The Hyperloop transportation system paradigm has gained increasing attention in the last years due to its potential advantages in technology, territory, and infrastructure. From an engineering point of view, it would lead to fast, safe, efficient transportation of passengers and cargo. The stability of the electrodynamic levitation system represents a key enabling aspect of Hyperloop. In this context, the state of the art presents numerous attempts to stabilize these systems without definitive guidelines on how to attain proper, stable behavior. Furthermore, research has provided extensive literature in the context of electrodynamic bearings, which requires proper interpretation and generalization into the translational domain. In this paper, we address the stabilization of levitation systems by reproducing the strong interaction between the electrodynamic phenomenon and the mechanical domain. A novel lumped-parameter model with a multiple-branch circuit is proposed and tuned through finite-element simulations to replicate the electrodynamic behavior. The multi-domain equations are linearized and the unstable nature of the levitation system is identified and discussed. Then, a suitable method to add damping and optimize stability is studied. Finally, the linearized model is compared with the nonlinear representation to validate the followed approach.