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ET2_Uebung_BEI/ET2_L_B14_A3.tex
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2022-02-24 12:16:45 +01:00

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\section{Brückenschaltung}
%\enlargethispage{0cm}
Zeichnen Sie zu der abgebildeten Brückenschaltung ein \underline{maßstäbliches} Zeigerdiagramm aller Ströme und Spannungen.\\
Entnehmen Sie dem Zeigerdiagramm die Spannung $U_a$ und geben Sie von dieser Spannung Betrag und Phasenwinkel an.\\[\baselineskip]
\uline{Maßstäbe: }$ 10\,\volt\,\widehat{=}\,1\centi\metre\qquad 0,2\,\ampere\,\widehat{=}\,1\,\centi\metre $ (Platzbedarf in x und y $15\,\centi\metre$)\\
$R_1=100\,\ohm \qquad R_2=80\,\ohm \qquad X_L=200\,\ohm \qquad X_C=-120\,\ohm\qquad\uline{U}=150\,\volt\cdot e^{j0\degree}$
\begin{align*}
\begin{tikzpicture}[very thick,scale=2]
\begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617
\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_1$};
\draw [<-,blue] (.3,.5)--(.7,.5)node at(.5,.5)[right]{\footnotesize$\uline{U}_{R_1}$};
\draw [<-,red] (.05,.0)--(.25,.0)node at(.15,.0)[right]{\footnotesize$\uline{I}_{1}$};
\end{scope}
\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617
\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$};
\draw [<-,blue] (.3,.5)--(.7,.5)node at(.5,.5)[right]{\footnotesize$\uline{U}_{R_2}$};
\end{scope}
\begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-1cm,rotate=90]%Spule |
\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_L$};
\fill (.3,-0.0667)rectangle(.7,0.0667);
\end{scope}
\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Kondensator |
\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$X_C$};
\draw [->,red] (.35,0)--(.15,0) node at (.25,0)[right]{\footnotesize$\uline{I}_2$};
\end{scope}
\begin{scope}[>=latex,very thick]%Knotenpunkte
\draw (1.5,.8)--(1.5,1)--(-.5,1) (1.5,-.8)--(1.5,-1)--(-.5,-1);
\fill (-.5,1)circle(.025) (-.5,-1)circle(.025);
\draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{a}$};
% \draw node at(-1,-.5)[left]{$\underline{Z}\Rightarrow$};
\fill (.5,0)circle(.025) node at (.5,0)[left]{A};
\fill (1.5,0)circle(.025)node at (1.5,0)[right]{B};
\draw [->,blue] (-.5,.8)--(-.5,-.8) node at (-.5,0)[left]{$\underline{U}$};
\end{scope}
\end{tikzpicture}
\end{align*}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%%begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:
\begin{align*}
Z_1&=\sqrt{R^2_1+X^2_L}=\sqrt{100^2+200^2}\,\ohm=223{,}6\,\ohm\\
Z_2&=\sqrt{R^2_2+X^2_C}=\sqrt{80^2+(-120)^2}\,\ohm=144{,}2\,\ohm\\
\varphi_1&=\arctan\frac{\Im}{\Re}=\arctan\frac{200\,\ohm}{100\,\ohm}=\arctan 2=63{,}4\,\degree\\
\varphi_2&=\arctan\frac{-120\,\ohm}{80\,\ohm}=\arctan (-1{,}5)=-56{,}3\,\degree\\
\uline{Z}_1&=Z_1\cdot e^{j\varphi_1}=223{,}6\,\ohm\cdot e^{j63{,}4\,\degree}\\
\uline{Z}_2&=Z_2\cdot e^{j\varphi_2}=144{,}2\,\ohm\cdot e^{-j56{,}3\,\degree}\\
\uline{I}_1&=\frac{U}{Z_1}=\frac{150\,\volt\cdot e^{j0\degree}}{223{,}6\,\ohm\cdot e^{j63{,}4\,\degree}}=0{,}67\,\ampere\cdot e^{-j63{,}4\,\degree} \approx 3{,}4\,\centi\metre\\
\uline{I}_2&=\frac{U}{Z_2}=\frac{150\,\volt\cdot e^{j0\degree}}{144{,}2\,\ohm\cdot e^{-j56{,}3\,\degree}}=1{,}04\,\ampere\cdot e^{j56{,}3\,\degree} \approx 5{,}2\,\centi\metre\\
\uline{I}&=\uline{I_1}+\uline{I_2}\\
\uline{U}_{R_1}&=I_1\cdot R_1= 0{,}671\,\ampere\cdot e^{j63{,}4\,\degree}\cdot 100\,\ohm=67{,}1\,\volt\cdot e^{-j63{,}4\,\degree}\\
\uline{U}_{R_2}&=I_2\cdot R_2=1{,}04\,\ampere\cdot 80\,\ohm=83{,}2\,\volt\cdot e^{j56{,}3\,\degree}\\
\uline{U}_a&=\uline{U}_{R_2}-\uline{U}_{R_1}\\
\uline{U}_L&=X_L\cdot \uline{I}_1=200\,\ohm\cdot e^{j90\degree}0{,}67\,\ampere\cdot e^{-j63{,}4\,\degree}=134\,\volt\cdot e^{j26{,}6\,\degree}\\
\uline{U}_C&=X_C\cdot \uline{I}_2=120\,\ohm\cdot e^{-j90\degree}\cdot 1{,}04\,\ampere\cdot e^{j56{,}3\,\degree}=124{,}8\,\volt\cdot e^{-j33{,}7\,\degree}
\end{align*}
\clearpage
Zeigerdiagramm Teil 1: (Maßstäblich)
\enlargethispage{2\baselineskip}
\begin{align*}
\begin{tikzpicture}[very thick,scale=.6]
\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
\draw [black!25!,very thin](0,-8)grid(15,8);
\draw [black!50!,thin,dashed](16,0)--(18,0);
\draw [black!50!,thin,dashed](3,-6)--(5,-6);
\draw [black!50!,thin](7.5,0)circle(7.5);
\draw [->,blue](0,0)--(15,0)node [right]{$\uline{U}$};
\draw [->,blue](0,0)--(-63.4:6.71)node [left]{$\uline{U}_{R_1}$};
\draw [->,blue](0,0)--(56.3:8.32)node [left]{$\uline{U}_{R_2}$};
\draw [->,blue!50!red](0,0)+(-63.4:6.71)--(15,0)node at(15,-1.5)[below]{$\uline{U}_L=134\,\volt\cdot e^{j26{,}63{,}4\,\degree}$};
\draw [->,blue!50!red](0,0)+(56.3:8.32)--(15,0)node at(15,2)[above]{$\uline{U}_C=124{,}8\,\volt\cdot e^{-j33{,}73\,\degree}$};
\draw [->,blue](0,0)+(-63.4:6.71)--(56.3:8.32)node at (4,2)[right]{$\uline{U}_{AB}=\uline{U}_a=130\,\volt\cdot e^{j83\,\degree}$};
\draw node at (3,-6)[below]{$A$};
\draw node at (4.8,7)[above]{$B$};
\draw node at (16,0)[above right]{$\Re (A)$};
\draw [black!50!](0,0)++(-63.4:6.71)+(26.6:1)arc(26.6:116.6:1);%Rechter Winkel bei UR1
\fill [black!50!](0,0)++(-63.4:6.71)+(71.6:.5)circle(0.1);
\draw [black!50!](0,0)++(56.3:8.32)+(-33.7:1)arc(-33.7:-123.7:1);%Rechter Winkel bei UR2
\fill [black!50!](0,0)++(56.3:8.32)+(-78.7:.5)circle(0.1);
\draw [<->,black!50!](0,0)+(-63.4:1)arc(-63.4:53.3:1);%Winkel UR1 UR2
\draw node at (0.8,0)[below right]{$\varphi_1=-63{,}4\,\degree$};
\draw node at (0.8,0)[above right]{$\varphi_2=53{,}3\,\degree$};
\draw [->,black!75!](3,-6)+(0:2)arc(0:83:2)node at (5,-5.5)[right]{$\varphi_{\uline{U}} =83\,\degree$};%U - UAB
\draw node at (19,5) [right]{$y=15\,\centi\metre$};
\draw node at (19,4) [right]{Kreis da};
\draw node at (19,3) [right]{$\uline{U}_{R_2}+\uline{U}_{C}=\uline{U}$};
\draw node at (19,2) [right]{$\uline{U}_{R_1}+\uline{U}_{L}=\uline{U}$};
\draw node at (19,1) [right]{$\uline{U}_{R_2}\bot \uline{U}_{C}$};
\draw node at (19,0) [right]{$\uline{U}_{R_1}\bot \uline{U}_{L}$};
\draw node at (19,-1) [right]{A und B einzeichnen};
\draw node at (19,-2) [right]{Messen $\uline{U}_{AB}=\uline{U}_a$};
\end{scope}
\end{tikzpicture}
\end{align*}
Zeigerdiagramm vollständig:
\begin{align*}
\begin{tikzpicture}[very thick,scale=.6]
\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]
\draw [black!25!,very thin](0,-8)grid(15,8);
\draw [black!50!,thin,dashed](16,0)--(18,0);
\draw [black!50!,thin,dashed](3,-6)--(5,-6);
\draw [black!50!, thin](7.5,0)circle(7.5);
\draw [->,blue](0,0)--(15,0)node [right]{$\uline{U}$};
\draw [->,blue](0,0)--(-63.4:6.71)node [left]{$\uline{U}_{R_1}$};
\draw [->,blue](0,0)--(56.3:8.32)node [left]{$\uline{U}_{R_2}$};
\draw [->,blue!50!red](0,0)+(-63.4:6.71)--(15,0)node at(12,-1.5)[below]{$\uline{U}_L$};
\draw [->,blue!50!red](0,0)+(56.3:8.32)--(15,0)node at(12,2)[above]{$\uline{U}_C$};
\draw [->,blue](0,0)+(-63.4:6.71)--(56.3:8.32)node at (4,2)[right]{$\uline{U}_{AB}=\uline{U}_a=130\,\volt\cdot e^{j83\,\degree}$};
\draw [black!50!](0,0)++(-63.4:6.71)+(26.6:1)arc(26.6:116.6:1);%Rechter Winkel bei UR1
\fill [black!50!](0,0)++(-63.4:6.71)+(71.6:.5)circle(0.1);
\draw [black!50!](0,0)++(56.3:8.32)+(-33.7:1)arc(-33.7:-123.7:1);%Rechter Winkel bei UR2
\fill [black!50!](0,0)++(56.3:8.32)+(-78.7:.5)circle(0.1);
\draw [->,red](0,0)--(-63.4:3.34)node [left]{$\uline{I}_1$};
\draw [->,red](0,0)--(56.3:5.2)node [left]{$\uline{I}_2$};
\draw [<-,red,thin](0:0)++(56.3:5.2)++(-63.4:3.34)--(-63.4:3.34)node at(3,-.5) [left]{$\uline{I}_2$};
\draw [<-,red,ultra thick](0:0)++(56.3:5.2)++(-63.4:3.34)--(0:0)node at (6,.6)[above]{$\uline{I}=\uline{I}_1+\uline{I}_2$};
\draw [<-,black](0,0)+(17.0:3)arc(17.0:0:3)node at (1.0,1.5)[right]{$\varphi_{\uline{I}}=17\,\degree$};%Winkel I
\draw node at (3,-6)[below]{$A$};
\draw node at (4.8,7)[above]{$B$};
\draw node at (16,0)[above right]{$\Re (A)$};
\draw [<->,black!50!](0,0)+(-63.4:1)arc(-63.4:53.3:1);%Winkel UR1 UR2
\draw node at (.7,-.35)[below right]{$\varphi_1$};
\draw node at (.7,.35)[above right]{$\varphi_2$};
\draw [->,black!75!](3,-6)+(0:2)arc(0:83:2)node at (5,-5.5)[right]{$\varphi_{\uline{U}} =83\,\degree$};%U - UAB
\draw node at (20.8,5) [right]{$\uline{I}_1 \approx 3{,}4\,\centi\metre$};
\draw node at (20.8,4) [right]{$\uline{I}_2 \approx 5{,}2\,\centi\metre$};
\draw node at (20.8,3) [right]{$\uline{I}=\uline{I_1}+\uline{I_2}$};
\end{scope}
\end{tikzpicture}
\end{align*}
\clearpage
}{}%
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