an investigation of dual stator winding induction machines

X
i
i
= X
Y = Y
i
i ( σ , ω)
( σ , ω)
are functions defined by the complex variable expression s = σ + jω
and
i
s X i +
i
389
(10.112)
= jY
(10.113)
X i and Y i can be obtained by applying the recurrence formulas as:
X
Y
i+
2
i+
2
− 2σ
X
− 2σ
Y
i+
1
i+
1
2
+ ω X
2
+ ω Y = 0
i
i
= 0
where, X 0 = 1,
1 0 = X , Y 0 = 0 , Y 01 = ω .
(10.114)
If the value of σ is zero in (10.111-10.114), the surface determined by (10.111)
corresponds to the zeros of F ( s)
that have zero real parts.
If the system parameters in the space are chosen from the set ( n,
0)
D , the stability of a
linear system with characteristic polynomial F ( s)
is assured. The points on the boundary
of the set ( n,
0)
D will satisfy the condition.
The general definition of D-decomposition is within an n dimension parameter
space, however it is convenient to apply it to two-parameter problems since the stability
boundary of two-parameter problems can be graphically shown. In the case of two
parameters, the boundary of the stable and unstable regions can be determined from
(10.110) and (10.111). The conditions for (10.111) are equivalent to the F ( s)
with
substituting s = jω
. Certain D-decomposition curve can be drawn by varying ω , in
which the two unknown parameters are the real and imaginary axis. The curve will divide
the system parameter plane into the stable and unstable regions. The stable region is
determined by following a certain shading rule [10.66]. In the s-plane, the stable area is
on the left half plane while the corresponding stable area in the parameter plane is