"Small parameter" problems in eddy-current theory: a review, and case-study on how to avoid meshing small air-gaps

Perturbation approaches in terms of a small dimensionless parameter are standard and very useful in field computations. For instance, if the width of an airgap can be considered "small", no mesh inside the gap will be required. However, it often happens that two such small parameters intervene in the modelling, and "compete" in some way. In such situations, two, very different, limit problems are candidates as simplified formulations, depending on which of the two small parameters is taken as dominant. A justified concern for ease of computation may then involuntarily lead to the selection of a wrong model, so this raises delicate modelling issues. Some methodological guidelines are offered in this respect. The case of the small airgap in a conductor is treated in detail, showing how the penetration depth intrudes in the modelling as a second small parameter, besides the airgap width. Implications as regards numerical schemes, in both main variants where either e or h is the primary unknown, are discussed.