Multi-layered Green function approach to 3D capacitance extraction with substrate loss

Compact Green function over a multi-layered substrate is derived by solving Poisson equation analytically in the z (vertical) coordinate and numerically in the x and y (planar) coordinates. This Green function is used to extract 3D capacitance by integrating it in z coordinate analytically. The discrete cosine transform (DCT) and its variations are used for rapid evaluation. Using this technique, the substrate coupling and loss in IC circuits can be analyzed. Comparisons to several other tools verify the advantage of the technique.