From 76781234f2c320ca71f308ada5a2f5b4d9154e59 Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Tue, 14 Apr 2020 20:32:26 -0400 Subject: update with a correction and more clarity --- execsumm/document.tex | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) (limited to 'execsumm') diff --git a/execsumm/document.tex b/execsumm/document.tex index 95146a4..930705b 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -12,7 +12,7 @@ is useful (this form was chosen to reduce fractions' usage): $$\x' = {1\over R_1C_1C_2\rload} \pmatrix{0&-C_2\rload &0 \cr - 0&-C_2(R_1+\rload)&-C_1R_1\cr + 0&-C_2(R_1+\rload)&C_1R_1\cr 0&C_2R_1 &-C_1R_1} \x + {1\over R_1} \pmatrix{\omega\cos(\omega t)\cr @@ -21,18 +21,20 @@ $$\x' = .$$ The characteristic polynomial is $$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1R_1) -+ C_2C_1R_1^2).$$ +- C_2C_1R_1^2).$$ Expanded, $$-\lambda( \lambda^2 + \lambda(C_2(R_1+\rload)+C_1R_1) -+ C_1C_2R_1(2R_1+\rload)).$$ ++ C_1C_2R_1\rload).$$ In terms of its roots (with $b=C_2(R_1+\rload)+C_1R_1$ and -$c = C_1C_2R_1(2R_1+\rload),$ +$c = C_1C_2R_1\rload,$ $$-\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-% @@ -56,9 +58,9 @@ $\displaystyle z = y{C_2R_1\over r+C_1R_1}.$ -{r_#1\over C_2\rload}\cr -{R_1r_#1\over \rload(r_#1+C_1R_1)}}} -\bu $\displaystyle\lambda_2 = {r_1\over C_1C_2R_1\rload}, v_2 = \num1.$ +\bu $\displaystyle\lambda_2 = {r_1C_1C_2R_1\rload}, v_2 = \num1.$ -\bu $\displaystyle\lambda_3 = {r_2\over C_1C_2R_1\rload}, v_3 = \num2.$ +\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 @@ -72,6 +74,6 @@ $g = C_1\cos{\re(\lambda)t}\re v + C_2\sin{\re(\lambda)t}\im v.$ %% DOUBLE CHECK 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 +should be a particular solution. \bye -- cgit