Question

Find gain of the following open loop system having phase cross-over frequency of 1.732 rad/sec. \[ G(s)H(s) = \frac{1}{s(1+s)} \]

• A. \( \sqrt{1/8} \)
• B. \( 3/\sqrt{10} \)
• C. \( 1/4 \)
• D. \( 8 \)

Hint:

At phase crossover frequency, use magnitude condition \( |G(j\omega)H(j\omega)| = 1 \) to find gain.

Answer 1

The Correct Option is C

At the phase crossover frequency \( \omega = 1.732 \), the phase of \( G(j\omega)H(j\omega) \) is -180°. Calculate magnitude at this frequency and equate to 1 for sustained oscillation condition. \[ |G(j\omega)H(j\omega)| = \left|\frac{1}{j\omega (1 + j\omega)}\right| = \frac{1}{\omega \sqrt{1 + \omega^2}} = \frac{1}{1.732 \cdot \sqrt{1 + 3}} = \frac{1}{1.732 \cdot 2} \approx \frac{1}{3.464} \Rightarrow K = \frac{1}{3.464} = 0.288 \approx \frac{1}{4} \]