SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

theory and will be very useful. Since it is vital to understand reference frame theory, and since the
distribution of the MMF SV coincides with the physical distribution, it is the author’s opinion
that studying the MMF’s distribution is the only connection point between the familiar physical
understanding of the machine and the abstract SV representation. With this opinion in mind it is
first shown how the stator MMF SV represents the actual MMF distribution produced by the
stator current. Then the previously-defined relationship between the stator current SV and the
stator MMF SV is discussed to reveal the interpretation of the current SV.
MMF Space Vector and its Distribution
This subsection demonstrates how the MMF SV represents the actual MMF distribution in the
machine. We will also gain a more solid understanding of reference frame theory.
In the CRT example the concept of a Cartesian voltage vector was reasonable because the force is
a linear one which acts in the plane perpendicular to the electron beam. But in a synchronous
machine the stator MMF does not simply act along a vector in a plane—it is a distribution about
the stator periphery. It is obvious from the development of the MMF wave earlier in the chapter
that the MMF distribution is cosinusoidal in θ, but this is perhaps not obvious from the math
presented thus far. To show that the MMF SV represents a cosinusoidal MMF distribution, return
to the real-valued expression of stator MMF developed earlier in the chapter (Equation 3.2) which
is reproduced here as Equation (3.51). Note that the currents are arbitrary and are not required
to be sinusoidal.
(3.2):
f( , t) fA( ) fB( ) fC(
)
Ne
iA( t) cos( ) iB( t) cos( 120 ) iC( t)
cos( 120
)
2
(3.51)
The cosine terms can be expanded into their complex equivalents using the identity in Equation
(3.52).
1 j
j
cos( )
ee 2
(3.52)
Substituting the resulting expressions into Equation (3.51) and simplifying yields Equation
(3.53).
Ne
1
j j j j j j
f , t
iA ie B ie C eiA ie B ie
C e
2 2
(3.53)
Recognizing that the first parenthesis contains definition of the current space vector (k=1) and the
second parenthesis contains its complex conjugate, this can be rewritten as Equation (3.54).
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