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.
The transform applied to three-phase currents, as used by Edith Clarke, is
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
which is a unitary matrix and the inverse coincides with its transpose.
In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is
Finally, the inverse transformation in this case is
Since in a balanced system and thus one can also consider the simplified transform
which is simply the original Clarke's transformation with the 3rd equation excluded, and
The 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 Ia + Ib + Ic = 0 is equivalent to the equation for in the transform. If the system is not balanced, then the term will contain the error component of the projection. Thus, a 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 Ia + Ib + Ic = 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 Ia + Ib + Ic = 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 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 . The - axis is shown with the axis aligned with phase 'A'. The current vector rotates with angular velocity . There is no component since the currents are balanced.
The transform is conceptually similar to the transform. Whereas the transform is the projection of the phase quantities onto a rotating two-axis reference frame, the transform can be thought of as the projection of the phase quantities onto a stationary two-axis reference frame.
^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. ISSN0096-3860.
^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.