Of following transfer function of second order linear time-invari

SOLUTION

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.