From e8daba748ede0e18af9841532086dd06ab7e57fa Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=ABl=20Leirer?= Date: Tue, 22 Nov 2022 13:16:24 +0100 Subject: [PATCH] update --- RegT1E_HS22.tex | 320 ++++++------------ RegT1E_HS22.xdv | Bin 147584 -> 196336 bytes img/Regelkreis.png | Bin 0 -> 352399 bytes include/Grundglieder/Grundglieder.tex | 299 ++++++++++++++++ .../Integraltransformationen.tex | 2 +- .../Integrieren und Differenzieren.tex | 86 +++-- include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex | 37 +- .../Wichtige Funktionen.tex | 202 +++++------ 8 files changed, 567 insertions(+), 379 deletions(-) create mode 100644 img/Regelkreis.png create mode 100644 include/Grundglieder/Grundglieder.tex diff --git a/RegT1E_HS22.tex b/RegT1E_HS22.tex index 9acc810..dc18ba6 100644 --- a/RegT1E_HS22.tex +++ b/RegT1E_HS22.tex @@ -16,6 +16,9 @@ \usepackage{tabularx} \usepackage{mathrsfs} \usepackage{tikz} +\usepackage{xcolor} +\usepackage{hyperref} + \usetikzlibrary{plotmarks} \usepackage{pgfplots} @@ -25,7 +28,11 @@ \usetikzlibrary{plotmarks} \definecolor{TabularBackgroundColor}{rgb}{0.83,0.96,0.96} +\definecolor{TabularTitleColor}{rgb}{0.89,0.94,0.94} +\definecolor{RefColor}{rgb}{0.59,0.88,0.72} +\newcommand{\skript}[1]{{\scriptsize \color{RefColor}Skript S.#1}} +\newcommand{\tiltewithref}[2]{\texorpdfstring{#1 {\scriptsize \color{RefColor}Skript S.#2}}{#1}} %% TODO: publish to CTAN \usepackage{tex/hsrstud} @@ -86,8 +93,9 @@ \section*{Lizenz} \clearpage \setcounter{page}{1} \pagenumbering{arabic} - -\section{Begriffe} +\section{Allgemeines} +\subsection{Begriffe} +\small \begin{itemize} \item Prozess \begin{itemize} @@ -126,254 +134,113 @@ \section{Begriffe} \end{itemize} \item Grundglieder \begin{itemize} - \item Kleinste Teilsysteme. Sie können weiter augeteilt werden in, siehe Abschnitt Grundglieder + \item Kleinste Teilsysteme. Sie können weiter augeteilt werden, siehe Abschnitt Grundglieder \end{itemize} \item Sprungantwort \begin{itemize} - \item Reaktion des Systems auf die Sprungfunktion. Siehe \refname{func} + \item Reaktion des Systems auf die Sprungfunktion. Siehe LTI-Systeme \end{itemize} \item Schrittantwort \item \begin{itemize} - \item Reaktion des Systems auf die Schrittfunktion. Siehe \refname{func} + \item Reaktion des Systems auf die Schrittfunktion. Siehe LTI-Systeme \end{itemize} \item Ausgleich \begin{itemize} \item Prozesse ohne Ausgleich: Sprungantwort wächst Grenzenlos an \item Prozess mit Ausgleich: Sprungantwort strebt endlichem Wert zu \end{itemize} - \end{itemize} +\normalsize + +\subsection{Regelkreis} +\includegraphics{img/Regelkreis.png} + +\subsection{Steuern und Regeln} +\begin{minipage}{0.5\textwidth} + \subsubsection*{Steuerung} +\end{minipage}% +\begin{minipage}{0.5\textwidth} + a +\end{minipage} +\input{include/Grundglieder/Grundglieder.tex} + + +\section{\tiltewithref{Klassifizierung von Systemen}{76,78,81}} +\begin{tabular}{|p{10cm}|p{10cm}|} + \hline + \rowcolor{TabularTitleColor} + \textbf{Statisch} & \textbf{Dynamisch} \\ + \begin{itemize} + \item System ohne Gedächtnis + \item Ausgang hängt nur von aktuellem Eingang ab. + \item $y(t) = f(x(t)) \forall t$ + \end{itemize} + & + \begin{itemize} + \item System mit Gedächtnis + \item Ausgang hängt von aktuellem sowie vergangenen und zukünftigen Eingängen ab. + \item $y(t) = f(x(t \pm t_0)) $ + \end{itemize} + \\ + \hline + \rowcolor{TabularTitleColor} + \textbf{Kausal} & \textbf{Akausal} \\ + \begin{itemize} + \item Ausgang \textbf{unabhängig} von zukünftigen Werten + \item $y(t) = f(x(t-t_0)) $ + \end{itemize} + & + \begin{itemize} + \item Ausgang \textbf{abhängig} von zukünftigem Eingang + \item $y(t) = f(x(t+t_0)) $ + \end{itemize} + \\ + \hline + \rowcolor{TabularTitleColor} + \textbf{Linear} & \textbf{Nicht-Linear}\\ + \begin{itemize} + \item Linearkombination am Eingang -> Linearkombination am Ausgang + \item Lineare DGL mit konstanten Koeffizenten + \item $\Phi(\alpha x_a(t)+\beta x_b(t)) =\alpha y_a(t) + \beta y_b(t)$ + \item $\Phi(0) = 0$ + \end{itemize} + & + \begin{itemize} + \item Ausgang ist nicht immer Proportional zum Eingang + \item Keine Superposition + \item Neue Frequenzanteile werden erzeugt + \end{itemize} -\begin{landscape} - \subsection{Grundglieder} - \begingroup - \scriptsize - \newcommand{\ImageWidth}{70pt} - \begin{tabularx}{\linewidth}{|p{100pt}|p{160pt}|p{60pt}|p{80pt}|p{120pt}|p{80pt}|} - \hline - \textbf{Benennung} - & - \textbf{Funktion} - & - \textbf{UTF\textsuperscript{1}} - & - \textbf{Symbol} - & - \textbf{Sprungantwort} - & - \textbf{Plot} - \\ - \hline - \hline - \textbf{P-Glied\textsuperscript{2}} - \newline Proportionalglied - & - $y = K \cdot u$ - & - $K$ - & - \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Proportionalglied.png}} - & - Sprungantwort - & - \raisebox{-.5\height}{ - \resizebox{\ImageWidth}{!}{% - \begin{tikzpicture} - % Grid - \draw[help lines,dashed] (0,0) grid (5,3); - - % Axes - \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} - |- (5.25,0) node[below]{$t$}; - - % Plot function - \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$y(0)$} - -- ++(0.5,0) - plot[domain=0:5, - samples = 50, - smooth]({\x}, {2}); - \end{tikzpicture} - } - } - \\ - \hline - \rowcolor{TabularBackgroundColor} - \textbf{I-Glied} - \newline(Idealer Integrierer) - & - $y(t) = K \cdot \int \limits _{t=0} ^{t} u(\tau) d\tau + y(0)$ - \newline $\dot{y}(t) = K \cdot u(t)$ - & - $K \frac{1}{s}$ - & - \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Integrator.png}} - - & - Sprungantwort - & - \raisebox{-.5\height}{ - \resizebox{\ImageWidth}{!}{% - \begin{tikzpicture} - % Grid - \draw[help lines,dashed] (0,0) grid (5,3); - - % Axes - \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} - |- (5.25,0) node[below]{$t$}; - - % Plot function - \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$y(0)$} - -- ++(0.5,0) - plot[domain=0:5, - samples = 50, - smooth]({\x},); - \end{tikzpicture} - } - } - \\ - \hline - \textbf{Totzeit-Glied} - & - $y(t) = u(t-T_t)$ - & - $e^{-s T_t}$ - & - \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Totzeitglied.png}} - & - Sprungantwort - & - \raisebox{-.5\height}{ - \resizebox{\ImageWidth}{!}{% - \begin{tikzpicture} - % Grid - \draw[help lines,dashed] (0,0) grid (5,3); - - % Axes - \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} - |- (5.25,0) node[below]{$t$}; - - % Plot function - \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$y(0)$} - -- ++(0.5,0) - plot[domain=0:5, - samples = 100, - ]({\x},{ (\x<=2) * 0 + (\x>2) * 2}); - \end{tikzpicture} - } - } - \\ - \hline - \rowcolor{TabularBackgroundColor} - \textbf{PT\textsubscript{1}-Glied} - { - \tiny \newline K: Verstärkung - \newline T: Zeitkonstante - } - & - $ T\dot{y} + y = K u(t)$ - \newline $T= \frac{1}{K_I \cdot K_P} $ - \newline $K = \frac{1}{K_P}$ - & - $\frac{K}{1+T_s}$ - & - \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/PT1-Glied.png}} - & - Sprungantwort - & - \raisebox{-.5\height}{ - \resizebox{\ImageWidth}{!}{% - \includegraphics{matlab/PT2.png} - - - \begin{tikzpicture} - % Grid - \draw[help lines,dashed] (0,0) grid (5,3); - - % Axes - \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} - |- (5.25,0) node[below]{$t$}; - - % Plot function - \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$y(0)$} - -- ++(0.5,0) - plot[domain=0:5, - samples = 50, - smooth]({\x},{2.5*(1- exp(-(\x)))}); - \end{tikzpicture} - } - } - \\ - \hline - \textbf{PT\textsubscript{2}-Glied} - {\tiny - \newline K: Verstärkung - \newline T: Zeitkonstante - \newline $\zeta$: Dämpfungskonstante - } - & - $T^2 \cdot \ddot{y}(t) + 2 \zeta T \cdot \dot{y}(t) + y(t) = K \cdot u(t)$ - \newline $y(t) = K \cdot x(t) + T^2 \cdot (-\ddot{y}(t)) - 2 \zeta T \cdot \dot{y}(t)$ - {\tiny - \newline $\zeta = \frac{h}{\sqrt{h^2 + \pi^2}}$ - \newline $h = \log(\frac{y_m}{y_{\infty}})$ - } - & - $\frac{K}{T^2 + s^2 + 2 \zeta T s + 1}$ - & - \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/PT2-Glied.png}} - & - $KA(1+e^{\sigma t}(-cos(\omega t) + \frac{\sigma}{\omega}sin(\omega t)))$ - {\tiny - \newline $\omega = \frac{2\pi}{T_m}$ - \newline $K = \frac{y_{\infty}}{A} $ - \newline $\sigma = \frac{h\omega}{\pi}$ - } - & - \raisebox{-.5\height}{ - \resizebox{\ImageWidth}{!}{% - \begin{tikzpicture} - % Grid - \draw[help lines,dashed] (0,0) grid (5,3); - - % Axes - \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} - |- (5.25,0) node[below]{$t$}; - - % Plot function - \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$y(0)$} - -- ++(0.5,0) - plot[domain=0:5, - samples = 50, - smooth]({\x},{2.5 * (1- exp(-3*(\x)) * (- cos(deg(1.9848*\x)) + (-3/1.9848)*sin(deg(1.9848*\x))) }); - %KA = 2.5, Sigma = -3, Omega = 1.9848 - \end{tikzpicture} - } - } - \end{tabularx} - \tiny{ - \\ - 1. UTF = Übertragungsfunktion (Laplace)\\ - 2. Proportionalglied ist einziges Statisches glied.\\ - } - \endgroup - \normalsize -\end{landscape} - -\section{Klassen von Systemen} + \\ + \rowcolor{TabularTitleColor} + \textbf{Zeitinvariant} & \textbf{Zeitvariant}\\ + \begin{itemize} + \item Zeitliche Verschiebung am Eingang -> gleiche Zeitliche Verschiebung am Ausgang + \item $\Phi(x(t-t_v)) = y(t-t_v)$ + \end{itemize} + & + \begin{itemize} + \item Zeitinvarianz ist nicht erfüllt + \item y(t) enthält t unabhägig von x(t) + \end{itemize} + \\ + \hline + +\end{tabular} \newpage \input{include/Integraltransformationen/Integraltransformationen.tex} \input{include/Wichtige Funktionen/Wichtige Funktionen.tex} -\input{include/LTI-Systeme/LTI-Systeme.tex} - +\input{include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex} +\input{include/Integrieren und Differenzieren/Integrieren und Differenzieren.tex} \section{Tabellen} -\subsection*{Blockschaltbilder MatLab} +\subsubsection*{Blockschaltbilder MatLab} \begin{tabular}{|c|c|c|c|c|} \hline \textbf{Summierer} & @@ -394,7 +261,10 @@ \subsection*{Blockschaltbilder MatLab} \textbf{} \\ \includegraphics[]{img/matlab/transport_delay_block_icon.png} & \includegraphics[]{img/matlab/derivative_block_icon.png} & + & + & \\ \hline \end{tabular} + \end{document} diff --git a/RegT1E_HS22.xdv b/RegT1E_HS22.xdv index 0b627b697e7ea9976fb58d9688c1e2073e8f06ff..66671dc0e0ffe47005a62eee1cb6c9cdf6394650 100644 GIT binary patch literal 196336 zcmeEv31C&#we}%_1VToV5FiO$2vdM0cODqSB(n&_7Dbc<0z?Q2Nl?(=)(-x5N+i(Y zP@lopTCJ_s4zI0kUq@>Pt)2c_JHG0(Q~H{|YWwYp=b&wbuUc>``wz@%f5H4zA2!Rr!|h$&13#d5_O$J8IkdNJ@#4CMmac=%b1w_UqWBlP%58-{Qqf zmkxxv_0R|P9nGC>M>-_ISGSTL+(r)$c@J+bHE#{c)90&0Tzkj9Cf|;_iyFJSnmZ2p z>Q*&h)v~*J>zWlieLj>DS9jkpOYiFL9sJP)10n8>=FTqPT7!>{#UmduLGR5FN*m9`Zx1(mKFOYJSEvIpD(ERXF z<59zdi2rJ#R30XL+gV+=99eCvThrFiR<~+yCq@m1Ky3hhkA69r{M}&6@2;RsIY0hn z2i@|Ac%br6bMl9dG?{PduMMeX>aQiuCH^i~`T%{GKkim_NCoZh2BnsFjjWd2DElr` zRi1GCA^s${sh-I0WZXl~>*!%ie&|fC6OvM1KeAG;F^!N^2<{qgC+-?9IhRiAaHbLv 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zim3pz4YlUI-iYM!o2N?+PqEQye3A2(myaX&Q~$a!f$}z%{Eaoe_D~tZDpaU2p{Va@ zD~uN~g5mrLT+vgf3Kc4Jj;ar3E|GG-aLox04 b5W@ctkd>cM8e4QF00000NkvXXu0mjfy=W@f literal 0 HcmV?d00001 diff --git a/include/Grundglieder/Grundglieder.tex b/include/Grundglieder/Grundglieder.tex new file mode 100644 index 0000000..c397d65 --- /dev/null +++ b/include/Grundglieder/Grundglieder.tex @@ -0,0 +1,299 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%Packages +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%Content +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{landscape} + \subsection{Grundglieder} + \begingroup + \scriptsize + \newcommand{\ImageWidth}{70pt} + \begin{tabularx}{\linewidth}{|p{100pt}|p{160pt}|p{60pt}|p{80pt}|p{120pt}|p{80pt}|} + \hline + \textbf{Benennung} + & + \textbf{Funktion} + & + \textbf{UTF\textsuperscript{1}} + & + \textbf{Symbol} + & + \textbf{Sprungantwort} + & + \textbf{Plot} + \\ + \hline + \hline + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{P-Glied\textsuperscript{2}} + \newline Proportionalglied + & + $y = K \cdot u$ + & + $K$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Proportionalglied.png}} + & + $K$ + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$0$} + -- ++(0.5,0) + plot[domain=0:5, + samples = 50, + smooth]({\x}, {2}); + \end{tikzpicture} + } + } + \\ + \hline + \rowcolor{TabularBackgroundColor} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{I-Glied} + \newline(Idealer Integrierer) + & + $y(t) = K \cdot \int \limits _{t=0} ^{t} u(\tau) d\tau + y(0)$ + \newline $\dot{y}(t) = K \cdot u(t)$ + & + $K \frac{1}{s}$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Integrator.png}} + + & + $K \cdot t$ + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$0$} + -- ++(0.5,0) + plot[domain=0:5, + samples = 50, + smooth]({\x},); + \end{tikzpicture} + } + } + \\ + \hline + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{Totzeit-Glied} + & + $y(t) = u(t-T_t)$ + & + $e^{-s T_t}$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/Totzeitglied.png}} + & + $\sigma(t-T) = \begin{cases} + 0, t2) * 2}); + \end{tikzpicture} + } + } + \\ + \hline + \rowcolor{TabularBackgroundColor} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{D-Glied} + \newline Idealer Differenzierer + & + $y(t) = K \cdot \frac{du(t)}{dt} = K\cdot \dot{u}(t)$ + & + $K \cdot S $ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/D-Glied.png}} + & + $K\cdot \delta(t)$ + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick ,teal] (2,0) -- node[left]{$\delta$} (2,2); + + \end{tikzpicture} + } + } + \\ + \hline + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{DT\textsubscript{1}-Glied} + { + \tiny + \newline Realisierbarer Differenzierer + \newline K: Verstärkung + \newline T: Zeitkonstante + } + & + $ y(t) + T \cdot \dot{y}(t) = K \dot{u}(t)$ + \newline $y(t) = K \dot{u}(t) - T \cdot \dot{y}(t)$ + + & + $\frac{K \cdot s}{1+T_s}$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/DT-Glied.png}} + & + $\frac{K}{T}\cdot e^{-\frac{t}{T}}$ + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$0$} + -- ++(0.5,0) + plot[domain=0:5, + samples = 50, + smooth]({\x},{2.5*exp(-(\x))}); + \end{tikzpicture} + } + } + \\ + \hline + \rowcolor{TabularBackgroundColor} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{PT\textsubscript{1}-Glied} + { + \tiny \newline K: Verstärkung + \newline T: Zeitkonstante + } + & + $ T\dot{y} + y = K u(t)$ + \newline $T= \frac{1}{K_I \cdot K_P} $ + \newline $K = \frac{1}{K_P}$ + & + $\frac{K}{1+T_s}$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/PT1-Glied.png}} + & + $K (1-e^{-\frac{t}{T}})$ + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$0$} + -- ++(0.5,0) + plot[domain=0:5, + samples = 50, + smooth]({\x},{2.5*(1- exp(-(\x)))}); + \end{tikzpicture} + } + } + \\ + \hline + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \textbf{PT\textsubscript{2}-Glied} + {\tiny + \newline K: Verstärkung + \newline T: Zeitkonstante + \newline $\zeta$: Dämpfungskonstante + } + & + $T^2 \cdot \ddot{y}(t) + 2 \zeta T \cdot \dot{y}(t) + y(t) = K \cdot u(t)$ + \newline $y(t) = K \cdot x(t) + T^2 \cdot (-\ddot{y}(t)) - 2 \zeta T \cdot \dot{y}(t)$ + {\tiny + \newline $\zeta = \frac{h}{\sqrt{h^2 + \pi^2}}$ + \newline $h = \log(\frac{y_m}{y_{\infty}})$ + } + & + $\frac{K}{T^2 + s^2 + 2 \zeta T s + 1}$ + & + \raisebox{-.5\height}{\includegraphics[width = \ImageWidth]{img/DIN-Symbole/PT2-Glied.png}} + & + $KA(1+e^{\sigma t}(-cos(\omega t) + \frac{\sigma}{\omega}sin(\omega t)))$ + {\tiny + \newline $\omega = \frac{2\pi}{T_m}$ + \newline $K = \frac{y_{\infty}}{A} $ + \newline $\sigma = \frac{h\omega}{\pi}$ + } + & + \raisebox{-.5\height}{ + \resizebox{\ImageWidth}{!}{% + \begin{tikzpicture} + % Grid + \draw[help lines,dashed] (0,0) grid (5,3); + + % Axes + \draw[very thick,latex-latex] (0,3.25) node[left]{$y(t)$} + |- (5.25,0) node[below]{$t$}; + + % Plot function + \draw[ultra thick,teal] (-0.5,0) node[left,black](s0){$0$} + -- ++(0.5,0) + plot[domain=0:5, + samples = 50, + %TODO: Fix this! + smooth]({\x},{2.5 * (1- exp(-3*(\x)) * (- cos(deg(1.9848*\x)) + (-3/1.9848)*sin(deg(1.9848*\x))) }); + %KA = 2.5, Sigma = -3, Omega = 1.9848 + \end{tikzpicture} + } + } + \\ + \hline + \end{tabularx} + \tiny{ + \\ + 1. UTF = Übertragungsfunktion (Laplacetransformierte Sprungantwort)\\ + 2. Proportionalglied ist einziges Statisches glied.\\ + } + \endgroup + \normalsize +\end{landscape} \ No newline at end of file diff --git a/include/Integraltransformationen/Integraltransformationen.tex b/include/Integraltransformationen/Integraltransformationen.tex index 28cf0a7..f4b8e94 100644 --- a/include/Integraltransformationen/Integraltransformationen.tex +++ b/include/Integraltransformationen/Integraltransformationen.tex @@ -113,7 +113,7 @@ \subsection{Laplace-Transformation} \includegraphics[width = 7cm]{include/Integraltransformationen/img/Zusammenhang_Laplace.png} \subsubsection*{Konvergenzhalbebene} - Die Konvergenzhalbebene beginnt bei der Polstelle der Laplace-Transformierten mit dem Grössete Realteil und geht bis unendlich. + Die Konvergenzhalbebene beginnt bei der Polstelle der Laplace-Transformierten mit dem Grössten Realteil und geht bis unendlich. \newline Bsp: Bei $\frac{1}{s-2}$ ist die Polstelle bei $+2$, daraus folgt: Konvergenzhalbebene $= [2,\infty)$ \end{multicols} diff --git a/include/Integrieren und Differenzieren/Integrieren und Differenzieren.tex b/include/Integrieren und Differenzieren/Integrieren und Differenzieren.tex index 8c7f220..4eb1879 100644 --- a/include/Integrieren und Differenzieren/Integrieren und Differenzieren.tex +++ b/include/Integrieren und Differenzieren/Integrieren und Differenzieren.tex @@ -16,46 +16,58 @@ \section{Integrieren und Differenzieren} \subsection{Integrationsregeln} \begin{tabular}{ll} - Linearit\"at & $\int{f(\alpha x+\beta )dx=\frac{1}{\alpha}\cdot F(\alpha x+\beta)+C}$ \\ - Partielle Integration & $\int\limits_a^b{u'(x)\cdot v(x)dx}=\biggl[ - u(x)\cdot v(x) \biggr]_a^b-\int\limits_a^b{u(x)\cdot v'(x)dx}$ - \tiny($v(x)$ = einfacheste Funktion wählen!) \normalsize\\ - - Substitution (Rationalisierung) & $t=\tan\frac{x}{2}, \qquad - dx=\frac{2dt}{1+t^2} \qquad \sin x=\frac{2t}{1+t^2} \qquad \cos x=\frac{1-t^2}{1+t^2} - \quad\int{R(\sin(x)\cos(x))dx}$\\ + Linearit\"at & $\int{f(\alpha x+\beta )dx=\frac{1}{\alpha}\cdot F(\alpha x+\beta)+C}$ \\ + + \rowcolor{TabularBackgroundColor} + Partielle Integration & $\int\limits_a^b{u'(x)\cdot v(x)dx}=\biggl[ + u(x)\cdot v(x) \biggr]_a^b-\int\limits_a^b{u(x)\cdot v'(x)dx}$ + \tiny($v(x)$ = einfacheste Funktion wählen!) \normalsize \\ - Allgemeine Substitution & - $\int\limits_{a}^{b}{f(x)dx}=\int\limits_{g(a)}^{g(b)}{f(g(t))\cdot - g'(t)dt}\qquad x=g(t)\qquad g'(t)=\frac{dt}{dx}\qquad dx=\frac{1}{g'(t)}\cdot dt$\\ - - Logarithmische Integration & $\int{\frac{f'(x)}{f(x)}dx}=\ln|f(x)|+C - \qquad{(f(x)\neq 1)}$\\ + Substitution (Rationalisierung) & $t=\tan\frac{x}{2}, \qquad + dx=\frac{2dt}{1+t^2} \qquad \sin x=\frac{2t}{1+t^2} \qquad \cos x=\frac{1-t^2}{1+t^2} + \quad\int{R(\sin(x)\cos(x))dx}$ \\ + + \rowcolor{TabularBackgroundColor} + Allgemeine Substitution & + $\int\limits_{a}^{b}{f(x)dx}=\int\limits_{g(a)}^{g(b)}{f(g(t))\cdot + g'(t)dt}\qquad x=g(t)\qquad g'(t)=\frac{dt}{dx}\qquad dx=\frac{1}{g'(t)}\cdot dt$ \\ - Spezielle Form des Integranden & $\int{f'(x)\cdot - (f(x))^{\alpha} dx}= f(x)^{\alpha +1}\cdot \frac{1}{\alpha+1}+C - \qquad{(\alpha \neq -1)}$\\ + Logarithmische Integration & $\int{\frac{f'(x)}{f(x)}dx}=\ln|f(x)|+C + \qquad{(f(x)\neq 1)}$ \\ - Differentiation & $\int \limits ^{b} _{a} {f'(t)dt}=f(b)-f(a)$\qquad - $\frac{d}{dx} \int \limits ^{x} _{1} {f(t)dt}=f(x)$ - \end{tabular} + \rowcolor{TabularBackgroundColor} + Spezielle Form des Integranden & $\int{f'(x)\cdot + (f(x))^{\alpha} dx}= f(x)^{\alpha +1}\cdot \frac{1}{\alpha+1}+C + \qquad{(\alpha \neq -1)}$ \\ -\subsection{Ableitungsregeln} -\begin{tabular}{ll} -Linerität & $(\lambda f + \mu g)'(x) = \lambda f'(x) + \mu g'(x) \; \forall \lambda, \mu \in \mathbb{R} $ \\ -Produktregel & $(f\cdot g)' = f'g + fg'$\\ -Quotientenregel & $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$\\ -Kettenregel & $ (f(g))' = f'(g) \cdot g' $\\ -Potez & $((x-a)^n)'= n\cdot(x-a)^{n-1}$\\ -Trigo & $sin'''' = cos''' = -sin'' = -cos' = sin$\\ + Differentiation & $\int \limits ^{b} _{a} {f'(t)dt}=f(b)-f(a)$\qquad + $\frac{d}{dx} \int \limits ^{x} _{1} {f(t)dt}=f(x)$ \end{tabular} +\\ - -\subsection{Partialbruchzerlegung} -Nenner Faktorisieren, bei mehreren gleichen Termen steigt exponent des Nenners -Zähler = Polynom mit einem Grad kleiner als Nenner -Zähler Gleichsetzen, Mit Gleichungssystem nach A,B,C.. auflösen. -Ansätze: -$$\frac{\dots}{x(x-3)^2} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{(x-3)^2}$$ -$$\frac{\dots}{(x-2)^3} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3}$$ -$$\frac{\dots}{x^4 + x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1}$$ \ No newline at end of file +\begin{minipage}{0.5\textwidth} + \subsection{Ableitungsregeln} + \begin{tabular}{ll} + Linerität & $(\lambda f + \mu g)'(x) = \lambda f'(x) + \mu g'(x)$ \\ + \rowcolor{TabularBackgroundColor} + Produktregel & $(f\cdot g)' = f'g + fg'$ \\ + Quotientenregel & $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ \\ + \rowcolor{TabularBackgroundColor} + Kettenregel & $ (f(g))' = f'(g) \cdot g' $ \\ + Potenz & $((x-a)^n)'= n\cdot(x-a)^{n-1}$ \\ + \rowcolor{TabularBackgroundColor} + Trigo & $sin'' = cos' = -sin$ \\ + Schrittfunktion & $\sigma'(t) = \delta(t)$ + \end{tabular} +\end{minipage}% +\begin{minipage}{0.5\textwidth} + \subsection{Partialbruchzerlegung} + Nenner Faktorisieren (Nennerexponent) + \\Zähler = Polynom (Nennergrad -1) + \\Gleichsetzen mit Originalbruch + \\Mit Gleichungssystem auflösen. + \\Ansätze: + \\ $\frac{\dots}{x(x-3)^2} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{(x-3)^2}$ + \\ $\frac{\dots}{(x-2)^3} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3}$ + \\ $\frac{\dots}{x^4 + x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 1}$ +\end{minipage} \ No newline at end of file diff --git a/include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex b/include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex index af6d7c3..d153856 100644 --- a/include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex +++ b/include/LTI-Systeme_kurz/LTI-Systeme_kurz.tex @@ -2,7 +2,8 @@ %\usepackage{bm} %\usepackage{multicol} \section{LTI-Systeme} -\begin{multicols}{2} +\begin{minipage}{0.5\textwidth} + \subsection*{Linearität und Zeitinvarianz} \begin{itemize} \item $\mathcal{T}[x_1(t) + x_2(t)] = y_1(t) + y_2(t)$ @@ -13,30 +14,35 @@ \section{LTI-Systeme} \subsection{Beschreibung von LTI Systemen} \subsubsection{Impulsantwort} - Impulsfunktion $\delta(t)$ wird am Eingang des Systems angelegt, - die Reaktion darauf am Ausgang nennt man die \textbf{Impulsantwort} \bm{$h(t)$}. - Sie beschreibt ein LTI-System vollständig. + Systemreaktion auf $\delta(t)$. - $$ y(t) = \mathcal{T}[x(t)] + $ y(t) = \mathcal{T}[x(t)] = \int \limits _{-\infty} ^{\infty} x(\tau) \cdot h(t-\tau)d\tau - = x(t) * h(t)$$ + = x(t) * h(t)$ \subsubsection{Frequenzantwort} - Die \textbf{Frequenzantwort} $\bm{H(\omega)}$ ist die Fouriertransformierte Impulsantwort. - Sie ist eine komplexwertige dimensionslose Gewichstsfunktion. - Auch sie beschreibt ein LTI-System vollständig. + Fouriertransformierte Impulsantwort. + \\ Auch Übertragungsfunktion genannt. - $$ Y(\omega) = X(\omega) \cdot H(\omega)$$ + $$ Y(\omega) = X(\omega) \cdot H(\omega) = X(\omega) \cdot G(j\omega)$$ \subsubsection{Berechnung des Ausgangssignals} - 1. Fourier-Transformation: $X(\omega) = \mathcal{F}[x(t)]$ \\ - 2. Berechnung in Frequenz: $Y(\omega) = X(\omega) \cdot H(\omega)$ \\ - 3. Rücktransformation: $y(t) = \mathcal{F}^{-1}[Y(\omega)]$ + 1. Integraltransformation: + \newline $X(\omega) = \mathcal{F}[x(t)] {\; \big / \;} + X(s) = \mathscr{L}[x(t)]$ \\ + 2. Berechnung in Bild / Frequenz: + \newline $Y(\omega) = X(\omega) \cdot H(\omega) {\; \big / \;} + Y(s) = X(s) \cdot G(s)$ \\ + 3. Rücktransformation: + \newline $y(t) = \mathcal{F}^{-1}[Y(\omega)] = \mathscr{L}^{-1}[Y(s)]$ +\end{minipage}% +\begin{minipage}{0.6\textwidth} \subsection{Bezeichnungen} - Übertragungsfunktion: $H(\omega)=|H(\omega)| \cdot e^{j\varphi_H(\omega)}$ \\ + Übertragungsfunktion: $H(\omega) = G(j\omega) =|H(\omega)| \cdot e^{j\varphi_H(\omega)}$ \\ + {\tiny auch: Frequenzgang}\\ Amplitudengang: $|H(\omega)|$ \\ Phasengang: $\varphi_H(\omega)$ \\ @@ -53,4 +59,5 @@ \section{LTI-Systeme} \begin{center} \includegraphics[width = 4cm]{include/Integraltransformationen/img/BiBo.png} \end{center} -\end{multicols} + +\end{minipage}% diff --git a/include/Wichtige Funktionen/Wichtige Funktionen.tex b/include/Wichtige Funktionen/Wichtige Funktionen.tex index 73c40d2..99d74f9 100644 --- a/include/Wichtige Funktionen/Wichtige Funktionen.tex +++ b/include/Wichtige Funktionen/Wichtige Funktionen.tex @@ -26,7 +26,7 @@ \subsubsection*{Diracimpuls \tiny (auch Impuls-/Deltafunktion,-Distribution)} {\footnotesize Unendlich kurzer, normierter Impuls mit unendlicher Amplitude. } - \resizebox{0.5\textwidth}{!}{% + \resizebox{0.45\textwidth}{!}{% \begin{tabular}{ccl} \hline \rowcolor{TabularBackgroundColor} 1. & $\delta(-t) = \delta(t) $ & gerade Funktion \\ @@ -66,104 +66,104 @@ \subsubsection*{Diracimpuls \tiny (auch Impuls-/Deltafunktion,-Distribution)} 18. & $\delta(t)$ \laplace $1(\omega)$ & Fourier \\ \end{tabular}} \end{multicols} - \begin{tabular}{|p{3.5cm}|p{3.5cm}|p{2cm}|p{3cm}|p{3.5cm}|} - \hline +\begin{tabular}{|p{3.5cm}|p{3.5cm}|p{2cm}|p{3cm}|p{3.5cm}|} + \hline - \textbf{Name} - & - \textbf{Definition} - & - \laplace - & - \textbf{Formelzeichen} - & - \textbf{plot} - \\ - \hline - Sprungfunktion - \newline (Heaviside) - & - $\begin{cases} - 0 \textrm{ für } t<0, \\ - [\frac{1}{2} \textrm{ für } t = 0,] \\ - 1 \textrm{ für } t >0. - \end{cases}$ - \newline \tiny(machmal: 1 für $t=0$) - & - $\frac{1}{j\omega} + \pi\delta(\omega)$ - & - $u(t), \sigma(t), h(t)$ - & - \raisebox{-.5\height}{\includegraphics[width = 3.5cm]{include/Wichtige Funktionen/img/Sprungfunktion.png}} - \\ - Signumfunktion - \newline (Vorzeichenfunktion) - & - $\begin{cases} - -1 \textrm{ für } t<0, \\ - 0 \textrm{ für } t = 0, \\ - 1 \textrm{ für } t >0. - \end{cases} $ - & - $\frac{2}{j\omega}$ - & - $sgn(t)$ - & - \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Signumfunktion.png}} - \\ - Rampenfunktion - & - $\begin{cases} - 0 \textrm{ für } t \leq 0, \\ - t \textrm{ für } t > 0. - \end{cases}$ - & - $\frac{1}{s^2}$ - & - $r(t)$ - & - \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Rampenfunktion.png}} - \\ - Rechteckimpuls - & - $\begin{cases} - 1 \textrm{ für } |t| < a, \\ - \frac{1}{2} \textrm{ für } |t| = a, \\ - 0 \textrm{ für } |t| > a. - \end{cases} $ - & - $\frac{1}{s}- e^{-as}\frac{1}{s} $ - & - $p_a(t), \beta(t)$ - \newline $\sigma(t+a)-\sigma(t-a)$ - & - \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Rechteckimpuls.png}} - \\ - Dreieckimpuls - & - $\begin{cases} - 1 - \frac{|t|}{a} \textrm{ für } |t| < a \\ - 0 \textrm{ für } |t| \geq a - \end{cases}$ - & - $sinc^2(\omega)$ - & - $\Lambda(t)$ - & - \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Dreieckimpuls.png}} - \\ - Sinc-Funktion - & - $\frac{sin(t)}{t}\; \forall t$ - \newline {\tiny $\lim _{t \to 0} sinc(t) = 1$} - \newline {\tiny wenn normalisiert: $t \to \pi t$} - & - $\beta(\omega)$ - \newline{\tiny(Rechteckimpuls)} - & - sinc(t) - & - \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/SincFunktion.png}} - \\ - \hline - \end{tabular} \ No newline at end of file + \textbf{Name} + & + \textbf{Definition} + & + \laplace + & + \textbf{Formelzeichen} + & + \textbf{plot} + \\ + \hline + Sprungfunktion + \newline (Heaviside) + & + $\begin{cases} + 0 \textrm{ für } t<0, \\ + [\frac{1}{2} \textrm{ für } t = 0,] \\ + 1 \textrm{ für } t >0. + \end{cases}$ + \newline \tiny(machmal: 1 für $t=0$) + & + $\frac{1}{j\omega} + \pi\delta(\omega)$ + & + $u(t), \sigma(t), h(t)$ + & + \raisebox{-.5\height}{\includegraphics[width = 3.5cm]{include/Wichtige Funktionen/img/Sprungfunktion.png}} + \\ + Signumfunktion + \newline (Vorzeichenfunktion) + & + $\begin{cases} + -1 \textrm{ für } t<0, \\ + 0 \textrm{ für } t = 0, \\ + 1 \textrm{ für } t >0. + \end{cases} $ + & + $\frac{2}{j\omega}$ + & + $sgn(t)$ + & + \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Signumfunktion.png}} + \\ + Rampenfunktion + & + $\begin{cases} + 0 \textrm{ für } t \leq 0, \\ + t \textrm{ für } t > 0. + \end{cases}$ + & + $\frac{1}{s^2}$ + & + $r(t)$ + & + \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Rampenfunktion.png}} + \\ + Rechteckimpuls + & + $\begin{cases} + 1 \textrm{ für } |t| < a, \\ + \frac{1}{2} \textrm{ für } |t| = a, \\ + 0 \textrm{ für } |t| > a. + \end{cases} $ + & + $\frac{1}{s}- e^{-as}\frac{1}{s} $ + & + $p_a(t), \beta(t)$ + \newline $\sigma(t+a)-\sigma(t-a)$ + & + \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Rechteckimpuls.png}} + \\ + Dreieckimpuls + & + $\begin{cases} + 1 - \frac{|t|}{a} \textrm{ für } |t| < a \\ + 0 \textrm{ für } |t| \geq a + \end{cases}$ + & + $sinc^2(\omega)$ + & + $\Lambda(t)$ + & + \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/Dreieckimpuls.png}} + \\ + Sinc-Funktion + & + $\frac{sin(t)}{t}\; \forall t$ + \newline {\tiny $\lim _{t \to 0} sinc(t) = 1$} + \newline {\tiny wenn normalisiert: $t \to \pi t$} + & + $\beta(\omega)$ + \newline{\tiny(Rechteckimpuls)} + & + sinc(t) + & + \raisebox{-.5\height}{\includegraphics[width=3.5cm]{include/Wichtige Funktionen/img/SincFunktion.png}} + \\ + \hline +\end{tabular} \ No newline at end of file