An efficient method based on a direct boundary element method is proposed for extracting frequency-dependent substrate coupling parameters. A frequency-independent real-valued linear equation system is first solved. Then, the solution is transformed into frequency-dependent parameters at a specified frequency with the Sherman-Morrison-Woodbury formula. The first step is performed only once for a given structure, and the method is very efficient for extraction with multiple frequencies. The proposed method is compared with the Green's function-based method and the approach in our earlier paper for typical substrate structures. Numerical results demonstrate its accuracy, efficiency, and versatility.

The couplings via realistic lossy substrates can be modeled as frequency-dependent coupling parameters. The fast extraction at multiple frequencies can be accomplished in two sequent steps. The first is to extract the coupling resistance using a direct boundary element method (DBEM). The second is to revise the resistance into the parameter at the frequency in an exact and rapid way. The first step is time-consuming, while it runs only one time; the second repeats at each frequency, but is much easier. For more frequency calculation, this method is more advanced. Numerical experiments illustrate that this method has high accuracy, and it can be hundreds of times faster than an advanced Green's function based method. Substrates with arbitrary doping profiles can also be easily handled, which is partly verified by experiment.

It is important to model substrate couplings for SoC/mixed-signal circuit designs. After introducing the continuation equation of full current in lossy substrates, we present a new direct boundary element method (DBEM), which can handle the substrates with arbitrary doping profiles. Three techniques can speed up the DBEM remarkably, which include reusing coefficient matrices for multiple-frequency calculation, condensing the linear system, and sparsifying coefficient matrix. Numerical experiments illustrate that DBEM has high accuracy and high efficiency, and is versatile for arbitrary doping profiles.

It is important to model the substrate coupling for mixed-signal or RF circuit designs. In this paper, a direct boundary element method (DBEM) and related efficient techniques are presented to calculate the coupling resistances for three-dimensional substrate structure. First, a nonuniform meshing scheme is presented to reduce boundary elements while preserving accuracy. Then, the unknowns on top medium surface are removed from the discretized linear system of DBEM with a matrix reduction technique. The third technique is applying the quasi-multiple medium idea (W. Yu, Z. Wang, and J. Gu, "Fast capacitance extraction of actual 3-D VLSI interconnects using quasi-multiple medium accelerated BEM," IEEE Trans. Microwave Theory Tech., vol. 51, no. 1, pp. 109-199, Jan. 2003), which greatly reduces the expense of matrix reduction and makes the final coefficient matrix much sparser. With these proposed techniques, the linear equation system is largely condensed and sparsified and then solved with a preconditioned generalized minimum residual solver for multiple right-hand sides to get the whole resistance matrix. Numerical experiments on typical substrates with various doping profiles show the high accuracy of the DBEM-based method. The authors also compared the DBEM method with the Green's function methods accelerated by discrete cosine transform or eigendecomposition techniques. The results show that the DBEM-based method is several times or tens of times faster than the other two. At the same time, the DBEM method has no difficulty in handling substrates with more complex than stratified doping profiles, which is a large advantage over the existing methods

It is important to model the substrate coupling for mixed-signal circuit designs today. This paper presents the direct boundary element method (BEM) for substrate resistance calculation, where only the boundary of substrate region is discretized. Firstly, an efficient scheme for non-uniform element partition is proposed. Secondly, a new technique is presented which can reduce the scale of produced linear system and then accelerate the equation solving, especially for the multiple right-hand sides problem like substrate resistance extraction. Experiments show that the proposed method has shown high efficiency compared with existing methods while preserving high accuracy.

It is important to model the substrate coupling for mixed-signal circuit designs today. This paper presents an improved direct boundary element method (DBEM) for substrate resistance calculation, where only the boundary of substrate volumes is discretized and only the free-space Green function is used. At first, we discard some inessential unknowns to compress the linear system without accuracy loss. Then we make the coefficient matrix sparser. In this way, solving the linear system is greatly accelerated. Experiments on various substrates validate that DBEM is several to tens of times faster than DCT-accelerated Green's function methods and the eigendecomposition method, while preserving high accuracy. Besides, another experiment shows that this method is versatile for irregular substrates.