SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

other methods, although it has close relationships to the other methods (phase-variable time-
domain analysis and phasor analysis).
A phasor can describe any sinusoidal scalar quantity. Although the phasor is used in three-phase
analysis, the phasor is used to describe the quantities associated with only one of the phases (the
reference phase, often phase-A) with the understanding that the system is balanced, hence
information about the other two phases can be found by adding or subtracting a 120° phase shift.
The analysis of one phase is often called “single-phase equivalent” analysis [39, p.547] and it is
almost always coupled with phasor analysis (that is, the single-phase equivalent circuit is
j
analyzed using phasors). The phasor ( X
Xe X)
is a complex-valued entity and the
real part represents the scalar quantity of interest; the complex entity does not necessarily have
any physical interpretation as a vector, it is just a convenient mathematical form. In contrast, the
()
SV ( ˆ j t
x Xt () e
) is a complex-valued notation that describes a vector quantity that exists in
the complex space; the entire complex entity has a physical interpretation as a vector (as opposed
to only the real part). Of course the complex plane has no physical meaning but it is mapped to a
real polar coordinate system that does have physical meaning. Yet, like a phasor, there is also
meaning in only the real part of the SV. The complex notation exists to aid in mathematical
manipulation (as with a phasor) but in addition the complex notation carries information about
quantities physically distributed in the machine.
Neither the SV nor the phasor can represent the zero-sequence component. Phasor analysis is
only valid for circuits with balanced sinusoidal excitation in steady state whereas SV analysis is
valid for arbitrary phase quantities and can describe transient behavior. When phasor analysis is
coupled with the single-phase equivalent circuit the phasor “represents” all three phases working
together, but only because the machine is balanced and the phase power can be multiplied by
three to obtain total power. In contrast, the SV directly accounts for (contains information about)
the total effect of each phase, even if the contributions of the phases are not equal.
It will be seen that there are several aspects of SV theory. (Compare this with the phasor.
Depending on the analysis, it may be treated as a stationary vector, a rotating vector, or a complex
number, yet these are all simply different ways of looking at the definition.) The vector nature of
the SV allows for intuitive graphical presentations similar to phasor diagrams. The SV is a
complex number that can be easily manipulated mathematically. The SV represents rotating
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