1 Interconnect Electromagnetic Modeling using Conduction Modes as Global Basis Functions Luca Daniel, Alberto Sangiovanni-Vincentelli, and Jacob White Abstract A new method is formulated for modeling current distributions inside conductors for a quasi-static or a full-wave electromagnetic field simulator. In our method, we model current distributions inside interconnects using a small number of conduction modes as global basis functions for the discretization of the Mixed Potential Integral Equation. A very simple example is presented to illustrate the potential of our method. I. I NTRODUCTION The past decade’s intense development of accelerated integral equation solvers has made it possible to perform electromagnetic analysis of packages or circuit boards with hundreds of conductors in just a few minutes on a workstation [1], [2], [3], [4]. The computational performance provided by these fast algorithms makes it now feasible to consider developing tools which can readily perform full-board analysis, for use in applications such as electromagnetic compatibility diagnosis and resolution. If the application requires many full-wave analyses of entire printed circuit boards, reducing computation time will remain critical, and therefore minimizing the number of unknowns used for each conductor remains an important problem. The most common approach to minimizing the number of unknowns used to discretize printed circuit board wires is to make a thin conductor, or 2 1 2 -d, approximation or a “skin-depth” approximation [5] using surface impedances. In addition, it has been recognized that the many conductor interiors can be decoupled into separate Helmholtz problems which can then be combined with a global exterior Helmholtz problem [6], [7]. The many Helmholtz equations can then be solved either by integral or by differential methods. In this paper we take a somewhat different approach, and make use of the interior Helmholtz equation to generate basis functions for a Galerkin-type solution of the Mixed Potential Integral Equation (MPIE). The paper is organized as follows: In Section II we summarize the classical integral equation method. In Section III-A, we derive possible “conduction modes” from the solution of the internal electric field Helmholtz equation. Based on such modes, we define in Section III-B global basis functions, that we use in Section III-C for the discretization of the MPIE. Finally, in Section IV a very simple example is used to illustrate the computational attractiveness of our method. II. BACKGROUND For a system of conductors embedded in a medium with constant dielectric permittivity ε, and magnetic permeability μ, the conductor current distribution, J, and the conductor surface charge, ρ, can be determined without computing any fields exterior to the conductors. In particular, the conductor currents can be related to the gradient of a scalar potential, φ, using the Mixed Potential Integral Equation (MPIE) Jr σ jω μ 4π V Jr e jk 0 r r r r dr ∇φ (1) where V is the union of the conductor volumes, r is a point in V , ω 2π f is the angular frequency of the conductor excitation, and k 0 ω με is the wave number. The scalar potential on the conductor surface can be related to the surface charge, ρ, through 1 4πε S ρ r s e jk 0 r s r s r s r s dr s φ r s (2) where S is the union of the conductor surfaces, and r s is a point in S. Since the charge in the interior of the conductor is zero, ∇ Jr 0 (3) for all points r in the interior of V . In addition, the current normal to the conductor surface is responsible for the accumulation of surface charge, ˆ nJr s jωρ r s (4) where ˆ n is the unit normal at the point r s on S. Luca Daniel (510-548-5800 dluca@eecs.berkeley.edu), and Alberto Sangiovanni-Vincentelli (510-642-4882 alberto@eecs.berkeley.edu) are with the Dept. of Electrical Eng. and Computer Science, University of California, 94720 Berkeley, CA. Fax: 510-642-2739. Jacob White (617-253-2543 white@mit.edu), is with the Dept. of Electrical Eng. and Computer Science, Massachusetts Institute of Technology, Bldg 36-817, 02139 Cambridge, MA. Fax: 617-258-5846. This work was supported by the MARCO Interconnect Focus Center, by the Semiconductor Research Corporation, and by the Hewlett-Packard Company.