Embed Size (px)
DESCRIPTION
arquivo ensinando alguma coisa que não lembro.
Citation preview
Finite Elements in
Electromagnetics
3. Eddy currents and skin effect
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24Graz, Austria
email: [email protected]
Overview
Eddy current problems
Formulations in eddy current free regions
Formulations in eddy current regions
Coupling of formulations
Skin effect problems
Voltage excitation, A,V-A formulation
Current excitation, T,F-F formulation
Wn: nonconducting region
Air
J(r,t) = 0
Coil
J(r,t) known
0
Wc: eddy current
region
J(r,t) unknown
Typical eddy current problem
Maxwell’s equations:
0JH curl ,
0Bdiv in Wn,
JH curl ,
BE jcurl ,
0Jdiv ,
0Bdiv in Wc.
BHHB ,
in Wn and in Wc,
EJ , JE in Wc.
Assumption: 00 JT curl .
Boundary conditions
n: outer boundary of Wn
c: outer boundary of Wc
0nH or 0nB onn,0nH or 0nE onc.
ornTnH 0 or nTnB 0 onn,
nTnH 0 or 0nE onc.
Interface conditions
nc: interface between Wn and Wc
nH and nB are continuous on nc.
Magnetic scalar potential in Wn
F grad0TH ,
en
k
kkt1
NT0
Differential equation:
)()( 0T divgraddiv F in Wn.
Boundary conditions:0F or nTn 0 F grad on n.
Finite element approximation
FFFnn
k
kk
n N1
)(
Galerkin equations:
W W
WWF
n n
dgradNdgradgradN i
n
i 0T )(,
i=1,2,...,nn
Magnetic vector potential in Wn
AB curl .
Differential equation
0TA curlcurlcurl )( in Wn
Boundary conditions:0nA or nTnA 0 curl on n.
Finite element approximation
en
k
kk
n a1
)(NAA
Galerkin equations:
W W
WW
n n
dcurldcurlcurl i
n
i 0TNAN)( ,
i=1,2,...,ne
Positive semidefinite matrix
Magnetic vector potential alone in Wc
*AB curl ,
*AE j .
Differential equation
0AA ** jcurlcurl in Wc.
Boundary conditions:
0nA * or nTnA 0 *curl on c.
Finite element approximation
en
k
kk
n a1
)(**NAA
Galerkin equations:
W W
WW
c c
djdcurlcurl n
i
n
i
)(*)(*ANAN
W
W
c
dcurl i 0TN , i=1,2,...,ne
Nonsingular but ill-conditioned matrix
Magnetic vector and electric
scalar potential in Wc
AB curl , gradVjj AE .
Differential equations:
0AA gradVjjcurlcurl ,
0)( gradVjjdiv A in Wc.
Boundary conditions:
0nA or nTnA 0 curl ,
0VV =constant or 0)( gradVjj An
on c.
Finite element approximation
en
k
kk
n a1
)(NAA ,
nn
k
kk
n NVVV1
)(.
Galerkin equations:
W W
WW
c c
djdcurlcurl n
i
n
i
)()(ANAN
WW
WW
cc
dcurldgradVj i
n
i 0TNN)( ,
i=1,2,...,ne,
W
W
c
dgradNj n
i
)(A
0)( W Wc
dgradVgradNj n
i , i=1,2,...,nn
Singular system but improved conditioning.
Current vector and magnetic
scalar potential in Wc
F gradTTH 0 , TTJ 0 curlcurl .
Differential equations:
F gradjjcurlcurl TT
00 TT jcurlcurl ,
)()( 0TT divjgraddivj F in Wc.
Boundary conditions:
0nT or nTnT 0 curlcurl ,
0FF =constant or 0)( TnTn F grad
on c.
Finite element approximation
en
k
kk
n t1
)(NTT ,
FFFnn
k
kk
n N1
)(.
Galerkin equations:
W W
WW
c c
djdcurlcurl n
i
n
i
)()(TNTN
W
WF
c
dgradj n
i
)(N
WW
WW
cc
djdcurlcurl ii 00 TNTN ,
i=1,2,...,ne,
W
W
c
dgradNj n
i
)(T
W
WF
c
dgradgradNj n
i
)(
W
W
c
dgradNj i 0T , i=1,2,...,nn
Singular system but good conditioning.
Coupling A,V in Wc to A in Wn:
A,V-A formulation
Interface conditions on nc:
Continuity of An nB is continuous
Continuity of nAcurl is a natural interface
condition nH is continuous
Galerkin equations remain unchanged
Coupling T,F in Wc to F in Wn:
T,F-F formulation
Interface conditions on nc:
Continuity of F and 0nT nH is
continuous
Continuity of nT F )( grad is a natural
interface condition nB is continuous
Galerkin equations remain unchanged
Typical skin effect problem
u(t)
i(t)
i(t) J(r,t)
B(r,t) Wn: =0
Wc: >0
E1: 0nE
E2: 0nE
cn: nBnH , cont.
n
n
n
HBEJ ,
,t
curl
BE
Wc:
,JH curl
Wn:
,JH curl
,0Bdiv
HB
Integral quantities, network
parameters
2
),()(
E
dtti nrJ
1
),(
E
dt nrJ
W
W
c
dt
tpv
2),(
)(rJ
)()( 2 titR
W
WW
dHdBtW
nc
tB
m
),(
0
)(
r
t
diLd
di
0
)()()(
dt
tdWtptitutp m
v
)()()()()(
Voltage excitation (1)
AB curl in nc WW and , gradUt
AE in cW
EH curl in cW , 0H curl in nW ,
)(, tuU 0nA on1E , 0, U0nA on
2E ,
0nA or 0nH on )( nc WW
nA and nH are continuous on cn 0 nJ
Voltage excitation (2)
W
W
c
dt
tp
2),(
)(rJ
W
WW
dHdBdt
d
nc
tB ),(
0
r
W
W
c
dtp JE)( WW
W
nc
dt
HB
W
WW nc
dt
HB
WW
W
nc
dt
curl HA
WW
W
nc
dcurlt
HA
0
)(
WW
nc
dt
nHA
W
W
c
dt
JA
Voltage excitation (3)
W
W
c
dt
tp JA
E)( W
W
c
dgradU J
W
W
c
dgradUtp J)(
W
W
cnEEc
dUdUdiv
21
0
nJJ
)()()()(
1
titudtutp
E
nJ
Boundary value problem for A,V (1)
Differential equations:
0A
A
t
Vgrad
tcurlcurl
1
t
VU
0)(
t
Vgrad
tdiv
A
0A
curlcurl
1
in Wc,
in Wn,
Boundary value problem for A,V (2)
Boundary conditions:
t
dutV0
)()(, 0nA on 1E ,
0)(, tV0nA on 2E ,
0nA or 0nA curl
1on )( nc WW .
Interface conditions:
nA and nAcurl
1are continuous on cn .
Current excitation (1)
in cW
incW ,
in nW .
on )( nc WW
and F are continuous and cn
,TTJ 0 curl F gradTTH 0,
F grad0TH
0B
J
tcurl
1 0Bdiv in nW ,
)0( nB0nE or
nTnH 0
nB 0nT on
,
.
Properties of T0
0T0 curl nWin ,
1E
dcurl nT0 2E
dcurl nT0 )(ti
nT0 is continuous on .cn
A possible choice of T0
Solve the static current field in Wc
i(t) 00 TJ curl
Wn: =0 Wc: >0 C
W
C
n
ti3
)(
4
)()()(:
Q
S0
rr
rrdsrHrT
nHnT S0 :cn
0T0
W curlcurlc
1:
0nT0 curlE
1:1
0nT0 curlE
1:2
Boundary value problem for T,F (1)
Differential equations:
in Wc,
in Wn,
0TTTT 00 F
grad
tcurlcurl
1
0)( F
graddiv
tTT0
0)( F
graddiv
t0T
Boundary value problem for T,F (2)
Boundary conditions:
2E
or on )( nc WW .
Interface conditions:
and
on 1E ,
are continuous on cn .
0,1
F nTT0nT 0 gradcurl
0nT oncn .
0F 0nT0 F grad
F nT0 F grad
