59 lines
3.2 KiB
TeX
59 lines
3.2 KiB
TeX
\section{Strom L-R-C}
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Berechnen Sie den Strom $\uline{I}$
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\begin{align*}
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\uline{U}&=15\,\volt \cdot e^{j20\,\degree}\quad f=1\,\kilo\hertz\\
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C_1&=9\,\micro\farad\quad C_2=4\,\micro\farad\quad R=20\,\ohm\quad L=2\,\milli\henry
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\end{align*}
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\ifthenelse{\equal{\toPrint}{Lösung}}{%
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%\begin{align}
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%\intertext{Formeln:}
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%\text{...noch einfügen...}
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%\end{align}
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\begin{align*}
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\begin{tikzpicture}[scale=2.5]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm,rotate=90]%Spannungsquelle |
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\draw (0,0)--(1,0) node at (.5,-.133) [right] {$\uline{U}$};
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\draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Kondensator |
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm,rotate=90]%Kondensator |
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\draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C_2$};
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\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$\uline{U}$};
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\draw [<-,red] (.75,0)--(.95,0) node at (.5,.2)[left]{$\uline{I}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Fehlstellen Eckverbindungen.
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\draw (0,.8)--(0,1)--(2,1) (.8,1)--(3,1)--(3,.9) (0,.2)--(0,0)--(3,0)--(3,.2);
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\begin{align*}
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\intertext{Berechnung:}
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C_1& \text{ unwirksam, da parallel zur Spannungsquelle.}\\
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\omega&=2\pi f=6283\,\frac{1}{\,\second}\\
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\uline{Z}_{RLC_2}&=X_L+\uline{Z}_{||}\\
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\uline{Y}_{||}&=G+jB_{C_2}\\
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G&=\frac{1}{R}=50\,\milli\siemens\\
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B_{C_2}&=\omega C_2=6283\frac{1}{\,\second}\cdot 4\,\micro\farad=25{,}13\,\milli\siemens\Rightarrow\\
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X_{C_2}&=-\frac{1}{B_{C_2}}=-39{,}79\,\ohm\\
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\uline{Y}_{||}&=G+jB_{C_2}=(50+j25{,}13)\,\milli\siemens=55{,}96\,\milli\siemens\cdot e^{j26{,}7\,\degree}\\
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\uline{Z}_{||}&=\frac{1}{\uline{Y}_{||}}=17{,}87\cdot e^{-j26{,}7\,\degree}=(15{,}97-j8{,}025)\,\ohm\\
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X_L&=\omega \cdot L=6283\frac{1}{\,\second}\cdot 2\,\milli\henry=12{,}57\,\ohm\\
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\uline{Z}_{LRC_2}&=\uline{Z}_{||}+jX_L=[15{,}97+j(-8{,}025+12{,}57)]\,\ohm
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=(15{,}97+j4{,}541)\,\ohm=16{,}6\,\ohm\cdot e^{j15{,}9\,\degree}\\
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\uline{U}_{C_2}&=\uline{U}\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{LRC_2}}=15\,\volt \cdot e^{j20\,\degree}\cdot \frac{17{,}87\cdot e^{-j26{,}7\,\degree}}{16{,}6\,\ohm\cdot e^{j15{,}9\,\degree}}=16{,}15\,\volt\cdot e^{-j22{,}6\,\degree}=(14{,}91-j6{,}21)\,\volt\\
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\uline{I}&=\uline{U}_{C_2}\cdot jB_{C_2}=16{,}15\,\volt\cdot e^{-j22{,}6\,\degree}\cdot 25{,}13\,\milli\siemens\cdot e^{j90\,\degree}\\
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&=\uuline{0{,}4058\,\ampere\cdot e^{j67{,}4\,\degree}}=\uuline{(0{,}156+j0{,}375)\,\ampere}\\
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\end{align*}
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\clearpage
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}{}%
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