SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

2
0
3
xA
1 1 1 x
x
B
k
3 3
x
x
C
0 0
(3.89)
Now that we have removed the linear dependence the transformation matrix and its inverse are of
rank 2 and each vector is only two-dimensional. However, phase-C’s value cannot be directly
obtained from Equation (3.89), thus we will use Equation (3.79 or 3.85) instead. Since the only
difference between the two is the presence or absence of phase-C output there is no penalty for
doing so. Our final forms of the Clarke transformation matrix and its inverse are then given by
Equation (3.90).
2
0
3
3
0 0
2
1
1 1
1
Ck ;
C
3
k
3 3
(3.90)
3 0
2
1 1
3 3
Scaling & Trajectories
Comparing Equations (3.83) and (3.85), it is apparent that scaling occurs when transforming
between coordinate systems. Since the purpose of the SV transformation is to work with the SV
instead of the phase variables, it is important to understand this scaling, and that is best
understood by studying the SV trajectory for balanced sinusoidal quantities.
The scaling has manifested itself several times already as the factor 3/2. This factor arose even
in real-valued expressions because the fields are described as a function of the angle around the
stator (a cross-section) 25 but are created by three phase values. Regardless of whether that crosssectional
plane is described by the complex plane, the αβ plane, or a cylindrical coordinate
system, the three phase axes “wrap around” as shown in Figure 3.29. Considering MMF for
example, each axis’ component MMF acts in a direction that has a component along the other
phase axes. The other element that contributes to the scaling is the isolated neutral connection.
25 The cross-section is sufficient for study because depth in the axial direction (rotor and stack length) has
no effect on this phenomenon. But for a given magnitude of radial quantity, increasing motor length
increases the flux linkage and hence the magnitudes of the torque and bEMF constants.
120