Question

For a unity feedback control with \(G\left( s \right) = \frac{9}{{s\left( {s + 3} \right)}}\), the damping ratio is
1. 0.5
2. 1
3. 0.707
4. 0.33

Answer 1

Correct Answer - Option 1 : 0.5

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:

Closed-loop transfer function,

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

\(= \frac{9}{{{s^2} + 3s + 9}}\)

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

ω2n = 9 ⇒ ωn = 3

⇒ 2ξ ωn = 3 ⇒ ξ = 0.5