ET2_Uebung_BEI/ET2_L_B17_A2.tex

\section{Leitwerts-, Widerstandsortskurve}
Konstruieren Sie graphisch fur den dargestellten Zweipol
die Leitwertsortskurve $\uline{Y}_1(p)$, die Widerstandsortskurve
$\uline{Z}_1(p)$, und schlieslich die Widerstandsortskurve $\uline{Z}(p)$.\\
Beziffern Sie jeweils die Punkte $p=0$; $p=1$; $p=3$ und den Grenzwert $p\rightarrow \infty$.\\[\baselineskip]
Parameter $p$: $\omega=p\cdot \omega_0$ mit $\omega_0=1000\,\frac{1}{\second}$\\[\baselineskip]
Maßstäbe: $2{,}5\,\milli\siemens\,\widehat{=}\,1\,\centi\metre$ : $10\,\ohm\,\widehat{=}\, 1\,\centi\metre$\\
(Platzbedarf in x: $12\,\centi\metre$; in y: $14\,\centi\metre$)\\
\begin{align*}
	\begin{tikzpicture}[very thick,scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=.75cm]%Widerstand - nach EN 60617
	\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$100\,\ohm$};
			\draw [blue] node at(.5,-.125){\footnotesize$\underbrace{\phantom{\uline{Y}_1\text{; } \uline{Z}_1}}$};
            \draw [blue] node at(.5,-.2)[below]{\footnotesize$\uline{Y}_1\text{; } \uline{Z}_1$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1.25cm]%Kondensator -		
			\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$5\,\micro\farad$};
		\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] {$20\,\milli\henry$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
        \end{scope}
		\begin{scope}[>=latex,very thick,xshift=0cm]%Knotenpunkte
			\draw (-.5,0)--(0,0) (.2,1.25)--(0,1.25)--(0,.75)--(.2,.75)  (.8,1.25)--(1,1.25)--(1,.75)--(.8,.75)(-.5,1)--(0,1) (-.5,0)--(2,0)--(2,1)--(1.8,1); 	
			\fill (-.5,0)circle(.025) (-.5,1)circle(.025);
			\draw [->,red] node at (-.5,.5)[left]{$\underline{Z}(p)\Rightarrow$};	
		\end{scope}	
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:
\begin{align*}
\uline{Y}_1(p)&=\frac{1}{R}+j\cdot p\cdot \omega_0\cdot  C=(10+j\cdot p\cdot 5)\,\milli\siemens\\
\uline{Z}_L(p)&=j\cdot  p\cdot \omega_0\cdot L=+j\cdot p\cdot 20\,\ohm\\
\uline{Z}(p)&=\uline{Z}_L(p)+\frac{1}{\uline{Y}_1(p)}
\intertext{Nicht gefragt: Kontrollrechnung: (konjugiert komplex erweitern)}
\uline{Z}(p)&=\uline{Z}_L(p)+\frac{1}{\uline{Y}_1(p)}=j\cdot  p\cdot 20\,\ohm +\frac{1}{(10+j\cdot p\cdot 5)\,\milli\siemens}\\
&=j\cdot  p\cdot 20\,\ohm +\frac{1}{(10+j\cdot p\cdot 5)\,\milli\siemens}\cdot \frac{(10-j\cdot p\cdot 5)\cancel{\,\milli\siemens}}{(10-j\cdot p\cdot 5)\cancel{\,\milli\siemens}}\\
&=\frac{10\cdot \power{10}{3}}{100+25\cdot p^2}\,\ohm+j\Big(20\cdot p-\frac{5\cdot \power{10}{3}\cdot p}{100+25\cdot p^2}\Big)\,\ohm
\end{align*}
\enlargethispage{1cm}
%\begin{align*}
    \begin{tikzpicture}[very thick,scale=1]
    %\centering
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
            \draw[ultra thin,black!50!](0,-8)grid(12,8);
            \draw[thin](5,-8)--(5,8)(10,-8)--(10,6)(0,5)--(12,5)(0,-5)--(12,-5);
            \draw[thin,->](0,0)--(12.5,0)node[right]{$\Re$};
            \draw[thin,->](0,-8)--(0,8.5)node[above]{$\Im$};
            \draw[red,->](0,0)--(4,0)node[above right]{$10\,\milli\siemens$};
            \draw[red](4,-8)--(4,8)node at (4,7.5)[right]{$\uline{Y}_1$};
            \draw[red]node at (4,-7.5)[right]{$\uline{Y}^*_1$};
            \draw [->](4.2,6.75)--(4.2,7.25)node at (4,6.5)[right]{$p$};
            \draw [->](4.2,-6.75)--(4.2,-7.25)node at (4,-6.5)[right]{$p$};
            \foreach \p in {-3,-2,...,3}
            \filldraw (4,2*\p)circle(.05cm)node at (4,2*\p)[below left]{$\p$};
            \draw[blue,thin](0,0)--(4,-6)(0,0)--(5,-5)(0,0)--(8,-4);
            \filldraw[blue](10,0)circle(0.05cm)node[above right]{$100\,\ohm$};
            \filldraw[blue](5,0)circle(0.05cm)node[below]{$M$};
            \draw[blue,thin](0:5cm)+(180:5cm)arc(180:360:5cm);%Mittelpunkt+Start arc Start:End:Radius
            \draw[blue,->](8,-4)--(8,-2)node[below left]{$\uline{Z}_L(1)$};
            \draw[blue,->](5,-5)--(5,-1)node[below left]{$\uline{Z}_L(2)$};
            \draw[blue,->](3.05,-4.6)--(3.05,1.4)node[below left]{$\uline{Z}_L(3)$};
            \filldraw node at (10,0)[below right]{$p=0$};
            \filldraw (8,-4)circle(.05cm)node at (8,-4)[below right]{$p=1$};
            \filldraw (5,-5)circle(.05cm)node at (5,-5)[below right]{$p=2$};
            \filldraw (3.05,-4.6)circle(.05cm)node at (3,-4.5)[below left]{$p=3$};
            \draw[color=blue!50!red, very thick,domain=0:5]  plot[parametric,samples=100,id=ortskurve17-2] function{1000/(100+25*t*t),2*t-500*t/(100+25*t*t)};% Ortskurve Faktor 1/10 Ohm in cm;
            \draw[color=blue!50!red, very thick] node at (2.5, 4.25){$\uline{Z}(p)$};
            \draw[very thick] node at (13, 3){induktiv};
            \draw[very thick] node at (13, -3){kapazitiv};
            \draw[very thick] node at (10, 1){$f = 0$ rein ohmisch};
            \draw[very thick] node at (2, 6.75){$f\rightarrow\infty$ rein induktiv};
		\end{scope}	
%		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
%            \draw[scale=0.5,domain=-3.141:3.141,smooth]
%plot[parametric,id=parametric-example] function{t*sin(t),t*cos(t)};
%		\end{scope}
	\end{tikzpicture}
%\end{align*}
\vspace{-.5\baselineskip}
\begin{enumerate}
    \item $\uline{Y}_1$ zeichnen mit $p$-Werten
    \item $\uline{Z}_{min}$, $\uline{Z}_{max}$ berechnen \\
        $\rightarrow \uline{Z}$-Halbkreis: $r=5\,\centi\metre$
    \item $p$-Werte auf $\uline{Z}$ einzeichnen
    \item $\uline{Z}_L(p)$ punktweise addieren
    \item $\uline{Z}(p)$ Kurve zeichnen
\end{enumerate}
\clearpage
}{}%