Time-Periodic Transmission Lines on Finite Intervals: A Hybrid Sturm--Liouville--Floquet Formulation

Abstract

This article presents a unified and rigorous framework for the analysis of finite-length transmission lines whose per-unit-length (PUL) parameters vary periodically in time. The spatial dependence is treated through a Sturm–Liouville (SL) modal decomposition that enforces exact terminal boundary conditions on a finite interval, while temporal periodicity is handled by a Floquet–harmonic-balance formulation based on block Toeplitz convolutions. The resulting frequency-domain description provides a structured hybrid modal–Floquet operator that generalizes the classical pole–residue expansion of the linear time invariant (LTI) case. In parallel, we derive a closed time-periodic state-space model for the truncated modal system and its Floquet-lifted counterpart for periodic steady-state analysis. The formulation incorporates the parametric terms $\dot {L}(t)$ and $\dot {C}(t)$ in a transparent way and provides a structured framework for the inclusion of nonlinear terminations through dedicated trace and injection operators. The resulting model supports transient simulation, energy accounting, and periodic steady-state analysis, while also providing a basis for future extensions to more general nonlinear configurations. The approach recovers the classical LTI theory exactly and offers a continuous-space counterpart to unit-cell periodic models of modulated structures.