SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Notice that the amplitude of the MMF SV seems to remain constant with the same amplitude of
the sum of instantaneous values of component MMF phasors shown in Figure 3.11. This can be
verified mathematically if the three-phase currents of Equation (3.3) are substituted into the
definition of the MMF space vector, Equation (3.41). After simplification the resulting MMF SV
is that given by Equation (3.49) and it is clear that the amplitude is indeed the same as that found
earlier using real-valued arithmetic.
3 Ne
j t
f Ie p
2 2
(3.49)
For simplicity the diagram in Figure 3.25 is usually drawn without showing the stator, as shown
in Figure 3.26. The instantaneous values of the space vectors are shown added together for the
two electrical positions used earlier.
Figure 3.26 – Addition of component MMF SVs to form total MMF SV (c.f. Figure 3.25).
Given the relationship in Equation (3.42) it is clear that MMF space vector and current space
vector are cophasal (they always have the same phase). This could also be seen if the three-phase
currents are substituted into the current SV (instead of substituting them into the MMF SV
equation). The resulting current SV would be the right-hand portion of Equation (3.49) and this
result again agrees with Equation (3.42). And in comparing Equation (3.40) with the second line
of Equation (3.48), it is seen that the component current space vectors add in the same way that
the component MMF space vectors do.
So far the comparison with the CRT example has worked well to illustrate a physical
interpretation of the space vector. It showed how each scalar quantity in υ (voltage, current) can
be multiplied by a constant of proportionality and basis vector in α that when summed represent a
vector with meaning in α (force in Cartesian space, MMF in complex space, Equation 3.32 and
101