Concept:
The transfer function of the standard secondorder 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 undamped natural frequency
Characteristic equation: \({s^2} + 2\zeta {\omega _n} + \omega _n^2 = 0\)
The roots of the characteristic equation are:
\( \zeta {\omega _n} \pm j{\omega _n}\sqrt {1  {\zeta ^2}} =  \alpha \pm j{\omega _d}\)
α is the damping factor
Additional Information
System 
Damping ratio 
Roots of the Characteristic equine. 
Root in the ‘S’ plane 
Undamped 
ξ = 0 
ξ = 0 Imaginary s = ±jωn 

Underdamped (Practical system) 
0 ≤ ξ ≤ 1 
\( \xi {\omega _n} \pm j{\omega _n}\sqrt {1  {\xi ^2}} \) Complex Conjugate 

Critically damped 
ξ = 1 
ωn Real and equal 

Overdamped 
ξ > 1 
\( \xi {\omega _n} \mp j{\omega _n}\sqrt {{\xi ^2}  1} \) Real and unequal 

Important Points: