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:

    gradN          gradF dW    gradN i  T0 dW ,
                             (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

         j gradN        gradV             dW  0 , i=1,2,...,nn
                                         (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

             jgradN  gradF               dW
                                        (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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