New closed-form complex proximity-effect permeability expression for characterizing litz-wire windings

The eddy-current losses in litz-wire windings are mostly due to the proximity effect, which can be macroscopically modelled via a complex frequency-dependent permeability. In this study, the authors propose a new closed-form expression for this complex permeability. Results are compared with approximations found in the literature in a wide-frequency range. For the sake of validation, a two-dimensional)-axisymmetric gapped transformer with litz-wire primary and secondary coils is studied.