1. Finite Elements in Electromagnetics 2. Static fields Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz, Austria email: biro@igte.tu-graz.ac.at

2. Overview Maxwell‘s equations for static fields Static current field Electrostatic field Magnetostatic field

3. Maxwell‘s equations for static fields

4. Static current field (1) or on n+1 electrodes  E  E0 +  E1 +  E2 +...+  Ei +...+  En on the interface  J to the nonconducting region n voltages between the electrodes are given: or n currents through the electrodes are given: i = 1, 2,..., n

5. Symmetry  E0 may be a symmetry plane A part of  J may be a symmetry plane Static current field (2)

6. Interface conditions Tangential E is continuous Normal J is continuous Static current field (3)

7. Network parameters (n>0) n=1:U 1 is prescribed and or I 1 is prescribed and n>1: ori = 1, 2,..., n Static current field (4)

8. Static current field (5) Scalar potential V

9. Static current field (6) Boundary value problem for the scalar potential V

10. Static current field (7) Operator for the scalar potential V

11. Static current field (8) Finite element Galerkin equations for V i = 1, 2,..., n

12. High power bus bar

13. Finite element discretization

14. Current density represented by arrows

15. Magnitude of current density represented by colors

16. Static current field (9) Current vector potential T

17. Static current field (10) Boundary value problem for the vector potential T

18. Static current field (11) Operator for the vector potential T

19. Static current field (12) Finite element Galerkin equations forT i = 1, 2,..., n

20. Current density represented by arrows

21. Magnitude of current density represented by colors

22. Electrostatic field (1) on n+1 electrodes  E  E0 +  E1 +  E2 +...+  Ei +...+  En on the boundary  D n voltages between the electrodes are given: or n charges on the electrodes are given: i = 1, 2,..., n

23. Symmetry  E0 may be a symmetry plane A part of  D (  =0) may be a symmetry plane Electrostatic field (2)

24. Interface conditions Tangential E is continuous Normal D is continuous Electrostatic field (3) Special case  =0:

25. Network parameters (n>0) n=1:U 1 is prescribed and orQ 1 is prescribed and n>1: ori = 1, 2,..., n Electrostatic field (4)

26. Electrostatic field (5) Scalar potential V

27. Electrostatic field (6) Boundary value problem for the scalar potential V

28. Electrostatic field (7) Operator for the scalar potential V

29. Electrostatic field (8) Finite element Galerkin equations for V i = 1, 2,..., n

30. 380 kV transmisson line

31. 380 kV transmisson line, E on ground

32. 380 kV transmisson line, E on ground in presence of a hill

33. Magnetostatic field (1) or on n+1 magn. walls  E  E0 +  E1 +  E2 +...+  Ei +...+  En on the boundary  B n magnetic voltages between magnetic walls are given: or n fluxes through the magnetic walls are given: i = 1, 2,..., n

34. Symmetry  H0 (K=0) may be a symmetry plane A part of  B (b=0) may be a symmetry plane Magnetostatic field (2)

35. Interface conditions Tangential H is continuous Normal B is continuous Magnetostatic field (3) Special case K=0:

36. Network parameters (n>0), J=0 n=1:U m1 is prescribed and or  1 is prescribed and n>1: ori = 1, 2,..., n Magnetostatic field (4)

37. Network parameter (n=0), b=0, K=0, J  0 Magnetostatic field (5) Inductance:

38. Magnetostatic field (6) Scalar potential , differential equation

39. Magnetostatic field (7) Scalar potential , boundary conditions

40. Magnetostatic field (8) Boundary value problem for the scalar potential  Full analogy with the electrostatic field

41. Magnetostatic field (9) Finite element Galerkin equations for  i = 1, 2,..., n

42. Magnetostatic field (10) In order to avoid cancellation errors in computing T 0 should be represented by means of edge elements: since and hence T 0 and grad   (n) are in the same function space

43. Magnetostatic field (11) Magnetic vector potential A

44. Magnetostatic field (12) Boundary value problem for the vector potential A

45. Magnetostatic current field (13) Operator for the vector potential A

46. Magnetostatic field (14) Finite element Galerkin equations for A i = 1, 2,..., n

47. Magnetostatic field (15) Consistence of the right hand side of the Galerkin equations Introduce T 0 as