The generalized standard form of transfer function of the Second-order System is
}}{{R(s)}} - \frac{{\omega _n-2}}{{{s-2} - 2\zeta {\omega _n}s - \omega _n-2}}.jpg "\frac{{C(s)}}{{R(s)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}")
where 
=damping factor (or damping ratio)
and 
=undamped natural frequency.
The characteristic equation of this system is
The roots of this characteristic equation are
(s - {s_2}).jpg "{s^2} + 2\zeta {\omega _n}s + \omega _n^2 = (s - {s_1})(s - {s_2})")
For ,
where 
is called the damped natural frequency.
For the unit step input -\frac{1}{s}.jpg "R(s)=\frac{1}{s}")
, the output response is given by
![C(s) = \frac{{\omega _n^2}}{{s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]\left[ {s + \zeta {\omega _n} + j{\omega _n}\sqrt {(1 -{\zeta ^2})} } \right]}}](../../images/_n} - j{\omega _n}\sqrt {(1 - {\zeta -2})} } \right]\left[ {s - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 -{\zeta -2})} } \right]}}_jpg_0jpvhiqpeg2mcsbbi0eqyi_cgi.jpg "C(s) = \frac{{\omega _n^2}}{{s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]\left[ {s + \zeta {\omega _n} + j{\omega _n}\sqrt {(1 -{\zeta ^2})} } \right]}}")
The Laplace Inverse of above equation is obatained by the method of residues as
![c(t) = {\left. {{{\omega _n^2} \over {{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}} \right|_{s = 0}} + 2{\rm{Re}}\left[ {{{\left. {{{\omega _n^2} \over {s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]}}} \right|}_{s = - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} }}{e^{\left[ { - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]t}}} \right]](../../images/j{\omega _n}\sqrt {(1 - {\zeta -2})} }}{e-{\left[ { - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta -2})} } \right]t}}} \right]_jpg_3a26gwqijgtgg2chh3pirc_cgi.jpg "c(t) = {\left. {{{\omega _n^2} \over {{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}} \right|_{s = 0}} + 2{\rm{Re}}\left[ {{{\left. {{{\omega _n^2} \over {s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]}}} \right|}_{s = - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} }}{e^{\left[ { - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]t}}} \right]")
![= 1 - \frac{{{e^{ - \zeta {\omega _n}{t_r}}}}}{{\sqrt {(1 - {\zeta ^2}} }}\sin \left[ {{\omega _n}\sqrt {(1 - {\zeta ^2})} {t_r} + {{\tan }^{ - 1}}\frac{{\sqrt{(1 - {\zeta ^2})} }}{\zeta }} \right]](../../images/a -2}} }}\sin \left[ {{\omega _n}\sqrt {(1 - {\zeta -2})} {t_r} - {{\tan }-{ - 1}}\frac{{\sqrt{(1 - {\zeta -2})} }}{\zeta }} \right]_jpg_zm0q5e2jsn8c2wucmp4h7r_cgi.jpg "= 1 - \frac{{{e^{ - \zeta {\omega _n}{t_r}}}}}{{\sqrt {(1 - {\zeta ^2}} }}\sin \left[ {{\omega _n}\sqrt {(1 - {\zeta ^2})} {t_r} + {{\tan }^{ - 1}}\frac{{\sqrt{(1 - {\zeta ^2})} }}{\zeta }} \right]")