ET2_Uebung_BEI/ET2_L_B18_A3.tex

\section {Übertrager mit Verbindung zum Eingang}
\enlargethispage{3cm}
Gegeben sind die Daten des abgebildeten Kreises:\\
$R_1=28\,\ohm$, $X_1=96\,\ohm$, $R_2=14\,\ohm$, $X_M=60\,\ohm$, $X_2=400\,\ohm$, $\uline{I}_1=2\,\ampere\cdot e^{j0\,\degree}$\\
Gesucht
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\item Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei offenem Schalter S.
\item Spannung $\uline{U}_S$ nach Betrag und Phase bei offenem Schalter S.
\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei geschlossenem Schalter S.
\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung bei geschlossenem Schalter S.
\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] (.3,0)--(.7,0) node at (.5,0)[left]{$\uline{U}_{ab}$};	
		\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] (0.15,.1)--(.3,.1)node at(.225,.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] (.85,.1)--(.7,.1)node at(.775,.1)[above]{\footnotesize$\uline{I}_2$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw (0,0)--(1,0)--(1,.2) (1.5,0)--(2.5,0)--(2.5,-.25)--(.1,-.25)--(.1,0) (.9,1)--(1,1)--(1,.9)(1.6,1)--(1.5,1)--(1.5,.9)
(2.4,1)--(2.5,1)--(2.5,1.5)--(1.4,1.5)(1.1,1.5)--(.1,1.5)--(.1,1);
            \filldraw (0,1)circle(0.025cm)node[left]{a};
            \filldraw (0,0)circle(0.025cm)node[left]{b};
            \fill(.1,0)circle(0.025cm)(.1,1)circle(0.025cm)            (1.1,1.5)circle(0.025cm)(1.4,1.5)circle(0.025cm);
			\draw [->,blue] (1,1.3)--(1.5,1.3) node at (1.25,1.3)[above]{$\uline{U}_S$};
            \draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.125)[below]{$X_M$};
            \draw[ultra thick](1.4,1.5)--+(150:.3cm)node at(1.4,1.7){$S$};
		\end{scope}	
	\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
\vspace{-.5cm}
Berechnung: Ersatzschaltbild Schalter offen
\begin{align*}
	\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$28$};
		\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$+j156$};				
        \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$U_M$};
			\draw node at(.25,.25){\footnotesize$-j60$};
		\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$+j460$};				
        \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$14$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw (0,0)--(4,0) (3.8,1)--(4,1)--(4,1.5)--(2.1,1.5)(1.9,1.5)--(.1,1.5)--(.1,1);
            \fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm)
            (2.1,1.5)circle(.025cm)(1.9,1.5)circle(.025cm)
            (0,1)circle(0.025cm);
            \filldraw (0,1)circle(0.025cm)node[left]{a};
            \filldraw (0,0)circle(0.025cm)node[left]{b};
			\draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$};
		    \draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$};
		    \draw [->,blue] (1.5,1.85)--(2.5,1.85) node at (2,1.85)[above]{$\uline{U}_S$};
		    \draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$0\,\volt$};
            \draw[ultra thick](2.1,1.5)--+(150:.2cm)node at(2,1.7){$S$};
\draw [->,red] (0.5,1.6)--(.75,1.6)node at(.625,1.6)[above]{\footnotesize$\uline{I}_S=0$};
            \end{scope}	
	\end{tikzpicture}
\end{align*}
\begin{enumerate}
\item Spannung $\uline{U}_{ab}$ Schalter offen\\
\vspace{-.5cm}
\begin{align*}
\uline{U}_{ab}&=[R_1+j(X_1-\cancel{X_M}+\cancel{X_M})]\cdot \uline{I}_1=(28+j96)\,\ohm\cdot 2\,\ampere=(56+j192)\,\volt\\
&=\uuline{200\,\volt\cdot e^{j73{,}74\,\degree}}
\end{align*}
\item Spannung $\uline{U}_S$  Schalter offen\\
\vspace{-.5cm}
\begin{align*}
&\text{Wegen gegensinnigem Wicklungssinn  $X_M=-60\,\ohm$ }\\
\uline{U}_S&=[R_1+j(X_1-X_M)]\cdot \uline{I}_1=(28+j156)\,\ohm\cdot 2\,\ampere=(56+j312)\,\volt\\
&=\uuline{317\,\volt\cdot e^{j79{,}8\,\degree}}
\end{align*}
\clearpage
\enlargethispage{3cm}
\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$: Schalter geschlossen:
\begin {align*}
	\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$28$};
			\draw [->,red] (0.15,.1)--(.3,.1)node at(.225,.1)[above]{\footnotesize$\uline{I}_1$};
			\draw [->,red] (0.15,.6)--(.3,.6)node at(.225,.6)[above]{\footnotesize$\uline{I}_2$};
		\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$+j156$};				
        \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$U_M$};
			\draw [->,red] (.25,-.1)--(.05,-.1)node at(.15,-.1)[right]{\footnotesize$\uline{I}=\uline{I}_1+\uline{I}_2$};
			\draw node at(.25,.25){\footnotesize$-j60$};
		\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$+j460$};				
        \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$14$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
			\draw (0,0)--(4,0) (3.8,1)--(4,1)--(4,1.5)--(.1,1.5)--(.1,1);
            \fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm)
            (0,1)circle(0.025cm);
            \filldraw (0,1)circle(0.025cm)node[left]{a};
            \filldraw (0,0)circle(0.025cm)node[left]{b};
			\draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$};
		    \draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$};
		    \draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$\uline{U}_S$};
		    \draw [->,blue] (1,.4)--(1,.1) node at (1,.25)[right]{$\uline{U}_1$};        
            \draw node at (1,2)[above]{$\uline{Z}_1$};
            \draw node at (3,2)[above]{$\uline{Z}_2$};
            \draw node at (1,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$};
            \draw node at (3,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$};
            \end{scope}	
	\end{tikzpicture}
\end {align*}
\vspace{-.5cm}
Entweder Stromteiler oder, weil Strom $\uline{I}_1$ wie in b)\\
\begin{align*}
\uline{U}_S&=\uline{Z}_1\cdot \uline{I}_1=\uline{Z}_2\cdot \uline{I}_2\\
\uline{I}_2&=\uline{I}_1\cdot \frac{\uline{Z}_1}{\uline{Z}_2}=2\,\ampere\cdot e^{j0\,\degree}\cdot \frac{(28+j156)\,\ohm}{(14+j460)\,\ohm}\\
&=(0{,}681-j0{,}101)\,\ampere=\uuline{0{,}689\,\ampere\cdot e^{-j8{,}43\,\degree}}\\
\uline{U}_{ab}&=\uline{U}_S+\uline{U}_M\\
&=\uline{Z}_1\cdot \uline{I}_1+jX_M\cdot (\uline{I}_1+\uline{I}_2)\\
&=(56+j312)\,\volt+\underbrace{\overbrace{(2+0{,}681-j0{,}101)\,\ampere}^{\uline{I}}\cdot (-j60)\,\ohm}_{\uline{U}_M=(-6{,}06-j160{,}81)\,\volt}\\
&=(50+j151)\,\volt=\uuline{159\,\volt\cdot e^{+j71{,}7\,\degree}}\\
\end{align*}
\vspace{-1.5cm}
\item Wirkleistung (über das Magnetfeld übertragen)
\begin{align*}
\uline{S}_1&=\uline{U}_1\cdot \uline{I}^*_1=[\uline{I}_1\cdot j(X_1-X_M)+\underbrace{\uline{I}\cdot jX_M}_{\uline{U}_M}]\cdot \uline{I}^*_1\\
&=\underbrace{[\underbrace{2\,\ampere\cdot j156\,\ohm}_{j312\,\volt}+(-6{,}1-j161)\,\volt]}_{(-6{,}1+j151)\,\volt}\cdot 2\,\ampere=\uuline{(\overbrace{-12{,}2}^{\text{Quelle}}+j302)\,\volt\ampere}\\
P&=\uuline{12{,}2\,\watt}\text{ Von Sekundär- nach Primärseite}
\end{align*}
\end{enumerate}
Kontrolle:
\begin{align*}
\uline{S}_2&=(\uline{U}_M+\uline{I}_2\cdot j460\,\ohm)\cdot \uline{I}^*_2=(12{,}1+j108)\,\volt\ampere\\
(P_2&=12{,}1\,\watt\text{ Verbraucher})
\end{align*}
\clearpage
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