SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

j j
j r
A B C
k x x e x e e
jr j( r) j(
r) A B C
k x e x e x e
(E.3)
Like the Clarke transform, the Park transform goes by many names, such as the dq0 transform,
Blondel transform, and commutator transform. The names most generally refer to Equation (E.2),
not Equation (E.1), but this is not always true. In Appendix D, when the Clarke transform was
discussed it was mentioned there are two primary axis conventions, and that the vector is
commonly ordered as either αβ0 or 0αβ. Unfortunately, with the Park transform comes a large
number of choices for conventions, and many are in use. Like the Clarke transform, one axis can
be defined in the negative direction, and the vector can be arranged as dq0 or 0dq. However, if
the axes directions for the αβ and dq axes are different this can lead to arrangements, such as qd0
or 0qd. Finally, the rotor position may be measured to the d or q axis (these are the two different
“correct” absolute rotor positions mentioned in Footnote (8) in Chapter 2). Finally, the rotor flux
may be defined to be coincident with either the d or q axis. Of course, all of these varieties have
different orderings of the elements of the transform matrix and the convention choice affects
many other aspects. There is again the issue of scaling constants, both for the dq components and
the ZS component. Since only the 2x2 Park transform is used in this report it is not necessary to
elaborate further, however some common variations are shown in Figure E.1 to illustrate the
issue. In most varieties the q axis leads the d axis in the direction of rotation, but this is not
always so.
Figure E.1 – Some common axes and rotor angle conventions.
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