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elektronische_schaltungstechnik:rechnung_betragundphase_umkehrintegrator [2020/05/21 19:21]
tfischer 		elektronische_schaltungstechnik:rechnung_betragundphase_umkehrintegrator [2021/06/17 04:06] (aktuell)
tfischer
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-~~REVEAL theme=whide&fade=fade&controls=1&show_progress_bar=1&build_all_lists=1&show_image_borders=1&horizontal_slide_level=2&enlarge_vertical_slide_headers=0&show_slide_details=0&open_in_new_window=1&size=2024x168~~+~~REVEAL theme=white&fade=fade&controls=1&show_progress_bar=1&build_all_lists=1&show_image_borders=1&horizontal_slide_level=2&enlarge_vertical_slide_headers=0&show_slide_details=0&open_in_new_window=1&size=2400x168~~
  
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 |$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_E(t)} \ dt + U_{A0}$|Sinusfunktion einsetzen|$ \color{blue}{U_E(t)}= \hat{U}_E \cdot sin(\omega \cdot t)$| |$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_E(t)} \ dt + U_{A0}$|Sinusfunktion einsetzen|$ \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\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$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}$|Stammfunktion mit Grenzen einsetzen|$\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$| +|$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}$|Stammfunktion mit \\ Grenzen einsetzen|$\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\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$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}$  |Konstante vor Integral setzen| | +|$U_A = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_E \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$  |Konstante vor \\ Integral setzen| | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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 |$U_A = { 1 \over {R\cdot C} }\cdot {\hat{U}_E \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$  |Grenzwerte einsetzen|$t_0=0$, $t_1=t$| |$U_A = { 1 \over {R\cdot C} }\cdot {\hat{U}_E \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$  |Grenzwerte einsetzen|$t_0=0$, $t_1=t$|
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot ( cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$  | |$\color{blue}{cos(0)}=1$| +|$U_A = {{{\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$  | |$\color{blue}{cos(0)}=1$| 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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 |$U_A = \color{blue}{{{ \hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$  |Ausmultiplizieren| | |$U_A = \color{blue}{{{ \hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$  |Ausmultiplizieren| |
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$  |Betrachtung der nicht-Kosinus-Terme|Dieser Teil ist zeitlich unabhängig. Da wir von rein sinusförmigen Größen ausgehen, muss die für die anfängliche Spannung des Kondensators gelten: $U_{C0} = U_{A0}={{\hat{U}_E} \over {\omega \cdot R\cdot C}}$| +|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$  |Betrachtung der \\ nicht-Kosinus-Terme| | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| 
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 +|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$  |Dieser Teil ist zeitlich unabhängig. Da wir von rein sinusförmigen Größen ausgehen, \\ muss die für die anfängliche Spannung des Kondensators gelten: $U_{C0} = U_{A0}={{\hat{U}_E} \over {\omega \cdot R\cdot C}}$|| 
 +|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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 |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$| |  |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$| | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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