Todai Entrance Exam: Subject 2009 – Problem 1

(1) Reference (F-matrix):https://kats.issp.u-tokyo.ac.jp/kats/circuit4/doc/presen/circuit_2.pdf

$F_{II} = \begin{pmatrix} 1 & 0\\ Cs & 1 \end{pmatrix}, F_{III} = \begin{pmatrix} 1 & Ls\\ 0 & 1 \end{pmatrix}, F_{IV} = \begin{pmatrix} 1 & 0\\ \frac{1}{R} & 1 \end{pmatrix}$

(2)

$\begin{pmatrix}1 + LCs^{2} & R + RLCs^{2} + Ls\\\frac{1 + LCs^{2}}{R} + Cs & 1 + LCs^{2} + \frac{Ls}{R} + RCs + 1\\ \end{pmatrix}$

(3)

This is an open circuit so $i_{r} = 0$ (reference: wikipedia & mit courseware).

$G(s) = \frac{1}{\tau^{2}s^{2} + 2\tau s + 2}$

(4)

$A_{1} = \frac{1}{2}, A_{2} = -\frac{1}{2}, A_{3} = -\frac{1}{2}$

(5)

$u_{r}(t) = \frac{1}{2} - \frac{1}{2}e^{-t}(\cos t + \sin t)$

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