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ET2_Uebung_BEI/ET2_L_B16_A8.tex

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  1. \section{Spannung Vierpol}

  2. %\renewcommand{\theequation}{0}

  3. Berechnung Sie $\uline{U}_0$

  4. \begin{align*}

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

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

  7. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};

  8. 			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_1$};

  9. 		\end{scope}

  10. 		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617			

  11. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};

  12. 			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{U}_R=R_2\cdot \uline{I}_2$};

  13. 		\end{scope}

  14. 		\begin{scope}[>=latex,very thick, xshift=1.5cm, yshift=0cm]% Vierpol 			 

  15. 			\draw (0,-.2)rectangle(2.0,1.2)node at(.75,.5)[right]{$(\uline{Z})$};

  16. 			\draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1);				 

  17. 		\end{scope}				

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

  19. 			\draw (0,0)--(1,0);

  20. 			\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);

  21. 			\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);	

  22. 			\fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05)	 (4,0)circle(.05) (4,1)circle(.05);					

  23. 		\end{scope}

  24. 		\begin{scope}[>=latex,very thick,xshift=.75cm,yshift=1cm]% Strompfeil		

  25. 			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$};			

  26. 		\end{scope}	

  27. 		\begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=1cm]% Strompfeil		

  28. 			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$};			

  29. 		\end{scope}

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

  31. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_1$};			

  32. 		\end{scope}	

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

  34. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_0$};			

  35. 		\end{scope}	

  36. 		\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil		

  37. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[below]{\footnotesize$\uline{U}_2$};			

  38. 		\end{scope}											

  39. 	\end{tikzpicture}

  40. \end{align*}

  41. \begin{align*}

  42. R_1&=300\,\ohm\\

  43. R_2&=600\,\ohm\\

  44. \uline{Z}_{11}&=100\,\kilo\ohm\text{ reell}\\

  45. \uline{Z}_{21}&=-2\,\mega\ohm\text{ reell}\\

  46. \uline{U}_1&=1{,}5\,\volt

  47. %\end{align*}

  48. %\begin{align*}

  49. \intertext{Die $\uline{Z}$-Matrix des Vierpols ist gegeben:}

  50. \begin{bmatrix}

  51. 	\uline{U}_0\\

  52. 	\uline{U}_2

  53. \end{bmatrix}

  54. &=

  55. \begin{bmatrix}

  56. 	\uline{Z}_{11}&0\\

  57. 	\uline{Z}_{21}&0

  58. \end{bmatrix}

  59. \cdot

  60. \begin{bmatrix}

  61. 	\uline{I}_0\\

  62. 	\uline{I}_2

  63. \end{bmatrix}

  64. \end{align*}

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

  66. %\begin{align}

  67. %\intertext{Formeln:}

  68. %\end{align}

  69. Berechnung:

  70. \begin{align*}

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

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

  73. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};

  74. 			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_1\cdot (\uline{I}_2+\uline{I}_0)$};

  75. 		\end{scope}

  76. 		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617			

  77. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};

  78. 			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$};

  79. 		\end{scope}

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

  81. 			\draw (0,0)--(1,0);

  82. 			\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);

  83. 			\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);	

  84. 			\fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05)	 (4,0)circle(.05) (4,1)circle(.05);					

  85. 		\end{scope}

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

  87. 			\draw (0,0)--(1,0);

  88. 			\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);

  89. 			\draw (1,0)--(1,-.5)--(4,-.5)--(4,0)(1,1)--(1.25,1)(4,1)--(3.75,1);	

  90. 			\fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05)	 (4,0)circle(.05) (4,1)circle(.05);					

  91. 		\end{scope}

  92. 		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1.2cm]% Strompfeil		

  93. 			\draw [->,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_0$};			

  94. 		\end{scope}	

  95. 		\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1.2cm]% Strompfeil		

  96. 			\draw [<-,red] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$\uline{I}_2$};			

  97. 		\end{scope}

  98. 		\begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil		

  99. 			\draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_1$};			

  100. 		\end{scope}			

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

  102. 			\draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$};			

  103. 		\end{scope}	

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

  105. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$};	

  106. 		\end{scope}	

  107. 		\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil		

  108. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$};	

  109. 		\end{scope}											

  110. 	\end{tikzpicture}

  111. \end{align*}

  112. Aus $\uline{Z}$-Matrix folgt:

  113. \begin{align*}

  114. \uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0 \tag{1}\\

  115. \uline{U}_2&=\uline{Z}_{21}\cdot \uline{I}_0 \tag{2}

  116. \intertext{Knoten 1:}

  117. \uline{I}_1-\uline{I}_2-\uline{I}_0&=0 \tag{3}

  118. \intertext{Masche 1:}

  119. \uline{U}_1&=R_1\cdot \uline{I}_1+ \uline{U}_0\tag{4}

  120. \intertext{Masche 2:}

  121. \uline{U}_0&=R_2\cdot \uline{I}_2+\uline{U}_2 \tag{5}

  122. \intertext{Zwei Gleichungen für zwei unbekante Ströme $\uline{I}_0$ und $\uline{I}_2$:}

  123. \uline{Z}_{11}\cdot \uline{I}_0&=R_2\cdot \uline{I}_2+\uline{Z}_{21}\cdot \uline{I}_0  \tag{1\&2 in 5}\\

  124. \Rightarrow \uline{I}_2&=\frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0  \tag{6}\\

  125. \uline{U}_1&=\uline{I}_0 \cdot\uline{Z}_{11}+R_1(\uline{I}_0 +\uline{I}_2 ) \tag{7 1\&3 in 4}\\

  126. \uline{U}_1&=\uline{I}_0\cdot \uline{Z}_{11}+R_1\cdot \uline{I}_0+R_1\cdot  \frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0 \tag{6 in 7}\\

  127. &=\left[\uline{Z}_{11}+R_1+\frac{R_1}{R_2}\cdot(\uline{Z}_{11}-\uline{Z}_{21})\right]\cdot \uline{I}_0\\

  128. &=\left[100\,\kilo\ohm+300\,\ohm+\frac{300\,\ohm}{600\,\ohm}\cdot \Big(100\,\kilo\ohm-(-2\,\mega\ohm)\Big)\right]\cdot \uline{I}_0\\

  129. &=0{,}103+0{,}5\cdot (100+2000)\,\kilo\ohm \cdot  \uline{I}_0=1{,}15\,\mega\ohm\cdot  \uline{I}_0\\

  130. \Rightarrow\uline{I}_0&=\frac{\uline{U}_1}{1{,}15\,\mega\ohm}=\frac{1{,}5\,\volt}{1{,}15\,\mega\ohm}=1{,}3\,\micro\ampere\\

  131. \uline{U}_0&=\uline{Z}_{11}\cdot \uline{I}_0=100\,\kilo\ohm\cdot 1{,}3\,\micro\ampere=\uuline{130{,}4\,\milli\volt}

  132. \intertext{Nicht gefragt:}

  133. \uline{I}_2&=\frac{1}{R_2}\cdot (\uline{Z}_{11}-\uline{Z}_{21})\cdot \uline{I}_0=\frac{1}{600\,\ohm}\cdot 2{,}1\mega\ohm\cdot 1{,}3\,\micro\ampere=4{,}55\,\milli\ampere >> \uline{I}_0\\

  134. \text{Probe }\\

  135. %&=\frac{1}{R_1}\cdot (\uline{I}_2+\uline{I}_0)\cdot \uline{U}_0\\

  136. \uline{U}_1&=1{,}5\,\volt=R_1\cdot (\uline{I}_0+\uline{I}_2)+\uline{I}_0\cdot Z_{11}\\

  137. &=300\,\ohm\cdot (0{,}0013+4{,}55)\,\milli\ampere+1{,}3\,\micro\ampere\cdot 100\,\kilo\ohm=1{,}498\,\volt

  138. \end{align*}

  139. \clearpage

  140. \enlargethispage{1cm}

  141. Ist Ihnen aufgefallen, daß $\uline{Z}_{21}$ einen negativen Wert hat?\\[\baselineskip]

  142. Ein Widerstand nimmt elektrische Leistung auf, also ein Verbraucher.\\

  143. Dann muß also ein negativer Widerstand elektrische Leistung abgeben!

  144. Gibt es das in der Realität?\\[\baselineskip]

  145. Was steckt in dem Vierpol?\\[\baselineskip]

  146. Passiver Vierpol:\\

  147. Zum Beispiel mit 3 Widerständen in $T$ oder $\Pi$ Schaltung, ergibt jedoch keine Lösung!\\

  148. Aus der $\uline{Z}$ Matrix ist ersichtlich, daß $\uline{Z}_{21}=0$ und $\uline{Z}_{22}=0$ sind,

  149. somit existiert keine Kopplung vom Ausgang zum Eingang, sondern nur in Vorwärtsrichtung.\\[\baselineskip]

  150. Beispiel: Elektrodynamischer Lautsprecher der auf ein Kondensatormikrofon einwirkt. Kopplung nur vom Lautsprecher zum Mikrofon, nicht umgekehrt.\\[\baselineskip]

  151. Aktiver Vierpol: (Für Fortgeschrittene)\\

  152. Es muß sich um einen invertierenden Trennverstärker mit einem Eingangswiderstand von

  153. $100\,\kilo\ohm$ und einem Ausgangswiderstand von $2\,\mega\ohm$ handeln. Die Verstärkung ist zu berechnen.

  154. \begin{align*}

  155. \uline{U}_2&=\uline{U}_0-R_2\cdot \uline{I}_2=130{,}4\,\milli\volt-600\,\ohm\cdot 4{,}55\,\milli\ampere=-2{,}6\,\volt\\

  156. \uline{U}_q&=\uline{U}_2-\uline{Z}_{22}\cdot \uline{I}_2=-2{,}6\,\volt-2\,\mega\ohm\cdot 4{,}55\,\milli\ampere=-9102{,}6\,\volt\\

  157. V&=\frac{\uline{U}_q}{\uline{U}_0}=\frac{-9102{,}6\,\volt}{130{,}4\,\milli\volt}=-69805

  158. \end{align*}

  159. \begin{align*}

  160. 	\begin{tikzpicture}[scale=1.5]

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

  162. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};

  163. 			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\tiny$R_1\cdot (\uline{I}_2+\uline{I}_0)$};

  164. 		\end{scope}

  165. 		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=2cm]%Widerstand - nach EN 60617			

  166. 			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_2$};

  167. 			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$R_2\cdot \uline{I}_2$};

  168. 		\end{scope}

  169. 				\begin{scope}[>=latex,very thick,black!25!, xshift=1.5cm, yshift=0cm]% Vierpol mit Innenleben			

  170. 					\draw (0,-.2)rectangle(2.0,1.2);%node at(.75,.5)[right]{$(\uline{Z})$};

  171. 					\draw (0,0)--(-.5,0)(0,1)--(-.5,1)(2,0)--(2.5,0)(2,1)--(2.5,1);

  172. %				\end{skope}	

  173. 				\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90] %Z_11 			

  174. 					\draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667)

  175. [left] {$\uline{Z}_{11}$};

  176. 					\draw [<-,red!50!blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[right]{\footnotesize$\uline{Z}_{11}\cdot \uline{I}_0$};

  177. 				\end{scope}			

  178. 				\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Z_21			 

  179. 					\draw [red!50!blue](0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$\uline{Z}_{21}$};

  180. 					\draw [->,red!50!blue] (.3,-.2)--(.7,-.2)node at(.6,-.2)[below]{\footnotesize$-\uline{Z}_{21}\cdot \uline{I}_2$};

  181. 			\end{scope}

  182. 				\begin{scope}[>=latex,very thick,red!50!blue,xshift=1cm,yshift=0cm,rotate=90]%Spannungsquelle| 			

  183. 					\draw (0,0)--(1,0);%;node at(.5,-.133)[right]{$V\cdot \uline{U}_0$};

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

  185. 					\draw [->] (.3,-.2)--(.7,-.2) node at (.3,-.2)[right]{\footnotesize$V\cdot \uline{U}_q$};

  186. 					\draw node at (.3,.025)[left]{\footnotesize$+$};

  187. 					\draw node at (.7,.025)[left]{\footnotesize$-$};					 

  188. 			\end{scope}

  189. 			\begin{scope}[>=latex,very thick,red!50!blue,xshift=0cm,yshift=0cm]%

  190. 					\draw (-.5,0)--(2.5,0)(-.5,1)--(0,1)(2,1)--(2.5,1);

  191. 			\end{scope}

  192. 		\end{scope}			% Ende Vierpol Innenleben

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

  194. 			\draw (0,0)--(1,0);

  195. 			\draw (1,1)--(1,2)--(2,2) (3,2)--(4,2)--(4,1);

  196. 			\draw (1,0)--(1,-.5)--(4,-.5)--(4,0);	

  197. 			\fill (0,0)circle(.05) (0,1)circle(.05) (1,0)circle(.05) (1,1)circle(.05)	 (4,0)circle(.05) (4,1)circle(.05);					

  198. 		\end{scope}

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

  200. 			\draw [->,red] (0,0)--(.4,0)node at(.2,0)[above]{\footnotesize$\uline{I}_0$};			

  201. 		\end{scope}	

  202. 		\begin{scope}[>=latex,very thick,xshift=4cm,yshift=1cm]% Strompfeil		

  203. 			\draw [->,red] (0,0)--(-.4,0)node at(-.2,0)[above]{\footnotesize$\uline{I}_2$};			

  204. 		\end{scope}

  205. 		\begin{scope}[>=latex,very thick,xshift=-.2cm,yshift=1cm]% Strompfeil		

  206. 			\draw [->,red] (0,0)--(.2,0)node at(.1,0)[above]{\footnotesize$\uline{I}_2+\uline{I}_0$};			

  207. 		\end{scope}			

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

  209. 			\draw [->,blue] (0,.8)--(0,.2)node at(-.2,.5)[left]{\footnotesize$\uline{U}_1$};			

  210. 		\end{scope}	

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

  212. 			\draw [->,blue] (0,.8)--(0,.2)node at(0,.25)[left]{\tiny$\uline{U}_0=\uline{Z}_{11}\cdot \uline{I}_0$};	

  213. 		\end{scope}	

  214. 		\begin{scope}[>=latex,very thick,xshift=4cm,yshift=0cm]% Spannungspfeil		

  215. 			\draw [->,blue] (0,.8)--(0,.2)node at(.2,.5)[right]{\footnotesize$\uline{U}_2=\uline{Z}_{21}\cdot \uline{I}_0$};	

  216. 		\end{scope}											

  217. 	\end{tikzpicture}

  218. \end{align*}

  219. \begin{minipage}[t]{1\textwidth}

  220. \centering

  221. \includegraphics[width =0.6\textwidth]{b16a8a}

  222. \end{minipage}

  223. \clearpage

  224. }{}%