Question

Given the unity feedback system with \( G(s) = \frac{K}{s(s+6)} \), the value of \( K \) for damping ratio of 0.75 is ________.

• A. 1
• B. 4
• C. 16
• D. 64

Hint:

Match coefficients with standard 2nd-order system \( s^2 + 2\zeta\omega_n s + \omega_n^2 \) to get damping ratio and natural frequency.

Answer 1

The Correct Option is B

For unity feedback, closed-loop transfer function is: \[ T(s) = \frac{K}{s^2 + 6s + K} \] Compare with standard second-order form: \[ T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \] Matching coefficients: \[ 2\zeta\omega_n = 6 \text{and} \omega_n^2 = K \] From first equation: \[ \omega_n = \frac{6}{2 \zeta} = \frac{6}{2 \cdot 0.75} = 4 \Rightarrow K = \omega_n^2 = 4^2 = 16 \] Wait! This contradicts the answer key. Let’s double-check.
Given: \[ 2\zeta\omega_n = 6, \zeta = 0.75 \Rightarrow \omega_n = \frac{6}{1.5} = 4 \Rightarrow K = \omega_n^2 = 16 \] Actually, the correct value should be: \[ K = \omega_n^2 = 16 \] % Correction Corrected Answer: (3) 16