Direct-quadrature-zero transformation

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, ${\displaystyle {\hat {u}}_{D}}$ and ${\displaystyle {\hat {u}}_{Q}}$ (the unit vectors, or axes, of the new reference frame from the perspective of the old reference frame), and a third, arbitrary, vector ${\displaystyle {\vec {v}}_{XY}}$. We can define the two unit vectors and the random vector in terms of their Cartesian coordinates in the old reference frame:

${\displaystyle {\hat {u}}_{D}=\cos {\left(\theta \right)}{\hat {u}}_{X}+\sin {\left(\theta \right)}{\hat {u}}_{Y}}$

${\displaystyle {\hat {u}}_{Q}=-\sin {\left(\theta \right)}{\hat {u}}_{X}+\cos {\left(\theta \right)}{\hat {u}}_{Y}}$

${\displaystyle {\vec {v}}_{XY}=v_{X}{\hat {u}}_{X}+v_{Y}{\hat {u}}_{Y}}$,

where ${\displaystyle {\hat {u}}_{X}}$ and ${\displaystyle {\hat {u}}_{Y}}$ are the unit basis vectors of the old coordinate system and ${\displaystyle \theta }$ is the angle between the ${\displaystyle {\hat {u}}_{X}}$ and ${\displaystyle {\hat {u}}_{D}}$ 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:

${\displaystyle v_{D}={\hat {u}}_{D}\cdot {\vec {v}}_{XY}}$

${\displaystyle \to \cos {\left(\theta \right)}v_{X}+\sin {\left(\theta \right)}v_{Y}}$

${\displaystyle v_{Q}={\hat {u}}_{Q}\cdot {\vec {v}}_{XY}}$

${\displaystyle \to -\sin {\left(\theta \right)}v_{X}+\cos {\left(\theta \right)}v_{Y}}$.

So, ${\displaystyle v_{D}}$ is the projection of ${\displaystyle {\vec {v}}_{XY}}$ onto the ${\displaystyle {\hat {u}}_{D}}$ axis, and ${\displaystyle v_{Q}}$ is the projection of ${\displaystyle {\vec {v}}_{XY}}$ onto the ${\displaystyle {\hat {u}}_{Q}}$ axis. These new vector components, ${\displaystyle v_{D}}$ and ${\displaystyle v_{Q}}$, together compose the new vector ${\displaystyle {\vec {v}}_{DQ}}$, the original vector ${\displaystyle {\vec {v}}_{XY}}$ in terms of the new DQ reference frame.

Projection of ${\displaystyle {\vec {v}}_{XY}}$ onto the DQ reference frame.

Notice that the positive angle ${\displaystyle \theta }$ 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:

${\displaystyle {\vec {v}}_{DQ}={\begin{bmatrix}\cos {\left(\theta \right)}&\sin {\left(\theta \right)}\\-\sin {\left(\theta \right)}&\cos {\left(\theta \right)}\end{bmatrix}}\cdot {\vec {v}}_{XY}}$.

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:

${\displaystyle K_{P}={\begin{bmatrix}\cos {\left(\theta \right)}&\sin {\left(\theta \right)}&0\\-\sin {\left(\theta \right)}&\cos {\left(\theta \right)}&0\\0&0&1\end{bmatrix}}}$.

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.

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.

ABC unit basis vectors.

Typically, in electrical engineering (or any other context that uses three-phase systems), 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.

Two-dimensional perspective of a three-dimensional reality.

Three-phase problems are typically described as operating within this plane. In reality, the problem is likely a balanced-phase problem (i.e., v_A + v_B + v_C = 0) and the net vector

${\displaystyle {\vec {v}}=v_{A}{\hat {u}}_{A}+v_{B}{\hat {u}}_{B}+v_{C}{\hat {u}}_{C}}$

is always on this plane.

To build the Clarke transform, we actually use the Park transform 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 transform requires rotating the ABC reference frame about the A axis. So, this time, the 1 will be in the first element of the Park transform:

${\displaystyle K_{1}={\begin{bmatrix}1&0&0\\0&\cos {\left(-{\frac {\pi }{4}}\right)}&\sin {\left(-{\frac {\pi }{4}}\right)}\\0&-\sin {\left(-{\frac {\pi }{4}}\right)}&\cos {\left(-{\frac {\pi }{4}}\right)}\end{bmatrix}}}$

${\displaystyle \to {\begin{bmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}}$

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 (i.e., the sphere did not grow or shrink).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.

AYC' unit basis vectors. The C' and Y axes now point to the edges of the box, but the magnitude has not changed.

Next, the following tensor rotates the vector about the new Y axis in a counter-clockwise direction with respect to the Y axis (The angle was chosen so that the C' axis would be pointed towards the corner of the box.):

${\displaystyle K_{2}={\begin{bmatrix}\cos {\left(\theta \right)}&0&-\sin {\left(\theta \right)}\\0&1&0\\\sin {\left(\theta \right)}&0&\cos {\left(\theta \right)}\end{bmatrix}}}$

${\displaystyle \theta =\cos ^{-1}\left({\sqrt {\frac {2}{3}}}\right)\to 35.26^{\circ }}$,

or

${\displaystyle K_{2}={\begin{bmatrix}{\sqrt {\frac {2}{3}}}&0&-{\frac {1}{\sqrt {3}}}\\0&1&0\\{\frac {1}{\sqrt {3}}}&0&{\sqrt {\frac {2}{3}}}\end{bmatrix}}}$.

Notice that the distance from the center of the sphere to the midpoint of the edge of the box is √2 but from the center of the sphere to the corner of the box is √3. That is where the 35.26° angle came from. The angle can be calculated using the dot product. Let ${\displaystyle {\vec {m}}=\left(0,{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)}$ be the unit vector in the direction of C' and let ${\displaystyle {\vec {n}}=\left({\frac {1}{\sqrt {3}}},{\frac {1}{\sqrt {3}}},{\frac {1}{\sqrt {3}}}\right)}$ be a unit vector in the direction of the corner of the box at ${\displaystyle {\vec {n}}=\left(1,1,1\right)}$. Because ${\displaystyle {\vec {m}}\cdot {\vec {n}}=|{\vec {m}}||{\vec {n}}|\cos \theta ,}$ where ${\displaystyle \theta }$ is the angle between ${\displaystyle {\vec {m}}}$ and ${\displaystyle {\vec {n}},}$ we have

${\displaystyle \left(0,{\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)\cdot \left({\frac {1}{\sqrt {3}}},{\frac {1}{\sqrt {3}}},{\frac {1}{\sqrt {3}}}\right)=\cos \theta }$

${\displaystyle \cos \theta =0+{\frac {\sqrt {2}}{2{\sqrt {3}}}}+{\frac {\sqrt {2}}{2{\sqrt {3}}}}={\sqrt {\frac {2}{3}}}}$

${\displaystyle \theta =\cos ^{-1}\left({\sqrt {\frac {2}{3}}}\right)}$

${\displaystyle \theta =35.26^{\circ }.}$

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.

XYZ unit basis vectors. The Z axis (rotated C' axis) now points into the corner of the box.

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 (a vector without a common mode) will travel about this plane. This plane will be called the zero plane and is shown below by the hexagonal outline.

Plane of the vectors without common mode indicated by the hexagonal outline. The Z axis is orthogonal to this plane, and the X axis is parallel to the projection of the A axis onto the zero plane.

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 √3/2. The arbitrary vector did not change magnitude through this conversion from the ABC reference frame to the XYZ reference frame (i.e., the sphere did not change size). This is true for the power-invariant form of the Clarke transform. The following figure shows the common two-dimensional perspective of the ABC and XYZ reference frames.

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 (i.e., the X and Y components are longer than the A, B, and C components). Perhaps this can be intuitively understood by considering that for a vector without common mode, what took three values (A, B, and C components) to express, now only takes 2 (X and Y components) since the Z component is zero. Therefore, the X and Y component values must be larger to compensate.

The power-invariant Clarke transformation matrix is a combination of the K₁ and K₂ tensors:

${\displaystyle K_{C}=\underbrace {\begin{bmatrix}{\sqrt {\frac {2}{3}}}&0&-{\frac {1}{\sqrt {3}}}\\0&1&0\\{\frac {1}{\sqrt {3}}}&0&{\sqrt {\frac {2}{3}}}\end{bmatrix}} _{K_{2}}\cdot \underbrace {\begin{bmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}} _{K_{1}}}$,

or

${\displaystyle K_{C}={\sqrt {\frac {2}{3}}}\cdot {\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}}$

${\displaystyle \to {\begin{bmatrix}{\frac {2}{\sqrt {6}}}&-{\frac {1}{\sqrt {6}}}&-{\frac {1}{\sqrt {6}}}\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}\end{bmatrix}}}$.

Notice that when multiplied through, the bottom row of the K_C matrix is 1/√3, not 1/3. (Edith Clarke did use 1/3 for the power-variant case.) 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.

K_C frame (blue) and the original Edith matrix (green).

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 transform.

It is easy to verify (by matrix multiplication) that the inverse of K_C is

${\displaystyle K_{C}^{-1}={\begin{bmatrix}{\frac {2}{\sqrt {6}}}&0&{\frac {1}{\sqrt {3}}}\\-{\frac {1}{\sqrt {6}}}&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {3}}}\\-{\frac {1}{\sqrt {6}}}&-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {3}}}\end{bmatrix}}}$

It is sometimes desirable to scale the Clarke transformation matrix so that the X axis is the projection of the A axis onto the zero plane. To do this, we uniformly apply a scaling factor of √2/3 and a ²√1/radical to the zero component to get the power-variant Clarke transformation matrix:

${\displaystyle K_{\hat {C}}={\sqrt {\frac {2}{3}}}\cdot \underbrace {{\sqrt {\frac {2}{3}}}\cdot {\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}} _{K_{C}}}$

${\displaystyle \to {\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}}$

or

${\displaystyle K_{\hat {C}}={\begin{bmatrix}{\frac {2}{3}}&-{\frac {1}{3}}&-{\frac {1}{3}}\\0&{\frac {1}{\sqrt {3}}}&-{\frac {1}{\sqrt {3}}}\\{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}}$.

This will necessarily shrink the sphere by a factor of √2/3 as shown below. Notice that this new X axis is exactly the projection of the A axis onto the zero plane.

The scaled XYZ reference frame of the power-variant Clarke transform.

With the power-variant Clarke transform, the magnitude of the arbitrary vector is smaller in the XYZ reference frame than in the ABC reference frame (the norm of the transform is √2/3), but the magnitudes of the individual vector components are the same (when there is no common mode). So, as an example, a signal defined by

${\displaystyle {\begin{bmatrix}A\\B\\C\end{bmatrix}}={\begin{bmatrix}\cos {\left(\omega t\right)}\\\cos {\left(\omega t-{\frac {2\pi }{3}}\right)}\\\cos {\left(\omega t+{\frac {2\pi }{3}}\right)}\end{bmatrix}}}$

becomes, in the XYZ reference frame,

${\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}={\begin{bmatrix}\cos {\left(\omega t\right)}\\\cos {\left(\omega t-{\frac {\pi }{2}}\right)}\\0\end{bmatrix}}}$,

a new vector whose components are the same magnitude as the original components: 1. In many cases, this is an advantageous quality of the power-variant Clarke transform.

The DQZ transformation uses the Clarke transform to convert ABC-referenced vectors into two differential-mode components (i.e., X and Y) and one common-mode component (i.e., Z) and then applies the Park transform to rotate the reference frame about the Z axis at some given angle. The X component becomes the D component, which is in direct alignment with the vector of rotation, and the Y component becomes the Q component, which is at a quadrature angle to the direct component. The DQZ transform is

${\displaystyle K_{CP}=K_{P}\cdot K_{C}}$

${\displaystyle \to {\begin{bmatrix}\cos {\left(\theta \right)}&\sin {\left(\theta \right)}&0\\-\sin {\left(\theta \right)}&\cos {\left(\theta \right)}&0\\0&0&1\end{bmatrix}}\cdot {\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&{\frac {-1}{2}}&{\frac {-1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}}$.

The transformation originally proposed by Park differs slightly from the one given above. In Park's transformation q-axis is ahead of d-axis, qd0, and the ${\displaystyle \theta }$ angle is the angle between phase-a and q-axis, as given below:

${\displaystyle P={\frac {2}{3}}{\begin{bmatrix}\cos(\theta )&\cos(\theta -{\frac {2\pi }{3}})&\cos(\theta +{\frac {2\pi }{3}})\\\sin(\theta )&\sin(\theta -{\frac {2\pi }{3}})&\sin(\theta +{\frac {2\pi }{3}})\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}}$

and

${\displaystyle P^{-1}={\begin{bmatrix}\cos(\theta )&\sin(\theta )&1\\\cos(\theta -{\frac {2\pi }{3}})&\sin(\theta -{\frac {2\pi }{3}})&1\\\cos(\theta +{\frac {2\pi }{3}})&\sin(\theta +{\frac {2\pi }{3}})&1\end{bmatrix}}}$

D. Holmes and T. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice, Wiley-IEEE Press, 2003, and

P. Krause, O. Wasynczuk and S. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2nd ed., Piscataway, NJ: IEEE Press, 2002.

The dqo transform is conceptually similar to the αβγ transform. Whereas the dqo transform is the projection of the phase quantities onto a rotating two-axis reference frame, the αβγ transform can be thought of as the projection of the phase quantities onto a stationary two-axis reference frame.