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| -rw-r--r-- | execsumm/document.tex | 20 |
1 files changed, 13 insertions, 7 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex index b5dfdc7..32d4a0a 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -8,20 +8,26 @@ \section{Matrix Representation} To determine the relevant properties of the linear system, matrix form -is useful: +is useful (this form was chosen to reduce fractions' usage): \def\x{{\bf x}} $$\x' = {1\over R_1C_1C_2\rload}\left( -\pmatrix{0&-C_2\rload&0 \cr - *&* &* \cr - 0&C_2R_1 &-C_2\rload} \x + +\pmatrix{0&-C_2\rload &0 \cr + 0&-C_2(R_1+\rload)&-C_1\rload\cr + 0&C_2R_1 &-C_1\rload} \x + \pmatrix{\omega\cos(\omega t)\cr \omega\cos(\omega t)\cr 0} \right). $$ - -{\it *Linearly dependent, meaning a trivial eigenvalue of 0} +The characteristic polynomial is +$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload) ++ C_2^2R_1\rload).$$ +Expanded, +$$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1\rload) ++ C_1C_2\rload(R_1+\rload) + C_2^2R_1\rload).$$ +In terms of its roots, +$$???$$ \def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}} \opt{.15fil} @@ -29,7 +35,7 @@ $$ \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$ -\def\num{\pmatrix{0\cr C_2\rload\cr C_2R_1}} +\def\num{\pmatrix{C_2\rload\cr C_2\rload\cr C_2R_1}} \bu $\lambda_2 = -C_2\rload, v_2 = \num$ \bu $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$ |