SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

0, and since the direction of integration used to get Equation (C.13) was from to , the
rotor-stator flux linkage integral will have to be taken over the span of conductors from only one
side ( ). This means the coil axis must swing from /2 to /2.
The total rotor-stator flux
linkage would be given by Equation (C.15).
/2
( ) n( ) (
, ) d
(C.15)
R r r
/2
However, the winding density of Equation (C.2) gives the density of conductors at θ as a function
of θ, not the density at /2 as a function of coil axis angle θc. This can be corrected by
c
modifying the density function to be Equation (C.16), which can be verified visually via Figure
C.12. It should be understood that the math is not changed—the winding density is simply
defined by the coil axis instead of the coilsides.
N
n( c) cos( c)
(C.16)
2
Now the rotor-stator flux linkage can be found as planned.
/2
( ) n( ) (
, ) d
R r c c r c
c/2 /2
N
cos( c) DY Bpcos( c r)
dc
2
c/2 /2
N 1
DY B cos( ) cos(2 ) d
2
2
p r c r c
c/2 /2
N 1 1
DY Bp cos( ) sin(2 )
2 2c r c r
2
c/2 N 1 1
1
DY Bp cos(
r)
sin(
) cos( ) sin( )
2 2
r r r
2 2
2 2
N 1 1 1
DY Bp cos( r) sin( r) cos( r) sin( r)
2 2
2 2 2 2
R( r) NDY Bpcos(
r)
(sine winding, sine rotor flux) (C.17)
4
Comparing Equation (C.17) with Equation (C.14) shows that the sinusoidal winding linked the
sinusoidal rotor flux but the amplitude was only /4 of the flux linked by the CFP.
295