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In [[electrical engineering]], the '''alpha-beta''' (<math>\alpha\beta\gamma</math>) '''transformation'''  (also known as the '''Clarke transformation''') is a mathematical [[Transform (mathematics)|transformation]] employed to simplify the analysis of [[Three-phase electric power|three-phase circuits]]. Conceptually it is similar to the [[dqo transformation]]. One very useful application of the <math>\alpha\beta\gamma</math> transformation is the generation of the reference signal used for space vector modulation control of three-phase [[Inverter (electrical)|inverters]].
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==Definition==
 
The <math>\alpha\beta\gamma</math> transform applied to three-phase currents, as used by [[Edith Clarke]], is<ref>{{cite journal| author=W. C. Duesterhoeft, Max W. Schulz and Edith Clarke| journal=Transactions of the American Institute of Electrical Engineers| title=Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components|date=july 1951|volume=70|issue=2|
pages=1248–1255|ISSN=0096-3860|doi=10.1109/T-AIEE.1951.5060554}}</ref>
 
:<math>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}</math>
where <math>i_{abc}(t)</math> is a generic three-phase current sequence and <math>i_{\alpha\beta\gamma}(t)</math> is the corresponding current sequence given by the transformation <math>T</math>.
The inverse transform is:
 
:<math>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}.</math>
 
The above Clarke's transformation preserves the amplitude of the electrical variables which is applied to. Indeed, consider a three-phase symmetric, direct, current sequence
:<math>  
\begin{align}
i_a(t)=&\sqrt{2}I\cos\theta(t),\\
i_b(t)=&\sqrt{2}I\cos\left(\theta(t)-\frac23\pi\right),\\
i_c(t)=&\sqrt{2}I\cos\left(\theta(t)+\frac23\pi\right),
\end{align}
</math>
where <math>I</math> is the rms of <math>i_a(t)</math>, <math>i_b(t)</math>, <math>i_c(t)</math> and <math>\theta(t)</math> is the generic time-varying angle that can also be set to <math>\omega t</math> without loss of generality. Then, by applying <math>T</math> to the current sequence, it results
:<math>
\begin{align}
i_{\alpha}=&\sqrt2 I\cos\theta(t),\\
i_{\beta}=&\sqrt2 I\sin\theta(t),\\
i_{\gamma}=&0,
\end{align}
</math>
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 <math>\alpha\beta\gamma</math> 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 <math>T</math> is not unitary. In order to preserve the active and reactive powers one has, instead, to consider
:<math>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}{\sqrt2} & \frac{1}{\sqrt2} & \frac{1}{\sqrt2} \\
\end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix},</math>
which is a unitary matrix and the inverse coincides with its transpose.<ref>{{cite journal|title = Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane|
author= S. CHATTOPADHYAY, M. MITRA, S. SENGUPTA|journal=Journal of Electrical Systems|year=2008|volume=04|issue=01|pages=62|url=http://www.doaj.org/doaj?func=fulltext&passMe=http://journal.esrgroups.org/jes/papers/4_1_5.pdf|accessdate=2012-04-26}}</ref>
In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is
:<math>
\begin{align}
i_{\alpha}=&\sqrt3 I\cos\theta(t),\\
i_{\beta}=&\sqrt3 I\sin\theta(t),\\
i_{\gamma}=&0.
\end{align}
</math>
 
=== Simplified transformation ===
Since in a balanced system <math>i_a(t)+i_b(t)+i_c(t)=0</math> and thus <math>i_\gamma(t)=0</math> one can also consider the simplified transform<ref name="Tahri">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.</ref>
 
:<math>i_{\alpha\beta}(t) = \frac23 \begin{bmatrix} 1 & -\frac12 & -\frac12\\
0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}
\end{bmatrix}\begin{bmatrix}i_a(t)\\i_b(t)\\i_c(t)\end{bmatrix}</math>
 
and
 
:<math>i_{abc}(t) = \frac32\begin{bmatrix} \frac23 & 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}.</math>
 
==Geometric Interpretation==
 
The <math>\alpha\beta\gamma</math> 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.
 
[[Image:AlphaBeta geometric interpretation.gif|center|frame|Shown above is the <math>\alpha\beta\gamma</math> 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 <math>\delta</math>. The <math>\alpha</math>-<math>\beta</math> axis is shown with the <math>\alpha</math> axis aligned with phase 'A'. The current vector <math>I_{\alpha\beta\gamma}</math> rotates with angular velocity <math>\omega</math>. There is no <math>\gamma</math> component since the currents are balanced.]]
 
===<math>dqo</math> transform===
 
==References==
{{reflist}}
 
==See also==
 
* [[Symmetrical components]]
* [[Y-Δ transform]]
* [[Field-oriented control]]
 
{{DEFAULTSORT:Alpha-beta transformation}}
[[Category:Electrical engineering]]

Revision as of 22:57, 3 March 2014

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