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