Scattered-Power Substructure Characteristic Mode Formulation via Hybrid Vector Spherical Wave–MoM Method: Comparative Analysis of Scattering-Matrix- and Impedance-Matrix-Based Formulations

In this paper, we introduce an alternative scattered-power substructure characteristic mode (CM) formulation based on a hybrid method combining vector spherical wave expansions and the method of moments (MoMs). Compared with the impedance-matrix-based eigenvalue problem, the proposed formulation involves matrices of smaller size, thereby reducing the computational burden associated with the eigenvalue decomposition. By applying the proposed hybrid approach, we also establish an equivalent form of the power-operator-based generalized eigenvalue equation. To further demonstrate the necessity of the hybrid method, this article investigates the impact of passive loading on several existing substructure CM formulations. The algebraic relations verify that these formulations are equivalent when the passive loading is lossless. In lossy passive-loading scenarios, however, the scattering-matrix-based formulations do not admit corresponding impedance-matrix-based formulations. In contrast, the CM formulations developed via the hybrid method remain equivalent to those based purely on the MoM. Several numerical examples are presented to validate the proposed formulations.