SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Back-EMF Generation
As the torque was developed using the BLi law the bEMF will now be developed using the BLv
law along with first principles. Using the conventions established earlier in the chapter the
voltage induced in the winding could be given by Equation (2.35).
e( t)
2N
B Y
v
(2.35)
Substituting in Equation (2.28) for B and simplifying yields Equation (2.36), where ω is the
rotational velocity.
D
e( r) 2 N[ Bpsin( r)] Y [ ]
2
e( ) NDYB sin( ) (
t)
(2.36)
r p r
Since the terms multiplying sine are constant, Equation (2.36) can be rewritten as Equation
(2.37).
e( ) K sin( ) (
t)
(2.37)
r e r
The bEMF produced in the sinusoidal motor, Equation (2.37), is seen to scale linearly with the
velocity, as expected from the discussion of the Faraday’s law. The velocity may vary arbitrarily
with time. To remove this dependence from the magnitude, Equation (2.37) can be rewritten as
the ratio of torque produced per rotational velocity, yielding Equation (2.38), which may be
called the per-phase bEMF function.
e(
r )
ke
( r ) K
e sin( r )
(2.38)
(
t)
Equation (2.38) is plotted in Figure 2.24; velocity is held constant. As with the torque function,
the bEMF function takes the same shape as the airgap flux density, the torque produced by the
trapezoidal motor is found graphically as shown in the figure.
Figure 2.24 – Back EMF functions of a sinusoidal and arbitrary trapezoidal motor.
40