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# ABSTRACT

Document source : repository.lib.ncsu.edu

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123
The stator-to-rotor and the rotor-to-stator inductances are mutual inductances and are a
function of the rotor position, as shown in Equation 11g.
[ ]
( )
( )
( )
-
+
+
-
-
+
=
=
r
r
r
r
r
r
r
r
r
sr
t
abc
rs
abc
sr
L
cos
3
2
cos
3
2
cos
3
2
cos
cos
3
2
cos
3
2
cos
3
2
cos
cos
L
L
(11g)
In Equations 11e and 11f, L
ls
is the per phase stator winding leakage inductance, L
lr
is the per
phase rotor winding leakage inductance, L
ss
is the self-inductance of the stator winding, L
rr
is
the self-inductance of the rotor winding, L
sm
is the mutual inductance between stator
windings, L
rm
is the mutual inductance between rotor windings, L
sr
is the peak value of the
stator-to-rotor mutual inductance, and
r
is the angle between the stator a-phase axis and the
rotor a-phase axis, as shown in Figure 18. The superscript t in Equation 11g represents the
transpose of the matrix.
Solving Equations 9 and 10 is extremely complex due to the dependence on the rotor
position. Stanley's transformation is well suited for this case, since the transformation
eliminates the dependence on the rotor position.
A.3.2 Induction Motor Equations in dq0 Frame
The stationary reference frame described by Stanley is the reference frame of choice for
induction motor transient analysis. In the stationary reference frame, the q-axis and the as-
axis shown in Figure 18 will be aligned, since
= 0. The voltage, current, and flux linkage
quantities may be transformed to the stationary dq0 frame by applying the transformation of
Equation 1 and Equation 8. The stator voltage equations in matrix form become
( )
( )
0
1
0
1
0
qd
s
s
abc
s
s
qd
s
s
s
qd
s
dt
d
i
K
r
K
K
K
v
-
-
+
=
(12)
Applying the transformations and re-writing Equation 12 in expanded form, the stator
voltage equations in the stationary reference frame become

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Summary :

## [ ] ( ) ( ) ( ) - + + - - + = = r r r r r r r r r sr t abc rs abc sr L cos 3 2 cos 3 2 cos 3 2 cos cos 3 2 cos 3 2 cos 3 2 cos cos L L (11g) In Equations 11e and 11f, L ls is the per phase stator winding leakage inductance, L lr is the per phase rotor winding leakage inductance, L ss is the self-inductance of the stator winding, L rr is the self-inductance of the rotor winding, L sm is the mutual inductance between stator windings, L rm is the mutual inductance between rotor windings, L sr is the peak value of the stator-to-rotor mutual inductance, and r is the angle between the stator a-phase axis and the rotor a-phase axis, as shown in Figure 18.

Tags : cos,rotor,stator,frame,inductance,equations,equation,mutual,winding,reference,stationary,transformation,between
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