Discrete complex image MPIE analysis of coax-fed coupled vertical monopoles in grounded dielectric substrate: two formulations

Two distinct mixed-potential integral equation (MPIE) formulations are combined with the discrete complex image method (DCIM) and applied to analyse isolated and coupled coax-fed vertical monopoles embedded in a grounded substrate. Both DCIM-MPIE approaches are shown to be accurate, efficient, and well suited for the modelling of vertical probe feeds, shorting posts and plates, and vias in microstrip structures. One of the two formulations uses the scalar potential associated with a horizontal Hertzian dipole, and thus may easily be integrated with existing MPIE-based algorithms for strictly planar microstrip geometries to extend them to also accommodate vertical elements