MÔ HÌNH HOÁ CỦA TỪ TRƯỜNG VÀ DÒNG ĐIỆN XOÁY TRONG MÀN CHẮN ĐIỆN TỪ BẰNG PHƯƠNG PHÁP LIÊN KẾT BÀI TOÁN NHỎ

Mục đích của bài bài báo được dựa trên phương pháp bài toán nhỏ với công thức véc tơ từ thế để tính toán từ trường, dòng điện xoáy và tổn hao công suất trong các màn chắn điện từ, mà kh[r]

MODELING OF MAGNETIC FIELDS AND EDDY CURRENT LOSSES IN ELECTROMAGNETIC SCREENS BY A SUBPROBLEM METHOD

Dang Quoc Vuong *

School of Electrical Engineering, Hanoi University of Science and Technology

ABSTRACT

The aim of this paper is based on a subproblem technique with the magnetic vector potential formulation to compute magnetic fields, eddy currents and Joule power losses in electromagnetic screens that are extreamly difficult to perform by a finite element method The subproblem method

is herein developed for coupling problems in two steps: A problem starting from simplified models with stranded inductors and thin screen models can be also first considered Then a correction problem with the actual volume thin regions is added to correct inaccuracies from previous problem All the steps are separately performed with different meshes and geometries, which facilitates meshing and speeding up the calculation of each problem

Keywords: Magnetic field, Eddy current, Joule power loss, Electromagnetic screen,

Magnetodynamics, Subproblem method (SPM), Magnetic vector potential

Received: 18/10/2018; Revised: 16/11/2018; Approved: 28/12/2018

MÔ HÌNH HOÁ CỦA TỪ TRƯỜNG VÀ DÒNG ĐIỆN XOÁY TRONG MÀN CHẮN ĐIỆN TỪ BẰNG PHƯƠNG PHÁP LIÊN KẾT BÀI TOÁN NHỎ

Đặng Quốc Vương *

Viện Điện - Trường Đại học Bách khoa Hà Nội

TÓM TẮT

Mục đích của bài bài báo được dựa trên phương pháp bài toán nhỏ với công thức véc tơ từ thế để tính toán từ trường, dòng điện xoáy và tổn hao công suất trong các màn chắn điện từ, mà khó có thể thực hiện trực tiếp bằng phương pháp phần tử hữu hạn Ở đây, phương pháp bài toán nhỏ được phát triển để liên kết các bài toán theo hai bước: Một bài toán với mô hình đơn giản (các cuộn dây

và màn chắn điện từ) được giải trước, sau đó một bài toán hiệu chỉnh được thêm vào để hiệu chỉnh sai số do bài toán trước gây ra Tất cả các bước đều được thực hiện độc lập với các lưới và miền hình học khác nhau, điều này tạo thuận lợi cho việc chia lưới cũng như tăng tốc độ tính toán của mỗi một bài toán

Keywords: Từ trường, dòng điện xoáy, tổn hao công suất, màn chắn điện từ, bài toán từ động,

phương pháp bài toán nhỏ (SPM), véc tơ từ thế

Ngày nhận bài: 18/10/2018; Ngày hoàn thiện: 16/11/2018;Ngày duyệt đăng: 28/12/2018

* Corresponding author: Tel: 0963286734, Email: vuong.dangquoc@hust.edu.vn

INTRODUCTION

Many papers have been applied a subproblem

method (SPM) for computing electromagnetic

fields (eddy current, magnetic flux density

and magnetic filed) and correcting

inaccuracies of fields in the vicinity of thin

shell models in three steps [1-5] In this paper,

the SPM is extended for coupling subprolems

(SPs) in two steps: A problem starting from

simplified models with stranded inductors

(Fig 1, top right) and thin screen models can

be also first considered, followed by a

correction problem (Fig 1, bottom) with the

actual volume thin regions

The key point of this method allows to benefit

from previous computations instead of

starting a new complete finite element (FE)

solution of thin shell model for any variation

of geometrical or physical characteristics

Thus, each SP is solved on its own domain

and mesh (Fig 2), which facilitates meshing

and may increase computational efficiency in

each step

The method is is validated on a test practical

problem Its main advantages are pointed out

Figure 1 Decomposition of a complete problem

into two subproblems

Figure 2 Decomposition of a complete mehs into

two sub-meshes: stranded inductor and thin shell mesh (top), actual volume mesh (bottom).

METHOD

Canonical Magnetodynamic problem

A canonical magnetodynamic problem i, to be solved at step i of the SPM, is defined in a Ω𝑖, with boundary 𝑖= Γℎ,𝑖∪ Γ𝑏,𝑖 Subscript i refers to the associated problem i The

equations, material relations, boundary conditions (BCs) and interface conditions (ICs) of SPs are [5-11]

curl 𝒉𝑖 = 𝒋𝑖, div 𝒃𝑖= 0, curl 𝒆𝑖 = −𝝏𝑡𝒃𝑖

(1a-b-c)

𝒉𝑖 = 𝜇𝑖−1𝒃𝑖+ 𝒉𝑠,𝑖, 𝒋𝑝= 𝜎𝑝𝒆𝑝+ 𝒋𝑠,𝑝 (2a-b)

𝒏 × 𝒉𝑖 = 𝒋𝑓,𝒊, 𝒏 × 𝒃𝑖|Γ𝑏,𝑖= 𝒇𝑓,𝑖 (3a-b)

where 𝒉𝑖 is the magnetic field, 𝒃𝑖 is the magnetic flux density, 𝒆𝑖 is the electric field,

𝒋𝑖 current density, 𝜇𝑖 is the magnetic permeability and 𝜎𝑖 is the electric

conductivity and n is the unit normal exterior

to Ω𝑝 The fields 𝒃𝑠,𝑖 and 𝒋𝑠,𝑝 in (2a-b) are volume sources (VSs) With the SPM, 𝒉𝑠,𝑖 is also used for expressing changes of permeability and 𝒋𝑠,𝑖 for changes of conductivity For changes in a region, from 𝜇𝑝 and 𝜎𝑝 for

problem (i =p) to 𝜇𝑘 and 𝜎𝑘 for problem (i =

𝒉𝑠,𝑘= (𝜇𝑘−1− 𝜇𝑝−1)𝒃𝑝, (4)

𝒋𝑠,𝑘= (𝜎𝑘− 𝜎𝑝)𝒆𝑝 (5) for the total fields to be related by 𝒉𝑝+ 𝒉𝑘 = (𝜇𝑘−1(𝒃𝑝+ 𝒃𝑘) and 𝒋𝑝+ 𝒋𝑘 = 𝜎𝑘(𝒆𝑝+ 𝒆𝑘)

=

+

c,k n

n

act ual volume

air

c

j s , hs

γt=γf

+

t hin shell

The surface fields 𝒋𝑓,𝑖 and 𝒇𝑓,𝑖 in (3a-b) are

defined possible SSs that account for

particular phenomena occurring in the thin

region between γ𝑖+ and γ𝑖− [2-7] This is the

case when some field traces in a SP 𝑝( 𝑖 = 𝑝)

are forced to be discontinuous The continuity

has to be recovered after a correction via a

SP 𝑘 (𝑖 = 𝑘) The SSs in SP 𝑘 are thus to be

fixed as the opposite of the trace solution of

SP 𝑝 [2-5]

Constraints between thin shell and

correction

The thin shell (TS) model [2-4] written with

the 𝒃𝑖-formulation, requires a free (unknown)

discontinuity 𝒂𝑑,𝑡,𝑖 of the tangential

component 𝒂𝑡,𝑖 = (𝒏 × 𝒂𝑖) × 𝒏 of 𝒂𝑖 through

the TS, i.e

[𝒏 × 𝒂𝑡,𝑖]Γ

with a fixed zero value along the TS border

∂Γ𝑡,𝑖, which neglects the magnetic flux

entering there To explicitly express this

discontinuity, one defines [2-5]

𝒂𝑖|Γ𝑡,𝑖 = 𝒂𝑐,𝑖+ 𝒂𝑑,𝑖, (7)

where 𝒂𝑐,𝑖 is the continuous component of 𝒂𝑖

In addition, the constraint for SPs are

respectively expressed via SSs and VSs SSs

in (3a-b) are defined via the BCs and ICs of

contributions from SP 𝑖 (𝑖 = 𝑝, 𝑘) [2-5] One

has [4],

[𝒏 × 𝒉] = [𝒏 × 𝒉𝑝]Γ

𝑡,𝑘+ [𝒏 × 𝒉𝑘]Γ𝑡,𝑘 =

−𝜎𝛽𝜕𝑡(2𝒂𝑐,𝑖+𝒂𝑑,𝑖),

(8)𝛽 = 𝛾𝑖−1tanh (𝑑𝑖 𝛾 𝑖

2 ) , 𝛾𝑖 =1+𝑗

𝛿 𝑖 ,

𝛿𝑖 = √ 2

𝜔𝜇𝜎

where d i is the local TS) thickness 𝛾𝑖, 𝛿𝑖 is the

skin depth in the TS, 𝜔 = 2𝜋𝑓 with f is the

frequency, j is the imaginary unit The VSs

can be defined in (4) and (5)

Finite element weak formulation

Equations (1b-c) are fulfilled via the

definition of a magnetic vector potential 𝒂𝑖

and an electric scalar potential 𝜈𝑖, leading to the 𝒂𝑖- formulation, with

curl 𝒂𝑖= 𝒃𝑖, 𝒆𝑖 = −𝝏𝑡𝒂𝑖− grad 𝜈𝑖, (9a-b)

𝒏 × 𝒂𝑖|Γ𝑏,𝑖 = 𝒂𝑓,𝑖 (10) The weak 𝒃𝑖-formulation (in terms of 𝒂𝑖) of

of the Ampère equation (1a), i.e [5] - [11] (𝜇𝑖−1curl 𝒂𝑖, curl 𝒂𝑖′)Ω

𝑖+ (𝒉𝑠,𝑖, curl 𝒂𝑖′)Ω

𝑖

+(𝜎𝑖𝜕𝑖𝒂𝑖, 𝒂𝑖′)Ω𝑐,𝑖+ (𝜎𝑖grad 𝜈𝑖, 𝒂𝑖′)Ω𝑐,𝑖

+< 𝒏 × 𝒉𝑖, 𝒂𝑖′ >Γℎ,𝑖−Γ𝑡,𝑖 + < [𝒏 × 𝒉𝑖]Γ𝑡,𝑖, 𝒂𝑖′ >Γ𝑡,𝑖

= (𝒋𝑖, 𝒂𝑖′)Ω𝑠,𝑖, ∀ 𝒂𝑖′ ∈ 𝐹𝑖1(Ω𝑖) (11)

𝑐,𝑖, gauged in 𝑐,𝑖𝐶 , and containing the basis functions for 𝒂𝑖 as well as for the test function 𝒂𝑖′ (at the discrete level, this space is defined by edge FEs; the gauge is based on the tree-co-tree technique); (·, ·) and

< ·, · > respectively denote a volume integral

in  and a surface integral on  of the product of their vector field arguments The surface integral term on Γℎ,𝑖-Γ𝑡,𝑖 accounts for natural BCs of type (3a), usually zero

The term < [𝒏 × 𝒉𝑖]Γ𝑡,𝑖, 𝒂𝑖′ > in (11) is defined via (8), that is

< [𝒏 × 𝒉𝑖]Γ𝑡,𝑖, 𝒂𝑖′ > =

< −𝜎𝛽𝜕𝑡(2𝒂𝑐,𝑖+ 𝒂𝑑,𝑖, 𝒂𝑖′ >, (12)

Once obtained, the TS solution is then corrected by a correction problem that overcomes the TS assumption [2-5]

At the discrete level, the source 𝒂𝑝(𝑖 = 𝑝),

initially in mesh of SP𝑢 has to be projected in

mesh of SP𝑘 (𝑖 = 𝑘) via a projection method

[2-3] For the TS problem, the mesh describes the details of the source and is simplified near the TS regions, whereas the correction problem mesh focuses on the actual volumic thin region, finely discretized in a homogeneous medium The required sources for the correction problem have to be transferred from the TS mesh to the correction mesh A rigorous expression of the

sources is crucial for the efficiency of the

method

Projections of Solutions between Meshes

Some parts of a previous solution 𝒂𝑝 serve as

sources in a subdomain 𝑝𝑘 of the

current problem SP𝑘 At the discrete level,

this means that this source quantity 𝒂𝑝 has to

be expressed in the mesh of problem SP𝑘,

while initially given in the mesh of problem

SP𝑝 This can be done via a projection method

[2-4] of its curl limited to 𝑘, i.e

(curl 𝒂𝑝, curl 𝒂𝑘′ )Ω

𝑘 = (curl 𝒂𝑘, curl 𝒂𝑘′ )Ω𝑘,

∀ 𝒂𝑘′ ∈ 𝐹𝑘1(Ω𝑘) (10) where 𝐹𝑘1(Ω𝑘) is a gauged curl-conform

function space for the k-projected source 𝒂 𝑝

(the projection of 𝒂𝑝 on mesh SP𝑘) and the

test function 𝒂𝑘′

APPLICATION TEST

The test problem is based on TEAM problem

21 (2D-model, page 4) [12], with two

inductors and a magnetic plate/screen (Fig 3),

with f = 50Hz, 𝜇𝑟 = 200, 𝜎 = 6.484MSm

Figure 3 Geometry of TEAM problem 21.

The problem is considered in two steps: the

distribution of magnetic flux density

generated by imposed electric currents

flowing in stranded inductors is pointed out in

Figure 4 (top) Then volume correction SP𝑘

replaces the TS plate with classical volume

FEs covering the plates and their

neighborhood, with an adequate refined mesh,

that does not include inductors and TS plate

anymore (Fig 4, bottom), to correct errors

arising from the TS plate [2-4] The distribution of eddy current density along the volume correction is shown in Figure 5 The computed results of magnetic flux density and Joule power loss in the plate checked to be close to the measured results for different parameters of exciting currents (proposed by author in [12]) are shown in Figure 6 and Figure 7 The everage errors between computed and measured methods on the magnetic flux density are lower than 7%, and are lower 10% for Joule power losses It can be shown that there is a very good agreement between two methods

This test problem has been successfully applied to standardize the type and material of the plate in practice

Figure 4 Magnetic flux density for stranded

inductors and TS plate SP 𝑝 (top), the correction

SP 𝑘 with an actual screen (bottom) (𝜇𝑟 =

200, 𝜎 = 6.484MSm, 𝑓 = 50Hz, 𝑑 = 5mm).

Figure 5 Eddy current along the plate/screen for

the correction SP𝑘 with an actual creen (𝜇𝑟 =

200, 𝜎 = 6.484MSm, 𝑓 = 50Hz, 𝑑 = 10mm)

Figure 6 Comparison of the computed and

measured Joule power loss in the plate (𝜇 𝑟 =

100, 𝜎 = 6.484MSm, 𝑓 = 50Hz, 𝑑 = 5mm)

Figure 7 Comparison of the computed and

measured magnetic flux density in the plate

(𝜇 𝑟 = 100, 𝜎 = 6.484MSm , 𝑓 = 50Hz, 𝑑 = 5mm)

CONCLUSIONS

All the steps of the subproblem technique

have been presented by coupling SPs in two

steps The accuracies of magnetic flux density

and Joule power losses are successfully

obtained in the plate/screen In particular,

they have been checked to be close with the

measured results [12] Moreover, The

correction is also directly linked to the volumic mesh of the plate/screen and its eneighboring, that permit to reduce the meshing efforts The method allows to use previous local meshes instead of starting a new complete mesh for any postion of the plate/screen

REFERENCES

1 Vuong Q Dang, P Dular R.V Sabariego, L

Krähenbühl, C Geuzaine, “Subproblem Approach for Modelding Multiply Connected Thin Regions with an h-Conformal Magnetodynamic Finite Element Formulation”, in EPJ AP., vol 63,

no.1, 2013

2 Vuong Q Dang, P Dular, R.V Sabariego, L

Krähenbühl, C Geuzaine, “Subproblem approach for Thin Shell Dual Finite Element Formulations,” IEEE Trans Magn., vol 48, no 2,

pp 407–410, 2012

3 P Dular, Vuong Q Dang, R V Sabariego,

L Krähenbühl and C Geuzaine, “Correction of thin shell finite element magnetic models via a subproblem method,” IEEE Trans Magn., Vol 47,

no 5, pp 158 –1161, 2011

4 Dang Quoc Vuong “Modeling of Electromagnetic Systems by Coupling of Subproblems – Application to Thin Shell Finite Element Magnetic Models,” PhD Thesis (2013/06/21), University of Liege, Belgium,

Faculty of Applied Sciences, June 2013

5 Dang Quoc Vuong “A Subproblem Method for Accurate Thin Shell Models between Conducting and Non-Conducting Regions,” The University of

Da Nang Journal of Science and Technology, no

12 (109).2016

6 Tran Thanh Tuyen, Dang Quoc Vuong, Bui

Duc Hung and Nguyen The Vinh “Computation of magnetic fields in thin shield magetic models via the Finite Element Method,” The University of Da

Nang Journal of Science and Technology, no 7 (104).2016

7 Dang Quoc Vuong, Bui Duc Hung and Khuong

Van Hai “Using Dual Formulations for Correction of Thin Shell Magnetic Models by a Finite Element Subproblem Method,” The University of Da Nang Journal of Science and Technology, no 6 (103).2016

8 Dang Quoc Vuong “Tính toán sự phân bố của

từ trường bằng phương pháp miền nhỏ hữu hạn -

0

50

100

150

200

250

300

350

400

Exciting currents (A)

Measured results Calculated results

0

0.5

1

1.5

2

-3 T

Exciting currents (A)

Measured results Calculated results

Ứng dụng cho mô hình cấu trúc vỏ mỏng,” Tạp chí

Khoa học và Công nghệ, Đại học Công nghiệp Hà

Nội, số 36, trang 18-21, 10/2016

9 Dang Quoc Vuong “An iterative subproblem

method for thin shell finite element magnetic

models," The University of Da Nang Journal of

Science and Technology, no 12 (121).2017

10 Tran Thanh Tuyen and Dang Quoc Vuong,

“Using a Magnetic Vector Potential Formulation

for Calculting Eddy Currents in Iron Cores of

Transformer by A Finite Element Method,” The

University of Da Nang Journal of Science and Technology, no 3 (112), 2017 (Part I)

11 S Koruglu, P Sergeant, R.V Sabarieqo,

Vuong Q Dang, M De Wulf “Influence of contact resistance on shielding efficiency of shielding gutters for high-voltage cables,” IET

Electric Power Applications, Vol.5, No.9, (2011),

pp 715-720

12 Zhiguang CHENG, Norio TKAHASHI, and Behzad Forghani “TEAM Problem 21 Family (V.2009),”-http://www.compumag.org/