From 725f8b2b2fc250dcd6efdf8ba4108b196ed6d6b3 Mon Sep 17 00:00:00 2001
From: Holden Rohrer <hr@hrhr.dev>
Date: Tue, 14 Apr 2020 22:42:59 -0400
Subject: merged latex into plain tex

---
 execsumm/ExecutiveSummaryDraft.tex | 87 --------------------------------------
 1 file changed, 87 deletions(-)
 delete mode 100644 execsumm/ExecutiveSummaryDraft.tex

(limited to 'execsumm/ExecutiveSummaryDraft.tex')

diff --git a/execsumm/ExecutiveSummaryDraft.tex b/execsumm/ExecutiveSummaryDraft.tex
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-\documentclass{article}
-\usepackage{hyperref}
-\def\rload{R_{\rm load}}
-\date{}
-\begin{document}
-\title{Project Executive Summary}
-\author{Holden Rohrer and Nithya Jayakumar}
-
-\maketitle
-\section{Matrix Representation and Homogeneous Solution}
-
-To determine the relevant properties of the linear system, matrix form
-is useful (this form was chosen to reduce fractions' usage):
-\def\x{{\bf x}}
-$$\x' =
-{1\over R_1C_1C_2\rload}
-\pmatrix{0&-C_2\rload      &0 \cr
-         0&-C_2(R_1+\rload)&C_1R_1\cr
-         0&C_2R_1          &-C_1R_1}   \x +
-{1\over R_1}
-\pmatrix{\omega\cos(\omega t)\cr
-         \omega\cos(\omega t)\cr
-         0}
-.$$
-The characteristic polynomial is
-$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1R_1)
-- C_2C_1R_1^2).$$
-Expanded,
-$$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1R_1)
-+ C_1C_2R_1\rload).$$
-In terms of its roots (with $b=C_2(R_1+\rload)+C_1R_1$ and
-$c = C_1C_2R_1\rload,$
-$$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
-(\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$
-%For reference,
-%$$b^2-4c = C_2^2(R_1+\rload)^2 + C_1^2\rload^2
-%           - 2C_1C_2\rload(3R_1+\rload).$$
-Let $r_1$ and $r_2$ designate these two non-zero roots.
-In the original matrix $A$, this being the transformed matrix $A/c$,
-$Av = cr_1$ because $A/c * v = r_1.$
-
-The trivial zero eigenvalue corresponds to a unit x-direction vector
-by inspection. The two remaining roots, in the general case of a non-%
-degenerate system, which hasn't been explicitly ruled out, the middle
-row can be ``ignored'' because it is linearly independent in the
-following system:
-$$
-\pmatrix{-r&-C_2\rload &0\cr
-         * &*          &*\cr
-         0 &C_2R_1     &-r-C_1R_1}
-$$
-With $x = 1$, $\displaystyle y = -{r\over C_2\rload}$ and
-$\displaystyle z = y{C_2R_1\over r+C_1R_1}.$
-
-\def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}}
-\noindent The eigenvalues and respective eigenvectors are:
-
-\bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$
-
-\def\num#1{\pmatrix{1\cr
-                    -{r_#1\over C_2\rload}\cr
-                    -{R_1r_#1\over \rload(r_#1+C_1R_1)}}}
-
-\bu $\displaystyle\lambda_2 = {r_1C_1C_2R_1\rload}, v_2 = \num1.$
-
-\bu $\displaystyle\lambda_3 = {r_2C_1C_2R_1\rload}, v_3 = \num2.$
-
-This corresponds to a solution of the form $f = Ce^{\lambda t}v,\cdots$
-where $C\in {\bf C},$ $f\in {\bf R}\to{\bf R}$. If the non-trivial
-eigenvectors are complex, %% TRY TO PROVE THIS!!
-\def\re{{\rm Re}}\def\im{{\rm Im}}
-their exponential solutions form, in the reals,
-$g = C_1\cos{\re(\lambda)t}\re v + C_2\sin{\re(\lambda)t}\im v.$ %% DOUBLE CHECK.
-
-\section{Nonhomogeneous System}
-
-Extending to the nonhomogeneous system will take slightly different
-paths depending on if the system has complex roots or has real roots.
-But in either case, $\cos x*{\rm polynomial}+\sin x*{\rm polynomial}$
-should be a particular solution. 
-
-We can apply the method of Variation of Parameters to this. Essentially, the solution is $\bf{x} = c_1\bf{\lambda_1}(t) + c_2\bf{\lambda_2}(t) + c_3\bf{\lambda_3}(t)\bf{\lambda_p}(t)$, where the particular solution $\bf{\lambda_p}(t)$ is:
-$$\bf{\lambda_p}(t) = \bf{X}(t)\int\bf{X^{-1}}(t)\bf{g}(t)dt,$$ where $\bf{X}(t)$ is the Fundamental matrix for the equation and $\bf{g}(t) = {1\over R_1}
-\pmatrix{\omega\cos(\omega t)\cr
-         \omega\cos(\omega t)\cr
-         0}.$
-\end{document}
\ No newline at end of file
-- 
cgit 

