6.0 Discussion and Alternative Potential Functions

dominant pole no longer is changing as we add more mutuals) at r 0 =4mm, we have a matrix sparsity of 95.02%. Because of the large size of the full dense partial inductance matrix, we are unable to compare the approximation result with the exact solution. However, we will assert correctness of our result in terms of eigen-value pattern of the partial inductance matrix in Fig.6. Fig.6 gives the eigenvalue distributions for the full matrix and the sparsified matrix. One interesting observation is that the eigenvalues of the sparsified matrix agree very well with those of the full matrix for all but the few largest eigenvalues. Noting that these large eigen-values of L correspond to the low frequency current assemblies, and since we are accurately approximating the dominant pole, we can assume that these eigenvalues correspond to impossible or negligible current assemblies given the connectivity of this circuit. We would further postulate that by matching the eigenvalues corresponding to the high frequency energies, we are providing an accurate approximation over all frequencies of interest. Next, we consider two 30-conductor packaging examples in Fig.7. The length of the lines is 0.5cm and the spacing between the lines is 0.2cm. Both circuits have the same horizontal topology but different connectivity. In order to consider the edge effect, we also model the vertical inductance effect in our extracted circuits. Referring to circuit Fig.7 (a), each signal line has its return path nearby. The r 0 for this circuit when moment ratio reaches convergence is 0.6cm, which translates to 84% sparsity for horizontal partial inductance matrix, and 90% sparsity for vertical partial inductance matrix. However, in circuit Fig.7 (b), all of the signal lines share one return path and the r 0 at moment ratio convergence is 1.2cm. The sparsity for horizontal partial inductance matrix and vertical partial inductance matrix are 50% and 65% respectively. Our methodology agrees with our understanding of good packaging design practice, namely, the circuit in Fig.7 (a) is a superior design. For the original partial inductance concept, the return current is assumed at infinity. However, this introduces a full matrix, which is a formidable task for simulation. A simple solution is to truncate the smallest terms in the partial inductance matrix to achieve sparsity, but this can cause an unstable circuit model, as shown in Section 3.0. Fig. 7: Two circuits with identical partial inductance matrices (in the horizontal …