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- \section{Blitzableiter}
- Eine $l=4\,\metre$ lange Verbindungsleitung, deren Leiter voneinander einen Abstand $d = 2\,\centi\metre$ haben, ist im Abstand $D=3\,\metre$ parallel zu einem Blitzableiter verlegt. (Siehe Skizze).\\
- $\mu_0=1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)$
- \renewcommand{\labelenumi}{\alph{enumi})}
- \begin{enumerate}
- \item Welche Spannung $u$ (Betrag !) wird induziert, wenn der
- Blitzstrom $i$ linear in $0{,}6\,\micro\second$ auf $15\,\kilo\ampere$ ansteigt?
- \item Ist die Spannung während dieser Zeit positiv oder negativ? (Begründung !)
- \end{enumerate}
- \begin{align*}
- \begin{tikzpicture}[scale=2]
- \begin{scope}[>=latex,red, ultra thick,xshift=0,yshift=0] % Blitzstrom
- \draw [->] (0,4) -- (0,0) node [below] {$i$};
- \end{scope}
- \begin{scope}[>=latex,blue,very thick, xshift=3cm, yshift=0.5cm] % Leitung
- \draw (0,3) circle (0.025) -- (0,0) -- (0.2,0) -- (0.2,3) circle (0.025); % node at (0.1,3) [above ] {$u$};
- \draw [<-] (0,3.1) -- (.2,3.1) node at (0.1,3.1) [above] {$u$};
- \end{scope}
- \begin{scope}[>=latex,thick,xshift=0cm,yshift=2cm] % Abstand D
- \draw [<->] (0,0) -- (3.1,0) node at (1.55,0) [above] {$D$};
- \draw [dashed] (3.1,.5)--(3.1,-.5);
- \end{scope}
- \begin{scope}[>=latex,thick,xshift=3cm,yshift=1cm] % Pfeile d
- \draw [->] (-.5,0) -- (0,0) node at (0.5,0) [above] {$d$};
- \draw [<-] (0.2,0) -- (0.7,0);
- \end{scope}
- \begin{scope}[black!75!,>=latex,thick,xshift=4.125cm,yshift=0.5cm] % Länge l
- \draw [<->] (-.125,3) -- (-.125,0) node at (0,1.5) [right] {$l$};
- \draw (0,0) -- (-0.5,0);
- \draw (0,3) -- (-0.5,3);
- \end{scope}
- \end{tikzpicture}
- \end{align*}
- \ifthenelse{\equal{\toPrint}{Lösung}}{%
- Formeln:
- \begin{align}
- % \intertext{Formeln:}
- u&=-N\cdot \frac{d\Phi}{dt}\\
- \Phi&=B\cdot A=\mu\cdot H\cdot A=\int{\vec{B}\cdot \vec{dA}}\\
- H&=\frac{i\cdot N}{l} \qquad \text{für Zylinder: }\qquad H=\frac{i\cdot N}{2\cdot \pi\cdot r}
- \end{align}
- Magnetischer Fluss $\Phi$; Flussdichte $B$; Feldstärke $H$;\\
- Magnetische Permeabilität $\mu=\mu_0\cdot \mu_r$; In Luft $\mu_r=1$\\
- \clearpage
- Berechnung:
- \begin{align*}
- u&=-N\cdot \frac{d\Phi}{dt} \qquad \text{mit $N=1$ }\qquad u=-\frac{d\Phi}{dt}\\
- \Phi&=\int{\vec{B}\cdot \vec{dA}} \qquad \text{mit $B\bot A$}\qquad \Phi=B\cdot A=\underbrace{\mu\cdot H}_{B}\cdot \underbrace{l\cdot d}_{A}\\
- H&=\frac{i}{2\cdot \pi\cdot D} \qquad \text{Abhängigkeit vom Abstand $d$ vernachlässigbar $d\ll D$. }\\
- \phantom{blablabla}bla\\
- \vphantom{u=-\frac{y}{x}}..\\
- u&=-\frac{d\Phi}{dt}\hspace{-.5cm}\underbrace{=}_{linearer Anstieg}\hspace{-.5cm}-\frac{\Phi}{t}=-\mu\cdot l\cdot d\cdot \frac{i}{2\cdot \pi\cdot D\cdot t}=-\frac{\mu\cdot i\cdot l\cdot d}{2\cdot \pi\cdot D\cdot t}\\
- % } %end pahntom
- &=-\frac{1{,}26\cdot \power{10}{-6}\,\volt\second\per(\ampere\metre)\cdot 15000\,\ampere\cdot 4\,\metre\cdot 0{,}02\,\metre}{2\cdot \pi \cdot 3\,\metre\cdot 0{,}6\cdot \power{10}{-6}\,\second}=\uuline{-134\,\volt}\\[\baselineskip]
- &\text{Leitung symmetrisch zur Mittellinie bei } 2{,}99\,\metre \text{ und } 3{,}01\,\metre\\
- \text{mit }&\Phi=\frac{\mu\cdot i\cdot l}{2\cdot \pi}\cdot\ln\frac{D+d/2}{D-d/2}\\
- u&=-\frac{\mu_0\cdot 15\cdot \power{10}{3}\,\ampere\cdot 4\,\metre}{0{,}6\cdot \power{10}{-6}\,\second\cdot 2\cdot \pi}\cdot \underbrace{\ln\frac{3{,}01\,\metre}{2{,}99\,\metre}}_{6{,}66\cdot \power{10}{-3}}=-133{,}7\,\volt\\
- \text{\uline{Lentzsche Regel:}}
- \end{align*}
- \begin{align*}
- \begin{tikzpicture}[scale=2.5]
- \begin{scope}[>=latex,very thick,xshift=0cm,yshift=0cm]%Widerstand - nach EN 60617
- \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$};
- \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$};
- \draw [<-,blue] (.2,.35)--(.8,.35)node at(.5,.35)[above]{\footnotesize$u$};
- \end{scope}
- \begin{scope}[>=latex,very thick,xshift=.25cm,yshift=-.25cm]% Leiterschleife
- \draw (0,0)--(0,-2)--(.5,-2)--(.5,0);
- \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25);
- \end{scope}
- \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=0cm]%Knotenpunkte
- \fill (0,0)circle(.05) node [above right] {\footnotesize$+$};
- \fill (1,0)circle(.05) node [above left] {\footnotesize$-$};
- \end{scope}
- \begin{scope}[>=latex,very thick,red,xshift=0cm,yshift=-1cm]% O.
- \draw (.5,0)circle(.133)node at (1,0)[right]{$\frac{d\Phi}{dt}$};
- \fill (.5,0)circle(.05);
- \end{scope}
- \begin{scope}[>=latex,very thick,green!50!black,xshift=0cm,yshift=-1.33cm]% Ox
- \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$};
- \draw [black]node at (1.5,-1)[below]{Wirkung $\frac{d\Phi_{ind}}{dt}$ entgegen Ursache $\frac{d\Phi}{dt}$};
- \draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133);
- \end{scope}
- \begin{scope}[>=latex,very thick,xshift=3cm,yshift=0cm]%Widerstand - nach EN 60617
- \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R$};
- \draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$u_{R}$};
- \end{scope}
- \begin{scope}[>=latex,very thick,red,xshift=3cm,yshift=0cm]%Knotenpunkte
- \fill (0,0)circle(.05) node [above right] {\footnotesize$+$};
- \fill (1,0)circle(.05) node [above left] {\footnotesize$-$};
- \end{scope}
- \begin{scope}[>=latex,very thick,xshift=3.25cm,yshift=-0.25cm]% Leiterschleife
- \draw (0,0)--(0,-2)--(.5,-2)--(.5,0);
- \draw [dashed] (0,0)--(-.25,.25)(.5,0)--(.75,.25);
- \end{scope}
- \begin{scope}[>=latex,very thick,xshift=3cm,yshift=-2.25cm]%Stromquelle
- \draw [green!50!black](.25,0)--(.75,0);
- \draw node at (.5,.133) [above] {$i_{}$};
- \draw [green!50!black](.5,0)circle(.133);
- \draw [<-,red] (.3,-.2)--(.7,-.2) node at (.5,-.2)[below]{\footnotesize$i_{ind}$};
- \end{scope}
- \begin{scope}[>=latex,very thick,green!50!black,xshift=3cm,yshift=-1.33cm]% Ox
- \draw (.5,0)circle(.133)node at (1,0)[right]{$-\frac{d\Phi_{ind}}{dt}$};
- \draw [very thick](.5,0)--+(45:.133) (.5,0)--+(135:.133)(.5,0)--+(225:.133)(.5,0)--+(315:.133);
- \end{scope}
- \begin{scope}[>=latex,very thick,xshift=2cm,yshift=-1.25cm]%
- \draw [->,red] (-2.5,.5)--(-2.5,-.5) node at(-2.5,0)[right]{$\frac{di}{dt}$};
- \draw [<-,green!50!black] (.5,.5)--(.5,-.5) node at(.5,0)[left]{$i_{ind}$};
- \draw [<-,red!75!black] (-1.75,.5)--(-1.75,-.5) node at(-1.75,0)[left]{$i_{ind}$};
- \end{scope}
- \end{tikzpicture}
- \end{align*}
- $u$ hat negatives Vorzeichen, da Polung von $i_{ind}$ entgegen Bezugspfeilen aus Skizze\\
- ($u_R = -u$)
- \clearpage
- }{}%
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