# What is the value of ωn in the given transfer function? $G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}$

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What is the value of ωn in the given transfer function?

$G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}$

1. 72
2. 4.2
3. 36
4. 6

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Correct Answer - Option 4 : 6

Concept:

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

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

ζ is the damping ratio

ωn is the natural frequency

Characteristic equation: ${s^2} + 2\zeta {\omega _n} + \omega _n^2$

Roots of the characteristic equation are: $- \zeta {\omega _n} + j{\omega _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{\omega _d}$

α is the damping factor

Calculation:

Transfer function $G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}$

By comparing the above transfer function with the standard second-order system,

$\omega _n^2 = 36 \Rightarrow {\omega _n} = 6$