SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

alanced machine only odd harmonics need be considered). When kw is written it is understood to
refer to the fundamental component. The harmonic winding factor for the n th harmonic would be
written kw,n and the amplitude of each harmonic of MMF would be given by Equation (C.9),
which follows directly from the Fourier series of a squarewave (see Figure C.9 and Equation
C.5). 48
Sinusoidal
41 N
Fn kw, n
i
n 2 (C.9)
Finally the MMF produced by a sinusoidal winding can be investigated. The MMF can be found
by using Equation (C.4) and the sinusoidal winding function given by Equation (C.2). θ’ is used
as a dummy variable of integration so that we can integrate beginning at an arbitrary angle θ.
H(2 g) N( ') id'
N
sin(
') id' 2
N
cos(
') i
2
N
cos(
) cos( )
i
2
f( ) Ni cos( )
(C.10)
Equation (C.10) is the total MMF along the path and due to symmetry it is split equally between
the two airgaps. The MMF per airgap is found by dividing each side by two, Equation (C.11). By
associating half the MMF with one airgap we get the traditional bipolar result that agrees with the
sign convention for magnetic quantities in the airgap.
N
f( ) icos(
)
(C.11)
2
As expected, the sinusoidal winding produces a cosinusoidal MMF with no other harmonics (the
harmonic winding factor kw,n is zero for all harmonics). Also as expected, the amplitude is not as
large as the fundamental amplitude produced by the CFP winding. It is in fact less by a factor of
( /4) , canceling the (4 / ) in Equation (C.6). In comparing Equation (C.11) with Equation
(C.8), it is clear that the value of kw for an ideal sinusoidal winding is ( /4) .
48 There are several possible definitions for the winding factors; those used here are just one example.
291