/
The self-induced E.M.F. is a sine function, and its
negative crest value occurs when wt = o, or an even
multiple of TT, the positive crest values when >/ is an
odd multiple of TT.
Since the equation of the current is
i=l sin >/
we see that for ^ = o, when the current is passing
84 TRANSFORMERS
through zero to become positive, the E.M.F. has its
greatest negative value, whilst for i I when a>t =
- the E.M.F. is passing through zero to become positive.
The phase of this E.M.F. lags therefore a quarter
period behind the current, and since it must be com-
pensated by an equal and opposite E.M.F. impressed
on the coil, this must lead a quarter period in advance of
the current. We write, therefore
E = o,LI
where E and I are crest values, or if we use effective
values we have
e = t*>Lz (17)
e being the E.M.F. which must be impressed on the
circuit in order to balance and overcome the self-induced
E.M.F. Since L is a length and to an angular velocity,
ft) L is a linear velocity and has therefore the same
dimensions as an ohmic resistance. It is called reactance.
If an alternating E.M.F. be applied to a condenser, a
charging current will flow. Let C be the capacity in
farads, then for E volt continuous pressure applied
to the terminals, the charge in coulomb or ampere-
seconds is
Q = CE
A change de in applied pressure taking place in the
time dt produces a change dQ in the charge, and since
dQ = idt
de
we have Cde = idt and i = C T
dt
Since e is a harmonic function with crest value E so that
e = E sin cot and de = >E cos totdt we find
z = Co)E cos wt
e = E sin wt
Since e is proportional to the sine and i to the cosine
of the same angle, it follows that the vectors of
these quantities are in quadrature, the current vector
leading. The charging current attains its positive crest
SELF-INDUCTION AND CAPACITY
value for wt = o, or an even multiple of 2?r, and is given
by-
if the capacity C is counted in microfarad and the
E.M.F. in volt. The effective value of the condenser
current is
* = o,C*io- (18)
If a circuit contains inductance L and resistance R, the
E.M.F. to be impressed in order to force the current
i through it must have two components : one Rz in
phase with the current and the other o>Le in advance of
the current by 90. The vectorial sum of these two is,
L
WUWUUlrj
[ooooooooooo_pooooo(T 1
" 4
SB,
uuuu
<
5
FIG. 45. Circuit containing resistance, inductance and capacity.
therefore, the hypotenuse of a right-angle triangle with
>Lz and R^ as cathets, and the angle of lag L) 2 is called the impedance of the
circuit. If two or more such circuits are fed from the
86 TRANSFORMERS
same source of E.M.F. we can thus determine the
position and magnitude of the current vector for each
separately, and combine them, as already shown, to get
the position and magnitude of the resultant current
vector. Alternating current problems are best solved
graphically. As an example we may take the circuit
shown on the left side of Fig. 45. On the right side is
the vector diagram. L and R being in series, we first
determine tg
= 27rv we find the natural frequency of the
crcut
1 60
C being given in Microfarad and L in Henry.
Influence of higher harmonics. Up to the present
we have assumed that current and E.M.F. follow a
simple sine law. If, however, their curves contain
upper harmonics, the E.M.F. of self-induction, as well
as the charging current, will be somewhat altered.
Since in one case e=\^, and in the other ^'=C^ the
investigations can be carried on in the same way provided
we put for i or e the expression
a = AX sin (o>/) + A 3 sin (3oi/) + A 5 sin (5>^)+ . . .
and determine the square root of mean squares. We then
find that besides the square of these terms, products of
two of them have to be integrated. The integral of
these products taken between the limits of W = o and
o>/ = 2?r is zero throughout, so that only the integrals of
the squares of the single terms remain, and we thus find
Ij being the crest value of the first harmonic, I 3 that of
the third, and so on.
In the same way we find
88
TRANSFORMERS
Ei, E 8 , E 5 , etc., being the crest values of the different
harmonics.
Power of an alternating current. In order to
investigate the working condition of a transformer we
must be able to determine the power given to the
primary and taken from the secondary terminals. It is
therefore necessary that we should be able to find, either
by direct measurement or in some other way, the power
conveyed by an alternating current. We assume for
the present that current and E.M.F. follow a sine law.
This assumption is made for the sake of simplicity. It
is not always correct, but we shall see later on that the
methods of measuring power which are based on this
assumption are also
applicable in the
general case where
the current as well
as the E.M.F. fol-
low any law, pro-
vided the frequency
of both is the same.
Let, in Fig. 46,
the sine line I re-
present the current
as a function of the
time, and the line E the E.M.F. impressed on any
two points of the circuit, say, for instance, the primary
terminals of a transformer. We count the time in the
direction to the right. At the time o the current is
negative (the ordinate of the current curve I being
below the axis), and the E.M.F. is zero, At time t^
the current is zero and the E.M.F. has a certain positive
value. The maximum E.M.F. occurs at time 4 and
the maximum current a little later at time 4- At
time / 4 the E.M.F. has decreased to zero, but the
current is still positive, though rapidly decreasing. It
reaches zero at time / 5 , when the E.M.F. has already
a negative value. Since both curves follow the same
law the horizontal distances between their maximum and
zero values must all be the same, that is to say, the time
interval between any two pairs of corresponding points is
FIG. 46. Curves of E.M.F. and current.
POWER OF AN ALTERNATING CURRENT 89
a constant. Thus 4 / 2 = 4 1 = t / 6 , etc. This time
difference between corresponding values of the E.M.F.
and current is called the lag or lead of current or
E.M.F. respectively. In our example, where the E.M.F.
passes its zero and maximum values before the current
passes through the corresponding values, we have a
lagging current as compared to the E.M.F., or a lead-
ing E.M.F. as compared to the current. The condition
under which this relation obtains is the existence, in
addition to the impressed E.M.F., of a second E.M.F.
which tends to oppose any and every change of current.
As already explained in this chapter, this is our E.M.F.
of self-induction, and is produced by the change in the
magnetic flux due to the current. If, however, instead
of this opposing E.M.F., there acts an E.M.F. in the
inverse sense, then every change in current strength is
thereby promoted, and the current attains its zero and
maximum values sooner than the impressed E.M.F., or
in other words, we have a current leading before the
impressed E.M.F. Such a second E.M.F. tending to
advance the current is produced by the insertion of a
condenser into the circuit. The condenser takes the
maximum positive charging current at the moment that
the impressed E.M.F. on its terminals passes through
zero in a positive sense. When the impressed E.M.F.
has attained its positive maximum the condenser is fully
charged, and the charging current is zero. When the
impressed E.M.F. now begins to decrease, it is still
positive, but the condenser begins already to discharge,
producing a negative current, which becomes a maximum
at the moment when the impressed E.M.F. passes through
zero, and so on. We see thus that the condenser current
leads over the impressed E.M.F. by a quarter period.
In addition to the two cases here considered, a
third case is possible in which no second E.M.F. either
advancing or retarding the current is acting ; in this case
(glow lamps fed from a transformer) the current will
have the same phase as the impressed E.M.F., and its
strength will be simply determined by Ohm's law.
The periodic variation in current and E.M.F. may
be conveniently represented by a clock diagram. Let,
TRANSFORMERS
in Fig. 47, the outermost circle be used to mark the
time (somewhat in the fashion of a clock-dial), and let
O/ be the hand of a clock revolving with constant
angular speed. We count the time from the moment in
which O^ stands horizontally to the left. Let in this
moment the E.M.F. be zero. Describe a circle the
radius of which represents to any convenient scale the
maximum or crest value of the E.M.F., then the pro-
jection of this radius on the vertical gives to the same
scale the instantaneous value of the E.M.F. at the time
to which the posi-
tion of O/ corre-
sponds. Thus at
the time t the
E. M. F. vector
occupies the posi-
tion OE, and the
instantaneous
value of the
E.M.F. is OE,.
We count the
E.M.F. as posi-
tive if E, is above,
and negative if
o
E, is below the
axis.
The instantane-
ous value of the
current may be represented in a similar manner, but
the current vector must be drawn with an angular lag
we denote the
angular speed, the following equations obtain
) = 27T
a = wl
da = wdt
da = 2
Since work is the product of power and time, we have
for the work performed by the current in the time dt the
expression
= Vdt
the curves E and I to
FIG. 48. Curves of E.M.F., current and power.
In Fig. 48 are drawn
represent respect-
ively E.M.F. and
current. By multi-
plying their ordin-
ates we obtain the
ordinates of a third
curve marked P,
which represents
the instantaneous
value of the power,
whilst the area en-
closed between P
and the horizontal
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 / = o and / = T. counting the small
O
shaded parts below the horizontal as negative. The
work corresponding to a complete cycle is
e =
The instantaneous power varies, as will be seen from
Fig. 48, 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
92 TRANSFORMERS
current, so that the work taken over the time T is the
same in both cases, then the constant power (which in
future we will call effective power) is the quotient of
work and time, or in symbols
p=l
T
Substituting for P, dt, and T the values given above,
we obtain also
ml ft 6TT
P = / E I sin a sin (a ) sin
/? .
= /(sin
J
. . \dmoi
cos m0 sm cos ;;za sin ma)
m
The integral of the second term in the bracket is zero,
and that of the first term is
27T
cos m$ ma 7
. / = TT cos mR
where R is the magnetic reluctance of the whole flux-
circuit, that is, the sum of the magnetic reluctances of
its individual parts.
R = '-si 1
O'47T S fJ.
//! i 4 i
O rv & I i i _ ^ i_
\Si fa S 2 fa
In this expression /j means the length of the magnetic
path in cm. in that part of the magnetic circuit which
has a cross-section of B! sq. cm. and permeability fa,
and so on with the other members. In this formula
it is assumed that the same flux passes through the
different cross-sections S 1? S 2 , etc., and this assumption
is justified in transformers which are generally so de-
signed as to reduce the leakage flux to a very small
fraction of the main flux.
By inserting the expression for R into the formula
for ampere-turns or exciting force X, and remembering
7 97
9 8
TRA NSFORMER S
that the flux equals the product of induction B and
cross-section S, we also have
If the dimensions of a magnetic circuit be known, we
can find the values of B 1} B 2 , etc., for any given value
of the total flux <. The corresponding values of the
permeability ja we can take from a magnetisation curve
of the material used, and the values of / can be
measured off on a drawing of the carcase. We have
thus all the data required to find the relation of X
and (f) or I and . In other words, we obtain < as a
function of I or as a function
of X. A curve which repre-
sents this function is called
the char-act eristic curve of the
magnetic circuit. Its general
shape is represented in Fig. 50.
In order to facilitate the
drawing of this curve, we may
use magnetisation curves,
which give the value
o'8B
FIG. 50.
X
that is, the ampere-turns required for i cm. of path and
various values of the inductions. We then find
If the magnetic path contains an air-gap of length S,
in which the induction is B, then this requires o'
ampere-turns, and the total exciting force is
Energy stored in a magnetic circuit* Let in Fig. 50
the current grow from zero to its final value Ij, and the
flux grow from zero to its final value fa. At a given
moment the current is i and the flux fa If the current
increases by di, the flux increases by tfy. If the increase
ENERGY STORED IN A MAGNETIC CIRCUIT 99
takes place in time dt, the E.M.F. generated in the
n turns of the magnetising coil is, in volt
C := I't ' I O
The energy d& given to the circuit is eidt, or
But id$ is the area of the shaded rectangle, and it is
therefore obvious that the total energy stored in the
magnetic circuit carrying the flux ( b by reason of the
exciting force \^n ampere-turns, is the area enclosed
between the characteristic curve and the ^ line,
multiplied with nio~ 8 . If the characteristic curve were
drawn with ampere-turns instead of ampere as abscissa?,
then the area multiplied with icr 8
would be the energy. It should be
noted that the energy is independent
of the number of turns in the exciting
coil. Since we used volt and ampere,
and e refers to the change of flux
per second, the energy is given in
watt-seconds or joule.
The more exciting force is re- FIG. 51.
quired to produce a given flux ^
the greater is the energy stored. If there were no
air space through which the flux has to pass, the
characteristic would for a moderate induction be very
steep, and the area enclosed between it and the axis
would be small. It follows from this that it is chiefly
that part of the magnetic circuit which lies in air which
forms the store of energy. For fairly low inductions,
that is, values of B which lie below the knee of the
magnetising curve and a long air space, the first term in
the equation for \n is enormously greater than the other
terms, so that we may neglect the latter and write
I=o*8B8
The characteristic then becomes a straight line, Fig.
51, and
ioo TRANSFORMERS
. rx =tga. being the slope of the characteristic to the
horizontal, and S the section of the air space. The
energy is the shaded area multiplied by io~ 8 , or
2
if we take as a unit for the flux the megaline, and as a
unit of exciting force 1000 ampere-turns, the energy is
in joule . . . 10 x shaded area
in km. . . .1*02 x shaded area
From X = o-8BS and = io~ 8 we find-
In this formula S is given in sq. cm. and in cm.; their
product is the volume of magnetised air given in cub. cm.
Let this volume be given in cub. dm. or litres, and call
it V, then the energy in joules can be written
e = 4 v(-B_y . (25)
Viooo/ v '
The energy contained in one litre of magnetised air
/By. i / B v
stores therefore I ) joule, or ^- ^- ) metre kilo-
gram, as shown in the following table
Energy stored in one litre of air traversed by a magnetic flux
with induction B.
= O'S I 3 8 12 IS 2O
1000
Joule =i 4 36 256 476 900 1600
Metre kg. = 1*02 4^08 3-67 26 48 92 163
The property of the magnetic circuit to act as a store
of energy is utilised in the construction of so-called choking
coils, as will be explained at the end of this chapter.
The hysteretic loop. It has already been stated that
the change of induction taking place continuously in the
carcase of a transformer involves a certain loss of energy
and a corresponding generation of heat. The energy lost
per cycle is the difference between that which has been
THE HYSTERETIC LOOP \>;&i
stored in one stage of the process and that which is
returned in another stage of the process. If the material
undergoing cyclic magnetisation is air, the whole of the
energy stored in magnetisation is again recovered in
demagnetisation ; but with iron this is not so. If we
determine experimentally the magnetisation curve of any
sample of iron whilst this is being carried through a
complete cycle (methods for such tests are given in
Chapter VIII), we find that the I/z-B curve follows one
path for increasing values of I
and another for decreasing
values, the two curves forming
a loop, the so-called hysteretic
loop. This is represented in
Fig. 52, where the magnetising
force is plotted horizontally and
the induction vertically. The
sense in which the cycle is
performed is shown by arrows.
Since the energy representing
the half cycle from - B to + B
is proportional to the area en-
closed between the right side of
the loop and the B axis, and
since the energy in the return FlG 52
half cycle from + B to -B is
represented by the area between the left side of the loop
and the same axis, the energy lost in one complete cycle
is proportional to the area of the loop, and is given by
the equation
6 = area x io~ 8 joule
Since we take ordinates to represent B and not < as
before, the loss refers to an element of the magnetic
circuit i sq. cm. in cross-section. Let / be the length of
this element, then the loss per cycle per cub. cm. will
be found by dividing the above expression by /, or in
mathematical language
+ E
*X
102 TRANSFORMERS
Now, _ 4^ = H, the magnetising force in C.G.S.
units, so that we may also write
-B
/"
= -^. /H^B
-B
= -. H^Bio- 7 joule
. (26)
-B
In this latter form the hysteretic loss per cycle for i
cub. cm. is generally given in text-books.
No-load current of a transformer. Since the cyclic
magnetisation is accompanied with certain losses, the idle
or no-load current must have a component in phase with
the induced voltage, a so-called watt component. It
must also have a wattless component, that is, one lagging
90 behind the induced voltage, and consequently co-
phasal with the flux. The idle current is very small in
comparison to the full-load current, only a few per cent.
of it, and as the ohmic drop with full-load current is
only a few per cent, (sometimes less than i per cent.)
of the working or impressed voltage, we may, in cal-
culating the idle current, assume equality between
induced and impressed voltage, and determine the two
components of this current on the assumption that one is
in phase and the other in quadrature with the impressed
E.M.F. We shall also assume that the impressed E.M.F.
and both components of the no-load current are sine
functions of their respective crest values. As will be
shown presently, this assumption is not strictly correct as
regards the wattless component, but we make it never-
theless in order to simplify the calculation. The error is
not important.
Let P 7j be the power wasted in the iron in hysteresis
and eddy currents, and e the effective value of the
NO-LOAD CURRENT OF A TRANSFORMER 103
E.M.F. ; then the effective value of the watt component
for a single-phase transformer is
P*
The effective value of the wattless component is
where B is the induction in the air-gaps of the butt-joints
and & their combined length. The symbols .r and / under
the summation sign have the meaning already explained,
FIG. 53. Diagram of idle
current and power.
FIG. 54. Magnetic path
in shell transformer.
whilst n is the total number of turns traversed by the
idle current. The effective value of the latter is
The relation between these quantities is shown in
ig. 53, which also shows in the shaded area the power
wasted by the idle current.
In a core transformer having the same cross-section
in core and yoke / is the mean length of the lines of force
taken round the rectangle of the carcase, and the sum-
mation in the formula for ^ has only one term. If the
yokes are of larger cross-section than the cores (they
would obviously not be made smaller) then the summation
has two terms, one for the two cores and the other for
the two yokes.
In a shell transformer / is taken round one window
as shown in Fig. 54. Since the flux divides in the
shell the cross-section of the latter may be half that of
the core. It may also be greater, but not smaller.
104
TRA NS FORMERS
To determine the idle current for a given impressed
voltage e we first calculate the flux < from
4'44
100
n
and then the various values of B = -
Ampere turns per centimeter
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
2 3
6 6 7 ft ft 10 11 12 13 14 15 10 17 18
Ampere turns per centimeter
FIG. 55. Characteristic of Transformer Plates.
The length of air-gap at each butt-joint, if any, may
be taken as at most mm., or 8= 0*025, and the values
of :r corresponding to each value of B may be taken from
the curves, Fig. 55. It will be seen that up to a certain
value of the induction alloyed iron requires a little less
exciting force than ordinary iron ; only if the induction
SHAPE OF EXCITING CURRENT 105
is very large is this relation reversed. A very high
induction is, however, hardly admissible, since, even if it
be possible to keep the temperature down by some special
cooling arrangements, the large magnetising current is as
a general rule objectionable.
In three-phase core transformers each limb serves
one phase, and since one phase is magnetically and
electrically always in series with one or two others, the
value of / and & is only half what it would be in a
single-phase transformer. Thus the wattless current of
one phase must be calculated on the assumption that
it has to propel the flux through its own core, two butt-
joints, and about half the yokes.
Shape of exciting current. A knowledge of the
hysteretic loop, sheered over so as to include the influence
of butt-joints if any, is necessary for the determination
of the shape of the exciting or magnetising current and
the resulting idle current. We assume, as before, a sine
wave of E.M.F., and consequently also a sine or rather
cosine wave of induction. To each value of B corre-
spond two values of z' M , the one given by the ascending,
the other by the descending branch of the loop. The
graphic construction of the z^-time curve is then as shown
in Fig. 56. The hysteretic loop on the left gives B as
a function of i^ the sine curve on the right gives B as a
function of the time. By combining the two it is easy
to find the curve giving ^ as a function of the time.
Thus a particular value of B, to which correspond the
currents zi and z 2 , occurs at the times ^ and t^. Then
plot o^ over / lf and oi 2 over t# giving z\ and i 2 as two
points of the curve required. The whole curve ^ may
be drawn in this way. The curve i h is simply a sine
curve in quadrature with B. The idle current curve z' is
the resultant of these two components.
Choking coil. A coil having large inductance and
small resistance, so that it will cause a lag of current of
nearly 90 behind the E.M.F. impressed on its terminals,
is called a choking coil. Such a coil might be obtained
by altering a transformer so that it will take a large no-
load current. The alteration would consist in enlarging
the butt-joints and suppressing the secondary winding as
io6
TRANSFORMERS
superfluous. Since now the current is large, the ohmic
loss is no longer negligible and the watt component of
the current will be increased. If R is the resistance of
the winding and the loss in iron is as before P //} we
have
CHOKING COIL 107
for calculating the wattless component we may neglect
and simply write
\/
n
The lag being intended to be nearly 90 we must so
design the coil that i h will be very small in comparison
with i^ and then i will be very nearly equal to i^. We
can therefore write
o-SBS
t = or i =
n\/2
TOO
if by P we denote the apparent power of the coil
expressed not in watt but in volt-ampere. The true
power taken by the coil is
P =P A +R; 2
and its power factor is
Pj
cos = - ^
ei
This is to be a minimum. It is obvious that maxi-
mum lag and therefore minimum power factor will be
reached with one particular value of i, which we find
from the condition that
d
this gives P A = Rt* ..... (27)
which means that greatest choking effect combined with
a minimum waste of power will be obtained if the coil is
so designed as to make the loss in iron equal to the
ohmic loss. For purely constructive or commercial
reasons it may sometimes be necessary to slightly depart
from this rule. Thus, it might be desirable to use
existing stampings for the carcase, or a particular gauge
of wire in stock, etc. If, by accommodating the design
to commercial requirements, the difference between the
two losses does not exceed 10 or 15 per cent, of their
1 08 TRANSFORMERS
sum, a departure to this extent from the rule of best
proportions is admissible.
The equation which connects current and induction
in the air-gap may be written in the form
if by I we denote the crest value of the current. Let
similarly E be the crest value of the E.M.F., then we have
the relation
E =
where > = ITTV. Since current and E.M.F. are in quadra-
ture, their simultaneous values at any instant are
i=\ sin a
e = E cos a
The instantaneous value of the power is-
ei= El sin a cos a
It is positive for all values of a between the limits o and
, or arand ; it is negative for all values of a between
the limits and TT, or TT and 2?r. We see thus that during
one complete period the choking coil takes in energy
twice and gives off energy twice. To find the amount
of energy stored in a quarter period we determine
= /E I sin a cos adt
7T
/~2 da.
S= /El sin a cos a-
J CO
i El i ei / Q x
e=- - = >- .... (28)
CO 2 CO 27TV
If then frequency, choking volt and ampere to be
passed are known, we can determine the energy to be
CHOKING COIL
109
stored in the air-gap. Inserting the values for E and I
we obtain
1"\ CV
I
~ 2(0 C
= o4r-
or if we take for the volume
*-4V(-5L\
\ i ooo/
the same expression which has already been found on
page 100. Each litre of magnetised air is the carrier of
the litre as unit
B \ 2
As an example for the applica-
tion of these formulae take the
case of a choking coil which is
required to let pass 10 ampere at
a choking pressure of 100 volt
when the frequency is 50. We
have then = 314 and ei=* 1000.
This gives
220
FIG. 57. Choking coil.
1000 .
e = = 3-2 joule
If we assume an induction in the air-gap of 5000,
one litre of air will store 200 joule, so that we require a
*"? * O
total volume of air-gaps of - cub. dm., or 32 cub. cm.,
100
which may conveniently be subdivided into four gaps, as
shown in Fig. 57. The dimensions are figured in mm.
We make the central core 5 cm. square, and the shell
of double the cross-section of the core. The latter is
2 i 7 sq. cm. The total weight of iron is 1 2 kg., and with
an induction of 11,600 in the core the total loss of
power in the carcase is 14 watt, alloyed iron being
used. The flux is 0*25 megalines, and this requires 180
turns of wire for 100 volts choking E.M.F. The winding
space allows of the use of 3-mm. wire, to which
corresponds a resistance of 0*14 ohm and an ohmic loss
110
TRANSFORMERS
of 14 watt. The excitation required for the air-gaps
is o'8 x 0*64 x 5000 = 2550 ampere-turns, or with 180
turns 1 4' i ampere crest value, corresponding to TO
ampere effective value. The power factor is
14+ 14
1000
= 0-028
FIG. 58. Ventilated choking coil.
so as to preserve the value of 8 correctly.
to which corre-
sponds an angle of
lag
< = 88 24'
The coil shown
in Fig. 57 being
of small power, no
special provision
need be made for
cooling. For large
power coils it is,
however, advisable
to provide air chan-
nels. Fig. 58 shows
a design which I
have found to take
up a large apparent
power per unit of
weight, and yet
keep fairly cool
when in continuous
use.
It is advisable to
put a hardwood
lining into the gaps
CHAPTER VI
DESIGN OF A CORE TRANSFORMERBEST DISTRI-
BUTION OF COPPER LOSSES AT DIFFERENT
LOADS TIME CONSTANT FOR HEATING-
WEIGHT AND COST OF ACTIVE MATERIAL-
BEST DISTRIBUTION OF LOSSES TRANS-
FORMERS FOR A SPECIAL SERVICE TRANS-
FORMERS FOR POWER TRANSFORMERS FOR
LIGHTING ANNUAL EFFICIENCY ECONOMIC
IMPORTANCE OF SMALL LOSSES CONSTRUC-
TIVE DETAILS
Design of a core transformer. As an example of the
practical application of the formulae developed in the
preceding chapters we will now get out the design of a
2O-kw. transformer of the air-cooled core type, for a
secondary pressure of 160 volt on open circuit at the
usual frequency of 50. The primary pressure is 3120
volt, giving a transforming ratio of 19*5 to i. A trans-
former of this pressure rnay be used to supply lighting
current to 5o-volt metallic filament lamps which are
arranged in three circuits and balanced by an autotrans-
former, as will be explained in Chapter XII. The lamps
being three in series, require 150 volt, so that 10 volt
remain for covering ohmic losses in the transformer and
lamp circuits.
The coefficient c in formula (16) for the weight of
iron is about 8, if we use alloyed iron. This gives
170 kg. To get the side, d, of the core we may use
the formula
G = 6o(^+ o*2) 3 = 1 70
This gives d= 1*22 dm., or if we chamfer the corners
d= 1 2 '5 cm. We can now design the carcase and
&>
in
I 12
TRANSFORMERS
410
8
determine its exact weight. The shape will be as shown
in Fig. 59.
To find approximately the cooling surface we assume
for the present that the outside diameter of the coils will
be 26 cm. and the inside diameter 15*6 cm. Of the
inside surface only about one-half can be considered as
available for cooling, whilst the whole of the outside
surface is of course effective. The total cooling surface
of the coils is 8800 sq. cm. The edgeways cooling
surface of the carcase is 2720, and to this has to be
added about 80 sq. cm. for the
flat surfaces, so that the total
cooling surface is 2800 sq. cm.
The area of the core, allowing
for the chamfered edges, is 1 30,
that of the yoke, which is
perfectly square, is 136. Allow-
ing an induction of 7900 in the
core (7550 in the yoke) we find
with alloyed iron a loss of 190
watt. This is permissible,
since it gives r> d r>
P v = o, or -, - P v = o
agi dqi
Selecting the first, we have
ii6 TRANSFORMERS
Since q^ = -- j- , we have also
4
d /K/i .g - K/ 2
-y- I- -- I? + 7 -- /
^i \ f i (^-A
from which we find
the .condition for minimum total copper heat. It is equal
current density in both circuits. But since the mean
perimeter is the same in both circuits, and the number
of turns is inversely as the current, the volume of copper
is the same in both circuits. The loss being proportional
to current density and volume, we find that also in this
case,, the same as in the case of concentric winding, the
total loss is a minimum for equal copper heat in primary
and secondary.
In the example under consideration, we have chosen
cylindrical coils, and have assumed as a first approxima-
tion that the available winding space will be divided in
the proportion of 40 per cent, for the inner (secondary)
cylinder and 60 per cent, for the outer (primary) cylinder.
We must now investigate whether this division is in
conformity with the law just passed. The flux is
1 30 x 7900 = i *O2 7 . i o 6 .
To get 1 60 volt on open circuit on the secondary
we require 70 turns
;^ 2 = 70
%=i365
If for these figures and the original assumption of
40 and 60 per cent, winding space we determine the size
of wire and the losses, we shall find that the loss in the
primary is greater than that in the secondary. The
40 : 60 division is, therefore, not correct. By a method
of trial and error, which need not be repeated here, we
find ultimately that the division in the ratio of 37:63
gives equality of losses. We thus find
Best radial depth of inner cylinder . 14 mm.
outer ,, . 24 ,,
BEST DISTRIBUTION OF COPPER iij
We may now draw the coils, and determine from
the drawing the exact mean perimeter of each. This
gives
^2 = 0*575 m. 77-1-0755 m.
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 cm.), and leave sufficient space for clearance "and 'the
end flanges of the cylinders. The total space required
for these purposes is about 3^ cm., leaving 41-5 cm. 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 after-
wards insulated by paper 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 1 8 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 xVth, but T Vth of the net winding space. This gives
415/19 = 21*8 mm. The thickness of the wire is already
determined by the depth of winding, which we found
must be 14 mm. Allowing 0*5 mm. for the thickness
of covering (or i mm. in all), we find that the section of
the wire will be 6 x 20*8 mm. 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 mm., so that the actual cross-section of
the wire becomes 6 x 20 = 120 sq. mm. The length
of winding is 70 x 0*575 40*5 m., and if we allow 0*5
m. for connections, we can take 4T m. as the basis oil
which to calculate the resistance of the secondary winding.
The formula for the resistance, taking rise of temperature
into account, is
D 0*02/2
R --^-
/, being the length in metre and q the cross-section in
square millimetre. We thus obtain
R 2 = 0*00682
1 1 8 TRA NSFORMER S
A similar calculation made for the primary winding
shows that we have to use round wire of 3*1 mm.
diameter (covered to 3*67 mm.) 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
R! = 2-8
Losses at different loads. We have now all the data
required for calculating the losses at different loads.
They are given in the following table
Output in kw. . . .10 12 15 20 25
Secondary current, ampere 63 76 96 128 161
Primary current, ampere . 3*4 3*9 5 6*6 8*4
Total loss in copper, watt 59 Si 133 224 374
Loss in iron, watt . . 190 190 190 190 190
Total losses, watt . . 249 271 323 414 564
Efficiency, per cent. . . 97*6 97*8 977 98 97*8
Specific cooling surface of ) ^ r
.1 s \ 149 IOQ 66 36 23
coils, sq. cm. J
Temperature rise of coils)
if air cooled, deg. C. } 15 24 45 /o
The specific cooling surface of the carcase is 147, to
which corresponds a temperature of 44 C.
Time constant for heating. The time constant is
found from-
CT
P
where P is the lost power, in our case for a load of
20 kw., 190+224 = 414 watt, and C the number of
watt-seconds required to raise the temperature of iron
and copper by i C. if radiation be neglected. We have
178 kg. of iron and 112 kg. of copper; hence
C 4200 (178 x o'n + 112 x 0*093) = J 26000
/= 126000 _._45 = T 3600 seconds
414
t = hours
WEIGHT AND COST OF ACTIVE MATERIAL 119
For the copper only the time constant is 2*4 hours.
If this transformer be required to work only four hours
every evening, the loss in the copper may be increased,
as shown on page 65, to
2-4
_
P =224-5-
e - i
P = 224 X I '21
The output may therefore be 20^1*21 = 22 kw. with-
out exceeding the temperature rise of 45 C.
Weight and cost of active material, In designing
this transformer we have used stampings of alloyed iron.
Had we used ordinary transformer sheets we should, in
order to remain within the same temperature limit, have
been obliged to reduce the induction to B = 5600, thus
reducing the secondary terminal pressure to about 1 10
volt, and the output to 14 kw. Alloyed iron costs about
twice as much as ordinary transformer sheet, or, say,
i6d. against &d. per kg. Copper may be taken on an
average at is. 9^. per kg. The weight of active material
and its cost is as under
Quality of iron.
Ordinary. Alloyed.
Shillings. Shillings.
Weight of carcase . 178 kg. cost . 120 240
Weight of coils . 112 ,., ,, . 196 196
290 316 436
Output, kw. . . . . 14 20
Cost per kw. output . . . . 227 217
It will be seen from this table that it pays to use the
more expensive iron.
Best distribution of losses. In designing this trans-
former we paid no heed to the question whether the
losses between iron and copper are correctly distributed.
We designed mainly with a view to moderate and equal
temperature rise in both parts, and it so happens that
the loss in iron is not very different from that in copper.
Theoretical conditions demand that they should be equal,
or nearly equal. The latter is the case in the transformer
120 TRANSFORMERS
under consideration. The proof of the rule for minimum
loss is as follows. Let a given transformer working on a
constant pressure circuit be loaded to different degrees
corresponding to various values of the primary current i.
Since primary and secondary current are at all loads
very nearly proportional, the total loss in the windings
will be correctly represented by Ri 2 , whilst the loss in
the iron is a constant quantity C. The efficiency is
ei cos -. = o. or
di
(ei cos - 1 -
j> j L
18 21 33
17 21 36
19 22 31
21 33 44
20 35 50
20 28 36
It will be seen that for cheap power current the
choice lies between G and P, whilst for the more expen-
sive lighting current L is the best type. The differences
in annual working charges are not very great. This is
due to the use of alloyed iron ; had we used ordinary
transformer sheet in these designs the annual charges
and their differences would have come out larger.
Economic importance of small losses. Station engineers
generally reckon the cost of lost energy not at the selling-
price of the current, as we have done here, but at the so-
called engine-room cost, which is, of course, much lower.
In this connection it is interesting to notice that even if
lost energy be only reckoned at engine-room figures (say
about id. per kw.-h.), a transformer with large iron loss
ECONOMIC IMPORTANCE OF SMALL LOSSES 129
FIG. 61. 20 kw. transformer. Scale I :6.
1 30 TRANSFORMERS
is a heavy charge on the working expenses. Most of
the transformers now in use for lighting and general
purposes have been made some years ago, and may be
assumed to have from 2 to 3 per cent, iron loss. This
means that each kw. of transformer installed uses up
250 kw.-h. in iron heat annually, whereas a modern
transformer made with alloyed iron would only use up
about 80 kw.-h. By replacing the old with modern
transformers a saving of 170 kw.-h., or 145-. worth of
current per kw. installed, would annually be made.
Against this must be set the annual charge of, say, 10
per cent, on the capital outlay, which on an average
may be taken at 2 per kw. Of the 14^. saved 4^.
must therefore be set aside for the annual charge,
leaving a clear saving of icxr. per annum, so that after
four years the cost of the new transformers could be
completely written off.
Constructive details. After this digression into what
may be termed the financial side of transformer design,
we return to the particular 2O-kw. transformer for general
purposes dealt with in the beginning of this chapter.
The core with the coils in place is shown in Fig. 61.
This transformer is put into a perforated sheet-metal
case, and is therefore air cooled by natural draught. It
can with such a covering only be used indoors, and then
only in dry places. For use in damp places and out of
doors a perforated covering is of course inadmissible.
We must put the transformer into a tight case of cast-
iron. As this would greatly diminish the effectiveness of
the enclosed air as a cooling medium, we use oil as the
internal cooling medium, and provide the case with ribs
to increase the cooling effect of the external air. See
Figs. 62, 63 and 64.
In building up the carcase, 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. Plates are now on the market
which have one side covered with a very thin insulating
film. These may be used without paper insertion. In
building up, the lower yoke and the two cores are first
CONSTRUCTIVE DETAILS
132
TRANSFORMERS
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 rect-
angular wire be used), the voltage, and the quality of the
FIG. 64. Plan of 20 kw. transformer.
cotton and number of coverings. There must at least
be two coverings, though treble covering with very fine
cotton is still better. For very stout wires braiding is
advisable. The thickness of the covering in millimetres
should not be less than
S = o'i3+o'o6^ (30)
when d is the diameter (or equivalent diameter) of the
naked wire in millimetres. The diameter of the covered
wire is then
CONSTRUCTIVE DETAILS 133
Wire of large rectangular section may also be wound
naked, suitable strips of fibre or other insulating material
being wound in, or afterwards inserted.
The resistance of the coil must be calculated with
reference to its temperature ; as a first approximation,
based on a temperature of 75 C, the following formula
may be used
r> CTO2/ i
R = .. _ ohm
where / is the length of wire in metres and q its area in
square millimetres.
To promote dissipation of heat, the casing may, as
already mentioned, be provided with external ribs or gills.
Small internal ribs are also provided to hold the trans-
former 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. 63, or they may be passed
through bushed holes which are afterwards cast out
with insulating compound. The latter arrangement is
generally adopted in large transformers. When the
pressure is very high the bushes take the form of long
glass or porcelain tubes.
CHAPTER VII
DESIGN OF A SHELL TRANSFORMER FILL
FACTOR WINDING EFFICIENCY, WEIGHT
AND COST ENLARGING A DESIGN
Design of a shell transformer. As an example of how
to design a small air-cooled shell transformer we take
a 7 kw. transformer to give at v = 50 with 2000 volt on
the primary terminals 64 ampere at no volt on the
secondary terminals, or allowing for a moderate drop, 113
to 114 volt on open circuit. If we make the height of
the windows equal to the thickness d of the core and
their width 70 per cent, of the height, the total weight of
iron in kg. is for a depth of core c
G =
d and c being given in dm. Using alloyed iron, the
formula giving the approximate weight is
G=io- -?- .... (16)
A/
V T r^r*
100
This gives for P = 7, G-8okg. and (Pc = 2'22. For a
core 15 cm. deep, the thickness would be 12*1 cm., or say
in round numbers 12
cm. Its area is 156
sq. cm.
Fig. 65 is a sketch
of the carcase, the
dimensions being
mm. The next point
to be determined is
__i the induction permis-
sible with regard to
o
?-600
O
'''
x
<^60-*
t
"4-84 >
_._120--i >
*--*
l*
i
i
- 408 * -
O
. 65.
FILL FACTOR 135
temperature rise. In a small transformer the windows
have to be closely packed with wire and cannot contribute
to the cooling surface. The flat surface of the plates is
nearly worthless for cooling, so that we can only reckon
on the outer edge surface. This is 1670 sq. cm. Allowing
a temperature rise of 55 C. we get a specific cooling
surface of corresponding to the frequency, so
that
ft) = 2TTV
From the definition it follows that a vector may be
displaced parallel to itself without ceasing to represent
the quantity correctly. Strictly considered, the length
of the vector should represent the crest value of the
magnitude ; its projection on the base line will then
represent the instantaneous value, and this will be
positive or negative accordingly as the projection lies to
one side or the other of that point on the base line which
corresponds to zero value. Sometimes it is convenient
to let the vector represent not the crest value, but the
effective value of the quantity.
This is permissible since the o
ratio between crest and effective 2
value is a constant for all quan- FlG> 6 8. -Conception of vectors.
tities, namely ^/ 2 > an d passing
from one to the other only means an alteration of the
scale in this ratio.
Vectors which represent different magnitudes having
the same phase must be drawn parallel to each other,
and if drawn from the same origin will lie upon each
other. Thus in Fig. 68 OI, ON, OX represent respectively
current, flux, and exciting force. They are co-phasal,
and for this reason drawn as parallel lines. They may
be drawn from the same origin so that all lie along OA.
The length of each will depend on the magnitude of the
quantity and the scale chosen to represent it. If we
select, for instance, the scale so that i mm. shall represent
i ampere, or i megaline or i ampere-turn, then the number
OX
of turns n will be represented by the ratio -=-=-> and the
OX
magnetic resistance R by the ratio -r- By altering
the scales for current and flux in these ratios it is obvious
that O A ~ OX may be made to represent not only ampere-
144 TRANSFORMERS
turns, but also current and flux. In order that OA may
represent current we shall use a scale, the divisions of
which are not i mm. but n mm. apart, and for the flux
we must use a scale, the divisions of which are R mm.
apart. The value of R is found from the following
consideration. The well-known law of the magnetic
circuit is
TT _
where H is the magnetic force, and / the length of path.
Let A be the area, B the induction, and p the perme
ability, then we have for the flux in megalines
N= " ! -X
o-8/i o 6 ~R
p. A
or for a magnetic circuit composed of different materials,
p. A
In a transformer the magnetic circuit consists of iron,
and if there are butt joints air. For
air /A is constant, namely i, but for iron
it varies with the induction. As, how-
ever, transformers are mostly worked at
a fixed pressure, and therefore at a
constant induction, R will also be a
constant, and we are therefore justified
in using the same vector for flux and
exciting force, provided we alter the
PIG. 69. E.M.F. , & 1-1 ir v u
and current vector, scale accordingly. If X be given not
as a crest value, but as an effective
value, we have
Let, in Fig. 69, OE represent the crest value of the
E.M.F. impressed on a circuit, for instance on the primary
terminals of a transformer, and OI the crest value of the
APPLICATION TO A TRANSFORMER 145
current. The phase difference is $, and if the vectors
rotate as shown by the arrow, the current lags and 2 > the difference being in the present case very
marked, because for the sake of greater clearness we
have exaggerated all losses and assumed too large an
exciting force. If the vectors represent effective values,
the following relations obtain :
o
Power supplied equals . . . e^ cos
2 .
This is given by the vector O^.
One component must be equal and opposite O^, and
one component must be provided to overcome the ohmic
resistance. Let the vector of the latter be Oa.
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 dia-
gram shows that e kl is greater than e k ^ the difference
being the more marked the greater are the ohmic resist-
ances and the E.M.F.s of self-induction in the two
windings. In both respects the diagram Fig. 78 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
^ 2 = o. We assume the primary E.M.F. to be so
adjusted that this amperemeter shows the normal second-
ary current corresponding to full load under normal
working conditions. The diagram then assumes the
DIAGRAM OF A TRANSFORMER
155
form shown in Fig. 78. The lettering is the same as in
FiV. 77. It will be seen from this diagram that although
o * o o
no pressure is obtained at the secondary terminals, a
pressure equal to e kl must be supplied to the primary
terminals in order that the current z* 2 may flow through
the short circuit.
If, as is always the case in modern transformers of
good design, the resistance of the windings is very small,
and the no-load current t is only a very small fraction of
/!, then z' 2 and i will lie very nearly in a straight line, and
e l and 2 will lie very nearly in a straight line.
With a symmetrical arrangement between the two
windings (and the assump-
tion that the number of
turns is the same in both)
we have e^ e^ and
The E.M.F. necessary
to overcome self-induction
and ohmic resistance can
thus be found by a very
simple experiment. We
short-circuit the secondary
terminals by means of an
amperemeter (having itself
as little induction as pos-
sible), and supply the
FIG. 78. Vector diagram for short-
circuited secondary.
primary terminals with current of normal frequency and
such E.M.F, that the normal secondary current is
indicated on the amperemeter. One-half the E.M.F.
supplied to the primary equals the E.M.F. required for
the primary winding. The E.M.F. required for the
secondary winding is equal to this value divided by the
transforming ratio. Take as an example the case of a 10
kw. transformer wound for a ratio of 2000 volt to 100 volt.
In testing this transformer, as above explained, it is
found that 100 volt must be supplied on the primary at
v=5o in order that 100 ampere may be driven through
the short-circuit. We have then e l = 50 and e^ = 2*5.
The E.M.F. has two components. One is a wattless
156 TRANSFORMERS
component at right angles to the current, and is due to
self-induction, and the other is in phase with the current,
and is due to resistance. There may be another watt
component due to eddy current losses in the copper or
other metal parts including the carcase. Such eddy
currents may be produced by the leakage field passing
laterally through the wires, plates, or other metal parts.
Not to complicate the investigation, we neglect the (in
any case small) influence of such eddies for the present.
We also assume for the present that on short-circuit
the currents are inversely proportional to the respective
numbers of turns. The watt component of impressed
primary E.M.F. is then simply the product of current
and resistance taken for both windings.
As we measure the pressure in the primary circuit
and the current in the secondary, R 2 z' 2 has to be reduced
to the primary. We have therefore the watt component
of the impressed E.M.F.
n l , -D n l
! - 2 + R 2 -
n, n 2
Since the wattless component must be at right angles
to e r , and since both together give the total impressed
E.M.F. e , we find the wattless component e^ that is, the
E.M.F. of self-induction
e^ may be divided into two parts inversely proportional
to the numbers of turns, so that we get the E.M.F. of
self-induction separately for each winding.
Voltage drop. If by making the experiment above
described we have found how much E.M.F. is produced
by magnetic leakage in each coil, we can use this informa-
tion to determine the voltage drop at various loads. In
this determination it is convenient and permissible to
assume exact opposition in the phases of primary and
secondary current. Modern transformers with closed
magnetic circuit require so little magnetising current,
that even at load this assumption is very nearly true.
Let, in Fig. 79, A represent the pressure at the secondary
terminals, AB the ohmic loss of pressure, BC = , 2 the
VOLTAGE DROP
157
E.M.F. due to self-induction; and therefore OC = 2 the
E.M.F. induced in the secondary. Let the transforming
ratio be reduced to unity, then OC = e l is also the E.M.F.
induced in the primary, and with symmetrical windings
CD = BC the E.M.F. of self-induction in the primary,
so that e sl = e s2 . The ohmic voltage loss in the primary
is DE = AB if the losses are equally divided between
the two windings as required by a good design. The
line joining A, C and E is therefore a straight line, and
its inclination to the vector of secondary terminal pressure
is the same for all loads. At a smaller load, for instance,
producing the ohmic loss A'B the terminal pressure
would be OA' in the secondary and OE' in the primary.
FIG. 80.
FIG. 79.
The ratio of the length of the lines AE and A'E' is
the same as that of the lines AB and A'B, and the
length of the line AE is directly proportional to the
current.
Let us now assume that we are able to vary the
primary E.M.F. in any way which may be required to
keep the pressure at the secondary terminals constant
for all loads. We draw the line AE (Fig. 80) for full
current, and make an ampere scale which corresponds
with this length, then we can, by using this scale, mark
off on the line AE the points E', E", etc., corresponding
to other currents, and thus find the primary E.M.F.
vector OE', OE", etc., corresponding to these currents.
It is thus possible to determine the primary E.M.F. as
i 5 8
TRANSFORMERS
a function of the load, if the secondary terminal pressure
is to be a constant.
This is however not the case generally met with
in practice. As a rule the E.M.F. in the primary or
supply-circuit is constant, and it is required to find the
secondary terminal pressure at various currents. This
problem can also be solved graphically in a very simple
manner.
Graphic determination of drop. It has already been
shown that in all the triangles OAE, OA'E', etc., the
obtuse angle at A, A', etc., is the same. The longest
side of the triangle represents the
E.M.F. impressed on the primary,
and the shortest side the current
in the secondary. We may now
imagine all the triangles in Fig. 80
so enlarged or reduced that all
the points E lie on a circle de-
scribed round O as centre, with
a radius equal to the impressed
E.M.F. Let OE in Fig. 81
represent this E.M.F. at full load
(current represented to a suitable
scale by the lengths AE) and
E', E" the positions of E for
smaller loads, then the length
OA, OA', OA", etc., gives the
corresponding pressures at the
secondary terminals. As a matter of convenience we
may also plot the secondary current on a horizontal o\
to a suitable scale, and find the points E by projection
from the points I, as shown by dotted lines.
Let us now apply this method to our previous
example of a 10 kw. transformer. We have assumed
that 100 volt must be impressed on the primary in order
to produce 100 ampere in the short-circuited secondary ;
that is, 5 per cent, of the normal primary voltage. Let
us further assume that the watt component as calculated
from resistance measurements has been found to be
2 per cent. The wattless component is therefore
2 2 = 4'58 percent.
FIG. 81.
VOLTAGE DROP 159
The slope of the line AE in Fig. 80 is 2 in 4*58,
and its length is 5, whilst the length OA is 100. The
angle at O is therefore very acute, and the difference
between OA and OE, that is, the drop at full current and
a non-inductive load, is 2 per cent. At half-load it would
be i per cent., and so on. If the transformer had con-
siderably more leakage, say 20 per cent, instead of 4*58
per cent, then the drop, even at non-inductive load, would
be appreciably greater than that given by ohmic resist-
ance. In such a case the construction shown in Fig. 81
may be applied. The ampere-load would be plotted on
the horizontal o\ by using a scale on which ol represents
100 ampere, and by projecting the corresponding points,
first to the circle and then to the vertical parallel to EA,
we find the terminal volt OA', OA", etc. This con-
struction, carried out for various loads, gives the
following results, the impressed E.M.F. being constant,
namely, 2076 volt.
Ampere in secondary o 25 50 75 100 200
Terminal pressure 103*8 103-2 102-35 101*3 100 92
The drop between no load and full load is thus 3*8 volt.
The drop between full load and 100 per cent, over load
(which the transformer is perfectly able to stand for a
short time) is 8 volt more, or a total between no load
and double the normal full load of 1 1 *8 volt. This drop
is of course too great for practical purposes. It is due
to the large inductance we have assumed, merely in order
to explain the graphic method.
Up to the present we have assumed that the load is
non-inductive. It remains yet to extend the investigation
to cases in which the secondary circuit has also self-
induction, or capacity, or both. Self-induction is intro-
duced, if the secondary current is used for feeding motors
or arc lamps, in which cases there is developed an E.M.F.
at right angles to the current. The pressure at the
secondary terminals must therefore have a component
equal and opposite to this E.M.F. of self-induction, and
this component must be in advance over the current by
90. Let in Fig. 82 OA represent the secondary current,
OB the power component of the secondary pressure, and
i6o
TRANSFORMERS
J.
FIG. 82.
OC the counter E.M.F. produced by self-induction.
The secondary pressure is then represented by the vector
OD, which advances over the current by the angle changes E
takes different positions on the circle of primary E.M.F.,
and the locus of B must therefore also be a circle of the
TRANSFORMERS
same radius, the centre of which has relatively to O the
same displacement as B has to E.
Let in Fig. 92 the vertical represent the current
vector, OS the E.M.F. of self-induction at full current,
and So the ohmic loss at full current in both windings ;
then O0 is equal and parallel with EB of Fig. 85, 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 OB is smaller
than OE, its value at no load. For a negative phase
difference ^ (current leading before the E.M.F.) the
secondary terminal pressure OB X is greater than its value
at no load. With a certain negative phase difference . The corresponding position of the vector of
E.M.F. is OE, 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 (p. The
result is given in the following table.
60 kw.-transformer 3000 : 200 volt on open circuit.
Pressure at secondary terminals with 300 ampere in
FIG. 93.
secondary and power factors varying from TOO 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 ij per cent. ; if used
on a circuit containing arc lamps or motors the power
factor of which is about 070 to o'So, the drop would be
approximately 6 per cent.
The diagram, Fig. 93, leads to some interesting deduc-
tions. In the majority of cases the circuit has, not
capacity, but inductance, and the following remarks apply
to these cases, that is to say, to the left-hand side of the
GRAPHIC DETERMINATION OF DROP 173
diagram. If we could build a transformer which has
absolutely no magnetic leakage, then OS would be zero,
and o would lie vertically above O. 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,
resistance. This case is, however, unattainable in practice,
for we can never reduce magnetic leakage to zero. The
inductance produced by magnetic leakage can, however,
with a careful design, be made very small, especially
for low periodicities. Imagine that we have reduced the
inductance so far as to be equal to the resistance, then
OS = So, and Oo includes with OA an angle of 45.
The distance between the two circles would then be
approximately the same for all values of q>. 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, greater 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 low frequency, the pressure at the secondary ter-
minals will at full current be lower in the former case.
The E.M.F. of self-induction is for both windings,
OS = 2 X
that is to say, OS is proportional to v. 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 small 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-synchronous motors.
174
TRANSFORMERS
Drop diagram simplified. The 60 kw. transformer
here chosen as an example has an inductive drop of over
8 per cent. This is rather more than usual in a good
design, but it was necessary to assume -so large a drop in
order to make the diagram, Fig. 93, distinctive. In well-
designed large transformers the inductive drop is generally
under 4 per cent., and then the graphic construction, Fig.
93, must be made on a very large-scale to get accurate
results. Even then the elasticity of the compasses with
which we draw the circles is a source of error. To
obtain the drop we can modify the construction so as
to make the drawing of
the circles superfluous.
If the sides of the tri-
angle OS0 are very small
as compared with OE,
then a line drawn from
o to E will be very nearly
parallel to OE, and the
drop will be very nearly
equal to the piece cut off
on 0E by a perpendicular
dropped from O on to it.
The triangle OS0 may
then be drawn to any
convenient scale, and the
drop found as shown in
Fi g- 94- O . . . . (31)
Correction for eddy current losses. There remains
still a slight correction to be made. On p. 156 it was
FIG. 94 .
CORRECTION FOR EDDY CURRENT LOSSES 175
mentioned that the watt component of the primary
impressed E.M.F. may not only be due to ohmic resist-
ance, but also to certain losses caused by eddy currents.
In consequence e r will be slightly greater than calculated
from the ohmic resistance. To find the true value of e r
we must use a wattmeter in the primary circuit, and
divide its indication by the secondary current. The
correction is small, and if a wattmeter is not available we
can approximate it by measuring, not only the secondary
current z' 2 , but also the primary current z lt and calculating
e r from
CHAPTER IX
CALCULATION OF INDUCTIVE DROP THE IN-
FLUENCE OF FREQUENCY ON DROP THE
INFLUENCE OF FREQUENCY ON OUTPUT-
EQUIVALENT COILS THE SELF-INDUCTION
OF A TRANSFORMER WORKING CONDITION
REPRESENTED BY VECTOR DIAGRAM CON-
STANT CURRENT TRANSFORMER
Calculation of inductive drop. The inductive drop being
due to the interlinking of the leakage field with the
windings, we can approximately pre-determine it from
the drawing of a transformer by mapping out the leakage
field in relation to the coils. Such a method can, how-
ever, only yield qualitative, not quantitative results, as
we have no means of determining exactly wh'at the flux
density is in any given point. By applying in a general
way the laws of magnetic circuits we can compare
different arrangements and say what details will influence
the drop and in what ratio, but we cannot calculate the
absolute value of the drop. To get quantitative results
we must fall back on experiments. The method of
investigation is then as follows. First we determine
the general principles of interlinkage between leakage
flux and winding without assigning to the resulting
E.M.F. a definite value; then we apply the formulae
thus developed to definite cases investigated experi-
mentally and obtain coefficients by which the formulae
become applicable quantitatively.
We investigate first cylindrical coils and then sand-
wiched coils. Let, in Pig. 95, I and II represent the
cross-section of the two co-axial coils, the radial depth
of winding being a^ and # 2 respectively, and the length of
176
CALCULATION OF INDUCTIVE DROP 177
the coils /. Let the secondary coil 1 1 be nearest the iron.
Leakage lines pass through the space b between the two
windings, and are of the general shape shown by the
dotted lines. The lines surrounding I pass wholly
through air, and have therefore to overcome a greater
magnetic reluctance than the lines surrounding II, which
pass partly through iron. The ampere-turns in both
coils being practically equal, the stray field of II will
therefore be stronger than that of I. In order to be
able to treat the problem mathematically we shall make
the assumption that the two currents have a phase
difference of 180 and that the ampere-turns are equal.
Both assumptions are very nearly correct. We shall
further assume that neither the
yoke nor the other core has a
material influence on the shape
of the stray field, which we take
to be distributed symmetrically
round the axis of the coils.
There must then be a boundary
surface of cylindrical shape be-
tween the two fields, which
passes through the space b, and
is distant ^ from the inner sur-
face of coil I, and A 2 from the
outer surface of coil II. Where
precisely this boundary is, we
cannot tell. All we know is
that b x < 4 because of the
presence of the iron on the right of II.
Let % and ;/, be the number of turns, / the perimeter
of the boundary, and 7 the number of turns per unit
radial depth of II. The ordinates of the shaded area
represent, according to the scale chosen, either ampere-
turns or induction. In the space b both are maxima,
and at the boundaries of the coils both are zero. In
an elementary strip of II having the radial depth da the
number of turns is
dn = yda
With these turns are interlinked all those lines of
FIG. 95. Leakage of cylindrical
winding.
12
1 78 TRANSFORMERS
force represented by the shaded area between Bj and B.
The corresponding flux is
and the E.M.F.-
de = 'v l
The total E.M.F. self-induced in II is the integral
of this expression taken between the limits a = o and
a = # 2 , or
Oo
e. =
&,Btf 2 +y (# 2 -
BTJ
i 15 T) a T1
bince - = or B 1 = B
d #2 ^2
we can write
B + B! i
2 \ 2
B / 2
-(/
Since ya. 2 is nothing else than the total number of turns
in II, and B is proportional to- 2 - = - 2 (X 2 being the
crest value of the ampere-turns in II), we find the
following expression for the self-induced E.M.F. in II
/
where / 2 is a coefficient to be found by experiment. In
CALCULATION OF INDUCTIVE DROP 179
the same way we have for the E.M.F. of self-induction
in I-
The total induced E.M.F. due to the main flux < being
E = 4'44v;z<
we find the ratio of leakae to useful E.M.F.
^ _ / 22 ,
-- 7 "
2 $
e l _ /^
ET -
It is obvious that /i?9 J-*l"
ioo ioo ;/!
1 T^ T T '^1
and rL S 2 = (oL 2 i }
n. 2
and o)L 2 = -^ / ] (-
100 li\#i
ioo
188 TRANSFORMERS
Thus all the electrical constants of a given
transformer can be determined.
We may still further simplify the conception of the
equivalent coils by assuming a transforming ratio of i : i
and the winding of the current receiving device so altered
that the primary voltage may be applied. The trans-
former may then be omitted from Fig. 97, and there will
only remain the equivalent coils, as shown in Fig. 98,
the primary and secondary being now combined into
one inductance coil L and one resistance coil R. The
apparatus receiving current is represented by the
inductance coil A and the resistance coil p.
The values of L^ and R 7 , have not altered. The
new values for the two remaining equivalent coils
are
L=.32_Ei and R=R 1 +
100 i!
The self-induction of a transformer. The self-induc-
tion of any apparatus through which current is passing
may be defined as the reactance (>L), that is, the ratio
between the wattless component of the E.M.F. (-- -]
V 100 /
and the current.
To get the reactance we short-circuit the terminals
2, 2 and measure the current supplied to i, i, and by
wattmeter the wattless component of the E.M.F. supplied
to these terminals. In this case the current I through
2, 2 will be very nearly the same as that supplied to i, i,
so that we can also write
lOOl
for the reactance of the transformer proper. Since i^
and i h are very small as compared with I at any but the
smallest loads, this expression will also hold good when
the transformer is under pressure. Only at very light loads
or on open circuit will the inductance L, and therefore
also the reactance o>L, be materially higher. In the latter
case we have I =o and l l = 7^ The electrical constants
THE SELF-INDUCTION OF A TRANSFORMER 189
of one equivalent coil containing both resistance and
inductance will then be
Take as an example a 2O-kw. transformer for 2000 volt
at 50 frequency on the primary side. Let it have 2 per
cent, iron loss and 2 per cent, copper loss, and let the
inductive drop be 3 per cent, and the wattless component
of the no-load current 0*5 ampere. We have for this
transformer under a moderate load
T 3 2000
100 10
314 L = 6 and L = 0*0192 Henry
The resistance of the equivalent coil is found from the
2 per cent, copper heat at 10 ampere primary current
from
R . 10= - . 2000, or R = 4 ohm
100
400 , .
** = - - = 0-2 and ^ =
2000
This gives
2000
- = roooo
0*2
T 2OOO IT/ T / O'Z \ 2
L = = 127 and L =ft>L u ( ) = 10*0
314.0-5 M vo'54/
We thus find that on open circuit the transformer has
a self-induction of 10*9 Henry, whereas on a moderate
load the self-induction is only 0*0192 Henry. On open
circuit the self-induction is 570 times as great as on load.
1 90 TRANSFORMERS
Working' condition represented by vector diagram.
We have already made use of vector diagrams to re-
present the relations between current, E.M.F., flux, exci-
tation and phase angles, assuming for sake of simplicity
that the transforming ratio is i : i. These vector dia-
grams can, however, be made more simple by introducing
the conception of equivalent coils, which conception be-
comes possible for a transformation ratio of unity. In
this case the consuming device receives from the second-
ary terminals the current which has passed through the
equivalent main coils L and R (Fig. 98) with its full
strength, but attenuated as regards pressure by the re-
action of these coils ; and to the primary terminals has
to be supplied, not only this current, but also the currents
taken by the equivalent coils L^ and R/,, which, being
placed as shunts, have no influence on the pressure.
They only influence the primary current. As far as
the working current in the load X, p, and the effect of R
and L are concerned, we may disregard the shunt coils
altogether, but we must take them into account if we wish
to determine the primary current. It should be noted
that at constant primary voltage the current taken by
the shunt coils and its phase are constant, whatever may
be the working condition of the transformer ; and herein
lies the advantage of this method of treatment. We can
determine the working condition of the main circuit
of the transformer by the use of a very simple vector
diagram, and when this is done, acid vectorially the
currents in the shunt coils.
Let, in Fig. 99, MD 2 be the vector of the E.M.F.
supplied to the load, and MI the current vector, then
D 2 D X = RI represents the loss of E.M.F. in R, and D X E =
>LI the E.M.F. required to overcome self-induction.
The E.M.F. to be impressed on the primary terminals
is therefore ME. If the load consists of a group of con-
suming devices coupled in parallel, all producing the
same lag, its angle p will not change if some of the con-
suming devices are switched on or off. The length
of the vector MI will change, but the phase angle p
will remain constant. The angle D^D 2 also remains
constant, although the sides of the triangle will vary
WORKING BY VECTOR DIAGRAM
191
proportionately with the current. We can therefore
regard
as a measure of the current^ and by using a scale the
divisions of which are >L times as long as those of the
volt scale, we can scale off the current on the line D X E.
We suppose the impressed voltage ME to be constant.
This is the usual case in
practice. To find the second-
ary E.M.F. and its phase
relation to the working cur-
rent I. if the latter be changed
by changing the load, we
reason as follows : Since the
angle DjED^j and the angle
L
and f is given by the character of the load. To find the
working condition for any current I at phase angle
L and rotate
the two components by 90, so that ^ falls in line with ME
and i h comes into quadrature with it. We make
then DjOi is a measure of the primary current, and
L of the equivalent coil must be large, and the
higher the frequency the better. Fig. 102 shows a
carcase specially designed to produce leakage.
The two coils are on two limbs of the carcase, and there
is a third limb without winding. Its object is simply to
form a magnetic by-pass to the useful flux, that is to say,
to increase leakage. Since the winding space is not re-
stricted as in an ordinary transformer, the ohmic resist-
ance of the coils can be made very small, so that the angle
a in the triangle D X ED 2 in Fig. 99 becomes very acute,
CONSTANT CURRENT TRANSFORMER 19;
and D X E =a>LI becomes large in comparison with DiD 2 .
We may assume a to be nearly zero ; then the centre of
the inner circle will be distant from ME by
The points D 2 and D x will nearly coincide. DjE is a
measure of the secondary current, and MD 1 is very nearly
the secondary E.M.F. On short circuiting the load
MD 2 becomes zero, and MDj so small that we may con-
sider ME a measure for the secondary current. The
smaller the power factor of the load the flatter will be
the inner circle (see Fig. 102) and the greater the varia-
tion of current D X E for a given range of secondary
E.M.F. of, say, MD X as a maximum down to zero.
If there be self-induction in the external circuit the
arrangement will therefore be very imperfect or only
applicable over a very small range of secondary E.M.F.,
the greatest attainable E.M.F. being many times smaller
than the primary E.M.F. The arrangement is therefore
useless for arc light circuits ; it can, however, ba used for
glow-lamp circuits. In this case 2 . The efficiency of the transformer is then
given by the expression
WDK'
If the supply voltage is high, it is advisable to so
connect the wattmeter that in the instrument itself no
great potential difference can arise. Otherwise there is
the risk of breaking down its insulation. Fig. 107 shows
the arrangement of connections which should on that
account be avoided. Theoretically Fig. 107 is equivalent
with Fig. 1 06 ; 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
inductance of the latter, and is
therefore only a small fraction of
the total pressure. On account
of safety in handling the instru-
ment, it is also advisable to earth,
if possible, that terminal of the
9 generator which is directly con-
nected with the terminal A of
the wattmeter.
The theory of the ordinary
dynamometer used as a watt-
meter given above is only correct if the assumption of
an inductionless shunt circuit is justified, by reason of
the inductionless resistance being very great as compared
with the reactance 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 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 induct-
ance which must produce some lag. If this inductance
is not negligible, a correction to the reading must be
made as shown in the following theory.
Let Oe in Fig. 108 represent the vector of the total
FIG. 107. Incorrect method
of connecting.
MEASUREMENT OF POWER
200
supply pressure, and O/ that of the main current through
the fixed coil, which has a lag 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
i//. The correcting factor is
then slightly less than unity, and attains its minimum if
,. Each
field produces eddy currents
lagging by 90, the strength of
which is proportional to the flux.
We thus get i m as the eddy due to m or I and i s as
the eddy due to fa or E. The turning moment exerted
is therefore proportional to
COS tp(l m (f) s -J- t s m )
E
FIG. no. Vector diagram of
induction wattmeter.
THE INDUCTION-WATTMETER 213
and since fluxes and eddies are proportional to I and E,
we have
Torque = cos the
FIG i i2.-vector diagram of
wattmeter.
MEASUREMENT OF POWER 215
phase angle 2 r z r^
where k is a constant. We have therv
a = A (e^k - e. 6 k] + B(^ - e^k)
Now, e^ e^ is nothing else than the mesh voltage
between A and C, and e. 2 e z is similarly the mesh voltage
between B and C. Call the former E ac and the latter
E 6c . The formula for a may then be written thus
Now, we know from the investigation previously carried
out on Behn-Eschenburg's connection that the expression
in brackets is a measure for the total power P, so that
The deflection shown on the dial of Franke's duplex
wattmeter is a true measure of the total power trans-
mitted by an unsymmetrically loaded three phase line.
Three-voltmeter method. Profs. Ayrton and Perry
have devised a method of measuring the power transmitted
to a consuming device which can be applied in all cases
where the available pressure is sensibly greater than that
required by the consuming device, and the necessary
reduction in pressure can be made by the interposition
of an induct ionless resistance. In this method it is
220 TRANSFORMERS
necessary to measure three voltages, hence the name.
We measure the total voltage and its two components.
The method will be understood from Fig. 115, where
~ is the generator, a an amperemeter, W an induc-
tionless resistance, 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.,
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 a
suitable switching arrange-
ment, as shown in Fig. 117.
This arrangement is pre-
ferable on account of its
greater simplicity, and be-
cause slight errors in the
FlG. I i 5 .-Ayrton and Perry three-volt- .
meter method of measuring power. Calibrations of the instru-
ment have less influence on
the result. The clock diagram of the method is shown
in Fig 116. OI is the current, OE^^ is the E.M.F.
impressed on the transformer, and EjE is the E.M.F.
absorbed in the resistance W. Since the latter is induc-
tionless, its vector EjE must be parallel to the current
vector OI. OE=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 OI x OA. Since W has no inductance, we
have AE 1 = BE= J , the E.M.F. of self-induction, due to
T, and the following equations obtain
from which we find
The power is given by the formula
THREE-VOLTMETER METHOD 221
^
To find the power we must take four readings, namely,
three voltmeter readings and one amperemeter reading.
If the resistance W is accurately known, the last reading
may be omitted and the power calculated according to
the formula
2\V
This is the power actually supplied to the apparatus
T, in our case a transformer. If
we want to know the power
2
supplied by the generator _ :? >
the power lost in the resistance
must of course be added, and we
obtain
Instead of calculating P, we can
find the watt-component of e l
graphically by drawing circles
with radii e l and e, and shifting
a vertical line parallel to itself Fl0 ' "Sl^^* 8 *""*
until a position is found in which
the piece contained between the two circles is exactly
equal to c z . This gives the position of the point E l in
Fig. 1 1 6, and therefore also the length of the vector
OA = e w . The power is then
The diagram shows at a glance that a small error in the
volt-measurements will produce the larger an error in
the determination of the power the nearer the circle
of e l is to O or e, and that the error will be least if e 1
is midway between O and e. To obtain an accurate
measurement of power by this method we must, therefore,
so choose the resistance W that e 2 does not sensibly differ
from e lt that is to say, that about the same pressure is
lost in the resistance as is used in the apparatus under
test. The total voltage e must then be considerably
222
TRANSFORMERS
greater (from ij to 2 times) than that required by the
apparatus under test.
Another difficulty is the necessity of using up in the
ballast resistance W approximately the same power as
in the consuming device itself. The method is therefore
specially applicable in cases where the power to be
measured is small such, for instance, as an open circuit
test of a transformer. The ballast resistance may con-
veniently be a lamp-board, and in order to use the same
voltmeter for all three readings the special switch shown
in Fig. 117 may be used. When testing small trans-
formers the power to be measured (being practically the
FIG. 117. Three-voltmeter method of measuring power.
equivalent of the iron losses) is so small that the power
required by a dynamometric voltmeter can in com-
parison not be considered as negligible. In such a case
the three-voltmeter method is specially useful. To
avoid the necessity of making a correction for the power
taken by the voltmeter, we may use an electrostatic
instrument.
A is an amperemeter, V a voltmeter having about
twice the range of the voltage necessary for the trans-
former, and S is the special switch. It is advisable to
make the supply voltage adjustable so that the test may
be made under the condition giving greatest accuracy, as
explained above. Where the supply voltage is no higher
THREE-AMPEREMETER METHOD 223
than that required by the consuming device the three-
voltmeter method is not applicable, but then we may
use the
Three - amperemeter method devised by Professor
Fleming. Current at fixed pressure is supplied at the
terminals K, Fig. 118, and is taken through an ampere-
meter a, at the other side of which it is "divided into two
circuits, one containing the transformer T to be tested,
and the other an inductionless resistance W. These
two currents are measured on the amperemeters a x and
a. 2 ; the pressure is measured on the voltmeter e. The
clock diagram of this combination is shown in Fig. 119,
where OE represents the pressure of the supply current, i
the primary current of T, and i w its power component ;
i 2 is the current flowing through the resistance W ; and
r ,m
^^ I\WVWWV
__r__/WVVWW\A
FT
i
'2
FIG. 118. Fleming's three-amperemeter FIG. 119. Vector diagram
method of measuring'power. to Fig. 118.
its vector must of course be parallel with OE. From
the diagram it will be seen that the following relation
obtains
The power is given by
If the resistance W is accurately known, the reading for
e need not be taken, and the power may be calculated
from
Also in this method accuracy depends upon the proper
choice of the resistance. It should be so adjusted that
/ 2 is not sensibly different from /, ; the total current i will
224
TRANSFORMERS
then be from i^ to 2 times the primary current zi taken
by the transformer. Fig. 1 20 shows an arrangement of
switches and connections whereby the same amperemeter
is used for all three readings. I is a single-lever and
1 1 a double-lever switch.
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 re-
sults 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
FIG. 1 20. Three-amperemeter method of measuring- power.
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
experimental 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-amperemeter 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
is valid at any time in the cycle. We also have at al!
times
THREE-AMPEREMETER METHOD 225
and the power at any moment is
f) = i^ l __f j or e
7 2 ,,2 i ^ ^, x, i ^2
The work done in the time T of a complete cycle \sf' r pdt,
and the effective power is
T
~
1 2 W o o o
It has been previously shown that the expression
= r r e^dt is simply the square of the effective pressure
indicated by the voltmeter ; if now we denote these
effective pressures by e, 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 applicable to currents of any form.
CHAPTER XI
TES TING TRA NS FORMER STES TING SHEE T-
IRON SPECIAL IMPLEMENTS BY DOLIVO
DOBROWOLSKY, KAPP, EPSTEIN, RICHTER,
E WING THE BALLIS TIC METHOD THE
FL UXOMETER SCOTT S METHOD KAPPS
METHOD
Testing transformers. -- By means of the various
methods above explained the output and efficiency of
transformers can be determined. It is of course neces-
sary to have a source of current capable of supplying all
the power wanted, and a load capable to absorb the full
output of the transformer. To obtain by this direct
method anything 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
kw., and let there be an error of i per cent, in each
measurement, the error being negative in the measure-
ment of the input and positive in the measurement of the
output. The measurements would then be 99 and 98
kw. 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 out-
put of No. i forms the input of No. 2, and the output of
this, supplemented by an external source of power, the
input of No. i. We obtain thus a circulation of power
through the two transformers, and need only supply as
226
TESTING TRANSFORMERS
227
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 arrangement of apparatus
is shown in Fig. 121. D and B are the two equal trans-
formers, and C is a small auxiliary transformer which
supplies the waste power and thus keeps the total power
in circulation. Into the primary of C we insert an in-
ductionless rheostat R, for the purpose of adjusting the
pressure supplied to C, so as to obtain in the ampere-
WWWW\A/W
FIG. 121. Testing transformers.
meter A the normal full load current of the big trans-
formers. The connections between the latter must,
of course, be so arranged that their E.M.Fs. oppose
each other. If the large transformers were only con-
nected to C, the full current could be obtained in
them, but not the pressure. To ensure that also the
right pressure is maintained in B and D, we connect
their primaries, shown in Fig. 121 as thick wire coils,
with the generator. The connections are taken through
the wattmeter W\ and through the electrical centre of the
auxiliary transformer. The object of the latter arrange-
ment is to ensure that the voltage on the primary of one
transformer shall be raised by the same amount as that
of the other is depressed, so that the induction in both
228 TRANSFORMERS
shall be as nearly alike the normal value as possible. If
the centre of the auxiliary transformer is not accessible,
the connection may be made on one of its terminals, and
then there will be some inequality in the working condi-
tion of the two transformers, but the error thereby intro-
duced is not very serious, since the difference in primary
voltage is comparatively small. The wattmeter W 1 is
introduced to measure iron losses. The copper losses
are measured on the wattmeter W 2 . If we short-circuit
the rheostat 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 lt Since both transformers 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 wattmeters W 1 and W 2 measure
all losses.
The voltage on each primary is measured by the volt-
meter V, which is provided with a change-over switch s.
Let e be the average of these two readings and i the
current indicated on A, then the combined power of both
transformers is iei and the total loss is the sum of the
two wattmeter readings
W = Wj + W 2
The efficiency of each transformer is therefore
ei
W
-
2
To get the true copper losses switch S should be
opened. If it remains closed whilst the contact on the
rheostat R is put down to the lowest contact so as to
short-circuit C, there will still be a small current (namely
the magnetising current of transformer D) flowing through
the wattmeter W 2 . If now the contact is raised so as to
produce a main current, this small current will, according
to the connection, either increase or diminish the current
flowing through the wattmeter, and to this extent W 2 will
indicate either a little more or a little less than the true
TESTING TRANSFORMERS 229
copper loss. The error may be avoided by repeating
the test with W 2 inserted in the primary of B and taking
the mean of the two readings. In practical work for
measuring efficiency this correction need, however, not
be made, since the error is very small ; whilst for measur-
ing copper losses only, the simple expedient of opening
switch S is sufficient to avoid the error.
The test illustrated in Fig. 121 can also be used to
determine the drop by opening S and moving the contact
of R to such a position that a predetermined (preferably
the normal working) current flows through the primaries.
The voltage must then be read on a second voltmeter
(not shown in the diagram), which is connected to the
primary terminals. Let e Q be this voltage, i the current,
and w the reading on W 2 , then the equivalent resistance
r of one transformer is
Its ohmic drop is
and its inductive drop is
(O
U= /5L-
)U^)being the reactance of the equivalent coil.
The advantage of this method of testing is not only
great accuracy, but also economy of power. The latter
point is of importance when testing for temperature rise,
since the final temperature is only reached after many
hours, and in the case of large transformers some days
of working at full power. The cost of power and the
difficulty of using it up in an artificial load become thus
serious obstacles, so that a test which only requires the
supply of the power wasted, as that shown in Fig. 121,
is also commercially advantageous.
It is, however, not always possible to test two equal
transformers together for temperature rise. In this case
the transformer may be preliminarily heated in a drying
room (most electrical engineering works are provided
230 TRANSFORMERS
with such a room), and then put to work under normal
load, whilst thermometric or resistance readings are taken
from time to time to find out when the final temperature
rise has been reached. Or the transformer may be
worked alternately on open circuit to heat the iron, and
have continuous currents passed through both coils to
heat the copper. This preliminary period of heating
may be shortened by working at increased voltage and
current density. When the probable final temperature
has been reached the transformer is put to work normally,
and kept at work until the final temperature has been
actually reached.
Another method is to heat the iron by alternating
current sent through one winding, and the other winding
at the same time by continuous current. This winding
must, of course, be opened in the middle, and the two
halves must be coupled up in opposition so that no alter-
nating E.M.F. is produced at the two free ends. By
keeping a record of the continuous current and E.M.F.
supplied to this winding the rise of its ohmic resistance,
and therefore the rise of its temperature, may be graphic-
ally represented as a function of the time. From this
curve the time constant for heating may be found, and
from that the final temperature rise and the time in which
it would be reached may be computed.
The insulation of a transformer should be tested when
hot. It is also advisable to flash the transformer, 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 connection should be made between
(a) a primary and secondary terminal ; (b) a primary
terminal and carcase ; (c) a secondary terminal and car-
case. Care must of course be taken that during these
tests both poles of the generator are well insulated from
earth, or the carcase must be insulated from earth.
Testing sheet-iron. An obvious way of testing any
particular batch of plates intended to be used in the
manufacture of transformers, is to select at random some
of the plates sufficient for the carcase of a small trans-
former (preferably a stock size), wind it in the usual way
and test for iron losses. If the test of this sample is
SPECIAL IMPLEMENTS
231
satisfactory the whole batch can' be passed as suitable.
The drawback to this method is that the building up of a
complete transformer takes too much time. What is
required is a method of testing samples which does not
involve the winding of coils, and where the test pieces
are of a simple form, so that but little time is required in
the preparation of samples and not too much material is
wasted.
Special implements. One of the oldest instruments
is the iron tester of Dolivo Dobrowolsky, shown in Fig.
122. It has now only historic interest. 1 It consists of
two | | shaped cores of sheet-iron, which can be laid
together either directly or placed on
either side of the sample AA to be
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 ; when 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
through the yoke. The current
magnets
is
FlG. 122. Dobrowolsky
iron tester.
measured by a dynamometer marked EL Dyn. 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 magnetisation. The power corre-
sponding to various values of the induction is then
measured, the induction being calculated from the fre-
quency, the pressure and the known data of the coils and
magnet cores. The sample is then inserted, the switch
changed over and the measurements repeated. The
1 First published in 1892 in the Elektrotechnische Zeitschrift, from
which Fig. 122 is copied.
232
TRANSFORMERS
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 drawback of this
method is the difference in magnetic leakage with and
without the sample. If the magnets are laid together
directly, and magnetised circularly, there is hardly any
leakage, and B can be calculated from E with great
accuracy. If the sample is inserted, the magnetic resist-
ance is increased, and leakage produced which diminishes
the 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 diffi-
| culty, the author has con-
structed the apparatus
shown in Fig. 123. The
i 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 longer sides are surrounded
by coils, the upper one being large enough to admit
the insertion of the sample without difficulty. The
connection is made for circular magnetisation, so that
only very little leakage takes place, and this is the
same for all samples. The sample must have approxi-
mately 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
FIG. 123. Kapp iron tester.
SPECIAL IMPLEMENTS
233
A slight error in the measure-
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 objection to this method of measuring the loss
in the sample is that the loss is obtained as the difference
between two measurements, both of which are larger
than the result desired,
ment of the total loss
may therefore mean
a large error in the
result. This draw-
back has been over-
come in the apparatus
shown in Fig. 124,
which has been de-
signed by the " Hys-
teresis Committee" of
the German Associa-
tion of Electrical
Engineers, and offici-
ally accepted by this
Association in 1902,
after having been on
trial in various works
for some years. The
instrument is also
known as the " Ep-
stein Iron Tester,"
Prof. Epstein having
been chairman of the
Committee. In this
method of testing no foreign iron is used, the whole of
the magnetic circuit being made up of sample plates in
the form of a square. Each side of the square is a bar
made up of strips with tissue-paper insertion. The bars
are 50 cm. long, and have a cross-section of 30 mm. by
about 25 mm. Each bar contains 2*5 kg. of plates, so
that for testing each batch a little over 10 kg. of plates
(allowing for waste) have to be cut up. Each side of
the square is surrounded by a magnetising coil, the four
FIG. 124. Epstein iron tester.
234
TRANSFORMERS
coils being fixtures of the apparatus and wide enough to
admit the samples. The samples abut at the corners,
where they are pressed together by screws, a thin sheet
of fibre being placed in the butt-joint to avoid loss of
power by eddies. Fig. 125 shows a diagram of con-
nections. The terminals k, k of the apparatus are con-
nected to the supply terminals K, K, from which the
magnetising current is taken. To find the "figure of
loss " the frequency to be used is 50 and the induction
10,000, which corresponds to about 85 volt. Each coil
has 1 50 turns, and the total resistance of the four coils
is 0*18 ohm. Since the magnetising current is only a
few ampere, the correction for copper loss is very small
in comparison with the figure of loss, which for alloyed
FIG. 125. Epstein iron tester.
iron is of the order of magnitude of 2 watt per kg. or
20 watt for the whole sample. It is advisable to use
a frequency indicator (not shown in the diagram) when
an accurate test is required. The magnetising current
is adjusted by the rheostat, and the power is measured
on the wattmeter W. The net area of the bars A is
calculated from the weight and density (about 777), and
their total length (2 m.) and the induction from the
formula
e being the E.M.F. induced in the four coils. This is
very nearly also the E.M.F. indicated on the voltmeter
V, but for very accurate work e may be found by correct-
ing the voltmeter reading for copper loss, the correction
SPECIAL IMPLEMENTS 235
being of course made vectorially. It is important to
open the voltmeter switch s when reading the wattmeter,
since the power taken by the voltmeter would otherwise
be counted as part of the iron loss. When measuring
the current it is advisable to open the switch s of the
pressure coil of the wattmeter so that only the true
magnetising current may pass through A. The machine
used as a source of E.M.F. should give as nearly as
possible a sine wave of E.M.F. The curve shown in
Fig. 1 1 gives total loss with alloyed plates of English
manufacture obtained by the author 'with this implement.
The power measured on W represents the combined
hysteretic and eddy current loss occurring in the sample,
and as the weight of the sample is known the loss per
unit weight can easily be found. We thus get the
quantity, which is of immediate interest to the designer,
but it is also possible to get hysteretic and eddy current
losses separately. From equations (8) and (So) it will be
seen that the total loss for any given sample of iron is
given by an expression of the form
P = /*vB*+/(vB) 2 (38)
where h and /"are constants depending on the quantity
and quality of iron under test and the thickness of the
plates. The exponent of B is usually taken as 1*6, but
to keep the investigation general we call it x. B is
found from the determination of e, as above explained,
and v is read on the frequency meter. We have thus
three unknown quantities, namely, h, f and x, and by
making three tests under different conditions we can
obtain three equations from which the three unknown
quantities are determined. The operation can be a little
simplified if we make all the tests for the same induction,
for which the condition is
e
- = constant
v
We need only vary the speed of the generator and
1 i i
keep its excitation as nearly constant as is required by
this condition. These two tests suffice. In these B
and therefore B^ and B 2 will remain constant, and the
236 TRANSFORMERS
equation for the lost power in the cases where the
frequency is v l and v 2 may be written in the simple
form
from which H and F may easily be found. We have
then for the fixed induction B
Hysteretic loss at frequency v x . . . .
v 2 . . . .
Eddy current loss at frequency v .
To get complete curves of losses the tests have to be
repeated with different values of B. To any two values
of B, say B and B', and the same frequency v correspond
two hysteretic losses, P^ and P/, so that
P AB
from which x may be found by taking logarithms
_
Having x, we find h from
and the coefficient^ may be found from
/ -Bl
The Epstein iron tester may thus be used not only
to find the total loss for any values of B and v, but also
to separate hysteretic and eddy current losses and deter-
mine the coefficients in their formulae.
Mr. Richter has devised an implement for the testing
SPECIAL IMPLEMENTS
237
of complete sheets as they are produced by the rolling-
mill. His object is to avoid the labour and waste of
material when cutting up sheets into sample strips. The
magnetising coils are long' and narrow rectangles, held
in a wooden frame with their long sides parallel to each
other and arranged circularly around the axis of the
frame. The coils are placed evenly round the axis, with
sufficient space between to allow the sheets to be slid
through their openings so as to form a closed cylinder,
FIG. 126. E wing's iron tester.
which is circularly magnetised. From the weight and
dimensions of the sheets the cross-sectional area of the
magnetic circuit can be calculated, the test being made
by voltmeter and amperemeter, as in the Epstein
apparatus.
An implement in which samples of only a few ounces'
weight can be tested has been devised by Prof. Ewing.
Its principle is the purely mechanical determination of
the hysteretic loss alone in a sample of very small
dimensions, namely 6 to 8 strips of 3 in. length and
238 TRANSFORMERS
-| in. width. The apparatus consists of a permanent
magnet e, Fig. 126, which is suspended on knife-edges, f t
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 produced when the sample a is
rotated between the poles. The sample is fastened by
screw-clamps b, b 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 the rotation
of the sample, and the work lost in hysteresis and eddy
currents per revolution is 2 TT x torque. The torque is
indicated on the scale by the pointer, and since 2?r is a
constant, we find that the deflection of the pointer gives
directly a measure for the loss per cycle, the speed of
rotation 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 over that of the sample itself. The object
of this arrangement 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 induction of about 4000 C.G.S. 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 seven strips of 0*37 mm. thickness.
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
determined by the ballistic method. Two such standard
samples are supplied with the apparatus, together with
THE BALLISTIC METHOD 239
tables giving the results of 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 hysteretic 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.
The ballistic method. The methods of testing iron
above described suffice for the immediate requirements
of the designer, but special circumstances may arise
when it is desirable to know not only the power lost in
hysteresis and eddies, but also the shape of the hysteretic
loop. This cannot be found by any of the methods
hitherto described, and to get a complete knowledge of
the magnetic qualities of any brand of iron, some method
must be used which gives the relation between exciting
force and induction throughout a complete magnetic cycle.
Such an investigation is also necessary for the calibration
of certain workshop implements, such as the Ewing iron
tester.
To find the B-H curve we may use a ballistic
galvanometer, and make use~of the well-known physical
law that the deflection of the moving system of such
an instrument is proportional to the total quantity of
electricity which has been suddenly discharged through
it. The moving system may be a little magnet or a coil
as first used by Deprey D' Arson val. In the first case
the system is only slightly, in the second more effectively,
damped ; a certain amount of damping is unavoidable,
and, indeed, necessary for rapid working.
The elongation of the spot of light of a damped
galvanometer is given by the well-known formula
where /, a and b are constants, and V Q is the initial
velocity with which the spot of light leaves its position
of rest. Counting the time / from x = o, then the interval
of time between two successive passages through zero
positions in the same direction is given by the condition
240 TRANSFORMERS
that the sine must be zero for both. We find thus the
periodic time
T _ 27r
: b
According to the theory of damped vacillations we have
where c is the controlling force, m the mass, and 8 a
coefficient which, multiplied by the velocity, gives the
damping force, all values being referred to the spot of
light. In a perfectly undamped galvanometer 8 = o and
b = 'Y , so that its periodic time is
Im
= 27TA/
1
the well-known pendulum equation. It is obvious that
T>T , that is to say, that damping lengthens the periodic
time. As a further result it will be seen that damping
reduces the first and all subsequent elongations.
Before entering into the question how the effect of
damping can be allowed, for let us assume that it were
possible to make a perfectly undamped instrument, and
consider how such an instrument could be used for testing
iron. The correction for damping can then be considered
later.
The first (and indeed every subsequent) elongation
X Q is then proportional to the initial velocity z/ , and this
again is proportional to the quantity discharged through
the galvanometer from a coil through which the flux <
is reversed or annulled. The first elongation is thus a
measure for the flux passing through the sample which
is surrounded by the coil. If the area of cross-section A
be known, and the magnetising force H be measured, we
can obtain the relation between B = ^ and H. Instead
A
of annulling or reversing the flux we can also change it
t> O O
suddenly, but by small increments (by changing H), and
thus get step by step the relation between B and H, that
is, the hysteresis loop. It is, of course, necessary to
THE BALLISTIC METHOD 24!
determine once for all the ratio between B and .% or in
other words to calibrate the galvanometer, and this
may be done in a variety of ways, which we shall now
consider.
Take a straight solenoid whose length / is at least
twenty diameters, and which has n^ turns, and place into
the centre a small co-axial pilot coil of area A and n turns,
then by sending l l ampere continuous current through
the solenoid there will be created within the pilot coil a
flux < = AH, where
H
The same holds good for a ring-shaped coil when /
is the mean circumference. The turns of the pilot coils
may be distributed all round the ring or placed in one
part only, but they should be below those of the magnet-
ising coil. The total flux AH which passes through the
n turns of the pilot coil is therefore known. If then we
observe the elongation, if the magnetising current is inter-
rupted (or reversed, which gives it twice as great), we can
determine the ratio between elongation and linkage flux.
As will be shown below, this ratio is constant for a given
resistance r in the galvanometer circuit, and we thus get
the equation
where b is the " ballistic constant." If we include in this
circuit not only the pilot coil of the standard solenoid,
but also that of the sample to be tested, the ballistic
constant need not be determined. All we need do is to
determine alternately the deflection obtained with the
two pilot coils and calculate from this the linkage flux of
the sample. This method of comparative observation
can be conveniently carried out by using the arrange-
ment shown in Fig 1 27.
S is the standard, A x an amperemeter to measure its
magnetising current l ly \J 1 a reversing switch, Rj a
rheostat, and B a battery. The sample is prepared in
form of a ring wound with two coils, one the magnetising
16
2 4 2
TRANSFORMERS
coil receiving current from the same battery through a
rheostat R 2 and reversing switch U 2 , and the other a
pilot coil in series with the galvanometer G, and the
pilot coil of the standard S. A resistance, r, is inserted
to reduce the deflection of the galvanometer to a con-
venient amount, and s is a damping key by which after
vWvWWv I /vWA/WV
FIG. 127. Ballistic test.
each reading the moving coil can be quickly brought to
rest.
The E.M.F. produced by a change of flux in the
standard is -^- 5f microvolt, and the corresponding
100 dt
current is . ^>JL *X . microampere. The quantity
r 100 dt
discharged through the galvanometer is, on reversal
of Ij
Q = fidt = . 2< P n = fa microcoulomb
o loor
The ballistic constant for microcoulomb is therefore
, I 2(t>n
=
XQ ioor
We may thus calibrate the galvanometer for micro-
coulomb or any other convenient unit of quantity. But
THE BALLISTIC METHOD 243
this calibration is not necessary if we wish to use a
merely -comparative method, as may be seen from the
following. Let the various quantities in the sample be
denoted by the same letters as in the standard, but
distinguished by a dash, thus
loor
<' = ^
If the flux linkages n$ and n f are not very widely
different, so that the deflections are of the same order of
magnitude, this method of testing is convenient, as the
constant of the instrument need not be known ; but if the
flux linkage of the sample is either very much greater
or very much smaller than that of the standard, it
becomes necessary to adjust r so as to get convenient
deflections, and then the simple proportionality between
quantity and deflection is lost. It is no longer admissible
to use a comparative method, and it becomes necessary
to determine the ballistic constant. One way of doing
this has already been shown. We found from a test on
the standard for microcoulomb
(39)
ioor
Another and very obvious method is to determine b by
discharging a condenser through the galvanometer and
observing the deflection. The arrangement is shown in
Fig. 125, where C is a standard condenser, Cl a Clark,
or other standard cell (*= 1*4323 5 volt at 15 C. for
the Clark, or 1-0196 volt at 15 C. for the cadmium cell)
and S a two-way key. The quantity discharged is then
with a condenser of C microfarad Ce microcoulomb, and
if a deflection of x scale dimensions is produced by this
discharge we have
b = (40)
244 TRANSFORMERS
A third method of finding b is as follows. Let k be
the constant of the galvanometer for steady currents, so
that
i = yx microampere
The deflecting force is proportional to the current and
also proportional to the deflection or
ex = at and a =
r
The acceleration on starting is produced by the force at,
and we have therefore
dv
y 00 -
/ aidt = mv
a Q = mv or abx = mv
From v' 2 = C -3? we find v=x*\ , and this inserted
171
gives
, , c , , b ./m
ab = Jem or - b = Jem or - = V
7 7 c
but for a completely undamped instrument the periodic
i m
time is 2irV = T, so that we can also write
27T X 27T
By sending a known fraction of a known current through
the galvanometer and observing the steady deflections
we determine 7, and by taking the current off and timing
the oscillations we determine T. From these two ob-
servations b can be found.
It should be noted that b depends on the damping
force, which may be considered constant, but not on the
resistance r, which may have to be raised between wide
THE BALLISTIC METHOD 245
limits so as to get convenient deflections at all values of
n(f>. We thus have the general formula to express a
sudden change in flux through the sample
lOOo:
n
In this formula x is the deflection which would have
been observed if the galvanometer had been absolutely
undamped. In reality the deflection is (because of damp-
ing) a little smaller, say x& We have from the theory
of harmonic motions for an undamped oscillation
and for a damped oscillation
7t ~ an
*>*>'&*
since the time t of a quarter period is . The ratio be-
n IT
T
20
air n IT
tween the two deflections is e~ . Write 8 instead of T ,
then
XQ = xe~ ft or x = xtfP
Let x be the next elongation in the same direction, x 2
the second next, and so on, then subsequent swings take
place in intervals of
T
=
27r
and the exponent of e in the equation for the deflections
x& x lt x. 2 . . . x n becomes
for x ....... y8
.;> ....... -j8-4)8
....... -p-sp
246 TRANSFORMERS
From this follows
= .... (42 )
4^ *
Since ft is a very small number we can in the series
neglect the third and subsequent members and write
* = *-o(i+) (43)
x$ is the first elongation actually observed with a
moderately damped galvanometer, x is that elongation
which would have been observed if the galvanometer had
been absolutely undamped. It is this value x and not X Q
which has to be used in the determination of the ballistic
constant and the calculation of the change of flux from
b i vorx
n
The number /3 is called the logarithmic decrement ; a
convenient value of it is 2 or 3 per cent.
The fluxometer. The use of a ballistic galvanometer
presupposes the ability to change the flux very suddenly,
for the whole discharge from the pilot coil must be
completed before the moving system has appreciably
changed its position of rest. The condition of a very
abrupt change of flux is not difficult to fulfil if dealing
with a sample of moderate size, but if we attempt to take
the hysteresis loop of the carcase of a large transformer in
this way we find that unless an enormous resistance is
put into the magnetising circuit the change of flux does
not take place rapidly enough for the galvanometer, and
it is preferable to use a method of investigation which is
independent of the time rate at which the flux changes.
Such methods have been devised by Mr. C. F. Scott, the
Author and Mr. Grassot, the latter using a special
instrument, termed by him a fliixometer. I take this
first, as being more akin to the ballistic galvanometer.
THE FLUXOMETER
24;
A delicately pivoted coil not subjected to any con-
trolling force swings in the strong field of a permanent
magnet, Fig. 128, the arrangement being similar to that
used in D'Arsonval instruments, but without a controlling
spring. The terminals, T, of this coil are connected to
the coil encircling the flux which is to be measured, say
the low-tension coil (or part of the low-tension coil) of a
transformer. The fluxometer coil is provided with a
pointer on one side and a mirror on the other, so that its
angular displacement may be observed either directly or
by a beam of light. The coil is set mechanically into its
zero position when its plane is parallel to the polar axis
FIG. 128. Principle of fluxometer.
of the permanent magnet and no flux threads it. If a
current is sent through it a torque is exerted and the
coil takes an angular position, the flux now threading it
being proportional to the angle of deflection.
The principle underlying the action of the instrument
is the physical law that any electric circuit has a tendency
to maintain its total linkage flux. If this quantity is
forcibly diminished in one part of the circuit another part
will try to restore it. Thus, if a pilot coil has been placed
over the middle of a bar magnet and is then stripped off
the linkage flux through the pilot coil is reduced to zero.
The fluxometer coil will then set itself at such an angle
as to thread the same linkage flux as that which has
vanished in the external part of the circuit.
Let M be the mass of the moving system,
c\ be the force exerted by the current through
the fluxometer coil,
2 4 8 TRANSFORMERS
Let Dz> be the damping force at speed v,
sv be the E.M.F. in the fluxometer coil generated
when moving at the speed v through the field
of the permanent magnet ;
the quantities M, c, D, and v referring to a point on the
indicating needle i cm. distant from the axis of rotation.
Let, further, L be the inductance and r the resistance
of the fluxometer coil, then an E.M.F. E applied to its
terminals will produce a current I and a displacement.
We have
at
T? T a ^
E - ev - L
di
dt
*''- w- L
dt dt
dv ^E csv ^L dfl
di~~ r r r dt
r \r ) r
Integrating this equation and remembering that both
I and r are zero at the beginning and at the end of the
process, we get
r /"x* //~ \ x* 00
- - / E* - (- + D )/ vdt
r \r J
Now the integral of vdt is simply the excursion of the
pointer 0. The damping coefficient D is so small that
it may be neglected, and we thus get
/*
= J Edt
o
Since E is produced by the change of the flux
through the pilot coil of n turns, we also have
THE FLUXOMETER 249
= n di
e / \
Yl~~Y^ = ~9 ' (44)
The change of flux is proportional to the deflection
9, and the latter is independent of the rapidity with which
the change takes place. We have defined s as a
coefficient which, multiplied by the speed of the point
to which all quantities refer, gives the E.M.F. induced in
the fluxometer coil. Since we have assumed this point
to be i cm. from the centre of rotation, 9 is not only a
length but also an angle, and v is not only a linear
speed but also an angular speed. The dimensions of
vz are therefore those of an E.M.F. L f M*T~ 2 , whilst
those of v are T" 1 , giving for s
L f M*T- 1
the dimensions of a magnetic flux ; this is in accordance
with the above formula (44), for 9 and n are simply numbers
having no dimensions. The factor e is therefore a
constant for each instrument, and is equal to the
product of the flux produced by the permanent magnet
with twice the number of turns in the fluxometer coil
divided by the angular length of the arc spanned by
each pole-piece. The calibration of the instrument is
done empirically, and in one specimen in the Author's
possession 5=12300 when 9 is reckoned not in radians
but in scale divisions. For this particular instrument
one scale division represents therefore a flux of
n
lines of force. The smallest number of turns we can
have in the exploring or pilot coil is n= i, so that the
total range (the scale has on either side 100 divisions)
is a little over a megaline. The method of using the
instrument for taking the hysteresis loop of the iron of
250
TRANSFORMERS
a transformer is shown diagrammatically in Fig. 129.
Two terminals of the transformer T are joined to the
supply terminals, K, K, of a source of continuous current
through a reversing switch A, amperemeter A, and
regulating resistance R. P is the exploring coil, con-
sisting of a simple loop, and this is connected with the
fluxometer F. If the transformer be very small, it may
be possible to use one of the windings as an exploring
coil. To take the hysteretic loop proceed as follows.
Set R to zero E.M.F. and put the fluxometer to zero
mechanically. Then shift R so that a current I is
indicated on A and observe the deflection of the fluxo-
FIG. 129. Testing transformer by fluxometer.
meter. Put fluxometer to zero again and then take another
step on R, observing again the new value of the current
and the new deflection on the fluxometer. Proceeding
thus step by step we get the positive rising branch of
the loop, the descending branch is found by bringing R
back in steps to zero. Then S is reversed and the whole
process repeated for the negative part of the loop.
When the loop is plotted take the area f\d$ by plani-
meter. The energy wasted in one cycle is obviously
T/jicryiflty, where n^ is the number of turns of the
magnetising coil. If there be butt-joints the hysteretic
loop will be sheared over, but its area will not be altered,
since air has no hysteresis. Let A be the area of cross-
THE FLUXOMETER
251
section of carcase, and K its weight in kg., then the
hysteretic loss at frequency v is for the whole carcase in
watt-
area of loop x IGT^/Z! = aior 8 vn 1
and the loss in watt per kg. is
for the induction
To calculate the loss per kg. is, of course, only possible
if the cross-section is con-
stant throughout the mag-
netic circuit.
The value of < is taken
from the loop ; as a check
it may also be taken directly
by reversing S and observ-
ing the fluxometer. The
deflection will then be twice FlG - 130. Varying range of fluxometer.
that corresponding to (, and
the direct method is with this particular instrument there-
fore only applicable for values of under o'6 megaline.
It is, however, possible to increase the range of the
instrument so that it may be used for measuring the
flux passing through any, even the largest transformer.
Let in Fig. 130 P be the pilot coil of one turn
encircling the flux <. Join its leads to an inductionless
resistance Rj, and from a small fraction of this resistance
take leads to the fluxometer whose resistance is r. In
the instrument mentioned above r is a little under 20
ohm. If we make R about o'l ohm then only ^ per
cent, of the current going through the coil will be shunted
through the fluxometer, so that practically the same
current will flow through the whole of the resistance R 1?
and the E.M.F. impressed on the fluxometer will be to
the total E.M.F. generated in P as R : R x . The resist-
ance of the fluxometer need not be accurately known.
All that we require to know accurately is the ratio
252 TRANSFORMERS
of R : Rj. Let Rj = ;;zR, then on reversing the magnet-
ising current and thus producing a change in the linkage
of P amounting to 2 (/> we have
...... (45)
Since only the ratio of R to R x , and not their absolute
values, are of importance, temperature does not affect the
accuracy of the method, and copper may be used for the
resistance coil, which should be wound bifilarly. Making
m = 2O the range of the fluxometer can be increased to
about 12 megalines.
Scotfs method. Mr. C. F. Scott l has devised a very
ingenious method for plotting the curve connecting excit-
ing current and flux in any magnetic circuit by making
use of the law that constant E.M.F. in the pilot coil means
proportionality between flux and time. In its simplest
form the test is carried out as follows : Through the fine
wire winding of a large transformer, which we will call
the primary winding, we send a current which can be
regulated between a positive and negative maximum
at any time rate that may be required to keep the
E.M.F. in the secondary winding constant. This
winding is connected to a sensitive voltmeter, and the
deflection must be kept constant whilst the primary
current is made to change from a positive maximum
value to an equal negative maximum value. The test
requires three observers ; one watches the voltmeter and
operates the regulating appliance for the. current, the
second marks time, and the third reads on an ampere-
meter the primary current and books it against the time.
The observations thus yield in the first instance merely
a time-current curve, but as by reason of the secondary
E.M.F. being constant B and t are proportional, the
curve may by a suitable change of scale, also be made to
represent the relation between induction and exciting
force, and if the carcase contains no butt-joints also the
true hysteretic loop.
Various appliances can be devised for current regula-
1 " On Testing Large Transformers," by I. S. Peck, EL World and
Engineer, 1901, p. 1083 ante.
SCOTT'S METHOD
253
tion, but I have found two filaments of mercury as shown
in Fig. 131 a very convenient form of rheostat. The
contacts are attached to a block of wood, which is provided
with a handle, and can be moved longitudinally on the
board ; the latter has two grooves planed out for the
reception of the mercury. Current is supplied by a few
secondary cells and measured in A. If the sliding
block is in the middle no current flows ; if it be shifted
to the right the magnetising coil of % turns receives
current in one direction, and if shifted to the left in the
other direction. V is a millivoltmeter connected to the
secondary winding of n turns. In the diagram the two
windings are shown on different limbs ; this is merely
FIG. 131. Scott's method.
done to avoid complication. In reality both windings
are on both limbs. Both instruments have central zero.
Assume a carcase without butt-joints and let A be
the section and / the length of the magnetic circuit, then
the current passing through the millivoltmeter of resist-
ance R will be in ampere
. nA dE
corresponding to the E.M.F. in volt
- r
at
R may be considered to include the resistance of the
254
TRANSFORMERS
O/d Current
secondary winding, which is, however, generally very
small in comparison with the 100 ohm or so of the
voltmeter. If, however, a milliamperemeter be used to
indicate the E.M.F., then its resistance is much lower
(with Siemens' instruments exactly i ohm), and the
resistance of the secondary winding must be included in R.
The ratio for changing the scale of / in seconds to B
in C.G.S. units is therefore
i :
The current i necessary to work the voltmeter is very
small, and when we are testing
a large transformer negligible
in comparison with the mag-
netising current I lf but when
testing a small transformer and
when for V a milliamperemeter
is used the current i may
produce a sensible magnetising
effect on the carcase, and has
to be considered in determin-
ing the ratio between excit-
ing current, which is directly
plotted, and the magnetic force
H, which we require, if we
wish to plot the hysteresis loop. We have
TT __ . (#1 1 + ni
I
where I takes all values between the maxima I .
Let in Fig. 132 dWd be the time-current curve
plotted from the original observations. Before moving
the slider the magnetising current is + \ Q = oa. Im-
mediately the movement begins the current passes to
T t T n
h = oa = 1 1
where the second term is the pilot current i reduced to
the primary winding. In the diagram it is a f a = i Q . If
we wish to leave off the process exactly at the same
FIG. 132. Scott test.
SCOTT S METHOD 255
negative induction as it was before we must continue to
move the slider until A indicates I 2 = (I + / )- As
soon as the movement stops z' becomes zero, and we have
I 2 = - I . From a preliminary test we find io. If then
we start the test with a current + Ii, we must continue
until the current is I 2 = (Ij + 2/o). This gives equal
positive and negative induction. Shifting the curve
a'b'c' originally plotted to the right by the distance z'
we get the curve abc, and symmetrically to this the curve
cda to complete the loop. This gives as yet only the
relation between time and current corrected for the dis-
turbing effect of the pilot current. To get the true
hysteretic loop H-B we must alter the scales as already
stated. The operation may be shown by the following
example. In a transformer having 670 primary and
100 secondary turns A is 70 sq. cm. and / 136 cm.
The resistance of the secondary is 0*038 and that of
the milliamperemeter used for V in Fig. 128 is i ohm.
The total time taken to perform the change from
4- B to B is 34 seconds, the instrument V showing
steady 30 millivolt or 0*03 ampere. This gives
0-03 =
1-038 at
B
Since t is one half 34 we find
To make Fig. 132 represent a true hysteretic loop we
must use such a scale for the ordinates that #0 = 7514.
To find the scale for the abscissae we determine
100
/o = ^ 0-03 = 0-0045 ampere
The test is started with 0#' = o*28 ampere and finished
with - (0-28 + 2 x 0*0045) ampere. We have
I = o*28 + 0*0045 =0*2845
i TJ 1*2^.670.0*284^
and H = - ^ - ' ^L=\-^^
136
2 5 6
TRA NS FORMERS
The scale for the abscissae must therefore be so chosen as
to make oa= 175. The area of the loop divided by 4?r
gives the energy in erg which is used up in hysteresis
by the whole carcase if this is taken through a complete
cycle between 6 = 7514.
This energy may also be found as follows. The
reactance E.M.F. in the primary is obviously
=e-
n
and the power absorbed by the iron at any moment is
0J. The total energy is the integral of e^dt taken
between the limits shown in the time current loop ; that
is to say, the energy is e l times the area of the loop. As
FIG. 133. Scott test as altered by Morris and Lister.
the latter is obtained in coulomb and ^ is given in watt
the product will be joule.
If the carcase has joints the loop obtained will not
be the true hysteretic loop, but its area will still give the
energy wasted in hysteresis, since air is not a hysteretic
substance.
The necessity of making a correction for the magne-
tising force of the pilot current can be avoided by adopt-
ing an arrangement proposed by Messrs. Morris and
Lister, 1 whereby the E.M.F. in the pilot coil is balanced
by an externally provided E.M.F., so that no pilot
current flows. The magnetisation of the carcase is then
due to the primary current only. In Fig. 133 the
external source for balancing the E.M.F. in the pilot
coil n is a battery B sending a heavy current through
1 Journal Inst. EL Eng., 1906, vol. 37.
KAPPS METHOD 257
the fixed resistance W and rheostat R. As soon as the
switch S is closed a P.D. will be maintained between the
terminals of W, and this may be read off on the milli-
amperemeter M. G is a detector showing whether
current is flowing through the pilot coil or not. During
the test the operator of the sliding contact watches G
and keeps its needle at zero. L is an inductance in-
serted to make this task easier by steadying the primary
current, and u is a reversing switch so that the operation
may be repeated in the opposite sense.
Kapp's method. In the author's method l a time-
current curve is also taken, but the current is not regu-
lated by an operator. It is simply allowed to flow under
a constant impressed E.M.F. Let < be the flux in
megalines produced by a continuous current of I amperes
through n turns of winding under an E.M.F. of e volts,
then
If now e be suddenly reversed, then I will pass from its
initial value I through zero to the final value -f I -
Any intermediate value of the current must obviously
satisfy the equation
=*.^+RI ..... (46)
TOO at
By observing t and I a time-current curve may be plotted,
and from this curve and the known values of e and n the
hysteresis loop giving < as a function of I may be drawn.
The arrangement of the test is shown in Fig. 134.
B is a source of current 2 capable of giving from 50
to 100 times the magnetising current I , which is passed
through the transformer coil T. This current is taken
off on the heavy shunt resistance S, between whose
1 Journal Inst. EL Eng., 1907, vol. 39.
2 In a modified arrangement due to Mr. Dennis Coales, two equal
batteries are used coupled up in opposition. One of the batteries is
shunted by a rheostat so that its terminal E.M.F. becomes lower than
that of the other. The balance being thus disturbed, the resultant
E.M.F. of the two batteries is no longer zero. It can be adjusted by
the rheostat to the same value as that obtained in the Author's original
arrangement between the terminals of S.
2 5 8
TRANSFORMERS
terminals the E.M.F. e is maintained and indicated on
the voltmeter V. A is an amperemeter with a central
zero and ^ a reversing switch. Care must be taken to
have the contacts of this switch in good order, so that
its resistance may be exactly the same in either position.
S may conveniently be the shunt belonging to V, so that
this is instrumental in indicating the main current given
by B. All connections should be of sufficiently stout
wire, and A should be of sufficiently low resistance to
FIG. 134. Kapp's test.
reduce the loss of E.M.F. between S and T as much as
possible.
To make the test, regulate r so that A indicates the
desired magnetising current I and note the E.M.F. e.
Then knock s sharply over, starting at the same time a
stop-watch and noting the current indicated by A as a
function of the time. The movement of the needle for
values of I lying between I and zero is fairly quick, so
that in this region only single observations can be taken
by stopping the watch at the moment that the pointer
passes a predetermined point on the scale. After the
zero has been passed the movement becomes sufficiently
slow for a continuous series of co-ordinate values of
KAPPS METHOD 259
current and time to be noted. For transformers of
similar type the speed of the needle is approximately
proportional to the f power of the output. Thus, if
with a lo-kw. transformer zero is reached in 4 seconds,
it would be reached in about 6^ seconds with a 2O-kw.
and in about 16 seconds with an 8o-kw. transformer.
The shape of the time-current curve is of the character
shown in Fig. 132. If there were no hysteretic loss, it
would be a true logarithmic curve, but owing to the
influence of hysteresis there is a depression in the upper
part as shown.
From (46) we have
n
looR/j
n
Now I I is the length of the ordinate between the
curve and the + I line, so that f(\Ql)dt is the area
enclosed between the curve and this line. Integrating
between the limits < and + < , to which correspond
the times o and t^ we find
i I OO -LV x"x / v
2 fa= -^ Qo (47)
if by Qo we denote the whole area between the curve and
its asymptote.
Integrating between the limits < and +<, to which
correspond the times o and /, we find
)o-Q) . '. . . ( 4 8)
By combining (47) and (48) we get
260
TRANSFORMERS
Q is the shaded area in Fig. 135. Having fixed on a
value of I, we find by planimeter the corresponding area
Q, and from (49) the corresponding value of the flux (.
It is thus easy to find by means of a planimeter corre-
sponding values of I and <, and to plot these as shown
in Fig. 134. The hysteretic energy per cycle is
obviously
E =
n
100
x area of loop
If there are no joints in the .carcase, and its cross-
FIG. 135. Kapp's test.
sectional dimensions are such as to make the induction
the same in any part, the true B-H loop can, of course,
be plotted, and the permeability as a function of the
induction may also be found. In most cases, however,
a knowledge of the exact shape of the B-H loop and of
the permeability is of secondary importance ; what we
require is a knowledge of the hysteretic loss in the whole
transformer, and this may be found graphically from
Fig. 135 without even drawing the loop.
The hysteretic energy absorbed by the carcase in one
half-cycle is obviously the difference between
r l
efldt
KAPPS METHOD 261
the total energy supplied, and
the energy lost in copper heat. The latter quantity may
be expressed in the form
'i T ^
Rio f \\-dt or e fl'dt
+s +J
ft *0
where I' = I can be determined graphically by the con-
struction shown by dotted lines in Fig. 135. The
hysteretic energy for one half-cycle is, therefore
o 4
i- = el (I - V\dt watt-second
2 <
The integral is the area (expressed in coulomb) between
the original time-current curve, and the new I' curve
shown in a dotted line. The area is to be taken with
reference to the sign of the current ; that is to say,
negative up to the point I = o and positive for I > o. By
plani metering the two areas and deducting that which
is negative, we find
This construction applies to any transformer, whether it
has joints or not, and whether the induction is the same
throughout the magnetic path or not.
CHAPTER XII
SAFETY APPLIANCES FOR TRANSFORMERS
SUB-STATION AND HOUSE TRANSFORMERS-
REDUCING IRON LOSSES TRANSFORMER FOR
THREE-WIRE SYSTEM BALANCING TRANS-
FORMERSA UTO - TRANSFORMERS SERIES
WORKING BOOSTERS 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 perfect. If this condition be not fulfilled, the
use of transformers may even become dangerous on
account of an unjustified feeling of security. The two
windings in a transformer must necessari-ly 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 network of primary conductors their insula-
tion 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 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 preventive is to place
262
SAFETY APPLIANCES FOR TRANSFORMERS 263
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 interven-
tion of this dividing sheet, and thus the potential 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, or between the terminals of the two
circuits. Even if by good workmanship the danger of a
leak in the transformer itself, or its terminal boards, could
be completely eliminated, there still remains the possi-
bility of a contact or leak between the supply wires. An
obvious case is that where both the high and low
pressure circuits are overhead, and so near each other
that a branch of a tree blown across them by the wind
bridges the two circuits. It is, of course, not good
practice to use the same posts for both circuits, but in
certain localities for instance, immediately outside a
transforming station proximity is sometimes unavoid-
able ; and to exclude any danger from such causes it is
wise to act on the principle that the line, rather than the
transformer, should be fitted with the safety appliance.
The transformer itself is the least vulnerable part of the
system, and requires protection less than the line, but if
the line is protected, the transformer is also protected.
One way of protecting line and transformer simul-
taneously is to earth some point of the secondary circuit,
preferably the middle of the winding in a single-phase
or the star point in a three-phase transformer, since then
the potential difference of the secondary mains to earth
becomes a minimum, namely, equal to half, or very little
more than half the line 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 expedient. If the whole
of the secondary circuit is insulated from earth, a fault
must occur at two places of different potential before a
264
TRANSFORMERS
danger in respect of fire can arise, but if one point of
the secondary circuit is permanently connected to earth,
a fault occurring in one place only is sufficient to create
danger. The margin of safety is therefore reduced by
one-half if we earth a point of the secondary winding.
There is also increased danger of damage by atmo-
spheric electricity. A system completely insulated from
earth is less liable to be struck by lightning than one
which has somewhere an earth-connection. Finally,
there is the objection that such a system may give rise
to capacity currents (the coils in the transformer as well
as those in the generator have capacity to each other
and to earth) which disturb telephonic work. For all
these reasons the simple expedient of permanently earth-
ing one point of the secondary
circuit cannot be considered
a generally applicable, or even
when applied, a satisfactory
way of protecting the low-
pressure circuits of trans-
formers against the infiltration
of high-pressure.
The trouble about in-
creased fire and lightning
danger and telephonic dis-
turbance can be overcome if the earth-connection is not
permanent, but only established at the moment when it
is wanted. This was the leading idea in a safety
appliance introduced quite early in the history of trans-
formers by the Thomson- Houston Company in America.
The appliance consists of an earth-plate and two
metal knobs, a, b, Fig. 136, which are connected to the
secondary mains. Between the knobs and the earth-
plate is inserted a thin sheet of insulating material
(paraffined paper or mica). As long as no fault between
primary and secondary occurs, the potential difference
between 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-
FIG. 136. Protection against rise
of pressure.
SAFETY APPLIANCES FOR TRANSFORMERS 265
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 fuses s, s go, and
the transformer is thereby automatically cut out of
circuit.
The same principle has more recently been revived
in an improved form by Prof. Goerges in his safety-plug,
which is being manufactured by the Siemens-Schnokert-
Werke, Berlin. Here, also, one electrode of the plug is
connected with the line to be protected, and the other
with earth, but the mica insertion is only used as a
distance-piece and not as a body which must be pierced
by the discharge to earth.
Externally the safety-plug resembles the well-known
fuse-plugs commonly used on the Continent for pro-
tection against excessive rise of current, but instead of
a fuse embedded in emery powder, the plug contains
two metal electrodes insulated from each other by the
body and screw of the plug, which are of porcelain, and
a thin mica disc pierced with four holes of 3*5 mm.
diameter. The electrodes are perfectly smooth circular
plates, and their distance apart is determined by the
thickness of the mica disc. In case of undue rise of
pressure between them, a spark passes through one or
more of the holes, and this welds the two discs together,
thus providing an efficient connection to earth, or the
other circuits similarly protected.
It is important to notice that in this plug even a very
small current suffices to produce the welding together of
the electrodes through the holes in the mica, so that
even an incipient fault will be detected and rendered
inocuous by this plug. The fact that welding takes
place already with a very minute current makes the
action independent of the goodness of the earth-con-
nection. It is well known that an " earth " good enough
to carry off large currents is generally very difficult to
provide, but so good an earth is not required for the
Goerges plug, since a current as small as 0*0345 ampere l
1 Elektrotechnische Zeitsckrift, 1905, p. 314.
266
TRANSFORMERS
is sufficient to produce welding. If then the earth is not
good enough to carry off sufficient current to lower the
pressure, the other plug will come into action, thus short-
circuiting the low-pressure leads, and causing the fuses
on the primary to blow, and thus removing all danger.
The pressure at which the plug acts depends on the
thickness of the mica insertion. With 0*12 mm. the
pressure is 800 volt. After a plug has acted it can be
put into working order again by cleaning off the welded
parts with emery paper, and turning the electrodes so as
to bring parts of the original surfaces facing each other
through the holes. The right length of spark-gap is
obtained by simply screwing the plug down tight on to
the mica.
A safety device invented by Major Cardew is shown
in Fig. 137. In this
arrangement the
action depends on
electrostatic attrac-
tion between a plate
E connected to the
secondary, and an
aluminium foil lying
on a plate connected
to earth. The alu-
minium 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 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 wind-
Earth
FIG. 137. Cardew's safely device.
SAFETY APPLIANCES FOR TRANSFORMERS 267
ing. In the safety device first described 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 trans-
former to be cut out of circuit. It has, however, been
found that when a good earth is obtainable 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 fuses go. The apparatus
can be set to come into action if the potential of the
secondary rises to 400 volt.
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. 138. The
secondary mains are con-
nected to the primaries
of two very small trans-
formers coupled in series,
whilst their secondaries are
coupled in parallel. The
secondaries are connected
to a fuse carrying a conical
weight over a correspond-
ing 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.Fs. of the secondary windings of the two small
auxiliary transformers, and no current passes through the
fuse. If, however, a fault occurs, the balance is disturbed,
a current passes through the fuse and melts it, and the
weight falling between the terminals short-circuits the
FIG. 138. Ferranti's safety device.
1 Journal Inst. EL Eng., Vol. XVII, p. 179.
268 TRANSFORMERS
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.
The safety devices here described and others on
similar principles are quite reliable where the secondary
mains are fed by one transformer only, and being within
a building or underground, are not subject to disturbance
from atmospheric electricity. The Goerges plug is even
applicable on an underground network fed by several
transformers in parallel, but when we have to protect
overhead low-pressure mains, all these devices, although
still ensuring safety, are liable to come into action by
reason of a passing disturbance through atmospheric
electricity. What is required is a safety device which
will discriminate between a rise of pressure due to an
atmospheric cause, and therefore lasting only a very
short time, and a permanent rise of pressure due to a
leak or short-circuit between the low-tension and high-
tension series. The device should not be of the nature
of a delicate physical apparatus, but rather of the nature
of a substantial appliance fit to be put into an engine-
room or sub-station, and should require no attention.
Up to the present no such implement has been put on
the market.
Sub-station and house transformers. It is convenient
to make a distinction between transformers placed into
a secondary distributing centre and large enough to
supply current to a number of distinct consumers and
transformers placed on the premises of each consumer.
In the first case we speak of sub-station transformers,
and in the second of house transformers. If we except
consumers of large powers where the pressure supplied
to the installation need not be limited by other considera-
tions than appertain to the wiring, switch-gear, and the
motors themselves, we may take it that for a general
lighting and power service, whether given from a sub-
station or a house transformer, considerations of personal
SUB-STATION AND HOUSE TRANSFORMERS 269
safety as well as the nature of glow-lamps impose a limit
on the pressure in the secondary circuit of the transformer.
With carbon filament lamps as at present made 220 volt,
or at most 250 volt may be considered as an upper limit
of working pressure, whilst with metallic filament glow-
lamps a still lower limit is as yet usual. Thus a low or
moderate pressure in the distributing circuit is a necessity,
whilst a high pressure in the transmission circuit is an
economic advantage, and, indeed, also a necessity, if the
transmission has to be effected over a considerable
distance. The transformer is then the intermediary
Li
r
i
FIG. 139. Distribution from sub-stations.
apparatus by which the two conditions, cheap mains and
moderate supply voltage, can be simultaneously fulfilled.
The typical arrangement of transformers for a sub-station
system is shown in Fig. 139. C denotes omnibus bars
in the central station ; S, s the primary transmission mains
or feeders ; T, T are transformers at two sub-stations, 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.
The diagram shows each transformer supplied with
current by its own feeder, whilst on the secondary side
each transformer supplies a network of distributing
2;o TRANSFORMERS
mains, which latter may be either separate from each
other, or they may be inter-connected, as shown by the
dotted lines. The inter-connection of secondary mains
has the advantage that a more nearly constant pressure
can be maintained throughout the secondary network,
and that at times of small demand some of the trans-
formers may be disconnected from the primary and
secondary 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 part of
the network may affect the whole system, and to mini-
mise this danger it is advisable to insert fuses into all
the important junctions of the secondary network.
Instead of using separate feeders to the different sub-
stations, we may also provide a primary network to
which the primary terminals of all the transformers are
connected in parallel.
When a district is supplied on the house-transformer
system a complete network of high-pressure feeding
and distributing mains conveys current to a large num-
ber 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), so that no network
of secondary or low-pressure street mains is required.
The weight of copper in the street mains is thereby
much reduced, which is an advantage. 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,
and the total capacity of the transformers connected must
be greater than in the sub-station system. The constant
losses are therefore also greater. One house wired for
100 lamps may have sometimes 80 per cent, of its lamps,
or say as many as 80 lamps, alight, though this will not
happen very often. Twenty houses wired collectively
SUB-STATION AND HOUSE TRANSFORMERS 271
for 2000 lamps will never use simultaneously 80 per cent,
that is, 1600 of the installed lamps but at most 1000,
or 50 per cent. In some cases, especially if the houses
are of widely different character (shops, offices, dwelling-
houses, restaurants), the maximum simultaneous load
may be even considerably less than 50 per cent, of the
total installed load. The ratio of the total maximum of
power supply observed in a given district to the sum of
the maxima observed at different times in each individual
house is called the diversity factor, and it is due to the
circumstance that this diversity factor is larger than
unity, that the total capacity of a sub-station may be
smaller than the collective capacity of house-transformers,
were the same districts supplied on the house-transformer
system.
Take a district in which 1000 kw. in motors and
lamps are installed. If it be supplied on the house-
transformer system the aggregate capacity of trans-
formers would be about 800 kw., made up of mostly
small sizes of, say, 2 to 10 kw. The aggregate iron
losses will be about 2^ per cent, or 20 kw., and the
copper losses about i|- per cent., or 12 kw. ; the latter
taking place, however, only during a short time daily.
The iron losses are going on all the year round, and the
energy wasted per annum is about 175,000 kw.-hrs.
The total copper loss is very much smaller ; we may
roughly estimate it at 5000 kw.-hrs. This is almost
negligible in comparison with the iron losses. If the
same district were supplied by two or three sub-stations,
the total capacity of sub-station transformers with a
diversity factor of 2 would only be 400 kw., and the
transformers at the sub-stations would be so large that
the iron losses need not exceed i per cent., or 4 kw.
The total annual iron loss would therefore be only 35,000
kw.-hrs. The copper losses will also be reduced, though
not in the same proportion, because, owing to the diversity
factor, each transformer will be working at a fair load
for a longer time daily. The annual copper loss may
be taken at about 3000 kw.-hrs. We thus find-
Energy wasted in house transformers . 180,000 kw.-hrs.
,, ,, sub-station transformers 38,000 ,,
272 TRANSFORMERS
If the total installed load is equally divided between
lighting and power, the energy sold per annum will be
about 400,000 kw.-hrs. for power, and 200,000 kw.-hrs.for
lighting, or 600,000 kw.-hrs. in all. The annual efficiency,
allowing 2 per cent, loss in the mains, will therefore
be
With house transformers ... 75 per cent.
With sub-station transformers . 92 ,,
The cost of house transformers, including terminal
boards and fuses, and a provision for housing them
safely that is, beyond the reach of unauthorised persons
is about ^4 a kw., whilst sub-station transformers, in-
cluding all accessory apparatus and their housing, may
be taken at half this amount. The initial outlay will
therefore respectively be ^3200 and ^800, showing a
saving of ^2400 in favour of the sub-station system.
At 10 per cent, for interest, repair, and amortisation, this
means an annual saving of ^240, to which must be
added the saving in energy wasted, which amounts to
1 42,000 kw.-hrs. Taking the engine-room cost at id. per
kw.-hr., this amounts to another ^590, making the total
saving ^830 annually. Against this has to be set the
increase in annual working expenses due to our having
to provide a secondary net-work. If the capital outlay
on this account exceeds % 6s. per installed kw., the
system of house transformers will be economically better ;
if the outlay for cables is less, it will be better to use
sub-stations.
These calculations have not been given as hard-and-
fast rules, but merely by way of example, how the com-
mercial advantages of the two systems may be compared.
A definite conclusion in any special case can, of course,
only be reached if all the conditions (such as annual
energy required per installed kw., diversity factor, extent
of district, cost of transformers and cables, and engine-
room cost of energy) are known, but from what has been
shown above we may postulate as a general principle
that the use of house transformers is commercially jus-
tified if power derived from a cheap source has to be
distributed in small parcels amongst widely scattered
REDUCING IRON LOSSES 273
customers in country districts. In towns customers are
fairly close together and the power has a greater value,
hence sub-stations are economically preferable.
Reducing iron losses. There are cases where a
consumer is not within easy reach of a secondary
network and yet requires low-pressure current. In such
cases (schools, hospitals, asylums lying outside the town),
the general system of supply from sub-stations has to be
supplemented by house transformers. If the supply is
given mainly for lighting, the load factor that is, the ratio
between the' total energy in kw.-hrs. actually supplied
during the year to the energy represented by the product
of maximum demand in kw. multiplied by the 8760 hours
of the year is very small. For a purely lighting load
the demand factor that is, the ratio between maximum
demand observed and lamps installed will seldom exceed
So per cent., whilst the load factor for an isolated
installation will only be from 4 to 6 per cent. If the
transformer capable of supplying the maximum demand
has 2 per cent, iron loss and is kept under pressure all
the year round it will consume annually 175 units
magnetising energy per kw. capacity, whilst the energy
actually delivered to the lamps is only from 350 to 520
units. The annual efficiency of the transformer is there-
fore only f to f . A better efficiency would be obtained
by installing two or more transformers and switching them
into and out of circuit in accordance with the demand
actually existing at any time. This, however, would
entail an amount of personal supervision which only large
establishments could afford to provide, and then there
would be the danger of overloading a small transformer
if the attendant forgets to switch in the big transformer
at the right time. For this reason it is safer to work the
switch-gear automatically. Several such systems (some of
them also applicable to sub-stations) have been suggested
from time to time, and by way of example I give here
the latest device, designed by Mr. A. F. Berry. This
inventor uses two transformers coupled in series, one
small, the other large. As long as the demand does not
exceed the current capacity of the small transformer, this
is doing the greater part of the work, but if the demand
18
274
TRA NSFORMER S
rises beyond a given limit, the primary and secondary of
the small transformer are simultaneously short-circuited,
and the large transformer is doing all the work alone.
The arrangement is shown diagrammatically, Fig. 140.
Tj is the small, and T the large transformer. The
secondary current is taken through on electromagnet E,
the armature of which rests on a lower contact as long
as the current is not strong enough to raise it. If the
demand has exceeded a predetermined limit, the attraction
of E is sufficient to raise the armature so that it comes
against the upper contact. The two contacts are
connected with the two coils of the solenoidal magnet E 1?
as shown, and accord-
ingly as one or the
other is in touch with
the armature the core
of this magnet is either
pulled down (light
load) or pulled up
(heavy load). When
in the latter position
the switch Si short-
circuits the primary,
and the switch S 2 the
secondary, of the small
transformer. A snap-
lock (not shown in the
diagram) is connected
with the core of the
solenoid which not only
holds the latter in position after each movement, but also
interrupts the current through the coils of the solenoid.
This is to prevent waste of current through these coils
during the time that they are not required to act. Mr.
Berry claims that the extra cost of this switch-gear and the
small transformer is compensated by the saving in capital
outlay on the large transformer. Since the latter is most
of the time only very slightly magnetised, it enters on the
period of its full load at a low temperature, and being
worked intermittently with long spells of rest between
short periods of load, it may, as was shown in Chapter VI,
FIG. 140. Berry's system of automatic control.
TRANSFORMER FOR THREE-WIRE SYSTEM 275
be made smaller than a transformer continuously under
pressure.
Transformer for three-wire system. The well-known
system of continuous current distribution by three wires
can also be employed in connection with transformers.
We need only connect the middle point o of the secondary
winding, Fig. 141, to the zero wire, and the outer
terminals m, n of this winding to the two outer wires.
The primary leads s consist
of two wires only connected
to the primary terminals^,
q. The lamps a, b are con- | &AAAA A
nected between the outer q-n, T I T T T
Wires and the zero wire 0. Fig< I4I< _ Se condary three-wire system.
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 dis-
tributing mains. Care must, however, be taken to group
the different coils of the secondary winding in such way
that the ampere-turns produced by the two secondary
currents have the same value in all parts of the magnetic
circuit. If this is not done, the leakage or inductive
drop would be greater on the more heavily loaded part
I 9 <*>
<><>
T
FIG. 142. Secondary three-wire system with balancing transformer.
of the system, and the supply voltage would be unevenly
divided between the two groups of lamps.
Balancing transformers. 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. 142,
if a balancing transformer Tj is established in some point
of the district to be lighted. The output of the balancing
transformer need not be larger than half the maximum
2 ;6 TRANSFORMERS
difference between the loads on the two sides a y b of the
system. Let i a be the maximum current in a and i b the
current which simultaneously obtains in b, then one coil
of the balancing transformer must take up the current
"7 __ 7
- and its other coil must give off an equal current.
If the lamp voltage is e, then the output of the balancing
transformer is given by the expression ( -- b je, the out-
put of the sub-station transformer at the same time being
= (i a + t b )e. Since it is, however, possible that
both sides of the system may occasionally carry the
maximum current, the sub-station transformer must be
designed for an output of 2t a e. If by / we denote the
ratio of load difference between the two sides to the
maximum load on one side, we have
The output of the balancing transformer must therefore be
2**
Since i a e is half the output of the sub-station trans-
former, we have the ratio between its size and that of
the balancing transformer given by the fraction 4 : p.
Thus for a load difference of 100, 50, 20, 10 per cent,
the balancing transformer would be respectively , ,
4 8
, the size of the sub-station transformer. These
20 40
figures show that a comparatively very small balancing
transformer may render it superfluous to carry the middle
wire of the system back to the sub-station.
Another application of balancing transformers may be
made in adapting a single continuous-current generator
to a three-wire system. Let in Fig. 143 the outer circle
represent the armature of an ordinary continuous-current
generator supplying current to the outer wires a, b of a
three-wire system. Then by taking from two tapping
A UTOTRANSFORMERS 277
points connections to the slip-rings (represented in the
diagram by the two inner circles) we obtain at their
brushes an alternating voltage whose crest value is equal
to the voltage on the outer mains. The brushes of the
slip-rings are connected to a balancing transformer such
as is shown in Fig. 142. From what has been explained
in connection with this diagram it will be obvious that
the middle point, o, of the winding of this trans-
former divides the pressure between a and b equally, pro-
vided both windings are as intimately mixed as the
primary and secondary of an ordinary transformer. The
zero wire may then be connected to the point o.
The balancing transformer must, of course, be de-
signed for the frequency
corresponding to the con-
tinuous-current machine.
This is
v =pu
where p is the number of
pairs of poles and u the
speed in revolutions per
second. As an instance
take a six-pole generator Fig " ^--Balancing transformer.
running at 120 revolutions
per minute. The frequency will be 6. Let the output be
200 kw. at 500 volt, then the full-load current will be 400
ampere. Let the greatest out of balance current be
10 per cent., then the transformer will have to take in
20 ampere on its primary and give out 20 ampere on
its secondary side, the terminal pressure on each side
being 250 volt crest value, or 180 volt effective value.
A transformer wound for an output of 3*6 kw. at 6 fre-
quency and transforming ratio of i : i will therefore
suffice for this purpose. It is important to design the
transformer for a very small copper loss so as to ensure
equal division of pressure, but it will be seen from this
example that the balancing transformer, even if designed
on a very liberal scale, will be only a small accessory to
the generator.
Autotransformers. Balancing transformers may also
278 TRANSFORMERS
be used for subdividing a given supply pressure between
a number of circuits, so that lamps requiring a lower
pressure than that supplied may be used individually on
these circuits. Originally used for arc lamps, this method
of subdividing pressure has, with the advent of the
metallic filament lamp, acquired additional importance.
Balancing transformers arranged for this purpose are
generally called autotransformers, because part of the
winding is traversed by the difference of the two currents,
and only the rest of the winding is traversed by the high-
pressure current only. This arrangement is instrumental
in a certain saving of material, so that an autotransformer
is smaller and cheaper than a
I r - 5 - r - 1 2 transformer with two distinct
windings. The extent to which
j 2 material may be saved can be
seen from the following con-
sideration-
Let in Fig. 144 oa be that
. _ I part of the winding which is
b transversed by the difference
FIG. i 44 . -Autotransformer. ! 2 - Ii of the two currents, and
ob the remainder which is tra-
versed by the current l l alone. Let oa consist of n^ and
ob of n 1 n z turns, and let the transforming ratio be
0tf
m = . Then I 2 = m\ ly and as equal current density gives
the best utilisation of the material we have qz = qi(m 0-
The total volume of copper will therefore be
m) ^ * 'm
m i
v =
m
The volume of copper in an ordinary transformer is
v l = &2qn l
Or, taking v l as the standard, we have
m i
v = v^
m
A UTOTRANSFORMERS 279
Since in transformers of the same type, but different
sizes, the ratio of volume of iron to volume of copper is
approximately constant, the fraction * indicates the
m
quantity of material required in an autotransformer rela-
tively to an ordinary transformer. The ratio of material
saved is the reciprocal of m, and hence for large trans-
forming ratios the auto-principle has very little advantage,
whilst the necessity of tying the two circuits electrically
together is a distinct disadvantage. This explains why
autotransformers are only used for low pressure and low
transforming ratio. They may be used with advantage
as starting devices for induction motors and for reducing
a moderate voltage to a still smaller value. Thus with a
transforming ratio of 1^5, 2, or 3 the weight of an auto-
transformer will be only 34 per cent., 50 per cent, or 67
per cent, respectively of an ordinary transformer.
Metallic filament lamps for no volt can now be
obtained. If, then, the supply pressure is 220 volt, we
could use such lamps by supplying them from an auto-
transformer, which need as regards weight and cost only
be equivalent to an ordinary transformer of half the out-
put. This is on the supposition that all the lamps are
fed from the same circuit, but if we can split up the lamps
into two circuits, each carrying half the number, the auto-
transformer need only be a quarter the size of an ordinary
transformer. Generally for m circuits each carrying th
m
of the total number of lamps at i th pressure we have
m
for the weight of the autotransformer
of the weights of an ordinary transformer. Thus, if at
220 volt supply pressure we wish to use osmium lamps
of 73 volt we can divide them into three circuits and use
an autotransformer which will only weigh one-third as
much as an ordinary transformer. Or, if we wish to use
arc lamps requiring a pressure of about 36 volt we can
280
TRANSFORMERS
group them in six circuits, and the autotransformer will
still be reasonably small, namely, a little less than half the
size of an ordinary transformer. The lamps will be quite
independent of each other, as if they were all in parallel
on one and the same circuit.
Series working. Transformers may be advantage-
ously used if it be required to work a number of lamps
in series off a circuit in which an alternatino; current of
o
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, how-
ever, 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 insula-
tion of the lamps need
only be good enough for
the voltage required by
each lamp. The arrange-
a~
/vwvwws
FIG. 145. Series working. merit is shown in Fig.
145. 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 con-
stant 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
secondary 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 increased. 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 alter-
SERIES WORKING 281
native 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" 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 paraffined
paper, which, under normal conditions, is sufficient to with-
stand the secondary voltage. If, however, the secondary
voltage rises considerably, in consequence of the lamp
circuit 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, allows a current to pass
through its winding proportional to the lamp voltage,
but lagging by nearly 90 behind it. The power lost in
the choking coil is the sum of hysteresis and ohmic loss ;
and by a proper design of choking coil it is thus possible
to minimise the loss of power, although the presence of
choking coils must worsen the power factor. This may
best be seen by an example. Let us assume that the
lamp requires 10 ampere at 35 volt, and that its power
factor is 80 per cent. The power actually supplied to
the lamp is therefore 280 watt. Let the choking coil be
so constructed that it takes 5 ampere if the pressure is
35 volt, and that the loss of power in it is 5 watt. Its
power factor is therefore - - = 0*0285. If we now
35 x 5
draw a vector diagram to represent these working con-
ditions, we find that the total secondary current supplied
by the transformer is 13-5 ampere. We also find from
this diagram the power factor of the combination lamp
plus choking coil is only 0*6. If now the lamp current
is interrupted the choking coil must pass the whole 13*5
ampere, and the voltage must rise to
35 = 95 volt
This is an excess of 170 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
282 TRANSFORMERS
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 transformer of the usual construction
having as little magnetic leakage as possible) would be
quite unsuitable. Such a 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 constant, whilst the voltage must
vary as nearly as possible in accordance with the number
of lamps alight at any time. As was already shown in
FIG. 146. Constant current transformer.
Chapter IX, this condition can be met at least approxi-
mately by shaping the transformer in such way as to
produce a large magnetic leakage. A construction of
this kind is shown in Fig. 146. It is a core transformer
with primary and secondary coils on separate limbs and
with expansions a, b of the two yokes arranged specially
to produce magnetic leakage. The primary coil is joined
to the primary constant-pressure lead s; and the secondary
coil to the circuit containing the glow lamps L in series.
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 b but
all over the transformer, will be much increased. The
BOOSTERS : :
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 which produces E.M.F. in the
secondary. The increase in current strength will there-
fore be considerably smaller than would obtain with an
ordinary transformer, and in this way it is possible to
keep the current at least approximately constant when
lamps are put out of action by being short-circuited.
For the exact determination of the working condition
see the vector diagram given at the end of Chapter IX.
Boasters. If some of the feeders between the central
station and the sub-stations are very long, it is some-
times advantageous to allow a greater voltage drop in
them than in the shorter feeders, and to raise the
1
147.:
pressure at the home end of these long feeders by an
amount corresponding to the extra drop. For this pur-
pose 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. 1 It is shown
diagrammatically in Fig. 147.
C are the bus bars in the station, S is a feeder
supplying current to the transformer T at a sub-station.
Y are the distributing mains connected to this trans-
former. The boosting transformer has its primary
permanently connected to the bus bars, whilst its
secondary is put in series with the feeder and is sub-
divided into sections, so that by using a switch , a
greater or lesser number of secondary turns can be
inserted. In this manner the additional voltage put into
11 British Patent, No. 4345, March 21,
284 TRANSFORMERS
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 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 the impedance of the feeder. 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
g worked automatically by a
~ small electro - motor con-
trolled by a relay. Since,
in passing from one contact
to the other, the switch-
lever, if it were made in one
solid piece, would short-
circuit, and possibly burn out
FIG. 148. Booster. 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 resistance, or preferably by a
choking coil. With such a construction 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
of the feeder current if this switch should get out of
order, has led the Author to design the modified arrange-
ment 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. 148.
The feeder circuit is permanently connected with the
BOOSTERS
285
bus bars through the secondary winding of the auxiliary
transformer, whilst the multiple contact switch is inserted
into its primary connection with the bus bars. The
primary winding is subdivided into groups a, b, 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. 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 over-
loaded. This kind of booster must therefore be larger
than that shown in Fig. 147, but as in any case the cost
of a booster is very small as compared 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.
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. 149. The con-
struction resembles 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. 148. By
means of worm gearing, the coil S may be placed at
various angles with reference to the 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 intermediate
positions any desired boosting effect may be obtained.
By turning the coil beyond its horizontal position the
FIG. 149. Booster.
286 TRANSFORMERS
action may also be reversed, that is to say, we can reduce
the E.M.F. at the home end of the feeder. The advan-
tages 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.
A booster constructed on the same principle may
also be used to regulate the alternating pressure supplied
to a rotary connector. In these machines the ratio
between the alternating pressure supplied to and the
continuous pressure derived from the armature is con-
stant whatever may be the excitation, so that no adjust-
ment of continuous pressure can be made by means of
a rheostat in the exciting circuit as is done in an
ordinary continuous-current generator. Yet it may be
necessary to adjust the pressure at which the continuous
current is delivered. This is done by adjusting the
alternating pressure of the driving current, a special
type of booster being used for the purpose.
This booster is a three-phase transformer with mov-
able secondary winding ; in construction it resembles an
ordinary induction motor, the primary being wound on
the stator to produce a rotating field, whilst the secondary
is wound on the part which usually is the rotor, but which
in this case is not allowed to rotate. The arrangement
is shown in Fig. 150.
U is the converter with its commutator K, from
which the continuous current is delivered, and its slip-
rings s, by which it relieves the alternating three-phase
current. Between the slip-rings and the source of alter-
nating current in this case a three-phase transformer T
is placed the adjustable booster B. Its primary wind-
ings are connected to the source, and produce a magnetic
field of constant strength revolving round the rotor with
a velocity corresponding to the frequency. The winding
of the rotor is represented by the three coils inside the
inner circle. For the sake of simplicity these are shown
parallel, but it must be understood that they are placed
with an electrical angular displacement of 120 to each
other, so that by being successively cut by the revolving
primary field, the E.M.Fs. induced in them follow each
BOOSTERS
287
other at intervals of a third period. The phase in rela-
tion to the primary at which the voltage of the second-
ary is injected must therefore depend on the angular
position at which the rotor is set, as shown by the little
vector diagram below the figure. In this E is the E.M.F.
of the source, e the E.M.F. induced in the secondary
coils of the booster, and E] the E.M.F. supplied to the
slip-rings of the converter. The position of the vector e
depends on the position to which the rotor is set, so that
E! may be made either larger or smaller than E. The
use of a booster of this kind alters slightly the power
factor, but as there are always two positions of e for each
FIG. 150. Booster applied to converter.
required value of E 1? we may choose that by which the
power factor is increased.
Since a considerable torque is exerted on the second-
ary, it is necessary to use worm gearing for setting the
rotor, and in large boosters it is advisable to couple two
mechanically together, the electrical connections being
made in such sense that the two torques eliminate each
other. The latter arrangement has been first used by
Messrs. Siemens, Schuckert Werke in the Paderno power
transmission. In this case the boosters were not used in
connection with converters, but simply for the purpose
of compensating the drop in long and heavy feeders.
TRA NSFORMER S
Scott's system. An interesting application of trans-
formers is the conversion of a two-phase into a three-
phase system, and vice versa, invented by Mr. C. F.
Scott. 1 The arrangement is diagrammatically repre-
sented in Fig. 151, where G is a two-phase generator
supplying current to the primaries of two transformers
Tj and T 2 . The secondaries of these transformers are
joined together, as shown in the figure, leaving three
terminals, A, B and C, free for connection to the secondary
circuit. Since the primary currents in T l and T 2 have a
phase difference of 90, there is also the same phase
difference in the E.M.Fs. generated in the two secondary
coils. The E.M.F. between terminals A and B is there-
fore the resultant of two components, one being the full
/wvwwwx
FIG. 151. Scott's system.
FIG. 152. Vector diagram of
Scott's system.
E.M.F. generated in the secondary of T 1} and the other
half the E.M.F. generated in the secondary of T 2 , the
latter component being moreover displaced by 90 as
regards the former component. Let, in Fig. 152, OA be
the E.M.F. of T\ and OB half the E.M.F. of T 2 , then
BA is the resultant E.M.F. which we measure between
the terminals A and B. In the same manner we find
CA as the resultant E.M.F. produced by Tj. and the left
half of T 2 , whilst CB is the E.M.F. produced by both
halves of T 2 . It will be obvious that, by a proper choice
of the number of turns in the secondaries, we may so
arrange matters that OB = lAB. Then AB = BC = CA,
and OA = AB->/ 3 , or OA = 0-867 AB-o'86; BC. The
1 The Electrician, April 6, 1894.
SCOTT S SYSTEM
winding must therefore be such that the secondary volt-
age of T! is 0*867 of the secondary voltage of T 2 . In
the clock diagram the vectors of terminal pressure pass
then through zero at intervals of 60, or in the same sense
at intervals of 120, which characterises a three-phase
current. We obtain thus from the terminals A,B,C a
three-phase current.
The advantage claimed by Mr. Scott for this system
is that the generation and utilisation of the current may
be effected by two-phase machinery, whilst the trans-
mission may be made in three phases. The former
condition he considers to be an advantage as regards the
independent working of motors and lamps, and especially
their regulation, whilst the latter condition is, of course,
1)
FIG. 153. Scott's system.
conducive to economy in copper on long lines of
transmissions.
A complete plant arranged according to Scott's
system is shown in Fig. 153. G is a two-phase generator
producing 100 volt, which pressure is transformed up to
2000 and 1730 volt in the two transformers shown. To
the three free terminals are joined the line wires, and
between each pair there is a pressure of 2000 volts. At
the points of consumption the three-phase current is
either transformed down and converted into a two-phase
current for working motors (A) or supplying light (B), or
it may be used as a three-phase current for working
motors (D). Although the circuits are inter-connected,
the regulation for constant pressure in the lamp circuits
causes, according to the inventor, no more difficulty
than if the lamps were connected directly with the
generator.
19
290 TRANSFORMERS
The mechanical construction of the carcase of the
loo-k.v.a. transformer is shown in Figs. 216 and 217. It
will be noticed that the frame is constructed in the form
of a grid, so as to allow the cooling medium direct access
to the plates of core and yoke.
CHAPTER XIII
THE TRANSFORMER IN RELATION TO ITS CIR-
CUITSEQUIVALENT COILS IN PARALLEL
AND SERIES CONNECTION RISE OF PRES-
SURE THROUGH RESONANCE DETERMINA-
TION OF THE DANGEROUS CONDITION RISE
OF PRESSURE ON LOADED LIGHTING SYS-
TEM IS SMALL INFLUENCE OF POWER
FACTOR BREAKDOWN OF CABLES IN LARGE
NETWORKS
The transformer in relation to its circuits. Up to the
present we have considered the transformer as an appar-
atus by itself, receiving energy from a source not in-
fluenced by -its presence and giving up energy to some
receiving device which, apart from its ability to absorb
energy, has no influence on the transformer. In other
words, we have assumed the primary current to be derived
from an inexhaustible source and the secondary current
to be given to an apparatus which can only absorb, but
not return energy. These conditions obtain in the
ordinary use of transformers. It has been shown in
Chapter IX that the reactance of a transformer working
under load is extremely small, and for this reason any
reactive effect of the consuming device is transmitted to
the source of current much in the same way as if the
transformer were not interposed. The transformer is
simply a means of linking the two circuits together and
adjusting the pressure, but is otherwise inert. There are,
however, cases when a transformer may cease to play
this passive role, and by reason of an interaction between
its inductance and the capacity of the circuit cause a rise
of pressure sufficient to break down itself or a cable,
generally the latter. Such special circumstances may
arise under the two extreme cases of a transformer work-
291
292 TRANSFORMERS
ing either at no-load or under short circuit. In the first
case the reactance is large because only the equivalent
coils representing excitation are acting ; in the second
tbe reactance voltage is large because the current has
enormously increased.
Equivalent coils in parallel and series connection.
When drawing the vector diagram of a transformer
under load we have made use of the conception of
equivalent coils in parallel across the primary terminals,
one of these coils having such an inductance as to let
pass the magnetising current, the other having such
a resistance as to let pass a current which, multiplied by
the primary pressure, represents iron losses. When
considering the effect of capacity in the circuits it is
convenient to substitute for these two parallel coils,
two coils in series with each other, and with the capacity,
as was already done in Chapter IX under the sub-
heading "The Self-induction of a Transformer." Let
R and >L be resistance and reactance of the two
equivalent coils in parallel, and Rj and ^Lj the respective
equivalent values for the coils in series, then we have
R
R sin L sin 2 L sin f cos
L and R, discussed previously
when introducing the conception of equivalent coils, we
can now substitute one coil containing wLj and Rj in
series, and the diagrammatic representation of a trans-
former having the transforming ratio i : i will be as shown
in Fig. 154, where coLj and Rj represent reactance and
RISE OF PRESSURE THROUGH RESONANCE 293
resistance of the exciting coil and L the load which is
supposed to be switched off.
Let B represent the bus bars at the station, and let
the transformer be joined to them by a concentric cable.
As long as both conductors remain connected to the bus
bars the pressure at the terminals of the transformer
cannot rise above the station voltage, but if the switch to
the outer combustion has opened the cable as well as the
transformer may, under certain circumstances, be subjected
to an excessive pressure due to exact or approximate
resonance between the inductance L x and the capacity of
the outer conductor to earth. This capacity is indicated
in the diagram by C, whilst the capacity of all the outer
conductors of other cables in the network fed from the
same bus bars is indicated by C . On opening the switch
s the circuit remaining is as follows : From the upper bus
Gable
nWWP-AAAAOi
coLi Hi
%%M%Z%Zf^
Earth
FIG. 154. Rise of pressure through resonance.
bar through the inner conductor to the equivalent coil
wLjRj, then to the outer conductor, from there through
C to earth and finally through C to the other bus bar.
We have thus two capacities and an inductance in series.
The two capacities in series are equivalent to a single
capacity c of the value
Now in a large distributing system the aggregate
capacity, C , of all the feeders and network connected with
them is enormously greater than the capacity of the
single feeder to the transformer under consideration, so
that we can write c = C, and we have thus a circuit as
shown in Fig. 155, whose natural frequency is
i i ooo / _ \
\~-- ..... (53)
27T
Ki
294 TRANSFORMERS
where Lj is given in Henry and C in microfarad. If
v l happens to be not very different from the frequency at
the bus bars we have approximate, if it happens to be
equal to this frequency we have exact, resonance, and the
current flowing through the circuit will be nearly or
exactly given by
the resistance of the cable being neglected because it is
very small as compared to R T .
This current is larger than the normal magnetising
current and produces a terminal pressure also larger than
the normal. How much larger will depend on the
relation between wLj and Rj. In a transformer having
little iron loss, but a large
magnetising current, that
is to say a low power factor
at no load, the excess pres-
sure thus produced by
approximate or complete
FIG. ^.-Resonating circuit. resonance may be several
times the normal working
pressure, and may cause a breakdown in the cable either
between the two conductors or between the outer
conductor and earth.
Determination of the dangerous condition. An
example will make the foregoing clear. Let the primary
feeders and networks of an electricity works have a total
length of 100 km. (63 miles), and let there be one feeder
leading to an isolated transformer. Assume a bus voltage
of 3000 and a frequency of 45. If concentric cables are
used, the capacity of outer conductor to earth will vary
according to the size of cable between 07 and 1*5 micro-
farad per km. Let in our case the capacity be i
microfarad per km. and let the cable feed a 2O-kw. trans-
former, which has i J per cent, or 300 watt iron loss, and
let the magnetising component of the no-load current be
3 per cent., or 0*196 ampere. The reactance of the
magnetising coil will then be 3000 : 0*196= 15,300 ohm,
and the resistance of the parallel coil representing iron
DANGEROUS CONDITION 295
losses will be 3000 : o* i = 30,000 ohm. For the equivalent
series arrangement we find from (50) and (51)
)L!= 1 2 100 ohm
1^ = 6200 ohm
Lj = 43 Henry
We have then a circuit consisting of a capacity C, a
resistance of 6200 ohm, and an inductance of 43 Henry,
all in series. It should be noted that Rj has no physical
existence ; it is a fictitious resistance corresponding to
the iron loss at normal excitation, that is to a no-load
current of ^/O'ig6 2 + O'i 2 = o'22 ampere. This current
will flow if on the terminals of the transformer 3000 volt
is impressed. To produce this E.M.F. the pressure on
the bus bars must be
where C is given in farad. For a certain value of C
(in our case about 0*3 microfarad), there will be reson-
ance, and the term in brackets will become zero, so that
a bus-bar voltage of o'22R 1 =1360 volt will suffice to
produce the full voltage on the transformer. Since the
bus-bar voltage is not 1360 but 3000, it will be obvious
that there must be a rise of terminal voltage on the
transformer. The question is, how large a rise? It
would not be correct to assume that the rise will be
simply in proportion of 1360 to 3000. This would be
the case if R! were a physical resistance, but as it is
only a fictitious resistance to represent iron loss, and as
the latter varies with the no-load current, it is obvious
that R! cannot be a constant. The problem is as follows :
Given a constant bus-bar voltage and a transformer of
known iron, find for various values of the capacity
between the outer conductor of cable and earth the
pressures between inner and outer conductor, and also
between outer conductor and earth.
The solution is as follows : From the known quality
of the iron calculate the iron loss P as a function of the
terminal pressure, and plot this as shown by the dotted
2 9 6
TRANSFORMERS
curve in Fig. 156. Plot in the same diagram, also as
functions of the terminal pressure, the total no-load
current i and its two components, i^ and i h . The copper
loss, being exceedingly small, need not be taken into
account. Now assume any terminal voltage, larger than
3000, say, for instance, 3500, and draw its vector in
Fig. 157 to an arbitrary volt scale. Let this be OA.
From Fig. 156 we find the corresponding magnetising
2000/
1-0
0-9
0-8
0-7
0-6
gO-5
*
| 04
o
0-3
i
0-2
0-1
Terminal Pressure. in
P/
1000
0123456789
FIG. 156. Characteristic curves of transformer.
current ^ = 0*23, and the current corresponding to the
iron loss / A = o'ii. Let OB and BC be the vectors of
these two components drawn to an arbitrary ampere
scale, then z* = OC is the resultant or no-load current.
Since this current is charging the condenser the terminal
E.M.F. of the latter must be at right angles to it. Draw
then from A the line AD at right angles to OC, and
determine its points of intersection with a circle, the
radius of which represents on the volt scale the pressure
DANGEROUS CONDITION
297
at the bus bars, namely, 3000 volt,
voltage e can then be scaled off on
the line AD. It is either of the
two values
AE = 5700
The condenser
No other value is possible at
the assumed terminal voltage of
3500. But in order that either
voltage may obtain the capacity C
must have a definite value, which
is found from
/o = >*Cio- . . (54)
C being given in microfarad. The
capacity is
for e= 575 . . C = i'6
,, = 5700 . . C = o'i6i
By repeating the construction here explained for
other values of terminal voltage we find other values
for e and C, and we are thus able to plot the relation
I $7' Determination of
dangerous capacity.
0-5 1-0 1-5
FIG. 158. Voltage due to resonance in an unloaded system.
between capacity to earth of outer conductor (or what
comes to the same thing), length of feeder and cor-
responding pressures between the two conductors and
298 TRANSFORMERS
between outer conductor and earth, as shown in
Fig. 158. In this figure the curve I gives the terminal
pressure on the transformer, which is, of course, equal
to the pressure between the two conductors of the
cable, and the lower curve 1 1 gives the terminal pressure
between the outer conductor and the lead sheath, that is,
earth. As will be seen, both pressures exceed 8000
volt if the capacity is 0*25 microfarad, which cor-
responds to a length of feeder of a quarter kilometer.
With a longer or a shorter feeder the excess of pressure
will be less than 5000 volt. We may thus consider
250 m. a dangerous length of feeder. For the insulation
between the two conductors the danger is not very great.
A 3OOO-volt cable will probably stand 8000 volt also,
but the outer conductor is not very heavily insulated
against the lead sheath, and for this light insulation
8000 volt is indeed a dangerous pressure, which in all
probability will produce a breakdown. All danger can,
however, be avoided if the switch gear is either so con-
structed that the inner conductor must be switched out
first, and the outer conductor must be switched in first, or
if only solid connections without any switches or fuses are
used for the outer conductors.
In exemplifying the rise of pressure by resonance
for a definite case, I have assumed that concentric cables
are used, as this is the usual practice in single-phase
working, but the same argument also applies to stranded
cables for either single or multiphase working. In such
cases the "dangerous length of feeder" is by reason of
the smaller capacity much greater, and as all the con-
ductors are equally well insulated the danger for each
is no greater than that for the inner conductor in the
case of a concentric cable. On the other hand, the
simple remedy of omitting all switches and fuses in one
of the conductors is no longer available, as it would
increase the danger for the others and displace the
electrical centre of the system. The remedy is, however,
simple enough ; it consists in arranging the switch gear
so that all the conductors of one feeder are switched on
and off together.
Rise of pressure on a loaded lighting system is small. It
PRESSURE ON LOADED LIGHTING SYSTEM 299
may be objected that this arrangement can only refer to
intentional switching, whilst the accidental blowing of a
fuse through overload may inter-
rupt one conductor and thus estab-
lish a dangerous condition. This
objection is not valid, because if
there be a load the rise of pressure
can only be very small. This
will be seen from Fig. 159, which
is plotted for the same transformer
as Fig. 158, but on the supposition
that the transformer has a load of
10 per cent, of its normal, and
that the power factor of the load
is 90 per cent. If the load were
non-inductive the rise of pressure
would be almost imperceptible,
but even at 90 per cent, power
factor it is quite moderate, and
the dangerous length of feeder is
now 600 m. For a feeder made
of stranded (instead of concentric)
conductors it would be 3 km. or
more and perfectly harmless.
Influence of power factor. I n
the case represented by Fig. 159
the slight rise of pressure is mainly
due to the fact that the power
factor of the load is only 0*9 instead
of unity, and the conclusion seems
plausible that a lower power factor
would result in a bigger rise of
pressure. This is indeed the case.
Let the transformer of the previous
example have an inductive drop of
4 per cent, and a copper loss of^
ij per cent., and let it be used to >
supply current to an induction FIG. 159. Rise of pressure on
Jri J ! r loaded system.
motor whose power factor at
starting at a pressure corresponding with 3000 volt on
the primary of a "perfect" transformer is 30 per cent.
300
TRANSFORMERS
with a starting current of 15 ampere in the primary. A
simple calculation, which need not be repeated here,
shows that the combination of motor and transformer
can be replaced by an equivalent coil of 79 ohm. resist-
ance and 230 ohm. reactance, or 244 ohm. impedance,
giving a starting current on the primary side of 12*3
ampere. Assume now that with this current one of the
fuses at the home end of the feeder happens to blow ; we
shall then have again a circuit as shown in Fig. 155,
only that now reactance and resistance are much smaller,
Volt
10,000
9000
8000
7000
6000
5000
4000
3000
2000
1000
s^
s*
V
^
/
/
\
/
1
\
/
1
\
\
/
\j
\
1
1
S
s
/
/
v
X
/
\
X
/
I
\
X
/
\
X.
,.,
/
s
>-.
-^
^___
/
1
<
*-**
***.
/
t
^
X
,
/
/
X
-
^
/
X
/
\
^^
I~
/
*
~->~
1
/
/
/
^x
Capa,
citj
in
M
F.
5 10 15 20 25 30
FIG. 160. Voltage due to resonance in a loaded system.
requiring a greater charging current, and therefore a
greater capacity that is to say, a greater length of feeder
to produce a dangerous condition. In this case it is not
the inductance of the transformer, but that of the receiving
apparatus which produces the rise of pressure. The
transformer may, however, break down in consequence.
The graphic treatment of this case is the same as
that already explained in the case of an unloaded
transformer, and need not be set out in detail. The
result is given in Fig. 160, which shows that with a
capacity of about 13 microfarad the pressure on the
transformer terminals, and therefore between the two
BREAKDOWN OF CABLES
301
conductors of the cable, as well as the pressure between
the disconnected conductor and earth, will be about
10,000 volt. The pressure on the terminals of the
transformer is represented by the curve I, and that
of the outer conductor to earth is represented by the
curve II.
Breakdown of cables in large networks. It is a
common experience that if a " dead earth " occurs at
some place in a high-pressure network, and even if
the faulty place is promptly isolated by the fuses
blowing, the insulation of the cables in some other part
Earth
FIG. 161. Illustrating break-down in large network.
of the network breaks down. This is also due to the
interaction of inductance and capacity. Let in Fig. 161
I be the inner and O the outer conductor leading from
the central station to a sub-station where a transformer
P, S supplies current to the secondary network. Ii d are
primary, and I 2 , O 2 secondary cables joining this sub-
station to others, which, in their turn, also receive
primary current from the central station, the arrangement
being that commonly in use of a complete secondary and
a complete primary network, the two interlinked by
transformers, and the primary network fed at many
points by high-pressure feeders. Let all the cables be
302 TRANSFORMERS
concentric, and the outer conductors neither fused nor
provided with switches. Owing to the great capacity of
the outer conductors to earth, the potential to earth of
d, as well as that of O 2 , will be very nearly zero, and
that of Ii to earth will be very nearly equal to the bus-
bar voltage, and that of 1 2 to the lamp voltage. A short-
circuit to earth on the outer conductor is therefore
unlikely, and, if it should nevertheless happen, inocuous,
but a short-circuit to earth of any part of the inner con-
ductor will result in a heavy current and blowing of the
nearest fuses. Let, for instance, a " dead earth " be
developed by some failure of insulation at point E, and
fusing together of the inner conductor with a metal part
well earthed, say the case of the transformer, then the
fuses f,fi, and/2 will promptly blow, but the current going
to earth will not be interrupted thereby, for P receives
E.M.F. by induction from S from the other sub-stations,
and it is only after the fuse f* has also blown that the
earth-current ceases. But/s, being in the secondary, is
necessarily a heavy fuse, and requires some time to come
into action. During that time there exists a dangerous
condition, for we have now the inductance of the trans-
former due to its magnetic leakage in series with the
capacity of O and d to lead sheath, that is to say, the
capacity of the whole of the network to earth. This capacity
in a larger system may be enormous, perhaps 100 micro-
farad or more. The charging current will now flow from E
(equivalent to the lead sheaths of all the primary cables)
through P to the totality of the outer conductors. Let
C be the capacity to lead of all the outer conductors,
then we have again a circuit as represented by Fig. 155,
but with this difference, that E is now given, not by the
bus bars, but by the primary of the transformer, and the
equivalent coils to Li and Ri represent now the effect of
magnetic leakage and true copper resistance. The
inductance and capacity being in series, there will be a
rise of pressure, but whether this will be a dangerous
rise will depend on the electrical constants of the trans-
former and network. In the first place it should be noted
that E is smaller than the bus-bar voltage, the reduction
depending on the resistance of those cables which bring
BREAKDOWN OF CABLES
303
the current from the neighbouring sub-stations. If the
capacity be very large and the transformer very small,
the latter is almost in the condition of short-circuit. It
may be burned up, but no great rise of pressure will be
produced on the cables, so that no breakdown of the
cables is likely to occur. Again, if the transformer is
very large, the charging current which the cables can
take will be insufficient to produce any considerable
E.M.F. of self-induction, and also in this case there is no
danger. Between these two extreme cases there may,
Volt
10,000
5000
10 20 30 40
FIG. 162. Rise of pressure in large network.
50K.V.A,
however, be others where the inductance due to
magnetic leakage and the capacity of the network are in
such proportion to produce perfect or approximate
resonance, and then the pressure of the outer conductors
to earth may rise sufficiently to break down the insula-
tion in one or more places.
After what has been already explained, the reader
will have no difficulty in determining for any given net-
work this dangerous size of transformers. As an example
I take a network of 100 km. having a capacity to earth of
100 microfarad and transformers with \\ per cent, ohmic
304 TRANSFORMERS
and 4 per cent, inductive drop. The resistance of the second-
ary network is such that at full load the ohmic drop between
a sub-station and a consumer midway between two sub-
stations is 1 1- per cent. The frequency is 4.5, and the working
pressure 3000 volt as before. On making the calcula-
tion for various sizes of transformers, we find that one of
about 10 k.v.a. will produce a pressure of about 8000
volt, both on its own terminals and between the outer
conductor and earth. With smaller and larger trans-
formers the pressures are less. Fig. 162 shows the
relation between capacity of transformer and pressure
for this particular case. Curve I shows the pressure on
the terminals of the transformer itself, and curve II
shows the pressure between the outer conductor and
earth. Taking a rise of pressure up to 3000 volt on the
outer conductor of the concentric cables as just on the
verge of danger, we see that transformers below 7 k.v.a.,
and above 22 k.v.a. may be used, but not transformers
between these two limits. This is with concentric cables.
With stranded cables the capacity of a loo-km. network
would be barely 20 microfarad, and then even a lo-kw.
transformer would already be outside of the danger limit.
CHAPTER XIV
SOME EXAMPLES OF MODERN TRANSFORMERS
THE practical development of any new piece of machinery
or apparatus is generally a matter of trial and error.
At first, whilst the scientific principles underlying the
new application of natural laws are but imperfectly
understood, we have a period during which inventors,
groping more or less in the dark, seek success in
abnormal designs or the special development of some
detail which later on is seen to be of minor importance ;
then comes a period where the really essential details are
recognised and receive consideration, and as these are
perfected we get to the final stage, characterised not by
divergence, but rather by uniformity of design. The first
period in the development of the transformer has not
been dealt with in this book. To the technical historian
it may be interesting to investigate the early designs of
Goulard and Gibbs, Lane Fox, Rankin Kennedy and
other pioneers, but such investigations will not help one
to either understand the working of a transformer or to
design one. The second stage, namely the conscious
improvement of details, has been treated in the previous
pages, and it now only remains to give the reader a
general survey of the last stage by placing before him a
few examples of the best modern practice.
The Brush Electrical Engineering Co., Ltd. Figs.
163 to 165 show a loo-k.v.a. single-phase oil-cooled
transformer, and Figs. 166 to 168 a 5<3'5
Length . . 465 ni.
Weight . . 645 kg.
EXAMPLES OF MODERN TRANSFORMERS 311
The current density in both windings is 2*55 ampere per
sq. mm. The induction is 13,600, and the total flux
20*6 megalines. The carcase weighs 7 tons, and the
calculated iron loss with alloyed plates is 21 kw. The
iron loss is therefore only 0*6 per cent, of the output.
The calculated copper loss is 10*35 kw. in the three
primary and 9' 15 kw. in the three secondary coils. The
total losses at full load are 40*5 kw., or 1*14 per cent, of
the full- load output. This makes the efficiency at full
load nearly 99 per cent. The insulation has been
tested with 50,000 volt for the high-pressure, and 6000
volt for the low-pressure coils to earth during fifteen
minutes.
Messrs. Ferranti, Ltd. An air-cooled core type
transformer for moderate power is represented in Figs.
172 to 174. The core is of rectangular section, with the
end plates stepped so as to more nearly fill the rounded
space within the coils. Core and yoke are bolted up
with strong gun-metal flanking plates, those of the lower
yoke being provided with flanges for attachment to the
cast-iron base. For outside protection a non-perforated
steel shell is used, and ventilation is provided by holes in
the baseband cap as shown in Fig. 172. For a primary
pressure of 2000 volt at 50 frequency and a trans-
forming ratio of about 10 : i the dimensions are as
follows
Output,
k.v.a.
A
B
C
D
E
F
30
3' 2"
i' 3"
10"
2' 10"
i' 7"
2' o"
50
3' 4"
i' 3"
10"
2' I 4"
,' 9"
2' o"
Fig. 175 shows the rise of temperature of the 30-
k.v.a. size, and Fig. 176 that of the 5O-k.v.a. size at
full load, whilst Figs. 177 and 178 represent the regula-
tion. In each diagram two curves are shown, the
ordinates representing percentage drop at the secondary
terminals as a function of the power factor in the
i_
fe
EXAMPLES OF MODERN TRANSFORMERS 313
secondary circuit, but with this difference, that the lower
curve refers to constant current and the upper to con-
Room Temperatur >
10.0 11.0 12.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time in Hours
FIG. 175. Heating curve of Ferranti 3o-k.v.a. transformer.
stant power. In the latter case the current must
increase with decreasing power factor, which accounts
for the greater drop.
35
Room
Temperature
9.0 10.0 11.0 12.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8-0
Time in Hours
FIG. 176. Heating curve of Ferranti 5O-k.v.a. transformer.
Fig. 179 shows the general design of single-phase
oil-cooled transformers for small and moderate out-
TRANSFORMERS
put. The transformers may be either placed on a level
floor or attached to a wall, for which purpose the case
1-0 0-95 0-9 0-85 0-8 0-75
FIG. 177. Regulation curve of Ferranti 3O-k.v.a. transformer.
is provided with lugs. The dimensions A to E in
inches, and the weight in cwt. are given in the following-
table.
The efficiency at full non-inductive load ranges from
1-0 0-95 0-9 0-85 0'8 0-75
> cos if
FIG. 178. Regulation curve of Ferranti 5o-k.v. a. transformer.
91*2 per cent, in the ^-k.v.a-. size to 97*9 per cent, in
the 5o-k.v.a. size; at quarter load from 777 to 97*1
per cent.
EXAMPLES OF MODERN TRANSFORMERS 315
K.v.a.
A
B
C
D
E
Weight
*
Mi
IO
Si
;i
15
I
I
i6f
"1
10
4
I/
i
2
i6|
"I
IO
si
17
2
5
20|
*5*
"I
1 1
*
3t
10
'25
18
i5i
H
30
7i
i5
25
18
5t
H
32
8*
20
25f
i8f
'6|
1 4!
35
*
25
27
20f
i7f
is!
36
M
30
29i
22 i
i9|
17
36
16
40
32
23i
22 J-
i9|
39
i7i
50
34
25i
24
21
39
20
For the transformers here described iron and copper
losses are as under
Output k v a
c
3O
^O
I 2O
J
o^
j v
Iron loss, watt ....
I 12
290
350
1670
Copper loss, watt . . .
131
400.
760
H85
The copper loss is given at full secondary current,
corresponding to the volt-ampere at which the trans-
former is rated, and after it has been at work sufficiently
long to have reached its final temperature.
TRANSFORMERS
MFiG. 179. Ferranti standard type
I oil-cooled transformer.
Fig. 1 80 shows the design of a small transformer for
a very high primary pressure. To secure good insula-
tion the primary circuit is arranged in 8 distinct coils,
which may be more easily handled and tested before
being assembled. In the present case the transformers
are intended for three-phase io,ooo-volt circuits.
The dimensions for a 50- and 3O-kw. transformer, the
output being obtained at a power factor of 075, are as
follows
Kw.
A
B
C
50
4 'o"
2' 7i"
2' IOJt"
30
3' 4"
2' 4"
I' 10"
EXAMPLES OF MODERN TRANSFORMERS 317
In star coupling each primary takes 5800 volt, or
725 volt on each individual coil. To protect the low-
pressure circuit an earthing shield is used.
A three-phase transformer, where the three phases
are combined in one apparatus, is shown in Figs. 181 and
182. It is a i2O-kw. transformer, star coupled for 5000
volt primary line pressure, and a secondary pressure,
which may be adjusted to either 370, 380, or 390 volt.
For this purpose tappings are taken on the secondary, and
by means of a three-pole switch the number of turns on
each limb may be changed from the normal of 49 to either
48 for the lower or 50 for the higher voltage. This is
shown diagrammatically in Fig. 183. The switch is seen
in Figs. 181 and 182 mounted on the top of the casing.
The core area is 350 sq. cm., the flux 4*45 megalines,
and the induction 12,700. The following table gives the
winding data
i2o-kw. Transformer
Primary
Secondary
Connection.
Star
Star
Line voltage .
5000
380
Number of coils . .
12
3
Wire, bare, mm. . .
4'30X2-55
1 2 '6 x 2-54, four in
parallel
Wire, covered, mm. .
4-8 x 3-05
13-2x3-05
Number of turns per
limb
6 3 6
50
Number of layers
5
i
Tappings ....
none
370 volt on 48th turn
380 volt on 49th turn
390 volt on 5Oth turn
INS.1Z 9 6 3
3 FT.
FIG. 1 8 1. I20-kw. 5000 to 380 volt 4O-frequency three-phas
made by Messrs. Ferranti, Ltd.
e transformer
320
TRANSFORMERS
SECONDARY TERMINALS
NEUTRAL
PRIMARY TERMINALS
FIG. 183. Diagram
of connections to
Fig. 181.
FIG. 182. End elevation to Fig. 181.
EXAMPLES OF MODERN TRANSFORMERS 321
r
FIG. 184. 5oo-k.v.a. 3 1200/2200- volt transformer made by the Bullock El. Mfg. Co.
The B^illock Electric Manufacturing Co. Figs. 1 84 to
188 are good examples of high-class American practice in
21
322 TRANSFORMERS
the design of transformers. Figs. 1 84 and 185 show a 500
k.v.a. oil-cooled shell-type transformer for 60 frequency
FIG. 185. 500-k.v.a. 3 1200/2200- volt transformer made by the Bullock
Electrical Manufacturing Co.
taking current at 31,200 volt, and delivering current at
2200 volt. On account of the high voltage, special
precaution has been taken to separate the coils by
EXAMPLES OF MODERN TRANSFORMERS 323
insulating partitions, shown by thick lines, and the
terminals are also immersed in oil. The secondary
winding is in two groups, so that the same type may be
,EP.
X?
g
-LP. Coil
-'Case
ater Outlet
Air
ater Inlet
^
^
==:
!
- .."
===^
=^
^
W
/
//
\
^
^
^-^
.
_/
/
X
5C
OK
w.
/I
/
_
__
_^ - -
_
^^=-
=rT
.
<-
^ =5 * fc -*
j
|!
<~ 112% Load ->
1 8 9 10 11 12 13 14 15 16 17
Hours Run
<- - 101% Load ----- ^j* 147%->j
Load
FIG. 186. Heating curves of 5oo-k.v. a. transformer.
coupled up for half the secondary pressure and double
current. The transformer is one of a group of three for
Load
FiG. 187. 500-k.v.a. 3 1 200/2200- volt transformer.
three-phase work, as is the usual practice in America, in
preference to building one three-phase transformer of
treble output. The use of three single transformers in
324
TRANSFORMERS
mesh connection has the advantage that a failure of one
transformer need not interrupt the three-phase service ;
the other two remaining at work are simply overloaded,
so as to do the work of three ; but as the time-constant of
large transformers is very great, no damage is done to
the overloaded transformers during the time required to
bring a spare transformer into service. This advantage
is, of course, lost if star coupling is adopted. The trans-
former here illustrated is, however, sufficiently well insu-
lated to be used in star-connection, when the line pressure
on the primary side is 54,000 volt. The oil is cooled by
FIG. 1 88. Plan to Fig. 189.
a cold-water worm placed in the upper part of the case,
and the terminals are brought through the cover by large
porcelain ferrules. The total weight of copper is 300 kg.
In Fig. 1 86 a heating test of this transformer is
recorded. To shorten the time, the test was started with
a 1 2 per cent, overload, and then the run was continued
with about full load. The final temperature-rise at
normal full load is 38 C. for the coils, and 35 C. for the
carcase (marked " case " in the diagram). The great
difference of temperature between water inlet and water
outlet should be noted, as also the small difference
between the temperature of the carcase and the out-
EXAMPLES OF MODERN TRANSFORMERS 325
flowing water. This indicates a very vigorous circulation
of the oil and efficient action of the worm.
r~
Fig. 187 shows the performance of this transformer
as regards efficiency and regulation.
A small transformer of the type frequently used in
326
TRANSFORMERS
America for private houses is shown in Figs. 188 to 190,
The secondary circuit is arranged in two groups, and
there are four secondary leads brought out of the case,
so that not only can the same type be used for full and
half voltage, but, by grouping the two windings in series
and joining the connecting joint to the neutral bar of the
distributing switchboard, a supply on the three-wire
system may be given. The dimensions refer to a 5-k.v.a.
transformer at 60 frequency for the voltage usual in a
house-to-house system of supply, namely, 1000 to 2000
volt on the primary and not over 200 volt on the
secondary.
The British Westinghouse Co., Ltd. Before entering
on a description of the various designs, it will be useful
to say a few words concerning the principles on which
this firm has standardised its transformers. Although
high efficiency is always desirable, there are cases where
it is especially important. Thus in a lighting transformer
high efficiency at low loads is far more important than
in a power transformer, because the latter is not worked
for a very long time at light load, and power current is,
as a rule, cheaper than lighting current. On the other
hand, capital outlay for the very reason that power
current must be supplied cheaply, is an important matter
in power supply undertakings, and for these reasons it
may be good policy to sacrifice a little in efficiency if
thereby the fixed charges can be reduced. To satisfy
the various conditions of working the British Westing-
house standardise two types, one for high, the other for
medium efficiencies.
The relation between these may be seen from the
following table, which refers to the two types of 4O-k.v.a.
transformer
Percentage of load . .
IOO
75
50
25
High Efficiency
97-80
9777
97'53
96-15
Medium Efficiency
97*50
97-40
96-80
9475
EXAMPLES OF MODERN TRANSFORMERS 327
The high-efficiency transformer is, of course, more
costly and also heavier. In the 4o-k.v.a. size the
complete weight with tank and oil is 660 kg. for the
high-efficiency transformer, and 540 kg. for the medium-
efficiency transformer.
FIG. 191. FIG. 192.
4O-k.v.a. 2ooo/2OO-volt transformer made by the British Westinghouse Co., Ltd.
Another matter which requires attention when
standardising a line of transformers is the question of
heating. An actual working temperature up to 85 C.
may be permitted, but a knowledge of this limit alone
is not sufficient to determine the cooling surfaces ; we
must also know the temperature of the room in which
the transformer will have to work and the character
of the load and the cooling conditions.
328 TRANSFORMERS
In large transformers this information is generally
available beforehand, and the designer can make his
calculations accordingly. Small transformers must, how-
ever, be made in quantities for stock, and the designer
cannot know in what localities and under what conditions
they will have to work. He must therefore design the
small transformer for a lesser temperature rise than might
be allowed in a medium-size transformer. If he designs
o
a large transformer with water-cooling he can calculate
still more closely, that is to say, allow a greater tempera-
ture rise, because all the conditions are of such a nature
that no great departures from known averages are likely
to occur.
The smaller transformers are rated for a temperature
rise of 40 C. over the surrounding air. An air tempera-
ture of 45 C., which might occasionally be reached in
badly-ventilated transformer chambers or pillars, would
then not cause damage. Larger transformers are rated
for a temperature rise of 45 C. as their load conditions
can more accurately be predetermined, and more care is
taken in placing them in properly-ventilated chambers.
Oil-insulated water-cooled transformers are normally
rated for 50 C. rise over the temperature of the
entering water ; as this will, under ordinary conditions,
not exceed 30 C., this rating should be safe.
Figs. 191 and 192 show a 4o-k.v.a. single-phase
oil-cooled shell transformer of the high-efficiency type.
Jt is designed for the standard frequency of 50 and a
primary voltage of 2000. The carcase weighs 220 kg.,
and the winding 125 kg. The coil area is 350 sq. cm.,
and the induction 6500 lines per sq. cm., giving a flux of
2*27 megalines. The window area is 216 sq. cm. The
quantity of oil required is 34 gallons, or 138 kg. No
cooling-worm is used, sufficient surface for air cooling
being provided by the corrugated sheet-iron case. In
smaller sizes up to 10 k.v.a. the case is of cast-iron,
and provided with lugs for fixing to hanger irons for
attachment to a wall.
In Fig. 193 is shown one of a set of three trans-
formers supplying current to a six-phase 1000 kw.
rotary converter. For this purpose each secondary
EXAMPLES OF MODERN TRANSFORMERS 329
6
&
I
s
I
330
TRANSFORMERS
EXAMPLES OF MODERN TRANSFORMERS 331
circuit must be in two parts, so that four secondary
terminals are required. To provide extra cooling
surface the heads of the coils are splayed out, the
carcase is built up of narrow packets separated by
circulation ducts, and the corrugations of the case are
very deep.
Fig. 194 shows a three-phase core-type oil-cooled
transformer made for the Castner-Kellner Alkali Works,
Newcastle-on-Tyne. The capacity is 1200 k.v.a. at
5750 volt on the primary side, and 40 frequency. The
voltage on the secondary side is 175, but by altering the
method of connecting up, the same winding may be used
for different voltages.
Fig"- J 95 shows three different arrangements of
terminals, marked A, B, and C. The winding is sub-
divided into discs as explained in Chapter VIII. For
sizes larger than 1200 k.v.a. the cooling by a corru-
gated case is supplemented by a cold-water worm.
For pressures up to 20,000 volt cooling by artificial
blast without oil may be used. Fig. 196 shows a trans-
former of this type for 11,000 volt on the primary and
400 volt on the secondary side. The output is 550
k.v.a. at 33 frequency. A number of these trans-
formers have been installed at the Baker Street sub-
station of the Metropolitan Railway. They serve for
supplying current to rotary converters. The trans-
formers are kept cool by a strong air-blast sent in
through a duct in the floor. Gratings, the opening of
which may be separately regulated, are provided to
suitably sub-divide and direct the stream of air through
carcase and winding, and in order that the attendant
may see at a glance whether the ventilation is in order
a little wind-mill, indicated at a in Fig. 196, is fitted to
each transformer over the top grating.
The advantages of cooling by air-blast over cooling
by oil are greater cleanliness and convenience in case of
repair, but care must be taken to have the air free from
soot or dust. On the other hand, oil (unless it must be
supplemented by a water-worm) requires no accessory
apparatus, such as a fan, and no expenditure of power.
It has also the advantage of increasing the time constant.
332
TRANSFORMERS
ft
ft ff
t t r t
FIG. 196. 55o-k.v.a. 1 1, 000/400- volt
air-blast transformer made by the
British Westinghouse Co., Ltd.
EXAMPLES OF MODERN TRANSFORMERS 333
r
V F
.A.
J.
FIG. 197. Lamp transformer made by the British
Westinghouse Co., Ltd. *
334
TRANSFORMERS
EXAMPLES OF MODERN TRANSFORMERS 335
In transformers connected to overhead lines which are
liable to atmospheric disturbances it is also sometimes
claimed for oil that it acts as a self-healing insulation,
closing the small hole punctured in the solid dielectric
when a static discharge occurs. Whatever may be the
value of this view, the fact remains that experience
has led most makers to use oil for high-pressure
transformers.
With the advent of the wire lamp a demand has
arisen for very small transformers for single lamps or
groups of a few lamps. Wire lamps are at present
made for moderate voltages, 25 to no, but many of the
existing supply systems exceed these
limits, so that a transformer becomes
necessary to burn the lamps inde-
pendently. It is, of course, possible to
use one transformer for the whole of
the lamps, but then the iron loss would
reduce the yearly efficiency of the in-
stallations, and it is also questionable
whether the existing leads would be
able to carry the larger currents with
a moderate ohmic drop. When this is
not the case, small transformers of the
type shown in Figs. 197 and 198 may be
used. The switch must, of course, be put
on the primary side of the transformer,
the latter with its wire lamp taking simply
the place of the previous carbon lamp. Fig. 199 shows
a lamp-transformer attached to a wall bracket.
Messrs. Brown, Boveri & Co. The largest trans-
former ever made is probably that supplied by Messrs.
Brown, Boveri & Co. to the Betznau Power Station
(Switzerland). Its capacity is 4600 k.v.a., and it serves
to transform up a machine current of 8000 volt to a
line pressure of 27,000 volt. It is a three-phase core
type star-coupled transformer. Its primary phase volt-
age is 4650, and its secondary 15,620 volt. Figs. 200
to 202 show the construction to a scale of i : 20*6.
Fig. 203 shows the method of winding, and Fig. 204 is a
general view of the transformer by the side of its tank.
FIG. 199. Lamp
transformer on wall
bracket.
336
TRANSFORMERS
FIG. 200. 46oo-k;V.a. three-phase transformer made by Messrs. Brown, Boveri & Co.
EXAMPLES OF MODERN TRANSFORMERS 337
FIG. 201. 46oo-k.v.a. transformer made by Messrs. Brown, Boveri & Co.
22
338
TRANSFORMERS
The following particulars will be of interest
Frequency * . . . "r ..-.." 50
Core area, sq. cm. , . . . . 2,380
Induction lines per sq. cm. . . . . 11,300
Flux in megalines . . . . ; . 26*9
Number of turns per phase in primary . 78
Number of turns per phase in secondary . 267
Area of primary conductor in sq. mm. '". 178
Area of secondary conductor in sq. mm. . 44
Current density in primary, ampere per sq. mm. 1*85
Current density in secondary, ampere per sq. mm. 2*14
Resistance of primary per phase in ohm . 0*030
Resistance of secondary per phase in ohm . 0^440
Iron loss measured in watt . ... . 40,500
Total copper loss measured in watt .. .-." 25,500
Efficiency at full load, per cent. . . . . 98*6
Maximum drop, per cent. . . . . 1*6
FIG. 202. 46cx)-k.v.a. three-phase transformer made by Messrs. Brown, Boveri & Co.
EXAMPLES OF MODERN TRANSFORMERS 339
The total weight of iron is 12 tons, so that with full
non-inductive load the weight of iron is only 2*6 kg.
per kw. The copper weight is 2400 kg. or 0*524 kg.
per kw. The total weight of active material is only
ABOVE
Presspahn
distance, pieces
The 40 top turns
separated by
1mm. presspahn
BELOW
FIG. 203. Method of separating the primary and
secondary coils.
SO
3*124 kg. per kw. It will be obvious that with
large a reduction in the weight of active material per kw.
a very perfect cooling device becomes necessary, and for
this reason the usual plain cooling pipe has been replaced
340 TRANSFORMERS
by an elaborate cooler provided with ribs in the manner
of a radiator as shown in Figs. 201 and 202. The
terminals are taken through long porcelain tubes, the
lower ends of which are well below the oil level.
FIG. 204. 46oo-k.v.a. three-phase transformer made by Messrs. Brown,
Boveri & Co.
Messrs. Siemens Schuckert Werke. Figs. 205 and 206
illustrate an air-cooled transformer of the shell type for
50 frequency, 3000 to 220 volt, and 6'86 to 92 ampere.
The dimensions inscribed are mm. The carcase is built
up of 0*3 mm. alloyed sheets insulated to 0^33 mm.,
and has butt-joints. The iron weight is 165 kg. and
the measured iron loss at the induction of 9700 is
335 watt > or at tne rate f 2 '3 watt P er kg- This
agrees within a few per cent, with the curve given on
page 26.
EXAMPLES OF MODERN TRANSFORMERS 341
-325
700
FIG. 205. 2O-k.v.a. 3OOO/22O-volt transformer made by Siemens Schuckert Werke.
342
TRANSFORMERS
The particulars of this transformer are as follows
Core area in sq. cm. . . ; . . 300
Flux in megalines . . . . . 2-9
Number of turns in primary . . V . 464
Number of turns in secondary . i . 34
Section of primary sq. mm. . . ".-'. ' 6
Section of secondary sq. mm. .. .. . . 80
Resistance of primary hot in ohm . . 2*25
Resistance of secondary hot in ohm , / 0*0126
Ohmic drop in per cent. ., . , . 1*03
Inductive drop in per cent. . . . ; . 0*90
Maximum possible drop in per cent. . j 1*37
Final temperature rise in iron, degree C. . / 55
Final temperature rise in copper, degree C. . 55
FIG. 206. Side view of Fig. 206.
FIG. 207. 14-k.v.a. three-phase transformer made by Siemens Schuckert Werke.
FIG. 208. Plan of Fig. 207.
344
TRANSFORMERS
The primary winding is subdivided into 4 coils of
116 turns. Each coil consists of 58 layers of 2 turns,
the section of copper being 1*3 mm. by 47 mm. wide.
The secondary winding is also subdivided into 4 coils, two
of which have each 8 turns, and the other two 9 turns.
The copper section is 10 mm. by 8 mm. wide. It is
T
^^TI
M
T^-N
~^ ~-~~
1
"
< 200
1
1-
|
A ^
--3--
- L^^^^,- -
I
Jt
i
~t ifz 4
I
I
i
i
t
1
I
i
/
V
u
FIG. 209. End view of Fig. 208.
made up of two strips, 5 by 8 mm. wound on together.
The weights are
Carcase . ... . , .165 kg.
Primary copper . V _, 37*3 ,,
Secondary copper . . -.; 36*1 ,,
Total active material . . 238*4 ,,
or at the rate of 11*92 kg. per kw. at full non-inductive
load. The efficiency is 97^ *
/o*
EXAMPLES OF MODERN TRANSFORMERS 345
A three-phase transformer, also of the shell type,
with butt joints, is illustrated in Figs. 207 to 209. The
dimensions are mm. It is a I4~k.v.a. oil-cooled trans-
FIG. 210. looo-k.v.a. three-phase transformer made by Siemens Schuckert Werke.
former for 50 frequency, 3000 to 200 volt, and 2*8 to 37*4
ampere. Both circuits are star coupled. The windows
are 87 mm. square, and each contains 518 primary and
346
TRANSFORMERS
38 secondary wires. Each primary is arranged in two
coils, and each of these has 28 layers of 9 turns and one
FIG. 2ii. Side view of Fig. 210.
layer of 7 turns. The two coils are placed side by side,
and outside of them are placed the two secondary coils,
each containing 19 turns. The primary conductor is
EXAMPLES OF MODERN TRANSFORMERS 347
round wire of 1*8 mm., the secondary is strip 3*4 mm.
by 9*4 mm. wide. The resistance cold is, for the primary
4*2, and fpr the secondary 0*0245 ohm. per phase. The
weight of carcase is 149 kg., and the measured iron
loss at an induction of 9600 is 276 watt or r86 watt per
kg., which corresponds exactly with the Author's tests
recorded in Fig. n. The primary copper weighs
38*8 kg., and the secondary 36*4 kg. The total copper
<$ j._ - .-.-.-.- p. - _ *._._ _.
1"- , -* -v ?n - t.-f.^'l: fflr.f--tr.--tr i.-^
FIG. 212. Plan of Fig. 210.
loss at full load when the transformer is hot is 250 watt.
The final temperature rise, both in iron and copper, is
55 C. The inductive drop is 1*8 % ; the ohmic drop is
i-5 % .
A three-phase transformer for 1000 k.v.a. at 50
frequency is illustrated in Figs. 210 to 212. It is also
of the shell type with butt joints, but with special
cooling appliances. The transformation ratio is 5000
to 10,000 volt, the iron loss amounts to 8*3 kw., and the
343
TRANSFORMERS
.
FIG. 213. 225O-k.v.a. three-phase transformer made by Siemens
Schuckert Werke.
EXAMPLES OF MODERN TRANSFORMERS 349
copper loss to 6*82 kw., corresponding to an efficiency of
98*5 % at full non-inductive load. Oil is used for cooling,
and to facilitate the flow of heat from oil to case the
latter is provided with internal ribs, as shown in
Fig. 212. There is no cold-water worm employed,
but to cool the case its external surface is played on
by water. For this purpose a water-supply pipe is laid
round the top of the case on the outside. This pipe is
provided with holes through which the water is squirted
against the outer surface of the case, and, flowing
o
D
D
FIG. 214. Carcase of three-phase Siemens Schuckert
225 . . . . = specific cooling surface.
w . . . = an angular speed.
v . . . = frequency, complete periods per second.
356
APPENDIX III
FORMULA USED
No.
Page
Formula
Subject
i
10
TT * g
Crest value of E.M.F.
2
3
4
12
12
12
*-4*-
Effective value of an alter-
nating current of sine
form
Effective value of an alter-
nating current of any
form
Effective value of an alter-
nating E.M.F. of any
form
,w;y;v,
5
14
e- E
Effective value of an alter-
nating E.M.F. of sine
form
6
14
,- 4 -44v*io-
Effective E.M.F. in coil
of transformer
7
!I 7
= /fcB r6
Energy per cycle
8 20
P -o-iQ/A " B V
Loss due to eddy currents
in ordinary transformer
plates.
JT w U 1 U 1 / \
\ 100 IOOO/
Sa
22
P *--"(lio- 8
E.M.F. of flat curve
10 30
e = 4'62vio- 8
E.M.F. of peaked curve
ii 55
c
Temperature rise
12
61
t=^j~
Time constant in seconds
13
14
61
61
/ T \
Heating time in seconds
Heating time in hours
2*3' / T \
3600 VT -y/
357
358
TRANSFORMERS
No.
Page
Formula
Subject
a+& b_
15
65
p _ PI(*~*~ ~ J ) - p o(^ ~ J )
Permissible load for inter-
mittent working
+
e^-e*
16
74
G-c P
Weight of carcase
/ v ff#p
\
IOO
i7
84
e = U
E.M.F. of self-induction
18
g
Z = w (>10- 6
Condenser or capacity
current
Current through resist-
in
Q
s^t^. o - r\ \t*kf\iir'\'CiY\r i '* in
j
VR 2 + W
cinCc dllLl UlU.U.i-'tcHix^C ill
series
20
86
/
Current through resist-
ance, inductance, and
\/ R 2 + (^L - j
capacity in series
21
8?
IOOO
d)
Natural angular velocity
u /
VCL
22
87
I 60
~VcE
Natural frequency
2 3
93
P = cos
100 o"i (t j -f- i
Inductive drop in cylin-
drical coils
e ^< \" 3 J I
34
181
ioo ^_ o X /z/ + 2\ /^ + # 1 + 2 \/
Inductive drop in flat
coils
J < \ A 6/1
35
182
e s , X/, a 1 + a z \p
Inductive drop in shell
transformer with sand-
e "^V 6 //
m
wiched coils
36
184
i/-r 3 D"
Relation between drop
and frequency for the
same iron heat
37
234
B ^
^ 7 " ^
Induction in sample when
tested by Epstein ap-
1000 J vA
paratus of normal con-
struction
38
235
P = //!i/B*+/(i/B) 2
General expression for
iron loss
39
243
, I 2^>
Ballistic constant
T O ioor
40
243
t> = ~
55 55
T i T
41
244
55 5J
27T X 27T
42
246
/3=2/4
Logarithmic decrement
43
246
.T = a- (i+' / 8)
Undamped elongation
44
249
n
Change of flux given by
fluxometer
45
252
(fr = ^mtQ
Total flux given by
shunted fluxometer
46
257
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