Alpha–beta transformation

In electrical engineering, the αβγ transformation (also known as the Clarke transformation) is a mathematical transformation employed to simplify the analysis of three-phase circuits. Conceptually it is similar to the dqo transformation. One very useful application of the αβγ transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.

Definition

The αβ transform applied to three-phase currents is shown below in matrix form^[1]:

$I_{\alpha\beta} = TI_{abc} = \frac{2}{3}\begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \end{bmatrix}\begin{bmatrix}I_a\\I_b\\I_c\end{bmatrix}$

The inverse transform is:

$I_{abc} = T^{-1}I_{\alpha\beta} = \begin{bmatrix} 1 & 0\\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{1}{2} & -\frac{\sqrt{3}}{2}\end{bmatrix} \begin{bmatrix}I_\alpha\\I_\beta\end{bmatrix}$

Geometric Interpretation

The αβγ transformation can be thought of as the projection of the three phase quantities (voltages or currents) onto two stationary axes, the alpha axis and the beta axis.

Shown above is the αβγ transform as applied to three symmetrical currents flowing through three windings separated by 120 physical degrees. The three phase currents lag their corresponding phase voltages by δ. The α-β axis is shown with the α axis aligned with phase 'A'. The current vector I_αβγ rotates with angular velocity ω. There is no γ component since the currents are balanced.

dqo transform

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.

References

1. ^ F. Tahri, A.Tahri, Eid A. AlRadadi and A. Draou Senior, "Analysis and Control of Advanced Static VAR compensator Based on the Theory of the Instantaneous Reactive Power," presented at ACEMP, Bodrum, Turkey, 2007.

See also

• dqo transformation
• Symmetrical components
• Y-Δ transform
• Three-phase electric power
• Field-oriented control