Analytical Modeling of High-Frequency Winding Loss in Round-Wire Toroidal Inductors

Abstract

Toroidal inductors are present in many different industrial applications, thus, still receive researchers' attention. AC winding loss in these inductors have become a major issue in the design process, since switching frequency is being continuously increased in power electronic converters. Finite element analysis software or analytical models such as Dowell's are the main existing alternatives for their calculation. However, the first one employs too much time if different designs are to be evaluated and the second one lacks accuracy when applied to toroidal inductor windings. Looking for an alternative that overcomes these drawbacks, this paper proposes an accurate, easy-to-use analytical model, specifically formulated for calculating high-frequency winding loss in round-wire toroidal inductors.

Analytical modelling of high‐frequency losses in toroidal inductors

“…A = D core /d 0 is the ratio coefficient of the core to the diameter of the round wire. λ can be defined in terms of the number of winding layers, geometrical parameters of the core, and the round wire as follows [27].…”

“…Therefore, a modified Ferreira's model considering the porosity factor was proposed by Bartoli et al. [27]. $$$$\begin{equation}F = \dfrac{\xi }{2}\left[ \def\eqcellsep{&}\begin{array}{l} \dfrac{{ber\xi bei^{\prime}\xi - bei\xi ber^{\prime}\xi }}{{be{{r^{\prime}}}^2\xi + be{{i^{\prime}}}^2\xi }}\\[9pt] - 2\pi {\eta }^2\dfrac{{4{m}^2 - 1}}{3}\dfrac{{be{r}_2\xi ber^{\prime}\xi + be{i}_2\xi bei^{\prime}\xi }}{{be{r}^2\xi + be{i}^2\xi }} \end{array} \right]\end{equation}$$$$…”

“…David proposed a high‐frequency winding loss model based on the Dowell method applicable to toroidal inductors. The model equates the toroidal winding layer to a rectangular copper conductor and introduces a porosity parameter to reduce the estimation error of the model [27]. $$$$\begin{equation}F = \dfrac{{\alpha ^{\prime}}}{2}\left[ \def\eqcellsep{&}\begin{array}{l} \dfrac{{\sinh 2\alpha ^{\prime} + \sin 2\alpha ^{\prime}}}{{\cosh 2\alpha ^{\prime} - \cos 2\alpha ^{\prime}}}\\[9pt] + \lambda \dfrac{{\sinh \alpha ^{\prime} - \sin \alpha ^{\prime}}}{{\cosh \alpha ^{\prime} + \cos \alpha ^{\prime}}} \end{array} \right]\end{equation}$$$$…”

“…The porosity of the winding is contained in α'. The porosity of the toroidal inductor is non‐linearly related to the number of winding layers, and the average porosity is used to describe the porosity of the winding layers [27]. $$$$\begin{equation} {\alpha}^{\prime}={d}_{0}/\sqrt{\pi f{\mu}_{0}\sigma {\eta}_{d}}\quad \mathrm{and}\quad {\eta}_{d}=\frac{n}{2\sqrt{\pi}m\left(A-m\right)} \end{equation}$$$$where n is the total number of turns of the inner winding.…”

“…λ can be defined in terms of the number of winding layers, geometrical parameters of the core, and the round wire as follows [27]. $$$$\begin{eqnarray} \lambda &=& \dfrac{{m\left( {4{A}^2 - 9Am + 5{m}^2} \right)}}{{8(A - m)}} - \dfrac{{{{(A - m)}}^2 + 3}}{8}\nonumber \\ &&+ \displaystyle\sum_{k = 0}^{m - 1} \dfrac{1}{{(A - 1 - 2k)}}\left[\dfrac{{{{\left( {{A}^2 - 1} \right)}}^2}}{{8m(A - m)}} + 2m(A - m)- {A}^2 + 1\right]\nonumber\\ \end{eqnarray}$$$$…”