Root locus fundamentals – definition of a pole In the context of the root locus method, a pole of a transfer function G(s) is the value of s for which G(s) tends to which limiting value?

Introduction / Context:
Root locus analysis studies how the closed-loop poles of a feedback system move in the complex plane as a gain parameter varies. A firm grasp of the underlying open-loop poles and zeros of G(s) is essential, since they anchor and shape the locus.

Given Data / Assumptions:

• Transfer function G(s) is rational with polynomial numerator and denominator.
• Poles are roots of the denominator; zeros are roots of the numerator.
• We consider the limiting behaviour of G(s) near its singularities.

Concept / Approach:
A pole is a value of s at which the denominator of G(s) is zero and the magnitude of G(s) becomes unbounded. In other words, as s approaches a pole, G(s) → ∞. Conversely, at a zero, the numerator is zero and G(s) → 0. These definitions underpin the angle and magnitude conditions for root locus construction and Bode/Nyquist interpretations.

Step-by-Step Solution:

Verification / Alternative check:
Partial fraction expansion shows terms like A/(s − s_p), which blow up as s → s_p, confirming unbounded behaviour at poles.

Why Other Options Are Wrong:

• 0: That is the limit at a zero, not a pole.
• 1 or −1: Arbitrary finite values; not definitions of poles.

Common Pitfalls:
Confusing open-loop poles/zeros of G(s) with closed-loop poles of 1 + K G(s) H(s); the former define asymptotes and segments of the locus, but the definitions are distinct.

Final Answer:
∞