Closed-loop relationship for a standard feedback system In a unity-feedback style block diagram where G is the forward (open-loop) transfer function and H is the feedback path transfer function, what is the output/input relation (closed-loop transfer function) for negative feedback?

Introduction / Context: Knowing how to combine blocks into a closed-loop transfer function is essential for predicting stability, gain, and transient response. In the standard single-loop configuration with negative feedback, the output/input relation is a compact algebraic expression involving the forward path G and the feedback path H.

Given Data / Assumptions:

• Single feedback loop, linear time-invariant dynamics.
• Negative feedback summing junction.
• Forward path transfer function: G; feedback path: H.

Concept / Approach: At the summing point, the error is e = r − H y. The forward block gives y = G e. Eliminating e and rearranging gives y = G (r − H y) ⇒ y + G H y = G r ⇒ y/r = G / (1 + G H). This is the canonical result for negative feedback.

Step-by-Step Solution:

Verification / Alternative check: The open-loop transfer L = G H appears in the denominator 1 + L, consistent with Nyquist/Bode formulations for negative feedback.

Why Other Options Are Wrong:

Common Pitfalls: Forgetting the sign convention: negative feedback always yields 1 + G H in the denominator; positive feedback gives 1 − G H.

Final Answer: Y/R = G / (1 + G H)