NUMERICALLY STABLE GREEN-FUNCTION FOR MODELING AND ANALYSIS OF SUBSTRATE COUPLING IN INTEGRATED-CIRCUITS

The Green function over a multilayer substrate is derived by solving P oisson's equation analytically in the z coordinate and numerically in the x and y coordinates. The x and y functional dependence is transfor med into a discrete cosine transform (DCT) representation for rapid ev aluation. The Green function is further transformed into a numerically stable form appropriate for finite-precision machine evaluation, This Green function is used to solve for the impedance matrix for an arbit rary three-dimensional arrangement of conductors placed anywhere in th e substrate. Using this technique, the substrate coupling and loss in IC circuits can be analyzed. A spiral inductor is presented as an exam ple. Experimental measurement results verify the accuracy of the techn ique,