13.3 SYNCHRONOUS MACHINES 583Equivalent Circuit of a Synchronous Machine
A review of the material about elementary synchronous machines presented in Section 13.1 is
very helpful at this stage to recall the principles of operation for synchronous machines.
To investigate the equivalent circuit of a synchronous machine, for the sake of simplicity, let
us begin by considering an unsaturated cylindrical-rotor synchronous machine. Effects of saliency
and magnetic saturation can be considered later. Since for the present we are only concerned with
the steady-state behavior of the machine, circuit parameters of the field and damper windings need
not be considered. The effect of the field winding, however, is taken care of by the flux produced
by the dc field excitation and the ac voltage generated by the field flux in the armature circuit.
Thus, letE ̄fbe the ac voltage, known asexcitation voltage, generated by the field flux. As for
the armature winding, under balanced conditions of operation we will analyze it on aper-phase
basis. The armature winding obviously has aresistanceRaand aleakage reactanceXlof the
armature per phase.
The armature currentI ̄aproduces the armature reaction flux, the effect of which can be
represented by an inductive reactanceXφ, known asmagnetizing reactanceorarmature reaction
reactance. Thus Figure 13.3.1 shows the equivalent circuits in phasor notation, with all per-phase
quantities, for a cylindrical-rotor synchronous machine working as either a generator or a motor.
The sum of the armature leakage reactance and the armature reaction reactance is known as the
synchronous reactance,
Xs=Xφ+Xl (13.3.1)andRa+jXsis called thesynchronous impedance Zs. Thus, the equivalent circuit for an
unsaturated cylindrical-rotor synchronous machine under balanced polyphase conditions reduces
to that shown in Figure 13.3.1(b), in which the machine is represented on a per-phase basis by
its excitation voltageE ̄fin series with the synchronous impedance. Note thatV ̄tis the terminal
per-phase rms voltage, usually taken as reference.
In all but small machines, the armature resistance is usually neglected except for its effect on
losses (and hence the efficiency) and heating. With this simplification, Figure 13.3.2 shows the
four possible cases of operation of a round-rotor synchronous machine, in which the following
relation holds:
V ̄t+jI ̄aXs=E ̄f (13.3.2)Observe that the motor armature current is taken in the direction opposite to that of the generator.
The machine is said to beoverexcitedwhen the magnitude of the excitation voltage exceeds
that of the terminal voltage; otherwise it is said to beunderexcited.The angleδbetween the
excitation voltageE ̄fand the terminal voltageV ̄tis known as thetorque angleorpower angleof
+−+−−Er(a)
+−Ef VtjXφ jXlIaRaMotorGeneratorIa+−
(b)+−Ef VtjXs = j(Xφ + XI)Ia
RaMotorGeneratorIaFigure 13.3.1Per-phase equivalent circuits of a cylindrical-rotor synchronous machine.