SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Part II – Space Vectors
Part II introduces the space vector, which is a complex-valued quantity defined by a linear
transformation of three-phase variables. The space vector is the basis of space vector analysis
which can be used to analyze three-phase circuits and machines. There are some intuitive aspects
of the space vector concept but other parts are more abstract and difficult to grasp upon first
encountering them. The vector nature of the space vector is easy to understand, so it is developed
first by using a simple example. Then the example is extended and the space vector’s
representation of distributed quantities is illustrated by building upon the understanding of stator
MMF distributions presented in Part I. (The MMF space vector is the “connection point” between
the familiar understanding of the machine and the more abstract representation by space vectors.)
Finally the more abstract nature of the space vector transform is studied. The concept of reference
frame theory will be encountered repeatedly and each time the understanding will be developed
further.
A concise introductory statement regarding space vector analysis is given in the foreword to [88]
by Dr. Donald Novotny, Professor Emeritus, University of Wisconsin–Madison:
“Part way through my academic lifetime I was introduced to the next phase of unified
theory, the use of complex notation to model the effective spatial orientation of quantities
within a machine. This concept, often called space vector theory, provides a much clearer
mathematical picture of what is happening in a machine, but at the expense of another
level of abstraction in the model. However, the insights provided to one initiated in the
method are so significant that today essentially all work in drive control is presented in
this format. And therein lies a problem. To the uninitiated these presentations appear
quite unintelligible. And a route to becoming initiated is generally hard to find and often
harder to follow once found.”
The author agrees heartily with these sentiments, particularly the last sentence. There are two
extremes of space vector usage in the literature. In the popular literature space vector theory is
used sparingly; many articles of popular literature incorrectly or inconsistently apply the theory. It
seems as if the authors “copy and paste” from the academic literature and do not explain the
equations they use. Perhaps this would be acceptable if space vector theory were a simpler and
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