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b/v0.1/ac_circuit.tex @@ -163,7 +163,7 @@ fließen l\"asst. Wir erhalten die Differenzialgleichung f\"ur $I$: \ddot{I} = -\frac{1}{LC} I \end{equation} mit der Randbedingung $I(t=0)=0$ l\"asst sich diese Gleichung (harmonischer Oszillator) -mit dem Ansatz l\"osen $I(t)=I_0 \sin(\omega t)$. Um die Kreisfrequenz $\omega$ +mit dem Ansatz l\"osen $I(t)=I_0 \sin(\omega t+\alpha)$. Um die Kreisfrequenz $\omega$ zu bestimmen, rechnen wir $$\ddot{I}=-\omega^2\,I_0\,\sin(\omega t)=-\omega^2\,I(t),$$ damit ergibt sich aus dem Vergleich mit Gl.~\ref{eqn:current} diff --git a/v0.1/figures/transformator.png b/v0.1/figures/transformator.png new file mode 100644 index 0000000000000000000000000000000000000000..802111120be6f70e1bf4a17ddd66a47b6a9556b0 GIT binary patch literal 41497 zcmeAS@N?(olHy`uVBq!ia0y~yU{zsYV4B6j#=yX!&Jq5Nfq{Xuz$3Dlfk96hgc&QA z+LtjfFi4iTMwA5Sr<If^7Ns(jmzV2h=4BTrCl;jY<rk&TerF@az@Wh3>EaktG3U+Q z@+nJ0=f3~{ef#dlHZ^utQ^x>@gAB^kmo#;F2v1dUWK`{Kz8rT<^4QM(+7IT=&vZL* ztU-fs%L@VC2BDn7DGDnz!W~%{Pg)+&v#NP7Uw!?`%SEeq?K<>1q+<UytJjw4^NZ(w zUK<+wb<Mvk<?m%@b}qT7s}#q7h9!=r?rM?cBo)s|OUithzOmI!p4q-Nw&%)hg9%gI zws*|niDBHc`fZeo=Oh(RWhbWmQ)h19(sPLtVfM@v>F&7NAkTRF`a<!>_D0sVqN*T8 zS+_17usRU*^<z%L&oguG$9V?5SKAufu%xlGgST-$!|ZDdH#h7*duHd`P>Av%mmKC8 zCr)Kyjt8!XJ$h4wZ|ON)n$+E~wozImF@;Gz{n^B}i6Hlx)VUmCZ&=-E+F*I0{MK5J zSMtoBnNvDDY8&P=uU{Bbz<2h{<S1P)kkTtFI%e?2aOiN`u)bmXv+wGnC3Qv<rnpV4 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ze7)h|iC2ey6>sf`P&NK<RT>Ll4e6OY<J;Z?``uiYr)e|#Gt_NeQ5@`b`Ngzf#Zz5* zG}gt|-^(&6dL;?<Gk<4CdO*{}˜{~Gx;{PJz(%!-*mZ+HAq+hDwO#pT=VvF1wC z*zU#3|GREdl79}i;&hf7ixuC+caPq@eATgK1|P#Y_Ipx-oHBRUZ}4@o=?eN;t^Ppg zf$imcr!7w`G6QFG&nP9(Vq5Em>&)Ak=Pwb-+NEUtesyjb(>&Ha92P7)xGUs8a4uJ# zAqQKJxW%QrBeXHLQMNIi={4ic9e>$&^a!o7(KmiC@9Y1(?LU7`^8HL6->qH@3=9mO Lu6{1-oD!M<Pjb_b literal 0 HcmV?d00001 diff --git a/v0.1/induktion.tex b/v0.1/induktion.tex index 4965977..bf405a9 100644 --- a/v0.1/induktion.tex +++ b/v0.1/induktion.tex @@ -117,7 +117,26 @@ elektromotischen Kraft (EMK) in einigen Lehrbüchern. weiter verstärkt. Das würde wiederum einen größeren Strom induzieren. Eine sich in dieser Weise selbst-verstärkende Reaktion nennen wir auch Instabilität. + + \textbf{Demonstrationsexperiment: Lenz'sche Regel mit Ring-Kanone, Gauss-Kanone} + +\section{Generator} +\label{section:generator} +Ein zeitlich sich ändernder magnetischer Fluss erzeugt eine induzierte Spannung. Die technische wichtigste Nutzung ist der Generator, bei +dem wir Bewegungsenergie in elektrische Energie umwandeln. Der einfachste Wechselstromgenerator besteht aus einer drehbaren Spule in einem +äußeren Magnetfeld $B$ (z.B. ein Hufeisenmagnet). Wenn sich die Spule mit $N$ Windungen und der Fläche $A$ im äußeren Magnetfeld dreht, +wird der magnetische Fluss durch +die rotierende Fläche, die von der Spule aufgespannt wird, periodisch mit der Kreisfrequenz $\omega$ moduliert: +\[ + \varphi_m = NBA\cos\omega t. +\] +In der Spule wird dann die Spannung $U_\mathrm{ind}=-\dot\varphi_m$, also +\[ + U_\mathrm{ind} = -NBA\omega \sin(\omega t) +\] +induziert. + \section{Wirbelströme} Bislang haben wir die induzierten Ströme in Kabeln betrachtet. Für den Fall, dass wir @@ -154,8 +173,7 @@ Anziehung dazu führen, dass eine Gesamtkraft der Bewegungsrichtung entgegenwirk Kraft ist proportional zur Geschwindigkeit und verhält sich also ganz ähnlich zu Newton'scher Reibung.} \end{figure} -\section{Anwendung: Transformator} -tbd. + \section{Selbst- und Gegeninduktion} \subsection{Selbstinduktion: Induktivität $L$} Jede Flussänderung induziert eine Spannung, die nach der Lenz'schen Regel der Flussänderung @@ -209,6 +227,42 @@ induziert eine Spannung $U_\mathrm{ind}$, die der Änderung entgegenwirkt, also \subsection{Gegeninduktion} TBD. + + \subsection{Anwendung: Transformator} + Wir betrachten eine Spule (Primärwicklung) mit $N_1$ Windungen auf einem Eisenjoch (z.B. laminiertes Harteisen) - siehe + Abbildung~\ref{fig:transformator}. Auf dem selben Eisenjoch + ist eine Sekundärwicklung mit $N_2$ Windungen angebracht. Wenn wir jetzt die Primärwicklung mit einem Generator (siehe Abschnitt + ~\ref{section:generator}) verbinden, erzeugen wir einen zeitlich variierenden magnetischen Fluss + \[ U_1 = - N_1 \dot\varphi.\] + Der magnetische Fluss $\varphi$ wird im idealen, verlustfreien Transformator durch das Eisenjoch geleitet, so dass die Sekundärspule + den selben Fluss sieht, so dass dort die Spannung + \[ U_2 = - N_2 \dot \varphi \] + induziert wird. + Für das Verhältnis der Spannungen im Falle eines unbelasteten Transformators ergibt sich + \[ + \frac{U_1}{U_2} = \frac{N_1}{N_2}. + \] + Andererseits kann die Leistung $P=U_1 I_1=U_2 I_2$ aufgrund der Energieerhaltung sich nicht ändern, so dass + \[ \frac{I_2}{I_1} = \frac{N_1}{N_2} \] + gelten muss. + \begin{figure} + \begin{center} + \includegraphics[width=0.7\linewidth]{figures/transformator.png} + \end{center} + \caption{\label{fig:transformator} Schematische Darstellung eines Transformators.} + + \end{figure} + + Mit einem Transformator lässt sich die Spannung an der Sekundärwicklung durch ein geeignet gewähltes Verhältnis + der Wicklungen entweder hochtransformieren ($N_2>N_1$) oder heruntertransformieren ($N_2<N_1$). Die Ströme in + der Sekundärspule werden beim Hochtransformieren um das selbe Verhältnis verkleinert. Beim Heruntertransformieren werden die Ströme + vergrößert. + + Im Detail ist beim Transformator zu berücksichtigen, dass ein Teil des magnetischen Flusses nicht in der Sekundärspule ankommt. Ein weiterer + wichtiger Punkt ist die Phasenlage der beiden Wechselströme, die durch eine Belastung der Sekundärspule modifizert wird. Interessant + ist auch die Orientierung der Wicklungen zueinander (siehe auch hier die Abbildung~\ref{fig:transformator}). + + \section{Maxwell'scher Verschiebungsstrom: Amp\`ere-Maxwell-Gleichung} \label{section:maxwell-strom} -- GitLab