Finite Elements in Electromagnetics 2. Static fields
description
Transcript of Finite Elements in Electromagnetics 2. Static fields
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