the load by fixing the total losses at 12,000 : 37 =
324 watt. The copper heat may therefore amount to
324-157 = 167 watt, or with the best distribution of
copper to 83*5 watt for each winding.
The next step is to more accurately determine the
windings. In doing so it is important to observe that
besides the loss of pressure due to ohmic resistance, there
is also loss of pressure due to magnetic leakage. The
latter cannot be calculated, but may be closely estimated
from experiments with other transformers of the same
type. In our design, where the coils are not placed upon,
but within, each other, the loss of pressure due to leakage
and with a noninductive load (such as glow lamps) is very
small, and may be roughly taken at 1 per cent. If we
allow an ohmic loss of pressure of 1J per cent., we have a
total loss of 2^ per cent., that is to say, the terminal pres-
sure on the secondary will be 2J per cent, higher at no
load than at full load.
Since N = 130 x 5000 = 650,000, and ~ = 50, we
obtain from the formula
E 2 = 4-44 ~n. 2 N 10~ 8
BEST DISTRIBUTION OF COPPER. 89
the number of turns n 2 in the secondary winding. This
must obviously be a whole number, and if we wish to get
the winding symmetrical on both limbs (so as to fully
utilize the available space) it must also be an even number.
The nearest number which satisfies these conditions is
n 2 = 70,
whereby for B = 5000 E. 2 = 101-23.
If there were no appreciable voltage drop either through
ohmic resistance or magnetic leakage, a condition fulfilled
when the transformer is working on open circuit, then the
number of primary turns would be 2000 : 100 = 20 times
as great as the number of secondary turns. This would
give n = 1400. With 2000 volts on the primary termin-
als, the pressure on the secondary terminals would at no
load be exactly 100 volts, but at full load 2J per cent, loss,
or only 97*5 volts. If then we wish to have 100 volts
pressure at full load, we must alter the transforming ratio
at no load by 2-J per cent., that is to say, we must reduce
,. , 2-5 x 1400 w
the primary winding by -~ = 3o turns. We have
then
% = 1365
On open circuit U 2 will then not be 101*23 volts, but
102'5 volts, arid the induction will increase in the same
ratio, namely, by 102'5 101'23 = T27 per cent. It is now
not 5000 but 5063, and the hysteresis loss will be increased
from 157 to 161 watt.
The cross section of wire may now be determined. For
fixing the length of the coils we have to consider the height
of the window in the iron frame (in our case 45 c.m.), and
leave sufficient space for clearance and the end flanges of
00 TRANSFORMERS.
the cylinders. The total space required for these purposes
is about 3J c.m., leaving 41*5 c.m. net length of coil. Each
secondary coil must contain 35 turns of wire. If these
were arranged in a single layer, the wire would have to be
wound on edge. Although this presents no difficulty with
naked wire which is afterwards insulated by fibre insertion,
it is not so easy with cotton-covered wire, and in this case
it would be better to wind the wire on the flat and make
two layers, one with 18 and the other with 17 turns. Since
the space of one turn is lost in crossing over from the
lower to the upper layer, we must arrange the width of the
wire to be not T Vth, but yy-th of the net winding space. This
gives 415/19 = 21*8 m.m. The thickness of the wire is
already determined by the depth of winding, which we
found must be 14 man. Allowing 0*5 m.m. for the thick-
ness of covering (or 1 m.m. in all) we find that the section
of the wire will be 6 X 20'8 m.m. Since it is, however,
scarcely possible to lay on succeeding turns with mathe-
matical accuracy, it will be advisable to take the width a
little less, say 20 m.m., so that the actual cross-section of
the wire becomes 6 X 20 120 square m.m. The length
of winding is 70 X 0*575 = 40'5 m., and if we allow 0*5 m.
for connections, we can take 41 m. as the basis on which
to calculate the resistance of the secondary winding. The
formula for the resistance, taking an average rise of
temperature into account, is
W. -
1 2 being the length in meter and q the cross-section in
square millimeter. We thus obtain
W, = 0-00682
A similar calculation made for the primary winding shows
BEST DISTEIBUTION OF COPPER. 91
that we have to use round wire of 3'1 m.m. diameter
(covered to 3'67 m.m.) in six layers of 122 turns, and
one layer of ten turns on one and eleven turns on the
other limb. The length of wire is
/! = 1030 m.
and its resistance warm is
Wi = 2-8
We have now all the data required for calculating the
losses at different loads. For convenience they are given
in the following table
Load in kilowatts 8 9 10 11 12 13 14 15
Secondary current ampere 80 90 100 110 120 130 140 150
Primary current 4'125 4'634 5'150 5-664 6-180 6'70 7'22 7'44
Copper heat watt ... 91 115 143 172 205 241 278 321
Hysteresis ... 161 161 161 161 161 161 161 161
Total losses ... 252 276 304 333 366 402 439 482
Percentage loss ... 3'05 3'06 3'04 3'03 3'06 3'09 3'15 3'22
Cooling surface a ... 47'6 43-5 39'5 36'0 32'8 30'0 27'4 25'0
Temperature rise in air ... 45'5 48'5 51'3 54'0 57'2 60'6 64 67
Temperature rise in oil ... 34 30'0 38'5 40'6 43 45 47 49'3
If the transformer is not to be cooled by oil, we see
from this table that the maximum load it can carry with
the heating limit assumed is 11 Kwt. The total loss is
then 333 watt, so that 11,333 watt must be supplied to the
primary, making the efficiency
If we use oil we may load the same transformer to 15
Kwt., and obtain approximately the same efficiency.
Cost of active material. The efficiency is however not
the only guide in judging the merit of a design. We must
also take into consideration at what cost in material the
92 TRANSFORMERS.
result is obtained. Assuming the finished or cut plates to
cost 9d. per kg., and the insulated copper wire Is. 9d. per
kg. (4fd. and 9^d. per Ib. respectively), then we have
Iron 179 kg. at 9d ...... 134s.
Copper 111*5 kg. at Is. 9d ...... 195s.
Total weight 290*5 Cost of active material 329s.
The merit of the transformer worked in air and oil is
then shown by the following figures
IN AIR. IN OIL.
Output in Kwt. ... ... 11 15
Weight of active material per Kwt. 26'4 kg. 19*4 k
Cost of active material per Kwt. 29'9s. 21'9s.
Best distribution of losses. If we use oil filling for the
transformer case we obtain a lighter and cheaper apparatus
per Kwt. output than if we leave the case empty. On the
other hand, the percentage loss is a trifle larger in the
transformer working in oil, namely, 3*22 per cent, against
3*03 per cent, for the transformer working in air. The
efficiency is a maximum with a load of 11 Kwt., when the
hysteresis loss is about the same as the ohmic loss. If the
output is less than 11 Kwt. we have a lower efficiency,
whilst the hysteresis loss is larger than the ohmic loss ; and
if we increase the output beyond 11 Kwt. we also lower
the efficiency, but now the ohmic loss is larger than the
hysteresis loss. These facts let it appear probable that we
obtain not only in this particular transformer, but generally
in all transformers, a maximum of efficiency at that load
at which the hysteresis loss and the ohmic loss are equal.
If this assumption is correct, then the design of our trans-
former when used with oil for 15 Kwt. output, must be
capable of improvement by increasing the hysteresis loss
BEST DISTRIBUTION OF LOSSES. 93
and decreasing the ohmic loss. The total loss in the
present design is 482 watt. We shall now alter the wind-
ings so as to increase the induction by such an amount as
will bring the hysteresis loss up to about half this figure,
or 241 watt. Retaining the same core, we may now allow
a loss of 1-34 watt, per kg., or 2'68 watt, at ~ = 100, to
which corresponds by the lower curve in Fig. 9, B =
6300.
To this induction corresponds n 2 = 56'5 ; since it is
however impossible to arrange for a fraction of a turn,
and since for the sake of symmetry we must have an even
number of turns, we make n 2 = 56. This gives N =
824,000 and B = 6350.
The hysteresis loss can now be calculated more exactly
as follows
2 Cores 116'8 kg. B = 6350 at T35 watt. ... 158 watt.
2 Yokes 61'2 B = 6080 1-25 ... J77
Total 235
Owing to the reduction in the number of turns we may
now make the secondary wire 26 X 6 m.m. The resistance
warm will be 0'00415 ohm. The primary will have
20 X 56 less 2J per cent. = 1092 turns. There is room
for round wire of 3*5 m.m. diameter (covered to 4 - 2 m.m.).
Of this wire 98 turns go to one layer, so that we shall
require five layers and 54 turns on each limb. The
resistance warm is 172 ohm. The weight of copper
in the secondary is 44 '5 kg. and in the primary 70*5 kg. ;
total 115 kg.
The following table gives the condition of working at
various loads
94 TRANSFORMERS.
LoadinKwt. ... 8 12 15 16 17
Secondary current ... 80 120 150 160 170
Primary current ... 4'15 618 772 8'23 8'7o
Ohmic loss ... 56 115 195 222 252
Hysteresis loss ... 235 235 235 235 235
Total loss ... 291 350 430 457 487
Percentage loss ... 3'64 2'92 2'87 2'86 2'86
We see from this table that the least percentage loss,
and therefore the highest efficiency, is obtained at that
load at which the ohmic loss is approximately equal to the
hysteresis loss. The weight of active material is in this
transformer 294 kg., and its cost determined on the same
basis as before is 335s. Since the load may be 16 Kwt.
we have 18'4 kg. and 20'9s. per Kwt. output. This is an
improvement on the previous design brought about by the
heavier magnetic stress on the iron which was rendered
possible through cooling with oil.
Let us now reduce the linear dimensions so far that
with B = 6350 the output is reduced from 16 to 15 Kwt.
3 / ~t ^
The ratio of reduction is "y -=-^ for the linear dimensions,
and 15 : 16 for the losses. The object of this reduction is
to bring both transformers (namely, that in which B = 5000,
and that in which B = 6350) to the same output of 15
Kwt., in order to have the same basis of comparison. Both
transformers are cooled by oil.
DESIGN. I. II.
Induction v 5000 6350
Output Kwt 15 15
Temperature rise degrees centigrade . . . 49*3 49
Hysteresis loss 161 220
Ohmic loss 321 208
ECONOMY IN WORKING. 95
DESIGN. I. II.
Total loss 482 428
Percentage loss ... ... ... ... 322 2*85
Weight per Kwt. output kg. ... ... 19*4 18'4
Cost per Kwt. output 6s. 21D 20'9
Economy in working. These figures show clearly that
design II. is preferable to design I. both as regards first
cost and economy in working, provided the transformer is
required to work permanently under full load. This is
generally the case in power transmission, for which purposes
transformers should therefore be so designed that the
hysteresis loss is approximately equal to the ohmic loss.
The case is however different with transformers required
chiefly for lighting. The average burning time of all
lamps installed is of course much shorter than the time
during which the transformer must be kept at work. The
transformer must be large enough to feed all the lamps
installed, as it may occasionally happen that all are wanted
at the same time, though as a rule the number of lamps
burning will be smaller than the number of lamps installed.
If a transformer for a single house is connected to a central
station it is at work day and night, that is for 8760 hours
during the year, whilst with an average burning time of
600 hours per lamp, the whole yearly output actually
obtained is only ^ times the possible yearly output.
The hysteresis loss goes on for 8760 hours ; the ohmic loss
only for the time that lamps are burning. The latter loss
must be less than corresponds to full output during 600
hours, since this loss varies as the square of the current, and
maximum current does not last the full 600 hours, but a
much shorter time.
Assume a house with 100 lamps installed, and let the
96
ohmic loss in the transformer when all the lamps are in
use be 100 watt. If all the lamps were simultaneously
lighted and extinguished, then for a yearly burning time
of 600 hours the loss would be 60 Board of Trade units.
In reality the loss will be smaller, because a smaller
number of lamps will be used for a longer time than 600
hours, notwithstanding the fact that the total yearly lamp-
hours will be 60,000. If the time-table for the lamps
alight is as given below, the loss is reduced to 22' 7 units.
LAMPS ALIGHT. HOURS. LAMP HOURS. WATT. LOSS IN UNITS.
100 50 5000 100 5
49 4-9
16 8
4 4
1 0-8
2450 60,000 227
The average burning time of all lamps is, as before
assumed, 600 hours ; some of the lamps are however in use
for a much longer time, with the result that the ohmic
loss in the transformer is reduced by about 38 per cent, as
compared with the loss which would obtain if all lamps
were always used simultaneously. Applying the above
reasoning to the comparison of the two transformer designs
(I. and II.) when used for lighting, we come to a totally
different conclusion to that we reached when using the
transformers for power purposes.
Since 100 watt, maximum ohmic loss means a yearly
loss of 22*7 units, the ohmic loss of design I. will be
321
227 AA = 73 units yearly ; and that of design II. will be
70
100
7000
40
500
20,000
20
1000
20,000
10
800
8000
208
22'7 -7: = 47'3 units. The hysteresis losses are for these
two designs, 1410 and 1930 units respectively.
ECONOMY IN WORKING 97
DESIGN.
I. II.
B= ., 5000 6350
Annual ohmic loss in units ... 73 47*3
Annual hysteresis loss in units ... 1410 1930
Total annual loss in units ... 1483 1977
Annual output in units 9000 9000
Annual efficiency per cent. ... 86 82
As regards annual efficiency, that is the ratio of the
units annually supplied to and obtained from the trans-
former, design I. is decidedly better than design II. On the
other hand, the first cost of design I. is somewhat greater,
namely 21*9s. against 20'9s. for active material alone. To
this must be added the cost of other materials, the case,
wages, shop and trade charges, and manufacturer's profit.
As a fair average we may take it that the net selling price
of the finished transformer amounts to 2 \ or 3 times the
cost of its active material. The net cost would therefore
be for
Design I. 45.
Design II 43.
The difference in cost is only 2, and against this we
have to set off a yearly waste of 494 units. Taking the
engine-room charge for a unit at the central station at
only Id., the units wasted represent 2 Is. Id. ; or in other
words, the more expensive design will pay for itself within
one year. We may now formulate the results of our
investigation as follows
Transformers for power should be so designed that the
stress in the iron is high and in the copper moderate;
iron and copper losses should be approximately equal.
Transformers for lighting should be so designed that the
stress in the iron is small, and in the copper large. The
98
TRANSFORMERS.
copper loss should be greater than the iron loss. It is
obvious that this condition must not be pushed beyond
the limits imposed by temperature rise and voltage drop.
Constructive details. Figs. 41 to 44 give details of the
design of the transformer here discussed. The output is
10 Kwt. with air cooling and 15 Kwt. with oil cooling. The
transformer is placed into a cast-iron case, and is therefore
adapted for use in a cellar or other damp place, or in the
open air in a damp climate. For outdoor use in a dry
climate the case need not be completely closed, but need
only be sufficient to protect the apparatus from rain, and
for dry interior situations the case may be replaced by
perforated covering (see Figs. 103, 110, 125). The ad van-
CONSTRUCTIVE DETAILS. 99
tage of a perforated case is that the cooling effect of the
air is greater. It may be taken as represented by the
lower curve in Fig. 29.
The plates for the core and yoke are cut to size and
punched for the bolt-holes, then laid together with an
insertion of very thin paper. Some makers use varnish
instead of paper, but this is not so reliable an insulation.
In building up, the lower yoke and the two cores are first
made up, the coils are then inserted, and lastly the plates
of the top yoke are put in. The coils are wound on paper
cylinders, which at their lower ends are provided with
flanges to prevent the coils slipping. In winding the
coils it is advisable to wrap each layer with a sheet of
thin paraffined calico, which is doubled back at the ends
so as to give additional insulation between adjacent layers.
The thickness of the cotton covering on the wire depends
on its diameter (or equivalent diameter if rectangular wire
be used), the voltage, and the quality of the cotton and
number of coverings. There must at least be two cover-
ings, though treble covering with very fine cotton is still
better. For very stout wires an additional braiding is ad-
visable. For work up to 3000 volts the thickness of the
covering in millimetres should not be less than
8 = 013 + 0-06 d,
when d is the diameter (or equivalent diameter) of the
naked wire in millimetres. The diameter of the covered
wire is then
di = d + 2 8,
d, = 0-26 + 012 d.
Wire of large rectangular section may also be wound
naked, suitable strips of fibre or other insulating material
being wound in, or afterwards inserted.
rcr^
iff"
<#>,
102; % , \ .* I : ;: TRANSFORMERS.
y^istairce o tfce'cfeil must be calculated with refer-
'ence to its' 'tem'perature' j as a first approximation, based
on a temperature of 75 C., the following formula may be
used
0-02 I .
w - ohm.,
2
where I is the length of wire in meters and q its area in
square millimetres.
To promote dissipation of heat, the casing may be pro-
vided with external ribs or gills. Small internal ribs are
also provided to hold the transformer securely. The
main cover is fitted with a small auxiliary cover to give
access to the terminals without the necessity of breaking
the joint of the main cover. The leading-in wires may be
taken through stuffing-boxes, as shown in Fig. 41, or
they may be simply passed through openings which are
afterwards cast out with insulating compound. The latter
arrangement is preferable for very large transformers.
CHAPTER VI
CLOCK DIAGRAMS WORKING ON OPEN CIRCUIT WORKING UNDER LOAD
MAGNETIC LEAKAGE WORKING DIAGRAM OF TRANSFORMER HAVING
LEAKAGE VOLTAGE DROP GRAPHIC DETERMINATION OF DROP.
Clock diagrams, The working conditions of a trans-
former can be represented in a very simple manner by
means of so-called vector or clock diagrams. It is thereby
convenient to assume for both circuits the same number
of turns of wire, which makes the transforming ratio equal
to unity. Such an assumption is permissible, as will be
seen from the consideration that without altering the
gauge of wire on the primary, we can by simply grouping
the turns differently bring a sufficient number of wires
into parallel connection to produce the effect of a winding
of fewer turns, but of stouter wires. If, for instance, the
transforming ratio is in reality 2000 to 100, and the high
pressure coil has 800 turns, we may assume these 800
turns to be grouped in 20 parallels of 40 turns each. The
current will now be 20 times as large as before, and the
electro-motive force will be reduced to the 20th .part ; that
is to say, it will not be 2000 volts but only 100 volts. In
assuming this alteration in the connections we have not
altered in any way the heating, the percentage of idle
current, the efficiency, etc., but we have obtained the ad-
vantage that the electromotive forces in both circuits are
104
TRANSFORMERS.
now of the same order of magnitude, and can therefore be
conveniently represented in our clock diagram to the
same scale. It is obvious that the current must be in-
creased in the same ratio as the electromotive force is
reduced, and that the resistance is reduced in the square of
this ratio.
Working on open circuit. As an introduction to the use
of clock diagrams we may investigate the simplest case,
Fig. 45.
namely, a transformer working on open circuit. Let in
Fig. 45 OI represent the idle current to any convenient
scale, and OI h and 01^ its two components, which can be
found by calculation as shown in chapter IV. The com-
ponent J M magnetizes the iron of the transformer, the total
magnetic flux being represented to any convenient scale
by the vector ON. The current and flux have of course
the same phase in the diagram. At the moment to which
CLOCK DIAG
the diagram refers the projection of the flux on the
vertical is zero and the E.M.F. has its maximum value,
namely 2 TT ~ Nn 10~ 8 volt. Since the E.M.F. in the
primary must try to prevent the growth of the current
(and flux) its direction in the diagram must be vertically
downwards. Let this be represented by the line OE
plotted to any convenient volt scale. It is obvious that the
impressed E.M.F. must be equal and opposite ; its vector
will therefore occupy the position OE. We neglect in
the diagram the influence of the resistance of the primary
coil since the error thereby committed is insignificant.
The diagram then gives OE as the E.M.F. impressed on
the primary terminals, OE as the E.M.F. obtained on the
secondary terminals, and 01 Q as the primary current, the
secondary current being zero.
The power supplied to the primary is 1 = ~ COS *** l ;
'
or, if instead of using maximum values, we use effective
values,
4 e l = cos $ i Q e lt
The apparent power supplied to the primary is i e v and
the ratio between effective and apparent power, that is
cos <, is called the power factor.
It is interesting to note that the flux ON which is
produced by the current i Q does neither in position nor
magnitude correspond with the flux that would be pro-
duced by a constant current of the strength i o . Yet the
equivalent alternating current i c passes actually through
the primary coils, and we should expect that it must pro-
duce the corresponding magnetization. This is however
not the case. The magnetization is less than corresponds
to the ampere turns of exciting power carried by the
coil ; and it lags behind the current by the angle <#>.
106
TRANSFORMERS.
This apparent contradiction can however be easily ex-
plained. The loss on open circuit is due to hysteresis and
eddy currents. If we could obtain a magnetically perfect
iron, and if it were possible to so design the transformer as
to be absolutely free from eddy currents, then i h zero
and i^ = i . The power factor will also be zero. Assuming
then that it were possible to obtain such a perfect trans-
former, we could by the addition of a third short circuited
winding of the proper resistance so alter it that its working
diagram would be the same as that of the transformer
practically possible. The only condition to get the two
transformers into the same state is that the power which
is transformed into heat in the third coil of the perfect
WORKING UNDER LOAD. 107
transformer equals the power wasted in hysteresis and eddy
currents in the imperfect transformer. It is obvious that
the current generated in the third coil is on the whole op-
posed to the primary current, and must therefore weaken
the magnetization; and this circumstance explains it
why only one component of the idle current (namely, that
which stands at right angles to the component wasting
power) is effective in producing magnetic flux.
Working under load. We may now consider the working
diagram of a loaded transformer. For the sake of simpli-
city we make at first the two following assumptions first,
the transformer has no magnetic leakage ; secondly, the
load is an absolutely non-inductive resistance. Under
these circumstances the secondary current must be pro-
portional to the E.M.F. induced in the secondary coil, and
must coincide with it in phase. If i^ (Fig. 46) represent
the magnetizing current, then e' 2 the E.M.F. in the second-
ary lags by 90 behind i^ and e\ the E.M.F. in the primary
is in advance of v by 90. In this, as in the following
diagrams, we assume clockwise rotation of the vectors.
The pressure at the secondary terminals is represented
by the line Oe$\ thus the distance e 2 4 represents the
voltage drop due to the resistance of the secondary coil.
To find the pressure at the primary terminals as regards
magnitude and .position in the diagram, we proceed as
follows. We plot to an arbitrary scale, the ampere turns in
the secondary on the vector of the secondary E.M.F. Let
this be the distance i. 2 n. 2 . To the same scale we plot Oa
which represents the ampere turns 'of the idle current in
the primary. It is obvious that Oa must be the resulting
ampere turns of the two coils, and since the ampere turns
of the secondary coil are known, we find those of the
primary coil by drawing the parallelogram as shown in the
108 TKANSFOEMEliS.
figure. We thus obtain ^ % and therefore also the primary
current %. This current is being driven through the ap-
paratus by an electromotive force which must obviously
have two components : first, the component Ob required to
overcome the ohmic resistance of the primary coil, and
secondly, the component Oe\ to balance the induced
electromotive force. The resultant, also obtained by con-
structing a parallelogram, is Oe lt which line therefore re-
presents the electromotive force that has to be supplied
at the primary terminals in order that the current i 2 may
be drawn from the secondary terminals under an electro-
motive force e z . A glance at the diagram shows that e l > e. 2)
the difference being in the present case very marked, be-
cause for the sake of greater clearness we have exagger-
ated all losses and assumed too large an exciting power.
If the vectors represent effective values, the following
relations obtain :
Power supplied equals e l i l cos
Power given off ....... e 2 \
Efficiency rj . _VV
6j j COS (j)
In good transformers the idle current is only -^ to - 6 V
of the primary current at full load. The distance Oa is
therefore, as compared to the distance i. z n. 2 . exceedingly
small when the transformer is working under full load,
It is obvious that the line i L n lt that is the vector of
the primary current, becomes nearly vertical, making the
angle so small that its cosine can, without appreciable
error, be considered equal to unity. The efficiency therefore
becomes r/ = ^-L 2
e 1 i l
Magnetic leakage. Up to the present we have as-
sumed that the transformer is free of magnetic leakage, so
MAGNETIC LEAKAGE.
109
that exactly the same magnetic flux passes through the
primary and secondary coil. This condition is as a rule
not realizable in practice, because the two circuits must be
separately wound and insulated from each other, whereby
intervening spaces are produced which admit of the
passage of leakage lines. The result of this leakage flux
is to increase the counter E.M.F. in the driving (primary)
coil, and to reduce the E.M.F. in the driven (secondary)
Fig. 47.
coil, an effect which will obviously be the greater the
heavier the transformer is loaded. If the transformer is
not loaded at all, that is to say, if it is worked on open
circuit, the effect vanishes completely and the ratio be-
tween the primary and secondary electro -motive force is
equal to the ratio between the primary and secondary
number of turns.* At load, however, the transforming
ratio rises, by reason of the leakage effect just mentioned,
and the secondary voltage drops below its value on open
circuit.
* A small change in this ratio may occur which is due to electro-
static capacity in the high pressure coil, if we transform from the
lower to the higher pressure.
110
TRANSFORMERS.
Let in Fig. 47 P and S represent the primary and
secondary coils of a shell transformer traversed at any
particular moment by the currents indicated by the dots
and crosses. The driving coil P produces a magnetic
field N! the lines of which flow in the sense indicated by
the arrows shown in full lines. At the same moment the
driven coil has the tendency to produce a field the lines
of which have the direction shown by the dotted arrows.
This field cannot in reality be produced, because the
Fig. 48.
\
\
driving power of the primary coil overpowers that of the
secondary, so that on the whole the direction of the lines
in the core within the coils is vertically downwards, but
the effect of the opposition offered by S, is to crowd lines
of force laterally outward through the sides AB, so that
the flux NI passing through the primary coil is larger than
the flux N z passing through the secondary coil, the
difference JV being squeezed laterally through the spaces
db. The E.M.F. induced in the secondary coil is there-
fore not proportional to N^ but to Ni N . When working
on open circuit, the secondary coil carries no current, and
there is consequently no tendency to force part of the flux
MAGNETIC LEAKAGE.
Ill
laterally outward through A and B. In this case JV =
and N 2 = N r The E.M.F. in the secondary is now pro-
portional to N L) and therefore greater than before, when we
assumed the transformer to have been loaded. It follows
that with a constant primary pressure, the secondary
Fte. 49.
pressure is a maximum on open circuit and decreases as
the load on the secondary increases. This so-called
Fig. 50.
" drop " of a transformer must in most cases be considered
as an imperfection, and the question arises as to what
112
TRANSFORMERS.
means can be used to minimize this imperfection. Since
the drop is due to the flux N Q) our aim must be to reduce
the latter, and this can be done in either or both of the
two following ways. We may reduce the magnitude of
the exciting power of each coil, and we may increase the
magnetic resistance of the leakage path. Thus the leakage
path in Fig. 47 is shorter and wider than in Fig. 48, and
we shall in the latter arrangement obtain a smaller drop
than in the former. We can carry this improvement a step
further by subdividing the primary coil into two parts, which
are separated by the secondary coil as in Fig. 49. In doing
so we have halved the ampere turns which produce the
leakage flux, whilst at the same time still further reducing
Fig. 51.
'O WAAAr-^
the width of their path. The same object is attained in
the arrangement of Fig. 50. Here we have subdivided
each coil into a number of parts, each carrying only a
corresponding fraction of the total exciting power. The
primary coils are sandwiched between the secondary coils,
and the leakage flux is produced by a much smaller
exciting power, and has a path of much higher resistance
than in Fig. 47. This arrangement is also convenient
because the coils need not be wound upon each other, but
may be wound insulated, and tested independently of
each other.
DIAGRAM OF TRANSFORMER HAVING LEAKAGE. 113
From what has been said above, it will be clear that the
effect of leakage can be represented by the addition of two
magnetic fields (one interlinked with a coil in the primary
and the other with a coil in the secondary circuit) to the
main field common to both coils. We may thus imagine
the practically possible transformer, which has leakage,
replaced by a perfect transformer T, Fig. 51, having no
leakage, to which however are added two choking coils I
and II, the terminals 11,22 being outside of the latter.
At full load the choking coil I produces the field Ni and the
choking coil II the field N 2 . For the sake of simplicity,
we assume the dimensions of the choking coils such that
each contains the same number of turns at the correspond-
ing transformer coil. The field common to both trans-
former coils is N. Then the E.M.F. induced in the
primary transformer coil is
l = 4,44 ~ Wi N 10~ 8
and that induced in the choking coil I is
e sl = 4,44 ~ % NI 10~ 8 .
Similarly we have for the secondary circuit
e. = 4,44 ~ n, N z 10~ 8 .
e s2 = 4,44 n a N z 10~ 8 .
In plotting these values in a clock diagram we must
not forget to place e s2 at right angles to i 2 and e sl at right
angles to %.
Working diagram of transformer having leakage, We
are now in a position to draw the clock diagram of a
transformer having leakage. Let for the present its load
be now inductive (glow-lamps for instance), and assume
that we have, as mentioned in the beginning of this
chapter, arranged the primary winding so as to have the
same number of turns as on the secondary coil.
Let Oi z in Fig. 52 represent the secondary current, Oe k2
114
TKANSFORMEltS.
the pressure at the secondary terminals, c k2 c' 2 the loss of
pressure due to the ohmic resistance of the secondary, then
Oe' 2 must be the resultant of the E.M.F. induced by the
main field N and the E.M.F. of self-induction due to the
leakage field N 2 . The vector of the latter must lie in such
Fig. 52.
position that c s2 tries to prevent the decrease of i 2 ; that is
to say, it must be drawn horizontally to the right as shown
at e s2 in the diagram, which is plotted to the same volt-
scale as Oc k2 . By constructing the parallelogram we obtain
Oe 2 the E.M.F. which must be induced in the secondary
coil in order that the current L may flow under the
terminal pressure e k2 .
DIAGRAM OF TRANSFORMER HAVING LEAKAGE. 115
The magnetizing current i^ must be at right angles to
c., and in advance of it, whilst the current representing
loss of power i h must be in line with e 2 e and of the same
sense as e^ The position and magnitude of the idle or no-
load current is thus completely denned. We obtain for it
in the diagram the vector i . The vector of the primary
current we find by combining ^and i 2 . This gives %. The
E.M.F. of self-induction (due to leakage) in the primary
must stand at right angles to % and must follow it. In our
diagram, e al must therefore be drawn to the left. The E.M.F.
supplied to the primary terminals must obviously contain
three components.
One component must be equal and opposite Oe 2 . This
is given by the vector Oe lt
One component must be equal and opposite Oe al , and one
component must be provided to overcome the ohmic
resistance. Let the vector of the latter be Oca.
By adding these three components graphically we
obtain the point e kl . Oe kl is the vector of the E.M.F.
supplied to the primary terminals. A glance at the
diagram shows that e kl is greater than e k2 , the difference
being the more marked the greater are the ohmic resist-
ances of and the E.M. Forces of self-induction in the two
windings. In both respects the diagram Fig. 52 has been
exaggerated, so that the influence of each part may be
more clearly seen.
It is interesting to investigate the case of a transformer
the secondary terminals of which are short-circuited by
a stout copper wire and amperemeter, thereby making
c kz ~ 0. We assume the primary E.M.F. to be so adjusted
that this amperemeter shows the normal secondary cur-
rent corresponding to full load under normal working
conditions. The diagram then assumes the form shown
11G
TRANSFORMERS.
in Fig. 53. The lettering is the same as in Fig. 52. It
will be seen from this diagram that although no pressure
is obtained at the secondary terminals, a pressure equal to
c kl must be supplied to the primary terminals in order
that the current i 2 may flow through the short circuit.
If, as is always the case in modern transformers of good
Fig. 53.
design, the resistance of the windings is very small, then
the vector of the
power factor of the motors or arc lamps worked by the
secondary current, and BD is the wattless component of
the secondary pressure.
It is conceivable that the secondary circuit contains
partly appliances having only resistance, and partly other
appliances having resistance and self-induction. This is
the case if to the same leads are joined glow-lamps and
arc-lamps or motors. In a distributing system of 100
Fig. C8.
volts the glow-lamps would be arranged in simple parallel
and the arc-lamps in parallel series of two or three lamps.
In Fig. 58 OE is the vector of secondary pressure, the
lag of that part of the circuit which has self-induction, and
GRAPHIC DETERMINATION OF DROP. 123
OA' the corresponding current. The current taken by the
glow-lamps is in phase with the pressure, and must be re-
presented by a vector which is parallel to OE. Let this
vector be A' A, then OA is the total current supplied to
the circuit and ^ its lag behind the pressure. A glance at
the diagram shows that OA < OA' + A' A and that ^ <
<. If then we measure separately the current supplied to
the glow-lamps and to the arc-lamps, and also the current
in the undivided part of the circuit, we shall find that the
latter is less than the sum of the other two currents. Let
for instance the power factor of the arc-light circuit be 0*71
(< = 45), and let there be 5 parallel series of lamps, each
taking 15 ampere, then OA' = 75 ampere. In addition to
the arc-lamps let us insert a number of glow-lamps, taking
collectively 32 ampere. The total current is then not 107,
but only 100 ampere, as will be found on drawing Fig. 58
to scale. The power factor of the whole circuit is then
cos \ff = 0*85. The transformer is thus apparently loaded
to 10 Kwt.; in reality, however, only to 8*5 Kwt. The re-
duction in the load is brought about by the phase difference
between pressure and current. We have now to investi-
gate what effect this phase difference has on the terminal
pressure or on the ratio of primary and secondary terminal
pressure.
Let in Fig. 59, OA represent the secondary current and
OB the terminal pressure. The E.M.F. induced in the
secondary winding must obviously contain the following
three components. First, OB ; secondly, BB', to cover the
ohmic loss ; and thirdly, B'C, to counteract the E.M.F. of
self-induction. The vector of the secondary E.M.F. is
therefore OC, and by our assumption of an equal number
of turns in both coils, OC is also the E.M.F. induced in the
primary coil t The E,M,F. impressed on the primary ter-
124
TBANSFORMEES.
minals must also have three components ; namely, OC, CD,
to counteract self-induction, and DE, to cover the olimic
loss. We thus obtain the vector of the E.M.F. impressed
on the primary terminals OE. The inclination of the
straight line BE is, as before, determined by the ratio
between resistance and reactance.
With a mixed load of glow-lamps and arcs the angle
\js varies with the number of lamps of each kind in use at
any time. If however the load consists only of arc-lamps
which are switched in or out in series of two or three,
then the lag (which we will call in this case, compare
Fig. 57) remains the same for all loads. The current may,
as before, be plotted on a line ol drawn parallel to BE (Fig.
GO), the scale being so chosen that ol represents full load.
The pressure on the secondary terminals is for full load OB.
For a smaller load o/ x the construction shown in the dia-
gram gives a greater pressure, namely OB f . By comparing
this diagram with Fig. 56 it will be seen that the drop is
greater when the load is inductive. This may also more
precisely be seen if we extend the example previously given
to a case where the load consists of arc-lamps. We investi-
gated the behaviour of a 10 Kwt, transformer, in which the
GRAPHIC DETERMINATION OF DROP.
125
ratio of reactance to resistance was 10 : 1. Let us now
connect the same transformer to an arc-light circuit in
which cos = >f 71 ( = 45), and construct the diagram
Fig. 60 for various loads. This gives the figures contained
in the following table. To facilitate the comparison the
figures of the previous table are repeated.
Secondary current in ampere 25 50 75 100
Pressure at secondary |" cos 1 103'8 103'2 102*35 101'3 100
terminals \cos $ = 071 103 '8 99'5 95'2 91 86'
This transformer (2 / Q resistance and 20 % reactance)
has on an inductionless resistance a full load drop of less
Fig. 60.
200
92
68
than 4 per cent. Although such a performance is not
quite as good as might be desired, it is tolerable ; for an
inductive load this transformer is however quite unfit,
since the drop amounts to as much as 17 per cent. To
render this transformer suitable for motor work, the geo-
metric arrangement of the winding would have to be so
altered as to considerably reduce magnetic leakage. If
tested on short-circuit, the full secondary current must be
obtained with a primary E.M.F., which at the outside must
126 TRANSFORMERS.
not exceed 10 per cent, of the normal supply voltage. Thus
the drop at full load with <$> = 45 would be about 9 per
cent., which is tolerable for motor-work, but already too
high for arc-lighting. Transformers for arc-lighting, or a
joint service of lighting and power purposes, must be so
designed that the product of current and reactance does
not exceed 5 per cent, of the normal supply voltage. This
result can be attained by proper grouping arid thorough
subdivision of the coils.
Our investigation thus far has shown that with a
constant E.M.F. impressed on the primary terminals, the
pressure on the secondary terminals drops as the load
increases, the drop being due to the ohmic resistance in
both coils, and to their reactance. We have also seen
that, other things being equal, the drop is the greater, the
smaller the power factor of the circuit to which the
transformer supplies current.
The case where the circuit has capacity must now
claim our attention. Capacity also reduces the power
factor, and it may perhaps be thought that also in this
case the reduction in the power factor would be accom-
panied by an increase of voltage drop. This, however, is
not the case. On the contrary, the introduction of capacity
into the circuit diminishes the drop, and may under
certain circumstances even convert it into a rise of
pressure. To facilitate the investigation, we assume at
first that the apparatus supplied with current by the
transformer has only resistance and capacity, but no
reactance. Let the capacity be a shunt to the resistance
(a concentric cable feeding glow-lamps is a practical
illustration of this case). If E is the maximum E.M.F., K
the capacity in Farad, then EK coulombs are charged into
and discharged from the condenser twice every period, the
GRAPHIC DETERMINATION OF DROP.
127
charges being alternately positive and negative. Let us
consider the moment when the E.M.F. has attained its
positive maximum value, and commences to decrease.
The condenser is then completely charged by the positive
current which has up to that time been flowing into it.
As soon as the E.M.F. begins to decrease, the period of
discharge begins, the current being now negative, although
the E.M.F. still remains positive for a quarter-period
longer. The negative current attains its maximum value
at the moment when the E.M.F. passes through zero. It
is therefore in advance over the E.M.F. by a quarter-
period.
Fig. 61.
In Fig. 61 OE represents the position of the E.M.F.
vector at time t corresponding to the angle a, and e the
instantaneous value of the E.M.F. After the lapse of an
infinitely short time dt the instantaneous value has
increased by de= Esm a d t, and the charge of the con-
(L t
denser has been increased by the amount i dt, i being the
current at time t which has flown under the potential
difference de. We have therefore
1 28 f
i d t = K d G
d c
The differential quotient -=- is determinable if the shape
cl t
of the E.M.F. curve is known. Let this be a sine function,
then
d e TT d a
=^008 0-
dt dt
and since d a = 2 TT - d t we have
~ =-#277 -cos a
oU
^ = KE 2 TT ~ cos a.
The condenser current attains its maximum value at all
values of a for which cos a= 1; that is to say,
for a = 0, a = TT, etc. It is zero for a = ^ a | TT, etc.
Since the opposite is the case with regard to the instant-
aneous value of the E.M.F., we see that the vector of the
condenser current must stand at right angles to the
E.M.F. vector, and be in advance over it by 90. The
maximum value of the condenser current is
and its effective value
. KE
ri
Lines j is the effective value of the E.M.F., which
v 2i
we may designate by e; we have also
In this formula i is given in Ampere, K in Farad, and e
in Yolt. The usual unit of capacity is, however, not the
Farad, but the Microfarad (a million times smaller), and
by adopting this unit we have
GRAPHIC DETERMINATION OF DROP.
129
the index k being added to the sign for the current, to
show that the latter is a condenser current, which, leading
over the E.M.F. by 90, carries, no power. Let i w be the
watt or power component of the current, the phase of
which coincides with the phase of the E.M.F., then
l = JF
W being the ohmic resistance of the apparatus to which
current is supplied. By reference to Fig. 62 it will be
seen that the total current i is given by the expression
Oe is the E.M.F. vector, Oi k the vector of the condenser
current, and Oe w the vector of the power current. It is
important to note that the diagram only represents the
case where capacity and resistance are in parallel, and the
condenser circuit has so little resistance that it may be
neglected. This is approximately the case of a concentric
cable connected at one end to the transformer and at the
other to glow-lamps. The two conductors form the inner
and outer coating of the condenser, which is being charged
and discharged by the condenser current i k . If all lamps
are switched out W = co and i w = 0. In this case
(f) = 90 and i = i k . The cable takes only the condenser
K
130
TRANSFORMERS.
current. As lamps are being switched on i w increases,
< decreases, i increases, and the power factor increases
also.
If arcs instead of glow-lamps are connected to the far
end of the cable, there will be reactance in addition to
resistance. The E.M.F. of self-induction is
2 TT ~ T i
v s it a j* i w
and its phase lags by 90 behind the phase of the current.
The corresponding component of the E.M.F. impressed on
the circuit must therefore have this value, but be 90 in
advance of the current. The watt component e w is of
Fig. 63..
course co-phasal with the current. In Fig. 63 i 1p repre-
sents the current flowing through the inductive resistance,
e w the watt component of the E.M.F., and e s the E.M.F. of
self-induction. By a parallelogram we find Oe, the vector
of the E.M.F. This not only drives the current through
the inductive resistance, but it also produces the condenser
current i k leading by 90. The total current supplied by
the transformer is the resultant of i w and i k and its vector
is Oi. Accordingly as reactance or capacity preponderates,
the current will lag behind or lead before the E.M.F. In
GRAPHIC DETERMINATION OF DROP.
131
the diagram the relations are so chosen that the current
leads.
We have up to the present assumed that the condenser
forms a shunt to the apparatus absorbing power. This is
indeed the most frequent case, but it is also possible that
the circuit is interrupted by a condenser, which is then in
series with the apparatus absorbing power. A case of
this kind occurs if a liquid resistance is used in testing
transformers. A barrel containing salt water or acidulated
water, and two sheets of lead as electrades, or an iron
trough containing an alkaline solution, and iron plates for
electrodes make very efficient resistances, capable of
taking up large amounts of power. Liquid resistances
are for these reasons very often used in lieu of solid
resistances. It is well known that a metal plate dipping
into a liquid acts as a condenser of very large capacity,
and in using a liquid resistance we introduce therefore
into the circuit a capacity in series with the resistance.
Fig. 64.
Let in Fig. 64 Oi represent the current passing through
the liquid resistance, and Oe m the watt component of the
E.M.F. The E.M.F. required to produce the condenser
current i is e k == i : K2 TT ~ in absolute measure. It lags
behind the current by 90, as shown in the diagram.
The resultant E.M.F., that is the E.M.F. which must be
132
TRANSFORMERS.
impressed on the terminals of the liquid resistance in
order that the current i may flow through it is Oc. It is
obvious that also in this case the current leads over the
E.M.F. by a certain angle (in the diagram the angle ),
and that the power factor of the liquid resistance is
smaller than unity.
In the foregoing it has been shown how the phase-differ-
ence between current and pressure can be determined for
every load if the electrical constants of the apparatus re-
ceiving power are known. In this manner the working con-
ditions of the transformer can in every case be determined.
Those cases where the terminal pressure leads as compared
with the current have already been investigated, and we
have now to extend the investigation to those cases where
the current leads over the pressure at the secondary term-
inals of the transformer, that is to say, where the phase
angle < is negative. The construction previously given can
obviously also be applied now. In Fig. 65 the current is
given by the vector OA, and the terminal pressure by the
vector OB. The ohmic drop in the secondary is BB',
and must obviously be parallel with OA. The E.M.F. of
GRAPHIC DETERMINATION OF DROP.
133
self-induction in the secondary is B'C, which is at right
angles to A. The E.M.F. of self-induction in the
primary is CD, and the ohmic drop in the primary is
DE. We thus obtain OE as the vector of the E.M.F.,
which must be impressed on the primary terminals of the
transformer. Since BB', D E, and B' D are all propor-
tional to the current, the inclination of the line BE
remains the same for all loads, and its length may to a
suitable scale be made to represent the secondary current.
Fig. 66.
We assume for the present that the power factor is
independent of the current. In this case the pressure at
the secondary terminals may be found for different currents
by the construction shown in Fig. 66, in which OA
represents the direction of the current vector, and Oo
that of the secondary terminal pressure. The inclination
of /o, on which we measure the secondary current, is
fixed by the electrical constants of the transformer. lo
must be parallel to E B in Fig. 65. From as centre we
describe a circle with a radius equal to the E.M.F.
impressed on the primary terminals (the transforming
ratio is supposed to be unity). To find the position of
134 TRANSFORMERS.
the primary E.M.F. vector we measure from o to the left
a distance representing the secondary current, and find
thus the point /. From / we draw a parallel to o 0, and
find K Drawing EB parallel to lo we find B. Then
OB is the pressure at the secondary terminals. For a
smaller current /' the same construction gives the
pressure OB'.
As will be seen from the diagram, the terminal pressure
increases with the current. We have therefore with in-
creasing load, not a drop, but a rise of terminal pressure.
The rise of pressure depends, amongst other things, on
the angle of lead $. If this is smaller the rise will also
be less marked, as will easily be seen from the diagram,
and it is obvious that there must be a certain phase angle,
for which there is neither rise nor drop of pressure, but
exactly the same pressure at full load and at no load.
For smaller values of there will be a drop, but this will
still be less marked than would be the case if the current
lagged behind the E.M.F. We thus come to the con-
clusion that the drop depends, not only on the magnitude,
but also on the nature of the load. It is greatest for
inductive loads, smaller for non-inductive loads, and
smallest (or even negative, i. e. a rise) on loads having
capacity. For this reason the use of a liquid resistance
in testing transformers for drop is misleading. The drop
so measured is always smaller than that which would be
found on a lamp-circuit, and it may even happen that the
drop on a liquid resistance is negative, that is to say, that
the pressure rises when the load is increased. This will
happen if the transformer has much magnetic leakage.
Thus the worse the transformer the more favourable will
it appear when tested on a liquid resistance.
Up to the present we have assumed that the angle of
GRAPHIC DETERMINATION OF DROP. 135
lead or lag is the same for all loads, and determined
the pressure at the secondary terminals as a function of
the secondary current. In most cases it is, however, only
of importance to know the drop or rise at full current,
since the fitness of a transformer for practical use must
naturally be judged by its behaviour at full, and not at
some intermediate, load. On the other hand, it is im-
portant to know how the secondary pressure depends on
the power factor of the. apparatus to which the trans-
former supplies current. The same transformer may be
intended for use in connection with apparatus of different
power factor, and it will depend on the change in terminal
pressure between full load and no load whether such use
is practically possible.
The problem we have to solve is therefore the following
Given a transformer of known resistance and reactance.
The primary or supply E.M.F. is constant. Find the
pressure at the secondary terminals with full secondary
current, but for different values of phase angle $. The
graphical solution of this problem is extremely simple,
and follows as a matter of course from what has been said
in connection with Figs. 59 and 65. For a constant
ampere-load the length of the line E B is constant, the
only variable in the diagram being the angle which the
vector of terminal pressure forms with the current vector.
The inclination of B E is also constant. As $ changes 12
takes different positions on the circle of primary E.M.F. ,
and the locus of B must therefore also be a circle of the
same radius, the centre of which has relatively to the
same displacement as B has to E.
Let in Fig. 67 the vertical represent the current vector,
S the E.M.F. of self-induction at full current, and S o
the ohmic loss at full current in both windings ; then o
13G
TRANSFORMERS.
is equal and parallel with E B of Fig. 60, and o is the
centre of the second circle just mentioned. For a positive
phase difference (current lagging behind E.M.F.) the
secondary terminal pressure B is smaller than E, its
value at no load. For a negative phase difference !
(current leading before the E.M.F.) the secondary term-
inal pressure B is greater than its value at no load.
With a certain negative phase difference $ 2 the terminal
pressure at full load is exactly the same as at no load,
and we have neither a drop nor a rise of voltage. This
phase angle is given by the point of intersection J9 9 of
Fig. 67.
the two circles. The drop or rise can be directly measured
in the diagram; it is the length included between the
two circles. If the load is an inductionless resistance
having no capacity = o and the drop is given by the
distance E% to B%. The diagram shows very clearly how
the drop increases as the lag of current behind E.M.F.
becomes greater in consequence of the greater reactance
in the apparatus to which the transformer supplies
current.
The length of the line Oo is, as already pointed out,
a measure for the secondary current. If the load is re-
GRAPHIC DETERMINATION OF DROP. 137
duced o is shifted accordingly, and the same construction
can be made for every load. We can thus determine the
secondary terminal pressure for every working condition
of the transformer. Such a determination is to be pre-
ferred to a direct measurement of the drop, for the
following reasons. In the first place, to make the direct
measurement reliable, the apparatus serving as a load to
the transformer must have exactly the same power factor
as obtains in the apparatus to which the transformer will
have to supply current. A liquid resistance can therefore,
as a rule, not be used in the test. Solid resistances and
lamps are not only more cumbersome, but their adjustment
to produce one particular lag requires the use of choking-
coils or similar appliances, and this complicates the test.
In the second place, the direct testing of large trans-
formers requires an amount of power which is not always
available ; and lastly, if these difficulties are overcome, we
have to face the further difficulty of having to determine
a comparatively small difference between two readings of
large absolute value, which cannot be done with great
accuracy. For these reasons it is better not to attempt any
direct determination of the drop, but to solve the problem
indirectly, in the manner described with reference to Fig.
67.* As an example to show the application of the in-
direct method, we take a 60 Kwt. transformer having a
ratio of 3000 to 200 volt. Resistance of primary 0'9
ohm ; ohmic loss 18 volt. Resistance of secondary 0*0036
ohm ; ohmic loss 1*08 volt. By reducing the transforming
ratio to unity we would have an ohmic loss in the primary
of 18 -OQQA = 1*20 volt. To determine the length Oo
* See the author's paper on this subject (Elektrotechnische
Zeitwhrift, 1895, p. 260), and discussion thereon.
138 TRANSFORMERS.
in Fig. 67 we liave therefore the following data :
Ohmic loss in primary T20
secondary 1-Q8
total , 2-28
We short-circuit the secondary through an ampere-
meter, and supply as much primary E.M.F. as will just
suffice to produce the full secondary current of 300 ampere.
This experiment shows that 255 volt must be supplied to
the primary terminals. This corresponds to 17 volt at
the transforming ratio of 1 : 1. We have therefore in Fig.
07 o = 17 and S o = 2'2S, whilst E is 200. We can
now design the diagram, Fig. 68. A is the current
vector. On this we mark off the power factor cos <. The
corresponding position of the vector of E.M.F. is E, and
the terminal pressure which we scale off on OE is 187
volt. In a similar manner we determine the terminal
pressure for all other values of cos $. The result is given
in the following table.
60 Kwt. -transformer 3000 : 200 volt on open circuit.
Pressure at secondary terminals with 300 ampere in
secondary and power factors varying from 100 to 50 per
cent.
Power factor in per cent. 100 99 90 80 70 60 50
With leading current . . 197 200 205 207 210 212 213
With lagging current . . 197 195 190 188 187 186 185
If used on a glow-lamp circuit this transformer would
at full load have a drop of only 1-J- per cent. ; if used on a
circuit containing arc lamps or motors the power factor
of which is about 0'70 to 0*80, the drop would be approxi-
mately 6 per cent.
The diagram Fig. 68 leads to some interesting deduc-
tions. In the majority of cases the circuit has not
capacity, but reactance, and the following remarks apply
GRAPHIC DETERMINATION OF DROP. 139
to these cases, that is to say, to the left-hand side of the
diagram. If we could build a transformer which has
o
absolutely no magnetic leakage, then OS would be zero,
and o would lie vertically above 0. The inner circle
would then approach the outer circle more closely as we
go to the left. In other words, the drop would be greatest
for an inductionless, and smaller for an inductive re-
sistance. This case is, however, unattainable in practice,
FIG. 68.
Scale for 1hf powcHhctar:
ir- ' ' -
for we can never reduce magnetic leakage to zero. The
reactance produced by magnetic leakage can, however,
with a very careful design, be made very small, especially
for low periodicities. Imagine that we have reduced the
reactance so far as to be equal to the resistance, then
S = S o, and o includes with A an angle of 45.
The distance between the two circles would then be ap-
proximately the same for all values of . We should
thus obtain a transformer which has approximately the
same drop for all values of the power factor.
As a rule, the reactance is, however, considerably greater
140 TRANSFORMERS.
than the resistance, and the two circles diverge towards
the left. As a consequence the drop increases as the
power factor decreases. If the same transformer is used
for a high and a low frequency, the pressure at the
secondary terminals will at full current be lower in the
former case. The E.M.F. of self-induction is for both
windings,
S = 2 x 2 TT ~ Z 2 i 2 ,
that is to say, S is proportional to ~. The higher the
frequency, the greater is the divergence between the two
circles. It must also be borne in mind that the power
factor of the apparatus to which the transformer supplies
current (motors or arc-lamps) is lower at the higher
frequency, and in consequence the vector of E.M.F. in our
diagram is shifted the more to the left, the higher the
frequency. Both causes conspire to increase the drop.
If then the transformer is intended to feed not only glow-
lamps, but also motors and arc-lamps, the frequency should
be chosen as smal] as compatible with the proper working
of alternating current arcs (45 to 50 complete cycles per
second). This frequency is also advisable on account of
certain reasons connected with the design of non-synchro-
nous motors.
CHAPTER VII
THE DYNAMOMETER THE WATTMETER MEASUREMENT OF IRREGULAR
CURRENTS OTHER METHODS OF POWER-MEASUREMENT TESTING
TRANSFORMERS TESTING SHEET-IRON.
The dynamometer. We have up to the present always
supposed that current and E.M.F. follow a sine law. This
assumption, although very convenient for the analytical
or graphic treatment of alternate current problems, is
rarely in accordance with fact, and the question therefore
arises what errors are introduced by assuming that current
or E.M.F. change according to a curve which is different
from that actually obtaining. This question will be
answered by investigating whether an amperemeter, if
applied to the measurement of a current of irregular curve,
gives the true effective value of the current, and a watt-
meter gives its true power. To investigate this matter, it
is in the first place necessary that we come to an under-
standing as to what is exactly meant by the term " effective
current." Let us assume that we have two glow-lamps
of exactly the same construction, and that one of these is
fed by a continuous current, and the other by an alternat-
ing current. If both lamps are brought to the same state
of incandescence, and absorb therefore an equal amount
of power, then the . effective strength of the alternating
current will obviously be equal to that of the continuous
142 TRANSFORMERS.
current. Since both lamps must have the same tempera-
ture, the resistances must also be equal. Let this be IF,
then the work done in the time T is / 2 W T, if 7 is the
strength of the continuous current. The work done on
the other lamp is obviously f r P W dt, if I denotes the
momentary value of the current, which is, of course, a
function of the time t, this function being graphically
represented by the current curve, which may be of any
shape. The effective value of the alternating current is
thus / , and is given by the equation
The question we have now to consider is whether the
amperemeter will show this or some other value. All
the instruments used for measuring alternating currents
are based upon some electro-dynamic or some heat effect
produced by the current. In either case the momentary
action is proportional to the square of the momentary
value of the current ; and all these instruments will there-
fore be equivalent as regards their suitability for measur-
ing alternating currents. We may thus restrict the
investigation to one particular type of instrument; the
result is valid for all other types.
For this purpose let us select the well-known dynamo-
meter invented by Weber. The movable coil is sus-
pended within the field of the fixed coil, and is thus
subjected to a deflecting force, which is proportional to
the square of the current, and which is balanced by the
torsional force of a spiral spring. In measuring a con-
tinuous current, / , the deflecting force is given by the
equation 7 2 -, K being a constant depending on the con-
THE DYNAMOMETER, 143
struction of the instrument. The balancing force exerted
by the spring is proportional to the angular deflection D,
which must be given to the torsion head to keep the
pointer at zero. We have thus the equations
which is the well-known formula showing the relation
between current and reading on a dynamometer.
The question we have to consider is whether the same
formula is applicable if the current is not continuous, but
alternating according to any irregular or regular function
of the time. Let this function be represented by a curve,
in which time is measured on the horizontal, and current
strength on the vertical. By squaring the ordinates we
obtain a second curve, and the area enclosed between the
axis of abscissae and this second curve represents the
expression/ 21 1 2 d t, whilst the height of a rectangle of
equal base and area represents the square of the effective
current. The movable coil of the dynamometer is acted
on by two forces; one being the tension of the spring,
and the other the electro-dynamic force of the current.
The former is constant, and the latter variable, but always
opposed to the former. If I t is the current at time t,
the force acting at that time upon the movable coil is
D K' 2 If. This force produces an acceleration given by the
expression - - ^ -, if by m we denote the mass of the
m
movable coil reduced to the radius at which the force
acts. If this mass were sufficiently small, then, this
acceleration, which is alternately positive and negative,
would produce a visible oscillatory movement of the coil,
the velocity of movement being represented by the formula
144 TRANSFORMERS.
whereby we assume that at time t = o the velocity is also
zero. The mass of the movable coil is, however, very
large in comparison with the forces acting on it, and the
frequency with which the acceleration changes from a
positive to a negative maximum is so great that no visible
oscillation is produced in the coil. This, moreover, is an
obvious condition for an exact measurement. The velocity
v must at all times be infinitely small, that is to say
m
From this condition we obtain
1
The expression on the right-hand side of this equation
is obviously nothing else than the height of the rectangle
previously mentioned, that is to say, the square of the
effective value of the current. We have therefore
and this is a proof that the dynamometer does really
measure the true effective value of the current, whatever
may be the shape of the current curve. The constant
K of the instrument may once for all be determined by
calibration with a continuous current, and then used for
all measurements with alternating currents, provided
always that the instrument does not contain any metal
masses in such proximity to the movable coil that a
disturbing effect through eddy currents is produced.
i* 4 v* **f&>*~* <
THE WATTMETER.
145
The wattmeter. We have now to investigate the
problem how a dynamometer may be used for measuring
power, and to determine whether such measurements are
accurate for currents of irregular shape. The arrange-
ment of the instrument is shown in Fig. 69. C is the
Fig. 69.
movable, and c the fixed coil. G is the alternator sup-
plying current to the apparatus 2\ which may be a
transformer.
The dynamometer, as usually constructed, has two
terminals, A and B (or three if the fixed coil is arranged
in two parts, so that the range may be increased). If the
instrument is intended for measuring power, then the
connection between the movable and the fixed coil must
be provided with a third terminal, D. The current at
which the power is to be measured is then led through
the fixed coil, whilst a shunt containing an inductionless
resistance, W, is arranged between the terminal B of the
movable coil and the return circuit. The shunt resistance
may consist of a ribbon of platinoid, constantan, or other
alloy having a small temperature co-efficient, and the ribbon
should be wound zigzag fashion, so as to reduce the re-
actance to a negligible quantity. A series of glow-lamps
may also be used instead of a metallic resistance, if the
TRANSFORMERS.
relation between current and resistance lias been previously
ascertained. In the shunt circuit we insert an ampere-
meter, o 1} although this is not absolutely necessary.
' If the .resistance W is very large, . we may without
sensible error assume that the shunt current is in phase
with the E.M.F. impressed upon the circuit, which is
shown by the voltmeter v t . In other words, the current
flowing through the movable coil has no lag, and is pro-
portional to the E.M.F. The main current in the fixed
coil may lag .or lead as compared to the E.M.F. If the
apparatus receiving power is a transformer, the current
will lag, and the lag will be the greater the more re-
actance the apparatus has. Let I be the main current
(maximum value / m ), and i is the shunt current (maximum
value i m ), then if E.M.F. and current follow a sine law,
the turning moment acting on the movable- coil is pro-
portional to
/ OT sin (o-^)^ sin a
a representing the phase of the E.M.F. at the time to
which the expression refers, and the angle of lag of the
current passing through c. If by W we denote the re-
sistance of the shunt, including that of the amperemeter
a x and connections, we have
i
i = *.
m W
and the turning moment may also be considered pro-
portional to
I m sin (a -<)~p sin a.
By twisting the torsion-head of the instrument we
apply a twisting couple to the movable coil, which is given
by the expression D K z , D being the angular deflection,
THE WATTMETER. 147
and K the constant of the instrument. Let m be the
mass of the movable coil reduced to the radius at which
the forces act, then the acceleration at the time to which
the phase angle refers is
D K 2 - I m ?s. sin (a - 0) sin a
m
and the velocity attained after the time t is
in (-*) sin a ) d *>
sn
provided at time t = o the velocity is. zero. The mass
of the coil is very great as compared with the forces, and
the changes in direction of the electro-dynamic force
occurs with great rapidity, so that the movable coil can-
not follow these changes, and remains, at rest. In other
words, v is -at all times zero, and this requires that
D K* =fl
sin (a - <) sin a
This equation may also be written as follows
-< sn a
The value of the integral is -i -? and we have therefore
or by introducing the effective values of current and
E.M.F.
It has already been shown that the product lecos is
the power carried by the current / flowing under a
potential difference c, and having a lag 0. The product
148
TRANSFORMERS.
W 1) K~ is therefore tlie power of the alternating current.
The symbols have the following meaning
1) is the angular deflection of the torsion-head,
K is the constant of the instrument which has
been determined by calibration with a known
continuous current i, so that K .,
\'D,
W is the total shunt resistance in Ohm.
We may use the instrument as above explained with
reference to Fig. 69, and also as an ordinary dynamometer,
using the terminals A and B, but not D. We measure
first the main current /, arranging the connections as
shown in Fig. 70, and then the shunt current e / W arrang-
ing the connections as shown in Fig. 71. If it is permissible
Fig. 71.
VVWMVWV-i
wwwwww^
to neglect the self-induction of the movable coil as com-
pared with the self-induction of the apparatus T, and if
it is also permissible to neglect the resistance of the
fixed coil as compared with the resistance of the shunt
circuit W, then the dynamometer does not increase the
lag nor waste power. Let the reading, when arranged
according to Fig. 70, be D lt and when arranged according
to Fig. 71, be D. 2 , then
CURRENTS OF IRREGULAR FORM. 149
By multiplying these equations we obtain
Previously we obtained the expression
K-D=Ico S = -- .
By thus taking three measurements with the same
instrument (one power measurement and two current
measurements), we may find the lag <. It is important
to note that only the readings enter into the equation for
the lag. In order to determine the latter, it is therefore
not necessary to know the constant of the instrument, nor
need we know the shunt resistance.
Measurement of the power carried by currents of
irregular form. Up to now we have assumed that current
and E.M.F. follow a sine law. If they follow any other
law, the measurement taken by the wattmeter, as ex-
plained with reference to Fig. 69, will still be perfectly
correct. The work done by the current in time T is
f r Iedt=Wf T Iidt.
The main current /, as well as the shunt current i, may
be any periodic function of the time, the only condition
being that the frequency must be the same for both.
This is obviously the case, since both are produced by the
same E.M.F. The turning moment acting on the movable
coil is DK^ Ii) and varies continuously. If the fre-
150 TRANSFORMERS.
quency were sufficiently small (T very large), and the
mass of the coil also very small, the coil would assume
an oscillatory movement. But the frequency is great, and
the mass of the coil is so considerable that no movement
takes place ; in other words, the integral of the oscilla-
tion taken over the time of a complete period, is zero.
f DK' 2 -Ii\dt = o
f(
m
TDK*=J T Iidt
Wf T
It was shown above that "TnJ lidt is the work clone
*
by the current in time T. Since power is work divided
by time, we have
that is to say, the reading of the wattmeter gives the true
power P, whatever may be the shape of the current and
E.M.F. curves.
To make a power measurement by means of a watt-
meter, we must know the constant of the instrument, and
the exact value of the shunt resistance W. If the latter
consists of platinoid or other alloy of small temperature
co-efficient, this condition causes no difficulty. If, how-
ever, a series of glow-lamps is taken as the shunt resist-
ance, the value of the latter is not constant, but varies
with the E.M.F. To find W we may proceed in either of
two ways : Read the deflection D on the wattmeter, the
current i on the amperemeter o 1} and the E.M.F. c on the
voltmeter v lm The two latter readings give W=c /i t and
the power is then given by the expression
CURRENTS OF IRREGULAR FORM.
151
The difficulty of this method lies in the necessity to take
three readings simultaneously. To obviate this, we may
do away with the reading of the current by calibrating
the resistance beforehand, which may be done with con-
tinuous currents. A curve may be plotted, giving W as
a function of e, and then it suffices to take the readings
of D and e t W being taken from the curve. The ampere-
meter a^ may be cut out of circuit, whereby the conditions
of an inductionless shunt is more easily fulfilled.
-WWVWWW-
/WWVWWw
If the instrument is connected as shown in Fig. CO, it
shows the power supplied to the transformer and that
wasted in the shunt, that is, the total power supplied by
the generator. If only the power given to the trans-
former is to be measured, the wattmeter must be con-
nected up as shown in Fig. 72, where the shunt is taken
from the wire which leads to terminal A. In this case,
the fixed coil c carries only the current passing through
the primary of the transformer, and the power wasted by
the current in the movable coil G is not measured. The
expression
152
TRANSFORMERS.
gives then exactly the power supplied to the transformer.
If the load on the latter consists entirely of glow-lamps,
then the product of secondary current and secondary
terminal pressure is the power delivered by the trans-
former. We measure on the amperemeter a 2 the secondary
current i 2 , and on the voltmeter v. 2 the secondary pressure
c, 2 . The efficiency of the transformer is then given by the
expression
WDK*
If the supply voltage is high, it is advisable to so con-
Fig. 73.
nect the wattmeter that in the instrument itself no great
potential difference can arise. Otherwise there is the
risk of breaking down its insulation. Fig. 73 shows the
arrangement of connections which should on that account
be avoided. Theoretically Fig. 73 is equivalent with Fig.
72 ; the latter arrangement is, however, from a practical
point of view, preferable, because the highest potential
difference which can arise between the fixed and movable
coil is only that due to the resistance and reactance of the
latter, and is therefore only a small fraction of the total
pressure. On account of safety in handling the instrument,
it is also advisable to earth that terminal of the generator
CURRENTS OF IRREGULAR FORM.
153
which is directly connected with the terminal A of the
wattmeter.
The theory of the wattmeter given above is only correct
if the assumption of a perfectly . inductionless shunt
circuit is justified, by reason of the inductionless resistance
being very great as compared with the self-induction of
the movable coil in series with it. There will then be
almost no lag of the current in this coil. To reduce the
lag absolutely to zero is, of course, impossible, since the
action of the instrument pre-supposes the existence of a
Fig. 74.
mechanical force, which can only be obtained by means
of a coil producing a magnetic field of its own, that is to
say, having a certain amount of self-induction which must
produce some lag. With careful workmanship, the self-
induction can, however, be brought down to such limits
that the condition ^ = -7^ is practically fulfilled. If
this condition is not fulfilled, a correction to the reading
must be made as shown in the following theory.
Let Oe in Fig. 74 represent the vector of the total
154 TRANSFORMERS.
supply pressure, and i that of the main current through
the fixed coil, which has a lag <^>, so that
2 77 ~ L is the reactance, and W the resistance of the
apparatus to which power is supplied. The power is
OixOa.
If there were absolutely no self-induction in the watt-
meter, the shunt current would be in phase with c. As
there is some self-induction, it lags behind by an angle
which we denote by v//-, and assumes in the diagram the
position Oi Q . If w is the shunt resistance, and I its co-
efficient of self-induction, we have
277~/
tan \k =
w
The wattmeter will then not indicate the true power
OixOa, but the apparent power OixOc. To find the
true power, we must therefore multiply the reading by the
ratio ~. Let P f be the apparent power shown by the
instrument, and P the true power, then
P=P X ~&c'
Since a = e x cos $ and c = 1} X cos (< ^) = e x
cos \ff cos ($ ^) we have also
PTV cos d)
j-* . /I .
COS ^r COS (< \|^)
Since ^ is a constant of the instrument, it may be
determined once for all. The phase difference between
main and shunt current may be found from
D
cos (cl> \l/) =
CURRENTS OF IRREGULAR FORM. 155
as previously explained. Since \jf is known, we know
also 0, and can determine the correcting factor. The
above expression may also be written in the form
T\ T~>/ J. "~i~" LclLL \s*
1=1 . .
1 + tan tan ^
Is y-<^, the correcting factor becomes smaller than
unity, that is to say, the true power is smaller than that
read off on the wattmeter ; is \js > (which may happen
if the power circuit has capacity, or only resistance, whilst
the wattmeter has too much self-induction), then the true
power is larger than that read off on the wattmeter.
There are two values of \^, for which the reading of the
wattmeter gives absolutely the true power. This will
obviously be the case if the correcting factor equals unity ;
namely if \js = O, and ^ = 0. In the former case we have
an instrument with negligible self-induction, and in the
latter case the self-induction in the shunt happens to be
of such a value as to produce the same lag as in the main
circuit. The case most frequently occurring in practice is
that there is some lag in the main circuit, and that <>Y''
especially with wattmeters of modern construction, where
^r is very small. The correcting factor is then slightly
less than unity, and attains its minimum if happens to be
equal to 2 x/f. This gives the greatest error possible, and the
correcting factor is then -?. For all other values of
0, either greater or smaller than 2 \l/, the error is less, and
the correcting factor approaches more nearly to unity.
The following table gives the maximum possible error for
different values of d>.
1 5f> TRANSFORMERS.
Power reading True The error is equal tn
^ on Wattmeter power or smaller than
5 1000 998-5 -15 per cent.
10 1000 994-G -54
15 1000 982-7 1-73
20 1000 968\S 312
25 1000 950-8 4'92
30" 1000 928-2 718
Other methods of measuring power. Measurement of
power may be made in a variety of ways. For con-
venience and accuracy the wattmeter is preferable, but
where such an instrument is not to hand, some other
method must be employed. We may broadly divide all
power measurements into two classes : (a) Absorption
methods, whereby the power to be measured is absorbed
by the apparatus itself which we use for measuring ; and
(b) Transmission methods, where the power merely passes
through the measuring apparatus, but is not absorbed
therein.
As an example of the first kind of measurement, we
may take the arrangement often used for alternators,
whereby the output of the machine is taken up in a
resistance which need not be absoluely inductionless. It
should, however, consist of a metal or alloy having a small
temperature co-efficient, such as platinoid, manganin, con-
stantan, etc. For the measurement we require an accurate
amperemeter and a Wheatstone bridge. After running
long enough to get the resistance heated up to its final
temperature, a reading of the amperemeter is taken, and
the current is then switched off the resistance, the time
being carefully noted. As quickly as possible the re-
sistance is connected to the Wheatstone bridge, and
resistance measurements are taken at frequent time-
OTHER METHODS OF MEASURING POWER. 157
intervals. A curve may then be plotted giving the re-
sistance as a function of the time, and this curve, pro-
longed backwards to the moment when the machine current
was switched off, gives the actual value of the resistance
when the current was passing through it. The power
taken up is then given by the product resistance x square
of current. By adjusting the bridge beforehand, and
arranging the switches so as to facilitate the changing
over from the power circuit to the bridge circuit, a high
degree of accuracy may be attained by this method.
Another example of the absorption method is the
measurement of the power wasted in a transformer by
means of temperature readings. The temperature of the
transformer is taken at normal load; the alternating
current is then switched off and a continuous current is
sent through the primary, which is so regulated as to keep
the temperature up to the same point. The power wasted
by the continuous current is the same as that previously
wasted in the transformer, working under normal con-
ditions, and the latter can therefore be accurately deter-
mined by volt and current measurements. It is advisable
to use a spirit thermometer, since a mercury thermometer
may show too high a reading, the error being due to eddy
currents set up in the mercury in the bulb by the stray
magnetic field. On the whole this method of measuring
power cannot be recommended ; it is at best only a crude
approximation, and requires much time and personal skill.
As examples of the transmission method of power
measurement, may be mentioned the use of the wattmeter
as already explained, the " Three- Voltmeter Method " of
Professor Ayton, and the " Three-Amperemeter Method "
of Professor Fleming.
Three-voltmeter method. The application of this
a 58 TRANSFORMED.
method will be understood from Fig. 75, where G is the
generator, a an amperemeter, W an inductionless resistance,
Fig. 75.
and T the transformer absorbing the power which we wish
to measure. A voltmeter is placed between the main
leads ; let its reading be e. Another voltmeter is used to
show the potential difference e 2 between the terminals of
the inductionless resistance, and a third instrument shows
the potential difference between the terminals of the
primary of the transformer. Instead of using three separate
voltmeters, we may of course use the same instrument for
taking all three readings by arranging a suitable system of
switches. This arrangement is preferable on account of its
greater simplicity, and because slight errors in the calibra-
tions of the instrument have less influence on the result.
The clock diagram of this arrangement is shown in Fig 76.
01 is the current, OJE i = e 1 is the E.M.F. impressed on
the transformer, and U^is the E.M.F. absorbed in the resist-
ance W. Since the latter is inductionless, its vector EE
must be parallel to the current vector 01. E=e is the
total E.M.F. The watt-component of the impressed
E.M.F. e l is e w = OA and the energy is 01 x A. Since
only the coil has inductance we have A E^ = B fi=e s , the
E.M.F. of self-induction, and the following equations
obtain
THREE-VOLTMETER METHOD.
from which we find
150
The power is given by the formula
P = i e " Cl ~ c ' 2
To find the power we must take four readings, namely,
three voltmeter readings and one amperemeter reading.
Fig. 76.
If the resistance W is accurately known, the last reading
may be omitted and the power calculated according to
the formula
^ __ o 2 _ r 2
P = _- _ * jL
2 W '
This is the power actually supplied to the apparatus T,
in our case a transformer. If we want to know the power
supplied by the generator G, e// W, the power lost in the
160 TRANSFORMERS
resistance must of course be added, and we obtain
0- I /> 2 '1
J y _ L - ~^^L
2W
Instead of calculating P, we can find the watt-component
of C L graphically by drawing circles with radii c and c, and
shifting a vertical line parallel to itself until a position is
found in which the piece contained between the two circles
is exactly equal to ^l~l2)
2 i. 2
If the resistance W is accurately known, the reading for e
Fig. 78.
^ 2
need not be taken, and the power may be calculated from
W (4% o 2 i' 2 \
p_ w (i i l ^ 2 ;.
2
162 TRANSFORMERS.
Also in this method accuracy depends upon the proper
choice of the resistance. It should be so adjusted that i 2
is not sensibly different from \ ; the total current i will
then be from 1J to 2 times the primary current ^ taken
by the transformer.
In considering both methods, we have tacitly assumed
that current and pressure follow a sine law ; the question
now arises, whether these methods will give accurate
results if this condition is not fulfilled, that is to say, if
the curves representing E.M.F. and current are of irregular
shape. That the wattmeter gives correct indications
also in such cases has already been shown, and since
simultaneous measurements by means of a wattmeter and
one or the other methods here described are always in
accord, we naturally conclude that these methods must
also be generally applicable. Apart from such experi-
mental proof, this can also be shown by theory. For
this purpose we shall consider the three-voltmeter method,
the application to the analogous case of the three-ampere-
meter method will then be self-evident. Let in the
following the letters e and i denote the instantaneous values
of E.M.F. and current respectively, then the expression
e = ei + e. 2
is valid at any time for t. We also have at all times
and the power at any moment is
w
TESTING TRANSFORMERS. 163
The work done in the time T of a complete cycle is f pdt }
i
and the effective power is
p =
1 C T
It has been previously shown that the expression --J o e 2 dt
is simply the square of the effective pressure indicated
by the voltmeter; if now we denote these effective
pressures by e t e lt e 2 respectively, we have
Since in arriving at this result (which is exactly the same
as that reached by the graphic method), we have made no
assumption whatever as regards the shape of the E.M.F.
curve, it follows that the three-voltmeter method is applic-
able to currents of any form.
Testing transformers. By means of the various methods
above explained the output and efficiency of transformers
can be determined. It is of course necessary to have a
source of current capable of supplying all the power wanted,
and some apparatus capable to absorb the full output of
the transformer. To obtain by this direct method any-
thing like a reliable figure for the efficiency, input and
output must be measured with extreme accuracy, the
reason being that the two are not very different, and a
small error in the determination of one or the other
causes a great error in their calculated ratio. Let for
instance the real input be 100 and the real output 97
Kwt., and let there be an error of 1 per cent, in each
measurement, the error being negative in the measurement
164 TRANSFORMERS.
of the input and positive in the measurement of the out-
put. The measurements would then be 99 and 98 Kwt.
respectively, and the calculated efficiency would be 99 per
cent, instead of 97 per cent., which it really is. To reduce
as much as possible the magnitude of the error in the
determination of the efficiency, it is advisable to make this
determination by an indirect method in the following way.
The test is made simultaneously on two equal transformers,
which are so connected that the output of No. 1 forms the
input of No. 2, and the output of this, supplemented by
an external source of power again, the input of No. 1.
I) 3
h/VAW\/v\\V_ A/VA\W/VW J
CVWWWVr-i W,
vVW
We obtain thus a circulation of power through the two
transformers, and need only supply as much power as is
wasted in both. This is a small amount, and need only
be measured with a moderate degree of accuracy. The
power circulating is also measured, and it will be obvious
that small or moderate errors in both measurements cannot
seriously affect the accuracy of the result. The arrange-
ment of apparatus is shown in Fig. 79. D and B are the
two equal transformers, and G is a small auxiliary trans-
former which supplies the waste power and thus keeps the
total power in circulation. Into the primary of C we
insert an inductionless rheostat R, for the purpose of
TESTING TRANSFORMERS. 1G5
adjusting the pressure supplied to C, so as to obtain in the
amperemeter a the normal full load secondary current
of the big transformers. The connection between the
latter must of course be so arranged that their E.M. Forces
oppose each other. If the large transformers were only
connected to (7, the full current could be obtained in them,
but not the pressure. To insure that also the right
pressure is maintained in B and D, we connect their
primaries with the generator Gf, as shown in the diagram.
If now we open the rheostat completely, and short-circuit
the secondary of C, then the generator has to supply only
the no load losses of B and D, which will be indicated on
the wattmeter W, provided the two-way switch $ is put
to contact a, as shown in the diagram. Since both trans-
formers are equal, no current will be indicated in a. Now
let us insert C and adjust the rheostat until a indicates
the full load current. Then the large transformers are
both working under full load and the wattmeter measures
all losses. These are : The ohmic loss in R and the losses
in C, B and D. Since we know the efficiency of the small
transformer, and can measure its primary current on the
amperemeter a lt we can calculate the power which the
small transformer supplies to the two large transformers.
If the switch is placed on its contact a, the wattmeter
indicates the total power given off by the generator. Let
this be P lt If the switch is placed on I it indicates only
that part of the power which flows to the rheostat and
auxiliary transformer. Let this be PC. The primary
current of C is measured in a lt Let this be i. If w is
the resistance of the rheostat, corresponding to the position
of its contact, then the power supplied to C is P c ihv.
Let rj'be the efficiency of C, then t] f (P c iho) is the power
given off by C. The generator delivers to the primaries
166 TRANSFORMERS.
of the two transformers the power 1\ P C and the auxiliary
transformer delivers rj' (P c i 2 w). The total power wasted
in B and D is therefore
If P is the output and rj the efficiency of transformer D,
then D receives an input of P watt and gives an output of
?; P watt. B receives an input of t] P watt, and gives an
output of rf P watt. From this follows
V
Since P v is very small as compared with P, it is obvious
that a moderate error in the determination of P v can only
produce a very small error in the determination of the
efficiency rj. According to our previous example, we
would have for P= 100, P v = 6. If we measure accurately,
we obtain
= 0-9695.
Now let us assume that we do not measure accurately,
but commit an error of 1 per cent, in the measurement of
P and 5 per cent, in that of P vt then the maximum
possible error which can occur in the determination of the
efficiency is only J per cent.
This indirect method of determining efficiency is there-
fore far and away more accurate than the direct method.
It has moreover also other advantages. In the first place,
no apparatus is required for taking up the output of the
transformer under test; and, secondly, only a moderate
amount of power need be employed. Both are import-
ant considerations when large transformers are to be
tested.
TESTING SHEET-IRON. 167
The heating of transformers can be investigated in the
following way. The transformers are first heated in a
stove or drying-room to that temperature which they will
probably assume in continuous work. Where a hot room
is not available, the heating may also be done by passing
through the fine wire winding a continuous current from
a secondary battery. When the transformers have been
brought up to near their probable working temperature,
they are loaded, the connection of Fig. 79 being preferably
employed, and the temperature is taken at stated time-
intervals by means of spirit thermometers. The readings
are plotted in a curve of which the abscissae represent
time and the ordinates temperature, and the test is
continued until the curve becomes horizontal. If an
uninterrupted supply of alternating current is available,
the preliminary heating can be dispensed with. The
transformers are then connected up from the first, as shown
in Fig. 79, and kept at work until they have attained their
maximum temperature.
The drop may be tested directly under normal working
conditions, or by the indirect method given in Chapter VI.
The latter is more simple and accurate.
The insulation test should be taken immediately after
the heating test. It is also advisable to flash the trans-
former, so that any weak spot in the insulation may be
found out and remedied before the apparatus is set
to work. For this purpose, temporary connections should
be made between (a) a primary and secondary terminal ;
(6) a primary terminal and earth ; (c) a secondary terminal
and earth. Care must of course be taken that during
these tests both poles of the generator are well insulated
from earth.
Testing sheet-iron. The methods in use for the deter-
168 TRANSFORMERS.
mi.nati.on of the permeability and the hysteresis loop of
sample bars cannot conveniently be applied for sample
sheets, because the attainment of a really reliable magnetic
contact at the ends of the sample is very difficult. It is
of course possible to find the hysteresis loop for sample
sheets prepared as rings, but since each ring has to be
specially wound with a magnetizing and a pilot coil, this
method is too cumbersome for practical work. Moreover,
the result is arrived at indirectly. From a practical point
of view we do not require to know the exact shape of the
hysteresis loop ; all we care to know is the power wasted
both in hysteresis and in eddy currents in a known weight
of sheet-iron at a given induction and a given frequency.
The most direct method is obviously to measure the power
lost in the sample by means of a wattmeter. The per-
meability cannot be found that way, bub can be estimated
with a fair degree of accuracy from the power factor of
actual transformers when working with open secondary.
Transformers used for this purpose must of course not have
butt-joints.
A very simple way for testing sheets is to make stamp-
ings of the shape used for transformers, and to insert them
in the usual way into a coil of known number of turns.
The induction is found from the terminal pressure, the
section of iron S inside the coil and the number of its
turns n.
The ohmic loss of pressure being very small may be
neglected. The total loss is due to hysteresis and eddy
currents in the iron. If the induction remains the same
whilst the frequency is changed, then the loss through
eddy currents varies with the square of the frequency,
\tfl r ERSITT)
^^^UFORrtiA^-^
TESTING SHEET-IRON. 169
whilst the hysteresis loss varies as the frequency. It is
thus possible to separate the two losses. For this purpose
we need only vary E proportionally with ~ and determine
the power lost in each case. Let us assume that we have
made two measurements, P l and P 2 for the power lost,
which correspond to the two frequencies ~ and ~ 2 re-
spectively. We have then
P 7) ^ _L f^ 2
*1 ltj \\J i
P Ti ~ _1_ /~ 2
ft 'f' 2~t~ t / 2
A and / being as yet unknown co-efficients relating to
hysteresis and eddy current losses. These co- efficients can
now be found from the two equations, and we obtain thus
a means to separate the two losses. The hysteresis loss
for the frequency ~ is then given by
p=*~,
and the eddy current loss is given by
Pf=f~?
In testing iron as here described, it is of course necessary
to employ the sample sheets in a form which permits the
building up of a transformer core. It is however not
always convenient to stamp or cut-out the sheets in this
particular form, and if a method can be found by which
the samples may have the shape of straight strips the
preparation of the batch would be easier, and no iron
need be wasted.
Such a method of testing has been devised by several
engineers. Herr von Dolivo Dobrowolsky uses the ap-
paratus shown in Fig. 80. 1 It consists of two I 1 shaped
cores of sheet-iron, which can be laid together either
directly or placed on either side of the sample A A to be
1 This illustration is taken from the Elektertechnische Zeitschrift,
1892, No. 30, page 406.
170
TRANSFORMERS.
tested. The sample is composed of rectangular sheets
and forms the common yoke to the electro-magnets n s.
When the magnets are placed directly in contact, the
direction of the current through the coils is such that both
drive the induction in the same sense ; Avhen the sample
is inserted the connections are charged by means of the
switch B, in such manner as to produce the polarity
indicated in the diagram. The flux now passes from both
magnets through the yoke. The current is measured by
Fig. 80.
a dynamometer marked EL Dijn. in the figure, and the
pressure by a Cardew voltmeter marked Card. The power
is measured by a wattmeter inserted as shown. In using
the apparatus the magnets are laid together and the switch
is put into the position which produces circular magnetiza-
tion. The power corresponding to various values of the
induction is then measured, the induction being calculated
from the frequency, the pressure and the known data of
the coils and magnet cores. The sample is then inserted,
TESTING SHEET-IRON.
171
the switch changed over and the measurements repeated.
The sectional area of the sample should be about double
that of the magnets. The difference between the two
sets of measurements is then the power wasted in the
sample at the various values of the induction. A draw-
back of this method is the difference in megnetic leakage
with and without the sample. If the magnets are laid
together directly, and magnetized circularly, there is hardly
any leakage, and B can be calculated from E with great
accuracy. If the sample is inserted, the magnetic resistance
is increased, and leakage produced which diminishes the
Fig. 81.
F
t
I/
/> 1
%
- f
[
1
1
t
value of B in the sample. At the same time there is a
difference in the value of the induction along the magnet
cores, the induction being a maximum in the centre of
each core. E can therefore no longer be regarded as an
exact measure for B, and an error is thus introduced.
To avoid this difficulty, the author has constructed the
apparatus shown in Fig. 81. The sample consists in this
apparatus also of a batch of rectangular plates, and forms
one of the two longer sides of a rectangular frame, the
three other sides being formed by | | shaped plates of
known magnetic quality. Both larger sides are sur-
172 TRANSFORMERS.
rounded by coils, the upper one being large enough to
admit the insertion of the sample without difficulty. The
connection is made for circular magnetization, so that only
very little leakage takes place, and this is the same for all
samples. The sample must have approximately the same
cross-section as the magnet. To calibrate the instrument,
a sample is prepared from the same iron as the magnet,
and after weighing the total amount of iron in the magnet,
the loss of power is determined for different values of B.
This loss is then allotted between magnet and sample
according to their relative weights, and a curve is plotted
showing the loss in the magnet as a function of B. If
now another sample is inserted, and the total loss measured,
we have only to deduct from it the loss as found from the
curve for the particular induction observed, and the rest
is the loss in the sample. The curves of Fig. 9 have been
partly plotted from tests made with this apparatus, and
partly from tests made on actual transformers.
Another apparatus for factory use has been devised by
Prof. Ewing. Its principle is the purely mechanical
determination of the loss in a sample of very small dimen-
sions, namely 6 to 8 strips of 3 in. length and f in. width.
The apparatus consists of a permanent magnet e, Fig. 82,
which is suspended on knife edges f, and weighted by a
screw g. For transport the magnet can be raised off the
knife edges by means of a rack and wheel h. A dashpot
below the magnet serves to steady its swing, and a pointer
moving over a scale at the top shows the deflection pro-
duced when the sample a is rotated between the poles.
The sample is fastened by screw clamps 1 1 to a carrier
which can be rotated by means of a handle, and the friction
wheels d c. The screw i serves to level the instrument.
The reversal of magnetism in the sample is produced by
TESTING SHEET-IRON.
173
the rotation, and the work lost in hysteresis and eddy
currents per revolution is 2 TT x Torque. The torque is indi-
cated on the scale by the pointer, and since 2ir is a constant,
we find that the deflection of the pointer gives directly
a measure for the loss per cycle, the speed of rotation
Fig. 82.
having no influence as long as it is not so high as to
sensibly augment eddy current losses.
The sample sheets are prepared to a gauge, the length
being sensibly less than the polar gap of the magnet, so
that the magnetic resistance of the air gap preponderates
174 TRANSFORMERS.
over that of the sample itself. The object of this arrange-
ment is to avoid the error which might otherwise be
introduced when samples of widely different permeability
are tested. The magnet produces in the sample an in-
duction of about 4000 CGS units, but this can be slightly
raised or lowered by taking less or more sample plates.
Prof. Ewing found that an accurate adjustment as regards
the weight of samples is not required, since the deflection
varies but slightly if the number of plates making up a
sample batch is varied. It suffices to adjust the weight
of the batch roughly to that which corresponds to 7 strips
of 13 to 14 mils, thickness. This is the thickness frequently
chosen for transformer sheets. When testing armature
plates, which are usually stouter, a correspondingly smaller
number of strips would be used to make up the sample
batch.
The apparatus is calibrated by using samples, the
hysteresis of which has previously been accurately deter-
mined by the ballistic method. Two such standard
samples are supplied with the apparatus, together with
tables giving the results of the ballistic tests. In testing
other samples, a reading is also taken with one of the
standards, and the ratio of the readings is taken as the
ratio of hysteresis losses between standard and sample.
By this method of testing, the accuracy of the instrument
is rendered independent of any possible change that may
have occurred in the strength of the permanent magnet.
CHAPTER VIII
SAFETY APPLIANCES FOE TRANSFORMERS SUB-STATIONS AND HOUSE-
TRANSFORMERS BOOSTERS CONNECTION IN SERIES CHOKING COILS
COMPENSATING COILS THREE-WIRE SYSTEM BALANCING TRANS-
FORMER SCOTT'S SYSTEM.
Safety appliances for transformers. The reason why we
use transformers is that we may carry the power under
high pressure, and distribute it under low or moderate
pressure. It is, however, an essential condition that the
insulation between the transmission circuit (primary)
and the distributing circuit (secondary) be absolutely per-
fect. If this condition be not fulfilled the advantage of
transforming becomes illusory, and the use of transformers
may even become dangerous on account of an unjustified
feeling of security. The two windings in a transformer
must necessarily lie in close proximity, and thus an injury
to the insulation may cause a leakage of current and a
transfer of pressure from the primary to the secondary
coil. Since in a widely distributed net-work of primary
conductors their insulation cannot be absolutely perfect, it
will be obvious that any leak between the primary and
secondary coil of any particular, transformer may raise
the absolute potential of the secondary to a dangerous
amount. This potential will depend on the position
of the leak in the transformer, on the position of the
equivalent leak in the general system of high pressure or
176 TRANSFORMERS.
primary circuits and on the insulation of the secondary
circuit. It may be a few hundred volts only, or it may be
equal to the full primary voltage. If in the latter case a
person touches any part of the secondary circuit he will
receive a dangerous or fatal shock. To avoid this danger
several expedients are possible. One very obvious pre-
ventive is to place between the two windings a metallic
dividing-sheet which is well earthed. If the insulation
between the two windings is damaged, contact is not made
between the primary and secondary direct, but through
the intervention of this dividing-sheet, and thus the po-
tential of the secondary is prevented from rising to any
dangerous extent. This appliance ensures safety only in
so far as regards a leak from one winding to the other, but
it is useless against a leak in any other part of the trans-
former, for instance, between the primary and secondary
leading-in wires. Although in a proper design, and with
careful workmanship, such a leak may be regarded as
almost impossible, yet we must admit the possibility that
design or fitting up may not be perfect, and for such cases
protectionary measures become necessary. The simplest
way of obtaining protection is to earth some point of the
secondary circuit, preferably the middle of the winding,
since then the potential difference of the secondary mains
to earth becomes a minimum, namely, equal to half the
voltage. If contact takes place anywhere between primary
and secondary, the former is thereby connected to earth,
and all danger of a fatal shock is avoided. The danger as
regards fire is, on the other hand, increased by this ex-
pedient. If the whole of the secondary circuit is insulated
from earth, a fault must occur at two places of different
potential before a danger in respect of fire can arise, but
if one point of the secondary circuit is permanently con-
SAFETY APPLIANCES FOE TRANSFORMERS. 177
nected to earth, a fault occurring in one place only is suffi-
cient to create danger. The margin of safety is therefore
reduced by one half if we earth the middle of the
secondary winding.
This drawback is overcome by the safety appliance
introduced a few years ago by the Thomson-Houston
Company. The appliance consists of an earth-plate and
two metal knobs a, b, Fig. 83, which are connected to
the secondary mains. Between the knobs and the earth-
plate is inserted a thin sheet of insulating material (paraf-
fined paper or mica). As long as no fault between
Fig. 83.
Earth
primary and secondary occurs, the potential difference be-
tween the knobs and the earth-plate remains within the
limit of the secondary voltage, and this is not sufficient to
break down the insulation between knobs and earth. If,
however, through a fault in the insulation between
secondary and primary, the secondary assumes the poten-
tial of the primary, the insulation between a and earth
and b and earth is broken down, thereby short-circuiting
the secondary winding. The primary current then rises to
such an amount that the safety forces s s go, and the
transformer is thereby automatically cut out of circuit.
178
TRANSFORMERS.
A safety device invented by Major Cardew, and much
used in England, is shown in Fig. 84. In this arrange-
ment the action depends on electrostatic attraction be-
tween a plate E connected to the secondary, and an alu-
minium foil lying on a plate connected to earth. The
aluminium foil has the form of two discs connected by a
narrow bridge, and is together with the two plates enclosed
in a box, provision being made by means of a screw thread
in the cover of the box to accurately adjust the distance
between the plate E and the aluminium foil. The latter is
Fig. 84.
Earth
permanently kept at the potential of the earth (zero),
whilst the plate E has under ordinary circumstances a
potential not exceeding the secondary voltage. The
electrostatic attraction corresponding to this potential
difference is insufficient to raise the foil; if, however, a
fault occurs between primary and secondary the potential
difference immediately rises to such an amount that the
electrostatic attraction suffices to raise the foil and bring
it into contact with the plate E, thereby earthing the
secondary winding. In the safety device first described
SAFETY APPLIANCES FOR TRANSFORMERS. 170
by Cardew 1 a fuse S was provided and arranged to hold up
a weight which, if the fuse melted, would short-circuit the
primary leads, and thus cause their fuses s s to go, and
the transformer to be cut-out of circuit. It has, however,
been found that this is a superfluous refinement, since the
short produced on the secondary by the lifting of the
aluminium foil is in itself sufficient to make the primary
Fig. 85,
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Earth
fuses go. The apparatus can be set to come into action
if the potential of the secondary rises to 400 volts. Hence
even an incipient fault in insulation between the two
circuits is sufficient to automatically disconnect the
transformer from the circuit.
Ferranti's safety device is shown in Fig. 85. The
Journal I r mt. EL Eng,^ vol. xvii. p. 179.
180 TRANSFORMERS.
secondary mains are connected to the primaries of two
very small transformers coupled in series, whilst their
secondaries are coupled in parallel. The secondaries are
connected to a fuse carrying a conical weight over a corre-
sponding set of terminals. The connection between the
two primaries is joined to earth, as is also one of the
terminals, the other two being joined to the secondary
mains. As long as the insulation between the primary
and secondary circuits of the main transformer is perfect,
there is absolute balance between the E.M. Forces of the
secondary windings of the two small auxiliary trans-
formers, and no current passes through the fuse. If, how-
ever, a fault occurs, the balance is disturbed, a current
passes through the fuse and melts it, and the weight fall-
ing between the terminals short-circuits the secondary
mains, and puts them to earth. The primary fuses s s are
thereby caused to blow, thus cutting the faulty transformer
completely out of circuit. It is important to note that
this safety device is a protection, not only against a real
short between primary and secondary, but even against an
incipient fault of insulation between the two circuits.
Sub-stations and house-transformers, The most import-
ant use of transformers is in connection with lighting or
power plants, where the pressure has to be raised or
lowered. The nature of the glow-lamp and the con-
dition of personal safety make it necessary to use a
moderate pressure in the distributing leads (say not ex-
ceeding 100 or 200 volts), whilst a high pressure in the
transmission leads is an economic advantage, and, indeed,
an absolute necessity, if the transmission has to be effected
over a considerable distance. The transformer is then the
intermediary apparatus by which the two conditions, cheap
mains and moderate supply voltage, can be simultaneously
SUB-STATIONS AND HOUSE-TRANSFORMERS. 181
fulfilled. The typical arrangement of transformers for this
purpose is shown in Fig. 86. C denotes omnibus bars in
the central station; Ss the primary transmission mains or
feeders ; T T are transformers, and V V the supply mains.
Measuring instruments, switches, and fuses are of course
also required, but have been omitted from the diagram to
avoid complication.
Fig. 86.
TTT
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re r
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The diagram shows each transformer supplied with
current by its own feeder, whilst on the secondary side
each transformer supplies a net- work of distributing mains,
which latter may be either separate from each other, or
they may be inter-connected, as shown by the dotted line.
The inter-connection of secondary mains has the advant-
age that at times of small demand some of the transformers
may be disconnected from the primary and secondary
182 TRANSFORMERS.
mains, whereby the power wasted by them when working
an open circuit is saved. On the other hand, there is the
danger that a defect in one of the small net-works may
affect all the others, and to minimize this danger it is
advisable to insert fuses into all the important junctions
of the secondary net-work. A system arranged as here
described is called a sub-station system, and is char-
acterized by the use of high-pressure feeders, and a com-
plete net-work of low-pressure distributing mains.
The system of house-transformers, on the other hand, is
characterized by a complete net-work of high-pressure
feeding and distributing mains, supplying current to a
large number of small transformers, each placed as near as
possible to the place where the low-pressure current is
required (i. e. one transformer to each house or group of
houses), so that no net-work of secondary or low-pressure
street mains is required. The weight of copper in the
street mains is thereby much reduced, which is an advant-
age. On the other hand, there are some drawbacks.
Owing to the greater length and the many junctions in
the system of high-pressure mains, the insulation is more
difficult, the high-pressure must be brought into the
houses of the consumers, and the loss of power in the
transformers is greater. Single transformers cannot be
disconnected, thus increasing the light-load loss, and even
at heavy load the loss of power is greater, since small
transformers cannot have as high an efficiency as large
transformers.
We may show the relative merits and faults of the two
systems by an example. Take a district in which 100,000
50-watt lamps are installed. If we use house -transformers
their joint output must be 5000 Kwt. Although all the
lamps installed will never be alight simultaneously, it may
THE SYSTEM OF HOUSE-TRANSFORMERS. 183
and will occasionally happen that all the lamps installed
in one house are simultaneously in use, and the output of
the transformer must therefore be equal to this demand.
This necessitates the employment of a large number of
small transformers (say from 2 to 10 Kwt. output) which
have collectively an output of 5000 Kwt. If sub-stations are
used we may take advantage of the fact, that of all the
lamps installed in a town only a certain percentage is in
use simultaneously. This percentage varies according to
the character of each district, but we may take 60 per cent.
as a very ample allowance. The sub-station transformers
need then only have an output of 5000 X 0'6 = 3000 Kwt.,
and may each be of a fairly large size, since only few of
them are required for this total output. As has already been
shown in Chapter V., the annual loss by ohmic resistance
is very small as compared with the hysteresis loss. We
may, therefore, without committing any great error, neg-
lect the copper loss for the present, and compare the two
systems as regards iron loss only. As a fair average we
may take 3| per cent, iron loss for the small house-trans-
formers, and 2 per cent, for the larger transformers at the
sub-stations. We shall also assume that in neither case
transformers are cut-out of circuit during the times of
light load.
Assuming 600 hours as the average annual burning time
of each lamp, we would then have an annual output of
3000 X 600 = 1'8 X 10 6 Units. If house-transformers are
used, the loss going on day and night would be 5000
X 0*035 = 175 Kwt., making in the course of a whole
year 175 x 8760 = T53 x 10 6 Units. The total work
(neglecting ohmic loss) which must be supplied during
the year is, therefore,
(18 + 1-53) 10 = 3-33 X 10 6 Units,
184 TEANSFOEMEES.
and the annual efficiency is
The efficiency is in reality slightly less than 54 per cent.,
because the loss due to ohmic resistance, which we neg-
lected, must be also covered by the work supplied.
If sub-stations are used the loss is considerably less,
namely, 3000 X 0'02 = 60 Kwt., or in one year 525,600
Units. The work put into the transformers is thus
(1-8 + 0-526) 10 6 = 2-326 x 10 C Units
annually, and the annual efficiency is
The efficiency is also in this case slightly lower because
of the ohmic loss. If we allow for the latter 2 per cent, in
both cases, we have the following comparison between the
two systems
House- Sub-
Transformers. Stations.
Annual efficiency per cent. 52 72
Annual output in units . . . 1,800,000 1,800,000
input ... 3,460,000 2,400,000
loss ... 1,660,000 600,000
If a unit costs l.2d. to produce in the central station,
the money loss chiefly occasioned by hysteresis in the
transformers is 8300 and 3000 respectively. The differ-
ence, namely 5300, shows by how much the working
expenses are increased if we employ house-transformers
instead of sub-stations. On the other hand, there is a
saving in capital outlay with the former system. To put
the comparison on a proper basis, we must now investigate
THE SYSTEM OF HOUSE-TRANSFORMERS. 185
by how much the capital outlay must be lessened in order
to make the system of house-transformers commercially
better than that of sub-stations. This will be the case if
the annual charges for interest, sinking fund, and mainten-
ance for cables and transformers show a difference of more
than 5300 in favour of the house-transformer system.
We may roughly estimate these charges at 10 per cent, of
the capital outlay for the transformers and 15 per cent, for
the cables. The capital outlay for small transformers may
be taken at 3 10s., and for large transformers at 2 15s.
per Kwt. The total cost of transformers is thus for the
system
Of house-transformers 17,500, or 1750 annually.
Of sub-stations 8250, or 825 annually.
The difference of 925 being in favour of the sub-station
system has to be added to the 5300 saved in power, so
that the saving in the annual charges for the cables must
exceed 6225 before house-transformers are commercially
preferable to sub-stations. The difference in the capital
outlay for the cables in the two systems must (with 15 per
cent, annual charge) come to or exceed
= 41,600
or a little more than 8s. per lamp installed. If then
estimates of the two systems show that with house-trans-
formers we save in cables alone more than 8s. per lamp as
compared with sub-stations, then the former system should
be adopted. This consideration is, of course, only valid for
the particular percentage charges on which the calculation
has been based. If, instead of allowing 15 per cent, on
cables, we had allowed a smaller charge, we would have
18G TRANSFORMERS.
obtained more than 8s. per lamp as the critical difference
in the capital outlay for cables. The same would have
been the case if we had estimated the works' cost of the
unit at more than l.Zd. It will also be obvious that the
cost of cables in both systems, and therefore the difference
in the cost of cables per lamp installed, must be greater for
smaller stations, especially if the lamps are distributed over
a wide area. Bearing in mind these various points, we
come to the following general principles.
The system of house-transformers is commercially
advantageous under the following conditions
Power cheap.
Lamps scattered over wide area.
Cables dear in first cost and upkeep.
The system of sub-stations is commercially advantageous
under the following conditions
Power dear.
Station large.
Lamps densely installed.
Cables cheap in first cost and upkeep.
Boosters. If some of the feeders between the central
station and the sub-stations are very long, it is sometimes
advantageous to allow a greater voltage drop in them
than in the shorter feeders, and to raise the pressure at
the home end of these long feeders by an amount corre-
sponding to the extra drop. For this purpose special
auxiliary transformers, so-called " boosters," may be used.
This system of boosting-up the pressure at the home end
of long feeders has been invented simultaneously and
independently by Mr. Stillwell in America, and by the
author in England. It is shown diagram matically in
Fig. 87,
BOOSTERS. 187
G are the bus bars in the station, 8 is a feeder sup-
plying current to the transformer T at a sub-station. V
are the distributing mains connected to this transformer.
The boosting transformer B has its primary permanently
connected to the bus bars, whilst its secondary is put in
series with the feeder and is subdivided into sections, so
that by using a switch s, a greater or lesser number of
secondary turns can be inserted. In this manner the
additional voltage put into the feeder at the home end
may be varied from zero to the full voltage given by all
the secondary turns of the booster. The full voltage is
added when the feeder carries its maximum load ; the
switch is then placed on its highest contact. As the load
Fig. 87.
decreases the switch is shifted to a lower contact, the
intention being to boost up by the amount corresponding
to the drop in pressure due to ohmic resistance. Since
this drop is proportional to the current, the adjustment of
the switch may be made in accordance with the readings
of an amperemeter in the feeder circuit, or pilot wires may
be brought back from the sub-station and connected to a
voltmeter. The switch is then adjusted so as to keep the
pressure indicated by the pilot voltmeter, constant. It
is obvious that in either case the switch-lever can be
worked automatically by a small electro-motor controlled
by a relais. Since, in passing from one contact to the
other, the switch-lever, if it were made in one solid piece,
188
TRANSFORMERS.
would short-circuit, and possibly burn out the section of
the secondary winding connected to the two corresponding
contacts, it is necessary to employ a lever consisting of two
parts, each smaller than the width of the gap between two
contacts, and having an insulating partition between them.
The two parts must of course be joined by a suitable resist-
ance, or preferably by a choking coil. With such a con-
struction there can occur neither a short-circuit in the
booster nor an interruption of the feeder current.
The necessity to send the whole feeder current through
the switch, and the drawback of a complete interruption
Fig. 88.
of the feeder current if this switch gets out of order, has
led the author to design the modified arrangement of
booster in which the switch is connected, not with the
secondary but with the primary circuit of the auxiliary
transformer. This arrangement is shown in Fig. 88. The
feeder circuit is permanently connected with the bus bars
through the secondary winding of the auxiliary trans-
former, whilst the multiple contact switch is inserted into
its primary connection with the bus bars. The primary
winding is subdivided into groups a, 1), c, etc. According to
the position of the switch-lever, more or less of these groups
are active, thus causing the magnetic flux and the E.M.F,
BOOSTERS,
189
in the secondary to be smaller or greater respectively.
The first group a must of course contain a sufficient
number of convolutions to prevent the auxiliary transformer
from being magnetically overloaded. This kind of booster
must therefore be larger than that shown in Fig. 87, but
as in any case the cost of a booster is very small as com-
pared with the saving in the cost of the feeder thereby
rendered possible, the extra outlay is insignificant, whilst
the possibility of keeping up the supply, even if the switch
should become deranged, is a distinct advantage.
Fig. 89.
In a third type of boosting apparatus there is no switch
of any kind, either in the secondary or primary circuit.
This type is shown in Fig. 89. The construction resem-
bles that of a two-pole dynamo with shuttle-wound
armature. The field is built up of sheet-iron plates,
and is provided with the primary winding P P, whilst
the armature carries the secondary winding S placed over
a core of sheet-iron discs in the usual manner. Both
windings are permanently connected, the primary with
the bus bars, and the secondary with bus bars and feeder
as in Fig. 88. By means of worm gearing, the coil S may
190
be placed at various angles with reference to tlie polar
surfaces. If the coil S is turned into a vertical position,
the flux of force passing through it is a maximum, and the
E.M.F. generated in this coil is a maximum. If the coil
be placed horizontally it is ineffective, whilst in inter-
mediate positions any desired boosting effect may be
obtained. By turning the coil beyond its horizontal
position the action may also be reversed, that is to say,
we can reduce the E.M.F. at the home end of the feeder.
The advantages of this type of booster are that no switches
of any kind are used and that the adjustment of the boosting
effect is made, not by definite steps, but as gradually as we
please, by means of the worm gear.
Connection in series. Transformers may be advantage-
ously used if it be required to work a number of lamps
in series off a circuit in which an alternating current of
constant strength is maintained. If we were to insert the
lamps themselves into such a circuit, the insulation of the
lamps to earth would have to be so perfect as to withstand
the full potential difference of the alternating current, a
condition not always easily fulfilled. If, however, we feed
the lamps from the secondaries of series-transformers, it is
only necessary to provide perfect insulation for the trans-
formers, which presents no difficulty ; the insulation of the
lamps need only be good enough for the voltage required
by each lamp. The arrangement is shown in Fig. 90. T T
are series-transformers supplied from a constant current
alternator, and L L are the lamps. The primary return
circuit is not shown. Since the current in the primary is
constant, the current in the secondary is also approximately
constant as long as the lamp is in circuit. There is, how-
ever, the drawback that if a carbon should fall out of a
lamp, or some other accident happen whereby the second-
CONNECTION IN SERIES.
191
ary current is interrupted, the induction in the core and
the E.M.F. in the secondary of that particular transformer
(if this is of the ordinary construction for parallel work)
would rise very considerably. Since the primary current
must, on account of the other lamps, be kept constant, the
pressure at the generator has, in such a case, to be in-
creased. The transformer with 'open secondary becomes
magnetically overloaded and must eventually burn out. To
avoid this danger we must make provision to give the
secondary current an alternative path in case the lamp
circuit should become interrupted. This may be done in
two ways. We may employ a kind of automatic " cut-in "
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as in a, or a choking coil as in b. The cut-in consists of
two electrodes separated by a thin sheet of mica or paraf-
fined paper, which, under normal conditions, is sufficient
to withstand the secondary voltage. If now the secondary
voltage rises considerably, in consequence of the lamp cir-
cuit being opened, the insulation between the electrodes
breaks -down, and the cut-in short-circuits the secondary
coil of the transformer. The choking coil, which may be
used instead of a cut-in, is a kind of small transformer with
only one winding on its core. The current passing through
this winding is proportional to the lamp voltage and lags
by nearly 90 behind it. The power lost in the choking
192 TRANSFORMERS.
coil is the sum of hysteresis and ohmic loss; it is of course
not equal to the product of current and voltage, but much
smaller. By a proper design of choking coil it is thus
possible to minimize the loss of power, although current
is apparently lost. This may best be seen by an example.
Let us assume that the lamp requires 20 amperes at 35
volts, and that its power factor is 80 per cent. The power
actually supplied to the lamp is therefore 560 watt. Let
the choking coil be so constructed that it also takes 20
ampere if the pressure is 35 volts, and that the loss of
power in it is 20 watt. Its power factor is therefore
20 : (35 x 20) = 0'0285. If we now draw a vector
diagram to represent these working conditions, we find
that the total secondary current supplied by the trans-
former is 36 ampere. If now the lamp current is inter-
rupted the choking coil must pass the whole 36 ampere,
and the voltage must rise to
Oft
35 x ~ = 63 volts.
This is an excess of 80 per cent, over the normal voltage,
and is accompanied by a similar rise in the magnetisation
of the iron core. It is of course always possible to so
design the choking coil that it can stand this increase of
magnetic load without danger for any length of time.
Sometimes it is convenient to use a transformer . for
feeding a circuit of lamps in series, which requires a nearly
constant current, although the number of lamps inserted
may be varied. This condition is of course fulfilled if the
primary current is constant, but if the primary voltage is
constant a transformer for parallel work (that is, a trans-
former of the usual construction having as little magnetic
leakage as possible) would be quite unsuitable. Such a
CONNECTION IN 8SK1B8.
193
transformer keeps the secondary voltage approximately
constant, but not the secondary current. When we have
lamps in series it is the current which must be kept con-
stant, whilst the voltage must vary as nearly as possible in
accordance with the number of lamps alight at any time.
This condition can be met at least approximately by
shaping the transformer in such way as to produce a
large magnetic leakage. A construction of this kind
is shown in Fig. 91. It is a core transformer with
primary and secondary coils on separate limbs and with
expansions a, 1) of the two yokes arranged specially to
produce magnetic leakage. The primary coil is joined
Fig. 91.
to the primary constant pressure lead s; and the secondary
coil to the lamp circuit L.
It will be obvious that with an open secondary or lamp
circuit the leakage field between a and b will be very
small, since the core of the secondary coil offers a ready
path for the magnetic flux. If, however, the lamp circuit
be closed, a current flows in the secondary coil, pushing
back part of the flux produced by the primary coil, and
the leakage field, not only between a and I but all over
the transformer, will be much increased. The larger the
secondary current the more lines are pushed back, and the
lower will be the secondary E.M.F. If a lamp is short-
circuited the current will at first increase. This increase
produces more magnetic leakage, and lessens thus the flux
194
TRANSFORMERS.
which produces E.M.F. in the secondary. The increase
in current strength will therefore be considerably smaller
than would obtain with an ordinary transformer, and in this
way it is possible to keep the current at least approxi-
mately constant when lamps are put out of action by being
short-circuited.
The limits within which this kind of regulation is
practically applicable can be seen from the vector diagram
Fig. 92. Let OA be the current and OJE 2 the secondary
voltage at full load (all lamps alight). In the diagram the
load is supposed to be without reactance. If E^E l is the
Fig. 92.
vector due to self-induction and resistance'(the latter being
assumed negligibly small) then OE l is the vector of the
primary voltage. The length of the line E 2 E l is of course
proportional to the current. If now we short-circuit half
the number of lamps, then only half the secondary voltage
is required, and J 2 in the diagram moves to H.J, and E l
moves to E{. The secondary current is now represented
by the length of the line E^E^ y that is to say, it is slightly
larger than before. If we short-circuit all the lamps and
the mains themselves, no E.M.F. is required, and E 2 moves
to 0, whilst E l moves to E". The current is now given
CHOKING COILS. 195
by the line OE". The ratio between the length of the
line E Z E 1 and OE' shows the proportion of increase in
current strength if the load is reduced from the full
number of lamps to zero, and it is obvious that we can
make this ratio as near unity as we please, if we only
provide sufficient magnetic leakage, making the angle in
the diagram sufficiently large. The transformer must,
however, be made correspondingly larger, and will be more
costly, whilst its efficiency will necessarily be somewhat
smaller than that of a transformer designed for a minimum
of magnetic leakage. It is also important to note that
with arc-lamps, which have considerable self-induction,
this kind of regulation for constant current is only possible
Fig. 93.
within narrow limits, since the line OJE 2 is inclined to the
left, and the difference in the length of E 2i E l and OE" is
much greater than in Fig. 92.
Choking coils. If lamps are worked in parallel from a
constant pressure circuit, the pressure at the lamp termi-
nals may conveniently be reduced by choking coils. This
is preferable to the use of resistances, since very much less
power is thereby wasted. Let, in Fig. 93, V be a pair
of constant pressure mains and L a lamp which requires
a reduced pressure. This may be obtained by inserting a
choking coil D as shown. Fig. 94 is the corresponding
clock diagram. 01 is the lamp current, OE W the watt
component of the pressure, and OE^ the total pressure at
196 TRANSFORMERS.
the lamp terminals ; the phase difference being given by
the angle <. For a glow-lamp < would be zero and E
would coincide with E w . With an arc-lamp there is a
lag between current and E.M.F., making OE^ > OE W . The
watt component of the E.M.F. in the choking coil is given
by the vector E^E Zt and the wattless component by E 2 E.
The vector E^E is therefore the pressure which exists
between the terminals of the choking coil and OE is the
vector of the supply E.M.F.
Compensating coils. The arrangement sketched in
Fig. 93 may be used if an arc -lamp has to be connected
to mains having a greater pressure than the lamp requires.
Fig. 95.
V 4
Alternating-current arc-lamps require from 30 to 35 volts
at their terminals. It is thus possible to work three
lamps in series from 110-volt mains. If only one lamp
is required the excess of pressure must be taken up by
a choking coil. Let us now assume that we have a pair
of 65-volt mains, then two lamps may be worked in series.
It is, however, not necessary to use both lamps together,
for by inserting compensating coils, as shown in Fig. 95,
we can render the lamps independent of each other.
D l D 2 are the two equal windings of a transformer placed
over the same core. In the diagram these windings are
shown side by side for greater clearness. The coils are
connected in o, and the direction of the winding is such
COMPENSATING COILS. 197
that a current flowing in D l from left to right produces
an E.M.F. in D 2 from right to left, and vice versa. Imagine
now one of the lamps, say L lt cut-out of circuit, then the
current will flow through D l to 0, and finds there two
paths; the one through D 2 and n, and the other through
in L 2 n to the other main. It is obvious that the first
path is impassable to the current, since in D 2 acts an E.M.F.
which is always opposed to the direction of the current.
This E.M.F. produces in fact itself a current, which added
to the current coming from D passes also through the
lamp. The two coils with their core act as a transformer
with the transforming ratio 1 : 1, D l being the primary
and Z> 2 the secondary winding. Let us assume, for the
sake of simplicity, that the efficiency of this transformer
is 100 per cent. If the lamps require each 12 ampere
then 6 ampere will pass through Z\ and 6 ampere will
pass in opposite direction through D 2 , the two currents
joining at and passing jointly through the lamp, which
thus receives a current of 12 ampere. At n the current
splits again, 6 ampere going to the other main V and
6 ampere going to D. 2 . Since the efficiency must be less
than 100 per cent., D 2 will contribute a little less and the
mains must contribute a little more than half the current
required by the lamp. The transformer has three
terminals, p, o, n, of which o is common to both coils and
is connected to the point m between the two lamps.
This principle of compensating coils may also be ex-
tended to more than two lamps. In Fig. 96 are shown
3 lamps and 3 coils. Let the lamp current be again
12 ampere and let the two lamps L 2) Z 3 be switched out.
Then D% and D 3 will- carry a current of a little over 4
ampere, which will produce in D l a current of a little under
8 ampere, so that Z x is fed by the sum of these currents
198
TRANSFORMERS.
and receives 12 ampere. These compensating transformers
are often used in house installations, because they combine
the advantage of the series arrangement of arc-lamps with
the further advantage that the lamps are independent of
each other. The output of these transformers is more-
over less than that of a corresponding number of single
Fig. 96.
transformers, as will be seen from the following. Let e
be the supply voltage, and P the power required by eacli
lamp when the current is i. Then we require, for the
arrangement shown in Fig. 95, an apparatus having an
output of
ex|=JWatt.
We assume hereby, for the sake of simplicity, cos = 1.
If we feed the two lamps by separate transformers, each
would have to be designed for an output of
The combined transformer for two lamps in series contains
no more material and costs no more than a single trans-
former for one lamp.
If we use three lamps in series, the combined trans-
former must be designed for the pressure e and the current
2
-i. The amount of material required for its construction
THREE-WIRE SYSTEM. 199
9
is, therefore, proportional to an output of ~ P Watt,
o
whereas the joint output of three single transformers
would be 3 x ^ X e = P Watt. We see thus that also
o
in this case there is some advantage in using a combined
transformer.
Three-wire system. If we connect the middle o of the
secondary winding (Fig. 97) with a third main, we obtain
a distributing system analogous to the so-called three-
wire system used in connection with continuous-current
stations. The primary coil pq receives high-pressure
current from the feeder s, whilst the secondary coil m n is
Fig. 97.
s ~jv rrv
35
if *ss
$r /i,
connected to the outer wires of the system in the usual
way. The lamps a, ~b are connected between these outer
wires and the zero wire o. The pressure between m and
n is double the lamp voltage, and we are thus able, exactly
as in the ordinary three-wire system, to effect considerable
economies in the cost of the distributing mains.
Balancing transformer. It may happen that the sub-
station must be placed at some distance from the district
to be lighted. In this case the middle wire need not be
brought back to the sub-station transformer T, Fig. 98, if
a balancing transformer 2\ is established in some point of
the district to be lighted. The output of the balancing
transform er need not be larger than half the maximum differ-
200
TRANSFORMERS,
ence between the loads on the two sides a, I of the system.
Let i a be the maximum current in a and i b the current
which simultaneously obtains in I, then one coil of the
balancing transformer must take up the current -^
and its other coil must give off an equal current. (Compare
also Fig. 95.) If the lamp voltage is e, then the output of
the balancing transformer is given by the expression
/fr so that the magnetic leakage
is exceedingly small. Fig. 129 shows a 10-Kwt. trans-
former, and two of its coils. A test made with this
transformer, to determine the drop according to the
author's method, as explained in Chapter VI., gave the
following results. The ohmic drop is 2 per cent., whilst
on short circuit 4 per cent, of the primary voltage is
required to produce the full secondary current. The
drop for full load determined from these data is for
= 2'3 per cent.
< = 60 4-0
4> = 90 3-9
Fig. 130 shows a 40-Kwt. three-phase transformer, of
the type used in the lighting plant at Strassburg. As it
was in this case very important to reduce the drop as
much as possible, the windings are not placed within each
other, but are arranged in small sections side by side.
The voltage on short circuit is in this design only 3 per
cent, of the full voltage, so that the drop does not exceed
3 per cent, even on a motor load. The wire for all the
sections is wound on formers of micanite. Both types
of transformers are protected by perforated side-plates,
not shown in the illustrations.
The Westinghouse Electric Manufacturing Co,, of Pitts-
burg, are building shell transformers, with short cores and
coils of special shape to facilitate cooling. Fig. 131 shows
the type used for lighting. This is built up to 30 Kwt.
The coils are wound in sections, and where they protrude
234
from the iron part they are bent out in fan-shape, so as to
give a greater cooling surface. The iron part is held
together by strong cast-iron frames and bolts, and the
Fig. 131.
coils are protected by a perforated casing of cast-iron.
This is partly shown in the illustration. Figs. 132 and
133 show the 100-Kwt. transformers used in the Niagara
installation. The transforming ratio is 2000 to 150 volts,
ILLUSTRATIONS Off TRANSFORMERS. 235
but provision is made by means of a second pair of
primary terminals to alter this ratio slightly in certain
cases. The primary and secondary winding consists each
Fig. 132.
of four sections, the secondary sections being connected
in parallel. In the middle of the core are two primary
sections, on either side are two secondary sections, and
finally a primary section on each outside. The exposed
236
parts of the sections are bent out fan-shaped to increase
the cooling surface. The cooling agent is, however, not
air, but oil, with which the case of the transformer is filled.
Fig. 133.
The connecting cables pass through insulated stuffing-
boxes as shown in Fig. 133. To cool the oil a wrought-
iron spiral tube is fitted to the inner surface of the case,
and a stream of cold water is sent through this tube.
INDEX
ACTION, principles of, 1
Allgemeine Elektricitatsgesellschaft
transformers, 231
Alternate and continuous currents
compared, 143
Alternating and pulsatory currents, 3
Alternating current, power of, 63
Alternators, coupling up of, 74
Apparatus, Dobrowolsky's, 169 ;
Ewing's, 172 ; Kapp's, 171
Appliances for transformers, safety,
175
Arc-lamps in series, 197
Arrangement of coils, 8 ; of trans-
formers, 181
Arrangements for cooling, 58
Avoidance of self-induction, 145
Balancing transformer, 199
Ballistic tests, 174
Bars, bus, 187
Best distribution of copper, 85 ; of
losses, 92
Boosters, 186
Breaking joints, 45
Brown, Boveri & Co.'s transformers,
215
Brush Electrical Engineering Co.'s
transformers, 219
Burning out, 191
Bus bars, 187
But-joints, formula for, 80 ; influence
of, 80
Calculation for correct proportions,
48 ; for voltage, 12
Calico, paraffined, for insulation, 99
Capacity, eifects of, 126
Cai dew's safety device, 178
Choking coils, 191, 195
Circuit, working on open, 104
Clearance, 85
Clock diagram, examples of use of,
103, 104, 105
Coils, arrangement of, 8 ; choking,
191, 195 ; compensating, 196 ; in-
fluence of shape of, 30 ; primary
and secondary, 9 ; resistance of,
102
Combinations of currents and pres-
sures, 71
Comparative losses, working and
idle, 96, 97
Comparative size tables, 60, 62
Comparison between alternate and
continuous currents, 143 ; core and
shell, 32, 33 ; of costs, 91, 92 ;
of magnetic flux to water flow, 4 ;
of measuring instruments, 142 ; of
systems, 182
Compensating coils, 196
Condenser, eifects of, 131
Connection in series, 190
Continuous and alternate currents
compared, 143
Constantan resistance, 145
Construction of a transformer, 83, 98 ;
of the iron part, 41
Constructive details, 98
Cooling arrangements, 58 ; spiral
water- tube for, 236 ; surface, 57
Copper, best distribution of, 85 ;
economy of, 83 ; wire covering,
238
INDEX.
Core and shell transformers, 32 ;
formula for shape of, 30 ; influence
of shape of, 30 ; taping, 84 ; thick-
ness of, 83
Cores, 99 ; advantage of continuous,
41 ; hysteresis in, 88 ; joints in,
41 ; weights of, 88
Correct proportions, calculations for,
48
Costs, comparison of, 91, 92
Coupling up alternators, 74
Covering copper wire, 99
Crompton's transformer, 43
Current, determination of the no
load, 75 ; effective power of, 69
"Current heat," 19
Current, heat of, 50 ; heat of eddy,
50 ; instantaneous power of, 68 ;
lagging, 64 ; leading, 64 ; measure-
ment of, 63, 64, 142
Currents, comparison between alter-
nating and continuous, 143 ; losses
by eddy, 19
Currents of irregular form, measure-
ment of power carried by, 149 ;
parallelograms of, 73
Currents and pressures, combinations
of, 71
Curve, formula for E.M.F., 24 ;
graphic, of heat, 51 ; E.M.F., in-
fluence on hysteresis loss, 23
Cut-in, 191
Designs, 39
Determination of "drop," 119; of
efficiency, 163 ; of lag, 149 ; of the
no load current, 75
Determining efficiency, indirect
method of, 166
Device, Cardew's safety, 178 ; earth-
ing, 176 ; Ferranti's safety, 179 ;
Thomson-Houston safety, 177
Diagram, clock, 103 ; example of use
of, 104, 105 ; (working) of trans-
former having leakage, 113
Dimensions, influence of linear, 53
Dissipation of heat by ribs, 102
Distribution (best) of copper, 85 ;
of losses, 92
Dividing sheet, metallic,- 176
Dobrowolsky's apparatus, 169
" Drop," graphic determination of,
119; of E.M.F., 117; of trans-
former, 111; testing, 167; to
remedy, 112 ; voltage, 117
Dynamometer, 141 ; formulae, 143
Earthing device, 176
Economy in working, 95 ; of copper,
83
Eddy currents, heat of, 50 ; losses by,
19
Effective current, power of, 69
Effective E.M.F., 13, 14, 17
Effects of capacity, 126 ; condenser,
131 ; induction, 121
Efficiency, determination of, 163 ;
indirect method of, 166
Electric Construction Co.'s trans-
formers, 226
Electric stress, 59
E.M.F. curve, formula for, 24; its
influence on hysteresis loss, 23 ;
" drop," 117 ; effective, 13, 14, 17 ;
impressed, 65 ; lag of, 64 ; lead of,
64 ; opposing, 65 ; phases of, 65 ;
set up, 11
Equation, fundamental, 10
Ewing's apparatus, 172
Examples of sizes, 60, 61 ; use of
clock diagram, 104, 105
Ferranti's safety device, 179 ; trans-
former, 43
Field, magnetic, 1
Flux, magnetic, 1 ; compared to
water flow, 4
Formula for but-joints, 80; E.M.F.
curve, 24 ; no load, 77 ; shape of
core, 30 ; thickness of plates, 22 ;
winding, 86, 87
Formulae, dynamometer, 143
Frequency, 83
Fundamental equation, 10
Ganz & Co.'s transformers, 222
Graphic curve of heat, 51 ; deter-
mination of drop, 119
Heat, current, 19 ; dissipation of, by
ribs, 102 ; graphic curve of, 51 ; of
current, 50 ; of eddy currents, 50 ;
INDEX.
239
of hysteresis, 50 ; readings, power
measurement by, 156 ; testing,
167
Heating of transformers, 49
Hedgehog transformer, 36
House-transformers and sub-stations,
180 ; the system of, 182
Hysteresis, heat of, 50 ; in cores, 88 ;
losses by, 19 ; losses by influence
of E.M.F. curve on, 23
Illustrations of transformers, 204
Impressed E.M.F. , 65
Indirect method of determining effici-
ency, 166
Induction, avoidance of self-, 145 ;
effects of, 121 ; testing, 168
Influence of hut-joints, 80; of E.M.F.
curve on hysteresis losses, 23 ;
of shape of core and coils, 30 ;
of the linear dimensions, 53
Installation of transformers at Nia-
gara, 234
Instantaneous power of current, 68 ;
value, 13, 66
Instruments, comparison of measur-
ing, 142
Insulation, mode of, 46 ; (perfect)
necessary, 175 ; paper, 99 ; par-
affined calico, 99 ; testing, 167 ;
varnish, 99
Iron part, construction of, 41 ; pro-
portions of, 47
Iron ring transformer, 2
Iron sheet, testing, 167
Johnson & Phillips' transformers, 222
Joints, breaking, 45 ; (but) formula
for, 80 ; (but) influence of, 80 ; in
cores, 41
Kapp's apparatus, 171 ; transformer,
42
Lag, determination of, 149 ; of
E.M.F., 64
Lagging current, 64
Lahmeyer & Co.'s transformers, 215
Lamps (arc), in series, 197 ; as re-
sistances, 145 ; (gloAv) in series,
192
Law of transformers, 10
Leading current, 64; of E.M.F., 64
Leakage, loss by, 3 ; magnetic, 3, 6,
7, 108, 139 ; working "diagram of
transformer having, 113
Linear dimensions, influence of, 53
Link, magnetic, 9
Liquid resistance, 131
Load, formula for no, 77
Load, working under, 107
Loss by leakage, 3
Losses, best distribution of, 92 ; by
eddy currents, 19 ; by hysteresis, 19
Losses comparative (working and idle),
96, 97 ; in transformers, 19 ; reduc-
tion of, 21 ; table of, 94
Low pressure winding, 55
Magnetic field, 1 ; flux, 1 ; flux com-
pared to water flow, 4 ; leakage, 3,
6, 7, 108, 139 ; link, 9 ; stress, 59
Maschinen-fabrik Oerlikon trans-
formers, 233
Measurement of current, 63, 64, 142 ;
of the power carried by currents
of irregular form, 149
Measurement, power, 150 ; by heat
readings, 156
Measuring instruments, comparison
of, 142 ; power, other methods of,
156 ; wattmeter for, 145
Metallic dividing sheet, 176
Method (indirect) of determining
efficiency, 166 ; three-amperemeter,
160 ; three-voltmeter, 157
Mode of insulation, 46
Mordey transformer, 44
Multiphase transformers, 204, 215
Necessity for perfect insulation, 175
Niagara installation transformers, 234
No load current, determination of
the, 75 ; formula for, 77
Ohmic resistance, 19
Oil, use of, 49
Paper insulation, 99
Paraffined calico insulation, 99
Parallelograms of currents, 73
Perfect insulation necessary, 175
240
INDEX.
Periodic variations, 65
Phases of E.M.F., 65
Plates, formula for thickness of, 22 ;
use of thin, 21
Platinoid resistance, 145
Polyphase work, transformer for, 40
Power carried by current of irregular j
form, measurement of, 149
Power of current (effective), 69 ; (in-
stantaneous), 68
Power measurement, 150 ; by heat
readings, 156
Power of alternating current, 63
Power, other methods of measuring,
156
Power, wattmeter for measuring, 145
Powers, table of true, 156
Pressures and currents, combinations
of, 71
Primary and secondary coils, 9 ;
winding, 84
Principles of action, 1
Proportions, calculations for correct,
48 ; of iron part, 47
Pulsatory and alternating currents, 3
Rate (time) of change, 1, 2
Reactance, 138
Reduction of losses, 21
Regulation, 194
Relation of size to efficiency, 53
Remedy (to) "drop," 112
Resistance, constantan, 145 ; liquid,
131 ; of coil, 102 ; ohmic, 19 ;
platinoid, 145
Resistances, lamps as, 145
Results of tests, 52
Ring (iron) transformer, 2
Rise in temperature, 49
Safety appliances for transformers,
175 ; device, Cardew's, 178 ; Fer-
ranti's, 179 ; Thomson-Houston's,
177
Schukert & Co.'s transformers, 210
Schwartykopff's transformers, 210
Scott's system, 201
Secondary and primary coils, 9
Series, arc-lamps in, 197 ; connection,
190 ; glow-lamps in, 192
Shape of core and coils, influence of,
30 ; formula for, 30
Sheet-iron, testing, 167 ; metallic
dividing, 176
Shell and core transformers, com-
parison between, 32, 33
Short-circuited transformer, 115
Siemens Bros. & Co.'s transformers,
214
Siemens & Halske's transformers, 204
Simple-phase transformer, 204, 210,
215, 223
Simple transformer, winding, 2
Sizes, comparative table of, 60, 62 ;.
examples of, 60, 61
Size, relation to efficiency, 53
Space for winding, 83
Spiral water tube for cooling, 236
Square wire, 99
Stress, electric, 59
Sub-stations and house -transformers,
180
Surface, cooling, 57
Systems compared, 182
System of house-transformers, 182 ;
Scott's, 201 ; three-wire, 199
Table of losses, 94 ; of tests, 53 ; of
true powers, 156
Taping core, 84
Temperature, rise in, 49
Testing "drop," 167; heat, 167;
induction, 168 ; insulation, 167 ;
sheet-iron, 167 ; transformer, 163
Tests, ballistic, 174 ; results of, 52 ;
tables of, 53
The dynamometer, 141 ; wattmeter,
145
Theory of wattmeter, 152
Thickness of core, 83
Transformer, balancing, 199 ; con-
struction of a, 83 ; Crompton's, 43 ;
"drop" of, 111; Ferranti's, 43;
for polyphase work, 40 ; heating
of, 49 ; hedgehog, 36 ; iron ring, 2 ;
Kapp's, 43 ; Mordey's, 44 ; short-
circuited, 115 ; Westinghouse, 42
Transformer, winding simple, 2
Transformers, Allgemeine Elektrici-
tatsgesellschaft's, 231 ; arrangement
of, 181 ; Brown, Boveri & Jpo.'s,
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APR, 17
FEB <^ 1934
FEB 26 1934
JUL 25 1936
FEB 271936
DEC 1 1937^
SEP 11
1939
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AUG 81 1942 /
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(D3279slO)476B
General Library
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UNIVERSITY OF CALIFORNIA LIBRARY