SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

phase variable axes; three axes in a plane cannot be independent of one another. Since this is the
case it is difficult to use the Clarke transform and its pseudoinverse as defined—we must remove
the linear dependence between the phase variable axes. To accomplish this we can first rewrite
the 0 condition as Equation (3.82) and substitute it into the α-portion of Equation (3.74) to
yield Equation (3.83). Then the complex form is given by Equation (3.84) and the inverse
transformation is the given by Equation (3.85) (which is incidentally the same as Equation 3.79).
1 1 1
xA xB xC
(3.82)
2 2 2
3 0 0 x A
x 2 k
x
B
x
3 3
0 x C
2 2
3 3
x k xA j xB xC
2 2
2 0
x 3
A
1 1 1 x
x
B
k
3
3 x
x
C
1 1
3 3
(3.83)
(3.84)
(3.85)
This makes the projection of the phase variable axes onto the α axis easier to see but it does not
remove the ambiguity. To finish removing the dependence the 0 condition is rewritten as
Equation (3.86) and substituted into the β-portion of Equation (3.83) to yield Equation (3.87).
Then the complex form is given by Equation (3.88) and the inverse transformation is given by
Equation (3.89) (which is now different from Equation 3.79).
3 3 3
xC xA xB
(3.86)
2 2 2
3
0 0 x A
x 2
k
x
B
x
3
3 0 x C
2
3 3
x k xA j ( xA 2
xB)
2 2
(3.87)
(3.88)
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