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 

Key Points