Mô hình hóa quá trình tự đốt nóng của cuộn cảm để nghiên cứu sự trao đổi điện từ - nhiệt

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.

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

1 INTRODUCTION

electromagnetic system is a key element

of an efficient energy conversion The

optimization of the magnetic circuit

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

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

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

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

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

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

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

k

M M dH

e

irr (  )





k

M M

dH

e



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 M rev and the irreversible

component M irr

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

increasing This increase is illustrated by

an expansion of the B (H) loop

The "flux tube" model [6] is build by

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, )

approximately by the equation:

e dH irr dM c e

dH an dM c

e dH an dM c e dH irr dM c dH

dM













1 1

1





) ( an irr

rev c M M

)

irr irr

M

dt

dB B

H

12

d2



 

(1)

(2)

(3)

(4)

(5)

(6)

(7)

with δ is the conductivity and d is the

sample thickness

The model has the advantage of being

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

H stat(B) Equation (4) expresses the static

model as a relation B(H stat) It may

equally well be placed under H stat(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

materials, preparation and knowledge of

measurement techniques are important to

have accurate baseline data A magnetic

material characterization bench has been

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

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

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

processes: either by an irreversible

evolution of their local composition

(aging) or by reversible changes of their

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)

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

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































edt S N

B dt

d N e

I L

N H R

U I

m shunt

1

2 2

1



(8)

increase for the MnZn N30 material

when the temperature increases [6]

Fig.4 B (H) loop measured and simulated at

23 0 C and 100 0 C, 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 23 0 C and 100 0 C, 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

whenthe temperature rises to85°C From this temperature, it tend to increase, we believe, to compensate the error made by the static model (OF1_115°C ≈ 2.5*

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

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; B mes and B sim are the

measured and simulated inductions

respectively; max (B mes) is the maximum

measurement

In static regime, the quality of the simulation is estimated via the relative error and the square error The results are

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

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

sim mes

B

j B j B N

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

(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;

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

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;

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:

R th1 = 8.43882°C/W; R th2 = 80.685°C/W;

J/°C.kg and C th2 = 1.5 J/°C.kg

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

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

sim mes j j N

OF







(10)











) max(

(j) (j)

max (%) Δ

mes

sim mes

max









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

correspondence between measurement

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