SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

called the per-phase torque function. As usual, the uppercase K represents a constant and the
lowercase k represents a function.
T ( r )
kt
( r ) K
t sin( r )
(2.33)
i(
t)
Equation (2.33) is plotted in Figure 2.22; current is held constant. Since it is obvious that the
torque function takes the same shape as the airgap flux density, 6 the torque produced by the
trapezoidal motor is found graphically as shown in the figure. Clearly the polarity of current will
need to match the polarity of the torque function to produce useful torque; this is discussed at the
end of this chapter.
Figure 2.22 – Torque functions of sinusoidal and arbitrary trapezoidal motors.
Expressing torque in terms of the torque function allows easy manipulation of the expression,
even if the torque function is not easily described mathematically. This is demonstrated in
Equation (2.34) and the corresponding simulation diagram is shown in Figure 2.23.
T t r
( t)
k ( ) i(
t)
(2.34)
Figure 2.23 – Simulation diagram for per-phase torque function.
6 To be clear, in this case the torque (a function of rotor position as measured with respect to the stator)
takes on the shape the rotor flux density (a function of the angle about the rotor axis), i.e. replace α with θr.
This is true for the simple concentrated full-pitch winding but is not true in general, as will be discussed
later.
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