SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

However, if the per-phase MMF distributions are not cosinusoidally distributed, SV theory cannot
be used as shown here to analyze the machine. (This is essentially the reason why many articles
in the literature assume a sinusoidal machine, even if the article specifically concerns a
trapezoidal machine.) The statement that SV theory applies only to sinusoidal machines is given
without proof in [87, p.34], [43, p.223]. It does not mean that the theory cannot be augmented to
describe nonsinusoidal machines [87, p.53]; it simply means that the theory as presented here
cannot be used to describe them unless modifications are made. From Appendix C, when a
nonsinusoidal machine is modeled using SV theory only the fundamental component of torque
and bEMF will be represented. The reason this is important is that the expression for torque
production involves the magnitude of the MMF (or flux, flux linkage, flux density, or field
strength) of the stator. If the phase MMF distributions are not sinusoidal, they will not sum to a
sinusoid of 3/2 amplitude, even for balanced sinusoidal current, thus the torque expression will be
invalid.
Before moving on, one important observation will be made regarding the MMF space vector’s
representation of a distributed quantity. This section began with the real-valued expression of
stator MMF, Equation (3.2), which was reproduced as Equation (3.51):
i( t)
cos( ) i ( t)
cos( 120
) i ( ) cos( 120
)
N e
f ( , t)
A
B
C t
2
This was reworked to give an equivalent complex form, Equation (3.56):
Ne
j
f , t Rei
e
2
,
which has provided satisfactory results in this discussion. One of those results was the
expression that describes the MMF resulting from balanced sinusoidal currents, Equation
(3.58):
3 N
f t Ipt
2 2
e
, cos
.
But earlier (in the section “Additivity of MMF and Current Space Vectors”), balanced
sinusoidal currents were substituted into the MMF SV definition, Equation (3.41):
Ne
jj f kiA()1 t i B() t e iC() t e
2
and the result was the SV representation of MMF due to balanced sinusoidal currents,
Equation (3.49):
.
108