Todai Entrance Exam: Subject 2012 – Problem 1

(1)

I make a reference to the previous problem:

    \[ \begin{Bmatrix} I_{1} = \frac{V_{1}}{R_{1} + \frac{1}{Cs}}\\  I_{2} = \frac{V_{1} - V_{OUT}}{R_{2}}\\  I_{1} + I_{2} = \frac{V_{IN} - V_{1}}{\frac{1}{Cs}}\\ V_{OUT} = I_{1}R_{1} \end{matrix} \]

I think using voltage divider would have been more straight-forward…

(2)

    \[ H(s) = \frac{V_{OUT}(s)}{V_{IN}(s)} = \frac{R_{1}R_{2}(Cs)^{2}}{2R_{2}Cs + 1 + (Cs)^{2}R_{1}R_{2}} \]

(3)

I don’t understand this question.

(4)

From (2), together with V_{IN} = \frac{1}{s} (Laplace transform of unit step function):

    \[ V_{OUT}(s) = \frac{R_{1}R_{2}(C)^{2}s}{2R_{2}Cs + 1 + (Cs)^{2}R_{1}R_{2}} = \frac{s+10}{(s+10)^{2}+100} - \frac{10}{(s+10)^{2} + 100} \]

So:

    \[ V_{OUT}(t) = L^{-1}\{V_{OUT}(s)\} = e^{-10t}(\cos(10t) - \sin(10t)) \]


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