| The increasing necessity of faster processing and transmitting of data is driving modern day analog and digital systems to operate at higher and higher frequencies. At the same time the cost of production is reduced significantly by tightly packing the devices on the integrated circuit (IC) chip. This double-pronged fact of modern IC design brings in a detrimental effect on circuit performance due to crosstalk between tightly packed electrical components. In such cases the simulation of electromagnetic (EM) effects in high-speed chips, packages and boards become essential for complete electrical characterization.; In this dissertation, we propose two schemes for the formulation of the combined circuit-EM problem and the time and memory efficient solution of the resultant system. In the first formulation approach, a novel coupling scheme based on charge-current continuity is presented which combined with surface based electric field integral equations (EFIE) and Kirchoff's current and voltage laws (KVL, KCL) formulates the hybrid problem in a single system matrix. Advantages of this method include a seamless transition across the EM and circuit interface, time efficient solution precluding multi-port S-parameter extraction and complete electric current and field visualization under circuit source excitations. In the second approach, the existing Partial Element Equivalent Circuit method has been extended from a volume only formulation to a surface-volume combined technique, capable of handling hybrid circuit-EM problems.; Integral equation based formulations give rise to a dense system matrix the solution of which present a time and memory bottleneck. To that end, we propose two efficient solution methods. In the first case, a fast iterative solution scheme is presented wherein the regular cube structure of the fast multipole method (FMM) and the QR compression scheme for interaction sub-matrices as in IES3 are combined to achieve superior memory and time performances as compared to existing solvers. In the second approach a fast iteration free technique is developed based on multilevel multi-pole expansions. Unlike its iterative counterpart this method does not suffer from poor convergence issues and is therefore more efficient for ill-conditioned systems specially for many right hand side (RHS) problems. |
