Tài liệu Mô hình hóa quá trình tự đốt nóng của cuộn cảm để nghiên cứu sự trao đổi điện - từ - nhiệt: TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
1
MODELING OF THE SELF-HEATING PROCESS
OF AN INDUCTANCE TO STUDY THERMAL - MAGNETIC
ELECTRIC EXCHANGES
MÔ HÌNH HÓA QUÁ TRÌNH TỰ ĐỐT NÓNG CỦA CUỘN CẢM
ĐỂ NGHIÊN CỨU SỰ TRAO ĐỔI ĐIỆN - TỪ - NHIỆT
Anh Tuan Bui - Tuan Anh Kieu
Electric Power University
Abstract:
This paper focuses on thermal stresses on magnetic materials under Curie temperature. The aim of
this article is to study the influence of temperature on all standard static magnetic properties. The
Jiles-Atherton model and “flux tube” model are used in order to reproduce static and dynamic
hysteresis loops for MnZn N30 (Epsco) alloy. For each temperature, the six model parameters are
optimized from measurements. The model parameters variations are also discussed. Finally, the
electromagnetic model is associated with a simple thermal model to simulate energy exchanges
among the three the...

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TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
1
MODELING OF THE SELF-HEATING PROCESS
OF AN INDUCTANCE TO STUDY THERMAL - MAGNETIC
ELECTRIC EXCHANGES
MÔ HÌNH HÓA QUÁ TRÌNH TỰ ĐỐT NÓNG CỦA CUỘN CẢM
ĐỂ NGHIÊN CỨU SỰ TRAO ĐỔI ĐIỆN - TỪ - NHIỆT
Anh Tuan Bui - Tuan Anh Kieu
Electric Power University
Abstract:
This paper focuses on thermal stresses on magnetic materials under Curie temperature. The aim of
this article is to study the influence of temperature on all standard static magnetic properties. The
Jiles-Atherton model and “flux tube” model are used in order to reproduce static and dynamic
hysteresis loops for MnZn N30 (Epsco) alloy. For each temperature, the six model parameters are
optimized from measurements. The model parameters variations are also discussed. Finally, the
electromagnetic model is associated with a simple thermal model to simulate energy exchanges
among the three thermal - magnetic - electric areas towards self-heating process of an inductance.
The simulation outcomes will be compared with experimental results.
Keywords:
Magnetic hysteresis; Magnetic materials; Modeling; Magneto-thermal coupling.
Tóm tắt:
Bài viết này tập trung vào các ứng suất nhiệt trên vật liệu từ dưới nhiệt độ Curie. Nghiên cứu ảnh
hưởng của nhiệt độ đến tất cả các thuộc tính từ tính của vật liệu từ. Mô hình Jiles-Atherton và mô
hình "ống từ thông" được sử dụng để mô phỏng các đường cong từ trễ ở chế độ ổn định tĩnh và
chế độ ổn định động của vật liệu từ ferit MnZn N30 (Epsco). Đối với mỗi nhiệt độ, sáu thông số
của hai mô hình mô phỏng trên được tối ưu hóa từ các phép đo. Sự thay đổi các thông số trong
hai mô hình mô phỏng sẽ được tìm hiểu. Cuối cùng, mô hình điện từ được kết hợp với một mô
hình nhiệt đơn giản mô phỏng quá trình tự trao đổi năng lượng giữa ba lĩnh vực: điện - từ - nhiệt
đối với hiện tượng tự đốt nóng của một cuộn cảm. Kết quả mô phỏng sẽ được so sánh với các kết
quả thực nghiệm.
Từ khóa:
Từ trễ, vật liệu từ, mô hình hóa, liên kết từ - nhiệt.1
1 Ngày nhận bài: 30/07/2015; Ngày chấp nhận: 03/08/2015; Phản biện: TS Nguyễn Đức Huy.
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
2
1. INTRODUCTION
The magnetic circuit in the
electromagnetic system is a key element
of an efficient energy conversion. The
optimization of the magnetic circuit
geometry, the control of energy
efficiency through the use of powerful
magnetic materials and a thorough
knowledge of their behavior, especially
under high stress as temperatures and
high frequencies that are meet more
today.
The temperature at which occurs the
disappearance of spontaneous
magnetization is called the Curie
temperature. The effect is not as brutal as
it seems. The temperature increase leads
to an evolution of the saturation
magnetization, coercive field, remanent
flux density, resistivity and magnetic
losses, etc [4], [5].
The objective of this study is to build a
model as complete as possible to cover a
wide class of samples of magnetic
materials. This model must take into
account several aspects of the
phenomena as the initial magnetization
curve and the major loop. The model
should allow further integration of the
evolution of the hysteresis loop based on
temperature and frequency. Finally, it
must be fast enough for inclusion in
design and simulation software.
The modeling of magnetic materials
plays an important role in modeling
systems in electromagnetism. Many
studies have shown that the mechanisms
at the origin of the phenomenon of
magnetization depends on many factors
[4]: the material, the excitation field, the
external conditions,... From an
experimental point of view, two
operating regimes can be distinguished:
the quasi-static and the dynamic one.
Below certain frequencies, the hysteresis
loop does not depend on frequency. The
material is in a quasi-static mode. Several
models are proposed to describe this
mode [1], [6]. To meet out our
objectives, we must have a model with a
basic mathematical and physical enough
flexibility and a complete
implementation for the integration of
additional parameters that take into
account the temperature and frequency.
One of these models is characterized by a
physical basis and theoretical particularly
comprehensive. This is the Jiles-
Atherton model [1], [2].
In dynamic regime, the hysteresis loop
expands with the frequency increase that
is the energy loss is high in dynamic
mode.
This paper presents first the static and
dynamic behaviors when the temperature
increases. It also presents the static
hysteresis model and the dynamic model
that can modelize the hysteresis
characteristics of magnetic materials as a
function of temperature. The “flux tube”
model [6] is used to model the dynamic
behavior. The MnZn N30 (Epcos)
magnetic material is used here because
this material has a low Curie temperature
(around 1300C), so we can clearly see the
change of factors: power loss, the
magnetization, temperature, resistance. In
addition, this material is widely used in
the fields of electrical, electronic,...
Finally, this material is used on self -
heating inductor to achieve a coupling
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
3
k
MM
dH
dM irran
e
irr )(
k
MM
dH
dM irran
e
irr )(
between three areas: electric - magnetic -
thermal.
2. THE “FLUX TUBE” MODEL
The Jiles-Atherton model, based on
physical considerations, is able to
describe the quasi-static hysteresis loops.
It assumes that the exchange energy per
unit volume is equal to the exchange of
magnetostatic energy added by hysteresis
loss. The magnetization M is separated
into two components: the reversible
component Mrev and the irreversible
component Mirr.
The irreversible component can be
written as follows [1]:
where the constant k is related to the
average energy density of Bloch walls.
The parameter δ takes the value 1
when dH/dt >0 and the value -1 when
dH/dt <0.
Jiles and Atherton show that the
reversible magnetization is proportional
to the difference (Mirr-Man):
with c is a coefficient of reversibility as c
[0,1].
So the total magnetization is the sum of
components reversible and irreversible
[3]:
The following differential equation is
obtained:
with
Equation (4) describes the behavior law
M(H). The five parameters c, a, k, α and
Ms are determined from measurements
(magnetization curve and major loop)
and by using an optimization algorithm
[6].
When the frequency increases, several
dynamic effects appear inside the
material, the eddy currents are
increasing. This increase is illustrated by
an expansion of the B (H) loop.
The "flux tube" model [6] is build by
considering the material as an
homogeneous flux tube. This can be
expressed in terms of flows through the
tube and parameter γ can be identified by
a first order ordinary differential equation
(6):
Hdyn is the excitation field, Hstat is a
fictitious field function of the flux
density, γ is a coefficient depending on
the material magnetic and electrical
properties (resistivity, permeability,...).
Its value may be calculated
approximately by the equation:
edH
irrdMc
edH
andMc
edH
andMc
edH
irrdMc
dH
dM
11
1
)( irranrev MMcM
)( irranirrirrrev MMcMMMM
dt
dB
BHH statdyn )(
12
. 2d
(1)
(2)
(3)
(4)
(5)
(6)
(7)
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
4
with δ is the conductivity and d is the
sample thickness.
The model has the advantage of being
simple because it requires the
identification of a single dynamic
parameter and have a very fast
computation time.
The "flux tube" model can use the Jiles-
Atherton model in order evaluate
Hstat(B). Equation (4) expresses the static
model as a relation B(Hstat). It may
equally well be placed under Hstat(B)
form which is done to solve the equation
(4).
The coefficient γ is optimized by
comparison between the measured and
simulated hysteresis loops.
The “flux tube” model therefore needs
the identification of six parameters (five
static parameters and one dynamic
parameter).
The “flux tube” model (6) has been
implemented in the Matlab Simulink
simulation software to test its accuracy
according to several criteria. The
Simulink scheme describing the model is
given in Fig.1.
Fig.1. Simulink diagram for the “flux tube” model
3. MEASUREMENTS AND
SIMULATIONS
In order to get a well suited hysteresis
model for various ferromagnetic
materials, preparation and knowledge of
measurement techniques are important to
have accurate baseline data. A magnetic
material characterization bench has been
developed for quickly measuring
hysteresis loops B(H) with high
accuracy. For our purpose, we need to
measure the B(H) loops for several
temperature values. Fig.2 shows a
scheme of the test bench used for these
measures.
The samples are placed in an oven that
increases the temperature (maximum
around 2500C). The samples are placed
in an aluminum box to obtain the
temperature stability on the sample
measurement (below 10C) after two
hours. We have used two thermocouples
to control the stability and the
homogenization of the temperature, one
is placed in the aluminum box space and
the other is fixed to the sample. The
process of measurement is realized when
the temperature of both thermocouples is
the same.
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Số 9 - tháng 10 năm 2015
5
Thanks to Ampere and Maxwell laws, H
and B are determined by the following
formulas:
Fig 2. Schematic of bench measurement
The temperature changes the magnetic
materials properties mainly by 2
processes: either by an irreversible
evolution of their local composition
(aging) or by reversible changes of their
electromagnetic constant with
temperature. The Fig.3 expresses the
evolution of hysteresis loops until the
Curie temperature. This clearly shows
that as the temperature increases, the
saturation induction density, the coercive
field density and the remanent induction
density decrease, as does the lower
hysteresis losses. Testing of the material
beyond the Curie temperature (135°C)
gave rise to complete thermal
demagnetization as expected for this
material.
The material is excited by a very low
frequency sinusoidal excitation field in
the static regime. In a first step, the static
model is identified and validated at 1Hz.
At each temperature value, the five Jiles-
Atherton model parameters are
optimized. The Fig.4 show good
agreement between the B(H) loops
obtained by the model with those
obtained by measurements for the same
input signal and each temperature.
Fig.3. Evolution of B(H) loop as a function
of temperature in statique regime (1 Hz)
The variation of each Jiles-Atherton
model parameter versus temperature is
shown on Fig.5. They tend to decrease
unless the parameter c, it tends to
edtSNBdt
d
Ne
I
L
N
H
R
U
I
mshunt
.
1
.
2
2
1
(8)
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
6
increase for the MnZn N30 material
when the temperature increases [6].
Fig.4. B (H) loop measured and simulated at
230C and 1000C, 1Hz
Fig.5. Evolution of five parameters
of Jiles-Atherton model as a function
temperature
When the frequency increases, several
dynamic effects appear inside the
material. The most visible effect is an
expansion of the B (H) loop. The “flux
tube" model is used to model this
behavior.
This model has the advantage of being
simple and having a computation time
very fast. The parameter γ is optimized
for the maximum excitation frequency
(here 10 kHz) until the error between
measured and simulated on iron losses is
below 10% for each temperature.
Fig.6. B (H) measured and simulated loops
at 230C and 1000C, 10 kHz
The Fig.6 show good agreement between
measured and simulated loops at 10 kHz
for each temperature. Once calibrated
parameter γ, we have all the comparison
criteria to estimate the performance of
this method. Then, the value of γ will be
used for other frequencies (lower). The γ
parameter variation versus temperature is
shown on Fig.7.
Fig.7. Evolution of the parameter γ
as a function of temperature
The γ parameter tends to decrease
when the temperature rises to 85°C. From
this temperature, it tend to increase, we
believe, to compensate the error made by
the static model (OF1_115°C ≈ 2.5*
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(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
7
OF1_23° C and ∆Bs_115°C≈
6.75*∆Bs_23°C) (Tab.1).
The simulation quality is estimated by
comparing B(H) measured loops and
simulated ones for the same input
signals. The criteria are the relative error
between maximum measured and
simulated induction, loops area and the
signal quality obtained for the same input
signal (H). These criteria give a quality
estimation of the model.
The signal quality is estimated by the
normalized mean square error (MSE)
between the measured and simulated
inductions [6]:
with N, the number of points in each of
the two vectors; Bmes and Bsim are the
measured and simulated inductions
respectively; max (Bmes) is the maximum
induction value obtained by
measurement.
In static regime, the quality of the
simulation is estimated via the relative
error and the square error. The results are
expressed by Tab.1. Within the
measurement interval, we obtain for any
temperature; the maximum induction
relative error is less than 0.7% and the
mean square error OF1 is less than
0.03%. These results represent a good
performance of the static model because
it is a wide range of temperature
variation. Moreover, the error is almost
constant over the entire temperature
range.
In dynamic regime, the quality of the
simulation is estimated by the maximum
induction relative error, the mean square
error OF1 and the relative error between
the measured and simulated loops area.
The model performance is summarized
by Tab.1. For the maximum measured
frequency, we get a mean relative error
for the iron losses of 0.12% and the mean
square error is 0.056%.
Tab.1. Models performance
θ(°C) Static regime Dynamic regime
10 kHz 5 kHz
∆Bs (%) OF1*10-4 ∆P (%) OF1*10-4 ∆P (%) OF1*10-4
23 0.36 2.1 1.8 3.9 8.8 4.3
65 0.29 1.9 5.7 7.1 0.8 5.6
85 0.98 2.1 5.9 4.3 2.9 2.6
100 0.51 2.7 8.9 6.6 6.9 4.
115 2.43 5.3 2.7 5.8 4.9 6.2
4. MODELING OF SELF -
HEATING OF AN INDUCTANCE
We use the previous simulation results to
achieve a coupling of the fields: electric -
magnetic - thermal of self - heating of an
inductance. The magnetic material of the
2
1
1
)max(
)()(1
N
j mes
simmes
B
jBjB
N
OF (9)
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(ISSN: 1859 - 4557)
Số 9 - tháng 10 năm 2015
8
magnetic circuit is MnZn N30 material.
The magnetic component is thermally
insulated (carton box + foam insulation).
A simple thermal model is first proposed
to estimate the operating temperature of
the transient component from Joule
losses and iron losses.
4.1. Development of a thermal
model
Many approaches are used to describe
heat transfer and to achieve a satisfactory
estimate of operating temperatures. Some
approaches lead to a temperature
mapping, computed at any point of the
component (numerical methods). Others
can only give the calculated temperature
in some parts of component
(conventional analytical methods, nodal
method.
In our work we use the nodal method to
model the transient heat transfer. This
method involves fixing insulated areas,
each zone forming a node. Several
simplifying assumptions are adopted:
Homogeneity of temperature inside
the magnetic core and copper winding.
Under these conditions, each element
(core and winding) corresponds to a node
and 2 thermocouple;
Capacity of thermal insulation
neglected due to its low mass;
Natural convection on the surface
of the box neglected, resulting in surface
temperature assumed equal to ambient
temperature. We checked that increasing
the temperature did not exceed 2°C.
These assumptions allow us to
define two thermal zones (Fig.8)
corresponding to the magnetic material
on the one hand, and the primary
winding, on the other. Both areas are
home to Joule heating due to losses in the
copper (Pj) and iron losses in the torus
(Pf). We assign the center of gravity of
each area and a source node representing
losses.
Thermal capacity Cth1 and Cth2
correspond to thermal energy storage:
Cth1 for the magnetic material and Cth2
for copper;
Rth1: between the core and winding,
which reflects the sum of the resistances
of conduction through the CT (resistance
between the center of the ferrite and the
surface), contact the torus - primary
winding and the winding (resistance
between the periphery and center
conductor);
Rth2: between the coil and the
ambient air, which reflects the sum of the
resistances of the contacts winding
conduction - insulator and insulator -
outer surface;
Rth3: between the core and
the surrounding air, which reflects
the sum of the resistances of conduction
contacts torus - insulation and insulation
- exterior surface.
The parameters of the thermal equivalent
circuit are determined in two steps:
Identification of thermal resistance
from the steady;
identification of thermal capacity with
the transitional regime.
We obtain the following results:
Rth1 = 8.43882°C/W; Rth2 = 80.685°C/W;
Rth3=45.2542°C/W; Cth1= 104
J/°C.kg and Cth2 = 1.5 J/°C.kg.
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9
Fig.8. Schematic of thermal model of the magnetic component studied
4.2. Algorithm coupled electro -
magneto - thermal
The corresponding coupling algorithm is
shown in Fig.9. At each change in
temperature Δθ, the model determines
the electromagnetic iron losses and Joule
losses.
The convergence of our model is not
very dependent on temperature Δθ, and
in particular as regards the last iteration.
After several tests, not adopted Δθ = 1°C
gives the best trade-precision
computation time. Two models,
electromagnetic and thermal, are
implemented in the Matlab environment.
4.3. Model validation
We validate our work by comparing the
results of measurements and simulations
in different conditions: sinusoidal voltage
sources and non-sinusoidal, various
frequencies.
To quantify the precision, the following
criteria are used:
The square error between measured
and simulated temperatures (OF1):
where:
θmes, θsim: temperatures measured
and simulated;
N: number of measurement points in
time and for θmes, θsim;
max (θmes): maximum temperature
reached.
The maximum relative error:
(11)
2
1
1
)max(
)()(1
N
j mes
simmes
jj
N
OF
(10)
.100
)max(
(j)(j)
max(%)Δ
mes
simmes
max
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Fig.9. Coupling algorithm
electro - magneto – thermal
Fig.10 - Fig.12 show the variations of
measured and simulated temperatures for
different excitation sources: sinusoidal,
rectangular and triangular at 10 kHz.
These results also show good
correspondence between measurement
and simulation, and confirm the
performance of our model (Tab.2).
Fig.10. Temperatures measured
and simulated for a sinusoidal source
at 10 kHz
Fig.11. Temperatures measured
and simulated for a rectangular source
at 10 kHz
Fig.12. Temperatures measured
and simulated for a triangular source
at 10 kHz
11 Số 9 - tháng 10 năm 2015
Criteria Excitation sources
Sinusoidal Rectangular Triangular
OF1_material 2.19E-4 4.71E-5 1.08E-4
OF1_winding 3.56E-5 1.77E-4 3.73E-4
Δθmax_material (%) 0.02 1.23 1.78
Δθmax_winding (%) 0.02 2.63 5.31
Computation time (s) 136 153 142
Tab.2. Performance of coupled models
Our model satisfies for different
sources of tension: the maximum
squared error is less than 3*10-4 for the
material, and 4*10-4 for the winding.
The maximum relative error is less than
2% for the material and 6% for
winding. The other advantage of our
model is short time calculate.
5. CONCLUSION
The results on the MnZn N30 material
depending on the temperature are very
encouraging. The model contains the
following benefits: rapid calculation
time, easy implementation. These
benefits provide users with a simple
model. Moreover, this model with the
input H and output B, easily invertible,
allows easy integration for modeling
electrical engineering systems. The
parameter γ in the “flux tube” model is
easily determined.
We realized a thermal-electromagnetic
coupling to study self-heating of
another single component, a coil with
magnetic core. We developed a simple
thermal model capable of estimating
the operating temperature of the
magnetic component from Joule losses
and iron losses. The performance of the
“flux tube” model coupled with the
thermal model allows to determine
with accuracy quite satisfactory self-
heating of different parts of component
(coil + magnetic circuit).
REFERENCES
[1] D.C. Jiles and D.L. Atherton, Ferromagnetic Hysteresis. IEEE Transactions on Magnetics,
Vol. 19, No 5, Sep 1983, 2183-2185.
[2] Jacek Izydorczyk, A New Algorithm for Extraction of Parameters Jiles and Atherton
Hysteresis Model. IEEE Transactions on Magnetics, pp. 3132-3134, Vol.42, No.10,
11/2006.
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC
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[3] DC.JILES, D.L ATHERTON, Theory of ferromagnetic hysteresis. Journal of magnetism
and magnetic materials, 1986, Vol.61, p48-60.
[4] Pierre Brissonneau, Magnétisme et matériaux magnétiques. Edition Hermès, pp86-87,
1997.
[5] Richard Lebourgebois, Ferrites doux pour l’électronique de puissance. Techniques de
l’Ingénieur, N 3 260.
[6] A.T. Bui, N. Burais, L. Morel, F. Sixdenier, Y. Zitouni, Characterization and modelling of
temperature influence on “flux tube” magnetic properties. 19th Soft Magnetic Materials
Conference, Turin : Italy (2009).
Biography:
Anh Tuan Bui, was born on 01/9/1978. Lecturer Faculty of Electrical
system Power University. Graduated from Hanoi University of
Technology in 2001, majoring in electrical systems. Completion of the
Master's program in 2006 with the same major. From 2007 to 2011,
the authors PhD student in the lab Ampere University Claude Bernard
Lyon 1 with the electrical materials field.

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