Method for 3-D eddy current analysis

Applied Mathematics and Mechanics (English Edition, Vol. 19, No. 11, Nov, 1998)

Published by SU, Shanghai, China

A,. ~o-I2 M E T H O D F O R 3-D E D D Y C U R R E N T A N A L Y S I S * Shi Zhanwei (SJ~{-tl)'

Zhao Xinghua ( ~ ) t

(Received June 20,1997; Revised June 5, 1998) Abstract After the field equations and the snonuHuo~ conditions between! theinterfaces for 3D eddy current problems under various gauges were discussed, it was pohTted out in this paper that ushlg the magnetic vector potential A, the electric scalar potential ~o and Coulomb gauge V "A =0 in eddy current regions and ushlg the magnetic scalar potential s hi the non-conducthTg regions are more suitable. All field equations, the boundary conditions,

the interface continuity conditions and the correspondhlg

variational prhwiple of this method are also given

Key words

I.

3-D eddy current field, A , 9 - O method, interface continuous conditions

Introduction

The analysis of 3-D eddy current field is an important problem in the electromagnetic induction heating and the electromagnctic design s. In recent fifteen years, numerous researches in this field have been done~ and A , 9 meth odt21, T42 method [4j and A* method ~ 51 etc. have been proposed early or late. But there exist two questions in their FEM calculations: (1) There are numerous, unknown quantities of the nodes for the whole region and the calculations are arduous; (2) On the interfaces or surfaces for the regions of different medium, the electromagnetic continuity conditions are not satisfied really or are not well dealt with, so the results of the calculations on the interfaces or surfaces are not good. To counter these ctuestions, in this paper, the magnetic vector potential A and the electric scalar potential 9 are used as field variables in the eddy current regions, and the magnetic scalar potential s is used as a field variable in the non-conducting regions, so the unknown quantities of the nodes in FEM cut down greatly. Based on analysis and comparison between the field equations and the interface continuity conditions under different gauges, Coulomb g a u g e V "A = 0 is selected, and the correct electromagnetic continuity conditions for the medium interfaces and the region surfaces and the corresponding variational principle with them are also obtained, so that various interface connective conditions can be well satisfied. II.

The Basic Equation

2.1

The Maxwell equations of eddy current problems

* Project supported by the National Natural Science Foundation of China (59375197) t Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P. K. China 1017

Shi Zhanwei and Zhao Xinghua

1018

Assume that the magnetic permeability tL. the conductivity cr and are is0tropic, and their values are different in different regions, but element. For sine time-varying field with middle or low frequency, frequency), the displacement current comparing with eddy density j , the Maxwell equations of eddy current problems can be written as:

the electric permittivity e are constants in a same r (c9 is the circular may be neglected. Thus,

7 x H = J , + J, OB/ 3--

7 xE=v

(2.1a) (2.1b)

9B = o

(2.1r

7 "D = 0

or

7 "J, = 0

(2.1d)

The constitute relationships are B = t~-I,

D = ~oE,

J, = ae

(2.2)

The electromagnetic continuity conditions of the interfaces between different mediums are B1 9 n = B 2 9 n ,

HI X n

= I-/2 • n

EL'n

= E2.n,

El •

= E2•

B,, = B%,

H, = H%

(2.3'j

The usu~.l boundary conditions are

J, = J%,

E, = E, ~

(2.4)

Here, H, B, D, and E are the magnetic field strength, the magnetic flux density, the electric flux density and the electric field strength, respectively. J, is a known source current density. n is an unit outer normal vector of the interfaces. B,,o,J,% and Ht0,E% are known normal and tangential components, respectiveIy. For practical 3-D eddy current problems, in general, there exist non-conducting regions. In these regions, there are not any current, so using magnetic scalar potential s as an unknown quantity is suitable. But in the eddy current regions and the source current regions, using magnetic vector potential A and electric scalar potential ~o are more effective for calculation. Here, we adopt the different variables for different regions. It not only can decrease greatly the unknown quantities of nodes in FEM, but also can obtain better results of electromagnetic calculation. The governing equations in two different regions are discussed respectively as follows: 2.2 Eddy current region and source current region Introducing a magnetic vector potential A , we take B

=

V x a

(2.5)

Substituting into Eq. (2.1b) and using 7 x 7 9' = 0 , we obtain e =-

OA 8--7- V 9

(2.6)

Substituting Eqs. (2.5) and (2.6) into Eqs. (2.1a, d) and using identical equation 7 x

/2

xA

= ~'(77

'A-

72A) +

7"7

x (7 xA)

A, ~p--Q Method for 3-D Eddy Current Analysis

1019

and in consideration of 7 1 = 0 for every element, we obtain ,u 1(7 ,u

OA

7 "A-

7 2 A ) + a - ~ -t + a V q ~ = J,

(2.7a)

This is a set equation of 3-D eddy current problems which is expressed by A and cp. In order to determine vector A, the only relationship B = 7 x A is not enough, its divergence must yet be complemented. The various definitions for divergence will lead to different field equations. Under three different gauges, Eq. (2.7) will become as: For Coulomb gauge ( 7 . A

=0),

.A

- 7ZA + / ~ a ~ - t + / ~ 7 ~

~72~ = 0 ,

= /a/,

}

(2.8)

V "A=O

ForLorentzgauge (7"A=-/~t), 0A - 72A + , u a ' ~ - t +,uaTq~ = ,uJ, -

72~o+1 ~

V "a

For generalized Lorentz gauge

aa.t ~2

=-I~

a--~ at

( 7 "A = -/.~q~ - / . ~ -

(2.9)

= 0

8t

V 2 A + t m - ~ 7t = / d ,

'

_

V2q~ + /m at V .A

(2 lO)

= _ /.~r.,o

32A and - ,ur ~at 2 are ne~_lected. Here, as ew ~ a, the terms - /.~-~-t2

2.3

Non-conducting region In non-conducting regions (air gap), there don't exist any current, and a = 0. So we take n

= -

Vn

(2.Xl)

Eq. (2.1) can be simplified as 7 2g] = 0 III. 3.1

(2.12)

The Interface Continuity Conditions

T h e i n t e r f a c e s b e t w e e n d i f f e r e n t m a t e r i a l s in t h e s a m e r e g i o n . (a) The eddy current region (include source region) Assume that AI~,AIr~,Aln, Tt and A2t,A2,~,A2n, ~2 are the tangential and the normal components of A and the value of r on both sides of the interfaces between two

Shi Zhanwei and Zhao Xinghua

1020

different materials, respectively, and' e t , P l , a t and e2,P2,0"2 are different material constants on both sides of the interfaces, respectively. Let 1, 177, n be tangential and normal directions of the interface, respectively. Substituting Eqs. (2.5) and (2.6) into the interface continuity condition Eq. (2.3). there results in Alt = A2I, A1,,, = A2rn q

-

"gAtn 'gA2n ,gt - e~"-gi-" t , 91 = 92

1 '9AI,~ t~tl

"91

1 '9A2,~ --~Z 2

1 'gAit 'gn

Ol

--tzl

1 '9A1,, 1 'gA2n F l 'gm - [A2 3 m '9 ~t ~1 =

'gn

1 ,gA,., - -

[A 1 On,

1 'gA2t tz2 'gn

(3.1)

1 '97n. [ ~2

`992 = ~2 'gn

To obtain the only determined value of A, the gauge condition on the interfaces must yet be complemented. For,Coulomb gauge, it has 'gAl. 'gA2n On - 'gn

(3.2a)

for Lorentz gauge, 'gAl,~ 391 'gA2n ,992 ,gn + ~Iel ,gt "- 'gn + ~'2e20t

(3.2b)

'gAin `9n + /zlal~t

(3.2e)

for generalized Lorentz gauge,

-

'gA2n 'gn + P2a2~2

.From Eqs. (3.1) and (3.2), we find that when the materials on both sides of the interfaces are all the same, the functions A , 9 and their normal derivative on the interfaces must be continued to satisfy the electromagnetic.continuity conditions and {he gauge constraint OO the interfaces, i. e. on the interfaces

A1 = A2, ~A1 ~A2 On - O n '

~! = ~2

]

3~1 ~2 ~n = On. J

(3.3)

In its finite element analysis, C' order continuity need to be satisfied. It will lead to considerable difficulies for constructing the interpolation functions of the elements. In addition, we see from above that t h e gauge condition (3.2) is a very important complement to the interface conditions, and the value of A can be only determined completely by it. As constructing finite elements, the gauge constraint on the interfaces must be satisfied. (b) Non-conducting regions In the non-conducting regions, using Eq. (2.11) and cofisidering all current in these

Method for 3:D Eddy Curt'ent Analysis

A, r

1021

regions vanish, we obtain from the interface continuity condition (2.3) 0 i = 02

] |

aI2, a122~ 5-; ~- 5-g j

(3.4)

If the materials on both sides of the inte?faces are the same, similarly the function .(2 and its normal derivative O0/On must also continue on the interfaces. 3~2 T h e i n t e r f a c e s b e t w e e n d i f f e r e n t r e g i o n s (eddy r e g i o n a n d n o n - c o n d u c t i n g region) As we adopt the variables A , 9 in the eddy (and source) regions and the variable s in the nonconducting region, in order to satisfy electromagnetic continuity conditions on the interfaces between two different regions, using Eqs. (2.5), (2.6) and '(2.11) and considering all current in the non-conducting regions vanish, the interface condition~ simplified from Eq. (2.3) can be written as

n x17

xa

=-nx

712]

n. 7 xA =-n.,u,,V~ 3A

I

(3.5)

0

where ,u,, e~ represent the constants in the eddy current regions, ,u, is a constant in the nonconducting regions. The component form of Eq. (3.5) is OAt

~

aA~

0__9_

aA,,

O_~

-at - + Ol : 0, -5-C + am = o , -~t + On = o

1 (OA,,

OA,,~ ~0

L, ~-

~J+

57 =0 ~n +~ =0

T2-

]

I

(3.~5)

OA,,, OAt ag2 a l - a m + #" ff-~n = 0

The complement conditions on the interface obtained from various gauge constraints are: for Coulomb gauge. OAt 3A., OA,, a--T + "~m 4 Tnn = 0

(3.7a)

for Lorentz gauge, OAt aA~ aA~ 8_~ 3-7 § ~ + ~ : - ~:" at

(3.7b)

for generalized Lorentz gauge,

3At

a A,,,

a A,,

Sl + "~m § ~

= - #:,9

(3.7c)

As constructing the models of the finite element, the conditions (3.6) and (3.7) should be both satisfied on the interfaces between the different regions.

Shi Zhanwei and Zhao Xinghua

1022 IV.

Choosing of Gauge and Establishment

of Functional

Using different gauge constraints, the obtained field equations and the interface continuity conditions are different. For the field equations, the equations obtained by generalized Lorentz gauge is very simple, because the variables A and ~p may be solved independently (see Eq. (2.10)), and for the equations (2.8) and (2.9) obtained under Coulomb gauge and Lorentz gauge, the variables A and q~ are relativeto each other.But, for the interface conditions (Eqs. (3.2) and(3.7)), we may see that the Coulomb gauge is very simple, because A and ~0 are independent of each other, and for other two gauges, they are relative to each other and t h e i r expressions are also rather complex. In previous finite element analysis of 3-1:) eddy current problems, since the interface continuity conditions (3.1), (3.2) and (3.6), (3.7) are not satisfied really when the models of FEM are constructed, the good result satisfied electromagnetic continuity conditions on the interfaces between the different mediums can not be obtained, so that the magnetic lines of force o n t h e interfaces present distortion. In previous references, it is not paid enough attention to. Comparing the field equations and the interface conditions under various gauges, we find that using Coulomb gauge is advisable, because it is much easier to satisfy Eqs. (3.2a) and (3.7a) than to satisfy complex coupling relations (3.2b), (3.7b) or (3.2c), (3.7c), when the models of the finite element analysis for 3-D eddy current problems are constructed. In this case, though A and ~o in field equations are coupling, first to solve q~ and then to solve A are suitable. So, in the finite element analysis of 3-D eddy current problems, we a d o p t A-~0 method under Coulomb gauge (7 "A = 0). From the field equations (2.8) and (2.11), the interface continuity condition (3.5) and the boundary condition (2.4), using the method Of weighted residuals, we can establish a divided region variational principle of A~-~o,.(2 method for 3-D eddy current problems under Coulomb gauge, i. e. among all permissible A, ~o and _(2 that satisfy the given boundary conditions on surfaces $1, $2, $5 of the regions, A Is1 = A0, q~ Is~ = q~0, ~ Is~ = ~o

(4.1)

the actual solution of A, q~ and s for sine steady problems must make following founctional F ( A , ~o,/'2 ) stationary value

lj

(7 xA)Z+

F ( A , ~o,g'2) = -~ v + +

:< ./:.o

7 q~)2- 2A 9 L

:5

E%TdS + s4

I,

1< Z 7

"A)2"+jwaA'A+2aT~o

} 'f: dV+~ -

f s

( 7 g2)9-dV --

:

s3

( 7 I"2 x A ) 9 dS

A

(H,o'A)dS (4.2)

where V, and V, are the volumes of the eddy current regions (include source region) and the non-conducting regions, respectively, and 5",, is the interface between this two regions.SI,S2, $3,$4,$5 and $6 are the boundaries that the given boundary values are Al0,q~0, Hto, E,,0,g20and B ~ , respectively, oJ is a circular frequency of sine variation. All field equations (2.8), (2.11), the interface continuity condition of the regions (3.5), the natural boundary conditions and the gauge constraing 7 " A = 0 can be obtained from the variation 6F = O.

A, ~o-I'2 Method for 3-D Eddy Current Analysis

1023

Using the functional (4.2) and the normal FEM, we can obtain all finite element formulas that the field variables represent as A , q~ and f2, respectively, and then may develop corresponding computational program. An analysis program of 3-D eddy current problems using this method has been developed, and we examine it by using typical and practical examples and obtain better results, especially the electromagnetic non-continuity on the medium interfaces that exists.in previous references of the eddy current problems is removed. References

[1] [2] [3] [4] [5]

Proc. Col~ oll tire Comptttation of Electromagnetic Fields (Graz. Austria, Aug. 25--28, 1987), also 1EEE Traits. Mag., 24, 1 (1988), 13~578. M . V . K . Chari, A. Konrad, M. A. Palmo and J. D. Angelo, Three dimensional vector potential analysis for machine field problem, IEEE Trans. Mag., 18, 2 (1982), 435--446. C. R. I. Emson and J. Somkin, An optimal method for 3-D eddy currents, IEEE Trans. Mag., 19, 6 (1983), 2450--2462. M. L. Brown, Calculation of 3 dimensional eddy currents at power frequencies, PIEE, 1119, I (1982), 46--53. R . I . Emson and C. W. Trowbridge, Transient 3D eddy currents using modified magnetic vector 'potentials and magnetic scalar potentials, IEEE Trans. Mag., 9.4, 1 (1988), 86--89.