Root locus basics for closed-loop design For a unity-feedback system, the number of branches (loci) in the root locus equals the number of open-loop poles. If the open-loop transfer function has four finite poles, how many root-locus branches will be plotted as the gain K varies from 0 to ∞?

Introduction / Context:
The root locus is a graphical method that shows how the closed-loop poles of a feedback system move in the complex plane as a real scalar gain K varies. It is a cornerstone tool for controller design and stability assessment in classical control theory.

Given Data / Assumptions:

• Unity-feedback configuration is implied.
• Open-loop transfer function G(s)H(s) possesses four finite poles.
• Zeros may be finite or at infinity; asymptotes accommodate any excess poles.

Concept / Approach:
A fundamental rule of root locus construction states: the number of branches (also called loci) equals the number of open-loop poles, because each closed-loop pole (for any K) originates from a distinct open-loop pole at K = 0 and traces a continuous path as K increases. If there are fewer finite zeros than poles, some branches terminate at zeros at infinity along asymptotes determined by standard angle and centroid formulas.

Step-by-Step Solution:

Verification / Alternative check:
Any textbook example with n poles and m zeros shows n branches; when m < n, (n − m) branches head to infinity, confirming the count.

Why Other Options Are Wrong:

• 1, 2, 3: Do not match the pole count; each open-loop pole must seed a branch.

Common Pitfalls:
Confusing the number of asymptotes (n − m) with the number of branches (n). Asymptotes are directions for branches that head to infinity; they do not reduce branch count.

Final Answer:
4