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attempt fixing latex
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Signed-off-by: Jean Alinei <[email protected]>
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jalinei committed Jan 10, 2025
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The Second Order Generalized Integrator based PLL `SOGI-PLL`:material-information-outline:{title="Mihai Ciobotaru, Remus Teodorescu and Frede Blaabjerg, « A New Single-Phase PLL Structure Based on Second Order Generalized Integrator », 2006."} is a filter used to retrieve phase information from a mono phase or three phased system.
It is included in the `filters.cpp` module.

Digital implementation of the SOGI PLL is done using the `impulse invariant`:material-information-outline:{title="Alejandro Gómez Yepes, « Digital Resonant Current Controllers for Voltage Source Converters », 2011."} using $$R_{1h}$$ discretization method.
Digital implementation of the SOGI PLL is done using the `impulse invariant`:material-information-outline:{title="Alejandro Gómez Yepes, « Digital Resonant Current Controllers for Voltage Source Converters », 2011."} using $R_{1h}$ discretization method.


![SOGI PLL block diagram](./images/sogi_pll.drawio){width=700px}
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The SOGI PLL is discretized form is the following:

$$R_{1hd} = Kr \times Ts \left(. \cos(\phi') . \dfrac{1- z^{-1}.cos(\omega .T_s)}{1 - 2.z^{-1}cos(\omega T_s)+z^{-2}} -sin(\phi').\dfrac{z^{-1}.sin(\omega .T_s)}{1 - 2.z^{-1}cos(\omega T_s)+z^{-2}}\right)$${width=700px}
$R_{1hd} = Kr \times Ts \left(. \cos(\phi') . \dfrac{1- z^{-1}.cos(\omega .T_s)}{1 - 2.z^{-1}cos(\omega T_s)+z^{-2}} -sin(\phi').\dfrac{z^{-1}.sin(\omega .T_s)}{1 - 2.z^{-1}cos(\omega T_s)+z^{-2}}\right)${width=700px}

Kr parameter augment SOGI PLL dynamic but decrease its selectivity.

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