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 and transform the system into a linear time-invariant system
Introduction
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 αβγ 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 H. Park) converts vectors in the XYZ reference frame to the DQZ reference frame. The Park transform's primary value is to rotate a vector's reference frame at an arbitrary frequency. The Park transform shifts the signal's frequency spectrum 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 , where θ is the instantaneous angle of an arbitrary ω frequency. To convert an XYZ-referenced vector to the DQZ reference frame, the column vector signal must be pre-multiplied by the Park transformation matrix: . And, to convert back from a DQZ-referenced vector to the XYZ reference frame, the column vector signal must be pre-multiplied by the inverse Park transformation matrix: .
The Clarke and Park transforms together form the DQZ transform:
The inverse transform is:
To convert an ABC-referenced vector to the DQZ reference frame, the column vector signal must be pre-multiplied by the DQZ transformation matrix: . And, to convert back from a DQZ-referenced vector to the ABC reference frame, the column vector signal must be pre-multiplied by the inverse DQZ transformation matrix: .
To understand this transform better, a derivation of the transform is included.
Derivation
The Park transform derivation
The Park transform is based on the concept of the dot product and projections of vectors onto other vectors. First, let us imagine two unit vectors, and (the unit vectors, or axes, of the new reference frame from the perspective of the old reference frame), and a third, arbitrary, vector . We can define the two unit vectors and the random vector in terms of their Cartesian coordinates in the old reference frame:
, where and are the unit basis vectors of the old coordinate system and is the angle between the and unit vectors (i.e., the angle between the two reference frames). The projection of the arbitrary vector onto each of the two new unit vectors implies the dot product:
. So, is the projection of onto the axis, and is the projection of onto the axis. These new vector components, and , together compose the new vector , the original vector in terms of the new DQ reference frame.