ET2_Uebung_BEI/ET2_L_B16_A7.tex

\section{Vierpol Y-Parameter}
\begin{align*}
	\begin{tikzpicture}[scale=3]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}			
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$};		 
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$};		 
		\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] {$2L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}
		\begin{scope}[>=latex,very thick]
			\fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05);
			\draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$};	
			\draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$};				 
		\end{scope}			
	\end{tikzpicture}
\end{align*}
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Berechnen Sie die $\uline{Y}$-Parameter des Vierpols in Abhängigkeit von $L$.\\
\item Bestimmen Sie die $\uline{Z}$-Parameter
\end{enumerate}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:
\begin{align*}
\intertext{a)\hspace{0.4cm} $Y$ Parameter; Einträge in Leitwertmatrix; Achtung Serienschaltung $\uline{Y}_{Serie}=\frac{\uline{Y}_1\cdot \uline{Y}_2}{\uline{Y}_1+\uline{Y}_2}!$}
\left[
  \begin{array}{c}
    \uline{I}_1 \\
    \uline{I}_2 \\
  \end{array}
\right]
&=
\left[
  \begin{array}{cc}
    \uline{Y}_{11} & \uline{Y}_{12} \\
    \uline{Y}_{21} & \uline{Y}_{22} \\
  \end{array}
\right]\cdot
\left[
  \begin{array}{c}
    \uline{U}_1 \\
    \uline{U}_2 \\
  \end{array}
\right]\\[\baselineskip]
\uline{I}_1&=\uline{Y}_{11}\cdot \uline{U}_1+\uline{Y}_{12}\cdot \uline{U}_2\\
\uline{I}_2&=\uline{Y}_{21}\cdot \uline{U}_1+\uline{Y}_{22}\cdot \uline{U}_2\\
\uline{Y}_L&=\uuline{\frac{1}{j\omega\cdot L}}=\uuline{-j\frac{1}{\omega\cdot L}}
\end{align*}
%\begin{center}
%\intertext{$Y_{11}$: $U_2=0$ d.h. Kurzschluß am Ausgang}
$Y_{11}$: $U_2=0$ d.h. Kurzschluß am Ausgang\\[\baselineskip]
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}			
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$};	
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$};	
		\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] {$2L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}
		\begin{scope}[>=latex,very thick]
			\fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05);
			\draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$};	
%			\draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$};	
			\draw node at (0,.5)[left]{$\uline{Y}_{11}\Rightarrow$};	
			\draw(2,0)--(2,1);
			\draw[red!50!blue,thick,dashed](.8,-.33)rectangle(2.2,1.33)node at (1.5,-.33)[below]{$L$};			
		\end{scope}	
		\begin{scope}[>=latex,thick,xshift=0cm]				
			\draw node at (0,-1)[right]{$\uline{Y}_{11}=\frac{\uline{I}_1}{\uline{U}_1}\Big{|}_{U_2=0}=\uuline{\frac{1}{3}\cdot \uline{Y}_L}$};
		\end{scope}				
	\end{tikzpicture}
\hspace{1cm}
% ----------------------------------
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}			
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
%			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$};	
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$Y_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
%			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$};	
		\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] {$\frac{1}{2} \cdot Y_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}
		\begin{scope}[>=latex,very thick]
			\fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05);
%			\draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$};
%			\draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[right]{$\uline{U}_{2}$};	
			\draw node at (0,.5)[left]{$\uline{Y}_{11}\Rightarrow$};	
			\draw(2,0)--(2,1);
%			\draw[red!50!blue,thick,dashed](.8,-.33)rectangle(2.2,1.33)node at (1.5,-.33)[below]{$Y_L$};			
		\end{scope}	
		\begin{scope}[>=latex,thick,xshift=0cm]				
			\draw[white] node at (0,-1)[right]{$\uline{Y}_{L}=\frac{1}{j\omega \cdot L}=-j\frac{1}{\omega\cdot L}$};
		\end{scope}	
		\begin{scope}[black!75!,>=latex,thick,xshift=0cm]
            \draw[black] node at (1.5,-.33)[above]{$\underbrace{\phantom{xxxxxxxx}}$};
            \draw node at (1.5,-.2)[below]{\footnotesize{$\frac{1}{2\cdot X_L}=\frac{1}{2}\cdot Y_L$}};
            \draw[black] node at (1.4,-.67)[above]{$\underbrace{\phantom{xxxxxxxxxxx}}$};
            \draw node at (1.4,-.54)[below]{\footnotesize{$\frac{1}{2}\cdot Y_L+\frac{1}{2}\cdot Y_L= Y_L$}};
            \draw[black] node at (1,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxx}}$};
            \draw node at (1,-.87)[below]{\footnotesize{$\frac{1}{3\cdot X_L}=\frac{1}{3}\cdot Y_L$}};
		\end{scope}				
	\end{tikzpicture}
% ----------------------------------
%\end{center}
\begin{align*}
\intertext{$Y_{12}$: $U_1=0$ d.h. Kurzschluß am Eingang}
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}			
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_1$};	
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{$\uline{I}_2$};	
		\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] {$2L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}
		\begin{scope}[>=latex,very thick]
			\fill (0,0)circle(.05)(0,1)circle(.05)(1,1)circle(.05)(1,0)circle(.05)(2,0)circle(.05)(2,1)circle(.05);
%			\draw [->,blue] (0,0.9)--(0,.1) node at (0,.5)[right]{$\uline{U}_{1}$};	
			\draw [->,blue] (2,0.9)--(2,.1) node at (2,.5)[left]{$\uline{U}_{2}$};				 
			\draw node at (2,.5)[right]{$\Leftarrow\uline{Y}_{L}$};	
			\draw(0,0)--(0,1);
			\draw[red!50!blue,thick,dashed](-.2,-.33)rectangle(1.2,1.33)node at (.5,-.33)[below]{$L$};			
		\end{scope}
		\begin{scope}[>=latex,thick,xshift=.5cm,yshift=.5cm]				
			\draw node at (2.5,.5)[right]{aus $\uline{I}_{1}=\uline{Y}_{11}\cdot \uline{U}_1+\uline{Y}_{12}\cdot \uline{U}_2$};
		\end{scope}
		\begin{scope}[>=latex,thick,xshift=.5cm,yshift=.25cm]				
			\draw node at (2.5,.5)[right]{folgt mit  $\uline{U}_{1}=0$};
		\end{scope}	
		\begin{scope}[>=latex,thick,xshift=.5cm,yshift=-.25cm]				
			\draw node at (2.5,.5)[right]{$\uline{Y}_{12}=\frac{\uline{I}_1}{\uline{U}_2}\big{|}_{U_1=0}$};
		\end{scope}				
	\end{tikzpicture}
\end{align*}
\begin{align*}
\text{Stromteiler: }
-\uline{I}_1&=\frac{2\cdot Z_L}{4\cdot Z_L}\cdot \uline{I}_2=\frac{1}{2}\cdot \uline{I}_2\\
%
%
\uline{I}_2&=\frac{\uline{U}_2}{3\cdot Z_L} = \uline{U}_2\cdot \frac{1}{3}\cdot \uline{Y}_L\\
\uline{I}_1&=-\frac{1}{2}\cdot \uline{U}_2 \cdot \frac{1}{3}\cdot \uline{Y}_L = -\frac{1}{6}\cdot \uline{U}_2\cdot \uline{Y}_L\\
\uline{Y}_{12}&=\frac{-\frac{1}{6}\cdot \uline{U}_2\cdot \uline{Y}_L}{\uline{U}_2}=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\\
%
%
%\uline{Y}_{12}&=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\text{ ??? 1/3 oder 1/6 ???}\\
\uline{Y}_{21}&=\uline{Y}_{12}=\uuline{-\frac{1}{6}\cdot \uline{Y}_L}\text{ da spiegelsymmetrisch}\\
\uline{Y}_{22}&=\frac{\uline{I}_2}{\uline{U}_2}\Big{|}_{_{U_1=I_1=0}}=\text{spiegelbildlich zu $\uline{Y}_{11}$ d.h. }\\
\uline{Y}_{22}&=\uline{Y}_{11}=\uuline{\frac{1}{3}\cdot \uline{Y}_L}\\
\intertext{b)\hspace{0.4cm} $Z$-Parameter Leerlauf}
\left[
  \begin{array}{c}
    \uline{U}_1 \\
    \uline{U}_2 \\
  \end{array}
\right]
&=
\left[
  \begin{array}{cc}
    \uline{Z}_{11} & \uline{Z}_{12} \\
    \uline{Z}_{21} & \uline{Z}_{22} \\
  \end{array}
\right]\cdot
\left[
  \begin{array}{c}
    \uline{I}_1 \\
    \uline{I}_2 \\
  \end{array}
\right]\\[\baselineskip]
\uline{U}_1&=\uline{Z}_{11}\cdot \uline{I}_1+\uline{Z}_{12}\cdot \uline{I}_2\\
\uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_1+\uline{Z}_{22}\cdot \uline{I}_2\\
\uline{Z}_{11}&=\frac{\uline{U}_{1}}{\uline{I}_{1}}\Big{|}_{_{I_2=0}}=4\cdot j\omega\cdot L=\uuline{\uline{Z}_{22}}\\[\baselineskip]
\uline{Z}_{12}&=\frac{\uline{U}_{1}}{\uline{I}_{2}}\Big{|}_{_{I_1=0}}\\
\uline{U}_{1}&=2\cdot X_L \cdot \uline{I}_{2}\\
\uline{Z}_{12}&=\frac{2\cdot X_L \cdot \uline{I}_{2}}{\uline{I}_{2}}=2\cdot X_L = 2\cdot j\omega\cdot L=\uuline{\uline{Z}_{21}}
\end{align*}
\clearpage
}{}%