SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 2.32 – Torque production in trapezoidal BPMS motor.
Although the exact same mechanism is responsible for torque production in the two types of
motors, comparing Figure 2.30 and Figure 2.32 makes it clear that the interaction of the current
and the torque function in each motor causes a very different per-phase contribution to torque in
the two types of motors. These are the desired modes of operation that typify sinusoidal and
trapezoidal motors.
Torque & Back-EMF Constants
In earlier subsections, Equations (2.42) or (2.52) demonstrated that the per-phase torque and
bEMF functions [kt(θr), ke(θr)] are identical for a given motor and thus the per-phase torque and
bEMF constants (Kt, Ke) are identical (Equation 2.48). Regardless of the motor type, the perphase
Ke is defined as the peak value of line-neutral bEMF per unit angular velocity and the perphase
Kt is defined as the peak value of torque per unit peak current. These definitions are always
true, but they are defined in terms of line-neutral quantities which are not always measurable
when a neutral connection is not available; they are rarely found on motor datasheets. The
previous section analyzed torque production of the overall motor and it was observed that when
the current is properly controlled, total torque in a sinusoidal and trapezoidal motor were given as
Equations (2.57) and (2.58), respectively. Since in practice one is only concerned with the total
torque, an “overall torque constant” can be defined for each motor by dividing each equation by
the peak of the phase current, Ip. These torque constants are defined by Equation (2.59), where
Ke=Kt. The uppercase subscript indicates that the constant describes total torque, not per-phase
torque.
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