78 lines
4.6 KiB
TeX
78 lines
4.6 KiB
TeX
\section{CLR Netzwerk}
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Von dem Netzwerk sind folgende Daten bekannt:\\[\baselineskip]
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$R_1=1,2\,\kilo\ohm\quad R_2=470\,\ohm\quad X_{C_0}=-906\,\ohm\quad X_{L_2}=628\,\ohm$\\
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$\uline{I}_2=12\,\milli\ampere\cdot e^{j20\,\degree}$\\
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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=-.5cm,rotate=90]%Spannungsquelle |
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\draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}_{0}$};
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\draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}_0$};
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\draw [->,red] (.8,.2)--(1.2,.2)node at(1,.2)[left]{$\uline{I}_0$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%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) [above] {$X_{C_0}$};
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\draw [->,blue] (0.3,-.25)--(0.7,-.25)node at(.25,-.4)[right]{\footnotesize$\uline{U}_{C_0}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=-.5cm,rotate=90]
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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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\draw [<-,blue] (0,-.5)--(1,-.5)node at(.5,-.5)[right]{\footnotesize$\uline{U}_2$};
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\draw [<-,red] (.8,.15)--(1.2,.15)node at(1,.1)[left]{\footnotesize$\uline{I}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]
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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] {$R_2$};
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\draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1cm,rotate=90]%Spule |
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_{L_2}$};
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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=0cm,yshift=-1cm]%Fehlstellen Eckverbindungen.
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\draw (0,1.5)--(0,2)--(.2,2) (1,2)--(2,2)--(2,1.75)(0,.5)--(0,0)--(2,0)--(2,.25)
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(1,0)--(1,.5) (1,2)--(1,1.5);
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\filldraw [red] (1,2)circle(0.02)node [above]{\footnotesize$KP$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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Berechnen Sie Schein- Wirk- und Blindleistung des Netzwerkes!\\[\baselineskip]
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\ifthenelse{\equal{\toPrint}{L<EFBFBD>sung}}{%
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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{align*}
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\uline{S}&=\uline{U}\cdot \uline{I}^*\qquad\text{mit }\uline{U}_0=\uline{U}_{C_0}+\uline{U}_2\qquad\uline{I}_0=\uline{I}_1+\uline{I}_2\\
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\intertext{Spannung $U_2$:}
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\uline{Z}_2&=R_2+jX_{L_2}=(470+j628)\,\ohm=784{,}4\,\ohm\cdot e^{j53{,}19\,\degree}\\
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\uline{U}_2&=\uline{Z}_2\cdot \uline{I}_2
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=9{,}413\,\volt\cdot e^{j73{,}19\,\degree}=(2{,}722+j9{,}011)\,\volt\\
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\intertext{Strom:}
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\uline{I}_1&=\frac{\uline{U}_2}{R_1}=\frac{9{,}413\,\volt\cdot e^{j73{,}19\,\degree}}{1,2\,\kilo\ohm}=7{,}844\,\milli\ampere\cdot e^{j73{,}19\,\degree}
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=(2{,}268+j7{,}5088)\,\milli\ampere\\
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\uline{I}_0&=\uline{I}_1+\uline{I}_2=(2{,}268+j7{,}5088+11{,}276+j4{,}104)\,\milli\ampere\\
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&=(13{,}544+j11{,}613)\,\milli\ampere=17{,}841\,\milli\ampere\cdot e^{j40{,}61\,\degree}\\
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\intertext{Spannung:}
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\uline{U}_{C_0}&=jX_{C_0}\cdot \uline{I}_0
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=-j906\,\ohm\cdot 17{,}841\,\milli\ampere\cdot e^{j40{,}61\,\degree}=16{,}164\,\volt\cdot e^{-j49{,}39\,\degree}\\
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&=(10{,}521-j12{,}271)\,\volt\\
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\uline{U}_0&=\uline{U}_{C_0}+\uline{U}_2=(10{,}521-j12{,}271+2{,}722+j9{,}011)\,\volt\\
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&=(13{,}243-j3{,}260)\,\volt=13{,}638\,\volt\cdot e^{-j13{,}83\,\degree}\\
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\end{align*}
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\begin{align*}
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\intertext{Scheinleistung:}
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\uline{S}&=\uline{U}_0\cdot \uline{I}_0^*=13{,}638\,\volt\cdot e^{-j13{,}83\,\degree}\cdot 17{,}841\,\milli\ampere\cdot e^{-j40{,}61\,\degree}=243{,}32\,\milli\volt\ampere\cdot e^{-j54{,}44\,\degree}\\[2\baselineskip]
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S&=|\uline{S}|=\uuline{243{,}32\,\milli\volt\ampere}\\
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P&=S\cdot \cos\varphi=243{,}32\,\milli\volt\ampere\cdot \underbrace{\cos(-54{,}44)}_{0{,}5816}=\uuline{141{,}50\,\milli\watt}\\
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Q&=S\cdot \sin\varphi=243{,}32\,\milli\volt\ampere\cdot \underbrace{\sin(-54{,}44)}_{-0{,}813}=\uuline{-197{,}93\,\milli\,\var}\\
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\intertext{Zweiter Weg:}
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\uline{S}&=\uline{U}_0\cdot \uline{I}^*_0=\uline{I}_0\cdot \uline{Z}\cdot \uline{I}^*_0=I^2_0\cdot \uline{Z}\\
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\uline{Z}&=jX_{C_0}+\frac{R_1\cdot (R_2-jX_{L_2})}{R_1+R_2-jX_{L_2}}=(445-j622)\,\ohm=764\,\ohm\cdot e^{-j54{,}44\,\degree}\\
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\uline{S}&=I^2_0\cdot \uline{Z}
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=(17{,}841\,\milli\ampere)^2\cdot 767\,\ohm\cdot e^{-j54{,}44\,\degree}=243{,}3\,\milli\volt\ampere\cdot e^{-j54{,}44\,\degree}\\
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S&=243{,}3\,\milli\volt\ampere\\
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P&=S\cdot\cos(-54{,}44\degree)=141{,}5\,\milli\watt\\
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Q&=S\cdot\sin(-54{,}44\degree)=-197{,}9\,\milli\,\var
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\end{align*}
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\clearpage
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}{}%
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