From e0a4aa29f8ccbc9e86ec202be199756aea2528b6 Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Tue, 28 May 2024 18:37:50 -0700 Subject: [PATCH] change emacs a little; more stuff --- config/emacs.el | 1 + config/emacs.org | 1 + journal/index.org | 15 ++++ mindmap/LRC circuit.org | 84 ++++++++++++++++++ mindmap/circuit_diagram.png | Bin 0 -> 2506 bytes mindmap/index.org | 21 +++-- mindmap/lrc_circuit.png | Bin 0 -> 2506 bytes ...7ebeb5ab4fe83e7cab0ad8efbdeac1710f2155.png | Bin 0 -> 1550 bytes ...6a53c3fae12bad57721e6974b25c31a762682b.png | Bin 0 -> 671 bytes ...e59d300345e6ef0cb993063f25aeea975bfe4f.png | Bin 0 -> 1498 bytes ...2e9640a765e905b0e23f2a0f476759774dcef6.png | Bin 0 -> 1255 bytes ...4dc05f2968a16631a950a08b9b5b067f6c36c5.png | Bin 0 -> 321 bytes ...a9cb8733a53c191de32948274e8cfcfb88223f.png | Bin 0 -> 2052 bytes ...0d5e5d4a0f0474bfd8bd30e6284edc4faffdfb.png | Bin 0 -> 2730 bytes ...6debe396d581ba32d3d0b561c815b1d4e734c0.png | Bin 0 -> 582 bytes 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'( (shell . t) (python . t) + (latex . t) ) ) diff --git a/config/emacs.org b/config/emacs.org index cf00924..9c2ce6e 100644 --- a/config/emacs.org +++ b/config/emacs.org @@ -162,6 +162,7 @@ For some reason, org-babel doesn't load these languages by default: '( (shell . t) (python . t) + (latex . t) ) ) #+end_src diff --git a/journal/index.org b/journal/index.org index ac98b34..b959e7a 100644 --- a/journal/index.org +++ b/journal/index.org @@ -35,6 +35,21 @@ done #+end_src #+RESULTS: +- [[file:20240524.org][20240524.org]] +- [[file:20240519.org][20240519.org]] +- [[file:20240512.org][20240512.org]] +- [[file:20240510.org][20240510.org]] +- [[file:20240503.org][20240503.org]] +- [[file:20240502.org][20240502.org]] +- [[file:20240416.org][20240416.org]] +- [[file:20240404.org][20240404.org]] +- [[file:20240330.org][20240330.org]] +- [[file:20240329.org][20240329.org]] +- [[file:20240324.org][20240324.org]] +- [[file:20240323.org][20240323.org]] +- [[file:20240321.org][20240321.org]] +- [[file:20240319.org][20240319.org]] +- [[file:20240318.org][20240318.org]] - [[file:20240317.org][20240317.org]] - [[file:20240316.org][20240316.org]] - [[file:20240315.org][20240315.org]] diff --git a/mindmap/LRC circuit.org b/mindmap/LRC circuit.org new file mode 100644 index 0000000..e28d9b0 --- /dev/null +++ b/mindmap/LRC circuit.org @@ -0,0 +1,84 @@ +:PROPERTIES: +:ID: 6dbe2931-cc18-48fc-8cc1-6c71935a6be3 +:END: +#+title: LRC circuit +#+author: Preston Pan +#+html_head: +#+html_head: +#+html_head: +#+options: broken-links:t + +* Introduction +LRC circuits are equivalent to mass-spring oscillation systems in terms of the differential equation generated. In other +words, they are an example of a wave generator. First we introduce the LRC circuit without a voltage source. Later, +another circuit diagram will include a possibly variable voltage source. +#+name: LRC Circuit +#+header: :exports results :file lrc_circuit.png +#+header: :imagemagick yes :iminoptions -density 600 :imoutoptions -geometry 400 +#+header: :fit yes :noweb yes :headers '("\\usepackage{circuitikz}") +#+begin_src latex :exports results :file + \documentclass{article} + \usepackage{circuitikz} + \begin{document} + \begin{center} + \begin{circuitikz} \draw + (0,0) to[resistor, l=\mbox{$R$}] (0,4) + to[inductor, l=\mbox{$L$}] (4,4) + to[capacitor, l=\mbox{$C$}] (4,0) + (4,0) -- (0,0) + (2,0) -- (2,-1) + to (2, -1) node[shape=ground]{}; + \end{circuitikz} + \end{center} + \end{document} +#+end_src + +#+RESULTS: LRC Circuit +#+begin_export latex +#+end_export + +* Mass-Spring Equation Equivalence +We know these relations for the given circuit elements above: +\begin{align} +v(t) = L\frac{di}{dt} \\ +i(t) = C\frac{dv}{dt} \\ +v = iR +\end{align} +if we analyze the current current signal, Kirchhoff's voltage law tells us that the total voltage +drop throughout this circuit is zero. We use the capacitor equation in integral form and sum the voltages: +\begin{align*} +L\frac{di}{dt} + \frac{1}{C}\int i(t)dt + iR = 0 +\end{align*} +We then take a derivative to remove the integral: +\begin{align*} +L\frac{d^{2}i}{dt^{2}} + R\frac{di}{dt} + \frac{1}{C}i = 0 \\ +(LD^{2} + RD + \frac{1}{C}) i(t) = 0 +\end{align*} +it is clear that the characteristic polynomial of this homogeneous linear differential equation is: +\begin{align*} +L\lambda^{2} + R\lambda + \frac{1}{C} = 0 +\end{align*} +which, utilizing the quadratic formula, has the solutions: +\begin{align*} +\lambda_{1} = \frac{-R + \sqrt{R^{2} - \frac{4L}{C}}}{2L}, +\lambda_{2} = \frac{-R - \sqrt{R^{2} - \frac{4L}{C}}}{2L} +\end{align*} +which implies the general solution to this differential equation is: +\begin{align*} +i(t) = \sum_{n=0}^{\infty} A_{n}e^{\lambda_{1} t} + B_{n}e^{\lambda_{2} t} +\end{align*} +We can make this nicer by setting $-\frac{R}{2L} = m$, $\frac{\sqrt{R^{2} - \frac{4L}{C}}}{2L} = p$, +then $\lambda_{1} = m + p$, $\lambda_{2} = m - p$. Then: +\begin{align*} +i(t) = \sum_{n=0}^{\infty} A_{n}e^{(m + p) t} + B_{n}e^{(m - p) t} \\ +i(t) = e^{m}(\sum_{n=0}^{\infty} A_{n}e^{pt} + B_{n}e^{-pt}) +\end{align*} +Then we can just recast our notation for the constants $A_{n}$ and $B_{n}$ to include this $e^{m}$ term: +\begin{align*} +i(t) = \sum_{n=0}^{\infty} A_{n}e^{pt} + B_{n}e^{-pt} +\end{align*} +** Dampened Oscillation +In the case $R^{2} < \frac{4L}{C}$, $p = i\frac{\sqrt{\sigma}}{2L}$ for some $\sigma > 0$. We re-case $\lambda = \frac{\sqrt{\sigma}}{2L}$ so $p = i\lambda$. Then: +\begin{align*} +i(t) = \sum_{n=0}^{\infty} A_{n}e^{i\lambda t} + B_{n}e^{-i\lambda t} +\end{align*} diff --git a/mindmap/circuit_diagram.png b/mindmap/circuit_diagram.png new file mode 100644 index 0000000000000000000000000000000000000000..bf5f638edad0356a4a4dce218942e1c6ae2265fc GIT binary patch literal 2506 zcmeAS@N?(olHy`uVBq!ia0y~yV4T3fz_5mcg@J+Lf2ENs0|SFXvPY0F14ES>14Ba# z1H&%{28MEr{OD#T@@;ftfTgvIWh;Qs~j_Z1VJDm2dx$`E! zcg(aGVx7j1+Tx>2g*c7&C$P?J67ll(_O8Bu?=8Dk4tM44*MWBPCqG}j|BK$1+Srqy zKX3j1^FT@Frvp`aGi&zDkq>@Q6CHW5bxWar=BFhZpBTbF3P(+p3X|$M!6+ygX?4S+ ztL|5$jODM66O-l%dA2cKIDO(txI-h)&6E8`?k6_2DhbOh?dP#nGIB{ddE{@>(x}wP zV^$9syd%!1Zzf-SB~6wZ=NZQ#FgF3#>5OYWQ{a~;JD4-STyttTg_Iope>X%uuYiWN?3O{wTy`owc%Pq{>b zpfeBS@r*}Yr*sy&xO6dpDH08}Tkha8@3PJVhU6s{ho|xcr08pV2pf9LQZC#dv$AX@ zcTHK?FQ)mv|7%`7{xaqLgjwJI28sXB@|>gsMoWG^l^0Mu8gy+!qQN^ulex>^Pg>OL zu5r}1CGFHP+Xgk=$I5p7)f{J^NU3P%Jz!RNEnlM6-KgU1qAow-DcgVdNk@Ixq(nH+ zD|`1qSnZLVYw%7d|0a>{jHD)=+IK6wvn1xRclul3evvYPcg<6g)jKB_M%_5(=)qBp! 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