SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

describe the flux density directly over a conductor of the coilside 5 in terms of the rotor
position (per the previous discussion, the flux takes the tooth and the wire is not actually in the
field but we use the equations anyway). For the sinusoidal motor the description is simple and is
given by Equation (2.28). Describing the flux density seen by the conductors of the trapezoidal
motor is best accomplished by considering the trapezoid approximation and using discontinuous
functions, but for simplicity the result is derived only graphically. In all of the following, the
analytical result will be provided for the sinusoidal case only; the results for the trapezoidal case
will be found graphically using the sinusoidal case as a guide.
B( r ) B
p sin( r )
(2.28)
This result will be used to examine first the torque production and then the bEMF generation.
Torque Production
The torque exerted on the rotor is found from first principles and the BLi law. Torque is produced
by both coilsides (each having N conductors) and is given by Equation (2.29), where F is the
force per coilside and D is the diameter of the coil centers. Substituting in the BLi law gives
Equation (2.30), where Y is the length of the conductors in the stator lamination stack and i(t) is
the current in the winding.
D T 2F
2 (2.29)
D T 2NBYi
2 (2.30)
Substituting Equation (2.28) for B in Equation (2.30) yields Equation (2.31),
( ) N
D Y
B sin(
) i(
t)
(2.31)
T r
p r
Since the terms premultiplying the sine are constant, Equation (2.31) can be rewritten as Equation
(2.32), where Kt is called the torque constant.
( ) K
sin( ) i(
t)
(2.32)
T r
t r
The torque produced in the sinusoidal motor, Equation (2.32), is seen to scale linearly with the
current, as expected from the discussion of the Lorentz force law. The current may vary
arbitrarily with time. To remove this dependence from the magnitude, Equation (2.32) can be
rewritten as the ratio of torque produced per unit current to yield Equation (2.33), which may be
5 This coilside is selected because the BLi law describes force on a conductor so the force on the rotor is in
the opposite direction. The stator is fixed thus the rotor rotates. Torque is written in terms of the force on
the rotor and its sign convention is the same as that for angles (Figure 2.12).
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