Surface impedance formulation for electric field integral equation in magneto-quasistatic and full-wave boundary element models of interconnects
AlQedra, Mohammed A.I.
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Today’s high-speed interconnects at the chip, package, and board levels of integration can be rigorously modeled with the boundary element method based on the surface discretization of the electric field integral equation (EFIE). The accuracy of such models critically depends on the surface impedance model, which has to accurately map the behavior of the electromagnetic field inside the wire volumes to their surfaces. This thesis proposes a surface impedance model, which casts the accurate but computationally intensive volumetric EFIE formulation to the boundary element framework. This is accomplished via approximating the volumetric current density as a product of the known exponential factor corresponding to the skin-effect behavior of the field inside the wires and the unknown surface current density on the conductor’s boundary. The reduction of the volumetric EFIE to its surface counterpart results in a physically consistent surface impedance model allowing to achieve the volumetric EFIE accuracy within the boundary element formulation. The method is initially introduced for lossy 2D interconnects and later generalized to 3D interconnects under magneto-quasistatic approximation. Finally, this work is extended to the Rao-Wilton-Glisson (RWG) method of moments (MoM) solution of the full-wave EFIE. The alternative models exhibit various limitations. For example, in the double-plane model the planar interconnect structure is replaced by two infinitely thin metal sheets at its top and bottom surfaces. This model succeeds for several practical scenarios where the conductor width is sufficiently larger than its thickness, or when the operating frequency is sufficiently low for the current distribution across the conductor cross section to be assumed uniform. The alternative”multi-sheet model” represents the interconnect by a number of infinitely thin metal sheets, which uniformly span its cross section such that the spacing between each two consecutive sheets is small compared to skin-depth. The model succeeds in accurately extracting conductor loss, however, it may require a large number of sheets, which makes the number of unknowns in MoM discretization of the same order as the number of unknowns in volumetric models.