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Implement *abc* to *αβ0*
transform

**Library:**Simscape / Electrical / Control / Mathematical Transforms

The Clarke Transform block converts the time-domain components of a
three-phase system in an *abc* reference frame to components in a
stationary *ɑβ0* reference frame. The block can preserve the active and
reactive powers with the powers of the system in the *abc* reference
frame by implementing a power invariant version of the Clarke transform. For a balanced
system, the zero component is equal to zero.

The figures show:

The direction of the magnetic axes of the stator windings in the

*abc*reference frame and the stationary*ɑβ0*reference frameEquivalent

*ɑ*,*β*, and zero components in the stationary reference frameThe time-response of the individual components of equivalent balanced

*abc*and*ɑβ0*systems

The block implements the Clarke transform as

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right],$

where:

*a*,*b*, and*c*are the components of the three-phase system in the*abc*reference frame.*α*and*β*are the components of the two-axis system in the stationary reference frame.*0*is the zero component of the two-axis system in the stationary reference frame.

The block implements the power invariant version of the Clarke transform as

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$

[1] Krause, P., O. Wasynczuk, S. D. Sudhoff, and S. Pekarek. *Analysis of
Electric Machinery and Drive Systems.* Piscatawy, NJ: Wiley-IEEE Press,
2013.