Finite Elements in Electromagnetics

2. Static fields

Oszkár BíróIGTE, TU Graz

Kopernikusgasse 24Graz, Austriaemail: biro@igte.tu-graz.ac.at

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