SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

3 Ne
j t
f Ie p
2 2
.
This was an acceptable result because is the same as that produced when the current SV
representing balanced sinusoidal currents, Equation (3.57):
3
i I pe
2
jt
is substituted into the relationship between the MMF SV and the current SV, Equation
(3.42):
N e
f i :
2
3 Ne
j t
f Ie p
2 2
.
However, the real part of Equation (3.49) (which was derived using only space vectors)
is:
3 N
f t Ipt
2 2
e
, cos
,
whereas Equation (3.58) (which was derived using only real-valued expressions) is:
3 N
f t Ipt
2 2
e
, cos
Comparing the two it is seen that the result derived using SV analysis is missing the θ term that
we have relied upon to prove that the MMF SV represents a distribution in θ. (To clarify, we
began with a real-valued expression, rewrote it in terms of a SV, and found that those particular
SV expressions (such as Equations 3.58 and 3.59) represent a cosinusoidal distribution. Per the
above, we have not yet explicitly found that the particular MMF SV derived solely from SV
equations represents this distribution and it appears that it does not since the θ term is missing.)
The author’s first encounter with this was met with much concern regarding the SV’s
representation of distributed quantities, but it turns out that any doubt is unfounded and the two
forms are indeed equivalent. In fact, examining this potential discrepancy reveals the concept of
reference frame theory that is requisite to understanding the space vector theory and FOC. Earlier
in the chapter the MMF wave was given as Equations (3.6) and (3.7).
3 N e
(3.6): f ( , t)
I
p cos( t
)
2 2
.
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