225 lines
12 KiB
TeX
225 lines
12 KiB
TeX
\section{Spannung Vierpol}
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%\renewcommand{\theequation}{0}
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Berechnung Sie $\uline{U}_0$
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%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] {$R_1$};
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\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%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] {$R_2$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{U}_R=R_2\cdot \uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=1.5cm, yshift=0cm]% Vierpol
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\draw (0,-.2)rectangle(2.0,1.2)node at(.75,.5)[right]{$(\uline{Z})$};
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\draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1);
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=0, yshift=0]
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\draw (0,0)--(1,0);
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\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);
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\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);
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\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);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=.75cm,yshift=1cm]% Strompfeil
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\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=1cm]% Strompfeil
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\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_2$};
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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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R_1&=300\,\ohm\\
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R_2&=600\,\ohm\\
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\uline{Z}_{11}&=100\,\kilo\ohm\text{ reell}\\
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\uline{Z}_{21}&=-2\,\mega\ohm\text{ reell}\\
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\uline{U}_1&=1{,}5\,\volt
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%\end{align*}
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%\begin{align*}
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\intertext{Die $\uline{Z}$-Matrix des Vierpols ist gegeben:}
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\begin{bmatrix}
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\uline{U}_0\\
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\uline{U}_2
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\end{bmatrix}
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&=
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\begin{bmatrix}
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\uline{Z}_{11}&0\\
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\uline{Z}_{21}&0
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\end{bmatrix}
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\cdot
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\begin{bmatrix}
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\uline{I}_0\\
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\uline{I}_2
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\end{bmatrix}
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\end{align*}
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\ifthenelse{\equal{\toPrint}{L<EFBFBD>sung}}{%
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%\begin{align}
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%\intertext{Formeln:}
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%\end{align}
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Berechnung:
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%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] {$R_1$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_1\cdot (\uline{I}_2+\uline{I}_0)$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%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] {$R_2$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=0, yshift=0]
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\draw (0,0)--(1,0);
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\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);
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\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);
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\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);
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=0, yshift=0]
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\draw (0,0)--(1,0);
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\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);
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\draw (1,0)--(1,-.5)--(4,-.5)--(4,0)(1,1)--(1.25,1)(4,1)--(3.75,1);
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\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);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1.2cm]% Strompfeil
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\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1.2cm]% Strompfeil
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\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil
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\draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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Aus $\uline{Z}$-Matrix folgt:
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\begin{align*}
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\uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0 \tag{1}\\
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\uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_0 \tag{2}
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\intertext{Knoten 1:}
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\uline{I}_1-\uline{I}_2-\uline{I}_0&=0 \tag{3}
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\intertext{Masche 1:}
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\uline{U}_1&=R_1\cdot \uline{I}_1+ \uline{U}_0\tag{4}
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\intertext{Masche 2:}
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\uline{U}_0&=R_2\cdot \uline{I}_2+\uline{U}_2 \tag{5}
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\intertext{Zwei Gleichungen f<>r zwei unbekante Str<74>me $\uline{I}_0$ und $\uline{I}_2$:}
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\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}\\
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\Rightarrow \uline{I}_2&=\frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0 \tag{6}\\
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\uline{U}_1&=\uline{I}_0 \cdot\uline{Z}_{11}+R_1(\uline{I}_0 +\uline{I}_2 ) \tag{7 1\&3 in 4}\\
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\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}\\
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&=\left[\uline{Z}_{11}+R_1+\frac{R_1}{R_2}\cdot(\uline{Z}_{11}-\uline{Z}_{21})\right]\cdot \uline{I}_0\\
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&=\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\\
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&=0{,}103+0{,}5\cdot (100+2000)\,\kilo\ohm \cdot \uline{I}_0=1{,}15\,\mega\ohm\cdot \uline{I}_0\\
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\Rightarrow\uline{I}_0&=\frac{\uline{U}_1}{1{,}15\,\mega\ohm}=\frac{1{,}5\,\volt}{1{,}15\,\mega\ohm}=1{,}3\,\micro\ampere\\
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\uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0=100\,\kilo\ohm\cdot 1{,}3\,\micro\ampere=\uuline{130{,}4\,\milli\volt}
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\intertext{Nicht gefragt:}
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\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\\
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\text{Probe }\\
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%&=\frac{1}{R_1}\cdot (\uline{I}_2+\uline{I}_0)\cdot \uline{U}_0\\
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\uline{U}_1&=1{,}5\,\volt=R_1\cdot (\uline{I}_0+\uline{I}_2)+\uline{I}_0\cdot Z_{11}\\
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&=300\,\ohm\cdot (0{,}0013+4{,}55)\,\milli\ampere+1{,}3\,\micro\ampere\cdot 100\,\kilo\ohm=1{,}498\,\volt
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\end{align*}
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\clearpage
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\enlargethispage{1cm}
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Ist Ihnen aufgefallen, da<64> $\uline{Z}_{21}$ einen negativen Wert hat?\\[\baselineskip]
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Ein Widerstand nimmt elektrische Leistung auf, also ein Verbraucher.\\
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Dann mu<6D> also ein negativer Widerstand elektrische Leistung abgeben!
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Gibt es das in der Realit<69>t?\\[\baselineskip]
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Was steckt in dem Vierpol?\\[\baselineskip]
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Passiver Vierpol:\\
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Zum Beispiel mit 3 Widerst<73>nden in $T$ oder $\Pi$ Schaltung, ergibt jedoch keine L<>sung!\\
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Aus der $\uline{Z}$ Matrix ist ersichtlich, da<64> $\uline{Z}_{21}=0$ und $\uline{Z}_{22}=0$ sind,
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somit existiert keine Kopplung vom Ausgang zum Eingang, sondern nur in Vorw<72>rtsrichtung.\\[\baselineskip]
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Beispiel: Elektrodynamischer Lautsprecher der auf ein Kondensatormikrofon einwirkt. Kopplung nur vom Lautsprecher zum Mikrofon, nicht umgekehrt.\\[\baselineskip]
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Aktiver Vierpol: (F<>r Fortgeschrittene)\\
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Es mu<6D> sich um einen invertierenden Trennverst<73>rker mit einem Eingangswiderstand von
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$100\,\kilo\ohm$ und einem Ausgangswiderstand von $2\,\mega\ohm$ handeln. Die Verst<73>rkung ist zu berechnen.
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\begin{align*}
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\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\\
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\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\\
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V&=\frac{\uline{U}_q}{\uline{U}_0}=\frac{-9102{,}6\,\volt}{130{,}4\,\milli\volt}=-69805
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\end{align*}
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\begin{align*}
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\begin{tikzpicture}[scale=1.5]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%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] {$R_1$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$R_1\cdot (\uline{I}_2+\uline{I}_0)$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%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] {$R_2$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,black!25!, xshift=1.5cm, yshift=0cm]% Vierpol mit Innenleben
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\draw (0,-.2)rectangle(2.0,1.2);%node at(.75,.5)[right]{$(\uline{Z})$};
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\draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1);
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% \end{skope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] %Z_11
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\draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667)
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[left] {$\uline{Z}_{11}$};
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\draw [<-,red!50!blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{Z}_{11}\cdot \uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Z_21
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\draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_{21}$};
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\draw [->,red!50!blue] (.3,-.2)--(.7,-.2)node at(.6,-.2)[below]{\footnotesize$-\uline{Z}_{21}\cdot \uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,red!50!blue,xshift=1cm,yshift=0cm,rotate=90]%Spannungsquelle|
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\draw (0,0)--(1,0);%;node at(.5,-.133)[right]{$V\cdot \uline{U}_0$};
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\draw (.5,0)circle(.133);
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\draw [->] (.3,-.2)--(.7,-.2) node at (.3,-.2)[right]{\footnotesize$V\cdot \uline{U}_q$};
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\draw node at (.3,.025)[left]{\footnotesize$+$};
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\draw node at (.7,.025)[left]{\footnotesize$-$};
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\end{scope}
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\begin{scope}[>=latex,very thick,red!50!blue,xshift=0cm,yshift=0cm]%
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\draw (-.5,0)--(2.5,0)(-.5,1)--(0,1)(2,1)--(2.5,1);
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\end{scope}
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\end{scope} % Ende Vierpol Innenleben
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\begin{scope}[>=latex,very thick, xshift=0, yshift=0]
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\draw (0,0)--(1,0);
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\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);
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\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);
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\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);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]% Strompfeil
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\draw [->,red] (0,0)--(.4,0)node at(.2,0)[above]{\footnotesize$\uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=4cm,yshift=1cm]% Strompfeil
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\draw [->,red] (0,0)--(-.4,0)node at(-.2,0)[above]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil
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\draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_2+\uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(0,.25)[left]{\tiny$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil
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\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\begin{minipage}[t]{1\textwidth}
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\centering
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\includegraphics[width =0.6\textwidth]{b16a8a}
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\end{minipage}
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\clearpage
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}{}%
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