SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Appendix E - Park Transforms
This appendix discusses a few forms of the Park transform and the issues related to their axis and
angle conventions.
In Chapter 3 and Appendix D it was explained that the Clarke transform defined in this report
(and found in the popular literature) is not the original Clarke transform. The original more
closely resembles the αβ0 transform which contained an extra row in the matrix to give the zerosequence
component. Similarly, the Park transform can be defined to transform the zero-sequence
component as well. This zero-sequence form is shown in Equation (E.1).
xd cos( r) sin( r)
0x
x
sin(
) cos( ) 0
x
q r r
x 0 0 0 1
x 0
By definition, the ZS component remains unchanged in the transformation.
(E.1)
Unlike the Clarke and αβ0 transforms, there is not only a ZS and non-ZS version, but also a
completely different version of the Park transform, which is a combination of the Park and αβ0
transforms. This is the “original” Park transform is the one used by Park and those before him
(although like the Clarke transform, the scaling and ordering of variables may be different) [22],
[30], [32], [34], [87]. It is obtained by multiplying the ZS-Park and αβ0 transforms. After
simplification the result is given by Equation (E.2).
x cos( ) sin( ) 0 1 1/2 1/2
d r r
x
x
sin( ) cos( ) 0
k 0 3 / 2 3 / 2
x
q r r
x0 0 0 1 1/2 1/2 1/2
x 0
xd cos( r) cos( r ) cos( r )
x
x
q k
sin( r) sin( r ) sin( r )
x
x 1/2 1/2 1/2 x 0 0
(E.2)
An easier derivation that avoids the algebraic simplification can be had by using SV notation.
When the real and imaginary parts of Equation (E.3) are taken and put into matrix form as usual,
the result is the same as Equation (E.2) (since the SV does not transform the ZS component it
must be manually added, as in the case of the αβ0 transform above.
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