SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

distributions that agree with the physical description of the motor’s operation. Finally, the SV is a
linear transformation of variables (LTV). Although the other aspects of the SV are obviously
important, this LTV aspect could be said to be the most important and is the basis of the method.
Simply stated, applying SV analysis consists of deriving the electrical model of a three-phase
electrical or electromechanical system (expressed in terms of phase variables) and then
transforming the phase variables to a different, two-dimensional orthogonal coordinate system
(the stationary reference frame). As with any LTV, the SV transform can be expressed as a
matrix. When the phase-variable model is expressed in matrix form, all of the standard linear
algebra techniques apply. The SV is essentially the complex-valued representation of the
traditional vector-matrix two-axis, two-reaction, or dq theories (though there are slight
differences between these terms and these will be explained as they are encountered).
The most fundamental goal in motor control is the control of torque. The understanding of torque
production requires a physical understanding of machine operation and SV theory can provide a
graphical representation of this understanding. Since the transformation of variables is the basis
of SV analysis and the motor control schemes that utilize it (such as FOC) it might make sense to
begin with an examination of this transformation. But the interpretation of the LTV aspect is
abstract compared to the physical aspect of the SV. Therefore the SV is developed using physical
concepts before the more abstract portions are presented.
Organization
The SV theory presented here in Part II is voluminous enough to warrant its own chapter, and this
may seem fitting since the theory is not specific to machine analysis. However, it is positioned at
this point to aid in learning and integration. SV theory could be thought of as being purely
mathematical (and this is the approach of some texts), but it seems to be best presented by
building upon the physical description of the machine that was presented in Part I, thus this
material should be located close to Part I. In this report, the primary purpose of using SV theory is
to model the machine (to be compatible with the control system and inverter schemes that are
likewise based on SV theory). Since the machine is modeled in Part I and SV theory is presented
in Part II, the SV theory applied to the machine is presented in Part III. This makes for a long
chapter but also facilitates integration that would be difficult otherwise. References in the
literature take different approaches but several texts ([73], [78], [87]) begin with an approach
similar to that used here. Some of the more technical references on SV math include [36], [84],
[85], [86], [87], [89].
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