nieblerch
/
ET2_Uebung_BEI
Watch 1
Star 0
Fork 0
Code Issues 0 Pull Requests 0 Releases 0 Wiki Activity
You can not select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
1 Commit
1 Branch
Branch: master
Branches Tags
${ item.name }
Create branch ${ searchTerm }
from 'master'
${ noResults }
ET2_Uebung_BEI/ET2_L_B17_A8.tex

ET2_L_B17_A8.tex 6.2KB
Raw Permalink Blame History

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103 
  1. \section{Wechselstrombrücke}

  2. a) Zustand abgeglichenen Brücke $\mathbf{(U_{ab}=0})$\\

  3. Welches Bauteil muss fur $X$ eingesetzt werden, um diese Voraussetzung zu erfüllen?\\

  4. Berechnen Sie den Wert des Bauteils als Funktion von $R$ und $X_L$.\\[\baselineskip]

  5. b) Zustand nicht abgeglichene Brücke: $\mathbf{(U_{ab}\neq 0)}$\\[\baselineskip]

  6. Zeichnen Sie ein qualitatives Zeigerdiagramm aller eingezeichneten Spannungen\\

  7. (Bezug $\uline{U}=U\cdot e^{j0\,\degree}$), so dass $\uline{U}_{ab}$ auf $\uline{U}$ senkrecht steht (rechte Winkel müssen gekennzeichnet werden).\\

  8. Welches Bauteil muss fur $X$ eingesetzt werden, um diese Voraussetzung zu erfüllen?\\ Berechnen Sie den Wert dieses Bauteils als Funktion von $R$ und $X_L$ anhand der Beziehung zwischen den Zeigerlängen.

  9. \begin{align*}

  10. 	\begin{tikzpicture}[very thick,scale=2.5]

  11. 		\begin{scope}[>=latex,very thick,xshift=0.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617			

  12. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};

  13. 			\draw [<-,blue] (.3,.35)--(.7,.35)node at(.5,.35)[right]{\footnotesize$\uline{U}_{R}$};

  14. 		\end{scope}

  15. 		\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=0cm,rotate=90]%Widerstand - nach EN 60617			

  16. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};

  17. 			\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{b}$};

  18. 		\end{scope}		

  19. 		\begin{scope}[>=latex,very thick,xshift=.5cm,yshift=-0cm,rotate=90]%Spule |

  20.             \draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$X_L$};

  21. 			\fill (.3,-0.0667)rectangle(.7,0.0667);

  22. 			\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{a}$};

  23. 		\end{scope}	

  24. 		\begin{scope}[>=latex,very thick,xshift=1.5cm,yshift=-1cm,rotate=90]%Widerstand - nach EN 60617			

  25. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$X$};

  26. 			\draw [<-,blue] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$\uline{U}_{X}$};

  27.             \draw node at (.5,0){?};

  28. 		\end{scope}	

  29. 		\begin{scope}[>=latex,very thick,xshift=-.5cm,yshift=-.5cm,rotate=90]%Spannungsquelle

  30. 			\draw (0,0)--(1,0)node at(.5,-.133)[right]{$\uline{U}$};

  31. 			\draw (.5,0)circle(.133);

  32. 			\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{\footnotesize$\uline{U}$};

  33. 		\end{scope}								

  34. 		\begin{scope}[>=latex,very thick]%Knotenpunkte

  35. 			\draw (1.5,.8)--(1.5,1)--(-.5,1) (1.5,-.8)--(1.5,-1)--(-.5,-1)(-.4,1)--(-.5,1)--(-.5,.5)(-.4,-1)--(-.5,-1)--(-.5,-.5);

  36. %			\fill (-.5,1)circle(.025) (-.5,-1)circle(.025);

  37. 			\draw [->,blue] (.6,0)--(1.4,0)node at(1,0)[below]{$\underline{U}_{ab}$};

  38. 			\fill (.5,0)circle(.025) node at (.5,0)[left]{a};

  39. 			\fill (1.5,0)circle(.025)node at (1.5,0)[right]{b};

  40. 		\end{scope}				

  41. 	\end{tikzpicture}

  42. \end{align*}

  43. \ifthenelse{\equal{\toPrint}{Lösung}}{%

  44. %\begin{align}

  45. %\intertext{Formeln:}

  46. %\end{align}

  47. Berechnung: (Platzbedarf in x: $9\,\centi\metre$; in y: $\pm 5\,\centi\metre$)\\

  48. \begin{align*}

  49. \intertext{a) Zum Brückenabgleich muß $X$ eine Kapazität sein.}\\

  50. \intertext{Abgleichbedingung:}\\

  51. \frac{jX_L}{R}&=\frac{R}{jX_C}\\

  52. X&=\uuline{-\frac{R^2}{X_L}} \qquad\text{ neg. VZ $\Rightarrow$ Kapazität $X=X_C$} \\

  53. -\frac{1}{j\omega C}&=-\frac{R^2}{j\omega L}\Rightarrow \quad\text{oder}\quad C=\frac{L}{R^2}

  54. \end{align*}

  55. \clearpage

  56. \begin{align*}

  57. \intertext{b) Bedingung: $\uline{U}_{ab}\,\bot\,\uline{U}$; gleiche Zeigerlängen.}

  58.     \begin{tikzpicture}[very thick,scale=1]

  59. 		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]

  60.             \draw[ultra thin,black!50!](0,-5)grid(8,5);

  61.             \draw[thin,->](0,0)--(8.5,0)node[right]{$\Re$};

  62.             \draw[thin,->](0,-5.5)--(0,5.5)node[above]{$\Im$};

  63.             \foreach \y in {5,4,...,-5}

  64.             \draw(0,\y)--(-.1,\y)node[left]{$\y$};

  65.             \draw[black!50!,dashed](0,0)+(20:4.2cm)arc(20:-20:4.2cm);

  66.             \draw[black!50!,dashed](8,0)+(160:4.2cm)arc(160:200:4.2cm);

  67.             \draw[black!50!,dashed](4,1.5)--(4,-1.5);

  68.             \fill(4,0)circle(0.075cm);

  69.             \draw[blue,->](0,0)--(8,0)node at(7.5,.25){$\uline{U}$};

  70.              \draw[black!50!](4,0)circle(4cm);

  71.              \draw[blue,->](0,0)--(30:6.93)node at(2.5,1.75){$\uline{U}_a$};

  72.              \draw[blue,->](0,0)--(-30:6.93)node at(2.5,-1.75){$\uline{U}_{b}$};             \draw[blue,->](30:6.93)--(-30:6.93)node at(5.5,-1.25){$\uline{U}_{ab}$};

  73.              \draw[blue,->](30:6.93)--(8,0)node at(6.75,1.5){$\uline{U}_{R}$};             \draw[blue,->](-30:6.93)--(8,0)node at(6.75,-1.5){$\uline{U}_{X}$};

  74.              \draw node at (6,3.75){a};

  75.              \draw node at (6,-3.75){b};

  76.              \draw[red](30:6.93)+(210:.5cm)arc(210:300:.5cm);

  77.              \fill[red](30:6.93)+(255:.25cm)circle(.05cm);

  78.              \draw[red](-30:6.93)+(150:.5cm)arc(150:60:.5cm);

  79.              \fill[red](-30:6.93)+(105:.25cm)circle(.05cm);

  80.              \draw[red](0:6)+(0:.5cm)arc(0:90:.5cm);

  81.              \fill[red](0:6)+(45:.25cm)circle(.05cm);

  82.              \draw node at(8.5,4.5)[right]{$\diamond$\ \ \footnotesize{$\uline{U}$}};

  83.              \draw node at(8.5,4)[right]{$\phantom{\diamond x}$\footnotesize{Thaleskreis, da $\uline{U}_q \bot \uline{U}_R$ und $\uline{U}_b \bot \uline{U}_X$}};

  84.              \draw node at(8.5,3.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_R=\uline{U}$}};

  85.              \draw node at(8.5,3)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_a$ und $\uline{U}_R$ zeichnen}};

  86.              \draw node at(8.5,2.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_a+\uline{U}_{ab}-\uline{U}_b=0$ und $\uline{U}_{ab} \bot\uline{U}$}};

  87.              \draw node at(8.5,2)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_{ab}$ und $\uline{U}_b$ zeichnen}};

  88.              \draw node at(8.5,1.5)[right]{$\diamond$\ \footnotesize{$\uline{U}_b+\uline{U}_X=\uline{U}$}};

  89.              \draw node at(8.5,1)[right]{$\phantom{\diamond x}$\footnotesize{$\uline{U}_X$ zeichnen}};

  90. 		\end{scope}	

  91. 	\end{tikzpicture}

  92. \end{align*}

  93. Bei der nicht abgeglichenen Brücke mit der Bedingung $\uline{U}_{ab}\,\bot\,\uline{U}$, muss $X$ eine Induktivität sein.\\

  94. 

  95. Rechnerisch:

  96. \begin{align*}

  97. |U_a|&=|U_b|\Rightarrow |X_L\cdot I_a|=|R\cdot I_b|\tag{1}\\

  98. |U_R|&=|U_X|\Rightarrow |R\cdot I_a|=|X\cdot I_b| \tag{2}\\

  99. \frac{X_L}{R}&=\frac{R}{X}\tag{1:2}\\

  100. X&=\uuline{+\frac{R^2}{X_L}}

  101. \end{align*}

  102. \clearpage

  103. }{}%