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in - Einheiten Farad\index{Farad}) beschreibt die - wesentliche Eigenschaft des Kondensators (z.B. Plattenkondensator mit Fläche $A$ und Plattenabstand $d$: $C=\epsilon_0\epsilon_r \frac{A}{d}$). Damit ist die Ladung $Q$ bei + Einheiten Farad\index{Farad}) gibt das Verhältnis von Ladung $Q$ zu Spannung $U$ an + (z.B. Plattenkondensator mit Fläche $A$ und Plattenabstand $d$: $C=\epsilon_0\epsilon_r \frac{A}{d}$). Damit ist die Ladung $Q$ bei einer Spannung $U$ gegeben als $$Q=CU$$ und die Energie im elektrischen Feld $W_E=\frac{Q}{2C^2}$. - \item \textit{Induktivität}:\index{Induktivität} Die Induktivität $L$ (abhängig von der Geometrie, Einheit Henry\index{Henry}) beschreibt die wesentliche Eigenschaft einer - Spule (z.B. Solenoidspule mit $N$ Windungen, Querschnittsfläche $A$ und + \item \textit{Induktivität}:\index{Induktivität} Die Induktivität $L$ (abhängig von der Geometrie, Einheit Henry\index{Henry}) gibt das Verhältnis von magnetischen Fluss $\phi_m$ zur + Stromstärke $I$ einer Spule an (z.B. Solenoidspule mit $N$ Windungen, Querschnittsfläche $A$ und der L\"ange $\ell$ hat $L=\mu_0\mu_r\frac{AN^2}{\ell}$). Der magnetische Fluss $\phi_m$ ergibt sich bei einem Strom $I$ zu $$\phi_m = LI$$ und die induzierte Spannung $U_\mathrm{ind}$: $\dot \phi_m=L\dot I=-U_\mathrm{ind}$ \end{itemize} - Die \textbf{Energie im magnetischen Feld} der stromdurchflossenen Spule l\"asst sich aus der Betrachtung eines einfachen Stromkreises (Abb.~\ref{fig:circuit1}) ableiten. + Die \textbf{Energie im magnetischen Feld} der stromdurchflossenen Spule l\"asst sich aus der Betrachtung eines einfachen Stromkreises (Abb.~\ref{fig:circuit1}) mit Spannnungsquelle $U_0$, Ohm'schen Widerstand $R$, Induktivität $L$ und + Schalter $S$ ableiten. \begin{figure} \begin{center} \begin{circuitikz} @@ -23,13 +25,13 @@ Nach Schließen des Schalters $S$ l\"asst sich f\"ur das geschlossene Wegintegral schreiben $$ \oint \bm{E}\cdot d\bm{s} = -\dot{\phi}_m=-L\dot{I}.$$ Wir integrieren entlang der Richtung, in der der Strom fließt (vom Pluspol zum Minuspol), wir beginnen beim geschlossenen Schalter $S$. Entlang des idealen Leiters (widerstandsfrei) f\"allt das elektrische Feld nicht ab. Ebenso nehmen wir an, dass die Spule aus dem selben -idealen Leiter besteht, so dass hier auch keine Spannung abf\"allt. Wenn wir \"uber den +idealen Leiter besteht, so dass auch hier keine Spannung abf\"allt. Wenn wir \"uber den Ohm'schen Widerstand hinwegintegrieren, f\"allt die Spannung entsprechend dem Ohm'schen Gesetz ab, so dass wir hier einen Beitrag von $R\cdot I$ erhalten. Wir schließen das Wegintegral, indem wir \"uber die Spannungsquelle vom negativen Pol zum positiven Pol integrieren; dort erhalten wir f\"ur das Wegintegral $-U_0$, also die negative Spannung. Insgesamt ergibt sich $$ \oint \bm{E}\cdot d\bm{s} = R\,I-U_0 = -L\,\dot{I}.$$ -Die Leistung $P$ ergibt sich dann aus $P=U_0\,I$ zu: +Die Leistung $P$ ergibt sich dann aus $P=U_0\,I$ und $U_0 = R\, I + L\,\dot{I}$ zu: $$ P=U_0\,I = (R\, I+L\,\dot{I})\, I = R\, I^2 + L\,\dot{I}\, I.$$ Der erste Term $RI^2$ ist die bereits bekannte Joule'sche Leistung (W\"armeleistung)\index{Joule'sche Leistung}, der zweite Term ist spezifisch f\"ur eine Induktivit\"at und entspricht der Leistung, die zum Aufbau des magnetischen Feldes der Spule ben\"otigt wird. @@ -41,12 +43,14 @@ der erste Term $W_R$ wird f\"ur großes $t$ divergieren, der zweite Term l\"asst $$ W_B =\int\limits_{0}^{\infty}dt L\,\dot{I}\,I=\frac{L\,I_0^2}{2}.$$ Dies ist die Energie im magnetischen Feld einer Induktivit\"at $L$. (Hinweis: Zur Integration nutzen Sie $dI^2 = 2 \dot{I}\,I dt$ und substituieren entsprechend -in der Integration, die Anfangsbedingung $I(t=0)=0$ und f\"ur $t\rightarrow\infty$ geht +in der Integration, die Anfangsbedingung $I(t=0)=0$ und f\"ur $t\rightarrow\infty$ geht $\dot I\rightarrow 0$ und damit $I\rightarrow U_0/R =I_0$)\\ (Zur \"Ubung: zeigen Sie, dass die Energiedichte $w_B=\frac{B^2}{2\mu_r\mu_0}$ ist.) \section{Ein- und Ausschaltvorg\"ange} -Wir betrachten den Verlauf von Strom $I(t)$ und $U(t)$ f\"ur einfache Schaltkreise mit Induktivit\"at $L$ und Kapazit\"at $C$. + Im vorigen Abschnitt haben wir schon das asymptotische Verhalten eines Stromkreises mit einer Induktivität $L$ untersucht. Jetzt wollen wir uns etwas genauer mit dem zeitabhängigen Verhalten von Strum und Spannung in einem + Stromkreis mit Kapazitäten und Induktivitäten nach dem Einschalten beschäftigen. + \subsection{Kapazit\"at $C$:} Wir betrachten den Stromkreis in Abbildung \ref{fig:circuit2} nach Schließen des Schalters $S$ und werten das Integral $\oint \bm{E}\cdot d\bm{s}=0$ aus. Wir integrieren erneut vom Schalter $S$ in Richtung des Stromes $I$. Aus der Beziehung $Q=CU$ f\"ur die Kapazit\"at und $U=RI$ ergibt sich $\oint \bm{E}\cdot d\bm{s} = 0=Q/C+R\,I-U_0$. Wir leiten die Gleichung nach der Zeit ab: $\frac{\dot{Q}}{C}+R\dot{I}=0$ ($\dot{U}_0=0$), @@ -147,7 +151,7 @@ $$\oint \bm{E}\cdot d\bm{s} = -L\dot{I} = -U_c=-\frac{Q_c}{C},$$ wobei $Q_c=Q_c(t)$ jetzt eine Funktion der Zeit ist mit $Q_c(t=0)=Q_{c,0}$ (Ladungen werden vom Kondensator abfließen und einen Strom im Leiter treiben). Wir leiten diese Gleichung nach der Zeit ab und stellen um: -$$-L\ddot{I} -\frac{\dot{Q}_c}{C} = L\ddot{I}+\frac{I}{C}.$$ +$$-L\ddot{I} = -\frac{\dot{Q}_c}{C}.$$% = L\ddot{I}+\frac{I}{C}.$$ Im letzten Schritt ber\"ucksichtigen wir, dass die zeitliche \"Anderung der Ladung auf dem Plattenkondensator $\dot{Q}_c=-I$ einen entsprechenden Strom (hier im Uhrzeigersinn) auf dem Leiter fließen l\"asst. Wir erhalten die Differenzialgleichung f\"ur $I$: @@ -264,10 +268,10 @@ Schaltkreis wie in Abbildung~\ref{fig:circuit5} dargestellt. \draw (0,-1) to[short,-o] (8,-1); \draw (4,1) to[R,l=$R_t$,-*] (4,-1); \end{circuitikz} -\caption{\label{fig:circuit5} Schema eines Demonstrationsexperiments: An den offenen Terminalen - werden Oszillographen angeschlossen, um den Spannungsverlauf über den + \caption{\label{fig:circuit5} Schema eines Demonstrationsexperiments: An den offenen Klemmen (kleine Kreise) + werden Oszillographen angeschlossen, um den zeitlichen Verlauf der Spannung über den Widerstand $R_t$ zu messen (proportional zum Strom $I(t)$. Die Spannung zwischen - dem obersten und dem unteren Terminal ist die eingespeiste Spannung $U(t)=U_0\cos(\omega\,t)$.} + der obersten und der unteren Klemme ist die eingespeiste Spannung $U(t)=U_0\cos(\omega\,t)$.} %\missingfigure{missing three figures with the $U(t)$, $I(t)$ indicating the phase shift} \end{figure} \begin{figure} @@ -280,9 +284,9 @@ Schaltkreis wie in Abbildung~\ref{fig:circuit5} dargestellt. \end{figure} Wenn der Schalter $S_R$ geschlossen ist (die anderen beiden Schalter bleiben ge\"offnet), messen wir Strom und Spannung \textit{in Phase} (Abb. \ref{fig:phase_shift}, links). \\ Wenn der Schalter $S_C$ geschlossen ist (die anderen beiden Schalter bleiben ge\"offnet), messen wir -einen Phasenunterschied: der Strom l\"auft der Spannung voraus (Abb.~\ref{fig:phase_shift}, mitte).\\ +einen Phasenunterschied: der Strom erreicht sein Maximum früher als die Spannung (Abb.~\ref{fig:phase_shift}, mitte).\\ Wenn der Schalter $S_L$ geschlossen ist (die anderen beiden Schalter bleiben ge\"offnet), messen wir -einen Phasenunterschied: der Strom l\"auft der Spannung hinterher (Abb.~\ref{fig:phase_shift}, rechts).\\ +einen Phasenunterschied: der Strom erreicht sein Maximum später als die Spannung (Abb.~\ref{fig:phase_shift}, rechts).\\ Wir betrachten zur Herleitung einfache (ideale) Stromkreise wie in Abbildung~\ref{fig:circuit6} dargestellt. \paragraph{Herleitung im Wechselstromkreis mit einer Induktivit\"at $L$:} Wir d\"urfen den Wechselstromkreis zu einem beliebigen konstanten Zeitpunkt $t_0$ anschauen und @@ -324,26 +328,25 @@ den Realteil, z.B. $U(t)=\Re[{U}=U_0\exp(i\omega\,t)]=U_0\,\cos(\omega\,t)$, erf \begin{enumerate} \item ${Z}=R$ (Ohm'scher Widerstand): $\phi=0$, also $Z=R$ - \item $|Z|=\omega\,L$ (Induktivit\"at): Wir wissen aus der Herleitung (s.o), dass - ${I}={U}\exp(-i\pi/2)$ gilt. Andererseits folgt aus Gl.~\ref{eqn:impedanz}, dass - $\phi=\pi/2$ und $|Z|=\omega\,L$, so dass + \item $|Z|=\omega\,L$ (Induktivit\"at): $\phi=\pi/2$ und $|Z|=\omega\,L$, so dass $${Z}=\omega\,L\,e^{i\pi/2}=i\omega\,L.$$ - \item $|Z|=(\omega\,C)^{-1}$ (Kapazit\"at): Aus der Herleitung (s.o) wissen wir, dass - ${I}={U}\exp(i\pi/2)$ gilt; zusammen mit Gl.~\ref{eqn:impedanz} - folgt, dass $\phi=-\pi/2$ ist und + \item $|Z|=(\omega\,C)^{-1}$ (Kapazit\"at): $\phi=-\pi/2$ ist und $${Z}=\frac{1}{\omega\,C}e^{-i\frac{\pi}{2}}=-\frac{i}{\omega\,C}=\frac{1}{i\,\omega\,C}.$$ \end{enumerate} \section{Einschub: Komplexwertige Str\"ome, Spannungen, Impedanzen im Zeigerdiagramm:} -Komplexwertige Zahlen lassen sich als Zeiger in der komplexen Ebene darstellen. Eine komplexwertige Zahl $z$ lässt sich in seinen -Real- und Imaginärteil separieren und damit in einer \textit{komplexen Ebene} dargestellt. Die komplexe Ebene wird durch den Realteil ($\Re(z)$), mit ${z}\in\mathbb{C}$, entlang der $x$-Achse und durch den + Eine komplexwertige Zahl $z$ lässt sich in seinen +Real- und Imaginärteil separieren und damit in einer \textit{komplexen Ebene} darstellen. Die komplexe Ebene wird durch den Realteil ($\Re(z)$), mit ${z}\in\mathbb{C}$, entlang der $x$-Achse und durch den Imagin\"arteil ($\Im({z})$) entlang der $y$-Achse aufgespannt.\\ -Entsprechend der Euler'schen Formel l\"asst sich eine komplexe Zahl auch folgendermaßen darstellen: +Entsprechend der Euler'schen Formel \begin{equation} \label{eqn:euler} z=|z|\,e^{i\varphi} = |z|(\cos\varphi + i\,\sin\varphi) \end{equation} +l\"asst sich eine komplexe Zahl $z$ auch folgendermaßen darstellen: +und +$$ |z| = \sqrt{\Re({z})^2 + \Im({z})^2} = z^*z$$ mit +$$z^* = |z| e^{-i\varphi}$$ und -$$ |z| = \sqrt{\Re({z})^2 + \Im({z})^2}$$ mit $$\Re({z})=|z|\cos\varphi\qquad\qquad\Im({z})=|z|\sin\varphi \qquad\qquad arg({z})=\varphi.$$ F\"ur harmonische Str\"ome und Spannungen mit einer Kreisfrequenz $\omega$ ist im Allgemeinen @@ -360,14 +363,14 @@ F\"ur eine Reihenschaltung von Impedanzen ${Z}_1$ und ${Z}_2$ ist die Gesamtimpe $${Z}={Z}_1 + {Z}_2.$$ Beispiel: F\"ur eine Reihenschaltung eines Ohm'schen Widerstands $R$, einer Kapazit\"at $C$ und einer Induktivit\"at $L$ (Schaltskizze \ref{fig:circuit7}, links) ergibt sich für die Impedanz -$${Z}=R+i\omega\,L -\frac{i}{\omega\,C} = R+i(\omega\,L-\frac{1}{\omega\,C}),$$ -so dass f\"ur den Winkel $\phi$ -$$\tan\phi = \frac{\Im({Z})}{\Re({Z})}=\frac{\omega\,L-\frac{1}{\omega\,C}}{R},$$ -im Grenzfall kleiner Frequenzen ($\omega\rightarrow 0$): $\phi\rightarrow-\pi/2$ und -f\"ur große Frequenzen ($\omega\rightarrow \infty$): $\phi\rightarrow +\pi/2$. +$${Z}=R+i\omega\,L -\frac{i}{\omega\,C} = R+i\left(\omega\,L-\frac{1}{\omega\,C}\right),$$ +so dass f\"ur den Winkel $varphi$ +$$\tan\varphi = \frac{\Im({Z})}{\Re({Z})}=\frac{\omega\,L-\frac{1}{\omega\,C}}{R},$$ +im Grenzfall kleiner Frequenzen ($\omega\rightarrow 0$): $\varphi\rightarrow-\pi/2$ und +f\"ur große Frequenzen ($\omega\rightarrow \infty$): $\varphi\rightarrow +\pi/2$. Der Betrag der Impedanz l\"asst sich hieraus berechnen: $$|Z|=\sqrt{R^2+\left(\omega\,L-\frac{1}{\omega\,C}\right)^2}=R\sqrt{1+\frac{L^2}{\omega^2\,R^2}(\omega^2-\omega_0^2)^2},$$ -(mit $\omega_0=(LC)^{-1}$), so dass offensichtlich f\"ur den Fall einer Frequenz $\omega=\omega_0$ +(mit $\omega_0^2=(LC)^{-1}$), so dass offensichtlich f\"ur den Fall einer Frequenz $\omega=\omega_0$ der Beitrag aus Induktivit\"at und Kapazit\"at sich kompensieren und nur noch ein Ohm'scher Widerstand $R$ zur Gesamtimpedanz beiträgt. \begin{figure} \begin{circuitikz} @@ -402,7 +405,7 @@ ${Z}_1$ und ${Z}_2$: $$\frac{1}{{Z}} = \frac{1}{{Z}_1} + \frac{1}{{Z}_2}.$$ Oft wird auch die \textit{Admittanz} \index{Admittanz} ${Y}:={Z}^{-1}$ verwendet. F\"ur die Schaltskizze in Abbildung~\ref{fig:circuit7}) rechts l\"asst sich zeigen ("Ubung), dass -$\Im({Y})=0$ wird, wenn $\omega=\omega_0$ ist. +$Y=0$ wird, wenn $\omega=\omega_0$ ist. \section{Frequenzfilter: Tiefpass, Hochpass, Sperrfilter, Bandfilter} Die frequenzabhängigen Eigenschaften von Kondensatoren und Spulen lassen sich in vielen Kombinationen dazu nutzen, bestimmte Frequenzbereiche eines @@ -482,9 +485,15 @@ als den Mittelwert \"uber eine Periode $T=2\pi/\omega$: Die zeitlich gemittelte Leistung kann also f\"ur die Phasenwinkel ($\phi\in[-\pi/2,\pi/2]$) zwischen $\langle P\rangle =0$ und $\langle P\rangle =U_\mathrm{eff}\,I_\mathrm{eff}$ variieren, wobei die maximale Leistung erzielt wird, wenn $\phi=0$ ist, in diesem Fall ist es ein \textit{angepasster} Stromkreis. F\"ur $\phi\ne0$, wird die zeitlich gemittelte Leistung um eine sogenannte Blindleistung reduziert, die in den Aufbau von Feldern in der Kapazit\"at oder -Induktivit\"at aufgebraucht wird. Im Falle von $\phi=\pm\pi/2$ wird keine Leistung im Ohm'schen -Widerstand umgesetzt. + Induktivit\"at aufgebraucht wird. Im (idealen) Fall von $\phi=\pm\pi/2$ sind keine Ohm'schen Lasten vorhanden. \end{itemize} +\section{Drehstrom} +Für die Zuführung von elektrischer Leistung mit Wechselströmen bietet sich eine elegante Lösung durch die Kombination von drei Strängen, deren Spannung jeweils um einen Phasenwinkel von $120^\circ$ zueinander verschoben sind. +\begin{figure} + \includegraphics[width=0.8\linewidth]{drawing_kreisstrom.png} +\end{figure} + + \section{Schwingkreise} \subsection{Serienschwingkreis:} Wir schauen uns einen \textit{Schwingkreis} an, in dem eine Spannungsquelle mit einstellbarer Kreisfrequenz $\omega$ einen Ohm'schen Widerstand $R$, eine Kapazit\"at $C$ und eine Induktivit\"at $L$ in Serie versorgt (Abb.~\ref{fig:circuit8}). Die zeitlich gemittelte Wirkleistung (Gleichung~\ref{eqn:pwirk}) ergibt sich mit $I_0 =U_0/|Z|$, @@ -537,4 +546,4 @@ Im Prinzip ist jede Frequenz $\omega_0$ möglich. 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