ET2_Uebung_BEI/ET2_L_B12_A1.tex

\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 $d\ll D$. }\\
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}\\
&=-\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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