Question

The transfer function of a phase-lead compensator is given by: $\frac{1 + 3Ts}{1 + Ts}$, $T>0$. The maximum phase shift provided is _______.

• A. $90^\circ$
• B. $60^\circ$
• C. $45^\circ$
• D. $30^\circ$

Hint:

Use: $\phi_{max} = \sin^{-1} \left( \frac{\alpha - 1}{\alpha + 1} \right)$ where $\alpha = \frac{\text{zero}}{\text{pole}}$

Answer 1

The Correct Option is D

Step 1: General form of phase-lead compensator:
\[ G(s) = \frac{1 + \alpha T s}{1 + T s}, \alpha>1 \] Given: $\alpha = 3$
Step 2: Formula for maximum phase shift:
\[ \phi_{max} = \sin^{-1} \left( \frac{\alpha - 1}{\alpha + 1} \right) \Rightarrow \sin^{-1} \left( \frac{3 - 1}{3 + 1} \right) = \sin^{-1} \left( \frac{2}{4} \right) = \sin^{-1}(0.5) = 30^\circ \] Conclusion: $\boxed{30^\circ}$