SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

torque will describe the torque produced by the fundamental components of stator MMF and
rotor-stator flux linkage.
Making these assumptions greatly simplifies the presentation and understanding of FOC but
leaves questions as to how a nonsinusoidal motor should be handled, but the answer is simple. A
motor with nonsinusoidal windings can be modeled using space vectors but the model will only
represent the fundamental components. Therefore the torque and bEMF space vector equations
will not be correct (there will be torque ripple even with sinusoidal current and there will be
bEMF harmonics).
Traditional texts work with the winding factor but to streamline the equations, modern texts
replace the number of turns N with the number of effective turns as defined by Equation (C.27)
[69, p.6.22], [42, pp.177,448].
4
Ne kwN (C.27)
The effective number of turns represents the number of turns used to produce fundamental MMF.
Using the effective number of turns allows the consideration of the fundamental MMF produced
by a nonsinusoidal winding. Although [87] states the space vector can only describe windings
that produce sinusoidal MMF, all references in the literature surveyed use something akin to
Equation (C.27).
NOTE
This appendix has analyzed the single-winding case only in order to show the mechanisms at
work. Every result is valid for a three phase machine. However, there is no point in computing
anything related to the third or triplen harmonics if the machine is wye-connected. Meaning that
even if a the third harmonic winding factor kw,3 is found to be nonzero, it would not contribute to
torque or bEMF [69, p.6.16]. As shown, the harmonic winding factor affects the rotor-stator flux
linkage, which affects both bEMF and torque. As far as bEMF is concerned, the triplen
components in the bEMF will simply appear at the isolated motor neutral, not affecting operation.
As far as torque is concerned, it is impossible for the third harmonic of flux to produce torque
without the third harmonic of current, which cannot be driven into a wye winding. Two cases
where the triplen harmonics are important are grid-connected generators (usually wye-connected
with a grounded neutral) and delta-connected machines (in which the third harmonic of bEMF
would cause circulating current).
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