Alpha–beta transformation

In electrical engineering, the alpha-beta (${\displaystyle \alpha \beta \gamma }$) 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 ${\displaystyle \alpha \beta \gamma }$ transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.

History

In 1937 and 1938, Edith Clarke published papers with modified methods of calculations on unbalanced three-phase problems, that turned out to be particularly useful.^[1]

Definition

The ${\displaystyle \alpha \beta \gamma }$ transform applied to three-phase currents, as used by Edith Clarke, is^[2]

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

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

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

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

${\displaystyle {\begin{aligned}i_{a}(t)=&{\sqrt {2}}I\cos \theta (t),\\i_{b}(t)=&{\sqrt {2}}I\cos \left(\theta (t)-{\frac {2}{3}}\pi \right),\\i_{c}(t)=&{\sqrt {2}}I\cos \left(\theta (t)+{\frac {2}{3}}\pi \right),\end{aligned}}}$

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

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

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 ${\displaystyle \alpha \beta \gamma }$ reference frame are the same of that in the natural reference frame.

Power invariant transformation

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

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

which is a unitary matrix and the inverse coincides with its transpose.^[3] In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is

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

Finally, the inverse transformation in this case is

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

Simplified transformation

Since in a balanced system ${\displaystyle i_{a}(t)+i_{b}(t)+i_{c}(t)=0}$ and thus ${\displaystyle i_{\gamma }(t)=0}$ one can also consider the simplified transform^[4][5]

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

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

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

which is the corresponding inverse transformation.

Geometric Interpretation

The ${\displaystyle \alpha \beta \gamma }$ 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. However, no information is lost if the system is balanced, as the equation ${\displaystyle I_{a}+I_{b}+I_{c}=0}$ is equivalent to the equation for ${\displaystyle I_{\gamma }}$ in the transform. If the system is not balanced, then the ${\displaystyle I_{\gamma }}$ term will contain the error component of the projection. Thus, a ${\displaystyle I_{\gamma }}$ of zero indicates that the system is balanced (and thus exists entirely in the alpha-beta coordinate space), and can be ignored for two coordinate calculations that operate under this assumption that the system is balanced. This is the elegance of the clarke transform as it reduces a three component system into a two component system thanks to this assumption.

Another way to understand this is that the equation ${\displaystyle I_{a}+I_{b}+I_{c}=0}$ defines a plane in a euclidean three coordinate space. The alpha-beta coordinate space can be understood as the two coordinate space defined by this plane, i.e. the alpha-beta axes lie on the plane defined by ${\displaystyle I_{a}+I_{b}+I_{c}=0}$.

This also means that in order the use the Clarke transform, one must ensure the system is balanced, otherwise subsequent two coordinate calculations will be erroneous. This is a practical consideration in applications where the three phase quantities are measured and can possibly have measurement error.

Shown above is the ${\displaystyle \alpha \beta \gamma }$ 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 ${\displaystyle \delta }$. The ${\displaystyle \alpha }$-${\displaystyle \beta }$ axis is shown with the ${\displaystyle \alpha }$ axis aligned with phase 'A'. The current vector ${\displaystyle I_{\alpha \beta \gamma }}$ rotates with angular velocity ${\displaystyle \omega }$. There is no ${\displaystyle \gamma }$ component since the currents are balanced.

dq0 transform

Main article: dq0 transformation

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

See also

• Symmetrical components
• Y-Δ transform
• Vector control (motor)

References

1. O'Rourke, Colm J. (December 2019). "A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park". IEEE Transactions on Energy Conversion. 34, 4 (4): 2070–2083. Bibcode:2019ITEnC..34.2070O. doi:10.1109/TEC.2019.2941175. hdl:1721.1/123557. S2CID 203113468 – via MIT Open Access Articles.
2. 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. S2CID 51636360.
3. S. CHATTOPADHYAY; M. MITRA; S. SENGUPTA (2008). "Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane". Journal of Electrical Systems. 04 (1): 62. Retrieved 2020-11-26.
4. 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.
5. "Clarke Transform". www.mathworks.com.

General references

• C.J. O'Rourke et al. "A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park," in IEEE Transactions on Energy Conversion, vol. 34, no. 4, pp. 2070-2083, Dec. 2019.