Finite element implementation of the generalized-Lorenz gauged A-Φ formulation for low-frequency circuit modeling

The A-Φ formulation with generalized Lorenz gauge is free of catastrophic breakdown at low frequencies. In the formulation, A and Φ are completely separated and Maxwell's equations are reduced into two independent equations pertinent to A and Φ, respectively. This, however, leads to more complicated equations in contrast to the traditional A-Φ formulation, which makes the numerical representation of the physical quantities challenging, especially for A. By virtue of the differential forms theory and Whitney elements, both A and Φ are appropriately represented. The condition of the resultant matrix system is wellcontrolled as frequency becomes low, even approaches to 0. The generalized-Lorenz gauged A-Φ formulation is applied to model low-frequency circuits at μm lengthscale.