diff --git a/README.md b/README.md index 381ead5..3bebcb4 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,17 @@ # ITET-Notes -My LaTeX Notes I took during my time at ETH \ No newline at end of file +Hier findest du meine Notizen zu den Fächern vom Studiengang ITET. + +Andere Dateien wie Lösungen zu Serien findest du auf meiner [ETH Website](https://eth.jirayuruh.ch). + +Die LaTeX Template ist inspiriert durch den Template von [Senior Mars](https://github.com/SeniorMars/dotfiles/tree/main/latex_template). + +## DISCLAIMER!!! + +Ich übernehme keine Haftung über mögliche Fehler in den Notizen und Zusammenfassungen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). + +Fehler können per Discord, WhatsApp, Mail und Moodle gemeldet werden. Sie sind in der Fehlerliste ersichtlich. + +--- + +Made by JirR02 in Switzerland 🇨🇭 diff --git a/fs25/README.md b/fs25/README.md new file mode 100644 index 0000000..e429b5d --- /dev/null +++ b/fs25/README.md @@ -0,0 +1,22 @@ +# ITET 2. Semester (FS 2025) + +Im zweiten Semester habe ich die folgenden Fächer belegt: + +- [Analysis II](https://eth.jirayuruh.ch/fs25/analysis/) +- [Grundzüge des Rechts](https://eth.jirayuruh.ch/fs25/grundzuege-recht/) +- [Informatik I](https://eth.jirayuruh.ch/fs25/informatik/) +- [Mathematische Methoden](https://eth.jirayuruh.ch/fs25/math_methoden/) +- [Netzwerke und Schaltungen II](https://eth.jirayuruh.ch/fs25/nus/) +- [Physik I](https://eth.jirayuruh.ch/fs25/physik/) + +Weitere Information findet ihr im VZZ oder auf eurer Moodle Seite. + +## DISCLAIMER!!! + +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). + +Fehler können per Discord, WhatsApp, Mail und Moodle gemeldet werden. Sie sind in der Fehlerliste ersichtlich. + +--- + +Made by JirR02 in Switzerland 🇨🇭 diff --git a/hs24/README.md b/hs24/README.md new file mode 100644 index 0000000..e5eeb7c --- /dev/null +++ b/hs24/README.md @@ -0,0 +1,21 @@ +# ITET 1. Semester (HS 2024) + +Im ersten Semester habe ich die folgenden Fächer belegt: + +- [Analysis I](https://eth.jirayuruh.ch/hs24/analysis/) +- [Digitaltechnik](https://eth.jirayuruh.ch/hs24/digitech/) +- [Lineare Algebra](https://eth.jirayuruh.ch/hs24/linalg/) +- [Netzwerke und Schaltungen I](https://eth.jirayuruh.ch/hs24/nus/) +- [Technische Mechanik](https://eth.jirayuruh.ch/hs24/techmech/) + +Weitere Information findet ihr im VZZ oder auf eurer Moodle Seite. + +## DISCLAIMER!!! + +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). + +Fehler können per Discord, WhatsApp, Mail und Moodle gemeldet werden. Sie sind in der Fehlerliste ersichtlich. + +--- + +Made by JirR02 in Switzerland 🇨🇭 diff --git a/hs24/analysis_I/.DS_Store b/hs24/analysis_I/.DS_Store new file mode 100644 index 0000000..f6c670c Binary files /dev/null and b/hs24/analysis_I/.DS_Store differ diff --git a/hs24/analysis_I/II_differentialrechnung_r/II_differentialrechnung_r.tex b/hs24/analysis_I/II_differentialrechnung_r/II_differentialrechnung_r.tex new file mode 100644 index 0000000..23b4a8c --- /dev/null +++ b/hs24/analysis_I/II_differentialrechnung_r/II_differentialrechnung_r.tex @@ -0,0 +1 @@ +\chapter{Differentialrechnung im $\mathbb{R}^{n}$} diff --git a/hs24/analysis_I/II_integration/II_integration.tex b/hs24/analysis_I/II_integration/II_integration.tex new file mode 100644 index 0000000..0b828d4 --- /dev/null +++ b/hs24/analysis_I/II_integration/II_integration.tex @@ -0,0 +1 @@ +\chapter{Mehrdimensionale Riemann-integration, Satz von Fubini über wiederholte Integration, Jordan-Mass, Substitutionsregel für mehrdimensionale Integrale} diff --git a/hs24/analysis_I/II_stetigkeit_topologie/II_stetigkeit_topologie.tex b/hs24/analysis_I/II_stetigkeit_topologie/II_stetigkeit_topologie.tex new file mode 100644 index 0000000..1d40d69 --- /dev/null +++ b/hs24/analysis_I/II_stetigkeit_topologie/II_stetigkeit_topologie.tex @@ -0,0 +1 @@ +\chapter{Topologie, Stetigkeit} diff --git a/hs24/analysis_I/I_differentialrechnung_r/I_differentialrechnung_r.tex b/hs24/analysis_I/I_differentialrechnung_r/I_differentialrechnung_r.tex new file mode 100644 index 0000000..063e7c9 --- /dev/null +++ b/hs24/analysis_I/I_differentialrechnung_r/I_differentialrechnung_r.tex @@ -0,0 +1,7 @@ +\chapter{Differentialrechnung auf $\mathbb{R}$} + +Intuitiv ist die Ableitung einer Funktion $f : \mathbb{R} \rightarrow \mathbb{R}$ an einer Stelle $x_0 \in \mathbb{R}$ die Steigung der Tangente an den Graphen von $f$ durch den Punkt $x_0, f(x_0)$. Genauer gesagt, ist die Ableitung der Grenzwert der Steigungen der Sekanten durch $(x_0, f(x_0)$ und $(x, f(x))$ für $x$ gegen $x_0$. Ableitungen sind allgegenwärtig in den Wissenschaften und im Ingenieurwesen. In der Mechanik ist die Geschwindigkeit eines Teilchens zum Beispiel die Ableitung seines Ortes als eine Funktion der Zeit. Als ein anderes Beispiel ist in einem elektrischen Schwingkreis die Stromstärke gleich der Ableitung der Ladung des Kondensators als eine Funktion der Zeit. + +\input{differential_differentiationsregel.tex} +\input{mittelwertsatz_folgerungen.tex} + diff --git a/hs24/analysis_I/I_differentialrechnung_r/differential_differentiationsregel.tex b/hs24/analysis_I/I_differentialrechnung_r/differential_differentiationsregel.tex new file mode 100644 index 0000000..b65b3a1 --- /dev/null +++ b/hs24/analysis_I/I_differentialrechnung_r/differential_differentiationsregel.tex @@ -0,0 +1,37 @@ +\section{Differential und Differentiationsregeln} + +\nt{ +\[ + dx \text{ und } dy \text{ sind Differentialen. } \Rightarrow f'(x_0) = \frac{dy}{dx}, dx \text{ nennt man Differentialquotient.} +.\] +} + +\nt{ + Je kleiner $\Delta x$ ist, desto näher kommt es an den Wert von $\Delta y$ für: + \[ + \Delta y \equiv f'(x_0) \Delta x + .\] + + % TODO: Add figure +} + +\exa{}{ +\[ + f : \mathbb{R} \rightarrow \mathbb{R}, f(x) := x ^2, f'(x_0 = 0) = 0 +.\] + +%TODO: Add figure +} + +%TODO: Add proof for Summen-, Produkt- und Quotioentenregel + +%TODO: Complete Proof + +%TODO: Complete Proof + +\nt{ + Die Menge $U$ ist offen, da $f$ stetig ist. Dies folgt aus dem Fakt, da das Urbild $f ^{-1}(V)$ offen ist $\forall V$ offen. +} + +%TODO: Add Examples and reference Exercises + diff --git a/hs24/analysis_I/I_differentialrechnung_r/mittelwertsatz_folgerungen.tex b/hs24/analysis_I/I_differentialrechnung_r/mittelwertsatz_folgerungen.tex new file mode 100644 index 0000000..d15d313 --- /dev/null +++ b/hs24/analysis_I/I_differentialrechnung_r/mittelwertsatz_folgerungen.tex @@ -0,0 +1,28 @@ +\section{Mittelwertsatz und Folgerungen} + +\nt{ + Für + \[ + f'(x_0) = \frac{f(b)-f(a)}{b-a} + .\] + + gilt, dass $a$ die Steigung der Sekante ist. + + %TODO: Add fig +} + +\nt{ + $\Rightarrow$ Die Funktion hat genau eine Lösung, die auch nach $f(0) = y_0$ erfüllt, mit $y_0 \in \mathbb{R}$ vorgegeben, nämlich $f(x) = ye ^{cx}$. +} + +\exa{}{ + \[ + \frac{x ^{n}}{e ^{(x ^{n})}} ? + .\] + + \[ + \frac{n \cdot x ^{n-1}}{e ^{(x ^{n})}} = \frac{e ^{x} - 1}{x} = \text{exp}(0) = 1 + .\] +} + +% TODO: Add Proof diff --git a/hs24/analysis_I/I_integration/I_integration.tex b/hs24/analysis_I/I_integration/I_integration.tex new file mode 100644 index 0000000..37bab18 --- /dev/null +++ b/hs24/analysis_I/I_integration/I_integration.tex @@ -0,0 +1 @@ +\chapter{Integration} diff --git a/hs24/analysis_I/I_stetigkeit_topologie/I_stetigkeit_topologie.tex b/hs24/analysis_I/I_stetigkeit_topologie/I_stetigkeit_topologie.tex new file mode 100644 index 0000000..d38312d --- /dev/null +++ b/hs24/analysis_I/I_stetigkeit_topologie/I_stetigkeit_topologie.tex @@ -0,0 +1,3 @@ +\chapter{Stetigkeit, Topologie} + +\newpage diff --git a/hs24/analysis_I/analysis_I.tex b/hs24/analysis_I/analysis_I.tex new file mode 100644 index 0000000..5688c35 --- /dev/null +++ b/hs24/analysis_I/analysis_I.tex @@ -0,0 +1,8 @@ +\part{Analysis I} + +\import{./grundlagen}{grundlagen.tex} +\import{./zahlen_und_vektoren}{zahlen_und_vektoren.tex} +\import{./folgen_und_reihen}{folgen_und_reihen.tex} +\import{./I_stetigkeit_topologie}{I_stetigkeit_topologie.tex} +\import{./I_differentialrechnung_r}{I_differentialrechnung_r.tex} +\import{./I_integration}{I_integration.tex} diff --git a/hs24/analysis_I/analysis_II.tex b/hs24/analysis_I/analysis_II.tex new file mode 100644 index 0000000..3490fe4 --- /dev/null +++ b/hs24/analysis_I/analysis_II.tex @@ -0,0 +1,8 @@ +\part{Analysis II} + +\import{./gdg}{gdg.tex} +\import{./II_stetigkeit_topologie}{II_stetigkeit_topologie.tex} +\import{./II_differentialrechnung_r}{II_differentialrechnung_r.tex} +\import{./umkehrsatz}{umkehrsatz.tex} +\import{./II_integration}{II_integration.tex} +\import{./vektorfeld}{vektorfeld.tex} diff --git a/hs24/analysis_I/disclaimer.tex b/hs24/analysis_I/disclaimer.tex new file mode 100644 index 0000000..9b42d66 --- /dev/null +++ b/hs24/analysis_I/disclaimer.tex @@ -0,0 +1,11 @@ +\section*{DISCLAIMER} +Diese Notizen wurden verfasst auf Basis der Vorlesung Analysis I (HS24) von F. Ziltener und dem Skript "Analysis für Informatik" von Michael Struwe. +\\ +\\ +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). +\\ +\\ +Alle Grafiken wurden eigenhändig mit Manim \cite{MCD2024} generiert. +\\ +\\ +Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden. diff --git a/hs24/analysis_I/fig/Fig_1.png b/hs24/analysis_I/fig/Fig_1.png new file mode 100644 index 0000000..f9594a1 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_1.png differ diff --git a/hs24/analysis_I/fig/Fig_10.png b/hs24/analysis_I/fig/Fig_10.png new file mode 100644 index 0000000..f49dd5b Binary files /dev/null and b/hs24/analysis_I/fig/Fig_10.png differ diff --git a/hs24/analysis_I/fig/Fig_11.png b/hs24/analysis_I/fig/Fig_11.png new file mode 100644 index 0000000..af4d1e9 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_11.png differ diff --git a/hs24/analysis_I/fig/Fig_12.png b/hs24/analysis_I/fig/Fig_12.png new file mode 100644 index 0000000..3c1938c Binary files /dev/null and b/hs24/analysis_I/fig/Fig_12.png differ diff --git a/hs24/analysis_I/fig/Fig_13.png b/hs24/analysis_I/fig/Fig_13.png new file mode 100644 index 0000000..fc4f186 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_13.png differ diff --git a/hs24/analysis_I/fig/Fig_14.png b/hs24/analysis_I/fig/Fig_14.png new file mode 100644 index 0000000..603b5fb Binary files /dev/null and b/hs24/analysis_I/fig/Fig_14.png differ diff --git a/hs24/analysis_I/fig/Fig_15.png b/hs24/analysis_I/fig/Fig_15.png new file mode 100644 index 0000000..0fad3c0 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_15.png differ diff --git a/hs24/analysis_I/fig/Fig_2.png b/hs24/analysis_I/fig/Fig_2.png new file mode 100644 index 0000000..80587a6 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_2.png differ diff --git a/hs24/analysis_I/fig/Fig_3.png b/hs24/analysis_I/fig/Fig_3.png new file mode 100644 index 0000000..136dece Binary files /dev/null and b/hs24/analysis_I/fig/Fig_3.png differ diff --git a/hs24/analysis_I/fig/Fig_4.png b/hs24/analysis_I/fig/Fig_4.png new file mode 100644 index 0000000..23d51b9 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_4.png differ diff --git a/hs24/analysis_I/fig/Fig_5.png b/hs24/analysis_I/fig/Fig_5.png new file mode 100644 index 0000000..12db570 Binary files /dev/null and b/hs24/analysis_I/fig/Fig_5.png differ diff --git a/hs24/analysis_I/fig/Fig_6.png b/hs24/analysis_I/fig/Fig_6.png new file mode 100644 index 0000000..eb9207f Binary files /dev/null and b/hs24/analysis_I/fig/Fig_6.png differ diff --git a/hs24/analysis_I/fig/Fig_7.png b/hs24/analysis_I/fig/Fig_7.png new file mode 100644 index 0000000..a46ff6b Binary files /dev/null and b/hs24/analysis_I/fig/Fig_7.png differ diff --git a/hs24/analysis_I/fig/Fig_8.png b/hs24/analysis_I/fig/Fig_8.png new file mode 100644 index 0000000..26b94fa Binary files /dev/null and b/hs24/analysis_I/fig/Fig_8.png differ diff --git a/hs24/analysis_I/fig/Fig_9.png b/hs24/analysis_I/fig/Fig_9.png new file mode 100644 index 0000000..0b0eb6c Binary files /dev/null and b/hs24/analysis_I/fig/Fig_9.png differ diff --git a/hs24/analysis_I/folgen_und_reihen/folgen_und_reihen.tex b/hs24/analysis_I/folgen_und_reihen/folgen_und_reihen.tex new file mode 100644 index 0000000..04c473a --- /dev/null +++ b/hs24/analysis_I/folgen_und_reihen/folgen_und_reihen.tex @@ -0,0 +1,2 @@ +\chapter{Folgen und Reihen} + diff --git a/hs24/analysis_I/gdg/gdg.tex b/hs24/analysis_I/gdg/gdg.tex new file mode 100644 index 0000000..4ffb42e --- /dev/null +++ b/hs24/analysis_I/gdg/gdg.tex @@ -0,0 +1 @@ +\chapter{Gewöhnliche Differentialgleichungen, Anwendung auf die Mechanik und die Elektrotechnik} diff --git a/hs24/analysis_I/grundlagen/grundlagen.tex b/hs24/analysis_I/grundlagen/grundlagen.tex new file mode 100644 index 0000000..2c1a392 --- /dev/null +++ b/hs24/analysis_I/grundlagen/grundlagen.tex @@ -0,0 +1,3 @@ +\chapter{Grundlagen: Logik, Mengen, Funktionen} + +\input{logik.tex} diff --git a/hs24/analysis_I/grundlagen/logik.tex b/hs24/analysis_I/grundlagen/logik.tex new file mode 100644 index 0000000..5d69865 --- /dev/null +++ b/hs24/analysis_I/grundlagen/logik.tex @@ -0,0 +1,54 @@ +\section{Logik} + +In der Logik werden (mathematische) Aussagen untersucht. Eine Aussage ist eine Äusserung, die entweder wahr oder falsch ist. \cite{Ziltner2024} (Wahr oder Falsch). +\\ +\\ +In der mathematischen Logik gelten die folgenden Sätze. + +\begin{itemize} + \item \textbf{Satz vom ausgeschlossenen Wiederspruch:} Eine Aussage ist nicht sowohl war als auch falsch. + \item \textbf{Satz vom ausgeschlossenen Dritten:} Jede Aussage ist wahr oder falsch. +\end{itemize} \cite{Ziltner2024} + +\nt{ + Es gibt gewisse Aussagen, als logische Aussage gelten könnte aber nicht zulässig ist. Solche Aussagen sind meisten rückbezügliche Äusserungen und sind deswegen keine sinnvollen Aussagen. (Siege Lügner-Paradox) +} + +Aussagen können verneint und miteinander verknüpft werden. + +\begin{center} + \begin{tabular}{| c | c | c |} + Notation & Bedeutung & Bezeichnung \\ + \hline + T & wahr & \\ + F & falsch & \\ + $\neg A$ & nicht A & Negation \\ + \end{tabular} +\end{center} + +Für Verknüpfungen verwenden wir folgende Notationen. + +\begin{center} + \begin{tabular}{| c | c | c |} + Notation & Bedeutung & Bezeichnung \\ + \hline + $A \land B$ & $A$ und $B$ & Konjunktion \\ + $A \lor B$ & $A$ oder $B$ & inklusive Disjunktion \\ + $A \dot{\lor} B$ & entweder $A$ oder $B$ & exklusive Disjunktion \\ + $A \Rightarrow B$ & wenn $A$, dann $B$ & Implikation \\ + $A \Leftrightarrow B$ & genau dann $A$, wenn $B$ & Äquivalenz \\ + \end{tabular} +\end{center} + +Die Wahrheitstablelle der vorher erwähnten Verknüpfungen sind wie folgt. + +\begin{center} + \begin{tabular}{| c | c | c | c | c | c | c |} + $A$ & $B$ & $A \land B$ & $A \lor B$ & $A \dot{\lor} B$ & $A \Rightarrow B$ & $A \Leftrightarrow B$ \\ + \hline + F & F & F & F & F & T & T \\ + F & T & F & T & T & T & F \\ + T & F & F & T & T & F & F \\ + T & T & T & T & F & T & T \\ + \end{tabular} +\end{center} diff --git a/hs24/analysis_I/letterfonts.tex b/hs24/analysis_I/letterfonts.tex new file mode 100644 index 0000000..3939d5b --- /dev/null +++ b/hs24/analysis_I/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/hs24/analysis_I/macros.tex b/hs24/analysis_I/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/hs24/analysis_I/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/hs24/analysis_I/notizen_analysis_ruh_jirayu.bbl b/hs24/analysis_I/notizen_analysis_ruh_jirayu.bbl new file mode 100644 index 0000000..d6c5f59 --- /dev/null +++ b/hs24/analysis_I/notizen_analysis_ruh_jirayu.bbl @@ -0,0 +1,11 @@ +\begin{thebibliography}{} + +\bibitem[{The Manim Community Developers}, 2024]{MCD2024} +{The Manim Community Developers} (2024). +\newblock {Manim - Mathematical Animation Framework}. + +\bibitem[Ziltner, 2024]{Ziltner2024} +Ziltner, Prof.~Dr., F. (2024). +\newblock {Notizen zur Vorlesung Analysis 1 für ITET und RW, Herbstsemester 2024}. + +\end{thebibliography} diff --git a/hs24/analysis_I/notizen_analysis_ruh_jirayu.pdf b/hs24/analysis_I/notizen_analysis_ruh_jirayu.pdf new file mode 100644 index 0000000..9de8577 Binary files /dev/null and b/hs24/analysis_I/notizen_analysis_ruh_jirayu.pdf differ diff --git a/hs24/analysis_I/notizen_analysis_ruh_jirayu.tex b/hs24/analysis_I/notizen_analysis_ruh_jirayu.tex new file mode 100644 index 0000000..85fcf36 --- /dev/null +++ b/hs24/analysis_I/notizen_analysis_ruh_jirayu.tex @@ -0,0 +1,43 @@ +\documentclass{report} + +\def\papertitle{Analysis I} + +\def\theorytitle{Theorie} +\def\corollarytitle{Daraus folgt} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Behauptung} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Beispiel} +\def\questiontitle{Frage} +\def\wrongctitle{Falscher Konzept} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[]{}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\import{./}{disclaimer.tex} +\import{./}{analysis_I.tex} +\import{./}{analysis_II.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% BIBLIOGRAPHY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\addcontentsline{toc}{chapter}{References} +\bibliographystyle{apalike} +\bibliography{sources} + +\end{document} diff --git a/hs24/analysis_I/preamble.tex b/hs24/analysis_I/preamble.tex new file mode 100644 index 0000000..23f5930 --- /dev/null +++ b/hs24/analysis_I/preamble.tex @@ -0,0 +1,778 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[german]{babel} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} 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\renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep +% \let\@ifnextchar\new@ifnextchar +% \array{*\c@MaxMatrixCols #1}} + + +%\usepackage{framed} +%\usepackage{titletoc} +%\usepackage{etoolbox} +%\usepackage{lmodern} + + +%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{} + +%\renewenvironment{leftbar} +%{\def\FrameCommand{\hspace{6em}% +% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}% +% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt% +%} +%{\endMakeFramed} + +%\titlecontents{chapter} +%[0em]{\vspace*{2\baselineskip}} +%{\parbox{4.5em}{% +% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}% +% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily} +%{}{\endleftbar} +%\titlecontents{section} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} +%\titlecontents{subsection} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage{tikz} +\definecolor{doc}{RGB}{0,60,110} +\usepackage{titletoc} +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Kapitel\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Seite \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/hs24/analysis_I/sources.bib b/hs24/analysis_I/sources.bib new file mode 100644 index 0000000..2fea1ad --- /dev/null +++ b/hs24/analysis_I/sources.bib @@ -0,0 +1,24 @@ +@misc{Ziltner2024, + author = {Ziltner, Prof. Dr., Fabian}, + month = dec, + title = {{Notizen zur Vorlesung Analysis 1 für ITET und RW, Herbstsemester + 2024}}, + year = {2024}, +} + +@misc{Struwe2010, + author = {Struwe, Prof. Dr., Michael}, + month = nov, + title = {{Analysis für Informatik}}, + year = {2010}, +} + +@software{MCD2024, + author = {{The Manim Community Developers}}, + license = {MIT}, + month = apr, + title = {{Manim - Mathematical Animation Framework}}, + url = {https://www.manim.community/}, + version = {v0.18.1}, + year = {2024}, +} diff --git a/hs24/analysis_I/umkehrsatz/umkehrsatz.tex b/hs24/analysis_I/umkehrsatz/umkehrsatz.tex new file mode 100644 index 0000000..137d2cb --- /dev/null +++ b/hs24/analysis_I/umkehrsatz/umkehrsatz.tex @@ -0,0 +1 @@ +\chapter{Umkehrsatz, Satz über implizite Funktionen, Untermannigfaltigkeit des Koordinatenraums, Tangentialraum} diff --git a/hs24/analysis_I/vektorfeld/vektorfeld.tex b/hs24/analysis_I/vektorfeld/vektorfeld.tex new file mode 100644 index 0000000..9e3a9ed --- /dev/null +++ b/hs24/analysis_I/vektorfeld/vektorfeld.tex @@ -0,0 +1 @@ +\chapter{Vektorfelder und die Sätze von Green, Stokes und Gauss} diff --git a/hs24/analysis_I/zahlen_und_vektoren/zahlen_und_vektoren.tex b/hs24/analysis_I/zahlen_und_vektoren/zahlen_und_vektoren.tex new file mode 100644 index 0000000..f3bf29d --- /dev/null +++ b/hs24/analysis_I/zahlen_und_vektoren/zahlen_und_vektoren.tex @@ -0,0 +1,3 @@ +\chapter{Zahlen und Vektoren} + +\newpage diff --git a/hs24/digitaltechnik/automaten/automaten.tex b/hs24/digitaltechnik/automaten/automaten.tex new file mode 100644 index 0000000..df1eda5 --- /dev/null +++ b/hs24/digitaltechnik/automaten/automaten.tex @@ -0,0 +1,7 @@ +\chapter{Automaten} + +\input{def.tex} +\input{entwurf_analyse.tex} +\input{garagenautomat_vertiefung.tex} +\input{fortgeschritten.tex} +\input{zaehler.tex} diff --git a/hs24/digitaltechnik/automaten/def.tex b/hs24/digitaltechnik/automaten/def.tex new file mode 100644 index 0000000..9c78aa7 --- /dev/null +++ b/hs24/digitaltechnik/automaten/def.tex @@ -0,0 +1,56 @@ +\section{Was sind Automaten?} \label{sec:auto} + +\subsection{Definition und Grundlagen} + +\dfn{Automat}{ + Ein Automat beschreibt ein System, welches auf sein Eingang reagiert und ein Ausgang produziert, der von dem Eingangssignal und von dem momentanen Zustand des Systems abhängt. \cite{Luisier2024} Ein typisches Beispiel für ein Automaten sind Schaltwerke. + \\ + Wenn wir von Automaten sprechen, reden wir hauptsächlich von endlichen Automaten. Endliche Automaten haben nur die möglichkeit endlich viele Eingänge, Ausgänge und gespeicherte Zustände zu haben. Unendliche Automaten gibt es nicht. +} + +Bei den Automaten unterscheiden wir zwischen synchrone und asynchrone Automaten. +\\ +\subsubsection*{Synchrone Automaten} + +Bei synchrone Automaten haben alle Speicherelemente (Flipflops) das gleiche Taktsignal. Daraus folgt, dass interne Zustandsänderungen synchron mit dem Taktsignal laufen. + +\includegraphics[width=\linewidth]{fig/Fig_33.png} \cite{Luisier2024} + +\subsubsection*{Asynchrone Automaten} + +In vergleich zu synchrone Automaten haben die Speicherelemente in asynchrone Automaten nicht das gleiche Taktsignal. Eine Zustandsänderung wird durch die Eingangssignale initiiert. +\\ +\\ +Ein endlicher Automat wird durch ein 6-Tupel charakterisiert: + +\begin{tabular}{ p{0.2\linewidth} p{0.8\linewidth} } + $X = (x_1, x_2, ... , x_e)$ & Eingabealphabet mit $e$ Eingängen $x_i$, die durch binäre Eingangsvariablen $\{ 0,1 \}$ repräsentiert werden. \\ + $Y = (y_1, y_2, ... , y_b)$ & Ausgabealphabet mit $b$ Ausgängen $y_i$, die als Bits mit Wert $\{ 0,1 \}$ dargestellt werden. \\ + $Z = (z_1, z_2, ... , z_m)$ & Zustandsmenge mit $m$ inneren Zustansvariablen $z_i$, die einen Wert $z_i$, die einen Wert $\{ 0,1 \}$ haben können. Insfesamt gibt es $n_m = 2 ^{m}$ Zustände. \\ + $Z_0 \in Z$ & Anfangszustand \\ + $f_{c 1}: (X_n, Z_n) \rightarrow Z_{n+1}$ & Übergangs-, Überführungsfunktion \\ + $f_{c 1}: (X_n, Z_n) \rightarrow Y_n$ & Ausgangs-, Ausgabefunktion \\ +\end{tabular} + +\subsection{Automatentypen: Mealy vs. Moore} + +\dfn{Mealy Automat}{ + Ein Mealy Automat ist ein endlicher Automat, dessen Ausgang abhängig ist vom Eingang $X_n$, sowie den momentanen Zustand $Z_n$. Der Folgezustand $Z_{n+1}$ ist abhängig vom momentanen Zustand sowie dem Eingang. + + \includegraphics[width=\textwidth]{fig/Fig_34.png} \cite{Luisier2024} +} + +\dfn{Moore Automat}{ + Im Vergleich zum Mealy Automat ist der Ausgang vom Moore Automat nur abhängig vom momentanen Zustand $Z_n$. Der Folgezustand $Z_{n+1}$ ist wie beim Mealy Automat abhängig vom momentanen Zustand, sowie dem Eingang. + + \includegraphics[width=\linewidth]{fig/Fig_35.png} \cite{Luisier2024} +} + +Eine weitere Art von Automat ist der Medwedjew Automat. Dieser unterscheidet sich vom Moore Automat dadurch, dass sein Ausgang gleich dem momentanen Zustand ist. Der Folgezustand ist wie beim Mealy Automat abhängig vom momentanen Zustand und dem Eingang. + +\includegraphics[width=\textwidth]{fig/Fig_36.png} \cite{Luisier2024} + +\nt{ + Medwedjew Automaten haben eine grosse Relevanz für Zähler. +} + diff --git a/hs24/digitaltechnik/automaten/entwurf_analyse.tex b/hs24/digitaltechnik/automaten/entwurf_analyse.tex new file mode 100644 index 0000000..b94efe8 --- /dev/null +++ b/hs24/digitaltechnik/automaten/entwurf_analyse.tex @@ -0,0 +1,44 @@ +\section{Entwurf und Analyse von Automaten} + +\subsection{Zustandsdiagramm, Folgezustandstabelle} + +Automaten können auf verschiedenen Weisen beschrieben werden. + +\begin{itemize} + \item Ausgangs- und Übergangsfunktionen + \item Zustandsfolgetabellen (Folgezustandstabelle) + \item Zustandsdiagramme oder Zustandsgraphen + \item Karnaugh Diagramme +\end{itemize} + +In den meisten Fällen fängt man mit dem Zustandsdiagramm, da dieser sehr schnell generiert werden kann. Aus dem Zustandsdiagramm kann die Folgezustandstabelle konstruiert werden. Wenn diese vorhanden ist, kann ein Karnaugh-Diagramm für jede innere Zustandsvariable und für jeden Ausgang aufgesetzt werden. Damit werden die Gleichungen der Ausgangs- und Übergangsfunktionen minimiert. Am Ende wird ein Schaltwerk erzeugt. \cite{Luisier2024} Es ist aber auch möglich, dass der umgekehrte Fall auftritt. + +\dfn{Zustandsdiagramm}{ + Ein Zustandsdiagramm ist eine graphische Darstellung der Funktion von Automaten, die aus Knoten und gericheteten Kanten besteht. + + \includegraphics[width=\linewidth]{fig/Fig_26.png} \cite{Luisier2024} + + Die Knoten bezeichnen die internen Zustände, die Kanten den Übergang zwischen zwei Zuständen. Die Eingangskombination $e$, die die Zustandsänderung bewirkt, und der Ausgang $a$ werden an der jeweiligen Kante vermerkt. \cite{Luisier2024} + + \includegraphics[width=\linewidth]{fig/Fig_27.png} \cite{Luisier2024} + + Wichtig zu erwähnen ist, dass bei Moore Automaten die Knoten sowohl die internen Zustände als auch die Ausgänge bezeichnen. +} + +\dfn{Folgezustandstabelle}{ + Folgezustandstabellen zeigen alle möglichen Kombinationen eines Automaten an. Die Anzahl Reihen einer Folgezustandstabelle ist $2 ^{e + m}$, wobei $e$ die Anzahl Eingangsbits und $m$ die Anzahl Zustandsbits ist. Die Anzahl Spalten beträgt $e + 2m + b$, wobei $b$ die Anzahl Ausgangsbits ist. +} + +\subsection{Eingangs-, Ausgangs-, Zustandsvariablen} + +Um einen Automaten zu entwerfen kann man das folgende Kochrezept befolgen. + +\begin{enumerate} + \item Zustandsmenge bestimmen, daraus folgt die Anzahl der Zustandsvariablen und der erforderlichen D-Flipflops + \item Definition der Ein- und Ausgangsvariablen, Kodierung + \item Darstellung der Zustandsfolge in einem Zustandsdiagramm + \item Aufstellung der Zustandsfolgetabelle + \item Bestimmung der minimierten Ausgangs- und Übergangsfunktionen mit der Hilfe von KV-Diagrammen + \item Prüfung auf unbenutzte Zustände + \item Konstruktion des Schaltplans anhand der Schaltfunktion +\end{enumerate} \cite{Luisier2024} diff --git a/hs24/digitaltechnik/automaten/fortgeschritten.tex b/hs24/digitaltechnik/automaten/fortgeschritten.tex new file mode 100644 index 0000000..ef583b3 --- /dev/null +++ b/hs24/digitaltechnik/automaten/fortgeschritten.tex @@ -0,0 +1,7 @@ +\section{Fortgeschrittene Automateneigenschaften} + +\subsection{Gekoppelte Automaten} + +In einigen Fällen ist es geeigneter komplexe Automaten aus mehreren Teilautomaten aufzubauen. Diese sind oft übersichtlicher. + +\includegraphics[width=\linewidth]{fig/Fig_29.png} \cite{Luisier2024} diff --git a/hs24/digitaltechnik/automaten/garagenautomat_vertiefung.tex b/hs24/digitaltechnik/automaten/garagenautomat_vertiefung.tex new file mode 100644 index 0000000..c0d2bb7 --- /dev/null +++ b/hs24/digitaltechnik/automaten/garagenautomat_vertiefung.tex @@ -0,0 +1,8 @@ +\section{Garagenautomat: Vertiefung} + +\subsection{Mealy $\leftrightarrow$ Moore Umwandlung} + +Mealy und Moore Automaten lassen sich \textbf{immer} ineinander umwanden. \cite{Vasylyev2024} Wichtig dabei zu erwähnen ist, dass sich das zeitliche Verhalten der Eingangs- und Ausgangssignale sich ändert aufgrund von ihren verschiedenen Eigenschaften (Kapitel \ref{sec:auto}). Mealy Automaten sind sehr anfällig für Störungen, da ihr Ausgang abhängig ist vom Eingang und momentanen Zustand. Der Eingang ist nicht abhängig vom Taktsignal. Bei einer Störung am Eingang besteht die Möglichkeit, dass ein Ausgang beschaltet wird aber der Zustand nicht aktualisiert wird. +\\ +\\ +Um dieses Problem zu bekämpfen könnte man den Eingang mit dem Taktsignal Synchronisieren. diff --git a/hs24/digitaltechnik/automaten/zaehler.tex b/hs24/digitaltechnik/automaten/zaehler.tex new file mode 100644 index 0000000..7850050 --- /dev/null +++ b/hs24/digitaltechnik/automaten/zaehler.tex @@ -0,0 +1,61 @@ +\section{Zähler} + +\subsection{Asynchronzähler} + +In Kapitel \ref{sec:ffcirc} haben wir gelernt, dass man T-Flipfops verketten kann, um $n$-Bit Dualzähler zu erstellen. Das Problem mit diesen sogenannten Asynchronzähler ist die Zeitverzögerung der T-Flipflops. Je mehr Bits der Dualzähler hat, desto mehr T-Flipflops haben wir und desto länger braucht der letzte T-Flipflop zum Schalten für den MSB. Deshalb müssen bei Asynchronzähler die maximale Taktfrequenz des Taktsignals berücksichtigt werden. Dies limitiert die Geschwindigkeit des Dualzählers. + +\subsection{Modulo-$n$ (mod-$n$) Zähler} + +In manchen Fällen möchte man nicht das volle Pottential von Asynchronzähler verwenden. Man möchte nur bis zu einer bestimmten Dualzahl hochzählen. In diesem Fall kommen Modulo-$n$ Zähler ins Spiel. Wie die $n$-Bit Dualzähler bestehen Modulo-$n$ Zähler aus T-Flipfops mit dem Unterschied, dass die T-Flipflops einen Reset Eingang haben. Die Ausgänge der T-Flipflops sind mit einem AND Gatter verbunden. Sobald die Zahl $n+1$ am Ausgang liegt, werden die Reset-Eingänge von den Flipflops beschaltet und der Zähler fängt wieder bei 0 an. Soll der Zähler bei einer anderen Zahl anfangen zu zählen, so muss bei den Flipflops mit einem Set-Eingang gearbeitet werden. + +\begin{minipage}{0.15\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_13.png} \cite{Luisier2024} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop T, dot on notQ, add async SR](tff1){} + (tff1.pin 3) --++(left:10mm) node[point, label=180:CLK] {} + (tff1.pin 6) --++(right:10mm) --++(down:17mm) --++(right:5mm) + node[flipflop T, dot on notQ, anchor=pin 3, add async SR](tff2){} + (tff2.pin 6) --++(right:10mm) --++(down:17mm) --++(right:5mm) + node[flipflop T, dot on notQ, anchor=pin 3, add async SR](tff3){} + (tff3.pin 6) --++(right:5mm) --++(up:20mm) node[point, label=90:$\text{Q}_{III}$] {} + (tff3.pin 6) --++(right:5mm) --++(up:5mm) --++(right:5mm) + node[and port, number inputs=3, anchor=in 3](and){} + node [notcirc, left] at (and.bin 1) {} + (and.in 1) --++(left:80mm) + (and.in 2) --++(left:42.5mm) + (and.out) --++(down:35mm) --++(left:115.5mm) + (tff1.down) --++(down:4mm) + (tff2.down) --++(down:4mm) + (tff3.down) --++(down:4mm); + \path[draw] (tff1.pin 6) --++(right:5mm) --++(up:20mm) node[point, label=90:$\text{Q}_I$] {}; + \path[draw] (tff2.pin 6) --++(right:5mm) --++(up:20mm) node[point, label=90:$\text{Q}_{II}$] {}; + \end{circuitikz} + \end{center} +\end{minipage} + +\subsection{Synchronzähler} + +Beim Asynchronzähler ist das Taktsignal der Flipflops immer vom vorherigen Ausgang des Flipflops abhängig (ausser beim ersten Flipflop). Im Gegensatz zu dem haben die Flipflops von Synchronzähler das gleiche Taktsignal und schalten gleichzeitig. Synchronzähler werden meistens als Medwedjew-Automaten implementiert. + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_13.png} \cite{Luisier2024} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_14.png} \cite{Luisier2024} +\end{minipage} + +\nt{ + Rückwärtszähler können auch durch Medwedjew-Automaten realisiert werden. Es muss zusätlich ein weiterer Eingang hinzugefügt werden, welcher die Zählrichtung bestimmt. + + \includegraphics[width=0.25\linewidth]{fig/Fig_15.png} \cite{Luisier2024} +} + +\nt{ + In Kapitel \ref{sec:ffcirc} haben wir gesehen, dass T-Flipflops als Freqenzteiler fungieren können. Dies kann auch mit einem Medwedjew-Automaten und ein AND Gatter realisiert werden. Das AND Gatter ist mit einem T-Flipflop verbunden, welcher beschaltet wird, sobald die Zahl erreicht wird. + + \includegraphics[width=\textwidth]{fig/Fig_16.png} \cite{Luisier2024} +} + diff --git a/hs24/digitaltechnik/cmos/cmos.tex b/hs24/digitaltechnik/cmos/cmos.tex new file mode 100644 index 0000000..4d0a65a --- /dev/null +++ b/hs24/digitaltechnik/cmos/cmos.tex @@ -0,0 +1,4 @@ +\chapter{CMOS Schaltungen} + +\input{funktion.tex} +\input{pullup_pulldown.tex} diff --git a/hs24/digitaltechnik/cmos/funktion.tex b/hs24/digitaltechnik/cmos/funktion.tex new file mode 100644 index 0000000..c231096 --- /dev/null +++ b/hs24/digitaltechnik/cmos/funktion.tex @@ -0,0 +1,151 @@ +\section{Wie funktionieren MOS-Transistoren?} + +\subsection{Kurzer Überblick} + +In diesem Kapitel geht es um CMOS Transistoren. In Allgemeinen werden Transistoren verwendet, um Gatter zu bauen. Diese werden, wie in den letzten paar Kapitel besprochen in sehr vielen logischen Schaltungen verwendet, welche fundamental sind für viele elektronischen Geräte, welche uns begleiten. +\\ +Wie durch den Namen zu erkennen fungieren die Transistoren durch die CMOS Technologie. CMOS steht für "Complementary Metal-Oxide-Semiconductor". Die Transistoren sind also Metalloxid-Halbleiter. +\\ +\\ +\dfn{MOS Transistoren}{ + MOS Transistoren sind steuerbare Widerstände bestehend aus 3 Kontakten, namentlich Source, Drain und Gate. Die Ladungsträger fliessen von Source zu Drain. Der Gate wird verwendet, um den Fluss von Ladungsträger zu beeinflussen. +} + +\includegraphics[width=0.7\linewidth]{fig/Fig_28.png} \cite{Luisier2024} + +MOS Transistoren verhalten sich wie Schalter. \cite{Luisier2024} Ihr Widerstand wird durch die Source-Gate Spannung ($V_{gs}$) gesteuert. Dabei spielt die Schwellspannung ($V_{th}$ $th$ für Threshhold) eine grosse Rolle. Ist $|V_{gs}| < |V_{th}|$ ist der Transistor hochohmig und der Schalter ist zu. Im umgekehrten Fall ist der Transistor niederohmig. Somit ist der Schalter auf. + +\nt{ + Die Schwellspannung $V_{th}$ wird in diesem Semester keine grosse Rolle spielen. Es ist aber hilfreich fürs Verständnis. +} + +Was sind jetzt aber CMOS Transistoren. Es gibt 2 CMOS Transistoren und beide sind ihr gegenseitiges Komplementär. Daher kommt auch ihr Name. Grundsätzlich unterscheidet man zwischen zwei verschiedene CMOS Transistoren. + +\dfn{NMOS-Transistor}{ + NMOS-Transistoren haben einen n-dotierten Innenwiderstand. Dadurch erhält er die Eigenschaft, dass der Transistor leitet, sobald eine Spannung am Gate liegt. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $G$ & NMOS & $Y$ \\ + \hline + 0 & Leitet nicht & 0 \\ + 1 & Leitet & 1 \\ + \end{tabular} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + + \begin{center} + \begin{circuitikz}[european] + \node[thick, nmos, anchor=D] (nmos1) at(0, -2) {} + (nmos1.gate) node[anchor=east] {G} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {S} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {D}; + \end{circuitikz} + \end{center} + \end{minipage} +} + +\dfn{PMOS-Transistor}{ + PMOS-Transistoren haben einen p-dotierten Innenwiderstand. Dadurch erhält er die Eigenschaft, dass der Transistor leitet, sobald keine Spannung am Gate liegt. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $G$ & PMOS & $Y$ \\ + \hline + 0 & Leitet & 1 \\ + 1 & Leitet nicht & 0 \\ + \end{tabular} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + + \begin{center} + \begin{circuitikz}[european] + \node[thick, pmos, anchor=D] (pmos1) at(0, -2) {} + (pmos1.gate) node[anchor=east] {G} + (pmos1.drain) node[anchor=west, yshift=-0.15cm] {D} + (pmos1.source) node[anchor=west,yshift=+0.15cm] {S}; + \end{circuitikz} + \end{center} + \end{minipage} +} + +\nt{ + Die Eigenschaft, dass NMOS- bzw. PMOS-Transistoren bei einer gewissen Gate Spannung leitet hat eine bestimmte Funktion bezüglich $V_{gs}$. NMOS Transistoren leiten, sobald eine positive Spannung am Eingang ist. Ist die Spannung negativ, leitet sie nicht. Umgekehrt ist es beim PMOS Transistor. +} + + +NMOS- und PMOS Transistoren haben vor allem eine bedeutende Verwendung bei Pull-up und Pull-Down Schaltungen. + +\dfn{Pull-down Schaltung}{ + Eine Pull-down ist eine Schaltung, welcher die Spannung $V_y$ von seinem Ausgang $Y$ in Abhängigkeit von der Gate Spannung definiert. ($V_{gs}$) + + \begin{minipage}{0.5\linewidth} + \begin{tabular}{| c | c | c |} + $G$ & NMOS & $Y$ \\ + \hline + 0 & Leitet nicht & 1 \\ + 1 & Leitet & 0 \\ + \end{tabular} + + $\Rightarrow$ Leitend falls Eingang $V_G = 1$, 0 liegt am Ausgang. + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \node[circ, label=90:{\small $V_{DD}$}](origin) at (0,0) {}; + \node[thick, nmos, anchor=D] (nmos1) at(0, -2) {} + (nmos1.gate) node[anchor=east] {G} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {D} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {S}; + \draw[thick] + (origin) + to[R=$R$] (0,-1.8) node[circ] (ybase) {} + to[] (0, -2); + \draw[thick] (nmos1.S) -- (0, -4) coordinate(gnd); + \path[draw] (ybase) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw, very thick] (-0.25, -4) -- (gnd) -- (0.25, -4); + \end{circuitikz} + \end{center} + \end{minipage} +} + +\dfn{Pull-up Schaltung}{ + Eine Pull-up ist eine Schaltung, welcher die Spannung $V_y$ von seinem Ausgang $Y$ in Abhängigkeit von der Gate Spannung definiert. ($V_{gs}$) + + \begin{minipage}{0.5\linewidth} + \begin{tabular}{| c | c | c |} + $G$ & NMOS & $Y$ \\ + \hline + 0 & Leitet & 1 \\ + 1 & Leitet nicht & 0 \\ + \end{tabular} + + $\Rightarrow$ Leitend falls Eingang $V_G = 0$, 1 liegt am Ausgang. + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \coordinate (gnd) at (0, -4); + \node[circ, label=90:{\small $V_{DD}$}] (vdd) at (0,0) {}; + \node[pmos, thick] (pmos) at (0, -1){} + (pmos.gate) node[anchor=east] {G} + (pmos.source) node[anchor=west,yshift=-0.15cm] {S} + (pmos.drain) node[anchor=west, yshift=+0.15cm] {D}; + + \draw[thick] (gnd) + to[R=$R$] (0, -2) + -- (0, -2) node[circ] (ybase) {} + -- (pmos.D); + + \draw[thick] (pmos.S) -- (vdd); + + \path[draw] (ybase) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw, very thick] (-0.25, -4) -- (gnd) -- (0.25, -4); + \end{circuitikz} + \end{center} + \end{minipage} +} diff --git a/hs24/digitaltechnik/cmos/pullup_pulldown.tex b/hs24/digitaltechnik/cmos/pullup_pulldown.tex new file mode 100644 index 0000000..25c232a --- /dev/null +++ b/hs24/digitaltechnik/cmos/pullup_pulldown.tex @@ -0,0 +1,186 @@ +\section{Pull-up und Pull-down Schaltungen} + +\subsection{NOT, NAND, und NOR} + +\dfn{NICHT Gatter in CMOS Technik}{ + Das NOT Gatter kann durch CMOS Gatter wie folgt dargestellt werden. + + \begin{center} + \begin{circuitikz}[european] + \node[circ, label=90:{\small $V_{DD}$}](origin) at (0,0) {}; + \node[thick, nmos, anchor=D] (nmos1) at(0, -2) {} + (nmos1.gate) node[anchor=east] {} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, pmos, anchor=D] (pmos1) at(0, -1.5) {} + (pmos1.gate) node[anchor=east] {} + (pmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (pmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \draw[thick] (nmos1.S) -- (0, -4) coordinate(gnd); + \draw[thick] (pmos1.D) -- (nmos1.D); + \draw[thick] (pmos1.G) -- (nmos1.G); + \path[draw] (0,-1.75) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw] (-1,-1.75) --++(left:10mm) node[point, label=180:A] {}; + \path[draw, very thick] (-0.25, -4) -- (gnd) -- (0.25, -4); + \end{circuitikz} + \end{center} +} + +\dfn{NAND Gatter in CMOS Technik}{ + Das NAND Gatter kann durch CMOS Gatter wie folgt dargestellt werden. + + \begin{center} + \begin{circuitikz}[european] + \node[circ, label=90:{\small $V_{DD}$}](origin) at (0,0) {}; + \node[thick, nmos, anchor=D] (nmos1) at(0, -2.25) {} + (nmos1.gate) node[anchor=east] {} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, nmos, anchor=D] (nmos2) at(0, -4.25) {} + (nmos2.gate) node[anchor=east] {} + (nmos2.drain) node[anchor=west, yshift=-0.15cm] {} + (nmos2.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, pmos, anchor=D] (pmos1) at(-1.5, -1.75) {} + (pmos1.gate) node[anchor=east] {} + (pmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (pmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, pmos, anchor=D] (pmos2) at(1.5, -1.75) {} + (pmos2.gate) node[anchor=east] {} + (pmos2.drain) node[anchor=west, yshift=-0.15cm] {} + (pmos2.source) node[anchor=west,yshift=+0.15cm] {}; + \draw[thick] (nmos2.S) -- (0, -6.25) coordinate(gnd); + \draw[thick] (pmos1.D) -- (pmos2.D); + \draw[thick] (pmos1.S) -- (pmos2.S); + \draw[thick] (origin) -- (0,-0.23); + \draw[thick] (nmos1.S) -- (nmos2.D); + \draw[thick] (0,-1.75) -- (nmos1.D); + \draw[thick] (pmos1.G) -- (-2.5,-3); + \draw[thick] (pmos2.G) -- (-1.25,-1); + \draw[thick] (-1.25,-1) -- (-1.25, -5); + \path[draw] (0,-2) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw] (nmos1.G) --++(left:20mm) node[point, label=180:A] {}; + \path[draw] (nmos2.G) --++(left:20mm) node[point, label=180:B] {}; + \path[draw, very thick] (-0.25, -6.25) -- (gnd) -- (0.25, -6.25); + \end{circuitikz} + \end{center} +} + +\dfn{NOR Gatter in CMOS Technik}{ + Das NOR Gatter kann durch CMOS Gatter wie folgt dargestellt werden. + + \begin{center} + \begin{circuitikz}[european] + \node[circ, label=90:{\small $V_{DD}$}](origin) at (0,0) {}; + \node[thick, nmos, anchor=D] (nmos1) at(-1.75, -3.5) {} + (nmos1.gate) node[anchor=east] {} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, nmos, anchor=D] (nmos2) at(1.75, -3.5) {} + (nmos2.gate) node[anchor=east] {} + (nmos2.drain) node[anchor=west, yshift=-0.15cm] {} + (nmos2.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, pmos, anchor=D] (pmos1) at(0, -1.5) {} + (pmos1.gate) node[anchor=east] {} + (pmos1.drain) node[anchor=west, yshift=-0.15cm] {} + (pmos1.source) node[anchor=west,yshift=+0.15cm] {}; + \node[thick, pmos, anchor=D] (pmos2) at(0, -3) {} + (pmos2.gate) node[anchor=east] {} + (pmos2.drain) node[anchor=west, yshift=-0.15cm] {} + (pmos2.source) node[anchor=west,yshift=+0.15cm] {}; + \draw[thick] (nmos1.S) -- (-1.75, -5.25) coordinate(gnd1); + \draw[thick] (nmos2.S) -- (1.75, -5.25) coordinate(gnd2); + \draw[thick] (nmos1.D) -- (nmos2.D); + \draw[thick] (pmos2.D) -- (0, -3.5); + \draw[thick] (nmos1.G) -- (-2.75, -0.75); + \draw[thick] (nmos2.G) -- (-1.5, -4.25); + \draw[thick] (-1.5, -4.25) -- (-1.5, -2.25); + \path[draw] (pmos2.D) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw] (pmos1.G) --++(left:30mm) node[point, label=180:A] {}; + \path[draw] (pmos2.G) --++(left:30mm) node[point, label=180:B] {}; + \path[draw, very thick] (-2, -5.25) -- (gnd1) -- (-1.5, -5.25); + \path[draw, very thick] (2, -5.25) -- (gnd2) -- (1.5, -5.25); + \end{circuitikz} + \end{center} +} + +\subsection{Komplexe Schaltungen} + +Jetzt wo wir wissen, wie logische Gatter durch CMOS Gatter dargestellt werden können, können wir komplexe logische Schaltungen durch CMOS Transistoren darstellen. + +\dfn{Pull-up / Pull-down Prinzip}{ + \begin{minipage}{0.5\linewidth} + Für die Darstellung von logischen Schaltungen durch CMOS Gattern gilt: + + \begin{itemize} + \item CMOS Gatter müssen aus genau so vielen NMOS und PMOS Transistoren bestehen. + \item Bei $m$ Eingängen gibt es $m$ NMOS und $m$ PMOS Transistoren + \end{itemize} + \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{minipage}{0.3\linewidth} + \begin{center} + \begin{tikzpicture} + \node[circ, label=90:{\small $V_{DD}$}] at (0,0) (origin) {}; + \node[draw, dotted](pmos) at(0,-0.5) {PMOS}; + \node[draw, dotted](nmos) at (0, -1.5) {NMOS}; + \coordinate(gnd) at (0, -2){}; + \draw[] (origin) -- (pmos) + (pmos) -- (nmos) + (nmos) -- (gnd); + \path[draw] (0,-1) --++(right:5mm) node[fill = white] {Y}; + \path[draw, thick] (-0.25, -2) -- (gnd) -- (0.25, -2); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.55\linewidth} + \begin{flushleft} + \begin{tabular}{l l} + Pull-u\textcolor{red}{p} Schaltung: & \textcolor{red}{P}MOS \\ + Pull-dow\textcolor{red}{n} Schaltung: & \textcolor{red}{N}MOS \\ + \end{tabular} + \end{flushleft} + \end{minipage} + \end{minipage} +} + +In den vorherigen Definitionen haben wir uns Pull-up und Pull-down Schaltungen von einzelnen logischen Gattern angeschaut. Wie wir aber in Kapitel \ref{sec:gat} gesehen haben, können logische Gatter miteinander verknüpft werden. Dies ist auch der Fall für CMOS Gatter. Um die verschiedenen logischen Gattern zu erkennen, kann man beim Pull-up bzw. beim Pull-down Pfad die Transistoren in Blöcke unterteilen. Dabei gelten die folgenden Regeln. + +\begin{minipage}{0.5\linewidth} + \subsection*{Pull-up} + + \begin{itemize} + \item 2 parallelgeschaltene PMOS Transistoren $\rightarrow$ ODER + \item 2 seriegeschaltene PMOS Transistoren $\rightarrow$ UND + \end{itemize} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \subsection*{Pull-down} + + \begin{itemize} + \item 2 parallelgeschaltene NMOS Transistoren $\rightarrow$ UND + \item 2 seriegeschaltene NMOS Transistoren $\rightarrow$ ODER + \end{itemize} +\end{minipage} + +Diese Regeln gelten nicht nur für einzelne Transistoren, sondern auch für Blöcke von Transistoren. + + +\subsection{Zeitverzögerung} + +CMOS Gatter können nicht unendlich schnell schalten. Ladungen (Elektronen und Löcher) müssen bewegt werden, was Laufzeit Verzögerungen verursacht. \cite{Luisier2024} In unserem Fall spielen der Propagation Delay von Low to High bzw. High to Low eine grosse Rolle. Aus der Tabelle kann man erkennen, dass es mehrere Faktoren hat, welche die Verzögerungszeit beeinflussen können. Wir fokussieren uns aber hauptsächlich auf die Propagation Delay. + +\begin{tabular}{| c | c | c |} + $t_{pHL}$ & Verzögerungszeit (Propagation Delay High Low) & Beim Übergang H $\rightarrow$ L Gemessen bei 50\% des Pegelhubs \\ + $t_{pLH}$ & Verzögerungszeit (Propagation delay Low High) & Beim Übergang L $\rightarrow$ H Gemessen bei 50\% des Pegelhubs \\ + $r_r$ & Anstieg- (Rise-) Zeit Transition Low High & Gemessen zwischen 10\% und 90\% des Pegelhubs \\ + $t_{tLH}$ & Transition Low High & \\ + $t_f$ & Abfall- (Fall-) Zeit & Gemessen zwischen 90\% und 10\% des Pegelhubs \\ + $t_{tHL}$ & Transition High Low & \\ +\end{tabular} + +In diesem Semester arbeiten wir hauptsächlich mit $t_d$ (Delay Time). Diese kann wie folgt bestimmt werden. + +\[ + t_d = \frac{t_{pLH} + t_{pHL}}{2} + .\] diff --git a/hs24/digitaltechnik/disclaimer.tex b/hs24/digitaltechnik/disclaimer.tex new file mode 100644 index 0000000..c4ca16f --- /dev/null +++ b/hs24/digitaltechnik/disclaimer.tex @@ -0,0 +1,12 @@ +\section*{DISCLAIMER} + +Diese Notizen wurden verfasst auf Basis der Vorlesung Digitaltechnik (HS24) von M. Luisier, sowie dem Übungsskript von M. Vasylyev. +\\ +\\ +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). +\\ +\\ +Falls nicht anders deklariert wurden alle Grafiken eigenhändig mit Manim \cite{MCD2024} oder CircuiTikz generiert. +\\ +\\ +Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden. diff --git a/hs24/digitaltechnik/einfuehrung/einfuehrung.tex b/hs24/digitaltechnik/einfuehrung/einfuehrung.tex new file mode 100644 index 0000000..ed095df --- /dev/null +++ b/hs24/digitaltechnik/einfuehrung/einfuehrung.tex @@ -0,0 +1,3 @@ +\chapter{Einführung in die digitale Welt} + +\input{grundlagen.tex} diff --git a/hs24/digitaltechnik/einfuehrung/grundlagen.tex b/hs24/digitaltechnik/einfuehrung/grundlagen.tex new file mode 100644 index 0000000..6d0ee26 --- /dev/null +++ b/hs24/digitaltechnik/einfuehrung/grundlagen.tex @@ -0,0 +1,54 @@ +\section{Grundlagen der Elektronik} \label{sec:grund} + +\subsection{Analog vs. Digital} + +Signale haben in der Elektrotechnik eine grosse Bedeutung. Wir unterscheiden zwischen zwei verschiedenen Signalen. + +\dfn{Das analoge Signal}{ + Analoge Signale haben kontinuierliche Werte und liefern deshalb unendlich genaue Informationen. \cite{Luisier2024} + + \includegraphics[width=\linewidth]{fig/Fig_1.png} \cite{Luisier2024} +} + +\dfn{Das digitale Signal}{ + Digitale Signale bestehen aus diskreten Werten und hängen von der Betrachtungsart ab. \cite{Luisier2024} + + \includegraphics[width=\textwidth]{fig/Fig_2.png} \cite{Luisier2024} +} + +In den meisten Fällen interessieren uns die Werte der digitalen Signale da diese ausreichend Daten liefern. Nun stellt sich die Frage welche Auflösung man haben möchte und wie sie dargestellt werden sollen. +\\ +\\ +Die Auflösung hängt nach Anwendungsbereich ab. Je nachdem braucht man eine höhere oder eine tiefere Auflösung. Für die Darstellung werden hauptsächlich Bits verwendet. + +\dfn{Bit}{ + Ein Bit (binäre Stelle, englisch: Binary digit) kann verwendet werden, um Zustände darzustellen. \cite{Luisier2024} +} + +Bits können zwei Zustände haben: 1 und 0. Mehrere Bits hintereinander nennt man Codierung. Das erste Bit wird "Most Significant Bit" (MSB, höchstes Gewicht) genannt, das letzte "Least Significant Bit" (LSB, niedriges Gewicht). Wie tut man nun mit Bits Daten darstellen? +\\ +\\ +Bits können verschiedene Zustände annehmen. Diese Zustände können zu bestimmten Daten korrespondieren. Je mehr Bits man hat, desto mehr Daten kann man darstellen. Genauer gesagt kann man die Anzahl Zustände $z$ mit den Anzahl Bits $n$ wie folgt berechnen. + +\[ + z = 2 ^{n} + .\] + +\subsection{Strom und Spannung} + +In NUS I haben wir das Konzept von Strom und Spannung angeschaut. In der Digitaltechnik werden wir nun diese verwenden, um die Zustände von Bits darzustellen. +\\ +\\ +Die Digitaltechnik verwendet zwei verschiedene Methoden, um Bits darzustellen. + +\dfn{Darstellung der Bits durch Spannung}{ + Der Zustand vom Bit kann durch die Menge der Spannung bestimmt werden. Ist die Spannung auf High (0.8 - 1 $V$), so hat der Bit den Zustand 1. Hat die Spannung einen Wert von 0 - 0.15 $V$ (Low), so hat der Bit den Zustand 0. + + \includegraphics[width=0.3\textwidth]{fig/Fig_3.png} \cite{Luisier2024} +} + +\dfn{Darstellung der Bits durch Schalter}{ + Bits können auch durch Schalter, genauer gesagt durch Relais dargestellt werden. Liegt beim Eingang vom Relais keine Spannung vor, so bleibt der Relais offen und der Zustand vom Bit ist 0. Im umgekehrten Fall schliesst sich der Relais und der Zustand vom Bit ist 1. + + \includegraphics[width=0.8\textwidth]{fig/Fig_4.png} \cite{Luisier2024} +} diff --git a/hs24/digitaltechnik/fig/Fig_1.png b/hs24/digitaltechnik/fig/Fig_1.png new file mode 100644 index 0000000..bf89a0f Binary files /dev/null and b/hs24/digitaltechnik/fig/Fig_1.png differ diff --git a/hs24/digitaltechnik/fig/Fig_10.png b/hs24/digitaltechnik/fig/Fig_10.png new file mode 100644 index 0000000..4d04146 Binary files /dev/null and b/hs24/digitaltechnik/fig/Fig_10.png differ diff --git a/hs24/digitaltechnik/fig/Fig_11.png b/hs24/digitaltechnik/fig/Fig_11.png new file mode 100644 index 0000000..ff6711e Binary files /dev/null and b/hs24/digitaltechnik/fig/Fig_11.png differ diff --git a/hs24/digitaltechnik/fig/Fig_12.png b/hs24/digitaltechnik/fig/Fig_12.png new file mode 100644 index 0000000..1aa67da Binary files /dev/null and b/hs24/digitaltechnik/fig/Fig_12.png differ diff --git a/hs24/digitaltechnik/fig/Fig_13.png 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a/hs24/digitaltechnik/latches_flipflops/einfuehrung.tex b/hs24/digitaltechnik/latches_flipflops/einfuehrung.tex new file mode 100644 index 0000000..4009347 --- /dev/null +++ b/hs24/digitaltechnik/latches_flipflops/einfuehrung.tex @@ -0,0 +1,49 @@ +\section{Einführung in die Flipflops} + +\subsection{SR- und D-Flipflops} + +In Vergleich zu taktgesteuerten Latches ändern taktflankengesteuerte Latches (Flipflops) den Zustand ihres Ausgangs, nur wenn der Eingang beschaltet wird und wenn das Taktsignal seinen Zustand ändert (entweder $T=0 \rightarrow 1$ oder $T = 1 \rightarrow 0$). Ändert sich das Taktsignal nicht, so ändert sich der Zustand des Ausgangs nicht. + +\dfn{D-Flipflops}{ + D-Flipflops haben die gleiche Funktion wie D-Latches. Der einzige Unterschied ist, dass D-Flipflops taktflankengesteuert sind. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop D, dot on notQ](tff1){}; + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_17.png} \cite{Luisier2024} + \end{minipage} +} + +\dfn{SR-Flipflops}{ + Wie beim D-Flipflop haben SR-Flipflops die gleiche Funktion wie SR-Latches mit dem Unterschied, dass SR-Flipflops taktflankengesteuert sind. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop mySR/.style={flipflop, + flipflop def={t1=S, t3=R, t6=Q, t4=$\bar{\text{Q}}$, c2=1, n4=1}}}; + \draw (0,0) node[flipflop mySR](tff1){}; + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_18.png} \cite{Luisier2024} + \end{minipage} +} + +\subsection{Dynamik von Flipflops} + +Wie bei logischen Gattern sind Flipflops limitiert durch Verzögerungszeiten. + +\begin{itemize} + \item $t_{pd}$: Verzögerungszeit $\rightarrow$ misst die Verzögerungszeit zwischen einer aktiven Taktflanke am Clock und ihrer Reaktion am Ausgang des Flipflops. + \item $t_s$: Setup-Zeit $\rightarrow$ bestimmt wie lange ein Daten-Signal (hier D) vor der aktiven Taktflanke unverändert anliegen muss, um sicher in das Flipflop übernommen zu werden. + \item $t_h$: Haltezeit $\rightarrow$ gibt an, wie lange ein Daten-Signal nach der aktiven Taktflanke unverändert anliegen muss, um sicher in das Flipflop übernommen zu werden. +\end{itemize} \cite{Luisier2024} + +Die Verzögerungszeiten von Flipflops beeinflusst die Frequenz vom Taktsignal. Ist die Frequenz zu hoch, so kann der Flipflop nicht rechtzeitig schalten. Die maximale Taktfrequenz eines Flipflops wird meistens durch den Hersteller gegeben. diff --git a/hs24/digitaltechnik/latches_flipflops/grundlagen.tex b/hs24/digitaltechnik/latches_flipflops/grundlagen.tex new file mode 100644 index 0000000..c0c8e8a --- /dev/null +++ b/hs24/digitaltechnik/latches_flipflops/grundlagen.tex @@ -0,0 +1,102 @@ +\section{Grundlagen der Latches} + +\subsection{Rückkopplungsprinzip} + +In den letzten paar Kapitel haben wir hauptsächlich kombinatorische Schaltungen untersucht. Kombinatorische Schaltungen nehmen mehrere Eingänge und kombinieren sie durch logische Bauelemente zu einem Ausgang. Im Gegensatz zu kombinatorische Schaltungen können sequentielle Schaltungen ihren Ausgang als Eingang wiederverwenden. Dieser wird auch Rückkopplung genannt. + +\includegraphics[width=\linewidth]{fig/Fig_10.png} \cite{Luisier2024} + +Dieses Prinzip der Rückkopplung ist eine essenzielle Basis für Bauelemente, deren Aufgabe es ist Informationen zu speichern. Diese fungieren meistens als Kippschaltungen (Latches). Wir unterscheiden zwischen zwei Verschiedene Latches. + +\begin{enumerate} + \item SR-Latch (Set/Reset Latch) + \item D-Latch (Data Latch) +\end{enumerate} + +\dfn{SR-Latch}{ + Der SR-Latch ist ein Kippschalter, welcher 2 Eingänge hat und 2 Ausgänge, wobei einer der Ausgänge die Inversion des anderen ist. Eines der Eingänge setzt den Zustand des Ausgangs auf 1 während der andere Eingang den Ausgang auf 0 setzt (zurücksetzt). Werden beide Eingänge nicht beschaltet, so bleiben die Eingänge in ihren vorherigen Zustand. + + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[nor gate] (nor1) at (0,0) {}; + \node[nor gate] (nor2) at (0,-2) {}; + \path[draw] (nor1.input 1) --++(left:10mm) node[point, label=180:S] {}; + \path[draw] (nor2.input 2) --++(left:10mm) node[point, label=180:R] {}; + \path[draw] (nor1.input 2) --++(left:5mm) --++(down:5mm); + \path[draw] (nor2.input 1) --++(left:5mm) --++(up:5mm); + \path[draw] (nor1.output) --++(right:10mm) --++(1,-2) --++(right:10mm) node[point, label=0:$\bar{\text{Q}}$] {}; + \path[draw] (nor2.output) --++(right:10mm) --++(1,2) --++(right:10mm) node[point, label=0:$\text{Q}$] {}; + \path[draw] (nor1.output) --++(right:5mm) --++(down:5mm) --++ (-1.75,-0.85); + \path[draw] (nor2.output) --++(right:5mm) --++(up:5mm) --++ (-1.75,0.85); + \end{tikzpicture} + \end{center} + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop srlatch/.style={flipflop, + flipflop def={t1=S, t3=R, t6=Q, t4=$\bar{\text{Q}}$, n4=1} + }} + \node[flipflop srlatch]{}; + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{tabular}{| c | c | c | c |} + S & R & $\text{Q}_{n+1}$ & $\bar{\text{Q}}_{n+1}$ \\ + \hline + 0 & 0 & $\text{Q}_n$ & $\bar{\text{Q}}_n$ \\ + 0 & 1 & 0 & 1 \\ + 1 & 0 & 1 & 0 \\ + 1 & 1 & X & X \\ + \end{tabular} + \end{minipage} + + \nt{ + Bei einem SR-Latch dürfen die Eingänge nicht gleichzeitig beschaltet werden, da dies zu einem Fehler führt. + } +} + +SR-Latches können auch taktgesteuert sein. Dies bedeutet, dass der Ausgang nur verändert wird, sobald der Takteingang beschaltet ist. + +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop srlatch/.style={flipflop, + flipflop def={t1=S, t3=R, t6=Q, t4=$\bar{\text{Q}}$, n4=1} + }} + \path[draw] node[flipflop srlatch](srlatch){} + (srlatch.pin 1)--++(left:5mm) + node[and port](and1){} + (and1.out) + (srlatch.pin 3)--++(left:5mm) + node[and port](and2){} + (and2.out); + \path[draw] (and1.in 1) --++(left:10mm) node[point, label=180:S] {}; + \path[draw] (and2.in 2) --++(left:10mm) node[point, label=180:R] {}; + \path[draw] (and2.in 1) --++(up:6mm)--++(left:10mm) node[point, label=180:T] {}; + \path[draw] (and1.in 2) --++(down:6mm); + \end{circuitikz} + \end{center} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop srlatch/.style={flipflop, + flipflop def={t1=S, t2=T, t3=R, t6=Q, t4=$\bar{\text{Q}}$, n4=1} + }} + \node[flipflop srlatch]{}; + \end{circuitikz} + \end{center} +\end{minipage} + + +\subsection{D-Latch (Data-Latch)} + +\dfn{D-Latch}{ + Der D-Latch ist ein Kippschalter, welcher 1 Eingang hat und 2 Ausgänge. Der Eingang ist in Vergleich zum SR-Latch allein zuständig für das Setzen und Rücksetzen des Ausgangs bzw. des invertierten Ausgangs. Wie beim SR-Latch können D-Latches auch Taktgesteuert sein. + + +} + +Taktgesteuerte Latches haben jedoch einen Nachteil. Sie sind sehr empfindlich zu Störungen. Deswegen eignet es sich eher, Flip-Flops zu verwenden, welche Taktflankengesteuert sind. diff --git a/hs24/digitaltechnik/latches_flipflops/latches_flipflops.tex b/hs24/digitaltechnik/latches_flipflops/latches_flipflops.tex new file mode 100644 index 0000000..235b60c --- /dev/null +++ b/hs24/digitaltechnik/latches_flipflops/latches_flipflops.tex @@ -0,0 +1,6 @@ +\chapter{Latches und Flipflops} + +\input{grundlagen.tex} +\input{einfuehrung.tex} +\input{mehr_grundlagen.tex} +\input{schaltungen.tex} diff --git a/hs24/digitaltechnik/latches_flipflops/mehr_grundlagen.tex b/hs24/digitaltechnik/latches_flipflops/mehr_grundlagen.tex new file mode 100644 index 0000000..f1f38bc --- /dev/null +++ b/hs24/digitaltechnik/latches_flipflops/mehr_grundlagen.tex @@ -0,0 +1,179 @@ +\section{Mehr über Flipflop Grundlagen} + +\subsection{JK und T-Flipflops} + +\dfn{JK-Flipflops (Jump-Kill Flipflops)}{ + \begin{minipage}{0.5\linewidth} + Der JK-Flipflop ist eine Erweiterung des SR-Flipflops. Beim SR-Flipflop war das Problem, dass beide Eingänge nicht gleichzeitig beschaltet werden dürfen, da dies zu einem Fehler führt. Dieses Problem existiert nicht mehr, da JK-Flipflops einfach die Ausgänge invertiert (Toggeln). + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{tabular}{| c | c | c | c |} + J & K & $\text{Q}_{1n+1}$ & $\text{Q}_{2n+1}$ \\ + \hline + 0 & 0 & $\text{Q}_{1n}$ & $\text{Q}_{2n}$ \\ + 0 & 1 & 0 & 1 \\ + 1 & 0 & 1 & 0 \\ + 1 & 1 & $\overline{\text{Q}_{1n}}$ & $\overline{\text{Q}_{2n}}$ \\ + \end{tabular} + + + \end{minipage} + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop JK, dot on notQ](tff1){}; + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_19.png} \cite{Luisier2024} + \end{minipage} + + \nt{ + JK-Latches existieren auch. + } + +} + +\dfn{T-Flipflops (Toggle Flipflops)}{ + Beim T-Flipflop gibt es zwei Varianten. + \\ + Bei der ersten Variante hat der T-Flipflop nur den Takteingang. Die Aufgabe dieses Flipflops ist es, den Zustand vom Ausgang zu invertieren, sobald das Taktsignal seinen Zustand ändert. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop T, dot on notQ, external pins width=0](tff){}; + \path[draw] (tff.bpin 3) --++(left:2.5mm); + \path[draw] (tff.bpin 4) --++(right:2.5mm); + \path[draw] (tff.bpin 6) --++(right:2.5mm); + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_20.png} \cite{Luisier2024} + \end{minipage} + + Bei der zweiten Variante hat der T-Flipflop 2 Eingänge. Bei diesem Flipflop wird der Ausgang invertiert, sobald der Toggle Eingang beschaltet ist und das Taktsignal seinen Zustand ändert. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop T, dot on notQ](tff1){}; + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_21.png} \cite{Luisier2024} + \end{minipage} +} + +\subsection{D-Flipflops in CMOS Technik} + +Um D-Flipflops durch Transistoren realisieren zu können, benötigen wir Transmission Gates. + +\dfn{Transmission Gates}{ + Transmission Gates sind Bauelemente, welche für den Aufbau von Logikgattern benötigt wird. Sie bestehen aus einem PMOS Transistor und einem NMOS Transistor. Dadurch können sie High- als auch Low-Pegel übertragen. \cite{Vasylyev2024} + + \begin{minipage}{0.5\linewidth} + \begin{tabular}{| c | c | c | c |} + T & IN & Wiederstand & OUT \\ + \hline + 0 & 0 & hochohmig & - \\ + 0 & 1 & hochohmig & - \\ + 1 & 0 & niederohmig & 0 \\ + 1 & 1 & niederohmig & 1 \\ + \end{tabular} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_22.png} \cite{Luisier2024} + \end{minipage} +} + +Das Verhalten von Transmission Gates hängt von der Eingangsspannung $V_{\text{in}}$ ab. Ist das Taktsignal beschaltet und die Eingangsspannung $<0$, so ist die Gate-Source Spannung $V_{\text{gs}}$ des PMOS Transistors $-V_{\text{in}}$ und der PMOS Transistor leitet. Dies ist anders beim NMOS Trasistor, welcher gesperrt bleibt. Wenn $V_{\text{in}} >0$ dann ist es die umgekehrte Situation und der NMOS Transistor leitet. + +\dfn{D-FlipFlops in CMOS Technik}{ + Um ein D-Flipflop mit Transmission Gates zu realisieren, benötigt man vier Transmission Gates und vier Inverter bzw. zwei Transmission Gates und zwei Inverter pro Latch. + + \includegraphics[width=\linewidth]{fig/Fig_23.png} \cite{Luisier2024} + + Mit CMOS Transistoren würde die vorherige Schaltung wie folgt aussehen. + + \includegraphics[width=\linewidth]{fig/Fig_24.png} \cite{Luisier2024} + + $\Rightarrow$ Für ein D-Flipflop werden acht PMOS- und acht NMOS-Transistoren benötigt. +} + +\subsection{Weitere Flipflop Eigenschaften} + +Ein JK-Flipflop kann immer durch ein D-Flipflop ersetzt werden. Neben den D-Flipflop selbst werden zwei AND und ein OR zusätzlich benötigt. + +\begin{center} + \begin{circuitikz}[european] + \path[draw] node[flipflop D, dot on notQ](dff){} + (dff.pin 4) --++(right:15mm) node[point, label=0:$\bar{\text{Q}}$]{} + (dff.pin 6) --++(right:15mm) node[point, label=0:Q]{} + (dff.pin 3) --++ (left:10mm) node[point, label=180:CLK]{} + (dff.pin 1) --++(left:5mm) + node[or port, anchor=out](or){} + (or.in 1) --++(left:5mm) + node[and port, anchor=out](and1){} + (and1.in 2) --++(left:10mm) node[point, label=180:J]{} + (or.in 2) --++(left:5mm) --++(down:20mm) --++(left:5mm) + node[and port, anchor=out](and2){} + node [notcirc, left] at (and2.bin 1) {} + (and2.in 1) --++(left:10mm) node[point, label=180:K]{} + (dff.pin 4) --++(right:5mm) --++(up:30mm) --++(left:77mm) --++(down:9mm) + (dff.pin 6) --++(right:10mm) --++(down:30mm) --++(left:86.5mm) --++(up:5.5mm); + \end{circuitikz} +\end{center} + +Flipflops können zusätzlich noch einen zusätzlichen Setz- bzw. Rücksetzkontakt haben. Dieser tut den Ausgang, egal in welchen Zustand das Taktsignal oder die Eingänge sind, setzen bzw. rücksetzen. + +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \path[draw] node[flipflop D, dot on notQ, add async SR](dff){}; + \end{circuitikz} + \end{center} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \path[draw] node[flipflop JK, dot on notQ, add async SR](dff){}; + \end{circuitikz} + \end{center} +\end{minipage} + +\dfn{JK-LF-Flipflops (Jump-Kill-Leader-Follower Flipflops)}{ + JK-LF-Flipflops (Zwischenspeicher Flipflops) sind in der Lage Signale zwischenzuspeichern und erst bei der nächsten fallenden Taktflanke zum Ausgang weiterzuleiten. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \path[draw] node[flipflop JK, dot on notQ](jkff1){} + (jkff1.pin 1) --++(left:10mm) node[point, label=180:J]{} + (jkff1.pin 2) --++(left:10mm) node[point, label=180:CLK]{} + (jkff1.pin 3) --++(left:10mm) node[point, label=180:K]{} + (jkff1.pin 6) --++(right:10mm) + node[flipflop JK, dot on notQ, anchor=pin 1](jkff2){} + node [notcirc, left] at (jkff2.bpin 2) {} + (jkff1.pin 4) --++(right:10mm) + (jkff2.pin 4) --++(right:10mm) node[point, label=0:Q]{} + (jkff2.pin 6) --++(right:10mm) node[point, label=0:$\bar{\text{Q}}$]{} + (jkff1.pin 2) --++(left:5mm) --++(down:15mm) --++(right:33mm) --++(up:15mm) --++(right:5mm); + \end{circuitikz} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz} + \tikzset{flipflop JKLF/.style={flipflop, + flipflop def={t1=J, t2=CLK, t3=K, t4=$\neg$, t6=$\neg$, c2=1, n4=1} + }} + \node [flipflop JKLF]{}; + \end{circuitikz} + \end{center} + \end{minipage} +} diff --git a/hs24/digitaltechnik/latches_flipflops/schaltungen.tex b/hs24/digitaltechnik/latches_flipflops/schaltungen.tex new file mode 100644 index 0000000..709b90b --- /dev/null +++ b/hs24/digitaltechnik/latches_flipflops/schaltungen.tex @@ -0,0 +1,18 @@ +\section{Schaltungen mit Flipflops} \label{sec:ffcirc} + +\subsection{Frequenzteiler und Zähler} + +Mithilfe von T-Flipflops kann die Frequenz vom Taktsignal verringert werden. Jeder T-Flipflop verringert die Frequenz um den Faktor 2. Dies bedeutet mit $n$ Flipflops wird die Frequenz um den Faktor $2 ^{n}$ verringert. + +\begin{center} + \begin{circuitikz}[european] + \draw (0,0) node[flipflop T, dot on notQ](tff1){} + (tff1.pin 3) --++(left:10mm) node[point, label=180:CLK] {} + (tff1.pin 6) --++(right:15mm) --++(down:17mm) --++(right:5mm) + node[flipflop T, dot on notQ, anchor=pin 3](tff2){} + (tff2.pin 6) --++(right:10mm) node[point, label=0:Q] {}; + \path[draw] (tff1.pin 6) --++(right:10mm) --++(up:10mm) node[point, label=90:M] {}; + \end{circuitikz} +\end{center} + +Mit zwei T-Flipflops kann auch ein zwei Bit Zähler erstellt werden. Dazu wird der Ausgang vom ersten Flipflop für den LSB verwendet und der Ausgang vom zweiten Flipflop als MSB verwendet. Die höchste Zahl hängt von der Anzahl Flipflops ab. Grundsätzlich kann man sagen, dass $n$ Flipflops von 0 bis $2 ^{n}-1$ zählen kann. diff --git a/hs24/digitaltechnik/letterfonts.tex b/hs24/digitaltechnik/letterfonts.tex new file mode 100644 index 0000000..3939d5b --- /dev/null +++ b/hs24/digitaltechnik/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/hs24/digitaltechnik/macros.tex b/hs24/digitaltechnik/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/hs24/digitaltechnik/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/hs24/digitaltechnik/mikroprozessoren/mikro.tex b/hs24/digitaltechnik/mikroprozessoren/mikro.tex new file mode 100644 index 0000000..ab0a75a --- /dev/null +++ b/hs24/digitaltechnik/mikroprozessoren/mikro.tex @@ -0,0 +1,38 @@ +\section{Mikroprozessoren} + +\subsection{Aufbau eines einfachen Rechners} + +Ein funktionierender Computer soll: + +\begin{itemize} + \item logische Verknüpfungen von binären Zahlen durchführen können. + \item Grundrechenarten beherschen können. + \item Abläufe steuern können. + \item sein Zustand andhand der Eingaben vom Benutzer ändern können. +\end{itemize} + +Aus diesem Grund benötigt ein Rechner + +\begin{enumerate} + \item Ein Speicherwerk, um das auszuführende Programm sowie Daten (z.B. Zwischenergebnisse) ablegen zu können + \item Ein Rechenwerk (Arithmetic Logic Unit, ALU), um die arithmetischen und logischen Operationen durchzuführen + \item Ein Steuerwerk, um den Programmablauf und die Kommunikation zwischen den verschiedenen Blöcken zu regeln + \item Ein Ein-/Ausgabewerk, um mit der Aussenwelt kommunizieren zu können. +\end{enumerate} + +Die meisten universellen Rechner haben eine Grundstruktur nach der "von Neumann" Architektur. Der universelle Rechner durchläuft ein Zyklus bestehend aus fünf Teilschritten. + +\begin{itemize} + \item FETCH (Befehlsabruf): das Steuerwerk holt aus dem Programmspeicher den nächsten Befehl. + \item DECODE (Dekodierung): ein Befehlswort, das aus einem Instruktionsteil (‚was soll getan werden‘) und aus einer Datenadresse (‚welche Daten sollen manipuliert werden‘) wird durch das Steuerwerk in Schaltinstruktionen aufgelöst. + \item FETCH OPERANDS (Operandenabruf): aus dem Speicher werden die Operanden geholt, die durch den Befehl verändert werden sollen oder als Parameter verwendet werden. + \item EXECUTE (Befehlsausführung): eine arithmetische oder logische Operation wird vom ALU ausgeführt. + \item WRITE BACK (Rückschreiben): das Ergebnis der Berechnung wird in Register oder Speicher zurückgeschrieben. +\end{itemize} \cite{Luisier2024} + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_8.png}\cite{Luisier2024} \cite{Luisier2024} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_9.png}\cite{Luisier2024} \cite{Luisier2024} +\end{minipage} diff --git a/hs24/digitaltechnik/mikroprozessoren/mikroprozessoren.tex b/hs24/digitaltechnik/mikroprozessoren/mikroprozessoren.tex new file mode 100644 index 0000000..6280ca5 --- /dev/null +++ b/hs24/digitaltechnik/mikroprozessoren/mikroprozessoren.tex @@ -0,0 +1,3 @@ +\chapter{Mikroprozessoren} + +\input{mikro.tex} diff --git a/hs24/digitaltechnik/notes.tex b/hs24/digitaltechnik/notes.tex new file mode 100644 index 0000000..594e811 --- /dev/null +++ b/hs24/digitaltechnik/notes.tex @@ -0,0 +1,10 @@ +\import{einfuehrung}{einfuehrung.tex} +\import{verknuepfung}{verknuepfung.tex} +\import{cmos}{cmos.tex} +\import{schaltalgebra}{schaltalgebra.tex} +\import{schaltungssynthese}{schaltungssynthese.tex} +\import{zahlen_codes}{zahlen_codes.tex} +\import{rechenschaltungen_datenpfadkomponenten}{rechenschaltungen_datenpfadkomponenten.tex} +\import{latches_flipflops}{latches_flipflops.tex} +\import{automaten}{automaten.tex} +\import{mikroprozessoren}{mikroprozessoren.tex} diff --git a/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.bbl b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.bbl new file mode 100644 index 0000000..90b40d6 --- /dev/null +++ b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.bbl @@ -0,0 +1,15 @@ +\begin{thebibliography}{} + +\bibitem[Luisier, 2024]{Luisier2024} +Luisier, D.~M. (2024). +\newblock Vorlesungsfolien. + +\bibitem[{The Manim Community Developers}, 2024]{MCD2024} +{The Manim Community Developers} (2024). +\newblock {Manim – Mathematical Animation Framework}. + +\bibitem[Vasylyev, 2024]{Vasylyev2024} +Vasylyev, M. (2024). +\newblock Übungen. + +\end{thebibliography} diff --git a/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.pdf b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.pdf new file mode 100644 index 0000000..20f7519 Binary files /dev/null and b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.pdf differ diff --git a/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.tex b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.tex new file mode 100644 index 0000000..bf0c916 --- /dev/null +++ b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.tex @@ -0,0 +1,43 @@ +\documentclass{report} + +\def\papertitle{Digitaltechnik} + +\def\theorytitle{Theorie} +\def\corollarytitle{Korollar} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Behauptung} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Beispiel} +\def\questiontitle{Frage} +\def\wrongctitle{Falscher Konzept} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\input{disclaimer.tex} + +\input{notes.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% BIBLIOGRAPHY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\addcontentsline{toc}{chapter}{References} +\bibliographystyle{apalike} +\bibliography{sources} + +\end{document} diff --git a/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.zip b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.zip new file mode 100644 index 0000000..8bf431d Binary files /dev/null and b/hs24/digitaltechnik/notizen_digitech_ruh_jirayu.zip differ diff --git a/hs24/digitaltechnik/preamble.tex b/hs24/digitaltechnik/preamble.tex new file mode 100644 index 0000000..65463cf --- /dev/null +++ b/hs24/digitaltechnik/preamble.tex @@ -0,0 +1,809 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[ngerman]{babel} +\selectlanguage{ngerman} +\usepackage{tabularx} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\definecolor{secondaryheader}{HTML}{f1935c} +\definecolor{darkgreen}{HTML}{008E00} +\usepackage{tikzsymbols} +\usepackage{tikz} +\usepackage{pgfplots} +\usetikzlibrary{positioning, calc, fit, matrix, decorations, angles, quotes, babel, 3d, arrows.meta, shadings, circuits.logic.IEC, circuits.ee.IEC, automata} +\usepackage{circuitikz} +\ctikzset{tripoles/pmos style/emptycircle} +\renewcommand\qedsymbol{$\Laughey$} +\tikzset{ + circ/.style = {circle, fill = black, minimum size=1mm, inner sep=0pt, outer sep=0pt}, + point/.style = {circle, draw = black, thick, minimum size=1mm, inner sep=0pt, outer sep=0pt}, + gnd/.style = {draw, fill, minimum width = 10mm, minimum height = 1pt}, + kvbinhead/.style = {draw = white, fill = secondaryheader!30, text = black}, + srLatch/.style = {flipflop, flipflop def = {t1 = S, t3 =R, t6={~}, n4 = 1}}, + srtLatch/.style = {flipflop, flipflop def = {t1 = S, t2 = T, t3 =R, t6={~}, n4 = 1}}, + dLatch/.style = {flipflop, flipflop def = {t1 = D, t2 = T, t6={~}, n4 = 1}}, + dFf/.style = {flipflop, flipflop def = {t1 = D, t2=\clktag, t6={~}, c2=1, n4=1}}, + srFf/.style = {flipflop, flipflop def = {t1 = S, t2=\clktag, t3=R, t6={~}, c2=1, n4=1}}, + jkFf/.style = {flipflop, flipflop def = {t1 = J, t2=\clktag, t3=K, t6={~}, c2=1, n4=1}}, + tFfv1/.style = {flipflop, flipflop def = {t2=\clktag, t6={~}, c2=1, n4=1}, label={[label distance =1.2mm]90:T}}, + tFfv2/.style = {flipflop, flipflop def = {t1=T, t2=\clktag, t6={~}, c2=1, n4=1}}, + tFfmin/.style = {flipflop, flipflop def = {t6={~}, c2=1}, label={[label distance =0mm]90:T}} +} +\pgfplotsset{compat=1.7} % Version? für Plots +\pgfdeclarelayer{bg} +\pgfdeclarelayer{l1} +\pgfdeclarelayer{l2} +\pgfdeclarelayer{l3} +\pgfdeclarelayer{tl1} +\pgfdeclarelayer{tl2} +\pgfdeclarelayer{tl3} +\pgfsetlayers{bg,l3,l2,l1,main,tl1,tl2,tl3} + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + Wrong Concept \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + + + + +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} + + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% + \begingroup + \sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% + \pdfsave + \pdfsetmatrix{1 0 0.2 1}% + \rlap{\usebox{\diffdbox}}% + \pdfrestore + \hskip\wd\diffdbox + \endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% + \slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% + {\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience +\DeclareMathOperator{\E}{E} % expected + +% Since the amsthm package isn't loaded + +% I prefer the slanted \leq +\let\oldleq\leq % save them in case they're every wanted +\let\oldgeq\geq +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +% % redefine matrix env to allow for alignment, use r as default +% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep +% \let\@ifnextchar\new@ifnextchar +% \array{*\c@MaxMatrixCols #1}} + + +%\usepackage{framed} +%\usepackage{titletoc} +%\usepackage{etoolbox} +%\usepackage{lmodern} + + +%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{} + +%\renewenvironment{leftbar} +%{\def\FrameCommand{\hspace{6em}% +% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}% +% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt% +%} +%{\endMakeFramed} + +%\titlecontents{chapter} +%[0em]{\vspace*{2\baselineskip}} +%{\parbox{4.5em}{% +% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}% +% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily} +%{}{\endleftbar} +%\titlecontents{section} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} +%\titlecontents{subsection} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage{tikz} +\definecolor{doc}{RGB}{0,60,110} +\usepackage{titletoc} +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Kapitel\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Page \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/addierer.tex b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/addierer.tex new file mode 100644 index 0000000..9a376d3 --- /dev/null +++ b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/addierer.tex @@ -0,0 +1,74 @@ +\section{Addierer} + +\subsection{Halb- und Volladdierer} + +\dfn{Halbaddierer}{ + Ein Halbaddierer ist ein Bauelement, welches in der Lage ist zwei Bits zu addieren. Es nimmt 2 Bits als Eingang und hat die Summe, sowie der Übertrag als Ausgang. + + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[and gate] (and) at (0,-1.5) {}; + \node[xor gate] (xor) at (0,0) {}; + \path[draw] (xor.input 1) --++(left:20mm) node[point, label=180:A] {}; + \path[draw] (xor.input 2) --++(left:20mm) node[point, label=180:B] {}; + \path[draw] (and.input 1) --++(left:5mm) --++(up:11.7mm); + \path[draw] (and.input 2) --++(left:10mm) --++(up:18.4mm); + \path[draw] (xor.output) --++(right:10mm) node[point, label=0:SUM] {}; + \path[draw] (and.output) --++(right:10mm) node[point, label=0:CO] {}; + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop hsum/.style={flipflop, + flipflop def={t1=A, t3=B, t6=SUM, t4=CO} + }} + \node[flipflop hsum]{$\sum$}; + \end{circuitikz} + \end{center} + \end{minipage} +} + +Für Zahlen im Binärsystem reichen Halbaddierer nicht aus, da sie den Übertrag nicht beachten. Deswegen kommt der Volladdierer ins Spiel. + +\dfn{Volladdierer}{ + Volladdierer sind Bauelemente, dessen Funktion es ist bei der Summe von 2 Bits den Übertrag mit einzubeziehen. + + \begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop fsum/.style={flipflop, + flipflop def={t1=A, t2=B, t3=CIN, t4=CO, t6=SUM} + }} + \node [flipflop fsum]{$\sum$}; + \end{circuitikz} + \end{center} +} + +\subsection{Mehrbit-Addierer und Subtrahierer} + +Wir haben nun die Basis zur Erstellung einer logischen Schaltung zum addieren von Zahlen im Binärsystem. Der Prozess vom Addieren von Dualzahlen kann entweder bitseriell oder bitparallel erfolgen. + +\begin{itemize} + \item Bitseriell: Pro Taktperiode wird nur ein Bit addiert + \item Bitparallel: In einer Taktperiode werden alle Bits addiert. +\end{itemize} + +Wird die Dualzahl seriell bzw. parallel addiert, so wird das Bauelement Serienaddierer bzw. Paralleladdierer. Bei den Paralleladdierer unterscheidet man zwischen den folgenden Methoden. + +\begin{itemize} + \item Paralleladdierer in der Normalform + \item Ripple-Carry Addierer + \item Carry-Look-Ahead Addierer +\end{itemize} + +Jedes dieser Paralleladdierer haben ihre Vor- und Nachteile. +\\ +Beim Paralleladdierer in der Normalform wird die Normalform für jedes Ausgangsbit verwendet. Der Vorteil davon ist, dass durch die Normalform das Bauelement eine sehr geringe Propagation Delay hat. Die Normalform bringt aber auch Nachteile, da diese eine sehr komplizierte kombinatorische Schaltung ist. +\\ +Im Vergleich zum Paralleladdierer in der Normalform ist der Ripple-Carry Addierer um einiges langsamer, da es die Bits nacheinander durch ein Halb- und mehrere Volladdierer addiert. Der Vorteil ist, dass die kombinatorische Schaltung einfacher zu realisieren ist. + +\includegraphics[width=\textwidth]{fig/Fig_25.png} \cite{Luisier2024} + +Der Carry-Look-Ahead Addierer ist ähnlich wie eine Ripple-Carry Addierer mit dem Unterschied, dass der parallel, während die Bits addiert werden der Carry-Out addiert wird und somit der Carry-Look-Ahead Addierer ein wenig schneller ist als der Ripple-Carry Addierer. Der Nachteil davon ist, dass die kombinatorische Schaltung immer komplizierter wird, wenn die Dualzahl grösser wird. diff --git a/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/datenpfadkomponenten.tex b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/datenpfadkomponenten.tex new file mode 100644 index 0000000..ba6eb6f --- /dev/null +++ b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/datenpfadkomponenten.tex @@ -0,0 +1,20 @@ +\section{Was sind Datenpfadkomponenten?} + +\subsection{Multiplexer und Code-Umsetzer} + +\dfn{Multiplexer}{ + Multiplexer sind Bauelemente, welche durch eine Codierung als Eingang Daten als Ausgang schaltet. Multiplexer können auch erweitert werden indem man mehrere Kanäle einführt für den Dateneingang. Diese werden Daten Selektion genannt. + + \includegraphics[width=\linewidth]{fig/Fig_38.png} \cite{Luisier2024} +} + +\dfn{Demultiplexer}{ + Demultiplexer haben die umgekehrte Funktion von Multiplexer. Demultiplexer sind Bauelemente, welche Daten als Eingang haben und eine Codierung als Ausgang haben. Wie beim Multiplexer kann auch eine Daten Selektion gebraucht werden für verschiedene Kanäle. + + \includegraphics[width=\linewidth]{fig/Fig_39.png} \cite{Luisier2024} +} + +\dfn{Code-Umsetzer}{ + Code-Umsetzer können eine Codierung als Eingang nehmen und diese dann umwandeln in eine andere Codierung umwandeln (Kapitel \ref{sec:code}). +} + diff --git a/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/hardware_multiplizierer.tex b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/hardware_multiplizierer.tex new file mode 100644 index 0000000..a533701 --- /dev/null +++ b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/hardware_multiplizierer.tex @@ -0,0 +1,35 @@ +\section{Hardware Multiplizierer} + +\subsection{Rechenregeln und Grundprinzipien} + +Die Multiplikation von Bits entspricht das logische AND. Doch wie geht man vor? Grundsätzlich geht man vor wie wenn man Zahlen im Dezimalsystem schriftlich multipliziert. + +\begin{enumerate} + \item Den Multiplikand mit dem LSB vom Multiplikator multiplizieren und aufschreiben. + \item Für die Codierung darunter eine Null als LSB einfügen. Für jede weitere Zeile eine Null mehr. + \item Den ersten Schritt mit den nächsten Bit wiederholen. +\end{enumerate} + +Ein Multiplikator Bauelement besteht aus mehreren Basiszellen. Diese bestehen aus einem AND und einem Volladdierer. + +\begin{center} + \begin{circuitikz}[european] + \tikzset{flipflop fsum/.style={flipflop, + flipflop def={t1=A, t2=B, t3=CI, t6=SUM, t4=CO} + }} + \path[draw] node[flipflop fsum](fsum){$\sum$} + (fsum.pin 1)--++(left:5mm) + node[and port](and){} + (and.out); + \path[draw] (and.in 1) --++(left:10mm) node[point, label=180:a] {}; + \path[draw] (and.in 2) --++(left:10mm) node[point, label=180:b] {}; + \path[draw] (fsum.pin 2) --++(left:10mm) node[point, label=180:$\text{E}_1$] {}; + \path[draw] (fsum.pin 3) --++(left:10mm) node[point, label=180:$\text{E}_2$] {}; + \path[draw] (fsum.pin 4) --++(right:10mm) node[point, label=0:$\text{A}_2$] {}; + \path[draw] (fsum.pin 6) --++(right:10mm) node[point, label=0:$\text{A}_1$] {}; + \end{circuitikz} +\end{center} + +Ein 4-Bit Multiplikator sieht zusammen mit allen Basiszellen wie folgt aus. + +\includegraphics[width=\linewidth]{fig/Fig_11.png} \cite{Luisier2024} diff --git a/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/rechenschaltungen_datenpfadkomponenten.tex b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/rechenschaltungen_datenpfadkomponenten.tex new file mode 100644 index 0000000..34faa00 --- /dev/null +++ b/hs24/digitaltechnik/rechenschaltungen_datenpfadkomponenten/rechenschaltungen_datenpfadkomponenten.tex @@ -0,0 +1,5 @@ +\chapter{Rechenschaltungen und Datenpfadkomponenten} + +\input{datenpfadkomponenten.tex} +\input{addierer.tex} +\input{hardware_multiplizierer.tex} diff --git a/hs24/digitaltechnik/schaltalgebra/demorgan.tex b/hs24/digitaltechnik/schaltalgebra/demorgan.tex new file mode 100644 index 0000000..760ed32 --- /dev/null +++ b/hs24/digitaltechnik/schaltalgebra/demorgan.tex @@ -0,0 +1,19 @@ +\section{Die De Morgan'schen Regeln} \label{sec:morgan} + +\subsection{Beziehung zwischen UND und OR} + +Es existieren Beziehungen zwischen den NOR/UND und zwischen den NAND/ODER Verknüpfungen, sodass ein gegenseitiger Austausch möglich ist: De Morgan‘sche Regeln. \cite{Luisier2024} + +\subsection*{De Morgan'sche Regeln} + +\begin{enumerate} + \item Die NAND-Funktion kann durch eine ODER-Funktion mit invertierten Eingängen ersetzt werden. ($\overline{A \land B} = \bar{A} \lor \bar{B}$) + \item Die NOR-Funktion kann durch eine UND-Funktion mit invertierten Eingängen ersetzt werden. ($\overline{A \lor B} = \bar{A} \land \bar{B}$) +\end{enumerate} \cite{Luisier2024} + +Mit den Grundgattern UND, ODER, NICHT kann jede logische Verknüpfung realisiert werden. Die De Morgan‘schen Regeln ermöglichen den Ersatz von Grundgattern. Jedes “Universalgatter„ NAND oder NOR allein ist ausreichend, um logische Verknüpfungen zu realisieren. \cite{Luisier2024} + +\nt{ + Die Beziehung zwischen der Pull-up und Pull-down Schaltung ist durch die De Morgan‘schen Regeln gegeben. \cite{Luisier2024} +} + diff --git a/hs24/digitaltechnik/schaltalgebra/normalformen.tex b/hs24/digitaltechnik/schaltalgebra/normalformen.tex new file mode 100644 index 0000000..71d1b6f --- /dev/null +++ b/hs24/digitaltechnik/schaltalgebra/normalformen.tex @@ -0,0 +1,35 @@ +\section{Normalformen} + +\subsection{Definition von Min- und Maxterm} + +\dfn{Minterm}{ + Ein Minterm ist eine AND-Verknüpfte logische Schaltung. Bei $n$ Eingangsvariablen gibt es $2 ^{n}$ mögliche Minterme. +} + +Der Minterm kann bestimmt werden, indem man die AND-Verknüpfung aller vorhandenen Variablen bildet und die Variablen, die 0 sind, invertiert. + +\nt{ + Beim Minterm haben die nicht-invertierten Variablen den Wert 1, die invertierten Variablen den Wert 0. +} + +\dfn{Maxterm}{ + Ein Maxterm ist eine OR-Verknüpfte logische Schaltung. Bei $n$ Eingangsvariablen gibt es $2 ^{n}$ mögliche Maxterme. +} + +Wie beim Minterm kann der Maxterm bestimmt werdem, indem man die OR-Verknüpfung aller vorhandenen Variablen bildet und die Variablen, die 1 sind, invertiert. + +\nt{ + Beim Maxterm haben die nicht-invertierten Variablen haben den Wert 0, die invertierten Variablen den Wert 1. +} + +\subsection{DNF und KNF} + +Ist die Wahrheitstabelle einer Schaltfunktion $f$ gegeben, so können zwei verschiedene Normalformen gebildet werden. Diese werden im späteren Teil des Semesters wichtig sein. + +\dfn{Disjunktive Normalform (DNF)}{ + Die Disjunktive Normalform ist die OR-Verknüpfung der Minterme. +} + +\dfn{Konjunktive Normalform (KNF)}{ + Die Konjunktive Normalform ist die AND-Verknüpfung der Maxterme. +} diff --git a/hs24/digitaltechnik/schaltalgebra/schaltalgebra.tex b/hs24/digitaltechnik/schaltalgebra/schaltalgebra.tex new file mode 100644 index 0000000..778b7c7 --- /dev/null +++ b/hs24/digitaltechnik/schaltalgebra/schaltalgebra.tex @@ -0,0 +1,5 @@ +\chapter{Schaltalgebra (Bool'sche Algebra)} + +\input{verknuepfungsgesetze.tex} +\input{demorgan.tex} +\input{normalformen.tex} diff --git a/hs24/digitaltechnik/schaltalgebra/verknuepfungsgesetze.tex b/hs24/digitaltechnik/schaltalgebra/verknuepfungsgesetze.tex new file mode 100644 index 0000000..1c50dd2 --- /dev/null +++ b/hs24/digitaltechnik/schaltalgebra/verknuepfungsgesetze.tex @@ -0,0 +1,32 @@ +\section{Verknüpfungsgesetze} + +\subsection{Basis- und Vereinfachungsregeln} \label{sec:verein} + +Logische Funktionen können sehr komplex werden. Glücklicherweise können sie vereinfacht werden durch Rechenoperationen, welche uns schon bekannt sind. + +\begin{itemize} + \item Die Reihenfolge der Variablen in UND-Verknüpfungen und in ODER-Verknüpfungen beeinflusst das Ergebnis nicht. + \item Variablen können zu Gruppen zusammengefasst werden. + \item Gemeinsame Variablen in UND- und ODER-Verknüpfungen können verteilt (ausmultipliziert, ausgeklammert) werden. + \item $\bar{\bar{A}} = A$ (Nicht) + \item $A \lor 0 = A$ / $A \land 0 = 0$ (Null-Theorem) + \item $A \lor 1 = 1$ / $A \land 1 = A$ (Eins-Theorem) + \item $A \lor A = A$ / $A \land A = A$ (Idempotenz) + \item $A \lor \bar{A} = 1$ / $A \land \bar{A} = 0$ (Verknüpfung mit Komplement) + \item $A \lor (\bar{A} \land B) = A \lor B$ / $A \land (\bar{A} \lor B) = A \land B$ (Adsorptionsgesetz) + \item $A \lor (A \land B) = A$ / $A \land (A \lor B) = A$ (Absorptionsgesetz) + \item $(A \land B) \lor (\bar{A} \land B) = B$ / $(A \lor B) \land (\bar{A} \lor B) = B$ (Nachbarschaftsgesetz) +\end{itemize} + +Für die Reihenfolge der Rechnungen gilt: + +\begin{enumerate} + \item Klammern + \item Negation + \item AND, NAND, OR, NOR vor XOR, XNOR +\end{enumerate} + +\nt{ + AND, NAND, OR, NOR, XOR und XNOR sind untereinander gleichwertig. +} + diff --git a/hs24/digitaltechnik/schaltungssynthese/dont_care.tex b/hs24/digitaltechnik/schaltungssynthese/dont_care.tex new file mode 100644 index 0000000..823b7c7 --- /dev/null +++ b/hs24/digitaltechnik/schaltungssynthese/dont_care.tex @@ -0,0 +1,11 @@ +\section{Don‘t Care Zustände (X)} + +\subsection{Nicht benutzte Wertekombinationen} + +Ab und zu bekommen wir mehr Codierungen als wir nötig haben. Aus diesem Grund verwenden wir Don't Care Zustände. + +\dfn{Don't Care Zustände}{ + Don't Care Zustände kennzeichnen Codierungen, welche nicht relevant sind für die logische Funktion. Die Codierung wird mit einem $X$ gekennzeichnet. Don't Care Zustände können den Wert 0 oder 1 annehmen. +} + +Don't Care Zustände haben vorallem eine grosse Bedeutung für die Bildung von Päckchen. Da Don't Care Zustände den Wert 1 und 0 annehmen können, können grössere Päckchen gebildet werden. diff --git a/hs24/digitaltechnik/schaltungssynthese/hazard.tex b/hs24/digitaltechnik/schaltungssynthese/hazard.tex new file mode 100644 index 0000000..d88eeee --- /dev/null +++ b/hs24/digitaltechnik/schaltungssynthese/hazard.tex @@ -0,0 +1,16 @@ +\section{Was sind Hazards?} + +\subsection{Identifizierung und Behandlung} + +\dfn{Hazard}{ + Ein Hazard ist eine kurzzeitige und unerwünschte Änderung von Signalwerten. \cite{Luisier2024} Diese tritt meistens aufgrund der Zeitverzögerung von Gattern auf. +} + +Hazards können aber frühzeitig erkannt und eliminiert werden mit Hilfe des Karnaugh Diagramms. Für Hazards gilt wenn sich Päckchen orthogonal berühren, markieren diese kritische Werteänderungen (statische Hazards). \cite{Luisier2024} Durch ein zusätzliches Päckchen, welches die zwei orthogonal berührenden Min- bzw. Maxtermen in den Päckchen umschliesst kann der Hazard eliminiert werden. + +\includegraphics[width=0.3\linewidth]{fig/Fig_37.png} \cite{Luisier2024} + +\nt{ + Wenn sich Päckchen diagonal berühren entsteht ein funktionaler Hazard. Dieser kann nicht leicht behoben werden. +} + diff --git a/hs24/digitaltechnik/schaltungssynthese/kv_diagramm.tex b/hs24/digitaltechnik/schaltungssynthese/kv_diagramm.tex new file mode 100644 index 0000000..c3ef44f --- /dev/null +++ b/hs24/digitaltechnik/schaltungssynthese/kv_diagramm.tex @@ -0,0 +1,151 @@ +\section{Karnaugh-Diagramme} + +\subsection{Einführung und Regeln} + +Karnaugh Diagramme werden zur Vereinfachung von logischen Gleichungen verwendet. Wir haben zwar in Kapitel \ref{sec:verein} und \ref{sec:morgan} eine Methode gelernt, logische Gleichungen zu vereinfachen. Die Karnaugh Diagramme sind eine andere Methode, welche in manchen Fällen schneller ist. + +\begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center + } + ]{ + $0$ & $1$ & $X$ & \\ + & & & \\ + & $\textcolor{blue}{1}$ & & \\ + & & & \\ + }; + + \path[draw, decorate, decoration=brace] (kv-1-1.north west) -- (kv-1-2.north east) node [midway, above] {$\bar{A}$}; + \path[draw, decorate, decoration=brace] (kv-1-3.north west) -- (kv-1-4.north east) node [midway, above] {$A$}; + \path[draw, decorate, decoration=brace] (kv-2-1.south west) -- (kv-1-1.north west) node [midway, left] {$\bar{C}$}; + \path[draw, decorate, decoration=brace] (kv-4-1.south west) -- (kv-3-1.north west) node [midway, left] {$C$}; + \path[draw, decorate, decoration=brace] (kv-2-4.north east) -- (kv-3-4.south east) node [midway, right] {$D$}; + \path[draw, decorate, decoration=brace] (kv-1-4.north east) -- (kv-1-4.south east) node [midway, right] {$\bar{D}$}; + \path[draw, decorate, decoration=brace] (kv-4-4.north east) -- (kv-4-4.south east) node [midway, right] {$\bar{D}$}; + \path[draw, decorate, decoration=brace] (kv-4-3.south east) -- (kv-4-2.south west) node [midway, below] {$B$}; + \path[draw, decorate, decoration=brace] (kv-4-1.south east) -- (kv-4-1.south west) node [midway, below] {$\bar{B}$}; + \path[draw, decorate, decoration=brace] (kv-4-4.south east) -- (kv-4-4.south west) node [midway, below] {$\bar{B}$}; + \end{tikzpicture} +\end{center} + +Karnaugh Diagramme können für logische Gleichungen mit mehreren Eingangsvariablen gebildet werden. Grundsätzlich gilt für $n$ Eingangsvariablen hat das Karnaugh Diagram $2 ^n$ Felder. Jedes Feld im Karnaugh Diagramm ist für ein Minterm oder ein Maxterm gedacht. Dabei gilt wie bei der Wahrheitstablle wenn die Ausgangsvariable 1 ist handelt es sich um ein Minterm. Ist die Ausgangsvariable 0, so ist es ein Maxterm. +\\ +Das interessante an Karnaugh Diagrammen ist, dass man mit den Min- bzw. Maxtermen "Päckchen" bilden kann. Päckchen können wie folgt gebildet werden. + +\begin{enumerate} + \item Päckchen haben eine Grösse von $2 ^{n}$ + \item Die Päckchen bestehen entweder nur aus Min- bzw. Maxtermen. + \item Es soll immer das grösstmögliche Päckchen gebildet werden. + \item Päckchen können sich schneiden. +\end{enumerate} + +Durch das Bilden von Päckchen können neue Min- bzw. Maxterme gebildet werden, da in Päckchen Variablen in ihrer negierten und nicht negierten Form vorkommen. Diese gleichen sich aus und verschwinden. + +\nt{ + Karnaugh Diagramme bestehend aus mehreren Eingangsvariablen können "eigenartige" Päckchen haben. + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_6.png} \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_7.png} \cite{Luisier2024} + \end{minipage} +} + +\nt{ + Karnaugh Diagramme können unendlich viele Eingangsvariablen haben, jedoch beschränken wir uns auf maximal fünf Eingangsvariablen, da Karnaugh Diagramme mit mehr als fünf Eingangsvariablen unübersichtlich wird. +} + +Um das Karnaugh Diagramm mit einer Wahrheitstablle auszufüllen, gibt es eine bestimmte Vorgehensweise. + +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center, + text = gray + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & & & & \\ + \node[kvbinhead] {01}; & & & & \\ + \node[kvbinhead] {11}; & & & & \\ + \node[kvbinhead] {10}; & & & & \\ + }; + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {AB}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {CD}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + + \begin{pgfonlayer}{tl3} + \draw[thick, blue, ->, > = stealth] (kv-2-2.center) -- (kv-3-2.center) to[out = 180, in = 180] (kv-5-2.center) -- (kv-4-2.center); + \draw[thick, blue, ->, > = stealth] (kv-2-3.center) -- (kv-3-3.center) to[out = 180, in = 180] (kv-5-3.center) -- (kv-4-3.center); + \draw[thick, blue, ->, > = stealth] (kv-2-5.center) -- (kv-3-5.center) to[out = 180, in = 180] (kv-5-5.center) -- (kv-4-5.center); + \draw[thick, blue, ->, > = stealth] (kv-2-4.center) -- (kv-3-4.center) to[out = 180, in = 180] (kv-5-4.center) -- (kv-4-4.center); + \end{pgfonlayer} + \begin{pgfonlayer}{tl2} + \draw[thick, darkgreen, ->, > = stealth] (kv-4-2.center) -- (kv-2-3.center); + \draw[thick, red, ->, > = stealth] (kv-4-3.center) -- (kv-2-5.center); + \draw[thick, darkgreen, ->, > = stealth] (kv-4-5.center) -- (kv-2-4.center); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center, + text = gray + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & & & & \\ + \node[kvbinhead] {01}; & & & & \\ + \node[kvbinhead] {11}; & & & & \\ + \node[kvbinhead] {10}; & & & & \\ + }; + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {CD}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {AB}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + + \begin{pgfonlayer}{tl3} + \draw[thick, blue, ->, > = stealth] (kv-2-2.center) -- (kv-2-3.center) to[out = 90, in = 90] (kv-2-5.center) -- (kv-2-4.center); + \draw[thick, blue, ->, > = stealth] (kv-3-2.center) -- (kv-3-3.center) to[out = 90, in = 90] (kv-3-5.center) -- (kv-3-4.center); + \draw[thick, blue, ->, > = stealth] (kv-5-2.center) -- (kv-5-3.center) to[out = 90, in = 90] (kv-5-5.center) -- (kv-5-4.center); + \draw[thick, blue, ->, > = stealth] (kv-4-2.center) -- (kv-4-3.center) to[out = 90, in = 90] (kv-4-5.center) -- (kv-4-4.center); + \end{pgfonlayer} + \begin{pgfonlayer}{tl2} + \draw[thick, darkgreen, ->, > = stealth] (kv-2-4.center) -- (kv-3-2.center); + \draw[thick, red, ->, > = stealth] (kv-3-4.center) -- (kv-5-2.center); + \draw[thick, darkgreen, ->, > = stealth] (kv-5-4.center) -- (kv-4-2.center); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} +\end{minipage} diff --git a/hs24/digitaltechnik/schaltungssynthese/schaltungssynthese.tex b/hs24/digitaltechnik/schaltungssynthese/schaltungssynthese.tex new file mode 100644 index 0000000..6c082d4 --- /dev/null +++ b/hs24/digitaltechnik/schaltungssynthese/schaltungssynthese.tex @@ -0,0 +1,5 @@ +\chapter{Schaltungssynthese} + +\input{kv_diagramm.tex} +\input{dont_care.tex} +\input{hazard.tex} diff --git a/hs24/digitaltechnik/sources.bib b/hs24/digitaltechnik/sources.bib new file mode 100644 index 0000000..3e3cffc --- /dev/null +++ b/hs24/digitaltechnik/sources.bib @@ -0,0 +1,24 @@ +@Software{MCD2024, + author = {{The Manim Community Developers}}, + license = {MIT}, + month = apr, + ranking = {rank5}, + title = {{Manim – Mathematical Animation Framework}}, + url = {https://www.manim.community/}, + version = {v0.18.1}, + year = {2024}, +} + +@Misc{Vasylyev2024, + author = {Mischa Vasylyev}, + title = {Übungen}, + year = {2024}, +} + +@Misc{Luisier2024, + author = {Dr. Mathieu Luisier}, + title = {Vorlesungsfolien}, + year = {2024}, +} + +@Comment{jabref-meta: databaseType:bibtex;} diff --git a/hs24/digitaltechnik/verknuepfung/basisfunktionen.tex b/hs24/digitaltechnik/verknuepfung/basisfunktionen.tex new file mode 100644 index 0000000..1c0da43 --- /dev/null +++ b/hs24/digitaltechnik/verknuepfung/basisfunktionen.tex @@ -0,0 +1,145 @@ +\section{Basisfunktion} + +\subsection{Nützliche Konzepte} + +\dfn{Schaltfunktion}{ + Eine Schaltfunktion $f$ nimmt mehrere Variablen $X_i$, z.B. $N$, als Eingang und produziert eine einzige Variable $Y$ als Ausgang. + + \begin{equation} + Y = f(X_0, X_1, X_2, ... , X_{N-1}) + \end{equation} + + Der Informationsgehalt der Variablen beträgt ein Bit (1 oder 0). \cite{Luisier2024} +} + +Schaltfunktionen sind nichts anderes als eine Codierung. Damit man die Verschiedenen Codierungen zu bestimmten Ausgängen zuteilen kann, verwendet man Wahrheitstabellen. +\\ +\\ +Wahrheitstabellen nehmen in den linken Spalten die Wertekombinationen der Variablen $X_i$. In der rechten Spalte befindet sich das Ergebnis der Ausgangsvariable $Y$. Die grösse der Wahrheitstabelle hängt von der Anzahl der Variablen $X_i$ ab. Grundsätzlich kann man sich folgendes Merken für die Anzahl Spalten $S$ und Anzahl Zeilen $Z$. + +\[ + S = i + 1 + .\] + +\[ + Z = 2 ^{i} + .\] + +\subsection{UND, ODER, NICHT Verknüpfung} \label{sec:verk} + +In diesem Kapitel schauen wir uns verschiedene Verknüpfungen an. Diese sind relevant für das Verständnis von Schaltfunktionen. + +\dfn{AND Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Wenn Aussage $A$ (Eingang) wahr \textbf{und} Aussage $B$ (Eingang) wahr sind, dann ist Aussage $Y$ (Ausgang) wahr. \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 0 \\ + 0 & 1 & 0 \\ + 1 & 0 & 1 \\ + 1 & 1 & 1 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das UND wie folgt gekennzeichnet. + \[ + A \land B = Y + .\] + Das UND wird mit dem folgenden Schaltzeichen gekennzeichnet. + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[and gate] (and) at (0,0) {}; + \node[] (iA) at (-1, 0.4) {A}; + \node[] (iB) at (-1, -0.4) {B}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (and.input 1); + \draw (iB.east) --++ (right:2mm) |- (and.input 2); + \draw (and.output) -- (oZ); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_30.png} \cite{Luisier2024} + \end{minipage} +} + +\dfn{OR Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Wenn Aussage $A$ (Eingang) wahr oder Aussage $B$ (Eingang) wahr ist, dann ist Aussage $Y$ (Ausgang) wahr. \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 0 \\ + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 1 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das ODER wie folgt gekennzeichnet. + \[ + Y = A \lor B + .\] + Das ODER wird mit dem folgenden Schaltzeichen gekennzeichnet. + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[or gate] (or) at (0,0) {}; + \node[] (iA) at (-1, 0.4) {A}; + \node[] (iB) at (-1, -0.4) {B}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (and.input 1); + \draw (iB.east) --++ (right:2mm) |- (and.input 2); + \draw (and.output) -- (oZ); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_31.png} \cite{Luisier2024} + \end{minipage} +} + +\dfn{NICHT Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Wenn Aussage $A$ (Eingang) wahr ist, dann ist Aussage $Y$ (Ausgang) falsch. \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c |} + $A$ & $Y$ \\ + \hline + 0 & 1 \\ + 1 & 0 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das NICHT wie folgt gekennzeichnet. + \[ + \bar{A} = Y + .\] + Das NICHT wird mit dem folgenden Schaltzeichen gekennzeichnet. + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[not gate] (not) at (0,0) {}; + \node[] (iA) at (-0.8, 0) {A}; + \node[] (oZ) at (0.8, 0) {Z}; + \draw (iA.east) -- (not.input); + \draw (not.output) -- (oZ); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \end{minipage} +} diff --git a/hs24/digitaltechnik/verknuepfung/schaltnetzanalyse.tex b/hs24/digitaltechnik/verknuepfung/schaltnetzanalyse.tex new file mode 100644 index 0000000..c78a557 --- /dev/null +++ b/hs24/digitaltechnik/verknuepfung/schaltnetzanalyse.tex @@ -0,0 +1,214 @@ +\section{Schaltnetzanalyse} + +\subsection{Zusammengesetzte Gatter} \label{sec:gat} + +Die Darstellung von Schaltfunktionen kann mithilfe von Schaltnetzen realisiert werden. Die in Kapitel \ref{sec:verk} erwähnten Schaltzeichen spielen in diesem Kapitel eine grosse Rolle. Wichtig ist vor allem die Kombination von Verknüpfungen. + +\dfn{NAND-Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Invertierung der UND-Funktion + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[and gate] (and) at (-1,0) {}; + \node[not gate] (not) at (1,0) {}; + \node[] (iA) at (-2, 0.4) {A}; + \node[] (iB) at (-2, -0.4) {B}; + \node[] (oY) at (2, 0) {Y}; + \draw (iA.east) --++ (right:2mm) |- (and.input 1); + \draw (iB.east) --++ (right:2mm) |- (and.input 2); + \draw (and.output) -- (not.input); + \draw (not.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 1 \\ + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 0 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das NAND wie folgt gekennzeichnet. + + \[ + \overline{A \land B} = Y + .\] + + Das NAND wird mit den folgenden Schaltzeichen gekennzeichnet. + + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nand gate] (nand) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oY) at (0.8, 0) {Y}; + \draw (iA.east) --++ (right:1.5mm) |- (nand.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nand.input 2); + \draw (nand.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_12.png} \cite{Luisier2024} + \end{minipage} +} + + +\dfn{NOR-Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Invertierung der ODER-Funktion + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[or gate] (or) at (-1,0) {}; + \node[not gate] (not) at (1,0) {}; + \node[] (iA) at (-2, 0.4) {A}; + \node[] (iB) at (-2, -0.4) {B}; + \node[] (oY) at (2, 0) {Y}; + \draw (iA.east) --++ (right:2mm) |- (or.input 1); + \draw (iB.east) --++ (right:2mm) |- (or.input 2); + \draw (or.output) -- (not.input); + \draw (not.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 1 \\ + 0 & 1 & 0 \\ + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das NOR wie folgt gekennzeichnet. + + \[ + \overline{A \lor B} = Y + .\] + + Das NOR wird mit den folgenden Schaltzeichen gekennzeichnet. + + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nor gate] (nor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oY) at (0.8, 0) {Y}; + \draw (iA.east) --++ (right:1.5mm) |- (nor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nor.input 2); + \draw (nor.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_32.png} \cite{Luisier2024} + \end{minipage} +} + + +\dfn{XNOR-Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Ein Gatter, das eine logische 1 liefert, wenn beide Eingänge gleich sind, sonst 0. \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 1 \\ + 0 & 1 & 0 \\ + 1 & 0 & 0 \\ + 1 & 1 & 1 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das XNOR wie folgt gekennzeichnet. + + \[ + \overline{A \oplus B} = Y + .\] + + Das XNOR wird mit den folgenden Schaltzeichen gekennzeichnet. + + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[xnor gate] (xnor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oY) at (0.8, 0) {Y}; + \draw (iA.east) --++ (right:1.5mm) |- (xnor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (xnor.input 2); + \draw (xnor.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \end{minipage} +} + +\dfn{XOR-Verknüpfung}{ + \begin{minipage}{0.5\linewidth} + Ein Gatter, das eine logische 1 liefert, wenn beide Eingänge ungleich sind, sonst 0. \cite{Luisier2024} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \begin{tabular}{| c | c | c |} + $A$ & $B$ & $Y$ \\ + \hline + 0 & 0 & 0 \\ + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 0 \\ + \end{tabular} + \end{center} + \end{minipage} + + \begin{minipage}{0.5\linewidth} + In einer logischen Gleichung wird das XOR wie folgt gekennzeichnet. + + \[ + A \oplus B = Y + .\] + + Das XOR wird mit den folgenden Schaltzeichen gekennzeichnet. + + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[xor gate] (xor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oY) at (0.8, 0) {Y}; + \draw (iA.east) --++ (right:1.5mm) |- (xor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (xor.input 2); + \draw (xor.output) -- (oY); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \end{minipage} +} + +\nt{ + Das XOR-Gatter wird auch EXCLUSIV-OR Gatter genannt. \cite{Luisier2024} +} + +\subsection{Schaltungen aus Grundgatter} + +Die Grundfunktionen und Schaltgatter sind nicht auf 2 Eingangsvariablen beschränkt. Grundsätzlich können Grundfunktionen und Schaltgatter $N$ Eingänge haben. +\\ +\\ +Um die Komplexität von Schaltgattern mit mehreren Eingängen zu vereinfachen, kann man sie in mehreren Schaltgattern verwandeln, sodass die Logik der Schaltung lesbarer ist. diff --git a/hs24/digitaltechnik/verknuepfung/verknuepfung.tex b/hs24/digitaltechnik/verknuepfung/verknuepfung.tex new file mode 100644 index 0000000..0642f49 --- /dev/null +++ b/hs24/digitaltechnik/verknuepfung/verknuepfung.tex @@ -0,0 +1,4 @@ +\chapter{Logische Verknüpfung} + +\input{basisfunktionen.tex} +\input{schaltnetzanalyse.tex} diff --git a/hs24/digitaltechnik/zahlen_codes/codes.tex b/hs24/digitaltechnik/zahlen_codes/codes.tex new file mode 100644 index 0000000..b9d5686 --- /dev/null +++ b/hs24/digitaltechnik/zahlen_codes/codes.tex @@ -0,0 +1,36 @@ +\section{Codes} \label{sec:code} + +\subsection{Was ist ein Code?} + +In Kapitel \ref{sec:grund} haben wir gelernt was eine Codierung ist. Codes kommen hauptsächlich in vierer Gruppen vor (Tetraden) und neben den Binär Codierung gibt es viele weitere Codes. + +\subsection*{Codes mit Tetraden} + +\begin{tabular}{| c | c | c | c | c | c | c |} + Binär & BCD & Excess-3 & Aiken & 4-2-2-1 & Gray & O'Brien \\ + \hline + 0000 & 0 & & 0 & 0 & 0 & \\ + 0001 & 1 & & 1 & 1 & 1 & \\ + 0010 & 2 & & 2 & 2 & 3 & 0 \\ + 0011 & 3 & 0 & 3 & 3 & 2 & \\ + 0100 & 4 & 1 & 4 & & 7 & 4 \\ + 0101 & 5 & 2 & & & 6 & 3 \\ + 0110 & 6 & 3 & & 4 & 4 & 1 \\ + 0111 & 7 & 4 & & 5 & 5 & 2 \\ + 1000 & 8 & 5 & & & & \\ + 1001 & 9 & 6 & & & & \\ + 1010 & & 7 & & & & 9 \\ + 1011 & & 8 & 5 & & & \\ + 1100 & & 9 & 6 & 6 & 8 & 5 \\ + 1101 & & & 7 & 7 & 9 & 6 \\ + 1110 & & & 8 & 8 & & 7 \\ + 1111 & & & 9 & 9 & & 8 \\ +\end{tabular} + +Codes werden hauptsächlich verwendet, um Informationen zu speichern, Daten zu komprimieren und Fehlerkorrekturen und Verschlüsselungen von Nachrichtenübertragungen zu machen. + +\subsection{Fehlererkennung und Fehlerkorrektur} + +Bei der Übertragung von Codes können Fehler auftauchen. Deswegen werden neben den Codes auch Parity Bits übertragen. Parity kennzeichnen ob die Codierung gerade oder ungerade ist. Falls bei der Übertragung ein Fehler entstanden ist und der Parity Bit nicht übereinstimmt, so wird ein Fehler gemeldet. +\\ +Eine weitere Stufe zur Fehlererkennung ist das Verwenden von einem Prüfwort. Das Prüfwort ist eine Codierung mit einem Parity Bit. Das Prüfwort wird gebildet durch die Addition von mehreren Codierungen, welche auf Fehler untersucht werden sollen. Dabei wird der Übertrag ignoriert. Hat das Prüfwort ein Fehler, so stimmt der Parity Bit nicht mehr überein und es wird nach der Falschen Codierung gesucht. diff --git a/hs24/digitaltechnik/zahlen_codes/zahlen_codes.tex b/hs24/digitaltechnik/zahlen_codes/zahlen_codes.tex new file mode 100644 index 0000000..fbe881e --- /dev/null +++ b/hs24/digitaltechnik/zahlen_codes/zahlen_codes.tex @@ -0,0 +1,4 @@ +\chapter{Zahlen und Codes} + +\input{zahlensysteme.tex} +\input{codes.tex} diff --git a/hs24/digitaltechnik/zahlen_codes/zahlensysteme.tex b/hs24/digitaltechnik/zahlen_codes/zahlensysteme.tex new file mode 100644 index 0000000..c43bd84 --- /dev/null +++ b/hs24/digitaltechnik/zahlen_codes/zahlensysteme.tex @@ -0,0 +1,56 @@ +\section{Zahlensysteme} + +\subsection{Was ist ein Zahlensystem?} + +\dfn{Zahlensystem}{ + Ein Zahlensystem ist eine Methode, um Zahlen darzustellen. Wir Menschen sind gewohnt, das Dezimalsystem zu verwenden. In der Digitaltechnik sind jedoch andere Zahlensysteme von grosser Bedeutung. +} + +\subsection*{Zahlensysteme} + +\begin{itemize} + \item Dezimalsystem stellt Zahlen in Zehnerpotenzen dar. (Basis = 10) + \item Oktalsystem stellt Zahlen in Achterpotenzen dar. (Basis = 8) + \item Dualsystem (Binärsystem) stellt Zahlen in Zweierpotenzen dar. (Basis = 2) + \item Hexadezimalsystem stellt Zahlen in sechszehnerpotenz dar. (Basis = 16) +\end{itemize} + +\nt{ + Da das Hexadezimalsystem über die Zahl 9 geht werden Buchstaben für die darauffolgenden Zahlen verwendet. +} + +\nt{ + Das Oktal- und Hexadezimalsystem wird verwendet um das Binärsystem kompakter zu machen. Sie können päckchenweise ineinander umgerechnet werden. \cite{Luisier2024} +} + +\subsection{Umwandlung zwischen Zahlensystemen} + +Wie vorher schon erwähnt können binär Zahlen sehr einfach in Oktalzahlen und Hexadezimalzahlen umgewandelt werden. Die Umwandlung von Dezimalzahlen in ein anderes Zahlensystem ist ein wenig komplizierter. +\\ +Grundsätzlich gilt, dass wenn man vom Dezimalsystem in ein anderes System umwandelt, dass man die Zahl durch die korrespondierende Basis dividiert. Der Rest entspricht jeweils die Ziffer in der Codierung. Bei der Division von Zahlen $>1$ entsteht bei der ersten Division der MSB und bei der letzten Division der LSB. Bei Zahlen $<1$ ist es umgekehrt. + +\subsection{Negative Dualzahlen (Zweierkomplement)} + +Für negative Zahlen ist es am einfachsten, die negative Zahl ins Binärsystem umzuwandeln und diese, wenn nötig in das gefragte Zahlensystem umzuwandeln. Dabei geht man wie folgt vor. + +\subsection*{Umwandlung von negativen Zahlen} + +\begin{enumerate} + \item Den Betrag der negativen Zahl ins Binärsystem umwandeln. + \item Die umgewandelte negative Zahl invertieren. ($0 \rightarrow 1$ und $1 \rightarrow 0$) + \item Addiere 1 zum LSB. + \item Vorzeichenbit als MSB einfügen. +\end{enumerate} + +\nt{ + Die negative Zahl im Binärsystem wird Zweierkomplement genannt. Ob das Binärsystem oder das Zweierkomplementsystem verwendet wird, wird meistens in der Aufgabe gesagt. +} + +Negative rationale Zahlen bestehen aus einem Vorzeichenbit, sowie $m$ Vorkomma- und $n$ Nachkommabits. Somit sind $1 + m + n$ Bits erforderlich. Das Komma wird vor den Vorkommabits eingefügt. + + +\subsection{Rechenoperationen mit Dualzahlen} + +Die Addition von zwei Dualzahlen erfolgt bitweise, beginnend mit den LSB, von rechts nach links \cite{Luisier2024}. Falls die Codierungen nicht die gleiche Anzahl an Bits haben werden Nullen hinzugefügt. +\\ +Für die Subtraktion ist es am einfachsten, die Zahl, welche subtrahiert wird ins Zweierkomplement umzuwandeln und anschliessend zur anderen Codierung zu addieren. diff --git a/hs24/lineare_algebra/.DS_Store b/hs24/lineare_algebra/.DS_Store new file mode 100644 index 0000000..2d94d9f Binary files /dev/null and b/hs24/lineare_algebra/.DS_Store differ diff --git a/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung.tex b/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung.tex new file mode 100644 index 0000000..a555af3 --- /dev/null +++ b/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung.tex @@ -0,0 +1,17 @@ +\chapter{Ausgleichsrechnung} + +\input{ausgleichsrechnung_least_squares.tex} +\input{normalengleichung.tex} +\input{qrzerlegung.tex} + +\nt{ + Nützliche Videos: + + \begin{itemize} + \item Least squares approximation $|$ Linear Algebra $|$ Khan Academy (\url{https://www.youtube.com/watch?v=MC7l96tW8V8}) + \item A=QR Factorizations: Part 4/5 Least Squares (\url{https://www.youtube.com/watch?v=B3_vu5WcBzg}) + \end{itemize} +} + + +\newpage diff --git a/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung_least_squares.tex b/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung_least_squares.tex new file mode 100644 index 0000000..958a7e0 --- /dev/null +++ b/hs24/lineare_algebra/ausgleichsrechnung/ausgleichsrechnung_least_squares.tex @@ -0,0 +1,36 @@ +\section{Ausgleichsrechnung ("Least Sqares")} + +Nehmen wir an, dass wir eine Datenbank haben mit Messungen, welches wir in einem Graphen eingefügt haben. Nun wollen wir eine Funktion finden, welches durch die Messwerte geht. Wir können das folgende Gleichungssystem bilden: + +\begin{equation} + \mathbf{A} \vec{x} = \vec{b} + \label{eq:least square prev} +\end{equation} + +Wobei die Matrix $\mathbf{A}$ die parameter von $\vec{x}$ sind und $\vec{b}$ die resultierende Gerade bildet. + +Wir schauen uns nun die folgende Grafik an: + +\includegraphics[width=\textwidth]{fig/Fig_3.png} + +Wie man anhand der Grafik sehen kann, sieht man, dass es $\vec{x}$ nicht existiert, da es keine Funktion existiert, welche durch alle Punkte geht. Was ist, wenn man trotz allem eine Funktion finden möchte? +\\ +Wir nehmen eine neue Betrachtungsweise. Nehmen wir an, dass die Matrix $\mathbf{A}$ eine Ebene bildet. Die Gleichung $\mathbf{A} \vec{x} = \vec{b}$ bildet dann einen Vector $\vec{b}$ welches in der Ebene ist, die von $\mathbf{A}$ gebildet wird. Nehmen wir an, dass $\vec{b}$ nicht in der Ebene ist. Daraus folgt, dass es kein $\vec{x}$ gibt, welches die Gleichung bildet. Was kann man stattdessen machen? +\\ +\\ +Man könnte ein $\vec{\hat{x}}$ wählen, welche die Gleichung zwar nicht bildet aber den Vektor $\vec{b}$ annähert. Daraus schliesst man, dass $\vec{\hat{x}}$ $\vec{b}$ am ehesten annähert, wenn die Distanz von $b'$, welches den $\vec{b'}$ in der Ebene bildet und $b$ am geringsten ist. + +\includegraphics[width=\textwidth]{fig/Fig_4.png} + +Anahand der Figur kann man erkennen, dass $\vec{\hat{x}}$ so gewählt werden sollte, dass es die Projektion von $\vec{b}$ bildet. Daraus kann man die folgende Definition ableiten. + +\dfn{Lineare Ausgleichsrechnung als lineares Gleichungssystem \cite{Gradinaru2024}}{ + Sei eine $m \times n$ Matrix $\mathbf{A}$ und der Vektor $\vec{b}$ mit $m$ Einträgen. Gesucht ist ein $\hat{x}$, so dass + + \[ + ||\mathbf{A}\vec{\hat{x}} - \vec{b}|| = \text{min} ||\mathbf{A}\vec{x} - \vec{b}|| + .\] + +} + +Es gibt zwei wege um $\vec{\hat{x}}$ zu berechnen, nähmlich die Lösung mit der Normalengleichung und Lösung mit der QR-Zerlegung. diff --git a/hs24/lineare_algebra/ausgleichsrechnung/normalengleichung.tex b/hs24/lineare_algebra/ausgleichsrechnung/normalengleichung.tex new file mode 100644 index 0000000..b3a2ec7 --- /dev/null +++ b/hs24/lineare_algebra/ausgleichsrechnung/normalengleichung.tex @@ -0,0 +1,21 @@ +\section{Lösung mit der Normalengleichung} + +\dfn{Normalengleichung \cite{Gradinaru2024}}{ + Die Gleichung + + \[ + \mathbf{A} ^{\text{T}} \mathbf{A} \vec{x} = \vec{b} \mathbf{A} ^{\text{T}} + \] + + heisst Normalengleichung. +} + +Mit dieser Funktion kann man ein $\vec{\hat{x}}$ finden, welche den Vektor $\vec{b}$ annähert. Dabei wird wie folgt vorgegangen: + +\begin{enumerate} + \item Man multipliziert die Matrix $\mathbf{A}$ mit ihrem Transponierten + \item Man multipliziert den Vector $\vec{b}$ mit $\mathbf{A} ^{\text{T}}$. + \item Man gausst die Matrix $\mathbf{A}$ um nach den Parametern von $\vec{x}$ aufzulösen +\end{enumerate} + +Wie man sieht, ist der Nachteil dieser Methode, dass man die Matrix $\mathbf{A}$ gaussen muss. Es gibt aber eine zweite Methode um $\vec{\hat{x}}$ herauszufinden. diff --git a/hs24/lineare_algebra/ausgleichsrechnung/qrzerlegung.tex b/hs24/lineare_algebra/ausgleichsrechnung/qrzerlegung.tex new file mode 100644 index 0000000..4f11d29 --- /dev/null +++ b/hs24/lineare_algebra/ausgleichsrechnung/qrzerlegung.tex @@ -0,0 +1,15 @@ +\section{Lösung mit der QR-Zerlegung} + +Eine einfachere Methode $\vec{\hat{x}}$ zu finden ist mit Hilfe der QR-Zerlegung. + +\dfn{Lösung mit der QR-Zerlegung}{ + Sei die QR-Zerlegung einer $m \times n$ Matrix $\mathbf{A}$ gegeben, so ist $\vec{\hat{x}}$ durch + + \[ + \mathbf{R} \vec{x} = \mathbf{Q} ^{\text{T}} \vec{b} + \] + + definiert. +} + +Dabei geht man wie bei der Normalengleichung vor, jedoch fällt das Gaussen weg, da $\mathbf{R}$ eine obere Dreiecksmatrix ist. diff --git a/hs24/lineare_algebra/determinante/anwendungen.tex b/hs24/lineare_algebra/determinante/anwendungen.tex new file mode 100644 index 0000000..ce90203 --- /dev/null +++ b/hs24/lineare_algebra/determinante/anwendungen.tex @@ -0,0 +1,15 @@ +\section{Anwendungen} + +Wie vorher erwähnt, wird die Determinante als ein Werkzeug für lineare Transformationen verwendet. Was wir bereits auch angeschaut haben anhand von Beispielen ist, dass det die Fläche ist welche eine $2 \times 2$ Matrix aufspannt. +\\ +Für eine $3 \times 3$ Matrix, ist det das Volumen des Körpers, welches von den Vektoren der Matrix gebildet wird. +\\ +Neben dem hat det eine Relevanz mit dem Kreuzprodukt: + +\[ + |(\mathbf{u} \times \mathbf{v}) \mathbf{w}| = |\text{det} \begin{bmatrix} + u_1 & u_2 & u_3\\ + v_1 & v_2 & v_3\\ + w_1 & w_2 & w_3\\ + \end{bmatrix}| +.\] diff --git a/hs24/lineare_algebra/determinante/definition_eigenschaften.tex b/hs24/lineare_algebra/determinante/definition_eigenschaften.tex new file mode 100644 index 0000000..b217ec2 --- /dev/null +++ b/hs24/lineare_algebra/determinante/definition_eigenschaften.tex @@ -0,0 +1,110 @@ +\section{Definition und Eigenschaften} + +\dfn{Determinante \cite{Gradinaru2024}}{ + Die Determinante ist eine Funktion + + \[ + \text{det : } \underbrace{\mathbb{R} ^{n} \times \mathbb{R} ^{n} \times ... \times \mathbb{R} ^{n}}_{n} \rightarrow \mathbb{R} + \] + + mit folgenden Eigenschaften: + + \begin{itemize} + \item[D1] $\text{det } \mathbf{I}_n = 1$ + \item[D2] det wechselt das Vorzeichen, wenn zwei Zeilen oder Spalten vertauscht werden (Antisymmetrie). + \item[D3] det ist linear in jeder Zeile und Spalte: + + \[ + \text{det} \begin{bmatrix} + ta & tb\\ + c & d\\ + \end{bmatrix} = t \cdot \text{det} \begin{bmatrix} + a & b\\ + c & d\\ + \end{bmatrix} + .\] + + \[ + \text{det} \begin{bmatrix} + a + \tilde{a} & b + \tilde{b}\\ + c & d\\ + \end{bmatrix} = \text{det} \begin{bmatrix} + a & b\\ + c & d\\ + \end{bmatrix} + \text{det} \begin{bmatrix} + \tilde{a} & \tilde{b}\\ + c & d\\ + \end{bmatrix} + .\] + + \end{itemize} +} + +Was ist genau die Determinante und was ist der Konzept dahinter? +\\ +\\ +Die Determinante ist ein Hilfsmittel um eine lineare Transformation besser verstehen zu können, genauer um welchen Faktor die Fläche sich vergrössert. Betrachten wir es anhand einer Grafik. +\\ +Wir wissen von D1, dass $\text{det } \mathbf{I}_2 = 1$. Versuchen wir mal dies in einer Grafik darzustellen. + +\includegraphics[width=\textwidth]{fig/Fig_5.png} + +$\mathbf{I}_2$ bildet im Koordinatensystem eine Fläche mit einer Grösse von 1. Wir nehmen nun einfachtshalber mal eine willkürliche Dreiecksmatrix $\mathbf{A}$. Diese Zeichnen wir auch in das Koordinatensystem ein. + +\includegraphics[width=\textwidth]{fig/Fig_6.png} + +Wir sehen, dass die Matrix $\mathbf{A}$ eine Fläche aufspannt mit Grösse 6. Somit ist det 6. Wir nehemen nun eine andere Matrix $\mathbf{B}$. + +\includegraphics[width=\textwidth]{fig/Fig_7.png} + +Wir sehen, dass die Matrix eine lineare Transformation ist, jedoch bleibt die Fläche gleich gross da det 1 ist. Somit würden die Flächen ihre Grösse nach der Transformation nicht verändern. +\\ +Was ist mit negativen det. Gibt es negative det und wie soll man sich negative det sich vorstellen? +\\ +Wir betrachten neben der $\mathbf{I}_2$ die Matrix $\mathbf{C}$. + +\includegraphics[width=\textwidth]{fig/Fig_8.png} + +Wir sehen, dass die Fläche, welche die Matrix $\mathbf{C}$ bildet in vergleich zu Matrix $\mathbf{B}$ gespiegelt ist. Daraus kann man ziehen, dass negative det eine Skalierung und eine Spiegelung sind. +\\ +Wie sieht es aus mit det die 0 sind? +\\ +Betrachten wir die Matrix $\mathbf{D}$. + +\includegraphics[width=\textwidth]{fig/Fig_9.png} + +Die Matrix $\mathbf{D}$ hat keine Fläche, da die Vektoren, welche die Fläche bilden übereinander sind. In manchen Fällen können Determinanten auch ein Punkt bilden. +\\ +\\ +Da wir das Konzept von Determinanten jetzt besser verstehen. Können wir mit ein paar wichtige Rechenregel für Determinanten fortfahren. Die Rechenregel sind ausführlich im Skript von Dr. Gradinaru \cite{Gradinaru2024} beschrieben. +\\ +\begin{itemize} + \item Falls zwei Spalten oder Zeilen einer Matrix identisch sind, so ist die Determinante der Matrix 0. + \item Lineare Kombinationen von Zeilen der Matrix ändert die Determinante dieser Matrix nicht. + \item Wenn eine Matrix eine Nullzeile oder eine Nullspalte hat, so ist ihre Determinante 0. + \item Falls $\mathbf{A}$ eine Dreiecksmatrix ist, so ist die Determinante von A das Produkt der Diagonaleinträge. + \item Falls A singulär ist, entsteht bei der Gauss-Elimination eine Nullzeile und die Determinante ist 0. + \item det $\mathbf{A} \mathbf{B}$ = det $\mathbf{A} \cdot$ det $\mathbf{B}$. + \item \[ + \text{det } \mathbf{A} ^{-1} = \frac{1}{\text{det } \mathbf{A}} + .\] + \item $\text{det } \mathbf{A} = a_{11} \cdot \text{det } \mathbf{A} _{11} - a_{12} \cdot \text{det } \mathbf{A} _{12} + ... + a_{1n} \cdot det \mathbf{A}_{1n}$ +\end{itemize} + +\nt{ + Es gibt noch ein paar Tricks um det zu berechnen: + + \begin{itemize} + \item det einer $3 \times 3$ Matrix lässt sich auch so rechnen: + + \begin{center} + \includegraphics[width=0.6\textwidth]{fig/Fig_10.png} + \end{center} + + \item Die Regel mit der Dreiecksmatrix lässt sich ein wenig erweitern. Falls man Blöcke bilden kann so lässt sich det durch das Multiplizieren der einzelnen det der Blöcke berechnen. + + \begin{center} + \includegraphics[width=0.3\textwidth]{fig/Fig_11.png} + \end{center} + \end{itemize} +} diff --git a/hs24/lineare_algebra/determinante/determinante.tex b/hs24/lineare_algebra/determinante/determinante.tex new file mode 100644 index 0000000..c35ce39 --- /dev/null +++ b/hs24/lineare_algebra/determinante/determinante.tex @@ -0,0 +1,15 @@ +\chapter{Determinante} + +\input{definition_eigenschaften.tex} +\input{anwendungen.tex} + +\nt{ + Nützliche Videos: + + \begin{itemize} + \item The determinant $|$ Chapter 6, Essence of linear algebra \url{https://www.youtube.com/watch?v=Ip3X9LOh2dk} + \end{itemize} +} + + +\newpage diff --git a/hs24/lineare_algebra/disclaimer.tex b/hs24/lineare_algebra/disclaimer.tex new file mode 100644 index 0000000..5d98be3 --- /dev/null +++ b/hs24/lineare_algebra/disclaimer.tex @@ -0,0 +1,12 @@ +\section*{DISCLAIMER} + +Diese Notizen wurden verfasst auf Basis der Vorlesung Lineare Algebra (HS24) von V. Gradinaru und dem Skript "Lineare Algebra" von Vasile Gradinaru. +\\ +\\ +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). +\\ +\\ +Alle Grafiken wurden eigenhändig mit Manim \cite{MCD2024} generiert. +\\ +\\ +Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden. diff --git a/hs24/lineare_algebra/eigenwertproblem/eigenwertproblem.tex b/hs24/lineare_algebra/eigenwertproblem/eigenwertproblem.tex new file mode 100644 index 0000000..dd07036 --- /dev/null +++ b/hs24/lineare_algebra/eigenwertproblem/eigenwertproblem.tex @@ -0,0 +1 @@ +\chapter{Eigenwertproblem} diff --git a/hs24/lineare_algebra/fig/Fig_1.png b/hs24/lineare_algebra/fig/Fig_1.png new file mode 100644 index 0000000..aed7e6d Binary files /dev/null and b/hs24/lineare_algebra/fig/Fig_1.png differ diff --git a/hs24/lineare_algebra/fig/Fig_10.png 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a/hs24/lineare_algebra/graphs/__pycache__/graph.cpython-312.pyc b/hs24/lineare_algebra/graphs/__pycache__/graph.cpython-312.pyc new file mode 100644 index 0000000..0d045a9 Binary files /dev/null and b/hs24/lineare_algebra/graphs/__pycache__/graph.cpython-312.pyc differ diff --git a/hs24/lineare_algebra/graphs/graph.py b/hs24/lineare_algebra/graphs/graph.py new file mode 100644 index 0000000..751e2b1 --- /dev/null +++ b/hs24/lineare_algebra/graphs/graph.py @@ -0,0 +1,391 @@ +from manim import * + + +class LSgraph1(Scene): + def construct(self): + # self.camera.background_color = "WHITE" + ax = Axes( + x_range=[-0.3, 10.3, 1], + y_range=[-0.3, 5.3, 1], + x_length=10.6, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + + self.camera.background_color = WHITE + + axes_lables = ax.get_axis_labels().set_color(BLACK) + + dot1 = Dot(point=ax.c2p(2, 1.3, 0), color=BLUE) + dot2 = Dot(point=ax.c2p(3, 1.6, 0), color=BLUE) + dot3 = Dot(point=ax.c2p(5, 2.0, 0), color=BLUE) + dot4 = Dot(point=ax.c2p(8, 4.7, 0), color=BLUE) + dot5 = Dot(point=ax.c2p(10, 4.2, 0), color=BLUE) + + function_graph = ax.plot(lambda x: 0.5 * x, color=BLACK) + + plot = VGroup(ax, axes_lables, dot1, dot2, dot3, dot4, dot5, function_graph) + self.add(plot) + + +class LSgraph2(ThreeDScene): + def construct(self): + # self.camera.background_color = "WHITE" + ax = ThreeDAxes( + x_range=[-0.3, 10.3, 1], + y_range=[-0.3, 10.3, 1], + z_range=[-0.3, 10.3, 1], + x_length=10.6, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + + self.camera.background_color = WHITE + self.set_camera_orientation( + phi=56 * DEGREES, theta=-30 * DEGREES, gamma=-7 * DEGREES, zoom=0.5 + ) + + axes_lables = ax.get_axis_labels().set_color(BLACK) + + plane = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[2, 8], + v_range=[2, 8], + checkerboard_colors=[BLUE, BLUE], + fill_opacity=0.75, + stroke_width=0, + ) + + vector = Arrow3D(ax.c2p(3, 3, 0), ax.c2p(6, 6, 3), color=RED) + vector1 = Arrow3D(ax.c2p(3, 3, 0), ax.c2p(6, 6, 0), color=GREEN) + + line = Line(ax.c2p(6, 6, 0), ax.c2p(6, 6, 3), color=ORANGE) + + self.add(ax, axes_lables, vector, vector1, plane, line) + + +class Detgraph1(Scene): + def construct(self): + ax = Axes( + x_range=[-5.3, 5.3, 1], + y_range=[-5.3, 5.3, 1], + x_length=7, + y_length=7, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + x = ax.get_x_axis() + x.numbers.set_color(BLACK) + y = ax.get_y_axis() + y.numbers.set_color(BLACK) + + self.camera.background_color = WHITE + + axes_lables = ax.get_axis_labels().set_color(BLACK) + + vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED) + vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE) + + plane = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 1], + v_range=[0, 1], + checkerboard_colors=[GREEN, GREEN], + fill_opacity=0.75, + stroke_width=0, + ) + + text = MathTex( + r"\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}", color=BLACK + ).move_to(ax.c2p(-2, 2, 0)) + + self.add(ax, axes_lables, vec1, vec2, plane, text) + + +class Detgraph2(Scene): + def construct(self): + ax = Axes( + x_range=[-5.3, 5.3, 1], + y_range=[-5.3, 5.3, 1], + x_length=7, + y_length=7, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + x = ax.get_x_axis() + x.numbers.set_color(BLACK) + y = ax.get_y_axis() + y.numbers.set_color(BLACK) + self.camera.background_color = WHITE + axes_lables = ax.get_axis_labels().set_color(BLACK) + + vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED) + vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE) + plane1 = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 1], + v_range=[0, 1], + checkerboard_colors=[GREEN, GREEN], + fill_opacity=0.75, + stroke_width=0, + ) + + vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 2, 0), color=PURPLE) + vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(3, 0, 0), color=PINK) + plane2 = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 3], + v_range=[0, 2], + checkerboard_colors=[YELLOW, YELLOW], + fill_opacity=0.75, + stroke_width=0, + ) + + text = MathTex( + r"\mathbf{A} \begin{bmatrix} 3 & 0\\ 0 & 2\end{bmatrix}", color=BLACK + ).move_to(ax.c2p(-2.5, 2, 0)) + + self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text) + + +class Detgraph3(Scene): + def construct(self): + ax = Axes( + x_range=[-5.3, 5.3, 1], + y_range=[-5.3, 5.3, 1], + x_length=7, + y_length=7, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + x = ax.get_x_axis() + x.numbers.set_color(BLACK) + y = ax.get_y_axis() + y.numbers.set_color(BLACK) + self.camera.background_color = WHITE + axes_lables = ax.get_axis_labels().set_color(BLACK) + + vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED) + vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE) + plane1 = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 1], + v_range=[0, 1], + checkerboard_colors=[GREEN, GREEN], + fill_opacity=0.75, + stroke_width=0, + ) + + vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 1, 0), color=PURPLE) + vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=PINK) + plane2 = Polygon( + ax.c2p(0, 0, 0), + ax.c2p(1, 0, 0), + ax.c2p(2, 1, 0), + ax.c2p(1, 1, 0), + fill_color=YELLOW, + fill_opacity=0.75, + color=YELLOW, + ) + + text = MathTex( + r"\mathbf{B} \begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix}", color=BLACK + ).move_to(ax.c2p(-2.5, 2, 0)) + + self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text) + + +class Detgraph4(Scene): + def construct(self): + ax = Axes( + x_range=[-5.3, 5.3, 1], + y_range=[-5.3, 5.3, 1], + x_length=7, + y_length=7, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + x = ax.get_x_axis() + x.numbers.set_color(BLACK) + y = ax.get_y_axis() + y.numbers.set_color(BLACK) + self.camera.background_color = WHITE + axes_lables = ax.get_axis_labels().set_color(BLACK) + + vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED) + vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE) + plane1 = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 1], + v_range=[0, 1], + checkerboard_colors=[GREEN, GREEN], + fill_opacity=0.75, + stroke_width=0, + ) + + vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 1, 0), color=PURPLE) + vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(2, -1, 0), color=PINK) + plane2 = Polygon( + ax.c2p(0, 0, 0), + ax.c2p(2, -1, 0), + ax.c2p(3, 0, 0), + ax.c2p(1, 1, 0), + fill_color=YELLOW, + fill_opacity=0.75, + color=YELLOW, + ) + + text = MathTex( + r"\mathbf{C} \begin{bmatrix} 1 & 2\\ 1 & -1\end{bmatrix}", color=BLACK + ).move_to(ax.c2p(-2.5, 2, 0)) + + self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text) + + +class Detgraph5(Scene): + def construct(self): + ax = Axes( + x_range=[-5.3, 5.3, 1], + y_range=[-5.3, 5.3, 1], + x_length=7, + y_length=7, + axis_config={ + "include_numbers": True, + "color": BLACK, + }, + ) + x = ax.get_x_axis() + x.numbers.set_color(BLACK) + y = ax.get_y_axis() + y.numbers.set_color(BLACK) + self.camera.background_color = WHITE + axes_lables = ax.get_axis_labels().set_color(BLACK) + + vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED) + vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE) + plane1 = Surface( + lambda u, v: ax.c2p(u, v, u * 0), + u_range=[0, 1], + v_range=[0, 1], + checkerboard_colors=[GREEN, GREEN], + fill_opacity=0.75, + stroke_width=0, + ) + + vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(4, 2, 0), color=PURPLE) + vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(2, 1, 0), color=PINK) + + text = MathTex( + r"\mathbf{C} \begin{bmatrix} 4 & 2\\ 2 & 1\end{bmatrix}", color=BLACK + ).move_to(ax.c2p(-2.5, 2, 0)) + + self.add(ax, axes_lables, plane1, vec3, vec4, vec2, vec1, text) + + +class Detgraph6(Scene): + def construct(self): + self.camera.background_color = WHITE + + text1 = MathTex( + r"\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{bmatrix} \begin{matrix} a & b\\ d & e\\ g & h\\ \end{matrix}", + color=BLACK, + ).scale(0.5) + text2 = ( + MathTex( + r"\Rightarrow a \cdot e \cdot i + b \cdot f \cdot g + c \cdot d \cdot h - g \cdot e \cdot c - h \cdot f \cdot a - i \cdot d \cdot b", + color=BLACK, + ) + .move_to([0, -1, 0]) + .scale(0.5) + ) + + arrow1 = Arrow( + [0, 0, 0], + [1.2, -0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=RED, + ).move_to([-0.4, -0.1, 0]) + arrow2 = Arrow( + [0, 0, 0], + [1.2, -0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=RED, + ).move_to([0, -0.1, 0]) + arrow3 = Arrow( + [0, 0, 0], + [1.2, -0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=RED, + ).move_to([0.4, -0.1, 0]) + + arrow4 = Arrow( + [0, 0, 0], + [1.2, 0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=BLUE, + ).move_to([-0.3, -0.1, 0]) + arrow5 = Arrow( + [0, 0, 0], + [1.2, 0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=BLUE, + ).move_to([0.1, -0.1, 0]) + arrow6 = Arrow( + [0, 0, 0], + [1.2, 0.9, 0], + max_tip_length_to_length_ratio=0.1, + stroke_width=0.7, + color=BLUE, + ).move_to([0.5, -0.1, 0]) + + self.add(text1, text2, arrow1, arrow2, arrow3, arrow4, arrow5, arrow6) + + +class Detgraph7(Scene): + def construct(self): + self.camera.background_color = WHITE + + text1 = MathTex( + r"\begin{bmatrix} a & b & 0 & 0\\ c & d & 0 & 0 \\ 0 & 0 & e & f \\ 0 & 0 & g & h \end{bmatrix}", + color=BLACK, + ).scale(0.5) + text2 = ( + MathTex( + r"\Rightarrow \text{det } \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \text{det } \begin{bmatrix} e & f \\ g & h \end{bmatrix}", + color=BLACK, + ) + .move_to([0, -1, 0]) + .scale(0.5) + ) + + box1 = Polygon([0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0]).move_to( + [-0.36, 0.3, 0] + ) + box2 = Polygon( + [0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0], color=RED + ).move_to([0.36, -0.3, 0]) + box3 = Polygon([0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0]).move_to( + [-0.3, -1, 0] + ) + box4 = Polygon( + [0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0], color=RED + ).move_to([1.13, -1, 0]) + + self.add(text1, text2, box1, box2, box3, box4) diff --git a/hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.svg b/hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.svg new file mode 100644 index 0000000..f38277e --- /dev/null +++ b/hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.svg @@ -0,0 +1,10 @@ +<?xml version='1.0' encoding='UTF-8'?> +<!-- This file was generated by dvisvgm 3.2.2 --> +<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='4.981339pt' height='6.420368pt' viewBox='169.364866 -10.006955 4.981339 6.420368'> +<defs> +<path id='g0-50' d='M1.265255-.767123L2.321295-1.793275C3.875467-3.16812 4.473225-3.706102 4.473225-4.702366C4.473225-5.838107 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b/hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.tex @@ -0,0 +1,9 @@ +\documentclass[preview]{standalone} +\usepackage[english]{babel} +\usepackage{amsmath} +\usepackage{amssymb} +\begin{document} +\begin{align*} +2 +\end{align*} +\end{document} \ No newline at end of file diff --git a/hs24/lineare_algebra/graphs/media/Tex/31d3165490bf3404.svg b/hs24/lineare_algebra/graphs/media/Tex/31d3165490bf3404.svg new file mode 100644 index 0000000..bebce29 --- /dev/null +++ b/hs24/lineare_algebra/graphs/media/Tex/31d3165490bf3404.svg @@ -0,0 +1,10 @@ +<?xml version='1.0' encoding='UTF-8'?> +<!-- This file was generated by dvisvgm 3.2.2 --> +<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='4.981339pt' height='6.420368pt' viewBox='169.364866 -10.006955 4.981339 6.420368'> +<defs> +<path id='g0-52' d='M2.929016-1.643836V-.777086C2.929016-.418431 2.909091-.308842 2.171856-.308842H1.96264V0C2.371108-.029888 2.889166-.029888 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letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} 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\newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/hs24/lineare_algebra/lineare_abbildungen/lineare_abbildungen.tex b/hs24/lineare_algebra/lineare_abbildungen/lineare_abbildungen.tex new file mode 100644 index 0000000..8cf3975 --- /dev/null +++ b/hs24/lineare_algebra/lineare_abbildungen/lineare_abbildungen.tex @@ -0,0 +1,3 @@ +\chapter{Lineare Abbildungen} + +\newpage diff --git a/hs24/lineare_algebra/lineare_gleichungssysteme/lineare_gleichungssysteme.tex b/hs24/lineare_algebra/lineare_gleichungssysteme/lineare_gleichungssysteme.tex new file mode 100644 index 0000000..bb2658c --- /dev/null +++ b/hs24/lineare_algebra/lineare_gleichungssysteme/lineare_gleichungssysteme.tex @@ -0,0 +1,3 @@ +\chapter{Lineare Gleichungssysteme} + +\input{vektoren_matrizen.tex} diff --git a/hs24/lineare_algebra/lineare_gleichungssysteme/vektoren_matrizen.tex b/hs24/lineare_algebra/lineare_gleichungssysteme/vektoren_matrizen.tex new file mode 100644 index 0000000..e915744 --- /dev/null +++ b/hs24/lineare_algebra/lineare_gleichungssysteme/vektoren_matrizen.tex @@ -0,0 +1,287 @@ +\section{Vektoren, Matrizen} + +\subsection{2 Dimensionen} + +Im Zweidimensionalen bestehen Vektoren aus zwei Komponenten, welche in einem Koordinatensystem dargestellt werden kann. + +\begin{figure}[h] + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_1.png} + \caption{Zweidimensionaler Vektor} + \label{fig:vek1} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \[ + \vec{x} = + \begin{bmatrix} + x_1 \\ + x_2 \\ + \end{bmatrix} + \in \mathbb{R} ^2 + .\] + \end{minipage} +\end{figure} + +Die Vektoren $\vec{e}_1$ und $\vec{e}_2$ aus Grafik \ref{fig:vek1} sind die Einheitsvektoren und können einen Vektor als Gleichung darstellen. Des Weiteren werden die Einheitsvektoren für die kanonische Basis wichtig sein. + +\[ + \vec{e}_1 = + \begin{bmatrix} + 1 \\ + 0 \\ + \end{bmatrix}, + \vec{e}_2 = + \begin{bmatrix} + 0 \\ + 1 \\ + \end{bmatrix} + .\] + +\[ + \vec{x} = + \begin{bmatrix} + x_1 \\ + x_2 \\ + \end{bmatrix} = + \begin{bmatrix} + x_1 \\ + 0 \\ + \end{bmatrix} + + \begin{bmatrix} + 0 \\ + x_2 \\ + \end{bmatrix} = + x_1 \cdot \begin{bmatrix} + 1 \\ + 0 \\ + \end{bmatrix} + x_2 \cdot + \begin{bmatrix} + 0 \\ + 1 \\ + \end{bmatrix} = x_1 \cdot \vec{e}_1 + x_2 \cdot \vec{e}_2 + .\] + +Unser Beispiel bestand aus reellen Zahlen $\mathbb{R}$ und befand sich im $\mathbb{R}^2$. Wir können dies jedoch erweitern zu den komplexen Zahlen $\mathbb{C}$. Somit können die Komponenten nun aus Komplexen Zahlen bestehen. Des Weiteren können wir die Dimension erweitern ins Dreidimensionale. + +\subsection{3 Dimensionen} + +Im Vergleich zum Zweidimensionalen bestehen Vektoren im Dreidimensionalen aus drei Komponenten, welche auch in einem Koordinatensystem dargestellt werden kann. + +\begin{figure}[h] + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_2.png} + \caption{Dreidimensionaler Vektor} + \label{fig:vek2} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \[ + \vec{x} = \begin{bmatrix} + x_1 \\ + x_2 \\ + x_3 \\ + \end{bmatrix} + .\] + \end{minipage} +\end{figure} + +Wie in Grafik \ref{fig:vek1} sind die Vektoren $\vec{e_1}, \vec{e_2}$ und $\vec{e_3}$ aus Grafik \ref{fig:vek2} Einheitsvektoren, welche dreidimensionale Vektoren als Gleichung darstellen können. + +\[ + \vec{e_1} = \begin{bmatrix} + 1 \\ + 0 \\ + 0 \\ + \end{bmatrix}, \vec{e_2} = \begin{bmatrix} + 0 \\ + 1 \\ + 0 \\ + \end{bmatrix}, \vec{e_3} = \begin{bmatrix} + 0 \\ + 0 \\ + 1 \\ + \end{bmatrix} + .\] + + +\[ + \vec{x} = \begin{bmatrix} + x_1 \\ + x_2 \\ + x_3 \\ + \end{bmatrix} = x_1 \cdot \begin{bmatrix} + 1 \\ + 0 \\ + 0 \\ + \end{bmatrix} + x_2 \cdot \begin{bmatrix} + 0 \\ + 1 \\ + 0 \\ + \end{bmatrix} + x_3 \cdot \begin{bmatrix} + 0 \\ + 0 \\ + 1 \\ + \end{bmatrix} + .\] + +\nt{ + Die Gleichungen, welche die Vektoren darstellen, nennt man auch eine lineare Kombination von Vektoren. +} + +\subsection{Lineare Kombination} + +\dfn{lineare Kombination \cite{Gradinaru2024}}{ + Die lineare Kombination von Vektoren $\vec{v_1}, ... , \vec{v_n}$ ist: + + \[ + c_1 \cdot \vec{v_1} + c_2 \cdot \vec{v_2} + ... + c_n \vec{v_n} + .\] + + wobei die $c_1, c_2, ... , c_n$ Skalare in $\mathbb{R}$ oder $\mathbb{C}$ sind. +} + +Die lineare Kombination ist eine Summe von Termen, wobei jeder Term ein gestreckter/gestauchter Vektor (d.h. die Multiplikation eines Skalars mit einem Vektor) ist. \cite{Gradinaru2024} + +\subsection{$n$ Dimensionen} + +Vektoren können natürlich mehr als zwei oder drei Komponenten haben. Wir können es auf $n$ Komponenten erweitern und deshalb auf $n$ Dimensionen. Dies bedeutet, dass wir unser Koordinatensystem auf $n$ Dimensionen erweitern. Somit hat der zugehörige Vektor $\vec{x} \in \mathbb{R} ^{n}$ die Komponenten $x_1, x_2, ... , x_{n}$. + +\[ + \vec{x} = \begin{bmatrix} + x_1 \\ + x_2 \\ + \vdots \\ + x_{n} \\ + \end{bmatrix} + .\] + +Vektoren in $n$-dimensionalen Räumen können auch als eine lineare Kombination dargestellt werden. + +\[ + \vec{x} = \begin{bmatrix} + x_1 \\ + x_2 \\ + \vdots \\ + x_{n} \\ + \end{bmatrix} = x_1 \cdot \vec{e_1} + x_2 \cdot \vec{e_2} + ... + x_{n} \cdot \vec{e_n} + .\] + +\dfn{Standardbasis \cite{Gradinaru2024}}{ + Sie Standardbasis (oder auch die kanonische Basis) in diesem $n$-dimensionalen Koordinatensystem besteht aus den folgenden $n$ Vektoren: + + \[ + \vec{e_1} = \begin{bmatrix} + 1 \\ + 0 \\ + \vdots \\ + 0 \\ + \end{bmatrix}, \vec{e_2} = \begin{bmatrix} + 0 \\ + 1 \\ + \vdots \\ + 0 \\ + \end{bmatrix}, ... , \vec{e_j} = \begin{bmatrix} + 0 \\ + \vdots \\ + 1 \\ + \vdots \\ + 0 \\ + \end{bmatrix}, ... , \vec{e_n} = \begin{bmatrix} + 0 \\ + 0 \\ + \vdots \\ + 1 \\ + \end{bmatrix} + .\] +} + +\subsection{Anwendung der linearen Kombination: Die Superposition von Feldern} + +Die lineare Kombination wir vor allem bei der Superposition bzw. der Überlagerung von Kräften verwendet. + +Nehmen wir an, dass 3 Vektoren $\in \mathbb{R} ^{3}$ gegeben sind. + +\[ + \vec{a_1} = \begin{bmatrix} + a_{11} \\ + a_{21} \\ + a_{31} \\ + \end{bmatrix}, \vec{a_2} = \begin{bmatrix} + a_{12} \\ + a_{22} \\ + a_{32} \\ + \end{bmatrix}, \vec{a_3} = \begin{bmatrix} + a_{13} \\ + a_{23} \\ + a_{33} \\ + \end{bmatrix} + .\] + +Jetzt wollen wir die Skalare herausfinden, welche den resultierenden Vektor $\vec{b}$ durch eine lineare Kombination mit den Vektoren $\vec{a_1}, \vec{a_2}, \vec{a_3}$ darstellen. + +\[ + \vec{b} = x_1 \cdot \vec{a_1} + x_2 \cdot \vec{a_2} + x_3 \cdot \vec{a_3} \text{ mit } x_1, x_2, x_3 \in \mathbb{R} + .\] + +Daraus können wir nun ein Gleichungssystem bilden. + +\[ + \left\{ + \begin{array}{lr} + a_{11} \cdot x_1 + a_{12} \cdot x_2 + a_{13} \cdot x_3 = b_1 \\ + a_{21} \cdot x_1 + a_{22} \cdot x_2 + a_{23} \cdot x_3 = b_2 \\ + a_{31} \cdot x_1 + a_{32} \cdot x_2 + a_{33} \cdot x_3 = b_3 \\ + \end{array} + \right. + .\] + +Das Gleichungssystem stellen wir nun wie folgt mit Matrizen und Vektoren dar. + +\[ + \begin{bmatrix} + a_{11} & a_{12} & a_{13} \\ + a_{21} & a_{22} & a_{23} \\ + a_{31} & a_{32} & a_{33} \\ + \end{bmatrix} \cdot \begin{bmatrix} + x_1 \\ + x_2 \\ + x_3 \\ + \end{bmatrix} = \begin{bmatrix} + b_1 \\ + b_2 \\ + b_3 \\ + \end{bmatrix} + .\] + +Die Vektoren $\vec{a_1}, \vec{a_2}, \vec{a_3}$ werden nebeneinander in einer Matrix geschrieben. Die Skalare $x_1, x_2, x_3$ werden als Vektor neben der Matrix geschrieben. Vereinfacht kann man auch $\vec{A} \cdot \vec{x} = \vec{b}$ schreiben. + +\dfn{Lineares Gleichungssystem (LGS) \cite{Gradinaru2024}}{ + Ein lineares Gleichungssystem (LGS) wird kurz geschrieben als: + + \[ + \vec{A} \cdot \vec{x} = \vec{b} + ,\] + + wobei $\vec{A}$ die Koeffizientenmatrix, $\vec{x}$ die Unbekannte und $\vec{b}$ die rechte Seite ist. +} + +\nt{ + Bei der Rechenoperation $\vec{A} \cdot \vec{x}$ handelt es sich um eine Matrix Vektor Multiplikation. +} + +Für lineare Gleichungssysteme gilt: + +\[ + \vec{A} \cdot \vec{x} = \vec{b} \Longleftrightarrow \vec{x} = \vec{A}^{-1} \cdot \vec{b} + .\] + +Die Inverse sowie die Multiplikation von Matrizen wird zu einem späteren Zeitpunkt besprochen. +\\ +\\ +Wichtig zu erwähnen ist, dass es nicht immer eine Inverse einer Matrix gibt, dies bedeutet aber nicht, dass das Gleichungssystem nicht lösbar ist. Um herauszufinden ob ein LGS lösbar ist, führen wir die Kompabilitätsbedingung (KB) ein. Dabei gilt folgendes: + +\begin{itemize} + \item falls $b_1 + b_2 + b_3 \neq 0$, dann gibt es keine Lösung; + \item falls $b_1 + b_2 + b_3 = 0$, dann gibt es unendlich viele Lösungen. +\end{itemize} + +Falls $b_1 + b_2 + b_3 = 0$ gilt, so kann $x_3$ beliebig gewählt werden. diff --git a/hs24/lineare_algebra/lineare_raeume/lineare_raeume.tex b/hs24/lineare_algebra/lineare_raeume/lineare_raeume.tex new file mode 100644 index 0000000..23ad4f0 --- /dev/null +++ b/hs24/lineare_algebra/lineare_raeume/lineare_raeume.tex @@ -0,0 +1,3 @@ +\chapter{Lineare Räume} + +\newpage diff --git a/hs24/lineare_algebra/macros.tex b/hs24/lineare_algebra/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/hs24/lineare_algebra/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/adjugierte_abbildung.tex b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/adjugierte_abbildung.tex new file mode 100644 index 0000000..d6845a4 --- /dev/null +++ b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/adjugierte_abbildung.tex @@ -0,0 +1,2 @@ +\section{Die adjugierte Abbildung} + diff --git a/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/darstellungssatz_von_riesz.tex b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/darstellungssatz_von_riesz.tex new file mode 100644 index 0000000..0b6bed4 --- /dev/null +++ b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/darstellungssatz_von_riesz.tex @@ -0,0 +1,2 @@ +\section{Darstellungssatz von Riesz} + diff --git a/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/gram_schmidt_algorithmus.tex b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/gram_schmidt_algorithmus.tex new file mode 100644 index 0000000..6940a0b --- /dev/null +++ b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/gram_schmidt_algorithmus.tex @@ -0,0 +1,2 @@ +\section{Gram-Schmidt-Algorithmus} + diff --git a/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/norm_und_skalarprodukt_in_linearen_raeumen.tex b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/norm_und_skalarprodukt_in_linearen_raeumen.tex new file mode 100644 index 0000000..5f8f6f7 --- /dev/null +++ b/hs24/lineare_algebra/norm_und_skalarprodukt_in_linearen_raeumen/norm_und_skalarprodukt_in_linearen_raeumen.tex @@ -0,0 +1,7 @@ +\chapter{Norm und Skalarprodukt in linearen Räumen} + +%\input{gram_schmidt_algorithmus.tex} +% \input{darstellungssatz_von_riesz.tex} +% \input{adjugierte_abbildung.tex} + +\newpage diff --git a/hs24/lineare_algebra/notes.tex b/hs24/lineare_algebra/notes.tex new file mode 100644 index 0000000..adcd845 --- /dev/null +++ b/hs24/lineare_algebra/notes.tex @@ -0,0 +1,8 @@ +\import{./lineare_gleichungssysteme}{lineare_gleichungssysteme.tex} +\import{./lineare_raeume}{lineare_raeume.tex} +\import{./lineare_abbildungen}{lineare_abbildungen.tex} +\import{./norm_und_skalarprodukt_in_linearen_raeumen}{norm_und_skalarprodukt_in_linearen_raeumen.tex} +\import{./ausgleichsrechnung}{ausgleichsrechnung.tex} +\import{./determinante}{determinante.tex} +\import{./eigenwertproblem}{eigenwertproblem.tex} +\import{./singulaerwertzerlegung}{singulaerwertzerlegung.tex} diff --git a/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.bbl b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.bbl new file mode 100644 index 0000000..f9eb9ac --- /dev/null +++ b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.bbl @@ -0,0 +1,11 @@ +\begin{thebibliography}{} + +\bibitem[Gradinaru, 2024]{Gradinaru2024} +Gradinaru, Prof.~Dr., V. (2024). +\newblock {Lineare Algebra}. + +\bibitem[{The Manim Community Developers}, 2024]{MCD2024} +{The Manim Community Developers} (2024). +\newblock {Manim - Mathematical Animation Framework}. + +\end{thebibliography} diff --git a/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.pdf b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.pdf new file mode 100644 index 0000000..d0dc96c Binary files /dev/null and b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.pdf differ diff --git a/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.tex b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.tex new file mode 100644 index 0000000..e187e9d --- /dev/null +++ b/hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.tex @@ -0,0 +1,43 @@ +\documentclass{report} + +\def\papertitle{Lineare Algebra} + +\def\theorytitle{Satz} +\def\corollarytitle{Korollar} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Behauptung} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Beispiel} +\def\questiontitle{Frage} +\def\wrongctitle{Falscher Konzept} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\input{disclaimer.tex} +\input{notes.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% BIBLIOGRAPHY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\addcontentsline{toc}{chapter}{References} +\bibliographystyle{apalike} +\bibliography{sources} + +\end{document} diff --git a/hs24/lineare_algebra/preamble.tex b/hs24/lineare_algebra/preamble.tex new file mode 100644 index 0000000..8b0461f --- /dev/null +++ b/hs24/lineare_algebra/preamble.tex @@ -0,0 +1,778 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[ngerman]{babel} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} +\usepackage{caption} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed 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size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + 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\begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/hs24/lineare_algebra/singulaerwertzerlegung/singulaerwertzerlegung.tex b/hs24/lineare_algebra/singulaerwertzerlegung/singulaerwertzerlegung.tex new file mode 100644 index 0000000..5796341 --- /dev/null +++ b/hs24/lineare_algebra/singulaerwertzerlegung/singulaerwertzerlegung.tex @@ -0,0 +1 @@ +\chapter{Singulärwertzerlegung} diff --git a/hs24/lineare_algebra/sources.bib b/hs24/lineare_algebra/sources.bib new file mode 100644 index 0000000..b4a856b --- /dev/null +++ b/hs24/lineare_algebra/sources.bib @@ -0,0 +1,16 @@ +@misc{Gradinaru2024, + author = {Gradinaru, Prof. Dr. ,Vasile}, + month = nov, + title = {{Lineare Algebra}}, + year = {2024}, +} + +@software{MCD2024, + author = {{The Manim Community Developers}}, + license = {MIT}, + month = apr, + title = {{Manim - Mathematical Animation Framework}}, + url = {https://www.manim.community/}, + version = {v0.18.1}, + year = {2024}, +} diff --git a/hs24/nus_I/.DS_Store b/hs24/nus_I/.DS_Store new file mode 100644 index 0000000..c0456b5 Binary files /dev/null and b/hs24/nus_I/.DS_Store differ diff --git a/hs24/nus_I/disclaimer.tex b/hs24/nus_I/disclaimer.tex new file mode 100644 index 0000000..6fefd23 --- /dev/null +++ b/hs24/nus_I/disclaimer.tex @@ -0,0 +1,12 @@ +\section*{DISCLAIMER} + +Diese Notizen wurden verfasst auf Basis der Vorlesung Netzwerke und Schaltungen (HS24) von C. Franck, sowie dem Übungsskript von L. Miller. +\\ +\\ +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). +\\ +\\ +Falls nicht anders deklariert wurden alle Grafiken eigenhändig mit Manim \cite{MCD2024} generiert. +\\ +\\ +Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden. diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/analyse.tex b/hs24/nus_I/einfache_elektrische_netzwerke/analyse.tex new file mode 100644 index 0000000..6f96dbf --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/analyse.tex @@ -0,0 +1,83 @@ +\section{Analyse umfangreicher Netzwerke} + +Wir nehmen an, dass das folgende Netzwerk gegeben ist. + +\includegraphics[width=\linewidth]{fig/Fig64.png} + +Um dieses Netzwerk zu analysieren verwendet man ein System, welches ein Gleichungssystem aufstellt um alle Grössen berechnen zu können. In unserem Fall benötigen wir mindestens 6 Gleichungen, da wir 6 Zweige haben. + +\begin{enumerate} + \item + \begin{minipage}{0.5\linewidth} + Zeichne den Netzwerkgraphen. (Zeichne das elektrische Netzwerk ohne jegliche Komponente) + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig65.png} + \end{minipage} + \item + \begin{minipage}{0.5\linewidth} + Definiere in jedem Zweig die Strom Richtung. Die Richtung kann willkürlich gewählt werden, da die Richtung durch ein Vorzeichen nach dem Auflösen vom Gleichungssystem korrigiert wird. + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig66.png} + \end{minipage} + \item Stelle die Knotengleichungen auf. Eine Knotengleichung kann ignoriert werden da sie linear abhängig ist zu einer der vorherigen Knotengleichungen. + \item Stelle die Maschengleichung auf. Auch hier ist es wichtig, dass alle Maschengleichungen linear unabhängig sind. Für die Maschengleichung gibt es 2 Methoden. + \begin{itemize} + \item + \begin{minipage}{0.5\linewidth} + Bei der Methode vom vollständigen Baum werden die Knoten mit $n$ Zweigen verbunden. ($n$ ist die Anzahl der Knoten) Danach werden die fehlenden Zweige eingesetzt um die Maschen zu bilden. + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig67.png} + \end{minipage} + \item Bei der Methode von der Auftrennung der Maschen wird aus dem Netzwerkgraph eine Masche gebildet und aus der Masche ein Zweig entfernt. Dieser Prozess wird so oft wiederholt, bis keine Maschen mehr gebildet werden können. + + \begin{minipage}{0.33\linewidth} + \includegraphics[width=\linewidth]{fig/Fig68.png} + \end{minipage} + \begin{minipage}{0.33\linewidth} + \includegraphics[width=\linewidth]{fig/Fig69.png} + \end{minipage} + \begin{minipage}{0.33\linewidth} + \includegraphics[width=\linewidth]{fig/Fig70.png} + \end{minipage} + \end{itemize} + \item Fasse die Gleichungen in einer Matrix zusammen. + + \[ + \left. + \begin{array}{rl} + 1 \cdot I_0 + 1 \cdot I_2 + -1 \cdot I_1 = 0 \\ + 1 \cdot I_1 + -1 \cdot I_3 + -1 \cdot I_4 = 0 \\ + 1 \cdot I_3 + 1 \cdot I_5 + -1 \cdot I_2 = 0 \\ + R_1 \cdot I_1 + R_3 \cdot I_3 + R_2 \cdot I_2 = 0 \\ + R_1 \cdot I_1 + R_4 \cdot I_4 = U_0 \\ + -R_2 \cdot I_2 + -R_5 \cdot I_5 = U_0 \\ + \end{array} + \right\} \begin{bmatrix} + 1 & -1 & 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & -1 & 1 & 0 \\ + 0 & 0 & -1 & 1 & 0 & 1 \\ + 0 & 0 & -1 & 1 & 0 & 1 \\ + 0 & R_1 & R_2 & R_3 & 0 & 0 \\ + 0 & 0 & -R_2 & 0 & 0 & -R_5 \\ + \end{bmatrix} + \begin{bmatrix} + I_0 \\ + I_1 \\ + I_2 \\ + I_3 \\ + I_4 \\ + I_5 \\ + \end{bmatrix} = + \begin{bmatrix} + 0 \\ + 0 \\ + 0 \\ + U_0 \\ + U_0 \\ + 0 \\ + \end{bmatrix} + \] +\end{enumerate} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/einfache_elektrische_netzwerke.tex b/hs24/nus_I/einfache_elektrische_netzwerke/einfache_elektrische_netzwerke.tex new file mode 100644 index 0000000..53bda75 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/einfache_elektrische_netzwerke.tex @@ -0,0 +1,11 @@ +\chapter{Einfache elektrische Netzwerke} + +\input{pfeil.tex} +\input{quellen.tex} +\input{kirchhoff.tex} +\input{widerstandsnetzwerke.tex} +\input{rquellen.tex} +\input{wechselwirkung.tex} +\input{netzwerkumwandlung.tex} +\input{superposition.tex} +\input{analyse.tex} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/kirchhoff.tex b/hs24/nus_I/einfache_elektrische_netzwerke/kirchhoff.tex new file mode 100644 index 0000000..592ec23 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/kirchhoff.tex @@ -0,0 +1,28 @@ +\section{Die Kirchhoff'schen Gleichungen} + +\dfn{Kirchhoff'schen Gleichungen}{ + Die Kirchhoff'schen Gleichungen werden vor allem bei der Schaltungstechnik und Netzwerkanalyse verwendet. Dabei spielen zwei Regeln eine sehr wichtige Rolle. + + \begin{itemize} + \item Maschenregel + \item Knotenregel + \end{itemize} +} + +\subsection{Maschen- und Knotenregel} + +Die Maschenregel wird für die Berechnung der Spannung in einer Masche eines komplexen elektrischen Netzwerks verwendet. Dabei gilt + +\begin{equation} + \sum_{k=0}^n U_k = 0 +\end{equation} + +\includegraphics[width=\linewidth]{fig/Fig43.png} + +Die Knotenregel, im Vergleich zur Maschenregel wird zur Berechnung des Stroms eines komplexen elektrischen Netzwerks verwendet. Dabei gilt + +\begin{equation} + \sum_{k=0}^n I_k = 0 +\end{equation} + +\includegraphics[width=\textwidth]{fig/Fig44.png} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/netzwerkumwandlung.tex b/hs24/nus_I/einfache_elektrische_netzwerke/netzwerkumwandlung.tex new file mode 100644 index 0000000..bcf0fdb --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/netzwerkumwandlung.tex @@ -0,0 +1,73 @@ +\section{Netzwerkumwandlungen} + +Zur Umwandlung von Netzwerken verwenden wir den Satz von Thevenin, den Satz von Norton und die Stern-Dreieck-Umwandlung. + +\subsection{Satz von Thevenin und Satz von Norton} + +\dfn{Satz von Thevenin}{ + Der Satz von Thevenin besagt, dass jede Anordnung von Widerständen und Quellen in eine reale Spannungsquelle umgewandelt werden kann. + + \includegraphics[width=\linewidth]{fig/Fig57.png} +} + +\dfn{Satz von Norton}{ + Der Satz von Norton besagt, dass jede Anordnung von Widerständen und Quellen in eine reale Stromquelle umgewandelt werden kann. + + \includegraphics[width=\linewidth]{fig/Fig58.png} +} + +Die Bestimmung der Spannungs- bzw. die Stromquelle benötigt nur die Quellspannung bzw. den Quellstrom, sowie der Innenwiderstand. + +\nt{ + Die Bestimmung vom Innenwiderstand geht mit drei Methoden: + + \begin{itemize} + \item Den Widerstand des elektrischen Netzwerks messen im Leerlauf + \item Thevenin: Kurzschlussstrom berechnen und damit den Innenwiderstand berechnen. + \item Norton: Leerlaufspannung berechnen und damit den Innenwiderstand berechnen. + \end{itemize} +} + +\subsection{Stern-Dreieck-Umwandlung} + +Die folgenden zwei elektrische Netzwerke sind identisch. + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig59.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig60.png} +\end{minipage} + +Für die Berechnung von den jeweiligen Widerständen können die folgenden Formeln von Nutzen sein. + +\begin{minipage}{0.5\linewidth} + + \begin{equation} + R_1 = \frac{R_{31} \cdot R_{12}}{R_{1 2} + R_{2 3} + R_{3 1}} + \end{equation} + + \begin{equation} + R_2 = \frac{R_{1 2} \cdot R_{2 3}}{R_{1 2} + R_{2 3} + R_{3 1}} + \end{equation} + + \begin{equation} + R_3 = \frac{R_{2 3} \cdot R_{3 1}}{R_{1 2} + R_{2 3} + R_{3 1}} + \end{equation} + +\end{minipage} +\begin{minipage}{0.5\linewidth} + + \begin{equation} + R_{1 2} = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_3} + \end{equation} + + \begin{equation} + R_{23} = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_1} + \end{equation} + + \begin{equation} + R_{31} = \frac{R_1 \cdot R_2 + R_2 \cdot R_3 + R_3 \cdot R_1}{R_2} + \end{equation} + +\end{minipage} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/pfeil.tex b/hs24/nus_I/einfache_elektrische_netzwerke/pfeil.tex new file mode 100644 index 0000000..2683db2 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/pfeil.tex @@ -0,0 +1,12 @@ +\section{Zählpfeil} + +\dfn{Zählpfeil}{ + Ein Zählpfeil veranschaulicht in einem elektrischen Schaltkreis die Richtung der Spannung und des Stroms. Dabei gelten die zwei folgenden Regel: + + \begin{itemize} + \item An Verbrauchern ist die Richtung der Spannung gleich der Richtung des Stroms. + \item Bei einer Quelle ist die Richtung der Spannung umgekehrt zur Richtung des Stroms. (Generatorzählpfeilsystem) + \end{itemize} + + \includegraphics[width=\linewidth]{fig/Fig40.png} +} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/quellen.tex b/hs24/nus_I/einfache_elektrische_netzwerke/quellen.tex new file mode 100644 index 0000000..13a8bad --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/quellen.tex @@ -0,0 +1,31 @@ +\section{Strom- und Spannungsquellen} \label{sec:quellen} + +Ein Kondensator, welcher parallel geschaltet ist, mit einem Widerstand kann als Spannungsquelle betrachtet werden, wobei die Energie vom Kondensator entnommen wird und als Wärme beim Widerstand abgegeben wird. + +\includegraphics[width=\linewidth]{fig/Fig41.png} + +Das Problem hierbei ist, dass die Spannung mit der Zeit abnimmt und die Leistung nur begrenzt abgegeben werden kann. Deshalb führen wir die Strom- und Spannungsquelle ein. + +\dfn{Strom- und Spannungsquelle}{ + Eine Strom- bzw. eine Spannungsquelle liefert in einem elektrischen Schaltkreis einen konstanten Strom bzw. eine konstante Spannung. Dabei gilt für die Stromquelle: + + \begin{itemize} + \item Der Ausgangsstrom der Stromquelle ist unabhängig von dem angeschlossenen Netzwerk. + \item Die Ausgangsspannung der Stromquelle hängt von dem angeschlossenen Netzwerk ab. + \end{itemize} + + \cite{Albach2020} + + Für die Spannungsquelle gilt. + + \begin{itemize} + \item Die Ausgangsspannung der Spannungsquelle ist unabhängig von dem angeschlossenen Netzwerk. + \item Ausgangsspannung der Spannungsquelle hängt von dem angeschlossenen Netzwerk ab + \end{itemize} + + \cite{Albach2020} + + Diese Regeln gelten nur für ideale Strom- und Spannungsquellen. +} + +\includegraphics[width=\linewidth]{fig/Fig42.png} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/rquellen.tex b/hs24/nus_I/einfache_elektrische_netzwerke/rquellen.tex new file mode 100644 index 0000000..6fc70b9 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/rquellen.tex @@ -0,0 +1,47 @@ +\section{Reale Strom- und Spannungsquellen} + +In Kapitel \ref{sec:quellen} haben wir uns ideale Strom- und Spannungsquellen angeschaut. Im Vergleich zu den idealen Strom- und Spannungsquellen haben reale einen Innenwiderstand aufgrund von den Strom- und Spannungsabfällen in den Quellen selbst. Dies macht das analysieren von elektrischen Netzwerken wesentlich schwieriger. + +\dfn{Reale Strom- und Spannungsquellen}{ + Reale Strom- und Spannungsquellen sind elektrische Energiequellen wobei die Verluste mit einberechnet werden durch Innenwiderstände. + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig50.png} + \end{minipage} + \begin{minipage}{0.49\linewidth} + $U_0$ ist die Lehrlaufspannung und kann durch die folgende Formel berechnet werden. + + \begin{equation} + U_0 = I_0 \cdot R_i + \end{equation} + + Wenn kein Verbraucher an der Stromquelle angeschlossen wird, so fliesst der ganze Strom $I_0$ über den Innenwiderstand und die Energie wird als Wärme an den Innenwiderstand abgegeben. Somit gilt $I_0 = I_K$ + \end{minipage} + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig51.png} + \end{minipage} + \begin{minipage}{0.49\linewidth} + $U = U_L = U_0$ ist die Lehrlaufspannung. Diese wird gemessen wenn kein Verbraucher an der Spannungsquelle verbunden ist. + \\ + Wenn die Spannungsquelle kurzgeschlossen wird, so fliesst ein Kurzschlussstrom $I_K$. + + \begin{equation} + I_K = \frac{U}{R_i} + \end{equation} + + Die Energie wird als Wärme an den Innenwiderstand abgegeben. + \end{minipage} +} + +\nt{ + Da ein Kurzschluss vermeidet werden soll gilt + + \begin{itemize} + \item Eine Stromquelle soll nicht ohne Verbraucher betrieben werden. + \item Eine Spannungsquelle soll nicht ohne Verbraucher kurzgeschlossen und betrieben werden. + \end{itemize} + + Natürlich ist das nicht realistisch, da in Realität Strom und Spannungsquellen Sicherheitsmechanismen haben um Schäden zu vermeiden. +} + diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/superposition.tex b/hs24/nus_I/einfache_elektrische_netzwerke/superposition.tex new file mode 100644 index 0000000..77df89f --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/superposition.tex @@ -0,0 +1,27 @@ +\section{Das Überlagerungsprinzip} + +Wir lernen nun ein Werkzeug kennen, welches für die elektrische Netzwerkanalyse mit mehreren Quellen eine signifikante Rolle spielt. Nehmen wir an, dass das folgende elektrische Netzwerk gegeben ist. + +\includegraphics[width=\linewidth]{fig/Fig61.png} + +Wir gehen nun die Schritte durch um eine Netzwerkanalyse zu machen. + +\begin{enumerate} + \item Wir teilen das elektrische Netzwerk auf. Im jeden aufgeteilten elektrischen Netzwerk ist jeweils nur eine Quelle enthalten. Dabei gehen wir wie folgt vor: + \begin{itemize} + \item Spannungsquellen werden mit einem Kurzschluss ersetzt + \item Stromquellen werden durch ein Leerlauf ersetzt + \end{itemize} + \includegraphics[width=0.5\linewidth]{fig/Fig62.png} + \includegraphics[width=0.5\linewidth]{fig/Fig63.png} + \item Wir berechnen in allen Teilnetzwerken Ströme und Spannungen. + \item Wir addieren alle Lösungen zusammen. +\end{enumerate} + +\nt{ + Das Überlagerungsprinzip funktioniert nur für den Strom und die Spannung. Die Leistung kann nicht mit dieser Methode berechnet werden da die Leistung keine lineare Beziehung zum Strom hat. +} + +\nt{ + Beachte beim Überlagerungsprinzip die Richtung vom Strom und von der Spannung! +} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/wechselwirkung.tex b/hs24/nus_I/einfache_elektrische_netzwerke/wechselwirkung.tex new file mode 100644 index 0000000..22f9166 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/wechselwirkung.tex @@ -0,0 +1,42 @@ +\section{Wechselwirkungen zwischen Quelle und Verbraucher} + +Im folgenden Kapitel werden elektrische Netzwerke mit verschiedenen Konfigurationen (hauptsächlich reale Quellen und Verbraucher) angeschaut. + +\subsection{Zusammenschaltung von Spannungsquellen} + +\includegraphics[width=\linewidth]{fig/Fig52.png} + +Spannungsquellen können in Serie oder Parallel geschaltet werden. Je nach dem haben sie eine andere Wirkung. + +\begin{itemize} + \item Spannungsquellen in Serie erhöhen die Gesamtspannung + \item Parallel geschaltete Spannungsquellen erhöhen den Strom oder die Kapazität +\end{itemize} + +Jedoch hat das parallelschalten von Spannungsquellen ihre Tücken. Einerseits kann die Leistungsabgabe der Spannungsquellen unterschiedlich sein, wenn die Spannungsquellen unterschiedlich sind. Andererseits kann eine Quelle auch als Verbraucher wirken. Dies kann aber eine willkommende Wirkung sein z.B. bei einer Batterie. + +\subsection{Leistungsanpassung} \label{subsec:LA} + +Man betrachte das folgende elektrische Netzwerk. + +\includegraphics[width=\linewidth]{fig/Fig53.png} + +Ziel ist es $R_L$ zu bestimmen, so dass die Leistung über $R_L$ am grössten ist. Wie kann man nun $R_L$ bestimmen? +\\ +Wie ersparen uns die Rechnerei. (Siehe \cite{Albach2020} S. 144) Aber damit eine Spannungsquelle die maximale Leistung ausgibt, muss $R_L = R_i$ sein. Ist $R_L$ zu gross, so fliesst kein Strom durch $R_L$. Ist $R_L$ zu klein, so haben wir einen Kurzschluss. Daraus entsteht der folgende Graf. + +\includegraphics[width=\linewidth]{fig/Fig54.png} + +In den meisten Fällen ist $R_L$ grösser als $R_i$ da sonst die Hälfte der Leistung über $R_i$ abgegeben wird und $R_i$ in den meisten Fällen nicht dafür gebaut ist. Ausserdem besteht keine Gefahr für einen Kurzschluss. + +\subsection{Wirkungsgrad} + +Der Wirkungsgrad hat eine hohe Relevanz in elektrischen Netzwerken. Sie gibt an, wie viel von der zugeführten Energie als Nutzenergie verwendet wird. Diese wird wie folgt berechnet. + +\begin{equation} + \eta = \frac{P_L}{P_{\text{ges}}} +\end{equation} + +Wenn wir das elektrische Netzwerk in Kapitel \ref{subsec:LA} anschauen so sehen wir, dass wenn $R_L$ zu klein ist die ganze Leistung in Hitze umgewandelt wird (Kurzschluss) und der Wirkungsgrad 0 ist. Wenn $R_L$ am grössten ist entstehen keine Energieverluste. Jedoch kann kein Strom mehr fliessen was auch nicht ideal ist. Deshalb wählt man ein $R_L$ welches den grössten Wirkungsgrad hat und die beste Leistungsanpassung. + +\includegraphics[width=\textwidth]{fig/Fig55.png} diff --git a/hs24/nus_I/einfache_elektrische_netzwerke/widerstandsnetzwerke.tex b/hs24/nus_I/einfache_elektrische_netzwerke/widerstandsnetzwerke.tex new file mode 100644 index 0000000..fa64c32 --- /dev/null +++ b/hs24/nus_I/einfache_elektrische_netzwerke/widerstandsnetzwerke.tex @@ -0,0 +1,102 @@ +\section{Einfache Widerstandsnetzwerke} + +Es gibt 2 verschiedene Arten von Widerstandsnetzwerke und jede davon hat verschiedene Eigenschaften. + +\dfn{Reihengeschaltene Widerstände}{ + \begin{minipage}{0.6\linewidth} + \includegraphics[width=\linewidth]{fig/Fig45.png} + \end{minipage} + \begin{minipage}{0.39\linewidth} + + \begin{equation} + R_{\text{Ges}} = \sum_{k=1}^n R_k + \end{equation} + + \begin{equation} + I_{\text{Ges}} = I_{R_1} = I_{R_2} = I_{R_i} + \end{equation} + + \begin{equation} + U_i = U_{\text{Ges}} \cdot \frac{R_i}{R_{\text{Ges}}} + \label{eq:Rvol} + \end{equation} + \end{minipage} +} + +\dfn{Parallelgeschaltene Widerstände}{ + \begin{minipage}{0.6\linewidth} + \includegraphics[width=\linewidth]{fig/Fig46.png} + \end{minipage} + \begin{minipage}{0.39\linewidth} + + \begin{equation} + \frac{1}{R_{\text{Ges}}} = \sum_{k=1}^n \frac{1}{R_k} + \end{equation} + + \begin{equation} + U_{\text{Ges}} = U_{R_1} = U_{R_2} = U_{R_i} + \end{equation} + + \begin{equation} + I_i = I_{\text{Ges}} \cdot \frac{R_{\text{Ges}}}{R_i} + \label{eq:Rcur} + \end{equation} + \end{minipage} +} + +\subsection{Spannungsteiler} + +Anhand Gleichung \ref{eq:Rvol} erkennt man, dass die Spannung sich bei reihengeschaltene Widerstände sich teilt. Aus dieser Eigenschaft kann man einen Spannungsteiler kreieren. + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig47.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + + \begin{equation} + \frac{U_1}{U_2} = \frac{R_1}{R_2} + \end{equation} + + \begin{equation} + \frac{U_2}{U} = \frac{R_2}{R_1 + R_2} + \end{equation} +\end{minipage} + +Die Gleichungen gelten nur wenn der gleiche Strom durch alle Widerstände fliesst. + +\subsection{Stromteiler} + +Mit Parallelgeschaltene Widerstände kann man ein Stromteiler kreieren. Dies erkennt man anhand der Gleichung \ref{eq:Rcur}. + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig48.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + + \begin{equation} + \frac{I_1}{I_2} = \frac{R_2}{R_1} + \end{equation} + + \begin{equation} + \frac{I_2}{I} = \frac{R_1}{R_1 + R_2} + \end{equation} +\end{minipage} + +Mit diesen Eigenschaften können auch Ströme und Spannungen gemessen werden. + +\nt{ + Ampèremeter und Voltmeter haben einen Innenwiderstand aufgrund der Bauteile und haben dadurch einen Messfehler. Dieser muss natürlich berücksichtigt werden. + + \includegraphics[width=\linewidth]{fig/Fig49.png} +} + +\subsection{Wheatstone Brücke} + +\dfn{Wheatstone Brücke}{ + Die Wheatstone Brücke ist eine Messeinrichtung zur Messung von elektrischen Widerständen und Widerstandsänderungen. + + \includegraphics[width=\linewidth]{fig/Fig56.png} + + Um den Widerstand zu bestimmen verändert man den variablen Widerstand, bis der Voltmeter eine Spannung von 0V anzeigt. Mit der Gleichung lässt sich denn der Widerstand messen. +} + diff --git a/hs24/nus_I/elektrostatische_feld/dielektrika.tex b/hs24/nus_I/elektrostatische_feld/dielektrika.tex new file mode 100644 index 0000000..f902d83 --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/dielektrika.tex @@ -0,0 +1,38 @@ +\section{Die dielektrische Polarisation} + +\dfn{Dielektrika}{ + Ein Dielektrika ist ein Stoff, welches aus Dipolen besteht. Beim Anlegen eines externen elektrischen Feldes richten sich die Dipole nach dem elektrischen Feld. Diese bilden wiederum ein elektrisches Feld, welches den externen elektrischen Feld abschwächt. + + \includegraphics[width=0.5\linewidth]{fig/Fig25.png} + \includegraphics[width=0.5\textwidth]{fig/Fig26.png} + + Diese Abschwächung ist je nach Material verschieden und kann anhand der Dielektrizitätszahl $\epsilon_r$ gekennzeichnet werden. +} + +Für Sprungstellen der Dielektrizitätskonstanten gilt folgendes: + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig27.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + \[ + D_{n 1} = D_{n 2} + .\] + + \[ + \epsilon_1 \cdot E_{n 1} = \epsilon_2 \cdot E_{n 2} \Rightarrow \frac{E_{n 1}}{E_{n 2}} = \frac{\epsilon_1}{\epsilon_2} + .\] + + \[ + E_{t 1} = E_{t 2} + .\] + + \[ + \frac{D_{t 1}}{D_{t 1}} = \frac{\epsilon_1}{\epsilon_2} + .\] +\end{minipage} + +\nt{ + Bei einer sprunghaften Änderung der Dielektrizitätskonstante sind die Normalkomponente der Flussdichte und die Tangentialkomponente der elektrischen Feldstärke stetig. \cite{Albach2020} Die Normalkomponente der elektrischen Feldstärke sowie die Tangentialkomponente der Flussdichte kann als Verhältnis der Dielektrizitätskonstanten berechnet werden. +} + diff --git a/hs24/nus_I/elektrostatische_feld/elektrisches_feld.tex b/hs24/nus_I/elektrostatische_feld/elektrisches_feld.tex new file mode 100644 index 0000000..9f004d8 --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/elektrisches_feld.tex @@ -0,0 +1,59 @@ +\section{Elektrisches Feld und Coulomb'sche Gesetze} + +\dfn{Elektrische Ladung}{ + Eine elektrische Ladung ist ein geladenes Teilchen. Dabei gilt folgendes: + + \begin{itemize} + \item Hat das Teilchen einen Protonenüberschuss, so ist das Teilchen positiv geladen. + \item Hat das Teilchen einen Elektronenüberschuss, so ist das Teilchen negativ geladen. + \end{itemize} + + \includegraphics[width=\linewidth]{fig/Fig15.png} +} + +Elektrische Ladungen können abstossende oder anziehende Kräfte haben. + +\dfn{Kräfte auf Elektrische Ladungen}{ + \includegraphics[width=\linewidth]{fig/Fig16.png} + Ist die Ladung der beiden Teilchen gleich, so ist die Kraft abstossend.\\ + \includegraphics[width=\linewidth]{fig/Fig17.png} + Ist die Ladung der beiden Teilchen unterschiedlich, so ist die Kraft anziehend. +} + +\newpage + +Die Kraft von Elektrischen Ladungen kann in einem Vektorfeld dargestellt werden, wobei an jedem Ort die Kraft eine Magnitude und eine Richtung hat. Dieses Feld nennen wir das Elektrische Feld. + +\includegraphics[width=\linewidth]{fig/Fig18.png} + +\dfn{Elektrisches Feld}{ + Das Elektrische Feld stellt die Kraft einer elektrischen Ladung als Feld dar, wobei man die Magnitude und die Richtung erkennen kann. Sie kann berechnet werden durch: + + \begin{equation} + \vec{E}_1 = \frac{Q_1}{4 \cdot \pi \cdot \epsilon} \cdot \frac{\vec{r}}{r ^{3}} = \frac{Q_1}{4 \cdot \pi \cdot \epsilon} \cdot \frac{\vec{e}_r}{r ^{2}} + \end{equation} + + wobei $\epsilon = \epsilon_0 \cdot \epsilon_r$, + + \includegraphics[width=\linewidth]{fig/Fig19.png} +} + +\newpage + +Die Kraft von Elektrische Ladungen kann auf andere elektrische Ladungen eine Kraft ausüben. Die Kraft, die auf eine Ladung $Q_2$ im Feld der Ladung $Q_1$ wirkt ist wir folgt definiert. \cite{Miller2024} + +\begin{equation} + \vec{F_2} = Q_2 \cdot \vec{E}_1 = \frac{1}{4 \cdot \pi \cdot \epsilon} \cdot \frac{Q_1 \cdot Q_2}{r ^{2}} \cdot \vec{e}_r +\end{equation} + +\includegraphics[width=\linewidth]{fig/Fig20.png} + +\nt{ + Da jede Ladung sein eigenes E-Feld erzeugt wirken auf zwei Punktladungen immer entgegengesetzte Kräfte. \cite{Miller2024} + + \includegraphics[width=\linewidth]{fig/Fig21.png} +} + +Wichtig zu bemerken ist, dass die Kraft der positive Ladungen entlang der Feldlinien wirken, während die Kraft der negativen Ladungen gegen die Feldlinien wirken. + +\includegraphics[width=\linewidth]{fig/Fig22.png} diff --git a/hs24/nus_I/elektrostatische_feld/elektrostatische_feld.tex b/hs24/nus_I/elektrostatische_feld/elektrostatische_feld.tex new file mode 100644 index 0000000..b4cf39d --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/elektrostatische_feld.tex @@ -0,0 +1,10 @@ +\chapter{Elektrostatische Feld} + +\input{elektrisches_feld.tex} +\input{elektrostatische_potenzial.tex} +\input{spannung.tex} +\input{flussdichte.tex} +\input{ladungsdichten.tex} +\input{maxwell.tex} +\input{dielektrika.tex} +\input{kondensator.tex} diff --git a/hs24/nus_I/elektrostatische_feld/elektrostatische_potenzial.tex b/hs24/nus_I/elektrostatische_feld/elektrostatische_potenzial.tex new file mode 100644 index 0000000..9ee9e19 --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/elektrostatische_potenzial.tex @@ -0,0 +1,14 @@ +\section{Elektrostatische Potenzial} + +Wenn wir eine Ladung im elektrischen Feld bewegen, wird Arbeit verrichtet. Diese kann mithilfe vom Integral berechnet werden. Man hat jedoch eine einfachere Methode gefunden. + +\dfn{Elektrostatische Potenzial}{ + Das elektrostatische Potenzial beschreibt die Fähigkeit von einem elektrischen Feld Arbeit an einer elektrischen Ladung zu verrichten. Das heisst, dass die Arbeit an jedem Punkt des elektrischen Feldes durch das Potenzial beschrieben werden kann.\\ + Somit vereinfacht sich die Berechnung der Arbeit von einem Integral zu einer Multiplikation. + + \begin{equation} + W = - \int_{P_0} ^{P_1} \vec{F} d \vec{s} = -Q \int_{P_0} ^{P_1} \vec{E} d \vec{s} = Q \cdot (\varphi_e(P_1) - \varphi_e(P_0)) + \end{equation} +} + +Für das elektrostatische Potenzial gilt des weiteren, dass das Potenzial mit der Distanz zur Ladung abnimmt. Daraus schliesst man, dass das Elektrische Feld die Änderung des elektrisches Potenzials darstellt. diff --git a/hs24/nus_I/elektrostatische_feld/flussdichte.tex b/hs24/nus_I/elektrostatische_feld/flussdichte.tex new file mode 100644 index 0000000..e0fe57c --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/flussdichte.tex @@ -0,0 +1,11 @@ +\section{Elektrische Flussdichte} + +\dfn{Elektrische Flussdichte}{ + Die elektrische Flussdichte beschreibt die die Dichte der elektrischen Feldlinien in Bezug auf eine Fläche. Sie ist eine vektorielle Quantitätsgrösse und zeigt immer in die gleiche Richtung wie $\vec{E}$. + + \begin{equation} + \vec{D}_1 = \epsilon \cdot \vec{E}_1 = \frac{Q_1}{4 \cdot \pi \cdot r ^2} \cdot \vec{e}_r + \end{equation} + + \includegraphics[width=\linewidth]{fig/Fig23.png} +} diff --git a/hs24/nus_I/elektrostatische_feld/kondensator.tex b/hs24/nus_I/elektrostatische_feld/kondensator.tex new file mode 100644 index 0000000..02b82e8 --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/kondensator.tex @@ -0,0 +1,144 @@ +\section{Kondensator} + +\dfn{Kondensator}{ + Der Kondensator kann als kurzzeitiger Energiespeicher betrachtet werden. \cite{Miller2024} Er besteht aus zwei entgegengesetzt geladene metallische Platten, welche ein elektrisches Feld bilden. + + \includegraphics[width=\linewidth]{fig/Fig28.png} +} + +Jeder Kondensator hat eine Kapazität. Diese wird wie folgt definiert. + +\dfn{Kapazität}{ + Die Kapazität ist ein Mass für die Fähigkeit eines Körpers, Ladungen zu speichern. \cite{Albach2020} Sie ist das Verhältnis aus der aufgenommenen Ladung $Q$ zu der angelegten Spannung $U$ und wird mit $C$ gekennzeichnet. + + \begin{equation} + C = \frac{Q}{U} + \label{eq:cap} + \end{equation} +} + +\newpage + +\subsection{Elektrische Flussdichte im Plattenkondensator} + +Wir nehmen immer einen idealen Kondensator an. Das heisst, dass die Felder im Kondensator homogen sind. + +\includegraphics[width=\linewidth]{fig/Fig29.png} + +Wendet man nun das Gauss'sche Gesetz (Kapitel \ref{sec:gauss}) an, so lässt sich die elektrische Flussdichte wie folgt herleiten. + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig30.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + \[ + Q = \oiint \vec{D} d \vec{A} = \iint D d A = D \cdot A + .\] + + \begin{equation} + \Rightarrow D = \frac{Q}{A} + \end{equation} +\end{minipage} + +\newpage + +Des Weiteren gilt für die Spannung + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig31.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + \[ + U = - \int \vec{E} d \vec{s} = E \cdot d + .\] + + \begin{equation} + \Rightarrow E = \frac{U}{d} + \end{equation} +\end{minipage} + +\subsection{Plattenkondensator mit Dielektrikum} + +Für Plattenkondensatoren mit einem Dielektrikum zwischen den geladenen Platten unterscheiden wir zwei Fälle. +\\ +\\ +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig32.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + Wenn das Dielektrikum parallel zur Platte ist wird die elektrische Flussdichte nicht beeinflusst. Da für das elektrische Feld $E$ die Distanz $d$ nicht mehr konstant ist, gelten die folgenden Formel. + + \[ + E = \left\{ + \begin{array}{lr} + \frac{D}{\epsilon_0 \cdot \epsilon_r} = \frac{Q}{A \cdot \epsilon_0 \cdot \epsilon_r} \text{ innerhalb des Dielektrikums} \\ + \frac{D}{\epsilon_0} = \frac{Q}{A \cdot \epsilon_0} \text{ ausserhalb des Dielektrikums} \\ + \end{array} + \right. + .\] +\end{minipage} +\\ +\\ +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig33.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + Wenn das Dielektrikum senkrecht zur Platte ist wird das elektrische Feld nicht beeinflusst. Da für den elektrischen Fluss $D$ die Fläche $A$ nicht mehr konstant ist, gelten die folgenden Formel. + + \[ + D = \left\{ + \begin{array}{lr} + \epsilon_0 \cdot \epsilon_r \cdot E = \frac{\epsilon_0 \cdot \epsilon_r \cdot U}{d} \text{ innerhalb des Dielektrikums} \\ + \epsilon_0 \cdot E = \frac{\epsilon_0 \cdot U}{d} \text{ ausserhalb des Dielektrikums} \\ + \end{array} + \right. + .\] +\end{minipage} +\\ +\\ +Die Formel der Kapazität (Gleichung \ref{eq:cap}) kann nun wie folgt umgeschrieben werden. + +\begin{equation} + C = \frac{Q}{U} = \frac{\oiint \vec{D} d \vec{A}}{\int \vec{E} d \vec{s}} = \frac{D \cdot A}{E \cdot d} = \frac{\epsilon_0 \cdot \epsilon_r \cdot A}{d} +\end{equation} + +Weitere Kapazitäten zeigt die folgende Tabelle. +\\ +\\ +\begin{tabular}{| c | c |} + Kondensator & Formel \\ + \hline + Zylinderkondensator & $\frac{2 \cdot \pi \cdot \epsilon_0 \cdot \epsilon_r}{\ln(\frac{r_2}{r_1})}$ \\ + Kugelkondensator & $4 \cdot \pi \cdot \epsilon_0 \cdot \epsilon_r \cdot \frac{r_1 \cdot r_2}{r_2 - r_1}$ \\ +\end{tabular} +\\ +\\ +Für die Energie, welche der Kondensator speichern kann wird die folgende Formel verwendet. + +\begin{equation} + W = \frac{1}{2} \cdot Q \cdot U = \frac{1}{2} \cdot C \cdot U ^2 +\end{equation} + +\subsection{Kondensator Netzwerke} + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig34.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + Für parallel geschaltete Kondensatoren gilt: + + \begin{equation} + C_{\text{ges}} = C_1 + C_2 + ... + C_n + \end{equation} +\end{minipage} + +\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig35.png} +\end{minipage} +\begin{minipage}{0.3\linewidth} + Für seriell geschaltete Kondensatoren gilt: + + \begin{equation} + \frac{1}{C_{\text{ges}}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n} + \end{equation} +\end{minipage} diff --git a/hs24/nus_I/elektrostatische_feld/ladungsdichten.tex b/hs24/nus_I/elektrostatische_feld/ladungsdichten.tex new file mode 100644 index 0000000..9990f1e --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/ladungsdichten.tex @@ -0,0 +1,15 @@ +\section{Ladungsdichten} + +\dfn{Ladungsdichten}{ + Die Ladungsdichte beschreibt die Verteilung der Ladung über eine Linie, eine Fläche oder ein Volumen. Da Ladungen positiv oder negativ sein können ist dem entsprechend die Ladungsdichte negativ oder positiv. + + \begin{tabular}{| c | c | c |} + Längenladungsdichte & Flächenladungsdichte & Volumenladungsdichte\\ + \hline + $\lambda(x)$ & $\sigma(x,y)$ & $\rho(x,y,z)$\\ + $\int_s \lambda(x) d x = Q$ & $\iint_A \sigma(x,y) d A = Q$ & $\iiint_V \rho(x,y,z) = Q$\\ + \end{tabular} + + \includegraphics[width=\linewidth]{fig/Fig24.png} +} + diff --git a/hs24/nus_I/elektrostatische_feld/maxwell.tex b/hs24/nus_I/elektrostatische_feld/maxwell.tex new file mode 100644 index 0000000..f11c5cc --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/maxwell.tex @@ -0,0 +1,10 @@ +\section{Maxwell Gleichung No. 1 (Gauss'sche Gesetz)} \label{sec:gauss} + +\dfn{Gauss'sche Gesetz}{ + Das Gauss'sche Gesetz beschreibt den elektrischen Fluss durch eine geschlossene Fläche und kann wie folgt berechnet werden. + + \begin{equation} + \oiint_A \vec{D} d \vec{A} = Q + \end{equation} +} + diff --git a/hs24/nus_I/elektrostatische_feld/spannung.tex b/hs24/nus_I/elektrostatische_feld/spannung.tex new file mode 100644 index 0000000..2785486 --- /dev/null +++ b/hs24/nus_I/elektrostatische_feld/spannung.tex @@ -0,0 +1,9 @@ +\section{Spannung} + +\dfn{Spannung}{ + Die elektrische Spannung ist ein Mass für wie viel Arbeit man pro Ladung benötigt, um eine Ladung im elektrischen Feld vom Referenzpotenzial an den gegebenen Ort zu bewegen. \cite{Miller2024} Da die Spannung die Differenz der elektrostatischen Potenziale zweier Punkte ist, kann sie wie folgt berechnet werden. + + \begin{equation} + W_{1 2} = - Q \int_{P_1} ^{P_2} \vec{E} d \vec{s} = Q \cdot U_{12} = Q \cdot (\varphi (P_1) - \varphi (P_2)) + \end{equation} +} diff --git a/hs24/nus_I/fig/.DS_Store b/hs24/nus_I/fig/.DS_Store new file mode 100644 index 0000000..145d542 Binary files /dev/null and b/hs24/nus_I/fig/.DS_Store differ diff --git a/hs24/nus_I/fig/Fig1.png b/hs24/nus_I/fig/Fig1.png new file mode 100644 index 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self.add(charge1, charge2) + + +class ef2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(3, 2 * LEFT) + charge2 = Charge(3, 2 * RIGHT) + + fv1 = LabeledArrow( + r"\vec{F}_e", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * LEFT, + end=4 * LEFT, + color=BLACK, + ) + fv2 = LabeledArrow( + r"\vec{F}_e", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * RIGHT, + end=4 * RIGHT, + color=BLACK, + ) + + self.add(charge1, charge2, fv1, fv2) + + +class ef3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(3, 2 * LEFT) + charge2 = Charge(-3, 2 * RIGHT) + + fv1 = LabeledArrow( + r"\vec{F}_e", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * LEFT, + end=ORIGIN, + color=BLACK, + ) + fv2 = LabeledArrow( + r"\vec{F}_e", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * RIGHT, + end=ORIGIN, + color=BLACK, + ) + + self.add(charge1, charge2, fv1, fv2) + + +class ef4(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge = Charge(3, ORIGIN) + field = ElectricField(charge) + + self.add(charge, field) + + +class ef5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge = Charge(3) + field = ElectricField(charge) + + p = LabeledDot(MathTex("P_1"), color=BLACK).move_to([3, 3, 0]) + + dv1 = LabeledArrow( + r"\vec{r}", + label_color=BLACK, + frame_fill_color=WHITE, + start=ORIGIN, + end=[3, 3, 0], + color=RED, + ) + dv2 = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=ORIGIN, + end=[1.3, 1.3, 0], + color=BLUE, + ) + + self.add( + field, + dv1, + dv2, + charge, + p, + ) + + +class ef7(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(3, 2 * LEFT) + charge2 = Charge(3, 2 * RIGHT) + field = ElectricField(charge1, charge2) + + fv1 = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * RIGHT, + end=4 * RIGHT, + color=BLACK, + ) + fv2 = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * LEFT, + end=4 * LEFT, + color=BLACK, + ) + + self.add( + field, + charge1, + charge2, + fv1, + fv2, + ) + + +class ef6(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(3, 2 * LEFT) + charge2 = Charge(3, 2 * RIGHT) + + fv = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * RIGHT, + end=4 * RIGHT, + color=BLACK, + ) + + self.add(charge1, charge2, fv) + + +class ef8(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(6) + charge2 = Charge(3, 2 * RIGHT) + charge3 = Charge(-3, 2 * LEFT) + field = ElectricField(charge1) + + fv1 = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * RIGHT, + end=4 * RIGHT, + color=BLACK, + ) + fv2 = LabeledArrow( + r"\vec{e}_r", + label_color=BLACK, + frame_fill_color=WHITE, + start=2 * LEFT, + end=ORIGIN, + color=BLACK, + ) + + self.add( + field, + charge1, + charge2, + charge3, + fv1, + fv2, + ) + + +class ef9(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + plate = Polygon( + [0, -2.5, 2.5], + [0, 3.5, 2.5], + [0, 3.5, -1.5], + [0, -2.5, -1.5], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + vec1 = Arrow3D(start=[-2, 0, 0], end=[2, 0, 0], color=BLUE) + vec2 = Arrow3D(start=[-2, -1, 0], end=[2, -1, 0], color=BLUE) + vec3 = Arrow3D(start=[-2, 1, 0], end=[2, 1, 0], color=BLUE) + vec4 = Arrow3D(start=[-2, 0, -1], end=[2, 0, -1], color=BLUE) + vec5 = Arrow3D(start=[-2, 0, 1], end=[2, 0, 1], color=BLUE) + + self.set_camera_orientation(PI / 3, PI / 4) + + self.add(vec1, vec2, vec3, vec4, vec5, plate) + + +class ef10(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + line = Line( + [4, -6, 0], + [2, -2, 0], + color=BLACK, + ) + chargel1 = Charge(1, [4, -6.3, 0]) + chargel2 = Charge(1, [4, -5.8, 0]) + chargel3 = Charge(-1, [2, -2.2, 0]) + chargel4 = Charge(-1, [2, -1.8, 0]) + + plane = Polygon( + [0, -2, 0], + [2, 0, 0], + [0, 2, 0], + [-2, 0, 0], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + chargep1 = Charge(1, [0.5, -1.5, 0]) + chargep2 = Charge(1, [1.5, -0.5, 0]) + chargep3 = Charge(-1, [-1.5, 0.5, 0]) + chargep4 = Charge(-1, [-0.5, 1.5, 0]) + + sphere = Sphere( + [-3, 3, 0], + fill_opacity=0.5, + checkerboard_colors=[BLUE, BLUE], + stroke_width=0, + ) + charges1 = Charge(1, [-2, 2, 0]) + charges2 = Charge(-1, [-4, 4, 0]) + + self.set_camera_orientation(PI / 3, PI / 4) + + self.add( + line, + chargel1, + chargel2, + chargel3, + chargel4, + plane, + chargep1, + chargep2, + chargep3, + chargep4, + sphere, + charges1, + charges2, + ) + + +class ef11(Scene): + def construct(self): + + self.camera.background_color = WHITE + + randc = VGroup() + + x1 = rn.randint(-4, 4) + y1 = rn.randint(-4, 4) + x2 = rn.randint(-4, 4) + y2 = rn.randint(-4, 4) + x3 = rn.randint(-4, 4) + y3 = rn.randint(-4, 4) + x4 = rn.randint(-4, 4) + y4 = rn.randint(-4, 4) + x5 = rn.randint(-4, 4) + y5 = rn.randint(-4, 4) + x6 = rn.randint(-4, 4) + y6 = rn.randint(-4, 4) + x7 = rn.randint(-4, 4) + y7 = rn.randint(-4, 4) + x8 = rn.randint(-4, 4) + y8 = rn.randint(-4, 4) + x9 = rn.randint(-4, 4) + y9 = rn.randint(-4, 4) + x10 = rn.randint(-4, 4) + y10 = rn.randint(-4, 4) + + charge1 = Charge( + 1, + [x1, y1, 0], + ) + charge2 = Charge( + 1, + [x2, y2, 0], + ) + charge3 = Charge( + 1, + [x3, y3, 0], + ) + charge4 = Charge( + 1, + [x4, y4, 0], + ) + charge5 = Charge( + 1, + [x5, y5, 0], + ) + charge6 = Charge( + 1, + [x6, y6, 0], + ) + charge7 = Charge( + 1, + [x7, y7, 0], + ) + charge8 = Charge( + 1, + [x8, y8, 0], + ) + charge9 = Charge( + 1, + [x9, y9, 0], + ) + charge10 = Charge( + 1, + [x10, y10, 0], + ) + field = ElectricField( + charge1, + charge2, + charge3, + charge4, + charge5, + charge6, + charge7, + charge8, + charge9, + charge10, + ) + + plane = Polygon( + [-2, 2, 0], + [2, 2, 0], + [2, -2, 0], + [-2, -2, 0], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + + self.add(field, plane) + + +class ef12(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge = Charge(1, [0, -1000, 0]) + field = ElectricField(charge, color=BLUE, x_range=[-2, 2], y_range=[-2, 1.5]) + + plane = Polygon( + [-2, 2, 0], + [2, 2, 0], + [2, -2, 0], + [-2, -2, 0], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + + vece = Arrow([-4, -2, 0], [-4, 2, 0], color=YELLOW) + + self.add(field, plane, vece) + + +class ef13(Scene): + def construct(self): + + self.camera.background_color = WHITE + + plane1 = Polygon( + [3, 10, 0], + [-10, 10, 0], + [-10, -10, 0], + [-3, -10, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=0.25, + ) + plane2 = Polygon( + [3, 10, 0], + [10, 10, 0], + [10, -10, 0], + [-3, -10, 0], + color=RED, + fill_color=RED, + fill_opacity=0.25, + ) + + vec1 = Arrow([-4, -2, 0], [0, 0, 0], color=GREEN) + vec2 = Arrow([0, 0, 0], [4, 1, 0], color=GREEN) + + text1 = MathTex(r"\epsilon_1", color=BLACK).move_to([-5, 3, 0]) + text2 = MathTex(r"\epsilon_2", color=BLACK).move_to([5, 3, 0]) + + self.add(plane1, plane2, vec1, vec2, text1, text2) + + +class ef15(Scene): + def construct(self): + + self.camera.background_color = WHITE + + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + evec1 = Arrow([-0.8, 2, 0], [0.8, 2, 0], color=RED) + evec2 = Arrow([-0.8, 1, 0], [0.8, 1, 0], color=RED) + evec3 = Arrow([-0.8, 0, 0], [0.8, 0, 0], color=RED) + evec4 = Arrow([-0.8, -1, 0], [0.8, -1, 0], color=RED) + evec5 = Arrow([-0.8, -2, 0], [0.8, -2, 0], color=RED) + + box = Polygon( + [-3, 3, 0], [3, 3, 0], [3, -3, 0], [-3, -3, 0], color=RED, fill_opacity=0 + ) + + self.add(plane1, plane2, evec1, evec2, evec3, evec4, evec5, box) + + +class ef14(Scene): + def construct(self): + + self.camera.background_color = WHITE + + cabell = Line([-5, 0, 0], [-1, 0, 0], color=BLACK) + cabelr = Line([1, 0, 0], [5, 0, 0], color=BLACK) + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + evec1 = Arrow([-0.8, 2, 0], [0.8, 2, 0], color=RED) + evec2 = Arrow([-0.8, 1, 0], [0.8, 1, 0], color=RED) + evec3 = Arrow([-0.8, 0, 0], [0.8, 0, 0], color=RED) + evec4 = Arrow([-0.8, -1, 0], [0.8, -1, 0], color=RED) + evec5 = Arrow([-0.8, -2, 0], [0.8, -2, 0], color=RED) + + text1 = MathTex(r"+", color=BLACK).move_to([-1, 3, 0]) + text2 = MathTex(r"-", color=BLACK).move_to([1, 3, 0]) + + self.add( + cabell, + cabelr, + plane1, + plane2, + evec1, + evec2, + evec3, + evec4, + evec5, + text1, + text2, + ) + + +class ef17(Scene): + def construct(self): + + self.camera.background_color = WHITE + + cabell = Line([-5, 0, 0], [-1, 0, 0], color=BLACK) + cabelr = Line([1, 0, 0], [5, 0, 0], color=BLACK) + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + evec1 = Arrow([-0.8, 2, 0], [0.8, 2, 0], color=RED) + evec2 = Arrow([-0.8, 1, 0], [0.8, 1, 0], color=RED) + evec3 = Arrow([-0.8, 0, 0], [0.8, 0, 0], color=RED) + evec4 = Arrow([-0.8, -1, 0], [0.8, -1, 0], color=RED) + evec5 = Arrow([-0.8, -2, 0], [0.8, -2, 0], color=RED) + + text1 = MathTex(r"+", color=BLACK).move_to([-1, 3, 0]) + text2 = MathTex(r"-", color=BLACK).move_to([1, 3, 0]) + + text3 = MathTex( + r"\vec{E}", + color=RED, + ).move_to([0, 0.5, 0]) + text4 = MathTex(r"\vec{s}", color=ORANGE).move_to([0, -3, 0]) + line = Arrow([1.3, 0, 0], [-1.3, 0, 0], color=ORANGE) + + self.add( + cabell, + cabelr, + plane1, + plane2, + evec1, + evec2, + evec3, + evec4, + evec5, + text1, + text2, + text3, + text4, + line, + ) + + +class ef16(Scene): + def construct(self): + + self.camera.background_color = WHITE + + cabell = Line([-5, 0, 0], [-1, 0, 0], color=BLACK) + cabelr = Line([1, 0, 0], [5, 0, 0], color=BLACK) + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + evec1 = Arrow([-0.8, 2, 0], [0.8, 2, 0], color=RED) + evec2 = Arrow([-0.8, 1, 0], [0.8, 1, 0], color=RED) + evec3 = Arrow([-0.8, 0, 0], [0.8, 0, 0], color=RED) + evec4 = Arrow([-0.8, -1, 0], [0.8, -1, 0], color=RED) + evec5 = Arrow([-0.8, -2, 0], [0.8, -2, 0], color=RED) + + text1 = MathTex(r"+", color=BLACK).move_to([-1, 3, 0]) + text2 = MathTex(r"-", color=BLACK).move_to([1, 3, 0]) + + text3 = MathTex( + r"\vec{D}", + color=RED, + ).move_to([0, 0.5, 0]) + text4 = MathTex(r"A", color=ORANGE).move_to([-0.75, -3, 0]) + line = Line([-0.75, 2, 0], [-0.75, -2, 0], color=ORANGE) + + self.add( + cabell, + cabelr, + plane1, + plane2, + evec1, + evec2, + evec3, + evec4, + evec5, + text1, + text2, + text3, + text4, + line, + ) + + +class ef18(Scene): + def construct(self): + + self.camera.background_color = WHITE + + cabell = Line([-5, 0, 0], [-1, 0, 0], color=BLACK) + cabelr = Line([1, 0, 0], [5, 0, 0], color=BLACK) + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + text1 = MathTex(r"+", color=BLACK).move_to([-1, 3, 0]) + text2 = MathTex(r"-", color=BLACK).move_to([1, 3, 0]) + + diel = Polygon( + [-1, 2, 0], + [0, 2, 0], + [0, -2, 0], + [-1, -2, 0], + color=GREEN, + fill_color=GREEN, + fill_opacity=0.75, + ) + + self.add(diel, cabell, cabelr, plane1, plane2, text1, text2) + + +class ef19(Scene): + def construct(self): + + self.camera.background_color = WHITE + + cabell = Line([-5, 0, 0], [-1, 0, 0], color=BLACK) + cabelr = Line([1, 0, 0], [5, 0, 0], color=BLACK) + plane1 = Line([-1, 2, 0], [-1, -2, 0], color=RED) + plane2 = Line([1, 2, 0], [1, -2, 0], color=BLUE) + + text1 = MathTex(r"+", color=BLACK).move_to([-1, 3, 0]) + text2 = MathTex(r"-", color=BLACK).move_to([1, 3, 0]) + + diel = Polygon( + [-1, 2, 0], + [1, 2, 0], + [1, 0, 0], + [-1, 0, 0], + color=GREEN, + fill_color=GREEN, + fill_opacity=0.75, + ) + + self.add(diel, cabell, cabelr, plane1, plane2, text1, text2) + + +class ef20(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c0 = ( + Capacitor( + label="", + ) + .move_to([0, 2, 0]) + .rotate(90 * DEGREES) + ) + c1 = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([-1, 0, 0]) + .set_color(BLACK) + ) + c2 = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([0, 0, 0]) + .set_color(BLACK) + ) + cn = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([1, 0, 0]) + .set_color(BLACK) + ) + gnd = Ground(ground_type="earth").move_to([0, -2, 0]) + + circuit = Circuit() + self.add(circuit) + circuit.add_components(c1, c2, cn) + + circuit.add_wire(c0.get_terminals("left"), c2.get_terminals("right")) + circuit.add_wire( + c1.get_terminals("right"), + [c1.get_terminals("right")[0], 1, 0], + ) + circuit.add_wire( + cn.get_terminals("right"), + [cn.get_terminals("right")[0], 1, 0], + ) + circuit.add_wire( + [c1.get_terminals("right")[0], 1, 0], [c2.get_terminals("right")[0], 1, 0] + ) + circuit.add_wire( + [c2.get_terminals("right")[0], 1, 0], [cn.get_terminals("right")[0], 1, 0] + ) + circuit.add_wire(c1.get_terminals("left"), [c1.get_terminals("left")[0], -1, 0]) + circuit.add_wire(c2.get_terminals("left"), gnd.get_terminals()) + circuit.add_wire(cn.get_terminals("left"), [cn.get_terminals("left")[0], -1, 0]) + circuit.add_wire( + [c1.get_terminals("left")[0], -1, 0], [c2.get_terminals("left")[0], -1, 0] + ) + circuit.add_wire( + [cn.get_terminals("left")[0], -1, 0], [c2.get_terminals("left")[0], -1, 0] + ) + + circuit.set_color(BLACK) + + +class ef21(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c0 = ( + Capacitor( + label="", + ) + .move_to([0, 2, 0]) + .rotate(90 * DEGREES) + ) + c1 = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([0, 1, 0]) + .set_color(BLACK) + ) + c2 = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([0, 0, 0]) + .set_color(BLACK) + ) + cn = ( + Capacitor( + label="", + polarized=False, + ) + .rotate(90 * DEGREES) + .move_to([0, -1, 0]) + .set_color(BLACK) + ) + gnd = Ground(ground_type="earth").move_to([0, -2, 0]) + + circuit = Circuit() + self.add(circuit) + circuit.add_components(c1, c2, cn) + + circuit.add_wire(c0.get_terminals("left"), c1.get_terminals("right")) + circuit.add_wire(c1.get_terminals("left"), c2.get_terminals("right")) + circuit.add_wire(c2.get_terminals("left"), cn.get_terminals("right")) + circuit.add_wire(cn.get_terminals("left"), gnd.get_terminals()) + + circuit.set_color(BLACK) + + +class magnet1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(-1, LEFT) + charge2 = Charge(1, RIGHT) + field = ElectricField(charge1, charge2) + fieldI = Arrow([-1, 0, 0], [1, 0, 0], color=YELLOW) + + magnetP = Polygon( + [-1.5, 0.5, 0], + [-1.5, -0.5, 0], + [0, -0.5, 0], + [0, 0.5, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + textP = Text("S").move_to([-1, 0, 0]) + magnetN = Polygon( + [1.5, 0.5, 0], + [1.5, -0.5, 0], + [0, -0.5, 0], + [0, 0.5, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textN = Text("N").move_to([1, 0, 0]) + + self.add(field, magnetP, magnetN, fieldI, textP, textN) + + +class ssf1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(1, 1 * LEFT) + charge2 = Charge(-1, 1 * RIGHT) + wire = Line(1 * LEFT, 1 * RIGHT, color=BLACK) + + vec = LabeledArrow( + "I", + label_color=BLACK, + frame_fill_color=WHITE, + start=0.75 * LEFT + 0.5 * UP, + end=0.75 * RIGHT + 0.5 * UP, + color=BLACK, + ) + + self.add(charge1, charge2, wire, vec) + + +class ssf2(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + plate = Polygon( + [0, -2.5, 2.5], + [0, 2.5, 2.5], + [0, 2.5, -2.5], + [0, -2.5, -2.5], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + vec1 = Arrow3D(start=[-2, 0, 0], end=[2, 0, 0], color=BLUE) + vec2 = Arrow3D(start=[-2, -1, 0], end=[2, -1, 0], color=BLUE) + vec3 = Arrow3D(start=[-2, 1, 0], end=[2, 1, 0], color=BLUE) + vec4 = Arrow3D(start=[-2, 0, -1], end=[2, 0, -1], color=BLUE) + vec5 = Arrow3D(start=[-2, 0, 1], end=[2, 0, 1], color=BLUE) + + label1 = ( + MathTex( + r"\Delta A", + color=BLACK, + ) + .rotate(160 * DEGREES, [0, 1, 1]) + .move_to([-2, 0, 2]) + ) + label2 = ( + MathTex(r"\vec{J}", color=BLUE) + .rotate(160 * DEGREES, [0, 1, 1]) + .move_to([0, 0, 0.3]) + ) + + self.set_camera_orientation(PI / 3, PI / 4) + + self.add(vec1, vec2, vec3, vec4, vec5, plate, label1, label2) + + +class ssf3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c0 = Charge(1, [1000, 0, 0]) + ef = ElectricField(c0) + + c1 = Charge(1, [2, 3, 0]) + c2 = Charge(1, [-2, 2, 0]) + c3 = Charge(1, [1, -3, 0]) + c4 = Charge(1, [-1, -1, 0]) + c5 = Charge(1, [2, -1, 0]) + + c6 = Charge(-1, [2, 0, 0]) + c7 = Charge(-1, [-4, -1, 0]) + c8 = Charge(-1, [0, -2, 0]) + c9 = Charge(-1, [3, -2, 0]) + c10 = Charge(-1, [4, 2, 0]) + + a1 = Arrow([2, 0, 0], [3, 2, 0], color=BLACK, path_arc=-90 * DEGREES) + a2 = Arrow([-4, -1, 0], [-3, 1, 0], color=BLACK, path_arc=-90 * DEGREES) + a3 = Arrow([4, 2, 0], [5, 4, 0], color=BLACK, path_arc=-90 * DEGREES) + a4 = Arrow([0, -2, 0], [1, 0, 0], color=BLACK, path_arc=-90 * DEGREES) + a5 = Arrow([3, -2, 0], [4, 0, 0], color=BLACK, path_arc=-90 * DEGREES) + + self.add(ef, a1, a2, a3, a4, a5, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10) + + +class ssf4(Scene): + def construct(self): + + self.camera.background_color = WHITE + + plane1 = Polygon( + [3, 10, 0], + [-10, 10, 0], + [-10, -10, 0], + [-3, -10, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=0.25, + ) + plane2 = Polygon( + [3, 10, 0], + [10, 10, 0], + [10, -10, 0], + [-3, -10, 0], + color=RED, + fill_color=RED, + fill_opacity=0.25, + ) + + vec1 = Arrow([-4, -2, 0], [0, 0, 0], color=GREEN) + vec2 = Arrow([0, 0, 0], [4, 1, 0], color=GREEN) + + text1 = MathTex(r"\kappa_1", color=BLACK).move_to([-5, 3, 0]) + text2 = MathTex(r"\kappa_2", color=BLACK).move_to([5, 3, 0]) + + self.add(plane1, plane2, vec1, vec2, text1, text2) + + +class een1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + r1 = Polygon( + [-0.5, 1, 0], [-0.5, 1.5, 0], [0.5, 1.5, 0], [0.5, 1, 0], color=BLACK + ) + r1c1 = Line([-2, 1.25, 0], [-0.5, 1.25, 0], color=BLACK) + r1c2 = Line([0.5, 1.25, 0], [2, 1.25, 0], color=BLACK) + i1 = Arrow([0.75, 1.25, 0], [1.75, 1.25, 0], color=RED) + vo1 = Arrow([-0.5, 1.75, 0], [0.5, 1.75, 0], color=BLUE) + i = MathTex(r"I", color=RED).move_to([1.25, 1.75, 0]) + v = MathTex(r"V", color=BLUE).move_to([0, 2.25, 0]) + + v1 = VoltageSource(direction=UP).rotate(-90 * DEGREES).move_to([0, -1, 0]) + i2 = Arrow( + [0.75, v1.get_terminals("positive")[1], 0], + [1.75, v1.get_terminals("positive")[1], 0], + color=RED, + ) + vo2 = Arrow([0.5, -0.25, 0], [-0.5, -0.25, 0], color=BLUE) + + circuit = Circuit() + + self.add(r1, r1c1, r1c2, i1, vo1, i, v, circuit, i2, vo2) + circuit.add_components(v1) + + circuit.add_wire( + [-2, v1.get_terminals("negative")[1], 0], v1.get_terminals("negative") + ) + circuit.add_wire( + [2, v1.get_terminals("positive")[1], 0], v1.get_terminals("positive") + ) + + circuit.set_color(BLACK) + + +class een2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = ( + Capacitor(label=False, direction=LEFT) + .move_to([-1, 0, 0]) + .rotate(90 * DEGREES) + ) + r1 = Polygon( + [0.75, 0.5, 0], + [1.25, 0.5, 0], + [1.25, -0.5, 0], + [0.75, -0.5, 0], + color=BLACK, + ) + + circuit = Circuit() + + self.add(circuit, r1) + + circuit.add_components(c1) + + circuit.add_wire( + c1.get_terminals("right"), + [c1.get_terminals("right")[0], 1.5, 0], + ) + circuit.add_wire( + [c1.get_terminals("right")[0], 1.5, 0], [1, 1.5, 0], junctions=False + ) + circuit.add_wire([1, 1.5, 0], [1, 0.5, 0], junctions=False) + circuit.add_wire([1, -0.5, 0], [1, -1.5, 0]) + circuit.add_wire( + [1, -1.5, 0], [c1.get_terminals("left")[0], -1.5, 0], junctions=False + ) + circuit.add_wire( + [c1.get_terminals("left")[0], -1.5, 0], + c1.get_terminals("left"), + junctions=False, + ) + + circuit.set_color(BLACK) + + +class een3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().move_to([-1.5, 0, 0]) + i1 = CurrentSource().move_to([1.5, 0, 0]) + vo1 = Arrow([-0.25, 1.5, 0], [-0.25, -1.5, 0], color=BLUE) + iv1 = Arrow( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 1.5, 0], + color=RED, + ) + + circuit = Circuit() + + self.add(circuit, vo1, iv1) + + circuit.add_components(v1, i1) + + circuit.add_wire( + v1.get_terminals("positive"), [v1.get_terminals("positive")[0], 1.5, 0] + ) + circuit.add_wire( + v1.get_terminals("negative"), [v1.get_terminals("negative")[0], -1.5, 0] + ) + circuit.add_wire( + i1.get_terminals("positive"), [i1.get_terminals("positive")[0], 1.5, 0] + ) + circuit.add_wire( + i1.get_terminals("negative"), [i1.get_terminals("negative")[0], -1.5, 0] + ) + + circuit.set_color(BLACK) + + +class een4(Scene): + def construct(self): + + self.camera.background_color = WHITE + + p = Dot([-1.5, 1.5, 0], radius=0.08, color=BLACK) + r1 = Polygon( + [-0.5, 1.25, 0], + [-0.5, 1.75, 0], + [0.5, 1.75, 0], + [0.5, 1.25, 0], + color=BLACK, + ) + r1c1 = Line([-1.5, 1.5, 0], [-0.5, 1.5, 0], color=BLACK) + r1c2 = Line([1.5, 1.5, 0], [0.5, 1.5, 0], color=BLACK) + r2 = Polygon( + [1.25, 0.5, 0], + [1.75, 0.5, 0], + [1.75, -0.5, 0], + [1.25, -0.5, 0], + color=BLACK, + ) + r2c1 = Line([1.5, 1.5, 0], [1.5, 0.5, 0], color=BLACK) + r2c2 = Line([1.5, -0.5, 0], [1.5, -1.5, 0], color=BLACK) + r3 = Polygon( + [-0.5, -1.25, 0], + [-0.5, -1.75, 0], + [0.5, -1.75, 0], + [0.5, -1.25, 0], + color=BLACK, + ) + r3c1 = Line([1.5, -1.5, 0], [0.5, -1.5, 0], color=BLACK) + r3c2 = Line([-1.5, -1.5, 0], [-0.5, -1.5, 0], color=BLACK) + r4 = Polygon( + [-1.25, -0.5, 0], + [-1.75, -0.5, 0], + [-1.75, 0.5, 0], + [-1.25, 0.5, 0], + color=BLACK, + ) + r4c1 = Line([-1.5, -1.5, 0], [-1.5, -0.5, 0], color=BLACK) + r4c2 = Line([-1.5, 1.5, 0], [-1.5, 0.5, 0], color=BLACK) + + pt = MathTex(r"P", color=BLACK).move_to([-1.5, 2, 0]) + u1t = MathTex(r"U_1", color=BLUE).move_to([0, 2.5, 0]) + u2t = MathTex(r"U_2", color=BLUE).move_to([2.5, 0, 0]) + u3t = MathTex(r"U_3", color=BLUE).move_to([0, -2.5, 0]) + u4t = MathTex(r"U_4", color=BLUE).move_to([-2.5, 0, 0]) + r1t = MathTex(r"R_1", color=BLACK).move_to([0, 1, 0]) + r2t = MathTex(r"R_2", color=BLACK).move_to([1, 0, 0]) + r3t = MathTex(r"R_3", color=BLACK).move_to([0, -1, 0]) + r4t = MathTex(r"R_4", color=BLACK).move_to([-1, 0, 0]) + + u1a = Arrow([-0.5, 2, 0], [0.5, 2, 0], color=BLUE) + u2a = Arrow([2, 0.5, 0], [2, -0.5, 0], color=BLUE) + u3a = Arrow([0.5, -2, 0], [-0.5, -2, 0], color=BLUE) + u4a = Arrow([-2, -0.5, 0], [-2, 0.5, 0], color=BLUE) + + self.add( + p, + r1, + r1c1, + r1c2, + r2, + r2c1, + r2c2, + r3, + r3c1, + r3c2, + r4, + r4c1, + r4c2, + pt, + u1t, + u2t, + u3t, + u4t, + r1t, + r2t, + r3t, + r4t, + u1a, + u2a, + u3a, + u4a, + ) + + +class een5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + p = Dot(ORIGIN, radius=0.08, color=BLACK) + c1 = Line([0, 2, 0], ORIGIN, color=BLACK) + c2 = Line([2, 0, 0], ORIGIN, color=BLACK) + c3 = Line([0, -2, 0], ORIGIN, color=BLACK) + c4 = Line([-2, 0, 0], ORIGIN, color=BLACK) + + ia1 = Arrow([0, 1.5, 0], [0, 0.5, 0], color=RED) + ia2 = Arrow([1.5, 0, 0], [0.5, 0, 0], color=RED) + ia3 = Arrow([0, -1.5, 0], [0, -0.5, 0], color=RED) + ia4 = Arrow([-0.5, 0, 0], [-1.5, 0, 0], color=RED) + + pt = MathTex(r"P", color=BLACK).move_to([0.5, -0.5, 0]) + it1 = MathTex(r"I_1", color=RED).move_to([0.5, 1, 0]) + it2 = MathTex(r"I_2", color=RED).move_to([1, 0.5, 0]) + it3 = MathTex(r"I_3", color=RED).move_to([0.5, -1, 0]) + it4 = MathTex(r"I_4", color=RED).move_to([-1, 0.5, 0]) + + self.add(p, c1, c2, c3, c4, ia1, ia2, ia3, ia4, pt, it1, it2, it3, it4) + + +class een6(Scene): + def construct(self): + + self.camera.background_color = WHITE + + r1 = Polygon( + [-0.25, 2.5, 0], + [0.25, 2.5, 0], + [0.25, 1.5, 0], + [-0.25, 1.5, 0], + color=BLACK, + ) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r3 = Polygon( + [-0.25, -2.5, 0], + [0.25, -2.5, 0], + [0.25, -1.5, 0], + [-0.25, -1.5, 0], + color=BLACK, + ) + c1 = Line([0, 3.5, 0], [0, 2.5, 0], color=BLACK) + c2 = Line([0, 1.5, 0], [0, 0.5, 0], color=BLACK) + c3 = Line([0, -0.5, 0], [0, -1.5, 0], color=BLACK) + c4 = Line([0, -2.5, 0], [0, -3.5, 0], color=BLACK) + + r1t = MathTex(r"R_1", color=BLACK).move_to([1, 2, 0]) + r2t = MathTex(r"R_2", color=BLACK).move_to([1, 0, 0]) + r3t = MathTex(r"R_3", color=BLACK).move_to([1, -2, 0]) + + self.add(r1, r2, r3, c1, c2, c3, c4, r1t, r2t, r3t) + + +class een7(Scene): + def construct(self): + + self.camera.background_color = WHITE + + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r2 = Polygon( + [-1.25, 0.5, 0], + [-0.75, 0.5, 0], + [-0.75, -0.5, 0], + [-1.25, -0.5, 0], + color=BLACK, + ) + r3 = Polygon( + [1.25, 0.5, 0], + [0.75, 0.5, 0], + [0.75, -0.5, 0], + [1.25, -0.5, 0], + color=BLACK, + ) + + c1 = Line([0, 2.5, 0], [0, 0.5, 0], color=BLACK) + c2 = Line([-1, 1.5, 0], [1, 1.5, 0], color=BLACK) + c3 = Line([-1, 1.5, 0], [-1, 0.5, 0], color=BLACK) + c4 = Line([1, 1.5, 0], [1, 0.5, 0], color=BLACK) + c5 = Line([0, -2.5, 0], [0, -0.5, 0], color=BLACK) + c6 = Line([-1, -1.5, 0], [1, -1.5, 0], color=BLACK) + c7 = Line([-1, -1.5, 0], [-1, -0.5, 0], color=BLACK) + c8 = Line([1, -1.5, 0], [1, -0.5, 0], color=BLACK) + j1 = Dot([0, 1.5, 0], radius=0.08, color=BLACK) + j2 = Dot([0, -1.5, 0], radius=0.08, color=BLACK) + + self.add(r1, r2, r3, c1, c2, c3, c4, c5, c6, c7, c8, j1, j2) + + +class een8(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().move_to([-2, 0, 0]).set_color(BLACK) + r1 = Polygon( + [1.75, 1.5, 0], [2.25, 1.5, 0], [2.25, 0.5, 0], [1.75, 0.5, 0], color=BLACK + ) + r2 = Polygon( + [1.75, -1.5, 0], + [2.25, -1.5, 0], + [2.25, -0.5, 0], + [1.75, -0.5, 0], + color=BLACK, + ) + c1 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 2.5, 0], + color=BLACK, + ) + c2 = Line([v1.get_terminals("positive")[0], 2.5, 0], [2, 2.5, 0], color=BLACK) + c3 = Line([2, 2.5, 0], [2, 1.5, 0], color=BLACK) + c4 = Line([2, 0.5, 0], [2, -0.5, 0], color=BLACK) + c5 = Line([2, -2.5, 0], [2, -1.5, 0], color=BLACK) + c6 = Line([v1.get_terminals("positive")[0], -2.5, 0], [2, -2.5, 0], color=BLACK) + c7 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("positive")[0], -2.5, 0], + color=BLACK, + ) + + r1t = MathTex(r"R_1", color=BLACK).move_to([1, 1, 0]) + r2t = MathTex(r"R_2", color=BLACK).move_to([1, -1, 0]) + u1t = MathTex(r"U_1", color=BLUE).move_to([3, 1, 0]) + u2t = MathTex(r"U_2", color=BLUE).move_to([3, -1, 0]) + + u1a = Arrow([2.5, 1.5, 0], [2.5, 0.5, 0], color=BLUE) + u2a = Arrow([2.5, -0.5, 0], [2.5, -1.5, 0], color=BLUE) + + self.add(v1, r1, r2, c1, c2, c3, c4, c5, c6, c7, r1t, r2t, u1t, u2t, u1a, u2a) + + +class een9(Scene): + def construct(self): + + self.camera.background_color = WHITE + + i1 = CurrentSource().move_to([-4, 0, 0]).set_color(BLACK) + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r2 = Polygon( + [3.75, 0.5, 0], + [4.25, 0.5, 0], + [4.25, -0.5, 0], + [3.75, -0.5, 0], + color=BLACK, + ) + + c1 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 1.5, 0], + color=BLACK, + ) + c2 = Line([i1.get_terminals("positive")[0], 1.5, 0], [4, 1.5, 0], color=BLACK) + c3 = Line([0, 1.5, 0], [0, 0.5, 0], color=BLACK) + c4 = Line([4, 1.5, 0], [4, 0.5, 0], color=BLACK) + c5 = Line([4, -0.5, 0], [4, -1.5, 0], color=BLACK) + c6 = Line([4, -1.5, 0], [i1.get_terminals("negative")[0], -1.5, 0], color=BLACK) + c7 = Line([0, -1.5, 0], [0, -0.5, 0], color=BLACK) + c8 = Line( + i1.get_terminals("negative"), + [i1.get_terminals("negative")[0], -1.5, 0], + color=BLACK, + ) + j1 = Dot([0, 1.5, 0], radius=0.08, color=BLACK) + j2 = Dot([0, -1.5, 0], radius=0.08, color=BLACK) + + i1a = Arrow([0, -0.5, 0], [0, -1.5, 0], color=RED) + i2a = Arrow([4, -0.5, 0], [4, -1.5, 0], color=RED) + + r1t = MathTex( + r"R_1", + color=BLACK, + ).move_to([1, 0, 0]) + r2t = MathTex( + r"R_2", + color=BLACK, + ).move_to([5, 0, 0]) + + self.add(i1, r1, r2, c1, c2, c3, c4, c5, c6, c7, c8, j1, j2, i1a, i2a, r1t, r2t) + + +class een10(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().move_to([-2, -2, 0]).set_color(BLACK) + mbody1 = Circle(radius=0.5, color=BLACK) + mbody2 = Circle(radius=0.5, color=BLACK) + marrow1 = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=BLACK) + marrow2 = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=BLACK) + m1 = VGroup(mbody1, marrow1).move_to([[-1.75, -2, 0]]) + m2 = VGroup(mbody2, marrow2).move_to([2.25, 0, 0]) + + r1 = Polygon( + [-2, 0.5, 0], [-2, 1.5, 0], [-1.5, 1.5, 0], [-1.5, 0.5, 0], color=BLACK + ) + i1 = CurrentSource().move_to([2, 0, 0]).set_color(BLACK) + r2 = Polygon( + [3.75, 0.5, 0], + [3.75, -0.5, 0], + [4.25, -0.5, 0], + [4.25, 0.5, 0], + color=BLACK, + ) + + c1 = Line( + [v1.get_terminals("negative")[0], -3, 0], + v1.get_terminals("negative"), + color=BLACK, + ) + c2 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 0.5, 0], + color=BLACK, + ) + c3 = Line( + [v1.get_terminals("positive")[0], 1.5, 0], + [v1.get_terminals("positive")[0], 3, 0], + color=BLACK, + ) + c4 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c5 = Line([i1.get_terminals("positive")[0], 2, 0], [5, 2, 0], color=BLACK) + c6 = Line([4, 2, 0], [4, 0.5, 0], color=BLACK) + c7 = Line([4, -0.5, 0], [4, -2, 0], color=BLACK) + c8 = Line([5, -2, 0], [i1.get_terminals("negative")[0], -2, 0], color=BLACK) + c9 = Line( + [i1.get_terminals("negative")[0], -2, 0], + i1.get_terminals("negative"), + color=BLACK, + ) + + r1t = MathTex(r"R_i", color=BLACK).move_to([-1.25, 1, 0]) + r2t = MathTex(r"R_i", color=BLACK).move_to([4.5, 0, 0]) + + self.add(m1, r1, m2, r2, c1, c2, c3, c4, c5, c6, c7, c8, c9, r1t, r2t) + + +class een11(Scene): + def construct(self): + + self.camera.background_color = WHITE + + i1 = CurrentSource().move_to([-1, 0, 0]).set_color(BLACK) + ri = Polygon( + [0.75, 0.5, 0], + [1.25, 0.5, 0], + [1.25, -0.5, 0], + [0.75, -0.5, 0], + color=BLACK, + ) + + c1 = Line( + [i1.get_terminals("negative")[0], -2, 0], + i1.get_terminals("negative"), + color=BLACK, + ) + c2 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c3 = Line([i1.get_terminals("positive")[0], 2, 0], [2, 2, 0], color=BLACK) + c4 = Line([1, 2, 0], [1, 0.5, 0], color=BLACK) + c5 = Line([1, -2, 0], [1, -0.5, 0], color=BLACK) + c6 = Line([i1.get_terminals("positive")[0], -2, 0], [2, -2, 0], color=BLACK) + + j1 = Dot(radius=0.08, color=BLACK).move_to([2, 2, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([2, -2, 0]) + + rit = MathTex(r"R_i", color=BLACK).move_to([1.5, 0, 0]) + + self.add(i1, ri, c1, c2, c3, c4, c5, c6, j1, j2, rit) + + +class een12(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().move_to([-0.25, -1, 0]).set_color(BLACK) + ri = Polygon( + [-0.25, 0.5, 0], + [-0.25, 1.5, 0], + [0.25, 1.5, 0], + [0.25, 0.5, 0], + color=BLACK, + ) + + c1 = Line([0, -3, 0], v1.get_terminals("negative"), color=BLACK) + c2 = Line(v1.get_terminals("positive"), [0, 0.5, 0], color=BLACK) + c3 = Line([0, 1.5, 0], [0, 3, 0], color=BLACK) + + j1 = Dot(radius=0.08, color=BLACK).move_to([0, 3, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([0, -3, 0]) + + rit = MathTex(r"R_i", color=BLACK).move_to([0.5, 1, 0]) + + self.add(v1, ri, c1, c2, c3, j1, j2, rit) + + +class een13(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().set_color(BLACK).move_to([-2, -1, 0]) + v2 = VoltageSource().set_color(BLACK).move_to([-2, 1, 0]) + v3 = VoltageSource().set_color(BLACK).move_to([1, 0, 0]) + v4 = VoltageSource().set_color(BLACK).move_to([3, 0, 0]) + + c1 = Line( + [v1.get_terminals("negative")[0], -3, 0], + v1.get_terminals("negative"), + color=BLACK, + ) + c2 = Line( + v2.get_terminals("negative"), v1.get_terminals("positive"), color=BLACK + ) + c3 = Line( + v2.get_terminals("positive"), + [v2.get_terminals("positive")[0], 3, 0], + color=BLACK, + ) + c4 = Line( + [v3.get_terminals("negative")[0], -2, 0], + v3.get_terminals("negative"), + color=BLACK, + ) + c5 = Line( + [v3.get_terminals("positive")[0], 2, 0], + v3.get_terminals("positive"), + color=BLACK, + ) + c6 = Line( + [v3.get_terminals("positive")[0], 2, 0], + [4, 2, 0], + color=BLACK, + ) + c7 = Line( + [v4.get_terminals("positive")[0], 2, 0], + v4.get_terminals("positive"), + color=BLACK, + ) + c8 = Line( + [v4.get_terminals("positive")[0], -2, 0], + v4.get_terminals("negative"), + color=BLACK, + ) + c9 = Line( + [4, -2, 0], + [v3.get_terminals("positive")[0], -2, 0], + color=BLACK, + ) + + j1 = Dot(radius=0.08, color=BLACK).move_to( + [v1.get_terminals("positive")[0], 3, 0] + ) + j2 = Dot(radius=0.08, color=BLACK).move_to( + [v1.get_terminals("positive")[0], -3, 0] + ) + j3 = Dot(radius=0.08, color=BLACK).move_to([4, 2, 0]) + j4 = Dot(radius=0.08, color=BLACK).move_to([4, -2, 0]) + + self.add(v1, v2, v3, v4, c1, c2, c3, c4, c5, c6, c7, c8, c9, j1, j2, j3, j4) + + +class een14(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().set_color(BLACK).move_to([-2, 0, 0]) + ri = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, 2, 0]) + rl = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([2, 0, 0]) + + c1 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c2 = Line([v1.get_terminals("positive")[0], 2, 0], [-0.5, 2, 0], color=BLACK) + c3 = Line([2, 2, 0], [2, 0.5, 0], color=BLACK) + c4 = Line([2, -2, 0], [2, -0.5, 0], color=BLACK) + c5 = Line([v1.get_terminals("positive")[0], -2, 0], [2, -2, 0], color=BLACK) + c6 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("positive")[0], -2, 0], + color=BLACK, + ) + c7 = Line([0.5, 2, 0], [2, 2, 0], color=BLACK) + + rit = MathTex(r"R_i", color=BLACK).move_to([0, 1.3, 0]) + rlt = MathTex(r"R_L", color=BLACK).move_to([1.3, 0, 0]) + + self.add(v1, ri, rl, c1, c2, c3, c4, c5, c6, c7, rit, rlt) + + +class een15(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ax = Axes( + y_range=[0, 10, 10], + x_range=[0, 5, 2.5], + tips=False, + ).set_color(BLACK) + labels = ax.get_axis_labels(Tex("R"), Tex("P")).set_color(BLACK) + + f1 = ax.plot(lambda x: -1.6 * ((x - 2.5) ** 2) + 10, x_range=[0, 5]).set_color( + BLACK + ) + + rit = MathTex(r"R_i", color=BLACK).move_to(ax.c2p(2.5, -0.5)) + + self.add(ax, labels, f1, rit) + + +class een16(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ax = Axes( + y_range=[0, 10, 10], + x_range=[0, 5, 2.5], + tips=False, + ).set_color(BLACK) + labels = ax.get_axis_labels(Tex("R"), Tex("")).set_color(BLACK) + + f1 = ax.plot(lambda x: -1.6 * ((x - 2.5) ** 2) + 10, x_range=[0, 5]).set_color( + BLACK + ) + f2 = Line(ax.c2p(0, 0), ax.c2p(5, 10), color=BLACK) + + rit = MathTex(r"R_i", color=BLACK).move_to(ax.c2p(2.5, -0.5)) + + self.add(ax, labels, f1, rit, f2) + + +class een17(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().move_to([2, 0, 0]).set_color(BLACK) + + vmb = Circle(radius=0.5, color=BLACK) + vma = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=BLACK) + vm = VGroup(vmb, vma).move_to([-1.5, 0, 0]) + + r = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, 2, 0]) + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([0, 2, 0]) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, -2, 0]) + + rvr = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + rva = Arrow([-0.75, -0.75, 0], [0.75, 0.75, 0], color=BLACK) + rv = VGroup(rvr, rva).move_to([0, -2, 0]) + + c1 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 3.5, 0], + color=BLACK, + ) + c2 = Line([v1.get_terminals("positive")[0], 3.5, 0], [0, 3.5, 0], color=BLACK) + c3 = Line([0, 3.5, 0], [0, 2.5, 0], color=BLACK) + c4 = Line([0, 1.5, 0], [0, -1.5, 0], color=BLACK) + c5 = Line([0, -2.5, 0], [0, -3.5, 0], color=BLACK) + c6 = Line([0, -3.5, 0], [-3, -3.5, 0], color=BLACK) + c7 = Line([-3, -3.5, 0], [-3, -2.5, 0], color=BLACK) + c8 = Line([-3, -1.5, 0], [-3, 1.5, 0], color=BLACK) + c9 = Line([-3, 2.5, 0], [-3, 3.5, 0], color=BLACK) + c10 = Line([-3, 3.5, 0], [0, 3.5, 0], color=BLACK) + c11 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("positive")[0], -3.5, 0], + color=BLACK, + ) + c12 = Line( + [v1.get_terminals("positive")[0], -3.5, 0], [0, -3.5, 0], color=BLACK + ) + c13 = Line([0, 0, 0], [-1, 0, 0], color=BLACK) + c14 = Line([-2, 0, 0], [-3, 0, 0], color=BLACK) + + rt = MathTex(r"R", color=BLACK).move_to([-2.5, 2, 0]) + r1t = MathTex(r"R_1", color=BLACK).move_to([0.5, 2, 0]) + r2t = MathTex(r"R_2", color=BLACK).move_to([-2.5, -2, 0]) + rvt = MathTex(r"R_v", color=BLACK).move_to([0.5, -2, 0]) + + self.add( + v1, + vm, + r, + r1, + r2, + rv, + c1, + c2, + c3, + c4, + c5, + c6, + c7, + c8, + c9, + c10, + c11, + c12, + c13, + c14, + rt, + r1t, + r2t, + rvt, + ) + + +class een18(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ic = Polygon( + [-2, 2, 0], [2, 2, 0], [2, -2, 0], [-2, -2, 0], color=BLACK + ).move_to([-4, 0, 0]) + + c1 = Line(ORIGIN, [1, 0, 0], color=BLACK).move_to([-1.5, 1.5, 0]) + c2 = Line(ORIGIN, [1, 0, 0], color=BLACK).move_to([-1.5, -1.5, 0]) + j1 = Dot(radius=0.08, color=BLACK).move_to([-1, 1.5, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([-1, -1.5, 0]) + + a = MathTex(r"\Rightarrow", color=BLACK) + + v1 = VoltageSource().move_to([1.5, -1, 0]).set_color(BLACK) + ri = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([1.75, 1, 0]) + + c3 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("negative")[0], -2.5, 0], + color=BLACK, + ) + c4 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("negative")[0], 0.5, 0], + color=BLACK, + ) + c5 = Line( + [v1.get_terminals("negative")[0], 1.5, 0], + [v1.get_terminals("negative")[0], 2.5, 0], + color=BLACK, + ) + + self.add(ic, c1, c2, j1, j2, a, v1, ri, c3, c4, c5) + + +class een19(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ic = Polygon( + [-2, 2, 0], [2, 2, 0], [2, -2, 0], [-2, -2, 0], color=BLACK + ).move_to([-4, 0, 0]) + + c1 = Line(ORIGIN, [1, 0, 0], color=BLACK).move_to([-1.5, 1.5, 0]) + c2 = Line(ORIGIN, [1, 0, 0], color=BLACK).move_to([-1.5, -1.5, 0]) + j1 = Dot(radius=0.08, color=BLACK).move_to([-1, 1.5, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([-1, -1.5, 0]) + + a = MathTex(r"\Rightarrow", color=BLACK) + + i1 = CurrentSource().move_to([1.75, 0, 0]).set_color(BLACK) + ri = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([4, 0, 0]) + + c3 = Line( + i1.get_terminals("negative"), + [i1.get_terminals("negative")[0], -1.5, 0], + color=BLACK, + ) + c4 = Line( + [i1.get_terminals("negative")[0], -1.5, 0], + [5, -1.5, 0], + color=BLACK, + ) + c5 = Line( + [4, -1.5, 0], + [4, -0.5, 0], + color=BLACK, + ) + c6 = Line( + [4, 1.5, 0], + [4, 0.5, 0], + color=BLACK, + ) + c7 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("negative")[0], 1.5, 0], + color=BLACK, + ) + c8 = Line( + [i1.get_terminals("negative")[0], 1.5, 0], + [5, 1.5, 0], + color=BLACK, + ) + j3 = Dot(radius=0.08, color=BLACK).move_to([5, 1.5, 0]) + j4 = Dot(radius=0.08, color=BLACK).move_to([5, -1.5, 0]) + + self.add(ic, c1, c2, j1, j2, a, i1, ri, c3, c4, c5, c6, c7, c8, j3, j4) + + +class een20(Scene): + def construct(self): + + self.camera.background_color = WHITE + + j1 = Dot(radius=0.08, color=BLACK).move_to([-3, -1.5, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([3, -1.5, 0]) + j3 = Dot(radius=0.08, color=BLACK).move_to([0, 1.5, 0]) + + r1 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([-1.5, -1.5, 0]) + r2 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([1.5, -1.5, 0]) + r3 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + c1 = Line([-3, -1.5, 0], [-2, -1.5, 0], color=BLACK) + c2 = Line([-1, -1.5, 0], [1, -1.5, 0], color=BLACK) + c3 = Line([3, -1.5, 0], [2, -1.5, 0], color=BLACK) + c4 = Line([0, -1.5, 0], [0, -0.5, 0], color=BLACK) + c5 = Line([0, 0.5, 0], [0, 1.5, 0], color=BLACK) + + r1t = MathTex(r"R_1", color=BLACK).move_to([0.5, 0, 0]) + r2t = MathTex(r"R_2", color=BLACK).move_to([-1.5, -1, 0]) + r3t = MathTex(r"R_3", color=BLACK).move_to([1.5, -1, 0]) + + self.add(j1, j2, j3, r1, r2, r3, c1, c2, c3, c4, c5, r1t, r2t, r3t) + + +class een21(Scene): + def construct(self): + + self.camera.background_color = WHITE + + j1 = Dot(radius=0.08, color=BLACK).move_to([-3, -1.5, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([3, -1.5, 0]) + j3 = Dot(radius=0.08, color=BLACK).move_to([0, 1.5, 0]) + + r1 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, -1.5, 0]) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, 0, 0]) + r3 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([3, 0, 0]) + c1 = Line([-3, -1.5, 0], [-0.5, -1.5, 0], color=BLACK) + c2 = Line([3, -1.5, 0], [0.5, -1.5, 0], color=BLACK) + c3 = Line([-3, -1.5, 0], [-3, -0.5, 0], color=BLACK) + c4 = Line([3, -1.5, 0], [3, -0.5, 0], color=BLACK) + c5 = Line([-3, 1.5, 0], [-3, 0.5, 0], color=BLACK) + c6 = Line([3, 1.5, 0], [3, 0.5, 0], color=BLACK) + c7 = Line([3, 1.5, 0], [-3, 1.5, 0], color=BLACK) + + r1t = MathTex(r"R_{23}", color=BLACK).move_to([0, -1, 0]) + r2t = MathTex(r"R_{12}", color=BLACK).move_to([-2.25, 0, 0]) + r3t = MathTex(r"R_{13}", color=BLACK).move_to([2.25, 0, 0]) + + self.add(j1, j2, j3, r1, r2, r3, c1, c2, c3, c4, c5, c6, c7, r1t, r2t, r3t) + + +class een22(Scene): + def construct(self): + + self.camera.background_color = WHITE + + i1 = CurrentSource().set_color(BLACK).move_to([-6.25, 0, 0]) + v1 = VoltageSource().set_color(BLACK).move_to([6.25, 0, 0]) + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, 0, 0]) + r3 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([3, 2, 0]) + + c1 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c2 = Line([i1.get_terminals("positive")[0], 2, 0], [2.5, 2, 0], color=BLACK) + c3 = Line([3.5, 2, 0], [v1.get_terminals("positive")[0], 2, 0], color=BLACK) + c4 = Line( + [v1.get_terminals("positive")[0], 2, 0], + v1.get_terminals("positive"), + color=BLACK, + ) + c5 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("negative")[0], -2, 0], + color=BLACK, + ) + c6 = Line( + [v1.get_terminals("negative")[0], -2, 0], + [i1.get_terminals("negative")[0], -2, 0], + color=BLACK, + ) + c7 = Line( + [i1.get_terminals("negative")[0], -2, 0], + i1.get_terminals("negative"), + color=BLACK, + ) + c8 = Line([0, 2, 0], [0, 0.5, 0], color=BLACK) + c9 = Line([0, -0.5, 0], [0, -2, 0], color=BLACK) + c10 = Line([-3, 2, 0], [-3, 0.5, 0], color=BLACK) + c11 = Line([-3, -0.5, 0], [-3, -2, 0], color=BLACK) + + self.add(v1, i1, r1, r2, r3, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11) + + +class een23(Scene): + def construct(self): + + self.camera.background_color = WHITE + + i1 = CurrentSource().set_color(BLACK).move_to([-6.25, 0, 0]) + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, 0, 0]) + r3 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([3, 2, 0]) + + c1 = Line( + i1.get_terminals("positive"), + [i1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c2 = Line([i1.get_terminals("positive")[0], 2, 0], [2.5, 2, 0], color=BLACK) + c3 = Line([3.5, 2, 0], [6, 2, 0], color=BLACK) + c4 = Line([6, 2, 0], [6, -2, 0], color=BLACK) + c5 = Line([6, -2, 0], [i1.get_terminals("negative")[0], -2, 0], color=BLACK) + c6 = Line( + i1.get_terminals("negative"), + [i1.get_terminals("negative")[0], -2, 0], + color=BLACK, + ) + c7 = Line([0, 2, 0], [0, 0.5, 0], color=BLACK) + c8 = Line([0, -0.5, 0], [0, -2, 0], color=BLACK) + c9 = Line([-3, 2, 0], [-3, 0.5, 0], color=BLACK) + c10 = Line([-3, -0.5, 0], [-3, -2, 0], color=BLACK) + + self.add(i1, r1, r2, r3, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10) + + +class een24(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().set_color(BLACK).move_to([6.25, 0, 0]) + r1 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ) + r2 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([-3, 0, 0]) + r3 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([3, 2, 0]) + + c1 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c2 = Line([v1.get_terminals("positive")[0], 2, 0], [3.5, 2, 0], color=BLACK) + c3 = Line([2.5, 2, 0], [-3, 2, 0], color=BLACK) + c5 = Line([-3, -2, 0], [v1.get_terminals("negative")[0], -2, 0], color=BLACK) + c6 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("negative")[0], -2, 0], + color=BLACK, + ) + c7 = Line([0, 2, 0], [0, 0.5, 0], color=BLACK) + c8 = Line([0, -0.5, 0], [0, -2, 0], color=BLACK) + c9 = Line([-3, 2, 0], [-3, 0.5, 0], color=BLACK) + c10 = Line([-3, -0.5, 0], [-3, -2, 0], color=BLACK) + + self.add(v1, r1, r2, r3, c1, c2, c3, c5, c6, c7, c8, c9, c10) + + +class een25(Scene): + def construct(self): + + self.camera.background_color = WHITE + + v1 = VoltageSource().set_color(BLACK).move_to([-3.25, 0, 0]) + r1 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, 2, 0]) + r2 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, -2, 0]) + r3 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, 1, 0]) + r4 = Polygon( + [-0.5, 0.25, 0], + [0.5, 0.25, 0], + [0.5, -0.25, 0], + [-0.5, -0.25, 0], + color=BLACK, + ).move_to([0, -1, 0]) + r5 = Polygon( + [-0.25, 0.5, 0], + [0.25, 0.5, 0], + [0.25, -0.5, 0], + [-0.25, -0.5, 0], + color=BLACK, + ).move_to([3, 0, 0]) + + c1 = Line( + v1.get_terminals("positive"), + [v1.get_terminals("positive")[0], 2, 0], + color=BLACK, + ) + c2 = Line([v1.get_terminals("positive")[0], 2, 0], [-0.5, 2, 0], color=BLACK) + c3 = Line([0.5, 2, 0], [3, 2, 0], color=BLACK) + c4 = Line([3, 2, 0], [3, 0.5, 0], color=BLACK) + c5 = Line( + v1.get_terminals("negative"), + [v1.get_terminals("positive")[0], -2, 0], + color=BLACK, + ) + c6 = Line([v1.get_terminals("positive")[0], -2, 0], [-0.5, -2, 0], color=BLACK) + c7 = Line([0.5, -2, 0], [3, -2, 0], color=BLACK) + c8 = Line([3, -2, 0], [3, -0.5, 0], color=BLACK) + c9 = Line([-2, 2, 0], [-2, 1, 0], color=BLACK) + c10 = Line([-2, 1, 0], [-0.5, 1, 0], color=BLACK) + c11 = Line([1, 1, 0], [0.5, 1, 0], color=BLACK) + c12 = Line([1, 1, 0], [1, -2, 0], color=BLACK) + c13 = Line([2, 2, 0], [2, -1, 0], color=BLACK) + c14 = Line([2, -1, 0], [0.5, -1, 0], color=BLACK) + c15 = Line([-2, -1, 0], [-0.5, -1, 0], color=BLACK) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=BLACK) + + j1 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, 2, 0]) + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j3 = Dot( + radius=0.08, + color=BLACK, + ).move_to([1, -2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + v1, + r1, + r2, + r3, + r4, + r5, + c1, + c2, + c3, + c4, + c5, + c6, + c7, + c8, + c9, + c10, + c11, + c12, + c13, + c14, + c15, + c16, + j1, + j2, + j3, + j4, + ) + + +class een26(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line( + [-3, -2, 0], + [-3, 2, 0], + color=BLACK, + ) + c2 = Line([-3, 2, 0], [3, 2, 0], color=BLACK) + c4 = Line([3, 2, 0], [3, -2, 0], color=BLACK) + c6 = Line([3, -2, 0], [-3, -2, 0], color=BLACK) + c7 = Line([-3, -2, 0], [-3, 2, 0], color=BLACK) + c9 = Line([-2, 2, 0], [-2, 1, 0], color=BLACK) + c10 = Line([-2, 1, 0], [1, 1, 0], color=BLACK) + c12 = Line([1, 1, 0], [1, -2, 0], color=BLACK) + c13 = Line([2, 2, 0], [2, -1, 0], color=BLACK) + c14 = Line([2, -1, 0], [-2, -1, 0], color=BLACK) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=BLACK) + + j1 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, 2, 0]) + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j3 = Dot( + radius=0.08, + color=BLACK, + ).move_to([1, -2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + c1, + c2, + c4, + c6, + c7, + c9, + c10, + c12, + c13, + c14, + c16, + j1, + j2, + j3, + j4, + ) + + +class een27(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line( + [-3, -2, 0], + [-3, 2, 0], + color=BLACK, + ) + c2 = Line([-3, 2, 0], [3, 2, 0], color=BLACK) + c4 = Line([3, 2, 0], [3, -2, 0], color=BLACK) + c6 = Line([3, -2, 0], [-3, -2, 0], color=BLACK) + c7 = Line([-3, -2, 0], [-3, 2, 0], color=BLACK) + c9 = Line([-2, 2, 0], [-2, 1, 0], color=BLACK) + c10 = Line([-2, 1, 0], [1, 1, 0], color=BLACK) + c12 = Line([1, 1, 0], [1, -2, 0], color=BLACK) + c13 = Line([2, 2, 0], [2, -1, 0], color=BLACK) + c14 = Line([2, -1, 0], [-2, -1, 0], color=BLACK) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=BLACK) + + j1 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, 2, 0]) + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j3 = Dot( + radius=0.08, + color=BLACK, + ).move_to([1, -2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + ia1 = Arrow([-0.5, 0, 0], [0.5, 0, 0], color=RED).move_to([0, 2, 0]) + ia2 = Arrow([-0.5, 0, 0], [0.5, 0, 0], color=RED).move_to([0, 1, 0]) + ia3 = Arrow([0.5, 0, 0], [-0.5, 0, 0], color=RED).move_to([0, -1, 0]) + ia4 = Arrow([0.5, 0, 0], [-0.5, 0, 0], color=RED).move_to([0, -2, 0]) + ia5 = Arrow([0, 0.5, 0], [0, -0.5, 0], color=RED).move_to([3, 0, 0]) + ia6 = Arrow([0, -0.5, 0], [0, 0.5, 0], color=RED).move_to([-3, 0, 0]) + + self.add( + c1, + c2, + c4, + c6, + c7, + c9, + c10, + c12, + c13, + c14, + c16, + j1, + j2, + j3, + j4, + ia1, + ia2, + ia3, + ia4, + ia5, + ia6, + ) + + +class een28(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line( + [-3, -2, 0], + [-3, 2, 0], + color=BLACK, + ) + c2 = Line([-3, 2, 0], [3, 2, 0], color=BLACK) + c3 = Line([3, -2, 0], [1, -2, 0], color=BLACK) + c4 = Line([3, 2, 0], [3, -2, 0], color=BLACK) + c5 = DashedLine([1, -2, 0], [-2, -2, 0], color=RED) + c6 = Line([-2, -2, 0], [-3, -2, 0], color=BLACK) + c7 = Line([-3, -2, 0], [-3, 2, 0], color=BLACK) + c9 = DashedLine([-2, 2, 0], [-2, 1, 0], color=BLUE) + c10 = DashedLine([-2, 1, 0], [1, 1, 0], color=BLUE) + c12 = DashedLine([1, 1, 0], [1, -2, 0], color=BLUE) + c13 = DashedLine([2, 2, 0], [2, -1, 0], color=GREEN) + c14 = DashedLine([2, -1, 0], [-2, -1, 0], color=GREEN) + c16 = DashedLine([-2, -1, 0], [-2, -2, 0], color=GREEN) + + j1 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, 2, 0]) + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j3 = Dot( + radius=0.08, + color=BLACK, + ).move_to([1, -2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + c1, + c2, + c3, + c4, + c5, + c6, + c7, + c9, + c10, + c12, + c13, + c14, + c16, + j1, + j2, + j3, + j4, + ) + + +class een29(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line([-3, -2, 0], [-3, 2, 0], color=BLACK) + c2 = Line([-3, 2, 0], [-2, 2, 0], color=BLACK) + c3 = Line([-2, 2, 0], [3, 2, 0], color=BLUE) + c4 = Line([3, 2, 0], [3, -2, 0], color=BLUE) + c6 = Line([3, -2, 0], [1, -2, 0], color=BLUE) + c7 = Line([1, -2, 0], [-3, -2, 0], color=BLACK) + c8 = Line([-3, -2, 0], [-3, 2, 0], color=BLACK) + c9 = Line([-2, 2, 0], [-2, 1, 0], color=BLUE) + c10 = Line([-2, 1, 0], [1, 1, 0], color=BLUE) + c12 = Line([1, 1, 0], [1, -2, 0], color=BLUE) + c13 = Line([2, 2, 0], [2, -1, 0], color=BLACK) + c14 = Line([2, -1, 0], [-2, -1, 0], color=BLACK) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=BLACK) + + j1 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, 2, 0]) + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j3 = Dot( + radius=0.08, + color=BLACK, + ).move_to([1, -2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + c1, c2, c3, c4, c6, c7, c8, c9, c10, c12, c13, c14, c16, j1, j2, j3, j4 + ) + + +class een30(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line( + [-3, -2, 0], + [-3, 2, 0], + color=BLACK, + ) + c2 = Line([-3, 2, 0], [3, 2, 0], color=RED) + c4 = Line([3, 2, 0], [3, -2, 0], color=RED) + c6 = Line([3, -2, 0], [-3, -2, 0], color=RED) + c7 = Line([-3, -2, 0], [-3, 2, 0], color=RED) + c13 = Line([2, 2, 0], [2, -1, 0], color=BLACK) + c14 = Line([2, -1, 0], [-2, -1, 0], color=BLACK) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=BLACK) + + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + c1, + c2, + c4, + c6, + c7, + c13, + c14, + c16, + j2, + j4, + ) + + +class een31(Scene): + def construct(self): + + self.camera.background_color = WHITE + + c1 = Line( + [-3, -2, 0], + [-3, 2, 0], + color=BLACK, + ) + c1 = Line([2, 2, 0], [3, 2, 0], color=BLACK) + c2 = Line([-3, 2, 0], [2, 2, 0], color=GREEN) + c3 = Line([-2, -2, 0], [-3, -2, 0], color=GREEN) + c4 = Line([3, 2, 0], [3, -2, 0], color=BLACK) + c6 = Line([3, -2, 0], [2, -2, 0], color=BLACK) + c7 = Line([-3, -2, 0], [-3, 2, 0], color=GREEN) + c13 = Line([2, 2, 0], [2, -1, 0], color=GREEN) + c14 = Line([2, -1, 0], [-2, -1, 0], color=GREEN) + c16 = Line([-2, -1, 0], [-2, -2, 0], color=GREEN) + + j2 = Dot( + radius=0.08, + color=BLACK, + ).move_to([2, 2, 0]) + j4 = Dot( + radius=0.08, + color=BLACK, + ).move_to([-2, -2, 0]) + + self.add( + c1, + c2, + c3, + c4, + c6, + c7, + c13, + c14, + c16, + j2, + j4, + ) + + +class slm1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wl = Line([0, 2.5, 0], [0, -2.5, 0], color=BLACK).move_to([-5, 0, 0]) + wr = Line([0, 2.5, 0], [0, -2.5, 0], color=BLACK).move_to([5, 0, 0]) + wb = Line([-5, 0, 0], [5, 0, 0], color=BLACK).move_to([0, -2.5, 0]) + + e1 = Polygon( + [-0.5, 1.5, 0], [0.5, 1.5, 0], [0.5, -1.5, 0], [-0.5, -1.5, 0], color=BLACK + ).move_to([-4, 1, 0]) + e2 = Polygon( + [-0.5, 1.5, 0], [0.5, 1.5, 0], [0.5, -1.5, 0], [-0.5, -1.5, 0], color=BLACK + ).move_to([4, 1, 0]) + + w = Polygon( + [-5, 1.5, 0], + [5, 1.5, 0], + [5, -2.5, 0], + [-5, -2.5, 0], + stroke_width=0, + fill_color=BLUE, + fill_opacity=0.75, + ) + + pt = MathTex(r"+\\+\\+", color=BLACK).move_to([4, 1, 0]) + nt = MathTex(r"-\\-\\-", color=BLACK).move_to([-4, 1, 0]) + + vmb = Circle(radius=0.5, color=BLACK) + vma = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=BLACK) + vm = VGroup(vmb, vma).move_to([0, 3, 0]) + + c1 = Line([4, 2.5, 0], [4, 3, 0], color=BLACK) + c2 = Line([4, 3, 0], [0.5, 3, 0], color=BLACK) + c3 = Line([-4, 3, 0], [-0.5, 3, 0], color=BLACK) + c4 = Line([-4, 2.5, 0], [-4, 3, 0], color=BLACK) + + self.add(wr, wl, wb, e1, e2, w, pt, nt, vm, c1, c2, c3, c4) + + +class slm2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wl = Line([0, 2.5, 0], [0, -2.5, 0], color=BLACK).move_to([-5, 0, 0]) + wr = Line([0, 2.5, 0], [0, -2.5, 0], color=BLACK).move_to([5, 0, 0]) + wb = Line([-5, 0, 0], [5, 0, 0], color=BLACK).move_to([0, -2.5, 0]) + + e1 = Polygon( + [-0.5, 1.5, 0], [0.5, 1.5, 0], [0.5, -1.5, 0], [-0.5, -1.5, 0], color=BLACK + ).move_to([-4, 1, 0]) + e2 = Polygon( + [-0.5, 1.5, 0], [0.5, 1.5, 0], [0.5, -1.5, 0], [-0.5, -1.5, 0], color=BLACK + ).move_to([4, 1, 0]) + + w = Polygon( + [-5, 1.5, 0], + [5, 1.5, 0], + [5, -2.5, 0], + [-5, -2.5, 0], + stroke_width=0, + fill_color=BLUE, + fill_opacity=0.75, + ) + + pt = MathTex(r"+\\+\\+", color=BLACK).move_to([4, 1, 0]) + nt = MathTex(r"-\\-\\-", color=BLACK).move_to([-4, 1, 0]) + + vmb = Circle(radius=0.5, color=BLACK) + vma = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=BLACK) + vm = VGroup(vmb, vma).move_to([0, 3, 0]) + + c1 = Line([4, 2.5, 0], [4, 3, 0], color=BLACK) + c2 = Line([4, 3, 0], [0.5, 3, 0], color=BLACK) + c3 = Line([-4, 3, 0], [-0.5, 3, 0], color=BLACK) + c4 = Line([-4, 2.5, 0], [-4, 3, 0], color=BLACK) + + nam = Circle(radius=0.5, color=BLACK) + nat = MathTex(r"Na^+", color=BLACK) + nav = Arrow([-0.25, 0, 0], [-1.75, 0, 0], color=BLACK) + na = VGroup(nam, nat, nav).move_to([-2, 0, 0]) + + fem = Circle(radius=0.5, color=BLACK) + fet = MathTex(r"Fe^-", color=BLACK) + fev = Arrow([0.25, 0, 0], [1.75, 0, 0], color=BLACK) + fe = VGroup(fem, fet, fev).move_to([2, 1, 0]) + + self.add(wr, wl, wb, e1, e2, w, pt, nt, vm, c1, c2, c3, c4, na, fe) + + +class slm3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + d1b = Polygon([-0.5, 1, 0], [-0.5, -1, 0], [0.5, 0, 0], color=BLACK) + d1i = Line([0.5, 1, 0], [0.5, -1, 0], color=BLACK) + d1 = VGroup(d1b, d1i) + + c1 = Line([-2.5, 0, 0], [-0.5, 0, 0], color=BLACK) + c2 = Line([2.5, 0, 0], [0.5, 0, 0], color=BLACK) + + j1 = Dot(radius=0.08, color=BLACK).move_to([2.5, 0, 0]) + j2 = Dot(radius=0.08, color=BLACK).move_to([-2.5, 0, 0]) + + i1a = Arrow([-1.5, 1.5, 0], [1.5, 1.5, 0], color=GREEN) + i2a = Arrow([1.5, -1.5, 0], [-1.5, -1.5, 0], color=RED) + + i1t = Tex("Leitet", color=GREEN).move_to([0, 2, 0]) + i2t = Tex("Sperrt", color=RED).move_to([0, -2, 0]) + + self.add(d1, c1, c2, j1, j2, i1a, i2a, i1t, i2t) + + +class magnet2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(-1, 3 * LEFT) + charge2 = Charge(1, 1 * LEFT) + + charge3 = Charge(-1, 1 * RIGHT) + charge4 = Charge(1, 3 * RIGHT) + + field = ElectricField(charge1, charge2, charge3, charge4) + + magnetP1 = Polygon( + [-3.5, 0.5, 0], + [-3.5, -0.5, 0], + [-2, -0.5, 0], + [-2, 0.5, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + textP1 = Text("S").move_to([-3, 0, 0]) + magnetN1 = Polygon( + [-0.5, 0.5, 0], + [-0.5, -0.5, 0], + [-2, -0.5, 0], + [-2, 0.5, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textN1 = Text("N").move_to([-1, 0, 0]) + + magnetP2 = Polygon( + [0.5, 0.5, 0], + [0.5, -0.5, 0], + [2, -0.5, 0], + [2, 0.5, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + textP2 = Text("S").move_to([1, 0, 0]) + magnetN2 = Polygon( + [3.5, 0.5, 0], + [3.5, -0.5, 0], + [2, -0.5, 0], + [2, 0.5, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textN2 = Text("N").move_to([3, 0, 0]) + + self.add( + field, + magnetP1, + magnetN1, + textP1, + textN1, + magnetP2, + magnetN2, + textP2, + textN2, + ) + + +class magnet3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + charge1 = Charge(1, 3 * LEFT) + charge2 = Charge(-1, 1 * LEFT) + + charge3 = Charge(-1, 1 * RIGHT) + charge4 = Charge(1, 3 * RIGHT) + + field = ElectricField(charge1, charge2, charge3, charge4) + + magnetP1 = Polygon( + [-3.5, 0.5, 0], + [-3.5, -0.5, 0], + [-2, -0.5, 0], + [-2, 0.5, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textP1 = Text("N").move_to([-3, 0, 0]) + magnetN1 = Polygon( + [-0.5, 0.5, 0], + [-0.5, -0.5, 0], + [-2, -0.5, 0], + [-2, 0.5, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + textN1 = Text("S").move_to([-1, 0, 0]) + + magnetP2 = Polygon( + [0.5, 0.5, 0], + [0.5, -0.5, 0], + [2, -0.5, 0], + [2, 0.5, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + textP2 = Text("S").move_to([1, 0, 0]) + magnetN2 = Polygon( + [3.5, 0.5, 0], + [3.5, -0.5, 0], + [2, -0.5, 0], + [2, 0.5, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textN2 = Text("N").move_to([3, 0, 0]) + + error = Polygon( + [-0.2, 0.2, 0], + [-0.2, -0.2, 0], + [0.2, -0.2, 0], + [0.2, 0.2, 0], + color=WHITE, + fill_color=WHITE, + fill_opacity=1, + ) + + self.add( + field, + magnetP1, + magnetN1, + textP1, + textN1, + magnetP2, + magnetN2, + textP2, + textN2, + error, + ) + + +class LF1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wire = Wire(Line([0, 0, -1000], [0, 0, 1000])) + magf = MagneticField(wire) + dot = Dot(point=ORIGIN, color=BLACK) + + self.add(magf, dot) + + +class LF2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + func = lambda pos: pos - DOWN * 1000 + vf = ArrowVectorField(func, color=BLUE) + + charge1 = Charge(3, ORIGIN) + velvec = LabeledArrow( + r"\vec{v}", + start=ORIGIN, + end=2 * RIGHT, + color=BLACK, + label_position=0.5, + label_color=BLACK, + frame_fill_color=WHITE, + ) + lkvec = LabeledArrow( + r"\vec{F}", + start=[0, 0, 0], + end=[-2, -2, 0], + color=RED, + label_position=0.5, + label_color=BLACK, + frame_fill_color=WHITE, + ) + + self.add(vf, charge1, velvec, lkvec) + + +class mf1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wire = Wire(Line([0, 0, -1000], [0, 0, 1000])) + magf = MagneticField(wire) + dot = Dot(point=ORIGIN, color=BLACK) + + fvec1 = Arrow(3 * RIGHT, 3 * RIGHT + 2 * UP, color=RED) + fvec2 = Arrow(5 * RIGHT, 5 * RIGHT + 1 * UP, color=RED) + + self.add(magf, dot, fvec1, fvec2) + + +class mf2(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + wire1 = Wire( + ParametricFunction( + lambda u: ( + u * 0.07, + 2 * np.cos(u), + 2 * np.sin(u), + ), + t_range=(-11 / 4 * TAU, 11 / 4 * TAU, 0.01), + ), + color=BLACK, + ) + mag_field = MagneticField( + wire1, + x_range=[-1, 1], + y_range=[-1, 1], + ) + self.set_camera_orientation(PI / 3, PI / 4) + # self.set_camera_orientation(PI / 2) + self.add(wire1, mag_field) + + +class mf3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wire = Wire(Circle(7).rotate(PI / 2, RIGHT)) + mag_field = MagneticField( + wire, + x_range=[-2, 1], + y_range=[-2, 3], + ) + + magnetP = Polygon( + [-2, 2, 0], + [-2, 3, 0], + [1, 3, 0], + [1, 2, 0], + color=RED, + fill_color=RED, + fill_opacity=1, + ) + textP = Text("N").move_to([-0.5, 2.5, 0]) + magnetM = Polygon( + [1, 3, 0], + [1, -3, 0], + [2, -3, 0], + [2, 3, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + textN = Text("S").move_to([-0.5, -2.5, 0]) + magnetN = Polygon( + [-2, -2, 0], + [-2, -3, 0], + [1, -3, 0], + [1, -2, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + + self.add(mag_field, magnetP, magnetM, magnetN, textP, textN) + + +class mf4(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + wire = Wire(Circle(10).rotate(PI / 2, UP)) + mag_field1 = MagneticField( + wire, + x_range=[-1, 1], + y_range=[-3, 3], + ).move_to([0, 0, 2]) + mag_field2 = MagneticField( + wire, + x_range=[-1, 1], + y_range=[-3, 3], + ).move_to([0, 0, 1]) + mag_field3 = MagneticField( + wire, + x_range=[-1, 1], + y_range=[-3, 3], + ) + mag_field4 = MagneticField( + wire, + x_range=[-1, 1], + y_range=[-3, 3], + ).move_to([0, 0, -1]) + mag_field5 = MagneticField( + wire, + x_range=[-1, 1], + y_range=[-3, 3], + ).move_to([0, 0, -2]) + + plate = Polygon( + [0, -2.5, 2.5], + [0, 3.5, 2.5], + [0, 3.5, -1.5], + [0, -2.5, -1.5], + color=GREY, + fill_color=GREY, + fill_opacity=0.25, + ) + + self.set_camera_orientation(PI / 3, PI / 4) + + self.add(plate, mag_field1, mag_field2, mag_field3, mag_field4, mag_field5) + + +class mf5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + tran = Line(start=[0, 5, 0], end=[0, -5, 0], color=BLACK) + + b1 = LabeledArrow( + r"\vec{B}_1", + label_color=BLACK, + frame_fill_color=WHITE, + start=[-3, -3, 0], + end=[0, 0, 0], + color=RED, + ) + + b2 = LabeledArrow( + r"\vec{B}_2", + label_color=BLACK, + frame_fill_color=WHITE, + start=[0, 0, 0], + end=[3, 1, 0], + color=BLUE, + ) + + mu1 = MathTex(r"\mu_1", color=BLACK).move_to([-1, 3, 0]) + + mu2 = MathTex(r"\mu_2", color=BLACK).move_to([1, 3, 0]) + + self.add(tran, b1, b2, mu1, mu2) + + +class zef1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + metal = Polygon( + [-1, 3, 0], + [1, 3, 0], + [1, -3, 0], + [-1, -3, 0], + color=BLACK, + fill_color=GREY, + fill_opacity=0.75, + ) + vv = Arrow3D([0, 0, 1], [0, 0, 3], color=RED) + c = Charge(1, [1000, 0, 0]) + ef = ElectricField(c) + c1 = Charge(-10, [0, 2.5, 0]) + c2 = Charge(10, [0, -2.5, 0]) + + self.add(metal, vv, ef, c1, c2) + + +class zef2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + l11 = Line([-1, 0, 0], [-0.5, 0, 0], path_arc=-180 * DEGREES) + l12 = Line([-0.5, 0, 0], ORIGIN, path_arc=-180 * DEGREES) + l13 = Line([1, 0, 0], [0.5, 0, 0], path_arc=180 * DEGREES) + l14 = Line([0.5, 0, 0], ORIGIN, path_arc=180 * DEGREES) + l1 = VGroup(l11, l12, l13, l14).move_to([-2, 0, 0]).set_color(BLACK) + l21 = Line([-1, 0, 0], [-0.5, 0, 0], path_arc=-180 * DEGREES) + l22 = Line([-0.5, 0, 0], ORIGIN, path_arc=-180 * DEGREES) + l23 = Line([1, 0, 0], [0.5, 0, 0], path_arc=180 * DEGREES) + l24 = Line([0.5, 0, 0], ORIGIN, path_arc=180 * DEGREES) + l2 = VGroup(l21, l22, l23, l24).move_to([2, 0, 0]).set_color(BLACK) + + c1 = Line([-4, -0.1, 0], [-3, -0.1, 0], color=BLACK) + c2 = Line([4, -0.1, 0], [3, -0.1, 0], color=BLACK) + c3 = Line([1, -0.1, 0], [-1, -0.1, 0], color=BLACK) + + j1 = Dot([-4, -0.1, 0], radius=0.08, color=BLACK) + j2 = Dot([4, -0.1, 0], radius=0.08, color=BLACK) + + lt1 = MathTex(r"L_1", color=BLACK).move_to([-2, 1, 0]) + lt2 = MathTex(r"L_2", color=BLACK).move_to([2, 1, 0]) + + self.add(l1, l2, c1, c2, c3, j1, j2, lt1, lt2) + + +class zef3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + l11 = Line([-1, 0, 0], [-0.5, 0, 0], path_arc=-180 * DEGREES) + l12 = Line([-0.5, 0, 0], ORIGIN, path_arc=-180 * DEGREES) + l13 = Line([1, 0, 0], [0.5, 0, 0], path_arc=180 * DEGREES) + l14 = Line([0.5, 0, 0], ORIGIN, path_arc=180 * DEGREES) + l1 = VGroup(l11, l12, l13, l14).move_to([0, 1, 0]).set_color(BLACK) + l21 = Line([-1, 0, 0], [-0.5, 0, 0], path_arc=-180 * DEGREES) + l22 = Line([-0.5, 0, 0], ORIGIN, path_arc=-180 * DEGREES) + l23 = Line([1, 0, 0], [0.5, 0, 0], path_arc=180 * DEGREES) + l24 = Line([0.5, 0, 0], ORIGIN, path_arc=180 * DEGREES) + l2 = VGroup(l21, l22, l23, l24).move_to([0, -1, 0]).set_color(BLACK) + + c1 = Line([-3, -0.1, 0], [-2, -0.1, 0], color=BLACK) + c2 = Line([3, -0.1, 0], [2, -0.1, 0], color=BLACK) + c3 = Line([2, 0.9, 0], [2, -1.1, 0], color=BLACK) + c4 = Line([2, 0.9, 0], [1, 0.9, 0], color=BLACK) + c5 = Line([-1, 0.9, 0], [-2, 0.9, 0], color=BLACK) + c6 = Line([2, -1.1, 0], [1, -1.1, 0], color=BLACK) + c7 = Line([-2, -1.1, 0], [-1, -1.1, 0], color=BLACK) + c8 = Line([-2, 0.9, 0], [-2, -1.1, 0], color=BLACK) + + j1 = Dot([-3, -0.1, 0], radius=0.08, color=BLACK) + j2 = Dot([3, -0.1, 0], radius=0.08, color=BLACK) + + lt1 = MathTex(r"L_1", color=BLACK).move_to([0, 2, 0]) + lt2 = MathTex(r"L_2", color=BLACK).move_to(ORIGIN) + + self.add(l1, l2, c1, c2, c3, c4, c5, c6, c7, c8, j1, j2, lt1, lt2) + + +class zef4(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ax = Axes( + y_range=[-1.5, 1.5, 3], + x_range=[0, (5 * PI) / 2, PI / 2], + tips=False, + ).set_color(BLACK) + labels = ax.get_axis_labels(Tex("t"), Tex("U")).set_color(BLACK) + + f1 = ax.plot(lambda x: np.sin(x), color=RED) + + self.add(ax, labels, f1) + + +class zef5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ax = Axes( + y_range=[-1.5, 1.5, 3], + x_range=[0, (5 * PI) / 2, PI / 2], + tips=False, + ).set_color(BLACK) + labels = ax.get_axis_labels(Tex("t"), Tex("U")).set_color(BLACK) + + f1 = ax.plot(lambda x: np.sin(x), color=RED) + f2 = ax.plot(lambda x: np.sin(x + 120 * DEGREES), color=BLUE) + + self.add(ax, labels, f1, f2) + + +class zef5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ax = Axes( + y_range=[-1.5, 1.5, 3], + x_range=[0, (5 * PI) / 2, PI / 2], + tips=False, + ).set_color(BLACK) + labels = ax.get_axis_labels(Tex("t"), Tex("U")).set_color(BLACK) + + f1 = ax.plot(lambda x: np.sin(x), color=RED) + f2 = ax.plot(lambda x: np.sin(x + 120 * DEGREES), color=BLUE) + f3 = ax.plot(lambda x: np.sin(x + 240 * DEGREES), color=GREEN) + + self.add(ax, labels, f1, f2, f3) diff --git a/hs24/nus_I/graphs/media/Tex/04cb5ea6058d2843.svg b/hs24/nus_I/graphs/media/Tex/04cb5ea6058d2843.svg new file mode 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y="30"/> +<use xlink:href="#glyph-0-2" x="44" y="30"/> +<use xlink:href="#glyph-0-3" x="50" y="30"/> +<use xlink:href="#glyph-0-4" x="54" y="30"/> +<use xlink:href="#glyph-0-5" x="58" y="30"/> +<use xlink:href="#glyph-0-6" x="64" y="30"/> +<use xlink:href="#glyph-0-7" x="67" y="30"/> +</g> +</svg> diff --git a/hs24/nus_I/letterfonts.tex b/hs24/nus_I/letterfonts.tex new file mode 100644 index 0000000..3939d5b --- /dev/null +++ b/hs24/nus_I/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/hs24/nus_I/macros.tex b/hs24/nus_I/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/hs24/nus_I/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/hs24/nus_I/notes.tex b/hs24/nus_I/notes.tex new file mode 100644 index 0000000..f05bd48 --- /dev/null +++ b/hs24/nus_I/notes.tex @@ -0,0 +1,6 @@ +\import{./elektrostatische_feld}{elektrostatische_feld.tex} +\import{./stationaere_elektrische_stroemungsfeld}{stationaere_elektrische_stroemungsfeld.tex} +\import{./einfache_elektrische_netzwerke}{einfache_elektrische_netzwerke.tex} +\import{./stromleitungsmechanismen}{stromleitungsmechanismen.tex} +\import{./stationaere_magnetfeld}{stationaere_magnetfeld.tex} +\import{./zeitlich_veraenderliche_elektromagnetische_feld}{zeitlich_veraenderliche_elektromagnetische_feld.tex} diff --git a/hs24/nus_I/notizen_nus_ruh_jirayu.bbl b/hs24/nus_I/notizen_nus_ruh_jirayu.bbl new file mode 100644 index 0000000..f1969e4 --- /dev/null +++ b/hs24/nus_I/notizen_nus_ruh_jirayu.bbl @@ -0,0 +1,16 @@ +\begin{thebibliography}{} + +\bibitem[Albach, 2020]{Albach2020} +Albach, M. (2020). +\newblock {\em Elektrotechnik}. +\newblock Pearson Deutschland GmbH. + +\bibitem[Miller, 2024]{Miller2024} +Miller, L. (2024). +\newblock Übungsskript. + +\bibitem[{The Manim Community Developers}, 2024]{MCD2024} +{The Manim Community Developers} (2024). +\newblock {Manim – Mathematical Animation Framework}. + +\end{thebibliography} diff --git a/hs24/nus_I/notizen_nus_ruh_jirayu.pdf b/hs24/nus_I/notizen_nus_ruh_jirayu.pdf new file mode 100644 index 0000000..f262ed2 Binary files /dev/null and b/hs24/nus_I/notizen_nus_ruh_jirayu.pdf differ diff --git a/hs24/nus_I/notizen_nus_ruh_jirayu.tex b/hs24/nus_I/notizen_nus_ruh_jirayu.tex new file mode 100644 index 0000000..4e093e6 --- /dev/null +++ b/hs24/nus_I/notizen_nus_ruh_jirayu.tex @@ -0,0 +1,47 @@ +\documentclass{report} + +% TODO: Change Variables +\def\papertitle{Netzwerke und Schaltungen I} + +% TODO: Change variables according to language preference +\def\theorytitle{Theorie} +\def\corollarytitle{Daraus folgt} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Behauptung} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Beispiel} +\def\questiontitle{Frage} +\def\wrongctitle{Falscher Konzept} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + +\maketitle +\newpage % or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\input{disclaimer.tex} + +\newpage + +\input{notes.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% BIBLIOGRAPHY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\addcontentsline{toc}{chapter}{References} +\bibliographystyle{apalike} +\bibliography{sources} + +\end{document} diff --git a/hs24/nus_I/notizen_nus_ruh_jirayu.zip b/hs24/nus_I/notizen_nus_ruh_jirayu.zip new file mode 100644 index 0000000..3837f8d Binary files /dev/null and b/hs24/nus_I/notizen_nus_ruh_jirayu.zip differ diff --git a/hs24/nus_I/preamble.tex b/hs24/nus_I/preamble.tex new file mode 100644 index 0000000..3de5e3a --- /dev/null +++ b/hs24/nus_I/preamble.tex @@ -0,0 +1,777 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[ngerman]{babel} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + Wrong Concept \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + + + + +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} + + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% +\begingroup +\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% +\pdfsave +\pdfsetmatrix{1 0 0.2 1}% +\rlap{\usebox{\diffdbox}}% +\pdfrestore +\hskip\wd\diffdbox +\endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% +\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% +{\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience 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\begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/hs24/nus_I/sources.bib b/hs24/nus_I/sources.bib new file mode 100644 index 0000000..40c2cb7 --- /dev/null +++ b/hs24/nus_I/sources.bib @@ -0,0 +1,25 @@ +@Book{Albach2020, + author = {Manfred Albach}, + publisher = {Pearson Deutschland GmbH}, + title = {Elektrotechnik}, + year = {2020}, + isbn = {978-3-86894-398-6}, +} + +@Software{MCD2024, + author = {{The Manim Community Developers}}, + license = {MIT}, + month = apr, + title = {{Manim – Mathematical Animation Framework}}, + url = {https://www.manim.community/}, + version = {v0.18.1}, + year = {2024}, +} + +@Misc{Miller2024, + author = {Louis Miller}, + title = {Übungsskript}, + year = {2024}, +} + +@Comment{jabref-meta: databaseType:bibtex;} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/bewegung.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/bewegung.tex new file mode 100644 index 0000000..265d1e6 --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/bewegung.tex @@ -0,0 +1,16 @@ +\section{Ladungsträgerbewegung in Leiter} \label{sec:ltb} + +Bei der Ladungsträgerbewegung spielt die Driftgeschwindigkeit eine grosse Rolle. Die Elektronen, welche sich in einem Medium befinden bewegen sich ohne Einfluss in jede Richtung. Wird jedoch ein Elektrisches Feld angebracht, so bewegen sich die Elektronen gegen das Feld. Dabei prallen sie aber gegen die Protonen, welche sich nicht bewegen können. Die Geschwindigkeit mit welche die Elektronen sich gegen das elektrische Feld bewegen ist die Driftgeschwindigkeit. + +\includegraphics[width=\linewidth]{fig/Fig38.png} + +\dfn{Driftgeschwindigkeit}{ + Die Driftgeschwindigkeit beschreibt die durchschnittliche Geschwindigkeit, mit welche Ladungsträger sich aufgrund eines externen elektrischen Feldes haben. Sie lässt sich wie folgt rechnen. + + \begin{equation} + \vec{v}_e = - \mu_e \cdot \vec{E} + \end{equation} + + wobei $\mu_e$ die Materialkonstante ist. +} + diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrische_stromdichte.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrische_stromdichte.tex new file mode 100644 index 0000000..439599e --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrische_stromdichte.tex @@ -0,0 +1,38 @@ +\section{Die elektrische Stromdichte} + +Die Stromdichte ist von grosser Bedeutung. Sie beschreibt den Stromfluss über eine bestimmte Querschnittsfläche. Eine sehr hohe Stromdichte würde bedeuten, dass des Medium wodurch der Strom fliesst sich sehr stark erhitzen würde und das möchte man vermeiden. + +\dfn{Elektrische Stromdichte}{ + Die elektrische Stromdichte beschreibt den Stromfluss über eine bestimmte Querschnittsfläche. Sie kann durch zwei Formel bestimmt werden. + + \begin{equation} + \vec{J} = \kappa \cdot \vec{E} + \end{equation} + + wobei $\kappa$ der spezifische Leitwert ist oder + + \begin{equation} + \vec{J} = \rho \cdot \vec{v} + \end{equation} + + wobei $\rho$ die Ladungsdichte und $\vec{v}$ die Geschwindigkeit der Elektronen sind. + + \includegraphics[width=\linewidth]{fig/Fig37.png} +} + +Die Stromdichte kann auch verwendet werden, um die Stromstärke zu berechnen. + +\begin{equation} + I = \iint_A \vec{J} d \vec{A} + \label{eq:stromdichte} +\end{equation} + +\nt{ + Für das stationäre Strömungsfeld wo gilt, dass die ortsabhängige Stromdichte konstant ist, hat die Stromstärke (Gleichung \ref{eq:stromdichte}) eine bestimmte Eigenschaft. + + \begin{equation} + I = \oiint_A \vec{J} d \vec{A} = 0 + \end{equation} + + Dies bedeutet dass der Strom, welche in die Fläche reinfliesst auch wieder rausfliessen muss. +} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrischer_strom.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrischer_strom.tex new file mode 100644 index 0000000..ac1a52d --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/elektrischer_strom.tex @@ -0,0 +1,13 @@ +\section{Der elektrische Strom} + +Wenn zwei Elektroden mit verschiedenen Potenzialen verbunden werden, so findet ein Ladungsausgleich statt. Es entsteht ein Fluss von elektrischer Ladung, welche wir als elektrischen Strom bezeichnen. + +\includegraphics[width=\textwidth]{fig/Fig36.png} + +\dfn{Elektrischer Strom}{ + Der elektrische Strom beschreibt die gerichtete Bewegung oder den Fluss von elektrischer Ladung. Er kann durch die Stromstärke beschrieben werden. Diese wird wie folgt berechnet. + + \begin{equation} + I = \frac{\Delta Q}{\Delta t} A + \end{equation} +} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/energie_leistung.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/energie_leistung.tex new file mode 100644 index 0000000..608ea52 --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/energie_leistung.tex @@ -0,0 +1,18 @@ +\section{Energie und Leistung} + +\dfn{Energie}{ + Energie ist eine wichtige, physikalische Grösse in vielen Gebieten der Naturwissenschaften und kann in Form von Wärme, Arbeit oder Strahlung vorkommen +} + +Wir haben gelernt, dass Elektronen durch ein externen elektrischen Feld in Bewegung gesetzt werden können. Die Energie, welche dabei aufgewendet wird um die Elektronen in Bewegung zu versetzen wird von dem elektrischen Feld entnommen. +\\ +\\ +Die aufgewendete Energie über einen bestimmten Zeitintervall kennen wir im allgemeinen als die Leistung. + +\dfn{Leistung}{ + Die Leistung ist über ein Zeitintervall umgesetzte Energie. Sie kann mit der folgenden Formel berechnet werden. + + \begin{equation} + P = U \cdot I = I ^2 \cdot R = \frac{U ^2}{R} + \end{equation} +} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/grenzflaechen.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/grenzflaechen.tex new file mode 100644 index 0000000..6cce6c8 --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/grenzflaechen.tex @@ -0,0 +1,24 @@ +\section{Das Verhalten der Feldgrössen an Grenzflächen} + +\begin{minipage}{0.6\linewidth} + \includegraphics[width=\textwidth]{fig/Fig39.png} +\end{minipage} +\begin{minipage}{0.4\linewidth} + + \[ + J_{n_1} = J_{n_2} + \] + + \[ + \frac{E_{n_1}}{E_{n_2}} = \frac{\kappa_2}{\kappa_1} + \] + + \[ + E_{t_1} = E_{t_2} + \] + + \[ + \frac{J_{t_1}}{J_{t_2}} = \frac{\kappa_1}{\kappa_2} + \] + +\end{minipage} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/leitfaehigkeit_wiederstand.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/leitfaehigkeit_wiederstand.tex new file mode 100644 index 0000000..1fa61cc --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/leitfaehigkeit_wiederstand.tex @@ -0,0 +1,34 @@ +\section{Die spezifische Leitfähigkeit und der spezifische Widerstand} + +\dfn{Spezifische Leitfähigkeit}{ + Die spezifische Leitfähigkeit ist eine Stoffeigenschaft und beschreibt, wie gut elektrischer Strom durch ein Stoff geleitet wird. Die spezifische Leitfähigkeit ist für jeden Stoff verschieden und wird mit $\kappa$ bezeichnet. +} + +Mithilfe vom elektrischen Feld $\vec{E}$ kann man die Stromdichte $\vec{J}$ mit der folgenden Gleichung berechnen. + +\begin{equation} + \vec{J} = (-ne)\cdot \vec{v}_e = ne \cdot \mu_e \cdot \vec{E} = \kappa \cdot \vec{E} +\end{equation} + +In vielen Fällen wird anstelle von der spezifischen Leitfähigkeit der spezifische Widerstand verwendet. + +\dfn{Spezifische Widerstand}{ + Der spezifische Widerstand ist eine temperaturabhängige Materialkonstante und wird zur Berechnung des elektrischen Widerstands eines Mediums verwendet. Der spezifische Widerstand ist der Kehrwert der spezifischen Leitfähigkeit $\kappa$. + + \begin{equation} + \rho_R = \frac{1}{\kappa} + \end{equation} +} + +Da die Materialkonstante temperaturabhängig ist, verändert sich der spezifische Widerstand je nach Temperatur. Um den spezifischen Widerstand eines Stoffes bei einer bestimmten Temperatur zu bekommen, kann die folgende Formel verwendet werden. + +\begin{equation} + \rho_{R,20 ^{\circ} C} \cdot (1 + \alpha \Delta T) +\end{equation} + +wobei $\alpha$ der Temperaturkoeffizient ist. + +\nt{ + In den meisten technischen Anwendungen ist der auftretende Temperaturbereich soweit begrenzt, dass die Temperaturabhängigkeit $\rho_R(T)$ durch eine lineare Näherung hinreichend genau beschrieben werden kann. \cite{Albach2020} +} + diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/stationaere_elektrische_stroemungsfeld.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/stationaere_elektrische_stroemungsfeld.tex new file mode 100644 index 0000000..8c5fbe9 --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/stationaere_elektrische_stroemungsfeld.tex @@ -0,0 +1,9 @@ +\chapter{Das stationäre elektrische Strömungsfeld} + +\input{elektrischer_strom.tex} +\input{elektrische_stromdichte.tex} +\input{bewegung.tex} +\input{leitfaehigkeit_wiederstand.tex} +\input{widerstand.tex} +\input{grenzflaechen.tex} +\input{energie_leistung.tex} diff --git a/hs24/nus_I/stationaere_elektrische_stroemungsfeld/widerstand.tex b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/widerstand.tex new file mode 100644 index 0000000..18a0f15 --- /dev/null +++ b/hs24/nus_I/stationaere_elektrische_stroemungsfeld/widerstand.tex @@ -0,0 +1,31 @@ +\section{Widerstand (Ohm'sche Gesetz)} + +\dfn{Widerstand}{ + Unter dem elektrischen Widerstand $R$ versteht man das Verhältnis von der angelegten Spannung $U$ zu dem Gesamtstrom $I$. \cite{Albach2020} Sie ist definiert wie folgt. + + \begin{equation} + R = \frac{U}{I} + \end{equation} +} + +Der elektrische Widerstand kann verwendet werden um den Gesamtwiderstand eines Bauteils zu berechnen, da dieser nicht abhängig ist von dem spezifischen Widerstand oder den Dimensionen des Stoffes. Umgekehrt kann der elektrische Widerstand durch den spezifischen Widerstand und den Dimensionen des Stoffes berechnet werden. Dies geht durch die folgende Formel. + +\begin{equation} + R = \frac{l}{\kappa \cdot A} = \frac{\rho_R \cdot l}{A} +\end{equation} + +Wie der spezifische Widerstand ist der elektrische Widerstand auch temperaturabhängig und definiert durch die folgende Gleichung. + +\begin{equation} + R = R_{20 ^{\circ}C} \cdot (1 + \alpha \Delta T) +\end{equation} + +\nt{ + In vielen Fällen wird die Berechnung von Schaltungen dadurch erleichtert, dass man nicht den Widerstand, sondern seinen Kehrwert verwendet. + + \begin{equation} + G = \frac{1}{R} + \end{equation} + + $G$ heisst elektrischer Leitwert. +} diff --git a/hs24/nus_I/stationaere_magnetfeld/induktivitaet.tex b/hs24/nus_I/stationaere_magnetfeld/induktivitaet.tex new file mode 100644 index 0000000..15f279c --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/induktivitaet.tex @@ -0,0 +1,44 @@ +\section{Induktivität} + +\dfn{Induktivität}{ + Die Induktivität beschreibt die Fähigkeit, magnetische Energie speichern zu können. \cite{Miller2024} + + \begin{equation} + \Phi = L \cdot I + \label{eq:inductivity} + \end{equation} +} + +Das meistgebrauchte induktive Bauteil ist die Spule, \cite{Miller2024} + +\section{Induktivitäten} + +Induktivitäten können vorallem bei schlaufenartigen Kabeln entdeckt werden. Sie kann berechnet werden durch das Umformen der Gleichung \ref{eq:inductivity}. + +\begin{equation} + L = \frac{\Phi}{I} +\end{equation} + +Bei einer Spule müssen wir zwischen dem Gesamtfluss und dem Fluss durch eine Windung unterscheiden. Die Induktivität lässt sich wie folgt berechnen. + +\dfn{Induktivität einer Spule}{ + \begin{equation} + \Phi_{ges} = N \cdot \Phi_A + \end{equation} + + \begin{equation} + L_{\text{Spule}} = \frac{N ^2 \cdot \mu_0 \cdot A}{l} + \end{equation} + + \nt{ + Für Toroidspulen kann man die folgende Formel verwenden + + \begin{equation} + L = \frac{N ^2}{R_m} = N ^2 \cdot A_L + \end{equation} + + wobei $A_L$ der magnetische Leitwert ist. + } + +} + diff --git a/hs24/nus_I/stationaere_magnetfeld/leiter.tex b/hs24/nus_I/stationaere_magnetfeld/leiter.tex new file mode 100644 index 0000000..dc84d3b --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/leiter.tex @@ -0,0 +1,14 @@ +\section{Stromdurchflossene Leiter} + +Es ist bekannt, dass stromdurchflossene Leiter in ihrer Umgebung ein Magnetfeld besitzen. Bei einem geraden stromdurchflossenen Draht sind die Feldlinien konzentrische Kreise mit dem Leiter as Mittelpunkt. \cite{Albach2020} Die Richtung der Feldlinie folgt der "rechten Hand Regel". + +\includegraphics[width=0.8\linewidth]{fig/Fig6.png} +\includegraphics[width=0.2\linewidth]{fig/Fig7.png} + +Der Betrag des Magnetfeldes nimmt mit zunehmendem Abstand vom Leiter ab. Der ist durch + +\begin{equation} + \vec{H} = \frac{1}{2 \pi} \cdot \frac{l}{\rho} \cdot \vec{e_{\varphi}} +\end{equation} + +gegeben. diff --git a/hs24/nus_I/stationaere_magnetfeld/lorenzkraft.tex b/hs24/nus_I/stationaere_magnetfeld/lorenzkraft.tex new file mode 100644 index 0000000..6f03455 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/lorenzkraft.tex @@ -0,0 +1,27 @@ +\section{Lorenzkraft} \label{sec:lk} + +Bemerkenswert ist der Fakt, dass stromdurchflossene Leiter nicht nur ein eigenes Magnetfeld erzeugt, er erfährt auch eine Kraftwirkung in einem externen Magnetfeld, das von anderen stromführenden Leitern oder Magneten hervorgerufen wird. \cite{Albach2020} Diese Kraftwirkung ist auch unter der Lorenzkraft bekannt. Diese Kraft gilt nicht nur für stromdurchflossene Leiter, sondern auch für geladen Teilchen. + +\dfn{Lorenzkraft \cite{Miller2024}}{ + Bewegt sich ein geladenes Teilchen duch ein Magnetfeld, wirkt eine sogenannte Lorenzkraft. Diese zeigt in die Richtung des Kreuzprodukts der Bewegung und des Magnetfelds. Sie ist deffiniert als: + + \[ + \vec{F} = Q \cdot (\vec{v} \times \vec{B}) + .\] + + Die Richtung kann mit der "rechten Hand Regel" bestimmt werden. + + \includegraphics[width=0.8\linewidth]{fig/Fig8.png} + \includegraphics[width=0.175\linewidth]{fig/Fig9.png} +} + +\nt{ + Die Kraft, welche ein stromdurchflossener Leiter erftährt kann mit der Formel für die Formel für die Lorenzkraft hergeleitet werden und lautet wie folgt. + + \[ + \vec{F} = I \cdot (\vec{s} \times \vec{B}) + .\] + + $\vec{s}$ ist die Länge des Leiters. +} + diff --git a/hs24/nus_I/stationaere_magnetfeld/magnete.tex b/hs24/nus_I/stationaere_magnetfeld/magnete.tex new file mode 100644 index 0000000..8dd1cc7 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/magnete.tex @@ -0,0 +1,21 @@ +\section{Magnete} + +\dfn{Magnete}{ + Magnete sind einfachtshalber bekannt als Stäbe aus Metall mit einem Nord- und einem Südpol. +} + +\includegraphics[width=\linewidth]{fig/Fig3.png} + +Anhand der Grafik kann man sehen, dass Magnete Kräfte auaüben. Man sieht, dass magnetische Feldlinien geschlossen sind und das Feld zeigt ausserhald des Magneten vom Nord- zum Südpol. \cite{Miller2024} Innerhalb des Magneten verlaufen die magnetische Feldlinien vom Süd- zum Nordpol. +\\ +\\ +Durch Versuche mit zwei Stabmegneten hat man festgestellt, dass Magnete abstossende oder anziehende Kräfte haben, je nachdem welche Enden sich gegenüberstehen. Dies ist auch in der folgenden zwei Grafiken zu erkennen. + +\includegraphics[width=0.5\linewidth]{fig/Fig4.png} +\includegraphics[width=0.5\linewidth]{fig/Fig5.png} + +\nt{ + Ungleichnamige Pole ziehen sich an. Gleichnahmige Pole stossen sich ab. \cite{Albach2020} +} + +Ein zerteilter Magnet besitzt wiederum ein jedem Teilstück ein Nord- und ein Südpol. Es gibt keine magnetische Einzelladung! \cite{Miller2024} diff --git a/hs24/nus_I/stationaere_magnetfeld/magnetische_polarisation.tex b/hs24/nus_I/stationaere_magnetfeld/magnetische_polarisation.tex new file mode 100644 index 0000000..6dcfa33 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/magnetische_polarisation.tex @@ -0,0 +1,25 @@ +\section{Magnetische Polarisation} + +\dfn{Magnetische Polarisation}{ + Die magnetische Polarisation beschreibt den Prozess der Ausrichtung magnetischer Momente. Die Stoffe, die den Prozess durchgehen werden nach ihren magnetischen Eigenschaften in drei Gruppen eingeteilt. \cite{Miller2024} Diese Eigenschaft wird durch die Permibilitätszahl gekennzeichnet. Dabei gilt: + + \begin{itemize} + \item $\mu_r < 1$ sind diamagnetische Stoffe + \item $\mu_r > 1$ sind paramagnetische Stoffe + \item $\mu_r >> 1$ sind ferromagnische Stoffe + \end{itemize} +} + +Während diamagnetische Stoffe das B-Feld geringfügig Schwächen, \cite{Albach2020} da sie gegen das magnetische Feld wirken und paramagnetische Stoffe das B-Feld geringfügig stärken, \cite{Albach2020} da sie mit das magnetische Feld wirken, haben ferromagnetische Stoffe eine besondere Eigenschaft. +\\ +Abgesehen von der Tatsache, dass ihre Permibilitätszahl sehr stark von 1 abweicht bilden ferromagnische Stoffe Weiss'sche Bezirke, in welchen die Polarisation einheitlich ist. Wird ein externes H-Feld angelegt, so richten sich diese Polarisationen langsam diesem H-Feld aus. Somit richten sich viele Weiss'sche Bezirke in die selbe Richtung und werden somit grösser. Diese Ausrichtung besteht auch, wenn das externe H-Feld ausgeschalten wird und folglich ist unser ferromagnetische Stoff selber ein Magnet geworden. \cite{Miller2024} + +%TODO: Add Graphics + +\dfn{Remanenz}{ + Die Remanenz ist die Restmagnetisierung eines ferromagnischen Stoffes nach dem ausschalten eines externen Feldes. Sie wird mit der Hysteresekurve beschrieben. + + %TODO: Add Graphics +} + +Man kann anhand der Hysteresekurve sehen, dass ferromagnische Stoffe die einen externen Feld angelegt sind irgendwann gesättigt sind und zu einem Magnet werden. Man kann diesen Vorgang aber auch rückgängig machen, indem man den Stoff in ein externen Gegenfeld anlegt. Damit es funktioniert muss das Gegenfeld die Koerzitivfeldstärke als Betrag haben. diff --git a/hs24/nus_I/stationaere_magnetfeld/oested_gesetz.tex b/hs24/nus_I/stationaere_magnetfeld/oested_gesetz.tex new file mode 100644 index 0000000..7fa1567 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/oested_gesetz.tex @@ -0,0 +1,12 @@ +\section{Oested'sches Gesetz} + +\dfn{Oested'sches Gesetz}{ + Um die Durchflutung, d.h. der resultierende Strom, welche durch eine Fläche fliesst zu berechnen verwendet man das Oestred'sche Gesetz. + + \begin{equation} + \oint_C \vec{H} ds = \Theta = \iint_A \vec{J}d \vec{A} = \sum I_k + \end{equation} + + \includegraphics[width=0.99\textwidth]{fig/Fig10.png} +} + diff --git a/hs24/nus_I/stationaere_magnetfeld/reluktanzmodel.tex b/hs24/nus_I/stationaere_magnetfeld/reluktanzmodel.tex new file mode 100644 index 0000000..3fc7dc9 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/reluktanzmodel.tex @@ -0,0 +1,30 @@ +\section{Reluktanzmodel} + +Es gibt Analogien zwischen dem elektrischen und dem magnetischen Kreis. Dies zeigt die Folgende Tabelle. + +\dfn{Analogie zwischen elektrischem und magnetischem Kreis \cite{Albach2020}}{ + \begin{tabular}{| c | c | c |} + Bezeichnung & Elektrisches Netzwerk & Magnetisches Netzwerk\\ + \hline + Leitfähigkeit & $\kappa$ & $\mu$\\ + Wiederstand & $R = \frac{l}{\kappa \cdot A}$ & $R_m = \frac{l}{\mu \cdot A}$\\ + Leitwert & $G = \frac{1}{R}$ & $\Lambda_m = \frac{1}{R_m}$\\ + Spannung & $U_{1 2} = \int_{P_1}^{P_2} \vec{E} \cdot d \vec{s}$ & $V_{m 1 2} = \int_{P_1}^{P_2} \vec{H} \cdot d \vec{s}$\\ + Strom bzw. Fluss & $I = \iint_A \vec{J} \cdot d \vec{A} = \kappa \iint_A \vec{E} \cdot d \vec{A}$ & $\Phi = \iint_A \vec{B} \cdot d \vec{A} = \mu \iint_A \vec{H} \cdot d \vec{A}$\\ + Ohm'sches Gesetz & $U = R \cdot I$ & $V_m = R_m \cdot \Phi$\\ + Maschengleichung & $U_0 = \sum_{\text{Masche}} R \cdot I$ & $\Theta = \sum_{\text{Masche}} R_m \cdot \Phi$\\ + Knotengleichung & $\sum_{\text{Knoten}} I = 0$ & $\sum_{\text{Knoten}} \Phi = 0$\\ + \end{tabular} +} + +Nun können wir unsere magnetische Netzwerke als ein äquivalentes Netzwerk darstellen und analysieren. \cite{Miller2024} +\\ +Dadurch bekommen wir die folgende definition für den Magnetischen Wiederstand. + +\dfn{Magnetischer Wiederstand}{ + Der magnetischer Wiederstand ist der Wiederstand, welches der magnetische Fluss wiederfährt wenn es durch ein Medium fliesst. Sie kann berechnet werden durch + + \begin{equation} + R_m = \frac{l}{\mu_r \cdot \mu_0} + \end{equation} +} diff --git a/hs24/nus_I/stationaere_magnetfeld/spannung_fluss.tex b/hs24/nus_I/stationaere_magnetfeld/spannung_fluss.tex new file mode 100644 index 0000000..2d93f87 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/spannung_fluss.tex @@ -0,0 +1,51 @@ +\section{Magnetische Spannung} + +\dfn{Magnetische Spannung}{ + Die magnetische Spannung beschreibt das Verhalten des magentischen Feldes entlang zwei Punkten. Sie ist definiert durch + + \begin{equation} + V_{m 12} = \int_2^1 \vec{H} d \vec{s} + \end{equation} + + \includegraphics[width=\linewidth]{fig/Fig12.png} +} + +\section{Magnetischer Fluss} \label{sec:mf} + +\dfn{Magnetischer Fluss \cite{Miller2024}}{ + Der magnetische Fluss bezeichnet den Fluss des B-Feldes durch eine Fläche. + + \begin{equation} + \Phi = \iint_A \vec{B} d \vec{A} + \end{equation} + + \includegraphics[width=\linewidth]{fig/Fig13.png} +} + +Da es keine magnetische Einzelladungen gibt, kann nirgendwo ein B-Feld entspringen oder verschwinden. Ähnlich wie bei der Stromdichte bedeutet dies, dass alles, was in eine Fläche hinein fliest, auch wieder herausfliesen muss. \cite{Miller2024} + +\begin{equation} + \oiint \vec{B} d \vec{A} = 0 +\end{equation} + +\dfn{Das magnetische Feld an $\mu$-Sprungstellen}{ + -\begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig14.png} + \end{minipage} + \begin{minipage}{0.29\linewidth} + \begin{equation} + B_{n 1} = B_{n 2} + \end{equation} + \begin{equation} + H_{t 1} = H_{t 2} + \end{equation} + Daraus folgt: + \begin{equation} + B_{t 2} = \frac{\mu_2}{\mu_1} \cdot B_{t 1} + \end{equation} + \begin{equation} + H_{n 2} = \frac{\mu 1}{\mu 2} \cdot H_{n 1} + \end{equation} + \end{minipage} +} + diff --git a/hs24/nus_I/stationaere_magnetfeld/spule.tex b/hs24/nus_I/stationaere_magnetfeld/spule.tex new file mode 100644 index 0000000..704d56f --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/spule.tex @@ -0,0 +1,18 @@ +\section{Zylinderspule} + +Wie sieht das Magnetfeld für eine Zylinderspule aus? Wenn man die Zylinderspule von vorne betrachtet kann man erkennen, dass diese aus mehreren Kreisförmigen Drähten besteht. Nach der "rechten Hand Regel" würde es ein Magnetfeld geben, welches durch die Spule geht. + +\includegraphics[width=\linewidth]{fig/Fig11.png} + +Wenn man die äusseren Feldlinien ignoriert und annimmt, dass der Betrag der Feldstärke konstant ist, so ist die Induktion wie folgt definiert. + +\dfn{Induktion in einer Zylinderspule}{ + Die Induktion in einer Zylinderspule kann mit der Formel + + \begin{equation} + \oint_C \vec{H} ds = H \cdot l = \sum I_k = N \cdot I \Rightarrow H = \frac{N \cdot I}{l} + \end{equation} + + wobei $l$ die Spulenlänge ist und $N$ die Wicklungszahl der Zylinderspule. +} + diff --git a/hs24/nus_I/stationaere_magnetfeld/stationaere_magnetfeld.tex b/hs24/nus_I/stationaere_magnetfeld/stationaere_magnetfeld.tex new file mode 100644 index 0000000..acda4c4 --- /dev/null +++ b/hs24/nus_I/stationaere_magnetfeld/stationaere_magnetfeld.tex @@ -0,0 +1,11 @@ +\chapter{Das stationäre Magnetfeld} + +\input{magnete.tex} +\input{leiter.tex} +\input{lorenzkraft.tex} +\input{oested_gesetz.tex} +\input{spule.tex} +\input{spannung_fluss.tex} +\input{magnetische_polarisation.tex} +\input{reluktanzmodel.tex} +\input{induktivitaet.tex} diff --git a/hs24/nus_I/stromleitungsmechanismen/gas.tex b/hs24/nus_I/stromleitungsmechanismen/gas.tex new file mode 100644 index 0000000..a3d98bf --- /dev/null +++ b/hs24/nus_I/stromleitungsmechanismen/gas.tex @@ -0,0 +1,9 @@ +\section{Stromleitung in Gasen} \label{sec:slg} + +Die Bewegung von Ladungen in Gasen ist sehr stark beeinträchtigt. Dennoch haben Stromleitungen in Gasen einen besonderen Stromleitungsmechanismus. +\\ +Gasen bestehen aus Molekülen, welche neutral geladen sind. Wenn die Moleküle ionisiert werden (d.h. sie werden zu Kationen wenn sie ein Elektron verlieren oder zu Anionen wenn sie ein Elektron aufnehmen) so tragen sie zum Stromfluss bei. Jedoch ist die Zunahme vom Stromfluss sehr gering, da die Moleküle eine viel grössere Masse haben als die Ladungen. + +\nt{ + Bei der Ionisierung von Gasen unterscheidet man zwischen einer selbstständigen Leitung und einer unselbstständigen Leitung. Bei einer selbstständigen Leitung werden die Moleküle durch die vorhandene kinetische Energie von Ionen ionisiert. Bei einer unselbstständigen Leitung werden die Moleküle durch äusserlichen Einwirkungen ionisiert. +} diff --git a/hs24/nus_I/stromleitungsmechanismen/ladungstransport.tex b/hs24/nus_I/stromleitungsmechanismen/ladungstransport.tex new file mode 100644 index 0000000..0de57ae --- /dev/null +++ b/hs24/nus_I/stromleitungsmechanismen/ladungstransport.tex @@ -0,0 +1,46 @@ +\section{Ladungstransport in Halbleitern} + +\dfn{Halbleiter}{ + Halbleiter sind Stoffe, deren elektrische Leitfähigkeit zwischen Leitern und Isolatoren liegen. Dies bedeutet, dass sie als Leiter, sowie Isolatoren fungieren können. +} + +Die Eigenschaft, dass Halbleiter als Leiter und Isolatoren fungieren können liegt am molekularen Aufbau von Halbleitern. Halbleiter bestehen aus frei beweglichen Elektronen und Löchern. Löcher sind freie Plätze im Kristallgitter, welche von Elektronen besetzt werden können. +\\ +Halbleiter welche einen Überschuss an Elektronen hat bezeichnet man als n-dotiert während Halbleiter mit einen Überschuss an Löchern p-dotiert nennt. Wird ein n-dotiereter Halbleiter mit einem p-dotierten Halbleiter in Kontakt gebracht, so Bewegen sich die Elektronen zu den Löchern und es entstehen Löcher beim n-dotierten Halbleiter. Dadurch entsteht eine Raumladungszone welche verhindert, dass ein weiterer Austausch von Elektronen und Löchern verhindert und ein elektrisches Feld bildet. Die Raumladung bildet auch eine Sperrschicht, welche als Isolator fungiert und den Stromfluss verhindert. Diese Eigenschaft von Halbleitern ist essentiell für Dioden. + +\subsection{Dioden} + +\dfn{Diode}{ + Eine Diode ist ein elektrisches Bauteil welches in der Lage ist den Stromfluss in eine Richtung zu sperren aber in die andere Richtung fliessen zu lassen. + + \includegraphics[width=\linewidth]{fig/Fig73.png} +} + +\begin{minipage}{0.5\linewidth} + Für die Analyse der Diode ist Diodenkennlinie von grosser Bedeutung. Neben der maximalen Durchlassstrom zeigt die Diodenkennlinie auch die Durchbruchsspannung. Überschreitet man den Durchlassstrom bzw. die Durchbruchsspannung, so geht die Diode kaputt. +\end{minipage} +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig74.png} + \cite{Albach2020} +\end{minipage} + +\nt{ + Die Durchbruchsspannung wird im gegensatz zu normalen Dioden bei Zenner Dioden verwendet. Diese werden vor allem bei der Spannungsstabilisierung verwendet sowie zum Schutz von Überspannung. +} + +\nt{ + Die Diodenkennlinie ist durch die folgenden Gleichungen definiert. + + \begin{equation} + I = \frac{U_R}{R} + \end{equation} + + \begin{equation} + U_R = U_0 - U_D + \end{equation} + + \begin{equation} + I = \frac{U_0 - U_D}{R} + \end{equation} +} + diff --git a/hs24/nus_I/stromleitungsmechanismen/liquiden.tex b/hs24/nus_I/stromleitungsmechanismen/liquiden.tex new file mode 100644 index 0000000..708335f --- /dev/null +++ b/hs24/nus_I/stromleitungsmechanismen/liquiden.tex @@ -0,0 +1,50 @@ +\section{Stromleitung in Flüssigkeiten} + +Wir haben in Kapitel \ref{sec:slg} gelernt, dass Moleküle zum Stromfluss beitragen können. Dies gilt auch für Flüssigkeiten. +\\ +Wenn zwei Elektroden in destillierten Wasser eingetaucht werden, so fliesst ein sehr geringer Strom, da es nicht genügend Ionen hat, so dass eine Ladungsübertragung stattfindet. + +\includegraphics[width=\linewidth]{fig/Fig71.png} + +Sobald das Wasser verunreinigt wird so steigt der Stromfluss. Dies wird sehr deutlich wenn Salz zum Wasser beigefügt wird. Das Salz trennt sich im destillierten Wasser zu Natrium-Kationen (positive geladene Ionen) und Chlor-Anion (negative geladene Ionen). Die Kationen wandern zur Kathode während die Anionen zur Anode sich bewegen. Dort angekommen geben bzw. nehmen sie Elektronen auf. Dadurch fliesst ein Strom. + +\includegraphics[width=\linewidth]{fig/Fig72.png} + +Dieser Prozess wird Elektrolyse genannt und wird sehr häufig fürs Galvanisieren verwendet. + +\dfn{Galvanisieren}{ + Unter Galvanisieren versteht man das elektrochemische Verfahren, bei dem die Elektrolyse eine dünne Metallschicht auf ein Material aufgebracht wird. +} + +Die Masse der Metallschicht, welche beim Galvanisieren entsteht kann durch die Farraday'sche Gesetze bestimmt werden. Die bilden zusammen die folgende Gleichung. + +\begin{equation} + m = \frac{A_r \cdot u}{z \cdot e} \cdot Q = \frac{A_r \cdot u}{z \cdot e} \cdot I \cdot t +\end{equation} + +Des weiteren gilt, dass das Verhältnis von $m \cdot z$ und $A_r$ konstant ist. + +\begin{equation} + \frac{m \cdot z}{A_r} = \text{konstant} +\end{equation} + +Der Strom welcher bei der Elektrolyse entsteht ist der Gesamtstrom, welcher von den negativen und positiven Ladungen induziert wird. Dadurch lässt sich die folgende Gleichung ableiten. + +\begin{equation} + I_+ = \eta \cdot z \cdot e \cdot v_+ \cdot A +\end{equation} + +\begin{equation} + I_- = \eta \cdot z \cdot e \cdot v_- \cdot A +\end{equation} + +\begin{equation} + I = \eta \cdot z \cdot e \cdot A \cdot (|v_+| + |v_-|) + \label{eq:il} +\end{equation} + +Aus der Gleichung \ref{eq:il} lässt sich der Widerstand berechnen. + +\begin{equation} + R = \frac{l}{\eta \cdot z \cdot e \cdot A \cdot (\mu_+ + \mu_-)} +\end{equation} diff --git a/hs24/nus_I/stromleitungsmechanismen/stromleitungsmechanismen.tex b/hs24/nus_I/stromleitungsmechanismen/stromleitungsmechanismen.tex new file mode 100644 index 0000000..2021855 --- /dev/null +++ b/hs24/nus_I/stromleitungsmechanismen/stromleitungsmechanismen.tex @@ -0,0 +1,6 @@ +\chapter{Stromleitungsmechanismen} + +\input{vakum.tex} +\input{gas.tex} +\input{liquiden.tex} +\input{ladungstransport.tex} diff --git a/hs24/nus_I/stromleitungsmechanismen/vakum.tex b/hs24/nus_I/stromleitungsmechanismen/vakum.tex new file mode 100644 index 0000000..7821e71 --- /dev/null +++ b/hs24/nus_I/stromleitungsmechanismen/vakum.tex @@ -0,0 +1,48 @@ +\section{Stromleitung im Vakuum} + +Wir wissen, dass Ladungen durch externe elektrische Felder in eine Richtung bewegt werden können. (Kapitel \ref{sec:ltb}) Dies bedeutet, dass Ladungen in elektrischen Feldern eine Beschleunigung erfährt. Mit dem Gesetz von Newton kann eine Formel für die Beschleunigung hergeleitet werden. + +\begin{equation} + \vec{a} = - \frac{e \cdot U }{m_e \cdot d} \cdot \vec{e_y} + \label{eq:acce} +\end{equation} + +Die Gleichung \ref{eq:acce} kann nun nach der Zeit abgeleitet werden um die Geschwindigkeit, sowie die Strecke berechnen zu können in Abhängigkeit zur Zeit. + +\begin{tabular}{| c | c | c | c |} + Beschleunigung & Geschwindigkeit & Position & Kinetische Energie \\ + \hline + $a(t) = \frac{e \cdot U}{m \cdot d}$ & $v(t) = \frac{e \cdot U}{m \cdot d} \cdot t$ & $v(t) = \frac{e \cdot U}{m \cdot d} \cdot \frac{t ^2}{2}$ & $W = \frac{1}{2} \cdot m \cdot v ^2 = e \cdot U$ \\ +\end{tabular} + +Falls die Position $y$ gegeben ist, kann durch die folgende Formel die Geschwindigkeit an dieser Position berechnet werden. + +\begin{equation} + v(y) = \sqrt{2 \cdot U \cdot \frac{e}{m_0} \cdot \frac{y}{d}} +\end{equation} + +\subsection{Relativitätstheorie} + +In Vakuum können Ladungen sehr hohe Geschwindigkeiten erreichen. Aus diesem Grund muss gegebenenfalls die relativistische Massenzunahme berücksichtigt werden. + +\begin{equation} + m = m_0 \cdot \frac{1}{\sqrt{1 - (\frac{v}{c}) ^2}} + \label{eq:mz} +\end{equation} + +\nt{ + Die Gleichung \ref{eq:mz} wird in diesem Semester nie gebraucht. +} + +\subsection{Raumladungsgesetz} + +\dfn{Raumladungsgesetz}{ + Das Raumladungsgesetz beschreibt den nichtlinearen Zusammenhang von Strom und Spannung in einer evakuierten Anordnung mit zwei Elektroden. (z.B. eine Vakuumröhre). Sie wird beschrieben durch die Gleichung + + \begin{equation} + I = \frac{4}{9} \cdot \frac{\epsilon_0 \cdot A}{d ^2} \cdot \sqrt{\frac{2 \cdot e}{m_0}} \cdot U ^{\frac{3}{2}} + \label{eq:ss} + \end{equation} +} + +Was sagt die Gleichung \ref{eq:ss} aus? Wenn eine Vakuumröhre bestehend aus zwei Elektroden unter Spannung liegt, so bewegen sich Ladungen von der Kathode zur Anode. An der Anode bildet sich eine Elektronenwolke aufgrund vom Elektronenüberschuss. Die Elektronenwolke beeinträchtigt den Fluss der Ladungen zur Anode. Dies beeinträchtigt auch den Strom. Die Gleichung \ref{eq:ss} beschreibt den Sättigungsstrom, also den maximalen Strom welcher durch diese Anordnung fliessen kann. diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/bewegungsinduktion.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/bewegungsinduktion.tex new file mode 100644 index 0000000..a5eac04 --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/bewegungsinduktion.tex @@ -0,0 +1,58 @@ +\section{Anwendung der Bewegungsinduktion} + +Die Bewegungsinduktion wird hauptsächlich zwischen zwei Anwendungen unterschieden. Bei der Umwandlung von mechanischer in elektrischer Energie wird das Prinzip eines Generators verwendet. Für die Umwandlung von elektrischer Energie in mechanische Energie wird das Prinzip eines Motors verwendet. Grundsätzlich ist der Aufbau beider Prinzipien gleich weshalb nur das Generatorprinzip angeschaut wird. + +\subsection{Das Generatorprinzip} + +In den bisherigen Kapiteln haben wir Gleichstrom und Gleichspannung behandelt. Der herkömmliche Strom, welche wir aus Steckdosen beziehen kommt in Form einer Wechselspannung. + +\includegraphics[width=\linewidth]{fig/Fig82.png} + +Wie man von der Abbildung sehen kann hat die Spannung eine sinusförmige Welle. Diese kann durch Induktion durch eine rotierende Leiterschleife in einem konstanten Magnetfeld erzeugt werden. + +\includegraphics[width=\linewidth]{fig/Fig83.png} \cite{Albach2020} + +Die induzierte Spannung kann dann wie folgt berechnet werden. + +\begin{equation} + U_{\text{ind}} = \omega \cdot B \cdot A \cdot \sin(\omega \cdot t) +\end{equation} + +Die Wechselspannung, welche wir verwenden kommt in Form von drei Phasen. + +\includegraphics[width=\linewidth]{fig/Fig84.png} + +Diese kann generiert werden indem drei Leiterschleifen, welche 120$^{\circ}$ räumlich verschoben sind rotiert werden. + +\includegraphics[width=\linewidth]{fig/Fig85.png} \cite{Albach2020} + +Doch welchen Vorteil verschafft ein Dreiphasensystem. Da die Wechselspannung nun aus drei Phasen besteht und jeder dieser Phasen bestromt werden kann, so kann eine grössere Leistung durch die Leitung geliefert werden. Dabei gilt für die Leistung. + +\begin{equation} + P = U \cdot I \cdot \sqrt{3} +\end{equation} + +Dies bedeutet, dass in einem Dreiphasensystem fast doppelt so viel Leistung zu den Verbrauchern transportiert werden kann in Vergleich zum Einphasensystem. +\\ +\\ +Für die Übertragung zu den Verbrauchern können zwei Schaltungen verwendet werden. + +\dfn{Sternschaltung}{ + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig86.png} \cite{Albach2020} + \end{minipage} + \begin{minipage}{0.5\linewidth} + Bei der Sternschaltung werden die Spulen an einem Anschluss dem Sternpunkt zusammengeschaltet. \cite{Albach2020} Die anderen Anschlüsse werden mit den Verbrauchern verbunden. Diese Leitungen werden Aussenleiter genannt. In den Aussenleitern fliesst der selbe Strom wie in den Strangleitungen während die Spannung um den Faktor $\sqrt{3}$ grösser ist. + \end{minipage} +} + +\dfn{Dreiecksschaltung}{ + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig87.png} \cite{Albach2020} + \end{minipage} + \begin{minipage}{0.5\linewidth} + Bei der Dreiecksschaltung werden die Spulen in Serie geschaltet. In Vergleich zur Sternschaltung ist die Leiterspannung und die Strangspannung gleich, während der Leiterstrom um den Faktor $\sqrt{3}$ grösser ist als der Strangstrom. + \end{minipage} +} + +In den meisten Fällen wird die Sternschaltung verwendet da diese einen Neutralleiter hat und aus Sicherheitsgründen verwendet wird. diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/energieinhalt_feld.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/energieinhalt_feld.tex new file mode 100644 index 0000000..eeb734c --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/energieinhalt_feld.tex @@ -0,0 +1,49 @@ +\section{Der Energieinhalt des Feldes} + +\dfn{Die magnetische Energie}{ + Unter der magnetischen Energie versteht man die Energie eines Magnetfeldes. Dies kann in Form eines Magneten oder, wie in den meisten, in Form einer Spule sein. +} + +\includegraphics[width=\linewidth]{fig/Fig80.png} \cite{Albach2020} + +Nehmen wir als Beispiel eine Ringkernspule. Wenn ein Spannung an der Spule angesetzt wird so fliesst ein Strom. Durch die Energie, welche der Spule zugeführt erhöht sich die Flussdichte im Ringkern. Diese gespeicherte Energie kann mit der folgenden Gleichung berechnet werden. + +\begin{equation} + W_m = \frac{1}{2} \cdot L \cdot I ^2 = \frac{1}{2} \cdot \Phi \cdot I +\end{equation} + +Für 2 Spulen gilt die Formel + +\begin{equation} + W_m = \frac{1}{2} \cdot L_1 \cdot I_1 ^2 + \frac{1}{2} \cdot L_2 \cdot I_2 ^2 + M \cdot I_1 \cdot I_2 +\end{equation} + +Für mehrere Spulen gilt die Formel + +\begin{equation} + W_m = \frac{1}{2} \sum_{i=1} ^{n} \sum_{k=1} ^{n} L_{ik} \cdot I_i \cdot I_k +\end{equation} + +\nt{ + Das Berechnen der magnetischen Energie von mehreren Spulen werden wir nur selten gebrauchen. +} + +Die magnetische Energie kann auch mit den Feldgrössen berechnet werden. + +\begin{equation} + W_m = \frac{1}{2} \iiint_V \vec{H} \cdot \vec{B} dV +\end{equation} + +\subsection{Hystereseverluste} + +Wie bei jedem Energie Prozess entstehen Energieverluste. Diese können anhand der Hysteresekurve veranschaulicht werden. + +\includegraphics[width=\linewidth]{fig/Fig81.png} \cite{Albach2020} + +Im Allgemeinen gilt: + +\nt{ + Der Energieverlust beim Umlaufen der Hystereseschleife entspricht dem Produkt aus der von der Schleife umfassten Fläche und dem Kernvolumen. \cite{Albach2020} +} + +Was bedeutet dies? Um den Ringkern zu magnetisieren und zu entmagnetisieren wird Energie aufgewendet. Da dieser Prozess nicht Verlustfrei ist entstehen Energieverluste. Der Verlust entspricht der Fläche innerhalb der Hysteresekurve. diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/gegeninduktion.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/gegeninduktion.tex new file mode 100644 index 0000000..6695900 --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/gegeninduktion.tex @@ -0,0 +1,52 @@ +\section{Die Gegeninduktion} \label{sec:ii} + +Der magnetische Fluss, welches aus einer Leiterschleife resultiert, kann eine andere Leiterschleife beeinflussen. In Kapitel \ref{sec:ig} haben wir gesehen, dass ein magnetischer Fluss eine Spannung in einer Leiterschleife induzieren kann. Dies ist auch hier der Fall. Betrachten wir die folgende Abbildung. + +\includegraphics[width=\linewidth]{fig/Fig78.png} \cite{Albach2020} + +Wie vorher beschrieben wird in der zweiten Leiterschleife ein Spannung induziert aufgrund des magnetischen Flusses von der ersten Leiterschleife. Nun beeinflusst der magnetische Fluss von der zweiten Leiterschleife die erste Leiterschleife. Die Spannungen in den jeweiligen Leiterschleifen kann wie folgt berechnet werden. + +\begin{equation} + u_1 = R_1 i_1 + L_{11} \cdot \frac{di_1}{dt} + L_{12} \cdot \frac{di_2}{dt} +\end{equation} + +\begin{equation} + u_2 = R_2 i_2 + L_{21} \cdot \frac{di_1}{dt} + L_{22} \cdot \frac{di_2}{dt} +\end{equation} + +$L_{11}$ und $L_{22}$ sind die Selbstinduktivitäten von der ersten bzw. der zweiten Leiterschleife während $L_{12}$ und $L_{21}$ die Gegeninduktivitäten von der ersten bzw. der zweiten Leiterschleife sind. Da $L_{12} = L_{21}$ (\cite{Albach2020} S. 274) bezeichnen wir die Gegeninduktion als $M$. Somit gilt für die induzierte Spannung die folgende Formel. + +\begin{equation} + u_{\text{ind}} = M \cdot \frac{di_1}{dt} +\end{equation} + +\subsection{Die Gegeninduktivität zweier Doppelleitungen} + +Für die Gegeninduktivität zweier Doppelleitungen gilt für den Faktor $M$ folgendes. + +\begin{minipage}{0.7\linewidth} + \begin{equation} + M = \frac{\mu \cdot l}{2 \cdot \pi} \cdot \ln(\frac{bc}{ad}) + \end{equation} +\end{minipage} +\begin{minipage}{0.3\linewidth} + \includegraphics[width=\linewidth]{fig/Fig79.png} \cite{Albach2020} +\end{minipage} + +\subsection{Die Koppelfaktoren} + +\dfn{Der Koppelfaktor}{ + Der Koppelfaktor ist ein Mass wie gut der magnetische Fluss einer Leiterschleife zur anderen übertragen lässt. Ist der Koppelfaktor hoch, so dringt fast der gesamte magnetische Fluss von der einen Spule zur anderen. Ist der Koppelfaktor niedrig, so dringt nur ein kleiner Teil des magnetischen Flusses von der einen Spule zur anderen. + + \begin{equation} + k_{12} = \frac{\Phi_{12}}{\Phi_{22}} = \frac{M}{L_{22}} + \end{equation} + + \begin{equation} + k_{21} = \frac{\Phi_{21}}{\Phi_{11}} = \frac{M}{L_{11}} + \end{equation} + + \begin{equation} + k = \pm \sqrt{k_{12} \cdot k_{21}} = \frac{M}{\sqrt{L_{11} \cdot L_{22}}} + \end{equation} +} diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/induktionsgesetz.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/induktionsgesetz.tex new file mode 100644 index 0000000..e243d9c --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/induktionsgesetz.tex @@ -0,0 +1,56 @@ +\section{Das Induktionsgesetz} \label{sec:ig} + +In Kapitel \ref{sec:lk} haben wir gelernt, dass auf elektrische Ladungen, welche durch ein magnetisches Feld sich bewegen eine Kraft ausgeübt wird. Wir betrachten nun grössere Anordnungen. +\\ +\\ +Nehmen wir an, dass sich ein Stück Metall durch ein magnetisches Feld bewegt. Aufgrund der Lorenzkraft bewegen sich die elektrische Ladungen innerhalb des Metalles. An einem Ende entsteht ein Elektronenüberschuss und am anderen Ende eine Elektronenmangel. + +\includegraphics[width=\linewidth]{fig/Fig75.png} + +Das Metallstück kann nun als Spannungsquelle agieren. Wird es mit einem Widerstand verbunden, so fliesst ein Strom. Die Spannung der Spannungsquelle kann mit der folgenden Gleichung berechnet werden. + +\begin{equation} + U_{12} = l \cdot v_x \cdot B = L \cdot \frac{dx}{dt} \cdot B +\end{equation} + +Da die Fläche der Leiterschleife mit der Zeit abnimmt, da das Metallstück immer näher zum Widerstand sich bewegt, können wir wir annehmen, dass + +\[ + l \cdot \frac{dx}{dt} = \frac{dA}{dt} + .\] + +Daraus folgt, dass die Induzierte Spannung einer Leiterschleife wie folgt berechnet werden kann. + +\begin{equation} + U_{\text{ind}} = -\frac{d \Phi}{dt} + \label{eq:uind} +\end{equation} + +\nt{ + Der in einer Leiterschleife induzierte Strom wirkt der ihn verursachenden Flussänderung entgegen. Deswegen ist das vorzeichen in Gleichung \ref{eq:uind} negativ. +} + +Was passiert wenn angenommen wird, dass anstelle von der Fläche der Leiterschleife das magnetische Feld sich ändert. Auch hier kann die Gleichung \ref{eq:uind} verwendet werden. (Rechung \cite{Albach2020} S. 261) + +\nt{ + Man unterscheidet zwischen der Bewegungsinduktion und der Ruheinduktion. + + \begin{itemize} + \item Bei der Bewegungsinduktion verändert sich die Fläche der Leiterschleife + \item Bei der Ruheinduktion verändert sich der magnetische Fluss + \end{itemize} + + Die Berechnung der induzierten Spannung ist die selbe. +} + +Für die Gleichung \ref{eq:uind} haben wir angenommen, dass die Fläche der Leiterschleife sich linear ändert, weshalb eine zeitlich konstante Spannung induziert wird. Sobald jedoch die Fläche sich nicht mehr linear ändert ist die induzierte Spannung auch nicht mehr zeitlich konstant. Die induzierte Spannung ist nun von der Zeit abhängig. + +\begin{equation} + u(t) = - \frac{d}{dt} \iint_A \vec{B} \cdot d \vec{A} +\end{equation} + +Dies folgt aus dem Faraday'sche Induktionsgesetz. + +\dfn{Faraday'sche Induktionsgesetz}{ + Das Faraday'sche Induktionsgesetz besagt, dass in einer Leiterschleife ein Strom fliesst, sobald der magnetischer Fluss, welcher abhängig ist mit der Leiterschleife sich zeitlich ändert. +} diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/punktkonvention.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/punktkonvention.tex new file mode 100644 index 0000000..3c7b97e --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/punktkonvention.tex @@ -0,0 +1,5 @@ +\section{Punktkonvention} + +In Schaltplänen kann man nicht erkennen, in welcher Richtung die Spule gewickelt ist. Durch die Punktkonvention kennzeichnet man mit einem Punkt wo der Draht der Spule vorne liegt. Dadurch kann man die Stromrichtung bestimmen. + +\includegraphics[width=\textwidth]{fig/Fig90.png} \cite{Albach2020} diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/ruheinduktion.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/ruheinduktion.tex new file mode 100644 index 0000000..2303924 --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/ruheinduktion.tex @@ -0,0 +1,52 @@ +\section{Anwendung der Ruheinduktion} + +Eine wichtige Anwendung für die Ruheinduktion ist der Transformator (Übertrager). Diese haben vor allem in der Starkstromtechnik und in der Leistungselektronik eine grosse Bedeutung. Ihre Funktion besteht darin Spannungen zu transformieren und Leistungen zwischen galvanisch getrennten Netzwerken zu übertragen. + +\includegraphics[width=\textwidth]{fig/Fig88.png} \cite{Albach2020} + +Ein Transformator besteht in den meisten Fällen aus einem Kern gewickelt von zwei Spulen. Der Kern führt den magnetischen Fluss von der einen Spule zur anderen. Dadurch wird eine Spannung induziert. +\\ +\\ +Bei den Transformatoren werden zwischen 3 verscheiden Vereinfachungen unterschieden. + +\dfn{Der Ideale Transformator}{ + Beim idealen Transformator vernachlässigen wir jegliche Verluste im Draht und im Eisenkern. Folglich gelten die gegebenen Zusammenhänge. + + \begin{equation} + \frac{I_1}{I_2} = \frac{U_1}{U_2} = \frac{N_2}{N_1} + \end{equation} + + Folglich ist diese Art von Transformator sehr stark vereinfacht, jedoch ungenau. +} + +\dfn{Verlustloser Transformator}{ + In Kapitel \ref{sec:ii} haben wir gelernt, dass Leiterschleifen, welche um ein Kern gewickelt sind sich gegenseitig beeinflussen können. Diese Gegeninduktion wird im Vergleich zum idealen Transformator berücksichtigt. Die induzierten Spannungen können wie in Kapitel \ref{sec:ii} berechnet werden. +} + +\dfn{Der verlustbehaftete Transformator}{ + Für den verlustbehafteten Transformator wird neben der Gegeninduktion der Widerstand der Spulen berücksichtigt. +} + +\subsection{Der Spartransformator} + +Falls man eine Spannungsumwandlung benötigt, jedoch die zwei Spannungsnetzwerke nicht galvanisch getrennt sein müssen, so kann man auch ein Spartransformator verwenden. Diese hat einen viel einfacheren Aufbau im Vergleich zum normalen Transformator. + +\dfn{Der Spartransformator}{ + Der Spartransformator ist eine kompakte Bauform des Transformators. Ein Teil der Wicklung wird sowohl von der Primär- als auch von der Sekundärseite verwendet. + + \includegraphics[width=\linewidth]{fig/Fig89.png} \cite{Albach2020} + + Es gelten die folgenden Gleichungen. + + \begin{equation} + \frac{u_p}{u_s} = \frac{N_p}{N_s} = \frac{N_1}{N_1 + N_2} = \frac{i_s}{i_p} + \end{equation} +} + +\nt{ + Beim Spartransformator ist die Eingangsleitung und die Ausgangsleistung zu jedem Zweitpunkt gleich. Ist die Ausgangsspannung grösser als die Eingangsspannung, so ist der Ausgangsstrom kleiner als der Eingangsstrom und umgekehrt. + \[ + u_p \cdot i_p = u_s \cdot i_s + .\] +} + diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/selbstinduktion.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/selbstinduktion.tex new file mode 100644 index 0000000..e6091e8 --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/selbstinduktion.tex @@ -0,0 +1,37 @@ +\section{Die Selbstinduktion} + +Nehmen wir an, dass durch die Leiterschleife ein zeitlich veränderlicher Strom fliesst. In Kapitel \ref{sec:mf} haben wir gelernt, dass der Strom zusammenhängt mit dem magnetischen Fluss. Dieser Zusammenhang haben wir als Induktivität bezeichnet. Daraus folgt, dass eine Veränderung des Stroms zu einer Veränderung des magnetischen Flusses beiträgt. Dieses Phänomen wird Selbstinduktion genannt. + +\dfn{Selbstinduktion}{ + Die Selbstinduktion ist die Induktionswirkung eines Stromes auf seinen eigenen Leitkreis. Die Induktionsspannung $u_L$ ist proportional zur Änderungsrate $\frac{di}{dt}$ und es gilt + + \begin{equation} + u_L = L \cdot \frac{di}{dt} + \end{equation} +} + +\subsection{Einfache Induktivitätsnetzwerke} + +Das Zusammensetzen von mehreren Spulen kann den Widerständen gleichgesetzt werden. Deshalb gelten die folgenden Definitionen. + +\dfn{Seriegeschaltene Induktivitäten}{ + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig76.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{equation} + L_{\text{ges}} = \sum_{k=1}^n L_k + \end{equation} + \end{minipage} +} + +\dfn{Parallelgeschaltene Induktivitäten}{ + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig77.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{equation} + \frac{1}{L_{\text{ges}}} = \sum_{k=1}^n \frac{1}{L_k} + \end{equation} + \end{minipage} +} diff --git a/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/zeitlich_veraenderliche_elektromagnetische_feld.tex b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/zeitlich_veraenderliche_elektromagnetische_feld.tex new file mode 100644 index 0000000..9582368 --- /dev/null +++ b/hs24/nus_I/zeitlich_veraenderliche_elektromagnetische_feld/zeitlich_veraenderliche_elektromagnetische_feld.tex @@ -0,0 +1,9 @@ +\chapter{Das Zeitlich veränderliche elektromagnetische Feld} + +\input{induktionsgesetz.tex} +\input{selbstinduktion.tex} +\input{gegeninduktion.tex} +\input{energieinhalt_feld.tex} +\input{bewegungsinduktion.tex} +\input{ruheinduktion.tex} +\input{punktkonvention.tex} diff --git a/hs24/technische_mechanik/disclaimer.tex b/hs24/technische_mechanik/disclaimer.tex new file mode 100644 index 0000000..b05b7ae --- /dev/null +++ b/hs24/technische_mechanik/disclaimer.tex @@ -0,0 +1,12 @@ +\section*{DISCLAIMER} + +Diese Notizen wurden verfasst auf Basis der Vorlesung Technische Mechanik (HS24) von Prof. Dr. P. Tiso, dem Skript Technische Mechanik von Prof. Dr. S. Kaufmann, sowie dem PVK Skrip von L. de Windt. +\\ +\\ +Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). +\\ +\\ +Falls nicht anders deklariert wurden alle Grafiken eigenhändig mit Manim \cite{MCD2024} generiert. +\\ +\\ +Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden. diff --git a/hs24/technische_mechanik/dynamik/beschleunigung.tex b/hs24/technische_mechanik/dynamik/beschleunigung.tex new file mode 100644 index 0000000..1fd6975 --- /dev/null +++ b/hs24/technische_mechanik/dynamik/beschleunigung.tex @@ -0,0 +1,28 @@ +\section{Beschleunigung} + +\dfn{Beschleunigung}{ + Die Beschleunigung gibt an, wie schnell sich die Geschwindigkeit eines Körpers sich ändert. \cite{Windt2023} Sie ist definiert als die Ableitung der Geschwindigkeit nach der Zeit $t$. + + \begin{equation} + \vec{a}(t) = \dot{\vec{v}}(t) = \ddot{\vec{r}}(t) + \end{equation} +} + +\subsection*{Karthesische Koordinaten} + +\begin{equation} + \vec{a} = \ddot{x} \cdot \vec{e}_x + \ddot{y} \cdot \vec{e}_y + \ddot{z} \cdot \vec{e}_z +\end{equation} + + +\subsection*{Zylindrischen Koordinaten} + +\begin{equation} + \vec{a} = (\ddot{\rho} - \rho \cdot \dot{\varphi}^2) \cdot \vec{e}_{\rho} + (\rho \cdot \ddot{\varphi} + 2 \cdot \dot{\rho} \cdot \dot{\varphi}) \cdot \vec{e}_{\varphi} + \ddot{z} \cdot \vec{e}_z +\end{equation} + +\subsection*{Polar Koordinaten} + +\begin{equation} + \vec{a} = (\ddot{\rho} - \rho \cdot \dot{\varphi}^2) \cdot \vec{e}_{\rho} + (\rho \cdot \ddot{\varphi} + 2 \cdot \dot{\rho} \cdot \dot{\varphi}) \cdot \vec{e}_{\varphi} +\end{equation} diff --git a/hs24/technische_mechanik/dynamik/drall_drallsatz.tex b/hs24/technische_mechanik/dynamik/drall_drallsatz.tex new file mode 100644 index 0000000..09cfadd --- /dev/null +++ b/hs24/technische_mechanik/dynamik/drall_drallsatz.tex @@ -0,0 +1,63 @@ +\section{Drall und Drallstatz} + +\dfn{Drall}{ + Der Drall ist eine physikalische Grösse, welche den Bewegungszustand eines rotierenden starren Körpers bestimmt. Den Drall kann man bezüglich eines inertialen Punktes (das heisst ein Punkt in Ruhe) oder bezüglich des Massenmittelpunktes bestimmen. \cite{Windt2023} + + \begin{minipage}{0.4\linewidth} + Bezüglich inertialer Punkt O: + + \begin{equation} + \vec{L}_O = \iiint_B \vec{r} \times \vec{v} d m + \end{equation} + + Bezüglich Massenmittelpunkt C: + + \begin{equation} + \vec{L}_C = \iiint_B \vec{r}' \times \vec{v}' d m + \end{equation} + + \end{minipage} + \begin{minipage}{0.6\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_45.png} + \end{minipage} +} + +\subsection*{Satz von Steiner} + +Wie beim Moment gibt es für den Drall eine Transformationsregele zwischen verschiedenen Punkten. + +\begin{equation} + \vec{L}_O = \vec{r}_C \times \vec{p} + \vec{L}_C +\end{equation} + +\dfn{Drallstatz}{ + Der Drallstatz ist ein physikalisches Gesetz, welches besagt, dass zur Änderung des Drehimpulses eines Körpers ein Drehmoment an ihm aufgebracht werden muss. + \\ + Drallstatz bezüglich des inertialen Punktes $O$: + \begin{equation} + \vec{L}_O = I_O \cdot \omega + \end{equation} + + Deallsatz bezüglich des Messenmittelpunktes $C$: + + \begin{equation} + \vec{L}_C = I_C \cdot \omega + \end{equation} +} + +\nt{ + Durch die Ableitung des Dralls erhält man das Moment. + + bezüglich O: + + \begin{equation} + \dot{\vec{L}}_O = \vec{M}_O ^{tot} + \end{equation} + + bezüglich C: + + \begin{equation} + \dot{\vec{L}}_C = \vec{M}_C ^{tot} + \end{equation} +} + diff --git a/hs24/technische_mechanik/dynamik/dynamik.tex b/hs24/technische_mechanik/dynamik/dynamik.tex new file mode 100644 index 0000000..078a3d5 --- /dev/null +++ b/hs24/technische_mechanik/dynamik/dynamik.tex @@ -0,0 +1,8 @@ +\chapter{Dynamik} + +\input{beschleunigung.tex} +\input{kinematische_relationen.tex} +\input{feder.tex} +\input{impuls.tex} +\input{drall_drallsatz.tex} +\input{massentraegheitsmoment.tex} diff --git a/hs24/technische_mechanik/dynamik/feder.tex b/hs24/technische_mechanik/dynamik/feder.tex new file mode 100644 index 0000000..5b6dfb4 --- /dev/null +++ b/hs24/technische_mechanik/dynamik/feder.tex @@ -0,0 +1,19 @@ +\section{Feder} + +\dfn{Feder}{ + Eine Feder ist ein Bauteil, welches eine entgegengesetzte Kraft ausübt, da sie in ihre ursprüngliche Lage zurück will. Diese Kraft kann durch das Hooksche Gesetz bestimmt werden. + + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_44.png} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \begin{equation} + F_{\text{Feder}} = \pm k \cdot \Delta x + \end{equation} + \end{minipage} +} + +\nt{ + Federn verringern den Freiheitsgrad eines Systems nicht. \cite{Windt2023} +} + diff --git a/hs24/technische_mechanik/dynamik/impuls.tex b/hs24/technische_mechanik/dynamik/impuls.tex new file mode 100644 index 0000000..60b64da --- /dev/null +++ b/hs24/technische_mechanik/dynamik/impuls.tex @@ -0,0 +1,20 @@ +\section{Impuls} + +\dfn{Impuls}{ + Der Impuls beschreibt den Bewegungszustand eines Körpers. Sie ist definiert durch die folgende Formel. + + \begin{equation} + \vec{p} = m \cdot \vec{v} + \end{equation} +} + +Durch die Ableitung vom Impuls durch die Zeit $t$ erhält man die resultierende Kraft, welche auf den Körper wirkt. Falls die Masse konstant ist, so gilt folgendes. + +\begin{equation} + \dot{\vec{p}} = m \cdot \dot{\vec{c}} = m \cdot \vec{a} = \vec{R} +\end{equation} + +\nt{ + Durch den Impuls können wir nun das Bewegungsgesetz, sowie den Massemittelpunktsatz bilden. Diese beschreiben die Bewegung eines Starrkörpers. +} + diff --git a/hs24/technische_mechanik/dynamik/kinematische_relationen.tex b/hs24/technische_mechanik/dynamik/kinematische_relationen.tex new file mode 100644 index 0000000..b63b341 --- /dev/null +++ b/hs24/technische_mechanik/dynamik/kinematische_relationen.tex @@ -0,0 +1,12 @@ +\section{Kinematische Relationen} + +Kinematische Relationen kennzeichnen meistens, dass eine Koordinate mit einer anderen Koordinate abhängt. Dies ist meistens der Fall bei Rollen. + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_43.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{equation} + x = R \cdot \varphi + \end{equation} +\end{minipage} diff --git a/hs24/technische_mechanik/dynamik/massentraegheitsmoment.tex b/hs24/technische_mechanik/dynamik/massentraegheitsmoment.tex new file mode 100644 index 0000000..0ce7de2 --- /dev/null +++ b/hs24/technische_mechanik/dynamik/massentraegheitsmoment.tex @@ -0,0 +1,21 @@ +\section{Massenträgheitsmoment} + +\dfn{Massenträgheitsmoment}{ + Das Massenträgheitsmoment gibt die Trägheit eines starren Körpers gegenüber einer Änderung seiner Rotationsgeschwindigkeit bei der Drehung um eine gegebene Achse an. \cite{Windt2023} + \\ + bezüglich O: + \begin{equation} + I_O = \iint_B r ^2 dm + \end{equation} + + bezüglich C: + \begin{equation} + I_C = \iint_B r^{\prime2} dm + \end{equation} +} + +Wie beim Moment und Drall gibt es eine Transformationsregel für den Massenträgheitsmoment. + +\begin{equation} + I_O = m \cdot r_{OC} ^2 + I_C +\end{equation} diff --git a/hs24/technische_mechanik/fig/Fig_1.png b/hs24/technische_mechanik/fig/Fig_1.png new file mode 100644 index 0000000..1458fc8 Binary files /dev/null and b/hs24/technische_mechanik/fig/Fig_1.png differ diff --git a/hs24/technische_mechanik/fig/Fig_10.png b/hs24/technische_mechanik/fig/Fig_10.png new file mode 100644 index 0000000..3a5447e Binary files /dev/null and b/hs24/technische_mechanik/fig/Fig_10.png differ diff --git a/hs24/technische_mechanik/fig/Fig_11.png b/hs24/technische_mechanik/fig/Fig_11.png new file mode 100644 index 0000000..6d1b224 Binary files /dev/null and b/hs24/technische_mechanik/fig/Fig_11.png differ diff --git a/hs24/technische_mechanik/fig/Fig_12.png b/hs24/technische_mechanik/fig/Fig_12.png new file mode 100644 index 0000000..97a50c0 Binary files /dev/null and b/hs24/technische_mechanik/fig/Fig_12.png differ diff --git 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differ diff --git a/hs24/technische_mechanik/graphs/__pycache__/graphs.cpython-312.pyc b/hs24/technische_mechanik/graphs/__pycache__/graphs.cpython-312.pyc new file mode 100644 index 0000000..1deb308 Binary files /dev/null and b/hs24/technische_mechanik/graphs/__pycache__/graphs.cpython-312.pyc differ diff --git a/hs24/technische_mechanik/graphs/graphs.py b/hs24/technische_mechanik/graphs/graphs.py new file mode 100644 index 0000000..0a77148 --- /dev/null +++ b/hs24/technische_mechanik/graphs/graphs.py @@ -0,0 +1,1065 @@ +from manim import * +from manim_physics import * +from manim.typing import Point3D +from manim.utils.rate_functions import sqrt +import numpy as np + + +class mov1(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + ax = ThreeDAxes( + x_range=[-0.3, 10.3, 1], + y_range=[-0.3, 10.3, 1], + z_range=[-0.3, 10.3, 1], + x_length=7.5, + y_length=7.5, + axis_config={ + "color": BLACK, + }, + ) + + self.set_camera_orientation( + phi=55 * DEGREES, theta=-45 * DEGREES, gamma=-7 * DEGREES, zoom=0.5 + ) + + axes_lables = ax.get_axis_labels().set_color(BLACK) + + graph = ax.plot_parametric_curve( + lambda u: (u, u, u**2), t_range=(0, 3, 0.01), color=RED + ) + + point = Dot3D(point=ax.c2p(2, 2, 4), color=BLUE, radius=0.3) + + self.add(ax, axes_lables, graph, point) + + +class mov2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + graph = FunctionGraph( + lambda t: sqrt(t), x_range=[0, 10, 0.01], color=RED + ).move_to([-0.5, 0, 0]) + + point = Dot(point=[0, 0.8, 0], color=BLUE, radius=0.08) + + vvec = Arrow([-0.2, 0.7, 0], [1, 1, 0], color=GREEN) + + self.add(graph, point, vvec) + + +class mov3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [1, 1, 0], + [-1, 1.5, 0], + [-2, 2, 0], + [-3, 1, 0], + [-1, 0, 0], + [0, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + dota = Dot(ORIGIN, color=RED, radius=0.08) + dotb = Dot([-2, 1, 0], color=BLUE, radius=0.08) + + vl = Line([2, -1, 0], [-4, 2, 0], color=GREEN) + + va = Arrow(ORIGIN, [1, 1, 0], color=PINK) + vb = Arrow([-2, 1, 0], [-2.7, -2, 0], color=PINK) + vap = Arrow([-0.2, 0.1, 0], [0.7, -0.35, 0], color=PURPLE) + vbp = Arrow([-2.2, 1.1, 0], [-1.2, 0.6, 0], color=PURPLE) + + ta = MathTex(r"A", color=BLACK).move_to([-2, 1.5, 0]) + tb = MathTex(r"B", color=BLACK).move_to([0, 0.5, 0]) + tva = MathTex(r"\vec{v}_A", color=BLACK).move_to([1, 1.5, 0]) + tvb = MathTex(r"\vec{v}_B", color=BLACK).move_to([-2.7, -2.5, 0]) + tvap = MathTex(r"\vec{v}_A '", color=BLACK).move_to([0.7, 0.15, 0]) + tvbp = MathTex(r"\vec{v}_B '", color=BLACK).move_to([-1.2, 1.1, 0]) + + self.add(body, dota, dotb, vl, va, vb, vap, vbp, ta, tb, tva, tvb, tvap, tvbp) + + +class mov4(ThreeDScene): + def construct(self): + + self.camera.background_color = WHITE + + ax = ThreeDAxes( + x_range=[-0.3, 10.3, 1], + y_range=[-0.3, 10.3, 1], + z_range=[-0.3, 10.3, 1], + x_length=7.5, + y_length=7.5, + axis_config={ + "color": BLACK, + }, + ) + + self.set_camera_orientation( + phi=50 * DEGREES, theta=5 * DEGREES, gamma=0 * DEGREES, zoom=0.5 + ) + + axes_lables = ax.get_axis_labels().set_color(BLACK) + + body = Sphere([0, 0, 2], radius=1, color=BLUE) + + vvec1 = Arrow3D([0, 1, 3], [0, 2, 3], color=RED) + vvec2 = Arrow3D([0, 1, 2], [0, 2, 2], color=RED) + vvec3 = Arrow3D([0, 1, 1], [0, 2, 1], color=RED) + + self.add(ax, axes_lables, body, vvec1, vvec2, vvec3) + + +class mov5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + dota = Dot(ORIGIN, color=RED, radius=0.08) + dotb = Dot([-1, 1, 0], color=BLUE, radius=0.08) + dotc = Dot([1, 0.5, 0], color=GREEN, radius=0.08) + + va = Arrow(ORIGIN, [1, 1, 0], color=PINK) + vb = Arrow([-1, 1, 0], [0, 2, 0], color=PINK) + vc = Arrow([1, 0.5, 0], [2, 1.5, 0], color=PINK) + + ta = MathTex(r"A", color=BLACK).move_to([0, 0.5, 0]) + tb = MathTex(r"B", color=BLACK).move_to([-1, 1.5, 0]) + tc = MathTex(r"C", color=BLACK).move_to([1, 1, 0]) + tva = MathTex(r"\vec{v}_A", color=BLACK).move_to([1, 1.5, 0]) + tvb = MathTex(r"\vec{v}_B", color=BLACK).move_to([0, 2.5, 0]) + tvc = MathTex(r"\vec{v}_C", color=BLACK).move_to([2, 2, 0]) + + self.add(body, dota, dotb, dotc, va, vb, vc, ta, tb, tc, tva, tvb, tvc) + + +class mov6(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + vvec1 = CurvedArrow([1.5, 1.5, 0], [0.5, 2.5, 0], color=PINK) + vvec2 = CurvedArrow([-1.5, 2.5, 0], [-2.5, 1.5, 0], color=PINK) + vvec3 = CurvedArrow([-1, 0, 0], [0, -1, 0], color=PINK) + + self.add(body, vvec1, vvec2, vvec3) + + +class mov7(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + pmz = Dot([1, 0, 0], color=RED, radius=0.08) + + vvec1 = CurvedArrow([1.5, 1.5, 0], [0.5, 2.5, 0], color=PINK) + vvec2 = CurvedArrow([-1.5, 2.5, 0], [-2.5, 1.5, 0], color=PINK) + vvec3 = CurvedArrow([-1, 0, 0], [0, -1, 0], color=PINK) + + tmz = MathTex(r"MZ", color=BLACK).move_to([1, 0.5, 0]) + + self.add(body, vvec1, vvec2, vvec3, pmz, tmz) + + +class mov8(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + vl = Line([-2, 2, 0], [2.5, -1, 0], color=ORANGE) + + pmz = Dot([1, 0, 0], color=RED, radius=0.08) + pp = Dot([-0.5, 1, 0], color=BLUE, radius=0.08) + + vvec1 = CurvedArrow([1.5, -0.5, 0], [0.5, 0.5, 0], color=PINK) + vvec2 = Arrow([-0.5, 1, 0], [-1.5, -0.5, 0], color=PINK) + + tmz = MathTex(r"MZ", color=BLACK).move_to([1, 0.5, 0]) + tp = MathTex(r"P", color=BLACK).move_to([-0.5, 1.5, 0]) + + tvvec2 = MathTex(r"\vec{v}_p", color=PINK).move_to([-1.5, -0.5, 0]) + + self.add(body, vl, vvec1, vvec2, pmz, tmz, tp, pp, tvvec2) + + +class mov9(Scene): + def construct(self): + + self.camera.background_color = WHITE + + sk = Polygon( + [-2, -1.5, 0], [1, -1.5, 0], [2, 1.5, 0], [-1, 1.5, 0], color=BLACK + ) + + pa = Dot([-2, -1.5, 0], color=RED, radius=0.08) + pb = Dot([1, -1.5, 0], color=BLUE, radius=0.08) + pc = Dot([2, 1.5, 0], color=GREEN, radius=0.08) + pd = Dot([-1, 1.5, 0], color=ORANGE, radius=0.08) + + va = CurvedArrow([-0.5, -2, 0], [-0.5, -1, 0], color=PINK) + vb = CurvedArrow([2, 0, 0], [1, 0, 0], color=PINK) + vc = CurvedArrow([0.5, 1, 0], [0.5, 2, 0], color=PINK) + vd = CurvedArrow([-1, 0, 0], [-2, 0, 0], color=PINK) + + ta = MathTex(r"A", color=BLACK).move_to([-2, -2, 0]) + tb = MathTex(r"B", color=BLACK).move_to([1, -2, 0]) + tc = MathTex(r"C", color=BLACK).move_to([2, 2, 0]) + td = MathTex(r"D", color=BLACK).move_to([-1, 2, 0]) + + tva = MathTex(r"\omega_1", color=BLACK).move_to([-0.5, -2.5, 0]) + tvb = MathTex(r"\omega_2", color=BLACK).move_to([2.5, 0, 0]) + tvc = MathTex(r"\omega_3", color=BLACK).move_to([0.5, 2.5, 0]) + tvd = MathTex(r"\omega_4", color=BLACK).move_to([-2.5, 0, 0]) + + self.add(sk, pa, pb, pc, pd, va, vb, vc, vd, ta, tb, tc, td, tva, tvb, tvc, tvd) + + +class mov10(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + vl1 = Line([-2, 2, 0], [2.5, -1, 0], color=ORANGE) + vl2 = Line([0, 2.5, 0], [1.5, -1.25, 0], color=ORANGE) + + pmz = Dot([1, 0, 0], color=RED, radius=0.08) + pa = Dot([-0.5, 1, 0], color=BLUE, radius=0.08) + pb = Dot([0.5, 1.25, 0], color=GREEN, radius=0.08) + + vvec1 = CurvedArrow([1.5, -0.5, 0], [0.5, 0.5, 0], color=PINK) + vvec2 = Arrow([-0.5, 1, 0], [-1.5, -0.5, 0], color=PINK) + vvec3 = Arrow([0.5, 1.25, 0], [-0.875, 0.5, 0], color=PINK) + + tmz = MathTex(r"MZ", color=BLACK).move_to([1, 0.5, 0]) + ta = MathTex(r"A", color=BLACK).move_to([-0.5, 1.5, 0]) + tb = MathTex(r"B", color=BLACK).move_to([0.5, 1.75, 0]) + + tvvec2 = MathTex(r"\vec{v}_A", color=PINK).move_to([-1.5, -0.5, 0]) + tvvec3 = MathTex(r"\vec{v}_B", color=PINK).move_to([-0.875, 0.5, 0]) + + self.add( + body, + vl1, + vl2, + vvec1, + vvec2, + vvec3, + pmz, + tmz, + ta, + tb, + pa, + pb, + tvvec2, + tvvec3, + ) + + +class mov11(Scene): + def construct(self): + + self.camera.background_color = WHITE + + vl = DashedLine([-1.5, -1.5, 0], [1.5, 1.5, 0], color=ORANGE) + + pp = Dot([-0.5, -0.5, 0], color=RED) + + fvec1 = Arrow([-0.5, -0.5, 0], [0.5, 0.5, 0], color=PINK) + + pt = MathTex(r"P \text{: Angriffspunkt}", color=BLACK).move_to([1.5, -0.5, 0]) + fvec1t = MathTex(r"\vec{F} = F \cdot \vec{e}_F", color=PINK).move_to( + [1.5, 0.5, 0] + ) + + self.add(vl, pp, fvec1, pt, fvec1t) + + +class mov12(Scene): + def construct(self): + + self.camera.background_color = WHITE + + fvec1 = Arrow([-2, -1.5, 0], [1, -1.5, 0], color=RED) + fvec2 = Arrow([-2, -1.5, 0], [-1, 1.5, 0], color=BLUE) + fvec3 = Arrow([-2, -1.5, 0], [2, 1.5, 0], color=GREEN) + + vl1 = DashedLine([-1, 1.5, 0], [2, 1.5, 0], color=ORANGE) + vl2 = DashedLine([1, -1.5, 0], [2, 1.5, 0], color=ORANGE) + + pp = Dot([-2, -1.5, 0], color=BLACK, radius=0.08) + + pt = MathTex(r"P", color=BLACK).move_to([-2, -2, 0]) + fvec1t = MathTex(r"\vec{F}_1", color=BLUE).move_to([-2, 0, 0]) + fvec2t = MathTex(r"\vec{F}_2", color=RED).move_to([-0.5, -2, 0]) + fvec3t = MathTex(r"\vec{F}_3", color=GREEN).move_to([2, 1.5, 0]) + + self.add(fvec1, fvec2, fvec3, vl1, vl2, pp, pt, fvec1t, fvec2t, fvec3t) + + +class mov13(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = Polygon([-1, -1, 0], [1, -1, 0], [1, 1, 0], [-1, 1, 0], color=BLACK) + floor = Line(config.left_side, config.right_side, color=BLACK).move_to( + [0, -1, 0] + ) + + fvec1 = Arrow([0, -1, 0], [0, -2, 0], color=RED) + fvec2 = Arrow([0, -1, 0], ORIGIN, color=BLUE) + + fvec1t = MathTex(r"\vec{F}_G", color=RED).move_to([0, -2, 0]) + fvec2t = MathTex(r"\vec{F}", color=BLUE).move_to(ORIGIN) + + self.add(body, floor, fvec1, fvec2, fvec1t, fvec2t) + + +class mov14(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = Polygon([-1, -1, 0], [1, -1, 0], [1, 1, 0], [-1, 1, 0], color=BLACK) + + fvec1 = Arrow([1, 0, 0], [2, 0, 0], color=RED) + fvec2 = Arrow([1, 0, 0], ORIGIN, color=BLUE) + + fvec1t = MathTex(r"\vec{F}_G", color=RED).move_to([2, 0, 0]) + fvec2t = MathTex(r"\vec{F}", color=BLUE).move_to(ORIGIN) + + self.add(body, fvec1, fvec2, fvec1t, fvec2t) + + +class mov15(Scene): + def construct(self): + + self.camera.background_color = WHITE + + vl = DashedLine([-1.25, -0.25, 0], [2.75, 1.75, 0], color=ORANGE) + + p0 = Dot([1.25, -1.5, 0], color=RED, radius=0.08) + pp = Dot([0.75, 0.75, 0], color=BLUE, radius=0.08) + ppp = Dot([0.25, 0.5, 0], color=GREEN, radius=0.08) + + vr = Arrow([1.25, -1.5, 0], [0.25, 0.5, 0], color=PURPLE) + vf = Arrow([0.75, 0.75, 0], [2.75, 1.75, 0], color=PINK) + + p0t = MathTex(r"O", color=BLACK).move_to([1.25, -2, 0]) + ppt = MathTex(r"P", color=BLACK).move_to([0.75, 1.25, 0]) + pppt = MathTex(r"P'", color=BLACK).move_to([-0.25, 0.5, 0]) + vrt = MathTex(r"\vec{r}", color=PURPLE).move_to([1.25, -0.5, 0]) + vft = MathTex(r"\vec{F}", color=PINK).move_to([2.75, 1.75, 0]) + + self.add(vl, p0, pp, ppp, vr, vf, p0t, ppt, pppt, vrt, vft) + + +class mov16(Scene): + def construct(self): + + self.camera.background_color = WHITE + + vl = DashedLine([-1.25, -0.25, 0], [2.75, 1.75, 0], color=ORANGE) + + p0 = Dot([1.25, -1.5, 0], color=RED, radius=0.08) + pp = Dot([0.25, 0.5, 0], color=GREEN, radius=0.08) + + vr = Arrow([1.25, -1.5, 0], [0.25, 0.5, 0], color=PURPLE) + vfp = Arrow([0.25, 0.5, 0], [2.25, 1.5, 0], color=PINK) + vf = Arrow([0.25, 0.5, 0], [1.75, 2.5, 0], color=BLUE) + + p0t = MathTex(r"O", color=BLACK).move_to([1.25, -2, 0]) + ppt = MathTex(r"P", color=BLACK).move_to([-0.25, 0.5, 0]) + vrt = MathTex(r"\vec{r}", color=PURPLE).move_to([1.25, -0.5, 0]) + vfpt = MathTex(r"F \cdot r \cdot \cos(\alpha)", color=PINK).move_to( + [2.75, 1.75, 0] + ) + vft = MathTex(r"F", color=BLUE).move_to([1.75, 2.5, 0]) + + self.add(vl, p0, pp, vr, vf, vfp, p0t, ppt, vrt, vfpt, vft) + + +class mov17(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + po = Dot([-2, -2, 0], color=BLACK, radius=0.06) + pi = Dot([1, 1, 0], color=BLACK, radius=0.06) + + vri = Arrow([-2, -2, 0], [1, 1, 0], color=BLUE) + vf1 = Arrow([1, 0, 0], [2, -1, 0], color=GREEN) + vf2 = Arrow([0, 1, 0], [0, 2.5, 0], color=GREEN) + vf3 = Arrow([-1, 1, 0], [-2, 0, 0], color=GREEN) + vfi = Arrow([1, 1, 0], [2, 3, 0], color=RED) + + tri = MathTex(r"\vec{r}_i", color=BLUE).move_to([-0.5, -1, 0]) + tf1 = MathTex(r"\vec{F}_1", color=GREEN).move_to([2, -1, 0]) + tf2 = MathTex(r"\vec{F}_2", color=GREEN).move_to([0, 2.5, 0]) + tf3 = MathTex(r"\vec{F}_3", color=GREEN).move_to([-2, 0, 0]) + tfi = MathTex(r"\vec{F}_i", color=RED).move_to([2, 3, 0]) + tpo = MathTex(r"O", color=BLACK).move_to([-1.5, -2, 0]) + tpi = MathTex(r"P_i", color=BLACK).move_to([1.5, 1, 0]) + + self.add( + body, po, pi, vri, tri, vf1, vf2, vf3, vfi, tf1, tf2, tf3, tfi, tpo, tpi + ) + + +class mov18(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + p = Dot([0, 1, 0], color=BLACK, radius=0.06) + + wl1 = DashedLine([0, 1, 0], [2, 3, 0], color=ORANGE) + wl2 = DashedLine([0, 1, 0], [2, 0, 0], color=ORANGE) + wl3 = DashedLine([0, 1, 0], [-3, 1, 0], color=ORANGE) + + vf1 = Arrow([1.75, 2.75, 0], [0.75, 1.75, 0], color=GREEN) + vf2 = Arrow([0.25, 0.875, 0], [1.25, 0.375, 0], color=GREEN) + vf3 = Arrow([-1, 1, 0], [-2, 1, 0], color=GREEN) + + tp = MathTex(r"P", color=BLACK).move_to([0.5, 1, 0]) + tf1 = MathTex(r"\vec{F_1}", color=GREEN).move_to([1.75, 2.75, 0]) + tf2 = MathTex(r"\vec{F_2}", color=GREEN).move_to([1.25, 0.375, 0]) + tf3 = MathTex(r"\vec{F_3}", color=GREEN).move_to([-2, 1, 0]) + + self.add(body, p, wl1, wl2, wl3, vf1, vf2, vf3, tp, tf1, tf2, tf3) + + +class mov19(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + p = Dot([0, 0.5, 0], color=BLACK, radius=0.06) + + vf1 = Arrow([0, 0.5, 0], [2, 1.5, 0], color=GREEN) + vf2 = Arrow([0, 1.5, 0], [-2, 0.5, 0], color=GREEN) + vr = Arrow([0, 0.25, 0], [0, 1.75, 0], color=BLUE) + + tp = MathTex(r"P", color=BLACK).move_to([0.5, 0.5, 0]) + tf1 = MathTex(r"\vec{F_1}", color=GREEN).move_to([2, 1.5, 0]) + tf2 = MathTex(r"\vec{F_2}", color=GREEN).move_to([-2, 0.5, 0]) + tr = MathTex(r"\vec{r}", color=BLUE).move_to([0, 1.75, 0]) + + self.add(body, p, vf1, vf2, vr, tp, tf1, tf2, tr) + + +class mov20(Scene): + def construct(self): + + self.camera.background_color = WHITE + + wl = DashedLine([-3, -3, 0], [3, 3, 0], color=ORANGE) + + vfa = Arrow([-2, -2, 0], [-1, -1, 0], color=GREEN) + vfb = Arrow([1, 1, 0], [2, 2, 0], color=GREEN) + + tfa = MathTex(r"\vec{F_A}", color=GREEN).move_to([-1, -1.5, 0]) + tfb = MathTex(r"\vec{F_B}", color=GREEN).move_to([2, 1.5, 0]) + + self.add(wl, vfa, vfb, tfa, tfb) + + +class stat1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = Polygon([-1, -1, 0], [1, -1, 0], [1, 1, 0], [-1, 1, 0], color=BLACK) + floor = Line(config.left_side, config.right_side, color=BLACK).move_to( + [0, -1, 0] + ) + + self.add(body, floor) + + +class stat2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + dot1 = Dot([-2, 0.5, 0], color=BLACK) + dot2 = Dot([2, 0.5, 0], color=BLACK) + dot3 = Dot([4, 0.5, 0], color=BLACK) + + sk11 = Line([-2, 0.5, 0], [-3, 0.5, 0], color=BLACK) + sk12 = DashedLine([-3, 0.5, 0], [-3.5, 0.5, 0], color=BLACK) + sk21 = Line([-2, 0.5, 0], [-3, -0.5, 0], color=BLACK) + sk22 = DashedLine([-3, -0.5, 0], [-3.5, -1, 0], color=BLACK) + sk31 = Line([-2, 0.5, 0], [-1, 0.5, 0], color=BLACK) + sk32 = DashedLine([-1, 0.5, 0], [-0.5, 0.5, 0], color=BLACK) + sk41 = Line([-2, 0.5, 0], [-1, -0.5, 0], color=BLACK) + sk42 = DashedLine([-1, -0.5, 0], [-0.5, -1, 0], color=BLACK) + sk51 = Line([2, 0.5, 0], [1, 0.5, 0], color=BLACK) + sk52 = DashedLine([1, 0.5, 0], [0.5, 0.5, 0], color=BLACK) + sk61 = Line([2, 0.5, 0], [1, -0.5, 0], color=BLACK) + sk62 = DashedLine([1, -0.5, 0], [0.5, -1, 0], color=BLACK) + sk71 = Line([4, 0.5, 0], [5, 0.5, 0], color=BLACK) + sk72 = DashedLine([5, 0.5, 0], [5.5, 0.5, 0], color=BLACK) + sk81 = Line([4, 0.5, 0], [5, -0.5, 0], color=BLACK) + sk82 = DashedLine([5, -0.5, 0], [5.5, -1, 0], color=BLACK) + + vf1 = Arrow([2, 0.5, 0], [2, 1.5, 0], color=GREEN) + vf2 = Arrow([4, 0.5, 0], [3, 0.5, 0], color=GREEN) + vf3 = Arrow([2, 0.5, 0], [3, 0.5, 0], color=GREEN) + vf4 = Arrow([4, 0.5, 0], [4, -0.5, 0], color=GREEN) + + tf1 = MathTex(r"A_x", color=GREEN).move_to([3, 1, 0]) + tf2 = MathTex(r"A_y", color=GREEN).move_to([2, 1.5, 0]) + tf3 = MathTex(r"A_y", color=GREEN).move_to([4, -0.5, 0]) + + ra = MathTex(r"\Rightarrow", color=BLACK).move_to([0, 0, 0]) + + self.add( + dot1, + dot2, + dot3, + sk11, + sk12, + sk21, + sk22, + sk31, + sk32, + sk41, + sk42, + sk51, + sk52, + sk61, + sk62, + sk71, + sk72, + sk81, + sk82, + vf1, + vf2, + vf3, + vf4, + tf1, + tf2, + tf3, + ra, + ) + + +class stat3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + sk1 = Line([-3, -1, 0], [-1, 1, 0], color=BLACK) + sk2 = Line([1, -1, 0], [3, 1, 0], color=BLACK) + + cvx1 = Arrow([-3, 0, 0], [-3, 1, 0], color=BLACK) + cvy1 = Arrow([-3, 0, 0], [-2, 0, 0], color=BLACK) + cvx2 = Arrow([1, 0, 0], [0, 1, 0], color=BLACK) + cvy2 = Arrow([1, 0, 0], [2, 1, 0], color=BLACK) + + tvx1 = MathTex(r"x", color=BLACK).move_to([-3, 0, 0]) + tvy1 = MathTex(r"y", color=BLACK).move_to([-3, 1, 0]) + tvx2 = MathTex(r"x", color=BLACK).move_to([1, 0, 0]) + tvy2 = MathTex(r"y", color=BLACK).move_to([0, 1, 0]) + + self.add(sk1, sk2, cvx1, cvx2, cvy1, cvy2, tvx1, tvx2, tvy1, tvy2) + + +class stat4(Scene): + def construct(self): + + self.camera.background_color = WHITE + + floor = Line([-6, -0.75, 0], [6, -0.75, 0], color=BLACK) + + pbody1 = Polygon( + [-1.25, -0.75, 0], [-0.75, -0.75, 0], [-1, -0.25, 0], color=BLACK + ) + ppivot1 = Dot([-1, -0.25, 0], color=BLACK) + pbody2 = Polygon([1.25, -0.75, 0], [0.75, -0.75, 0], [1, -0.25, 0], color=BLACK) + ppivot2 = Dot([1, -0.25, 0], color=BLACK) + + bsk1 = Line([-1, -0.25, 0], [1, -0.25, 0], color=BLACK) + bsk2 = Line([1, -0.25, 0], [0.5, 0.75, 0], color=BLACK) + bsk3 = Line([0.5, 0.75, 0], [-0.5, 0.75, 0], color=BLACK) + bsk4 = Line([-1, -0.25, 0], [-0.5, 0.75, 0], color=BLACK) + bsk5 = Line([0.5, 0.75, 0], [0, -0.25, 0], color=BLACK) + bsk6 = Line([-0.5, 0.75, 0], [0, -0.25, 0], color=BLACK) + bp1 = Dot([0, -0.25, 0], color=BLACK) + bp2 = Dot([0.5, 0.75, 0], color=BLACK) + bp3 = Dot([-0.5, 0.75, 0], color=BLACK) + + vv = Arrow([0.5, 0.75, 0], [0, 1.75, 0], color=RED) + vo = CurvedArrow([-0.75, -0.75, 0], [-1.25, 0.25, 0], color=RED) + + tv = MathTex(r"\tilde{v}", color=RED).move_to([0.5, 1.25, 0]) + to = MathTex(r"\tilde{\omega}", color=RED).move_to([-0.5, -0.75, 0]) + + pivot1 = VGroup(pbody1, ppivot1) + pivot2 = VGroup(pbody2, ppivot2) + sk = VGroup(bsk1, bsk2, bsk3, bsk4, bsk5, bsk6, bp1, bp2, bp3) + + self.add(floor, pivot1, pivot2, sk, vv, vo, tv, to) + + +class stat5(Scene): + def construct(self): + + self.camera.background_color = WHITE + + floor = Line([-6, -1.25, 0], [0, -1.25, 0], color=BLACK) + + pbody1 = Polygon( + [-5.25, -1.25, 0], [-4.75, -1.25, 0], [-5, -0.75, 0], color=BLACK + ) + ppivot1 = Dot([-5, -0.75, 0], color=BLACK) + pbody2 = Polygon( + [-1.25, -1.25, 0], [-0.75, -1.25, 0], [-1, -0.75, 0], color=BLACK + ) + ppivot2 = Dot([-1, -0.75, 0], color=BLACK) + + bsk1 = Line([-5, -0.75, 0], [-1, -0.75, 0], color=BLACK) + bsk2 = Line([-1, -0.75, 0], [-2, 1.25, 0], color=BLACK) + bsk3 = Line([-2, 1.25, 0], [-4, 1.25, 0], color=BLACK) + bsk4 = Line([-4, 1.25, 0], [-5, -0.75, 0], color=BLACK) + bsk5 = Line([-4, 1.25, 0], [-3, -0.75, 0], color=BLACK) + bp1 = Dot([-3, -0.75, 0], color=BLACK) + bp2 = Dot([-2, 1.25, 0], color=BLACK) + bp3 = Dot([-4, 1.25, 0], color=BLACK) + + vsf1 = Arrow([-3, -0.75, 0], [-2.5, 0.25, 0], color=GREEN) + vsf2 = Arrow([-2, 1.25, 0], [-2.5, 0.25, 0], color=GREEN) + + bsk6 = Line([2.5, -0.75, 0], [3.5, 1.25, 0], color=BLACK) + + vsf3 = Arrow([2.5, -0.75, 0], [2, -1.75, 0], color=GREEN) + vsf4 = Arrow([3.5, 1.25, 0], [4, 2.25, 0], color=GREEN) + + pivot1 = VGroup(pbody1, ppivot1) + pivot2 = VGroup(pbody2, ppivot2) + sk = VGroup(bsk1, bsk2, bsk3, bsk4, bsk5, bp1, bp2, bp3, vsf1, vsf2, vsf3, vsf4) + + self.add(floor, pivot1, pivot2, sk, bsk6) + + +class stat6(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ground = Polygon( + [-8, -1, 0], + [8, -1, 0], + [7, -7, 0], + [-7, -7, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + + body = Polygon( + [-1, 1, 0], + [1, 1, 0], + [1, -1, 0], + [-1, -1, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + + velvec = Arrow( + start=[-1, 1.5, 0], + end=[1, 1.5, 0], + color=RED, + ) + fricvec = Arrow( + start=[-0.75, -1, 0], + end=[-3, -1, 0], + color=YELLOW, + ) + forcevec = Arrow( + start=[0.75, 0, 0], + end=[3, 0, 0], + color=GREEN, + ) + + tvv = MathTex(r"\vec{v}", color=RED).move_to([0, 2, 0]) + tfrv = MathTex(r"\vec{F}", color=YELLOW).move_to([-2, -0.5, 0]) + tfov = MathTex(r"\vec{F}_R", color=GREEN).move_to([3.5, 0, 0]) + + self.add(ground, body, velvec, fricvec, forcevec, tvv, tfrv, tfov) + + +class stat7(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ground = Polygon( + [-8, -1, 0], + [8, -1, 0], + [7, -7, 0], + [-7, -7, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + + slope = Line([-2, 1, 0], [2, -1, 0], color=BLACK) + + body = ( + Polygon( + [-1, 1, 0], + [1, 1, 0], + [1, -1, 0], + [-1, -1, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + .rotate(-26.56 * DEGREES, axis=[0.0, 0.0, 1.0], about_point=None) + .move_to([0, 1.1, 0]) + ) + + velvec = Arrow(start=[-0.5, 2.75, 0], end=[2, 1.5, 0], color=RED) + fricvec = Arrow(start=[-1, 1, 0], end=[-3, 2, 0], color=YELLOW) + forcevec = Arrow(start=[0.5, 0, 0], end=[2.5, -1, 0], color=GREEN) + + tvv = MathTex(r"\vec{v}", color=RED).move_to([2, 1.5, 0]) + tfrv = MathTex(r"\vec{F}_R", color=YELLOW).move_to([-3, 2, 0]) + tfov = MathTex(r"\vec{F}", color=GREEN).move_to([2.5, -1, 0]) + tv0 = MathTex(r"\vec{v} = 0", color=RED).move_to([2, 0.5, 0]) + + self.add(ground, slope, body, velvec, fricvec, forcevec, tvv, tfrv, tfov, tv0) + + +class stat8(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ground = Polygon( + [-8, -1, 0], + [8, -1, 0], + [7, -7, 0], + [-7, -7, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + + slope = Line([-2, 1, 0], [2, -1, 0], color=BLACK) + + body = ( + Polygon( + [-1, 1, 0], + [1, 1, 0], + [1, -1, 0], + [-1, -1, 0], + color=BLUE, + fill_color=BLUE, + fill_opacity=1, + ) + .rotate(-26.56 * DEGREES, axis=[0.0, 0.0, 1.0], about_point=None) + .move_to([0, 1.1, 0]) + ) + + velvec = Arrow(start=[-0.5, 2.75, 0], end=[2, 1.5, 0], color=RED) + fricvec = Arrow(start=[-1, 1, 0], end=[-3, 2, 0], color=YELLOW) + forcevec = Arrow(start=[0.5, 0, 0], end=[2.5, -1, 0], color=GREEN) + + tvv = MathTex(r"\vec{v}", color=RED).move_to([2, 1.5, 0]) + tfrv = MathTex(r"\vec{F}_R", color=YELLOW).move_to([-3, 2, 0]) + tfov = MathTex(r"\vec{F}", color=GREEN).move_to([2.5, -1, 0]) + tv0 = MathTex(r"\vec{v} \neq 0", color=RED).move_to([2, 0.5, 0]) + + self.add(ground, slope, body, velvec, fricvec, forcevec, tvv, tfrv, tfov, tv0) + + +class stat9(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ground = Polygon( + [-8, -1, 0], + [8, -1, 0], + [7, -7, 0], + [-7, -7, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + + slope = Line([-2, 1, 0], [2, -1, 0], color=BLACK) + + body = Circle(radius=1, color=BLACK).move_to([0.25, 1, 0]) + + velvec = Arrow(start=[-0.5, 2.75, 0], end=[2, 1.5, 0], color=RED) + fricvec = Arrow(start=[0, 0, 0], end=[-2, 1, 0], color=YELLOW) + forcevec = Arrow(start=[0, 0, 0], end=[2, -1, 0], color=GREEN) + + tvv = MathTex(r"\vec{v}", color=RED).move_to([2, 1.5, 0]) + tfrv = MathTex(r"\vec{F}_R", color=YELLOW).move_to([-2, 1, 0]) + tfov = MathTex(r"\vec{F}", color=GREEN).move_to([2, -1, 0]) + + self.add(ground, slope, body, velvec, fricvec, forcevec, tvv, tfrv, tfov) + + +class dyn1(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = Circle(radius=1, color=BLACK).move_to([0, 2, 0]) + + pbody1 = Polygon([-0.25, 1.5, 0], [0.25, 1.5, 0], [0, 2, 0], color=BLACK) + ppivot1 = Dot([0, 2, 0], color=BLACK) + + line = Line([1, 2, 0], [1, -2, 0], color=BLACK) + m = Dot([1, -2, 0], color=BLACK, radius=0.1) + + vr = Arrow([0, 2, 0], [1, 2, 0], color=GREEN) + vphi = CurvedArrow([0, 3.25, 0], [1.25, 2, 0], angle=-90 * DEGREES, color=RED) + vx = Arrow([1, -2, 0], [1, -3, 0], color=BLUE) + + tr = MathTex(r"R", color=GREEN).move_to([0.5, 2.5, 0]) + tphi = MathTex(r"\phi", color=RED).move_to([1.5, 3, 0]) + tx = MathTex(r"x", color=BLUE).move_to([1.25, -2.5, 0]) + + pivot1 = VGroup(pbody1, ppivot1) + + self.add(body, pivot1, line, m, vr, vphi, vx, tr, tphi, tx) + + +class dyn2(Scene): + def construct(self): + + self.camera.background_color = WHITE + + ground = Polygon( + [-8, 6, 0], + [-8, -6, 0], + [-5, -6, 0], + [-5, 6, 0], + color=GREY, + fill_color=GREY, + fill_opacity=1, + ) + shold = Line([-5, 0, 0], [-3, 0, 0], color=BLACK) + ss1 = Line([-3, 0, 0], [-2.5, 0.5, 0], color=BLACK) + ss2 = Line([-2.5, 0.5, 0], [-1.5, -0.5, 0], color=BLACK) + ss3 = Line([-1.5, -0.5, 0], [-0.5, 0.5, 0], color=BLACK) + ss4 = Line([-0.5, 0.5, 0], [0.5, -0.5, 0], color=BLACK) + ss5 = Line([0.5, -0.5, 0], [1.5, 0.5, 0], color=BLACK) + ss6 = Line([1.5, 0.5, 0], [2.5, -0.5, 0], color=BLACK) + ss7 = Line([2.5, -0.5, 0], [3, 0, 0], color=BLACK) + + m = Dot([3, 0, 0], color=BLACK, radius=0.3) + + eb1 = DashedLine([1, 2, 0], [1, -2, 0], color=BLACK) + eb2 = DashedLine([3, 2, 0], [3, -2, 0], color=BLACK) + tdx = MathTex(r"\Delta x", color=BLACK).move_to([2, -2, 0]) + + spring = VGroup(shold, ss1, ss2, ss3, ss4, ss5, ss6, ss7) + extension = VGroup(eb1, eb2, tdx) + + self.add(ground, spring, m, extension) + + +class dyn3(Scene): + def construct(self): + + self.camera.background_color = WHITE + + body = ( + Polygram( + [ + [2, 1, 0], + [0, 1.5, 0], + [-1, 2, 0], + [-2, 1, 0], + [0, 0, 0], + [1, -1, 0], + ], + ) + .set_color(BLACK) + .round_corners(radius=0.73) + ) + + po = Dot([-2, -2, 0], color=BLACK, radius=0.06) + pdm = Dot([1, 1, 0], color=BLACK, radius=0.06) + pc = Dot([0, 0.75, 0], color=BLACK, radius=0.06) + + vr = Arrow( + [-2, -2, 0], [1, 1, 0], color=BLUE, max_tip_length_to_length_ratio=0.05 + ) + vroc = Arrow( + [-2, -2, 0], [0, 0.75, 0], color=BLUE, max_tip_length_to_length_ratio=0.07 + ) + vv = Arrow([0, 0.75, 0], [1, 1, 0], color=RED) + vrp = Arrow([1, 1, 0], [2, 1.5, 0], color=GREEN) + vvoc = Arrow([0, 0.75, 0], [-1, 0.25, 0], color=GREEN) + + tr = MathTex(r"\vec{r}", color=BLUE).move_to([-0.5, -1, 0]) + troc = MathTex(r"\vec{r}_{OC}", color=BLUE).move_to([-1.5, -0.5, 0]) + tvoc = 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b/hs24/technische_mechanik/graphs/media/Tex/fd82f95e3e649a68.tex @@ -0,0 +1,9 @@ +\documentclass[preview]{standalone} +\usepackage[english]{babel} +\usepackage{amsmath} +\usepackage{amssymb} +\begin{document} +\begin{align*} +\vec{v}_A +\end{align*} +\end{document} \ No newline at end of file diff --git a/hs24/technische_mechanik/kinematik/aequivalenz_reduktion_kraeftegruppen.tex b/hs24/technische_mechanik/kinematik/aequivalenz_reduktion_kraeftegruppen.tex new file mode 100644 index 0000000..dd261b5 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/aequivalenz_reduktion_kraeftegruppen.tex @@ -0,0 +1,92 @@ +\section{Äquivalenz und Reduktion von Kräftegruppen} + +\dfn{Kräftegruppe}{ + Eine Kräftegruppe ist eine Sammlung an Kräften, welche an einem Starrkörper wirken. + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_19.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{equation} + \{ \vec{F}_i \} = \{ \vec{F_1}, \vec{F_2}, ... , \vec{F_N} \} + \end{equation} + \end{minipage} + + \nt{ + Um die resultierende Kraft an einem Punkt $P_i$ zu bestimmen, wären sehr rechenaufwendige Prozesse nötig, welche mit grosser Wahrscheinlichkeit hier nicht besprochen wurden. + } + +} + +\dfn{Zentrale Kräftegruppe}{ + Wenn alle Wirkungslinien aller Kräfte einer Kräftegruppe durch einen Punkt gehen, so kann der Punkt als Angriffspunkt gesehen werden. In Kapitel \ref{sec:kraft} haben wir gelernt, dass eine Kraft entlang ihrer Wirkungslinie verschoben werden kann, weshalb die Kräfte auf einen Punkt wirken. Aus diesem Grund kann die resultierende Kraft $\vec{R}$ sehr einfach berechnet werden. + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_20.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \[ + \vec{R} = \sum_i \vec{F_i} = \vec{F_1} + \vec{F_2} + \vec{F_3} \neq 0 + .\] + + \[ + \vec{M_p} = \vec{0} + .\] + \end{minipage} +} + +\dfn{Kräftepaar}{ + Bei einem Kräftepaar haben die zwei Kräfte den gleichen Betrag aber die entgegengesetzte Richtung. Ihre Wirkungslinie müssen sich nicht schneiden. + + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_21.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \[ + \vec{R} = 0 + .\] + + \[ + \vec{M} = \vec{r} \times \vec{F} + .\] + + \[ + |\vec{F_1}| = |\vec{F_2}| = |F| + .\] + + \[ + \vec{F_1} = - \vec{F_2} = \vec{F} + .\] + \end{minipage} +} + +\dfn{Nullsystem}{ +Ein Nullsystem bezeichnet ein Starrkörper oder eine Kräftegruppe, welche im Gleichgewicht ist. Dies setzt die folgenden Bedingungen vorraus. + +\[ + \vec{R} = \vec{0} + .\] + +\[ + \vec{M}_0 ^{tot} = \vec{0} + .\] +} + +\dfn{Statische Äquivalenz}{ + Statische Äquivalenz bezeichnet zwei Kräftegruppen, deren Gesamtleistung gleich ist. + + \[ + P(\{ \vec{F_i} \}) = P(\{ \vec{G_i} \}) + .\] + + Dies bedeutet, dass wenn zwei Kräftegruppen in einem beliebigen Punkt die gleiche Dyname haben, sind sie statisch äquivalent. \cite{Windt2023} +} + +Wenn wir die Statische Äquivalenz von zwei Kräften betrachten, so muss folgendes gelten. + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_22.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + $\vec{F_A}$ und $\vec{F_B}$ sind statisch äquivalent. +\end{minipage} diff --git a/hs24/technische_mechanik/kinematik/dyname.tex b/hs24/technische_mechanik/kinematik/dyname.tex new file mode 100644 index 0000000..3699628 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/dyname.tex @@ -0,0 +1,4 @@ + +\section{Die Dyname} + +Die Dyname hat eine Ähnlichkeit wie die Kinemate. (Kapitel \ref{sec:kin}) Die Dyname beschreibt die Bewegung eines Starrkörpers anhand der Resultierende Kraft und dem Moment. Wie bei der Kinemate gibt es zwei Invarianten. $I_1 = \vec{R}$ und $I_2 = \vec{M}_P \cdot \vec{\omega}$. diff --git a/hs24/technische_mechanik/kinematik/ebene_bewegung.tex b/hs24/technische_mechanik/kinematik/ebene_bewegung.tex new file mode 100644 index 0000000..50566cc --- /dev/null +++ b/hs24/technische_mechanik/kinematik/ebene_bewegung.tex @@ -0,0 +1,45 @@ +\section{Ebene Bewegungen} \label{sec:evenm} + +\dfn{Ebene Bewegung}{ + Als ebene Bewegung bezeichnet man ein Körper, welches nur in unserem Fall in der xy-Ebene sich bewegt. Deswegen gilt die folgende Beziehung. + + \begin{minipage}{0.5\linewidth} + \[ + \left\{ + \begin{array}{lr} + v_z = 0 \\ + v_x = v_x(x,y) \text{ und } v_y = v_y(x,y) \\ + \end{array} + \right. + .\] + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_4.png} + \end{minipage} +} + +Für starre Körper unterscheiden wir zwischen zwei verschiedene ebene Bewegungen. + +\subsection*{Translation} + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_5.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{equation} + \vec{v}_A = \vec{v}_B + \end{equation} + $\Rightarrow$ Alle Punkte, welche innerhalb des Starrkörpers sind haben die gleiche Geschwindigkeit. +\end{minipage} + +\subsection*{Rotation} + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_6.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \begin{equation} + \vec{v}_p = \vec{\omega} \times \vec{r}_{MP} + \label{eq:rot} + \end{equation} +\end{minipage} diff --git a/hs24/technische_mechanik/kinematik/freiheitsgrad_bindung.tex b/hs24/technische_mechanik/kinematik/freiheitsgrad_bindung.tex new file mode 100644 index 0000000..1ac8f16 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/freiheitsgrad_bindung.tex @@ -0,0 +1,31 @@ +\section{Freiheitsgrad und Bindung} + +\dfn{Freiheitsgrad}{ + Unter den Freiheitsgrad versteht man die minimale Anzahl an Koordinaten, welche benötigt werden um die Lage eines Systems bestimmen zu können. \cite{Tiso2024} Dabei gilt die folgende Formel. + + \begin{equation} + f = n - b + \end{equation} + + $f$ ist der Freiheitsgrad, $n$ der Freiheitsgrad des ungebundenen Systems und $b$ die Anzahl unabhängiger Bindungen. +} + +Im zweidimensionalen hat ein Starkörper immer den Freiheitsgrad 3. Im dreidimensionalen hat ein Starkörper den Freiheitsgrad 6. +\\ +Für die Bindungen kann die folgende Tabelle betrachtet werden. + +\begin{center} + \begin{tabular}{| c | c | c |} + & Name & Bindung\\ + \hline + \includegraphics[width=0.25\linewidth]{fig/Fig_24.png} & Auflager & 1\\ + \includegraphics[width=0.25\linewidth]{fig/Fig_25.png} & Gelenk oder Slider & 2\\ + \includegraphics[width=0.25\linewidth]{fig/Fig_26.png} & Einspannung & 3\\ + \end{tabular} +\end{center} + +Für Gelenke, welche Starkörper verbindet kann die folgende Formel verwendet werden. + +\begin{equation} + b = (\text{Anzahl Starkörper am Gelenk} - 1) \cdot 2 +\end{equation} diff --git a/hs24/technische_mechanik/kinematik/geschwindigkeit.tex b/hs24/technische_mechanik/kinematik/geschwindigkeit.tex new file mode 100644 index 0000000..7fbc172 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/geschwindigkeit.tex @@ -0,0 +1,68 @@ +\section{Geschwindigkeit} + +\dfn{Geschwindigkeit}{ + Die Geschwindigkeit ist ein Vektor, welcher die Richtung sowie die Geschwindigkeit des Punktes beschreibt, mit welcher der Punkt die Bahnkurve durchläuft. Sie kann berechnet werden durch die zeitliche Ableitung der Ortsfunktion und verläuft immer tangential zur Bahnkurve. + + \begin{minipage}{0.5\linewidth} + \[ + \vec{v}(t) = \dot{\vec{r}}(t) = \frac{d}{dt} \vec{r}(t) + .\] + \end{minipage} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_2.png} + \end{minipage} +} + +Die Ableitung der Ortsfunktion geschieht komponentenweise. + +\subsection*{2D} + +\[ + \vec{v}(t) = \dot{\vec{r}}(t) = \frac{d}{dt} \vec{r}(t) = \frac{d}{dt} + \begin{pmatrix} + x \\ + y \\ + \end{pmatrix} = + \begin{pmatrix} + \frac{d}{dt}x \\ + \frac{d}{dt}y \\ + \end{pmatrix} = + \begin{pmatrix} + \dot{x} \\ + \dot{y} \\ + \end{pmatrix} = + \begin{pmatrix} + v_x \\ + v_y \\ + \end{pmatrix} = + v_x \cdot \vec{e}_x + v_y \cdot \vec{e}_y + .\] + +\subsection*{3D} + +\[ + \vec{v}(t) = \dot{\vec{r}}(t) = \frac{d}{dt} \vec{r}(t) = \frac{d}{dt} + \begin{pmatrix} + x \\ + y \\ + z \\ + \end{pmatrix}= + \begin{pmatrix} + \frac{d}{dt}x \\ + \frac{d}{dt}y \\ + \frac{d}{dt}z \\ + \end{pmatrix}= + \begin{pmatrix} + \dot{x} \\ + \dot{y} \\ + \dot{z} \\ + \end{pmatrix}= + \begin{pmatrix} + v_x \\ + v_y \\ + v_z \\ + \end{pmatrix}= + v_x \cdot \vec{e}_x + v_y \cdot \vec{e}_y + v_z \cdot \vec{e}_z + .\] + + diff --git a/hs24/technische_mechanik/kinematik/kinemate.tex b/hs24/technische_mechanik/kinematik/kinemate.tex new file mode 100644 index 0000000..7aee3ae --- /dev/null +++ b/hs24/technische_mechanik/kinematik/kinemate.tex @@ -0,0 +1,12 @@ +\section{Die Kinemate} \label{sec:kin} + +Wir haben in Kapitel \ref{sec:skf} gelernt, dass die Bewegung eines Starrkörpers durch zwei Vektoren eindeutig bestimmt werden kann. Dies wird bei den Kinematen eine wichtige Rolle spielen, denn die Kinematen beschreiben die Bewegung eines Starrkörpers. +\\ +Die Kinemate ist bestimmt durch zwei Invarianten. $I_1 = \vec{\omega} \text{ und } I_2 = \vec{\omega} \cdot \vec{v}_P$. + +\begin{tabular}{| c | c | c |} + & $I_2 = 0$ & $I_2 \neq 0$ \\ + \hline + $I_1 = 0$ & Translation oder Stillstand & - \\ + $I_1 \neq 0$ & Rotation & Kreiselung \\ +\end{tabular} diff --git a/hs24/technische_mechanik/kinematik/kinematik.tex b/hs24/technische_mechanik/kinematik/kinematik.tex new file mode 100644 index 0000000..47851fe --- /dev/null +++ b/hs24/technische_mechanik/kinematik/kinematik.tex @@ -0,0 +1,20 @@ +\chapter{Kinematik} + +\input{ortsfunktion.tex} +\input{geschwindigkeit.tex} +\input{schnelligkeit.tex} +\input{spdg.tex} +\input{ebene_bewegung.tex} +\input{svm.tex} +\input{freiheitsgrad_bindung.tex} +\input{raeumliche_bewegung.tex} +\input{starrkoerperformel.tex} +\input{kinemate.tex} +\input{kraft.tex} +\input{reaktionsprinzip.tex} +\input{moment.tex} +\input{transformationsregel.tex} +\input{leistung.tex} +\input{dyname.tex} +\input{aequivalenz_reduktion_kraeftegruppen.tex} + diff --git a/hs24/technische_mechanik/kinematik/kraft.tex b/hs24/technische_mechanik/kinematik/kraft.tex new file mode 100644 index 0000000..372df1a --- /dev/null +++ b/hs24/technische_mechanik/kinematik/kraft.tex @@ -0,0 +1,41 @@ +\section{Kraft} \label{sec:kraft} + +\dfn{Kraft}{ + Die Kraft ist eine physikalische Grösse, welche eine Einwirkung auf einen Körper hat. Die Kraft hat einen Betrag, eine Richtung, sowie ein Angriffspunkt. + + \includegraphics[width=\linewidth]{fig/Fig_13.png} + + \nt{ + Die Kraft kann entlang der Wirkungslinie verschoben werden ohne dabei ihre Wirkung zu verlieren. + } + +} + +\nt{ + Man unterscheidet zwischen innere und äussere Kräfte. Äussere Kräfte sind Kräfte, welche von aussen auf den Starkörber Wirken, während innere Kräfte innerhalb des Starkörpers wirken. +} + + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_14.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + Falls die Kräft den gleichen Angriffspunkt haben, so können die Kräft vektoriell addiert werden. +\end{minipage} + +\nt{ + Negative Kräft haben entweder einen negativen Betrag oder zeigen in die entgegengesetzte Richtung. +} + +\subsection{Resultierende Kraft} + +Die resultierende Kraft eines Starrkörpers kann durch die Summe aller Kräfte berechnet werden. + +\begin{equation} + \vec{R} = \sum \vec{F}_i +\end{equation} + +\nt{ + Die Addition von Kraftvektoren geschieht komponentenweise. +} + diff --git a/hs24/technische_mechanik/kinematik/leistung.tex b/hs24/technische_mechanik/kinematik/leistung.tex new file mode 100644 index 0000000..2db402b --- /dev/null +++ b/hs24/technische_mechanik/kinematik/leistung.tex @@ -0,0 +1,28 @@ +\section{Leistung} + +Die Leistung beschreibt die Kraft, welche aufgewendet wurde in einer bestimmten Zeitspanne. Es gilt die folgende Formel. + +\begin{equation} + P = \vec{F} \cdot \vec{v}_Q +\end{equation} + +\subsection{Gesamtleistung} + +Für die Gesamtleistung gilt die folgende Gleichung. + +\begin{equation} + P_{tot} = \sum P_i = \sum \vec{F}_i \cdot \vec{v}_Q + \sum \vec{M}_i \cdot \vec{\omega}_i +\end{equation} + +\nt{ + Wenn es sisch um eine Rotation handelt, dann fällt die Summe mit den Kräften weg. +} + +\nt{ + Falls alle Kräfte nur auf einem Starrkörper angreifen, dann gilt die folgende Formel. + + \begin{equation} + P_{tot} = \vec{R} \cdot \vec{v}_A + \vec{M}_A \cdot \vec{\omega} + \end{equation} +} + diff --git a/hs24/technische_mechanik/kinematik/moment.tex b/hs24/technische_mechanik/kinematik/moment.tex new file mode 100644 index 0000000..2cf31eb --- /dev/null +++ b/hs24/technische_mechanik/kinematik/moment.tex @@ -0,0 +1,50 @@ +\section{Moment} + +\dfn{Moment}{ + Eine Kraft, welche ein Starrkörper zum rotieren bringt wird als Moment bezeichnet. Im Vergleich zum Satz vom Momentanzentrum kann das Moment von irgendeinem Punkt innerhalb des Starrkörpers berechnet werden. Das Moment kann durch die folgende Formel berechnet werden. + + \begin{equation} + \vec{M}_A = \vec{r}_{AP} \times \vec{F}_i + \end{equation} + + Der Betrag vom Moment kann durch das verschieben der Kraft entlang der Wirkungslinie berechnet werden. Dazu berschiebt man die Kraft, bis sie orthogonal zu $\vec{r}$ ist. Daraus folgt + + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_17.png} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \begin{equation} + M_A = r_{AP'} \cdot F + \end{equation} + \end{minipage} +} + +\nt{ + Für das Vorzeichen von $\vec{M}$ kann man vorgehen wie bei $\vec{\omega}$. (Kapitel \ref{sec:wige}) +} + +\includegraphics[width=\textwidth]{fig/Fig_18.png} + +Für Momente in zweidimensionalen Räumen gilt, dass sie nur eine Komponente haben. Diese ist immer senkrecht auf $\vec{r}$. Aus diesem Grund kann der Betrag vom Moment wie folgt berechnet werden. + +\[ + M_A = r \cdot F_A \cdot \cos(\alpha) + .\] + +Im Fall vom zweidimensionalen Raum wäre der berechnete Betrag die z Komponente. + +\[ + \vec{M}_A = \begin{pmatrix} + 0 \\ + 0 \\ + r \cdot F \cdot \cos(\alpha) \\ + \end{pmatrix} + .\] + +\subsection{Das resultierende Moment} + +Das resultierende Moment beschreibt alle Momente die auf einem Punkt wirken. Diese kann durch die Addition von allen Momenten, welche auf ein Punkt wirken berechnet werden. + +\begin{equation} + \vec{M}_A ^{tot} = \sum \vec{M}_A +\end{equation} diff --git a/hs24/technische_mechanik/kinematik/ortsfunktion.tex b/hs24/technische_mechanik/kinematik/ortsfunktion.tex new file mode 100644 index 0000000..53d3842 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/ortsfunktion.tex @@ -0,0 +1,23 @@ +\section{Ortsfunktion eines materiellen Punktes} \label{sec:ortf} + +\dfn{Ortsfunktion}{ + Eine Ortsfunktion beschreibt die Position eines materiellen Punktes in Abhängigkeit mit der Zeit in einem Raum. Wir unterscheiden zwischen drei verschiedenen Koordinaten Systemen. + + \begin{itemize} + \item Karthesische Koordinaten + \item Zylindrische Koordinaten + \item Polar Koordinaten + \end{itemize} + + \includegraphics[width=\linewidth]{fig/Fig_1.png} +} + +Der Ortsfunktion wird als Vektor dargestellt. Die Komponenten des Ortsvektors beschreiben die Position im Raum. + +\[ + \begin{bmatrix} + \cos(t) \\ + \sin(t) \\ + t \\ + \end{bmatrix} + .\] diff --git a/hs24/technische_mechanik/kinematik/raeumliche_bewegung.tex b/hs24/technische_mechanik/kinematik/raeumliche_bewegung.tex new file mode 100644 index 0000000..f58780b --- /dev/null +++ b/hs24/technische_mechanik/kinematik/raeumliche_bewegung.tex @@ -0,0 +1,12 @@ +\section{Räumliche Bewegungen} + +In diesem Kapitel werden wir die ebene Bewegung (Kapitel \ref{sec:evenm}) erweitern zur räumlichen Bewegung. Die räumliche Bewegung ist, in Vergleich zur ebenen Bewegung die Bewegung eines Starrkörpers in dreidimensionalen Raum. Eine solche Bewegung ist die Kreiselung. + +\subsection*{Kreiselung} + +\begin{minipage}{0.7\linewidth} + Eine Kreiselung unterscheidet sich von der Rotation dadurch, dass $\vec{\omega}$ sich auch "rotiert". Betrachten wir jedoch den momentanen Zustand, so ist eine Kreiselung nichts anderes als eine Rotation und somit gilt der Satz vom Momentanzentrum. Die Rotation von $\vec{\omega}$ darf aber natürlich nicht ignoriert werden, weshalb eine neue Formel eingeführt wird, welche für allmögliche Bewegungen von Starrkörpern gilt. +\end{minipage} +\begin{minipage}{0.3\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_11.png} +\end{minipage} diff --git a/hs24/technische_mechanik/kinematik/reaktionsprinzip.tex b/hs24/technische_mechanik/kinematik/reaktionsprinzip.tex new file mode 100644 index 0000000..24c4e57 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/reaktionsprinzip.tex @@ -0,0 +1,10 @@ +\section{Reaktionsprinzip} + +Wenn Kräfte auf ein Körper wirken so gibt es immer eine entgegengesetzte Kraft. Dieses Phänomen wird Reaktionsprinzip genannt. Würde es dieses Prinzip nicht geben, würde der Körper sich dofomieren oder ein Körper sich durch den Boden bewegen. + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_15.png} +\end{minipage} +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_16.png} +\end{minipage} diff --git a/hs24/technische_mechanik/kinematik/schnelligkeit.tex b/hs24/technische_mechanik/kinematik/schnelligkeit.tex new file mode 100644 index 0000000..75c41a6 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/schnelligkeit.tex @@ -0,0 +1,27 @@ +\section{Schnelligkeit} \label{sec:spee} + +Die Schnelligkeit ist der Betrag des Geschwindigkeit Vektors. Sie kann wie folgt berechnet werden. + +\subsection*{2D} + +\[ + v = |v(t)| = \sqrt{v_x ^2 + v_y ^2} + .\] + +\subsection*{3D} + +\[ + v = |v(t)| = \sqrt{v_x ^2 + v_y ^2 + v_z ^2} + .\] + +\section{Übersicht} + +In Kapitel \ref{sec:ortf} gesehen, dass die Ortsfunktion in verschiedenen Koordinatensystemen berschrieben werden können. In der folgenden Tablle werden die Ortsfunktion, die Geschwindigkeit und die Schnelligkeit in Abhängigkeit von der Ortsfunktion in den verschiedenen Koordinatensystemen beschrieben. + +\begin{tabular}{| c | c | c | c |} + KS & Ortsfunktion & Geschwindigkeit & Schnelligkeit \\ + \hline + Kartesisch & $x(t) \cdot \vec{e}_x + y(t) \cdot \vec{e}_y + z(t) \cdot \vec{e}_z$ & $\dot{x}(t) \cdot \vec{e}_x + \dot{y}(t) \cdot \vec{e}_y + \dot{z}(t) \cdot \vec{e}_z$ & $\sqrt{\dot{x}(t) ^2 + \dot{y}(t) ^2 + \dot{z}(t) ^2}$ \\ + Zylindrisch & $\rho(t) \cdot \vec{e}_{\rho}(t) + z \cdot \vec{e}_z$ & $\dot{\rho}(t) \cdot \vec{e}_{\rho}(t) + \rho(t) \cdot \dot{\varphi}(t) \cdot \vec{e}_{\varphi}(t) + \dot{z} \cdot \vec{e}_z(t)$ & $\sqrt{\dot{\rho}(t) ^2 + \rho(t) ^2 \cdot \dot{\varphi}(t) ^2 + \dot{z}(t) ^2}$ \\ + Polar & $\rho(t) \cdot \vec{e}_{\rho}(t)$ & $\dot{\rho}(t) \cdot \vec{e}_{\rho}(t) + \rho(t) \cdot \dot{\varphi}(t) \cdot \vec{e}_{\varphi}(t)$ & $\sqrt{\dot{\rho}(t) ^2+ \rho(t) ^2 \cdot \dot{\varphi}(t) ^2}$ \\ +\end{tabular} diff --git a/hs24/technische_mechanik/kinematik/spdg.tex b/hs24/technische_mechanik/kinematik/spdg.tex new file mode 100644 index 0000000..6153109 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/spdg.tex @@ -0,0 +1,21 @@ +\section{Satz der projizierten Geschwindigkeiten (SdpG)} + +\dfn{Satz der projizierten Geschwindigkeiten}{ + Der Satz der projizierten Geschwindigkeit besagt, dass die Geschwindigkeiten von 2 Punkten, welche auf die Verbindungslinie der Punkte projiziert wird, gleich sind. + + \begin{minipage}{0.3\linewidth} + \begin{equation} + \vec{v}_A ' = \vec{v}_B ' + \label{eq:sdpg} + \end{equation} + \end{minipage} + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_3.png} + \end{minipage} + + Aus Gleichung \ref{eq:sdpg} lässt sich die Formel unformen zu eine weiteren essentiellen Gleichung umformen. + + \begin{equation} + (\vec{v}_A - \vec{v}_B) \cdot (\vec{r}_B - \vec{r}_A) = 0 + \end{equation} +} diff --git a/hs24/technische_mechanik/kinematik/starrkoerperformel.tex b/hs24/technische_mechanik/kinematik/starrkoerperformel.tex new file mode 100644 index 0000000..257d5c6 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/starrkoerperformel.tex @@ -0,0 +1,32 @@ +\section{Starrkörperformel} \label{sec:skf} + +\dfn{Starrkörperformel (ABBA-Formel)}{ + Die Starrkörperformel beschreibt die Bewegung von Starrkörper im allgemeinen Fall. Dabei gilt die folgende Formel. + + \begin{minipage}{0,7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_12.png} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \begin{equation} + \vec{v}_A = \vec{v}_B + \vec{\omega} \times \vec{r}_{BA} + \label{eq:skf} + \end{equation} + \end{minipage} +} + +Aus der Gleichung \ref{eq:skf} lassen sich die Gleichungen der Ebenen Bewegungen herleiten. + +\begin{minipage}{0.5\linewidth} + Wenn $\vec{\omega} = 0$, dann gilt: + \[ + \vec{v}_A = \vec{v}_B + .\] +\end{minipage} +\begin{minipage}{0.5\linewidth} + Wenn $\vec{v}_B$ das Momentanzentrum ist d.h. $\vec{v}_B = 0$, dann gilt: + \[ + \vec{v}_A = \vec{\omega} \times \vec{r}_{BA} + .\] +\end{minipage} + +Man sieht, dass die Bewegung eines Starrkörpers durch zwei Vektoren eindeutig bestimmt werden kann. diff --git a/hs24/technische_mechanik/kinematik/svm.tex b/hs24/technische_mechanik/kinematik/svm.tex new file mode 100644 index 0000000..4681169 --- /dev/null +++ b/hs24/technische_mechanik/kinematik/svm.tex @@ -0,0 +1,80 @@ +\section{Satz vom Momentanzentrum} + +In Kapitel \ref{sec:evenm} haben wir einen wichtigen Satz in der Technischen Mechanik kennengelernt. Dabei haben wir zwischen Translation und Rotation geredet. Dabei kam die Gleichung \ref{eq:rot} ins Spiel. Wir werden nun diese Gleichung genauer betrachten. +\\ +Bevor wir weiterfahren klären wir zuerst, was das Momentanzentrum ist. + +\dfn{Momentanzentrum}{ + Das Momentanzentrum ist der Punkt eines Starkörpers, welcher momentan in Ruhe ist d.h. keine Geschwindigkeit besitzt. + + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_7.png} + \end{minipage} + \begin{minipage}{0.3\linewidth} + \[ + \vec{v}_{MZ} = 0 + .\] + \end{minipage} + + \nt{ + Ein Punkt, der den Boden oder die Wand berührt, ist ein Momentanzentrum. Im Dreidimensionalen ist das Momentanzentrum eine Kontaktlinie. + } + +} + +\dfn{Satz vom Momentanzentrum}{ + \begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{fig/Fig_8.png} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \[ + \vec{v}_p = \vec{\omega} \times \vec{r}_p + .\] + \end{minipage} + + Die Formel besagt, dass die Geschwindigkeit bei einer Rotation an einem Punkt durch das Kreuzprodukt der Rotationsgeschwindigkeit und dem Vektor vom Momentanzentrum zum Punkt bestimmt werden kann. + + \nt{ + Das Kreuzprodukt darf nicht vertauscht werden. + } + + Aufgrund des Kreuzproduktes und SdpG sind die Geschwindigkeitsvektoren immer senkrecht zum Verbindungsvektor zwischen dem Momentanzentrum und dem Punkt. +} + +Die Rotationsschnelligkeit kann durch den Betrag der Rotationsgeschwindigkeit berechnet werden (Kapitel \ref{sec:spee}) oder durch die zeitliche Ableitung des Rotationswinkels $\Theta$. ($\omega = \dot{\Theta} = \frac{d \Theta}{dt}$) +\\ +Die Schnelligkeit eines Punktes kann durch die Multiplikation der Winkelschnelligkeit und der Distanz vom Momentanzentrum zum Punkt. ($v_p = \omega \cdot r_p$) + +\nt{ + Es hat immer nur ein $\omega$ pro Starrkörper. +} + +\subsection{Die Parallelogrammregel} + +\dfn{Parallelogrammregel}{ + Wenn Starrkörper ein Parallelogramm bilden, so haben die parallel stehenden Starrkörper dieselbe Winkelgeschwindigkeit. + + \begin{minipage}{0.3\linewidth} + \[ + \omega_1 = \omega_3 \text{ und } \omega_2 = \omega_4 + .\] + \end{minipage} + \begin{minipage}{0.7\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_9.png} + \end{minipage} +} + +\nt{ + Die Vorzeichen der x und y Komponente des Geschwindigkeitsvektors kann man entweder grafisch oder über das Kreuzprodukt bestimmen. +} + +\subsection{Winkelgeschwindigkeit $\omega$} \label{sec:wige} + +$\omega$ kann durch die rechte Hand bestimmt werden. Dies folgt aus dem Fakt, dass $\vec{\omega}$ nur eine z Komponente besitzt. +\\ +\begin{minipage}{0.3\linewidth} + \includegraphics[width=\linewidth]{fig/Fig_10.png} +\end{minipage} +\begin{minipage}{0.7\linewidth} + Die ursprüngliche Richtung von $\omega$ kann selbst bestimmt werden. Diese Richtung bestimmt imnachhinein die Richtung von $\vec{\omega}$. +\end{minipage} diff --git a/hs24/technische_mechanik/kinematik/transformationsregel.tex b/hs24/technische_mechanik/kinematik/transformationsregel.tex new file mode 100644 index 0000000..8a751db --- /dev/null +++ b/hs24/technische_mechanik/kinematik/transformationsregel.tex @@ -0,0 +1,18 @@ +\section{Transformationsregel} + +Die Transformationsregel kann durch eine Gleichung das Moment an einem anderen Punkt beschreiben. + +\begin{equation} + \vec{M}_A = \vec{M}_O + \vec{r}_{AO} \times \vec{R} +\end{equation} + +Wenn das Moment, sowie die Resultierende Kraft eines Punktes bekannt ist, so kann man die soeben genannte Formel anwenden. + +\nt{ + Wenn $\vec{R} = 0$ so sind die Momente in beiden Punkten gleich. +} + +\nt{ + Wenn die Wirkungslinie von $\vec{R}$ durch den Bezugspunkt geht, dann ist das Moment 0. +} + diff --git a/hs24/technische_mechanik/letterfonts.tex b/hs24/technische_mechanik/letterfonts.tex new file mode 100644 index 0000000..3939d5b --- /dev/null +++ b/hs24/technische_mechanik/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/hs24/technische_mechanik/macros.tex b/hs24/technische_mechanik/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/hs24/technische_mechanik/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/hs24/technische_mechanik/notes.tex b/hs24/technische_mechanik/notes.tex new file mode 100644 index 0000000..0206db2 --- /dev/null +++ b/hs24/technische_mechanik/notes.tex @@ -0,0 +1,4 @@ +\import{kinematik}{kinematik.tex} +\import{statik}{statik.tex} +\import{dynamik}{dynamik.tex} + diff --git a/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.bbl b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.bbl new file mode 100644 index 0000000..543de37 --- /dev/null +++ b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.bbl @@ -0,0 +1,19 @@ +\begin{thebibliography}{} + +\bibitem[de~Windt, 2023]{Windt2023} +de~Windt, L. (2023). +\newblock {TechMech PVK HS23}. + +\bibitem[Kaufmann, 2017]{Kaufmann2017} +Kaufmann, D. P.~S. (2017). +\newblock {Technische Mechanik}. + +\bibitem[{The Manim Community Developers}, 2024]{MCD2024} +{The Manim Community Developers} (2024). +\newblock {Manim – Mathematical Animation Framework}. + +\bibitem[Tiso, 2024]{Tiso2024} +Tiso, D. P.~P. (2024). +\newblock Vorlesungen. + +\end{thebibliography} diff --git a/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.pdf b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.pdf new file mode 100644 index 0000000..bf44209 Binary files /dev/null and b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.pdf differ diff --git a/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.tex b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.tex new file mode 100644 index 0000000..ea06235 --- /dev/null +++ b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.tex @@ -0,0 +1,44 @@ +\documentclass{report} + +% TODO: Change Variables +\def\papertitle{Technische Mechanik} + +% TODO: Change variables according to language preference +\def\theorytitle{Theorie} +\def\corollarytitle{Daraus folgt} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Behauptung} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Beispiel} +\def\questiontitle{Frage} +\def\wrongctitle{Falscher Konzept} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +\input{disclaimer.tex} +\input{notes.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% BIBLIOGRAPHY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\addcontentsline{toc}{chapter}{References} +\bibliographystyle{apalike} +\bibliography{sources} + +\end{document} diff --git a/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.zip b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.zip new file mode 100644 index 0000000..8cc7ab7 Binary files /dev/null and b/hs24/technische_mechanik/notizen_technische_mechanik_ruh_jirayu.zip differ diff --git a/hs24/technische_mechanik/preamble.tex b/hs24/technische_mechanik/preamble.tex new file mode 100644 index 0000000..bdac800 --- /dev/null +++ b/hs24/technische_mechanik/preamble.tex @@ -0,0 +1,777 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage[ngerman]{babel} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + +%================================ +% Corollery +%================================ +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + Wrong Concept \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + + + + +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} + + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% + \begingroup + \sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% + \pdfsave + \pdfsetmatrix{1 0 0.2 1}% + \rlap{\usebox{\diffdbox}}% + \pdfrestore + \hskip\wd\diffdbox + \endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% + \slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% + {\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience +\DeclareMathOperator{\E}{E} % expected + +% Since the amsthm package isn't loaded + +% I prefer the slanted \leq +\let\oldleq\leq % save them in case they're every wanted +\let\oldgeq\geq +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +% % redefine matrix env to allow for alignment, use r as default +% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep +% \let\@ifnextchar\new@ifnextchar +% \array{*\c@MaxMatrixCols #1}} + + +%\usepackage{framed} +%\usepackage{titletoc} +%\usepackage{etoolbox} +%\usepackage{lmodern} + + +%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{} + +%\renewenvironment{leftbar} +%{\def\FrameCommand{\hspace{6em}% +% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}% +% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt% +%} +%{\endMakeFramed} + +%\titlecontents{chapter} +%[0em]{\vspace*{2\baselineskip}} +%{\parbox{4.5em}{% +% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}% +% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily} +%{}{\endleftbar} +%\titlecontents{section} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} +%\titlecontents{subsection} +%[8.4em] +%{\sffamily\contentslabel{3em}}{}{} +%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage{tikz} +\definecolor{doc}{RGB}{0,60,110} +\usepackage{titletoc} +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Kapitel\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries Page \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/hs24/technische_mechanik/sources.bib b/hs24/technische_mechanik/sources.bib new file mode 100644 index 0000000..257b51c --- /dev/null +++ b/hs24/technische_mechanik/sources.bib @@ -0,0 +1,27 @@ +@misc{Tiso2024, + author = {Dr. Prof. Pablo Tiso}, + title = {Vorlesungen}, + year = {2024}, +} + +@misc{Windt2023, + author = {Lina de Windt}, + title = {{TechMech PVK HS23}}, + year = {2023}, +} + +@software{MCD2024, + author = {{The Manim Community Developers}}, + license = {MIT}, + month = apr, + title = {{Manim – Mathematical Animation Framework}}, + url = {https://www.manim.community/}, + version = {v0.19.0}, + year = {2024}, +} + +@misc{Kaufmann2017, + author = {Dr. Prof. Stephan Kaufmann}, + title = {{Technische Mechanik}}, + year = {2017}, +} diff --git a/hs24/technische_mechanik/statik/freischneiden.tex b/hs24/technische_mechanik/statik/freischneiden.tex new file mode 100644 index 0000000..089a8f7 --- /dev/null +++ b/hs24/technische_mechanik/statik/freischneiden.tex @@ -0,0 +1,26 @@ +\section{Freischneiden} + +Für die Analyse von Systemen ist es sinnvoll, den Starrkörper freizuschneiden. Dazu werden am Starrkörper alle äusseren Kräfte eingezeichnet, welche auf den Starrkörper wirken. +\\ +Beim Freischneiden können auch der Starrkörper selbst in einzelne Starrkörper geschnitten werden. Hier ist es einfach wichtig, dass man die Gegenkräfte einzeichnet. + +\includegraphics[width=\textwidth]{fig/Fig_27.png} + +Des weiteren kann für jeden Starrkörper ein beliebiges Koordinatensystem gewählt werden. + +\includegraphics[width=\textwidth]{fig/Fig_28.png} + +\subsection{Lager- und Bindungskräfte} + +\begin{center} + \begin{tabular}{| c | c | c |} + & Lager vor dem Freischnitt & nach dem Freischnitt \\ + \hline + Auflager (einseitig) & \includegraphics[width=0.25\textwidth]{fig/Fig_29.png} & \includegraphics[width=0.25\textwidth]{fig/Fig_30.png} \\ + Gelenk Festlager & \includegraphics[width=0.25\textwidth]{fig/Fig_31.png} & \includegraphics[width=0.25\textwidth]{fig/Fig_32.png} \\ + Einspannung & \includegraphics[width=0.25\textwidth]{fig/Fig_33.png} & \includegraphics[width=0.25\textwidth]{fig/Fig_34.png} \\ + Faden / Seil & \includegraphics[width=0.25\textwidth]{fig/Fig_35.png} & \includegraphics[width=0.25\textwidth]{fig/Fig_36.png} \\ + \end{tabular} +\end{center} + +\cite{Kaufmann2017} diff --git a/hs24/technische_mechanik/statik/hauptsatz_statik.tex b/hs24/technische_mechanik/statik/hauptsatz_statik.tex new file mode 100644 index 0000000..e0ded4f --- /dev/null +++ b/hs24/technische_mechanik/statik/hauptsatz_statik.tex @@ -0,0 +1,14 @@ +\section{Hauptsatz der Statik} \label{sec:hs} + +\dfn{Hauptsatz der Statik}{ + Der Hauptsatz der Statik besagt, dass in einer Ruhelage eines Starkörpers müssen alle (äusseren) Kräfte im Gleichgewicht sein. \cite{Tiso2024} Deshalb gilt + + \[ + \vec{R} = \vec{0} , \vec{M}_0 = \vec{0} + .\] +} + +\nt{ + Wenn der Hauptsatz der Statik bei einem Starkörper gilt, so ist der Starkörper in Ruhelage. Dies gilt nicht für Systeme. Wenn der Hauptsatz der Statik gilt, muss dies nicht umbeding bedeuten, dass das System un Ruhelage ist. +} + diff --git a/hs24/technische_mechanik/statik/kippen.tex b/hs24/technische_mechanik/statik/kippen.tex new file mode 100644 index 0000000..147a703 --- /dev/null +++ b/hs24/technische_mechanik/statik/kippen.tex @@ -0,0 +1,5 @@ +\section{Kippen} + +In der realen Welt können Körper kippen. Um zu prüfen, wann ein Körper kippt, muss die Normalkraft eingeführt werden. +\\ +Ein System ist standfest (kein Kippen), wenn $\vec{N}$ innerhalb der Standfläche liegt. \cite{Tiso2024} Liegt $\vec{N}$ ausserhalb der Standfläche, so kippt der Körper. diff --git a/hs24/technische_mechanik/statik/knotengleichgewicht.tex b/hs24/technische_mechanik/statik/knotengleichgewicht.tex new file mode 100644 index 0000000..08680ed --- /dev/null +++ b/hs24/technische_mechanik/statik/knotengleichgewicht.tex @@ -0,0 +1,4 @@ +\section{Knotengleichgewicht} + + +Beim Knotengleichgewicht schneiden wir einen Knoten (Gelenk) eines Starrkörpers frei und berechnen dort alle Kräfte, welche auf den Knoten wirken. Dadurch kann man die Kräfte bestimmen. Das Knotengleichgewicht verwendet man, wenn man mehrere Stabkräfte bestimmen möchte. diff --git a/hs24/technische_mechanik/statik/kraefteschnitt.tex b/hs24/technische_mechanik/statik/kraefteschnitt.tex new file mode 100644 index 0000000..7db7dbf --- /dev/null +++ b/hs24/technische_mechanik/statik/kraefteschnitt.tex @@ -0,0 +1,3 @@ +\section{Kräfteschnitt} + +Beim Kräfteschnitt ''schneidet'' man das System frei, um die Stabkräfte der durchschnittenen Pendelstäbe zu bestimmen. Den Kräfteschnitt verwendet man vor allem, um mehrere Stabkräfte bestimmen zu können. diff --git a/hs24/technische_mechanik/statik/prinzip_virtuellen_leistungen.tex b/hs24/technische_mechanik/statik/prinzip_virtuellen_leistungen.tex new file mode 100644 index 0000000..603227f --- /dev/null +++ b/hs24/technische_mechanik/statik/prinzip_virtuellen_leistungen.tex @@ -0,0 +1,21 @@ +\section{Prinzip der virtuellen Leistungen (PdvL)} + +\dfn{Prinzip der virtuellen Leitungen}{ + Ein System befindet sich genau dann in einer Ruhelage, wenn die virtuelle Gesamtleistung der inneren und äusseren Kräfte bei jedem virtuellen Bewegungszustand verschwindet (und die Eigenschaften des Systems und seiner Lagerung diese Kräfte zulassen). \cite{Kaufmann2017} + + \begin{equation} + \tilde{P} = \tilde{P} ^{(i)} + \tilde{P} ^{(a)} = 0 + \end{equation} +} + +Das Prinzip der virtuellen Leistungen stellt sicher, dass das vorliegende System die für die Ruhe nötigen Kräfte tatsächlich aufnehmen kann. \cite{Windt2023} + +\nt{ + Das Prinzip der virtuellen Leistungen ermöglicht uns Stabkräfte zu bestimmen. +} + +\subsection{PdvL vs HS} + +Wir wissen von Kapitel \ref{sec:hs}, dass wenn der Hauptsatz der Statik gilt, dass der Körper in Ruhe ist. Dies gilt aber nicht für ein System. +\\ +Beim PdvL gilt, wenn das Prinzip der virtuellen Leitungen gilt, so ist der Starrkörper oder das System in Ruhe. Zusätzlich gilt auch, wenn das Prinzip der virtuellen Leistungen gilt, so gilt auch der Hauptsatz der Statik. diff --git a/hs24/technische_mechanik/statik/reibung.tex b/hs24/technische_mechanik/statik/reibung.tex new file mode 100644 index 0000000..bb3d9fa --- /dev/null +++ b/hs24/technische_mechanik/statik/reibung.tex @@ -0,0 +1,41 @@ +\section{Reibung} + +\dfn{Reibung}{ + Die Reibung ist eine Kraft, welche gegen die Kraft wirkt, welche die Bewegung eines Starrkörpers hervorruft. Sie wirkt immer gegen die Bewegungsrichtung und ist senkrecht zur normalen. + + \includegraphics[width=\textwidth]{fig/Fig_39.png} +} + +\nt{ + Wenn die Reibung angegeben wird, so muss sie beim HS und PdvL berüchsichtigt werden. +} + +Bei der Reibung unterscheiden wir zwischen 2 verschiedenene Arten. + +\dfn{Haftreibung}{ + Bei der Haftreibung findet keine Bewegung der Körper zueinander statt. \cite{Windt2023} + + \includegraphics[width=\textwidth]{fig/Fig_40.png} + + \begin{equation} + |\vec{F}_R| \leq \mu_0 \cdot |\vec{N}| + \end{equation} + +} + +\dfn{Gleitreibung}{ + Bei der Gleitreibung bewegen sich die Oberflächen der Körper relativ zueinander. Dabei gleitet der Körber und erfährt eine konstante Reibungskraft. \cite{Windt2023} + + \includegraphics[width=\textwidth]{fig/Fig_41.png} + + \begin{equation} + |\vec{F}_R| = \mu_0 \cdot |\vec{N}| + \end{equation} + + \nt{ + Bei der Gleitreibung wird meistens $\mu_1$ angegeben. Sei dir einfach bewust, dass $\mu_1 = \mu_0$ ist. + } + +} + + diff --git a/hs24/technische_mechanik/statik/rollreibung.tex b/hs24/technische_mechanik/statik/rollreibung.tex new file mode 100644 index 0000000..76bf2c5 --- /dev/null +++ b/hs24/technische_mechanik/statik/rollreibung.tex @@ -0,0 +1,30 @@ +\section{Rollreibung} + +\dfn{Rollreibung}{ + Rollreibung tritt auf, wenn ein Körper über einen anderen Körper rollt. \cite{Windt2023} Wie bei der Reibungskraft ist die Rollreibung entgegen der Bewegung gerichtet. Der Unterschied ist jedoch, dass die Rollreibung keine Kraft, sondern ein Moment ist. + + \includegraphics[width=\textwidth]{fig/Fig_42.png} +} + +Wie bei der Reibung unterscheiden wir zwischen zwei verschiedenen Rollreibungen. + +\subsection*{Ruhe} + +Im Fall von Ruhe oder ''Haftreibung'': + +\[ + |\vec{M}_R \leq \mu_2 \cdot |\vec{N}| + .\] + +\subsection*{Bewegung} + +Im Fall von Bewegung oder ''Gleitreibung'': + +\[ + |\vec{M}_R = \mu_2 \cdot |\vec{N}| + .\] + +\nt{ + Verwechsele nicht Rollreibung mit Reibung. Manchmal muss beides eingefürht werden. +} + diff --git a/hs24/technische_mechanik/statik/ruhe_ruhelage.tex b/hs24/technische_mechanik/statik/ruhe_ruhelage.tex new file mode 100644 index 0000000..39be196 --- /dev/null +++ b/hs24/technische_mechanik/statik/ruhe_ruhelage.tex @@ -0,0 +1,12 @@ +\section{Ruhe und Ruhelage} + +\dfn{Ruhe}{ + Wenn alle Geschwindigkeiten eines Systems 0 sind so befindet sich das System in Ruhe. + + \includegraphics[width=\textwidth]{fig/Fig_23.png} +} + +\dfn{Ruhelage}{ + Die Ruhelage ist ein Zustand eines Systems, in dem keine Änderung mehr durchgemacht werden. +} + diff --git a/hs24/technische_mechanik/statik/stabkraefte.tex b/hs24/technische_mechanik/statik/stabkraefte.tex new file mode 100644 index 0000000..3fb9b4e --- /dev/null +++ b/hs24/technische_mechanik/statik/stabkraefte.tex @@ -0,0 +1,11 @@ +\section{Stabkräfte} + +Bei den Stabkräften beschäftigen wir uns hauptsächlich mit Pendelstäben. Diese sind an den Enden gelenkig gelagert und können nur Kräfte aufnehmen, deren Wirkungslinie mit dem Pendelstab verläuft. Das heisst, dass der Stab nur Zug- und Druckkräfte aufnehmen kann. Externe Kräfte wirken nur auf den Gelenken. +\\ +Wenn wir ein Fachwerk anschauen und die Kräfte einzeichnen, welche durch den Pendelstab auf das Fachwerk wirkt, so sind sie immer entgegengesetzt zu den Stabkräften, welche auf den Pendelstab wirken. + +\includegraphics[width=\textwidth]{fig/Fig_38.png} + +Anhand der vorherigen Grafik kann man sehen, dass die Stabkräfte an den Gelenken des Pendelstabes immer entgegengesetzt sind. Dies gilt immer für alle Pendelstäben. +\\ +Pendelstäbe, deren Stabkräfte nach innen zeigen, nennen wir Druckstäbe. Wenn die Stabkraft nach aussen zeigt nennen wir sie Zugstäbe. diff --git a/hs24/technische_mechanik/statik/statik.tex b/hs24/technische_mechanik/statik/statik.tex new file mode 100644 index 0000000..9de66f1 --- /dev/null +++ b/hs24/technische_mechanik/statik/statik.tex @@ -0,0 +1,14 @@ +\chapter{Statik} + +\input{ruhe_ruhelage.tex} +\input{stische_kinematische_unbestimmtheit.tex} +\input{hauptsatz_statik.tex} +\input{freischneiden.tex} +\input{virtueller_bewegungszustand.tex} +\input{prinzip_virtuellen_leistungen.tex} +\input{stabkraefte.tex} +\input{knotengleichgewicht.tex} +\input{kraefteschnitt.tex} +\input{reibung.tex} +\input{rollreibung.tex} +\input{kippen.tex} diff --git a/hs24/technische_mechanik/statik/stische_kinematische_unbestimmtheit.tex b/hs24/technische_mechanik/statik/stische_kinematische_unbestimmtheit.tex new file mode 100644 index 0000000..f93af75 --- /dev/null +++ b/hs24/technische_mechanik/statik/stische_kinematische_unbestimmtheit.tex @@ -0,0 +1,13 @@ +\section{Statische und kinematische (Un-)bestimmtheit} + +\dfn{Statische und kinematische Bestimmtheit}{ + Die statische / kinematische Bestimmtheit kann das mechanische / kinematische Gleichgewicht eines Systems prüfen. Dazu wird der Freiheitsgrad verwendet. + \\ + Wenn $f = 0$, dann ist das System statisch bestimmt und ein Gleichgewicht ist möglich. + \\ + Wenn $f < 0$, dann ist das System statisch unbestimmt und die Bindungskräfte können nicht durch die Gleichgewichtsbedingung bestimmt werden. + \\ + Wenn $f > 0$, dann ist das System statisch und kinematisch unbestimmt und das Gleichgewicht ist nur für spezielle Belastungsfälle möglich. + + \cite{Tiso2024} +} diff --git a/hs24/technische_mechanik/statik/virtueller_bewegungszustand.tex b/hs24/technische_mechanik/statik/virtueller_bewegungszustand.tex new file mode 100644 index 0000000..d2ec5ed --- /dev/null +++ b/hs24/technische_mechanik/statik/virtueller_bewegungszustand.tex @@ -0,0 +1,24 @@ +\section{Virtueller Bewegungszustand} + +\dfn{Virtueller Bewegungszustand}{ + Ein virtueller Bewegungszustand ist ein gedachter Bewegungszustand, der keinen Bezug zu den wirlich möglichen Bewegungszuständen haben muss. \cite{Windt2023} + + \includegraphics[width=\textwidth]{fig/Fig_37.png} + + Um virtuelle Bewegungszustände von ''normalen'' Bewegungszustände unterscheiden zu können, markiert man virtuelle Bewegungszustände mit einem Tilde. ($\tilde{v}$) +} + +Wir unterscheiden zwischen 2 verschiedenen virtuellen Bewegungszuständen. + +\subsection*{Zulässiger Bewegungszustand} + +Bei einem zulässigen Bewegungszustand werden keine Bindungen verletzt. + +\subsection*{Unzulässiger Bewegungszustand} + +Bei einem unzulässigen Bewegungszustand werden mindestens eine Bindung verletzt. + +\nt{ + Wir werden uns vor allem mit zulässigen Bewegungszuständen beschäftigen. +} + diff --git a/template/README.md b/template/README.md new file mode 100644 index 0000000..ea9697c --- /dev/null +++ b/template/README.md @@ -0,0 +1,18 @@ +# Template + +Die LaTeX Template ist inspiriert durch den Template von [Senior Mars](https://github.com/SeniorMars/dotfiles/tree/main/latex_template). Sie kann von dort runtergeladen werden. + +Die Template, welche ich hier verwende ist ein wenig auf meine Bedürfnisse modifiziert, und darf verwendet werden. + +## Beispiele + +- [Meine Notizen](https://eth.jirayuruh.ch) +- [Rares seine Übungsnotizen](https://github.com/RaresBares/Networks-and-Circuits-II-Exercise-Materials) + +## DISCLAIMER!!! + +Dieser Template soll nicht für Arbeiten verwendet werden. Verwendet bitte die [Template der ETH](https://www.overleaf.com/latex/templates/tagged/ethz). + +--- + +Made by JirR02 in Switzerland 🇨🇭 diff --git a/template/letterfonts.tex b/template/letterfonts.tex new file mode 100644 index 0000000..3939d5b --- /dev/null +++ b/template/letterfonts.tex @@ -0,0 +1,178 @@ +% Things Lie +\newcommand{\kb}{\mathfrak b} +\newcommand{\kg}{\mathfrak g} +\newcommand{\kh}{\mathfrak h} +\newcommand{\kn}{\mathfrak n} +\newcommand{\ku}{\mathfrak u} +\newcommand{\kz}{\mathfrak z} +\DeclareMathOperator{\Ext}{Ext} % Ext functor +\DeclareMathOperator{\Tor}{Tor} % Tor functor +\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group +\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6 + +% More script letters etc. +\newcommand{\SA}{\mathcal A} +\newcommand{\SB}{\mathcal B} +\newcommand{\SC}{\mathcal C} +\newcommand{\SF}{\mathcal F} +\newcommand{\SG}{\mathcal G} +\newcommand{\SH}{\mathcal H} +\newcommand{\OO}{\mathcal O} + +\newcommand{\SCA}{\mathscr A} +\newcommand{\SCB}{\mathscr B} +\newcommand{\SCC}{\mathscr C} +\newcommand{\SCD}{\mathscr D} +\newcommand{\SCE}{\mathscr E} +\newcommand{\SCF}{\mathscr F} +\newcommand{\SCG}{\mathscr G} +\newcommand{\SCH}{\mathscr H} + +% Mathfrak primes +\newcommand{\km}{\mathfrak m} +\newcommand{\kp}{\mathfrak p} +\newcommand{\kq}{\mathfrak q} + +% number sets +\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}} +\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}} +\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}} +\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}} +\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}} +\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}} +\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}} +\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}} +% expected value +\newcommand{\EE}{\ensuremath{\mathbb{E}}} +\newcommand{\charin}{\text{ char }} +\DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} % General linear group +\DeclareMathOperator{\SL}{SL} % Special linear group + +%--------------------------------------- +% BlackBoard Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}} +\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}} +\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}} +\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}} +\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}} +\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}} +\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}} +\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}} +\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}} +\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}} +\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}} +\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}} +\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}} + +%--------------------------------------- +% MathCal Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}} +\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}} +\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}} +\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}} +\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}} +\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}} +\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}} +\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}} +\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}} +\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}} +\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}} +\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}} +\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}} + + +%--------------------------------------- +% Bold Math Fonts :- +%--------------------------------------- + +%Captital Letters +\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}} +\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}} +\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}} +\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}} +\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}} +\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}} +\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}} +\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}} +\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}} +\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}} +\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}} +\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}} +\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}} +%Small Letters +\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}} +\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}} +\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}} +\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}} +\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}} +\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}} +\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}} +\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}} +\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}} +\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}} +\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}} +\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}} +\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}} + +%--------------------------------------- +% Scr Math Fonts :- +%--------------------------------------- + +\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}} +\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}} +\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}} +\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}} +\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}} +\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}} +\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}} +\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}} +\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}} +\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}} +\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}} +\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}} +\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}} + + +%--------------------------------------- +% Math Fraktur Font +%--------------------------------------- + +%Captital Letters +\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}} +\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}} +\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}} +\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}} +\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}} +\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}} +\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}} +\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}} +\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}} +\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}} +\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}} +\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}} +\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}} +%Small Letters +\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}} +\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}} +\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}} +\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}} +\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}} +\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}} +\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}} +\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}} +\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}} +\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}} +\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}} +\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}} +\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}} diff --git a/template/macros.tex b/template/macros.tex new file mode 100644 index 0000000..8bbdb96 --- /dev/null +++ b/template/macros.tex @@ -0,0 +1,88 @@ +%From M275 "Topology" at SJSU +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +%From M170 "Introduction to Graph Theory" at SJSU +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} + +%From the USAMO .tex files +\newcommand{\ts}{\textsuperscript} +\newcommand{\dg}{^\circ} +\newcommand{\ii}{\item} + +% % From Math 55 and Math 145 at Harvard +% \newenvironment{subproof}[1][Proof]{% +% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% +% {\end{proof}} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} +\newcommand{\On}{\mathrm{On}} % ordinals +\DeclareMathOperator{\img}{im} % Image +\DeclareMathOperator{\Img}{Im} % Image +\DeclareMathOperator{\coker}{coker} % Cokernel +\DeclareMathOperator{\Coker}{Coker} % Cokernel +\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\Spec}{Spec} % spectrum +\DeclareMathOperator{\Tr}{Tr} % trace +\DeclareMathOperator{\pr}{pr} % projection +\DeclareMathOperator{\ext}{ext} % extension +\DeclareMathOperator{\pred}{pred} % predecessor +\DeclareMathOperator{\dom}{dom} % domain +\DeclareMathOperator{\ran}{ran} % range +\DeclareMathOperator{\Hom}{Hom} % homomorphism +\DeclareMathOperator{\Mor}{Mor} % morphisms +\DeclareMathOperator{\End}{End} % endomorphism + +\newcommand{\eps}{\epsilon} +\newcommand{\veps}{\varepsilon} +\newcommand{\ol}{\overline} +\newcommand{\ul}{\underline} +\newcommand{\wt}{\widetilde} +\newcommand{\wh}{\widehat} +\newcommand{\vocab}[1]{\textbf{\color{blue} #1}} +\providecommand{\half}{\frac{1}{2}} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} +\newcommand{\arc}[1]{\wideparen{#1}} +\DeclareMathOperator{\cis}{cis} +\DeclareMathOperator*{\lcm}{lcm} +\DeclareMathOperator*{\argmin}{arg min} +\DeclareMathOperator*{\argmax}{arg max} +\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} +\newcommand{\symsum}{\sum_{\mathrm{sym}}} +\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} +\newcommand{\symprod}{\prod_{\mathrm{sym}}} +\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}} +\newcommand{\parinn}{\setlength{\parindent}{1cm}} +\newcommand{\parinf}{\setlength{\parindent}{0cm}} +% \newcommand{\norm}{\|\cdot\|} +\newcommand{\inorm}{\norm_{\infty}} +\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}} +\newcommand{\oset}{V_{\alpha}} +\newcommand{\opset}[1]{V_{\alpha_{#1}}} +\newcommand{\lub}{\text{lub}} +\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}} +\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}} +\newcommand{\lm}{\lambda} +\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}} +\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}} +\newcommand{\lt}{\left} +\newcommand{\rt}{\right} +\newcommand{\bs}[1]{\boldsymbol{#1}} +\newcommand{\exs}{\exists} +\newcommand{\st}{\strut} +\newcommand{\dps}[1]{\displaystyle{#1}} + +\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} } +\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed} diff --git a/template/main.tex b/template/main.tex new file mode 100644 index 0000000..4a5d277 --- /dev/null +++ b/template/main.tex @@ -0,0 +1,47 @@ +\documentclass{report} + +% TODO: Change Variables +\def\papertitle{Title} + +% TODO: Change variables according to language preference +\def\theorytitle{Theory} +\def\theorytitle{Theorie} +\def\corollarytitle{Corollary} +\def\corollarytitle{Daraus folgt} +\def\proposaltitle{Proposal} +\def\proposaltitle{Vorschlag} +\def\claimtitle{Claim} +\def\claimtitle{Behauptung} +\def\exercisetitle{Exercise} +\def\exercisetitle{Aufgabe} +\def\exampletitle{Example} +\def\exampletitle{Beispiel} +\def\questiontitle{Question} +\def\questiontitle{Frage} +\def\wrongctitle{Wrong Concept} +\def\wrongctitle{Falscher Konzept} +\def\pagetitle{Page} +\def\pagetitle{Seite} +\def\chaptertitle{Chapter} +\def\pagetitle{Kapitel} + +\input{preamble} +\input{macros} +\input{letterfonts} + +\title{\huge{\papertitle}} +\author{\huge{Jirayu Ruh}} +\date{} + +\begin{document} + +\maketitle +\newpage% or \cleardoublepage +% \pdfbookmark[<level>]{<title>}{<dest>} +\pdfbookmark[section]{\contentsname}{toc} +\tableofcontents +\pagebreak + +% TODO: Begin Paper here. + +\end{document} diff --git a/template/preamble.tex b/template/preamble.tex new file mode 100644 index 0000000..1655d64 --- /dev/null +++ b/template/preamble.tex @@ -0,0 +1,741 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PACKAGE IMPORTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% TODO: Uncomment if the paper is in German +% \usepackage[ngerman]{babel} +\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry} +\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools} +\usepackage[varbb]{newpxmath} +\usepackage{xfrac} +\usepackage[makeroom]{cancel} +\usepackage{mathtools} +\usepackage{bookmark} +\usepackage{enumitem} +\usepackage{hyperref,theoremref} +\hypersetup{ + pdftitle={Assignment}, + colorlinks=true, linkcolor=doc!90, + bookmarksnumbered=true, + bookmarksopen=true +} +\usepackage[most,many,breakable]{tcolorbox} +\usepackage{xcolor} +\usepackage{varwidth} +\usepackage{varwidth} +\usepackage{etoolbox} +%\usepackage{authblk} +\usepackage{nameref} +\usepackage{multicol,array} +\usepackage{tikz-cd} +\usepackage[ruled,vlined,linesnumbered]{algorithm2e} +\usepackage{comment} % enables the use of multi-line comments (\ifx \fi) +\usepackage{import} +\usepackage{xifthen} +\usepackage{pdfpages} +\usepackage{transparent} + +\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}} +\SetCommentSty{mycommfont} +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} + +\usepackage{tikzsymbols} +\renewcommand\qedsymbol{$\Laughey$} + + +%\usepackage{import} +%\usepackage{xifthen} +%\usepackage{pdfpages} +%\usepackage{transparent} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COLORS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\definecolor{myg}{RGB}{56, 140, 70} +\definecolor{myb}{RGB}{45, 111, 177} +\definecolor{myr}{RGB}{199, 68, 64} +\definecolor{mytheorembg}{HTML}{F2F2F9} +\definecolor{mytheoremfr}{HTML}{00007B} +\definecolor{mylenmabg}{HTML}{FFFAF8} +\definecolor{mylenmafr}{HTML}{983b0f} +\definecolor{mypropbg}{HTML}{f2fbfc} +\definecolor{mypropfr}{HTML}{191971} +\definecolor{myexamplebg}{HTML}{F2FBF8} +\definecolor{myexamplefr}{HTML}{88D6D1} +\definecolor{myexampleti}{HTML}{2A7F7F} +\definecolor{mydefinitbg}{HTML}{E5E5FF} +\definecolor{mydefinitfr}{HTML}{3F3FA3} +\definecolor{notesgreen}{RGB}{0,162,0} +\definecolor{myp}{RGB}{197, 92, 212} +\definecolor{mygr}{HTML}{2C3338} +\definecolor{myred}{RGB}{127,0,0} +\definecolor{myyellow}{RGB}{169,121,69} +\definecolor{myexercisebg}{HTML}{F2FBF8} +\definecolor{myexercisefg}{HTML}{88D6D1} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TCOLORBOX SETUPS +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\setlength{\parindent}{0cm} + +%================================ +% THEOREM BOX +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{theorem}{\theorytitle} +{% + enhanced, + breakable, + colback = mytheorembg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mytheoremfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mytheoremfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mytheoremfr}, +} +{th} + + +\tcbuselibrary{theorems,skins,hooks} +\newtcolorbox{Theoremcon} +{% + enhanced + ,breakable + ,colback = mytheorembg + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{mytheoremfr} + ,sharp corners + ,description font = \mdseries + ,separator sign none +} + + +%================================ +% Corollery +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle} +{% + enhanced + ,breakable + ,colback = myp!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myp!85!black} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myp!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myp!85!black} +} +{th} + + +%================================ +% LENMA +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{lenma}{Lenma} +{% + enhanced, + breakable, + colback = mylenmabg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mylenmafr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mylenmafr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mylenmafr}, +} +{th} + + +%================================ +% PROPOSITION +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{prop}{\proposaltitle} +{% + enhanced, + breakable, + colback = mypropbg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{mypropfr}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = mypropfr, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, mypropfr}, +} +{th} + + +%================================ +% CLAIM +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{claim}{\claimtitle} +{% + enhanced + ,breakable + ,colback = myg!10 + ,frame hidden + ,boxrule = 0sp + ,borderline west = {2pt}{0pt}{myg} + ,sharp corners + ,detach title + ,before upper = \tcbtitle\par\smallskip + ,coltitle = myg!85!black + ,fonttitle = \bfseries\sffamily + ,description font = \mdseries + ,separator sign none + ,segmentation style={solid, myg!85!black} +} +{th} + + + +%================================ +% Exercise +%================================ + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=section]{Exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +\tcbuselibrary{theorems,skins,hooks} +\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle} +{% + enhanced, + breakable, + colback = myexercisebg, + frame hidden, + boxrule = 0sp, + borderline west = {2pt}{0pt}{myexercisefg}, + sharp corners, + detach title, + before upper = \tcbtitle\par\smallskip, + coltitle = myexercisefg, + fonttitle = \bfseries\sffamily, + description font = \mdseries, + separator sign none, + segmentation style={solid, myexercisefg}, +} +{th} + +%================================ +% EXAMPLE BOX +%================================ + +\newtcbtheorem[number within=section]{Example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +\newtcbtheorem[number within=chapter]{example}{\exampletitle} +{% + colback = myexamplebg + ,breakable + ,colframe = myexamplefr + ,coltitle = myexampleti + ,boxrule = 1pt + ,sharp corners + ,detach title + ,before upper=\tcbtitle\par\smallskip + ,fonttitle = \bfseries + ,description font = \mdseries + ,separator sign none + ,description delimiters parenthesis +} +{ex} + +%================================ +% DEFINITION BOX +%================================ + +\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} +\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced, + before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm, + attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm, + boxed title style={frame code={ + \path[fill=tcbcolback] + ([yshift=-1mm,xshift=-1mm]frame.north west) + arc[start angle=0,end angle=180,radius=1mm] + ([yshift=-1mm,xshift=1mm]frame.north east) + arc[start angle=180,end angle=0,radius=1mm]; + \path[left color=tcbcolback!60!black,right color=tcbcolback!60!black, + middle color=tcbcolback!80!black] + ([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east) + [rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east) + -- (frame.south east) -- (frame.south west) + -- ([xshift=-1mm,yshift=-1mm]frame.north west) + [sharp corners]-- cycle; + },interior engine=empty, + }, + fonttitle=\bfseries, + title={#2},#1}{def} + + + +%================================ +% Solution BOX +%================================ + +\makeatletter +\newtcbtheorem{question}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=myb!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +%================================ +% SOLUTION BOX +%================================ + +\makeatletter +\newtcolorbox{solution}{enhanced, + breakable, + colback=white, + colframe=myg!80!black, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + title=Solution, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, +} +\makeatother + +%================================ +% Question BOX +%================================ + +\makeatletter +\newtcbtheorem{qstion}{\questiontitle}{enhanced, + breakable, + colback=white, + colframe=mygr, + attach boxed title to top left={yshift*=-\tcboxedtitleheight}, + fonttitle=\bfseries, + title={#2}, + boxed title size=title, + boxed title style={% + sharp corners, + rounded corners=northwest, + colback=tcbcolframe, + boxrule=0pt, + }, + underlay boxed title={% + \path[fill=tcbcolframe] (title.south west)--(title.south east) + to[out=0, in=180] ([xshift=5mm]title.east)-- + (title.center-|frame.east) + [rounded corners=\kvtcb@arc] |- + (frame.north) -| cycle; + }, + #1 +}{def} +\makeatother + +\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{ + breakable, + enhanced, + colback=white, + colframe=myr, + arc=0pt, + outer arc=0pt, + fonttitle=\bfseries\sffamily\large, + colbacktitle=myr, + attach boxed title to top left={}, + boxed title style={ + enhanced, + skin=enhancedfirst jigsaw, + arc=3pt, + bottom=0pt, + interior style={fill=myr} + }, + #1 +}{def} + + + +%================================ +% NOTE BOX +%================================ + +\usetikzlibrary{arrows,calc,shadows.blur} +\tcbuselibrary{skins} +\newtcolorbox{note}[1][]{% + enhanced jigsaw, + colback=gray!20!white,% + colframe=gray!80!black, + size=small, + boxrule=1pt, + title=\textbf{Bemerkung:-}, + halign title=flush center, + coltitle=black, + breakable, + drop shadow=black!50!white, + attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2}, + minipage boxed title=2.5cm, + boxed title style={% + colback=white, + size=fbox, + boxrule=1pt, + boxsep=2pt, + underlay={% + \coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$); + \coordinate (dotB) at ($(interior.east) + (0.5pt,0)$); + \begin{scope} + \clip (interior.north west) rectangle ([xshift=3ex]interior.east); + \filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east); + \end{scope} + \begin{scope}[gray!80!black] + \fill (dotA) circle (2pt); + \fill (dotB) circle (2pt); + \end{scope} + }, + }, + #1, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% SELF MADE COMMANDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}} +\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}} +\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}} +\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}} +\newcommand{\clm}[3]{\begin{claim}{#1}{#2}#3\end{claim}} +\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}} +\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}} +\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}} +\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}} +\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}} +\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}} +\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}} +\newcommand{\nt}[1]{\begin{note}#1\end{note}} + +\newcommand*\circled[1]{\tikz[baseline=(char.base)]{ + Wrong Concept \node[shape=circle,draw,inner sep=1pt] (char) {#1};}} +\newcommand\getcurrentref[1]{% + \ifnumequal{\value{#1}}{0} + {??} + {\the\value{#1}}% +} +\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}} +\newenvironment{myproof}[1][\proofname]{% + \proof[\bfseries #1: ]% +}{\endproof} + +\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}} +\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{} + +\newcounter{mylabelcounter} + +\makeatletter +\newcommand{\setword}[2]{% + \phantomsection + #1\def\@currentlabel{\unexpanded{#1}}\label{#2}% +} +\makeatother + + + + +\tikzset{ + symbol/.style={ + draw=none, + every to/.append style={ + edge node={node [sloped, allow upside down, auto=false]{$#1$}}} + } +} + + +% deliminators +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} + +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\round}{\lfloor}{\rceil} + +\newsavebox\diffdbox +\newcommand{\slantedromand}{{\mathpalette\makesl{d}}} +\newcommand{\makesl}[2]{% + \begingroup + \sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}% + \pdfsave + \pdfsetmatrix{1 0 0.2 1}% + \rlap{\usebox{\diffdbox}}% + \pdfrestore + \hskip\wd\diffdbox + \endgroup +} +\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{% + \slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}% + {\slantedromand\hspace{0.2ex}^{#1}}}} +\ProvideDocumentCommand\dv{o m g}{% + \ensuremath{% + \IfValueTF{#3}{% + \IfNoValueTF{#1}{% + \frac{\dd #2}{\dd #3}% + }{% + \frac{\dd^{#1} #2}{\dd #3^{#1}}% + }% + }{% + \IfNoValueTF{#1}{% + \frac{\dd}{\dd #2}% + }{% + \frac{\dd^{#1}}{\dd #2^{#1}}% + }% + }% + }% +} +\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}} +% - others +\DeclareMathOperator{\Lap}{\mathcal{L}} +\DeclareMathOperator{\Var}{Var} % varience +\DeclareMathOperator{\Cov}{Cov} % covarience +\DeclareMathOperator{\E}{E} % expected + +% Since the amsthm package isn't loaded + +% I prefer the slanted \leq +\let\oldleq\leq % save them in case they're every wanted +\let\oldgeq\geq +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% TABLE OF CONTENTS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\usepackage{tikz} +\definecolor{doc}{RGB}{0,60,110} +\usepackage{titletoc} +\contentsmargin{0cm} +\titlecontents{chapter}[3.7pc] +{\addvspace{30pt}% + \begin{tikzpicture}[remember picture, overlay]% + \draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);% + \pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries \chaptertitle\ \thecontentslabel};% + \end{tikzpicture}\color{doc!60}\large\sc\bfseries}% +{} +{} +{\;\titlerule\;\large\sc\bfseries \pagetitle \thecontentspage + \begin{tikzpicture}[remember picture, overlay] + \draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt); + \end{tikzpicture}}% +\titlecontents{section}[3.7pc] +{\addvspace{2pt}} +{\contentslabel[\thecontentslabel]{2pc}} +{} +{\hfill\small \thecontentspage} +[] +\titlecontents*{subsection}[3.7pc] +{\addvspace{-1pt}\small} +{} +{} +{\ --- \small\thecontentspage} +[ \textbullet\ ][] + +\makeatletter +\renewcommand{\tableofcontents}{% + \chapter*{% + \vspace*{-20\p@}% + \begin{tikzpicture}[remember picture, overlay]% + \pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};% + \draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);% + \clip (13,-.75) rectangle (20,1); + \pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};% + \end{tikzpicture}}% + \@starttoc{toc}} +\makeatother diff --git a/zusammenfassung/README.md b/zusammenfassung/README.md new file mode 100644 index 0000000..de14686 --- /dev/null +++ b/zusammenfassung/README.md @@ -0,0 +1,18 @@ +# Zusammenfassung + +Ich habe für die folgenden Fächer eine Zusammenfassung geschrieben **ODER** eine von der AMIV Plattform modifiziert. + +- [Digitaltechnik](https://eth.jirayuruh.ch/hs24/digitech/) +- [Technische Mechanik](https://eth.jirayuruh.ch/hs24/techmech/) + +Für die Fächer welche nicht aufgelistet wurden, habe ich entweder die Zusammenfassung 1:1 von AMIV genommen oder mir wurde eine Zusammenfassung zur Verfügung gestellt oder es war nicht erlaubt eine Zusammenfassung zu verwenden. + +## DISCLAIMER!!! + +Ich übernehme keine Haftung über mögliche Fehler in den Zusammenfassungen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!). + +Fehler können per Discord, WhatsApp, Mail und Moodle gemeldet werden. Sie sind in der Fehlerliste ersichtlich. + +--- + +Made by JirR02 in Switzerland 🇨🇭 diff --git a/zusammenfassung/digitaltechnik/.DS_Store b/zusammenfassung/digitaltechnik/.DS_Store new file mode 100644 index 0000000..5008ddf Binary files /dev/null and b/zusammenfassung/digitaltechnik/.DS_Store differ diff --git a/zusammenfassung/digitaltechnik/images/AddSub.jpg b/zusammenfassung/digitaltechnik/images/AddSub.jpg new file mode 100644 index 0000000..6a56725 Binary files /dev/null and b/zusammenfassung/digitaltechnik/images/AddSub.jpg differ diff --git a/zusammenfassung/digitaltechnik/images/Addition.jpg b/zusammenfassung/digitaltechnik/images/Addition.jpg new file mode 100644 index 0000000..1c1e601 Binary files /dev/null and b/zusammenfassung/digitaltechnik/images/Addition.jpg differ diff --git a/zusammenfassung/digitaltechnik/images/AsyncCounter.jpg b/zusammenfassung/digitaltechnik/images/AsyncCounter.jpg new file mode 100644 index 0000000..05a8459 Binary files /dev/null and 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differ diff --git a/zusammenfassung/digitaltechnik/images/sram_write.JPG b/zusammenfassung/digitaltechnik/images/sram_write.JPG new file mode 100644 index 0000000..0f43032 Binary files /dev/null and b/zusammenfassung/digitaltechnik/images/sram_write.JPG differ diff --git a/zusammenfassung/digitaltechnik/images/tg.jpeg b/zusammenfassung/digitaltechnik/images/tg.jpeg new file mode 100644 index 0000000..60ddd00 Binary files /dev/null and b/zusammenfassung/digitaltechnik/images/tg.jpeg differ diff --git a/zusammenfassung/digitaltechnik/sections/EinfacheHardwarekomp.tex b/zusammenfassung/digitaltechnik/sections/EinfacheHardwarekomp.tex new file mode 100644 index 0000000..3fd1105 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/EinfacheHardwarekomp.tex @@ -0,0 +1,59 @@ +\section{Einfache Hardwarekomponente} + +\subsection{Multiplexer und Demultiplexer} + +Multiplexer ermöglichen das durch Steuersignale gewählte Aufschalten eines Eingangssignal aus mehreren möglichen: + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Multiplexer.jpg} +\end{center} + +Demultiplexer nehmen Daten aus einem einzigen Kanal und verteilen es auf einen beliebigen Ausgang. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Demultiplexer.jpg} +\end{center} + + +\subsection{Halb- und Volladdierer} + +Halbaddierer sind Rechenschaltungen, die zwei Dualzahlen addieren. Ausgänge: SUM (Summe), CO (Carry, Übertrag) + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Halbaddierer.jpg} +\end{center} + +Volladdierer haben einen zusätzlichen Eingang CI (Carry in), dieser ermöglicht das Bilden von Mehrbit-Addierer. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Volladdierer.jpg} +\end{center} + + +\subsection{Mehrbit-Addierer (Paralleladdierer)} + +Ein \emph{Paralleladdierer} in Normalform ist sehr aufwendig zu realisieren, da $\sim n \cdot 2^{2n - 1}$ Min-/Maxterme verknüpft werden müssen. \emph{Vorteil}: Laufzeit unabhängig von Stellenanzahl \medskip + +Ein \emph{Ripple-Carry Addierer} ist eine Kaskadierung von Volladdierer. Einfach skalierbar, leidet aber am 'ripple' Effekt, d.h. Laufzeiten addieren sich auf. \medskip + +Der \emph{Carry-Look-Ahead Addierer} kombiniert die Vorteil der beiden, d.h. man kaskadiert die Addierer, aber berechnet die Überträge parallel zur Summenbildung. (Berechnungsaufwand linear zur Stellenanzahl, aber Laufzeit konstant) + + +\subsubsection{Ripple-Carry Addierer mit Subtraktion} + +Die Subtraktion erfolgt durch Bildung des 2er-Komplement: + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/AddSub.jpg} +\end{center} + + +\subsection{Hardware Multiplizierer} + +Folgt dem Prinzip der Bitweisen Multiplikation. Besteht aus folgender Basiszelle: + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/HardwareMult.jpg} +\end{center} + +Die Multiplikation mit negativen Zahlen im 2er-Komplement ist eher schwierig. Eine Möglichkeit ist der iterative Booth Algorithmus. diff --git a/zusammenfassung/digitaltechnik/sections/automaten.tex b/zusammenfassung/digitaltechnik/sections/automaten.tex new file mode 100644 index 0000000..72f3462 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/automaten.tex @@ -0,0 +1,196 @@ +\section{Automaten} + +Ein Automat beschreibt ein System, welches auf seine Eingänge reagiert und ein Ausgangsignal produziert, dass vom Eingangssignal und momentanen Zustand abhängt. \medskip + +Endliche Automaten, bzw. Finite state machines (FSM), können nur vorprogrammierte Lagen passieren. Schaltungen mit Rückkopplungen sind typische Beispiele für FSM. \medskip + +Bei synchronen Automaten sind alle FF gleich getaktet, d.h. Zustandsänderungen sind synchron mit dem Takt. \medskip + +Mit $n$-D-FF kann eine FSM $2^n$ innere Zustände speichern. + + +\subsection{Beschreibung von Automaten} + +\subsubsection{Formelle Beschreibung von Automaten} + +\begin{tabular}{l p{50mm}} + $X_n = (x_1, \dots, x_e)$ & Eingangsalphabet \\ + $Y_n = (y_1, \dots, y_b)$ & Ausgangsalphabet \\ + $Z_n = (z_1, \dots, z_m)$ & Zustandsmenge \\ + $Z_0 \in Z$ & Anfangszustand \\ + $f_{c1}:(X_n, Z_n) \to Z_{n+1}$ & Folgezustandsfunktion \\ + $f_{c2}:(X_n, Z_n) \to Y_n$ & Ausgangsfunktion \\ +\end{tabular} + +\subsubsection{Zustandsfolgetabellen (Folgezustandstabellen)} + +Listet in Form einer Wahrheitstabelle alle Kombinationen des aktuellen Zustandvektors $Z_n$ und Eingangsvektor $X_n$, aus welchen der Ausgangsvektor $Y_n$ und Folgezustandvektor $Z_{n+1}$ entsteht. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Automat_1.jpg} +\end{center} + +Man kann die verschiedenen Vektoren abstrahieren durch Dualzahlen oder ihnen sinnvolle Namen geben, welche den Zustand/Eingang/Ausgang erklären. +\\ +Um die Anzahl Zeilen und Spalten einer Zustandsfolgetabelle zu bestimmen kann man die folgenden Formeln verwenden. + +\eqbox{$Z = 2 ^{eb + zb}$} + +\eqbox{$S = eb + (2 \cdot zb) + ab$} + +$eb$ ist die Anzahl Eingangsbits, $zb$ die Anzahl Zustandsbits und $ab$ die Anzahl Ausgangsbits. + +\subsubsection{Zustandsdiagramme (Zustandsgraphen)} + +Äquivalente graphische Darstellung der Folgezustandstabelle. Zustandsdiagramme bestehen aus Kanten und Knoten, siehe spätere Abbildungen bei den verschiedenen Automatentypen. + + +\subsection{Mealy-Automat} + +Bei einem Mealy-Automat beinflussen Eingangsveränderungen das Ausgangsignal jederzeit, d.h. das Ausgangssignal ist Störungsanfällig. Ein Mealy-Automat hat folgende Struktur: + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Mealy_1.jpg} \medskip + + \eqbox{$Y_n = f_{C2}(X_n, Z_n) \qquad Z_{n+1} = f_{C1}(X_n, Z_n)$} +\end{center} + +Erst bei einem Taktwechsel $t_n \to t_{n + 1}$ schalten die D-FF, d.h. $Z_{n + 1} \to Z_n$. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Mealy_2.jpg} +\end{center} + +\vfill + +\subsection{Moore-Automat} + +Sonderfall des Mealy-Automat, wo $Y_n$ nur von getaktetem $Z_n$ abhängt. Aus diesem Grund ist er weniger Störungsanfällig. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Moore_1.jpg} \medskip + + \eqbox{$Y_n = f_{C2}(Z_n) \qquad Z_{n+1} = f_{C1}(X_n, Z_n)$} +\end{center} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Moore_2.jpg} +\end{center} + +Eine \emph{komplett störungssichere Automatenfunktion} ist jedoch nur möglich, wenn die Eingangswerte $X_n$ mit gleich getakteten D-FF synchronisiert werden. + + +\subsection{Medwedjew-Automat} + +Ausgangsvektor $Y_n$ ist identisch mit dem Zustandsvektor $Z_n$. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Medwedjew.jpg} \medskip + + \eqbox{$Y_n = Z_n \qquad Z_{n+1} = f_{C1}(X_n, Z_n)$} +\end{center} + +\vfill + +\subsection{Entwurf eines Automaten} + +Die Normale Entwurfsabfolge ist, wie folgt: + +\begin{enumerate} + \item Auftrag lesen und analysieren + \item Zustandsmenge bestimmen, daraus folgt Anzahl Zustandsvariablen und D-FF. + \item Kodierung: Eingangs- und Ausgangsvariablen definieren + \item Darstellung der Zustandsfolge in einem \emph{Zustandsdiagram} + \item \emph{Zustandsfolgetabelle} aufstellen. + \item Minimierung der Ausgangs- und Übergangsfunktionen mit \emph{KV-Diagrammen} + \item Einfluss unbenutzter Zustände (Don't Cares) überprüfen! + \item Schaltplan anhand der Schaltfunktion konstruieren +\end{enumerate} + +Wenn das Schaltwerk eines Automaten vorgegeben ist, wird die Reihenfolge vertauscht. + + +\subsection{Umwandlung Mealy $\Leftrightarrow$ Moore} + +Achtung: Bei der Umwandlung verändert sich das Zeitverhalten der Ausgänge. + +\subsubsection{Moore $\to$ Mealy} + +Die Moore zu Mealy Umwandlung ist einfach, da einfach die Ausgänge von den Knoten der Folgezustände auf die entsprechenden Kanten gewechselt werden müssen. + +\subsubsection{Mealy $\to$ Moore} + +Ein Mealy-Automat lässt sich immer in einen Moore-Automat mit gleicher Funktion umwandeln, dieser besitzt aber in der Regel mehr Zustände. + +Eine direkte Moore Implementierung ist nur möglich, wenn jeder Zustand $Z_n$, unabhängig von $X_n$, immer das gleiche Ausgangssignal $Y_n$ produziert. Ist dies nicht der Fall, dann muss man neue Zustände definieren. + + +\subsection{Asynchronzähler} + +Ein einfacher Asynchron Vorwärtszähler aus T-FF sieht, wie folgt, aus: + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/AsyncCounter.jpg} +\end{center} + +Will man einen Rückwärtszähler realisieren, dann muss man auf die positive Flanke gesteuerte FFs verwenden ($0 \to 1$) und statt $Q_i$ die invertierten Ausgänge $\not{Q}_i$ benutzen. \medskip + +Die Problematik vom Asynchronzähler ist, dass die Zustandsänderungen einen 'ripple' Effekt aufzeigen, d.h. die FF-Verzögerungszeiten kumulieren sich entlang der Schaltung. Die maximale Taktfrequenz, für welche ein Asynchronzähler 'theoretisch' noch funktioniert, ist: + +\begin{center} + \eqbox{$f_{max} = \dfrac{1}{\sum t_{pd,FFs}}$} +\end{center} + + +\subsection{Modulo-n Zähler} + +Ein Modulo-n Zähler zählt bis zu einem bestimmten Zustand $n$ und springt dann zum vorgegebenen Anfangszustand zurück. Dafür benötigt man FFs mit Asychronen Reset und Set Eingängen, welche benutzt werden um den Rücksprung zum Anfangszustand auszuführen. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/AsyncCounter2.jpg} +\end{center} + +Der Zustand $n+1$ ist kurzzeitig für die Durchlaufzeit des AND-Gatters vorhanden an den Ausgängen, bis der Rücksprung zum Anfangszustand durchgeführt wurde. +\vfill + + +\subsection{Synchronzähler} + +Bei einem Synchronzähler haben alle FF den gleichen Takt. Synchronzähler sind meist Medwedjew-Automanten, wo der Steuereingang benutzt wird um zwischen Vorwärts- und Rückwartszählen zu wechseln. Beispiel anhand eines Modulo-6 Zählers mit Anfangszustand '1': + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/SyncCounter.jpg} +\end{center} + +Falls JK-FF verwendet werden müssen, dann muss man bei der Minimierung mit dem KV-Diagramm aufpassen, dass man die charakteristische Gleichung des JK-FF einhält. + + +\subsection{Vorwärts-Rückwärtszähler (Reversible Zähler)} + +Ein grosser reversibler Zähler ist relativ aufwendig zu realisieren als Automat. Einfacher ist es D-FFs geschickt mit einem Addierer zu kombinieren. + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/SyncCounter2.jpg} +\end{center} + +Für das Wechseln zwischen Vorwärts- und Rückwärtszählen wechselt man beim Addierer einfach zwischen Addition und Subtraktion (2er Komplement). +\vfill + + +\subsection{Frequenzteiler mit einem Zähler} + +Gegeben ist ein Zähler mit $N$ Zuständen und einem Anfangszustand $k$. Wenn der Zustand $N$ erreicht wird, dann wird bei der nächsten Taktflanke der Anfangszustand $k$ geladen und der T-FF wechselt seinen Zustand. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Frequenzteiler2.jpg} +\end{center} + +Mit einer solchen Schaltung kann man einen Frequenzteiler realisieren, welcher flexibler ist in seinem Teilungsverhältnis als ein simpler Frequenzteiler aus nur T-FFs. \medskip + +Der Ausgang vom T-FF hat die Frequenz: + +\begin{center} + \eqbox{$f_{out} = \dfrac{f_{in}}{2 (N - k + 1)}$} +\end{center} + + diff --git a/zusammenfassung/digitaltechnik/sections/boolsche_algebra.tex b/zusammenfassung/digitaltechnik/sections/boolsche_algebra.tex new file mode 100644 index 0000000..020419f --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/boolsche_algebra.tex @@ -0,0 +1,96 @@ +\section{Boolsche Algebra} + +Es gelten folgende Grundgesetze, wie in der Algebra: + +\begin{center} + \begin{tabular}{r l l} \toprule + Kommutativität & $A \land B = B \land A$ \\ + & $A \lor B = B \lor A$ \\ + Assoziativität & $A \land (B \land C) = (A \land B) \land C$ \\ + & $A \lor (B \lor C) = (A \lor B) \lor C$ \\ + Distributivität & $(\textcolor{blue}{A\,\land}~B) \lor (\textcolor{blue}{A\,\land}\,C) = \textcolor{blue}{A\,\land}\,(B \lor C)$ \\ + & $(\textcolor{blue}{A\,\lor}~B) \land (\textcolor{blue}{A\,\lor}\,C) = \textcolor{blue}{A\,\lor}\,(B \land C)$ \\ \bottomrule + \end{tabular} +\end{center} + +Aus diesem Grund gilt folgende für die Logikminimierung sehr nützliche Umwandlung: + +\begin{center} + \eqboxf{$\land \rightleftharpoons \cdot$ \qquad $\lor \rightleftharpoons +$} +\end{center} + +Zusätzlich zu den Grundregeln gelten folgende Regeln: + +\begin{center} + \begin{tabular}{r l l} \toprule + Nicht & $\overline{\overline{A}} = A$ & \\ + Null-Theorem & $A \lor 0 = A$ & $A \land 0 = 0$ \\ + Eins-Theorem & $A \lor 1 = 1$ & $A \land 1 = A$ \\ + Idempotenz & $A \lor A = A$ & $A \land A = A$ \\ + Verknüpfung mit Komplement & $A \lor \overline{A} = 1$ & $A \land \overline{A} = 0$ \\ \midrule + Adsorptionsgesetze & \multicolumn{2}{l}{$A \lor (\overline{A} \land B) = A \lor B$} \\ + & \multicolumn{2}{l}{$A \land (\overline{A} \lor B) = A \land B$} \\ + Absorptionsgesetze & \multicolumn{2}{l}{$A \lor (A \land B) = A$} \\ + & \multicolumn{2}{l}{$A \land (A \lor B) = A$} \\ + Nachbarschafts Gesetze & \multicolumn{2}{l}{$(A \land B) \lor (\overline{A} \land B) = B$} \\ + & \multicolumn{2}{l}{$(A \lor B) \land (\overline{A} \lor B) = B$} \\ \bottomrule + \end{tabular} +\end{center} +\vfill + + +\subsection{De Morgan'sche Gesetze} + +Beziehungen zwischen NAND/NOR und AND/OR: + +\begin{center} + \begin{tabular}{r l} \toprule + Erstes Gesetz: & $\overline{A \land \dots \land B} = \not{A} \lor \dots \lor \not{B}$ \\ + Zweites Gesetz: & $\overline{A \lor \dots \lor B} = \not{A} \land \dots \land \not{B}$ \\ \bottomrule + \end{tabular} +\end{center} + +Daraus folgen zwei sehr nützliche Umwandlungen: + +\begin{center} + \eqboxf{$A \land \dots \land B = \overline{\not{A} \lor \dots \lor \not{B}} \qquad A \lor \dots \lor B = \overline{\not{A} \land \dots \land \not{B}}$} +\end{center} + + +\subsection{Normalformen} + +Der \emph{Minterm} ist eine AND-Verknüpfung, welcher $'1'$ ergibt für nur eine Kombination der Schaltungsvariablen. \medskip + +Der \emph{Maxterm} ist eine OR-Verknüpfung, welcher $'0'$ ergibt für nur eine Kombination der Schaltungsvariablen. Bei der Bildung der Maxterme \emph{müssen die Variablen invertiert werden}!\medskip + +\begin{center} + \renewcommand{\arraystretch}{1.25} + \begin{tabular}{|c c|c|c|c|} \hline + A & B & Z & Minterme & Maxterme \\ \hline + 0 & 0 & 1 & $\overline{A} \land \overline{B}$ & \\ + 0 & 1 & 0 & & $A \lor \overline{B}$ \\ + 1 & 0 & 0 & & $\overline{A} \lor B$ \\ + 1 & 1 & 1 & $A \land B$ & \\ \hline + \end{tabular} +\end{center} + +\subsubsection{Disjunktive Normalform DNF (ODER-Normalform)} + +Besteht aus einer ODER-Verknüpfung aller Minterme: + +\begin{center} + \eqbox{$Z = (\not{A} \land \not{B}) \lor (A \land B)$} +\end{center} + +\subsubsection{Konjunktive Normalform KNF (UND-Normalform)} + +Besteht aus einer UND-Verknüpfung aller Maxterme: + +\begin{center} + \eqbox{$\not{Z} = (A \lor \not{B}) \land (\not{A} \lor B)$} +\end{center} + +\subsubsection{Kanonische Normalform} + +Die kanonische Normalform ist die unvereinfachte Normalform einer Wahrheitstabelle. Sie gibt also nicht notwendigerweise die einfachsten Funktionsgleichungen an. +\vfill diff --git a/zusammenfassung/digitaltechnik/sections/cmos.tex b/zusammenfassung/digitaltechnik/sections/cmos.tex new file mode 100644 index 0000000..463de47 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/cmos.tex @@ -0,0 +1,134 @@ +\section{CMOS Gatter} + +High Pegel ''H'': 0.9-0.7V \qquad Low Pegel ''L'': 0.15-0V \medskip + +\begin{minipage}{0.48\linewidth} + \subsubsection{NMOS} + \begin{center} + \begin{circuitikz}[european] + \node[circ, label=90:{\small $V_{DD} = 0.8 \si{\volt}$}](origin) at (0,0) {}; + \node[thick, nmos, anchor=D] (nmos1) at(0, -2) {} + (nmos1.gate) node[anchor=east] {G} + (nmos1.drain) node[anchor=west, yshift=-0.15cm] {D} + (nmos1.source) node[anchor=west,yshift=+0.15cm] {S}; + \draw[thick] + (origin) + to[R=$R$] (0,-1.8) node[circ] (ybase) {} + to[] (0, -2); + + + \draw[thick] (nmos1.S) -- (0, -4) coordinate(gnd); + + \path[draw] (ybase) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw, very thick] (-0.25, -4) -- (gnd) -- (0.25, -4); + \end{circuitikz} \medskip + + \begin{tabular}{c|c|c} + G & Schalter & Y \\ + \hline + 0 & offen & 1 \\ + 1 & zu & 0 + \end{tabular} + \end{center} +\end{minipage} +\vline +\begin{minipage}{0.48\linewidth} + \subsubsection{PMOS} + \begin{center} + \begin{circuitikz}[european] + \coordinate (gnd) at (0, -4); + \node[circ, label=90:{\small $V_{DD} = 0.8 \si{\volt}$}] (vdd) at (0,0) {}; + \node[pmos, thick] (pmos) at (0, -1){} + (pmos.gate) node[anchor=east] {G} + (pmos.source) node[anchor=west,yshift=-0.15cm] {S} + (pmos.drain) node[anchor=west, yshift=+0.15cm] {D}; + + \draw[thick] (gnd) + to[R=$R$] (0, -2) + -- (0, -2) node[circ] (ybase) {} + -- (pmos.D); + + \draw[thick] (pmos.S) -- (vdd); + + \path[draw] (ybase) --++(right:10mm) node[point, label=0:Y] {}; + \path[draw, very thick] (-0.25, -4) -- (gnd) -- (0.25, -4); + \end{circuitikz} \medskip + + \begin{tabular}{c|c|c} + G & Schalter & Y \\ + \hline + 0 & zu & 1 \\ + 1 & offen & 0 + \end{tabular} + \end{center} +\end{minipage} \medskip + + +Zustand NN: Potential an Source unbestimmt, ''free floating'' + + +\subsection{Konstruktion von CMOS-Gatter} + +CMOS-Gatter benötigen \emph{pro} Eingang 1 NMOS + 1 PMOS. \medskip + +Sie bestehen aus zwei ergänzenden Schaltungsteilen: + +\begin{center} + \begin{minipage}{0.3\linewidth} + \begin{center} + \begin{tikzpicture} + \node[circ, label=90:{\small $V_{DD} = 0.8 \si{\volt}$}] at (0,0) (origin) {}; + \node[draw, dotted](pmos) at(0,-0.5) {PMOS}; + \node[draw, dotted](nmos) at (0, -1.5) {NMOS}; + \coordinate(gnd) at (0, -2){}; + \draw[] (origin) -- (pmos) + (pmos) -- (nmos) + (nmos) -- (gnd); + \path[draw] (0,-1) --++(right:5mm) node[fill = white] {Y}; + \path[draw, thick] (-0.25, -2) -- (gnd) -- (0.25, -2); + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.55\linewidth} + \begin{flushleft} + \begin{tabular}{l l} + Pull-u\textcolor{red}{p} Schaltung: & \textcolor{red}{P}MOS \\ + Pull-dow\textcolor{red}{n} Schaltung: & \textcolor{red}{N}MOS \\ + \end{tabular} + \end{flushleft} + \end{minipage} +\end{center} + +\subsubsection{Funktionsgleichung CMOS-Gatter} + +\begin{tabular}{l l l} + Pull-Up: & $Y_{\text{pu}} = 1$ & Eingänge Invertiert \\ + Pull-Down: & $Y_{\text{pd}} = 0$ & Eingänge nicht Invertiert \\ +\end{tabular} + +\begin{center} + \eqbox{$Y_{\text{pu}} = \overbrace{\underbrace{(\not{A} \land \not{B})}_{\text{Seriell} } \lor \not{C}}^{\text{Paralell} } \quad \Leftrightarrow \quad$ + $Y_{\text{pd}} = \overbrace{\overline{\underbrace{(A \lor B)}_{\text{Paralell} } \land C}}^{\text{Seriell} }$} + + \includegraphics[width = 0.32\textwidth]{images/pnmosDetConv.png} +\end{center} + + +\subsubsection{Umwandlung zwischen Pull-up und Pull-down} + +\begin{enumerate} + \item Schaltung in Parallele und Serielle Blöcke zerlegen + \item Umwandeln: Parallele $\rightleftharpoons$ Serielle Blöcke + \item 1),2) wiederholen bis einzelnen Transistoren übrig sind + \item Umwandeln: PMOS $\rightleftharpoons$ NMOS + \item Schaltungstyp entsprechend Y, GND bzw. VDD setzen +\end{enumerate} \medskip + +\includegraphics[width = 0.3\textwidth]{images/PU_to_PD.jpg} +\vfill + +\subsection{Zeitverhalten CMOS Gatter} + +\includegraphics[width = 0.32\textwidth]{images/Laufzeit.jpg} + +Durchschnittliche Verzögerung: \eqbox{$t_d = \dfrac{t_{pHL} + t_{pLH}}{2}$} diff --git a/zusammenfassung/digitaltechnik/sections/codes.tex b/zusammenfassung/digitaltechnik/sections/codes.tex new file mode 100644 index 0000000..749e4da --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/codes.tex @@ -0,0 +1,79 @@ +\section{Codes} + +\subsection{Tetraden-Codes} + +\begin{itemize} + \item \textbf{BCD}: Häufig benutzt, keine Rundungsfehler! + \item \textbf{Excess-3 und Aiken}: Ziffern liegen symmetrisch im Binärfeld, günstige Verteilung für dezimale Rechenwerke + \item \textbf{4-2-2-1}: Interessante Gewichtung für A/D Wandler + \item \textbf{Gray und O'Brien}: Einschrittige Codes (Schwächere Auswirkung von Übertragungsfehlern), keine Fehlinformation bei Übergängen (Winkelkodierung) +\end{itemize} + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/TetradenCodes.jpg} +\end{center} + + +\subsection{Parity-Bits} + +Redundante Kodierung, welche Übertragungsfehler erkennen kann, solange \emph{höchstens} ein Fehler pro Bitgruppe geschieht. \medskip + +Bitgruppe wird durch ein Parity-Bit ergänzt, welches die Bitgruppe auf geradzähligkeit ($P_E$) oder ungeradzähligkeit ($P_O$) überprüft. Der Datenempfänger kann so die Richtigkeit der Datenübertragung überprüfen. + +\begin{center} + \begin{minipage}{0.45\linewidth} + \subsubsection{Korrekt mit $P_E$} + \begin{center} + \begin{tikzpicture} + \begin{pgfonlayer}{l1} + \matrix (pm) [ + matrix of nodes, + nodes in empty cells + ]{ + 0 & 1 & 0 & 1 & 0 \\ + 1 & 1 & 0 & 1 & 1 \\ + 1 & 0 & 1 & 1 & 1 \\ + 0 & 0 & 1 & 0 & 1 \\ + 0 & 0 & 0 & 1 & 1 \\ + }; + \end{pgfonlayer} + \begin{pgfonlayer}{l1} + \draw[] (pm-4-1.south west) -- (pm-4-4.south east) -- (pm-1-4.north east); + \end{pgfonlayer} + \fill[draw, thick, darkgreen!80, rounded corners = 3pt, fill opacity = 0.3] ($(pm-5-1.north west) + (1.5pt, -1.5pt)$) rectangle ($(pm-5-5.south east) + (-1.5pt, 1.5pt)$); + \begin{pgfonlayer}{bg} + \fill[draw, thick, blue!80, rounded corners = 3pt, fill opacity = 0.3] ($(pm-1-5.north west) + (1.5pt, -1.5pt)$) rectangle ($(pm-5-5.south east) + (-1.5pt, 1.5pt)$); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{0.45\linewidth} + \subsubsection{Fehler mit $P_E$} + \begin{center} + \begin{tikzpicture} + \begin{pgfonlayer}{l2} + \matrix (pm) [ + matrix of nodes, + nodes in empty cells + ]{ + 0 & 1 & 0 & 1 & 0 \\ + 1 & 1 & 1 & 1 & \node[text = blue] {\bfseries 1}; \\ + 1 & 0 & 1 & 1 & 1 \\ + 0 & 0 & 1 & 0 & 1 \\ + 0 & 0 & \node[text = blue] {\bfseries 0}; & 1 & 1 \\ + }; + \end{pgfonlayer} + \begin{pgfonlayer}{l1} + \draw[] (pm-4-1.south west) -- (pm-4-4.south east) -- (pm-1-4.north east); + \end{pgfonlayer} + \begin{pgfonlayer}{bg} + \fill[mred, draw, thick, fill opacity = 0.3, rounded corners = 3pt] (pm-1-3.north west) rectangle (pm-5-3.south east); + \fill[mred, draw, thick, fill opacity = 0.3, rounded corners = 3pt] (pm-2-1.north west) rectangle (pm-2-5.south east); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} + \end{minipage} +\end{center} +\vfill + +Für eine Fehlerkorrektur muss zusätzlich ein Prüfwort übertragen werden, welches Spaltenweise ein Parity-Bit bildet. Der Empfänger kann so das fehlerbehaftete Bit in der Matrixdarstellung (siehe Bsp.) erkennen und korrigieren. \ No newline at end of file diff --git a/zusammenfassung/digitaltechnik/sections/gates.tex b/zusammenfassung/digitaltechnik/sections/gates.tex new file mode 100644 index 0000000..0d4b02e --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/gates.tex @@ -0,0 +1,306 @@ +\section{Gatter} + +\subsection{AND- und OR-Gatter} + +\begin{minipage}{0.27\linewidth} + \subsubsection{AND} + \begin{equation*} + Z = A \land B + \end{equation*} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[and gate] (and) at (0,0) {}; + \node[] (iA) at (-1, 0.4) {A}; + \node[] (iB) at (-1, -0.4) {B}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (and.input 1); + \draw (iB.east) --++ (right:2mm) |- (and.input 2); + \draw (and.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vline \, +\begin{minipage}{0.27\linewidth} + \subsubsection{OR} + \begin{equation*} + Z = A \lor B + \end{equation*} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[or gate] (or) at (0,0) {}; + \node[] (iA) at (-1, 0.4) {A}; + \node[] (iB) at (-1, -0.4) {B}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (or.input 1); + \draw (iB.east) --++ (right:2mm) |- (or.input 2); + \draw (or.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{minipage}{0.4\linewidth} + \begin{tabular}{|c c|c|c|} \hline + A & B & AND & OR \\ \hline + 0 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 1 \\ + 1 & 0 & 0 & 1 \\ + 1 & 1 & 1 & 1 \\ \hline + \end{tabular} +\end{minipage} + + +\subsection{NAND- und NOR-Gatter} + +\begin{minipage}{0.25\linewidth} + \subsubsection{NAND} + \begin{equation*} + Z = \overline{A \land B} + \end{equation*} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nand gate] (nand) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oZ) at (0.8, 0) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nand.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nand.input 2); + \draw (nand.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vline \, +\begin{minipage}{0.25\linewidth} + \subsubsection{NOR} + \begin{equation*} + Z = \overline{A \lor B} + \end{equation*} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nor gate] (nor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oZ) at (0.8, 0) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nor.input 2); + \draw (nor.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{minipage}{0.45\linewidth} + \begin{tabular}{|c c|c|c|} + \hline + A & B & NAND & NOR \\ + \hline + 0 & 0 & 1 & 1 \\ + 0 & 1 & 1 & 0 \\ + 1 & 0 & 1 & 0 \\ + 1 & 1 & 0 & 0 \\ + \hline + \end{tabular} +\end{minipage} + +\begin{minipage}{0.47\linewidth} + \subsubsection{NAND} + \includegraphics[height = 45mm]{images/nand.png} +\end{minipage} +\vline \, +\begin{minipage}{0.47\linewidth} + \subsubsection{NOR} + \includegraphics[height = 45mm]{images/nor.png} +\end{minipage} + + +\subsection{NOT} + +\begin{minipage}{0.3\linewidth} + \begin{equation*} + Z = \overline{A} + \end{equation*} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[not gate] (not) at (0,0) {}; + \node[] (iA) at (-0.8, 0) {A}; + \node[] (oZ) at (0.8, 0) {Z}; + \draw (iA.east) -- (not.input); + \draw (not.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{minipage}{0.25\linewidth} + \begin{tabular}{|c|c|} \hline + A & NOT \\ \hline + 0 & 1 \\ + 1 & 0 \\ \hline + \end{tabular} +\end{minipage} +\hfill +\begin{minipage}{0.3\linewidth} + \includegraphics[height = 25mm]{images/not.png} +\end{minipage} + +\begin{minipage}{0.46\linewidth} + \subsubsection{NOT aus NOR} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[nor gate] (gate) at (0,0) {}; + \node[] (iA) at (-1, 0) {A}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (gate.input 1); + \draw (iA.east) --++ (right:2mm) |- (gate.input 2); + \draw (gate.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vline \, +\begin{minipage}{0.46\linewidth} + \subsubsection{NOT aus NAND} + \begin{center} + \begin{tikzpicture}[circuit logic IEC] + \node[nand gate] (gate) at (0,0) {}; + \node[] (iA) at (-1, 0) {A}; + \node[] (oZ) at (1, 0) {Z}; + \draw (iA.east) --++ (right:2mm) |- (gate.input 1); + \draw (iA.east) --++ (right:2mm) |- (gate.input 2); + \draw (gate.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vfill + + +\subsection{XOR und XNOR} + +\begin{minipage}{0.47\linewidth} + \subsubsection{XOR} + \begin{tabular}{l l} + $Z$ & $= A \oplus B$ \\ + & $= (A \land \overline{B}) \lor (\overline{A} \land B)$ \\ + \end{tabular} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[xor gate] (xor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oZ) at (0.8, 0) {E}; + \draw (iA.east) --++ (right:1.5mm) |- (xor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (xor.input 2); + \draw (xor.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vline \, +\begin{minipage}{0.47\linewidth} + \subsubsection{XNOR} + \begin{tabular}{l l} + $Z$ & $= \overline{A \oplus B}$ \\ + & $= (A \land B) \lor (\overline{A} \land \overline{B})$ \\ + \end{tabular} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[xnor gate] (xnor) at (0,0) {}; + \node[] (iA) at (-0.8, 0.4) {A}; + \node[] (iB) at (-0.8, -0.4) {B}; + \node[] (oZ) at (0.8, 0) {F}; + \draw (iA.east) --++ (right:1.5mm) |- (xnor.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (xnor.input 2); + \draw (xnor.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\begin{center} + \begin{tabular}{|c c|c|c|}\hline + A & B & XOR & XNOR \\\hline + 0 & 0 & 0 & 1 \\ + 0 & 1 & 1 & 0 \\ + 1 & 0 & 1 & 0 \\ + 1 & 1 & 0 & 1 \\\hline + \end{tabular} +\end{center} + +\subsection{Gatter aus NAND- und NOR-Gatter} +\begin{minipage}{0.47\linewidth} + \subsubsection{AND-Gatter aus NOR-Gatter} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nor gate] (nor1) at (0,0) {}; + \node[nor gate] (nor2) at (0,-1.1) {}; + \node[nor gate] (nor3) at (1.1,-0.55) {}; + \node[] (iA) at (-0.8, 0) {A}; + \node[] (iB) at (-0.8, -1.1) {B}; + \node[] (oZ) at (2, -0.55) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nor1.input 1); + \draw (iA.east) --++ (right:1.5mm) |- (nor1.input 2); + \draw (iB.east) --++ (right:1.5mm) |- (nor2.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nor2.input 2); + \draw (nor1.output) --++ (right:1.5mm) |- (nor3.input 1); + \draw (nor2.output) --++ (right:1.5mm) |- (nor3.input 2); + \draw (nor3.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\vline \, +\begin{minipage}{0.47\linewidth} + \subsubsection{OR-Gatter aus NAND-Gatter} + \begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nand gate] (nand1) at (0,0) {}; + \node[nand gate] (nand2) at (0,-1.1) {}; + \node[nand gate] (nand3) at (1.1,-0.55) {}; + \node[] (iA) at (-0.8, 0) {A}; + \node[] (iB) at (-0.8, -1.1) {B}; + \node[] (oZ) at (2, -0.55) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nand1.input 1); + \draw (iA.east) --++ (right:1.5mm) |- (nand1.input 2); + \draw (iB.east) --++ (right:1.5mm) |- (nand2.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nand2.input 2); + \draw (nand1.output) --++ (right:1.5mm) |- (nand3.input 1); + \draw (nand2.output) --++ (right:1.5mm) |- (nand3.input 2); + \draw (nand3.output) -- (oZ); + \end{tikzpicture} + \end{center} +\end{minipage} +\subsubsection{XOR Gatter} +\begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nand gate] (nand2) at (1.3,0) {}; + \node[nand gate] (nand3) at (1.3,-1.1) {}; + \node[nand gate] (nand1) at (0.2,-0.55) {}; + \node[nand gate] (nand4) at (2.4,-0.55) {}; + \node[] (iA) at (-0.8, 0.17) {A}; + \node[] (iB) at (-0.8, -1.27) {B}; + \node[] (oZ) at (3.3, -0.55) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nand1.input 1); + \draw (iA.east) --++ (right:1.5mm) |- (nand2.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nand1.input 2); + \draw (iB.east) --++ (right:1.5mm) |- (nand3.input 2); + \draw (nand1.output) --++ (right:1.5mm) |- (nand2.input 2); + \draw (nand1.output) --++ (right:1.5mm) |- (nand3.input 1); + \draw (nand2.output) --++ (right:1.5mm) |- (nand4.input 1); + \draw (nand3.output) --++ (right:1.5mm) |- (nand4.input 2); + \draw (nand4.output) -- (oZ); + \end{tikzpicture} +\end{center} +\subsubsection{XNOR Gatter} +\begin{center} + \begin{tikzpicture}[circuit logic IEC, thick] + \node[nand gate] (nand2) at (1.3,0) {}; + \node[nand gate] (nand3) at (1.3,-1.1) {}; + \node[nand gate] (nand1) at (0.2,-0.55) {}; + \node[nand gate] (nand4) at (2.4,-0.55) {}; + \node[nand gate] (nand5) at (3.5,-0.55) {}; + \node[] (iA) at (-0.8, 0.17) {A}; + \node[] (iB) at (-0.8, -1.27) {B}; + \node[] (oZ) at (4.4, -0.55) {Z}; + \draw (iA.east) --++ (right:1.5mm) |- (nand1.input 1); + \draw (iA.east) --++ (right:1.5mm) |- (nand2.input 1); + \draw (iB.east) --++ (right:1.5mm) |- (nand1.input 2); + \draw (iB.east) --++ (right:1.5mm) |- (nand3.input 2); + \draw (nand1.output) --++ (right:1.5mm) |- (nand2.input 2); + \draw (nand1.output) --++ (right:1.5mm) |- (nand3.input 1); + \draw (nand2.output) --++ (right:1.5mm) |- (nand4.input 1); + \draw (nand3.output) --++ (right:1.5mm) |- (nand4.input 2); + \draw (nand4.output) --++ (right:1.5mm) |- (nand5.input 1); + \draw (nand4.output) --++ (right:1.5mm) |- (nand5.input 2); + \draw (nand5.output) -- (oZ); + \end{tikzpicture} +\end{center} +\vfill diff --git a/zusammenfassung/digitaltechnik/sections/karnaugh.tex b/zusammenfassung/digitaltechnik/sections/karnaugh.tex new file mode 100644 index 0000000..a557fa0 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/karnaugh.tex @@ -0,0 +1,261 @@ +\section{Karnaugh Diagramme (KVD)} + +KV-Diagramme sind eine äquivalente Darstellungsform von Schaltfunktionen in Matrizenform. KVD ermöglichen die unkomplizierte Bildung der \emph{vereinfachten} DNF/KNF. \medskip + +Schema zum ausfüllen eines 4 Variablen KV-Diagramm: + +\begin{center} + \includegraphics[width = 0.4\linewidth]{images/KVDia.jpg} +\end{center} + +\begin{center} + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center + } + ]{ + $0$ & $1$ & $X$ & \\ + & & & \\ + & $\textcolor{blue}{1}$ & & \\ + & & & \\ + }; + + \path[draw, decorate, decoration=brace] (kv-1-1.north west) -- (kv-1-2.north east) node [midway, above] {$\not{A}$}; + \path[draw, decorate, decoration=brace] (kv-1-3.north west) -- (kv-1-4.north east) node [midway, above] {$A$}; + \path[draw, decorate, decoration=brace] (kv-2-1.south west) -- (kv-1-1.north west) node [midway, left] {$\not{C}$}; + \path[draw, decorate, decoration=brace] (kv-4-1.south west) -- (kv-3-1.north west) node [midway, left] {$C$}; + \path[draw, decorate, decoration=brace] (kv-2-4.north east) -- (kv-3-4.south east) node [midway, right] {$D$}; + \path[draw, decorate, decoration=brace] (kv-1-4.north east) -- (kv-1-4.south east) node [midway, right] {$\not{D}$}; + \path[draw, decorate, decoration=brace] (kv-4-4.north east) -- (kv-4-4.south east) node [midway, right] {$\not{D}$}; + \path[draw, decorate, decoration=brace] (kv-4-3.south east) -- (kv-4-2.south west) node [midway, below] {$B$}; + \path[draw, decorate, decoration=brace] (kv-4-1.south east) -- (kv-4-1.south west) node [midway, below] {$\not{B}$}; + \path[draw, decorate, decoration=brace] (kv-4-4.south east) -- (kv-4-4.south west) node [midway, below] {$\not{B}$}; + \end{tikzpicture} + \end{center} + \end{minipage} + \hfill + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead, text = red] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & $0$ & $1$ & $X$ & \\ + \node[kvbinhead] {01}; & & & & \\ + \node[kvbinhead, text = darkgreen] {11}; & & $\textcolor{blue}{1}$ & & \\ + \node[kvbinhead] {10}; & & & & \\ + }; + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {AB}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {CD}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + \end{tikzpicture} + \end{center} + \end{minipage} +\end{center} + +\begin{center} + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center, + text = gray + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & & & & \\ + \node[kvbinhead] {01}; & & & & \\ + \node[kvbinhead] {11}; & & & & \\ + \node[kvbinhead] {10}; & & & & \\ + }; + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {AB}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {CD}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + + \begin{pgfonlayer}{tl3} + \draw[thick, blue, ->, > = stealth] (kv-2-2.center) -- (kv-3-2.center) to[out = 180, in = 180] (kv-5-2.center) -- (kv-4-2.center); + \draw[thick, blue, ->, > = stealth] (kv-2-3.center) -- (kv-3-3.center) to[out = 180, in = 180] (kv-5-3.center) -- (kv-4-3.center); + \draw[thick, blue, ->, > = stealth] (kv-2-5.center) -- (kv-3-5.center) to[out = 180, in = 180] (kv-5-5.center) -- (kv-4-5.center); + \draw[thick, blue, ->, > = stealth] (kv-2-4.center) -- (kv-3-4.center) to[out = 180, in = 180] (kv-5-4.center) -- (kv-4-4.center); + \end{pgfonlayer} + \begin{pgfonlayer}{tl2} + \draw[thick, darkgreen, ->, > = stealth] (kv-4-2.center) -- (kv-2-3.center); + \draw[thick, red, ->, > = stealth] (kv-4-3.center) -- (kv-2-5.center); + \draw[thick, darkgreen, ->, > = stealth] (kv-4-5.center) -- (kv-2-4.center); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} + \end{minipage} + \hfill + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center, + text = gray + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & & & & \\ + \node[kvbinhead] {01}; & & & & \\ + \node[kvbinhead] {11}; & & & & \\ + \node[kvbinhead] {10}; & & & & \\ + }; + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {CD}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {AB}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + + \begin{pgfonlayer}{tl3} + \draw[thick, blue, ->, > = stealth] (kv-2-2.center) -- (kv-2-3.center) to[out = 90, in = 90] (kv-2-5.center) -- (kv-2-4.center); + \draw[thick, blue, ->, > = stealth] (kv-3-2.center) -- (kv-3-3.center) to[out = 90, in = 90] (kv-3-5.center) -- (kv-3-4.center); + \draw[thick, blue, ->, > = stealth] (kv-5-2.center) -- (kv-5-3.center) to[out = 90, in = 90] (kv-5-5.center) -- (kv-5-4.center); + \draw[thick, blue, ->, > = stealth] (kv-4-2.center) -- (kv-4-3.center) to[out = 90, in = 90] (kv-4-5.center) -- (kv-4-4.center); + \end{pgfonlayer} + \begin{pgfonlayer}{tl2} + \draw[thick, darkgreen, ->, > = stealth] (kv-2-4.center) -- (kv-3-2.center); + \draw[thick, red, ->, > = stealth] (kv-3-4.center) -- (kv-5-2.center); + \draw[thick, darkgreen, ->, > = stealth] (kv-5-4.center) -- (kv-4-2.center); + \end{pgfonlayer} + \end{tikzpicture} + \end{center} + \end{minipage} +\end{center} + +Bei 5 Schaltvariablen wird die 5te Schaltvariable mit einer ABCD Matrix für $E$ und einer für $\not{E}$ dargestellt (2D!). + + +\subsection{Päckchen} + +Orthogonal benachbarte Minterme (bzw. Maxterme) werden zu einem ''Päckchen'' zusammengefasst. + +\begin{minipage}{0.48\linewidth} + \hfill \includegraphics[width = 0.4\textwidth]{images/Packchen.jpg} +\end{minipage} +\begin{minipage}{0.48\linewidth} + \begin{center} + Implizite Anwendung des Nachbarschaftsgesetz: \medskip + + $\Rightarrow (\not{A} \land \not{B}) \lor (A \land \not{B}) = \not{B}$ + \end{center} +\end{minipage} \medskip + +Es gelten folgende Regeln für Päckchen: + +\begin{itemize} + \item Päckchen sind rechteckig (Ausnahme: über Ecken) + \item \emph{Umfassen möglichst grosse Zweierpotenz}. + \item Dürfen über Ecken und Grenzen hinausgehen und sich überlappen. +\end{itemize} +\vfill + + +\subsection{Don't Care Zustände} + +Redundante oder unmögliche Kombinationen der Eingangsvariablen werden mit einem 'X' im KVD markiert. Man \emph{darf} solche Zustände benutzen um Päckchen zu bilden! + + +\subsection{Hazard} + +Hazards sind kurzzeitige, unerwünschte Änderung der Signalwerte, die durch Zeitverzögerung der Gatter entstehen. + +\subsubsection{Strukturhazard (Statischer Hazard)} + +Statische Hazards sind im KVD Stellen, an denen sich Päckchen orthogonal berühren, aber nicht überlappen. + +\begin{center} + \begin{minipage}{0.48\linewidth} + \begin{center} + \begin{tikzpicture} + \begin{pgfonlayer}{bg} + \def\ppad{1.5pt} + \matrix (kv) [ + matrix of nodes, + nodes in empty cells, + column sep=-\pgflinewidth, row sep=-\pgflinewidth, + nodes = { + rectangle, + draw = black, + text width = 5mm, + text height = 5mm, + align = center + } + ]{ + \node[kvbinhead] {}; & \node[kvbinhead] {00}; & \node[kvbinhead] {01}; & \node[kvbinhead] {11}; & \node[kvbinhead] {10}; \\ + \node[kvbinhead] {00}; & 0 & 1 & 1 & 0 \\ + \node[kvbinhead] {01}; & 0 & 1 & 1 & 0 \\ + \node[kvbinhead] {11}; & 1 & 1 & 0 & 0 \\ + \node[kvbinhead] {10}; & 1 & 1 & 0 & 0 \\ + }; + + + \node[] at ($(kv-1-1.north east) + (-6mm, +2mm)$) {AB}; + \node[] at ($(kv-1-1.south west) + (-1mm, +5mm)$) {CD}; + \draw[] ($(kv-1-1.north west) + (-3mm, 3mm)$) -- ($(kv-1-1.south east) + (-3.5mm, 3.5mm)$); + \end{pgfonlayer} + + \draw[blue] ($(kv-2-3.north west) + (1.5pt, -1.5pt)$) rectangle ($(kv-3-4.south east) + (-1.5pt, 1.5pt)$); + \draw[blue] ($(kv-4-2.north west) + (1.5pt, -1.5pt)$) rectangle ($(kv-5-3.south east) + (-1.5pt, 1.5pt)$); + \draw[darkgreen, thick, dashed] ($(kv-2-3.north west) + (1.5pt, -1.5pt)$) rectangle ($(kv-5-3.south east) + (-1.5pt, 1.5pt)$); + + \begin{pgfonlayer}{tl3} + \path[draw, <->, > = stealth, red] ($(kv-3-3.center) + (2pt, 0)$) to[in=45, out = -45] ($(kv-4-3.center) + (2pt, 0)$); + \path[draw, <-, > = stealth, red] ($(kv-5-3.center) + (2pt, 0)$) to[in=90, out = -45] ($(kv-5-3.center) + (5pt, -3mm)$); + \path[draw, <-, > = stealth, red] ($(kv-2-3.center) + (2pt, 0)$) to[in=270, out = 45] ($(kv-2-3.center) + (5pt, 3mm)$); + \end{pgfonlayer} + \begin{pgfonlayer}{tl2} + \node[text = red, fill = white, fill opacity = 0.8, text opacity = 1, rounded corners = 4pt] at ($(kv-3-3.south east) + (4mm, 0mm)$) {\small Hazard}; + \node[text = red, fill = white, fill opacity = 0.8, text opacity = 1, rounded corners = 4pt] at ($(kv-5-3.south east) + (4mm, 0mm)$) {\small Hazard}; + \end{pgfonlayer} + \end{tikzpicture} + \end{center} + \end{minipage} + \hfill + \begin{minipage}{0.5\linewidth} + \emph{Lösung:} Berührendes \textcolor{blue}{Päck-} + + \textcolor{blue}{chen} mit zusätzlichem \textcolor{darkgreen}{Päckchen} (möglichst gross) verbinden. + \end{minipage} +\end{center} + +\subsubsection{Funktionale Hazards} + +Wenn sich Päckchen diagonal berühren, dann wechseln mindestens zwei Variablen. Dies ist ein funktionaler Hazard und kann \emph{nicht} leicht behoben werden. \ No newline at end of file diff --git a/zusammenfassung/digitaltechnik/sections/latches_flipflops.tex b/zusammenfassung/digitaltechnik/sections/latches_flipflops.tex new file mode 100644 index 0000000..beca333 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/latches_flipflops.tex @@ -0,0 +1,337 @@ +\section{Latches und Flipflops} + +Sequentielle Schaltungen welche Rückkopplungen enthalten. + + +\subsection{Zustandgesteurte Latches} + +\emph{Zustandsgesteuert}: Das Verhalten eines Latches hängt nicht nur von den aktuellen Eingangsvariablen ab, sondern auch von den intern gespeicherten Zuständen. + + +\subsubsection{SR-Latch} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/SR-Latch.jpg} \medskip + + \eqbox{$Q_{n+1} = S \lor (Q_n \land \not{R}) \quad \text{Bedingung: } R \land S = 0$} +\end{center} + +\begin{tabular}{c c|c l} + S & R & $Q_{n + 1}$ & \\ \cline{1-3} + $0$ & $0$ & $Q_n$ & speichern \\ + $0$ & $1$ & $0$ & reset \\ + $1$ & $0$ & $1$ & set \\ + $1$ & $1$ & - & unzulässig \\ +\end{tabular} + + +\subsubsection{$\overline{SR}$-Latch} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/SR-Latch_AND.jpg} +\end{center} + +\begin{tabular}{c c|c l} + $\not{S}$ & $\not{R}$ & $Q_{n + 1}$ & \\ \cline{1-3} + $0$ & $0$ & - & unzulässig \\ + $0$ & $1$ & $1$ & set \\ + $1$ & $0$ & $0$ & reset \\ + $1$ & $1$ & $Q_n$ & speichern \\ +\end{tabular} + + +\subsection{Taktzustandgesteuerte Latches} + +\emph{Taktzustandgesteuert}: Änderungen am Eingang werden nur wahrgenommen, wenn das Taktsignal $T = 1$ ist. \medskip + +Taktzustandgesteuerte Latches sind gegenüber Störimpulsen empfindlich, da bei $T=1$ jede Änderung am Eingang übernommen wird. + + +\subsubsection{SRT-Latch} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/SRT-Latch.jpg} +\end{center} + +\begin{tabular}{c c c c c l} + T & & $\text{S}_{\text{int}}$ & $\text{R}_{\text{int}}$ & & \\ \cline{1-4} + $0$ & $\rightarrow$ & $0$ & $0$ & $\rightarrow$ & Datenspeicherung \\ + $1$ & $\rightarrow$ & S & R & $\rightarrow$ & Normales SR-Latch \\ +\end{tabular} + +\vfill +\subsubsection{D-Latch} + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/D-Latch.jpg} \medskip + + \eqbox{$Q_{n+1} = (Q_n \land \not{T}) \lor (D \land T)$} +\end{center} + +\begin{tabular}{c c c l} + T & $Q_{n + 1}$ & & \\ \cline{1-2} + $0$ & $Q_n$ & $\rightarrow$ & alter Ausgang gespeichert \\ + $1$ & D & $\rightarrow$ & Input übernommen \\ +\end{tabular} + +\subsubsection{JK-Latch} + +\begin{center} + \begin{circuitikz}[european] + \ctikzset{tripoles/european not symbol=circle} + \tikzset{flipflop srlatch/.style={flipflop, + flipflop def={t1=S, t3=R, t6=Q, t4=$\bar{\text{Q}}$, n4=1} + }} + \path[draw] node[flipflop srlatch](srlatch){} + (srlatch.pin 1)--++(left:5mm) + node[nand port, number inputs=3](nand1){} + (nand1.out) + (srlatch.pin 3)--++(left:5mm) + node[nand port, number inputs=3](nand2){} + (nand2.out); + \path[draw] (nand1.in 2) --++(left:10mm) node[point, label=180:J] {}; + \path[draw] (nand2.in 2) --++(left:10mm) node[point, label=180:K] {}; + \path[draw] (nand2.in 1) --++(up:6mm)--++(left:10mm) node[point, label=180:T] {}; + \path[draw] (nand1.in 3) --++(down:6mm); + \path[draw] (srlatch.pin 4) --++(right:10mm); + \path[draw] (srlatch.pin 4) --++(right:6mm) --++(up:23mm) --++(left:53mm) --++(down:3.2mm) --++(right:2mm); + \path[draw] (srlatch.pin 6) --++(right:10mm); + \path[draw] (srlatch.pin 6) --++(right:3mm) --++(down:23mm) --++(left:50mm) --++(up:3.2mm) --++(right:2mm); + \end{circuitikz} +\end{center} + +\begin{center} + \begin{circuitikz}[european] + \ctikzset{tripoles/european not symbol=circle} + \tikzset{flipflop jklatch/.style={flipflop, + flipflop def={t1=J, t2=T, t3=K, t6=Q, t4=$\bar{\text{Q}}$, n4=1} + }} + \path[draw] node[flipflop jklatch](jklatch){}; + \end{circuitikz} +\end{center} + +\begin{tabular}{c c c|c l} + T & J & K & $Q_{n + 1}$ & \\ \cline{1-4} + $0$ & $X$ & $X$ & $Q_n$ & speichern \\ + $1$ & $0$ & $0$ & $Q_n$ & speichern \\ + $1$ & $0$ & $1$ & $0$ & reset \\ + $1$ & $1$ & $0$ & $1$ & set \\ + $1$ & $1$ & $1$ & $\bar{Q_n}$ & kippen \\ +\end{tabular} + +\subsection{Flipflops} + +\emph{Taktflankensteuerung}: Serieschaltung von zwei mit gegenphasigem Takt gesteuerten Latches(Master-Slave Aufbau). + +\begin{center} + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[thick] (-0.75, 0) -- (0,0) + (0,0.4) -- (0,-0.4) + (0,0.25) -- (0.5,0) -- (0,-0.25); + \node[] at (1, 0) {CLK}; + \end{tikzpicture} + \end{center} + Input beim Übergang von $0 \to 1$ von CLK wirksam. + \begin{center} + \begin{tikzpicture} + \draw (0,0) -- (0.5, 0) -- (0.5, 0.3) -- (1, 0.3) -- (1, 0) -- (1.5, 0) -- (1.5, 0.3) -- (2, 0.3) -- (2, 0) -- (2.5, 0); + \draw[very thick, red] (0.5, 0) -- (0.5, 0.3) (1.5, 0) -- (1.5, 0.3); + \node[text width = 30mm] at (1.25, -0.5) {\small Positive/steigende Taktflanke}; + \end{tikzpicture} + \end{center} + \end{minipage} + \hfill + \begin{minipage}{0.45\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[thick] (-0.75, 0) -- (0,0) + (0,0.4) -- (0,-0.4) + (0,0.25) -- (0.5,0) -- (0,-0.25); + \draw[thick, fill = white] (-0.1,0) circle [radius = 0.75mm]; + \node[] at (1, 0) {CLK}; + \end{tikzpicture} + \end{center} + Input beim Übergang von $1 \to 0$ von CLK wirksam. + \begin{center} + \begin{tikzpicture} + \draw (0,0) -- (0.5, 0) -- (0.5, 0.3) -- (1, 0.3) -- (1, 0) -- (1.5, 0) -- (1.5, 0.3) -- (2, 0.3) -- (2, 0) -- (2.5, 0); + \draw[very thick, blue] (1, 0.3) -- (1, 0) (2, 0.3) -- (2, 0); + \node[text width = 30mm] at (1.25, -0.5) {\small Negative/fallende Taktflanke}; + \end{tikzpicture} + \end{center} + \end{minipage} +\end{center} + +Vorteil: Sehr robust gegenüber Störimpulsen + +\subsubsection{D-Flipflop} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/D-FF.jpg} \medskip + + \eqbox{$Q_{n+1} = D_n \quad \text{Wenn CLK }0 \to 1$} +\end{center} + + +\subsubsection{SR-Flipflop} + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/SR-FF.jpg} \medskip + + \eqbox{$Q_{n+1} = [S \lor (Q_1 \land \not{R})]_n \quad \text{Reqs: } R \land S = 0, \text{CLK }0 \to 1$} +\end{center} + + +\subsubsection{JK-Flipflop} + +Beim JK-Flipflop gibt es keinen unzulässigen Zustand mehr, dieser wurde durch eine Toggle Funktionalität ersetzt. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/JK-FF.jpg} \medskip + + \eqbox{$Q_{n + 1} = \left(J \land \overline{Q_n}\right) \lor \left(\overline{K} \land Q_n\right) \quad \text{Wenn CLK }0 \to 1$} +\end{center} + +\begin{tabular}{c c | c c l} + J & K & $Q_{n + 1}$ & $\not{Q}_{n + 1}$ & \\ + \cline{1-4} + $0$ & $0$ & $Q_{n}$ & $\not{Q}_{n}$ & speichern \\ + $0$ & $1$ & $0$ & $1$ & reset \\ + $1$ & $0$ & $1$ & $0$ & set \\ + $1$ & $1$ & $\not{Q}_n$ & $Q_n$ & toggle \\ +\end{tabular} \medskip + +Es gibt natürlich auch (takt)zustandsgesteuerte JK-Latches! + +\vfill + +\subsubsection{Toggle-Flipflop} + +Schaltung welche bei jeder aktiven Taktflanke kippt. + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/T-FF.jpg} + + \eqbox{$Q_{n + 1} = \overline{Q_n} \quad \text{Wenn CLK } 0 \to 1$} +\end{center} + +Folgende Schaltung kippt nur bei Taktflanken wenn T=1: + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/T-FF2.jpg} +\end{center} + +\subsection{Asynchroner Set/Reset Input} + +Können gespeicherte Zustände asynchron zu CLK überschreiben, d.h. jederzeit auch ohne ein Taktflankensignal. + +\begin{center} + \includegraphics[width = 0.075\textwidth]{images/Asynchron.JPG} +\end{center} + +\pagebreak +\subsection{D-Flipflop in CMOS-Technik} + +In den meisten Anwendungen werden D-Flipflops verwendet, da sie mit CMOS Technik effizient realisierbar sind. + +\subsubsection{Transmission Gates (TG)} + +TGs bestehen aus 2 Transistoren, einem NMOS und einem PMOS. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Transmissiongates.jpg} +\end{center} + +Pro D-Latch sind 2 TG und 2 Inverter notwendig. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/d_ff_cmos.jpeg} +\end{center} + +Insgesamt sind also 8NMOS und 8PMOS notwendig. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/d_ff_cmos_2.jpeg} +\end{center} +\vfill + + +\subsubsection{D-Flipflop $\Rightarrow$ JK-Flipflop} + +Ein JK-FF kann \emph{nur} mit einem D-FF realisiert werden, wenn: + +\begin{center} + \eqbox{$D_n = (J_n \land \not{Q}_n) \lor (\not{K}_n \land Q_n)$} +\end{center} + + +\subsection{Verzögerungszeiten} + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Dynamik-FF.jpg} +\end{center} + +\begin{flushleft} + \begin{tabular}{r p{50mm}} + Setup-Zeit ($t_s$) & Solange muss Signal an FF \underline{vor} aktiver Taktflanke stabil anliegen. \\ + Hold-Zeit ($t_h$) & Solange muss Signal an FF \underline{nach} aktiver Taktflanke stabil anliegen. \\ + Verzögerungszeit ($t_{pd}$) & Durchlaufzeit + \end{tabular} +\end{flushleft} + +Bei Verletzung der Zusatzbedingungen $t_s, t_h$ kann der Zustand des FF unbestimmt oder metastabil werden. + + +\subsubsection{Maximale Taktfrequenz bei seriellen FFs} + +In einem Schaltnetz mit mindestens zwei Flipflops in Serie (kann auch der gleiche FF sein), ist die maximale Taktfrequenz des Clocks begrenzt. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/FF_Delay.jpg} \medskip + + \eqbox{$T_{\text{min}} \geq t_{pd,ff1} + t_{pd,ks} + t_{s,ff2} \qquad f_{max} = \frac{1}{T_{min}}$} + + wobei $t_{pd,ks}$ Verzögerungszeit kombinatorische Schaltung +\end{center} + +Bei komplizierten kombinatorischen Schaltungen begrenzt der Pfad mit der \emph{längsten} Zeitverzögerung die Taktfrequenz. +\vfill + + +\subsection{Master-Slave Flipflops (Zwischenspeicher FFs)} + +\begin{center} + \begin{tabular}{r p{235pt}} + - & Übernehmen Eingangssignal mit der steigenden (bzw. fallenden) Taktflanke. \\ + - & Geben das Ausgangssignal mit der nächsten fallenden (bzw. steigenden) Taktflanke aus. \\ + \end{tabular} +\end{center} + +JK-Master-Slave FF (mit SR-,D-,T-FF auch möglich): + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/JK-MS-FF.JPG} + + \includegraphics[width = 0.3\textwidth]{images/JK-MS-FF2.JPG} +\end{center} + + +\subsection{Frequenzteiler und Zähler} + +T-FF können gut verwendet werden, um die Periode T eines periodischen Signals zu verlängern. Mit $n$-T-FFs kann die Frequenz durch den Faktor $2^n$ geteilt werden. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Frequenzteiler.JPG} +\end{center} + +Eine andere wichtige Anwendung von FFs ist als digitale Dualzähler. Dafür müssen T-FFs mit Rückflankensteuerung ($1 \to 0$) verwendet werden. Mit $n$-T-FFs kann man von 0 bis $2^{n-1}$ zählen. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Zahler.JPG} +\end{center} +\vfill + +\pagebreak + diff --git a/zusammenfassung/digitaltechnik/sections/misc.tex b/zusammenfassung/digitaltechnik/sections/misc.tex new file mode 100644 index 0000000..580dcdb --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/misc.tex @@ -0,0 +1,11 @@ +\vfill\null +\columnbreak + +%\section{Diverses} + +\section{Gate Varianten} + +\begin{center} + \rotatebox{90}{\includegraphics[width = 0.65\textwidth]{images/Logic-gate-index.png}} +\end{center} + diff --git a/zusammenfassung/digitaltechnik/sections/speicher.tex b/zusammenfassung/digitaltechnik/sections/speicher.tex new file mode 100644 index 0000000..5e71ebc --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/speicher.tex @@ -0,0 +1,126 @@ +\section{Speicher} + + +\subsection{Schieberegister} + +Ein Schieberegister ist ein Seriell-Parallel Wandler. Es ist eine Kette von D-FFs, durch welche die Seriellen Daten getaktet 'durchgeschoben' werden. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/Schieberegister.jpg} +\end{center} + +Schieberegister finden unter anderem Verwendung als Speicherregister um Daten zwischen 2 'Orten' zu speichern. + + +\subsection{Halbleiterspeicher} + +Bei Halbleiterspeicher muss unterschieden werden zwischen: + +\begin{tabular}{r p{170pt}} + ROM: & 'Read only memory' (Nur lesen) \\ + RAM: & 'Random access memory' (Schreiben \& Lesen, Wahlfreier Zugang) \\ + Flüchtig: & Abhängig von Versorgungsspannung \\ + Nicht flüchtig: & Unabhängig von Versorgungsspannung \\ +\end{tabular} \medskip + +Halbleiterspeicher werden oft als Zellenarray oder Matrizen organisiert, die von einem Zeilen-Decoder und einem Spalten-Decoder angesteuert werden. + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/SpeicherOrg.jpg} +\end{center} + +Dank den Decoder ist jede Speicherzelle einzeln ansprechbar mit nur linearem Wachstum in Adressleitungen bei höherer Zeilen-/Spaltenanzahl. Das Konstruktionsprinzip ist ein 'AND-Baum', wo jedes AND ein Minterm bildet. So besitzt jede Speicherzelle eine \emph{eindeutige} Adresse. +\vfill + + +\subsection{SRAM (Statische-RAM)} + +SRAMs sind flüchtige Speicher, welche überall Anwendung finden, wo schneller Datenzugriff notwendig ist. Eine SRAM Zelle besteht aus 6 Transistoren. Mit der Adressleitung selektiert man die Speicherzelle. + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/SRAM.jpg} +\end{center} + +An $V_L$ liegt der gespeicherte Wert der Speicherzelle und an $V_R$ der invertierte Wert. \medskip + +CMOS sind symmetrische Bauteile, Source und Drain hängen vom Elektronenstrom ab. Dementsprechend schaltet der CMOS, wenn vom Gate zu einem der beiden symmetrischen Anschlüssen ein Spannungsabfall vorhanden ist. + + +\subsubsection{Schreibevorgang} + +Die Bitleitung wird beim Schreibevorgang fest auf GND/VCC geschaltet. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/SRAM_Write.jpg} +\end{center} +\vfill + + +\subsubsection{Lesevorgang} + +Die Bitleitung ist \emph{nicht} fest auf GND/VCC geschaltet und verhält sich wie ein Kondensator. Sie übernimmt deswegen den Wert der Speicherzelle. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/SRAM_Read.jpg} +\end{center} + +Links wird eine 1 ausgelesen und Rechts eine 0. + + +\subsection{DRAM (Dynamische-RAM)} + +DRAMs sind flüchtige Speicher, welche periodisch (20ms) wieder aufgefrischt werden müssen aufgrund des Leckstroms vom Kondensator. DRAMs besitzen eine höhere Dichte als SRAMs aber langsämere Zugriffszeiten. + +\begin{center} + \includegraphics[width = 0.15\textwidth]{images/DRAM.jpg} +\end{center} + +Der Kondensator agiert als Speicher. Beim Schreibevorgang wird die Bitleitung fest auf GND/VCC geschaltet und der Kondensator aufgeladen (1) bzw. entladen (0). \medskip + +Beim Lesevorgang ist die Bitleitung \emph{nicht} fest auf GND/VCC geschaltet und verhält sich wie ein Kondensator. Sie übernimmt deswegen den in der Zelle gespeicherten Wert. + + +\subsection{MROMs (Maskable ROM)} + +ROMs sind nichtflüchtige Speicher. Sie werden überall dort benötigt, wo ein System unveränderliche Daten-/Programmstrukturen benötigt. \medskip + +MROM Speicherzellen werden durch spezielle 'Masken' bei der Herstellung als 0 oder 1 programmiert (Ideal für Massenproduktion). + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/MROM.jpg} +\end{center} + + +\subsection{PROMs (Programmable ROM)} + +PROMs sind für kleinere Produktionsvolumen geeignet, weil sie durch den Benutzer einmalig programmiert werden können. + +\begin{center} + \includegraphics[width = 0.15\textwidth]{images/PROM.jpg} +\end{center} + +Die Programmierung erfolgt durch die Schmelzsicherung. Für eine '0' muss man nichts machen und für eine '1' wird die Schmelzsicherung durch eine hohe Spannung durchgebrannt. + + +\subsection{EPROM (Erasable PROM)} + +EPROMs lassen sich durch UV-Bestrahlung optisch löschen und werden durch Anlegen einer hohen Spannung programmiert. Sie sind also mehrfach beschreibbar ($\approx$1'000-Mal). + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/EPROM.jpg} +\end{center} + +Floating-Gate Transistoren können negative Ladungen speichern, was die Kennline der für das Schalten notwendigen Gate-Source Spannung nach rechts verschiebt, als Folge reicht die $V_{DD} \approx 0.8V$ Spannung nicht mehr aus für das Schalten des Transistors. + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/EPROM_2.jpg} +\end{center} + +Nach etwa 20 minütiger Bestrahlung durch UV-Licht sind die negativen Ladungen wieder aus dem Floating-Gate entfernt und der Transistor zeigt erneut normales Verhalten. + + +\subsection{EEPROM (Electrically erasable PROM)} + +Mit einer dünneren Oxidschicht beim Floating-Gate Transistor können die Gate Ladungen auch elektrisch gelöscht werden. Solche EEPROMs können ohne Probleme $10^6$-Mal neubeschrieben werden. \medskip + +EEPROMs sind, unter anderem, die Grundbausteine von den heutzutage weit verbreiteten FLASH Speicher. diff --git a/zusammenfassung/digitaltechnik/sections/zahlensysteme.tex b/zusammenfassung/digitaltechnik/sections/zahlensysteme.tex new file mode 100644 index 0000000..bf8afa7 --- /dev/null +++ b/zusammenfassung/digitaltechnik/sections/zahlensysteme.tex @@ -0,0 +1,128 @@ +\section{Zahlensysteme} + +\subsection{Polyadische Systeme} + +Die Position einer Ziffer innerhalb einer Zahl gibt den Wert an, mit der die Basis des Zahlensystems an dieser Stelle potenziert wird. Umwandlung einer Zahl $D_{(R)}$ ins Dezimalsystem: + +\begin{center} + \begin{minipage}{0.55\linewidth} + \begin{center} + \begin{tabular}{c l} + $R$ & Basis/Radix von $D_{R}$ \\ + $b_i$ & Koeffizienten (''Ziffern'') \\ + \end{tabular} + \end{center} + \end{minipage} + \begin{minipage}{0.40\linewidth} + \begin{center} + \eqbox{$D_{(10)} = \sum\limits_{i = -\infty}^{\infty} b_i \cdot R^i$} + \end{center} + \end{minipage} +\end{center} +Darstellung von $D_{(R)}$ in Basis $R$: $\dots b_2 b_1 b_0 . b_{-1} b_{-2} \dots _{R}$ +\begin{flushleft} + \begin{tabular}{l c l} + Dezimal & $10$ & $b_i \in \{0, 1, \dots, 9\}$ \\ + Dual/Binär & $2$ & $b_i \in \{0, 1\}$ \\ + Oktal & $8$ & $b_i \in \{0, 1, \dots, 7\}$ \\ + Hexa (0x) & $16$ & $b_i \in \{0, 1, \dots, 9, A, B, C, D, E, F\}$ \\ + \end{tabular} +\end{flushleft} +\vfill + + +\subsection{Umwandlung Dezimal $\to$ Zahlensystem R} + +Für den Teil \emph{vor} dem Komma: + +Wiederholte ganzzahlige Division durch Basis $R$ mit Rest $r$, der jeweilige Rest $r$ entspricht der Ziffer im neuen +Zahlensystem, beginnend mit dem \textbf{L}east \textbf{S}ignificant \textbf{D}igit (LSD). + +\begin{center} + \eqbox{$\dfrac{D_{(10)}}{R} = Q_0 + r_0$ und dann Rekursiv: $\dfrac{Q_i}{R} = Q_{i+1} + r_{i+1}$} +\end{center} + +Für den \emph{'Fraktionalteil'} $a_o$ hinter dem Komma: + +Wiederholte Multiplikation von $a_0$ mit der Zahlenbasis $R$ und ausführen der Abrundungsfunktion floor(x) zur Berechnung von $K$, welches der Ziffer im neuen Zahlensystem entspricht, beginnend mit dem \textbf{M}ost \textbf{S}ignificant \textbf{D}igit (MSD). + +\begin{center} + \eqbox{$a_i \cdot R = P_i \quad \to \text{floor}(P_i) = K_{i - 1}$, $a_{i-1} = P_i - K_{i - 1}$} +\end{center} + +Ein Zahlenbeispiel: + +\begin{center} + \includegraphics[width = 0.32\textwidth]{images/UmwandlungZahlensys.jpg} +\end{center} + + +\subsubsection{Binär zu Dezimal} + +\begin{center} + \begin{tabular}{c|c|c|c|c|c|c|c} + $2^7$ & $2^6$ & $2^5$ & $2^4$ & $2^3$ & $2^2$ & $2^1$ & $2^0$ \\ + $128$ & $64$ & $32$ & $16$ & $8$ & $4$ & $2$ & $1$ \\ + \end{tabular} \medskip + + \begin{tabular}{c|c|c|c} + $2^{-1}$ & $2^{-2}$ & $2^{-3}$ & $2^{-4}$ \\ + $0.5$ & $0.25$ & $0.125$ & $0.0625$ \\ + \end{tabular} +\end{center} + + +\subsection{Zweierkomplement} + +Für negative Dualzahlen wird das Zweierkomplement verwendet. Das MSB (sign Bit) hat eine \emph{negative} Wertigkeit. \medskip + +Bildung Zweierkomplement für eine negative Zahl: + +\begin{tabular}{r l} + i) & Absolutbetrag der Zahl in Dualzahl umwandeln \\ + ii) & Dualzahl bitweise invertieren und beim LSB '+1' \\ + iii) & Zuvorderst das sign Bit hinzufügen \\ +\end{tabular} +\vfill + + +\subsection{Darstellung Rationaler Zahlen als Dualzahl (Q-Format)} + +Vorzeichenbehaftete Zahlen mit rationalem Anteil werden im Q-Format dargestellt. Bei $m$ Vorkomma- und $n$ Nachkommabits sind insgesamt $k = 1 + m + n$ Binärstellen erforderlich. + +\begin{center} + \eqbox{$\displaystyle D_{(10)} = \underbrace{- b_m \cdot 2^m}_{\text{sign Bit}} + \sum_{i = 0}^{m - 1} b_i \cdot 2^i + \sum_{i = 1}^{n} b_i \cdot 2^{-i}$} +\end{center} + + +\subsection{Binäre Rechenoperationen} + + +\subsubsection{Addition} + +Schriftliche bitweise Addition mit Übertrag. + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/Addition.jpg} +\end{center} + + +\subsubsection{Subtraktion} + +Addition, aber in 2er-Komplement Darstellung. + +\begin{center} + \includegraphics[width = 0.2\textwidth]{images/Subtraktion.jpg} +\end{center} + +Minuend und Subtrahend müssen über gleiche Stellenanzahl verfügen, sonst kürzere Zahl linksbündig mit Vorzeichenbit erweitern. Überträge nach dem Vorzeichenbit ignorieren! + + +\subsubsection{Multiplikation} + +Multiplikation zweier vorzeichenloser Dualzahlen: + +\begin{center} + \includegraphics[width = 0.25\textwidth]{images/Multiplikation.jpg} +\end{center} +\vfill diff --git a/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.pdf b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.pdf new file mode 100644 index 0000000..15633c5 Binary files /dev/null and b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.pdf differ diff --git a/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.tex b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.tex new file mode 100644 index 0000000..66f456f --- /dev/null +++ b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.tex @@ -0,0 +1,174 @@ +\documentclass[a4paper, fontsize = 10pt, landscape]{scrartcl} +\usepackage[utf8]{inputenc} +\usepackage[ngerman]{babel} 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+\usepackage{circuitikz} +\ctikzset{tripoles/pmos style/emptycircle} +\usepackage{wrapfig} +%% Formal tables +\usepackage{booktabs} + +% pgfplot layers +\pgfplotsset{compat=1.7} % Version? für Plots +\pgfdeclarelayer{bg} +\pgfdeclarelayer{l1} +\pgfdeclarelayer{l2} +\pgfdeclarelayer{l3} +\pgfdeclarelayer{tl1} +\pgfdeclarelayer{tl2} +\pgfdeclarelayer{tl3} +\pgfsetlayers{bg,l3,l2,l1,main,tl1,tl2,tl3} + +% custom colors +\definecolor{darkgreen}{HTML}{008E00} +\definecolor{darkred}{HTML}{990000} +\definecolor{mred}{HTML}{e60000} +\definecolor{burntorange}{HTML}{ff9a00} +\definecolor{darkblue}{HTML}{000099} +\definecolor{lightblue}{HTML}{99d6ff} +\definecolor{lightgray}{HTML}{A9A9A9} +\definecolor{darkyellow}{HTML}{FED600} +\definecolor{limegreen}{HTML}{3ae32b} +\definecolor{pastelmoss}{HTML}{A0C1B8} +\definecolor{pastelaqua}{HTML}{709FB0} +\definecolor{pastelviolet}{HTML}{726a95} + +\definecolor{primaryheader}{HTML}{30475e} +\definecolor{secondaryheader}{HTML}{f1935c} 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port/height=0.3} + \ctikzset{tripoles/european not port/width=0.5} +}{ + \end{circuitikz} + \end{center} +} + +\newcommand{\uunits}[2]{$\left[#1\right] = \si{#2}$} +\newcommand{\floor}[1]{\text{floor}\left(#1\right)} + +\newcommand{\cemph}[2][red]{\textcolor{#1}{\sffamily\bfseries\boldmath #2}} +\newcommand{\cbmath}[2]{\textcolor{#1}{\bfseries\boldmath #2}} +\newcommand{\twocom}{$2^{\text{er}}$} +\newcommand{\nibp}[2]{#1\textcolor{blue}{#2}} + + + +\newcommand{\overrideplainstyle}{ + \fancypagestyle{plain}{ + \fancyhf{} + \rfoot{Seite \thepage \protect{ }von \pageref{LastPage}} + \renewcommand{\headrulewidth}{0pt} + \renewcommand{\footrulewidth}{0pt} + } +} + + +\setlength{\columnseprule}{1pt} +\def\columnseprulecolor{\color{black!20!white}} + +\newcommand{\raisedrulefill}[2][0ex]{\leaders\hbox{\rule[#1]{1pt}{#2}}\hfill} + +\titleformat{\section}{\sffamily\bfseries\boldmath\large}{}{-5pt}{\colorbox{primaryheader}{\parbox[c][3mm]{\linewidth + 4pt}{\textcolor{white}{#1}}}} +\titleformat{\subsection}{\sffamily\bfseries\boldmath}{}{-5pt}{\colorbox{secondaryheader}{\parbox[c][2mm]{\linewidth + 4pt}{#1}}} +\titleformat{\subsubsection}{\sffamily\bfseries\boldmath}{}{0pt}{\textcolor{thirdheader}{#1 \raisedrulefill[0.4ex]{0.5pt}}} +\titleformat{\paragraph}[runin]{\sffamily\bfseries\boldmath}{}{-5pt}{\uline{#1}} + +\parindent 0pt + +\definecolor{formulabox}{RGB}{182,225,189} +\renewcommand{\not}[1]{\overline{#1}} +% equation box +\newcommand{\eqbox}[1]{\fcolorbox{black}{white}{\hspace{0.5em}#1\hspace{0.5em}}} +\newcommand{\eqboxf}[1]{\fcolorbox{black}{formulabox}{\hspace{0.5em}#1\hspace{0.5em}}} + + +\begin{document} +\begin{multicols*}{3} + \begin{center} + \Large{Digitaltechnik} \\ + \tiny{von Jirayu Ruh, \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch}} + \end{center} + + \input{sections/gates.tex} + \input{sections/cmos.tex} + \input{sections/boolsche_algebra.tex} + \input{sections/karnaugh.tex} + \input{sections/zahlensysteme.tex} + \input{sections/codes.tex} + \input{sections/EinfacheHardwarekomp.tex} + \input{sections/latches_flipflops.tex} + \input{sections/automaten.tex} + \input{sections/speicher.tex} + \input{sections/misc.tex} +\end{multicols*} +\end{document} diff --git a/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.zip b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.zip new file mode 100644 index 0000000..53a0ce6 Binary files /dev/null and b/zusammenfassung/digitaltechnik/zusammenfassung_digitech_ruh_jirayu.zip differ diff --git a/zusammenfassung/technische_mechanik/Figures/2D_Moment.jpg b/zusammenfassung/technische_mechanik/Figures/2D_Moment.jpg new file mode 100644 index 0000000..4add582 Binary files /dev/null and b/zusammenfassung/technische_mechanik/Figures/2D_Moment.jpg differ diff --git a/zusammenfassung/technische_mechanik/Figures/Bindungskraefte.jpg b/zusammenfassung/technische_mechanik/Figures/Bindungskraefte.jpg new file mode 100644 index 0000000..5d19455 Binary files /dev/null and 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- \vect{v}(t)}{dt} = \vect{\ddot r}(t) = \begin{pmatrix}\ddot x(t) \\ \ddot y(t) \\ \ddot z(t) \end{pmatrix}$ +\end{center} + +In Zylinderkoordinaten ist die Beschleunigung: + +\begin{center} + \eqbox{$\vect{a} = \underbrace{(\ddot \rho - \rho \dot\varphi^2)}_{\text{Radiale Beschl.}} \vect{e}_\rho + \underbrace{(2\dot\rho \dot\varphi + \rho \ddot \varphi)}_{\text{Azimutale Beschl.}} \vect{e}_\varphi + \ddot z \vect{e}_z$} +\end{center} + + +\subsection{Beschleunigte Kreisbewegung} + +Für eine Kreisbewegung muss man $r$ konstant halten, d.h. eine nach innen gerichtete radiale Beschleunigung ist notwendig. + +\begin{minipage}{0.35\linewidth} + \begin{center} + \includegraphics[width=1\textwidth]{Figures/Kreisbewegung.jpg} + \end{center} +\end{minipage} +\begin{minipage}{0.60\linewidth} + \begin{center} + \eqbox{$\vect{v} = r \dot \varphi \vect{e}_\varphi \qquad \vect{a} = -\underbrace{r \dot\varphi^2}_{a_r} \vect{e}_r + r \ddot \varphi \vect{e}_\varphi$} \medskip + + \eqboxf{$a_r = r \dot \varphi^2 = \dfrac{v^2}{r}$} + \end{center} +\end{minipage} + + + + + + +\subsection{Impulssatz} + +Der Impuls eines Massenpunktes: \eqbox{$\vect{P}(t) = m(t) \cdot \vect{v}(t)$} \medskip + +\begin{center} + \eqboxf{$\dfrac{d\vect{P}}{dt} = \dot{\vect{P}} = \vect{R}$ \qquad Falls $m(t) = m \,\,\, \forall t$: $m \cdot \vect{a} = \vect{R}$} + + wobei $\vect{R}$ die Resultierende der äusseren Kräfte ist +\end{center} + +Um die Lösung der Bewegungsdifferentialgleichung eindeutig festzulegen, benötigt man noch die Anfangsbedingungen: + +\begin{center} + \eqbox{$\vect{x}(0) = \vect{x}_0$ \quad und \quad $\vect{\dot x}(0) = \vect{v_0}$} +\end{center} + +Für die Geschwindigkeit (bzw. Ortsvektor) integriert man die Beschleunigung (bzw. Geschwindigkeit) zwischen null und einem beliebigen Zeitpunkt $t$. + + +\subsection{Mathematisches Pendel} + +Annahmen: Undehnbares und massenloses Seil, ebene Bewegung ohne Reibung. + +\begin{center} + \begin{minipage}{0.35\linewidth} + \begin{center} + \includegraphics[width=0.5\textwidth]{Figures/MathPendel.jpg} + \end{center} + \end{minipage} + \begin{minipage}{0.60\linewidth} + Homogene lineare DGL 2ter Ordnung: + + \begin{center} + \eqbox{$\ddot \varphi + \underbrace{\dfrac{g}{l}}_{=\omega^2} \overbrace{\varphi}^{\approx \, \sin\varphi} = 0$} + + unter der Annahme $\varphi << 1: \varphi \approx \sin\varphi$ + \end{center} + \end{minipage} +\end{center} +\vfill \null +\columnbreak + + +\subsection{Federschwinger} + +Eine lineare Feder übt bei einer Verlängerung um $\delta l$ aus der ungespannten Lage eine Kraft von folgendem Betrag aus: + +\begin{center} + \begin{minipage}{0.4\linewidth} + \begin{center} + \eqbox{$|F| = c \,|\delta l|$} ($\delta l = x - l$) + + \begin{small} wobei $c$ die Federkonstante ist \end{small} + \end{center} + \end{minipage} + \begin{minipage}{0.58\linewidth} + \begin{tabular}{l} + $\delta l > 0$: Feder zieht am Körper \\ + $\delta l < 0$: Feder drückt auf den Körper \\ + \end{tabular} + \end{minipage} +\end{center} + +Annahme: Der Massenpunkt $m$ gleitet reibungsfrei. + +\begin{center} + \begin{minipage}{0.4\linewidth} + \begin{center} + \includegraphics[width=1\textwidth]{Figures/Federschwinger.jpg} + \end{center} + \end{minipage} + \begin{minipage}{0.58\linewidth} + Inhomogene lineare DGL 2ter Ordnung: + + \begin{center} + \eqbox{$\ddot x + \underbrace{\dfrac{c}{m}}_{\omega^2} x = \underbrace{\dfrac{c}{m}}_{\omega^2} l$} + \end{center} + \end{minipage} +\end{center} + + +\subsection{Lineare DGLs 2ter Ordnung} + +Bei harmonischen Schwingungen (Mathematische Pendel, Federschwinger, \dots) ist die homogene Lösung folgende Linearkombination: + +\begin{center} + \begin{minipage}{0.45\linewidth} + Homogene DGL 2ter Ordnung: + + \begin{center} + \eqbox{$\ddot x + a \dot x + b x(t) = 0$} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{center} + \eqboxf{$x_h(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)$} + \end{center} + \end{minipage} +\end{center} + +Bei inhomogenen DGL, d.h. es gibt eine Störfunktion $g(t)$, gilt: + +\begin{center} + \begin{minipage}{0.5\linewidth} + Inhomogene DGL 2ter Ordnung: + + \begin{center} + \eqbox{$\ddot x + a \dot x + b x(t) = g(t)$} + \end{center} + \end{minipage} + \begin{minipage}{0.45\linewidth} + \begin{center} + \eqbox{$x(t) = x_h(t) + x_p(t)$} + \end{center} + \end{minipage} +\end{center} + +Die partikuläre Lösung $x_p(t)$ der DGL ist abhängig von der Störfunktion $g(t)$. Hier sind es (nur?) Polynomfunktionen von Grad 0. Lösungansatz: + +\begin{center} + $x_p(t) = \begin{cases} + P_n(t) & \text{falls } b \neq 0 \\ + x \cdot P_n(t) & \text{falls } a \neq 0, b = 0 \\ + x^2 \cdot P_n(t) & \text{falls } a = b = 0 \\ + \end{cases}$ \quad [$P_n(t)$ gleicher Grad wie $g(t)$] +\end{center} + +\begin{tabular}{r l} + i) & $P_n, \dot P_n, \ddot P_n$ für $x, \dot x, \ddot x$ in die DGL einsetzen \\ + ii) & Koeffizientenvergleich mit der Störfunktion $g(t)$ \\ + iii) & $x_p(t) = P_n$ mit den herausgefunden Koeffizienten \\ +\end{tabular} \medskip + +Eine \textbf{gekoppelte DGL}, d.h. die Gleichung ist abhängig von zwei Funktionen, ist unmöglich zu lösen. In einem solchen Fall muss man sie zuerst entkoppeln, indem man eine neue Funktion einführt, welche eine linearkombination der ursprünglichen zwei Funktionen ist. + + +\subsection{Eigenschaften der Harmonischen Schwingungen} + +\begin{minipage}{0.6\linewidth} + \begin{tabular}{r l} \toprule + A,B & Amplitude \\ + $\omega$ & Kreisfrequenz \\ + T & Periode \\ \bottomrule + \end{tabular} wobei \eqbox{$T = \dfrac{2\pi}{\omega}$} +\end{minipage} +\begin{minipage}{0.35\linewidth} + \begin{center} + \includegraphics[width=1\textwidth]{Figures/Harmonisch.jpg} + \end{center} +\end{minipage} +\vfill \null +\columnbreak + +\subsection{Prinzip der virtuellen Leistungen} + +Trägheitskraft bei homogener Beschleunigung: \eqbox{$\vect{F}^T = m \vect{a}$} \medskip + +Die Gesamtleistung aller wirklichen Kräfte und aller (für die Beschleunigungen der wirklichen Bewegung berechneten) Trägheitskräfte verschwindet für jeden virtuellen Bewegungszustand. + +\begin{center} + \eqboxf{{$\tilde{P}^{i} + \tilde{P}^{a} + \underbrace{\tilde{P}^{T}}_{\vect{F}^T \vect{\tilde{v}}} = 0 \quad \forall \{\tilde{\vect{v}}\}$}} +\end{center} + +Das PdvL gilt auch für Mehrkörpersysteme! + + +\subsection{Kinematische Relationen} + +Gibt es mehr Koordinaten als Freiheitsgrade, so sind diese voneinander abhängig. Der Freiheitsgrad ist die Anzahl der Minimalkoordinaten, die nötig sind, um die Lage des Systems eindeutig zu beschreiben. \medskip + +Die Kinematischen Relationen erstellt man in solchen Fällen durch Anwendung der Gesetze der Kinematik. + + +\subsection{Massenmittelpunktsatz} + +Bei einem Körper/System gilt: + +\begin{center} + $m \cdot \ddot{\vect{r}}_C = \vect{R} \Rightarrow$ \eqboxf{$m \cdot \vect{a}_C = \vect{R}$} + + wobei $\vect{a}_C$ die Beschleunigung im Schwerpunkt des Körpers ist +\end{center} + + +\subsection{Drallsatz für inertialen Punkt $O$} + +Drall eines Körpers bezüglich einem Festpunkt $O$ (Punkt $O$ ist in Ruhe): + +\begin{center} + \begin{minipage}{0.5\linewidth} + \begin{center} + \includegraphics[width=0.75\textwidth]{Figures/Drall.jpg} + \end{center} + \end{minipage} + \begin{minipage}{0.45\linewidth} + \begin{center} + \eqbox{$\vect{L}_O = \displaystyle\iiint \vect{r}_{OP} \times \vect{v} \d m$} + \end{center} + \end{minipage} +\end{center} + +Drallsatz für einen Starrkörper: \quad \eqboxf{$\dot{\vect{L}}_O = \vect{M}_O$} + + +\subsection{Drallsatz für körperfesten Punkt $C$} + +Oft ist es praktisch den Drall bezüglich Schwerpunkt $C$ anzuwenden: + +\begin{center} + \begin{minipage}{0.5\linewidth} + \begin{center} + \includegraphics[width=0.75\textwidth]{Figures/Drall2.jpg} + \end{center} + \end{minipage} + \begin{minipage}{0.45\linewidth} + \begin{center} + \eqbox{$\vect{L}_C = \displaystyle\iiint \vect{r}' \times \vect{v}' \d m$} + \end{center} + \end{minipage} +\end{center} + +Drallsatz bezüglich dem Schwerpunkt $C$: \quad \eqboxf{$\dot{\vect{L}}_C = \vect{M}_C$} \medskip + +Transformationsformel des Dralls: + +\begin{center} + \eqbox{$\vect{L}_O = \vect{r}_{OC} \times \vect{P} + \vect{L}_C$} +\end{center} +\vfill\null +\columnbreak + + +\subsection{Physikalischer Pendel} + +Annahmen: Gelenk reibungsfrei, Starrer Stab und Masse homogen verteilt. + +\begin{center} + \begin{minipage}{0.35\linewidth} + \begin{center} + \includegraphics[width=\textwidth]{Figures/PhysPendel.jpg} + \end{center} + \end{minipage} + \begin{minipage}{0.60\linewidth} + Homogene lineare DGL 2ter Ordnung: + + \begin{center} + \eqbox{$\ddot \varphi + \underbrace{\dfrac{3g}{2l}}_{=\omega^2} \overbrace{\varphi}^{\approx \, \sin\varphi} = 0$} + + unter der Annahme $\varphi << 1: \varphi \approx \sin\varphi$ + \end{center} + \end{minipage} +\end{center} + + +\subsection{Ebene Kinetik} + +Das Massenträgheitsmoment gibt an, wie ''schwer'' es ist, den Körper um den Punkt zu drehen. + +\begin{center} + \begin{minipage}{0.45\linewidth} + $O$ ist ein Fixpunkt: + + \begin{center} + Trägheitsmoment: \eqbox{$I_O = \displaystyle \iint r^2 \d m$} \medskip + + \eqboxf{$\vect{L}_O = \omega \, I_O \, \vect{e}_z$} + + $\dot{\vect{L}}_O = I_O \dot \omega = I_O \ddot \varphi$ + \end{center} + \end{minipage} + \vline \qquad + \begin{minipage}{0.45\linewidth} + $C$ ist der Schwerpunkt: + + \begin{center} + Trägheitsmoment: \eqbox{$I_C = \displaystyle \iint r'^2 \d m$} \medskip + + \eqboxf{$\vect{L}_C = \omega \, I_C \, \vect{e}_z$} + + $\dot{\vect{L}}_C = I_C \dot \omega = I_C \ddot \varphi$ + \end{center} + \vfill + \end{minipage} +\end{center} + +\subsection{Satz von Steiner} + +\begin{center} + \eqboxf{$I_O = mr^2_{OC} + I_C$} +\end{center} + +\begin{center} + \begin{tabular}{r l l} \toprule + Massepunkt & $I_C = 0$ & $I_O = m L^2$ \\ + Stange & $I_C = \dfrac{1}{12} m L^2$ & $I_O = \dfrac{1}{3} m L^2$ \\ + Gleichmässige Scheibe & $I_C = m \dfrac{R^2}{2}$ & $I_O = \dfrac{3}{2} m R^2$ \\ + Kreisring & $I_C = m R^2$ & $I_O = 2 mR^2$ \\ \bottomrule + \end{tabular} \medskip + + \includegraphics[width=1\linewidth]{Figures/Steiner.jpg} +\end{center} +\vfill\null +\columnbreak \ No newline at end of file diff --git a/zusammenfassung/technische_mechanik/Sections/Kinematik.tex b/zusammenfassung/technische_mechanik/Sections/Kinematik.tex new file mode 100644 index 0000000..f0832ce --- /dev/null +++ b/zusammenfassung/technische_mechanik/Sections/Kinematik.tex @@ -0,0 +1,347 @@ + +\section{Basics} + + + +\subsection{Trigonometrie} + +\begin{minipage}{0.25\linewidth} + \includegraphics[width=1\textwidth]{Figures/Trigo_2.jpg} +\end{minipage} +\begin{minipage}{0.73\linewidth} + \eqbox{$\sin(\alpha) = \dfrac{G}{H}, \qquad \cos(\alpha) = \dfrac{A}{H}, \qquad \tan(\alpha) = \dfrac{G}{A}$} +\end{minipage} \medskip + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \begin{tabular}{r c c c c c} \toprule + deg/rad & 0$^\circ$/0 & 30$^\circ$/$\frac{\pi}{6}$ & $45^\circ$/$\frac{\pi}{4}$ & 60$^\circ$/$\frac{\pi}{3}$ & 90$^\circ$/$\frac{\pi}{2}$ \\ \midrule + sin & $0$ & $\frac{\sqrt{1}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{3}}{2}$ & $1$ \\ + cos & $1$ & $\frac{\sqrt{3}}{2}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{1}}{2}$ & $0$ \\ + tan & $0$ & $\frac{\sqrt{3}}{3}$ & 1 & $\sqrt{3}$ & - \\ \bottomrule + \end{tabular} +\end{center} + +Trigonometrische Identitäten: \eqbox{$\tan = \dfrac{\sin}{\cos}, \qquad \sin^2 + \cos^2 = 1$} \medskip + +Ausserdem gilt: $\sin(-x) = -\sin(x), \quad \cos(-x) = \cos(x)$ + + +\subsection{Nützliche Geometrien} + +Die normierten Einheitsvektoren für häufig vorkommende Winkel: + +\begin{center} + \includegraphics[width=0.32\textwidth]{Figures/Trigo_1.jpg} +\end{center} + +Ähnliche Dreiecke (Dreiecke mit gleichen Winkeln): \eqbox{$\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c}$} + +\begin{center} + \includegraphics[width=0.13\textwidth]{Figures/Similar.jpg} +\end{center} + + +\subsection{Vektorgeometrie} + +Man normiert einen Vektor, d.h. $|\vect{e}_v | = 1$, wie folgt: \eqbox{$\vect{e}_v = \dfrac{\vect{v}}{| \vect{v} |}$} \medskip + +Trick in 2D: Orthogonaler Vektor zu einem Vektor $\begin{pmatrix} a \\ b \end{pmatrix}$ ist $\begin{pmatrix} b \\ -a \end{pmatrix}$. \medskip + +Verbindungsvektor zwischen 2 Punkten: \eqbox{$\vect{r}_{OA} = \vect{r}_A - \vect{r}_O$} +\vfill + + + +\subsection{Analysis} + +Konvention: \, +\begin{tabular}{l l} + $y'(x) = y'$ & Ableitung nach $x$ \\ + $y'(t) = \dot y$ & Ableitung nach $t$ \\ +\end{tabular} \medskip + +Kettenregel: \eqbox{$[g(f(x))]' = g'(f(x)) \cdot f'(x)$} \medskip + +Ist $\dot{y}$ abhängig von $x(t)$: \eqbox{$\dot{y} = \dfrac{dy(t)}{dt} = \dfrac{dy(t)}{dx} \cdot \dfrac{dx}{dt} = \dfrac{dy(t)}{dx} \cdot \dot{x}$} + + +\section{Kinematik} + + +\subsection{Bahnkurve} + +Die Parametrisierung der Lage eines Punktes nach Zeit: + +\begin{center} + Formell: $\mathbb{R} \to \mathbb{R}^3 \quad t \mapsto \vect{r}(t)$ wobei $\vect{r}(t) = \begin{pmatrix}x(t) \\ y(t) \\ z(t) \end{pmatrix}$ +\end{center} + + +\subsection{Geschwindigkeit und Schnelligkeit (in kart. Koordinaten)} + +Die Geschwindigkeit beschreibt, wie sich der Ortsvektor ändert: + +\begin{center} + $\vect{v}(t) := \dfrac{d \vect{r}(t)}{d t} = \lim\limits_{dt \to 0} \dfrac{\vect{r}(t + dt) - \vect{r}(t)}{dt} = \vect{\dot r}(t) = \begin{pmatrix}\dot x(t) \\ \dot y(t) \\ \dot z(t) \end{pmatrix}$ +\end{center} + +Geometrisch ist es die Tangente zur Bahnkurve. Achtung, die Geschwindigkeit ist \emph{nicht} immer senkrecht zum Ortsvektor (Gegenbsp. Gerade). + +\begin{center} + Schnelligkeit: \eqbox{$v = |\vect{v}| = \sqrt{\dot x^2 + \dot y^2 + \dot z^2}$} +\end{center} + + +\subsection{Zylinderkoordinaten ($\rho, \varphi, z$)} + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \begin{tabular}{r l} \toprule + Ortsvektor: & $\vect{r} = \rho \vect{e}_\rho(\varphi) + z \vect{e}_z$ \\ + Geschwindigkeit: & $\vect{v} = \dot \rho \vect{e}_\rho + \rho \dot \varphi \vect{e}_\varphi + \dot z \vect{e}_z$ \\ + Schnelligkeit: & $v = \sqrt{\dot{\rho}^2 + \rho^2 \dot{\varphi}^2 + \dot{z}^2}$ \\ \bottomrule + \end{tabular} + + \includegraphics[width=0.30\textwidth]{Figures/KoordinatenSys.jpg} +\end{center} + +Transformationsregeln: + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \begin{tabular}{r l l l } \toprule + $Z \to C$ & $x = \rho \cos(\varphi)$ & $y = \rho \sin(\varphi)$ & $z = z$ \\ + $C \to Z$ & $\rho = \sqrt{x^2 + y^2}$ & $\varphi = \arctan\frac{y}{x}$ & $z = z$ \\ \bottomrule + \end{tabular} +\end{center} + +Alternativ, Einheitsvektoren transformieren: + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \begin{tabular}{r l l} \toprule + $C \to Z$ & $\vect{e}_\rho = \cos(\varphi)\vect{e}_x + \sin(\varphi)\vect{e}_y$ & $\vect{e}_\varphi = -\sin(\varphi)\vect{e}_x + \cos(\varphi)\vect{e}_y$ \\ + $Z \to C$ & $\vect{e_x} = \cos(\varphi)\vect{e}_\rho - \sin(\varphi)\vect{e}_\varphi$ & + $\vect{e_y} = \sin(\varphi)\vect{e}_\rho + \cos(\varphi)\vect{e}_\varphi$ \\ \bottomrule + \end{tabular} +\end{center} +\vfill + + + +\subsection{Satz der projizierten Geschwindigkeiten (SdpG)} + +Charakterisierung starrer Körper: \eqbox{$\forall P,Q \in SK: |\vect{r}_Q - \vect{r}_P| = \text{const.}$} \medskip + +Die Projektionen $\mathbf{v}'_p, \mathbf{v}'_Q$ der Geschwindigkeit von zwei beliebigen Punkten $P, Q$ eines SKs auf ihre Verbindungsgerade sind gleich: + +\begin{center} + \eqboxf{$v'_P = v'_Q \qquad (\vect{r}_P - \vect{r}_Q)(\vect{v}_P - \vect{v}_Q) = 0$} + + \includegraphics[width=0.15\textwidth]{Figures/SdPG.jpg} +\end{center} + + +\subsection{Momentane Bewegungsarten eines Starrkörpers} + +Solange der Körper \emph{wenigstens} momentan ein SK ist, gilt momentan: + +\begin{tabular}{r l} \toprule + i) & Starre Bewegung: SdpG immer erfüllt! \\ + ii) & Translation: $\vect{v}_p = \vect{v} \quad \forall P \in SK$ \quad (Alle Geschw. parallel) \\ + iii) & Rotation: Starre Bewegung mit ruhender Rotationsachse \\ + iv) & Ebene Bewegung: - Alle $\vect{v}$ sind zur Ebene $E$ parallel. \\ + & - Alle $P$ auf einer Normalen zur Ebene $E$ haben gleiches $\vect{v}$. \\ \bottomrule +\end{tabular} + +Momentan bedeutet zu einem bestimmten Zeitpunkt $t$ ('Snapshot'). + + +\subsection{Satz vom Momentanzentrum (Rotation)} + +Das Momentanzentrum $M$ eines SK ist beim Punkt (kann ausserhalb SK sein), wo momentan $v = 0$ gilt. Die Geschwindigkeit $\vect{v}_P$ eines beliebigen Punktes im SK steht stehts sekrecht auf der Verbindungsgerade mit $M$. + +\begin{center} + \includegraphics[width=0.25\textwidth]{Figures/Momentanzentrum.jpg} +\end{center} + +Jeder SK besitzt ein \emph{eigenes} $M$ und mit eigener Rotationsschnelligkeit $\omega$, welche stehts CCW dreht (Rechte Hand Regel). + +\begin{center} + \eqboxf{$\vect{v}_P = \vect{\omega} \times \vect{r}_P \qquad (\text{2D}: v_P = \omega \cdot r_P)$} +\end{center} + +Die \emph{Rotationsachse} ist eine Gerade in Richtung der Rotationsgeschwindigkeit $\vect{\omega}$, auf welcher alle Punkte momentan $v = 0$ besitzen. + +\begin{minipage}{0.45\linewidth} + \eqbox{\begin{tabular}{l} + Parallelogrammregel: \\ + $\vect{\omega}_2 = \vect{\omega}_4 \qquad \vect{\omega}_1 = \vect{\omega}_3$ \\ + \end{tabular}} +\end{minipage} +\begin{minipage}{0.45\linewidth} + \includegraphics[width=1\textwidth]{Figures/Parallelogramm.jpg} +\end{minipage} + +\begin{minipage}{0.35\linewidth} + \includegraphics[width=0.8\textwidth]{Figures/RollenOhneGleiten.jpg} +\end{minipage} +\begin{minipage}{0.6\linewidth} + \eqbox{\begin{tabular}{l} + Rollen ohne gleiten: \\ + - $\vect{v}_A = 0 \Rightarrow$ Momentanzentrum \\ + - Der Körper \emph{haftet} am Boden \\ + \end{tabular}} +\end{minipage} +\vfill + + + +\pagebreak + + + +\subsection{Freiheitsgrad (FG)} + +Der Freiheitsgrad ist die minimale Anzahl Koordinaten für die eindeutige Bestimmung der Lage eines bestimmten Systems. Ein unbehinderter Starrkörper besitzt in 2D einen FG von 3 und in 3D einen FG von 6. + +\begin{center} + \begin{minipage}{0.2\linewidth} + \eqboxf{$f = n - b$} + \end{minipage} + \begin{minipage}{0.75\linewidth} + \begin{tabular}{l} + $f$: Freiheitsgrad gebundenes System \\ + $n$: Anzahl SK $\cdot$ deren FG (2D: FG 3, 3D: FG 6) \\ + $b$: Anzahl der linear unabhängigen Bindungen \\ + \end{tabular} + \end{minipage} +\end{center} + + +\includegraphics[width=0.32\textwidth]{Figures/FG.jpg} + + +\subsection{Kreiselung} + +Die Kreiselung ist eine spezielle Starrkörperbewegung, die dadurch charakterisiert ist, dass \emph{nur ein} Punkt des Körpers fixiert bleibt. \medskip + +Eine Kreiselung ist \emph{momentan} eine Rotation. + + +\subsection{Starrkörperformel und Kinemate} + +Für jeden Punkt \emph{innerhalb} einem SK gilt für die Geschwindigkeit: + +\begin{center} + \eqboxf{$\vect{v}_P = \vect{v}_B + \vect{\omega} \times \vect{r}_{BP} \quad \forall P,B \in SK$} + + \includegraphics[width=0.20\textwidth]{Figures/SKFormel.jpg} +\end{center} + +Die Kinemate von Punkt $B$ ist \eqboxf{$\{\vect{v}_B,\vect{\omega}\}$} \medskip + +Mit den \emph{Invarianten}, d.h. der Wert ist unabhängig vom Punkt im SK: + +\begin{center} + \eqbox{$\vect{I}_1 = \vect{\omega}$} und \eqbox{$I_2 = \vect{v}_P \cdot \vect{\omega}$} \medskip + + \begin{tabular}{r l l } \toprule + Translation: & $\vect{I}_1 = \vect{0}$ und $I_2 = 0$ & $\Rightarrow \vect{\omega} = \vect{0}$ \\ + Rotation: & $\vect{I}_1 \neq \vect{0}$ und $I_2 = 0$ & $\Rightarrow \vect{\omega} \perp \vect{v}_P$ \\ + Schraubung: & $\vect{I}_1 \neq \vect{0}$ und $I_2 \neq 0$ \\ \bottomrule + \end{tabular} +\end{center} + +Alle Punkte auf der \emph{Zentralachse} besitzen die gleiche Geschwindigkeit. + +\begin{center} + \includegraphics[width=0.1\textwidth]{Figures/Zentralachse.jpg} +\end{center} +\vfill + + + +\subsection{Kräfte} + +Kräfte werden durch einen Vektor dargestellt, welcher einen Angriffspunkt und eine Wirkungslinie besitzt. + +\begin{minipage}{0.5\linewidth} + \begin{center} + \includegraphics[width=0.75\textwidth]{Figures/Kraft.jpg} +\end{center} +\end{minipage} +\begin{minipage}{0.45\linewidth} + Einheit: $[F] = N = \frac{m \cdot kg}{s^2}$ +\end{minipage} + +\textbf{Reaktionsprinzip}: Wenn ein Körper $K_1$ eine Kraft auf einen anderen Körper $K_2$ ausübt, dann übt $K_2$ auch auf $K_1$ eine gleich grosse, aber entgegengesetze Kraft aus. \medskip + +Kräfte werden \emph{abhängig vom betrachteten System} unterteilt in: + +\begin{center} + \begin{tabular}{r l} \toprule + \textbf{Äussere Kräfte}: & Reaktionskraft \emph{nicht} im betrachteten System. \\ + \textbf{Innere Kräfte}: & Reaktionskraft \emph{auch} im betrachteten System. \\ \bottomrule + \end{tabular} +\end{center} + + +\subsection{Leistung} + +Die Leistung ist: \eqboxf{$P = \vect{F} \cdot \vect{v}$} \quad Einheit: $[P] = W = \frac{J}{s} = \frac{N \cdot m}{s}$ \medskip + +Eine \emph{leistungslose Kraft} ($P = 0$) hat die Eigenschaft $\vect{F} \perp \vect{v}$. + + +\subsection{Moment} + +Das Moment beschreibt die Drehwirkung einer Kraft $F$ auf einen Körper in einem Punkt $P$ und ist vom Bezugspunkt $O$ abhängig: + +\begin{center} + \eqboxf{$\vect{M}_O(P) = \vect{r}_{OP} \times \vect{F}_P$} \quad Einheit: $[M] = Nm$ + + $F$ darf man auf der Wirkungslinie verschieben! +\end{center} + +Transformationsregel vom Moment: \eqbox{$\vect{M}_B = \vect{M}_O + \vect{r}_{BO} \times \vect{R}$} \medskip + +Bei einer reinen Rotation gilt: $P = \vect{M}_O \cdot \vect{\omega}$ + +\begin{center} + \includegraphics[width=0.32\textwidth]{Figures/2D_Moment.jpg} +\end{center} + +Der Betrag von Moment $M_0$ einer Kraft $F$ mit Angriffspunkt $P$ ist: + +\begin{center} + \eqbox{$|\vect{M}_O| = F \cdot r_{OP} \cdot \sin\gamma = \pm F \cdot d \qquad d = | \vect{r}_{OP} \times \vect{e}_{F} |$} + + Wobei $\pm$ durch die Rechte Hand Regel bestimmt wird +\end{center} + +Hier bezeichnet $d$ den kürzesten Abstand der Wirkungslinie der Kraft vom Bezugspunkt $O$. Diese Länge nennt man \emph{Hebelarm}. +\vfill + + + +\subsection{Resultierende} + +\begin{center} + \renewcommand{\arraystretch}{2} + \begin{tabular}{r l} \toprule + Resultierende Kraft: & $\vect{R} = \sum\limits_{i = 1}^n \vect{F}_i$ \\ + Resultierendes Moment: & $\vect{M}_O^{tot} = \sum\limits_{i = 1}^n \vect{M}^i_O$ \\ + Gesamtleistung: & $P_{tot} = \sum\limits_{i = 1}^n P_i = \sum\limits_{i = 1}^n \vect{F}_i \cdot \vect{v}_i$ \\ \bottomrule + \end{tabular} +\end{center} + +Die Gesamtleistung eines Starrkörpers ist: + +\begin{center} + \eqbox{$P_{tot} = \vect{R} \cdot \vect{v}_B + \vect{M}_B \cdot \vect{\omega}$} + + mit Kinemate $\{ \vect{v}_B, \vect{\omega} \}$ und Dyname $\{\vect{R}, \vect{M}_B \}$ +\end{center} +\vfill \ No newline at end of file diff --git a/zusammenfassung/technische_mechanik/Sections/Statik.tex b/zusammenfassung/technische_mechanik/Sections/Statik.tex new file mode 100644 index 0000000..ccddf1f --- /dev/null +++ b/zusammenfassung/technische_mechanik/Sections/Statik.tex @@ -0,0 +1,335 @@ +\section{Statik} + + +\subsection{Statische Äquivalenz} + +Zwei Kräftegruppen $\{ \vect{G}_1 \}, \{ \vect{G}_2 \}$ sind statisch äquivalent, wenn + +\begin{center} + \eqbox{$P_{tot}(\{ \vect{G}_1 \}) = P_{tot}(\{ \vect{G}_2 \})$} +\end{center} + +für alle Starrkörperbewegungen gilt. Dies ist der Fall, wenn + +\begin{center} + \eqbox{$\vect{R}_2 = \vect{R}_1 \qquad (\vect{M}_B)_2 = (\vect{M}_B)_1$} +\end{center} + +Zwei \emph{Kräfte} sind statisch äquivalent, wenn sie vektoriell gleich sind und ihre Wirkungslinien übereinstimmen. \textbf{Gilt nur innerhalb vom SK!} + + +\subsection{Dyname} + +Die Dyname einer Kräftegruppe ist \eqboxf{$\{ \vect{R},\vect{M}_B \}$} \medskip + +Mit den \emph{Invarianten} ($\forall P \in SK$): \eqbox{$\vect{I}_1 = \vect{R}$} und \eqbox{$ I_2 = \vect{R} \cdot \vect{M}_P$} \medskip + +Eine Kräftegruppe (KG) ist statisch äquivalent zu: + +\begin{center} + \begin{tabular}{R{2cm} l l} \toprule + Gleichgewicht (Nullsystem): & $\vect{R} = \vect{0}, \, \vect{M}_P = \vect{0}$ & $\vect{I}_1 = \vect{0}$ und $I_2 = 0$ \\[0.5mm] + Kräftepaar: & $\vect{R} = \vect{0}, \, \vect{M}_P \neq \vect{0}$ & $\vect{I}_1 = \vect{0}$ und $I_2 = 0$ \\[0.5mm] + Einzelkraft: & $\vect{R} \neq \vect{0}, \, \vect{M}_P = \vect{0}$ & $\vect{I}_1 \neq \vect{0}$ und $I_2 = 0$ \\[0.5mm] + Schraube: & & $\vect{I}_1 \neq \vect{0}$ und $I_2 \neq 0$ \\ + \bottomrule + \end{tabular} +\end{center} + +Ergänzung: Bei $\vect{R} \perp \vect{M_P}$ gibt es einen Punkt $O$, so dass $\vect{M}_O = \vect{M}_P + \vect{R} \times \vect{r}_{PO} = \vect{0}$. \medskip + +\begin{minipage}{0.6\linewidth} + \textbf{Kräftepaar}: + + \begin{tabular}{r p{0.8\linewidth}} + - & $\vect{M}_O = \vect{r} \times \vect{F} \qquad M_O = d F$ \\ + - & Moment ist unabhängig von Bezugspunkt ($M_O = M$) \\ + \end{tabular} +\end{minipage} +\begin{minipage}{0.35\linewidth} + \includegraphics[width=1\textwidth]{Figures/Kraeftepaar.jpg} +\end{minipage} + + +\subsection{Parallele Kräftegruppen} + +Parallele Kräftegruppen (KG) lassen sich auf ein Moment oder eine Einzelkraft reduzieren. Es gibt zwei Fälle: \medskip + +\textbf{a)} Falls $\vect{R} = 0$, kann man die KG auf ein vom Bezugspunkt unabhängiges Moment $\vect{M}$ reduzieren. Man definiert das Dipolmoment der KG als: + +\begin{center} + \eqbox{$\vect{N} = \sum\limits_{i = 1}^n F_i \vect{r_i}$} \qquad $\vect{M} = \vect{N} \times \vect{e}$ + + wobei $\vect{e}$ in Richtung der Kräftegruppen zeigt. +\end{center} + +In der \emph{Elektrostatik} ($\vect{R} = 0 \Leftrightarrow Q = \sum q_i = 0$): + +\begin{center} + \eqbox{\begin{tabular}{r l} + Auf die Ladungen wirkende Kraft: & $\vect{F}_i = q_i \vect{E}$ \\ + Dipolmoment Punktladungsgruppe: & $\vect{P} = \sum\limits_{i = 1}^n q_i \vect{r}_i \quad \vect{N} = E \vect{P}$ \\ + \end{tabular}} +\end{center} + +\textbf{b)} Falls $\vect{R} \neq 0$, kann man die Kräftegruppe auf eine Einzelkraft reduzieren mit dem Kräftemittelpunkt $C$ der Kräftegruppe: + +\begin{center} + \eqbox{$\displaystyle\vect{r}_{OC} = \dfrac{1}{\sum F_i} \sum F_i \cdot \vect{r}_i$} +\end{center} + + +\subsection{Schwerpunkt (= Massenmittelpunkt)} + +Der Kräftemittelpunkt der Gewichtskraft heisst Schwerpunkt. Allg: + +\begin{center} + \eqbox{$\vect{r}_{OC} = \dfrac{1}{m} \displaystyle\iiint \vect{r} \d m$ wobei $m = \displaystyle\iiint \d m$} +\end{center} + +Für homogene Körper, d.h. überall gleiche Dichte, gilt: + +\begin{center} + $\d m = \gamma \d V$ \qquad Dichte: $[\gamma] = \frac{kg}{m^3}$ \medskip + + \eqbox{$\vect{r}_{OC} = \dfrac{1}{\gamma V} \displaystyle\iiint \vect{r} \gamma \d V = \dfrac{1}{V} \displaystyle\iiint \vect{r} \d V$} +\end{center} + +Bei homogenen 2D-Körper vereinfacht sich das Integral zu: + +\begin{center} + \eqboxf{$\vect{r}_{OS} = \dfrac{1}{\sum A_i} \sum A_i \vect{r}_{OS_i}$} + + \eqboxf{$\displaystyle x_s = \dfrac{\sum x_i \cdot A_i}{\sum A_i} \qquad y_s = \dfrac{\sum y_i \cdot A_i}{\sum A_i}$} +\end{center} + +$A_i$ kann auch die Masse des Körpers sein. +\\ +Mit folgenden Teilkörpern und ihren Schwerpunkten: + +\begin{center} + \begin{tabular} {r l l} \toprule + Kreis: & $\vect{r}_{OS} = \begin{bmatrix}R \\ R \\ \end{bmatrix}$ & $A = \pi r^2$ \\ + Halbkreis: & $\vect{r}_{OS} = \begin{bmatrix}0 \\ 4R / (3\pi) \\ \end{bmatrix}$ & $A = \frac{1}{2}\pi r^2$ \\ + Rechteck: & $\vect{r}_{OS} = \begin{bmatrix}a / 2 \\ b / 2 \\ \end{bmatrix}$ & $A = a \cdot b$ \\ + Dreieck & $\vect{r}_{OS} = \dfrac{\vect{r}_A + \vect{r}_B + \vect{r}_C}{3}$ & $A = \dfrac{b \cdot h_b}{2}$ \\ \bottomrule + \end{tabular} \medskip + + \includegraphics[width=0.31\textwidth]{Figures/Schwerpunkt.jpg} +\end{center} + + +\subsection{Hauptsatz der Statik} + +Ein System ist in Ruhe, wenn alle Geschwindigkeiten \emph{null} sind. Notwendig dafür (aber nicht hinreichend) ist, dass die äusseren Kräfte im Gleichgewicht sind. Gleichgewichtsbedingungen: + +\begin{center} + \begin{tabular}{r l} \toprule + Komponentenbedingungen (KB): & \eqboxf{$\vect{R} = \vect{0}$} \\ + Momentenbedingung (MB): & \eqboxf{$\vect{M}_O = \vect{0}$} \\ \bottomrule + \end{tabular} + + Der Bezugspunkt für das Moment ist frei wählbar! +\end{center} + +Im Raum kriegt man so 6 Gleichung und in der Ebene 3 Gleichungen. \medskip + +\textbf{Freischnitt}: Um die Gleichgewichtsbedingungen zu überprüfen muss man zuerst einen Freischnitt des Systems machen, d.h. man erstellt eine Skizze mit allen am System angreifenden Kräften (Bindungen $\Rightarrow$ Bindungskräfte!). \medskip + +\textbf{Zusatzbedingungen}: \begin{tabular}{l l} \toprule + i) & Seilkraft: $S > 0$ (nur auf Zug) \\ + ii) & Auflager: $F_y > 0$ (kein Abheben) \\ + iii) & Reibung, Standfestigkeit \\ \bottomrule +\end{tabular} +\vfill\null + + +Falls man alle Lager- und Bindungskräfte bestimmen will, so lohnt sich die Auftrennung des Systems in Teile und die Anwendung des Hauptsatzes auf alle Teile. Ist aber nur eine Kraft gesucht, dann ist das PdvL am besten geeignet. \medskip + +Sind mehrere Stabkräfte benötigt in einem System, dann kann es sich lohnen einen Kräfteschnitt auszuführen, d.h. das System geschickt für die gesuchten Stabkräften in zwei aufzuteilen. + + +\subsection{Bindungskräfte} + +Allgemein versteht man unter einer Bindung eine Einschränkung der Bewegungsfreiheit eines Körpers. Bindungen erzeugen Kräfte oder Momente, die in die behinderte Richtung der Bewegung gerichtet sind. + +\begin{center} + \includegraphics[width=0.31\textwidth]{Figures/Bindungskraefte.jpg} +\end{center} +\vfill + + +\subsection{Prinzip der virtuellen Leistungen (PdvL)} + +Ein System befindet sich \emph{genau dann} in einer Ruhelage, \emph{wenn} die virtuelle Gesamtleistung der inneren und äusseren Kräfte bei \emph{jedem} virtuellen Bewegungszustand verschwindet (und die Eigenschaften des Systems und seiner Lagerung diese Kräfte zulassen). Formell: + +\begin{center} + \eqbox{$\tilde{P} = \tilde{P}^{(i)} + \tilde{P}^{(a)} = 0 \quad \forall \{\tilde{\vect{v}}\}$} + + Wobei $\tilde{\vect{v}}, \, \tilde{\vect{\omega}}$ virtuelle Bewegungszustände sind. +\end{center} + +Im Klartext kann man sagen: \eqboxf{$\tilde{P}_{tot} = \sum \vect{F}_i \cdot \tilde{\vect{v}}_i = 0$} \medskip + +Zulässige virtuelle Bewegungszustände sind Zustände, die keine Bindungen verletzt, d.h. man muss sich an die Systemeigenen Einschränkungen der Bewegungsmöglichkeiten halten! + + +\subsection{Fachwerke} + +Fachwerke bestehen aus gelenkig gelagerten Pendelstäben. Annahmen: + +\begin{center} + \begin{tabular}{r l p{120pt}} \toprule + i) & Gelenke Reibungsfrei & $\Rightarrow$ keine Reibungskräfte \\ + ii) & Stäbe gewichtlos & $\Rightarrow$ keine Gewichtskräfte \\ + iii) & Knoten nur an Stabenden \\ + iv) & Alle lasten in den Knoten & $\Rightarrow$ $\vect{F}^{(a)}$ greifen an Gelenken an \\ \bottomrule + \end{tabular} +\end{center} + +Ausserdem können die Stäbe nur axial belastet werden, d.h. die Stabkräfte sind ausschliesslich parallel zur Stabrichtung. + + +\subsection{Stabkräfte} + +Pendelstäbe können nur Kräfte parallel zur Stabrichtung aufnehmen. Lösungsverfahren wenn einzelne Stabkräfte gesucht sind: + +\begin{center} + \begin{tabular}{r l} \toprule + i) & Lager/Stab entfernen (FG + 1), und entsprechende \\ + & Stabkraft $S$ in Skizze einsetzen \\ + ii) & Einen virtuellen Bewegungszustand einführen und alle \\ + & für das PdvL nötige $\tilde{\vect{v}}_i$ berechnen \\ + iii) & PdvL anwenden und nach Stabkraft $S$ auflösen \\ + & Bindungskräfte sind nicht erforderlich! ($\vect{v} = 0$ oder $\vect{v} \perp \vect{F}$) \\ \bottomrule + \end{tabular} +\end{center} + +\begin{minipage}{0.5\linewidth} + \includegraphics[width=\textwidth]{Figures/Zug-Druckstabe.jpg} +\end{minipage} +\begin{minipage}{0.45\linewidth} + \begin{center} + \eqbox{\begin{tabular}{r l} + $s > 0$ & $\Rightarrow$ Zugstab \\ $s < 0$ & $\Rightarrow$ Druckstab \\ + \end{tabular}} + \end{center} +\end{minipage} + + +\subsection{Statische und Kinematische Bestimmtheit} + +\textbf{Statische Bestimmtheit}: Ein System ist statisch bestimmt, wenn die Anzahl der Bindungskräfte und Momente (unabhängig von Belastungen) gleich der Anzahl der Gleichgewichtsbedingungen ist. \medskip + +\textbf{Kinematische Bestimmtheit}: Wenn $FG > 0$ ist, dann ist das System kinematisch unbestimmt, ansonsten ist es kinematisch bestimmt. \medskip + +\begin{tabular}{r l l} \toprule + $FG = 0$ & Statisch bestimmt & Kinematisch bestimmt \\ + $FG < 0$ & Statisch unbestimmt & Kinematisch bestimmt \\ + $FG > 0$ & Statisch unbestimmt & Kinematisch unbestimmt \\ \bottomrule +\end{tabular} +\vfill + + +\subsection{Knotengleichgewicht} + +Pendelstäbe können nur Kräfte parallel zur Stabrichtung aufnehmen. Lösungsverfahren wenn alle Stabkräfte gesucht sind: \medskip + +\begin{tabular}{r l} \toprule + i) & Knoten freischneiden und Hauptsatz der Statik anwenden \\ + ii) & Pro Knoten 2 Gleichgewichtsbedingungen \\ + & $\Rightarrow$ An Knoten starten mit $\leq 2$ Kräften \\ + iii) & Erforderliche Bindungskräfte berechnen \\ \bottomrule +\end{tabular} + + +\subsection{Flaschenzug} + +In der Statik triff man beim Flaschenzug folgende Annahmen: + +\begin{center} + \begin{tabular}{r l} \toprule + i) & Undehnbares Seile $\Rightarrow$ Überträgt Geschw. \& Kraft 1:1 \\ + ii) & Flaschenzüge masselos \\ + iii) & Seile um Flaschenzüge gewickelt und rutschen nicht darüber \\ + iv) & Bewegung in horizontaler Richtung nicht möglich \\ \bottomrule + \end{tabular} +\end{center} + +Wenn ein Seil ohne äussere Einwirkungen eine frei drehbare Rolle umläuft, bleibt die Seilkraft vor und nach der Rolle gleich. \medskip + +Lösungsansatz: Je nachdem entweder PdvL oder Hauptsatz der Statik. + + +\subsection{Standfestigkeit} + +Die Standfestigkeit gilt als Zusatzbedingung für ein Ruhesystem. + +\begin{center} + \eqboxf{\begin{tabular}{p{230pt}} + Ein System ist standfest (kein Kippen), wenn der Durchstosspunkt $A$ der Normalkraft $N$ innerhalb der Standfläche liegt. \\ + \end{tabular}} +\end{center} + +Die Standfläche ist die kleinste konvexe Fläche, die die Berührungsfläche umschliesst. \medskip + +Normalkraft: \eqbox{$N = \iint \d N$} wobei $\vect{N} \perp$ Standfläche + +\begin{center} + \includegraphics[width=0.3\textwidth]{Figures/Standfestigkeit.jpg} +\end{center} + +Um die Standfestigkeit zu überprüfen führt man $N$ mit einer Distanzvariable $d$ ein, welche einen höchstwert nicht überschreiten darf (Durchstosspunkt muss innerhalb der Standfläche sein). \medskip + +Ist Standfestigkeit vorgegeben und die Bedingung damit das System nicht kippt gesucht, dann kann man auch direkt die Momentbedingung als Ungleichung aufstellen. +\vfill\null +\columnbreak + + +\subsection{Reibung} + + + +\begin{minipage}{0.5\linewidth} + Materialabhängige Koeffizienten: +\end{minipage} +\begin{minipage}{0.47\linewidth} + \begin{tabular}{r l} \toprule + $\mu_0$ & Haftreibungskoeffizient \\ + $\mu_1$ & Gleitreibungskoeffizient \\ + $\mu_2$ & Rollwiderstandslänge \\ \bottomrule + \end{tabular} +\end{minipage} + +\begin{center} + \includegraphics[width=0.32\textwidth]{Figures/Reibung_2.jpg} +\end{center} + +Zusatzbedingungen für ein ruhendes System: + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \eqboxf{\begin{tabular}{r l} + Haftreibungsgesetz & $|\vect{F}_r| \leq \mu_0 |\vect{N}|$ \\ + Rollwiderstandsgesetz bei Ruhe & $|\vect{M}_f| \leq \mu_2 |\vect{N}|$ \\ + \end{tabular}} +\end{center} + +Ist das System in Bewegung, dann gibt es zusätzliche Gleichung(en): + +\begin{center} + \renewcommand{\arraystretch}{1.5} + \eqbox{\begin{tabular}{r l} + Gleitreibungsgesetz & $| \vect{F}_r | = \mu_1 | \vect{N} |$ \\ + & $\vect{F}_r = -\mu_1 |\vect{N}| \frac{\vect{v}_B}{|\vect{v}_B|}$ \\ + Rollwiderstandsgesetz bei Bewegung & $|\vect{M}_f| = \mu_2 |\vect{N}|$ \\ + & $\vect{M}_f = - \mu_2 |\vect{N}|\frac{\vect{\omega}}{|\vect{\omega}|}$ \\ + \end{tabular}} +\end{center} + +Ideal rau (beim Zahnrad) ist: $\mu_0 = \infty \quad \mu_2 = 0$ + +\begin{center} + \includegraphics[width=0.32\textwidth]{Figures/Reibung.jpg} +\end{center} +\vfill diff --git a/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.pdf b/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.pdf new file mode 100644 index 0000000..68cd831 Binary files /dev/null and b/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.pdf differ diff --git a/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.tex b/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.tex new file mode 100644 index 0000000..c97e7e8 --- /dev/null +++ b/zusammenfassung/technische_mechanik/zusammenfassung_techmech_ruh_jirayu.tex @@ -0,0 +1,117 @@ +\documentclass[a4paper, fontsize = 8pt, landscape]{scrartcl} +\usepackage[german]{babel} +\usepackage[landscape, margin=0.5cm]{geometry} +\usepackage[dvipsnames]{xcolor} +% Hyperref links, pdf metadata +\usepackage{hyperref} +\usepackage{amscd, amsmath, amssymb, blindtext, empheq, enumitem, multicol} +\usepackage{mathtools} +\usepackage{graphicx} +\usepackage{tikz} +\usepackage{array} %for bigger tabular spacings +\usepackage{multirow} % for multirow cells in tabular +\usepackage{booktabs} +\usepackage{comment} +%% Formal tables +\usepackage{booktabs} + +% make document compact + +\parindent 0pt +\pagestyle{empty} +\setlength{\unitlength}{1cm} +\setlist{leftmargin = *} + +% define some colors +\definecolor{title}{RGB}{229,140,140} +\definecolor{subtitle}{RGB}{239,185,185} +\definecolor{text}{RGB}{0,0,0} +\definecolor{formulabox}{RGB}{182,225,189} + +% section color box +\setkomafont{section}{\mysection} +\newcommand{\mysection}[1]{% + \Large\normalfont \sffamily \bfseries% + \setlength{\fboxsep}{0cm}%already boxed + \colorbox{title}{% + \begin{minipage}{\linewidth}% + \vspace*{2pt}%Space before + \leftskip2pt %Space left + \rightskip\leftskip %Space right + {\color{text} #1} + \vspace*{1pt}%Space after + \end{minipage}% + }} +%subsection color box +\setkomafont{subsection}{\mysubsection} +\newcommand{\mysubsection}[1]{% + \normalsize \normalfont \sffamily \bfseries% + \setlength{\fboxsep}{0cm}%already boxed + \colorbox{subtitle}{% + \begin{minipage}{\linewidth}% + \vspace*{2pt}%Space before + \leftskip2pt %Space left + \rightskip\leftskip %Space right + {\color{text} #1} + \vspace*{1pt}%Space after + \end{minipage}% + }} + + +% equation box +\newcommand{\eqbox}[1]{\fcolorbox{black}{white}{\hspace{0.5em}#1\hspace{0.5em}}} +% equation box +\newcommand{\eqboxf}[1]{\fcolorbox{black}{formulabox}{\hspace{0.5em}#1\hspace{0.5em}}} + +%Laplace Operator +\newcommand{\LAPB}[1]{\text{\textbf{L}}\left\{ #1 \right\}} +\newcommand{\LAPZ}[1]{\text{\textbf{L}}^{-1}\left\{ #1 \right\}} + + +% tabular collum types +\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}} + +%macro for vectors +\newcommand{\vect}[1]{\mathbf{\underline{#1}}} +\renewcommand{\d}[1]{\text{ d}#1} + + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\newcolumntype{R}[1]{>{\raggedleft\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} + + + +\begin{document} + +\setcounter{secnumdepth}{0} %no enumeration of sections +\begin{multicols*}{3} + \begin{center} + \Large{Technische Mechanik} \\ + \tiny{von Jirayu Ruh, \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch}} + \end{center} + + \input{sections/Kinematik.tex} + \pagebreak + \input{sections/Statik.tex} + \pagebreak + \input{sections/Dynamik.tex} + \pagebreak +\end{multicols*} +\end{document} + + + + + + + + + + + + + + + +