Determination of Impulse Stresses withinTransformer Windings by Computers

J. H. McWHIRTERMEMBERAIEE

C. D. FAHRNKOPFMEMBERAIEE

Synopsis: A theoretical basis fordetermina-tion of the short-time transient charac-teristics of transformer windings is pre-sented. This theory was used to developan accurate, economical, and convenientmethod of determining impulse voltagestresses within transformer windings throughthe use of modern digital and analogue com-puters. The method has been developedfor and applied to the shell form of con-struction, but the theory is equally applica-ble to other forms of construction.

Purpose

THE CONTINUING trend towardhigher transmission voltages, the de-

sire to ship transformers with higherpower ratings without increasing the sizeand weight, and the increased com-plexity of transformer designs have justi-fied increased and more detailed con- that

the voltage distribution was almost thesame whether or not the iron core was in-cluded in the measurements. For thisreason, the inductance parameters werecalculated on the basis of an air core.Various mathematical and empirical im-provements were made over the years.One of these which reached the literaturewas the work by Bellaschi and Palermo.Although there were errors imposed bythe basic assumption of air core induct-

J. H. STEELEASSOCIATE MEMBERAIEE

ances, these were obscured by the ap-proximations of the simplified and oftenempirical design methods which werenecessarily employed to make practicaluse of the theory. A valuable tool wasdeveloped in the form of a low-voltagerepetitive surge generator. This wasused to measure the voltage distributionon a winding which had already beenbuilt and it could also be used to testmodels of winding arrangements. Thedevice has advantages in determining thecause of a transformer failure on impulsetest and it is also useful as a functiongenerator in the equivalent circuit type ofanalogue computer.

In 1953, one of the writers formulateda theory which was the basis for a prac-tical solution of the problem by means ofmodern digital and analogue computers.This theory forms the basis fpr the workreported in this paper.

At about this same time Abetti2 re-ported his work on electromagneticmodels,which represents a definite achievementin this field and which is certainlythe most theoretically satisfying methodproposed to date. However, it was con-sidered too expensive and it required anexcessive amount of time delay in the de-sign as compared with the methods al- inductances are small compared

to the self-inductances and mutual in-ductances. This is true to a large degreein closely coupled transformer windingseven if the core permeability is unity.It is certainly valid if the effective perme-ability is more than one. The secondstatement is true to the degree that theleakage flux path is through air or to thedegree that the core permeability isgreater than one. Calculations of fluxpenetration of the core at frequencies cor-responding to impulse voltages show thatthe effective permeability is high. This isverified by tests.

A consideration of these statements dis-closes why the use of air core inductanceshas met with such success. An interest-ing and perhaps surprising conclusion isthat it does not make a great deal of dif-ference in most cases whether unity orinfinite permeability is assumed.

Assumptions are made to neglect all ofthe winding, dielectric, and core losses,and to assume linearity in the remainingcapacitance and inductance elements.Finally, the usual liberty of lumping dis-tributed parameters is taken.

EQUIVALENT CIRCUITTo introduce the method by which the

problem is solved in terms of familiarconcepts, a simplified example willbeused.Considering the schematic winding con-figuration in Fig. 1(A), a circuit can befound which will be approximatelyequivalent to the transformer.

Consider this as a 3-winding trans-former by breaking the series connection.The usual inductance diagram can thenbe drawn on a reference turns basis. Toput these inductances on the proper turnsbasis, "perfect" transformers are used inthe equivalent circuit. The terminals ofthese perfect transformers are nowequivalent in inductance to the terminalsof the transformer. By reconnecting theseries connection and placing the windingcapacitances in the circuit, the completeequivalent circuit is as shown in Fig. 1 (B).This circuit is equivalent to the trans-former under impulse conditions exceptfor the approximation brought about by

Paper 56-742, recommended by the AIEE Trans-formers Committee and Computing Devices Com-mittee of the Communication and Electronics Di-vision, and approved by the AIEE Committee onTechnical Operations for presentation at the AIEESummer and Pacific General Meeting, San Fran-cisco, Calif., June 25-29, 1956. Manuscript sub-mitted March 27, 1956; made available for print-ing May 10, 1956.J. H. MCWHIRTERis with the Westinghouse Elec-tric Corporation, Sharon, Pa.; and C. D. FAHRN-Kops and J. H. STEELE are with the WestinghouseElectric Corporation, East Pittsburgh, Pa.

McWhirter, Fahrnkopf, Steele-Determining Impulse StressesFEBRUARY1 957 1267

LV HV HV LV(3) (I) (2 (4)

LV SERIESCONNECTION

LOW HIGH LOW (5)VIOL- VOLTAGE VOL-TAGE (b) TAGE(a) (a)

(A) WINDING CONFIGURATION

LV HV HV LV

L(3) (I07 Hf2) Lg4) SERIES CONNECTION(5)/ CAPACITANCES

Csb Cbc Csc CcdCck Csk CkO-- ~_ HF HY-i I_ H---------f

switches, and potentiometers. The num-ber of these elements and their layout onthe machine are designed to solve simul-taneously for five unknown impulse volt-ages with the impulse voltage being ap-plied at a sixth node. Thus, the capaci-tance and inductance matrices are 5 by 6.The computer circuit also has provisionfor generating a standard impulse voltage.The problem is permanently wired on aremovable patch panel so that onlyminor wiring changes are required to setup an individual problem. This patch

COL.

COL. 2

COL. 3

COL. 4

COL. 5

COL. 6

00z t

z z~~~~Ic.

z ox: (fZ

INDUCTANCECOEFFICIENTS

panel is shown in Fig. 6. All of the com-ponents which were added to the basiccomputer are terminated at the patchpanel so that they may conveniently beused on any problem other than the onefor which they were especially designed.

Following is an explanation of Fig. 7.1. Pot bank 1 and switch bank 2 are usedto set the magnitudes of the inductancematrix. 2. Switch bank 1 is used for set-ting the signs of the inductance matrix.3. Pot bank 2 is used to set the capaci-tance matrix values. 4. The selector

LOOP GAIN WIRING(SCALE FACTOR)

OFF3 -4 0-

5 ea-0

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switch bank is used to select the nonzeroelements of the capacitance matrix.5. Pot bank 3 contains the potentiom-eters used for generating the impulsefunction and a number of potentiometersused for miscellaneous purposes or spares.

The coefficients of the integrodifferen-tial equations are modified in such a wayas to facilitate setting up the switch andpotentiometer matrices. Consider thefirst of the equations. The digital com-puter provides the coefficients for the fol-lowing form of the equationsp2V, - Cl2p2v . - -C]5PV5-C16p2V6+

(a1,rl, v1- a.T12v! . ..-aj,r15v5-a16rj6v16)bjci =0

Since in almost all cases a given nodewill be capacitively coupled with onlythree or fewer other nodes, there willusually be a total of only four, or fewer,p2 terms in the equation. If the threenodes which are capacitively coupled withnode 1 are designated as i, j, and k, thisequation is written asp2V, -C, i p2Vi-Clj p2Vj-Cc P2Vk +

(a,Fitiv1 -aj,r12V,. -a 1,r,5v5 -a,6r716v6) bic1 = 0

The following restrictions are imposedon these coefficients:1. The b and c coefficients are either oneor ten. They are chosen so that the largest(ar) term in an equaton is between one andten.2. The a coefficient is one or ten.3. The r1 terms are then less than one.

The C and r terms will be potentiom-eter settings and the a, b, and c termswill be summer or integrator gains of oneor ten.

The analogue computer wiring for thesolution of this equation is shown in Fig.8. The signs of the r coefficients are seton switch bank 1, row 1. The r termsare set on pot bank 1, row 1. The aterms are set on switch bank 2, row 1.Thus, the output of the first integrator is

Fig. 8 (left). Portion of analogue computer pre-patchpanel wiring

Fig. 9 (above). Schematic cross section of core open-ing of transformer used for comparison of measured

and computer results

McWhirter, Fahrnkopf, Steele-Determining Impulse Stresses

Fig. 7. Potentiometer and switch panels of analogue computer

1270 FEBRUARY1957

TIME- MICROSECONDS

TIME- MICROSECONDS

TIME- MICROSECONOS

l.Iiw NODE-9

101q

U.0 >

or -20

gna

e0

J

10 20 30 40 50 60 70 80 90TIME- MICROSECONDS

NO1DE-8 7 if-- -- -- - ------ - - --- - --:; -% IL) .1 .1 N I

-i - - N-...U. 00 >OR -20 - i L10 20 30 40 50 60 70

TIME - MICROSECONDS80 90

TIME- MICROSECONDS

100 k-ft

TRANSFORMER hlEASUIRFMIkENS-

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~rw_svo10 20 30 40 50 60

TIME- MICROSECONDS70 80 S

TIME- MICROSECONDS

TIME - MICROSECONDS

Fig. 1 0. Comparison of transformer measurements and computer results

-alPllvl/p+al2]l2v2/p. . +alr,,r5v,;,p +0 161i6VI6/p

This is equal to

(pV1i ClPV2 CljPVJ - Clk pvp) /bic,

This term is integrated in integrator 2with a gain of (b1), producing an output of

(-VI + Cli Vi + C1Jvj + ClkVk) /cI

This term is put into summer four withthe capacitance terms. The actual nodescorresponding to the subscripts i, j, and kare selected on selector switches, row 1,and the C values are set on pot bank 2.The output of summer 3 is then v1 and

FMcWhirter, Fahtrnkopf, Steele-Determining Impulse Stresses

a

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0.

41

IL0

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41

IL.0

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ANALOG COMPUTER---

-NODE- [ _I__ __ __ __ __

FERUARY1957 1271