Question

A second-order real system has the following properties:

The damping ratio \(\xi = 0.5\) and undamped natural frequency \({\omega _n} = 10\;rad/s\), the steady state value at zero is 1.02.

The transfer function of the system is
1. \(\frac{{1.02}}{{{s^2} + 5s + 100}}\)
2. \(\frac{{102}}{{{s^2} + 10s + 100}}\)
3. \(\frac{{100}}{{{s^2} + 10s + 100}}\)
4. \(\frac{{102}}{{{s^2} + 5s + 100}}\)

Answer 1

Correct Answer - Option 2 : \(\frac{{102}}{{{s^2} + 10s + 100}}\)

Standard 2^nd order system is \({\rm{T}}\left( {\rm{s}} \right) = \frac{{{\rm{K\omega }}_{\rm{n}}^2}}{{{{\rm{s}}^2} + 2{\rm{\xi }}{{\rm{\omega }}_{\rm{n}}}{\rm{s}} + {\rm{\omega }}_{\rm{n}}^2}}\) 

Given \({\rm{\xi \;}} = {\rm{\;}}0.5{\rm{\;and\;}}{{\rm{\omega }}_{\rm{n}}}{\rm{\;}} = {\rm{\;}}10\)

Steady state value \({\left. {{\rm{T}}\left( {\rm{s}} \right)} \right|_{{\rm{s}} = 0}} = {\rm{K}} = 1.02\)

\(\therefore {\rm{T}}\left( {\rm{s}} \right) = \frac{{\left( {1.02} \right)\left( {100} \right)}}{{{{\rm{s}}^2} + 10{\rm{s}} + 100}} = \frac{{102}}{{{{\rm{s}}^2} + 10{\rm{s}} + 100}}\)