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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{Gegenkopplung und Mitkopplung}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_13_mitkopplung}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- % Rückkopplungsfaktor %
- \begin{basicbox}
- $\underline{k}
- = \frac{-\underline{u}_{id}\vert_{u_1 = 0}}{\underline{u}_2} \newline
- = \frac{\underline{u}(-)-\underline{u}(+)}{\underline{u}_2}\vert_{u_1 = 0} \newline
- = \underline{k}^{(-)} - \underline{k}^{(+)}$
- \end{basicbox}
-
- \begin{emphbox}
- $\underline{k} = \frac{\underline{Z}_1}{\underline{Z}_1 + \underline{Z}_2} - \frac{\underline{Z}_3}{\underline{Z}_3 + \underline{Z}_4} > 0!$
- \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}
-
- \section{Filter}
-
- \begin{sectionbox}
-
- % Filter Grundlagen
- % ----------------------------------------------------------------------
- \subsection{Grundlagen}
- \subsubsection{TP1 / HP1}
- % Tiefpass 1. Ordnung %
- \begin{bluebox}
- $\underline{H}_{TP1}(\omega) = \frac{K_P}{1 + j\omega \cdot \tau_1}$ \quad\
- $\underline{H}_{TP1}(f) = \frac{K_P}{1 + j\cdot f / f_C}$ \quad\
- $f_C = f_{3dB} = \frac{1}{2\pi \cdot \tau_1}$ \newline
-
- $\underline{H}_{HP1}(\omega) = \frac{K_P \cdot j\omega \cdot \tau_1}{1 + j\omega \cdot \tau_1}$ \quad\
- $\underline{H}_{HP1}(f) = \frac{K_P \cdot \frac{j\cdot f}{f_C}}{1+ \frac{j\cdot f}{f_C}}$ \quad\
- $f_C = f_{3dB} = \frac{1}{2\pi \cdot \tau_1}$ \newline
- \end{bluebox}
-
- \subsubsection{TP2 / HP2}
- % Tiefpass 2. Ordnung %
- \begin{bluebox}
- $\underline{H}_{TP2}(\omega) = \frac{K_P}{1 + j\omega \cdot \tau_1 + (j\omega \cdot \tau_2)^2}$ \quad\
- $\underline{H}_{TP2}(f) = \frac{K_P}{1 + j \cdot \frac{f}{f_C}/Q + (j \cdot \frac{f}{f_C})^2}$ \newline
-
- $\underline{H}_{HP2}(\omega) = \frac{K_P \cdot (j\omega \cdot \tau_2)^2}{1 + j\omega \cdot \tau_1 + (j\omega \cdot \tau_2)^2}$ \quad\
- $\underline{H}_{HP2}(f) = \frac{K_P\cdot (\frac{j \cdot f}{f_C})^2}{1 + j \cdot \frac{f}{f_C}/Q + (j \cdot \frac{f}{f_C})^2}$
- \end{bluebox}
-
- $f_C = \frac{1}{2\pi \cdot \tau_2}$ \quad \quad \quad\
- $Q = \frac{\tau_2}{\tau_1}$ % Güte %
-
- % Grenzfrequenz %
- \begin{emphbox}
- \item{$\frac{f_{3dB(TP2)}}{f_C} = \sqrt{\sqrt{1+(\frac{1}{2\cdot Q^2}-1)^2} - (\frac{1}{2\cdot Q^2}-1)}$}
- \item{$\frac{f_{3dB(HP2)}}{f_C} = 1 / \sqrt{\sqrt{1+(\frac{1}{2\cdot Q^2}-1)^2} - (\frac{1}{2\cdot Q^2}-1)}$}
- \end{emphbox}
- Bei $Q = 1/\sqrt{2}$ : $f_C = f_{3dB}$
-
- Resonanzüberhöhung:
- \begin{bluebox}
- \begin{center}
- \item{$|\underline{H}_{TP2,HP2}(F_C)| = K_P \cdot Q$}
- \item{$\varphi(\underline{H}_{TP2}(f_C)) = -90\degree$ \quad\
- $\varphi(\underline{H}_{HP2}(f_C)) = +90\degree$}
- \end{center}
- \end{bluebox}
-
- \subsubsection{TP - HP Transformation}
- % TP - HP Transformation %
- \begin{emphbox}
- $\underline{H}_{HP}(j\cdot f) = \underline{H}_{TP}(\frac{j\cdot f}{f_C} \to \frac{f_C}{j\cdot f})$
- \end{emphbox}
-
- \subsubsection{Dimensionierungshinweise}
- % Dimensionierungshinweise
- \begin{bluebox}
- \item{GBW-Reserve ($V_{ud}$-Abstand):
- $|\underline{V}_{ud}(f_C)|/|H(f_C)| > 20dB ... \emph{40dB}$}
- \end{bluebox}
- für $f_C >> f_1$ : \emph{$|\underline{V}_{ud}(f_C)| = GBW / f_C$}
- \begin{bluebox}
- \item{Grenze des Normalbetriebs gemäß Vorgabe; für HP relevant:
- \quad $f_g \approx GBW \cdot |\underline{k}(f_g)|$}
- \item{Stabilität im Normalbetrieb: $Q > 0$}
- \item{Scharfer Übergang: $Q > 0,5 ... < 3$}
- \end{bluebox}
-
- \subsection{Prinzip der Bandpass-, Bandsperren-Realisierung}
- % Bandpass / Bandsperre
- \parbox{0.5\textwidth}{
- Bandpass ($f_{3dB,HP} \leq f_{3dB,TP}$):
- \begin{center}
- \includegraphics[width = 0.5\columnwidth]{img_02_23_bandpass}
- \end{center}
- \begin{emphbox}
- \item{$\underline{H}_{BP}(f) = \underline{H}_{HP}(f) \cdot \underline{H}_{TP}(f)$}
- \end{emphbox}
- Für $f_{3dB,HP} << f_{3dB,TP}$ gilt: \newline $f_{3dB,u} \approx f_{3dB,HP}$ \newline $f_{3dB,o} \approx f_{3dB,TP}$
- }
- \parbox{0.5\textwidth}{
- Bandsperre ($f_{3dB,TP} < f_{3dB,HP}$):
- \begin{center}
- \includegraphics[width = 0.5\columnwidth]{img_02_24_bandsperre}
- \end{center}
- \begin{emphbox}
- \item{$\underline{H}_{BS}(f) = \underline{H}_{TP}(f) + \underline{H}_{HP}(f)$}
- \end{emphbox}
- Für $f_{3dB,TP} << f_{3dB,HP}$ gilt: \newline $f_{3dB,u} \approx f_{3dB,TP}$ \newline $f_{3dB,o} \approx f_{3dB,HP}$
- }
- \begin{emphbox}
- \item{$f_{3dB,o} = f_m \cdot (\sqrt{1+(\frac{1}{2\cdot Q}^2)}+\frac{1}{2\cdot Q})$}
- \item{$f_{3dB,u} = f_m \cdot (\sqrt{1+(\frac{1}{2\cdot Q}^2)}-\frac{1}{2\cdot Q})$}
-
- \end{emphbox}
- \end{sectionbox}
-
- \setlength{\imagewidth}{4cm}
-
- \begin{sectionbox}
- % Filter Grundschaltungen
- % ----------------------------------------------------------------------
- \subsection{Filter-Grundschaltungen mit OPV}
- \subsubsection{TP1, nichtinv.}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_15_tp1_nichtinv}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- \begin{basicbox}
- $K_P = \frac{R_2+R_3}{R_2} (=\frac{1}{k})$ \quad \quad\
- $f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
- \end{basicbox}
- $f_g = GBW \cdot k$
- }
- \subsubsection{TP1, inv.}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_16_tp1_inv}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- \begin{basicbox}
- $K_P = -\frac{R_2}{R_1}$ \quad \quad\
- $f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_2 \cdot C_2}$
- \end{basicbox}
- für typ. $f_g >> f_C \cdot (R_1 + R_2) / R_1$ gilt: \newline
- $f_g \approx GBW \cdot k(=1)$
- }
- \subsubsection{HP1, nichtinv.}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_17_hp1_nichtinv}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- \begin{basicbox}
- $K_P = \frac{R_2+R_3}{R_2} (=\frac{1}{k})$ \quad \quad\
- $f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
- \end{basicbox}
- $f_g = GBW \cdot k$
- }
- \subsubsection{HP1, inv.}
- \pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_18_hp1_inv}}
- \parbox{\textwidth - \imagewidth + 1cm}{
- \begin{basicbox}
- $K_P = -\frac{R_2}{R_1}$ \quad \quad\
- $f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
- \end{basicbox}
- Allg. gilt: $\underline{k} = \frac{1+j\cdot f/f_C}{1+\frac{j\cdot f}{f_C}\cdot \frac{R_1+R_2}{R_1}}$ \newline
- für typ. $f_g >> f_C \to |\underline{k}(f_g)| \approx \frac{R_1}{R_1+R_2}$ \newline
- $f_g \approx GBW \cdot |\underline{k}(f_g)|$
- }
- \end{sectionbox}
-
- \begin{sectionbox}
- % Höhere Filter
- % ----------------------------------------------------------------------
- \subsection{Sallen-Key (nichtinv.)}
- \pbox{0.5\textwidth}{TP2:\newline \includegraphics[width = \imagewidth]{img_02_19_sallenkey_tp2}}
- \pbox{0.5\textwidth}{HP2:\newline \includegraphics[width = \imagewidth]{img_02_20_sallenkey_hp2}}
- \begin{multicols*}{2}
- \begin{basicbox}
- $K_P = 1 + \frac{R_4}{R_3} $ (a)
- \end{basicbox}
- \begin{basicbox}
- $f_C = \frac{1}{2\pi \cdot \sqrt{R_1 \cdot C_1 \cdot R_2 \cdot C_2}} $ (b)
- \end{basicbox}
- \begin{basicbox}
- $Q = \frac{1/(2\pi \cdot f_C)}{R_1 \cdot (C_2 - (K_P - 1) \cdot C_1) + R_2 \cdot C_2}$ (c)
- \end{basicbox} \newpage
- \begin{basicbox}
- $R_1 = \frac{1}{(2\pi \cdot f_C)^2 \cdot C_1 \cdot R_2 \cdot C_2} $ (d)
- \end{basicbox}
- \begin{basicbox}
- $R_2 = \frac{\frac{1}{2Q}\pm \sqrt{\frac{1}{(2Q)^2}+(K_P-1)-\frac{C_2}{C_1}}}{2\pi \cdot f_C \cdot C_2}$ (e)
- \end{basicbox}
- \begin{basicbox}
- $\frac{C_2}{C_1} \leq \frac{1}{(2Q)^2} + (K_P -1)$ (f)
- \end{basicbox}
- \end{multicols*}
- \begin{bluebox}
- \item{1. $K_P$ Vorgabe \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
- 4. $C_1, C_2$ wählen (nF Bereich)}
- \item{2. $R_3$ wählen, (a) $\to R_4 = R_3 \cdot (K_P - 1)$ \quad \quad\
- 5. (e) $\to R_2$, eingesetzt in (d) $\to R_1$}
- \item{3. $f_C, Q$ Vorgabe, (f) auswerten}
- \end{bluebox}
- Grenze des Normalbetriebs (TP, HP): $f_g \approx GBW \cdot k (= \frac{R_3}{R_3 + R_4})$ \newline
- $V_{ud}$ - Abstand (typ $>$ (20dB ... 40dB)): $\frac{GBW / f_C}{K_P \cdot Q} > 10 ... 100$?
- \end{sectionbox}
-
- \begin{sectionbox}
- \subsection{Multifeedback (MFB) (inv.)}
- \pbox{0.5\textwidth}{TP2:\newline \includegraphics[width = \imagewidth]{img_02_21_mfb_tp2}}
- \pbox{0.5\textwidth}{HP2:\newline \includegraphics[width = \imagewidth]{img_02_22_mfb_hp2}}
- \begin{multicols*}{2}
- \begin{basicbox}
- $K_P = -\frac{R_2}{R_1} $ (a)
- \end{basicbox}
- \begin{basicbox}
- $f_C = \frac{1}{2\pi \cdot \sqrt{R_3 \cdot C_1 \cdot R_2 \cdot C_2}} $ (b)
- \end{basicbox}
- \begin{basicbox}
- $Q = \frac{1/(2\pi \cdot f_C)}{C_1 \cdot (R_2+R_3+R_2\cdot R_3 / R_1)}$ (c)
- \end{basicbox} \newpage
- \begin{basicbox}
- $R_1 = \frac{1}{(2\pi \cdot f_C)^2 \cdot C_1 \cdot R_2 \cdot C_2} $ (d)
- \end{basicbox}
- \begin{basicbox}
- $R_2 = \frac{\frac{1}{2Q}\pm \sqrt{\frac{1}{(2Q)^2}-(1-K_P)\cdot \frac{C_1}{C_2}}}{2\pi \cdot f_C \cdot C_1}$ (e)
- \end{basicbox}
- \begin{basicbox}
- $\frac{C_1}{C_2} \leq \frac{1}{(2Q)^2 \cdot (1-K_P)}$ (f)
- \end{basicbox}
- \end{multicols*}
- \begin{bluebox}
- \item{1. $f_C, Q, K_P$ Vorgabe \quad \quad \quad \quad \quad\ \
- 4. (e) $\to R_2$, eingesetzt in (d) $\to R_3$}
- \item{2. (f) auswerten \quad \quad \quad \quad \quad \quad \quad \quad \quad \
- 5. (a) $\to R_1 = R_2/(-K_P)$}
- \item{3. $C_1, C_2$ wählen (nF Bereich)}
- \end{bluebox}
- Grenze des Normalbetriebs (TP): $f_g \approx GBW \cdot k(=1)$ \newline
- Grenze des Normalbetriebs (HP): $f_g \approx GBW \cdot k (= \frac{C_2}{C_1 + C_2})$ \newline
- $V_{ud}$ - Abstand (typ $>$ (20dB ... 40dB)): $\frac{GBW / f_C}{K_P \cdot Q} > 10 ... 100$?
-
- %\end{sectionbox}
-
- %\begin{sectionbox}
-
- % Bandpass / Bandsperre
- % ----------------------------------------------------------------------
- \subsection{Bandpass 2. Ordnung}
- Allgemeine Normalform (BP2):
- \begin{emphbox}
- \item{$\underline{H}_{BP2}(f) = \frac{H_m \cdot j \cdot \frac{f}{f_m}/Q}{1+j\cdot \frac{f}{f_m}/Q + (j \cdot\frac{f}{f_m})^2}$}
- \end{emphbox}
- \subsubsection{HP1, TP1 kaskdiert (nichtinv.)}
- \pbox{6cm}{\includegraphics[width = 6cm - 1cm]{img_02_25_bp2_kaskadiert}}
- \parbox{\textwidth - 6cm + 1cm}{
- $f_m = \sqrt{f_{3dB,u} \cdot f_{3dB,o}}$ \newline
- $Q = \frac{f_m}{f_{3dB,TP1} - f_{3dB,HP1}} = \frac{f_m}{B}$ \newline
- $H_m \approx K_{P,HP1} \cdot K_{P,TP1}$
- }
- \subsubsection{BP2, Invertierende Standardstruktur}
- \pbox{4cm}{\includegraphics[width = 4cm - 1cm]{img_02_26_bp2_inv}}
- \parbox{\textwidth - 4cm + 1cm}{
- $f_m = \frac{1}{2\pi \cdot \sqrt{(R_1||R_3)\cdot C_1 \cdot R_2 \cdot C_2}}$ \newline
- Für typ. $C_1 = C_2 = C$ : $Q = f_m \cdot \pi \cdot R_2 \cdot C$ \newline
- $H_m = -\frac{R_2}{2\cdot R_1}$
- }
-
- \subsubsection{BP2, MFB (inv.)}
- \pbox{4cm}{\includegraphics[width = 4cm - 1cm]{img_02_27_bp2_mfb_inv}}
- \parbox{\textwidth - 4cm + 1cm}{
- $f_m = \frac{1}{2\pi \cdot \sqrt{(R_1||R_3)\cdot C_1 \cdot R_2 \cdot C_2}}$ \newline
- Für typ. $C_1 = C_2 = C$ : $Q = f_m \cdot \pi \cdot R_2 \cdot C$ \newline
- $H_m = -\frac{R_2}{2\cdot R_1}$
- }
- \begin{bluebox}
- \item{1. $f_m$ Vorgabe \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
- 5. $-H_m (<<GBW \cdot f_m !) < 2 \cdot Q^2$ festlegen!}
- \item{2. $C_1 = C_2 = C$ wählen (nF-Bereich) \quad \
- 6. $R_1 = R_2/(-2\cdot H_m)$}
- \item{3. $Q = \frac{f_m}{B}$ festlegen \quad \quad \quad \quad \quad \quad \quad \quad \
- 7. $R_3 = 1 / (2\cdot 2\pi \cdot Q \cdot f_m \cdot C \cdot (1+\frac{H_m}{2\cdot Q^2}))$}
- \item{4. $R_2 = \frac{Q}{\pi \cdot f_m \cdot C}$}
- \end{bluebox}
- \end{sectionbox}
-
- \begin{sectionbox}
-
- \subsection{Bandsperre 2. Ordnung}
- Allgemeine Normalform (BS2):
- \begin{emphbox}
- \item{$\underline{H}_{BS2}(f) = \frac{H_0\cdot (1+\frac{H_m}{H_0}\cdot j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2)}{1+j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2}$}
- \item{$\underline{H}_{BS2(ideal)}(f) = \frac{H_0\cdot (1+(j\cdot \frac{f}{f_m})^2)}{1+j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2} = H_0 \cdot (1-\underline{H}_{BP2,Kp=1}(j\cdot f))$}
- \end{emphbox}
- \subsubsection{HP1 + TP1 in Summe \emph{($K_{P,HP} = K_{P,TP}$! (=-1))}}
- \begin{center} \includegraphics[width = \imagewidth + 1cm]{img_02_28_bs2} \end{center}
-
- Güte (Polqualität): $Q = \frac{f_m}{B}$ (wie bei BP2) \newline
- Durchlassverstärkung: $H_0 = K_{P,TP} (K_{P,TP}) \cdot K_{P,Addierer}$ \newline
- Resonanzverstärkung: $H_m = \frac{H_0 \cdot 2 \cdot f_{3dB,TP1}}{f_{3dB,TP1} + f_{3dB,HP1}}$ \newline
- Grenze des Normalbetriebs:\newline \newline
- $\underline{k}_1(f_{g1}) \approx \frac{R_{11}}{R_{11}+R_{12}} \to f_{g1} = GBW1 \cdot \underline{k}_1(f_{g1})$ \newline
- $\underline{k}_2(f_{g2}) \approx 1 \to f_{g2} = GBW2 \cdot \underline{k}_2(f_{g2})$ \newline
- $k_3 \approx \frac{R_{31} || R_{32}}{R_{31} || R_{32} + R_{33}} \to f_{g3} = GBW3 \cdot \underline{k}_3$
-
- \subsubsection{BS2, Kerb- (Notch-) Filter}
- \pbox{5cm}{\includegraphics[width = 5cm - 1cm]{img_02_29_bs2_kerb}}
- \parbox{\textwidth - 5cm + 1cm}{
- $\underline{H}_{BS2(Notch)} = \underline{H}_{BS2(ideal)}$ \newline
- $Q = \frac{f_m}{B}$ \newline
- $H_0 = 1 + \frac{R_2}{R_1} \geq 1$ \newline
- $|\underline{H}(f_m)| = 0$ \newline
- Stabilitätsbedingungen: $1 \leq H_0 < 2$ ; $2 \geq 1/Q > 0$
- }
- \begin{bluebox}
- \item{1. $f_m$ festlegen \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
- 4. $Q = f_m / B(3dB)$ festlegen}
- \item{2. C wählen (nF-Bereich) \quad \quad \quad \quad \quad \quad \quad\
- 5. $H_0 = 2 - \frac{1}{2\cdot Q}$}
- \item{3. $R = 1/(2\pi \cdot f_m \cdot C)$ \quad \quad \quad \quad \quad \quad \quad \
- 6. $R_2 / R_1 = (H_0 - 1)$}
- \end{bluebox}
-
- \end{sectionbox}
-
- \begin{sectionbox}
- \subsection{Zur OPV-Auswahl}
- Kleinsignalmäßig: GBW-Reserve
- \begin{emphbox}
- $\frac{GBW/f}{|\underline{H}(f)|} (spez. \frac{GBW/f_C}{|\underline{H}(f_C)|} bzw. \frac{GBW/f_m}{|\underline{H}(f_m)|}) > (typ. 10...100(=40dB)!)$
- \end{emphbox}
- Großsignalmäßig: Slew-Rate SR (Def. $\Delta U_{out} / \Delta t$)
- \begin{emphbox}
- $SR > \pi \cdot f_{3dB,max} \cdot U_{out,pp} !$
- \end{emphbox}
- \end{sectionbox}
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