SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

The potential discrepancy (p.108) will first be recounted. The real-valued stator MMF
i( t)
cos( ) i ( t)
cos( 120
) i ( ) cos( 120
)
N e
( , t)
A
B
t
2
f C
was put into complex form
f , t
Ne
Re
2
j
i e
(A):
and found to be equivalent to the real-valued MMF description found in the beginning of
the chapter
3 N
2 2
e
(B): f , t I cost
p
.
Thus, all that had been done was to put the traditional real-valued result into complex
form and check that it was correct. Then the stator MMF SV was derived using only SV
analysis
N e
f i
2
and when the SV corresponding to balanced sinusoidal currents
3 j t
i Ipe 2
was substituted in
3 Ne
j t
f Ie p
2 2
it was found that when the real part (c.f. Equation A) was taken
f , t Re
f
Ne
Re 2
i
3 Ne
Ipcos
t
2 2
(C):
the result did not match Equation (B).
First we should ask why Re is present in Equation (A). The reason is that we had an
expression for MMF (Equation 3.53, p.104) which contained the sum of complex conjugates and
an identity (Equation 3.55) was used to eliminate one conjugate. But we have seen that the SV
ˆ j
x Xe
is defined with respect to the phase-A axis (that is, ξ is measured CCW from 0° which
is defined to be coincident with the phase-A axis). When the SV is one that represents balanced
j t
sinusoidal quantities, x (3 / 2) X e
, the projection onto the real axis is
p
135