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diff --git a/execsumm/document.tex b/execsumm/document.tex index 4db9e4a..b5dfdc7 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -1,4 +1,43 @@ +\def\rload{R_{\rm load}} +\def\opt#1{\vskip0pt plus #1\vskip 0pt plus -#1} \input ../format \titlesub{Part 3: Executive Summary}{Mystery Circuit} +\input ../com + +\section{Matrix Representation} + +To determine the relevant properties of the linear system, matrix form +is useful: +\def\x{{\bf x}} +$$\x' = +{1\over R_1C_1C_2\rload}\left( +\pmatrix{0&-C_2\rload&0 \cr + *&* &* \cr + 0&C_2R_1 &-C_2\rload} \x + +\pmatrix{\omega\cos(\omega t)\cr + \omega\cos(\omega t)\cr + 0} +\right). +$$ + +{\it *Linearly dependent, meaning a trivial eigenvalue of 0} + +\def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}} +\opt{.15fil} +\noindent The eigenvalues and respective eigenvectors are: + +\bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$ + +\def\num{\pmatrix{0\cr C_2\rload\cr C_2R_1}} +\bu $\lambda_2 = -C_2\rload, v_2 = \num$ + +\bu $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$ + +This gives the solution to the homogenous system +$$D_1\pmatrix{1\cr0\cr0} + D_2e^{-C_2\rload t}\num ++ D_3te^{-C_2\rload t}\pmatrix{0\cr0\cr1}.$$ + +Extending to the nonhomogenous system, + \bye |