340 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 3, MARCH 2008

Characteristics of Jiles–Atherton Model Parameters and Their Applicationto Transformer Inrush Current Simulation

X. Wang1, D. W. P. Thomas1, M. Sumner2, J. Paul1, and S. H. L. Cabral3

George Green Institute of Electromagnetic Research, University of Nottingham, Nottingham NG7 2RD, U.K.School of Electrical and Electronic Engineering, University of Nottingham, Nottingham NG7 2RD, U.K.

Department of Electrical Engineering, Universidade Regional de Blumenau, Rua Sao Paulo 3250, Blumenau/SC Brazil

We describe a time-domain simulation technique for nonlinear hysteretic transformer behavior based on the Jiles–Atherton model butwith variable parameters. The normal Jiles–Atherton model assumes its parameters are independent of the applied maximum (peak)magnetic field intensity

max. In this paper, we demonstrate that these parameters vary with

maxfor a ferromagnetic material com-

monly found in transformers and we also propose a technique for modelling transformer behavior whenmax

is rapidly changing. Wepresent examples of inrush current simulation for transformer protection research and compare them with experimental measurements,demonstrating that the proposed algorithm is robust and accurate.

Index Terms—Hysteresis, inrush current, Jiles–Atherton model, parameter estimation.

I. INTRODUCTION

I N THE past few years, many researchers have realized thedrawback of the original Jiles–Atherton hysteresis model [1]

and, hence, the difficulty of its application in simulation. Mostproblems happen when the original model is applied to sim-ulate minor hysteresis loops or dynamic/transient conditions.For example, J. H. B. Deane [2] made some progress in mod-eling the simple nonlinear inductor circuit, but also found thatthe Jiles–Atherton model parameters should be revised signif-icantly if compared with the results from experimental mea-surement in [3]. Therefore, various modifications of the originalJiles–Atherton model were proposed. K. H. Carpenter [4] and D.Lederer et al. [5] suggested that scaling of either the irreversibleor the total magnetization should be done for minor hysteresisloops. By this way, the agreement between experimental mea-surement and model is good for large loops and only slightlyworse for smaller loops [5]. On the other hand, U.D. Annakkageet al. [6] replaced the modified Langevin function in the orig-inal model for more precision. R. Wilson [7] did much workon parameter optimization for hysteresis loops at different fieldstrengths. In addition, Lederer [5] also illustrated the changingtrend of the Jiles–Atherton model parameters, although onlythree parameters were allowed to be varied.

In this paper, the modified Langevin function is still used asthe description of anhysteretic magnetization. Then, the char-acteristics of the Jiles–Atherton model parameters for the samemagnetic material with different field strengths are investigated.To demonstrate how such characteristics can facilitate the appli-cation of the Jiles–Atherton model, examples of inrush current,which is one of the important phenomena in transformer protec-tion, are simulated and compared with experimental results.

In Section II, the problem is defined. In Section III, thecharacteristics of the Jiles–Atherton model parameters are

Digital Object Identifier 10.1109/TMAG.2007.914671

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

described. In Section IV, we show how inrush current can beaccurately simulated with these characteristics.

II. PROBLEM DESCRIPTION

The Jiles–Atherton hysteresis model, which is derived fromthe physical views of the magnetization process, decomposesthe whole magnetization into reversible component andirreversible component [1].

(1)

They are linked with magnetic field intensity by the mod-ified Langevin function (3) and the differential equation (4)

(2)

(3)

(4)

where and is given by

if andif andotherwise

, , , , and are five model parameters to be estimated.However, some problems happen if the uniform Jiles–Athertonmodel parameters, which are determined by only one hysteresisloop, are used to simulate all the other loops. The problems areillustrated as follows.

A. Estimate the Model Parameters From a Single ExperimentalHysteresis Loop

The Jiles–Atherton model parameters listed in Table I arecalculated from the experimentally measured hysteresis loopshown as the solid curve in Fig. 1 using the methods described

0018-9464/$25.00 © 2008 IEEE

WANG et al.: JILES–ATHERTON MODEL PARAMETERS AND THEIR APPLICATION TO TRANSFORMER INRUSH CURRENT SIMULATION 341

TABLE IMODEL PARAMETERS DETERMINED BY FIG. 1 (H = 394:4 A/m)

M : Saturation magnetizationa: Anhysteretic form factor�: Interdomain coupling coefficientk: Coercive field magnitudec: Magnetization weighting factor

Fig. 1. Determination of Jiles–Atherton model parameters from a single hys-teresis loop (H = 394:4 A/m)

in either [3] or [8]. Then, the simulated hysteresis loop by thetransmission-line modeling (TLM) technique [9], [10] incor-porated with Jiles–Atherton model using these parameters isgiven in Fig. 1 as the dashed curve. It indicates that a set ofJiles–Atherton model parameters can match the hysteresis loopfrom which such model parameters are calculated.

B. When the Model Parameters in Table I are Used to Simulatethe Other Hysteresis Loops, the Problem Happens

Four hysteresis loops with different maximum (peak) fieldstrengths are simulated using the TLM technique and the sameJiles–Atherton model parameters given in Table I and thenshown compared with the experimental results in Fig. 2. Fig. 2indicates that the uniform Jiles–Atherton model parameterscannot match all the hysteresis loops. The bigger the differencewith the maximum (peak) field strength of the hysteresisloop from which the model parameters are calculated, thebigger the error produced.

III. CHARACTERISTICS OF JILES–ATHERTON

MODEL PARAMETERS

Considering the problem described in the previous section,different Jiles–Atherton model parameters are calculated fromeach experimental hysteresis loop respectively and given inTable II. Table II shows how the Jiles–Atherton model parame-ters change with different field strengths.

In Fig. 3, the TLM simulations of the hysteresis loops usingthe model parameters given in Table II are compared with the

Fig. 2. Difference between measure and model if uniform parametersare used to simulate all hysteresis loops. (a) H = 222:8 A/m;(b)H = 160:4 A/m; (c) H = 118:7 A/m; (d) H = 92:48 A/m.

TABLE IIJILES–ATHERTON MODEL PARAMETERS FOR DIFFERENT HYSTERESIS LOOPS

experimental results. It indicates that the simulated hysteresisloops can have excellent agreement with the experimental re-sults when the model parameters are varied according to themaximum (peak) field intensity that the loop reaches.

IV. APPLICATION OF PARAMETER CHARACTERISTICS TO

SIMULATION OF TRANSFORMER INRUSH CURRENT

Inrush current is one of the common phenomena intransformer protection. The characteristics of the variableJiles–Atherton model parameters have to be applied to thesimulation of the transformer inrush current, because the prop-erty of inrush current guarantees that it goes through severalhysteresis loops with different degrees of saturation and, hence,different maximum (peak) magnetic field strengths. The firstfew cycles of inrush current may reach deep saturation andthe peak current decreases in the following cycles with time.Therefore, if the uniform/fixed Jiles–Atherton model parame-ters are used to simulate the whole inrush current, considerableerrors will be produced as shown in Fig. 4.

The experimental measurements of inrush current inSection IV were done on a 400 VA, 60 Hz, single-phaseshell-type custom-built transformer by switching on the powersupply at the specific moment. The circuit for measurement isshown in Fig. 5(a), where the secondary of the transformer is

342 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 3, MARCH 2008

Fig. 3. Results of measure and model match with each other if differentJiles–Atherton model parameters for different field strengths are used.(a) H = 394:4 A/m; (b) H = 298:6 A/m; (c) H = 222:8 A/m;(d) H = 160:4 A/m; (e) H = 118:7 A/m; (f) H = 92:48A/m.

Fig. 4. Inrush current by measure and by simulation with uniformJiles–Atherton model parameters.

open circuit. , , and are the resistance for time constantrestraint, the primary winding, and the measurement of current,respectively.

The details of the transformer and the circuit are listed asfollows and the geometrical size is shown in Fig. 5(b).

Fig. 5. (a) Circuit for inrush current measurement. (b) Geometrical size of thetransformer on which inrush current is measured.

• The rated voltage ratio of the transformer is 440 V/440 V.• The turn ratio of the transformer is 727/727.• The transformer core is made of nonoriented grain and the

thickness of the lamination is 0.5 mm.• is recorded by Tektronix TDS3012 oscilloscope.• is precise resistance with 1.072 ; is 95.46 ; is

approximately 11.3 .The dashed curve in Fig. 4 is the TLM simulation with the

uniform Jiles–Atherton model parameters that are calculatedfrom the hysteresis loop in Fig. 3(e) with A/m.But in the first few cycles, the actual inrush current makesmagnetic field far exceed 118.7 A/m, so an observable errorcan be found if such uniform Jiles–Atherton model parametersare used. To control this kind of error, the appropriate variablemodel parameters with respect to the actual field intensityshould be adopted. The way to do this is described in thefollowing.

First, the variation of the Jiles–Atherton model parameters isrepresented by the functions with respect to maximum (peak)magnetic field intensity of the hysteresis loops. Such functionscan be obtained by any suitable fitting technique. Here, we rec-ommend rational functions because of their simplicity.

As shown in Fig. 6, the following functions give a good fit tothe changing trend of the Jiles–Atherton model parameters. Onthe other hand, there are obvious knees in Fig. 6(a), (b), and (d).The knees in the curves appear around the area where isapproximately 250 A/m. It seems to be the point over which the

WANG et al.: JILES–ATHERTON MODEL PARAMETERS AND THEIR APPLICATION TO TRANSFORMER INRUSH CURRENT SIMULATION 343

Fig. 6. Fitting for Jiles–Atherton model parameters. (a) Parameter a versusH ; (b) parameter� versusH ; (c) parameter k versusH ; (d) param-eter c versus H .

saturation becomes more apparent

(5)

Meanwhile, because all the model parameters must be posi-tive, the constraints in (6) are added to (5)

(6)

Second, because the actual magnetic field intensity isunknown before the start of the simulation, an algorithmbased on an iterative process is proposed. First, a set of initialJiles–Atherton model parameters are used, then the maximum(peak) is obtained after the simulation of the first cycle.The model parameters are then updated according to (5). Ifthe model parameters have not changed significantly comparedwith those before the update, then the simulation moves tothe next half cycle. The algorithm is depicted in Fig. 7. Here,“significantly” is defined as the change of more than 1%.

To validate and interpret this algorithm, two cases of the in-rush current simulation on that custom-built transformer are il-lustrated and compared with experiments.

A. Case 1

Conditions under which such inrush current takes place areV (RMS) with initial phase 0.527 rad and rema-

nent magnetization A/m.Two cycles are taken as the examples. In the first cycle,

three iterations are performed before the model parameters are

Fig. 7. Process to get correct Jiles–Atherton model parameters in inrush currentsimulation.

Fig. 8. Procedure for model parameters to converge to correct values. (a) Cycleno. 1. (b) Cycle no. 2.

deemed to have converged to a reasonable accuracy. In thesecond cycle, four iterations are performed before convergenceis reached. Fig. 8 shows the details of the iterations and Fig. 9

344 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 3, MARCH 2008

Fig. 9. Procedure for correct inrush current simulation to build up.

Fig. 10. (a) Inrush current and (b) magnetizing procedure.

indicates how the simulated current changes with each iter-ation. In Fig. 9, the currents within the last two iterations ofany cycle almost match with each other because convergenceis reached. In Fig. 10, the complete simulated inrush current iscompared with the experimental measurement. A maximum offour iterations is needed in any cycle.

Note that because of the remanent magnetization, each halfcycle has a different .

B. Case 2

Conditions under which such inrush current takes place areV (RMS) with initial phase 3.073 rad and rema-

nent magnetization A/m.Again, two cycles are taken as the examples. In the first cycle,

three iterations are performed and in the second cycle there arefive iterations as depicted in Fig. 11. Fig. 12 shows how the sim-ulated inrush current is built up after the iterations. In Fig. 12, thecurrents within the last two iterations of any cycle almost matchwith each other because convergence is reached. In Fig. 13 thecomplete simulated inrush current is compared with the mea-sured current.

Fig. 11. Procedure for model parameters to converge to correct values.(a) Cycle no. 1. (b) Cycle no. 2.

Fig. 12. Procedure for correct inrush current simulation to build up.

V. CONCLUSION

A modified Jiles–Atherton model has been described inwhich the model parameters are varied according to the max-imum (peak) magnetic field intensity in any given powerfrequency half cycle. First, the careful estimation of the modelparameters variation with maximum (peak) magnetic fieldintensity is performed and rational functions are fitted to thesevariations. Inrush current is then simulated using the time-do-main TLM method where the Jiles–Atherton model parametersare repeatedly adjusted in each cycle until they have convergedto a reasonable tolerance. It is shown that this approach isstable and produces excellent agreement with the measured

WANG et al.: JILES–ATHERTON MODEL PARAMETERS AND THEIR APPLICATION TO TRANSFORMER INRUSH CURRENT SIMULATION 345

Fig. 13. (a) Inrush current and (b) magnetizing procedure.

current inrush transients. The technique was demonstrated ona small transformer but it should be valid and scalable to anytransformer type and most transformers.

On the other hand, we only investigate the application of theproposed methods on the simulation of inrush current, which isone of the important phenomena in transformer protection. In-rush current is basically periodic and the excitation is mainlysinusoidal with power frequency. More work is required to ex-pand to higher frequencies and more complex applications, e.g.,in power electronic systems.

ACKNOWLEDGMENT

X. Wang would like to thank the High Voltage Laboratory inUniversidade Regional de Blumenau Brazil for kindly providingexperiment devices.

REFERENCES

[1] D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,”J. Magn. Magn. Mater., vol. 61, no. 1-2, pp. 48–60, Sep. 1986.

[2] J. H. B. Deane, “Modeling the dynamics of nonlinear inductor circuits,”IEEE Trans. Magn., vol. 30, no. 5, pp. 2795–2801, Sep. 1994.

[3] D. C. Jiles, J. Thoelke, and M. Devine, “Numerical determination ofhysteresis parameters for the modeling of magnetic properties usingthe theory of ferromagnetic hysteresis,” IEEE Trans. Magn., vol. 28,no. 1, pp. 27–35, Jan. 1992.

[4] K. H. Carpenter, “A differential equation approach to minor loops inthe Jiles-Atherton hysteresis model,” IEEE Trans. Magn., vol. 27, no.6, pp. 4404–4406, Nov. 1991.

[5] D. Lederer, H. Igarashi, A. Kost, and T. Honma, “On the parameteridentification and application of the Jiles-Atherton hysteresis model fornumerical modelling of measured characteristics,” IEEE Trans. Magn.,vol. 35, no. 3, pp. 1211–1214, May 1999.

[6] U. D. Annakkage, P. G. McLaren, E. Dirks, R. P. Jayasinghe, and A.D. Parker, “A current transformer model based on the Jiles-Athertontheory of ferromagnetic hysteresis,” IEEE Trans. Power Del., vol. 15,no. 1, pp. 57–61, Jan. 2000.

[7] P. R. Wilson and J. N. Ross, “Definition and application of magneticmaterial metrics in modeling and optimization,” IEEE Trans. Magn.,vol. 37, no. 5, pp. 3774–3780, Sep. 2001.

[8] X. Wang, D. W. P. Thomas, M. Sumner, J. Paul, and S. H. L. Cabral,“Numerical determination of Jiles–Atherton model parameters,” IEEETrans. Magn., submitted for publication.

[9] 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. 182–183.

[10] D. W. P. Thomas, J. Paul, O. Ozgonenel, and C. Christopoulos, “Time-domain simulation of nonlinear transformers displaying hysteresis,”IEEE Trans. Magn., vol. 42, no. 7, pp. 1820–1827, Jul. 2006.

Manuscript received August 23, 2007; revised November 27, 2007. Corre-sponding author: X. Wang (e-mail: xiaohui.wang@ieee.org).

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