Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| circuit_design:rechnung_betragundphase_umkehrintegrator [2022/01/10 00:24] – tfischer | circuit_design:rechnung_betragundphase_umkehrintegrator [2023/03/28 14:52] (aktuell) – mexleadmin | ||
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| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_E(t)} \ dt + U_{A0}$| | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} \color{blue}{U_{\rm I}(t)} \ {\rm d}t + U_{\rm O0}$| |
| - | | $\;$ \\ $\;$ \\ $\;$ | insert sine function: \\ $ \color{blue}{U_E(t)}= \hat{U}_E \cdot sin(\omega \cdot t)$| | + | | $\;$ \\ $\;$ \\ $\;$ | insert sine function: \\ $ \color{blue}{U_{\rm I}(t)}= \hat{U}_{\rm I} \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$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_E \cdot sin(\omega \cdot t) \ dt} + U_{A0}$| | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\color{blue}{\int_{t_0}^{t_1} \hat{U}_{\rm I} \cdot \sin(\omega \cdot t) \ {\rm d}t} + U_{\rm O0}$| |
| - | | $\;$ \\ $\;$ \\ $\;$ |insert root function with limits \\ $\color{blue}{\int_{x_0}^{x_1} sin(a \cdot x) \ dx} = [- {1 \over a} \cdot cos(a \cdot x) ]_{x_0}^{x_1}$| | + | | $\;$ \\ $\;$ \\ $\;$ |insert root function with limits \\ $\color{blue}{\int_{x_0}^{x_1} |
| | |$\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$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_E \over \omega} \cdot cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{A0}$ | | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot [- \color{blue}{\hat{U}_{\rm I} \over \omega} \cdot \cos(\omega \cdot t) ]_{t_0}^{t_1} + U_{\rm O0}$ | |
| | $\;$ \\ $\;$ \\ $\;$ | put constant before integral| | | $\;$ \\ $\;$ \\ $\;$ | put constant before integral| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 20: | Zeile 20: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = { 1 \over {R\cdot C} }\cdot {\hat{U}_E \over \omega} \cdot \color{blue}{[ cos(\omega \cdot t) ]_{t_0}^{t_1}} + U_{A0}$| | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = { 1 \over {R\cdot C} }\cdot {\hat{U}_{\rm I} \over \omega} \cdot \color{blue}{[ |
| | $\;$ \\ $\;$ \\ $\;$ |insert limits: $t_0=0$, $t_1=t$| | | $\;$ \\ $\;$ \\ $\;$ |insert limits: $t_0=0$, $t_1=t$| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 26: | Zeile 26: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = {{{\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - \color{blue}{cos(0)} ) + U_{A0}$| | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = {{{\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot (} \cos(\omega \cdot t) - \color{blue}{\cos(0)} ) + U_{\rm O0}$| |
| - | | $\;$ \\ $\;$ \\ $\;$ | $\color{blue}{cos(0)}=1$| | + | | $\;$ \\ $\;$ \\ $\;$ | $\color{blue}{\cos(0)}=1$| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = \color{blue}{{{ \hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot (} cos(\omega \cdot t) - 1 \color{blue}{)} + U_{A0}$ | | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = \color{blue}{{{ \hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot (} \cos(\omega \cdot t) - 1 \color{blue}{)} + U_{\rm O0}$ | |
| | $\;$ \\ $\;$ \\ $\;$ |multiply| | | $\;$ \\ $\;$ \\ $\;$ |multiply| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 38: | Zeile 38: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ | | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C}} + U_{\rm O0}}$ | |
| | $\;$ \\ $\;$ \\ $\;$ |consider the non-cosine terms: \\ The blue part is independent in time. \\ We assume purely sinusoidal quantities! | | | $\;$ \\ $\;$ \\ $\;$ |consider the non-cosine terms: \\ The blue part is independent in time. \\ We assume purely sinusoidal quantities! | | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 44: | Zeile 44: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_E } \over {\omega \cdot R\cdot C}} + U_{A0}}$ | | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t) \color{blue}{-{ {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C}} + U_{\rm O0}}$ | |
| - | | $\;$ \\ $\;$ \\ $\;$ | $\rightarrow$ initial voltage of the capacitor: \\ $U_{C0} = U_{A0}={{\hat{U}_E} \over {\omega \cdot R\cdot C}}$|| | + | | $\;$ \\ $\;$ \\ $\;$ | $\rightarrow$ initial voltage of the capacitor: \\ $U_{C0} = U_{\rm O0}={{\hat{U}_{\rm I}} \over {\omega \cdot R\cdot C}}$|| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ \\ $\;$ |$U_A = { {\hat{U}_E } \over {\omega \cdot R\cdot C} } \cdot cos(\omega \cdot t)$| | + | | $\;$ \\ $\;$ \\ $\;$ |$U_{\rm O} = { {\hat{U}_{\rm I} } \over {\omega \cdot R\cdot C} } \cdot \cos(\omega \cdot t)$| |
| | $\;$ \\ $\;$ \\ $\;$ | | | | $\;$ \\ $\;$ \\ $\;$ | | | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||