SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

However, Equation (D.2) is incorrect. Since the fundamental component of phase-B and phase-C
are displaced by 120
from the fundamental component of phase-A, the harmonic components
in phase-B and phase-C must be similarly displaced. That is, for each harmonic set the n th
component in each phase must be displaced by
120 / 360 2 / 3 of a fundamental electrical
cycle from the n th component in the other phases, not 120° of that harmonic electrical cycle as
Equation (D.2) would imply. The period of the n th
harmonic is 360 / n , thus the angle
corresponding to a 120° delay in harmonic measure is equal to a 120 / n
delay in fundamental
measure. Therefore to make the phase-B and phase-C 120
harmonic delays equal to the phase-
B and phase-C 120
fundamental delays, the 120
harmonic delay must be multiplied by n,
as shown in Equation (D.3). In other words, there is noting special about the harmonic
components themselves. If they were indeed defined by Equation (D.2) their phase offsets would
all be 0, ,
, exactly like those of the fundamental, and they would all sum to zero just the
same. It is the fact that harmonics are 120 fundamental-degrees apart that gives them different
properties in a three-phase system. (Again, the reason that the harmonic components are defined
to be 120 fundamental degrees apart, instead of 120 harmonic degrees, is because the harmonics
of each phase are given by Equation (D.1), thus the entire set of harmonics in a phase are
displaced by 120 fundamental degrees from the set of each other phase.)
xnA() t Xncosnt
xnB
() t Xncosntn
xnC
() t Xncosntn
xnA() t Xncosn t
= xnB
() t Xncosnt
xnC () t Xncosnt Equation (D.3) is now evaluated for the fundamental and the first two harmonics.
(D.3)
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