Question:
The value of ultimate period of oscillation \(P_u\) is 3 minutes and that of the ultimate controller gain \(K_u\) is 2. Select the correct set of tuning parameters (controller gain \(K_c\), the derivative time constant \(T_D\) in minutes and the integral time constant \(T_I\) in minutes) for a PID controller using Ziegler-Nichols controller settings
The value of ultimate period of oscillation \(P_u\) is 3 minutes and that of the ultimate controller gain \(K_u\) is 2. Select the correct set of tuning parameters (controller gain \(K_c\), the derivative time constant \(T_D\) in minutes and the integral time constant \(T_I\) in minutes) for a PID controller using Ziegler-Nichols controller settings
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Ziegler-Nichols Closed Loop Tuning Reference Matrix:
tabular|c|c|c|c|
Controller Type & \(K_c\) & \(T_I\) & \(T_D\)
P & \(0.5 K_u\) & — & —
PI & \(0.45 K_u\) & \(P_u / 1.2\) & —
PID & \(0.6 K_u\) & \(P_u / 2\) & \(P_u / 8\)
tabular
P & \(0.5 K_u\) & — & —
PI & \(0.45 K_u\) & \(P_u / 1.2\) & —
PID & \(0.6 K_u\) & \(P_u / 2\) & \(P_u / 8\)
tabular
Updated On: Jul 4, 2026
- \(K_c = 1.1, T_I = 2.1, T_D = 1.31\)
- \(K_c = 1.2, T_I = 1.5, T_D = 0.375\)
- \(K_c = 1.5, T_I = 1.8, T_D = 0.51\)
- \(K_c = 0.9, T_I = 1.2, T_D = 0.25\)
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The Correct Option is B
Solution and Explanation
Concept:
The Ziegler-Nichols (Z-N) closed-loop tuning technique is an empirical method based on finding the ultimate gain \(K_u\) and ultimate period \(P_u\) where the system exhibits sustained oscillations under pure proportional control. Fixed mathematical design ratios are then used to calculate P, PI, or PID controller parameters.
Step 1: Recalling the classic Z-N tuning formulas for a PID controller.
For a full Proportional-Integral-Derivative (PID) loop architecture, the standard Ziegler-Nichols rules specify the following tuning criteria:
• Controller Gain: \( K_c = 0.6 \cdot K_u \)
• Integral Time Constant: \( T_I = \frac{P_u}{2} = 0.5 \cdot P_u \)
• Derivative Time Constant: \( T_D = \frac{P_u}{8} = 0.125 \cdot P_u \)
Step 2: Calculating parameter values using the given data.
The problem provides the following operating parameters: \[ K_u = 2, \quad P_u = 3 \text{ minutes} \] Let's compute each tuning constant individually:
• Calculate \(K_c\): \[ K_c = 0.6 \times K_u = 0.6 \times 2 = 1.2 \]
• Calculate \(T_I\): \[ T_I = \frac{P_u}{2} = \frac{3}{2} = 1.5 \text{ minutes} \]
• Calculate \(T_D\): \[ T_D = \frac{P_u}{8} = \frac{3}{8} = 0.375 \text{ minutes} \]
Reviewing the available choices, the calculated array matches option (B) exactly.
Step 1: Recalling the classic Z-N tuning formulas for a PID controller.
For a full Proportional-Integral-Derivative (PID) loop architecture, the standard Ziegler-Nichols rules specify the following tuning criteria:
• Controller Gain: \( K_c = 0.6 \cdot K_u \)
• Integral Time Constant: \( T_I = \frac{P_u}{2} = 0.5 \cdot P_u \)
• Derivative Time Constant: \( T_D = \frac{P_u}{8} = 0.125 \cdot P_u \)
Step 2: Calculating parameter values using the given data.
The problem provides the following operating parameters: \[ K_u = 2, \quad P_u = 3 \text{ minutes} \] Let's compute each tuning constant individually:
• Calculate \(K_c\): \[ K_c = 0.6 \times K_u = 0.6 \times 2 = 1.2 \]
• Calculate \(T_I\): \[ T_I = \frac{P_u}{2} = \frac{3}{2} = 1.5 \text{ minutes} \]
• Calculate \(T_D\): \[ T_D = \frac{P_u}{8} = \frac{3}{8} = 0.375 \text{ minutes} \]
Reviewing the available choices, the calculated array matches option (B) exactly.
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