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Vol. 136 (2019) ACTA PHYSICA POLONICA A No. 5
Proc. of the 13th Symposium of Magnetic Measurements and Modeling SMMM’2018, Wieliczka, Poland, October 8–10, 2018
Experimental Determination of the Preisach Model
for Grain Oriented Steel
J. Eichler∗, M. Novak and M. Kosek
Technical University of Liberec, Studentská 2, 461 17 Liberec, The Czech Republic
The full material characteristics in the Preisach model of hysteresis is a two-dimensional weighting function. It
can be determined experimentally from systematic measurement of partial hysteresis loops followed by derivation
of their decreasing parts. Because of measurement errors, the derivation is not correct. Nevertheless, basic
material features can be obtained either from incomplete measurement that uses the Preisach triangle respecting
the measurement errors. The non-uniform grid in the Preisach model is used in order to get approximately the
same steps in changes of the flux density. Simulation shows that this withdrawal of some rows and columns does
not change the hysteresis loop considerably, only the steps of the flux density may be higher.
DOI: 10.12693/APhysPolA.136.713
PACS/topics: 75.60.–d, 75.40.Mg, 75.50.Bb, 85.70.Ay
1. Introduction
The Preisach model [1] is very suitable for a complete
description of the hysteresis. Because of the hysteresis,
the model must be two-dimensional (2D). The Preisach
model uses fictive hysterons, exhibiting the rectangular
hysteresis loop, and arranges them systematically in the
triangle grid; see the last chapter of the paper for some
details. The simulated magnetic material is character-
ized by its weighting function. Although its analytical
description is possible, in practice, the digital form is
used. If the weighting function is well-known, the mate-
rial can be analyzed by numerical methods efficiently and
correctly. There are two basic ways for finding this func-
tion: using an estimated analytical form, or determining
it from an experiment [2]. The analytical estimation has
several serious limitations. Therefore, theoretically, the
best choice is to use the experiment. The experiment
requires a measurement of a lot of elementary hysteresis
loops, starting from the negative saturation. In practice,
the experimentally accessible negative level of excitation
is used. The excitation magnetic field has a harmonic
waveform with negative bias and its amplitude gradu-
ally increases. The decreasing branches of the loops are
used to form the 2D Everett surface [1]. Usually, they
are termed the first order reverse curves (FORC). The
weighting function is given by partial derivations of the
Everett surface both horizontally and vertically. Due to
the experimental inaccuracy the derivation exhibits seri-
ous errors [3]. The accuracy of the weighting function and
the Preisach model application is a subject of the paper.
2. Measurement in the time domain
The current source was used to ensure the har-
monic field strength excitation. The standard apparatus,
∗corresponding author; e-mail: jakub.eichler@tul.cz
described in [4] was used. The measured material was a
grain oriented steel with 4% of silicon that was used to
create the toroid transformer. The core of the trans-
former is 24.3 mm high, inner diameter was 125 mm,
outer diameter was 135 mm and was used 500 turns
for both the primary and the secondary winding. The
well defined starting level was the negative saturation.
The time varying field strength for partial loop is Hd,
while its systematically increasing amplitude (up to the
positive saturation) is Hu. Only decreasing part of the
loops (FORC) is used to form the Everett surface. Since
Hd≤Hu, the Everett surface is defined on the Preisach
triangle. Its horizontal axis is Hd, while the vertical one
is Hu. The amplitude Hushould increase by the smallest
possible steps because of precise determination of values,
especially in the neighborhood of the weighting function
main peak, typical for magnetically soft materials. Basic
experimental limitation of the procedure is demonstrated
in Fig. 1. The total of 1800 loops were measured. The
illustrative choice of the loops in the time domain is given
in Fig. 1. The increment of four was used to improve the
curve visibility. The excitation is in the upper graphs,
in the lower one is the response. In the left hand graphs
of Fig. 1, there are wave parts near the maximum ex-
citation, which is the time t=T/2. In the right hand
graphs the situation near the excitation zero crossing at
time t= (3/4)Tis shown. Symbol Tis used for the pe-
riod of excitation. For each curve set (both excitation
and response) the points in its central part are taken and
presented in the summarizing graphs on its right hand
side. The curve number is on the horizontal axis of the
graph. These graphs allow to judge the measurement
quality of both the excitation and the response.
For the excitation at FORC starting point (the max-
imum of excitation), the excitation curves are well de-
fined, the amplitude increase is linear. The excitation
that corresponds to three quarters of period, which is
centre of the FORC, has only a small deviation from
the expected linear shape (upper right hand graph of
Fig. 1). The excitation, therefore, exhibits a high quality.
(713)
714 J. Eichler, M. Novak, M. Kosek
Fig. 1. Excitation and response in the time domain for the maximum excitation and points where the excitation crosses
time axis.
The response is measured with a lower accuracy. The
curve numbers are preserved at FORC start and the de-
viation from the linear course is acceptable. Although,
at the response in the FORC centre, three quarters of
period, the monotonic increase does not exist. It means
that the impossible negative derivation with respect to
Hutakes a place. The selected excitation and response
curves originate at the area of rapidly increasing flux den-
sity. Therefore, the distance between response curves is
relatively high. Near the saturation, the curve disorder is
much higher and appears also in the start of FORC [5].
Several cuts of the Everett surface are in Fig. 2. The
selected area is close to the central part of the hysteresis
loop. The change speed of flux density reaches maximum
in this area. The cut curves along the rows in Fig. 2a are
flat and increasing. Therefore, the partial derivation with
respect to Hdwill be positive. The decrease followed by
constant value at the end part of several curves in Fig. 2a
was explained in [4]. The vertical cuts, in Fig. 2b, behave
quite differently. In this case, the magnetic field strength
Huis a variable and Hdis a parameter. In this case small
noise is superposed to the expected flat and increasing
shapes. Although the noise is small, the derivation with
respect to Huwill change the sign, giving incorrect values
of the weighting function. It should be also stressed that
there is different number of points forming the curves. In
Fig. 2a each curve contains million of points, because the
sample rate was 500 kHz and the sample time was 2 s.
While curves in Fig. 2b contain only 1800 points, that
is restricted by the minimum step of the power supply
current magnitude. Other reasons are current accuracy,
stability, noise from the current source, random effects
in a material, and total time of measurement. Therefore,
we achieved almost the practical limit.
The noise is caused by the incorrect order of the re-
sponse curves as it is explained in Fig. 1. The main
problem is that the curves cannot be shifted to the cor-
rect position. Their orders change along the curves.
The result of numeric derivation of the Everett surface,
weighting function, is given in Fig. 3. It was obtained
using several corrections of data. The only cut at the
vertical plane in Hu=−5A/m is shown for clarity.
Experimental Determination of the Preisach Model for Grain Oriented Steel 715
Fig. 2. Cuts of the Everett surface (a) horizontal, (b)
vertical.
Fig. 3. Weighting function determined by the numeric
derivation in the cut of Hd=−5A/m.
It contains the main sharp peak and long area with
values near zero. The first inset in Fig. 3 shows details
of the main peak with its vicinity and the second inset
reveals typical area near zero values.
It corresponds to simple simulations. Since the hys-
teresis curve is narrow, the maximum should be at the
Preisach triangle axis close to the hypotenuse. The dis-
tance of the maximum from the hypotenuse can be esti-
mated from the coercitive force. Areas distant from the
central part exhibit noise containing negative values, see
the second inset in Fig. 3. It was due to the noise in the
Everett surface presented in its cuts in Fig. 2b.
3. Reduction of weighting function
Response curve disorder causes errors in the weighting
function determination. The best solution should be to
increase the experiment accuracy. It is a very difficult
task. Furthermore, the disorder may be due to the ma-
terial itself, at least partially. Therefore, another way of
experiment and data processing should be used.
The simplest way is a change of the experiment. The
step of the magnetic excitation field increase is con-
stant. Due to the strong nonlinearity, the increase step
of flux density will differ strongly. In the central part
the step will be large, while near the saturation it will
be small. The reason is that the response curve or-
der is not kept. The better experiment should use al-
most constant steps of the flux density. In this case the
grid for the Everett surface should be also non-uniform
and the weighting function will be reduced. We simu-
lated the effect of reduction of the weighting function.
In the central part, near the maximum, the grid is not
changed, but at the distant parts, every second row and
column are withdrawn. The grid is sketched in Fig. 4.
Of course, in practice, the number of rows and columns
is significantly higher.
Fig. 4. Non-uniform grid used in simulation.
Hysteresis loops reconstructed from the reduced
weighting function are in Fig. 5. The loop shape is not
considerably changed by the grid reduction, except for
quantization. The inset shows that the reduction causes
major deviations of steps in the central part. According
to the second inset of Fig. 5, in the area of saturation,
the reduction effect is negligible. The legend contains the
number of nodes in the Preisach triangle. The steps are
made predominantly by the low number of nodes and not
by the row and column reduction.
716 J. Eichler, M. Novak, M. Kosek
Fig. 5. Hysteresis loops by the Preisach model with
a strong reduction of number of hysterons (rows and
columns).
4. Proposal of an ideal non-uniform excitation
Usually a constant step in the excitation field is used
as the simplest way. But the corresponding flux density
that responses the excitation steps vary in a large extent.
Therefore an approximately constant step in response is
the best solution [6]. In the design of variable steps in
excitation, we use the approximation of the major hys-
teresis loop.
There are a lot of approximations of the hysteresis
loop branches: functions arctan, arctanh, erf, step re-
sponse, etc. We used the simplest function arctan with
several variable parameters for the approximation of the
flux density in one branch of the hysteresis loop
Ba(H) = Aarctan(k(H−H0)) + cH, (1)
were Ais the amplitude of approximation that defines
the limit values, kis the coefficient of argument that de-
termines the slope of approximation especially in the zero
area, and H0is the coercitive magnetic field strength of
the selected branch that is either positive or negative.
The linear term with constant cis added, since the sat-
uration does not appear practically. Since the excitation
is symmetric, the response given by (1) is symmetric,
too. Therefore, absolute term in (1) is zero and omit-
ted. All the four parameters were found by the use of
fminsearch function in MATLAB. This function is called
many times with different initial values of parameters.
As the optimum, the set of parameters with minimum
deviation from experiment is selected.
The approximation is presented in Fig. 6. The loop
in Fig. 6a is well approximated. For a better evaluation
of the approximation, the increasing part of the loop to-
gether with experimental points is in Fig. 6b. The ac-
cordance is good with an exception of the initial part.
Only the positive part is shown in graph for clarity. The
deviation of experimental data and the approximation
is less than about 20 mT with an exception of the area
near the zero excitation. Therefore, in practice, the ap-
proximation is quite acceptable. The current, necessary
for excitation of the hysteresis loop with the constant
step in the flux density magnitude of 38 mT, is given by
the inverse function to (1), practically by tangent func-
tion. The total scale is in Fig. 7a. The very large step
of more than 2 A is in the initial part, at the half time
of period, and at the end of period. Otherwise, the cur-
rent change is small, usually several mA or less. The
current difference of flat areas is the effect of hystere-
sis. Figure 7b shows the detail of the current near the
half of period, where the abrupt current change takes
place. In this critical part the current change is several
mA only. In practice, a big current change will be im-
possible experimentally and could lead to other negative
effects. In the saturation area, a smaller flux density step
should be projected.
Fig. 6. Approximation of a maximum hysteresis loop by arctan function in the whole positive range (a), and the detail
(b) with experimental points.
Experimental Determination of the Preisach Model for Grain Oriented Steel 717
Fig. 7. Time dependence of the current for excita-
tion of the hysteresis curve (a) with a constant step
of flux density (b) zoomed detail (dt= 5.051 ms,
∆B= 38.21 mT).
5. Model calculation
The important part of the model processing is to cal-
culate the response from the model for any excitation.
The geometrical presentation of the Preisach model is
used in this case. Basics are shown in Fig. 8. The hys-
teron in Fig. 8a has a rectangular loop with unit mag-
netic momentum and switching levels Huand Hd. In
the model, the hysterons are systematically arranged ac-
cording to switching levels. The principle of the model
is sketched in Fig. 8b. It contains a lot of hysterons,
hundreds in the row. The effect of external field consists
of switching the hysterons into a positive or a negative
magnetization M. When the excitation field increases,
its level in the model moves from the bottom to the top.
All the hysterons under the excitation level are switched
up to the positive magnetization +M, all other remain
unchanged, as Fig. 9a explains. In the case of the ex-
citation decrease, the corresponding level is vertical and
moves from the right to the left, see Fig. 9b. In the fig-
ure the previous increasing excitation remains. All the
hysterons on the right hand side of excitation level are
switched down, and the magnetic momentum of all other
remains unchanged. Now it is clear that 2D model pro-
vides the hysteresis, while the 1D model can be used only
for modeling nonlinearity.
Fig. 8. Basics of the Preisach model: (a) hysteron, (b)
principle of the model — distribution of hysterons in
the Preisach plane.
Fig. 9. Application of the Preisach model — bound-
ary in the Preisach plane: (a) increasing excitation, (b)
decreasing field.
718 J. Eichler, M. Novak, M. Kosek
Fig. 10. Excitation levels in the Preisach plane for a
decreasing (a) and an increasing (b) excitation.
The setting of the hysteron momentum and their sum-
ming can be made by a very simple function in MATLAB.
But the simplicity is connected with a higher computa-
tion time, since the function is used for each excitation
level. However, only the level that meets the row or col-
umn with hysterons makes the change of the total mo-
mentum. Since the constant time increment is used for
excitation, the inactive levels exist for both the direc-
tions of excitation, see Fig. 10. The case in Fig. 10b cor-
responds to the harmonic excitation near its maximum
or minimum. Ignoring the inactive levels increases the
computation speed several times.
6. Discussion
The weighting function is the complete characteriza-
tion of a material when using the Preisach model. There-
fore, it has a key meaning in the model application and
should be found with a maximum accuracy. There are
two basic ways of its determination: analytical approx-
imation and experimental determination. The analyti-
cal approximation uses the fact that the weighting func-
tion exhibits sharp maximum for soft magnetic mate-
rials. The 2D analytical functions with their features
are used, usually in combination. The unknown coeffi-
cients are found numerically by the best fit of approxi-
mation and experiment. The advantage of the method is
that only one hysteresis loop for the maximum excitation
is sufficient. That is why an experiment is simpler and
faster. However, for lower excitations the accordance of
approximation and experiment can decrease.
The experimental method needs to form the Everett
surface by a complex experiment. Therefore, all the ma-
terial information is included in the Everett surface. The
disadvantage is that the numeric derivation must be used
to get the weighting function from the Everett surface.
Unfortunately, the derivation enhances experimental er-
rors that were shown for the case of cuts of the Everett
surface by several values of Huin Fig. 2b, for instance.
There are several sources of the curve imperfection. The
integration is used to get the flux density from the sec-
ondary voltage. In principle, this method reduces the
errors. On the other hand, a small positive or negative
constant exists that shifts the resulting curve monotoni-
cally up or down. It is well-known as the drift in practice.
The sign and value of the constant can change during the
experiment that leads to a curve disorder. The improve-
ment of integration is a very complicated technical task.
A simple way of drift reduction is to decrease the period.
However, the frequency increase leads to the increase of
eddy currents. The elimination of the eddy current ef-
fect in the loop is almost impossible. The frequency of
1 Hz or little less was used as the limit. Probably, the
material feature also can take place in errors. Existing
thermal effects can slowly change magnetic momentum
of the magnetic domain. Also a small change of the flux
density for a constant magnetic field strength is reported
and it is known as the after-effect [7, 8]. Unfortunately,
it is a very difficult task to find whether the complication
is of experimental nature or a material feature, or both
of them with different contribution.
Since the calculation is numeric, the steps of response
corresponding to the steps of excitation appear at the
output in every case. The simulation showed that the
grid reduction leads only to the increase of the response
steps. The average curve shape is preserved. The steps
are highest in the area of the abrupt change of the flux
density.
An ideal case is the use of an excitation that leads to
approximately constant steps of the flux density. Such
theoretical excitation was found, however, there is a ques-
tion if it can be realized experimentally for all its points.
7. Conclusion
Experimental inaccuracy limits the number of points
in the Preisach triangle. The distance between rows and
columns should be greater than the estimated experi-
mental error. Fortunately, the simulation confirms that
this limitation does not considerably affect the Preisach
model prediction.
The excitation for the constant step of the flux density
magnitude was found. The speed of repeated calcula-
tions can increase by two orders at the expense of lower
algorithm robustness.
Experimental Determination of the Preisach Model for Grain Oriented Steel 719
Acknowledgments
The result of the work was obtained thanks to
the financial support the European Union within
the scope of the project “Modular platform for
autonomous chassis of specialized electric vehicles
for freight and equipment transportation”, Reg. No.
CZ.02.1.01/0.0/0.0/16_025/0007293.
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