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

http://archive.org/details/theoryoftransforOOjame

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

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