From 63e27f21bd4a8bb8d7cb98579f4483d561f871ec Mon Sep 17 00:00:00 2001
From: SpaceOddity404 <40719093+SpaceOddity404@users.noreply.github.com>
Date: Mon, 30 Mar 2020 11:43:05 -0400
Subject: Add files via upload

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+\documentclass{article}
+\usepackage{hyperref}
+\date{}
+\begin{document}
+\title{Project Progress Report}
+\author{Holden Rohrer and Nithya Jayakumar}
+
+\maketitle
+
+\section{Project Topic}
+Our group will be working on the \href{https://simiode.org/resources/3187/download/4-23-S-MysteryCircuit-StudentVersion.pdf}{Mystery Circuit Modelling
+Scenario from SIMIODE}. This applies Kirchhoff's Voltage
+and Current Laws to the given circuit, which describe, respectively,
+that the sum of all voltages in a closed loop is zero and the sum of all
+currents at a node is zero. The circuit we're examining is an RLC
+(resistor, inductor, capacitor) circuit, with zeroed initial conditions.
+The specific circuit has two linked loops of resistors and capacitors,
+in which ``gain,'' the ratio between chosen voltage differentials in the
+circuit can be modeled mathematically.
+
+\section{Progress}
+We have the two equations that are results of Kirchhoff's Voltage and Current Laws:
+$$\sin(100t) = x(t)R_1 + \frac{1}{C_1}\int y(t)dt,$$
+$$\frac{1}{C_1}\int y(t)dt = \frac{1}{C_2} \int z(t) dt + z(t)R_{load}$$
+Differentiating these and rearranging gives:
+$$x'(t) = -\frac{y(t)}{R_1C_1} + \frac{100\cos(100t)}{R_1},$$
+$$z'(t) = \frac{y(t)}{C_1R_{load}} - \frac{z(t)}{C_2R_{load}}$$
+
+Through Kirchhoff's laws, we know that $y + z = x$ (current law), $y = z$ (voltage law), and $y + x = $ Electromotive Force (voltage law). Thus, we know that 
+$$y' = x' - z' = -\frac{y(t)}{R_1C_1} + \frac{100\cos(100t)}{R_1} - \frac{y(t)}{C_1R_{load}} + \frac{z(t)}{C_2R_{load}},$$
+and we now have a system of differential equations.
+
+\section{Further Exploration}
+We will use the method of Eigenvalues and Eigenvectors to solve the system of differential equations we have developed, and accordingly model the circuit. Using this, we can find out the purpose of the circuit and do further work on similar circuits.
+
+\end{document}
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