MATHEMATICAL MODELLING OF FERROMAGNETIC
HYSTERESIS IN SOFT MAGNETIC MATERIALS
K. Chwastek1, J. Szczygłowski1, W. Wilczyński2
1
Częstochowa University of Technology, Częstochowa, Poland
2
Electrotechnical Institute, Warsaw, Poland
Abstract
The paper discusses some physical concepts important for mathematical modelling of
ferromagnetic hysteresis and their implementation in chosen macroscopic descriptions of the
phenomenon.
Introduction
Recent advances in the design and emerging applications of electric machines
stimulate the development of new engineering descriptions of hysteresis phenomenon and
core losses [1-4]. Hysteresis is also the field of intensive research for solid state physicists.
The paper focuses on several concepts advanced by physicists and implemented in practical
descriptions of the phenomenon, used by electrical engineers in computer simulations of
ferromagnetic cores.
The effective field and the anhysteretic curve
The cornerstone for contemporary understanding of ferromagnetism is the idea of
effective field introduced by P. Weiss in 1907 [5]. According to the well known monograph
[6, p. 130] “ … it is of invaluable importance in giving a simple and at the same time deep
physical interpretation of the existence of spontaneous magnetization …”. Briefly speaking,
the effective field is interpreted as the internal magnetic field experienced by magnetic
moments within the material, different from the externally applied magnetic field. In the
language of control theory, the effective field is the sum of the applied field and a certain
positive feedback in the system, H eff  H  M , where the second term is referred to as the
mean (or molecular) field [7,8]. It is remarkable that a similar concept of molecular
interaction forces has been considered by J. D. van der Waals [9]. The effective field may be
perceived as a manifestation of cooperative interactions between a large number of identical
elements [10-14]. The concept has evolved during the years and many sophisticated
descriptions useful for solid state physics have been derived from it, just to mention the
Heinsenberg exchange model, the Ising model, the XY model etc. Some of the advances on
the subject are summarized in a number of monographs, review papers and textbooks,
cf. [15-17]. Their discussion is beyond the scope of the present paper. The seminal works by
L. D. Landau and E. M. Lifschitz [18] have initiated the intensive research on the paramount
role of the effective field in contemporary micromagnetics, what resulted in the development
of magnetic memories [6, 19-24]. In the seventies and eighties of the last century one of the
“hot” topics in physics was the study of universality, scaling and renormalization in the
context of critical phenomena and phase transitions [25-27]. Connections of effective field
concept with the theory of phase transitions are discussed in detail e.g. in Refs. [28, 29].
A generally accepted opinion is that the effective field theory might be an useful aproximation
for the description of complex systems in those cases when fluctuations may be neglected.
In macroscopic hysteresis models, which are of interest to engineers, the effective field
concept is also useful, as the description may be expressed using a self-consistent relationship
M (t )   H eff (t )  , where  is the hysteresis operator and M is magnetization [30].
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In practical calculations it is convenient to invert the abovegiven relationship, thus to
calculate the effective field from magnetization, subsequently external field strength is
obtained. This approach is consistent with the conditions used for material characterization
defined by international standards (ASTM, IEC 404), which assumes that ferromagnetic
materials are examined for sine waveform of flux density rate (for soft magnetic materials this
condition applies to magnetization rate as well, because M (t )  H (t ) and
B(t )  0 H (t )  M (t ) ).
There exists a number of useful descriptions of hysteresis phenomenon, cf. [31]. The
most important and widespread model is due to F. Preisach [32]. Its extensions including
effective field have been introduced and discussed by E. Della Torre [33] and
I. D. Mayergoyz [34]. This model is however purely phenomenological, it relies on a
weighted superposition of outputs from abstract elementary relay components (“hysterons”),
thus it is quite difficult to ascribe any physical meaning to hysterons. Some attempts to
correlate or even interpret them to grains within the magnetic material may be found in the
literature [35-37], but this conjecture is not fully justified. Another important model, but
operating on a smaller length scale (closer to micromagnetics, so more interesting to
physicists [12]) is the Stoner-Wohlfarth description [38]. The effective field concept has been
examined for this description e.g. in Refs. [39-40].
The macroscopic model advanced by D. C. Jiles and D. L. Atherton [41] has gained
a considerable interest in the last 25 years. The concept of effective field is inherent in the
description, as H eff is the independent variable in the fundamental ODE of the model [42].
The basic idea behind the description is to offset magnetization from a theoretical sigmoidshaped curve, which describes the state of global thermodynamical equillibrium, cf. Fig. 1.
Hysteresis is due to energy dissipation on the inclusions, voids, dislocations of crystalline
structure in the material, commonly termed as pinning sites.
Fig. 1. Major hysteresis loop and anhysteretic curve in the Jiles-Atherton model
There is a number of problems with the seemingly simple description. Minor loops,
i.e. those, which do not reach saturation (either symmetrical or asymmetrical ones) and more
complex magnetization cycles are usually poorly represented, if the same parameter set as for
the major loop is used [43]. The original parameter estimation procedure [44] is error-prone,
so other methods have to be aplied [45]. Some problems have been caused by the introduction
of supplementary relationships for the description of reversible magnetization phenomena
[41, 45, 46]. The model developers did not notice the fact that the introduced relationships are
merely approximations resulting from a Taylor expansion of more complicated formulas. An
attempt to fix this problem has been proposed [47], where the concept of modulated
contribution of reversible susceptibility, borrowed from the product Preisach model [48], has
been incroporated in the description. The modified description resulted in an improved
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description of minor loops. Anyhow there is yet another subtle problem with the reversibility
issue, which cannot be so easily fixed. This is due to the very idea of D. C. Jiles and D. L.
Atherton, that the magnetization (or its irreversible component) may be given with an ODE
with respect to the effective field.
Let us refer again to Figure 1.  denotes simply the sign of variation of input variable
(   sign(dH / dt ) for the simple model or   sign(dM / dt ) for the inverse model, which
corresponds to the conditions used for material characterization in normalized conditions).
But  M is introduced in order to remove the non-physical model behaviour after a sudden
change of sign of the input signal. The original model formulation resulted in the appearance
of negative susceptibility regions after the reversal point. The existence of this term may be
justified with the fact, that right after the applied field changes its sign, only the reversible
process contributes, because the irreversible process does not return any energy to the applied
field [49].  M is aimed at switching off the irreversible component of susceptibility and it is
given most consisely as  M  0.51  signM an  M irr  dx / dt  , where x denotes the input
variable. Note, however, that in order to use the model, we have to track the  M sign instantly.
There are parts of the modelled hysteresis loop, which are described with the fundamental
Jiles-Atherton model equation dM irr / dH eff  M an  M irr  / k and there are other parts,
which are not. In a certain sense the parameter  M is introduced post-hoc into the formal
derivation of the final relationship for dM / dH .
The effective field in the Jiles-Atherton model has been extended in order to take into
account some important physical phenomena: magnetostriction [42, 50-52] and eddy currents
[53]. There were some attempts to consider and describe the effect of texture [54-56],
anistropy of magnetic materials [37, 55-61] and steel microstructure [37, 62, 63]. A general
conclusion may be drawn that the popularity of the description stems from the possibility to
consider a number of physical phenomena and technological parameters like strain. A great
merit of the formalism is the noticing of the role of the effective field and anhysteretic
magnetization (or, more generally, reversible magnetization processes) in modelling. The
simplicity of the description makes it a favourite “playground” for hysteresis modellers.
However the Jiles-Atherton model is not necessarily useful for applications in electrical
engineering calculations, what is due to the aforementioned low accuracy in representation of
complex magnetization cycles and the necessity to introduce artificial “patches” to improve
model behaviour.
A much more promising approach is the one proposed by Bergqvist [64, 65] or its later
modifications advanced by Henrotte et al. [66-68]. In the aforementioned descriptions the
offset of loop branches from the anhysteretic curve is accomplished along the H-axis, not
along the M-axis like in the Jiles-Atherton model. This eliminates the need to introduce  M .
Bergqvist has shown the compliance of the model with the first and the second laws of
 
thermodynamics, expressed using the Clausius-Duhem inequality dQ / dt  H  M  F  0
[64]. In the description he has proposed a smooth curve for the anhysteretic and a number of
pseudoparticles to describe potential wells. Figure 2 depicts the basic concepts in the
Bergqvist model. F(M) is the Helmholtz free energy. Its profile is convex, close to parabolic.
The derivative of the free energy with respect to magnetization for the ideal parabolic case
would yield the single-valued inverse anhysteretic curve. If the system were disrupted by a
small potential well, the hysteresis would emerge. In the Figure a number of such potential
wells is depicted. The author has investigated some formulas for calculation of distributions
of pseudoparticles and has found that even using crude approximations it is possible to obtain
realistic modelling results for the alternating and rotating excitation.
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The hysteresis loop modelled using the Bergqvist model
is a bit “jagged”, in order to obtain a smooth
representation it is necessary to take a sufficient number
of elementary pseudoparticles. An alternative is to avail
of some other model, which is based on offseting the
loop branches from the anhysteretic curve, a good
candidate might be the Takács description, which shall
be discussed in more detail in the subsequent section.
It is interesting to take stock of the concept of
anhysteretic curve in more detail. D. C. Jiles and D. L.
Atherton have proposed to describe it with the modified
Langevin function (“modified” means that its argument
is
the
effective
field).
Langevin
function
m  coth( x)  1 / x is in fact valid for gases. The
Brillouin
function
given
as
m  2 J  1 /(2 J ) coth2 J  1x /(2 J )  1 /(2 J ) coth x /(2 J )
should be more appropriate for solids [47]. Jiles has
considered a number of different formulas for
anhysteretic curve in dependence whether the magnetic
material reveals isotropic properties ( m  coth( x)  1 / x ),
axial ( m  tanh( x) ) or planar anisotropy (
m  I 0' ( x ) / I 0 ( x) , I 0 is the modified Bessel function of
the zeroth order [12, 59]. Using the identity
I 0' ( x )  I1 ( x ) (formula 9.6.27 in [69]), the relationship
may be further simplified. The three abovegiven
functions are depicted in Figure 3. The Padé approximation to the inverse Langevin function
3  m2
proposed by A. Cohen [70], given as x  m
, which seems to be the simplest one for
1  m2
practical applications, is also shown.
Fig. 2. Profile of free energy,
its derivative with respect to
magnetization and hysteresis loop
described using the Bergqvist model
Fig. 3. Functions appropriate for description of anhysteretic curve
Yet another approach to describe the effect of anisotropy on the shape of the
magnetization curves has been proposed in Ref. [48] and introduced into the Jiles-Atherton
model in Ref. [60]. Gy. Kádár has considered the Brillouin function with a properly chosen
quantum number J for the description of anhysteretic curve. Moreover a modulation factor
R(m) dependent on the chosen form of anhysteretic has been introduced into the relationship
1-65
dM irr 
dm
dM

 R(m)  
, where R(m)  an expressed as the

dH
dH 
dx

function of man itself. Analytical calculation of R(m) is possible only for man  tanh( x) , i.e.
J  0.5 , which corresponds to the extreme anisotropic, two-state case. In all other cases one
has to resort to numerical methods to calculate R(m) for arbitrarily chosen values of J and x.
for total susceptibility,
Fig. 4. Dependence of R(m)/R(0) on m for several values of quantum number J [60]
It is important to remark that the notion of anhysteretic curve is useful in practical
applications e.g. removal of magnetic signatures of Navy vessels and submarines (the
so-called “deperming” procedure). For a brief introduction to the subject, the Readers are
referred to Refs. [71, 72].
Modelling
We have chosen the phenomenological description advanced by J. Takács for
modelling hysteresis loops in chosen core materials used in electrical engineering [73, 74].
The Takács approach is based on the assumption that there is a hyperbolic tangent mapping
between the variables involved in the description of hysteresis loop. The model has a number
of advantages: it is purely phenomenological in nature, therefore its users may easily adjust it
to their needs, it is able to describe a number of important phenomena observed
experimentally by solid state physicists: minor loops, transients, accomodation, existence of a
unique anhysteretic curve [75], etc. The variable presented at the abscissa axis has been
identified previously as equivalent to the effective field [76, 77], this concept has been
extended recently in Refs. [78, 79]. It was assumed that the effective field might include an
additional term related to magnetization rate (which, generally, might be fractional [80, 81]).
Therefore the extended description is able to describe dynamic hysteresis loops, which are
affected by the effect of eddy currents generated in the conductive core material [82]. The
modified description has been identified [79] as a specific case of the famous Chua model1.
The relationship describing the major hysteresis loop (and other symmetric loops) using
the Takács model may be written as follows:
 H (t )  H c 0 
  bTIP
M (t )  M s tanh  eff
A


1
(1)
L. Chua is the pioneer of research on dynamic hysteresis modelling [83, 84].
1-66
where M (t ) denotes the instant value of magnetization, M s is saturation magnetization,
A is a model parameter, H c 0 is coercive field strength in quasi-static conditions, whereas bTIP
is introduced in order to match the ascending and the descending loop branches at loop tip

 H TIP  H c 0 
 H TIP  H c 0 
  tanh  eff
 ,
bTIP  0.5M s  tanh  eff



A
A





(2)
where H effTIP denotes the effective field at loop tip, when the dynamic effects vanish. Generally,
we assume the dynamic effects may be incorporated in the definition of the effective field
H eff (t )  H (t )  M (t )   dM / dt  ,

(3)
where  is the Weiss coupling coefficient, whereas  is a new model parameter.
The exponent  may be in most cases considered fractional (an estimate may be obtained
from the fitting of H c  H c ( f ) dependence, cf. Fig. 5. This piece of information may be
considered useful during estimation of some model parameters. Actually the values of
fractional exponent vary in dependence on flux density level, thus the word ,,multifractional”
might be more appropriate for the description of energy dissipation due to eddy currents in
soft magnetic materials).
A number of formulas similar to (1) for other more complicated magnetization cycles
have been derived and discussed in detail by the model developer [73, 74].
Fig. 5. Fitting of the dependence H c  H c ( f ) for the amorphous Metglas core. Dots – measurement
In Figures 6 and 7 some exemplary results depicting the behaviour of the aforedescribed
Takács model with effective field, which includes additional fractional term, are shown.
Modelling was carried out for two chosen core materials used in electrical engineering: an
amorphous glass (Metglas 2605 TCA) core and a sheet of grain-oriented electrical steel, grade
ET 120-27. Model parameters were determined in quasi-static excitation conditions. Their
values were as follows:
  8.2 105 [-], A  71.3 [A/m], H c 0  2.14 [A/m], M s  9.55 105 [A/m] for Metglas
  2.4 105 [-], A  35.9 [A/m], H c 0  9.29 [A/m], M s  1.44 106 [A/m] for the GO steel
(   1.17 [A/m/Hz ],  0.68 @ Bm  1.8 [T] ).
These values were next used for prediction of loop shapes under dynamic excitation
conditions. Despite some slight discrepancies in the shapes of the measured and the modelled
1-67
loop shapes, in particular for the grain-oriented steel sheet, it may be stated that the model
predicts the change of loop shape upon increase in excitation frequency with an accuracy
acceptable for engineering purposes.
Fig. 6. Modelling results for the amorphous Metglas core
Fig. 6. Modelling results for the grain-oriented steel sheet
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Conclusions
Some concepts important in hysteresis modelling (effective field, anhysteretic magnetization)
have been discussed. Some of deficiencies of the Jiles-Atherton description in this context
have been pointed out. As an alternative to the Jiles-Atherton model our proposal is to focus
on the Takács approach. Exemplary modelling of hysteresis loops for chosen core materials
used in electrical engineering has been carried out.
Acknowledgements
The work of Krzysztof Chwastek has been carried out within the framework of research grant
N N510 702540 from National Research Centre.
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