Complementary variational formulation for eddy-current problems using the field variables E and H directly

Abstract

Complementary functionals, expressed in terms of either E and V x E or H and V x W , applying to linear eddy-current problems are given and are shown to be stationary subject to simple, and sometimes natural, boundary constraints. The analysis is carried through to a late stage before the pre-Maxwell zerodisplacement current approximation is applied, in order not to obscure the principal results. The extension to a laminated media is considered. Representation of the quasistatic //-field in any current-free region of the problem by a scalar potential is allowable. The results are verified by perturbing the known 'skin effect' quasistatic solution for parallel busbars. It is suggested how real eddy-current problems might be tackled by using the finite-element matrices for tetrahedral elements which have been derived by previous authors to apply to high-frequency problems and to the scalar Helmholtz equation.

Upper and lower bounds in eddy-current calculations

“…In this paper, we restrict ourselves to single regions having uniform conductivity, because we wish to investigate the energy distribution in such regions. Ferrari [7] shows how such regions can be linked to nonconducting regions.…”

“…Rodger [15] combines the use of the magnetic vector potential inside eddy-current regions, with the use of the scalar magnetic potential outside such regions. Ferrari [7] points out that the use of potentials requires the insertion of difficult boundary conditions. He therefore favours the use of the electric and magnetic field vectors themselves and we follow his advice.…”

“…In an early paper written with Dr. J. Penman [5], I put forward the view that bounded values for the equivalent resistance and reactance could be obtained, and this view was restated and amplified in a book [6]. More recently, Ferrari [7] has given support to this view, in general terms, by stating that bounded values should be obtainable, but Rikabi* [8] has expressed doubts about the existence of bounded solutions at least as far as his own variational procedure is concerned. Penman and Fraser give dual bounds for static but not for eddy-current problems [9].…”

Finite element and boundary element analysis of electromagnetic NDE phenomena

“…The major considerations for a good three dimensional model are the establishment of the correct formulation and derivation of the quantities of interest such as eddy current densities, flux densities and coil impedance from the solution. Many different formulations [21][22][23][24][25] have been presented and used, each having its own advantages and disadvantages. Three dimen sional modeling is concerned not only with the problem of discretization difficulties, but also with the vector nature of the application.…”

“…This is accomplished by solving Maxwell's equation using potential functions and then computing the field quantity. Ferrari [24] and others show that certain problems with simple boundary and interface conditions can be solved by directly solving Maxwell's equations for the É and H fields.…”