Three‐dimensional finite element modeling of inductive and capacitive effects in micro‐coils

The paper seeks to develop dual 3D finite element (FE) formulations for modeling both inductive and capacitive effects in massive inductors, in particular micro‐coils. The paper aims to build circuit relations relating the voltages and the currents in such inductors to be used in circuit coupling.

A circuit relation involving a unique voltage and complementary inductive and capacitive currents is defined for each inductor. The inductive circuit relation is first classically obtained by a FE magnetodynamic model. Then, the capacitive relation is obtained through a FE electric model, using sources evaluated from the first model. The conformity is defined on one hand for the magnetic flux density and the electric field, and on the other hand for the magnetic field and the electric flux density. Mixed FE, i.e. nodal, edge and face elements, are used to satisfy each chosen conformity level for the unknown fields and to naturally define the involved global quantities, i.e. the voltages, currents and charges.

This contribution points out the interest of satisfying conformity properties for the coupled magnetic and electric problems. An accurate computation of these effects is obtained in the critical frequency range of their strong interaction. In addition, the complementarity of dual solutions gives the possibility to estimate the discretisation error.

The mathematical and discretisation tools for any wished conformity level are unified for naturally coupling magnetic and electric problems. The global quantities basis functions involved in the FE circuit relations benefit from a significant support reduction, which facilitates their evaluation and gives them direct physical interpretations.