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# Clarke Transform

Implement abc to αβ0 transform

Libraries:
Simscape / Electrical / Control / Mathematical Transforms

## Description

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 frame

• Equivalent ɑ, β, and zero components in the stationary reference frame

• The time-response of the individual components of equivalent balanced abc and ɑβ0 systems

### Equations

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

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

## Ports

### Input

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Components of the three-phase system in the abc reference frame.

Data Types: `single` | `double`

### Output

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Alpha-axis component,α, beta-axis component β, and zero component in the stationary reference frame.

Data Types: `single` | `double`

## Parameters

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Preserve the active and reactive power of the system in the abc reference frame.

## References

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

## Version History

Introduced in R2017b