Correct Answer - Option 3 : 24.5 and 1.43
Concept:
The transfer function of the standard second-order system is:
\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2\zeta {ω _n}s + ω _n^2}}\)
ζ is the damping ratio
ωn is the undamped natural frequency
The characteristic equation is given as:
\({s^2} + 2\zeta {ω _n} + ω _n^2 = 0\)
Roots of the characteristic equation are:
\(- \zeta {ω _n} + j{ω _n}\sqrt {1 - {\zeta ^2}} = - \alpha \pm j{ω _d}\)
α is the damping factor
Calculation:
Given:
R(t) = u(t)
c(t) = 1 + 0.2 e-60t – 1.2 e-10t.
Taking the Laplace transform of c(t),
\(C(s)=\frac{1}{s} + \frac{0.2}{s+60}-\frac{1.2}{s+10}\)
\(R(s)=\frac{1}{s}\)
The transfer function of the standard second-order system is:
\( \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{(s+60)(s+10)+0.2s(s+10)-1.2s(s+60)}{(s+60)(s+10)}\)
The characteristic equation q(s) is:
q(s) = (s+60)(s+10) = 0
q(s) = s2 + 70s + 600 =0
Compare it with a standard characteristic equation is,
ωn2 = 600
ωn = 24.5 rad/sec
2ξωn = 70
ξ = 1.43
