Local Updates of Thevenin Equivalents in Linear Networks via Rank-One Perturbations (Sherman–Morrison)

We derive an exact, practical method to update Thévenin parameters (open-circuit voltage and equivalent resistance) of a linear network under a single internal branch modification (open/short/resistance change), without recomputing the full nodal solution from scratch. The change is modeled as a rank-one perturbation of the nodal admittance matrix, and the Sherman–Morrison identity yields closed-form port updates in terms of three physically interpretable scalars: local self-coupling, port–branch coupling, and state projection across the modified branch. We discuss limiting cases (open and short), include a brief note on complex admittances (phasors/Laplace), and provide a reproducible Python check.