Electromagnetic Repulsion Systems: Exact Co-Energy Force Law, Stability Closure, Identification Protocol, and Numerical Verification

This manuscript provides a complete mathematical and engineering closure for electromagnetic repulsion systems based on magnetic co-energy. Rather than treating repulsion as a primitive force, the framework models it strictly as a controlled energy-gradient device parameterized by the mutual inductance kernel of two electromagnetic bodies. The paper derives the exact quasi-static force law directly from both field energy and co-energy principles, proving that the actuation law is inherently quadratic in current. Furthermore, it demonstrates structurally that unconstrained passive static ferromagnetic repulsion cannot furnish generic asymptotic stability, mandating the use of constrained motion and nonlinear feedback control. Alongside the theoretical derivations, the manuscript provides a complete engineering blueprint. It defines the necessary nonlinear control inversion laws, establishes the coupled electrical and thermal dynamics to define a strict safety envelope, and outlines a comprehensive experimental protocol for system identification. A numerical verification pathway using Biot-Savart field synthesis is also provided to bypass dipole approximations and extract geometry-faithful force curves. This work serves as a foundational blueprint for designing, controlling, and validating active electromagnetic separation and repulsion systems.