Derivation of optimum winding thickness for duty cycle modulated current waveshapes

Increased switching frequencies in magnetic components have resulted in renewed attention to the problem of proximity effect losses in layered transformer windings. The ideal situation is to design at the point of minimum AC winding resistance. This paper provides a unified approach which gives exact AC resistance formulas for pulsed, rectangular and triangular waveforms, with variable duty cycle. In every case an approximation to the AC resistance versus layer thickness curve is derived and the optimum point can be found with a simple calculation involving the number of harmonics (related to rise time), the duty cycle and the number of layers. This process yields a result that is at least as accurate as reading the point from a generated graph (without the considerable effort involved in generating the graph),