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Park transform


The direct-quadrature-zero (DQZ or DQ0 or DQO, sometimes lowercase) transformation or zero-direct-quadrature (0DQ or ODQ, sometimes lowercase) transformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.

   The DQZ transform is often used in the context of electrical engineering with three-phase circuits. The transform can be used to rotate the reference frames of ac waveforms such that they become dc signals. Simplified calculations can then be carried out on these dc quantities before performing the inverse transform to recover the actual three-phase ac results. As an example, the DQZ transform is often used in order to simplify the analysis of three-phase synchronous machines or to simplify calculations for the control of three-phase inverters. In analysis of three-phase synchronous machines the transformation transfers three-phase stator and rotor quantities into a single rotating reference frame to eliminate the effect of time-varying inductances.

The DQZ transform is made of the Park and Clarke transformation matrices. The Clarke transform (named after Edith Clarke) converts vectors in the ABC reference frame to the XYZ (often αβz) reference frame. The primary value of the Clarke transform is isolating that part of the ABC-referenced vector which is common to all three components of the vector; it isolates the common-mode component (i.e., the Z component). The power-invariant, right-handed, uniformly-scaled Clarke transformation matrix is

   To convert an ABC-referenced column vector to the XYZ reference frame, the vector must be pre-multiplied by the Clarke transformation matrix:

And, to convert back from an XYZ-referenced column vector to the ABC reference frame, the vector must be pre-multiplied by the inverse Clarke transformation matrix:

   The Park transform (named after Robert Park) converts vectors in the XYZ reference frame to the DQZ reference frame. The primary value of the Park transform is to rotate the reference frame of a vector at an arbitrary frequency. The Park transform shifts the frequency spectrum of the signal such that the arbitrary frequency now appears as "dc" and the old dc appears as the negative of the arbitrary frequency. The Park transformation matrix is


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