Voltage Equation of an Alternator

In a d.c. generators, we have seen that due to the armature resistance drop and brush drop it is not possible to have all the induced e.m.f. available across the load. The voltage available to the load is called terminal voltage. The concept is same in case of alternators. The entire induced e.m.f. can not be made available to the load due to the various internal voltage drops. So the voltage available to the load is called terminal voltage denoted as. In case of three phase alternators as all the phases are identical, the equations and the phasor diagrams are expressed on per phase basis.
So if Eph is the induced e.m.f. per phase in the alternator, there are following voltage drops occur in an alternator.
i) The drop across armature resistance Ia Ra both Ia and Ra are per phase values.
ii) The drop across synchronous reactance  Ia Xs, both  Ia and Xs are per phase values.
After supplying these drops, the remaining voltage of Eph is available as the terminal voltage Vph.
Note : Now drop Ia Ra is always in phase with Ia due to a resistive drop while current Ia lags by 90o with respect to drop  Ia Xs as it is a drop across purely inductive reactance.
Hence all these quantities can not be added or subtracted algebraically but must be added or subtracted vectorially considering their individual phases. But we can write a voltage equation in its phasor from as,

This is called voltage equation of an alternator.
From this equation, we can draw the phasor diagram for various load power factor conditions and establish the relationship between Eph and Vph, in terms of armature current i.e. load current and the power factor cos(Φ).