Question

A phase-lead compensator is primarily used to _______.

• A. decrease the steady-state error
• B. increase the damping and improve transient response
• C. reduce the bandwidth of the system
• D. increase the rise time

Hint:

Associate compensator functions with their primary effects: - Phase-Lead $\rightarrow$ Functions like a High-Pass Filter; improves transient response, speed, and damping (shifts root locus left). - Phase-Lag $\rightarrow$ Functions like a Low-Pass Filter; improves steady-state accuracy.

Answer 1

The Correct Option is B

Concept: A phase-lead compensator introduces a dominant zero and a pole into the loop transfer function, with the zero positioned closer to the origin in the complex $s$-plane than the pole. Its transfer function is given by: \[ G_c(s) = \frac{s + \frac{1}{\tau}}{s + \frac{1}{\alpha\tau}} \quad (0 < \alpha < 1) \] This configuration injects a positive phase shift into the system over a specified frequency band.

Step 1: Structural effects on the root locus.
Because the zero lies closer to the origin than the pole, the phase-lead network produces a dominant pulling effect that shifts the system's root locus branches to the left in the complex $s$-plane. Shifting the poles further into the left-half plane increases the absolute value of the real part of the dominant closed-loop poles ($\sigma = \zeta\omega_n$), which directly enhances the system's damping factor.

Step 2: Impact on transient performance parameters.
This leftward shift improves the transient response parameters as follow:
• Damping ratio ($\zeta$): Appreciably increased, which minimizes peak overshoot.
• System Bandwidth ($\text{BW}$): Increased, which accelerates the speed of response.
• Rise time ($t_r$) and Settling time ($t_s$): Reduced, enabling faster stabilization. Thus, a phase-lead compensator is primarily used to increase damping and improve the transient response.