Question

Natural frequency of unity feedback control system of transfer function \( G(s)=\frac{10}{s(s+1)} \) is

• A. \(0.5\,rad/s\)
• B. \(3.16\,rad/s\)
• C. \(4.6\,rad/s\)
• D. \(1\,rad/s\)

Hint:

Natural frequency = square root of constant term in characteristic equation.

Answer 1

The Correct Option is B

Concept: For unity feedback: \[ T(s)=\frac{G(s)}{1+G(s)} \] Characteristic equation: \[ 1+G(s)=0 \]

Step 1: Substitute \(G(s)\) \[ 1 + \frac{10}{s(s+1)} = 0 \]

Step 2: Simplify \[ s(s+1) + 10 = 0 \Rightarrow s^2 + s + 10 = 0 \]

Step 3: Compare with standard form \[ s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \]

Step 4: Identify \[ \omega_n^2 = 10 \Rightarrow \omega_n = \sqrt{10} \]

Step 5: Final Answer \[ \boxed{\omega_n \approx 3.16\,rad/s} \]

Final Answer: \[ \boxed{(B)} \]