An Improved Six-Port Equivalent-Circuit Model for Millimeter-Wave On-Chip Transformers With Accurate Coupling Factor Modeling

This article presents an improved equivalent-circuit model for millimeter-wave six-port transformers on silicon with accurate coupling factor modeling. The lossy silicon substrate eddy current effects have a significant impact on the mutual resistive coupling factor ( $k_{\mathrm {re}}$ ) of the primary and secondary coils, which in turn affects the maximum available gain ( $G_{\mathrm {max}}$ ) of the transformer. To characterize eddy currents’ loss in the silicon substrate, “effective substrate current loops,” which are magnetically coupled by the primary and secondary coils simultaneously, are presented. The mutual inductances and coupling capacitances are used to model coupling effects between all segments of the coils. The proposed model can accurately predict the frequency-dependent mutual reactive coupling factor ( $k_{\mathrm {im}}$ ) and mutual resistive coupling factor ( $k_{\mathrm {re}}$ ). A corresponding methodology to parameter extraction based on electromagnetic (EM) simulations is demonstrated. A two-way current-combining-based power amplifier (PA) is presented for fifth-generation (5G) communication in the 65-nm CMOS process. Using the developed transformer model, the baluns and inter-stage transformers in the PA are modeled and analyzed. The power gain of the PA is 22.8 dB, the saturated output power ( $P_{\mathrm {sat}}$ ) is 18.7 dBm, and the power-added efficiency (PAE) is 21.88%.