Finite Elements in Electromagnetics
2. Static fields
Oszkár BíróIGTE, TU Graz
Kopernikusgasse 24Graz, Austriaemail: [email protected]
Overview
• Maxwell‘s equations for static fields• Static current field• Electrostatic field• Magnetostatic field
Maxwell‘s equations for static fields
DB
0EJJH
divdivcurl
divcurl
0
0
EDJEEJBHHB ;,;,
Static current field (1)
E 1
E i
E 2
E 0
E n
J
J
J
J
J
0
0
I 1
I i
I 2
I n
I 0 = I 1 + I 2 + . . . + I i + . . . + I nU 1
U 2
U i
U n
C 1
C 2
C i
C n
n
0Ecurl0JdivEJ
JE or
0nE on n+1 electrodes EE0+E1+E2+ ...+ Ei+ ...+ En
0nJ on the interface J to the nonconducting region
n voltages between the electrodes are given:
iC
iUdlE
orn currents through the electrodes are given:
i
iIdE
nJ
i = 1, 2, ..., n
Symmetry
2I1
I2 I2
U1
U2
J
2(I1+I2)
U1
U2
I1 I1
I2 I2
2U1
U2- U1 U2- U1
E0
E0 may be a symmetry plane
A part of J may be a symmetry plane
Static current field (2)
Interface conditions
nEnE 21
nJnJ 21
Tangential E is continuousNormal J is continuous
1
2>
1
J1
J2
n1
2>
1
E1
E2
n
Static current field (3)
Network parameters (n>0)
n=1:1
1
IUR U1 is prescribed and
1E
1 dI nJ
or I1 is prescribed and 1
1C
U dlE
n>1:
n
jjiji IrU
1
n
jjiji UgI
1
or i = 1, 2, ..., n
jkIj
iij
kIUr
,0 jkUj
iij
kUIg
,0
i = 1, 2, ..., n
Static current field (4)
Static current field (5)Scalar potential V
0Ecurl gradVE
EJJ ,0div in 0)( gradVdiv
0nE EiV on constant
. auf , auf 0 0
Eii
E
UV EUV on 0
0nJ JnVgradV on 0
n
Static current field (6)Boundary value problem for the scalar potential V
0)( gradVdiv in , (1)
V U 0 on E , (2)
gradV Vn
n 0 on J. (3)
div gradV div gradVD( ) ( ) in ,
V 0 auf E,
Vn
VnD auf J.
VVV D arbitrary otherwise ,on 0 ED UV
Static current field (7)
Operator for the scalar potential V
ngraddivA
J )(
EAA VDVD on 0 :
nVgradVdivVA D
D J )(
Static current field (8)Finite element Galerkin equations for V
n
kkkD
n NVVVV1
)( )()()()( rrrr
nn
nkkkD NVV
1
)()( rr
,1
dgradVgradNdgradNgradNV Di
n
kkik
i = 1, 2, ..., n bVA definite positive is A
High power bus bar
Finite element discretization
Current density represented by arrows
Magnitude of current density represented by colors
Static current field (9)
0Jdiv
Current vector potential T
TJ curl
JE0E ,curl in )( 0Tcurlcurl 0nJ 0nTcurl J on tTn
0tdiv 0 TnTn curldiv
iIEi
dlnt )(
0nE Ecurl on 0nT
Static current field (10)Boundary value problem for the vector potential T
0T )( curlcurl in , (1 )
tTn o n J , (2 )
0nT curl o n E . (3 )
TTT D arbitrary otherwise ,on JD tTn
)()( Dcurlcurlcurlcurl TT i n ,
n T 0 o n J ,
nTnT Dcurlcurl o n E .
Static current field (11)Operator for the vector potential T
n curlcurlcurlAE )(
JAA DD on : 0TnT
nTTT DD curlcurlcurlAE )(
Static current field (12)Finite element Galerkin equations forT
n
kkkD
n t1
)( )()()()( rNrTrTrT
en
nkkkD t
1
)()( rNrT
dcurlcurldcurlcurlt Di
n
kkik TNNN
1
i = 1, 2, ..., n bTA definite semi positive is A
Current density represented by arrows
Magnitude of current density represented by colors
Electrostatic field (1)0EcurlDdiv
ED
0nE on n+1 electrodes EE0+E1+E2+ ...+ Ei+ ...+ En
nD on the boundary D
n voltages between the electrodes are given:
iC
iUdlE
orn charges on the electrodes are given:
i
iQdE
nD
i = 1, 2, ..., n
E1
Ei
E2
E0
En D
D D
D
D
Q1
Qi
Q2
Qn
Q0=-Q1-Q2-...-Qi-...-Qn U1
U2
Ui
Un
C1
C2
Ci
Cn
n
Symmetry
E0 may be a symmetry plane
A part of D (=0) may be a symmetry plane
Electrostatic field (2)
Q1 -Q1
Q2 -Q2
2U1
U2- U1 U2- U1
E0
2Q1
Q2 Q2
U1
U2
D
-2(Q1+Q2)
U1
U2
Interface conditions
nEnE 21
nDnD 21
Tangential E is continuous
Normal D is continuous
Electrostatic field (3)
nDnD 12 Special case =0:
1=0
2>
1
D1
D2
n
1
=0
2>
1
E1
E2
n
0
D 1
D 2
n
Network parameters (n>0)
n=1:1
1
UQC U1 is prescribed and
1E
1 dQ nD
or Q1 is prescribed and 1
1C
U dlE
n>1:
n
jjiji QpU
1
n
jjiji UcQ
1
or i = 1, 2, ..., n
jkQj
iij
kQUp
,0 jkUj
iij
kUQc
,0
i = 1, 2, ..., n
Electrostatic field (4)
Electrostatic field (5)Scalar potential V
0Ecurl gradVE
EDD ,div in )( gradVdiv
0nE EiV on constant
. auf , auf 0 0
Eii
E
UV EUV on 0
nD DnVgradV on
n
Electrostatic field (6)Boundary value problem for the scalar potential V
VVV D arbitrary otherwise ,on 0 ED UV
div gradV( ) in , (1)
V U 0 on E , (2)
gradV Vn
n on D . (3)
div gradV div gradVD( ) ( ) in ,
V 0 on E,
Vn
VnD on D.
Electrostatic field (7)
Operator for the scalar potential V
ngraddivA
D )(
EAA VDVD on 0 :
)()]([n
VgradVdivVA DD D
Electrostatic field (8)Finite element Galerkin equations for V
n
kkkD
n NVVVV1
)( )()()()( rrrr
nn
nkkkD NVV
1
)()( rr
D
dNdNdgradNgradNV ii
n
kkik
1
i = 1, 2, ..., n
bVA definite positive is A
,
dgradVgradN Di
380 kV transmisson line
380 kV transmisson line, E on ground
380 kV transmisson line, E on ground in presence of a hill
Magnetostatic field (1)
JH curl0BdivHB
BH or
KnH on n+1 magn. walls EE0+E1+E2+ ...+ Ei+ ...+ En
bnB on the boundary B
n magnetic voltages between magnetic walls are given:
iC
miUdlH
orn fluxes through the magnetic walls are given:
Hi
idnB
i = 1, 2, ..., n
B/T2.0
1.8
1.6
1.4
1.2
1.0
0.6
0.4
0.8
0.2
0.0140120100 80 60 40 20 0
H/Am-1
Iron
Air
H1
Hi
H2
H0
Hn
B
B
B
B
B
1
i
2
n
0=1+2+...+i+...+nUm1
Um2
Umi
Umn
C1
C2
Ci
Cn
n
J
Symmetry
H0 (K=0) may be a symmetry plane
A part of B (b=0) may be a symmetry plane
Magnetostatic field (2)
1 2Um1
Um2- Um1
H0
1
2 2
Um2- Um1 Jx Jx Jy Jy
Jz Jz
21
2
Um1
Um2
B
2( 1+ 2)
Um1
Um2
2 Jx
Jy
Jz Jx
Jy
Jz
Interface conditions
nHnH 21
nBnB 21
Tangential H is continuousNormal B is continuous
Magnetostatic field (3)
Special case K=0:KnHnH 21
1=0
2>
1
B1
B2
n
1
=0
2>
1
H1
H2
n
K 0
H 1
H 2
n
Network parameters (n>0), J=0
n=1:1
1
m
mUR Um1 is prescribed and
1
1
H
dnB
or 1 is prescribed and 1
1C
mU dlH
n>1:
n
jjmijmi rU
1
n
jmjmiji Ug
1
or i = 1, 2, ..., n
jkj
mimij
k
Ur
,0 jkUmj
imij
mkU
g
,0
i = 1, 2, ..., n
Magnetostatic field (4)
Network parameter (n=0), b=0, K=0, J0
Magnetostatic field (5)
Inductance:
dI
L 22
1 H
dI
22
1 B
Magnetostatic field (6)Scalar potential , differential equation
JHcurl grad0TH
HBB ,0div
arbitrary otherwise ,: JTT 00 curl
Q
QP
QP dQ
PP 2
)(41)()( :e.g.
r
eJHT S0
)()( 0T divgraddiv
Magnetostatic field (7)Scalar potential , boundary conditions
KnH H on 0
HiC
i
P
P on 0 dsnTKn 0
.on ,on 0
m
0
Hii
Hi U
bnB Bbn
on nT0
Magnetostatic field (8)
Boundary value problem for the scalar potential
d i v g r a d d i v( ) ( ) T 0 i n , ( 1 )
0 o n H , ( 2 )
g r a dn
b n T n0 o n B . ( 3 )
Full analogy with the electrostatic field
,V , 0 0U , div( ) T0 , b T n0 ,
Magnetostatic field (9)Finite element Galerkin equations for
n
kkkD
n N1
)( )()()()( rrrr
nn
nkkkD N
1
)()( rr
B
dbNdgradNdgradNgradN ii
n
kkik 0T
1
i = 1, 2, ..., n
bA definite positive is A
,
dgradgradN Di
Magnetostatic field (10)In order to avoid cancellation errors in computing
)(ngrad 0TH
T0 should be represented by means of edge elements:
en
iiit
1
NT0 iedge
it dlT0
since
en
kkiki cgradN
1
N and hence T0 and grad(n)
are in the same function space
Magnetostatic field (11)
0Bdiv
Magnetic vector potential A
AB curl
BHJH ,curl in )( JAcurlcurl
bnB bcurl nA B on aAnbdiva bcurldiv AnAn
i
Hi
dlna )(
KnH Hcurl on KnA
Magnetostatic field (12)Boundary value problem for the vector potential A
JA )( curlcurl i n , ( 1 )
aAn o n B , ( 2 )
KnA curl o n H . ( 3 )
AAA D arbitrary otherwise ,on BD aAn
)()( Dcurlcurlcurlcurl AJA i n ,
0An o n B ,
nAKnA Dcurlcurl o n H .
Magnetostatic current field (13)Operator for the vector potential A
n curlcurlcurlAH )(
BAA DD on : 0AnA
)()(( nAKAJA DD curlcurlcurlAE
Magnetostatic field (14)Finite element Galerkin equations for A
n
kkkD
n a1
)( )()()()( rNrArArA
en
nkkkD a
1
)()( rNrA
H
dddcurlcurla ii
n
kkik KNJNNN
1
i = 1, 2, ..., n
bAA definite semi positive is A
dcurlcurl Di AN
Magnetostatic field (15)Consistence of the right hand side of the
Galerkin equations
,in 0 JTcurlIntroduce T0 as .on 0 H KnT
bi
dcurli 0TN
H
di )( 0 nTN
dcurlcurl Di AN
N T ni dH
( )0
( )n N T i dH
0
( )N T ni d 0
div di( )N T0
dcurliNT0
dcurli 0TN .
Bi on 0Nn
drotcurldcurlb Diii ANNT 0