an investigation of dual stator winding induction machines

t
2bi
⎛ rsiLri
K iL
⎞ ⎛ ri
K iL
⎞ ⎛ mi rriLsi
K iL
⎞ ⎛ mi K iL
⎞
11
22
21
12 ri
= − ⎜ +
ei ri
ei
Di
D ⎟ ⋅ ⎜
⎜ω
− ω +
i
D ⎟ + ⎜ −
i Di
D ⎟ ⋅ ⎜ − ω
i D ⎟
⎝
⎠ ⎝
⎠ ⎝
⎠ ⎝ i ⎠
⎛ rsiLmi
K iL
⎞ mi K iL
⎛ ri rriLmi
K iL
⎞
11
22
21 ri K12iLmi
− ⎜
⎜−
−
Di
D ⎟ ⋅ +
i D ⎜
⎜−
+
i Di
D ⎟
⎝
⎠ ⎝
i ⎠ Di
⎛ rsiLri
K iL
⎞
⎛
11 ri
rriLsi
K 21iL
⎞ mi rriLsiK12iLri
rriK12iL
= − ⎜ + ( ei ri ) ⋅ ( ei ) +
− 2
2
Di
D ⎟ ⋅ ω − ω − ⎜ −
i
Di
D ⎟ ω
⎝
⎠
⎝
i ⎠
Di
Di
⎛ rsiLri
K11iL
⎞
⎛
ri
rriLsi
K 21iL
⎞ mi rriK12i
= − ⎜ + ⋅ ( ei − ri ) −
⋅ ( ei ) +
Di
D ⎟ ω ω ⎜ −
i
Di
D ⎟ ω
⎝
⎠
⎝
i ⎠ Di
(10.80)
The observer gain design is based on the characteristic equation of the observer so
that the selected observer gains ensure the stability within the entire possible operating
speed range. It should be noted that both the constant rotor speed with variable slip
frequency condition and the constant slip frequency with variable rotor speed condition
should be considered in the stability analysis.
Two pole placement methods have been used to design the observer gain. The first
one is Butterworth method, which has been used in the controller design and has been
described in [10.39]. The other method is called pole-shift method, which has been
proposed in [10.25] and has also been adopted for observer gain design in [10.34]. The
results from both methods have been compared and the Butterworth method is finally
used to design the observer gain.
10.7.1 Butterworth Method
The Butterworth polynomial is used to locate the poles of the characteristic function.
Since the complex form is used in the transfer function, there are only two poles.
However the complex conjugates of these two poles yield other two poles [10.39]. The
375
2
mi