SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 3.32 – Arbitrary SV referenced to axes of phases –A, -B, and –C.
It is seen that multiplying a SV by
j
e
rotated the SV (in the αβ plane) by 120
while the
phase axes remained in place (in Figure 3.31-b x was rotated to the position of y ). This result
could also be interpreted as rotating the phase axes while leaving the SV in place (from the axes
in Figure 3.32-a to the axes in Figure 3.32-b and Figure 3.32-c). This is an important result and is
a demonstration of a change in reference frame. It has been presented here to aid the discussion of
projections but will be shown in the next section in its more traditional and useful form.
Before concluding this subsection it is worth verifying the action of the inverse Clarke transform
on a SV that represents balanced sinusoidal phase variables. Substituting Equation (3.93) into
Equation (3.100) yields Equation (3.102), as expected.
2 3kjt
xA()
t Re X pe
1
3k 2
X p cos t
2 3kjt j
xB() t Re X pe e X pcost
3k 2
X p cos t
2 3k
jt j
xC() t Re
X pe
e
3k 2
(3.102)
Park Transform; Reference Frame Theory
As shown in the previous section, multiplying a SV by a complex exponential can be interpreted
as rotating a SV in fixed coordinates or rotating the coordinates while the SV remains fixed. This
concept applies to all regular vectors in linear algebra; it is not specific to the SV or to complex
vectors. Previously the phase axes were studied to understand projections but now the same
concept will be applied to coordinate systems of only two variables. (In the literature, coordinate
system and reference frame are used synonymously.) The 0 condition means only two phase
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