From 8e0088399dfb3252ddf0af0de367592f2af1d589 Mon Sep 17 00:00:00 2001 From: Paul Thiel Date: Sat, 9 Feb 2019 06:29:07 +0100 Subject: [PATCH] feature: add reactances (#7) --- Schaltungstechnik.tex | 28 +++++++++++++++++++++++++++- img/reactance_overview.pdf | Bin 0 -> 20468 bytes 2 files changed, 27 insertions(+), 1 deletion(-) create mode 100644 img/reactance_overview.pdf diff --git a/Schaltungstechnik.tex b/Schaltungstechnik.tex index 3b42ae7..eb39c55 100644 --- a/Schaltungstechnik.tex +++ b/Schaltungstechnik.tex @@ -689,7 +689,9 @@ \section{Allgemeines Reaktiver Elemente} \textbf{Magnetischer Fluss \textit{$\Phi$}}: Grundeigenschaft von elektr. magn. Feldern\\ \subsubsection{Allgemeine Zusammenhänge $u,i,q,\Phi$} Ladung und Strom beschreiben den Zustand der Materie.\\ - Spannung und magn. Fluss beschreiben den Zustand des elekt. magn. Feldes.\\ \\ + Spannung und magn. Fluss beschreiben den Zustand des elekt. magn. Feldes.\\ + Kondensator ist u-gesteuert (q-gesteuert), falls für ein u (q) nur ein q (u) existiert. \\ + Induktivität ist i-gesteuert ($\phi$-gesteuert), falls für ein i ($\phi$) nur ein $\phi$ (i) existiert. \\ \begin{tabular}{l|l} $i(t) = \dot q(t)$ & $[i]=A$\\ $q(t) = q(t_0) + \int_{t_0}^t i(\tau) \mathrm d\tau$ & $[q]=As=C$ \\ \hline @@ -708,6 +710,30 @@ \section{Allgemeines Reaktiver Elemente} Induktivität & \includegraphics[height=0.4cm]{./img/Induktivitat.pdf} & $f_L(i,\Phi)$ & $\Phi = \Phi_0 + L \cdot i$\\ Memristivität & \includegraphics[height=0.4cm]{./img/Memristivitat.pdf} & $f_M(q,\Phi)$ & $\Phi = \Phi_0 + M \cdot q$\\ \end{tabular} + \subsubsection{Zusammenhang der Bauelemente} + \begin{center} + \includegraphics[scale=0.3]{./img/reactance_overview.pdf} + \end{center} + \subsubsection{Eigenschaften von Reaktanzen} + \textbf{Linearität}: siehe Eintore\\ + \textbf{Differentialgleichung}: $i(t) = C \frac{\mathrm du(t)}{\mathrm dt}, u(t) = L \frac{\mathrm di(t)}{\mathrm dt}$\\ + \textbf{Gedächtnis}: Verhalten durch vorhergehende Klemmengrößen bestimmt.\\ + \textbf{Stetigkeit}: $u_C(t)$, $i_L(t)$ stetig in $(t_a, t_b)$, wenn Torgrößen endlich\\ + \textbf{Verlustfreiheit}: $W_C(t_1, t_2) = \int_{t_1}^{t_2}\! u(t)i(t)\,\mathrm dt = \int_{q_1}^{q_2}\! \mathrm{X}(q)\,\mathrm{d}t$ (Arbeit)\\ + Falls linear: $W = \frac{Cu^2}{2} = \frac{Li^2}{2}$\\ + Periodisch: $u(t+T) = u(t)$, $q(t+T) = q(t)$\\ + Graphisch: Falls keine geschlossenen Schleifen in q/u, $\Phi$/i-Diagramm existiert (Hystenesefrei)\\ + \textbf{Energie (nicht linearer Fall)}:\\ + - Kapazitiv: $W_C(t_1, t_2) = \int_{t_1}^{t_2} \! u(t)i(t)\, \mathrm dt = \int_{q_1}^{q_2} \! u(q)\, \mathrm dq$\\ + - Induktiv: $W_C(t_1, t_2) = \int_{t_1}^{t_2} \! u(t)i(t)\, \mathrm dt = \int_{\Phi_1}^{\Phi_2} \! i(\Phi)\, \mathrm d\Phi$\\ + \textbf{Energie (linearer Fall)}:\\ + - Kapazitiv: $W_C = \frac{C}{2}u^2 = \frac{1}{2C}q^2$\\ + - Induktiv: $W_L = \frac{L}{2}i^2 = \frac{1}{2L} \Phi^2$\\ + Graphisch: Fläche zwischen der Kennlinie und der q-/$\Phi$-Achse\\ + \textbf{Relaxationspunkte (=Ruhepunkte)}: Betriebspunkte, in dem die in einer Reaktanz gespeicherte Energie minimal ist. Kandidaten sind: Extremwerte, Wendepunkte, Knicke, Schnittpunkte mit q-/$\Phi$-Achse\\ + \subsubsection{Verschaltung von Reaktanzen} + - Parallelschaltung: $C_p = C_1 + C_2$, $L_p = L_1 || L_2 = \frac{L_1L_2}{L_1+L_2}$\\ + - Serienschaltung: $C_p = C_1 || C_2 = \frac{C_1C_2}{C_1+C_2}$, $L_p = L_1 + L_2$\\ Merke: Am Kondensa\textsl{tor}, eilt der Strom \textsl{vor}, bei Induktivi\textsl{täten}, wird er sich ver\textsl{späten}\\ diff --git a/img/reactance_overview.pdf b/img/reactance_overview.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7baab55c4e597c7d17cc259f6b2359bf3a165060 GIT binary patch literal 20468 zcmbrlb9`jm)~KD1ZL?zAwr$(C?R0G0>DW#>ww-jxM#t$mH@)}S=bU@r_xtOsRkhX{ zV~*)r^&4Z=Gf5Rh#Aq4mSfNRqFKS+(SpW6qA008L4 zENxs&oj$KNhAyTerpERrrqFzR(9SMSriQlA9@$e`vT?YrNZm8F$3-Rv>ok5SkWLQq zpbY21)!k5BG;XAx7g$5S zir9DO9h`soZeqQi9Q7h>>1FNg4af~=2RoWRg7Evs`S2Ox_b4wd6 zGIJ3HC--v^YZH?x3fv?c?-FxdZe^X+fM%<&EU9ac6U8DuB8ye38v6PyH8+@e7^;Kb 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