SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

(3.7):
3 Ne
f ( , t) Ipcos(
t)
2 2
At that time the argument of the cosine was briefly discussed and now we return to that
discussion. It was shown that Equation (3.7) gives the traveling wave as expected, while
Equations (3.6) is interpreted as giving the value of MMF at some observation point θ, as a
function of time. In the simplest interpretation, taking the real part of Equation (3.49) yields
Equation (3.6) for 0 .
3 Ne jt 3
Ne
Re Ie p Ipcos( t), 0
2
2
2
2
Therefore, it can be interpreted that the real part of the MMF SV represents the value of MMF at
the point of observation 0 , ensuring us that the MMF SV does indeed represent the
cosinusoidal distribution even if the θ terms is not present. But this might then raise questions as
to how Equation (3.6) represents a cosinusoidal distribution in θ without having θ in the
argument, or how it can remain cosinusoidal even if the currents are not. Assume the currents do
not form a balanced sinusoidal set, but let the angle of this arbitrary current SV be defined as
as usual. Whatever it is, it will vary with time: () t . Substitution into Equation (3.6) and
evaluating at 0 gives Equation (3.62).
3 Ne
f( , t) Ipcos
( t)
2 2
3 Ne
f( , t) Ipcos
( t)
2 2
(3.62)
As before, the MMF is known to exist in the airgap, thus Equation (3.62) must describe some
function of the angle around the stator. This shows then that the currents can be arbitrary and the
distribution is still cosinusoidal, although it is perhaps not immediately obvious unless the form
of Equation (3.7) is used. Of course, we should certainly expect this to be the case since
Equations (3.6) and (3.7) are equal to one another.
Now that the differences are resolved, an additional interpretation is presented. It was just shown
that the equations with θ were the same as those without, simply because in the latter case, 0 .
Figure 3.9 shows the distribution in θ, but if we set 0 it is clear that the equations give the
peak of the MMF wave, measured from 0 . The reason that 0 in the SV equations is that
the SV is defined in the stationary reference frame affixed to the stator. That means angles are
defined as being measured from 0 . The importance of this is that if we can set 0 , we can
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