K. Chwastek, J. Szczygłowski
Czestochowa University of Technology, Czestochowa, Poland
Abstract
The paper describes an approach to modelling hysteresis loops in grain-oriented steel.
The model comprises ideas inherent in the Jiles-Atherton description and the product model proposed by Gy. Kádár. For estimation of model parameters a robust optimization routine - DIRECT sampling algorithm is used. Some of model parameters are found to obey power laws with respect to the relative magnetization level. A good agreement between the measured and modelled loops is obtained.
Introduction
Modelling of hysteresis loops are important in electrical engineering for optimal design of magnetic circuits used in cores of electric devices.
The knowledge of such macroscopic material properties (Fig. 1) as saturation flux density, coercivity, magnetization at remanence point, loss density, obtained by numerical integration of loop area, as well on their dependence on processing conditions, variations of ambient temperature, external stress etc., are helpful for tailoring the working point of a magnetic circuit.
Fig. 1. A family of hysteresis loops and some related quantities
A number of possible approaches to model magnetization curves has been developed in the past. The description proposed by Prof. David Jiles and Prof. David Atherton
[1] is one of the most popular ones, as it is based on physical premises and it is relatively easy to be implemented. The Jiles-Atherton theory was initially developed to describe saturated hysteresis loops in isotropic materials. Its further extensions allowed to model the behaviour of minor loops [2], anisotropic and textured materials
[3-6] and the influence of eddy currents on the shape of hysteresis loop [7-11]. The magnetomechanical effects, important from the practical point of view (NDT, sensors) have also been included into the description [12-15]. The scalar model has been vectorized [16-18] and incorporated in finite element calculations [17-23]. Some commercial circuit simulation packages like PSpice (at present part of
Orcad/Cadence) [24-27] or Saber [28] avail of Jiles-Atherton description as the standard hysteresis modelling tool.
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The approach, initially proposed for description of ferromagnetic hysteresis, was later used to model the phenomenon in ferroelectric, ferroelastic and piezoceramic materials [29-31]. The unquestionable popularity of the description, its wide application range and simplicity of numerical implementation are serious reasons for studying its properties.
Model description
Jiles-Atherton model considers the hysteresis phenomenon as a result of energy dissipation during domain wall motion on structural defects (inhomogeneities, impurities, dislocations, inclusions, voids etc.), termed as pinning sites. Hysteresis loop is obtained by offsetting the irreversible magnetization from the anhysteretic curve, which describes a theoretical structure devoid of pinning sites.
(1) dM irr dH e


M

M an k


M irr

In Equation (1) M denotes irreversible magnetization, irr
M is anhysteretic an magnetization, within material,
H

H
 
M is the so-called effective field, which actually exists e

is a parameter, which is a measure of internal feedback, k is a model parameter, which is related to pinning site density and its value is roughly equal to coercivity,

is introduced to distinguish the ascending and descending parts of the loop, whereas

M

0 , 5

1
 sign
 
M an

M irr

 dH / dt
 
eliminates non-physical model behaviour (negative slopes of hysteresis loop after a sudden change of sign of external field).
Fig. 2. Hysteresis loop and anhysteretic curve
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The fundamental model equation (1) has to be supplemented with additional relationships, which link irreversible differential susceptibility dM irr
/ dH to total differential susceptibility dM / dH and define anhysteretic magnetization M an
.
For further considerations we assume the form of equations proposed recently in Ref.
[32]. In order to describe anhysteretic magnetization we choose the Brillouin function
(2) M an

M s


2 J
2 J

1 coth
H a e

1
2 J a
H e


M s is saturation magnetization, a is one of model parameters, whereas J

0 .
5 is assumed for grain-oriented steels. The choice of Brillouin function is justified on physical grounds [33].
The total differential susceptibility may be given with the following equation:
(3) dM dH



1



M
M s


2




 dM dH irr

 .
A similar formula was proposed for the first time in a modification of another popular hysteresis model – product Preisach model [34, 35].

is a model parameter.
Estimation of model parameters
The values of the following model parameters

,

, a , k , M s
should be estimated. For their estimation a number of possible approaches, including artificial intelligence methods, has been developed. A recent review is given in Ref. [36]. For estimation of model parameters we have chosen the robust optimization algorithm DIRECT, described in detail in Refs. [37, 38] and implemented in Matlab by Daniel E. Finkel
[39]. The choice of estimation method, different from the original procedure proposed by model author [40], was justified with the following reasons:
 the alternative formulation is robust and insensitive to noise introduced e.g. by random measurement errors,
 the method is easily implemented numerically and may be extended to include additional variables (degrees of freedom),
 the calculation time is reasonably low (several minutes on a weaker PC),
 the algorithm is deterministic, so there is no need to repeat the calculations many times, like for example in the case of genetic algorithms [40] or particle swarm optimization [41].
The idea behind the method is to transform the search space, where the values of model parameters are sought, into a unit hypercube. The number of model parameters
(here equal to five) is the number of dimensions of unit hypercube. The optimization process is carried out iteratively in such a way, that in successive steps the space of unit hypercube is being shrunk according to a strategy, that identifies the possible global minima on the basis of fitness values in some sampled data points. The choice of data points to be sampled is precisely determined ahead.
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Fig. 3. The idea behind the DIRECT algorithm: the darker field indicates the region of identified global minimum [43].
The fitness value is given as the sum of squared errors in magnetization for a number of points on the measured and the modelled hysteresis loops. The details on the application of DIRECT algorithm to the issue of estimation of Jiles-Atherton model parameters are disclosed in [43].
In the course of our forthcoming research we have found, that it is necessary to assume some functional dependencies for two model parameters a , k in order to obtain a good agreement between the modelled and the measured minor loops (loops which do not reach saturation). It was assumed, that these parameters could obey power laws [44, 45]
(4) k minor
  k major b
 a minor
  a major b

Basic properties of the examined steel, measurements, estimation
Several measurements for different grades of steel used in electrical engineering have been carried out in MALET (Materials for Low-Energy Consuming Technologies in
Electrotechnics) Centre of Excellence, located at Institute of Electrical Engineering,
Wrocław, Poland. These included cold rolled fully processed non-oriented steel sheets for use in alternators and cold rolled grain-oriented steel sheets used as core material in power transformers. The steel samples were supplied by the leading Polish producer Stalprodukt S.A. from Bochnia. Stalprodukt S.A. produces cold rolled electrical sheets and strips for more than 25 years. It has introduced Quality
Management System (compatible with EN ISO 9000 standard), Environmental
Management System (according to PN - EN ISO 14001 standard) and has been certified by TÜV CERT Anlagentechnik GmbH as compatible with PN - EN ISO
9001:2000 standard. Table 1 includes the catalogue reference data for grain-oriented steel produced in Bochnia.
Grain-oriented steel is commonly used as core material in transformer laminations. It features a strong favourable texture (110)[001] (Goss texture). The maximum dispersion of different grains with respect to the rolling direction in industrially produced steels does not usually exceed 7 % [46]. The remarkable texture of grainoriented alloys, together with a large grain size (from a few millimeters to a few centimeters) and a low content of impurities, lead to coercive fields as low as
4-10 [A/m] and maximum permeabilities around 5

10
4
. These values differ by about an order of magnitude from those typically found in non-oriented steels. Wide-grained highly textured laminations are obtained during a complex metallurgical processing, whose basic steps are [46]:
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 melting of the master alloy (Si concentration around 2.9-3.2 %, Al, Mn, Sb, S, N additions in concentrations around few hundred ppm)
 slab reheating (1250-1350 o
C)
 hot rolling
 annealing (900-1000 o
C)
 fast cooling (down to approximately 50 o
C)
 cold reduction to final thickness
 decarburizing and primary recrystallization (800-850 o
C)

MgO coating and coiling (50 o
C)
 box annealing and secondary recrystallization (1200 o
C for many hours)
 phosphate coating and thermal flattening (50 o
C)
Table 1
Magnetic properties of cold rolled grain oriented sheets from Stalprodukt S.A.
Thickness
Commercial designation
Standard designation according to
EN 10107
Maximum specific total core loss at
50Hz,
1.7T
Typical specific total core loss at
50Hz,
1.7T
Minimum magnetic polarization for
H = 800A/m
Typical magnetic polarization for
H = 800A/m mm - - W/kg W/kg T T
0.27
0.30
ET 114-27
ET 120-27
1.14
1.20
ET 130-27 M130-27S 1.30
ET 140-27 M089-27N 1.40
ET 117-30 M117-30P 1.17
ET 122-30 1.22
ET 130-30 1.30
ET 140-30 M140-30S 1.40
1.11
1.17
1.23
1.34
1.15
1.19
1.25
1.34
1.83
1.82
1.80
1.77
1.85
1.83
1.82
1.80
1.88
1.86
1.85
1.84
1.88
1.87
1.86
1.84
0.35
ET 150-30 M097-30N 1.50
ET 130-35 1.30
ET 140-35 1.40
ET 150-35 M150-35S 1.50
1.44
1.27
1.34
1.44
1.75
1.84
1.83
1.80
1.81
1.87
1.86
1.84
ET 160-35 M111-35N 1.65 1.55 1.75 1.78
Present trends in the development of low loss grain-oriented laminations are to combine high-purity and sharp (110)[001] orientation (what increases saturation flux density and leads to low coercivity) with low thickness. Further improvement of properties, applied in top-class high-permeability GO materials, may be achieved by mechanical, plasma jet or laser scribing [46-49], which lead to a substantial domain refinement. The challenges for metallurgy and demands for electrical engineering in optimizing the magnetic properties of contemporary grain-oriented alloys, whose global production is to reach 2 .
07

10 6 tones in 2010, were presented in an interesting panel discussion during the 17th SMM (Soft Magnetic Materials) in 2007 held in
Cardiff, Wolfson Centre for Magnetics [50].
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The measurements of hysteresis loops and associated loss were carried out using a computer-aided laboratory stand in Single Sheet Tester device, in conformance with the requirements of IEC 60404-3 standard, i.e. for sine wave of flux density. The extended type B uncertainty of loss measurement related to errors introduced by the measurement system was lower than 1.5 %.
The steel grade under examination for the purpose of this paper was ET 122-30
(0.3 mm thick). The grades ET 120-27 and ET 130-35 are examined in greater detail in Ref. [45]. The frequency and the amplitude of flux density were equal to 5 Hz and
1.8 T, respectively. It was assumed, that for this frequency and gauge of examined steel the dynamic effects from eddy currents could be neglected.
The estimated set of model parameters is given in Table 2, whereas the measured and the modelled major hysteresis loops are depicted in Figure 4. Figure 5 depicts the course of the estimation process. The algorithm has reached the final fitness value
3.86

10
10
[(A/m)
2
] determined as the sum of squared errors in 33 reference points because of exceeding the assumed number of iterations. The number of data points used was redundant in comparison to the problem dimensions in order to diminish the possible effect of measurement errors.
Table 2
Estimated set of model parameters for ET 122-30 steel sample
Parameter

Value
1.26

10
-5 a
 k
M s
17.3
2546.5
17.3
1.82

10
6
Fig. 4. The measured and modelled major hysteresis loops.
137

Fig. 5. The course of the estimation process.
Keeping the values of the parameters
 and

M s
fixed, the scaling coefficients for the dependencies a=a(b) and k=k(b) were determined, again with the use of DIRECT algorithm. The dependencies are depicted in Figures 6 and 7. It can be stated, that these two parameters could indeed obey scaling laws with respect to relative magnetization level. Similar power laws are recently the subject of intensive research in reference to e.g. plastic deformation [51] and other physical phenomena [52-53].
Fig. 6. A fitting for the k =k (b) dependence.
138

Fig. 7. A fitting for the a =a (b) dependence.
Figure 8 depicts an exemplary modelled minor loop, whose parameters were updated according to the abovegiven power laws (solid line). For comparison a modelled minor loop, whose parameters were kept the same as for the major loop is shown (dot line). The discrepancy between model and experiment is much larger in the latter case. Flat regions of the modelled loops after field reversal are caused by the introduced parameter



, which eliminates non-physical negative susceptibilities and also makes the model equations well conditioned to be introduced into finite element codes.
Fig. 8. The measured and modelled minor hysteresis loops: solid line – model with updated parameters, dot line – model with the same parameters as for the major loop.
139

Conclusions
In the paper an approach to modelling hysteresis loops based on Jiles-Atherton description is proposed. The model is applied to magnetization curves of a grainoriented anisotropic steel ET 122-30, used in electrical engineering. For estimation of model parameters a deterministic DIRECT algorithm was used. Some of model parameters were expressed as power laws with respect to relative magnetization level.
It was found that this approach yields much better modelling results for minor loops compared to the case, when the values of model parameters were kept the same as for the major loop.
Acknowledgements
Acknowledgements are due to Stalprodukt S.A., Bochnia, Poland for supplying samples of electrical steel sheets. Authors are grateful to Prof. Wiesław Wilczyński from Institute of Electrical Engineering for help with magnetic measurements.
Authors acknowledge a seminal discussion on the description with the model creator,
Prof. David Jiles, which took place during the last 1 & 2 D Magnetic Measurement
Symposium at Wolfson Centre for Magnetics, Cardiff in September 2008.
References
[1] Jiles D.C., Atherton D.L., “Theory of ferromagnetic hysteresis”, J. Magn. Magn.
Mater., 1986, 61, 48-60.
[2] Jiles D.C., “A self consistent generalized model for the calculation of minor loop excursions in the theory of hysteresis”, IEEE Trans. Magn., 1992, 28 (5), 2602-4.
[3] Ramesh A., Jiles D.C., Roderick J.M., “A model of anisotropic anhysteretic magnetization”, IEEE Trans. Magn., 1996, 32 (5), 4234-6.
[4] Ramesh A., Jiles D.C., Bi Y., “Generalization of hysteresis modeling to anisotropic materials”, J. Appl. Phys., 1997, 81 (8), 5585-7.
[5] Jiles D.C., Ramesh A., Shi Y., Fang X., “Application of the anisotropic extension of the theory of hysteresis to the magnetization curves of crystalline and textured magnetic materials”, IEEE Trans. Magn., 1997, 33 (5), 3961-3.
[6] Shi Y.M., Jiles D.C., Ramesh A., “Generalization of hysteresis modeling to anisotropic and textured materials”, J. Magn. Magn. Mater., 1998, 187, 75-8.
[7] Jiles D.C., “Frequency dependence of hysteresis curves in conducting magnetic materials”, J. Appl. Phys., 1994, 76 (10), 5849-55.
[8] Jiles D.C., “Modelling the effect of eddy current losses on frequency dependent hysteresis in electrically conducting media”, IEEE Trans. Magn., 1994, 30 (6),
4326-8.
[9] Szczygłowski J., “Influence of eddy currents on magnetic hysteresis loops in soft magnetic materials”, J. Magn. Magn. Mater., 2001, 233, 97-102.
[10] Kis P., Iványi A., “Parameter identification of Jiles-Atherton model with nonlinear least-square method”, Physica B, 2004, 343, 59-64.
[11] Chwastek K., “Modelling of dynamic hysteresis loops using the Jiles-Atherton approach”, Math. Comp. Model. Dynam., 2009, 15 (1), 95-105.
[12] Sablik M.J, Kwun H., Burkhardt G.L., Jiles D.C., “Model for the effect of tensile and compresive stress of ferromagnetic hysteresis”, J. Appl. Phys., 1987, 61 (8),
3799-801.
[13] Garikepati P., Chang T.T., Jiles D.C., “Theory of ferromagnetic hysteresis: evaluation of stress from hysteresis curves”, IEEE Trans. Magn., 1988, 24 (6),
2922-4.
140

[14] Jiles D.C., “Theory of the magnetomechanical effect”, J.Phys. D: Appl. Phys.,
1995, 28, 1537-46.
[15] Stevens K.J., “Stress dependence of ferromagnetic hysteresis loops for two grades of steel”, NDT&E Int., 2000, 33, 111-21.
[16] Bergqvist A.J., “A simple vector generalization of the Jiles-Atherton model of hysteresis”, IEEE Trans. Magn., 1996, 32 (5), 4213-5.
[17] Iványi A., “Hysteresis in rotation magnetic field”, Physica B, 2000, 275, 107-13.
[18] Szymański G., Waszak M, “Vectorized Jiles-Atherton hysteresis model”,
Physica B, 2004, 343, 26-9.
[19] Toms H.L., Colclaser R.G., Krefta M.P., “Two-dimensional finite element magnetic modeling for scalar hysteresis effects”, IEEE Trans. Magn., 2001, 37 (2),
982-88.
[20] Sadowski N., Batistela N.J., Bastos J.P.A., Lajoie-Mazenc M., „An inverse Jiles-
Atherton model to take into account hysteresis in time-stepping finite-element calculations”, IEEE Trans. Magn,, 2002, 38 (2), 797-800.
[21] Benabou A., Clénet S., Piriou F., “Comparison of Preisach and Jiles-Atherton models to take into account hysteresis phenomenon for finite element analysis”,
J. Magn. Magn. Mater., 2003, 261, 139-60.
[22] Lee J.Y., Lee S.J., Melikhov Y. Jiles D.C., Garton M., Lopez R., Brasche L.,
“Incorporation of hysteresis effects into magnetic finite element modeling”,
AIP Conf. Proc. ,,Qualitative nondestructive evaluation”, 2004, vol. 700, 429-436
[23] Gyselink J., Dular P., Sadowski N., Leite J., Bastos J.P.A., “Incorporation of a
Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations”,
Compel, 2004, 23 (3), 685-93.
[24] Prigozy S., “PSPICE computer modeling of hysteresis effects”, IEEE Trans.
Educat., 1993, 36 (1), 2-5.
[25] Fujiwara T., Tahara R., “Eddy current modeling of silicon steel for use on
SPICE”, IEEE Trans. Magn., 1995, 31 (6), 4059-61.
[26] Cundeva S., “A transformer model based on Jiles-Atherton theory of ferromagnetic hysteresis”, Serb. J. Electr. Eng., 2008, 5 (1), 21-30.
[27] Ben-Yaakov S., Peretz M.M., “Simulation bits: a SPICE behavioral model of non-linear inductors”, IEEE Power Electronics Society Newsletter, 2003, 9-10.
[28] Wilson P.R., Neil Ross J., Brown A.D., “Magnetic material model characterization and optimization software”, IEEE Trans. Magn., 2002, 38 (2),
1049-52.
[29] Smith R.C., “Smart structures: model development and control applications”,
Technical Report CRSC-TR-01-01, North Carolina State University, Raleigh 2001.
[30] Smith R.C., Seelecke S., Ounaides Z., Smith J., “A free energy model for hysteresis in ferroelectric materials”, Technical Report CRSC-TR-03-01, North
Carolina State University, Raleigh 2001.
[31] Smith R.C., Massad J.E., “A unified methodology for modelling hysteresis in ferroelectric, ferromagnetic and ferroelastic materials”, Technical Report
CRSC-TR-01-10, North Carolina State University, Raleigh 2001.
[32] Chwastek K., “Frequency behaviour of the modified Jiles-Atherton model”,
Physica B, 2008, 403, 2484-7.
[33] Bouktache S., Azoui B., Féliachi M., “A novel model for magnetic hysteresis of silicon-iron sheets”, Eur. Phys. J. Appl. Phys., 2006, 34, 201-4.
[34] Kádár Gy., “On the Preisach function of ferromagnetic hysteresis”,
J. Appl. Phys., 1987, 61, 4013-5.
141

[35] Kádár Gy., “On the product Preisach model of hysteresis”,
Physica B 275, 2000, 40-4.
[36] Chwastek K., Szczygłowski J., “Estimation methods for the Jiles-Atherton model parameters - a review”, Przegl. Elektr. (Electrical Review), 2008, 12, 145-8.
[37] Jones D.R., Perttunen C.D., Stuckman B.E., “Lipschitzian optimization without the lipschitz constant”, J. Optimiz. Theory Appl., 1993, 79(1), 157-81.
[38] Jones D.R., “The DIRECT global optimization algorithm”, In Encyclopedia of
Optimization, vol. 1, Kluwer Academic Publishers, Boston, 2001, 431-40.
[39] Finkel D.E., “Global optimization with the DIRECT algorithm”, PhD Thesis,
North Carolina State University, Raleigh 2005.
[40] Jiles D.C., Thoelke J.B., Devine M.K., “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. Magn.,1992, 28, 27-35.
[41] Chwastek K., Szczygłowski J., “Identification of a hysteresis model parameters with genetic algorithms”, Math. Comput. Simulat., 2006, 71, 206-11.
[42] Marion R., Siauve N., Raulet M.-A., Krähenbühl L., Chwastek K.,
Szczygłowski J., Wilczyński W., “A comparison of identification techniques for the
Jiles-Atherton model of hysteresis”, XX Symposium Electromagnetic
Phenomena in Nonlinear Circuits EPNC’2008, 2-4.06.2008, Lille, 55-6.
[43] Chwastek K., Szczygłowski J., “An alternative method to estimate the parameters of Jiles-Atherton model”, J. Magn. Magn. Mater., 2007, 314, 47-51.
[44] Chwastek K., Szczygłowski J., Wilczyński W., Marion R., Raulet M.-A.,
Zitouni Y., Krähenbühl L.: “Modelling minor hysteresis loops of high silicon steel using the modified Jiles-Atherton approach”, Przegl. Elektr., 2009, 1, 68-70.
[45] Chwastek K., Szczygłowski J., Wilczyński W., “Dynamic hysteresis loops of steel sheets modelled with the modified Jiles-Atherton approach”, Compel, 2009, 3
(28), 603-12.
[46] Fiorillo F., “Highlights on soft magnetic materials” in “Magnetic properties of matter” Proceedings of the National School “New Developments and Magnetism’s
Applications”, L. Lanotte, F. Lucari and L. Pareti (Eds.), S. Agnello di Sorrento,
Napoli, 23-28.10.1995, World Scientific, 26-55.
[47] Verbrugge B., Jiles D.C., “Core loss reduction in electrical steels through material processing”, J. Appl. Phys., 1999, 85 (8), 4895-7.
[48] Dragoshanskii Yu.N., Gubernatorov V.V., Sokolov B.K., Ovchinnikov V.V.,
“Structural inhomogenenity and magnetic properties of soft magnetic materials”,
Doklady Physics, 2002, 47 (4), 302-4. Translated from Доклады Акад. Наук, 2002,
383 (6), 761-3.
[49] Dragoshanskii Yu.N., Sokolov B.K., Pudov V.I., Reutov Yu. Ya., Tiunov V.F. and Gubernatorov V.V., “Improvement of the properties of anisotropic soft magnetic materials by laser treatment and monitoring of its efficiency”, Doklady Physics, 2003,
48 (7), 340-2.
[50] Davies H.A. et al.
, “Challenges in optimizing the magnetic properties of bulk soft magnetic materials”, J. Magn. Magn. Mater., 2008, 320, 2411-22.
[51] Koslowski M., “Scaling laws in plastic deformation”, Phil. Mag., 2007, 87, 1175-
84.
[52] Stanley H. E., “Scaling, university and renormalization: three pillars of modern critical phenomena”, Rev. Mod. Phys., 1999, 71, S358-66.
[53] Bunde A., Havlin Sh. (Eds.), “Fractals and disordered systems”, Springer Verlag,
Berlin, 1991.
142