Finite Elements in Electromagnetics 2. Static fields Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz, Austria

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

Maxwell‘s equations for static fields

Static current field (1) or on n+1 electrodes  E  E0 +  E1 +  E  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

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

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

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)

Static current field (5) Scalar potential V

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

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

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

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Finite element discretization

Current density represented by arrows

Magnitude of current density represented by colors

Static current field (9) Current vector potential T

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

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

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

Current density represented by arrows

Magnitude of current density represented by colors

Electrostatic field (1) on n+1 electrodes  E  E0 +  E1 +  E  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

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

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

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)

Electrostatic field (5) Scalar potential V

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

Electrostatic field (7) Operator for the scalar potential V

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

380 kV transmisson line

380 kV transmisson line, E on ground

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

Magnetostatic field (1) or on n+1 magn. walls  E  E0 +  E1 +  E  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

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

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

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)

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

Magnetostatic field (6) Scalar potential , differential equation

Magnetostatic field (7) Scalar potential , boundary conditions

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

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

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

Magnetostatic field (11) Magnetic vector potential A

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

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

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

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