ET2_Uebung_BEI/ET2_L_B17_A6.tex

\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
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