A linear system with H(s) = 1/s is excited by a unit step function input. The output for t > 0 is given by

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]

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

So that transfer function of the system is used to calculate the output for a given input.

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

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

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

Transfer function = L[IR]

IR = L-1 [TF]

Application:

Given-

TF = H(s) = 1/s,

Input is unit step, R(s) = 1/s

Now output for t > 0 can be calculated as

C(s) = H(s) x R(s)

C(s) = (1/s) x (1/s)

C(s) = 1 / s²

L^-1[C(s)] = t

C(t) = t = ramp function