Overcoming eddy current solvers: on the full-wave modelling of quasi-static scenarios

Low-frequency electromagnetic compatibility and interference (EMC/EMI) circuit simulations are often based on quasi-static, eddy-current approximations of Maxwell's system and require ad-hoc solvers to be employed. The need for relying on approximate physics stems from the instability of standard full-wave Maxwell solvers when the frequency decreases. The focus of this thesis has been to work on and extend a formulation capable of overcoming this limitation and performing exact modelling of circuits with a full-wave Maxwell solver. Based on the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation, this full-wave formulation is well-conditioned, stable and accurate over a broad frequency range. This result is obtained by exploiting primal and dual quasi-Helmholtz projectors that allow for an adequate frequency rescaling of the solenoidal and quasi-irrotational components of the system. The eddy current regime is included in the range of applicability of this solver which answers to the strong industrial need for efficient modelling of eddy currents with the significant advantage that, unlike standard eddy current solvers, the stabilized formulation is capable of handling multi-scale scenarios in which incompatible approximations of the physics would need to co-exist. The resulting scheme is also compatible with multiply connected geometries, allowing modelling of complex and realistic circuits. One of the most promising applications of this technology is the study of the behaviour of an electronic equipment in terms of emission, immunity to radiation, and coupling. These electromagnetic compatibility properties are known to be of crucial importance in the design phase of electronic systems. In this work, the formulation has been validated against analytic scattering models, both in near and in far field, for exciting plane waves oscillating in a wide range of frequencies to ensure that it remains accurate and stable under the most extreme modelling conditions. After this preliminary check, we tackled the problem of modelling of the electromagnetic fields on some canonical circuital structures for which analytical solutions are known, in order to further verify the performances of the solver. To this end, a different kind of excitation has been implemented, capable of modelling the enforcement of a potential difference. We successfully managed to extract system-based parameters, such as voltages and currents, that made it possible to estimate the impedance of the structures under test, both inductive and capacitive. The results obtained are in excellent agreement with what expected from circuit theory. Finally, a new line of investigation into strategies for the numerical integration of the Green’s function dampened behaviour in highly lossy media has been opened. The promising results achieved from this analysis will allow to further broaden the range of frequencies and conductivities over which the model can be efficiently exploited and ensure that the solver can perform even on the most challenging-to-model media that can be encountered.