SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Historically, AC machines were designed to produce sinusoidal electrical quantities so the
fundamental component was of greatest concern. The winding’s effect on fundamental amplitude
was determined by accounting for the distribution (breadth) and pitch by defining reduction
factors whose values could be found for each particular winding. These were called the
distribution- (or breadth-) factor kd (or kb) and the pitch factor kp. Then to obtain the value of
fundamental MMF amplitude, the maximum value (produced by the CFP winding) could be
multiplied by these reduction factors per Equation (C.7).
4 N
F1 kbkp i
2 (C.7)
The effects of the breadth and pitch factors are often combined to form the winding factor:
kw kb kp.
46 There are a variety of ways to define these factors ([27], [69], [40]) but the
definitions are not required here. 47 We are not concerned with the winding specifics (kb, kp) so we
will only use the overall winding factor kw. The important point is that deviation from the CFP
winding will reduce the value of the amplitude of the fundamental component of MMF by a
certain fraction (kw) for a particular winding configuration. The amplitude of fundamental MMF
is thus given by Equation (C.8).
4 N
F1kw i
2 (C.8)
It may be that traditionally the winding factor was defined for only the fundamental component.
But it is clear that distributing the winding reduces the amplitudes of other harmonics as well and
modern texts [68], [69] also define a harmonic winding factor for each harmonic (though in a
46 In addition to breadth and pitch factors, a skew factor is required for skewed rotor magnets or stator slots
[69, p.6-21]. This advanced topic is not considered here.
47 Traditional texts do a fair job explaining the winding factor, its usage, and its derivation. A typical key
result is that a shortening the winding or magnet pitch by 1/n of a pole pitch will eliminate the n th harmonic
from flux linkage so long as fringing and slotting is ignored [69, p.5-28]. It seems that this was a good rule
for traditional synchronous machines which had a steel rotor; the airgap was small and the assumption
could be made that the field was radial in the airgap [27, p.117]. However, since the effective airgap of a
permanent magnet machine is larger (because the relative permeability of the magnet is close to that of air),
fringing is more pronounced and the former rule cannot be used reliably. (For example a 5/6 pitch coil and
full-pitch magnet should create a 150° flat top bEMF but in practice it is closer to 120° and is significantly
rounded [69, p.5-5].) Indeed, one designer states that the traditional breadth and pitch factors are no longer
used in motor design—design is done on a computer and the Fourier series components of various key
waveforms can be adjusted directly [68, p.140]. Further, basic magnetic analysis (such as using the shortpitch
rule described) cannot accurately predict the flux linkage in a brushless permanent magnet motor and
more advanced techniques must be used [68, p.151]. As with all other construction-specific details, this
result is not of concern here but the author feels it is worth pointing that not all standard results found in
traditional machine texts are directly applicable to understanding brushless permanent magnet motors.
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