Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| circuit_design:rechnung_umkehrintegrator [2022/01/10 00:01] – tfischer | circuit_design:rechnung_umkehrintegrator [2023/03/28 14:44] (aktuell) – mexleadmin | ||
|---|---|---|---|
| Zeile 3: | Zeile 3: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = f(U_E)$ | | + | | $\;$ \\ $\;$ |$U_{\rm O} = f(U_{\rm I})$ | |
| | $\;$ \\ $\;$ | with III.| | | $\;$ \\ $\;$ | with III.| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 9: | Zeile 9: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A=\color{blue}{-U_D}-U_C$ | + | | $\;$ \\ $\;$ |$U_{\rm O}=\color{blue}{-U_{\rm D}}-U_C$ |
| - | | $\;$ \\ $\;$ |with II. and I.:$ \color{blue}{U_D} = { 1 \over A_D } \cdot U_A \overset{A_D -> \infty}\longrightarrow 0$| | + | | $\;$ \\ $\;$ |with II. and I.:$ \color{blue}{U_{\rm D}} = { 1 \over A_D } \cdot U_{\rm O} \overset{A_D -> \infty}\longrightarrow 0$| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A= \quad 0 \quad -\color{blue}{U_C}$| | + | | $\;$ \\ $\;$ |$U_{\rm O}= \quad 0 \quad -\color{blue}{U_C}$| |
| - | | $\;$ \\ $\;$ |with V.: $\color{blue}{U_C}={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ dt+ Q_0(t_0))$| | + | | $\;$ \\ $\;$ |with V.: $\color{blue}{U_C}={ 1 \over C }\cdot(\int_{t_0}^{t_1} I_C \ {\rm d} t+ Q_0(t_0))$| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = {-{ 1 \over C }\cdot}(\int_{t_0}^{t_1} \color{blue}{I_C} \ dt+ Q_0(t_0)) $| | + | | $\;$ \\ $\;$ |$U_{\rm O} = {-{ 1 \over C }\cdot}(\int_{t_0}^{t_1} \color{blue}{I_C} \ {\rm d} t+ Q_0(t_0)) $| |
| | $\;$ \\ $\;$ |with IV.: $\color{blue}{I_C}=I_R$| | | $\;$ \\ $\;$ |with IV.: $\color{blue}{I_C}=I_R$| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 27: | Zeile 27: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = \color{blue}{-{ 1 \over C }\cdot(}\int_{t_0}^{t_1} I_R \ dt+ Q_0(t_0)\color{blue}{)} $| | + | | $\;$ \\ $\;$ |$U_{\rm O} = \color{blue}{-{ 1 \over C }\cdot(}\int_{t_0}^{t_1} I_R \ {\rm d} t+ Q_0(t_0)\color{blue}{)} $| |
| | $\;$ \\ $\;$ |Factor out| | | $\;$ \\ $\;$ |Factor out| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 33: | Zeile 33: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} I_R \ dt - \color{blue}{ Q_0(t_0) \over C } $| | + | | $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} I_R \ {\rm d} t - \color{blue}{ Q_0(t_0) \over C } $| |
| - | | $\;$ \\ $\;$ |consider the integration constant: $\color{blue}{ Q_0(t_0) \over C }= U_C(t_0) = -U_{A0}$| | + | | $\;$ \\ $\;$ |integration constant: $\color{blue}{ Q_0(t_0) \over C }= U_C(t_0) = -U_{\rm O0}$| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{I_R} \ dt + U_{A0}$| | + | | $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{I_R} \ {\rm d} t + U_{\rm O0}$| |
| - | | $\;$ \\ $\;$ |with VI. and II.: $\color{blue}{I_R}={ U_R \over R}={ U_E \over R} $| | + | | $\;$ \\ $\;$ |with VI. and II.: $\color{blue}{I_R}={ U_R \over R}={ U_{\rm I} \over R} $| |
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| <---- | <---- | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{1 \over R} \cdot U_E \ dt + U_{A0}$| | + | | $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over C }\cdot\int_{t_0}^{t_1} \color{blue}{1 \over R} \cdot U_{\rm I} \ {\rm d} t + U_{\rm O0}$| |
| | $\;$ \\ $\;$ |move constant ahead| | | $\;$ \\ $\;$ |move constant ahead| | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 51: | Zeile 51: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} | + | | $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {R\cdot C} }\cdot\int_{t_0}^{t_1} |
| | $\;$ \\ $\;$ | insert time constant | | $\;$ \\ $\;$ | insert time constant | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||
| Zeile 57: | Zeile 57: | ||
| ----> | ----> | ||
| - | | $\;$ \\ $\;$ |$U_A = -{ 1 \over {\tau} }\cdot\int_{t_0}^{t_1} | + | | $\;$ \\ $\;$ |$U_{\rm O} = -{ 1 \over {\tau} }\cdot\int_{t_0}^{t_1} |
| | $\;$ \\ $\;$ | | | | $\;$ \\ $\;$ | | | ||
| | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | | |$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$| | ||