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

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

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 u_{a}, u_{b},
u_{c} quantities represent three sinusoidal balanced
currents:

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

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

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

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

The `power_Transformations`

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