Direct and Quadrature Axis Synchronous Reactance

Blondel's Two Reaction Theory (Theory of Salient Pole Machine) : part 2 

We know that, the armature reaction flux Φ_AR has two components, Φ_d along direct axis and Φ_q along quadrature axis. These fluxes are proportional to the respective m.m.f. magnitudes and the permeance of the flux path oriented along the respective axes.

.^..                          Φ_d  = P_d F_d

            where        P_d = permeance alomng the direct axis

       Permeance is the reciprocal of reluctance and indicates ease with which flux can travel along the path.

       But                 F_d = m.m.f. = K_ar I_d in phase with  I_d 

       The m.m.f. is always proportional to current. While K_ar is the armature reaction coefficient.

.^..                          Φ_d  = P_d K_ar I_d 

       Similarly          Φ_q = P_q K_ar I_q

       As the reluctance along direct axis is less than that along quadrature axis, the permeance  P_d along direct axis is more than that along quadrature axis, (P_d < P_q ).

        Let E_d and E_q be the induced e.m.f.s due to the fluxes Φ_d  and Φ_q respectively. Now E_d lags Φ_d  by 90^o while E_q  lags Φ_q by 90^o .

       where K_e  = e.m.f. constant of armature winding

       The resultant e.m.f. is the phasor sum of E_f, E_d  and E_q.

       Substituting expressions for Φ_d  and Φ_q 

       Now       X_ard  = Equivalent reactance corresponding to the d-axis component of armature reaction

                               = K_e  P_d  K_ar 

       and         X_arq    = Equivalent reactance corresponding to the q-axis component of armature reaction

                                =  K_e  P_q  K_ar 

       For a realistic alternator we know that the voltage equation is,

        where       V_t = terminal voltage

                         X_L  = leakage reactance

       Substituting in expression for Ē_R ,

        where  X_d    = d-axis synchronous reactance = X_L  + X_ard                                   .............(2)

        and      X_q    = q-axis synchronous reactance =  X_L  +  X_arq                                     .........(3)

It can be seen from the above equation that the terminal voltage V_t is nothing but the voltage left after deducing ohmic drop I_a R_a, the reactive drop I_d X_d in quadrature with I_d and the reactive drop I_q X_q  in quadrature with I_d, from the total e.m.f. E_f.

The phasor diagram corresponding to the equation (1) can be shown as in the Fig. 1. The current I_a lags terminal voltage V_t by Φ. Then add I_a R_a in phase with I_a to V_t. The drop I_d X_d leads I_d by 90^o as in case purely reactive circuit current lags voltage by 90^o i.e. voltage leads current by 90^o . Similarly the drop I_q X_q  leads X_q  by 90^o . The total e.m.f. is E_f.