This letter presents an accurate magnetic coupling coefficient ( kk ) model for a through-silicon via (TSV)-based 3-D transformer. The kk 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.

中文翻译：

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基于 TSV 的 3D 变压器磁耦合系数精确建模


这封信介绍了基于硅通孔 (TSV) 的 3D 变压器的精确磁耦合系数 (kk) 模型。 kk 因子可以从自感和互感精确推导出来，自感和互感是通过基于物理几何的各种解析公式计算得出的。该模型的结果与Q3D提取器和高频结构模拟器（HFSS）的结果吻合较好，最大误差分别为3.8%和4.4%。基于 TSV 的变压器的等效电路模型用于进一步验证。电路模型获得的 S 参数与测量结果吻合良好。