Discussion
For the system shown in the following figures, the damping ratio of closed loop poles is :
Answer: C
The damping ratio is 0.5.
With unity negative feedback, the closed loop characteristic equation is \(1 + G(s) = 0\):
\[ \begin{aligned} s(s + 2) + 4 &= 0 \\ s^{2} + 2s + 4 &= 0 \end{aligned} \]
Comparing with the standard form \(s^{2} + 2\zeta\omega_{n}s + \omega_{n}^{2} = 0\):
\[ \begin{aligned} \omega_{n}^{2} &= 4, \quad \omega_{n} = 2 \\ 2\zeta\omega_{n} &= 2, \quad \zeta = 0.5 \end{aligned} \]
The system is underdamped. A value of 1 would mean critical damping, 1.5 overdamping, and 0.25 comes from taking \(\omega_{n}\) as 4, the value of \(\omega_{n}^{2}\), instead of its square root.
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