ET2_Uebung_BEI/ET2_L_B13_A1.tex

\section{Scheinersatzwiderstände}
Der Eingangswiderstand eines linearen Zweipols beträgt bei der Frequenz $f=800\,\hertz$\\
$Z=600\,\ohm$, sein Phasenwinkel ist $\varphi=30\,\degree$ induktiv.
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Berechnen Sie die Schaltungselemente $R_r$ und $L_r$ der gleichwertigen Reihenersatzschaltung!
\item Berechnen Sie die Schaltungselemente $R_p$ und $L_p$ der gleichwertigen Parallelersatzschaltung!
\item Wie ändern sich die Scheinersatzwiderstände (Betrag und Phase) beider Ersatzschaltungen, wenn die Frequenz $f'= 600\,\hertz$ beträgt?
\end{enumerate}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
\begin{align}
\intertext{Formeln:}
\underline{Z}& &\text{Scheinwiderstand (Impedanz)}\\
Z&=|\underline{Z}| &\text{Betrag des Scheinwiderstandes}\\
X&=\omega\cdot L &\text{Blindwiderstand (Reaktdanz)}\\
B&=-\frac{1}{\omega\cdot L} &\text{Blindleitwert (Suszepdanz)}
\end{align}
\begin{align*}
\intertext{Berechnung:}
\intertext{a) Widerstandsebene:}
R_r&=Z\cdot \cos(\varphi_r)=600\,\ohm\cdot \cos(30\degree)=\uuline{520\,\ohm}\\
X_r&=Z\cdot \sin(\varphi_r)=600\,\ohm\cdot \sin(30\degree)=300\,\ohm\\
L_r&=\frac{X_r}{\omega}=\frac{X_r}{2\pi f}=\frac{300\,\ohm}{2\pi\cdot 800\,\frac{1}{\second}}=\uuline{60\,\milli\henry}
\end{align*}
\begin{align*}
	\begin{tikzpicture}[scale=3]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%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] {$R_r$};
			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R_r}$};
		\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] {$L_r$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{L_r}$};		 
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Knotenpunkte		
			\fill (0,0)circle(.025) node  [above left] {\footnotesize$1$};
			\fill (2,0)circle(.025) node  [above right] {\footnotesize$2$};			
		\end{scope}
\begin{scope}[>=latex,very thick, xshift=3cm, yshift=-.25cm]
		\draw [->,thin](0,0)--(1,0)node[right]{$R$};
		\draw [->](0,0)--(.52,0)node at (.26,0)[below]{$R_r$};
		\draw [->,thin](0,0)--(0,.5)node[above]{$X$};		
		\draw [->](.52,0)--(.52,.3)node at (.52,.15)[right]{$X_r$};
		\draw [->](0:0)--(30:.6)node at (30:.3)[above left]{$Z$};
			\draw [->,red,thin] (0:.26) arc (0:30:.26cm) node at (15:.26) [right] {$\varphi_r$};						 
\end{scope}						
	\end{tikzpicture}
\end{align*}
\begin{align*}
\intertext{b) Leitwertebene:}
Y&=\frac{1}{Z}=\frac{1}{600\,\ohm}=1{,}667\,\milli\siemens \text{; }\qquad\varphi_p=-\varphi_r=-30\degree\\
G_p&=Y\cdot \cos(\varphi_p)=1{,}667\,\milli\siemens\cdot \cos(-30\degree)
=1{,}443\,\milli\siemens\Rightarrow R_p=\frac{1}{G_p}=\uuline{693\,\ohm}\\
B_p&=Y\cdot \sin(\varphi_p)=1{,}667\,\milli\siemens\cdot \sin(-30\degree)
=-0{,}833\,\milli\siemens\Rightarrow X_p=-\frac{1}{B_p}=1200\,\ohm\\
%&\text{mit }B_p=-\frac{1}{\omega\cdot L_p}\Rightarrow \\
L_p&=-\frac{1}{\omega\cdot B_p}=-\frac{1}{2\pi f\cdot B_p}=\frac{-1}{2\pi\cdot 800\,\frac{1}{\second}\cdot (-0{,}833\cdot \power{10}{-3}\,\frac{1}{\ohm})}=\uuline{239\,\milli\henry}
\end{align*}
\begin{align*}
	\begin{tikzpicture}[scale=3]
		\begin{scope}[>=latex,very thick,xshift=0cm,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] {$G_p$};
			\draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{G_p}$}; 		
		\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] {$B_p$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{B_p}$};			 
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%End Knoten 			
			\draw (2,.5)--(2,0)--(0,0);
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=1cm]%End Knoten 			
			\draw (2,-.5)--(2,0)--(0,0);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%Knotenpunkte		
			\fill (0,1)circle(.025) node  [above left] {\footnotesize$1$};
			\fill (0,0)circle(.025) node  [above left] {\footnotesize$2$};			
		\end{scope}	
\begin{scope}[>=latex,very thick, xshift=2cm, yshift=.5cm]
		\draw [->,thin](0,0)--(1,0)node[right]{$G$};
		\draw [->](0,0)--(.52,0)node at (.26,0)[above]{$G_p$};
		\draw [->,thin](0,-.5)--(0,.5)node[above]{$jB$};		
		\draw [->](.52,0)--(.52,-.3)node at (.52,-.15)[right]{$B_p$};
		\draw [->](0:0)--(-30:.6)node at (-30:.3)[below left]{$Y$};
			\draw [->,red,thin] (0:.26) arc (0:-30:.26cm) node at (-15:.26) [right] {$\varphi_p$};						
\end{scope}											
	\end{tikzpicture}
\end{align*}
\begin{align*}
\intertext{c) Frequenz $f'$ \newline Reihenschaltung:}
R'_r&\stackrel{!}{=} R_r=520\,\ohm\\
X'_r&=\omega'\cdot L_r=2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}06\,\ohm\second=226\,\ohm\\
Z'_r&=\sqrt{R'^2_r+X'^2_r}=\sqrt{520^2+226^2}\,\ohm=\uuline{567\,\ohm}\\
%\varphi'&=\arctan\frac{\Im\mathfrak m}{\Re\mathfrak e}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree}
\varphi'_r&=\arctan\frac{\Im}{\Re}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree}
\intertext{Parallelschaltung:}
G'_p&\stackrel{!}{=} G_p=1{,}443\,\milli\siemens\\
B'_p&=\frac{-1}{\omega'\cdot L_p}=\frac{-1}{2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}239\,\ohm\second}=\frac{-1}{901\,\ohm}=-1{,}11\,\milli\siemens\\
Y'_p&=\sqrt{G'^2_r+B'^2_r}=\sqrt{1{,}443^2+(-1{,}11)^2}\,\milli\siemens=1{,}82\,\milli\siemens\\
Z'_p&=\frac{1}{Y'_p}=\uuline{549\,\ohm}\\
\varphi'_p&=\arctan\frac{-1{,}11\,\milli\siemens}{1{,}443\,\milli\siemens}=\uuline{-37{,}6\,\degree}
\end{align*}
\clearpage
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