\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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