Alternative Expression for Power Developed by a Synchronous Motor

 Consider the phasor diagram of a synchronous motor running on leading power factor cosΦ as shown in the Fig. 1.

Fig. 1

       The line CD is drawn at an angle θ to AB.

       The lines AC and DE are perpendicular to CD and AE.

       and angle between AB = E_bph and I_aph is also ψ.

      
The mechanical per phase power developed is given by,

       In triangle OBD,

                  BD = OB cosψ = I_a  Zs cosψ

                  OD = OB sin ψ = Ia  Zs sin

       Now    BD = CD - BC = AE - BC

      Substituting in (2),

                   I_a  Zs cosψ = V_ph  cos (θ-δ) - E_b  cosθ

       All values are per phase values

       Substituting (3) in (1),

       This is the expression for the mechanical power developed interms of the load angle δ and the internal machine angle θ, for constant voltage V_ph  and constant E_ph i.e. excitation.