107 lines
5.8 KiB
TeX
107 lines
5.8 KiB
TeX
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\section{Scheinersatzwiderst<EFBFBD>nde}
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Der Eingangswiderstand eines linearen Zweipols betr<74>gt bei der Frequenz $f=800\,\hertz$\\
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$Z=600\,\ohm$, sein Phasenwinkel ist $\varphi=30\,\degree$ induktiv.
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\renewcommand{\labelenumi}{\alph{enumi})}
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\begin{enumerate}
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\item Berechnen Sie die Schaltungselemente $R_r$ und $L_r$ der gleichwertigen Reihenersatzschaltung!
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\item Berechnen Sie die Schaltungselemente $R_p$ und $L_p$ der gleichwertigen Parallelersatzschaltung!
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\item Wie <20>ndern sich die Scheinersatzwiderst<73>nde (Betrag und Phase) beider Ersatzschaltungen, wenn die Frequenz $f'= 600\,\hertz$ betr<74>gt?
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\end{enumerate}
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\ifthenelse{\equal{\toPrint}{L<EFBFBD>sung}}{%
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\begin{align}
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\intertext{Formeln:}
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\underline{Z}& &\text{Scheinwiderstand (Impedanz)}\\
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Z&=|\underline{Z}| &\text{Betrag des Scheinwiderstandes}\\
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X&=\omega\cdot L &\text{Blindwiderstand (Reaktdanz)}\\
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B&=-\frac{1}{\omega\cdot L} &\text{Blindleitwert (Suszepdanz)}
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\end{align}
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\begin{align*}
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\intertext{Berechnung:}
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\intertext{a) Widerstandsebene:}
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R_r&=Z\cdot \cos(\varphi_r)=600\,\ohm\cdot \cos(30\degree)=\uuline{520\,\ohm}\\
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X_r&=Z\cdot \sin(\varphi_r)=600\,\ohm\cdot \sin(30\degree)=300\,\ohm\\
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L_r&=\frac{X_r}{\omega}=\frac{X_r}{2\pi f}=\frac{300\,\ohm}{2\pi\cdot 800\,\frac{1}{\second}}=\uuline{60\,\milli\henry}
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\end{align*}
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\begin{align*}
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\begin{tikzpicture}[scale=3]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617
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\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_r$};
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R_r}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_r$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{L_r}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Knotenpunkte
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\fill (0,0)circle(.025) node [above left] {\footnotesize$1$};
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\fill (2,0)circle(.025) node [above right] {\footnotesize$2$};
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=3cm, yshift=-.25cm]
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\draw [->,thin](0,0)--(1,0)node[right]{$R$};
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\draw [->](0,0)--(.52,0)node at (.26,0)[below]{$R_r$};
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\draw [->,thin](0,0)--(0,.5)node[above]{$X$};
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\draw [->](.52,0)--(.52,.3)node at (.52,.15)[right]{$X_r$};
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\draw [->](0:0)--(30:.6)node at (30:.3)[above left]{$Z$};
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\draw [->,red,thin] (0:.26) arc (0:30:.26cm) node at (15:.26) [right] {$\varphi_r$};
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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{b) Leitwertebene:}
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Y&=\frac{1}{Z}=\frac{1}{600\,\ohm}=1{,}667\,\milli\siemens \text{; }\qquad\varphi_p=-\varphi_r=-30\degree\\
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G_p&=Y\cdot \cos(\varphi_p)=1{,}667\,\milli\siemens\cdot \cos(-30\degree)
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=1{,}443\,\milli\siemens\Rightarrow R_p=\frac{1}{G_p}=\uuline{693\,\ohm}\\
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B_p&=Y\cdot \sin(\varphi_p)=1{,}667\,\milli\siemens\cdot \sin(-30\degree)
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=-0{,}833\,\milli\siemens\Rightarrow X_p=-\frac{1}{B_p}=1200\,\ohm\\
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%&\text{mit }B_p=-\frac{1}{\omega\cdot L_p}\Rightarrow \\
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L_p&=-\frac{1}{\omega\cdot B_p}=-\frac{1}{2\pi f\cdot B_p}=\frac{-1}{2\pi\cdot 800\,\frac{1}{\second}\cdot (-0{,}833\cdot \power{10}{-3}\,\frac{1}{\ohm})}=\uuline{239\,\milli\henry}
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\end{align*}
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\begin{align*}
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\begin{tikzpicture}[scale=3]
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\begin{scope}[>=latex,very thick,xshift=0cm,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] {$G_p$};
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\draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{G_p}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90]%Spule |
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,-.0667) [right] {$B_p$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw [<-,red] (.3,.2)--(.7,.2)node at(.5,.2)[left]{\footnotesize$i_{B_p}$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%End Knoten
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\draw (2,.5)--(2,0)--(0,0);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=1cm]%End Knoten
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\draw (2,-.5)--(2,0)--(0,0);
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=-1cm,yshift=0cm]%Knotenpunkte
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\fill (0,1)circle(.025) node [above left] {\footnotesize$1$};
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\fill (0,0)circle(.025) node [above left] {\footnotesize$2$};
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\end{scope}
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\begin{scope}[>=latex,very thick, xshift=2cm, yshift=.5cm]
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\draw [->,thin](0,0)--(1,0)node[right]{$G$};
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\draw [->](0,0)--(.52,0)node at (.26,0)[above]{$G_p$};
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\draw [->,thin](0,-.5)--(0,.5)node[above]{$jB$};
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\draw [->](.52,0)--(.52,-.3)node at (.52,-.15)[right]{$B_p$};
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\draw [->](0:0)--(-30:.6)node at (-30:.3)[below left]{$Y$};
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\draw [->,red,thin] (0:.26) arc (0:-30:.26cm) node at (-15:.26) [right] {$\varphi_p$};
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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{c) Frequenz $f'$ \newline Reihenschaltung:}
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R'_r&\stackrel{!}{=} R_r=520\,\ohm\\
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X'_r&=\omega'\cdot L_r=2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}06\,\ohm\second=226\,\ohm\\
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Z'_r&=\sqrt{R'^2_r+X'^2_r}=\sqrt{520^2+226^2}\,\ohm=\uuline{567\,\ohm}\\
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%\varphi'&=\arctan\frac{\Im\mathfrak m}{\Re\mathfrak e}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree}
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\varphi'_r&=\arctan\frac{\Im}{\Re}=\arctan\frac{X'_r}{R'_r}=\arctan\frac{226\,\ohm}{520\,\ohm}=\uuline{23{,}5\,\degree}
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\intertext{Parallelschaltung:}
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G'_p&\stackrel{!}{=} G_p=1{,}443\,\milli\siemens\\
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B'_p&=\frac{-1}{\omega'\cdot L_p}=\frac{-1}{2\pi \cdot 600\,\frac{1}{\second}\cdot 0{,}239\,\ohm\second}=\frac{-1}{901\,\ohm}=-1{,}11\,\milli\siemens\\
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Y'_p&=\sqrt{G'^2_r+B'^2_r}=\sqrt{1{,}443^2+(-1{,}11)^2}\,\milli\siemens=1{,}82\,\milli\siemens\\
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Z'_p&=\frac{1}{Y'_p}=\uuline{549\,\ohm}\\
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\varphi'_p&=\arctan\frac{-1{,}11\,\milli\siemens}{1{,}443\,\milli\siemens}=\uuline{-37{,}6\,\degree}
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\end{align*}
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\clearpage
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}{}%
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