Correct Answer - Option 4 :
\(H\left( s \right) = \frac{1}{{{s^2} + 3s + 4}}\)
Concept:
The standard second-order system is given by \(\frac{{\omega _n^2}}{{{s^2} + 2\xi {\omega _n}s + \omega _n^2}}\)
Where ξ is the damping ratio.
If ξ = 1, then the system is critically damped.
If ξ < 1, then the system is underdamped.
If ξ > 1, then the system is order damped.
Calculation:
1. \(H\left( s \right) = \frac{1}{{{s^2} + 4s + 4}}\)
By comparing with a standard second-order transfer function,
ωn2 = 4 ⇒ ωn = 2
\(2\xi {{\rm{\omega }}_n} = 4 \Rightarrow \xi = \frac{4}{{2\times2 }} = 1\)
So, it is a critically damped system.
2. \(H\left( s \right) = \frac{1}{{{s^2} + 5s + 4}}\)
ωn2 = 4 ⇒ ωn = 2
2 ξ ωn = 5 ⇒ ξ > 1
So, it is over damped system.
3. \(H\left( s \right) = \frac{1}{{{s^2} + 4.5s + 4}}\)
ωn2 = 4 ⇒ ωn = 2
\(2\xi {{\rm{\omega }}_n} = 4.5 \Rightarrow \xi > 1\)
So, it is overdamped system.
4. \(H\left( s \right) = \frac{1}{{{s^2} + 3s + 4}}\)
ωn2 = 4 ⇒ ωn = 2
2 ξ ωn = 3 ⇒ ξ < 1
So, it is under damped system.
