Thomas 2006

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1820 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 7, JULY 2006

Time-Domain Simulation of Nonlinear TransformersDisplaying Hysteresis

D. W. P. Thomas , John Paul , Okan Ozgonenel , and Christos Christopoulos , Fellow, IEEE

George Green Institute of Electromagnetics Research, University of Nottingham, Nottingham NG7 2RD, U.K.Electrical and Electronic Engineering Department, Ondokuz Mayis University, Kurupelit-55139-Samsun, Turkey

This paper introduces a novel technique for modeling, in the time domain, a power transformer with nonlinear and hysteretic behavior.The model is particularly suitable for harmonic or protection studies where the transformer is driven into the nonlinear regime. Thetechnique uses a single-phase two-winding transformer model based on the JilesAtherton model of ferromagnetic hysteresis. It includeseddy-current loss by adding an extra single-turn winding so that the transients are modeled as fully as possible.

Index TermsHysteresis, JilesAtherton, modeling, nonlinear magnetics, transformers.

I. INTRODUCTION

POWER transformers are an important and universalcompo-nentof power systems,forwhichvariousmodelingschemes

have been developed for many years. Leibfried and Feser[1]in-

troduced the transfer function concept for power transformers.

This method is well known as an additional method of evalu-

atingthe impulse testof power transformers. Wilcoxintroduceda

time-domain modeling and modal analysis which described how

a new form of transformer model can be converted from the fre-

quency domain into the time domain for Electromagnetic Tran-

sients Program (EMTP) implementation [2]. To date none of the

models include an accurate representation of hysteresis as given

by theJilesAtherton (JA) model [3].TheJ-Amodelisaninter-

mediate solution of ferromagnetic hysteresis with some physical

basisandiscomputationallytractableforpracticalproblems.The

J-A model is well suited to transmission-line modeling (TLM)

for lumped components because it is formulated in terms of a

first-order differential equation.

This paper demonstrates a novel simulation program based

on the TLM method for simulating a power transformer which

include nonlinear hysteresis and eddy-current loss effects [4].

The TLM algorithm is discrete in nature and is compatible with

other time-domain simulation programs such as ATP, PSCAD-

EMTDC, or MATLAB[5]. This is an extension on a previous

published work on modeling of nonlinear inductors displaying

hysteresis[6].

II. TLM MODELING ANDAPPLICATION TOTRANSFORMERS

The basic TLM technique models linear reactive components

as transmission line segments (called stubs). The stub model

representing the inductor is terminated by a short circuit, to em-

phasize inductive behavior, current, and hence storage in the

magnetic field must be maximized. The TLM model for a capac-

itor is a stub with its far end open circuit. It emphasizes voltage

differences, storage in the electric field, and hence mainly ca-

pacitive behavior.

Digital Object Identifier 10.1109/TMAG.2006.874183

Fig. 1. (a) Inductance, (b) equivalent circuit, (c) stub model of inductance,

TLM equivalent circuit

In many applications, it is necessary to account for the non-

linear behavior of inductors and capacitors. These can be mod-

eled as nonlinear stubs. Magnetic coupling between components

may also be described using TLM. Mutual coupling is modeled

by a current-controlled voltage source[7],[8].

A. Linear Inductor Model

An inductor modeled using a transmission line stub is shown

in Fig. 1. The stub model representing the inductor is terminated

by a short circuit. For an inductor the voltage across it is

equal to . The differential term is thenreplaced in TLM by the discrete transform ,

where is the time step, and is the incident

pulse at thestub terminals. It is assumed that it takes onetime step

for the pulse to make a round-trip to travel to the end and be

reflectedback astheincident pulse inthe nexttimestep . Thus,

at any time step the voltage across the inductor is

(1)

Also, if the reflected amplitude at the inductor terminals is

then

(2)

0018-9464/$20.00 2006 IEEE

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THOMASet al.: TIME-DOMAIN SIMULATION OF NONLINEAR TRANSFORMERS DISPLAYING HYSTERESIS 1821

Fig. 2. TLM model of a nonlinear inductor.

where the subscript donates the th time step. For a short-

circuited transmission line stub, the incident pulse in the next

time step becomes

(3)

In a similar manner, capacitance can be simulated. The algo-

rithm can be shown to be equivalent to the trapezoidal integra-

tion method adopted by ATP and PSCAD [8].

B. Nonlinear Inductor[6]

A full description of the modeling of an inductor can be found

in[6].Theflux density in an inductor core of area is given by

, where is the coreflux. The constitutive relation fortheflux density is , where is the perme-

ability of free space, is the magneticflux density, and is

the magnetization intensity. From Amperes law ,

where is the number of winding turns and is the meanlength

of the magnetic path. Also, . These

relationships can then be combined using and

, where is the normalized magnetization giving

(4)

Thus, in TLM we can model the nonlinear inductance voltageas the sum of two voltages given by

(5)

(6)

At each time step , the incident waves are updated for the next

time step with

(7)

(8)

The equivalent circuit is then as given inFig. 2.

Fig. 3. Schematic diagram of a two-winding transformer.

C. JilesAtherton Model

The classical J-A model[3], [6]is described in the following

subsections.

1) Weighting Coefficient: The magnetization is split into two

parts, the anhysteretic magnetization and the irreversible mag-

netization. In normalized form, this is expressed by

(9)

where is the weighting coefficient with is

the normalized anhysteretic magnetization, and is the nor-

malized irreversible magnetization.

2) Modified Langevin Function: The anhysteretic magneti-

zation dependence is given by a modified Langevin function,

i.e.,

(10)

where is the normalized saturation magnetization, is

the interdomain coupling coefficient, and is the normal-

ized anhysteretic magnetization form factor. The coefficientsare positive constants. Also, denotes the mod-

ified Langevin function with argument .

To avoid difficulties with the modified Langevin function for

small arguments, a linear approximation is used where for

we put .

3) Differential Equation for the Irreversible Magnetization:

In the J-A model, the derivitive of the normalized irreversible

magnetization w.r.t. the inductor current is

(11)

where the migrationflag is given by

if and

if and

otherwise.

(12)

D. Modeling of Transformers

Consider a nonlinear lumped two winding transformer as

shown inFig. 3where the primary winding has turns, the

secondary winding has turns, the area of the core is , andthe magnetic path length is .

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Fig. 4. A general ideal transformer equivalent circuits for TLM modeling. (a) Linear transformer. (b) Nonlinear hysteretic transformer core.

The primary and secondary voltages ( and ) can be de-

rived in terms of the primary and secondary currents ( and )

and the magnetization intensity as for the lumped inductance

to give

(13)

where

and .

1) Ideal Linear and Lossless Transformers: For linear loss-

less transformers ,(14)then reduces to

(14)

where

, and .

A linear lossless transformer equivalent circuit is then as

given inFig. 4(a). The controlled sources representing mutual

terms of the type are as follows:

(15)

(16)

where .

The following simultaneous equations are solved for and

:

(17)

(18)

where ,

and is the source voltage. The incident TLM stub voltages

are calculated as for the linear inductor.

2) Ideal Transformer With Nonlinear Hysteretic Core: For an

ideal transformer with nonlinear and hysteretic core inductance,

the TLM model is as given inFig. 4(b). There is an extra source

term representing the magnetization and is given by

(19)

The magnetization is nonlinear so that an iterative solution

for the following simultaneous equations has to be found:

(20)

(21)

where and . We have chosen theNewtonRaphson technique for its efficiency and stability so a

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solution is found through the following iterative procedure:

(22)

where is the iteration number. is the Jacobian matrix given

by

(23)

and is given by the average solution to(20)and(21)which

is

(24)

The iteration is started with initial values taken from the pre-

vious time step and continued until suitable convergence criteria

are met. In this work, this is set as

and

and (25)

where is the convergence parameter ( in the simu-lations presented).

3) Full Transformer Model With Copper Losses Leakage

Inductance and Nonlinear Hysteretic Core: In the complete

model, we also include transformer leakage inductance in the

primary and secondary winding and resistive winding losses

and eddy-current losses. In this model, we have used a scrapless

[9]lamination core geometry as shown inFig. 5.

The copper loss resistance can be calculated for each

winding from[9]

(26)

where is the copper conductivity, is the mean turn

length, is the bare wire area, and is the number of turns

for winding .

The winding leakage inductance for each winding of a

scraplesslamination core and assuming concentric secondary

and primary windings as shown inFig. 5is given by[9]

(27)

The eddy-current losses are due to circulating currents in the

iron core and behave as a third winding the resistance of whichcan be calculated from the fundamental of power frequency .

Fig. 5. Typical scrapless core transformer geometry.

is the mean windinglength,

is the mean core length, and

is the lamination thickness.

For a linear magnetization, the eddy-current power loss at is

given by[9]

(28)

where is the primary rms input voltage, is the lamination

factor of the iron core, is the volume of the iron core, is

the iron core area, is the iron core conductivity, is the lam-

ination thickness, and is the number of turns in the primary

winding. It is normal to then simply represent the eddy-current

loss as due to a shunt impedance across the primary windinggiven by

(29)

A more complete and representative model of eddy-current loss

is obtained by adding a third single-turn winding loaded by a

resistance given by

(30)

The full TLM equivalent circuit for a nonlinear transformer in-

cluding leakage reactance, iron, and copper loss is as shown in

Fig. 6.

There are now three nonlinear simultaneous equations to

solve for and which have the form

(31)

(32)

(33)

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Fig. 6. Full transformer TLM model equivalent circuit with nonlinear hysteretic core, leakage inductance, conductor losses, eddy-current losses, and source andload impedances.

where

and the Jacobian matrix is now

(34)

III. RESULTS

To demonstrate the modeling procedure, a small 25 kVA,

11 kV/220 V power transformer with the parameters as given

in the Appendix was modeled using the TLM method described

inSection II. The JilesAtherton parameters were typical of a

core made of FeSi sheets[10]. The transformer was loaded with

a series resistor and inductor which would provide a 25 kW load

of 0.9 p.f. at 220 V.

As a comparison, the same transformer parameters and

electrical supply and load were modeled using the MATLABpower system block set with a transformer hysteretic core. The

MATLAB program uses a shunt linear resistance to represent

eddy-current losses and a semi-empirical hysteresis character-

istic curvefitting the empirical data defining the major loop and

single-valued saturation. The user defines: the remanent flux,

saturationflux, saturation current, coercive current (current at

zeroflux),flux slope at coercive current, and pairs of values of

currents and fluxes in the saturation region. This is a standard

approach which is also available in the EMTP simulationprogram[5].

Fig. 7shows the curve for a soft start simulation where

the supply voltage increases to rated value with a time constant

of 0.0434 s. Fig. 8 shows the curves produced by the

MATLAB simulation. It can be seen that, although the major

loop at rated voltage have been constructed to be similar for

both simulations, the loops at reduced supply voltages are dif-

ferent with the MATLAB simulation producing a large coercive

current at small supply voltages. The JilesAtherton model is

believed to be a more accurate representation[10],particularly

for transients.

Fig. 9 shows the simulated primary currents obtained using

the TLM with the JilesAtherton model compared with thosefrom the MATLAB simulink simulation. Fig. 10 shows the

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Fig. 7. TLM simulated curve under full load and the supply voltage in-

creasing to rated voltage

s.

Fig. 8. MATLAB simulink simulated curve under full load and thesupply voltage increasing to rated voltage s.

Fig. 9. Comparison of the simulated primary current using the TLM model or

the MATLAB simulink model for rated load as supply voltage increases to ratedvalue

s.

simulated secondary currents obtained using the TLM with the

JilesAtherton model compared with those from the MATLAB

simulink simulation. Fig. 11 shows the simulated primary

currents at no-load. The power losses due to hysteresis, copper,

and eddy current have been calculated and the transformer

efficiency is derived, the results of which are given in Table I.

The results obtained are typical of comparable commercial

power transformers[9]. Note that, for the steady state, the two

models have been designed to give comparable results.

Figs. 12and13compare the minor loops modeled using theJilesAtherton model and the MATLAB power system block

Fig. 10. Comparison of the simulated secondary current using the TLM modelor the MATLAB simulink model for rated load as supply voltage increases torated value s.

Fig. 11. Comparison of the simulated primary current using the TLM model orthe MATLAB simulink model for no load as supply voltage increases to ratedvalue s.

TABLE IRESULTS FORPOWER LOSSES, POWER TRANSFERRED, AND EFFICIENCY

(P INPUT/P OUTPUT %) AT RATED LOAD WITH 0.9 P.FANDRATEDSUPPLYVOLTAGE

set. The waveforms were created by initially operating at 0.3

p.u. of the rated supply voltage and then adding a small Gaussian

pulse of amplitude 0.05 p.u and width 0.02 s to give a dc offset as

given inFig. 14. The transient properties of the two methods are

clearly different. It is expected that the JilesAtherton model is

more accurate although more experimental measurements and

comparisons with other models are needed to confirm the tran-

sient properties presented in this work[11], [12].

IV. CONCLUSION

This paper introduces a novel technique for modeling, in thetime domain, a power transformer with nonlinear and hysteretic

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Fig. 12. TLM simulated curve minor loops for an initial 0.3 p.u. supply

with a dc pulse is inFig. 14.

Fig. 13. MATLAB simulink simulated curve minor loops for an initial

0.3 p.u. supply with a dc pulse as in Fig. 14.

Fig. 14. Supply voltage waveform used to reproduce hysteretic curves given inFigs. 12and 13.

behavior. A single-phase two winding transformer TLM model

is developed. The hysteretic model is based on JilesAtherton

model of ferromagnetic hysteresis and eddy-current loss is in-

cluded as an extra single-turn winding so that the transients are

modeled as fully as possible. The simulations produce a more

accurate transformer transient response than that currently avail-able in commercial power system simulation programs.

APPENDIXA

Data for 11 kV/220 V, 25 kVA power transformer

m Core dimension.

Magnetic path length.

Core area.m Lamination thickness.

S m Iron conductivity.

S m Copper conductivity.

Number of turns in primary.

Number of turn in secondary.

Lamination factor.

Window utilization factor.

a Mean turn length.

Primary copper loss resistance.

Secondary copper lossresistance.

H Primary leakage inductance.

H Secondary leakage inductance.

A m Saturation magnetization.

A m Anhysteretic form factor.

Interdomain coupling

coefficient.

A m Coercivefield magnitude.

Magnetization weighting

factor.

H m Permeability of the free space.

ACKNOWLEDGMENT

This work was supported in part by the UK EPSRC research

council.

REFERENCES

[1] T. Leibfried and K. Feser, Monitoring of power transformers usingthe transfer function method,IEEE Trans. Power Del., vol. 14, no. 4,pp. 13331341, Oct. 1999.

[2] D. J. Wilcox, Time-domain modelling of power transformers usingmodal analysis, IEE Proc.Elect. Power Appl. , vol. 144, no. 2, pp.7784, Mar. 1997.

[3] D.C. Jiles andD. L.Atherton, Ferromagnetic hysteresis,IEEE Trans.Magn., vol. MAG-19, no. 5, pp. 21832185, Sep. 1983.

[4] C. Christopoulos, The transmission line modeling method TLM,IEEE/OUP Series on Electromagnetic Wave Theory, 1995.

[5] ATP Rule Book, European EMTP-ATP Users Group.[6] J. Paul, C. Christopoulos, and D. W. P. Thomas, Time-domain sim-

ulation of nonlinear inductors displaying hysteresis,in COMPUMAG2003, Saratoga Springs, NY, Jul. 2003, pp. 182183.

[7] S.Y.R. Hui and C. Christopoulos, Discrete transform technique forsolving nonlinear circuits and equations, IEE Proc. ASci., Meas.Technol., vol. 139, no. 6, pp. 321328, Nov. 1992.

[8] S. Y. R. Hui and C. Christopoulos,Non-linear transmission line mod-eling technique for modeling power electronic circuits,in Proc. Eur.

Power Electronics Conf., Florence, Italy, 1991, vol. 1, pp. 8084.[9] W. T. McLyman, Transformer and Inductor Design Handbook. NewYork: Marcel Dekker, 2004.

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[10] A. Benabou, S. Clenet, and F. Piriou, Comparison of Preisach andJilesAtherton models to take into account of hysteresis phenomenonforfinite element analysis,J. Magn. Magn. Mater., vol. 261, no. 1, pp.139160, 2003.

[11] S.E. Zirka, Y.I. Moroz, P.Marketos, and A.J. Moses, A viscous-typedynamic hysteresis model as a tool for loss separation in conductingferromagnetic laminiations, IEEE Trans. Magn., vol. 41, no. 3, pp.11091111, Mar. 2005.

[12] G. Stumberger, B. Polajzer, B. Stumberger, M. Toman, and D. Dolinar,Evaluation of experimental methods for determining the magneticallynonlinear characteristics of electromagnetic devices, IEEE Trans.

Magn., vol. 41, no. 10, pp. 40304032, Oct. 2005.

Manuscript received March 1, 2005; revised December 30, 2005. Corre-sponding author: D. W. P. Thomas (e-mail: [email protected]).

David W. P. Thomas(M95) was born in Padstow, U.K., in 1959. He receivedthe B.Sc. degree in physics from Imperial College of Science and Technology,London, U.K., in 1981, the M.Phil. degree in space physics from Sheffield Uni-versity, Sheffield, U.K., in 1987, and the Ph.D. degree in electrical engineeringfrom Nottingham University, Nottingham, U.K., in 1990.

In 1990, he joined the Department of Electrical and Electronic Engineering,University of Nottingham, where he is now a Senior Lecturer. His research in-terests are in electromagnetic compatibility, electrostatic precipitation, and the

protection and simulation of power networks.

John Paul wasborn in Peterborough U.K., in 1960. He received the M.Eng. and

the Ph.D. degrees in electrical and electronic engineering from the Universityof Nottingham, Nottingham, U.K., in 1994 and 1999 respectively. His Ph.D.dissertation involved the application of signal processing and control system

techniquesto thesimulation of general material propertiesin time-domain TLM.He is currentlya Research Associatewith theGeorge Green Institutefor Elec-

tromagnetics Research at the University of Nottingham. His research interestsare in the application of signal processing techniques for material modeling in

time-domain computational electromagnetics, the simulation of complete sys-tems for electromagnetic compatibility studies, and the interaction of electro-magnetic waves with biological tissues.

Okan Ozgonenelwas born in Samsun, Turkey. He received the M.Sc. degreein electrical education from Marmara University in 1992 and the Ph.D. degree

in electrical engineering from Sakarya University in 2001.He has been with Ondokuz Mayis University, Samsun, Turkey, since 1991,

where he is a Lecturer in theElectrical andElectronics Engineering Department.Hismain research interestsare digital algorithms, digital signalprocessing, sim-ulation methods for power transformers, power system control and protection,and wavelet techniques.

Christos Christopoulos(F05) was born in Patras, Greece, on September 17,1946. He received the Diploma in electrical and mechanical engineering from

the National Technical University of Athens, Athens, Greece, in 1969 and theM.Sc. and D.Phil. degrees from the University of Sussex, Sussex, U.K., in 1974

and 1979, respectively.In 1974, he joined the Arc Research Project, University of Liverpool, Liver-

pool, U.K., and spent two years working on vacuum arcs and breakdown while

on attachment at the UKAEA Culham Laboratory. In 1976, he joined the Uni-versity of Durham, Durham, U.K., as a Senior Demonstrator in Electrical En-gineering Science. In October 1978, he joined the Department of Electrical andElectronic Engineering, University of Nottingham, Nottingham, U.K., where heis now Professor of Electrical Engineering. His research interests are in compu-tational electromagnetics, electromagnetic compatibility, signal integrity, pro-tection and simulation of power networks, and electricaldischargesand plasmas.He is the author of over 250 research publications and five books.

Dr. Christopoulos has received the Electronics Letters and the Snell Pre-miums from the Institute of Electrical Engineers and several conference BestPaper awards. He is a member of the Institute of Electrical Engineers (IEE),

U.K., and IoP. He is the Executive Team Chairman of the IEE ProfessionalNetwork in EMC, member of the CIGRE Working Group 36.04 on EMC,and Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY.

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