Abstract-Based on the cavity-mode model, we have developed a fast algorithm for calculating power bus impedance in multilayer printed circuit boards. The fast algorithm is based on a closed-form expression for the impedance matrix of a rectangular power bus structure; this expression was obtained by reducing the original double infinite series into a single infinite series under an approximation. The convergence of the single series is further accelerated analytically. The accelerated single summation enables much faster computation, since use of only a few terms is enough to obtain good accuracy. In addition, we propose two ways to compensate for the error due to the approximation involved in the process of reducing the double series to the single series, and have demonstrated that these two techniques are almost equivalent.Index Terms-Cavity-mode model, closed-form expression for fast calculation, power bus impedance, power bus resonance.

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SUMMARY This study focuses on the three‐dimensional (3‐D) characteristics of wave propagation in pipe‐pile using elastodynamic finite integration technique. First, a real 3‐D pile‐soil model in cylindrical coordinate system is presented. Then, the governing equations are established. With the boundary and initial conditions, the numerical solution is obtained. The accuracy and feasibility of the self‐written code are further verified via comparing with the measured data. Velocity histories at different angles of pile top and pile tip are illustrated, and the snapshots reflecting the 3‐D characteristics of wave propagation are given. It shows that the interferences of Rayleigh waves can confuse the result interpretation for pile integrity testing. The increase of hammer contact time can effectively mitigate the interferences, and the interferences of Rayleigh waves are weakest at an angle of 90° from where hammer hits. Besides, surrounding soil can partly mitigate the wave interferences. Copyright © 2012 John Wiley & Sons, Ltd.