SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

As an example, find the SV that corresponds to unit-amplitude balanced sinusoidal
quantities at an electrical position of zero. From Figure 3.6 or Equation (3.91), the values
are given by Equation (3.96). Substituting them into the SV definition gives Equation
(3.97), assuming k 1 for simplicity.
x
A 1
xB
1/2
xC
1/2
3 j0
3
x e
2 2
(3.96)
(3.97)
Clearly the SV is aligned to the phase-A and α axes and the projection onto either is
given by Equation (3.97). The scaling of 3/2 is present as expected. But obtaining the
value of x A from Equation (3.85) gives
2 23 xAx 1.
This shows that the
3 32 phase variables obtained using an inverse transform (such as Equation 3.85) are equal to
the projected value reduced by a factor of 2/(3 k ) . The Clark transformation
(equivalently, SV transformation) maps x abc to x thus any SV exists in the αβ plane or
with scaling thus it makes sense that when we project back onto the phase axes we
should have to reverse the scaling. The matrix in Equation (3.85) is the pseudoinverse of
the Clarke transfomation matrix and it inherently scales the result back.
The issue of scaling and the difference between projection and the inverse transformation is not
treated well in the popular literature. In most cases the “inverse Clarke” transform is presented as
nothing other than a projection of a SV or an algorithm. Understanding the true inverse
transformation provides much needed insight, especially when working with modulation
techniques based on space vectors. Although the inverse transformation is the more rigorous
perspective, the projection perspective is obviously useful and is correct when scaled properly. In
fact, further studying the projections will provide further insight into reference frame theory.
Projections
Projecting the SV onto the α axis is simple because the projection is given by the real part of the
SV. The projection onto the phase-A axis is given by the same. Similarly, projection onto the β
axis is given by the imaginary part of the SV. However, projection onto the phase-B and phase-C
axes is not that simple because they are not perpendicular to the real or imaginary axes. There are
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