\section {3-Phasen System mit unsymmetrischem Verbraucher} Ein symmetrisches 3-Phasen System mit der Phasenlage 1-2-3 speist einen unsymmetrischen Verbraucher. Gegeben sind:\\ $\uline{U}_{31}=380\,\volt\cdot e^{j0\,\degree}$; $R_1=60\,\ohm$; $\uline{X}_{L_1}=20\,\ohm$; $R_2=100\,\ohm$; $\uline{X}_{C_2}=-80\,\ohm$; $X_{L_3}=50\,\ohm$;\\[\baselineskip] Es ist die Anzeige des Leistungsmessinstrumentes zu berechnen! \begin{align*} \begin{tikzpicture}[very thick,scale=4] \begin{scope}[>=latex,xshift=0cm,yshift=1.5cm]%Wattmeter \draw (0,0)--(.367,0) (.633,0)--(1,0) node at (.5,0) {$W$}; \draw (.5,0)circle(.133); \draw(.25,0)--(.25,.25)--(.5,.25)--(.5,.133)(.5,-.133)--(.5,-1); \end{scope} \begin{scope}[xshift=1cm,yshift=.5cm,rotate=60,scale=.577] \begin{scope}[>=latex,xshift=0cm,yshift=0cm]%Widerstand \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.1330667) [left] {$R_1$}; \draw [<-,red] (.8,.1)--(1.2,.1)node at(1,.14)[left]{$\uline{I}_{12}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule - \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.1330667) [ left] {$L_1$}; \fill (.3,-0.0667)rectangle(.7,0.0667); \end{scope} \end{scope} \begin{scope}[>=latex,very thick,xshift=2.01cm,yshift=.75cm,rotate=120,scale=.577]%Spule \draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,-.1330667) [ right] {$L_3$}; \fill (.3,-0.0667)rectangle(.7,0.0667); \draw [<-,red] (1.2,-.1)--(.8,-.1)node at(1,-.14)[right]{$\uline{I}_{31}$}; \end{scope} \begin{scope}[>=latex,xshift=1.289cm,yshift=.5cm,scale=.577]%Widerstand \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=1.289cm,yshift=.25cm,scale=.577]%Kondensator - \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,.1133) [above] {$C_2$}; % \draw [->,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$U_{C2}$}; \end{scope} \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen. \draw (0,0)--(2.154,0)--(2.154,.5)(.9,.5)--(1.289,.5) (1.289,.5)--(1.289,.25)(1.867,.5)--(1.867,.25); \filldraw(0,0)circle(0.025cm)node [left]{$L3$}; \filldraw(0,.5)circle(0.025cm)(1,.5)--(0,.5)node [left]{$L2$}; \filldraw(0,1.5)circle(0.025cm)node at(0,1.5) [left]{$L1$}; \fill(.5,.5)circle(0.025cm)(1,.5)circle(0.025cm)(1.289,.5)circle(0.025cm)(1.867,.5)circle(0.025cm)(2.154,.5)circle(0.025cm)(1.577,1.5)circle(0.025cm); \draw [->,blue] (.8,1.4)--(.8,0.6) node at (.8,1)[right]{$\uline{U}_{12}$}; \draw [->,red] (.9,1.6)--(1.1,1.6) node at (1,1.6)[above]{$\uline{I}_1$}; \draw +(1,1.5)--(1.577,1.5)--(2.154,.5cm)--(1.75,.5); \end{scope} \end{tikzpicture} \end{align*} \ifthenelse{\equal{\toPrint}{Lösung}}{% %\begin{align} %\intertext{Formeln:} %\end{align} Berechnung:\\ %\begin{align*} % \begin{tikzpicture}[scale=2,rotate=-150] % \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm] % \draw[black!25!,very thin,step=.5cm](-1,-1)grid(1,1); % \draw[->]++(-180:1cm)--++(30:1.732cm)node [above right]{$2$}; % \draw[->]++(-180:1cm) ++(30:1.732cm)-- ++(270:1.732cm)node [below]{$3$}; % \draw[->]++(-180:1cm) ++(330:1.732cm)--(-180:1cm)node [left]{$1$}; % \draw[->](-180:1cm)--(0,0)node [right]{$N$}; % \draw[->](60:1cm)--(0,0); % \draw[->](300:1cm)--(0,0); % \draw[blue] node at(-.35,.6){$\uline{U}_{12}$}; % \draw[blue] node at(-.35,-.6){$\uline{U}_{31}$}; % \draw[blue] node at(.75,0){$\uline{U}_{23}$}; % \draw[blue] node at(-.35,.15){$\uline{U}_{1}$}; % \draw[blue] node at(.35,.25){$\uline{U}_{2}$}; % \draw[blue] node at(.35,-.25){$\uline{U}_{3}$}; % \end{scope} % \end{tikzpicture} %\end{align*} %%\clearpage \begin{minipage}[c]{0.25\textwidth} \begin{align*} \begin{tikzpicture}[very thick,scale=1] \draw[->,blue](0:0)--(0:1cm)node [right]{$\uline{U}_{31}$}; \draw[->,blue](0:0)--(120:1cm)node [above]{$\uline{U}_{23}$}; \draw[->,blue](0:0)--(-120:1cm)node [below]{$\uline{U}_{12}$}; \draw[->,red](0:0)--(141.2:.577cm)node [above left]{$\uline{I}_1$}; \end{tikzpicture} \end{align*} \end{minipage} %\hfill \begin{minipage}[l]{0.75\textwidth} \begin{align*} \uline{U}_{31}&=380\,\volt\cdot e^{j0\,\degree}\\ \uline{U}_{12}&=380\,\volt\cdot e^{-j120\,\degree}\\ \end{align*} \end{minipage} \begin{align*} P&=\Re\{\uline{U}_{12}\cdot I^*_1\}\\ \uline{I}_1&=\uline{I}_{12}-\uline{I}_{31}\\ \end{align*} \begin{align*} \uline{I}_{12}&=\frac{\uline{U}_{12}}{R_1+jX_{L_1}} =\frac{\overbrace{380\,\volt\cdot e^{-j120\,\degree}}^{(-190-j329{,}09)\,\volt}}{\underbrace{63,246\,\ohm\cdot ^{j18{,}435\,\degree}}_{(60+j20)\,\ohm}}=6{,}01\,\ampere\cdot e^{-j138{,}4\,\degree}=(-4{,}495-j3{,}986)\,\ampere\\ \uline{I}_{31}&=\frac{\uline{U}_{31}}{jX_{L_3}}=\frac{380\,\volt\cdot e^{j0\,\degree}}{j50\,\ohm}=-j\frac{380\,\volt}{50\,\ohm}=-j7{,}6\,\ampere \qquad\tag{$\frac{1}{j}=-j$}\\ \uline{I}_1&=[(-4{,}495-j3{,}986)-(-j7{,}6)]\,\ampere\\ &=(-4{,}495+j3{,}614)\,\ampere=5{,}77\,\ampere\cdot e^{j141{,}2\,\degree}\\[\baselineskip] \end{align*} \begin{align*} \begin{tikzpicture}[very thick,scale=2.5] \draw[->,blue](0:0)--(0:1cm)node [right]{$\uline{U}_{31}$}; \draw[->,blue](0:0)--(120:1cm)node [above]{$\uline{U}_{23}$}; \draw[->,blue](0:0)--(-120:1cm)node [below]{$\uline{U}_{12}$}; \draw[->,red!50!blue](0:0)--(-138.4:.601cm)node [below]{$\uline{I}_{12}$}; \draw[->,red!50!blue](0:0)--(-90:.76cm)node [below]{$\uline{I}_{31}$}; \draw[<-,red!75!blue](-138.4:.601cm)-- ++(90:.76cm)node [below left]{$\uline{I}_{31}$}; \draw[->,red](0:0)--(141.2:.577cm)node [above left]{$\uline{I}_1$}; \end{tikzpicture} \end{align*} \begin{align*} \uline{S}&=\uline{U}_{12}\cdot \uline{I}^*_1=\underbrace{380\,\volt\cdot e^{-j120\,\degree}}_{(-190-j329{,}09)\,\volt}\cdot \underbrace{5{,}77\,\ampere\cdot e^{-j141{,}2\,\degree}}_{(-4{,}495-j3{,}614)\,\ampere}\\ &=2192{,}6\,\volt\ampere\cdot e^{j98{,}8\,\degree}=(\underbrace{-335{,}28}_{P}+\underbrace{j2165{,}92}_{jQ})\,\volt\ampere\\ P&=S\cdot \cos(\varphi)=2192{,}6\,\volt\ampere\cdot \underbrace{\cos(98{,}8\,\degree)}_{-0{,}153}=\uuline{-335{,}44\,\watt}\\ \end{align*} \clearpage }{}%