SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

the same magnitude, the projection x B ( t1)
definition for x and y shows they are related by
must be identical to A 1
y x e
j120
y ( t ) . Comparing the
. Thus, y was obtained by
rotating x
by 120
. The benefit is that the projection can then be obtained by simply taking the
real part of the rotated vector as shown in Equation (3.98).
j120
x Re x e
B (3.98)
However it must be understood that rotating the SV does nothing magic. For the moment we will
say that we are not attempting to change or “control” the SV (this will be rectified shortly).
Rather, we simply note that the projection of a SV onto the phase-B axis is numerically equal to
the projection of (any identical, separate SV delayed by 120°) onto the phase-A axis. Since the
phase-A axis is collinear with the real axis we can then take the real part to obtain the projection.
The phase-C projection follows in a similar manner except the SV must be advanced by 120° (or
equivalently, delayed by 240°). The projections are summarized in Equation (3.99).
xARex1
j
xBRexe
(3.99)
j
xCRexe
Finally, the relationship between projections and the phase variables obtained from the inverse
Clarke transform is summarized in Equation (3.100). Note that by definition if the magnitudeinvariant
version of the Clarke transform and its inverse are used the coefficients in Equation
(3.100) are unity thus the projection and inverse transformation are the same.
2 2
xA() t xA Rex1 3k 3k
2 2
xB()
t xB Rexe
3k 3k
2 2
xC()
t xC Rexe
3k 3k
j
j
(3.100)
In Figure 3.31-a and Figure 3.31-b the phase axes are exactly the same because the SV x was
fixed in place and the SV y was used for a demonstration. It must be understood that the
projection was not onto the phase-A axis—it was the projection of the rotated SV onto the phase-
A axis; that rotated SV would have been collinear with y . Earlier it was said in pretense that in
obtaining the projection we are not attempting to change the SV, even though we did rotate the
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