SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

epresent any physically-distributed quantity; this conclusion holds for all other space vectors
except MMF. What its “distribution” instead represents is the meaning of reference frame theory
discussed in the previous subsection; that interpretation is ultimately the most useful view of
space vector theory. The meaning of the current SV will be further examined using Equation
(3.68).
This direct phase relationship between current and MMF has already been encountered: the
sinusoidal case was examined in Part I, and both the sinusoidal and arbitrary cases have been
examined here in Part II. In Part I when the MMF wave was developed mathematically (Equation
3.6, θ=0), it was obvious that the angular position of the stator MMF is identical to the electrical
position (ωt) of the balanced sinusoidal currents (refer again to Figure 3.9). In Part II when the
SV was developed (for the case of balanced sinusoidal currents) it was shown that the MMF SV
describes the same MMF with the peak again at ωt and that this is shown very clearly in the SV
itself, as in Equation (3.49) for example.
3 Ne
j t
(3.49): f Ie p
2 2
When the currents are arbitrary the SV theory gains an advantage over the real-valued
descriptions of MMF since it is not restricted to balanced sinusoidal current. As discussed before,
the MMF always retains its cosinusoidal distribution. Even in the arbitrary case, the SV of stator
MMF is still cophasal with the SV of stator current, per Equation (3.42). Thus the angle ξ of the
MMF SV represents the angular position of the peak of the MMF distribution, as measured from
the phase-A axis and this is the same angle of the current SV. Further, the magnitude ˆ F of the
physical MMF SV is proportional to the amplitude of the MMF wave and proportional to the
magnitude of the current SV. Again from Part II, this was given by Equation (3.50).
Ne
(3.50): ˆ j ˆ j
f Ie Fe
2
When Equation (3.50) was developed it was shown that the stator MMF SV was cophasal with
the current SV in Figure 3.27. But at that time the primary interest was the addition of the SVs
and the distributions were ignored. The figure is now redrawn as Figure 3.28 and the equivalent
sinusoidal amp-turn distribution is shown. This leads to a better interpretation of the current SV.
When Figure 3.27 was discussed it was said that the resulting MMF could be thought of as being
produced by a coil whose magnetic axis was oriented along the direction of the total MMF
produced by the three phase currents; this was compared to the CRT example of Figure 3.24. The
situation is the same here except now we can say that the MMF could be thought of as being
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