Question

The unit step response of the transfer function \( G(s) = \frac{1}{s^2 + 2s + 3} \) has:

• A. a damped oscillatory characteristic
• B. is overdamped
• C. has a non-zero slope at the origin
• D. is unstable

Hint:

For second-order systems with a positive damping ratio, the step response exhibits damped oscillatory characteristics.

Answer 1

The Correct Option is A

Step 1: Analyzing the transfer function.
The transfer function \( G(s) = \frac{1}{s^2 + 2s + 3} \) represents a second-order system. The step response of such a system depends on the nature of its poles.

Step 2: Explanation of options.
- (A) The characteristic equation indicates underdamped behavior, leading to a damped oscillatory response. - (B) Overdamped systems do not exhibit oscillatory behavior. - (C) The response does not have a non-zero slope at the origin. - (D) The system is stable because its poles have negative real parts.

Final Answer: \[ \boxed{\text{A) a damped oscillatory characteristic}} \]