Début de la boite de navigation du chapitre
fin de la boite de navigation du chapitre
En raison de limitations techniques, la typographie souhaitable du titre, «
Amplificateur opérationnel : Sommateur et soustracteur Amplificateur opérationnel/Sommateur et soustracteur », n'a pu être restituée correctement ci-dessus.
modifier
Nous rappelons que
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{\displaystyle v^{+}=v^{-}}
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{\displaystyle {\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}}}
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{\displaystyle 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}}}}}}
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{\displaystyle 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}}}}
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{\displaystyle -{\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}}}}
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{\displaystyle {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)}
modifier
{
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{\displaystyle {\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}}}
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{\displaystyle v^{-}=-{\frac {V_{s}}{A}}}
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{\displaystyle {\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}}}}})}
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{\displaystyle 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)}
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{\displaystyle 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)}
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{\displaystyle 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)}
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{\displaystyle 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)}
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{\displaystyle 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
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{\displaystyle 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}}}}}}
A
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{\displaystyle A\gg 1}
ce qui nous donne
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{\displaystyle 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}}}}}}
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{\displaystyle 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}}}}}
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{\displaystyle V_{s}=-R_{f}\times \left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)}
Nous rappelons que
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{\displaystyle v^{+}=v^{-}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}}
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{\displaystyle 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)}
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{\displaystyle 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)}
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{\displaystyle 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)}}}
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{\displaystyle 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}}}}
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{\displaystyle 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}}}}
modifier
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