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GATE 2026 CH – Question 50

Instrumentation and Process Control · Controller modes, control valves and tuning · 2 marks · Multiple choice

Consider three identical non-interacting first order processes in series, each having unit gain and a time constant of 2 min. Tuning of proportional controller using closed loop Ziegler-Nichols technique is considered. Which one of the following is the ultimate period of sustained cycling (in min per cycle)?

  1. $4\sqrt{3}\pi$
  2. $\frac{\sqrt{3}\pi}{4}$
  3. $\frac{4\pi}{\sqrt{3}}$
  4. $\frac{\pi}{4\sqrt{3}}$

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Correct answer: (C) $\frac{4\pi}{\sqrt{3}}$

Explanation

The open loop transfer function is $\frac{K_c}{(2s + 1)^3}$. At the ultimate frequency the phase lag is $180°$, so $3\tan^{-1}(2\omega_u) = 180°$, which gives $\tan^{-1}(2\omega_u) = 60°$, that is $2\omega_u = \sqrt{3}$ and $\omega_u = \frac{\sqrt{3}}{2}$ rad/min. The ultimate period is $P_u = \frac{2\pi}{\omega_u} = \frac{4\pi}{\sqrt{3}} \approx 7.26$ min.