Actuator-Oriented Analytical Efficiency Modelling of Planetary Gear Transmissions Under Reduction and Multiplication Modes

Planetary gear trains (PGTs) are core transmission units in high-ratio mechanical systems, including vehicle drivetrains, industrial reducers, robotic joints, and servo mechanisms, where transmission efficiency affects power demand, thermal loading, actuator sizing, housing reaction, and self-locking risk. This study develops a unified analytical framework for estimating the efficiency of involute PGTs used in actuator-oriented systems. The formulation covers common 2k–h and k–h–v layouts, single- and double-rim satellites, reduction and multiplication modes, and different assignments of driving, driven, and stationary members. Starting from power balance, Willis’ relation, and torque equilibrium, compact closed-form expressions are derived by representing mesh losses through the efficiency of the reverted mechanism. The resulting baselines quantify the effects of transmission ratio, mesh efficiency, operating mode, self-locking boundary, and housing-reaction torque. The results show that multiplication mode becomes more sensitive to loss amplification as mesh efficiency decreases, especially in k–h–v configurations approaching the self-locking boundary. Double-rim 2k–h configurations also exhibit branch-dependent efficiency and housing-torque behavior. For representative duty-cycle calculations, increasing gear-train efficiency from 0.85 to 0.92 reduces motion energy by approximately 7–8%. The framework provides a lightweight tool for early-stage actuator-transmission screening before detailed modelling, dynamic simulation, or prototype testing.