187 lines
10 KiB
TeX
187 lines
10 KiB
TeX
\section {Übertrager mit Verbindung zum Eingang}
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\enlargethispage{3cm}
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Gegeben sind die Daten des abgebildeten Kreises:\\
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$R_1=28\,\ohm$, $X_1=96\,\ohm$, $R_2=14\,\ohm$, $X_M=60\,\ohm$, $X_2=400\,\ohm$, $\uline{I}_1=2\,\ampere\cdot e^{j0\,\degree}$\\
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Gesucht
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\renewcommand{\labelenumi}{\alph{enumi})}
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\begin{enumerate}
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\item Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei offenem Schalter S.
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\item Spannung $\uline{U}_S$ nach Betrag und Phase bei offenem Schalter S.
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\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$ nach Betrag und Phase bei geschlossenem Schalter S.
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\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung bei geschlossenem Schalter S.
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\end{enumerate}
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle
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% \draw (0,0)--(1,0) node at (.5,-.133) [right] {$U_1$};
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% \draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,0)--(.7,0) node at (.5,0)[left]{$\uline{U}_{ab}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
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\draw [->,red] (0.15,.1)--(.3,.1)node at(.225,.1)[above]{\footnotesize$\uline{I}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule|
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_1$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\fill(.825,.075)circle(.033);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Spule|
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_2$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\fill(.175,-.075)circle(.033);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};
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\draw [->,red] (.85,.1)--(.7,.1)node at(.775,.1)[above]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
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\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)
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(2.4,1)--(2.5,1)--(2.5,1.5)--(1.4,1.5)(1.1,1.5)--(.1,1.5)--(.1,1);
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\filldraw (0,1)circle(0.025cm)node[left]{a};
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\filldraw (0,0)circle(0.025cm)node[left]{b};
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\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);
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\draw [->,blue] (1,1.3)--(1.5,1.3) node at (1.25,1.3)[above]{$\uline{U}_S$};
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\draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.125)[below]{$X_M$};
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\draw[ultra thick](1.4,1.5)--+(150:.3cm)node at(1.4,1.7){$S$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\ifthenelse{\equal{\toPrint}{Lösung}}{%
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%\begin{align}
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%\intertext{Formeln:}
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%\end{align}
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\vspace{-.5cm}
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Berechnung: Ersatzschaltbild Schalter offen
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
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\draw node at(.5,-.2){\footnotesize$28$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$+j156$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule|
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$U_M$};
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\draw node at(.25,.25){\footnotesize$-j60$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$+j460$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};
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\draw node at(.5,-.2){\footnotesize$14$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
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\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);
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\fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm)
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(2.1,1.5)circle(.025cm)(1.9,1.5)circle(.025cm)
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(0,1)circle(0.025cm);
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\filldraw (0,1)circle(0.025cm)node[left]{a};
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\filldraw (0,0)circle(0.025cm)node[left]{b};
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\draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$};
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\draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$};
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\draw [->,blue] (1.5,1.85)--(2.5,1.85) node at (2,1.85)[above]{$\uline{U}_S$};
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\draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$0\,\volt$};
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\draw[ultra thick](2.1,1.5)--+(150:.2cm)node at(2,1.7){$S$};
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\draw [->,red] (0.5,1.6)--(.75,1.6)node at(.625,1.6)[above]{\footnotesize$\uline{I}_S=0$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\begin{enumerate}
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\item Spannung $\uline{U}_{ab}$ Schalter offen\\
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\vspace{-.5cm}
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\begin{align*}
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\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\\
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&=\uuline{200\,\volt\cdot e^{j73{,}74\,\degree}}
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\end{align*}
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\item Spannung $\uline{U}_S$ Schalter offen\\
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\vspace{-.5cm}
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\begin{align*}
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&\text{Wegen gegensinnigem Wicklungssinn $X_M=-60\,\ohm$ }\\
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\uline{U}_S&=[R_1+j(X_1-X_M)]\cdot \uline{I}_1=(28+j156)\,\ohm\cdot 2\,\ampere=(56+j312)\,\volt\\
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&=\uuline{317\,\volt\cdot e^{j79{,}8\,\degree}}
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\end{align*}
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\clearpage
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\enlargethispage{3cm}
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\item Strom $\uline{I}_2$ und Spannung $\uline{U}_{ab}$: Schalter geschlossen:
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\begin {align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
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\draw node at(.5,-.2){\footnotesize$28$};
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\draw [->,red] (0.15,.1)--(.3,.1)node at(.225,.1)[above]{\footnotesize$\uline{I}_1$};
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\draw [->,red] (0.15,.6)--(.3,.6)node at(.225,.6)[above]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_1-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$+j156$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule |
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw [<-,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$U_M$};
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\draw [->,red] (.25,-.1)--(.05,-.1)node at(.15,-.1)[right]{\footnotesize$\uline{I}=\uline{I}_1+\uline{I}_2$};
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\draw node at(.25,.25){\footnotesize$-j60$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_2-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$+j460$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};
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\draw node at(.5,-.2){\footnotesize$14$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
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\draw (0,0)--(4,0) (3.8,1)--(4,1)--(4,1.5)--(.1,1.5)--(.1,1);
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\fill (4,0)circle(0.025cm)(.1,1)circle(0.025cm)
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(0,1)circle(0.025cm);
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\filldraw (0,1)circle(0.025cm)node[left]{a};
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\filldraw (0,0)circle(0.025cm)node[left]{b};
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\draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_{ab}$};
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\draw [->,blue] (.5,.7)--(1.5,.7) node at (1,.7)[below]{$\uline{U}_S$};
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\draw [->,blue] (3.5,.7)--(2.5,.7) node at (3,.7)[below]{$\uline{U}_S$};
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\draw [->,blue] (1,.4)--(1,.1) node at (1,.25)[right]{$\uline{U}_1$};
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\draw node at (1,2)[above]{$\uline{Z}_1$};
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\draw node at (3,2)[above]{$\uline{Z}_2$};
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\draw node at (1,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$};
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\draw node at (3,2)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxx}}_{}$};
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\end{scope}
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\end{tikzpicture}
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\end {align*}
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\vspace{-.5cm}
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Entweder Stromteiler oder, weil Strom $\uline{I}_1$ wie in b)\\
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\begin{align*}
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\uline{U}_S&=\uline{Z}_1\cdot \uline{I}_1=\uline{Z}_2\cdot \uline{I}_2\\
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\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}\\
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&=(0{,}681-j0{,}101)\,\ampere=\uuline{0{,}689\,\ampere\cdot e^{-j8{,}43\,\degree}}\\
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\uline{U}_{ab}&=\uline{U}_S+\uline{U}_M\\
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&=\uline{Z}_1\cdot \uline{I}_1+jX_M\cdot (\uline{I}_1+\uline{I}_2)\\
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&=(56+j312)\,\volt+\underbrace{\overbrace{(2+0{,}681-j0{,}101)\,\ampere}^{\uline{I}}\cdot (-j60)\,\ohm}_{\uline{U}_M=(-6{,}06-j160{,}81)\,\volt}\\
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&=(50+j151)\,\volt=\uuline{159\,\volt\cdot e^{+j71{,}7\,\degree}}\\
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\end{align*}
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\vspace{-1.5cm}
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\item Wirkleistung (über das Magnetfeld übertragen)
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\begin{align*}
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\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\\
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&=\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}\\
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P&=\uuline{12{,}2\,\watt}\text{ Von Sekundär- nach Primärseite}
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\end{align*}
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\end{enumerate}
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Kontrolle:
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\begin{align*}
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\uline{S}_2&=(\uline{U}_M+\uline{I}_2\cdot j460\,\ohm)\cdot \uline{I}^*_2=(12{,}1+j108)\,\volt\ampere\\
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(P_2&=12{,}1\,\watt\text{ Verbraucher})
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\end{align*}
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\clearpage
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}{}%
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