ET2_Uebung_BEI/ET2_L_B18_A5.tex

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