Exact Operational Solution of Short-Circuit Transient Currents via SON

This document develops an application of the SON nilpotent operational method to the calculation of short-circuit transient currents in equivalent R-L models. The work takes as its starting point the framework used for RLC circuits with polynomial-exponential inputs, where the differentiation operator D = d/dt is nilpotent on the space Vₙ of polynomials of degree at most n. The short-circuit current is studied as the sum of a forced component and a decaying natural component. When the equivalent source is expressed as P(t)eᵃᵗsin(ωt+θ), P(t)eᵃᵗcos(ωt+θ) or a combination of these forms, the particular solution is obtained via complexification and a finite inversion of the shifted operator D+λ, with λ=a+iω. The decaying DC component appears automatically upon imposing the initial condition, as the natural mode of the R-L circuit. We present the physical model, the main theorem, the classical sinusoidal case, the case with a polynomial envelope, numerical examples, a comparison with the traditional method, an interpretation of the worst closing angle, three-phase extensions, and a discussion of scope and limitations.