SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

given in Equation (3.47). The result is Equation (3.59), where
f , ( t) f( ,
t)
.
Ne
, Reˆjj
f t Ie e
2
N
f t I
2
e , ˆcos
f , ( t) f( ,
t)
just as
(3.59)
The current SV describes the combined effect of all three phase currents and the orientation of the
windings. Regardless of the amplitude and angle of the current SV, it is seen that the MMF
retains its cosinusoidal distribution; this matches the description from Part I.
It has been shown that the MMF SV represents a cosinusoidal distribution of the stator MMF
regardless of the nature of the phase currents; for the special case of balanced sinusoidal currents
the MMF SV is simply the complex representation of the same real-valued MMF expression from
Part I. Now to further discuss the cosinusoidal distribution it would be useful to show how each
component MMF SV represents a cosinusoidal MMF distribution about its basis vector. Then the
special case of balanced sinusoidal currents will be examined again in this light. Returning to
Equation (3.56), the current space vector can be expanded and the constant of proportionality can
be redistributed as in previous discussions, such as those that developed Equation (3.36),
Equation (3.37), and Equation (3.48).
Ne
j
f , t Re
2
i e
Ne
j j j
Re A( ) 1 B( ) C(
)
2
i t i t e i t e
e
fA fB
fC
Ne j Ne j j Ne
j j
Re iA( t) 1 e iB( t) e e iC( t) e e
2
2
2
f A f B f C
Ne Ne Ne
iA( t) cos(0 ) iB( t) cos(120 ) iC( t)
cos( 120 )
2
2
2
Ne
f , t iA( t) cos( ) iB( t) cos( 120 ) iC( t)
cos( 120
)
2
(3.60)
Equation (3.60) is seen to be the same as Equation (3.2). Comparison of the third and fifth lines
that developed Equation (3.60) shows clearly that each component MMF SV represents an MMF
106