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Impulse

For the compe...

Question

For the compensated attenuator of figure, the impulse response under the condition $R_{1} C_{1} = R_{2} C_{2}$ is

A

\frac{R_{2}}{R_{1} + R_{2}} [1 - e^{\frac{1}{R_{1} C_{1}}}] u (t)

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B

$\frac{R_{2}}{R_{1} + R_{2}} δ (t)$

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C

$\frac{R_{2}}{R_{1} + R_{2}} u (t)$

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D

\frac{R_{2}}{R_{1} + R_{2}} e^{\frac{1}{R_{1} C_{1}}} u (t)

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Solution

The correct option is B
$\frac{R_{2}}{R_{1} + R_{2}} δ (t)$

$Z_{2} (s) = \frac{R_{2} \times \frac{1}{C_{2} s}}{R_{2} + \frac{1}{C_{2} s}} = \frac{R_{2}}{R_{2} C_{2} S + 1}$

$Z_{1} (s) = \frac{R_{1} \times \frac{1}{C_{1} s}}{R_{1} + \frac{1}{C_{1} s}} = \frac{R_{1}}{R_{1} C_{1} s + 1}$

$\frac{V_{2} (s)}{V_{1} (s)} = \frac{Z_{2} (s)}{Z_{1} (s) + Z_{2} (s)}$

$R_{1} C_{1} = R_{2} C_{2}$

$\frac{V_{2} (s)}{V_{1} (s)} = \frac{\frac{R_{2}}{R_{2} C_{2} s + 1}}{\frac{R 1}{R_{2} C_{2} s + 1} + \frac{R 2}{R_{2} C_{2} s + 1}} = \frac{R_{2}}{R_{1} + R_{2}}$

$V_{2} (s) = \frac{R_{2}}{R_{1} + R_{2}} V_{1} (s)$

For impulse response.

$V_{1} (s) = 1$

$v_{1} (t) = δ (t)$

$v_{2} (t) = \frac{R_{2}}{R_{1} + R_{2}} δ (t)$

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