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Finite Elements in Electromagnetics 2. Static fields

Finite Elements in Electromagnetics 2. Static fields. Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz, Austria email: biro@igte.tu-graz.ac.at. Overview. Maxwell‘s equations for static fields Static current field Electrostatic field Magnetostatic field.

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Finite Elements in Electromagnetics 2. Static fields

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  1. Finite Elements in Electromagnetics2. 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. on n+1 electrodes GE =GE0+GE1+GE2+ ...+ GEi+ ...+ GEn on the interface GJto the nonconducting region Static current field (1) n voltages between the electrodes are given: or or n currents through the electrodes are given: i = 1, 2, ..., n

  5. Static current field (2) Symmetry GE0 may be a symmetry plane A part of GJ may be a symmetry plane

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

  7. Static current field (4) Network parameters (n>0) n=1: U1 is prescribed and or I1 is prescribed and n>1: or i = 1, 2, ..., n i = 1, 2, ..., n

  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) n voltages between the electrodes are given: or n charges on the electrodes are given: i = 1, 2, ..., n on n+1 electrodes GE =GE0+GE1+GE2+ ...+ GEi+ ...+ GEn on the boundary GD

  23. Electrostatic field (2) Symmetry GE0 may be a symmetry plane A part of GD (s=0) may be a symmetry plane

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

  25. Electrostatic field (4) Network parameters (n>0) n=1: U1 is prescribed and or Q1 is prescribed and n>1: or i = 1, 2, ..., n i = 1, 2, ..., n

  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) n magnetic voltages between magnetic walls are given: or or n fluxes through the magnetic walls are given: i = 1, 2, ..., n on n+1 magn. walls GE =GE0+GE1+GE2+ ...+ GEi+ ...+ GEn on the boundary GB

  34. Magnetostatic field (2) Symmetry GH0 (K=0) may be a symmetry plane A part of GB (b=0) may be a symmetry plane

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

  36. Magnetostatic field (4) Network parameters (n>0), J=0 n=1: Um1 is prescribed and or Y1 is prescribed and n>1: or i = 1, 2, ..., n i = 1, 2, ..., n

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

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

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

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

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

  42. Magnetostatic field (10) In order to avoid cancellation errors in computing T0 should be represented by means of edge elements: since and hence T0 and gradF(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. Introduce T0 as Magnetostatic field (15) Consistence of the right hand side of the Galerkin equations

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