'(
(shell . t)
(python . t)
+ (latex . t)
)
)
'(
(shell . t)
(python . t)
+ (latex . t)
)
)
#+end_src
#+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]]
--- /dev/null
+:PROPERTIES:
+:ID: 6dbe2931-cc18-48fc-8cc1-6c71935a6be3
+:END:
+#+title: LRC circuit
+#+author: Preston Pan
+#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
+#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+#+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*}
- [[file:recursion.org][recursion.org]]
- [[file:python.org][python.org]]
- [[file:physics.org][physics.org]]
+- [[file:philosophy.org][philosophy.org]]
- [[file:partial derivative.org][partial derivative.org]]
+- [[file:Newtonian mechanics.org][Newtonian mechanics.org]]
- [[file:natural number.org][natural number.org]]
+- [[file:Maxwell's Equations.org][Maxwell's Equations.org]]
- [[file:mathematics.org][mathematics.org]]
+- [[file:mass-spring system.org][mass-spring system.org]]
- [[file:magnetostatics.org][magnetostatics.org]]
+- [[file:ltximg][ltximg]]
+- [[file:lrc_circuit.png][lrc_circuit.png]]
+- [[file:LRC circuit.org][LRC circuit.org]]
+- [[file:Lorentz Force.org][Lorentz Force.org]]
- [[file:logic.org][logic.org]]
+- [[file:limit.org][limit.org]]
+- [[file:Legendre Transformation.org][Legendre Transformation.org]]
+- [[file:Lagrangian mechanics.org][Lagrangian mechanics.org]]
- [[file:inverse square.org][inverse square.org]]
- [[file:inheritance.org][inheritance.org]]
- [[file:infinity.org][infinity.org]]
- [[file:induction.org][induction.org]]
- [[file:group.org][group.org]]
- [[file:function.org][function.org]]
+- [[file:framework.org][framework.org]]
+- [[file:Fourier Transform.org][Fourier Transform.org]]
- [[file:factorial.org][factorial.org]]
- [[file:everything.org][everything.org]]
- [[file:emergence.org][emergence.org]]
- [[file:electrostatics.org][electrostatics.org]]
+- [[file:egoism.org][egoism.org]]
- [[file:duality.org][duality.org]]
- [[file:divergence theorem.org][divergence theorem.org]]
- [[file:dirac delta.org][dirac delta.org]]
- [[file:del operator.org][del operator.org]]
- [[file:continuity equation.org][continuity equation.org]]
- [[file:conservative force.org][conservative force.org]]
+- [[file:circuit_diagram.png][circuit_diagram.png]]
- [[file:central force.org][central force.org]]
-- [[file:Newtonian mechanics.org][Newtonian mechanics.org]]
-- [[file:Maxwell's Equations.org][Maxwell's Equations.org]]
-- [[file:Lorentz Force.org][Lorentz Force.org]]
-- [[file:Legendre Transformation.org][Legendre Transformation.org]]
-- [[file:Lagrangian mechanics.org][Lagrangian mechanics.org]]
-- [[file:Fourier Transform.org][Fourier Transform.org]]
@@html: </div>@@
--- /dev/null
+:PROPERTIES:
+:ID: 5004f2b4-f8f7-46e6-87b5-e547184f82ef
+:END:
+#+title: mass-spring system
+#+author: Preston Pan
+#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
+#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+#+options: broken-links:t
+* Introduction
+A mass-spring model uses Hooke's law $F = kx$ in order to model a mass attached to a spring with a differential equation.
projects / monorepo.git / commitdiff
