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Implement *αβ0* to *abc*
transform

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

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

The figures show:

Balanced

*ɑ*,*β*, and zero components in a stationary reference frameThe direction of the magnetic axes of the stator windings in the stationary

*ɑβ0*reference frame and the*abc*reference frameThe time-response of the individual components of equivalent balanced

*ɑβ0*and*abc*systems

The block implements the inverse Clarke transform as

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

where:

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

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

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

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