REVOLVING REFERENCE FRAME (Induction Motor)

vectors, but the vectors keep revolving nevertheless. Consequently, their d and q components are ac variables, which are less convenient to analyze and utilize in a control system than the dc signals commonly used in control theory. Therefore, in addition to the static, abc—>dq and dq-»abc, transformations, the dynamic, dq->DQ and DQ—»dq, transformations from the stator reference frame to a revolving frame and vice versa are often employed. Usually, the revolving reference frame is so selected that it moves in synchronism with a selected space vector.
The revolving reference frame, DQ, rotating with the frequency coe (the subscript “e” comes from the commonly used term “excitation frame”), is shown in Figure 6.4 with the stator reference frame in the background. The stator voltage vector, vs, revolves in the stator frame with the angular velocity of a>, remaining stationary in the revolving frame if coe = (o. Consequently, the vDS and vQS components of that vector in the latter frame are dc signals, constant in the steady state and varying in transient states. Considering the same stator voltage vector, its dq—»DQ transformation is given by

FIGURE 6.4 Space vector of stator voltage in the stationary and revolving reference frames.

To indicate the reference frame of a space vector, appropriate superscripts are used. For instance, the stator voltage vector in the stator reference frame can be expressed as

and the same vector in the revolving frame as

where ®e denotes the angle between the frames. Angles s and ©e are given by

Motor equations in a reference frame revolving with the angular velocity of coe can be obtained from those in the stator frame by replacing the differentiation operator p with p + jioe. In particular,

Equations that do not involve differentiation or integration, such as the torque equations, are the same in both frames.