SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

power invariant form, k c.
In this report the αβ0 transform is primarily used when studying
inverters so the magnitude-invariant variety is used.
1 1/2 1/2
C 0 0 3/2 3/2 ;
k c1 (D.31)
1/2 1/2 1/2
2/3 1/3 1/3
k c2/3 C 0m
0 1/ 3 1/
3 ;
(D.32)
(magnitude-invariant)
1/3 1/3 1/3
2/3 1/ 6 1/ 6 k 2/3
C 0 p 0 1/ 2 1/ 2 ;
c 2/
3
(D.33)
1/ 6 1/ 6 1/ 6
(power-invariant)
The α axis has been defined to be coincident with the real axis and the β with the imaginary axis.
However, for some reason a sizeable portion of the literature aligns the β axis in the negative
direction of the imaginary axis. The order of the variables in the vector are also sometimes
rearranged (for example, the 0αβ transform uses the order 0,α,β). Both of these changes affect the
way the SV and Clarke transform are defined and this change obviously propagates to other
definitions and expressions, thus care is required in interpreting the literature. Finally, it should be
noted that that α and β axes may be labeled D,Q or d,q.
The Zero Sequence Component
The chapter on inverters shows that the magnitude-invariant form of the Clarke transform has
direct correlation to the physical values (such as voltage). The interpretation is the same for the
ZS component of the magnitude-invariant αβ0 transform. Clearly, if Equation (D.32) were used
on a set of voltages, v 0 would be the same term defined by Equation (D.16). If the voltages were
across an isolated-neutral wye load, the v 0 term would be the neutral voltage, in agreement with
Equation (D.15). Thus the 0-component is the same as the general ZS component encountered
throughout this appendix. As mentioned throughout the report, the SV (or αβ components) simply
cannot encode the ZS component; this is now shown. Consider the example of phase variables
(Equation D.34) that contain the first PS, ZS, and NS harmonics (which were defined by
Equations D.4–D.6). Writing the phase variables in complex form (Equation D.35) and
substituting them into the SV definition (Equation D.36, where k 2/3 to achieve magnitude
invariance) yields Equation (D.37) which simplifies to Equation (D.38).
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