Question

Overall transfer function of a system is given as \(G(s) = \frac{{(2s + 1)}}{{({s^2} + 2s + 5)}}\).The system is characterized as
1. Underdamped system
2. Undamped system
3. Critically damped system
4. Over damped system

Answer 1

Correct Answer - Option 1 : Underdamped system

Concept:

The type of system is determined by the value of the damping factor (ζ), i.e.

1) If  0 < ζ < 1, the system is underdamped.

2) If ζ = 1, the system is critically damped.

3) If ζ > 1, the system is overdamped.

Analysis:

\(G(s) = \frac{{(2s + 1)}}{{({s^2} + 2s + 5)}}\)

Characteristic equation of the system is the denominator of the overall transfer function.

s² + 2s + 5 = 0    ---(1)

Compare it with the standard characteristic equation of 2^nd order system:

1 + G(s)H(s) = 0

s² + 2ζω_ns + ω²_n = 0.....(ii)

On comparing equation (i) and (ii), we get:

ω2n = 5

ω_n = √5 rad/sec = 2.23 rad/sec

2ζωn = 2

ζ = \(\frac{1}{{{\omega _n}}}\) = \(\frac{1}{{{2.23}}}\)

=0.44

So, 0 < ζ < 1 

Hence the system is underdamped.