\def\rload{R_{\rm load}}
\def\opt#1{\vskip0pt plus #1\vskip 0pt plus -#1}
\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}}
$$\x' =
{1\over R_1C_1C_2\rload}\left(
\pmatrix{0&-C_2\rload      &0 \cr
         0&-C_2(R_1+\rload)&-C_1\rload\cr
         0&C_2R_1          &-C_1\rload}   \x +
\pmatrix{\omega\cos(\omega t)\cr
         \omega\cos(\omega t)\cr
         0}
\right).
$$
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
$c = C_1C_2\rload(2R_1+\rload),$
$$-\lambda{(\lambda-{-b+\sqrt{b^2-4c}\over2})
(\lambda-{-b-\sqrt{b^2-4c}\over 2})}$$

\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 $\lambda_3 = -C_2\rload, v_2 = \pmatrix{0\cr0\cr1}$

This gives the solution to the homogenous system
$$D_1\pmatrix{1\cr0\cr0} + D_2e^{-C_2\rload t}\num
+ D_3te^{-C_2\rload t}\pmatrix{0\cr0\cr1}.$$

Extending to the nonhomogenous system, 

\bye