Question

The transfer function of a first-order controller is given as: \[ G_c(s) = K \frac{(s + a)}{(s + b)} \] where \( K, a, b \) are positive real numbers. The condition for this controller to act as a phase lag compensator is:

• A. \( a<b \)
• B. \( a>b \)
• C. \( K<ab \)
• D. \( K>ab \)

Hint:

For a compensator to introduce phase lag, the pole must be closer to the origin than the zero, ensuring \( a<b \).

Answer 1

The Correct Option is A

Step 1: A phase lag compensator is used to improve the steady-state accuracy of a control system while reducing bandwidth and increasing stability margins. 
Step 2: The general form of a phase lag compensator is: \[ G_c(s) = K \frac{(s + a)}{(s + b)} \] where: - \( a \) represents the zero of the compensator. - \( b \) represents the pole of the compensator. 
Step 3: Phase lag compensators are characterized by:
- A pole (\( b \)) closer to the origin than the zero (\( a \)).
- This ensures that at lower frequencies, the compensator reduces the phase angle, introducing a negative phase shift. 
Step 4: The condition for a phase lag compensator is: \[ a<b \] which ensures that the pole is dominant and the system experiences phase lag.