SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

distribution that is cosinusoidal in θ with the peak located at the basis vector corresponding to the
magnetic axis of that phase; this matches the physical understanding from Part I and is what we
set out to show. Now we examine the special case of balanced sinusoidal currents by again
considering Equation (3.56). Substituting the balanced sinusoidal currents of Equation (3.3) for
the general currents (or equivalently, substituting Equation (3.57) in for the current SV) yields
Equation (3.61), which is exactly Equation (3.4) found in the beginning of the chapter.
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
N Ipcos( t)cos( ) Ipcos( t120 )cos( 120
)
e f( ,
t)
2 Ipcos( t120 )cos( 120
)
(3.61)
Thus, not only does each component MMF SV represent a component distribution about its basis
vector, but for the special case of balanced cosinusoidal currents, each one of those component
distributions is a cosinusoidally-modulated standing wave cosinusoidally distributed in space,
exactly as in Part I.
In summary, this section has shown that in SV theory the component MMF SV of each phase
represents a cosinusoidally distributed MMF wave and these three component MMF SVs will
sum to yield an MMF SV that also represents a cosinusoidally distributed MMF; this has direct
correspondence to the real-valued descriptions of MMF developed earlier. That these SVs
represent cosinusoidal distributions is true regardless of the nature of the currents. It was also
shown that if the currents do form a balanced sinusoidal set, the distribution represented by the
MMF SV is a traveling wave of constant amplitude whose peak aligns with the electrical position,
just as the real-valued expressions developed in the beginning of the chapter. Since the MMF
distributions can be described accurately by the SV, SV theory can be used to analyze machines
whose currents do not form a balanced sinusoidal set. This stands in contrast to the single-phase
equivalent approach and is one reason why SV theory is more useful.
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