SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

a number of ways to project the SV onto these axes but only the commonly used method is
discussed here. Since the projecting is done in the αβ plane it seems reasonable to find a way to
make the phase-B and phase-C projections equal to the real part of a complex expression; this is
exactly what the common method does.
If CCW rotation is to be defined to match the positive-sequence order (phase-A, phase-B, phase-
C) with increasing ωt, arranging the phase axes as shown in Figure 3.29 forces the space vector to
be defined as Equation (3.69). (There are various other combinations of definitions and
conventions used in the literature; these obviously lead to different SV definitions.)
jj (3.69): x k xA1xBe xC e
A space vector ˆ j
x( t1) Xe
is at the position shown in Figure 3.31-a at some time t1. Its
projection onto the phase-B axis is also shown; the magnitude of the projection is denoted
x ( t ) , where the subscript indicates that it is a projection onto phase-B, not the actual value of
B
1
the phase variable xB ( t 1)
that produced that SV.
Figure 3.31 – SV projection onto phase axes: (a) actual; (b) alternative.
( 120 )
At the same instant t1 another SV ( ˆ j
yt1) Xe is shown in Figure 3.31-b that has a
magnitude equal to that of x( t1)
but an angle that is 120° less. Its projection onto the phase-A
axis is denoted y A(
t1)
. Since the phase axes are 120° apart, the angle between x and the phase-
B axis is the same as that between y and the phase-A axis. Therefore, because x and y have
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