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On the Response of Interleaved Transformer Windings to Surge Voltages

Pedersen, A.

Published in:I E E E Transactions on Power Systems

Link to article, DOI:10.1109/TPAS.1963.291363

Publication date:1963

Document VersionPublisher's PDF, also known as Version of record

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Citation (APA):Pedersen, A. (1963). On the Response of Interleaved Transformer Windings to Surge Voltages. I E E ETransactions on Power Systems, 82(66), 349-356. DOI: 10.1109/TPAS.1963.291363

cos[(N2+ Ni)x-NV2ntl, the ratio of travelingwave and rotor speeds is given by:

Speed of permeance wave N2Rotor speed N2+ N1

If this harmonic permeance wave is to lockin with the fundamental mmf wave at3,600 rpm, the rotor speed should be:

Rotor speed= N X3,600N2

24 ±2222 X3,600= 7,527 rpm

It is obvious that this high operating speedhas academic interest only. As mentioned

by Mr. Koch, external power would probablyhave to be supplied over the rotor shaft tomaintain that rotor speed.

Equation 38 includes only fundamentalterms of slot waves. In calculating motortorque, two components should be included.The main component of torque comes fromthe interaction between the 2-pole perme-ance wave and the 2-pole mmf wave. Theother component of torque comes from theinteraction between the 46-pole permeancewave (the third term of equation 38) and the23rd harmonic of the mmf wave. At a rotorspeed of 327 rpm backward, the speed of the46-pole permeance wave for the samplemotor is

Speed of 46-pole permeance wave

-N2+N X(-327) =-156 rpm

and

Speed of 23rd mmf harmonic-13,600- 36 = _156 rpm23

Therefore, these two waves will react andproduce useful reluctance torque.

REFERENCE1. THE NATURE OF POLYPHASE INDUCTIONMACHINES (book), P. L. Alger. John Wiley &Sons, Inc., New York, N. Y., 1951.

On the Response of Interleaved

Transformer Windings to Surge VoltagesA. PEDERSEN

Summary: The high series capacitancetheory for the response of interleaved trans-former windings to surge voltages is criti-cized from the point of view that an increasedseries capacitance as a result of interleavingis incompatible with the concept of a purecapacitive initial voltage distribution. Anew theory is proposed according to whichthe distributed earth capacitances arecharged up during an initial period by heavycurrents flowing into the winding throughsurge impedances formed by the coils as aresult of interleaving. Formulas are derivedfor the initial voltage distribution and forthe maximum axial voltage gradient.

ZOR A TRANSFORMER winding withFuniformly distributed earth capaci-tance C and series capacitance K, andthe neutral directly grounded, the initialelectrostatic voltage distribution set upinstantaneously by a rectangular wave ofamplitude E is given by

() =Esinh (ax) (1)sinh (a)

where a =\ C/K and x is the ratio be-tween the axial distance from the neutraland the axial length of the winding, ande(x) is the voltage to earth at the point

Paper 63-17, recommended by the AIEE Trans-formers Committee and approved by the AIEETechnical Operations Department for presentationat the IEEE Winter General Meeting, New York,N. Y., January 27-February 1, 1963. Manuscriptsubmitted April 30, 1962; made available forprinting October 22, 1962.

A. PEDERSEN is with the Technical University ofDenmark, Lyngby, Denmark.

x. Maximum initial voltage gradientoccurs at the line end of the winding andis approximately a times greater than itwould be for a uniform distribution. Fora conventional cylindrical disk-type wind-ing the value of a normally lies between10 and 20. A low a value requires a lowearth capacitance or a high series capaci-tance, the latter being the only feasiblepossibility for a disk-type winding. Theinterleaved disk winding, which is thesubject of this paper, is one attempt atachieving this. However, it will be shownthat the improved surge characteristic ofthis type of winding can be explainedwithout recourse to the hypothesis ofan increased series capacitance.

The Interleaved Winding

In the interleaved disk winding twoconsecutive electrical turns are separatedphysically by a turn which is electricallymuch further along the winding. Fig.1(A) shows a cross section of a double-section interleaved coil. It is wound asa conventional double-section disk coil,but with two wires in parallel. Thewires are transposed at the inside and theappropriate wires joined together at theoutside, thus forming a single-circuitdouble-section coil. Each electrical turnmay consist of two or more wires in par-allel, as shown in Fig. 1(B).The interleaved disk-type winding was

introduced in two papers1'2 by Chadwick,Ferguson, Ryder, and Stearn in 1950.

No quantitative theory was given. Itwas merely assumed that a large increasein the effective series capacitance wasobtained by interleaving the turns. Itwas stated that interleaving of the turnsof a double section coil 5 feet in diametercontaining 24 turns increased the seriescapacitance about 30 times. Later, theprinciple of interleaving was discussed inpapers by Grimmer and Teague,3 FerrariBardile,4 Brechna,5 Price,6 Zambardino,7Gorio,8 and Jayaram.9

The High Series Capacitance TheoryFormulas for the calculation of the

effective series capacitance of interleavedwindings have been proposed in references5, 8, and 9. They define the series capaci-tance as the capacitance which, whencharged to the same voltage as the coil,stores an electrostatic energy equal tothat stored in the interturn capacitances.It is further assumed that the voltage willbe distributed uniformly between theturns of each coil. These assumptionslead to the following approximate formula

9 |l~~~~8|6 |1171| 5 116 |14 115 | 3 114 | 2 113 | 1 |

AX~ ~~~I

IBI I

Fig. 1. Section through an interleaved double-section coil

A--n =24, p=1B--n=12, p=2

Pedersen-Response of Transformer Windings to Surge VoltagesJUNE 1963 349

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for the series capacitance K of an inter-leaved winding.

4N (2)

in which p is the number of parallel paths,n the number of electricalturns per double-section coil, ct the capacitance betweentwo physically adjacent turns, and N thenumber of series-connected double-sectioncoils. This equation leads to seriescapacitances which are considerablyhigher than they would be for a similarnoninterleaved winding. The parametera becomes

4 1 N 1 da= t_ -C (3)

l7r eeOLnp-2 D b

in which e is the relative dielectric con-stant of the interturn insulation, 80 thepermittivity of free space, 8.854X10-12farads per meter, D the mean diameter ofthe coil, d the thickness of the interturninsulation, b the axial width of the copperwire, and C the earth capacitance of thewinding. Price8 states that at lies between1 and 3 for a normal interleaved winding.

Analysis of the High SeriesCapacitance Theory

All the evidence in favor of the highseries capacitance theory is indirect inthe sense that it is not based on any directexperimental determination of the seriescapacitance. What is known is that in-terleaving improves the response of thewinding to surge voltages, and that thisapparently can be accounted for by as-suming that interleaving increases theseries capacitance.

It is inherent in the concept of a pureelectrostatic initial voltage distributionthat initially no current is allowed toflow through the inductances. It is,therefore, difficult to see that interleavingcan have any effect on the equivalentcapacitive network. The pure capaci-tive network of two disk-type windings ofexactly the same dimensions, one inter-leaved and the other noninterleaved,must be identical. Therefore any differ-ence between two such windings can onlybe accounted for if currents are allowed-to flow through the electrical turns of thecoils. However, in that case the physicalprocesses cannot be described mathe-matically by the formulas of the purecapacitance network theory. The ideathat the interleaving connections betweenthe two disks of an interleaved doublesection coil should have any effect uponthe capacitive network of the coil cannotbe valid. There is no principal differ-

ence between such connections and theconnections between any two consecutiveelectrical turns within a coil.An attempt at assessing experimentally

the equivalent series capacitance of aninterleaved winding is described byGorio.8 It is based on measurements ofthe frequency of the free oscillations ina winding which has been removed fromthe core in order to make the earthcapacitance as small as possible. Thisfrequency depends on the inductance andthe self-capacitance, which Gorio assumesis identical to the series capacitance. If aknown capacitance is connected acrossthe winding, the frequency is reducedand the self-capacitance can be evaluated.Gorio found that the results agreed within10% with those predicted from the highseries capacitance hypothesis, and this isnot surprising. The equivalent capacitivecircuit which determines the free oscilla-tion of a winding is by definition veryclosely linked with the distributed in-ductances, and the effect of interleavingmust necessarily be very pronounced.This is, however, an entirely differentsituation from that which arises instan-taneously when a steep impulse wave isapplied to a winding and the electrostaticvoltage distribution is set up with nocurrents flowing in the inductances. Thehigh self-capacitance of an interleavedwinding explains the pronounced smooth-ing effect of interleaving upon the tran-sients between initial and final voltagedistribution.

The Initial Voltage Distribution inInterleaved Windings

Each of the two disks of an interleaveddouble-section coil forms, when theinterleaving joints are disconnected, aspiral-wound parallel plate capacitorwith the capacitance

1 bC=- (np-2)eoorD

b(4)2 d(4

The disk will, however, only behave likea capacitor if the voltage applied to theterminals varies so slowly that the finitecharging time of the spiral electrodes canbe ignored. When this is not the case thedisk will respond like a transmissionline characterized by a surge impedance Zand a transit time T, which we will de-fine as twice the time taken for a signalto travel thorugh the line. Z and T aregiven by

ILZ=* C .(5)

T=2x/LC (6)

where C is the capacitance of the line andL its self-inductance given by

1 d n4 b P

And we obtain for Z and T

d 1 1 1Z = s.AoC.-

b p 2

np

(7)

(8)

T= I _ (9)

in which po is the permeability of freespace, 4rX lO-7 henrys per meter, andc is the velocity of light in free space,3X 108 meters per second.

Let a rectangular wave of amplitude Ebe applied across an interleaved double-section coil at the time t= 0. The coilmay be visualized, apart from the insidecrossover, as two series-connected surgeimpedances of the type previously de-scribed. The effect of the inside cross-over is not felt at the outside terminalsuntil t= T and the current flowing intothe coil will be constant during this inter-val. At t= 1/2 T the two surges of ampli-tude 1/2 E penetrating the two disks willmeet at the inside crossover. By con-sidering the electric fields set up in theinterturn insulation by the flow of electriccharges along the copper wires it is seenthat the meeting of the two waves at t=1/2 T results in a doubling of the inter-turn voltage to E in the form of wavesmovingback to the outside terminals. Att= T these waves have reached the outsideterminals and are reflected back into thedisks reducing the interturn voltage to1/2 E, i.e., the coil will oscillate with thefrequencyf= 1/T. These oscillationswillbe superimposed upon the voltages inducedby the magnetic field linked with the in-ductive current which will start flowingthrough the electrical turns. Due to thelosses the current will finally become adirect current determined by the total re-sistance. We may thus divide the wholesequence into three periods: the initialperiod from t= 0 to t= T where the coil be-haves as two series-connected surge im-pedances, followed by a transition periodin which an inductive current flowsthrough the turns, and the final periodwhere the current is a constant directcurrent.A transformer winding consisting of N

series-connected double-section inter-leaved coils will initially respond to arectangular wave like 2N series-connectedsurge impedances. Neglecting the time(a few millimicroseconds) it takes thecurrent to travel through all the jointsbetween the coils at the outside of thecoil stack, a surge current flowing into the

Pedersen -Response of Transformer Windings to Surge Voltages350 JUNE, 1963

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w>D4ti-J

Fig. 2. Equivalent diagram of an interleavedwinding for the initial voltage distribution

Fig. 3. Idealized surge wave with finitefront time

winding at the line terminal would, butfor the effect of the distributed capaci-tances, immediately appear at the neutralwith the same magnitude, and the voltagedistribution along the coil stack would beuniform. However, the earth capaci-tances will be charged up and the dis-tribution becomes nonuniform.

Calculation of the VoltageDistribution During the InitialPeriod in an Interleaved Winding

The equivalent diagram of an inter-leaved winding will, as far as the initialperiod is concerned, be an RC (resistancecapacitance) network of the type shownin Fig. 2. For a wave front of finitesteepness we may neglect the effect ofthe series capacitances which normallyrepresent much higher impedances thanthe series-connected surge impedances.For a large number of coils the voltagee(x, t) to earth is given by the partialdifferential equation

a2e(x, t) ae(x, t)=RC (10)aX2 At

in which x is the relative axial distancealong the coil stack from the neutral, C isthe total capacitance to earth, and R=2NZ is the resultant surge impedance ofthe N series-connected double-section in-terleaved coils. All voltages are assumedto be zero for t <0. At t= 0 a voltagee(l, t) is applied to the line terminal atx= 1. The neutral at x= 0 is directlygrounded. Let E(x, s) be the Laplacetransform of e(x, t) defined by

E(x, S)= f0o exp (-st)e(x, t)dt (11)

The partial differential equation can thenbe transformed into the ordinary differ-ential equation

(, )=RCsE(x, s) (12)dx2

which, with the given boundary condi-tions, has the solution

E(x, s)=E(1, s) sinh (VRCsx) (13)sinh (VIRCs)

where E(1, s) is the Laplace transform ofthe applied voltage. This equation canbe expanded in the following way:

E(x, s) = E(1, s)X_co

> {exp [-V\RCs(2v+1-x)]-

exp [-V\RCs(2v+1+x)]} (14)

We assume for simplicity that the front ofthe applied wave is linear and of a dura-tion TF which is smaller than, or equal to,the duration T of the initial period. Thewave tail is infinite and of constant volt-age E; see Fig. 3. This wave is givenanalytically by

E Ee(1, t)=-t--(t-TF)H(t-TF), t_O

TF TF(15)

in which H(t-TF) is Heaviside's unitfunction

H(t-TF) =0 for t< TF

H(t-TF)= 1 for t> TF

(16)(17)

The Laplace transform of e(1, t) is

El1E(1, s) = - - 2[1-exp (-TFs)I (18)

TF S2

The inverse Laplace transform of equa-tion 14 then becomes1o

te(x, t)=4E >X

TF

0

4ifierfc [ t(2 +l1-x)].V=O

iierf [ / (2v+l+x)] ?-

4E H(t-TF)XTF

co

iierfc [(2Y ]xv = O

iierfc TF) (2V+1+X)

(19)

where iierfc(t) is the second repeatedintegral of the error function complement.The axial voltage gradient along the coilstack is given by

ae(x, t) t IRC=2E- -Xax TF t

Z ierr [ IRC (2v+1-x) +

ierfc [ (2 + 1 +x)

2Et- H(t-Tp) XTF t-TF

Z ierfc Li4(tT)(2v+1-x) +V =o

*K!RCierfc L t _-TF) (2v+1+x)

(20)

in which ierfc(t) is the first repeatedintegral of erfc(t). The voltage gradientis maximum at the line end, i.e., for x= 1I

ae(x, t) 2E t i XAX X=1 TF t

ao

{ +2 ierfc [1tvv = 1

V~=1TF t-TF

{ +_+ ierfc t v]

(21)

The error function complement and itsrepeated integrals are defined by

2 co

erfc(O)= 7 J exp (-Mu2)du (22)

ierfc(t)= fc erfc(,u)du (23)

iierfc(t)=A ierfc(u)du (24)

ierfc(Q) and iierfc(t) converge veryrapidly towards zero for increasing valuesof t and for our purpose may be takenas zero for t greater than two. Furtherdetails can be found in the paper byHartree'° from which the values in thetable are quoted.

It is seen from equation 21 that thevoltage gradient has its maximum valuefor t=TF. Let a(TF) denote the ratiobetween the actual voltage gradient at theline end at t= TF and the average axialgradient along the coil stack, then

JRCF 1aC(TFx)2TF

co

ierf1 5

The contribution from the infinite serieswill normally be negligible, and we get

Pedersen-Response of Transformer Windings to Surge Voltages

it I if i I. If

I I I I T--- I I

351JUNE 1963

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O i RC. The axial voltage distribution whichforms the boundary condition for thetrain of transients which leads to the finaldistribution is that given by equation19 for t=T. It will be still more uni-form than that for t= TF.The voltage distributions calculated

D _ \ \ \ \ \from the theory outlined here do notdiffer much from those calculated from

1.0 the formulas derived from the high series° -

0.1 capacitance hypothesis, except for waves

\K.01\\ \ \ with a very steep front. As it is knoWn. that the high series capacitance theory

1007% 80 60 40 20 0 normally gives results which are reason-

DISTANCE FROM NEUTRAL ably close to those measured on inter-leaved transformers, this must be equally

4. Voltage distributions in interleaved true for the theory proposed here. Theings at t = TF for various values of high stresses predicted by the present

r=TF/RC theory for very steep wave fronts are re-laxed so rapidly that they may easily beleft unnoticed in recurrent surge oscillo-graph measurements. It is inherent in

following approximate expression for the theory that it will be most difficult tomaximum relative axial voltage gra- observe these increased stresses in wind-t: ings of relatively small geometrical dimen-

sions such as are often used in laboratory8N= C (26) investigations.= 7rTF The equations derived above do not

equation may be written as hold for TF= 0 because the effect of theseries capacitances has been neglected.

F j2T A. _ N C. This is permissible for a finite front time.7rTF 7r (ao np-2 D b For a rectangular wave, however, the

(27) series capacitances will, under quasi-sta-tionary conditions, instantaneously carry

has a striking resemblance to equation an infinite current and the initial electro-iich gives the maximum relative volt- static voltage distribution is given bygradient at the line e-nd as calculated equation 1 in which the series capacitancel rectangular wave from the high series is the same as that for a similar non-icitance hypothesis. For TF= T the interleaved winding. This initial electro-formulas agree within 20%. For an static voltage distribution will, however,rleaved double-section coil 5 feet be smoothed out extremely rapidly by theiameter containing 24 electrical turns, mechanism previously described.

I will be about ().45 ,usec (microseconcQ).For this front time equation 27 gives amaximum axial gradient which is 20%less than that calculated from the highseries capacitance hypothesis. If thefront time is reduced to 0.2 Asec the volt-age gradient will be about 20% higher,and for a 0.1-,usec front about 70% higherthan that calculated from the high seriescapacitance theory. It will, however, beseen from equation 21 that these highstresses will be relaxed extremely rapidly.The voltage distribution at t= TF repre-

sents the maximum axial stresses

e(x, TF)=4EXco

RCTiierfc (2v+1-x)

V=o

iierfc T (2v+1+x)]} (28)

This distribution is shown in Fig. 4 forvarious values of the parameter T= TF/

The Surge Capacitance of anInterleaved Winding

It is a necessary consequence of thehigh series capacitance hypothesis that an

interleaved winding must have a very higheffective surge capacitance. If the seriescapacitance is increased about 100 timesby the effect of interleaving, as claimed byPrice,6 the surge capacitance should beincreased by a factor of about 10. Thecurrent flowing into the winding duringthe interval from t= 0 to t= Tp for theidealized wave of Fig. 3 would, accordingto the high series capacitance theory, beconstant and equal to

EIC=- Cs (29)

TF

where Cs is the surge capacitance of thewinding, which should be approximately

Table 1. Table of the First and the SecondRepeated Integral of the Error Function

Complement

The Functions 2 ierfc (Q) dnd 4 iierfc (t)

t 2ierfc (t) 4iierfc (t)

0.00. 1.1284.... 1.00000.05. 1.0312.... 0.89210.10. 0.9396.... 0.79360.15. 0.8537 . ...0.70400.20. 0.7732.... 0.62270.25. 0.6982.... 0.54910.30. 0.6284.... 0.48280.35. 0.5639.... 0.42330.40. 0.5043.... 0.36990.45. 0.4495.... 0.32230.50. 0.3993.... 0.27990.55. 0.3535.... 0.24230.60. 0.3119.... 0.20900.65. 0.2742.... 0.17980.70. 0.2402.... 0.15410.75. 0.2097.... 0.13160.80. 0.1823.... 0.11200.85. 0.1580.... 0.09500.90. 0.1364.... 0.08030.95. 0.1173.... 0.06771.00. 0.1005.... 0.05681.10. 0.0729.... 0.03961.20. 0.0521.... 0.02721.30. 0.0366.... 0.01841.40. 0.0253.... 0.01221.50. 0.0172.... 0.00801.60.0.0115.... 0.00521.70. 0.0076.... 0.00331.80. 0.0049.... 0.00211.90. 0.0031.... 0.00132.00. 0.0020.... 0.00082.10. 0.0012.... 0.00052.20. 0.0007.... 0.00032.30. 0.0004.... 0.00022.40. 0.0002.... 0.00012.50. 0.0001.... 0.0000

d (30)

It is clear from the mode of operationproposed in this paper that the concept ofa surge capacitance cannot be applied toan interleaved winding except in relationto a true rectangular wave. For such a

wave the winding will initially at t= 0 be-have like a surge capacitance which, how-ever, will be precisely the same as thatfor a similar conventional disk windingwith the same physical dimensions. Thecurrent flowing into an interleaved wind-ing during the initial period is

i(t)1 oe(x, t)

R ax ~,=i(31)

which reaches its maximum value i(TF)at t= TF

i(TF)=- N -C (32)

This current is very much higher than itwould be had the winding not been inter-leaved. It is of the same order ofmagnitude as that predicted from the highseries capacitance hypothesis. This ex-

plains why an interleaved winding super-

ficially will appear to have a relative highsurge capacitance.

Pedersen-Response of Transformer Windings to Surge Voltages

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Conclusions

The high series capacitance theory forthe impulse behavior of interleaved wind-ings is based on the assumption that inter-leaving of turns which are electricallyabout half a coil apart results in a largeincrease in the electrostatic series capaci-tance. This is, however, logically in-compatible with the idea of an initialelectrostatic voltage distribution. Asfar as capacitances are concerned, theeffect of interleaving must be restricted tosuch cases where currents are actuallyflowing through the turns. This is thecase for the transients leading from theinitial to the final distribution, and it ex-

plains the pronounced smoothing effectof interleaving on these transients.The interleaving of the turns results in

the formation of surge impedancesthrough which a high current flows intothe winding during an initial periodthereby charging up the distributed earthcapacitances. This leads to a voltage dis-tribution which is much more uniformthan it would be for a similar noninter-leaved winding. The conclusions whichcan be drawn from this hypothesis differin some important ways from thosederived from the high series capacitancehypothesis. For front times shorterthan the duration of the initial period themaximum axial voltage gradient willincrease inversely proportional to thesquare root of the front time. For ex-

tremely short front times or for steeplychopped waves the axial voltage gradientswill be as high as for a similar conventional

disk-type winding. These high stresseswill, however, be relaxed very rapidly.Considering the risk for failures due con-sideration must, therefore, be given to theshort-time impulse breakdown character-istics of the insulation, and a design basedon the high series capacitance hypothesiswill probably not involve any risks.Another important difference is related

to the maximum interturn voltages.These will, according to the high seriescapacitance theory, be equal to half theinitial voltage across a double section coil.According to the theory proposed in thispaper the interturn voltages will have atendency to oscillate if the rise time of thevoltage across a coil is shorter than theduration of the initial period. This will,ignoring the attenuation, lead to a dou-bling of the interturn voltages. Theseoscillations should be more pronouncedin coils with a large number of electricalturns per disk.

In order to achieve the best possibleresponse to surge voltages an interleavedtransformer winding should be designedso that the initial period T is as small aspossible and the ratio between T and2NZC as large as possible. T and Z aregiven by equations 8 and 9, N is thenumber of double-section coils, and C isthe total capacitance to earth.The theory outlined here is, necessarily,

very approximate, and many of theassumptions made are, rigorously speak-ing, not true. Similar limitations do,however, also hold for the high seriescapacitance theory. It has merely beenthe intention to show that the effect of

interleaving the electrical turns of a trans-former winding on the response of thewinding to surge voltages can be ex-plained without recourse to the conceptof a high series capacitance.

References

1. A NEW TYPE OF TRANSFORMER WINDINGGIVING IMPROVED IMPULSE VOLTAGE DISTRIBU-TION, A. T. Chadwick, J. M. Ferguson, D. H.Ryder, G. F. Stearn. Report no. 107, CIGRE,Paris, France, 1950.

2. DESIGN OF POWER TRANSFORMERS TO WITH-STAND SURGES DUE TO LIGHTNING, WITH SPECIALREFERENCE TO A NEW TYPE OF WINDING, A. T.Chadwick, J. M. Ferguson, D. H. Ryder, G. F.Stearn. Proceedings, Institution of ElectricalEngineers, London, England, vol. 97, pt. II,1950, p. 737.

3. IMPROVED CORE FORM TRANSFORMER WINDING,E. J. Grimmer, W. L. Teague. AIEE Transactions,vol. 70, pt. I, 1951, pp. 962-67.

4. L'AVVOLGIMENTO CON BOBINE A SPIRE INTER-CALATE NEI TRASPORMATORI AD ALTA TENSIONE,M. Ferrari Bardile. Rendiconti della 58a riunioneannuale dell' Associazione Elettrotecnica Italiana,Rome, Italy, 1957, paper no. 519.

5. STOSSSPANNUNGSSICHERE TRANSFORMATORWICK-LUNGEN, H. Brechna. Bulletin Oerlikon, Zurich,Switzerland, no. 328/329, 1958, p. 89.

6. INTERLEAVED TRANSFORMER WINDINGS, J.B. Price. Electrical Review, London, England,vol. 165, 1959, p. 927.

7. IMPULSE STRESSES IN TRANSFORMER WINDINGS,R. A. Zambardino. Electrical Times, London,England, vol. 137, 1960, pp. 3, 81.

8. LA SOLLECITAZIONE A IMPULSO NEI TRAS-FORMATORI CON AVVOLGIMENTI A SPIRE INTER-CALATE, V. Gorio. L'Elettrotecnica, vol. 47, 1960,p. 78.

9. BESTIMMUNG DER STOSSSPANNUNGSVERTEILUNGIN TRANSFORMATOREN MIT DIGITALRECHNER,Bangalore Narayanamurti Jayaram. ETZ-A,Berlin, Germany, vol. 82, 1961, p. 1.

10. SOME PROPERTIES AND APPLICATIONS OF THEREPEATED INTEGRALS OF THE ERROR FUNCTION,D. R. Hartree. Memoirs and Proceedings of theManchester Literary and Philosophical SocietyManchester, England, vol. 80, 1936, p. 85.

DiscussionR. A. Zambardino (English Electric Com-pany Ltd., London, England): The dis-cusser has found the paper presented byMr. Pedersen extremely interesting. Manypoints in his theoretical approach and inhis mathematical treatment of the inter-leaved winding will be a valuable con-tribution to any future complete study ofthe impulse behavior of transformer wind-ings.

However, my company, as originator andmost experienced user of the interleavedwinding, has comments and reservations tomake regarding the premises and conclusionsof this study.

COMPARISON OF THE SURGE-IMPEDANCEAND SERIES CAPACITANCE THEORIES

The first important point is to clarify therelationship between the surge-impedanceapproach presented in this paper in regardto the series capacitance theory.The paper might give the impression,

see the Summary, that the series capacitancetheory is to be criticized and is in conflict

with the new method discussed therein.It should, in fact, be noted that the value

of capacitance assumed by the author forhis method of calculation is absolutelyidentical to that of the series capacitancetheory (see equation 4) which gives thecapacitance per section in his theory, andthis is exactly 2N times the total capaci-tance in the series capacitance theory asgiven in equation 2.The author is simply noting that this

capacitance, which value he fully accepts,cannot be charged at once but has a finitecharging time, determined mainly by itscharging inductance, for which formulas aregiven (see equations 6 and 9). This is, ofcourse, true, not only for the interleavedwinding but for any winding and, indeed,for any capacitance whatsoever.Once the problem is seen in this light the

following consequences will logically follow:

1. This series capacitance theory is per-fectly valid whenever the charging time issmall compared with the front time of theapplied wave. This, we can assume, isagreed to by the author considering hisstatements in the section entitled "The

Initial Voltage Distribution in InterleavedWindings," and in the "Conclusions." Thestriking resemblance of the gradient's equa-tions 7 and 3 is in fact perfectly natural,since both are derived from identicalvalues of capacitance.2. The new method presented is not analtemative to the series capacitance theory,but an attempt to extend its applicationto applied waves with front times which areshort in comparison with the charging timeof the series capacitances.3. The basic approach presented in thispaper is of more general application thanto the interleaved windings only. Theprinciples expounded are relevant to thetheoretical study of transformer windings ingeneral.4. The introduction of the charging in-ductance associated with the series capaci-tance of the interleaved winding emphasizeswhat is in fact a major advantage of thesewindings, namely, their extremely shortcharging time. The fact that the charginginductance is that of the two strands ofinterleaved conductors makes this parametervery small, and ensures that the charging

Pedersen-Response of Transformer Windings to Surge Voltages 353JUNE l1963

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'\ ct /Ct/(- 2) -

\: c

Ha |:l q N

l~~~

Fig. 5. Equivalent circuit

Fig. 6. Current path and equivalent circuitof an interleaved winding for hypotheticalwaves with fronts of a few millimicroseconds

APPLIED VOLTAGE

FULLWAVE

CHOPPEDWAVE

time of an interleaved coil is a fraction of 1,usec.

5. In all practical cases the charging time ofinterleaved coils will be well below the fronttime of the fastest surge wave which couldstrike a transformer in service or test condi-tions. This is self-evident for an appliedfull wave, its standard front of 1 or 1.5 ,seebeing much larger than the largest chargingtime that can conceivably be obtained in aninterleaved coil. For sharply choppedwaves (or front of wave) the following pointsshould be considered:

(a) In the paper, surges are considered asimpinging directly on the winding. Ofcourse this is possible only in laboratoryconditions. In practice surges will reachthe winding only after traveling alongsome length of line, busbar, bushing, andleads. Even in the extreme case of a wavechopped by a rod gap mounted on thebushing the surge corresponding to theactual chopping will have to travel alongthe inductance represented by the bushingand leads before reaching the winding.Even assuming as fully correct the wind-ing's equivalent circuit shown in Fig. 2 ofthe paper, this additional inductancewould slow down the front time, i.e.,chopping duration appearing at windingline end to values comparable with thecharging time of the interleaved coil. Infact, for normal designs even this veryworst case would give effective choppingdurations larger than the charging time ofthe coil.

(b) On an interleaved winding the maxi-mum stresses during a chopped wave are

Fig. 7. Equivalentcircuit of two inter-

leaved sections

VOLTAGE ACROSS LINE END COIL

produced by the front of the wave and notduring the chopping. The gradients pro-duced by the chopping subtract fromthose already established by the front ofthe incoming wave (see Fig. 5). Even iffor some special transformer the choppingwere so sharp as to be faster than thecharging time of the coil, this would atleast need to double the gradient cal-culated with the series capacitance theorybefore the over-all actual stress, i.e., thedifference between the gradient due to thefront and that due to the chopping, be-came greater than that produced by thefront of the wave. An increase by thisamount is out of the question, even onthe basis of the theory produced in thispaper (see equation 27 giving an increasein stress of 70% for a 0.1-,usec front).Therefore, the maximum stress duringchopped waves is produced by the front,i.e., before chopping, for which the seriescapacitance theory is fully valid.

BEHAVIOR OF INTERLEAVED COILSUNDER HYPOTHETICAL SURGES HAVINGEXTREMELY SHORT FRONTS, OF THEORDER OF A FEW MILLIMICROSECONDS

We have already discussed the principleson which is based the theory developed inthe paper. This has shown that they areapplicable to surges having a front muchfaster than could be produced in practiceunder test or service conditions. Suchhypothetical conditions are, however, oftheoretical interest and we would now liketo make some comments on the method ofcalculation developed in the paper. Wefoundr ather baffling the statement that"for extremely short front time, . ., theaxial voltage gradients will be as high as fora conventional disk-type winding" (see"Conclusions"). Since the author hasnoted that it takes only a few millimicro-seconds for an applied wave to travel alongthe interleaving joints at the outside of thecoil stack, the statement quoted is obviouslyuntenable, even for completely imaginarywave with fronts of a few millimicroseconds.For such rather metaphysical waves thecapacitive circuit of the interleavedwould be as in Fig. 6. The series capacitancewould be that given by the first pair of inter-leaved turns (for p= 1) and the groundcapacitance C, being generally at the insideof the coil, would be reduced by being inseries with the remaining turn capacitancesin series with each other. The same resultcould be obtained in a much more compli-cated way, using the surge-impedancetheory, but taking into account the fullequivalent circuit of the coil as shown inFig. 7. The fact that the surge may nothave traveled round a full turn is im-material, since this would vary by the sameamount both series and ground capacitance.If the original a (calculated with the fullseries capacitance of the coil) is between 1and 3 the aX calculated with the circuit ofFig. 6 would be approximately between 3and 6, respectively.The voltage distribution given by the ax

is the worst which could be produced byimaginary fronts of a few millimicro.-seconds.

Formulas derived from the surge imped-ance theory should give this distribution as alimit for Tf tending to zero (or, to be moreprecise, Tf tending to 1 or 2 millimicro-

Pedersen-Response of Transformer Windings to Surge Voltages

Ir

Fig. 8. Applied voltage and voltage across an interleaved line end coil

C--

JUNE 1963354

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seconds). The voltage distributions derivedin the paper (shown in Fig. 4) go well belowthe above distribution and, therefore, theformulas given in the paper must be con-sidered as unduly pessimistic, even for thehypothetical range of front times for whichthey could apply. The circuit shown inFig. 2 of the paper should be modified sothat for Tf tending to practically zero (a fewmillimicroseconds) it should reduce to a cir-cuit equivalent to that shown in our Fig. 6.If this is done the method of calculationbased on surge impedances should give, evenfor the hypothetical conditions where itapplies, results considerably nearer to thosederived from the series capacitance theory.The approach to the calculation of surge

voltage distribution, presented by Mr.Pedersen, is of great theoretical interest andits principles could usefully be extended tothe study of transformer windings of anytype.

Its application to interleaved windings islimited by the fact that the range of appliedwave front times to which it would apply isin practice below the range of front timeswhich occur in service or test conditions.For the latter front times the series capaci-tance theory hitherto used is fully valid.

For the hypothetical very-steep-frontedwaves to which the theory would apply, theresults derived in the paper are undulypessimistic, as the equivalent circuit usedneglects elements which play an importantpart. If it were objected that, strictlyspeaking, a Tf of 1 millimicrosecond isstill not zero, it should be noted that, apartfrom the physical impossibility of getting azero rise time, all the impulse calculationtheories would break down for such condi-tions and all windings, however built, wouldlook the same. If the rise time is zero thewave reaches the winding before any currenthas flown through it. The field distributionat t = 0 is identical with that which would beobtained if a horizontal electrode (the bush-ing lead) was brought up to but not con-nected to the winding. The field conditionswould be 3-dimensional with a concen-trated field radiating from the point wherethe bushing lead enters the winding. Theactual characteristics of any winding wouldbe immaterial in such conditions, which- areentirely a figment of the imagination.

G. M. Stein and J. M. McWhirter (West-inghouse Electric Corporation, Sharon, Pa):The author's contention that the interleavedwinding has not a higher series capacitancethan the ordinary disk winding cannot beaccepted in a practical sense for the follow-ing reasons:The author postulates that, for establish-

ing a pure electrostatic voltage distribution,no current is allowed to flow through in-ductances, that is, in this case in the conduc-tors of a transformer winding. Since thecross section of the wires supplying voltageto each end of the coil is usually very smalland their distance very large, as comparedwith the spacing and size of the capacitorsurfaces formed by the conductors, the seriescapacitances accounting for that pure elec-trostatic voltage distribution are practicallyzero.

In order to create the field and to justifythe use of the series capacitances appearingin conventional calculations of the initial,

that is, short-time, surge distribution in anordinary disk winding, current must pass atleast through the turn adjacent to eachline terminal and through any static platesused, that is, current must flow in conductorswith inductances. After charging the staticplate and the first conductor, this currentseems to prefer the shorter and less inductivepath through the dielectric between turnsand between coils to the high inductivepath through the other conductors, becausesuch a flow places the capacitances betweenall conductors in series and surge calcula-tions based on this condition are borne outby test results.

In an interleaved winding, however, dis-tant conductors are connected together sothat the current, after passing through thefirst turn seems to prefer the path throughthe other conductors to the path through thedielectrics between them since this flowcharges the capacitances between adjacentconductors of an interleaved group in paral-lel, and calculations of the initial surge dis-tribution made under this condition agreewith test results. It is this paralleling effectwhich permits the author to treat the wind-ing like a transmission line and thus, to intro-duce the time element into the analysis, be-cause the capacitances of such a line appearparallel connected in its wave resistance. Ifone quarter period T of the oscillation oc-curring during the charging period becomeslonger than the time TF to maximum peakin the applied voltage wave, he finds theparalleling effect on the series capacitancesreduced to the extent that these capacitancestend to be the same in ordinary and inter-leaved disk windings for sufficiently shortrise times TF.To summarize, a conductor current has to

flow only in the turn adjacent to each lineterminal in order to explain the series capaci-tance in ordinary disk windings while cur-rent has to pass through the whole windingfor obtaining the series capacitance corre-sponding to surge phenomena observed whenthe coils are interleaved. Consequently,inductive currents have to be present inboth cases which are distinguished from eachother only by the extent of this currentflow.By introducing this inductive effect, and

thus, the time element, into the analysisof interleaved disk coils, the present paperbecomes a major contribution to our knowl-edge of the functioning of this type of wind-ing.

A. Pedersen: I am very grateful to Mr.Zambardino, and to Mr. Stein and Mr.McWhirter, for their discussions whichclarify some problems which I have dealtwith too superficially.The concepts of series and earth capaci-

tance originate from studies of the impulseresponse of a -uniformly wound single-layerhelical coil for which these capacitances havea simple geometrical interpretation. In con-ventional calculations of voltage distribu-tions in actual transformer windings theformulas derived for the simple helical coilare used without modifications. To do this,equivalent series and earth capacitancesmust be introduced. In the publishedversions of the higher series capacitancetheory the equivalent series capacitance isfound by considering the electrostatic energyin the interturn insulation. To find this

energy a linear voltage distribution along theelectrical turns is assumed (a distributionwhich, by the way, cannot be achieved elec-trostatically). In this way equation 2 ofthe paper is derived. Mr. Zambardinopoints out that this is the same capacitanceper coil as that involved in the surge capaci-tances. This is, however, entirely acci-dental because equation 2 depends' on anarbitrarily chosen voltage distribution withina coil.

I disagree with Mr. Zambardino's state-ment that "the series capacitance theory isperfectly valid whenever the charging timeis small compared with the front time of theapplied wave." In this case the mutual andself-inductances must be taken into account,and the distribution is thus not even approxi-mately electrostatic, and the use of equa-tion 1 is, consequently, meaningless. Forsuch slow wave fronts interleaving will, asstated in the paper, greatly influence thecapacitance of the winding. This is, how-ever, inherently associated with currentsflowing through the main inductances.The difference between. the ordinary diskwinding and the interleaved winding undersuch conditions will be as described by Mr.Stein and Mr. McWhirter. But the highcapacitance thus achieved by interleaving isnot the electrostatic series capacitance in-volved in equation l. And the fact that thisequation, when applied to cases for which itdoes not hold, appears to give reasonableresults if one of its fundamental parametersis replaced by a quantity which is derivedfrom an entirely different situation as thatdescribed by equation 1, can only emphasizethat the high series capacitance theory isbased on assumptions which are funda-mentally wrong.

I do, of course, agree with Mr. Zambardinothat a strictly rectangular wave is a hypo-thetical condition, but I think that he isunduly optimistic in asserting that the finitecharging time of the transmission linesformed by the coils will, in all practical cases,be small compared with the front timeswhich could occur in service or test condi-tions. With data quoted from reference 1,the transit time is half a Aisec, which cer-tainly is not small compared to what couldreasonably be expected during serviceconditions.

Concerning the distribution for very steepfronted waves, Mr. Zambardino views theproblem qualitatively in the same way as Ido (see section B in Fig. 6). However,when it comes to the quantitative estimationof the distribution, Mr. Zambardino iswrong in assuming that the equivalent cir-cuit is that shown in Fig. 6 for the fol-lowing reason. The current flowing intothe "series capacitance" formed by the firstpair of interleaved turns will be strictly pro-portional to the voltage applied across thispair of turns (it is a current through asurge impedance) and not to the derivativeof this voltage with respect to the time, i.e.,the differential equation for the distributionis equation 10 of the paper and not

d2e(x)-a2e(x)dx2

as assumed by Mr. Zambardino. For a fronttime of 10 to 15 millimicroseconds and atransit time of 0.5 ,usec, equation 10 will givea maximum axial stress which is approxi-mately 4 to 5 times the stress predicted from

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the conventional high series capacitancetheory; see equation 27. The apparentvalue of a would thus approach 15 and not 6as stated by Mr. Zambardino, i.e., the dis-tribution would approach the pure electro-static distribution for a rectangular wave.I cannot, therefore, accept Mr. Zambardino's

statement that the results derived in thepaper are unduly pessimistic. I would,however, like to emphasize that both thetheory in the paper and the high seriescapacitance theory imply quasi-stationaryconditions, and this will not be the case forextremely short front times. In such cases

the circuit theoretical approach shouldbe replaced by field theoretical considera-tions.

In conclusion I wish to thank Mr.Zambardino and Mr. Stein and Mr. Mc-Whirter for their valuable and stimulatingdiscussions of my paper.

Compressed Gas Insulation in the

Million-Volt Range: A Comparison

or SF6 with N2 and CO2

S. F. PHILPSENIOR MEMBER IEEE

Summary: Maximum voltage which canbe insulated between a sphere and a planehas been measured as a function of gaspressure and gap. It is found to be ap-proximately three times higher in SF6than in N2+ C02, up to pressures of roughly9 atm (atmosphere). For higher pressuresthe relative superiority of SF6 over N2+ C02diminishes. Gradients of more than 100Mv (million volts) per m (meter) were insu-lated on a 19-mm (millimeter)-diameter elec-trode in 20 atm of SF6.

THE maximum voltage which can beI insulated between a sphere and a plane

has been measured as a function of gaspressure and gap. The objective hasbeen particularly to compare the insulat-ing ability of sulfur hexafluoride withthat of a mixture of equal parts of N2and C02, a commonly used insulating gasmixture. It is found that the maximumvoltage is approximately three timeshigher in SF6 than in N2+CO2, up topressures of roughly 9 atm. For higherpressures the relative superiority of SF6over N2+CO2 diminishes. Gradients ofmore than 100 Mv per m were insulatedon - 19-mm-diameter electrode in 20atm of SF6.

Gases whose molecular structure in-

Paper 63-27, recommended by the AIEE ElectricalInsulation Committee and approved by the AIEETechnical Operations Department for presentationat the IEEE Winter General Meeting, New York,N. Y., January 27-February 1, 1963. Manuscriptsubmitted May 1, 1961; made available forprinting October 23, 1962.

S. F. PHILP is with the Massachusetts Institute ofTechnology, Cambridge, Mass.

The author wishes to express his appreciation toProfessor John G. Trump for his help and guidance.It is a pleasure to acknowledge also the support ofthe National Science Foundation.

volves a large number of atoms oftenhave good insulating properties.'-4 Inaddition, many of the heavy gases willattach electrons to form negative ions, aprocess which augments their insulatingproperties.5'7'8To be practically useful for insulating

purposes, these heavy gases should:1. Be chemically inert, at least to the

extent that they will not attack plastics,metals, and other substances which arecommonly used in high-voltage equip-ment.

2. Be readily available and not tooexpensive.

3. For very-high-voltage applicationsit is sometimes desirable that the gashave sufficiently high vapor pressure to beused at pressures of 10 atm or more atordinary temperatures.

Sulfur hexafluoride (SF6) is an out-standing example of a gas which fulfillsthese requirements and has particularlygood insulating properties. SF6 is veryinert and satisfies the first requirement;however, it should be noted that the mo-lecular fractionsformed during the passageof an electric discharge are corrosive andhighly toxic.9 The high vapor pressureof SF6 (24 atm at standard temperature)makes it of greater interest for insulationof voltages in the million-volt range thangases such as carbon tetrachloride orfreon whose vapor pressures are 0.13atm and 5 atm, respectively. However, amixture of these heavy gases with a lightergas such as nitrogen or hydrogen results inan increase of insulating strength for thesame total gas pressure, particularly forsmall amounts of added carbon tetra-chloride or freon.3'4'10'11'24

The measurements reported here areconcerned only with a comparison be-tween pure SF6 and a gas consisting ofequal parts of N2 and CO2. This mixturewas chosen for comparison because it iscommonly used in insulation, havingabout the same insulating properties ascompressed air without the concomitantcombustion hazard.

ApparatusThe voltage source used in these experi-

ments is a Van de Graaff electrostaticgenerator. This machine, identical toone described in detail elsewhere,12 hasan insulating column 24 inches long and15 inches in diameter. The high-voltageterminal is a polished stainless-steel spin-ning also 15 inches in diameter and hasthe form of a cylinder surmounted by ahemisphere. The generator is enclosedin a cylindrical pressure vessel of 30-inchdiameter.The test gas was in all cases technical

grade gas.The experiments were performed within

the pressure vessel of the generator. Thetest electrodes, polished steel ball bear-ings, were introduced through a port inthe vessel and supported on metal rodswhich could be moved from outside. Thegeometry is approximately that of asphere facing an infinite plane, since thehigh-voltage terminal is large comparedto either the gap or the spherical electrode.A generating voltmeter is used to

measure the terminal potential. Thisvoltmeter is in principle a linear device,and it is found in practice to be veryaccurately linear except for a small regionnear zero voltage.13 The voltmeter iscalibrated for terminal potentials up to80 kv by introducing a potential, meas-ured with a laboratory standard high-voltage resistor, from an external d-cpower supply. In addition, a calibrationpoint in the Mv range is obtained by usingthe bremsstrahlung produced by electronsaccelerated in the tube of the Van deGraaff generator, in an experimentaldetermination of the threshold for agamma-ray-induced nuclear reaction. 14This calibration was completed before

Philp-Compressed Gas Insulation in the Million- Volt Range35-6 JUNE 1963

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