a calculation method for impulse voltage distribution in transformer

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, no. 3, May/June 1978

A CALCULATION METHOD FOR IMPULSE VOLTAGE DISTRIBUTIONAND TRANSFERRED VOLTAGE IN TRANSFORMER WINDINGS

A. Miki T. HosoyaHitachi Research Laboratory, Hitachi, Ltd.

Ibaraki, Japan

ABSTRACT

This paper presents a calculation method for im-pulse voltage distribution in the complex winding ar-rangement encountered in large power transformers. Thismethod takes into account the electrostatic and elec-tromagnetic combinations of windings, and, therefore,permits a precise analysis of voltage response in thewindings to which an impulse voltage is applied and ofthe transferred voltage in other windings to which animpulse voltage is not applied directly.

The necessary multi-winding equivalent network ismade up by adding mutual inductance and capacitancebetween windings to the equivalent networks of individ-ual windings which are based on the traditional net-work of a single winding. The multi-winding network ismarked by the precise subdivision of transformer wind-ings according to the unit coil. A unit coil may bedefined as one pair of disk coils or one layer coil.

The results of experimental investigations in thedetermination of winding constants and comparison be-tween the calculated and measured impulse voltage re-sponses for various types of transformers demonstratethat this calculation method for impulse voltage dis-tribution and transferred voltage in transformer wind-ings can be applied to the transformer winding designwith satisfactory accuracy.

INTRODUCTION

The distribution of impulse voltage in transformerwindings has an important effect on the design of highvoltage transformers. In the past, this phenomena waselucidated by theoretical analyses which assumed thattransformer windings have uniformly distributed capaci-tance and inductance1. Calculations with this assump-tion were not applicable to the transformer windingdesign in practice.

After Lewis 2 proposed a ladder network having afinite number of uniform sections for the transformerwinding, others advanced Lewis' studies and, usingdigital computers, attempted numerical solutions fora network having non-uniformly distributed winding con-stants . Thus, calculation of impulse voltage dis-tribution in a transformer winding having arbitrarilyselected winding constants became possible. This methodwas very effective in obtaining the impulse voltagedistribution in a transformer winding in a very shorttime and in having the flexibility necessary for designwork, as compared with experimental analyses using ascale-model transformer6.

F 77 583-8. A paper recaurended and approved bythe IEEE Trensfor!rs Candittee of the IEEE Pcwer En-gineering Society for presentation at the ITEEE PESSunrr Meeting, Mexico City, Mex., July 17-22, 1977.Manuscript submitted January 28, 1977; made availablefor printing April 6, 1977.

K. OkuyamaKokubu Works, Hitachi, Ltd.

Ibaraki, Japan

However, recently higher voltage and larger capaci-ty power systems have been developed, so that a greaterreliability of transformer windings against applied im-pulse voltage is required. Therefore, a more preciseanalysis of the impulse voltage distribution in trans-former windings becomes important, in order to analyzethe voltage distribution in an impulsed winding, and inorder to predict transferred voltage in a non-im-pulsed winding. The latter point is especially impor-tant for two reasons. It is necessary to determine themost suitable insulation composition and constructionfeatures which take into account the secondary oscil-lation in the winding facing the impulsed winding. Andsecondly, it is necessary to optimize the insulationcoordination with other power apparatus which ar.e con-nected to the non-impulsed winding. This means that acalculation method for impulse voltage distribution inmulti windings must be established. An approach to thisproblem was proposed by Fergestad and Henriksen7.

In this work, a more precise calculation method forimpulse voltage distribution in impulsed windings andfor transferred voltage in non-impulsed windings inlarge power transformers which is directly applicableto the routine design work of transformer winding insu-lation is presented. The development of this methodwas conducted through the following steps:

(1) to determine a precise multi-winding equivalentnetwork composed of such subdivided elements as onepair of disk coils or one layer coil;

(2) to determine the appropriate value of inductanceof the transformer winding by systematically in-vestigating whether the existence of an iron coreinfluences impulse voltage response;

(3) to determine the optimum calculation method for se-ries capacitance by comparing measured impulsevoltage response with calculations obtained fromvarious calculation methods; and

(4) to verify the accuracy of the method for impulsevoltage distribution and for transferred voltage bycomparing the calculated and measured impulse volt-age responses of various types of transformer wind-ings.

EQUIVALENT NETWORK

Typical winding constructions used in core-typetransformers are shown in Fig. 1. In general, impulsevoltage distribution can be analyzed using a laddernetwork which is composed of lumped constants such asself and mutual inductances, series capacitance and ca-pacitance to earth8. An equivalent network for a multi-winding transformer can be established by using mutual-inductance and capacitance between windings to combinea few traditional networks with each other. This multi-winding network must satisfy the following conditionsin order to determine the impulse voltage distributionin each winding for various types of winding con-"tructions.

(a) The network must be applicable not only to disktype but also to layer type windings.

0018-9510/78/0500-0930$00.75 @ 1978 IEEE

930

Authorized licensed use limited to: MADHAV INSTITUTE OF TECHNOLOGY AND SCIENCE. Downloaded on February 8, 2010 at 08:08 from IEEE Xplore. Restrictions apply.

Fig. 1. Typical winding construction used in core typetransformers.

(b) The network must be suitable for analyzing impulsevoltage distribution in impulsed wiuidings andtransferred voltage in associated windings, takinginto account the effects of other windings.

(c) It must be possible to calculate precisely thevoltages between adjacent coils in a winding andthe voltages between windings facing each other.

An example of an equivalent network for a trans-

former with three disk windings obtained under theabove conditions is shown in Fig. 2. This network hasthe feature that any node can be joined with any othernode in another winding network by the capacitance be-

tween the windings. Therefore, each winding in a

transformer can be subdivided into the arbitrary number

of elements.

IRON

CORE LV mV HV TANK

931

Then one inductance element in the proposed equivalentnetwork represents one pair of coils in a disk windingor one layer coil in a layer winding and each induc-tance element is combined with a mutual inductanceof any other inductance element. This allows for amore accurate representation of multi-winding trans-formers. Analysis of this network enables calculationof the precise impulse voltage distribution in eachwinding of multi-winding transformers. It also enablescalculation of the potential differences between wind-ings facing each other which is of primary concern tothe design engineers in determining suitable insulationconstruction of transformer windings and optimum in-sulation coordination with other power apparatus. Inthis network, damping of oscillation caused by copperloss, core loss and dielectric loss is not taken intoaccount.

NETWORK ANALYSIS

Assuming that [V] is a nodal voltage vector and [I]a current vector through an inductance element, equa-tions for a multi-winding equivalent network are obta-ined in matrix form as follows.

d (1)

[Bt [I] (2)=-[E] d v(t) + [F] dd [V

where [A] and [B] are the transformation matrices de-termined by network conditions; [B]t is the transposedmatrix of [B]; [M] is inductance matrix; [El and [F]are capacitance matrices; and v(t) is applied voltage,which is shown as

v(t) =V0(e -at-e -/Jt) : for full wave (3)'

v( t) =V0( e - t - e -#t) -V0(e - (t-t-) -e -6(( t-)

: for chopped wave (4)

and VO, a, S, VIo,y, 6, and to are constants which de-termine the shape of the wave.

Eliminating [I] from Eqs. (1) and (2), a lineardifferential equation of the second order for [V] isobtained as written in Eq. (5).

(5)[VI -([PI dt2+ EQ] )v( t)-[ER] [VIdt02 L.dt

where

[P]=[FI-1 [El

[Q]=[F]- I[ B]t [MI-' [A]

[R]= [F] -1 [B]' [M]-' [B]

The initial conditions are given by Eq. (6).

[V] =O

dt [V] = p] dd v t)lt=((6)

Fig. 2. Equivalent network of a multi-winding trans-former. Ci = series capacitance, Gi= capacitance to

earth, Ki. = capacitance between windings, Li = selfinductanc , MN. = mutual inductance.

ij

Although Eq. (5) can be solved in various ways,Milne's method18, an iterative technique based on a

numerical integration method is used in this paper be-cause of its stability in solving large simultaneousequations of over one hundred unknown variables.

1 2 33 WINDING SPLIT WtJDING Au TRANs

DISKTYPE DEUjDEI

LV MV HV TAP TAPHV LV HV LV MV W TAP

LAYER

TYPELV MV HV TAP

IA VHV LV MV HV


932

DETERMINATION OF WINDING CONSTANTS

Formulas to determine the winding constants havealready been proposed in a number of papers9'13. But,the calculation of exact inductance for the transfor-mer winding is very difficult because of the non--line-arity of the permeability of an iron core, and the cal-culation of series capacitance is also difficult be-cause many capacitances which are continuously dis-tributed along the winding, such as capacitance betweenturns or between coils, must be lumped together in cal-culations. For these reasons a more precise investi-gation in the determination of winding constants wasrequired.

In this paper, more appropriate formulas for wind-ing inductance are derived after the effect of an ironcore on impulse voltage distribution in transformerwindings is examined through systematic experiments.A formula which produces the optimum series capaci-tance values is selected after comparing calculated andmeasured voltage responses in several model windings.

EFFECT OF IRON CORE

In an ordinary lightning impulse voltage test, mainflux in the iron core will be cancelled by the fluxcaused from a short-circuited non-impulsed winding, sothat only leakage flux contributes to impulse voltageresponse in the windings. As a result, it has been as-sumed that the inductance of a transformer winding withan iron core was approximately similar to that of awinding without an iron core,and that the impulse volt-age distribution in transformer windings could be ana-lyzed using the inductance values of a winding withoutan iron core. But, in order to obtain accurately thepotential differences between coils and transferredvoltage in non-impulsed windings, it is necessary tomake clear the difference between the impulse voltageresponse in windings with and without an iron core.This is investigated systematically with various typesof model windings.

A comparison between measured waveshapes of the im-pulse voltage response in a single winding with andwithout an iron core made with the neutral terminalgrounded is shown in Fig. 3. The difference betweenthe voltage responses in two different conditions isnegligible. On the other hand, a comparison made withthe neutral terminal isolated is shown in Fig. 4. Inthis case, the voltage response in the winding with aniron core is different from that in the winding withoutan iron core. This seems to indicate that magneticflux exists in the iron core when the neutral terminalis isolated. The damping effect caused by iron losskeeps the peak value of the voltage response in thewinding with the iron core lower than that in the wind-ing without the iron core.

When an inner winding, such as a l.v. winding, isprovided and is short-circuited, the voltage responsein an impulsed h.v. winding with an iron core is simi-lar to that in the winding without an iron core, evenwhen the neutral terminal of the impulsed winding isisolated, as shown in Fig. 5. Since the capacitancedistributions in both cases shown in Figs. 4 and 5,either with or without iron cores, are approximatelysame, the similarlity of voltage responses shown inFigs.5(a) and (b) indicates that the magnetic flux inan iron core is cancelled by the short-circuited wind-ing and the impulse voltage distribution is affected byleakage flux through air.

PU.1.0

o.o0 90

TIME-P(a) WITH IRON COI

10~II6~II

* PU.

m 0040 0 20 40S TIME-P S

IRE (b) WITHOUT IRON CORE

Fig. 3. Comparison of voltage responses in a windingwith and without an iron core when neutral is ground-ed.

PU.1.0

0.0

12

PU.2.0

1.0

0.00 20 40

TIME-PS(a) WITH IRON CORE

0

(b)

20 40TIME- P S

WITHOUT IRON CORE

Fig. 4. Comparison of voltage responses inwith and without an iron corelated.

when neutrala windingis iso-

rio112

RU.2.0

1.0

QO0 20 40

TIME-PS(a) WITH IRON CORE

0 20 40TIME-PS

(b) WITHOUT IRON CORE

A comparison between transferred voltages in a

winding with and without an iron core is shown in Fig.

Fig. 5. Comparison of voltage responses in a windingwith and without an iron core when l.v. winding isshort-circuited and neutral is isolated.


13~

WY2

1.

0 20 40PU020Q

0.0

0 20 40TIME-IJS

(a) WITH IRON CORE

I I3.

Is

3. 4PU.1.0

0.0

PU.

0.20

PU.0 20 401.0

0.0PU.0 20 400.2 r l _Q0o

0 20 40TIME-pS

(b) WITHOUT IRON COREFig. 6. Comparison of transferred voltage in a wind-

ing with and without an iron core.1. Neutral is grounded and secondary is isolated2. Neutral and secondary are grounded3. Neutral and secondary are isolated4. Neutral is isolated and secondary is grounded

6. A summary of this comparison i3 given by the fol-lowing three ponts.

933

The above-mentioned results show that Impulse volt-age distribution in transformer windings under the ter-minal connections for an ordinary impulse voltage testcan be analyzed using the 'air' inductance without se-rious error.

CALCULATION OF INDUCTANCE

Formulas for the calculation of selfwithout an iron core are well known. Theycable to the calculation of impulse voltagetion in any transformer windings.

inductancesare appli-

distribu-

Mutual inductance between the coils which have anycross-sectional size and shape must be calculated accu-rately. Mutual inductance of two coaxial circularfilaments of negligible cross-section as shown in Fig.7is calculated with the following formulal .

M22= k k)K(k)- E(k)|I

where

(7)

/4ab1/( T- b )2 yd2

th0 is permeability in a vacuum, a, b, and d arethe dimensions shown in Fig. 7 and K(k) and E(k) arethe complete elliptic integrals of the first and secondkinds respectively. However, if a cross-section of a

A COIL

8 COIL(-

(1) When both terminals of the l.v. winding are ground-ed, the difference between the transferred voltagesin the windings with and without an iron core isnegligible, independent of whether the neutral ter-minal of the h.v. winding is grounded or isolated.

(2) When both terminals of the l.v. winding are isolat-ed and the neutral terminal of the h.v. winding isgrounded, the peak valves of transferred voltagenear the terminals of the l.v. winding with aniron core tend to be slightly larger than thosein the winding without an iron core. Butthis distinction is probably caused by the differ-ence of capacitances to earth due to the elec-trostatic fringing effect of a yoke, and not by thedifference in the inductances of windings with andwithout an iron core. This is indicated from thefact that the oscillation frequency of the trans-ferred voltage in the winding with an iron core isalmost the same as that in the winding without aniron core.

(3) When both terminals of the l.v. winding and theneutral terminal of the h.v. winding are isolated,the difference between the transferred voltage inthe winding with and without an iron core increaseswith increasing distance from the line terminal ofthe h.v. winding. The peak values of transferredvoltage in the winding with an iron core is smallerthan that in the winding without an iron core.

Fig. 7. Mutual inductance between twolar filaments.

coaxial circu-

NO.1 COIL

NO.2 COIL

CENTER LLNE

Fig. 8. Calculation method for mutual inductance be-tween two coils of arbitrary cross-sectional size andshape.

d

i fH-E-_-,1I. I

I IjIm


934

coil is not negligible as shown in Fig. 8, mutual in-ductance between No. 1 and No. 2 coils must be calcu-lated by sulmming the mutual inductance between smallelementsof No. 1 coil and of No. 2 coil. In Fig.8, No.1 coil is subdivided into m' x n' elements and No. 2coil is subdivided into m x n. Since the cross sec-

tion of each element is small, the mutual inductancebetween the (k, Q) element of No. 1 coil and the (i, j)element of No. 2 coil is written as follows.

NlmN2

M(k, i~, ) _'ao J. (2-~ -k)K(k) _- E k (8)

where

P. U.1.0

0.5

0.0

/4R1 Rjk (RP= II4(Ri )+Hki

N1 N2ml x n' and m x n are the number of turns in each

element of No. 1 and No. 2 coils respectively, and R.,R. and Hki are the dimensions shown in Fig. 8.J ki

Consequently, summing Eq. (8) for all elements ofNo. 1 and No. 2 coils, mutual inductance between No. 1and No. 2 coil can be written as follows.

N1 N2 in 2 2M =,U,-, E E N E T/R1Rj 1(--k)K(k)--2E(k)l (9)m n mn 1i=Ij=k= 1= k

where

P.U.0.2

0.0

MEASUREDSTEI NlO

-- J A YARAM19WALDVOGEL3

I

0 2 4 6 8 10

(a )T I M E- pS

-31

4

-5

-61

7

0 2 4 6 8 10

(b) TIME - pS

Fig. 9. Comparison of voltage responses calculated byseveral calculation formulas for series capacitance.(a) The voltage to earth at section 2(b) The voltage across sections 1 and 2r--

/4R1 Rjk=R2d(Rl+Rj)2+Hk~~~~~~~i

Eq. (9) is available to calculate mutual inductancebetween two coils which have any cross-sectional sizeand shape.

WINDING CONDUCTOR SHIELDING CONDUCTOR

CALCULATION OF CAPACITANCE

Series capacitance, capacitance to earth and capa-citance between windings are necessary for the calcula-tion of impulse voltage distribution in transformerwindings. Capacitance to earth and capacitance betweenwindings can be calculated as the capacitance betweencoaxial cylindrical electrodes.

A method of calculation of series capacitance hasbeen proposed in several papers3, 9, 1 However, it isvery difficult to obtain an exact value for series ca-pacitance in a winding as described before. So, in thispaper, an optimum calculation method for series capaci-tance is selected by comparing the voltage responsescalculated by using several proposed calculation for-mulas for series capacitance with measured voltage re-sponse in several transformer windings. As shown inFig. 9, it results that Stein's formulal which calcu-lates series capacitance by summing electrostaticallystored energy between turns and between coils of aunit coil , has sufficient accuracy for the estima-tion of series capacitance in the calculation of im-pulse voltage responses.

In our practice, the high voltage disk winding ofpower transformers has shielding conductors to improveits impulse voltage distribution. A typicaldisk winding with shielding conductors is shown in Fig.10. Shielding conductors have an effect of making se-ries capacitance of disk windings increase equivalent-ly. Therefore, uniform impulse voltage distribution inh.v. windings can be easily accomplished by changingthe number of turns of shielding conductors or theirconnections s, 16 . The series capacitance of shieldedwinding is mainly determined by the capacitance betweenshielding conductor and winding conductor, which can becalculated as the capacitance between coaxial cylindri-cal electrodes.

A COIL

B COIL

Fig. 10. Cross section of disk coilsconductors and connections.

with shielding

VERIFICATION OF THE CALCULATION METHOD

IMPULSE VOLTAGE DISTRIBUTION IN MULTI-WINDINGTRANSFORMERS

A comparison between calculated and measuresi volt-age responses in various types of experimental modelsand actual transformers was made.

Fig. 11 shows a comparison between calculated andmeasured impulse voltage distributions irn both h.v.winding to which an impulse voltage is applied and l.v.winding to which an impulse voltage is not directly ap-plied. The winding in Fig. 11 is split in that the h.v.winding is constructed of a series connection of 5 lay-er coils and 20 pairs of disk coils, and the l.v. wind-ing of a series connection of 20 pairs of disk coils.This model aims to confirm the accuracy of calculationof impulse voltage distribution in such a complex

-IIz

I

-I -

IEl El113 0 El 0 13 9

1 113113101111010 13 El 0 0 13 0

1 1El 113101010113

\ /


5 5

HV LV HV

935

transformer winding arrangement. The results of thecalculations of impulse voltage response in the h.v.winding and transferred voltage response in the l.v.winding agree well with the measurements as shown inFig. 12.

Fig. 13 shows the calculatedshapes of voltage responses in atransformer composed of 60 pairs

and measuredthree-windingof disk coils.

x x CALCULATED- MEASURED

VOLTAGE TO EARTH

P.U.1.0o

Q5-

o.o

5 10

(a)30 40

TIME -PS

02

0.0e25 30 35 4- ..

P.U.

0.0

0 10 2 0 30 40TIME - Ps

(a) MEASURED

SECTION NUMBER( b)

Fig. 11. Full wave impulse voltage distribution in awinding with secondary grounded.(a) Impulse voltage distribution throughout the h.v.

winding(b) Transferred voltage distribution throughout the

l.v. winding

P.U.

0.5

.0 10 20 30 40

P.U. TIME- pS0.5

0.0 ~`_vf~- ,o0 10 20 30 40

TIME- pS

eU

0 10

TIME- pS

(b) CALCULATED

Fig. 12. Comparison of waveshapes of voltage to earthfor the case shown in Fig. 11.

TIME - &

P.U.0.5 ~4

0.0'0 10 20 3 40

TIME- pj

(b) CALCULATED

Fig. 13. Comparison of waveshapes of voltage to earthin a winding of three disk winding construction.

Figs. 14-16 indicate good agreement between cal-culated and measured voltage responses in actualmulti-winding transformers under various conditions ofimpulse voltage application.

The calculated peak values and oscillation frequen-cies of impulse voltage responses in various types oftransformer windings agree with the measured oneswithin ±15%. These good agreements demonstrate thatthe calculation method in this paper can be used withsatisfactory accuracy for determining not only thevoltage distribution in impulsed windings, but also thetransferred voltage in non-impulsed windings, no matterwhat the structural design of the multi-winding trans-former.

Analysis of impulse voltage distribution in multi-winding transformers is very important for insulationdesign of transformer windings, because it now becomespossible to predict the maximum potential differencebetween the windings facing each other, and consequ-ently, to establish more suitable insulation construc-tion features, especially between h.v. and l.v. wind-ings.

For instance, the maximum potential differencebetween the h.v. and l.v. windings exceeds the ap-plied voltage as shown in Fig. 16, because the trans-ferred voltage at the midpoint of the l.v. winding os-cillates when both terminals of the l.v. winding aregrounded. In this case, the potential difference comesup to 1.2 p.u. of the applied voltage.

1.0

w(D1"-

-J0

a

wave-model

P[] VI/LTAI. TO FADrTh

0.0

P.U0.5

RU.0.1[0.0

TIME -iS

TIME - pS(a) MEASURED

TIME- uS

I

L

_%_4\f_20 .10 .0


PU.1.0

10 II,

LV MV HV

- CALCLATEDME~WASURED

- - I I IU

D 5 10 5 20 25 29SECTION NUMBER

(a)

I

----- CALCULATED7 MEASURED

e10 20 TIME-1ISI %

L0MLV MV HV

I

w

0

I--J0

(b)

Fig. 14. Full wave impulse voltage410 kV autotransformer.(a) Throughout h.v. winding(b) Throughout m.v. winding

PU.I rI

Icrw

01

-J0

distribution in a

Fig. 16. Maximum potential difference between windingsin a 262.5 kV transformer.(a) The potential difference between sections U and

21 for 1 x 40 microsec full wave(b) The potential difference between sections U and

21 for 1 x 40 microsec 3.5 microsec chopped wave

TRANSFERRED VOLTAGE IN A THREE-PHASE CONNECTION

Transferred voltage in non-impulsed windings is afactor of significance for the insulation design of thewinding itself. And transferred voltage in a three-phase connection also becomes important in order toachieve the optimum insulation coordination with otherpower apparatus which are connected to the winding ter-minals.

A three-phase connection for ordinary impulse volt-age tests is shown in Fig. 17 (a). Since the magneticlinkage between the windings of different phases is ne-gligible during an application of impulse voltage, itcan be assumed that the l.v. winding of non-impuilsedphases is substituted for a parallel circuit wh1ich iscomposed of one self inductance and one capacitanceelements as shown in Fig. 17 (b).

-CALCULATEDMEASURED

(a) (b)

Fig. 17. Approximate calculation method for transfer-red voltage in three phase. connection.

Fig. 15. Full wave impulse voltage distribution in a262.5 kV transformer .

936

PU.1.0

I

LU 0.80

W 0.6

> 0.4

-02

0 1 2 3 4 5 6 7 8 910SECTION NUMBER

I.v


Therefore a three-phase equivalent network can beconstructed by adding self inductance and capacitanceto the terminals of a single phase network as shown inFig. 2. The transferred voltage for each phase can becalculated by analyzing this network.

For example, calculated results of transferredvoltage on the l.v. winding in a three-phase connectionare compared with measured transferred voltage in Figs.18 and 19. The calculated, transferred voltages have asimilarity to the measured voltages in spite of thelumping together the winding constants of non-impulsedphases, so that the transferred voltage on the l.v.winding in a three-phase connection can be calculatedwith the accuracy necessary for practical usage withthe above-mentioned equivalent network.

937

(2) In ordinary impulse voltage tests, impulse voltageresponse in transformer windings can be analyzedby using the inductances of a winding without aniron core.

(3) A method of calculating mutual inductance betweencoils which have any cross sectional size and shapewas developed.

(4) Stein's series capacitance formula was confirmed asthe optimum method from comparing measured impulsevoltage responses with the calculated values obta-ined using that and several other well known for-mulas.

(5) The calculated peak values and oscillation frequ-encies of impulse voltage responses in varioustypes of transformer windings agree with the me-asured ones within ±15%.

PU. u0.5-

,00. ,° ° 0~~~~~1

0.2 w0.0 _

0 20\_40 60-TIME- P S

CALCULATED-~MEASURED

0 uU _ _

(6) Transferred voltage in a three-phase connection canbe analyzed using the multi-winding network withlumped self inductance and capacitance for non-im-pulsed phases.

ACKNOWLEDGEMENTS

The authors wish to thank Mr. M. Moriyama and Mr.S. Akimaru of Kokubu Works, Hitachi, Ltd., for theirsupport of this work, and Dr. Y. Kako, Mr. Y. Kamataand Mr. M. Higaki of Hitachi Research Laboratory,Hitachi, Ltd., for their continued encouragement.

Fig. 18. Transferred voltage in a three phase 275 kVtransformer.

CALCULATED-~MEASURED

PU. %,,~

020 -0-I

0 20 40 6T 80~~~TIME- P S

7\, v

U 0_ u_

U w

APPENDIX

CALCULATION FORMULAS FOR INDUCTANCES 17

The self inductance of a disk coil (Fig. 20)calculated by Eq. (10).

L=,u, RN2(In 8R _2)

where

In Rl=2 ln(a2 +b2 ) 2n( 1 2

-a22 bl+2) 23b aa-+ 2a TYn~ 3a b

+2a ba1 253b a 12

is

( 10)

N is the number of turns in a pair of disk coils, anda, b, and R are the dimensions shown in Fig. 20.

Fig. 19. Transferred voltage in a three phase 275 kVtransformer.

CONCLUSIONS

A calculation method for the impulse voltage dis-tribution and the transferred voltage in transformerwindings was developed.

(1) An equivalent network of multi-winding transformerswas composed of self and mutual inductances, seriescapacitance, capacitance to earth and capacitancebetween windings. The inductance in the networkrepresents one pair of disk coils or one layercoil, and any node in the network was joined withany other node by a capacitance between windings,so that the precise impulse voltage response inmulti-winding transformers can be calculated pre-cisely, regardless of the winding constructions.

VA/X

Fig. 20. Disk coil with rectangular cross section.

The self inductance of a layer coil (Fig. 21) iscalculated by Eq. (11).

L=,u,o KRN2 ( 11)

ia 4

-a- b


938

where

2 K(k) +(tan2 a-l)E(k)K= 2 1 -tan2 at3 sina

k= 14 2+R2

2R

N is the number of turns in one layer coil, and Q and Rare the dimensions shown in Fig. 21.

Fig. 21. Cross section of layer coil.

The mutual inductance between layer coils which arearranged as shown in Fig. 22 is calculated by Eq. (12).

M°7(R N N2 R R31 2

2d T~~3-)Iwhere

12d:= 4 2 + I

N1 and N2 are the numberlayer coils respectivelydimensions shown in Fig.

(12)

of turns in No. 1 and No. 2and Qi, k2, R1 and R2 are the22.

works of the Type Representing Transformer andMachine Windings", Proc. IEE, vol. 101, Pt. II,1954, pp. 541-553.

[3] P. Waldvogel and R. Rouxel, "A New Method of Calcu-lating the Electric Stresses in a Winding Subjectedto a Surge Voltage", Brown Boveri Review, vol. 43,No. 6, 1956, pp. 206-213.

[4] J.H. McWhirter, C.D. Fahrnkopf and J.H. Steele,"Determination of Impulse Stresses within Trans-former Windings by Computers", AIEE Trans., vol.75, pt. III, 1957, pp. 1267-1279.

[5] B.M. Dent, E.R. Hartill and J.G. Miles, "A Methodof Analysis of Transformer Impulse Voltage Distri-bution Using a Digital Computer", Proc. IEE, vol.105, pt. A, 1958, pp. 445-459.

[6] P.A. Abetti, "Transformer Models for the Determi-nation of Transient Voltages", AIEE Trans., 1953,pp. 468-480.

[7] P.I. Fergestad and T. Henriksen, "Transient Oscil-lations in Multiwinding Transformers", IEEE Trans.,vol. PAS-93, 1974, pp. 500-509.

[8] K. Okuyama, "A Numerical Analysis of Impulse Volt-age Distribution in Transformer Windings", Elect.Eng. Japan, vol. 87, No. 1, 1967, pp. 80-88.

[9] B.N. Jayaram., "Bestimmung der StosspannungsverteLung in Transformatoren mit Digitalrechner", ETZ-AHeft 1, 1961, pp. 1-9.

[10] G.M. Stein, "A Study of Initial Surge Distributionin Concentric Transformer Windings", IEEE Trans.,vol. PAS-83, 1964, pp. 877-893.

[11] P.I. Fergestad and T. Henriksen, "Inductances forthe Calculation of Transient Oscillations in Trans-formers", IEEE Trans., vol. PAS-93, 1974, pp. 510-517.

[12] K.A. Wirgau, "Inductance Calculation of an Air-CoreDisk Winding", IEEE Trans., vol. PAS-95, 1976, pp.394-400.

[13] M.F. Beavers, J.E. Holcomb, L.C. Leoni, "Magnetisa-tion of Transformer Cores during Impulse Testing",AIEE Trans., vol. 74, pt. III, 1955, pp. 118-124.

[14] J.C. Maxwell, A Treatise on Electricity and Magne-tism, Oxford at the Clarendon Press, 1904.

Fig. 22. Disposition of coaxial layer coils.

REFERENCES

[1] R. Rudenberg, "Performance of Travelling Waves inCoils and Windings", AIEE Trans., vol. 59, 1940,pp. 1031-1040.

[2] T.J. Lewis, "The Transient Behavior of Ladder Net-

[15] K. Okuyama, "Effect of Series Capacitance onpulse Voltage Distribution in Transformerings", Elect. Eng. Japan, vol. 87, No. 12,pp. 27-34.

Im-Wind-196 7,

[16] K.Okuyama, United States Patent, No. 3,691,494

[17] A. Gray, Absolute Measurements in Electricity andMagnetism, Macmillian and Co. Ltd., 1921.

[18] K.S. Kunz, Numerical Analysis, McGraw-Hill BookCompany, Inc., 1957.


Discussion

J. H. McWhirter (Westinghouse Electric Corp., Pittsburgh, PA): Thework reported in the paper by Mr. Miki and his co-authors is well doneand quite interesting. I am particularly interested in their experienceregarding the effect of the iron core. This experience is consistent withthe analysis reported by my co-authors and me in [4]. Our approachis also suitable for the analysis of multi-winding transformers and thecalculation of induced voltages. Although our approach resolves manydifficulties and inconsistencies, it appears not to be understood in theindustry.

I will discuss the inductance aspect of the problem first in amanner which is not inconsistent with the treatment by Mr. Miki, butmay differ in viewpoint or emphasis. Finally, I restate the computa-tional approach used in [4] which I believe is more accurate than thatused by Mr. Miki, as well as essentially all other investigators of thissubject.

The keys to the analysis are the experimental facts that1) Under most conditions, the transient response is the same whether

or not the core is present during the experiment.2) Under some other conditions, notably voltages induced from one

winding into another, the experimental response will depend uponthe presence or absence of the core.How can the first result be true? We consider the explanation that

the self and mutual inductances are unaffected by the core. Such asuggestion is, obviously, ridiculous at 60 Hz, as the self and mutualinductances associated with a winding or a part of a winding (all otherwindings or parts of windings being open-circuited, of course) willincrease by several orders of magnitude when the core is inserted. Thisis the purpose of the core. For rapid impulses, I believe that the selfand mutual inductances are similarly affected by the core. This hasbeen demonstrated by calculation of flux penetration into the lamina-tions and, also, by direct experimental determination of the inductanceparameters.

A better explanation of 1) is(a) If there is an iron core present, the ampere turns under transient

conditions will add to zero for the same reasons that they add tozero in a transformer operating at 60 Hz.

(b) If there is no iron core but all the involved turns are closelycoupled because of proximity, the ampere turns may again beapproximated as adding to zero.In either (a) or (b), the concept of leakage inductance is logically

introduced and the inductive part of the model can be treated as wasdone in reference [4]. In order to explain the experimental results,it is necessary to add one additional observation:(c) The leakage inductance values are affected only to a small degree(say, 10%) by the presence of the iron core.

in the cases where the presence of the core does affect the impulsedistribution, I suggest that the coils largely involved are not closelycoupled in the air core case, and it is not valid to assume that theampere turns add to zero. When the iron core is added, the ampereturns will again add to zero and the experimental results will differfrom the air core case. It still may be valid to calculate the iron coreleakage inductances from air core values and use these values for acalculation which is based on the use of leakage inductances. It maybe very inaccurate to use these same inductance values in an analysiswhich directly uses air core self and mutual inductances and where theampere turns are not required to sum to zero.

Most or all of the above does not seem inconsistent with theviewpoint of Mr. Miki and, also, that of Fergestad and Henriksen [7].My final point is a suggestion which, I believe, will simplify and im-prove the accuracy of the calculations. This suggestion is: calculateleakage inductances directly and use them in the matrix form of re-ference [4].

If we can agree that the leakage inductances are the importantparameters whether we use them directly or indirectly, their direct

Manuscript received October 27, 1977.

939

calculation is likely to be more accurate. To illustrate this point,suppose that a leakage inductance is ideally calculated as

L2=LI +L2-2M12= 100+ 100-2x95= 10

In the actual calculation, errors are introduced in the calculationsof the self and mutual inductances so that the calculation may proceedas

105+ 105-2x90=30The error is 200o. On the other hand, it is usually possible to

make a direct calculation of leakage inductance using simple formulaewhich are accurate to well within, say, 20% which may be sufficient.The example, the leakage inductances between coils 1 and 2 of Fig. 8could be calculated directly. This comparison involves a well knownpoint of computational accuracy when large numbers are differencedto give a relatively small result.

The mathematics of this approach are in reference [4]. It is un-fortunate that some of the matrix equations of Appendix I are mis-printed. However, knowing this, it may be possible for the reader tofollow the derivations and make the necessary corrections.

It would be of interest for Mr. Miki to comment on these ideasin the light of his extensive experience and understanding of the im-pulse distribution phenomenon.

A. Miki, T. Hosoya, and K. Okuyama: The authors greatly appreciatethe comment of Mr. McWhirter. His method of determining impulsevoltage distributions in [4] is based on the assumption of infinite corepermeability under impulse conditions. He stated it in two parts:

1. The summation of ampere-turns around a magnetic circuit iszero.

2. The leakage flux paths and magnitudes are the same underimpulse conditions as they are at low frequencies.

In order to investigate the core effect, we measured the transientlresponses under various conditions. These results show that in ordinaryimpulse voltage tests, transient responses are the same whether the coreis present or not (See Figs. 3,5, and 6). This fact means that the tran-sient response in transformer windings can be analyzed by using aircore self and mutual inductances neglecting the iron core.

Fergestad and Henriksen analyzed the transient response as-suming the relative permeability of the core is about 60 in [ 11].

For the analysis of transient phenomenon in transformer wind-ings, there are various calculation methods and the most suitable wind-ing constants to be used in each method according to the winding con-nections, the conditions of impulse voltage application, and the loca-tions to be analyzed in the windings. Even if the different windingconstants are selected for each calculation method, each method maybe considered to be correct in the case when it give sufficient results.

Our study presented in the paper showed that under most condi-tions, the transient responses in the windings can be calculated by usingair core self and mutual inductances with satisfactory accuracy. Butthere is also the fact this method can not give sufficient results undersome other conditions.

For these conditions the winding constants have to be estimatedby precise magnetic field analysis taking into account the permeabilitycharacteristics of the core, and the core loss has to be introduced inthe calculation method. Although it is evident that these problemsrequire us to spend a considerable amount of time and effort, we wishto continue further study.

Manuscript received October 27, 1977.


a calculation method for impulse voltage distribution in transformer

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