SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

j
set θ to anything. Taking the real part of (a current SV multiplied by e
) gave cos( t ) and
θ disappears when it is set to zero. But in either the SV or real-valued equations, if we set θ to
anything else, the equations define the peak of the MMF wave relative to that point θ, rather than
relative to zero; this is the meaning of reference frame theory. While the use of the phase-A axis
as a reference was declared in the beginning of the chapter (see Figure 3.1), this discussion
provides a new insight into the MMF equation that relates the position of the peak of MMF to the
distribution of the MMF. This discussion may fail to illuminate the concept, but as we revisit it
throughout the remainder of Part II it should eventually become clear.
Current Space Vector and its Interpretation
The previous subsection demonstrated how the MMF SV f represents the actual MMF
distribution in the machine; the MMF is the only true distribution represented by the space vector.
In this section it is shown that the current SV i does not represent a distribution but that it does
have close association with f . This relationship is the connection between the physical
understanding of the machine and the more abstract space vector model; in other words, the
relationship is the basis for using SV theory to describe a machine (this basis will be finalized in
Part III).
From Appendix C, the sinusoidal winding density for phase-A is given by Equation (3.63) and
the number of ampere-turns for phase-A is given by Equation (3.64).
Ne
nA( ) sin( )
(3.63)
2
Ne
dAiAnA( ) iA
sin
(3.64)
2
For a single phase each function is fixed to the stator but when all phases are considered the total
effect of current in those windings is the same as if an equivalent winding were rotating about the
stator. This has direct similarity to the way the MMF component of each phase is fixed in space
but together they produce a rotating MMF. This concept is sometimes referred to as the “rotating
ampere-turn distribution” [69, p.6.8]. Similar to the way the MMF wave was developed we can
find an expression for the overall amp-turn distribution due to the winding currents, Equation
(3.65), where balanced sinusoidal currents are used.
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