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-rw-r--r--execsumm/document.tex20
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 $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$