Volume 03 Issue 02-2023
38
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
A
BSTRACT
The subject of research in the article is ferromagnetic soft magnetic amorphous materials and alloys, which
are used in power sources of automated control systems, and in vehicle electrical installations. The aim of
the work is an objective comparison of two magnetization models for soft magnetic amorphous alloys - the
main magnetization curve using approximation functions and the Giles-Atherton hysteresis loop model. In
the study, the least squares method was used to optimize the main magnetization curve and the Giles-
Atherton hysteresis loop optimization method with the aim of its maximum coincidence with the
experimentally obtained loop. Experimental and reference data for common types of magnetically soft
amorphous alloys were used for modeling. As a criterion of model accuracy for both modeling methods,
the relative error in determining the magnetic induction value was chosen, and the experimental value
obtained from the real hysteresis loop of a magnetically soft amorphous alloy was taken as its exact value,
and the approximate value was taken from calculations of the magnetic induction value using the methods
Journal
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Research Article
COMPARISON OF MODELS OF MAGNETIZATION CURVES AND
HYSTERESIS LOOPS ACCORDING TO THE GILES-ATHERTON
MODEL FOR SOFT MAGNETIC AMORPHOUS ALLOYS
Submission Date:
February 14, 2023,
Accepted Date:
February 19, 2023,
Published Date:
February 24, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-02-06
Bedritsky I.M.
Tashkent State Transport University (Tashkent, Uzbekistan)
Bazarov L.Kh.
Tashkent State Transport University (Tashkent, Uzbekistan)
Zhuraeva K.K.
Tashkent State Transport University (Tashkent, Uzbekistan)
Mirasadov M.Zh.
Tashkent State Transport University (Tashkent, Uzbekistan)
Volume 03 Issue 02-2023
39
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
of approximating the magnetization curve and simulation of the hysteresis loop using the Giles-Atherton
method. As a result of the research, it was revealed that both models of magnetization of magnetically soft
amorphous materials give simulation results similar in accuracy. The obtained results of the study can be
used to select an appropriate magnetization model for the mathematical description of ferromagnetic
devices using magnetically soft amorphous metals and alloys. The final conclusion about the advantages of
a particular model can only be made on the basis of the ultimate goals of the analysis.
K
EYWORDS
Vehicle power supply, magnetization curve, approximating function, Giles-Atherton hysteresis model,
method error.
I
NTRODUCTION
Among the means of technical, mathematical,
linguistic and other types of support for
automated control systems (ACS), the technical
support that ensures the functioning of various
technical devices of the system due to their
various power sources is of particular
importance. Especially effective is the use of soft
magnetic amorphous alloys in transformer power
supplies, the energy efficiency of which is 70-85%
higher compared to similar transformers using
ordinary electrical steel. Since the devices for
supplying automatic control systems usually
operate in a continuous mode, the use of
transformers with low losses for magnetization
reversal and hysteresis can significantly save the
consumed electrical energy. As the cost of
electricity increases, the use of soft magnetic
amorphous alloys becomes justified in power
distribution networks, where transformers with a
power of up to 1600 kVA are used. For this
reason, the use of a suitable magnetization model
for amorphous materials, which arises in the
calculation of the cores of ferromagnetic devices,
in particular transformers, is an urgent scientific
problem.
To approximate the hysteresis loop in
ferromagnetic materials, the Giles-Atherton [4, 5,
6, 7, 10], Chan [8, 9, 10, 11, 14, 15] and other
models are most often used. However, if the
ferromagnetic elements in these devices operate
at high values of magnetic induction, then in this
case the main magnetization curve is used, which
is approximated by a suitable algebraic
expression. Most often, to approximate the
magnetization curve, the hyperbolic sinus, arc
tangent, complete and incomplete polynomials of
the n-th degree, where n is an odd integer [1, 2, 3,
17, 18], are used. The use of one or another
method for creating mathematical models of
ferromagnetic devices depends on the goals set
and the depth of study of the processes occurring
in them.
Volume 03 Issue 02-2023
40
International Journal of Advance Scientific Research
(ISSN
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2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
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FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
In the qualitative analysis of ferromagnetic
devices, the requirement of simplicity of
analytical transformations is decisive; in this case,
an analytical description of the magnetization
curve is usually used. However, with a deeper
study of the ongoing processes, for example,
when studying the quantitative parameters of the
device operation, it may be necessary to describe
the magnetization process using one of the
existing models of the hysteresis loop. Therefore,
of significant scientific interest is a comparative
analysis of the description of magnetization using
magnetization curves and using hysteresis loops,
in order to identify the optimal method for a
particular problem being solved, as well as an
objective assessment of the error when using
both methods of mathematical description of
hysteresis.
Research methods. As models for the study, cores
made of magnetically soft amorphous steels and
amorphous iron-based alloys were used, the
experimental magnetization curve of which was
recorded at an alternating current with a
frequency of 50 Hz according to the methods
described in [1, 2], in particular, for the AMAG 492
alloy, for other amorphous alloys, data on
magnetization are taken from the literature [12,
13]. The appearance of the main magnetization
curves is shown in fig. 1. It can be seen from the
curves that for the majority of soft magnetic
amorphous alloys, saturation occurs already at
low values of the magnetic field intension
compared to cold-rolled electrical steel, for which
the saturation strength is , which indicates a high
value of relative magnetic permeability for
amorphous alloys.
Figure 1. Magnetization curves of common types of amorphous steels and alloys
Volume 03 Issue 02-2023
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(ISSN
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VOLUME
03
I
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02
Pages:
38-52
SJIF
I
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FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
The linear coefficients in the approximating expressions were calculated based on the minimum total
quadratic error using the least squares method, the transition from nonlinear to linear functions was
carried out using the appropriate substitutions [16, 20] and using the 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 is N - the number of experimental points on the magnetization curve;
i
–
point number;
i
B
,
i
H
- are
the experimental values of the magnetic induction and the magnetic field intension at the
i
-th point,
respectively. For cores made of a soft amorphous alloy based on iron grade AMAG 492 in the range of
inductions from 0 to 1.6 T (saturation induction), the following approximating expressions were obtained:
−
hyperbolic sinus
)
552
,
11
(
10
892
,
1
5
B
sh
H
=
−
;
−
arc tangent
)
049
,
0
(
022
,
1
H
arctg
B
=
;
−
incomplete polynomial of the ninth degree
9
66
,
14
B
H
=
−
incomplete polynomial of the eleventh degree
11
22
,
5
B
H
=
.
Graphs of the main curve of magnetization of the amorphous alloy AMAG 492 and functions
approximating it are shown in Fig.2.
(experiment)
Tl
Figure 2. Magnetization curve and its approximating functions for AMAG 492 alloy
Volume 03 Issue 02-2023
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(ISSN
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VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
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FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
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 AMAT 492 alloy. However, the
expressions for the hyperbolic sine and arc tangent 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 most suitable for the
criterion of simplicity and accuracy is the approximation by incomplete polynomials of the ninth and
eleventh degrees.
To estimate the errors, we study the nature of the change in the relative approximation error with a change
in the magnetic field intension. The relative approximation error for each of the experimental points can
be calculated from the expression
%
100
(%)
−
=
i
iA
i
B
B
B
, where
i
B
–
is an experimental value of
magnetic induction in
i
-th point;
iA
B
is the value of the magnetic induction calculated from the
approximating function. Dependence curves
)
(
(%)
B
f
=
for incomplete polynomials of degrees from 9 to
11, as well as for the functions of the hyperbolic sine and arc tangent for the core of the AMAT 492
amorphous alloy are shown in Fig. 3.
1-incomplete polynomial of degree 11, 2-incomplete polynomial of degree 9,
3-hyperbolic sine, 4- arctangent
Figure 3. Approximation errors
It can be seen from the graphs that the errors in
the approximation by polynomials with degrees
of 9 and 11 give errors not exceeding 9%, which
can be considered acceptable when calculating
Volume 03 Issue 02-2023
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(ISSN
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VOLUME
03
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02
Pages:
38-52
SJIF
I
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FACTOR
(2021:
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)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
ferromagnetic elements based on amorphous
alloys.
The above methods for approximating the
magnetization curve are approximate, since in
reality any ferromagnetic material is magnetized
along a hysteresis loop. Therefore, it is of interest
to mathematically describe the process of
material magnetization taking into account
hysteresis. For modeling, we will use the
hysteresis loop of the AMAG 492 material, using
the Giles-Atherton model [4, 5, 19], as the most
commonly used in commercial programs for
calculating ferromagnetic devices. Due to the lack
of parameters of this model in the reference data,
it is necessary to apply its optimization, which
makes it possible to calculate the model
parameters using known experimental and
reference data.
The essence of the Giles-Atherton model is that
the total magnetization consists of three
components: hysteresis-free magnetization
an
M
,
reversible (reversible) magnetization
rev
M
,
irreversible magnetization
irr
M
, and the
relationship between the magnetization
,
M
magnetic field intension
H
and magnetic
induction value
B
is described by the expression
).
(
0
H
M
B
+
=
Magnetization
M
ferromagnetic in an external
magnetic field depends on the magnitude of the
internal field
e
H
, equal to
M
H
H
e
+
=
, where
–
coefficient taking into account the effect of
interaction between the external and internal
magnetic fields. Due to small value
, equal to
5
10
6
4
−
−
in the sources [4] it is recommended to
take it equal to zero, thus it turns out
H
H
e
.
The
value
of
the
hysteresis-free
magnetization
an
M
can be written in the form of
)
(
H
f
M
M
s
an
=
, where
s
M
–
saturation
magnetization,, and
)
(
H
f
–
function equal to
zero at
0
=
H
and unit at
H
, tending to infinity.
In the Giles-Atherton model, as a function
)
(
H
f
the Langevin function is used in the form
£(x)=coth(x)-1/x
,
with
considering,
the
hysteresis-free magnetization curve is described
by the function
−
=
H
A
A
H
M
M
s
an
coth
,
where
А
–
a scale factor ranging from 0.1 to 10000
is chosen according to the appearance of the
hysteresis loop so that the curve
an
M
passed
through the points (0,0) and
)
,
(
r
c
B
H
hysteresis
curve, where
c
H
и
r
B
–
respectively coercive
force and residual magnetic induction.
It is known from [4] that the total
magnetization
M
is the sum of two components
–
the irreversible magnetization
irr
M
and the
reversible magnetization
rev
M
.
rev
irr
M
M
M
+
=
. (1)
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VOLUME
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Pages:
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SJIF
I
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(2021:
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)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
The derivatives with respect to
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)
whence, after transformations and taking into account (1), a differential equation can be obtained
that describes the hysteresis in the Giles-Atherton model
dH
dM
c
c
M
M
k
M
M
c
dH
dM
an
an
an
)
1
(
)
(
)
(
)
1
(
1
0
+
+
−
−
−
+
=
. (3)
Here:
–
sign function,
1
=
if
,
0
dt
dH
1
−
=
if
,
0
dt
dH
c
H
k
0
- coefficient, approximately equal
to the coercive force; c - is the weight 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;
–
the coefficient taking
into account the effect of interaction between the external and internal magnetic fields, previously its
value was taken equal to zero.
With these notations, expression (3) can be rewritten as
dH
dM
c
H
M
M
c
dH
dM
an
c
an
+
−
−
=
)
(
)
1
(
. (4)
Integrating the left and right sides of (4) over
dH
, we obtain
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 into (5), we finally obtain
)
H
A
A
H
(M
c
M)dH
H
A
A
H
(M
H
c
δ
M
s
s
c
−
+
−
−
−
=
coth
coth
1
(6)
We perform the integration of equation (6) by the numerical method of Gauss-Kronrod- [16] as giving
the highest algebraic accuracy with the following initial parameters characteristic of the AMAG 492
alloy, which are given in Table 1.
Tab. 1 Calculation parameters of the Giles-Atherton model for the amorphous alloy
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(ISSN
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VOLUME
03
I
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02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
AMAG 492
Parameter
Size
Unit of measurement
В
s
0,75
Tl
M
s
1,27*10
4
А/м
Н
с
8
А
/
м
Δ
+1, -1
-
Α
32
-
c
0,58
-
α
0
-
Based on the results of numerical integration, we obtain a series of values of the magnetic field intension
H
and the corresponding induction
B
, and we will take the integral within the range of the magnetic field
intension from -1000 to +1000 A/m. On fig. 4 shows plots of the hysteresis curves of dependence
)
(
H
f
B
=
for the AMAG 492 alloy, obtained experimentally and calculated from the results of solving equation (6)
for a steady state at a magnetization reversal frequency of 50 Hz.
Figure 4. Calculated and experimental graphs of hysteresis curves of dependence
)
(
H
f
B
=
for AMAG 492 alloy
From those shown in Fig. 4 graphs show a good
agreement between the calculated and
experimental curves, which at the reference
points (the exact origin of coordinates, the point
Volume 03 Issue 02-2023
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(ISSN
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2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
with the coercive force
с
H
and the residual
magnetic induction
r
B
and the point with the
limiting value of the magnetic field intension, in
our case equal to 800 A/m) coincide completely.
The greatest difference between the experimental
and calculated plots of hysteresis loops is
observed in the area of the greatest bend in the
magnetization curve. In the sections of the linear
dependence of
)
(
H
f
B
=
and the saturation
section of the magnetization curve, the
calculation errors are minimal.
R
ESEARCH RESULTS
We compare the magnetization curves of
amorphous materials obtained by their
approximation by an algebraic expression and
their hysteresis loops obtained using the Giles-
Atherton model. As a comparison criterion, the
value of the relative
modeling error can be used,
calculated
by
the
expression
%
B
B
B
(%)
i
iA
i
100
−
=
, where
В
i
is the
experimental value of the magnetic induction at
the
i
-th point;
В
iA
is the value of the magnetic
induction calculated from the approximating
function and using the Giles-Atherton model at
the same point.
On fig. Figure 5 shows the graphs of dependence
В
=f(H)
for the amorphous alloy AMAG 492,
constructed for various modeling methods: the
experimental dependence
В
=f(H),
taken on a full-
scale sample, the calculated dependence
В
=f(H)
,
obtained by use of approximation by an
incomplete polynomial of the form
H=14,66B
9
and
the calculation model of the hysteresis loop
obtained from the Giles-Atherton model. It can be
seen from the graphs that the adopted methods
give approximately the same modeling accuracy.
(experiment)
the Giles-Atherton
Tl
A/m
Volume 03 Issue 02-2023
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VOLUME
03
I
SSUE
02
Pages:
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SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
Figure 5. Graphs of dependence
В
=f(H)
with various modeling methods
On fig. 6 shows plots of the relative simulation error
)
B
(
f
(%)
=
for the AMAG 492 alloy using the
simulation methods discussed above.
Figure. 6. Graphs of dependence
)
B
(
f
(%)
=
for alloy AMAG 492:
1
–
approximation by the function
H=14,66B
9
, 2
–
direct branch of the hysteresis loop of the Giles-
Atherton model, 3
–
reverse branch of the Giles-
Atherton hysteresis loop model
From the graphs shown in fig. Figure 6 shows that
the relative errors of approximation of the
magnetization curve and the hysteresis loop of
the Giles-Atherton model in their largest value
differ little from each other.
From the graphs shown in fig.6 shows that the
relative errors of approximation of the
magnetization curve and the hysteresis loop of
the Giles-Atherton model in their largest value
differ little from each other.
As an example, let us consider the calculation
using the models of the magnetization curve and
the Giles-Atherton hysteresis loop model of the
values of magnetic inductions in the stabilizer
rods using the amorphous alloy AMAG 492, the
scheme of which is shown in Fig. 7 [14, p.96].
Volume 03 Issue 02-2023
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VOLUME
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02
Pages:
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SJIF
I
MPACT
FACTOR
(2021:
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)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
Figure 7. Model of a parametric stabilizer with an AMAG 492 amorphous alloy core
The electrical state of the stabilizer can be described by a system of algebraic equations for
instantaneous values of electrical and magnetic quantities
+
=
+
=
=
−
+
3
3
2
2
2
2
3
3
1
1
1
1
3
2
1
0
L
h
L
h
W
i
L
h
L
h
W
i
, (7)
Where :
u, i
–
instantaneous supply voltage and current,
φ
1
,
φ
1
,
φ
1
–
instantaneous values of magnetic
fluxes;
L
1
,
L
2
,
L
3
are the lengths of the average magnetic lines of the magnetic circuit;
h
1
,
h
2
,
h
3 -
are the
instantaneous values of the magnetic field in the rods of the magnetic circuit;
i
1
,
i
2
–
instantaneous currents
in the coils of the outermost rods of the magnetic circuit, respectively, with the number of turns
W
1
,
W
2
;
s
1
,
s
2
,
s
3
–
are the sections of the magnetic core rods; C - is the capacitance of the capacitor. We transform
expression (7) in such a way that the instantaneous values of magnetic inductions become unknown, for
which we use the known ratio
φ
=b*s
;
;
*
W
L
h
i
=
M
b
h
−
=
0
. (8)
Taking into account (8) expression (7) can be rewritten in the form of
Volume 03 Issue 02-2023
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VOLUME
03
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02
Pages:
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SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
−
+
−
=
−
+
−
=
=
−
+
3
0
3
2
0
2
2
2
3
0
3
1
0
1
1
1
3
3
2
2
1
1
0
L
)
M
b
(
L
)
M
b
(
L
*
h
L
)
M
b
(
L
)
M
b
(
L
*
h
s
b
s
b
s
b
. (9)
The solution of the system of algebraic equations (9) was carried out in the following sequence:
1) by setting the values of the magnetic field intension in the range from -1000 to +1000 A / m through
the
value
of
10
A
/
m,
we
solve
the
differential
equation
))
A
H
(
coth
*
M
(
*
)
)
A
H
(
coth
*
M
(
1
S
1000
1000
S
H
A
c
dH
M
H
A
H
c
M
c
−
+
−
−
−
=
+
−
, finding the values of the magnetization
M
corresponding to the magnetic field intension
H
, Values of unknown quantities
δ
,
с
,
А
,
Н
с
,
М
с
should be
taken from Table 1;
2) after substituting the found values of
M
in (9), this system of equations turns into a system of
equations for the instantaneous values of magnetic inductions with linear coefficients;
3) solving the system of equations (3) we find three values of magnetic induction in the extreme rods
b
1
,
b
2
and the middle rod
b
3
. After that, we set the next value of
H
and repeat the calculations until all
calculations are completed in the range of values of the magnetic field intension
H
from -1000 to +1000
A/m
The results of the calculations are shown in Fig. 8, the results of calculations using the magnetization
curve model and theoretical calculations are taken from [14, p. 96-108 ] for the model with parameters:
core material - amorphous alloy AMAG 492, magnetization curve of the material
–
H
=14,66
B
9
, lengths of
the middle lines
L
1
=
L
2
=0,245
м
,
L
3
=0,15
м
, core cross-sections
S
1
=
S
2
=0,00085
м
2
,
S
3
=0,0017
м
2
, number of
turns W
1
=300, W
2
=350, C=25
мкФ
, current through the coil
W
2
is determined from the expression
2
3
0
3
2
0
2
2
)
(
)
(
W
L
M
b
L
M
b
i
−
+
−
=
–
current through the capacitor
C -
from the expression
2
2
2
2
2
2
2
*
*
*
dt
b
d
S
C
W
i
C
=
, and the supply voltage with the parameters of the stabilizer is related by the
Volume 03 Issue 02-2023
50
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
expression
)
6
sin(
312
*
*
2
2
2
1
1
1
+
=
+
t
dt
db
S
W
dt
db
S
W
, the parameters for the Giles-Atherton model are taken
from Table 1.
Magnetization curve model
Giles-Atherton model
Experiment
Fig. 8. Dependence of the amplitudes of inductions in the rods on the amplitude of the input
voltage of the stabilizer
From the graphs in Fig. 8 it can be seen that in the
zone of existence of parametric oscillations (the
range of change in the amplitudes of the supply
voltages is between points a and b), the deviations
of the magnetic induction in rod 2 do not exceed
7-9% of the calculated values obtained from the
model of the magnetization curve and the Giles-
Atherton hysteresis loop model , for inductions in
rods 1 and 3 (not shown on the graphs), the
deviations are approximately in the same range
C
ONCLUSION
1.
Thus, the model of the magnetization
curve,
obtained
by
approximating
the
magnetization curve of a ferromagnetic material,
and the Giles-Atherton model can be taken to
analyze devices based on magnetically soft
amorphous materials, including those operating
in the saturation mode, and the maximum
calculation error does not exceed 7
–
nine%
2.
The errors in the calculations of magnetic
inductions obtained in the Giles-Atherton
hysteresis loop model and by approximating the
magnetization curve in the saturation zone of the
cores have approximately the same values
Volume 03 Issue 02-2023
51
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
I
SSUE
02
Pages:
38-52
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
3.
The final conclusion about the advantages
of a particular model can be made only on the
basis of the final goals of the analysis.
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VOLUME
03
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SSUE
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