Finite Element Implementation of the Generalized-Lorenz Gauged A- <tex-math notation='LaTeX'>$\Phi $ </tex-math> Formulation for Low-Frequency Circuit Modeling

The A- $\Phi $ formulation with generalized-Lorenz gauge is free of catastrophic breakdown in low-frequency regime. In the formulation, A and $\Phi $ are completely separated and Maxwell’s equations are reduced into two independent equations pertinent to A and $\Phi $ . This, however, leads to more complicated equations in contrast to the traditional E formulation. The numerical dicretization of the equations is challenging, especially for the equation pertinent to A. By virtue of the differential forms theory and Whitney elements, the direct action of divergence operator on A is bypassed. Thus, the equations can be discretized compatibly using regular finite element method. The condition of the resultant matrix system is much better than that of the E formulation as frequency becomes low, and even approaches to zero. The generalized-Lorenz gauged A- $\Phi $ formulation is verified to be accurate and efficient for low-frequency circuit problems.