Question

For a linear stable second order system, if the unit step response is such that the peak time is twice the rise time, the system is \(\underline{\hspace{2cm}}\).

• A. underdamped
• B. undamped
• C. overdamped
• D. critically damped

Hint:

In second-order systems, when peak time is twice the rise time, the system is undamped. This is a useful relation for quick analysis of step responses.

Answer 1

The Correct Option is B

For a linear second-order system, the relationship between rise time and peak time is governed by the damping ratio \( \zeta \) of the system. The rise time and peak time are related by the following formulas: \[ T_r = \frac{1.8}{\omega_n \sqrt{1 - \zeta^2}} \text{(rise time)} \] \[ T_p = \frac{\pi}{\omega_n \sqrt{1 - \zeta^2}} \text{(peak time)} \] Where:
- \( \omega_n \) is the natural frequency.
- \( \zeta \) is the damping ratio.
If the peak time is twice the rise time, this implies: \[ T_p = 2 T_r. \] Using the above formulas and solving for the damping ratio, we find that this condition holds true for an undamped system. Therefore, the system is undamped. Hence, the correct answer is (B).