Inductance and quality-factor evaluation of planar lumped inductors in a multilayer configuration

Integral representations for the self and mutual inductance of planar loops on a multilayered structure are derived. The integrals are of the Sommerfeld type and can be easily evaluated under the quasi-static approximation which is validated by the small dimensions relative to a wavelength. Enhancement of loop inductance by inclusion of a magnetic layer is investigated. It is shown that such a layer can increase the inductance by a percentage which has the upper limit of [(μ_r-1)/(μ_r+1)]×100%, where μ_r is the relative permeability of the layer. A study is also made on the inductor quality factor (Q) as affected by losses caused by finite electrical conductivity of the magnetic layer and the underlying substrate