\documentclass[11pt,a4paper]{report} \usepackage[ansinew]{inputenc} \usepackage[margin=2.5cm,nohead]{geometry} \usepackage{color} \definecolor{grau}{rgb}{0.5,0.5,0.5} \usepackage{amssymb} \usepackage{amsmath} \usepackage{empheq} \usepackage{cancel} %\usepackage{makeidx} %\makeindex \usepackage{wasysym} \usepackage{ngerman,latexsym,alltt,graphicx,textcomp} \usepackage[bookmarks, colorlinks=false, pdftitle={Mitschrift Analoge Schaltungen Prof. Dr.-Ing. Schwarz}, pdfauthor={Fabian Kurz}, pdfsubject={Elektrotechnik}, pdfkeywords={Mathematik Elektrotechnik TU Dresden}, linkbordercolor={1 1 1}]{hyperref} %\usepackage{xy} %\input xy %\xyoption{all} \date{Zuletzt aktualisiert:\\\today} \author{Fabian~Kurz\\\href{http://fkurz.net/}{http://fkurz.net/}} \title{Analoge Schaltungen -- WS 05/06\\Prof. Dr.-Ing. Schwarz, TU Dresden\\Mitschrift} \begin{document} \def\thesubsection{\arabic{chapter}.\arabic{section}.\alph{subsection}} \setcounter{secnumdepth}{3} \def\thesubsubsection{\arabic{subsubsection}.} \setlength{\parindent}{0pt} \maketitle \pagenumbering{Roman} \tableofcontents \newpage \chapter{Bauelementemodelle} \pagenumbering{arabic} Ein Bauelementemodell ist ein Netzwerk, das zu einem gegebenen Bauelement ein "aquivalentes Klemmenverhalten hat. \section{Modelle} \begin{minipage}{8cm} \center Eintore (Zweipole) \begin{picture}(110,45) \put(0,25){\vector(1,0){15}} \put(7.5,28){\makebox(0,0)[b]{$I$}} \put(19,23){\vector(0,-1){16}} \put(17,15){\makebox(0,0)[r]{$U$}} \multiput(19,5)(0,20){2}{\circle{2}} \multiput(20,5)(0,20){2}{\line(1,0){20}} \multiput(40,0)(0,30){2}{\line(1,0){30}} \multiput(40,0)(30,0){2}{\line(0,1){30}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center Zweitore (Dreipole) \begin{picture}(110,45) \put(0,25){\vector(1,0){15}} \put(7.5,28){\makebox(0,0)[b]{$I_1$}} \put(19,23){\vector(0,-1){16}} \put(17,15){\makebox(0,0)[r]{$U_1$}} \multiput(19,5)(0,20){2}{\circle{2}} \multiput(20,5)(0,20){2}{\line(1,0){20}} \multiput(40,0)(0,30){2}{\line(1,0){30}} \multiput(40,0)(30,0){2}{\line(0,1){30}} \multiput(91,5)(0,20){2}{\circle{2}} \multiput(70,5)(0,20){2}{\line(1,0){20}} \put(110,25){\vector(-1,0){15}} \put(102.5,28){\makebox(0,0)[b]{$I_2$}} \put(91,23){\vector(0,-1){16}} \put(94,15){\makebox(0,0)[l]{$U_2$}} \end{picture} \end{minipage}% \subsubsection*{Beispiele} \bigskip \begin{minipage}{8cm} \center \begin{picture}(120,60) \multiput(5,0)(0,60){2}{\circle{2}} \multiput(5,1)(0,39){2}{\line(0,1){19}} \multiput(0,20)(0,20){2}{\line(1,0){10}} \multiput(0,20)(10,0){2}{\line(0,1){20}} \multiput(40,0)(0,60){2}{\circle{2}} \multiput(40,1)(0,31){2}{\line(0,1){27}} \thicklines \multiput(33,28)(0,4){2}{\line(1,0){14}} \thinlines \multiput(75,0)(0,60){2}{\circle{2}} \multiput(75,1)(0,36.5){2}{\line(0,1){21.5}} \multiput(67.5,22.5)(0,15){2}{\line(1,0){15}} \put(67.5,37.5){\line(1,-2){7.5}} \put(82.5,37.5){\line(-1,-2){7.5}} \multiput(110,0)(0,60){2}{\circle{2}} \multiput(110,1)(0,31){2}{\line(0,1){27}} \thicklines \put(105,28){\line(1,0){10}} \thinlines \put(100,32){\line(1,0){20}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(200,60) \multiput(0,0)(37,0){2}{\multiput(5,0)(0,60){2}{\circle{2}}} \multiput(0,0)(0,60){2}{\multiput(6,0)(25,0){2}{\line(1,0){10}}} \multiput(0,0)(15,0){2}{\multiput(16,0)(0,40){2}{\line(0,1){20}}} \multiput(16,20)(0,5){4}{\qbezier(0,0)(5,0)(5,2.5)\qbezier(5,2.5)(5,5)(0,5)} \multiput(31,20)(0,5){4}{\qbezier(0,0)(-5,0)(-5,2.5)\qbezier(-5,2.5)(-5,5)(0,5)} \put(69,30){\circle{2}} \put(121,60){\circle{2}} \multiput(69,0)(52,0){2}{\circle{2}} \put(70,30){\line(1,0){20}} \thicklines \put(90,20){\line(0,1){20}} \thinlines \put(90,30){\line(1,1){10}} \put(97,23){\vector(1,-1){0}} \put(90,30){\line(1,-1){10}} \put(100,0){\line(0,1){20}} \put(70,0){\line(1,0){50}} \put(100,0){\circle*{2}} \put(100,40){\line(0,1){20}} \put(100,60){\line(1,0){20}} \put(90,0){ % das ganze verschoben \put(59,30){\circle{2}} \put(111,60){\circle{2}} \multiput(59,0)(52,0){2}{\circle{2}} \put(60,30){\line(1,0){20}} \put(80,20){\line(0,1){20}} \thicklines \put(82,20){\line(0,1){20}} \thinlines \multiput(82,20)(0,20){2}{\line(1,0){8}} \put(90,0){\line(0,1){20}} \put(60,0){\line(1,0){50}} \put(90,0){\circle*{2}} \put(90,40){\line(0,1){20}} \put(90,60){\line(1,0){20}} } % bis hier \end{picture} \end{minipage}% \subsubsection*{Beschreibung} \begin{minipage}{8cm} \center 1 Gleichung $\rightarrow$ Zweig im Netzwerk \begin{picture}(50,80) \multiput(19,0)(0,60){2}{\circle{2}} \multiput(20,0)(0,60){2}{\line(1,0){20}} \multiput(40,0)(0,40){2}{\line(0,1){20}} \multiput(40,20)(-10,10){2}{\line(1,1){10}} \multiput(40,20)(10,10){2}{\line(-1,1){10}} \put(19,57){\vector(0,-1){54}} \put(17,30){\makebox(0,0)[r]{$U$}} \put(0,60){\vector(1,0){16}} \put(8,63){\makebox(0,0)[b]{$I$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center 2 Gleichungen $\rightarrow$ 2 gekoppelte Zweige im Netzwerk \begin{picture}(105,80) \multiput(19,0)(0,60){2}{\circle{2}} \multiput(20,0)(0,60){2}{\line(1,0){20}} \multiput(65,0)(0,60){2}{\line(1,0){20}} \multiput(86,0)(0,60){2}{\circle{2}} \multiput(0,0)(25,0){2}{ \multiput(40,0)(0,40){2}{\line(0,1){20}} \multiput(40,20)(-10,10){2}{\line(1,1){10}} \multiput(40,20)(10,10){2}{\line(-1,1){10}} } \put(19,57){\vector(0,-1){54}} \put(17,30){\makebox(0,0)[r]{$U_1$}} \put(86,57){\vector(0,-1){54}} \put(89,30){\makebox(0,0)[l]{$U_2$}} \put(0,60){\vector(1,0){16}} \put(8,65){\makebox(0,0)[b]{$I_1$}} \put(105,60){\vector(-1,0){16}} \put(97,65){\makebox(0,0)[b]{$I_2$}} \end{picture} \end{minipage}% \bigskip \begin{description} \item[Gro"ssignal-Modell:] Modelliert das Klemmenverhalten in einem weiten Arbeitsbereich, i.d.R. nichtlinear. \item[Kleinsignal-Modell:] Modelliert das Klemmenverhalten in der Umgebung eines Arbeitspunktes $\rightarrow$ linear. \end{description} \newpage \section{Zweipole} \subsection*{Gro"ssignalmodelle} \begin{center} \begin{picture}(80,30) \put(0,5){\vector(1,0){16}} \multiput(19,5)(62,0){2}{\circle{2}} \multiput(20,5)(40,0){2}{\line(1,0){20}} \multiput(40,0)(0,10){2}{\line(1,0){20}} \multiput(40,0)(20,0){2}{\line(0,1){10}} \multiput(40,0)(5,0){3}{\line(1,1){10}} \qbezier(30,10)(50,21)(70,10) \put(70,10){\vector(2,-1){0}} \put(51,18){\makebox(0,0)[bc]{$U$}} \put(8,8){\makebox(0,0)[b]{$I$}} \end{picture} \qquad $f(U,I) = 0$ \bigskip $\swarrow$ \hspace{6cm} $\searrow$ \end{center} \begin{minipage}{8cm} \center $I = Y(U)$ \begin{picture}(100,55) \put(0,25){\vector(1,0){16}} \multiput(19,25)(62,0){2}{\circle{2}} \multiput(20,25)(40,0){2}{\line(1,0){20}} \multiput(40,25)(10,-10){2}{\line(1,1){10}} \multiput(40,25)(10,10){2}{\line(1,-1){10}} \qbezier(30,30)(50,45)(70,30) \put(70,30){\vector(3,-2){0}} \put(51,40){\makebox(0,0)[bc]{$U$}} \put(8,28){\makebox(0,0)[b]{$I$}} \put(50,15){\line(0,1){20}} \thicklines\put(35,12){\vector(1,0){30}} \put(50,2){\makebox(0,0){$Y(U)$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center $U = Z(I)$ \begin{picture}(100,55) \put(0,25){\vector(1,0){16}} \multiput(19,25)(62,0){2}{\circle{2}} \multiput(20,25)(40,0){2}{\line(1,0){20}} \multiput(40,25)(10,-10){2}{\line(1,1){10}} \multiput(40,25)(10,10){2}{\line(1,-1){10}} \qbezier(30,30)(50,45)(70,30) \put(70,30){\vector(3,-2){0}} \put(51,40){\makebox(0,0)[bc]{$U$}} \put(8,28){\makebox(0,0)[b]{$I$}} \put(40,25){\line(1,0){20}} \thicklines\put(35,12){\vector(1,0){30}} \put(50,2){\makebox(0,0){$Z(I)$}} \end{picture} \end{minipage}% \subsection*{Kleinsignalmodelle} \begin{minipage}{8cm} \center \begin{picture}(150,100) \put(20,15){\vector(0,1){75}} \put(15,20){\vector(1,0){130}} \put(16,83){\makebox(0,0)[r]{$I$}} \put(17,60){\makebox(0,0)[r]{$I_A$}} \put(140,17){\makebox(0,0)[t]{$U$}} \put(100,17){\makebox(0,0)[t]{$U_A$}} \multiput(19,60)(4,0){21}{\line(1,0){2}} \multiput(100,19)(0,4){11}{\line(0,1){2}} \qbezier(20,20)(80,20)(113,80) \put(100,60){\vector(1,0){20}} \put(100,60){\vector(0,1){20}} \put(98,75){\makebox(0,0)[r]{$i$}} \put(115,57){\makebox(0,0)[t]{$u$}} \put(100,60){\line(4,5){10}} \put(100,60){\line(-4,-5){10}} \put(100,60){\circle*2} \put(93,65){\makebox(0,0)[c]{$A$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \[U = U_A + u, \qquad I = I_A + i\] (Koordinatenverschiebung) \bigskip $U_A$, $I_A$: Spannung/Strom im Arbeitspunkt \end{minipage}% \begin{minipage}{8cm} \begin{align*} I &= I_A + i = Y(U_A + u)\\ &= \underbrace{Y(U_A)}_{I_A} + \left.\frac{dY}{dU}\right|_{U_A} \cdot u + \cdots \end{align*} \end{minipage}% \begin{minipage}{8cm} \begin{align*} U &= U_A + u = Z(I_A + i)\\ &= \underbrace{Z(I_A)}_{U_A} + \left.\frac{dZ}{dI}\right|_{I_A} \cdot i + \cdots \end{align*} \end{minipage}% \begin{center}lineare N"aherung \end{center} \begin{minipage}{8cm} \begin{align*} I &= I_A + i = I_A + g_D \cdot u\\[1ex] g_D &= \left.\frac{dI}{dU}\right|_{U_A} = \left.\frac{dY(U)}{dU}\right|_{U_A} \end{align*} \end{minipage}% \begin{minipage}{8cm} \begin{align*} U &= U_A + u = U_A + r_D \cdot i\\[1ex] r_D &= \left.\frac{dU}{dI}\right|_{I_A} = \left.\frac{dZ(I)}{dI}\right|_{I_A} \end{align*} \end{minipage}% \begin{center}Schaltungsmodell \end{center} \begin{minipage}{8cm} \center \begin{picture}(190,75) \put(20,30){\makebox(0,0)[r]{$I_A\Big\uparrow$}} \multiput(0,0)(40,0){2}{ \multiput(30,0)(0,40){2}{\line(0,1){20}} \put(30,30){\circle{20}} \put(20,30){\line(1,0){20}} } \put(60,30){\makebox(0,0)[r]{$i\Big\uparrow$}} \multiput(30,0)(0,60){2}{\line(1,0){140}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \put(130,30){\circle{20}} \put(120,30){\line(1,0){20}} \multiput(170,0)(0,40){2}{\line(0,1){20}} \multiput(165,20)(10,0){2}{\line(0,1){20}} \multiput(165,20)(0,20){2}{\line(1,0){10}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow I_A$}} \put(178,30){\makebox(0,0)[l]{$g_D$}} \multiput(70,0)(60,0){2}{\circle*2} \multiput(70,60)(60,0){2}{\circle*2} \put(100,58){\vector(0,-1){56}} \put(103,30){\makebox(0,0)[l]{$u$}} \put(100,66){\makebox(0,0)[b]{$I$}} \put(92.5,63){\vector(1,0){15}} \end{picture} $\ \ \ \Downarrow$ \begin{picture}(190,65) \multiput(70,0)(0,40){2}{\line(0,1){20}} \put(70,30){\circle{20}} \put(60,30){\line(1,0){20}} \put(60,30){\makebox(0,0)[r]{$i\Big\uparrow$}} \multiput(70,0)(0,60){2}{\line(1,0){60}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \multiput(125,20)(10,0){2}{\line(0,1){20}} \multiput(125,20)(0,20){2}{\line(1,0){10}} \put(138,30){\makebox(0,0)[l]{$g_D$}} \put(100,58){\vector(0,-1){56}} \put(103,30){\makebox(0,0)[l]{$u$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(120,75) \multiput(30,25)(60,0){2}{\line(0,1){10}} \multiput(30,15)(0,30){2}{\circle{20}} \multiput(30,0)(0,60){2}{\line(1,0){60}} \put(90,0){\line(0,1){35}} \put(90,55){\line(0,1){5}} \put(90,15){\circle{20}} \multiput(85,35)(10,0){2}{\line(0,1){20}} \multiput(85,35)(0,20){2}{\line(1,0){10}} \put(30,0){\line(0,1){60}} \put(20,15){\makebox(0,0)[r]{$U_A\Big\downarrow$}} \put(20,45){\makebox(0,0)[r]{$u\Big\downarrow$}} \put(100,15){\makebox(0,0)[l]{$\Big\downarrow U_A$}} \put(98,45){\makebox(0,0)[l]{$r_D$}} \put(60,66){\makebox(0,0)[b]{$i$}} \put(52.5,63){\vector(1,0){15}} \end{picture} $\Downarrow$ \begin{picture}(120,65) %\multiput(30,25)(60,0){2}{\line(0,1){10}} \put(30,30){\circle{20}} \multiput(30,0)(0,60){2}{\line(1,0){60}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(85,20)(10,0){2}{\line(0,1){20}} \multiput(85,20)(0,20){2}{\line(1,0){10}} \put(30,0){\line(0,1){60}} \put(20,30){\makebox(0,0)[r]{$u\Big\downarrow$}} \put(98,30){\makebox(0,0)[l]{$r_D$}} \put(60,6){\makebox(0,0)[b]{$i$}} \put(67.5,3){\vector(-1,0){15}} \end{picture} \end{minipage}% \paragraph{Beispiel:} Diode \begin{picture}(100,35) \put(0,5){\vector(1,0){16}} \multiput(19,5)(53.5,0){2}{\circle2} \multiput(20,5)(31.5,0){2}{\line(1,0){20}} \multiput(40,-2.5)(11.5,0){2}{\line(0,1){15}} \put(40,-2.5){\line(3,2){11.5}} \put(40,12.5){\line(3,-2){11.5}} \put(8,8){\makebox(0,0)[b]{$I$}} \qbezier(27,10)(46.75,22)(64.5,10) \put(64.5,10){\vector(2,-1){0}} \put(46,19){\makebox(0,0)[b]{$U$}} \end{picture} $I = I_s \left(e^{\frac U {U_T}} - 1\right) \approx I_s \cdot e^{\frac U {U_T}}$ \quad f"ur\quad $I \gg I_s$ \bigskip \[g_D = \left.\frac{dI}{dU}\right|_{U_A} = \frac{I_s \cdot e^{\frac{U_A}{U_T}}}{U_T} = \frac{I_A}{U_T}, \qquad r_D = \frac 1 {g_D} = \frac{U_T}{I_A} \] \bigskip zugeschnittene Gr"o"sengleichungen f"ur $I_s = 1\,\mathrm{pA}$, $U_T = 26\,\mathrm{mV}$: \[g_D/\mathrm{S} = \frac{I_A/\mathrm{mA}}{26} \qquad r_D/\Omega = \frac{26}{I_A/\mathrm{mA}}\] \section{Zweitore} \begin{minipage}{5cm}\center \begin{picture}(110,45) \put(0,25){\vector(1,0){15}} \put(7.5,28){\makebox(0,0)[b]{$I_1$}} \put(19,23){\vector(0,-1){16}} \put(17,15){\makebox(0,0)[r]{$U_1$}} \multiput(19,5)(0,20){2}{\circle{2}} \multiput(20,5)(0,20){2}{\line(1,0){20}} \multiput(40,0)(0,30){2}{\line(1,0){30}} \multiput(40,0)(30,0){2}{\line(0,1){30}} \multiput(91,5)(0,20){2}{\circle{2}} \multiput(70,5)(0,20){2}{\line(1,0){20}} \put(110,25){\vector(-1,0){15}} \put(102.5,28){\makebox(0,0)[b]{$I_2$}} \put(91,23){\vector(0,-1){16}} \put(94,15){\makebox(0,0)[l]{$U_2$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \begin{align*} Y_1 & = F_1(X_1, X_2)\\ Y_2 & = F_2(X_1, X_2) \end{align*} \end{minipage}% \begin{minipage}{5cm}\center \begin{picture}(105,80) \multiput(19,0)(0,60){2}{\circle{2}} \multiput(20,0)(0,60){2}{\line(1,0){20}} \multiput(65,0)(0,60){2}{\line(1,0){20}} \multiput(86,0)(0,60){2}{\circle{2}} \multiput(0,0)(25,0){2}{ \multiput(40,0)(0,40){2}{\line(0,1){20}} \multiput(40,20)(-10,10){2}{\line(1,1){10}} \multiput(40,20)(10,10){2}{\line(-1,1){10}} } \put(19,57){\vector(0,-1){54}} \put(17,30){\makebox(0,0)[r]{$U_1$}} \put(35,40){\makebox(0,0)[r]{$F_1$}} \put(86,57){\vector(0,-1){54}} \put(89,30){\makebox(0,0)[l]{$U_2$}} \put(70,40){\makebox(0,0)[l]{$F_2$}} \put(0,60){\vector(1,0){16}} \put(8,65){\makebox(0,0)[b]{$I_1$}} \put(105,60){\vector(-1,0){16}} \put(97,65){\makebox(0,0)[b]{$I_2$}} \end{picture} \end{minipage} \paragraph{Kleinsignalmodell:} $X_i = X_{i1} + x_i$, \quad $Y_i = Y_{i1} + y_i$, \quad $i = 1,2$ \begin{align*} Y_1 &= \cancel{Y_{1A}} + y_1 = \cancel{\underbrace{F_1(X_{1A},X_{2A})}_{Y_{1A}}} + \left.\frac{\partial F_1}{\partial X_1}\right|_A \cdot x_1 + \left.\frac{\partial F_1}{\partial X_2}\right|_A \cdot x_2 + \cdots \\ Y_2 &= \cancel{Y_{2A}} + y_2 = \cancel{\underbrace{F_2(X_{1A},X_{2A})}_{Y_{2A}}} + \left.\frac{\partial F_2}{\partial X_1}\right|_A \cdot x_1 + \left.\frac{\partial F_2}{\partial X_2}\right|_A \cdot x_2 + \cdots \end{align*} \begin{minipage}{8cm} \begin{empheq}[innerbox=\fbox]{align*} \quad y_1 &= k_{11} \cdot x_1 + k_{12} \cdot x_2 \vphantom{\Big|} \quad \\ \quad y_2 &= k_{21} \cdot x_1 + k_{22} \cdot x_2 \vphantom{\Big|} \quad \end{empheq} \end{minipage}% \begin{minipage}{8cm} $\rightarrow$ lineare Zweipolbeschreibung \end{minipage} \subsubsection*{Leitwertbeschreibung} \begin{minipage}{6cm} \begin{align*} I_1 & = Y_1(U_1, U_2)\\ I_2 & = Y_2(U_1, U_2) \end{align*} \end{minipage}% \begin{minipage}{4cm} \begin{picture}(50,45) \put(9,20){\circle{2}} \put(10,20){\line(1,0){20}} \thicklines \put(30,10){\line(0,1){20}} \thinlines \put(30,20){\line(1,1){10}} \put(37,13){\vector(1,-1){0}} \put(30,20){\line(1,-1){10}} \put(40,1){\line(0,1){9}} \put(40,00){\circle{2}} \put(40,40){\circle{2}} \put(40,30){\line(0,1){9}} \qbezier(20,15)(20,5)(35,5) \put(35,5){\vector(1,0){0}} \put(08,0){$U_{BE}$} \put(0,25){\vector(1,0){15}} \put(7.5,27){\makebox(0,0)[b]{$I_B$}} \put(45,45){\vector(0,-1){15}} \put(48.5,37.5){\makebox(0,0)[l]{$I_B$}} \qbezier(62,40)(70,20)(62,0) \put(62,0){\vector(-1,-3){0}} \put(68,20){\makebox(0,0)[l]{$U_{CE}$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \begin{align*} I_B & = Y_1(U_{BE}, U_{CE})\\ I_C & = Y_2(U_{BE}, U_{CE}) \end{align*} \end{minipage} \bigskip Kleinsignalmodell \begin{align*} i_1 &= y_{11} \cdot u_1 + y_{12} \cdot u_2 & i_B &= g_{BE} \cdot u_{BE} + g_{\nu} \cdot u_{CE} \\ i_2 &= y_{21} \cdot u_1 + y_{22} \cdot u_2 & i_{C} &= g_m \cdot u_{BE} + g_{CE} \cdot u_{CE} \end{align*} \begin{align*} g_{BE} &= \left.\frac{\partial I_B}{\partial U_{BE}}\right|_{A} & g_{\nu} &= \left.\frac{\partial I_B}{\partial U_{CE}}\right|_{A} \\ g_{m} &= \left.\frac{\partial I_C}{\partial U_{BE}}\right|_{A} & g_{CE} &= \left.\frac{\partial I_C}{\partial U_{CE}}\right|_{A} \end{align*} \begin{center} \begin{picture}(245,80) \multiput(19,0)(0,60){2}{\circle{2}} \multiput(20,0)(0,60){2}{\line(1,0){90}} \multiput(135,0)(0,60){2}{\line(1,0){90}} \multiput(226,0)(0,60){2}{\circle{2}} \multiput(70,0)(25,0){2}{ \multiput(40,0)(0,40){2}{\line(0,1){20}} \multiput(40,20)(-10,10){2}{\line(1,1){10}} \multiput(40,20)(10,10){2}{\line(-1,1){10}} \put(30,30){\line(1,0){20}} } \multiput(50,0)(0,60){2}{\circle*2} \multiput(50,0)(0,40){2}{\line(0,1){20}} \multiput(45,20)(0,20){2}{\line(1,0){10}} \multiput(45,20)(10,0){2}{\line(0,1){20}} \put(43,30){\makebox(0,0)[r]{$g_{BE}$}} \put(100,30){\makebox(0,0)[r]{$g_{\nu} u_{CE}\Big\downarrow$}} \put(19,57){\vector(0,-1){54}} \put(17,30){\makebox(0,0)[r]{$u_{BE}$}} \put(226,57){\vector(0,-1){54}} \put(228,30){\makebox(0,0)[l]{$u_{CE}$}} \put(0,60){\vector(1,0){16}} \put(8,63){\makebox(0,0)[b]{$i_B$}} \put(245,60){\vector(-1,0){16}} \put(238,63){\makebox(0,0)[b]{$i_{C}$}} \put(145,30){\makebox(0,0)[l]{$\Big\downarrow g_mu_{BE}$}} \multiput(195,0)(0,40){2}{\line(0,1){20}} \multiput(195,0)(0,60){2}{\circle*2} \multiput(190,20)(0,20){2}{\line(1,0){10}} \multiput(190,20)(10,0){2}{\line(0,1){20}} \put(203,30){\makebox(0,0)[l]{$g_{CE}$}} \end{picture} \end{center} s. Folie: "`BPT/FET-Modelle"' \bigskip \begin{minipage}{8cm} \center \textbf{Y-Beschreibung} \[i_B = g_{BE} \cdot u_{BE}\] \[\boxed{\vphantom{\int}i_C = g_m \cdot u_{BE} } + g_{CE} \cdot u_{CE}\] Hauptsteuergleichung \hspace*{2cm} \end{minipage}% \begin{minipage}{8cm} \center \textbf{H-Beschreibung} \[u_{BE}= r_{BE} \cdot i_{B}\] \[\boxed{\vphantom{\int}i_C = b \cdot i_{B} } + g_{CE} \cdot u_{CE}\] Hauptsteuergleichung \hspace*{2cm} \end{minipage} \bigskip \[\text{Zusammenhang: \quad }g_m = \frac{b}{r_{BE}},\quad g_{BE} = \frac 1 {r_{BE}}\] \begin{minipage}{8cm} \begin{picture}(215,80) \multiput(29,0)(0,60){2}{\circle2} \multiput(60,0)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} \put(30,60){\line(1,0){30}} \put(5,60){\vector(1,0){20}} \put(15,67){\makebox(0,0){$i_B$}} \put(29,58){\vector(0,-1){56}} \put(30,0){\line(1,0){155}} \multiput(125,0)(0,40){2}{\line(0,1){20}} \multiput(125,20)(-10,10){2}{\line(1,1){10}} \multiput(125,20)(10,10){2}{\line(-1,1){10}} \put(115,30){\makebox(0,0)[r]{$g_mu_{BE}\Big\downarrow$}} \put(53,30){\makebox(0,0)[r]{$g_{BE}$}} \put(27,30){\makebox(0,0)[r]{$u_{BE}$}} \put(115,30){\line(1,0){20}} \multiput(155,0)(0,40){2}{\line(0,1){20}} \multiput(150,20)(10,0){2}{\line(0,1){20}} \multiput(150,20)(0,20){2}{\line(1,0){10}} \put(125,60){\line(1,0){60}} \put(162,30){\makebox(0,0)[l]{$g_{CE}$}} \put(186,58){\vector(0,-1){56}} \multiput(186,0)(0,60){2}{\circle2} \multiput(155,0)(0,60){2}{\circle*2} \multiput(60,0)(65,00){2}{\circle*2} \put(188,30){\makebox(0,0)[l]{$u_{CE}$}} \put(210,60){\vector(-1,0){20}} \put(202.5,67){\makebox(0,0){$i_C$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{picture}(215,80) \multiput(29,0)(0,60){2}{\circle2} \multiput(60,0)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} \put(30,60){\line(1,0){30}} \put(5,60){\vector(1,0){20}} \put(15,67){\makebox(0,0){$i_B$}} \put(29,58){\vector(0,-1){56}} \put(30,0){\line(1,0){155}} \multiput(125,0)(0,40){2}{\line(0,1){20}} \multiput(125,20)(-10,10){2}{\line(1,1){10}} \multiput(125,20)(10,10){2}{\line(-1,1){10}} \put(115,30){\makebox(0,0)[r]{$b\cdot i_B\Big\downarrow$}} \put(53,30){\makebox(0,0)[r]{$r_{BE}$}} \put(27,30){\makebox(0,0)[r]{$u_{BE}$}} \put(115,30){\line(1,0){20}} \multiput(155,0)(0,40){2}{\line(0,1){20}} \multiput(150,20)(10,0){2}{\line(0,1){20}} \multiput(150,20)(0,20){2}{\line(1,0){10}} \put(125,60){\line(1,0){60}} \put(162,30){\makebox(0,0)[l]{$r_{CE}$}} \put(186,58){\vector(0,-1){56}} \multiput(186,0)(0,60){2}{\circle2} \multiput(155,0)(0,60){2}{\circle*2} \multiput(60,0)(65,00){2}{\circle*2} \put(188,30){\makebox(0,0)[l]{$u_{CE}$}} \put(210,60){\vector(-1,0){20}} \put(202.5,67){\makebox(0,0){$i_C$}} \end{picture} \end{minipage}% \bigskip s. Folie: "`Transferkennlinien"' \section{Anwendung der Modelle} \paragraph{Beispiel:}Emitterschaltung \begin{minipage}{11cm} \center \begin{picture}(295,160) \put(0,25){ %%%%%%%%%%%%%%%%%%%% von hier bis zur vorletzten Zeile \put(50,0){\line(0,1){60}} \put(50,30){\circle{20}} \put(38,30){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \multiput(50,60)(32,0){2}{\line(1,0){28}} \multiput(78,53)(4,0){2}{\thicklines\line(0,1){14}} \multiput(110,60)(0,40){2}{\line(0,1){20}} \multiput(105,80)(10,0){2}{\line(0,1){20}} \multiput(105,80)(0,20){2}{\line(1,0){10}} \put(110,60){\circle*2} \put(110,60){\line(1,0){30}} \put(140,50){\thicklines\line(0,1){20}} \put(140,60){\line(1,1){10}} \put(140,60){\line(1,-1){10}} \put(148,52){\vector(1,-1){0}} \put(150,70){\line(0,1){10}} \put(150,100){\line(0,1){20}} \multiput(145,80)(10,0){2}{\line(0,1){20}} \multiput(145,80)(0,20){2}{\line(1,0){10}} \put(150,00){\line(0,1){50}} \put(145,00){\line(1,0){10}} \put(45,00){\line(1,0){10}} \put(150,70){\line(1,0){38}} \put(192,70){\line(1,0){38}} \multiput(188,63)(4,0){2}{\thicklines\line(0,1){14}} \put(230,0){\line(0,1){20}} \put(230,40){\line(0,1){30}} \put(220,30){\line(1,0){20}} \put(225,0){\line(1,0){10}} \put(230,30){\circle{20}} \put(110,120){\line(1,0){60}} \put(171,120){\circle{2}} \multiput(150,70)(0,50){2}{\circle*{2}} \put(175,120){\makebox(0,0)[l]{$U_{0C}$}} \put(103,90){\makebox(0,0)[r]{$R_{B}$}} \put(80,75){\makebox(0,0)[c]{$C_{1}$}} \put(190,85){\makebox(0,0)[c]{$C_{2}$}} \put(143,90){\makebox(0,0)[r]{$R_{C}$}} \put(240,30){\makebox(0,0)[l]{$\Big\uparrow i_a$}} \qbezier(95,55)(80,45)(65,55) \put(65,55){\vector(-3,2){0}} \put(80,42){\makebox(0,0)[c]{$U_{BEA}$}} \put(130,56){\vector(0,-1){56}} \put(128,30){\makebox(0,0)[r]{$U_{BE}$}} \put(165,66){\vector(0,-1){66}} \put(167,30){\makebox(0,0)[l]{$U_{CE}$}} \put(215,66){\vector(0,-1){66}} \put(213,30){\makebox(0,0)[r]{$u_a$}} \put(110,10){ \qbezier(95,55)(80,45)(65,55) \put(80,42){\makebox(0,0)[c]{$U_{CEA}$}} } \put(205,65){\vector(3,2){0}} % u_E \put(10,70){\vector(0,1){25}} \put(8,80){\vector(1,0){25}} \put(8,90){\makebox(0,0)[r]{$\scriptscriptstyle u_e$}} \multiput(10,80)(10,0){2}{ \qbezier(0,0)(1.25,8)(2.5,8) \qbezier(2.5,8)(3.75,8)(5,0) } \put(10,80){ \qbezier(5,0)(6.25,-8)(7.5,-8) \qbezier(7.5,-8)(8.75,-8)(10,0) } \put(25,75){\vector(2,-1){25}} % u_BE \put(75,95){\vector(1,-1){30}} \put(50,90){\vector(0,1){25}} \put(48,92){\vector(1,0){25}} \multiput(49,103)(2,0){10}{\line(1,0){1}} \put(48,103){\makebox(0,0)[r]{$\scriptscriptstyle U_{BEA}$}} \multiput(50,103)(10,0){2}{ \qbezier(0,0)(1.25,3)(2.5,3) \qbezier(2.5,3)(3.75,3)(5,0) } \put(50,103){ \qbezier(5,0)(6.25,-3)(7.5,-3) \qbezier(7.5,-3)(8.75,-3)(10,0) } % u_CE \put(195,101){\vector(-3,-2){40}} \put(210,95){\vector(0,1){25}} \put(208,97){\vector(1,0){25}} \multiput(209,108)(2,0){10}{\line(1,0){1}} \put(208,108){\makebox(0,0)[r]{$\scriptscriptstyle U_{CEA}$}} \multiput(210,108)(10,0){2}{ \qbezier(0,0)(1.25,-3)(2.5,-3) \qbezier(2.5,-3)(3.75,-3)(5,0) } \put(210,108){ \qbezier(5,0)(6.25,3)(7.5,3) \qbezier(7.5,3)(8.75,3)(10,0) } % u_a \put(270,70){\vector(0,1){25}} \put(268,80){\vector(1,0){25}} \put(268,90){\makebox(0,0)[r]{$\scriptscriptstyle u_a$}} \multiput(270,80)(10,0){2}{ \qbezier(0,0)(1.25,-11)(2.5,-11) \qbezier(2.5,-11)(3.75,-11)(5,0) } \put(270,80){ \qbezier(5,0)(6.25,11)(7.5,11) \qbezier(7.5,11)(8.75,11)(10,0) } \put(260,75){\vector(-2,-1){25}} } % bis hier \put(100,7){\makebox(0,0)[c]{$\scriptstyle U_{BE} = U_{BEA} + \underbrace{\scriptstyle u_{BE}}_{u_e}$}} \put(190,7){\makebox(0,0)[c]{$\scriptstyle U_{CE} = U_{CEA} + u_{a} \vphantom{\underbrace{}_{u_e}}$}} \end{picture} \end{minipage}% \hfill \begin{minipage}{4.5cm} \begin{description} \item[gegeben:] \hspace*{\fill} \\ \raggedright $U_{0C} = 12\,\mathrm V$, $U_{CEA} = 6\,\mathrm V$, $B = 100$, $U_{BE0} = 0{,}6\,\mathrm V$, $R_C = 10\,\mathrm k\Omega$,\\ $C_1$, $C_2 \to \infty$ \item[gesucht:]\hspace*{\fill} \\ $R_B$, $r_e$, $u_a(u_e,\, i_a)$ \end{description} \end{minipage} \newpage % :-/ \subsection{Gro"ssignalmodell (zur Arbeitspunkteinstellung)} \paragraph{Anwendung:} Das Bauelement wird durch das Modell (Ersatzschaltung) ersetzt $\to$ Netzwerk (i.d.R. nichtlinear) $\to$ Analyse \bigskip \begin{minipage}{7cm} \center \begin{picture}(180,100) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$U_{BE0}\Big\downarrow$}} \multiput(0,0)(60,0){2}{ \multiput(25,60)(10,0){2}{\line(0,1){20}} \multiput(25,60)(0,20){2}{\line(1,0){10}} \put(30,80){\line(0,1){20}} } \multiput(30,0)(0,100){2}{\line(1,0){130}} \multiput(90,0)(0,100){2}{\circle*{2}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(90,20)(-10,10){2}{\line(1,1){10}} \multiput(90,20)(10,10){2}{\line(-1,1){10}} \put(80,30){\line(1,0){20}} \put(23,70){\makebox(0,0)[r]{$R_B$}} \put(83,70){\makebox(0,0)[r]{$R_C$}} \put(35,17.5){\vector(0,-1){15}} \put(38,10){\makebox(0,0)[l]{$I_{BA}$}} \put(11,50){\line(1,0){19}} \put(10,50){\circle{2}} \put(30,50){\circle*{2}} \put(7,50){\makebox(0,0)[r]{$B$}} \put(71,50){\line(1,0){19}} \put(70,50){\circle{2}} \put(90,50){\circle*{2}} \put(67,50){\makebox(0,0)[r]{$C$}} \put(75,48){\vector(0,-1){46}} \put(72,30){\makebox(0,0)[r]{$\scriptstyle U_{CEA}$}} \put(141,15){\line(1,0){19}} \put(140,15){\circle{2}} \put(160,15){\circle*{2}} \put(137,15){\makebox(0,0)[r]{$E$}} \put(100,30){\makebox(0,0)[l]{$\Big\downarrow B\cdot I_{BA}$}} \put(160,0){\line(0,1){100}} \put(160,50){\circle{20}} \put(170,50){\makebox(0,0)[l]{$\Big\downarrow U_{0C}$}} \end{picture} \end{minipage}% \hfill \begin{minipage}{8cm} \begin{minipage}{3.5cm} \begin{empheq}[innerbox=\fbox]{align*} \quad U_{BE} &= U_{BE0} \vphantom{\Big|} \quad \\ \quad I_C &= B \cdot I_B \vphantom{\Big|} \quad \end{empheq} \end{minipage}% \begin{minipage}{4.5cm} \center \bigskip \textbf{Transistormodell} \footnotesize (Vernachl"a"sigung des Early-Effekts) \end{minipage} \bigskip \end{minipage} % naja ;) \bigskip \bigskip \[I_{CA} = \frac{U_{0C}-U_{CEA}}{R_C} = \frac{6\,\mathrm V}{12\,\mathrm k\Omega},\qquad I_{BA} = \frac{U_{0C} - U_{BE0}}{R_B} = \frac{I_{CA}}{B} = 6\,\mu\mathrm A \] \[\boxed{R_B = \frac{U_{0C} - U_{BEA}}{I_{BA}} = \frac{U_{0C} - U_{BE0}}{U_{0C} - U_{CEA}} \cdot B \cdot R_C = B\cdot R_C \cdot \frac{1 - \frac{U_{BC0}}{U_{0C}}}{1-\frac{U_{CEA}}{U_{0C}}}}\] \[R_B = B \cdot R_C \cdot \frac{1-\frac{0{,}6\,\mathrm V}{12\,\mathrm V}}{1-\frac{6\,\mathrm V}{12\,\mathrm V}} = 1{,}9 \cdot B \cdot R_C = 1{,}9\,\mathrm M\Omega\] \paragraph{Spezialfall:} $U_{0C} \gg U_{CE0} \rightarrow \frac{U_{CEA}}{U_{0C}} = \frac 1 2 \quad \Rightarrow \quad \boxed{\ R_B \approx 2 \cdot B \cdot R_C \ \vphantom{|}}$ \subsection{Kleinsignalmodell (Analyse bei kleinen Aussteuerungen)} \paragraph{Anwendung:}\quad \begin{enumerate} \item Ersetzen des Bauelements durch sein Kleinsignal-Ersatzschaltbild \item Alle Konstantspannungs(-strom)quellen werden durch Kurzschluss (Leerlauf) ersetzt\\ $\to$ Lineares Netzwerk mit Kleinsignalgr"o"sen \end{enumerate} \paragraph{Begr"undung:}\quad \begin{minipage}{8cm} \center \begin{picture}(150,80) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$U_{0C}$\Big\downarrow}} \put(94,30){\makebox(0,0)[l]{$u = U_{0C} + RI$}} \put(60,71){\makebox(0,0)[c]{$R$}} \multiput(25,0)(61,0){2}{\line(1,0){10}} \multiput(30,60)(40,0){2}{\line(1,0){20}} \multiput(50,55)(0,10){2}{\line(1,0){20}} \multiput(50,55)(20,0){2}{\line(0,1){10}} \put(91,60){\circle{2}} \put(91,58){\vector(0,-1){56}} \put(110,60){\vector(-1,0){15}} \put(105,65){\makebox(0,0)[c]{$I$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(150,80) \put(0,-10){ % naja.. \put(80,10){\vector(0,1){70}} \put(00,15){\vector(1,0){120}} \put(20,10){\line(3,2){80}} \put(78,71){\makebox(0,0)[r]{$U$}} \put(110,8){\makebox(0,0)[c]{$I$}} \put(78.5,50){\line(1,0){3}} \put(84,50){\makebox(0,0)[l]{$U_{0C}$}} \put(78.5,30){\line(1,0){3}} \put(48,30){\vector(1,0){25}} \put(50,28){\vector(0,1){25}} \put(84,30){\makebox(0,0)[l]{$U_{A}$}} \put(50,13.5){\line(0,1){3}} \put(50,7){\makebox(0,0)[c]{$I_{A}$}} \put(48,50){\makebox(0,0)[r]{$u$}} \put(70,28){\makebox(0,0)[t]{$i$}} \put(50,30){\circle*2} \put(45,32){\makebox(0,0)[r]{$A$}} } \end{picture} \end{minipage}% \bigskip \bigskip \[U = U_A + u = U_{0C} + (I_A + i)\cdot R = \underbrace{U_{0C} + I_A \cdot R}_{U_A} + i\cdot R\] \bigskip \begin{minipage}{3cm} $\boxed{\ \vphantom{|} u = i \cdot R \ } \quad \Rightarrow$ \end{minipage}\hfill \begin{minipage}{12cm} \center nach Koordinatenverschiebung reiner Widerstand f"ur das Kleinsignal $\to$ Strom/Spannungsquellen deaktivieren \end{minipage}% \paragraph{Kleinsignalmodell:} \begin{center} \begin{picture}(450,70) \put(100,0){ % verschoben in die Mitte bis %%%%%%%%%%%% \multiput(29,0)(0,60){2}{\circle2} \multiput(60,0)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} \put(30,60){\line(1,0){30}} %\put(5,60){\vector(1,0){20}} %\put(15,67){\makebox(0,0){$i_B$}} \put(29,58){\vector(0,-1){56}} \put(30,0){\line(1,0){155}} \multiput(125,0)(0,40){2}{\line(0,1){20}} \multiput(125,20)(-10,10){2}{\line(1,1){10}} \multiput(125,20)(10,10){2}{\line(-1,1){10}} \put(115,30){\makebox(0,0)[r]{$g_mu_{e}\Big\downarrow$}} \put(53,30){\makebox(0,0)[r]{$r_{BE}$}} \put(27,30){\makebox(0,0)[r]{$u_{e}$}} \put(115,30){\line(1,0){20}} \multiput(155,0)(0,40){2}{\line(0,1){20}} \multiput(150,20)(10,0){2}{\line(0,1){20}} \multiput(150,20)(0,20){2}{\line(1,0){10}} \put(125,60){\line(1,0){60}} \put(162,30){\makebox(0,0)[l]{$g_{CE}$}} \put(230,58){\vector(0,-1){56}} \multiput(186,0)(0,60){2}{\circle2} \multiput(155,0)(0,60){2}{\circle*2} \multiput(60,0)(65,00){2}{\circle*2} \put(232,30){\makebox(0,0)[l]{$u_{a}$}} %\put(210,60){\vector(-1,0){20}} %\put(202.5,67){\makebox(0,0){$i_C$}} } % hier %%%%%%%%%%%%%%%%%%% \multiput(128,0)(0,60){2}{\line(-1,0){70}} \multiput(100,0)(0,40){2}{\line(0,1){20}} \multiput(100,0)(0,60){2}{\circle*2} \multiput(95,20)(10,0){2}{\line(0,1){20}} \multiput(95,20)(0,20){2}{\line(1,0){10}} \put(93,30){\makebox(0,0)[r]{$R_B$}} \put(58,0){\line(0,1){60}} \put(58,30){\circle{20}} \put(47,30){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \put(68,63){\vector(1,0){15}} \put(75.5,65){\makebox(0,0)[b]{$i_e$}} \multiput(287,0)(0,60){2}{\line(1,0){70}} \multiput(300,0)(0,40){2}{\line(0,1){20}} \multiput(300,0)(0,60){2}{\circle*2} \multiput(295,20)(10,0){2}{\line(0,1){20}} \multiput(295,20)(0,20){2}{\line(1,0){10}} \put(307,30){\makebox(0,0)[l]{$R_C$}} \multiput(357,0)(0,40){2}{\line(0,1){20}} \put(357,30){\circle{20}} \put(347,30){\line(1,0){20}} \put(367,30){\makebox(0,0)[l]{$\Big\uparrow i_a$}} \end{picture} \end{center} \textbf{gegeben:} $u_e$, $i_a$,\quad \textbf{gesucht:} $i_c$, $u_a$ \hfill NR: $r_{BE} = \frac{U_T}{I_{BA}} = %\frac{26\,\mathrm{mV}}{6\,\mu\mathrm A} = 4{,}3\,\mathrm k\Omega, \quad g_m = \frac{I_{CA}}{U_T} = %\frac{0{,}6\,\mathrm{mA}}{26\,\mathrm{mV}} = 23\,\mathrm{mS}$ \bigskip \[i_e = \frac{u_e}{R_B\parallel r_{BE}} \quad \Rightarrow \quad r_e = \frac{u_e}{i_e} = \frac{R_B \cdot r_{BE}}{R_B + r_{BE}} = r_{BE}\] \begin{align*} u_a &= (-g_m \cdot u_e + i_a) \cdot R_C \parallel r_{CE} & \text{("Uberlagerungssatz)}\\ &= -g_m u_e \cdot R_C\parallel r_{CE} + i_a \cdot R_C\parallel r_{CE} & \text{Annahme: } U_Y \to \infty \Rightarrow r_{CE} \to \infty \\ &= -g_m u_e R_C + i_a R_C = v \cdot u_e + r_a \cdot i_a & i_e = \frac 1 {r_e} u_e \end{align*} \begin{description} \item[Verst"arkung:] $v = -g_m \cdot R_C = -23\,\mathrm{mS}\cdot 10\,\mathrm k\Omega = - 230$ \item[Ausgangswiderstand:] $r_a \approx R_C \approx 10\,\mathrm k\Omega$ (\emph{Ohne} Early-Effekt!) \end{description} Die nat"urliche Beschreibung des Verst"arkers ist die \emph{inverse Hybridbeschreibung}. \chapter{Methoden der Schaltungsanalyse} \section{Netzwerkanalyse} Da die wesentlichen aktiven Bauelemente durch gesteuerte Stromquellen modelliert werden, wird die Knotenspannungsanalyse bevorzugt. \paragraph{Voraussetzung:} Alle $U$-$I$-Relationen m"ussen in der Form \begin{minipage}{7cm} \center $\boxed{\ \vphantom{\Big|} I = Y(U) \ }$ \qquad (Zweipole) \end{minipage}% \begin{minipage}{2cm} \center bzw. \end{minipage}% \begin{minipage}{7cm} \center \begin{minipage}{4.5cm} \begin{empheq}[innerbox=\fbox]{align*} \quad I_1 &= Y_1(U_1,U_2) \vphantom{\Big|} \quad\\ \quad I_2 &= Y_2(U_1,U_2) \vphantom{\Big|} \end{empheq} \end{minipage}% \begin{minipage}{2.5cm} \center (Vierpole) \end{minipage}% \end{minipage}% vorliegen. \begin{itemize} \item[$\rightarrow$] Im Netzwerk nur noch unabh"angige oder spannungsgesteuerte Stromquellen \item[$\rightarrow$] Spannungssteuermodelle bevorzugt \end{itemize} \subsubsection*{Algorithmus} \begin{enumerate} \item Festlegen des Bezugsknoten (meist Masse) \item Knotenspannungen einf"uhren \item Knotengleichungen (Strombilanzen) unter Verwendung der $U$-$I$-Relationen der Zweige aufstellen \item Knotengleichungen l"osen \item gesuchte Gr"o"sen berechnen \end{enumerate} \subsubsection*{Beispiel: Diodenschalter} %(vlg. "Ubungsaufgabe 1.27) \begin{minipage}{6cm} \begin{picture}(150,130) \put(19,75){\circle{2}} \put(20,75){\line(1,0){20}} \multiput(40,75)(24,24){2}{\line(1,1){14}}%ol \multiput(49,94)(10,10){2}{\line(1,-1){10}} \put(59,104){\line(-1,-3){5}} \put(69,94){\line(-3,-1){14.5}} \multiput(40,75)(24,-24){2}{\line(1,-1){14}}%ul \multiput(49,56)(10,-10){2}{\line(1,1){10}} \put(49,56){\line(3,-1){14.5}} \put(59,66){\line(1,-3){5}} \multiput(116,75)(-24,24){2}{\line(-1,1){14}}%or \put(38,38){ \multiput(49,56)(10,-10){2}{\line(1,1){10}} \put(49,56){\line(3,-1){14.5}} \put(59,66){\line(1,-3){5}} } \multiput(116,75)(-24,-24){2}{\line(-1,-1){14}}%ol \put(38,-38){ \multiput(49,94)(10,10){2}{\line(1,-1){10}} \put(59,104){\line(-1,-3){5}} \put(69,94){\line(-3,-1){14.5}} } \multiput(78,37)(0,48){2}{\line(0,1){28}} \put(78,75){\circle{20}} \put(68,75){\line(1,0){20}} \put(87,75){\makebox(0,0)[l]{$\Big\uparrow I_0$}} \multiput(40,75)(76,0){2}{\circle*2} \multiput(78,37)(0,76){2}{\circle*2} \put(137,75){\circle{2}} \put(116,75){\line(1,0){20}} \put(0,75){\vector(1,0){15}} \put(7.5,78){\makebox(0,0)[b]{$I_1$}} \put(35,85){\makebox(0,0)[c]{\textcircled{\raisebox{-1pt}{1}}}} \put(78,120){\makebox(0,0)[c]{\textcircled{\raisebox{-1pt}{2}}}} \put(78,28){\makebox(0,0)[c]{\textcircled{\raisebox{-1pt}{3}}}} \put(120,85){\makebox(0,0)[c]{\textcircled{\raisebox{-1pt}{0}}}} \put(78,0){\makebox(0,0)[b]{$I_D = I_S\left(e^{U_D/U_T} -1\right)$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \begin{description} \item[Gesucht:] $r_1 = \dfrac{dU_{10}}{dI_1}$ \item[Knotenspannungsanalyse:] Knotengleichungen \begin{align*} \text{\textcircled{\raisebox{-1pt}{1}}} && I_S\left(e^{\frac{U_1-U_3}{U_T}}-1\right) - I_S\left(e^{\frac{U_2-U_1}{U_T}}-1\right) &= I_1\\ \text{\textcircled{\raisebox{-1pt}{2}}} && I_S\left(e^{\frac{U_2-U_1}{U_T}}-1\right) - I_S\left(e^{\frac{U_2}{U_T}}-1\right) &= I_0\\ \text{\textcircled{\raisebox{-1pt}{3}}} && I_S\left(e^{\frac{U_1-U_3}{U_T}}-1\right) - I_S\left(e^{-\frac{U_3}{U_T}}-1\right) &= I_0 \end{align*} \end{description} \end{minipage}% Substitution: $e^{\frac{U_i}{U_T}} = x_i$, $i = 1,\,2,\,3$ \begin{align*} \text{\textcircled{\raisebox{-1pt}{1}}} && \frac{x_1}{x_3} - \frac{x_2}{x_1} &= \frac{I_1}{I_2} \\ \text{\textcircled{\raisebox{-1pt}{2}}} && \frac{x_2}{x_1} + x_2 &= \frac{I_0}{I_S} + 2 & \Rightarrow \quad x_2 &= \frac{\frac{I_0}{I_S} + 2}{1 + \frac 1 {x_1}} \\ \text{\textcircled{\raisebox{-1pt}{3}}} && \frac{x_1}{x_3} + \frac{1}{x_3} &= \frac{I_0}{I_S} + 2 & \Rightarrow\quad \frac{1}{x_3} &= \frac{\frac{I_0}{I_S} + 2}{1 + {x_1}} \end{align*} Einsetzen in \textcircled{\raisebox{-1pt}{1}} f"uhrt zu $x_1 = \dfrac{1+\frac{I_1}{I_0 + 2I_S}}{1 - \frac{I_1}{I_0 +2I_S}}$. Nach R"ucksubstitution und $\ln \frac{1+x}{1-x} \approx 2x$ f"ur $x \ll 1$: \[U_1 = U_T \cdot \ln \frac{1+\frac{I_1}{I_0 + 2I_S}}{1 - \frac{I_1}{I_0 +2I_S}} \approx U_T \frac{2I_1}{I_0 + 2I_S} = \left\{\begin{array}{ll} \frac{2U_T}{I_0}\cdot I_1 & \text{f"ur } I_0 \gg I_S,\ I_1\\[2ex] \frac{U_T}{I_S}\cdot I_1 & \text{f"ur } I_0=0 \end{array}\right.\] Somit ergibt sich $r_1$ zu: \[r_1 = \frac{dU_{10}}{dI_1} = \left\{\begin{array}{ll} 2\frac{U_1}{I_0} & \text{f"ur } I_0 \gg I_S,\ I_1\\[2ex] \frac{U_T}{I_S} & \text{f"ur } I_3 = 0 \end{array} \right.\] \section{Signalflu"sanalyse} Siehe \href{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/analyse/sigfl.pdf}{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/analyse/sigfl.pdf} (11 Seiten). % juppie. scheissarbeit gespart ;) \chapter{Grundschaltungen} \section{Grundmodell Spannungsverst"arker} \begin{minipage}{6cm}\center \begin{picture}(110,55) \put(0,25){\vector(1,0){15}} \put(7.5,28){\makebox(0,0)[b]{$I_1=I_e$}} \put(19,23){\vector(0,-1){16}} \put(17,15){\makebox(0,0)[r]{$U_1 = U_e$}} \multiput(19,5)(0,20){2}{\circle{2}} \multiput(20,5)(0,20){2}{\line(1,0){20}} \multiput(40,0)(0,30){2}{\line(1,0){30}} \multiput(40,0)(30,0){2}{\line(0,1){30}} \multiput(91,5)(0,20){2}{\circle{2}} \multiput(70,5)(0,20){2}{\line(1,0){20}} \put(110,25){\vector(-1,0){15}} \put(102.5,28){\makebox(0,0)[b]{$I_2 = I_a$}} \put(91,23){\vector(0,-1){16}} \put(94,15){\makebox(0,0)[l]{$U_2 = U_a$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \center \textbf{Gro"ssignalmodell} (inv. Hybridbeschreibung) \begin{empheq}[innerbox=\fbox]{align*} \quad I_1 &= H_1'(U_1,\,I_2) \vphantom{|} \quad\\ \quad U_2 &= H_2' (U_1,\,I_2) \vphantom{|} \end{empheq} \end{minipage}% \paragraph{Kleinsignalmodell:} \begin{empheq}[innerbox=\fbox]{align*} \quad i_e &= g_e \cdot u_e + \alpha \cdot i_a \quad\\ \quad u_a &= v \cdot u_e + r_a \cdot i_a \end{empheq} \bigskip \smallskip \begin{tabular}{@{}p{2.2cm}p{4.5cm}p{2.3cm}p{5.6cm}@{}} $v = \left.\dfrac{u_a}{u_e}\right|_{i_a=0}$ & Leerlaufverst"arkung & $r_a = \left.\dfrac{u_a}{i_a}\right|_{u_e=0}$ & Kurzschluss-Ausgangswiderstand\\[5ex] $g_e = \left.\dfrac{i_e}{u_e}\right|_{i_a=0}$ & Leerlauf-Eingangsleitwert & $\alpha = \left.\dfrac{i_e}{i_a}\right|_{u_e=0}$ & Kurzschluss-Stromr"uckwirkung \end{tabular} \begin{minipage}{11cm} \begin{picture}(200,100) \color{grau} \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$u_G\Big\downarrow$}} \put(30,0){\line(1,0){60}} \multiput(30,60)(40,0){2}{\line(1,0){20}} \multiput(50,55)(0,10){2}{\line(1,0){20}} \multiput(50,55)(20,00){2}{\line(0,1){10}} \put(60,68){\makebox(0,0)[b]{$R_G$}} \color{black} \multiput(91,0)(0,60){2}{\circle2} \put(91,58){\vector(0,-1){56}} \put(88,30){\makebox(0,0)[r]{$u_e$}} \put(75,65){\vector(1,0){15}} \put(82.5,68){\makebox(0,0)[b]{$i_e$}} \multiput(92,0)(0,60){2}{\line(1,0){58}} \multiput(120,0)(0,40){2}{\line(0,1){20}} \multiput(115,20)(10,0){2}{\line(0,1){20}} \multiput(115,20)(00,20){2}{\line(1,0){10}} \put(113,30){\makebox(0,0)[r]{$g_e$}} \multiput(150,0)(0,40){2}{\line(0,1){20}} \multiput(150,20)(-10,10){2}{\line(1,1){10}} \multiput(150,20)(10,10){2}{\line(-1,1){10}} \put(140,30){\line(1,0){20}} \put(158,30){\makebox(0,0)[l]{$\Big\downarrow \alpha \cdot i_a$}} \multiput(200,20)(-10,10){2}{\line(1,1){10}} \multiput(200,20)(10,10){2}{\line(-1,1){10}} \put(200,0){\line(0,1){60}} \put(208,30){\makebox(0,0)[l]{$\Big\downarrow v\cdot u_e$}} \put(200,0){\line(1,0){60}} \multiput(200,60)(40,0){2}{\line(1,0){20}} \multiput(220,55)(0,10){2}{\line(1,0){20}} \multiput(220,55)(20,00){2}{\line(0,1){10}} \put(230,68){\makebox(0,0)[b]{$r_a$}} \multiput(261,0)(0,60){2}{\circle2} \put(261,58){\vector(0,-1){56}} \put(260,30){\makebox(0,0)[r]{$u_a$}} \put(280,65){\vector(-1,0){15}} \put(272.5,68){\makebox(0,0)[b]{$i_e$}} \color{grau} \multiput(262,0)(0,60){2}{\line(1,0){18}} \multiput(280,0)(0,40){2}{\line(0,1){20}} \multiput(275,20)(10,0){2}{\line(0,1){20}} \multiput(275,20)(00,20){2}{\line(1,0){10}} \put(288,30){\makebox(0,0)[l]{$R_L$}} \end{picture} \end{minipage}% \begin{minipage}{5cm} \begin{picture}(140,100) \multiput(30,00)(0,60){2}{\circle*3} \multiput(120,00)(0,60){2}{\circle*3} \put(28,0){\makebox(0,0)[r]{$i_a$}} \put(30,63){\makebox(0,0)[b]{$u_e$}} \put(124,0){\makebox(0,0)[l]{$u_a$}} \put(124,60){\makebox(0,0)[l]{$i_e$}} \multiput(0,0)(0,60){2}{ \put(30,0){\vector(1,0){48}} \put(30,0){\line(1,0){90}} } \put(30,0){\line(3,2){90}} \put(30,0){\vector(3,2){60}} \put(30,60){\line(3,-2){43}} \put(30,60){\vector(3,-2){35}} \put(120,00){\line(-3,2){43}} \put(75,3){\makebox(0,0)[b]{$r_a$}} \put(75,56){\makebox(0,0)[t]{$g_e$}} \put(90,30){\makebox(0,0)[b]{$\alpha$}} \put(60,30){\makebox(0,0)[b]{$v$}} \color{grau} \qbezier(30,60)(75,80)(120,60) \put(72,70){\vector(-1,0){0}} \put(75,72){\makebox(0,0)[b]{$R_G$}} \qbezier(30,00)(75,-20)(120,0) \put(72,-10){\vector(-1,0){0}} \put(75,-12){\makebox(0,0)[t]{$-1/R_L$}} \put(10,60){\line(1,0){20}} \put(10,60){\vector(1,0){12}} \put(10,60){\circle*{3}} \put(10,63){\makebox(0,0)[b]{$u_G$}} \color{black} \multiput(30,00)(0,60){2}{\circle*3} \multiput(120,00)(0,60){2}{\circle*3} \end{picture} \end{minipage}% \bigskip \subsubsection*{Betriebsparameter} \begin{itemize} \item Betriebsverst"arkung: $v_B = \dfrac{u_a}{u_e} = \dfrac{1}{1+\frac{r_a}{R_L}}$, bzw. $v_{B2} = \dfrac{u_a}{u_G}$ (ber"ucksichtigt $R_G$) \item Betriebs-Eingangswiderstand: $r_{eB} = \dfrac{u_e}{i_e}$ \item Betriebs-Ausgangswiderstand: $r_a =\left.\dfrac{u_a}{i_a}\right|_{u_G=0}$ \end{itemize} \section{Emitterschaltung} \subsection{Gro"ssignalverhalten} \begin{minipage}{5cm} \begin{picture}(110,105) \put(29,40){\circle{2}} \put(30,40){\line(1,0){20}} \put(50,30){\thicklines\line(0,1){20}} \put(50,40){\line(1,1){10}} \put(50,40){\line(1,-1){10}} \put(58,32){\vector(1,-1){0}} \put(60,0){\line(0,1){30}} \put(55,0){\line(1,0){10}} \put(60,50){\line(1,0){20}} \put(81,50){\circle{2}} \put(60,50){\circle*{2}} \multiput(60,50)(0,30){2}{\line(0,1){10}} \multiput(55,60)(10,0){2}{\line(0,1){20}} \multiput(55,60)(0,20){2}{\line(1,0){10}} \put(60,91){\circle{2}} \put(60,94){\makebox(0,0)[b]{$U_{0C}$}} \put(29,38){\vector(0,-1){38}} \put(27,20){\makebox(0,0)[r]{$U_{e}$}} \put(5,40){\vector(1,0){20}} \put(15,42){\makebox(0,0)[b]{$I_{e}$}} \put(81,48){\vector(0,-1){38}} \put(83,25){\makebox(0,0)[l]{$U_{a}$}} \put(105,50){\vector(-1,0){20}} \put(95,52){\makebox(0,0)[b]{$I_{a}=0$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \center \textbf{Transferkennlinie:} $U_a = f(U_C)$ \[I_C = I_S \cdot e^{\frac{U_{BE}}{U_T}}\] \[U_{BE} = U_e, \ U_a = U_{0C} - I_C \cdot R_C\] \[\boxed{\quad U_a = U_{0C} - I_S \cdot R_C \cdot e^{\frac{U_e}{U_T}} \vphantom{\int^b_a}\quad}\] \end{minipage}% \begin{minipage}{5cm} \hfill \begin{picture}(110,110) \put(20,10){\vector(0,1){90}} \put(10,15){\vector(1,0){100}} \put(17,90){\makebox(0,0)[r]{$U_a$}} \put(17,70){\makebox(0,0)[r]{$U_{0C}$}} \put(100,06){\makebox(0,0)[c]{$U_{e}$}} \multiput(19,70)(4,0){20}{\line(1,0){2}} \qbezier(20,70)(25,70)(30,69.5) \qbezier(20,70)(30,70)(35,69.5) \multiput(0,0)(5,0){2}{ \qbezier(30,69.5)(50,69)(58,40) \qbezier(58,40)(65,17)(80,16) } \put(70,45){\vector(-3,-1){20}} \put(73,48){\makebox(0,0)[l]{$R_C$}} \end{picture} \end{minipage} \paragraph{Eingangskennlinie:} \quad $\boxed{\quad I_e = I_B = \frac{I_S}{B_F} \cdot e^{\frac{U_{e}}{U_T}} \quad}$ \subsubsection*{Aussteuerdiagramm:} \begin{minipage}{4.5cm} \center \begin{picture}(110,90) \put(20,10){\vector(0,1){80}} \put(10,15){\vector(1,0){100}} \put(17,80){\makebox(0,0)[r]{$U_a$}} \put(17,45){\makebox(0,0)[r]{$U_{aA}$}} \put(100,06){\makebox(0,0)[c]{$U_{e}$}} \put(57,06){\makebox(0,0)[c]{$U_{eA}$}} \multiput(19,45)(4,0){20}{\line(1,0){2}} \multiput(19,70)(4,0){20}{\line(1,0){2}} \qbezier(20,70)(25,70)(30,69.5) \qbezier(30,69.5)(53,69)(58,40) \qbezier(58,40)(65,17)(80,16) \multiput(57,14)(0,4){15}{\line(0,1){2}} \multiput(53,14)(0,4){11}{\line(0,1){2}} \multiput(61,14)(0,4){5}{\line(0,1){2}} \multiput(53,56)(4,0){8}{\line(1,0){2}} \multiput(61,33)(4,0){6}{\line(1,0){2}} \put(90,45){\vector(0,1){25}} \put(90,70){\vector(0,-1){25}} \put(92,56.5){\makebox(0,0)[l]{$U_{R_CA}$}} \multiput(0,0)(0,8){2}{ \qbezier(57,15)(53,16)(53,17) \qbezier(53,17)(53,18)(57,19) \qbezier(57,19)(61,20)(61,21) \qbezier(61,21)(61,22)(57,23) } \multiput(0,0)(8,0){2}{ \qbezier(62,45)(63,33)(64,33) \qbezier(64,33)(65,33)(66,45) \qbezier(66,45)(67,56)(68,56) \qbezier(68,56)(69,56)(70,45) } \end{picture} \end{minipage}% \hfill \begin{minipage}{10.5cm} \[\boxed{\quad v = \left.\frac{dU_a}{dU_e}\right|_{AP} = -\frac{R_C\cdot I_S}{U_T} \cdot e^{\frac{U_eA}{U_T}} = -\frac{R_C \cdot I_{CA}}{U_T} = - \frac{U_{R_CA}}{U_T} \quad}\] \end{minipage}% \subsection{Kleinsignalverhalten} \begin{minipage}{6cm} \center \begin{picture}(160,125) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \multiput(30,60)(22,0){2}{\line(1,0){18}} \multiput(48,53)(4,0){2}{\thicklines\line(0,1){14}} \multiput(70,60)(0,40){2}{\line(0,1){20}} %RB \multiput(65,80)(10,0){2}{\line(0,1){20}} \multiput(65,80)(0,20){2}{\line(1,0){10}} \put(070,60){\circle*2} \put(070,60){\line(1,0){20}} \put(90,50){\thicklines\line(0,1){20}} \put(90,60){\line(1,1){10}} \put(90,60){\line(1,-1){10}} \put(98,52){\vector(1,-1){0}} \put(100,70){\line(0,1){10}} \put(100,100){\line(0,1){20}} \multiput(95,80)(10,0){2}{\line(0,1){20}} \multiput(95,80)(0,20){2}{\line(1,0){10}} \put(100,00){\line(0,1){50}} \put(95,00){\line(1,0){10}} \put(25,00){\line(1,0){10}} \multiput(100,70)(22,0){2}{\line(1,0){18}} \multiput(118,63)(4,0){2}{\thicklines\line(0,1){14}} \put(140,0){\line(0,1){20}} \multiput(135,20)(10,0){2}{\line(0,1){20}} \multiput(135,20)(0,20){2}{\line(1,0){10}} \put(140,40){\line(0,1){30}} \put(135,0){\line(1,0){10}} \put(070,120){\line(1,0){50}} \put(121,120){\circle{2}} \multiput(100,70)(0,50){2}{\circle*{2}} \put(125,120){\makebox(0,0)[l]{$U_{0C}$}} \put(063,90){\makebox(0,0)[r]{$R_{B}$}} \put(133,30){\makebox(0,0)[r]{$R_{L}$}} \put(107,90){\makebox(0,0)[l]{$R_{C}$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \begin{picture}(150,60) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$\Rightarrow \ u_e\Big\downarrow$}} \multiput(30,0)(0,60){2}{\line(1,0){60}} \multiput(0,0)(30,0){2}{ \multiput(60,00)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} } \multiput(60,0)(0,60){2}{\circle*2} \put(67,30){\makebox(0,0)[l]{$R_{B}$}} \put(97,30){\makebox(0,0)[l]{$r_{BE}$}} \multiput(130,00)(0,40){2}{\line(0,1){20}} \multiput(130,20)(-10,10){2}{\line(1,1){10}} \multiput(130,20)(10,10){2}{\line(-1,1){10}} \put(120,30){\line(1,0){20}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow g_mu_{E}$}} \multiput(130,0)(0,60){2}{\line(1,0){115}} \multiput(125,0)(30,0){3}{ \multiput(60,00)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} } \put(191,30){\makebox(0,0)[l]{$r_{CE}$}} \put(221,30){\makebox(0,0)[l]{$R_{C}$}} \put(251,30){\makebox(0,0)[l]{$R_{L}$}} \qbezier(260,60)(275,30)(260,0) \put(260,0){\vector(-1,-2){0}} \put(270,30){\makebox(0,0)[l]{$u_{a}$}} \put(75,-13){\makebox(0,0){$\underbrace{\hphantom{lalalalala}}_{\displaystyle R_1}$}} \put(215,-13){\makebox(0,0){$\underbrace{\hphantom{lalallalalaalala}}_{\displaystyle R_2}$}} \end{picture} \end{minipage}% \bigskip \begin{align*} v_{B_1} &= - \frac{g_m \cdot u_e \cdot R_2}{u_2} = -g_m \cdot R_2 = - g_m(R_{CE}\parallel R_C \parallel R_L) && (\text{Gr"o"senordnung }10^2)\\ r_{e_B} &= R_B\parallel r_{BE} \approx r_{BE} && (11^0\,\mathrm k\Omega)\\ r_{a_B} &= r_{CE} \parallel R_C && (10^1\mathrm k\Omega) \end{align*} \subsection{Arbeitspunkteinstellung} \begin{minipage}{5cm} \begin{picture}(110,105) \put(29,40){\circle{2}} \put(30,40){\line(1,0){20}} \put(50,30){\thicklines\line(0,1){20}} \put(50,40){\line(1,1){10}} \put(50,40){\line(1,-1){10}} \put(58,32){\vector(1,-1){0}} \put(60,0){\line(0,1){30}} \put(55,0){\line(1,0){10}} \put(60,50){\line(1,0){20}} \put(81,50){\circle{2}} \put(60,50){\circle*{2}} \multiput(60,50)(0,30){2}{\line(0,1){10}} \multiput(55,60)(10,0){2}{\line(0,1){20}} \multiput(55,60)(0,20){2}{\line(1,0){10}} \put(60,91){\circle{2}} \put(60,94){\makebox(0,0)[b]{$U_{0C}$}} \put(29,38){\vector(0,-1){38}} \put(27,20){\makebox(0,0)[r]{$U_{e}$}} \put(5,40){\vector(1,0){20}} \put(15,42){\makebox(0,0)[b]{$I_{e}$}} \put(81,48){\vector(0,-1){38}} \put(83,25){\makebox(0,0)[l]{$U_{a}$}} \put(105,50){\vector(-1,0){20}} \put(95,52){\makebox(0,0)[b]{$I_{a}=0$}} \end{picture} \end{minipage}% \begin{minipage}{5cm} \center \begin{picture}(110,110) \put(20,10){\vector(0,1){90}} \put(10,15){\vector(1,0){100}} \put(17,90){\makebox(0,0)[r]{$U_a$}} \put(17,70){\makebox(0,0)[r]{$U_{0C}$}} \put(100,06){\makebox(0,0)[c]{$U_{e}$}} \multiput(19,70)(4,0){20}{\line(1,0){2}} \qbezier(20,70)(25,70)(30,69.5) \qbezier(30,69.5)(50,69)(70,20) \qbezier(70,20)(73,16)(90,16) \put(69.3,22){\circle*{2}} \put(72,23){\textcircled{\raisebox{-1.5pt}{1}}} \put(59.8,42){\circle*{2}} \put(47,35){\textcircled{\raisebox{-1.5pt}{2}}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \center \begin{picture}(110,110) \color{grau} \qbezier(30,90)(30,25)(100,25) \qbezier(45,90)(45,40)(100,40) \qbezier(60,90)(60,55)(100,55) \put(75,73){$P$} \qbezier(45,90)(45,40)(100,40) \put(60,50){\vector(1,1){25}} \color{black} \put(20,10){\vector(0,1){90}} \put(10,15){\vector(1,0){100}} \put(17,92){\makebox(0,0)[r]{$I_C$}} \put(100,06){\makebox(0,0)[c]{$U_{CE}$}} \multiput(23,30)(2,10){6}{\line(5,1){60}} \put(20,15){\line(1,5){13}} \put(115,60){\makebox(0,0)[c]{$\Big\uparrow U_{BE},\,I_{B}$}} \put(20,75){\line(1,-1){60}} \put(18,75){\makebox(0,0)[r]{$\frac{U_{0C}}{R_C}$}} \put(34,61){\circle*{2}} \put(36,63){\textcircled{\raisebox{-1.5pt}{1}}} \put(50,45){\circle*{2}} \put(47,49){\textcircled{\raisebox{-1.5pt}{2}}} \end{picture} \end{minipage}% \begin{tabular}{lp{4cm}p{4cm}p{5cm}} & \bf Forderung & \bf Arbeitspunkt & \bf Aussteuerdiagramm \tabularnewline\tabularnewline 1. & \raggedright Hohe Verst"arkung bei kleiner Aussteuerung & $U_{R_CA}$ m"oglichst gro"s & \\ & & &\qquad\begin{picture}(100,65) \put(15,10){\vector(0,1){70}} \put(10,15){\vector(1,0){90}} \put(14,73){\makebox(0,0)[r]{$\scriptstyle U_{CE}$}} \put(14,32){\makebox(0,0)[r]{$\scriptstyle U_{CEA}$}} \put(14,23){\makebox(0,0)[r]{$\scriptstyle U_{CE_{min}}$}} \put(95,9){\makebox(0,0)[r]{$\scriptstyle t$}} \put(60,50){\makebox(0,0)[c]{$\scriptstyle U_{CEA} = U_{CE_{min}} + \,\widehat{u}_a$}} \multiput(14,25)(4,0){20}{\line(1,0){2}} \multiput(14,30)(4,0){20}{\line(1,0){2}} \multiput(0,-5)(20,0){3}{ \qbezier(15,35)(17.5,40)(20,40) \qbezier(20,40)(22.5,40)(25,35) \qbezier(25,35)(27.5,30)(30,30) \qbezier(30,30)(32.5,30)(35,35) } \end{picture} \\ 2. & \raggedright Maximale Aussteuerung & \raggedright Mitte des aktiven Bereichs & \\ & & & \qquad\begin{picture}(100,65) \put(15,10){\vector(0,1){70}} \put(10,15){\vector(1,0){90}} \put(14,73){\makebox(0,0)[r]{$\scriptstyle U_{CE}$}} \put(14,40){\makebox(0,0)[r]{$\scriptstyle U_{CEA}$}} \put(95,9){\makebox(0,0)[r]{$\scriptstyle t$}} \multiput(14,40)(4,0){20}{\line(1,0){2}} \multiput(14,62)(4,0){20}{\line(1,0){2}} \multiput(14,20)(4,0){20}{\line(1,0){2}} \put(14,20){\makebox(0,0)[r]{$\scriptstyle U_{CE_{min}}$}} \put(95,59){$\scriptstyle U_{0C}$} \multiput(0,0)(20,0){3}{ \qbezier(15,40)(17.5,60)(20,60) \qbezier(20,60)(22.5,60)(25,40) \qbezier(25,40)(27.5,20)(30,20) \qbezier(30,20)(32.5,20)(35,40) } \end{picture}\\ 3. & Thermische Stabilit"at & $U_{CEA} \geq \frac{U_{0C}}{2}$ \end{tabular} \subsection{Schaltungen zur Arbeitspunkteinstellung} \paragraph*{Basisstromeinstellung:} siehe 1.4 \subsubsection*{Emitterstromregelung} \begin{minipage}{4.5cm} \center \begin{picture}(120,125) \put(-40,0){ \multiput(70,60)(0,40){2}{\line(0,1){20}} %R2 \multiput(65,80)(10,0){2}{\line(0,1){20}} \multiput(65,80)(0,20){2}{\line(1,0){10}} \multiput(70,00)(0,40){2}{\line(0,1){20}} %R1 \multiput(65,20)(10,0){2}{\line(0,1){20}} \multiput(65,20)(0,20){2}{\line(1,0){10}} \put(070,60){\circle*2} \put(070,60){\line(1,0){20}} \put(90,50){\thicklines\line(0,1){20}} \put(90,60){\line(1,1){10}} \put(90,60){\line(1,-1){10}} \put(98,52){\vector(1,-1){0}} \put(100,70){\line(0,1){10}} \put(100,100){\line(0,1){20}} \multiput(95,80)(10,0){2}{\line(0,1){20}} \multiput(95,80)(0,20){2}{\line(1,0){10}} \put(95,00){\line(1,0){10}} \put(100,0){\line(0,1){20}} \multiput(95,20)(10,0){2}{\line(0,1){20}} \multiput(95,20)(0,20){2}{\line(1,0){10}} \put(100,40){\line(0,1){10}} \put(65,0){\line(1,0){10}} \put(070,120){\line(1,0){50}} \put(121,120){\circle{2}} \multiput(100,70)(0,50){2}{\circle*{2}} \put(125,120){\makebox(0,0)[l]{$U_{0C}$}} \put(063,90){\makebox(0,0)[r]{$R_{B}$}} \put(122,30){\makebox(0,0)[r]{$R_{E}$}} \put(107,90){\makebox(0,0)[l]{$R_{C}$}} \put(50,60){\line(1,0){20}} \put(100,70){\line(1,0){20}} \put(121,70){\circle{2}} \put(49,60){\circle{2}} } \end{picture} \end{minipage}% \begin{minipage}{5cm} \begin{picture}(130,125) \multiput(30,60)(0,40){2}{\line(0,1){20}} %R2 \multiput(25,80)(10,0){2}{\line(0,1){20}} \multiput(25,80)(0,20){2}{\line(1,0){10}} \multiput(30,00)(0,40){2}{\line(0,1){20}} %R1 \multiput(25,20)(10,0){2}{\line(0,1){20}} \multiput(25,20)(0,20){2}{\line(1,0){10}} \multiput(30,120)(40,0){2}{\line(1,0){20}} \multiput(50,115)(0,10){2}{\line(1,0){20}} \multiput(50,115)(20,0){2}{\line(0,1){10}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(90,60)(0,40){2}{\line(0,1){20}} \multiput(85,20)(10,0){2}{\line(0,1){20}} \multiput(85,20)(0,20){2}{\line(1,0){10}} \multiput(90,80)(10,10){2}{\line(-1,1){10}} \multiput(90,80)(-10,10){2}{\line(1,1){10}} \put(30,60){\line(1,0){60}} \put(60,60){\circle{20}} \multiput(25,0)(60,0){2}{\line(1,0){10}} \put(20,120){\line(1,0){10}} \put(19,120){\circle{2}} \multiput(30,60)(60,0){2}{\circle*{2}} \put(80,90){\line(1,0){20}} \put(17,120){\makebox(0,0)[r]{$U_{0C}$}} \put(23,30){\makebox(0,0)[r]{$R_{1}$}} \put(23,90){\makebox(0,0)[r]{$R_{2}$}} \put(60,80){\makebox(0,0)[c]{$U_{BE0}$}} \put(50,73){\vector(1,0){20}} \put(60,113){\makebox(0,0)[t]{$R_C$}} \put(100,90){\makebox(0,0)[l]{$\Big\downarrow B\cdot I_B$}} \put(98,30){\makebox(0,0)[l]{$R_E$}} \put(80,50){\makebox(0,0){$I_{BA}$}} \put(83,60){\vector(1,0){0}} \put(40,58){\vector(0,-1){58}} \put(43,30){\makebox(0,0)[l]{$U_{BA}$}} \put(0,60){\makebox(0,0){$\Rightarrow$}} \end{picture} \end{minipage}% \hfill \begin{minipage}{6cm} \subsubsection*{Entwurfsbeispiel:}\raggedright gegeben: $U_{BE0} = 0{,}6\,\mathrm V$, $B = 100$, $I_{CA} = 1\,\mathrm{mA}$, $R_C = 500\,\Omega$, $U_{0C} = 10\,\mathrm V$ \bigskip $U_{BA} = U_{EA} + U_{BE0} \approx I_{CA} \cdot R_E + U_{BE0} = 1{,}1\,\mathrm V$ \bigskip muss mit dem "`Spannungsteiler"' $R_1$/$R_2$ eingestellt werden. \end{minipage}% \smallskip \paragraph{Annahme:} Um den Spannungsteiler m"oglichst wenig zu belasten, wird $I_{R_1} = \alpha \cdot I_{BA}$ mit $\alpha = 10$ gew"ahlt. \[R_1 = \frac{U_{BA}}{\alpha \cdot I_{BA}} = 11\,\mathrm k\Omega \qquad R_2 = \frac{U_{0C} - U_{BA}}{(1+\alpha)\cdot I_{BA}} = 81\,\mathrm k\Omega\] \subsubsection*{Nachrechnen des Spannungsteilerverh"altnisses} \[\frac{R_1}{R_1 + R_2} = \frac{11\,\mathrm k\Omega }{11\,\mathrm k\Omega + 81\,\mathrm k\Omega} \qquad \frac{U_{BA}}{U_{0C}} = \frac{1{,}1\,\mathrm V}{10\,\mathrm V} = 0{,}11\] \smallskip Relative "Anderung: \ 9\%. \subsubsection*{Nachrechnen des Eingangsleitwertes} \[g_e = \frac 1 {R_1} + \frac 1 {R_2} + \frac{1}{r_{BE}} = \underbrace{0{,}09\,\mathrm{mS} + 0{,}12\,\mathrm{mS}}_{9{,}7\,\mathrm k\Omega} + \underbrace{0{,}38\,\mathrm{mS}}_{2{,}6\,\mathrm k\Omega}\] Relative "Anderung: \ $g_e:$ 27\% $\uparrow$, $r_e:$ 21\% $\downarrow$ \subsubsection*{Arbeitspunkteinstellung bei Sourcestufen} \begin{minipage}{6cm} \center \begin{picture}(120,130) \multiput(0,0)(0,60){2}{ \multiput(5,0)(0,40){2}{\line(0,1){20}} \multiput(0,20)(10,0){2}{\line(0,1){20}} \multiput(0,20)(0,20){2}{\line(1,0){10}} } \put(50,100){\line(0,1){20}} \put(50,68){\line(0,1){12}} \put(50,0){\line(0,1){52}} \multiput(45,80)(10,0){2}{\line(0,1){20}} \multiput(45,80)(0,20){2}{\line(1,0){10}} \put(5,120){\line(1,0){50}} \put(56,120){\circle2} \put(59,120){\makebox(0,0)[l]{$U_{0D}$}} \put(59,90){\makebox(0,0)[l]{$R_{D}$}} \put(5,60){\circle*2} \put(5,60){\line(1,0){30}} \put(35,50){\thicklines\line(0,1){20}} \multiput(38,52)(0,16){2}{\line(1,0){12}} \put(50,60){\vector(-1,0){12}} \put(50,60){\line(0,-1){10}} \put(50,52){\circle*2} \multiput(38,50)(0,8){3}{\line(0,1){4}} \multiput(0,0)(45,0){2}{\line(1,0){10}} \put(95,0){\vector(0,1){130}} \put(90,0){\line(1,0){10}} \put(92.50,10){\line(1,0){5}} \put(92.50,30){\line(1,0){5}} \put(92.50,75){\line(1,0){5}} \put(92.50,120){\line(1,0){5}} \put(99,10){\makebox(0,0)[l]{$U_t$}} \put(99,30){\makebox(0,0)[l]{$U_G$}} \put(99,75){\makebox(0,0)[l]{$U_D$}} \put(99,120){\makebox(0,0)[l]{$U_{0D}$}} \end{picture} \end{minipage}% \hfill Enhancement FET \hfill \begin{minipage}{6cm} \center \begin{picture}(145,130) \put(30,10){\vector(0,1){100}} \put(25,15){\vector(1,0){120}} \put(26,100){\makebox(0,0)[r]{$I_D$}} \put(26,70){\makebox(0,0)[r]{$I_{DA}$}} \put(135,7){\makebox(0,0){$U_{GS}$}} \put(101,7){\makebox(0,0){$U_{GSA}$}} \multiput(101,14)(0,4){14}{\line(0,1){2}} \multiput(29,70)(4,0){18}{\line(1,0){2}} \put(45,7){\makebox(0,0){$U_{t}$}} \put(45,14){\line(0,1){2}} \thicklines \qbezier(45,15)(80,28)(115,95) \qbezier(30,15)(30,15)(45,15) \end{picture} \end{minipage} \bigskip \bigskip \begin{minipage}{6cm} \center \begin{picture}(120,130) \multiput(5,0)(0,40){2}{\line(0,1){20}} \multiput(0,20)(10,0){2}{\line(0,1){20}} \multiput(0,20)(0,20){2}{\line(1,0){10}} \put(50,100){\line(0,1){20}} \put(50,68){\line(0,1){12}} \put(50,0){\line(0,1){20}} \multiput(45,20)(10,0){2}{\line(0,1){20}} \multiput(45,20)(0,20){2}{\line(1,0){10}} \put(50,40){\line(0,1){10}} \multiput(45,80)(10,0){2}{\line(0,1){20}} \multiput(45,80)(0,20){2}{\line(1,0){10}} \put(50,121){\circle2} \put(50,124){\makebox(0,0)[b]{$U_{0D}$}} \put(59,90){\makebox(0,0)[l]{$R_{D}$}} \put(42,30){\makebox(0,0)[r]{$R_{S}$}} \put(5,60){\line(1,0){30}} \put(35,50){\thicklines\line(0,1){20}} \multiput(38,52)(0,16){2}{\line(1,0){12}} \put(50,60){\vector(-1,0){12}} \put(50,60){\line(0,-1){10}} \put(50,52){\circle*2} \put(38,50){\line(0,1){20}} \multiput(0,0)(45,0){2}{\line(1,0){10}} \put(95,0){\vector(0,1){130}} \put(90,0){\line(1,0){10}} \put(92.50,60){\line(1,0){5}} \put(92.50,120){\line(1,0){5}} \put(103,0){\makebox(0,0)[l]{$U_G$}} \put(99,60){\makebox(0,0)[l]{$U_S$}} \put(99,120){\makebox(0,0)[l]{$U_{0D}$}} \put(70,0){\line(0,1){28}} \put(70,32){\line(0,1){20}} \put(50,52){\line(1,0){20}} \put(65,0){\line(1,0){10}} \multiput(63,28)(0,4){2}{\thicklines\line(1,0){14}} \end{picture} \end{minipage}% \hfill Depletion FET, R"ohre \hfill \begin{minipage}{6cm} \center \begin{picture}(145,130) \put(80,10){\vector(0,1){100}} \put(25,15){\vector(1,0){120}} \put(76,100){\makebox(0,0)[r]{$I_D$}} \put(83,28){\makebox(0,0)[l]{$I_{DA}$}} \put(135,7){\makebox(0,0){$U_{GS}$}} \put(67,7){\makebox(0,0){$U_{GSA}$}} \multiput(67,14)(0,4){4}{\line(0,1){2}} \multiput(81,28)(-4,0){4}{\line(-1,0){2}} \put(45,7){\makebox(0,0){$U_{T}$}} \put(45,14){\line(0,1){2}} \thicklines \qbezier(45,15)(80,28)(115,95) \qbezier(30,15)(30,15)(45,15) \thinlines \put(80,15){\line(-1,1){30}} \put(48,50){\makebox(0,0)[b]{$I_{D} = \frac{U_{DS}}{R_S}$}} \end{picture} \end{minipage}% \subsection{Emitterschaltung mit Laststromquelle} \begin{picture}(70,120) \put(9,30){\circle2} \put(10,30){\line(1,0){20}} \put(30,20){\thicklines\line(0,1){20}} \put(30,30){\line(1,-1){10}} \put(38,22){\vector(1,-1){0}} \put(30,30){\line(1,1){10}} \multiput(40,40)(0,40){2}{\line(0,1){20}} \put(40,70){\circle{20}} \put(30,70){\line(1,0){20}} \multiput(40,50)(0,40){2}{\line(-1,0){20}} \multiput(20,50)(0,30){2}{\line(0,1){10}} \multiput(15,60)(10,0){2}{\line(0,1){20}} \multiput(15,60)(0,20){2}{\line(1,0){10}} \multiput(40,50)(0,40){2}{\circle*2} \put(40,101){\circle2} \put(40,105){\makebox(0,0)[b]{$U_{0C}$}} \put(13,70){\makebox(0,0)[r]{$r_{CE2}$}} \put(40,0){\line(0,1){20}} \put(35,0){\line(1,0){10}} \put(9,28){\vector(0,-1){28}} \put(7,15){\makebox(0,0)[r]{$U_e$}} \put(40,50){\line(1,0){20}} \put(61,50){\circle{2}} \put(61,48){\vector(0,-1){48}} \put(63,25){\makebox(0,0)[l]{$U_a$}} \put(50,70){\makebox(0,0)[l]{$\Big\downarrow I_0$}} \end{picture} \hfill \begin{picture}(70,120) \put(30,30){\line(1,0){20}} \put(50,20){\thicklines\line(0,1){20}} \put(50,30){\line(1,1){10}} \put(50,30){\line(1,-1){10}} \put(58,22){\vector(1,-1){0}} \put(30,70){\line(1,0){20}} \put(50,60){\thicklines\line(0,1){20}} \put(50,70){\line(1,1){10}} \put(50,70){\line(1,-1){10}} \put(53,73){\vector(-1,-1){0}} \put(60,40){\line(0,1){20}} \put(55,57.5){\vector(0,-1){15}} \put(60,50){\circle*2} \put(81,50){\circle2} \put(29,30){\circle2} \put(29,70){\circle2} \put(60,50){\line(1,0){20}} \put(60,0){\line(0,1){20}} \put(55,0){\line(1,0){10}} \put(53,50){\makebox(0,0)[r]{$I_C$}} \put(27,30){\makebox(0,0)[r]{$U_e$}} \put(27,70){\makebox(0,0)[r]{$U_{ref}$}} \put(60,80){\line(0,1){20}} \put(60,101){\circle2} \put(60,105){\makebox(0,0)[b]{$U_{0C}$}} \end{picture} \hfill \begin{picture}(190,120) \put(25,10){\vector(0,1){100}} \put(20,15){\vector(1,0){150}} \put(23,100){\makebox(0,0)[r]{$I_C$}} \put(160,7){\makebox(0,0)[c]{$U_{CE}$}} \put(24,80){\line(1,0){130}} \multiput(15,68.75)(0,15){2}{\line(0,1){7.5}} \put(15,80){\circle{7.4}} \put(11.25,80){\line(1,0){7.5}} \put(25,15){\line(2,5){25}} \multiput(0,0)(-6,-15){3}{\qbezier(49.8,77.5)(52,81)(60,82)} \qbezier(60,82)(60,82)(140,90) \qbezier(54,67)(54,67)(140,75) \qbezier(48,52)(48,52)(140,60) \put(143,63){$\Big\uparrow U_e$} \qbezier(45,105)(50,90)(50,90) \qbezier(50,90)(52,86)(60,84) \qbezier(60,84)(140,67)(140,67) \put(50,100){\line(1,0){5}} \put(55,95){\thicklines\line(0,1){10}} \put(55,100){\qbezier(0,0)(0,0)(5,5)} \put(55,100){\qbezier(0,0)(0,0)(5,-5)} \put(55,100){\vector(-1,-1){0}} \end{picture} \subsubsection*{Kleinsignal-Ersatzschaltbild} \smallskip \begin{center} \begin{picture}(365,60) \multiput(19,0)(0,60){2}{\circle2} \multiput(351,0)(0,60){2}{\circle2} \put(354,30){\makebox(0,0)[l]{$u_a$}} \multiput(55,0)(0,40){2}{\line(0,1){20}} \multiput(50,20)(10,0){2}{\line(0,1){20}} \multiput(50,20)(0,20){2}{\line(1,0){10}} \put(48,30){\makebox(0,0)[r]{$r_{BE}$}} \multiput(80,0)(0,40){2}{\line(0,1){20}} \multiput(80,20)(-10,10){2}{\line(1,1){10}} \multiput(80,20)(10,10){2}{\line(-1,1){10}} \put(70,30){\line(1,0){20}} \put(90,30){\makebox(0,0)[l]{$\Big\downarrow g_{m1}u_e$}} \put(20,0){\line(1,0){330}} \put(20,60){\line(1,0){35}} \put(19,58){\vector(0,-1){56}} \put(351,58){\vector(0,-1){56}} \put(17,30){\makebox(0,0)[r]{$u_e$}} \put(80,60){\line(1,0){270}} \multiput(140,0)(0,40){2}{\line(0,1){20}} \multiput(135,20)(10,0){2}{\line(0,1){20}} \multiput(135,20)(0,20){2}{\line(1,0){10}} \put(148,30){\makebox(0,0)[l]{$r_{CE1}$}} \put(180,0){\line(0,1){50}} \multiput(195,0)(0,35){2}{\line(0,1){15}} \multiput(190,15)(10,0){2}{\line(0,1){20}} \multiput(190,15)(0,20){2}{\line(1,0){10}} \put(180,50){\line(1,0){15}} \put(203,25){\makebox(0,0)[l]{$r_{BE2}$}} \multiput(240,0)(0,40){2}{\line(0,1){20}} \multiput(240,20)(-10,10){2}{\line(1,1){10}} \multiput(240,20)(10,10){2}{\line(-1,1){10}} \put(230,30){\line(1,0){20}} \put(250,30){\makebox(0,0)[l]{$\Big\downarrow g_{m2}u_{BE2}$}} \multiput(310,0)(0,40){2}{\line(0,1){20}} \multiput(305,20)(10,0){2}{\line(0,1){20}} \multiput(305,20)(0,20){2}{\line(1,0){10}} \put(318,30){\makebox(0,0)[l]{$r_{CE2}$}} \multiput(55,0)(25,0){2}{\circle*2} \multiput(140,0)(0,60){2}{\circle*2} \multiput(180,0)(15,0){2}{\circle*2} \multiput(240,0)(0,60){2}{\circle*2} \multiput(310,0)(0,60){2}{\circle*2} \end{picture} \end{center} \begin{description} \item[Leerlaufverst"arkung:] $v = \left.\dfrac{u_a}{u_e}\right|_{i_a=0} = -g_{m1}(r_{CE1}\parallel r_{CE1})$ \item[Beispiel:] $I_0 = I_{C1/2A} = I_{CA} = 200\mu\mathrm A$. $U_{Y1/2} = 100\,\mathrm V$, $U_{CE1/2A} = U_{CE} = 5\,\mathrm V$, $U_T = 26\,\mathrm{mV}$ \begin{align*} v =& -g_{m1} \cdot (r_{CE1}\parallel r_{CE1}) = -g_{m1} \cdot \frac{r_{CE}}{2} = - \frac{I_{CA}}{U_T} \cdot \frac{U_Y + U_{CEA}}{2\cdot I{CA}}\\ &= - \frac{U_Y + U_{CEA}}{2\cdot U_T} \end{align*} \end{description} \section{Basisschaltung und Gateschaltung} \subsection{Gro"ssignalverhalten} \begin{minipage}{8cm} \center \begin{picture}(120,100) \put(19,30){\circle2} \put(20,30){\line(1,0){20}} \put(50,20){\line(1,1){10}} \put(50,20){\line(-1,1){10}} \put(42,28){\vector(-1,1){0}} \put(40,20){\thicklines\line(1,0){20}} \put(45,0){\line(1,0){10}} \put(50,0){\line(0,1){20}} \put(60,30){\line(1,0){40}} \multiput(80,30)(0,40){2}{\line(0,1){20}} \multiput(75,50)(10,0){2}{\line(0,1){20}} \multiput(75,50)(0,20){2}{\line(1,0){10}} \put(80,30){\circle*2} \put(80,91){\circle2} \put(101,30){\circle2} \put(83,91){\makebox(0,0)[l]{$U_{0C}$}} \multiput(19,28)(82,0){2}{\vector(0,-1){26}} \multiput(14,0)(82,0){2}{\line(1,0){10}} \put(17,15){\makebox(0,0)[r]{$U_e$}} \put(103,15){\makebox(0,0)[l]{$U_a$}} \put(0,30){\vector(1,0){15}} \put(7.5,32){\makebox(0,0)[b]{$I_e$}} \put(120,30){\vector(-1,0){15}} \put(112.5,32){\makebox(0,0)[b]{$I_a$}} \put(72,60){\makebox(0,0)[r]{$R_C$}} \put(75,33){\vector(-1,0){15}} \put(70,35){\makebox(0,0)[b]{$I_C$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(120,100) \put(19,30){\circle2} \put(20,30){\line(1,0){30}} \multiput(40,22)(8,0){3}{\line(1,0){4}} \put(42,30){\line(0,-1){8}} \put(58,30){\line(0,-1){8}} \put(50,30){\vector(0,-1){8}} \put(40,20){\thicklines\line(1,0){20}} \put(45,0){\line(1,0){10}} \put(50,0){\line(0,1){20}} \put(58,30){\line(1,0){42}} \multiput(80,30)(0,40){2}{\line(0,1){20}} \multiput(75,50)(10,0){2}{\line(0,1){20}} \multiput(75,50)(0,20){2}{\line(1,0){10}} \put(80,30){\circle*2} \put(80,91){\circle2} \put(101,30){\circle2} \put(83,91){\makebox(0,0)[l]{$U_{0D}$}} \multiput(19,28)(82,0){2}{\vector(0,-1){26}} \multiput(14,0)(82,0){2}{\line(1,0){10}} \put(17,15){\makebox(0,0)[r]{$U_e$}} \put(103,15){\makebox(0,0)[l]{$U_a$}} \put(0,30){\vector(1,0){15}} \put(7.5,32){\makebox(0,0)[b]{$I_e$}} \put(120,30){\vector(-1,0){15}} \put(112.5,32){\makebox(0,0)[b]{$I_a$}} \put(72,60){\makebox(0,0)[r]{$R_D$}} \put(75,33){\vector(-1,0){15}} \put(70,35){\makebox(0,0)[b]{$I_D$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{align*} I_C &= I_S \cdot e^{\frac{U_{BE}}{U_T}}\\[1ex] U_{BE} &= - U_e \\[1ex] U_a &= U_{0C} - I_C \cdot R_C\\ &= U_{0C} - R_C \cdot I_S \cdot e^{\frac{U_{BE}}{U_T}} \end{align*} \end{minipage}% \begin{minipage}{8cm} \begin{align*} I_D &= k (U_{GS} - U_t)^2 \\[1ex] U_{GS} &= -U_e\\[1ex] U_a &= U_{0D} - I_D \cdot R_D \\ &= U_{0D} - R_D \cdot k (U_{GS} - U_t)^2 \end{align*} \end{minipage}% \subsubsection*{Transferkennlinien} \begin{minipage}{8cm} \center \begin{picture}(150,100) \put(75,0){\vector(0,1){100}} \put(0,30){\vector(1,0){150}} \put(140,23){\makebox(0,0)[c]{$U_e$}} \put(73,90){\makebox(0,0)[r]{$U_a$}} \put(74,20){\line(1,0){2}} \put(78,20){\makebox(0,0)[l]{$\approx-0{,}5\,\mathrm{V}$}} \put(10,20){\line(1,0){20}} \qbezier(30,20)(40,70)(75,70) \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(150,100) \put(75,0){\vector(0,1){100}} \put(0,30){\vector(1,0){150}} \put(140,23){\makebox(0,0)[c]{$U_e$}} \put(73,90){\makebox(0,0)[r]{$U_a$}} \put(74,20){\line(1,0){2}} \put(78,20){\makebox(0,0)[l]{$- U_t$}} \put(45,20){\line(-5,-1){30}} \qbezier(45,20)(45,70)(75,70) \end{picture} \end{minipage}% \subsubsection*{Kleinsignalverst"arkung} \begin{minipage}{8cm} \begin{align*} v &= \frac{I_S \cdot R_C}{U_T} \cdot e^{-\frac{U_{BA}}{U_T}} = \frac{I_{CA}}{U_T} \cdot R_C = g_m \cdot R_C \end{align*} \end{minipage}% \begin{minipage}{8cm} \begin{align*} v = g_m \cdot R_D = 2R_D \sqrt{k\cdot I_{DA}} \vphantom{\frac{I}{U_T}} \end{align*} \end{minipage}% \subsubsection*{Eingangskennlinien} \begin{minipage}{8cm} \[I_e = -I_E = -I_S \cdot \frac{B_F + 1}{B_F}\cdot e^{-\frac{U_e}{U_T}}\] \end{minipage}% \begin{minipage}{8cm} \[I_e = -I_S = -I_D = -k(U_e + U_t)^2\] \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(120,120) \put(90,0){\vector(0,1){110}} \put(0,90){\vector(1,0){120}} \put(110,83){\makebox(0,0)[c]{$U_e$}} \put(93,104){\makebox(0,0)[l]{$I_e$}} \qbezier(40,10)(50,90)(90,90) \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(120,120) \put(90,0){\vector(0,1){110}} \put(0,90){\vector(1,0){120}} \put(110,83){\makebox(0,0)[c]{$U_e$}} \put(93,104){\makebox(0,0)[l]{$I_e$}} \qbezier(30,10)(40,90)(75,90) \put(75,89){\line(0,1){2}} \put(75,93){\makebox(0,0)[b]{$-U_t$}} \end{picture} \end{minipage}% \newpage \paragraph{Eingangsleitwert:} $g_e = \left.\frac{dI_e}{dU_e}\right|_{\mathrm{AP}} = \frac{1}{r_e}$ \begin{minipage}{8cm} \[ g_e = \frac{B_F + 1}{B_F} \cdot \frac{I_S \cdot e^{-\frac{U_e}{U_T}}}{U_T} = = \underbrace{\frac{B_F + 1}{B_F}}_{\approx 1} \cdot \underbrace{\frac{I_{CA}}{U_T}}_{g_m} \] \end{minipage}% \begin{minipage}{8cm} \[ g_e = 2k(U_e + U_t) = 2\sqrt{k I_{DA}} = g_m \] \end{minipage}% \subsection{Kleinsignalverhalten} \begin{center} \begin{picture}(300,75) \put(0,30){\makebox(0,0)[l]{$U_G\!\Big\downarrow$}} \put(30,30){\circle{20}} \put(25,0){\line(1,0){10}} \put(30,0){\line(0,1){60}} \multiput(30,60)(40,0){2}{\line(1,0){20}} \multiput(50,55)(0,10){2}{\line(1,0){20}} \multiput(50,55)(20,0){2}{\line(0,1){10}} \put(60,67){\makebox(0,0)[b]{$R_G$}} \multiput(90,53)(4,0){2}{\thicklines\line(0,1){14}} \put(94,60){\line(1,0){56}} \multiput(110,0)(0,40){2}{\line(0,1){20}} \multiput(105,20)(10,0){2}{\line(0,1){20}} \multiput(105,20)(0,20){2}{\line(1,0){10}} \put(110,60){\circle*2} \put(105,0){\line(1,0){10}} \put(103,30){\makebox(0,0)[r]{$R_E$}} \put(80,58){\vector(0,-1){58}} \put(77,30){\makebox(0,0)[r]{$u_e$}} \put(150,60){\line(1,-1){10}} \put(160,50){\line(1,1){10}} \put(150,50){\thicklines\line(1,0){20}} \put(152,58){\vector(-1,1){0}} \put(160,40){\line(0,1){10}} \put(140,40){\line(1,0){40}} \put(160,40){\circle*2} \put(140,0){\line(0,1){10}} \multiput(135,10)(10,0){2}{\line(0,1){20}} \multiput(135,10)(0,20){2}{\line(1,0){10}} \put(140,30){\line(0,1){10}} \multiput(135,0)(20,0){2}{\line(1,0){10}} \multiput(160,0)(0,22){2}{\line(0,1){18}} \multiput(153,18)(0,4){2}{\thicklines\line(1,0){14}} \put(133,20){\makebox(0,0)[r]{$R_1$}} \put(187,20){\makebox(0,0)[l]{$R_2$}} \put(180,0){\line(0,1){10}} \multiput(175,10)(10,0){2}{\line(0,1){20}} \multiput(175,10)(0,20){2}{\line(1,0){10}} \put(180,30){\line(0,1){10}} \put(180,0){\line(1,0){40}} \put(221,0){\circle2} \put(170,60){\line(1,0){40}} \multiput(210,0)(0,40){2}{\line(0,1){20}} \multiput(205,20)(10,0){2}{\line(0,1){20}} \multiput(205,20)(0,20){2}{\line(1,0){10}} \put(217,30){\makebox(0,0)[l]{$R_C$}} \put(224,0){\makebox(0,0)[l]{$U_{0C}$}} \multiput(210,60)(32,0){2}{\line(1,0){28}} \multiput(238,53)(4,0){2}{\thicklines\line(0,1){14}} \multiput(210,0)(0,60){2}{\circle*2} \multiput(270,0)(0,40){2}{\line(0,1){20}} \multiput(265,20)(10,0){2}{\line(0,1){20}} \multiput(265,20)(0,20){2}{\line(1,0){10}} \put(265,0){\line(1,0){10}} \put(263,30){\makebox(0,0)[r]{$R_L$}} \qbezier(275,55)(285,30)(275,5) \put(275,5){\vector(-1,-3){0}} \put(282,30){\makebox(0,0)[l]{$u_a$}} \end{picture} \end{center} \begin{minipage}{8cm} \center \begin{picture}(170,100) \put(0,30){\makebox(0,0)[l]{$u_G^*\Big\downarrow$}} \put(30,30){\circle{20}} \put(25,0){\line(1,0){10}} \put(30,0){\line(0,1){60}} \multiput(30,60)(40,0){2}{\line(1,0){20}} \multiput(50,55)(0,10){2}{\line(1,0){20}} \multiput(50,55)(20,0){2}{\line(0,1){10}} \put(60,67){\makebox(0,0)[b]{$R_G^*$}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(85,20)(10,0){2}{\line(0,1){20}} \multiput(85,20)(0,20){2}{\line(1,0){10}} \put(85,0){\line(1,0){10}} \put(97,30){\makebox(0,0)[l]{$r_{BE}$}} \qbezier(85,55)(75,30)(85,5) \put(85,5){\vector(1,-3){0}} \put(77,30){\makebox(0,0)[r]{$u_{e}$}} \multiput(90,60)(40,0){2}{\line(1,0){20}} \multiput(110,60)(10,-10){2}{\line(1,1){10}} \multiput(110,60)(10,10){2}{\line(1,-1){10}} \put(120,50){\line(0,1){20}} \put(130,48){\vector(-1,0){20}} \put(120,38){\makebox(0,0)[b]{$g_m u_{BE}$}} \multiput(90,60)(60,0){2}{\circle*2} \multiput(90,60)(60,0){2}{\line(0,1){20}} \multiput(90,80)(40,0){2}{\line(1,0){20}} \multiput(110,75)(0,10){2}{\line(1,0){20}} \multiput(110,75)(20,0){2}{\line(0,1){10}} \put(120,88){\makebox(0,0)[b]{$r_{CE}$}} \multiput(150,00)(0,40){2}{\line(0,1){20}} \multiput(145,20)(10,0){2}{\line(0,1){20}} \multiput(145,20)(0,20){2}{\line(1,0){10}} \put(145,0){\line(1,0){10}} \put(158,30){\makebox(0,0)[l]{$R_{L}^*$}} \put(80,63){E} \put(153,63){C} \put(93,3){B} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{align*} u_G^* &= U_G \cdot \frac{R_E}{R_E+R_G} & g_CE &= \frac 1 {r_{CE}}\\ R_L^* &= R_C \parallel R_L & g_{BE} &= \frac{1}{r_{BE}} \\ R_G^* &= R_E \parallel R_G & G_L^* &= \frac{1}{R_L^*} \end{align*} \end{minipage}% \subsubsection*{Betriebsparameter} $\boxed{v_{B1} = \dfrac{u_a}{u_e}}$ \quad KSA: \textcircled{\raisebox{-1.5pt}{C}}: \quad $(u_e - u_a) \cdot g_{CE} + g_m \cdot u_e - G_L^*\cdot u_a = 0$ \[u_a = \frac{g_{CE} + g_m}{g_{CE} + G_L^*}\cdot u_e \quad \Rightarrow \quad v_{B1} = \frac{g_{CE} + g_m}{g_{CE} + G_L^*} \approx \frac{g_m}{G_L^*}\] $\boxed{r_{eB} = \dfrac{u_e}{i_e}}$ \quad $g_{eB} = \dfrac{i_e}{u_e}$ \[i_e = g_{BE} \cdot u_e + g_m \cdot u_e - (\underbrace{u_a}_{v_{B1}u_e} - u_e)\cdot g_{CE} = (g_{BE} + g_m - (v_{B1}-1)g_{CE}) \cdot u_e \] \[ g_{eB} = g_{BE} + g_m - (v_{B1}-1)g_{CE} = \underbrace{g_{BE}}_{g_m/b} + g_m - \frac{g_mR^*_L \cancel{-1}}{ g_{CE} R^*_L +1}g_{CE} \approx g_m \left(1 + \cancel{\frac 1 b} - \frac{g_mR^*_L }{ g_{CE} R^*_L +1}\right) \] \begin{minipage}{8cm} \[g_e \approx g_m \frac 1 {g_{CE} R_L^* +1}\] \[\boxed{\quad r_e = \frac 1 {g_e} = \frac 1 {g_m} \left(1 + \frac{R_L^*}{r_{CE}}\right) \quad}\] \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(170,110) \put(30,10){\vector(0,1){90}} \put(25,15){\vector(1,0){140}} \put(28,93){\makebox(0,0)[r]{$r_e$}} \put(28,75){\makebox(0,0)[r]{$\frac{2}{g_m}$}} \put(28,50){\makebox(0,0)[r]{$\frac{1}{g_m}$}} \put(155,5){\makebox(0,0)[c]{$R_L^*$}} \put(105,8){\makebox(0,0)[c]{$r_{CE}$}} \multiput(105,14)(0,4){16}{\line(0,1){2}} \multiput(29,75)(4,0){19}{\line(1,0){2}} \put(29,50){\line(1,0){2}} \put(30,50){\line(3,1){85}} \put(130,10){\vector(0,1){90}} \put(133,93){\makebox(0,0)[l]{$r_e/\Omega$}} \multiput(129,50)(0,25){2}{\line(1,0){2}} \put(133,50){\makebox(0,0)[l]{20}} \put(133,75){\makebox(0,0)[l]{40}} \end{picture} \end{minipage}% \newpage \begin{minipage}{8cm} $\boxed{r_{aB} = \left.\dfrac{u_a}{i_a}\right|_{U_G=0} }$ \begin{align*} u_a &= u_e + u_{CE} = u_e + (i_a + g_mu_e)\\ &= (1+g_mr_{CE})u_e + i_a r_{CE}\\[1ex] u_e &= i_a \cdot r_{BE} \parallel R_G^* = \frac{r_{BE}\cdot R^*_G}{r_{BE}+R^*_G}i_a \end{align*} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(200,95) \put(25,0){\line(1,0){10}} \put(30,0){\line(0,1){60}} \multiput(30,60)(40,0){2}{\line(1,0){20}} \multiput(50,55)(0,10){2}{\line(1,0){20}} \multiput(50,55)(20,0){2}{\line(0,1){10}} \put(60,67){\makebox(0,0)[b]{$R_G^*$}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(85,20)(10,0){2}{\line(0,1){20}} \multiput(85,20)(0,20){2}{\line(1,0){10}} \put(85,0){\line(1,0){10}} \put(97,30){\makebox(0,0)[l]{$r_{BE}$}} \qbezier(85,55)(75,30)(85,5) \put(85,5){\vector(1,-3){0}} \put(77,30){\makebox(0,0)[r]{$u_{e}$}} \multiput(90,60)(40,0){2}{\line(1,0){20}} \multiput(110,60)(10,-10){2}{\line(1,1){10}} \multiput(110,60)(10,10){2}{\line(1,-1){10}} \put(120,50){\line(0,1){20}} \put(110,48){\vector(1,0){20}} \put(120,38){\makebox(0,0)[b]{$g_m u_{e}$}} \multiput(90,60)(60,0){2}{\circle*2} \multiput(90,60)(60,0){2}{\line(0,1){20}} \multiput(90,80)(40,0){2}{\line(1,0){20}} \multiput(110,75)(0,10){2}{\line(1,0){20}} \multiput(110,75)(20,0){2}{\line(0,1){10}} \put(120,88){\makebox(0,0)[b]{$r_{CE}$}} \put(150,30){\circle{20}} \multiput(150,0)(0,40){2}{\line(0,1){20}} \put(140,30){\line(1,0){20}} \put(145,0){\line(1,0){10}} \put(160,30){\makebox(0,0)[l]{$\Big\uparrow i_a$}} \qbezier(175,55)(185,30)(175,5) \put(175,5){\vector(-1,-3){0}} \put(182,30){\makebox(0,0)[l]{$u_a$}} \put(80,63){E} \put(153,63){C} \put(93,3){B} \end{picture} \end{minipage}% \bigskip \[r_{aB} = (\cancel 1 + g_m\cdot r_CE) \cdot \frac{r_{BE} R_G^*}{r_{BE}+R_G^*} + r_{CE} \approx r_{CE} \left(1 + b\frac{\frac{R_G^*}{r_{BE}}}{1+\frac{R_G^*}{r_{BE}}}\right)\] mit $g_m \cdot r_{CE} = \frac{I_{CA}}{U_T}\frac{U_Y}{I_{CA}} = \frac{U_Y}{U_T} \gg 1$ und $g_m \cdot r_{BE} = \frac{I_{CA}}{U_T} \frac{U_T}{I_B} = B_F \approx b$. \bigskip \begin{minipage}{8cm} \center \begin{picture}(200,140) \setlength{\unitlength}{1.25pt} % groesse flasch eingeschaetzt... \put(30,10){\vector(0,1){90}} \put(25,15){\vector(1,0){125}} \put(28,91){\makebox(0,0)[r]{$\frac{r_{aB}}{r_{CE}}$}} \put(145,05){\makebox(0,0)[r]{$\frac{R_G^*}{r_{BE}}$}} \multiput(55,14)(0,4){9}{\line(0,1){2}} \multiput(29,48)(4,0){7}{\line(1,0){2}} \put(29,25){\line(1,0){2}} \multiput(29,70)(4,0){30}{\line(1,0){2}} \put(27,25){\makebox(0,0)[r]{$1$}} \put(27,48){\makebox(0,0)[r]{$\scriptstyle (1 + \frac{b}{2})$}} \put(27,70){\makebox(0,0)[r]{$1 + b$}} \qbezier(30,25)(60,69)(145,69) \put(70,45){$r_{aB} > r_{CE}$} \end{picture} \end{minipage}% \begin{minipage}{8cm} \paragraph{Anwendung:} Stromquelle \center \begin{picture}(100,100) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(25,0){\line(1,0){10}} \put(65,0){\line(1,0){10}} \put(30,60){\line(1,0){30}} \put(40,60){\line(1,0){20}} \put(60,50){\thicklines\line(0,1){20}} \put(60,60){\line(1,1){10}} \put(60,60){\line(1,-1){10}} \put(68,52){\vector(1,-1){0}} \put(70,0){\line(0,1){20}} \put(70,40){\line(0,1){10}} \multiput(65,20)(10,0){2}{\line(0,1){20}} \multiput(65,20)(0,20){2}{\line(1,0){10}} \put(20,30){\makebox(0,0)[r]{$u_{ref}\Big\downarrow$}} \put(77,30){\makebox(0,0)[l]{$R_E$}} \put(70,70){\line(0,1){20}} \put(70,91){\circle{2}} \put(74,90){\vector(0,-1){15}} \put(78,82.5){\makebox(0,0)[l]{$i$}} \end{picture} \end{minipage}% \section{Kollektorstufe (Emitterfolger), Drainstufe (Sourcefolger)} \subsection{Gro"ssignalverhalten} \begin{minipage}{5cm} \begin{picture}(100,120) \put(24,0){\line(1,0){10}} \put(65,0){\line(1,0){10}} \put(30,70){\line(1,0){30}} \put(40,70){\line(1,0){20}} \put(60,60){\thicklines\line(0,1){20}} \put(60,70){\line(1,1){10}} \put(60,70){\line(1,-1){10}} \put(68,62){\vector(1,-1){0}} \put(70,0){\line(0,1){15}} \put(70,35){\line(0,1){25}} \multiput(65,15)(10,0){2}{\line(0,1){20}} \multiput(65,15)(0,20){2}{\line(1,0){10}} \put(28,35){\makebox(0,0)[r]{$U_{e}$}} \put(29,65){\vector(0,-1){60}} \put(29,70){\circle2} \put(63,25){\makebox(0,0)[r]{$R_E$}} \put(85,25){\makebox(0,0)[l]{$U_a$}} \put(70,80){\line(0,1){20}} \put(70,101){\circle{2}} \put(70,105){\makebox(0,0)[b]{$U_{0C}$}} \qbezier(50,65)(50,55)(65,55) \put(65,55){\vector(1,0){0}} \put(58,55){\makebox(0,0)[tr]{$U_{BE}$}} \qbezier(80,12.5)(85,22.5)(80,37.5) \put(80,12.5){\vector(-1,-3){0}} \put(75,60){\vector(0,-1){15}} \put(78,52.5){\makebox(0,0)[l]{$I_E$}} \end{picture} \end{minipage}% \begin{minipage}{11cm} \paragraph{Transferkennlinie:} $U_a = f(U_e)$ \[U_a = R_E \cdot I_E = R_E (I_C + I_B) = \frac{B+1}{B} \cdot I_C \cdot R_E\] \[I_C = I_S \cdot e^{\frac{U_{BE}}{U_T}}, \quad U_{BE} = U_e - U_a\] \[U_a = \frac{B+1}{B} \cdot I_S \cdot R_E \cdot e^{\frac{U_e-U_a}{U_T}}\] kann nicht nach $U_a$ aber nach $U_e$ aufgel"ost werden. \end{minipage}% \bigskip \bigskip \begin{minipage}{10cm} \[\Rightarrow \quad U_e = U_a + \underbrace{U_T \mathop\mathrm{ln}\,\frac{U_a}{\frac{B+1}{B}R_E I_s}}_{ U_{BE}} \qquad\] \begin{description} \item[Beispiel:] $U_a = 1\ldots 10\,\mathrm V$, $R_E = 1\,\mathrm k\Omega$, $I_S = 100\,\mathrm{pA}$ $\Rightarrow U_{BE} = 0{,}48\mathrm V \ldots 0{,}55 \mathrm V \to U_e \approx U_a + \underbrace{U_{BE0}}_{0{,}5\,\mathrm V}$ \end{description} \end{minipage}% \begin{minipage}{6cm} \begin{picture}(130,100) \put(20,10){\vector(0,1){90}} \put(15,15){\vector(1,0){115}} \put(18,90){\makebox(0,0)[r]{$U_a$}} \put(125,8){\makebox(0,0)[r]{$U_e$}} \put(20,15){\line(1,1){70}} \thicklines \qbezier(20,15)(35,20)(45,30) \qbezier(45,30)(45,30)(95,80) \end{picture} \end{minipage}% \bigskip $\Rightarrow$ Die Ausgangsspannung folgt der Eingangsspannung im Abstand von $U_{BE0}$ \subsubsection*{Verst"arkung:} \begin{minipage}{9cm} \[ \frac 1 v = \left.\frac{dU_e}{dU_a}\right|_A = 1 + \frac{U_T}{U_aA}, \quad v = \frac{1}{1+\frac{U_T}{U_{aA}}} \approx 1 - \frac{U_T}{U_{aA}}\] \end{minipage}% \begin{minipage}{7cm} \center \begin{tabular}{c|c|c|c} $\frac{U_{aA}}{v}$ & 1 & 5 & 10 \\ \hline $1-v$ & 0{,}03 & 0{,}006 & 0{,}0003 \end{tabular} \end{minipage}% \bigskip $\Rightarrow$ \textbf{Transferkennlinie} $\to$ Gerade, $v \approx 1$ \subsubsection*{Signalverarbeitungsstruktur} \begin{center} \begin{picture}(260,60) \put(0,40){\vector(1,0){20}} \put(25,40){\circle{10}} \put(30,40){\vector(1,0){30}} \put(60,25){\line(0,1){30}} \multiput(60,25)(0,30){2}{\line(1,0){25}} \qbezier(85,25)(95,25)(100,40) \qbezier(85,55)(95,55)(100,40) \put(80,40){\makebox(0,0){$\scriptstyle I_S\cdot e^{\frac{U_{BE}}{U_T}}$}} \multiput(100,40)(60,0){2}{\vector(1,0){30}} \multiput(130,25)(0,30){2}{\line(1,0){30}} \multiput(130,25)(30,0){2}{\line(0,1){30}} \put(145,40){\makebox(0,0){$\frac{B+1}{B}$}} \multiput(190,25)(0,30){2}{\line(1,0){30}} \multiput(190,25)(30,0){2}{\line(0,1){30}} \put(205,40){\makebox(0,0){$R_E$}} \put(220,40){\line(1,0){40}} \put(240,40){\circle*2} \put(240,40){\line(0,-1){35}} \put(25,5){\line(1,0){215}} \put(25,5){\vector(0,1){30}} \put(10,45){\makebox(0,0)[b]{$U_e$}} \put(45,45){\makebox(0,0)[b]{$U_{BE}$}} \put(115,45){\makebox(0,0)[b]{$I_{C}$}} \put(175,45){\makebox(0,0)[b]{$I_{E}$}} \put(250,45){\makebox(0,0)[b]{$U_{a}$}} \put(20,30){\makebox(0,0)[r]{$-$}} \end{picture} \end{center} \subsection{Kleinsignalverhalten} \begin{minipage}{7cm} \center \begin{picture}(190,125) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$U_G\Big\downarrow$}} \put(30,60){\line(1,0){10}} \put(70,56){\vector(0,-1){56}} \put(68,30){\makebox(0,0)[r]{$u_e$}} \multiput(40,55)(0,10){2}{\line(1,0){20}} \multiput(40,55)(20,0){2}{\line(0,1){10}} \put(50,68){\makebox(0,0)[b]{$R_G$}} \multiput(60,60)(22,0){2}{\line(1,0){18}} \multiput(78,53)(4,0){2}{\thicklines\line(0,1){14}} \multiput(100,60)(0,40){2}{\line(0,1){20}} %RB \multiput(95,80)(10,0){2}{\line(0,1){20}} \multiput(95,80)(0,20){2}{\line(1,0){10}} \put(100,60){\circle*2} \put(100,60){\line(1,0){20}} \put(120,50){\thicklines\line(0,1){20}} \put(120,60){\line(1,1){10}} \put(120,60){\line(1,-1){10}} \put(128,52){\vector(1,-1){0}} \put(130,70){\line(0,1){50}} \put(130,00){\line(0,1){20}} \put(130,40){\line(0,1){10}} \multiput(125,20)(10,0){2}{\line(0,1){20}} \multiput(125,20)(0,20){2}{\line(1,0){10}} \put(125,00){\line(1,0){10}} \put(25,00){\line(1,0){10}} \multiput(130,50)(22,0){2}{\line(1,0){18}} \multiput(148,43)(4,0){2}{\thicklines\line(0,1){14}} \put(170,0){\line(0,1){20}} \multiput(165,20)(10,0){2}{\line(0,1){20}} \multiput(165,20)(0,20){2}{\line(1,0){10}} \put(170,40){\line(0,1){10}} \put(165,0){\line(1,0){10}} \put(100,120){\line(1,0){50}} \put(151,120){\circle{2}} \multiput(130,50)(0,70){2}{\circle*{2}} \put(155,120){\makebox(0,0)[l]{$U_{0C}$}} \put(093,90){\makebox(0,0)[r]{$R_{2}$}} \put(163,30){\makebox(0,0)[r]{$R_{L}$}} \put(70,60){\color{white}\circle*{2}} % *hust* \put(70,60){\circle{2}} % *kotz* \multiput(100,0)(0,40){2}{\line(0,1){20}} \multiput(95,20)(10,0){2}{\line(0,1){20}} \multiput(95,20)(0,20){2}{\line(1,0){10}} \put(95,0){\line(1,0){10}} \put(094,30){\makebox(0,0)[r]{$R_{1}$}} \put(124,30){\makebox(0,0)[r]{$R_{E}$}} \qbezier(175,50)(185,30)(175,10) \put(175,10){\vector(-1,-3){0}} \put(182,30){\makebox(0,0)[l]{$u_{a}$}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \begin{picture}(200,80) \put(30,0){\line(0,1){60}} \put(30,30){\circle{20}} \put(25,0){\line(1,0){10}} \put(20,30){\makebox(0,0)[r]{$U_G^*\Big\downarrow$}} \multiput(30,60)(35,0){2}{\line(1,0){15}} \multiput(45,55)(0,10){2}{\line(1,0){20}} \multiput(45,55)(20,0){2}{\line(0,1){10}} \put(55,68){\makebox(0,0)[b]{$R_G^*$}} \put(81,60){\circle2} \put(81,68){\makebox(0,0)[b]{B}} \put(120,5){\makebox(0,0)[b]{C}} \put(132,68){\makebox(0,0)[b]{E}} \multiput(82,60)(35,0){2}{\line(1,0){15}} \multiput(97,55)(0,10){2}{\line(1,0){20}} \multiput(97,55)(20,0){2}{\line(0,1){10}} \put(107,68){\makebox(0,0)[b]{$r_{BE}$}} \multiput(132,0)(0,40){2}{\line(0,1){20}} \multiput(132,20)(-10,10){2}{\line(1,1){10}} \multiput(132,20)(10,10){2}{\line(-1,1){10}} \put(122,30){\line(1,0){20}} \put(124,30){\makebox(0,0)[r]{$g_mu_{BE}\!\Big\uparrow $}} \put(81,56){\vector(0,-1){56}} \put(80,30){\makebox(0,0)[r]{$u_e$}} \put(127,0){\line(1,0){10}} \put(132,60){\circle*2} \put(132,60){\line(1,0){88}} \multiput(10,0)(30,0){3}{ \multiput(150,0)(0,40){2}{\line(0,1){20}} \multiput(145,20)(10,0){2}{\line(0,1){20}} \multiput(145,20)(0,20){2}{\line(1,0){10}} \put(145,0){\line(1,0){10}} } \put(160,60){\circle*{2}} \put(190,60){\circle*{2}} \put(166,30){\makebox(0,0)[l]{$r_{CE}$}} \put(196,30){\makebox(0,0)[l]{$R_{E}$}} \put(226,30){\makebox(0,0)[l]{$R_{L}$}} \qbezier(237.5,5)(247.5,30)(237.5,55) \put(237.5,5){\vector(-1,-3){0}} \put(245,30){\makebox(0,0)[l]{$u_a$}} \put(190,75){\makebox(0,0){$\overbrace{\hspace{2.5cm}}^{\displaystyle R_L^*}$}} \end{picture} \end{minipage}% \bigskip \bigskip \[U_G^* = \frac{R_1\parallel R_2}{R_G + R_1\parallel R_2} U_G, \quad R_G^* = R_1\parallel R_2 \parallel R_G, \quad R_L^* = r_{CE}\parallel R_L \parallel R_E\] \subsubsection*{Signalflu"sgraph} \begin{minipage}{8cm} \center \begin{picture}(200,75) \put(0,-15){ % verschaetzt, oben links angefangen :) \put(26,60){\makebox(0,0)[r]{$u_e$}} \put(30,60){\circle*3} \put(30,60){\vector(1,0){22}} \put(50,64){\makebox(0,0)[b]{$1$}} \put(30,60){\line(1,0){40}} \put(70,60){\circle*3} \put(70,66){\makebox(0,0)[b]{$u_{BE}$}} \qbezier(70,60)(80,70)(110,70) \put(93,69){\vector(4,1){0}} \put(90,73){\makebox(0,0)[b]{$\frac{1}{r_{BE}}$}} \put(110,70){\circle*3} \put(110,74){\makebox(0,0)[b]{$i_{e}$}} \qbezier(110,70)(140,70)(150,60) \put(132,68){\vector(4,-1){0}} \put(132,73){\makebox(0,0)[b]{$1$}} \put(150,60){\circle*3} \put(150,64){\makebox(0,0)[b]{$i_{E}$}} \put(150,60){\line(1,0){40}} \put(150,60){\vector(1,0){22}} \put(170,64){\makebox(0,0)[b]{$R_{L}^*$}} \put(190,60){\circle*3} \put(194,60){\makebox(0,0)[l]{$u_{a}$}} \qbezier(70,60)(80,50)(110,50) \put(93,51){\vector(4,-1){0}} \put(90,47){\makebox(0,0)[t]{${g_{m}}$}} \put(110,50){\circle*3} \put(110,46){\makebox(0,0)[t]{$i_{C}$}} \qbezier(110,50)(140,50)(150,60) \put(132,52){\vector(4,1){0}} \put(132,47){\makebox(0,0)[t]{$1$}} \qbezier(70,60)(80,25)(130,25) \qbezier(130,25)(180,25)(190,60) \put(128,25){\vector(-1,0){0}} \put(130,22){\makebox(0,0)[t]{$-1$}} } \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{align*} u_{BE} &= u_e - u_a & i_e &= \frac{u_e}{r_{BE}} = i_B \\[1ex] i_E & = i_e + i_C & u_a & = R_L^* \cdot i_E \end{align*} \end{minipage}% \subsubsection*{Verst"arkung} \[v_{B1} = \frac{u_a}{u_e} = \frac{\left(g_m + \frac 1{r_BE}\right)R^*_L}{1+ \left(g_m + \frac 1{r_BE}\right)R^*_L} = \frac{\left(1 + \frac 1 b\right) g_m R_L^*}{1+ \left(1 + \frac 1 b\right) g_m R_L^*} \approx \frac{g_mR_L^*}{1+g_mR^*_L} \approx 1\] \subsubsection*{Eingangswiderstand} \[g_{eB} = \frac{i_e}{u_e} = \frac{\frac 1{r_BE}}{1+\left(\frac 1{r_BE} + g_m\right)R^*_L}, \quad r_{eB} = \frac{1}{g_{eB}} = r_{BE} + (1 + \overbrace{g_mr_{BE}}^{b}) R_L^* \approx r_{BE} + b \cdot r_L^* \] \subsubsection*{Ausgangswiderstand} \begin{minipage}{3cm} \center \[r_{aB} = \left.\frac{u_a}{i_a}\right|_{U_G^*=0}\] \end{minipage}% \begin{minipage}{6cm} \center \begin{picture}(130,60) \put(2,45){\makebox(0,0)[l]{$i_a$}} \multiput(15,45)(50,0){3}{\circle*3} \multiput(15,45)(50,0){2}{\line(1,0){50}} \multiput(15,45)(50,0){2}{\vector(1,0){27}} \put(65,48){\makebox(0,0)[b]{$i_a$}} \put(40,48){\makebox(0,0)[b]{$1$}} \put(90,48){\makebox(0,0)[b]{$\scriptstyle R_G^*+r_{BE}$}} \put(120,45){\makebox(0,0)[l]{$u_a$}} \put(65,20){\line(0,1){25}} \put(65,20){\vector(0,1){14.5}} \put(65,20){\circle*3} \qbezier(65,20)(100,20)(115,45) \put(93,25){\vector(-2,-1){0}} \put(65,15){\makebox(0,0)[t]{$u_{BE}$}} \put(95,26){\makebox(0,0)[tl]{$\frac{-r_{BE}}{R_G^*+r_{BE}}$}} \end{picture} \end{minipage}% \begin{minipage}{7cm} \[u_a = i_1 (R_G^*+r_{BE}) \qquad i_1 = g_mu_{BE} + i_a\] \[u_{BE} = - \frac{r_{BE}}{R_G^*+r_{BE}}u_a, \qquad i_1 = -i_e\] \end{minipage}% \bigskip \bigskip \[r_{aB} = \frac{R_G^* + r_{BE}}{11 + r_{BE} g_m} = \frac{R_G^* + r_{BE}}{1+b} \approx \frac 1 {g_m} + \frac{R_G^*}{b}\] \section{Stromspiegel} \subsection{Grundmodell} \begin{minipage}{5cm} \begin{picture}(130,60) \put(30,0){\line(1,0){60}} \put(60,0){\line(0,1){10}} \multiput(40,10)(0,25){2}{\line(1,0){40}} \multiput(40,10)(40,0){2}{\line(0,1){25}} \multiput(30,45)(45,0){2}{\line(1,0){15}} \multiput(45,35)(30,0){2}{\line(0,1){10}} \multiput(29,0)(0,45){2}{\circle2} \put(60,0){\circle*2} \multiput(91,0)(0,45){2}{\circle2} \multiput(45,30)(30,0){2}{\vector(0,-1){15}} \put(50,22.5){\vector(1,0){20}} \multiput(29,40)(62,0){2}{\vector(0,-1){35}} \put(27,22.5){\makebox(0,0)[r]{$U_1$}} \put(93,22.5){\makebox(0,0)[l]{$U_2$}} \put(5,45){\vector(1,0){20}} \put(15,47){\makebox(0,0)[b]{$I_1$}} \put(115,45){\vector(-1,0){20}} \put(105,47){\makebox(0,0)[b]{$I_2$}} \end{picture} \end{minipage}% \begin{minipage}{11cm} \[I_2 = M \cdot I_1 \qquad M: \text{ Spiegelverh"altnis, ganzzahlig}\] \begin{minipage}{0.5\textwidth} \begin{empheq}[innerbox=\fbox]{align*} \quad U_1 &= H_1 (I_1,\,U_2) \quad\\ \quad I_2 &= H_2 (I_1,\, U_2) \end{empheq} \end{minipage}% \begin{minipage}{0.5\textwidth} Hybriddarstellung \end{minipage}% \end{minipage}% \subsubsection*{Kleinsignalmodell} \begin{minipage}{8cm} \center \begin{picture}(150,75) \put(0,0){\line(1,0){165}} \multiput(-1,0)(0,60){2}{\circle2} \multiput(166,0)(0,60){2}{\circle2} \multiput(140,0)(0,60){2}{\circle*2} \multiput(0,60)(40,0){2}{\line(1,0){20}} \multiput(20,55)(0,10){2}{\line(1,0){20}} \multiput(20,55)(20,0){2}{\line(0,1){10}} \multiput(60,20)(-10,10){2}{\line(1,1){10}} \multiput(60,20)(10,10){2}{\line(-1,1){10}} \put(60,00){\line(0,1){60}} \put(50,30){\makebox(0,0)[r]{$\nu\cdot u_2\Big\downarrow$}} \multiput(90,0)(0,40){2}{\line(0,1){20}} \multiput(90,20)(-10,10){2}{\line(1,1){10}} \multiput(90,20)(10,10){2}{\line(-1,1){10}} \put(80,30){\line(1,0){20}} \put(100,30){\makebox(0,0)[l]{$\Big\downarrow mi_1$}} \put(90,60){\line(1,0){75}} \multiput(140,0)(0,40){2}{\line(0,1){20}} \multiput(135,20)(10,0){2}{\line(0,1){20}} \multiput(135,20)(0,20){2}{\line(1,0){10}} \put(30,68){\makebox(0,0)[b]{$r_e$}} \put(148,30){\makebox(0,0)[l]{$g_a$}} \multiput(60,0)(30,0){2}{\circle*2} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{align*} u_1 &= r_e \cdot i_1 + \nu \cdot u_2\\ i_2 &= m \cdot i_1 + g_a \cdot u_2 \end{align*} \vspace{-1.2cm} \[\underbrace{\hphantom{\ i_2 = m \cdot i_1\ }}_{\displaystyle\text{Hauptsteuergleichung}} \hphantom{+ g_a \cdot u_2}\] \end{minipage}% \subsection{Grundschaltung} \begin{minipage}{7cm} \center \begin{picture}(175,70) \put(30,60){\circle2} \put(30,55){\vector(0,-1){55}} \put(28,30){\makebox(0,0)[r]{$U_1$}} \put(15,62){\makebox(0,0)[b]{$I_1$}} \put(5,60){\vector(1,0){20}} \put(70,30){\line(1,0){30}} \put(70,20){\thicklines\line(0,1){20}} \put(70,30){\line(-1,1){10}} \put(70,30){\line(-1,-1){10}} \put(62,22){\vector(-1,-1){0}} \put(90,30){\line(1,0){20}} \put(110,20){\thicklines\line(0,1){20}} \put(110,30){\line(1,1){10}} \put(110,30){\line(1,-1){10}} \put(118,22){\vector(1,-1){0}} \put(31,60){\line(1,0){59}} \put(90,30){\line(0,1){30}} \put(60,40){\line(0,1){20}} \put(55,55){\vector(0,-1){15}} \put(52,47.5){\makebox(0,0)[r]{$I_{C1}$}} \put(125,55){\vector(0,-1){15}} \put(128,47.5){\makebox(0,0)[l]{$I_{C2}$}} \put(95,60){\vector(0,-1){20}} \put(98,50){\makebox(0,0)[l]{$2I_{B}$}} \put(95,25){\vector(1,0){10}} \put(100,23){\makebox(0,0)[t]{$\scriptstyle I_{B}$}} \put(85,25){\vector(-1,0){10}} \put(80,23){\makebox(0,0)[t]{$\scriptstyle I_{B}$}} \put(60,60){\circle*2} \multiput(60,00)(60,0){2}{\line(0,1){20}} \multiput(55,00)(60,0){2}{\line(1,0){10}} \put(90,30){\circle*2} \put(120,40){\line(0,1){20}} \put(120,60){\line(1,0){30}} \put(151,60){\circle{2}} \put(151,55){\vector(0,-1){55}} \put(154,30){\makebox(0,0)[l]{$U_2$}} \put(175,60){\vector(-1,0){20}} \put(165,62){\makebox(0,0)[b]{$I_2$}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \[I_B = \frac{I_S}{B_F}\cdot e^{\frac{U_{BE}}{U_T}}, \quad I_{C1/2} = I_S \cdot e^{\frac{U_{BE}}{U_T}}\left(1+\frac{U_{1/2}}{U_Y}\right) \] \[I_1 = I_{C1} + 2I_B = I_S \cdot e^{\frac{U_{BE}}{U_T}}\left(1+ \frac{U_{1}}{U_Y} + \frac 2 {B_F}\right)\] \end{minipage}% \bigskip \[I_2 = I_{C2} = I_S \cdot e^{\frac{U_{BE}}{U_T}}\left(1+\frac{U_2}{U_Y}\right) \qquad M = \frac{I_2}{I_1} \approx \frac{1 + \frac{U_2}{U_Y}}{1+\cancel{\frac{U_{BE0}}{U_Y}}+\frac{2}{B_F}} \approx \frac{1 + \frac{U_2}{U_Y}}{1+\frac{2}{B_F}} \] \subsection{Verbesserung der Symmetrie (Bipolar-Stromspiegel)} \begin{minipage}{7cm} \center \begin{picture}(175,95) \put(30,60){\circle2} \put(30,55){\vector(0,-1){55}} \put(28,30){\makebox(0,0)[r]{$U_1$}} \put(15,62){\makebox(0,0)[b]{$I_1$}} \put(5,60){\vector(1,0){20}} \put(70,30){\line(1,0){30}} \put(70,20){\thicklines\line(0,1){20}} \put(70,30){\line(-1,1){10}} \put(70,30){\line(-1,-1){10}} \put(62,22){\vector(-1,-1){0}} \put(90,30){\line(1,0){20}} \put(110,20){\thicklines\line(0,1){20}} \put(110,30){\line(1,1){10}} \put(110,30){\line(1,-1){10}} \put(118,22){\vector(1,-1){0}} \put(31,60){\line(1,0){49}} \put(90,30){\line(0,1){20}} \put(90,70){\line(0,1){10}} \put(90,81){\circle{2}} \put(90,84){\makebox(0,0)[b]{$U_{0C}$}} \put(60,40){\line(0,1){20}} \put(55,55){\vector(0,-1){15}} \put(52,47.5){\makebox(0,0)[r]{$I_{C1}$}} \put(125,55){\vector(0,-1){15}} \put(128,47.5){\makebox(0,0)[l]{$I_{C2}$}} \put(95,50){\vector(0,-1){15}} \put(97,43){\makebox(0,0)[l]{$\scriptstyle 2I_{B1}$}} \put(95,25){\vector(1,0){10}} \put(100,23){\makebox(0,0)[t]{$\scriptstyle I_{B2}$}} \put(85,25){\vector(-1,0){10}} \put(80,23){\makebox(0,0)[t]{$\scriptstyle I_{B1}$}} \put(60,60){\circle*2} \multiput(60,00)(60,0){2}{\line(0,1){20}} \multiput(55,00)(60,0){2}{\line(1,0){10}} \put(90,30){\circle*2} \put(120,40){\line(0,1){20}} \put(120,60){\line(1,0){30}} \put(151,60){\circle{2}} \put(151,55){\vector(0,-1){55}} \put(154,30){\makebox(0,0)[l]{$U_2$}} \put(175,60){\vector(-1,0){20}} \put(165,62){\makebox(0,0)[b]{$I_2$}} \put(60,60){\line(1,0){20}} \put(80,50){\thicklines\line(0,1){20}} \put(80,60){\line(1,1){10}} \put(80,60){\line(1,-1){10}} \put(88,52){\vector(1,-1){0}} \put(90,60){\makebox(0,0)[l]{3}} \put(120,30){\makebox(0,0)[l]{2}} \put(60,30){\makebox(0,0)[r]{1}} \multiput(90,00)(0,20){2}{\line(0,1){10}} \multiput(87.5,0)(0,10){3}{\line(1,0){5}} \multiput(87.5,10)(5,0){2}{\line(0,1){10}} \put(60,65){\vector(1,0){15}} \put(67.5,67){\makebox(0,0)[b]{$I_{B3}$}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \[B_1 = B_2 = B, \quad I_{B1} = I_{B2}\] \[M = \frac{I_2}{I_1} = \frac{1}{1+\frac{2}{B(1+B_3)}} \approx 1 - \frac{2}{B(1+B_3)}\] \end{minipage}% \bigskip \paragraph{Zahlenbeispiel:} $B_1 = B_2 = 100$, $B_3 = 50$, $M = 1-0{,}4\cdot 10^{-3}$ \subsection{Strombank} \begin{picture}(175,70) \put(30,60){\circle2} \put(15,62){\makebox(0,0)[b]{$I_1$}} \put(5,60){\vector(1,0){20}} \put(70,30){\line(1,0){30}} \put(70,20){\thicklines\line(0,1){20}} \put(70,30){\line(-1,1){10}} \put(70,30){\line(-1,-1){10}} \put(62,22){\vector(-1,-1){0}} \put(90,30){\line(1,0){100}} \multiput(0,0)(40,0){3}{ \put(110,20){\thicklines\line(0,1){20}} \put(110,30){\line(1,1){10}} \put(110,30){\line(1,-1){10}} \put(118,22){\vector(1,-1){0}} \put(120,40){\line(0,1){20}} \put(120,61){\circle2} \put(123,53){\makebox(0,0)[l]{$\Big\downarrow I_1$}} \put(120,00){\line(0,1){20}} \put(115,00){\line(1,0){10}} } \put(31,60){\line(1,0){59}} \put(90,30){\line(0,1){30}} \put(60,40){\line(0,1){20}} \put(60,60){\circle*2} \put(60,00){\line(0,1){20}} \put(55,00){\line(1,0){10}} \put(90,30){\circle*2} \put(220,30){\makebox(0,0){$\cdots$}} \end{picture} \subsection{Verbesserung des Ausgangsverhaltens (Wilson-Stromspiegel)} \begin{minipage}{7cm} \begin{picture}(197,100) \put(0,40){\makebox(0,0)[l]{$I_1\!\Big\uparrow$}} \multiput(25,0)(0,50){2}{\line(0,1){30}} \put(25,40){\circle{20}} \put(15,40){\line(1,0){20}} \put(25,80){\line(1,0){40}} \put(65,0){\line(0,1){10}} \put(65,30){\line(0,1){50}} \multiput(65,10)(0,20){2}{\line(1,0){10}} \put(75,10){\line(0,1){20}} \put(77,10){\thicklines\line(0,1){20}} \multiput(120,0)(0,70){2}{\line(0,1){10}} \put(120,30){\line(0,1){20}} \multiput(110,10)(0,20){2}{\line(1,0){10}} \multiput(110,50)(0,20){2}{\line(1,0){10}} \multiput(110,10)(0,40){2}{\line(0,1){20}} \multiput(108,10)(0,40){2}{\thicklines\line(0,1){20}} \put(65,60){\line(1,0){43}} \put(65,60){\circle*2} \put(77,20){\line(1,0){31}} \put(100,20){\circle*2} \put(100,20){\line(0,1){20}} \put(100,40){\line(1,0){20}} \put(120,40){\circle*2} \put(20,0){\line(1,0){10}} \multiput(60,0)(55,0){2}{\line(1,0){10}} \put(120,80){\line(1,0){30}} \put(150,00){\line(0,1){80}} \put(150,40){\circle{20}} \put(145,0){\line(1,0){10}} \put(160,40){\makebox(0,0)[l]{$\Big\downarrow U_2$}} \put(155,70){\makebox(0,0)[l]{$\Big\uparrow I_2$}} \put(65,20){\makebox(0,0)[r]{$1$}} \put(120,20){\makebox(0,0)[l]{$2$}} \put(120,60){\makebox(0,0)[l]{$3$}} \put(39,75){\vector(0,-1){75}} \put(43,40){\makebox(0,0)[l]{$U_1$}} \put(90,09){\makebox(0,0)[l]{$U_3$}} \put(85,18){\vector(0,-1){18}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \begin{picture}(150,100) \put(0,40){\makebox(0,0)[l]{$i\Big\uparrow$}} \put(25,40){\circle{20}} \put(15,40){\line(1,0){20}} \multiput(25,0)(0,50){2}{\line(0,1){30}} \put(25,80){\line(1,0){50}} \put(75,0){\line(0,1){10}} \put(75,30){\line(0,1){50}} \multiput(75,10)(-10,10){2}{\line(1,1){10}} \multiput(75,10)(10,10){2}{\line(-1,1){10}} \put(65,20){\line(1,0){20}} \put(65,20){\makebox(0,0)[r]{$g_mu_3\Big\downarrow$}} \multiput(20,0)(50,0){2}{\line(1,0){10}} \multiput(95,0)(0,40){2}{\line(0,1){20}} \multiput(90,20)(10,0){2}{\line(0,1){20}} \multiput(90,20)(0,20){2}{\line(1,0){10}} \put(103,30){\makebox(0,0)[l]{$g_{DS1}$}} \put(90,0){\line(1,0){10}} \put(75,60){\line(1,0){20}} \put(75,60){\circle*2} \multiput(145,10)(-10,10){2}{\line(1,1){10}} \multiput(145,10)(10,10){2}{\line(-1,1){10}} \multiput(145,50)(-10,10){2}{\line(1,1){10}} \multiput(145,50)(10,10){2}{\line(-1,1){10}} \multiput(145,0)(0,70){2}{\line(0,1){10}} \multiput(135,20)(0,40){2}{\line(1,0){20}} \put(145,30){\line(0,1){20}} \multiput(165,10)(10,0){2}{\line(0,1){20}} \multiput(165,10)(0,20){2}{\line(1,0){10}} \multiput(165,50)(10,0){2}{\line(0,1){20}} \multiput(165,50)(0,20){2}{\line(1,0){10}} \multiput(170,0)(0,70){2}{\line(0,1){10}} \multiput(170,30)(0,15){2}{\line(0,1){5}} \multiput(145,35)(0,10){2}{\line(1,0){25}} \multiput(145,35)(0,10){2}{\circle*2} \put(170,80){\circle*2} \put(145,80){\line(1,0){60}} \multiput(140,0)(25,0){2}{\line(1,0){10}} \put(205,0){\line(0,1){80}} \put(205,40){\circle{20}} \put(215,40){\makebox(0,0)[l]{$\Big\downarrow u_2$}} \put(210,70){\makebox(0,0)[l]{$\Big\uparrow i_2$}} \put(179,60){\makebox(0,0)[l]{$g_{DS3}$}} \put(179,20){\makebox(0,0)[l]{$g_{DS2}$}} \put(200,0){\line(1,0){10}} \put(114,68){\makebox(0,0)[c]{$\scriptstyle g_m\cdot$}} \put(135,60){\makebox(0,0)[r]{$\scriptstyle (u_1-u_3)\!\Big\downarrow$}} \put(135,20){\makebox(0,0)[r]{$g_mu_3\Big\downarrow$}} \put(143,40){\makebox(0,0)[r]{$i_2\!\downarrow$}} \put(80,70){\textcircled{\raisebox{-1.5pt}{1}}} \put(175,85){\textcircled{\raisebox{-1.5pt}{2}}} \put(170,37){\textcircled{\raisebox{-1.5pt}{3}}} \end{picture} \end{minipage}% \bigskip Alle Transistoren f"uhren den gleichen Arbeitspunkt-Strom. $g_{m1} = g_{m2} = g_{m3} = g_m$ \begin{align*} \text{\textcircled{\raisebox{-1.5pt}{1}}} && i_1 - g_{DS1}\cdot u_1 - g_m \cdot u_3 &= 0 &&&&&& \\ % baaaaaaaaaaaaaaaaaaaaah \text{\textcircled{\raisebox{-1.5pt}{2}}} && i_2 - g_{DS3}(u_2-u_3) - g_m(u_1-u_3) &= 0\\ \text{\textcircled{\raisebox{-1.5pt}{3}}} && i_2 - g_{DS2}\cdot u_3 - g_m\cdot u_3 &=0 \end{align*} \textbf{gegeben:} $u_2$, $i_1$ \qquad \textbf{gesucht:} $i_2$, ($u_1$, $u_3$ ebenso unbekannt) \[i_2 = \underbrace{\frac{g_m}{g_{DS1}\left(2+\frac{g_m}{g_{DS1}}\right)}}_{\displaystyle m} \cdot i_1 + \underbrace{\frac{g_{DS3}}{1+\frac{g_m}{g_{DS1}}}}_{\displaystyle g_a}\cdot u_2 \qquad \Rightarrow \qquad m \approx 1 \quad (1\text{\textperthousand}), \qquad g_a \approx \frac{g_{DS3}}{\frac{g_m}{g_{DS1}}} \approx 10^{-1}\,\mathrm S \] \subsection{Widlar-Stromquelle} \begin{minipage}{8cm} \center \begin{picture}(185,110) \put(70,70){\line(1,0){30}} \put(70,60){\thicklines\line(0,1){20}} \put(70,70){\line(-1,1){10}} \put(70,70){\line(-1,-1){10}} \put(62,62){\vector(-1,-1){0}} \put(90,70){\line(1,0){20}} \put(110,60){\thicklines\line(0,1){20}} \put(110,70){\line(1,1){10}} \put(110,70){\line(1,-1){10}} \put(118,62){\vector(1,-1){0}} \put(31,100){\line(1,0){59}} \put(90,70){\line(0,1){30}} \put(60,80){\line(0,1){20}} \put(60,100){\circle*2} \put(120,40){\line(0,1){20}} \put(115,40){\line(1,0){10}} \put(115,20){\line(1,0){10}} \put(115,20){\line(0,1){20}} \put(125,20){\line(0,1){20}} \put(120,00){\line(0,1){20}} \multiput(55,0)(60,0){2}{\line(1,0){10}} \put(60,0){\line(0,1){60}} \put(90,70){\circle*2} \put(120,80){\line(0,1){20}} \put(120,100){\line(1,0){40}} \multiput(30,0)(0,60){2}{\line(0,1){40}} \put(30,50){\circle{20}} \put(20,50){\line(1,0){20}} \put(25,0){\line(1,0){10}} \put(20,50){\makebox(0,0)[r]{$I_1\Big\uparrow$}} \put(60,70){\makebox(0,0)[r]{$1$}} \put(120,70){\makebox(0,0)[l]{$2$}} \qbezier(80,65)(80,50)(65,50) \put(65,50){\vector(-1,0){0}} \put(65,43){\makebox(0,0)[l]{$\scriptstyle U_{BE1}$}} \put(115,43){\makebox(0,0)[r]{$\scriptstyle U_{BE2}$}} \put(113,30){\makebox(0,0)[r]{$R_{E}$}} \qbezier(100,65)(100,50)(115,50) \put(115,50){\vector(1,0){0}} \qbezier(125,50)(135,30)(125,10) \put(125,10){\vector(-1,-2){0}} \put(132,30){\makebox(0,0)[l]{$U_{E2}$}} \put(160,0){\line(0,1){100}} \put(155,0){\line(1,0){10}} \put(160,50){\circle{20}} \put(170,50){\makebox(0,0)[l]{$\Big\downarrow U_{2}$}} \put(125,89){\makebox(0,0)[l]{$\Big\downarrow I_{2}$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[I_1 \approx I_S \cdot e^{\frac{U_{BE1}}{U_T}} \qquad I_2 = I_S\cdot e^{\frac{U_{BE2}}{U_T}}\] \[U_{BE2} = U_{BE1} - U_{E2} \approx U_{BE1} - I_2\cdot R_E\] \[I_2 = I_S \cdot e^{\frac{U_{BE1}- I_2R_E }{U_T}} = \underbrace{I_S\cdot e^{\frac{U_{BE1}}{U_T}}}_{I_1} \cdot e^{\frac{-I_2R_E}{U_T}} \] \[\boxed{\quad I_2 = I_1 \cdot e^{\frac{-I_2R_E}{U_T}}\quad \vphantom{\int_a^a}}\] \end{minipage}% % Trivialgraphik I2/I1 mit veraenderlichem R_E gespart \subsubsection*{Ausgangswiderstand} \begin{minipage}{4cm} \center \begin{picture}(100,75) \put(15,0){\line(1,0){10}} \put(20,0){\line(0,1){60}} \put(20,30){\circle{20}} \multiput(20,60)(40,0){2}{\line(1,0){20}} \multiput(40,55)(0,10){2}{\line(1,0){20}} \multiput(40,55)(20,0){2}{\line(0,1){10}} \put(50,68){\makebox(0,0)[b]{$g_a$}} \put(81,60){\circle{2}} \put(81,55){\vector(0,-1){55}} \put(83,30){\makebox(0,0)[l]{$u_{BE1}$}} \put(92.5,63){\makebox(0,0)[b]{$i_{B1}$}} \put(100,60){\vector(-1,0){15}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \hfill \begin{picture}(130,75) \multiput(30,0)(0,40){2}{\line(0,1){20}} \multiput(30,20)(-10,10){2}{\line(1,1){10}} \multiput(30,20)(10,10){2}{\line(-1,1){10}} \put(20,30){\line(1,0){20}} \put(20,30){\makebox(0,0)[r]{$g_a u_{BE1}\Big\downarrow$}} \put(25,0){\line(1,0){10}} \put(50,0){\line(1,0){10}} \put(30,60){\line(1,0){60}} \put(55,60){\circle*2} \multiput(55,0)(0,40){2}{\line(0,1){20}} \multiput(50,20)(10,0){2}{\line(0,1){20}} \multiput(50,20)(0,20){2}{\line(1,0){10}} \put(91,60){\circle2} \put(91,55){\vector(0,-1){55}} \put(93,30){\makebox(0,0)[l]{$u_{BE1}$}} \put(63,30){\makebox(0,0)[l]{$r_{BE}$}} \end{picture} \end{minipage}% \begin{minipage}{5cm} \[i_{B1} = g_mu_{BE1} + \frac{u_{BE1}}{r_{BE}} \] \[\to \quad g_a = g_m + \frac 1 {r_{BE}}\] \end{minipage}% \bigskip $\to$ Transistor 2 in Basisschaltung \[r_a = (1 + g_{m2} R_E) \cdot r_{CE2} \approx g_m \cdot r_{CE2} \cdot R_{E} = \frac{U_Y}{U_T}\cdot R_E\] \paragraph{Bemessungsbeispiel:} gegeben: $I_1 = 1\,\mathrm{mA}$, $I_2 = 200\,\mu\mathrm A$, $U_Y = 100\,\mathrm V$, gesucht: $R_E$, $r_a$ \[R_E = \frac{U_T}{I_2}\ln\frac{I_1}{I_2} = \frac{25\,\mathrm{mV}}{200\,\mu\mathrm A} \cdot \ln \frac{1\,\mathrm{mA}}{200\,\mu\mathrm A} = 201\,\Omega, \qquad r_a = \frac{U_Y}{U_T}\cdot R_E = 805\,\mathrm k\Omega\] \section{Differenzstufe} \subsection{Gro"ssignalverhalten} \begin{minipage}{8cm} \center \begin{picture}(155,110) \multiput(29,70)(102,0){2}{\circle{2}} \multiput(30,70)(80,0){2}{\line(1,0){20}} \put(50,60){\thicklines\line(0,1){20}} \put(50,70){\line(1,1){10}} \put(50,70){\line(1,-1){10}} \put(58,62){\vector(1,-1){0}} \put(60,60){\line(1,0){40}} \put(80,60){\circle*2} \multiput(80,0)(0,40){2}{\line(0,1){20}} \put(75,0){\line(1,0){10}} \put(70,30){\line(1,0){20}} \put(80,30){\circle{20}} \put(110,60){\thicklines\line(0,1){20}} \put(110,70){\line(-1,1){10}} \put(110,70){\line(-1,-1){10}} \put(102,62){\vector(-1,-1){0}} \multiput(60,80)(40,0){2}{\line(0,1){10}} \multiput(60,91)(40,0){2}{\circle{2}} \multiput(60,110)(40,0){2}{\vector(0,-1){15}} \put(57,102.5){\makebox(0,0)[r]{$I_{C1}$}} \put(103,102.5){\makebox(0,0)[l]{$I_{C2}$}} \put(5,70){\vector(1,0){20}} \put(15,72.5){\makebox(0,0)[b]{$I_{e1}$}} \put(155,70){\vector(-1,0){20}} \put(145,72.5){\makebox(0,0)[b]{$I_{e2}$}} \multiput(29,65)(102,0){2}{\vector(0,-1){65}} \put(27,35){\makebox(0,0)[r]{$U_{e1}$}} \put(135,35){\makebox(0,0)[l]{$U_{e2}$}} \put(90,30){\makebox(0,0)[l]{$\Big\downarrow I_{E}$}} \qbezier(70,55)(60,30)(70,5) \put(70,5){\vector(1,-4){0}} \put(62,30){\makebox(0,0)[r]{$U_{E}$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{description} \item[gesucht:] $I_{C1}$ und $I_{C2}$ un Abh"angigkeit von $U_{e1}$ und $U_{e2}$. \item[Transistormodell:] \[I_C = I_S \cdot e^{\frac{U_{BE}}{U_T}}\] \[I_B = \frac{I_C}{B_F}\] \end{description} \end{minipage}% \bigskip \bigskip \[I_{C1} = I_S \cdot e^{\frac{U_{e1} - U_E}{U_T}}, \qquad I_{C2} = I_S \cdot e^{\frac{U_{e2} - U_E}{U_T}}, \qquad (I_{C1} - I_{C2}) \cdot \underbrace{(1 + \scriptstyle \frac 1 B\displaystyle)}_{\approx 1} = I_E \] \[\begin{array}{rlclll} I_{C1} - I_{C2} &=& I_S \cdot \left( e^{\frac{U_{e1} - U_E}{U_T}}- e^{\frac{U_{e2} - U_E}{U_T}}\right) &= I_S \cdot e^{-\frac{U_E}{U_T}}\left(e^{\frac{U_{e1}}{U_T}} - e^{\frac{U_{e2}}{U_T}}\right) \\[2ex] I_E = I_{C1} + I_{C2} &=& \cdots &=I_S \cdot e^{\frac{- U_E}{U_T}} \left(e^{\frac{U_{e1}}{U_T}} + e^{\frac{U_{e2}}{U_T}}\right) \end{array}\] \[\Rightarrow I_{C1} - I_{C2} = I_E \cdot \frac{e^{\frac{U_{e1}}{U_T}} - e^{\frac{U_{e2}}{U_T}}}{e^{\frac{U_{e1}}{U_T}} + e^{\frac{U_{e2}}{U_T}}} = I_E \cdot \tanh \frac{U_{e1}-U_{e2}}{2U_T} \] \paragraph{Definitionen:} $I_{ad} = I_{C1} - I_{C2}$, $U_{ed} = U_{e1} - U_{e2}$ \[\Rightarrow \quad \boxed{\quad I_{ad} = I_E \cdot \tanh \frac{U_{ed}}{2U_T} \quad}\] $\left.\begin{array}{ll} I_{C1} - I_{C2} &= I_{ad}\\ I_{C1} + I_{C2} &= I_{E} \end{array}\right\} \quad I_{C1} = \frac{I_{ad} + I_E}{2} = \frac{I_E}{2}\left(1 + \tanh\frac{U_{ed}}{2U_T}\right), \ I_{C2} = \frac{I_E - I_{ad}}{2} = \frac{I_E}{2}\left(1 - \tanh\frac{U_{ed}}{2U_T}\right) $ \begin{center} \begin{picture}(150,150) \put(0,75){\vector(1,0){160}} \put(75,0){\vector(0,1){150}} \qbezier(5,75)(65,75)(75,100) \qbezier(75,100)(85,125)(145,125) \qbezier(145,75)(85,75)(75,100) \qbezier(5,125)(65,125)(75,100) \qbezier(5,25)(65,25)(75,75) \qbezier(75,75)(85,125)(145,125) \put(10,128){\makebox(0,0)[b]{$I_{C2}$}} \put(140,128){\makebox(0,0)[b]{$I_{C1}$}} \put(28,40){\makebox(0,0)[b]{$I_{C1}-I_{C2}$}} \put(155,79){\makebox(0,0)[b]{$U_{ed}$}} \put(145,74){\line(0,1){2}} \put(145,72){\makebox(0,0)[t]{$\scriptstyle 100\,\mathrm{mV}$}} \put(5,74){\line(0,1){2}} \put(5,72){\makebox(0,0)[t]{$\scriptstyle -100\,\mathrm{mV}$}} \multiput(6,25)(4,0){18}{\line(1,0){2}} \put(76,25){\makebox(0,0)[l]{$\scriptstyle -1\,\mathrm{mA}$}} \put(72,125){\makebox(0,0)[r]{$\scriptstyle 1\,\mathrm{mA}$}} \put(78,145){\makebox(0,0)[l]{$I_{ad}$}} \multiput(74,125)(4,0){18}{\line(1,0){2}} \end{picture} \end{center} \subsection{Kleinsignalverhalten} \begin{minipage}{6cm} \center \begin{picture}(155,150) \multiput(29,70)(102,0){2}{\circle{2}} \multiput(30,70)(80,0){2}{\line(1,0){20}} \put(50,60){\thicklines\line(0,1){20}} \put(50,70){\line(1,1){10}} \put(50,70){\line(1,-1){10}} \put(58,62){\vector(1,-1){0}} \put(60,60){\line(1,0){40}} \put(80,60){\circle*2} \multiput(80,0)(0,40){2}{\line(0,1){20}} \put(75,0){\line(1,0){10}} \put(70,30){\line(1,0){20}} \put(80,30){\circle{20}} \put(110,60){\thicklines\line(0,1){20}} \put(110,70){\line(-1,1){10}} \put(110,70){\line(-1,-1){10}} \put(102,62){\vector(-1,-1){0}} \multiput(0,0)(40,0){2}{ \multiput(55,90)(0,20){2}{\line(1,0){10}} \multiput(55,90)(10,0){2}{\line(0,1){20}} } \multiput(60,110)(40,0){2}{\line(0,1){10}} \multiput(60,80)(40,0){2}{\line(0,1){10}} \put(60,120){\line(1,0){40}} \put(80,120){\circle*2} \put(80,120){\line(0,1){10}} \put(80,131){\circle2} \put(80,134){\makebox(0,0)[b]{$U_{0C}$}} \multiput(55,130)(50,0){2}{\vector(0,-1){15}} \put(52,122.5){\makebox(0,0)[r]{$i_{C1}$}} \put(108,122.5){\makebox(0,0)[l]{$i_{C2}$}} \put(52,100){\makebox(0,0)[r]{$R_{C}$}} \put(108,100){\makebox(0,0)[l]{$R_{C}$}} \put(5,70){\vector(1,0){20}} \put(15,72.5){\makebox(0,0)[b]{$i_{e1}$}} \put(155,70){\vector(-1,0){20}} \put(145,72.5){\makebox(0,0)[b]{$i_{e2}$}} \multiput(29,65)(102,0){2}{\vector(0,-1){65}} \put(27,35){\makebox(0,0)[r]{$u_{e1}$}} \put(135,35){\makebox(0,0)[l]{$u_{e2}$}} \put(90,30){\makebox(0,0)[l]{$\Big\downarrow i_{E}$}} \qbezier(70,55)(60,30)(70,5) \put(70,5){\vector(1,-4){0}} \put(62,30){\makebox(0,0)[r]{$u_{E}$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \center \begin{picture}(245,150) \put(30,0){\line(0,1){40}} \put(30,20){\circle{20}} \put(25,0){\line(1,0){10}} \multiput(30,40)(40,0){2}{\line(1,0){20}} \multiput(50,35)(0,10){2}{\line(1,0){20}} \multiput(50,35)(20,0){2}{\line(0,1){10}} \put(80,70){\makebox(0,0)[r]{$g_mu_{BE1}\Big\downarrow$}} \put(180,70){\makebox(0,0)[l]{$\Big\downarrow g_mu_{BE2}$}} \multiput(0,0)(80,0){2}{ \multiput(90,40)(0,40){2}{\line(0,1){20}} \multiput(90,60)(-10,10){2}{\line(1,1){10}} \multiput(90,60)(10,10){2}{\line(-1,1){10}} \put(80,70){\line(1,0){20}} \multiput(90,80)(0,40){2}{\line(0,1){20}} \multiput(85,100)(10,0){2}{\line(0,1){20}} \multiput(85,100)(0,20){2}{\line(1,0){10}} \put(85,140){\line(1,0){10}} \put(90,90){\circle*2} } \multiput(90,90)(60,0){2}{\line(1,0){20}} \multiput(111,90)(38,0){2}{\circle2} \multiput(111,87)(38,0){2}{\vector(0,-1){20}} \multiput(106,65)(38,0){2}{\line(1,0){10}} \put(113,78){\makebox(0,0)[l]{$u_{a2}$}} \put(147,78){\makebox(0,0)[r]{$u_{a1}$}} \put(115,90){\vector(1,0){30}} \put(130,93){\makebox(0,0)[b]{$u_{ad}$}} \put(90,40){\line(1,0){80}} \multiput(170,40)(40,0){2}{\line(1,0){20}} \multiput(190,35)(0,10){2}{\line(1,0){20}} \multiput(190,35)(20,0){2}{\line(0,1){10}} \put(230,0){\line(0,1){40}} \put(230,20){\circle{20}} \put(225,0){\line(1,0){10}} \put(82,110){\makebox(0,0)[r]{$R_{C}$}} \put(178,110){\makebox(0,0)[l]{$R_{C}$}} \put(60,47){\makebox(0,0)[b]{$r_{BE}$}} \put(200,47){\makebox(0,0)[b]{$r_{BE}$}} \put(40,20){\makebox(0,0)[l]{$\Big\downarrow u_{e1}$}} \put(220,20){\makebox(0,0)[r]{$u_{e2} \Big\downarrow$}} \multiput(90,40)(80,0){2}{\circle*{2}} \put(37.5,47){\makebox(0,0)[b]{$i_{e1}$}} \put(30,45){\vector(1,0){15}} \put(222.5,47){\makebox(0,0)[b]{$i_{e2}$}} \put(230,45){\vector(-1,0){15}} \multiput(130,0)(0,30){2}{\line(0,1){10}} \multiput(125,10)(10,0){2}{\line(0,1){20}} \multiput(125,10)(0,20){2}{\line(1,0){10}} \put(125,00){\line(1,0){10}} \put(130,40){\circle*2} \put(137,20){\makebox(0,0)[l]{$r_{E}$}} \end{picture} \end{minipage}% \bigskip \bigskip \[u_{a1/2} = -i_{C1/2} \cdot R_C \qquad i_{e1/2} = (u_{e1/2}-u_{E})\cdot g_{BE} \qquad i_{C1/2} = g_m \cdot (u_{e1/2} - u_E) \] \[u_E = r_E (i_{E1} + i_{E2}) = \frac{b+1}{b} \cdot r_E (i_{C1} + i_{C2}) \approx r_E (i_{C1} + i_{C2})\] \subsubsection*{Signalflussplan} \begin{center} \begin{picture}(190,120) \put(27,10){\makebox(0,0)[r]{$u_{e2}$}} \put(27,110){\makebox(0,0)[r]{$u_{e1}$}} \multiput(29,10)(0,100){2}{\circle*{3}} \multiput(80,10)(0,100){2}{\circle*{3}} \multiput(180,10)(0,100){2}{\circle*{3}} \multiput(30,10)(0,100){2}{\line(1,0){150}} \multiput(30,10)(0,100){2}{\vector(1,0){27}} \multiput(80,10)(0,100){2}{\vector(1,0){27}} \multiput(130,10)(0,100){2}{\vector(1,0){27}} \put(80,10){\line(-1,1){40}} \put(80,10){\vector(-1,1){22}} \put(80,110){\line(-1,-1){40}} \put(80,110){\vector(-1,-1){22}} \put(58,93){\makebox(0,0)[r]{$g_{BE}$}} \put(58,28){\makebox(0,0)[r]{$g_{BE}$}} \multiput(40,50)(0,20){2}{\circle*{3}} \put(38,70){\makebox(0,0)[r]{$i_{e1}$}} \put(38,50){\makebox(0,0)[r]{$i_{e2}$}} \put(80,113){\makebox(0,0)[b]{$u_{BE1}$}} \put(80,7){\makebox(0,0)[t]{$u_{BE2}$}} \put(80,10){\line(0,1){100}} \multiput(80,60)(50,0){2}{\circle*{3}} \multiput(130,10)(0,100){2}{\circle*{3}} \put(77,60){\makebox(0,0)[r]{$u_{E}$}} \put(80,60){\line(1,0){50}} \put(130,60){\vector(-1,0){27}} \put(80,60){\vector(0,1){27}} \put(80,60){\vector(0,-1){27}} \put(77,35){\makebox(0,0)[r]{$-1$}} \put(77,85){\makebox(0,0)[r]{$-1$}} \put(105,107){\makebox(0,0)[t]{$g_{m}$}} \put(105,13){\makebox(0,0)[b]{$g_{m}$}} \put(105,63){\makebox(0,0)[b]{$r_{E}$}} \put(130,10){\line(0,1){100}} \put(130,10){\vector(0,1){27}} \put(130,110){\vector(0,-1){27}} \put(133,35){\makebox(0,0)[l]{$1$}} \put(133,85){\makebox(0,0)[l]{$1$}} \put(130,113){\makebox(0,0)[b]{$i_{C1}$}} \put(130,7){\makebox(0,0)[t]{$i_{C2}$}} \put(155,107){\makebox(0,0)[t]{$-R_{C}$}} \put(155,13){\makebox(0,0)[b]{$-R_{C}$}} \put(183,10){\makebox(0,0)[l]{$u_{a2}$}} \put(183,110){\makebox(0,0)[l]{$u_{a1}$}} \end{picture} \end{center} \subsubsection*{Symmetrische Differenzverst"arkung} \begin{minipage}{6cm} \center \begin{picture}(110,50) \multiput(19,10)(0,30){2}{\circle{2}} \multiput(20,10)(0,30){2}{\line(1,0){20}} \put(40,0){\line(0,1){50}} \put(40,0){\line(3,2){37.5}} \put(40,50){\line(3,-2){37.5}} \put(16,10){\makebox(0,0)[r]{$u_{e2}$}} \put(16,25){\makebox(0,0)[r]{$u_{ed}$}} \put(19,35){\vector(0,-1){20}} \put(16,40){\makebox(0,0)[r]{$u_{e1}$}} \put(42,40){\makebox(0,0)[l]{$-$}} \put(42,10){\makebox(0,0)[l]{$+$}} \multiput(55,10)(0,30){2}{\line(1,0){35}} \multiput(91,10)(0,30){2}{\circle{2}} \put(95,10){\makebox(0,0)[l]{$u_{a2}$}} \put(95,25){\makebox(0,0)[l]{$u_{ad}$}} \put(91,35){\vector(0,-1){20}} \put(95,40){\makebox(0,0)[l]{$u_{a1}$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \begin{align*} u_{ad} &= u_{a1} - u_{a2} = - g_m R_C (u_{e1} - u_{e2}) = -g_m R_C u_{ed} \\ &= v_{ds} \cdot u_{ed}\\[1ex] v_{ds} &= -g_m \cdot R_C = \frac{-I_{CA}R_C}{U_T} \approx - \frac{I_E R_C}{2U_T} \end{align*} \end{minipage}% \subsubsection*{Unsymmetrische Differenzverst"arkung} \begin{minipage}{5cm} \center \begin{picture}(110,50) \multiput(19,10)(0,30){2}{\circle{2}} \multiput(20,10)(0,30){2}{\line(1,0){20}} \put(40,0){\line(0,1){50}} \put(40,0){\line(3,2){37.5}} \put(40,50){\line(3,-2){37.5}} \put(16,10){\makebox(0,0)[r]{$u_{e2}$}} \put(16,40){\makebox(0,0)[r]{$u_{e1}$}} \put(42,40){\makebox(0,0)[l]{$-$}} \put(42,10){\makebox(0,0)[l]{$+$}} \put(77.5,25){\line(1,0){20}} \put(98.5,25){\circle{2}} \put(102,25){\makebox(0,0)[l]{$u_{a2}$}} \end{picture} \end{minipage}% \hfill \begin{minipage}{5cm} \center \begin{picture}(120,100) \put(19,50){\circle{2}} \put(20,50){\line(1,0){10}} \put(30,40){\thicklines\line(0,1){20}} \put(30,50){\line(1,1){10}} \put(30,50){\line(1,-1){10}} \put(38,42){\vector(1,-1){0}} \put(40,40){\line(1,0){20}} \multiput(50,0)(0,30){2}{\line(0,1){10}} \put(45,0){\line(1,0){10}} \put(50,40){\circle*2} \put(40,20){\line(1,0){20}} \put(50,20){\circle{20}} \put(81,50){\circle{2}} \put(70,50){\line(1,0){10}} \put(70,40){\thicklines\line(0,1){20}} \put(70,50){\line(-1,1){10}} \put(70,50){\line(-1,-1){10}} \put(62,42){\vector(-1,-1){0}} \multiput(60,60)(0,30){2}{\line(0,1){10}} \multiput(55,70)(10,0){2}{\line(0,1){20}} \multiput(55,70)(0,20){2}{\line(1,0){10}} \put(40,60){\line(0,1){40}} \put(40,100){\line(1,0){70}} \put(60,65){\line(1,0){30}} \put(60,65){\circle*2} \put(90,55){\thicklines\line(0,1){20}} \put(90,65){\line(1,1){10}} \put(90,65){\line(1,-1){10}} \put(98,57){\vector(1,-1){0}} \put(100,75){\line(0,1){25}} \multiput(60,100)(40,0){2}{\circle*2} \put(100,0){\line(0,1){20}} \put(100,40){\line(0,1){15}} \multiput(95,20)(10,0){2}{\line(0,1){20}} \multiput(95,20)(0,20){2}{\line(1,0){10}} \put(100,55){\line(1,0){10}} \put(111,55){\circle{2}} \put(111,100){\circle{2}} \put(100,55){\circle*{2}} \put(95,0){\line(1,0){10}} \end{picture} \end{minipage}% \hfill \begin{minipage}{5cm} \center \begin{picture}(100,100) \put(50,0){\vector(0,1){100}} \put(0,50){\vector(1,0){100}} \put(50,50){\vector(1,1){40}} \put(50,50){\vector(1,-1){40}} \put(50,50){\line(-1,1){40}} \put(48,90){\makebox(0,0)[r]{$u_{e2}$}} \put(95,47){\makebox(0,0)[t]{$u_{e1}$}} \put(87,80){\makebox(0,0)[l]{$u_{egl}$}} \put(82,10){\makebox(0,0)[r]{$u_{ed}$}} \multiput(75,49)(0,4){7}{\line(0,1){2}} \multiput(49,75)(4,0){7}{\line(1,0){2}} \put(75,47){\makebox(0,0)[t]{$1$}} \put(47,75){\makebox(0,0)[r]{$1$}} \put(75,78){\makebox(0,0)[b]{$1$}} \end{picture} \end{minipage}% \begin{minipage}{3.5cm} \center \[\boxed{u_{ed} = u_{e1} - u_{e2} \vphantom{\dfrac u u}}\] \end{minipage}% \begin{minipage}{4.5cm} \center \bigskip Differenz- Eingangsspannung \end{minipage}% \begin{minipage}{3.5cm} \center \[\boxed{u_{egl} = \frac{u_{e1} + u_{e2}}{2}}\] \end{minipage}% \begin{minipage}{4.5cm} \center \bigskip Gleichtakt-Eingangsspannung \end{minipage} \begin{minipage}{4cm} \center \begin{picture}(100,60) \multiput(20,10)(0,30){2}{\circle{15}} \multiput(00,10)(0,30){2}{\line(1,0){40}} \multiput(00,5)(0,30){2}{\line(0,1){10}} \put(40,0){\line(0,1){50}} \put(40,0){\line(3,2){37.5}} \put(40,50){\line(3,-2){37.5}} \put(20,25){\makebox(0,0)[c]{$u_{e2}$}} \put(20,55){\makebox(0,0)[c]{$u_{e1}$}} \multiput(0,0)(0,30){2}{ \qbezier(30,15)(20,25)(10,15) \put(9.9,15){\vector(-3,-2){0}} } \put(42,40){\makebox(0,0)[l]{$-$}} \put(42,10){\makebox(0,0)[l]{$+$}} \put(77.5,25){\line(1,0){20}} \put(98.5,25){\circle{2}} \end{picture} \end{minipage}% \hfill \begin{minipage}{0.5cm} \center $\Rightarrow$ \end{minipage}% \hfill \begin{minipage}{5cm} \center \begin{picture}(120,60) \multiput(60,10)(0,30){2}{\circle{15}} \multiput(40,10)(0,30){2}{\line(1,0){40}} \put(40,10){\line(0,1){30}} \put(80,0){\line(0,1){50}} \put(80,0){\line(3,2){37.5}} \put(80,50){\line(3,-2){37.5}} \put(60,25){\makebox(0,0)[c]{$u_{ed2}$}} \put(60,55){\makebox(0,0)[c]{$u_{ed1}$}} \multiput(40,0)(0,30){2}{ \qbezier(30,15)(20,25)(10,15) \put(9.9,15){\vector(-3,-2){0}} } \put(82,40){\makebox(0,0)[l]{$-$}} \put(82,10){\makebox(0,0)[l]{$+$}} \put(117.5,25){\line(1,0){20}} \put(138.5,25){\circle{2}} \put(0,20){\line(0,1){10}} \put(0,25){\line(1,0){40}} \put(20,25){\circle{15}} \put(20,40){\makebox(0,0)[c]{$u_{egl}$}} \qbezier(30,30)(20,40)(10,30) \put(9.9,30){\vector(-3,-2){0}} \put(40,25){\circle*2} \end{picture} \end{minipage}% \hfill \hfill \hfill \begin{minipage}{5cm} \center \begin{picture}(120,60) \put(0,0){\line(1,0){120}} \put(0,60){\line(5,-1){120}} \put(0,60){\circle*2} \put(70,46){\circle*2} \put(120,36){\circle*2} \put(0,55){\vector(0,-1){50}} \put(70,40){\vector(0,-1){35}} \put(120,30){\vector(0,-1){25}} \put(120,60){\vector(0,-1){20}} \put(3,30){\makebox(0,0)[l]{$u_{e1}$}} \put(67,23){\makebox(0,0)[r]{$u_{egl}$}} \put(117,17.5){\makebox(0,0)[r]{$u_{e2}$}} \put(117,50){\makebox(0,0)[r]{$u_{ed}$}} \end{picture} \end{minipage}% \subsubsection*{Verst"arkerkenngr"o"sen} \begin{align*} u_a & = v_1 \cdot u_{e1} + v_2 \cdot u_{e2} = v_1 \left(u_{egl} + \frac{u_{ed}}2\right) + v_2 \left(u_{egl} - \frac{u_{ed}}2\right) \\ &= \underbrace{\frac{v_1 - v_2}{2}}_{\displaystyle v_d} \cdot u_{ed} + \underbrace{(v_1 + v_2)}_{\displaystyle v_{gl}} \cdot u_{egl}\\[3ex] v_d &= \left.\frac{u_a}{u_{ed}}\right|_{u_{egl}=0} = \frac{v_1-v_2}{2} \qquad \text{Differenzverst"arkung}\\[2ex] v_{gl} &= \left.\frac{u_a}{u_{gl}}\right|_{u_{ed}=0} = v_1 + v_2 \qquad\ \text{ Gleichtaktverst"arkung} \end{align*} \begin{minipage}{8cm} \[\boxed{\quad CMRR = 20\,\lg \frac{v_d}{v_{gl}} \, \mathrm{dB} \vphantom{\frac{d}{d}} \quad }\] \end{minipage}% \begin{minipage}{5.5cm} \center Gleichtaktunterdr"uckung Common Mode Rejection Ratio \end{minipage}% \subsubsection*{Berechnung der Kenngr"o"sen} \[u_{a2} = v_1 \cdot u_{e1} + v_2 \cdot u_{e2} = v_d \cdot u_{ed} + v_{gl} \cdot u_{egl} \] Aus Signalflu"sgraph (\textsc{Mason}): \[u_{a2} = \frac{1}{1+2g_mr_E} \Big({g_m}^2 r_ER_Cu_{e1} - g_mR_C(1+g_mr_E)u_{e2}\Big)\] Koeffizientenvergleich: \[v_1 = \frac{{g_m}^2 r_E R_C}{1+2g_mr_E} \qquad\qquad v_2 = - \frac{g_mR_C(1+g_mr_E)}{1+2g_mr_E}\] Differenzverst"arkung: \[v_d = \frac{v_1 - v_2}{2} = \frac{g_mR_C(g_mr_E + 1 + g_mr_E)}{2(1+2g_mr_E} = \frac{g_mR_C}{2}\] Gleichtaktverst"arkung: \[v_{gl} = v_1 + v_2 = \frac{g_mR_C(g_mr_E-1-g_mr_E)}{1+2g_mr_E} = \frac{-g_mR_C}{1+2g_mr_E} \approx - \frac{-R_C}{2r_E}\] Gleichtaktunterdr"uckung: \[CMMR = 20\log \left|\frac{v_d}{v_{gl}}\right| \approx 20\log (g_m r_E) = 20\log \frac{I_E r_E}{2U_T} \] Zahlenbeispiel: $I_E = 1\,\mathrm{mA}$, $g_m = \frac{0{,}5\,\mathrm{mA}}{25\,\mathrm{mV}} = 20\,\mathrm{mS}$, $R_C = 10\,\mathrm k\Omega$ \begin{center} \begin{tabular}{l|c|c|c} $r_E/\mathrm k\Omega$ & 1 & 10 & 1000 \\ \hline $CMRR/\mathrm{dB}$ & 26 & 46 & 86 \end{tabular} \end{center} \subsubsection*{Eingangsverhalten} \begin{minipage}{8cm} \center \begin{picture}(190,70) \multiput(30,0)(120,0){2}{\line(0,1){60}} \multiput(25,0)(120,0){2}{\line(1,0){10}} \multiput(55,0)(60,0){2}{\line(1,0){10}} \multiput(30,30)(120,0){2}{\circle{20}} \put(20,30){\makebox(0,0)[r]{$u_{e1}\Big\downarrow$}} \put(160,30){\makebox(0,0)[l]{$\Big\downarrow u_{e2}$}} \multiput(30,60)(90,0){2}{\line(1,0){30}} \multiput(60,60)(60,0){2}{\circle*2} \multiput(60,0)(0,40){2}{\line(0,1){20}} \multiput(55,20)(10,0){2}{\line(0,1){20}} \multiput(55,20)(0,20){2}{\line(1,0){10}} \multiput(60,60)(40,0){2}{\line(1,0){20}} \multiput(80,55)(0,10){2}{\line(1,0){20}} \multiput(80,55)(20,0){2}{\line(0,1){10}} \multiput(120,0)(0,40){2}{\line(0,1){20}} \multiput(115,20)(10,0){2}{\line(0,1){20}} \multiput(115,20)(0,20){2}{\line(1,0){10}} \put(5,60){\vector(1,0){20}} \put(175,60){\vector(-1,0){20}} \put(15,63){\makebox(0,0)[b]{$i_{e1}$}} \put(165,63){\makebox(0,0)[b]{$i_{e2}$}} \put(67,30){\makebox(0,0)[l]{$r_{egl}$}} \put(113,30){\makebox(0,0)[r]{$r_{egl}$}} \put(90,68){\makebox(0,0)[b]{$r_{ed}$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} $i_{e1}$ aus Signalflu"sgraph: \[i_{e1} = \frac{g_{BE}}{1+2g_mr_E} \left((1+g_mr_E)u_{e1} - g_mr_Eu_{e2} \right)\] $i_{e1}$ aus Ersatzschaltung \[i_{e1} = (g_{egl} + g_{ed}) u_{e1} - g_{ed} u_{e2}\] \end{minipage}% Koeffizientenvergleich: \[r_{ed} = \frac{1+2g_m r_E}{g_mr_E} \cdot r_{BE} \approx 2 r_{BE} , \qquad r_{egl} = r_{BE} \cdot (1+2g_mr_E) \approx 2br_E\] % gespart: T-ersatzschaltung mit 2* r_BE und b*r_E ; sollte klar sein, wenn man % keine Flachzange ist. \paragraph{Anwendung: OTA} (Operational Transconductance Amplifier) \begin{minipage}{8cm} \center \begin{picture}(110,60) \multiput(19,10)(0,30){2}{\circle{2}} \put(19,35){\vector(0,-1){20}} \multiput(20,10)(0,30){2}{\line(1,0){20}} \put(40,0){\line(0,1){50}} \put(40,0){\line(3,2){25}} \put(40,50){\line(3,-2){25}} \put(65,16.5){\line(0,1){17}} \put(16,10){\makebox(0,0)[r]{$u_{e2}$}} \put(16,25){\makebox(0,0)[r]{$u_{ed}$}} \put(16,40){\makebox(0,0)[r]{$u_{e1}$}} \put(42,40){\makebox(0,0)[l]{$-$}} \put(42,10){\makebox(0,0)[l]{$+$}} \put(65,25){\line(1,0){20}} \put(86,25){\circle{2}} \put(97.5,28){\makebox(0,0)[b]{$i_{a}$}} \put(90,25){\vector(1,0){15}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \[i_a = g_m \cdot u_{ed}\] \end{minipage}% \bigskip \begin{minipage}{8cm} \center \begin{picture}(190,180) \multiput(0,0)(0,90){2}{ \multiput(0,0)(80,0){2}{ \multiput(30,30)(40,0){2}{\line(0,1){25}} \multiput(30,30)(0,25){2}{\line(1,0){40}} \multiput(35,50)(30,0){2}{\vector(0,-1){15}} \put(40,42.5){\vector(1,0){20}} } } \multiput(50,15)(80,0){2}{\line(0,1){15}} \multiput(50,14)(80,0){2}{\circle{2}} \multiput(50,10)(80,0){2}{\makebox(0,0)[t]{$-U_{0C}$}} \multiput(50,165)(80,0){2}{\makebox(0,0)[b]{$U_{0C}$}} \multiput(50,145)(80,0){2}{\line(0,1){15}} \multiput(50,161)(80,0){2}{\circle{2}} \put(0,65){\makebox(0,0)[l]{$I_E$}} \put(15,65){\circle2} \put(16,65){\line(1,0){19}} \put(0,87.5){\makebox(0,0)[l]{$U_d$}} \put(15,87.5){\circle2} \put(16,87.5){\vector(1,0){26.5}} \put(35,65){\vector(0,-1){10}} \put(50,87.5){\circle{15}} \put(50,87.5){\makebox(0,0)[c]{$\times$}} \put(65,70){\vector(0,-1){15}} \put(67,63){\makebox(0,0)[l]{$I_E$}} \put(65,70){\line(-1,1){11}} \put(35,120){\line(0,-1){15}} \put(35,105){\line(1,-1){11}} \put(65,120){\line(0,-1){10}} \put(64,104){\line(1,0){51}} \put(115,104){\line(0,1){16}} \put(54,94){\line(1,1){10}} \put(75,113){\vector(1,0){15}} \put(82.5,115){\makebox(0,0)[b]{$I_1$}} \put(85,100){\vector(-1,0){15}} \put(78.5,98){\makebox(0,0)[t]{$I_2$}} \put(65,110){\line(1,0){25}} \put(90,110){\line(0,-1){12}} \put(90,98){\line(1,0){25}} \put(115,98){\line(0,-1){43}} \put(145,120){\line(0,-1){65}} \put(145,87.5){\circle*2} \put(145,87.5){\line(1,0){20}} \put(166,87.5){\circle2} \put(145,87.5){\vector(1,0){13}} \put(168,87.5){\makebox(0,0)[l]{$I_a$}} \multiput(149,112.25)(0,-32.5){2}{\vector(0,-1){15}} \put(152,104.75){\makebox(0,0)[l]{$I_2$}} \put(152,71.25){\makebox(0,0)[l]{$I_1$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \[I_a = I_E \tanh \frac{U_{ed}}{2U_T} \approx \underbrace{\frac{I_E}{2U_T}} _{g_m} U_{ed} \] \end{minipage}% \chapter{Leistungsverst"arker} \section{Leistungsbilanz} [siehe Folie] \section{A-Verst"arker} \begin{minipage}{6.5cm} \begin{picture}(150,110) \put(9,30){\circle{2}} \put(10,30){\line(1,0){20}} \put(30,20){\thicklines\line(0,1){20}} \put(30,30){\line(1,1){10}} \put(30,30){\line(1,-1){10}} \put(38,22){\vector(1,-1){0}} \put(40,00){\line(0,1){20}} \put(35,00){\line(1,0){10}} \multiput(40,40)(0,40){2}{\line(0,1){20}} \multiput(35,60)(10,0){2}{\line(0,1){20}} \multiput(35,60)(0,20){2}{\line(1,0){10}} \put(40,101){\circle2} \put(37,101){\makebox(0,0)[r]{$U_{0C}$}} \put(32,70){\makebox(0,0)[r]{$R$}} \put(55,70){\makebox(0,0)[l]{$U_{a}$}} \put(55,30){\makebox(0,0)[l]{$U_{CE}$}} \qbezier(48,85)(58,70)(48,55) \put(48,55){\vector(-2,-3){0}} \qbezier(48,45)(58,30)(48,15) \put(48,15){\vector(-2,-3){0}} \multiput(120,0)(0,40){2}{\line(0,1){20}} \multiput(115,20)(10,0){2}{\line(0,1){20}} \multiput(115,20)(0,20){2}{\line(1,0){10}} \put(90,70){\line(1,0){20}} \put(110,60){\thicklines\line(0,1){20}} \put(110,70){\line(1,1){10}} \put(110,70){\line(1,-1){10}} \put(118,62){\vector(1,-1){0}} \put(120,80){\line(0,1){20}} \put(120,101){\circle2} \put(89,70){\circle2} \put(117,101){\makebox(0,0)[r]{$U_{0C}$}} \put(115,0){\line(1,0){10}} \put(80,0){ \put(55,70){\makebox(0,0)[l]{$U_{CE}$}} \put(55,30){\makebox(0,0)[l]{$U_{a}$}} \qbezier(48,85)(58,70)(48,55) \put(48,55){\vector(-2,-3){0}} \qbezier(48,45)(58,30)(48,15) \put(48,15){\vector(-2,-3){0}} } \end{picture} \end{minipage}% \begin{minipage}{9.5cm} \subsubsection*{Annahmen:} \begin{itemize} \item symmetrische Aussteuerung \[U_{CEA} = \frac{U_{0C}}{2} = \widehat{U}_{a,max}, \quad \widehat{I}_{C,max} = I_{CA}\] \item $U_{CES}$ und $I_B$ vernachl"assigt \end{itemize} \end{minipage}% \bigskip \begin{minipage}{6.5cm} \center \begin{picture}(175,120) \put(25,10){\vector(0,1){100}} \put(20,15){\vector(1,0){130}} \put(23,100){\makebox(0,0)[r]{$I_C$}} \put(23,77.5){\makebox(0,0)[r]{$2I_{CA}$}} \put(23,47){\makebox(0,0)[r]{$I_{CA}$}} \put(140,7){\makebox(0,0)[c]{$U_{CE}$}} \put(107.5,7){\makebox(0,0)[c]{$U_{0}$}} \put(66.25,7){\makebox(0,0)[c]{$U_{CEA}$}} \put(25,77.5){\line(4,-3){82.5}} \put(24,77.5){\line(1,0){2}} \put(24,47){\line(1,0){2}} \put(107.5,14){\line(0,1){2}} \multiput(25,47)(4,0){21}{\line(1,0){2}} \multiput(107.5,14)(0,3.85){9}{\line(0,1){2}} \multiput(66.25,14)(0,3.85){9}{\line(0,1){2}} \qbezier(25,15)(26,35)(30,40) % kennlinie \qbezier(30,40)(31,44)(120,53) \qbezier(45,100)(45,30)(120,35) % hyperbel \put(48,90){\makebox(0,0)[l]{$\scriptstyle \frac{U^2_{0C}}{4R} = P_{TR=}$}} \put(45.625,31){\makebox(0,0)[c]{$P_{TR=}$}} \put(86.875,31){\makebox(0,0)[c]{$P_{R=}$}} \end{picture} \end{minipage}% \begin{minipage}{9.5cm} \subsubsection*{Gr"o"senordnungen:} \[\left.\begin{array}{l@{\ }l} U_{0C} &= 20 \ldots 40 \, \mathrm V \\ I_{CA} &= 1 \ldots 10\,\mathrm A \end{array}\right\} \Rightarrow R = 2 \ldots 40\,\Omega \] \end{minipage}% \subsubsection*{Transistorverlustleistung in Abhängigkeit von $U_{CEA}$} \begin{minipage}{8cm} \center \begin{picture}(175,130) \put(30,40){\vector(0,1){90}} \put(25,45){\vector(1,0){130}} \put(27,120){\makebox(0,0)[r]{$\frac{P_{Tr=}}{P_0}$}} \put(27,90){\makebox(0,0)[r]{$\frac 1 4$}} \put(75,42){\makebox(0,0)[t]{$\frac 1 2$}} \put(120,42){\makebox(0,0)[t]{$1$}} \put(150,41){\makebox(0,0)[t]{$x$}} \qbezier(30,45)(52.5,90)(75,90) \qbezier(75,90)(97.5,90)(120,45) \multiput(29,90)(4,0){30}{\line(1,0){2}} \multiput(75,44)(0,4){12}{\line(0,1){2}} \put(120,44){\line(0,1){2}} \multiput(75,0)(0,4){7}{\line(0,1){2}} \qbezier(75,3)(60,5.5)(60,8) \qbezier(60,8)(60,10.5)(75,13) \qbezier(90,18)(90,20.5)(75,23) \qbezier(75,13)(90,15.5)(90,18) \multiput(90,0)(0,4){23}{\line(0,1){2}} \multiput(60,0)(0,4){23}{\line(0,1){2}} \multiput(29,85)(4,0){30}{\line(1,0){2}} \qbezier(120.5,89)(122.5,85)(125,85) \qbezier(125,85)(127.5,85)(129.5,89) \qbezier(130.5,89)(132.5,85)(135,85) \qbezier(135,85)(137.5,85)(139.5,89) \qbezier(129.5,89)(130,90)(130.5,89) \end{picture} \end{minipage}% \begin{minipage}{8cm} \[P_{Tr=} = U_{CEA} \cdot I_{CA},\qquad I_{CA} = \frac{U_{0C} - U_{CEA}}{R}\] \begin{align*} P_{TR=} &= U_{CEA} \cdot \frac{U_{0C} - U_{CEA}}{R} \\ &= \frac{U_{0C}^2}{R} \cdot \frac{U_{CEA}}{U_{0C}} \left(1-\frac{U_{CEA}}{U_{0C}}\right)\\ &= P_0 \cdot x(1-x) \end{align*} \[P_0 = \frac{{U_{0C}}^2}{R}, \qquad x = \frac{U_{CEA}}{U_{0C}}\] \end{minipage}% \begin{minipage}{8cm} \begin{align*} I_C(t) &= I_{CA} + \widehat{I}_C \cos \omega t \\ U_a(t) &= U_{aA} + \widehat{U}_a \cos \omega t = U_R \end{align*} \begin{align*} P_= &= \overline{U_{0C} \cdot I_{C}(t)} = U_{0C} \cdot I_{CA}\\ &= 2 \cdot U_{CEA} \cdot I_{CA} = \frac{{U_{0C}}^2}{2R} = \frac{2{U_{CEA}}^2}{R} \end{align*} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(175,130) \put(30,40){\vector(0,1){90}} \put(25,45){\vector(1,0){130}} %\put(27,120){\makebox(0,0)[r]{$\frac{P_{Tr=}}{P_0}$}} \put(75,42){\makebox(0,0)[t]{$\scriptstyle U_{CEA}$}} \put(120,45){\line(-3,2){90}} \put(30,45){\line(3,2){45}} \multiput(75,44)(45,0){2}{\line(0,1){2}} \multiput(75,10)(0,4){7}{\line(0,1){2}} \qbezier(75,13)(60,15.5)(60,18) \qbezier(60,18)(60,20.5)(75,23) \qbezier(90,28)(90,30.5)(75,33) \qbezier(75,23)(90,25.5)(90,28) \multiput(90,10)(0,4){17}{\line(0,1){2}} \multiput(60,10)(0,4){17}{\line(0,1){2}} \put(75,13){\vector(1,0){15}} \put(90,13){\vector(-1,0){15}} \put(82.5,10){\makebox(0,0)[t]{$\scriptstyle \widehat{U}_{a}$}} \put(123,78){\makebox(0,0)[bl]{$\mathbf P_=$}} \put(35,50){\makebox(0,0)[bl]{$\mathbf P_{Tr}$}} \put(115,50){\makebox(0,0)[br]{$\mathbf P_{R=}$}} \put(83,90){\makebox(0,0)[l]{$\mathbf P_{S}$}} \put(57,69.5){\makebox(0,0)[r]{$\scriptstyle\widehat{I}_{a}$}} \put(67.5,64){\makebox(0,0)[t]{$\scriptstyle\widehat{U}_{a}$}} \thicklines \put(30,75){\line(1,0){90}} \multiput(30,45)(45,0){3}{\line(0,1){30}} \put(75,75){\line(-3,-2){15}} \put(60,65){\line(1,0){15}} \thinlines \qbezier(67.5,68)(68,90)(80,90) \end{picture} \end{minipage}% \bigskip $\displaystyle P_R = P_{R=} + P_S, \qquad P_{R=} = (U_{0C} - U_{CEA})I_{CA} = U_{CEA} \cdot I_{CA} = \frac{U_{=}}{2}$ \bigskip $\displaystyle P_S = \frac{\widehat{U}_a \cdot \widehat{I}_a}{2} = \frac{{\widehat{U}_a}^2}{2R} = \frac{{\widehat{U}_a}^2}{2R} \cdot \frac{{U_{CEA}}^2}{{U_{CEA}}^2} = \frac{{U_{CEA}}^2}{2R}\cdot m^2 = \frac{P_=}{4} \cdot m^2$ \bigskip $\boxed{\quad m = \frac{\widehat U_a}{U_{CEA}} \quad\vphantom{\int^a_a}}$ \qquad Aussteuerungsgrad \qquad $0 \leq m \leq 1$ \bigskip $\displaystyle P_{Tr} = P_= - P_R = \overbrace{P_= - P_{R=}}^{P_=/2} - P_S = P_S\left(\frac 1 2 - \frac{m^2}{4}\right)$ \bigskip $\eta = \dfrac{P_S}{P_=} = \frac{m^2}{4}$ \begin{minipage}{7cm} \center \begin{picture}(180,130) \put(25,10){\vector(0,1){100}} \put(20,15){\vector(1,0){145}} \put(22,95){\makebox(0,0)[r]{$P_=$}} \put(22,55){\makebox(0,0)[r]{$\frac{P_=}2$}} \multiput(24,55)(4,0){29}{\line(1,0){2}} \multiput(24,95)(4,0){29}{\line(1,0){2}} \qbezier(25,15)(100,15)(140,35) \qbezier(25,55)(100,55)(140,75) \put(140,14){\line(0,1){81}} \put(143,35){\makebox(0,0)[l]{$\frac{P_=}4$}} \put(120,35){\makebox(0,0)[c]{$P_S$}} \put(90,15){\vector(0,1){45}} \put(90,60){\vector(0,-1){45}} \put(90,60){\vector(0,1){35}} \put(90,95){\vector(0,-1){35}} \put(87,37){\makebox(0,0)[r]{$P_R$}} \put(87,77){\makebox(0,0)[r]{$P_{Tr}$}} \put(140,12){\makebox(0,0)[t]{$1$}} \put(160,10){\makebox(0,0)[t]{$m$}} \end{picture} \end{minipage}% \begin{minipage}{9cm} Mit steigender Aussteuerung nimmt die Transistorverlustleistung ab. \bigskip Niedriger Wirkungsgrad ($\leq 25\%$): \begin{itemize} \item Gleichleistung an $R$ \item Gleichleistung am Transistor \end{itemize} \end{minipage}% \section{Gegentakt-B- und AB-Verstärker} \subsection{Leistung und Wirkungsgrad} \begin{minipage}{5cm} \centering \begin{picture}(120,100) \put(30,30){\line(1,0){20}} \put(50,20){\thicklines\line(0,1){20}} \put(50,30){\line(1,1){10}} \put(50,30){\line(1,-1){10}} \put(54,34){\vector(-1,-1){0}} % \put(30,60){\line(1,0){20}} \put(50,50){\thicklines\line(0,1){20}} \put(50,60){\line(1,1){10}} \put(50,60){\line(1,-1){10}} \put(58,52){\vector(1,-1){0}} \put(30,30){\line(0,1){30}} \put(30,45){\line(-1,0){14}} \put(15,45){\circle2} \put(15,43){\vector(0,-1){41}} \put(10,0){\line(1,0){10}} \put(13,22.5){\makebox(0,0)[r]{$U_e$}} \multiput(60,2)(0,68){2}{\line(0,1){18}} \multiput(60,1)(0,88){2}{\circle2} \put(60,40){\line(0,1){10}} \multiput(30,45)(30,0){2}{\circle*2} \multiput(60,45)(40,0){2}{\line(1,0){20}} \multiput(80,40)(0,10){2}{\line(1,0){20}} \multiput(80,40)(20,0){2}{\line(0,1){10}} \put(120,40){\line(0,1){10}} \put(63,1){\makebox(0,0)[l]{$-U_{0C}$}} \put(63,89){\makebox(0,0)[l]{$U_{0C}$}} \put(90,57){\makebox(0,0)[b]{$R$}} \qbezier(75,40)(90,30)(105,40) \put(105,40){\vector(3,2){0}} \put(90,32){\makebox(0,0)[t]{$U_a$}} \put(60,30){\makebox(0,0)[l]{2}} \put(60,60){\makebox(0,0)[l]{1}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \centering \begin{picture}(140,110) \put(70,0){\vector(0,1){100}} \put(0,50){\vector(1,0){140}} \put(73,90){\makebox(0,0)[l]{$U_a$}} \put(130,48){\makebox(0,0)[t]{$U_e$}} \qbezier(70,50)(75,50)(80,53) \qbezier(80,53)(110,83)(110,83) \qbezier(70,50)(65,50)(60,47) \qbezier(60,47)(30,17)(30,17) % "sinus" \qbezier(70,5)(55,7.5)(55,10) \qbezier(55,10)(55,12.5)(70,15) \qbezier(70,15)(85,17.5)(85,20) \qbezier(85,20)(85,22.5)(70,25) \put(35,90){\makebox(0,0)[c]{$T2$}} \put(35,78){\makebox(0,0)[c]{arbeitet}} \put(105,22){\makebox(0,0)[c]{$T1$}} \put(105,10){\makebox(0,0)[c]{arbeitet}} \end{picture} \end{minipage}% \begin{minipage}{5cm} \center \begin{picture}(130,110) \put(15,0){\vector(0,1){100}} \put(10,50){\vector(1,0){120}} \put(13,90){\makebox(0,0)[r]{$U_a$}} \put(120,47){\makebox(0,0)[t]{$t$}} \qbezier(20,50)(23,50)(25,53) \multiput(0,0)(50,0){2}{ \qbezier(25,53)(28.75,80)(32.5,80) \qbezier(32.5,80)(36.25,80)(40,53) \qbezier(40,53)(42,50)(45,50)} \qbezier(50,47)(48,50)(45,50) \qbezier(50,47)(53.75,20)(57.5,20) \qbezier(57.5,20)(61.25,20)(65,47) \qbezier(65,47)(68,50)(70,50) \qbezier(70,50)(73,50)(75,53) \put(57.5,99){\makebox(0,0)[c]{\small "Ubernahme-}} \put(57.5,85){\makebox(0,0)[c]{\small verzerrung}} \put(55,80){\vector(-1,-3){8}} \put(60,80){\vector(1,-3){8}} \end{picture} \end{minipage}% \bigskip \[U_e(t) = \widehat{U}_e \cdot \cos \omega t, \quad U_a(t) = \widehat{U}_a \cdot \cos \omega t, \qquad \widehat{U}_a = \widehat{U}_e, \quad \widehat{I}_a = \widehat{U}_a/R, \quad \widehat{U}_{a,\mathrm{max}} = U_{0C} \] \subsubsection*{Signalleistung} $ P_S = \dfrac{{\widehat{U}_a}^2}{2R} = \dfrac{{U_{0C}}^2}{2R}\cdot m^2$, davon $\frac 1 2 P_S$ oben und $\frac 1 2 P_S$ unten erzeugt. \qquad $\boxed{\quad m = \dfrac{\widehat{U}_a}{\vphantom{\widehat{U}_a}U_{0C}} \quad}$ \subsubsection*{Gleichleistung} \vspace*{-1em} % :-) \begin{align*} P_= &= 2\overline{U_{0C} \cdot I_a(t)} = \underbrace{2}_{\text{\tiny 2 Quellen}} \cdot U_{0C} \cdot \frac{1}{T} \int\limits_0^{T/2} \widehat{I}_a \sin \omega t\,dt = \frac{2U_{0C}\widehat{I}_a}{2\pi}\int\limits_0^{\pi} \sin u \, du = \frac{2}{\pi} U_{0C} \widehat{I}_a = \frac{2}{\pi} U_{0C} \frac{\widehat{U}_a}{R}\\ & = \frac{2}{\pi} \frac{{U_{0C}}^2}{R} \cdot m \end{align*} davon $\frac 1 2 P_=$ oben und $\frac 1 2 P_=$ unten erzeugt. \subsubsection*{Wirkungsgrad} $\eta = \frac{P_S}{P_=} = \frac{\pi }{4} \cdot m$, maximal $\frac{\pi}{4} \approx 0{,}78$ \subsubsection*{Verlustleistung} Ein Zweig: $\displaystyle P_{Tr} = \frac{P_=}{2} - \frac{P_S}{2} = \left(\frac{m}{\pi} - \frac{m^2}{4}\right) \frac{{U_{0C}}^2}{R}$ \bigskip \begin{minipage}{8cm} \center \begin{picture}(160,120) \put(30,10){\vector(0,1){100}} \put(25,15){\vector(1,0){135}} \put(27,100){\makebox(0,0)[r]{$\frac{P}{{U_{0C}^2}/{R}}$}} \put(27,80){\makebox(0,0)[r]{$\frac 2 {\pi}$}} \multiput(29,80)(4,0){25}{\line(1,0){2}} \multiput(127.5,14)(0,4){17}{\line(0,1){2}} \put(30,15){\line(3,2){97.5}} \qbezier(30,15)(90,15)(127.5,60) \put(75,55){\makebox(0,0)[c]{$P_=$}} \put(110,30){\makebox(0,0)[c]{$P_S$}} \put(93,48){\makebox(0,0)[l]{$\scriptstyle 2P_{Tr}$}} \put(90,29){\vector(0,1){26}} \put(90,55){\vector(0,-1){26}} \put(126.5,60){\line(1,0){2}} \put(132,59){\makebox(0,0)[l]{$0{,}5$}} \put(127.5,11){\makebox(0,0)[t]{$1$}} \put(150,11){\makebox(0,0)[t]{$m$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(160,120) \put(30,10){\vector(0,1){100}} \put(25,15){\vector(1,0){135}} \put(27,100){\makebox(0,0)[r]{$\frac{P_{Tr}}{{U_{0C}^2}/{R}}$}} \put(27,80){\makebox(0,0)[r]{$\frac 1 {\pi^2}$}} \multiput(29,80)(4,0){25}{\line(1,0){2}} \multiput(127.5,14)(0,4){17}{\line(0,1){2}} \qbezier(30,15)(75,80)(90,80) \qbezier(90,80)(110,80)(127.5,60) \put(126.5,60){\line(1,0){2}} \put(132,59){\makebox(0,0)[l]{$\frac 1{\pi}-\frac 1 4 \approx 0{,}07$}} \put(132,80){\makebox(0,0)[l]{$0{,}1$}} \put(127.5,11){\makebox(0,0)[t]{$1$}} \put(150,11){\makebox(0,0)[t]{$m$}} \multiput(90,14)(0,4){17}{\line(0,1){2}} \put(90,11){\makebox(0,0)[t]{$\frac{2}{\pi}$}} \end{picture} \end{minipage}% \paragraph{Bemessungsbeispiel:} Leistungsverst"arker $P_S = 20\,\mathrm W$, $R = 6\,\Omega$ gesucht: $U_{0C}$, $P_{Tr}$ \smallskip $P_S = \dfrac{{\widehat{U}_a}^2}{2R}, \quad \widehat{U}_{a,\mathrm{max}} = U_{0C}$ \smallskip $P_{S,\mathrm{max}} = \dfrac{{U_{0C}}^2}{2R}\quad \Rightarrow \quad U_{0C} = \sqrt{2R \cdot P_{max}} = 15{,}5\,\mathrm V \quad \longrightarrow \quad 17\,\mathrm V$ gew"ahlt \smallskip $P_{Tr,\mathrm{max}} = \dfrac{1}{\pi^2} \dfrac{{U_{0C}}^2}{R} = \dfrac{1}{\pi^2} \cdot \dfrac{(17\,\mathrm V)^2}{6\,\Omega} = 4{,}88\,\mathrm W \quad \longrightarrow \quad 5\,\mathrm W$ \subsection{AB-Betrieb} \begin{minipage}{8cm} \centering \begin{picture}(160,140) \put(19,20){\circle{2}} \put(20,20){\line(1,0){20}} \put(40,10){\thicklines\line(0,1){20}} \put(40,20){\line(1,1){10}} \put(40,20){\line(1,-1){10}} \put(48,12){\vector(1,-1){0}} \put(50,10){\line(0,-1){8}} \put(50,1){\circle{2}} \put(53,1){\makebox(0,0)[l]{$-U_{0C}$}} \put(50,30){\line(0,1){10}} \put(50,40){\line(1,0){20}} \put(70,40){\line(1,0){20}} \put(90,30){\thicklines\line(0,1){20}} \put(90,40){\line(1,1){10}} \put(90,40){\line(1,-1){10}} \put(98,32){\vector(1,-1){0}} %2 \put(70,90){\line(1,0){20}} \put(90,80){\thicklines\line(0,1){20}} \put(90,90){\line(1,1){10}} \put(90,90){\line(1,-1){10}} \put(98,82){\vector(1,-1){0}} \put(100,50){\line(0,1){30}} \put(100,65){\circle*{2}} \multiput(100,65)(40,0){2}{\line(1,0){20}} \multiput(120,60)(0,10){2}{\line(1,0){20}} \multiput(120,60)(20,0){2}{\line(0,1){10}} \multiput(70,40)(0,20){3}{\line(0,1){10}} \multiput(65,50)(0,10){4}{\line(1,0){10}} \multiput(70,50)(0,20){2}{\qbezier(0,0)(0,0)(5,10)\qbezier(0,0)(0,0)(-5,10)} \put(100,2){\line(0,1){28}} \put(100,1){\circle{2}} \put(103,1){\makebox(0,0)[l]{$-U_{0C}$}} \put(70,40){\circle*2} \put(70,90){\circle*2} \multiput(70,90)(0,30){2}{\line(0,1){10}} \multiput(65,100)(10,0){2}{\line(0,1){20}} \multiput(65,100)(0,20){2}{\line(1,0){10}} \put(70,130){\line(1,0){40}} \put(100,130){\circle*2} \put(111,130){\circle2} \put(100,100){\line(0,1){30}} \put(114,130){\makebox(0,0)[l]{$U_{0C}$}} \put(63,110){\makebox(0,0)[r]{$R_3$}} \put(130,73){\makebox(0,0)[b]{$R$}} \put(63,65){\makebox(0,0)[r]{$* \left\{\vphantom{\int\limits_b^j}\right.$}} \put(160,60){\line(0,1){10}} \put(16,20){\makebox(0,0)[r]{$U_e$}} \qbezier(115,60)(130,50)(145,60) \put(145,60){\vector(3,2){0}} \put(130,52){\makebox(0,0)[t]{$U_a$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \centering \begin{picture}(150,110) \put(70,0){\vector(0,1){100}} \put(0,50){\vector(1,0){140}} \put(73,90){\makebox(0,0)[l]{$U_a$}} \put(130,48){\makebox(0,0)[t]{$U_e$}} \qbezier(70,50)(71,50)(73,53) \qbezier(73,53)(103,83)(103,83) \qbezier(70,50)(69,50)(67,47) \qbezier(67,47)(37,17)(37,17) \end{picture} \end{minipage}% \subsubsection*{* $U_{BE}$-Vervielfacher} \begin{minipage}{5cm} \centering \begin{picture}(100,130) \put(-40,0){ \multiput(70,60)(0,40){2}{\line(0,1){20}} %R2 \multiput(65,80)(10,0){2}{\line(0,1){20}} \multiput(65,80)(0,20){2}{\line(1,0){10}} \multiput(70,00)(0,40){2}{\line(0,1){20}} %R1 \multiput(65,20)(10,0){2}{\line(0,1){20}} \multiput(65,20)(0,20){2}{\line(1,0){10}} \put(070,60){\circle*2} \put(100,120){\circle*2} \put(070,60){\line(1,0){20}} \put(90,50){\thicklines\line(0,1){20}} \put(90,60){\line(1,1){10}} \put(90,60){\line(1,-1){10}} \put(98,52){\vector(1,-1){0}} \put(100,70){\line(0,1){30}} \put(100,100){\line(0,1){20}} \put(95,00){\line(1,0){10}} \put(100,0){\line(0,1){50}} \put(100,40){\line(0,1){10}} \put(65,0){\line(1,0){10}} \put(070,120){\line(1,0){30}} \put(100,120){\line(0,1){10}} \put(103,120){\makebox(0,0)[l]{$\Big\downarrow I$}} \put(063,90){\makebox(0,0)[r]{$R_{1}$}} \put(063,30){\makebox(0,0)[r]{$R_{2}$}} \put(105,45){\vector(-2,-3){0}} \qbezier(105,75)(115,60)(105,45) \put(113,60){\makebox(0,0)[l]{$U$}} } \end{picture} \end{minipage}% \begin{minipage}{5cm} \center \begin{picture}(140,140) \put(25,10){\vector(0,1){110}} \put(20,15){\vector(1,0){115}} \put(22,110){\makebox(0,0)[r]{$I$}} \put(125,7){\makebox(0,0)[c]{$U$}} \put(25,15){\line(3,1){80}} \put(107,40){$R_1+R_2$} \qbezier(25,15)(65,29)(65,29) \qbezier(65,29)(68,30)(70,35) \qbezier(70,35)(75,100)(75,100) \put(72,14){\line(0,1){2}} \put(72,6){\makebox(0,0)[c]{$U_g$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \[\frac{R_2}{R_1+R_2} U_g = U_{BE0}\] \[U_g= \left(1+\frac{R_1}{R_2}\right)\] \end{minipage}% \subsection{Strombegrenzung} Siehe Folie: \href{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/leistv/pa1.pdf}{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/leistv/pa1.pdf} \chapter{R"uckkopplung} \section{Grundprinzip} \subsection{Gro"ssignalverhalten} \begin{minipage}{8cm} \centering \begin{picture}(170,85) \put(0,60){\makebox(0,0)[l]{$X$}} \put(159,60){\makebox(0,0)[l]{$Y$}} \put(14,60){\circle{2}} \put(15,60){\vector(1,0){22.5}} \put(45,60){\circle{15}} \put(52.5,60){\vector(1,0){22.5}} \multiput(75,45)(40,0){2}{\line(0,1){30}} \multiput(75,45)(0,30){2}{\line(1,0){40}} \put(95,60){\makebox(0,0)[c]{$f(U)$}} \put(65,63){\makebox(0,0)[b]{$U$}} \put(115,60){\line(1,0){39}} \put(155,60){\circle2} \put(135,60){\circle*2} \put(135,20){\line(0,1){40}} \multiput(45,20)(55,0){2}{\line(1,0){35}} \put(45,20){\vector(0,1){32.5}} \multiput(80,10)(20,0){2}{\line(0,1){20}} \multiput(80,10)(0,20){2}{\line(1,0){20}} \put(90,20){\makebox(0,0)[c]{$k$}} \multiput(25,5)(0,4){19}{\line(0,1){2}} \multiput(145,5)(0,4){19}{\line(0,1){2}} \multiput(25,5)(4,0){30}{\line(1,0){2}} \multiput(27,80)(4,0){30}{\line(1,0){2}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[Y = f(U), \qquad U = X + kY\] \[\boxed{\quad Y = f(X + kY) = \varphi(X) \vphantom{\int} \quad}\] i.a. nicht nach $Y$ aufl"osbar. inverse Beziehung: $X = \varphi^{-1}(Y) = f^{-1}(Y) - kY$ \end{minipage}% \bigskip \bigskip \begin{minipage}{8cm} \center \begin{picture}(200,120) \put(100,0){\vector(0,1){120}} \put(0,60){\vector(1,0){200}} \put(100,60){\line(1,3){17}} \put(100,60){\line(-1,-3){17}} \put(120,115){\makebox(0,0)[l]{$X=kY$}} \put(183,90){\makebox(0,0)[l]{$f$}} \thicklines \qbezier(100,60)(110,90)(120,90) \qbezier(100,60)(90,30)(80,30) \qbezier(120,90)(120,90)(180,90) \qbezier(80,30)(20,30)(20,30) \thinlines % k > 0 \qbezier(100,60)(100,90)(110,90) \qbezier(110,90)(120,90)(120,90) \qbezier(100,60)(100,30)(90,30) \qbezier(90,30)(80,30)(80,30) %k < 0 \qbezier(100,60)(120,90)(130,90) \qbezier(100,60)(80,30)(70,30) \put(117,75){\makebox(0,0)[l]{$k < 0$}} \put(97,90){\makebox(0,0)[r]{$k > 0$}} \qbezier(90,85)(90,80)(101.4,80) \put(97,112){\makebox(0,0)[r]{$Y$}} \put(190,56){\makebox(0,0)[t]{$X$}} \end{picture} \smallskip $f$ monoton steigend (nicht invertierender Verst"arker) \end{minipage}% \begin{minipage}{8cm} \center \begin{picture}(200,120) \put(100,0){\vector(0,1){120}} \put(0,60){\vector(1,0){200}} \put(100,60){\line(1,3){17}} \put(100,60){\line(-1,-3){17}} \put(120,115){\makebox(0,0)[l]{$X=kY$}} \put(17,90){\makebox(0,0)[r]{$f$}} \thicklines \qbezier(100,60)(110,30)(120,30) \qbezier(100,60)(90,90)(80,90) \qbezier(120,30)(120,30)(180,30) \qbezier(80,90)(20,90)(20,90) \thinlines % k > 0 \qbezier(100,60)(100,30)(110,30) \qbezier(110,30)(120,30)(120,30) \qbezier(100,60)(100,90)(90,90) \qbezier(90,90)(80,90)(80,90) %k < 0 \qbezier(100,60)(120,30)(130,30) \qbezier(100,60)(80,90)(70,90) \put(83,75){\makebox(0,0)[r]{$k > 0$}} \put(103,20){\makebox(0,0)[l]{$k < 0$}} \put(97,112){\makebox(0,0)[r]{$Y$}} \put(190,56){\makebox(0,0)[t]{$X$}} \end{picture} \smallskip $f$ monoton fallend (invertierender Verst"arker) \end{minipage}% \paragraph{Beispiel:} Emitterschaltung mit Stromgegenkopplung \begin{minipage}{6cm} \center \begin{picture}(85,110) \put(19,50){\circle{2}} \put(19,45){\vector(0,-1){40}} \put(14,0){\line(1,0){10}} \put(45,0){\line(1,0){10}} \put(20,50){\line(1,0){20}} \put(40,40){\thicklines\line(0,1){20}} \put(40,50){\line(1,1){10}} \put(40,50){\line(1,-1){10}} \put(48,42){\vector(1,-1){0}} \multiput(50,0)(0,30){2}{\line(0,1){10}} \multiput(45,10)(10,0){2}{\line(0,1){20}} \multiput(45,10)(0,20){2}{\line(1,0){10}} \multiput(50,60)(0,30){2}{\line(0,1){10}} \multiput(45,70)(10,0){2}{\line(0,1){20}} \multiput(45,70)(0,20){2}{\line(1,0){10}} \put(17,25){\makebox(0,0)[r]{$U_e$}} \put(43,20){\makebox(0,0)[r]{$R_E$}} \put(43,80){\makebox(0,0)[r]{$R_C$}} \qbezier(55,5)(65,20)(55,35) \put(55,5){\vector(-2,-3){0}} \put(63,20){\makebox(0,0)[l]{$U_{RE}$}} \put(50,101){\circle{2}} \put(47,101){\makebox(0,0)[r]{$U_{0C}$}} \put(50,60){\circle*2} \put(50,60){\line(1,0){20}} \put(71,60){\circle2} \put(66,40){\line(1,0){10}} \put(71,58){\vector(0,-1){16}} \put(73,50){\makebox(0,0)[l]{$U_a$}} \end{picture} \begin{picture}(160,80) \put(0,40){\makebox(0,0)[l]{$U_e$}} \put(14,40){\circle2} \put(15,40){\vector(1,0){10}} \put(30,40){\circle{10}} \put(45,44){\makebox(0,0)[b]{$\scriptstyle U_{BE}$}} \put(35,40){\vector(1,0){20}} \multiput(55,30)(0,20){2}{\line(1,0){20}} \multiput(55,30)(20,0){2}{\line(0,1){20}} \put(65,40){\makebox(0,0)[c]{$f$}} \put(75,40){\vector(1,0){15}} \multiput(90,30)(0,20){2}{\line(1,0){20}} \multiput(90,30)(20,0){2}{\line(0,1){20}} \put(100,40){\makebox(0,0)[c]{$\scriptstyle -R_C$}} \put(82.5,40){\line(0,-1){30}} \put(30,10){\vector(0,1){25}} \put(30,10){\line(1,0){15}} \put(82.5,10){\line(-1,0){17.5}} \multiput(45,0)(0,20){2}{\line(1,0){20}} \multiput(45,0)(20,0){2}{\line(0,1){20}} \put(55,10){\makebox(0,0)[c]{$\scriptstyle -R_E$}} \put(82.5,40){\circle*{1.5}} \put(82.5,44){\makebox(0,0)[b]{$\scriptstyle I_{C}$}} \put(110,40){\vector(1,0){10}} \put(125,40){\circle{10}} \put(125,55){\vector(0,-1){10}} \put(125,57){\makebox(0,0)[b]{$U_{0C}$}} \put(130,40){\line(1,0){9}} \put(140,40){\circle{2}} \put(143,40){\makebox(0,0)[l]{$U_{a}$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \begin{align*} I_C &= I_S \cdot e^{\frac{U_{BE}}{U_T}} \\ U_{BE} &= U_e - U_{RE} = U_e - \frac{B+1}{B}I_C R_E \approx U_e - I_CR_E \\ I_C &= I_S \cdot e^{\frac 1 {U_T}\left(U_e -ICR_E\right)}\\ U_a &= U_{0C} - I_C R_C \quad \Rightarrow \quad I_C = \frac{U_{0C} - U_a}{R_E} \end{align*} \[ \boxed{\vphantom{\int} \quad U_a = U_{0C} - R_C I_S \cdot e^{\frac 1{U_T} \left(U_e - \frac{U_{0C} - U_a}{R_E}\cdot R_E\right)} \quad} \] \end{minipage}% \begin{minipage}{6cm} \center \begin{picture}(120,90) % das ist die haesslichste Grafik aller Zeiten \put(25,10){\vector(0,1){80}} \put(20,15){\vector(1,0){100}} \put(23,80){\makebox(0,0)[r]{$U_a$}} \put(110,7){\makebox(0,0)[c]{$U_e$}} \put(24,60){\line(1,0){2}} \put(22,60){\makebox(0,0)[r]{$\scriptstyle U_{0C}$}} \put(25,60){\line(1,-1){45}} \put(70,14){\line(0,1){2}} \put(70,13){\makebox(0,0)[t]{$\scriptstyle U_{0C}\frac{R_E}{R_C}$}} \qbezier(25,60)(35,60)(40,58) \qbezier(40,58)(48,56)(48,15) \qbezier(48,15)(48,15)(110,30) \thicklines \qbezier(25,60)(35,60)(43,58) \qbezier(43,58)(73,25)(73,25) \qbezier(73,25)(75,23)(110,33) \thinlines \put(50,68){\makebox(0,0)[r]{$\scriptstyle U_{RE}$}} \put(60,45){\makebox(0,0)[l]{$\sim\frac{R_C}{R_E} = v'$}} \qbezier(38,63)(38,58)(28,58) \end{picture} \end{minipage}% \begin{minipage}{10cm} \[u_e = \underbrace{\frac{(U_{0C} - U_a)R_E}{R_C}}_{U_{RE}} + \underbrace{U_T \ln \frac{U_{0C}-U_a}{R_C \cdot I_S}}_{U_{BE}}\] \end{minipage}% \subsection{Kleinsignalverhalten} \begin{minipage}{8cm} \centering \begin{picture}(170,85) \put(0,60){\makebox(0,0)[l]{$x$}} \put(159,60){\makebox(0,0)[l]{$y$}} \put(14,60){\circle{2}} \put(15,60){\vector(1,0){22.5}} \put(45,60){\circle{15}} \put(52.5,60){\vector(1,0){22.5}} \multiput(75,45)(40,0){2}{\line(0,1){30}} \multiput(75,45)(0,30){2}{\line(1,0){40}} \put(95,60){\makebox(0,0)[c]{$v$}} \put(65,63){\makebox(0,0)[b]{$u$}} \put(115,60){\line(1,0){39}} \put(155,60){\circle2} \put(135,60){\circle*2} \put(135,20){\line(0,1){40}} \multiput(45,20)(55,0){2}{\line(1,0){35}} \put(45,20){\vector(0,1){32.5}} \multiput(80,10)(20,0){2}{\line(0,1){20}} \multiput(80,10)(0,20){2}{\line(1,0){20}} \put(90,20){\makebox(0,0)[c]{$k$}} \multiput(25,5)(0,4){19}{\line(0,1){2}} \multiput(145,5)(0,4){19}{\line(0,1){2}} \multiput(25,5)(4,0){30}{\line(1,0){2}} \multiput(27,80)(4,0){30}{\line(1,0){2}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[y = v \cdot u, \qquad u = x + ky\] \[\boxed{\quad y = v(x + ky) \quad \vphantom{\int}}\] \[y(1-kv) = v \cdot x \quad \Rightarrow \quad y = \frac{c}{1-kv}x = v'x\] \end{minipage}% \bigskip \bigskip \begin{minipage}{8cm} \textbf{Begriffe:} \smallskip \begin{tabular}{ll} $k$ & R"uckkopplungsfaktor \\[1ex] $k\cdot v$ & Schleifenverst"arkung\\[1ex] $g = 1 - k\cdot v$ & R"uckkopplungsgrad\\[1ex] $r = \frac{1}{g} = \frac{1}{1 - k\cdot v} = \frac{v'}{v}$ &Regelfaktor \end{tabular} \end{minipage}% \begin{minipage}{8cm} \centering \begin{picture}(170,125) \put(0,70){\vector(1,0){170}} \put(75,10){\vector(0,1){110}} \qbezier(10,75)(95,73)(95,110) \qbezier(95,30)(95,65)(140,65) \multiput(95,30)(0,4){22}{\line(0,1){2}} \put(160,66){\makebox(0,0)[t]{$kv$}} \put(78,115){\makebox(0,0)[l]{$r$}} \put(52,100){\makebox(0,0)[r]{$g$}} \put(95,70){\line(-3,2){40}} \put(95,70){\line(3,-2){40}} \put(0,30){\line(1,0){30}} \put(47.5,31){\makebox(0,0){stabil}} \put(66,30){\vector(1,0){29}} \put(132.5,31){\makebox(0,0){instabil}} \put(110,30){\vector(-1,0){15}} \put(155,30){\line(1,0){15}} \put(37.5,11){\makebox(0,0){\small Gegenkopplung}} \put(122.5,11){\makebox(0,0){\small Mitkopplung}} \end{picture} \end{minipage}% \subsection{Mit- und Gegenkopplung} \begin{center} \begin{tabular}{p{7cm}|p{7cm}} \textbf{Gegenkopplung} & \textbf{Mitkopplung} \\ \hline \hline $kv < 0$ & $kv > 0$ \\ $g = 1 - kv > 1 $ & $g = 1 - kv < 1 $\\ $|v'| < |v|$ & $|v'| > |v|$ \\ \hline \multicolumn{2}{c}{\textbf{Anwendungen}}\\ \hline Verst"arker: Reduzierung der Verzerrungen, Ver"anderung von Ein- und Ausgangswiderstand, Stabilisierung & Kippschaltungen, Oszillatoren, Erzeugung negativer Widerst"ande, G"uteverbesserung von resonanten Schaltungen \end{tabular} \end{center} \subsubsection*{Spezialf"alle} \begin{tabular}{lll@{\qquad}l} $kv= 1$ & $\Rightarrow$ & $v' \to \infty$ & Stabilit"atsgrenze (Schwingungserzeugung)\\ $kv \gg 1$ & $\Rightarrow$ & $v' = \frac{v}{1-kv} \approx -\frac 1 k$ & OPV-Schaltungen \end{tabular} \section{Gegenkopplungsarten} \subsection{Allgemein} Siehe Folie: {\small \href{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/feedb/feedb1.pdf}{http://www.iee.et.tu-dresden.de/iee/ge/student/materialien/ST/folien/feedb/feedb1.pdf}} \subsection{Bei Transistorgrundstufen} \begin{minipage}{8cm} \centering \begin{picture}(70,120) \put(19,50){\circle{2}} \put(19,45){\vector(0,-1){40}} \put(14,0){\line(1,0){10}} \put(45,0){\line(1,0){10}} \put(20,50){\line(1,0){20}} \put(40,40){\thicklines\line(0,1){20}} \put(40,50){\line(1,1){10}} \put(40,50){\line(1,-1){10}} \put(48,42){\vector(1,-1){0}} \multiput(50,0)(0,30){2}{\line(0,1){10}} \multiput(45,10)(10,0){2}{\line(0,1){20}} \multiput(45,10)(0,20){2}{\line(1,0){10}} \multiput(50,60)(0,30){2}{\line(0,1){10}} \multiput(45,70)(10,0){2}{\line(0,1){20}} \multiput(45,70)(0,20){2}{\line(1,0){10}} \put(50,101){\circle{2}} \put(57,20){\makebox(0,0)[l]{$R_E$}} \put(57,80){\makebox(0,0)[l]{$R_C$}} \put(16,25){\makebox(0,0)[r]{$U_e$}} \put(50,104){\makebox(0,0)[b]{$U_{0C}$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \centering \begin{picture}(120,120) \put(30,0){\line(0,1){50}} \put(25,0){\line(1,0){10}} \put(30,25){\circle{20}} \put(20,25){\makebox(0,0)[r]{$U_{G}\Big\downarrow$}} \put(50,57){\makebox(0,0)[b]{$R_{G}$}} \multiput(30,50)(30,0){2}{\line(1,0){10}} \multiput(40,45)(0,10){2}{\line(1,0){20}} \multiput(40,45)(20,0){2}{\line(0,1){10}} \put(70,50){\line(1,0){20}} \put(90,40){\thicklines\line(0,1){20}} \put(90,50){\line(1,1){10}} \put(90,50){\line(1,-1){10}} \put(98,42){\vector(1,-1){0}} \put(100,0){\line(0,1){40}} \put(95,0){\line(1,0){10}} \put(100,60){\line(0,1){20}} \put(100,100){\line(0,1){10}} \multiput(95,80)(10,0){2}{\line(0,1){20}} \multiput(95,80)(0,20){2}{\line(1,0){10}} \put(90,70){\line(1,0){10}} \put(100,70){\circle*2} \multiput(70,65)(0,10){2}{\line(1,0){20}} \multiput(70,65)(20,0){2}{\line(0,1){10}} \put(70,70){\line(-1,0){5}} \put(65,70){\line(0,-1){20}} \put(65,50){\circle*2} \put(80,78){\makebox(0,0)[b]{$R$}} \put(108,90){\makebox(0,0)[l]{$R_C$}} \put(100,111){\circle2} \put(97,111){\makebox(0,0)[r]{$U_{0C}$}} \end{picture} \end{minipage}% \bigskip \begin{minipage}{8cm} \centering Serien-Stromgegenkopplung Unwirksam bei Stromeinspeisung! \end{minipage}% \begin{minipage}{8cm} \centering Parallel-Spannungsgegenkopplung Unwirksam bei \emph{reiner} Spannungsansteuerung \end{minipage}% \section{Effekte der Gegenkopplung} \subsection{Parameterempfindlichkeit (Sensitivity)} \begin{minipage}{7cm} \subsubsection*{Definition:} \smallskip \hfill \begin{picture}(150,30) \put(7,15){\makebox(0,0)[r]{$x$}} \put(10,15){\vector(1,0){20}} \multiput(30,0)(30,0){2}{\line(0,1){30}} \multiput(30,0)(0,30){2}{\line(1,0){30}} \put(60,15){\vector(1,0){20}} \put(83,15){\makebox(0,0)[l]{$y$}} \put(40,15){\makebox(0,0)[c]{$v$}} \put(50,10){\makebox(0,0)[c]{$a$}} \put(46,6){\vector(1,1){10}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{tabular}{l@{\qquad\qquad }l@{\ }l} $y = v(a) \cdot x$ &$v$: & beeinflu"sende Gr"o"se\\ & $a$: & Einflu"sgr"o"se \end{tabular} \[\boxed{\quad S_a^v = \frac{a\,dv}{v\,da} = \frac{d\ln v}{d \ln a}\quad } \qquad\qquad \] \end{minipage}% \bigskip \bigskip \begin{minipage}{7cm} \centering \begin{picture}(170,85) \put(0,60){\makebox(0,0)[l]{$x$}} \put(159,60){\makebox(0,0)[l]{$y$}} \put(14,60){\circle{2}} \put(15,60){\vector(1,0){22.5}} \put(45,60){\circle{15}} \put(52.5,60){\vector(1,0){22.5}} \multiput(75,45)(40,0){2}{\line(0,1){30}} \multiput(75,45)(0,30){2}{\line(1,0){40}} \put(95,60){\makebox(0,0)[c]{$v(a)$}} \put(65,63){\makebox(0,0)[b]{$u$}} \put(115,60){\line(1,0){39}} \put(155,60){\circle2} \put(135,60){\circle*2} \put(135,20){\line(0,1){40}} \multiput(45,20)(55,0){2}{\line(1,0){35}} \put(45,20){\vector(0,1){32.5}} \multiput(80,10)(20,0){2}{\line(0,1){20}} \multiput(80,10)(0,20){2}{\line(1,0){20}} \put(90,20){\makebox(0,0)[c]{$k$}} \put(130,15){\makebox(0,0)[c]{$v'$}} \multiput(25,5)(0,4){19}{\line(0,1){2}} \multiput(145,5)(0,4){19}{\line(0,1){2}} \multiput(25,5)(4,0){30}{\line(1,0){2}} \multiput(27,80)(4,0){30}{\line(1,0){2}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \begin{description} \item[gegeben:] $S_a^v$, $v' = \frac{v(a)}{1-kv(a)} = v'(a)$ \item[gesucht:] $\displaystyle S_a^{v'} = \frac{dv'}{da} \frac{a}{v'} = \frac{a}{v'}\frac{dv'}{dv}\frac{dv}{da} = \underbrace{\frac v {v'} \frac{dv'}{dv}}_{S_v^{v'}} \underbrace{\frac a v \frac{dv}{da}}_{S_a^v}$ \end{description} \end{minipage}% \[\Rightarrow \quad \boxed{\quad \vphantom{\int} S_a^{v'} = S_v^{v'} \cdot S_a^v \quad} \qquad \text{Kettenregel f"ur Empfindlichkeiten}\] \[S_v^{v'} = \frac{v}{v'} \frac{dv'}{dv} = \frac{\cancel{v}}{\frac{\cancel{v}}{1-kv}} \frac{(1-\cancel{kv})+\cancel{vk}}{(1-kv)^{\cancel{2}}} = \frac{1}{1-kv} = \frac 1 g\,, \qquad S_a^{v'} = S_a^v \frac{1}{1-kv} \] \subsubsection*{Beispiel:} \begin{minipage}{8cm} \centering \begin{picture}(150,100) \multiput(40,0)(0,30){2}{\line(0,1){10}} \multiput(35,10)(10,0){2}{\line(0,1){20}} \multiput(35,10)(0,20){2}{\line(1,0){10}} \multiput(40,40)(40,0){2}{\line(1,0){20}} \multiput(60,35)(0,10){2}{\line(1,0){20}} \multiput(60,35)(20,0){2}{\line(0,1){10}} \put(35,0){\line(1,0){10}} \put(40,40){\line(0,1){20}} \put(40,40){\circle*2} \multiput(40,60)(0,30){2}{\line(1,0){20}} \put(60,50){\line(0,1){50}} \put(60,50){\line(3,2){37.5}} \put(60,100){\line(3,-2){37.5}} \put(62,90){\makebox(0,0)[l]{$+$}} \put(62,60){\makebox(0,0)[l]{$-$}} \put(97.5,75){\line(1,0){20}} \put(100,40){\line(1,0){10}} \put(110,40){\line(0,1){35}} \put(110,75){\circle*2} \put(110,75){\line(1,0){15}} \put(126,75){\circle2} \put(126,73){\vector(0,-1){71}} \put(121,0){\line(1,0){10}} \put(20,90){\line(1,0){20}} \put(19,90){\circle{2}} \put(19,88){\vector(0,-1){86}} \put(14,0){\line(1,0){10}} \put(16,45){\makebox(0,0)[r]{$u_e$}} \put(38,75){\makebox(0,0)[r]{$u_d$}} \put(40,88){\vector(0,-1){26}} \put(129,37.5){\makebox(0,0)[l]{$u_a$}} \put(47,20){\makebox(0,0)[l]{$R_1$}} \put(70,33){\makebox(0,0)[t]{$R_2$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \centering \begin{picture}(180,85) \put(0,60){\makebox(0,0)[l]{$u_+$}} \put(159,60){\makebox(0,0)[l]{$u_a$}} \put(14,60){\circle{2}} \put(15,60){\vector(1,0){22.5}} \put(45,60){\circle{15}} \put(52.5,60){\vector(1,0){22.5}} \multiput(75,45)(40,0){2}{\line(0,1){30}} \multiput(75,45)(0,30){2}{\line(1,0){40}} \put(95,60){\makebox(0,0)[c]{$v$}} \put(62,63){\makebox(0,0)[b]{$u_d$}} \put(58,40){\makebox(0,0)[b]{$u_-$}} \put(115,60){\line(1,0){39}} \put(155,60){\circle2} \put(135,60){\circle*2} \put(135,20){\line(0,1){40}} \multiput(45,20)(55,0){2}{\line(1,0){35}} \put(45,20){\vector(0,1){32.5}} \multiput(80,10)(20,0){2}{\line(0,1){20}} \multiput(80,10)(0,20){2}{\line(1,0){20}} \put(90,20){\makebox(0,0)[c]{$k$}} \put(35,50){\makebox(0,0)[c]{$-$}} \multiput(25,5)(0,4){19}{\line(0,1){2}} \multiput(145,5)(0,4){19}{\line(0,1){2}} \multiput(25,5)(4,0){30}{\line(1,0){2}} \multiput(27,80)(4,0){30}{\line(1,0){2}} \end{picture} \end{minipage}% \bigskip \[u_a = v\cdot u_d = v(u_+ - u_-) \qquad\qquad k = \frac{R_1}{R_1+R_2}, \ u_+ = u_e,\ u_- = ku_a\] \[v' = \frac{v}{1+kv}, \qquad S_a^{v'} = \frac{1}{1+kv}S_a^v, \qquad v = 10^4,\ k = \frac 1 2 \] \[v' = \frac{10^4}{1+0{,}5\cdot 10^4} \approx \frac 1 k = 2\] \paragraph{Empfindlichkeit:} $\displaystyle S_a^{v'} = \frac{1}{1+0{,}5\cdot 10^4}\cdot s_a^v = 2 \cdot 10^{-4} \cdot S_a^v$ \subsection{Eingangswiderstand} \subsubsection{Parallel-Gegenkopplung (Miller-Effekt)} \begin{center} \begin{picture}(450,110) \put(30,10){\line(0,1){60}} \put(30,40){\circle{20}} \put(20,40){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \multiput(30,10)(0,60){2}{\line(1,0){50}} \multiput(80,0)(0,80){2}{\line(1,0){80}} \multiput(80,0)(80,0){2}{\line(0,1){80}} \put(80,10){\line(1,0){120}} \put(160,70){\line(1,0){40}} \multiput(55,70)(125,0){2}{\line(0,1){25}} \multiput(55,70)(125,0){2}{\circle*2} \multiput(55,95)(72.5,0){2}{\line(1,0){52.5}} \multiput(107.5,90)(0,10){2}{\line(1,0){20}} \multiput(107.5,90)(20,0){2}{\line(0,1){10}} \put(117.5,102){\makebox(0,0)[b]{$R$}} \put(140,10){\line(0,1){60}} \put(140,70){\line(1,0){20}} \put(140,10){\circle*2} \multiput(140,30)(-10,10){2}{\line(1,1){10}} \multiput(140,30)(10,10){2}{\line(-1,1){10}} \put(130,40){\makebox(0,0)[r]{$vu_e\Big\uparrow$}} \put(42.5,78){\makebox(0,0)[b]{$i_e$}} \put(35,75){\vector(1,0){15}} \put(67.5,78){\makebox(0,0)[b]{$i_{ev}$}} \put(60,75){\vector(1,0){15}} \put(70,103){\makebox(0,0)[b]{$i$}} \put(77.5,100){\vector(-1,0){15}} \put(170,78){\makebox(0,0)[b]{$i_{av}$}} \put(177.5,75){\vector(-1,0){15}} \put(190,78){\makebox(0,0)[b]{$i_{a}$}} \put(197.5,75){\vector(-1,0){15}} \multiput(201,10)(0,60){2}{\circle{2}} \put(201,67){\vector(0,-1){54}} \put(204,40){\makebox(0,0)[l]{$u_{a}$}} %%% \put(226.25,40){\makebox(0,0){$\widehat{=}\vphantom{j}$}} %%% \put(265,10){\line(0,1){60}} \put(265,40){\circle{20}} \put(255,40){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \multiput(265,10)(0,60){2}{\line(1,0){50}} \multiput(315,0)(0,80){2}{\line(1,0){80}} \multiput(315,0)(80,0){2}{\line(0,1){80}} \put(315,10){\line(1,0){120}} \put(395,70){\line(1,0){40}} \multiput(290,10)(0,60){2}{\circle*2} \put(375,10){\line(0,1){60}} \put(375,70){\line(1,0){20}} \put(375,10){\circle*2} \multiput(375,30)(-10,10){2}{\line(1,1){10}} \multiput(375,30)(10,10){2}{\line(-1,1){10}} \put(365,40){\makebox(0,0)[r]{$vu_e\Big\uparrow$}} \put(277.5,78){\makebox(0,0)[b]{$i_e$}} \put(270,75){\vector(1,0){15}} \put(302.5,78){\makebox(0,0)[b]{$i_{ev}$}} \put(295,75){\vector(1,0){15}} \put(405,78){\makebox(0,0)[b]{$i_{av}$}} \put(412.5,75){\vector(-1,0){15}} \put(425,78){\makebox(0,0)[b]{$i_{a}$}} \put(432.5,75){\vector(-1,0){15}} \multiput(436,10)(0,60){2}{\circle{2}} \put(436,67){\vector(0,-1){54}} \put(441,40){\makebox(0,0)[l]{$u_{a}$}} \multiput(290,10)(0,40){2}{\line(0,1){20}} \multiput(285,30)(10,0){2}{\line(0,1){20}} \multiput(285,30)(0,20){2}{\line(1,0){10}} \put(295,40){\makebox(0,0)[l]{$\frac{R}{1+v}$}} \put(287,60){\makebox(0,0)[r]{$-i\!\downarrow$}} \multiput(415,10)(0,40){2}{\line(0,1){20}} \multiput(410,30)(10,0){2}{\line(0,1){20}} \multiput(410,30)(0,20){2}{\line(1,0){10}} \multiput(415,10)(0,60){2}{\circle*{2}} \put(410,40){\makebox(0,0)[r]{$\frac{vR}{1\!+\!v}$}} \put(412,60){\makebox(0,0)[r]{$i\!\downarrow$}} \end{picture} \end{center} Verst"arker: $u_a = - v \cdot u_e, \quad v > 0$ \[i = \frac{u_a - u_e}{R} = - \frac{1+v}{R} u_e = \frac{1+\frac 1 v}{R}u_a\] \subsubsection*{Beispiel:} \begin{minipage}{6cm} \centering \begin{picture}(130,80) \multiput(40,20)(0,30){2}{\line(1,0){20}} \put(60,10){\line(0,1){50}} \put(60,10){\line(3,2){37.5}} \put(60,60){\line(3,-2){37.5}} \put(62,50){\makebox(0,0)[l]{$-$}} \put(62,20){\makebox(0,0)[l]{$+$}} \put(97.5,35){\line(1,0){20}} \put(40,0){\line(0,1){20}} \put(35,0){\line(1,0){10}} \multiput(40,70)(40,0){2}{\line(1,0){20}} \multiput(60,65)(0,10){2}{\line(1,0){20}} \multiput(60,65)(20,0){2}{\line(0,1){10}} \put(40,70){\line(0,-1){20}} \put(20,50){\line(1,0){20}} \put(40,50){\circle*2} \put(19,50){\circle{2}} \put(19,46){\vector(0,-1){42}} \put(14,0){\line(1,0){10}} \put(16,25){\makebox(0,0)[r]{$u_e$}} \put(70,78){\makebox(0,0)[b]{$R$}} \put(7.5,53){\makebox(0,0)[b]{$i_e$}} \put(0,50){\vector(1,0){15}} \put(40,45){\vector(1,0){15}} \put(47.5,42){\makebox(0,0)[t]{$i_{ev}$}} \put(100,70){\line(1,0){10}} \put(110,35){\line(0,1){35}} \put(110,35){\circle*2} \put(118.5,35){\circle2} \put(118.5,32){\vector(0,-1){29}} \put(113.5,0){\line(1,0){10}} \put(121,17.5){\makebox(0,0)[l]{$u_a$}} \end{picture} \bigskip $R = 10\,\mathrm k\Omega$, $v = 10^5$, $i_{ev} = 0$ \end{minipage}% \begin{minipage}{10cm} \vspace{-1em} \begin{align*} u_a &= v \cdot u_d = - v \cdot u_e\\ u_e &= u_a + i_e R && r_e = \frac{u_e}{i_e} = \frac{R}{1+v}\\ u_e(1+v) &\approx i_e R \end{align*} \[\boxed{\quad r_e = \frac{10^4\,\Omega}{1 + 10^5} = 0{,}1\,\Omega \quad}\] \end{minipage}% \subsubsection*{Anwendungen} \begin{enumerate} \item \mbox{} \begin{picture}(70,70) \put(11,25){\makebox(0,0)[c]{$\underline Z_e \to$}} \multiput(19,10)(0,30){2}{\circle{2}} \multiput(20,10)(0,30){2}{\line(1,0){20}} \put(40,0){\line(0,1){50}} \put(40,0){\line(3,2){37.5}} \put(40,50){\line(3,-2){37.5}} \put(42,40){\makebox(0,0)[l]{$-$}} \put(42,10){\makebox(0,0)[l]{$+$}} \put(77.5,25){\line(1,0){20}} \multiput(30,60)(32,0){2}{\line(1,0){28}} \multiput(58,53)(4,0){2}{\thicklines\line(0,1){14}} \put(30,40){\circle*2} \put(30,40){\line(0,1){20}} \put(90,25){\line(0,1){35}} \put(90,25){\circle*2} \put(98.5,25){\circle2} \put(120,25){\makebox(0,0)[l]{$\underline Z_e = \dfrac{1}{sC(1+v)} \quad \Rightarrow$}} \multiput(240,-5)(0,32){2}{\line(0,1){28}} \multiput(233,23)(0,4){2}{\thicklines\line(1,0){14}} \put(253,25){\makebox(0,0)[l]{$C(1+v)$}} \end{picture} \item \mbox{} \begin{minipage}{4.5cm} \begin{picture}(90,100) \put(44,40){\circle{2}} \put(55,40){\circle*{2}} \put(45,40){\line(1,0){20}} \put(65,30){\thicklines\line(0,1){20}} \put(65,40){\line(1,1){10}} \put(65,40){\line(1,-1){10}} \put(73,32){\vector(1,-1){0}} \multiput(55,0)(0,22){2}{\line(0,1){18}} \multiput(48,18)(0,4){2}{\thicklines\line(1,0){14}} \put(75,0){\line(0,1){30}} \multiput(50,0)(20,0){2}{\line(1,0){10}} \put(75,50){\line(0,1){20}} \multiput(70,70)(10,0){2}{\line(0,1){20}} \multiput(70,70)(0,20){2}{\line(1,0){10}} \put(83,80){\makebox(0,0)[l]{$R_C$}} \put(73,101){\makebox(0,0)[r]{$U_{0C}$}} \put(75,90){\line(0,1){10}} \put(75,101){\circle{2}} \multiput(62,53)(2,2){2}{\thicklines\qbezier(0,0)(0,0)(-7,7)} \qbezier(55,40)(55,50)(58.5,56.5) \qbezier(75,60)(60,60)(60.5,58.5) \put(75,60){\circle*{2}} \put(38,65){$C_{CB}$} \put(5,17){$\scriptstyle (1+v)C_{CB}$} \qbezier(50,58)(30,40)(45,25) \put(45,25){\vector(1,-1){0}} \end{picture} \end{minipage}% \begin{minipage}{10cm} $C_{CB}$ \ldots parasit"are Kapazit"at \bigskip $C_{CB} = 0{,}5\,\mathrm{pF}$, \quad $v = 100$ \bigskip $C_{CE} = (1+v) \cdot C_{BE} = 50\,\mathrm{pF} \widehat{=} \frac{1}{\omega C_{CE}} = 32\,\Omega$ bei $f = 100\,\mathrm{MHz}$ \end{minipage}% \end{enumerate} \subsubsection{Seriengegenkopplung (Bootstrap-Effekt)} \begin{minipage}{8cm} \centering \begin{picture}(200,100) \put(30,0){\line(0,1){90}} \put(30,45){\circle{20}} \put(20,45){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \put(30,90){\line(1,0){60}} \multiput(90,50)(0,30){2}{\line(0,1){10}} \multiput(85,60)(10,0){2}{\line(0,1){20}} \multiput(85,60)(0,20){2}{\line(1,0){10}} \put(104,79){\makebox(0,0)[c]{$r_{ev}$}} \put(70,70){\makebox(0,0)[r]{$u_{ev}\Big\downarrow$}} \put(60,50){\line(1,0){30}} \put(60,00){\line(0,1){10}} \multiput(55,10)(10,0){2}{\line(0,1){20}} \multiput(55,10)(0,20){2}{\line(1,0){10}} \put(60,30){\line(0,1){20}} \put(60,40){\line(1,0){30}} \put(60,40){\circle*2} \multiput(90,40)(40,0){2}{\line(1,0){20}} \multiput(110,35)(0,10){2}{\line(1,0){20}} \multiput(110,35)(20,0){2}{\line(0,1){10}} \multiput(140,60)(-10,10){2}{\line(1,1){10}} \multiput(140,60)(10,10){2}{\line(-1,1){10}} \put(140,50){\line(0,1){40}} \put(135,50){\line(1,0){10}} \put(130,70){\makebox(0,0)[r]{$vu_{ev}\Big\downarrow$}} \multiput(80,55)(4,0){19}{\line(1,0){2}} \multiput(80,85)(4,0){19}{\line(1,0){2}} \multiput(80,55)(0,4){8}{\line(0,1){2}} \multiput(154,55)(0,4){8}{\line(0,1){2}} \put(140,90){\line(1,0){40}} \put(160,90){\circle*2} \put(160,40){\line(0,1){50}} \put(150,40){\line(1,0){10}} \put(25,0){\line(1,0){10}} \put(55,0){\line(1,0){10}} \put(67,20){\makebox(0,0)[l]{$R_1$}} \put(120,33){\makebox(0,0)[t]{$R_2$}} \qbezier(80,35)(90,20)(80,5) \put(80,5){\vector(-2,-3){0}} \put(87,20){\makebox(0,0)[l]{$k\cdot u_a$}} \put(181,90){\circle{2}} \put(181,86){\vector(0,-1){82}} \put(176,0){\line(1,0){10}} \put(184,45){\makebox(0,0)[l]{$u_a$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[i_e = \frac{u_{ev}}{r_{ev}} = \frac{u_e - k \cdot u_a}{r_{ev}}, \qquad k \approx \frac{R_1}{R_1+R_2}\] \[u_a = v\cdot u_{ev} = v(u_e - ku_a), \qquad R_1 \parallel R_2 \ll r_{ev} \vphantom{\frac a a}\] \[u_a(1+kv) = vu_e, \qquad u_a = \frac{v}{1+kv}u_e = v' u_e\] \end{minipage}% \bigskip \bigskip \[i_e = \frac{1}{r_{ev}} \left(1 - \frac{kv}{1+kv}\right) u_e = \frac{1}{r_{ev}}\frac{u_e}{1+kv} \qquad \Rightarrow \qquad r_e = r_{ev} (1+kv) = r_{ev} \cdot g \] \paragraph{Beispiel:} $k = 1$ ($R_1 \to \infty$), $r_{ev} = 10\,\mathrm{k}\Omega$, $v = 10^{4} \Rightarrow r_e = 10^8\,\Omega$ \subsubsection*{Anwendungen:} \begin{minipage}{6cm} \centering \begin{picture}(70,95) \put(19,50){\circle{2}} \put(19,45){\vector(0,-1){40}} \put(14,0){\line(1,0){10}} \put(45,0){\line(1,0){10}} \put(20,50){\line(1,0){20}} \put(40,40){\thicklines\line(0,1){20}} \put(40,50){\line(1,1){10}} \put(40,50){\line(1,-1){10}} \put(48,42){\vector(1,-1){0}} \multiput(50,0)(0,30){2}{\line(0,1){10}} \multiput(45,10)(10,0){2}{\line(0,1){20}} \multiput(45,10)(0,20){2}{\line(1,0){10}} \put(50,81){\circle{2}} \put(43,20){\makebox(0,0)[r]{$R_E$}} \put(16,25){\makebox(0,0)[r]{$r_e \Rightarrow \quad u_e$}} \put(50,84){\makebox(0,0)[b]{$U_{0C}$}} \put(50,60){\line(0,1){20}} \qbezier(55,5)(65,20)(55,35) \put(55,5){\vector(-2,-3){0}} \put(63,20){\makebox(0,0)[l]{$u_a$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[r_{ev} = r{BE}, \qquad v = \frac{u_a}{u_{ev}} = g_m \cdot R_E\] \[k = 1 \quad \Rightarrow r_e = r_{ev}(1+g_mR_E) = {r_{BE} + bR_E} \] \end{minipage}% \bigskip \begin{center} \fbox{ \begin{minipage}{12.5cm} \centering \begin{tabular}{llll} Parallel- & & verkleinert \\[-1ex] & Gegenkopplung & & den Eingangswiderstand\\[-1ex] Serien- & & vergr"o"sert \end{tabular} \end{minipage}% } \end{center} \subsection{Ausgangswiderstand} \begin{center} \fbox{ \begin{minipage}{12.5cm} \centering \begin{tabular}{llll} Spannungs- & & verkleinert \\[-1ex] & Gegenkopplung & & den Ausgangswiderstand\\[-1ex] Strom- & & vergr"o"sert \end{tabular} \end{minipage}% } \end{center} \paragraph{Beispiel 1:} Verst"arker mit Spannungsgegenkopplung \begin{minipage}{8cm} \begin{picture}(220,95) \put(30,10){\line(0,1){40}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$u_e\Big\downarrow$}} \multiput(30,50)(30,0){2}{\line(1,0){10}} \multiput(40,45)(0,10){2}{\line(1,0){20}} \multiput(40,45)(20,0){2}{\line(0,1){10}} \put(30,10){\line(1,0){50}} \multiput(70,50)(100,0){2}{\line(0,1){20}} \multiput(70,50)(100,0){2}{\circle*2} \multiput(70,70)(80,0){2}{\line(1,0){20}} \put(70,50){\line(1,0){10}} \multiput(80,0)(80,0){2}{\line(0,1){60}} \multiput(80,0)(0,60){2}{\line(1,0){80}} \multiput(90,70)(40,0){2}{\line(1,0){20}} \multiput(110,65)(0,10){2}{\line(1,0){20}} \multiput(110,65)(20,0){2}{\line(0,1){10}} \multiput(120,50)(30,0){2}{\line(1,0){10}} \multiput(130,45)(0,10){2}{\line(1,0){20}} \multiput(130,45)(20,0){2}{\line(0,1){10}} \multiput(120,20)(-10,10){2}{\line(1,1){10}} \multiput(120,20)(10,10){2}{\line(-1,1){10}} \put(120,10){\line(0,1){40}} \put(120,10){\line(1,0){80}} \put(160,50){\line(1,0){40}} \put(110,30){\makebox(0,0)[r]{$vu_{ev}\Big\downarrow$}} \put(140,43){\makebox(0,0)[t]{$r_{av}$}} \put(120,78){\makebox(0,0)[b]{$R$}} \put(50,58){\makebox(0,0)[b]{$R_1$}} \put(75,30){\makebox(0,0)[r]{$u_{ev}\Big\downarrow$}} \put(165,30){\makebox(0,0)[l]{$\Big\downarrow u_{a}$}} \multiput(200,10)(0,30){2}{\line(0,1){10}} \put(190,30){\line(1,0){20}} \put(200,30){\circle{20}} \put(210,30){\makebox(0,0)[l]{$\Big\uparrow i_{a}$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \[u_a = - v \cdot u_{ev} + (i_a - i)r_a \approx -v u_{ev} + i_a r_a, \quad i \ll i_a \] \[u_{ev} = \frac{R}{R+R_1} u_e + \frac{R_1}{R_1 + R}u_a\] \[u_a = \underbrace{\frac{R}{R+R_1} \cdot \frac{-v}{1+\frac{R_1}{R_1+R}v}}_{v'}u_e + \underbrace{\frac{r_{av}}{1+\frac{R_1}{R_1+R}v}}_{r_a'} i_a\] \end{minipage}% \begin{minipage}{7cm} \center \begin{picture}(110,100) \put(0,40){\makebox(0,0)[l]{$u_e$}} \put(15,40){\circle*3} \put(15,40){\line(1,0){40}} \put(15,40){\vector(1,0){22}} \put(55,40){\circle*3} \put(55,43){\makebox(0,0)[b]{$u_{ev}$}} \put(35,37){\makebox(0,0)[t]{$\frac{R}{R_1+R}$}} \put(55,40){\line(1,0){40}} \put(55,40){\vector(1,0){22}} \put(75,43){\makebox(0,0)[b]{$-v$}} \put(95,40){\circle*3} \qbezier(95,40)(75,20)(55,40) \put(73,30){\vector(-1,0){0}} \put(75,27){\makebox(0,0)[t]{$\frac{R_1}{R_1+R}$}} \put(95,80){\line(0,-1){40}} \put(95,80){\vector(0,-1){22}} \put(95,80){\circle*3} \put(98,60){\makebox(0,0)[l]{$r_{av}$}} \put(95,83){\makebox(0,0)[b]{$i_{a}$}} \put(98,40){\makebox(0,0)[l]{$u_{a}$}} \end{picture} \end{minipage}% \begin{minipage}{9cm} \[r_a' = \frac{r_a}{1+vk} = \frac{r_a}{g}, \qquad v' \approx -\frac{R}{R_1+R} \frac{R_1+R}{R_1} = - \frac{R}{R_1}\] \paragraph{Beispiel:} $k = 0{,}1 \to v' = 10$, $v= 10^5$, $ r_{av} = 100\,\Omega$ \[\Rightarrow r_{a}' = \frac{100\,\Omega}{1+10^4} \approx 0{,}01\,\Omega = 10\,\mathrm m\Omega\] \end{minipage}% \paragraph{Beispiel 2:} MOS-Stromquelle \begin{minipage}{6cm} \center \begin{picture}(80,130) \multiput(5,0)(0,40){2}{\line(0,1){20}} \put(50,68){\line(0,1){12}} \put(50,0){\line(0,1){20}} \multiput(45,20)(10,0){2}{\line(0,1){20}} \multiput(45,20)(0,20){2}{\line(1,0){10}} \put(42,30){\makebox(0,0)[r]{$R_S$}} \put(5,0){\line(0,1){60}} \put(5,60){\line(1,0){30}} \put(35,50){\thicklines\line(0,1){20}} \multiput(38,52)(0,16){2}{\line(1,0){12}} \put(50,60){\vector(-1,0){12}} \put(50,60){\line(0,-1){10}} \put(50,40){\line(0,1){10}} \put(38,50){\line(0,1){20}} \multiput(0,0)(45,0){2}{\line(1,0){10}} \put(50,105){\vector(0,-1){20}} \put(53,95){\makebox(0,0)[l]{$I_a$}} \put(5,30){\circle{20}} \put(-5,30){\makebox(0,0)[r]{$U_G\Big\downarrow$}} \qbezier(55,75)(70,35)(55,5) \put(55,5){\vector(-1,-2){0}} \put(66,35){\makebox(0,0)[l]{$U_a$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \centering \begin{picture}(160,130) \put(30,0){\line(0,1){60}} \put(25,0){\line(1,0){10}} \put(30,30){\circle{20}} \put(20,30){\makebox(0,0)[r]{$u_G\Big\downarrow$}} \put(30,60){\line(1,0){20}} \put(75,0){\line(1,0){10}} \multiput(80,0)(0,40){2}{\line(0,1){20}} \multiput(75,20)(10,0){2}{\line(0,1){20}} \multiput(75,20)(0,20){2}{\line(1,0){10}} \multiput(80,40)(0,40){2}{\line(0,1){20}} \multiput(80,60)(-10,10){2}{\line(1,1){10}} \multiput(80,60)(10,10){2}{\line(-1,1){10}} \put(70,70){\line(1,0){20}} \put(110,50){\line(0,1){10}} \put(110,80){\line(0,1){20}} \multiput(105,60)(10,0){2}{\line(0,1){20}} \multiput(105,60)(0,20){2}{\line(1,0){10}} \put(80,100){\line(1,0){70}} \put(80,50){\line(1,0){30}} \put(80,50){\circle*{2}} \put(110,100){\circle*{2}} \put(87,30){\makebox(0,0)[l]{$R_S$}} \put(73,30){\makebox(0,0)[r]{$i_a\Big\downarrow$}} \put(110,45){\vector(0,-1){45}} \put(113,25){\makebox(0,0)[l]{$u_S$}} \put(118,70){\makebox(0,0)[l]{$r_{DS}$}} \put(73,70){\makebox(0,0)[r]{$g_mu_{GS}\Big\downarrow$}} \qbezier(53,60)(55,50)(76,50) \put(76,50){\vector(1,0){0}} \put(60,49){\makebox(0,0)[t]{$u_{GS}$}} \put(150,0){\line(0,1){100}} \put(145,0){\line(1,0){10}} \put(150,50){\circle{20}} \put(160,50){\makebox(0,0)[l]{$\Big\downarrow u_a$}} \put(153,80){\makebox(0,0)[l]{$\Big\uparrow i_a$}} \end{picture} \end{minipage}% \paragraph{gesucht:} $i_a$ in Abh"angigkeit von $u_G$, $u_a$. \begin{minipage}{6cm} \center \begin{picture}(120,120) \put(0,60){\makebox(0,0)[l]{$u_G$}} \put(35,63){\makebox(0,0)[b]{$1$}} \put(55,63){\makebox(0,0)[b]{$u_{GS}$}} \put(75,63){\makebox(0,0)[b]{$g_m$}} \multiput(15,60)(40,0){3}{\circle*3} \multiput(15,60)(40,0){2}{\line(1,0){40}} \multiput(15,60)(40,0){2}{\vector(1,0){22}} \put(55,20){\circle*3} \put(55,20){\line(0,1){40}} \put(55,20){\vector(0,1){22}} \put(55,17){\makebox(0,0)[t]{$u_S$}} \put(52,40){\makebox(0,0)[r]{$-1$}} \put(55,20){\line(1,1){40}} \put(55,20){\vector(1,1){22}} \put(75,40){\makebox(0,0)[br]{$\frac{-1}{r_{DS}}$}} \qbezier(55,20)(95,20)(95,60) \put(83,28){\vector(-1,-1){0}} \put(84,27){\makebox(0,0)[tl]{$R_S$}} \put(97,60){\makebox(0,0)[l]{$i_a$}} \put(95,100){\circle*3} \put(95,100){\line(0,-1){40}} \put(95,100){\vector(0,-1){22}} \put(95,103){\makebox(0,0)[b]{$u_a$}} \put(98,80){\makebox(0,0)[l]{$\frac{1}{r_{DS}}$}} \end{picture} \end{minipage}% \begin{minipage}{10cm} \[i_a = g_m \cdot u_{GS} + \frac{u_a-u_S}{r_{DS}}\] \[u_s = R_S \cdot i_a\] \[u_{GS} = u_G - U_S\] \[i_a = \underbrace{\frac{g_m}{1+g_mR_s+\frac{R_S}{r_{DS}}}}_{g_m'}u_G + \underbrace{\frac{1}{r_{DS}\left(1+g_mR_S+\frac{R_S}{r_{DS}} \right)}}_{g'_a = \frac{1}{r'_a}}u_a\] \end{minipage}% \paragraph{Beispiel:} $g_m = 5\,\mathrm{mS}$, $R_S = 10\,\mathrm k\Omega$, $r_{DS} = 20\,\mathrm k\Omega \quad \Rightarrow \quad r_a' = r_{DS} + R_S(g_mr_{DS} + 1) = 1{,}03\,\mathrm{M}\Omega$ \subsection{Arbeitspunktstabilisierung} Temperaturabh"angigkeit von Bipolartransistoren. \paragraph{Modellgleichungen:} \begin{align*} I_B = \cancelto{}{I_S} \frac{1}{\cancelto{}{B_F}} \ e^{\frac{U_{BE}}{\cancelto{}{U_T}}} &= Y_1 (U_{BC}, T)\\ I_C = \cancelto{}{I_S} \ e^{\frac{U_{BE}}{\cancelto{}{U_T}}} &= Y_2 (U_{BE}, T) \end{align*} \paragraph{Kleinsignaldarstellung:} \begin{align*} i_B &= g_{BE} u_{BE} + \alpha_B \vartheta \qquad \vartheta = T - T_0\\[1ex] g_{BE} &= \left.\frac{\partial I_B}{\partial U_{BE}}\right|_A = \frac{I_{CA}}{B_FU_T}\\[1ex] \alpha_B &= \left.\frac{\partial I_B}{\partial T}\right|_{T_0} = \left.\frac{\partial}{\partial T}\left(\frac{I_C}{B_F}\right)\right|_{T_0} \frac{I_{BA}}{U_T} = \frac{1}{B_F} \left.\frac{\partial I_C}{\partial T}\right|_{T_0} - \left.\frac{I_{CA}}{{B_F}^2}\frac{\partial B_F}{\partial T}\right|_{T_0} \\[1ex] &\approx 0{,}07\frac{I_{BA0}}{\mathrm K} \qquad I_{BA0}\ldots\text{AP-Strom bei Bezugstemperatur }T_0\\[1ex] i_C &= g_m u_{BE} + \alpha_C \vartheta\\[1ex] g_m &= \left.\frac{\partial I_C}{\partial U_{BE}}\right|_A = \frac{I_{CA}}{U_T}\\[1ex] \alpha_C &= \left.\frac{\partial I_C}{\partial T}\right|_{T_0} \approx 0{,}08\cdot \frac{I_{CA0}}{\mathrm K} \approx B_F \cdot \alpha_B \end{align*} \subsubsection*{Netzwerkmodell} \begin{minipage}{10cm} \centering \begin{picture}(250,80) \multiput(19,0)(0,60){2}{\circle{2}} \multiput(20,0)(0,60){2}{\line(1,0){80}} \multiput(125,0)(0,60){2}{\line(1,0){105}} \multiput(231,0)(0,60){2}{\circle{2}} \multiput(60,0)(25,0){2}{ \multiput(40,0)(0,40){2}{\line(0,1){20}} \put(30,30){\line(1,0){20}} } \put(100,30){\circle{20}} \multiput(125,20)(-10,10){2}{\line(1,1){10}} \multiput(125,20)(10,10){2}{\line(-1,1){10}} \put(90,30){\makebox(0,0)[r]{$\alpha_B\vartheta\Big\downarrow$}} \multiput(50,0)(0,60){2}{\circle*2} \multiput(50,0)(0,40){2}{\line(0,1){20}} \multiput(45,20)(0,20){2}{\line(1,0){10}} \multiput(45,20)(10,0){2}{\line(0,1){20}} \put(43,30){\makebox(0,0)[r]{$g_{BE}$}} \put(19,57){\vector(0,-1){54}} \put(17,30){\makebox(0,0)[r]{$u_{BE}$}} \put(231,57){\vector(0,-1){54}} \put(233,30){\makebox(0,0)[l]{$u_{CE}$}} \put(0,60){\vector(1,0){16}} \put(8,63){\makebox(0,0)[b]{$i_B$}} \put(250,60){\vector(-1,0){16}} \put(243,63){\makebox(0,0)[b]{$i_{C}$}} \put(135,30){\makebox(0,0)[l]{$\Big\downarrow g_mu_{BE}$}} \multiput(190,0)(0,40){2}{\line(0,1){20}} \multiput(190,0)(0,60){2}{\circle*2} \put(190,30){\circle{20}} \put(180,30){\line(1,0){20}} \put(200,30){\makebox(0,0)[l]{$\Big\downarrow\alpha_C\vartheta$}} \end{picture} \end{minipage}% \begin{minipage}{6cm} \[\boxed{\begin{array}{ccccc} i_B &=& g_{BE} \cdot u_{BE}& +& \alpha_B \vartheta \\ i_C &=& g_{m} \cdot u_{BE} &+& \alpha_C \vartheta \end{array}} \] \end{minipage}% \bigskip \begin{align*} i_B &= g_{BE} \left(u_{BE} + \frac{\alpha_B}{g_{BE}}\vartheta \right) = g_{BE} (u_{BE} + d_B \vartheta), & d_B &= \frac{\alpha_B}{g_{BE}} \approx 2\, \frac{\mathrm{mV}}{\mathrm K}\\[1ex] i_C &= g_m \left(u_{BE} + \frac{\alpha_C}{g_{m}}\vartheta \right) = g_m (\underbrace{u_{BE} + d_C \vartheta}_{u_{BET}}), & d_C &= \frac{\alpha_C}{g_{m}} \approx 2{,}3\, \frac{\mathrm{mV}}{\mathrm K} \end{align*} \bigskip \begin{minipage}{7.5cm} \centering \begin{picture}(195,80) \multiput(0,0)(0,60){2}{\circle2} \put(1,0){\line(1,0){190}} \put(40,0){\line(0,1){10}} \multiput(35,10)(10,0){2}{\line(0,1){20}} \multiput(35,10)(0,20){2}{\line(1,0){10}} \put(40,30){\line(0,1){30}} \put(40,45){\circle{15}} \put(1,60){\line(1,0){80}} \put(82,60){\circle2} \multiput(40,0)(0,60){2}{\circle*2} \put(60,60){\circle{15}} \put(32,20){\makebox(0,0)[r]{$g_{BE}$}} \put(28,45){\makebox(0,0)[r]{$d_B\vartheta$}} \put(30,35){\vector(0,1){20}} \put(60,73){\makebox(0,0)[b]{$d_C\vartheta$}} \put(70,70){\vector(-1,0){20}} \put(82,55){\vector(0,-1){50}} \put(85,30){\makebox(0,0)[l]{$u_{BET}$}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \multiput(130,20)(-10,10){2}{\line(1,1){10}} \multiput(130,20)(10,10){2}{\line(-1,1){10}} \put(120,30){\line(1,0){20}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow g_m u_{BET}$}} \put(130,60){\line(1,0){61}} \multiput(192,0)(0,60){2}{\circle2} \end{picture} \end{minipage}% \begin{minipage}{1cm} \centering \bigskip \smallskip $\widehat{=}$ \end{minipage}% \begin{minipage}{7.5cm} \centering \begin{picture}(195,80) \multiput(0,0)(0,60){2}{\circle2} \put(1,0){\line(1,0){190}} \multiput(40,0)(0,40){2}{\line(0,1){20}} \multiput(35,20)(10,0){2}{\line(0,1){20}} \multiput(35,20)(0,20){2}{\line(1,0){10}} \put(1,60){\line(1,0){80}} \put(82,60){\circle2} \multiput(40,0)(0,60){2}{\circle*2} \multiput(0,0)(-40,0){2}{ \put(60,60){\circle{15}} \put(70,70){\vector(-1,0){20}} } \put(32,20){\makebox(0,0)[r]{$g_{BE}$}} \put(60,73){\makebox(0,0)[b]{$\scriptstyle (d_C-d_B)\vartheta$}} \put(20,73){\makebox(0,0)[b]{$d_B\vartheta$}} \put(82,55){\vector(0,-1){50}} \put(85,30){\makebox(0,0)[l]{$u_{BET}$}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \multiput(130,20)(-10,10){2}{\line(1,1){10}} \multiput(130,20)(10,10){2}{\line(-1,1){10}} \put(120,30){\line(1,0){20}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow g_m u_{BET}$}} \put(130,60){\line(1,0){61}} \multiput(192,0)(0,60){2}{\circle2} \end{picture} \end{minipage}% \bigskip \hfill $d_C-d_B \approx 0{,}26\frac{\mathrm{mV}}{\mathrm K}$ \paragraph{Beispiel 1:} Basisspannungseinspeisung \smallskip $U_{0C}=10\,\mathrm V$, $U_{BE0} = 0{,}65\,\mathrm V$, $R_C = 10\,\mathrm k\Omega$, $I_{CA} = 0{,}5\,\mathrm{mA}$, $B_F = b = 100$ \bigskip \begin{minipage}{5cm} \centering \begin{picture}(100,100) \put(30,0){\line(0,1){40}} \put(25,0){\line(1,0){10}} \put(30,20){\circle{20}} \put(20,20){\makebox(0,0)[r]{$U_{BE}\Big\downarrow$}} \put(30,40){\line(1,0){20}} \put(50,30){\thicklines\line(0,1){20}} \put(50,40){\line(1,1){10}} \put(50,40){\line(1,-1){10}} \put(58,32){\vector(1,-1){0}} \put(55,0){\line(1,0){10}} \put(60,0){\line(0,1){30}} \multiput(60,50)(0,30){2}{\line(0,1){10}} \multiput(55,60)(10,0){2}{\line(0,1){20}} \multiput(55,60)(0,20){2}{\line(1,0){10}} \put(60,91){\circle{2}} \put(60,50){\line(1,0){20}} \put(60,50){\circle*2} \put(81,50){\circle2} \put(68,70){\makebox(0,0)[l]{$R_C$}} \put(57,91){\makebox(0,0)[r]{$U_{0C}$}} \put(81,45){\vector(0,-1){40}} \put(83,25){\makebox(0,0)[l]{$U_a$}} \put(76,0){\line(1,0){10}} \end{picture} \bigskip \end{minipage}% \begin{minipage}{11cm} \centering \begin{picture}(310,60) \put(0,0){\line(0,1){60}} \multiput(40,0)(0,40){2}{\line(0,1){20}} \put(30,30){\line(1,0){20}} \put(40,30){\circle{20}} \put(30,30){\makebox(0,0)[r]{$\alpha_B\vartheta\Big\downarrow$}} \multiput(0,0)(0,60){2}{\line(1,0){80}} \multiput(40,0)(0,60){2}{\circle*2} \multiput(80,0)(0,40){2}{\line(0,1){20}} \multiput(75,20)(10,0){2}{\line(0,1){20}} \multiput(75,20)(0,20){2}{\line(1,0){10}} \put(73,30){\makebox(0,0)[r]{$g_{BE}$}} \qbezier(85,10)(95,30)(85,50) \put(85,10){\vector(-1,-2){0}} \put(93,30){\makebox(0,0)[l]{$u_{BE}$}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \multiput(130,20)(-10,10){2}{\line(1,1){10}} \multiput(130,20)(10,10){2}{\line(-1,1){10}} \put(120,30){\line(1,0){20}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow g_mu_{BE}$}} \multiput(200,0)(0,40){2}{\line(0,1){20}} \put(200,30){\circle{20}} \multiput(200,0)(0,60){2}{\circle*2} \put(190,30){\line(1,0){20}} \put(210,30){\makebox(0,0)[l]{$\Big\downarrow \alpha_C\vartheta$}} \multiput(255,0)(0,40){2}{\line(0,1){20}} \multiput(250,20)(10,0){2}{\line(0,1){20}} \multiput(250,20)(0,20){2}{\line(1,0){10}} \multiput(255,0)(0,60){2}{\circle*2} \put(263,30){\makebox(0,0)[l]{$R_C$}} \multiput(130,0)(0,60){2}{\line(1,0){160}} \multiput(291,0)(0,60){2}{\circle2} \put(291,55){\vector(0,-1){50}} \put(294,30){\makebox(0,0)[l]{$u_a$}} \end{picture} \[u_a = - \alpha_C \vartheta R_C = - 0{,}08 \frac{0{,}5\,\mathrm{mA}}{\mathrm K}\cdot 10\,\mathrm k\Omega = - 0{,}4\,\frac{\mathrm V}{\mathrm K} \cdot \vartheta\] \end{minipage} \paragraph{Beispiel 2:} Basisstromeinspeisung \bigskip \begin{minipage}{5cm} \centering \begin{picture}(100,100) \put(-15,0){ \put(30,20){\line(1,0){20}} \put(50,10){\thicklines\line(0,1){20}} \put(50,20){\line(1,1){10}} \put(50,20){\line(1,-1){10}} \put(58,12){\vector(1,-1){0}} \put(55,0){\line(1,0){10}} \put(60,0){\line(0,1){10}} \multiput(60,30)(0,40){2}{\line(0,1){20}} \multiput(55,50)(10,0){2}{\line(0,1){20}} \multiput(55,50)(0,20){2}{\line(1,0){10}} \multiput(30,20)(0,40){2}{\line(0,1){20}} \multiput(25,40)(10,0){2}{\line(0,1){20}} \multiput(25,40)(0,20){2}{\line(1,0){10}} \put(30,20){\line(0,1){20}} \put(30,80){\line(1,0){30}} \put(60,80){\circle*2} \put(60,91){\circle{2}} \put(60,30){\line(1,0){20}} \put(60,30){\circle*2} \put(81,30){\circle2} \put(68,60){\makebox(0,0)[l]{$R_C$}} \put(23,50){\makebox(0,0)[r]{$R_B$}} \put(63,91){\makebox(0,0)[l]{$U_{0C}$}} \put(81,25){\vector(0,-1){20}} \put(83,12.5){\makebox(0,0)[l]{$U_a$}} \put(76,0){\line(1,0){10}} } \end{picture} \end{minipage}% \begin{minipage}{11cm} \centering \begin{picture}(310,60) \multiput(0,0)(0,40){2}{\line(0,1){20}} \multiput(-5,20)(10,0){2}{\line(0,1){20}} \multiput(-5,20)(0,20){2}{\line(1,0){10}} \put(-8,30){\makebox(0,0)[r]{$R_B$}} \multiput(40,0)(0,40){2}{\line(0,1){20}} \put(30,30){\line(1,0){20}} \put(40,30){\circle{20}} \put(30,30){\makebox(0,0)[r]{$\alpha_B\vartheta\!\Big\downarrow\!$}} \multiput(0,0)(0,60){2}{\line(1,0){80}} \multiput(40,0)(0,60){2}{\circle*2} \multiput(80,0)(0,40){2}{\line(0,1){20}} \multiput(75,20)(10,0){2}{\line(0,1){20}} \multiput(75,20)(0,20){2}{\line(1,0){10}} \put(73,30){\makebox(0,0)[r]{$g_{BE}$}} \qbezier(85,10)(95,30)(85,50) \put(85,10){\vector(-1,-2){0}} \put(93,30){\makebox(0,0)[l]{$u_{BE}$}} \multiput(130,0)(0,40){2}{\line(0,1){20}} \multiput(130,20)(-10,10){2}{\line(1,1){10}} \multiput(130,20)(10,10){2}{\line(-1,1){10}} \put(120,30){\line(1,0){20}} \put(140,30){\makebox(0,0)[l]{$\Big\downarrow g_mu_{BE}$}} \multiput(200,0)(0,40){2}{\line(0,1){20}} \put(200,30){\circle{20}} \multiput(200,0)(0,60){2}{\circle*2} \put(190,30){\line(1,0){20}} \put(210,30){\makebox(0,0)[l]{$\Big\downarrow \alpha_C\vartheta$}} \multiput(255,0)(0,40){2}{\line(0,1){20}} \multiput(250,20)(10,0){2}{\line(0,1){20}} \multiput(250,20)(0,20){2}{\line(1,0){10}} \multiput(255,0)(0,60){2}{\circle*2} \put(263,30){\makebox(0,0)[l]{$R_C$}} \multiput(130,0)(0,60){2}{\line(1,0){160}} \multiput(291,0)(0,60){2}{\circle2} \put(291,55){\vector(0,-1){50}} \put(294,30){\makebox(0,0)[l]{$u_a$}} \end{picture} \end{minipage} \begin{align*} u_a &= - (g_m u_{BE} + \alpha_C \vartheta) \cdot R_C \hspace{5cm} u_{BE} = -\frac{\alpha_B \vartheta}{\frac{1}{R_B}+\frac{1}{r_{BE}}}\\ &= \left(\frac{g_m\alpha_B\vartheta r_{BE}}{1+\frac{r_{BE}}{R_B}} - \alpha_C \vartheta\right) R_C = \left(\frac{1}{1+\frac{r_{BE}}{R_B}}\right)\alpha_C\vartheta R_C = - \frac{\frac{r_{BE}}{R_B}}{1+\frac{r_{BE}}{R_B}} R_C \alpha_C\vartheta \end{align*} \[\frac{r_{BE}}{R_B} = \frac{U_T}{I_{BA}} \cot \frac{I_{BA}}{U_{0C}-U_{BE0}} = 2{,}8\cdot 10^{-3}\] \[ \Rightarrow\quad u_a = -1\,\mathrm{mV} \frac{\vartheta}{\mathrm K}\] \begin{center} \begin{picture}(190,70) \put(10,40){\makebox(0,0)[l]{$\vartheta$}} \put(185,40){\makebox(0,0)[r]{$u_a$}} \multiput(20,40)(50,0){4}{\circle*{3}} \put(20,40){\line(1,0){150}} \multiput(20,40)(50,0){3}{\vector(1,0){28}} \put(45,45){\makebox(0,0)[b]{$-R_C\alpha_C$}} \put(95,45){\makebox(0,0)[b]{$\frac{r_{BE}}{R_B}$}} \qbezier(120,40)(95,15)(70,40) \put(93,27.5){\vector(-1,0){0}} \put(95,24){\makebox(0,0)[t]{$-1$}} \end{picture} \end{center} \chapter{Operationsverst"arker} \section{Aufbau und Grundmodell} \begin{center} \begin{picture}(235,70) \multiput(19,10)(0,30){2}{\circle{2}} \multiput(20,10)(0,30){2}{\line(1,0){20}} \multiput(0,0)(60,0){3}{ \put(80,25){\line(1,0){20}} \multiput(40,5)(40,0){2}{\line(0,1){40}} \multiput(40,5)(0,40){2}{\line(1,0){40}} } \put(43,10){\makebox(0,0)[l]{$-$}} \put(43,40){\makebox(0,0)[l]{$+$}} \put(19,43){\makebox(0,0)[b]{$u_+$}} \put(19,7){\makebox(0,0)[t]{$u_-$}} \put(19,35){\vector(0,-1){20}} \put(16,25){\makebox(0,0)[r]{$u_d$}} \put(224,25){\makebox(0,0)[l]{$u_a$}} \put(60,55){\makebox(0,0)[c]{\small Diff.-Verst"arker}} \put(120,55){\makebox(0,0)[c]{\small Treiber}} \put(180,55){\makebox(0,0)[c]{\small Endstufe}} \put(112.5,15){\line(0,1){20}} \put(112.5,15){\line(3,2){15}} \put(112.5,35){\line(3,-2){15}} \thicklines \multiput(180,10)(0,20){2}{\line(0,1){10}} \thinlines \multiput(0,0)(0,20){2}{ \qbezier(180,15)(185,10)(185,10) \qbezier(180,15)(185,20)(185,20) } \put(180,15){\vector(-1,-1){0}} \put(185,30){\vector(1,-1){0}} \put(185,20){\line(0,1){10}} \put(185,25){\line(1,0){15}} \put(221,25){\circle2} \multiput(185,0)(0,40){2}{\line(0,1){10}} \end{picture} \end{center} \paragraph{Grundmodell:} $u_a = v \cdot u_d + r_a i_a= v(u_+ - u_-) + r_ai_a$ \begin{minipage}{8cm} \center \begin{picture}(70,90) \put(13,50){\makebox(0,0)[r]{$u_+$}} \put(13,10){\makebox(0,0)[r]{$u_-$}} \multiput(15,10)(0,40){2}{\circle*3} \multiput(35,30)(30,0){2}{\circle*3} \put(15,10){\line(1,1){20}} \put(15,10){\vector(1,1){11}} \put(15,50){\line(1,-1){20}} \put(15,50){\vector(1,-1){11}} \put(65,60){\circle*3} \put(35,30){\line(1,0){30}} \put(35,30){\vector(1,0){17}} \put(65,60){\vector(0,-1){17}} \put(65,60){\line(0,-1){30}} \put(25,20){\makebox(0,0)[tl]{$-1$}} \put(25,40){\makebox(0,0)[bl]{$\,1$}} \put(50,35){\makebox(0,0)[b]{$v$}} \put(68,45){\makebox(0,0)[l]{$r_a$}} \put(65,64){\makebox(0,0)[b]{$i_a$}} \end{picture} \end{minipage}% \begin{minipage}{8cm} \begin{picture}(100,90) \multiput(15,0)(20,0){2}{\line(1,0){10}} \multiput(20,60)(20,-10){2}{\circle2} \put(20,55){\vector(0,-1){50}} \put(40,45){\vector(0,-1){40}} \put(17,60){\makebox(0,0)[r]{$u_+$}} \put(43,50){\makebox(0,0)[l]{$u_-$}} \put(23,58){\vector(2,-1){14}} \put(31,58){\makebox(0,0)[b]{$u_d$}} \put(70,0){\line(0,1){60}} \multiput(70,20)(-10,10){2}{\line(1,1){10}} \multiput(70,20)(10,10){2}{\line(-1,1){10}} \put(65,0){\line(1,0){10}} \multiput(70,60)(40,0){2}{\line(1,0){20}} \multiput(90,55)(0,10){2}{\line(1,0){20}} \multiput(90,55)(20,0){2}{\line(0,1){10}} \put(131,60){\circle{2}} \put(131,55){\vector(0,-1){50}} \put(126,0){\line(1,0){10}} \put(80,30){\makebox(0,0)[l]{$\Big\downarrow v\cdot u_d$}} \put(134,30){\makebox(0,0)[l]{$u_a$}} \put(100,68){\makebox(0,0)[b]{$r_a$}} \put(142.5,63){\makebox(0,0)[b]{$i_a$}} \put(150,60){\vector(-1,0){15}} \end{picture} \end{minipage}% \bigskip im Bildbereich: \[\underline U_a(s) = \frac{v}{3+s\tau} \underline U_d + r_a \cdot \underline I_a, \qquad \tau = \frac{1}{2\pi f_{\infty}} = \frac{1}{\omega_{\infty}}\] \section{Analyseverfahren (idealer OPV)} $r_e \to \infty$, $I_+ = I_- = 0$, $r_a \to 0$, $v \to \infty$ \end{document}