Electrical Power Systems

Synchronous Machine: Steady State and Transient Operations 81
where Emax = N fow = 2pf Nfo ...(4.3)
Therefore, the rms value of the generated voltage is
Erms = 4.44 f Nfo ...(4.4)
Actually, the armature coil of each phase is distributed in a number of slots and the emfs
induced in different slots are not in phase and hence their phasor sum is less than their
numerical sum. Therefore, a reduction factor Kw, called the winding factor, must be considered.
or most of the machines Kw varies from 0.85 to 0.95. Therefore, for a distributed phase
winding,
Erms = 4.44 Kw f Nfo ...(4.5)
The frequency of the armature voltage is given by
PI n
f = HG 2KJ60 sI
HG KJ
...(4.6)
where ns is the rotor speed in rpm, referred to as synchronous speed.
Under normal operating conditions, three phase armature currents are balanced. Assuming
current in phase a is lagging the generated emf Ea by an angle f, which is indicated by line pq
in ig. 4.1, the instantaneous armature currents are
HG
HG
I a = I max sin(wt – f)
2p
Ib = Imax sin wt - f -
3
I
KJ
I
KJ
...(4.7)
4p
Ic = Imax sin wt - f -
3
rom eqn. (4.2), it is seen that the generated emf Ea is maximum when rotor magnetic axis
is under phase a. In ig. 4.1, when line pq reaches the axis of coil aa¢, current in phase a reaches
it maximum value. At any instant of time, each phase winding of synchronous machine produces
a sinusoidally distributed mmf wave with its peak along the axis of the phase winding. These
mmf amplitudes are proportional to the instantaneous value of the phase current, i.e.,
a = K.Ia b = K.Ib ...(4.8)
c = K.Ic Using eqns. (4.8) and (4.7) we get,
a = m sin (wt – f)
HG
HG
2p
b = m sin wt - f -
3
4p
c = m sin wt - f -
3
I
KJ
I
KJ
...(4.9)
where m = K.Imax ...(4.10)
where K is proportional to the number of armature turns per phase and is a function of the
winding type. Now the resultant mmf may be obtained by projecting these mmf on line pq and
obtain the resultant in-phase and quadrature-phase components. The resultant in-phase
components are