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ET2_Uebung_BEI/ET2_L_B13_A4.tex

ET2_L_B13_A4.tex 6.7KB
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  1. \section{Energieübertragung}

  2. Die Skizze zeigt ein System zur elektrischen Energieübertragung bestehend aus Quelle,

  3. Leitung und Verbraucher. Das System soll mit einem parallel geschalteten Kondensator $X_C$ so

  4. optimiert werden, dass die Leitungsverluste $P_{VRL}$ minimal werden.\\

  5. \begin{align*}

  6. 	\begin{tikzpicture}[scale=3]

  7. 		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle |

  8. 			\draw (0,0)--(1,0)node at(.5,-.133)[right]{\footnotesize{$\underline{U}_q=100\,\volt\cdot e^{j0}$}};

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

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

  11. 		\end{scope}

  12. 		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617			

  13. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {\footnotesize{$\underline{Z}_i=(1+2j)\,\ohm$}};

  14. 		\end{scope}

  15. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%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) [above] {\footnotesize{$R_L=1\,\ohm$}};

  17. 		\end{scope}						

  18. 		\begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=0cm,rotate=90] 			

  19. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [above right] {\footnotesize{$\underline{Z}_V=(10+j5)\,\ohm$}};

  20. 		\end{scope}

  21. 		\begin{scope}[>=latex,very thick,xshift=3.5cm,yshift=0cm,rotate=90]%Kondensator | 			

  22. 			\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {\footnotesize{$jX_C$}};

  23. 		\end{scope}	

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

  25. 			\draw [dashed](2.5,0)--(3.5,0)--(3.5,.2) (2.5,1)--(3.5,1)--(3.5,.8);

  26. 			\draw [dashed](0,0)--(2.5,0) (2.5,1)--(3.5,1)--(3.5,.8);

  27. 			\draw (0,0)--(2.5,0) (2,1)--(2.5,1);			

  28. 			\draw (0,.2)--(0,0)--(0,.2) (0,.8)--(0,1)--(.2,1);

  29. 			\draw  node at(.5,0) [below] {\footnotesize Quelle};

  30. 			\draw  node at(1.5,0) [below] {\footnotesize Leitung};

  31. 			\draw  node at(2.5,0) [below] {\footnotesize Verbraucher};

  32. 			\draw [very thin, dashed] (1,-.2)--(1,1.2)(2,-.2)--(2,1.2);

  33. 			\draw [->,red](2.125,1.125)--(2.375,1.125) node [right] {\footnotesize{$\underline{I}$}};

  34. 		\end{scope}						

  35. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Knotenpunkte		

  36. 			\draw (0,0)circle(.035); 

  37. 			\fill [white](0,0)circle(.025); 

  38. 			\draw (1,0)circle(.035);

  39. 			\fill [white](1,0)circle(.025);

  40. 			\fill (1.5,0)circle(.025);				

  41. 			\fill (2.5,0)circle(.025);			 

  42. 		\end{scope}

  43. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Knotenpunkte		

  44. 			\draw (0,0)circle(.035); 

  45. 			\fill [white](0,0)circle(.025); 

  46. 			\draw (1,0)circle(.035);

  47. 			\fill [white](1,0)circle(.025);

  48. 			\fill (1.5,0)circle(.025);			

  49. 			\fill (2.5,0)circle(.025);				 

  50. 		\end{scope}				

  51. 	\end{tikzpicture}

  52. \end{align*}

  53. \renewcommand{\labelenumi}{\alph{enumi})}

  54. \begin{enumerate}

  55. \item Bestimmen Sie $X_C$ so, dass der Blindleistungsbedarf des Verbrauchers verschwindet.

  56. \item  Berechnen Sie die Verlustleistung $P_{VRL}$ der Leitung und die Wirkleistung $P_W$ im Verbraucher.

  57. \end{enumerate}

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

  59. %\begin{align}

  60. %\intertext{Formeln:}

  61. %\end{align}

  62. Berechnung:\\[\baselineskip]

  63. a) Verbraucher $\uline{Z}_V\,||\,X_C$, daher Ersatzschaltbild für $\uline{Z}_V$ (ESB) in Parallelform erforderlich

  64. \begin{align*}

  65. \underline{Z}_V&=R_V+jX_V=(10+j5)\,\ohm\,\,\quad\text{ Scheinwiderstand, entspricht einer Reihenschaltung}\\[\baselineskip]

  66. Z^2_V&=R_V\cdot R_p=R^2_V+X^2_V\,\qquad\qquad\text{Umwandlung in Parallel-ESB}\\

  67. R_p&=R_V+\frac{X^2_V}{R_V}=(10+\frac{25}{10})\,\ohm=12{,}5\,\ohm\\[\baselineskip]

  68. Z^2_V&=X_V\cdot X_{L_p}=R^2_V+X^2_V\,\,\quad\qquad\text{Umwandlung in Parallel-ESB}\\

  69. X_{L_p}&=X_V+\frac{R^2_V}{X_V}=(5+\frac{100}{5})\,\ohm=25\,\ohm

  70. \end{align*}

  71. \begin{align*}

  72. 	\begin{tikzpicture}[scale=2]

  73. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] 			

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

  75. 		\end{scope}

  76. 		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule | 			

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

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

  79. 		\end{scope}		

  80. 		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator | 			

  81. 			\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$jX_C$};

  82. 		\end{scope}	

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

  84. 			\draw (0,0)--(3,0)--(3,.2) (0,1)--(3,1)--(3,.8);			

  85. 		\end{scope}						

  86. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Knotenpunkte		

  87. 			\fill (0,0)circle(.025); 

  88. 			\fill (1,0)circle(.025);

  89. 		\end{scope}

  90. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Knotenpunkte		

  91. 			\fill (0,0)circle(.025); 

  92. 			\fill (1,0)circle(.025);

  93. 		\end{scope}	

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

  95. 			\draw (0,0)circle(.05); 

  96. 			\fill (0,0)circle(.025); 

  97. 		\end{scope}

  98. 		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Knotenpunkte		

  99. 			\draw (0,0)circle(.05);

  100. 			\fill (0,0)circle(.025);			 

  101. 		\end{scope}

  102. 		\begin{scope}[>=latex,very thick]		

  103. 			\draw node at (0,0.5) [left] {$\underline{Z}'\Rightarrow$};	

  104. 		\end{scope}

  105. 		\draw node at(1.5,0)[below]{Verbraucher $Z_V$};

  106. %	\end{tikzpicture}

  107. %		\begin{tikzpicture}[scale=.5,xshift=15cm,yshift=-2cm]

  108. 		\begin{scope}[>=latex,very thick,scale=.25,xshift=18cm,yshift=2cm] 			

  109. 			\draw [->](0,0)--(1,0)node [right]{$R_p$};

  110. 			\draw [->](1,0)--(1,2)node [above]{$jX_{Lp}$};

  111. 			\draw [->](1,0)--(1,-2)node[below]{$jX_C$};	

  112. 			\draw [->,red!50!blue](0,0)--(1,2)node at (.5,1)[left]{$Z_V$};								

  113. 		\end{scope}

  114. 	\end{tikzpicture}

  115. \end{align*}

  116. \begin{align*}

  117. \underline{Z}'&=\underline{Z}_V || jX_C\\

  118. \frac{1}{\underline{Z}'}&=\frac{1}{R_p}+\frac{1}{jX_{Lp}}+\frac{1}{jX_C}\\

  119. \intertext{Leitungsverluste sind minimal, wenn die Blindleistung $=0$ wird (Kompensation)}

  120. \frac{1}{\underline{Z}'}&=\frac{1}{R_p}+\cancel{\frac{1}{jX_{Lp}}}+\cancel{\frac{1}{jX_C}}\qquad\Rightarrow \underline{Z}'=R_p\\

  121. \Im(\underline{Z}')&=0  \quad\text{oder}\quad |X_C| \stackrel{!}{=} |X_{L_p}|\text{ also}\\

  122. %\Re(\underline{Z}')&=R_p\\

  123. X_C&=-X_{L_p}=\uuline{-25\,\ohm}\\

  124. \underline{Z}_{ges}&=\underline{Z}_i+R_L+\underline{Z}'\\

  125. \underline{Z}_{ges}&=\underline{Z}_i+R_L+R_p=(1+j2+1+12{,}5)\,\ohm=(14{,}5+j2)\,\ohm\\

  126. |\underline{Z}_{ges}|&=\sqrt{14{,}5^2+2^2}\,\ohm=14{,}64\,\ohm\\

  127. I&=\frac{U}{|Z_{ges}|}=\frac{100\,\volt}{14{,}64\,\ohm}=6{,}83\,\ampere\\

  128. \intertext{b) Verlust- und Wirkleistung}

  129. P_{VR_L}&=I^2\cdot R_L=(6{,}83\,\ampere)^2\cdot 1\,\ohm=\uuline{46{,}7\,\watt}\\

  130. P_W&=I^2\cdot R_p=(6{,}83\,\ampere)^2\cdot 12{,}5\,\ohm=\uuline{583\,\watt}

  131. \end{align*}

  132. \clearpage

  133. }{}%