SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

common choice is to force the power in the equivalent two-phase network to match the power of
the original three-phase network. For this to be true requires that k 2/3;
this yields a power
invariant transformation [87, p.34], [88, p.87]. Different choices are made by different authors
working in different areas. This report primarily uses k 1 because using a value other than
unity requires keeping track of the value of k used. The Clarke transformation matrices for these
different choices are summarized below.
2
0
3
3
0 0
2
1
1
1
C ;
C
3
3 3
(k=1)
3 0
2
1 1
3 3
1 0
1 0 0
1
1
3
Cm ;
C m (k=2/3 : magnitude invariant)
1 2 2 2
0
3 3
1 3
2 2
2
0
3
3
0 0
2
1
1
1
Cp ;
C p ( k 2/3 : power invariant)
1
6 2
2 0
2
1 1
6 2
It should be noted that these various forms of C could have been derived such that only phase-B
and phase-C (or only phase-A and phase-C) quantities were used. When only two phase variables
are desired the choice most common in the literature is to use phase-A and phase-B as shown
here.
Now to address the case when the phase variables are nonsinusoidal (even in this case they cannot
have a zero-sequence component because the SV cannot represent it). In the sinusoidal case the
phase variables have constant amplitudes and the SV trajectory has a constant radius—this
allowed us to take the ratio and the factor 3/2 was immediately obvious. When the phase
variables contain harmonics the trajectory deviates from the circle. In this arbitrary case both the
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