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Journal of Electrostatics 71 (2013) 533e539

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Arrangements of transformer winding with respect to impulse stress

Jan Mike�s a,*, Dalibor Koke�s b

aCzech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Dejvice, Czech RepublicbCzech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, B�rehová 7, 115 19 Prague 1, Czech Republic

a r t i c l e i n f o

Article history:Received 31 August 2012Received in revised form10 December 2012Accepted 10 December 2012Available online 21 December 2012

Keywords:High voltage transformersPower transformer protectionTransformer windingsAtmospheric dischargeVoltage distributionLongitudinal capacitance of transformerwindingLateral capacitance of transformer winding

* Corresponding author.E-mail addresses:mikes.jan@fel.cvut.cz (J. Mike�s), ko

0304-3886/$ e see front matter � 2012 Elsevier B.V.http://dx.doi.org/10.1016/j.elstat.2012.12.015

a b s t r a c t

Danger and stress posed to transformer winding through overvoltage still represent a hotly debated andhitherto unresolved technical issue the designers and operators of high-performance equipment havebeen grappling with. The actual impossibility of accepting all the real parameters of a transformer in itssubstitute model leaves considerable space for its constant improvement and modifications. Acceptingthe surrounding phenomena and properties of the transformers gives rise to complex situations anddifficulties in the process of solving the model. Models tackling some of the issues pertaining to circuitmodels or electromagnetic field models have been developed on a long-term basis. Another issue in handis the very complexity of the process of solving a model. This study introduces a model accepting sol-ely the capacitance influences of transformer components, using the methods derived from the theory ofLaxeWendroff’s and LaxeFriedrich’s approximation of differential equations of the hyperbolic type forthe solution of the respective equations. It does not represent solutions for all the parameters ofa transformer, but provides an overview of the size of the initial impulse stress of transformer winding,doing so with adequate accuracy.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Dynamic interactions in transformer winding follow either thedistribution of the electric field and overvoltage phenomena in thewinding, at the entry of the surge, or the distribution of the powerfield and the mechanical behaviour of transformer winding duringvarious types of short-circuit. The first type is designated as fast, theother one as slow. The study deals with the first type of interactions.Examination of overvoltage relations in the transformer windinghas been the subject of innumerable studies. Modern findings inthe field of mathematical analysis and numerical mathematics havefacilitated specification of the physical model under scrutiny.

The problems concerning a substitution transformer diagramcover its discretemodel analogous, incircuitmodels, to electricwiringwhere longitudinal capacitance, eventually resistivity of theconductor used (that is, however, frequently ignored) operatesbetween the turns. This fundamentalmodelwas published in 1915 byK.W.Wagner and all the subsequent theories proceed therefrom [1].

The first works stemmed from the model of a single-layer coilwithout iron, whichmade it possible to perform certain predictions

kesdal@fjfi.cvut.cz (D. Koke�s).

All rights reserved.

analytically, further studies stemmed from the gradually morecomplex physical models, with numerical methods being incor-porated into their behaviour step by step. In ref. [2] authorsexplained that the role played by the iron core in the response of animpulse-stressed winding is negligible. However, even modernstudies proceed from a relatively heavily simplified configuration ofthe physical model.

In methodological terms, two approaches may be distinguishedin the physicalemathematical description of the overvoltagephenomena. The first consists in the construction of the so calledfield model, i.e. in the formulation of the electromagnetic field inthe sphere of winding as a marginal assignment for the partialdifferential equations of the type of wave equations in which thevector magnetic potential, less often the vector electric potential,figures most frequently as the unknown quantity. In numericalterms, the solution of this 3D, eventually 2D, assignment can bemanaged quite satisfactorily by applying a suitable commercialprogram (evidently based on FEM e Finite Element Method), but itis not easy to determine the appropriate boundary conditions. Theother concept is based on the layout of the so called circuit model,i.e. formulation of a system of ordinary differential equations fora locally discretized circuit comprising elements R, L, C for thenumerical solution of which one can use some standard numericalmethod. This type of solution poses the problem of precisely

J. Mike�s, D. Koke�s / Journal of Electrostatics 71 (2013) 533e539534

determining the parameter R, L, C, and also the process of setting upa 3D circuit model runs up against certain difficulties. However, inboth types the numerical solution itself is accompanied by thenecessity of coping with the reliability of the obtained results(especially stability, convergence etc.).

2. Justification of the choice of the problems under scrutinyand current status of solution

Considerable attention has been devoted to impulse stress ofelectric machines on a long-term basis. More precise, morecomputationally complex as well as less challenging and also lessaccurate mathematical models have been developed, based on theavailable mathematical theories (since the early 20th century).Owing to what is a frequently inaccurate and less credible way ofstipulating the input parameters of the winding in the model therehave emerged diverse simplified notions of voltage distribution ina transformer. Quite frequently, this issue is resolved on the basis ofsingle-layer coil. The resulting solutions are then extended to coverthe entire winding as well as multiple-winding transformers. Interms of voltage distribution, transformer’s multiple-layer coil ewithout adequately thorough knowledge of its parameters e

constitutes such a complex task that very small congruence may beanticipated if such parameters are neglected. Former Czechoslo-vakia (from the 1950s until 1989) figured among the leadingcountries generating valuable mathematical models of single- andmultiple-layer coils e see works by A. Veverka, B. Heller, �S. Mat�ena.Since these studies could not be published abroad before 1989 theynow represent a valuable biography for the topic that has not yetbeen reflected abroad. The most important specialist studies on thesubject are as follows [3e5]: (in Czech). In addition, a large numberof contributions was published in the journal Elektrotechnickýobzor (active in 1910e1989) by A. Veverka and B. Heller. There aredetails that were not possible to publish in ref. [3].

However, both academic, business and manufacturing centreshave been returning to the subject of impulse stress in the pastdecade. For their part, themajormanufacturers of transformers andreactance coils strive to have the possibility of predicting compu-tations through which they could declare e prior to designinga transformer e its resistance to impulse stress (most frequently animpulse of 1.2/50 ms). Up to now resistance has been verified solelyby recording oscilloscopic response from the taps of the trans-former winding when leading the impulse in to its input clamp.This gives rise to temporalespatial behaviour of the voltagewave inwhich the spot of maximum coil stress is monitored. In view of thelimitations of the possibility of calculating all the parameters of thewinding, of reckoning precisely with own andmutual inductance inthe model and of respecting the impact of the iron core, we havedecided to verify the validity of the numerical solution solely whilerespecting the capacitance reserve model. With this particularmodel we want to single out the necessity of devoting great carewhen making conclusions from incomplete or very little validcircuits which represent only limited properties transferred to theentire winding, eventually to multiple-winding systems e three-phase transformer, while respecting the impact of all the inter-links involved.

As for the entry of voltage impulse to the transformer winding,we can state that voltage distribution along the winding isdependent solely on the winding’s capacitance conditions sinceinductance in a time interval close to zero may be neglected. Thedesign parameters of the coils of the transformer itself havea highly decisive influence on the initial voltage distribution.Assuming that we know initial and terminal voltage distribution onthe transformer winding, it is quite easy to determine the freeoscillations envelope, which represents a theoretical maximum

voltage of the insulation in any point of the winding. Much smallerattention has been given to the calculations of the parameters ofthe longitudinal and earth capacitance themselves, and own andmutual inductance of the coils of transformers than, for instance, tothe theories of calculation themselves, and to the proposednumerical solution of wave phenomena in the winding. Sucha model yields more objective results than endeavours to capturesimultaneously all the influences within a transformer. Among thekey studies for the computation of C, K, L andMwe canmention, forinstance, the following refs. [6e12].

It is possible to trace in the literature two approaches to thestudied issues of impulse voltage distribution in a transformer. Oneof them is the design of a model with concentric parameters; theother consists in observing the significance of distributed param-eters. Authors in ref. [13] tend to distinguish models into Fasttransient overvoltages (FTO) and Very fast transient overvoltage(VFTO). As for the FTO models, in which a frequency range from10 kHz � f � 1 MHz is assumed, many models have already beenpublished, based on the theory of quadrupole [14,15] which arecascaded and calculated by means of the respective computationalinstruments. Subsequently, most [16,17] of other models are basedon the solution of the telegraph equation, solved either analyticallyor numerically. For the VFTO models, hence models witha frequency over 1 MHz, it is no longer possible to neglect thewavelength of the input high-frequency impulse, and the circuitsare resolved by means of distributed parameters e the mostfrequently used methods are then the hybrid calculation methodswhere parts of the winding are calculated as concentric e forinstance for lower frequencies where the influence of conductivityprevails, and as distributed ones for moments when capacity is ofexplicit importance within a circuit. The models with concentricparameters reduce the calculation solely to predetermined pointsin the winding; it is impossible to monitor voltage behaviour in anyrandom spot of the winding.

The types of the used windings themselves, too, have consid-erable impact on the design of the relevant models e the mostfrequently used is the simplified method via simple single-layercoils, but the transformer winding tends to be much morecomplex; multiple-layer disk and cylindrical windings are oftenused, or special adjustments, for instance interlaced windings, areutilized. Relevant studies dealing with these subjects may be foundin: refs. [18,19]. The last approach to modelling resistant trans-formers is the application of various limiting elements, overvoltagearresters or the use of an older method involving voltage-dependent varnishes [20,21].

Transition from the transformer’s real winding (single-layer) toits discrete model is described in Fig. 1.The classical theory oftransformermodel for the effects of overvoltage has been discussedin great detail, for instance, in refs. [3,4,22]. For the purpose of ourstudy, we proceed from a simplified solution of the equationdescribing the overall diagram given in Fig. 2. This set of equationsoriginated as a result of solving the Kirchhoff laws in the describedFig. 2.

v2uvx2

þ LKv4u

vx2vt2� LC

v2uvt2

¼ 0 (1)

vivx

¼ �Cvuvt

(2)

i ¼ �Kv2uvxvt

(3)

In case of initial voltage distribution at voltage impulse, a single-layer transformer winding may be approximated by means of

Fig. 1. Capacitance location in transformer winding.

J. Mike�s, D. Koke�s / Journal of Electrostatics 71 (2013) 533e539 535

a substitute capacitance diagram, given specifically in Fig. 2.Inductivities from Wagner’s diagram have no impact on eitherinitial or terminal voltage distribution. See refs. [3,4,22]. Under theassumption of knowing initial and terminal voltage distribution onthe transformer winding it is easy to establish free oscillationsenvelope, which constitutes a theoretical maximum voltage stressof the insulation at any random point of the winding. This proce-dure is suitable for simple verification of thewinding’s resistance toimpulse stress. When accepting only capacitance links, as assumedby Fig. 2, the problems are simplified into two hyperbolic differ-ential equations [5,23,24]. Application in the Matlab programmingenvironment thus offers a tentative idea for checking the correct-ness of any designed winding.

v2uvxvt

�����l

k

¼vuvt

����l

kþ1� vu

vt

����l

k�12Dx

z

12Dt

�ukþ1;lþ1 � ukþ1;l�1

� � 12Dt

�uk�1;jþ1 � uk�1;l�1

�2Dx

z1

4DxDt�ukþ1;lþ1 � ukþ1;l�1 � uk�1;lþ1 þ uk�1;l�1

�

(7)

3. Simulations and results

The equations (2) and (3) can be solved analytically, but underthe assumption that a nominal impulse is formed by input voltage.Solution in an analytical manner is difficult to attain for other inputimpulses. That is why methods of numerical calculation respecting,for instance, impulse in the form of 1,2/50 ms have been sought.Analytical solution is derived in great detail in ref. [3].

To solve the set of equations (2) and (3) we will createa temporalespatial network with spatial step Dx and temporal stepDt by means of the method of finite differences. There are severalnumerical methods to calculate partial differential equations of thehyperbolic type, a case in point may be provided by the FTCS, LaxeFriedrichs, LaxeWendroff and the more complex methodUpwinding, which, however, best succeeds in responding toa dramatic change in the function gradient. To calculate thefollowing set of equations we have employed the LaxeWendroff

method or rather the implicit Wendroff differential approxima-tion set up according to refs. [23,24].

Wewill intersperse the continuous definition areaU of the set ofequations (2) and (3) evenly with a temporalespatial network withsteps Dx and Dt so that we will discretize a winding d long with anequivalent step Dx, obtaining a one-dimensional geometricnetwork containing N elements, delineated by N þ 1 nodes whereN ¼ d/Dx.

We discretize the semi-delineated time coordinate t withequivalent step Dt, obtaining a set of discrete time levels t1 ¼ lDt,where l ¼ 0, 1, 2, ... All the spatial derivations contained in theequations (2) and (3) will be replaced with central differences inthe following forms.

vivx

����l

k¼ 1

2Dx�ikþ1;lþ1 � ik;lþ1 þ ikþ1;l � ik;l

�(4)

vuvt

����l

k¼ 1

2Dt�uk;lþ1 � uk;l þ ukþ1;lþ1 � ukþ1;l

�(5)

Equation (3) is a mixed partial derivation. For its approximationwe have used the following procedure according to the relations (4)and (5), mentioned, for instance, in refs. [24,25].

v2uvxvt

¼ v

vx

�vuvt

�¼ v

vt

�vuvx

�(6)

vuvt

����l

kþ1¼ 1

2Dt�ukþ1;lþ1 � ukþ1;l�1

�(6a)

vuvt

����l

k�1¼ 1

2Dt

�uk�1;jþ1 � uk�1;l�1

�(6b)

After substituting (4), (5), (6) and (7) into the equations (2) and(3), we obtain the following relations (8) and (9), and adjust thegiven expressions into the form (10) and (11).

vuvt

¼ 1Cvivxz

12Dt

�uk;lþ1 � uk;l þ ukþ1;lþ1 � ukþ1;l

�

¼ 1C

12Dx

�ikþ1;lþ1 � ik;lþ1 þ ikþ1;l � ik;l

�(8)

iK

¼ � v2uvxvt

z14K

�ik;l þ ikþ1;l þ ik;lþ1 þ ikþ1;lþ1

�

¼ � 14DtDx

�ukþ1;lþ1 � ukþ1;l�1 � uk�1;lþ1 þ uk�1;l�1

�(9)

Fig. 2. Initial substitute model for numerical solution.

J. Mike�s, D. Koke�s / Journal of Electrostatics 71 (2013) 533e539536

12Dt

�uk;lþ1�uk;l þukþ1;lþ1�ukþ1;l

�

�1C

12Dx

�ikþ1;lþ1� ik;lþ1þ ikþ1;l � ik;l

�

¼ 12Dt

�uk;lþ1þukþ1;lþ1

� �1C

12Dx

�ikþ1;lþ1� ik;lþ1

�

¼ 12Dt

�uk;lþukþ1;l

� þ1C

12Dx

�ikþ1;l� ik;l

�

(10)

Fig. 3. Obtained behaviour of voltage distribution along the axis of winding for a change inK ¼ 20 � 10�12 and C ¼ 17 � 10�11 c) K ¼ 20 � 10�12 and C ¼ 17 � 10�12 d) K ¼ 20 � 10

14K

�ik;l þ ikþ1;l þik;lþ1 þ ikþ1;lþ1

�

¼ � 14DtDx

�ukþ1;lþ1 � ukþ1;l�1 �uk�1;lþ1 þuk�1;l�1

�

¼ 1DtDx

�ukþ1;lþ1

� þ 1K�ik;lþ1 þ ikþ1;lþ1

�

¼ 1DtDx

�ukþ1;l

� � 1K�ik;l þ ikþ1;l

�(11)

Sign change corresponds recurrent LaxeWendroff trans-formation and underlined elements correspond:

Boundary value conditions

t ¼ 0; x ¼ 0 : Fðu; i; tÞ ¼ 0; x ¼ d : Fðu; i; tÞ ¼ 0

Initial value conditions

t ¼ 0; 0 � x � d : iðx; 0Þ ¼ 0; uðx;0Þ ¼ 0

It will be written as equations (10) and (11) for all k elements ofthe differential network (k ¼ 1, ., N), obtaining a set of equationswith 2.(N) unknowns. We will supplement these equations withalgebraic approximation of equations proceeding from marginalconditions. We will obtain a set of 2.(N þ 1) linear algebraicequation (12) [24,25].

K (F/m) and C (F m) e values C and K in order a) K ¼ 20 � 10�12 and C ¼ 17 � 10�10 b)�11 and C ¼ 17 � 10�12.

J. Mike�s, D. Koke�s / Journal of Electrostatics 71 (2013) 533e539 537

AXlþ1 ¼ BXl (12)

Fig. 4. Behaviour of voltage distribution when using a shield at a) 30%, b) 60%, c) 90% ofthe winding’s axial length for K ¼ 20 � 10�12 and C ¼ 17 � 10�10.

where vector Xlþ1 contains elements uk,lþ1, ik,lþ1 and vector Xl

contains elements uk,l, ik,l pro k ¼ Nþ1. By solving equation (12) insoftware Matlab we will obtain voltage values and flow of timelevel on the basis of known quantities from the previous time level.Calculation commences from the time level for t ¼ 0, i.e. at themoment of the entry of voltage impulse on the winding. Matrix Aand B without boundary conditions were generated from Matlabsoftware.

The charts in Fig. 3 contain initial voltage distribution (in red),terminal voltage distribution (in green) and approximate freeoscillations envelopes (in blue). An earthed end of the winding isinvolved in case of selected following courses.

The values C and K given in the description of Fig. 3 weregradually inserted, and, using a program written in the Matlabcode, we have obtained the relevant graphic outputs. In all thecases, the normalized impulse 1.2/50 ms, modelled by means ofexponential curves, was used as an input signal. In our instance, wemodelled the impulse according to the exponential relationconfirmed by the standards:

uðtÞ ¼ 1;03$U0$�e�a$i$dt � e�b$i$dt

�; where constants a

¼ 14400 and b ¼ 3500000 (13)

similarly it is possible to model impulses in the form: a) for theleading edge, b) for the trailing edge

a�uðtÞ ¼ Umaxð1� cos2;6$tÞ

2for 0 � t � tforehead (14)

and

b�uðtÞ ¼ Umax$e½�0;014$ðt�tforeheadÞ� (15)

It is evident in Fig. 3 that if the value of longitudinal capacitanceK is lower than the value of the earth capacitance C by less than anorder, a virtually problem-free initial voltage distribution occurs,and thus the shape of the free oscillation envelopes will be highlyfavourable. The model was constructed for depicting initial andterminal voltage distribution for homogenous winding at the entryof nominal voltage impulse amounting to 500 V on the windingwith axial length of l ¼ 1.5 m. The program serves for tentativeverification of maximum possible overvoltage generated within thewinding. This is highly demanded for practical purposes, eventhough it is only an approximate method. Parameters of longitu-dinal and earth capacitance may be established either bymeasurement (with high performance transformers this is possibleonly very approximately indeed) or numerically on the basis ofworks derived, for instance, from such authors as Massarini andKazimierczuk [26], who determine longitudinal and earth capaci-tance by means of the relation derived on the basis of the magneticfield theory. This particular approach is frequently employed indetermining parasite transformers [7] and thanks to this procedurevalues with an accuracy of up to 20% may be obtained [7,9].

One of the most widely used methods of ensuring linear voltagedistribution along thewinding is to compensate the impact of earthcapacitance C bymeans of metal shields and shades. The function ofa capacitance shield consists in mutual compensation of thedischarge on the earth capacitance C and discharge on the insertedexternal capacitance Cex. These discharges then have no impact onthe distribution of the residual discharge in the chain of lateralcapacitances K/dx. Under the assumption that the lateral capaci-tance K/dx is equally large in all the high-voltage points of the

J. Mike�s, D. Koke�s / Journal of Electrostatics 71 (2013) 533e539538

winding, voltage distribution along the winding will be linear andfree oscillations will be eliminated.

This shielding may be performed by means of a beam trap. Themost suitable behaviour of the initial voltage distribution can beobtained under the conditions whereby the shield is placed aroundthe entire circumference of the winding. However, in a real situa-tion, the beam trap usually does not surround the winding’s entirecircumference, but only its part, and that is primarily due to reasonsof dimension. Such a shield located along the entire length of thewinding must be insulated at the maximum voltage. This conditionleads to an increase in the dimension of transformers to such anextent that only parts of the winding are normally shielded. Asa rule, a shield is placed at the entry of the winding to increase thelongitudinal capacitance K in view of the earth capacitance C. If theshield eliminates the influence of the capacitance C, or the wind-ing’s earth capacitance, then there occurs a linear voltage distri-bution in the shielded part. Fig. 4 depicts simulated shielding of theearth capacitance C at 30, 60 and 90% of the winding’s axial lengthfrom the entry (values K are given in F/m and C in F m).

Fig. 5 describes the case when, in the first part of the wiring, wemarkedly increase the value of the lateral capacitance K by meansof external capacities, for instance by a chain of condensers. Thefirst part of the winding’s insulation is then stressed by constantvoltage of 500 V and in the remaining section of the windinginsulation stress will be lower than in case of the same windingwithout an external capacity (see Fig. 4a). It is evident that this isa little efficient mode of protecting insulation against puncture.

4. Objective facts

The generated model can be used for monitoring initial distri-bution of a single-layer winding, which e according to prerequi-sites e would correspond to the relation (16) at the evenlydistributed capacitance K and C [3].

u0 ¼ sinhaðl� xÞsinhal

(16)

When reducing the winding’s earth capacitance in a realtransformer by using a capacitance shield, it is possible also byincreasing the capacitance K in the individual segments of thewinding to linearize initial voltage and thus prevent emergence ofoscillations. If a single condenser is used the capacitance increases

Fig. 5. Behaviour of voltage distribution when using increased lateral capacitance K at30% of the winding’s axial length from the entry for K ¼ 20 � 10�12, Kex ¼ 20 � 10�10

and C ¼ 17 � 10�10.

but in terms of the free oscillations envelope the terminal voltagedistribution simultaneously deteriorates. The resulting free oscil-lations envelopes do not make it possible to determine the idealcapacitance connected between the winding’s input and winding’sbranch. However, the program created in the Matlab programmingenvironment allows e for the purpose of designing a transformerresistant to overvoltage impulsee to stipulate an optimumvalue ofthe inserted capacitance (capacitance chain).

5. Conclusion

In our article we have concentrated on an alternative electricalmodel of the transformer which is easier for calculating, however, itrespects all important phenomena which during approachingimpulse overvoltage stress the winding. Such a model itself isanalytically very complicated to solve. For the solution we haveapplied and verified generally known numerical mathematicalmethod. In theMatlab programming environment we have created,out of the capacitance model, graphic outputs of dependence ofvoltage at a given time and position in a VN transformer winding.Owing to disregarding inductivity it is impossible to establish fromthe model at which particular point maximum stress of longitu-dinal insulation occurs. This results from the fact that it is preciselyinductivity which affects the oscillation period, or rather theamplitude of voltage oscillation. This means that voltage behaviourat time levels between the first and last level will be considerablyaffected by the absence of inductivity in the approximate equiva-lent circuit. Their patterns thus do not correspond with the realstatus, or rather no free oscillations will occur. With this model wehave stipulated initial and terminal voltage distribution, which isnot dependent on induction, and the impact of secondary windingis minimal for these cases. Thanks to that we are in a position tostipulate the free oscillations envelope of the so-called values ofmaximum possible voltage, which may appear in the individualspots of the winding during transitional process depending on thecapacitance relations.

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