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.