Effect of finite thickness of magnetic substrate on planar inductors

The effect of finite thickness of magnetic substrate or film on the inductance of planar inductors is studied using the method of current images. The results obtained reduce to previous results in the limit of infinite thickness. It is found that the thickness and permeability play complementary roles in the enhancement of inductance, with the normalized inductance determined roughly by the product of permeability and thickness in the most interesting region of parameter space. The results derived allow determination of the required minimum thickness of the substrate which will produce maximum inductance enhancement for a given permeability of the substrate material. The same results can also be used to determine the permeability of a magnetic film or substrate quickly and are directly applicable to the design of planar inductors in cofireable ceramic, hybrid, and thin-film technologies