Question

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

• A. \(4\sqrt{3}\,\pi\)
• B. \(\dfrac{\sqrt{3}\,\pi}{4}\)
• C. \(\dfrac{4\pi}{\sqrt{3}}\)
• D. \(\dfrac{\pi}{4\sqrt{3}}\)

Hint:

At the ultimate frequency the total phase lag of the three lags equals \(-180^\circ\); each lag contributes \(-60^\circ\), giving \(\omega_u\tau = \tan 60^\circ = \sqrt{3}\).

Answer 1

The Correct Option is C

Step 1: Open loop transfer function.

\[ G(s) = \frac{1}{(1+\tau s)^3} \]

Step 2: Sustained oscillation condition.

Total phase lag = -180 degrees at the ultimate frequency: \[ 3\arctan(\omega_u\tau) = 180^\circ \Rightarrow \arctan(\omega_u\tau)=60^\circ \]

Step 3: Solve for ultimate frequency.

\[ \omega_u\tau = \sqrt{3} \Rightarrow \omega_u = \frac{\sqrt{3}}{2}\ \mathrm{rad/min} \]

Step 4: Convert to period.

\[ P_u = \frac{2\pi}{\omega_u} = \frac{4\pi}{\sqrt{3}}\ \mathrm{min/cycle} \]

\[ \boxed{P_u = \frac{4\pi}{\sqrt{3}}\ \mathrm{min/cycle}} \]