Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies

TLDR

3‑D eddy‑current calculations are difficult because flux and current must be described by coupled vector functions, with coupling driven by material interfaces and boundary conditions, and magnetic‑vector‑potential formulations are less attractive due to the need to compute the vector throughout the domain and complex interface conditions. The study examines how selecting different magnetic or electric vector potentials, such as ξ and H, and their gauge choices affect the formulation, noting that scalar potentials can be introduced to control coupling and that a generalized H formulation allows the vector to vanish outside conductors, simplifying the problem. The choice of gauge or vector‑potential divergence significantly alters the problem formulation.

Abstract

The numerical calculation of 3-dimensional eddy currents is complicated by the need to use vector functions to describe the interlinked flux and current distributions. The three vector components are coupled, particularly by the conditions imposed by the copper and iron surfaces, and by other changes in material properties. The range of choice of magnetic and electric vector potentials is examined, including the ξ and H vectors as particular examples, and it is shown that the problem formulation is greatly affected by the choice of gauge, or vector-potential divergence. This introduces scalar potentials which can be used in various ways. They control the coupling conditions between the vector components. One advantage of an electric vector potential, or generalised H, formulation is that the gauge can be chosen so that the vector is zero outside the conductors, and is replaced by a scalar. Magnetic vector-potential formulations are generally less attractive, both because the vector has to be computed throughout the region of interest and because of the greater complexity of the magnetic-and electric-interface conditions.