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| -rw-r--r-- | execsumm/document.tex | 23 |
1 files changed, 11 insertions, 12 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex index 8ed5e21..f3cf093 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -12,22 +12,21 @@ is useful (this form was chosen to reduce fractions' usage): $$\x' = {1\over R_1C_1C_2\rload} \pmatrix{0&-C_2\rload &0 \cr - 0&-C_2(R_1+\rload)&-C_1\rload\cr - 0&C_2R_1 &-C_1\rload} \x + + 0&-C_2(R_1+\rload)&-C_1R_1\cr + 0&C_2R_1 &-C_1R_1} \x + {1\over R_1} \pmatrix{\omega\cos(\omega t)\cr \omega\cos(\omega t)\cr 0} -. -$$ +.$$ The characteristic polynomial is -$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload) -+ C_2C_1R_1\rload).$$ +$$-\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_1\rload) -+ C_1C_2\rload(2R_1+\rload)).$$ -In terms of its roots (with $b=C_2(R_1+\rload)+C_1\rload$ and -$c = C_1C_2\rload(2R_1+\rload),$ +$$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1R_1) ++ C_1C_2R_1(2R_1+\rload)).$$ +In terms of its roots (with $b=C_2(R_1+\rload)+C_1R_1$ and +$c = C_1C_2R_1(2R_1+\rload),$ $$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2}) (\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$ %For reference, @@ -40,9 +39,9 @@ Let $r_1$ and $r_2$ designate these two non-zero roots. \bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$ -\bu $\lambda_2 = r_1, v_2 = ??$ +\bu $\lambda_2 = {r_1\over C_1C_2R_1\rload}, v_2 = ??$ -\bu $\lambda_3 = r_2, v_3 = ??$ +\bu $\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = ??$ \section{Nonhomogenous System} |