ET2_Uebung_BEI/ET2_L_B14_A5.tex

\section{Blind- Wirk- und Scheinleistung}
Von dem untenstehenden Schaltbild ist gegeben:\\
$R_L = X_L  = 20\,\ohm \quad R_C=200\,\ohm \quad X_C=-100\,\ohm\quad\uline{U}=230\,\volt\cdot e^{j0\,\degree}$\\
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Der Eingangswiderstand Z der Schaltung nach Betrag und Phasenwinkel
\item  Sämtliche Ströme und Spannungen nach Betrag und Phasenwinkel
\item  Wirk- Blind- und Scheinleistungsaufnahme der Schaltung
\item  Qualitatives Zeigerdiagramm aller Ströme und Spannungen unter der Annahme, daß sich
die Gesamtschaltung induktiv verhält.
\end{enumerate}
\begin{align*}
	\begin{tikzpicture}[very thick,scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,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] {$R_L$};
%			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$};
		\end{scope}	
%		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617			
%			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$};
%		\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] {$X_L$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
		\end{scope}		
		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617			
			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_C$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=3cm,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] {$X_C$};
		\end{scope}		
		\begin{scope}[>=latex,very thick]%Knotenpunkte
			\draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.,0)--(3,0)--(3,.2) (.8,1)--(3,1)--(3,.8);
			\draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}$};	
			\fill (-.5,0)circle(.025) (-.5,1)circle(.025);
			\draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}$};	
			\draw [->,red] (2.1,.9)--(2.1,.7) node at (2.1,.8)[right]{$\underline{I}_{R_C}$};
			\draw [->,red] (3.1,.9)--(3.1,.7) node at (3.1,.8)[right]{$\underline{I}_C$};
			\draw [->,black!50!] (.3,.8)--(.7,.8) node at (.5,.8)[below]{$\underline{U}_{RL}$};
			\draw [->,black!50!] (1.3,.8)--(1.7,.8) node at (1.5,.8)[below]{$\underline{U}_L$};
			\draw [->,black!50!]  node at (-.5,.5)[left]{$\uline{Z} =>$};
			\draw [|-|,black!50!] (2.,-.2)--(3,-.2) node at (2.5,-.2)[below]{$\uline{Z}_{||}$};					
		\end{scope}	
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
\begin{align}
\intertext{Formeln:}
e^{j\varphi}&=\cos\varphi+j\sin\varphi \quad\text{Eulersche Formel}\\
\uline{Z}&=Z\cdot e^{\pm j\varphi}=R\pm jX\\
\cos\varphi&=\frac{R}{Z}\\
\sin\varphi&=\frac{X}{Z}
\end{align}
Berechnung:
\begin{align*}
\intertext{a) Eingangswiderstand (ist ohmisch-kapazitiv)}
\uline{Z}_{||}&=\frac{R_C\cdot jX_C}{R_C+jX_C}=\frac{200\cdot (-j100)}{200-j100}\,\ohm=(40-j80)\,\ohm\\
\uline{Z}&=\uline{Z}_{||}+R_L+X_L=[20+40+j(20-80)]\,\ohm=\uuline{(60-j60)\,\ohm}=\uuline{84{,}85\,\ohm\cdot e^{-j45\,\degree}}\\
&(\Rightarrow \varphi=-45\,\degree) 
\end{align*}
\clearpage
\begin{align*}
\intertext{b) Ströme}
\uline{I}&=\frac{\uline{U}}{\uline{Z}}=\frac{230\,\volt\cdot e^{j0\,\degree}}{84{,}5\,\ohm\cdot e^{-j45\,\degree}}=\uuline{2{,}71\,\ampere\cdot e^{j45\,\degree}}=\uuline{(1{,}916+j1{,}916)\,\ampere}
% \end{align*}
% \clearpage
% \begin{align*}
\intertext{Stromteiler}
\uline{I}_{R_C}&=\uline{I}\cdot \frac{jX_C}{R_C+jX_C}=\uline{I}\cdot \frac{-j100}{200-j100}
=\uline{I}\cdot  \frac{-j}{2-j}\cdot \frac{2+j}{2+j}=\uline{I}\cdot \frac{1-j2}{4+1}=\uline{I}\cdot (0{,}2-j0{,}4)\\
&=2{,}71\,\ampere\cdot e^{j45\,\degree}\cdot 0{,}447\cdot e^{-j63{,}4\,\degree}=\uuline{1{,}21\,\ampere\cdot e^{-j18{,}4\,\degree}}=\uuline{(1{,}150-j0{,}383)\,\ampere}\\%Ende IRC
\uline{I}_{C}&=\uline{I}-\uline{I}_{R_C}=(1{,}916+j1{,}916)\,\ampere -(1{,}150-j0{,}383)\,\ampere=\uuline{(0{,}766+j2{,}30)\,\ampere}\\
&=\uuline{2{,}42\,\ampere\cdot e^{+j71{,}6\,\degree}} \\
\text{ alternativ}&\\
\uline{I}_{C}&=\uline{I}\cdot \frac{R_C}{R_C+jX_C}=\uline{I}\cdot \frac{200}{200-j100}=\uline{I}\cdot \frac{1}{1-j0{,}5}\quad\text{ konjugiert komplex erweitern}\\
&=\uline{I}\cdot \frac{1}{1-j0{,}5}\cdot \frac{1+j0{,}5}{1+j0{,}5}
=\uline{I}\cdot \frac{1+j0{,}5}{1+0{,}5^2}
=\uline{I}\cdot (0{,}8+j0{,}4)=\uline{I}\cdot 0{,}894\cdot e^{0{,}5j}\\
&=2{,}71\,\ampere\cdot e^{j45\,\degree}\cdot 0{,}894\cdot e^{j26{,}6\,\degree}=\uuline{2{,}42\,\ampere\cdot e^{+j71{,}6\,\degree}}=\uuline{(0{,}766+j2{,}30)\,\ampere}
\intertext{Spannungen}
\uline{U}_{R_C}&=R_C\cdot \uline{I}_{R_C}=200\,\ohm\cdot 1{,}21\,\ampere\cdot e^{-j18{,}4\,\degree}=\uuline{242\,\volt\cdot e^{-j18{,}4\,\degree}}\\
\uline{U}_{R_L}&=R_L\cdot \uline{I}=20\,\ohm\cdot 2{,}71\,\ampere \cdot e^{j45\,\degree}=\uuline{54{,}2\,\volt\cdot e^{j45\,\degree}}\\
\uline{U}_L&=X_L\cdot \uline{I}=20\,\ohm\cdot e^{j90\,\degree} \cdot 2{,}71\,\ampere \cdot e^{j45\,\degree}=\uuline{54.2\,\volt\cdot e^{j135\,\degree}}
\end{align*}
\clearpage
\begin{align*}
\intertext{Zeigerdiagramm: Beginne mit $\uline{U}$ und $\uline{I}\qquad (50\,\volt\per\centi\metre$; $1\,\ampere\per\centi\metre$)}
&\uline{I} \text{ um Winkel $\varphi=45\,\degree$ voreilend, kapazitives Gesamtverhalten.}\\
&\uline{I}=\uline{I}_{R_C}+\uline{I}_C \qquad(\uline{I}_C \,\bot\, \uline{I}_{R_C}) \qquad (\uline{I}_C \text{ voreilend})\\
&\uline{U}_{R_L}\,||\,\uline{I}\\	
&\uline{U}_L\,\bot\, \uline{I} \qquad (\uline{U}_L\text{ voreilend})\\
&\uline{U}_{R_C}\,||\,\uline{I}_{R_C}\\
&\uline{U}_{R_L}=\uline{U}_L+\uline{U}_{R_C}=\uline{U}\ \widehat{=}\ 4{,}6\,\centi\metre	
\end{align*}
 %Trennzeile
\begin{align*}	
	\begin{tikzpicture}[very thick,scale=2.]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
		  \draw [black!25!,very thin](0,-1)grid(5,2);
		  \draw [->](0,0)--(5.2,0)node[right]{$\Re$};
		  \draw [->](0,-1)--(0,2.2)node[above]{$\Im$};
			\draw [->,blue] (0,0)--(0:4.6)node [below left]{$\underline{U}$};
			\draw [->,red] (0,0)--(45:2.71)node at(45:2.3) [above left]{$\underline{I}$};
			\draw [->,red] (0,0)--(-18.4:1.21)node at(1.5,.5)[right]{$\underline{I}_C$};
			\draw [->,red] (0,0)++(-18.4:1.21)--+(71.6:2.42)node at(.6,-.3)[below]{$\underline{I}_{R_C}$};				
			\draw [black!50!] (0,0)++(-18.4:1.21)+(71.6:.25)arc(71.6:161.6:.25);%Rechter Winkel
			\fill [black!50!] (0,0)++(-18.4:1.21)+(116.6:.125)circle(.025);%Rechter Winkel	
			\draw [->,blue] (0,0)--(45:1.084)node at(0,.6)[right]{$\underline{U}_{R_L}$};			
			\draw [black!50!] (0,0)++(45:1.084)+(225:.25)arc(225:135:.25);%Rechter Winkel
			\fill [black!50!] (0,0)++(45:1.084)+(180:.125)circle(.025);%Rechter Winkel			
			\draw [->,blue] (0,0)++(45:1.084)--+(135:1.084)node at(3,.5)[above right]{$\underline{U}_{R_C}$};	
			\draw [<-,blue] (0,0)++(0:4.6)--+(161.6:4.84)node at(0,1)[right]{$\underline{U}_L$};		
			\draw [->,black!75!] (0,0)+(0:.5)arc(0:45:.5)node at(22.5:.5) [right]{$\varphi =45\,\degree$};				
		\end{scope}	
		\begin{scope}[>=latex,xshift=0.55cm,yshift=-.1cm]%Parallelen
			\draw [double](0:-.1)--+(-108.4:.1);		
		\end{scope}		
		\begin{scope}[>=latex,xshift=2.45cm,yshift=.8cm]%Parallelen
			\draw [double](0:-.1)--+(-108.4:.1);		
		\end{scope}							
	\end{tikzpicture}
\end{align*}
 %Trennzeile
\begin{align*}
\intertext{c) Scheinleistung}
S&=U\cdot I=230\,\volt\cdot 2{,}71\,\ampere=\uuline{623\,\volt\ampere} &\text{Scheinleistung}\\
P&=S\cdot \cos \varphi=623\,\volt\ampere\cdot \cos (-45\,\degree)=\uuline{447\,\watt} &\text{Wirkleistung}\\
Q&=S\cdot \sin \varphi=623\,\volt\ampere\cdot \sin (-45\,\degree)=\uuline{-447\,\volt\ampere r} &\text{Blindleistung}\\
\end{align*}
\clearpage
d) Annahme, daß sich die Gesamtschaltung induktiv verhält.
\begin{align*}
% \intertext{d) Annahme, daß sich die Gesamtschaltung induktiv verhält.}
&\text{Reihenfolge:  $\qquad (25\,\volt\per\centi\metre$; $1\,\ampere\per\centi\metre$)}\\
&\text{Beginne mit $\uline{U}=230\,\volt\cdot e^{j0\,\degree}$ und $\uline{I}=2{,}71\,\ampere\cdot e^{-j45\,\degree}$}\\
&\uline{I} \text{ um Winkel $\varphi=-45\,\degree$ nacheilend, da induktives Gesamtverhalten.}\\
&\uline{I}=\uline{I}_{R_C}+\uline{I}_C \qquad(\uline{I}_C \,\bot\, \uline{I}_{R_C}) \qquad (\uline{I}_C \text{ voreilend}) \qquad [\text{Thaleskreis}]\\
&\uline{U}_{R_C}\,||\,\uline{I}_{R_C}  \hspace{14em}   [\text{Gerade von der Spitze }\underline{U}]\\
&\uline{U}_{R_L}\,||\,\uline{I}  \hspace{15em}\, [\text{ Gerade } \bot\ \underline{U}_{RL}]\\	
&\uline{U}_L\,\bot\, \uline{I} \qquad (\uline{U}_L\text{ voreilend})\\
\end{align*}
\begin{align*}	
	\begin{tikzpicture}[very thick,scale=1.5]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
		  \draw [black!25!,very thin,scale=.5](0,-6)grid(10,4);
		  \draw [->](0,0)--(5.2,0)node[right]{$\Re$};
		  \draw [->](0,-3)--(0,2.2)node[above]{$\Im$};		  
			\draw [->,blue] (0,0)--(0:4.6)node [below right]{$\underline{U}$};
			\draw [->,red] (0,0)--(-45:2.71)node [above right]{$\underline{I}$};
			\draw [->,red] (0,0)++(-71.6:2.42)--+(18.4:1.21)node at(1.5,-2.2)[below]{$\underline{I}_{C}$};
			\draw [black!50!] (0,0)++(-71.6:2.42)+(18.4:.25)arc(18.4:108.4:.25);%Rechter Winkel
			\fill [black!50!] (0,0)++(-71.6:2.42)+(63.4:.125)circle(.02);%Rechter Winkel					
			\draw [->,red] (0,0)--(-71.6:2.42)node at(.5,-1.5)[left]{$\underline{I}_{R_C}$};	
			\draw [->,blue] (0,0)--(-45:2.168)node at(1,-1)[above right]{$\underline{U}_{R_L}$};
			\draw [black!50!] (0,0)++(-45:2.418)arc(-45:45:.25);%Rechter Winkel
			\fill [black!50!] (0,0)++(-45:2.168)+(0:.125)circle(.02);%Rechter Winkel			
			\draw [->,blue] (0,0)++(-45:2.168)--+(45:3.8)node at(3.5,.5)[left]{$\underline{U}_L$};			
			\draw [<-,blue] (0,0)++(0:4.6)--+(-251.6:1.21)node at(4.5,.5)[right]{$\underline{U}_{R_C}$};
			\draw [->,black!75!] (0,0)+(0:.5)arc(0:-45:.5)node at(-22.5:.5) [right]{$\varphi =-45\,\degree$};		
		\end{scope}	
		\begin{scope}[>=latex,xshift=.38cm,yshift=-1cm]%Parallelen
			\draw [double](0:-.1)--+(18.4:.1);		
		\end{scope}		
		\begin{scope}[>=latex,xshift=4.38cm,yshift=.8cm]%Parallelen
			\draw [double](0:-.1)--+(18.4:.1);		
		\end{scope}							
			\fill [black!60!](-45:1.355)circle(.05); %Mittelpunkt Thaleskreis	2		
			\draw [very thin](-45:2.71)arc(-45:-225:1.355); %Tahleskreis 2	
	\end{tikzpicture}
\end{align*}
\begin{align*}
\uline{U}&=\uuline{230\,\volt \cdot e^{j0\,\degree}} \ \widehat{=}\ 9{,}2\,\centi\metre\\
\uline{I}&=\uuline{2{,}71\,\ampere\cdot e^{-j45\,\degree}}\ \widehat{=}\ 2{,}71\,\centi\metre\\
\uline{I}_{R_C}&=\uuline{2{,}42\,\ampere\cdot e^{-j71{,}6\,\degree}}\ \widehat{=}\ 2{,}42\,\centi\metre\\
\uline{I}_C&=\uuline{1{,}21\,\ampere\cdot e^{j18{,}4\,\degree}}\ \widehat{=}\ 1{,}21\,\centi\metre\\
\uline{U}_{R_L}&=\uuline{108{,}4\,\volt \cdot e^{-j45\,\degree}}\ \widehat{=}\ 4{,}3\,\centi\metre\\
\uline{U}_{R_C}&=\uuline{60{,}4\,\volt \cdot e^{-j71{,}6\,\degree}}\ \widehat{=}\ 2{,}4\,\centi\metre\\
\uline{U}_L&=\uuline{190\,\volt \cdot e^{j45\,\degree}}\ \widehat{=}\ 7{,}6\,\centi\metre
\end{align*}
\clearpage
}{}%