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An efficient analysis using the contour integral method (CIM) for the modeling of inhomogeneous substrates with circular or arbitrarily shaped dielectric inclusions in planar structures is presented. It is shown how the segmentation method is applied to obtain circuit parameters as well as field values and how different boundary conditions can be implemented. Such an analysis is especially useful for the modeling of photonic crystals power/ground layers (PCPLs) in power delivery networks (PDNs) on printed circuit boards (PCBs). For circular inclusions, an analytical solution is presented, which improves efficiency and accuracy of the proposed method. An equivalent circuit model for cylindrical inclusions is derived, in order to obtain an effective capacitance for low frequencies and an estimate for parasitic effect at higher frequencies. It is shown how circular inclusions can be compared to decoupling capacitors on a basis of impedances. The method is validated with full-wave simulations showing a reduction in simulation time of about one order of magnitude.Index Terms-Contour integral method (CIM), inhomogeneous substrate, printed circuit board (PCB), photonic crystals power/ ground layers (PCPL), segmentation method.

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