Accurate Magnetic Coupling Coefficient Modeling of 3-D Transformer Based on TSV

This letter presents an accurate magnetic coupling coefficient ( $k$ ) model for a through-silicon via (TSV)-based 3-D transformer. The $k$ factor can be precisely derived from the self-inductance and mutual inductance, which are calculated by various analytical formulas based on physical geometries. The results of this model correspond well with those of Q3D extractor and high-frequency structural simulator (HFSS), with maximum errors of 3.8% and 4.4%, respectively. An equivalent circuit model of a TSV-based transformer is used for further verification. The S-parameters obtained by the circuit model match well with the measurements.