IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 15, NO. 12, DECEMBER 2007 1289 Efficient Modeling of Transmission Lines With Electromagnetic Wave Coupling by Using the Finite Difference Quadrature Method Qinwei Xu, Member, IEEE, and Pinaki Mazumder, Fellow, IEEE Abstract—This paper proposes an efficient numerical technique, called the finite difference quadrature (FDQ) method, to model the transmission line with radiated electromagnetic (EM) wave noise coupling. A discrete modeling approach, the FDQ method adapts coarse grid points along the transmission line to compute the finite difference between adjacent grid points. A global ap- proximation scheme is formulated in the form of a weighted sum of quantities beyond the local grid points. Unlike the Gaussian quadrature method that computes numerical integrals by using global approximation framework, the FDQ method uses a global quadrature method to construct the approximation schemes for the computation of, however, numerical finite differences. As a global approximation technique, the FDQ method has superior numerical dispersion to the finite difference (FD) method, and, therefore, needs much sparser grid points than the FD method to achieve comparable accuracy. Equivalent voltage and current sources are derived, exciting the transmission line at the grid points. Equivalent circuit models are consequently derived to rep- resent the transmission line subject to radiated electromagnetic wave noise. The FDQ-based equivalent models can be integrated into a simulator like SPICE. Index Terms—Electromagnetic (EM) wave illuminating, ex- ternal field coupling, finite difference quadrature (FDQ) method, interconnect modeling, transient simulation, transmission lines (TL). I. INTRODUCTION E LECTROMAGNETIC interference (EMI) problems have been a great concern in high-speed digital systems and plenty of works have been done to handle the electromagnetic compatibility (EMC). Depending on different propagation ap- proaches, there are conducted EMI noise, capacitive/inductive coupling EMI noise, and radiated electromagnetic (EM) wave noise. Conducted and coupling EMI problems have been studied most in the literature since fast operation and large integration scale started making the interconnect effect an important issue in high-speed systems two decades ago. In addition to having the on-chip and on-board effects of delay, crosstalk, and re- flection, electrically long interconnects pose the antenna effect, Manuscript received July 10, 2004. This work was supported in part by a Multidisciplinary University Research Initiative (MURI) grant and also by an Office of Naval Research (ONR) grant under the Dual-Use Program. Q. Xu is with Research and Development Team of Electronic Design Au- tomation Tools, Cadence Design Systems, San Jose, CA 95134 USA (e-mail: qwxu@umich.edu). P. Mazumder is with the Department of Electrical Engineering and Com- puter Science, the University of Michigan, Ann Arbor, MI 48109 USA (e-mail: mazum@eecs.umich.edu). Digital Object Identifier 10.1109/TVLSI.2007.904105 when they receive considerable dose of incident electromag- netic (EM) waves emitted by other electronic devices [1]. Fast clocking rate and short rise time result in signals with wave- lengths comparable to interconnect sizes that increase the ra- diation efficiency of the conducting traces, while the shrinking feature size and the increasing integration scale lead to higher EMI susceptibility among the circuit parts. The antenna-effect EMI problem becomes a more serious challenge to signal in- tegrity with progressive down-scaling. With the traditional EMC problems being involved most with cabling, there are a few scenarios in which the external EM waves are coupled to transmission lines (TL) in the high-speed systems. The interconnects on printed circuit board (PCB) il- luminated by EM waves are in the first scenario. As the PCB routing generally involves long interconnects, the EM wave cou- pling most likely happens in this case, which is in the category of PCB level EMI problem. In the second scenario the pack- aging structures, like the pad/pin and the leadframe, are sub- ject to EM wave coupling. The packaging structures have elec- trically large sizes to pick up the illuminating EM wave and the resultant EMI noise travels inside the chips and interferes with signals. This situation is in the packaging level EMI prob- lems. In the third scenario, the on-chip long interconnects, like power/ground lines and clocking lines, have the antenna effect in which the induced voltages and currents due to EM wave pose the sources of noise. Also, in the large array structures, like RAM/ROM, the horizontal and vertical data tracks are long and, therefore, may be sensitive to the external interference. These cases are the on-chip level EMI problems. EM wave coupling to transmission lines can happen at any level as discussed before. The problem has been handled by using 3-D full-wave solvers like finite-difference time-domain (FDTD) methods. In the circuit oriented EMI application, FDTD models the lumped devices as grid or subgrid elements having the explicit integration scheme as FDTD required [2]. Following the Courant constraints, the FDTD theoreti- cally gives accurate results as it directly simulates the wave phenomena represented by electric field and magnetic field. However, in view of circuit design and circuit simulation, direct full-wave technique, usually having 3-D scale, is com- putationally expensive and, therefore, prohibitive in most of the cases. This is especially true when handling the on-chip problem where the operation frequencies are much higher than those on off-chip or PCB. Furthermore, full-wave solvers may suffer from numerical instability when incorporated into circuit simulators, due to its tiny step size determined by the Courant condition. 1063-8210/$25.00 © 2007 IEEE Authorized licensed use limited to: University of Michigan Library. Downloaded on October 25, 2008 at 21:42 from IEEE Xplore. Restrictions apply.