Question

Which of the following is not correct with respect to Phase Lead Compensator?

• A. Bandwidth increases
• B. High frequency gain decreases
• C. Dynamic response becomes faster
• D. Susceptible to high frequency noise

Hint:

Phase lead compensator: \(G_c(s) = K_c \frac{s+z}{s+p}\) with pole \(p\) further from origin than zero \(z\) (i.e., \(|p|>|z|\)). Or \(G_c(s) = \alpha \frac{1+sT}{1+s\alpha T}\) with \(\alpha<1\).
Adds positive phase, increases bandwidth, improves transient response (faster), increases gain margin.
Increases high-frequency gain (relative to its DC gain), which can amplify noise.

Answer 1

The Correct Option is B

Phase Lead Compensator

A phase lead compensator adds positive phase shift to the open-loop frequency response over a certain frequency range. Its typical transfer function is:

\[ G_c(s) = K_c \frac{s+z}{s+p} \quad \text{where } p > z \]

Alternatively, a normalized form is often used:

\[ G_c(s) = \alpha \frac{1 + sT}{1 + s\alpha T} \quad \text{with } \alpha < 1 \]

This means:

• Zero at \( s = -1/T \)
• Pole at \( s = -1/(\alpha T) \)
• Since \( \alpha < 1 \), \( 1/(\alpha T) > 1/T \) ⇒ pole is further left

Effects of a Phase Lead Compensator

• (a) Bandwidth increases: ✅ True. The gain crossover frequency increases, expanding the bandwidth.
• (b) High frequency gain decreases: ❌ False. The gain at high frequencies tends to be equal to or higher than the DC gain. For example: \[ G_c(s) = \alpha \frac{1 + sT}{1 + s\alpha T} \Rightarrow \lim_{s \to \infty} G_c(s) = 1 \] \[ \lim_{s \to 0} G_c(s) = \alpha < 1 \Rightarrow \text{High-frequency gain > DC gain} \] Hence, the gain at high frequencies increases or stays constant compared to DC. This statement is incorrect. 
• (c) Dynamic response becomes faster: ✅ True. Greater bandwidth generally implies improved transient characteristics, such as reduced rise and settling times.
• (d) Susceptible to high frequency noise: ✅ True. The boost in gain at higher frequencies can amplify noise, making the system more sensitive to high-frequency disturbances.

Conclusion

\[ \boxed{\text{Incorrect statement: (b) High frequency gain decreases}} \]