Page 491 - Handbook of Electrical Engineering

GENERALISED THEORY OF ELECTRICAL MACHINES 481

Leakage reactance of the d-axis rotor damping winding.
X lkd
Leakage reactance of the q-axis rotor damping winding.
X lkq
Leakage reactance of the d-axis rotor field winding.
X lfd
Reactance of a component connected in series with the
X ext
stator winding, one in each phase.
d) Frequencies

f n nominal cyclic frequency of the power system in cycles per second or hertz.
ω n nominal angular frequency of the power system in radians per
second = 2πf n .
f any cyclic frequency of the power system within its normal operating range
in cycles per second.
ω any angular frequency of the power system within its normal operating range
in radius per second = 2πf .

Note: ω and f could be the frequency of the system when speed governor action is present,
or the fundamental frequency of a variable frequency e.g. as used in the speed control of
synchronous or induction motors.

e) Leakage inductances are due to flux which only links with its own winding and is caused by its
own current.
f) Mutual inductances are due to flux which links two windings that share the same magnetic circuit.
The same flux is created by either of the currents in the two windings. Mutual inductances are
defined between two windings, not three or more, even though there may be several windings
sharing the same magnetic circuit e.g. the d-axis of a synchronous machine, a three-winding three-
phase transformer. The mutual inductance between the three pairs of windings in a magnetic circuit
of three windings can often be assumed to be equal e.g. M 12 = M 13 = M 23 = M.
g) Self or total inductances are the addition of the leakage inductance of a particular winding and
the mutual inductance between it and another winding. See Figure 20.1.

h) Reactances at frequencies other than the nominal frequency.
Each of the reactances X md through X ext and others yet to be defined or that exist in the power
system could be modified as the frequency of the system changes, e.g. during a long disturbance
such as starting a large motor with a high inertia load. The necessary modification is simply to
apply the ratio of the disturbance frequency (ω) to the nominal frequency (ω n ) as a multiplying
factor e.g., X md changes to X md ω/ω n . This modification applies especially to machines supplied
from variable frequency power sources. In systems where the frequency deviations are small
during a disturbance the modification is usually ignored. The difference in computed results will
be small compared with the tolerances on the data used in the program.
i) Flux linkages
A coil or winding carrying a current I will produce a proportional amount of flux ∅, provided the
permeability of the magnetic circuit remains constant for all values of the current. The winding
will usually consist of N closed loops of conductor connected in series, with each loop being one
turn. Hence the winding has N turns. The total amount of flux linking the N turns of the winding