2192 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 6, NOVEMBER 1985

A NEW FORMULATION OF THE MAGNETIC VECTOR POTENTIAL METHOD FOR THREE DIMENSIONAL MAGNETOSTATIC FIELD PROBLEMS

Toshiya Morisue

Abstract - A novel formulation of the magnetic vector p o t e n t i a l method for three dimensional magnetostatic f i e l d c a l c u l a t i o n s i s derived. Rigorously defining the interface and boundary condi t ions of the gauge o f t he veczor po ten t ia l , the new methodgivesaunique solution t o t h e problem. The new f i e ld equa t ion does not con- t a i n t h e gauge condition against the usual formula- t i o n s [ l ] , [ 2 ] , [ 3 ] , and t akes t he form of t h e d i f f u s i o n equation. Computed resu l t s a re favorably compared with the analytic soluzion of a t e s t problem. This formula- t i o n i s d i r e c t l y a p p l i c a b l e t o t h r e e d i m e n s i o n a l eddy cux-en-c problems.

INTRODUCTION

To formulate the three dimensional magnetostatic f i e l d problem us ing t he magnetic vector potential as a boundary value problem, the gauge of the vector poten- t i a l and t h e i n t e r f a c e and boundary conditions of t h e gauge sho.dd f i r s t of a l l be def ined c lear ly . Without them t h e boundarg- value problem could not be solved uniquely.

This paper presents a new method fo r ca l cu la t ing the three dimensional magnetostatic f ield problem, rigorously defining zhe interface and boundary condi- t i o n s of t h e gauge of the vec tor po ten t ia l . The new field equasior, does not contain the gauge condition against the us;lal formulations [l] , [2], [ 3 ] , and takes t h e form of the diffusion equat ion. The gaugecondition appears in the interface and bomdary condi t ions. The method gives a unique solution t o t h e problem.

corresponding to the equat ion does not include the Lagrange mul t io l i e r . The re fo re , t he f i n i t e element formulation based on t h i s me-chod may be more e f f e c t i v e than the usual ones.

analy-cic solution of a t e s t problem.

dimensional eddy current problems.

THE A FORMULATION

By Trirtue of t h e new formulation, the weak form

CompEted resu l t s a re favorably compared wi th t he

This formulation i s d i r e c t l y a p p l i c a b l e t o t h r e e

The problem space X i s an unbounded region. R contains several regions. Source current regions and mater ia l regions are bomded, respect ively.

Maxwell's Equations

c u r l X = , div = 0

Const i t - t ive Relat ion

- 3 = k(B) 2 The r e l u c t i v i t y k i s a monotonically increasing

posizive-valued sca1a.r function of the magnitude of g. i n o the r words, 3 - H curve i s convex as shown in F ig .1 .

B I

The author i s with Faculty of Engineering, Uni- v e r s i t y of Tokushima, Minami-josanjima 2-1, Tokushima, 770, Japan

Uniqueness of Field Intensity

from (I) t h e r e e x i s t s a sca la r func t ion f such that grad f = zl - 22, and i s continuous over R . (grad f may be discont inuous a t in terfaces . ) The cons t i t u t ive r e l a - t i on g ives t ha t (51 - g 2 ) . ( 2 1 - g 2 ) 0, since

- B1.22) = kl.(B;! -21I2 + (k2 - k1).($22 - g2.Bl ) . %et C be a loop represent ing some f l u x l i n e of El - 22. Tnen,

Suppose we have two so lu t ions , gl and g2. Then,

(gl - g 2 ) . ( g l - 2 2 ) = k2. (Zl - B ; ? ) 2 + ( k l - k2) . (B12 -

This i s a contradict ion and therefore the value of g i s unique.

Magnetic Vector P o t e n t i a l

t o r p o t e n t i a l such t h a t : Since i s solenoidal over R , t h e r e e x i s t s a vec-

- S = c u r l & ( 3 ) From (1) and (31,

c u r l ( k c u r l 4) - 2 = 0 (4) where k = k ( l c u r 1 A I ) . Gauge Condition

introduce the Coulomb gauge: To ensure uniqueness of the vector potential , we

d iv A = 0 ( 5 ) Boundary Condition of the Vector Potential

following manner: The vec to r po ten t i a l van i shes a t i n f in i ty i n t he

where = (x ,y,z) , s ince the source current and mater i - a l regions are bounded, and tne vec to r po ten t i a l due t o a loop current I i s expressed as follows:

where S i s a surface s-oanndd by the loop current , and - n ' i s a uni t vec tor normal t o t h e s u r f a c e element d s ' .

Mater ia l Interface Condi t ions The cont inui ty of the normal component of f lux

density and tangential component of f i e l d i n t e n s i t y form the mater ia l in terface condi t ions, and are ex- pressed in terms of the vec tor po ten t ia l as fo l lows:

where g i s a uni t vector normal t o t h e i n t e r f a c e .

Gauge Interface Conditions

t e r f a c e i s expressed as follows: The continuity of the gauge(div A) across the in-

- n.Al = g.&2 (9) - n.grad(g.&l) = n.grad(q.fi2) i I C )

together with (7). Uniqueness of the Vector Potent ia l

y i e l d s a unique solution over X . Proof. Let &a and Ab be two so lu t ions t o the system. From t h e c o n s t i t u t i v e r e i a t i o n and the uniqueness Of t h e f i e l d i n t e n s i t y f o l l o w s t h a t c u r 2 n a = c u r l g o . Then, t h e r e e x i s t s a sca la r func t ion g such that gradg = Aa - &b. Using (5) gives that d iv grad g = 0 . By v i r - tu; of t h e gauge in te r face condi t ions , th i s Laplace ' s equation holds over R (even on t h e i n t e r f a c e ) . From t h i s f a c t and ( 6 ) , g vanishes over R. Therefore, &a = Ab.

The system of the equat ions (4) through (10)

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0018-9464/85/1100-2192801.00O1985 IEEE

2193

THE ALTERNATIVE FORMULATION

The Diffusion Equation

the following diffusion equation: Using t h e Coulomb gauge ( 5 ) , (4) can be writ ten as

( d i v k grad)A - ( (grad k) .grad)A

- ( g r a d k ) x c u r l + 2 = 0 (11)

The r e l u c t i v i t y k can be considered as the "thermal conductivity", and the term: -1 (grad k) .grad)A - (g rad k ) x c u r l & + 5 as the " internal heat source".

The Interface Conditions

wr i t ten as follows: The interface condi t ions ( 7 ) through (10) can be

- A 1 = 42 (12)

k l (g.grad)& = k2 (g.grad)A;?

+ ( k l - k2) grad(g.h2) (13)

( see Appendix 1. ) The term: (kl - k2) grad(c.,2) can be considered as the "surface heat source".

The Boundary Condition The boundary condition i s t h e same a s ( 6 ) :

- A(L) - O(l/r2) (14) The diffusion system (11) through (14) forms a

"well-posed" problem, t h a t i s , additional information i s not necessary t o solve the problem.

By t h i s f a c t and the preceeding discussion, the diffusion system has a unique solution. Furthermore, t h i s s o l u t i o n a u t o m a t i c a l l y s a t i s f i e s t h e Coulomb gage.

The f low diagram for the calculat ion of the mag- ne t i c vec to r po ten t i a l & i s shown in F ig .2 for a case of two media.

4 (n.grad),l I 1

( d i v k l g r a d ) @ - ( ( g r a d k l ) . grad)&l

- ( g r a d k l ) x c u r l &l + 21 = 0

3 A 1 - 1 '

A2 = 41 - c.

( d i v k2 grad),2 - ( (grad k2) .grad)A2

- (grad k2) x c u r l A2 + 52 = 0

1

1 ( n . grad)A2

k l ( g . g r a d ) & l = k2 (n.grad)&2 - + ( k l - k 2 ) g r a d ( 2 . g )

Fig.2 Flow-diagram fo r ca l cu la t ing

The Weak Form The weak form of the diffusion system i s obtained

using the usual Galerkin method. For s impl ic i ty , we assume t h a t R = R1 U R2, where R1 i s t h e bounded i ron region and R 2 i s t h e unbounded a i r region surrounding R1. Since t he r e luc t iv i ty i s constant over R2, t h e boundary integral formulat ion i s p r e f e r a b l e f o r t h e region R2. The weak form i s as follows:

where El and W:, are weighting (vector) functions and S i s t h e i n t e r f a c e between X 1 and R2. Repeated indices imply summation. In (15 ) and' (16) , & and 9 a r e unknown var iab les . For fu r the r de t a i l s s ee Appendix 2.

It seems reasonable t o choose the weighting func- t i ons such t ha t :

over the in te r face .

RESULTS

The new formulation was a p p l i e d t o a t e s t prob- lem, an inf in i te l ength square i ron-bar wi th a con- s tan t permeabi l i ty exc i ted by an in f in i t e l eng th so l e - noid, which has an analytical solution against which the numerical resul ts could be compared.

The problem geometry i s shown i n Fig. 3.

-7

Y current Solenoid

/ -

- 7 x

A i r

Fig.3 The problem geometry

For t h i s problem, the vec to r po ten t i a l had two

- A(x,y,z) = (Ax(x,y), A ( x , y ) , 0 )

components :

Y (18) Since the permeabili ty of i r o n was cons tan t , the boundary i n t e g r a l method was used for bo th t he i ron . and air regions.

wr i t ten as fol lows: The two dimensional boundary integral equation i s

where 1 = t h e i r o n , 2 = t h e a i r , and S = t he i n t e r f ace . Note t h a t t h e boundary integral vanishes at i n f i n i t y , s ince - O(l/r) for t h e two dimensional f ield and the o rde r of t h e boundary i n t e g r a l i s e i t h e r l/r or ( l o g r ) / r .

Four boundary in t eg ra l equa t ions : ( lg ) and fo,ur

2194

i n t e r f ace cond i t ions : (12 ) , (13) uniquely determine the boundary values of Alx, Aly, ---, ( n . grad)A2y.

ing method. The boundary elements used were zero-order and variable-length. The d i v i s i o n o f t h e i n t e r f a c e i s shown in Fig.4.

Discre t iza t ion was ca r r i ed ou t by the po in t match-

Y

I L node

H element

I 32 x 4 = 128 nodes I 0 15 .x

I

Fig.4 Division of t h e i n t e r f a c e

Short elements were used where the vector poten- t i a l changed grea t ly . Rounding the sharp edges of the i r o n was necessary for the normal vector to be cont inu- ous on the in te r face and for ob ta in ing good r e s u l t s using the reasonable number of the elements.

t i a l i s shown in Fig.5. (For t h i s problem, Ay(x,y) = -Ax(y,x), and there was a four fo ld symmetry in Ax(x ,y ) )

The computed resu l t o f the magnet ic f lux dens i ty i s shown in Fig.6. (There was a e ight fo ld symmetry i n Bz(x,y).)

The computed r e s u l t of t h e magnetic vector poten-

The re la t ionship between the computat ion error and the permeabi l i ty i s shown in Fig.7.

It turns out from the computed r e s u l t s t h a t t h e computation error remains within 1% f o r t h e r e l a t i v e permeability of 1 through 1000.

t h a t t h e new formulation i s accurate and valid. From t h e above r e s u l t s we come to t he conc lus ion

CONCLUSIONS

This formulation of the three dimensional magneto- s t a t i c f i e l d problem appears t o be promising. The major advantages are :

- The f i e ld eqda t ion i s a well-posed diffusion equation. It au tomat ica l ly sa t i s f ies the Coulomb gauge so t h a t t h e Lagrange m u l t i p l i e r i s not n e c e s s a r y i n t h e f i n i t e element computation.

- The method i s d i r e c t l y a p p l i c a b l e t o t h e t h r e e dimensional eddy current problem.

-3000

-2000

AX

-1000

1 2 3 x o r y ,

4 5

Fig.5 The computed r e s u l t of the vector p o t e n t i a l ( AX )

- - - - - - - - py = 200 - ~ -.-

0.984 - ____ ~ ~ ....... . . . . . . ..~. ~. . .

- ~. I 1

0 ~ . I.-.. 2 3 4 5 Distance from the mid-point of the interface

Fig.6 The computed r e s u l t of the f lux dens i ty a long t he a i r - s ide i n t e r f ace

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Appendix 2

stant in each finite element, them grad k = 0 and the weak form is expressed as follows:

If the magnetic flux density is assumed to be con-

0 . - . 200 400 600 800 .. _ _ 1000 Relative permeability

Fig.7 Relationship between the computation error of the flux density and the relative per- meability

REFERENCES

R.D. Pillsbury, Jr.: "A three dimensional eddy current formulation using two potentials: the magnetic vector potential and total magnetic scalar potential, '1IEEE Transactions on Magnetics,

M.V.K. Chari, A. Konrad, J.D'Angelo, M.A. Palmo: "Finite element computation of three-dimensional electrostatic and magnetostatic field problems", IEEE Transactions on Magnetics, Vol.MAG-19, No.6,

C.J. Carpenter: "Comparison of alternative formu- lations of 3-dimensional magnetic-field and eddy- current problems at power frequencies", PRO. IEE,

vol. MAG-I~, N0.6, xov. 1.983, pp.22a4-22a7.

NOV. 1983, pp. 2321-2324.

V01.124, N0.11, NOV: 1977, pp.1026-1034.

APPENDIX

Appendix 1 - Al'= A2 follows from that 2 x A1 = g x A2 and

_ _ n.Al = ;.E. Let (n, u, v) be the local Descartes coordinates

on the interface such that n-direction coincides with the direction of the unit vector normal to the inter- face. (8) can be written as follows:

kl ( (2. grad) (2.g) - (2. grad) (2.g) )

= k2 ( (e. grad) (g.A2) - (2. grad) (;.E) ) , kl ( (,%.grad) (v.Al) - (v.grad) (g.41) )

, = k2 ((~.grad)(v.&2) - (v.grad)(g.A2)) (A-1) From (A-11, (101, and (12) is obtained that: kl (g.grad)Al = k2 (2. grad)&2

+ (kl - k2) grad(2.g)

k grad W .grad A dv P - SUM is.$.(ki - k )grad(n. A) asij

?

(id) IJ j -1 j .-

where S. is the common interface between i-element I J and j-element, and n is the normal surface vector

which is taken to point into j-element. Repeated indi- ces: p imply summation.

-i j