The Copper Losses of Gapped Inductors with Litz-Wire Windings

In this paper, an analytical 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 the real as well as the imaginary part of the complex permeability can be determined with the copper filling factor as a parameter. An analytical 2D model is deduced to describe the air gap fringing fields of gapped inductors. Accordingly, the 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. The novelty of the approach presented in this paper is given by the fact that an exact 2D analytical solution is conducted to describe the air gap fringing field. Furthermore, the distance between air gap(s) and winding can be considered as a degree of freedom as well as the number of single gaps to realize a distributed air gap.