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Finite Elements in Electromagnetics 3. Eddy currents and skin effect

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

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Finite Elements in Electromagnetics 3. Eddy currents and skin effect

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  1. Finite Elements in Electromagnetics3. 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,F-F formulation

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

  4. Maxwell’s equations:

  5. Boundary conditions

  6. Interface conditions

  7. Magnetic scalar potential in Wn

  8. Finite element approximation

  9. Magnetic vector potential in Wn

  10. Finite element approximation

  11. Magnetic vector potential alone in Wc

  12. Finite element approximation

  13. Magnetic vector and electric scalar potential in Wc

  14. Finite element approximation

  15. Current vector and magnetic scalar potential in Wc

  16. Finite element approximation

  17. Coupling A,V in Wc to A in Wn: A,V-A formulation

  18. Coupling T,F in Wc to F in Wn: T,F-F formulation

  19. Typical skin effect problem

  20. Integral quantities, network parameters

  21. Voltage excitation (1) , in in , , in in , , on on on or and are continuous on

  22. Voltage excitation (2)

  23. Voltage excitation (3)

  24. Boundary value problem for A,V (1) Differential equations: in Wc, in Wn,

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

  26. Current excitation (1) , in . in , in , in or , on . on are continuous and and F

  27. Properties of T0 , in is continuous on

  28. A possible choice of T0 Solve the static current field in Wc

  29. Boundary value problem for T,F (1) Differential equations: in Wc, in Wn,

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

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