A shell element for computing 3D eddy currents-application to transformers

General Integral Formulation for the 3D Thin Shell Modeling

“…Because of the jump of the magnetic field tangential component through the thickness, a double layer surface element is necessary [2]. The expressions related to the tangential magnetic fields on both sides and outside of the shell are: (5) where and are the tangential source fields generated by inductors at each side of the shell and are the both associated scalar reduced magnetic potentials.…”

“…Equations (2) and (3) represent the electromagnetic behavior of the conductive shell itself. In order to take into account the surrounding air region, several authors choose the Finite Element Method [2]- [8]. Thus, the air region has to be also discretized.…”

“…The integration of tangential magnetization through the depth can be expressed by [1], [2]: (9) where . The equivalent shell current distribution can be also expressed by [1], [2]: (10) Using (9) and (10), (8) becomes: (11) The equation on the side "2" of shell is obtained by permuting the subscripts "1" and "2" as (12) If we consider that has no variation through the thickness of the shell, we obtain: (13) Equations (7) and (13) have to be discretized.…”

“…The field variation of the tangential component across the thickness of shell can be approximated by the analytical solution of the problem for an infinite plane [1], [2]: (1) where , and are the tangential values of the magnetic field on both sides of the shell. Applying Galerkin's method on the Maxell-Faraday equation for the side "1" of the shell, we have [2], [9]: (2) where , ; w is a set of nodal surface weighting functions; is the normal vector corresponding to the side "1" of the shell and is the line region delimiting the surface. The symmetrical equation corresponding to the other side of the shell is obtained by simply exchanging the indices "1" and "2":…”

“…It increases the size of the needed mesh if this effect is properly modeled. Shell element formulations have been developed to overcome such difficulties, e.g., with the boundary element method (BEM) [1], with the finite-element method [2]- [8] and with an integral method recently proposed by the authors of this paper [9].…”