Modeling of the self-heating process of an inductance to study thermal - Magnetic electric exchanges

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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. 
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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 
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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) 
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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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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) 
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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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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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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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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. 
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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.