A salient difficulty in designing high-frequency inductors and
transformers is eddy-current effects in windings. These effects include
skin-effect losses and proximity-effect losses. Both effects can be controlled by the use of conductors
made up of multiple, individually insulated strands, twisted or woven
together. Sometimes the term *litz wire* is reserved for conductors
constructed according to a carefully prescribed pattern, and strands simply
twisted together are called bunched wire. We will use the term *litz
wire* for any insulated grouped strands, but will discuss the effect of
different constructions.

This paper addresses the choice of the degree of stranding in litz wire for a transformer winding. The number of turns and the winding cross-sectional area are assumed to be fixed. The maximum cross-sectional area of each turn is thus fixed, and as the number of strands is increased, the cross-sectional area of each strand must be decreased. This typically leads to a reduction in eddy-current losses. However, as the number of strands increases, the fraction of the window area that is filled with copper decreases and the fraction filled with insulation increases.This results in an increase in dc resistance. Eventually, the eddy-current losses are made small enough that the increasing dc resistance offsets any further improvements in eddy-current loss, and the losses start to increase. Thus, there is an optimal number of strands that results in minimum loss. This paper presents a method of finding that optimum, using standard methods of estimating the eddy-current losses.

Optimizations on magnetics design may be done to minimize volume, loss, cost, weight, temperature rise, or some combination of these factors. For example, in the design of magnetic components for a solar-powered race vehicle [1] (the original impetus for this work) an optimal compromise between loss and weight is important. Although we will explicitly minimize only winding loss, the results are compatible with and useful for any minimization of total loss (including core loss), temperature rise, volume or weight. This is because the only design change considered is a change in the degree of stranding, preserving the overall diameter per turn and overall window area usage. This does not affect core loss or volume, and has only a negligible effect on weight. However, the degree of stranding does significantly affect cost. Although we have not attempted to quantify or optimize this, additional results presented in Section V are useful for cost-constrained designs. The analysis of eddy-current losses used here does not differ substantially from previous work [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] ([10] gives a useful review). Although different descriptions can be used, most calculations are fundamentally equivalent to one of three analyses. The most rigorous approach uses an exact calculation of losses in a cylindrical conductor with a known current, subjected to a uniform external field, combined with an expression for the field as a function of one-dimensional position in the winding area [12]. Perhaps the most commonly cited analysis [11] uses "equivalent" rectangular conductors to approximate round wires, and then proceeds with an exact one-dimensional solution. Finally, one may use only the first terms of a series expansion of these solutions, e.g. [9].All of these methods give similar results for strands that are small compared to the skin depth [12]. (See Appendix B for a discussion of one minor discrepancy.) The solutions for optimal stranding result in strand diameters much smaller than a skin depth. In this region the distinctions between the various methods evaporate, and the simplest analysis is adequate. More rigorous analysis (e.g. [12]) is important when strands are not small compared to a skin depth. In this case, losses are reduced relative to what is predicted by the analysis used here, due to the self-shielding effect of the conductor.

Previous work, such as [2] , [3], [4] has addressed optimal wire diameter for
single-strand windings. The approach in [2], [3], [4] is also applicable for
litz-wire windings in the case that the number of strands is fixed, and
the strand diameter for lowest loss is desired. As discussed in Section V,
this can be useful for cost-sensitive applications, if the number of strands
is the determining factor in cost, and the maximum cost is constrained.
However, this will, in general, lead to higher-loss designs than are possible
using the optimal number of strands.