SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

x Re x (3
/ 2) X cos( t)
. There is an obvious connection between this projection and
p
Equation (C). By taking the real part of a SV we are asking for the projection onto the real axis,
which is a cosine (projection onto the imaginary axis would give a sine). The way to use a
complex SV to represent the MMF (which we know to be real-valued and cosinusoidal) is to
f ( ,
t) Re f . This principle is not much different than that of the phasor—we define
define
the signal of interest to be associated with the real part of the inverse transform because that is
xt () Re
j t
X e
. This is the reason that the complex
what is mathematically valid:
conjugates appeared in the identity (that identity is the same rearrangement of Euler’s formula
that gives us the complex-valued equivalents for the cosine and sine functions).
Thus, the Re in Equations (A) and Equation (C) are present for the same reason, though they
were derived in different ways. With that out of the way we can then see that the discrepancy
j
would boil down to the question as to why Equation (A) has Rei
e
while Equation (C) has
Rei
. The simple answer is that Rei
j
is equal to Rei
e
when 0 . When Equation
(A) was developed, θ was left explicit. Equation (C) was developed using SV theory and at that
time only the stationary reference frame had been used, thus 0 . The two equations are
j
therefore equivalent. Finally, this shows that the multiplication by e
is the very same change
of reference frame we have been working with in this subsection. In Part I the argument of
Equation (B) was discussed and it was said that when we substitute a value for θ we are asking
for the value of MMF at that angle about the stator. Further discussion revealed that it was the
same as asking where the peak is in relationship to any position θ on the stator. This concept is
j
Re i e I cos( t) melds
exactly the reference frame change. That is, the relationship
the physical understanding with the SV perspective. What is described here is the general
reference frame (i.e., an arbitrary standpoint in θ from which we can observe something). When
the MMF SV defined in the stator reference frame, f , is multiplied by
j
e
p
, the MMF becomes
described in whichever reference frame is defined as being displaced by θ from the phase-A. If
r
, the reference frame is the rotor frame.
Perhaps obviously, the inverse Park transform is found by inverting P. Since P is orthogonal the
inverse is simply the transpose and the inverse transform is given by Equation (3.107).
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