Question

A step change of magnitude is introduced to a system having the transfer function \[ G(s) = \frac{2}{s^2 + 2s + 4} \] The percent overshoot is:

• A. 50
• B. 33.6
• C. 16.3
• D. 0

Hint:

Percent overshoot in a second-order system depends on damping ratio \( \zeta \).

Answer 1

The Correct Option is C

Step 1: Identify damping ratio.
Standard 2nd-order denominator: \( s^2 + 2 \zeta \omega_n s + \omega_n^2 \). Here, \( \omega_n = 2 \), \( 2 \zeta \omega_n = 2 \implies \zeta = 0.5 \).

Step 2: Formula for percent overshoot.
\[ PO = e^{-\zeta \pi / \sqrt{1 - \zeta^2}} \times 100% \] Substituting \( \zeta = 0.5 \): \[ PO = e^{-0.5 \pi / \sqrt{0.75}} \times 100 \approx 16.3%. \]

Final Answer: \[ \boxed{\text{C) 16.3}} \]