commit 4212916e1ac29c4890bf05e11900530d2feaea2f Author: Niebler Date: Thu Feb 24 12:16:45 2022 +0100 first commit diff --git a/ET2_Deckblatt.tex b/ET2_Deckblatt.tex new file mode 100644 index 0000000..90e4718 --- /dev/null +++ b/ET2_Deckblatt.tex @@ -0,0 +1,46 @@ +\pagenumbering{roman} + +%\ifthenelse{\equal{\toPrint}{Lösung}}{\input{ET2_Deckblatt}}{\input{ET2_Deckblatt_A}} + +\begin{titlepage} +\vspace*{3cm} +\begin{center} +{\bfseries \large {Technische Hochschule Nürnberg \\ Georg Simon Ohm}\\[\baselineskip] + Übung Grundlagen der Elektrotechnik 2}\\[\baselineskip] + + + % Prüfen ob Musterlösung oder nicht, wenn ja, dann mit Dozentenkommentar, ansonsten nur Aufgabenstellung für Studierende + \ifthenelse{\equal{\toPrint}{Lösung}}% +{% +{\normalsize Prof.\,Dr.\,C.\,Niebler}\\[2\baselineskip] + +\textcolor{red}{Nur für Dozentengebrauch, nicht zur Weitergabe an Studenten.}\\[\baselineskip] +{\large \textbf{Dozenten Exemplar mit Lösungen für}}\\[2\baselineskip] + +\textbf{\ProfName}\\[2\baselineskip] + +%\copyright\ Dr. Christine Niebler\\ + +}% Dozentenausgabe +{}% Studentenausgabbe +\end{center} +\end{titlepage} + + + + + +\thispagestyle{empty} +\cleardoublepage + + + + +\clearpage +\pagenumbering{arabic} + +\enlargethispage{1cm} +%\chead{Inhaltsverzeichnis} +\footnotesize{\tableofcontents} + +\clearpage \ No newline at end of file diff --git a/ET2_Deckblatt_A.log b/ET2_Deckblatt_A.log new file mode 100644 index 0000000..a3ffd24 --- /dev/null +++ b/ET2_Deckblatt_A.log @@ -0,0 +1,47 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9 64-bit) (preloaded format=pdflatex 2014.2.26) 17 MAR 2014 17:43 +entering extended mode +**C:/Users/Niebler/Documents/Vorlesungen/ETechnik/TeX/UeET2_20130923/ET2_Deckbl +att_A.tex + +(C:/Users/Niebler/Documents/Vorlesungen/ETechnik/TeX/UeET2_20130923/ET2_Deckbla +tt_A.tex +LaTeX2e <2011/06/27> +Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar +abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, +croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga +lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland +ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, + lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n +german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman +ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm +an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl +ishmax, welsh, loaded. + +! LaTeX Error: Environment titlepage undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2 \begin{titlepage} + +? +! Emergency stop. + ... + +l.2 \begin{titlepage} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Here is how much of TeX's memory you used: + 9 strings out of 493921 + 477 string characters out of 3147276 + 49031 words of memory out of 3000000 + 3401 multiletter control sequences out of 15000+200000 + 3640 words of font info for 14 fonts, out of 3000000 for 9000 + 841 hyphenation exceptions out of 8191 + 5i,0n,4p,116b,16s stack positions out of 5000i,500n,10000p,200000b,50000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/ET2_Deckblatt_A.tex b/ET2_Deckblatt_A.tex new file mode 100644 index 0000000..eb5e98c --- /dev/null +++ b/ET2_Deckblatt_A.tex @@ -0,0 +1,27 @@ +\pagenumbering{roman} +\begin{titlepage} +\vspace*{3cm} +\begin{center} +{\bfseries \Large {Technische Hochschule Nürnberg\\ +Georg Simon Ohm\\[\baselineskip] +{Übung Grundlagen der Elektrotechnik 2}}\\[\baselineskip] +\normalsize{Prof.\,Dr.\,C.\,Niebler}\\[\baselineskip] +\Large\textbf{Aufgabenstellung}}\\[\baselineskip] +% SS 2013}\\[\baselineskip] +\end{center} +\end{titlepage} +\thispagestyle{empty} +\cleardoublepage + + + +\clearpage +\pagenumbering{arabic} + +\enlargethispage{1cm} +%\chead{Inhaltsverzeichnis} +%\footnotesize{ +\tableofcontents +%} + +\clearpage \ No newline at end of file diff --git a/ET2_LS_B12_A1.tex b/ET2_LS_B12_A1.tex new file mode 100644 index 0000000..f0b4061 --- /dev/null +++ b/ET2_LS_B12_A1.tex @@ -0,0 +1,116 @@ +\section{Blitzableiter} +Eine $l=4\,\metre$ lange Verbindungsleitung, deren Leiter voneinander einen Abstand $d = 2\,\centi\metre$ haben, ist im Abstand $D=3\,\metre$ parallel zu einem Blitzableiter verlegt. (Siehe Skizze).\\ +$\mu_0=1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)$ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Welche Spannung $u$ (Betrag !) wird induziert, wenn der +Blitzstrom $i$ linear in $0{,}6\,\micro\second$ auf $15\,\kilo\ampere$ ansteigt? +\item Ist die Spannung während dieser Zeit positiv oder negativ? (Begründung !) +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] % Blitzstrom + \draw [->] (0,4) -- (0,0) node [below] {$i$}; + \end{scope} + \begin{scope}[>=latex,blue,very thick, xshift=3cm, yshift=0.5cm] % Leitung + \draw (0,3) circle (0.025) -- (0,0) -- (0.2,0) -- (0.2,3) circle (0.025); % node at (0.1,3) [above ] {$u$}; + \draw [<-] (0,3.1) -- (.2,3.1) node at (0.1,3.1) [above] {$u$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=0cm,yshift=2cm] % Abstand D + \draw [<->] (0,0) -- (3.1,0) node at (1.55,0) [above] {$D$}; + \draw [dashed] (3.1,.5)--(3.1,-.5); + \end{scope} + \begin{scope}[>=latex,thick,xshift=3cm,yshift=1cm] % Pfeile d + \draw [->] (-.5,0) -- (0,0) node at (0.5,0) [above] {$d$}; + \draw [<-] (0.2,0) -- (0.7,0); + \end{scope} + \begin{scope}[black!75!,>=latex,thick,xshift=4.125cm,yshift=0.5cm] % Länge l + \draw [<->] (-.125,3) -- (-.125,0) node at (0,1.5) [right] {$l$}; + \draw (0,0) -- (-0.5,0); + \draw (0,3) -- (-0.5,3); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +Formeln: +\begin{align} +% \intertext{Formeln:} +u&=-N\cdot \frac{d\Phi}{dt}\\ +\Phi&=B\cdot A=\mu\cdot H\cdot A=\int{\vec{B}\cdot \vec{dA}}\\ +H&=\frac{i\cdot N}{l} \qquad \text{für Zylinder: }\qquad H=\frac{i\cdot N}{2\cdot \pi\cdot r} +\end{align} +Magnetischer Fluss $\Phi$; Flussdichte $B$; Feldstärke $H$;\\ +Magnetische Permeabilität $\mu=\mu_0\cdot \mu_r$; In Luft $\mu_r=1$\\ +\clearpage +Berechnung: +\begin{align*} +u&=-N\cdot \frac{d\Phi}{dt} \qquad \text{mit $N=1$ }\qquad u=-\frac{d\Phi}{dt}\\ +\Phi&=\int{\vec{B}\cdot \vec{dA}} \qquad \text{mit $B\bot A$}\qquad \Phi=B\cdot A=\underbrace{\mu\cdot H}_{B}\cdot \underbrace{l\cdot d}_{A}\\ +H&=\frac{i}{2\cdot \pi\cdot D} \qquad \text{Abhängigkeit vom Abstand $d$ vernachlässigbar $d\ll D$. }\\ + \phantom{blablabla}bla\\ + \vphantom{u=-\frac{y}{x}}..\\ +u&=-\frac{d\Phi}{dt}\hspace{-.5cm}\underbrace{=}_{linearer Anstieg}\hspace{-.5cm}-\frac{\Phi}{t}=-\mu\cdot l\cdot d\cdot \frac{i}{2\cdot \pi\cdot D\cdot t}=-\frac{\mu\cdot i\cdot l\cdot d}{2\cdot \pi\cdot D\cdot t}\\ +% } %end pahntom +&=-\frac{1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)\cdot 15000\,\ampere\cdot 4\,\metre\cdot 0{,}02\,\metre}{2\cdot \pi \cdot 3\,\metre\cdot 0{,}6\cdot \power{10}{-6}\,\second}=\uuline{-134\,\volt}\\[\baselineskip] +&\text{Leitung symmetrisch zur Mittellinie bei } 2{,}99\,\metre \text{ und } 3{,}01\,\metre\\ +\text{mit }&\Phi=\frac{\mu\cdot i\cdot l}{2\cdot \pi}\cdot\ln\frac{D+d/2}{D-d/2}\\ +u&=-\frac{\mu_0\cdot 15\cdot \power{10}{3}\,\ampere\cdot 4\,\metre}{0{,}6\cdot \power{10}{-6}\,\second\cdot 2\cdot \pi}\cdot \underbrace{\ln\frac{3{,}01\,\metre}{2{,}99\,\metre}}_{6{,}66\cdot \power{10}{-3}}=-133{,}7\,\volt\\ +\text{\uline{Lentzsche Regel:}} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$}; + \draw [<-,blue] (.2,.35)--(.8,.35)node at(.5,.35)[above]{\footnotesize$u$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.25cm,yshift=-.25cm]% Leiterschleife + \draw (0,0)--(0,-2)--(.5,-2)--(.5,0); + \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25); + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.05) node [above right] {\footnotesize$+$}; + \fill (1,0)circle(.05) node [above left] {\footnotesize$-$}; + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=-1cm]% O. + \draw (.5,0)circle(.133)node at (1,0)[right]{$\frac{d\Phi}{dt}$}; + \fill (.5,0)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,green!50!black,xshift=0cm,yshift=-1.33cm]% Ox + \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$}; + \draw [black]node at (1.5,-1)[below]{Wirkung $\frac{d\Phi_{ind}}{dt}$ entgegen Ursache $\frac{d\Phi}{dt}$}; + \draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=3cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.05) node [above right] {\footnotesize$+$}; + \fill (1,0)circle(.05) node [above left] {\footnotesize$-$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=-0.25cm]% Leiterschleife + \draw (0,0)--(0,-2)--(.5,-2)--(.5,0); + \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=-2.25cm]%Stromquelle + \draw [green!50!black](.25,0)--(.75,0); + \draw node at (.5,.133) [above] {$i_{}$}; + \draw [green!50!black](.5,0)circle(.133); + \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$i_{ind}$}; + \end{scope} + \begin{scope}[>=latex,very thick,green!50!black,xshift=3cm,yshift=-1.33cm]% Ox + \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$}; +\draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133); +\end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1.25cm]% + \draw [->,red] (-2.5,.5)--(-2.5,-.5) node at(-2.5,0)[right]{$\frac{di}{dt}$}; + \draw [<-,green!50!black] (.5,.5)--(.5,-.5) node at(.5,0)[left]{$i_{ind}$}; + \draw [<-,red!75!black] (-1.75,.5)--(-1.75,-.5) node at(-1.75,0)[left]{$i_{ind}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +$u$ hat negatives Vorzeichen, da Polung von $i_{ind}$ entgegen Bezugspfeilen aus Skizze\\ +($u_R = -u$) +\clearpage +}{}% diff --git a/ET2_L_B12_A1.tex b/ET2_L_B12_A1.tex new file mode 100644 index 0000000..efc0cc4 --- /dev/null +++ b/ET2_L_B12_A1.tex @@ -0,0 +1,113 @@ +\section{Blitzableiter} +Eine $l=4\,\metre$ lange Verbindungsleitung, deren Leiter voneinander einen Abstand $d = 2\,\centi\metre$ haben, ist im Abstand $D=3\,\metre$ parallel zu einem Blitzableiter verlegt. (Siehe Skizze).\\ +$\mu_0=1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)$ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Welche Spannung $u$ (Betrag !) wird induziert, wenn der +Blitzstrom $i$ linear in $0{,}6\,\micro\second$ auf $15\,\kilo\ampere$ ansteigt? +\item Ist die Spannung während dieser Zeit positiv oder negativ? (Begründung !) +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] % Blitzstrom + \draw [->] (0,4) -- (0,0) node [below] {$i$}; + \end{scope} + \begin{scope}[>=latex,blue,very thick, xshift=3cm, yshift=0.5cm] % Leitung + \draw (0,3) circle (0.025) -- (0,0) -- (0.2,0) -- (0.2,3) circle (0.025); % node at (0.1,3) [above ] {$u$}; + \draw [<-] (0,3.1) -- (.2,3.1) node at (0.1,3.1) [above] {$u$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=0cm,yshift=2cm] % Abstand D + \draw [<->] (0,0) -- (3.1,0) node at (1.55,0) [above] {$D$}; + \draw [dashed] (3.1,.5)--(3.1,-.5); + \end{scope} + \begin{scope}[>=latex,thick,xshift=3cm,yshift=1cm] % Pfeile d + \draw [->] (-.5,0) -- (0,0) node at (0.5,0) [above] {$d$}; + \draw [<-] (0.2,0) -- (0.7,0); + \end{scope} + \begin{scope}[black!75!,>=latex,thick,xshift=4.125cm,yshift=0.5cm] % Länge l + \draw [<->] (-.125,3) -- (-.125,0) node at (0,1.5) [right] {$l$}; + \draw (0,0) -- (-0.5,0); + \draw (0,3) -- (-0.5,3); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +Formeln: +\begin{align} +% \intertext{Formeln:} +u&=-N\cdot \frac{d\Phi}{dt}\\ +\Phi&=B\cdot A=\mu\cdot H\cdot A=\int{\vec{B}\cdot \vec{dA}}\\ +H&=\frac{i\cdot N}{l} \qquad \text{für Zylinder: }\qquad H=\frac{i\cdot N}{2\cdot \pi\cdot r} +\end{align} +Magnetischer Fluss $\Phi$; Flussdichte $B$; Feldstärke $H$;\\ +Magnetische Permeabilität $\mu=\mu_0\cdot \mu_r$; In Luft $\mu_r=1$\\ +\clearpage +Berechnung: +\begin{align*} +u&=-N\cdot \frac{d\Phi}{dt} \qquad \text{mit $N=1$ }\qquad u=-\frac{d\Phi}{dt}\\ +\Phi&=\int{\vec{B}\cdot \vec{dA}} \qquad \text{mit $B\bot A$}\qquad \Phi=B\cdot A=\underbrace{\mu\cdot H}_{B}\cdot \underbrace{l\cdot d}_{A}\\ +H&=\frac{i}{2\cdot \pi\cdot D} \qquad \text{Abhängigkeit vom Abstand $d$ vernachlässigbar $d\ll D$. }\\ +u&=-\frac{d\Phi}{dt}\hspace{-.5cm}\underbrace{=}_{linearer Anstieg}\hspace{-.5cm}-\frac{\Phi}{t}=-\mu\cdot l\cdot d\cdot \frac{i}{2\cdot \pi\cdot D\cdot t}=-\frac{\mu\cdot i\cdot l\cdot d}{2\cdot \pi\cdot D\cdot t}\\ +&=-\frac{1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)\cdot 15000\,\ampere\cdot 4\,\metre\cdot 0{,}02\,\metre}{2\cdot \pi \cdot 3\,\metre\cdot 0{,}6\cdot \power{10}{-6}\,\second}=\uuline{-134\,\volt}\\[\baselineskip] +&\text{Leitung symmetrisch zur Mittellinie bei } 2{,}99\,\metre \text{ und } 3{,}01\,\metre\\ +\text{mit }&\Phi=\frac{\mu\cdot i\cdot l}{2\cdot \pi}\cdot\ln\frac{D+d/2}{D-d/2}\\ +u&=-\frac{\mu_0\cdot 15\cdot \power{10}{3}\,\ampere\cdot 4\,\metre}{0{,}6\cdot \power{10}{-6}\,\second\cdot 2\cdot \pi}\cdot \underbrace{\ln\frac{3{,}01\,\metre}{2{,}99\,\metre}}_{6{,}66\cdot \power{10}{-3}}=-133{,}7\,\volt\\ +\text{\uline{Lentzsche Regel:}} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$}; + \draw [<-,blue] (.2,.35)--(.8,.35)node at(.5,.35)[above]{\footnotesize$u$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.25cm,yshift=-.25cm]% Leiterschleife + \draw (0,0)--(0,-2)--(.5,-2)--(.5,0); + \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25); + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.05) node [above right] {\footnotesize$+$}; + \fill (1,0)circle(.05) node [above left] {\footnotesize$-$}; + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=-1cm]% O. + \draw (.5,0)circle(.133)node at (1,0)[right]{$\frac{d\Phi}{dt}$}; + \fill (.5,0)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,green!50!black,xshift=0cm,yshift=-1.33cm]% Ox + \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$}; + \draw [black]node at (1.5,-1)[below]{Wirkung $\frac{d\Phi_{ind}}{dt}$ entgegen Ursache $\frac{d\Phi}{dt}$}; + \draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,red,xshift=3cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.05) node [above right] {\footnotesize$+$}; + \fill (1,0)circle(.05) node [above left] {\footnotesize$-$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=-0.25cm]% Leiterschleife + \draw (0,0)--(0,-2)--(.5,-2)--(.5,0); + \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=-2.25cm]%Stromquelle + \draw [green!50!black](.25,0)--(.75,0); + \draw node at (.5,.133) [above] {$i_{}$}; + \draw [green!50!black](.5,0)circle(.133); + \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$i_{ind}$}; + \end{scope} + \begin{scope}[>=latex,very thick,green!50!black,xshift=3cm,yshift=-1.33cm]% Ox + \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$}; +\draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133); +\end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1.25cm]% + \draw [->,red] (-2.5,.5)--(-2.5,-.5) node at(-2.5,0)[right]{$\frac{di}{dt}$}; + \draw [<-,green!50!black] (.5,.5)--(.5,-.5) node at(.5,0)[left]{$i_{ind}$}; + \draw [<-,red!75!black] (-1.75,.5)--(-1.75,-.5) node at(-1.75,0)[left]{$i_{ind}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +$u$ hat negatives Vorzeichen, da Polung von $i_{ind}$ entgegen Bezugspfeilen aus Skizze\\ +($u_R = -u$) +\clearpage +}{}% diff --git a/ET2_L_B12_A2.tex b/ET2_L_B12_A2.tex new file mode 100644 index 0000000..d84ce20 --- /dev/null +++ b/ET2_L_B12_A2.tex @@ -0,0 +1,123 @@ +\section{Drahtschleife} +Eine Drahtschleife $N=1$ wird von einem zeitlich veränderlichen Fluss durchsetzt.\\ +$\Phi(t)=\Phi_0\cdot (1-e^{-t/T})$ mit $\Phi_0=60\cdot \power{10}{-6 +}\,\volt\second$ und $T=1\,\milli\second$. +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie die Spannung $u$ für $t=0{,}5\cdot T$! +\item Geben Sie die Polarität der Anschlussklemmen der Drahtschleife a-b für diesen Zeitpunkt an und begründen Sie ihre Angabe! +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,red,ultra thick,xshift=0,yshift=0] + \draw (0,0) -- (0,1); + \draw [dashed] (0,1) -- (0,2); + \end{scope} + \begin{scope}[>=latex,blue,very thick,xshift=0,yshift=2cm] + \draw (0,0) ellipse (2cm and 1cm); + \end{scope} + \begin{scope}[>=latex,blue,very thick,xshift=1.9cm,yshift=1.75cm] + \filldraw [black!0!] (0,0.02) rectangle (0.2,0.48); + \draw [fill] (1pt,0) -- (1,0) circle (2pt) node [right] {b}; + \draw [fill] (1pt,0.5cm) -- (1,0.5) circle (2pt) node [right] {a}; + \draw node at (1.5,0.25)[right] {$u$}; + \end{scope} + \begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] + \draw [->] (0,2) -- (0,4) node [above] {$\Phi(t)$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +u&=-N\cdot \frac{d\Phi}{dt} +\end{align} +\begin{align*} +\intertext{Berechnung:} +\intertext{a) Spannung $u$ für $t=0{,}5\cdot T$} +u&=-N\cdot \frac{d\Phi}{dt} \qquad \text{mit $N=1$ }\\ +&=-\frac{d\left(\Phi_0\cdot \left(1-e^{-\frac{t}{T}}\right)\right)}{dt}\\ +&=-\Phi_0\cdot \frac{d}{dt} \left(1-e^{-\frac{t}{T}}\right)\\ +&=-\Phi_0\cdot \Bigg[-e^{-\frac{t}{T}}\cdot \hspace{-.7cm}\underbrace{ \left(-\frac{1}{T}\right)}_{\mathrm{Nachdifferenzieren}}\hspace{-.5cm}\Bigg]\\ +&\text{mit $t=0{,}5\cdot T\qquad T=1\,\milli\second$}\\ +u&=-60\cdot \power{10}{-6}\,\volt\second\cdot \left[-e^{-0{,}5}\cdot \left(-\frac{1}{1\,\milli\second}\right)\right]=\uuline{-36{,}39\,\milli\volt} +\end{align*} +\begin{align*} +\intertext{b) Polarität der Spannung $u$ für $t=0{,}5\cdot T$} +\frac{d\Phi}{dt}&>0 \Rightarrow \text{Linke-Hand für Stromrichtung}\\ +\end{align*} +\begin{align*} +\begin{tikzpicture}[scale=1] +\foreach \xs in {0} { % Enter Start value of x label +\foreach \ys in {0} { % Enter Start value of y label +\foreach \ii in {10} { % Enter Number of Decades in x +\foreach \jj in {6} { % Enter Number of Decades in y +\foreach \xe in {5} { % Enter End value of x label +\foreach \ye in {60} { % Enter End value of y label + \foreach \i in {1,2,...,\ii} { + \foreach \j in {1,2,...,\jj} { +}}% End Log Grid +\draw[black!80!] (0,0) grid (\ii,\jj); % Draw Linear grid +\draw [<->,thick] (0,\jj+.2) node (yaxis) [above] {$\Phi\,[\micro\volt\second]$} |- (\ii+.2,0) node (xaxis) [right] {$t\,[\milli\second]$}; % Draw axes +\draw [thick] (-1.25,\jj+.2) node (yaxis) [above] {$u\,[\milli\volt]$} ; % Draw axes +\foreach \x in {\xs,.5,...,\xe}% x Axis Label: +\node [black,anchor=north] at(\x*2-\xs,0){$\x$}; +\foreach \y in {\ys,10,...,\ye}% y Axis Label: +\node [red,anchor=east] at(0,\y/10-\ys){$\y$}; +\foreach \y in {\ys,10,...,\ye}% y Axis Label: +\node [blue,anchor=east] at(-1,\y/10-\ys){$-\y$}; +}}}}}} +\draw [black,thick,dashed] (0,0) -- (2,6) -- (2,0.3); +\draw [blue,thick,dashed] (1,3.64) -- (-.75,3.64) node [left]{$-36{,}39\,\milli\volt$}; +\draw[color=red, very thick,domain=0:10] plot[id=tau] function{6*(1-exp(-x/2))}; +\draw[color=blue, very thick,domain=0:10] plot[id=tau] function{6*(exp(-x/2))}; +\draw [black] node at (2,0)[above] {$T$}; +\draw [red] node at (2,3.5) [right]{$\Phi=\Phi_0\cdot \left(1-e^{-t/T}\right)$}; +\draw [blue] node at (2,2.5) [right]{$u=-\Phi_0\cdot \left(\frac{e^{-t/T}}{T}\right)$}; +\draw node at (0,-1) [right]{Induzierte Spannung bei zunehmendem Fluss}; +\end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{\uline{Lenzsche Regel:}} +\text{Klemme a $\Rightarrow$ $+$}\\ +\text{Klemme b $\Rightarrow$ $-$}\\ +%F_{\sqcup\sqcap}\\ %-------------------Versuch mit Symbolen-------------- +%\ding{43}\\ +%\oplus\ominus\odot\\ +%\circlearrowright\\ +%\lightning\\ +%\curvearrowright\\ +%\leftthumbsup\\ +%\SmallSquare\\ +%\SmallTriangleUp\\ %-------------------Versuch mit Symbolen-------------- +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,red,ultra thick,xshift=0,yshift=0] + \draw (0,0) -- (0,1); + \draw [dashed] (0,1) -- (0,2); + \end{scope} + \begin{scope}[>=latex,blue,very thick,xshift=0,yshift=2cm] + \draw (0,0) ellipse (2cm and 1cm); + \end{scope} + \begin{scope}[>=latex,blue,very thick,xshift=1.9cm,yshift=1.75cm] + \filldraw [black!0!] (0,0.02) rectangle (0.2,0.48); + \draw [fill] (1pt,0) -- (1,0) circle (2pt) node [below] {b $-$}; + \draw [fill] (1pt,0.5cm) -- (1,0.5) circle (2pt) node [above] {a $+$}; + \draw node at (1.5,0.25)[right] {$u$}; + \end{scope} + \begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] + \draw [->] (0,2) -- (0,4) node [above] {$\Phi(t)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=5cm,yshift=1.5cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$U_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=2cm] + \draw [dashed] (0,.25)--(2,.5) (0,-.25)--(2,-.5); + \draw [->,red] (0,.25) +(7.2:.75) -- +(7.2:1.25) node [above left]{$i$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B12_A3.tex b/ET2_L_B12_A3.tex new file mode 100644 index 0000000..2c50ad5 --- /dev/null +++ b/ET2_L_B12_A3.tex @@ -0,0 +1,125 @@ +\section{Metallstab} +Ein Metallstab $M$ rotiert um die Achse $A$ mit der Winkelgeschwindigkeit $\omega$.\\ +Der Metallstab schleift auf dem Metallring $S$. Senkrecht zu dem Metallstab und Metallring wirkt eine homogene magnetische Flussdichte $\vec{B}=B_z\cdot \vec{e}_z$\\ +Zwischen Schleifring und Achse wird eine Gleichspannung $U_{SA}$ gemessen.\\[\baselineskip] +Berechnen Sie die Flussdichte $B_z$.\\[\baselineskip] +$R=2\,\centi\metre$; $\omega =100\cdot \pi\cdot\frac{1}{\second}$ ; $U_{SA}=25\,\milli\volt$.\\ +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,thick,xshift=0,yshift=0] + \draw [->] (0,0) -- (0,5) node [above] {$y$}; + \draw [->] (0,0) -- (5,0) node [right] {$x$}; + \draw [draw=black,fill=white,very thick](0,0) circle (0.1) node at (0,-.25) [below]{$z$}; + \fill [fill=black] (0,0) circle (0.04); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=2.5cm] + \draw [red,ultra thick](-2,-2) -- (2,2) node [below right]{M}; + \draw [->](0,0) -- (0,-2) node at (0,-1)[right]{$R$}; + \draw [->](-1.414,1.414) -- (0,0)node at (-0.707,0.707) [right]{$U_{SA}$}; + \draw [blue](0,0) circle (2)node at (0,2)[above] {S}; + \draw [->,red](0,0) -- (45:1.414) arc (45:30:1.5cm) node [below] {$\omega$}; + \fill [blue](0,0) circle (0.1) node [below right] {A}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.5cm,yshift=2.5cm] + \draw [red] (0,0) circle (0.1) node [below]{$\vec{B}$}; + \fill [red] (0,0) circle (0.04); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +u_{ind}&=-N\cdot \Bigg(\underbrace{\frac{dB(t)}{dt}\cdot A(t)}_{\mathrm{Ruheinduktion}}+\underbrace {\frac{B(t)}{dt}\cdot dA(t)}_{\mathrm{Bewegungsinduktion}}\Bigg) +\end{align} +\begin{align*} +\intertext{Berechnung:} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,thick,xshift=0,yshift=0] + \draw [->] (0,0) -- (0,5) node [above] {$y$}; + \draw [->] (0,0) -- (5,0) node [right] {$x$}; + \draw [draw=black,fill=white,very thick](0,0) circle (0.1) node at (0,-.25) [below]{$z$}; + \fill [fill=black] (0,0) circle (0.04); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=2.5cm] + \fill [black!15!](0,0) -- (45:2) arc (45:30:2cm) -- (0,0); + \draw node at (30:2)[above right]{$dt$}; + \draw [red,ultra thick](-2,-2) -- (2,2) node [below right]{M}; + \draw [->](0,0) -- (0,-2) node at (0,-1)[right]{$R$}; + \draw [->](-1.414,1.414) -- (-0.05,0.05)node at (-0.707,0.707) [right]{$U_{SA}$}; + \draw [blue](0,0) circle (2)node at (0,2)[above] {S}; + \draw [->,red](0,0) -- (45:1.414) arc (45:30:1.5cm) node [below] {$\omega$}; + \fill [blue](0,0) circle (0.1) node [below right] {A}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.5cm,yshift=2.5cm] + \draw [red] (0,0) circle (0.1) node [below]{$\vec{B}$}; + \fill [red] (0,0) circle (0.04); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=2.5cm]% Voltmeter + \draw [red!50!blue](-.5,0)--(.25,0) (.75,0)--(1.5,0) node at (.5,.25) [above] {$V$}; + \draw [red!50!blue](.5,0)circle(.25); + \draw [->,red!50!blue](.25,-.5)--(.75,.5)node at (.2,0) [above] {$+$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Bewegungsinduktion! $N=1$, homogenes zeitlich unverändertes Feld $B$ $\bot$ zu $\omega$} +u_{ind}&=-\frac{d\Phi}{dt}=-B\cdot \frac{dA(t)}{dt}\\ +\intertext{Der Metallstab überstreicht im Zeitintervall $dt$ den vom Fluß $\Phi$ durchsetzten Kreissektor mit der Fläche} +dA(t)&=\underbrace{R^2\cdot \pi}_{\text{Kreisfläche}}\cdot \underbrace{\frac{\omega\cdot dt}{2\pi}}_{\text{Segment}}\\ +dA(t)&=\frac{1}{2}\cdot R^2\cdot \omega\cdot dt\\ +\frac{dA(t)}{dt}&=\frac{1}{2}\cdot R^2\cdot \omega\\ +\frac{d\Phi}{dt}&=B\cdot \frac{1}{2}\cdot R^2\cdot \omega \\ +u_{ind}&=-\frac{d\Phi}{dt}=-B\cdot \frac{1}{2}\cdot R^2\cdot \omega\\ +|u_{ind}|&=|B_z|\cdot \frac{1}{2}\cdot R^2\cdot \omega\\ +|B_z|&=\frac{2\cdot U_{SA}}{R^2\cdot \omega}=\frac{2\cdot 25\,\milli\volt}{(2\,\centi\metre)^2\cdot 100\cdot \pi\cdot \frac{1}{\second}}=\uuline{0{,}398\,\frac{\volt\second}{\square\metre}} +\end{align*} +\clearpage +\textbf{Alternativ:}\\ +Bewegungsinduktion mit $v(r)=\omega\cdot r$ (radiusabhängig) +\begin{align*} +% \intertext{Bewegungsinduktion mit $v(r)=\omega\cdot r$ (radiusabhängig)} +|u_{ind}|&=\int{(\vec{v} \times \vec{B})\cdot \vec{dl}}=\int{v\cdot |B_z|\cdot dr}=\omega\cdot |B_z|\int_{r=0}^{R}{r\cdot dr}=\omega\cdot |B_z|\left[\frac{r^2}{2}\right]_{0}^{R}=\frac{1}{2}\cdot \omega\cdot |B_z|\cdot R^2\\ +|B_z|&=\uuline{0{,}398\,\frac{\volt\second}{\square\metre}} +% \intertext{Richtung der Lorenzkraft $\vec{F}_L$ wirkt so, daß positive Ladungsträger $q$ zum Zentrum $(A)$ gedrückt werden (entspricht der technischen Stromrichtung $I$).} +\end{align*} +Richtung der Lorenzkraft $\vec{F}_L$ wirkt so, daß positive Ladungsträger $q$ zum Zentrum $(A)$ gedrückt werden (entspricht der technischen Stromrichtung $I$).\\ + +$\vec{F}_L=q(\vec{v}\times \vec{B});\qquad (F_L=q\cdot v\cdot B\text{, wenn }v\bot B)$\\ + +Rechte Hand Regel:\\ +Der \textbf{Daumen} zeigt in Richtung der Ursache:\\ +a.) Bewegter Leiter im Magnetfeld: Die Relativbewegung $\vec{v}$ des Leiters im Magnetfeld\\ +b.) Strom durch Leiter im Magnetfeld: Die technische Stromrichtung $I$ bzw. Bewegungsrichtung der positiven Ladung $q$\\ +Der \textbf{Zeigefinger} zeigt senkrecht zum Daumen in Richtung der magnetischen Feldlinien, also der Vermittlung (auch Verknüpfung), also dem Magnetfeld $\vec{B}$\\ +Der \textbf{Mittelfinger} zeigt senkrecht zu Daumen und Zeigefinger in Richtung der Wirkung, der Lorentzkraft $\vec{F}_L$ \\ + +a.) Bewegter Leiter im Magnetfeld +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex, xshift=0, yshift=0] + \draw [->](0,0)--(1,0,0) node [right]{$\vec{v}$\,\text{(Daumen)}}; + \draw [->](0,0)--(0,1,0) node [above]{$\vec{B}$\,\text{(Zeigefinger)}}; + \draw [->](0,0)--(0,0,1.41) node [below]{$\vec{F}_L$ \,\text{(Mittelfinger) Kraft auf pos. Ladung} $q\Rightarrow I$}; +% \draw node at (0,2.25)[left]{Rechte Hand Regel}; + \end{scope} + \end{tikzpicture}\\ +% \hspace{1cm} +\end{align*} +\begin{align*} +&\text{Da $U_{SA}$ positiv, muß $B_z$ negativ sein. } +\Rightarrow B_z=\uuline{-0{,}398\,\frac{\volt\second}{\square\metre}}\\ +\end{align*} + +b.) Strom durch Leiter im Magnetfeld: +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex, xshift=0, yshift=1cm] + \draw [->](0,0)--(1,0,0) node [right]{$\vec{F}_L$ \,\text{(Mittelfinger)}}; + \draw [->](0,0)--(0,-1,0) node [below]{$\vec{B}$ \,\text{(Zeigefinger)}}; + \draw [->](0,0)--(0,0,1.41) node [left]{$i=\frac{dq}{dt}$\,\text{(Daumen)}}; %\ \sim\ \vec{v} + \end{scope} + \end{tikzpicture} +\end{align*} + +\clearpage +}{}% diff --git a/ET2_L_B12_A4.tex b/ET2_L_B12_A4.tex new file mode 100644 index 0000000..1ebef20 --- /dev/null +++ b/ET2_L_B12_A4.tex @@ -0,0 +1,50 @@ +\section{Spannungsverlauf} +Gegeben ist die dargestellte Spannung: +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,thick] + \draw [->](0,0) -- (6,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,-1.25) -- (0,1.25) node [above] {$u\,[\volt]$}; + \draw [red,very thick](0,0)--(1,1)--(1,0)--(1.5,0)--(1.5,-1) + --(2.5,-1)--(2.5,0)--(3.5,1)--(3.5,0)--(4,0)--(4,-1)--(5,-1)--(5,0)--(6,1); + \foreach \x in {10,20,...,50} + \draw (\x/10,0) -- (\x/10,-0.2) node[anchor=north] {$\x$}; + \foreach \y in {-10,0,10} + \draw (0,\y/10) -- (-0.2,\y/10) node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Ermitteln Sie die Frequenz der Grundschwingung! +\item Berechnen Sie den Gleichrichtwert der Spannung! +\item Berechnen Sie den Effektivwert der Spannung! +\item Berechnen Sie den Formfaktor der Spannung! +\item Nun wird die dargestellte Spannung an einen Ohmschen Widerstand von $100\,\ohm$ angelegt. Welche Verlustleistung tritt im Widerstand auf?\\ +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +\overline{|u|}&=\frac{1}{T}\cdot \int_{t=0}^{T}{|u(t)|\cdot dt}&\text{Gleichrichtwert}\\ +U&=U_{\textrm{eff}}=\sqrt{\frac{1}{T}\cdot \int_{t=0}^{T}{u^2(t)\cdot dt}}&\text{Effektivwert}&\\ +F&=\frac{U}{\overline{|u|}}=\frac{\text{Effektivwert}}{\text{Gleichrichtwert}} +\end{align} +\begin{align*} +\intertext{Berechnung:} +\intertext{a) Grundschwingung mit $T=25\,\milli\second$:} +f&=\frac{1}{T}=40\,\hertz\\ +\intertext{b) Gleichrichtwert der Spannung:} +\overline{|u|}&=\frac{1}{T}\cdot (F_{\triangle} +F_{\sqcup\hspace{-.2cm}\sqcap})=\frac{1}{25\,\milli\second}\cdot (\frac{1}{2}\cdot 10\,\volt\cdot 10\,\milli\second+10\,\volt\cdot 10\,\milli\second)=\frac{150\,\volt\cdot \milli\second}{25\milli\second}=\uuline{6\,\volt}\\ +\intertext{c) Effektivwert der Spannung:} +U&=\sqrt{\frac{1}{T}\int_{0}^{T}{u^2}\cdot dt}\\ +U^2&=\frac{1}{T}\left(\int_{0}^{10\,\milli\second}{\left(\frac{10\,\volt}{\power{10}{-2}\,\second}\cdot t\right)^2\cdot dt}+\int_{15\,\milli\second}^{25\,\milli\second}{(-10\,\volt)^2\cdot dt}\right)\\ +&=\frac{1}{25\,\milli\second}\left(\frac{100\,\square\volt}{\power{10}{-4}\,\square\second}\cdot \left[\frac{t^3}{3}\right]_{0}^{10\,\milli\second}+100\,\square\volt\cdot \big[t\big]_{15\,\milli\second}^{25\,\milli\second}\right)\\ +&=\frac{100\,\square\volt}{25\,\milli\second}\left(\power{10}{4}\frac{1}{\,\square\second}\cdot \frac{1}{3}\cdot \power{10}{-6}\,\cubic\second+10\,\milli\second\right)=53{,}33\,\square\volt\\ +U&=\sqrt{53{,}33}\,\volt=\uuline{7{,}30\,\volt} +\intertext{d) Formfaktor der Spannung:} +F&=\frac{U}{\overline{|u|}}=\frac{7{,}30\,\volt}{6\,\volt}=\uuline{1{,}22} +\intertext{e) Verlustleistung im Widerstand:} +P&=\frac{U^2}{R}=\frac{53{,}33\,\square\volt}{100\,\ohm}=\uuline{0{,}533\,\watt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B12_A5.tex b/ET2_L_B12_A5.tex new file mode 100644 index 0000000..acd17a7 --- /dev/null +++ b/ET2_L_B12_A5.tex @@ -0,0 +1,70 @@ +\section{Phasenanschnitt} +Berechnen Sie den Effektivwert dieser sinusförmigen Spannung mit Phasenanschnitt.\\ +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,thick] + \draw [ultra thin] (0,-2)grid(6,2); + \draw [->](0,0) -- (6.5,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,-2.25) -- (0,2.25) node [above] {$u\,[\volt]$}; + \draw[color=red,very thick,domain=0.6:2,smooth,samples=100](0,0)--(0.6,0)--(0.6,1.618) + plot[id=sina]function{2*sin(.5*3.14*x)}; + \draw[color=red,very thick,domain=0.6:2,smooth,samples=100]plot[id=sinb]function{2*sin(0.5*3.14*x)}; + \draw[color=red,very thick,domain=2.6:4,smooth,samples=100] (2,0)--(2.6,0)-- (2.6,-1.618) plot[id=sinc] function{2*(sin(0.5*3.14*x))}; + \draw[color=red,very thick,domain=4.6:6,smooth,samples=100] (4,0)--(4.6,0)--(4.6,1.618)plot[id=sind]function{2*sin(0.5*3.14*x)}; + \foreach \x in {10,20,30} + \draw(\x/5,0)--(\x/5,-0.2) node[anchor=north]{$\x$}; + \draw (3/5,0)--(3/5,-0.2)node[anchor=north]{$3$}; + \foreach \y in {-400,-200,...,400} + \draw (0,\y/200)--(-0.2,\y/200)node[anchor=east]{$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +U&=U_{\textrm{eff}}=\sqrt{\frac{1}{T}\cdot \int_{t=0}^{T}{u^2(t)\cdot dt}}&\text{Effektivwert}&\\ +\sin^2\alpha&=\frac{1}{2}(1-\cos 2\alpha) +\end{align} +\begin{align*} +\intertext{Berechnung:} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,thick] + \draw [ultra thin] (0,-2)grid(6,2); + \draw [->](0,0) -- (6.5,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,-2.25) -- (0,2.25) node [above] {$u\,[\volt]$}; + \draw[color=red,very thick,domain=0.6:2,smooth,samples=100] (0,0)--(0.6,0)--(0.6,1.618) plot[id=sin12a53] function{2*sin(0.5*3.14*x)}; + \draw[color=red,very thick,domain=2.6:4,smooth,samples=100] (2,0)--(2.6,0)--(2.6,-1.618) plot[id=sin12a54] function{2*(sin(0.5*3.14*x))}; + \draw[color=red,very thick,domain=4.6:6,smooth,samples=100] (4,0)--(4.6,0)--(4.6,1.618) plot[id=sin12a55] function{2*sin(0.5*3.14*x)}; + \draw[color=blue,very thick,dashed, domain=0:4,smooth,samples=100] plot[id=sin12a56] function{2*sin(0.5*3.14*x)}; + \foreach \x in {10,20,30} + \draw (\x/5,0) -- (\x/5,-0.2) node[anchor=north] {$\x$}; + \draw (3/5,0) -- (3/5,-0.2)node[anchor=north] {$3$}; + \foreach \y in {-400,-200,0,200,400} + \draw (0,\y/200) -- (-0.2,\y/200) node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Periodendauer $T=20\,\milli\second$, da Symmetrie in einer Periode. Betrachtung nur einer Sinus-Halbwelle mit $\frac{1}{2}\cdot T=10\,\milli\second$} +\omega&=2\pi f=\frac{2\pi}{T}=\frac{2\pi}{20\,\milli\second}=314\,\frac{1}{\second}\\ +u(t)&= +\begin{cases} +0&\text{ für }t=0\ldots 3\,\milli\second\\ +400\,\volt\cdot \sin(\omega t)=400\,\volt\cdot \sin(314\,\frac{1}{\second}\cdot t) &\text{ für }t=3\,\milli\second\ldots 10\,\milli\second\\ +\end{cases}\\[\baselineskip] +U^2&=\frac{1}{T/2}\int_{3\,\milli\second}^{10\,\milli\second}{(400\,\volt\cdot \sin(\omega t))^2\cdot dt}\\ +&=\frac{1}{T/2}\cdot (400\,\volt)^2\int_{3\,\milli\second}^{10\,\milli\second}{\sin^2(\omega t)\cdot dt}\\[\baselineskip] +&\text{mit }\sin^2\alpha=\frac{1}{2}(1-\cos 2\alpha)\\[\baselineskip] +U^2&=\frac{(400\,\volt)^2}{10\,\milli\second}\cdot\frac{1}{2}\cdot \left(\int_{3\,\milli\second}^{10\,\milli\second}{1\, dt} -\int_{3\,\milli\second}^{10\,\milli\second}{\cos(2 \omega t)\cdot dt}\right)\\ +&=\frac{(400\,\volt)^2}{10\,\milli\second}\cdot \frac{1}{2}\cdot \left(\Big[t\Big]_{3\,\milli\second}^{10\,\milli\second}-\left[\sin(2 \omega t)\cdot \frac{1}{2\omega} \right]_{3\,\milli\second}^{10\,\milli\second}\right)\\ +&=\frac{(400\,\volt)^2}{20\,\milli\second}\cdot \left(7\,\milli\second-\frac{1}{2\cdot \omega}\cdot \underbrace{\left(sin(\cancel{2}\cdot \frac{2\pi}{\cancel{20\,\milli\second}}\cdot \cancel{10\,\milli\second}\right)}_{sin(2\pi)=0}-sin\left(2\cdot \frac{\pi}{20\,\milli\second}\cdot 3\,\milli\second\right)\right)\\ +&=\frac{(400\,\volt)^2}{20\,\milli\second}\cdot \left(7\,\milli\second-\frac{20\,\milli\second}{4\pi}\cdot \underbrace{(-sin(0{,}6\pi))}_{-0{,}951}\right)\\ +&=\frac{(400\,\volt)^2}{20\,\milli\second}\cdot 8{,}51\,\milli\second=68109\,\square\volt\\[\baselineskip] +U&=400\,\volt\cdot \sqrt{\frac{8{,}51\,\milli\second}{20\,\milli\second}}=\uuline{261\,\volt}\\ +\intertext{Zum Vergleich: Sinus ohne Phasenanschnitt hätte einen Effektivwert von} +U_{(sin)}&=\frac{400\,\volt}{\sqrt{2}}=283\,\volt +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B12_A6.tex b/ET2_L_B12_A6.tex new file mode 100644 index 0000000..a573f40 --- /dev/null +++ b/ET2_L_B12_A6.tex @@ -0,0 +1,80 @@ +\section{Rechteckspannung} +Gegeben ist eine periodische Rechteckspannung mit der Periodendauer von $10\,\milli\second$.\\ +Berechnen Sie den Effektivwert, wenn der arithmetische Mittelwert gleich Null ist.\\ +\begin{align*} + \begin{tikzpicture}[scale=0.5] + \begin{scope}[>=latex,thick] + \draw [->](0,-2) -- (10.75,-2) node [right] {$t\,[\milli\second]$}; + \draw [->](0,0) -- (0,5.5) node [above] {$u\,[\volt]$}; + \draw [<->,blue, very thick] (-.5,0)--(-.5,5) node at (0,2.5)[right]{$5\,\volt$}; + \draw [red,very thick](-0.5,0)--(0,0)--(0,5)--(7,5)--(7,0)--(10,0) + --(10,5)--(10.5,5); + \foreach \x in {0,1,...,10} + \draw (\x,-2) -- (\x,-2.2) node[anchor=north] {$\x$}; + \foreach \y in {0,1,...,5} + \draw (0,\y) -- (-0.2,\y);% node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +\overline{u}&=\frac{1}{T}\cdot \int_{t=0}^{T}{u(t)\cdot dt}&\text{Arithmetischer Mittelwert}\\ +U&=U_{\textrm{eff}}=\sqrt{\frac{1}{T}\cdot \int_{t=0}^{T}{u^2(t)\cdot dt}}&\text{Effektivwert}& +\end{align} +\begin{align*} +\intertext{Berechnung:} +\intertext{a) Arithmetischer Mittelwert $\overline{u}=0$} +&\text{$\Rightarrow$ Fläche ober- und unterhalb der Nulllinie muß gleich sein!}\\ +&\text{$\Rightarrow$ Wo ist die Nulllinie?} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=0.5] + \begin{scope}[>=latex,thick] + \draw [->](0,0) -- (10.75,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,0) -- (0,5.5) node [above] {$u\,[\volt]$}; + \fill [black!15!] (0,3.5)rectangle(7,5) (7,3.5)rectangle(10,0); + \draw [red,very thick](-0.5,0)--(0,0)--(0,5)--(7,5)--(7,0)--(10,0) + --(10,5)--(10.5,5); + \draw [blue,very thick] (0,3.5)--(10,3.5) node [right]{NULL}; + \draw [->,blue,very thick] (3.5,3.5)--(3.5,5) node at(3.5,4.25)[right]{$u_1$}; + \draw [<-,blue,very thick] (8.5,0)--(8.5,3.5) node at(8.5,1.75)[right]{$u_2$}; + \foreach \x in {0,1,...,10} + \draw (\x,0) -- (\x,-0.2) node[anchor=north] {$\x$}; + \foreach \y in {-3.5,-2.5,...,1.5} + \draw (0,\y+3.5) -- (-0.2,\y+3.5) node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\enlargethispage{1cm} +\begin{align*} +&u_1+(-u_2)=5\,\volt\rightarrow u_2=-(5\,\volt-u_1)\\ +&\text{Fläche:}\\ +&\frac{1}{T}\int_{0}^{T}{u(t)\cdot dt}\stackrel{!}{=}0\\ +&\frac{1}{T}\cdot \left(\int_{0}^{7\,\milli\second}{u_1\cdot dt}-\int_{7\,\milli\second}^{10\,\milli\second}{u_2\cdot dt}\right)\stackrel{!}{=}0\\ +&u_1\cdot 7\,\milli\second-u_2\cdot 3\,\milli\second=0\\ +&u_1\cdot 7\,\milli\second-(5\,\volt-U_1)\cdot 3\,\milli\second=0\\ +&(7\,\milli\second+3\,\milli\second)\cdot u_1=15\,\volt\cdot \milli\second\\ +&u_1=\frac{15\,\volt\cdot \milli\second}{10\,\milli\second}=\uuline{1{,}5\,\volt}\\ +&u_2=-(5\,\volt-1{,}5\,\volt)=\uuline{-3{,}5\,\volt} +\intertext{Alternativ mit Beträgen} +&|u_1|+|u_2|=5\,\volt\rightarrow |u_2|=5\,\volt-|u_1|\\ +&\text{Fläche:}\\ +&\frac{1}{T}\int_{0}^{T}{u(t)\cdot dt}\stackrel{!}{=}0\\ +&\frac{1}{T}\cdot \left(\int_{0}^{7\,\milli\second}{u_1\cdot dt}-\int_{7\,\milli\second}^{10\,\milli\second}{u_2\cdot dt}\right)\stackrel{!}{=}0\\ +&u_1\cdot 7\,\milli\second-u_2\cdot 3\,\milli\second=0\\ +&u_1\cdot 7\,\milli\second-(5\,\volt-|u_1|)\cdot 3\,\milli\second=0\\ +&(7\,\milli\second+3\,\milli\second)\cdot |u_1|=15\,\volt\cdot \milli\second\\ +&|u_1|=\frac{15\,\volt\cdot \milli\second}{10\,\milli\second}=\uline{1{,}5\,\volt}\qquad +|u_2|=5\,\volt-1{,}5\,\volt)=\uline{3{,}5\,\volt}\\ +&\text{Da $u_1$ positives Vorzeichen in der Skizze hat, muss $u_2$ ein negatives Vorzeichen erhalten.}\\ +\Rightarrow& u_1=\uuline{1{,}5\,\volt} \qquad u_2=\uuline{-3{,}5\,\volt} +\intertext{b) Effektivwert} +U^2&=U^2_{\textrm{eff}}=\frac{1}{T}\int{\left(u(t)\right)^2\cdot dt}\\ +&=\frac{1}{10\,\milli\second}\left(\int_{0}^{7\,\milli\second}{(1{,}5\,\volt)^2\cdot dt}+\int_{7\,\milli\second}^{10\,\milli\second}{(-3{,}5\,\volt)^2\cdot dt}\right)\\ +&=\frac{1}{10\,\milli\second}\cdot \left(2{,}25\,\volt^2\cdot \big[t\big]_{0}^{7\,\milli\second}+12{,}25\,\volt^2\cdot \big[t\big]_{7\,\milli\second}^{10\,\milli\second}\right)\\ +&=\frac{1}{10\,\milli\second}\cdot \left(2{,}25\,\volt^2\cdot 7\,\milli\second+12{,}25\,\volt^2\cdot (10\,\milli\second-7\,\milli\second)\right)=5{,}25\,\volt^2\\ +U&=\sqrt{5{,}25\,\volt^2}=\uuline{2{,}29\,\volt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B13_A1.tex b/ET2_L_B13_A1.tex new file mode 100644 index 0000000..7cdfd7a --- /dev/null +++ b/ET2_L_B13_A1.tex @@ -0,0 +1,106 @@ +\section{Scheinersatzwiderstände} +Der Eingangswiderstand eines linearen Zweipols beträgt bei der Frequenz $f=800\,\hertz$\\ +$Z=600\,\ohm$, sein Phasenwinkel ist $\varphi=30\,\degree$ induktiv. +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie die Schaltungselemente $R_r$ und $L_r$ der gleichwertigen Reihenersatzschaltung! +\item Berechnen Sie die Schaltungselemente $R_p$ und $L_p$ der gleichwertigen Parallelersatzschaltung! +\item Wie ändern sich die Scheinersatzwiderstände (Betrag und Phase) beider Ersatzschaltungen, wenn die Frequenz $f'= 600\,\hertz$ beträgt? +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +\underline{Z}& &\text{Scheinwiderstand (Impedanz)}\\ +Z&=|\underline{Z}| &\text{Betrag des Scheinwiderstandes}\\ +X&=\omega\cdot L &\text{Blindwiderstand (Reaktdanz)}\\ +B&=-\frac{1}{\omega\cdot L} &\text{Blindleitwert (Suszepdanz)} +\end{align} +\begin{align*} +\intertext{Berechnung:} +\intertext{a) Widerstandsebene:} +R_r&=Z\cdot \cos(\varphi_r)=600\,\ohm\cdot \cos(30\degree)=\uuline{520\,\ohm}\\ +X_r&=Z\cdot \sin(\varphi_r)=600\,\ohm\cdot \sin(30\degree)=300\,\ohm\\ +L_r&=\frac{X_r}{\omega}=\frac{X_r}{2\pi f}=\frac{300\,\ohm}{2\pi\cdot 800\,\frac{1}{\second}}=\uuline{60\,\milli\henry} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_r$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R_r}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_r$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{L_r}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.025) node [above left] {\footnotesize$1$}; + \fill (2,0)circle(.025) node [above right] {\footnotesize$2$}; + \end{scope} +\begin{scope}[>=latex,very thick, xshift=3cm, yshift=-.25cm] + \draw [->,thin](0,0)--(1,0)node[right]{$R$}; + \draw [->](0,0)--(.52,0)node at (.26,0)[below]{$R_r$}; + \draw [->,thin](0,0)--(0,.5)node[above]{$X$}; + \draw [->](.52,0)--(.52,.3)node at (.52,.15)[right]{$X_r$}; + \draw [->](0:0)--(30:.6)node at (30:.3)[above left]{$Z$}; + \draw [->,red,thin] (0:.26) arc (0:30:.26cm) node at (15:.26) [right] {$\varphi_r$}; +\end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{b) Leitwertebene:} +Y&=\frac{1}{Z}=\frac{1}{600\,\ohm}=1{,}667\,\milli\siemens \text{; }\qquad\varphi_p=-\varphi_r=-30\degree\\ +G_p&=Y\cdot \cos(\varphi_p)=1{,}667\,\milli\siemens\cdot \cos(-30\degree) +=1{,}443\,\milli\siemens\Rightarrow R_p=\frac{1}{G_p}=\uuline{693\,\ohm}\\ +B_p&=Y\cdot \sin(\varphi_p)=1{,}667\,\milli\siemens\cdot \sin(-30\degree) +=-0{,}833\,\milli\siemens\Rightarrow X_p=-\frac{1}{B_p}=1200\,\ohm\\ +%&\text{mit }B_p=-\frac{1}{\omega\cdot L_p}\Rightarrow \\ +L_p&=-\frac{1}{\omega\cdot B_p}=-\frac{1}{2\pi f\cdot B_p}=\frac{-1}{2\pi\cdot 800\,\frac{1}{\second}\cdot (-0{,}833\cdot \power{10}{-3}\,\frac{1}{\ohm})}=\uuline{239\,\milli\henry} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$G_p$}; + \draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{G_p}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$B_p$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{B_p}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%End Knoten + \draw (2,.5)--(2,0)--(0,0); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-1cm,yshift=1cm]%End Knoten + \draw (2,-.5)--(2,0)--(0,0); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%Knotenpunkte + \fill (0,1)circle(.025) node [above left] {\footnotesize$1$}; + \fill (0,0)circle(.025) node [above left] {\footnotesize$2$}; + \end{scope} +\begin{scope}[>=latex,very thick, xshift=2cm, yshift=.5cm] + \draw [->,thin](0,0)--(1,0)node[right]{$G$}; + \draw [->](0,0)--(.52,0)node at (.26,0)[above]{$G_p$}; + \draw [->,thin](0,-.5)--(0,.5)node[above]{$jB$}; + \draw [->](.52,0)--(.52,-.3)node at (.52,-.15)[right]{$B_p$}; + \draw [->](0:0)--(-30:.6)node at (-30:.3)[below left]{$Y$}; + \draw [->,red,thin] (0:.26) arc (0:-30:.26cm) node at (-15:.26) [right] {$\varphi_p$}; +\end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{c) Frequenz $f'$ \newline Reihenschaltung:} +R'_r&\stackrel{!}{=} R_r=520\,\ohm\\ +X'_r&=\omega'\cdot L_r=2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}06\,\ohm\second=226\,\ohm\\ +Z'_r&=\sqrt{R'^2_r+X'^2_r}=\sqrt{520^2+226^2}\,\ohm=\uuline{567\,\ohm}\\ +%\varphi'&=\arctan\frac{\Im\mathfrak m}{\Re\mathfrak e}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree} +\varphi'_r&=\arctan\frac{\Im}{\Re}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree} +\intertext{Parallelschaltung:} +G'_p&\stackrel{!}{=} G_p=1{,}443\,\milli\siemens\\ +B'_p&=\frac{-1}{\omega'\cdot L_p}=\frac{-1}{2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}239\,\ohm\second}=\frac{-1}{901\,\ohm}=-1{,}11\,\milli\siemens\\ +Y'_p&=\sqrt{G'^2_r+B'^2_r}=\sqrt{1{,}443^2+(-1{,}11)^2}\,\milli\siemens=1{,}82\,\milli\siemens\\ +Z'_p&=\frac{1}{Y'_p}=\uuline{549\,\ohm}\\ +\varphi'_p&=\arctan\frac{-1{,}11\,\milli\siemens}{1{,}443\,\milli\siemens}=\uuline{-37{,}6\,\degree} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B13_A2.tex b/ET2_L_B13_A2.tex new file mode 100644 index 0000000..24591b1 --- /dev/null +++ b/ET2_L_B13_A2.tex @@ -0,0 +1,68 @@ +\section{Verbraucherleistung} +An einem Verbraucher liegt die Spannung $u(t)=310\,\volt \cdot \sin(\omega t+55\,\degree)$ an, er nimmt einen Strom von $i(t)=8,5\,\ampere\cdot \cos (\omega t)$ auf. +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie den zeitlichen Verlauf des Momentanwertes der Verbraucherleistung! +\item Berechnen Sie die Schein-, Wirk- und Blindleistung! +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Merksatz:\\ +Kondensat\textbf{o}r, Strom eilt v\textbf{o}r\\ +Induktivit\textbf{ä}t, Strom ist zu sp\textbf{ä}t\\[\baselineskip] +Berechnung:\\ +\begin{align*} +\begin{tikzpicture}[scale=2] +\begin{scope}[>=latex, xshift=0cm, yshift=0] +\foreach \ii in {5} { % Enter Number of Decades in x +\foreach \jj in {2} { % Enter Number of Decades in y + \foreach \i in {1,2,...,\ii} { + \foreach \j in {1,2,...,\jj} { +\draw[black!50!, step=0.5] (0,0) grid (\ii,\jj); % Draw Sub Linear grid +}}% End Log Grid +\draw[black!80!] (0,0) grid (\ii,\jj); % Draw Linear grid +\draw [->,blue,thick] (5,0)--(5,\jj+.25) node (yaxis) [above] {$u\,[\volt]$}; +\draw [->,red,thick] (0,0)--(0,\jj+.25) node (yaxis) [above] {$i\,[\ampere]$}; +\draw [->,thick] (0,0)--(\ii+.25,0) node (xaxis) [right] {$\omega t\,[\degree]$}; % Draw axes +\foreach \x in {-90,0,90,180,270,360}% x Axis Label: +\node [blue,anchor=north] at(\x/90+1,0){$\x$}; +\foreach \y in {0,10}% y Axis Label: +\node [red,anchor=east] at(0,\y/10+1){$\y$}; +\foreach \y in {0,500}% y Axis Label: +\node [blue,anchor=west] at(5,\y/500+1){$\y$}; +}} +\draw[very thick](1,0)--(1,2); +\draw[->,very thick](1,1)--(.389,1) node [below right]{ $-55\degree$}; +\draw[<-,very thick](2,1)--(2.389,1) node at (2,1.2)[above right]{$-35\degree$ $i$ vor $u \Rightarrow$ kapazitiv}; +\end{scope} +\begin{scope}[>=latex, xshift=0cm, yshift=1cm] +\draw[color=red,thick,domain=0:5,smooth,samples=100] plot[id=i] function{.85*cos(.5*3.14*x-1.57)}; +\draw[color=blue,thick,domain=0:5,smooth,samples=100] plot[id=u] function{.62*sin(.5*3.14*x+.96-1.57)}; +%\draw[->,blue, very thick] (1.5,1) -- (2.5,1); +\draw[red] node at (1.5,1.25) {{\footnotesize $i(t)=8,5\,\ampere\cdot \cos (\omega t)$}}; +\draw[blue] node at (3.5,1.25) {{\footnotesize $u(t)=310\,\volt \cdot \sin(\omega t+55\,\degree)$}}; +\end{scope} +\end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{a) Leistungsverlauf} +p(t)&=u(t)\cdot i(t) \qquad\text{Momentane Leistung}\\ +p(t)&=310\,\volt\cdot 8{,}5\,\ampere\cdot \sin x\cdot \cos y \\ +&\text{mit $x=\omega t+55\degree=\omega t+0{,}96\,\text{rad} \qquad y=\omega t$}\\ +&\text{und } \sin x\cdot \cos y=\frac{1}{2}[\sin(x-y)+\sin(x+y)]\Rightarrow\\ +p(t)&=\widehat{u}\cdot \widehat{i}\cdot \frac{1}{2}\cdot [\sin(x-y)+\sin(x+y)]\\ +&=310\,\volt\cdot 8{,}5\,\ampere\cdot \frac{1}{2}\cdot \big[\sin(\cancel{\omega t}+0{,}96-\cancel{\omega t})+\sin(\omega t+0{,}96+\omega t)\big]\\ +&=\underbrace{310\,\volt\cdot 8{,}5\,\ampere\cdot \frac{1}{2}\vphantom{\frac{1}{2}}}_{S=1318\,\volt\ampere}\cdot \big[\underbrace{\sin(0{,}96)\vphantom{\frac{1}{1}}}_{0{,}819}+\sin(2\omega t+0{,}96)\big]\\ +p(t)&=\uuline{1079\,\watt+1318\,\volt\ampere\cdot \sin(2\omega t+0{,}96)} +\intertext{b) $S$ Schein-, $P$ Wirk- und $Q$ Blindleistung} +\cos(\omega t)&=\sin(\omega t+90\degree)\\ +\varphi_i&=+90\degree\quad\varphi_u=+55\degree\\ +\varphi_u-\varphi_i&=+55\degree-90\degree=-35\degree\\ +S&=U\cdot I=\frac{\widehat{u}}{\sqrt{2}}\cdot \frac{\widehat{i}}{\sqrt{2}}=\frac{1}{2}\cdot \widehat{u}\cdot \widehat{i}=\frac{2635}{2}\,\volt\ampere=\uuline{1318\,\volt\ampere}\\ +P&=S\cdot \cos(-35\degree)=S\cdot 0{,}819=\uuline{1079\,\watt}\\ +Q&=S\cdot \sin(-35\degree)=S\cdot (-0{,}576)=\uuline{-756\,\mathrm{var}}\qquad\text{Lies: Volt-Ampere-reaktiv}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B13_A3.tex b/ET2_L_B13_A3.tex new file mode 100644 index 0000000..570d2a2 --- /dev/null +++ b/ET2_L_B13_A3.tex @@ -0,0 +1,74 @@ +\section{Blindleistungskompensation} +Die Daten der beiden Verbraucher am Einphasen-Wechselstromnetz sind:\\ +Heizwiderstand $R_H$: +Aufgenommene Leistung $P_H=1,5\,\kilo\watt$\\ +Motor $M$: +Aufgenommene Leistung $P_{auf}=2,5\,\kilo\watt$\\ +Leistungsfaktor $\cos\varphi=0,7$\\ +$U_N=230\,\volt$; $f=50\,\hertz$ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Wie groß ist der dem Netz entnommene Strom $I_N$? +\item Welche Phasenverschiebung hat der Strom $\uline{I}_N$ zu der Spannung $\uline{U}_N$? +\item Welche Kapazität muss ein Kondensator, parallel zu den Verbrauchern geschaltet, haben, damit der Blindstrom voll kompensiert wird? +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=.75] + \begin{scope}[>=latex,very thick] + \draw (0,0) circle (1) node [above]{$M$}; + \draw node at (0,0) [below]{$1 \approx$}; % F E H L E R + \draw (180:1)--(-2,0)--(-2,5)circle(.025) (0:1)--(2,0)--(2,5)circle(.025) ; + \draw (-1,2.5)--(-2,2.5)circle(.025) (1,2.5)--(2,2.5)circle(.025) (-1,2.2 )rectangle (1,2.8); + \draw (0,2.5)node{$R_H$}; + \draw [->,blue](-1.8,5)--(1.8,5) node at (0,5) [above]{$\underline{U}_N$;$f$}; + \draw [->,red](-1.8,4.8)--(-1.8,2.7) node at (-1.8,3.75) [right]{$\underline{I}_N$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +\begin{align*} +\intertext{Berechnung:} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,very thick] + \draw [->,red!50!blue](0:0)--(45.6:3.571)node [above left]{$\underline{S}_M$}; + \draw [->,red!50!blue](0:0)--(0:2.5)node at (1.25,0)[above]{$P_M$}; + \draw [->](0:0)--(32.52:4.744)node [right]{$\underline{S}_{gesamt}$}; + \draw [->](45.6:3.571)--+(0:1.5)node at (3.25,2.551)[above]{$P_H$}; + \draw [->,red!50!blue](2.5,0)--+(90:2.551)node at (2.5,1.25)[right]{$Q_M$}; + \draw [red!50!blue]node at (1.5,.75)[below]{Motor}; + \draw [->](2.5,0)--(4,0)node at (3.25,0)[above]{$P_H$}; + \draw [->,red!50!blue](4,0)--+(90:2.551)node at (4,1.25)[right]{$Q_M$}; + \end{scope} +\begin{scope}[>=latex, xshift=0, yshift=0] + \draw [black!50!] (0,0)grid(4,3); + \foreach \x in {0,1,...,4}% x Axis Label: + \node [anchor=north] at(\x,0){$\x$}; + \foreach \y in {0,1,...,3}% y Axis Label: + \node [anchor=east] at(0,\y){$\y$}; + \draw [->,thick](0,0)--(4.25,0) node [right]{$P\,[\kilo\watt]$}; + \draw [->,thick](0,0)--(0,3.25) node [above]{$Q\,[\kilo\var]$}; +\end{scope} + \end{tikzpicture}\\ +\centering \underline{S}_{gesamt}=\sqrt{(P_M+P_H)^2+Q_M^2} +\end{align*} +\begin{align*} +\intertext{a) Nennstrom} +\varphi&=\arccos(0{,}7)=45{,}6\degree\\ +S_M&=\frac{P_{auf}}{0{,}7}=\uline{3571\,\volt\ampere} &\text{Motor Scheinleistung}\\ +Q_M&=S_M\cdot \sin\varphi=3571\,\volt\ampere\cdot 0{,}7141=\uline{2551\,\var} &\text{Motor Blindleistung}\\ +P&=P_{auf}+P_H=\uline{4\,\kilo\watt} &\text{Gesamte Wirkleistung}\\ +S&=\sqrt{P^2+Q^2_M}=\uline{4744\,\volt\ampere} &\text{Gesamte Scheinleistung}\\ +I_N&=\frac{S}{U_N}=\frac{4744\,\volt\ampere}{230\,\volt}=\uuline{20{,}63\,\ampere} +\intertext{b) Phasenverschiebung} +\varphi_N&=\arccos\frac{P}{S}=\arccos\frac{4000\,\watt}{4744\,\volt\ampere}=\uuline{32,52\degree} +\intertext{c) Kompensation} +|Q_C|&=Q_M=2551\,\var=U^2_N\cdot \omega\cdot C\\ +C&=\frac{2551\,\var}{2\pi\cdot 50\,\frac{1}{\second}\cdot (230\,\volt)^2}=\uuline{153{,}6\,\micro\farad} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B13_A4.tex b/ET2_L_B13_A4.tex new file mode 100644 index 0000000..81e62e0 --- /dev/null +++ b/ET2_L_B13_A4.tex @@ -0,0 +1,133 @@ +\section{Energieübertragung} +Die Skizze zeigt ein System zur elektrischen Energieübertragung bestehend aus Quelle, +Leitung und Verbraucher. Das System soll mit einem parallel geschalteten Kondensator $X_C$ so +optimiert werden, dass die Leitungsverluste $P_{VRL}$ minimal werden.\\ +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{\footnotesize{$\underline{U}_q=100\,\volt\cdot e^{j0}$}}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2)node at (.5,.2)[left]{\footnotesize$\underline{U}_q$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {\footnotesize{$\underline{Z}_i=(1+2j)\,\ohm$}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {\footnotesize{$R_L=1\,\ohm$}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [above right] {\footnotesize{$\underline{Z}_V=(10+j5)\,\ohm$}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.5cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {\footnotesize{$jX_C$}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [dashed](2.5,0)--(3.5,0)--(3.5,.2) (2.5,1)--(3.5,1)--(3.5,.8); + \draw [dashed](0,0)--(2.5,0) (2.5,1)--(3.5,1)--(3.5,.8); + \draw (0,0)--(2.5,0) (2,1)--(2.5,1); + \draw (0,.2)--(0,0)--(0,.2) (0,.8)--(0,1)--(.2,1); + \draw node at(.5,0) [below] {\footnotesize Quelle}; + \draw node at(1.5,0) [below] {\footnotesize Leitung}; + \draw node at(2.5,0) [below] {\footnotesize Verbraucher}; + \draw [very thin, dashed] (1,-.2)--(1,1.2)(2,-.2)--(2,1.2); + \draw [->,red](2.125,1.125)--(2.375,1.125) node [right] {\footnotesize{$\underline{I}$}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Knotenpunkte + \draw (0,0)circle(.035); + \fill [white](0,0)circle(.025); + \draw (1,0)circle(.035); + \fill [white](1,0)circle(.025); + \fill (1.5,0)circle(.025); + \fill (2.5,0)circle(.025); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Knotenpunkte + \draw (0,0)circle(.035); + \fill [white](0,0)circle(.025); + \draw (1,0)circle(.035); + \fill [white](1,0)circle(.025); + \fill (1.5,0)circle(.025); + \fill (2.5,0)circle(.025); + \end{scope} + \end{tikzpicture} +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Bestimmen Sie $X_C$ so, dass der Blindleistungsbedarf des Verbrauchers verschwindet. +\item Berechnen Sie die Verlustleistung $P_{VRL}$ der Leitung und die Wirkleistung $P_W$ im Verbraucher. +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\[\baselineskip] +a) Verbraucher $\uline{Z}_V\,||\,X_C$, daher Ersatzschaltbild für $\uline{Z}_V$ (ESB) in Parallelform erforderlich +\begin{align*} +\underline{Z}_V&=R_V+jX_V=(10+j5)\,\ohm\,\,\quad\text{ Scheinwiderstand, entspricht einer Reihenschaltung}\\[\baselineskip] +Z^2_V&=R_V\cdot R_p=R^2_V+X^2_V\,\qquad\qquad\text{Umwandlung in Parallel-ESB}\\ +R_p&=R_V+\frac{X^2_V}{R_V}=(10+\frac{25}{10})\,\ohm=12{,}5\,\ohm\\[\baselineskip] +Z^2_V&=X_V\cdot X_{L_p}=R^2_V+X^2_V\,\,\quad\qquad\text{Umwandlung in Parallel-ESB}\\ +X_{L_p}&=X_V+\frac{R^2_V}{X_V}=(5+\frac{100}{5})\,\ohm=25\,\ohm +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_p$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$jX_{L_p}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$jX_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw (0,0)--(3,0)--(3,.2) (0,1)--(3,1)--(3,.8); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Knotenpunkte + \fill (0,0)circle(.025); + \fill (1,0)circle(.025); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Knotenpunkte + \fill (0,0)circle(.025); + \fill (1,0)circle(.025); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Knotenpunkte + \draw (0,0)circle(.05); + \fill (0,0)circle(.025); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Knotenpunkte + \draw (0,0)circle(.05); + \fill (0,0)circle(.025); + \end{scope} + \begin{scope}[>=latex,very thick] + \draw node at (0,0.5) [left] {$\underline{Z}'\Rightarrow$}; + \end{scope} + \draw node at(1.5,0)[below]{Verbraucher $Z_V$}; +% \end{tikzpicture} +% \begin{tikzpicture}[scale=.5,xshift=15cm,yshift=-2cm] + \begin{scope}[>=latex,very thick,scale=.25,xshift=18cm,yshift=2cm] + \draw [->](0,0)--(1,0)node [right]{$R_p$}; + \draw [->](1,0)--(1,2)node [above]{$jX_{Lp}$}; + \draw [->](1,0)--(1,-2)node[below]{$jX_C$}; + \draw [->,red!50!blue](0,0)--(1,2)node at (.5,1)[left]{$Z_V$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\underline{Z}'&=\underline{Z}_V || jX_C\\ +\frac{1}{\underline{Z}'}&=\frac{1}{R_p}+\frac{1}{jX_{Lp}}+\frac{1}{jX_C}\\ +\intertext{Leitungsverluste sind minimal, wenn die Blindleistung $=0$ wird (Kompensation)} +\frac{1}{\underline{Z}'}&=\frac{1}{R_p}+\cancel{\frac{1}{jX_{Lp}}}+\cancel{\frac{1}{jX_C}}\qquad\Rightarrow \underline{Z}'=R_p\\ +\Im(\underline{Z}')&=0 \quad\text{oder}\quad |X_C| \stackrel{!}{=} |X_{L_p}|\text{ also}\\ +%\Re(\underline{Z}')&=R_p\\ +X_C&=-X_{L_p}=\uuline{-25\,\ohm}\\ +\underline{Z}_{ges}&=\underline{Z}_i+R_L+\underline{Z}'\\ +\underline{Z}_{ges}&=\underline{Z}_i+R_L+R_p=(1+j2+1+12{,}5)\,\ohm=(14{,}5+j2)\,\ohm\\ +|\underline{Z}_{ges}|&=\sqrt{14{,}5^2+2^2}\,\ohm=14{,}64\,\ohm\\ +I&=\frac{U}{|Z_{ges}|}=\frac{100\,\volt}{14{,}64\,\ohm}=6{,}83\,\ampere\\ +\intertext{b) Verlust- und Wirkleistung} +P_{VR_L}&=I^2\cdot R_L=(6{,}83\,\ampere)^2\cdot 1\,\ohm=\uuline{46{,}7\,\watt}\\ +P_W&=I^2\cdot R_p=(6{,}83\,\ampere)^2\cdot 12{,}5\,\ohm=\uuline{583\,\watt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B13_A5.tex b/ET2_L_B13_A5.tex new file mode 100644 index 0000000..842161b --- /dev/null +++ b/ET2_L_B13_A5.tex @@ -0,0 +1,59 @@ +\section{Wechselstrommotor} +Ein Einphasen- Wechselstrommotor liegt an einer Spannung von $230\,\volt - 50\,\hertz$ und gibt eine Leistung von $2\,\kilo\watt$ ab, wobei sein Wirkungsgrad $\eta= 80\%$ und sein Leistungsfaktor $\cos\varphi=0,7$ beträgt. +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Wie groß ist die Stromaufnahme des Motors? +\item Welche Kapazität muss parallelgeschaltet werden, um eine Blindstromkompensation auf +$cos\varphi=0{,}9$ zu erreichen? +\item Wie groß ist der dem Netz bei $cos\varphi=0{,}9$ entnommene Strom? +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +S&=U\cdot I\qquad Scheinleistung\\ +P_{el}&=S\cdot\cos(\varphi)\qquad Wirkleistung\\ +P_{ab}&=\eta\cdot P_{el}\\ +\cos\varphi&=\frac{P}{S} +\end{align} +\begin{align*} +\intertext{Berechnung:} +\intertext{a) Stromaufnahme:} +P_{el}&=\frac{P_{ab}}{\eta}=\frac{2\,\kilo\watt}{0{,}8}=2{,}5\,\kilo\watt\\ +S&=\frac{P_{el}}{\cos(\varphi)}=\frac{2{,}5\,\kilo\watt}{0{,}7}=3{,}571\,\kilo\volt\ampere\\ +I&=\frac{S}{U}=\frac{3{,}571\,\kilo\volt\ampere}{230\,\volt}=\uuline{15{,}5\,\ampere} +\end{align*} +\clearpage +b) Kapazität: +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex, xshift=0, yshift=0] + \draw [black!50!,very thin](0,0)grid(2,2); + \draw [->,red,very thick] (2.525,0)--(2.525,2.55)node [right]{$Q$}; + \draw [->,thick](0:0)--(0:2.5)node [right]{$P$}; + \draw [->,thick] (0:0)--(90:2.55)node [above]{$Q$}; + \draw [->,very thick](0:0)--(25.8:2.778) node [above left] {$\underline{S}'$}; + \draw [->,very thick] (0,0)--(45.6:3.571) node [above left] {$\underline{S}$}; + \draw [->,blue,ultra thick] (45.6:3.571)--(25.8:2.778)node [above right] {$Q_C$}; + \draw [->,black!50!green, very thick] (2.5,0)--(2.5,1.209)node [below right]{$Q'$}; + \draw (0:1)arc(0:45.6:1)node [right]{$45{,}6\degree$}; + \draw (0:1.75)arc(0:25.8:1.75)node [below right]{$25{,}8\degree$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\varphi&=\arccos(0{,}7)=45{,}6\degree\\ +{\color{red}Q}&=S\cdot \sin(45{,}6\degree)=3{,}571\,\kilo\volt\ampere \cdot 0{,}714=2{,}55\,\kilo \var\\ +\varphi'&=\arccos(0{,}9)=25{,}86\degree\\ +S'&=\frac{P}{\cos \varphi'}=\frac{2{,}5\,\kilo\watt}{0{,}9}=2{,}778\,\kilo\volt\ampere\\ +{\color{black!50!green}Q'}&=S'\cdot \sin \varphi '=2{,}778\,\kilo\volt\ampere \cdot 0{,}435=1{,}209\,\kilo \var\\ +\intertext{für Kompensation muß gelten:} +Q+Q_C-Q'&=0 \\ +\Rightarrow Q_C=Q'-Q&=1{,}209\,\kilo \var -2{,}55\,\kilo \var=-1{,}341\,\kilo \var\\ +{\color{blue}|Q_C|}&=\frac{U^2}{|X_C|}\quad\Rightarrow |X_C|=\frac{U^2}{|Q_C|}=\frac{(230\,\volt)^2}{1341\,\var}=39{,}4\,\ohm=\frac{1}{\omega C}\Rightarrow\\ +C&=\frac{1}{\omega |X_C|}=\frac{1}{2\pi\cdot 50\frac{1}{\,\second}\cdot 39{,}4\,\frac{\,\volt}{\ampere}}=8{,}06\cdot\power{10}{-5}\,\frac{\ampere\second}{\volt}=\uuline{80{,}6\,\micro\farad}\\ +\intertext{c) Stromaufnahme bei $\cos\varphi=0{,}9$:} +S'&=U\cdot I'\Rightarrow I'=\frac{S'}{U}=\frac{2778\,\volt\ampere}{230\,\volt}=\uuline{12{,}1\,\ampere} +\end{align*} +Nicht auf $cos(\varphi)=1$ kompensieren, da dann Schwingkreis ! +\clearpage +}{}% diff --git a/ET2_L_B13_A6.tex b/ET2_L_B13_A6.tex new file mode 100644 index 0000000..4251726 --- /dev/null +++ b/ET2_L_B13_A6.tex @@ -0,0 +1,88 @@ +\section{Parallelschaltung von L und C} +An der Parallelschaltung von $L$ und $C$ liegt die Spannung $u(t)$ (siehe Diagramm).\\ +Bei $t=0$ ist $i_L=0$.\\ +Berechnen Sie den Strom $i$ bei $t=t_2$! +\begin{align*} +U_0&=5\,\volt\\ +t_1&=3\milli\second\\ +t_2&=5\milli\second\\ +L&=6\,\milli\henry\\ +C&=100\,\micro\farad +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule + \draw (0,0)--(.2,0) (.2,-0.1)rectangle(.8,0.1) (.8,0)--(1,0)node at (.5,.1) [above] {$L$}; + \fill (.2,-0.1)rectangle(.8,0.1); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Kondensator + \draw (0,0)--(.45,0) (.45,-.2)--(.45,.2) (.55,-.2)--(.55,.2) (.55,0)--(1,0)node at(.75,.05)[above]{$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw (0,2)--(0,0)--(.1,0) (1,2)--(1,0)--(.9,0); + \fill (0,2)circle(.05) (1,2)circle(.05); + \draw [->,blue] (.3,2)--(.7,2) node at (.5,2)[below]{$u(t)$}; + \draw [->,red] (.2,1.75)--(.2,1.25) node at (.2,1.5)[right]{$i(t)$}; + \end{scope} + \begin{scope}[>=latex,thick,scale=.5,xshift=4cm,yshift=.5cm] + \draw [very thin,step=0.5cm](0,0)grid(5,2.5); + \draw [->](0,0) -- (5.5,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,0) -- (0,3) node [above] {$u\,[\volt]$}; + \draw [blue,very thick](0,0)--(3,2.5)--(5,2.5)node at (1.5,1.25)[right]{$u(t)$}; + \foreach \x in {1,2,...,5} + \draw (\x,0) -- (\x,-0.2) node[anchor=north] {$\x$}; + \foreach \y in {0,5} + \draw (0,\y/2) -- (-0.2,\y/2) node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +i_C&=C\cdot \frac{du}{dt}\\ +u_L&=L\cdot \frac{di}{dt}\\ +\text{KNP: }\sum i&=0 +\end{align} +Berechnung:\\ +\begin{align*} +i(t)&=i_C(t)+i_L(t)\\ +i_C(t)&=C\cdot \frac{du}{dt}\tag{1}\label{eq:paralleschaltung-ic}\\ +u_L(t)&=L\cdot \frac{di}{dt}\tag{2}\label{eq:paralleschaltung-ul}\\ +\end{align*} +\begin{align*} +\text{aus \ref{eq:paralleschaltung-ic}}\quad\, i_C(t_2)&=0 \quad \text{zum Zeitpunkt }t_2=5\,\milli\second \quad \frac{du}{dt}=0\\ +&\Rightarrow i(t_2)=i_L(t_2)\\ +\text{aus \ref{eq:paralleschaltung-ul}}\qquad\, di_L&=\frac{1}{L}\cdot u_L\cdot dt \qquad \Big|\int \\ +\big[i_L(t)\big]_{t_a}^{t_b}&=i_L(t_b)-i_L(t_a)=\frac{1}{L}\int_{t_a}^{t_b}{u_L\cdot dt}\\[\baselineskip] +\text{für }0&\leq t \leq t_1\\ +i_L(t_1)-\underbrace{i_L(t=0)}_{0}&=\frac{1}{L}\int_{0}^{t_1}{\frac{U_0}{t_1}\cdot t\cdot dt}=\frac{U_0}{L\cdot t_1}\left[\frac{t^2}{2}\right]_{0}^{t_1}=\frac{U_0\cdot t_1}{2\cdot L}\\ +i_L(t_1)&=\frac{5\,\volt\cdot 3\cdot \,\milli\second}{2\cdot 6\cdot \frac{\,\milli\volt\second}{\ampere}}=1{,}25\,\ampere\\[\baselineskip] +\text{für }t_1&\leq t \leq t_2\\ +i_L(t_2)-i_L(t_1)&=\frac{1}{L}\int_{t_1}^{t_2}{U_0\cdot dt}=\frac{U_0}{L}\cdot (t_2-t_1)\\ +i_L(t_2)&=1{,}25\,\ampere+\frac{5\,\volt\cdot 2\cdot \power{10}{-3}\,\second}{6\cdot \power{10}{-3}\frac{\,\volt\second}{\ampere}}=1{,}25\,\ampere+1{,}67\,\ampere=\uuline{2{,}92\,\ampere} +\end{align*} +\begin{align*} +\intertext{Alternativ Graphisch: $i_L \textrm{ ist proportional zur \textit{Fläche}}\cdot \frac{1}{L}+\textrm{const.}$} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,thick,scale=.5,xshift=4cm,yshift=.5cm] + \fill [black!10!](0,0)--(3,2.5)--(3,0)--(0,0); + \fill [black!25!](3,2.5)--(5,2.5)--(5,0)--(3,0); + \draw [very thin,step=0.5cm](0,0)grid(5,2.5); + \draw [->](0,0) -- (5.5,0) node [right] {$t\,[\milli\second]$}; + \draw [->](0,0) -- (0,3) node [above] {$u\,[\volt]$}; + \draw [blue,very thick](0,0)--(3,2.5)--(5,2.5)node at (1.5,1.25)[right]{$u(t)$}; + \foreach \x in {1,2,...,5} + \draw (\x,0) -- (\x,-0.2) node[anchor=north] {$\x$}; + \foreach \y in {0,5} + \draw (0,\y/2) -- (-0.2,\y/2) node[anchor=east] {$\y$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +i(t_2)&=\frac{1}{L}\cdot \underbrace{\left(\frac{U_0\cdot t_1}{2}+U_0(t_2-t_1)\right)}_{\textrm{Fläche}} =\frac{1}{6\,\milli\henry}\cdot \left(\frac{5\,\volt\cdot 3\,\milli\second}{2}+ 5\,\volt\cdot (5-3)\,\milli\second\right)\\ +&=\frac{(7{,}5+10)\cdot \power{10}{-3}\,\volt\second}{6\cdot \power{10}{-3}\,\frac{\volt\second}{\ampere}}= \uuline{2{,}92\,\ampere} +\end{align*} +\footnotesize{Warum ist $i_L(t_2)-i_L(t_1)\not= 0$? Strom ändert sich noch, nur Spannung ist konstant.\\ +Wenn $u=konstant \rightarrow$ Strom steigt unendlich an. $u=L\frac{di}{dt} \Rightarrow i(t)=1/L\int u(t)dt$} +\clearpage +}{}% diff --git a/ET2_L_B13_A7.tex b/ET2_L_B13_A7.tex new file mode 100644 index 0000000..76bbb36 --- /dev/null +++ b/ET2_L_B13_A7.tex @@ -0,0 +1,101 @@ +\section{Werte $R_L$ und $L$ einer Spule} +Aus den drei gemessenen sinusförmigen Spannungen $U$, $U_N$, und $U_{SP}$ lassen sich die Werte $R_L$ und $L$ einer Spule bestimmen. +\begin{align*} +U=100\,\volt\\ +U_N=60\,\volt\\ +U_{SP}=70\,\volt\\ +R_N=60\,\ohm\\ +f = 50\,\hertz +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Zeichnen Sie ein qualitatives Zeigerdiagramm der Spannungen! +\item Bestimmen Sie $R_L$ und $L$! +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand + \draw (0,0)--(0.2,0) (.2,-0.1)rectangle(.8,0.1) (.8,0)--(1,0)node at (.5,.1) [above] {$R_N$}; + \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$U_N$}; + \draw (0,1)--(0,0)--(.1,0) (3,1)--(3,0)--(2.9,0);%anschuß und Füllt die Ecken der Verbindung! + \fill (0,1)circle (0.025) (3,1)circle (0.025); + \draw [->,blue](.2,1)--(2.8,1)node at (1.5,1)[below]{\footnotesize$U$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Widerstand + \draw (0,0)--(0.2,0) (.2,-0.1)rectangle(.8,0.1) (.8,0)--(1,0)node at (.5,.1) [above] {$R_L$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm]%Spule + \draw (0,0)--(.2,0) (.2,-0.1)rectangle(.8,0.1) (.8,0)--(1,0)node at (.5,.1) [above] {$L$}; + \fill (.2,-0.1)rectangle(.8,0.1); + \draw [->,blue] (-.7,-.2)--(.7,-.2) node at (0,-.2)[below]{\footnotesize$U_{SP}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +\begin{align*} +\intertext{Berechnung:} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=.5] + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw [black!25!,very thin](0,0)grid(8,7); + \draw [->,red, thick](0,0)--(8,0)node [right] {$\underline{I}$}; + \draw [->](0,0)--(6,0)node at(3,0)[below] {$\underline{U}_N$}; + \draw [->](6,0)--(7.25,6.887)node at(6.5,3.5)[left] {$\underline{U}_{SP}$}; + \draw [->](0,0)--(7.25,6.887)node at(3.5,3.5)[left] {$\underline{U}$}; + \draw [->,blue](6,0)--(7.25,0)node at(6.5,0)[below] {$\underline{U}_{R_L}$}; + \draw [->,blue](7.25,0)--(7.25,6.887)node at(7,3.5)[right] {$\underline{U}_{L}$}; + \draw [black!50!](33:10)arc(33:53:10)node [left]{$\underline{U}=100\,\volt\widehat{=}10\,\centi\metre$}; + \draw [black!50!](6,0)+(90:7)arc(90:70:7)node [right]{$\uline{U}_{SP}=70\,\volt\widehat{=}7\,\centi\metre$}; + \draw [black!50!](3,-1)node[below]{$\underline{U}_N=60\,\volt\widehat{=}6\,\centi\metre$}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=12cm, yshift=2cm] + \draw node at(0,2)[right]{$I$ zeichnen}; + \draw node at(0,1)[right]{$u_N || I$}; + \draw node at(0,0)[right]{Mit Zirkel $U$ und $U_{SP}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +I&=\frac{U_N}{R_N}=\frac{60\,\volt}{60\,\ohm}=\uuline{1\,\ampere}\\ +\underline{U}_{SP}&=70\,\volt=\sqrt{U^2_{RL}+U^2_L}\\ +\end{align*} +\clearpage +Widerstandsoperatoren:\\ + +\footnotesize{Impedanzdreieck wie Spannungsdreieck} +\begin{align*} + \begin{tikzpicture}[scale=.5] + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw [black!25!,very thin](0,0)grid(8,7); + \draw [->](0,0)--(6,0)node at(3,0)[below] {$R_N$}; + \draw [->](6,0)--(7.25,6.887)node at(6.5,3.5)[left] {$\underline{Z}_{SP}$}; + \draw [->](0,0)--(7.25,6.887)node at(3.5,3.5)[left] {$\underline{Z}$}; + \draw [->,blue](6,0)--(7.25,0)node at(6.5,0)[below] {$R_L$}; + \draw [->,blue](7.25,0)--(7.25,6.887)node at(7,3.5)[right] {$X_L$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +Z_{SP}&=\frac{U_{SP}}{I}=\frac{70\,\volt}{1\,\ampere}=70\,\ohm \quad \text{\footnotesize{(Nur Effektivwerte - ohne Winkel)}}\\%=\sqrt{R^2_L+X^2_L}\\ +Z^2_{SP}&=R^2_L+X^2_L=(70\,\ohm)^2\\ +X^2_L&=(70\,\ohm)^2-R^2_L \tag{1}\\[\baselineskip] +Z&=\frac{U}{I}=\frac{100\,\volt}{1\,\ampere}=100\,\ohm\\ +Z^2&=(R_N+R_L)^2+X^2_L=(100\,\ohm)^2\\ +X^2_L&=(100\,\ohm)^2-(R_N+R_L)^2\\ +&=(100\,\ohm)^2-(R^2_N+2\cdot R_N\cdot R_L+R^2_L) \tag{2}\\[\baselineskip] +(70\,\ohm)^2-\cancel{R^2_L}&=(100\,\ohm)^2-R^2_N-2\cdot R_N\cdot R_L-\cancel{R^2_L}\tag{$1$ in $2$}\\ +2\cdot R_N\cdot R_L&=(100\,\ohm)^2-R^2_N-(70\,\ohm)^2\\ +R_L&=\frac{(100\,\ohm)^2-R^2_N-(70\,\ohm)^2}{2\cdot R_N}=\frac{(100\,\ohm)^2-(60\,\ohm)^2-(70\,\ohm)^2}{2\cdot 60\,\ohm}\\ +&=\frac{1500\,\ohm^2}{2\cdot 60\,\ohm}=\uuline{12{,}5\,\ohm}\\ +%(100\,\ohm)^2&=(60\,\ohm)^2+2\cdot 60\,\ohm\cdot R_L+\cancel{R^2_L}+(70\,\ohm)^2 -\cancel{R^2_L}\\ +%2\cdot 60\,\ohm\cdot R_L&=(100\,\ohm)^2-(60\,\ohm)^2-(70\,\ohm)^2=1500(\,\ohm)^2\\ +%R_L&=\frac{1500(\,\ohm)^2}{2\cdot 60\,\ohm}=\uuline{12{,}5\,\ohm} \tag{in $1$}\\ +\text{in (1) }\qquad X_L&=\sqrt{(70\,\ohm)^2-(12{,}5\,\ohm)^2}=68{,}87\,\ohm\\ +L&=\frac{X_L}{\omega}=\frac{68{,}87\,\ohm}{2\pi\cdot 50\,\frac{1}{\second}}=\uuline{0{,}219\,\henry} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B14_A1.tex b/ET2_L_B14_A1.tex new file mode 100644 index 0000000..16cfbfa --- /dev/null +++ b/ET2_L_B14_A1.tex @@ -0,0 +1,130 @@ +\section{Zeigerdiagramm} +Gegeben ist die Ausgangsspannung $U_a = 5\,\volt \cdot e^{j0\degree}$ und $R_1=R_2=\frac{1}{\omega C_1}=\frac{1}{\omega C_2}=1\,\kilo\ohm$\\ +Zeichnen Sie ein maßstäbliches Zeigerdiagramm aller Spannungen und aller Ströme!\\ +Maßstäbe: $1\,\centi\metre\,\widehat{=}\, 1\,\volt \text{;}\quad 1\,\centi\metre\,\widehat{=}\, 1\,\milli\ampere$ +Entnehmen Sie dem Zeigerdiagramm Betrag und Phasenwinkel der Spannung $U_e$ ! +\begin{align*} + \begin{tikzpicture}[very thick,scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$\underline{U}_{R_1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_1$}; + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\tiny$\underline{U}_{1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$\underline{U}_{R_2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (0,0)--(2.5,0) (2,1)--(2.5,1); + \fill (0,0)circle(.025) (2.5,0)circle(.025) (0,1)circle(.025) (2.5,1)circle(.025); + \draw [->,blue] (0,.9)--(0,.1) node at (0,.5)[right]{$\underline{U}_{e}$}; + \draw [->,blue] (2.5,.9)--(2.5,.1) node at (2.5,.5)[right]{$\underline{U}_{a}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +e^{j\varphi}&=\cos\varphi+j\sin\varphi \quad\text{Eulersche Formel} +\end{align} +\footnotesize{R-C Ketten sind u.a. ein Ersatzbild für Leitungen (Kapazität pro Längeneinheit)}\\ +Berechnung: +\begin{align*} + \begin{tikzpicture}[very thick,scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$\underline{U}_{R_1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_1$}; + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\tiny$\underline{U}_{1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$\underline{U}_{R_2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (0,0)--(2.5,0) (2,1)--(2.5,1); + \fill (0,0)circle(.025) (2.5,0)circle(.025) (0,1)circle(.025) (2.5,1)circle(.025); + \draw [->,blue] (0,.9)--(0,.1) node at (0,.5)[right]{$\underline{U}_{e}$}; + \draw [->,blue] (2.5,.9)--(2.5,.1) node at (2.5,.5)[right]{$\underline{U}_{a}$}; + \draw [->,red] (.05,1.1)--(.25,1.1) node at (.1,1.1)[above]{$\underline{I}_{e}$}; + \draw [->,red] (2.2,1.1)--(2.4,1.1) node at (2.3,1.1)[above]{$\underline{I}_{a}=0$}; + \draw [->,red] (1.1,.9)--(1.1,.7) node at (1.1,.8)[right]{$\underline{I}_{1}$}; + \draw [->,red] (2.1,.9)--(2.1,.7) node at (2.1,.8)[right]{$\underline{I}_{2}$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=.4cm,yshift=.5cm]%Masche + \draw [->,red!50!blue] (270:.15)arc(270:-60:.15) node at (0,0){$M_1$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\underline{U}_a&=5\,\volt\cdot e^{j0\degree}\\ +\underline{I}_a&=0\text{, da kein Lastwiderstand angeschlossen ist!}\\ +\underline{I}_2&=\frac{\underline{U}_a}{\underline{jX}_2} +=\frac{5\,\volt\cdot e^{j0\degree}}{1\kilo\ohm\cdot e^{-j90\degree}} +=5\,\milli\ampere\cdot e^{j90\degree}=j5\,\milli\ampere +\quad\text{(Strom eilt vor)}\quad 5\,\milli\ampere\;\angle +90\degree\\ +\underline{U}_{R_2}&=R_2\cdot \underline{I}_2=1\,\kilo\ohm\cdot e^{-j90\degree}=1\,\kilo\ohm\cdot 5\,\milli\ampere\;\angle +90\degree=5\,\volt\;\angle +90\,\degree\\ +\underline{U}_1&=\underline{U}_a+\underline{U}_{R_2}=5\,\volt+j5\,\volt\quad\text{(Vektoren addieren)}\quad =\sqrt{2}\cdot 5\,\volt\;\angle +45\degree \sqrt{2}\cdot 5\,\volt\cdot e^{j45\degree}\\\underline{I}_1&=\frac{\underline{U}_1}{\underline{jX}_1}=\frac{\sqrt{2}\cdot 5\,\volt\cdot e^{j45\degree}}{ 1\kilo\ohm\cdot e^{-j90\degree}}=\sqrt{2}\cdot 5\,\milli\ampere\cdot e^{j135\degree}=5\cdot (-1+j)\,\milli\ampere\quad\text{(Strom eilt $90\degree$ vor) } 5\,\milli\ampere\;\angle +135\degree\\ +\underline{I}_e&=\underline{I}_1+\underline{I}_2=\sqrt{2}\cdot 5\,\milli\ampere\;\angle +135\degree+5\,\milli\ampere\;\angle +90\degree\quad\text{(Vektoren addieren)}\\[\baselineskip] +\end{align*} +\begin{align*} +\intertext{Jetzt zeichnen oder rechnerisch: (jedoch aufwendiger)} +\underline{I}_e&=I_1\cdot (\cos\varphi+j\sin\varphi)+I_2\cdot (\cos\varphi+ j\sin\varphi)\\ +&=I_1\cdot (\cos135\degree+j\sin 135\degree)+I_2\cdot (\cos 90\degree+ j\sin 90\degree)\\ +&=|\underline{I}_1|\cdot (-\frac{1}{\sqrt{2}}+j\frac{1}{\sqrt{2}})+|\underline{I}_2|\cdot (0+j)\\ +&=[\cancel{\sqrt{2}}\cdot 5\cdot (-\frac{1}{\cancel{\sqrt{2}}}+j\frac{1}{\cancel{\sqrt{2}}})+j\cdot 5]\,\milli\ampere=(-5+j10\,\milli\ampere)\\ +&\quad |\underline{I}_e|= \sqrt{10^2+5^2}\,\milli\ampere=11{,}18\,\milli\ampere \\ +&\quad\tan\varphi=\frac{\Im}{\Re}=\tan\frac{10}{-5}=\tan-2\Rightarrow\varphi=\arctan\frac{-2}=-1{,}107\,\rad\,\widehat{=}\,-63{,}435\degree\\ +&\quad\text{(Definitionsbereich $\tan\varphi [-\pi/2\cdots\pi/2]$ beachten!)}\\ +&=11{,}18\cdot e^{j116{,}565\degree}\,\milli\ampere\\[\baselineskip] +\underline{U}_{R_1}&=R_1\cdot \underline{I}_e\\ +&\quad\text{(Nur zur Vollständigkeit) }\underline{U}_{R_1}=1\,\kilo\ohm\cdot 11{,}18\cdot e^{j116{,}565\degree}\,\milli\ampere=11{,}18\cdot e^{j116{,}565\degree}\,\volt\\ +\underline{U}_e&=\underline{U}_{R_1}+\underline{U}_1=\uuline{15\,\volt\cdot e^{+j90\degree}}\quad\text{(Vektoren addieren)} +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=.5] + \begin{scope}[>=latex] + \draw [very thin,black!50!](-5,0)grid(5,15); + \end{scope} + \begin{scope}[>=latex,very thick] + \draw [->,blue] (0:0)--(90:15) node at (90:7.5)[right]{$\underline{U}_e$}; + \draw [->,blue] (0:0)--(0:5) node at (0:2.5)[below]{$\underline{U}_a$}; + \draw [->,blue] (0:0)--(45:7.07) node at (45:3.54)[right]{$\underline{U}_1$}; + \draw [->,blue,ultra thick] (0:0)--(-5,10) node at (-2.5,5)[right]{$\underline{U}_{R_1}$}; + \draw [->,blue] (-5,10)--(0,15) node at (-2.5,12.5)[left]{$\underline{U}_{1}$}; + \draw [->,blue,ultra thick] (0:0)--(90:5) node at (90:3.5)[right]{$\underline{U}_{R_2}$}; + \draw [->] (5,0)--(5,5) node at (5,3.5)[right]{$\underline{U}_{R_2}$}; + \draw [->,red] (0:0)--(135:7.07) node at (135:3.54)[left]{$\underline{I}_1$}; + \draw [->,red,thick] (0:0)--(90:5) node at (90:3.5)[left]{$\underline{I}_2$}; + \draw [->] (135:7.07)--+(90:5) node at (-5,7.5)[right]{$\underline{I}_2$}; + \draw [->,red,thick] (0:0)--(-5,10) node at (-2.5,5)[left]{$\underline{I}_{e}$}; + \end{scope} + \draw node at (8,10)[right]{Reihenfolge$U_a\, I_2, U_{R_2},U_1, I_1, I_e, U_{R_1})$}; + \draw node at (8,9)[right]{$U_a\widehat{=}\,5\,\centi\metre\angle 0\,\degree\quad(5+j0)$}; + \draw node at (8,8)[right]{$U_{R_2}\widehat{=}\,5\,\centi\metre\angle 90\,\degree\quad(0+j5)$}; + \draw node at (8,7)[right]{$U_1\widehat{=}\,7{,}07\,\centi\metre\angle 45\,\degree\quad(5+j5)$}; + \draw node at (8,6)[right]{$I_1\widehat{=}\,7{,}07\,\centi\metre\angle 135\,\degree\quad(-5+j5)$}; + \draw node at (8,5)[right]{$I_2\widehat{=}\,5\,\centi\metre\angle 90\,\degree\quad(0+j5)$}; + \draw node at (8,4)[right]{$I_2\widehat{=}\,5\,\centi\metre\angle 90\,\degree\quad(-5+j5)\rightarrow(0+j5)$ addiert zu $I_1$}; + \draw node at (8,3)[right]{$I_e\widehat{=}\,11{,}18\,\centi\metre\angle 116{,}5\,\degree(-5+j10)$}; + \draw node at (8,2)[right]{$U_{R_1}\widehat{=}\,11{,}18\,\centi\metre\angle 116{,}5\,\degree\quad(-5+j10)$}; + \draw node at (8,1)[right]{$U_1\widehat{=}\,7{,}07\,\centi\metre\angle 45\,\degree\quad(-5+j10)\rightarrow(5+j5)$ addiert zu $U_{R_1}$}; + \end{tikzpicture} +\end{align*} +\begin{align*} +\sum M_1=0=U_{R_1}+U_1-U_e\\ +U_e=15\,\volt \cdot e^{j90\degree} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B14_A2.tex b/ET2_L_B14_A2.tex new file mode 100644 index 0000000..aeb50d1 --- /dev/null +++ b/ET2_L_B14_A2.tex @@ -0,0 +1,143 @@ +\section{Gesamtwiderstand} +Von der Schaltung (Bild 1) sind die Zeiger $\uline{U}_0$ und $\uline{I}_0$ gegeben (Bild 2). +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Ist der Gesamtwiderstand $\uline{Z}$ induktiv, ohmsch oder kapazitiv? (Stichwortartige Begründung !) +\item Vervollständigen Sie Bild 2 zu einem qualitativen Zeigerdiagramm \uline{aller} Ströme und Spannungen.\\ +(Rechte Winkel oder Parallelen sind zu kennzeichnen. Alle Ströme und Spannungen müssen im Schaltbild (Bild 1) und im Zeigerbild unmissverständlich benannt werden.)\\ +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-0.5cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-2cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (-.5,.2)--(-0.5,0)--(.5,0)--(.5,.2) (-1,1)--(.5,1)--(.5,.8) (-1,-2)--(0,-2)--(0,-1.8); + \draw [->,blue] (-1,.9)--(-1,-1.9)node at(-1,-.5)[right]{$\underline{U}_{0}$}; + \draw node at(-1,-.5)[left]{$\underline{Z}\Rightarrow$}; + \fill (-1,1)circle(.025) (-1,-2)circle(.025); + \draw [->,red] (-.9,1.1)--(-.6,1.1) node at (-.75,1.1)[above]{$\underline{I}_{0}$}; + \draw node at (0,-2.2)[below]{Bild 1}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm] + \draw [->,blue] (0,0)--(3,0)node [right]{$\underline{U}_{0}$}; + \draw [->,red] (0,0)--(-33:1.5)node [right]{$\underline{I}_{0}$}; + \draw node at (1.5,-2.2)[below]{Bild 2}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Merksätze:} +&\text{Ohm'scher Widerstand: Strom und Spannung in Phase}\\ +&\text{Kondensator: Strom eilt $90\degree$ vor}\\ +&\text{Induktivität: Spannung eilt $90\degree$ vor} +\end{align} +Berechnung:\\[\baselineskip] +a) Induktiv, da nacheilender Strom.\\ +\begin{align*} +\intertext{b) Schaltbild mit Strom- und Spannungspfeilen ergänzen, Zeigerdiagramm erstellen.} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_C$}; + \draw [<-,blue] (.3,.5)--(.7,.5)node at(.5,.5)[right]{\footnotesize$\uline{U}_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-0.5cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$\uline{U}_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-2cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{L}$}; + \draw [<-,blue] (.5,-.5)--(1.5,-.5)node at(1,-.5)[right]{\footnotesize$\uline{U}_{RL}$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (-.5,.2)--(-0.5,0)--(.5,0)--(.5,.2) (-1,1)--(.5,1)--(.5,.8) (-1,-2)--(0,-2)--(0,-1.8); + \draw [->,blue] (-1,.9)--(-1,-1.9)node at(-1,-.5)[right]{$\underline{U}_{0}$}; + \draw node at(-1,-.5)[left]{$\underline{Z}\Rightarrow$}; + \fill (-1,1)circle(.025) (-1,-2)circle(.025); + \draw [->,red] (-.9,1.1)--(-.6,1.1) node at (-.75,1.1)[above]{$\underline{I}_{0}$}; + \draw [<-,red] (-.7,.3)--(-.7,.7) node at (-.7,.5)[left]{$\underline{I}_{C}$}; + \draw [<-,red] (1.,.3)--(1.,.7) node at (1.,.5)[right]{$\underline{I}_{RC}$}; +% \draw node at (0,-2.2)[below]{Bild 1}; + \end{scope} + \end{tikzpicture} +\end{align*} +%\begin{align*} +%\uline{I}_0&=\uline{I}_{RC}+\uline{I}_C\\ +%\uline{I}_C\,& \bot \,\uline{I}_{RC}\\ +%\uline{I}_C &\text{ eilt vor } [\text{ für }\uline{I}_{RC}\text{ und }\uline{I}_{C}] +%\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [->,blue] (0,0)--(3,0)node [above left]{$\underline{U}_{0}$}; + \draw [->,red] (0,0)--(-33:1.5)node at (-33:1)[above right]{$\underline{I}_{0}$}; + \fill [black!50!](-33:.75)circle(.02); + \draw [very thin](-33:.0)arc(147:327:.75); + \draw (-66:1.258)+(24:.15)arc(24:114:.15) ; %Rechter Winkel + \fill (-66:1.258)+(69:.075)circle(.02) ; %Punkt Rechter Winkel + \draw [->,red] (0,0)--(-66:1.258)node at(-68:.8)[above left]{$\underline{I}_{RC}$}; + \draw [->,red] (-66:1.258)--(-33:1.5)node at (-55:1.2)[above right]{$\underline{I}_{C}$}; + \draw (4,-.3)node [right] {$\uline{I}_0=\uline{I}_{RC}+\uline{I}_C$ und}; + \draw (4,-.6)node [right] {$\uline{I}_C\, \bot \,\uline{I}_{RC}$ (Thaleskreis) und}; + \draw (4,-.9)node [right] {$\uline{I}_C$ eilt $\uline{I}_{RC}$ vor.}; +% \draw node at (1.5,-2.2)[below]{Bild 2}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [->,blue] (0,0)--(3,0)node at(2,0)[above left]{$\underline{U}_{0}$}; + \draw [->,red] (0,0)--(-33:1.5)node at (-33:1)[above right]{$\underline{I}_{0}$}; + \fill [black!50!](-33:.75)circle(.02); + \draw [very thin](-33:.0)arc(147:327:.75); + \draw [->,red] (0,0)--(-66:1.258)node at(-68:.8)[above left]{$\underline{I}_{RC}$}; + \draw [->,red] (-66:1.258)--(-33:1.5)node at (-55:1.2)[above right]{$\underline{I}_{C}$}; +% \draw node at (1.5,-2.2)[below]{Bild 2}; + \draw [->,blue] (0,0)--(-66:2)node at(-68:1.5)[above left]{$\underline{U}_{C}$}; + \draw [->,blue] (-66:2)--(3,0)node at (2.5,-.2)[below left]{$\underline{U}_{RL}$}; +% \draw [red,very thin,dashed](-66:.5)arc(114:-66:1.5) (0:3)+(114:1.5)arc(114:294:1.5);%Seitenhalbierende + \draw [very thin,dashed](-66:2.0)+(70:1.5)arc(70:10:1.5) (0:3)+(190:1.5)arc(190:250:1.5);%Seitenhalbierende Teilkreise + \fill [black!60!](-25.6:2.115)circle(.02); %Mittelpunkt Thaleskreis 2 + \draw [very thin](0:3)arc(40:-140:1.425); %Tahleskreis 2 + \draw [very thin](-14.3:1.5)--(-31.9:2.8); %Seitenhalbierende + \draw (-66:1.258)+(24:.15)arc(24:114:.15) ; %Rechter Winkel + \fill (-66:1.258)+(69:.075)circle(.02) ; %Punkt Rechter Winkel + \draw [->,blue!50!red](-66:2)--+(-30:1)node at(1.2,-2.3)[below right]{$\underline{U}_{R}$}; + \draw [->,blue!50!red](-66:2)--+(-30:1)--(0:3)node at(2.6,-.7)[below right]{$\underline{U}_{L}$}; + \draw (-66:2)++(-30:1)++(60:0.15)arc(60:150:.15) ; %Rechter Winkel + \fill (-66:2)++(-30:1)++(106:.075)circle(.02) ; %Punkt Rechter Winkel + \draw (4,-.6)node [right] {$\uline{U}_C\, ||\, \uline{I}_{RC}$}; + \draw (4,-.9)node [right] {$\uline{U}_{0}=\uline{U}_C+\uline{U}_{RL}$}; + \draw (4,-1.5)node [right] {$\uline{U}_R\, ||\, \uline{I}_{0}$}; + \draw (4,-1.8)node [right] {$\uline{U}_{RL}=\uline{U}_{R}+\uline{U}_{L}$}; + \draw (4,-2.1)node [right] {$\uline{U}_L\, \bot \,\uline{U}_R$ (oder $\uline{U}_L\, \bot \,\uline{I}_0$)}; + \draw (4,-2.4)node [right] {$\uline{U}_L\,$ eilt $\uline{U}_R\,$ vor}; + \end{scope} + \begin{scope}[>=latex,xshift=.86cm,yshift=-.552cm]%Parallelen + \draw [double](0:-.1)--+(57:.1); + \end{scope} + \begin{scope}[>=latex,xshift=1.4cm,yshift=-2.167cm]%Parallelen + \draw [double](0:-.1)--+(57:.1); + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B14_A3.tex b/ET2_L_B14_A3.tex new file mode 100644 index 0000000..a57c7fb --- /dev/null +++ b/ET2_L_B14_A3.tex @@ -0,0 +1,133 @@ +\section{Brückenschaltung} +%\enlargethispage{0cm} +Zeichnen Sie zu der abgebildeten Brückenschaltung ein \underline{maßstäbliches} Zeigerdiagramm aller Ströme und Spannungen.\\ +Entnehmen Sie dem Zeigerdiagramm die Spannung $U_a$ und geben Sie von dieser Spannung Betrag und Phasenwinkel an.\\[\baselineskip] +\uline{Maßstäbe: }$ 10\,\volt\,\widehat{=}\,1\centi\metre\qquad 0,2\,\ampere\,\widehat{=}\,1\,\centi\metre $ (Platzbedarf in x und y $15\,\centi\metre$)\\ +$R_1=100\,\ohm \qquad R_2=80\,\ohm \qquad X_L=200\,\ohm \qquad X_C=-120\,\ohm\qquad\uline{U}=150\,\volt\cdot e^{j0\degree}$ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; + \draw [<-,blue] (.3,.5)--(.7,.5)node at(.5,.5)[right]{\footnotesize$\uline{U}_{R_1}$}; + \draw [<-,red] (.05,.0)--(.25,.0)node at(.15,.0)[right]{\footnotesize$\uline{I}_{1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \draw [<-,blue] (.3,.5)--(.7,.5)node at(.5,.5)[right]{\footnotesize$\uline{U}_{R_2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$}; + \draw [->,red] (.35,0)--(.15,0) node at (.25,0)[right]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (1.5,.8)--(1.5,1)--(-.5,1) (1.5,-.8)--(1.5,-1)--(-.5,-1); + \fill (-.5,1)circle(.025) (-.5,-1)circle(.025); + \draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{a}$}; +% \draw node at(-1,-.5)[left]{$\underline{Z}\Rightarrow$}; + \fill (.5,0)circle(.025) node at (.5,0)[left]{A}; + \fill (1.5,0)circle(.025)node at (1.5,0)[right]{B}; + \draw [->,blue] (-.5,.8)--(-.5,-.8) node at (-.5,0)[left]{$\underline{U}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%%begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} +Z_1&=\sqrt{R^2_1+X^2_L}=\sqrt{100^2+200^2}\,\ohm=223{,}6\,\ohm\\ +Z_2&=\sqrt{R^2_2+X^2_C}=\sqrt{80^2+(-120)^2}\,\ohm=144{,}2\,\ohm\\ +\varphi_1&=\arctan\frac{\Im}{\Re}=\arctan\frac{200\,\ohm}{100\,\ohm}=\arctan 2=63{,}4\,\degree\\ +\varphi_2&=\arctan\frac{-120\,\ohm}{80\,\ohm}=\arctan (-1{,}5)=-56{,}3\,\degree\\ +\uline{Z}_1&=Z_1\cdot e^{j\varphi_1}=223{,}6\,\ohm\cdot e^{j63{,}4\,\degree}\\ +\uline{Z}_2&=Z_2\cdot e^{j\varphi_2}=144{,}2\,\ohm\cdot e^{-j56{,}3\,\degree}\\ +\uline{I}_1&=\frac{U}{Z_1}=\frac{150\,\volt\cdot e^{j0\degree}}{223{,}6\,\ohm\cdot e^{j63{,}4\,\degree}}=0{,}67\,\ampere\cdot e^{-j63{,}4\,\degree} \approx 3{,}4\,\centi\metre\\ +\uline{I}_2&=\frac{U}{Z_2}=\frac{150\,\volt\cdot e^{j0\degree}}{144{,}2\,\ohm\cdot e^{-j56{,}3\,\degree}}=1{,}04\,\ampere\cdot e^{j56{,}3\,\degree} \approx 5{,}2\,\centi\metre\\ +\uline{I}&=\uline{I_1}+\uline{I_2}\\ +\uline{U}_{R_1}&=I_1\cdot R_1= 0{,}671\,\ampere\cdot e^{j63{,}4\,\degree}\cdot 100\,\ohm=67{,}1\,\volt\cdot e^{-j63{,}4\,\degree}\\ +\uline{U}_{R_2}&=I_2\cdot R_2=1{,}04\,\ampere\cdot 80\,\ohm=83{,}2\,\volt\cdot e^{j56{,}3\,\degree}\\ +\uline{U}_a&=\uline{U}_{R_2}-\uline{U}_{R_1}\\ +\uline{U}_L&=X_L\cdot \uline{I}_1=200\,\ohm\cdot e^{j90\degree}0{,}67\,\ampere\cdot e^{-j63{,}4\,\degree}=134\,\volt\cdot e^{j26{,}6\,\degree}\\ +\uline{U}_C&=X_C\cdot \uline{I}_2=120\,\ohm\cdot e^{-j90\degree}\cdot 1{,}04\,\ampere\cdot e^{j56{,}3\,\degree}=124{,}8\,\volt\cdot e^{-j33{,}7\,\degree} +\end{align*} +\clearpage +Zeigerdiagramm Teil 1: (Maßstäblich) +\enlargethispage{2\baselineskip} +\begin{align*} + \begin{tikzpicture}[very thick,scale=.6] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [black!25!,very thin](0,-8)grid(15,8); + \draw [black!50!,thin,dashed](16,0)--(18,0); + \draw [black!50!,thin,dashed](3,-6)--(5,-6); + \draw [black!50!,thin](7.5,0)circle(7.5); + \draw [->,blue](0,0)--(15,0)node [right]{$\uline{U}$}; + \draw [->,blue](0,0)--(-63.4:6.71)node [left]{$\uline{U}_{R_1}$}; + \draw [->,blue](0,0)--(56.3:8.32)node [left]{$\uline{U}_{R_2}$}; + \draw [->,blue!50!red](0,0)+(-63.4:6.71)--(15,0)node at(15,-1.5)[below]{$\uline{U}_L=134\,\volt\cdot e^{j26{,}63{,}4\,\degree}$}; + \draw [->,blue!50!red](0,0)+(56.3:8.32)--(15,0)node at(15,2)[above]{$\uline{U}_C=124{,}8\,\volt\cdot e^{-j33{,}73\,\degree}$}; + \draw [->,blue](0,0)+(-63.4:6.71)--(56.3:8.32)node at (4,2)[right]{$\uline{U}_{AB}=\uline{U}_a=130\,\volt\cdot e^{j83\,\degree}$}; + \draw node at (3,-6)[below]{$A$}; + \draw node at (4.8,7)[above]{$B$}; + \draw node at (16,0)[above right]{$\Re (A)$}; + \draw [black!50!](0,0)++(-63.4:6.71)+(26.6:1)arc(26.6:116.6:1);%Rechter Winkel bei UR1 + \fill [black!50!](0,0)++(-63.4:6.71)+(71.6:.5)circle(0.1); + \draw [black!50!](0,0)++(56.3:8.32)+(-33.7:1)arc(-33.7:-123.7:1);%Rechter Winkel bei UR2 + \fill [black!50!](0,0)++(56.3:8.32)+(-78.7:.5)circle(0.1); + \draw [<->,black!50!](0,0)+(-63.4:1)arc(-63.4:53.3:1);%Winkel UR1 UR2 + \draw node at (0.8,0)[below right]{$\varphi_1=-63{,}4\,\degree$}; + \draw node at (0.8,0)[above right]{$\varphi_2=53{,}3\,\degree$}; + \draw [->,black!75!](3,-6)+(0:2)arc(0:83:2)node at (5,-5.5)[right]{$\varphi_{\uline{U}} =83\,\degree$};%U - UAB + \draw node at (19,5) [right]{$y=15\,\centi\metre$}; + \draw node at (19,4) [right]{Kreis da}; + \draw node at (19,3) [right]{$\uline{U}_{R_2}+\uline{U}_{C}=\uline{U}$}; + \draw node at (19,2) [right]{$\uline{U}_{R_1}+\uline{U}_{L}=\uline{U}$}; + \draw node at (19,1) [right]{$\uline{U}_{R_2}\bot \uline{U}_{C}$}; + \draw node at (19,0) [right]{$\uline{U}_{R_1}\bot \uline{U}_{L}$}; + \draw node at (19,-1) [right]{A und B einzeichnen}; + \draw node at (19,-2) [right]{Messen $\uline{U}_{AB}=\uline{U}_a$}; + \end{scope} + \end{tikzpicture} +\end{align*} +Zeigerdiagramm vollständig: +\begin{align*} + \begin{tikzpicture}[very thick,scale=.6] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [black!25!,very thin](0,-8)grid(15,8); + \draw [black!50!,thin,dashed](16,0)--(18,0); + \draw [black!50!,thin,dashed](3,-6)--(5,-6); + \draw [black!50!, thin](7.5,0)circle(7.5); + \draw [->,blue](0,0)--(15,0)node [right]{$\uline{U}$}; + \draw [->,blue](0,0)--(-63.4:6.71)node [left]{$\uline{U}_{R_1}$}; + \draw [->,blue](0,0)--(56.3:8.32)node [left]{$\uline{U}_{R_2}$}; + \draw [->,blue!50!red](0,0)+(-63.4:6.71)--(15,0)node at(12,-1.5)[below]{$\uline{U}_L$}; + \draw [->,blue!50!red](0,0)+(56.3:8.32)--(15,0)node at(12,2)[above]{$\uline{U}_C$}; + \draw [->,blue](0,0)+(-63.4:6.71)--(56.3:8.32)node at (4,2)[right]{$\uline{U}_{AB}=\uline{U}_a=130\,\volt\cdot e^{j83\,\degree}$}; + \draw [black!50!](0,0)++(-63.4:6.71)+(26.6:1)arc(26.6:116.6:1);%Rechter Winkel bei UR1 + \fill [black!50!](0,0)++(-63.4:6.71)+(71.6:.5)circle(0.1); + \draw [black!50!](0,0)++(56.3:8.32)+(-33.7:1)arc(-33.7:-123.7:1);%Rechter Winkel bei UR2 + \fill [black!50!](0,0)++(56.3:8.32)+(-78.7:.5)circle(0.1); + \draw [->,red](0,0)--(-63.4:3.34)node [left]{$\uline{I}_1$}; + \draw [->,red](0,0)--(56.3:5.2)node [left]{$\uline{I}_2$}; + \draw [<-,red,thin](0:0)++(56.3:5.2)++(-63.4:3.34)--(-63.4:3.34)node at(3,-.5) [left]{$\uline{I}_2$}; + \draw [<-,red,ultra thick](0:0)++(56.3:5.2)++(-63.4:3.34)--(0:0)node at (6,.6)[above]{$\uline{I}=\uline{I}_1+\uline{I}_2$}; + \draw [<-,black](0,0)+(17.0:3)arc(17.0:0:3)node at (1.0,1.5)[right]{$\varphi_{\uline{I}}=17\,\degree$};%Winkel I + \draw node at (3,-6)[below]{$A$}; + \draw node at (4.8,7)[above]{$B$}; + \draw node at (16,0)[above right]{$\Re (A)$}; + \draw [<->,black!50!](0,0)+(-63.4:1)arc(-63.4:53.3:1);%Winkel UR1 UR2 + \draw node at (.7,-.35)[below right]{$\varphi_1$}; + \draw node at (.7,.35)[above right]{$\varphi_2$}; + \draw [->,black!75!](3,-6)+(0:2)arc(0:83:2)node at (5,-5.5)[right]{$\varphi_{\uline{U}} =83\,\degree$};%U - UAB + \draw node at (20.8,5) [right]{$\uline{I}_1 \approx 3{,}4\,\centi\metre$}; + \draw node at (20.8,4) [right]{$\uline{I}_2 \approx 5{,}2\,\centi\metre$}; + \draw node at (20.8,3) [right]{$\uline{I}=\uline{I_1}+\uline{I_2}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B14_A4.tex b/ET2_L_B14_A4.tex new file mode 100644 index 0000000..19d18c2 --- /dev/null +++ b/ET2_L_B14_A4.tex @@ -0,0 +1,87 @@ +\section{Zeigerdiagramm Netzwerk} +Zeichnen Sie für das abgebildete Netzwerk ein maßstäbliches Zeigerdiagramm aller Spannungen und Ströme.\\ +Wie groß muß der Widerstand $R_1$ sein damit der Strom $I$ der Spannung $U$ um $30\,\degree$ nacheilt?\\[\baselineskip] +$\varphi_u - \varphi_i = 30\,\degree$\\ +$R_2 = |X_L| = |X_C| = 10\,\kilo\ohm$\\ +$\uline{I}_C = 1\,\milli\ampere\cdot e^{j90\,\degree}$\\[\baselineskip] +\uline{Maßstäbe:}\\ +$1\,\volt \,\widehat{=}\,0{,}8\centi\metre$\\ +$1\,\milli\ampere \,\widehat{=}\,5\centi\metre$ +\begin{align*} + \begin{tikzpicture}[very thick,scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.,0)--(2,0)--(2,.2) (.8,1)--(2,1)--(2,.8); + \draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}$}; + \fill (-.5,0)circle(.025) (-.5,1)circle(.025); + \draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}$}; + \draw [->,red] (1.1,.9)--(1.1,.6) node at (1.1,.75)[right]{$\underline{I}_C$}; + \draw [->,black!50!] (1.5,.9)--(1.5,.1)node at(1.5,.5)[right]{$\underline{U}_{R_2}$}; + \draw [->,black!50!] (1.4,1)--(1.6,1) node at (1.5,1)[above]{$\underline{I}_2$}; + \draw [->,black!50!] (.3,.9)--(.7,.9) node at (.5,.9)[below]{$\underline{U}_L$}; + \draw [->,black!50!] (.75,1)--(.95,1) node at (.85,1)[above]{$\underline{I}_L$}; + \draw [->,black!50!] (0,.95)--(0,.75) node at (0,.85)[left]{$\underline{I}_{R_1}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align*} +\intertext{Berechnung: (Platz in $x=\pm10\,\centi\metre$ und $x=10\,\centi\metre$} +R_2=|X_C|&=10\,\kilo\ohm\text{ (Stomteiler, mit gleichem Betrag des Stroms)}\\ +\text{mit }|\uline{I}_{R_2}| &=|\uline{I}_C|=1\,\milli\ampere\\ +\uline{I}_C &= 1\,\milli\ampere\cdot e^{j90\,\degree}\Rightarrow\uline{I}_{R_2}=1\,\milli\ampere\cdot e^{j0\,\degree}\text{ $(\uline{I}_C$ eilt vor)}\\ +\uline{U}_{R_2}&=R_2\cdot \uline{I}_{R_2}=10\,\cancel{\kilo}\ohm\cdot 1\,\cancel{\milli}\ampere\cdot e^{j0\,\degree}=10\,\volt\cdot e^{j0\,\degree}\,\widehat{=}\,8\centi\metre\\ +\uline{I}_L&=\uline{I}_{R_2}+\uline{I}_C =1\,\milli\ampere\cdot e^{j0\,\degree}+1\,\milli\ampere\cdot e^{j90\,\degree}=1{,}41\,\milli\ampere\cdot e^{j45\,\degree}\\ +\uline{U}_L&=\uline{I}_L\cdot j\cdot X_L=1{,}41\,\milli\ampere\cdot e^{j45\,\degree}\cdot j10\,\kilo\ohm =14{,}1\,\volt\cdot e^{j135\,\degree}\,\widehat{=}\,11{,}3\centi\metre\text{ $(\uline{U}_L$ voreilend)}\\ +\uline{U}&=\uline{U}_{L}+\uline{U}_{R_2}=14{,}1\,\volt\cdot e^{j135\,\degree}+10\,\volt\cdot e^{j0\,\degree} +=(-10+j10+10)\,\volt=10\,\volt\cdot e^{j90\,\degree}\\ +\text{Zeichnen: }\varphi_u-\varphi_i&=30\,\degree\text{ deshalb $30\,\degree$, Linie zeichnen, Schnittpunkt mit } \uline{I}_L+\uline{I}_{R_1}\Rightarrow\uline{I}=2\,\milli\ampere\cdot e^{j30\,\degree}\\ +\text{Ablesen: }\qquad\,\,\uline{I}_{R_1}&=0{,}72\,\milli\ampere\cdot e^{j90\,\degree}\\ +R_1&=\frac{\uline{U}}{\uline{I}_{R_1}}=\frac{10\,\volt\cancel{\cdot e^{j90\,\degree}}}{0{,}72\,\milli\ampere\cancel{\cdot e^{j90\,\degree}}}=\uuline{13{,}89\,\kilo\ohm}\\ +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=.7] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [black!15!,very thin](-10,0)grid(10,10); + \draw [->,black!25!](-10,0)--(10.5,0); + \draw [->,black!25!](0,0)--(0,10.5); + \draw node at (10.5,0)[right]{$\Re$}; + \draw node at (0,10.5)[above]{$\Im$}; + \draw [black!75!,thick](0:0)--(60:11)node [right]{$30\,\degree$ Linie}; + \draw [->,blue] (0,0)--(90:8)node [below right]{$\underline{U}$}; + \draw [->,red] (0,0)--(90:5)node [below right]{$\underline{I}_C$}; + \draw [->,red] (0,0)--(0:5)node [above left]{$\underline{I}_R$}; + \draw [->,red] (0,0)--(45:7.07)node [below right]{$\underline{I}_L$}; + \draw [->,red!50!blue] (5,5)--(5,8.65)node [below right]{$\underline{I}_{R_1}=0{,}72\,\milli\ampere\,\widehat{=}\,3{,}6\centi\metre$}; + \draw [->,red!50!blue](0:0)--(60:10)node at (3,7.8) [above]{$\uline{I}=2\,\milli\ampere$}; + \draw [->,blue] (0,0)--(135:11.28)node [below left]{$\underline{U}_L$}; + \draw [->,blue] (-8,8)--(0,8)node [above left]{$\underline{U}_{R_2}$}; + \draw [->,black!75!](0:0)+(90:2)arc(90:60:2)node at(.6,2)[above]{$30\,\degree$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{I}_{R_2}&=1\,\milli\ampere\cdot e^{j0\,\degree} \,\widehat{=}\ 5\,\centi\metre\\ +\uline{I}_{C}&=1\,\milli\ampere\cdot e^{j90\,\degree} \,\widehat{=}\ 5\,\centi\metre\\ +\uline{I}_L&=\uline{I}_{R_2}+\uline{I}_C \,\widehat{=}\ 7{,}05\,\centi\metre\\ +\uline{U}_{L}&=14{,}1\,\volt\cdot e^{j135\,\degree}\,\widehat{=}\ 11{,}3\centi\metre \quad \uline{U}_{L}\bot \uline{I}_{L}\\ +\uline{R}_{R_2}&=10\,\volt\cdot e^{j0\,\degree} \,\widehat{=}\ 8\centi\metre\\ +\uline{U}&=\uline{U}_{L}+\uline{U}_{R_2}\\ +&\text{Gerade für I, $30\degree $ nacheilend}\\ +\uline{I}_{L}&+\uline{I}_{R_1}=\uline{I};\qquad \uline{I}_{R_1}||\uline{U}\\ +\rightarrow\ &\text{ ablesen } 3{,}6\,\centi\metre\ \widehat{=}\ \uline{I}_{R_1}=0{,}72\,\milli\ampere\cdot e^{j90\,\degree}\\ +\end{align*} +\clearpage +}{}% \ No newline at end of file diff --git a/ET2_L_B14_A5.tex b/ET2_L_B14_A5.tex new file mode 100644 index 0000000..960b072 --- /dev/null +++ b/ET2_L_B14_A5.tex @@ -0,0 +1,180 @@ +\section{Blind- Wirk- und Scheinleistung} +Von dem untenstehenden Schaltbild ist gegeben:\\ +$R_L = X_L = 20\,\ohm \quad R_C=200\,\ohm \quad X_C=-100\,\ohm\quad\uline{U}=230\,\volt\cdot e^{j0\,\degree}$\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Der Eingangswiderstand Z der Schaltung nach Betrag und Phasenwinkel +\item Sämtliche Ströme und Spannungen nach Betrag und Phasenwinkel +\item Wirk- Blind- und Scheinleistungsaufnahme der Schaltung +\item Qualitatives Zeigerdiagramm aller Ströme und Spannungen unter der Annahme, daß sich +die Gesamtschaltung induktiv verhält. +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_L$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$}; + \end{scope} +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 +% \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; +% \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.,0)--(3,0)--(3,.2) (.8,1)--(3,1)--(3,.8); + \draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}$}; + \fill (-.5,0)circle(.025) (-.5,1)circle(.025); + \draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}$}; + \draw [->,red] (2.1,.9)--(2.1,.7) node at (2.1,.8)[right]{$\underline{I}_{R_C}$}; + \draw [->,red] (3.1,.9)--(3.1,.7) node at (3.1,.8)[right]{$\underline{I}_C$}; + \draw [->,black!50!] (.3,.8)--(.7,.8) node at (.5,.8)[below]{$\underline{U}_{RL}$}; + \draw [->,black!50!] (1.3,.8)--(1.7,.8) node at (1.5,.8)[below]{$\underline{U}_L$}; + \draw [->,black!50!] node at (-.5,.5)[left]{$\uline{Z} =>$}; + \draw [|-|,black!50!] (2.,-.2)--(3,-.2) node at (2.5,-.2)[below]{$\uline{Z}_{||}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +e^{j\varphi}&=\cos\varphi+j\sin\varphi \quad\text{Eulersche Formel}\\ +\uline{Z}&=Z\cdot e^{\pm j\varphi}=R\pm jX\\ +\cos\varphi&=\frac{R}{Z}\\ +\sin\varphi&=\frac{X}{Z} +\end{align} +Berechnung: +\begin{align*} +\intertext{a) Eingangswiderstand (ist ohmisch-kapazitiv)} +\uline{Z}_{||}&=\frac{R_C\cdot jX_C}{R_C+jX_C}=\frac{200\cdot (-j100)}{200-j100}\,\ohm=(40-j80)\,\ohm\\ +\uline{Z}&=\uline{Z}_{||}+R_L+X_L=[20+40+j(20-80)]\,\ohm=\uuline{(60-j60)\,\ohm}=\uuline{84{,}85\,\ohm\cdot e^{-j45\,\degree}}\\ +&(\Rightarrow \varphi=-45\,\degree) +\end{align*} +\clearpage +\begin{align*} +\intertext{b) Ströme} +\uline{I}&=\frac{\uline{U}}{\uline{Z}}=\frac{230\,\volt\cdot e^{j0\,\degree}}{84{,}5\,\ohm\cdot e^{-j45\,\degree}}=\uuline{2{,}71\,\ampere\cdot e^{j45\,\degree}}=\uuline{(1{,}916+j1{,}916)\,\ampere} +% \end{align*} +% \clearpage +% \begin{align*} +\intertext{Stromteiler} +\uline{I}_{R_C}&=\uline{I}\cdot \frac{jX_C}{R_C+jX_C}=\uline{I}\cdot \frac{-j100}{200-j100} +=\uline{I}\cdot \frac{-j}{2-j}\cdot \frac{2+j}{2+j}=\uline{I}\cdot \frac{1-j2}{4+1}=\uline{I}\cdot (0{,}2-j0{,}4)\\ +&=2{,}71\,\ampere\cdot e^{j45\,\degree}\cdot 0{,}447\cdot e^{-j63{,}4\,\degree}=\uuline{1{,}21\,\ampere\cdot e^{-j18{,}4\,\degree}}=\uuline{(1{,}150-j0{,}383)\,\ampere}\\%Ende IRC +\uline{I}_{C}&=\uline{I}-\uline{I}_{R_C}=(1{,}916+j1{,}916)\,\ampere -(1{,}150-j0{,}383)\,\ampere=\uuline{(0{,}766+j2{,}30)\,\ampere}\\ +&=\uuline{2{,}42\,\ampere\cdot e^{+j71{,}6\,\degree}} \\ +\text{ alternativ}&\\ +\uline{I}_{C}&=\uline{I}\cdot \frac{R_C}{R_C+jX_C}=\uline{I}\cdot \frac{200}{200-j100}=\uline{I}\cdot \frac{1}{1-j0{,}5}\quad\text{ konjugiert komplex erweitern}\\ +&=\uline{I}\cdot \frac{1}{1-j0{,}5}\cdot \frac{1+j0{,}5}{1+j0{,}5} +=\uline{I}\cdot \frac{1+j0{,}5}{1+0{,}5^2} +=\uline{I}\cdot (0{,}8+j0{,}4)=\uline{I}\cdot 0{,}894\cdot e^{0{,}5j}\\ +&=2{,}71\,\ampere\cdot e^{j45\,\degree}\cdot 0{,}894\cdot e^{j26{,}6\,\degree}=\uuline{2{,}42\,\ampere\cdot e^{+j71{,}6\,\degree}}=\uuline{(0{,}766+j2{,}30)\,\ampere} +\intertext{Spannungen} +\uline{U}_{R_C}&=R_C\cdot \uline{I}_{R_C}=200\,\ohm\cdot 1{,}21\,\ampere\cdot e^{-j18{,}4\,\degree}=\uuline{242\,\volt\cdot e^{-j18{,}4\,\degree}}\\ +\uline{U}_{R_L}&=R_L\cdot \uline{I}=20\,\ohm\cdot 2{,}71\,\ampere \cdot e^{j45\,\degree}=\uuline{54{,}2\,\volt\cdot e^{j45\,\degree}}\\ +\uline{U}_L&=X_L\cdot \uline{I}=20\,\ohm\cdot e^{j90\,\degree} \cdot 2{,}71\,\ampere \cdot e^{j45\,\degree}=\uuline{54.2\,\volt\cdot e^{j135\,\degree}} +\end{align*} +\clearpage +\begin{align*} +\intertext{Zeigerdiagramm: Beginne mit $\uline{U}$ und $\uline{I}\qquad (50\,\volt\per\centi\metre$; $1\,\ampere\per\centi\metre$)} +&\uline{I} \text{ um Winkel $\varphi=45\,\degree$ voreilend, kapazitives Gesamtverhalten.}\\ +&\uline{I}=\uline{I}_{R_C}+\uline{I}_C \qquad(\uline{I}_C \,\bot\, \uline{I}_{R_C}) \qquad (\uline{I}_C \text{ voreilend})\\ +&\uline{U}_{R_L}\,||\,\uline{I}\\ +&\uline{U}_L\,\bot\, \uline{I} \qquad (\uline{U}_L\text{ voreilend})\\ +&\uline{U}_{R_C}\,||\,\uline{I}_{R_C}\\ +&\uline{U}_{R_L}=\uline{U}_L+\uline{U}_{R_C}=\uline{U}\ \widehat{=}\ 4{,}6\,\centi\metre +\end{align*} + %Trennzeile +\begin{align*} + \begin{tikzpicture}[very thick,scale=2.] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [black!25!,very thin](0,-1)grid(5,2); + \draw [->](0,0)--(5.2,0)node[right]{$\Re$}; + \draw [->](0,-1)--(0,2.2)node[above]{$\Im$}; + \draw [->,blue] (0,0)--(0:4.6)node [below left]{$\underline{U}$}; + \draw [->,red] (0,0)--(45:2.71)node at(45:2.3) [above left]{$\underline{I}$}; + \draw [->,red] (0,0)--(-18.4:1.21)node at(1.5,.5)[right]{$\underline{I}_C$}; + \draw [->,red] (0,0)++(-18.4:1.21)--+(71.6:2.42)node at(.6,-.3)[below]{$\underline{I}_{R_C}$}; + \draw [black!50!] (0,0)++(-18.4:1.21)+(71.6:.25)arc(71.6:161.6:.25);%Rechter Winkel + \fill [black!50!] (0,0)++(-18.4:1.21)+(116.6:.125)circle(.025);%Rechter Winkel + \draw [->,blue] (0,0)--(45:1.084)node at(0,.6)[right]{$\underline{U}_{R_L}$}; + \draw [black!50!] (0,0)++(45:1.084)+(225:.25)arc(225:135:.25);%Rechter Winkel + \fill [black!50!] (0,0)++(45:1.084)+(180:.125)circle(.025);%Rechter Winkel + \draw [->,blue] (0,0)++(45:1.084)--+(135:1.084)node at(3,.5)[above right]{$\underline{U}_{R_C}$}; + \draw [<-,blue] (0,0)++(0:4.6)--+(161.6:4.84)node at(0,1)[right]{$\underline{U}_L$}; + \draw [->,black!75!] (0,0)+(0:.5)arc(0:45:.5)node at(22.5:.5) [right]{$\varphi =45\,\degree$}; + \end{scope} + \begin{scope}[>=latex,xshift=0.55cm,yshift=-.1cm]%Parallelen + \draw [double](0:-.1)--+(-108.4:.1); + \end{scope} + \begin{scope}[>=latex,xshift=2.45cm,yshift=.8cm]%Parallelen + \draw [double](0:-.1)--+(-108.4:.1); + \end{scope} + \end{tikzpicture} +\end{align*} + %Trennzeile +\begin{align*} +\intertext{c) Scheinleistung} +S&=U\cdot I=230\,\volt\cdot 2{,}71\,\ampere=\uuline{623\,\volt\ampere} &\text{Scheinleistung}\\ +P&=S\cdot \cos \varphi=623\,\volt\ampere\cdot \cos (-45\,\degree)=\uuline{447\,\watt} &\text{Wirkleistung}\\ +Q&=S\cdot \sin \varphi=623\,\volt\ampere\cdot \sin (-45\,\degree)=\uuline{-447\,\volt\ampere r} &\text{Blindleistung}\\ +\end{align*} +\clearpage +d) Annahme, daß sich die Gesamtschaltung induktiv verhält. +\begin{align*} +% \intertext{d) Annahme, daß sich die Gesamtschaltung induktiv verhält.} +&\text{Reihenfolge: $\qquad (25\,\volt\per\centi\metre$; $1\,\ampere\per\centi\metre$)}\\ +&\text{Beginne mit $\uline{U}=230\,\volt\cdot e^{j0\,\degree}$ und $\uline{I}=2{,}71\,\ampere\cdot e^{-j45\,\degree}$}\\ +&\uline{I} \text{ um Winkel $\varphi=-45\,\degree$ nacheilend, da induktives Gesamtverhalten.}\\ +&\uline{I}=\uline{I}_{R_C}+\uline{I}_C \qquad(\uline{I}_C \,\bot\, \uline{I}_{R_C}) \qquad (\uline{I}_C \text{ voreilend}) \qquad [\text{Thaleskreis}]\\ +&\uline{U}_{R_C}\,||\,\uline{I}_{R_C} \hspace{14em} [\text{Gerade von der Spitze }\underline{U}]\\ +&\uline{U}_{R_L}\,||\,\uline{I} \hspace{15em}\, [\text{ Gerade } \bot\ \underline{U}_{RL}]\\ +&\uline{U}_L\,\bot\, \uline{I} \qquad (\uline{U}_L\text{ voreilend})\\ +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [black!25!,very thin,scale=.5](0,-6)grid(10,4); + \draw [->](0,0)--(5.2,0)node[right]{$\Re$}; + \draw [->](0,-3)--(0,2.2)node[above]{$\Im$}; + \draw [->,blue] (0,0)--(0:4.6)node [below right]{$\underline{U}$}; + \draw [->,red] (0,0)--(-45:2.71)node [above right]{$\underline{I}$}; + \draw [->,red] (0,0)++(-71.6:2.42)--+(18.4:1.21)node at(1.5,-2.2)[below]{$\underline{I}_{C}$}; + \draw [black!50!] (0,0)++(-71.6:2.42)+(18.4:.25)arc(18.4:108.4:.25);%Rechter Winkel + \fill [black!50!] (0,0)++(-71.6:2.42)+(63.4:.125)circle(.02);%Rechter Winkel + \draw [->,red] (0,0)--(-71.6:2.42)node at(.5,-1.5)[left]{$\underline{I}_{R_C}$}; + \draw [->,blue] (0,0)--(-45:2.168)node at(1,-1)[above right]{$\underline{U}_{R_L}$}; + \draw [black!50!] (0,0)++(-45:2.418)arc(-45:45:.25);%Rechter Winkel + \fill [black!50!] (0,0)++(-45:2.168)+(0:.125)circle(.02);%Rechter Winkel + \draw [->,blue] (0,0)++(-45:2.168)--+(45:3.8)node at(3.5,.5)[left]{$\underline{U}_L$}; + \draw [<-,blue] (0,0)++(0:4.6)--+(-251.6:1.21)node at(4.5,.5)[right]{$\underline{U}_{R_C}$}; + \draw [->,black!75!] (0,0)+(0:.5)arc(0:-45:.5)node at(-22.5:.5) [right]{$\varphi =-45\,\degree$}; + \end{scope} + \begin{scope}[>=latex,xshift=.38cm,yshift=-1cm]%Parallelen + \draw [double](0:-.1)--+(18.4:.1); + \end{scope} + \begin{scope}[>=latex,xshift=4.38cm,yshift=.8cm]%Parallelen + \draw [double](0:-.1)--+(18.4:.1); + \end{scope} + \fill [black!60!](-45:1.355)circle(.05); %Mittelpunkt Thaleskreis 2 + \draw [very thin](-45:2.71)arc(-45:-225:1.355); %Tahleskreis 2 + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{U}&=\uuline{230\,\volt \cdot e^{j0\,\degree}} \ \widehat{=}\ 9{,}2\,\centi\metre\\ +\uline{I}&=\uuline{2{,}71\,\ampere\cdot e^{-j45\,\degree}}\ \widehat{=}\ 2{,}71\,\centi\metre\\ +\uline{I}_{R_C}&=\uuline{2{,}42\,\ampere\cdot e^{-j71{,}6\,\degree}}\ \widehat{=}\ 2{,}42\,\centi\metre\\ +\uline{I}_C&=\uuline{1{,}21\,\ampere\cdot e^{j18{,}4\,\degree}}\ \widehat{=}\ 1{,}21\,\centi\metre\\ +\uline{U}_{R_L}&=\uuline{108{,}4\,\volt \cdot e^{-j45\,\degree}}\ \widehat{=}\ 4{,}3\,\centi\metre\\ +\uline{U}_{R_C}&=\uuline{60{,}4\,\volt \cdot e^{-j71{,}6\,\degree}}\ \widehat{=}\ 2{,}4\,\centi\metre\\ +\uline{U}_L&=\uuline{190\,\volt \cdot e^{j45\,\degree}}\ \widehat{=}\ 7{,}6\,\centi\metre +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A1.tex b/ET2_L_B15_A1.tex new file mode 100644 index 0000000..d1c6f5d --- /dev/null +++ b/ET2_L_B15_A1.tex @@ -0,0 +1,52 @@ +\section{Komplexe Wechselstromrechnung Netzwerk Strom} +Berechnen sie den Strom $I_2$\\ +$I_0=3\,\milli\ampere \cdot e^{-j30\,\degree}\qquad C_1= 1{,}2\,\nano\farad \qquad L=100\,\micro\henry$\\ +$R=9,3\,\ohm \qquad C_2=820\,\pico\farad \qquad f=570\,\kilo\hertz$\\ +Hinweis: Rechnung nur mit komplexen Größen\\ +Kann $I_2$ größer als $I_0$ sein? Wenn ja, warum?\\ +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-.5cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$I_0$}; + \draw (.5,0)circle(.133); + \draw [<-,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{I}_{0}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=-1cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [->,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{$\uline{I}_{1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-.5cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$}; + \draw [<-,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{I}_{2}$}; + \draw [->,red] (-.2,.2)--(.2,.2) node at (0,.2)[left]{$\uline{I}_{2'}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-.25cm]%Fehlstellen Eckverbindungen. + \draw (0,.75)--(0,1.25)--(.2,1.25) (2,.75)--(2,1.25)--(.8,1.25) (0,0)--(0,-1)-- (2,-1)--(2,0) (1,-1)--(1,-.5); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Berechnung:} +C_1&\text{ spielt keine Rolle, da in Reihe mit Stromquelle.}\\ +\uline{I_2}&=-\uline{I}_2'\quad\text{, wegen Knotenpunkt $\uline{I}_0-\uline{I}_1'-\uline{I}_2'=0$}\\ +\uline{I_2}&=-\uline{I}_2'=-\frac{R+jX_L}{R+j(X_L+X_{C_2})}\cdot \uline{I}_0\\ +\text{mit }X_{C_2}&=\frac{-1}{\omega C_2}=-\frac{1}{2\pi\cdot 570\cdot \power{10}{3}\cdot 820\cdot \power{10}{-12}}\,\ohm=-340{,}51\,\ohm\\ +X_L&=\omega \cdot L=2\pi\cdot 570\cdot \power{10}{3}\cdot \power{10}{-4}\,\ohm= +358{,}14\,\ohm\\ +X_{C_2}+X_L&=(-340{,}51+358{,}14)\,\ohm=17{,}63\,\ohm\\ +\uline{I_2}&=-\frac{9{,}3\,\ohm +j 358{,}14\,\ohm}{9{,}3\,\ohm +j 17{,}63\,\ohm}\cdot 3\,\milli\ampere \cdot e^{-j30\,\degree}=(-16{,}11+j7{,}97)\cdot 3\,\milli\ampere\cdot e^{-j30\,\degree}\\ +&=17{,}97\cdot e^{-j153{,}7\,\degree}\cdot 3\,\milli\ampere\cdot e^{-j30\,\degree}=\uuline{53{,}92\,\milli\ampere\cdot e^{j176{,}32\,\degree}}\\ +\uline{I}_2&>\uline{I}_0\text{, da sehr nahe an Resonanz $(X_L\approx |X_C|$)} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A2.tex b/ET2_L_B15_A2.tex new file mode 100644 index 0000000..ab94847 --- /dev/null +++ b/ET2_L_B15_A2.tex @@ -0,0 +1,73 @@ +\section{Übergang Zeitabhängige zu Komplexen Größen} +Bestimmen Sie den momentanen Strom $i_{L}(t)$ zum Zeitpunkt $t=T$\\[\baselineskip] +$u(t)=U_0+\widehat{U}_1\cdot cos(\omega t+\varphi)$\\[\baselineskip] +$U_0 = 2\,\volt \qquad \widehat{U}_1=3\,\volt \qquad f=20\,\kilo\hertz \qquad \varphi= 50\,\degree$\\[\baselineskip] +$C = 130\,\nano\farad \qquad R = 60\,\ohm \qquad L = 480\,\micro\henry \qquad T= 26\,\micro\second$\\[\baselineskip] +\uline{Hinweise:} An dieser Aufgabe sollen Sie den Übergang von der realen, zeitabhängigen Größe +$u(t)$ in komplexe Größen $\uline{U}$, $\uline{I}$ und $\uline{I}_L$ und wieder zurück in die reale Größe $i_L(t)$ lernen. (Sehr grundsätzliche und wichtige Übung !)\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Wie wirkt sich der Gleichanteil der Spannung $U_0$ aus? +\item Übergang vom Zeitbereich in komplexen \glqq Bild\grqq -Bereich $u(t)\Rightarrow \uline{U}$ +\item Berechnen Sie als Zwischenschritt den komplexen Strom $\uline{I}_L$! +\item Übergang von komplexen Bereich in Zeitbereich $\uline{I}_L\Rightarrow i_L(t)$ +\item Achten Sie auf die Darstellung des Phasenwinkels $\omega t+\varphi$. (in Grad oder rad?!!) +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\text{...noch einfügen...} +%\end{align} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$u(t)$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$u(t)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$u_{1}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.75,.0)--(.95,.0)node at(.85,.0)[left]{$i_L(t)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,.9)--(0,1)--(.2,1) (.8,1)--(2,1)--(2,.9) (0,.2)--(0,0)--(2,0)--(2,.2); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Berechnung:} +&\text{a) $U_0$ spielt wegen $C$ keine Rolle - kein Gleichstrom!}\\ +\intertext{Transformation in komplexen \glqq Bildbereich\grqq $u(t)\rightarrow\uline{U}$} +\uline{U}&=\frac{\widehat{U_1}}{\sqrt{2}}\cdot e^{j\varphi}=\frac{3\,\volt}{\sqrt{2}}\cdot e^{j50\,\degree}=2{,}12\,\volt\cdot e^{j50\,\degree}\\ +\uline{Z}_C&=\frac{-1}{\omega C}=\frac{-j1}{2\pi\cdot 20\cdot \power{10}{3}\frac{1}{\,\second}\cdot 130\cdot \power{10}{-9}\,\frac{\ampere\second}{\volt}}=-j61{,}21\,\ohm\\ +\uline{Z}_L&=j\omega L=j2\pi\cdot 20\cdot \power{10}{3}\frac{1}{\,\second}\cdot 480\cdot \power{10}{-6}\,\frac{\volt\second}{\ampere}=+j60{,}32\,\ohm\\ +\uline{Y}_{||}&=\frac{1}{R}+\frac{1}{j\omega L}=(\frac{1}{60}+\frac{1}{j60{,}32})\,\siemens=(16{,}67-j16{,}58)\,\milli\siemens\\ +\uline{Z}_{||}&=\frac{1}{Y}_{||}=(30{,}16+j30)\,\ohm\\ +\uline{U}_1&=\uline{U}_L=\uline{U}\cdot\frac{\uline{Z}_{||}}{\uline{Z}_C+\uline{Z}_{||}} +=2{,}12\,\volt\cdot e^{j50\,\degree}\cdot \frac{(30{,}16+j30)\,\ohm}{-j61{,}21\,\ohm+(30{,}16+j30)\,\ohm}\\ +&=2{,}12\,\volt\cdot e^{j50\,\degree}\cdot \frac{(30{,}16+j30)\,\ohm}{(30{,}16-j31{,}21)\,\ohm}\\ +&=2{,}12\,\volt\cdot e^{j50\,\degree}\cdot (-0{,}0142+j0{,}98)=2{,}12\,\volt\cdot e^{j50\,\degree}\cdot 0{,}98\cdot e^{j90{,}83\,\degree}\\ +&=2{,}078\,\volt\cdot e^{j140{,}83\,\degree} +\intertext{c) komplexer Strom} +\uline{I}_L&=\frac{\uline{U}_{||}}{\uline{Z}_L}=\frac{2{,}078\,\volt\cdot e^{j140{,}83\,\degree}}{60{,}32\,\ohm\cdot e^{j90\,\degree}}=\uuline{34{,}45\,\milli\ampere\cdot e^{j50{,}83\,\degree}} +\intertext{d) Übergang in den Zeitbereich} +i_L(t)&=\sqrt{2}\cdot |\uline{I}_L|\cdot \cos(\omega T+\varphi_{i_L}) +=\sqrt{2}\cdot 34{,}45\,\milli\ampere\cdot \cos(2\pi \cdot 20\cdot \power{10}{3}\frac{1}{\,\second}\cdot t+50{,}83\,\degree)\\ +&=48{,}72\,\milli\ampere\cdot \cos(4 \cdot \power{10}{4}\frac{1}{\,\second}+50{,}83\,\degree) +\intertext{e)} +&\text{mit } T=26\,\micro\second\\ +i_L(T)&=48{,}72\,\milli\ampere\cdot \cos(\underbrace{3{,}267}_{\rad}+50{,}83\,\degree)\qquad\text{Achtung! }\\ +&=48{,}72\,\milli\ampere\cdot \cos(3{,}267\cdot \frac{360\,\degree}{2\pi}+50{,}83\,\degree)\\ +&=48{,}72\,\milli\ampere\cdot \cos(187{,}2\,\degree+50{,}83\,\degree)=48{,}72\,\milli\ampere\cdot \cos(238\,\degree)\\ +i_L(T)&=48{,}72\,\milli\ampere\cdot \cos(238\,\degree)=48{,}72\,\milli\ampere\cdot (-0{,}529)=\uuline{-25{,}79\,\milli\ampere}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A2rekap.tex b/ET2_L_B15_A2rekap.tex new file mode 100644 index 0000000..303b33e --- /dev/null +++ b/ET2_L_B15_A2rekap.tex @@ -0,0 +1,71 @@ +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\section*{Rekapitulieren} +\textbf{Zusammenfassung} An Tafel rekapitulieren\\[\baselineskip] +\uline{1. Transformation:}$\quad u(t) \Rightarrow \uline{U}$\\[\baselineskip] +\uline{2. Lösung komplexer algebraischer Gleichungen:} Rechnung mit komplexem Effektivwert.\\[\baselineskip] +\uline{3. Rücktransformation:}$\quad\uline{I}_L\Rightarrow i_L(t)$\\[\baselineskip] +%&\uline{U}\qquad \text{komplexer Effektivwert}\\ +%&U\qquad \text{Betrag $=$ Effektivwert $(U=\frac{\widehat{u}}{\sqrt{2}})$} +\begin{align*} + \begin{tikzpicture}[scale=1.25] + \begin{scope}[>=latex, xshift=0cm, yshift=3.5cm] + \draw node at (7,.5)[right]{$\uline{U}$ komplexer Effektivwert}; + \draw node at (7,0)[right]{$U=|\uline{U}|=\frac{\widehat{u}}{\sqrt{2}}$ Effektivwert}; + \end{scope} + \begin{scope}[>=latex, xshift=0cm, yshift=-7cm] + \draw node at (0,0)[right]{$\widehat{i}_L\qquad$ Scheitelwert $(\,\widehat{i}_L={\sqrt{2}}\cdot I_L)$}; + \draw node at (0,-.5)[right]{$\uline{I_L}\qquad$ komplexer Effektivwert}; + \end{scope} + \begin{scope}[>=latex, xshift=0cm, yshift=0cm] + \draw (0,0)rectangle(4,2)node at(2,2)[above] {Zeitbereich}; + \draw node at (2,1){$u(t)=\widehat{u}\cdot \cos(\omega t+\varphi_u)$}; + \end{scope} + \begin{scope}[>=latex, xshift=7cm, yshift=0cm] + \draw (0,0)rectangle(4,2)node at(2,2)[above] {Komplexer Bildbereich}; + \draw node at (2,1.25){$\uline{U}=U\cdot e^{j\varphi_u}$}; + \draw node at (2,.75){mit $U=\frac{\widehat{u}}{\sqrt{2}}$}; + \end{scope} + \begin{scope}[>=latex, xshift=0, yshift=-3cm] + \draw (0,0)rectangle(4,2); + \draw node at (2,1.5){Lösen von}; + \draw node at (2,1){Differentialgleichungen}; + \draw node at (2,.5){ist schwierig}; + \end{scope} + \begin{scope}[>=latex, xshift=7cm, yshift=-3cm] + \draw (0,0)rectangle(4,2); + \draw node at (2,1.5){Lösen von}; + \draw node at (2,1){komplexen algebraischen}; + \draw node at (2,.5){Gleichungen}; + \end{scope} + \begin{scope}[>=latex, xshift=0, yshift=-6cm] + \draw (0,0)rectangle(4,2); + \draw node at (2,1.5){Beachte $\omega t$ [rad] $\varphi$ [\degree]}; + \draw node at (2,1){$i_L(t)=\widehat{i}_L\cdot \cos(\omega t+\varphi_{I_L})$}; + \draw node at (2,.5){mit $\widehat{i}_L=\sqrt{2}\cdot \uline{I}_L$}; + \end{scope} + \begin{scope}[>=latex, xshift=7cm, yshift=-6cm] + \draw (0,0)rectangle(4,2); + \draw node at (2,1){$\uline{I}_L=I_L\cdot e^{j\varphi_{I_L}}$}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0cm] + \draw [->,red,dashed](2,0)--(2,-1); + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0cm] + \draw [->](4,1)--(7,1)node at (5.5,1)[above]{Transformation}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=-3cm] + \draw [->,red,dashed](2,0)--(2,-1); + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=-6cm] + \draw [<-](4,1)--(7,1)node at (5.5,1)[above]{Rücktransformation}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=7cm, yshift=0cm] + \draw [->](2,0)--(2,-1); + \end{scope} + \begin{scope}[>=latex,very thick, xshift=7cm, yshift=-3cm] + \draw [->](2,0)--(2,-1); + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +}{}% \ No newline at end of file diff --git a/ET2_L_B15_A3.tex b/ET2_L_B15_A3.tex new file mode 100644 index 0000000..1ce3330 --- /dev/null +++ b/ET2_L_B15_A3.tex @@ -0,0 +1,42 @@ +\section{Leitwert} +Der Widerstand des abgebildeten Netzwerkes soll $\uline{Z}=1\,\kilo\ohm\cdot e^{j60\,\degree}$ sein.\\[\baselineskip] +Wie groß müssen $R$ und $B_C$ sein, wenn $B_L=-3{,}33\,\milli\siemens$ ist? +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\text{...noch einfügen...} +%\end{align} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$B_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$B_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (.8,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%End Knoten + \fill (0,0)circle(.02); + \fill (0,1)circle(.02); + \draw node at (0,.5)[left] {$\uline{Z}\Rightarrow$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Berechnung:} +\uline{Z}&\stackrel{!}{=}\power{10}{3}\,\ohm\cdot e^{j60\,\degree}\quad\,=(500+j866{,}25)\,\ohm\\ +\uline{Z}&=R+\frac{1}{j(B_C+B_L)}=\,\,\, R\,\,\, -j\frac{1}{(B_C+B_L)}\\[\baselineskip] +\intertext{$\Re$} +R&=\uuline{500\,\ohm}\\ +\intertext{$\Im$} +B_C+B_L&=\frac{-1}{866{,}25}\,\siemens=-1{,}1547\,\milli\siemens\\ +B_C&=-1{,}1547\,\milli\siemens -B_L=-1{,}1547\,\milli\siemens +3{,}33\,\milli\siemens =\uuline{2{,}175\,\milli\siemens}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A4.tex b/ET2_L_B15_A4.tex new file mode 100644 index 0000000..c00fc30 --- /dev/null +++ b/ET2_L_B15_A4.tex @@ -0,0 +1,58 @@ +\section{Strom L-R-C} +Berechnen Sie den Strom $\uline{I}$ +\begin{align*} +\uline{U}&=15\,\volt \cdot e^{j20\,\degree}\quad f=1\,\kilo\hertz\\ +C_1&=9\,\micro\farad\quad C_2=4\,\micro\farad\quad R=20\,\ohm\quad L=2\,\milli\henry +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\text{...noch einfügen...} +%\end{align} +\begin{align*} + \begin{tikzpicture}[scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$\uline{U}$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$}; + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}$}; + \draw [<-,red] (.75,0)--(.95,0) node at (.5,.2)[left]{$\uline{I}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,.8)--(0,1)--(2,1) (.8,1)--(3,1)--(3,.9) (0,.2)--(0,0)--(3,0)--(3,.2); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\intertext{Berechnung:} +C_1& \text{ unwirksam, da parallel zur Spannungsquelle.}\\ +\omega&=2\pi f=6283\,\frac{1}{\,\second}\\ +\uline{Z}_{RLC_2}&=X_L+\uline{Z}_{||}\\ +\uline{Y}_{||}&=G+jB_{C_2}\\ +G&=\frac{1}{R}=50\,\milli\siemens\\ +B_{C_2}&=\omega C_2=6283\frac{1}{\,\second}\cdot 4\,\micro\farad=25{,}13\,\milli\siemens\Rightarrow\\ +X_{C_2}&=-\frac{1}{B_{C_2}}=-39{,}79\,\ohm\\ +\uline{Y}_{||}&=G+jB_{C_2}=(50+j25{,}13)\,\milli\siemens=55{,}96\,\milli\siemens\cdot e^{j26{,}7\,\degree}\\ +\uline{Z}_{||}&=\frac{1}{\uline{Y}_{||}}=17{,}87\cdot e^{-j26{,}7\,\degree}=(15{,}97-j8{,}025)\,\ohm\\ +X_L&=\omega \cdot L=6283\frac{1}{\,\second}\cdot 2\,\milli\henry=12{,}57\,\ohm\\ +\uline{Z}_{LRC_2}&=\uline{Z}_{||}+jX_L=[15{,}97+j(-8{,}025+12{,}57)]\,\ohm +=(15{,}97+j4{,}541)\,\ohm=16{,}6\,\ohm\cdot e^{j15{,}9\,\degree}\\ +\uline{U}_{C_2}&=\uline{U}\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{LRC_2}}=15\,\volt \cdot e^{j20\,\degree}\cdot \frac{17{,}87\cdot e^{-j26{,}7\,\degree}}{16{,}6\,\ohm\cdot e^{j15{,}9\,\degree}}=16{,}15\,\volt\cdot e^{-j22{,}6\,\degree}=(14{,}91-j6{,}21)\,\volt\\ +\uline{I}&=\uline{U}_{C_2}\cdot jB_{C_2}=16{,}15\,\volt\cdot e^{-j22{,}6\,\degree}\cdot 25{,}13\,\milli\siemens\cdot e^{j90\,\degree}\\ +&=\uuline{0{,}4058\,\ampere\cdot e^{j67{,}4\,\degree}}=\uuline{(0{,}156+j0{,}375)\,\ampere}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A5.tex b/ET2_L_B15_A5.tex new file mode 100644 index 0000000..4291f4d --- /dev/null +++ b/ET2_L_B15_A5.tex @@ -0,0 +1,120 @@ +\section{Überlagerungsmethode} +Berechnen Sie mit der Überlagerungsmethode den Strom $\uline{I}_C$ +\begin{align*} +R&=R_L=10\,\ohm\quad L=50\,\milli\henry\quad C=100\,\micro\farad\\ +f&=50\,\hertz\quad \uline{U}_q=5\,\volt \cdot e^{j20\,\degree}\quad \uline{I}_q=2\,\ampere \cdot e^{-j60\,\degree}\\ +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=.333cm,rotate=90,scale=.75]%Spule | + \draw (.2,0)--(.3,0) (.7,0)--(0.9,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-.083cm,rotate=90,scale=.75] + \draw (0.1,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_L$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$\uline{U}_q$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_q$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$I_q$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{I}_q$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \draw [<-,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{I}_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,.98)--(0,1)--(1,1) (1.8,1)--(3,1)--(3,.9) (0,.02)--(0,0)--(3,0)--(3,.2); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\text{...noch einfügen...} +%\end{align} +Berechnung:\\[\baselineskip] +$R_L+jX_L \text{ unwirksam, da parallel zur Spannungsquelle.}$\\[\baselineskip] +$\omega=2\pi f=2\pi\cdot 50\,\frac{1}{\second}=314{,}2\,\frac{1}{\second}$ +\begin{align*} +\intertext{a) nur Spannungsquelle $\Rightarrow\uline{I}_q=0$} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$\uline{U}_q$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_q$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \draw [<-,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{I}'_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (1,.98)--(1,1)--(1.1,1) (1.8,1)--(2,1)--(2,.9) (1,.02)--(1,0)--(2,0)--(2,.2); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +B_C&=\omega \cdot C=314{,}2\frac{1}{\,\second}\cdot 100\,\micro\farad=31{,}42\,\milli\siemens\\ +X_C&=-\frac{1}{B_C}=-\frac{1}{31{,}42\,\milli\siemens}=-31{,}831\,\ohm\\ +\uline{I}'_C&=\frac{\uline{U}_q}{R+jX_C}=\frac{4{,}698+j1{,}710}{10-j31{,}83}\,\ampere=\uline{(-6{,}693+j149{,}7)\,\milli\ampere}=149{,}8\,\milli\ampere\cdot e^{j92{,}3\,\degree} +\end{align*} +\begin{align*} +\intertext{b) Nur Stromquelle $\Rightarrow\uline{U}_q=0$} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$G=\frac{1}{R}$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}''$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$I_q$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{I}_q$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \draw [<-,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{I}''_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (1,.98)--(1,1)--(2,1) (1.8,1)--(3,1)--(3,.9) (1,.02)--(1,0)--(3,0)--(3,.2); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{I}''_C&=\uline{U}''\cdot jB_C\qquad \uline{U}''=\uline{I}_q\cdot \uline{Z}_{||}\\ +&\text{oder Stromteiler } \uline{I}''_C=\uline{I_q}\cdot\frac{R}{R-jX_C}\\[\baselineskip] +\uline{I}_q&=2\,\ampere \cdot e^{-j60\,\degree}=(1-j1{,}732)\,\ampere\\ +G&=\frac{1}{R}=100\,\milli\siemens\\ +\uline{Y}_{||}&=G+jB_C=(100+j31{,}42)\,\milli\siemens=0{,}105\,\siemens \cdot e^{j17{,}44\,\degree}\\ +\uline{Z}_{||}&=\frac{1}{\uline{Y}_{||}}=9{,}54\,\ohm\cdot e^{-j17{,}44\,\degree}\\ +\uline{U}''&=\uline{I}_q\cdot \uline{Z}_{||} +%\frac{\uline{I}_q}{Y_{||}}=\frac{\uline{I}_q}{G+jB_C}= +%\frac{2\,\ampere \cdot e^{-j60\,\degree}}{(0{,}1+j0{,}03142)\,\siemens}\\ +%&=\frac{2\,\ampere \cdot e^{-j60\,\degree}}{0{,}105\,\siemens \cdot e^{-j17{,}44\,\degree}} +=2\,\ampere \cdot e^{-j60\,\degree}\cdot 9{,}54\,\ohm \cdot e^{-j17{,}44\,\degree}\\ +&=19{,}08\,\volt\cdot e^{-j77{,}44\,\degree}=(4{,}15-j18{,}62)\,\volt\\[\baselineskip] +%jB_C&=31{,}42\,\milli\siemens=31{,}42\,\milli\siemens\cdot e^{j90\,\degree}\\ +\uline{I}''_C&=\uline{U}''\cdot jB_C=19{,}08\,\volt\cdot e^{-j77{,}44\,\degree} +\cdot 31{,}42\,\milli\siemens \cdot e^{j90\,\degree}\\ +&=600\,\milli\ampere\cdot e^{j12{,}56\,\degree}=\uline{(585{,}1+j130{,}3)\,\milli\ampere}\\ +%&=\frac{\uline{I}_q\cdot X_C}{G+jB_C}\\ +%=\frac{(1-j1{,}732)\,\ampere\cdot (-j31{,}42\cdot \power{10}{-3})\,\ampere\second\per\volt}{(0{,}1+j31{,}42\cdot \power{10}{-3})\,\ampere\second\per\volt}\\ +%&=\frac{(54{,}42+j31{,}42)\cdot \power{10}{-3}}{(0{,}1+j31{,}42\cdot \power{10}{-3})}\,\ampere +%=\frac{(54{,}42+j31{,}42)}{(100+j31{,}42)\cdot }\,\ampere=(585{,}1+j130{,}3)\,\milli\ampere +&\text{oder alternativ:}\\ +\uline{I}''_C&=\uline{I}_q\cdot \frac{R}{R+jX_C}\\ +\intertext{Überlagerung} +\uline{I}_C&=\uline{I}'_C+\uline{I}''_C=(-6{,}69+j149{,}7+585{,}1+j130{,3})\,\milli\ampere\\ +&=\uuline{(578{,}41+j279{,}99)\,\milli\ampere}=\uuline{642{,}6\,\milli\ampere\cdot e^{+j25{,}38\,\degree}}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B15_A6.tex b/ET2_L_B15_A6.tex new file mode 100644 index 0000000..9570e08 --- /dev/null +++ b/ET2_L_B15_A6.tex @@ -0,0 +1,91 @@ +\section{Momentan Leistung} +In der Reihenschaltung fließt der Strom\\ +\begin{align*} +i(t)&= +\begin{cases} +&0 \text{ für }t<0\\ +&I_0\cdot \sin (\omega t) \text{ für }t\geq 0\\ +\end{cases} +\end{align*} +Der Kondensator ist zur Zeit $t=0$ entladen.\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie die momentanen Spannungen an $R$, $L$ und $C$ zur Zeit $t_1=350\,\micro\second$. +\item Welche Leistung nimmt die Schaltung in diesem Moment auf? +\item Hinweis: Rechnung mit komplexen Größen wäre hier falsch. Warum? +\end{enumerate} +\begin{align*} +R&=12\,\ohm\qquad L=1{,}3\,\milli\henry\qquad C=8{,}7\,\micro\farad\\ +f&=2\,\kilo\hertz\qquad I_0=10\,\milli\ampere\\ +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{L}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$U_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Strompfeil + \draw [->,red] (-.2,-.2)--(.2,-.2)node at(0,-.2)[below]{\footnotesize$I_0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\text{...noch einfügen...} +%\end{align} +Berechnung: +\begin{align*} +\omega&=2\pi f=1{,}257\cdot\power{10}{4}\,\frac{1}{\second}\\ +\omega t_1&=1{,}257\cdot \power{10}{4}\,\frac{1}{\second}\cdot 350\cdot \power{10}{-6}\,\second=4{,}398\,[rad]\,\widehat{=}\,252\,\degree\\ +\end{align*} +Zur Erklärung wie die Schwingung aussieht: +\begin{align*} +\begin{tikzpicture}[scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw [->] (0,2)--(4.2,2)node[right] {$\omega t$}; % Draw x axis; + \draw [->] (0,0)--(0,4)node[above] {$i$}; % Draw y axis; + \draw [red, very thick](0,2) sin (1,4) cos (2,2) sin (3,0) cos (4,2); + \draw node at(1.8,3)[right] {$i(t)$}; + \draw node at(2,2)[below] {$\pi$}; + \draw node at(4,2)[below] {$2\pi$}; + \draw node at(2.8,2)[below] {$\omega t_1$}; + \filldraw (2.8,2)--(2.8,.10)circle (2pt)node [below] {$t_1=350\,\micro\second$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=10cm,yshift=2cm] + \draw [thin](-2.2,0)--(2.2,0) (0,-2.2)--(0,2.2); + \draw (0,0) circle (2); + \draw [->,red](0:0)--(252:2)node [below]{$252\,\degree$}; + \draw [red!70!blue](270:1.92)--(0:0)node at (270:.951)[right]{$\sin (\omega t_1)=-0{,}951$}; + \draw [red!70!blue](252:2)--(270:1.92)node at (261:2.4)[below]{$\cos (\omega t_1)=-0{,}309$}; + \end{scope} +\end{tikzpicture} +\end{align*} +a) Berechnung der momentanen Spannung bei $t=t_1$ +\begin{align*} +i(t)&=I_0\cdot \sin(\omega t)=10\,\milli\ampere\cdot \sin{(\underbrace{1{,}257\cdot \power{10}{4}\,\frac{1}{\,\second}\cdot 350\,\micro\second}_{4{,}3995 rad})}\\ +&=10\,\milli\ampere\cdot (-0{,}951)=-9{,}51\,\milli\ampere +\intertext{Für $t=t_1$:} +u_R(T)&=R\cdot i(t)=12\,\ohm\cdot (-9{,}51\,\milli\ampere)=\uuline{-114{,}1\,\milli\volt}\\ +u_L(T)&=L\cdot \frac{di}{dt}\Bigg{|}_{t=T}=L\cdot I_0\cdot \hspace{-.9cm}\underbrace{\omega}_{\mathrm{nachdifferenzieren}}\hspace{-.9cm}\cdot \cos(\omega t)\\ +&=1{,}3\,\milli\henry\cdot 10\,\milli\ampere\cdot 1{,}257\cdot \power{10}{4}\,\frac{1}{\second}\cdot (-0{,}309)=\uuline{-50{,}5\,\milli\volt}\\ +u_C(T)&=\underbrace{U_0}_0+\frac{1}{C}\int_{t=0}^{T}{I_0\cdot \sin(\omega t)\cdot dt} =\frac{I_0}{C}\cdot \frac{1}{\omega}\cdot \hspace{-1,5cm}\underbrace{\Big{|}-\cos(\omega t)\Big{|}_{t=0}^{T}}_{-\cos(\omega T)+\cos(0)=-(-0{,}309)+1=1,309}\\ +&=\frac{0{,}01\,\ampere\cdot 1{,}309}{8{,}7\cdot \power{10}{-6}\,\farad\cdot 1{,}257\cdot \power{10}{4}\,\frac{1}{\second}}=\uuline{119{,}7\,\milli\volt}\\[\baselineskip] +\intertext{b) Momentanleistung bei $t=t_1$} +p(t)&=u(t)\cdot i(t)=[u_R(t)+u_L(t)+u_C(t)]\cdot i(t) +\intertext{Für $t=T$:} +p(T)&=u(T)\cdot i(T)=(-114{,}1-50{,}5+119{,}7)\,\milli\volt\cdot (-9{,}51)\,\milli\ampere\\ +&=(-44{,}9)\,\milli\volt\cdot (-9{,}51)\,\milli\ampere=\uuline{0{,}427\,\milli\watt} +\end{align*} +c) Komplexe Größen verwenden den Effektivwert der Schwingung! +\clearpage +}{}% \ No newline at end of file diff --git a/ET2_L_B16_A1.tex b/ET2_L_B16_A1.tex new file mode 100644 index 0000000..2d7bc76 --- /dev/null +++ b/ET2_L_B16_A1.tex @@ -0,0 +1,77 @@ +\section{CLR Netzwerk} +Von dem Netzwerk sind folgende Daten bekannt:\\[\baselineskip] +$R_1=1,2\,\kilo\ohm\quad R_2=470\,\ohm\quad X_{C_0}=-906\,\ohm\quad X_{L_2}=628\,\ohm$\\ +$\uline{I}_2=12\,\milli\ampere\cdot e^{j20\,\degree}$\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-.5cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}_{0}$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}_0$}; + \draw [->,red] (.8,.2)--(1.2,.2)node at(1,.2)[left]{$\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$X_{C_0}$}; + \draw [->,blue] (0.3,-.25)--(0.7,-.25)node at(.25,-.4)[right]{\footnotesize$\uline{U}_{C_0}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=-.5cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; + \draw [<-,blue] (0,-.5)--(1,-.5)node at(.5,-.5)[right]{\footnotesize$\uline{U}_2$}; + \draw [<-,red] (.8,.15)--(1.2,.15)node at(1,.1)[left]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_{L_2}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-1cm]%Fehlstellen Eckverbindungen. + \draw (0,1.5)--(0,2)--(.2,2) (1,2)--(2,2)--(2,1.75)(0,.5)--(0,0)--(2,0)--(2,.25) + (1,0)--(1,.5) (1,2)--(1,1.5); + \filldraw [red] (1,2)circle(0.02)node [above]{\footnotesize$KP$}; + \end{scope} + \end{tikzpicture} +\end{align*} +Berechnen Sie Schein- Wirk- und Blindleistung des Netzwerkes!\\[\baselineskip] +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} +\uline{S}&=\uline{U}\cdot \uline{I}^*\qquad\text{mit }\uline{U}_0=\uline{U}_{C_0}+\uline{U}_2\qquad\uline{I}_0=\uline{I}_1+\uline{I}_2\\ +\intertext{Spannung $U_2$:} +\uline{Z}_2&=R_2+jX_{L_2}=(470+j628)\,\ohm=784{,}4\,\ohm\cdot e^{j53{,}19\,\degree}\\ +\uline{U}_2&=\uline{Z}_2\cdot \uline{I}_2 +=9{,}413\,\volt\cdot e^{j73{,}19\,\degree}=(2{,}722+j9{,}011)\,\volt\\ +\intertext{Strom:} +\uline{I}_1&=\frac{\uline{U}_2}{R_1}=\frac{9{,}413\,\volt\cdot e^{j73{,}19\,\degree}}{1,2\,\kilo\ohm}=7{,}844\,\milli\ampere\cdot e^{j73{,}19\,\degree} +=(2{,}268+j7{,}5088)\,\milli\ampere\\ +\uline{I}_0&=\uline{I}_1+\uline{I}_2=(2{,}268+j7{,}5088+11{,}276+j4{,}104)\,\milli\ampere\\ +&=(13{,}544+j11{,}613)\,\milli\ampere=17{,}841\,\milli\ampere\cdot e^{j40{,}61\,\degree}\\ +\intertext{Spannung:} +\uline{U}_{C_0}&=jX_{C_0}\cdot \uline{I}_0 +=-j906\,\ohm\cdot 17{,}841\,\milli\ampere\cdot e^{j40{,}61\,\degree}=16{,}164\,\volt\cdot e^{-j49{,}39\,\degree}\\ +&=(10{,}521-j12{,}271)\,\volt\\ +\uline{U}_0&=\uline{U}_{C_0}+\uline{U}_2=(10{,}521-j12{,}271+2{,}722+j9{,}011)\,\volt\\ +&=(13{,}243-j3{,}260)\,\volt=13{,}638\,\volt\cdot e^{-j13{,}83\,\degree}\\ +\end{align*} +\begin{align*} +\intertext{Scheinleistung:} +\uline{S}&=\uline{U}_0\cdot \uline{I}_0^*=13{,}638\,\volt\cdot e^{-j13{,}83\,\degree}\cdot 17{,}841\,\milli\ampere\cdot e^{-j40{,}61\,\degree}=243{,}32\,\milli\volt\ampere\cdot e^{-j54{,}44\,\degree}\\[2\baselineskip] +S&=|\uline{S}|=\uuline{243{,}32\,\milli\volt\ampere}\\ +P&=S\cdot \cos\varphi=243{,}32\,\milli\volt\ampere\cdot \underbrace{\cos(-54{,}44)}_{0{,}5816}=\uuline{141{,}50\,\milli\watt}\\ +Q&=S\cdot \sin\varphi=243{,}32\,\milli\volt\ampere\cdot \underbrace{\sin(-54{,}44)}_{-0{,}813}=\uuline{-197{,}93\,\milli\,\var}\\ +\intertext{Zweiter Weg:} +\uline{S}&=\uline{U}_0\cdot \uline{I}^*_0=\uline{I}_0\cdot \uline{Z}\cdot \uline{I}^*_0=I^2_0\cdot \uline{Z}\\ +\uline{Z}&=jX_{C_0}+\frac{R_1\cdot (R_2-jX_{L_2})}{R_1+R_2-jX_{L_2}}=(445-j622)\,\ohm=764\,\ohm\cdot e^{-j54{,}44\,\degree}\\ +\uline{S}&=I^2_0\cdot \uline{Z} +=(17{,}841\,\milli\ampere)^2\cdot 767\,\ohm\cdot e^{-j54{,}44\,\degree}=243{,}3\,\milli\volt\ampere\cdot e^{-j54{,}44\,\degree}\\ +S&=243{,}3\,\milli\volt\ampere\\ +P&=S\cdot\cos(-54{,}44\degree)=141{,}5\,\milli\watt\\ +Q&=S\cdot\sin(-54{,}44\degree)=-197{,}9\,\milli\,\var +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B16_A1a.tex b/ET2_L_B16_A1a.tex new file mode 100644 index 0000000..2eb53c7 --- /dev/null +++ b/ET2_L_B16_A1a.tex @@ -0,0 +1,14 @@ +\section{Wirkleistung vs. Blindleistung} +Was sind die Unterschiede von \\ +\textbf{Wirkleistungsanpassung} und \textbf{Blindleistungskompensation}\\[\baselineskip] + +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\textbf{Wirkleistungsanpassung}\\[\baselineskip] + +Maximale Wirkleistung bei Anpassung! $\uline{Z}^*_i \stackrel{!}{=} \uline{Z}_v$\\ + +\textbf{Blindleistungskompensation}\\[\baselineskip] + +Maximale Blindleistungskompensation bei $\cos(\varphi=0)$ +\clearpage +}{}% diff --git a/ET2_L_B16_A2.tex b/ET2_L_B16_A2.tex new file mode 100644 index 0000000..2181138 --- /dev/null +++ b/ET2_L_B16_A2.tex @@ -0,0 +1,72 @@ +\section{Wirkleistung} +Welche Werte müssen $R$ und $C$ annehmen, damit im Verbraucher die maximale Wirkleistung umgesetzt wird?\\ +Wie groß ist diese Wirkleistung ?\\[\baselineskip] +$R_1=20\,\ohm\quad C_1=3{,}18\,\micro\farad\quad L=0{,}6\,\milli\henry\quad \uline{U}=1\,\volt\cdot e^{j20\,\degree}\quad f=1000\,\hertz$\\[\baselineskip] +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-1cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=.5cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$U_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; +% \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$U_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); +% \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$U_{L}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; +% \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$U_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick] + \draw(0,0)--(0,1)--(0.2,1) (.9,.5)--(1,.5)--(1,1) (1,1)--(2,1)--(2,.9) (1.5,-.1)--(1.5,0)--(2,0)(0,-.9)--(0,-1)--(2,-1)--(2,-.9); + \draw node at(.5,-1)[below]{Quelle}; + \draw node at(2,-1)[below]{Verbraucher}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.25cm,yshift=-1cm,]%Knoten | + \filldraw (0,0)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.25cm,yshift=1cm,]%Knoten | + \filldraw (0,0)circle(.05); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\[\baselineskip] +Phase von $\uline{U}$ ohne Bedeutung! (Berechnung über Impedanzen)\\ +Maximale Wirkleistung bei Anpassung! $\uline{Z}^*_i \stackrel{!}{=} \uline{Z}_v$ +\begin{align*} +\omega&=2\pi\cdot f=2\pi \cdot 1000\,\hertz=6283\,\frac{1}{\second}\\ +B_1&=\omega\cdot C_1=6283\,\frac{1}{\second}\cdot 3{,}18\,\micro\farad=0{,}02\,\siemens\\ +X_L&=\omega\cdot L=6283\,\frac{1}{\second}\cdot 0{,}6\,\milli\henry=3{,}77\,\ohm\\ +\intertext{Quelle:} +\uline{Y}_i&=G_1+jB_1=\frac{1}{R_1}+jB_1=(0{,}05+j0{,}02)\,\siemens\\ +\uline{Z}_i&=\frac{1}{\uline{Y}_i}=(17{,}24-j6{,}897)\,\ohm\\ +R_i&=17{,}24\,\ohm\qquad X_i=-6{,}897\,\ohm\\ +\intertext{Anpassung, wenn $\uline{Z}_v=\uline{Z}^*_i$} +\uline{Z}_v&=R_v+jX_v \stackrel{!}{=} (17{,}24+j6{,}897)\,\ohm\quad\Rightarrow\\ +R_v&=R=\uuline{17{,}24\,\ohm}\\ +X_v&=6{,}897\,\ohm\\ +B_v&=B_L+B_C=\frac{-1}{X_v}=\frac{-1}{6{,}897\,\ohm}=-0{,}145\,\siemens\\ +B_L&=\frac{-1}{X_L}=\frac{-1}{3{,}77\,\ohm}=-0{,}2653\,\siemens\\ +B_C&=B_v-B_L=(-0{,}145+0{,}2653)\,\siemens=0{,}1203\,\siemens\\ +C&=\frac{B_C}{\omega}=\frac{0{,}1203\,\siemens}{6283\,\frac{1}{\second}}=\uuline{19{,}14\,\micro\farad}\\[\baselineskip] +P_{v\text{,}max}&=\frac{U^2}{4\cdot R_v}=\frac{1\,\volt^2}{4\cdot 17{,}24\,\ohm}=\uuline{14{,}5\,\milli\watt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B16_A3.tex b/ET2_L_B16_A3.tex new file mode 100644 index 0000000..c0dca2d --- /dev/null +++ b/ET2_L_B16_A3.tex @@ -0,0 +1,104 @@ +\section{Abgebbare Wirkleistung} +Um wieviel Prozent weicht die in dem passiven Zweipol umgesetzte Wirkleistung von der in dem aktiven Zweipol maximal abgebbaren Wirkleistung ab?\\ +$C_1=2\,\nano\farad\quad +C_2=3\,\nano\farad\quad +L_1=2{,}5\,\micro\henry\quad +R_1=20\,\ohm\quad +L_2=3\,\micro\henry\quad +R_1=15\,\ohm\quad +f=3\,\mega\hertz$\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=.5cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$\uline{I}$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{I}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=2cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=.5cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0cm] + \draw(0,1.5)--(0,2)--(.1,2)(1,2)--(3,2)--(3,1.9)(1,1.5)--(1,2); + \draw(0,.5)--(0,0)--(3,0)--(3,.1)(1,0)--(1,.5); + \fill (2.5,2)circle(.05); + \fill (2.5,0)circle(.05); + \draw [thin,dashed](-.45,-.2)rectangle(2.4,2.5); + \draw [thin,dashed](2.6,-.2)rectangle(3.45,2.5); + \draw node at(1,-.25)[below]{Aktiver Zweipol}; + \draw node at(3,-.25)[below]{Passiver}; + \draw node at(3,-.5)[below]{Zweipol}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%%\begin{align} +%%\intertext{Formeln:} +%%\end{align} +Berechnung:\\[\baselineskip] +$C_1$ spielt für die Berechnung der Wirkleistung keine Rolle, da in Reihe zu Stromquelle.\\[\baselineskip] +ESB: +\begin{align*} +\begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$U_q$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$U_{q}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_i$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Z}_v$}; + \end{scope} +\begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw(0,.9)--(0,1)--(.1,1)(0,.1)--(0,0)--(1,0)--(1,.1); + \fill (1,1)circle(.05); + \fill (1,0)circle(.05); + \end{scope} + \end{tikzpicture} +\end{align*} +Als ESB ist eine Stromquelle $I_q$ mit parallelem $\underline{Z}_i$ und dazu parallelem $\underline{Z}_v$ möglich. +\clearpage +\begin{align*} +Z_{i}&=jX_C || (R_1 + jX_{L1})\\ +\omega&=2\pi\cdot f=2\pi\cdot 3\,\mega\hertz=18{,}85\cdot \power{10}{6}\,\frac{1}{\second}\\ +X_{C_2}&=\frac{-1}{\omega\cdot C_2}=\frac{-1}{18{,}85\cdot \power{10}{6}\,\cdot \frac{1}{\second}\cdot 3\,\nano\farad}=-17{,}68\,\ohm\\ +X_{L_1}&=\omega\cdot L_1=18{,}85\cdot \power{10}{6}\,\frac{1}{\second}\cdot 2{,}5\,\micro\henry=47{,}12\,\ohm\\ +X_{L_2}&=\omega\cdot L_2=18{,}85\cdot \power{10}{6}\,\frac{1}{\second}\cdot 3\,\micro\henry=56{,}55\,\ohm\\ +\uline{Z}_i&=jX_{C_2}||(R_1+jX_{L_1}) +=\frac{-j17{,}68\,\ohm\cdot (20\,\ohm+j47{,}12\,\ohm)}{-j17{,}68\,\ohm + 20\,\ohm+j47{,}12\,\ohm} +=\frac{-j17{,}68\cdot (20+j47{,}12)}{20+j(47{,}12-17{,}68)}\\ +&=\frac{833{,}08-j353{,}6}{20+j29{,}44}\,\ohm=\frac{905{,}02\cdot e^{-j23{,}0\,\degree}}{35{,}59\,\ohm\cdot e^{j55{},81\,\degree}}=25{,}428\,\ohm\cdot e^{-j78{,}81\,\degree}=(4{,}935-j24{,}94)\,\ohm +\end{align*} +Verbraucherwiderstand: +\begin{align*} +\uline{Z}_v&=(15+j56{,}55)\,\ohm\\ +\uline{Z}_{ges}&=\uline{Z}_i+\uline{Z}_v=(4{,}935-j24{,}94)\,\ohm+(15+j56{,}55)\,\ohm=(19{,}94+j31{,}61)\,\ohm\\ +\intertext{Anmerkung: $\uline{U}_q$ ist unbekannt, kürzt sich später heraus.} +P_{v,max}&=\frac{U^2_q}{4\cdot R_i}=\frac{U^2_q}{4\cdot 4{,}935\,\ohm}=\frac{U^2_q}{19{,}94\,\ohm}\\ +P_v&=I^2\cdot R_v\\ +I&=\frac{U_q}{Z_{ges}}\qquad \text{Anmerkung: $I$ und $U_q$ Effektivwert; $Z$ Betrag}\\ +P_v&=\left(\frac{U_q}{Z_{ges}}\right)^2 \cdot R_v =\frac{U^2_q}{(19{,}94^2+31{,}61^2)\,\ohm^2}\cdot 15\,\ohm=U^2_q\cdot \frac{15}{1397\,\ohm}\\ +F_ \% &=100\,\%\cdot \frac{P_v-P_{v,max}}{P_{v,max}}=100\,\%\cdot \left(\frac{P_v}{P_{v,max}}-1\right)\\ +&=100\, \% \cdot \left(\frac{\cancel{U^2_q}\cdot 15}{1397\,\ohm}\cdot \frac{19{,}94\,\ohm}{\cancel{U^2_q}}-1\right)\\ +&=100\, \% \cdot (0{,}212-1)=\uuline{-78{,}8\, \%} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B16_A4.tex b/ET2_L_B16_A4.tex new file mode 100644 index 0000000..3250323 --- /dev/null +++ b/ET2_L_B16_A4.tex @@ -0,0 +1,115 @@ +\section{Wirkleistung Spannungsquelle} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie die Wirkleistung der Spannungsquelle $\uline{U}_2$. +\item Wird Wirkleistung aufgenommen oder abgegeben? +\end{enumerate} +$R=200\,\ohm\quad L=80\,\micro\henry\quad C=500\,\pico\farad \quad f=1\,\mega\hertz$\\ +$\uline{I}_1=10\,\milli\ampere\cdot e^{j60\,\degree}\quad \uline{U}_2=3\,\volt\cdot e^{-j30\,\degree}$ +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$\uline{I}_1$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize $\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}_2$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[very thick](0,.9)--(0,1)--(2,1)--(2,.9)(0,.1)--(0,0)--(.1,0)(2,.1)--(2,0)--(1.9,0); + \draw [->,red] (1.3,1.1)--(1.7,1.1)node at (1.5,1.1)[above]{\footnotesize$\uline{I}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\[\baselineskip] +\begin{align*} +\intertext{$C$ spielt keine Rolle, da in Reihe zu Stromquelle.} +\text{Gesucht: } P&=\Re(\uline{U}_2\cdot \uline{I}^*) +\end{align*} +Lösung mit Überlagerungsverfahren:\\[\baselineskip] +Nur Stromquelle: +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0)node at(.5,-.133)[right]{$\uline{I}_1$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[very thick](0,.9)--(0,1)--(2,1)--(2,.9)(0,.1)--(0,0)--(1,0)(1.9,1)--(2,1)--(2,0)--(1.9,0); + \draw [->,red] (1.3,1.1)--(1.7,1.1)node at (1.5,1.1)[above]{\footnotesize$\uline{I}'$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +Stromteiler:\\ +\begin{align*} +X_L&= \omega \cdot L = 2\cdot \pi\cdot 1\,\mega\hertz\cdot 80\,\micro\henry= 503\,\ohm\\ +\uline{I}'&=\uline{I}_1\cdot \frac{jX_L}{R+jX_L} +=10\,\milli\ampere\cdot e^{j60\,\degree}\cdot \frac{j503}{200+j503}\\ +&=10\,\milli\ampere\cdot e^{j60\,\degree}\cdot \underbrace{\frac{503\cdot e^{j90\,\degree}}{541{,}3\cdot e^{j68{,}3\,\degree}}}_{0{,}929\,\milli\ampere\cdot e^{j21{,}7\,\degree}}\\ +&=9{,}29\,\milli\ampere\cdot e^{j81{,}7\,\degree} = (1{,}34 + j9{,}19)\,\milli\ampere\\ +\end{align*} +Nur Spannungquelle: +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}_2$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[very thick](1,.9)--(1,1)--(2,1)--(2,.9)(1,.1)--(1,0)(2,.1)--(2,0)--(1.9,0); + \draw [<-,red] (1.3,1.1)--(1.7,1.1)node at (1.5,1.1)[above]{\footnotesize$\uline{I}''$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{I}''&=\frac{\uline{U}_2}{R+jX_L} +=\frac{3\,\volt\cdot e^{-j30\,\degree}}{(200+j503)\,\ohm} +=\frac{3\,\volt\cdot e^{-j30\,\degree}}{541{,}3\,\ohm \cdot e^{j68{,}32\,\degree} +}\\ +&=5{,}54\,\milli\ampere\cdot e^{-j98{,}32\,\degree} = (-0{,}80-j5{,}48)\,\milli\ampere\\ +\intertext{Überlagerung - vorzeichenrichtg:} +\uline{I}&=\uline{I}'-\uline{I}''= (1{,}34 + j9{,}19+0{,}80+j5{,}48)\,\milli\ampere \\ +&=(2{,}14+j14{,}67)\,\milli\ampere=\uline{14{,}83\,\milli\ampere\cdot e^{j81{,}7\,\degree}}\\ +S&=\uline{U}_2\cdot \uline{I}^*=3\,\volt\cdot e^{-j30\,\degree} +\cdot 14{,}83\,\milli\ampere\cdot e^{-j81{,}7\,\degree} +=44{,}49\,\milli\volt\ampere\cdot e^{-j111{,}7\,\degree}\\ +&=(\underbrace{-16{,}31}_{P}-j\underbrace{40{,}97}_{Q})\,\milli\volt\ampere\Rightarrow\\ +P&=\uuline{-16{,}31\,\milli\watt} +\intertext{Verbraucher-Zählpfeilsystem $\Rightarrow$ \uuline{Spannungsquelle gibt Leistung ab.}} +\end{align*} +\clearpage +}{}% \ No newline at end of file diff --git a/ET2_L_B16_A5.tex b/ET2_L_B16_A5.tex new file mode 100644 index 0000000..59e8145 --- /dev/null +++ b/ET2_L_B16_A5.tex @@ -0,0 +1,90 @@ +\section{Dualitätskonstante} +Berechnen Sie zu der gegebenen Schaltung die duale Schaltung mit der Dualitätskonstanten.\\[\baselineskip] +$R^2_D=10000\,\ohm^2$\\ +$R_1=80\,\ohm$\\ +$L_1=50\,\milli\henry$\\ +$C_1=10\,\micro\,\farad$ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=.5cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=-.5cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw (1.1,.5)--(1,.5)--(1,-.5)--(1.1,-.5)(1.9,.5)--(2,.5)--(2,-.5)--(1.9,-.5)(2,0)--(3,0); + \fill(0,0)circle(.05) (3,0)circle(.05); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +\uline{Z}_1(\omega)&=R^2_D \cdot \uline{Y}_2(\omega) +\end{align} +Berechnung:\\[\baselineskip] +Parallel $\Leftrightarrow$ Serie\\ +Leitwert $\Leftrightarrow$ Widerstand\\ +Kapazität $\Leftrightarrow$ Induktivität\\ +Parallelschaltung $R_1||L_1$ in Serienschaltung $R_2+C_2$ +Serienschaltung $C_1+(R_2+C_2)$ in Parallelschaltung $L_2 ||(R_2+C_2)$ +\begin{align*} + \begin{tikzpicture}[scale=2] +\draw node at (0,0){\phantom{.}}; +\draw node at (0,1.5){Zwischenschritt:}; + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0.5cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0.5cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0.5cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0.5cm] +% \draw (1.1,.5)--(1,.5)--(1,-.5)--(1.1,-.5)(1.9,.5)--(2,.5)--(2,-.5)--(1.9,-.5)(2,0)--(3,0); +% \draw (0,.5)--(.5,.5) (.5,.5)--(.5,.5)--(.5,-.5)--(2,-.5) +% (2.5,.5)--(2.5,-.5)--(2,-.5)(2.5,.5)--(3,.5); + \fill(0,0)circle(.05) (3,0)circle(.05); + \end{scope} + \end{tikzpicture} + \begin{tikzpicture}[scale=2] +\draw node at (0,1.0){Duale Schaltung:}; + \begin{scope}[>=latex,very thick,xshift=.5cm,yshift=.5cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=.5cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=-.5cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] +% \draw (1.1,.5)--(1,.5)--(1,-.5)--(1.1,-.5)(1.9,.5)--(2,.5)--(2,-.5)--(1.9,-.5)(2,0)--(3,0); + \draw (0,.5)--(.5,.5) (.5,.5)--(.5,.5)--(.5,-.5)--(2,-.5) + (2.5,.5)--(2.5,-.5)--(2,-.5)(2.5,.5)--(3,.5); + \fill(0,.5)circle(.05) (3,.5)circle(.05); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +R_1\cdot R_2&=R^2_D\Rightarrow R_2=\frac{R^2_D}{R_1}=\frac{10000\,\ohm^2}{80\,\ohm}=\uuline{125\,\ohm}\\ +\text{(C zu L) aus: }\frac{L_2}{C_1}&=R^2_D\Rightarrow L_2=R^2_D\cdot C_1 +=\power{10}{4}\,\ohm^{{\cancel{2}}} +\cdot \power{10}{-5}\,\frac{\second}{\cancel{\ohm}} +=0{,}1\,\frac{\volt\second}{\ampere} +=\uuline{100\,\milli\henry}\\ +\text{(L zu C) aus: }\frac{L_1}{C_2}&=R^2_D\Rightarrow C_2=\frac{L_1}{R^2_D} +=\frac{50\,\milli\henry}{10000\,\ohm^2} +=5\cdot \power{10}{-6}\,\frac{\cancel{\volt}\second}{\cancel{\ampere}} +\cdot \frac{\ampere^{\cancel{2}}}{\volt^{\cancel{2}}} +=\uuline{5\,\micro\farad} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B16_A6.tex b/ET2_L_B16_A6.tex new file mode 100644 index 0000000..f82150b --- /dev/null +++ b/ET2_L_B16_A6.tex @@ -0,0 +1,81 @@ +\section{Dualitätskonstante verlustbehaftete Bauelemente} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=.5cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$G$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-.5cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-.5cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw (0,0)--(1,0) (1.1,.5)--(1,.5)--(1,0) (.6,-.5)--(.5,-.5)--(.5,0) (1.9,.5)--(2,.5)--(2,0)(2.4,-.5)--(2.5,-.5)--(2.5,0) (2,0)--(3,0); + \fill(.5,0)circle(.03) (1,0)circle(.03)(2,0)circle(.03) (2.5,0)circle(.03); + \fill(0,0)circle(.05)node[left]{$a$}; + \fill(3,0)circle(.05)node[right]{$b$}; + \end{scope} + \end{tikzpicture} + \end{align*} +$\hspace{3cm} R=20\,\ohm\quad L=1\,\milli\henry\quad C=100\,\nano\,\farad\quad G=2\,\milli\siemens$\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Geben Sie das duale Schaltbild für den Zweipol $a-b$ an und berechnen Sie dessen Elemente für $R^2_D=(100\,\ohm)^2$. +\item Welches verlustbehaftete Bauelement stellt die Reihenschaltung $R$ und $L$ dar? +\item Welches verlustbehaftete Bauelement stellt die Parallelschaltung $G$ und $C$ dar? +\item Interpretieren Sie das Ergebnis der Bauelementegrößen der beiden dualen Schaltungen in Bezug auf verlustbehaftete Bauelemente. +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +%\uline{Z}_1(\omega)&=R^2_D \cdot \uline{Y}_2(\omega) +R\cdot R_{dual}&=R^2_D \quad = \quad\text{Dualitätskonstante}\\ +R^2_D&=\frac{L}{C}=\frac{L_{dual}}{C}=\frac{L}{C_{dual}} +\end{align} +Berechnung:\\[\baselineskip] +Duales Netzwerk:\\ +Parallel $\Leftrightarrow$ Serie\\ +Leitwert $\Leftrightarrow$ Widerstand\\ +Kapazität $\Leftrightarrow$ Induktivität\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=.25cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$G_R$}; +\draw node at (.75,.35) [above] {\footnotesize{Verlustbehafteter Kondensator}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-.25cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_L$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_G$}; +\draw node at (.75,-.5) [above] {\footnotesize{Verlustbehaftete Spule}}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_C$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw (2.1,.25)--(2,.25)--(2,-.25)--(2.1,-.25) (2.9,.25)--(3,.25)--(3,-.25)--(2.9,-.25) (3,0)--(4,0); + \fill(2,0)circle(.03)(3,0)circle(.03); + \fill(0,0)circle(.05)node[left]{$a$}; + \fill(4,0)circle(.05)node[right]{$b$}; + \end{scope} + \end{tikzpicture} + \end{align*} +\begin{align*} +\intertext{a) Duale Bauelemente:} +R_G&=\frac{R^2_D}{R}=R^2_D\cdot G=(100\,\ohm)^2\cdot 2\,\milli\siemens=\uuline{20\,\ohm}\quad(=R\text{ !})\\ +L_C&=C\cdot R^2_D=100\cdot \power{10}{-9}\frac{\,\ampere\second}{\volt}\cdot (100\frac{\,\volt}{\ampere})^2=\uuline{1\,\milli\henry}=\quad(=L\text{ !})\\ +G_R&=\frac{R}{R^2_D}=\frac{20\,\ohm}{(100\,\ohm)^2}=\uuline{2\,\milli\siemens} \quad(=G\text{ !)}\\ +C_L&=\frac{L}{R^2_D}=\frac{1\,\milli\henry}{(100\,\ohm)^2}=\uuline{100\,\nano\farad} \quad(=C\text{ !)}\\ +\intertext{b) Verlustbehaftete Spule.} +\intertext{c) Verlustbehafteter Kondensator.} +\intertext{d) Die duale Schaltung ist die Reihenschaltung der gleichen verlustbehafteten Bauteile Spule und Kondensator.} +\end{align*} +\clearpage +}{}% \ No newline at end of file diff --git a/ET2_L_B16_A7.tex b/ET2_L_B16_A7.tex new file mode 100644 index 0000000..f46fc56 --- /dev/null +++ b/ET2_L_B16_A7.tex @@ -0,0 +1,239 @@ +\section{Vierpol Y-Parameter} +\begin{align*} + \begin{tikzpicture}[scale=3] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$2L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick] + \fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05); + \draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$}; + \draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie die $\uline{Y}$-Parameter des Vierpols in Abhängigkeit von $L$.\\ +\item Bestimmen Sie die $\uline{Z}$-Parameter +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} +\intertext{a)\hspace{0.4cm} $Y$ Parameter; Einträge in Leitwertmatrix; Achtung Serienschaltung $\uline{Y}_{Serie}=\frac{\uline{Y}_1\cdot \uline{Y}_2}{\uline{Y}_1+\uline{Y}_2}!$} +\left[ + \begin{array}{c} + \uline{I}_1 \\ + \uline{I}_2 \\ + \end{array} +\right] +&= +\left[ + \begin{array}{cc} + \uline{Y}_{11} & \uline{Y}_{12} \\ + \uline{Y}_{21} & \uline{Y}_{22} \\ + \end{array} +\right]\cdot +\left[ + \begin{array}{c} + \uline{U}_1 \\ + \uline{U}_2 \\ + \end{array} +\right]\\[\baselineskip] +\uline{I}_1&=\uline{Y}_{11}\cdot \uline{U}_1+\uline{Y}_{12}\cdot \uline{U}_2\\ +\uline{I}_2&=\uline{Y}_{21}\cdot \uline{U}_1+\uline{Y}_{22}\cdot \uline{U}_2\\ +\uline{Y}_L&=\uuline{\frac{1}{j\omega\cdot L}}=\uuline{-j\frac{1}{\omega\cdot L}} +\end{align*} +%\begin{center} +%\intertext{$Y_{11}$: $U_2=0$ d.h. Kurzschluß am Ausgang} +$Y_{11}$: $U_2=0$ d.h. Kurzschluß am Ausgang\\[\baselineskip] + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$2L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick] + \fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05); + \draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$}; +% \draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$}; + \draw node at (0,.5)[left]{$\uline{Y}_{11}\Rightarrow$}; + \draw(2,0)--(2,1); + \draw[red!50!blue,thick,dashed](.8,-.33)rectangle(2.2,1.33)node at (1.5,-.33)[below]{$L$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=0cm] + \draw node at (0,-1)[right]{$\uline{Y}_{11}=\frac{\uline{I}_1}{\uline{U}_1}\Big{|}_{U_2=0}=\uuline{\frac{1}{3}\cdot \uline{Y}_L}$}; + \end{scope} + \end{tikzpicture} +\hspace{1cm} +% ---------------------------------- + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); +% \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); +% \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$\frac{1}{2} \cdot Y_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick] + \fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05); +% \draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$}; +% \draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$}; + \draw node at (0,.5)[left]{$\uline{Y}_{11}\Rightarrow$}; + \draw(2,0)--(2,1); +% \draw[red!50!blue,thick,dashed](.8,-.33)rectangle(2.2,1.33)node at (1.5,-.33)[below]{$Y_L$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=0cm] + \draw[white] node at (0,-1)[right]{$\uline{Y}_{L}=\frac{1}{j\omega \cdot L}=-j\frac{1}{\omega\cdot L}$}; + \end{scope} + \begin{scope}[black!75!,>=latex,thick,xshift=0cm] + \draw[black] node at (1.5,-.33)[above]{$\underbrace{\phantom{xxxxxxxx}}$}; + \draw node at (1.5,-.2)[below]{\footnotesize{$\frac{1}{2\cdot X_L}=\frac{1}{2}\cdot Y_L$}}; + \draw[black] node at (1.4,-.67)[above]{$\underbrace{\phantom{xxxxxxxxxxx}}$}; + \draw node at (1.4,-.54)[below]{\footnotesize{$\frac{1}{2}\cdot Y_L+\frac{1}{2}\cdot Y_L= Y_L$}}; + \draw[black] node at (1,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxx}}$}; + \draw node at (1,-.87)[below]{\footnotesize{$\frac{1}{3\cdot X_L}=\frac{1}{3}\cdot Y_L$}}; + \end{scope} + \end{tikzpicture} +% ---------------------------------- +%\end{center} +\begin{align*} +\intertext{$Y_{12}$: $U_1=0$ d.h. Kurzschluß am Eingang} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$2L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick] + \fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05); +% \draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$}; + \draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[left]{$\uline{U}_{2}$}; + \draw node at (2,.5)[right]{$\Leftarrow\uline{Y}_{L}$}; + \draw(0,0)--(0,1); + \draw[red!50!blue,thick,dashed](-.2,-.33)rectangle(1.2,1.33)node at (.5,-.33)[below]{$L$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=.5cm,yshift=.5cm] + \draw node at (2.5,.5)[right]{aus $\uline{I}_{1}=\uline{Y}_{11}\cdot \uline{U}_1+\uline{Y}_{12}\cdot \uline{U}_2$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=.5cm,yshift=.25cm] + \draw node at (2.5,.5)[right]{folgt mit $\uline{U}_{1}=0$}; + \end{scope} + \begin{scope}[>=latex,thick,xshift=.5cm,yshift=-.25cm] + \draw node at (2.5,.5)[right]{$\uline{Y}_{12}=\frac{\uline{I}_1}{\uline{U}_2}\big{|}_{U_1=0}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\text{Stromteiler: } +-\uline{I}_1&=\frac{2\cdot Z_L}{4\cdot Z_L}\cdot \uline{I}_2=\frac{1}{2}\cdot \uline{I}_2\\ +% +% +\uline{I}_2&=\frac{\uline{U}_2}{3\cdot Z_L} = \uline{U}_2\cdot \frac{1}{3}\cdot \uline{Y}_L\\ +\uline{I}_1&=-\frac{1}{2}\cdot \uline{U}_2 \cdot \frac{1}{3}\cdot \uline{Y}_L = -\frac{1}{6}\cdot \uline{U}_2\cdot \uline{Y}_L\\ +\uline{Y}_{12}&=\frac{-\frac{1}{6}\cdot \uline{U}_2\cdot \uline{Y}_L}{\uline{U}_2}=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\\ +% +% +%\uline{Y}_{12}&=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\text{ ??? 1/3 oder 1/6 ???}\\ +\uline{Y}_{21}&=\uline{Y}_{12}=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\text{ da spiegelsymmetrisch}\\ +\uline{Y}_{22}&=\frac{\uline{I}_2}{\uline{U}_2}\Big{|}_{_{U_1=I_1=0}}=\text{spiegelbildlich zu $\uline{Y}_{11}$ d.h. }\\ +\uline{Y}_{22}&=\uline{Y}_{11}=\uuline{\frac{1}{3}\cdot \uline{Y}_L}\\ +\intertext{b)\hspace{0.4cm} $Z$-Parameter Leerlauf} +\left[ + \begin{array}{c} + \uline{U}_1 \\ + \uline{U}_2 \\ + \end{array} +\right] +&= +\left[ + \begin{array}{cc} + \uline{Z}_{11} & \uline{Z}_{12} \\ + \uline{Z}_{21} & \uline{Z}_{22} \\ + \end{array} +\right]\cdot +\left[ + \begin{array}{c} + \uline{I}_1 \\ + \uline{I}_2 \\ + \end{array} +\right]\\[\baselineskip] +\uline{U}_1&=\uline{Z}_{11}\cdot \uline{I}_1+\uline{Z}_{12}\cdot \uline{I}_2\\ +\uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_1+\uline{Z}_{22}\cdot \uline{I}_2\\ +\uline{Z}_{11}&=\frac{\uline{U}_{1}}{\uline{I}_{1}}\Big{|}_{_{I_2=0}}=4\cdot j\omega\cdot L=\uuline{\uline{Z}_{22}}\\[\baselineskip] +\uline{Z}_{12}&=\frac{\uline{U}_{1}}{\uline{I}_{2}}\Big{|}_{_{I_1=0}}\\ +\uline{U}_{1}&=2\cdot X_L \cdot \uline{I}_{2}\\ +\uline{Z}_{12}&=\frac{2\cdot X_L \cdot \uline{I}_{2}}{\uline{I}_{2}}=2\cdot X_L = 2\cdot j\omega\cdot L=\uuline{\uline{Z}_{21}} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B16_A8.tex b/ET2_L_B16_A8.tex new file mode 100644 index 0000000..ebf747b --- /dev/null +++ b/ET2_L_B16_A8.tex @@ -0,0 +1,224 @@ +\section{Spannung Vierpol} +%\renewcommand{\theequation}{0} +Berechnung Sie $\uline{U}_0$ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{U}_R=R_2\cdot \uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=1.5cm, yshift=0cm]% Vierpol + \draw (0,-.2)rectangle(2.0,1.2)node at(.75,.5)[right]{$(\uline{Z})$}; + \draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1); + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw (0,0)--(1,0); + \draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1); + \draw (1,0)--(1,-.5)--(4,-.5)--(4,0); + \fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05) (4,0)circle(.05) (4,1)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.75cm,yshift=1cm]% Strompfeil + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=1cm]% Strompfeil + \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_2$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +R_1&=300\,\ohm\\ +R_2&=600\,\ohm\\ +\uline{Z}_{11}&=100\,\kilo\ohm\text{ reell}\\ +\uline{Z}_{21}&=-2\,\mega\ohm\text{ reell}\\ +\uline{U}_1&=1{,}5\,\volt +%\end{align*} +%\begin{align*} +\intertext{Die $\uline{Z}$-Matrix des Vierpols ist gegeben:} +\begin{bmatrix} + \uline{U}_0\\ + \uline{U}_2 +\end{bmatrix} +&= +\begin{bmatrix} + \uline{Z}_{11}&0\\ + \uline{Z}_{21}&0 +\end{bmatrix} +\cdot +\begin{bmatrix} + \uline{I}_0\\ + \uline{I}_2 +\end{bmatrix} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_1\cdot (\uline{I}_2+\uline{I}_0)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw (0,0)--(1,0); + \draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1); + \draw (1,0)--(1,-.5)--(4,-.5)--(4,0); + \fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05) (4,0)circle(.05) (4,1)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw (0,0)--(1,0); + \draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1); + \draw (1,0)--(1,-.5)--(4,-.5)--(4,0)(1,1)--(1.25,1)(4,1)--(3.75,1); + \fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05) (4,0)circle(.05) (4,1)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1.2cm]% Strompfeil + \draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1.2cm]% Strompfeil + \draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil + \draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +Aus $\uline{Z}$-Matrix folgt: +\begin{align*} +\uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0 \tag{1}\\ +\uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_0 \tag{2} +\intertext{Knoten 1:} +\uline{I}_1-\uline{I}_2-\uline{I}_0&=0 \tag{3} +\intertext{Masche 1:} +\uline{U}_1&=R_1\cdot \uline{I}_1+ \uline{U}_0\tag{4} +\intertext{Masche 2:} +\uline{U}_0&=R_2\cdot \uline{I}_2+\uline{U}_2 \tag{5} +\intertext{Zwei Gleichungen für zwei unbekante Ströme $\uline{I}_0$ und $\uline{I}_2$:} +\uline{Z}_{11}\cdot \uline{I}_0&=R_2\cdot \uline{I}_2+\uline{Z}_{21}\cdot \uline{I}_0 \tag{1\&2 in 5}\\ +\Rightarrow \uline{I}_2&=\frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0 \tag{6}\\ +\uline{U}_1&=\uline{I}_0 \cdot\uline{Z}_{11}+R_1(\uline{I}_0 +\uline{I}_2 ) \tag{7 1\&3 in 4}\\ +\uline{U}_1&=\uline{I}_0\cdot \uline{Z}_{11}+R_1\cdot \uline{I}_0+R_1\cdot \frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0 \tag{6 in 7}\\ +&=\left[\uline{Z}_{11}+R_1+\frac{R_1}{R_2}\cdot(\uline{Z}_{11}-\uline{Z}_{21})\right]\cdot \uline{I}_0\\ +&=\left[100\,\kilo\ohm+300\,\ohm+\frac{300\,\ohm}{600\,\ohm}\cdot \Big(100\,\kilo\ohm-(-2\,\mega\ohm)\Big)\right]\cdot \uline{I}_0\\ +&=0{,}103+0{,}5\cdot (100+2000)\,\kilo\ohm \cdot \uline{I}_0=1{,}15\,\mega\ohm\cdot \uline{I}_0\\ +\Rightarrow\uline{I}_0&=\frac{\uline{U}_1}{1{,}15\,\mega\ohm}=\frac{1{,}5\,\volt}{1{,}15\,\mega\ohm}=1{,}3\,\micro\ampere\\ +\uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0=100\,\kilo\ohm\cdot 1{,}3\,\micro\ampere=\uuline{130{,}4\,\milli\volt} +\intertext{Nicht gefragt:} +\uline{I}_2&=\frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0=\frac{1}{600\,\ohm}\cdot 2{,}1\mega\ohm\cdot 1{,}3\,\micro\ampere=4{,}55\,\milli\ampere >> \uline{I}_0\\ +\text{Probe }\\ +%&=\frac{1}{R_1}\cdot (\uline{I}_2+\uline{I}_0)\cdot \uline{U}_0\\ +\uline{U}_1&=1{,}5\,\volt=R_1\cdot (\uline{I}_0+\uline{I}_2)+\uline{I}_0\cdot Z_{11}\\ +&=300\,\ohm\cdot (0{,}0013+4{,}55)\,\milli\ampere+1{,}3\,\micro\ampere\cdot 100\,\kilo\ohm=1{,}498\,\volt +\end{align*} +\clearpage +\enlargethispage{1cm} +Ist Ihnen aufgefallen, daß $\uline{Z}_{21}$ einen negativen Wert hat?\\[\baselineskip] +Ein Widerstand nimmt elektrische Leistung auf, also ein Verbraucher.\\ +Dann muß also ein negativer Widerstand elektrische Leistung abgeben! +Gibt es das in der Realität?\\[\baselineskip] +Was steckt in dem Vierpol?\\[\baselineskip] +Passiver Vierpol:\\ +Zum Beispiel mit 3 Widerständen in $T$ oder $\Pi$ Schaltung, ergibt jedoch keine Lösung!\\ +Aus der $\uline{Z}$ Matrix ist ersichtlich, daß $\uline{Z}_{21}=0$ und $\uline{Z}_{22}=0$ sind, +somit existiert keine Kopplung vom Ausgang zum Eingang, sondern nur in Vorwärtsrichtung.\\[\baselineskip] +Beispiel: Elektrodynamischer Lautsprecher der auf ein Kondensatormikrofon einwirkt. Kopplung nur vom Lautsprecher zum Mikrofon, nicht umgekehrt.\\[\baselineskip] +Aktiver Vierpol: (Für Fortgeschrittene)\\ +Es muß sich um einen invertierenden Trennverstärker mit einem Eingangswiderstand von +$100\,\kilo\ohm$ und einem Ausgangswiderstand von $2\,\mega\ohm$ handeln. Die Verstärkung ist zu berechnen. +\begin{align*} +\uline{U}_2&=\uline{U}_0-R_2\cdot \uline{I}_2=130{,}4\,\milli\volt-600\,\ohm\cdot 4{,}55\,\milli\ampere=-2{,}6\,\volt\\ +\uline{U}_q&=\uline{U}_2-\uline{Z}_{22}\cdot \uline{I}_2=-2{,}6\,\volt-2\,\mega\ohm\cdot 4{,}55\,\milli\ampere=-9102{,}6\,\volt\\ +V&=\frac{\uline{U}_q}{\uline{U}_0}=\frac{-9102{,}6\,\volt}{130{,}4\,\milli\volt}=-69805 +\end{align*} +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$R_1\cdot (\uline{I}_2+\uline{I}_0)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,black!25!, xshift=1.5cm, yshift=0cm]% Vierpol mit Innenleben + \draw (0,-.2)rectangle(2.0,1.2);%node at(.75,.5)[right]{$(\uline{Z})$}; + \draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1); +% \end{skope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] %Z_11 + \draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) +[left] {$\uline{Z}_{11}$}; + \draw [<-,red!50!blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{Z}_{11}\cdot \uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Z_21 + \draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_{21}$}; + \draw [->,red!50!blue] (.3,-.2)--(.7,-.2)node at(.6,-.2)[below]{\footnotesize$-\uline{Z}_{21}\cdot \uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,red!50!blue,xshift=1cm,yshift=0cm,rotate=90]%Spannungsquelle| + \draw (0,0)--(1,0);%;node at(.5,-.133)[right]{$V\cdot \uline{U}_0$}; + \draw (.5,0)circle(.133); + \draw [->] (.3,-.2)--(.7,-.2) node at (.3,-.2)[right]{\footnotesize$V\cdot \uline{U}_q$}; + \draw node at (.3,.025)[left]{\footnotesize$+$}; + \draw node at (.7,.025)[left]{\footnotesize$-$}; + \end{scope} + \begin{scope}[>=latex,very thick,red!50!blue,xshift=0cm,yshift=0cm]% + \draw (-.5,0)--(2.5,0)(-.5,1)--(0,1)(2,1)--(2.5,1); + \end{scope} + \end{scope} % Ende Vierpol Innenleben + \begin{scope}[>=latex,very thick, xshift=0, yshift=0] + \draw (0,0)--(1,0); + \draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1); + \draw (1,0)--(1,-.5)--(4,-.5)--(4,0); + \fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05) (4,0)circle(.05) (4,1)circle(.05); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]% Strompfeil + \draw [->,red] (0,0)--(.4,0)node at(.2,0)[above]{\footnotesize$\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=1cm]% Strompfeil + \draw [->,red] (0,0)--(-.4,0)node at(-.2,0)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil + \draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_2+\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(0,.25)[left]{\tiny$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil + \draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{minipage}[t]{1\textwidth} +\centering +\includegraphics[width =0.6\textwidth]{b16a8a} +\end{minipage} +\clearpage +}{}% diff --git a/ET2_L_B17_A1.tex b/ET2_L_B17_A1.tex new file mode 100644 index 0000000..281570e --- /dev/null +++ b/ET2_L_B17_A1.tex @@ -0,0 +1,186 @@ +\section{Stromortskurve} +Konstruieren Sie die Stromortskurve $\uline{I}=\uline{g}(f)$ von dem abgebildeten Netzwerk!\\ +Entnehmen Sie der Stromortskurve den Strom $\uline{I}$ für $f_0=2,5\,\kilo\hertz$!\\[\baselineskip] +Gegeben sind: $R_1=15\,\ohm$; $R_2=50\,\ohm$; $C=1{,}59\,\micro\farad$;\\ $0{,}5\,\kilo\hertz \leq f \leq 2{,}5\,\kilo\hertz$;\\ +$\uline{U}=10\,\volt$ = konstant\\[\baselineskip] +Maßstäbe: $5\,\milli\siemens\,\widehat{=}\,1\,\centi\metre$ : $ 10\,\ohm\,\widehat{=}\, 1\,\centi\metre$ (Platzbedarf in x: $14\,\centi\metre$; in y: $12\,\centi\metre$)\\ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$}; + \end{scope} +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 +% \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; +% \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.,0)--(2,0)--(2,.2) (.8,1)--(2,1)--(2,.8); + \draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}$}; + \fill (-.5,0)circle(.025) (-.5,1)circle(.025); + \draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}$}; +% \draw [->,red] (2.1,.9)--(2.1,.7) node at (2.1,.8)[right]{$\underline{I}_{R_C}$}; +% \draw [->,red] (3.1,.9)--(3.1,.7) node at (3.1,.8)[right]{$\underline{I}_C$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} +\intertext{Parallelschaltung $\uline{Y}_P$} +\uline{Y}_P&=G_2+jB\\ +G_2&=\frac{1}{R_2}=\frac{1}{50\,\ohm}=20\,\milli\siemens\\ +f_u&=0{,}5\,\kilo\hertz\qquad f_o=2{,}5\,\kilo\hertz \qquad\text{(Anmerkung)}\\ +B_u&=\omega_u\cdot C=2\cdot \pi\cdot f_u=2\cdot \pi\cdot 0{,}5\,\kilo\hertz\cdot 1{,}59\,\micro\farad =5\,\milli\siemens\\ +B_o&=\omega_o\cdot C=25\,\milli\siemens\qquad\text{(5-facher Wert von $B_u$)}\\ +\uline{Y}_P&=20\,\milli\siemens+j(5\ldots 25)\,\milli\siemens\\ +\end{align*} +\vspace{-1cm} +\begin{align*} +\begin{aligned} +\uline{Y}_{P,min}&=G_2 & (\text{für } & f=0) \qquad +&\uline{Z}_{P,min}&=0 \quad & (\text{für } & f\negmedspace\rightarrow\negmedspace\infty)\\ +\uline{Y}_{P,max}&=\infty & (\text{für } & f\negmedspace\rightarrow\negmedspace\infty) \qquad +&\uline{Z}_{P,max}&=R_2 \quad &(\text{für } & f=0)\\ +\end{aligned} +\end{align*}\\ +Zeichnen der Leitwertgeraden $\uline{Y}_P$; Konstruktion des Halbkreises für $\uline{Z}_P$ +\begin{align*} +(G_2=20\,\milli\siemens\,\widehat{=}\,4\,\centi\metre : R_2 = 50\,\ohm\,\widehat{=}\, 5\,\centi\metre) +\end{align*} +Einzeichnen der $\uline{Z}_{P_o}$, $\uline{Z}_{P_u}$ Linien (Zeiger).\\ +\clearpage +Berechnen der Gesamtschaltung $\uline{Y}$\\ +Graphisch wird der Widerstand $R_1$ addiert, durch verschieben der $\Im$ Achse um $1{,}5\,\centi\metre$ nach links.\\ +\begin{align*} +\uline{Y}_{min}&=\frac{1}{R_1+R_2}=\frac{1}{65\,\ohm}=15{,}4\,\milli\siemens\,\widehat{=}\,3{,}08\,\centi\metre \qquad (f=0)\\ +\uline{Y}_{max}&=\frac{1}{R_1}=\frac{1}{15\,\ohm}=66{,}7\,\milli\siemens\,\widehat{=}\,13{,}33\,\centi\metre\qquad (f\rightarrow\infty)\\ +\end{align*} + +Zeichnen des Leitwertkreises $\uline{Y}$ mit Radius $r$: +\begin{align*} +r&=\frac{\uline{Y}{max}-\uline{Y}_{min}}{2}\cdot \hspace{-1cm} \underbrace{\frac{1\,\centi\metre}{5\,\milli\siemens}}_{\text{Maßstabsumwandlung}}\hspace{-1cm}=5{,}12\,\centi\metre\\ +\end{align*} +Der Mittelpunkt ergibt sich aus: +\begin{align*} +\uline{Y}_{min}+r=3{,}08\,\centi\metre+5{,}12\,\centi\metre=8{,}2\,\centi\metre +\end{align*} +zu messen ab neuer imaginärer Achse.\\ + +Einzeichnen der $\uline{Z}_o$, $\uline{Z}_u$ Linien (Zeiger).\\ + +Spiegeln der Zeiger $\uline{Z}_o$, $\uline{Z}_u$ an der reellen Achse liefert Schnittpunkt mit $\uline{Y}$. Diese sind entsprechend $\uline{Y}_o$, $\uline{Y}_u$.\\ +\clearpage +$x=14\,\centi\metre; y=12\,\centi\metre$ +\begin{align*} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](-2,-6)grid(12,6); + \draw[thin](5,-6)--(5,6)(10,-6)--(10,6)(-2,5)--(12,5)(-2,5)--(12,5); + \draw[thin,->](0,0)--(12.5,0)node[right]{$\Re$}; + \draw[thin,->](0,-6.5)--(0,6.5)node[above]{$\Im$}; + \draw[blue,->,ultra thick](0,0)--(5,0)node[below left]{$R_2$}; + \draw[blue,thin](2.5,0)circle(2.5cm); + \fill[blue](2.5,0)circle(.075cm); + \draw[red,->](0,0)--(4,0)node[below left]{$G_2$}; + \draw[red!50!blue,thin](6.67,0)circle(5.125cm);% Y-Kreis + \fill[red!50!blue](6.67,0)circle(.075cm); + \draw[red!50!blue](1.525,-.1)--(1.525,.1)(1.525,0)node[below]{$\uline{Y}_{min}$}; + \draw[red!50!blue](11.807,-.1)--(11.807,.1)(11.807,0)node[below]{$\uline{Y}_{max}$}; + \draw[red,thin](4,-6)--(4,6); + \draw[red,|-|](4,1)--(4,5)node at (4.5,5){$f_o$}; + \draw[red]node at (4.5,1){$f_u$}; + \draw[red]node at (4.5,3){$\uline{Y}_P$}; + \draw[red]node at (4.5,-3){$\uline{Y}^*_P$}; + \draw[red]node at (6.5,4.5){$\uline{Y}$}; + \draw[red,|-|](4,-1)--(4,-5)node at (4.5,-5){$f_o$}; + \draw[red]node at (4.5,-0.75){$f_u$}; + \draw[blue,thin](0,0)--(8,-2)node at (3.0,-1.75){$\uline{Z}_P$; $\uline{Z}$}; + \draw[blue,thick,->](0,0)--(-14.036:4.851cm)node at(3,-.5){$\uline{Z}_{P_u}$}; + \draw[green!50!black,->](-1.5,0)--(-14.036:4.851cm)node at(3,-1.25){$\uline{Z}_u$}; + \draw[->,green!50!black](-1.5,0)--(1.9,-2.4)node at (-.5,-1){$\uline{Z}_o$}; + \draw[blue,thin](0,0)--(4,-5); + \draw[->,blue](0,0)--(1.9,-2.4)node at(1.2,-1){$\uline{Z}_{P_o}$}; + \draw[blue,ultra thick](0:2.5cm)+(257:2.5cm)arc(257:332:2.5cm);%Mittelpunkt+Start arc Start:End:Radius + \draw[blue]node at (5.5,-1.25){$f_u$}; + \draw[blue]node at (2,-2.75){$f_o$}; + \draw[blue!50!red,thin](-1.5,0)--(9.5,2)node at (2.5,1.5){$\uline{Y}$}; \draw[blue!50!red,thin](-1.5,0)--+(35.2:8.5cm); + \draw[blue!50!red,ultra thick](0:6.67cm)+(148:5.125cm)arc(148:174.5:5cm);%Mittelpunkt+Start arc Start:End:Radius +% \draw[blue!50!red]node at (1.25,.75){$f_u$}; + \draw[blue!50!red]node at (2,2.825){$f_o$}; + \draw[blue!20!red,ultra thick,->](-1.5,0)--+(35.2:4.72cm)node at (.5,2){$\uline{Y}(f_o)$}; + \draw[blue!50!red,ultra thick,->](-1.5,0)--+(10.3:3.15cm)node at (.5,.625){$\uline{Y}(f_u)$}; + \draw[blue!50!red]node at (5.5,1.25){$f_u$}; +% \draw[blue!50!red]node at (2.5,3.25){$f_o$}; + \draw[blue!50!red,thin,->](-1.5,-6.5)--(-1.5,6.5)node[above]{$\Im$ neu}; + \draw[blue!50!red,->](-1.5,0)--(0,0)node[below left]{$R_1$}; + \end{scope} + \end{tikzpicture} +\end{align*} + + +Ablesen der $\uline{Y}$ Werte und Maßstabsumrechnung $M=5\,\milli\siemens/\centi\metre$ ergibt Bereich zwischen $f_u$ und $f_o$. +\begin{align*} +\text{ablesen: } \uline{Y}(f_o)&=23{,}6\,\milli\siemens\cdot e^{(j35{,}2\,\degree)}\quad\text{aus Zeichnung $4{,}72\,\centi\metre$}\\ +\uline{I}&=\uline{Y}\cdot \uline{U}=\uuline{236\,\milli\ampere\cdot e^{(j35{,}2\,\degree)}}\\ +\uline{I}&=\uuline{(193+j136)\,\milli\ampere}\\ +\uline{Y}_P \text{ Gerade } \Rightarrow \uline{Z}_P \text{ Kreis } \\ +\uline{Z}_{P_u}, \uline{Z}_{P_o} \text{ Linie }\\ +R_1 \text{ addieren }\\ +\uline{Y}_{min} \text{ Kreis } \\ +\Rightarrow \uline{Y}(f_u); \uline{Y}(f_o) +\end{align*} +\clearpage +\subsubsection*{Kurzanleitung} + (Siehe Merksätze zu Inversion im Script, hier graphisch veranschaulicht:) +\begin{align*} + \begin{tikzpicture}[very thick,scale=1.25] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw(0,-1)--(0,1)node[above]{$\Im$}; + \draw(0,0)--(2,0); + \draw[red](1,-1)--(1,1); + \draw[red,->](0,0)--(1,.75)node at(1.25,.75){$\uline{Y}_p$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm] + \draw(0,-1)--(0,1)node[above]{$\Im$}; + \draw(0,0)--(2,0); + \draw[blue](.5,0)circle(.5cm); + \draw[blue,->](0,0)--(.7,-.5cm)node at(1.25,-.5){$\uline{Z}_p$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=6cm,yshift=0cm] + \draw(0,-1)--(0,1); + \draw(-.5,0)--(2,0); + \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu}; + \draw[blue](.5,0)circle(.5cm); + \draw[green!50!black,->](-.5,0)--(.7,-.5cm)node at(1,-.5){$\uline{Z}$}; \end{scope} + \begin{scope}[>=latex,very thick,xshift=9cm,yshift=0cm] + \draw(0,-1)--(0,1); + \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu}; + \draw(-.5,0)--(2.5,0); + \draw[blue](1.25,0)circle(1cm); + \draw[blue!50!red,->](-.5,0)--(2.1,.5)node at(2.5,.5){$\uline{Y}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{enumerate} +\item Leitwertortskuve der Parallelschaltung von $R_2$ und $C$ ergibt eine Gerade die im Abstand $G_2$ parallel zur imaginären Achse liegt.\\ +$G_2=20\,\milli\siemens\,\widehat{=}\, 4\,\centi\metre$ +\item Grenzen $f_u: \uline{Y}_P=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre$, $f_o: \uline{Y}_P=25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre$ einzeichnen, auch für $\uline{Y}^*_P$ +%Grenzen eintragen für $\uline{Y}_P=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre (f_u)\quad 25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre (f_o)\quad\text{ auch für $\uline{Y}^*_P$}$\\ +\item Inversion von $\uline{Y}_P$ liefert die Widerstandsortskurve $\uline{Z}_P$ ein Kreis durch den Ursprung.\\ +$\uline{Z}_{P_{min}}=0;\quad \uline{Z}_{P_{max}}=R_2=50\,\ohm\,\widehat{=}\, 5\,\centi\metre \quad\rightarrow\text{Kreis um P(2.5,0) $r=2{,}5\,\centi\metre$}$ +\item Der Serienwiderstand $R_1$ wird durch verschieben der imaginären Achse addiert $R_1=15\,\ohm\,\widehat{=}\,1{,}5\,\centi\metre$ (Verschiebung nach links). +\item Punktweise Inversion von $\uline{Z}$ liefert $\uline{Y}$ mit\\ $\uline{Y}_{min}=15{,}4\,\milli\siemens\,\widehat{=}\,3{,}08\,\centi\metre$ und $\uline{Y}_{max}=66{,}7\,\milli\siemens\,\widehat{=}\,13{,}3\,\centi\metre$,\\ wieder einen Kreis der jetzt nicht mehr durch den Ursprung geht. +\item Ablesen von $\uline{Y}(f_o)=4{,}72\,\centi\metre$ und Winkel $35{,}2\,\degree \\ \Rightarrow \uline{Y}(f_o)=23{,}6\,\milli\siemens\cdot e^{(j35{,}2\,\degree)}$. +\item Berechnen des Stroms \ldots \quad ;-) +\end{enumerate} +\clearpage +}{}% diff --git a/ET2_L_B17_A2.tex b/ET2_L_B17_A2.tex new file mode 100644 index 0000000..033d5ba --- /dev/null +++ b/ET2_L_B17_A2.tex @@ -0,0 +1,94 @@ +\section{Leitwerts-, Widerstandsortskurve} +Konstruieren Sie graphisch fur den dargestellten Zweipol +die Leitwertsortskurve $\uline{Y}_1(p)$, die Widerstandsortskurve +$\uline{Z}_1(p)$, und schlieslich die Widerstandsortskurve $\uline{Z}(p)$.\\ +Beziffern Sie jeweils die Punkte $p=0$; $p=1$; $p=3$ und den Grenzwert $p\rightarrow \infty$.\\[\baselineskip] +Parameter $p$: $\omega=p\cdot \omega_0$ mit $\omega_0=1000\,\frac{1}{\second}$\\[\baselineskip] +Maßstäbe: $2{,}5\,\milli\siemens\,\widehat{=}\,1\,\centi\metre$ : $10\,\ohm\,\widehat{=}\, 1\,\centi\metre$\\ +(Platzbedarf in x: $12\,\centi\metre$; in y: $14\,\centi\metre$)\\ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=.75cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$100\,\ohm$}; + \draw [blue] node at(.5,-.125){\footnotesize$\underbrace{\phantom{\uline{Y}_1\text{; } \uline{Z}_1}}$}; + \draw [blue] node at(.5,-.2)[below]{\footnotesize$\uline{Y}_1\text{; } \uline{Z}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1.25cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$5\,\micro\farad$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$20\,\milli\henry$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (-.5,0)--(0,0) (.2,1.25)--(0,1.25)--(0,.75)--(.2,.75) (.8,1.25)--(1,1.25)--(1,.75)--(.8,.75)(-.5,1)--(0,1) (-.5,0)--(2,0)--(2,1)--(1.8,1); + \fill (-.5,0)circle(.025) (-.5,1)circle(.025); + \draw [->,red] node at (-.5,.5)[left]{$\underline{Z}(p)\Rightarrow$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} +\uline{Y}_1(p)&=\frac{1}{R}+j\cdot p\cdot \omega_0\cdot C=(10+j\cdot p\cdot 5)\,\milli\siemens\\ +\uline{Z}_L(p)&=j\cdot p\cdot \omega_0\cdot L=+j\cdot p\cdot 20\,\ohm\\ +\uline{Z}(p)&=\uline{Z}_L(p)+\frac{1}{\uline{Y}_1(p)} +\intertext{Nicht gefragt: Kontrollrechnung: (konjugiert komplex erweitern)} +\uline{Z}(p)&=\uline{Z}_L(p)+\frac{1}{\uline{Y}_1(p)}=j\cdot p\cdot 20\,\ohm +\frac{1}{(10+j\cdot p\cdot 5)\,\milli\siemens}\\ +&=j\cdot p\cdot 20\,\ohm +\frac{1}{(10+j\cdot p\cdot 5)\,\milli\siemens}\cdot \frac{(10-j\cdot p\cdot 5)\cancel{\,\milli\siemens}}{(10-j\cdot p\cdot 5)\cancel{\,\milli\siemens}}\\ +&=\frac{10\cdot \power{10}{3}}{100+25\cdot p^2}\,\ohm+j\Big(20\cdot p-\frac{5\cdot \power{10}{3}\cdot p}{100+25\cdot p^2}\Big)\,\ohm +\end{align*} +\enlargethispage{1cm} +%\begin{align*} + \begin{tikzpicture}[very thick,scale=1] + %\centering + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,-8)grid(12,8); + \draw[thin](5,-8)--(5,8)(10,-8)--(10,6)(0,5)--(12,5)(0,-5)--(12,-5); + \draw[thin,->](0,0)--(12.5,0)node[right]{$\Re$}; + \draw[thin,->](0,-8)--(0,8.5)node[above]{$\Im$}; + \draw[red,->](0,0)--(4,0)node[above right]{$10\,\milli\siemens$}; + \draw[red](4,-8)--(4,8)node at (4,7.5)[right]{$\uline{Y}_1$}; + \draw[red]node at (4,-7.5)[right]{$\uline{Y}^*_1$}; + \draw [->](4.2,6.75)--(4.2,7.25)node at (4,6.5)[right]{$p$}; + \draw [->](4.2,-6.75)--(4.2,-7.25)node at (4,-6.5)[right]{$p$}; + \foreach \p in {-3,-2,...,3} + \filldraw (4,2*\p)circle(.05cm)node at (4,2*\p)[below left]{$\p$}; + \draw[blue,thin](0,0)--(4,-6)(0,0)--(5,-5)(0,0)--(8,-4); + \filldraw[blue](10,0)circle(0.05cm)node[above right]{$100\,\ohm$}; + \filldraw[blue](5,0)circle(0.05cm)node[below]{$M$}; + \draw[blue,thin](0:5cm)+(180:5cm)arc(180:360:5cm);%Mittelpunkt+Start arc Start:End:Radius + \draw[blue,->](8,-4)--(8,-2)node[below left]{$\uline{Z}_L(1)$}; + \draw[blue,->](5,-5)--(5,-1)node[below left]{$\uline{Z}_L(2)$}; + \draw[blue,->](3.05,-4.6)--(3.05,1.4)node[below left]{$\uline{Z}_L(3)$}; + \filldraw node at (10,0)[below right]{$p=0$}; + \filldraw (8,-4)circle(.05cm)node at (8,-4)[below right]{$p=1$}; + \filldraw (5,-5)circle(.05cm)node at (5,-5)[below right]{$p=2$}; + \filldraw (3.05,-4.6)circle(.05cm)node at (3,-4.5)[below left]{$p=3$}; + \draw[color=blue!50!red, very thick,domain=0:5] plot[parametric,samples=100,id=ortskurve17-2] function{1000/(100+25*t*t),2*t-500*t/(100+25*t*t)};% Ortskurve Faktor 1/10 Ohm in cm; + \draw[color=blue!50!red, very thick] node at (2.5, 4.25){$\uline{Z}(p)$}; + \draw[very thick] node at (13, 3){induktiv}; + \draw[very thick] node at (13, -3){kapazitiv}; + \draw[very thick] node at (10, 1){$f = 0$ rein ohmisch}; + \draw[very thick] node at (2, 6.75){$f\rightarrow\infty$ rein induktiv}; + \end{scope} +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] +% \draw[scale=0.5,domain=-3.141:3.141,smooth] +%plot[parametric,id=parametric-example] function{t*sin(t),t*cos(t)}; +% \end{scope} + \end{tikzpicture} +%\end{align*} +\vspace{-.5\baselineskip} +\begin{enumerate} + \item $\uline{Y}_1$ zeichnen mit $p$-Werten + \item $\uline{Z}_{min}$, $\uline{Z}_{max}$ berechnen \\ + $\rightarrow \uline{Z}$-Halbkreis: $r=5\,\centi\metre$ + \item $p$-Werte auf $\uline{Z}$ einzeichnen + \item $\uline{Z}_L(p)$ punktweise addieren + \item $\uline{Z}(p)$ Kurve zeichnen +\end{enumerate} +\clearpage +}{}% diff --git a/ET2_L_B17_A3.tex b/ET2_L_B17_A3.tex new file mode 100644 index 0000000..9164894 --- /dev/null +++ b/ET2_L_B17_A3.tex @@ -0,0 +1,130 @@ +\section{Ortskurve} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\setlength{\itemsep}{-0.2\baselineskip} + \item Zeichnen Sie maßstäblich die Ortskurve für das Spannungsverhältnis $U_2/U_1$ in Abhängigkeit von der Frequenz $f$. + \item Geben Sie die Grenzfrequenz der Schaltung an. + \item Wie groß ist die Dämpfung des Vierpols für die Frequenz $f=1{,}2\,\kilo\hertz$ (falls in Vorlesung behandelt: in $\deci\bel$) +\end{enumerate} +$\qquad \,\, R=16\,\kilo\ohm$; $C=12\,\nano\farad$; $200\,\hertz\leq f \leq 1{,}2\,\kilo\hertz$ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.5,0)--(1.5,0) (.8,1)--(1.5,1); + \draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}_1$}; \fill (-.5,0)circle(.025) (-.5,1)circle(.025); + \draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}_1$}; + \draw [->,blue] (1.5,.9)--(1.5,.1)node at(1.5,.5)[right]{$\underline{U}_2$}; \fill (1.5,0)circle(.025) (1.5,1)circle(.025); + \draw [->,red] (1.4,1.1)--(1.1,1.1) node at (1.25,1.1)[above]{$\underline{I}_2=0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: (Platzbedarf in x: $5\,\centi\metre$; in y: $15\,\centi\metre$)\\[0.5\baselineskip] +a) Spannungsverhältnis $U_2/U_1$ in Abhängigkeit von der Frequenz $f$ +\begin{align*} +\uline{I}_1&=\uline{U}_1 \cdot \uline{Y} = \uline{U}_1 \cdot \frac{j\omega\cdot C}{1+j\omega\cdot C\cdot R}=\frac{\uline{U}_1}{R+\frac{1}{j\omega\cdot C}} \\ +\uline{U}_2&= \uline{I}_1\cdot \frac{1}{j\omega\cdot C}= \frac{\uline{U}_1}{(R+\frac{1}{j\omega\cdot C})\cdot j\omega\cdot C}= +\frac{\uline{U}_1}{1+j\omega\cdot R\cdot C}\\ +\frac{\uline{U}_2}{\uline{U}_1}&=\frac{1}{1+j\omega\cdot R \cdot C} +=\frac{1-j\omega RC}{1+(\omega RC)^2}\\[0.5\baselineskip] +&\begin{tabular}{|l|l|l|} + \hline + % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ... + $f/Hz$&$2\pi\cdot f\cdot R\cdot C$&$\uline{U}_2/\uline{U}_1$\\ + \hline + 200 & 0{,}241&0{,}945-j0{,}228\\ + 828{,}9&1&0{,}5-j0{,}5\\ + 1200&1{,}448&0{,}323-j0{,}468\\ + \hline +\end{tabular} +\intertext{b) Grenzfrequenz} +\omega_g\cdot R\cdot C&=1\\ +f_g&=\frac{1}{2\pi\cdot R\cdot C}=\uuline{828{,}9\,\hertz} +\intertext{c) Dämpfung} +f&=1{,}2\,\kilo\hertz\qquad a=20\cdot \lg\frac{\uline{U}_2}{\uline{U}_1}\\ +\frac{\uline{U}_2}{\uline{U}_1}&=0{,}323-j0{,}468=0{,}569\cdot e^{-j50{,}56\,\degree}\\ +a&=20\cdot \lg(0{,}569)=\uuline{-4{,}9\,\deci\bel} +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,-7.5)grid [step=.5cm](5,7.5); + \draw[thin,->](0,0)--(5.5,0)node[right]{$\Re \, \left\{\frac{\uline{U}_2}{\uline{U}_1}\right\}$}; + \draw[thin,->](0,-8)--(0,8)node[above]{$\Im \, \left\{\frac{\uline{U}_2}{\uline{U}_1}\right\}$}; + \foreach \y in {1.5,1,...,-1.5} + \draw(0,5*\y)--(-.1,5*\y)node[left]{$\y$}; + \draw[red,thin](5,-7.5)--(5,7.5); + \draw[red](4.9,5)--(5.1,5)node [right]{$f_g=828{,}9\,\hertz$}; + \draw[red](4.9,-5)--(5.1,-5)node [right]{$f_g=828{,}9\,\hertz$}; + \draw[red](4.9,1.205)--(5.1,1.205)node [right]{$f=200\,\hertz$}; + \draw[red](4.9,-1.205)--(5.1,-1.205)node [right] {$f=200\,\hertz$}; + \draw[red](4.9,7.24)--(5.1,7.24)node [right] {$f=1200\,\hertz$}; + \draw[red](4.9,-7.24)--(5.1,-7.24)node [right] {$f=1200\,\hertz$}; + \draw node at(6.125,6.2){$(1+j\omega RC)$}; + \foreach \x in {.5,1} + \draw(5*\x,.1)--(5*\x,-.1)node at (5*\x+.1,-.25){$\x$}; + \fill[blue](2.5,0)circle(.05cm); + \draw[blue](0:2.5cm)+(180:2.5cm)arc(180:360:2.5cm);%Mittelpunkt+Start arc Start:End:Radius + \draw[blue!50!red,thin](0,0)--(5,-1.205)(0,0)--(5,-5)(0,0)--(5,-7.24); + \filldraw[blue!50!red](4.73,-1.14)circle(.05)node [below left]{$200\,\hertz$}; + \filldraw[blue!50!red](2.5,-2.5)circle(.05)node [below]{$f_g$}; + \filldraw[blue!50!red](1.62,-2.35)circle(.05)node [below left]{$1{,}2\,\kilo\hertz$}; + \draw [blue] node at(4,-2.5){$\frac{\underline{U}_2}{\underline{U}_1}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +% (Siehe Merksätze zu Inversion im Script, hier graphisch veranschaulicht:) +%\begin{align*} +% \begin{tikzpicture}[very thick,scale=1.25] +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] +% \draw(0,-1)--(0,1)node[above]{$\Im$}; +% \draw(0,0)--(2,0); +% \draw[red](1,-1)--(1,1); +% \draw[red,->](0,0)--(1,.75)node at(1.25,.75){$\uline{Y}_p$}; +% \end{scope} +% \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm] +% \draw(0,-1)--(0,1)node[above]{$\Im$}; +% \draw(0,0)--(2,0); +% \draw[blue](.5,0)circle(.5cm); +% \draw[blue,->](0,0)--(.7,-.5cm)node at(1.25,-.5){$\uline{Z}_p$}; +% \end{scope} +% \begin{scope}[>=latex,very thick,xshift=6cm,yshift=0cm] +% \draw(0,-1)--(0,1); +% \draw(-.5,0)--(2,0); +% \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu}; +% \draw[blue](.5,0)circle(.5cm); +% \draw[green!50!black,->](-.5,0)--(.7,-.5cm)node at(1,-.5){$\uline{Z}$}; \end{scope} +% \begin{scope}[>=latex,very thick,xshift=9cm,yshift=0cm] +% \draw(0,-1)--(0,1); +% \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu}; +% \draw(-.5,0)--(2.5,0); +% \draw[blue](1.25,0)circle(1cm); +% \draw[blue!50!red,->](-.5,0)--(2.1,.5)node at(2.5,.5){$\uline{Y}$}; +% \end{scope} +% \end{tikzpicture} +%\end{align*} +%Leitwertsortskuve der Parallelschaltung von $R_2$ und $C$ ergibt eine Gerade die im Abstand $G_2$ parallel zur imaginären Achse liegt.\\ +%$G_2=20\,\milli\siemens\,\widehat{=}\, 4\,\centi\metre$\\ +%Grenzen eintragen für $\uline{Y}_P=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre (f_u)\quad 25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre (f_o)\quad\text{ auch für $\uline{Y}^*_P$}$\\ +%Inversion von $\uline{Y}_P$ liefert die Widerstandsortskurve $\uline{Z}_P$ ein Kreis durch den Ursprung.\\ +%$R_2=50\,\ohm\,\widehat{=}\, 5\,\centi\metre \quad\text{Kreis um P(2.5,0) $r=2{,}5\,\centi\metre$}$\\ +%Der Serienwiderstand $R_1$ wird durch verschieben der Koordinaten hinzugefügt.\\ +%Punktweise Inversion von $\uline{Z}$ ein Kreis der jetzt nicht mehr durch den Ursprung geht, liefert $\uline{Y}$, wieder einen Kreis.\\ +%$\uline{Y}\quad\text{Kreis um P(6.67,0) $r=5\,\centi\metre$, da $\uline{Y}_{min}$ und $\uline{Y}_{max}$ bekannt sind}$\\ +%\begin{align*} +%R_2&=50\,\ohm\,\widehat{=}\, 5\,\centi\metre \quad\text{Kreis um P(2.5,0) $r=2{,}5\,\centi\metre$}\\ +%G_2&=20\,\milli\siemens\,\widehat{=}\, 4\,\centi\metre\\ +%\uline{Y}_P&=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre (f_u)\quad 25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre (f_o)\quad\text{zeichnen, auch $\uline{Y}^*_P$}\\ +%\uline{Y}&\quad\text{Kreis um P(6.67,0) $r=5\,\centi\metre$, da $\uline{Y}_{min}$ und $\uline{Y}_{max}$ bekannt sind}\\ +%\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B17_A4.tex b/ET2_L_B17_A4.tex new file mode 100644 index 0000000..7dbc835 --- /dev/null +++ b/ET2_L_B17_A4.tex @@ -0,0 +1,104 @@ +\section{Stromortskurve} +Konstruieren Sie die Stromortskurve $\uline{I}=f(p)$ zu der abgebildeten Schaltung\\ +für $0\leq p\leq 1$ !\\ +Es ist $Z_{RL}=p(R_0+jX_{L_0})$. Die Parameterwerte $p=0$; $0{,}25$; $0{,}5$; $0{,}75$ und $1$ sind zu markieren.\\ +Für welches $p$ wird $I=I_{max}$? Geben Sie diesen Stromwert an.\\ +Gegeben sind: $\uline{U}=U=10\,\volt$; $X_C=-3\,\kilo\ohm$; $R_0=6\,\kilo\ohm$; $X_{L_0}=8\,\kilo\ohm$.\\ +Maßstäbe: $1\,\kilo\ohm\,\widehat{=}\,1\,\centi\metre $; $50\,\micro\second\,\widehat{=}\, 1\,\centi\metre$\\ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.10) [above] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_{\phantom{L}}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.45,.0667) [above] {$X_{L}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (0,0)--(3,0)--(3,1)--(2.9,1); + \draw [->,blue] (0,.9)--(0,.1)node at(0,.5)[right]{$\underline{U}$}; \fill (0,0)circle(.025) (0,1)circle(.025); + \draw [->,red] (0,1.1)--(.4,1.1) node at (.25,1.1)[above]{$\underline{I}$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Variablen Pfeile + \draw[->] (1.3,.75)--(1.7,1.25); + \draw[->] (2.3,.75)--(2.7,1.25); + \draw[dashed] (1.3,.75)--(2.3,.75); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: (Platzbedarf in x: $11\,\centi\metre$; in y: $12\,\centi\metre$)\\[0.5\baselineskip] +\begin{align*} +\uline{I}(p)&=\uline{Y}(p)\cdot \uline{U}\\ +\intertext{$\uline{Z}(p)$ durch Vektoraddition der Widerstände zeichnen,} +Z(p)&=\sqrt{R_O^2+X^2_{LO}}=\sqrt{6^2+8^2}\,\kilo\ohm=10\,\kilo\ohm\qquad \text{für }p=1 +\intertext{$\uline{Z}^*(p)$ durch Spiegelung an der reellen Achse zeichnen und Parameter $p$ einzeichnen.} +X_C&=-3\,\kilo\ohm\,\widehat{=}\,-3\,\centi\metre\\ +R_0&=6\,\kilo\ohm\,\widehat{=}\,6\,\centi\metre\\ +X_{L_0}&=8\,\kilo\ohm\,\widehat{=}\,8\,\centi\metre\\ +\intertext{Senkrechte zu $\uline{Z}^*(p)$ durch den Ursprung zeichnen} +\overline{0N}&=1{,}8\,\centi\metre\,\widehat{=}\,1{,}8\,\kilo\ohm\\ +\intertext{Invertieren ergibt Durchmesser des Kreises} \overline{0D}&=\frac{1}{\overline{0N}}=\frac{1}{1{,}8\,\kilo\ohm}=555{,}5\,\micro\siemens +\,\widehat{=}\,11{,}1\,\centi\metre\\ +\intertext{Mittelpunkt bestimmen} \overline{0M}&=\frac{1}{2}\,\,\overline{0D}\,\widehat{=}\,5{,}55\,\centi\metre\\ +\text{$\uline{Y}(p)$ Kreis zeichnen. Max. Strom bei größtem Leitwert im Punkt D\newline (Durchmesser des Kreises = max. Abstand vom Ursprung)} +\intertext{Ablesen von $p=0{,}24$ (Abstand zwischen $N(OD\,\cap\,\uline{Z}^*(p))$ und $\uline{Z}^*(p)|_{p=0}$)} +%\uline{Z}^*(p)\text{ gibt }\Delta p=0{,}1\,\kilo\ohm\,\widehat{=}1\,\centi\metre,\text{ auf } +%I_{max}\text{ für }p&=0{,}24\\ +I_{max}&=\uline{Y}(p)\cdot \uline{U}=555{,}5\,\micro\siemens\cdot 10\,\volt=\uuline{5{,}55\,\milli\ampere} +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,-5)grid(10,10); + \draw[thin,->](0,0)--(10.5,0)node[right]{$\Re$}; + \draw[thin,->](0,-5.5)--(0,10.5)node[above]{$\Im$}; + \foreach \y in {10,9,...,-5} + \draw(0,\y)--(-.1,\y)node[left]{$\y$}; + \draw[red,->](0,0)--(0,-3)node at(.5,-1.5){$-jX_C$}; + \draw[red,->](0,-3)--(6,-3)node at(3,-2.75){$R$}; + \draw[red,->](6,-3)--(6,5)node at(6.5,1){$jX_{LO}$}; + \draw[red,->](0,-3)--(6,5)node at(4,1.25){$\uline{Z}(p)$}; + \draw[red,->](0,3)--(6,-5)node at(6,-4){$\uline{Z}^*(p)$}; + \draw[black!35!,thin](0,1.5)circle(1.5cm); + \draw[blue,thin](0,0)--(36.87:11.1cm)node at(9.1,6.75){D}; + \draw[blue]node at(9,7.5){$\uline{Y}(p)$}; + \filldraw[blue](36.87:5.55)circle(0.05cm)node[left]{M}; + \filldraw[blue](0,0)--(8,6)node at(1,1.1){N}; + \draw[blue](36.87:5.55)circle(5.55cm); + \filldraw[red!50!blue](0,3)circle(0.05cm)node [right]{\footnotesize{$p=0$}}; + \filldraw[red!50!blue](0,3)++(-53.13:2.5cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}25$}}; + \filldraw[red!50!blue](0,3)++(-53.13:5cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}5$}}; + \filldraw[red!50!blue](0,3)++(-53.13:7.5cm)circle(0.05cm)node [above right]{\footnotesize{$p=0{,}75$}}; + \filldraw[red!50!blue](0,3)++(-53.13:10cm)circle(0.05cm)node [right]{\footnotesize{$p=1$}}; + \filldraw[green!50!black](0,-3)circle(0.05cm)node [below right]{\footnotesize{$p=0$}}; + \filldraw[green!50!black](0,-3)++(53.13:2.5cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}25$}}; + \filldraw[green!50!black](0,-3)++(53.13:5cm)circle(0.05cm)node [below right]{\footnotesize{$p=0{,}5$}}; + \filldraw[green!50!black](0,-3)++(53.13:7.5cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}75$}}; + \filldraw[green!50!black](0,-3)++(53.13:10cm)circle(0.05cm)node [left]{\footnotesize{$p=1$}}; + \draw[red!50!blue,very thin](0,0)--(6,-5); + \draw[red!50!blue,very thin](0,0)--(4.5,-3); + \draw[red!50!blue,very thin](0,0)--(6,-2); + \draw[red!50!blue,very thin](0,0)--(12,8); + \draw[red!50!blue,very thin](0,0)--(0,6.7); + \filldraw[red!50!blue,very thin](-40.5:2.45cm)circle(0.05cm)node [right]{\footnotesize{$p=1$}}; + \filldraw[red!50!blue,very thin](-34.4:3.6cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}75$}}; + \filldraw[red!50!blue,very thin](6,-2)circle(0.05cm)node [right]{\footnotesize{$p=0{,}5$}}; + \filldraw[red!50!blue,very thin](33.69:11.1cm)circle(0.05cm)node [right]{\footnotesize{$p=0{,}25$}}; + \filldraw[red!50!blue,very thin](90:6.666cm)circle(0.05cm)node [right]{\footnotesize{$p=0$}}; + \foreach \x in {0,1,...,10} + \filldraw(\x,.1)--(\x,-.1)node at (\x,-.33){$\x$}; + \end{scope} + \end{tikzpicture} +\end{align*} +Reihenfolge: $j\underline{X}_C; R; j\underline{X}_L; \underline{Z}(p)$; p-Werte; $\bot \underline{Z^*}(p) +\Rightarrow \overline{ON}; \\ +\overline{OD}$ Durchmesser; Kreis um $\overline{OM} \Rightarrow \underline{Y(p)}$ +\clearpage +}{}% diff --git a/ET2_L_B17_A5.tex b/ET2_L_B17_A5.tex new file mode 100644 index 0000000..b4c5bf0 --- /dev/null +++ b/ET2_L_B17_A5.tex @@ -0,0 +1,123 @@ +\section{Widerstandstransformation} +Ein Verbraucher mit $\uline{Z}_v=(6+j4)\,\kilo\ohm$ soll mit Hilfe von zwei Blindwiderständen so an eine Spannungsquelle mit dem Innenwiderstand $\uline{Z}_i=(3+j1{,}5)\,\kilo\ohm$ angepasst werden, dass er die größtmögliche Wirkleistung aufnimmt.\\ +Bestimmen Sie zeichnerisch die hinzuzuschaltenden Blindwiderstände (Art und Größe) einer möglichen Schaltung und skizzieren Sie ihre Zusammenschaltung mit $\uline{Z}_v$.\\ +Maßstab: $1\,\kilo\ohm \,\widehat{=}\,1\,\centi\metre$\\[\baselineskip] +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: (Platzbedarf in x: $10\,\centi\metre$; in y: $20\,\centi\metre$) \\ +($14\,\centi\metre$ in y: reichen auch, wenn eine Linie durch die Rechnung geht)\\[0.5\baselineskip] +Anpassung: Der transformierte Widerstand muss gleich dem konjugiert komplexen Innenwiderstand sein.\\[.5\baselineskip] +$\uline{Z}^*_i=(3-j1{,}5)\,\kilo\ohm$\\[\baselineskip] +\begin{minipage}[t]{0.45\textwidth} +1. Möglichkeit +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_s$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_p$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Z}_v$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (0,0)--(2,0)--(2,.1)(.9,1)--(2,1)--(2,.9); + \draw node at(-.5,.5)[right]{$\underline{Z}^*_i\Rightarrow$}; + \fill (0,0)circle(.025) (0,1)circle(.025) (1,0)circle(.025) (1,1)circle(.025); + \end{scope} + \draw node at(1.5,-.25){$\underbrace{\phantom{xxxxxxxxxx}}_{\uline{Z}_p'}$}; + \end{tikzpicture} +\end{align*} +Imaginärteil anpassen: +\begin{align*} +\frac{1}{\underline{Z}'}&=\frac{1}{\underline{Z}_{v}}+\frac{1}{X_L{_p}}\\ +\intertext{Jetzt Zeichnung anfertigen um $X_{vp}$ und $X'_p$ zu bestimmen.} +\frac{1}{X_L{_p}}&=\left(\overbrace{\frac{1}{6{,}3\,\kilo\ohm}}^{1/X'_p}-\overbrace{\frac{1}{13\,\kilo\ohm}}^{1/X_{vp}}\right)\\ +X_L{_p}&=\uuline{12{,}2\,\kilo\ohm}\\ +X_{C_s}&=\uuline{-5{,}6\,\kilo\ohm} +\end{align*} +\end{minipage} +\hfill +\begin{minipage}[t]{0.45\textwidth} +2. Möglichkeit +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_s$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_p$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Z}_v$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (0,0)--(2,0)--(2,.1)(.9,1)--(2,1)--(2,.9); + \draw node at(-.5,.5)[right]{$\underline{Z}^*_i\Rightarrow$}; + \fill (0,0)circle(.025) (0,1)circle(.025) (1,0)circle(.025) (1,1)circle(.025); + \end{scope} + \draw node at(1.5,-.25){$\underbrace{\phantom{xxxxxxxxxx}}_{\uline{Z}_p''}$}; + \end{tikzpicture} +\end{align*} + +\begin{align*} +\frac{1}{\underline{Z}''_p}&=\frac{1}{\underline{Z}_{v}}+\frac{1}{X_{C_p}}\\ +\intertext{Im weiteren Schritt Zeichnung vervollständigen um $X''_p$ zu bestimmen.} +\frac{1}{X_{C_p}}&=\left(\overbrace{\frac{1}{-6{,}3\,\kilo\ohm}}^{1/X''_p}-\overbrace{\frac{1}{13\,\kilo\ohm}}^{1/X_{vp}}\right)\\ +{X_{L_s}}&=\uuline{+2{,}6\,\kilo\ohm}\\ +X_{C_p}&=\uuline{-4{,}2\,\kilo\ohm} +\end{align*} +\end{minipage} +%\vfill +\newpage +\begin{align*} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,-7)grid(10,13); + \draw[thin,->](0,0)--(10.5,0)node[right]{$\Re$}; + \draw[thin,->](0,-5.5)--(0,13.5)node[above]{$\Im$}; + \foreach \y in {13,12,...,-7} + \draw(0,\y)--(-.1,\y)node[left]{$\y$}; + \foreach \x in {0,1,...,10} + \draw(\x,0)--(\x,0)node[below]{$\x$}; + \draw[red,->](0,0)--(6,4)node at(3.5,2){$\uline{Z}_v$}; + \draw[black!50!,thin](6,4)+(214:.5cm)arc(214:304:.5cm); + \fill[black!50!,thin](6,4)+(259:.25cm)circle(.05cm); + \draw[red,->](0,0)--(3,-1.5)node at(1.75,-1.25){$\uline{Z}^*_i$}; + \filldraw[blue](4.3,0)circle(0.05cm)node[above]{M $\footnotesize(4.33)$}; + \draw(4.33,0)circle(4.33cm); + \draw[blue](8.66,0)--(0,13)node [below right]{$X_{vp}\approx{13\,\kilo\ohm}$}; + \draw[blue,thick](8.66,0)--(0,6.3)node [right]{$X'_p=6{,}3\,\kilo\ohm$}; + \draw[blue,thick](8.66,0)--(0,-6.3)node [right]{$X''_p=-6{,}3\,\kilo\ohm$}; + \draw[red!50!blue,->,ultra thick](4.33,0)+(67.3:4.33cm)arc(67.3:108:4.33cm)node at(4.5,4.75){$X_{L_p}$}; + \draw[green!50!blue,->,ultra thick](4.33,0)+(67.3:4.33cm)arc(67.3:-108:4.33cm)node at(4.5,-4.75){$X_{C_p}$}; + \draw[red!50!blue,->,ultra thick](4.33,0)+(108:4.33cm)--(3,-1.5)node at(2.5,2.5){$X_{C_s}$}; + \draw[green!50!blue,->,ultra thick](4.33,0)+(-108:4.33cm)--(3,-1.5)node at(2.5,-2.5){$X_{L_s}$}; + \draw[magenta,->,ultra thick](0,0)--(3,4.14)node at(1,2){$\uline{Z}'$}; + \draw[green!75!red,->,ultra thick](0,0)--(3,-4.14)node at(1,-2){$\uline{Z}''$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=-.5cm] + \fill[white](5,7)rectangle(10,13.5); + \draw node at (5,13.25)[right]{1. Möglichkeit:}; + \draw[red] node at (5,12.75)[right]{1) $Z_v=(6+j4)\,\kilo\ohm$ und $\uline{Z}^*_i=(3-j1{,}5)\,\kilo\ohm$}; + \draw[blue] node at (5,12.25)[right]{2) $X_{vp}\bot Z_v\rightarrow X_{vp}\cap \Im\rightarrow X_{vp}\approx 13\,\kilo\ohm$}; + \draw node at (5,11.75)[right]{3) Kreis mit {\O}: $[0; X_{vp}\cap\Re]\qquad M(4.33)$}; + \draw[red!50!blue] node at (5,11.25)[right]{4) $L ||$ geschaltet $\rightarrow$ Kreissegment bis $\Re(\uline{Z}^*_i)$}; + \draw[red!50!blue] node at (5,10.75)[right]{5) $X_{C_s}$}; + \draw[blue] node at (5,10.25)[right]{6) $X'_p=6{,}3\,\kilo\ohm$}; + \draw[magenta] node at (5,9.75)[right]{7) $Z'$}; + \draw node at (5,9.25)[right]{2. Möglichkeit: (1--3 wie 1. Möglichkeit)}; + \draw[green!50!blue] node at (5,8.75)[right]{8) $X_{C_p}$ Kreissegment bis $\Re(\uline{Z}^*_i)$}; + \draw[green!50!blue] node at (5,8.25)[right]{9) $X_{L_s}=2{,}6\,\kilo\ohm$ abgelesen}; + \draw[blue] node at (5,7.75)[right]{10) $X''_p=-6{,}3\,\kilo\ohm$}; + \draw[green!75!red] node at (5,7.25)[right]{11) $Z''$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B17_A6.tex b/ET2_L_B17_A6.tex new file mode 100644 index 0000000..3485e65 --- /dev/null +++ b/ET2_L_B17_A6.tex @@ -0,0 +1,97 @@ +\section{Brückenschaltung} +Gegeben:\\[\baselineskip] +$R_1=R_2=R_3=|X_C|=1\,\kilo\ohm$; $U=200\,\volt\cdot e^{j0\degree}$\\[\baselineskip] +Gesucht: +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} + \item Spannung $U_{ab}$ zwischen den Klemmen $a-b$ nach Betrag und Phasenwinkel. + \item Qualitatives Zeigerdiagramm aller Ströme und Spannungen.\\ + (Qualitativ, d.h. alle Bauteilwerte verschieden) +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$}; + \draw [->,red] (.95,-.1)--(.75,-.1) node at (.85,-.1)[right]{\footnotesize$I_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_3$}; + \draw [->,red] (.95,-.1)--(.75,-.1) node at (.85,-.1)[right]{\footnotesize$I_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{R_2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Kondensator + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{C}$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (1.5,.8)--(1.5,1)--(-.5,1) (1.5,-.8)--(1.5,-1)--(-.5,-1); + \fill (-.5,1)circle(.025) (-.5,-1)circle(.025); + \draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{ab}$}; + \fill (.5,0)circle(.025) node at (.5,0)[left]{a}; + \fill (1.5,0)circle(.025)node at (1.5,0)[right]{b}; + \draw [->,blue] (-.5,.8)--(-.5,-.8)node at (-.5,0)[left]{$\underline{U}$}; + \draw node at (-.75,1){$+$}; + \draw node at (-.75,-1){$-$}; +\end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: (Platzbedarf in x: $11\,\centi\metre$; in y: $6\,\centi\metre$)\\ +a) Spannung $U_{ab}$ +\begin{align*} +%R_i&=|X_c|=1\,\kilo\ohm\\ +%\uline{U}&=200\,\volt\\ +\uline{U}_{ab}&+\uline{U}_C-\uline{U}_{R_2}=0 \qquad \text{Masche}\\ +\uline{U}_{ab}&=\uline{U}_{R_2}-\uline{U}_C +\intertext{Spannungsteiler} +\uline{U}_{R_2}&=\uline{U}\cdot \frac{R}{2\cdot R}=\frac{\uline{U}}{2}=100\,\volt\\ +\uline{U}_C&=\uline{U}\cdot \frac{jX_C}{R+jX_C}=200\,\volt\cdot \frac{-j\cdot 1}{1-j\cdot 1}=200\,\volt(0{,}5-j0{,}5)=(100-j100)\,\volt\\ +\Rightarrow\uline{U}_{ab}&=100\,\volt-(100-j100)\,\volt=+j100\,\volt=\uuline{100\,\volt\cdot e^{+j90\degree}} +\end{align*} +\begin{align*} +\intertext{b) Qualitatives Zeigerdiagramm (zu Schaltbild, beliebiges $X_C$)} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,0)grid(10,5); + \draw[thin,->](0,0)--(10.5,0)node[right]{$\Re$}; + \draw[thin,->](0,0)--(0,5.5)node[above]{$\Im$}; + \foreach \x in {10,9,...,0} + \draw(\x,0)--(\x,-.1)node[below]{$\x$}; + \foreach \y in {5,4,...,0} + \draw(0,\y)--(-.1,\y)node[left]{$\y$}; + \draw[blue,->](0,0)--(10,0)node at(11.5,.375){$\uline{U}=\uline{U}_{R_1}+\uline{U}_{R_2}$}; + \draw[blue,->](0,0)--(5,0)node at(3.75,.25){$\uline{U}_{R_1}$}; + \draw[blue,->]node at(7.5,.25){$\uline{U}_{R_2}$}; + \draw[black!50!,thin](5,0)+(0:5cm)arc(0:180:5cm); + \draw[black!50!,thin](2,4)+(-116.565:.5cm)arc(-116.565:-26.565:.5cm); + \fill[black!50!,thin](2,4)+(-71.565:.25cm)circle(.05cm); + \draw[blue!50!red,->](0,0)--(2,4)node at (1,3.5){$\uline{U}_{R_3}$}; + \draw[blue!50!red,->](2,4)--(10,0)node at (7,2){$\uline{U}_C$}; + \draw[blue!50!red,->](5,0)--(2,4)node at (4.5,1.5){$\uline{U}_{ab}$}; + \draw[red,->](0,0)--(1.5,3)node at (.45,1.5){$\uline{I}_{C}$}; + \draw[red,->](1.5,3)--(4,3)node at (2.5,2.75){$\uline{I}_{R}$}; + \draw[red,->](0,0)--(2.5,0)node at (1.5,.25){$\uline{I}_{R}$}; + \draw[magenta,->](0,0)--(4,3)node at (2.5,1.5){$\uline{I}$}; + \draw node at (-1.25,0){\textbf{+}}; + \draw node at (11.25,0){\textbf{--}}; + \draw node at (5.25,.375){\textbf{a}}; + \draw node at (2,4.25){\textbf{b}}; + \draw [blue]node at (10,5)[right]{\footnotesize{$\uline{U}=200\,\volt\,\widehat{=}\,10\,\centi\metre$}}; + \draw [blue]node at (10,4.5)[right]{\footnotesize{$\uline{U}_{R_1}=100\,\volt, \uline{U}_{R_2}=100\,\volt$}}; + \draw [red]node at (10,4)[right]{\footnotesize{$\uline{I}_{R}$}}; + \draw [red!50!blue]node at (10,3.5)[right]{\footnotesize{$\uline{U}_{C}\bot\,\uline{U}_{R_3}$ addiert sich zu $\uline{U}$}}; + \draw [red]node at (10,3)[right]{\footnotesize{$\uline{I}_{C}$ in Phase zu $\uline{U}_{R_3}$}}; + \draw [magenta]node at (10,2.5)[right]{\footnotesize{$\uline{I}=\uline{I}_{C}+\uline{I}_{R}$}}; + \end{scope} + \end{tikzpicture} +\end{align*} +Anmerkung:\\ +Wenn wie angegeben $|X_C|=R\Rightarrow \uline{U}_{R3}\bot\uline{U}_{C}\Rightarrow$ gleichschenkliges, rechtwinkliges Dreieck $\Rightarrow\uline{U}_{ab}\bot \Re$-Achse. +\clearpage +}{}% diff --git a/ET2_L_B17_A7.tex b/ET2_L_B17_A7.tex new file mode 100644 index 0000000..a17acf4 --- /dev/null +++ b/ET2_L_B17_A7.tex @@ -0,0 +1,93 @@ +\section{Wechselstrombrücke} +Gegeben ist die dargestellte Wechselstrombrücke, die zum Messen der Größe von $R_2$ +und $L_2$ dient. Dabei ist $R_1=1\,\kilo\ohm$, $R_3=R_4=2\,\kilo\ohm$ und $L_1=1\,\milli\henry$.\\ +Die Brücke ist bei einer Kreisfrequenz von $\omega=\power{10}{6}\,\power{\second}{-1}$ und $C_1=2\,\nano\farad$ abgeglichen.\\[\baselineskip] +Berechnen Sie $R_2$ und $L_2$! +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_1$}; + \draw[->](.3,-.2)--(.7,.2); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_3$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=.25cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=-.25cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_4$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%V-meter + \draw (0,0)--(.367,0) (.633,0)--(1,0); + \draw (.5,0)circle(.133); + \draw[->](.4,.1)--(.6,-.1); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte + \draw (.1,1)--(0,1)--(0,-.5)(0,0)--(1,0) + (1.01,-.25)--(1,-.25)--(1,.25)--(1.01,.25) + (1.99,-.25)--(2,-.25)--(2,.25)--(1.99,.25) + (2,0)--(3,0) + (3.9,1)--(4,1)--(4,-.5); + \draw [->,blue] (.2,-.5)--(3.8,-.5)node at(3,-.5)[above]{$\underline{U}=20\,\volt$}; + \fill (0,-.5)circle(.025) (4,-.5)circle(.025); + \end{scope} + \draw node at (1.5,1.25)[above]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxxxxxxx}}$}; + \draw node at (3.5,1.25)[above]{$\overbrace{\phantom{xxxxx}}$}; + \draw node at (1.5,1.5)[above]{$\uline{Z}_1$}; + \draw node at (3.5,1.5)[above]{$\uline{Z}_3$}; + \draw node at (1.5,-.5)[below]{$\underbrace{\phantom{xxxxxxxxxxx}}$}; + \draw node at (3.5,-.5)[below]{$\underbrace{\phantom{xxxxx}}$}; + \draw node at (1.5,-.75)[below]{$\uline{Z}_2$}; + \draw node at (3.5,-.75)[below]{$\uline{Z}_4$}; +% \begin{scope}[>=latex,very thick]%Variablen Pfeile +% \draw[->] (1.3,.75)--(1.7,1.25); +% \draw[->] (2.3,.75)--(2.7,1.25); +% \draw[dashed] (1.3,.75)--(2.3,.75); +% \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +R_P&=R_S+\frac{X^2_S}{R_S}\label{eq:rp177}\\ +X_P&=X_S+\frac{R^2_S}{X_S}\label{eq:xp177} +\end{align} +Berechnung: +\begin{align*} +\uline{Z}_1&=R_1+jX_{L_1}-jX_{C_1} &\uline{Z}_3&=R_3\\ +\quad\uline{Z}_2&=R_2 ||X_{L_2} &\uline{Z}_4&=R_4 +\end{align*} +\begin{align*} +\intertext{Brücke abgeglichen, wenn:} +\frac{\uline{Z}_1}{\uline{Z}_2}&=\frac{\uline{Z}_3}{\uline{Z}_4}=\frac{R_3}{R_4}=1\\ +\uline{Z}_2&=\uline{Z}_1=R_1+j\omega L_1 -j\frac{1}{\omega C_1}\\ +&=1\,\kilo\ohm+j\Big(\underbrace{\power{10}{6} \,\cancel{\second^{-1}} \cdot \power{10}{-3}\,\ohm\cancel{\second}}_{1\,\kilo\ohm} -\underbrace{\frac{1}{\power{10}{6} \,\cancel{\second^{-1}}\cdot 2 \cdot \power{10}{-9}\,\frac{\cancel{\second}}{\ohm}}}_{\frac{1}{2}\,\kilo\ohm}\Big)=\uuline{(1+j0{,}5)\,\kilo\ohm}\\ +\end{align*} +Um $R_2$ und $L_2$ zu bestimmen, die Reihenschaltung $\uline{Z}_2=R_{2S}+jX_{2S}=(1+j0{,}5)\,\kilo\ohm$ in Parallelwiderstände umrechnen. +\begin{align*} +\uline{Y_2}&=\frac{1}{\uline{Z_2}}=\frac{1}{(1+j0{,}5)\,\kilo\ohm} +=(\underbrace{0{,}8}_{G_2}-\underbrace{j0{,}4}_{B_{L_2}})\,\milli\siemens\\ +R_{2}&=\frac{1}{G_2}=\frac{1}{0{,}8\,\milli\siemens}=\uuline{1{,}25\,\kilo\ohm}\\ +X_{L_2}&=\frac{-1}{B_{L_2}}=\frac{-1}{-0{,}4\,\milli\siemens}=\uline{+2{,}5\,\kilo\ohm}\\ +\text{aus }X_{L_2}&=\omega\cdot L \Rightarrow \\ +L_2&=\frac{X_{L_2}}{\omega}=\frac{2{,}5\,\kilo\ohm}{\power{10}{6}\cdot \power{\,\second}{-1}}=\uuline{2{,}5\,\milli\henry}\\ +\intertext{Alternativ mit Formeln \ref{eq:rp177} und \ref{eq:xp177}} +R_{2P}&=R_{2S}+\frac{X^2_{2S}}{R_{2S}}\\ +X_{2P}&=X_{2S}+\frac{R^2_{2S}}{X_{2S}}\\ +R_2&=R_{2S}+\frac{X^2_{2S}}{R_{2S}}=(1+\frac{0{,}5^2}{1})\,\kilo\ohm=1{,}25\,\kilo\ohm\\ +X_{L_2}&=X_{2S}+\frac{R^2_{2S}}{X_{2S}}=(0{,}5+\frac{1^2}{0{,}5})\,\kilo\ohm=2{,}5\,\kilo\ohm +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B17_A8.tex b/ET2_L_B17_A8.tex new file mode 100644 index 0000000..adec1bf --- /dev/null +++ b/ET2_L_B17_A8.tex @@ -0,0 +1,103 @@ +\section{Wechselstrombrücke} +a) Zustand abgeglichenen Brücke $\mathbf{(U_{ab}=0})$\\ +Welches Bauteil muss fur $X$ eingesetzt werden, um diese Voraussetzung zu erfüllen?\\ +Berechnen Sie den Wert des Bauteils als Funktion von $R$ und $X_L$.\\[\baselineskip] +b) Zustand nicht abgeglichene Brücke: $\mathbf{(U_{ab}\neq 0)}$\\[\baselineskip] +Zeichnen Sie ein qualitatives Zeigerdiagramm aller eingezeichneten Spannungen\\ +(Bezug $\uline{U}=U\cdot e^{j0\,\degree}$), so dass $\uline{U}_{ab}$ auf $\uline{U}$ senkrecht steht (rechte Winkel müssen gekennzeichnet werden).\\ +Welches Bauteil muss fur $X$ eingesetzt werden, um diese Voraussetzung zu erfüllen?\\ Berechnen Sie den Wert dieses Bauteils als Funktion von $R$ und $X_L$ anhand der Beziehung zwischen den Zeigerlängen. +\begin{align*} + \begin{tikzpicture}[very thick,scale=2.5] + \begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,blue] (.3,.35)--(.7,.35)node at(.5,.35)[right]{\footnotesize$\uline{U}_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{b}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{a}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$X$}; + \draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{X}$}; + \draw node at (.5,0){?}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=-.5cm,yshift=-.5cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick]%Knotenpunkte + \draw (1.5,.8)--(1.5,1)--(-.5,1) (1.5,-.8)--(1.5,-1)--(-.5,-1)(-.4,1)--(-.5,1)--(-.5,.5)(-.4,-1)--(-.5,-1)--(-.5,-.5); +% \fill (-.5,1)circle(.025) (-.5,-1)circle(.025); + \draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{ab}$}; + \fill (.5,0)circle(.025) node at (.5,0)[left]{a}; + \fill (1.5,0)circle(.025)node at (1.5,0)[right]{b}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: (Platzbedarf in x: $9\,\centi\metre$; in y: $\pm 5\,\centi\metre$)\\ +\begin{align*} +\intertext{a) Zum Brückenabgleich muß $X$ eine Kapazität sein.}\\ +\intertext{Abgleichbedingung:}\\ +\frac{jX_L}{R}&=\frac{R}{jX_C}\\ +X&=\uuline{-\frac{R^2}{X_L}} \qquad\text{ neg. VZ $\Rightarrow$ Kapazität $X=X_C$} \\ +-\frac{1}{j\omega C}&=-\frac{R^2}{j\omega L}\Rightarrow \quad\text{oder}\quad C=\frac{L}{R^2} +\end{align*} +\clearpage +\begin{align*} +\intertext{b) Bedingung: $\uline{U}_{ab}\,\bot\,\uline{U}$; gleiche Zeigerlängen.} + \begin{tikzpicture}[very thick,scale=1] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[ultra thin,black!50!](0,-5)grid(8,5); + \draw[thin,->](0,0)--(8.5,0)node[right]{$\Re$}; + \draw[thin,->](0,-5.5)--(0,5.5)node[above]{$\Im$}; + \foreach \y in {5,4,...,-5} + \draw(0,\y)--(-.1,\y)node[left]{$\y$}; + \draw[black!50!,dashed](0,0)+(20:4.2cm)arc(20:-20:4.2cm); + \draw[black!50!,dashed](8,0)+(160:4.2cm)arc(160:200:4.2cm); + \draw[black!50!,dashed](4,1.5)--(4,-1.5); + \fill(4,0)circle(0.075cm); + \draw[blue,->](0,0)--(8,0)node at(7.5,.25){$\uline{U}$}; + \draw[black!50!](4,0)circle(4cm); + \draw[blue,->](0,0)--(30:6.93)node at(2.5,1.75){$\uline{U}_a$}; + \draw[blue,->](0,0)--(-30:6.93)node at(2.5,-1.75){$\uline{U}_{b}$}; \draw[blue,->](30:6.93)--(-30:6.93)node at(5.5,-1.25){$\uline{U}_{ab}$}; + \draw[blue,->](30:6.93)--(8,0)node at(6.75,1.5){$\uline{U}_{R}$}; \draw[blue,->](-30:6.93)--(8,0)node at(6.75,-1.5){$\uline{U}_{X}$}; + \draw node at (6,3.75){a}; + \draw node at (6,-3.75){b}; + \draw[red](30:6.93)+(210:.5cm)arc(210:300:.5cm); + \fill[red](30:6.93)+(255:.25cm)circle(.05cm); + \draw[red](-30:6.93)+(150:.5cm)arc(150:60:.5cm); + \fill[red](-30:6.93)+(105:.25cm)circle(.05cm); + \draw[red](0:6)+(0:.5cm)arc(0:90:.5cm); + \fill[red](0:6)+(45:.25cm)circle(.05cm); + \draw node at(8.5,4.5)[right]{$\diamond$\ \ \footnotesize{$\uline{U}$}}; + \draw node at(8.5,4)[right]{$\phantom{\diamond x}$\footnotesize{Thaleskreis, da $\uline{U}_q \bot \uline{U}_R$ und $\uline{U}_b \bot \uline{U}_X$}}; + \draw node at(8.5,3.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_R=\uline{U}$}}; + \draw node at(8.5,3)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_a$ und $\uline{U}_R$ zeichnen}}; + \draw node at(8.5,2.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_{ab}-\uline{U}_b=0$ und $\uline{U}_{ab} \bot\uline{U}$}}; + \draw node at(8.5,2)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_{ab}$ und $\uline{U}_b$ zeichnen}}; + \draw node at(8.5,1.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_b+\uline{U}_X=\uline{U}$}}; + \draw node at(8.5,1)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_X$ zeichnen}}; + \end{scope} + \end{tikzpicture} +\end{align*} +Bei der nicht abgeglichenen Brücke mit der Bedingung $\uline{U}_{ab}\,\bot\,\uline{U}$, muss $X$ eine Induktivität sein.\\ + +Rechnerisch: +\begin{align*} +|U_a|&=|U_b|\Rightarrow |X_L\cdot I_a|=|R\cdot I_b|\tag{1}\\ +|U_R|&=|U_X|\Rightarrow |R\cdot I_a|=|X\cdot I_b| \tag{2}\\ +\frac{X_L}{R}&=\frac{R}{X}\tag{1:2}\\ +X&=\uuline{+\frac{R^2}{X_L}} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B18_A1.tex b/ET2_L_B18_A1.tex new file mode 100644 index 0000000..8e2a96c --- /dev/null +++ b/ET2_L_B18_A1.tex @@ -0,0 +1,88 @@ +\section {Übertrager im Leerlauf} +$U_1=230\,\volt$, $R_1=5\,\ohm$, $I_1=10\,\ampere$, $U_2=100\,\volt$\\ +ausgangsseitiger Leerlauf\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie den Eigangswiderstand $\uline{Z}_1=\frac{\uline{U}_1}{\uline{I}_1}$ nach Betrag und Phase sowie die aufgenommene Wirk- und Blindleistung. +\item Berechnen sie $\omega L_1$, $\omega L_2$ und $\omega M$ unter Annahme einer idealen Kopplung. +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,-.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,red] (1,.1)--(.75,.1)node at(.875,.1)[above right]{\footnotesize$\uline{I}_2=0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(1,0)--(1,.2) (2.5,0)--(1.5,0)--(1.5,.2) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9); + \fill (2.5,0)circle(0.025cm)(2.5,1)circle(0.025cm); + \draw [->,blue] (2.5,.8)--(2.5,.2) node at (2.5,.5)[right]{$\uline{U}_2$}; + \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$M$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{enumerate} +\item Komplexer Eingangswiderstand\\ +Begriffe:\\ +$\uline{Z}$ Impedanz oder komplexer Widerstand\\ +$Z=|\uline{Z}|$ Scheinwiderstand \\ +\begin{minipage}[c]{0.25\textwidth} + \begin{tikzpicture}[scale=.2] + \draw[->](0,0)--(5,0)node[right]{$R_1$}; + \draw[->](5,0)--(5,22.45)node at(5,10)[right]{$\omega L_1$}; + \draw[->](0,0)--(77.44:23cm)node at(0,10)[left]{$\uline{Z}_1$}; + \draw[->](2,0)arc(0:77.4:2cm)node at(1,2)[right]{$\varphi_1$}; + \end{tikzpicture} +\end{minipage} +\begin{minipage}[c]{0.75\textwidth} +\begin{align*} +Z_1&=\frac{U_1}{I_1}=\frac{230\,\volt}{10\,\ampere}=23\,\ohm\\ +\text{mit }R_1&=Z_1\cdot \cos\varphi_1\Rightarrow\\ +\varphi_1&=\arccos\frac{R_1}{Z_1}=\arccos\frac{5\,\ohm}{23\,\ohm}=77{,}44\,\degree\\ +\uline{Z}_1&=Z_1\cdot e^{j\varphi_1}=\uuline{23\,\ohm\cdot e^{j77{,}44\,\degree}}\\ +P&=U_1\cdot I_1\cdot \cos\varphi_1=230\,\volt\cdot 10\,\ampere\cdot \cos(77{,}44\,\degree)=\uuline{500\,\watt}\\ +Q&=U_1\cdot I_1\cdot \sin\varphi_1=230\,\volt\cdot 10\,\ampere\cdot \sin(77{,}44\,\degree)=\uuline{2245\,var}\\ +\text{oder }\uline{S}&=P+jQ=\frac{\uline{U}_1^2}{\uline{Z_1}}=\frac{(230\,\volt)^2}{23\,\ohm\cdot e^{j77{,}44\,\degree}}=(\underbrace{500}_{P}-j\underbrace{2245}_{Q})\,\volt\ampere +\end{align*} +\end{minipage} +\clearpage +\item Blindwiderstände und Kopplungswiderstand\\ +Begriffe:\\ +$L$ Selbstinduktivität\\ +$M$ Gegeninduktivität\\ +$X_L=\omega L$ Reaktanz oder Blindwiderstand\\ +$X_M=\omega M$ Kopplungswiderstand\\ +\begin{align*} +\omega L_1&=Z_1\cdot \sin\varphi_1=23\,\ohm\cdot \sin(77{,}44\,\degree)=\uuline{22{,}45\,\ohm}\\ +U_2&=\omega M\cdot I_1\Rightarrow\\ +\omega M&=\frac{U_2}{I_1}=\frac{100\,\volt}{10\,\ampere}=\uuline{10\,\ohm}\\ +M&=\sqrt{L_1\cdot L_2}\\ +\omega M&=\sqrt{\omega L_1\cdot \omega L_2}\\ +\Rightarrow\omega L_2&=\frac{(\omega M)^2}{\omega L_1}=\frac{(10\,\ohm)^2}{22{,}45\,\ohm}=\uuline{4{,}45\,\ohm} +\end{align*} +\end{enumerate} +\clearpage +}{}% diff --git a/ET2_L_B18_A2.tex b/ET2_L_B18_A2.tex new file mode 100644 index 0000000..1f61f31 --- /dev/null +++ b/ET2_L_B18_A2.tex @@ -0,0 +1,204 @@ +\section {Übertrager mit kapazitiver Last} +Von nebenstehender Schaltung ist gegeben:\\ +$\uline{U}_1=1\,\volt\cdot e^{j0\,\degree}$, $R_1=10\,\ohm$, $X_1=100\,\ohm$, $R_2=40\,\ohm$, $X_2=400\,\ohm$, $X_C=-200\,\ohm$, $X_M=40\,\ohm$\\ + +Gesucht\\ +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Spannung $\uline{U}_2$ nach Betrag und Phase bei offenem Schalter S. +\item Spannung $\uline{U}_2$ nach Betrag und Phase bei geschlossenem Schalter S. +\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung für den Fall b). +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.175,-.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator| + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(1,0)--(1,.2) (3,.2)--(3,0)--(1.5,0)--(1.5,.2) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9)(3,.8)--(3,1)--(2.8,1); + \fill (2.5,0)circle(0.025cm)(2.5,1)circle(0.025cm)(2.8,1)circle(0.025cm); + \draw [->,blue] (2.5,.8)--(2.5,.2) node at (2.5,.5)[left]{$\uline{U}_2$}; + \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$X_M$}; + \draw[ultra thick](2.8,1)--+(150:.3cm)node at(2.7,1.2){$S$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{enumerate} +\item Offener Schalter\\ +\begin{align*} +\uline{I}_2&=0\quad\\ +&\text{Wegen Wicklungssinn $\uline{U}_2$ negativ (bzw. $X_M$ negativ)}\\ +\uline{U}_2&=jX_M\cdot \uline{I}_1=jX_M\cdot \frac{\uline{U}_1}{R+jX_1}=\frac{(-j40\cancel{\,\ohm})\cdot 1\,\volt}{(10+j100)\cancel{\,\ohm}}=(0{,}396-j0{,}04)\,\volt\\ +&=\uuline{0{,}398\,\volt\cdot e^{-j174{,}3\,\degree}}\\ +\end{align*} +\clearpage +\item Ersatzschaltbild: Schalter geschlossen\\ + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw node at(.5,-.2){\footnotesize$10$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j140$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{U}_M$}; + \draw node at(.25,.1)[left]{\footnotesize$-j40$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j440$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw node at(.5,-.2){\footnotesize$40$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$}; + \draw node at(.25,-.05)[right]{\footnotesize$-j200$}; + \draw[->,red](.9,-.1)--(.6,-.1)node at(.75,-.1)[right]{$I_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0)--(4,.2) (3.8,1)--(4,1)--(4,.8); + \fill (0,0)circle(0.025cm)(0,1)circle(0.025cm); + \draw [->,blue] (3.8,.7)--(3.8,.3) node at (3.75,.5)[left]{$\uline{U}_2$}; + \draw [->,blue!50!red] (3.5,.7)--(2.5,.7) node at (3,.7)[above]{$\uline{U}'_2$}; + \draw [->,blue!50!red,thick] (3,.075)arc(270:0:.25cm) node at (3,.325){\scriptsize{Masche}}; + \draw [->,blue!50!red] node at (5,.7)[right]{$\uline{U}'_2$ \footnotesize{Kontrollrechnung}}; + \draw node at (0,.5)[left]{$\uline{Z}_{ges}\Rightarrow$}; + \draw node at (1,1.5)[above]{$\uline{Z}_1$}; + \draw node at (3,1.5)[above]{$\uline{Z}_2$}; + \draw node at (1,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.5)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.25)[below]{$\uline{Z}'=(40+j240)\,\ohm=243{,}3\,\ohm\cdot e^{j80{,}5\,\degree}$}; + \draw node at (3,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.75)[below]{$\uline{Z}_{||}=\frac{(-j40)\cdot (40+j240)}{40+j200}\,\ohm=(1{,}54-j47{,}7)\,\ohm=47{,}72\,\ohm\cdot e^{-j88{,}2\,\degree}$}; + \end{scope} +% \draw[dashed,blue!50!red](3,.9)--(3,.1)node at (3,.5)[right]{$P_2$}; + \end{tikzpicture} +\begin{align*} +\uline{Z}_{ges}&=\uline{Z}_1+\uline{Z}_{||}=(11{,}54+j92{,}3)\,\ohm=93{,}03\,\ohm\cdot e^{j82{,}87\,\degree} +\intertext{Spannungsteiler} +\uline{U}_M&=\uline{U}_1\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{ges}}=1\,\volt\cdot e^{j0\,\degree}\cdot \frac{47{,}72\,\ohm\cdot e^{-j88{,}2\,\degree}}{93{,}03\,\ohm\cdot e^{j82{,}87\,\degree}}=(-0{,}51-j0{,}08)\,\volt=0{,}51\,\volt\cdot e^{-j171{,}1\,\degree}\\ +\uline{U}_2&=\uline{U}_M\cdot \frac{jX_C}{\uline{Z}'}=0{,}51\,\volt\cdot e^{-j171{,}1\,\degree}\cdot \frac{200\,\ohm\cdot e^{-j90\,\degree}}{243{,}3\,\ohm\cdot e^{j80{,}5\,\degree}}\\ +&=(0{,}4+j0{,}13)\,\volt=\uuline{0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}} +\end{align*} +\item Wirkleistung +\begin{align*} +P_2&=I^2_2\cdot R_2\\ +\uline{I}_2&=\frac{\uline{U}_2}{jX_C}=\frac{0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}}{-j200\,\ohm}=2{,}108\,\milli\ampere\cdot e^{+j108{,}4\,\degree}=(-0{,}666+j2)\,\milli\ampere\\ +P_2&=(2{,}108\,\milli\ampere)^2\cdot 40\,\ohm=1{,}78\cdot \power{10}{-4}\,\watt=\uuline{178\,\micro\watt}\\ +&\text{Von Primär- nach Sekundärseite}\\ +\intertext{Alternativ, in 2 Schritten} +U_{R_2}&=I_2\cdot R_2=2{,}108\,\milli\ampere\cdot 40\,\ohm=84{,}32\,\milli\volt\\ +P_2&=U_{R_2}\cdot I_2=84{,}32\,\milli\volt\cdot 2{,}108\,\milli\ampere=178\micro\watt\\ +\end{align*} +\end{enumerate} +Kontrolle: (über das Magnetfeld übertragenen Wirkleistung) +\begin{align*} +\uline{U}'_2&=\uline{U}_2 - \uline{U}_M=(0{,}4+j0{,}13)-(-0{,}51-j0{,}08))\,\volt=(0{,}91+j0{,}21)\,\volt=0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\\ +&\text{oder auch aus den Impedanzen berechnet}\\ +\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)]=-(-0{,}666+j2)\,\milli\ampere\cdot (40+j440)\,\ohm\\ +&=(0{,}907+j0{,}213)\,\volt=0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\\[\baselineskip] +&\text{mit }S_2=\uline{U}'_2\cdot \uline{I}^*_2\\[\baselineskip] +P_2&=\Re\{\uline{U}'_2\cdot \uline{I}^*_2\}\\ +&=\Re\{(0{,}91+j0{,}21)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\ +&=\Re\{0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\ +&=\Re\{1{,}965\cdot e^{-j95{,}18\,\degree}\}\,\micro\volt\ampere +=\Re\{\underbrace{-177{,}4}_{\text{EPS$\rightarrow$Verbraucher}}-j1{,}957\}\,\micro\volt\ampere\\ +&\text{\footnotesize{Erzeuger-Pfeil-System (EPS)}} +\end{align*} + +\begin{align*} + \begin{tikzpicture}[very thick,scale=10] + \draw[thin,black!50!,step=.1cm](-.1,-.3001)grid(1,.6); + \draw[black!25!]+(18.42:.934cm)arc(18.42:198.42:.467cm); % Thales +% \draw[black!25!](18.42:.467cm)circle(.467cm); % Thales + \draw[<-,blue](0,0)++(18.4:.42cm)--+(8.9:.51cm)node at(.42,.175)[right]{$\uline{U}_M$}; + \draw[->,magenta](0,0)--(12.99:.934cm)node at(.7,.15)[below left]{$\uline{U}'_2$}; + \draw[->,red](0,0)--(108.46:.21cm)node [above]{$\uline{I}_2$}; + \draw[->,red](0,0)--(-108.4:.21cm)node [below]{$\uline{I}^*_2$}; + \draw[->,black!50!green](0,0)++(12.99:.934cm)--+(108.46:.084cm)node [below right]{$\uline{U}_{R2}=I_2\cdot R_2$}; + \draw[->,black!50!green](0,0)--(18.4:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$}; + \draw[->,blue](0,0)--++(18.4:.42cm)node [above left]{$\uline{U}_2$}; + \draw node at(1.05,.55)[right]{$\uline{U}'_2=\uline{U}_2 - \uline{U}_M=0{,}932\cdot \,\volt\cdot e^{j13{,}22\,\degree}$}; + \draw node at(1.05,.45)[right]{$\uline{U}_{R_2}\quad||\quad\uline{I}_2$}; + \draw node at(1.05,.35)[right]{$\uline{U}_{(X_2-X_M)}\quad||\quad\uline{U}_2 \quad\bot\quad\uline{I}_2$}; + \draw node at(1.05,-.05)[right]{$S=\uline{U}'_2\cdot \uline{I}^*_2=1965\,\micro\volt\ampere$}; + \draw node at(1.05,-.15)[right]{$P=S\cdot \cos(-95{,}18\,\degree)=-177\micro\watt$}; + \draw node at(1.05,-.25)[right]{$Q=S\cdot \sin(-95{,}18\,\degree)=-1957\micro\volt\ampere r$}; + \draw[black!75!](0,0)++(108.4:.1cm)arc(108.4:12.99:.1cm)node at(90:.1cm) [above right]{$\varphi=-95{,}18\,\degree$}; + \end{tikzpicture} +\end{align*} +\clearpage +}{}% +%%%%delete +%\uline{U}'_2&=-\uline{I}_2\cdot(\uline{X}_2-\uline{X}_M)=-(-0{,}666+j2)\,\milli\ampere\cdot (j440)\,\ohm=(0{,}88+j0{,}29)\,\volt\\ +%\uline{U}'_2+\uline{U}_M&=(0{,}88+j0{,}29)\,\volt+(-0{,}51-j0{,}08)\,\volt=(0{,}37+j0{,}21)\,\volt=0{,}425\,\volt\cdot e^{j29{,}58\,\degree}\\ +%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\ +%&=\Re\{(0{,}37+j0{,}21)\,\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\}=\Re\{0{,}425\,\volt\cdot e^{j29{,}58\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\\ +%&=\Re\{0{,}896\cdot e^{-j78{,}82\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{+174}_{\text{EPS$\rightarrow$Quelle}}-j879\}\,\micro\volt\ampere\\[\baselineskip] +%% +%&\text{Langer Pfeil auch über XC}\\ +%\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)+\uline{X}_C]=-(-0{,}666+j2)\,\milli\ampere\cdot (40+j240)\,\ohm\\ +%&=(0{,}507+j0{,}08)\,\volt=0{,}513\,\volt\cdot e^{j8{,}96\,\degree}\\ +%\uline{U}'_2+\uline{U}_M&=(0{,}507+j0{,}08)\,\volt+(-0{,}51-j0{,}08)\,\volt=0\\[\baselineskip] +%&\text{Kurzer Pfeil + C2 ohne R2}\\ +%\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)+\uline{X}_C]=-(-0{,}666+j2)\,\milli\ampere\cdot (+j240)\,\ohm\\ +%&=(0{,}480+j0{,}160)\,\volt=0{,}505\,\volt\cdot e^{j18{,}42\,\degree}\\ +%\uline{U}'_2+\uline{U}_M&=(0{,}507+j0{,}08)\,\volt+(-0{,}51-j0{,}08)\,\volt=(0{,}480+j0{,}160)\,\volt=0{,}505\,\volt\cdot e^{j18{,}44\,\degree}\\[\baselineskip] +%&\text{Kurzer Pfeil}\\ +%\uline{U}'_2&=-\uline{I}_2\cdot (\uline{X}_2-\uline{X}_M)=-(-0{,}666+j2)\,\milli\ampere\cdot (+j440)\,\ohm\\ +%&=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{-j18{,}42\,\degree}\\ +%\uline{U}'_2+\uline{U}_M&=(0{,}880+j0{,}293)\,\volt+(-0{,}51-j0{,}08)\,\volt +%=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{j18{,}42\,\degree}\\[\baselineskip] +%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\ +%&=\Re\{(0{,}880+j0{,}293)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\ +%&=\Re\{0{,}927\,\milli\volt\cdot e^{j18{,}42\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\ +%&=\Re\{1{,}954\cdot e^{-j89{,}98\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{785}_{\text{EPS$\rightarrow$Quelle}}+j1954\}\,\micro\volt\ampere\\[\baselineskip] +%&\text{seine alte Lösung }(0{,}4+j0{,}13)=0{,}4206\,\volt\cdot e^{-j18{,}0\,\degree}\\ +%\uline{U}''_x&=\uline{U}_2-\uline{U}_M=(0{,}4+j0{,}13)\,\volt-(-0{,}51-j0{,}08)\,\volt=(0{,}91+j0{,}21)\,\volt\\ +%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}=(-0{,}91-j0{,}21)\,\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\\ +%&=\Re\{\underbrace{+178}_{\text{EPS$\rightarrow$Quelle}}+j1960\}\,\micro\volt\ampere\\[\baselineskip] +% \draw[->,black!50!green](0,0)++(18.4:.42cm)--+(-171.1:.51cm)node[left]{$\uline{U}_M$}; +% \draw[->,black!75!green](0,0)++(18.4:.42cm)++(-171.1:.51cm)--(0:0cm)node[left]{$\uline{U}_2 - \uline{U}_M$}; +% \draw[->,black!85!green](0,0)--(8.4:.89cm)node[right]{$\uline{U}''_x$}; +% \draw[->](0,0)--(8.9:.51cm);%{$\uline{U}_M$}; +% \draw[->,black!50!green](0,0)--(-171.1:.51cm)node[left]{$\uline{U}_M$}; +% \draw[->,magenta](0,0)--(.91,.21)node at(.7,.15)[below left]{$\uline{U}'_2$}; +% \draw[->,black](0,0)++(12.99:.934cm)--+(108.46:.15cm)node [below right]{$\uline{U}_{R2}$}; +% \draw[->,black](0,0)--(22:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$}; \ No newline at end of file diff --git a/ET2_L_B18_A3.tex b/ET2_L_B18_A3.tex new file mode 100644 index 0000000..85109c8 --- /dev/null +++ b/ET2_L_B18_A3.tex @@ -0,0 +1,186 @@ +\section {Übertrager mit Verbindung zum Eingang} +\enlargethispage{3cm} +Gegeben sind die Daten des abgebildeten Kreises:\\ +$R_1=28\,\ohm$, $X_1=96\,\ohm$, $R_2=14\,\ohm$, $X_M=60\,\ohm$, $X_2=400\,\ohm$, $\uline{I}_1=2\,\ampere\cdot e^{j0\,\degree}$\\ +Gesucht +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei offenem Schalter S. +\item Spannung $\uline{U}_S$ nach Betrag und Phase bei offenem Schalter S. +\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei geschlossenem Schalter S. +\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung bei geschlossenem Schalter S. +\end{enumerate} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle +% \draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$}; +% \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,0)--(.7,0) node at (.5,0)[left]{$\uline{U}_{ab}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (0.15,.1)--(.3,.1)node at(.225,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule| + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule| + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.175,-.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,red] (.85,.1)--(.7,.1)node at(.775,.1)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(1,0)--(1,.2) (1.5,0)--(2.5,0)--(2.5,-.25)--(.1,-.25)--(.1,0) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9) +(2.4,1)--(2.5,1)--(2.5,1.5)--(1.4,1.5)(1.1,1.5)--(.1,1.5)--(.1,1); + \filldraw (0,1)circle(0.025cm)node[left]{a}; + \filldraw (0,0)circle(0.025cm)node[left]{b}; + \fill(.1,0)circle(0.025cm)(.1,1)circle(0.025cm) (1.1,1.5)circle(0.025cm)(1.4,1.5)circle(0.025cm); + \draw [->,blue] (1,1.3)--(1.5,1.3) node at (1.25,1.3)[above]{$\uline{U}_S$}; + \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.125)[below]{$X_M$}; + \draw[ultra thick](1.4,1.5)--+(150:.3cm)node at(1.4,1.7){$S$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +\vspace{-.5cm} +Berechnung: Ersatzschaltbild Schalter offen +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw node at(.5,-.2){\footnotesize$28$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j156$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule| + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$U_M$}; + \draw node at(.25,.25){\footnotesize$-j60$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j460$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw node at(.5,-.2){\footnotesize$14$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0) (3.8,1)--(4,1)--(4,1.5)--(2.1,1.5)(1.9,1.5)--(.1,1.5)--(.1,1); + \fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm) + (2.1,1.5)circle(.025cm)(1.9,1.5)circle(.025cm) + (0,1)circle(0.025cm); + \filldraw (0,1)circle(0.025cm)node[left]{a}; + \filldraw (0,0)circle(0.025cm)node[left]{b}; + \draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$}; + \draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$}; + \draw [->,blue] (1.5,1.85)--(2.5,1.85) node at (2,1.85)[above]{$\uline{U}_S$}; + \draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$0\,\volt$}; + \draw[ultra thick](2.1,1.5)--+(150:.2cm)node at(2,1.7){$S$}; +\draw [->,red] (0.5,1.6)--(.75,1.6)node at(.625,1.6)[above]{\footnotesize$\uline{I}_S=0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{enumerate} +\item Spannung $\uline{U}_{ab}$ Schalter offen\\ +\vspace{-.5cm} +\begin{align*} +\uline{U}_{ab}&=[R_1+j(X_1-\cancel{X_M}+\cancel{X_M})]\cdot \uline{I}_1=(28+j96)\,\ohm\cdot 2\,\ampere=(56+j192)\,\volt\\ +&=\uuline{200\,\volt\cdot e^{j73{,}74\,\degree}} +\end{align*} +\item Spannung $\uline{U}_S$ Schalter offen\\ +\vspace{-.5cm} +\begin{align*} +&\text{Wegen gegensinnigem Wicklungssinn $X_M=-60\,\ohm$ }\\ +\uline{U}_S&=[R_1+j(X_1-X_M)]\cdot \uline{I}_1=(28+j156)\,\ohm\cdot 2\,\ampere=(56+j312)\,\volt\\ +&=\uuline{317\,\volt\cdot e^{j79{,}8\,\degree}} +\end{align*} +\clearpage +\enlargethispage{3cm} +\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$: Schalter geschlossen: +\begin {align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw node at(.5,-.2){\footnotesize$28$}; + \draw [->,red] (0.15,.1)--(.3,.1)node at(.225,.1)[above]{\footnotesize$\uline{I}_1$}; + \draw [->,red] (0.15,.6)--(.3,.6)node at(.225,.6)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j156$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$U_M$}; + \draw [->,red] (.25,-.1)--(.05,-.1)node at(.15,-.1)[right]{\footnotesize$\uline{I}=\uline{I}_1+\uline{I}_2$}; + \draw node at(.25,.25){\footnotesize$-j60$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j460$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw node at(.5,-.2){\footnotesize$14$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0) (3.8,1)--(4,1)--(4,1.5)--(.1,1.5)--(.1,1); + \fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm) + (0,1)circle(0.025cm); + \filldraw (0,1)circle(0.025cm)node[left]{a}; + \filldraw (0,0)circle(0.025cm)node[left]{b}; + \draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$}; + \draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$}; + \draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$\uline{U}_S$}; + \draw [->,blue] (1,.4)--(1,.1) node at (1,.25)[right]{$\uline{U}_1$}; + \draw node at (1,2)[above]{$\uline{Z}_1$}; + \draw node at (3,2)[above]{$\uline{Z}_2$}; + \draw node at (1,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$}; + \draw node at (3,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$}; + \end{scope} + \end{tikzpicture} +\end {align*} +\vspace{-.5cm} +Entweder Stromteiler oder, weil Strom $\uline{I}_1$ wie in b)\\ +\begin{align*} +\uline{U}_S&=\uline{Z}_1\cdot \uline{I}_1=\uline{Z}_2\cdot \uline{I}_2\\ +\uline{I}_2&=\uline{I}_1\cdot \frac{\uline{Z}_1}{\uline{Z}_2}=2\,\ampere\cdot e^{j0\,\degree}\cdot \frac{(28+j156)\,\ohm}{(14+j460)\,\ohm}\\ +&=(0{,}681-j0{,}101)\,\ampere=\uuline{0{,}689\,\ampere\cdot e^{-j8{,}43\,\degree}}\\ +\uline{U}_{ab}&=\uline{U}_S+\uline{U}_M\\ +&=\uline{Z}_1\cdot \uline{I}_1+jX_M\cdot (\uline{I}_1+\uline{I}_2)\\ +&=(56+j312)\,\volt+\underbrace{\overbrace{(2+0{,}681-j0{,}101)\,\ampere}^{\uline{I}}\cdot (-j60)\,\ohm}_{\uline{U}_M=(-6{,}06-j160{,}81)\,\volt}\\ +&=(50+j151)\,\volt=\uuline{159\,\volt\cdot e^{+j71{,}7\,\degree}}\\ +\end{align*} +\vspace{-1.5cm} +\item Wirkleistung (über das Magnetfeld übertragen) +\begin{align*} +\uline{S}_1&=\uline{U}_1\cdot \uline{I}^*_1=[\uline{I}_1\cdot j(X_1-X_M)+\underbrace{\uline{I}\cdot jX_M}_{\uline{U}_M}]\cdot \uline{I}^*_1\\ +&=\underbrace{[\underbrace{2\,\ampere\cdot j156\,\ohm}_{j312\,\volt}+(-6{,}1-j161)\,\volt]}_{(-6{,}1+j151)\,\volt}\cdot 2\,\ampere=\uuline{(\overbrace{-12{,}2}^{\text{Quelle}}+j302)\,\volt\ampere}\\ +P&=\uuline{12{,}2\,\watt}\text{ Von Sekundär- nach Primärseite} +\end{align*} +\end{enumerate} +Kontrolle: +\begin{align*} +\uline{S}_2&=(\uline{U}_M+\uline{I}_2\cdot j460\,\ohm)\cdot \uline{I}^*_2=(12{,}1+j108)\,\volt\ampere\\ +(P_2&=12{,}1\,\watt\text{ Verbraucher}) +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B18_A4.tex b/ET2_L_B18_A4.tex new file mode 100644 index 0000000..460a45d --- /dev/null +++ b/ET2_L_B18_A4.tex @@ -0,0 +1,150 @@ +\section {Impedanzmatrix} +%\enlargethispage{3cm} +Geben Sie die Impedanzmatrix $(Z)$ des Vierpols an.\\ +$R_1=10\,\ohm$, $X_1=100\,\ohm$, $R_2=20\,\ohm$, $X_2=200\,\ohm$, $X_C=-200\,\ohm$, $X_M=120\,\ohm$\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \fill(.825,-.075)circle(.033); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,red] (1.5,.1)--(1.25,.1)node at(1.375,.1)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [left] {$X_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(1,0)--(1,.2) (3,0)--(1.5,0)--(1.5,.2) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9)(3,1)--(2.5,1); + \fill (2.5,0)circle(0.025cm)(2.5,1)circle(0.025cm)(3,1)circle(0.025cm)(3,0)circle(0.025cm); + \draw [->,blue] (3,.8)--(3,.2) node at (3,.5)[right]{$\uline{U}_2$}; + \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$X_M$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +Umwandlung in T-Ersatzschaltbild\\ +%\begin{enumerate} +%\item Offener Schalter\\ +%\begin{align*} +%\uline{I}_2&=0\quad\text{Wegen Wicklungssinn $\uline{U}_2$ negativ (bzw. $X_M$ negativ)}\\ +%\uline{U}_2&=jX_M\cdot \uline{I}_1=jX_M\cdot \frac{\uline{U}_1}{(R+jX_1)}=\frac{(-j40)\,\ohm\cdot 1\,\volt}{(10+j100)\,\ohm}=(0{,}396-j0{,}04)\,\volt\\ +%&=\uuline{0{,}398\,\volt\cdot e^{-j174{,}3\,\degree}}\\ +%\end{align*} +%\clearpage +%\item Ersatzschaltbild\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw node at(.5,-.2){\footnotesize$10$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$-j20$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$U_M$}; + \draw node at(.25,.25){\footnotesize$j120$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j80$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw node at(.5,-.2){\footnotesize$20$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$}; + \draw node at(.25,-.2){\footnotesize$-j200$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0)--(4,.2) (3.8,1)--(4,1)--(4,.8)(4,0)--(4.5,0)(4,1)--(4.5,1); + \fill (0,0)circle(0.025cm)(0,1)circle(0.025cm)(4.5,0)circle(0.025cm)(4.5,1)circle(0.025cm); + \draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_1$}; + \draw [->,blue] (4.5,.7)--(4.5,.3) node at (4.5,.5)[right]{$\uline{U}_2$}; +% \draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$\uline{U}'_2$}; +% \draw node at (0,.5)[left]{$\uline{Z}_{ges}\Rightarrow$}; + \draw node at (1,1.5)[above]{$\uline{Z}_1$}; + \draw node at (3,1.5)[above]{$\uline{Z}_2$}; + \draw node at (1,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.5)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.25)[below]{$\uline{Z}'=(20-j120)\,\ohm$}; + \draw node at (3,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxxxxxx}}_{}$}; + \draw node at (3.25,-.75)[below]{$\uline{Z}_{||}=X_M||\uline{Z}'=(720+j120)\,\ohm$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{Z}_{1}&=(10-j20)\,\ohm &\uline{Z}_{2}&=(20+j80)\,\ohm \\ +\uline{Z}'&=(20-j120)\,\ohm &\uline{Z}_{||}&=X_M||\uline{Z}'=\frac{j120\cdot (20-j120)}{\cancel{j120}+20-\cancel{j120}}\,\ohm=(720+j120)\,\ohm \\ +\end{align*} +\clearpage +Vierpol: +\begin{align*} +\begin{bmatrix} +\uline{U}_{1} \\ +\uline{U}_{2} +\end{bmatrix}= +\begin{bmatrix} +\uline{Z}_{11} & \uline{Z}_{12} \\ +\uline{Z}_{21} & \uline{Z}_{22} +\end{bmatrix}\cdot +\begin{bmatrix} +\uline{I}_{1} \\ +\uline{I}_{2} +\end{bmatrix} +\end{align*} + +\begin{align*} +\uline{U}_1&=\uline{Z}_{11}\cdot \uline{I}_1+\uline{Z}_{12}\cdot \uline{I}_2\\ +\uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_1+\uline{Z}_{22}\cdot \uline{I}_2\\ +\uline{Z}_{11}&=\frac{\uline{U}_1}{\uline{I}_1}\Big|_{\uline{I}_2=0}\\ +&=\uline{Z}_1+\uline{Z}_{||}=\uuline{(730+j100)\,\ohm}\\ +\uline{Z}_{12}&=\frac{\uline{U}_1}{\uline{I}_2}\Big|_{\uline{I}_1=0}\\[\baselineskip] +\uline{I}_1&=0 \Rightarrow \text{ Leerlauf an Primärseite; Spannung an }\uline{U}_M=\uline{U}_1\\[\baselineskip] +&\underline{I}_{X_M}=\uline{I}_2\cdot \frac{jX_C}{R_2+j(X_2-\cancel{X_M})+\cancel{jX_M}+jX_C}\quad\text{( Stromteiler)}\\ +\uline{U}_1&=jX_M\cdot \uline{I}_{X_M}=\uline{I}_2\cdot \underbrace{jX_M\cdot \frac{jX_C}{R_2+j(X_2+X_C)}}_{\uline{Z}_{12}}\\ +\uline{Z}_{12}&=\frac{-X_M\cdot X_C}{R_2+j(X_2+X_C)}=\frac{-120\cdot (-200)}{20-j0}\,\ohm=\uuline{+1200\,\ohm}\\ +\uline{Z}_{21}&=\uuline{\uline{Z}_{12}}\\ +\uline{Z}_{22}&=\frac{\uline{U}_2}{\uline{I}_2}\Big|_{\uline{I}_1=0}\\ +&=jX_C||(\uline{Z}_2)+jX_M)=\frac{-j200\cdot (20+j(80+120))}{20+\underbrace{j(80+120-200)}_{=0}}\,\ohm=\uuline{(2000-j200)\,\ohm} +\end{align*} +Impedanzmatrix: +\begin{align*} +\uuline{Z}&= +\left[ + \begin{array}{cc} + 730+j100 & 1200 \\ + 1200 & 2000-j200 \\ + \end{array} +\right]\,\ohm +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B18_A5.tex b/ET2_L_B18_A5.tex new file mode 100644 index 0000000..9ff62c2 --- /dev/null +++ b/ET2_L_B18_A5.tex @@ -0,0 +1,152 @@ +\section {Netztransformator} +Von einem Netztransformator sind folgende Daten gegeben:\\[.5\baselineskip] +Primärspannung $U_1=230\,\volt$; Frequenz $f=50\,\hertz$;\\ +Primärwindungszahl $N_1=784$ Windungen.\\[.5\baselineskip] +Induktivität der Primärwicklung $L_1=5{,}66\,\henry$; \\ +Induktivität der Sekundärwicklung $L_2=1{,}42\,\henry$; \\[.5\baselineskip] +Widerstand der Primärwicklung $R_1=800\,\ohm$;\\ +Widerstand der Sekundärwicklung $R_2=150\,\ohm$.\\[.5\baselineskip] +Eisenquerschnitt $A_{Fe}=11\,\centi\square\metre$. Das Feld ist über dem Querschnitt $A_{Fe}$ homogen, die Streuung ist Null! +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Welche Spannung $U_2$ tritt an der Sekundärwicklung auf, wenn sie unbelastet ist, d.h. $I_2=0$ ist? +\item Welchen Strom nimmt der Transformator bei sekundärseitigem Leerlauf auf? +\item Welche Flussdichte $\widehat{B}_{Fe}$ tritt bei sekundärseitigem Leerlauf im Eisen auf? +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0);% node at (.5,-.133) [right] {$U_1$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$j\omega M\cdot \uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0);% node at (.5,-.133) [right] {$U_1$}; + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$j\omega M\cdot \uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_2$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3.5cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw [->,red] (1,.1)--(.75,.1)node at(.875,.1)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(2,0)--(2,.2) (4.5,0)--(2.5,0)--(2.5,.2) (2,.8)--(2,1)--(1.8,1)(2.5,.8)--(2.5,1)--(2.6,1); + \fill (0,0)circle(0.025cm)(0,1)circle(0.025cm)(4.5,0)circle(0.025cm)(4.5,1)circle(0.025cm); + \draw [->,blue] (0,.8)--(0,.2) node at (0,.5)[left]{$\uline{U}_1$}; + \draw [->,blue] (4.5,.8)--(4.5,.2) node at (4.5,.5)[right]{$\uline{U}_2$}; + \end{scope} + \end{tikzpicture} +\end{align*} +Ersatzschaltbild +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; + \draw node at(.5,-.2){\footnotesize$800$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_{L_1}-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$+j887$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{U}_M$}; + \draw node at(.25,.25){\footnotesize$j891$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_{L_2}-X_M$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw node at(.5,-.2){\footnotesize$-j445$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \draw node at(.5,-.2){\footnotesize$150$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0); + \fill (0,0)circle(0.025cm)(0,1)circle(0.025cm)(4,0)circle(0.025cm)(4,1)circle(0.025cm); + \draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_1$}; + \draw [->,blue] (4,.7)--(4,.3) node at (4,.5)[right]{$\uline{U}_2$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{enumerate} +\item Spannung $U_2$ +\begin{align*} +\uline{U}_1&=\uline{I}_1\cdot (R_1+j\omega\cdot L_1) +j\omega\cdot M\cdot \uline{I}_2\\ +\uline{U}_2&=\uline{I}_2\cdot (R_2+j\omega\cdot L_2)+j\omega\cdot M\cdot \uline{I}_1 +\end{align*} +\footnotesize{Hinweis: Es kann auch $X_{L_1}=\omega\cdot L_1$ bzw. $X_M=\omega\cdot M$ verwendet werden.}\\ +\normalsize +\clearpage +\enlargethispage{2cm} +\begin{align*} +\text{mit }\uline{I}_2&=0\\ +\uline{U}_1&=\uline{I}_1\cdot (R_1+j\omega\cdot L_1)\\ +\uline{U}_2&=j\omega\cdot M\cdot \uline{I}_1 \\ +\Rightarrow\frac{\uline{U}_2}{\uline{U}_1}&=\frac{j\omega\cdot M}{R_1+j\omega\cdot L_1}\\ +U_2&=U_1\cdot \frac{\sqrt{(\omega\cdot M)^2}}{\sqrt{R^2_1+(\omega\cdot L_1)^2}}\qquad\text{(Betrag: }U_2=|\uline{U}_2|)\qquad \text{(1)}\\[.5\baselineskip] +&\text{Streuung ist Null } \Rightarrow \text{Kopplungsfaktor }K=1=\frac{|M|}{\sqrt{L_1\cdot L_2}}\\ +\Rightarrow M&=\sqrt{L_1\cdot L_2}=\sqrt{5{,}66\cdot 1{,}42}\,\henry=2{,}835\,\henry\\ +X_{L_1}&=\omega\cdot L_1=2\cdot \pi\cdot 50\,\cancel{\power{\second}{-1}}\cdot 5{,}66\,\volt\cancel{\second}\per\ampere=1{,}778\,\kilo\ohm\quad\text{\footnotesize{(Zur Vollständigkeit $X_{L_2}=446\,\ohm$)}}\\ +X_{M}&=\omega\cdot M=2\cdot \pi\cdot 50\,\cancel{\power{\second}{-1}}\cdot 2{,}835\,\volt\cancel{\second}\per\ampere=891\,\ohm\\ +&\text{aus (1):} \Rightarrow U_2=\uline{U}_1\cdot \frac{X_M}{\sqrt{R^2_{1}+X^2_{L_1}}}=230\,\volt\cdot \frac{891}{\sqrt{800^2+1778^2}}=\uuline{105{,}1\,\volt} +\end{align*} +\item Stromaufnahme bei Leerlauf +\vspace{-.25cm} +\begin{align*} +\uline{I}_1&=\frac{\uline{U}_1}{R_1+j\omega L_1}\\ +I_1&=\frac{U_1}{\sqrt{R^2_{1}+X^2_{L_1}}}=\frac{230\,\volt}{\sqrt{800^2+1778^2}}\,\frac{1}{\ohm}=\uuline{118\,\milli\ampere}\quad\text{(Betrag)}\\ +\end{align*} +\begin{minipage}[c]{.62\textwidth} +\item Flussdichte, Sekundärspule spielt keine Rolle bei $I_2=0$ +\vspace{-.25cm} +\begin{align*} +\psi&=L_1\cdot i=N_1\cdot \phi=N_1\cdot B\cdot A\\ +B&=\frac{L_1\cdot i}{N_1\cdot A}\\ +\widehat{B}&=\frac{L_1\cdot \widehat{I}}{N_1\cdot A_{Fe}}=\frac{5{,}66\,\volt\second\per\cancel{\ampere}\cdot \sqrt{2}\cdot 118\cdot \power{10}{-3}\,\cancel{\ampere}}{784\cdot 11\cdot \underbrace{\power{10}{-4}}_{(\power{10}{-2})^2}\,\metre^2}=\uuline{1{,}095\,\tesla} +\end{align*} +\end{minipage}% +\begin{minipage}[c]{.33\textwidth} +\begin{align*} + \begin{tikzpicture}[scale=1.0] + \draw(0,0)rectangle(1.5,1.5); + \draw(.25,.25)rectangle(1.25,1.25); + \draw[->,red](.25,1.375)--(.5,1.375)node at(.375,1.5)[above]{$\phi$}; + \draw[black!75!](1,1.25)--(1,1.5)node [above]{$A_{Fe}$}; + \draw[red!70!blue](0,.5)--(-.5,.5)(0,1)--(-.5,1)node [left]{$\uline{I}_1$}; + \fill[red!70!blue](0,.5)rectangle(.25,1)node at(.25,.75)[right]{$N_1$}; + \draw[red!70!blue](1.5,.5)--(2,.5)(1.5,1)--(2,1)node [right]{$\uline{I}_2=0$}; + \fill[red!70!blue](1.25,.5)rectangle(1.5,1)node at(1.25,.75)[left]{$N_2$}; + \end{tikzpicture} +\end{align*} +\end{minipage} +Bemerkung: +\begin{align*} +U_{L_1}&=\frac{2\pi}{\sqrt{2}}\cdot f\cdot N_1\cdot \widehat{B}\cdot A_{Fe}=I_1\cdot \omega\cdot L_1=209{,}7\,\volt\\ +\widehat{B}&=\frac{2\pi}{\sqrt{2}}\cdot f\cdot \frac{U_1}{N_1\cdot A_{Fe}}=1{,}2\,\tesla \text{ ist falsch, gilt nur für idealen Transformator! } +\end{align*} +\footnotesize{Warum? Hier ist der Widerstand $R_1$ nicht berücksichtigt!\\ +Rechnung oben nur mit Beträgen, nicht im Komplexen.\\ $\uline{U}_1=\uline{U}_{R_1}+\uline{U}_{L_1}=(94{,}4+j209{,}7)\,\volt=230\,\volt\cdot e^{-j114{,}3}$} +\end{enumerate} +\clearpage +}{}% diff --git a/ET2_L_B18_A6.tex b/ET2_L_B18_A6.tex new file mode 100644 index 0000000..c10302f --- /dev/null +++ b/ET2_L_B18_A6.tex @@ -0,0 +1,75 @@ +\section {3-Phasen Spannungssystem} +Ein symmetrisches 3-Phasen Spannungssystem mit der Phasenlage 1-2-3 speist einen unsymmetrischen +Verbraucher mit den Impedanzen\\ +$\uline{Z}_1=R_1$; $\quad\uline{Z}_2=R_2+jX_{L2}$; $\quad\uline{Z}_3=jX_{L3}$;\\ +$U_{12}=400\,\volt\cdot e^{j30\degree}$; $\quad R_2=50\,\ohm$; $\quad X_{L2}=30\,\ohm$; $\quad X_{L3}=25\,\ohm$\\ +Der Leistungsmesser zeigt $1323\,\watt$ an. Berechnen Sie den Strom $\uline{I}_N$. +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,xshift=0cm,yshift=1.5cm]%Wattmeter + \draw (0,0)--(.367,0) (.633,0)--(1,0) node at (.5,0) {W}; + \draw (.5,0)circle(.133); + \draw(.25,0)--(.25,.25)--(.5,.25)--(.5,.133)(.5,-.133)--(.5,-1.5); + \end{scope} + \begin{scope}[>=latex,xshift=1cm,yshift=1.5cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_1$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,xshift=1cm,yshift=1.cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_2$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,xshift=1cm,yshift=.5cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_3$}; + \draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_3$}; + \end{scope} + \begin{scope}[>=latex,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(2,0)--(2,1.5)--(1.8,1.5);% (4.5,0)--(2.5,0)--(2.5,.2) (2,.8)--(2,1)--(1.8,1)(2.5,.8)--(2.5,1)--(2.6,1); + \filldraw(0,0)circle(0.025cm)node [left]{$N$}; + \filldraw(0,.5)circle(0.025cm)(1,.5)--(0,.5)node [left]{$L3$}; + \filldraw(0,1)circle(0.025cm)(1,1)--(0,1)node [left]{$L2$}; + \filldraw(0,1.5)circle(0.025cm)node at(0,1.5) [left]{$L1$}; + \fill(.5,0)circle(0.025cm)(2,.5)circle(0.025cm)(2,1)circle(0.025cm); + \draw [->,red] (1.25,.1)--(1,.1) node at (1.125,.1)[above]{$\uline{I}_N$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] + \draw[black!25!,very thin,step=.5cm](-1,-1)grid(1,1); + \draw[->]++(-180:1cm)--++(30:1.732cm)node [above right]{$2$}; + \draw[->]++(-180:1cm) ++(30:1.732cm)-- ++(270:1.732cm)node [below]{$3$}; + \draw[->]++(-180:1cm) ++(330:1.732cm)--(-180:1cm)node [left]{$1$}; + \draw[->](-180:1cm)--(0,0)node [right]{$N$}; + \draw[->](60:1cm)--(0,0); + \draw[->](300:1cm)--(0,0); + \draw[blue] node at(-.35,.6){$\uline{U}_{12}$}; + \draw[blue] node at(-.35,-.6){$\uline{U}_{31}$}; + \draw[blue] node at(.75,0){$\uline{U}_{23}$}; + \draw[blue] node at(-.35,.15){$\uline{U}_{1}$}; + \draw[blue] node at(.35,.25){$\uline{U}_{2}$}; + \draw[blue] node at(.35,-.25){$\uline{U}_{3}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +%\clearpage +\enlargethispage{2\baselineskip} +\begin{align*} +\uline{I}_N&=\uline{I}_1+\uline{I}_2+\uline{I}_3\qquad \text{(Um Ströme zu berechnen, Spannungen ermitteln)}\\ +\uline{U}_1&=\frac{\uline{U}_{12}}{\sqrt{3}}=\frac{400\,\volt}{\sqrt{3}}=230{,}9\,\volt\cdot e^{j0\,\degree}\\ +\uline{U}_2&=230{,}9\,\volt\cdot e^{-j120\,\degree}\\ +\uline{U}_3&=230{,}9\,\volt\cdot e^{+j120\,\degree}\\ +P_{Anzeige}&=I_1\cdot U_1\cdot \cos(\varphi_1)\qquad\text{ mit }\varphi_1=0\text{, da }\uline{Z}_1=R_1\\ +I_1&=\frac{P_{Anzeige}}{U_1}=\frac{1323\,\watt}{230{,}9\,\volt}=5{,}73\,\ampere \qquad\Rightarrow\uline{I}_1=(5{,}73+j0)\,\ampere\\[.5\baselineskip] +\uline{I}_2&=\frac{\uline{U}_2}{\uline{Z}_2}=\frac{230{,}9\,\volt\cdot e^{-j120\,\degree}}{(50+j30)\,\ohm}=\frac{230{,}9\,\volt\cdot e^{-j120\,\degree}}{58{,}31\,\ohm\cdot e^{j30{,}96\,\degree}}=3{,}96\,\ampere\cdot e^{-j150{,}96\,\degree}=(-3{,}463-j1{,}923)\,\ampere\\ +\uline{I}_3&=\frac{\uline{U}_3}{\uline{Z}_3}=\frac{230{,}9\,\volt\cdot e^{j120\,\degree}}{25\,\ohm\cdot e^{j90\,\degree}}=9{,}24\,\ampere\cdot e^{j30\,\degree}=(8+j4{,}62)\,\ampere\\[.5\baselineskip] +\uline{I}_N&=\uline{I}_1+\uline{I}_2+\uline{I}_3=(10{,}3+j2{,}7)\,\ampere=\uuline{10{,}6\,\ampere\cdot e^{j14{,}7\,\degree}} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B18_A7.tex b/ET2_L_B18_A7.tex new file mode 100644 index 0000000..7cc164c --- /dev/null +++ b/ET2_L_B18_A7.tex @@ -0,0 +1,116 @@ +\section {3-Phasen System mit unsymmetrischem Verbraucher} +Ein symmetrisches 3-Phasen System mit der Phasenlage 1-2-3 speist einen unsymmetrischen +Verbraucher. Gegeben sind:\\ +$\uline{U}_{31}=380\,\volt\cdot e^{j0\,\degree}$; $R_1=60\,\ohm$; $\uline{X}_{L_1}=20\,\ohm$; $R_2=100\,\ohm$; $\uline{X}_{C_2}=-80\,\ohm$; $X_{L_3}=50\,\ohm$;\\[\baselineskip] +Es ist die Anzeige des Leistungsmessinstrumentes zu berechnen! +\begin{align*} + \begin{tikzpicture}[very thick,scale=4] + \begin{scope}[>=latex,xshift=0cm,yshift=1.5cm]%Wattmeter + \draw (0,0)--(.367,0) (.633,0)--(1,0) node at (.5,0) {$W$}; + \draw (.5,0)circle(.133); + \draw(.25,0)--(.25,.25)--(.5,.25)--(.5,.133)(.5,-.133)--(.5,-1); + \end{scope} + \begin{scope}[xshift=1cm,yshift=.5cm,rotate=60,scale=.577] + \begin{scope}[>=latex,xshift=0cm,yshift=0cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.1330667) [left] {$R_1$}; + \draw [<-,red] (.8,.1)--(1.2,.1)node at(1,.14)[left]{$\uline{I}_{12}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.1330667) [ left] {$L_1$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.01cm,yshift=.75cm,rotate=120,scale=.577]%Spule + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,-.1330667) [ right] {$L_3$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (1.2,-.1)--(.8,-.1)node at(1,-.14)[right]{$\uline{I}_{31}$}; + \end{scope} + \begin{scope}[>=latex,xshift=1.289cm,yshift=.5cm,scale=.577]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1.289cm,yshift=.25cm,scale=.577]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.1133) [above] {$C_2$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$U_{C2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(2.154,0)--(2.154,.5)(.9,.5)--(1.289,.5) (1.289,.5)--(1.289,.25)(1.867,.5)--(1.867,.25); + \filldraw(0,0)circle(0.025cm)node [left]{$L3$}; + \filldraw(0,.5)circle(0.025cm)(1,.5)--(0,.5)node [left]{$L2$}; + \filldraw(0,1.5)circle(0.025cm)node at(0,1.5) [left]{$L1$}; + \fill(.5,.5)circle(0.025cm)(1,.5)circle(0.025cm)(1.289,.5)circle(0.025cm)(1.867,.5)circle(0.025cm)(2.154,.5)circle(0.025cm)(1.577,1.5)circle(0.025cm); + \draw [->,blue] (.8,1.4)--(.8,0.6) node at (.8,1)[right]{$\uline{U}_{12}$}; + \draw [->,red] (.9,1.6)--(1.1,1.6) node at (1,1.6)[above]{$\uline{I}_1$}; + \draw +(1,1.5)--(1.577,1.5)--(2.154,.5cm)--(1.75,.5); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +%\begin{align*} +% \begin{tikzpicture}[scale=2,rotate=-150] +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] +% \draw[black!25!,very thin,step=.5cm](-1,-1)grid(1,1); +% \draw[->]++(-180:1cm)--++(30:1.732cm)node [above right]{$2$}; +% \draw[->]++(-180:1cm) ++(30:1.732cm)-- ++(270:1.732cm)node [below]{$3$}; +% \draw[->]++(-180:1cm) ++(330:1.732cm)--(-180:1cm)node [left]{$1$}; +% \draw[->](-180:1cm)--(0,0)node [right]{$N$}; +% \draw[->](60:1cm)--(0,0); +% \draw[->](300:1cm)--(0,0); +% \draw[blue] node at(-.35,.6){$\uline{U}_{12}$}; +% \draw[blue] node at(-.35,-.6){$\uline{U}_{31}$}; +% \draw[blue] node at(.75,0){$\uline{U}_{23}$}; +% \draw[blue] node at(-.35,.15){$\uline{U}_{1}$}; +% \draw[blue] node at(.35,.25){$\uline{U}_{2}$}; +% \draw[blue] node at(.35,-.25){$\uline{U}_{3}$}; +% \end{scope} +% \end{tikzpicture} +%\end{align*} +%%\clearpage +\begin{minipage}[c]{0.25\textwidth} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1] +\draw[->,blue](0:0)--(0:1cm)node [right]{$\uline{U}_{31}$}; +\draw[->,blue](0:0)--(120:1cm)node [above]{$\uline{U}_{23}$}; +\draw[->,blue](0:0)--(-120:1cm)node [below]{$\uline{U}_{12}$}; +\draw[->,red](0:0)--(141.2:.577cm)node [above left]{$\uline{I}_1$}; + \end{tikzpicture} +\end{align*} +\end{minipage} +%\hfill +\begin{minipage}[l]{0.75\textwidth} +\begin{align*} +\uline{U}_{31}&=380\,\volt\cdot e^{j0\,\degree}\\ +\uline{U}_{12}&=380\,\volt\cdot e^{-j120\,\degree}\\ +\end{align*} +\end{minipage} +\begin{align*} +P&=\Re\{\uline{U}_{12}\cdot I^*_1\}\\ +\uline{I}_1&=\uline{I}_{12}-\uline{I}_{31}\\ +\end{align*} +\begin{align*} +\uline{I}_{12}&=\frac{\uline{U}_{12}}{R_1+jX_{L_1}} +=\frac{\overbrace{380\,\volt\cdot e^{-j120\,\degree}}^{(-190-j329{,}09)\,\volt}}{\underbrace{63,246\,\ohm\cdot ^{j18{,}435\,\degree}}_{(60+j20)\,\ohm}}=6{,}01\,\ampere\cdot e^{-j138{,}4\,\degree}=(-4{,}495-j3{,}986)\,\ampere\\ +\uline{I}_{31}&=\frac{\uline{U}_{31}}{jX_{L_3}}=\frac{380\,\volt\cdot e^{j0\,\degree}}{j50\,\ohm}=-j\frac{380\,\volt}{50\,\ohm}=-j7{,}6\,\ampere \qquad\tag{$\frac{1}{j}=-j$}\\ +\uline{I}_1&=[(-4{,}495-j3{,}986)-(-j7{,}6)]\,\ampere\\ &=(-4{,}495+j3{,}614)\,\ampere=5{,}77\,\ampere\cdot e^{j141{,}2\,\degree}\\[\baselineskip] +\end{align*} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2.5] +\draw[->,blue](0:0)--(0:1cm)node [right]{$\uline{U}_{31}$}; +\draw[->,blue](0:0)--(120:1cm)node [above]{$\uline{U}_{23}$}; +\draw[->,blue](0:0)--(-120:1cm)node [below]{$\uline{U}_{12}$}; +\draw[->,red!50!blue](0:0)--(-138.4:.601cm)node [below]{$\uline{I}_{12}$}; +\draw[->,red!50!blue](0:0)--(-90:.76cm)node [below]{$\uline{I}_{31}$}; +\draw[<-,red!75!blue](-138.4:.601cm)-- ++(90:.76cm)node [below left]{$\uline{I}_{31}$}; +\draw[->,red](0:0)--(141.2:.577cm)node [above left]{$\uline{I}_1$}; + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{S}&=\uline{U}_{12}\cdot \uline{I}^*_1=\underbrace{380\,\volt\cdot e^{-j120\,\degree}}_{(-190-j329{,}09)\,\volt}\cdot \underbrace{5{,}77\,\ampere\cdot e^{-j141{,}2\,\degree}}_{(-4{,}495-j3{,}614)\,\ampere}\\ +&=2192{,}6\,\volt\ampere\cdot e^{j98{,}8\,\degree}=(\underbrace{-335{,}28}_{P}+\underbrace{j2165{,}92}_{jQ})\,\volt\ampere\\ +P&=S\cdot \cos(\varphi)=2192{,}6\,\volt\ampere\cdot \underbrace{\cos(98{,}8\,\degree)}_{-0{,}153}=\uuline{-335{,}44\,\watt}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B18_A8.tex b/ET2_L_B18_A8.tex new file mode 100644 index 0000000..4dd33de --- /dev/null +++ b/ET2_L_B18_A8.tex @@ -0,0 +1,108 @@ +\section {Strangströme 3-Phasen System mit unsymmetrischem Verbraucher} +Ein symmetrisches 3-Phasen System mit der Phasenlage 1-2-3 speist einen unsymmetrischen +Verbraucher. Gegeben sind:\\[\baselineskip] +$\uline{U}_{12}=400\,\volt\cdot e^{j30\,\degree}$; $\quad \uline{X}_C=-100\,\ohm$; $\quad \uline{X}_{L_1}=125\,\ohm$; $\quad X_{L_2}=60\,\ohm$;$\quad R=80\,\ohm$\\[\baselineskip] +Berechnen Sie die $3$ Strangströme, den Leiterstrom $\uline{I}_1$ und die Anzeige des Leistungsmessers! +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] +\draw[black!15!,very thin](0,0)grid(4,3); + \begin{scope}[>=latex,very thick,xshift=2.268cm,yshift=1.995cm,scale=1,rotate=30]%Kondensator + \draw (-1,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(2,0)node at (.5,.1133) [left] {$X_C$}; + \draw [<-,red] (1.15,.1)--(1.55,.1)node at(1.25,.1)[left]{$\uline{I}_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2.268cm,yshift=1.005cm,rotate=-30,scale=1]%Spule + \draw (-1,0)--(.3,0) (.7,0)--(2,0)node at (.5,-.1330667) [ left] {$X_{L_1}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (-.3,-.1)--(-.7,-.1)node at(-.5,-.14)[left]{$\uline{I}_L$}; + \end{scope} + \begin{scope}[>=latex,xshift=4cm,yshift=2cm,rotate=90]%Wattmeter + \draw (0,0)--(.367,0) (.633,0)--(1,0) node at (.5,0) {W}; + \draw (.5,0)circle(.133); + \draw(.45,-.125)--(.45,-1)--(0,-1)--(-2,-1)--(-2,0)(.55,-.125)--(.55,-1)--(1,-1)--(1,0); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=1cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_{L_2}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [->,red] (.8,-.1)--(1.2,-.1)node at(1,-.1)[right]{\footnotesize$\uline{I}_{RL}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(4,0)(0,1.5)--(1.433,1.5)(0,3)--(4,3); + \filldraw(0,0)circle(0.033cm)node [left]{L3}; + \filldraw(0,1.5)circle(0.033cm)node [left]{L2}; + \filldraw(0,3)circle(0.033cm)node [left]{L1}; + \draw [->,blue] (0,2.8)--(0,1.7)node at(0,2.25)[right]{\footnotesize$\uline{U}_{12}$}; + \draw [->,blue] (0,1.3)--(0,.2)node at(0,.75)[right]{\footnotesize$\uline{U}_{23}$}; + \fill(1.433,1.5)circle(0.033cm)(4,3)circle(0.033cm)(4,2)circle(0.033cm)(4,1)circle(0.033cm)(4,0)circle(0.033cm); + \draw [->,red] (.8,3.1)--(1.2,3.1)node at(1,3.1)[above]{\footnotesize$\uline{I}_1$}; + \draw [red!50!blue,very thick]node at (4,3)[above]{u}; + \draw [red!50!blue,very thick]node at (4,0)[below]{w}; + \draw [red!50!blue,very thick]node at (1.433,1.5)[above left]{v}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{minipage}[c]{0.35\textwidth} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1.5] +\draw[thin](0,0)--(1,0); +\draw[->,blue](0:0)--(30:1cm)node [right]{$\uline{U}_{12}$}; +\draw[->,blue](0:0)--(-90:1cm)node [below]{$\uline{U}_{23}$}; +\draw[->,blue](0:0)--(150:1cm)node [left]{$\uline{U}_{31}$}; +\draw[->,red](0:0)--(-153.2:.48cm)node [left]{$\uline{I}_1$}; +\draw[thin](0:.7cm)arc(0:30:.7cm)node at(10:1cm){\footnotesize$30\,\degree$}; +\draw[thin](30:.7cm)arc(30:150:.7cm)node at(90:1cm){\footnotesize$120\,\degree$}; + \end{tikzpicture} +\end{align*} +\end{minipage} +\begin{minipage}[c]{0.35\textwidth} +\begin{align*} + \begin{tikzpicture}[very thick,scale=1.5] +\draw[thin](0,0)--(1,0); +\draw[thin](0:0cm)--+(0:1cm);%node [right]{$\uline{U}_{12}$}; +\draw[thin](0:1cm)--+(-60:1cm);%node [below]{$\uline{U}_{23}$}; +\draw[thin](0:1cm)--+(60:1cm);%node [left]{$\uline{U}_{31}$}; +\draw[->,blue](0:0cm)--+(30:1.732cm)node [right]{$\uline{U}_{12}$}; +\draw[->,blue](30:1.732cm)--+(-90:1.732cm)node [below]{$\uline{U}_{23}$}; +\draw[->,blue](-30:1.732cm)--(0:0cm)node [left]{$\uline{U}_{31}$}; +\draw[thin](0:.7cm)arc(0:30:.7cm)node at(15:1cm){\footnotesize$30\,\degree$}; + \end{tikzpicture} +\end{align*} +\end{minipage} +%%\hfill +\begin{minipage}[l]{0.3\textwidth} +\begin{align*} +\uline{U}_{12}&=400\,\volt\cdot e^{j30\,\degree}\\ +\uline{U}_{31}&=400\,\volt\cdot e^{j150\,\degree}\\ +\uline{U}_{31}&=400\,\volt\cdot e^{j270\,\degree}=400\,\volt\cdot e^{-j90\,\degree}\\ +\end{align*} +\end{minipage} +Strangströme: +\begin{align*} +\uline{I}_C&=\frac{\uline{U}_{12}}{jX_C}=\frac{400\,\volt\cdot e^{j30\,\degree}}{100\,\ohm\cdot e^{-j90\,\degree}}=\uuline{4\,\ampere\cdot e^{j120\,\degree}}=\uuline{(-2+j3{,}46)\,\ampere}\\ +\uline{I}_L&=\frac{\uline{U}_{23}}{jX_{L_1}}=\frac{400\,\volt\cdot e^{-j90\,\degree}}{125\,\ohm\cdot e^{j90\,\degree}}=\uuline{3{,}2\,\ampere\cdot e^{-j180\,\degree}}=-3{,}2\,\ampere\\ +\uline{I}_{RL}&=\frac{\uline{U}_{31}}{R+jX_{L_2}}=\frac{400\,\volt\cdot e^{j150\,\degree}}{(80+j60)\,\ohm}=\frac{400\,\volt\cdot e^{j150\,\degree}}{100\,\ohm\cdot e^{j36{,}9\,\degree}}\\ +&=\uuline{4\,\ampere\cdot e^{j113{,}1\,\degree}}=\uuline{(-1{,}57+j3{,}68)\,\ampere}\\ +\intertext{Leiterströme:} +\uline{I}_1&=\uline{I}_C-\uline{I}_{RL}=(-0{,}439-j0{,}214)\,\ampere=\uuline{0{,}48\,\ampere\cdot e^{-j153\,\degree}}\\ +\text{Zur Vollständigkeit}\\ +\uline{I}_2&=\uline{I}_L-\uline{I}_C=(-1{,}2-j3{,}46)\,\ampere=\uuline{3{,}66\,\ampere\cdot e^{j109{,}1\,\degree}}\\ +\uline{I}_3&=\uline{I}_{RL}-\uline{I}_L=(1{,}63+j3{,}68))\,\ampere=\uuline{4{,}03\,\ampere\cdot e^{j66{,}1\,\degree}} +\intertext{Anzeige der Wirkleistung:} +P&=\Re\{\uline{U}_{31}\cdot I^*_{RL}\}\\ +&=\Re\{400\,\volt\cdot e^{j150\,\degree}\cdot 4\,\ampere\cdot e^{-j113{,}1\,\degree}\}\\ +&=\Re\{1600\,\volt\ampere\cdot e^{j37\,\degree}\}\\ +&=\Re\{(1280-j960)\,\volt\ampere\}=\uuline{1280\,\watt}\\ +\intertext{oder} +P&=U_{31}\cdot I_{31}\cdot \cos(\varphi_{_{31}})=400\,\volt\cdot 4\,\ampere\cdot \cos(36{,}9\,\degree)=\uuline{1280\,\watt}\\ +\intertext{oder} +P&=I^2_{RL}\cdot R=(4\,\ampere)^2\cdot 80\,\ohm=\uuline{1280\,\watt}\qquad \text{(Betrag von $\uline{I}_{RL}$!)} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B19_A1.tex b/ET2_L_B19_A1.tex new file mode 100644 index 0000000..7282579 --- /dev/null +++ b/ET2_L_B19_A1.tex @@ -0,0 +1,51 @@ +\section {Resonanzfrequenz Zweipol} +Berechnen Sie die Resonanzfrequenz des abgebildeten Zweipols +$L=12\,\milli\henry$, $C_1=2\,\micro\farad$, $R=160\,\ohm$ +\vspace{-.5cm} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,0)--(2,0)--(2,.2) (1,1)--(2,1)--(2,.9); + \fill (0,0)circle(0.05cm)(0,1)circle(0.05cm); +% \draw [->,blue] (2.5,.8)--(2.5,.2) node at (2.5,.5)[right]{$\uline{U}_2$}; +% \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$M$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{align*} +\text{Falsch ist: } f_{res}&=\frac{1}{2\pi\sqrt{L\cdot C}}=\frac{1}{2\pi\sqrt{12\cdot \power{10}{-3}\,\ohm\second\cdot 2\cdot \power{10}{-6}\,\frac{\second}{\ohm}}}=\uline{1027\,\frac{1}{\second}}\\ +&\text{Gilt nur für } R\rightarrow\infty \text{, reine Reihen- oder Parallelschaltung.}\\[.5\baselineskip] +\text{Bei Resonanz: }\Im\, (\uline{Z})&=0 \\ +\uline{Z}=jX_L+(R||jX_C)&=j\omega L+\frac{R\cdot \frac{-j}{\omega C}}{R-j\cdot \frac{1}{\omega C}}\underbrace{\cdot \frac{R+j\frac{1}{\omega C}}{R+j\frac{1}{\omega C}}}_{\text{konj. komplex erweitern}}\hspace{-.75cm}=j\omega L+\frac{R^2\cdot \frac{-j}{\omega C}+R\frac{1}{(\omega C)^2}}{R^2+\frac{1}{(\omega C)^2}}\\ +&=\underbrace{\frac{\frac{R}{(\omega C)^2}}{R^2+\frac{1}{(\omega C)^2}}}_{\Re\,= \text{ Widerstand bei Resonanz, }\omega_{res}}+j\underbrace{\Big(\omega L-\frac{R^2}{\omega C\big(R^2+(\frac{1}{\omega C})^2\big)}\Big)}_{\Im =0\text{ bei }\omega_{res}}\\ +&\omega L-\frac{R^2}{\omega C\left(R^2+\frac{1}{(\omega C)^2}\right)}=0\\ +\Im\quad\Rightarrow\, &\, \omega_{res}\cdot \left[L(\omega_{res}\cdot C\cdot \left(R^2+\frac{1}{(\omega_{res}\cdot C)^2}\right)\right]=R^2\\ +&\omega^2_{res}\cdot L\cdot C\cdot R^2+\frac{L}{C}=R^2\\ +\Rightarrow\, &\omega^2_{res}=\frac{R^2-\frac{L}{C}}{L\cdot C\cdot R^2}=\frac{1}{L\cdot C}-\frac{1}{(R\cdot C)^2}=\frac{1}{12\,\milli\henry\cdot 2\,\micro\farad}-\frac{1}{(160\,\ohm\cdot 2\,\micro\farad)^2}\\ +&=\frac{1}{12\cdot \power{10}{-3}\,\cancel{\ohm}\second\cdot 2\cdot \power{10}{-6}\,\frac{\second}{\cancel{\ohm}}}-\frac{1}{160\,\cancel{\ohm}\cdot 2\cdot \power{10}{-6}\,\frac{\second}{\cancel{\ohm}}}\\ +&=4{,}17\cdot \power{10}{7}\power{\second}{-2}-9{,}77\cdot \power{10}{6} \power{\second}{-2} +=3{,}19\cdot \power{10}{7}\cdot \power{\second}{-2}\\ +\omega_{res}&=\sqrt{\omega^2_{res}}=5648\cdot \power{\second}{-1}\\ +f_{res}&=\frac{\omega_{res}}{2\pi}=\uuline{899\,\hertz}\\ +\end{align*} +\begin{align*} +\intertext{Nicht gefragt Resonanzwiderstand: }\\ +Z&=\frac{R}{1+(\omega \cdot C\cdot R)^2}=\frac{160\,\ohm}{(5648\cdot \cancel{\power{\second}{-1}}\cdot 2\cdot \power{10}{-6}\,\cancel{\frac{\second}{\ohm}}\cdot 160\,\cancel{\ohm})^2+1}=\uuline{37{,}5\,\ohm} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B19_A2.tex b/ET2_L_B19_A2.tex new file mode 100644 index 0000000..f7fb60a --- /dev/null +++ b/ET2_L_B19_A2.tex @@ -0,0 +1,105 @@ +\section {RLC-Reihenschwingkreis} +Von einem RLC-Reihenschwingkreis ist die Abhängigkeit $I(f)$ gegeben, siehe Kennlinie. Der Schwingkreis wird von einer konstanten sinusförmigen Spannung gespeist mit \\ +$U=100\,\volt$.\\ +Bestimmen Sie die Bauelemente $R$, $L$, und $C$!\\ +Sie dürfen auch mit der Näherung Güte $\gg 1$ rechnen. +\begin{align*} +\tikzstyle{every pin}=[fill=white,draw=black,font=\footnotesize] +\tikzstyle{every axis legend}+=[at={(0.935,0.1)},anchor=south east,inner sep=0pt]% + \begin{tikzpicture}[scale=1] + \begin{axis}[xlabel=$f (Hz)$,ylabel=$I (A)$,title={Kennlinie },grid=major,xminorgrids=false,yminorgrids=false,xmin=300,xmax=600,ymin=0,ymax=10] +\addplot[color=red,smooth,thick] +plot coordinates { +(300,1.15) +(350,1.85) +(400,3.63) +(432.2,7.07) +%(445,9.26) +%(450,9.84) +(452,9.96) +(454,10) +(456,9.96) +(477.15,7.07) +(500,4.57) +(550,2.49) +(600,1.73) +}; +\addplot[color=blue] +plot coordinates { +(432.2,7.07) +(477.15,7.07) +}; +\axispath\node[coordinate,pin=below right:f res] +at (axis cs:454,10) {}; +\axispath\node[coordinate,pin=left:7.07A] +at (axis cs:430,7.07) {}; +\axispath\node[coordinate,pin=below:B] +at (axis cs:454,7.07) {}; + \end{axis} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \fill (0,0)circle(0.025cm)(3,0)circle(0.025cm); + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\text{Ablesen: } I_{max}&=10\,\ampere\qquad f_{res}=454\,\hertz\\ +R&=\frac{U}{I_{res}}=\frac{100\,\volt}{10\,\ampere}=\uuline{10\,\ohm}\qquad\text{Bei Resonanz: Nur Spannung über R, da $\Im=0$}\\ +I&=\frac{I_{max}}{\sqrt{2}}=\frac{10\,\ampere}{\sqrt{2}}=7{,}07\,\ampere\Rightarrow\Delta f=B\approx 45\,\hertz\qquad\text{Bandbreite-Grenzfrequenzen}\\ +\end{align*} +\clearpage +\begin{align*} +&C \text{ und } L \text{ bestimmen}\\ +f_{res}&=\frac{1}{2\pi\sqrt{L\cdot C}}=454\,\hertz\tag{1}\\ +\text{Näherung }B&\approx \frac{f_{res}}{Q_S}\Rightarrow Q_S\approx \frac{f_{res}}{B}=\frac{454\,\hertz}{45\,\hertz}=10{,}1\\ +Q_S&=\frac{1}{R}\cdot \sqrt{\frac{L}{C}}=10{,}1\qquad\text{Gleichung für L und C}\tag{2}\\ +\sqrt{L}&=\underbrace{\frac{1}{2\pi \cdot f_{res}\cdot \sqrt{C}}}_{\text{aus (1)}} +=\underbrace{Q_S\cdot \sqrt{C}\cdot R}_{\text{aus (2)}}\\ +C&=\frac{1}{2\pi \cdot f_{res}\cdot Q_S\cdot R}=\frac{1}{2\pi \cdot 454\,\hertz\cdot 10{,}1\cdot 10\,\ohm}=3{,}47\cdot \power{10}{-6}\,\frac{\ampere\second}{\volt}=\uuline{3{,}47\,\micro\farad}\\ +L&=(\sqrt{L})^2=Q^2_S\cdot C\cdot R^2=10{,}1^2\cdot 3{,}47\,\micro\frac{\ampere\second}{\volt}\cdot (10\,\ohm)^2=\uuline{35{,}4\,\milli\henry} +\intertext{Alternative: 2. Punkt auf der Kurve z.B. $4\,\ampere$ bei $405\,\hertz$ ergibt 2 Gleichungen.} +\omega&=2\pi\cdot f\qquad\Rightarrow\omega=\omega_{405}=2545;\qquad\omega_{res}=2853\\ +Z_{405}&=\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}=\frac{U}{I_{405}}=\frac{100\,\volt}{4\,\ampere}=25\,\ohm\\ +&\Rightarrow \,(\omega L-\frac{1}{\omega C})^2=Z^2-R^2\\ +&\omega L-\frac{1}{\omega C}=\pm \sqrt{Z^2-R^2}\\ +&\Rightarrow L=\frac{1}{\omega}\cdot \Big(\pm \sqrt{Z^2-R^2}+\frac{1}{\omega C}\Big)\\ +L&=\frac{1}{\omega^2_{res}\cdot C}=\Big(\pm \frac{\sqrt{Z^2-R^2}}{\omega}+\frac{1}{\omega^2 C}\Big)\quad\text{(Formelsammlung: $\omega_{res}=\frac{1}{\sqrt{LC}}$)}\\ +&\Big(\frac{1}{\omega^2_{res}}-\frac{1}{\omega^2}\Big)\cdot \frac{1}{C} +=\pm\frac{\sqrt{Z^2-R^2}}{\omega}\quad\text{(nach C auflösen)}\\ +\end{align*} +\begin{align*} +C&=\frac{\Big(\frac{1}{\omega^2_{res}} +-\frac{1}{\omega^2}\Big)\cdot \omega}{\pm \sqrt{Z^2-R^2}} +=\frac +{\bigg( +\overbrace{ +\frac{1}{\Large(2835\cdot \frac{1}{\second}\Large)^2} +-\frac{1}{\Large(2545\cdot \frac{1}{\second}\Large)^2} +}^{31{,}54\cdot \power{10}{-9}} +\bigg) +\cdot 2545\cdot \frac{1}{\second}} +{\pm\sqrt{(25\,\ohm)^2-(10\,\ohm)^2}} +=\frac{-80{,}3\cdot \power{10}{6}}{\underbrace{\pm}_{- gilt} \sqrt{525}}=\uuline{3{,}5\,\micro\farad}\\ +L&=\frac{1}{\omega^2_{res}\cdot C}=\frac{1}{(2835\,\frac{1}{\second})^2\cdot 3{,}5\,\micro\farad}=\frac{1}{28{,}07\,\frac{\ampere}{\volt\second}}=\uuline{35{,}6\milli\henry} +\end{align*} +Rundungsfehler durch Ablesung und Näherung +\clearpage +}{}% diff --git a/ET2_L_B19_A3.tex b/ET2_L_B19_A3.tex new file mode 100644 index 0000000..84b33b4 --- /dev/null +++ b/ET2_L_B19_A3.tex @@ -0,0 +1,157 @@ +\section {Effektivwert und Klirrfaktor} +\begin{minipage}[c]{.7\textwidth} +Bild 1 zeigt einen Teil aus dem Ersatzschaltbild eines Transformators, aus dem hervorgeht, +dass sich der Leerlaufstrom $i_0(t)$ zusammensetzt aus dem (verzerrten) Magnetisierungsstrom +$i_\mu(t)$ und dem Strom $i_{Fe}(t)$, der die Eisenverluste repräsentiert. +\end{minipage} +\begin{minipage}[c]{.3\textwidth} +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_{1h}$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.75,.1)--(.95,.1)node at(.85,.1)[left]{\footnotesize$i_{\mu}(t)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_{Fe}$}; + \draw [<-,red] (.75,-.1)--(.95,-.1)node at(.85,-.1)[right]{\footnotesize$i_{Fe}(t)$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw(0,.1)--(0,0)--(1,0)--(1,.1)(0,.9)--(0,1)--(1,1)--(.9,1)(.5,1)--(.5,1.5)(.5,0)--(.5,-.5); + \fill (0.5,0)circle(0.025cm)(.5,1)circle(0.025cm); + \draw [<-,red] (.6,-.35)--(.6,-.15)node at(.6,-.25)[right]{\footnotesize$i_{0}(t)$}; + \end{scope} +\draw node at (.5,-1){Bild 1}; + \end{tikzpicture} +\end{align*} +\end{minipage} +Bild 2 zeigt die zeitlichen Verläufe von $i_\mu(t)$ und $i_{Fe}(t)$, welche durch folgende Fourier-Reihen approximiert werden können:\\ +$i_\mu(t)=10\,\milli\ampere\cdot \cos(\omega t) + 2,88\,\milli\ampere\cdot \cos(3\omega t)$;\\ $i_{Fe}(t)=-4\,\milli\ampere\cdot \sin(\omega t)$\\[\baselineskip] +Der resultierende, in Bild 3 dargestellte Leerlaufstrom ist die Summe:\\ +$i_0(t)= i_\mu(t)+i_{Fe}(t)$ +\begin{align*} +% \begin{tikzpicture}[scale=1.5] +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spule | +% \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L_{1h}$}; +% \fill (.3,-0.0667)rectangle(.7,0.0667); +% \draw [<-,red] (.75,.1)--(.95,.1)node at(.85,.1)[left]{\footnotesize$i_{\mu}(t)$}; +% \end{scope} +% \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] +% \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_{Fe}$}; +% \draw [<-,red] (.75,-.1)--(.95,-.1)node at(.85,-.1)[right]{\footnotesize$i_{Fe}(t)$}; +% \end{scope} +% \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. +% \draw(0,.1)--(0,0)--(1,0)--(1,.1)(0,.9)--(0,1)--(1,1)--(.9,1)(.5,1)--(.5,1.5)(.5,0)--(.5,-.5); +% \fill (0.5,0)circle(0.025cm)(.5,1)circle(0.025cm); +% \draw [<-,red] (.6,-.35)--(.6,-.15)node at(.6,-.25)[right]{\footnotesize$i_{0}(t)$}; +% \end{scope} +%\draw node at (.5,-1){Bild 1}; +% \end{tikzpicture} +\begin{tikzpicture}[scale=1.25] +\begin{scope}[>=latex, xshift=0cm, yshift=0] +\foreach \ii in {5} { % Enter Number of Decades in x +\foreach \jj in {2.5} { % Enter Number of Decades in y + \foreach \i in {1,2,...,\ii} { + \foreach \j in {0,1,2,...,\jj} { +\draw[black!50!, step=0.5] (0,-.5) grid (\ii,\jj); % Draw Sub Linear grid +}}% End Log Grid +\draw[black!80!] (0,-.5) grid (\ii,\jj); % Draw Linear grid +\draw [->,thick] (0,-.5)--(0,\jj+.25) node (yaxis) [above] {$i\,[\milli\ampere]$}; +\draw [->,thick] (0,1)--(\ii+.25,1) node (xaxis) [right] {$\omega t$}; % Draw axes +\draw node at (0,-.5)[below]{$0$}; +\draw node at (2,-.5)[below]{$\pi$}; +\draw node at (4,-.5)[below]{$2\pi$}; +\foreach \y in {-10,0,10}% y Axis Label: +\node [anchor=east] at(0,\y/10+1){$\y$}; +}} +\end{scope} +\begin{scope}[>=latex, xshift=0cm, yshift=1cm] +\draw[color=red,thick,domain=0:5,smooth,samples=100] plot[id=iomega] function{1*cos(.5*3.14*x)+.288*cos(1.5*3.14*x)}; +\draw[color=blue,thick,domain=0:5,smooth,samples=100] plot[id=iFe] function{-.4*sin(.5*3.14*x)}; +\draw[red] node at (.825,1.25) {{\footnotesize $i_{\mu}(t)$}}; +\draw[blue] node at (2.5,.75) {{\footnotesize $i_{Fe}(t)$}}; +\end{scope} +\draw node at (2.25,-1)[below]{Bild 2}; +\end{tikzpicture} +\begin{tikzpicture}[scale=1.25] +\begin{scope}[>=latex, xshift=0cm, yshift=0] +\foreach \ii in {5} { % Enter Number of Decades in x +\foreach \jj in {2.5} { % Enter Number of Decades in y + \foreach \i in {1,2,...,\ii} { + \foreach \j in {0,1,2,...,\jj} { +\draw[black!50!, step=0.5] (0,-.5) grid (\ii,\jj); % Draw Sub Linear grid +}}% End Log Grid +\draw[black!80!] (0,-.5) grid (\ii,\jj); % Draw Linear grid +\draw [->,thick] (0,-.5)--(0,\jj+.25) node (yaxis) [above] {$i\,[\milli\ampere]$}; +\draw [->,thick] (0,1)--(\ii+.25,1) node (xaxis) [right] {$\omega t$}; % Draw axes +\draw node at (0,-.5)[below]{$0$}; +\draw node at (2,-.5)[below]{$\pi$}; +\draw node at (4,-.5)[below]{$2\pi$}; +\foreach \y in {-10,0,10}% y Axis Label: +\node [anchor=east] at(0,\y/10+1){$\y$}; +}} +\end{scope} +\begin{scope}[>=latex, xshift=0cm, yshift=1cm] +\draw[color=red!50!blue,thick,domain=0:5,smooth,samples=100] plot[id=iomega] function{1*cos(.5*3.14*x)+.288*cos(1.5*3.14*x)-.4*sin(.5*3.14*x)}; +\draw[red!50!blue] node at (.825,1.25) {{\footnotesize $i_0(t)$}}; +\end{scope} +\draw node at (2.25,-1)[below]{Bild 3}; +\end{tikzpicture} +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie Effektivwert und Klirrfaktor von $i_\mu(t)$ +\item Berechnen Sie Effektivwert und Klirrfaktor von $i_0(t)$ +\end{enumerate} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +k&=\frac{\sqrt{\sum \limits_{n=2}^{\infty}A^2_n}}{\sqrt{\sum \limits_{n=1}^{\infty}A^2_n}}=\frac{\text{Effektivwert der Oberschwingungen}}{\text{Effektivwert des Gesamtsignals}} +\end{align} +\clearpage +Berechnung: +\begin{align*} +\intertext{a) Effektivwert $I_{\mu}$ und Klirrfaktor $k_{\mu}$} +I_{\mu}&=\sqrt{I_{\mu,\omega}^{\phantom{\mu}2}+I_{\mu,3\omega}^{\phantom{\mu}2}} +=\sqrt{\left(\frac{\widehat{i}_{\mu,\omega}}{\sqrt{2}}\right)^2 ++\left(\frac{\widehat{i}_{\mu,3\omega}}{\sqrt{2}}\right)^2} +=\frac{1}{\sqrt{2}}\cdot \sqrt{(10\,\milli\ampere)^2+(2{,}88\,\milli\ampere)^2}=\uuline{7{,}36\,\milli\ampere}\\ +%I_{\mu}&=\sqrt{I_{\mu,\omega}^{\phantom{\mu}2}+I_{\mu,3\omega}^{\phantom{\mu}2}}=\frac{1}{\sqrt{2}}\cdot \sqrt{\widehat{i}_{\mu,\omega}^{\phantom{\mu}2}+\widehat{i}_{\mu,3\omega}^{\phantom{\mu}2}}=\frac{1}{\sqrt{2}}\cdot \sqrt{(10\,\milli\ampere)^2+(2{,}88\,\milli\ampere)^2}=\uuline{7{,}36\,\milli\ampere}\\ +k_{\mu}&=\frac{I_{\mu,3\omega}}{I_{\mu}} +=\frac{\widehat{i}_{\mu,3\omega}\cancel{/\sqrt{2}}}{\sqrt{\widehat{i}_{\mu,\omega}^{\phantom{\mu}2} ++\widehat{i}_{\mu,3\omega}^{\phantom{\mu}2}}\cancel{/\sqrt{2}}} +=\frac{2{,}88\,\milli\ampere}{\sqrt{(10\,\milli\ampere)^2+(2{,}88\,\milli\ampere)^2}}=\uuline{0{,}277}=\uuline{27{,}7\%}\\ +\end{align*} +\begin{minipage}[c]{.8\textwidth} +\begin{align*} +\intertext{b) Effektivwert $I_0$ und Klirrfaktor $k_0$} +\widehat{i}_{0,3\omega}&=\widehat{i}_{\mu,3\omega}=2{,}88\,\milli\ampere\\ +\intertext{Nulldurchgang von $\widehat{i}_{Fe}$ bei den Spitzenwerten}\\ +\widehat{i}_{0,\omega}&=\sqrt{\widehat{i}_{\mu,\omega}^{\phantom{\mu}2}+\widehat{i}_{Fe,\omega}^{\phantom{Fe}2}}\\ +&=\sqrt{(10\,\milli\ampere)^2+(4\,\milli\ampere)^2}=10{,}77\,\milli\ampere\\[\baselineskip] +I_0&=\frac{1}{\sqrt{2}}\cdot \sqrt{\widehat{i}_{0,\omega}^{\phantom{\mu}2}+\widehat{i}_{0,3\omega}^{\phantom{\mu}2}}=\frac{1}{\sqrt{2}}\cdot \sqrt{(10{,}77\,\milli\ampere)^2+(2{,}88\,\milli\ampere)^2}=\uuline{7{,}88\,\milli\ampere}\\ +k_0&=\frac{\widehat{i}_{0,3\omega}\cancel{/\sqrt{2}}}{\sqrt{\widehat{i}_{0,\omega}^{\phantom{\mu}2}+\widehat{i}_{0,3\omega}^{\phantom{\mu}2}}\cancel{/\sqrt{2}}} +=\frac{2{,}88\,\milli\ampere}{\sqrt{(10{,}77\,\milli\ampere)^2+(2{,}88\,\milli\ampere)^2}}=\uuline{0{,}258}=\uuline{25{,}8\%} +\end{align*} +\end{minipage} +\begin{minipage}[c]{.2\textwidth} +\begin{tikzpicture}[scale=1.2] + \begin{scope}[>=latex, xshift=2cm, yshift=5] + \draw[dashed](0,-2)rectangle(1,0); + \draw[->,blue](0,0)--(1,0)node[right]{$\widehat{i}_{Fe,\omega}$}; + \draw[->,red](0,0)--(0,-2)node[below]{$\widehat{i}_{\mu,\omega}$}; + \draw[->,red!50!blue](0,0)--(1,-2)node[below right]{$\widehat{i}_{0,\omega}$}; + \end{scope} +\end{tikzpicture} +\end{minipage}\\[\baselineskip] + +\uline{Nicht gefragt}\\[\baselineskip] +Typische Klirrfaktoren:\\ +Rechteckschwingung $33\%$\\ +Sprache noch verständlich $10\%$\\ +Max. HiFi Verstärker $1\%$\\ +Guter HiFi Verstärker $0{,}1\%$\\ + +Weiteres unter \url{http://de.wikipedia.org/wiki/Klirrfaktor} +\clearpage +}{}% diff --git a/ET2_L_B19_A4.tex b/ET2_L_B19_A4.tex new file mode 100644 index 0000000..6d209e8 --- /dev/null +++ b/ET2_L_B19_A4.tex @@ -0,0 +1,60 @@ +\section {Klirrfaktor} +Am Eingang liegt die Spannung\\ +$u_e(t)=2\,\volt+3\,\volt\cdot \sin(2\pi\cdot 50\,\power{\second}{-1}\cdot t) + 4\,\volt\cdot \sin(2\pi\cdot 100\,\power{\second}{-1}\cdot t)$ +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,xshift=0cm,yshift=1cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); +% \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{L}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; +% \draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$U_{C1}$}; + \end{scope} + \draw(0,0)--(1.5,0)(1,1)--(1.5,1); + \fill(0,0)circle(.05cm)(0,1)circle(.05cm)(1.5,0)circle(.05cm)(1.5,1)circle(.05cm); + \draw[->,blue](0,.9)--(0,.1)node at(0,.5)[left]{$u_e$}; + \draw[->,blue](1.5,.9)--(1.5,.1)node at(1.5,.5)[right]{$u_a$}; + \draw[->,red](1.5,1.1)--(1.3,1.1)node at(1.2,1.1)[above]{$i_a=0$}; + \end{tikzpicture} +\end{align*} +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie den Klirrfaktor $k_a$ der Ausgangsspannung $u_a$ +\item Berechnen Sie den Effektivwert $U_a$ +\end{enumerate} +$L=100\,\milli\henry$, $C=250\,\micro\farad$ +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +k&=\frac{\sqrt{\sum \limits_{n=2}^{\infty}A^2_n}}{\sqrt{\sum \limits_{n=1}^{\infty}A^2_n}}=\frac{\text{Effektivwert der Oberschwingungen}}{\text{Effektivwert des Gesamtsignals}}\\ +k&=\sqrt{\frac{A^2_2+A^2_3+\cdots+A^2_n}{A^2_1+A^2_2+\cdots+A^2_n}}\\ +A_0&\quad\text{Gleichanteil}\notag\\ +A_1&\quad\text{Grundschwingung}\notag\\ +A_2\cdots A_n&\quad\text{Oberwellen}\notag +\end{align} +\clearpage +Berechnung: +\begin{align*} +\intertext{a) Klirrfaktor am Ausgang\ $k_a$} +U_{a0}&=U_{e0}=2\,\volt\qquad\text{Gleichanteil ist ohne Einfluß auf den Klirrfaktor}\\[\baselineskip] +\frac{\uline{U}_a}{\uline{U}_e}&=\frac{jX_C}{jX_L+jX_C}=\frac{X_C}{X_L+X_C}\qquad\text{Wechselspannung}\\[\baselineskip] +\text{Für }&50\,\hertz\text{ Grundwelle}\\ +X_{C1}&=-\frac{1}{2\pi f\cdot C}=\frac{-1}{2\pi\cdot 50\,\cancel{\frac{1}{\second}}\cdot 250\cdot \power{10}{-6}\,\frac{\ampere\cancel{\second}}{\volt}}=-12{,}73\,\ohm\\ +X_{L1}&=2\pi f\cdot L=2\pi\cdot 50\,\cancel{\frac{1}{\second}}\cdot 0{,}1\cdot \,\frac{\volt\cancel{\second}}{\ampere}=31{,}42\,\ohm\\ +U_{a1}&=\frac{-12{,}73\,\ohm}{31{,}42\,\ohm-12{,}73\,\ohm}\cdot U_{e1}=-0{,}6811\cdot U_{e1}\\ +U_{a1}&=|-0{,}6811\cdot U_{e1}|=0{,}6811\cdot \frac{3\,\volt}{\sqrt{2}}=1{,}445\,\volt\quad \text{Effektivwert}\\ +U^2_{a1}&=2{,}088\,\volt^2\\[\baselineskip] +\text{Für }&100\,\hertz\text{ 1. Oberwelle}\\ +X_{C2}&=\frac{1}{2}\cdot X_{C1}=-6{,}365\,\ohm\\ +X_{L2}&=2X_{L1}=62{,}84\,\ohm\\ +U_{a2}&=\left|\frac{-6{,}365\,\ohm}{62{,}84\,\ohm-6{,}365\,\ohm}\right|\cdot \frac{4\,\volt}{\sqrt{2}}=0{,}1127\cdot \frac{4\,\volt}{\sqrt{2}}=0{,}3188\,\volt\quad \text{Effektivwert}\\ +U^2_{a2}&=0{,}1016\,\volt^2\\[\baselineskip] +k_a&=\sqrt{\frac{U^2_{a2}}{U^2_{a1}+U^2_{a2}}}=\sqrt{\frac{0{,}1016\,\volt^2}{2{,}088\,\volt^2+0{,}1016\,\volt^2}} +=\uuline{0{,}215}=\uuline{21{,}5\%} +\intertext{b) Effektivwert $U_a$} +U_a&=\sqrt{U^2_{a0}+U^2_{a1}+U^2_{a2}}=\sqrt{(2\,\volt)^2+2{,}088\,\volt^2+0{,}1016\,\volt^2}=\uuline{2{,}49\,\volt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B19_A5.tex b/ET2_L_B19_A5.tex new file mode 100644 index 0000000..adeed8b --- /dev/null +++ b/ET2_L_B19_A5.tex @@ -0,0 +1,93 @@ +\section {Momentanspannung} +An die Schaltung wird die Spannung +$u_E(t)=U_0+\widehat{u}_1\cdot \cos(\omega t+\varphi_1) + \widehat{u}_2\cdot \cos(2\omega t+\varphi_2)$ +angelegt.\\[.5\baselineskip] +Berechnen Sie dies Spannung $u_C$ zur Zeit $t=T$\\ +\begin{minipage}[c]{.5\textwidth} +\begin{align*} + \begin{tikzpicture}[very thick,scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$}; +% \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \draw(0,0)--(2.5,0)(1,1)--(2.5,1); + \fill(0,0)circle(.025cm)(0,1)circle(.025cm)(2.5,0)circle(.025cm)(2.5,1)circle(.025cm); + \draw[->,blue](0,.9)--(0,.1)node at(0,.5)[left]{$u_E$}; + \draw[->,blue](2.5,.9)--(2.5,.1)node at(2.5,.5)[right]{$u_C$}; + \end{tikzpicture} +\end{align*} +\end{minipage} +\begin{minipage}[c]{.5\textwidth} +\begin{align*} +R_1&=12\,\ohm\quad R_2=20\,\ohm\\ +C&=25\,\nano\farad\quad \omega=961\cdot \power{10}{3}\,\frac{1}{\second}\\ +U_0&=6\,\volt\quad \widehat{u}_1=7\,\volt\quad \widehat{u}_2=3\,\volt\\ +\varphi_1&=60\,\degree\quad \varphi_2=135\,\degree\quad T=2\,\micro\second +\end{align*} +\end{minipage} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\[.5\baselineskip] +Jede Frequenz für sich betrachten; Überlagerung der Momentanwerte\\[.5\baselineskip] +$u_E(t)=6\,\volt+7\,\volt\cdot \cos(961\cdot 10^3\,\cdot 1\per\second\cdot t+60\degree)+3\,\volt\cdot \cos(2\cdot 961\cdot 10^3\,\cdot 1\per\second\cdot t+135\degree)$\\ +a) Gleichspannung - Spannungsteiler +\begin{align*} +%\intertext{a) Gleichspannung - Spannungsteiler} +u_{C0}&=U_0\cdot \frac{R_2}{R_1+R_2}=6\,\volt\cdot \frac{20\,\ohm}{32\,\ohm}=3{,}75\,\volt +\intertext{b) allgemein:} +%\vspace{-1.5cm} +\uline{U_C}&=\uline{U}_E\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{||}+R_1}\\ +\uline{Z}_{||}&=\frac{R_2\cdot jX_C}{R_2+jX_C} +\intertext{c) $1\cdot \omega$: Transformation in komplexe Ebene: $u(t)\rightarrow \uline{U}$} +\uline{U}_{E1}&=\frac{7\,\volt}{\sqrt{2}}\cdot e^{j60\,\degree}=4{,}95\,\volt\cdot e^{j60\,\degree}=(2{,}47+j4{,}29)\,\volt\\ +X_{C1}&=\frac{-1}{\omega C}=-41{,}6\,\ohm\\ +\uline{Z}_{||1}&=\frac{20\,\ohm\cdot (-j41{,}6\,\ohm)}{20\,\ohm-j41{,}6\,\ohm}=\frac{-j832\,\ohm^2}{20\,\ohm-j41{,}6\,\ohm}\\ +&=\frac{828\,\ohm\cancel{^2}\cdot e^{-j90}}{45{,}98\cancel{\,\ohm}\cdot e^{-j64{,}22}}=18{,}05\,\ohm\cdot e^{-j25{,}67}=(16{,}27-j7{,}82)\,\ohm\\ +\uline{Z}_{||1}+R1&=(16{,}27-j7{,}82+12)\,\ohm=(28{,}27-j7{,}82)\,\ohm=29{,}33\,\ohm\cdot e^{-j15{,}52}\\ +\uline{U}_{C1}&=\uline{U}_{E1}\cdot \frac{\uline{Z}_{||1}}{\uline{Z}_{||1}+R_1} +%&= 4{,}95\,\volt\cdot e^{j60\,\degree}\cdot \frac{(16{,}2-j7{,}834)\cancel{\,\ohm}}{(28{,}2-j7{,}834)\cancel{\,\ohm}} +=4{,}95\,\volt\cdot e^{j60\,\degree}\cdot \frac{18{,}05\cancel{\,\ohm}\cdot e^{-j25{,}67}}{29{,}33\cancel{\,\ohm}\cdot e^{-j15{,}46\,\degree}}\\ +&=3{,}04\,\volt\cdot e^{j49{,}79\,\degree}=(1{,}97+j2{,}32)\,\volt\\ +U_{C1}&=3{,}04\,\volt\quad\text{Effektivwert}\\ +\varphi_{C1}&=49{,}79\,\degree=0{,}868\,\radian\quad\text{Umwandlung wegen } (\omega t+\varphi_{C1}) +\end{align*} +\enlargethispage{2cm} +\clearpage +\begin{align*} +\intertext{d) $1\cdot \omega$ Rücktransformation: mit $t=T$} +\omega t+\varphi_{C1}&=\omega \cdot T+\varphi_{C1}=961\cdot \power{10}{3}\,\cancel{\frac{1}{\second}}\cdot 2\cdot \power{10}{-6}\,\cancel{\second}+0{,}868\,\radian=2{,}791\,\radian =159{,}9\,\degree\\ +u_{C1}(t=T)&=\sqrt{2}\cdot U_{C1}\cdot \cos(\omega T+\varphi_{C1})=\sqrt{2}\cdot 3{,}04\,\volt\cdot \cos(159{,}9\,\degree)\\ +&=4{,}3\,\volt\cdot (-0{,}939)=-4{,}037\,\volt +\intertext{e) $2\cdot \omega$: Transformation in komplexe Ebene: $u(t)\rightarrow \uline{U}$} +\uline{U}_{E2}&=\frac{3\,\volt}{\sqrt{2}}\cdot e^{j135\,\degree}=2{,}121\,\volt\cdot e^{j135\,\degree}=(-1{,}5+j1{,}5)\,\volt\\ +X_{C2}&=\frac{1}{2}\cdot X_{C1}=-20{,}8\,\ohm\\ +\uline{Z}_{||2}&=\frac{20\,\ohm\cdot (-j20{,}8\,\ohm)}{20\,\ohm-j20{,}8\,\ohm}=\frac{-j414\,\ohm^2}{20\,\ohm-j20{,}8\,\ohm}\\ +&=\frac{414\,\ohm\cancel{^2}\cdot e^{-j90}}{28{,}78\cancel{\,\ohm}\cdot e^{-j45{,}99}}=14{,}41\,\ohm\cdot e^{-j43{,}87}=(10{,}39-j9{,}992)\,\ohm\\ +\uline{Z}_{||2}+R1&=(10{,}39-j9{,}992+12)\,\ohm=(22{,}34-j9{,}992)\,\ohm=24{,}51\,\ohm\cdot e^{-j24\,\degree}\\ +\uline{U}_{C2}&=\uline{U}_{E2}\cdot \frac{\uline{Z}_{||2}}{\uline{Z}_{||2}+R_1}\\ +%&= 2{,}121\,\volt\cdot e^{j135\,\degree}\cdot +&=\uline{U}_{E2}\cdot +\frac{(10{,}39-j9{,}992)\cancel{\,\ohm}}{(22{,}39-j9{,}992)\cancel{\,\ohm}} +=2{,}121\,\volt\cdot e^{j135\,\degree}\cdot \frac{14{,}38\cdot e^{-j44{,}01}}{24{,}48\cdot e^{-j24{,}1\,\degree}}\\ +&=1{,}247\,\volt\cdot e^{j115{,}1\,\degree}=(-0{,}530+j1{,}129)\,\volt\\ +U_{C2}&=1{,}247\,\volt\quad\text{Effektivwert}\\ +\varphi_{C2}&=115{,}1\,\degree=2{,}009\,\radian\quad\text{Umwandlung wegen } (2\cdot \omega t+\varphi_{C2}) +\intertext{f) $2\cdot \omega$ Rücktransformation: mit $t=T$} +2\cdot \omega t+\varphi_{C2}&=2\omega\cdot T+\varphi_{C2}=2\cdot 961\cdot \power{10}{3}\,\cancel{\frac{1}{\second}}\cdot 2\cdot \power{10}{-6}\,\cancel{\second}+2{,}009\,\radian\\ +&=5{,}854\,\radian =335{,}4\,\degree\\ +u_{C2}(t=T)&=\sqrt{2}\cdot U_{C2}\cdot \cos(2\cdot \omega T+\varphi_{C2})=\sqrt{2}\cdot 1{,}247\,\volt\cdot \cos(335{,}4\,\degree)\\ +&=1{,}762\,\volt\cdot 0{,}909=1{,}602\,\volt +\intertext{g) Überlagerung:} +u_C(t=T=2\,\micro\second)&=u_{C0}(T)+u_{C1}(T)+u_{C2}(T)=(3{,}75\,\volt-4{,}037\,\volt+1{,}602\,\volt)\\ +&=\uuline{1{,}315\,\volt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B19_A6.tex b/ET2_L_B19_A6.tex new file mode 100644 index 0000000..fe100a3 --- /dev/null +++ b/ET2_L_B19_A6.tex @@ -0,0 +1,29 @@ +\section {Nichtlinears Bauelement} +Für ein nichtlinears Bauelement gilt: $i=a\cdot u^2$ \\ +Berechnen Sie den Klirrfaktor des Stromes, wenn die Spannung $u(t)=U_0+\widehat{u}\cdot \sin(\omega t)$ angelegt wird.\\[\baselineskip] +$a=20\,\milli\ampere\per\volt^2;\quad U_0=1{,}5\,\volt;\quad \widehat{u}=1{,}2\,\volt; \quad \omega=1500\,\power{\second}{-1}$\\[\baselineskip] +Es gilt $\sin^2\alpha=0{,}5\cdot (1-\cos(2\alpha))$\\[\baselineskip] +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{align*} +k_i&=\frac{I_2}{\sqrt{I^2_1+I^2_2}}\qquad\text{$\omega$ ohne Bedeutung}\\ +(\alpha&=\omega t)\\ +i(t)&=a\cdot (U_0+\widehat{u}\cdot \sin\alpha)^2=a\cdot (U^2_0+2\cdot U_0\cdot \widehat{u}\cdot \sin\alpha+\widehat{u}^2 \overbrace{\cdot \sin^2\alpha)}^{0,5-0,5\cdot cos(2\alpha)}\\ +&=\overbrace{a\cdot U^2_0}^{DC}+\underbrace{2\cdot a\cdot U_0\cdot \widehat{u}}_{\widehat{i}_1}\cdot \sin\alpha+\overbrace{0{,}5\cdot a\cdot \widehat{u}^2}^{DC}-\underbrace{0{,}5\cdot a\cdot \widehat{u}^2}_{\widehat{i}_2}\cdot \cos(2\alpha)\qquad\text{DC=Gleichanteil, ohne Einfluß}\\[\baselineskip] +k_i&=\frac{\frac{1}{\sqrt{2}}\cdot 0{,}5\cdot a\cdot \widehat{u}^2}{\sqrt{\frac{1}{2}\cdot 4\cdot a^2\cdot U^2_0\cdot \widehat{u}^2+\frac{1}{2}\cdot \frac{1}{4}\cdot a^2\cdot \widehat{u}^4}} +=\frac{\cancel{\frac{1}{\sqrt{2}}}\cdot 0{,}5\cdot \cancel{a}\cdot \widehat{u}\cancel{^2}} +{\cancel{\frac{1}{\sqrt{2}}\cdot a\cdot \widehat{u}}\cdot \sqrt{4\cdot U^2_0\cdot +\frac{1}{4}\cdot \widehat{u}^2}}\\ +&=\frac{0{,}5\cdot \widehat{u}}{\sqrt{4\cdot U^2_0+\frac{1}{4}\cdot \widehat{u}^2}}=\frac{0{,}5\cdot 1{,}2\,\volt}{\sqrt{4\cdot 1{,}5^2\,\volt^2+\frac{1}{4}\cdot 1{,}2^2\,\volt^2}}=\uuline{0{,}1961}=\uuline{19{,}61\%} +\end{align*} +alternativ: +\begin{align*} +I_1&=\frac{\widehat{i}_1}{\sqrt{2}}=50{,}91\,\milli\ampere\\ +I_2&=\frac{\widehat{i}_2}{\sqrt{2}}=10{,}18\,\milli\ampere\\ +k_i&=\frac{I_2}{\sqrt{I^2_1+I^2_2}}=\frac{10{,}18\,\milli\ampere}{\sqrt{(50{,}91\,\milli\ampere)^2 ++(10{,}18\,\milli\ampere)^2}}=\uuline{0{,}1961}=\uuline{19{,}61\%} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B19_A7.tex b/ET2_L_B19_A7.tex new file mode 100644 index 0000000..774594c --- /dev/null +++ b/ET2_L_B19_A7.tex @@ -0,0 +1,68 @@ +\section {Wirkleistung Zweipol} +An einem Zweipol liegt die dargestellte Spannung $u$. Es fließt der dargestellte Rechteckstrom $i$. Spannung und Strom sind periodisch und haben die gleiche Periodendauer. +\begin{align*} +\begin{tikzpicture}[scale=1] +\begin{scope}[>=latex, xshift=-4cm, yshift=0] +\draw(0,-1)rectangle(2,1); +\draw(-.5,.75)--(0,.75); +\draw(-.5,-.75)--(0,-.75); +\draw[->,blue,thick](-.5,.5)--(-.5,-.5)node at(-.5,0)[left]{$u$}; +\draw[->,red,thick](-.5,.9)--(-.1,.9)node at(-.3,.9)[above]{$i$}; +\draw node at(1,0){Zweipol}; +\end{scope} +\begin{scope}[>=latex, xshift=0cm, yshift=0] +\draw[thin,black!50!](0,-1)grid(8,1); +\draw[color=blue,thick,domain=0:8,samples=100] plot[id=sin19_7] function{sin(1.047*x)} node at (2,1) [above] {$u$}; +\draw[red, very thick] (0,-.5)--(1,-.5)--(1,.5)--(4,.5)--(4,-.5)--(7,-.5)--(7,.5)--(7,.5)--(8,.5)node at(4,.5)[above right]{i}; +\draw node at(0,1)[left]{$u_{max}$}; +\draw node at(0,.5)[left]{$iI_{max}$}; +\draw node at(0,-.5)[left]{$-i_{max}$}; +\draw node at(0,-1)[left]{$-u_{max}$}; +\draw node at(1,-1)[below]{$\frac{\pi}{3}$}; +\draw node at(4,-1)[below]{$\frac{4}{3}\pi$}; +\draw node at(7,-1)[below]{$\frac{7}{3}\pi$}; +\draw[->,thick,black](0,-1)--(0,1.25); +\draw[->,thick,black](0,0)--(8.25,0)node [right]{$\omega t$}; +\draw node at(4,-1.75)[below]{$u_{max}=300\,\volt\qquad i_{max}=6\,\ampere$}; +\end{scope} +\end{tikzpicture} +\end{align*} +Berechnen Sie die Wirkleistung, die der Zweipol aufnimmt. \\[\baselineskip] +Hinweis: Unterschiedliche Lösungsverfahren sind möglich. +falls benötigt: Der Strom $i$ kann durch folgende Fourier Reihe dargestellt werden: +\begin{align*} +i&=\frac{24\,\ampere}{\pi}\left[\sin\left(\omega t-\frac{\pi}{3}\right)+\frac{1}{3}\sin\left( 3\left(\omega t-\frac{\pi}{3}\right)\right)+\frac{1}{5}\sin\left(5\left(\omega t-\frac{\pi}{3}\right)\right)+\cdots\right] +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\[\baselineskip] +Elegante Lösung mit Fourier-Reihe:\\ +Für die Leistungsaufnahme des Zweipols ist nur der Stromanteil entscheidend, der die gleiche Frequenz wie die Spannung besitzt, d.h. nur +\begin{align*} +P&=U\cdot I_\omega\cdot \cos\varphi\\ +i_\omega(t)&=\frac{24\,\ampere}{\pi}\sin(\omega t\underbrace{-\frac{\pi}{3}}_{\varphi})\\ +I_\omega&=\frac{24\,\ampere}{\pi\cdot \sqrt{2}}\\ +\Rightarrow\quad P&=U\cdot I_\omega\cdot \cos\varphi=\frac{300\,\volt}{\sqrt{2}}\cdot \frac{24\,\ampere}{\pi\sqrt{2}}\cdot \underbrace{\cos\left(-\frac{\pi}{3}\right)}_{0{,5}}=\uuline{573\,\watt}\\ +\end{align*} +oder Standardlösung +\begin{align*} +p(t)&=u(t)\cdot i(t)\\ +P&=\frac{1}{T}\int_0^Tp(t)\cdot dt=\frac{1}{T}\int_0^Tu(t)\cdot i(t)\cdot dt=\frac{1}{2\pi}\int_0^{2\pi}u(\omega t)\cdot i(\omega t)\cdot d(\omega t)\\ +&\text{(Verschiebung Start- und Endwert für Integration)}\\ +&=\frac{1}{2\pi}\left[\int_{\frac{\pi}{3}}^{\frac{4}{3}\pi}6\,\ampere\cdot 300\,\volt\cdot \sin(\omega t)d(\omega t)+\int_{\frac{4}{3}\pi}^{\frac{7}{3}\pi}-6\,\ampere\cdot 300\,\volt\cdot \sin(\omega t)d(\omega t)\right]\\ +&=\frac{1}{2\pi}\Big[1800\,\volt\ampere\cdot \Big(\underbrace{-\cos\Big(\frac{4}{3}\pi\Big)}_{0{,}5}+\underbrace{\cos\Big(\frac{\pi}{3}\Big)}_{0{,}5}\Big) +-1800\,\volt\ampere\cdot \Big(\underbrace{-\cos\Big(\frac{7}{3}\pi\Big)}_{-0{,}5}+\underbrace{\cos\Big(\frac{4}{3}\pi\Big)}_{-0{,}5}\Big)\Big]\\ +&=\frac{1800\,\volt\ampere}{\pi}=\uuline{573\,\watt}\\[\baselineskip] +\end{align*} +oder sinusförmiger Strom unter Berücksichtigung des Formfaktors +\begin{align*} +&\text{Für sinusförmigen Strom:}\\ +P_{\sin}&=U\cdot I\cdot \cos\varphi=\frac{300\,\volt}{\sqrt{2}}\cdot \frac{6\,\ampere}{\sqrt{2}}\cdot \cos(60\,\degree)=450\,\watt (\sin) =636\,\watt\\ +&\text{Für den rechteckförmigen Strom muß der Formfaktor berücksichtigt werden:}\\ +F&=\frac{\pi}{2\sqrt{2}}=1{,}11\\ +P&=\frac{P_{\sin}}{F}=\frac{636\,\watt}{1{,}11}=\uuline{573\,\watt}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B20_A1.tex b/ET2_L_B20_A1.tex new file mode 100644 index 0000000..b5ab333 --- /dev/null +++ b/ET2_L_B20_A1.tex @@ -0,0 +1,92 @@ +\section {Ringspule} +Durch das Zentrum einer Ringspule mit rechteckigem Querschnitt (Länge +$l=12\,\milli\metre$, $r_i=10\,\milli\metre$, $r_a=20\,\milli\metre$; +Windungszahl $N= 1800$) wird ein Leiter geführt, in dem ein Wechselstrom fließt. +($I_{eff}=40\,\ampere$; $f=50\,\hertz$).\\ +Berechnen Sie den Effektivwert $U$ der Spannung $u$.\\ +$\mu=\mu_0=1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)$ +\begin{align*} + \begin{tikzpicture}[scale=2.5, z=0.2pt] + \begin{scope}[>=latex, xshift=0, yshift=0] + \foreach \z in {0,0.2,...,1} + \fill [black!25!](0,0,\z cm) circle (0.7cm);% Zylinder außen + \draw [black!75!](0,0,1 cm) circle (0.7cm);%Hintere äußere Ringkante +%\foreach \z in {0,0.2,...,0.5} +%\fill [black!50!](0,0,\z cm) circle (0.5cm);% Zylinder innen + \fill [black!25!](0,0) circle (0.7cm);% Stirnseitenfüllung + \fill [black!35](0,0) circle (0.5cm); %Füllung innen + \begin{scope}%Clip Füllung + \clip (0,0,1cm) circle (0.5cm);%Clip mit hinterer inneren Ringkante + \fill [black!0!](0,0) circle (0.5cm);%Innere Ringkante + \end{scope} + \begin{scope}%Clip hintere innere Kante + \clip (0,0) circle (0.5cm);%Clip mit hinterer inneren Ringkante + \draw [black!75!](0,0,1cm) circle (0.5cm);%Hintere innere Ringkante + \end{scope} + \draw [black!75!](0,0) circle (0.5cm);% Innere Ringkante + \draw [black!75!](0,0) circle (0.7cm);% Äußere Ringkante + \foreach \a in {-45,-30,-15,15,30,45,60,75,90,105,120,135} %Rechte Drähte + \draw [red!60!black, ultra thick] (\a:0.5)--(\a:0.7)--+(45:.3); + \foreach \a in {150,165,...,300} %Linke Drähte + \draw [red!60!black, ultra thick] (\a:0.7)--(\a:0.5)--+(45:.3); + \draw [red!60!black, ultra thick] (0.5,0,0)--(1.5,0,0)circle(0.01);%Anschußdrähte + \draw [red!60!black, ultra thick] (0.7,0,1cm)--(1.5,0,1cm)circle(0.01);%Anschußdrähte + \draw [->,blue, thick] (1.6,0,1cm) -- (1.6,0,0) node at (1.6,0,.5cm)[right] {$u$}; + \draw [blue, ultra thick] (0,0,3.5cm) -- (0,0,7cm); + \draw [->,red, thick] (0,0,7.5cm) -- (0,0,8cm)node [right] {$i$}; + \draw [blue, ultra thick] (0,0,1.75cm) -- (0,0,-5cm)node [above left]{Leiter}; + \end{scope} + \end{tikzpicture} +\begin{tikzpicture}[scale=1.4]%Querschnitt + \begin{scope}[>=latex,xshift=0,yshift=0] + \fill [black!25!] (0,0)rectangle(1,1); + \end{scope} + \begin{scope}[>=latex,xshift=0,yshift=3cm] + \fill [black!25!] (0,0)rectangle(1,1); + \end{scope} + \begin{scope}[>=latex,xshift=0,yshift=0cm] + \draw [very thick] (0,0)rectangle(1,4); + \end{scope} + \begin{scope}[>=latex,xshift=-1cm,yshift=1.9cm] + \fill [blue!50!] (0,0)rectangle(3,.2); + \end{scope} + \begin{scope}[>=latex,xshift=0cm,yshift=4cm]%Hilfslinien + \draw [very thin] (-1,0)--(0,0)--(0,.5) (1,0)--(1,.5)(-.5,-1)--(0,-1); + \draw [very thin, dashed] (-1.5,-2)--(2.5,-2); + \draw [->,very thin] (-.9,-2)--(-.9,0)node at(-.9,-1)[right]{$r_a$}; + \draw [->,very thin] (-.4,-2)--(-.4,-1)node at(-.4,-1.5)[right]{$r_i$}; + \draw [<->,very thin] (0,0.4)--(1,0.4)node at(.5,0.4)[above]{$l$}; + \draw node at(.5,-.5){$\mu=\mu_0$}; + \draw [->,red,thin] (1.25,-1.75)--(1.5,-1.75) node [right]{$i$}; + \end{scope} +\end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +\begin{align} +\intertext{Formeln:} +u&=N\cdot \frac{d\phi(t)}{dt} +\end{align} +Berechnung: +Strom im Leiter bewirkt Fluss durch die Spule: +\begin{align*} +\phi&=\widehat{\phi}_0\cdot \sin(\omega t)\\ +\widehat{\phi}_0&=\int_{r_i}^{r_a}l\cdot \widehat{B}(r)\cdot dr=\frac{\mu_0\cdot l\cdot \widehat{i}}{2\pi}\cdot \int_{r_i}^{r_a}\frac{1}{r}\cdot dr=\frac{\mu_0\cdot l\cdot \widehat{i}}{2\pi}\cdot \ln\frac{r_a}{r_i}\\ +\text{mit }\widehat{i}&=\sqrt{2}\cdot I\\ +\Rightarrow \widehat{\phi}_0&=\frac{1{,}26\cdot \power{10}{-6}\, \frac{\volt\second}{\ampere\metre}\cdot 12\cdot \power{10}{-3}\,\metre\cdot \sqrt{2}\cdot 40\,\ampere}{2\pi}\cdot \ln(2)=9{,}44\cdot \power{10}{-8}\,\volt\second\\ +\phi_{eff}&=\frac{\widehat{\phi}_0}{\sqrt{2}}=6{,}672\cdot \power{10}{-8}\,\volt\second\\ +\intertext{Veränderlicher Fluss induziert Spannung:} +u&=N\cdot \frac{d\phi(t)}{dt}=N\cdot \widehat{\phi}_0\cdot \omega\cdot \cos(\omega t)=\widehat{u}\cdot \cos(\omega t)\\ +\widehat{u}&=N\cdot \phi_0\cdot \omega=1800\cdot 9{,}44\cdot \power{10}{-8}\,\volt\second\cdot 2\pi\cdot 50\,\power{\second}{-1}=53{,}36\,\milli\volt\\ +U&=\frac{\widehat{u}}{\sqrt{2}}=\frac{53{,}36\,\milli\volt}{\sqrt{2}}=\uuline{37{,}73\,\milli\volt} +\intertext{Anmerkung:} +u_i&=2\cdot a\cdot r\cdot N\cdot B\cdot \omega\cdot \sin(\omega t)\\ +&=2\cdot a\cdot r\cdot N\cdot \frac{\Phi}{A}\cdot \omega\cdot \sin(\omega t)\\ +&\text{mit $2\cdot a\cdot r=A$}\\ +&=N\cdot \Phi \cdot \omega\cdot \sin(\omega t)\\ +&=\widehat{u}_i\cdot \sin(\omega t)\\ +\intertext{oder} +U&=\mu_0\cdot l\cdot \ln\frac{r_a}{r_i}\cdot N\cdot f\cdot I\\ +&=1{,}26\cdot \power{10}{-6}\,\frac{\volt\cancel{\second}}{\cancel{\ampere}\cancel{\metre}}\cdot 12\cdot \power{10}{-3}\,\cancel{\metre}\cdot \ln\frac{20\,\cancel{\milli\metre}}{10\,\cancel{\milli\metre}}\cdot 1800\cdot 50\,\cancel{\frac{1}{\second}}\cdot 40\,\cancel{\ampere}=\uuline{37{,}73\,\milli\volt} +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B20_A2.tex b/ET2_L_B20_A2.tex new file mode 100644 index 0000000..be93068 --- /dev/null +++ b/ET2_L_B20_A2.tex @@ -0,0 +1,192 @@ +\section {Netzwerk Wirk- und Blindanteil} +$\uline{U}=4{,}2\,\volt\cdot e^{j30\,\degree};\quad f=120\,\kilo\hertz ;\quad \uline{I}_0= 2{,}7\milli\ampere \cdot e^{j110\,\degree}$\\ +$R=1{,}3\,\kilo\ohm; \quad L=3{,}6\,\milli\henry; \quad C=1{,}5\,\nano\farad$. +\renewcommand{\labelenumi}{\alph{enumi})} +\begin{enumerate} +\item Berechnen Sie den Strom \uline{I} und seinen Wirk- und Blindanteil $\uline{I}_{\textrm{w}}$ bzw. $\uline{I}_{\textrm{b}}$. +\item Berechnen Sie die Wirk und Blindleistung an den Anschlussklemmen und an der Stromquelle $\uline{I}_0$. +\end{enumerate} +\vspace{-.5cm} +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] + \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_L$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); + \draw [->,blue](.8,.2)--(.2,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{RL}$}; + \draw (.5,0)circle(.133); + \draw [->,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{I}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (.8,1)--(3,1)--(3,.9) (0,0)--(3,0)--(3,.2); + \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); + \end{scope} + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung:\\ +\begin{align*} +\text{a)}\quad X_L&=\omega L=2\pi\cdot f\cdot 3{,}6\,\milli\henry=2714\,\ohm\\ +X_C&=\frac{-1}{\omega C}=\frac{-1}{2\pi\cdot f\cdot 1{,}5\,\nano\farad}=-884\,\ohm\\ +\uline{Z}_{RL}&=\frac{R\cdot jX_L}{R+jX_L}=\frac{1{,}3\,\kilo\ohm\cdot j2714\,\ohm}{1{,}3\,\kilo\ohm+j2714\,\ohm}=(1057+j506)\,\ohm=1172\,\ohm\cdot e^{j25{,}59\,\degree} +\end{align*} +Serienersatzschaltbild: +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] + \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$X_C$}; + \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; + \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$\uline{U}_C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_{RL}$}; + \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{RL}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle + \draw (0,0)--(1,0); + \draw (.5,0)circle(.133); + \draw (.5,0)circle(.133); + \draw [<-,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{U}_0$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (1.8,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2); + \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); + \end{scope} + \end{tikzpicture} + \begin{tikzpicture}[scale=.5] + \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] + \draw[black!25!,very thin](-3,0)grid(5,5); + \draw[thick](-3,0)--(5,0) (0,0)--(0,5); + \draw[->,blue](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; + \draw[->,red](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; + \draw[->,blue](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; + \draw[->,red](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\uline{U}_0&=\uline{I}_0\cdot \uline{Z}_{RL}=2{,}7\milli\ampere \cdot e^{j110\,\degree}\cdot 1172\,\ohm\cdot e^{j25{,}59\,\degree}\\ +&=3{,}164\,\volt\cdot e^{j135{,}6\,\degree}=(-2{,}26+j2{,}21)\,\volt\\ +\uline{Z}_{RLC}&=\uline{Z}_{RL}+jX_C=(1060+j507)\,\ohm-j884\,\ohm\\ +&=(1057-j378)\,\ohm=1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}\\ +\intertext{Masche:} +\uline{I}&\cdot \uline{Z}_{RLC}+\uline{U}_0-\uline{U}=0\\ +\uline{I}&=\frac{\uline{U}-\uline{U}_0}{\uline{Z}_{RLC}}=\frac{\overbrace{(3{,}637+j2{,}1)\,\volt}^{4{,}2\,\volt\cdot e^{j30\,\degree}}-(-2{,}26+j2{,}21)\,\volt}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}=\frac{5{,}90\,\volt\cdot e^{-j1{,}06\,\degree}}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}\\ +&=\uuline{5{,}25\milli\ampere\cdot e^{j18{,}5\,\degree}}=\uuline{(4{,}98+j1{,}67)\milli\ampere} +\end{align*} +\clearpage +Parallelersatzschaltbild: (Wirkstrom in Phase, Blindstrom senkrecht zur Spannung) +\begin{align*} + \begin{tikzpicture}[scale=2] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] + \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - + \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_W$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$X$}; + \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_B$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2); + \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); + \end{scope} + \end{tikzpicture} + \begin{tikzpicture}[scale=.5] + \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] + \draw[black!25!,very thin](-3,0)grid(5,5); + \draw[thick](-3,0)--(5,0) (0,0)--(0,5); + \draw[->,blue!45!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; + \draw[->,red!45!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; + \draw[->,blue!45!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; + \draw[->,red!45!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; + \draw[->,red](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$}; + \draw[->,red](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +|\uline{I}_{\textrm{w}}|&=I\cdot \cos(\underbrace{\varphi_u}_{30\,\degree}-\underbrace{\varphi_i}_{18{,}5\,\degree})=5{,}14\,\milli\ampere\\ +\Rightarrow & \uline{I}_{\textrm{w}}=\uuline{5{,}14\,\milli\ampere\cdot e^{j30\,\degree}}\\ +|\uline{I}_b|&=I\cdot \sin(\varphi_u-\varphi_i)=1{,}05\,\milli\ampere\\ +\Rightarrow & \uline{I}_b=\uuline{1{,}05\,\milli\ampere\cdot e^{-j60\,\degree}}\\ +\end{align*} +\begin{minipage}[b]{.5\textwidth} +b) +\begin{align*} +\uline{U}_C&=\uline{I}\cdot jX_C=4{,}64\,\volt\cdot e^{-j71{,}5\,\degree}\\ +\uline{U}_{RL}&=\uline{U}-\uline{U}_C=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\\ +\uline{I}_R&=\frac{\uline{U}_{RL}}{R}=5{,}27\,\milli\ampere\cdot e^{j71{,}6\,\degree}\\ +\uline{I}_L&=\frac{\uline{U}_{RL}}{jX_L}=2{,}52\,\milli\ampere\cdot e^{-j18{,}4\,\degree}\\ +\end{align*} +Probe: +\begin{align*} +I+I_0&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\ +I_R+I_L&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\ +&\text{stimmt überein.} +\end{align*} +\end{minipage} +\begin{minipage}[c]{.5\textwidth} + \begin{tikzpicture}[scale=.75] + \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] + \draw[black!25!,very thin](-3,-5)grid(5,7); + \draw[thick](-3,0)--(5,0) (0,-5)--(0,7); + \draw[->,red!66!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; + \draw[->,blue!25!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; + \draw[->,red!66!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; + \draw[->,red!25!](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$}; + \draw[->,red!25!](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$}; + \draw[->,blue!66!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; + \draw[->,blue](0,0)--(-71.5:4.64cm)node[right]{$\uline{U}_C$}; + \draw[->,blue](30:4.25cm)--+(108.5:4.64cm)node[below right]{$-\uline{U}_C$}; + \draw[->,blue](0,0)--(71.6:6.85cm)node[below left]{$\uline{U}_{RL}$}; \draw[->,red](0,0)--(71.6:5.27cm)node[below left]{$\uline{I}_R$}; + \draw[->,red](71.6:5.27cm)--(46:5.85cm)node[above left]{$\uline{I}_L$}; + \draw[<-,red!50!blue]+(46:5.85cm)--(18.5:5.25cm)node at(5,3){$\uline{I}_0$}; \draw[->,red!50!blue](0,0)--(46:5.85cm)node at(2,3.25){\footnotesize$\uline{I}+\uline{I}_0$}; + \end{scope} + \end{tikzpicture} +\end{minipage}\\ +\clearpage +\begin{align*} +\intertext{Leistung an Klemme:} +\uline{S}_K&=\uline{U}\cdot \uline{I}^*=4{,}2\,\volt\cdot e^{j30\,\degree}\cdot 5{,}25\milli\ampere\cdot e^{-j18{,}5\,\degree}=(\underbrace{21{,}6}_{P_K}+\underbrace{j4{,}4}_{Q_K})\,\milli\volt\ampere\\ +P_K&=\uuline{21{,}6\,\milli\watt}\\ +Q_K&=\uuline{4{,}4\,\milli\var}\\ +\intertext{Leistung der Stromquelle (Pfeilsystem wie oben):} +S_I&=\uline{U}_{RL}\cdot I^*_0=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\cdot 2{,}7\milli\ampere \cdot e^{-j110\,\degree}=(\underbrace{14{,}5}_{P_I}\underbrace{-j11{,}5}_{Q_I})\,\milli\volt\ampere\\ +P_I&=\uuline{14{,}5\,\milli\watt}\\ +Q_I&=\uuline{-11{,}5\,\milli\var}\\ +\intertext{Gesamtleistung: Summe aus Quell- und Klemmleistung} S&=S_K+S_I=(21{,}6+j4{,}4+14{,}5-j11{,}5)\,\milli\volt\ampere=36{,}1\,\milli\watt-j7{,}1\,\milli\volt\ampere +\intertext{Probe: Wirkleistung: } +P&=\frac{U^2_{RL}}{R}=\frac{(6{,}85\,\volt)^2}{1300\,\ohm}=36{,}1\,\milli\watt \quad\text{(Übereinstimmung)} +\intertext{Probe: Blindleistung: } +C:\quad Q_C&=X_C\cdot I^2=-884\,\ohm\cdot(5{,}25\,\milli\ampere)^2=-24{,}4\,\milli\var\\ +L:\quad Q_L&=\frac{U^2_{RL}}{X_L}= \frac{(6{,}85\,\volt)^2}{2714\,\ohm}=+17{,}3\,\milli\var\\ +Q&=Q_C+Q_L=(-24{,}4+17{,}3)\,\milli\var=-7{,}1\,\milli\var \quad\text{(Übereinstimmung)}\\ +\end{align*} +\clearpage +}{}% diff --git a/ET2_L_B20_A3.tex b/ET2_L_B20_A3.tex new file mode 100644 index 0000000..7972903 --- /dev/null +++ b/ET2_L_B20_A3.tex @@ -0,0 +1,185 @@ +\section {Gleichungen in Matrizenschreibweise} +Die Spannungen am gegebenen Netzwerk +sollen mit Hilfe des Knotenpotenzialverfahrens +berechnet werden. Stellen Sie die Gleichungen +in Matrizenschreibweise auf.\\ +$R=2\,\kilo\ohm;\quad C=5\,\nano\farad \quad L=2{,}5\,\milli\henry; $\\ +$f=63{,}662\,\kilo\hertz;$\\ +$\uline{I}_1=7\,\milli\ampere;\quad \uline{U}_2=3\,\volt\cdot e^{j90\,\degree} \quad \uline{I}_3= 5\milli\ampere \cdot e^{-j90\,\degree}$ +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spule | + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=2cm,rotate=90]%Spannungsquelle | + \draw (0,0)--(1,0); + \draw (.5,0)circle(.133); + \draw [->,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_{2}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=3cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=2cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=2cm]%Kondensator - + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Spule - + \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$}; + \fill (.3,-0.0667)rectangle(.7,0.0667); + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0.5cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=2.5cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=.5cm,rotate=90]%Kondensator | + \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=2.5cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); + \draw (.5,0)circle(.133); + \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$\uline{I}_3$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,.2)--(0,0)--(4,0)--(4,.7) (0,3.8)--(0,4)--(4,4)--(4,3.4)(4,1.2)--(4,2.7) +(3,0)--(3,.7)(3,1.2)--(3,2.7)(3,2)--(4,2)(3,3.2)--(3,4)(1,0)--(1,-.2)(.9,-.2)--(1.1,-.2); + \fill(0,2)circle(0.05cm)(3,2)circle(0.05cm)(4,2)circle(0.05cm)(3,0)circle(0.05cm) + (4,0)circle(0.05cm)(3,4)circle(0.05cm)(4,4)circle(0.05cm)(0,4)circle(0.05cm)(1,0)circle(0.05cm); + \draw(0,4)--+(135:.5cm)(4,0)--+(225:.5cm); + \draw[<-,red](0,4)--+(135:.25cm)node[above right]{$\uline{I}_1$}; + \draw[->,red](4,0)--+(225:.25cm)node[below right]{$\uline{I}_0$}; + \end{scope} + \draw[magenta](0,2)node[left]{$1$}; + \draw[magenta](4,2)node[right]{$3$}; + \draw[magenta](3,4)node[above]{$2$}; + \draw[magenta](3,0)node[below]{$0$}; + \end{tikzpicture} +\end{align*} +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\begin{align} +%\intertext{Formeln:} +%\end{align} +Berechnung: +\begin{align*} + \begin{tikzpicture}[scale=1.5] + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=.5cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); + \draw (.5,0)circle(.133); + \draw [->,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$\uline{I}_1$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Y}_{10}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); + \draw (.5,0)circle(.133); + \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$\uline{I}_2$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Y}_{12}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Y}_{23}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=1cm,rotate=90]%Stromquelle + \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); + \draw (.5,0)circle(.133); + \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$\uline{I}_3$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Widerstand + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [below] {$\uline{Y}_{13}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm,rotate=90] + \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$\uline{Y}_{30}$}; + \end{scope} + \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. + \draw (0,.7)--(0,0)--(4,0)--(4,.2) (0,1.3)--(0,2)--(4,2)--(4,1.8)(1,1)--(2,1) +(3,1)--(4,1); + \draw[magenta](1,1)node[left]{$1$}; + \draw[magenta](4,1)node[right]{$3$}; + \draw[magenta](2.5,2)node[above]{$2$}; + \draw[magenta](2.5,0)node[below]{$0$}; + \end{scope} + \end{tikzpicture} +\end{align*} +\begin{align*} +\left[\begin{array}{c}\uline{Y}\\ \end{array} \right]\cdot +\left[\begin{array}{c}\uline{U}\\ \end{array}\right] +&=\left[\begin{array}{c} -\sum{\uline{I}_q}\\ \end{array}\right]\\ +\end{align*} +$\uline{I}_0\stackrel{!}{=}\uline{I}_1$ wegen Gleichgewicht. Abfließende Quellströme positiv. +\clearpage +\begin{align*} +G&=\frac{1}{R}=0{,}5\,\milli\siemens\\ +\omega&=2\pi\cdot f=0{,}4\cdot \power{10}{-6}\cdot \frac{1}{\second}\\ +X_C&=\frac{-1}{\omega C}=-0{,}5\,\kilo\ohm\\ +\Rightarrow\quad B_C&=\omega C=2{,}0\,\milli\siemens\\ +X_L&=\omega L=1{,}0\,\kilo\ohm\\ +\Rightarrow\quad B_L&=\frac{-1}{\omega L}=-1{,}0\,\milli\siemens\\[\baselineskip] +\uline{Y}_{10}&=\frac{1}{R+jX_L}=\frac{1}{(2+j1)\,\kilo\ohm}&=(0{,}4-j0{,}2)\,\milli\siemens\\ +\uline{Y}_{30}&=G+jB_C&=(0{,}5+j2{,}0)\,\milli\siemens\\ +\uline{Y}_{13}&=\frac{1}{R+j(X_L+X_C)}=\frac{1}{[2+j(1-0{,}5)]\,\kilo\ohm}&=(0{,}4706-j0{,}1176)\,\milli\siemens\\ +\uline{Y}_{12}&=jB_C=2\,\milli\siemens\cdot e^{j90\,\degree}&=(0+j2{,}0)\,\milli\siemens\\ +\uline{Y}_{23}&=G&=(0{,}5+j0)\,\milli\siemens\\[\baselineskip] +\uline{I}_1&=7\,\milli\ampere \cdot e^{j0\,\degree}&=(7-j0)\,\milli\ampere\\ +\uline{I}_2&=\uline{Y}_{12}\cdot \uline{U}_2=2\,\milli\siemens\cdot e^{j90\,\degree}\cdot 3\,\volt\cdot e^{j90\,\degree}=6\,\milli\ampere \cdot e^{j180\,\degree}&=(-6+j0)\,\milli\ampere\\ +\uline{I}_3&=5\,\milli\ampere \cdot e^{-j90\,\degree}&=(0-j5)\,\milli\ampere\\ +\end{align*} +%\scriptsize +\begin{align*} +&\left( + \begin{array}{ccc} + \uline{Y}_{10}+\uline{Y}_{12}+\uline{Y}_{13} & -\uline{Y}_{12} & -\uline{Y}_{13} \\ + -\uline{Y}_{12} & \uline{Y}_{12}+\uline{Y}_{23} & -\uline{Y}_{23} \\ + -\uline{Y}_{13} & -\uline{Y}_{23} & \uline{Y}_{30}+\uline{Y}_{13}+\uline{Y}_{23} \\ + \end{array} +\right) +\left( + \begin{array}{c} + \uline{U}_{10} \\ + \uline{U}_{20} \\ + \uline{U}_{30} \\ + \end{array} +\right)= +\left( + \begin{array}{c} + \uline{I}_2\\ + \uline{I}_1-\uline{I}_2-\uline{I}_3\\ + \uline{I}_3\\ + \end{array} +\right)\\ +\end{align*} +\scriptsize +\begin{align*} +&\left( + \begin{array}{ccc} + (0{,}8706+j1{,}6824) & (0-j2) & (-0{,}4706+j0{,}1176) \\ + (0-j2) & (0{,}5+j2{,}0) & (0{,}5+j0) \\ + (-0{,}4706-j0{,}1176) & (0{,}5+j0) & (1{,}4706+j1{,}8823) \\ + \end{array} +\right)\,\milli\siemens +\left( + \begin{array}{c} + \uline{U}_{10} \\ + \uline{U}_{20} \\ + \uline{U}_{30} \\ + \end{array} +\right)= +\left( + \begin{array}{c} + (6-j0)\\ + (13+j5)\\ + (0-j5)\\ + \end{array} +\right)\,\milli\ampere +\end{align*} +\normalsize +\clearpage +}{}% diff --git a/ET2_L_Ergebnisse.tex b/ET2_L_Ergebnisse.tex new file mode 100644 index 0000000..cfddde2 --- /dev/null +++ b/ET2_L_Ergebnisse.tex @@ -0,0 +1,254 @@ +% \ifthenelse{\equal{\toPrint}{Lösung}}{% +\newcounter{acounter} +\setcounter{acounter}{1} +\newcounter{bcounter} +\newcounter{bsubcounter} +\setcounter{bcounter}{12} +\setcounter{bsubcounter}{0} +\section{Ergebnisse} +%% Blatt 12 +\textbf{\arabic{acounter} {Blitzableiter}} %\textbf{\arabic{acounter} Blitzableiter} \textbf{\arabic{acounter} {Blitzableiter}} +\stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$u=-133{,}7\,\volt$\\ +\textbf{\arabic{acounter} Drahtschleife} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$u=-36{,}39\,\milli\volt$\\ +\textbf{\arabic{acounter} Metallstab} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$|B_z|=0{,}398\,\frac{\volt\second}{\square\metre}$\\ +\textbf{\arabic{acounter} Spannungsverlauf} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$f=40\,\hertz$\\ +$\overline{|u|}=6\,\volt$\\ +$U=7{,}30\,\volt$\\ +$F=1{,}22$\\ +$P=0{,}533\,\watt$\\ +\textbf{\arabic{acounter} Phasenanschnitt} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$U=261\,\volt$\\ +\textbf{\arabic{acounter} Rechteckspannung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$u_1=1{,}5\,\volt \qquad u_2=-3{,}5\,\volt\\ +U=2{,}29\,\volt$\\ +%% Blatt 13 +\textbf{\arabic{acounter} Scheinersatzwiderstände} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter} \setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$R_r=520\,\ohm\\ +L_r=60\,\milli\henry +G_p=693\,\ohm\\ +L_p=239\,\milli\henry\\ +\varphi'_r=23{,}5\,\degree\\ +\varphi'_p=-37{,}6\,\degree$\\ +\textbf{\arabic{acounter} Verbraucherleistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$p(t)=1079\,\watt+1318\,\volt\ampere\cdot \sin(2\omega t+0{,}96)\\ +S=1318\,\volt\ampere\\ +P=1079\,\watt\\ +Q=-756\,\mathrm{var}$\\ +\textbf{\arabic{acounter} Blindleistungskompensation} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$I_N=20{,}63\,\ampere\\ +\varphi_N=32,52\degree\\ +C=153{,}6\,\micro\farad$\\ +\textbf{\arabic{acounter} Energieübertragung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$X_C=-25\,\ohm\\ +P_{VR_L}=46{,}7\,\watt\\ +P_W=583\,\watt $\\ +\textbf{\arabic{acounter} Wechselstrommotor} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$I=15{,}5\,\ampere\\ +C=80{,}6\,\micro\farad\\ +I'=12{,}1\,\ampere $\\ +\clearpage +\textbf{\arabic{acounter} Parallelschaltung von L und C} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$i_L(t_2)=2{,}92\,\ampere $\\ +\textbf{\arabic{acounter} Werte $R_L$ und $L$ einer Spule} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$I=1\,\ampere\\ +R_L=12{,}5\,\ohm\\ +L=0{,}219\,\henry $\\ +%% Blatt 14 +\textbf{\arabic{acounter} Zeigerdiagramm} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$\underline{U}_e=15\,\volt\cdot e^{+j90\degree}$\\ +\textbf{\arabic{acounter} Gesamtwiderstand} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +Induktiv\\ +\textbf{\arabic{acounter} Brückenschaltung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{U}_a=130\,\volt\cdot e^{j83\,\degree}$\\ +\textbf{\arabic{acounter} Zeigerdiagramm Netzwerk} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$R_1=13{,}89\,\kilo\ohm$\\ +\textbf{\arabic{acounter} Blind- Wirk- und Scheinleistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{Z}=84{,}85\,\ohm\cdot e^{-j45\,\degree}=(60-j60)\,\ohm\\ +\uline{I}=2{,}71\,\ampere\cdot e^{j45\,\degree}=(1{,}916+j1{,}916)\,\ampere\\ +\uline{I}_{R_C}=1{,}21\,\ampere\cdot e^{-j18{,}4\,\degree}=(1{,}150-j0{,}383)\,\ampere\\ +\uline{I}_{C}=2{,}42\,\ampere\cdot e^{+j71{,}6\,\degree}=(0{,}766+j2{,}30)\,\ampere\\ +\uline{U}_{R_C}=242\,\volt\cdot e^{-j18{,}4\,\degree}\\ +\uline{U}_{R_L}=54{,}2\,\volt\cdot e^{j45\,\degree}\\ +\uline{U}_L=54.2\,\volt\cdot e^{j135\,\degree}\\ +S=623\,\volt\ampere\\ +P=447\,\watt\\ +Q=-447\,\volt\ampere r $\\ +%% Blatt 15 +\textbf{\arabic{acounter} Komplexe Wechselstromrechnung Netzwerk Strom} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$\uline{I_2}=53{,}92\,\milli\ampere\cdot e^{j176{,}32\,\degree}$\\ +\textbf{\arabic{acounter} Übergang Zeitabhängige zu Komplexen Größen} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}_L=34{,}45\,\milli\ampere\cdot e^{j50{,}83\,\degree}\\ +i_L(T)=-25{,}79\,\milli\ampere$\\ +\textbf{\arabic{acounter} Leitwert} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$R=500\,\ohm\\ +B_C=2{,}175\,\milli\siemens$\\ +\textbf{\arabic{acounter} Strom L-R-C} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}=0{,}4058\,\ampere\cdot e^{j67{,}4\,\degree}=(0{,}156+j0{,}375)\,\ampere $\\ +\textbf{\arabic{acounter} Überlagerungsmethode} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}_C=(578{,}41+j279{,}99)\,\milli\ampere=642{,}6\,\milli\ampere\cdot e^{+j25{,}38\,\degree}$\\ +\textbf{\arabic{acounter} Momentan Leistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$u_R(T)=-114{,}1\,\milli\volt\\ +u_L(T)=-50{,}5\,\milli\volt\\ +u_C(T)=119{,}7\,\milli\volt\\ +p(T)=0{,}427\,\milli\watt$\\ +\clearpage +%% Blatt 16 +\textbf{\arabic{acounter} CLR Netzwerk} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$S=243{,}32\,\milli\volt\ampere\\ +P=141{,}50\,\milli\watt\\ +Q=-197{,}93\,\milli\,\var$\\ +\textbf{\arabic{acounter} Wirkleistung vs. Blindleistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +--\\ +\textbf{\arabic{acounter} Wirkleistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$R_v=17{,}24\,\ohm\\ +C=19{,}14\,\micro\farad\\ +P_{v\text{,}max}=14{,}5\,\milli\watt$ +\textbf{\arabic{acounter} Abgebbare Wirkleistung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$F_ \% =-78{,}8\, \%$\\ +\textbf{\arabic{acounter} Wirkleistung Spannungsquelle} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$P=-16{,}31\,\milli\watt$\\ +\textbf{\arabic{acounter} Dualitätskonstante} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$L_2=100\,\milli\henry\\ +C_2=5\,\micro\farad$ +\textbf{\arabic{acounter} Dualitätskonstante verlustbehaftete Bauelemente} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$R_G=1\,\milli\henry\\ +G_R=2\,\milli\siemens\\ +C_L=100\,\nano\farad$\\ +\textbf{\arabic{acounter} Vierpol Y-Parameter} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{Y}_L=\frac{1}{j\omega\cdot L}=-j\frac{1}{\omega\cdot L}\\ +\uline{Y}_{11}=\frac{1}{3}\cdot \uline{Y}_L\\ +\uline{Y}_{12}=-\frac{1}{6}\cdot \uline{Y}_L\\ +\uline{Y}_{22}=\uline{Y}_{11}=\frac{1}{3}\cdot \uline{Y}_L=\\ +\uline{Z}_{11}=\uline{Z}_{24}= 4\cdot j\omega\cdot L\\ +\uline{Z}_{12}=\uline{Z}_{21}= 2\cdot j\omega\cdot L$\\ +\textbf{\arabic{acounter} Spannung Vierpol} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{U}_0=130{,}4\,\milli\volt$\\ +%% Blatt 17 +\textbf{\arabic{acounter} Stromortskurve} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$\uline{I}=236\,\milli\ampere\cdot e^{(j35{,}2\,\degree)}=(193+j136)\,\milli\ampere$\\ +\textbf{\arabic{acounter} Leitwerts-, Widerstandsortskurve} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +-- +\textbf{\arabic{acounter} Ortskurve} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$f_g=828{,}9\,\hertz\\ +a=-4{,}9\,\deci\bel$\\ +\textbf{\arabic{acounter} Stromortskurve} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$I_{max}=5{,}55\,\milli\ampere $\\ +\textbf{\arabic{acounter} Widerstandstransformation} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$X_L{_p}=12{,}2\,\kilo\ohm\\ +X_{C_s}=-5{,}6\,\kilo\ohm\\ +{X_{L_s}}=+2{,}6\,\kilo\ohm\\ +X_{C_p}=-4{,}2\,\kilo\ohm $\\ +\clearpage +\textbf{\arabic{acounter} Brückenschaltung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{U}_{ab}=+j100\,\volt=100\,\volt\cdot e^{+j90\degree} $ +\textbf{\arabic{acounter} Wechselstrombrücke} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$ \uline{Z}_2=(1+j0{,}5)\,\kilo\ohm\\ +R_2=1{,}25\,\kilo\ohm\\ +L_2=2{,}5\,\milli\henry$\\ +\textbf{\arabic{acounter} Wechselstrombrücke} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$X=+\frac{R^2}{X_L}$\\ +%% Blatt 18 +\textbf{\arabic{acounter} Übertrager im Leerlauf} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): }\\ +$Z_1=23\,\ohm\cdot e^{j77{,}44\,\degree}\\ +P=500\,\watt\\ +Q=2245\,var\\ +\omega L_1=22{,}45\,\ohm\\ +\omega M=10\,\ohm\\ +\omega L_2=4{,}45\,\ohm$\\ +\textbf{\arabic{acounter} Übertrager mit kapazitiver Last} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{U}_2=0{,}398\,\volt\cdot e^{-j174{,}3\,\degree}\\ +\uline{U}_2=0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}\\ +P_2=178\,\micro\watt $\\ +\textbf{\arabic{acounter} Übertrager mit Verbindung zum Eingang} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{U}_{ab}=200\,\volt\cdot e^{j73{,}74\,\degree}\\ +\uline{U}_S=317\,\volt\cdot e^{j79{,}8\,\degree}\\ +\uline{I}=0{,}689\,\ampere\cdot e^{-j8{,}43\,\degree}\\ +\uline{U}_{ab}=159\,\volt\cdot e^{+j71{,}7\,\degree}\\ +P=12{,}2\,\watt$\\ +\textbf{\arabic{acounter} Impedanzmatrix} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{Z}_{11}=(730+j100)\,\ohm\\ +\uline{Z}_{12}=\uline{Z}_{21}=1200\,\ohm\\ +\uline{Z}_{22}=(2000-j200)\,\ohm $\\ +\textbf{\arabic{acounter} Netztransformator} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$U_2=105{,}1\,\volt\\ +I_1=118\,\milli\ampere\\ +\widehat{B}=1{,}095\,\tesla$\\ +\textbf{\arabic{acounter} 3-Phasen Spannungssystem} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}_N=10{,}6\,\ampere\cdot e^{j14{,}7\,\degree} $\\ +\textbf{\arabic{acounter} 3-Phasen System mit unsymmetrischem Verbraucher} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$P=-335{,}44\,\watt $\\ +\textbf{\arabic{acounter} Strangströme 3-Phasen System mit unsymmetrischem +Verbraucher} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}_C=4\,\ampere\cdot e^{j120\,\degree}=(-2+j3{,}46)\,\ampere\\ +\uline{I}_{RL}=4\,\ampere\cdot e^{j113{,}1\,\degree}=(-1{,}57+j3{,}68)\,\ampere\\ +\uline{I}_1=0{,}48\,\ampere\cdot e^{-j153\,\degree}\\ +\uline{I}_2=3{,}66\,\ampere\cdot e^{j109{,}1\,\degree}\\ +\uline{I}_3=4{,}03\,\ampere\cdot e^{j66{,}1\,\degree}\\ +P=1280\,\watt $ +\clearpage +%% Blatt 19 +\textbf{\arabic{acounter} Resonanzfrequenz Zweipol} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): } +$f_{res}=899\,\hertz\\ +Z=37{,}5\,\ohm $\\ +\textbf{\arabic{acounter} RLC-Reihenschwingkreis} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$R=10\,\ohm\\ +C=3{,}47\,\micro\farad\\ +L=35{,}4\,\milli\henry$\\ +\textbf{\arabic{acounter} Effektivwert und Klirrfaktor} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$I_{\mu}=7{,}36\,\milli\ampere\\ +k_{\mu}=27{,}7\%\\ +I_0=7{,}88\,\milli\ampere\\ +k_0=25{,}8\%$\\ +\textbf{\arabic{acounter} Klirrfaktor} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$k_a=21{,}5\%\\ +U_a=2{,}49\,\volt$\\ +\textbf{\arabic{acounter} Momentanspannung} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$u_C(t=T=2\,\micro\second)=1{,}315\,\volt $\\ +\textbf{\arabic{acounter} Nichtlinears Bauelement} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$k_i=19{,}61\%$\\ +\textbf{\arabic{acounter} Wirkleistung Zweipol} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$P=573\,\watt$\\ + +%% Blatt 20 +\textbf{\arabic{acounter} Ringspule} \stepcounter{bcounter}\stepcounter{acounter}\textbf{(B\arabic{bcounter}\setcounter{bsubcounter}{1}A\arabic{bsubcounter}): } +$U=37{,}73\,\milli\volt$\\ +\textbf{\arabic{acounter} Netzwerk Wirk- und Blindanteil} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +$\uline{I}=5{,}25\milli\ampere\cdot e^{j18{,}5\,\degree}=(4{,}98+j1{,}67)\milli\ampere\\ +\uline{I}_{\textrm{w}}=5{,}14\,\milli\ampere\cdot e^{j30\,\degree}\\ +\uline{I}_b=1{,}05\,\milli\ampere\cdot e^{-j60\,\degree}\\ +P_K=21{,}6\,\milli\watt\\ +Q_K=4{,}4\,\milli\var\\ +P_I=14{,}5\,\milli\watt\\ +Q_I=-11{,}5\,\milli\var $\\ +\textbf{\arabic{acounter} Gleichungen in Matrizenschreibweise} \stepcounter{acounter}\textbf{(B\arabic{bcounter}\stepcounter{bsubcounter}A\arabic{bsubcounter}): }\\ +\scriptsize +\begin{align*} +&\left( + \begin{array}{ccc} + (0{,}8706+j1{,}6824) & (0-j2) & (-0{,}4706+j0{,}1176) \\ + (0-j2) & (0{,}5+j2{,}0) & (0{,}5+j0) \\ + (-0{,}4706-j0{,}1176) & (0{,}5+j0) & (1{,}4706+j1{,}8823) \\ + \end{array} +\right)\,\milli\siemens +\left( + \begin{array}{c} + \uline{U}_{10} \\ + \uline{U}_{20} \\ + \uline{U}_{30} \\ + \end{array} +\right)= +\left( + \begin{array}{c} + (6-j0)\\ + (13+j5)\\ + (0-j5)\\ + \end{array} +\right)\,\milli\ampere +\end{align*}\\ +\clearpage +% }{}% + diff --git a/ET2_Style.tex b/ET2_Style.tex new file mode 100644 index 0000000..17fe725 --- /dev/null +++ b/ET2_Style.tex @@ -0,0 +1,98 @@ +\documentclass[twoside,a4paper]{scrartcl} +\usepackage[T1]{fontenc} +\usepackage[ansinew]{inputenc} +\usepackage[ngerman]{babel} +%\usepackage{textcomp} % dazugeladen wegen Spezial Zeichen +\usepackage{pifont} % dazugeladen wegen Spezial Zeichen +%\usepackage{ifsym} % dazugeladen wegen Spezial Zeichen +%\usepackage{dingbat} % dazugeladen wegen Spezial Zeichen +%\usepackage{mathabx} % nicht vorhanden +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage[amssymb,thickspace, thickqspace]{SIunits}% see SIunits manual.. Load SIunits package after amssymb package. +\usepackage{graphicx} +\usepackage{xcolor} +\usepackage{xspace} +\usepackage{scrpage2} + +\usepackage{tikz} +\usetikzlibrary{decorations.pathmorphing,decorations.pathreplacing,decorations.shapes} +\usepackage{xkeyval} %see tikzpgf manual page 54 +%\usetikzlibrary{patterns} +\usepackage{pgf} +%\usepackage{pgfbaseplot} +%\usepackage{pgfmath} +\usepackage{pgfplots}% neu Feuersänger +\usetikzlibrary {arrows} +\SIunits[thickspace] % , thickqspace (geht nicht) +\usepackage{sistyle} %% wegen amssymb gestrichen !? +\usepackage{slashed} +\usepackage{cancel} +\usepackage {ulem} +\numberwithin{equation}{section} +\usepackage{ifthen} +% \newcommand\toPrint{xLösung}% print Lösung (Lösung) ein (xLösung) druckt nur die Aufgaben +\newcommand{\var}{\mathrm{var}} +\usepackage{pdfpages} +\usepackage{hyperref} +\hypersetup{colorlinks=true, + pdfpagemode=none, + pdftitle={ET2 Übungen}, + pdfsubject={ET2 Übungen}, + pdfauthor={Prof.\,Dr.\,C.\,Niebler}, + pdfkeywords={Elektrotechnik Uebung}, + pdffitwindow=true, + pdfstartview=FitH + } +%%\setkomafont{pagehead}{% +%%\footnotesize\sffamily +%%} +%% +%%\pagestyle{scrheadings} +%%\lehead{\pagemark} +%%\cehead{Georg-Simon-Ohm Hochschule Nürnberg} +%%\rehead{} +%%\lohead{} +%%\cohead{Georg-Simon-Ohm Hochschule Nürnberg} +%%\rohead{\pagemark} +%%\lefoot{} +%%\cefoot{C. Niebler -- 2008} +%%\refoot{} +%%\lofoot{} +%%\cofoot{C. Niebler -- 2008} +%%\rofoot{} +%%%\cfoot{\pagemark} + +%\usepackage{geometrie} +%{geometry:a4paper} +%{geometry:left=5cm} +%{geometry:right=3cm} + +\usepackage{scrtime} +%\newcommand{\footlinetext}{\footnotesize \textsf{\color{gray} Niebler, C.: +% Grundlagen Elektrotechnik -- Übung 2, GSO-Hochschule, SS\,2008;\hfill\today\ \thistime \normalsize}} +% \newcommand{\footlinetext}{\footnotesize \textsf{\color{gray} +% Technische Hochschule Nürnberg Georg Simon Ohm\hfill Übung Grundlagen Elektrotechnik 2\hspace{3em} SS\,2013 \\ +% Dr. Christine Niebler \hfill\today \normalsize}} +\pagestyle{scrheadings} +\ohead{\input{ohmlogoTH}} %% mit TIKZ +\chead{} +\ihead{\footnotesize{Fachbereich Elektro-\\ +Feinwerk- und Informationstechnik}} +\cfoot{\tiny{Übung Grundlagen Elektrotechnik 2 (\myVersion)}} +\ifoot{\tiny{Prof.\,Dr.\,C.\,Niebler}} +\ofoot{\pagemark} +\setlength{\parindent}{0pt} +\setlength{\parsep}{\baselineskip} + +\usepackage[titles]{tocloft} +\setlength{\cftbeforesecskip}{0.5ex} +% \ohead{\headmark} +% \chead{} +% \ihead{\pagemark} +% \ifoot{\footlinetext} +% \cfoot{} +% \ofoot{} +% \setlength{\parindent}{0pt} +% \setlength{\parsep}{\baselineskip} diff --git a/ET2_Versionierung.tex b/ET2_Versionierung.tex new file mode 100644 index 0000000..5fe4fa1 --- /dev/null +++ b/ET2_Versionierung.tex @@ -0,0 +1,13 @@ +\thispagestyle{empty} +{\Large Vorwort:} +\paragraph{1. Ausgabe --- SoSe 2008} +Ein wichtiges Element beim Studium ist es, sich die Grundlagen zu verinnerlichen, um diese dann in der Praxis fachgerecht anwenden zu können. + +Diese Unterlage wurde aus einer handschriftlichen Loseblattsammlung entwickelt. Die Lösungen wurden in logische Schritte eingeteilt, damit es den Studenten das Verständnis erleichtert. Wichtig ist es auch um die Ãœbungen effizient vorbereiten zu können. +% Die Aufgaben wurden mit SPICE simuliert und die Lösungen verifiziert.\\ + +\paragraph{2. Ausgabe --- SoSe 2013} +Die Erfahrungen aus der Vorlesung/Ãœbung wurde in der Ausgabe eingebracht. + +\paragraph{3. Ausgabe --- SoSe 2014} +Anpassung der Ãœberschriftennummern an Blatt und Aufgabennummer, zur leichteren Gruppierung in Themenfelder. \ No newline at end of file diff --git a/ET2_Wasser.tex b/ET2_Wasser.tex new file mode 100644 index 0000000..786db02 --- /dev/null +++ b/ET2_Wasser.tex @@ -0,0 +1,13 @@ +\usepackage{graphicx,type1cm,eso-pic,color} + +\makeatletter + \AddToShipoutPicture{ + \setlength{\@tempdimb}{.75\paperwidth} + \setlength{\@tempdimc}{.25\paperheight} + \setlength{\unitlength}{1pt} + \put(\strip@pt\@tempdimb,\strip@pt\@tempdimc){ + \makebox(0,0){\rotatebox{55}{\textcolor[gray]{0.9} + {\fontsize{3cm}{3cm}\selectfont{\ProfName}}}} + } + } +\makeatother \ No newline at end of file diff --git a/Grafik/b13a4.PNG b/Grafik/b13a4.PNG new file mode 100644 index 0000000..4e5cdaf Binary files /dev/null and b/Grafik/b13a4.PNG differ diff --git a/Grafik/b16a8.PNG b/Grafik/b16a8.PNG new file mode 100644 index 0000000..d4f397a Binary files /dev/null and b/Grafik/b16a8.PNG differ diff --git a/Grafik/b16a8a.PNG b/Grafik/b16a8a.PNG new file mode 100644 index 0000000..800819b Binary files /dev/null and b/Grafik/b16a8a.PNG differ diff --git a/Leerblatt.tex b/Leerblatt.tex new file mode 100644 index 0000000..b4924d3 --- /dev/null +++ b/Leerblatt.tex @@ -0,0 +1 @@ +\cleardoublepage \ No newline at end of file diff --git a/Main_ET2_V2.aux b/Main_ET2_V2.aux new file mode 100644 index 0000000..6707d11 --- /dev/null +++ b/Main_ET2_V2.aux @@ -0,0 +1,90 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\catcode`"\active +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\select@language{ngerman} +\@writefile{toc}{\select@language{ngerman}} +\@writefile{lof}{\select@language{ngerman}} +\@writefile{lot}{\select@language{ngerman}} +\tocstyle@set@width {unum}{toc}{}{32.72185pt} +\tocstyle@set@width {num}{toc}{1}{32.72185pt} +\tocstyle@set@width {skip}{toc}{1}{0.0pt} +\@writefile{toc}{\contentsline {section}{\numberline {12/1{}}Blitzableiter}{3}{section.1}} +\@writefile{toc}{\contentsline {section}{\numberline {12/2{}}Drahtschleife}{5}{section.2}} +\@writefile{toc}{\contentsline {section}{\numberline {12/3{}}Metallstab}{7}{section.3}} +\@writefile{toc}{\contentsline {section}{\numberline {12/4{}}Spannungsverlauf}{11}{section.4}} +\@writefile{toc}{\contentsline {section}{\numberline {12/5{}}Phasenanschnitt}{13}{section.5}} +\@writefile{toc}{\contentsline {section}{\numberline {12/6{}}Rechteckspannung}{15}{section.6}} +\@writefile{toc}{\contentsline {section}{\numberline {13/1{}}Scheinersatzwiderst\"ande}{17}{section.7}} +\@writefile{toc}{\contentsline {section}{\numberline {13/2{}}Verbraucherleistung}{19}{section.8}} +\@writefile{toc}{\contentsline {section}{\numberline {13/3{}}Blindleistungskompensation}{21}{section.9}} +\@writefile{toc}{\contentsline {section}{\numberline {13/4{}}Energie\"ubertragung}{23}{section.10}} +\@writefile{toc}{\contentsline {section}{\numberline {13/5{}}Wechselstrommotor}{25}{section.11}} +\@writefile{toc}{\contentsline {section}{\numberline {13/6{}}Parallelschaltung von L und C}{27}{section.12}} +\newlabel{eq:paralleschaltung-ic}{{{1}}{27}{Parallelschaltung von L und C}{AMS.4}{}} +\newlabel{eq:paralleschaltung-ul}{{{2}}{27}{Parallelschaltung von L und C}{AMS.5}{}} +\@writefile{toc}{\contentsline {section}{\numberline {13/7{}}Werte $R_L$ und $L$ einer Spule}{29}{section.13}} +\@writefile{toc}{\contentsline {section}{\numberline {14/1{}}Zeigerdiagramm}{31}{section.14}} +\@writefile{toc}{\contentsline {section}{\numberline {14/2{}}Gesamtwiderstand}{33}{section.15}} +\@writefile{toc}{\contentsline {section}{\numberline {14/3{}}Br\"uckenschaltung}{35}{section.16}} +\@writefile{toc}{\contentsline {section}{\numberline {14/4{}}Zeigerdiagramm Netzwerk}{37}{section.17}} +\@writefile{toc}{\contentsline {section}{\numberline {14/5{}}Blind- Wirk- und Scheinleistung}{39}{section.18}} +\@writefile{toc}{\contentsline {section}{\numberline {15/1{}}Komplexe Wechselstromrechnung Netzwerk Strom}{43}{section.19}} +\@writefile{toc}{\contentsline {section}{\numberline {15/2{}}\"Ubergang Zeitabh\"angige zu Komplexen Gr\"o\IeC {\ss }en}{45}{section.20}} +\@writefile{toc}{\contentsline {section}{\numberline {15/3{}}Leitwert}{48}{section.21}} +\@writefile{toc}{\contentsline {section}{\numberline {15/4{}}Strom L-R-C}{49}{section.22}} +\@writefile{toc}{\contentsline {section}{\numberline {15/5{}}\"Uberlagerungsmethode}{51}{section.23}} +\@writefile{toc}{\contentsline {section}{\numberline {15/6{}}Momentan Leistung}{53}{section.24}} +\@writefile{toc}{\contentsline {section}{\numberline {16/1{}}CLR Netzwerk}{55}{section.25}} +\@writefile{toc}{\contentsline {section}{\numberline {16/1{}}Wirkleistung vs. Blindleistung}{57}{section.26}} +\@writefile{toc}{\contentsline {section}{\numberline {16/2{}}Wirkleistung}{58}{section.27}} +\@writefile{toc}{\contentsline {section}{\numberline {16/3{}}Abgebbare Wirkleistung}{60}{section.28}} +\@writefile{toc}{\contentsline {section}{\numberline {16/4{}}Wirkleistung Spannungsquelle}{62}{section.29}} +\@writefile{toc}{\contentsline {section}{\numberline {16/5{}}Dualit\"atskonstante}{64}{section.30}} +\@writefile{toc}{\contentsline {section}{\numberline {16/6{}}Dualit\"atskonstante verlustbehaftete Bauelemente}{65}{section.31}} +\@writefile{toc}{\contentsline {section}{\numberline {16/7{}}Vierpol Y-Parameter}{67}{section.32}} +\@writefile{toc}{\contentsline {section}{\numberline {16/8{}}Spannung Vierpol}{69}{section.33}} +\@writefile{toc}{\contentsline {section}{\numberline {17/1{}}Stromortskurve}{73}{section.34}} +\@writefile{toc}{\contentsline {section}{\numberline {17/2{}}Leitwerts-, Widerstandsortskurve}{77}{section.35}} +\@writefile{toc}{\contentsline {section}{\numberline {17/3{}}Ortskurve}{79}{section.36}} +\@writefile{toc}{\contentsline {section}{\numberline {17/4{}}Stromortskurve}{81}{section.37}} +\@writefile{toc}{\contentsline {section}{\numberline {17/5{}}Widerstandstransformation}{83}{section.38}} +\@writefile{toc}{\contentsline {section}{\numberline {17/6{}}Br\"uckenschaltung}{85}{section.39}} +\@writefile{toc}{\contentsline {section}{\numberline {17/7{}}Wechselstrombr\"ucke}{87}{section.40}} +\newlabel{eq:rp177}{{17/7{}.1}{87}{Wechselstrombrücke}{equation.40.1}{}} +\newlabel{eq:xp177}{{17/7{}.2}{87}{Wechselstrombrücke}{equation.40.2}{}} +\@writefile{toc}{\contentsline {section}{\numberline {17/8{}}Wechselstrombr\"ucke}{89}{section.41}} +\@writefile{toc}{\contentsline {section}{\numberline {18/1{}}\"Ubertrager im Leerlauf}{91}{section.42}} +\@writefile{toc}{\contentsline {section}{\numberline {18/2{}}\"Ubertrager mit kapazitiver Last}{93}{section.43}} +\@writefile{toc}{\contentsline {section}{\numberline {18/3{}}\"Ubertrager mit Verbindung zum Eingang}{97}{section.44}} +\@writefile{toc}{\contentsline {section}{\numberline {18/4{}}Impedanzmatrix}{99}{section.45}} +\@writefile{toc}{\contentsline {section}{\numberline {18/5{}}Netztransformator}{101}{section.46}} +\@writefile{toc}{\contentsline {section}{\numberline {18/6{}}3-Phasen Spannungssystem}{103}{section.47}} +\@writefile{toc}{\contentsline {section}{\numberline {18/7{}}3-Phasen System mit unsymmetrischem Verbraucher}{105}{section.48}} +\@writefile{toc}{\contentsline {section}{\numberline {18/8{}}Strangstr\"ome 3-Phasen System mit unsymmetrischem Verbraucher}{107}{section.49}} +\@writefile{toc}{\contentsline {section}{\numberline {19/1{}}Resonanzfrequenz Zweipol}{109}{section.50}} +\@writefile{toc}{\contentsline {section}{\numberline {19/2{}}RLC-Reihenschwingkreis}{111}{section.51}} +\@writefile{toc}{\contentsline {section}{\numberline {19/3{}}Effektivwert und Klirrfaktor}{113}{section.52}} +\@writefile{toc}{\contentsline {section}{\numberline {19/4{}}Klirrfaktor}{115}{section.53}} +\@writefile{toc}{\contentsline {section}{\numberline {19/5{}}Momentanspannung}{117}{section.54}} +\@writefile{toc}{\contentsline {section}{\numberline {19/6{}}Nichtlinears Bauelement}{119}{section.55}} +\@writefile{toc}{\contentsline {section}{\numberline {19/7{}}Wirkleistung Zweipol}{120}{section.56}} +\@writefile{toc}{\contentsline {section}{\numberline {20/1{}}Ringspule}{122}{section.57}} +\@writefile{toc}{\contentsline {section}{\numberline {20/2{}}Netzwerk Wirk- und Blindanteil}{124}{section.58}} +\@writefile{toc}{\contentsline {section}{\numberline {20/3{}}Gleichungen in Matrizenschreibweise}{127}{section.59}} +\@writefile{toc}{\contentsline {section}{\numberline {}Ergebnisse}{129}{section.60}} diff --git a/Main_ET2_V2.i.gnuplot b/Main_ET2_V2.i.gnuplot new file mode 100644 index 0000000..f43a3cb --- /dev/null +++ b/Main_ET2_V2.i.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.i.table"; set format "%.5f" +set samples 100; plot [x=0:5] .85*cos(.5*3.14*x-1.57) diff --git a/Main_ET2_V2.iFe.gnuplot b/Main_ET2_V2.iFe.gnuplot new file mode 100644 index 0000000..902cea6 --- /dev/null +++ b/Main_ET2_V2.iFe.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.iFe.table"; set format "%.5f" +set samples 100; plot [x=0:5] -.4*sin(.5*3.14*x) diff --git a/Main_ET2_V2.iomega.gnuplot b/Main_ET2_V2.iomega.gnuplot new file mode 100644 index 0000000..001cb1f --- /dev/null +++ b/Main_ET2_V2.iomega.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.iomega.table"; set format "%.5f" +set samples 100; plot [x=0:5] 1*cos(.5*3.14*x)+.288*cos(1.5*3.14*x)-.4*sin(.5*3.14*x) diff --git a/Main_ET2_V2.log b/Main_ET2_V2.log new file mode 100644 index 0000000..b31ca71 --- /dev/null +++ b/Main_ET2_V2.log @@ -0,0 +1,3395 @@ +This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9 64-bit) (preloaded format=pdflatex 2014.2.26) 17 MAR 2014 18:24 +entering extended mode +**C:/Users/Niebler/Documents/Vorlesungen/ETechnik/TeX/UeET2_20130923/Main_ET2_V +2.tex + +(C:/Users/Niebler/Documents/Vorlesungen/ETechnik/TeX/UeET2_20130923/Main_ET2_V2 +.tex +LaTeX2e <2011/06/27> +Babel and hyphenation patterns for english, afrikaans, ancientgreek, ar +abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic, +croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga +lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland +ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian, + lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n +german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman +ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm +an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl +ishmax, welsh, loaded. + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_Style.t +ex ("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrartcl.cls" +Document Class: scrartcl 2013/12/19 v3.12 KOMA-Script document class (article) +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrkbase.sty" +Package: scrkbase 2013/12/19 v3.12 KOMA-Script package (KOMA-Script-dependent b +asics and keyval usage) + +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrbase.sty" +Package: scrbase 2013/12/19 v3.12 KOMA-Script package (KOMA-Script-independent +basics and keyval usage) + +("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\keyval.sty" +Package: keyval 1999/03/16 v1.13 key=value parser (DPC) +\KV@toks@=\toks14 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrlfile.sty" +Package: scrlfile 2013/12/19 v3.12 KOMA-Script package (loading files) + +Package scrlfile, 2013/12/19 v3.12 KOMA-Script package (loading files) + Copyright (C) Markus Kohm + +))) ("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\tocbasic.sty" +Package: tocbasic 2013/12/19 v3.12 KOMA-Script package (handling toc-files) +) +Package tocbasic Info: omitting babel extension for `toc' +(tocbasic) because of feature `nobabel' available +(tocbasic) for `toc' on input line 115. +Package tocbasic Info: omitting babel extension for `lof' +(tocbasic) because of feature `nobabel' available +(tocbasic) for `lof' on input line 116. +Package tocbasic Info: omitting babel extension for `lot' +(tocbasic) because of feature `nobabel' available +(tocbasic) for `lot' on input line 117. +Package scrbase Info: Unknown processing state. +(scrbase) Processing option `twoside' +(scrbase) of member `.scrartcl.cls' of family +(scrbase) `KOMA' doesn't set +(scrbase) a valid state. This will be interpreted +(scrbase) as \FamilyKeyStateProcessed on input line 1150. +Class scrartcl Info: File `scrsize11pt.clo' used instead of +(scrartcl) file `scrsize11.clo' to setup font sizes on input line 153 +1. + +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrsize11pt.clo" +File: scrsize11pt.clo 2013/12/19 v3.12 KOMA-Script font size class option (11pt +) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\typearea.sty" +Package: typearea 2013/12/19 v3.12 KOMA-Script package (type area) + +Package typearea, 2013/12/19 v3.12 KOMA-Script package (type area) + Copyright (C) Frank Neukam, 1992-1994 + Copyright (C) Markus Kohm, 1994- + +\ta@bcor=\skip41 +\ta@div=\count79 +Package typearea Info: You've used standard option `a4paper'. +(typearea) This is correct! +(typearea) Internally I'm using `paper=a4'. +(typearea) If you'd like to set the option with \KOMAoptions, +(typearea) you'd have to use `paper=a4' there +(typearea) instead of `a4paper', too. +\ta@hblk=\skip42 +\ta@vblk=\skip43 +\ta@temp=\skip44 +\footheight=\skip45 +Package typearea Info: These are the values describing the layout: +(typearea) DIV = 10 +(typearea) BCOR = 0.0pt +(typearea) \paperwidth = 597.50793pt +(typearea) \textwidth = 418.25555pt +(typearea) DIV departure = -6% +(typearea) \evensidemargin = 47.2316pt +(typearea) \oddsidemargin = -12.5192pt +(typearea) \paperheight = 845.04694pt +(typearea) \textheight = 595.80026pt +(typearea) \topmargin = -25.16531pt +(typearea) \headheight = 17.0pt +(typearea) \headsep = 20.40001pt +(typearea) \topskip = 11.0pt +(typearea) \footskip = 47.6pt +(typearea) \baselineskip = 13.6pt +(typearea) on input line 1330. +) +\c@part=\count80 +\c@section=\count81 +\c@subsection=\count82 +\c@subsubsection=\count83 +\c@paragraph=\count84 +\c@subparagraph=\count85 +\abovecaptionskip=\skip46 +\belowcaptionskip=\skip47 +\c@pti@nb@sid@b@x=\box26 +\c@figure=\count86 +\c@table=\count87 +\bibindent=\dimen102 +) ("C:\Program Files\MiKTeX 2.9\tex\latex\base\fontenc.sty" +Package: fontenc 2005/09/27 v1.99g Standard LaTeX package + +("C:\Program Files\MiKTeX 2.9\tex\latex\base\t1enc.def" +File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file +LaTeX Font Info: Redeclaring font encoding T1 on input line 43. +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\base\inputenc.sty" +Package: inputenc 2008/03/30 v1.1d Input encoding file +\inpenc@prehook=\toks15 +\inpenc@posthook=\toks16 + +("C:\Program Files\MiKTeX 2.9\tex\latex\base\ansinew.def" +File: ansinew.def 2008/03/30 v1.1d Input encoding file +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\babel\babel.sty" +Package: babel 2008/07/08 v3.8m The Babel package + +************************************* +* Local config file bblopts.cfg used +* +("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\bblopts.cfg" +File: bblopts.cfg 2006/07/31 v1.0 MiKTeX 'babel' configuration +) +("C:\Program Files\MiKTeX 2.9\tex\generic\babel\ngermanb.ldf" +Language: ngermanb 2008/07/06 v2.6n new German support from the babel system + +("C:\Program Files\MiKTeX 2.9\tex\generic\babel\babel.def" +File: babel.def 2008/07/08 v3.8m Babel common definitions +\babel@savecnt=\count88 +\U@D=\dimen103 +) +\l@naustrian = a dialect from \language\l@ngerman +Package babel Info: Making " an active character on input line 92. +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\pifont.sty" +Package: pifont 2005/04/12 PSNFSS-v9.2a Pi font support (SPQR) +LaTeX Font Info: Try loading font information for U+pzd on input line 63. + +("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\upzd.fd" +File: upzd.fd 2001/06/04 font definitions for U/pzd. +) +LaTeX Font Info: Try loading font information for U+psy on input line 64. + +("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\upsy.fd" +File: upsy.fd 2001/06/04 font definitions for U/psy. +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\amsfonts\amsfonts.sty" +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\@emptytoks=\toks17 +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +) +("C:\Program Files\MiKTeX 2.9\tex\latex\amsfonts\amssymb.sty" +Package: amssymb 2013/01/14 v3.01 AMS font symbols +) +("C:\Program Files\MiKTeX 2.9\tex\latex\amsmath\amsmath.sty" +Package: amsmath 2013/01/14 v2.14 AMS math features +\@mathmargin=\skip48 + +For additional information on amsmath, use the `?' option. +("C:\Program Files\MiKTeX 2.9\tex\latex\amsmath\amstext.sty" +Package: amstext 2000/06/29 v2.01 + +("C:\Program Files\MiKTeX 2.9\tex\latex\amsmath\amsgen.sty" +File: amsgen.sty 1999/11/30 v2.0 +\@emptytoks=\toks18 +\ex@=\dimen104 +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\amsmath\amsbsy.sty" +Package: amsbsy 1999/11/29 v1.2d +\pmbraise@=\dimen105 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\amsmath\amsopn.sty" +Package: amsopn 1999/12/14 v2.01 operator names +) +\inf@bad=\count89 +LaTeX Info: Redefining \frac on input line 210. +\uproot@=\count90 +\leftroot@=\count91 +LaTeX Info: Redefining \overline on input line 306. +\classnum@=\count92 +\DOTSCASE@=\count93 +LaTeX Info: Redefining \ldots on input line 378. +LaTeX Info: Redefining \dots on input line 381. +LaTeX Info: Redefining \cdots on input line 466. +\Mathstrutbox@=\box27 +\strutbox@=\box28 +\big@size=\dimen106 +LaTeX Font Info: Redeclaring font encoding OML on input line 566. +LaTeX Font Info: Redeclaring font encoding OMS on input line 567. +\macc@depth=\count94 +\c@MaxMatrixCols=\count95 +\dotsspace@=\muskip10 +\c@parentequation=\count96 +\dspbrk@lvl=\count97 +\tag@help=\toks19 +\row@=\count98 +\column@=\count99 +\maxfields@=\count100 +\andhelp@=\toks20 +\eqnshift@=\dimen107 +\alignsep@=\dimen108 +\tagshift@=\dimen109 +\tagwidth@=\dimen110 +\totwidth@=\dimen111 +\lineht@=\dimen112 +\@envbody=\toks21 +\multlinegap=\skip49 +\multlinetaggap=\skip50 +\mathdisplay@stack=\toks22 +LaTeX Info: Redefining \[ on input line 2665. +LaTeX Info: Redefining \] on input line 2666. +) +("C:\Program Files\MiKTeX 2.9\tex\latex\siunits\SIunits.sty" +Package: SIunits 2007/12/02 v1.36 Support for the International System of units + (MH) + + +\@qskwidth=\skip51 +\symgreek=\mathgroup6 +("C:\Program Files\MiKTeX 2.9\tex\latex\siunits\SIunits.cfg") +LaTeX Info: Redefining \usk on input line 576. + +Option `thickspace' provided! +LaTeX Info: Redefining \@qsk on input line 576. +Option `thickqspace' provided! +Option `amssymb' provided! +) ("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\graphicx.sty" +Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) + +("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\graphics.sty" +Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR) + +("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\trig.sty" +Package: trig 1999/03/16 v1.09 sin cos tan (DPC) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\graphics.cfg" +File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive +) +Package graphics Info: Driver file: pdftex.def on input line 91. + +("C:\Program Files\MiKTeX 2.9\tex\latex\pdftex-def\pdftex.def" +File: pdftex.def 2011/05/27 v0.06d Graphics/color for pdfTeX + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\infwarerr.sty" +Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\ltxcmds.sty" +Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO) +) +\Gread@gobject=\count101 +)) +\Gin@req@height=\dimen113 +\Gin@req@width=\dimen114 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\xcolor\xcolor.sty" +Package: xcolor 2007/01/21 v2.11 LaTeX color extensions (UK) + +("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\color.cfg" +File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive +) +Package xcolor Info: Driver file: pdftex.def on input line 225. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1337. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1341. +Package xcolor Info: Model `RGB' extended on input line 1353. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1355. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1356. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1357. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1358. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1359. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1360. +) +("C:\Program Files\MiKTeX 2.9\tex\latex\tools\xspace.sty" +Package: xspace 2009/10/20 v1.13 Space after command names (DPC,MH) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrpage2.sty" +Package: scrpage2 2013/12/19 v3.12 KOMA-Script package (page head and foot) +LaTeX Info: Redefining \pagemark on input line 174. +) +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\frontendlayer\tikz.sty" +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\basiclayer\pgf.sty" +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\utilities\pgfrcs.sty" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfutil-common.tex" +\pgfutil@everybye=\toks23 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfutil-latex.def" +\pgfutil@abb=\box29 + +("C:\Program Files\MiKTeX 2.9\tex\latex\ms\everyshi.sty" +Package: everyshi 2001/05/15 v3.00 EveryShipout Package (MS) +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfrcs.code.tex" +Package: pgfrcs 2010/10/25 v2.10 (rcs-revision 1.24) +)) +Package: pgf 2008/01/15 v2.10 (rcs-revision 1.12) + +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\basiclayer\pgfcore.sty" +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\systemlayer\pgfsys.sty" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgfsys.code.tex" +Package: pgfsys 2010/06/30 v2.10 (rcs-revision 1.37) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfkeys.code.tex" +\pgfkeys@pathtoks=\toks24 +\pgfkeys@temptoks=\toks25 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfkeysfiltered.code.te +x" +\pgfkeys@tmptoks=\toks26 +)) +\pgf@x=\dimen115 +\pgf@y=\dimen116 +\pgf@xa=\dimen117 +\pgf@ya=\dimen118 +\pgf@xb=\dimen119 +\pgf@yb=\dimen120 +\pgf@xc=\dimen121 +\pgf@yc=\dimen122 +\w@pgf@writea=\write3 +\r@pgf@reada=\read1 +\c@pgf@counta=\count102 +\c@pgf@countb=\count103 +\c@pgf@countc=\count104 +\c@pgf@countd=\count105 + ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgf.cfg" +File: pgf.cfg 2008/05/14 (rcs-revision 1.7) +) +Package pgfsys Info: Driver file for pgf: pgfsys-pdftex.def on input line 900. + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgfsys-pdftex.def" +File: pgfsys-pdftex.def 2009/05/22 (rcs-revision 1.26) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgfsys-common-pdf.def +" +File: pgfsys-common-pdf.def 2008/05/19 (rcs-revision 1.10) +))) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgfsyssoftpath.code.t +ex" +File: pgfsyssoftpath.code.tex 2008/07/18 (rcs-revision 1.7) +\pgfsyssoftpath@smallbuffer@items=\count106 +\pgfsyssoftpath@bigbuffer@items=\count107 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\systemlayer\pgfsysprotocol.code.t +ex" +File: pgfsysprotocol.code.tex 2006/10/16 (rcs-revision 1.4) +)) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcore.code.tex" +Package: pgfcore 2010/04/11 v2.10 (rcs-revision 1.7) + ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmath.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathcalc.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathutil.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathparser.code.tex" +\pgfmath@dimen=\dimen123 +\pgfmath@count=\count108 +\pgfmath@box=\box30 +\pgfmath@toks=\toks27 +\pgfmath@stack@operand=\toks28 +\pgfmath@stack@operation=\toks29 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.basic.code. +tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.trigonometr +ic.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.random.code +.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.comparison. +code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.base.code.t +ex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.round.code. +tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfunctions.misc.code.t +ex"))) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\math\pgfmathfloat.code.tex +" +\c@pgfmathroundto@lastzeros=\count109 +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorepoints.code.tex +" +File: pgfcorepoints.code.tex 2010/04/09 (rcs-revision 1.20) +\pgf@picminx=\dimen124 +\pgf@picmaxx=\dimen125 +\pgf@picminy=\dimen126 +\pgf@picmaxy=\dimen127 +\pgf@pathminx=\dimen128 +\pgf@pathmaxx=\dimen129 +\pgf@pathminy=\dimen130 +\pgf@pathmaxy=\dimen131 +\pgf@xx=\dimen132 +\pgf@xy=\dimen133 +\pgf@yx=\dimen134 +\pgf@yy=\dimen135 +\pgf@zx=\dimen136 +\pgf@zy=\dimen137 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorepathconstruct.c +ode.tex" +File: pgfcorepathconstruct.code.tex 2010/08/03 (rcs-revision 1.24) +\pgf@path@lastx=\dimen138 +\pgf@path@lasty=\dimen139 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorepathusage.code. +tex" +File: pgfcorepathusage.code.tex 2008/04/22 (rcs-revision 1.12) +\pgf@shorten@end@additional=\dimen140 +\pgf@shorten@start@additional=\dimen141 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorescopes.code.tex +" +File: pgfcorescopes.code.tex 2010/09/08 (rcs-revision 1.34) +\pgfpic=\box31 +\pgf@hbox=\box32 +\pgf@layerbox@main=\box33 +\pgf@picture@serial@count=\count110 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoregraphicstate.co +de.tex" +File: pgfcoregraphicstate.code.tex 2008/04/22 (rcs-revision 1.9) +\pgflinewidth=\dimen142 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoretransformations +.code.tex" +File: pgfcoretransformations.code.tex 2009/06/10 (rcs-revision 1.11) +\pgf@pt@x=\dimen143 +\pgf@pt@y=\dimen144 +\pgf@pt@temp=\dimen145 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorequick.code.tex" +File: pgfcorequick.code.tex 2008/10/09 (rcs-revision 1.3) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoreobjects.code.te +x" +File: pgfcoreobjects.code.tex 2006/10/11 (rcs-revision 1.2) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorepathprocessing. +code.tex" +File: pgfcorepathprocessing.code.tex 2008/10/09 (rcs-revision 1.8) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorearrows.code.tex +" +File: pgfcorearrows.code.tex 2008/04/23 (rcs-revision 1.11) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoreshade.code.tex" +File: pgfcoreshade.code.tex 2008/11/23 (rcs-revision 1.13) +\pgf@max=\dimen146 +\pgf@sys@shading@range@num=\count111 +) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoreimage.code.te +x" +File: pgfcoreimage.code.tex 2010/03/25 (rcs-revision 1.16) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoreexternal.code.t +ex" +File: pgfcoreexternal.code.tex 2010/09/01 (rcs-revision 1.17) +\pgfexternal@startupbox=\box34 +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorelayers.code.tex +" +File: pgfcorelayers.code.tex 2010/08/27 (rcs-revision 1.2) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcoretransparency.co +de.tex" +File: pgfcoretransparency.code.tex 2008/01/17 (rcs-revision 1.2) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\basiclayer\pgfcorepatterns.code.t +ex" +File: pgfcorepatterns.code.tex 2009/07/02 (rcs-revision 1.3) +))) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\modules\pgfmoduleshapes.code.tex" +File: pgfmoduleshapes.code.tex 2010/09/09 (rcs-revision 1.13) +\pgfnodeparttextbox=\box35 +) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\modules\pgfmoduleplot.code.tex" +File: pgfmoduleplot.code.tex 2010/10/22 (rcs-revision 1.8) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\compatibility\pgfcomp-version-0-65. +sty" +Package: pgfcomp-version-0-65 2007/07/03 v2.10 (rcs-revision 1.7) +\pgf@nodesepstart=\dimen147 +\pgf@nodesepend=\dimen148 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\compatibility\pgfcomp-version-1-18. +sty" +Package: pgfcomp-version-1-18 2007/07/23 v2.10 (rcs-revision 1.1) +)) ("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\utilities\pgffor.sty" +("C:\Program Files\MiKTeX 2.9\tex\latex\pgf\utilities\pgfkeys.sty" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgfkeys.code.tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\utilities\pgffor.code.tex" +Package: pgffor 2010/03/23 v2.10 (rcs-revision 1.18) +\pgffor@iter=\dimen149 +\pgffor@skip=\dimen150 +\pgffor@stack=\toks30 +\pgffor@toks=\toks31 +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\tikz.code.tex" +Package: tikz 2010/10/13 v2.10 (rcs-revision 1.76) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\pgflibraryplothandlers. +code.tex" +File: pgflibraryplothandlers.code.tex 2010/05/31 v2.10 (rcs-revision 1.15) +\pgf@plot@mark@count=\count112 +\pgfplotmarksize=\dimen151 +) +\tikz@lastx=\dimen152 +\tikz@lasty=\dimen153 +\tikz@lastxsaved=\dimen154 +\tikz@lastysaved=\dimen155 +\tikzleveldistance=\dimen156 +\tikzsiblingdistance=\dimen157 +\tikz@figbox=\box36 +\tikz@tempbox=\box37 +\tikztreelevel=\count113 +\tikznumberofchildren=\count114 +\tikznumberofcurrentchild=\count115 +\tikz@fig@count=\count116 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\modules\pgfmodulematrix.code.tex" +File: pgfmodulematrix.code.tex 2010/08/24 (rcs-revision 1.4) +\pgfmatrixcurrentrow=\count117 +\pgfmatrixcurrentcolumn=\count118 +\pgf@matrix@numberofcolumns=\count119 +) +\tikz@expandcount=\count120 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +librarytopaths.code.tex" +File: tikzlibrarytopaths.code.tex 2008/06/17 v2.10 (rcs-revision 1.2) +))) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +librarydecorations.pathmorphing.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +librarydecorations.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\modules\pgfmoduledecorations.code +.tex" +\pgfdecoratedcompleteddistance=\dimen158 +\pgfdecoratedremainingdistance=\dimen159 +\pgfdecoratedinputsegmentcompleteddistance=\dimen160 +\pgfdecoratedinputsegmentremainingdistance=\dimen161 +\pgf@decorate@distancetomove=\dimen162 +\pgf@decorate@repeatstate=\count121 +\pgfdecorationsegmentamplitude=\dimen163 +\pgfdecorationsegmentlength=\dimen164 +) +\tikz@lib@dec@box=\box38 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\decorations\pgflibraryd +ecorations.pathmorphing.code.tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +librarydecorations.pathreplacing.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\decorations\pgflibraryd +ecorations.pathreplacing.code.tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +librarydecorations.shapes.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\decorations\pgflibraryd +ecorations.shapes.code.tex")) +("C:\Program Files\MiKTeX 2.9\tex\latex\xkeyval\xkeyval.sty" +Package: xkeyval 2012/10/14 v2.6b package option processing (HA) + +("C:\Program Files\MiKTeX 2.9\tex\generic\xkeyval\xkeyval.tex" +\XKV@toks=\toks32 +\XKV@tempa@toks=\toks33 +\XKV@depth=\count122 +File: xkeyval.tex 2012/10/14 v2.6b key=value parser (HA) +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\pgfplots\pgfplots.sty" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.revision.tex") +Package: pgfplots 2013/10/03 v1.9 Data Visualization (1.9) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotscore.code.tex" +\t@pgfplots@toka=\toks34 +\t@pgfplots@tokb=\toks35 +\t@pgfplots@tokc=\toks36 +\pgfplots@tmpa=\dimen165 +\c@pgfplots@coordindex=\count123 +\c@pgfplots@scanlineindex=\count124 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\sys\pgfplotssysgeneric.code. +tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\libs\pgfplotslibrary.code.te +x") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_loader.code.tex" +Package pgfplots: loading complementary code for your PGF version... + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_misc.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgfkeys.code.tex" +\pgfkeys@pathtoks=\toks37 +\pgfkeys@temptoks=\toks38 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgfkeysfiltered.code.tex" +\pgfkeys@tmptoks=\toks39 +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgfmathfloat.code.tex" +\c@pgfmathroundto@lastzeros=\count125 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgflibraryplothandlers.code.tex" +File: pgflibraryplothandlers.code.tex 2012/09/11 v2.10 (rcs-revision 1.17) +\pgf@plot@mark@count=\count126 +\pgfplotmarksize=\dimen166 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgflibraryfpu.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgfcorescopes.code.tex" +File: pgfcorescopes.code.tex 2012/11/30 (rcs-revision 1.38) +\pgfpic=\box39 +\pgf@hbox=\box40 +\pgf@layerbox@main=\box41 +\pgf@picture@serial@count=\count127 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\oldpgfcompatib\pgfplotsoldpg +fsupp_pgfcorelayers.code.tex" +File: pgfcorelayers.code.tex 2012/11/07 (rcs-revision 1.5) +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotsutil.code.tex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\liststructure\pgfplotslistst +ructure.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\liststructure\pgfplotslistst +ructureext.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\liststructure\pgfplotsarray. +code.tex" +\c@pgfplotsarray@tmp=\count128 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\liststructure\pgfplotsmatrix +.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\numtable\pgfplotstableshared +.code.tex" +\c@pgfplotstable@counta=\count129 +\t@pgfplotstable@a=\toks40 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\liststructure\pgfplotsdeque. +code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotsbinary.code.tex +" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotsbinary.data.cod +e.tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotsutil.verb.code. +tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\libs\pgflibrarypgfplots.surf +shading.code.tex" +\c@pgfplotslibrarysurf@no=\count130 + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\sys\pgflibrarypgfplots.surfs +hading.pgfsys-pdftex.def"))) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotscolormap.code.t +ex" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\util\pgfplotscolor.code.tex" +)) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotsstackedplots.code.te +x") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotsplothandlers.code.te +x" +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotsmeshplothandler.code +.tex")) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.scaling.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotscoordprocessing.code +.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.errorbars.code.tex" +) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.markers.code.tex" +) ("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplotsticks.code.tex") +("C:\Program Files\MiKTeX 2.9\tex\generic\pgfplots\pgfplots.paths.code.tex") +\pgfplots@numplots=\count131 +\pgfplots@xmin@reg=\dimen167 +\pgfplots@xmax@reg=\dimen168 +\pgfplots@ymin@reg=\dimen169 +\pgfplots@ymax@reg=\dimen170 +\pgfplots@zmin@reg=\dimen171 +\pgfplots@zmax@reg=\dimen172 +) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +libraryplotmarks.code.tex" +File: tikzlibraryplotmarks.code.tex 2008/01/09 v2.10 (rcs-revision 1.1) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\pgflibraryplotmarks.cod +e.tex" +File: pgflibraryplotmarks.code.tex 2010/10/15 v2.10 (rcs-revision 1.12) +))) +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\frontendlayer\tikz\libraries\tikz +libraryarrows.code.tex" +File: tikzlibraryarrows.code.tex 2008/01/09 v2.10 (rcs-revision 1.1) + +("C:\Program Files\MiKTeX 2.9\tex\generic\pgf\libraries\pgflibraryarrows.code.t +ex" +File: pgflibraryarrows.code.tex 2008/10/27 v2.10 (rcs-revision 1.9) +\arrowsize=\dimen173 +)) +LaTeX Info: Redefining \usk on input line 28. + +runtime option `thickspace' provided! +("C:\Program Files\MiKTeX 2.9\tex\latex\SIstyle\sistyle.sty" +Package: sistyle 2008/07/16 v2.3a SI units and numbers (DNJ Els) +\ttoks@A=\toks41 +\ttoks@B=\toks42 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\carlisle\slashed.sty" +Package: slashed 1997/01/16 v0.01 Feynman Slashed Character Notation (DPC) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\cancel\cancel.sty" +Package: cancel 2013/04/12 v2.2 Cancel math terms +) +("C:\Program Files\MiKTeX 2.9\tex\latex\ulem\ulem.sty" +\UL@box=\box42 +\UL@hyphenbox=\box43 +\UL@skip=\skip52 +\UL@hook=\toks43 +\UL@height=\dimen174 +\UL@pe=\count132 +\UL@pixel=\dimen175 +\ULC@box=\box44 +Package: ulem 2012/05/18 +\ULdepth=\dimen176 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\base\ifthen.sty" +Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\pdfpages\pdfpages.sty" +Package: pdfpages 2013/08/25 v0.4v Insert pages of external PDF documents (AM) + +("C:\Program Files\MiKTeX 2.9\tex\latex\tools\calc.sty" +Package: calc 2007/08/22 v4.3 Infix arithmetic (KKT,FJ) +\calc@Acount=\count133 +\calc@Bcount=\count134 +\calc@Adimen=\dimen177 +\calc@Bdimen=\dimen178 +\calc@Askip=\skip53 +\calc@Bskip=\skip54 +LaTeX Info: Redefining \setlength on input line 76. +LaTeX Info: Redefining \addtolength on input line 77. +\calc@Ccount=\count135 +\calc@Cskip=\skip55 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\eso-pic\eso-pic.sty" +Package: eso-pic 2013/10/06 v2.0d eso-pic (RN) + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\atbegshi.sty" +Package: atbegshi 2011/10/05 v1.16 At begin shipout hook (HO) + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\ifpdf.sty" +Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO) +Package ifpdf Info: pdfTeX in PDF mode is detected. +))) +("C:\Program Files\MiKTeX 2.9\tex\latex\pdfpages\pppdftex.def" +File: pppdftex.def 2013/08/25 v0.4v Pdfpages driver for pdfTeX (AM) +) +\AM@pagebox=\box45 +\AM@toc@title=\toks44 +\c@AM@survey=\count136 +\AM@templatesizebox=\box46 +) +("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\hyperref.sty" +Package: hyperref 2012/11/06 v6.83m Hypertext links for LaTeX + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\hobsub-hyperref.sty" +Package: hobsub-hyperref 2012/04/25 v1.12 Bundle oberdiek, subset hyperref (HO) + + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\hobsub-generic.sty" +Package: hobsub-generic 2012/04/25 v1.12 Bundle oberdiek, subset generic (HO) +Package: hobsub 2012/04/25 v1.12 Construct package bundles (HO) +Package hobsub Info: Skipping package `infwarerr' (already loaded). +Package hobsub Info: Skipping package `ltxcmds' (already loaded). +Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO) +Package ifluatex Info: LuaTeX not detected. +Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. +Package: intcalc 2007/09/27 v1.1 Expandable calculations with integers (HO) +Package hobsub Info: Skipping package `ifpdf' (already loaded). +Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO) +Package etexcmds Info: Could not find \expanded. +(etexcmds) That can mean that you are not using pdfTeX 1.50 or +(etexcmds) that some package has redefined \expanded. +(etexcmds) In the latter case, load this package earlier. +Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO) +Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO) +Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO +) +Package pdftexcmds Info: LuaTeX not detected. +Package pdftexcmds Info: \pdf@primitive is available. +Package pdftexcmds Info: \pdf@ifprimitive is available. +Package pdftexcmds Info: \pdfdraftmode found. +Package: pdfescape 2011/11/25 v1.13 Implements pdfTeX's escape features (HO) +Package: bigintcalc 2012/04/08 v1.3 Expandable calculations on big integers (HO +) +Package: bitset 2011/01/30 v1.1 Handle bit-vector datatype (HO) +Package: uniquecounter 2011/01/30 v1.2 Provide unlimited unique counter (HO) +) +Package hobsub Info: Skipping package `hobsub' (already loaded). +Package: letltxmacro 2010/09/02 v1.4 Let assignment for LaTeX macros (HO) +Package: hopatch 2011/06/24 v1.1 Wrapper for package hooks (HO) +Package: xcolor-patch 2011/01/30 xcolor patch +Package: atveryend 2011/06/30 v1.8 Hooks at the very end of document (HO) +Package atveryend Info: \enddocument detected (standard20110627). +Package hobsub Info: Skipping package `atbegshi' (already loaded). +Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO) +Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO) +) +("C:\Program Files\MiKTeX 2.9\tex\generic\ifxetex\ifxetex.sty" +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional +) +("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\auxhook.sty" +Package: auxhook 2011/03/04 v1.3 Hooks for auxiliary files (HO) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\kvoptions.sty" +Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO) +) +\@linkdim=\dimen179 +\Hy@linkcounter=\count137 +\Hy@pagecounter=\count138 + +("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\pd1enc.def" +File: pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO) +) +\Hy@SavedSpaceFactor=\count139 + +("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\hyperref.cfg" +File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive +) +Package hyperref Info: Hyper figures OFF on input line 4443. +Package hyperref Info: Link nesting OFF on input line 4448. +Package hyperref Info: Hyper index ON on input line 4451. +Package hyperref Info: Plain pages OFF on input line 4458. +Package hyperref Info: Backreferencing OFF on input line 4463. +Package hyperref Info: Implicit mode ON; LaTeX internals redefined. +Package hyperref Info: Bookmarks ON on input line 4688. +\c@Hy@tempcnt=\count140 + +("C:\Program Files\MiKTeX 2.9\tex\latex\url\url.sty" +\Urlmuskip=\muskip11 +Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc. +) +LaTeX Info: Redefining \url on input line 5041. +\XeTeXLinkMargin=\dimen180 +\Fld@menulength=\count141 +\Field@Width=\dimen181 +\Fld@charsize=\dimen182 +Package hyperref Info: Hyper figures OFF on input line 6295. +Package hyperref Info: Link nesting OFF on input line 6300. +Package hyperref Info: Hyper index ON on input line 6303. +Package hyperref Info: backreferencing OFF on input line 6310. +Package hyperref Info: Link coloring OFF on input line 6315. +Package hyperref Info: Link coloring with OCG OFF on input line 6320. +Package hyperref Info: PDF/A mode OFF on input line 6325. +LaTeX Info: Redefining \ref on input line 6365. +LaTeX Info: Redefining \pageref on input line 6369. +\Hy@abspage=\count142 +\c@Item=\count143 +\c@Hfootnote=\count144 +) + +Package hyperref Message: Driver (autodetected): hpdftex. + +("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\hpdftex.def" +File: hpdftex.def 2012/11/06 v6.83m Hyperref driver for pdfTeX +\Fld@listcount=\count145 +\c@bookmark@seq@number=\count146 + +("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\rerunfilecheck.sty" +Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO) +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 +82. +) +\Hy@SectionHShift=\skip56 +) +Package hyperref Info: Option `colorlinks' set `true' on input line 47. + + +Package hyperref Warning: Values of option `pdfpagemode': +(hyperref) * `UseNone' +(hyperref) * `UseOutlines' +(hyperref) * `UseThumbs' +(hyperref) * `FullScreen' +(hyperref) * `UseOC' (PDF 1.5) +(hyperref) * `UseAttachments' (PDF 1.6) +(hyperref) * An empty value disables the option. +(hyperref) Unknown value `none' on input line 47. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) \kern 1.81503pt +(hyperref) removed on input line 47. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) \kern 1.81503pt +(hyperref) removed on input line 47. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) \kern 1.81503pt +(hyperref) removed on input line 47. + +Package hyperref Info: Option `pdffitwindow' set `true' on input line 47. +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\scrtime.sty" +Package: scrtime 2011/03/31 v3.08b KOMA-Script package (time of LaTeX run) +) +("C:\Program Files\MiKTeX 2.9\tex\latex\tocloft\tocloft.sty" +Package: tocloft 2013/05/02 v2.3f parameterised ToC, etc., typesetting +Package tocloft Info: The document has section divisions on input line 44. +\cftparskip=\skip57 +\cftbeforetoctitleskip=\skip58 +\cftaftertoctitleskip=\skip59 +\cftbeforepartskip=\skip60 +\cftpartnumwidth=\skip61 +\cftpartindent=\skip62 +\cftbeforesecskip=\skip63 +\cftsecindent=\skip64 +\cftsecnumwidth=\skip65 +\cftbeforesubsecskip=\skip66 +\cftsubsecindent=\skip67 +\cftsubsecnumwidth=\skip68 +\cftbeforesubsubsecskip=\skip69 +\cftsubsubsecindent=\skip70 +\cftsubsubsecnumwidth=\skip71 +\cftbeforeparaskip=\skip72 +\cftparaindent=\skip73 +\cftparanumwidth=\skip74 +\cftbeforesubparaskip=\skip75 +\cftsubparaindent=\skip76 +\cftsubparanumwidth=\skip77 +\cftbeforeloftitleskip=\skip78 +\cftafterloftitleskip=\skip79 +\cftbeforefigskip=\skip80 +\cftfigindent=\skip81 +\cftfignumwidth=\skip82 +\c@lofdepth=\count147 +\c@lotdepth=\count148 +\cftbeforelottitleskip=\skip83 +\cftafterlottitleskip=\skip84 +\cftbeforetabskip=\skip85 +\cfttabindent=\skip86 +\cfttabnumwidth=\skip87 +)) +("C:\Program Files\MiKTeX 2.9\tex\latex\koma-script\tocstyle.sty" +Package: tocstyle 2013/08/11 v0.2e-alpha LaTeX2e KOMA-Script package (versatile + toc styles) + + +Package tocstyle Warning: THIS IS AN ALPHA VERSION! +(tocstyle) USAGE OF THIS VERSION IS ON YOUR OWN RISK! +(tocstyle) EVERYTHING MAY HAPPEN! +(tocstyle) EVERYTHING MAY CHANGE IN FUTURE! +(tocstyle) THERE IS NO SUPPORT, IF YOU USE THIS PACKAGE! +(tocstyle) Maybe it would be better, not to load this package. + +\tocstyle@indentstyle=\count149 +Package tocstyle Info: no tocstyle.cfg found on input line 856. +) +exclude: + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_Wasser. +tex ("C:\Program Files\MiKTeX 2.9\tex\latex\type1cm\type1cm.sty" +Package: type1cm 2002/09/05 v0.04 BlueSky/Y&Y Type1 CM font definitions (DPC, p +atched RF) +)) +\c@blatt=\count150 +\c@aufgabe=\count151 + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Main_ET2_V2 +.aux) +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 50. +LaTeX Font Info: Try loading font information for TS1+cmr on input line 50. + ("C:\Program Files\MiKTeX 2.9\tex\latex\base\ts1cmr.fd" +File: ts1cmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +LaTeX Font Info: ... okay on input line 50. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 50. +LaTeX Font Info: ... okay on input line 50. + +Option `amssymb' provided! + Command \square redefined by SIunits package! + +("C:\Program Files\MiKTeX 2.9\tex\context\base\supp-pdf.mkii" +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count152 +\scratchdimen=\dimen183 +\scratchbox=\box47 +\nofMPsegments=\count153 +\nofMParguments=\count154 +\everyMPshowfont=\toks45 +\MPscratchCnt=\count155 +\MPscratchDim=\dimen184 +\MPnumerator=\count156 +\makeMPintoPDFobject=\count157 +\everyMPtoPDFconversion=\toks46 +) ABD: EveryShipout initializing macros + +Package pgfplots Warning: running in backwards compatibility mode (unsuitable t +ick labels; missing features). Consider writing \pgfplotsset{compat=1.9} into y +our preamble. + on input line 50. + +\AtBeginShipoutBox=\box48 +Package hyperref Info: Link coloring ON on input line 50. +("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\nameref.sty" +Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section + +("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\gettitlestring.sty" +Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO) +) +\c@section@level=\count158 +) +LaTeX Info: Redefining \ref on input line 50. +LaTeX Info: Redefining \pageref on input line 50. +LaTeX Info: Redefining \nameref on input line 50. + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Main_ET2_V2 +.out) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Main_ET2_V2 +.out) +\@outlinefile=\write4 +LaTeX Font Info: Try loading font information for T1+cmss on input line 50. + ("C:\Program Files\MiKTeX 2.9\tex\latex\base\t1cmss.fd" +File: t1cmss.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +Package tocstyle Info: prepare \l@part for redefinition on input line 50. +Package tocstyle Info: prepare \l@section for redefinition on input line 50. +Package tocstyle Info: prepare \l@subsection for redefinition on input line 50. + +Package tocstyle Info: prepare \l@subsubsection for redefinition on input line +50. +Package tocstyle Info: prepare \l@table for redefinition on input line 50. +Package tocstyle Info: prepare \l@figure for redefinition on input line 50. + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_Deckbla +tt.tex + +LaTeX Font Warning: Font shape `T1/cmr/m/n' in size <85.35826> not available +(Font) size <35.83> substituted on input line 27. + +[1{C:/ProgramData/MiKTeX/2.9/pdftex/config/pdftex.map} + +] [2 + + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Main_ET2_V2 +.toc +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex +LaTeX Font Info: Try loading font information for T1+phv on input line 16. + ("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\t1phv.fd" +File: t1phv.fd 2001/06/04 scalable font definitions for T1/phv. +) +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <6> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 16. +) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[1 + +] +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +number indent by \l@... (toc, 1): 0.0pt +text indent by \l@... (toc, 1): 16.33182pt +number indent by parent (toc, 1): 0.0pt +text indent calculated (toc, 1): 32.72185pt +number indent calculated (toc, 1): 0.0pt +) +\tf@toc=\write5 + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[2]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +1.tex +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <9> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 3. + +Underfull \hbox (badness 10000) in paragraph at lines 113--113 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[3 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[4 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +2.tex +runsystem(gnuplot Main_ET2_V2.tau.gnuplot)...disabled (restricted). + + + +Package pgf Warning: Plot data file `Main_ET2_V2.tau.table' not found. on input + line 123. + +runsystem(gnuplot Main_ET2_V2.tau.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.tau.table' not found. on input + line 123. + +runsystem(gnuplot Main_ET2_V2.tau.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.tau.table' not found. on input + line 123. + +runsystem(gnuplot Main_ET2_V2.tau.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.tau.table' not found. on input + line 123. + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[5 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[6]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +3.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--7 + + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[7 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[8] +Underfull \hbox (badness 10000) in paragraph at lines 125--125 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 125--125 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 125--125 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[9 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [10 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +4.tex +Underfull \hbox (badness 10000) in paragraph at lines 23--24 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[11] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[12]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +5.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--3 + + [] + +runsystem(gnuplot Main_ET2_V2.sina.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sina.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sinb.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sinb.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sinc.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sinc.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sind.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sind.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sina.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sina.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sinb.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sinb.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sinc.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sinc.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sind.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sind.table' not found. on inpu +t line 21. + +runsystem(gnuplot Main_ET2_V2.sin12a53.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a53.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a54.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a54.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a55.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a55.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a56.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a56.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a53.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a53.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a54.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a54.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a55.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a55.table' not found. on +input line 70. + +runsystem(gnuplot Main_ET2_V2.sin12a56.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin12a56.table' not found. on +input line 70. + + +Underfull \vbox (badness 1668) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[13 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[14]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B12_A +6.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--4 + + [] + + +Underfull \vbox (badness 1342) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[15 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[16]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +1.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[17 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[18]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 68--68 + + [] + +runsystem(gnuplot Main_ET2_V2.i.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.i.table' not found. on input l +ine 68. + +runsystem(gnuplot Main_ET2_V2.u.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.u.table' not found. on input l +ine 68. + +runsystem(gnuplot Main_ET2_V2.i.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.i.table' not found. on input l +ine 68. + +runsystem(gnuplot Main_ET2_V2.u.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.u.table' not found. on input l +ine 68. + + +Underfull \vbox (badness 6188) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[19 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[20]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +3.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[21 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[22]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +4.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--5 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[23 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[24]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +5.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[25 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[26 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +6.tex +Underfull \hbox (badness 10000) in paragraph at lines 88--88 + + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[27 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[28]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B13_A +7.tex + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 1. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `subscript' on input line 1. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 1. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 1. + + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 1. + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[29 + +] +Underfull \hbox (badness 10000) in paragraph at lines 101--101 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[30 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B14_A +1.tex +Overfull \hbox (8.53903pt too wide) in paragraph at lines 130--130 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[31 + +] +Overfull \hbox (8.52571pt too wide) in paragraph at lines 130--130 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[32]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B14_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 6--8 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 143--143 + + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[33 + +] +Overfull \hbox (1.31601pt too wide) in paragraph at lines 143--143 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[34]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B14_A +3.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[35 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[36 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B14_A +4.tex +Overfull \hbox (4.0845pt too wide) in paragraph at lines 87--87 +[] + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[37 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[38]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B14_A +5.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--5 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[39 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[40 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[41 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[42 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +1.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--52 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[43 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [44 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--9 + + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[45] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[46]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +2rekap.tex +Underfull \hbox (badness 10000) in paragraph at lines 71--71 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[47 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +3.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[48 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +4.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[49 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [50 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +5.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[51] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[52]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B15_A +6.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--3 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 9--12 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[53 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[54]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +1.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--5 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[55 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[56]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +1a.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--4 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 14--14 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 14--14 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 14--14 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[57 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--5 + + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[58 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[59]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +3.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--10 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[60 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[61 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +4.tex +Underfull \hbox (badness 10000) in paragraph at lines 36--115 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[62 + +] +Underfull \hbox (badness 10000) in paragraph at lines 115--115 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[63 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +5.tex +Overfull \hbox (2.20787pt too wide) in paragraph at lines 90--90 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[64 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +6.tex +Underfull \hbox (badness 10000) in paragraph at lines 24--27 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 81--81 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[65 + + + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[66]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +7.tex +Underfull \hbox (badness 10000) in paragraph at lines 35--36 + + [] + + +Underfull \vbox (badness 4713) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[67 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[68]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B16_A +8.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[69 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[70] +File: Grafik/b16a8a.png Graphic file (type png) + + +Package pdftex.def Info: Grafik/b16a8a.png used on input line 224. +(pdftex.def) Requested size: 250.95587pt x 127.13837pt. + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[71 + + ]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [72 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +1.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--7 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[73] +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[74 + +] +Overfull \hbox (8.28606pt too wide) in paragraph at lines 186--186 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[75 + +] +Overfull \hbox (3.9696pt too wide) in paragraph at lines 186--186 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[76 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--9 + + [] + +runsystem(gnuplot Main_ET2_V2.ortskurve17-2.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.ortskurve17-2.table' not found +. on input line 94. + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[77 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[78]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +3.tex +Underfull \vbox (badness 3492) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[79 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[80]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +4.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--8 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 31--104 + + [] + + +Overfull \hbox (310.51369pt too wide) in paragraph at lines 104--104 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[81 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[82]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +5.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[83 + +] + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 12 +3. + + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 12 +3. + + +Overfull \vbox (25.79439pt too high) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[84]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +6.tex +Overfull \hbox (28.08589pt too wide) in paragraph at lines 97--97 +[] + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[85 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[86]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +7.tex +Underfull \vbox (badness 1584) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[87 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[88]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B17_A +8.tex +Underfull \hbox (badness 10000) in paragraph at lines 42--103 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[89 + +] +Overfull \hbox (11.72293pt too wide) in paragraph at lines 103--103 +[] + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 103--103 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[90 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +1.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--5 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 41--88 + + [] + + +Overfull \hbox (29.43011pt too wide) in paragraph at lines 88--88 +$[]$ $[]$ + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[91 + +] +Underfull \hbox (badness 10000) in paragraph at lines 88--88 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[92 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +2.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--4 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 5--7 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 48--168 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 168--168 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[93 + + + +] +Overfull \hbox (21.75961pt too wide) in paragraph at lines 168--168 +[] + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[94 + +] +Overfull \hbox (62.61896pt too wide) in paragraph at lines 168--168 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[95]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [96 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +3.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[97] +Underfull \hbox (badness 10000) in paragraph at lines 186--186 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[98 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +4.tex +Underfull \hbox (badness 10000) in paragraph at lines 3--5 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 40--150 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[99 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[100 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +5.tex +Underfull \hbox (badness 10000) in paragraph at lines 152--152 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 152--152 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[101 + +] +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 152. + +Overfull \hbox (3.22795pt too wide) in paragraph at lines 152--152 +[] + [] + + +Overfull \hbox (9.68893pt too wide) in paragraph at lines 152--152 +$[]$$[]$ + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[102 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +6.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[103 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [104 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +7.tex +Overfull \hbox (2.05505pt too wide) in paragraph at lines 46--116 +$[]$ $[]$ + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[105] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[106]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B18_A +8.tex +Overfull \hbox (22.0161pt too wide) in paragraph at lines 108--108 +[] + [] + + +Overfull \hbox (26.13255pt too wide) in paragraph at lines 45--108 +$[]$ $[]$ $[]$ + [] + + +Underfull \vbox (badness 1810) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[107 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[108]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +1.tex +Underfull \hbox (badness 10000) in paragraph at lines 24--51 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[109 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex [110 + + + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +2.tex + +Package pgfplots Warning: Loading deprecated style option /tikz/every axis lege +nd. Please replace '\tikzstyle{every axis legend}' with '\pgfplotsset{every axi +s legend/.style={}}' (or '\pgfplotsset{every axis legend/.append style={}}'). o +n input line 40. + + +Package pgfplots Warning: Loading deprecated style option /tikz/every axis lege +nd. Please replace '\tikzstyle{every axis legend}' with '\pgfplotsset{every axi +s legend/.style={}}' (or '\pgfplotsset{every axis legend/.append style={}}'). o +n input line 40. + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[111] + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + +LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <14.4> not available +(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 105. + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +LaTeX Font Warning: Command \Large invalid in math mode on input line 105. + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[112 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +3.tex +Overfull \hbox (2.05504pt too wide) in paragraph at lines 2--32 +$[]$ $[]$ + [] + +runsystem(gnuplot Main_ET2_V2.iomega.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iomega.table' not found. on in +put line 100. + +runsystem(gnuplot Main_ET2_V2.iFe.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iFe.table' not found. on input + line 100. + +runsystem(gnuplot Main_ET2_V2.iomega.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iomega.table' not found. on in +put line 100. + +runsystem(gnuplot Main_ET2_V2.iomega.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iomega.table' not found. on in +put line 100. + +runsystem(gnuplot Main_ET2_V2.iFe.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iFe.table' not found. on input + line 100. + +runsystem(gnuplot Main_ET2_V2.iomega.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.iomega.table' not found. on in +put line 100. + + +Overfull \hbox (32.76923pt too wide) in paragraph at lines 100--100 +[] + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[113 + +] +Overfull \hbox (2.05504pt too wide) in paragraph at lines 157--157 +$[]$ $[]$ + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 157--157 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 157--157 + + [] + +LaTeX Font Info: Try loading font information for T1+cmtt on input line 157. + +("C:\Program Files\MiKTeX 2.9\tex\latex\base\t1cmtt.fd" +File: t1cmtt.fd 1999/05/25 v2.5h Standard LaTeX font definitions +) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[114 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +4.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[115 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[116 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +5.tex +Overfull \hbox (2.05505pt too wide) in paragraph at lines 2--93 +$[]$ $[]$ + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[117 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[118 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +6.tex +Underfull \hbox (badness 10000) in paragraph at lines 2--29 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[119 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B19_A +7.tex +runsystem(gnuplot Main_ET2_V2.sin19_7.gnuplot)...disabled (restricted). + + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin19_7.table' not found. on i +nput line 29. + +runsystem(gnuplot Main_ET2_V2.sin19_7.gnuplot)...disabled (restricted). + + +Package pgf Warning: Plot data file `Main_ET2_V2.sin19_7.table' not found. on i +nput line 29. + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[120 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[121]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B20_A +1.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[122 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[123]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B20_A +2.tex + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 39 +. + + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 39 +. + + +Underfull \hbox (badness 10000) in paragraph at lines 39--192 + + [] + + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 19 +2. + + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 19 +2. + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[124 + +] + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 19 +2. + + +LaTeX Font Warning: Command \footnotesize invalid in math mode on input line 19 +2. + + +Overfull \hbox (2.05505pt too wide) in paragraph at lines 192--192 +[] $[]$ + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 192--192 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[125 + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[126 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Leerblatt.t +ex) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_B20_A +3.tex +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[127 + + + +] +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[128 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ET2_L_Ergeb +nisse.tex +\c@acounter=\count159 +\c@bcounter=\count160 +\c@bsubcounter=\count161 + +Underfull \hbox (badness 10000) in paragraph at lines 10--53 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[129 + + +] +Underfull \hbox (badness 10000) in paragraph at lines 54--98 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[130 + +] +Underfull \hbox (badness 10000) in paragraph at lines 100--145 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[131 + +] +Overfull \hbox (11.12962pt too wide) in paragraph at lines 146--192 +\T1/cmr/bx/n/10.95 49 Strangströ-me 3-Phasen Sys-tem mit un-sym-me-tri-schem Ve +r-brau-cher (B18A8): + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[132 + +] +Underfull \hbox (badness 10000) in paragraph at lines 194--215 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 217--229 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 251--252 + + [] + + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ohmlogoTH.t +ex) +Overfull \vbox (13.55199pt too high) has occurred while \output is active [] + + +[133 + +]) +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\ZeichenHilf +e.tex) +Overfull \hbox (37.78833pt too wide) in paragraph at lines 3--269 +[] + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 3--269 + + [] + +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 288. +[134 + + +] +Package atveryend Info: Empty hook `AfterLastShipout' on input line 288. + +(C:\Users\Niebler\Documents\Vorlesungen\ETechnik\TeX\UeET2_20130923\Main_ET2_V2 +.aux) +Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 288. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 288. +Package rerunfilecheck Info: File `Main_ET2_V2.out' has not changed. +(rerunfilecheck) Checksum: 45B2AE80582320ADB8F3BA9E1ADD1887;3626. + + +LaTeX Font Warning: Size substitutions with differences +(Font) up to 49.52826pt have occurred. + +Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 288. + ) +Here is how much of TeX's memory you used: + 28988 strings out of 493921 + 641929 string characters out of 3147276 + 780943 words of memory out of 3000000 + 30862 multiletter control sequences out of 15000+200000 + 50687 words of font info for 123 fonts, out of 3000000 for 9000 + 841 hyphenation exceptions out of 8191 + 67i,28n,78p,817b,1842s stack positions out of 5000i,500n,10000p,200000b,50000s +{C:/Program Files/MiKTeX 2.9/fonts/enc/dvips/cm-super/cm-super-t1.enc}{C:/Pro +gram Files/MiKTeX 2.9/fonts/enc/dvips/fontname/8r.enc} +Output written on Main_ET2_V2.pdf (136 pages, 1244949 bytes). +PDF statistics: + 1287 PDF objects out of 1440 (max. 8388607) + 353 named destinations out of 1000 (max. 500000) + 498 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/Main_ET2_V2.ortskurve17-2.gnuplot b/Main_ET2_V2.ortskurve17-2.gnuplot new file mode 100644 index 0000000..c9d0064 --- /dev/null +++ b/Main_ET2_V2.ortskurve17-2.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.ortskurve17-2.table"; set format "%.5f" +set samples 100; set parametric; plot [t=0:5] 1000/(100+25*t*t),2*t-500*t/(100+25*t*t) diff --git a/Main_ET2_V2.out b/Main_ET2_V2.out new file mode 100644 index 0000000..702c63c --- /dev/null +++ b/Main_ET2_V2.out @@ -0,0 +1,60 @@ +\BOOKMARK [1][-]{section.1}{Blitzableiter}{}% 1 +\BOOKMARK [1][-]{section.2}{Drahtschleife}{}% 2 +\BOOKMARK [1][-]{section.3}{Metallstab}{}% 3 +\BOOKMARK [1][-]{section.4}{Spannungsverlauf}{}% 4 +\BOOKMARK [1][-]{section.5}{Phasenanschnitt}{}% 5 +\BOOKMARK [1][-]{section.6}{Rechteckspannung}{}% 6 +\BOOKMARK [1][-]{section.7}{Scheinersatzwiderst\344nde}{}% 7 +\BOOKMARK [1][-]{section.8}{Verbraucherleistung}{}% 8 +\BOOKMARK [1][-]{section.9}{Blindleistungskompensation}{}% 9 +\BOOKMARK [1][-]{section.10}{Energie\374bertragung}{}% 10 +\BOOKMARK [1][-]{section.11}{Wechselstrommotor}{}% 11 +\BOOKMARK [1][-]{section.12}{Parallelschaltung von L und C}{}% 12 +\BOOKMARK [1][-]{section.13}{Werte RL und L einer Spule}{}% 13 +\BOOKMARK [1][-]{section.14}{Zeigerdiagramm}{}% 14 +\BOOKMARK [1][-]{section.15}{Gesamtwiderstand}{}% 15 +\BOOKMARK [1][-]{section.16}{Br\374ckenschaltung}{}% 16 +\BOOKMARK [1][-]{section.17}{Zeigerdiagramm Netzwerk}{}% 17 +\BOOKMARK [1][-]{section.18}{Blind- Wirk- und Scheinleistung}{}% 18 +\BOOKMARK [1][-]{section.19}{Komplexe Wechselstromrechnung Netzwerk Strom}{}% 19 +\BOOKMARK [1][-]{section.20}{\334bergang Zeitabh\344ngige zu Komplexen Gr\366\337en}{}% 20 +\BOOKMARK [1][-]{section.21}{Leitwert}{}% 21 +\BOOKMARK [1][-]{section.22}{Strom L-R-C}{}% 22 +\BOOKMARK [1][-]{section.23}{\334berlagerungsmethode}{}% 23 +\BOOKMARK [1][-]{section.24}{Momentan Leistung}{}% 24 +\BOOKMARK [1][-]{section.25}{CLR Netzwerk}{}% 25 +\BOOKMARK [1][-]{section.26}{Wirkleistung vs. Blindleistung}{}% 26 +\BOOKMARK [1][-]{section.27}{Wirkleistung}{}% 27 +\BOOKMARK [1][-]{section.28}{Abgebbare Wirkleistung}{}% 28 +\BOOKMARK [1][-]{section.29}{Wirkleistung Spannungsquelle}{}% 29 +\BOOKMARK [1][-]{section.30}{Dualit\344tskonstante}{}% 30 +\BOOKMARK [1][-]{section.31}{Dualit\344tskonstante verlustbehaftete Bauelemente}{}% 31 +\BOOKMARK [1][-]{section.32}{Vierpol Y-Parameter}{}% 32 +\BOOKMARK [1][-]{section.33}{Spannung Vierpol}{}% 33 +\BOOKMARK [1][-]{section.34}{Stromortskurve}{}% 34 +\BOOKMARK [1][-]{section.35}{Leitwerts-, Widerstandsortskurve}{}% 35 +\BOOKMARK [1][-]{section.36}{Ortskurve}{}% 36 +\BOOKMARK [1][-]{section.37}{Stromortskurve}{}% 37 +\BOOKMARK [1][-]{section.38}{Widerstandstransformation}{}% 38 +\BOOKMARK [1][-]{section.39}{Br\374ckenschaltung}{}% 39 +\BOOKMARK [1][-]{section.40}{Wechselstrombr\374cke}{}% 40 +\BOOKMARK [1][-]{section.41}{Wechselstrombr\374cke}{}% 41 +\BOOKMARK [1][-]{section.42}{\334bertrager im Leerlauf}{}% 42 +\BOOKMARK [1][-]{section.43}{\334bertrager mit kapazitiver Last}{}% 43 +\BOOKMARK [1][-]{section.44}{\334bertrager mit Verbindung zum Eingang}{}% 44 +\BOOKMARK [1][-]{section.45}{Impedanzmatrix}{}% 45 +\BOOKMARK [1][-]{section.46}{Netztransformator}{}% 46 +\BOOKMARK [1][-]{section.47}{3-Phasen Spannungssystem}{}% 47 +\BOOKMARK [1][-]{section.48}{3-Phasen System mit unsymmetrischem Verbraucher}{}% 48 +\BOOKMARK [1][-]{section.49}{Strangstr\366me 3-Phasen System mit unsymmetrischem Verbraucher}{}% 49 +\BOOKMARK [1][-]{section.50}{Resonanzfrequenz Zweipol}{}% 50 +\BOOKMARK [1][-]{section.51}{RLC-Reihenschwingkreis}{}% 51 +\BOOKMARK [1][-]{section.52}{Effektivwert und Klirrfaktor}{}% 52 +\BOOKMARK [1][-]{section.53}{Klirrfaktor}{}% 53 +\BOOKMARK [1][-]{section.54}{Momentanspannung}{}% 54 +\BOOKMARK [1][-]{section.55}{Nichtlinears Bauelement}{}% 55 +\BOOKMARK [1][-]{section.56}{Wirkleistung Zweipol}{}% 56 +\BOOKMARK [1][-]{section.57}{Ringspule}{}% 57 +\BOOKMARK [1][-]{section.58}{Netzwerk Wirk- und Blindanteil}{}% 58 +\BOOKMARK [1][-]{section.59}{Gleichungen in Matrizenschreibweise}{}% 59 +\BOOKMARK [1][-]{section.60}{Ergebnisse}{}% 60 diff --git a/Main_ET2_V2.pdf b/Main_ET2_V2.pdf new file mode 100644 index 0000000..f455ebc Binary files /dev/null and b/Main_ET2_V2.pdf differ diff --git a/Main_ET2_V2.sin12a53.gnuplot b/Main_ET2_V2.sin12a53.gnuplot new file mode 100644 index 0000000..91af28b --- /dev/null +++ b/Main_ET2_V2.sin12a53.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sin12a53.table"; set format "%.5f" +set samples 100; plot [x=0.6:2] 2*sin(0.5*3.14*x) diff --git a/Main_ET2_V2.sin12a54.gnuplot b/Main_ET2_V2.sin12a54.gnuplot new file mode 100644 index 0000000..5dbb07c --- /dev/null +++ b/Main_ET2_V2.sin12a54.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sin12a54.table"; set format "%.5f" +set samples 100; plot [x=2.6:4] 2*(sin(0.5*3.14*x)) diff --git a/Main_ET2_V2.sin12a55.gnuplot b/Main_ET2_V2.sin12a55.gnuplot new file mode 100644 index 0000000..e688805 --- /dev/null +++ b/Main_ET2_V2.sin12a55.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sin12a55.table"; set format "%.5f" +set samples 100; plot [x=4.6:6] 2*sin(0.5*3.14*x) diff --git a/Main_ET2_V2.sin12a56.gnuplot b/Main_ET2_V2.sin12a56.gnuplot new file mode 100644 index 0000000..60e05e8 --- /dev/null +++ b/Main_ET2_V2.sin12a56.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sin12a56.table"; set format "%.5f" +set samples 100; plot [x=0:4] 2*sin(0.5*3.14*x) diff --git a/Main_ET2_V2.sin19_7.gnuplot b/Main_ET2_V2.sin19_7.gnuplot new file mode 100644 index 0000000..ad41308 --- /dev/null +++ b/Main_ET2_V2.sin19_7.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sin19_7.table"; set format "%.5f" +set samples 100; plot [x=0:8] sin(1.047*x) diff --git a/Main_ET2_V2.sina.gnuplot b/Main_ET2_V2.sina.gnuplot new file mode 100644 index 0000000..03e90f1 --- /dev/null +++ b/Main_ET2_V2.sina.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sina.table"; set format "%.5f" +set samples 100; plot [x=0.6:2] 2*sin(.5*3.14*x) diff --git a/Main_ET2_V2.sinb.gnuplot b/Main_ET2_V2.sinb.gnuplot new file mode 100644 index 0000000..0734ba2 --- /dev/null +++ b/Main_ET2_V2.sinb.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sinb.table"; set format "%.5f" +set samples 100; plot [x=0.6:2] 2*sin(0.5*3.14*x) diff --git a/Main_ET2_V2.sinc.gnuplot b/Main_ET2_V2.sinc.gnuplot new file mode 100644 index 0000000..acecd74 --- /dev/null +++ b/Main_ET2_V2.sinc.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sinc.table"; set format "%.5f" +set samples 100; plot [x=2.6:4] 2*(sin(0.5*3.14*x)) diff --git a/Main_ET2_V2.sind.gnuplot b/Main_ET2_V2.sind.gnuplot new file mode 100644 index 0000000..0b2d8c8 --- /dev/null +++ b/Main_ET2_V2.sind.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.sind.table"; set format "%.5f" +set samples 100; plot [x=4.6:6] 2*sin(0.5*3.14*x) diff --git a/Main_ET2_V2.synctex.gz b/Main_ET2_V2.synctex.gz new file mode 100644 index 0000000..08411c3 Binary files /dev/null and b/Main_ET2_V2.synctex.gz differ diff --git a/Main_ET2_V2.tau.gnuplot b/Main_ET2_V2.tau.gnuplot new file mode 100644 index 0000000..97f3cfd --- /dev/null +++ b/Main_ET2_V2.tau.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.tau.table"; set format "%.5f" +set samples 25; plot [x=0:10] 6*(exp(-x/2)) diff --git a/Main_ET2_V2.tex b/Main_ET2_V2.tex new file mode 100644 index 0000000..0d37371 --- /dev/null +++ b/Main_ET2_V2.tex @@ -0,0 +1,288 @@ +\input{ET2_Style} + +\newcommand{\myVersion}{V2.1} + +% 2008- v1.0 initial version +% 2013-09-23 v2.0 verbesserte Ausgabe 2013 +% 2014-03-17 v2.1 added Blatt und Aufgabennummer in Überschriften, manuelle Kopfzeilenbeschriftung entfernt + +% Abstand nach Nummerierung in TOC anpassen +\usepackage{tocstyle} +\settocfeature{spaceafternumber}{0.5cm} + +\newcommand\toPrint{Lösung}% print Lösung (Lösung) ein (xLösung) druckt nur die Aufgaben +%\newcommand\toPrint{xLösung}% print Lösung (Lösung) ein (xLösung) druckt nur die Aufgaben + +\newcommand{\ProfName}{} %% Watermark Text +\renewcommand{\ProfName}{Prof.\,Dr.\,C.\,Niebler} +% \renewcommand{\ProfName}{Prof.\,Dr.\,T.\,Giesler} +% \renewcommand{\ProfName}{Prof.\,Dr.\,G.\,Sztefka} +% \renewcommand{\ProfName}{Prof.\,Dr.\,R.\,Janker} +% \renewcommand{\ProfName}{Prof.\,Dr.\,M.\,Zwanger} +% \renewcommand{\ProfName}{Prof.\,Dr.\,W.\,Kühnel} +% \renewcommand{\ProfName}{Prof.\,Dr.\,S.\,May} +% \renewcommand{\ProfName}{Prof.\,Dr.\,M.\,Chowanetz} +% \renewcommand{\ProfName}{Prof.\,Dr.\,S.\,Walter} +% \renewcommand{\ProfName}{Prof.\,Dr.\,J.\,Wohlrab} +% \renewcommand{\ProfName}{Prof.\,Dr.\,A.\,Eizenhöfer} +% \renewcommand{\ProfName}{Prof.\,Dr.\,G.\,Siegmund} +% \renewcommand{\ProfName}{Dipl.-Ing.\,P.\,Wiegner} +% \renewcommand{\ProfName}{Prof.\,Dr.\,K.\,Schmidt} +% \renewcommand{\ProfName}{Prof.\,Dr.\,B.\,Klehn} +% \renewcommand{\ProfName}{Dr.\,H.\,Brünner} +\ifthenelse{\equal{\toPrint}{Lösung}}{\input{ET2_Wasser}}{} + +%Anlgegen der Zaehlervariablen +\newcounter{blatt} +\newcounter{aufgabe} +% Vorbelegung ab welchem Blatt die Nummerierung beginnt. +\setcounter{blatt}{11} +\setcounter{aufgabe}{0} + +% Befehle zum Ausführen einer automatischen Zählung. wenn ein neues Blatt beginnt, muss die Aufgabenzählung wieder bei 1 beginnen. +\newcommand{\Aufgabennr}{\arabic{blatt}/\arabic{aufgabe}} +\newcommand{\incrementAufgabennr}{\stepcounter{aufgabe}} +\newcommand{\incrementBlatt}{\stepcounter{blatt}\setcounter{aufgabe}{0}} + +\renewcommand{\thesection}{\Aufgabennr{}} % Fortlaufender Zähler \arabic{section} + +% % % \usepackage{draftwatermark} +\begin{document} +\graphicspath{{Grafik/}{../}} + +% % % % \SetWatermarkText{niebler} +% % % % \SetWatermarkText{chowanetz} +% % % % \SetWatermarkText{giesler} +% % % % \SetWatermarkText{sztefka} +% % % \SetWatermarkText{ } +% % % % \SetWatermarkText{ revision } +% % % % +% % % \SetWatermarkAngle{75} +% % % % \SetWatermarkColor[rgb]{0.95,0.95,0.95} +% % % \SetWatermarkLightness{.98} %% dark=0 light=1 %% alternativ bei Grau +% % % \SetWatermarkFontSize{5cm} +% % % \SetWatermarkScale{2.5} + + + + +% %\part{Übungsaufgaben ET2} + +% \incrementAufgabennr\input{ET2_Deckblatt_v2} %% vorschlag draft erweiterung +%\ifthenelse{\equal{\toPrint}{Lösung}}{\input{ET2_Deckblatt}{\input{ET2_Deckblatt_A}} + +\input{ET2_Deckblatt} + +\incrementBlatt +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 12 -- Aufgabe 1}} %% Blatt 12 komplett Revision 19. 05. 2013 +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{Blatt 12 -- Aufgabe 1 -- 2}}{\chead{Blatt 12 -- Aufgabe 1}} +\incrementAufgabennr\input{ET2_L_B12_A1} % Blitzableiter +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 12 -- Aufgabe 2}} +\incrementAufgabennr\input{ET2_L_B12_A2} % Drahtschleife +%\chead{Blatt 12 -- Aufgabe 3} +% \ifthenelse{\equal{\toPrint}{Lösung}}{\chead{Blatt 12 -- Aufgabe 3}}{} +\incrementAufgabennr\input{ET2_L_B12_A3} % Metallstab +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{ \chead{Blatt 12 -- Aufgabe 4}} +% \ifthenelse{\equal{\toPrint}{Lösung}}{}{\chead{Blatt 12 -- Aufgabe 4 -- 5}} +\incrementAufgabennr\input{ET2_L_B12_A4} % Spannungsverlauf +%\chead{Blatt 12 -- Aufgabe 5} +\incrementAufgabennr\input{ET2_L_B12_A5} % Phasenanschnitt +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 12 -- Aufgabe 6}} +% \ifthenelse{\equal{\toPrint}{Lösung}}{xxxx}{\chead{Blatt 12 -- Aufgabe 6}} +\incrementAufgabennr\input{ET2_L_B12_A6} % Rechteckspannung + +\incrementBlatt +%\chead{Blatt 13 -- Aufgabe 1} %% Blatt 16 komplett Revision 19. 05. 2013 +\incrementAufgabennr\input{ET2_L_B13_A1} % Scheinersatzwiderstände +%\chead{Blatt 13 -- Aufgabe 2} +\incrementAufgabennr\input{ET2_L_B13_A2} % Verbraucherleistung +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 13 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B13_A3} % Blindleistungskompensation + % \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 13 -- Aufgabe 4}} +\incrementAufgabennr\input{ET2_L_B13_A4} % Energieübertragung +%\chead{Blatt 13 -- Aufgabe 5} +\incrementAufgabennr\input{ET2_L_B13_A5} % Wechselstrommotor +%\chead{Blatt 13 -- Aufgabe 6} +\incrementAufgabennr\input{ET2_L_B13_A6} % Parallelschaltung von $L$ und $C$ +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 13 -- Aufgabe 7}} +\incrementAufgabennr\input{ET2_L_B13_A7} % Werte $R_L$ und $L$ einer Spule +%\chead{Blatt 14 -- Aufgabe 1} + +\incrementBlatt +\incrementAufgabennr\input{ET2_L_B14_A1} % Zeigerdiagramm + % \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 14 -- Aufgabe 2}} +\incrementAufgabennr\input{ET2_L_B14_A2} % Gesamtwiderstand +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 14 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B14_A3} % Brückenschaltung +%\chead{Blatt 14 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B14_A4} % Zeigerdiagramm Netzwerk +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 14 -- Aufgabe 5}} +\incrementAufgabennr\input{ET2_L_B14_A5} % Blind- Wirk- und Scheinleistung + +\incrementBlatt +%\chead{Blatt 15 -- Aufgabe 1} +\incrementAufgabennr\input{ET2_L_B15_A1} % Komplexe Wechselstromrechnung Netzwerk Strom +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 15 -- Aufgabe 2}} +\incrementAufgabennr\input{ET2_L_B15_A2} % Übergang Zeitabhängige zu Komplexe Größen +%\chead{Blatt 15 -- Rekapitulieren} + \input{ET2_L_B15_A2rekap} % An Tafel rekapitulieren +%\chead{Blatt 15 -- Aufgabe 3} +\incrementAufgabennr\input{ET2_L_B15_A3} % Leitwert +%\chead{Blatt 15 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B15_A4} % Strom L-R-C +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +%\chead{Blatt 15 -- Aufgabe 5} +\incrementAufgabennr\input{ET2_L_B15_A5} % Überlagerungsmethode +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 15 -- Aufgabe 6}} +\incrementAufgabennr\input{ET2_L_B15_A6} % Momentan Leistung +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 16 -- Aufgabe 1}} %% Blatt 16 komplett Revision 20. 05. 2013 + +\incrementBlatt +\incrementAufgabennr\input{ET2_L_B16_A1} % CLR Netzwerk +\input{ET2_L_B16_A1a} +%\chead{Blatt 16 -- Aufgabe 2} +\incrementAufgabennr\input{ET2_L_B16_A2} % Wirkleistung + % \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{ \chead{Blatt 16 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B16_A3} % Abgebbare Wirkleistung +%\chead{Blatt 16 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B16_A4} % Wirkleistung der Spannungsquelle +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 16 -- Aufgabe 5}} +\incrementAufgabennr\input{ET2_L_B16_A5} % Dualitatskonstanten +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +%\chead{Blatt 16 -- Aufgabe 6} +\incrementAufgabennr\input{ET2_L_B16_A6} % Dualitätskonstante verlustbehaftete Bauelemente +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 16 -- Aufgabe 7}} +\incrementAufgabennr\input{ET2_L_B16_A7} % Vierpol Y-Parameter +%\chead{Blatt 16 -- Aufgabe 8} +\incrementAufgabennr\input{ET2_L_B16_A8} % Spannung Vierpol +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 17 -- Aufgabe 1}} + +\incrementBlatt +\incrementAufgabennr\input{ET2_L_B17_A1} % +%\chead{Blatt 17 -- Aufgabe 2} +\incrementAufgabennr\input{ET2_L_B17_A2} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 17 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B17_A3} % +%\chead{Blatt 17 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B17_A4} % +%\chead{Blatt 17 -- Aufgabe 5} +\incrementAufgabennr\input{ET2_L_B17_A5} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 17 -- Aufgabe 6}} %% Blatt 17 ab A6 komplett Revision 20. 05. 2013 +\incrementAufgabennr\input{ET2_L_B17_A6} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 17 -- Aufgabe 7}} +\incrementAufgabennr\input{ET2_L_B17_A7} % +%\chead{Blatt 17 -- Aufgabe 8} +\incrementAufgabennr\input{ET2_L_B17_A8} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{ \chead{Blatt 18 -- Aufgabe 1}} %% Blatt 18 alle komplett Revision 20. 05. 2013 + +\incrementBlatt +\incrementAufgabennr\input{ET2_L_B18_A1} % +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +%\chead{Blatt 18 -- Aufgabe 2} +\incrementAufgabennr\input{ET2_L_B18_A2} % +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 18 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B18_A3} % +%\chead{Blatt 18 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B18_A4} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 18 -- Aufgabe 5}} +\incrementAufgabennr\input{ET2_L_B18_A5} % +%\chead{Blatt 18 -- Aufgabe 6} +\incrementAufgabennr\input{ET2_L_B18_A6} % +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 18 -- Aufgabe 7}} +\incrementAufgabennr\input{ET2_L_B18_A7} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 18 -- Aufgabe 8}} +\incrementAufgabennr\input{ET2_L_B18_A8} % + +\incrementBlatt +%\chead{Blatt 19 -- Aufgabe 1} %% Blatt 19 komplett Revision 19. 05. 2013 +\incrementAufgabennr\input{ET2_L_B19_A1} % +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 19 -- Aufgabe 2}} +\incrementAufgabennr\input{ET2_L_B19_A2} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 19 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B19_A3} % +%\chead{Blatt 19 -- Aufgabe 4} +\incrementAufgabennr\input{ET2_L_B19_A4} % +%\chead{Blatt 19 -- Aufgabe 5} +\incrementAufgabennr\input{ET2_L_B19_A5} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 19 -- Aufgabe 6}} +\incrementAufgabennr\input{ET2_L_B19_A6} % +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +}{}% +%\chead{Blatt 19 -- Aufgabe 7} +\incrementAufgabennr\input{ET2_L_B19_A7} % +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 20 -- Aufgabe 1}} %% Blatt 20 komplett Revision 19. 05. 2013 + +\incrementBlatt +\incrementAufgabennr\input{ET2_L_B20_A1} % Ringspule +%\chead{Blatt 20 -- Aufgabe 2} +\incrementAufgabennr\input{ET2_L_B20_A2} % Netzwerk +\ifthenelse{\equal{\toPrint}{Lösung}}{% +%\chead{Leerseite} +\input{Leerblatt} % Leerseite +}{}% +% \ifthenelse{\equal{\toPrint}{xLösung}}{\chead{}}{\chead{Blatt 20 -- Aufgabe 3}} +\incrementAufgabennr\input{ET2_L_B20_A3} % Knotenpotential +% \SetWatermarkText{ } +\clearpage + +%\chead{Ergebnisse} +\renewcommand{\thesection}{} +\input{ET2_L_Ergebnisse} +\ifthenelse{\equal{\toPrint}{xLösung}}{} +{\chead{Anhang} +\incrementAufgabennr\input{ZeichenHilfe} +}% + +% % % Diverses +% %\incrementAufgabennr\input{NONsmoker} +% %\incrementAufgabennr\input{NI} +% %\incrementAufgabennr\input{RLC} +% %\incrementAufgabennr\input{Specials} % Besondere Formatierungen +% %\incrementAufgabennr\input{Test_A3} +% %\incrementAufgabennr\input{LinLin}% Lineares Koordinatensystem +% %\incrementAufgabennr\input{LogLog} % Doppelt Logarithmisches Koordinatensystem +% %\incrementAufgabennr\input{LogLin} % Einfach y Logarithmisches Koordinatensystem +% %\incrementAufgabennr\input{LinLog} % Einfach x Logarithmisches Koordinatensystem +% %\incrementAufgabennr\input{LogLinPhase} % Einfach Logarithmisches Koordinatensystem +% %\input {ppr} +% %\input {tikz_axes} +% %\input {intro} +% %\input {tikz} +% %\input {X_3D} +% %\input {pgf_anleitung} +% %\incrementAufgabennr\input{ET2_A_B2} +\end{document} diff --git a/Main_ET2_V2.toc b/Main_ET2_V2.toc new file mode 100644 index 0000000..9e15541 --- /dev/null +++ b/Main_ET2_V2.toc @@ -0,0 +1,61 @@ +\select@language {ngerman} +\contentsline {section}{\numberline {12/1{}}Blitzableiter}{3}{section.1} +\contentsline {section}{\numberline {12/2{}}Drahtschleife}{5}{section.2} +\contentsline {section}{\numberline {12/3{}}Metallstab}{7}{section.3} +\contentsline {section}{\numberline {12/4{}}Spannungsverlauf}{11}{section.4} +\contentsline {section}{\numberline {12/5{}}Phasenanschnitt}{13}{section.5} +\contentsline {section}{\numberline {12/6{}}Rechteckspannung}{15}{section.6} +\contentsline {section}{\numberline {13/1{}}Scheinersatzwiderst\"ande}{17}{section.7} +\contentsline {section}{\numberline {13/2{}}Verbraucherleistung}{19}{section.8} +\contentsline {section}{\numberline {13/3{}}Blindleistungskompensation}{21}{section.9} +\contentsline {section}{\numberline {13/4{}}Energie\"ubertragung}{23}{section.10} +\contentsline {section}{\numberline {13/5{}}Wechselstrommotor}{25}{section.11} +\contentsline {section}{\numberline {13/6{}}Parallelschaltung von L und C}{27}{section.12} +\contentsline {section}{\numberline {13/7{}}Werte $R_L$ und $L$ einer Spule}{29}{section.13} +\contentsline {section}{\numberline {14/1{}}Zeigerdiagramm}{31}{section.14} +\contentsline {section}{\numberline {14/2{}}Gesamtwiderstand}{33}{section.15} +\contentsline {section}{\numberline {14/3{}}Br\"uckenschaltung}{35}{section.16} +\contentsline {section}{\numberline {14/4{}}Zeigerdiagramm Netzwerk}{37}{section.17} +\contentsline {section}{\numberline {14/5{}}Blind- Wirk- und Scheinleistung}{39}{section.18} +\contentsline {section}{\numberline {15/1{}}Komplexe Wechselstromrechnung Netzwerk Strom}{43}{section.19} +\contentsline {section}{\numberline {15/2{}}\"Ubergang Zeitabh\"angige zu Komplexen Gr\"o\IeC {\ss }en}{45}{section.20} +\contentsline {section}{\numberline {15/3{}}Leitwert}{48}{section.21} +\contentsline {section}{\numberline {15/4{}}Strom L-R-C}{49}{section.22} +\contentsline {section}{\numberline {15/5{}}\"Uberlagerungsmethode}{51}{section.23} +\contentsline {section}{\numberline {15/6{}}Momentan Leistung}{53}{section.24} +\contentsline {section}{\numberline {16/1{}}CLR Netzwerk}{55}{section.25} +\contentsline {section}{\numberline {16/1{}}Wirkleistung vs. Blindleistung}{57}{section.26} +\contentsline {section}{\numberline {16/2{}}Wirkleistung}{58}{section.27} +\contentsline {section}{\numberline {16/3{}}Abgebbare Wirkleistung}{60}{section.28} +\contentsline {section}{\numberline {16/4{}}Wirkleistung Spannungsquelle}{62}{section.29} +\contentsline {section}{\numberline {16/5{}}Dualit\"atskonstante}{64}{section.30} +\contentsline {section}{\numberline {16/6{}}Dualit\"atskonstante verlustbehaftete Bauelemente}{65}{section.31} +\contentsline {section}{\numberline {16/7{}}Vierpol Y-Parameter}{67}{section.32} +\contentsline {section}{\numberline {16/8{}}Spannung Vierpol}{69}{section.33} +\contentsline {section}{\numberline {17/1{}}Stromortskurve}{73}{section.34} +\contentsline {section}{\numberline {17/2{}}Leitwerts-, Widerstandsortskurve}{77}{section.35} +\contentsline {section}{\numberline {17/3{}}Ortskurve}{79}{section.36} +\contentsline {section}{\numberline {17/4{}}Stromortskurve}{81}{section.37} +\contentsline {section}{\numberline {17/5{}}Widerstandstransformation}{83}{section.38} +\contentsline {section}{\numberline {17/6{}}Br\"uckenschaltung}{85}{section.39} +\contentsline {section}{\numberline {17/7{}}Wechselstrombr\"ucke}{87}{section.40} +\contentsline {section}{\numberline {17/8{}}Wechselstrombr\"ucke}{89}{section.41} +\contentsline {section}{\numberline {18/1{}}\"Ubertrager im Leerlauf}{91}{section.42} +\contentsline {section}{\numberline {18/2{}}\"Ubertrager mit kapazitiver Last}{93}{section.43} +\contentsline {section}{\numberline {18/3{}}\"Ubertrager mit Verbindung zum Eingang}{97}{section.44} +\contentsline {section}{\numberline {18/4{}}Impedanzmatrix}{99}{section.45} +\contentsline {section}{\numberline {18/5{}}Netztransformator}{101}{section.46} +\contentsline {section}{\numberline {18/6{}}3-Phasen Spannungssystem}{103}{section.47} +\contentsline {section}{\numberline {18/7{}}3-Phasen System mit unsymmetrischem Verbraucher}{105}{section.48} +\contentsline {section}{\numberline {18/8{}}Strangstr\"ome 3-Phasen System mit unsymmetrischem Verbraucher}{107}{section.49} +\contentsline {section}{\numberline {19/1{}}Resonanzfrequenz Zweipol}{109}{section.50} +\contentsline {section}{\numberline {19/2{}}RLC-Reihenschwingkreis}{111}{section.51} +\contentsline {section}{\numberline {19/3{}}Effektivwert und Klirrfaktor}{113}{section.52} +\contentsline {section}{\numberline {19/4{}}Klirrfaktor}{115}{section.53} +\contentsline {section}{\numberline {19/5{}}Momentanspannung}{117}{section.54} +\contentsline {section}{\numberline {19/6{}}Nichtlinears Bauelement}{119}{section.55} +\contentsline {section}{\numberline {19/7{}}Wirkleistung Zweipol}{120}{section.56} +\contentsline {section}{\numberline {20/1{}}Ringspule}{122}{section.57} +\contentsline {section}{\numberline {20/2{}}Netzwerk Wirk- und Blindanteil}{124}{section.58} +\contentsline {section}{\numberline {20/3{}}Gleichungen in Matrizenschreibweise}{127}{section.59} +\contentsline {section}{\numberline {}Ergebnisse}{129}{section.60} diff --git a/Main_ET2_V2.u.gnuplot b/Main_ET2_V2.u.gnuplot new file mode 100644 index 0000000..0c08be2 --- /dev/null +++ b/Main_ET2_V2.u.gnuplot @@ -0,0 +1,2 @@ +set table "Main_ET2_V2.u.table"; set format "%.5f" +set samples 100; plot [x=0:5] .62*sin(.5*3.14*x+.96-1.57) diff --git a/ZeichenHilfe.tex b/ZeichenHilfe.tex new file mode 100644 index 0000000..0d5444e --- /dev/null +++ b/ZeichenHilfe.tex @@ -0,0 +1,18 @@ +\clearpage +\pagestyle{empty} +Zeichenhilfe:\\ + \begin{tikzpicture}[thin,scale=1] + \draw[step=0.5cm,ultra thin,black!50!](-8,-8)grid(8,8); + \draw[thick](0,-8)--(0,8)(-8,0)--(8,0); + \fill[blue](0,0)circle(0.075cm); + \foreach \r in {.25,.75,...,7.75} + \draw[orange](0,0)circle(\r cm); + \foreach \r in {.5,1.5,...,7.5} + \draw[blue](0,0)circle(\r cm); + \foreach \r in {1,3,...,7} + \draw[red](0,0)circle(\r cm); + \foreach \r in {2,4,...,8} + \draw[red!50!blue](0,0)circle(\r cm); + \foreach \w in {0,10,...,350} + \draw[orange](\w:1)--(\w:8); + \end{tikzpicture} \ No newline at end of file diff --git a/ohmlogoTH.tex b/ohmlogoTH.tex new file mode 100644 index 0000000..16d81da --- /dev/null +++ b/ohmlogoTH.tex @@ -0,0 +1,18 @@ +% Reduzierte Papier Größe 6.7*1cm + \setlength{\paperwidth}{6.7cm} + \setlength{\paperheight}{1cm} + \hoffset-81pt + \voffset-70pt + \oddsidemargin=0pt + \topmargin=0pt + \headheight=0pt + \headsep = 0pt +% \fontfamily{phv}\selectfont\tiny % Helvetica Font +% \fontfamily{phv}\selectfont\footnotesize{ % Helvetica Font +\xdefinecolor{ohmblue}{rgb}{0,.33,.98} % Logo Farbe +\begin{tikzpicture}[scale=.33] +\filldraw[ohmblue](0,0)--(0,.65)--(.45,.65)--(0,1.15)--(0,2.55)--(2.55,2.55)--(2.55,1.15) +--(2.1,.65)--(2.55,.65)--(2.55,0)--(1.4,0)--(1.4,2)--(1.17,2)--(1.17,0)--(0,0); +\draw node at(2.7,1.2)[right]{\fontfamily{phv}\fontseries{b}\fontshape{n}\selectfont\tiny\textbf{TECHNISCHE HOCHSCHULE NÜRNBERG}}; +\draw node at(2.7,.38)[right]{\fontfamily{phv}\fontseries{m}\fontshape{n}\selectfont\tiny {GEORG SIMON OHM}}; +\end{tikzpicture}