\def\rload{R_{\rm load}}
\input ../format
\titlesub{Part 3: Executive Summary}{Mystery Circuit}

\input ../com

\section{Matrix Representation}

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}}
$$\x' =
{1\over R_1C_1C_2\rload}
\pmatrix{0&-C_2\rload      &0 \cr
         0&-C_2(R_1+\rload)&-C_1\rload\cr
         0&C_2R_1          &-C_1\rload}   \x +
{1\over R_1}
\pmatrix{\omega\cos(\omega t)\cr
         \omega\cos(\omega t)\cr
         0}
.
$$
The characteristic polynomial is
$$-\lambda( (-\lambda-C_2(R_1+\rload))(-\lambda-C_1\rload)
+ C_2C_1R_1\rload).$$
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
$c = C_1C_2\rload(2R_1+\rload),$
$$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
(\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$
%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}}}
\noindent The eigenvalues and respective eigenvectors are:

\bu $\lambda_1 = 0, v_1 = \pmatrix{1\cr0\cr0}$

\bu $\lambda_2 = r_1, v_2 = ??$

\bu $\lambda_3 = r_2, v_3 = ??$

Extending to the nonhomogenous system, 

\bye