Optimising planar inductors

A numerical simulation program, QvalueC, was developed for the design of planar inductors on lossy substrates in two-dimensional geometry. Integral equations are used for the estimation of the overall impedance of a circular inductor at a given frequency. The program computes the current density distribution in the wire that has a finite electric conductivity. This so called magnetic computation gives the series inductance and series resistance of the coil. By using Green functions, integral equations are derived for equivalent surface charges that represent the conductor in the electric problem. From the surface charges the parallel capacitance and the parallel resistance are obtained. The code is written with the view to designing planar coils in four dielectric layers, but also solenoids with multilayered conductors can be computed. The approach taken allows an accurate simulation of the following dissipation mechanisms affecting the Q-value: ohmic losses in the wire, skin-depth effect, proximity (eddy current) effect, and losses in supporting dielectric but conductive layers via capacitive coupling. Simulated and measured results from planar inductors on lossy silicon and other substrates show a good agreement.