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- \setlength{\imagewidth}{6cm}
-
- % ============================================================================================
- \section{Lineare OPV-Schaltungen, Gegengekoppelte Strukturen}
- % ============================================================================================
-
- \begin{sectionbox}
-
- % OPV Modelle
- % ----------------------------------------------------------------------
- \subsection{Allgemines Modell}
- \begin{center}
- \includegraphics[width = 0.5\columnwidth]{img_02_00_modell_opv}
- \end{center}
-
- % OPV Formeln
- % ----------------------------------------------------------------------
- \subsection{Operationsverstärker}
-
- % Differenzverstärkung %
- $A_{VD}(=V_{UD})=\frac{U_{OUT}}{U_{ID}}(typ.>100k)>>1$
-
- % GLeichtaktverstärkung %
- $A_{VC}=\frac{U_{OUT}}{U_{CM}} \approx 0$
-
- % Common Mode Rejection Ratio %
- $CMMR=\frac{A_{VD}}{A_{VC}}>>1$ \quad\
- $CMMR/dB=20\cdot log(\frac{A_{VD}}{A_{VC}})$
-
- % Frequenzgang %
- $\underline{V}_{ud}(f)=\frac{V_{UD}}{\cancel{1}+\frac{j\cdot f}{f_1}}$
- \begin{emphbox}
- $f_1(=f_{1,3dB})=\frac{f_T(=GBW)}{V_{ud}}$
- \end{emphbox}
-
- % Standard-Rückkopplungsstruktur
- % ----------------------------------------------------------------------
- \subsection{Standardstruktur}
-
- \pbox{5cm}{\includegraphics[width = 5cm - 1cm]{img_02_01_Standardstruktur}}
- \parbox{\textwidth - 5cm + 1cm}{
- % Rückkopplungsfaktor %
- \begin{bluebox}
- $\underline{k} = \frac{\underline{u}_k}{\underline{u}_2}\vert_{u_1 = 0} = \frac{-\underline{u}_{id}}{\underline{u}_2}\vert_{u_1 = 0}$
- \end{bluebox}
-
- % Schleifenverstärkung %
- Schleifenverstärkung: $\underline{g} = \underline{V}_{ud} \cdot \underline{k}$
-
- % Ausgangsspannung %
- $\underline{u}_2 = \underline{a}_V^+ \cdot \underline{u}_1^+ + \underline{a}_V^- \cdot \underline{u}_1^-$
-
- % Spannungsverstärkung %
- \begin{emphbox}
- $\underline{a}_V^+ = \frac{\underline{V}_{ud}}{1+\underline{k}\cdot\underline{V}_{ud}}$ \newline
- $\underline{a}_V^- = -\frac{\underline{V}_{ud}\cdot(1-\underline{k})}{1+\underline{k}\cdot\underline{V}_{ud}}$\newline
- \end{emphbox}
- }
-
- \subsubsection{Betriebsmodi}
-
- % Nichtinvertierender Betrieb %
- \underline{Nichtinvertierender Betrieb:}
- \begin{bluebox}
- \begin{center}
- $\underline{u}_1^- = 0!$ \quad\
- $\underline{u}_1 = \underline{u}_1^+$ \quad\
- $\underline{g} = \underline{k} \cdot \underline{V}_{ud}$
- \end{center}
- \end{bluebox}
-
- Normalbetrieb: $|\underline{k} \cdot \underline{V}_{ud}| >> 1$
- \begin{emphbox}
- $\underline{a}_V = +\frac{1}{\underline{k}} = 1 + \frac{\underline{Z}_2}{\underline{Z}_1}$
- \end{emphbox}
-
- OPV-Vorwärtsbertrieb: $|\underline{k} \cdot \underline{V}_{ud}| << 1$
- \begin{emphbox}
- $\underline{a}_V = \underline{V}_{ud}$
- \end{emphbox}
-
- % Invertierender Betrieb %
- \underline{Invertierender Betrieb:}
- \begin{bluebox}
- \begin{center}
- $\underline{u}_1^+ = 0!$ \quad\
- $\underline{u}_1 = \underline{u}_1^-$ \quad\
- $\underline{g} = \underline{k} \cdot \underline{V}_{ud}$
- \end{center}
- \end{bluebox}
-
- Normalbetrieb: $|\underline{k} \cdot \underline{V}_{ud}| >> 1$
- \begin{emphbox}
- $\underline{a}_V = -\frac{1-\underline{k}}{\underline{k}} = 1 - \frac{1}{\underline{k}}
- = -\frac{\underline{Z}_2}{\underline{Z}_1}$
- \end{emphbox}
-
- OPV-Vorwärtsbertrieb: $|\underline{k} \cdot \underline{V}_{ud}| << 1$
- \begin{emphbox}
- $\underline{a}_V = -\underline{V}_{ud} \cdot (1 - \underline{k})$
- \end{emphbox}
-
- \subsubsection{Betriebsfrequenzgrenze der Schaltung}
- Betriebsfrequenzgrenze $f_g$ (= Durchtrittsfreq. $f_D$)
- \begin{bluebox}
- \begin{center}
- $|\underline{g}(f_g (= f_D))| = |\underline{k}(f_g) \cdot \underline{V}_{ud}(f_g)| = 1$
- \end{center}
- \end{bluebox}
-
- \begin{emphbox}
- $f_g \approx \frac{GBW}{1/|\underline{k}(f_g)|}$
- \end{emphbox}
-
- \end{sectionbox}
- \begin{sectionbox}
-
- % Standard-Rückkopplungsstruktur
- % ----------------------------------------------------------------------
- \subsection{Stabilität von gegengekoppelten OPV-Schaltungen}
- $\varphi_R = \varphi(\underline{g}(f_D)) - (-180\degree)$
- \begin{bluebox}
- \item Bei negativer Schleifenverstärkung (= Mitkopplung): $\underline{g} < 1$
- \item Robust stabile Schaltung: $\varphi_R > 45 \degree$
- \end{bluebox}
-
- % Testschaltung zur Ermittlung der Schleifenverstärkung
- % ----------------------------------------------------------------------
- \subsection{Testschaltung zur Ermittlung der Schleifenverstärkung}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_12_testschaltung_schleifenverstaerkung}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{g} = - \frac{\underline{v}(g\_out)}{\underline{v}(g\_in)}$
- }
-
- % Kompensation der Ausgangs-Offset-Spannung
- % ----------------------------------------------------------------------
- \subsection{Kompensation der Ausgangs-Offset-Spannung}
- \pbox{5cm}{\includegraphics[width = 4cm]{img_02_13_ruhestromkompensation}}
- \pbox{6cm}{\includegraphics[width = 5cm]{img_02_14_uio_kompensation}}
- \newline
- \parbox{4cm}{\begin{emphbox} $R^+ = R^-$ \end{emphbox}} \quad\quad\quad
- \parbox{4cm}{\begin{emphbox} $U_{ID} = U_{IO}$ \end{emphbox}}
-
-
- % Gegenkopplung und Mitkopplung
- % ----------------------------------------------------------------------
- \subsection{Kompensation der Ausgangs-Offset-Spannung}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_13_mitkopplung}}
- \parbox{\textwidth - \imagewidth}{
- % Rückkopplungsfaktor %
- \begin{bluebox}
- $\underline{k}
- = \frac{-\underline{u}_{id}\vert_{u_1 = 0}}{\underline{u}_2} \newline
- = \frac{\underline{u}(-)-\underline{u}(+)}{\underline{u}_2}\vert_{u_1 = 0} \newline
- = \underline{k}^{(-)} - \underline{k}^{(+)}$
- \end{bluebox}
-
- \begin{emphbox}
- $\underline{k} = \frac{\underline{Z}_1}{\underline{Z}_1 + \underline{Z}_2} - \frac{\underline{Z}_3}{\underline{Z}_3 + \underline{Z}_4}$
- \end{emphbox}
- }
-
- \end{sectionbox}
- \newpage
- \begin{sectionbox}
-
- % Standard lineare OPV-Schaltungen
- % ----------------------------------------------------------------------
- \subsection{Standard Linearverstärker mit OPV}
- \subsubsection{Invertierender Standard Verstärker}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_02_invertierender_verstaerker}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{a}_V = - \frac{R_2}{R_1}$ \newline
- $\underline{z}_{in} = R_1$ \newline
- $\underline{z}_a = (R_1+R_2)||\frac{\underline{z}_{a,OPV}}{1+\underline{k} \cdot \underline{V}_{ud}}$
- }
-
- \subsubsection{Nichtinvertierender Standard Verstärker}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_03_nichtinvertierender_verstaerker}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{a}_V = 1 + \frac{R_2}{R_1}$ \newline
- $\underline{z}_{in} = \underline{z}_{id} \cdot (1+\underline{k} \cdot \underline{V}_{ud})$ \newline
- $\underline{z}_a = (R_1+R_2)||\frac{\underline{z}_{a,OPV}}{1+\underline{k} \cdot \underline{V}_{ud}}$
- }
-
-
- \subsubsection{Spannungsfolger, Impedanzwandler}
- \pbox{\imagewidth}{\includegraphics[width = {\imagewidth - 2cm}]{img_02_04_impedanzwandler}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{a}_V = 1$ \newline
- $\underline{z}_{in} = \underline{z}_{id} \cdot (1 + 1 \cdot \underline{V}_{ud})$ \newline
- $\underline{z}_a = \frac{\underline{z}_{a,OPV}}{1 + 1 \cdot \underline{V}_{ud}}$
- }
-
- \subsubsection{Integrierer}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_05_integrierer}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- $U_2(t)= -\frac{1}{R \cdot C} \cdot \int_0^t U_1(t) \cdot dt + U_2(0)$ \newline
- $\frac{U_2(s)}{U_1(s)} = - \frac{1}{s \cdot R \cdot C}$ \newline
- $\underline{a}_V = - \frac{1}{j\omega \cdot R \cdot C}$ \newline
- $\underline{z}_{in} = R$ \newline
- $\underline{z}_a = (\frac{1}{j\omega \cdot C}+R)||\frac{\underline{z}_{a,OPV}}{1+\underline{k} \cdot \underline{V}_{ud}}$
- }
-
- \subsubsection{Differentiator (Differenzierer)}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_06_differenzierer}}
- \parbox{\textwidth - \imagewidth}{
- $U_2(t) \approx - R_2 \cdot C_1 \cdot \frac{U_1(t)}{dt}$ \newline
- $\frac{U_2(s)}{U_1(s)} = - s \cdot R_2 \cdot C_1$ \newline
- $\underline{a}_V \approx - j\omega \cdot R_2 \cdot C_1$ \newline
- $\underline{z}_{in} \approx \frac{1}{j\omega \cdot C_1}$
- }
- \begin{emphbox}
- für $\varphi_R = 45\degree$ : $R_1 = \frac{1}{f_D\cdot 2 \pi \cdot C_1} = \frac{1}{2\pi \cdot C_1 \cdot \sqrt{\frac{GBW}{2\pi \cdot R_2 \cdot C_1}}}$
- \end{emphbox}
-
- \end{sectionbox}
- %Force column break
- \begin{sectionbox}
-
-
- \subsubsection{Summierer (Invertierend)}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_07_summierer}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{a}_{V,i} = \frac{R_2}{R_1}$ \newline
- $\underline{z}_{in,i} = R_1$ \newline
- $\underline{z}_a = (R_2+\frac{R_1}{n})||\frac{\underline{z}_{a,OPV}}{1 + \underline{k} \cdot \underline{V}_{ud}}$
- }
-
- \subsubsection{Differenzverstärker (aktiver Subtrahierer, einfache Struktur)}
- \pbox{5cm}{\includegraphics[width = 5cm - 1cm]{img_02_08_differenzverstaerker}}
- \parbox{\textwidth - 5cm + 1cm}{
- $\underline{u}_2 = -\frac{R_2}{R_1}\cdot \underline{u}_{in2}$ \newline $+ \frac{R_1+R_2}{R1}\cdot\frac{R_4}{R_3+R_4}\cdot\underline{u}_{in1}$ \newline
- $\underline{z}_{in1} = R_3 + R_4$ \newline
- $\underline{z}_{in2} = R_1 \big \vert _{\underline{u}_{in1}=0} = R_1$ \newline
- \begin{emphbox}
- Für $R_3=R_1$ und $R_4=R_2$ : \newline
- $\underline{u}_2 = \frac{R_2}{R_1}\cdot(\underline{u}_{in1} - \underline{u}_{in2})$
- \end{emphbox}
- }
-
- \subsubsection{Instrumentenverstärker (verbesserter Differenzverstärker)}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_09_instrumentenverstaerker}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- $\underline{u}_{out1} = (1+\frac{R_2}{R_1}) \cdot \underline{u}_{in1} - \frac{R_2}{R_1} \cdot \underline{u}_{in2}$ \newline
- $\underline{u}_{out2} = (1+\frac{R_2}{R_1}) \cdot \underline{u}_{in2} - \frac{R_2}{R_1} \cdot \underline{u}_{in1}$ \newline
- $\underline{u}_2 = \frac{R_4}{R_3} \cdot (1+2\cdot \frac{R_2}{R_1})\cdot (\underline{u}_{in1} - \underline{u}_{in2})$ \newline \newline
- $\underline{z}_{in1,2} \to \infty$
- }
-
- \subsubsection{Spannungsgesteuerte Stromquelle ($G_m$)}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 2cm]{img_02_10_stromquelle}}
- \parbox{\textwidth - \imagewidth}{
- $\underline{u}_1 = \underline{u}_2 \cdot (1-\frac{R_4 \cdot R_1}{R_3 \cdot R_2}+\frac{R_1}{R_L})$ \newline
- \begin{emphbox}
- Für $\frac{R_4}{R_3} = \frac{R_2}{R_1}$: \quad\
- $i_2 = \frac{1}{R_1} \cdot u_1$
- \end{emphbox}
- }
-
- \subsubsection{Negativ-Impedanz-Konverter (NIC)}
- \pbox{0.7\imagewidth}{\includegraphics[width = 0.7\imagewidth - 1cm]{img_02_11_NIC}}
- \parbox{\textwidth - 0.7\imagewidth}{
- $\underline{z}_1 = -\underline{Z} \cdot \frac{R_1}{R_2}$
- \begin{emphbox}
- Für $R_1 = R_2$: \quad\
- $\underline{z}_1 = -\underline{Z}$
- \end{emphbox}
- }
-
- \end{sectionbox}
-
- \begin{sectionbox}
-
- % Filter
- % ----------------------------------------------------------------------
- \subsection{Filter-Grundschaltungen mit OPV}
- %TODO
- \end{sectionbox}
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