ET2_Uebung_BEI/ET2_L_B17_A5.tex

\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
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