$U_A = f(U_E)$ mit III.
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$U_A=\color{blue}{-U_D}-U_C$ mit II. und I.$ \color{blue}{U_D} = { 1 \over A_D } \cdot U_A \overset{A_D \rightarrow \infty}\longrightarrow 0$
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$U_A= \quad 0 \quad -\color{blue}{U_C}$mit V.$\color{blue}{U_C}={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ dt+ Q_0(t_0))$
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$U_A = {-{ 1 \over C }\cdot}(\int_{t_0}^{t_1} \color{blue}{I_C} \ dt+ Q_0(t_0)) $mit IV.$\color{blue}{I_C}=I_R$
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$U_A = \color{blue}{-{ 1 \over C }\cdot(}\int_{t_0}^{t_1} I_R \ dt+ Q_0(t_0)\color{blue}{)} $Ausklammern
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$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} I_R \ dt - \color{blue}{ Q_0(t_0) \over C } $Integrationskonstante
betrachten
$\color{blue}{ Q_0(t_0) \over C }= U_C(t_0) = -U_{A0}$
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$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{I_R} \ dt + U_{A0}$mit VI. und II.$\color{blue}{I_R}={ U_R \over R}={ U_E \over R} $
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$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{1 \over R} \cdot U_E \ dt + U_{A0}$Konstante vorziehen
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$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} U_E \ dt + U_{A0}$ Zeitkonstante
$\tau = R \cdot C$ einfügen
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$U_A = -{ 1 \over {\tau} }\cdot\int_{t_0}^{t_1} U_E \ dt + U_{A0}$
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