SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

x
x a b c (3.77)
A
k
x
B
x
x C
Equation (3.77) clearly shows how the α and β components are related to the value of the phase
variables acting along their phase axes (basis vectors). Equations (3.75 or 3.77) clearly show a
linear transformation in the form of xNEW Tx OLD . The inverse of the transform is given by
x T x . A 2x3 matrix is singular but the inverse can be found several ways (Appendix
1
OLD NEW
D) and is given by Equation (3.78) and Equation (3.79).
1
k
1
xabc C x
(3.78)
2 0
x 3
A
1 1 1 x
x
B
k
3 3
x
x
C
1 1
3 3
(3.79)
As before, defining the columns of the matrix as basis vectors (Equation 3.80) allows the
transformation (Equation 3.78) to be rewritten in terms of basis vectors (3.81).
2
0
3
1 1
α ;
β
3
(3.80)
3
1 1 3
3
x A
x
B
x C
1 x α β (3.81)
k x As before, it is clear that the phase variable components are given by the values of the stationary
variables along their axes, but these basis vectors are more difficult to visualize. The
transformation matrix is singular thus its columns (the phase variable basis vectors) are not
linearly dependent. Thus there is not a one-to-one mapping between the phase variables and the
stationary reference frame. This should be exceptionally clear when examining Figure 3.29.
Action along any phase variable axis has a component along the direction of each of the other two
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