diff options
Diffstat (limited to 'execsumm/document.tex')
| -rw-r--r-- | execsumm/document.tex | 20 |
1 files changed, 9 insertions, 11 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex index c308d05..45d6546 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -1,5 +1,4 @@ \def\rload{R_{\rm load}} -\def\opt#1{\vskip0pt plus #1\vskip 0pt plus -#1} \input ../format \titlesub{Part 3: Executive Summary}{Mystery Circuit} @@ -11,14 +10,15 @@ 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}} $$\x' = -{1\over R_1C_1C_2\rload}\left( +{1\over R_1C_1C_2\rload} \pmatrix{0&-C_2\rload &0 \cr 0&-C_2(R_1+\rload)&-C_1\rload\cr 0&C_2R_1 &-C_1\rload} \x + +{1\over R_1} \pmatrix{\omega\cos(\omega t)\cr \omega\cos(\omega t)\cr 0} -\right). +. $$ The characteristic polynomial is $$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload) @@ -30,21 +30,19 @@ In terms of its roots (with $b=C_2(R_1+\rload)+C_1\rload$ and $c = C_1C_2\rload(2R_1+\rload),$ $$-\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).$$ +Let $r_1$ and $r_2$ designate these two non-zero roots. \def\bu{\par\leavevmode\llap{\hbox to \parindent{\hfil $\bullet$ \hfil}}} -\opt{.15fil} \noindent The eigenvalues and respective eigenvectors are: \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$ -\def\num{\pmatrix{C_2\rload\cr C_2\rload\cr C_2R_1}} -\bu $\lambda_2 = -C_2\rload, v_2 = \num$ +\bu $\lambda_2 = r_1, v_2 = ??$ -\bu $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$ - -This gives the solution to the homogenous system -$$D_1\pmatrix{1\cr0\cr0} + D_2e^{-C_2\rload t}\num -+ D_3te^{-C_2\rload t}\pmatrix{0\cr0\cr1}.$$ +\bu $\lambda_3 = r_2, v_3 = ??$ Extending to the nonhomogenous system, |