IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 815 Electromagnetic Field Computation of Simple Structures on Printed Circuit Boards by the Finite-Element Method K. Hollaus , O. Bíró , P. Caldera , G. Matzenauer , G. Paoli , K. Preis , C. Stockreiter , and B. Weiss Institut for Fundamentals and Theory in Electrical Engineering, IGTE, A-8010 Graz, Austria Infineon Technologies, Microelectronic Design Centers, Austria GmbH, Villach A-9500, Austria The aim of this paper is to study some problems arising in simulating structures on printed circuit boards. Hexahedral edge finite elements of second order using different potential formulations have been employed. Some simplifications are proposed to reduce the computational effort. The input impedance of a micro strip on a test board is computed and compared with measurement data. Some numerical issues are investigated. A semi-analytical method to compute the losses in the microstrip and in the board is presented and results are shown. Index Terms—Finite-element methods, numerical analysis, printed circuit, wave propagation. I. INTRODUCTION T HE MOTIVATIONof the present work has been to study the feasibility of applying edge finite elements to simulate the electromagnetic field of structures on a printed circuit board (PCB) considering the full set of Maxwell’s equations. Due to steadily growing demands on the electromagnetic compatibility, i.e., humans being unintentionally exposed to electromagnetic fields on one hand and the signal integrity on the other hand [1], [2], it is absolutely necessary to have a tool allowing a fast and accurate assessment of PCBs already in the design phase. To this end, a test board with simple structures has been manufactured and simulated by the finite-element method. In the present work and the potential formula- tions [3] were used to study different excitation models and the computational effort. Since all material properties are linear, in- vestigations were advantageously carried out in the frequency domain. The vector potentials are approximated by hexahedral edge basis functions of second order, the scalar potentials are represented by nodal ones. Neither formulation is gauged. The arising singular linear complex algebraic equation systems were solved by the incomplete Cholesky conjugate gradient (ICCG) method iteratively. Nodal elements to approximate the vector potentials have not been considered here. The potential is not capable of repre- senting the electric field intensity at sharp and perfectly con- ducting edges [4]. On the other hand, leads to large errors in the vicinity of reentrant corners in the conducting region [5]. Since the field is concentrated close to the conductor and decays rapidly with the distance from the board no absorbing boundary conditions have been necessary on the far boundary. The input impedance of a microstrip [6] obtained by a comprehensive finite-element model using , and , respectively, is compared with measurement data. Next, approx- imate models are studied and their accuracy is investigated by Digital Object Identifier 10.1109/TMAG.2006.871959 Fig. 1. One fourth of the microstrip; all dimensions are in millimeters, thickness of the conductor is equal to mm. Different methods of excitation are indicated in grey. comparing loss values with those obtained by a comprehensive model. A semi-analytical method to compute the losses in the microstrip and in the board is given and results are presented. II. NUMERICAL SIMULATIONS A. Microstrip Since and are planes of symmetry, only one fourth of the entire problem region has been modeled as shown in Fig. 1. The conductivity of the microstrip is assumed to be 5.8 10 S/m. The microstrip is mounted on a board symmetric with respect to the plane and is considered to be infinite. According to the specifications of the manufacturer of the board, the relative electric permittivity decreases from 4.475 at a frequency of 100 MHz to 4.4 at 1 GHz practically linearly. The loss factor , wherein means the electric permittivity of vacuum, varies from 0.0145 to 0.0245 [7]. 0018-9464/$20.00 © 2006 IEEE