Dies ist eine alte Version des Dokuments!
| $\;$ $\;$ $\;$ | $U_O = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_I(t)} \ dt + U_{A0}$ |
| $\;$ $\;$ $\;$ | insert sine function: $ \color{blue}{U_I(t)}= \hat{U}_I \cdot sin(\omega \cdot t)$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_I \cdot sin(\omega \cdot t) \ dt} + U_{A0}$ |
| $\;$ $\;$ $\;$ | insert root function with limits $\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_I \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$ |
| $\;$ $\;$ $\;$ | put constant before integral |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = { 1 \over {R\cdot C} }\cdot {\hat{U}_I \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$ |
| $\;$ $\;$ $\;$ | insert limits: $t_0=0$, $t_1=t$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = {{{\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$ |
| $\;$ $\;$ $\;$ | $\color{blue}{cos(0)}=1$ |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = \color{blue}{{{ \hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$ |
| $\;$ $\;$ $\;$ | multiply |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_I } \over {\omega \cdot R\cdot C}} + U_{A0}}$ |
| $\;$ $\;$ $\;$ | consider the non-cosine terms: The blue part is independent in time. We assume purely sinusoidal quantities! |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |
| $\;$ $\;$ $\;$ | $U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_I } \over {\omega \cdot R\cdot C}} + U_{A0}}$ | |
| $\;$ $\;$ $\;$ | $\rightarrow$ initial voltage of the capacitor: $U_{C0} = U_{A0}={{\hat{U}_I} \over {\omega \cdot R\cdot C}}$ |
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| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ | ||
| $\;$ $\;$ $\;$ | $U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$ |
| $\;$ $\;$ $\;$ | |
| $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ |