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

Using the Routh array on the expanded characteristic polynomial $(1+\tau s)^3+K_c=0$, the s^1 row vanishing gives $K_{cu}=8$, and the auxiliary equation $3\tau^2s^2+9=0$ gives $\omega_u=\sqrt3/\tau$. With $\tau=2$ min, $P_u=4\pi/\sqrt3$ min/cycle, matching option (C).