Inductor Size vs. Q: A Dimensional Analysis

Equations are derived and curves are plotted showing the change in Q of an inductor as a function of a change in size. The approach used is to assume the existence of a model having a fixed inductance and operating under a fixed set of conditions. A change in size is introduced by a magnification-type enlargement or shrinkage of all outline dimensions by the factor "a." Assuming a constant effective permeability and readjusting the number of turns to maintain inductance constant results in the elimination of "n," the relative number of turns. With the further assumption of a constant copper space factor, the relative copper losses (in the absence of core losses) determine the relative Q as a function of relative size. Copper losses with skin effect and with solid conduction are considered. The same approach is used to derive equations of relative size vs relative Q for a pure core-loss inductor. For this purpose, core losses are approximated by a straight-line equation between the logarithm of flux density and the logarithm of core-loss density. This turns out to depend upon the properties of magnetic materials. Although under certain conditions core losses may actually go down as size goes down, the broader considerations (not treated mathematically in this paper) of combined copper and core losses make such a condition untenable. At least, the equations and curves can provide the basis for a judicious estimate of the effect of miniaturization on inductor Q in any specific application.