Augmentation of Volume-Surface Integral Equations for Low-Frequency Modeling of Interconnect Structures

Interconnect structures include both conducting and penetrable media and electromagnetic (EM) modeling and analysis for them require the use of volume-surface integral equations (VSIEs) in the integral equation approach. The VSIEs include the $\mathcal{L}$ operator in both surface integral equations (SIEs) and volume integral equations (VIEs), and it will produce a low-frequency-breakdown (LFB) problem for low-frequency modeling. Although the augmentations of SIEs for both conducting and dielectric media have been proposed for remedying the LFB problem, the augmentation for the VSIEs has not been addressed yet. This paper aims to this problem and proposes an augmented VSIEs (AVSIEs) by introducing the magnetic charge density and continuity equation of magnetic current density. A numerical example is provided to demonstrate the approach and its effectiveness has been validated.