104 lines
6.2 KiB
TeX
104 lines
6.2 KiB
TeX
\section{Wechselstrombrücke}
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a) Zustand abgeglichenen Brücke $\mathbf{(U_{ab}=0})$\\
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Welches Bauteil muss fur $X$ eingesetzt werden, um diese Voraussetzung zu erfüllen?\\
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Berechnen Sie den Wert des Bauteils als Funktion von $R$ und $X_L$.\\[\baselineskip]
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b) Zustand nicht abgeglichene Brücke: $\mathbf{(U_{ab}\neq 0)}$\\[\baselineskip]
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Zeichnen Sie ein qualitatives Zeigerdiagramm aller eingezeichneten Spannungen\\
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(Bezug $\uline{U}=U\cdot e^{j0\,\degree}$), so dass $\uline{U}_{ab}$ auf $\uline{U}$ senkrecht steht (rechte Winkel müssen gekennzeichnet werden).\\
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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.
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\begin{align*}
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\begin{tikzpicture}[very thick,scale=2.5]
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\begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};
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\draw [<-,blue] (.3,.35)--(.7,.35)node at(.5,.35)[right]{\footnotesize$\uline{U}_{R}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};
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\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{b}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-0cm,rotate=90]%Spule |
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_L$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{a}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$X$};
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\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{X}$};
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\draw node at (.5,0){?};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-.5cm,yshift=-.5cm,rotate=90]%Spannungsquelle
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\draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}$};
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\draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}$};
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\end{scope}
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\begin{scope}[>=latex,very thick]%Knotenpunkte
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\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);
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% \fill (-.5,1)circle(.025) (-.5,-1)circle(.025);
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\draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{ab}$};
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\fill (.5,0)circle(.025) node at (.5,0)[left]{a};
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\fill (1.5,0)circle(.025)node at (1.5,0)[right]{b};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\ifthenelse{\equal{\toPrint}{Lösung}}{%
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%\begin{align}
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%\intertext{Formeln:}
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%\end{align}
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Berechnung: (Platzbedarf in x: $9\,\centi\metre$; in y: $\pm 5\,\centi\metre$)\\
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\begin{align*}
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\intertext{a) Zum Brückenabgleich muß $X$ eine Kapazität sein.}\\
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\intertext{Abgleichbedingung:}\\
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\frac{jX_L}{R}&=\frac{R}{jX_C}\\
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X&=\uuline{-\frac{R^2}{X_L}} \qquad\text{ neg. VZ $\Rightarrow$ Kapazität $X=X_C$} \\
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-\frac{1}{j\omega C}&=-\frac{R^2}{j\omega L}\Rightarrow \quad\text{oder}\quad C=\frac{L}{R^2}
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\end{align*}
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\clearpage
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\begin{align*}
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\intertext{b) Bedingung: $\uline{U}_{ab}\,\bot\,\uline{U}$; gleiche Zeigerlängen.}
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\begin{tikzpicture}[very thick,scale=1]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
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\draw[ultra thin,black!50!](0,-5)grid(8,5);
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\draw[thin,->](0,0)--(8.5,0)node[right]{$\Re$};
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\draw[thin,->](0,-5.5)--(0,5.5)node[above]{$\Im$};
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\foreach \y in {5,4,...,-5}
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\draw(0,\y)--(-.1,\y)node[left]{$\y$};
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\draw[black!50!,dashed](0,0)+(20:4.2cm)arc(20:-20:4.2cm);
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\draw[black!50!,dashed](8,0)+(160:4.2cm)arc(160:200:4.2cm);
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\draw[black!50!,dashed](4,1.5)--(4,-1.5);
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\fill(4,0)circle(0.075cm);
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\draw[blue,->](0,0)--(8,0)node at(7.5,.25){$\uline{U}$};
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\draw[black!50!](4,0)circle(4cm);
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\draw[blue,->](0,0)--(30:6.93)node at(2.5,1.75){$\uline{U}_a$};
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\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){$\uline{U}_{ab}$};
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\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){$\uline{U}_{X}$};
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\draw node at (6,3.75){a};
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\draw node at (6,-3.75){b};
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\draw[red](30:6.93)+(210:.5cm)arc(210:300:.5cm);
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\fill[red](30:6.93)+(255:.25cm)circle(.05cm);
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\draw[red](-30:6.93)+(150:.5cm)arc(150:60:.5cm);
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\fill[red](-30:6.93)+(105:.25cm)circle(.05cm);
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\draw[red](0:6)+(0:.5cm)arc(0:90:.5cm);
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\fill[red](0:6)+(45:.25cm)circle(.05cm);
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\draw node at(8.5,4.5)[right]{$\diamond$\ \ \footnotesize{$\uline{U}$}};
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\draw node at(8.5,4)[right]{$\phantom{\diamond x}$\footnotesize{Thaleskreis, da $\uline{U}_q \bot \uline{U}_R$ und $\uline{U}_b \bot \uline{U}_X$}};
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\draw node at(8.5,3.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_R=\uline{U}$}};
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\draw node at(8.5,3)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_a$ und $\uline{U}_R$ zeichnen}};
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\draw node at(8.5,2.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_{ab}-\uline{U}_b=0$ und $\uline{U}_{ab} \bot\uline{U}$}};
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\draw node at(8.5,2)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_{ab}$ und $\uline{U}_b$ zeichnen}};
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\draw node at(8.5,1.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_b+\uline{U}_X=\uline{U}$}};
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\draw node at(8.5,1)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_X$ zeichnen}};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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Bei der nicht abgeglichenen Brücke mit der Bedingung $\uline{U}_{ab}\,\bot\,\uline{U}$, muss $X$ eine Induktivität sein.\\
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Rechnerisch:
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\begin{align*}
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|U_a|&=|U_b|\Rightarrow |X_L\cdot I_a|=|R\cdot I_b|\tag{1}\\
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|U_R|&=|U_X|\Rightarrow |R\cdot I_a|=|X\cdot I_b| \tag{2}\\
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\frac{X_L}{R}&=\frac{R}{X}\tag{1:2}\\
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X&=\uuline{+\frac{R^2}{X_L}}
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\end{align*}
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\clearpage
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}{}%
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