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 Poisson's equation analytically in the coordinate and numerically in the z and y coordinates. The x and y functional dependence is transformed into a discrete cosine transform (DCT) representation for rapid evaluation. 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 arbitrary three-dimensional arrangement of conductors placed anywhere in the substrate. Using this technique, the substrate coupling and loss in IC circuits can be analyzed. A spiral inductor is presented as an example. Experimental measurement results verify the accuracy of the technique.