Physics
Electricity & Magnetism fields
Okayama University
Year 1985
Computaton of 3-dimensional Eddy
current problems by using boundary
element method
T. Misaki
H. Tsuboi
Okayama University
Okayama University
This paper is posted at eScholarship@OUDIR : Okayama University Digital Information
Repository.
http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/143
IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 6, NOVEMBER1985
2221
COMPUTATION O F 3-DIMENSIONAL! EDDY CURRENT PROBLEMS
BY USING BOUNDARY ELEMENT METHOD
k. Misakiand
H. Tsuboi
-
The boundaryelement
method f o r computing
Abstract
3-dimensionaleddy
c u r r e n td i s t r i b u t i o n s i s p r e s e n t e d .
on
Vector
Green's
Theorem,
and
This method i s based
unknown
e l e c t r fi ice v
l de c t o r s
and
magnetic
flux
two
d e n s i t yv e c t o r sa r e
assumed on t h eb o u n d a r i e so f
m a t e r i a lasn, d
unknown e l e c t r ifci e lvde c t o rasr e
assumed in
the
conductor
regions.
After
determining
t h e s e unknown vectors,
3-dimensional
eddy
current
distribution
ti h
nc
s eo n d u c t o ar sr e
computed.
The
computation
results
of
conducting
a
sphere
model by
t o thoseofa
t h i s method were examined i nc o n t r a s t
c o u p l e d c i r c u i t model.
INTRODUCTION
Fig. 1 Typical Eddy C u r r e n t Problem
The computation
3-dimensional
of
eddy
current
d i s t r i b u t i o n sh a sb e e n
becoming i n c r e a s i n q l yi m p o r t a n t
i no r d e rt oa c h i e v et h er a t i o n a ld e s i g n
of e l e c t r i c a l
machineryandapparatus.Recently,therehavebeen
many
u s e f u lp a p e r sw r i t t e nc o n c e r n i n gn u m e r i c a l
methods t o
computeeddy c u r r e n t d i s t r i b u t i o n s .
i s tdoe s c r i bae
The p u r p o soteh
f ipsa p e r .
boundary
element
method f o r computing
3-dimensional
eddy c u r r e ndt i s t r i b u t i o n sT
. his
method i s based on
Vector
Green's
Theorem[lI,
this
[21.
method,
In
two m a t e r i a l sa r e
conductorregionsandboundariesof
a number otfe t r a h e d r a
and t r i a n g u l a r
d i v i d e idn t o
elements,
respectively,
and
unknown e l e c t r ifci e l d
v e c t o r s andmagnetic
flux d e n s i t yv e c t o r sa r e
assumed
on boundariesof
two m a t e r i a l s , and unknown e l e c t r i c
f i e l dv e c t o r s
are assumed i nt h ec o n d u c t orre q i o n s .
Electric f i e v
l de c t o r s
an
mda g n e tfilcu
dx
ensity
v e c t o r s on t h et r i a n g u l a re l e m e n t
and e l e c t r i cf i e l d
the
vectorsinthetetrahedronareapproximatedthrough
u s eo fl i n e a rf u n c t i o nc o o r d i n a t e s .A f t e rd e t e r m i n i n g
t h e s e unknown vectors,
3-dimensional
eddy c u r r e n t
Here, a
d i s t r i b u t i o n si n the conductorsare.computed.
conductingsphere
model and as q u a r ep l a t e
model were
chosen
as examples.
The computed r e s u l t s a of
conducting sphere model u s i n g t h i s method were examined
i n c o n t r a s t t o t h o s e ofacoupledcircuit
model.
2
is
Js is
u is
11 is
E
is
w is
j is
t h ec u r r e n td e n s i t y
t h es o u r c ec u r r e n td e n s i t y
t h ec o n d u c t i v i t y
t h ep e r m e a b i l i t y
t h ep e r m i t t i v i t y
t h ea n g u l a rf r e q u e n c y
t h e complex o p e r a t o r
-+
The s e c o n dt e r mo nr i g h ts i d eo f
( 6 ) , a , represents
t h e eddy c u r r e n t d e n s i t y .
VectorGreen'sTh.eorern[ll,
I21 is given by
I,
(c .VxVx$
V x E =
+
- j W
B
+
+
v x H = j W E E + J
v . I ; = o
+
y . B = O
- + - +
-+
B = L l H , D = E E
- + - +
J =
where
Js+
-+
4
B
3
H
is
is
is
is
+
U E
=
Is ( 6 x V x 6 -
(7)
6 x V X f )d- s:
where
P
and Q a r et h ev e c t o fr u n c t i o n s
of p o s i t i o n
h a v i ntghceo n t i n u o uf si r s t
and second d e r i v a t i v e s
t h er e g i o n , VI,
w i t h i n and on t h es u r f a c e , S , bywhich
is enclosed, and
i s t h e normal u nvietc t o r
at
integration point.
Substituting
(7),
v e c t o r )i n t o
t h ee l e c t r i cf i e l d ,
o b t a i n e da sf o l l o w s :
-+
CilEil
=
&@
(2 i s an a r b i t r a r yu n i t
and using (1), ( 2 ) ,..., ( 5 ) and ( 6 ) ,
%il, a t c o m p u t a t i o np o i n t , il, i s
$=$,
ss1O +
++
- I v l (j W 1 1
-Is {-jW(Axzf)
-+
jWVU E ) d v
(?ixB)x~~
+ (it.$) V$} d s
Formulation
of
a boundary
element
method
for
computing3-dimensional
eddy c u r r e n t d i s t r i b u t i o n s a r e
u s e of t h et y p i c a l
eddy c u r r e n t
i l l u s t r a t e db yt h e
1.
Maxwell's
equations
for
problem shown iFni g .
time dependence
and
tchoen s t i t u t i v e
sinusoidal
r e l a t i o n s are given by
-+
s- VxVx6) d v
-+
-+
FORMULATION O F BOUNDARY ELEMENT METHOD
-+
-
+
where
i s a w e i g h t i n gf u n c t i o n ( $ = 1 / 4 7 r r )S. i m i l a r l y ,
t h em a g n e t i cf l u xd e n s i t y
,
a t computation p i n t ,
il, i n v1 i s o b t a i n e d .T h a t
is
-+
C i l B i l
E;,,,
=
(1)
(2)
(3)
(4)
(5)
(6)
t h ee l e c t r i cf i e l d
t h em a g n e t i cf l u xd e n s i t y
t h ee l e c t r i cf l u xd e n s i t y
themagneticfiel.d
(8)
I v l ( L l 3slXvO + ) l U $ X v @ ) d v
-1s { jWV&( Z x 3 ) o + (;x6) ~ V
+
O
(;.&)V$]ds
(9)
where c o e f f i c i e n t , C i l , i s d e c i d e db yt h ep o s i t i o no f
t h ec o m p u t a t i o np o i n t ,
il,
Cil
where
=
R i l /4n
(10)
nil
is thesolidangleat
il a s shown i n F i g . 1.
I nas i m i l a r
way, t h e e l e c t r i c f i e l d ,
& 2 , and t h e
a t c o m p u t a t i o np o i n t ,
i2,
m a g n e t i cf l u xd e n s i t y ,
i n V2 are given 'by
Si2,
Electrical
The
a u t h oaw
rrsei t h
Department
of
Engineering, Okayama U n i v e r s i t y , Okayama 700, Japan.
Ci,2
&
=
-Iv2jWV0 3 s ~
@
-++
-1s { j W ( n x B ) 9
0018-9464/85/1100-2227$01.0001985 IEEE
dv
-
($x$)
Xv@
2228
so
i s the external magnetic flux density.
The b o u n d a r y c o n d i t i o n s f o r t h e e l e c t r i c f i e l d
m a g n e t i cf l u xd e n s i t y
on the s u r f a c e , S , a r e g i v e n
where
(14),(15),(16) and ( 1 7 ) a t each
formed
by
using
computation
point
on
the
boundary
surface,
S,
and
p i n ti nt h er e g i o n ,
by using ( 8 ) a t eachcomputation
VI,
and
each
element
the
final
ofsimultaneous
equations i s o b t a i n e db yu s i n gn u m e r i c a li n t e g r a t i o n s .
A f t ed
r e t e r m i n i n gt h e
unknown e l e c t r i cf i e l dv e c t o r s
andmagneticfluxdensityvectors,3-dimensional,eddy
currentdistributionsintheconductorsare
computed by
(6).
using ( 8 ) and t h e secondterm on t h e r i g h t s i d e o f
and
by
COMPUTATION
RESULTS
A conducting sphere
were
method
chosen
.
as
model and a s q u a r e p l a t e model
examples
of
t h ae p p l i c a t i o n
otfh i s
A Conductinq Sphere
Model
A conducting sphere model i n a uniform a l t e r n a t i n g
magnetic
field131
and
the
arrangement
of
tetrahedra
of
the
and t r i a n g u l aer l e m e n tfso r
one e i g h t h
part
where
and
atth
raeen g e n tui av
nlei tc t o r s
at.
i,
on
S,
which
intersect
computation
point,
p e r p e n d i c u l a r lty
eoa coht h e rN. e x t ,
from ( 8 ) , (91,
(ll), (12), and (13;,
t h sei m u l t a n e o ues q u a t i o nfso r
theelectricfields,
~ iand
, t h em a g n e t i cf l u xd e n s i t y ,
-+
~ ia ,t computationpoint, i, on S i n t h e s i d e o f
V1 a r e
o b t a i n e da sf o l l o w s :
Fig. 2
Arrangement ofTetrahedraandTriangular
Element of a Conducting Sphere Model
/L
x
.T-.
(a)imaginarycomponents
2
.
where CL=EO/(&-ju/W)
Eni, is t h e normal component of
8,. E U i and E v ia r et h et a n g e n t i a l
components
of
Bni i s t h e normal component of
Bui
and
Bvi a r e
t h et a n g e n t i a l components of 3 5 . Here, t h er e g i o n , V1,
andtheboundarysurface,
S , a r ed i v i d e di n t o
a number
otfe t r a h e d ra n tdr i a n g u l aerl e m e n t sr,e s p e c t i v e l y ,
a n de l e c t r i cf i e l dv e c t o r s
andmagneticfluxdensity
v e c t o r s on t h et r i a n g u l a re l e m e n t
and e l e c t r i c f i e l d
v e c t o r si nt h et e t r a h e d r o na r ea p p r o x i m a t e dt h r o u g ht h e
l io
unfseueanr c tci o no r d i n a t e s .
The
final
simultaneousequationsforcomputingtheelectricfield
a nmda g n e tfi lcudxe n s i v
t ye c t onr su m e r i c a l al yr e
si.
Ei.
( b )r e a l
Fig. 3
components
D i s t r i b u t i o n so f
Eddy Current
i n ConductinqSphere
2229
sphere
region
i s shown iFni g .
2.
The number of
t e t r a h e d r a and t h aottfr i a n g u l a r
elements
are 702
and 54, r e s p e c t i v e l y .
The c o m p u t a t i o nr e s u l t s
of t h e
eddy c u r r e n td i s t r i b u t i o n s and t h o s eo fm a g n e t i cf l u x
are shown i nF i g .
3 andFig.
4,
d e n s i t yd i s t r i b u t i o n s
r e s p e c t i v e l y .I no r d e r
t o v e r i f yt h ea c c u r a c yo ft h e
abovemethod, a comparisonbetween the boundary element
model and a c o u p l e dc i r c u i t
model[41 a s shown i nF i g .
5
was made., I nt h i sc a s e t, h es p h e r er e g i o no ft h e
above c i r c u i t model was d i v i d e d i n t o 210 c i r c u i t s , w i t h
t h es i m u l t a n e o u se q u a t i o n s
formed a s f o l l o w s :
Z
,r
(Ri+jwLi)Ii+
( a )i m a g i n a r y
2 10
C j w ~ ~ ~ I ~ = - j w @ i
j=1
jSi
components
i=1,2,
(18)
...,210
where Ii and Ij are e d d y c u r r e n t s i n c o i l
i and c o i l j ,
respectively.
and ti atrh
reees i s t a n c e
and
the
s e lifn d u c t a n c e
for c o i l i, r e s p e c t i v e l y ,
Mi) is t h e
i and c o i l j and Qi is
mutualinductancebetweencoil
t h eq u a n t i t y
of t h ei n t e r l i n k e df l u xf o rc o i l
i.
Li
and Mij f o rc o i l
i and c o i l j , shown i nF i g .
6,
are
obtained by
I
( b ) real Components
Fig. 4
DistributionsofMagneticFluxDensity
i n Conductingsphere
0
-:
: Boundary
Element
NestedCoupled
Model
C i r c u i t Model
2.0r
I
Fig. 5
(a) imaginarycomponents
A Coupled C i r c u i t Model
k3
Coil j
(b) r e a l components
Fig. 7
Fig. 6
Arrangementof
TWO
Coils
Electric F i e l d i n
alonq x-axis
a ConductingSphere
2230
where K ( k ) and E ( k ) are t hfei skt i n o
dcfo m p l e t e
e l l i p t iicn t e g r a l
and t h e second
kind
complete
of
ellipticintegral,respectively,
and k i s givenby
k =
I
J a i 4+aaija, +h2
(
'T I=+--.-
"1I
(22)
)
A s shown
i nF i g . 7 , t h er e s p e c t i v ec o m p u t a t i o nr e s u l t s
proved t o b e c o i n c i d e n t .
A Square Plate
Model
The arrangement
tetrahedra
of
and
triangular
plate i s
e l e m e n t sf o ro n ef o u r t hp a r o
t ft h es q u a r e
shown i nF i g .
8. The number o ft e t r a h e d r aa n dt h a to f
t r i a n g u l a re l e m e n t sa r e
288 and 1 1 2 , r e s p e c t i v e l y . The
c o m p u t a t i o nc o n d i t i o n so ft h i s
model a r e shown i n F i g .
9.
The c o m p u t a t i o nr e s u l t so ft h es q u a r ep l a t e
model
and near a current
i n a unit alternating magnetic field
10.
carrying
c o n d u c t oar r e
shown i nF i g .
(a) lying in a unit alternate magnetic field
Yt
2
't
o=5.92x107
I
I
1
I
I
Y
X
( b )l y i n gn e a r
Fig. 1 0
Fig.8ArrangementofTetrahedraandTriangular
Elements of a S q u a r e P l a t e Model
a c u r r e n tc a r r y i n gc o n d u c t o r
D i s t r i b u t i o n so f Eddy Current
i n a Square Plate
CONCLUSION
Z
I
.02
a unit alternate magnetic field
(a) lying in
I
I tnh ips a p e rt,hb
eoundare
yl e m e n t
msthod f o r
computing3-dimensionaleddycurrentdistributionsand
thecomputationresultsweredescribed.
The computation
resulto
s ac
f o n d u c t i n gs p h e r e
model by t h i s method
almostagreedwiththoseofacoupledcircuitmodel.
The a d v a n t a g e o f t h i s method i s t h a t t h e f o r m u l a t i o n o f
t h i s method i s performedwithoutgaugecondition,which
a p p e a ristnhfeo r m u l a t i o o
nffi n i teel e m e n t
method
u s i n g p o t e n t i a l s , andboundaryconditionsofelectric
f i e l d a n dm a g n e t i cf l u xd e n s i t y . a r ep h y s i c a l l yc l e a r .
This method can baep p l i etd
coo m p u t a t i oonf
eddy
curre
dn
i st t r i b u t imoi angs n em
t iact e r i a l s
and
conductorswith3-dimensionalshapes.
REFERENCES
00050
-ld+
I
.02
( b )l y i n gn e a rac u r r e n tc a r r y i n gc o n d u c t o r
Fig. 9
ComputationConditions
P l a t e Model
o f aSquare
J.
A. S t r a t t o nE:l e c t r o m a q n e t iTch e o r y ,
McGrawH i l l , New York,1941.
Computer
T e c h n i q uE
fo
el sre c t r o [2] R. Mittra:
1973.
maqnetics, Pergamon Press,Oxford,
[3] D.
Rodger
and
J.F.
Eastham:
"A Formulation
for
Low
Frequency Eddy
C u r r eSnot l u t i o n s ,
"
Transactions on Maqnetics,
Vol.
MAG-19,
NO. 6 ,
November 1983,pp.
2443-2446.
[4] S . J. Salon, B . Mathewson and S . Uda: "An IntegroD i f f e r e n t i a l Approach t o Eddy Currents
in
Thin
Maqnetics,
on
Vol.
P l a t e s , " IEEE Transactions
MAG-19, No. 6 , November 1983,pp.
2405-2408.
111