This question was previously asked in

ESE Electronics 2016 Paper 2: Official Paper

Option 2 : 1.04 and 2.67 r/s

ST 1: General Awareness

6945

15 Questions
15 Marks
15 Mins

__Concept:__

The transfer function of the standard 2^{nd} 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 M_{0} is given by:

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

Calculation:

Given:

\(G(s)=\frac{25}{s(s+6)}\)

The given feedback is a unity feedback system,

\(CLTF=\frac{G\left( s \right)}{1+G\left( s \right)}\)

\(CLTF=\frac{25}{{{s}^{2}}+6s+25}\)

The characteristic equation is given by;

s2 + 6s + 25 = 0

Comparing this with the standard 2nd order characteristic equation:.

ω n = 5

2ξωn = 6

ξ = 0.6

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

So, the resonant frequency is given by;

\({{ω }_{r}}={{ω }_{n}}\sqrt{1-2{{\xi }^{2}}}\)

\({{ω }_{r}}=5\sqrt{1-2{{\left(0.6 \right)}^{2}}}\)

ω_{r} = 2.67 rad/sec

the resonant frequency is given by;

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

M_{0} = 1.04