Alpha–beta transformation

In electrical engineering, the alpha-beta ( $\alpha \beta \gamma$ ) 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 dq0 transformation. One very useful application of the $\alpha \beta \gamma$ transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.

Definition

The $\alpha \beta \gamma$ transform applied to three-phase currents, as used by Edith Clarke, is^[1]

i_{{\alpha \beta \gamma }}(t)=Ti_{{abc}}(t)={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {{\sqrt {3}}}{2}}&-{\frac {{\sqrt {3}}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}

where $i_{{abc}}(t)$ is a generic three-phase current sequence and $i_{{\alpha \beta \gamma }}(t)$ is the corresponding current sequence given by the transformation $T$ . The inverse transform is:

i_{{abc}}(t)=T^{{-1}}i_{{\alpha \beta \gamma }}(t)={\begin{bmatrix}1&0&1\\-{\frac {1}{2}}&{\frac {{\sqrt {3}}}{2}}&1\\-{\frac {1}{2}}&-{\frac {{\sqrt {3}}}{2}}&1\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.

The above Clarke's transformation preserves the amplitude of the electrical variables which it is applied to. Indeed, consider a three-phase symmetric, direct, current sequence

{\begin{aligned}i_{a}(t)=&{\sqrt {2}}I\cos \theta (t),\\i_{b}(t)=&{\sqrt {2}}I\cos \left(\theta (t)-{\frac 23}\pi \right),\\i_{c}(t)=&{\sqrt {2}}I\cos \left(\theta (t)+{\frac 23}\pi \right),\end{aligned}}

where $I$ is the RMS of $i_{a}(t)$ , $i_{b}(t)$ , $i_{c}(t)$ and $\theta (t)$ is the generic time-varying angle that can also be set to $\omega t$ without loss of generality. Then, by applying $T$ to the current sequence, it results

{\begin{aligned}i_{{\alpha }}=&{\sqrt 2}I\cos \theta (t),\\i_{{\beta }}=&{\sqrt 2}I\sin \theta (t),\\i_{{\gamma }}=&0,\end{aligned}}

where the last equation holds since we have considered balanced currents. As it is shown in the above, the amplitudes of the currents in the $\alpha \beta \gamma$ reference frame are the same of that in the natural reference frame.

Power invariant transformation

The active and reactive powers computed in the Clark's domain with the transformation shown above are not the same of those computed in the standard reference frame. This happens because $T$ is not unitary. In order to preserve the active and reactive powers one has, instead, to consider

i_{{\alpha \beta \gamma }}(t)=Ti_{{abc}}(t)={\sqrt {{\frac {2}{3}}}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {{\sqrt {3}}}{2}}&-{\frac {{\sqrt {3}}}{2}}\\{\frac {1}{{\sqrt 2}}}&{\frac {1}{{\sqrt 2}}}&{\frac {1}{{\sqrt 2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}},

which is a unitary matrix and the inverse coincides with its transpose.^[2] In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is

{\begin{aligned}i_{{\alpha }}=&{\sqrt 3}I\cos \theta (t),\\i_{{\beta }}=&{\sqrt 3}I\sin \theta (t),\\i_{{\gamma }}=&0.\end{aligned}}

Finally, the inverse transformation in this case is

i_{abc}(t)={\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&0&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.

Simplified transformation

Since in a balanced system $i_{a}(t)+i_{b}(t)+i_{c}(t)=0$ and thus $i_{\gamma }(t)=0$ one can also consider the simplified transform^[3]

i_{{\alpha \beta }}(t)={\frac 23}{\begin{bmatrix}1&-{\frac 12}&-{\frac 12}\\0&{\frac {{\sqrt {3}}}{2}}&-{\frac {{\sqrt {3}}}{2}}\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}

which is simply the original Clarke's transformation with the 3rd equation thrown away, and

i_{{abc}}(t)={\frac 32}{\begin{bmatrix}{\frac 23}&0\\-{\frac {1}{3}}&{\frac {{\sqrt {3}}}{3}}\\-{\frac {1}{3}}&-{\frac {{\sqrt {3}}}{3}}\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\end{bmatrix}}.

Geometric Interpretation

The $\alpha \beta \gamma$ 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.

$dq0$ transform

The $dq0$ transform is conceptually similar to the $\alpha \beta \gamma$ transform. Whereas the $dq0$ transform is the projection of the phase quantities onto a rotating two-axis reference frame, the $\alpha \beta \gamma$ transform can be thought of as the projection of the phase quantities onto a stationary two-axis reference frame.

References

↑ W. C. Duesterhoeft; Max W. Schulz; Edith Clarke (July 1951). "Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components". Transactions of the American Institute of Electrical Engineers. 70 (2): 1248–1255. doi:10.1109/T-AIEE.1951.5060554. ISSN 0096-3860.
↑ S. CHATTOPADHYAY; M. MITRA; S. SENGUPTA (2008). "Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane" (PDF). Journal of Electrical Systems. 04 (01): 62. Retrieved 2012-04-26.
↑ 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.

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[1] W. C. Duesterhoeft; Max W. Schulz; Edith Clarke (July 1951). "Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components". Transactions of the American Institute of Electrical Engineers. 70 (2): 1248–1255. doi:10.1109/T-AIEE.1951.5060554. ISSN 0096-3860.

[2] S. CHATTOPADHYAY; M. MITRA; S. SENGUPTA (2008). "Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane" (PDF). Journal of Electrical Systems. 04 (01): 62. Retrieved 2012-04-26.

[Tahri-3] 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.