153 lines
8.3 KiB
TeX
153 lines
8.3 KiB
TeX
\section {Netztransformator}
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Von einem Netztransformator sind folgende Daten gegeben:\\[.5\baselineskip]
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Primärspannung $U_1=230\,\volt$; Frequenz $f=50\,\hertz$;\\
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Primärwindungszahl $N_1=784$ Windungen.\\[.5\baselineskip]
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Induktivität der Primärwicklung $L_1=5{,}66\,\henry$; \\
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Induktivität der Sekundärwicklung $L_2=1{,}42\,\henry$; \\[.5\baselineskip]
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Widerstand der Primärwicklung $R_1=800\,\ohm$;\\
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Widerstand der Sekundärwicklung $R_2=150\,\ohm$.\\[.5\baselineskip]
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Eisenquerschnitt $A_{Fe}=11\,\centi\square\metre$. Das Feld ist über dem Querschnitt $A_{Fe}$ homogen, die Streuung ist Null!
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\renewcommand{\labelenumi}{\alph{enumi})}
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\begin{enumerate}
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\item Welche Spannung $U_2$ tritt an der Sekundärwicklung auf, wenn sie unbelastet ist, d.h. $I_2=0$ ist?
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\item Welchen Strom nimmt der Transformator bei sekundärseitigem Leerlauf auf?
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\item Welche Flussdichte $\widehat{B}_{Fe}$ tritt bei sekundärseitigem Leerlauf im Eisen auf?
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\end{enumerate}
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\ifthenelse{\equal{\toPrint}{Lösung}}{%
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%\begin{align}
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%\intertext{Formeln:}
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%\end{align}
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Berechnung:\\
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
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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] {$R_1$};
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\draw [->,red] (0,.1)--(.25,.1)node at(.125,.1)[above]{\footnotesize$\uline{I}_1$};
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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] {$L_1$};
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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]%Spannungsquelle
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\draw (0,0)--(1,0);% node at (.5,-.133) [right] {$U_1$};
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\draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,.2)--(.7,.2) node at (.5,.2)[left]{$j\omega M\cdot \uline{I}_2$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=0cm,rotate=90]%Spannungsquelle
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\draw (0,0)--(1,0);% node at (.5,-.133) [right] {$U_1$};
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\draw (.5,0)circle(.133);
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\draw [<-,blue] (.3,-.2)--(.7,-.2) node at (.5,-.2)[right]{$j\omega M\cdot \uline{I}_1$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2.5cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$L_2$};
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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=3.5cm,yshift=1cm]%Widerstand
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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] {$R_2$};
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\draw [->,red] (1,.1)--(.75,.1)node at(.875,.1)[above]{\footnotesize$\uline{I}_2$};
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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 (0,0)--(2,0)--(2,.2) (4.5,0)--(2.5,0)--(2.5,.2) (2,.8)--(2,1)--(1.8,1)(2.5,.8)--(2.5,1)--(2.6,1);
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\fill (0,0)circle(0.025cm)(0,1)circle(0.025cm)(4.5,0)circle(0.025cm)(4.5,1)circle(0.025cm);
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\draw [->,blue] (0,.8)--(0,.2) node at (0,.5)[left]{$\uline{U}_1$};
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\draw [->,blue] (4.5,.8)--(4.5,.2) node at (4.5,.5)[right]{$\uline{U}_2$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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Ersatzschaltbild
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\begin{align*}
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\begin{tikzpicture}[scale=2]
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\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand
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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] {$R_1$};
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\draw node at(.5,-.2){\footnotesize$800$};
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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] {$X_{L_1}-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$+j887$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Spule |
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at(.5,.0667) [left] {$X_M$};
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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)[right]{\footnotesize$\uline{U}_M$};
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\draw node at(.25,.25){\footnotesize$j891$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=2cm,yshift=1cm]%Spule -
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\draw (0,0)--(.3,0) (.7,0)--(1,0)node at (.5,.0667) [above] {$X_{L_2}-X_M$};
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\fill (.3,-0.0667)rectangle(.7,0.0667);
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\draw node at(.5,-.2){\footnotesize$-j445$};
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\end{scope}
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\begin{scope}[>=latex,very thick,xshift=3cm,yshift=1cm]%Widerstand
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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] {$R_2$};
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\draw node at(.5,-.2){\footnotesize$150$};
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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 (0,0)--(4,0);
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\fill (0,0)circle(0.025cm)(0,1)circle(0.025cm)(4,0)circle(0.025cm)(4,1)circle(0.025cm);
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\draw [->,blue] (0,.7)--(0,.3) node at (0,.5)[left]{$\uline{U}_1$};
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\draw [->,blue] (4,.7)--(4,.3) node at (4,.5)[right]{$\uline{U}_2$};
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\end{scope}
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\end{tikzpicture}
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\end{align*}
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\begin{enumerate}
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\item Spannung $U_2$
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\begin{align*}
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\uline{U}_1&=\uline{I}_1\cdot (R_1+j\omega\cdot L_1) +j\omega\cdot M\cdot \uline{I}_2\\
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\uline{U}_2&=\uline{I}_2\cdot (R_2+j\omega\cdot L_2)+j\omega\cdot M\cdot \uline{I}_1
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\end{align*}
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\footnotesize{Hinweis: Es kann auch $X_{L_1}=\omega\cdot L_1$ bzw. $X_M=\omega\cdot M$ verwendet werden.}\\
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\normalsize
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\clearpage
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\enlargethispage{2cm}
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\begin{align*}
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\text{mit }\uline{I}_2&=0\\
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\uline{U}_1&=\uline{I}_1\cdot (R_1+j\omega\cdot L_1)\\
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\uline{U}_2&=j\omega\cdot M\cdot \uline{I}_1 \\
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\Rightarrow\frac{\uline{U}_2}{\uline{U}_1}&=\frac{j\omega\cdot M}{R_1+j\omega\cdot L_1}\\
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U_2&=U_1\cdot \frac{\sqrt{(\omega\cdot M)^2}}{\sqrt{R^2_1+(\omega\cdot L_1)^2}}\qquad\text{(Betrag: }U_2=|\uline{U}_2|)\qquad \text{(1)}\\[.5\baselineskip]
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&\text{Streuung ist Null } \Rightarrow \text{Kopplungsfaktor }K=1=\frac{|M|}{\sqrt{L_1\cdot L_2}}\\
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\Rightarrow M&=\sqrt{L_1\cdot L_2}=\sqrt{5{,}66\cdot 1{,}42}\,\henry=2{,}835\,\henry\\
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X_{L_1}&=\omega\cdot L_1=2\cdot \pi\cdot 50\,\cancel{\power{\second}{-1}}\cdot 5{,}66\,\volt\cancel{\second}\per\ampere=1{,}778\,\kilo\ohm\quad\text{\footnotesize{(Zur Vollständigkeit $X_{L_2}=446\,\ohm$)}}\\
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X_{M}&=\omega\cdot M=2\cdot \pi\cdot 50\,\cancel{\power{\second}{-1}}\cdot 2{,}835\,\volt\cancel{\second}\per\ampere=891\,\ohm\\
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&\text{aus (1):} \Rightarrow U_2=\uline{U}_1\cdot \frac{X_M}{\sqrt{R^2_{1}+X^2_{L_1}}}=230\,\volt\cdot \frac{891}{\sqrt{800^2+1778^2}}=\uuline{105{,}1\,\volt}
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\end{align*}
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\item Stromaufnahme bei Leerlauf
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\vspace{-.25cm}
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\begin{align*}
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\uline{I}_1&=\frac{\uline{U}_1}{R_1+j\omega L_1}\\
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I_1&=\frac{U_1}{\sqrt{R^2_{1}+X^2_{L_1}}}=\frac{230\,\volt}{\sqrt{800^2+1778^2}}\,\frac{1}{\ohm}=\uuline{118\,\milli\ampere}\quad\text{(Betrag)}\\
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\end{align*}
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\begin{minipage}[c]{.62\textwidth}
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\item Flussdichte, Sekundärspule spielt keine Rolle bei $I_2=0$
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\vspace{-.25cm}
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\begin{align*}
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\psi&=L_1\cdot i=N_1\cdot \phi=N_1\cdot B\cdot A\\
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B&=\frac{L_1\cdot i}{N_1\cdot A}\\
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\widehat{B}&=\frac{L_1\cdot \widehat{I}}{N_1\cdot A_{Fe}}=\frac{5{,}66\,\volt\second\per\cancel{\ampere}\cdot \sqrt{2}\cdot 118\cdot \power{10}{-3}\,\cancel{\ampere}}{784\cdot 11\cdot \underbrace{\power{10}{-4}}_{(\power{10}{-2})^2}\,\metre^2}=\uuline{1{,}095\,\tesla}
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\end{align*}
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\end{minipage}%
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\begin{minipage}[c]{.33\textwidth}
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\begin{align*}
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\begin{tikzpicture}[scale=1.0]
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\draw(0,0)rectangle(1.5,1.5);
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\draw(.25,.25)rectangle(1.25,1.25);
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\draw[->,red](.25,1.375)--(.5,1.375)node at(.375,1.5)[above]{$\phi$};
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\draw[black!75!](1,1.25)--(1,1.5)node [above]{$A_{Fe}$};
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\draw[red!70!blue](0,.5)--(-.5,.5)(0,1)--(-.5,1)node [left]{$\uline{I}_1$};
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\fill[red!70!blue](0,.5)rectangle(.25,1)node at(.25,.75)[right]{$N_1$};
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\draw[red!70!blue](1.5,.5)--(2,.5)(1.5,1)--(2,1)node [right]{$\uline{I}_2=0$};
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\fill[red!70!blue](1.25,.5)rectangle(1.5,1)node at(1.25,.75)[left]{$N_2$};
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\end{tikzpicture}
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\end{align*}
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\end{minipage}
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Bemerkung:
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\begin{align*}
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U_{L_1}&=\frac{2\pi}{\sqrt{2}}\cdot f\cdot N_1\cdot \widehat{B}\cdot A_{Fe}=I_1\cdot \omega\cdot L_1=209{,}7\,\volt\\
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\widehat{B}&=\frac{2\pi}{\sqrt{2}}\cdot f\cdot \frac{U_1}{N_1\cdot A_{Fe}}=1{,}2\,\tesla \text{ ist falsch, gilt nur für idealen Transformator! }
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\end{align*}
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\footnotesize{Warum? Hier ist der Widerstand $R_1$ nicht berücksichtigt!\\
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Rechnung oben nur mit Beträgen, nicht im Komplexen.\\ $\uline{U}_1=\uline{U}_{R_1}+\uline{U}_{L_1}=(94{,}4+j209{,}7)\,\volt=230\,\volt\cdot e^{-j114{,}3}$}
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\end{enumerate}
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\clearpage
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}{}%
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