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 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.

Definition

transform applied to three-phase currents, as used by Edith Clarke, is

. 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

is the RMS without loss of generality to the current sequence, it results

reference frame are the same of that in the natural reference frame.

Power invariant transformation

is not unitary. In order to preserve the active and reactive powers one has, instead, to consider

is a (real) 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

Simplified transformation

one can also consider the simplified transform

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

which is the corresponding inverse transformation.

Geometric Interpretation

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.

.

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.

dq0 transform

The dq0 transform transform can be thought of as the projection of the phase quantities onto a rotating two-axis reference frame.

See also

• Symmetrical components

• Y-Δ transform

• Vector control (motor)

References

1. General references

2. 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.