Magnetless rotary transformer excitation systems enable fully brushless operation of electrically excited synchronous machines (EESMs) and offer a scalable alternative to permanent-magnet drives. This review presents a control-oriented synthesis of rotary-transformer-excited EESMs by linking machine topology, modeling methods, resonant compensation networks, inverter architectures, and excitation-control strategies within a unified framework. The paper outlines a complete modeling chain, from fundamental-harmonic approximation to dynamic direct-quadrature-axis and coupled electromagnetic-thermal analysis, to support accurate field-current regulation and efficiency optimization. Inverter and compensation networks that enable soft switching are compared in terms of controllability, voltage regulation, efficiency, and implementation complexity. Emerging self-excitation methods, including common-mode voltage injection, harmonic excitation, and carrier-phase modulation are also assessed with respect to implementation simplicity and control flexibility. The review concludes by identifying key challenges in sensorless excitation control, thermal coupling, and high-frequency operation, and by highlighting research directions that can further enhance the performance, reliability, and scalability of brushless EESM drives.