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Implement *ab* to *αβ* transformation

**Library:**Motor Control Blockset / Controls / Math Transforms

The Clarke Transform block computes the Clarke transformation of balanced
three-phase components in the *abc* reference frame and outputs the balanced
two-phase orthogonal components in the stationary *αβ* reference
frame.

The block accepts two signals out of the three phases (*abc*),
automatically calculates the third signal, and outputs the corresponding components in the
*αβ* reference frame.

For example, the block accepts *a* and *b* input values
where the phase-*a* axis aligns with the *α*-axis.

This figure shows the direction of the magnetic axes of the stator windings in the

*abc*reference frame and the stationary*αβ*reference frame.This figure shows the equivalent

*α*and*β*components in the stationary*αβ*reference frame.The time-response of the individual components of equivalent balanced

*abc*and*αβ*systems.

The following equation describes the Clarke transform computation:

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

For balanced systems like motors, the zero sequence component calculation is always zero. For example, the currents of the motor can be represented as,

$${i}_{a}+{i}_{b}+{i}_{c}=0$$

Therefore, you can use only two current sensors in three-phase motor drives, where you can calculate the third phase as,

$${i}_{c}=-({i}_{a}+{i}_{b})$$

By using these equations, the block implements the Clarke transform as,

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

where:

${f}_{a}$, ${f}_{b}$, and ${f}_{c}$ are the balanced three-phase components in the

*abc*reference frame.${f}_{\alpha}$ and ${f}_{\beta}$ are the balanced two-phase orthogonal components in the stationary

*αβ*reference frame.${f}_{0}$ is the zero component in the stationary

*αβ*reference frame.