Which of the following is the correct pair of natural angular frequency ωn and damping ratio ζ for the second-order system with the transfer function G(s) = 25/(s² + 4s + 25)?

Answer: A. ωn = 5 rad/s, ζ = 0.4

Comparing with the standard form ωn²/(s² + 2ζωn s + ωn²), ωn² = 25 gives ωn = 5 rad/s. The coefficient of s is 2ζωn = 4, so ζ = 4/(2 × 5) = 0.4. The poles are at s = −2 ± j4.58; the real part −ζωn = −2 gives the rate of decay and the imaginary part ωn√(1 − ζ²) ≈ 4.58 rad/s the angular frequency of oscillation.

• A. Correct. ωn comes from the constant term and ζ from the coefficient of s.
• B. This only divides the coefficient 4 of s by ωn and forgets to divide by 2.
• C. The constant term 25 is ωn², not ωn.