Theory of cylindrical rotor machine

 Let us consider the phasor diagram for alternator as shown in the Fig. 1.

Fig. 1

       Let          E = E.M.F. induced in each phase

                      V = Terminal voltage

                      Φ = Phase angle between voltage and current 

                      δ = Power angle 

                      R_a  = Resistance of armature 

                      X_s  = Synchronous reatance of alternator.

.^..                   tanθ = X_s/R_a 

.^..                   θ = tan^-1 (X_s/R_a)

                     α = Φ + δ

       Power output per phase = V . I cos (V ^ I)

                                            = VI cos

       Power input,   P_i = EI cos (E ^ I)

                                 = E I cos (Φ + δ) = E I cosα = I (E cos α)

                                 = I (V cos Φ + I R_a)

                                 = VI cosΦ + I² R_a 

       The voltage equation of alternator is given by,

        P_i  is the power converted internally into electrical power from mechanical power which is given by,

        P_i  = E^o . I

Note : With increase in δ power increases and with decrease in δ power decreases. Power in case of synchronous machines depends on the angle δ. This angle δ is called power angle.

In case of large synchronous machines,  X_s >>> R

.^..                θ = tan^-1(X_s/R)= 90^o (if resistance is neglected)

                   θ = 90^o

       Substituting this value in above expression for power