Transactionson Power Systems on Modelling Iron Core Nonlinearities

8/19/2019 Transactionson Power Systems on Modelling Iron Core Nonlinearities

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IEEE Transactions on Power Systems, Vol. 8, No. 2 May 1993

O N MODELLING IRON CORE NONLINEARITIES

Washington L. A. Neves Hermann W . DommelStudent Member, IEEE Fellow, IEEE

Department of Electrical EngineeringUniversityof British Columbia

Vancouver, B. C.,Canada V6T 124

Abstract

An algorithm is presented for the comp utation of thesaturation characteristics of transformer iron coresbased on supplied conventional V,,, - I curvesand nd o a d losses at rated frequency. Laboratorymeasurements on a steel sample were carried out. Itis shown tha t the iron core losses are a nonlinear func-tion of the applied voltage. Taking these losses intoaccount improves the nonlinear flux-current charac-teristic.

1 Introduction

The simulation of electromagnetic transients in powersystems is essential for insulation coordination stud-ies and for the adequate design of equipment and itsprotection. To carry out these studies on digital com-puters, mathematical models are needed for the var-ious components. Models for transformers and re-actors are especially important for studying inrushcurrents, ferroresonance, harmonics and subharmon-

ics. In these types of studies, iron core nonlinearitiesplay an important role.

The major nonlinear effects in iron cores are:

Saturation

Eddy Currents

Hysteresis

92 176-8 PWRSby the IEEE Power System Engi neering Committee ofthe IEEE Power Engineering Society for presentationat the IEEE/PES 1992 Winter Ueeting, New York, NewYork, January 26 - 30, 1992. Uanuscript submittedAugust 28, 1991; made available for printingDecember 23, 1991.

A pap er recommended and approved

417

Saturation is the predominant effect in power trans-formers, followed by eddy current and hysteresis ef-fects[l]. Thus, the instantaneous saturation charac-teristic, which gives flux linkage A as a function ofcurrent i , is an essential part for modelling the ironcore nonlinearities.

Santesmases et al.[2] represent transformer coresby a simple equivalent circuit consisting of a nonlin-

ear inductanceA - t

curve) in parallel with a non-linear resistance v - , curve). These elements areobtained from functions derived from the dynamichysteresis loops. This is essentially the same modelproposed by Chua and Stromsmoe[3]. The resistancein this model accounts for the energy losses due tothe loops. Chua and Stromsmoe did make compar-isons between simulations and laboratory tests for asmall audio transformer, and for a supermdloy coreinductor as well. A family of flux curren t loops for60, 120 and 180 Hz sinusoidal excitations of variousamplitudes were obtained as well as minor dynamichysteresis loops. The agreement between simulationsand measurements was very good.

In this paper we use the same model proposed inreferences [2] and [3]. However, the nonlinear param-eters are calculated in a simpler way directly fromthe transformer test da ta. The nonlinear resistance(piecewise linear v - , curve) is found from the no-load (excitation) losses, and this information is thenused to compute the current thro ugh the nonlinear in-ductance and to construct the piecewise linear A - lcurve.

Transformer manufactur ers usually supply th e sat-uration curves in the form of rms voltages as a func-tion of rms currents. Numerical methods have beenused for some time to convert these V,,,-I,,, curvesinto peak flux - peak current curves[4,5]. As shown inthis pap er, these methods can be modified to take theiron core losses into account, thereby producing thenonlinear inductance as well as the parallel nonlinearresistance.

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2 Saturation Curves

Figure l(a) shows a voltage source connected to atransformer whose excitation branch is representedby a nonlinear inductance in parallel with a nonlinearresistance. Their nonlinear characteristics are com-puted according to the following assumptions:

the - , and X - curves (Figures l(b) andl(c)) are symmetric with respect to the origin( and Lk are the slopes of segment k of the

- , and A - , curves, respectively);

the no-load test is performed with a sinusoidalvoltage source;

the winding resistances and leakage inductancesare ignored.

The conversion algorithm works as follows:

1. For the construction of the - , curve (Section2.1):

compute the peak values of the currenti, t ) oint by point from the no-load losses,and subsequently compute their rms valuesI , r ma .

Figure 1: (a) Voltage source connected to trans-former; (b) Nonlinear w - , characteristic; (c) Non-linear A - r characteristic.2. For the construction of the X - l (Section 2.2):

obtain the rms values I~-,.,, of the currenti f ( t ) through the nonlinear inductance fromI,-,,,, the total r m s current It-,,, andthe applied voltage v t ) .

compute the peak values of the inductivecurrent i l t ) point by point from their rmsvalues and T m s voltages.

2.1

Let us assume that the no-load losses PI ,Pz , . . Pare available as a function of the applied voltageVrmdt , Vrm,,, . . Vrmb,,, , as shown in Figure 2. Fromthese data points we want to construct a piecewiselinear resistance curve, as shown in Figure 3(b),which would produce these voltage dependent no-loadlosses. Let us first explain how the no-load losses canbe obtained from a given - , curve, before describ-ing the reverse problem of constructing the v - ,curve from th e given no-load losses at ra ted frequency.For instance, assume tha t t he applied voltage is V,.,,,and varies sinusoidally as a function of time, as shownin Figure 3(a) , with

Computation of the w - Curve

v2(d) = V, sin d (1)

P

Figure 2: V - Average Power curve.



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~ - 7.....

419

where V2 = V,,,,& Because of the sy mmetry of thev - , curve with respect to the origin, it is sufficientto observe 1/4 of a cycle, to 8 = 5. From Figure 3,it can be seen that :

if 0 < 810 V l ) / R 2 if el 5 6 5 5

In general, (e)can be found for each .(e) throughthe nonlinear w - , characteristic, either graphically(as ndicated by the dotted lines in Figure 3), or withequations. This will give us the curve (e) over 1/4of a cycle, Gom which the no-load losses are found as

sP = 'J w e ) i , e )de . (2)

s o

Let us now address the reverse problem, i.e ., con-structing the w - i, curve from the given no-loadlosses. Obtain ing th e points V1,VZ ,..,, V, on thevertical axis of Figure 3(a) is simply a re-scaling pro-cedure from r m s o peak values,

Vk = v?77lSk h (3)

for IC = 1 , 2 , 3 . . m. For the first linear segment inthe w - , curve, the calculation of the peak currentI rk , n the horizontal axis is straightforward. SincePI = V,,,, I,,,, in the linear case,

Figure 3: (a) Sinusoidal voltage input signal; (b ) v- i ,curve to be computed; (c) Output current.

For the following segments I C 2 2) , we must usethe power definition of equation (2) , with the appliedvoltage .(e) = v sin e (Figure 3a). Then

(5)

The break points 81,82, . . . e k - l in equation (5)are known from

0, = arcsin(Vj/Vk), 6)

for j = 1 , 2 ,. . . , - . The only unknown in equation(5) is the slope Rk in the last segment. The average

power can therefore be rewritten in the form



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with a,.,, br, and P k known values. Rk is then easilycomputed and Irk s calculated from

This computation is done segment by segment,starting with I,., and ending with the last point I.,.Whenever a point Irk as been found for the horizon-tal axis in Figure 3(b), its rrns value is calculated aswell because it is needed l ater for th e construction ofthe X - r curve. is found from the definitionof the rms value,

2.2 Computation of the X - l curve

The X - r curve is computed using the rms currentinformation from the U - ,. curve. Peak voltagesare converted to peak fluxes and the rrns values ofthe current throug h the nonlinear inductance are con-verted to peak values.

The conversion of peak values of v to flux A is againa re-scaling procedure. Hence, for each linear segmentin the A - t curve,

where w is the angular frequency.

Let us now compute th e peak values of t he induc-tive current. At first, their rms values are eval uated.It can be shown that for sinusoidal input voltages,the harmonic components of the resistive current areorthogonal to their respective harmonic componentsof the inductive current (see Appendix A). Then,

with the resistive current I,.-,,, already computedfrom equation (10) and the total current It-,.msknown from the transformer test dat a.

For the first linear segment in the X - l curve,

For the following segments k 2 2), the peak cur-rents are obtained by evaluating I~I- . or each seg-ment k , using equation (9). Thus , assuming Xk(0) =Xk sin e, we have'

Here, similarly to the case of the v curve com-putatio n, only the last segment (Lk) of equation (14)is unknown. Equation (14) can be rewritten in th eform

01 yt+br, Yk +C l k = 0, (15)with constants a rk ,blk and cl, known, and y k = 1/Lkto be computed. I t can be shown tha t a / > 0, br, > 0and clk < 0. Since Yk must be positive, the n

The peak current I/, s computed from

In this fashion, the peak values of th e inductive cur-rent are computed directly for every segment in theX - r curve.

3 Comparisons Between Ex-periment s and Simulations

Laboratory experiments were performed with a sili-con iron steel core assembled in an Epstein frame[6].

'For computation of the Tms value of the inductive current,it does not matter what the flux phase is, owing to the factthat the voltage (or flux) s assumed to be sinusoidal and theX t curve symmetric with respect to the or igin. Here, forcomputing purposes only it is assumed Xk 0) = Xk sine. Thishas the advantage that the limits of integration in equation(14) are the same as hose in equation (5 ) . The same procedureapplied in Figure 3 for the computation of the , curve canthen be used for the X t curve computation.



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Table I: Laboratory measurements

X(F.'S)0.00000.01200.02520.03330.04450.05440.06100.06980.07850.08920.09460.1057

L o s s e s ( W )0.00000.0727

0.26280.42230.69090.97231.18501.50901.88302.46202.83704.0100

&(A)0.00000.05790.05990.08580.09500.13130.15140.19970.26740.43490.58191.0251

Table 11: Computed ZI - , and X - l curves

4.53119.4880

12.569516.783920.500422.988026.327029.596733.635735.653739.8497

0.03210.05240.06300.07760.08940.09740.10980.12400.14840.16840.2371

No-load losses and rrns current at 60 He were mea-sured for different voltage levels (Table I). For com-parison purposes, the initial magnetization curve[7]for the core material was measured as well (AppendixB). Table I1 shows the computed w - r and X lpoints (including core losses). Figure 4 shows themeasured and the calculated points (connected bystraight line segments) with and without includingthe core losses. Figure 5 shows the computed w rpoints connected by straight line segments (the firsttwo columns of Table 11) .

It can be seen that the computed X - l curve iscloser to the measured one if we consider the corelosses. The w - curve (Figure 5) is nonlinear andthis may be important when modelling transformersand reactors for transients or harmonic studies.

0.10

0.08

0.06

0.04

0.02

421

osses not included4 Losses ncluded

Measuredpoints

0.20 0.40 0.60 0.80 1.00

Current A)

Figure 4: X - l curve

0 0.05 0.1 0.15 0.2 0.25

Cwrent A)

Figure 5: Computed v - , curve



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4 Conclusions

A direct method for the computation of iron core sat-uration curve A - l ) has been presented. It is basedon the transformer test data. It is a modificationof previous methods, with core losses taken into ac-count. Besides the A- i r curve, it produces a nonlinear

v - , curve as well.

Comparisons between measqrements and simula-tions were made. As shown ip the paper, more ac-curate A - r curves can be obtained if losses are in-cluded .

Once the A - i l and v - i , curves have been obtained,they can easily be used for modelling transformersand iron core reactors in electromagnetic transientsand harmonic loadflow programs.

5 Acknowledgements

The financial supp ort of Mr. Washington Nevesfrom the National Research Council CNPq of Braziland from Universidade Federal da Paraiba, CampinaGrande, PB - Brazil, is gratefully acknowledged. Theauthors would also like to thank Dr. Jose Marti €orvaluable discussions on this project.

6 Bibliography

1. Glenn W . Swift, Power Transformer Core Be-havior Under Transient Condi t ion s ,IEEE Trans.Power App. Sys t, vol. PAS-90, No 5 , Septem-ber/October, 1971, pp. 2206-2210.

2. J . G . Santesmases, J . Ayala, S. H. Cachero,Analytical Approximationof Dynamic Hysteres isLoops and its Applicatio n to a Series Ferroreso-nant Circuit , Proc. IEE 117, No. 1, January1970,pp. 234240 .

3. L. 0 Chua and K . A . Stromsmoe, Lumped Cir-cuit Models for Nonlinear Inductors ExhibitingHysteresis Loops, IEEE Trans. on Circuit The-ory, vol. CT-17, No. 4, Nov. 1970 pp. 564-574.

4. S. Prusty and M. V . S. b o , A Direct Piece-wise Linearized Approach to Convert rms Sat-uration Characteristic to Instantaneous Satura-t i on Curve , IEEE Tkans. Mag., vol. Mag-16,No. 1 , January 1980, pp. 156-160.

5. H. W . Dommel, Electromagnetic TransientsPro-gram Reference Manual (Section 6 ) , Bonneville

Power Administration, Portland, Oregon, Au-gust 1986.

6 . S. L. Burgwin, Measurement of Core Loss andA .C. Perme abili ty with the25 cm Epste in Frame,Proceedings, Am. Soc. Testing Mats., ASTEAVol 41,1941pp. 779-796.

7 . Melville B. Stout, Basic Electrical Measurements(Section 16-8), New York, Preptice Hall, Inc.,1950.

Appendix A - Orthogonality Be-tween i , and i l

Consider the circuit of Figure l(a). The voltageacross the transformer terminals and its correspon-dent flux linkage can be written in the form

and

respectively.

Let us use Fourier analysis to represent the currenti , (B) through the nonlinear resistance and the currentI r ( e ) through the nonlinear inductance. Due to theodd symmetry of the v - , and A r curves, (e)and i l 8 ) will have only odd harmonic components inthe form

i,.(O) = a1 sin 6 f us sin 36 + ... + a , sin ne,and

(A.3)

ir(6) = bl COS 6 + b3 COS 38 + . . . + b, cos ne , (A.4)where n = 1 , 3 , . ..The total current (e) s then:

(e) = (e)+ i@ , 64.5)i.e.,

(e) = d G s i n ( 6 +71) m s i n ( 3 6 + 73)+ . . . + J-sin(n6 +y,,), (A.6)

where 7 = arctan(b,/a,).

Evaluating the T ~ Salues of i,(f?), l 8 ) and (e),we have

I,-,,, = Jas + U + ... + U : , (A.7)Ir-,,, = JbT +b + . . . + bg ( A 4



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and

It-rmS = Ju +b: +a i +b; + . . . + U: +b : (A.9)respectively. From equat ions (A.7), (A.8)and (A.9) ,it can be seen th at

2K r m s = I r - r r n s + K r r n s . ( ~ . 1 0 )

Appendix B - Measurement of theInitial Magnetization Curve

The initial magnetization curve is a plot of the lo-cus of the D.C. symmetrical hysteresis loop tips fordifferent peak values of magnetization. Figure B . l isthe circuit used to measure it.

R

Figure B . l :ment.

Initial magnetization curve measure-

'I\A

Figure B.2: Hysteresis loop locus.

Th e magnetizing winding of the Epstein frame (pri-mary winding) is connected to a D.C. power supplythrough a reversing switch SI ammeter and a decaderesistance box R. The secondary winding is con-nected to a digital waveform analyzer where the volt-age waveform is obtained and numerically integratedin order to give the flux linkage across the secondary

winding.Th e Epstein frame is demagnetized before any mea-surement is taken. This is accomplished by driving

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the core into saturation using alternating current atpower frequency and gradually reducing the core ex-citation to zero.

After demagnetization, R is set to provide a lowcurrent, and S is reversed several times t o assure thesample is in a definite hysteresis cycle (AA' and A'Atrajectories of Figure B.2). Then, the first readingtakes place. The voltage across the secondary wind-

ing of the Epstein frame is integrated and the fluxdifference between AA' is obtained. This value is di-vided by two and segment O A is plotted.

After the first reading, R is changed to give aslightly greater value of the current in the primarywinding and t he process is repeated up t o the desiredlimit.

Biographies

Washington L . A . Neves wasborn in Brazil on March

1, 1957. He received theBS c. and M.Sc. de-grees in Electrical Engineer-ing from Universidade Fed-eral da Paraiba in 1979 and1982,respectively. From 1982to 1985 he was with the De-partment of Electrical Engi-neering of Faculdade de En-genharia de Joinville, SantaCat arina, Brazil.

Since November 1985 he has been with the Depart-ment of Electrical Engineering of Universidade Fed-eral da Paraiba, Campina Grande - PB , Brazil. Heis currently a Ph. D candidate at th e University ofBritish Columbia, Vancouver, Canada.

Hennann W . Dommel w a s born in Germany in1933. He received the Dip1.-Ing. and Dr.-Ing. de-grees in electrical engineer ing from the Technical Uni-versity, Munich, Germany in 1959 and 1962, respec-tively. From 1959 to 1966 he was with the Techni-cal University Munich, and from 1966 to 1973 withBonneville Power Administration, Portland, Oregon.Since July 1973 he has been with the University ofBritish Columbia in Vancouver, Canada. Dr. Dom-me1 is a Fellow of IEEE and a registered professionalengineer in British Columbia, Canada.



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Discussion

R . Meredith (New York Power Authority, White Plains, NY): Thesubject of the pa per is well presented, but the flexibility of the modelseems too restricted to be of practical application.

Is it not true that the model is valid for only one frequency? Doesnot the loss model produce the sameloss for all frequencies? If so itwould not be applicable for transients of another frequencyor forsuperposed transient frequencies. The single nonlinear inductancealso appears to assume uniform flux density within the core Iamina-tions. More detailed core models suchas those presented and dis-cussed in 92 W M 177-6 PWRS confirm that such an assumption isinvalid at ev en low order harmo nic situations.

There are also problems with obtaining meaningful informationfrom tested rms current vs voltage tests of transformers. Newly manu-factured transformers have such high permeability steels that magne-tizing currents ar e less than capacitive charging currents. The result isthat the rms exciting curren ts for voltages below 90 arecapacitive/resistive in nature. The rms current at100 is often lowerthan the rms current at 90 , due to cancellation of fundamentalfrequency reactive components. It would seem impossible to derivemeaningful information from such test results unless either the capaci-tance or the B -H curve is already known.

Ano ther major restriction would occur for three phase transformers,which by virtue of either embedded delta windings, three-leg coreconstruction, or the usual testing from the delta winding result inremoval of triple harmonics from the measured rms values.Is itpossible to obtain any useful information fromrms curren ts when theyare so confused by capacitive effects, interphase coupling and lack of

knowledge of the harmonic content of the current?Manuscript received January30, 1992.

Y. Lhghmuz (University of Nevada,Las Vegas, NV : The authors areto be commended for proposing a simple and elegant procedure todetermine the nonlinear transformer - , and A - , curves fromopen-circuit test data. Th e method is perfectly valid under the follow-ing assumptions:1. The sup ply voltage is sinusoidal and instruments used to measure

the rms current and active power are accurate under distortedcurrents. Have the authors examined these conditions prior torecording the measured values in TableI?

2. The two curves extend as the rms value of the supply voltageincreases. However, recent laboratory measurements (w ith the aidof microcomputer software) ona 120/60V , 60V A shell-type trans-former indicate that the A - , and - , curves do not simply

350.00

3cQ.m

250.00

m OO

150.00

100.00

50.00

0.00~~

0.00 20.00 40.00 60.00 80.00 100.00

;I (mA)Fig. 1. A - , Curves

I I I I I

J I I I I I

m OO

180.00

160.00

140.00

120.00

F 100.00a

80.00

60.00

40.00

20.00

0.00

v = l o 0 Vv = 1 2 0 vV = 140V

= l o 0 Vv = 1 2 0 vV = 140V

-I I I II I I

0.00 10.00 m.00 30.00

ic (mA)Fig. 2. U - , Curves

extend, but rather change in shape. These curves are shown forthree different supply voltages below.

The answer the following additional questions wouldbe appreciated.

It would be desirable to elabo rate more on th e curves shown in Fig.4. It is understood that the A - , curve, with losses included,corresponds to the lasttwo columns ofTable 11. How is i computedwithout considering losses? Did the authors measurei, by readingthe peak value of the excitation current on a scope?. igure 5 implies that the core loss resistance seen by the supply-frequency current component decreasesas the supply voltage in-creases. However, the curves in the figure above, and results re-ported by othe r investigators [A] contradict th e authors’ findings.. he end results are in tabulated form (i.e., Table11 . From theapplications point of view, it may not be convenient to representtransformer nonlinearity in such a form. For example, in powersystem harmo nic studies, t best to express the peak values of eachharmonic component of i, and i, in analytical form (pe rhaps poly-nomial functions of the supply voltage).

Reference

[A] B. Szabados and J. Lee, “Harmonic Impedance Measurement ofTransformers,” IEEE Trans. Power App. and Syst., Vol. PAS-100,NO. 12, 1981, pp. 5020-6.

Manuscript received February 24, 1992.

Washington Neves’ and H erman n Dommel (University of BritishColumbia, B.C., Canada):

We would like to thank the discussers for their comments andquestions. We will address each discusser at a time.

Yahia Baghzouz: The AC measurements (TableI were performedusing a digital waveform analyzer. Analog voltmeter and amp ereme terwere used just for checking purposes. The current sample waveformwas taken from a standard0.1 ohm resistance connected in series withthe Epstein frame primary winding. Current and voltage waveforms(512 points) were obtained in the scope. The digital analyzer calcu-lated the r m s currents, voltages and no loadlosses.

The discusser has found multi-valuedA - , and U , peak curves.The vertical axis of the A - , curve should be related to the verticalaxis of the - , curve according to the equation:

’On leave from Universidade Federal da Paraiba, Campina Grande-PB,Brazil



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Where I s the peak value of the voltage,A p the peak value of theflux and o he angular frequency. If we take the lastU - , curve,V = 140.00\/2= 197.99 volts. We computed A for 60 Hz and ob-t&ed A p = 197.99/(2~ X 60) = 525.18 mWb _ P t .The axis in thecurve provided by th e discusser does not extend up to525.18 mWb - .A scale factor of \/2 is probably m issing.

In a transient, the flux and voltage may vary from small values tolarger ones. It is appropriate to represent them as single-valuedfunctions of current which give the best overall response. TheA ,curve obtained by our approach is theoretically identical to the DC

initial magnetization curve and extends as t he peak value of theflux

linkage increases. OurA - , curve is also single-valued and extendsas the peak value of the voltage increases. Although the method is notexact, it is very useful. We have used curves obtained by this approachto represent magnetizing branches of large transformers and carriedout simulations of inrush currents and ferroresonance. The simula-tions showed close agreement with field tests.

The last two columns of Table I1 show the computedA - , curvewhen losses are included in the model. To computei, without consid-ering the losses we just set losses equal to zero. The program assumesi , = 0 and computes i,. We calculated i , using our routine andcompared to t he initial magnetization curve. The measured curve wasthen obtained by DC measurements (Appedix B).

The shape of the - , and - , will depend on the core materialand transformer design. We have found distribution transformers inwhich the core loss resistance increases as the supply voltage in-creases. Unfortunately, we do not have information about the prow-dure used by the discusser to obtain thev - , and A - , curves, aswell as the core material used. We do not see any contradiction heresince we may be dealing with different materials.

Reference [A] presents an experimental method to obtain theimpedance for each harmonic component in a transformer. In ourwork we compute peak v - , and - curves. It seems there is nodirect relationship between reference[Aiand our paper.

The data presented in Ta ble I1 is suitable to be used in an Electro-magnetic Transients Program.If applications require analytical formsto represent U - , and A , curves, curve fittings of the resultsofTable I1 may be appropriate.

Robert J. Meredith: Although the parameters of the - , and- , curves were obtained from single frequency tests, we think we

can apply this model for different frequencies with reasonable accu-racy. Chua and Stromsmoe[3] had also obtained the iron core circuitparameters from single frequency tests. They made comparisons be-tween measurements and simu lation for different frequencies60, 20and 180 Hz) sing both sinusoidal voltage and cu rrent waveforms asinputs. A sinusoidal voltage signal applied to a nonlinear elementproduces a distorted current waveform. In the same way,if a sinu-soidal current is applied to a nonlinear element, the voltage will haveharmonic components. Chua and Stromsmoe had also carried outmeasurements in which a sinusoidal field was superimposed by aD.C.field. The agreement between simulations and measurement wasex-cellent.

There are uncertainties on iron cores modelling. Theflux distribu-

tion in the core may not be uniform even at low frequencies [I]. Itsaccurate calculation is very complex[II]. The core loss mechanism isalso complicated [III, IV]. We assumed the lu distribution do notchange in frequency. This modelis suitable for low frequency range(few wiz pplications. For situations in which the frequencies in-volved are mainly the natural frequencies of the system (for instance,switching off a transformer whena fault nearby is cleared), thefrequency dependent winding impedances andstray capacitances playa very important role. They should be modelled as well. We arecurrently dealing with distribution transformers that shows resonantpeaks as

lowas 4.0

khz during short circuit frequency response tests.We think our model is also useful for newly manufactured trans-formers. We are aware of the high permeability of the new magneticsteel laminations and the capacitive effect in the excitation current.Most of the transformers inservice are not new and probably do notshow this effect. The computation of t he nonlinear resistance does notpresent any problem since the capacitance does not affect powerlosses. However, winding capacitances mustbe included in the algo-rithm when computing the , curve for these transformers. Theyneed to be known somehow, either from tests or calculations. Theireffects can easilybe included in our algorithm. The knowledge of B-Hcurves from steel manufacturers does not help. Theyare usuallydifferent from B-H curves of transformers due toeffects such as airgaps and b utt joints.

It may be difficult to apply the model for deltaconnected threephase transformers whose dataare not suppliedby the manufacturerand in which th e delta connected w inding cannotbe opened for tests.

The application of the modelis not restricted. In power systemsthere are many single-phase transformers (transformer banks), severalfive-legged and shell-type three phase transformers, not to mentionthe large quantity of current and voltage transformers. We may useour mod el for those situations. Besides, the n umber of d ata required issmall. In our simulations, threelinear segments of the A - andU - , curves have successfully represented the magnetizing branch oftransformers.

We would like to thank the discussers again for their valuablecomments.

References

I. A. Basak and A. A. Abdul Qader, Fundamental and HarmonicFlux Behaviorin a 100 kVA Distribution Tmnsjormer IEEE Trans-actions on Magnetics, Vol.MAG-19, No. 5, September 1983, pp.2100-2102.

11. T. H. ODell , F m m a g n e t ~ n a m i c s ,hapter 1, The MacmillanPress LTD, ondon 1981.

111. C. W. Chen, Magnetism an dMetauwgVof Sofr Magnetic Materials,Chapter 4, Section 2.2, Dover Publications, Inc., New York,1986.

IV. S. Hill and K. . Overshott, rite 0tigi.n of the Anomalous Loss inGrain-oriented Silicon-iron, IEE Pub. 142, September 1976, pp.25-28.

Manuscript receivedApril 10 1992.

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Transactionson Power Systems on Modelling Iron Core Nonlinearities