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

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

The Specialized Power Systems library will be removed in R2026a. Use the Simscape™ Electrical™ blocks and functions instead. For more information on updating your models, see Upgrade Specialized Power System Models to use Simscape Electrical Blocks.

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Libraries:
Simscape / Electrical / Specialized Power Systems / Control

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 u_a, u_b, u_c quantities represent three sinusoidal balanced currents:

$$\begin{array}{l}{i}_a=I\sin (\omega t)\\ i_b=I\sin \left(\omega t-\frac{2\pi }{3}\right)\\ i_c=I\sin \left(\omega t+\frac{2\pi }{3}\right)\end{array}$$

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:

$$I_s=i_a+j\cdot i_{\beta }=\frac{2}{3}\left(i_a+i_b\cdot e^{\frac{j2\pi }{3}}+i_c\cdot e^{-\frac{j2\pi }{3}}\right)$$

Ports

Input

abc — Vectorized abc signal

vector

Vectorized abc signal.

Output

αβ0 — Vectorized αβ0 signal

vector

Vectorized αβ0 signal.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013a