This paper presents a knowledge-aided automated synthesis method to design broadband power amplifiers (PAs) integrated with transformer-coupled resonators (TCRs). Different from the traditional TCR-based PA designs that focus solely on broadband load-pull matching or TCR transimpedance, the intrinsic design knowledge of the proposed method utilizes the mechanism of joint-design between PA active circuits and TCR networks. By combining this knowledge with an optimization-oriented genetic algorithm (GA), an initial TCR solution that achieves low-ripple and broadband PA output response can be rapidly synthesized, in the sense that the superior initial population, not as random as that in traditional GA, can improve the GA convergence and reduce the algorithm iterations. Simulated results show that this work achieves about 4 times speed improvement in TCR synthesis than the traditional GA. The proposed method can efficiently synthesize the optimal TCR parameters for multi-stage PA to obtain the user-defined output power within a target bandwidth. As a proof of this work, a cascode-based PA for FMCW radar application is designed in 110-nm CMOS process, aiming to realize a peak 14-dBm output power and 50% 1-dB fractional bandwidth (FBW). The measured PA large-signal performance versus frequency shows a wide 1-dB BW for ${P} {_{\text {sat}}}$ over 8.5-14.5 GHz (52.2% FBW) with 15.7-dBm peak value and 27% peak PAEmax at 11.5 GHz, demonstrating the efficacy of the proposed broadband synthesis mothod for PAs.