blob: b5dfdc7907c3acfcb184a251425c117725b5fad2 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
|
\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:
\def\x{{\bf x}}
$$\x' =
{1\over R_1C_1C_2\rload}\left(
\pmatrix{0&-C_2\rload&0 \cr
*&* &* \cr
0&C_2R_1 &-C_2\rload} \x +
\pmatrix{\omega\cos(\omega t)\cr
\omega\cos(\omega t)\cr
0}
\right).
$$
{\it *Linearly dependent, meaning a trivial eigenvalue of 0}
\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{0\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
|
