Electrical Power Systems

92 Electrical Power Systems
Similar equivalent circuits can be obtained for reactances along the quadrature axis. These
reactances, x² q, x¢ q, and x q may be considered for cases when the circuit resistance results in a
power factor above zero and the armature reaction is not totally on the direct axis.
The armature current following sudden short circuit to the armature of an initially unloaded
machine can be expressed as:
Where
i a (t) = 2
E
o
L
NMHG
I
HG
- t t¢¢ - t t¢
I
d d
1 1 1 1 1
e
e
wt
Xd¢¢ Xd
Xd Xd
X d
- +
¢ KJ ¢ - + sin( + d)
...(4.35)
KJ P
t² d = direct axis short-circuit subtransient time constant
t¢ d = direct axis short-circuit transient time constant.
4.8 DC COMPONENTS O STATOR CURRENTS
The expression for the armature current as given by eqn. (4.35), the unidirectional transient
component has not been considered. As we know from the consideration of the simple R-L
circuit, there will be a dc offset depending on when the voltage is applied. Similarly, in a
synchronous machine, the dc offset component depends on the instantaneous value of the stator
voltage at the time of the short circuit. The time constants associated with the decay of the dc
component of the stator current is defined as the armature short circuit time constant, t a .
Most of the decay of the dc component current occurs during the subtransient period and
because of this reason, the average value of the direct axis and quadrature axis subtransient
reactances is used for calculating t a . It’s approximate value is given by,
( Xd¢¢ + Xq¢¢
)
ta =
2 Ra
Typical value of ta is around 0.045 to 0.18 second.
The dc component current for phase ‘a’ is given by
idc = 2 Eo
Xd¢¢
sin. d e
-t/
ta
Combination of eqns. (4.35) and (4.37) will give an asymmetrical wave form, i.e.,
i asy (t) = 2
E
o
L
NMHG
I
HG
1 1 1 1 1
X X e
X X e
t
t
d¢¢ d
d d X
-
+
¢ KJ ¢ -
+ KJ
I
O
QP
O
Q
Eo
sin( wt + d) + 2 sin d.
e
X ¢¢
...(4.36)
...(4.37)
- / td¢¢ - / td¢ -t/
ta
d
d
The maximum possible initial magnitude of the dc component current is
max Eo
idc = 2
Xd
¢¢
Therefore, the maximum rms current at the beginning of the short circuit is
...(4.38)
...(4.39)