\section {Netzwerk Wirk- und Blindanteil} $\uline{U}=4{,}2\,\volt\cdot e^{j30\,\degree};\quad f=120\,\kilo\hertz ;\quad \uline{I}_0= 2{,}7\milli\ampere \cdot e^{j110\,\degree}$\\ $R=1{,}3\,\kilo\ohm; \quad L=3{,}6\,\milli\henry; \quad C=1{,}5\,\nano\farad$. \renewcommand{\labelenumi}{\alph{enumi})} \begin{enumerate} \item Berechnen Sie den Strom \uline{I} und seinen Wirk- und Blindanteil $\uline{I}_{\textrm{w}}$ bzw. $\uline{I}_{\textrm{b}}$. \item Berechnen Sie die Wirk und Blindleistung an den Anschlussklemmen und an der Stromquelle $\uline{I}_0$. \end{enumerate} \vspace{-.5cm} \begin{align*} \begin{tikzpicture}[scale=2] \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$C$}; \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_R$}; \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) [right] {$L$}; \fill (.3,-0.0667)rectangle(.7,0.0667); \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_L$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Stromquelle \draw (0,0)--(.367,0) (.5,-.133)--(.5,.133) (.633,0)--(1,0); \draw [->,blue](.8,.2)--(.2,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{RL}$}; \draw (.5,0)circle(.133); \draw [->,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{I}_0$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. \draw (.8,1)--(3,1)--(3,.9) (0,0)--(3,0)--(3,.2); \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); \end{scope} \end{tikzpicture} \end{align*} \ifthenelse{\equal{\toPrint}{Lösung}}{% %\begin{align} %\intertext{Formeln:} %\end{align} Berechnung:\\ \begin{align*} \text{a)}\quad X_L&=\omega L=2\pi\cdot f\cdot 3{,}6\,\milli\henry=2714\,\ohm\\ X_C&=\frac{-1}{\omega C}=\frac{-1}{2\pi\cdot f\cdot 1{,}5\,\nano\farad}=-884\,\ohm\\ \uline{Z}_{RL}&=\frac{R\cdot jX_L}{R+jX_L}=\frac{1{,}3\,\kilo\ohm\cdot j2714\,\ohm}{1{,}3\,\kilo\ohm+j2714\,\ohm}=(1057+j506)\,\ohm=1172\,\ohm\cdot e^{j25{,}59\,\degree} \end{align*} Serienersatzschaltbild: \begin{align*} \begin{tikzpicture}[scale=2] \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.133) [above] {$X_C$}; \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$\uline{U}_C$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Widerstand - nach EN 60617 \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_{RL}$}; \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{RL}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spannungsquelle \draw (0,0)--(1,0); \draw (.5,0)circle(.133); \draw (.5,0)circle(.133); \draw [<-,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{\footnotesize$\uline{U}_0$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. \draw (1.8,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2); \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); \end{scope} \end{tikzpicture} \begin{tikzpicture}[scale=.5] \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] \draw[black!25!,very thin](-3,0)grid(5,5); \draw[thick](-3,0)--(5,0) (0,0)--(0,5); \draw[->,blue](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; \draw[->,red](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; \draw[->,blue](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; \draw[->,red](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; \end{scope} \end{tikzpicture} \end{align*} \begin{align*} \uline{U}_0&=\uline{I}_0\cdot \uline{Z}_{RL}=2{,}7\milli\ampere \cdot e^{j110\,\degree}\cdot 1172\,\ohm\cdot e^{j25{,}59\,\degree}\\ &=3{,}164\,\volt\cdot e^{j135{,}6\,\degree}=(-2{,}26+j2{,}21)\,\volt\\ \uline{Z}_{RLC}&=\uline{Z}_{RL}+jX_C=(1060+j507)\,\ohm-j884\,\ohm\\ &=(1057-j378)\,\ohm=1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}\\ \intertext{Masche:} \uline{I}&\cdot \uline{Z}_{RLC}+\uline{U}_0-\uline{U}=0\\ \uline{I}&=\frac{\uline{U}-\uline{U}_0}{\uline{Z}_{RLC}}=\frac{\overbrace{(3{,}637+j2{,}1)\,\volt}^{4{,}2\,\volt\cdot e^{j30\,\degree}}-(-2{,}26+j2{,}21)\,\volt}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}=\frac{5{,}90\,\volt\cdot e^{-j1{,}06\,\degree}}{1122{,}6\,\ohm\cdot e^{-j19{,}67\,\degree}}\\ &=\uuline{5{,}25\milli\ampere\cdot e^{j18{,}5\,\degree}}=\uuline{(4{,}98+j1{,}67)\milli\ampere} \end{align*} \clearpage Parallelersatzschaltbild: (Wirkstrom in Phase, Blindstrom senkrecht zur Spannung) \begin{align*} \begin{tikzpicture}[scale=2] \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] \draw [->,blue](.8,0)--(.2,0)node at(.5,0)[left]{\footnotesize$\uline{U}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Kondensator - \draw [<-,red] (.25,0)--(.05,0)node at(.15,0)[above]{$\footnotesize\uline{I}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$}; \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_W$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90] \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$X$}; \draw [<-,red] (.75,0)--(.95,0)node at(.85,0)[right]{\footnotesize$\uline{I}_B$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. \draw (0,1)--(2,1)--(2,.9) (0,0)--(2,0)--(2,.2); \fill(0,0)circle(0.05cm)(0,1)circle(0.05cm); \end{scope} \end{tikzpicture} \begin{tikzpicture}[scale=.5] \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] \draw[black!25!,very thin](-3,0)grid(5,5); \draw[thick](-3,0)--(5,0) (0,0)--(0,5); \draw[->,blue!45!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; \draw[->,red!45!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; \draw[->,blue!45!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; \draw[->,red!45!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; \draw[->,red](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$}; \draw[->,red](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$}; \end{scope} \end{tikzpicture} \end{align*} \begin{align*} |\uline{I}_{\textrm{w}}|&=I\cdot \cos(\underbrace{\varphi_u}_{30\,\degree}-\underbrace{\varphi_i}_{18{,}5\,\degree})=5{,}14\,\milli\ampere\\ \Rightarrow & \uline{I}_{\textrm{w}}=\uuline{5{,}14\,\milli\ampere\cdot e^{j30\,\degree}}\\ |\uline{I}_b|&=I\cdot \sin(\varphi_u-\varphi_i)=1{,}05\,\milli\ampere\\ \Rightarrow & \uline{I}_b=\uuline{1{,}05\,\milli\ampere\cdot e^{-j60\,\degree}}\\ \end{align*} \begin{minipage}[b]{.5\textwidth} b) \begin{align*} \uline{U}_C&=\uline{I}\cdot jX_C=4{,}64\,\volt\cdot e^{-j71{,}5\,\degree}\\ \uline{U}_{RL}&=\uline{U}-\uline{U}_C=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\\ \uline{I}_R&=\frac{\uline{U}_{RL}}{R}=5{,}27\,\milli\ampere\cdot e^{j71{,}6\,\degree}\\ \uline{I}_L&=\frac{\uline{U}_{RL}}{jX_L}=2{,}52\,\milli\ampere\cdot e^{-j18{,}4\,\degree}\\ \end{align*} Probe: \begin{align*} I+I_0&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\ I_R+I_L&=5{,}85\,\milli\ampere\cdot e^{j46\,\degree}\\ &\text{stimmt überein.} \end{align*} \end{minipage} \begin{minipage}[c]{.5\textwidth} \begin{tikzpicture}[scale=.75] \begin{scope}[>=latex,very thick,xshift=8cm,yshift=0cm] \draw[black!25!,very thin](-3,-5)grid(5,7); \draw[thick](-3,0)--(5,0) (0,-5)--(0,7); \draw[->,red!66!](0,0)--(110:2.7cm)node[above]{$\uline{I}_0$}; \draw[->,blue!25!](0,0)--(135.6:3.173cm)node[above left]{$\uline{U}_0$}; \draw[->,red!66!](0,0)--(18.5:5.25cm)node[below right]{$\uline{I}$}; \draw[->,red!25!](0,0)--(30:5.14cm)node at (2.5,2){$\uline{I}_W$}; \draw[->,red!25!](30:5.14cm)--+(-60:1.05cm)node at (5.5,2.25){$\uline{I}_B$}; \draw[->,blue!66!](0,0)--(30:4.25cm)node[above right]{$\uline{U}$}; \draw[->,blue](0,0)--(-71.5:4.64cm)node[right]{$\uline{U}_C$}; \draw[->,blue](30:4.25cm)--+(108.5:4.64cm)node[below right]{$-\uline{U}_C$}; \draw[->,blue](0,0)--(71.6:6.85cm)node[below left]{$\uline{U}_{RL}$}; \draw[->,red](0,0)--(71.6:5.27cm)node[below left]{$\uline{I}_R$}; \draw[->,red](71.6:5.27cm)--(46:5.85cm)node[above left]{$\uline{I}_L$}; \draw[<-,red!50!blue]+(46:5.85cm)--(18.5:5.25cm)node at(5,3){$\uline{I}_0$}; \draw[->,red!50!blue](0,0)--(46:5.85cm)node at(2,3.25){\footnotesize$\uline{I}+\uline{I}_0$}; \end{scope} \end{tikzpicture} \end{minipage}\\ \clearpage \begin{align*} \intertext{Leistung an Klemme:} \uline{S}_K&=\uline{U}\cdot \uline{I}^*=4{,}2\,\volt\cdot e^{j30\,\degree}\cdot 5{,}25\milli\ampere\cdot e^{-j18{,}5\,\degree}=(\underbrace{21{,}6}_{P_K}+\underbrace{j4{,}4}_{Q_K})\,\milli\volt\ampere\\ P_K&=\uuline{21{,}6\,\milli\watt}\\ Q_K&=\uuline{4{,}4\,\milli\var}\\ \intertext{Leistung der Stromquelle (Pfeilsystem wie oben):} S_I&=\uline{U}_{RL}\cdot I^*_0=6{,}85\,\volt\cdot e^{j71{,}6\,\degree}\cdot 2{,}7\milli\ampere \cdot e^{-j110\,\degree}=(\underbrace{14{,}5}_{P_I}\underbrace{-j11{,}5}_{Q_I})\,\milli\volt\ampere\\ P_I&=\uuline{14{,}5\,\milli\watt}\\ Q_I&=\uuline{-11{,}5\,\milli\var}\\ \intertext{Gesamtleistung: Summe aus Quell- und Klemmleistung} S&=S_K+S_I=(21{,}6+j4{,}4+14{,}5-j11{,}5)\,\milli\volt\ampere=36{,}1\,\milli\watt-j7{,}1\,\milli\volt\ampere \intertext{Probe: Wirkleistung: } P&=\frac{U^2_{RL}}{R}=\frac{(6{,}85\,\volt)^2}{1300\,\ohm}=36{,}1\,\milli\watt \quad\text{(Übereinstimmung)} \intertext{Probe: Blindleistung: } C:\quad Q_C&=X_C\cdot I^2=-884\,\ohm\cdot(5{,}25\,\milli\ampere)^2=-24{,}4\,\milli\var\\ L:\quad Q_L&=\frac{U^2_{RL}}{X_L}= \frac{(6{,}85\,\volt)^2}{2714\,\ohm}=+17{,}3\,\milli\var\\ Q&=Q_C+Q_L=(-24{,}4+17{,}3)\,\milli\var=-7{,}1\,\milli\var \quad\text{(Übereinstimmung)}\\ \end{align*} \clearpage }{}%