Analysis of Frequency- and Temperature-Dependent Substrate Eddy Currents in On-Chip Spiral Inductors Using the Complex Image Method

Using the complex image method (CIM), we have analyzed the frequency and temperature dependencies of substrate eddy currents for single-ended and differential spiral inductors on a lossy silicon substrate. From our analysis, we have derived a set of accurate closed-form expressions for calculating inductances and substrate losses due to substrate eddy currents. Here, we propose a frequency-dependent eleven-element equivalent circuit model based on these formulas. We established the validity of the model by comparing the simulated and measured results, which are in good agreement.