Page 493 - Handbook of Electrical Engineering

GENERALISED THEORY OF ELECTRICAL MACHINES 483

produces the flux linkages is kept constant but its winding is rotated at an angular velocity ω r then
the emf induced is,
e = ω r ψ volts

This process is called ‘induction by rotating action’ or ‘rotationally induced emf’.
These two processes are fundamental to the induction of emfs in all the windings of a motor
or generator.

20.2.1 Basic Mathematical Transformations

The generalised theory when applied in a suitable manner has the very convenient effect of removing
the sinusoidal variations that are at the frequency of the power system. The frequency variations are
those which are associated with the instantaneous currents, voltages and emfs. Their removal occurs,
when these variables are transformed to the d and q axes. In effect the d and q-axes stator currents
and voltages become envelope values of their corresponding stator three-phase sinusoidal quantities.
This is very advantageous when digital computers are used to solve single machine and especially
multi-machine transient problems. This is similar to using rms quantities in circuit analysis instead of
instantaneous quantities. The labour and calculation times are greatly reduced. Two commonly used
matrix transformations for currents, voltages and emfs are:-
a) Transform a, b, c variables to d, q, o variables

   ◦ ◦   
v d = cos θ cos(θ − 120 ) cos(θ + 120 ) v a
◦
◦
= k sin θ sin(θ − 120 ) sin(θ + 120 ) (20.2)
v q v b
     
v o = 0.5 0.5 0.5 v c
b) Transform d, q, o variables to a, b, c variables
     
v a = cos θ sin θ 1.0 v d
◦
◦
cos(θ − 120 ) sin(θ − 120 ) 1.0 (20.3)
v b = k i v q
     
◦
◦
v c = cos(θ + 120 ) sin(θ + 120 ) 1.0 v c
Where (20.3) is the inverse transformation of (20.2) and the lower-case letter ‘v’represent
the instantaneous variation of the corresponding peak value of voltage ‘v’. The same transfor-
mations apply to the instantaneous currents i a through i o . The suffices ‘o’ are attached to the
zero sequence instantaneous quantities, which are essentially added to the matrices to make them
invertable. Under balanced circuit conditions and balanced disturbances the zero sequence com-
ponents have no effect on the computed results. Their use in the ‘generalised theory’ to study
line-to-ground faults and single-phase unbalanced loading should be approached with some cau-
tion. The combining of the symmetrical component theory with the ‘general theory’ should be
undertaken with care, the additional mathematics becomes formidable, see Reference 5, Chapters 9
and 10, Reference 13, and Reference 3, Chapter X.
The two constants k and k i have different values in the literature and occur as interrelated
√ √
pairs e.g. where k = 2/3, k i = 1.0 see References 5, 7, 8 and 13, when k = 2/3, k i = 2/3and