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