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- \section{Verbraucherleistung}
- An einem Verbraucher liegt die Spannung $u(t)=310\,\volt \cdot \sin(\omega t+55\,\degree)$ an, er nimmt einen Strom von $i(t)=8,5\,\ampere\cdot \cos (\omega t)$ auf.
- \renewcommand{\labelenumi}{\alph{enumi})}
- \begin{enumerate}
- \item Berechnen Sie den zeitlichen Verlauf des Momentanwertes der Verbraucherleistung!
- \item Berechnen Sie die Schein-, Wirk- und Blindleistung!
- \end{enumerate}
- \ifthenelse{\equal{\toPrint}{Lösung}}{%
- %\begin{align}
- %\intertext{Formeln:}
- %\end{align}
- Merksatz:\\
- Kondensat\textbf{o}r, Strom eilt v\textbf{o}r\\
- Induktivit\textbf{ä}t, Strom ist zu sp\textbf{ä}t\\[\baselineskip]
- Berechnung:\\
- \begin{align*}
- \begin{tikzpicture}[scale=2]
- \begin{scope}[>=latex, xshift=0cm, yshift=0]
- \foreach \ii in {5} { % Enter Number of Decades in x
- \foreach \jj in {2} { % Enter Number of Decades in y
- \foreach \i in {1,2,...,\ii} {
- \foreach \j in {1,2,...,\jj} {
- \draw[black!50!, step=0.5] (0,0) grid (\ii,\jj); % Draw Sub Linear grid
- }}% End Log Grid
- \draw[black!80!] (0,0) grid (\ii,\jj); % Draw Linear grid
- \draw [->,blue,thick] (5,0)--(5,\jj+.25) node (yaxis) [above] {$u\,[\volt]$};
- \draw [->,red,thick] (0,0)--(0,\jj+.25) node (yaxis) [above] {$i\,[\ampere]$};
- \draw [->,thick] (0,0)--(\ii+.25,0) node (xaxis) [right] {$\omega t\,[\degree]$}; % Draw axes
- \foreach \x in {-90,0,90,180,270,360}% x Axis Label:
- \node [blue,anchor=north] at(\x/90+1,0){$\x$};
- \foreach \y in {0,10}% y Axis Label:
- \node [red,anchor=east] at(0,\y/10+1){$\y$};
- \foreach \y in {0,500}% y Axis Label:
- \node [blue,anchor=west] at(5,\y/500+1){$\y$};
- }}
- \draw[very thick](1,0)--(1,2);
- \draw[->,very thick](1,1)--(.389,1) node [below right]{ $-55\degree$};
- \draw[<-,very thick](2,1)--(2.389,1) node at (2,1.2)[above right]{$-35\degree$ $i$ vor $u \Rightarrow$ kapazitiv};
- \end{scope}
- \begin{scope}[>=latex, xshift=0cm, yshift=1cm]
- \draw[color=red,thick,domain=0:5,smooth,samples=100] plot[id=i] function{.85*cos(.5*3.14*x-1.57)};
- \draw[color=blue,thick,domain=0:5,smooth,samples=100] plot[id=u] function{.62*sin(.5*3.14*x+.96-1.57)};
- %\draw[->,blue, very thick] (1.5,1) -- (2.5,1);
- \draw[red] node at (1.5,1.25) {{\footnotesize $i(t)=8,5\,\ampere\cdot \cos (\omega t)$}};
- \draw[blue] node at (3.5,1.25) {{\footnotesize $u(t)=310\,\volt \cdot \sin(\omega t+55\,\degree)$}};
- \end{scope}
- \end{tikzpicture}
- \end{align*}
- \begin{align*}
- \intertext{a) Leistungsverlauf}
- p(t)&=u(t)\cdot i(t) \qquad\text{Momentane Leistung}\\
- p(t)&=310\,\volt\cdot 8{,}5\,\ampere\cdot \sin x\cdot \cos y \\
- &\text{mit $x=\omega t+55\degree=\omega t+0{,}96\,\text{rad} \qquad y=\omega t$}\\
- &\text{und } \sin x\cdot \cos y=\frac{1}{2}[\sin(x-y)+\sin(x+y)]\Rightarrow\\
- p(t)&=\widehat{u}\cdot \widehat{i}\cdot \frac{1}{2}\cdot [\sin(x-y)+\sin(x+y)]\\
- &=310\,\volt\cdot 8{,}5\,\ampere\cdot \frac{1}{2}\cdot \big[\sin(\cancel{\omega t}+0{,}96-\cancel{\omega t})+\sin(\omega t+0{,}96+\omega t)\big]\\
- &=\underbrace{310\,\volt\cdot 8{,}5\,\ampere\cdot \frac{1}{2}\vphantom{\frac{1}{2}}}_{S=1318\,\volt\ampere}\cdot \big[\underbrace{\sin(0{,}96)\vphantom{\frac{1}{1}}}_{0{,}819}+\sin(2\omega t+0{,}96)\big]\\
- p(t)&=\uuline{1079\,\watt+1318\,\volt\ampere\cdot \sin(2\omega t+0{,}96)}
- \intertext{b) $S$ Schein-, $P$ Wirk- und $Q$ Blindleistung}
- \cos(\omega t)&=\sin(\omega t+90\degree)\\
- \varphi_i&=+90\degree\quad\varphi_u=+55\degree\\
- \varphi_u-\varphi_i&=+55\degree-90\degree=-35\degree\\
- S&=U\cdot I=\frac{\widehat{u}}{\sqrt{2}}\cdot \frac{\widehat{i}}{\sqrt{2}}=\frac{1}{2}\cdot \widehat{u}\cdot \widehat{i}=\frac{2635}{2}\,\volt\ampere=\uuline{1318\,\volt\ampere}\\
- P&=S\cdot \cos(-35\degree)=S\cdot 0{,}819=\uuline{1079\,\watt}\\
- Q&=S\cdot \sin(-35\degree)=S\cdot (-0{,}576)=\uuline{-756\,\mathrm{var}}\qquad\text{Lies: Volt-Ampere-reaktiv}\\
- \end{align*}
- \clearpage
- }{}%
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