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A Contribution to the Frequency Analysis and TransientResponse of Power Transformer WindingsMohamed Mostafa Saieda

a Electrical Engineering Department, Kuwait University, Kuwait City, KuwaitPublished online: 30 Jul 2014.

To cite this article: Mohamed Mostafa Saied (2014) A Contribution to the Frequency Analysis and Transient Response of PowerTransformer Windings, Electric Power Components and Systems, 42:11, 1143-1151, DOI: 10.1080/15325008.2014.921952

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Electric Power Components and Systems, 42:1143–1151, 2014Copyright C© Taylor & Francis Group, LLCISSN: 1532-5008 print / 1532-5016 onlineDOI: 10.1080/15325008.2014.921952

A Contribution to the Frequency Analysisand Transient Response of PowerTransformer WindingsMohamed Mostafa SaiedElectrical Engineering Department, Kuwait University, Kuwait City, Kuwait

CONTENTS

1. Introduction

2. Method of Analysis

3. Sample Results for the Frequency Response

4. Sample Results for the Transient Response

5. Conclusions

References

Keywords: power transformers, equivalent circuit, power system transients,transformer winding, distributed parameters, simulation, high-frequencymodel, frequency response, resonance

Received 24 February 2013; accepted 1 May 2014

Address correspondence to Mohamed Mostafa Saied, Electrical EngineeringDepartment, Kuwait University, Kuwait City, Kuwait. E-mail: m.saied@ieee.orgColor versions of one or more of the figures in the article can be found onlineat www.tandfonline.com/uemp.

Abstract—This article proposes a new approach to study the fre-quency response and the transient analysis of power transformer wind-ings. For improved accuracy, the suggested model includes, amongother equivalent circuit elements, the mutual magnetic couplings be-tween any winding turn and all others. This implies that the equivalentinductance of any considered turn will be a location-dependent pa-rameter. Accordingly, the winding will be analyzed as a non-uniformtransmission line. Through the application of a recursive circuit re-duction technique, a closed-form Laplace s-domain analytical ex-pression for the winding’s input impedance can be obtained for anyneutral treatment. The resulting expression can be used to determinethe winding’s series and parallel resonance frequencies. The s-domainexpression for the input impedance, in connection with the numericalinverse Laplace transform, will be utilized for determination of thewinding’s time-domain transient response for any input voltage orcurrent time waveform. Accuracy increases with the assumed num-ber of winding sections, which can be even increased to the actualnumber of turns, limited only by the available computation resources.The results of case studies are in good agreement with those availablein the literature using the time-domain solution of the simultaneousdifferential equations in the state variables.

1. INTRODUCTION

The transient analysis of power transformer windings and theirresonance phenomena has attracted the attention of numerousresearchers [1–21]. Two main procedures have been typicallyapplied to perform these analyses. The first is the time-domainanalysis for identifying the time response of the different volt-ages and currents within the windings due to the applicationof different input waveforms [1, 4, 9]. This approach can dealwith eventually existing non-linear phenomena, such as coresaturation, inrush currents, and ferroresonance [8, 16, 19].The second approach applies frequency and Laplace s-domainanalyses. Among other features, they enable the distributed-parameter representation of transformer windings. Linearityof the different circuit elements is a prerequisite to these pro-cedures [3, 6, 7, 12, 17].

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1144 Electric Power Components and Systems, Vol. 42 (2014), No. 11

In both of the above procedures, the proper representationof all relevant elements is crucial to the accuracy of the result-ing solutions [14, 15, 19]. For example, in [19], it was indicatedthat the accurate determination of the transformer’s inrush cur-rent is important for the discrimination between internal andexternal faults. Usually, for each winding turn or section, theseries resistance, self-inductance, capacitance to ground, andcapacitance to the two immediately neighboring turns are con-sidered. In some studies, the dielectric losses of both the inter-turn and the winding-to-earth insulation were also considered.This rather approximate treatment could be successfully usedfor the derivation of closed-form expressions for the voltagesand currents within the transformer based on a distributed pa-rameter approach. The distributions within other eventuallyconnected elements, such as transmission lines and/or cablesections, could be determined [7, 10]. These models couldalso be used effectively for the determination of the windingresonance frequencies. In [18], a method was presented forthe efficient use of the transformer winding’s characteristicimpedance or, equivalently, the transmission line diagnosticsfor the condition monitoring of transformers. Several types ofmechanical deformations were addressed. It should be notedthat the so-defined “intrinsic” characteristic impedance is aglobal quantity describing the entire winding and is indepen-dent on the external testing circuit. A comparison is also madewith the frequency response analysis (FRA) techniques re-cently described [20, 21].

As a further step, the model was refined by including themutual inductances between any winding turn or section andall other turns [2]. As will be seen later, neglecting them canlead to a significant error.

There are no closed-form analytical solutions based on dis-tributed s-domain analysis available in the literature with allthe inter-turn mutual inductances taken into account. This isbecause the inductively induced voltages in a winding turnwill include components proportional to the rate of change ofthe still-unknown currents in the other turns. This led to anapproximate procedure based on dividing the winding into afinite number of identical sections for which a system of si-multaneous equations in the s-domain is formulated and solvednumerically at each frequency [2, 12], where the number ofsections in the coils representation should be somewhat largerthan the number of required resonance frequencies. Accord-ing to [2], the maximum frequency for which this techniqueis accurately applicable increases with the assumed numberof sections. For a particular transformer, the approximate fre-quency limits of 56.5 and 93.6 kHz for 10 and 20 sectionsmodels, respectively, were given.

This article presents a direct procedure for solving the aboveproblem analytically, based on a complete winding model that

includes all self-inductances and mutual inductances and ca-pacitances. To take all the inter-turn mutual magnetic cou-plings into account, the equivalent inductance of any turn orsection will be treated as a location-dependent circuit param-eter.

Applying a recursive circuit reduction technique to the re-sulting non-uniform transmission line, a closed-form Laplaces-domain analytical expression for the winding’s inputimpedance can be found. It can be efficiently used for

a) identifying the winding’s series and parallel resonancefrequencies, and

b) determination of the winding’s time-domain transientresponse for any input voltage or current time wave-form. This implies the application of numerical inverseLaplace transform techniques.

Needless to say, the accuracy of the simulation increases withthe assumed number of sections, which can even be increasedto the actual number of turns, limited only by the availablecomputational resources.

In [13], a completely different approach to the winding fre-quency response and transient analysis based on state-spaceanalysis was applied. The state variables in this case werethe different inductor currents and capacitor voltages appear-ing in the winding’s equivalent circuit. The s-domain expres-sions for currents, voltages, and the input impedance werefound through the solution of the set of simultaneous equa-tions. This is contrary to the method suggested herein, whichis based on a different network analysis technique involvingcascade-connected winding sections. In terms of Mathematicaprogramming, the computational burden of the old techniquepresented in [13] is larger. Moreover, there is a maximummemory allocated to each computation cell of the Mathemat-ica notebook. This is the reason why the number of sectionsin [13] was limited to about N = 5. This should be comparedwith the larger numbers N = 18 and N = 30 (presented later)as a result of applying the approach suggested here for lossyand lossless analysis using the recursive network reductiontechnique, respectively.

In summary, the objectives of this article are therefore

1. To introduce a recursive circuit reduction technique ca-pable of providing an analytical closed-form s-domainexpression for the winding input impedance for anytransformer neutral treatment.

2. To use the above expression for identifying the windingresonance and anti-resonance frequencies.

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Saied: A Contribution to the Frequency Analysis and Transient Response of Power Transformer Windings 1145

3. To apply the numerical Laplace transform inversion toget the time waveforms of the winding voltages andcurrents in response to typical input signals’ waveforms.

2. METHOD OF ANALYSIS

Figure 1 shows the schematic representation of the trans-former winding under consideration. The winding is com-posed of N turns or sections. The turn location is identifiedby the coordinate n ranging between n = 1 near the neutralpoint to n = N at the source terminal. The general neutralimpedance Zo can simulate the four cases of the solid earth-ing (Zo = 0), isolated neutral (Zo = ∞), inductive earthingvia a Petersen coil (Zo = s.L Petersen), and resistive earthing(Zo = RNeutral ).

The four thick arrows in Figure 1 indicate the definition ofthe input impedance at four different points along the coildescribed by the coordinate values 0, n, (n + 1), and N.The equivalent circuit of the indicated section between n and(n + 1) is depicted in Figure 2, with the following symbols[12, 13]:

Cg = capacitance to ground/section = Cg(total)/N ,Go = conductance to ground/section = Go(total)/N ,L(n) = equivalent inductance of the nth coil section =

L (total)(n)/N ,R = series resistance/section = R(total)/N ,Cs = series (self-) capacitance/section = Cs(total) N , andGs = insulation conductance/section= Gs(total) N .

The equivalent inductance of the nth coil section L(n) takescare of the location-independent self-inductance of the sectionas well as the location-dependent magnetic coupling betweenthis section and all the others. Figure 3 depicts the value of L(n)versus the coordinate n for the transformer described in [12].The curve is symmetrical around the winding’s midpoint andranges between the least values of Lmin(total) = 0.016 H at both

FIGURE 1. Schematic representation of transformer winding.

FIGURE 2. Assumed equivalent circuit for the winding sec-tion between boundaries n and (n + 1) indicated in Figure 1[12, 13].

winding terminals to the highest value of Lmax(total) = 0.028H at its midpoint. The difference can be explained as follows.In the first case, the separation between the considered sectionand the others ranges between zero and the entire windinglength, whereas in the second case, the separation lies betweenzero and half the winding length.

The curve in Figure 3 is adopted from [12]. It was ob-tained by evaluating the integrals of Neumann’s formula forthe mutual inductive coupling between any two winding turns.By inspection, this curve can be considered as the sum of aconstant component Lmin(total), which is independent on thecoordinate n, and a component proportional to the cubic rootof sin(πn/N ). Accordingly, the dependence of L (total)(n) onthe coordinate n can be approximated by

L(n) = [L (total)(n)/N ] =[

Lmin(total) + (Lmax(total)

−Lmin(total)). 3

√sin(π

n

N)

]/N . (1)

FIGURE 3. Value of L (total)(n) versus coordinate n along thewinding [12].

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It should be noted, however, that other candidate functionsfor expressing L(n) can be adopted through curve-fitting tech-niques.

Considering Figure 2, the equivalent impedance in theLaplace s-domain of the three branches including the fourelements L(n), R, Cs , and Gs will be

Zeq (n) =(

sCs + Gs + 1

[R + sL(n)]

)−1

. (2)

The equivalent admittance of the two series-connectedimpedances Zeq (n) and Z (n) is 1/Zeq (n) + Z (n). Accordingly,the input impedance Z (n + 1) of the entire circuit is given by

Z (n + 1) =[

1

Zeq (n) + Z (n)+ Go + sCg

]−1

. (3)

Starting from the neutral point (n = 0) and with the initialvalue Z(0) = Zo, Eq. (3) can be applied repeatedly to getthe values of Z(1), Z(2), Z(3), etc. up to the required totalwinding input impedance Zinput (s) = Z(N) as a function ofthe windings data and the complex frequency s. The accuracycan be increased by dividing the winding into a larger numberof sections N. In the proposed recursive technique, there isbasically no upper bound for N , which can even assume theactual number of winding turns. The only limitation will be theavailable computational resources. Nevertheless, there will beno technical significance for using a larger number of sectionsN.

The frequency response of the winding input impedancecan then be found by substituting s = jω.

In the Laplace domain, the following equation applies:

Is(s) = E(s)/[Zinput (s) + Rsource], (4)

where Is(s) and E(s) are the Laplace transforms of the sourcecurrent i(t) and the source voltage e(t), respectively, accordingto Figure 1.

The Laplace transform of the voltage at the source-sidewinding terminal v(t)ter min al is given by

V (s)ter min al = E(s) − Is(s).Rsource

= E(s).Zinput (s)/[Zinput (s) + Rsource]. (5)

Using the equivalent circuit in Figures 1 and 2, starting fromthe source terminal backward, the s-domain expressions forthe voltage and current pertinent to any location n along thewinding can easily be determined. Applying Kirchhoff’s lawsto the circuit depicted in Figure 2, the following two equationscan be obtained:

I(n+1) = In + (Go + sCg).V(n+1), (6)

V(n+1) = I(n+1).Z (n + 1). (7)

Combining these two equations yields the following recursiveequation for the current distribution:

In = I(n+1)[1 − (Go + sCg).Z (n + 1)]. (8)

If this equation is evaluated once, the current at a point onesection apart from the source terminal can be determined. Re-peating the recursive equation(N/2) times through Do loops inthe Mathematica program will yield the current at the windingmiddle point, while an evaluation N times will determine thecurrent at the neutral point. Since Vn = In.Z (n), Eq. (8) canbe modified as follows to get the voltage distribution along thewinding:

Vn = V(n+1)[1 − (Go + sCg).Z (n + 1)].Z (n)

Z (n + 1). (9)

3. SAMPLE RESULTS FOR THE FREQUENCYRESPONSE

The above procedure was applied to investigate the frequencyand transient responses of the transformer winding describedin [12]. In addition to the data pertinent to the non-uniformlydistributed inductance L (total)(n) shown in Figure 3, the equiv-alent circuit parameters are given by:

Cs = 2.07 pF, N ,and Gs = 0.15 pS, N ,and Cg = 20.7 nF/N ,Go = 0.15 nS/N ,and R is assumed 2.198 �/N .

here N is the assumed number of sections.It should be noted that due to the limited memory in-

herently allocated to each computation cell in the Math-ematica notebook, the maximum number of sections ofN = 18 could be handled for the determination of an ex-pression for the input impedance, with all losses included.Neglecting all lossy elements of the equivalent circuit, i.e.,Gs , Go, and R, can simplify the impedance expressionand, thus, allow the division of the winding to a largernumber of sections, N = 30 in this case study. For thiscase, exact frequency values could be directly obtainedusing the Mathematica software for solving polynomialequations.

First, the results of a solidly earthed winding will bepresented. The winding was divided into N = 18 sections.The neutral impedance assumes the special value Zo = 0.Figure 4 depicts the winding’s frequency response in therange of 0–400 kHz. The impedance starts with the purely

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Saied: A Contribution to the Frequency Analysis and Transient Response of Power Transformer Windings 1147

FIGURE 4. Frequency response of winding with solidlyearthed neutral point: (a) impedance magnitude, (b) impedancereal part (resistance), and (c) impedance imaginary part(reactance).

resistive DC value of about 2.2 �, which is close to theassumed total winding resistance R(total) to a purely capac-itive value of approximately 500 � at 400 kHz. The lat-ter value agrees with the fact that the winding at extremelyhigh frequency will be equivalent to a pure capacitance ofabout

√CsCg . The plots exhibit several series and parallel

frequencies at which the winding impedance magnitude as-sumes minimum or maximum values, respectively. At thesefrequencies, there will be sign reversals in the impedanceangle.

Series resonance frequencies based on the lossless analysisusing N = 30 winding sections are 26.2059, 51.561, 76.559,100.9756, 124.630, 147.365 kHz, etc.; parallel resonancefrequencies are 12.763, 37.919, 62.750, 87.094, 110.782,133.663, 155.603, 196.256 kHz, etc. These values could beobtained using the feature of the solution of polynomial equa-tions in the software Mathematica.

To assess the importance of taking the non-uniform dis-tributed nature of the winding inductance into account,the winding was analyzed once more assigning a con-stant uniform value to L(n) = [Lmin(total) + Lmax(total)]/2 =0.021 H. The resonance frequencies for the case of solidlyearthed neutral were found as now given. Series reso-nance frequencies based on the lossless analysis of 30winding sections are 28.7384, 57.2425, 85.2855, 112.6557,139.1616, 164.6372 kHz, etc.; parallel resonance frequenciesare 14.1021, 42.1954, 69.9595, 97.1865, 123.6846, 149.2832,173.8357, 197.2220 kHz, etc. These values are about 10%larger than the corresponding resonance frequencies that re-sulted earlier from analyzing the same winding but withthe non-uniformity of the inductance distribution taken intoconsideration.

The results for the case of an isolated neutral, i.e., Zo =∞, are shown in Figure 5. The input impedance at very lowfrequencies is very large and capacitive. The performance atvery high frequencies is similar to that of the solidly earthedneutral discussed in the previous section. The first series res-onance frequency is about 10 kHz, which is lower than thefirst parallel one of 20.1 kHz. This situation is opposite to thecase of solid earthing, which indicates some duality betweenthe series and parallel resonance frequencies of a transformerwith solidly earthed neutral and the parallel and series reso-nance frequencies of the same transformer if operated withan isolated neutral. In other words, the series resonance fre-quencies in the case of solid earthing are numerically closeto parallel resonance ones in the case of an isolated neutralpoint. This can be explained if the transformer winding isvisualized as a long transmission line. A winding with solidearthing is analogous to a line with a short-circuited receiv-ing end, while an isolated neutral resembles the case of anopen-circuited line. The product of the two winding’s inputimpedances under these two extreme conditions are equal tothe square of the line or winding surge impedance. Assum-ing, as a first approximation, that the surge impedance is fre-quency independent, it follows that the zeroes of the winding’sinput impedance with solid earthing (determining the series

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FIGURE 5. Frequency response of winding with isolated neu-tral point: (a) impedance magnitude, (b) impedance real part(resistance), and (c) impedance imaginary part (reactance).

resonance frequencies) are close to the poles of the windingimpedance with an isolated neutral (determining the parallelresonance frequencies).

For the sake of completeness, the computed resonance fre-quencies for the case of an inductively earthed neutral via a15.92-mH Petersen coil (corresponding to a reactance of 5 �

at 50 Hz) with the non-uniformity of the winding inductance

included are now given. Series resonance frequencies basedon the lossless analysis of 30 winding sections are 17.6851,41.075, 66.0160, 90.8576, 115.1242, 138.5623 kHz, etc.;parallel resonance frequencies are 7.850, 28.1476, 52.3124,76.8821, 101.0803, 124.5968, 147.2354, 168.8518 kHz, etc.More results on the transformer’s frequency analysis could notbe presented due to space constraints.

4. SAMPLE RESULTS FOR THE TRANSIENTRESPONSE

Sample results of analyzing the transient response of the trans-former’s winding are now presented, which were obtained us-ing an efficient technique for the numerical determination ofthe inverse Laplace transform f (t) for any s-domain functionF(s). For more details, the reader can consult [3, 7, 10, 17].

The plots in Figure 6 show the winding’s input current is(t)and terminal voltage vter min al(t) following the application ofa step-shaped source voltage of magnitude 1000 V, i.e., e(t) =1000.u(t), and an internal resistance Rsource = 20 � for a solid

FIGURE 6. Winding input current and voltage in response toa 1000-volt step voltage source with an internal resistance of20 � winding’s neutral is solidly earthed (plots in time rangefrom 0–1000 μs).

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Saied: A Contribution to the Frequency Analysis and Transient Response of Power Transformer Windings 1149

neutral earthing. u(t) denotes a unit-step function. The twocurves exhibit superimposed high-frequency oscillations. Thedominant frequency is around 25 kHz, which is close to the firstseries resonance frequency (26.206 kHz) mentioned earlier forthis case. The transient current response in Figure 6(b) for thesame signals is similar to that of an RL series circuit. The timeconstant is about 1 ms, which is very close to τ = L/R = 0.021H/(2.2 + 20) � ≈ 0.95 ms. The final values of the results arein full agreement with the DC analysis of an equivalent RLseries circuit comprising the 1000-V source of 20-� internalresistance and the winding total resistance of 2.198 �. Thecurrent and terminal voltage should therefore approach thevalues of 45 A and 100 V.

Applying the same source to the winding operating withan isolated neutral point yields the transient responses of theinput current is(t) and terminal voltage vter min al(t) shown inFigure 7 over the time range 0 to 1500 μs. Both signals includecomponents of high frequencies depicting internal winding os-cillations as well as multiple wave reflections. The decayingamplitudes reflect the damping effect due to the winding cop-

FIGURE 7. Winding input current is(t) and voltage vter min al (t)for a 1000-volt step voltage source with internal resistance of20 � winding neutral is isolated.

per losses, its dielectric losses, as well as the source resistance.The final values of the current and voltage are 0 and 1000 V,respectively. They are in full agreement with simple DCanalysis.

To validate the suggested approach, the results of two morecase studies are presented in Figures 8(a) and 8(b). Their so-lutions are already available in the literature, obtained throughapplying a completely different technique in the time domain.The technique is based on solving a set of simultaneous differ-ential equations in the state variables (capacitor voltages andinductor currents) of the lumped parameter equivalent circuitcomprising several cascaded sections [13].

The following total values of the transformer parameters(for the entire winding) are assumed: Cs = 0.34 pF, Gs = 0.606μS, Cg = 8.25 nF, and Go = 47.62 pS, and R is assumed as 226�. The individual self and mutual inductances of the differentwinding sections are discussed in details in [13].

A curve similar to Figure 3 for the location-dependent in-ductance per unit length is adopted for the solutions based

FIGURE 8. Results of two case studies obtained from thesuggested procedure for which the corresponding solutions,based on time-domain analysis, are available in [13]: (a) inputwinding impedance and (b) developed voltage due to vertical1-A current chopping.

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1150 Electric Power Components and Systems, Vol. 42 (2014), No. 11

on the proposed method using the recursive formula givenby Eq. (3). The maximum and minimum inductances areLmax(total) = 0.60 and Lmin(total) = 0.30 H. Figure 8(a) showsthe input impedance of the transformer with the neutral earthedvia a resistance of 226 �. The trace exhibits several paralleland series resonances over the considered frequency range0 ≤ f ≤ 50 kHz. The plot is in a good agreement with thecorresponding one in [13].

Figure 8(b) illustrates the voltage developed at the trans-former input terminals due to vertical current chopping of 1 Aduring the time range 0 ≤ t ≤ 4000 μsec. The transformer isassumed to be solidly earthed. It can be seen that the transientvoltage can reach values as high as 10 kV per chopped am-pere. Again, a reasonable agreement with the correspondingtime domain results of [13] could be noticed.

More results on the transformer’s transient response couldnot be presented due to space constraints.

5. CONCLUSIONS

1. A new approach to the study of frequency response andtransient analysis of transformer windings is presented.The model takes into account the mutual magnetic cou-plings between any winding turn and all others. Theequivalent inductance of any considered turn is there-fore a location-dependent parameter, and the windingwill be analyzed as a non-uniform transmission line.

2. Applying a suggested recursive circuit reduction tech-nique, a closed-form Laplace s-domain expression forthe winding input impedance can be obtained for anyneutral treatment. It can be used to determine the wind-ing’s resonance frequencies. In connection with the nu-merical inverse Laplace transform, it will be also uti-lized for determining the winding’s transient response.

3. The impedance plots of a solidly earthed winding ex-hibit several series and parallel resonances. The seriesresonance frequencies based on the lossless analysisare also given. For this case, exact frequency valuescould be directly obtained using Mathematica softwarefor solving polynomial equations.

4. In the case of an isolated neutral, the input impedanceat extremely low frequencies is large and capacitive.The performance at high frequencies is similar to thatof a solid earthing. The first series resonance frequencyis about 10 kHz, which is lower than the first parallelone of 20.1 kHz. This situation is opposite to the caseof a solid earthing.

5. To assess the importance of taking the non-uniformlydistributed nature of the winding inductance into ac-count, the winding was analyzed once again assuminga uniformly distributed inductance. The resonance fre-quencies for the case of a solidly earthed neutral areabout 10% higher than those brought earlier with theconsideration of the non-uniformity of the inductance.The computed resonance frequencies for the case ofan inductively earthed neutral via a Petersen coil werealso presented.

6. Results for the input current and terminal voltage fol-lowing the application of a step source voltage to asolidly earthed transformer are presented. They exhibita combination of high-frequency oscillations with adominant frequency of around 25 kHz. Over the ex-tended time range of 0–5000 μs, and apart from fast-decaying high-frequency components, the current re-sponse is similar to that of an RL series circuit.

7. The response to the same source to the winding operat-ing with an isolated neutral point is presented that indi-cates internal winding oscillations and multiple wavereflections. The rapidly decaying current and voltageamplitudes reflect the damping effect due to windinglosses. The final values of the current and voltage arezero and the value of the source, respectively, as ex-pected.

8. The accuracy can be increased by using a larger num-ber of sections N. There is no upper bound for N , whichcan even assume the actual number of winding turns.The only limitation is the available computational re-sources.

9. The results of two case studies are compared and foundin to be in good agreement with those available inthe literature from applying the time-domain solutionof the simultaneous differential equations governingthe state variables of the winding’s lumped-parameterequivalent circuit.

10. Future investigations are planned to extend the pro-posed method to transformers with winding axial dis-placement and other mechanical deformations.

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[8] Chakravarthy, S., “Nonlinear oscillations due to spurious ener-gization of transformers,” IEE Proc. Elect. Power Appl., Vol.145, No. 6, pp. 585–592, 1998.

[9] Mombello, E., and Moeller, K., “New power transformer modelfor the calculation of electromagnetic resonant transient phe-nomena including frequency-dependent losses,” IEEE Trans.Power Delivery, Vol. 15, No. 1, pp. 167–174, 2000.

[10] Saied, M. M., and AlFuhaid, A. S., “Frequency response oftwo-winding transformers obtained by a distributed-parameters-domain method,” Elect. Power Compon. Syst., Vol. 32, No. 8,pp. 755–766, 2004.

[11] Miki, A., Hosoya, T., and Okuyama, K., “A calculation methodfor impulse voltage distribution and transferred voltage intransformer windings,” IEEE Trans. PAS, Vol. 97, No. 3, pp.930–939, May/June 1978.

[12] Predota, A., Benesova, Z., and Koudela, L., “Analysis of tran-sients in transformer winding respecting space-varying in-ductance,” Przegl

↪ad Elektrotechniczny, Vol. 88, No. 7b, pp.

220–222, 2012.[13] Saied, M. M., “New solution technique for the frequency and

transient response of transformer windings with all inter-turnmutual inductances and capacitances included,” J. Trends Elect.Eng., Vol. 3, No. 1, Paper No. 3143, 2013.

[14] Rashtchi, V., Rahimpour, E., and Shahrouzi, H., “Model reduc-tion of transformer detailed R-C-L-M model using the impe-rialist competitive algorithm,” IET Elect. Power Appl., Vol. 6,No. 4, pp. 233–242, April 2012.

[15] Garcia-Gracia, M., Villen, M., Cova, M., and El-Halabi, N.,“Detailed three-phase circuit model for power transformers overwide frequency range based on design parameters,” Elect. PowerSyst. Res., Vol. 92, pp. 115–122, November 2012.

[16] Zhang, X., and He, J., “Lightning overvoltage suppression toUHV power transformer by ferromagnetic ring,” Elect. PowerSyst. Res., Vol. 94, pp. 122–128, 2013.

[17] Saied, M., “On the transient response and frequency analy-sis of transmission line towers”, ISRN Power Eng., Vol. 2013,Article 874647, available at: http://dx.doi.org/10.1155/2013/874647

[18] Ghanizadeh, A. J., and Gharehpetian, G. B., “Applicationof characteristic impedance and wavelet coherence techniqueto discriminate mechanical defects of transformer winding,”Elect. Power Compon. Syst., Vol. 41, No. 9, pp. 868–878,2013.

[19] Fani, B., Hamedani Golshan, M. E., and Saghaian-nejad, M.,“Transformer differential protection using geometrical struc-ture analysis of waveforms,” Elect. Power Compon. Syst., Vol.39, No. 3, pp. 204–224, 2011.

[20] Al-Shaher, M., and Saied, M., “Recognition and location oftransformer winding faults using input impedance,” Elect.Power Compon. Syst., Vol. 35, pp. 785–802, 2007.

[21] Saied, M., and Al-Shaher, M., “Recognition of power trans-former winding movement and deformation using FRA,” COM-PEL, Vol. 26, No. 5, pp. 1293–1410, 2007.

BIOGRAPHY

Mohamed Mostafa Saied Received the B.Sc. degree (Hon-ors) in Electrical Engineering from Cairo University, Egypt in1965, then the Dipl.-Ing. and Dr-Ing. Degrees from RWTHAachen, Germany, in 1970 and 1974. From 1974 to 1983, hewas at Assiut University, Egypt. In 1983, he joined KuwaitUniversity where he served as a full professor (1983-2009)and Department Chairman (2002-2007). He spent one-yearsabbatical leave (1998) as a Visiting Professor at Cairo Uni-versity. His main areas of interest are: electric power transients,supply quality and power systems economy.

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