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| -rw-r--r-- | execsumm/ExecutiveSummaryDraft.tex | 87 | ||||
| -rw-r--r-- | execsumm/document.tex | 9 |
2 files changed, 9 insertions, 87 deletions
diff --git a/execsumm/ExecutiveSummaryDraft.tex b/execsumm/ExecutiveSummaryDraft.tex deleted file mode 100644 index a9c133b..0000000 --- a/execsumm/ExecutiveSummaryDraft.tex +++ /dev/null @@ -1,87 +0,0 @@ -\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 diff --git a/execsumm/document.tex b/execsumm/document.tex index 85d2e87..2354637 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -78,6 +78,15 @@ should be a particular solution. This would allow usage of method of undetermined coefficients, which may be attempted, but variation of parameters will be used so that a general form for both systems with both complex and real eigenvalues may be found. +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 +${\bf g}(t) = {1\over R_1} +\pmatrix{\omega\cos(\omega t)\cr + \omega\cos(\omega t)\cr + 0}.$ \section{Possible Generalization} |