Question

The characteristic equation of a closed-loop system using a proportional controller with gain \(K_C\) is \( 12s^3 + 19s^2 + 8s + 1 + K_C = 0 \). At the onset of instability, the value of \( K_C \) is —————.

• A. \( \frac{35}{3} \)
• B. 10
• C. \( \frac{25}{3} \)
• D. \( \frac{20}{3} \)

Hint:

Routh–Hurwitz criterion is a powerful tool for stability analysis of linear systems.

Answer 1

The Correct Option is C

Routh–Hurwitz stability criterion is applied to find the critical gain at the onset of instability. 
Conclusion: The correct answer is option (C).