SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

lumped-parameter model in this report and in most references. [104] provides details and
advanced theory regarding the interpretation of the SV, including the distribution aspect.
The SV as a Linear Transformation; Reference Frame Theory
The third and final facet of the SV to study is the fact that the definition of the SV is a linear
transformation of variables (LTV). The transformed variables are the basis of reference frame
theory thus the two are covered together. The previous two sections in Part II already touched on
the LTV concept. The development of the vector aspect showed that the definition of the SV was
3
to a complex variable in (the SV). The
a mapping (a LTV) from phase variables in
development of the distribution aspect mentioned the concept of reference frame theory in which
quantities were defined with respect to the phase-A axis. Now the LTV aspect of the SV can be
pursued further.
Clarke Transform
The complex SV is defined [42, p.396], [87, p.33], [88, p.84] as either the polar form of Equation
(3.69) or the equivalent rectangular form of Equation (3.70).
jj x k
xA1xBe xC e
(3.69)
1 1 3
x kxA xB xC
j xB xC
2 2 2
(3.70)
Regardless of the definition, an arbitrary SV (shown in Figure 3.29) can be described in terms of
its magnitude and angle, Equation (3.71).
ˆ j
x Xe
(3.71)
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