1. 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

2. 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,  -  formulation

3.  n : nonconducting region Air J(r,t) = 0 Coil J(r,t) known  c : eddy current region J(r,t) unknown Typical eddy current problem

4. Maxwell’s equations:

5. Boundary conditions

6. Interface conditions

7. Magnetic scalar potential in  n

8. Finite element approximation

9. Magnetic vector potential in  n

10. Finite element approximation

11. Magnetic vector potential alone in  c

12. Finite element approximation

13. Magnetic vector and electric scalar potential in  c

15. Finite element approximation

17. Current vector and magnetic scalar potential in  c

19. Finite element approximation

22. Coupling A,V in  c to A in  n : A,V-A formulation

23. Coupling T,  in  c to  in  n : T,  -  formulation

28. Typical skin effect problem

29. Integral quantities, network parameters

30. Voltage excitation (1) in,,, on,, or on andare continuous on

31. Voltage excitation (2)

32. Voltage excitation (3)

33. Boundary value problem for A,V (1) Differential equations: in  c, in  n,

34. Boundary value problem for A,V (2) Boundary conditions: on,, or on. Interface conditions: and are continuous on.

35. Current excitation (1) in,. on and  are continuous and, in, or on,.

36. Properties of T 0 in, is continuous on

37. A possible choice of T 0 Solve the static current field in  c

38. Boundary value problem for T,  (1) Differential equations: in  c, in  n,

39. Boundary value problem for T,  (2) Boundary conditions: or on. Interface conditions: and on, are continuous on. on.