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ET2_Uebung_BEI/ET2_L_B17_A3.tex

ET2_L_B17_A3.tex 7.5KB
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  1. \section{Ortskurve}

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

  3. \begin{enumerate}

  4. \setlength{\itemsep}{-0.2\baselineskip}

  5.   \item Zeichnen Sie maßstäblich die Ortskurve für das Spannungsverhältnis $U_2/U_1$ in Abhängigkeit von der Frequenz $f$.

  6.   \item Geben Sie die Grenzfrequenz der Schaltung an.

  7.   \item  Wie groß ist die Dämpfung des Vierpols für die Frequenz $f=1{,}2\,\kilo\hertz$ (falls in Vorlesung behandelt: in $\deci\bel$)

  8. \end{enumerate}

  9. $\qquad \,\, R=16\,\kilo\ohm$; $C=12\,\nano\farad$; $200\,\hertz\leq f \leq 1{,}2\,\kilo\hertz$

  10. \begin{align*}

  11. 	\begin{tikzpicture}[very thick,scale=2]

  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] {$R$};

  14. 		\end{scope}	

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

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

  17. 		\end{scope}		

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

  19. 			\draw (-.5,0)--(0,0) (-.5,1)--(0,1) (-.5,0)--(1.5,0) (.8,1)--(1.5,1);

  20. 			\draw [->,blue] (-.5,.9)--(-.5,.1)node at(-.5,.5)[right]{$\underline{U}_1$};				 \fill (-.5,0)circle(.025) (-.5,1)circle(.025);

  21. 			\draw [->,red] (-.4,1.1)--(-.1,1.1) node at (-.25,1.1)[above]{$\underline{I}_1$};

  22. 			\draw [->,blue] (1.5,.9)--(1.5,.1)node at(1.5,.5)[right]{$\underline{U}_2$};				 \fill (1.5,0)circle(.025) (1.5,1)circle(.025);

  23. 			\draw [->,red] (1.4,1.1)--(1.1,1.1) node at (1.25,1.1)[above]{$\underline{I}_2=0$};

  24. 		\end{scope}	

  25. 	\end{tikzpicture}

  26. \end{align*}

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

  28. %\begin{align}

  29. %\intertext{Formeln:}

  30. %\end{align}

  31. Berechnung: (Platzbedarf in x: $5\,\centi\metre$; in y: $15\,\centi\metre$)\\[0.5\baselineskip]

  32. a) Spannungsverhältnis $U_2/U_1$ in Abhängigkeit von der Frequenz $f$

  33. \begin{align*}

  34. \uline{I}_1&=\uline{U}_1 \cdot \uline{Y} = \uline{U}_1 \cdot \frac{j\omega\cdot C}{1+j\omega\cdot C\cdot R}=\frac{\uline{U}_1}{R+\frac{1}{j\omega\cdot C}} \\

  35. \uline{U}_2&= \uline{I}_1\cdot \frac{1}{j\omega\cdot C}= \frac{\uline{U}_1}{(R+\frac{1}{j\omega\cdot C})\cdot j\omega\cdot C}=

  36. \frac{\uline{U}_1}{1+j\omega\cdot R\cdot C}\\

  37. \frac{\uline{U}_2}{\uline{U}_1}&=\frac{1}{1+j\omega\cdot R \cdot C}

  38. =\frac{1-j\omega RC}{1+(\omega RC)^2}\\[0.5\baselineskip]

  39. &\begin{tabular}{|l|l|l|}

  40.   \hline

  41.   % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...

  42.   $f/Hz$&$2\pi\cdot f\cdot R\cdot C$&$\uline{U}_2/\uline{U}_1$\\

  43.   \hline

  44.   200 & 0{,}241&0{,}945-j0{,}228\\

  45.   828{,}9&1&0{,}5-j0{,}5\\

  46.   1200&1{,}448&0{,}323-j0{,}468\\

  47.   \hline

  48. \end{tabular}

  49. \intertext{b) Grenzfrequenz}

  50. \omega_g\cdot R\cdot C&=1\\

  51. f_g&=\frac{1}{2\pi\cdot R\cdot C}=\uuline{828{,}9\,\hertz}

  52. \intertext{c) Dämpfung}

  53. f&=1{,}2\,\kilo\hertz\qquad a=20\cdot \lg\frac{\uline{U}_2}{\uline{U}_1}\\

  54. \frac{\uline{U}_2}{\uline{U}_1}&=0{,}323-j0{,}468=0{,}569\cdot e^{-j50{,}56\,\degree}\\

  55. a&=20\cdot \lg(0{,}569)=\uuline{-4{,}9\,\deci\bel}

  56. \end{align*}

  57. \begin{align*}

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

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

  60.             \draw[ultra thin,black!50!](0,-7.5)grid [step=.5cm](5,7.5);

  61.             \draw[thin,->](0,0)--(5.5,0)node[right]{$\Re \, \left\{\frac{\uline{U}_2}{\uline{U}_1}\right\}$};

  62.             \draw[thin,->](0,-8)--(0,8)node[above]{$\Im \, \left\{\frac{\uline{U}_2}{\uline{U}_1}\right\}$};

  63.             \foreach \y in {1.5,1,...,-1.5}

  64.             \draw(0,5*\y)--(-.1,5*\y)node[left]{$\y$};

  65.             \draw[red,thin](5,-7.5)--(5,7.5);

  66.             \draw[red](4.9,5)--(5.1,5)node [right]{$f_g=828{,}9\,\hertz$};

  67.             \draw[red](4.9,-5)--(5.1,-5)node [right]{$f_g=828{,}9\,\hertz$};

  68.             \draw[red](4.9,1.205)--(5.1,1.205)node [right]{$f=200\,\hertz$};

  69.             \draw[red](4.9,-1.205)--(5.1,-1.205)node [right] {$f=200\,\hertz$};

  70.             \draw[red](4.9,7.24)--(5.1,7.24)node [right] {$f=1200\,\hertz$};

  71.             \draw[red](4.9,-7.24)--(5.1,-7.24)node [right] {$f=1200\,\hertz$};

  72.             \draw node at(6.125,6.2){$(1+j\omega RC)$};

  73.             \foreach \x in {.5,1}

  74.             \draw(5*\x,.1)--(5*\x,-.1)node at (5*\x+.1,-.25){$\x$};

  75.             \fill[blue](2.5,0)circle(.05cm);

  76.             \draw[blue](0:2.5cm)+(180:2.5cm)arc(180:360:2.5cm);%Mittelpunkt+Start arc Start:End:Radius

  77.             \draw[blue!50!red,thin](0,0)--(5,-1.205)(0,0)--(5,-5)(0,0)--(5,-7.24);

  78.             \filldraw[blue!50!red](4.73,-1.14)circle(.05)node [below left]{$200\,\hertz$};

  79.             \filldraw[blue!50!red](2.5,-2.5)circle(.05)node [below]{$f_g$};

  80.             \filldraw[blue!50!red](1.62,-2.35)circle(.05)node [below left]{$1{,}2\,\kilo\hertz$};

  81. 	    \draw [blue] node at(4,-2.5){$\frac{\underline{U}_2}{\underline{U}_1}$};

  82. 		\end{scope}	

  83. 	\end{tikzpicture}

  84. \end{align*}

  85. % (Siehe Merksätze zu Inversion im Script, hier graphisch veranschaulicht:)

  86. %\begin{align*}

  87. %    \begin{tikzpicture}[very thick,scale=1.25]

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

  89. %            \draw(0,-1)--(0,1)node[above]{$\Im$};

  90. %            \draw(0,0)--(2,0);

  91. %            \draw[red](1,-1)--(1,1);

  92. %            \draw[red,->](0,0)--(1,.75)node at(1.25,.75){$\uline{Y}_p$};

  93. %		\end{scope}	

  94. %		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm]

  95. %            \draw(0,-1)--(0,1)node[above]{$\Im$};

  96. %            \draw(0,0)--(2,0);

  97. %            \draw[blue](.5,0)circle(.5cm);

  98. %            \draw[blue,->](0,0)--(.7,-.5cm)node at(1.25,-.5){$\uline{Z}_p$};

  99. %		\end{scope}	

  100. %		\begin{scope}[>=latex,very thick,xshift=6cm,yshift=0cm]

  101. %            \draw(0,-1)--(0,1);

  102. %            \draw(-.5,0)--(2,0);

  103. %            \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu};

  104. %            \draw[blue](.5,0)circle(.5cm);

  105. %            \draw[green!50!black,->](-.5,0)--(.7,-.5cm)node at(1,-.5){$\uline{Z}$}; 		 \end{scope}

  106. %		\begin{scope}[>=latex,very thick,xshift=9cm,yshift=0cm]

  107. %            \draw(0,-1)--(0,1);

  108. %            \draw[blue!50!red](-.5,-1)--(-.5,1)node[above]{$\Im$ neu};

  109. %            \draw(-.5,0)--(2.5,0);

  110. %            \draw[blue](1.25,0)circle(1cm);

  111. %            \draw[blue!50!red,->](-.5,0)--(2.1,.5)node at(2.5,.5){$\uline{Y}$};

  112. %		\end{scope}	

  113. %	\end{tikzpicture}

  114. %\end{align*}

  115. %Leitwertsortskuve der Parallelschaltung von $R_2$ und $C$ ergibt eine Gerade die im Abstand $G_2$ parallel zur imaginären Achse liegt.\\

  116. %$G_2=20\,\milli\siemens\,\widehat{=}\, 4\,\centi\metre$\\

  117. %Grenzen eintragen für $\uline{Y}_P=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre (f_u)\quad 25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre (f_o)\quad\text{ auch für $\uline{Y}^*_P$}$\\

  118. %Inversion von $\uline{Y}_P$ liefert die Widerstandsortskurve $\uline{Z}_P$ ein Kreis durch den Ursprung.\\

  119. %$R_2=50\,\ohm\,\widehat{=}\, 5\,\centi\metre \quad\text{Kreis um P(2.5,0) $r=2{,}5\,\centi\metre$}$\\

  120. %Der Serienwiderstand $R_1$ wird durch verschieben der Koordinaten hinzugefügt.\\

  121. %Punktweise Inversion von $\uline{Z}$ ein Kreis der jetzt nicht mehr durch den Ursprung geht, liefert $\uline{Y}$, wieder einen Kreis.\\

  122. %$\uline{Y}\quad\text{Kreis um P(6.67,0) $r=5\,\centi\metre$, da $\uline{Y}_{min}$ und $\uline{Y}_{max}$ bekannt sind}$\\

  123. %\begin{align*}

  124. %R_2&=50\,\ohm\,\widehat{=}\, 5\,\centi\metre \quad\text{Kreis um P(2.5,0) $r=2{,}5\,\centi\metre$}\\

  125. %G_2&=20\,\milli\siemens\,\widehat{=}\, 4\,\centi\metre\\

  126. %\uline{Y}_P&=5\,\milli\siemens\,\widehat{=}\, 1\,\centi\metre (f_u)\quad 25\,\milli\siemens\,\widehat{=}\, 5\,\centi\metre  (f_o)\quad\text{zeichnen, auch $\uline{Y}^*_P$}\\

  127. %\uline{Y}&\quad\text{Kreis um P(6.67,0) $r=5\,\centi\metre$, da $\uline{Y}_{min}$ und $\uline{Y}_{max}$ bekannt sind}\\

  128. %\end{align*}

  129. \clearpage

  130. }{}%