SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

The component MMFs act away from the center of the machine thus they could possibly all
cancel, but since the currents sum to zero their polarity is such that the component MMFs always
sum to a nonzero value. This is seen clearly in Figure 3.7 and Figure 3.8 by examining the dashed
lines which represent maximum MMF components—in each case at least one instantaneous
component MMF had a polarity opposite the others.
We should expect the scaling to occur regardless of the nature of the currents. For the case of
balanced sinusoidal currents the summation results in a constant amplitude traveling wave 3/2
larger than the amplitude of a standing wave created by one phase alone. It turns out that this
factor of 3/2 does exist for nonsinusoidal currents but it is more difficult to see and more
difficult to define, though it will be treated here.
From a linear algebra point of view we should expect the possibility of scaling in Equations
(3.75) and (3.78). However the scaling is not easy to understand in terms of the traditional
eigenvalues and eigenvectors for several reasons. Instead the scaling could be studied by
examining the magnitude of the SV and the magnitude of its projections onto the αβ axes. It is
easiest to begin with balanced sinusoidal phase variables; most of these results have already
appeared earlier in the chapter but are collected here for clarity. Balanced sinusoidal phase
variables are given by Equation (3.91).
xA X pcos(
t)
xB
X pcos(
t)
xC
X pcos(
t) (3.91)
The phase variables may be substituted into any of the SV transform definitions. It is easiest to
rewrite the phase variables as their complex equivalents, Equation (3.92),
1 j j
xA
2 e e
1 j( ) j(
) xB e e
2
1 j( ) j(
) xC e e
2
(3.92)
and substitute them into the complex polar definition of the SV (Equation 3.69) and simplify to
get Equation (3.93).
3 j t
x k X pe
2
(3.93)
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