ET2_Uebung_BEI/ET2_L_B18_A2.tex

\section {Übertrager mit kapazitiver Last}
Von nebenstehender Schaltung ist gegeben:\\
$\uline{U}_1=1\,\volt\cdot e^{j0\,\degree}$, $R_1=10\,\ohm$, $X_1=100\,\ohm$, $R_2=40\,\ohm$, $X_2=400\,\ohm$, $X_C=-200\,\ohm$, $X_M=40\,\ohm$\\

Gesucht\\
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Spannung $\uline{U}_2$ nach Betrag und Phase bei offenem Schalter S.
\item Spannung $\uline{U}_2$ nach Betrag und Phase bei geschlossenem Schalter S.
\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung für den Fall b).
\end{enumerate}
\begin{align*}
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle
			\draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$};
			\draw (.5,0)circle(.133);	
			\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_1$};	
		\end{scope}				
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
            \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
			\draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$};
		\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) [left] {$X_1$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);	
            \fill(.825,.075)circle(.033);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule |	
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_2$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);	
            \fill(.175,-.075)circle(.033);
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand
            \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};
			\draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_2$};
		\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,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw (0,0)--(1,0)--(1,.2) (3,.2)--(3,0)--(1.5,0)--(1.5,.2) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9)(3,.8)--(3,1)--(2.8,1);
            \fill (2.5,0)circle(0.025cm)(2.5,1)circle(0.025cm)(2.8,1)circle(0.025cm);
			\draw [->,blue] (2.5,.8)--(2.5,.2) node at (2.5,.5)[left]{$\uline{U}_2$};
        \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$X_M$};
        \draw[ultra thick](2.8,1)--+(150:.3cm)node at(2.7,1.2){$S$};
		\end{scope}	
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:\\
\begin{enumerate}
\item Offener Schalter\\
\begin{align*}
\uline{I}_2&=0\quad\\
&\text{Wegen Wicklungssinn $\uline{U}_2$ negativ (bzw. $X_M$ negativ)}\\
\uline{U}_2&=jX_M\cdot \uline{I}_1=jX_M\cdot \frac{\uline{U}_1}{R+jX_1}=\frac{(-j40\cancel{\,\ohm})\cdot 1\,\volt}{(10+j100)\cancel{\,\ohm}}=(0{,}396-j0{,}04)\,\volt\\
&=\uuline{0{,}398\,\volt\cdot e^{-j174{,}3\,\degree}}\\
\end{align*}
\clearpage
\item Ersatzschaltbild: Schalter geschlossen\\
	\begin{tikzpicture}[scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
            \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
			\draw node at(.5,-.2){\footnotesize$10$};
		\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] {$X_1-X_M$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw node at(.5,-.2){\footnotesize$+j140$};				
        \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) [left] {$X_M$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);	
			\draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{U}_M$};
			\draw node at(.25,.1)[left]{\footnotesize$-j40$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule -			
			\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$};
			\fill (.3,-0.0667)rectangle(.7,0.0667);
			\draw node at(.5,-.2){\footnotesize$+j440$};				
        \end{scope}
		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand
            \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};
			\draw node at(.5,-.2){\footnotesize$40$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=4cm,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] {$X_C$};
 			\draw node at(.25,-.05)[right]{\footnotesize$-j200$};
            \draw[->,red](.9,-.1)--(.6,-.1)node at(.75,-.1)[right]{$I_2$};
		\end{scope}		
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw (0,0)--(4,0)--(4,.2) (3.8,1)--(4,1)--(4,.8);
            \fill (0,0)circle(0.025cm)(0,1)circle(0.025cm);
			\draw [->,blue] (3.8,.7)--(3.8,.3) node at (3.75,.5)[left]{$\uline{U}_2$};
		    \draw [->,blue!50!red] (3.5,.7)--(2.5,.7) node at (3,.7)[above]{$\uline{U}'_2$};	
		    \draw [->,blue!50!red,thick] (3,.075)arc(270:0:.25cm) node at (3,.325){\scriptsize{Masche}};
        \draw [->,blue!50!red]  node at (5,.7)[right]{$\uline{U}'_2$ \footnotesize{Kontrollrechnung}};
            \draw node at (0,.5)[left]{$\uline{Z}_{ges}\Rightarrow$};
            \draw node at (1,1.5)[above]{$\uline{Z}_1$};
            \draw node at (3,1.5)[above]{$\uline{Z}_2$};
            \draw node at (1,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$};
            \draw node at (3,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$};
            \draw node at (3.25,-.5)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxx}}_{}$};
            \draw node at (3.25,-.25)[below]{$\uline{Z}'=(40+j240)\,\ohm=243{,}3\,\ohm\cdot e^{j80{,}5\,\degree}$};
            \draw node at (3,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxxxxxx}}_{}$};
            \draw node at (3.25,-.75)[below]{$\uline{Z}_{||}=\frac{(-j40)\cdot (40+j240)}{40+j200}\,\ohm=(1{,}54-j47{,}7)\,\ohm=47{,}72\,\ohm\cdot e^{-j88{,}2\,\degree}$};
            \end{scope}	
%            \draw[dashed,blue!50!red](3,.9)--(3,.1)node at (3,.5)[right]{$P_2$};
	\end{tikzpicture}
\begin{align*}
\uline{Z}_{ges}&=\uline{Z}_1+\uline{Z}_{||}=(11{,}54+j92{,}3)\,\ohm=93{,}03\,\ohm\cdot e^{j82{,}87\,\degree}
\intertext{Spannungsteiler}
\uline{U}_M&=\uline{U}_1\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{ges}}=1\,\volt\cdot e^{j0\,\degree}\cdot \frac{47{,}72\,\ohm\cdot e^{-j88{,}2\,\degree}}{93{,}03\,\ohm\cdot e^{j82{,}87\,\degree}}=(-0{,}51-j0{,}08)\,\volt=0{,}51\,\volt\cdot e^{-j171{,}1\,\degree}\\
\uline{U}_2&=\uline{U}_M\cdot \frac{jX_C}{\uline{Z}'}=0{,}51\,\volt\cdot e^{-j171{,}1\,\degree}\cdot \frac{200\,\ohm\cdot e^{-j90\,\degree}}{243{,}3\,\ohm\cdot e^{j80{,}5\,\degree}}\\
&=(0{,}4+j0{,}13)\,\volt=\uuline{0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}}
\end{align*}
\item Wirkleistung
\begin{align*}
P_2&=I^2_2\cdot R_2\\
\uline{I}_2&=\frac{\uline{U}_2}{jX_C}=\frac{0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}}{-j200\,\ohm}=2{,}108\,\milli\ampere\cdot e^{+j108{,}4\,\degree}=(-0{,}666+j2)\,\milli\ampere\\
P_2&=(2{,}108\,\milli\ampere)^2\cdot 40\,\ohm=1{,}78\cdot \power{10}{-4}\,\watt=\uuline{178\,\micro\watt}\\
&\text{Von Primär- nach Sekundärseite}\\
\intertext{Alternativ, in 2 Schritten}
U_{R_2}&=I_2\cdot R_2=2{,}108\,\milli\ampere\cdot 40\,\ohm=84{,}32\,\milli\volt\\
P_2&=U_{R_2}\cdot I_2=84{,}32\,\milli\volt\cdot 2{,}108\,\milli\ampere=178\micro\watt\\
\end{align*}
\end{enumerate}
Kontrolle: (über das Magnetfeld übertragenen Wirkleistung)
\begin{align*}
\uline{U}'_2&=\uline{U}_2 - \uline{U}_M=(0{,}4+j0{,}13)-(-0{,}51-j0{,}08))\,\volt=(0{,}91+j0{,}21)\,\volt=0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\\
&\text{oder auch aus den Impedanzen berechnet}\\
\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)]=-(-0{,}666+j2)\,\milli\ampere\cdot (40+j440)\,\ohm\\
&=(0{,}907+j0{,}213)\,\volt=0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\\[\baselineskip]
&\text{mit }S_2=\uline{U}'_2\cdot \uline{I}^*_2\\[\baselineskip]
P_2&=\Re\{\uline{U}'_2\cdot \uline{I}^*_2\}\\
&=\Re\{(0{,}91+j0{,}21)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\
&=\Re\{0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\
&=\Re\{1{,}965\cdot e^{-j95{,}18\,\degree}\}\,\micro\volt\ampere
=\Re\{\underbrace{-177{,}4}_{\text{EPS$\rightarrow$Verbraucher}}-j1{,}957\}\,\micro\volt\ampere\\
&\text{\footnotesize{Erzeuger-Pfeil-System (EPS)}}
\end{align*}

\begin{align*}
	\begin{tikzpicture}[very thick,scale=10]
        \draw[thin,black!50!,step=.1cm](-.1,-.3001)grid(1,.6);
        \draw[black!25!]+(18.42:.934cm)arc(18.42:198.42:.467cm); % Thales
%       \draw[black!25!](18.42:.467cm)circle(.467cm); % Thales
        \draw[<-,blue](0,0)++(18.4:.42cm)--+(8.9:.51cm)node at(.42,.175)[right]{$\uline{U}_M$};
        \draw[->,magenta](0,0)--(12.99:.934cm)node at(.7,.15)[below left]{$\uline{U}'_2$};
        \draw[->,red](0,0)--(108.46:.21cm)node [above]{$\uline{I}_2$};
        \draw[->,red](0,0)--(-108.4:.21cm)node [below]{$\uline{I}^*_2$};
        \draw[->,black!50!green](0,0)++(12.99:.934cm)--+(108.46:.084cm)node [below right]{$\uline{U}_{R2}=I_2\cdot R_2$};
        \draw[->,black!50!green](0,0)--(18.4:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$};
        \draw[->,blue](0,0)--++(18.4:.42cm)node [above left]{$\uline{U}_2$};
        \draw node at(1.05,.55)[right]{$\uline{U}'_2=\uline{U}_2 - \uline{U}_M=0{,}932\cdot \,\volt\cdot e^{j13{,}22\,\degree}$};
        \draw node at(1.05,.45)[right]{$\uline{U}_{R_2}\quad||\quad\uline{I}_2$};
        \draw node at(1.05,.35)[right]{$\uline{U}_{(X_2-X_M)}\quad||\quad\uline{U}_2 \quad\bot\quad\uline{I}_2$};
        \draw node at(1.05,-.05)[right]{$S=\uline{U}'_2\cdot \uline{I}^*_2=1965\,\micro\volt\ampere$};
        \draw node at(1.05,-.15)[right]{$P=S\cdot \cos(-95{,}18\,\degree)=-177\micro\watt$};
        \draw node at(1.05,-.25)[right]{$Q=S\cdot \sin(-95{,}18\,\degree)=-1957\micro\volt\ampere r$};
        \draw[black!75!](0,0)++(108.4:.1cm)arc(108.4:12.99:.1cm)node at(90:.1cm) [above right]{$\varphi=-95{,}18\,\degree$};
	\end{tikzpicture}
\end{align*}
\clearpage
}{}%
%%%%delete
%\uline{U}'_2&=-\uline{I}_2\cdot(\uline{X}_2-\uline{X}_M)=-(-0{,}666+j2)\,\milli\ampere\cdot (j440)\,\ohm=(0{,}88+j0{,}29)\,\volt\\
%\uline{U}'_2+\uline{U}_M&=(0{,}88+j0{,}29)\,\volt+(-0{,}51-j0{,}08)\,\volt=(0{,}37+j0{,}21)\,\volt=0{,}425\,\volt\cdot e^{j29{,}58\,\degree}\\
%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\
%&=\Re\{(0{,}37+j0{,}21)\,\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\}=\Re\{0{,}425\,\volt\cdot e^{j29{,}58\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\\
%&=\Re\{0{,}896\cdot e^{-j78{,}82\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{+174}_{\text{EPS$\rightarrow$Quelle}}-j879\}\,\micro\volt\ampere\\[\baselineskip]
%%
%&\text{Langer Pfeil auch über XC}\\
%\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)+\uline{X}_C]=-(-0{,}666+j2)\,\milli\ampere\cdot (40+j240)\,\ohm\\
%&=(0{,}507+j0{,}08)\,\volt=0{,}513\,\volt\cdot e^{j8{,}96\,\degree}\\
%\uline{U}'_2+\uline{U}_M&=(0{,}507+j0{,}08)\,\volt+(-0{,}51-j0{,}08)\,\volt=0\\[\baselineskip]
%&\text{Kurzer Pfeil + C2 ohne R2}\\
%\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)+\uline{X}_C]=-(-0{,}666+j2)\,\milli\ampere\cdot (+j240)\,\ohm\\
%&=(0{,}480+j0{,}160)\,\volt=0{,}505\,\volt\cdot e^{j18{,}42\,\degree}\\
%\uline{U}'_2+\uline{U}_M&=(0{,}507+j0{,}08)\,\volt+(-0{,}51-j0{,}08)\,\volt=(0{,}480+j0{,}160)\,\volt=0{,}505\,\volt\cdot e^{j18{,}44\,\degree}\\[\baselineskip]
%&\text{Kurzer Pfeil}\\
%\uline{U}'_2&=-\uline{I}_2\cdot (\uline{X}_2-\uline{X}_M)=-(-0{,}666+j2)\,\milli\ampere\cdot (+j440)\,\ohm\\
%&=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{-j18{,}42\,\degree}\\
%\uline{U}'_2+\uline{U}_M&=(0{,}880+j0{,}293)\,\volt+(-0{,}51-j0{,}08)\,\volt
%=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{j18{,}42\,\degree}\\[\baselineskip]
%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\
%&=\Re\{(0{,}880+j0{,}293)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\
%&=\Re\{0{,}927\,\milli\volt\cdot e^{j18{,}42\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\
%&=\Re\{1{,}954\cdot e^{-j89{,}98\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{785}_{\text{EPS$\rightarrow$Quelle}}+j1954\}\,\micro\volt\ampere\\[\baselineskip]
%&\text{seine alte Lösung }(0{,}4+j0{,}13)=0{,}4206\,\volt\cdot e^{-j18{,}0\,\degree}\\
%\uline{U}''_x&=\uline{U}_2-\uline{U}_M=(0{,}4+j0{,}13)\,\volt-(-0{,}51-j0{,}08)\,\volt=(0{,}91+j0{,}21)\,\volt\\
%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}=(-0{,}91-j0{,}21)\,\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\\
%&=\Re\{\underbrace{+178}_{\text{EPS$\rightarrow$Quelle}}+j1960\}\,\micro\volt\ampere\\[\baselineskip]
%        \draw[->,black!50!green](0,0)++(18.4:.42cm)--+(-171.1:.51cm)node[left]{$\uline{U}_M$};
%        \draw[->,black!75!green](0,0)++(18.4:.42cm)++(-171.1:.51cm)--(0:0cm)node[left]{$\uline{U}_2 - \uline{U}_M$};
%        \draw[->,black!85!green](0,0)--(8.4:.89cm)node[right]{$\uline{U}''_x$};
%        \draw[->](0,0)--(8.9:.51cm);%{$\uline{U}_M$};
%        \draw[->,black!50!green](0,0)--(-171.1:.51cm)node[left]{$\uline{U}_M$};
%        \draw[->,magenta](0,0)--(.91,.21)node at(.7,.15)[below left]{$\uline{U}'_2$};
%        \draw[->,black](0,0)++(12.99:.934cm)--+(108.46:.15cm)node [below right]{$\uline{U}_{R2}$};
%        \draw[->,black](0,0)--(22:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$};