On the passivity of the quasi-static partial element equivalent circuit method

The partial element equivalent circuit (PEEC) electromagnetic method has attracted a lot of attention for its capability to give a circuit interpretation to Maxwell's equations. The PEEC equivalent circuits are usually connected with terminations such as drivers and receivers in a time-domain circuit simulator. Passivity is a fundamental property for the time-domain simulations of circuit models connected to terminations at their electrical ports. Stable, but nonpassive, models can produce unstable systems when connected to other stable, even passive, loads. The so-called quasi-static PEEC formulations leads to a descriptor state-space circuital representation of the electromagnetic phenomena. Multiple state-space representations are possible. In this paper, we study the passivity property of different quasi-static PEEC representations in detail. A novel analytical additive decomposition is proposed concerning the admittance formulation, which allows extracting the polynomial part of the transfer function that represents the behavior at infinity in the Laplace domain. This decomposition is discussed from a mathematical and a physical point of view. Such a detailed study for the passivity of quasi-static PEEC models and the novel analytical decomposition is not available in the literature. Numerical results support the theoretical analysis.