Analytical Double-2D Models of Leakage Inductance and Frequency-Dependent Resistance for Litz-Wire Medium Frequency Transformer

Traditional one-dimensional (1D) analytical models of leakage inductance and winding loss for litz-wire transformers fail to deliver accurate results under strong two-dimensional (2D) fringing effects, and the inaccuracy of analytical 2D models is attributed to the shape approximation (SA) from round litz wires to rectangular foil blocks. This paper identifies the source of error in current analytical models and proposes analytical double-2D models based on the no-shape-approximation (NSA) concept, aiming at accurately predicting the leakage inductance and frequency-dependent resistance for medium-frequency transformers (MFTs) with round litz-wire windings. Without changing the outermost shape of the litz wire, the NSA models can capture the fringing effects not only at the top and bottom of the windings, but also in the irregular inter-turn regions between adjacent litz conductors. This paper derives analytical solutions for the magnetic vector potential of a round conductor. On this basis, equations describing the contributions of magnetic vector potential, magnetic energy, and magnetic field between two conductors are solved, yielding clear analytical expressions. Considering the rotationally asymmetric transformer structure, the magnetic energy and winding loss, both inside and outside the magnetic core window, are independently calculated using the method of images and the superposition principle. The models are validated through experimental measurements of transformer prototypes with different geometry parameters. Owing to their accuracy, flexibility, and computational efficiency, the proposed double-2D NSA models are suitable for the optimization design of litz-wire MFTs.