Amplificateur opérationnel/Sommateur et soustracteur

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Additionneur

Amplificateur opérationnel idéal

Nous rappelons que $v^{+}=v^{-}$

${\begin{cases}v^{+}=0\\v^{-}={\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{f}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}}}\\\end{cases}}$

$0={\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{f}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}}}$

$0={\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{f}}}$

$-{\frac {V_{s}}{R_{f}}}={\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}$

${V_{s}}=-R_{f}\left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)$

Amplificateur opérationnel non idéal

${\begin{cases}v^{+}=0\\v^{-}={\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+{\frac {V_{3}}{R_{3}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{f}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}}}\\V_{s}=A\left(v^{+}-v^{-}\right)\end{cases}}$

$v^{-}=-{\frac {V_{s}}{A}}$

${\frac {V_{s}}{A}}=-({\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{s}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}}})$

$V_{s}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}\right)=-A\left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}+{\frac {V_{s}}{R_{f}}}\right)$

$V_{s}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}\right)=-\left({\frac {A\times V_{1}}{R_{1}}}+{\frac {A\times V_{2}}{R_{2}}}+\cdots +{\frac {A\times V_{n}}{R_{n}}}+{\frac {A\times V_{s}}{R_{f}}}\right)$

$V_{s}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}\right)+{\frac {A\times V_{s}}{R_{f}}}=-\left({\frac {A\times V_{1}}{R_{1}}}+{\frac {A\times V_{2}}{R_{2}}}+\cdots +{\frac {A\times V_{n}}{R_{n}}}\right)$

$V_{s}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}+{\frac {A}{R_{f}}}\right)=-\left({\frac {A\times V_{1}}{R_{1}}}+{\frac {A\times V_{2}}{R_{2}}}+\cdots +{\frac {A\times V_{n}}{R_{n}}}\right)$

$V_{s}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1+A}{R_{f}}}\right)=-A\times \left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)$

Conclusion

$V_{s}=-A\times {\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1+A}{R_{f}}}}}$

Pour rejoindre le résultat précédent on pose :

$A\gg 1$

ce qui nous donne

$V_{s}=-A\times {\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots +{\frac {1}{R_{n}}}+{\frac {1}{R_{f}}}+{\frac {A}{R_{f}}}}}$

$V_{s}=-A\times {\frac {{\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}}{\frac {A}{R_{f}}}}$

$V_{s}=-R_{f}\times \left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)$

Impédance de sortie

Soustracteur

Ampli OP soustracteur idéal

Nous rappelons que $v^{+}=v^{-}$

${\begin{cases}v^{+}={\frac {V_{2}\times R_{g}}{R_{2}+R_{g}}}\\v^{-}={\frac {V_{1}\times R_{f}+V_{s}\times R1}{R_{f}+R_{1}}}\\\end{cases}}$

${\frac {V_{2}\times R_{g}}{R_{2}+R_{g}}}={\frac {V_{1}\times R_{f}+V_{s}\times R_{1}}{R_{f}+R_{1}}}$

$V_{2}\times R_{g}\times \left(R_{f}+R_{1}\right)=\left(V_{1}\times R_{f}+V_{s}\times R_{1}\right)\times \left(R_{2}+R_{g}\right)$

$V_{2}\times R_{g}\times \left(R_{f}+R_{1}\right)=V_{1}\times R_{f}\times \left(R_{2}+R_{g}\right)+V_{s}\times R_{1}\times \left(R_{2}+R_{g}\right)$

$V_{s}={\frac {V_{2}\times R_{g}\times \left(R_{f}+R_{1}\right)-V_{1}\times R_{f}\times \left(R_{2}+R_{g}\right)}{R_{1}\times \left(R_{2}+R_{g}\right)}}$

$V_{s}=V_{2}\times {\frac {R_{g}\times \left(R_{f}+R_{1}\right)}{R_{1}\times \left(R_{2}+R_{g}\right)}}-V_{1}\times {\frac {R_{f}}{R_{1}}}$

$V_{s}=V_{2}\times {\frac {R_{g}\times \left(R_{f}+R_{1}\right)}{R_{1}\times \left(R_{2}+R_{g}\right)}}-V_{1}\times {\frac {R_{f}}{R_{1}}}$

Impédance de sortie