SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 3.29 – Arbitrary space vector in the complex/αβ plane.
For convenience the real and imaginary axes could be relabeled as the α and β axes, respectively
as shown in Figure 3.29. The component along the real axis is x and the component along the
imaginary axis is x . The space vector can therefore be decomposed as shown in Equation
(3.72), where both x and x are time varying—this is nothing but the standard parametric
decomposition of a vector (the SV and its components are instantaneous quantities that are
functions of time).
x Rex jImx (3.72)
x x jx
Putting the α- and β- components into a vector would allow a SV quantity to be expressed as a
real vector (Equation 3.73) and the SV transform itself to be expressed as a linear transformation
matrix (as in Equation 3.74).
x x
x
(3.73)
1 1
1
x A
x 2 2
k
x
B
x
3 3
0 x C
2 2
(3.74)
When the groups of variables are treated as vectors we can work with the complex SV by using
standard linear algebra techniques and rewrite Equation (3.74) as Equation (3.75).
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