SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Returning to Figure C.18 it is clear that a sinusoidal torque function can be obtained using any of
three basic winding-rotor combinations:
1. CFP winding, sinusoidal rotor
2. sinusoidal winding, sinusoidal rotor
3. sinusoidal winding, squarewave rotor
We know the CFP winding passes all harmonics of rotor flux thus combination (1) works because
the rotor flux has no harmonic content. Combinations (2) and (3) both use a sine winding and we
know a sine winding does not link flux harmonics, thus flux harmonics do not produce torque in
a sine winding. Examining the coefficients for these three combinations (Figure C.19) shows that
combination (2) links less flux than combination (3) because the fundamental of rotor flux for the
squarewave is larger than unity. As long as a sinusoidal winding is used the rotor-stator flux
linkage will be sinusoidal—the rotor flux profile is does not affect the shape of R( r)
but the
amplitude of its fundamental does. Since the torque and bEMF functions are equal to the
derivative of R( r)
they are sinusoidal as well (and have a +90° phase shift).
In the CFP winding with squarewave rotor flux the amplitudes of rotor flux harmonics are
multiplied by the harmonic winding factors to produce a R( r)
that is triangular. But the
derivative of a triangle wave is a squarewave therefore the torque and bEMF functions are as
shown in Figure C.15-b. If the MMF and rotor flux harmonic content were not exactly equal the
shape of the flux linkage would not have been “recovered” in the rotor-stator flux linkage. This
combination shows that nonsinusoidal motors are essentially sinusoidal motors with harmonic
content in their rotor-stator flux linkage.
Using this information the torque production in a sinusoidal motor will be discussed. Torque is
given by Equation (C.21).
d
(C.21): Tt () R( r)
it ()
d r
For the CFP winding and sinusoidal rotor R( r)
was given by Equation (C.14).
(C.14): ( ) NDY B cos( ) (CFP winding, sine rotor flux)
R r p r
This expression can be reduced by the winding factor and substituted to give the torque as
Equation (C.24), where B1 is the fundamental of rotor flux.
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