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.

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]

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.

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

Zdroj:https://en.wikipedia.org?pojem=Alpha–beta_transformation
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