# Finite Elements in Electromagnetics 3. Eddy Currents by dffhrtcv3

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```									      Finite Elements in
Electromagnetics
3. Eddy currents and skin effect
Oszkár Bíró
IGTE, TU Graz
Kopernikusgasse 24Graz, Austria
email: biro@igte.tu-graz.ac.at
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
Typical eddy current problem
Wn: nonconducting region
Air                                   Coil
J(r,t) = 0                              J(r,t) known

 0

Wc: eddy current
region
J(r,t) unknown
Maxwell’s equations:
curlH  J 0 ,        B   H , H  B
divB  0      in Wn,            in Wn and in Wc,

curl H  J ,        J  E , E  J in Wc.
curlE   jB ,
divJ  0 ,
divB  0     in Wc. Assumption: curlT0  J 0 .
Boundary conditions
n: outer boundary of Wn
c: outer boundary of Wc

H  n  0 or B  n  0   onn,
H  n  0 or E  n  0 onc.
or
H  n  T0  n or B  n  T0  n   onn,
H  n  T0  n or E  n  0         onc.
Interface conditions
nc: interface between Wn and Wc

H  n and B  n are continuous on nc.
Magnetic scalar potential in Wn
ne
H  T0  gradF ,          T0   tk N k
k 1

Differential equation:
div ( gradF )  div ( T0 ) in Wn.

Boundary conditions:
F  0 or gradF  n  T0  n on n.
Finite element approximation
nn
FF      (n)
  Fk Nk
k 1

Galerkin equations:

(n)
i
Wn                                Wn

i=1,2,...,nn
Magnetic vector potential in Wn
B  curl A .

Differential equation
curl (curlA )  curlT0 in Wn

Boundary conditions:
A  n  0 or curlA  n  T0  n on n.
Finite element approximation
ne
AA     (n)
  ak N k
k 1

Galerkin equations:
 curlN i curlA dW       curlN i  T0 dW ,
(n)

Wn                         Wn

i=1,2,...,ne
Positive semidefinite matrix
Magnetic vector potential alone in Wc
B  curlA ,
*
E   j A .
*

Differential equation
curlcurlA  j A  0 in Wc.
*         *

Boundary conditions:
A  n  0 or curlA*  n  T0  n on c.
*
Finite element approximation
ne
A A *     *( n )
  ak N k
k 1
Galerkin equations:
 curlN i curlA dW               j N       A            dW
*( n )                               *( n )
i
Wc                               Wc

  curlN i  T0 dW ,                 i=1,2,...,ne
Wc

Nonsingular but ill-conditioned matrix
Magnetic vector and electric
scalar potential in Wc
B  curl A ,   E   jA  jgradV .

Differential equations:

curlcurlA  j A  j gradV  0 ,

 div ( j A  j gradV )  0 in Wc.
Boundary conditions:

A  n  0 or curlA  n  T0  n ,

V  V0 =constant or n  (  j A  j gradV )  0

on c.
Finite element approximation
ne
AA    (n)
  ak N k ,
k 1

nn
V V   (n)
 Vk N k .
k 1
Galerkin equations:
 curlNi curlA dW   j Ni  A dW
(n)               (n)

Wc                                  Wc

    j N  gradV            dW   curlN i  T0 dW ,
(n)
i
Wc                                     Wc

i=1,2,...,ne,
 j gradN          A dW
(n)
i
Wc

(n)
i
Wc

Singular system but improved conditioning.
Current vector and magnetic
scalar potential in Wc
H  T0  T  gradF , J  curlT0  curlT .

Differential equations:

curlcurlT  j T  j gradF
 curlcurlT0  j T0 ,

jdiv ( T  gradF )   jdiv ( T0 ) in Wc.
Boundary conditions:

T  n  0 or curlT  n   curlT0  n ,

F  F 0 =constant or n  ( T  gradF )  n  T0

on c.
Finite element approximation
ne
TT   (n)
  tk N k ,
k 1

nn
FF   (n)
  Fk Nk .
k 1
Galerkin equations:

 curlN  curlT             dW     j N  T         dW
(n)                        (n)
i                                i
Wc                               Wc

    j N  gradF            dW
(n)
i
Wc

   curlN i  curlT0 dW            j N       i    T0 dW ,
Wc                          Wc

i=1,2,...,ne,
    jgradN  T            dW
(n)
i
Wc

(n)
i
Wc

   
Wc
jgradNi  T0 dW ,          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 n  A  B  n is continuous

Continuity of curl A  n is a natural interface
condition  H  n 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 T  n  0  H  n is
continuous
Continuity of ( gradF  T)  n is a natural
interface condition  B  n is continuous

Galerkin equations remain unchanged
Typical skin effect problem
n
E1: E  n  0                    B(r,t)       Wn: =0

cn: H  n, B  n cont.
Wc:                           J(r,t)
curl H  J ,     i(t)                    Wc: >0
B                                                E2: E  n  0
curlE   ,         Wn:                                      n
t       curl H  J ,            n
J  E, B  H     div B  0,
i(t)
B  H
u(t)
Integral quantities, network
parameters
i (t )      J (r, t )  nd    J (r, t )  nd
E 2                             E 1
2
J (r, t )
pv (t )                             dW  R(t )i (t )
2

Wc

            
B ( r ,t )            t
d
Wm (t )        
Wc  W n 
 HdB dW   i( ) d L( )i( )d
0

      0

dWm (t )
p(t )  u (t )i (t )  pv (t ) 
dt
Voltage excitation (1)
A
B  curlA in W c and W n , E       gradU in W c
t
curlH  E in W c , curlH  0 in W n ,

A  n  0,U  u (t ) on E1 , A  n  0,U  0 on E 2 ,

A  n  0 or H  n  0 on  (W c  W n )

A  n and H  n are continuous on cn  J  n  0
Voltage excitation (2)
2
J (r, t )        d         B (r ,t ) 
p(t )              dW         W   HdB dW
Wc
            dt Wc  n  0



B
p (t )   E  JdW                HdW
Wc            Wc  W n
t
B                        A
W t  HdW  W W curl t  HdW
Wc  n                  c n

A                             A  H   nd
               curlHdW                       
Wc  W n
t              ( Wc  W n ) 
t     

A                                     0
           JdW
Wc
t
Voltage excitation (3)
p(t )    E  A   JdW   gradU  JdW
Wc 

t           
Wc

p(t )    gradU  JdW   UdivJdW              UJ  nd
Wc                    Wc         E 1  E 2  cn
 
 
0

p(t )  u (t )  J  nd  u (t )i (t )
E 1
Boundary value problem for A,V (1)
V
U
t
Differential equations:
1           A           V
curl  curlA         grad     0
           t           t
A         V
 div (     grad     )  0 in Wc,
t         t
1      
curl  curlA   0   in Wn,
      
Boundary value problem for A,V (2)
Boundary conditions:
t
A  n  0,V (t )   u ( )d on E1,
0
A  n  0,V (t )  0 on E 2 ,
1
A  n  0 or curlA  n  0 on  (W c  W n ) .

Interface conditions:
1
A  n and curlA  n are continuous on cn .

Current excitation (1)
J  curl T0  T , H  T0  T  gradF in W c ,
H  T0  gradF in W n .
1      B
curl J          0 in W c , divB  0 in W n ,
      t

E  n  0 (B n  0) or
H  n  T0  n on  (W c  W n ) ,
B  n and F are continuous and T  n  0 on cn .
Properties of T0
curlT0  0 in W n ,

  curlT0  nd  curlT0  nd  i (t )
E 1             E 2

T0  n is continuous on cn .
A possible choice of T0
Solve the static current field in Wc

W c : curl curlT0   0
1
1                                                      
E1 : curlT0  n  0                                      
                                              cn : T0  n  H S  n
1
J 0  curl T0              E 2 : curlT0  n  0
i(t)                                           
C                         Wc: >0
Wn: =0
i (t ) ds Q  (r  rQ )
W n : T0 (r )  H S (r ) 
4 C   r r 3 
Q
Boundary value problem for T,F (1)
Differential equations:

 1 curl T  T    T  T  gradF   0
curl 
         0     t

0


div (T0  T  gradF)  0 in Wc,
t


div (T0  gradF)  0 in Wn,
t
Boundary value problem for T,F (2)
Boundary conditions:
1
curlT  n  0,  T0  T  gradF   n  0 on E1, E 2

Tn  0       on cn .

F  0 or  T0  gradF   n  0 on  (W c  W n ) .
Interface conditions:

F and  T0  gradF   n are continuous on cn .

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