SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Method of Symmetrical Components
The method of symmetrical components (MSC) [29], [32], [26] is a linear transformation of
variables that is used to study unbalanced three-phase circuits. A discussion is beyond scope but
because it bears much similarity to the αβ0 transform and space vector it is mentioned briefly.
The MSC (phasor) transforms the three-phase variables X a, Xb, Xc
components X0, X1, X2
into three sequence
(ZS, PS, and NS, respectively). The transformation is defined by
Equation (D.23) and the inverse transform by Equation (D.24).
X 1 1 1 X
X
a a X
X
a a
X
a
0
2
b
1
1 xabc Ax012
2 c 1 2
X 1 1 1 X
X
a a X
X
a a
X
0
a
1 2 1
1
1
b 012
abc
3
x A
x
2 2 1 c
(D.23)
(D.24)
In single-phase equivalent (SPE) analysis we analyze the phasor diagram for phase-A only;
phase-B and phase-C have diagrams of identical magnitude and phase but are displaced by 120°.
Unlike SPE analysis, the MSC is concerned with only one quantity (such as a load voltage) but all
three phasors Xa, Xb, Xc
of that quantity are considered because they are unbalanced (and
SPE is not applicable). Each sequence component X 0, X1, X2
is a single phasor for only one
phase (phase-A) but there is also a phase-B and a phase-C phasor for each component. Thus the
phasors are
X , X , X , X , X , X , X , X , X
a0 a1 a2 b0 b1 b2 c0 c1 c2
but we choose to only work with the ones for phase-A. Like any set of phasors, those for a
sequence component (such as 0 X ) are displaced by 120°. 55 The PS phasors are made to have the
PS order, the NS the NS order, and the ZS order as was shown in Figure D.1. This is the purpose
of the A matrix—it creates the various sequence phasors from those of phase-A simply by
rotating each by 0 or 120
. For illustration, the PS row in Equation (D.24) is broken out as
55 Since each of the three phasors for a sequence component are of equal magnitude but displaced by 120°
they are not linearly dependent. Thus the MSC takes three variables to three variables, not three-to-nine.
320