theory of the transformer - illinois: ideals home
TRANSCRIPT
THE UNIVERSITY
OF ILLINOIS
LIBRARY
THEORY OF THE TRANSFORMER
BY
LEONARD VAUGHAN JAMES
B. S. UNIVERSITY OF ILLINOIS. 1906
M.S. UNIVERSITY OF ILLINOIS, 1012
THESIS
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
ELECTRICAL ENGINEER
IN
THE GRADUATE SCHOOL
OF THE
UNIVERSITY OF ILLINOIS
1913
UNIVERSITY OF ILLINOIS
THE GRADUATE SCHOOL
1 HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY
ENTITLED 7Xe
BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE
DEGREE OF ^
In Charge/of Major Work
Head of Department
Recommendation concurred in:
Committee
on
Final Examination
Digitized by the Internet Archive
in 2013
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141
V
TABLE OF COUTEiJTS
PageIntroduction X
Development of Differential Equations 3
Transformer Characteristics 14
Predetermination of Characteristics:
Solution by Differential Equation Method 17
Graphical Solution by Waves and "by
Vector Diagrams ^Solution by Complex Quantities 29
Summary 44
Appendix 46
-1-
TEEORY OF EBB ERANSFOBMER
INTRODUCTION
The predetermination of the operating charac-
teristics of the transformer, from a knowledge of its con-
stants and of the load conditions, is largely a study of
simple physical relations. This is true in the trans-
former "because of the actual constancy in value of the so-
called "constants", a condition due especially to the pres-
ence of a low permeability path for the main or exciting
field and of high and constant permeability paths for the
leakage fields, as well as to the fact that there is no
shifting of these fields between paths of different per-
meabilities when the load changes. Because of these
things, the physical relations when . determined from simple
short circuit and open circuit tests, may be considered as
the general relations for any condition of loading. This
makes possible, therefore, the use of comparatively simple
mathematical equations and a correspondingly simple solution
of the problem.
As will appear later, the most essential principle
upon which the operation of the transformer is based is that
of mutual induction, requiring differential equations in its
general solution. Thus the machine will first be considered
-2-
from this standpoint. The solution will then he ob-
tained graphically by the substitution of equivalent sine
waves for the several more or less irregular waves repre-
senting the alternating voltages and currents. It will
he seen that the vector diagram presents the graphical
solution more clearly than the waves hut that both waves and
vectors have the disadvantage of being inaccurate. When
the vectors are expressed as general numbers, however, and
the problem solved by the algebra of the complex quantity,
the clearness of the vector analysis is combined with the
accuracy of exact analytical solution. The only limita-
tion in this method is that equivalent sine waves must be
substituted for the original waves. Whenever this is
impossible, the problem can only be solved by resorting
to the use of differential equations.
-3-
DEVEIOPMEIIT OP DIFFERENTIAL EQUATIONS
Principles and Construction of the Transformer
A transformer is a machine in which the principle
of mutual induction "between stationary coils is utilized in
transferring electrical energy from one circuit to another.
In general, the number of turns is not the same in the two
coils so that the energy will appear in the second coil at
a different voltage from that in the first9this transfor-
mation being responsible for the name, transformer. Depend-
ing upon induction, this machine is used only in alternating
current circuits and can be used only where the energy sup-
plied is alternating or oscillatory in nature.
In mechanical construction, the transformer con-
sists of two coils, a primary and a secondary, connected by
an interlinking magnetic circuit. Due to the resistances
of the electric circuits and to any hysteretic effects in
the magnetic circuits, there will be a certain loss of
energy, while the periodic storage and release of energy
in the leakage magnetic fields will affect the phase dis-
placement between the voltage and current.
A survey of the physical phenomena involved in this
machine shows that, when an alternating voltage is impress-
ed upon the primary coil a current flows in that coil, caus-
ing an alternating flux to be set up in the magnetic circuit.
_4_
This flux induces an electro-motive-force "both in the pri-
mary and in the secondary winding, opposing that impressed
in the primary and providing an electro-motive-force in
the secondary which is available to send a current through
a second circuit. Should this current be permitted to
flow, it will react upon the magnetic field according to
Lenz's Law in such a way as to decrease the same and
finally to cause an increase in the current in the primary
winding. The currents thus appearing in the primary and
secondary coils are known by the names of the respective
windings as primary and secondary currents . Likewise the
primary voltage is the voltage impressed upon the terminals
of the primary winding and the secondary voltage is that
which appears at the terminals of the secondary.
Mathematical Relation, Referred to the Primary Coil
In developing the mathematics expressing the
fundamental relations in the transformer, the primary coil
will first be considered alone. As before stated, if a
voltage be impressed upon this coil a current will flow
through it. The relation between this voltage and cur-
rent is expressed in the fundamental differential equation
of the electric circuit,dil
e 1 • ii *i 3L« H- (1)
where
e-^ m impressed voltage at any instant, t,
i^ = current in primary at the same instant, t.
-5-
r^ v resistance of primary coil,
L'-^= total inductance of primary coil.
The inductance, L 1
-^, is here the total inductance of the
primary coil and is a quantity depending upon the dimen-
sions and material of the magnetic circuit. One of thesel
dimensions is seen to be the entire cross-sectional area
of the coil.
It is apparent that a part of the total flux due
to the primary current will induce a voltage in the secon-
dary coil so that, if this coil he closed through a resis^
tance, a current ig will flow. The problem now becomes
one of imperfect mutual induction, and
di-. dipe
n« U XS * L'-, --- - M --- (2)
dt dt
where M is the coefficient of mutual induction. (See note, p. 13.)
In (2), L'^ is still the inductance due to the
entire area of the primary coil. If the flux in any coil
is separated into a number of parts, in order to consider
the various paths or for any other reason, the sum of the
voltages induced by each part must be the e. m. f. which
would be induced by the total flux.
That is, if
* - *a * *b*
or if *
* = Ki - V + V + V + —*Paths in air having constant inductance for different densi-ties are here intended. Were an iron path introduced thismathematics would not apply exactly as shown but the reason-ing and final conclusions would be similar.
-6-
then
d? di di di di-n — = -nk — as -nZa - nK-^ — - nKc —
dt dt dt dt dt
Where <P is flux, i is current, n is turns and K is a con-
stant depending upon the dimensions of the coil and the
part of the area under consideration. Then, by the defini-
tion of the coefficient of self-induction,
L = La + 1^ + Lc +— (3)
where the values of L are calculated for their respective
parts of the area.
Prom (3) it appears that (1) may "be written as
follows
,
di l dile i - h r
i + Li :;- + L
0l--- ^
dt dt
where LQ^ is the inductance factor for that part of the
flux which threads "both coils and which will here he termed
the useful or exciting flux because it induces the secondary
e. m. f., while L-^ is the inductance factor for the rest of
the flux threading the primary coil, which, since it per-
forms no useful function, is termed the leakage flux. This
idea may also be expressed in (2), so that
di-, di-, di2el =11*1+ Li — + L0n M — . (5)
dt 1 dt dt
"induced Voltage" and Reactance Drop
In (5) it is seen that the total voltage con-
sumed by induced e. m. f.'s is
-7-
din
di-, dip
dt 1 dt dt
Of this, that voltage consumed by the e. m. f. induced by
the useful flux or what is commonly termed the "induced
voltage" in the primary is
di-. di ?eu - L - M --- . (6)
1 dt dt
di-,The remaining part, L-i-r^, is consumed by the e. m. f. in-
xdt
duced by the leakage flux and is known as the reactance drop
or leakage reactance drop.
By substitution from (6), (5) now becomes
di-
h * i-, r-, + Li — - + e± (7)dt
This equation is very suggestive. If perfect mutual in-
duction and zero resistance may be assumed, it becomes
di-. dipe-i = e. m Ln — - - M —1 x l °1 dt dt
which shows that, in this case and for the same impressed
voltage, the exciting flux will be the same whatever the
value of the secondary resistance and current, the primary
current changing to maintain the equilibrium. Under the
natural or imperfect conditions it is seen that e. will bexl
less than e^ by
dil
i! J* * Li ---
or by the resistance and reactance drops in the primary.
-8-
By substituting for M, (5) may be re-written so
as to show two very important relations. The mutual in-
ductance is
(8)M°2
where 1 is the inductance of the secondary coil for the
dimensions of the exciting field. From this definition it
follows that
and so by substitution
n.
_2
*i(9)
nx °1
Substituting this value of M into (5) and collecting,
di-
1 = il
rl
+ Ll 7"" + Lo-dt
di
dt
n_2_ **2
n]_ dt(10)
Relative Magnitudes of Induced and Impressed Voltages
In (10) it first appears that, if the induced
primary e. m. f. is to approach the impressed voltage, the
resistance and reactance drops must be very small when com-
pared with which requires that must be very small
when compared with L ^. This relation is obtained by pro-
viding an iron core for the exciting flux and by winding the
coils closely over this core. In many cases, the primary
and secondary windings are made up of several coils each and
these coils are "sandwiched" between each other so as to
provide a still greater ratio of LQ^
to L^. In some cases,
-9 -
however, definite provision is made for a large value of L-j_
as a protective feature against surges or short-cirouits.
The JExciting Current
The second relation to be found in (10) is that,
if there is to he an exciting flux to induce e. ,
1
nl dil> n2 di2
or
ni ii> n
2 h •
Interpreted, this means that the primary m. m. f. must he
greater than the secondary m. m. f. "by enough to produce the
exciting flux. The additional primary current necessary
is known as the exciting current and is designated by i00 ,
where
For an air core this exciting current will have the same
wave form as the flux it produces and will be in phase with
that flux; for an iron core its wave form and phase relation
are distorted when compared with those of the flux, due to
the hysteresis and saturation effects in the iron. To be
exact in the case of the iron core, then, the exciting cur-
rent can not be assumed to vary directly either with the
flux or with the induced e. m. f. Its wave shape will also
be likely to change for different values of maximum flux
density even though the flux waves have the same shape.
-10-
Since, however, the transformer is usually worked well be-
low the knee of the saturation curve and since the maximum
range of variation of the effective value of e-j^is from e-^
on open circuit to half of e^ on short circuit, when the ex-
citing current is a small percent of the rated current, the
approximation
ij.-x.ij (11*)
can he made without material error in the results. When
this is done, the exciting current is usually resolved into
two components, the core loss current in phase with the in-
duced voltage being that required to supply the hysteresis
and eddy current losses in the iron, and the magnetizing
current in phase with the flux being that required to mag-
netize the core.
Final Equation for the Primary
If the relation of currents as expressed in (11)
is substituted into (10) and the equation made to depend
only upon the primary and exciting currents, the final ex-
pression isdi111 1*+. °1
di1
ng
d [Xy i 00 )
dt n-j_ dtdt
Mathematical Relations in the Secondary
(12)
A similar study can be made of the conditions
in the secondary coil. It is evident that the e. m. f.
induced by the exciting field must be
= M - L0p H2 dt 2 dt. M ||3= - L02 US. , 13 )
-11-
which may "be written
ni
dil
di.
112 dt dt(14)
orni d(i2 + i00 )
n2 dt
as shown in (10) and (12) when considering the e. m. f.
induced in the primary due to the same flux. The secondary
terminal voltage, e^ , then is
dioo.
•2~ L
2 dt"" i
2r2
di, di2 L2 —
-
M —± - Ldt '2 dt
di,
dt*2 r2
(15)
(16)
!i *lV+i*fti .^2
n2 dt
It also appears that
dt
di.
-L2 —* - i2 r2dt
di,
% =e2+
*2 r2+ L
2'
dt(17)
The constants used in the secondary correspond to the sim-
ilar ones in the primary, both in their relative values and
in their physical significance.
Combination of Equations through Ratio of Turns
That the electro-motive-force induced in the two
coils "by the same exciting flux should be in the ratio of the
number of turns is perfectly evident. It may be checked
from the equations evolved here if desired, by comparing
-12-
(14) with (10), bearing in mind the relation (9), Since
this relation is true it is apparent that any voltage in
either coil may he represented in equivalent torms with res-
pect to the other coil by multiplying that voltage by the
inverse ratio of turns. Then from this relation and (17),
(7) may be written
die l 38 *1 rl + Ll H"
+ K* i2
din
di= i
xrx
* 1^ —i + Ke i
2r2
4 K«L2
+ Kte2
(18)dt dt
nlwhere Kt
= -=,
and the difference between the terminal equivalent voltages
is
el - " *1 rl + Kt*2 r2 + Ll ll'+^jr* U9)
!Dhe Constants
The almost absolute constancy in value of r^,
r2 , and L2 is noteworthy. On,ly a variation in tempera-
ture will change the resistances, r^ and r2 , and as tests
and calculations are supposed to be made for "hot" tempera-
tures, they may be considered constant. Since the iron
core passes through both coils, no leakage line can have a
complete iron path for its circuit but must bridge a consid-
erable airgap. The effect of this high reluctance, constant
permeability gap is to make the magnetization curve for this
flux a comparatively straight line throughout a very great
-13-
range of density, assuring a similarly constant value for
L. Tlie limits of variation of the exciting current from
a constant value or a constant times the induced e. m. f.
have already been considered.
(Note:- The differential equation development as given inthe preceding chapter is not according to standard practicein problems involving imperfect mutual induction, the casebeing treated rather as a combination of self-inductionand perfect mutual induction in series. (See page 5) Theinductive effect of the exciting field is separated fromthat of the leakage field because the m. m. f.'s of primaryand secondary combine to keep the exciting field practicallyconstant, whatever the load, while the leakage flux varieswith the load. As appears later, this treatment makes itpossible to eliminate the induced e. m. f. and the mutualor exciting flux except in the determination of the ex-citing current, and permits the study of the entire machineas a simple electric circuit.)
-14-
TRANSFORMER CHARACTERISTICS
The Constant Potential Machine
Transformers are designed to give varied charac-
teristics according to the manner in which they are to be
used. The most common type* is the constant potential
transformer which receives energy from one circuit at a
constant effective voltage and delivers it to a second cir-
cuit at as nearly constant an effective voltage as is possible.
Thus far, the development has applied in a general way to all
types. Prom here on, while much of the work will apply in
the same way to the general transformer, it will he devel-
oped essentially for the constant potential machine. Con-
stant potential should be construed to mean a constant wave
shape of the same effective value and not a constant instan-
taneous voltage.
The Essential Characteristics
As with all machines, the first requirement of
the transformer is that it be efficient. Due to the fact
that modern lamps and motors operate to best effect only at
voltages very near the values for which they are designed,
the transformer must also have a very small percent regula-
Other types are current transformers used mostly for instru-ment3, constant current transformerrj for arc circuits, etc.
-15-
tion, that is the voltage delivered must he practically
constant as the load varies from full load to no load.
Again, since the regulation of the system is materially
affected by the power factor, being worse for inductive
load, the reactance of the transformer must "be small enough
so that the current will lag hut little more on the primary
side than it does in the secondary, -heing inductive in
itself, the machine cannot he made to improve the power
factor, except for leading current.
The Solution; Methods of Attack
In order to determine the essential characteris-
tics as noted ahove , it is sufficient to know the voltages
and currents with their phase relations, at the primary and
secondary terminals, for the various loads to he considered.
The natural solution of the problem with constant impressed
primary voltage would he to find the primary current by com-
bining the secondary and exciting currents (equivalent values)
and then, by substituting these and the primary voltage into
(18), solve for the secondary voltage. The difficulties
are that the secondary voltage must be known in order to de-
termine the load current, and the induced e. m. f. in order
to determine the exciting current. If the secondary volt-
age may be assumed, the secondary current can be found and
from these values, with the constants r and L and the relation
between the induced e. m. f. and the exciting current, the
primary voltage and current can be determined. While this
-16-
makes the primary voltage dependent upon the secondary
value and is therefore not natural, relative results deter-
mined in this way will be found to approximate normal test
results very closely, in fact they will be fully as accurate
as are the measured constants.
-17-
PREDETERMINATION OF CHARACTERISTICS:
SOLUTION BY DIFFERENTIAL EQUATION METHOD
General Solution
From the mathematical relations as previously
established, the character istios may now "be predetermined.
The secondary voltage and the constants of the circuits will
he assumed as has just "been suggested. The general solution
is then as follows
:
Let e2= function of time,
and let i2 m f2 function of time.
Then, by substitution into (17),
e, . f, (t)
and from the development for (18)
From (11a)
,
e i = Kt e* » f . (t)
.
xl *
x2 4
± oo - K e i - f5 ^)
where K is determined by test,
and from (11)
i n = - i + i „ = f - (t).1 K
t2 oo 5
Finally, by substitution into (7)
ex = f7 (t)
.
-18-
The average power output is now
where T is the period of the wave, the average power
input is z
1 /T
and the efficiency is
PEff . = - .
*o
The effective values of voltage and current may
he obtained in the ordinary way as from
and from these and the values of average power the power
factor may be determined,
pPF = --- .
E I
From an inspection of (19) and (11), it appears
that for no load the equivalent primary and secondary volt-
ages are practically equal. Thus, if the load be removed,
the secondary voltage will change from eg or Eg (effective)
to \ e n or 1 E-, and the percent change will be the regulation,
or- En - Bp
% Reg. = 100.Eg
Application to Sine Wave Conditions
By way of partial illustration of the solution, let
-19-
e2m ]fz E2 sin cot (20)
andi2 =VrFl
2sin (cot + «*) (21)
where I2
and °<z are determined "by the load. E2 ^2
are effective values of the secondary voltage and current
respectively.
Substituting (20) and (21) into (17) and simplifying, the
induced e. m. f. in the secondary is
ei = |/2" sin (cot + «) , (22)2 2
and, multiplying (22) by the ratio of turns, the primary
induced e. m. f. is
e n- =K\[2"En. sin (cot + <*)
11 *
x2
= tf2~Ei sin (cot + a) . (23)1
From test, when
ei
' = tfTE^ sin cot
is impressed upon the primary (secondary open circuited)
or when
-e±
' = -V2Ei
' 3 in cot
is induced, the exciting current has been found to be
where oc' is the angle by which this current is out of phase
-20-
with the impressed voltage. This equation' is an approx-
imation but is sufficiently accurate where final results are
to be obtained in terms of effective values.* Then for
the induced voltage in (23) , the exciting current is
i 0Q m - p" I E^sin (cot + « + «')
m - \[2~ I0Q sin (cot + oc J . (24)
Aside from the exciting current, the primary-
must supply a current to overcome the counter m. m. f . from
the secondary load current. This is
1 .
and the total primary current is then
1Axl " i oo ~ ^*2
= sin (cot +oc# ). (25)
The primary voltage may now be determined by substituting
(24) and (25) into (7) and simplifying, or
e±
= - V2 E sin (cot )
,
(26)
the negative sign in (26) showing that, when fi m , e-^
will be 180° out of phase with e^.
From the mathematics of the sine wave, the final
results can now be obtained as follows:
P~E2Effective value of eo = = ED .
See page 10
-21-
Vs i2Effective value of i 9 = m I P •2
V2
Effective value of e-^ = E-^
Effective value of i-, = 1^
Phase angle between e2and i^ = oc
z .
Phase angle "between and i-j_ = ( - cc,) .
/I
eg ig dt » Eg Ig cos oc2
1
Output = P = -
T
Input = P = - e1 i2_ dt = E1 Ii cos ( /3 - cc,;
I
E2 Io cos <xzEfficiency =
Ei I x cos (A -*,)
Eg I2 cos of2P F
2» — = cos oc
2.
E2 I2
P?1
a COS (>9 - CC,) .
RegulationE2
The calculation where sine waves of secondary volt-
age and current can be assumed is thus seen to be comparative-
ly simple, the only difficulty appearing in the trigonometric
transformations necessary in combining the various waves. It
will later be shown that, by using the general number and by
resolving all waves into component sine and cosine waves with
respect to a given zero angle, much of the labor of the ordin-
-22-
ary analytical method oan "be dispensed with. Where the
waves are not sine waves, no such short cut method can be
resorted to; furthermore it is probable that the assumpt-
ions made regarding i 00 would then have to he modified, and
the problem thus made more difficult. It is therefore
fortunate that at least equivalent sine waves can be used in
practically all cases except where instantaneous or trans-
ient conditions are under consideration.
-23-
GRAPEICAL SOLUTION BY WAVES AM) BY VECTOR DIAGRAMS
When desirable for clearness or for checking,
the relations just expressed analytically may be shown
graphically by waves as in figures 1, 2 and 3. Also where
the sine wave may be assumed, the waves may be represented
by equivalent systems of vectors, as in figures 4, 5 and 6.
In each case in the diagrams the secondary voltage and cur-
rent and the power factor of the receiving circuit have been
assumed. (See pages 25 to 28 for diagrams.)
The graphical methods really need no explanation.
For the sake of clearness a ratio of 1:1 has been chosen
and the resistance and reactance drops have been assumed to
be about ten times normal in all but figure 3 and figure 6,
showing inductive load. In these, actual values for a
small lighting transformer are shown so as to illustrate
real conditions. The induced voltages are shown as E n-
1and E-^ while that part of the impressed primary voltage
which is consumed by is shown as EiQ
« A normal phase
angle has been chosen between E^Q
and I00 and the magnitude
of I 00 has been permitted to vary with E^ . I00 ,also,
o
is shown too large in all but figure 3 and figure 5. Even
by considering all voltages and currents as percentage quan-
tities with reference to normal full-load non-inductive
values in each coil, so that the ratio will appear as 1:1,
-24-
it is evident that errors of measurement will make the
graphical methods unsatisfactory, except as mere expressions
of physical relations. The useful flux, 9, is shown in
the vector diagrams, leading the induced e. m. f. "by 90* •
-Z5-
/ \/ V
/ \
rig. /.
-26-
-29-
SOLUTIOII BY COHPIEX QUANTITIES
Substitution of General Numbers for Vectors
There oan be no doubt about the distinct advan-
tage of graphical methods over analytical, from the stand-
point of clearness, but they are at best very inaccurate
and it is necessary to return to mathematical expressions
for dependable results. It has already been demonstrated
that the combination or solution of a system of waves by
means of the equations representing them is likely to be a
long and involved process. It is fortunate, therefore ,that
a more simple method is available in the mathematics of
the general number which may be applied directly to the
vector diagram representing this system of waves. The
general number method has the added advantage of the vector
diagram over the waves; that of clearness and ease of man-
ipulation. It is, however, limited to the conditions
where the vector diagram can be used and is therefore appli-
cable only in those cases where sine waves are found or
may be substituted for the waves in the original system.
The general number, a, + j a2 , or as is usually
given in algebra*, a-j_ + a2 i , represents a line or vector
^Because of the universal' use of i for current in electri-
"
cal calculations, j is generally used in place of themathematical i.
-30-
(See Fig. 7) of length
A + a/,
extending from the origin into the first quadrant and
making an angle oc with the horizontal axis, where
agtan oc = — •
al
ft
«2
7.
The general number may he represented by its scalar length
with a dot, as
A = &1 + j a2 .
The line represented "by A may also be described
as a vector of length A , rotated through a positive angle
from the horizontal axis, the mathematical operation of
doing this being expressed by the operator or coefficient of
A in
A = A (cos«+ j sin a).
Rotation into the fourth quadrant would be through an angle
(-<*) and such a vector would be
31-
B « B [cos (-of) + j sin (-«fj
m B (cos oc - j sin «)
= bl " j b2*
where -b2
tan = .
*i
Note that the sign preceding the j indicates direction in the
vertical or j axis, but that, when discussing the components
of the vector, it is desirable to retain this sign as a co-
efficient of the component.
It is interesting to note that the operators given
above are reciprocals of each other, or
1cos or - j since =
cos oc + j sin oc
so that if oc is the same for A and B and if B = 1, then
B » - • 1 TA A( cos a + 3 sin oc
)
= j (cos oc - j sin oc )
.
No attempt will be made here to explain more in detail the
the general application of the complex quantity to the vector
diagram, that being done in many texts written by Dr. Stein-
metz and others and in many magazine articles. The direct
application to alternating current problems will be considered
very briefly for the same reasons.
-32-
Application of General lumbers to
Alternating Current Problems
By definition, the impedance of a circuit is
that factor "by which the current is multiplied to determine
the total drop across the circuit, or
EZ = -.
I
If oc is the angle of phase difference betv/een E and I and
and the current leads* the horizontal axis by 9, then
E =B Qcos (0 +«) + j sin (9 +«F] f
1 =1 (cos 9 + j sin 9)
and E e2 = - = - (cos <* + j sin oc)
I I
E cos oc E sin a- 3
I I
* r + j x,
Also by definition, the admittance of a circuit
is the factor by which the voltage impressed is multiplied
to obtain the current flowing, or
I
and vectorially, using the above values of voltage and
current
,
I
Y s= *
E•
*By leading is meant separation by a positive an<?le in thedirection'of mathematical rotation or counter-clockwise.
-33-
I (cos © + 3 sin © )
E [cos (© +«) + j sin (0
= - (cos + 3 (-sin « ) )
E
- g * 3 i>.
Again
Y =|z
1~ r~+~j~x
r -xss + 3
Thus it is seen that the conductance or in phase component
of the admittance is
r
Sr2 4
and the susceptance or quadrature component is
-x
r2 + x2
where r and x are the resistance and reactance respectively.
Some writers preferx
r2 + x2
and
Y = g - 3 h.
This is satisfactory for special cases but where the prob-
lem is perfectly general and the sign of h is not known, it
-34-
is an advantage to develop all the literal mathematics for
positive quantities so that, when numerical values are sub-
stituted, the signs in the results will immediately deter-
mine their physical relations. This plan is followed
closely in this thesis. Another standard form adopted is
the use of a prime (' ) to indicate the j component of volt-
age or current, permitting the use of all subscripts to des-
ignate the various parts of the circuits.
As a result of the relations just discussed, there
are four standard general numhers, representing the four
vectors or complex quantities used in electrical calculations,
namely:
E * e + j e*
I - i + 3 if
z - r + j x
The mathematics and diagrams for a circuit
whose total impedance is
Z = r + j x
and where either the current or the voltage is known, are
as follows.
Current Known
Let I =s i *» j i1 be the current flowing, referred
to i as reference vector.
Then
E x= I u
where E = \/e2 + e' 2
where I ^i2i'
2
where X = \/r2 + x2
where Y + b2
-35-
m (i + j i*)(r * ^x)
= (i r - i 1 z) + j (i 1 r + i x)
m e + j e f,
where the vectors and their components are as shown in
figures 8 and 9. (See page 36.)
Voltage Known
Let E = e + j e 1 he the voltage impressed.
Then e + j e 1
I =Z.
= (o - j e') Y
«(•+-£•») ig * j b)
> (e g - e 1 b) + j (e 1 g + e b)
= i + j i'.
Mathematically this solution is complete and
general. To interpret it for a special case, the signs
of g and b must be determined from the physics of the prob-
lem. To make the diagrams in figures 10 and 11 conform
to those in figures 8 and 9, the same physical relations
must exist, or g must be positive and b negative because r
and x are both positive. For these diagrams then, the
numerical result may be written
I = g - e' (-bfj + j £e' g + e f-bfj
= (e g + e T b) + j (e» g - e b)
,
from which figures 10 and 11 are obtained.
-36-
X = £Y
x=er
-37-
Equation for Power
The power in any circuit is
P m E I cos oc
where oc is the angle of phase difference "between the volt-
age B and the current I. When the voltage and current are
given as
E = e j e f
and
the power equation may also be written
P = e i + e f i'
.
This can be proven as follows. The power is given as
P = E I cos «
= E I cos (oc, - eg (a)
where cxt» arc cos | and <x
g= arc cos
Expanding (a) and collecting terms,
P s E I (cos or cos or2 4 sin a, sin <*£ )
b E cos oc I cos oc^ + E sin a, I sin ac£
= e i + e r i' . Q. E. D.
38-
Application to the Transformer
Prom the physical relations as explained for the
transformer it will he seen that the machine may he repre-
sented "by an equivalent circuit as shown in figure 12,
Either percentage calculation or a 1:1 ratio is assumed.
The admittance YQ0 is the factor "by which to multiply the
voltage consumed hy the induced e. m. f . , to obtain the ex-
citing current in its proper magnitude and phase relation
with that voltage.
r.
E, r/.
L
E.
Let the constants, voltages and currents he as
indicated in figure 12 and let J2g he the reference vector.
Then
?2 . " e2+
j 0.
Also let the secondary current required hy the load be
***** *i•
Then the impedance drop in the secondary is
I2 Z2 = (i2 + 3 i'
2^ r2 +
J x2 )
-59-
- (12r2 " V 2 x2 } +
jfi,
2r2 + h x
2 }
and the induced voltage is
?i - h +is ?8
- (ez+ *2 r2 - i's *g > + J li'8»a + ^ *S )
- e. + J .
While the primary voltage consumed by this induced voltageo
is really 180 out of phase with it, as seen in figure 6,
it is not necessary to encumber the mathematics with these
negative signs, if the fact that all primary vectors are
opposite in sign to the corresponding secondary vectors is
borne in mind. Then the voltage E. may be considered as• l
impressed upon the exciting admittance, and the exciting
current is
I m E. Y.00 .1 .00
- (e± + j e'
i )(g00+
I "boo*
" {ei ^oo - e
'iboo )+ 3 f# f
i S00 * e±
b0Q )
= *oo +3 1 'oo
•
The primary current is the sum of the secondary and exciting
currents considered in their phase relations, or
.1 .2 .oo
- fig * 3 i»2 ) (i00 + 3 i' 00 >
- f£| * i 00 )+
3 U>* i f
00)
= i1
+ j i^.
-40-
Then the primary drop is
- (ilrl " l*i*l'
+ 3 (i'lrl
+ h *1>
and the primary impressed voltage is
?i " fx * !i ?i
Thus from the load voltage and current, the
groupings as shown above and the constants of the machine
as determined from test, the following values have been
obtained
:
Vectorial Scalar
«2Assumed
fa2
= e2
+ j o E2
-
|J2 = i 2 3 i'2 12 -\
I *i +ji' I =\.00 00 00 00\
h - il +j 1*1 ii -
e2^ e ? 2 -2
"
»/* i'22
1 |+ l'o
2o
li2
K2 ^
1E = e + j e* E =.11 I 1
From these values and the following formulas, it is now
possible to obtain the regulation, the power factor at
primary and secondary terminals and the efficiency.
El " E
2Regulation = •
E3
-41-
Watts Output m P = e2
ig
+ e'2
iT
£= e
2 *2
Voltamperes Output = Eg Ip
Power factor of Load = P P = --— - = --
Watts Input = P =8, L i' noil 11Voltamperes Input * E 1^
e i + e' i»
Power factor at Intake =PP * -1—1 1 1
h hOutput e? i ?
Efficiency = = -—- .
Input e^ i^ + e'-^ i f
i
To obtain the operating characteristics for a
definite machine under a specific load it is merely necess-
ary to substitute the proper numerical values in the above
expressions. Such calculations are shown in the appendix
for varying load at unity power factor and leading and lag-
ging load current and for varying power factor at several
loads. Since the impedances of the primary and secondary
coils and of the exciting branch are inductive, the numer-
ical test values to be substituted for r, x and g should be
positive and that for b should be negative. The sign of
i1 is fixed by the character of the load, positive for lead-
ing current and negative for lagging current.
Complete Vector Diagrams, Showing Components
Representative diagrams for leading and lagging
load currents are given in figures 13 and 14, showing all
the vectors and their components. Ratios of 1:1 are
-43-
assumed. The symbols are as already used, except that <p
represents only the useful or exciting flux and that Ig^ is
that part of the primary current which overcomes the counter
m. m. f. of the secondary or load current. The diagrams
are exaggerated so as to show method and comparative results
rather than exact magnitudes and phase relations.
SUMMARY
The purpose of this thesis has been two-fold,
first to explain the physical relations existing in the
transformer by means of mathematical expressions and then
to develop from these expressions more or less convenient
methods for predetermining: the operating characteristics
of the machine. In describing the transformer, nothing
more complicated than very simple differentials has been
encountered. Equations like (6), (12) and (18) have been
developed in which definite relations are shown to exist be-
tween the several voltages and currents. The only approx-
imation has been that the exciting current varies directly
with the voltage induced by the useful field and bears a
constant phase relation with the same, the accuracy of this
approximation being discussed on page 10.
In applying this mathematics, the specific oases
have been divided into two classes, one including all cases
where sine waves of voltage and current may be assumed and
the other including those where this assumption is not poss-
ible. The solution of problems of the second class is
likely to be long and involved, but where sine waves may be
used, the solution is comparatively simple, even when per-
formed analytically or by means of the differential equations.
Also, since sine waves lend themselves readily to graphical
-45-
representation by curves or "by vectors, problems of this
class may be solved graphically. The ohief advantages
of the graphical method are clearness both in showing the
physical relations in the problem and in comparing the re-
sults when certain of these relations are varied. (See
figures 4, 5 and 6.) The disadvantage is the inaccuracy
of the results.
The complex quantity or general number method,
where the vectors are represented analytically by general
numbers, combines the clearness of the graphical method
with the accuracy of the purely analytical method. Like
the graphical, it can be used only where sine waves may be
considered. It has an additional advantage in the ease
with which constants or loads may be varied in the calcula-
tion without greatly increasing the labor of solving beyond
that for the single set of values. (See Appendix.)
In closing it should be said that the conditions
as set forth in the problem and the accuracy desired in the
results must determine the method to be used. It is for-
tunate, however, that sine waves are approximated in most
cases and may be assumed, and where this is possible the
complex quantity method of calculation is recommended.
-45-
APPE1TDIX
CALCULATIONS BY THE COMPLEX QUANTITY METHOD
Method of Determination of Constants from Tests
The constants used in the transformer calculations
are the primary and secondary impedances and the exciting
admittance, and are readily obtained from the ordinary res-
istance, short circuit and open circuit tests. The short
circuit data gives at once the equivalent impedance of the
machine from which the resistance may be taken leaving the
total equivalent reactance, half of which is allotted each
winding. The admittance in the open circuit test is the
proportionality factor "between the voltage and the exciting
current, and may he separated into conductance and suscept-
ance, or if the exciting current has first been separated
into the coreloss and magnetizing currents the conductance
and susceptance may be obtained directly from these values
and the voltage. The constants as thus determined are ex-
pressed in their ordinary units, -ohms and mhos.
The problem is simplified if reference quantities
of primary and secondary voltage, current and power are
chosen and the constants expressed as factors having refer-
ence to these quantities. Percent impedance would thus be
defined a3 the percent drop with respect to 100% voltage
-47-
which would be consumed if 100% current is passed through
the impedance, and percent admittance would "be defined in a
similar way except that the current due to 100% voltage is
here taken in terms of 100% current. It is convenient to
choose the rated values as reference quantities in the secon-
dary, and the same values v/hen corrected "by the ratio of
turns as those in the primary. It should be noted that
100% voltage and current output will give 100% power output
only when the power factor is unity.
Specifically, then, the percent constants may be
determined by substitution into the following expressions.
100 (100% I2 ) r21° T2 =
(100$ E2
)
100 (100$ IP ) x% x S 2
fc(100$ E2 )
100 (100% I ) t
% r mf
(100% Bx )
100 (100$ I-, ) xo/o x ± JL
A(100$ Ex )
100 (100% E) g$ 8 .
100% I
100 (100$ E) b% t> . ... _ .
100$ I
In determining percent g and b, the reference voltage and
current to be used are those for the winding on which the
data was ta3cen, -usually the secondary. When normal volt-
age is used in the test, the numerators are the coreloss
-4 8-
and magnetizing currents and the constants are
100.
1
% gm -5J&(100% I)
and100.
1
°/ b _ sasi.(100$ I)
The short circuit or impedance drop may be considered in the
same way if the current under test is normal.
numerical Constants from Test Data
Consider a transformer with the rating
50 K. V. A., 2200:220 volts, 60 cycles»
on which the following test data is assumed to have been
taken
:
Resistance
R„ = .97 , R„ = .0097.p * s
Short circuit, voltage applied to primary,
-
E m 98.5 , I = 22.7 = normal.P P
Open circuit, voltage applied to secondary,
-
Sg
* 220, I Q0 = 12.25, Watts = 1000.
The percent resistances are therefore
r * i22:§§iZ:i2Z = i% m .01 in percent.p 2200
100»227- .0097 m . „ .
rs m = l/t> = .01 m percent.220
The percent impedance is directly obtained from the test data,
100«98.52 * = 4.48% = .0448 in percent.
2200
-49-
The total equivalent resistance is .02, hence the total
equivalent reactance is
\ / 2 2x m W.0448 - .t52 = .04 in percent
and the primary and secondary reactances are,
Xp = xg = - = .02 in percent.
Thus the primary and secondary impedances in percent and
complex quantities are
2p » rp
j Xp - .01 + j .02,
and
is
zs " rs +^ xs • 01 +
«j- 02 «
From the open circuit data, the coreloss current
W 1000I, = - s as 4.55 amperesCL E 220
and the magnetizing current is
V ooIm- V
r oo"j2CL
\ / 2 J2» y 12.25 - 3755 m 11.35 amperes.
The percent conductance and susceptance are now directly
obtained as100*4.55
g m , = 2% = .02 in percent227
and-100*11.35
b = = -5% m -.05 in percent.227
These values also represent the percent coreloss and magne-
tizing currents respectively.
3
-50-
GALCULATIOUS
The literal solution of the transformer is
given on pages 38 to 44 in the body of the thesis. All
that is necessary then, in the calculation under any condi-
tion of loading, is that the proper constants and load
values be substituted into the equations and the indicated
operations performed. Two sets of calculations
are given here, one showing the effect upon the efficiency,
power factor, regulation etc. of the character of the load
and the other showing the effect upon the same values of a
variation in the inherent constants of the machine under
consideration. Each column in each table is really a
complete problem in itself, determining the characteristics
for one set of constants and one condition of load.
-5/-
CftLCULRTtON OF THE TRANSFORMEREffect of the Variation of the Power Factor and Amount of Load
Upon its Operating Characteristics.
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- 52 -
CALCULATION OF THE TRANSFORMEREffect of the Variation of the Resistance of the Transformer
Upon its Operating Characteristics.
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-<5J-
CHLCULflTlON OF THE TRANSFORMEREffect of the Variation of the Reactance of the Transformer
Upon its Operating Characteristics.
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