characteristics of tranformer.pdf

.EXCHANGE

-H AR 1 1913

UNIVERSITY OF ILLINOIS BULLETINVol. X OCTOBER 28, 1912 No. 9

[Entered Feb.l4,1902,atUrbana,IH.,as second-class matter under Act of Congress of July 16,1894J

BULLETIN No. 61

CHARACTERISTICS ANDLIMITATIONS OF THE SERIES

TRANSFORMER

BY

A. R. ANDERSONAND

H. R. WOODROW

UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

URBANA, ILLINOISPUBLISHED BY THE UNIVERSITY

PRICE: TWENTY-FIVE: CENTSEUROPEAN AGENT

CHAPMAN AND HALL, LTD., LONDON

THEEngineering Experiment Station was established by act of

the Board of Trustees, December 8,1903. It is the purpose ofthe Station to carry on investigations along various lines of en-

gineering and to study problems of importance to professional engi-neers and to the manufacturing, railway, mining, constructional andindustrial interests of the State.

The control of the Engineering Experiment Station is vested in theheads of the several departments of the College of Engineering. Theseconstitute the Station Staff, and with the Director, determine the char-acter of the investigations to be undertaken. The work is carried onunder the supervision of the Staff, sometimes by research fellows as

graduate work, sometimes by members of the instructional staff of theCollege of Engineering, but more frequently by investigators belongingto the Station corps.

The results of these investigations are published in the form ofbulletins, which record mostly the experiments of the Station's ownstaff of investigators. There will also be issued from time to time inthe form of circulars, compilations giving the results of the experimentsof engineers, industrial works, technical institutions, and governmentaltesting departments.

The volume and number at the top of the title page of the coverare merely arbitrary numbers and refer to the general publications of the

University of Illinois; above the title is given the number of the Engi-neering Experiment Station bulletin or circular, which should be usedin referring to these publications.

For copies of bulletins, circulars or other information address the

Engineering Experiment Station, Urbana, Illinois.

LT

UNIVERSITY OF ILLINOIS

ENGINEERING EXPERIMENT STATION

No. 61 OCTOBER, 1912

CHARACTERISTICS AND LIMITATIONS OF THESERIES TRANSFORMER

BY

*A. R. ANDERSON, E^ECTRICAI, ENGINEER,THE JEFFREY MANUFACTURING Co., COLUMBUS, O.

and*H. R. WOODROW, ASSISTANT ELECTRICAI, ENGINEER, NEW YORK EDISON Co.

CONTENTSI. INTRODUCTION PAGE

1. Uses of the Series Transformer 32. Scope of Bulletin 3

II. CHARACTERISTICS OF THE SERIES TRANSFORMER3. General Features of the Series Transformer 44. Vector Diagrams for the Series Transformer 55. Formulas for Current Ratio and Phase Angle Use

of Complex Quantities 86. Formulas for Current Ratio and Phase Angle of

Series Transformers with Iron Cores 167. Discussion of Curves Plotted from Formulas 22

III. THEORY o$ THE SERIES TRANSFORMER BASED ON^ INSTANTANEOUS CURRENT VALUES

8. Symmetrical Currents in Series Transformer withAir Core 24

9. Unsymmetrical Currents in Series Transformerwith Air Core 28

10. Effect of Constants of Series Transformer on In-stantaneous Current Values 30

11. Series Transformer with Iron Core and NegligibleSecondary Leakage Reactance 32

12. Series Transformer with Iron Core and AppreciableSecondary Leakage Reactance 34

13. Comparison of Methods of Computation for SeriesTransformers 37

IV. CONCLUSIONS

formerly graduate students in Electrical Engineering at the Universityof Illinois.

264344

CHARACTERISTICS AND LIMITATIONS OF THE SERIESTRANSFORMER

I. INTRODUCTION

1. Uses of the Series Transformer. High potential distributionand the large currents carried by feeders have made the use of seriestransformers, or so-called "current transformers," imperative. Wherehigh voltages are used it would be a source of considerable danger tobring the potential of the distribution system to the switchboard andcontrolling apparatus. The use of the series transformer, in connec-tion with the potential transformer, makes it possible to meter the powerand control such system without handling voltages which are dangerousto life. Furthermore, if live potentials were brought to the switch-

board, the proper insulation of instruments and controlling apparatuswould be accomplished only with considerable difficulty and expense.Again in the present day system of distribution, even where the voltagemay be so low as to overrule the above objection, the feeders leadingfrom the station may carry such large currents as to make the use ofseries transformers a decided convenience. Obviously it is muchcheaper and easier to run small wires to instruments requiring but alow value of current than to bring large feeders to a board on whichare mounted cumbersome instruments of sufficient capacity to carry thefull live current. The demands of convenience and safety, then, haveplaced the series transformer in its present position of importance.

In general, the function of the series transformer is to furnish acurrent in a circuit not metallically connected with the main circuit,which current shall bear a definite constant ratio to the main current,and shall be 180 out of phase with it. The uses to which this second-ary current is put may be divided into two classes: (1) to give indica-tions in metering instruments, and (2) to furnish power for the opera-tion of regulators, time-limit relays, and like controlling apparatus. Butwith the series transformer the ideal is never completely realized, and

consequently the above function is but imperfectly performed.2. Scope of Bulletin. It is the purpose of this bulletin to study

the imperfections of the series transformer, to determine how and towhat extent certain constants influence its operation, and to deduce cer-tain general characteristics. The first part of the bulletin will be de-voted to a discussion of the fundamental principles of the series trans-former and the representation by vector diagrams of its operation. Adeduction of current relations by the method of complex quantities, and

3

ILLINOIS ENGINEERING EXPERIMENT STATION

a discussion of conclusions that may be derived therefrom will follow.The derivation of current relations by the use of instantaneous currentvalues will then be given. In this last named part particular stress willbe laid upon the application of the current transformer for the purposeof recording transient phenomena. A comparison of the results obtainedfor stable condition by the two methods and a general summary will begiven in conclusion.

r x

wzINSTRUMENTS

FIG. 1. SYMBOLS USED FOR CONSTANTS OF CIRCUIT

II. CHARACTERISTICS OF THE SERIES TRANSFORMER

3. General Features of the Series Transformer. In constructionthe series transformer does not greatly differ from the ordinary powerpotential transformer, inasmuch as both consist essentially of two elec-trical circuits interlinked by a magnetic circuit. In the potential trans-former for furnishing power there is, of course, a variation of primarycurrent with secondary current; in fact, the primary current is afunction of the secondary current. In the current transformer, how-

ever, the effect of the secondary current upon the primary current isso very small as to be negligible. The primary current is determinedby the constants of the main circuit, and is taken to be independent ofvariations in secondary load and current. Herein then, lies the basicelement of distinction between the power potential transformer and thecurrent transformer. This dependence of the primary current upon theconstants of the main circuit only, and its independence of variationsin secondary current, is the starting point for the theory of the currenttransformer which follows. Naturally, this difference in operation be-tween the two types of transformers makes a difference in their design,and just how and why some of the design constants differ will be pointed

ANDERSON-WOODROW SERIES TRANSFORMERS 5

out later after a development of the theory has made their bearing onthe operation of the transformer clearer.

There are certain facts about the series transformer which are gend-erally known among men who have anything at all to do with it. It isknown that the series transformer consists of two coils metallically sep-arate, wound on a laminated iron core; that the coil connected in theline generally has fewer turns than the secondary coil across which theinstruments are connected, and that the ratio of the turns is about ininverse proportion to the ratio of the currents. It is known that it isdangerous to leave the secondary open-circuited because if a .currentflows in the primary, high voltages will be induced in the open-circuitedsecondary and there will be high heating of the iron core which maycause disaster. It is also known that slight variations in secondary load(or resistance and reactance) do not change the transformation ratioto a very great extent, and that the ratio is nearly constant for all valuesof primary current within a given range. It is also known that whenused in connection with wattmeters annoyance is often caused by a cer-tain angle of phase displacement from the ideal position. These verynearly constant factors, however, such as current ratio and phase angle,are very often subject to great variations, and in the following discussionit is proposed to investigate the relations between these factors and theconstants of the transformer and load; to determine how and to whatextent they are influenced by these constants.

4. Vector Diagrams for the Series Transformer. At the outset itis thought advantageous to study the vector diagram representation ofthe current transformer in order to gain familiarity with its operation.Fig. 1 is given to indicate the constants represented by the different

symbols. Fig. 2 is a simple vector diagram of a series transformer hav-

ing no core loss. This would be the case with a transformer havingan air core. The voltage e across the load is taken as the reference vec-

tor, and the primary current 7' lags behind it an angle dependent uponVthe power-factor of the load. ( tan

6 ILLINOIS ENGINEERING EXPERIMENT STATION

This flux is, of course, in phase with the primary current. Since it isestablished in a path external to the secondary coil, it is very properlycalled leakage flux, and if subtracted from the total primary flux, givesas a resultant what might be called the useful primary flux, in phasewith the primary current, and represented in the diagram by the vector'-(f>i!. Now in order that a current may flow in a secondary circuithaving impedance, there must be induced in the secondary an e.m.f.,and to accomplish this there must exist in the core a real flux leadingthe induced e.m.f. by 90. This flux, indicated in the diagram by < M isthe only flux really existing in the core and is the resultant of the useful

primary flux and the useful secondary flux " < L", as indicated inthe diagram. " is the total secondary flux, and < L

"

is the secondaryleakage flux, corresponding to


It is oftentimes convenient to draw the vector diagram using currentsin place of fluxes, as is done in Fig. 3. If the ratio of turns is otherthan 1 to 1, it will be necessary to consider the magnitude of the currentvectors given in ampere turns, or what amounts to the same thing, ifthe primary current vector is drawn to scale the secondary currentshould be multiplied by the ratio of the number of secondary turns toprimary turns to give the magnitude of the secondary current vector.The resulting current vectors will then be in terms of primary currentdirectly, and it will not be necessary to divide by the number of turnsas would be the case if the vectors were scaled off as ampere-turns.

TIME

FIG. 3. VECTOR DIAGRAM FOR CURRENTS IN TRANSFORMERWITH ZERO CORE Loss

i

Fig. 3 is a vector diagram of a current transformer with an iron core,having consequently some core loss. It is very similar to the diagramin Fig. 2 with the exception that the flux vectors are plotted as currents,and the core loss component 7 C is taken into consideration. Core loss

represents power dissipated in a secondary circuit, and requires in phasecomponents of current and electromotive force. Since the dissipationof energy takes place in a secondary circuit and we have already drawna vector for secondary e.m.f. the current vector Ic in phase with E" will

give the required representation of energy dissipation. This current /is consumed in the core, and hence is not available in the useful second-

ary circuit. Subtracting 7C from i" gives the true secondary current I"as represented in the diagram. The angles oc and (3 correspond to the

angles oc and ft of Fig. 2. In both Fig. 1 and Fig. 2, El is the electro-motive force across the primary and Y the angle that it makes with the

primary current. From Fig. 3 the effect of core loss is readily seen. Itdecreases the secondary current 7", and with an inductive secondary cir-cuit it decreases also the phase angle ft. Since it is impossible to con-struct a transformer which has not at least the secondary reactance due


to leakage flux, it must follow that in a commercial transformer core losshas both of the above effects. If these were its only effects it might bethought, ignoring efficiency, that core loss is desirable. But it will beseen later that hysteresis loss is accompanied by a variation in corereluctance that is extremely undesirable so far as regulation is concerned.

5. Formulas for Current Ratio and Phase Angle Use of ComplexQuantities. With these vector diagrams in mind we may proceed witha mathematical discussion of the current transformer by the use of

complex quantities. In the following derivation it will be consideredfor simplicity, that permeability of the iron is constant. The error thusintroduced is not so great as might at first be expected, due to the factthat the iron in a current transformer is generally worked at a verylow density on the straighter portion of the saturation curve. However,the assumption of constant permeability still makes it possible to dis-cuss from the resulting equations the effects of variation in permeability.If the permeability is taken constant the hysteresis loop vanishes, andthe only remaining core loss is the eddy current loss. With sufficientlycareful lamination and at the low value of 'flux density used, this loss

may be very small. Indeed in a commercial current transformer thecore losses are very small. Neglecting then for the present all core

losses, the following derivation will correspond to the vector diagramof Fig. 2 and will be rigid for a transformer with an air core. Let asine wave of e.ni.f . e instantaneous value = ey^~sin wts be impressedupon the circuit as indicated in Fig. 1. Then since the effect of thesecondary current upon the primary current is neglected

where R jXis the load impedance.eR eX

where t = and %\ =r>2 I V* Z?2 I V%i

-\- JL n -\- ^

The flux set up by the primary current or the total primary flux isproportional to the primary current and in phase with it or

== Ki Nil' X 108

where K\= :- x io~8 ; (f*i= combined reluctance of iron circuit*In all these following equations involving complex quantity, time is con-

sidered the independent variable and is represented by counter clock-wiserotation. The inductive impedance is R JX and thus there is a laggingcurrent I=

'ANDERSON-WOODROW SERIES TRANSFORMERS 9

and leakage path ) , and Ni is the number of primary turns. Of


Combining (6) and (7) an expression for magnetizing flux in termsof secondary current and impedance is obtained as follows :

=

108

Whence,Tir pit*=

-y -=-io8 .................... (8)

The total secondary flux is proportional to the secondary currentand in phase with it, or

$" = fa -#2 1" X 108 . .................. (9)

where Ki = X 10"8 ; (f*2= combined reluctance of iron circuit

and leakage path).In a manner similar to that for obtaining equation (5) for the pri-

mary leakage flux, the following expression for the secondary leakageflux is obtained,

*L" = ?-- 108 ..................... (10)

If we subtract from the total primary flux the primary leakage fluxwe obtain what may be called the useful primary flux. Similarly thedifference between the total secondary flux and the secondary leakageflux gives the useful secondary flux. The sum of these two usefulfluxes then gives the magnetizing flux, or expressed in the form of an

equation,


or (Kl Nl _ _.) r + ( Kt N, _ -L + j ^- ) I" =0. ..(13')to-V2

Whence, /* =__.-x p .........

coJVi ( Ki ]\T22 to xt + / Z" )

But ^ = ( r2-f r ) j ( xi + )

And / ^ = ( x^ + x ) -f j ( n + r ) ......... (15)

Substituting (15) in (14),

Ni [ JT2 Nf to -f x + j ( r^ -f- r ) ]

which might better be written

(Ki JVi2 to a*) r~-,ro , -x ,) 1j?>( l7 )

The ratio of secondary current to primary current then becomes

/" N ( K\ N-? w xi }^ \ -"-A~**_* */ /1Q^

= C'o)I f Ni V ( K* N# to -f- x )2 + ( n + r )2

Equation (18) may be reduced as follows: Kl and K2 representthe primary and secondary magnetic conductances respectively. If

F = X 10~8 represents the conductance of the iron path, andJc

land Jc2 represent the magnetic conductances of primary and secondary

leakage paths respectively, then for ^T1 and K2 in equation (18) maybe substituted :


(19)/' N (F N


^(Xi-si)=XM .................. (23)

Now substituting (21), (22) and (23), in (18), the following expres-sion is obtained,

I" X M(24)

r vT^ST* )2 + ( ** + r )a'

From (24) it is seen that with zero resistance of the secondary coil andload, the ratio of transformation is as the ratio of the mutual inductivereactance to the total reactance of the secondary circuit.

From equation (20) it is seen that the effect of increasing the sec-

ondary resistance or reactance is to decrease the secondary current; thatthe effect of increasing the frequency or increasing F, which meansdecreasing the magnetic reluctance, is to increase the secondary current.

Equation (20) further shows that a large number of secondary turnsmakes variations in frequency, permeability, and secondary resistanceand reactance less effective in influencing the transformation ratio, andhence points to the desirability of a large number of secondary turns.

Beferring again to equation (17) it is seen that the phase anglebetween the secondary current and the projected primary current is

P = arc tan- _ - _ are tan- _Ki N


As an example a transformer having the following constants maybe considered :

T-J=

.0058 ohms.

Zj = .058 ohms at 60-

K = .000015F 5= .00001425

'N2 = 133

r2= 1 ohm

OTj = Z^ 2 co = 1.11 ohms at 60~x2= KN2 2 o> = 100 ohms at 60 -

,

= ( J7! xi ) 10 ohms at 60~

^ 0.09

a_"0.08

2 0.07C 0.06

O 0.05


Taking the secondary load reactance equal to 1 ohm, we may studythe effect of secondary load resistance upon the transformation ratio.Fig. 4 Curve I. shows the relation between the transformation ratio andsecondary load resistance. It is seen that for reasonable values of re-sistance (say up to 5 ohms) the ratio is but little affected. Beyond 10ohms, however, an increase in the resistance results in quite an appre-ciable decrease in the value of secondary current, and the rate of de-crease becomes greater with increased resistance. If the secondary loadresistance be taken as 1 ohm, the effect of secondary load reactance uponthe transformation ratio may be studied. Fig. 4 Curve II. shows thevariation of the transformation ratio with secondary load reactance. Itis seen that the ratio is very appreciably decreased by increasing the re-actance and that the ratio of decrease is greatest for low values ofreactance. Comparing then secondary resistance and reactance as re-gards their effect upon the transformation ratio, it may be said that forreasonable values the effect of increasing the former is to decrease butnegligibly the ratio, whereas an increase in the latter results in a veryconsiderable decrease of the ratio.

Taking the secondary load reactance equal to 1 ohm at 60 cycles, andchoosing three values of secondary load resistance, viz., 1 ohm, 10 ohms,and 50 ohms, three curves were plotted, one for each value of resistance,showing the effect of varying the frequency upon the transformationratio. These curves are given in Fig. 5. An inspection of these curvesshows that for high values of frequency the effect of variation in frequencyupon the transformation ratio becomes negligible, and also that thelower the secondary resistance, the lower the frequency at which theratio becomes practically constant. With 1 ohm secondary load resist-ance it is seen that the ratio is practically constant above 25 cycles.With reasonably low secondary resistance then it may be concluded thatthe effect of frequency variation, within commercial limits, upon thetransformation ratio is quite small. It should not be forgotten, however,that below a certain critical frequency for a given value of secondaryresistance decreasing the frequency will result in a very rapid decreaseof the transformation ratio.

The effect of secondary load resistance on the phase angle (3 will beconsidered next. Taking the secondary load reactance x equal to 1,the curve shown in Fig. 6, was plotted showing the variation of phaseangle with the secondary load resistance. It is seen that the increase inphase angle is practically proportional to the increase In resistance forreasonable values of r. By taking the secondary load resistance equalto unity, the variation of phase angle with secondary load reactance maybe studied. Fig. 7 shows the relation between these two quantities. It is


noted that increasing the secondary reactance decreases the phase angle,the ratio of decrease being almost constant for reasonable values of x.The effect of x in decreasing the phase angle, however, is not nearly asgreat as the effect of r in increasing it. As shown by the curves, a givenchange in x changes the phase angle by less than 2% of that caused byan equivalent change in r.

By taking secondary load resistance equal to 1 ohm, and secondaryload reactance equal to 1 ohm at 60 cycles, Fig. 8 was drawn showingthe effect of frequency on the magnitude of the phase angle. This curveshows that the phase angle is very sensitive to changes in frequencywithin the range of commercial frequencies, the phase angle at 25 cyclesbeing more than twice as great as at 60 cycles and over five times asgreat as at 133 cycles.

6. Formulas for Current Ratio and Phase Angle of Series Trans-formers with Iron Cores: -Throughout the above discussion zero coreloss and constant magnetic reluctance have been assumed. In otherwords the above discussion applies rigidly to a transformer with an aircore. The commercial series transformer, however, has an iron core.As was pointed out previously, due to the low flux density in the coreand careful lamination, the core loss in such a transformer Is very small ;so small in fact, as to be of no consequence in affecting the validity ofthe foregoing discussion. In using an iron core, however, there is aneffect more objectionable than the hysteretic core loss, viz., the accom-panying variation in core reluctance.

An examination of equation (19) shows that with given constantsand frequency, the ratio of secondary current to primary current is con-stant, independent of the value of primary current. This is true of anair core transformer where the permeability of the core, and consequentlythe factor F, are constant. But in a transformer with an iron core the

permeability and consequently the factor F varies with the flux density.Since the flux density varies with the primary current, it must followthat the factor F is a function of the primary current, and consequentlyit is not to be expected that the ratio of secondary current to primarycurrent will be constant for all values of primary current in a trans-former with an iron core. In order to determine the nature of thevariation in F and the consequent variation in the current ratio, it willbe necessary to consider that portion of the saturation curve aroundwhich the transformer operates. In Fig. 9 Curve I. is given an assumedsaturation curve. The scales of the co-ordinates are not given becausethey are not important for our purpose. The essential thing that con-cerns us is the shape of the saturation curve.

ANDERSON-WOODROW SERIES TRAN SFORMERS 17

0-02If{....SECONDARY LOAD RESISTANCE, r = I OHM

mill.-- r= 10 OHMS

100 200 300 400 500

FREQUENCY-CYCLES PER SECONDFIG. 5. EFFECT OF FREQUENCY ON

TRANSFORMATION RATIO

(3

PHASE

ANGLE

8

5

20 40 60 80 100

RESISTANCE OF SECONDARY LOAD (r)

FIG. 6. EFFECT OF SECONDARY L^OAD RESISTANCEOF PHASE ANGI,E


It will be remembered that the factor F is given by the expression:

F = ^L X 10-8f

where p is the reluctance of the iron circuit. Then we may write

.477 fl ic _8

I

where a is the average cross section and I the length of the iron circuit.Again for a given transformer, this may be written

F = Constant X j* .................... (27)From the saturation curve a curve may be derived whose ordinates areproportional to the permeability /u, by dividing the ordinates of the sat-uration curve by corresponding abscissas, and plotting the quotients asordinates of the new curve. From (27) the ordinates of this new curvewill also be proportional to F, and by properly choosing the scale it willbe possible to so plot it that the value of F corresponding to any pointon the saturation curve may be directly read from it. It will be seenhow this may be done.

The point of operation on the saturation curve depends upon thevoltage that must be induced in the secondary to send the secondarycurrent through the secondary impedance. This voltage is of course di-

rectly proportional to the secondary current, and to the impedance ofthe secondary circuit. For a given frequency the flux is proportional tothe voltage. If the normal secondary load then is known and the fullload secondary current, the flux for this condition may be computedfrom the relation.

T" 7"* M = - 108 .................... (28)

which is obtained from equation (8) by dropping the complex quantitysymbols. This establishes a point on the saturation curve, as denoted

by A in Fig. 4. From the saturation curve the value of permeability /*at this point may be found, and knowing the dimension of the iron cir-cuit F may be calculated from equation (26). This value of F is thenlaid off to scale as an ordinate through point A, and the other ordinatesare plotted proportionally, as explained above from equation (27). This

gives Curve II of Fig. 9. As an example, let the normal secondary loadfor the transformer under consideration be r = 1 ohm and x = 1 ohm,and let the value given for F apply to the point of normal full load


SECONDARY LOAD REACTANCE (X)00

OHMS AT 60 CYCLESFIG. 7. EFFECT OF SECONDARY LOAD REACTANCE

ON PHASE ANGI/E

3LUo:

aQI

I

I

I

LJ

d

LUCO

14

12

10

100 200 300 400 500

FREQUENCY CYCLES PER SECONDFIG. 8. EFFECT OF FREQUENCY ON

PHASE ANGI


operation indicated by A in Fig. 9. Curve II of Fig. 9 gives the valuesF at other points on the saturation curve. Let the problem of plottinga curve showing the variation of current ratio with primary current beconsidered.

It is known that with constant secondary load the variations insecondary current and induced voltage are proportional, and that there-fore, the flux is proportional to the secondary current. For any per-centage of full load secondary current then the corresponding pointon the saturation curve may be found from A, Fig. 9 by direct propor-tion. Going up along the ordinate from the point thus found to CurveII, the corresponding value of F is obtained. This value of F is substi-

I"tuted in equation (19), and the ratio - calculated. From this ratio

and the value of secondary current the corresponding primary currentin per cent of full load primary current may be determined, and thus apoint on the required curve established. This is done for a sufficientnumber of values of secondary current to enable a smooth curve to bedrawn through the points obtained. Table 1 gives a convenient tabula-tion for these computations, and Fig. 10 gives the result. It is seenthat above 60% of full load primary current the ratio is very nearlyconstant, and that below 40% it falls off very rapidly. The variationin current ratio with primary current may be more clearly shown byplotting as ordinates the difference between the ratio for a given valueof primary current and the ratio for full load primary current expressedin percent of full load primary current. In Fig. 11 such a curve, cor-

responding to Fig. 10, is shown by a full line. For the purpose of gen-eral comparison with the theoretical curve, there is also shown in Fig.11, by a dotted line, a curve obtained experimentally. No unusual pre-cision was used in obtaining the data for this curve, the currents beingmeasured simply by two ammeters. The very close similarity of the twocurves is apparent. It is to be expected that the magnitude of theordinates, and even the shape of such curves from different transform-ers will differ even more than these do, inasmuch as they depend notonly upon the point of operation on the saturation curve, but also uponthe shape of the saturation curve, especially upon the prominence of thefirst bend, indicated by B in Fig. 9. Since there is very considerablevariation in saturation curves for different qualities of iron, close agree-ment between curves derived from them cannot be expected. The curvesin Fig. 10 and Fig. 11, however, point very truly to the general way inwhich transformation ratio and primary current vary.


I. SATURATION CURVEII. CURVE GIVING VALUES OF K

MAGNETIZING CURRENTFIG. 9. SATURATION CURVE AND CURVE GIVING

VALUES OF K

ILLINOIS ENGINEERING EXPERIMENT STATION

With values of F already determined for various values of primarycurrent, it is an easy matter to substitute these in equation (25), andfind the corresponding phase angles. Fig. 12 shows the variation ofthe phase angle (3 with the primary current. It is noted that above 60%full load primary current the variation in phase angle in small, but be-low 40% it increases quite rapidly.

Curves similar to those shown in Fig. 10, Fig. 11, and Fig. 12 maybe determined for any other secondary load by making use of the factthat the ordinate of point A Fig. 9 is directly proportional to the sec-ondary impedance. Knowing the new values of secondary resistance andreactance, and having established the new point A, the method of de-termining the new curves is exactly the same as that given above.

7. Discussion of Curves Plotted from Formulas. The curves whichhave been plotted, showing the variations of transformation ratio andphase angle with various constants of the transformer are largely self-

explanatory. Perhaps the most general observation that can be madefrom them is that the factor which least affects the transformation ratiois likely to greatly affect the phase angle, and vice versa. Thus fromCurve I. Fig. 4 it is seen that for reasonable values the effect of sec-

ondary resistance upon the transformation ratio is practically negligible,whereas Fig. 6 shows that its effect upon the phase angle is very great.Again the effect of secondary reactance upon the ratio is very consid-erable as shown by Curve II. Fig. 4 whereas its effect upon the phaseangle is quite small, as shown by Fig. 7. The curves in Fig. 5 showthat the effect of changes in frequency becomes less as the secondary re-sistance is decreased. This then together with the fact that the phaseangle increases almost in direct proportion to the secondary resistance

points to the desirability of a low value of secondary resistance.On the other hand, the secondary reactance, neglecting its effect uponthe point of operation on the saturation curve, is not a matter of such

great consequence, for increasing it decreases but little the phase angle,and the effect of changes in its value upon the transformation ratio is

practically constant for all reasonable values of reactance. The effectof changing either secondary resistance or reactance is to change the

point of operation on the saturation curve and, as a consequence, to

change the shapes of the curves in Fig. 10, Fig. 11, and Fig. 12. Thedesirability of using iron having as nearly constant permeability at lowdensities as possible is made evident by Fig. 10, Fig. 11, and Fig. 12.These figures also indicate that the reliable working range lies above 50or 60 per cent full load current.


it

,

*.

< 7.

1z sin ( ^ - 5 + => ) ........ (36)

Where ^ = arc tan ^*- ................................ (36'

Knowing the instantaneous values of i' and i" and the ratio of theJV 2

ndary to primary turns , the instant

current i is given by the following equation,

secondary to primary turns , the instantaneous value of magnetizing

(37)

Substituting equation (34) and (30) in (37)

i = A \ sin ( 9 s ) --sin ( 9 l s ) $-*&'-

ANDERSON-WOODROW SERIES TRANSFORMERS 27/

m JV2 sin ( * + ft ) a

l2 + l a

/l )

J) (38)

-MRepresenting by the constant K and replacing and combining

-A 2 -/Vl

the trigonometric functions into one, the equation becomes

i== A \\l(l U ~ **} sin ( s * ^ )(x V ^ + j ;

(I + r--

) sin ( 0i - 8 ) e-a*

- k Cz-W \ (39)QJ )

kbWhere u = arc tan

,

which is the angle of lag of the( #

2-J- 1 k )

stable magnetizing current behind the primary current.

Example 1. As an example of the application of this method thetransformer for which data have already been given may be considered,with the additional data

Ol = 175

The following data, previously given, will be needed,

= II =0.1X\

l = ^ = 0.01X2

XK === 10

A"2 100

^ = 9.5Jfi

5 = 85 (calculated)

Substituting these constants in equation (30)


i' =r sin ( B 85 ) e-1^'

.... ........... (A')

p from equation (36') is

^_

P= a,rc tan - - = arc tan ( .09) = - 51

-f- ab

The value of C from equation (36) becomes

C= .111ft= arc tan Z> 35 minutes

Hence, from equation (34)

t" - .1 [ sin ( 9 84.4 ) 1.11 e-ld/

-f .111 e-01*'

] . . (B')

z.

tt = arc tan- - == 1045'W + 1 - -

That is, the stable condition of the magnetizing current lags 10 45'behind the primary current.

i from equation (39) becomes

i = [ .05 sin (0 95.75 ) + .055 e-10 .1055 e-01^ ]. . . (C')

In Fig. 13 equations (A'), (B'), and (C') are plotted with B asabscissas and i', i", and i as ordinates. The ordinates derived from equa-

Vtion (B') are multiplied by the transformation ratio

- :

10 and-^M

turned 180 so as to be superimposed upon the primary current for bet-ter comparison. The curves thus plotted show clearly the error in sec-ondary current due to the air core transformer with a 5% magnetizingcurrent.

9. Unsymmetrical Currents in Series Transformer with Air Core.As a second problem a series transformer traversed by an unsymmetricalprimary current might be considered. If the current lies entirely abovethe zero line, the equation would be

i' == A [ sin B -f 1 C -*#' ]...., ........ (40)Where C = sin Bl + 1

Differentiating (40) and substituting in equation (29)

And the solution of equation (41) is


CURRENT IN PRIMARY COIL (f,)CURRENT IN SECONDARY COIL (/2)

TMAGNETIZING CURRENT

FIG. 13. CURRENT DIAGRAM FOR

-a6*' + C'- 1

FT ( * - )

Where /? = arc tan &, and since when = #r, 0' = and i;/ =

(42)

"

( *- )

If the time constant~~

of the primary circuit is exceedingly small,a

then e~a^' will practically be zero in a very short time; (a} will be largein comparison to (#) and equation (42) becomes, after an infinitesimaltime.

i"=A I sin(#+/3) rsin(^i-h/3) #\\ , , sia ^i 1 'b hI i/^2

-i- i L Vv + i 11 43)And equation (40) becomes

t' = A [ sin ^ + 1 ] (44)


Substituting equations (44) and (43) in (37)

i = A sin-f 1

- -

sin ( 9 -f p ) -f C t*e'

(45)V W 4- 1-f

Where

[sin

( 0i -f- B ) 1-V^ sin 0i 1 (46)V & + 1

Equation (45) may be further reduced to the form

\ ^fc K ) / a n'\ l_ 1 I n c-b&' ( f?\- fcin ( ey s ) -\- 1 -\- O e 147)

Where s f = arc tan

Example 2. Assume that the same transformer is used as in Ex-ample 1, but that the constant (a) is very large and the primary waveof the form,

t" = sin + 1 (A")Then from equation (43) m

i" = .1 [ sin ( 9 + 35' ) + s"01^'

] (B")And from (47)

i = .05 sin ( 9 10.75 ) .95 e"01^' + 1 (C")

10. Effect of Constants of Transformer on Instantaneous CurrentValues. The curves on Fig. 14, plotted from equations A", B", and C"show clearly how the current in the secondary of the transformer grad-ually becomes symmetrical in reference to the zero line, while the mag-netizing current creeps up to a line which is symmetrical in respect tothe primary current.

The reactance of the secondary, X2 , as has previously been indicated,contains two factors, X'2 and x2 , of which X't represents that portion ofthe flux that interlinks with the primary coil, and x2 represents that

portion of the flux that does not interlink with the primary coil, and isknown as leakage reactance. X'2 and X^ hold a definite relation to eachother, which ratio, is that of the secondary turns N2 to the primaryturns Nr The transformation ratio of the transformer for zero second-ary impedance is that ratio of the mutual inductive reactance XM to thetotal secondary self inductive reactance X2 , which ratio can only beconsidered equal to the ratio of the respective turns in so far as x2 canbe neglected in comparison to X'2 . The resistance in the secondary cir-


CURRENT IN PRIMARY COIL (,)CURRENT IN SECONDARY COIL (t2)

~ A A0.2

I \W17 \7

MAGNETIZING CURRENT (im)FIG. 14. CURRENT DIAGRAM FOR EXAMPLE 2

cuit has little effect on the transformation ratio, but does produce a dis-

placement in phase in the stable condition, and introduces large errorsin the transient term, since 6 is directly proportional to r2 . The smallerthe factor I, the more closely the secondary current follows the primarycurrent and therefore, for observing transient currents, the secondaryleakage reactance improves the accuracy; while resistance, though small,introduces large errors. The value of x2 is limited only in so far as itseffect upon the primary current can be neglected. As increase of x2 doesincrease the magnetizing current, but this increase is in phase with the

primary current and therefore no displacement results, and the trans-formation ratio is only diminished. Therefore, a transformer designedto be used for transient phenomena should have an exceedingly largesecondary reactance and a very small secondary resistance.

Under stable conditions, equation (34) becomes

i" =fl)

V X? + r22sin (0 8 (48)


Since X'2 is large in comparison to either r2 or x%, it is evident fromthe above equation that an increase of r2 has little effect on the trans-formation ratio, while an increase in x2 has an appreciable effect. Also,

T-2

since S is small (arc tan ) the angle of displacement J3A 2-f" X2

is directly proportional to r2 , while increasing x2 decreases the displace-ment. In other words, the magnetizing current required to force the

secondary current through the resistance r2 lags ninety degrees behindthat portion of the primary current that is transferred to the secondary,while the magnetizing current required to force the secondary throughthe reactance x2 is in phase with the said current.

11. Series Transformer wiih Iron Gore and Negligible SecondaryLeakage Reactance. This gradual increase, or creeping of the magnetiz-ing current of the series transformer with unsymmetrical currents, has amuch greater and more disastrous effect in the case of an ordinarytransformer with an iron core. In the case of the air core transformerthe magnetizing current was a direct function of the primary current inall stable conditions ; but this, as has already been shown, does not holdtrue in the case of the transformer with an iron core. The mutual in-ductive reactance XM and the corresponding self-inductive reactance X'2depend, in exactly the same way, upon the reluctance of the core circuit,and consequently their ratio is a constant, with a value the same as thatof the ratio of the number of primary and secondary turns. The leakagereactance x2 is more or less independent of the permeability of the core,

XMand consequently the transformation ratio - is not a constant

J 2 + #2but depends upon the magnitude of the magnetizing current. Further-more as X'2 decreases it necessarily approaches the value of r2 and theerror due to the increasing factor b becomes very large.

It is unfortunate that the saturation curve of the iron cannot be rep-resented by a mathematical equation and therefore the only solution ofthe problem is the tedious step by step method. Although this methodobviously is not absolutely correct, it does give a close approximation.Consider first the problem where the secondary leakage reactance x2 is

negligible. From the terminal conditions and constants of the circuit,the primary current at any instant may be determined, as by equation(30). From equation (37),

where the quantities are expressed arithmetically instead of algebraically.


The nature of what follows makes it more convenient to express therelation in this manner. Since the secondary leakage reactance is negli-gible, the change of flux need give only the e.m.f. necessary to force the

secondary current through the resistance r2 , or

(50)

where D is a proportionality constant. Equation (50) may be written

D tl

Substituting (51) in (49)JV2 D

*-~

Changing from differential to difference, that is, replacing as approxi-mation d by A gives

N D A2Whence A4> = :

~

- 175 [if

- -

i] ............. (54)

If the remanent magnetism in the core of the transformer at thetime the circuit is made be denoted by , the flux at any instant isgiven by the expression,

Substituting (54) in (55),

-|- ^> 77

~.175 [*' i] (56)

For convenience let r2 have such a value that .175 becomes equal

to unity. Then (56) becomes

(57)


As indicated above, the primary current may be determined from theconditions of the main circuit. Assuming a value for < , we may takeas a first approximation

(58)

From the saturation curve a value of i corresponding to


0.02 0.04 0.06 0.12 0.14

MAGNETIZING CURRENT

FIG. 15. SATURATION CURVE FOR EXAMPLE 3

And A*"] (61)

Substituting (61) in (49)

N* 5.73

Which reduces to.175r2 Ni

A t" (62)


Again, let r2 have such a value that .175 becomes equal to unity.

Then D = .175 r2 , and equation (62) becomes

A

Taking the secondary reactance equal to the secondary resistance,equation (63) becomes

Ni= : *'--

*] + 5.73 A*'"

JJTlBut from equation (40) A*" :

~^~CA*' - - A*J

Substituting (65) in (64) it becomes

(66)

Now substituting equation (66) in equation (55), an expressionfor the flux is obtained

Ni \= =

'-- i 5.73 t' - -

/

(< + 5.73 A *'') - ( + 5.73 A[

. . (67)

As a first approximation, the parenthesis containing i may be omitted,and equation (67) becomes

Ni4> == to + ^ ^ (i

f

+ 5.73 At') ............ (68)

With this value of an approximate value i is obtained from the sat-

uration curve. By this value of i the parenthesis of equation (67) isthen filled in, and a second approximate value of obtained, whence

by means of the saturation curve, a second very close approximate valueof i is obtained.

Table 4 is a convenient form for tabulation. Column 7 of Table 4gives ' as obtained by the first approximation, and column 8 gives thecorresponding magnetizing current. Column 12 gives A as obtainedwhen using i, and column 13 gives the second approximation of .Column 14 gives the corresponding value of magnetizing current, andfrom 15 the value of secondary current may be obtained.


CURRENT IN PRIMARY COIL (,',)CURRENT IN SECONDARY COIL (/2)

MAGNETIZING CURRENT (im)

FIG. 16. CURRENT DIAGRAM FOR EXAMPLE 3

Fig. 16 and Fig. 17 (corresponding to examples 3 and 4), show

clearly the errors introduced by the use of the series transformer withan iron core in recording transient phenomena. The "steadying" effectof secondary reactance is made evident by a comparison of the twocurves.

The oscillograph record, shown in Fig. 18, is that of the currentthrough the secondary of the series transformer together with the cur-rent through the primary, which is the starting current of an inductive

/v

circuit with an electrical time constant (--). of 10. These experi-mental curves are very similar to those as calculated from Examples 3and 4 although the constants are somewhat different.

13. Comparison of Methods of Computation for Series Transform-ers. The agreement between the two methods of solution, namely, the"complex quantity method" and the "differential equation method" maybe pointed out briefly as follows. If transient terms be dropped, and themaximum value of equation (34) be divided by the maximum value of


equation (30) this will be the same as the quotient of the effective values,or the transformation ratio will be

4 - - *M . (69)^

f Xi y tf + i V rf + Xi2It will be remembered that secondary load resistance and reactance

were taken equal to zero in the second method. If therefore, r and xin equation (24) be replaced by zero, the agreement with equation (69)is evident. Again a comparison of equations (34) and (30) shows thatthe angle of phase difference between the two currents i" and i' understable conditions is ft and

7*2

(^= arc tan ~b = arc tan

~~

-&2

If r and x in equation (25') be replaced by zero, the agreement inphase angle as obtained by the two methods is evident.

IV. CONCLUSIONS

The conclusions that may be drawn from the preceding discussionin regard to the behavior of current transformers, are of a qualitativerather than a quantitative nature. The examples chosen, however, havebeen such as to give a fair and reasonable representation of magnitude,but the main purpose has been to derive and explain the general char-acteristics of the current transformer, and to point out some of itslimitations.

To enumerate again in detail all the results of this investigation is

thought unnecessary. An examination of the figures will show most ofthem. In conclusion a few of the most general and important resultsare given :

1. The transformation ratio and phase angle of a series transformerhaving a core of constant permeabilty (as air) are constant under givenconditions for all values of primary current; but this is not so with atransformer having an iron core. With an iron-cored series transformerthe form of variation depends upon the shape of the saturation curve,and upon the range over which the transformer operates. The rangeover which the permeability remains most nearly constant is the rangeover which the ratio remains most nearly constant. In a transformer ofconstant core reluctance the so-called magnetizing current is propor-tional to the primary current, and its phase position is constant.

2. The introduction of resistance in the secondary circuit of a seriestransformer has the effect of increasing the phase angle, and this increasein phase angle is practically proportional to the secondary resistance for


CURRENT IN PRIMARY COIL (/,)


reasonable values. Increasing the secondary resistance decreases but

slightly the transformation ratio. Hence it may be said that in generalthe introduction of secondary resistance is very objectionable when thetransformer supplies current for a wattmeter, but is not seriously ob-

jectionable when the transformer supplies current for an ammeter.3. The effect of secondary reactance, and the equivalent effect of

magnetic leakage is to reduce the phase angle slightly, and the trans-formation ratio very considerably.

4. The phase angle increases with decreased permeability, and conse-quently in a transformer with an iron core the phase angle increases asthe line current decreases.

o. The effect of changes in frequency within a range of 10 cycles isnot generally serious. It should be pointed out, however, that in addi-tion to the effects of frequency shown in the curves, in a transformer withan iron core, decreasing the frequency raises the point of operation onthe saturation curve, and hence increases the core loss and alters theform of variation of transformation ratio and phase angle with primarycurrent.

6. The desirability of a high number of turns was pointed out. Witha reasonably high number of turns and a not excessive value of second-ary resistance the effect of frequency over a considerable range is

negligible.7. The effect of core loss is to decrease the secondary current, this

effect being lessened by inductive secondary load. Increased core lossdecreases the phase angles, and this effect is increased by inductive sec-

ondary load.8. In an iron-core series transformer the value of flux density should

be low (say B = 2,000 at full load). This means a low value of mag-netizing current. To this end excessive secondary impedance should beavoided, as increased impedance requires an increase in flux in prac-tically direct proportion to the impedance. Since the effect of magneticleakage is equivalent to the effect of secondary reactance, the transformershould be designed with a view to minimum magnetic leakage. Thisrequires a well closed iron circuit.

9. For recording instantaneous values of current in transient or

unsymmetrical systems, the commercial series transformer with an ironcore is quite inadequate, and cannot be relied upon.

10. If necessity demands the use of a series transformer in recordingtransient or unsymmetrical currents, an air-core transformer, designedto have a very small secondary resistance and a large secondary reactancewill be found to give results nearer to those desired than can be obtainedwith an iron core transformer.


TABLE 2DETERMINATION OF VALUES OF *'

1


TABLE 3SOLUTION OF EXAMPLE 3

1

vr>II

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wK4 2PQ H^ RH

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