SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

x cos( r) sin( r)
xd
x
sin( r) cos( r)
x
q
1
x P x dq
(3.107)
In terms of the SV, since we rotated by r
to transform to the rotor frame, we will now rotate
by r
to transform from the rotor frame, as shown in Equation (3.108). 26
R j r x x e
(3.108)
There are many other reference frames used and each is suited to a particular purpose. But since
they only differ in the speed or position that they are referenced to, the math is always the same,
hence the usefulness. Induction motor analysis is laden with several different frames because the
rotor flux is not fixed to the rotor. These various reference frames are responsible in part for the
wide variety of vector control schemes in the literature (rotor-flux-oriented, stator-flux-oriented,
magnetizing-flux-oriented) and each serves its own purpose. Thankfully, only the rotor frame is
required when working with the synchronous motor (again, other frames can be used).
26
Per Equation (3.105), transforming to the rotor frame requires a multiplication by exp( jr
) . Per
Equation (3.108), transforming back from the rotor frame requires a multiplication by exp( jr
) as one
would expect. Only transformation to the rotor frame was discussed from both the perspective of
coordinate rotation and SV rotation. It was found that the former required exp( j r ) and the later
required exp( j r ) , but both were for transforming to the rotor. Yet we have just seen that transforming
from the rotor also required exp( j r ) . Thus we can correctly infer that Equation (3.108) is like
Equation (3.105)—it is written for the coordinate rotation perspective. And also that if transformation from
the rotor were performed from the perspective of SV rotation, it would require exp( j ) .
r
137