SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Trapezoidal and Sinusoidal BPMS Motors
This chapter has developed the per-phase model of a motor and the previous section described
how three of these can be combined to give the complete model of a three-phase motor. Now that
we have this model, the torque production of three-phase trapezoidal and sinusoidal motors will
be examined. This provides an explanation of each motor’s operation, which facilitates a
discussion of whole-motor torque constants and the ideal type of winding current control
(electronic commutation) required of each motor. Finally, the chapter concludes with a
commentary regarding the differences between the trapezoidal and sinusoidal motor. While that
discussion may not be useful to the general reader the conclusions are crucial to the remainder of
the report.
Torque Production
The per-phase torque functions for the sinusoidal and trapezoidal motor types were shown in
Figure 2.22. The per-phase torque produced as a function of rotor position will match the torque
function in shape when a DC current is forced through the winding. In this case if a hand crank
were attached to the shaft and rotated through one revolution, the operator would have to exert
positive (CCW) torque on the crank from 0°-180°, whereas the operator would have to resist the
force of the crank (by applying negative (CW) torque) from 180°-360° as it seeks to return to the
equilibrium point from which it started (both the 0° and 180° positions are equilibrium points).
The force would vary with position according to Figure 2.22, depending on the type of motor.
The concept of a torque function is an important one because it shows this bipolar symmetry that
is present in every PM or wound-field machine. Since the product of bEMF and current divided
by speed is torque, this makes it clear that both DC and AC motors require the polarity of the
armature current to reverse each electrical cycle in order to produce always-positive (but not
constant) torque.
In a traditional brush DC motor the torque function of each winding is made to look something
like that of the trapezoidal motor. The commutator reverses the current flow in each armature
winding every 180°. There are several armature windings, so as one winding is energized another
is de-energized, thereby producing a nearly-constant torque. AC motors do not have a brushcommutator
system and therefore require that the terminal current be reversed every half cycle by
electrical or electronic means. For this reason the action of controlling the current relative to the
shaft position in a brushless motor is sometimes called electronic commutation. There are two
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