SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Although Equations (3.6) or (3.7) are a standard result, the argument of the cosine is one of the
more interesting aspects of AC machine theory. The angle θ is a spatial angle around the stator
and the angle t is an electrical angle of a scalar circuit variable, yet the two appear together in
the argument. The MMF exists in the airgap around the stator so the cosine describes the MMF as
a function of this angle, irregardless of the t term.
In Equation (3.6) ωt is usually thought of as the independent variable along the abscissa and the
variable in the position of θ is a delay. If an observer were stationed at the position θ, the equation
would describe the magnitude of MMF at that position as a function of time, as shown in Figure
3.10. The observer at θ sees the peak / seconds later than an observer at zero (black line);
hence the delay. It is helpful to mentally verify this result. Assume the currents are at zero so the
peak of MMF wave is at zero as well. If the currents increase in angle the peak will move CCW
with time. It can be seen that the observer at θ will indeed see the peak at a time greater than zero.
Figure 3.10 – Observing MMF value at fixed position.
In contrast to the interpretation of Equation (3.6), in Equation (3.7) the independent variable is θ
and the delay is t . Instead of fixing the observation point, the point in time is fixed and the
equation describes the traveling wave shown in Figure 3.9. As t increases the wave is delayed
in θ. A delay makes the peak occur later in θ (at a larger positive value) thus this correctly
describes the traveling wave (rotating CCW). Equation (3.6) describes the peak MMF from at an
observation point and Equation (3.7) describes the instantaneous distribution at a point in time,
yet they are the same equation and describe the same MMF. This concept seems universally
accepted in the literature without consequence or mention (a good sentence on it is found in [43,
p.748], followed by, “This basic tenet is essential for the understanding of AC motor control”).
This important concept is further investigated in Part II and we will revisit this discussion at that
time.
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