SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

which are required to understand modern motor control. An attempt has been made to provide
explanations that are as simple as possible and easy to understand but are comprehensive enough
to give the reader an understanding of space vector theory that can assist in migrating between
references in the literature. If there is any one thing the reader should realize, it is that the theory
can be approached from many different angles, all of which may be correct; therefore, one should
not absorb or stick to one “interpretation” too rigidly (including the interpretation in this report).
Introduction
The meaning of the space vector and the nature of the application of its associated theory should
become clear in reading this report. The theory and its application are intertwined and for this
reason the material is difficult to organize and present; it is even more difficult to provide a
concise overview. Oddly enough, when the details are correctly understood the application is easy
to grasp and an overview is not as necessary as might be expected, but an attempt has been made
to provide one here. A good starting point is to compare the space vector with the phasor (a more
detailed comparison with the phasor is given in the last subsection of Part II).
Similar to the phasor, the space vector (SV) is simply a mathematical construct designed to ease
analysis of machines and circuits; whereas the phasor exists to analyze sinusoidal quantities, the
SV exists to analyze three-phase quantities. The phasor is a complex-valued representation of a
scalar sinusoidal signal. The sinusoid is the basis of linear system analysis since it is the only
signal that can pass through a linear time-invariant system without a change in shape. Further, a
sinusoid is completely described by its amplitude, frequency, and phase; since the phasor contains
this information, it is a convenient tool to represent a sinusoid. The phasor’s vector nature is
convenient for representing magnitude and phase, while its complex-exponential nature make it
amenable to mathematical manipulation. If the voltages and currents in a circuit are represented
using phasors, it is then possible to define a complex impedance and a complex power such that
linear circuits may be analyzed using only complex numbers. This analysis is so compact and
rapid that it might be termed an entirely new type of analysis, such as “complex AC analysis,” to
distinguish it from time-domain analysis. In addition to the speed and ease of use, the method
provides a graphical presentation that helps visualize the magnitude and phase relationships
between quantities. However, the entire analysis is built upon nothing more than a definition of
the phasor and some supporting assumptions and conventions. In a similar way, the space vector
is simply a mathematical contrivance with associated conventions, but an entire method of
analysis has been built upon it; it might be called “space vector analysis” to distinguish it from
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