ET2_Uebung_BEI/ET2_L_B20_A2.tex

\section {Netzwerk Wirk- und Blindanteil}
$\uline{U}=4{,}2\,\volt\cdot e^{j30\,\degree};\quad f=120\,\kilo\hertz ;\quad \uline{I}_0= 2{,}7\milli\ampere \cdot e^{j110\,\degree}$\\
$R=1{,}3\,\kilo\ohm; \quad L=3{,}6\,\milli\henry; \quad C=1{,}5\,\nano\farad$.
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Berechnen Sie den Strom \uline{I} und seinen Wirk- und Blindanteil $\uline{I}_{\textrm{w}}$ bzw. $\uline{I}_{\textrm{b}}$.
\item  Berechnen Sie die Wirk und Blindleistung an den Anschlussklemmen und an der Stromquelle $\uline{I}_0$.
\end{enumerate}
\vspace{-.5cm}
\begin{align*}
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] 			
            \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$};
		\end{scope}				
		\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] {$C$};
			\draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,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] {$R$};
			\draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_R$};
		\end{scope}		
		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | 			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);	
			\draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_L$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Stromquelle 			
			\draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0);
            \draw [->,blue](.8,.2)--(.2,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{RL}$};
			\draw (.5,0)circle(.133);
			\draw [->,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{I}_0$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw  (.8,1)--(3,1)--(3,.9) (0,0)--(3,0)--(3,.2);
            \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm);
		\end{scope}	
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:\\
\begin{align*}
\text{a)}\quad X_L&=\omega L=2\pi\cdot f\cdot 3{,}6\,\milli\henry=2714\,\ohm\\
X_C&=\frac{-1}{\omega C}=\frac{-1}{2\pi\cdot f\cdot 1{,}5\,\nano\farad}=-884\,\ohm\\
\uline{Z}_{RL}&=\frac{R\cdot jX_L}{R+jX_L}=\frac{1{,}3\,\kilo\ohm\cdot j2714\,\ohm}{1{,}3\,\kilo\ohm+j2714\,\ohm}=(1057+j506)\,\ohm=1172\,\ohm\cdot e^{j25{,}59\,\degree}
\end{align*}
Serienersatzschaltbild:
\begin{align*}
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] 			
            \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$};
		\end{scope}				
		\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] {$X_C$};
			\draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$};
			\draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$\uline{U}_C$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%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] {$\uline{Z}_{RL}$};
			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{RL}$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle 			
			\draw (0,0)--(1,0);
			\draw (.5,0)circle(.133);
			\draw (.5,0)circle(.133);
			\draw [<-,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{U}_0$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw  (1.8,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2);
            \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm);
		\end{scope}							
	\end{tikzpicture}
	\begin{tikzpicture}[scale=.5]
		\begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm]
            \draw[black!25!,very thin](-3,0)grid(5,5);
            \draw[thick](-3,0)--(5,0) (0,0)--(0,5);
            \draw[->,blue](0,0)--(30:4.25cm)node[above right]{$\uline{U}$};
            \draw[->,red](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$};
            \draw[->,blue](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$};
            \draw[->,red](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$};
		\end{scope}
	\end{tikzpicture}
\end{align*}
\begin{align*}
\uline{U}_0&=\uline{I}_0\cdot \uline{Z}_{RL}=2{,}7\milli\ampere \cdot e^{j110\,\degree}\cdot 1172\,\ohm\cdot e^{j25{,}59\,\degree}\\
&=3{,}164\,\volt\cdot e^{j135{,}6\,\degree}=(-2{,}26+j2{,}21)\,\volt\\
\uline{Z}_{RLC}&=\uline{Z}_{RL}+jX_C=(1060+j507)\,\ohm-j884\,\ohm\\
&=(1057-j378)\,\ohm=1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}\\
\intertext{Masche:}
\uline{I}&\cdot \uline{Z}_{RLC}+\uline{U}_0-\uline{U}=0\\
\uline{I}&=\frac{\uline{U}-\uline{U}_0}{\uline{Z}_{RLC}}=\frac{\overbrace{(3{,}637+j2{,}1)\,\volt}^{4{,}2\,\volt\cdot e^{j30\,\degree}}-(-2{,}26+j2{,}21)\,\volt}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}=\frac{5{,}90\,\volt\cdot e^{-j1{,}06\,\degree}}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}\\
&=\uuline{5{,}25\milli\ampere\cdot e^{j18{,}5\,\degree}}=\uuline{(4{,}98+j1{,}67)\milli\ampere}
\end{align*}
\clearpage
Parallelersatzschaltbild: (Wirkstrom in Phase, Blindstrom senkrecht zur Spannung)
\begin{align*}
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] 			
            \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$};
		\end{scope}				
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator -		
			\draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,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] {$R$};
			\draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_W$};
		\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] {$X$};
			\draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_B$};
		\end{scope}		
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw  (0,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2);
            \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm);
		\end{scope}							
	\end{tikzpicture}
	\begin{tikzpicture}[scale=.5]
		\begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm]
            \draw[black!25!,very thin](-3,0)grid(5,5);
            \draw[thick](-3,0)--(5,0) (0,0)--(0,5);
            \draw[->,blue!45!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$};
            \draw[->,red!45!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$};
            \draw[->,blue!45!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$};
            \draw[->,red!45!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$};
            \draw[->,red](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$};
            \draw[->,red](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$};
		\end{scope}
	\end{tikzpicture}
\end{align*}
\begin{align*}
|\uline{I}_{\textrm{w}}|&=I\cdot \cos(\underbrace{\varphi_u}_{30\,\degree}-\underbrace{\varphi_i}_{18{,}5\,\degree})=5{,}14\,\milli\ampere\\
\Rightarrow & \uline{I}_{\textrm{w}}=\uuline{5{,}14\,\milli\ampere\cdot e^{j30\,\degree}}\\
|\uline{I}_b|&=I\cdot \sin(\varphi_u-\varphi_i)=1{,}05\,\milli\ampere\\
\Rightarrow & \uline{I}_b=\uuline{1{,}05\,\milli\ampere\cdot e^{-j60\,\degree}}\\
\end{align*}
\begin{minipage}[b]{.5\textwidth}
b)
\begin{align*}
\uline{U}_C&=\uline{I}\cdot jX_C=4{,}64\,\volt\cdot e^{-j71{,}5\,\degree}\\
\uline{U}_{RL}&=\uline{U}-\uline{U}_C=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\\
\uline{I}_R&=\frac{\uline{U}_{RL}}{R}=5{,}27\,\milli\ampere\cdot e^{j71{,}6\,\degree}\\
\uline{I}_L&=\frac{\uline{U}_{RL}}{jX_L}=2{,}52\,\milli\ampere\cdot e^{-j18{,}4\,\degree}\\
\end{align*}
Probe:
\begin{align*}
I+I_0&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\
I_R+I_L&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\
&\text{stimmt überein.}
\end{align*}
\end{minipage}
\begin{minipage}[c]{.5\textwidth}
	\begin{tikzpicture}[scale=.75]
		\begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm]
            \draw[black!25!,very thin](-3,-5)grid(5,7);
            \draw[thick](-3,0)--(5,0) (0,-5)--(0,7);
            \draw[->,red!66!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$};
            \draw[->,blue!25!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$};
            \draw[->,red!66!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$};
            \draw[->,red!25!](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$};
            \draw[->,red!25!](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$};
            \draw[->,blue!66!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$};
            \draw[->,blue](0,0)--(-71.5:4.64cm)node[right]{$\uline{U}_C$};
            \draw[->,blue](30:4.25cm)--+(108.5:4.64cm)node[below right]{$-\uline{U}_C$};
            \draw[->,blue](0,0)--(71.6:6.85cm)node[below left]{$\uline{U}_{RL}$};                        \draw[->,red](0,0)--(71.6:5.27cm)node[below left]{$\uline{I}_R$};
            \draw[->,red](71.6:5.27cm)--(46:5.85cm)node[above left]{$\uline{I}_L$};
            \draw[<-,red!50!blue]+(46:5.85cm)--(18.5:5.25cm)node at(5,3){$\uline{I}_0$};                        \draw[->,red!50!blue](0,0)--(46:5.85cm)node at(2,3.25){\footnotesize$\uline{I}+\uline{I}_0$};
		\end{scope}
	\end{tikzpicture}
\end{minipage}\\
\clearpage
\begin{align*}
\intertext{Leistung an Klemme:}
\uline{S}_K&=\uline{U}\cdot \uline{I}^*=4{,}2\,\volt\cdot e^{j30\,\degree}\cdot 5{,}25\milli\ampere\cdot e^{-j18{,}5\,\degree}=(\underbrace{21{,}6}_{P_K}+\underbrace{j4{,}4}_{Q_K})\,\milli\volt\ampere\\
P_K&=\uuline{21{,}6\,\milli\watt}\\
Q_K&=\uuline{4{,}4\,\milli\var}\\
\intertext{Leistung der Stromquelle (Pfeilsystem wie oben):}
S_I&=\uline{U}_{RL}\cdot I^*_0=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\cdot 2{,}7\milli\ampere \cdot e^{-j110\,\degree}=(\underbrace{14{,}5}_{P_I}\underbrace{-j11{,}5}_{Q_I})\,\milli\volt\ampere\\
P_I&=\uuline{14{,}5\,\milli\watt}\\
Q_I&=\uuline{-11{,}5\,\milli\var}\\
\intertext{Gesamtleistung: Summe aus Quell- und Klemmleistung} S&=S_K+S_I=(21{,}6+j4{,}4+14{,}5-j11{,}5)\,\milli\volt\ampere=36{,}1\,\milli\watt-j7{,}1\,\milli\volt\ampere
\intertext{Probe: Wirkleistung: }
P&=\frac{U^2_{RL}}{R}=\frac{(6{,}85\,\volt)^2}{1300\,\ohm}=36{,}1\,\milli\watt \quad\text{(Übereinstimmung)}
\intertext{Probe: Blindleistung: }
C:\quad  Q_C&=X_C\cdot I^2=-884\,\ohm\cdot(5{,}25\,\milli\ampere)^2=-24{,}4\,\milli\var\\
L:\quad  Q_L&=\frac{U^2_{RL}}{X_L}= \frac{(6{,}85\,\volt)^2}{2714\,\ohm}=+17{,}3\,\milli\var\\
Q&=Q_C+Q_L=(-24{,}4+17{,}3)\,\milli\var=-7{,}1\,\milli\var \quad\text{(Übereinstimmung)}\\
\end{align*}
\clearpage
}{}%