Skin and proximity effects in multi-layer transformer windings of finite thickness

Losses due to skin and proximity effects within power transformer windings were previously calculated using the assumption that the curvature and thickness of the winding could be neglected and the magnetic field boundary conditions were known. This paper presents formulae suitable for a wider range of dimensions which accurately account for differences in the winding radii and do not use magnetic fields as an input parameter. The results show the conditions under which the AC resistance loss can be significantly different than in prior publications. Thus, the expressions presented can be used with a much higher level of confidence to predict transformer performance. Transformers with solenoidal windings and transformers with coaxial windings are structural duals of each other and both are subject to equivalent skin and proximity effects. Presented in detail are the relationship between solenoidal and coaxial windings, the key analytic expressions for both winding types, normalized AC resistance curves with and without accounting for curvature and the effects of winding layers. Additionally, the results are verified by numerical and expert data. A useful appendix is included which provides a practical, straight-forward procedure to numerically evaluate the seemingly complicated expressions