81 lines
4.4 KiB
TeX
81 lines
4.4 KiB
TeX
\section{Dualitätskonstante verlustbehaftete Bauelemente}
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=.5cm]%Kondensator -
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%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) [above] {$G$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-.5cm]%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) [above] {$R$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-.5cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
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\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);
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\fill(.5,0)circle(.03) (1,0)circle(.03)(2,0)circle(.03) (2.5,0)circle(.03);
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\fill(0,0)circle(.05)node[left]{$a$};
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\fill(3,0)circle(.05)node[right]{$b$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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$\hspace{3cm} R=20\,\ohm\quad L=1\,\milli\henry\quad C=100\,\nano\,\farad\quad G=2\,\milli\siemens$\\
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\renewcommand{\labelenumi}{\alph{enumi})}
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\begin{enumerate}
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\item Geben Sie das duale Schaltbild für den Zweipol $a-b$ an und berechnen Sie dessen Elemente für $R^2_D=(100\,\ohm)^2$.
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\item Welches verlustbehaftete Bauelement stellt die Reihenschaltung $R$ und $L$ dar?
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\item Welches verlustbehaftete Bauelement stellt die Parallelschaltung $G$ und $C$ dar?
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\item Interpretieren Sie das Ergebnis der Bauelementegrößen der beiden dualen Schaltungen in Bezug auf verlustbehaftete Bauelemente.
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\end{enumerate}
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\ifthenelse{\equal{\toPrint}{Lösung}}{%
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\begin{align}
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\intertext{Formeln:}
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%\uline{Z}_1(\omega)&=R^2_D \cdot \uline{Y}_2(\omega)
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R\cdot R_{dual}&=R^2_D \quad = \quad\text{Dualitätskonstante}\\
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R^2_D&=\frac{L}{C}=\frac{L_{dual}}{C}=\frac{L}{C_{dual}}
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\end{align}
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Berechnung:\\[\baselineskip]
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Duales Netzwerk:\\
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Parallel $\Leftrightarrow$ Serie\\
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Leitwert $\Leftrightarrow$ Widerstand\\
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Kapazität $\Leftrightarrow$ Induktivität\\
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=.25cm]%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) [above] {$G_R$};
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\draw node at (.75,.35) [above] {\footnotesize{Verlustbehafteter Kondensator}};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=-.25cm]%Kondensator -
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C_L$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%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) [above] {$R_G$};
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\draw node at (.75,-.5) [above] {\footnotesize{Verlustbehaftete Spule}};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_C$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
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\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);
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\fill(2,0)circle(.03)(3,0)circle(.03);
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\fill(0,0)circle(.05)node[left]{$a$};
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\fill(4,0)circle(.05)node[right]{$b$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\begin{align*}
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\intertext{a) Duale Bauelemente:}
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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{ !})\\
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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{ !})\\
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G_R&=\frac{R}{R^2_D}=\frac{20\,\ohm}{(100\,\ohm)^2}=\uuline{2\,\milli\siemens} \quad(=G\text{ !)}\\
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C_L&=\frac{L}{R^2_D}=\frac{1\,\milli\henry}{(100\,\ohm)^2}=\uuline{100\,\nano\farad} \quad(=C\text{ !)}\\
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\intertext{b) Verlustbehaftete Spule.}
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\intertext{c) Verlustbehafteter Kondensator.}
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\intertext{d) Die duale Schaltung ist die Reihenschaltung der gleichen verlustbehafteten Bauteile Spule und Kondensator.}
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\end{align*}
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\clearpage
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}{}% |