ET2_Uebung_BEI/ET2_L_B19_A5.tex

\section {Momentanspannung}
An die Schaltung wird die Spannung 
$u_E(t)=U_0+\widehat{u}_1\cdot \cos(\omega t+\varphi_1) + \widehat{u}_2\cdot \cos(2\omega t+\varphi_2)$ 
angelegt.\\[.5\baselineskip] 
Berechnen Sie dies Spannung $u_C$ zur Zeit $t=T$\\
\begin{minipage}[c]{.5\textwidth}
\begin{align*}
	\begin{tikzpicture}[very thick,scale=2]
		\begin{scope}[>=latex,very thick,xshift=0cm,yshift=1cm]%Widerstand - nach EN 60617
			 \draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,.0667) [above] {$R_1$};
%			\draw [->,blue] (.3,-.2)--(.7,-.2)node at(.5,-.2)[below]{\footnotesize$U_{R}$};
		\end{scope}
		\begin{scope}[>=latex,very thick,xshift=1cm,yshift=0cm,rotate=90] 			
			\draw (0,0)--(.3,0) (.3,-0.0667)rectangle(.7,0.0667) (.7,0)--(1,0)node at (.5,-.0667) [right] {$R_2$};
		\end{scope}	
		\begin{scope}[>=latex,very thick,xshift=2cm,yshift=0cm,rotate=90]%Kondensator |
 		     \draw (0,0)--(.475,0) (.475,-.125)--(.475,.125) (.525,-.125)--(.525,.125) (.525,0)--(1,0)node at (.5,-.133) [right] {$C$};
		\end{scope}	
            \draw(0,0)--(2.5,0)(1,1)--(2.5,1);
            \fill(0,0)circle(.025cm)(0,1)circle(.025cm)(2.5,0)circle(.025cm)(2.5,1)circle(.025cm);
            \draw[->,blue](0,.9)--(0,.1)node at(0,.5)[left]{$u_E$};
            \draw[->,blue](2.5,.9)--(2.5,.1)node at(2.5,.5)[right]{$u_C$};
    \end{tikzpicture}
\end{align*}
\end{minipage}
\begin{minipage}[c]{.5\textwidth}
\begin{align*}
R_1&=12\,\ohm\quad R_2=20\,\ohm\\
C&=25\,\nano\farad\quad \omega=961\cdot \power{10}{3}\,\frac{1}{\second}\\
U_0&=6\,\volt\quad \widehat{u}_1=7\,\volt\quad \widehat{u}_2=3\,\volt\\
\varphi_1&=60\,\degree\quad \varphi_2=135\,\degree\quad T=2\,\micro\second
\end{align*}
\end{minipage}
\ifthenelse{\equal{\toPrint}{Lösung}}{%
%\begin{align}
%\intertext{Formeln:}
%\end{align}
Berechnung:\\[.5\baselineskip]
Jede Frequenz für sich betrachten; Überlagerung der Momentanwerte\\[.5\baselineskip]
$u_E(t)=6\,\volt+7\,\volt\cdot \cos(961\cdot 10^3\,\cdot 1\per\second\cdot t+60\degree)+3\,\volt\cdot \cos(2\cdot 961\cdot 10^3\,\cdot 1\per\second\cdot  t+135\degree)$\\
a) Gleichspannung - Spannungsteiler
\begin{align*}
%\intertext{a) Gleichspannung - Spannungsteiler}
u_{C0}&=U_0\cdot \frac{R_2}{R_1+R_2}=6\,\volt\cdot \frac{20\,\ohm}{32\,\ohm}=3{,}75\,\volt
\intertext{b) allgemein:}
%\vspace{-1.5cm}
\uline{U_C}&=\uline{U}_E\cdot \frac{\uline{Z}_{||}}{\uline{Z}_{||}+R_1}\\
\uline{Z}_{||}&=\frac{R_2\cdot jX_C}{R_2+jX_C}
\intertext{c) $1\cdot \omega$: Transformation in komplexe Ebene: $u(t)\rightarrow \uline{U}$}
\uline{U}_{E1}&=\frac{7\,\volt}{\sqrt{2}}\cdot e^{j60\,\degree}=4{,}95\,\volt\cdot e^{j60\,\degree}=(2{,}47+j4{,}29)\,\volt\\
X_{C1}&=\frac{-1}{\omega C}=-41{,}6\,\ohm\\
\uline{Z}_{||1}&=\frac{20\,\ohm\cdot (-j41{,}6\,\ohm)}{20\,\ohm-j41{,}6\,\ohm}=\frac{-j832\,\ohm^2}{20\,\ohm-j41{,}6\,\ohm}\\
&=\frac{828\,\ohm\cancel{^2}\cdot e^{-j90}}{45{,}98\cancel{\,\ohm}\cdot e^{-j64{,}22}}=18{,}05\,\ohm\cdot e^{-j25{,}67}=(16{,}27-j7{,}82)\,\ohm\\
\uline{Z}_{||1}+R1&=(16{,}27-j7{,}82+12)\,\ohm=(28{,}27-j7{,}82)\,\ohm=29{,}33\,\ohm\cdot e^{-j15{,}52}\\
\uline{U}_{C1}&=\uline{U}_{E1}\cdot \frac{\uline{Z}_{||1}}{\uline{Z}_{||1}+R_1}
%&= 4{,}95\,\volt\cdot e^{j60\,\degree}\cdot \frac{(16{,}2-j7{,}834)\cancel{\,\ohm}}{(28{,}2-j7{,}834)\cancel{\,\ohm}}
=4{,}95\,\volt\cdot e^{j60\,\degree}\cdot \frac{18{,}05\cancel{\,\ohm}\cdot e^{-j25{,}67}}{29{,}33\cancel{\,\ohm}\cdot e^{-j15{,}46\,\degree}}\\
&=3{,}04\,\volt\cdot e^{j49{,}79\,\degree}=(1{,}97+j2{,}32)\,\volt\\
U_{C1}&=3{,}04\,\volt\quad\text{Effektivwert}\\
\varphi_{C1}&=49{,}79\,\degree=0{,}868\,\radian\quad\text{Umwandlung wegen } (\omega t+\varphi_{C1})
\end{align*}
\enlargethispage{2cm}
\clearpage
\begin{align*}
\intertext{d) $1\cdot \omega$ Rücktransformation: mit $t=T$}
\omega t+\varphi_{C1}&=\omega \cdot T+\varphi_{C1}=961\cdot \power{10}{3}\,\cancel{\frac{1}{\second}}\cdot 2\cdot \power{10}{-6}\,\cancel{\second}+0{,}868\,\radian=2{,}791\,\radian =159{,}9\,\degree\\
u_{C1}(t=T)&=\sqrt{2}\cdot U_{C1}\cdot \cos(\omega T+\varphi_{C1})=\sqrt{2}\cdot 3{,}04\,\volt\cdot \cos(159{,}9\,\degree)\\
&=4{,}3\,\volt\cdot (-0{,}939)=-4{,}037\,\volt
\intertext{e) $2\cdot \omega$: Transformation in komplexe Ebene: $u(t)\rightarrow \uline{U}$}
\uline{U}_{E2}&=\frac{3\,\volt}{\sqrt{2}}\cdot e^{j135\,\degree}=2{,}121\,\volt\cdot e^{j135\,\degree}=(-1{,}5+j1{,}5)\,\volt\\
X_{C2}&=\frac{1}{2}\cdot X_{C1}=-20{,}8\,\ohm\\
\uline{Z}_{||2}&=\frac{20\,\ohm\cdot (-j20{,}8\,\ohm)}{20\,\ohm-j20{,}8\,\ohm}=\frac{-j414\,\ohm^2}{20\,\ohm-j20{,}8\,\ohm}\\
&=\frac{414\,\ohm\cancel{^2}\cdot e^{-j90}}{28{,}78\cancel{\,\ohm}\cdot e^{-j45{,}99}}=14{,}41\,\ohm\cdot e^{-j43{,}87}=(10{,}39-j9{,}992)\,\ohm\\
\uline{Z}_{||2}+R1&=(10{,}39-j9{,}992+12)\,\ohm=(22{,}34-j9{,}992)\,\ohm=24{,}51\,\ohm\cdot e^{-j24\,\degree}\\
\uline{U}_{C2}&=\uline{U}_{E2}\cdot \frac{\uline{Z}_{||2}}{\uline{Z}_{||2}+R_1}\\
%&= 2{,}121\,\volt\cdot e^{j135\,\degree}\cdot 
&=\uline{U}_{E2}\cdot 
\frac{(10{,}39-j9{,}992)\cancel{\,\ohm}}{(22{,}39-j9{,}992)\cancel{\,\ohm}}
=2{,}121\,\volt\cdot e^{j135\,\degree}\cdot \frac{14{,}38\cdot e^{-j44{,}01}}{24{,}48\cdot e^{-j24{,}1\,\degree}}\\
&=1{,}247\,\volt\cdot e^{j115{,}1\,\degree}=(-0{,}530+j1{,}129)\,\volt\\
U_{C2}&=1{,}247\,\volt\quad\text{Effektivwert}\\
\varphi_{C2}&=115{,}1\,\degree=2{,}009\,\radian\quad\text{Umwandlung wegen } (2\cdot \omega t+\varphi_{C2})
\intertext{f) $2\cdot \omega$ Rücktransformation: mit $t=T$}
2\cdot \omega t+\varphi_{C2}&=2\omega\cdot T+\varphi_{C2}=2\cdot 961\cdot \power{10}{3}\,\cancel{\frac{1}{\second}}\cdot 2\cdot \power{10}{-6}\,\cancel{\second}+2{,}009\,\radian\\
&=5{,}854\,\radian =335{,}4\,\degree\\
u_{C2}(t=T)&=\sqrt{2}\cdot U_{C2}\cdot \cos(2\cdot \omega T+\varphi_{C2})=\sqrt{2}\cdot 1{,}247\,\volt\cdot \cos(335{,}4\,\degree)\\
&=1{,}762\,\volt\cdot 0{,}909=1{,}602\,\volt
\intertext{g) Überlagerung:}
u_C(t=T=2\,\micro\second)&=u_{C0}(T)+u_{C1}(T)+u_{C2}(T)=(3{,}75\,\volt-4{,}037\,\volt+1{,}602\,\volt)\\
&=\uuline{1{,}315\,\volt}
\end{align*}
\clearpage
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