A second-order system has

\(\frac{C(s)}{R(s)}=\frac{ω_n^2}{(s^2+2ξω_n s+ω_n^2 )}\)

Its frequency response will have a maximum value at the frequency:

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ESE Electronics 2013 Paper 2: Official Paper
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  1. \(ω_n \sqrt{(1-ξ^2 )}\)
  2. ωnξ
  3. \(ω_n \sqrt{(1-2ξ^2 )}\)
  4. Zero

Answer (Detailed Solution Below)

Option 3 : \(ω_n \sqrt{(1-2ξ^2 )}\)
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Detailed Solution

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The transfer function of the standard 2nd order system is defined as

\(T.F = \frac{{ω _n^2}}{{{s^2} + 2ξ {ω _n}s + ω _n^2}}\)

The resonant frequency ω0 is given by

\({ω _0} = {ω _n}\sqrt {1 - 2{ξ ^2}} \)

Where ωn = undamped natural frequency

ξ = Damping ratio

Resonant peak M0 is given by:

\({M_0} = \frac{1}{{2ξ \sqrt {1 - {ξ ^2}} }},\;0 \le ξ \le \frac{1}{{\sqrt 2 }}\)

Resonant frequency (ωr) is the frequency at which the magnitude characteristic of frequency response curve has its peak value.

The above equation is meaningful only for 1 – 2ζ2 ≥ 0, i.e.

\(zeta \le \frac{1}{\sqrt{2}}~or~\zeta \le 0.707\)

Note: If ζ > 0.707, then ωr = 0

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