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structure/tex/latex/packages/latex4ei.sty delete mode 100644 packages/intern/miktex structure/tex/latex/packages/scientific.sty delete mode 100644 packages/intern/scientific.sty diff --git a/.DS_Store b/.DS_Store deleted file mode 100644 index 4a558c3d10eb7cf388eec50ed89ad5ffb7c9a421..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6148 zcmeHK%}T>S5T3PFQ&i|h^tgB~(o&%nJcLrgqmV)mRx~lC4VI)dsXf$ad?8Ohi_hcC z?nWpL9!2a7?0&QJvzz%K`vU-?Gxr++RRFNC5eiZYgv^DmEfY*A)Eq91U@9bKYinTxJ2caJ=I?+7z;~Qrj zEu6p~dhYC2>4}@nm%)&_&j?kCqU7yKcCL}?f-V%h;plvYIh#nN7 zQxSElFi#Aj)6p(XoNX}EsMA5Hm2n=ka{hP`YIU?r6%NAI$UQT_46HIx(!(m9|Cjj7 z6h87-Q+UJ-Fa!UL0a5Nc-3}II&epZ^=&Y64?y-?jTt*5C>MNH39MC>;ten~}Nyj+b XV5X5~A-hUP 0:$ System stabil \\ $\tau < 0:$ System instabil \\ } \qquad \pbox{5.0cm}{ \includegraphics{./img/char/char_Schaltung1Grad.pdf} } \\ \boxed{ \text{Lösung:}\quad x(t) = x_\infty + (x_0 - x_\infty) \exp\left(- \frac{t-t_0}{\tau}\right) }\\ + Mit Parameter aus ESB: C: $x_\infty = U_0(t)$, L: $x_\infty = I_0(t)$ \\ Beispiele: \hspace{3cm} $u_C(t) = A \cdot \sin(\omega t)$\\ \pbox{5.0cm}{ \includegraphics[scale = 0.8]{./img/Schaltung1Grad.pdf} \\ \\ $\dot u(t) = - \frac{u(t)}{RC} + \frac{U_0 + RI_0}{RC}$ } @@ -293,7 +294,7 @@ \section{Dynamische Schaltungen} \lambda_{1/2} = \frac{\mathrm{sp} \ma A}{2} \pm \sqrt{ \left( \frac{\mathrm{sp} \ma A}{2} \right)^2 - \det \ma A } } \quad \boxed{ \begin{array}{ll} \text{Wähle} \\ \bigl| \lambda_1 \bigr| < \bigl|\lambda_2 \bigr| \end{array}}\\[0.5em] $\mathrm{Sp } \ma A = T = a_{11} + a_{22}$, \quad $\det \ma A = \Delta = a_{11} a_{22} - a_{12} a_{21}$\\ \\ - Zeitkonstante $\tau = -\frac{1}{\lambda}$ + Zeitkonstante $\tau = RC = GL$ % = -\frac{1}{\lambda}$ \subsection{Eigenvektoren(EV) $\vec q$ bestimmen} @@ -308,7 +309,8 @@ \section{Dynamische Schaltungen} \end{tabular}\\ \\ Falls $a_{12} = a_{21} = 0:$ $q_1 = \mvect{1 \\ 0} \quad q_2 = \mvect{0 \\ 1}$ - \fbox{ Eigenschaften EV: $\vec q_1 \perp \vec q_2 $} + \fbox{ Eigenschaften EV (falls gefragt kann man das prüfen): $\vec q_1 \perp \vec q_2 $} + \subsection{Gleichgewichtspunkte(GGP) $\vec x_\infty$ bestimmen} $\vec x_\infty$: $\bs{ \dot {\vec x}} = 0 = \ma A \vec x_\infty + \ma B \vec v \quad \Ra \quad$ \boxed { \vec x_\infty = - \ma A^{-1} \ma B \vec v }\\[0.5em] Oder aus Schaltbild berechnen:\\ @@ -531,10 +533,57 @@ \section{Lösen von homogenen DGLs} \end{tabular} + % Ende der Spalten \end{multicols} +\hrule \hrule + +\begin{multicols}{3} +\section{Anhang} + + \subsection{Mathematik} + Matritzen:\\ + $\ma{A} = \frac{1}{\det \ma{A} } \begin{bmatrix} + a_{22} & -a_{12}\\ + -a_{21} & a_{11} + \end{bmatrix}$ + \\ \\ + %\subsection{Quadratische Lösungsformel} + Quadratische Lösungsformel: \\ + $ax^2+ bx + c = 0$ :\\ + \boxed{x_{1/2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} } \\ \\ + Einfache quadratische Lösungsformel: \\ + $x^2 + px + q = 0$ :\\ + \boxed{x_{1/2} = -\frac{p}{2} \pm \sqrt{\left ( \frac{p}{2}\right )^2 - q}} \\ + + \subsection{Ohmsches Gesetz} + \begin{tabular}{ll} + Basis: & \\ + $U = R \cdot I$ & $I = G \cdot U$ \\ + & \\ + Spannungsteiler: & Stromteiler: \\ + $u_2 = \frac{R_2}{R_1+R_2} \cdot u_1$ & $\frac{i_1}{i_ges} = \frac{R-ges}{ R_1} = \frac{G_1}{u}$ \\ + \end{tabular} + + \subsection{Zweitorbeschreibungen} + \begin{tabular}{cll} + I & Widerstandsbeschreibung:\\ + & $\left.\begin{matrix} u_1\\ u_2\end{matrix}\right]=\left.\begin{matrix} r_1(i_1,i_2)\\ r_2(i_1,i_2) \end{matrix}\right] = \ma{R} \cdot \left.\begin{matrix} i_1\\ i_2 \end{matrix}\right]$\\ + II & Leitwertsbeschreibung:\\ + & $\left.\begin{matrix} i_1\\ i_2\end{matrix}\right]=\ma{G} \cdot \left.\begin{matrix} u_1\\ u_2 \end{matrix}\right]$\\ + III & Hybridebeschreibung:\\ + & $\left.\begin{matrix} u_1\\ i_2\end{matrix}\right]=\ma{H} \cdot \left.\begin{matrix} i_1\\ u_2 \end{matrix}\right]$\\ + IV & Inverse Hybridbeschreibung:\\ + & $\left.\begin{matrix} i_1\\ u_2\end{matrix}\right]=\ma{H'} \cdot \left.\begin{matrix} u_1\\ i_2 \end{matrix}\right]$\\ + V & Kettenbeschreibung:\\ + & $\left.\begin{matrix} u_1\\ i_1\end{matrix}\right]=\ma{A} \cdot \left.\begin{matrix} u_2\\ -i_2 \end{matrix}\right]$\\ + VI & Inverse Kettenbeschreibung:\\ + & $\left.\begin{matrix} u_2\\ i_2\end{matrix}\right]=\ma{A'} \cdot \left.\begin{matrix} u_1\\ -i_1 \end{matrix}\right]$\\ + \end{tabular} +% Ende der Spalten +\end{multicols} % Dokumentende % ====================================================================== diff --git a/Schaltungstechnik.tex b/Schaltungstechnik.tex index 7af30b2..5c9c239 100644 --- a/Schaltungstechnik.tex +++ b/Schaltungstechnik.tex @@ -1,5 +1,10 @@ % .:: Laden der LaTeX4EI Formelsammlungsvorlage \documentclass[fs, footer]{latex4ei} +\usepackage[european]{circuitikz} + +\usepackage{multirow} + + % Dokumentbeginn % ====================================================================== @@ -10,12 +15,16 @@ \vspace{-4mm} \begin{multicols*}{4} \fstitle{Schaltungstechnik 1} - + + \emphbox{ + \textbf{Wichtiger Hinweis} + \\ Diese Formelsammlung ist noch in der Entwicklung und nicht Prüfungstauglich ! \\ Allerdings würden wir uns über Unterstützung freuen das zu ändern. Wer Lust hat kann uns über das Kontaktformular auf www.latex4ei.de erreichen. + } % ------------------------------------------- % | Schaltungstechnik | % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ % SECTION ==================================================================================== -\section{Mathematische Grundlagen} +\section{Grundlagen} % ============================================================================================ \sectionbox{ @@ -32,6 +41,24 @@ \subsection{Sinus, Cosinus \quad $\sin^2(x) \bs + \cos^2(x) = 1$} } +Verbraucherpfeilsystem: Spannung und Strom sind assoziiert. Positive Leistung bedeutet Leistungsaufnahme oder Verbrauch von Leistung. +Negative Leistung bedeutet Leistungsabgabe oder Erzeugung von Leistung.\\ +Das Gegenteil wäre das Erzeugersystem.\ +In den meisten Fachgebieten wird im Verbrauchersystem gerechnet. + +\sectionbox{ + \subsection{Begriffserklärung, Glossar} + \begin{description}%\itemsep0pt + \item[Zählpfeile:] Zeigen die gemeinsame(assoziierte) Zählrichtung von Stromfluss und Spannungsabfall zwischen zwei Knoten an, unabhängig von den tatsächlichen Richtungen (Vorzeichen). + \item[Masse (Signale) $\perp$:] Gemeinsamer el. Bezugspunkt mit Potential $0V$ + \item[Erdung (Fehlstrom):] Schutz vor Kurzschluss, führt nur im Fehlerfall Strom. + \item[Kurzschluss (KS):] ideal leitender Draht. $u_{\ir KS} = 0$, $i_{\ir KS}=\; \mathrm{beliebig}$ + \item[Leerlauf (LL):] ideal isolierende Luft. $u_{\ir LL}=\; \mathrm{beliebig}$, $i_{\ir LL} = 0$ + \item[Tor:] Ein Tor bilden zwei Anschlüsse bei denen der Stromzufluss des einen Anschluss gleich dem Stromabfluss des anderen Anschluss entspricht. $i_{\ir in} = i_{\ir out}$ + \item[Arbeitspunnkt (AP):] Betriebspunkt bei dem alle Kleinsignalquellen Null sind. + \end{description} +} + % SECTION ==================================================================================== \section{Netzwerktheorie} % ============================================================================================ @@ -42,8 +69,20 @@ \subsection{Kirchhoff-Gesetze} Konzentriertheitshypothese: $d << \lambda$ mit \\ $d = $ Größe der Schaltung, Wellenlänge $\lambda = c T$ \\ -Knotenregel \emph{KCL}: $ \sum \limits_{\text{Knoten}}i_{\ir j} (t) = 0$ (heraußfließende Ströme positiv) \\ -Maschenregel \emph{KVL}: $ \sum \limits_{\text{Umlauf}} u_{\ir j} (t) = 0$ (Spannungen in Umlaufrichtung positiv) +%Knotenregel \emph{KCL}: $ \sum \limits_{\text{Knoten}}i_{\ir j} (t) = 0$ (heraußfließende Ströme positiv) \\ +%Maschenregel \emph{KVL}: $ \sum \limits_{\text{Umlauf}} u_{\ir j} (t) = 0$ (Spannungen in Umlaufrichtung positiv) + +\tablebox{ + \begin{tabular*}{\columnwidth}{@{\extracolsep\fill}ll@{}} \trule + $\underset{\text{Kirchoff's Current Law}}{\text{\large Stromgesetz \textbf{KCL}}}$ & $\underset{\text{Kirchoff's Voltage Law}}{\text{\large Spannungsgesetz \textbf{KVL}}}$ \\ \mrule + Knotenregel & Maschenregel\\[0.5em] + $\displaystyle \sum\limits_{\ir Knoten} i_k (t) = 0 $ & $\displaystyle \sum\limits_{\ir Masche} u_m (t) = 0 $ \\[1.5em] + % ToDo: Bild!! + rausfließende Ströme positiv & Spannungen in Umlaufrichtung positiv \\ + Maxwell: $\mathrm{div}\, \vec j = 0$ & Maxwell: $\mathrm{rot}\, \vec E = 0$ \\ + $(n-1)$ Gleichungen & $b-(n-1)$ Gleichungen\\ \brule + \end{tabular*} +} Knoteninzidenzmatrix: $\ma A' = \mat{\alpha_{11} & \ldots & \alpha_{1b} \\ \vdots & & \\ \alpha_{n1} & \ldots & \alpha_{nb}}$ $n$ Knoten @@ -68,20 +107,10 @@ \subsection{Kirchhoff-Gesetze} Um die Betriebspunkte einer Schaltung zu bestimmen sind $2b$ linear unabhängige Gleichungen nötig. Man erhält diese $2b$ Gleichungen aus den Beschreibungen der Bauelemente und den Kirchoff Gleichungen. - \subsubsection{Wichtige Begriffe} - \begin{description}%\itemsep0pt - \item[Zählpfeile:] Zeigen die gemeinsame(assoziierte) Zählrichtung von Stromfluss und Spannungsabfall zwischen zwei Knoten an, unabhängig von den tatsächlichen Richtungen(Vorzeichen). - \item[Masse(Erdung) $\perp$:] Bezugspunkt des elektr. Potentials mit Potential $0V$ - \item[Kurzschluss(KS):] ideal leitender Draht. $u_{KS} = 0$, $i_{KS}=$beliebig - \item[Leerlauf(LL):] ideal isolierende Luft. $u_{LL}=$beliebig, $i_{LL} = 0$ - \item[Tor:] Ein Tor bilden zwei Anschlüsse bei denen der Stromzufluss des einen Anschluss gleich dem Stromabfluss des anderen Anschluss entspricht. $i_{in} = i_{out}$ - \end{description} - } +\sectionbox{ \subsection{Eintorverschaltungen} - - \tablebox{ \begin{tabular*}{\columnwidth}{@{\extracolsep\fill}ll@{\hspace{1em}}|ll@{}} \ctrule \multicolumn{2}{c}{\large{Serienschaltung}} & \multicolumn{2}{c}{\large{Parallelschaltung}} \\ \ctrule @@ -92,6 +121,25 @@ \subsection{Kirchhoff-Gesetze} $\cx Z=\sum \cx Z_i$ & $\frac{1}{\cx Y} = \sum \frac{1}{\cx Y_i}$ & $\frac{1}{\cx Z} = \sum \frac{1}{\cx Z_i}$ & $\cx Y=\sum \cx Y_i$\\ \cbrule \end{tabular*} } + \emphbox{$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \quad \Ra \quad R = R_1 \parallel R_2 = \frac{R_1 \cdot R_2}{R_1 + R_2}$} +} + +\sectionbox{ + \subsection{Linearisierung} + \begin{tabular}{ll} + Großsignal & Kleinsignal\\ + $i=I_{\ir AP}+\Delta i$ \ & \ $\Delta i= i - I_{\ir AP}$\\ + $u=U_{\ir AP}+\Delta u$ \ & \ $\Delta u= u - U_{\ir AP}$\\ + \end{tabular} + \\ + $\Delta \vec i \approx \ma G \cdot \Delta \vec u$ \qquad $\ma G$ ist die Jakobimatrix $\left. \frac{\partial g_i(\vec u)}{\partial u_j} \right|_{U_{\ir AP}}$\\ + $\Delta \vec u \approx \ma R \cdot \Delta \vec u$\\ + Großsignal: $i \approx I_{\ir AP} + G(U_{\ir AP}) \cdot (u - U_{\ir AP})$\\[1em] + + Implizite Liniarisierung: $\Delta \vec f(\Delta \vec u, \Delta \vec i) = \ma M \Delta \vec u + \ma N \Delta \vec i$\\ + $\ma M := \mat{ \frac{\partial f_1}{\partial u_1} & \frac{\partial f_1}{\partial u_2} \\[0.5em] \frac{\partial f_2}{\partial u_1} & \frac{\partial f_2}{\partial u_2} }$ \quad + $\ma N := \mat{ \frac{\partial f_1}{\partial i_1} & \frac{\partial f_1}{\partial i_2} \\[0.5em] \frac{\partial f_2}{\partial i_1} & \frac{\partial f_2}{\partial i_2} }$ +} \sectionbox{ @@ -130,17 +178,269 @@ \subsection{Resistive Eintore} } + + + +% SECTION ==================================================================================== +\section{Resistive Eintore} +% ============================================================================================ + + + \begin{tabular}{@{}p{1.1cm}p{2.9cm}l} + \parbox{1.3cm}{ Leerlauf \\[5pt] \includegraphics{./img/opencircuit.pdf} } & \parbox{2.5cm}{$u=\mathrm{beliebig}$ \\ $i=0$ \\ u, sl, v} & \parbox{1.8cm}{\includegraphics{./img/char/char_opencircuit.pdf} } \\ \mrule + \parbox{1.3cm}{ Kurzschluss \\[5pt] \includegraphics{./img/shortcircuit.pdf} } & \parbox{2.5cm}{$u = 0$ \\ $i = \mathrm{beliebig}$ \\ u, sl, v} & \parbox{1.8cm}{ \includegraphics{./img/char/char_shortcircuit.pdf} } \\ \mrule + \parbox{1.3cm}{ Nullator \\[5pt] \includegraphics{./img/symb/symb_nullator.pdf} } & \parbox{2.5cm}{$u = 0 \quad i = 0$ \\ u, sl, v, dual zu sich selbst} & \parbox{1.8cm}{\includegraphics{./img/char/char_nullator.pdf} } \\ \mrule + \parbox{1.3cm}{ Norator \\[5pt] \includegraphics{./img/symb/symb_norator.pdf} } & \parbox{2.5cm}{$u = \mathrm{beliebig}$ \\ $i=\mathrm{beliebig}$ \\[0.5em] u, sl, a, dual zu sich selbst} & \parbox{1.8cm}{\includegraphics{./img/char/char_norator.pdf} } \\ \mrule + \parbox{1.3cm}{ Widerstand \\[5pt] \includegraphics{./img/symb/symb_resistor.pdf} }& $\begin{array}{@{}cc} u = R \cdot i \\ i = G \cdot u \end{array} \quad G = \frac{1}{R}$ & \parbox{1.8cm}{\includegraphics{./img/char/char_resistor.pdf} } \\ \mrule + \parbox{1.3cm}{ Spannungsq. \\[3pt] \includegraphics{./img/symb/symb_source_voltage.pdf} }& $\begin{array}{@{}ll} u = U_0 \\ i = \mathrm{beliebig}\end{array}$ & \parbox{1.8cm}{\includegraphics{./img/char/char_source_voltage.pdf} } \\ \mrule + \parbox{1.3cm}{ Stromquelle \\[3pt] \includegraphics{./img/symb/symb_source_current.pdf} }& $\begin{array}{@{}ll} u = \mathrm{beliebig} \\ i = I_0 \end{array}$ & \parbox{1.8cm}{\includegraphics{./img/char/char_source_current.pdf} } \\ \mrule + \parbox{1.3cm}{ ideale Diode \\[5pt] \includegraphics{./img/symb/symb_diode.pdf} }& $\begin{array}{@{}ll} u = 0 & \text{falls } i > 0 \\ i = 0 & \text{falls } u < 0\end{array}$ & \parbox{1.8cm}{\includegraphics{./img/char/char_diode_ideal.pdf} } \\ \mrule + \parbox{1.3cm}{ reale Diode \\[5pt] \includegraphics{./img/symb/symb_diode_real.pdf} }& $\begin{array}{@{}ll} u_D = u_T \cdot \ln \left(\frac{i_D}{I_S} + 1 \right) \\[0.7em] i_D = I_S \cdot \left( \exp \left(\frac{u_D}{U_T}\right) -1 \right) \end{array}$ & \parbox{1.8cm}{\includegraphics{./img/char/char_diode_real.pdf} } \\ \mrule + \parbox{1.3cm}{ Photodiode \\ \includegraphics{./img/symb/symb_diode_photo.pdf} }& $i = I_{\ir S} \left( e^{\frac{u_{\ir D}}{U_{\ir T}}} -1 \right) - i_L$ & \parbox{1.8cm}{\includegraphics{./img/char/char_diode_photo.pdf} } \\ + \end{tabular} + + + + + + + + +% SECTION ==================================================================================== +\section{Resistive Zweitore} +% ============================================================================================ +Ein Zweitor besteht aus zwei Eintoren.\\ + + \subsection{Beschreibungsformen von Zweitoren} + \begin{tabular}{@{}p{1.3cm}|ll@{}} + Beschreibung & nicht linear & linear \\ \mrule + \textbf{Implizit} & $f_{\mathcal F}( \ul u, \underline i) = \ul 0$ & $\mat{ \utilde{ \bf M}\ \utilde{ \bf N}} \cdot \left. \begin{matrix} \ul u \\ \ul i \end{matrix} \right] = 0$ \\ \mrule + \textbf{Parametrisch} & $\begin{array}{@{}l} \ul u = u_{\mathcal F}(\ul \lambda) \\ \ul i=i_{\mathcal F}(\ul \lambda) \end{array}$ & $\mvect{ \ul u \\ \ul i } = \begin{bmatrix} \utilde{ \bf U} \\ \utilde{ \bf I} \end{bmatrix} \cdot \ul \lambda$ \vspace{2em} \\ + \textbf{Explizit} & nicht linear & linear \\\mrule + Widerstand-beschreibung & $\mvect{ u_1\\ u_2} = \mvect{ r_1(i_1,i_2)\\ r_2(i_1,i_2) }$ & $= \mvect{ U_0 \\ U_0} + \utilde{ \bf R} \mvect{ i_1 \\ i_2} $ \\ \mrule + Leitwert-beschreibung & $\mvect{ i_1\\ i_2} = \mvect{g_1(u_1,u_2) \\ g_2(u_1,u_2)}$ & $= \mvect{ I_0 \\ I_0} + \utilde{ \bf G} \mvect{ u_1 \\ u_2} $ \\ \mrule + Hybrid-beschreibung & $\mvect{ u_1\\ i_2} = \mvect{h_1(i_1,u_2) \\ h_2(i_1,u_2)} $ & $= \ma{H} \cdot \mvect{ i_1\\ u_2 }$\\ \mrule + Inver. Hybrid- beschreibung & $\mvect{ i_1\\ u_2} = \mvect{h'_1(u_1,i_2) \\ h'_2(u_1,i_2)}$ & $= \ma{H'} \cdot \mvect{ u_1\\ i_2 }$ \\ \mrule + Ketten-beschreibung & $\mvect{ u_1\\ i_1} = \mvect{a_1(u_2,-i_2) \\ a_2(u_2,-i_2)}$ & $= \ma{A} \cdot \mvect{ u_2\\ -i_2 }$\\ \mrule + Inver. Ketten- beschreibung & $\mvect{ u_2\\ i_2} = \mvect{a'_1(u_1,-i_1) \\ a'_2(u_1,-i_1)}$ & $= \ma{A'} \cdot \mvect{ u_1\\ -i_1 }$ + + \end{tabular}\\ + + + + + \subsection{Verschaltung von Zweitoren} + % ============================================================================================================ + Es gibt sechs mögliche Verschaltungen.\\ + \begin{tabular}{@{}ll} + Verschaltung & Bild\\ \hline + \parbox{3cm}{Parallel: $g_{ges}=g_{\mathcal F1}+g_{\mathcal F2}$ \\ linear: $\ma G_{\ir ges} = \ma G_1 + \ma G_2$ \\ Umrechnung: } & \parbox{2.5cm}{\includegraphics{./img/ztv_leitwert.pdf}}\\[2em] + \parbox{3cm}{Serie: $r_{ges}=r_{\mathcal F1}+r_{\mathcal F2}$ \\ linear: $\ma R_{\ir ges} = \ma R_1 + \ma R_2$} & \parbox{2.5cm}{\includegraphics{./img/ztv_widerstand.pdf}}\\[2em] + \parbox{3cm}{Hybrid: $h_{ges}=h_{\mathcal F1}+h_{\mathcal F2}$ \\ linear: $\ma H_{\ir ges} = \ma H_1 + \ma H_2$} & \parbox{2.5cm}{\includegraphics{./img/ztv_hybrid.pdf}}\\[2em] + \parbox{3.5cm}{Hybrid Inv.: $h'_{ges}=h_{\mathcal F1}+h'_{\mathcal F2}$ \\ linear: $\ma H'_{\ir ges} = \ma H'_1 + \ma H'_2$} & \parbox{2.5cm}{\includegraphics{./img/ztv_hybrid_inv.pdf}}\\[2em] + \parbox{3cm}{Kette: $a_{ges}=a_{\mathcal F1} \cdot a_{\mathcal F2}$ \\ linear: $\ma A_{\ir ges} = \ma A_1 \cdot \ma A_2$} & \parbox{2.5cm}{\includegraphics{./img/ztv_kette.pdf}}\\[2em] + \parbox{3cm}{Kette Inv: $a'_{ges}=a'_{\mathcal F2} \cdot a'_{\mathcal F1}$ \\ linear: $\ma A'_{\ir ges} = \ma A'_2 \cdot \ma A'_1$} & \parbox{2.5cm}{\includegraphics{./img/ztv_kette_inv.pdf}}\\ + \end{tabular} + + + + \subsection{Liste von Zweitoren} + + \subsubsection{Übertrager (z.B. Transformator} + $A=\begin{bmatrix} \text{ü} & 0 \\ 0 & \frac{1}{\text{ü}} \end{bmatrix}$\\ + $R_{in}=\frac{u_1}{i_1}=\text{ü}^2 R_L$\\ + Eigenschaften: verlustlos(ideal) + + \subsubsection{Gyrator} + Der Gyrator wandelt das an Tor 1 geschaltete Bauteil in das duale Bauteil an Tor 2 um.\\ + \includegraphics{./img/symb/symb_gyrator.pdf} \qquad $\mathcal F_{in}=\mathcal F^d$ + $A=\begin{bmatrix} 0 & R_1 \\ \frac{1}{R_2} & 0 \end{bmatrix}$\\ + Eigenschaften: streng linear, verlustlos für $R_1=R_2$\\ + + \subsubsection{Negativ-Immitanz-Konverter} + $A=\begin{bmatrix} -k & 0 \\ 0 & \frac{1}{k} \end{bmatrix}$\\ + $R_{in}=\frac{u_1}{i_1}=-k^2R$ \quad $-k^2 R:$\ negativer Widerstand(et voilà xD)\\ + Eigenschaften: streng linear, aktiv + +\sectionbox{ + \subsection{NIK allgemein (Polung beachten)} + \includegraphics[scale = 0.85]{./img/NIK.pdf} \qquad \includegraphics{./img/char/char_NIK.pdf}\\ + \begin{tabular}{cll} + I & negative Sättigung $u_d < 0 \Leftrightarrow u_{out}=-U_{sat}$\\ + & $u = R_0 i - U_{sat}$\\ % < \frac{R_L}{R_L + R_1} \cdot U_{sat}$\\ + II & linearer Bereich $u_d = 0$\\ + & $u = - \frac{R_0}{R_1} R_L \cdot i$ \qquad \quad $-U_{sat} < \frac{R_L + R_1}{R_L}u < U_{sat}$\\ + III & positive Sättigung $u_d > 0 \Leftrightarrow u_{out}=U_{sat}$\\ + & $u = R_0 i + U_{sat}$ % < \frac{R_L}{R_L + R_1} \cdot U_{sat}$\\ + \end{tabular} +} + + +% SECTION ==================================================================================== +\section{Eigenschaften von Ein- und Mehrtoren} +% ============================================================================================ +Ein Mehrtor $\mathcal F(\vec u, \vec i)$ ist ... + + Resistiv: Gedächtnislos; nur von $u$ und $i$ abhängig\\ + Zeitvariant: Betriebsraum kann sich ändern\\ + Reziprok: + + Rezeprozität, Matrix transponierbar(r12=r21). Stärker: Symmetrie: r11=r22 + Merke: Alle Mehrtore die nur aus R, L, C bestehen sind reziprok! + +\tablebox{ + \begin{tabular*}{\columnwidth}{@{\extracolsep\fill}ll@{}} + \ctrule + Eigenschaft & Bedingung\\ \mrule + F quellenfrei & $\vec 0 \in \mathcal F$; enthält den Ursprung \\ + F streng linear & $(ku, ki) \in F$ \quad $ (u_1 + u_2, i_1 + i_2) \in F$ \\ + \cbrule + \\ + Nur für Eintore:\\ + ungepolt & Kennlinie punktsymm. zum Ursprung \\ + & $\mathcal F(u,i)=\mathcal F(-u,-i)$ + \end{tabular*} +} + + \subsection{Liniarität} + Linear: $(ku, ki) \in F$ \quad $(u_1 + u_2, i_1 + i_2) \in F$ (Kennlinie gerade) \\ + Streng Linear: linear + quellenfrei, (Gerade durch Ursprung)\\ + + \subsection{Zeitvarianz} + Ein Mehrtor heißt zeitvariant, wenn sich sein Betriebsraum mit der Zeit ändern kann, ansonsten + ist es Zeitinvariant. + + \subsection{Steuerung} + Ein Bauelement ist von einer Größe gesteuert, wenn die jeweilige explizite Beschreibung existiert.\\ + \begin{tabular*}{\columnwidth}{@{\extracolsep\fill}llll@{}} + Stromgesteuert: & $u=\mathcal R(i)$ & Spannungsgestuert: & $i=\mathcal G(i)$\\ + Ladungsgesteuert: & $u=\mathcal C^{-1}(q)$ & Flussgesteuert: & $i=\mathcal L^{-1}(\Phi)$ + \end{tabular*} + + \subsection{Leistungsbetrachtung} + Leistung: $P(t)=\ul u^T \cdot \ul i=u_1 i_1 + \dotsc + u_n i_n$ \\ + \begin{tabular*}{\columnwidth}{@{\extracolsep\fill}lll@{}} + Passiv: & $\forall \mathcal F(u,i): P(t)\ge 0$ & Kennlinie nur I. oder III. Quadrant\\ + Aktiv: & $\exists \mathcal F(u,i): P(t)<0$ & Kennlinie im II. oder IV. Quadrant\\ + Verlustlos: & $\forall \mathcal F(u,i): P(t)=0$ & Kennlinie nur auf Koordinatenachsen\\ + \end{tabular*}\\ + Merke: Alle Mehrtore die nur aus passiven Bauelementen(R,C,L,...) bestehen, sind selbst passiv! + inkremental passiv:\\ + letztendlich passiv: $\exists U,I>0 \ \forall (u,i) \in \mathcal F : (|u| > U \lor |i| > I \ \Rightarrow \ ui > 0)$\\ + Alle realen Bauteile sind letztendlich passiv, da sonst unendlich viel Energie entstehen würde.\\ + + + + + +% SECTION ==================================================================================== +\section{Transistoren} +% ============================================================================================ +Ein \textbf{Trans}fer Res\textbf{istor} ist ein elektronisch gesteuerter Widerstand zum verstärken bzw. schalten von Strömen. +Eigenschaften: passiv, + +\sectionbox{ + \subsection{Emitterschaltung} + \includegraphics{./img/npn-emitter.pdf} \quad \includegraphics{./img/pnp-emitter.pdf}\\[1em] + + \emphbox{ + $i_{\ir b} = I_{\ir S} \left( \exp\left( \frac{u_{\ir be}}{U_{\ir T}} \right) - 1 \right)$ \qquad\qquad $u_{\ir be} = U_{\ir T} \ln\left( \frac{i_{\ir b}}{I_{\ir S}} + 1 \right)$ \\[0.5em] + $i_{\ir c} = \beta i_{\ir b}$\\ + + } +} + +% SECTION ==================================================================================== +\section{Operationsverstärker (OpAmp)} +% ============================================================================================ + Der Operationsverstärker ist ein elektronischer Verstärker. + +\sectionbox{ + \subsection{Idealer Operationsverstärker} + + \includegraphics{./img/opamp.pdf} \qquad \includegraphics{./img/char/char_opamp.pdf}\\[2em] + \begin{tabular}{lll} + \parbox{3cm}{\textbf{\large ESB I} \\ Sättigungsbereich \\ $u_{\ir d} < 0$ \\ $u_{\ir out} = - U_{\ir sat}$} & \parbox{3cm}{\includegraphics[scale = 0.9]{./img/opamp_esb_1.pdf} }\\[4em] + \parbox{3cm}{\large \textbf{ESB II} \\ streng linearer Bereich \\ $u_{\ir d} = 0$ \\ $\abs{ u_{\ir out} } \le \abs{ U_{\ir sat}}$} & \parbox{3cm}{\includegraphics[scale = 0.9]{./img/opamp_esb_2.pdf} }\\[4em] + \parbox{3cm}{\textbf{\large ESB III} \\ Sättigungsbereich \\ $u_{\ir d} > 0$ \\ $u_{\ir out} = + U_{\ir sat}$} & \parbox{3cm}{\includegraphics[scale = 0.9]{./img/opamp_esb_3.pdf} } + + \end{tabular} +} + + + \begin{tabular}{lll} + Invertierender Verstärker & Nichtinvert. Verstärker & Spannungsfolger\\ + \includegraphics[scale = 0.9]{./img/invamp.pdf} & \includegraphics[scale = 0.9]{./img/notinvamp.pdf} & \includegraphics[scale = 0.9]{./img/voltagefollower.pdf}\\ + $u_{\ir out} = \frac{-R_0}{R_1} u_{in}$ & $u_{\ir out} = \left( 1+ \frac{R_0}{R_1} \right) u_{in}$ & $u_{\ir out} = u_{in}$ \quad $i_{in} = 0$\\[0.5em] \mrule + Differenzierer & Integrierer & Addierer \\ + \includegraphics[scale = 0.9]{./img/differenzierer.pdf} & \includegraphics[scale = 0.9]{./img/integrierer.pdf} & \\ + $u_{\ir out} = -RC \cdot \dot u_{in}$ & $u_{\ir out} = -\frac{1}{RC} \int\limits^{t_1}_{t_0} u_{in} \diff t$ & $u_{\ir out} = -R_0 \sum \frac{u_i}{R_i}$\\ \mrule + Logarithmierer & Exponenzierer & NIK \\ + \includegraphics[scale = 0.9]{./img/logarithmierer.pdf} & \includegraphics[scale = 0.9]{./img/exponenzierer.pdf} & \includegraphics[scale = 0.8]{./img/NIK.pdf}\\ + $u_{\ir out} = -U_T \ln \frac{u_{in}}{R \cdot I_S}$ & $u_{\ir out} = -R \cdot I_S \exp \left( \frac{u_{in}}{U_T} \right)$ & $u = -R_L \cdot i$ \\ + \end{tabular} + + + + + + +% SECTION ==================================================================================== +\section{Allgemeine Analyseverfahren} +% ============================================================================================ + + + \subsection{Die Tableau-Gleichung} + ... beschreibt ein Netzwerk vollständig bezüglich Verschaltung und Bauteilverhalten. + $\ma T = \mat{ \ma A & \ma 0 \\ \ma 0 & \ma B \\ \ma M & \ma N}$ + + + + \subsection{Knotenspannungsanalyse} +\boxed{ \underset{\text{Knotenleitwertsmatrix}}{\ma Y_{\ir k}} \cdot \underset{\text{Spannungsvektor}}{\vec u_{\ir k}} = \underset{\text{Stromquellenvektor}}{\vec i_{\ir q}} }\\ +Vorgehen: +\begin{enumerate}\itemsep0pt + \item Nicht lineare Elemente linearisieren + \item Nicht spannungsgesteuerte Elemente (dual)wandeln + \item Aufstellen der Leitwertsmatrix $\ma Y_k$ + \item Bestimmung des Stromquellenvektors $\vec i_q$ +\end{enumerate} + + + +% SECTION ==================================================================================== +\section{Digitale Logik} +% ============================================================================================ + +\sectionbox{ + \subsection{CMOS Logik} + Komplementäre Logik durch pMos und nMos Transistoren auf einem Substrat.\\ + \begin{tabular}{ccc} + NOT (2 Trans.) & NAND (4 Trans.) & NOR (4 Trans.)\\ + \includegraphics[scale = 0.9]{./img/mosfet_not.pdf} & \includegraphics[scale = 0.9]{./img/mosfet_nand.pdf} & \includegraphics[scale = 0.9]{./img/mosfet_nor.pdf} \\ + \end{tabular}\\ +} + + % -- Seite 1 + + + + \newpage % ------------------------------------------------------- % | S C H A L T U N G S T E C H N I K 2 | % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ %######################################################################################################################################################################################################## -{\huge{\textbf{Schaltungstechnik 2}}} +\fstitle{Schaltungstechnik 2} +\emphbox{ + \textbf{Wichtiger Hinweis} + \\ Diese Formelsammlung ist noch in der Entwicklung und nicht Prüfungstauglich ! \\ Allerdings würden wir uns über Unterstützung freuen das zu ändern. Wer Lust hat kann uns über das Kontaktformular auf www.latex4ei.de erreichen. + } \section{Allgemeines} % =============================================================================================== @@ -214,6 +514,85 @@ \section{Allgemeines} +Merke: Am Kondensa\textsl{tor}, eilt der Strom \textsl{vor}, bei Induktivi\textsl{täten}, wird er sich ver\textsl{späten}\\ +Merke: Ist das Mädchen brav, bleibt der Bauch konkav, hat das Mädchen Sex, wird der Bauch konvex.\\ +% Liste mit Eselsbrücken für Ingenieure +% selbstausdenken begriffspaare + + +\sectionbox{ + \subsection{Schaltungen ersten Grades} + \pbox{5.0cm}{ Zustandsgleichung: \\[0.2em] \boxed{ \dot x(t) = - \frac{x(t)}{\tau} + \frac{x_\infty}{\tau} } \\[0.5em] $\tau > 0:$ System stabil \\ $\tau < 0:$ System instabil \\ } \qquad + \pbox{5.0cm}{ \includegraphics{./img/char/char_Schaltung1Grad.pdf} } \\ + \boxed{ \text{Lösung:}\quad x(t) = x_\infty + (x_0 - x_\infty) \exp\left(- \frac{t-t_0}{\tau}\right) }\\ + Mit Parameter aus ESB: C: $x_\infty = U_0(t)$, L: $x_\infty = I_0(t)$ + +\ctikzset{voltage/distance from node=.25}% in \pgf@circ@Rlen units +\ctikzset{voltage/distance from line=.25}% pos. between 0 and 1 + + \ctikzset{bipoles/length=0.7cm} + +\subsubsection{Helmholz / Thévenin ESB} + \begin{circuitikz} + \draw (0,0) to[R, l=R, i_<=$i$, -o] (2,0) to[short, i_>=$i_c$] (3,0) to[C, v^>=$u_C$] (3,-2) -- (3,-2) to[short, o-] (0, -2) to[V, v<=$U_0$] (0,0); + \draw (1.8,0) to[open, v^>=$u$] (1.8, -2); + \end{circuitikz} + + $i_C = -i$ \quad $i_C = C \cdot \dot u_C$ \quad + $u_C (t_\infty) = U_0$ \\ \\ + Zeitkonstante: $\tau = R \cdot C$ \\ + + \subsubsection{Mayer / Norton ESB} + + \begin{circuitikz} + \draw (0,0) -- (1,0) to[short, i^<=$i$, -o] (2,0) to[short, i^<=$i_L$] (3,0) to[L, l_=L, v^<=$u_L$] (3,-2) to (2, -2) to[short, o-] (1, -2) --(0, -2) to[I, i<^=$I_0$] (0,0); + \draw (1,0) to[R, l = G] (1, -2); + \draw (1.8,0) to[open, v^>=$u$] (1.8,-2); + \end{circuitikz} + + $u_L = -u$ \quad $u_L = L \cdot \dot i_L$ \quad $u_L = L \cdot \dot i_L$ \\ \\ + $i_L (t_\infty) = I_0$ \quad + Zeitkonstante: $\tau = G \cdot L$ + + +} + + +\sectionbox{ + \subsection{Dynamischer Pfad} + \begin{tabular}{l|l} + kapazitiv & induktiv \\ \mrule + $u_C$ stetig, $i_C$ springt & $i_L$ stetig, $u_L$ springt \\ + $\dot u(t) = - \frac{1}{C} \cdot i(t)$ & $\dot i(t) = - \frac{1}{L} \cdot u(t)$ \\ \mrule + $i > 0 \Ra \dot u < 0 \Ra u$ fällt & $u > 0 \Ra \dot i < 0 \Ra i$ fällt \\ + $i < 0 \Ra \dot u > 0 \Ra u$ steigt & $u < 0 \Ra \dot i > 0 \Ra i$ steigt \\ + $i = 0 \Ra \dot u = 0 \Ra$ GGP & $u = 0 \Ra \dot i = 0 \Ra$ GGP\\ + \end{tabular}\\ + \\[0.5em] + \pbox{6.0cm}{ + Zeitdauer auf linearen Pfaden:\\ + \boxed{\Delta t = t_1 - t_0 = \tau \ln \left( \frac{x(t_0) - x_\infty}{x(t_1) - x_\infty} \right)} + } \quad + \pbox{3.0cm} { + \includegraphics{./img/pfaddauer.pdf} }\\ + $x(t_0)$: Startwert,\quad $x(t_1)$: Zielwert,\quad $x_\infty$: (gedachter) GGP +} + + + +\sectionbox{ + \subsection{Übertragungsfunktion} + + + + +} + + + + + + @@ -221,6 +600,91 @@ \section{Allgemeines} % Ende der Spalten \end{multicols*} + + + + +\hrule \hrule + +\begin{multicols}{3} +\section{Lösen von homogenen DGLs 2. Ordnung} +Gegeben: Homogene Differnetialgleichungen der Form $\vec{\dot x} = \ma A \vec x$ \\ +mit Anfangswerten $x_{0,1}$ und $x_{0,2}$\\[2em] +\boxed{ \boldsymbol{\lambda_1 \ne \lambda_2 \in \R} } \qquad \boxed{ \bigl| \lambda_1 \bigr| < \bigl|\lambda_2 \bigr| \quad \Ra \vec q_2 \text{,,schneller''} } \\[1.5em] +$\vec x(t) = x_{0,1} \cdot \exp(\lambda_1 t) \cdot \vec q_1 + x_{0,2} \cdot \exp(\lambda_2 t) \cdot \vec q_2$\\[1.5em] +\begin{tabular}{l|l|l|l|l} + Matrix $\Lambda$ & Eigenwerte & $\vec x = 0$ & Name & Portrait\\ \mrule \mrule + {$\mat{\lambda_1 & 0 \\ 0 & \lambda_2}$} & $\lambda_1 < 0 < \lambda_2$ & instabil & Sattelpunkt & \pbox{3.0cm}{ \includegraphics{./img/char/PP_Sadle.pdf} }\\ \mrule + & $\lambda_2 < 0, \lambda_1 < 0 $ & stabil & Knoten 2 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableNode.pdf} }\\ \mrule + & $0 < \lambda_1, 0 < \lambda_2 $ & instabil & Knoten 2 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableNode.pdf} }\\ \mrule + \multirow{2}{*}{$\mat{0 & 0 \\ 0 & \lambda_2}$} & $\lambda_1 = 0,\ \lambda_2 < 0$ & stabil & Kamm & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableLine.pdf} }\\ \mrule + & $\lambda_1 = 0,\ \lambda_2 > 0$ & instabil & Kamm & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableLine.pdf} }\\ +\end{tabular} +\vspace{2em}\\ + +%\vspace{5em}\\ + +\boxed{ \boldsymbol{\lambda_1 = \lambda_2 \in \R} }\\ +$\ma Q' = \mat{ \vec q'_1 & \vec q'_2} = \mat{ -a_{12} & -a_{12} \\ \frac{a_{11} - a_{22}}{2} & \frac{a_{11} - a_{22}}{2} -1 } = \mat{ \vect{\text{Eigen-} \\ \text{vektor}} & \vect{\text{Haupt-} \\ \text{vektor}}}$\\ +$\vec x(t) = \left[ \ma 1 + (\ma A - \lambda \ma 1) \cdot t \right] \cdot \exp(\lambda t) \cdot \mvect{x_{0,1} \\ x_{0,2}}$\\[0.5em] +\begin{tabular}{l|l|l|l|l} + Matrix $\Lambda$ & Eigenwerte & $\vec x = 0$ & Name & Portrait\\ \mrule \mrule + \multirow{2}{*}{$\mat{\lambda & 0 \\ 0 & \lambda}$} & $\lambda < 0$ & stabil & Knoten 1 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableStarNode.pdf} }\\ \mrule + & $\lambda > 0$ & instabil & Knoten 1 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableStarNode.pdf} }\\ \mrule + \multirow{2}{*}{$\mat{\lambda & 1 \\ 0 & \lambda}$} & $\lambda < 0$ & stabil & Knoten 3 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableWhirlNode.pdf} }\\ \mrule + & $\lambda > 0$ & instabil & Knoten 3 & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableWhirlNode.pdf} }\\ \mrule + $\mat{0 & 0 \\ 0 & 0}$ & $\lambda = 0$ & stabil & Ruheebene & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableZero.pdf} }\\ \mrule + $\mat{0 & 1 \\ 0 & 0}$ & $\lambda = 0$ & instabil & Ruhegerade & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableZero.pdf} }\\ +\end{tabular} + + +\boxed{ \cx \lambda_1 = \cx \lambda^*_2 = \alpha + \beta j \in \C } \qquad \boxed{ \ma Q' = \mat{\Re{\vec q_1} & \Im{\vec q_1}} = \mat{ \vec q_r & \vec q_j} }\\ + +$\begin{array}{rl} \vec x(t) & = x_{0,1} \cdot e^{\alpha t} \cdot \left[ \cos (\beta t) \vec q_r - \sin(\beta t) \cdot \vec q_j \right] + \\ & +\, x_{0,2} \cdot e^{\alpha t} \cdot \left[ \sin (\beta t) \vec q_r + \cos(\beta t) \cdot \vec q_j \right] \end{array}$\\ +\begin{tabular}{l|l|l|l|l} + Matrix $\Lambda$ & Eigenwerte & $\vec x = 0$ & Name & Portrait\\ \mrule \mrule + \multirow{2}{*}{$\mat{\alpha & -\beta \\ \beta & \alpha}$} & $\alpha < 0,\ \beta \ne 0$ & stabil & Strudel & \pbox{3.0cm}{ \includegraphics{./img/char/PP_stableFocus.pdf} }\\ \mrule + & $\alpha > 0,\ \beta \ne 0$ & instabil & Strudel & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableFocus.pdf} }\\ \mrule + $\mat{0 & -\beta \\ \beta & 0}$ & $\alpha = 0,\ \beta \ne 0$ & stabil & Wirbel & \pbox{3.0cm}{ \includegraphics{./img/char/PP_Whirl.pdf} }\\ +\end{tabular} +\vspace{1em}\\ +Zeitverlauf immer von $\vec q_j$ nach $\vec q_r$ bzw. von $\vec q_r$ nach $-\vec q_j$\\[4em] +Lösung für \emph{inhomogene DGL}($\vec v \ne 0$) mit singulärer Matrix $\ma A$ (nicht entkoppelbar):\\ +\begin{tabular}{l|l|l|l|l} +$\mat{0 & 0 \\ 0 & \lambda_2}$ & $\lambda_1 = 0,\ \lambda_2 < 0$ & instabil & Kamm & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableLine2.pdf} }\\ +$\mat{0 & 1 \\ 0 & 0}$ & $\lambda = 0$ & instabil & Knoten & \pbox{3.0cm}{ \includegraphics{./img/char/PP_unstableZero2.pdf} }\\ +\end{tabular} + + + + + + + + +% Ende der Spalten +\end{multicols} + + +\setlength{\tabcolsep}{2pt} +\renewcommand{\arraystretch}{1.3} +\def\mrule{\noalign{\vspace{3pt}\hrule\vspace{5pt}}} % middle rule +\arraycolsep=3pt +\tabcolsep=5pt + \section{Zweitormatrizen} + \vspace{1em} + \begin{tabular}{ccccccc} % {c|c|c|c|c|c|c} % + In $\rightarrow$ & $\ma R$ & $\ma G$ & $\ma H$ & $\ma H'$ & $\ma A$ & $\ma A'$ \\ \mrule + $\ma R$ & $\mat{r_{11} & r_{12} \\ r_{21} & r_{22} }$ & $\frac{1}{\det \ma G} \mat{ g_{22} & -g_{12} \\ -g_{21} & g_{11} }$ & $\frac{1}{h_{22}} \mat{ \det \ma H & -h_{12} \\ -h_{21} & 1 }$ & $\frac{1}{h'_{11}} \mat{ 1 & -h'_{12} \\ -h'_{21} & \det \ma H' }$ & $\frac{1}{a_{21}} \mat{ a_{11} & \det \ma A \\ 1 & a_{22} }$ & $\frac{1}{a'_{21}} \mat{ a'_{22} & 1 \\ \det \ma A & a'_{11} }$ \\ \mrule + $\ma G$ & $\frac{1}{\det \ma R} \mat{ r_{22} & -r_{12} \\ -r_{21} & r_{11} }$ & $\mat{g_{11} & g_{12} \\ g_{21} & g_{22} }$ & $\frac{1}{h_{11}} \mat{ 1 & -h_{12} \\ -h_{21} & \det \ma H }$ & $\frac{1}{h'_{22}} \mat{ \det \ma H' & -h'_{12} \\ -h'_{21} & 1 }$ & $\frac{1}{a_{12}} \mat{ a_{22} & -\det \ma A \\ -1 & a_{11} }$ & $\frac{1}{a'_{12}} \mat{ a'_{11} & -1 \\ -\det \ma A & a'_{22} }$ \\ \mrule + $\ma H$ & $\frac{1}{r_{22}} \mat{ \det \ma R & r_{12} \\ -r_{21} & 1 }$ & $\frac{1}{g_{11}} \mat{ 1 & -g_{12} \\ -g_{21} & \det \ma G }$ & $\mat{h_{11} & h_{12} \\ h_{21} & h_{22} }$ & $\frac{1}{\det \ma H'} \mat{ h'_{22} & -h'_{12} \\ -h'_{21} & h'_{11} }$ & $\frac{1}{a_{22}} \mat{ a_{12} & \det \ma A \\ -1 & a_{21} }$ & $\frac{1}{a'_{11}} \mat{ a'_{12} & 1 \\ -\det \ma A & a'_{21} }$ \\ \mrule + $\ma H'$ & $\frac{1}{r_{11}} \mat{ 1 & -r_{12} \\ -r_{21} & \det \ma R }$ & $\frac{1}{g_{22}} \mat{ \det \ma G & g_{12} \\ -g_{21} & 1 }$ & $\frac{1}{\det \ma H} \mat{ h_{22} & -h_{12} \\ -h_{21} & h_{11} }$ & $\mat{h'_{11} & h'_{12} \\ h'_{21} & h'_{22} }$ & $\frac{1}{a_{11}} \mat{ a_{21} & -\det \ma A \\ 1 & a_{12} }$ & $\frac{1}{a'_{22}} \mat{ a'_{21} & -1 \\ \det \ma A & a'_{12} }$ \\ \mrule + $\ma A$ & $\frac{1}{r_{21}} \mat{ r_{11} & \det \ma R \\ 1 & r_{22} }$ & $\frac{1}{g_{21}} \mat{ -g_{22} & -1 \\ -\det \ma G & -g_{11} }$ & $\frac{1}{h_{21}} \mat{- \det \ma H & -h_{11} \\ -h_{22} & -1 }$ & $\frac{1}{h'_{21}} \mat{-\det H' & -h'_{22} \\ -h'_{11} & -1 }$ & $\mat{a_{11} & a_{12} \\ a_{21} & a_{22} }$ & $\frac{1}{\det \ma A'} \mat{ a'_{22} & a'_{12} \\ a'_{21} & a'_{11} }$ \\ \mrule + $\ma A'$ & $\frac{1}{r_{12}} \mat{r_{22} & \det \ma R \\ 1 & r_{11}}$ & $\frac{1}{g_{12}} \mat{-g_{11} & 1 \\ -\det \ma G & -g_{22}}$ & $\frac{1}{h_{12}} \mat{1 & h_{11} \\ h_{22} & \det \ma H}$ & $\frac{1}{h'_{12}} \mat{-\det \ma H' & -h'_{22} \\ -h'_{11} & -1}$ & $\frac{1}{\det \ma A} \mat{a_{22} & a_{12} \\ a_{21} & a_{11} }$ & $\mat{a'_{11} & a'_{12} \\ a'_{21} & a'_{22} }$\\ + + \end{tabular} + + % Dokumentende % ====================================================================== \end{document} diff --git a/Symbols.pdf b/Symbols.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7dab684227154332bcf27c932ea94855fcd4af67 GIT binary patch literal 261478 zcmbTdbxy6%m;7k77ecXuuB?(Xhx#ogWA3I&S0ySuwn;G^$aJ-d7MJ~Qh({6mr_ z$(@85o?q@<7m2*EC=CNG6BH9ZJ-)4h1wJ=76rHrOjj59vJ`)oYGyXq6P;{c^R!+ta z_;jLH`cB5e#)h^=#!$SxP>xOx#`@M!ZY%wo&#|imDDU1MP%P$HWzpypB5=^iN89Y+ zimtobC1BXV`-)oO+PI7q?w!2vk6qhW=n=68Rt>U<-Bb#(v9?{8SwA7wTd(lc5iYpq zNT4@wv4{5vM<(7~+#lu*2~iWdy6(_Y7MqUt5BhIH=SN2`?`^Ah-KF0SIBfXQ=Hi7H z7=6!;+h4;^LG1L`rL-x_K=Weg_6%sS2ZX)fPA@oNQQ!6tD_9(lZ}7JD(M-1q!h4mq z2}UD3tVBBZ=Ph7AP^s#rsouQWATs$iTS@y|iGvTn9Ehiw1G_Mo>3OB0kgP3uaU4)# zQpe8MEle)eYg7&2z=3Qr_`IR}s&_M498a|}(ZRjMM0bfM^K}IMJmlUu!uzK6(HRh_ zU&Xe7hNp8zFBPfMG0O;}kQOtigX^1i^= zCHQsk8dS{*5HGw7{UEyExFb>ZYCH9)rU@tWE(8S7ZDs_$Z)N#K 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