Which one of the following transfer functions represents the critically damped system?

Which one of the following transfer functions represents the critically damped system? Correct Answer H<sub style="">1</sub>(s) = 1/s<sup style="">2</sup> + 4s + 4

Concept:

The characteristic equation of the standard second-order system is given by

s2 + 2ξ ωn s + ω2n = 0

The system is said to be

a) undamped if ξ = 0

b) critically damped if ξ = 1

c) underdamped if ξ < 1

d) overdamped if ξ > 1

Analysis:

1. s2 + 4s + 4 = 0

ω_n = 2

2ξω_n = 4 

ξ = 1 (Critically damped)

2. s2 + 3s + 4 = 0

ωn = 2

2ξωn = 3 

ξ = 0.75 (under damped)

3. s2 + 2s + 4 = 0

ωn = 2

2ξωn = 2

ξ = 0.5 (under damped)

4. s2 + s + 4 = 0

ωn = 2

2ξωn = 1

ξ = 0.25 (under damped)

Hence, Option 1 is correct.