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 ▿ × E or H and ▿ × H, 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 zero-displacement current approximation is applied, in order not to obscure the principal results. The extension to a laminated media is considered. Representation of the quasistatic H-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’ quasi-static 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.

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