The analytical expression of the efficiency of planetary gear trains

Two procedures are presented for obtaining the analytical expression for the efficiency of any planetary gear train. The first employs the speed and torque equations and the concept of virtual gear teeth ratios. The second is based on an relationship between the gearing power and the speed ratio. This relationship, which is satisfied by any spur gear train, allows one to obtain the efficiency expression in a surprisingly simple manner. Two proofs of that relationship are presented.