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Finite Elements in Electromagnetics
3. Eddy currents and skin effect
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24Graz, Austria
email: [email protected]
OverviewEddy current problemsFormulations in eddy current free regionsFormulations in eddy current regionsCoupling of formulationsSkin effect problemsVoltage excitation, A,V-A formulationCurrent excitation, T,- formulation
n: nonconducting region
Air
J(r,t) = 0
Coil
J(r,t) known
0
c: eddy current regionJ(r,t) unknown
Typical eddy current problem
Maxwell’s equations:0JH curl ,
0Bdiv i n n ,
JH curl ,BE jcurl ,
0Jdiv ,0Bdiv i n
c .
BHHB ,i n
n a n d i n c ,
EJ , JE i n c .
A s s u m p t i o n : 00 JT curl .
Boundary conditions
n : o u t e r b o u n d a r y o f n
c : o u t e r b o u n d a r y o f
c
0nH o r 0 nB o n n ,
0nH o r 0nE o n c .
o rnTnH 0 o r nTnB 0 o n
n ,nTnH 0 o r 0nE o n
c .
Interface conditions
nc: interface betw een n and c
nH and nB are continuous on nc.
Magnetic scalar potential in n
grad0TH ,
en
kkkt
1
NT 0
D i f f e r e n t i a l e q u a t i o n :)()( 0T divgraddiv i n
n .
B o u n d a r y c o n d i t i o n s :0 o r nTn 0 grad o n
n .
Finite element approximation
nn
kkk
n N1
)(
G a l e r k i n e q u a t i o n s :
n n
dgradNdgradgradN in
i 0T )( ,
i = 1 , 2 , . . . , n n
Magnetic vector potential in n
AB curl .
D i f f e r e n t i a l e q u a t i o n
0TA curlcurlcurl )( i n n
B o u n d a r y c o n d i t i o n s :0nA o r nTnA 0 curl o n
n .
Finite element approximation
en
kkk
n a1
)( NAA
G a l e r k i n e q u a t i o n s :
n n
dcurldcurlcurl in
i 0TNAN )( ,
i = 1 , 2 , . . . , n e
P o s i t i v e s e m i d e f i n i t e m a t r i x
Magnetic vector potential alone in c
*AB curl , *AE j .
D i f f e r e n t i a l e q u a t i o n0AA ** jcurlcurl i n
c .
B o u n d a r y c o n d i t i o n s :0nA * o r nTnA 0 *curl o n
c .
Finite element approximation
en
kkk
n a1
)(** NAA
G a l e r k i n e q u a t i o n s :
c c
djdcurlcurl ni
ni
)(*)(* ANAN
c
dcurl i 0TN , i = 1 , 2 , . . . , n e
N o n s i n g u l a r b u t i l l - c o n d i t i o n e d m a t r i x
Magnetic vector and electric scalar potential in c
AB curl , gradVjj AE .
D i f f e r e n t i a l e q u a t i o n s :
0AA gradVjjcurlcurl ,
0)( gradVjjdiv A i n c .
B o u n d a r y c o n d i t i o n s :
0nA o r nTnA 0 curl ,
0VV = c o n s t a n t o r 0)( gradVjj An
o n c .
Finite element approximation
en
kkk
n a1
)( NAA ,
nn
kkk
n NVVV1
)( .
G a l e r k i n e q u a t i o n s :
c c
djdcurlcurl ni
ni
)()( ANAN
cc
dcurldgradVj in
i 0TNN )( ,
i = 1 , 2 , . . . , n e ,
c
dgradNj ni
)(A
0)( c
dgradVgradNj ni , i = 1 , 2 , . . . , n n
S i n g u l a r s y s t e m b u t i m p r o v e d c o n d i t i o n i n g .
Current vector and magnetic scalar potential in c
gradTTH 0 , TTJ 0 curlcurl .
D i f f e r e n t i a l e q u a t i o n s :
gradjjcurlcurl TT
00 TT jcurlcurl ,
)()( 0TT divjgraddivj i n c .
B o u n d a r y c o n d i t i o n s :
0nT o r nTnT 0 curlcurl ,
0 = c o n s t a n t o r 0)( TnTn grad
o n c .
Finite element approximation
en
kkk
n t1
)( NTT ,
nn
kkk
n N1
)( .
G a l e r k i n e q u a t i o n s :
c c
djdcurlcurl ni
ni
)()( TNTN
c
dgradj ni
)(N
cc
djdcurlcurl ii 00 TNTN ,
i = 1 , 2 , . . . , n e ,
c
dgradNj ni
)(T
c
dgradgradNj ni
)(
c
dgradNj i 0T , i = 1 , 2 , . . . , n n
S i n g u l a r s y s t e m b u t g o o d c o n d i t i o n i n g .
Coupling A,V in c to A in n: A,V-A formulation
I n t e r f a c e c o n d i t i o n s o n n c :
C o n t i n u i t y o f An nB i s c o n t i n u o u s
C o n t in u i t y o f nA curl i s a n a tu r a l i n t e r f a c ec o n d i t i o n nH i s c o n t i n u o u s
G a le r k in e q u a t i o n s r e m a in u n c h a n g e d
Coupling T, in c to in n: T,- formulation
In terface conditions on nc:
C ontinuity of and 0nT nH iscontinuous
C ontinuity of nT )( grad is a naturalin terface condition nB is continuous
G alerkin equations rem ain unchanged
Typical skin effect problem
u (t )
i ( t )
i ( t ) J ( r , t )
B ( r , t ) n : = 0
c : > 0
E 1 : 0nE
E 2 : 0nE
c n : nBnH , c o n t .
n
n
n HBEJ ,
,t
curl B
E
c : ,JH curl
n : ,JH curl
,0Bdiv
HB
Integral quantities, network parameters
2
),()(E
dtti nrJ
1
),(E
dt nrJ
c
dt
tpv
2),(
)(rJ
)()( 2 titR
dHdBtWnc
tB
m
),(
0
)(r
t
diLdd
i0
)()()(
dttdW
tptitutp mv
)()()()()(
Voltage excitation (1)
AB curl in nc and , gradUt
A
E in c
EH curl in c , 0H curl in n ,
)(, tuU 0nA on 1E , 0, U0nA on 2E ,
0nA or 0nH on )( nc
nA and nH are continuous on cn 0 nJ
Voltage excitation (2)
c
dt
tp
2),(
)(rJ
dHdBdtd
nc
tB ),(
0
r
c
dtp JE)(
nc
dt
HB
nc
dt
HB
nc
dt
curl HA
nc
dcurlt
HA
0
)(
nc
dt
nHA
c
dt
JA
Voltage excitation (3)
c
dt
tp JA
E)(
c
dgradU J
c
dgradUtp J)(
cnEEc
dUdUdiv21
0
nJJ
)()()()(1
titudtutpE
nJ
Boundary value problem for A,V (1)
Differential equations:
0A
A
tV
gradt
curlcurl 1
tV
U
0)(
tV
gradt
div A
0A
curlcurl1
in c,
in n,
Boundary value problem for A,V (2)Boundary conditions:
t
dutV0
)()(, 0nA on 1E ,
0)(, tV0nA on 2E ,
0nA or 0nA curl1
on )( nc .
Interface conditions:
nA and nAcurl1
are continuous on cn .
Current excitation (1)
in c
in c ,
in n .
on )( nc
and are continuous and cn
,TTJ 0 curl gradTTH 0,
grad0TH
0B
J
tcurl
1 0Bdiv in n ,
)0( nB0nE or
nTnH 0
nB 0nT on
,
.
Properties of T0
0T0 curl nin ,
1E
dcurl nT0 2E
dcurl nT0 )(ti
nT0 is continuous on .cn
A possible choice of T0
Solve the static current field in c
i ( t ) 00 TJ curl
n : = 0 c : > 0 C
C
n
ti3
)(
4)(
)()(:Q
QQS0
rr
rrdsrHrT
nHnT S0 :cn
0T 0
curlcurlc
1:
0nT 0 curlE 1
:1
0nT 0 curlE 1
:2
Boundary value problem for T, (1)
Differential equations:
in c,
in n,
0TTTT 00
grad
tcurlcurl
1
0)(
graddivt
TT0
0)(
graddivt 0T
Boundary value problem for T, (2)Boundary conditions:
2E
or on )( nc .
Interface conditions:
and
on 1E ,
are continuous on cn .
0,1 nTT0nT 0 gradcurl
0nT on cn .
0 0nT0 grad
nT0 grad
