SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure C.3 – Sinusoidal winding density.
To derive the ideal sinusoidal winding function, assume that the radial winding density (which is
always nonnegative) is sinusoidal about the stator as shown in Equation (C.1); [26, p.460], [78,
p.37], [42, p.173], [73, p.9.2].
n( ) N sin( )
(C.1)
p
Np is the peak value of this distribution and it clearly must be less than N because the N turns are
distributed around one-half the stator. To determine the value of Np, the winding density can be
integrated over one side of the stator to obtain the total number of windings. Since we stipulate
that there be N total turns, the result is set equal to N.
set
N N sin( ) d
0
p
N 2N
p
N
N
N p
2
cos( )
p
0
Therefore the winding density is given by Equation (C.2). The density is negative from π to 2π
and this may be interpreted as indicating that the direction of the winding on that side of the stator
is opposite that of the top [42, p.173]. 45
N
n( ) sin(
)
2
(C.2)
45 The reader is alerted to a conflict in the literature for which the author has no explanation. The majority
of references surveyed agree with the MMF expression given here. However, there are several authoritative
references whose expression lacks the factor ½, such as [87] and [36].
283