SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

j
j
ee 1
cos( )
(3.45)
2
3
j t
x k X pe
2
(3.46)
If the three phase quantities do not form a balanced sinusoidal set then they are called arbitrary.
Using the notation listed in the beginning of the report, the SV is represented as Equation (3.47).
exp ˆ j
x x j x Xe
(3.47)
In other words, ˆ X and are the magnitude and angle of any arbitrary SV; if the SV happens to
represent sinusoidal quantities then we specifically know that
ˆ 3
X k X p and t
. Note
2
that the factor of 3/2 is not present in the arbitrary case (Equation 3.47) as is for the sinusoidal
case (Equation 3.46). The reason the factor 3/2 appears in Equation (3.46) is that we have a
known amplitude of the phase values which we can see as being scaled. Equation (3.47) lacks the
factor because we do not have a phase amplitude that is fixed. In other words, the scaling
apparent in the sinusoidal case is also present in the arbitrary case but is not so obvious. The topic
of scaling is important but it will be played down until the end of Part II (and generally the
scaling will be left as k 1).
The take-away from this subsection is that the space vector is an instantaneous quantity that
contains information from each phase variable and its orientation. Those variables do not need to
be sinusoidal (but they cannot contain any zero-sequence component). As the values of the phase
variables change the instantaneous magnitude and angle of the SV will change. If the phase
variables happen to form a balanced sinusoidal set the magnitude and angle are easily related to
the amplitude and phase of the variables.
Additivity of MMF and Current Space Vectors
When the rotating stator MMF wave was developed earlier in the chapter it was shown how the
current in each phase creates a component MMF that is distributed cosinusoidally in space with
the peak aligned with the magnetic axis of that phase. These component MMFs add to form a
total MMF with the same distribution. The distributed nature of the MMF will continue to be
ignored. Here we will assume that the SVs can be added as standard vectors and seek only to
understand the basic vector nature of that addition. The MMF space vector (Equation 3.41) is
99