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In electrical engineering, the alpha-beta () 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 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 transform applied to three-phase currents, as used by Edith Clarke, is[2]
where is a generic three-phase current sequence and is the corresponding current sequence given by the transformation . The inverse transform is:
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
where is the RMS of , , and is the generic time-varying angle that can also be set to without loss of generality. Then, by applying to the current sequence, it results
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 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 is not unitary. In order to preserve the active and reactive powers one has, instead, to consider
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