Characteristic equation location: the characteristic equation used for stability analysis is the denominator of which transfer function?

Introduction / Context:In feedback systems, the characteristic equation defines the closed-loop poles and thus stability. Knowing where it appears in the transfer function structure is foundational for applying Routh, root locus, or frequency-domain techniques.

Given Data / Assumptions:

• Unity or non-unity feedback; the algebraic form changes but the idea is the same.
• Characteristic equation set by 1 + L(s) = 0, where L(s) is the loop transfer function.

Concept / Approach:The closed-loop transfer function T(s) = G(s) / [1 + L(s)] has its denominator equal to the characteristic equation. The roots of that denominator are the closed-loop poles; stability requires all to lie in the left half-plane. Therefore, the characteristic equation belongs to the closed-loop denominator.

Step-by-Step Solution:Write T(s) = G(s) / [1 + L(s)].Identify the denominator: 1 + L(s) → characteristic equation.Conclude: denominator of the closed-loop transfer function.

Verification / Alternative check:Standard derivations for root-locus plots begin from 1 + K G(s) H(s) = 0, explicitly the closed-loop characteristic equation.

Why Other Options Are Wrong:Open-loop: Does not contain the stability-defining denominator.Both/neither: Contradicts the mathematical structure.Plant-only numerator/denominator does not account for feedback closure.

Common Pitfalls:Confusing numerator zeros with poles; using the open-loop denominator for Routh instead of the closed-loop characteristic polynomial.

Final Answer:Closed-loop transfer function