Direct-quadrature-zero transformation

The direct-quadrature-zero or Park '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 transformation combines a Clarke transformation with a new rotating reference frame.
The Park transformation is often used in the context of electrical engineering with three-phase circuits. The transformation 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 transformation to recover the actual three-phase AC results. As an example, the Park transformation 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 transforms the system into a linear time-invariant system

Introduction

The Park transformation is equivalent to the product of the rotation and Clarke transformation matrices. The Clarke transformation converts vectors in the ABC reference frame to the XYZ reference frame. The primary value of the Clarke transformation 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. 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 rotation converts vectors in the XYZ reference frame to the DQZ reference frame. The rotation's primary value is to rotate a vector's reference frame at an arbitrary frequency. The rotation 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 rotation 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 rotation 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 rotation matrix:
The Park transformation is equivalent to the product of the Clarke transformation and a rotation:
The inverse transformation is:
To convert an ABC-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 ABC reference frame, the column vector signal must be pre-multiplied by the inverse Park transformation matrix:
To understand this transformation better, a derivation of the transformation is included.

Derivation

Rotation matrix derivation

A rotation matrix is based on the concept of the dot product and projections of vectors onto other vectors. First, let us imagine two unit vectors, and , 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. 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.
Notice that the positive angle above caused the arbitrary vector to rotate backward when transitioned to the new DQ reference frame. In other words, its angle concerning the new reference frame is less than its angle to the old reference frame. This is because the reference frame, not the vector, was rotated forwards. Actually, a forward rotation of the reference frame is identical to a negative rotation of the vector. If the old reference frame were rotating forwards, such as in three-phase electrical systems, then the resulting DQ vector remains stationary.
A single matrix equation can summarize the operation above:
This tensor can be expanded to three-dimensional problems, where the axis about which rotation occurs is left unaffected. In the following example, the rotation is about the Z axis, but any axis could have been chosen:
From a linear algebra perspective, this is simply a clockwise rotation about the z-axis and is mathematically equivalent to the trigonometric difference angle formulae.

Clarke transformation derivation

The ABC unit basis vectors

Consider a three-dimensional space with unit basis vectors A, B, and C. The sphere in the figure below is used to show the scale of the reference frame for context and the box is used to provide a rotational context.
Typically, in electrical engineering, the three-phase components are shown in a two-dimensional perspective. However, given the three phases can change independently, they are by definition orthogonal to each other. This implies a three-dimensional perspective, as shown in the figure above. So, the two-dimensional perspective is really showing the projection of the three-dimensional reality onto a plane.
Three-phase problems are typically described as operating within this plane. In reality, the problem is likely a balanced-phase problem and the net vector
is always on this plane.

The AYC' unit basis vectors

To build the Clarke transformation, we actually use the Park transformation in two steps. Our goal is to rotate the C axis into the corner of the box. This way the rotated C axis will be orthogonal to the plane of the two-dimensional perspective mentioned above. The first step towards building the Clarke transformation requires rotating the ABC reference frame about the A axis. So, this time, the 1 will be in the first element of the Park transformation:
The following figure shows how the ABC reference frame is rotated to the AYC' reference frame when any vector is pre-multiplied by the K₁ matrix. The C' and Y axes now point to the midpoints of the edges of the box, but the magnitude of the reference frame has not changed.This is due to the fact that the norm of the K₁ tensor is 1: ||K₁|| = 1. This means that any vector in the ABC reference frame will continue to have the same magnitude when rotated into the AYC' reference frame.

The XYZ unit basis vectors

Next, the following tensor rotates the vector about the new Y axis in a counter-clockwise direction with respect to the Y axis :
or
Notice that the distance from the center of the sphere to the midpoint of the edge of the box is but from the center of the sphere to the corner of the box is. That is where the 35.26° angle came from. The angle can be calculated using the dot product. Let be the unit vector in the direction of C' and let be a unit vector in the direction of the corner of the box at
. Because
where is the angle between and we have
The norm of the K₂ matrix is also 1, so it too does not change the magnitude of any vector pre-multiplied by the K₂ matrix.

The zero plane

At this point, the Z axis is now orthogonal to the plane in which any ABC vector without a common-mode component can be found. Any balanced ABC vector waveform will travel about this plane. This plane will be called the zero plane and is shown below by the hexagonal outline.
The X and Y basis vectors are on the zero plane. Notice that the X axis is parallel to the projection of the A axis onto the zero plane. The X axis is slightly larger than the projection of the A axis onto the zero plane. It is larger by a factor of. The arbitrary vector did not change magnitude through this conversion from the ABC reference frame to the XYZ reference frame. This is true for the power-invariant form of the Clarke transformation. The following figure shows the common two-dimensional perspective of the ABC and XYZ reference frames.
It might seem odd that though the magnitude of the vector did not change, the magnitude of its components did. Perhaps this can be intuitively understood by considering that for a vector without common mode, what took three values to express, now only takes 2 since the Z component is zero. Therefore, the X and Y component values must be larger to compensate.

Combination of tensors

The power-invariant Clarke transformation matrix is a combination of the K₁ and K₂ tensors:
or
Notice that when multiplied through, the bottom row of the K_C matrix is 1/, not 1/3. The Z component is not exactly the average of the A, B, and C components. If only the bottom row elements were changed to be 1/3, then the sphere would be squashed along the Z axis. This means that the Z component would not have the same scaling as the X and Y components.
As things are written above, the norm of the Clarke transformation matrix is still 1, which means that it only rotates an ABC vector but does not scale it. The same cannot be said for Clarke's original transformation.
It is easy to verify that the inverse of K_C is