SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

That the two functions are equivalent is readily apparent when Figure 2.22 and Figure 2.24 are
compared. It must be noted that Kt and Ke are per-phase values. These are not necessarily the
torque and bEMF constants found on a motor datasheet. This will be discussed toward the end of
this chapter.
Back-EMF and Torque in Terms of Rotor-Stator Flux Linkage
Earlier, the concept of flux linkage was developed in order to obtain an electrical model and gain
an understanding of inductance in a BPMS motor. The connection between stator-rotor flux
linkage and bEMF was shown. And previously, the bEMF and torque functions were derived
using simple first principles for the case of a concentrated full-pitch winding. Now the bEMF and
torque functions will be derived again, this time using the flux linkage. The understanding
provided by this derivation will be requisite to further understanding the difference between
motor types and in understanding FOC.
Returning to Figure 2.20, it is clear that the amount of flux linked by the winding of either motor
will be a function of rotor position. As before, the sinusoidal case will be derived analytically and
the trapezoidal case will be analyzed graphically. The flux linkage for the sinusoidal motor is
found as follows, where α and θr are defined in Figure 2.20,
Y is length of the rotor and stator lamination stack, and D is the diameter of the N-turn winding.
Since the motor uses a concentrated full-pitch winding, N is simply a constant.
( ) N NBA R r
/2
N B dA
/2
N
Bpcos r
Y D/ 2 d
N( D/2) YB sin
p r
/2
/2 N( D/ 2) YB [sin( / 2 ) sin( / 2
)]
p r r
( ) NDY B cos( )
(2.43)
R r p r
( ) cos( )
(2.44)
R r R r
42