Magnetic hysteresis of soft magnetic material under an applied continuous external magnetic field

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
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© I A E M E
MAGNETIC HYSTERESIS OF SOFT MAGNETIC
MATERIAL UNDER AN APPLIED CONTINUOUS
EXTERNAL MAGNETIC FIELD
Hamza Houassine*, Djelloul Moussaoui**, S. Noureddine**, S. Moulahoum*
*Electrical engineering Departement, University of Medea Algeria
**Laboratory Of Electromagnetic Systems, Emp (former Enita)
Bp 17, 16111 Bordj El Bahri, Algiers, Algeria
1
ABSTRACT
Electromagnetic systems such as AC machines have to support supply voltages containing a
DC component which induces both an increase of the total magnetic losses and the premature
saturation of the magnetic core. In the present paper, we present an approach for predicting the
hysteresis loop of a magnetic material such as non oriented FeSi 3% which is subjected to a DC bias.
The measurements are carried out with a bench test built around an Epstein frame. The material is
excited with a damped sinusoidal flux density superimposed to a known continuous field. We obtain
superimposed asymmetrical hysteresis loops. The cycles are modeled via the Preisach Model (PM)
[1], which provides both a mathematical model for the B(H) curve and an analytical approach which
identifies and predicts the parameter behaviour needed by the PM.
Keywords: Hysteresis, Preisach Model, Continuous Field, Magnetic.
INTRODUCTION
The magnetic properties of soft material such as Fesi 3% are very sensitive to both applied
mechanical and thermal stresses as well as the external electromagnetic field. Magnetic materials
such as Fesi 3% are commonly used in the manufacturing of electromagnetic devices which are
supplied by a sinusoidal electrical network. The flux density is symmetrical, a very important
property for the good functioning of this type of device.
Many devices operating in the low- and medium- frequency range, ranging from rotating
machinery (dc machines, permanent magnet motor) to magnetic cores in electronic drives, are
employed under special supply conditions, which may include dc bias and generally non-sinusoidal
induction with local minima. The pre- diction of the core losses under such conditions, which are far

2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
removed from the standard conditions required for conventional magnetic testing of materials [1],
may present special difficulties, inherent to the general features of hysteresis (e.g. Non-local memory
effects) and their evolution with the magnetizing frequency [2].
This work attempts to investigate the function b(h) take account the continue component of
the excitation. The pm model is chosen because, on one hand, it merely accounts for all the
parameters involved in process of magnetization of magnetic materials such as saturation, minor
cycles, etc, and on the other hand, the simplicity of its implementation.
In order to validate the simulations, we carried out tests on two types of sheets: Fesi 3% with
oriented-grains 0.5mm thickness on the Epstein bench.
, H(t) ( ) = , F (1)
2
I. Preisach Model
The classical Preisach model together with quite a few generalizations has been efficiently
applied to describe hysteresis phenomena under various conditions [1, 2]. It consists in a set of
hysteresis operators (ideal switches with rectangular characteristics) and features some very
favourable properties: fast and memory sparing numerical implementation based on the everett
integrals, well defined and reliable experimental identification procedure based on measurement of
first order reversal curves. It can adequately describe the static magnetic behavior of various
ferromagnetic materials. Combined with eddy current computation, the dynamic operation of
homogeneous materials or thick parts having randomly oriented, tiny domains can be simulated with
acceptable accuracy [3]. This method fails however to encompass the effects of domain wall
displacement [4], important in the case of electrical steel sheets and it is also costly—in terms of
computing time and memory requirement. Preisach represented the magnetic state of magnetic
material, at any time, has two possible states of magnetization (m=1 and m= -1), defined by a
rectangular elementary cycle on the input-output diagram. The latter is characterized by the inversion
ields of a and b (with a³b) for which there is an irreversible transition from the high state (m=1) to
the low state m= -1 and vice versa; that is, a and b correspond to up and down switching values of
the input, respectively. The calculation of total magnetization requires knowledge of the statistical
distribution of the elementary cycles. This function is called Preisach function [1], [2]. Assuming the
input and output variables as function of time, the Preisach function of magnetization resulting from
the application of an h(t) field is given by:
r (a b ) a b [ ] a b
M t d d
With: (the value is (+1) if h=a and (-1) if h=b)
r(a,b): density function –also referred as Preisach measure-. It depends on the nature of the material
[1 ].
Fa,b[H(t)] : operator associated to the magnetic entities referred to as elementary Preisach hysteron
operator.
I.1. Geometrical Interpretation of the Model
There is a known one-to-one correspondence between the operator Fa,b[H(t)] and the points
(a,b) located in the half plana³b . Geometrically, s can be subdivided into two parts, which are
separated by the border l(t), which is itself time dependent. The surface s+(t) represents all the
entities whose state of magnetization is (+1), while s-(t) represents those with state of magnetization
(-1). Model (1) can then be written in the following form:

3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
S S (2)
r a b (4)
+ +
3
=r(a b) a b −r(a b) a b
M(t) , d d , d d
+ −
I.2. The Distribution Function
The complete determination of the Preisach model requires the knowledge of the density
function r(a,b), which is the basis for the calculation of the total magnetization. For this purpose and
at a given time, intuitively two approaches contrast.
The first method relies on extensive experimental hysteretic loops while the second method
consists of approximating real loops by means of some analytical function. In our study, we consider
the second approach.
I.3. Analytical Approach
Several analytical expressions can be used. One of these approximations is the Lorentz
function given by [1], [2]:
( )
+ +

4. + −

5. =
2
1 0.5
2
1 0.5
,
c
H
c
H
K
a b
r a b
(3)
With
K: constant of standardization adjusted to haveM(Hs(t))=Ms ,
Hc: the coercitive field.
For a better approximation of the experimental loop, the Lorentzian function is modified by adding
parameters and takes the form:
( )
2
a b
+ +

6. + −

7. =
2 2
,
b
H
b a
H
a
K a
c c
Experimental measurements show that the two parameters a and b depend on the the nature
of the material, i.e. remanent induction, coercitive field and permeability of the material. On the
other hand, the area of B(H) increases with the frequency [ 3 ]. An adjustment of the parameters a
and b is necessary to have a correct modelling of the B(H) behaviour. The parameters a and b are
defined as follow: a Î Â * + and b Î
Hs
Hc
1 ,
I.4. Mathematical Modified Formulation
With the pm model and the modified Lorentz function, we obtain the following expression.
2
( ) ( ) 2 (5)
1 2 2
a b
a b
d d
b
H
b a
+ −
H
a
K a
M t M t
c c
D
i

8. = ± −

9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
Where Mi-1 stands for the previous magnetization moment. This formulation makes the
1.5
1
0.5
0
-0.5
-1
-100 -50 0 50 100 150 200
H[A/m ]
Fig.1: Hysteresis loop under the continuous component of
magnetic field
0 0.02 0.04 0.06 0.08 0.1 0.12
t[s]
4
calculation easier.
Figure. 1 shows the influence of the continuous component of the magnetic field on the
hysteresis loop of the ferromagnetic material.
B[T]
As we see, the hysteresis loop is asymmetrical. Hence, the magnetic properties of the material
are modified under the solicitation field with constant component.
II. Simulation of the Hysteresis Loops under Solicitation with Constant Component
The solicitation considered is a combination of dc bias and ac magnetization with linearly
diminishing amplitude. In order to modelize the hysteresis, we subdivide the considered signal into
other periodical signal, and we use the Preisach Model for the simulation.
The combined signal from dc bias and ac magnetization as well as the linearly diminishing
amplitude and the hysteresis loops obtained using PM are shown in Fig.2 and Fig.3 respectively.
500
400
300
200
100
0
-100
-200
-300
H[A/m]
Fig.2: DC bias combined whit AC magnetization field excitation

10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-300 -200 -100 0 100 200 300 400 500
H[A/m]
B[T]
Fig.3: Hysteresis loop under DC bias combined whit AC magnetization field excitation
Fig.4 shows the anysteretic curve obtained by varying the amplitude of the dc bias
1.5
1
0.5
0
-0.5
-1
-1.5
-500 -400 -300 -200 -100 0 100 200 300 400 500
H[A/m]
5
magnetization
B[T]
Fig.4: The Anysteretic curve obtained by varying the amplitude of the dc bias magnetization
III. Experimental Bench
Fig.5 shows the experimental measurement setup composed from:
- An Epstein frame standardized
- A power amplifier KEPCO
- A numerical generator programmable WAVETEK 39.
- A digital scope (MXOX2000) for the visualisation and for data acquisition.
- A PC for data acquisition and processing.
- A current sensor.

11. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
Fig.5
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-100 -50 0 50 100 150 200
6
IV. EXPERIMENTAL VALIDATION
The Epstein test method is used to perform measurements for an FeSi 3% sheet. The primary
winding is supplied with a sinusoidal voltage superimposed to a known DC voltage component. The
amplitude of the maximum alternating voltage is decreasing linearly (Figure 4). We keep constant
the parameter b in the Lorentz distribution function while the parameter a is adjusted to draw
hysteresis loops as is described in Fig.6. We obtain a linear evolution for the parameter a as a
function of the magnetisation and the loops are converging to the magnetic state determined by the
dc field, as shown in Fig.7.
The same process is reproduced by considering an exponential decrease of the maximum
amplitude of the alternating voltage. Figures 8 and 9 show the comparison between predicted and
experimental hysteresis loops and the dependence of parameter a wiht maximal induction Bmax .
1
0.5
0
-0.5
Fig.6: Experimental and modelled loops under sinusoidal field superimposed to a known DC voltage
component
-1
H[A/m]
B[T]
-4 -3 -2 -1 0 1 2 3 4 5
H[A/m]
B[T]
Zoom

12. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
1.4
1.2
1
0.8
0.6
0.4
0.2
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Fig.7. Dependence of “a “parameter of the Lorentz function with the magnetic induction
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0.08
0.07
0.06
0.05
0.04
0.03
-200 0 200 400 600 800 1000
Fig.8: Experimental and modelled loops an exponential decrease of the maximum amplitude of the
alternating voltage
0.8
0.7
0.6
0.5
T)
B(0.4
0.3
0.2
0.1
0
a Fig.9: Dependence of “a “parameter of the Lorentz function with the magnetic induction
7
0
a
B[T]
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-0.4
H[A/m]
B[T]
10 12 14 16 18 20
0.02
Zoom

13. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print),
ISSN 0976 – 6553(Online) Volume 5, Issue 10, October (2014), pp. 01-08 © IAEME
By repeating the process for different DC voltage values corresponding to the different DC
magnetisation, we obtain a comparison between the experimental and predicted values of the DC
magnetisation state (table 1). We notice that there is no significant difference between the two.
Table 1: Comparison between experiments and predictions
0.2587 0.22
0.3235 0.35
0.4038 0.45
0.5265 0.55
0.6612 0.7
8
DC component of the Experimental
CONCLUSION
magnetisation
DC component of the predicted
magnetisation
Measurements have been carried out in 0.5mm thick non-oriented FeSi 3% laminations. The
material is excited with a deadened sinusoidal field superimposed on a continuous component. The
obtained asymmetric loops are modelled through the proposed approach based on the Preisach
Model (PM) combined with a mathematical model which predicts the parameters that the PM needs.
The experiment validation results are in agreement with several experiments. As a perspective for
this work, the study of the modelling of the B(H) curve through the new approach could be extended
to predict magnetic losses.
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