Application of the tree-cotree splitting for improving matrix conditioning in the full-wave finite-element analysis of high-speed circuits

In this study the tree-cotree splitting technique is applied for improving the finite-element matrix conditioning for the analysis of high-speed circuits. A well-known issue is that at low frequencies a full-wave solver yields less accurate solutions and may even breakdown due to ill-conditioned system matrices. To enhance the capability and reliability of the conventional finite element method in broadband full-wave analyses, we apply the tree-cotree splitting to edge elements to account for the decoupling between the electric and magnetic fields at low frequencies, which is the main reason for the low-frequency problem. The algorithm for finding a minimum spanning tree when there exist wave ports, lumped ports, or for a PEC-free structure are described. Besides, a model order reduction method, called the solution space projection, is applied for a fast broadband analysis. We further propose an expansion to available solution bases for a better approximation to low-frequency fields, so that a simulation can be extended to extremely low frequencies. The application is focused on the simulation of high-speed circuits, of which both low and high frequency characteristics are of equal importance.