porosity_factor

# Porosity factor

 Stan Zurek, Porosity factor, Encyclopedia-Magnetica.com, {accessed 2021-01-28}

Porosity factor or layer factor1) (also winding porosity) often referred to by the Greek letter η - a factor related to the distance between the wires in a given winding. It is used in analysis of proximity loss in windings.

 Support us with just $1.00 through PayPal or a credit card: Porosity factor reduces calculation errors of proximity loss, especially at lower frequencies2) S. Zurek, Encyclopedia Magnetica, CC-BY-4.0 At high frequencies the errors remain high for most methods3) S. Zurek, Encyclopedia Magnetica, CC-BY-4.0 The introduction of the factor improves accuracy of proximity loss calculations, but only when the conductors are closely packed. Introduction of the porosity correction only reduces the calculation errors. At lower frequencies the calculations follow closely the expected curve, but at high frequency (when wire diameter is much greater that the skin depth) the errors still remain relatively high for most of the calculation methods (see graph). However, the comparison is made to the 2D simulation results, and not to experimentally measured data, so the absolute accuracy of using the porosity factor is not exactly known. The porosity factor was first proposed by P.L. Dowell.4)5) In the analysis Dowell replaced round with square wires, which were then brought together to represent an equivalent “foil”. Such hypothetical foil is then stretched so that it has the same thickness and width as the original winding. The porosity factor allows matching the DC resistance value to the original winding, and is defined as:  (1) $$\eta_{Dowell} = \frac{N·a}{b}$$ (unitless) where: N - number of square conductors, a - width of an individual conductor, b - width of the winding. A different approach was later taken by M. Bartoli, et al.6) who defined the porosity factor as:  (2) $$\eta_{Bartoli} = \frac{d}{t} · \sqrt{\frac{\pi}{4}}$$ (unitless) where: d - diameter of round wire, t - distance between the centres of adjacent conductors.7) The definition depends on the kind of assumptions made in a given analytical model. Once defined the porosity factor can be then used in a much more complex equations, which are used for calculation of the increase of AC resistance (as compared to the DC resistance). For instance, Xi Nan and Sullivan8) use the following equations:  (3) $$R_{ac} = R_{dc} · \zeta · \left( \frac{sinh(2 · \zeta) + sin(2 · \zeta)}{cosh(2 · \zeta) - cos(2 · \zeta)} + \frac{2}{3} · (m^2 - 1) · \frac{sinh(2 · \zeta) - sin(2 · \zeta)}{cosh(2 · \zeta) + cos(2 · \zeta)} \right)$$ (Ω) where: m - number of layers,$\zeta = \frac{\pi · d}{2 · \delta} · \sqrt{\eta_{Dowell}}\$ (in which: δ - skin depth, d - wire diameter)

The equation (3) is just an example for a given type of analysis. Other authors, (for instance Ferreira 9)), can use differently formulated equations. This would also change depending on the system of coordinates used.

## References

4) P.L. Dowell, Effects of eddy currents in transformer windings, Proceedings of the IEE, vol. 113 (8), Aug 1966, p. 1387
6) Bartoli, M., Noferi, N., Reatti A., (1995), Modelling winding losses in high-frequency power inductors, Journal of Circuits, Systems and Computers, Vol.5, no. 4, pp. 607
9) J. A. Ferreira, Improved analytical modeling of conductive losses in magnetic components, IEEE Transactions on Power Electronics, Vol. 9 (1), Jan 1994, p. 127