# The characteristic equation of a simple servo system is s2 + 6s + 25 = 0. Damping factor of the system is

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The characteristic equation of a simple servo system is s2 + 6s + 25 = 0. Damping factor of the system is
1. 3.2
2. 2.4
3. 1.8
4. 3.0

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Correct Answer - Option 4 : 3.0

Concept:

The general expression of the transfer function of the standard second-order system is:

$TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2ζ {ω _n}s + ω _n^2}}$

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Characteristic equation: ${s^2} + 2ζ {ω _n} + ω _n^2 = 0$

The roots of the characteristic equation are:

$- ζ {ω _n} + j{ω _n}\sqrt {1 - {ζ ^2}} = - \alpha \pm j{ω _d}$

α is the damping factor

Calculation:

Given, the characteristic equation s2 + 6s + 25 = 0

Comparing this with the general expression of the transfer function of the second-order control system, we can write:

2 ζ ωn = 6

⇒ ζ ωn = 3

The nature of the system is described by its ‘ζ’ value

 ζ Nature ζ = 0 Undamped 0 < ζ < 1 Underdamped ζ = 1 Critically damped ζ > 1 Overdamped