Alpha–beta transformation

The Clarke transformation, named after Edith Clarke and also as α, β - transformation called, serves three phase parameters as for a three-phase machine with the axes U, V, W in a simpler two-axis coordinate system α, β to transfer to the axles. The Clarke transformation, together with the d / q- transform is a mathematical basis for the vector control of three-phase machines and describes one of several possible space vector images.


In the Clarke transformation of the underlying rectangular coordinate system is chosen to be equal to the stationary stator and ready α and the imaginary part of β in the complex plane by the real part, the sum of the three phase currents at any one time is always zero. In the three- phase system, the three coils of the stator of an induction machine are offset by an angle of 120 °, where by definition the U axis with the real axis coincides α, as shown in the adjacent figure.

The Clarke transformation leads the three phase currents and, in to two equal streams and over. In element -wise matrix notation it is:

Due to the condition that at any time the sum of the three phase currents is always zero, this equation can be simplified to:

In practice, the simplification means that the current has to be actually measured by, for example, only two current transformers, and not with three strands.

The inverse Clarke transformation is:

The transformation is not limited to the electric currents, but may be applied by analogy to all other electrical parameters such as the occurring voltage or the magnetic flux density.


In a three phase system is not in equilibrium, by adding a third parameter in the theory of symmetrical components of the α, β - transformation to the α, β, γ - transformation will be expanded. is the sum of the three phase currents:

With the then-current for non-equilibrium systems the three-phase α, β, γ - transformation:

And the inverse to α, β, γ - transformation:


  • Edith Clarke: Circuit Analysis of AC Power Systems. Vol I. J. Wiley & Sons, New York, 1943.
  • Electromagnetic Theory
  • Electric machine
  • Digital Signal Processing
  • Transformation