αβγ to ABC

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Description

This component performs the αβγ to ABC transformation, also known as Clarke's inverse transformation. This transformation projects the two stationary axis back onto the three-phase quantities axis. This component has the same three modes as the ABC to αβγ transformation. See ABC to αβγ for more details.

The parameter Power transformation form allows the selection of the transformation mode.

Alpha-beta-gamma to ABC Variant power - Clarke's original transformation matrix:

$\left[\begin{array}{c}A\\ B\\ C\end{array}\right] = \left[\begin{array}{ccc}1& 0& 1\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 1\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$-\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 1\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right]$

Alpha-beta-gamma to ABC Variant power - uniform transformation matrix:

$\left[\begin{array}{c}A\\ B\\ C\end{array}\right] = \left[\begin{array}{ccc}1& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$-\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right]$

Alpha-beta-gamma to ABC Invariant power transformation matrix:

$\left[\begin{array}{c}A\\ B\\ C\end{array}\right] = \left[\begin{array}{ccc}\raisebox{1ex}{$\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{6}$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.\\ \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{6}$}\right.& \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.\end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right]$

The matrix computation is implemented with and optimized code as follows:

$A = \gamma \bullet k_1$

$B = A-\alpha \bullet k_2$

$C = B-\beta \bullet k_3$

$B += \beta \bullet k_3$

$A += \alpha \bullet k_4$

Where for Variant power - Clarke's original $k_1 = 1$ , $k_2 = 1/2$ , $k_3 = \sqrt{3}/2$ and ${\text{k}}_{\text{4}}\text{ = }{k}_1$ ;

for Variant power - uniform $k_1 = 1/\sqrt{2}$ , $k_2 = 1/2$ , $k_3 = \sqrt{3}/2$ and ${\text{k}}_{\text{4}}\text{ = }\text{1}$ ;

and for Invariant power $k_1 = 1/\sqrt{3}$ , $k_2 = 1/\sqrt{6}$ , $k_3 = 1/\sqrt{2}$ and ${\text{k}}_{\text{4}}\text{ = }\sqrt{\text{2}}/\sqrt{\text{3}}$ .

Figure below illustrates the transformation of alpha beta system to a three-phase abc frame using amplitude invariant transformation.

Ports

• • α (in)
    • Input signal of the component related to the alpha signal of the alpha-beta-gamma sequence frame.
      • Supported types: real, int, uint.
      • Vector support: no.
• • β (in)
    • Input signal of the component related to the beta signal of the alpha-beta-gamma sequence frame.
      • Supported types: real, int, uint.
      • Vector support: no.
• • γ (in)
    • Input signal of the component related to the gamma signal of the alpha-beta-gamma sequence frame.
      • Supported types: real, int, uint.
      • Vector support: no.
• a (out)
  • Output a of the component, corresponding to the three-phase abc system.
    • Supported types: real.
    • Vector support: no.

• b (out)
  • Output b of the component, corresponding to the three-phase abc system.
    • Supported types: real.
    • Vector support: no.

• c (out)
  • Output c of the component, corresponding to the three-phase abc system.
    • Supported types: real.
    • Vector support: no.

Properties

• Power transformation form

  • Variant – Clarke's original: Use this method when you want the resulting three-phase abc frame to be amplitude invariant. That is, the amplitude of the alpha-beta-gamma rotating system will be preserved in the abc frame.

    Variant – uniform: Use this method when the input alpha-beta-gamma frame signal is a balanced system and you want the resulting three-phase abc frame to be amplitude invariant. That is, the amplitude of the original alpha-beta-gamma frame will be preserved in the abc frame.

    Invariant: Use this method when you want the resulting three-phase abc frame to be power invariant. That is, the power of the of the alpha-beta-gamma rotating system will be preserved in the abc frame.

• Execution rate

  • Type in the desired signal processing execution rate. This value must be compatible with other signal processing components of the same circuit: the value must be a multiple of the fastest execution rate in the circuit. There can be up to four different execution rates, but they must all be multiple of the basic simulation timestep. To specify the execution rate, you can use either decimal (e.g. 0.001) or exponential values (e.g. 1e-3) in seconds. Alternatively, you can type in ‘inherit’ in which case the component will be assigned execution rate based on the execution rate of the components it is receiving input from.