Copper losses of litz-wire windings due to an air gap

In this paper, a systematic procedure is presented, how to predict the AC resistance of litz-wire windings considering air gap fringing fields. For this purpose, an equivalent complex permeability model is derived for hexagonally packed wires. It is shown, how finite element method (FEM) can be used to determine the real as well as the imaginary part of the complex permeability with the copper filling factor as a parameter. A further FEM-model is deduced to describe the air gap fringing fields of gapped inductors. Accordingly, the exact proximity losses of the litz-wire winding are determined correctly and the AC resistance of practical inductors can be predicted over a wide frequency range with high accuracy. This offers the opportunity to optimize such components. Finally, the influence of various parameters on the copper losses is investigated and verified by means of experimental data drawn from impedance measurements.