Analytical Calculation of Copper Losses in Litz-Wire Windings of Gapped Inductors

In this paper, we present an analytical procedure 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. We show how the real as well as the imaginary parts of the complex permeability can be determined with the copper filling factor as a parameter. An analytical 2-D 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.