Question

Which of the following is the transfer function of:

\(\frac{{dc\left( t \right)}}{{dt}} + 2c\left( t \right) = r\left( t \right)\)

Where, r(t) is the unit impulse signal

1. \(G\left( s \right) = \frac{s}{{s - 2}}\)
2. \(G\left( s \right) = \frac{1}{{s - 2}}\)
3. \(G\left( s \right) = \frac{s}{{s + 2}}\)
4. \(G\left( s \right) = \frac{1}{{s + 2}}\)

Answer 1

Correct Answer - Option 4 : \(G\left( s \right) = \frac{1}{{s + 2}}\)

Concept:

A transfer function is defined as the ratio of the 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:

The given differential equation is,

\(\frac{{dc\left( t \right)}}{{dt}} + 2c\left( t \right) = r\left( t \right)\)

By applying the Laplace transform, we get

⇒ s C(s) + 2 C(s) = R(s)

\( \Rightarrow G\left( s \right) = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{1}{{s + 2}}\)