# Overall transfer function of a system is given as $G(s) = \frac{{(2s + 1)}}{{({s^2} + 2s + 5)}}$.The system is characterized as

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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

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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 2nd order system:

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

s2 + 2ζωns + ω2n = 0.....(ii)

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

ω2n = 5

2ζω= 2

ζ = $\frac{1}{{{\omega _n}}}$ = $\frac{1}{{{2.23}}}$

=0.44

So, 0 < ζ < 1

Hence the system is underdamped.