Question

If the characteristic equation of a closed-loop system is $s^2 + 2s + 2 = 0$, then the system is:

• A. overdamped
• B. critically damped
• C. underdamped
• D. un-damped

Hint:

For second-order systems, if $0<\zeta<1$, the system is underdamped. If $\zeta = 1$, it's critically damped. If $\zeta>1$, it's overdamped.

Answer 1

The Correct Option is C

The characteristic equation is given as: $s^2 + 2s + 2 = 0$
Compare this with the standard second-order form: $s^2 + 2\zeta \omega_n s + \omega_n^2 = 0$
Here, $2\zeta \omega_n = 2$ and $\omega_n^2 = 2$
Solving for $\omega_n$: $\omega_n = \sqrt{2}$
Now, $2\zeta \cdot \sqrt{2} = 2 ⇒ \zeta = \dfrac{1}{\sqrt{2}} \approx 0.707$
Since $0<\zeta<1$, the system is classified as underdamped.