SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

transformation matrix must be composed of row vectors that are the basis vectors of the old
coordinate system (αβ), which are found by inspection to be Equation (3.103). The transform is
therefore defined by Equation (3.104).
cos( r) sin( r)
α ;
sin( r) β
cos( r)
(3.103)
T α cos( r) sin( r)
T T
sin(
r) cos( r)
β
xd cos( r) sin( r)
x
dq
x
q sin( r) cos( r)
x
x Px
(3.104)
The transformation matrix in Equation (3.104) is the same as Givens rotation and in the literature
it goes by many names. In the popular literature and in this report it is called the Park
transformation (after R.H. Park) even though this is not the original transformation (see Appendix
E). It is clear that the complex equivalent of the Park transform is given by Equation (3.105),
where the superscript R denotes that the SV is defined in the rotor reference frame (in this report
any SV no so marked is understood to be in the stationary frame).
R j r x xe
(3.105)
It is worth showing how Equations (3.104) and (3.105) are equivalent.
R j r x xe
cos( ) sin( )
xd jxq x jx r j r
xd jxq x cos( r) jx sin( r) jx cos( r) x
sin( r)
x cos( r) x
sin( r)
xd jxq
jx sin( r) jx
cos( r)
xd cos( r)
x
q sin( r) sin( r)
x
cos( r)
x
R
x x jx
d q
(3.106)
As previously discussed the transform can be interpreted as rotating the vector or rotating the
coordinate system, but they are the same. However, a survey of the popular literature indicates
that there is some general confusion on the subject and it is indeed easy to get “disoriented,”
S
therefore the different perspectives will be discussed. In Figure 3.35 the SV is at from the
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