Skin effect is the tendency for high-frequency currents to flow on the surface of a conductor. Proximity effect is the tendency for current to flow in other undesirable patterns---loops or concentrated distributions---due to the presence of magnetic fields generated by nearby conductors. In transformers and inductors, proximity-effect losses typically dominate over skin-effect losses. In litz-wire windings, proximity effect may be further divided into internal proximity effect (the effect of other currents within the bundle) and external proximity effect (the effect of current in other bundles) [14], [15]. However, the distinction is useful only as a form of bookkeeping. The actual losses in one strand of a litz bundle are simply a result of the total external field, due to the currents in all the other strands present. Another approach to calculating the loss in a litz winding is to look at it as a single winding, made up of nN turns of the strand wire, each with current i/n flowing in it, where n is the number of strands, N is the number of turns of litz wire, and i is current flowing in the overall litz bundle. The loss in the litz winding will be the same as in the equivalent single-strand winding as long as the currents flowing in all the strands are equal [16]. Other methods of calculating loss in litz wire also assume equal current in all strands [14], [12], [17]].

The objective of twisting or weaving litz wire, as opposed to just grouping fine conductors together, is to ensure that the strand currents are equal. Simple twisted bunched-conductor wire can accomplish this adequately in situations where proximity effect would be the only significant problem with solid wire. Where skin effect would also be a problem, more complex litz constructions can be used to ensure equal strand currents. Thus, in a well-designed construction, strand currents are very close to equal. However, our results remain valid even when simple twisting results in significant skin effect at the litz-bundle level. This is because the the bundle-level skin-effect loss is independent of the number of strands, and is orthogonal [15] to the strand-level eddy-current losses.

We represent winding losses by


where Fr is a factor relating dc resistance to an ac resistance which accounts for all winding losses, given a sinusoidal current with rms amplitude Iac. As shown in Appendix A, we can approximate Fr by


where ; is the radian frequency of a sinusoidal current, n is the number of strands, N is the number of turns, dc is the diameter of the copper in each strand, is the resistivity of the copper conductor, bc is the breadth of the window area of the core, and k is a factor accounting for field distribution in multi-winding transformers, normally equal to one (see Appendix A). For waveforms with a dc component, and for some non-sinusoidal waveforms, it is possible to derive a single equivalent frequency that may be used in this analysis (Appendix C). In an inductor, the field in the winding area depends on the gapping configuration, and this analysis is not directly applicable.

The analysis described here considers the strands of all litz bundles to be uniformly distributed in the window, as they would be in a single winding using N turns of wire the diameter of the litz strands. In fact, the strands are arranged in more or less circular bundles. In this sense, the analysis of [15] may be more accurate, but this difference has very little effect on the results. The most important difference between the model used here and the model in [15] is the greater accuracy of [15] for strands that are large compared to a skin depth. The simpler model is used because it is accurate for the small strand diameters that are found to be optimal, and because its simplicity facilitates finding those optimal diameters. Other models (such as [14] and the similar analysis in [17]) also model large strand diameters and circular bundle configurations accurately, but they fully calculate only internal, and not external, proximity effect, and so are not useful for the present purposes.