SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

T( ) k NDYB sin( ) i( t)
(C.24)
r w 1 r
This expression gives the torque produced by the fundamental of stator MMF interacting with the
fundamental of rotor flux. Substituting in the appropriate winding factor for CFP (kw=1) and
sinusoidal (kw= /4)
windings yields the torque produced in each case.
T( ) NDYB sin( ) i( t)
(CFP winding, sine rotor flux) (C.25)
r 1 r
T( r) NDYB1sin( r)
i( t)
(sine winding, sine rotor flux) (C.26)
4
Equation (C.25) is identical to the expression for torque found in Chapter 2 for the same motor. It
must be emphasized that Equation (C.25) describes only that torque produced by the fundamental
of rotor flux. If the rotor flux has harmonics, torque ripple will be produced. Equation (C.26)
correctly describes the torque regardless of rotor flux because the sinusoidal winding filters the
harmonics. Since any practical winding will function somewhere between the two it would be
possible to use the harmonic winding factor (in Equation C.24) to compute the torque produced
by each harmonic. (The harmonic winding factors can be obtained from spectral analysis of the
bEMF.)
Conclusions
It is clear that the shape of the rotor-stator flux linkage is determined by the winding function and
the rotor flux profile. Several simple windings and two simple rotor profiles have been presented
to demonstrate how R( r)
can be changed. In addition there are numerous ways to manipulate
the harmonic content of the rotor flux [69], [68]. These include varying the magnet thickness,
magnetization strength function, magnetization direction, and span/pitch of the magnet (among
others). Sinusoidal motors do not typically have full-pitch windings [69, ch.6], [68, p.78] and
often have fractional pitch magnets or may be of fractional-slot design [68, p.180].
The focus of this report is on sinusoidal motors. A sinusoidal motor could be defined as one in
which R( r)
(and consequently the torque and bEMF functions) have a sinusoidal shape. By
this definition a sinusoidal motor is not required to have sinusoidal windings. However, space
vector theory can only describe the MMF produced by a sinusoidal winding. For this reason most
of the academic literature concerning FOC makes the assumption of a sinusoidal winding and a
sinusoidal rotor flux. This report will use the same assumptions and the resulting expression for
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