# Alpha-Beta-Zero to dq0, dq0 to Alpha-Beta-Zero

Perform transformation from αβ0 stationary reference frame to dq0 rotating reference frame or the inverse

## Library

Simscape / Electrical / Specialized Power Systems / Control & Measurements / Transformations

## Description

The Alpha-Beta-Zero to dq0 block performs a transformation of αβ0 Clarke components in a fixed reference frame to dq0 Park components in a rotating reference frame.

The dq0 to Alpha-Beta-Zero block performs a transformation of dq0 Park components in a rotating reference frame to αβ0 Clarke components in a fixed reference frame.

The block supports the two conventions used in the literature for Park transformation:

• Rotating frame aligned with A axis at t = 0. This type of Park transformation is also known as the cosine-based Park transformation.

• Rotating frame aligned 90 degrees behind A axis. This type of Park transformation is also known as the sine-based Park transformation. Use it in Simscape™ Electrical™ Specialized Power Systems models of three-phase synchronous and asynchronous machines.

Knowing that the position of the rotating frame is given by ω.t (where ω represents the frame rotation speed), the αβ0 to dq0 transformation performs a −(ω.t) rotation on the space vector Us = uα + j· uβ. The homopolar or zero-sequence component remains unchanged.

Depending on the frame alignment at t = 0, the dq0 components are deduced from αβ0 components as follows:

When the rotating frame is aligned with A axis, the following relations are obtained:

`$\begin{array}{l}{U}_{s}={u}_{d}+j\cdot {u}_{q}=\left({u}_{a}+j\cdot {u}_{\beta }\right)\cdot {e}^{-j\omega t}\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\left(\omega t\right)& \mathrm{sin}\left(\omega t\right)& 0\\ -\mathrm{sin}\left(\omega t\right)& \mathrm{cos}\left(\omega t\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{\beta }\\ {u}_{0}\end{array}\right]\end{array}$`

The inverse transformation is given by

`$\begin{array}{l}{u}_{\alpha }+j\cdot {u}_{\beta }=\left({u}_{d}+j\cdot {u}_{q}\right)\cdot {e}^{j\omega t}\\ \left[\begin{array}{c}{u}_{\alpha }\\ {u}_{\beta }\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\left(\omega t\right)& -\mathrm{sin}\left(\omega t\right)& 0\\ \mathrm{sin}\left(\omega t\right)& \mathrm{cos}\left(\omega t\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{d}\\ uq\\ {u}_{0}\end{array}\right]\end{array}$`

When the rotating frame is aligned 90 degrees behind A axis, the following relations are obtained:

`$\begin{array}{l}{U}_{s}={u}_{d}+j\cdot {u}_{q}=\left({u}_{\alpha }+j\cdot {u}_{\beta }\right)\cdot {e}^{-j\left(\omega t-\frac{\pi }{2}\right)}\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{sin}\left(\omega t\right)& \mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)\\ \mathrm{cos}\left(\omega t\right)& \mathrm{cos}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{cos}\left(\omega t+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]\end{array}$`

The inverse transformation is given by

`${u}_{\alpha }+j\cdot {u}_{\beta }=\left({u}_{d}+j\cdot {u}_{q}\right)\cdot {e}^{j\left(\omega t-\frac{\pi }{2}\right)}$`

The abc-to-Alpha-Beta-Zero transformation applied to a set of balanced three-phase sinusoidal quantities ua, ub, uc produces a space vector Us whose uα and uβ coordinates in a fixed reference frame vary sinusoidally with time. In contrast, the abc-to-dq0 transformation (Park transformation) applied to a set of balanced three-phase sinusoidal quantities ua, ub, uc produces a space vector Us whose ud and uq coordinates in a dq rotating reference frame stay constant.

## Parameters

Rotating frame alignment (at wt=0)

Select the alignment of rotating frame, when wt = 0, of the dq0 components of a three-phase balanced signal:

(positive-sequence magnitude = 1.0 pu; phase angle = 0 degree)

When you select `Aligned with phase A axis`, the dq0 components are d = 0, q = −1, and zero = 0.

When you select `90 degrees behind phase A axis`, the default option, the dq0 components are d = 1, q = 0, and zero = 0.

## Inputs and Outputs

αβ0

The vectorized αβ0 signal.

`dq0`

The vectorized dq0 signal.

`wt`

The angular position, in radians, of the dq rotating frame relative to the stationary frame.

## Example

The `power_Transformations` example shows various uses of blocks performing Clarke and Park transformations.