K. Chwastek, J. Szczygłowski
Czestochowa University of Technology, Czestochowa, Poland
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.
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.
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.
dM irr  M M an  M irr 
dH e
In Equation (1) M irr denotes irreversible magnetization, M an is anhysteretic
magnetization, H e  H  M is the so-called effective field, which actually exists
within material,  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
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
 2J 1
H  1 a 
M an  M s 
coth  e  
 a  2J He 
 2J
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:
dM   M   
dM irr 
   
 1  
dH   M s   
dH 
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.
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]
kminor   kmajor b
aminor   amajor 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  104 . 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]:
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 oC)
hot rolling
annealing (900-1000 oC)
fast cooling (down to approximately 50 oC)
cold reduction to final thickness
decarburizing and primary recrystallization (800-850 oC)
MgO coating and coiling (50 oC)
box annealing and secondary recrystallization (1200 oC for many hours)
phosphate coating and thermal flattening (50 oC)
Table 1
Magnetic properties of cold rolled grain oriented sheets from Stalprodukt S.A.
specific Minimum
total core
core polarization polarization
loss at
loss at
EN 10107
50Hz, H = 800A/m H = 800A/m
ET 114-27
ET 120-27
ET 130-27 M130-27S
ET 140-27 M089-27N
ET 117-30 M117-30P
ET 122-30
ET 130-30
ET 140-30 M140-30S
ET 150-30 M097-30N
ET 130-35
ET 140-35
ET 150-35 M150-35S
ET 160-35 M111-35N
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 106 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].
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  1010 [(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
1.82  10 6
Fig. 4. The measured and modelled major hysteresis loops.
Fig. 5. The course of the estimation process.
Keeping the values of the parameters andMs 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.
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
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.
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 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.
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