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