The effects of eddy currents in transformer windings are considered, and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings. The method consists in dividing the winding into portions, calculating the d.c. resistances and d.c. leakage inductances of each of these portions, and then multiplying the d.c. values by appropriate factors to obtain the corresponding a.c. values. These a.c. values are then referred to, say, the primary winding and summed to give the total winding resistance and leakage inductance of the transformer. Formulas are derived and quoted for calculating the d.c. resistances and leakage inductances of the winding portions. Theoretical expressions are derived for the variation with frequency etc. of the factors by which the d.c. values must be multiplied to obtain the corresponding a.c. values. These expressions are presented in the form of graphs, permitting the factors to be read as required.

Effects of eddy currents in transformer windings

The one well-known one-dimensional method for calculating the AC resistance of multilayer transformer windings contains a built-in orthogonality which has not been reported previously. Orthogonality between skin effect and proximity effect makes a more generalized approach for the analytical solution of AC resistance in windings possible. This includes a method to calculate the AC resistance of round conductor windings which is not only convenient to use, but gives more accurate answers than the basic one-dimensional method because the exact analytical equations for round conductors can be used. >

Improved analytical modeling of conductive losses in magnetic components

AC losses due to nonsinusoidal current waveforms have been found by calculating the losses at harmonic frequencies when the Fourier coefficients are known. An optimized foil or layer thickness in a winding may be found by applying the Fourier analysis over a range of thickness values. This paper presents a new formula for the optimum foil or layer thickness, without the need for Fourier coefficients and calculations at harmonic frequencies. The new formula requires the RMS value of the current waveform and the RMS value of its derivative. It is simple, straightforward and applies to any periodic waveshape.

Optimizing the AC resistance of multilayer transformer windings with arbitrary current waveforms

The two best-known methods for calculating high-frequency winding loss in round-wire windings-the Dowell method and the Ferreira method-give significantly different results at high frequency We apply 2-D finite-element method (FEM) simulations to evaluate the accuracy of each method for predicting proximity-effect losses We find that both methods can have substantial errors, exceeding 60% The Ferreira method, which is based on the exact Bessel-function solution for the eddy current in an isolated conducting cylinder subjected to a time-varying magnetic field, is found to be most accurate for loosely packed windings, whereas the Dowell method, which approximates winding layers comprising multiple turns of round wire with a rectangular conducting sheet, is most accurate for closely-packed windings To achieve higher accuracy than is possible with either method alone, we introduce a new formula, based on modifying the Dowell method Parameters in the new formula are chosen based on fitting our FEM simulation data By expressing the results in terms of normalized parameters, we construct a model that can be used to determine proximity-effect loss for any round-wire winding with error under 2%

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An improved calculation of proximity-effect loss in high-frequency windings of round conductors

This study presents an approximate model for multi-strand wire winding, including litz-wire winding. The proposed model is evaluated using Dowell's equation. The model takes into account the existence of proximity effect within the litz-wire bundle between the strands and between the bundles, as well the skin effect. The expressions for optimum strand diameter and number of strands at which minimum winding AC resistance is obtained for the litz-wire windings are derived. The boundary frequency between the low-frequency and the medium-frequency ranges are given for both litz-wire and solid-round wire inductors. Hence, the low-frequency ranges of both wire windings are determined. It is shown that litz-wire is better than the solid wire only in specific frequency range. The model has been verified by the measurements, and the theoretical results were in good agreement with those experimentally measured. Comparison of the theoretical predictions of the proposed approximate litz-wire model with models proposed in other publications and with experimental results is given.

Winding resistance of litz-wire and multi-strand inductors

IN any multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers. Eddy currents are set up in each layer in a direction to partly neutralize the magnetic intensities in the interior of the copper wire in each layer. As a result of the eddy-current losses in the copper, the effective resistance of the winding to the alternating current it carries may be many times its resistance to continuous currents.

Sidney C. Larson

Papers

Effective resistance to alternating currents of multilayer windings