Volume 02 Issue 12-2022
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American Journal Of Applied Science And Technology
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2771-2745)
VOLUME
02
I
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12
Pages:
06-16
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I
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5.
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1121105677
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ABSTRACT
Two models of remagnetization of soft-magnetic amorphous alloys are considered: the Jiles-Atherton hysteresis
model and a model of the magnetization curve. The aim of the research is to evaluate models according to the criteria
of simplicity of the mathematical expressions obtained and the adequacy of the description of the magnetization
phenomenon. The relative error of modeling is chosen as a criterion for the accuracy of the model. In the study, the
least squares method was used to model the main magnetization curve and the method of optimizing the Jiles-
Atherton hysteresis curve using experimental and reference data. It is concluded that both models of magnetization
of magnetically soft amorphous alloys give approximately the same modeling accuracy.
KEYWORDS
Research Article
MODELS OF JILES-ATHERTON HYSTERESIS LOOPS AND MODELS OF
MAGNETIZATION CURVES FOR MAGNETICALLY SOFT AMORPHOUS
ALLOYS
Submission Date:
December 10, 2022,
Accepted Date:
December 15, 2022,
Published Date:
December 20, 2022
Crossref doi:
https://doi.org/10.37547/ajast/Volume02Issue12-02
Bedritskiy I.M.
Researcher Tashkent State Transport University (Tashkent, Uzbekistan)
Jurayeva K.K.
Researcher Tashkent State Transport University (Tashkent, Uzbekistan)
Bazarov L.Kh.
Researcher Tashkent State Transport University (Tashkent, Uzbekistan)
Mirasadov M.J.
Researcher Tashkent State Transport University (Tashkent, Uzbekistan)
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 02 Issue 12-2022
7
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
12
Pages:
06-16
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
Magnetization curve of magnetically soft amorphous material, Jiles-Atherton hysteresis model, approximating
function, method error.
INTRODUCTION
The analysis of devices with ferromagnetic elements
involves the approximation of the magnetization
characteristics of ferromagnetic materials, for the
approximation of the hysteresis loop, the Jiles-
Atherton, [4, 5, 6, 7, 10] or Chan models are most often
used [8, 9, 10, 11, 14, 15]. However, if the ferromagnetic
core in the devices operate in saturation mode, and the
hysteresis loop has an insignificant width, then in this
case the main magnetization curve is used, the
approximation of which is carried out using suitable
mathematical expressions.
Most often, hyperbolic sine, arctangent, full and
incomplete polynomials of the n - th degree are used to
approximate the magnetization curve, where n is an
odd integer [1, 2, 3, 17, 18]. The use of one or another
method
to
create
mathematical
models
of
ferromagnetic devices depends on the goals set and
the depth of study of the processes occurring in them.
Therefore, a comparative analysis of the description of
magnetization using magnetization curves and using
hysteresis loops is of scientific interest in order to
identify the optimal method for a particular problem
being solved, as well as an error estimate when using
both methods of mathematical description of
hysteresis.
METHODS
Cores made of magnetically soft amorphous steels and
amorphous iron-based alloys were used as models for
the study, the experimental magnetization curve of
which was taken at alternating current with a
frequency of 50 Hz according to the methods
described in [1, 2], in particular for the AMAG 492 alloy
(a close analogue of the Metalglass alloy described in
[13]),
for
the
remaining
amorphous
alloys
magnetization data are taken from literature sources
[12, 13]. The appearance of the main magnetization
curves is shown in Fig. 1. It can be seen from the curves
that for most soft-magnetic amorphous materials,
saturation occurs already at values of the magnetic
field strength , which indicates a high value of relative
magnetic permeability for materials of this type.
Volume 02 Issue 12-2022
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American Journal Of Applied Science And Technology
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VOLUME
02
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12
Pages:
06-16
SJIF
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FACTOR
(2021:
5.
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(2022:
5.
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)
OCLC
–
1121105677
METADATA
IF
–
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Publisher:
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Fig. 1. Magnetization curves of common types of amorphous steels and alloys.
Linear coefficients in approximating expressions were calculated based on the minimum of the total quadratic
error by the least squares method, the transition from nonlinear to linear functions was carried out using appropriate
substitutions [16] and using an expression
=
=
=
=
=
−
−
=
N
i
n
i
N
i
n
i
i
N
i
n
i
N
i
i
N
i
n
i
B
N
B
H
B
N
H
B
k
1
2
2
1
1
1
1
modified for the condition of passing
the curve through the origin, where
N
- the number of experimental points on the magnetization curve;
i
–
the
number of points;
i
B
,
i
H
,
–
experimental values, respectively, of magnetic induction and magnetic field strength at
the -th point. For cores made of a magnetically soft amorphous iron
i
- based alloy of the AMG 492 brand in the range
of induction variation from 0 to 1.6 Tl (saturation induction , the following approximating expressions were obtained:
hyperbolic sine
)
552
,
11
(
10
892
,
1
5
B
sh
H
=
−
;
arctangent
)
049
,
0
(
022
,
1
H
arctg
B
=
;
an incomplete polynomial of the ninth degree
9
66
,
14
B
H
=
an incomplete polynomial of the eleventh degree
11
22
,
5
B
H
=
.
Volume 02 Issue 12-2022
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VOLUME
02
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06-16
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I
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5.
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(2022:
5.
705
)
OCLC
–
1121105677
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IF
–
5.582
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Graphs of the main magnetization curve of the amorphous AMAG 492 alloy and its approximating functions
are shown in Fig. 2.
Fig. 2. Magnetization curve and its approximating functions for AMAG 492 alloy
It can be seen from the graphs of the functions that, according to the accuracy criterion, all of them are
sufficiently suitable for approximating the main magnetization curve of the AMAG 492 alloy. However, expressions
for hyperbolic sine and arctangent are inconvenient for subsequent transformations, in particular, expressions with
hyperbolic functions are inconvenient for obtaining inverse dependencies (
H
from
B
or
B
from
H
), which is necessary
when analyzing circuits. Obviously, the approximation by incomplete polynomials of the ninth and eleventh degrees
is the most suitable by the criterion of simplicity and accuracy.
The relative approximation error for each of the experimental points can be calculated by the expression
%
100
(%)
−
=
i
iA
i
B
B
B
, where
i
B
- is the experimental value of magnetic induction at the
i
–
th point;
iA
B
- is
the value of magnetic induction calculated by the approximating function. The dependence curves for incomplete
polynomials of degrees from 9 to 11, as well as for the hyperbolic sine and arctangent functions for the amorphous
AMAG 492 alloy core are shown in Fig. 3.
Volume 02 Issue 12-2022
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VOLUME
02
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I
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(2021:
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5.
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)
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1121105677
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It can be seen from the graphs that errors in approximation by polynomials with degrees 9 and 11 give errors
not exceeding 9%, which can be considered acceptable when calculating ferromagnetic elements based on amorphous
alloys.
It is obvious that the methods of approximation of the magnetization curve discussed above are approximate,
since in reality any ferromagnetic material is magnetized by a hysteresis loop. Therefore, a mathematical description
of the magnetization process of the material, taking into account the hysteresis, is of interest. For modeling, we will
use the hysteresis loop of the AMAG 492 material, using for these purposes the Jiles-Atherton hysteresis loop model
[4, 5, 19], often used for modeling and calculations of ferromagnetic devices. To obtain the best accuracy of the model,
it is necessary to apply its optimization, which makes it possible to calculate the optimal parameters through known
experimental and reference data.
Fig. 3. Approximation errors
1
–
incomplete polynomial of degree 11, 2
–
incomplete polynomial of degree 9,
3
–
hyperbolic sine, 4
–
arctangent
The essence of the Jiles-Atherton model is that the total magnetization
M
consists of three components:
hysteresis-free magnetization
an
M
, reversible magnetization
rev
M
, irreversible magnetization
irr
M
, and the
relationship between the magnetization
,
M
of the magnetic field strength
H
and the magnitude of magnetic
induction
B
is described by the expression
).
(
0
H
M
B
+
=
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VOLUME
02
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(2022:
5.
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The magnetization
M
of a ferromagnet in an external magnetic field depends on the magnitude of the
internal field
i
H
, equal to
M
H
H
i
+
=
, where
- is a coefficient that takes into account the effect of the
interaction of the external and internal magnetic field. Due to the small value
equal
5
10
6
4
−
−
in the sources
[4], it is recommended to take it equal to zero, thus it turns ou
H
H
e
.
The magnitude of the hysteresis
–
free magnetization
an
M
can be written in the form
)
(
H
f
M
M
s
an
=
,
where
s
M
- is the saturation magnetization, and
)
(
H
f
- is a function equal to zero at
0
=
H
and one at
H
,
tending to infinity. In the Jiles-Atherton model as a function
)
(
H
f
, the Lanjevin function is used as a function in the
form
£(x)=coth(x)-1/x
, with this in mind, the hysteresis
–
free magnetization curve is described by a function
−
=
H
A
A
H
M
M
s
an
coth
, where
A
- is a scale factor ranging from 0.1 to 10000, selected by the
appearance of the hysteresis loop so that the curve
an
M
passes through the points (0,0) and the
)
,
(
r
c
B
H
hysteresis curve, where
c
H
and
r
B
–
accordingly, the coercive force and the residual magnetic induction of the
investigated ferromagnetic material.
It is known from [4] that the total magnetization
M
is the sum of two components
–
irreversible
magnetization
irr
M
and reversible magnetization
rev
M
rev
irr
M
M
M
+
=
. (1)
The derivatives
H
of the irreversible and reversible components are determined , respectively , by the
expressions
)
(
0
irr
an
irr
an
irr
M
M
k
M
M
dH
dM
−
−
−
=
;
−
=
dH
dM
dH
dM
c
dH
dM
an
rev
, (2)
from where, after transformations and taking into account (1), a differential equation describing the hysteresis in
the Jiles-Atherton model can be obtained
dH
dM
c
c
M
M
k
M
M
c
dH
dM
an
an
an
)
1
(
)
(
)
(
)
1
(
1
0
+
+
−
−
−
+
=
. (3)
Here:
–
the sign function,
1
=
if
,
0
dt
dH
1
−
=
if
,
0
dt
dH
c
H
k
0
- a coefficient
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approximately equal to the coercive force; c
–
a weighting coefficient equal to the ratio of the differential
susceptibilities of the initial and hysteresis
–
free magnetization curves, determined experimentally by the best
approximation of the calculated and experimental hysteresis curves, is in the range from 0 to 1;
- a coefficient that
takes into account the effect of the interaction of external and internal magnetic fields, previously its value was
assumed to be zero.
Taking into account these notations, expression (3) will be rewritten as
dH
dM
c
H
M
M
c
dH
dM
an
c
an
+
−
−
=
)
(
)
1
(
. (4)
Integrating the left and right parts of (4) by
dH
, we get
an
an
c
M
c
dH
M
M
H
c
M
+
−
−
=
)
(
1
. (5)
Since,
−
=
H
A
A
H
M
M
s
an
coth
, after substituting this expression in (5), we finally get
−
+
−
−
−
=
H
A
A
H
M
c
dH
M
H
A
A
H
M
H
c
δ
M
s
s
c
coth
coth
1
(6)
Let us perform the integration of equation (6) by the numerical Gauss-Kronrod method - [16] as giving the
highest algebraic accuracy with the following initial parameters for the AMAG 492 alloy, given below:
Tl
B
s
75
,
0
=
;
m
A
M
s
/
10
27
,
1
4
=
;
m
A
H
s
/
8
=
;
1
,
1
−
=
;
32
=
A
;
58
,
0
=
c
;
0
=
a
.
Based on the results of numerical integration, we obtain a number of values of the magnetic field strengths
H
and the corresponding inductions
B
, and we will take the integral within the range of the change in the magnetic
field strength from
-1000
to
+1000 A/m
. Figure 4 shows graphs of the hysteresis curves of the dependence
)
(
H
f
B
=
for the AMAG 492 alloy, obtained experimentally and calculated based on the results of solving
equation (6) for the steady-state mode at a magnetization reversal frequency equal to 50 Hz.
Volume 02 Issue 12-2022
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VOLUME
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SJIF
I
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FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
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1121105677
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IF
–
5.582
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Fig. 4. Calculated and experimental graphs of hysteresis curves of dependence
)
(
H
f
B
=
for AMAG 492 alloy
From the graphs shown in Fig. 4, a good coincidence of the calculated and experimental curves can be seen,
which at the reference points (the origin is exact, the point with the coercive force
H
c
and the residual magnetic
induction
B
r
and the point with the limiting value of the magnetic field strength, in our case equal to 800 A/m)
coincide completely. The greatest difference between experimental and calculated graphs of hysteresis loops is
observed in the area of the greatest bend of the magnetization curve. In the areas of linear dependence
)
(
H
f
B
=
and the saturation area of the magnetization curve, the calculation errors are minimal.
Results
Let's compare the magnetization curves of amorphous materials obtained by approximating them with an
algebraic expression and their hysteresis loops obtained using the Jiles-Atherton model. As a comparison criterion,
the value of the relative modeling error calculated by the expression
%
100
(%)
−
=
i
iA
i
B
B
B
can be used,
where
B
i
- is the experimental value of magnetic induction at the
i
–
th point;
B
iA
- is the value of magnetic induction
calculated by the approximating function and using the Jiles-Atherton model at the same point.
Volume 02 Issue 12-2022
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06-16
SJIF
I
MPACT
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(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
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In figure 5 shows the graphs of the dependence
)
(
H
f
B
=
for the amorphous AMAG 492 alloy,
constructed for various modeling methods: the experimental dependence
)
(
H
f
B
=
, taken on a full-scale sample,
the calculated dependence
)
(
H
f
B
=
, obtained by using an approximation by an incomplete polynomial of the
form
9
66
,
14
B
H
=
and a computational model of the hysteresis loop derived from the Gills-Atherton model. It can
be seen from the graphs that the accepted methods give approximately the same modeling accuracy.
Fig. 5. Graphs of the dependence
)
(
H
f
B
=
for various modeling methods
Figure 6 shows graphs of the dependence of the relative modeling error
)
B
(
f
(%)
=
for the AMAG 492
alloy when using the modeling methods discussed above.
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Fig. 6. Dependence graphs
)
B
(
f
(%)
=
for AMAG 492 alloy:
1 - approximation by the function
9
66
,
14
B
H
=
, 2
–
the direct branch of the hysteresis loop of the Jiles-Atherton
model, 3
–
the reverse branch of the Jiles-Atherton hysteresis loop model
CONCLUSION
1. It is evident from the graphs shown in Fig. 6 that the
relative errors of approximation of the magnetization
loop and the hysteresis loop of the Jiles-Atherton
model in their greatest magnitude differ little from
each other, and on this basis both the model of the
magnetization curve obtained by approximating the
magnetization curve of a ferromagnetic material and
the Jiles-Atherton model can be accepted for analysis
devices based on magnetically soft amorphous
materials, including those operating in saturation
mode.
2. The final conclusion about the advantages of a
particular model can be made only on the basis of the
final goals of the analysis, since the permissible errors
differ by no more than the magnitude of the
measurement error and are not sufficiently reliable.
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