Question

For the standard second order system, with two poles lying at 60° , damping ratio is _______.

• A. 0
• B. 1
• C. 0.707
• D. 0.5

Hint:

In second-order systems, use geometry of poles in s-plane to find damping ratio.

Answer 1

The Correct Option is D

In a standard second-order system: \[ \text{Poles: } s = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2} \] If poles lie at an angle \( \theta = 60^\circ \) from the negative real axis: \[ \tan(\theta) = \frac{\sqrt{1 - \zeta^2}}{\zeta} \Rightarrow \tan(60^\circ) = \sqrt{3} \] \[ \frac{\sqrt{1 - \zeta^2}}{\zeta} = \sqrt{3} \Rightarrow 1 - \zeta^2 = 3\zeta^2 \Rightarrow 4\zeta^2 = 1 \Rightarrow \zeta = 0.5 \]