Finite Elements in Electromagnetics 3. Eddy currents and skin effect Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz, Austria

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

 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

Maxwell’s equations:

Boundary conditions

Interface conditions

Magnetic scalar potential in  n

Finite element approximation

Magnetic vector potential in  n

Finite element approximation

Magnetic vector potential alone in  c

Finite element approximation

Magnetic vector and electric scalar potential in  c

Finite element approximation

Current vector and magnetic scalar potential in  c

Finite element approximation

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

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

Typical skin effect problem

Integral quantities, network parameters

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

Voltage excitation (2)

Voltage excitation (3)

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

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

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

Properties of T 0 in, is continuous on

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

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

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