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circuit_design:rechnung_betragundphase_umkehrintegrator [2021/09/26 19:19]
tfischer 		circuit_design:rechnung_betragundphase_umkehrintegrator [2023/01/31 18:37]
mexleadmin
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-~~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~~+~~REVEAL~~
  
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-|$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_E(t)} \ dt + U_{A0}$|insert sine function|$ \color{blue}{U_E(t)}= \hat{U}_E \cdot sin(\omega \cdot t)$| +| $\;$ \\ $\;$ \\ $\;$ |$U_O = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_I(t)} \ dt + U_{A0}$| 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ | 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$|
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-|$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}$|insert root function with \ limits|$\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$| +| $\;$ \\ $\;$ \\ $\;$ |$U_O = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_I \cdot sin(\omega \cdot t) \ dt} + U_{A0}$| 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ |insert root function with limits \\  $\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_E \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$ |put constant before \\ integral| | +| $\;$ \\ $\;$ \\ $\;$ |$U_O = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_I \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$ | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ | put constant before integral| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { 1 \over {R\cdot C} }\cdot {\hat{U}_E \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$ |insert limits|$t_0=0$, $t_1=t$| +| $\;$ \\ $\;$ \\ $\;$ |$U_O = { 1 \over {R\cdot C} }\cdot {\hat{U}_I \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$| 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ |insert limits: $t_0=0$, $t_1=t$| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = {{{\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$  | |$\color{blue}{cos(0)}=1$| +| $\;$ \\ $\;$ \\ $\;$ |$U_O = {{{\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$| 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ | $\color{blue}{cos(0)}=1$| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = \color{blue}{{{ \hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$ |multiply| | +| $\;$ \\ $\;$ \\ $\;$ |$U_O = \color{blue}{{{ \hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$ | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ |multiply| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ |consider the \\ non-cosine terms| | +| $\;$ \\ $\;$ \\ $\;$ |$U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_I } \over {\omega \cdot R\cdot C}} + U_{A0}}$ | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ |consider the non-cosine terms: \\ The blue part is independent in time. \\ We assume purely sinusoidal quantities! | 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ |This part is independent in time. Since we assume purely sinusoidal quantities, \the for the initial voltage of the capacitor must be: $U_{C0} = U_{A0}={{\hat{U}_E} \over {\omega \cdot R\cdot C}}$|| +| $\;$ \\ $\;$ \\ $\;$ |$U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_I } \over {\omega \cdot R\cdot C}} + U_{A0}}$ | 
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ | $\rightarrow$ initial voltage of the capacitor: \\ $U_{C0} = U_{A0}={{\hat{U}_I} \over {\omega \cdot R\cdot C}}$|| 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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-|$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$| | +| $\;$ \\ $\;$ \\ $\;$ |$U_O = { {\hat{U}_I } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$|  
-|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad$|$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|+| $\;$ \\ $\;$ \\ $\;$ | | 
 +| |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$|
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