# abc to Alpha-Beta-Zero, Alpha-Beta-Zero to abc

Perform transformation from three-phase (abc) signal to αβ0 stationary reference frame or the inverse

## Library

Control and Measurements/Transformations

## Description

The abc to Alpha-Beta-Zero block performs a Clarke transform on a three-phase abc signal. The Alpha-Beta-Zero to abc block performs an inverse Clarke transform on the αβ0 components.

`$\left[\begin{array}{c}{u}_{\alpha }\\ {u}_{\beta }\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\frac{2}{3}& -\frac{1}{3}& -\frac{1}{3}\\ 0& \frac{1}{\sqrt{3}}& \frac{-1}{\sqrt{3}}\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]$`

The inverse transformation is given by

`$\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}& 1\\ -\frac{1}{2}& -\frac{\sqrt{3}}{2}& 1\end{array}\right]\left[\begin{array}{c}{u}_{\alpha }\\ {u}_{\beta }\\ {u}_{0}\end{array}\right]$`

Assume that ua, ub, uc quantities represent three sinusoidal balanced currents:

`$\begin{array}{l}{i}_{a}=I\mathrm{sin}\left(\omega t\right)\\ {i}_{b}=I\mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)\\ {i}_{c}=I\mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)\end{array}$`

These currents are flowing respectively into windings A, B, C of a three-phase winding, as the figure shows.

In this case, the iα and iβ components represent the coordinates of the rotating space vector Is in a fixed reference frame whose α axis is aligned with phase A axis. Is amplitude is proportional to the rotating magnetomotive force produced by the three currents. It is computed as follows:

`${I}_{s}={i}_{a}+j\cdot {i}_{\beta }=\frac{2}{3}\left({i}_{a}+{i}_{b}\cdot {e}^{\frac{j2\pi }{3}}+{i}_{c}\cdot {e}^{-\frac{j2\pi }{3}}\right)$`

## Examples

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