Todai Entrance Exam: Subject 2009 – Problem 1

(1)

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

(2)

$\begin{pmatrix} 1 - RCs - RLs - R^{2} + L^{2} & Ls + R - LRCs^{2}\\Cs - RLCs^{2} - R & LCs^{2} + 1 \end{pmatrix}$

(3)

$U_{r}(s) = (1 - RCs - RLs - R^{2} + L^{2})U_{s}(s) + (Ls + R - LRCs^{2})\frac{U_{s}(s)}{R}$

So:

$G(s) = (L^{2} - LC)s^{2} + (\sqrt{LC} - RL - \tau)s + 2 - R^{2}$

(4)

$G(s)\frac{u_{0}}{s} = \left ((L^{2} - LC)\frac{s}{\tau}+ \frac{\sqrt{LC} - RL - \tau}{\tau} + (2 - R^{2})\frac{1}{s\tau} \right )u_{0}\tau$

$G(s)\frac{u_{0}}{s} = \left (\frac{(L^{2} - LC)s^{2}((s\tau + 1)^{2}+1)}{s\tau((s\tau + 1)^{2}+1)}+ \frac{(\sqrt{LC} - RL - \tau)s((s\tau + 1)^{2}+1)}{s\tau((s\tau + 1)^{2}+1)} + \frac{(2 - R^{2})((s\tau + 1)^{2}+1)}{s\tau((s\tau + 1)^{2}+1)} \right )u_{0}\tau$

Not sure if I’m doing this correctly…

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