Todai Entrance Exam: Subject 2012 – Problem 1

(1)

I make a reference to the previous problem:

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

(2)

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

(3)

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

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

    \[ V_{OUT}(s) = \frac{s+\frac{1}{R_{1}C}}{\left ( s+\frac{1}{R_{1}C} \right )^{2} + \frac{1}{C^{2}R_{1}}\frac{R_{1} - R_{2}}{R_{1}R_{2}}} - \sqrt{\frac{R_{2}}{R_{1} - R_{2}}}\frac{\frac{1}{R_{1}C}\sqrt{\frac{R_{1} - R_{2}}{R_{2}}}}{\left ( s+\frac{1}{R_{1}C} \right )^{2} + \frac{1}{C^{2}R_{1}}\frac{R_{1} - R_{2}}{R_{1}R_{2}}} \]

    \[ V_{OUT}(t) = e^{-\frac{1}{R_{1}C}t}\cos\left (  \frac{1}{R_{1}C}\sqrt{\frac{R_{1} - R_{2}}{R_{2}}}t \right ) - e^{-\frac{1}{R_{1}C}t}\sqrt{\frac{R_{2}}{R_{1} - R_{2}}}\sin\left (  \frac{1}{R_{1}C}\sqrt{\frac{R_{1} - R_{2}}{R_{2}}}t \right ) \]

If R_{2} < R_{1}: V_{OUT}(t) is oscillatory.

If R_{2} = R_{1}: V_{OUT}(t) = e^{-\frac{1}{R_{1}C}t} is critical.

If R_{2} > R_{1}: V_{OUT}(t) is non-oscillatory because t is always a real number with zero imaginary part (\sinh(x) = -i\sin(ix), \cosh(x) = \cos(ix)).

(4)

    \[ V_{OUT}(t) = e^{-0.01t}\left [ \cos\left ( \frac{\sqrt{1999}}{100}t \right ) -\frac{1}{\sqrt{1999}}\sin\left ( \frac{\sqrt{1999}}{100}t \right ) \right ] \]


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