204 lines
14 KiB
TeX
204 lines
14 KiB
TeX
\section {Übertrager mit kapazitiver Last}
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Von nebenstehender Schaltung ist gegeben:\\
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$\uline{U}_1=1\,\volt\cdot e^{j0\,\degree}$, $R_1=10\,\ohm$, $X_1=100\,\ohm$, $R_2=40\,\ohm$, $X_2=400\,\ohm$, $X_C=-200\,\ohm$, $X_M=40\,\ohm$\\
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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 $\uline{U}_2$ nach Betrag und Phase bei offenem Schalter S.
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\item Spannung $\uline{U}_2$ nach Betrag und Phase bei geschlossenem Schalter S.
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\item Größe und Richtung der über das Magnetfeld übertragenen Wirkleistung für den Fall b).
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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,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}_1$};
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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,.1)--(.25,.1)node at(.125,.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] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator|
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$};
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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) (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);
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\fill (2.5,0)circle(0.025cm)(2.5,1)circle(0.025cm)(2.8,1)circle(0.025cm);
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\draw [->,blue] (2.5,.8)--(2.5,.2) node at (2.5,.5)[left]{$\uline{U}_2$};
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\draw[<->](1.5,1.125)arc(60:120:.5cm)node at(1.25,1.25)[above]{$X_M$};
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\draw[ultra thick](2.8,1)--+(150:.3cm)node at(2.7,1.2){$S$};
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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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Berechnung:\\
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\begin{enumerate}
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\item Offener Schalter\\
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\begin{align*}
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\uline{I}_2&=0\quad\\
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&\text{Wegen Wicklungssinn $\uline{U}_2$ negativ (bzw. $X_M$ negativ)}\\
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\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\\
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&=\uuline{0{,}398\,\volt\cdot e^{-j174{,}3\,\degree}}\\
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\end{align*}
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\clearpage
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\item Ersatzschaltbild: Schalter geschlossen\\
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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$10$};
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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$+j140$};
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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$\uline{U}_M$};
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\draw node at(.25,.1)[left]{\footnotesize$-j40$};
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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$+j440$};
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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$40$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm,rotate=90]%Kondensator |
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$};
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\draw node at(.25,-.05)[right]{\footnotesize$-j200$};
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\draw[->,red](.9,-.1)--(.6,-.1)node at(.75,-.1)[right]{$I_2$};
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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)--(4,0)--(4,.2) (3.8,1)--(4,1)--(4,.8);
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\fill (0,0)circle(0.025cm)(0,1)circle(0.025cm);
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\draw [->,blue] (3.8,.7)--(3.8,.3) node at (3.75,.5)[left]{$\uline{U}_2$};
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\draw [->,blue!50!red] (3.5,.7)--(2.5,.7) node at (3,.7)[above]{$\uline{U}'_2$};
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\draw [->,blue!50!red,thick] (3,.075)arc(270:0:.25cm) node at (3,.325){\scriptsize{Masche}};
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\draw [->,blue!50!red] node at (5,.7)[right]{$\uline{U}'_2$ \footnotesize{Kontrollrechnung}};
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\draw node at (0,.5)[left]{$\uline{Z}_{ges}\Rightarrow$};
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\draw node at (1,1.5)[above]{$\uline{Z}_1$};
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\draw node at (3,1.5)[above]{$\uline{Z}_2$};
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\draw node at (1,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$};
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\draw node at (3,1.5)[below]{$\overbrace{\phantom{xxxxxxxxxxxxxxxxx}}_{}$};
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\draw node at (3.25,-.5)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxx}}_{}$};
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\draw node at (3.25,-.25)[below]{$\uline{Z}'=(40+j240)\,\ohm=243{,}3\,\ohm\cdot e^{j80{,}5\,\degree}$};
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\draw node at (3,-1)[above]{$\underbrace{\phantom{xxxxxxxxxxxxxxxxxxxxx}}_{}$};
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\draw node at (3.25,-.75)[below]{$\uline{Z}_{||}=\frac{(-j40)\cdot (40+j240)}{40+j200}\,\ohm=(1{,}54-j47{,}7)\,\ohm=47{,}72\,\ohm\cdot e^{-j88{,}2\,\degree}$};
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\end{scope}
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% \draw[dashed,blue!50!red](3,.9)--(3,.1)node at (3,.5)[right]{$P_2$};
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\end{tikzpicture}
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\begin{align*}
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\uline{Z}_{ges}&=\uline{Z}_1+\uline{Z}_{||}=(11{,}54+j92{,}3)\,\ohm=93{,}03\,\ohm\cdot e^{j82{,}87\,\degree}
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\intertext{Spannungsteiler}
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\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}\\
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\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}}\\
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&=(0{,}4+j0{,}13)\,\volt=\uuline{0{,}42\,\volt\cdot e^{+j18{,}4\,\degree}}
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\end{align*}
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\item Wirkleistung
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\begin{align*}
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P_2&=I^2_2\cdot R_2\\
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\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\\
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P_2&=(2{,}108\,\milli\ampere)^2\cdot 40\,\ohm=1{,}78\cdot \power{10}{-4}\,\watt=\uuline{178\,\micro\watt}\\
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&\text{Von Primär- nach Sekundärseite}\\
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\intertext{Alternativ, in 2 Schritten}
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U_{R_2}&=I_2\cdot R_2=2{,}108\,\milli\ampere\cdot 40\,\ohm=84{,}32\,\milli\volt\\
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P_2&=U_{R_2}\cdot I_2=84{,}32\,\milli\volt\cdot 2{,}108\,\milli\ampere=178\micro\watt\\
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\end{align*}
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\end{enumerate}
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Kontrolle: (über das Magnetfeld übertragenen Wirkleistung)
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\begin{align*}
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\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}\\
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&\text{oder auch aus den Impedanzen berechnet}\\
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\uline{U}'_2&=-\uline{I}_2\cdot [R_2+(\uline{X}_2-\uline{X}_M)]=-(-0{,}666+j2)\,\milli\ampere\cdot (40+j440)\,\ohm\\
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&=(0{,}907+j0{,}213)\,\volt=0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\\[\baselineskip]
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&\text{mit }S_2=\uline{U}'_2\cdot \uline{I}^*_2\\[\baselineskip]
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P_2&=\Re\{\uline{U}'_2\cdot \uline{I}^*_2\}\\
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&=\Re\{(0{,}91+j0{,}21)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\
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&=\Re\{0{,}932\,\volt\cdot e^{j13{,}22\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\
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&=\Re\{1{,}965\cdot e^{-j95{,}18\,\degree}\}\,\micro\volt\ampere
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=\Re\{\underbrace{-177{,}4}_{\text{EPS$\rightarrow$Verbraucher}}-j1{,}957\}\,\micro\volt\ampere\\
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&\text{\footnotesize{Erzeuger-Pfeil-System (EPS)}}
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\end{align*}
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\begin{align*}
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\begin{tikzpicture}[very thick,scale=10]
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\draw[thin,black!50!,step=.1cm](-.1,-.3001)grid(1,.6);
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\draw[black!25!]+(18.42:.934cm)arc(18.42:198.42:.467cm); % Thales
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% \draw[black!25!](18.42:.467cm)circle(.467cm); % Thales
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\draw[<-,blue](0,0)++(18.4:.42cm)--+(8.9:.51cm)node at(.42,.175)[right]{$\uline{U}_M$};
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\draw[->,magenta](0,0)--(12.99:.934cm)node at(.7,.15)[below left]{$\uline{U}'_2$};
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\draw[->,red](0,0)--(108.46:.21cm)node [above]{$\uline{I}_2$};
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\draw[->,red](0,0)--(-108.4:.21cm)node [below]{$\uline{I}^*_2$};
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\draw[->,black!50!green](0,0)++(12.99:.934cm)--+(108.46:.084cm)node [below right]{$\uline{U}_{R2}=I_2\cdot R_2$};
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\draw[->,black!50!green](0,0)--(18.4:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$};
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\draw[->,blue](0,0)--++(18.4:.42cm)node [above left]{$\uline{U}_2$};
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\draw node at(1.05,.55)[right]{$\uline{U}'_2=\uline{U}_2 - \uline{U}_M=0{,}932\cdot \,\volt\cdot e^{j13{,}22\,\degree}$};
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\draw node at(1.05,.45)[right]{$\uline{U}_{R_2}\quad||\quad\uline{I}_2$};
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\draw node at(1.05,.35)[right]{$\uline{U}_{(X_2-X_M)}\quad||\quad\uline{U}_2 \quad\bot\quad\uline{I}_2$};
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\draw node at(1.05,-.05)[right]{$S=\uline{U}'_2\cdot \uline{I}^*_2=1965\,\micro\volt\ampere$};
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\draw node at(1.05,-.15)[right]{$P=S\cdot \cos(-95{,}18\,\degree)=-177\micro\watt$};
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\draw node at(1.05,-.25)[right]{$Q=S\cdot \sin(-95{,}18\,\degree)=-1957\micro\volt\ampere r$};
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\draw[black!75!](0,0)++(108.4:.1cm)arc(108.4:12.99:.1cm)node at(90:.1cm) [above right]{$\varphi=-95{,}18\,\degree$};
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\end{tikzpicture}
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\end{align*}
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\clearpage
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}{}%
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%%%%delete
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%\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\\
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%\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}\\
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%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\
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%&=\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}\}\\
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%&=\Re\{0{,}896\cdot e^{-j78{,}82\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{+174}_{\text{EPS$\rightarrow$Quelle}}-j879\}\,\micro\volt\ampere\\[\baselineskip]
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%%
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%&\text{Langer Pfeil auch über XC}\\
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%\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\\
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%&=(0{,}507+j0{,}08)\,\volt=0{,}513\,\volt\cdot e^{j8{,}96\,\degree}\\
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%\uline{U}'_2+\uline{U}_M&=(0{,}507+j0{,}08)\,\volt+(-0{,}51-j0{,}08)\,\volt=0\\[\baselineskip]
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%&\text{Kurzer Pfeil + C2 ohne R2}\\
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%\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\\
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%&=(0{,}480+j0{,}160)\,\volt=0{,}505\,\volt\cdot e^{j18{,}42\,\degree}\\
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%\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]
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%&\text{Kurzer Pfeil}\\
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%\uline{U}'_2&=-\uline{I}_2\cdot (\uline{X}_2-\uline{X}_M)=-(-0{,}666+j2)\,\milli\ampere\cdot (+j440)\,\ohm\\
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%&=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{-j18{,}42\,\degree}\\
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%\uline{U}'_2+\uline{U}_M&=(0{,}880+j0{,}293)\,\volt+(-0{,}51-j0{,}08)\,\volt
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%=(0{,}880+j0{,}293)\,\volt=0{,}927\,\volt\cdot e^{j18{,}42\,\degree}\\[\baselineskip]
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%P_2&=\Re\{(\uline{U}'_2+\uline{U}_M)\cdot \uline{I}^*_2\}\\
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%&=\Re\{(0{,}880+j0{,}293)\,\milli\volt\cdot (-0{,}666-j2)\cdot \power{10}{-3}\,\ampere\} \quad\text{Rechtwinklige Koordinaten oder}\\
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%&=\Re\{0{,}927\,\milli\volt\cdot e^{j18{,}42\,\degree}\cdot 2{,}108\,\milli\ampere\cdot e^{-j108{,}4\,\degree}\}\quad\text{Polar Koordinaten}\\
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%&=\Re\{1{,}954\cdot e^{-j89{,}98\,\degree}\}\,\micro\volt\ampere=\Re\{\underbrace{785}_{\text{EPS$\rightarrow$Quelle}}+j1954\}\,\micro\volt\ampere\\[\baselineskip]
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%&\text{seine alte Lösung }(0{,}4+j0{,}13)=0{,}4206\,\volt\cdot e^{-j18{,}0\,\degree}\\
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%\uline{U}''_x&=\uline{U}_2-\uline{U}_M=(0{,}4+j0{,}13)\,\volt-(-0{,}51-j0{,}08)\,\volt=(0{,}91+j0{,}21)\,\volt\\
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%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\\
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%&=\Re\{\underbrace{+178}_{\text{EPS$\rightarrow$Quelle}}+j1960\}\,\micro\volt\ampere\\[\baselineskip]
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% \draw[->,black!50!green](0,0)++(18.4:.42cm)--+(-171.1:.51cm)node[left]{$\uline{U}_M$};
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% \draw[->,black!75!green](0,0)++(18.4:.42cm)++(-171.1:.51cm)--(0:0cm)node[left]{$\uline{U}_2 - \uline{U}_M$};
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% \draw[->,black!85!green](0,0)--(8.4:.89cm)node[right]{$\uline{U}''_x$};
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% \draw[->](0,0)--(8.9:.51cm);%{$\uline{U}_M$};
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% \draw[->,black!50!green](0,0)--(-171.1:.51cm)node[left]{$\uline{U}_M$};
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% \draw[->,magenta](0,0)--(.91,.21)node at(.7,.15)[below left]{$\uline{U}'_2$};
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% \draw[->,black](0,0)++(12.99:.934cm)--+(108.46:.15cm)node [below right]{$\uline{U}_{R2}$};
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% \draw[->,black](0,0)--(22:.935cm)node [above left]{$\uline{U}_{(X_2-X_M)}$}; |