The overall transfer function of a second order control system is given by

\(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{2}{{{s^2} + 3s + 2}}\)

The time response of this system, when subjected to a unit step response isThis question was previously asked in

ESE Electrical 2014 Paper 1: Official Paper

Option 3 : 1 – 2e^{-t} + e^{-2t}

CT 3: Building Materials

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Concept:

A transfer function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}}\)

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Calculation:

\(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{2}{{{s^2} + 3s + 2}} = \frac{2}{{\left( {s + 1} \right)\left( {s + 2} \right)}}\)

Time response when subjected to unit step response is,

\(C\left( s \right) = \frac{2}{{\left( {s + 1} \right)\left( {s + 2} \right)}}\frac{1}{s}\;\)

By using partial fraction method,

\(C\left( s \right) = - \frac{2}{{s + 1}} + \frac{1}{{s + 2}} + \frac{1}{s}\;\)

By applying inverse Laplace transform,

⇒ c(t) = 1 – 2e^{-t} + e^{-2t}