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