GATE EC 2017 Set 1 — Question 20

A phase-lag compensator (or lag compensator) is used to improve the steady-state error of a system. It introduces a pole and a zero, with the pole being closer to the origin than the zero. This configuration provides a negative phase shift, hence 'lag'.The transfer function of a lag compensator is typically given by $G_c(s) = K \frac{s + z}{s + p}$, where $|z| > |p|$. This means the zero is further away from the origin than the pole.Let's analyze the given pole-zero plots:Option (A) shows a pole at $s=0$ (which is not part of the compensator itself but often represents the system type), a zero at a negative real value, and a pole at a negative real value further to the left than the zero. This configuration, specifically the pole-zero pair on the negative real axis, has the pole closer to the origin than the zero. This is characteristic of a phase-lag compensator. For example, if the zero is at $-z_1$ and the pole is at $-p_1$, then $p_1 < z_1$.Option (B) shows a pole at $s=0$, a pole at a negative real value, and a zero at a negative real value further to the left than the pole. This configuration, where the zero is further from the origin than the pole, is characteristic of a phase-lead compensator.Option (C) is similar to (A) but the pole-zero pair is located differently. However, the relative positions of the pole and zero for the compensator part are consistent with a lag compensator (pole closer to origin than zero).Option (D) is similar to (B) and represents a lead compensator.Comparing (A) and (C), both show a pole closer to the origin than the zero for the compensator part. However, the standard representation of a lag compensator places the pole and zero on the negative real axis, with the pole being to the right of the zero (closer to the origin). Option (A) correctly depicts this relationship for the compensator pole-zero pair (excluding the pole at origin which is likely part of the plant).Therefore, option (A) is the correct representation of a phase-lag controller's pole-zero configuration.The final answer is $\boxed{A}$