[Truncated introduction] A major contribution of this thesis is the observation that the N port impedance parameters for the distribution planes can be modelled as simple LC series elements in the frequency range over which the interesting interactions between the loading elements and the planes occur. Loosely speaking, the C represents the inter-plane capacitance and the L is associated with a first order frequency trend of the transfer and input impedances associated with the planes. In the literature, values for L have been obtained for power ground plane structures using curve fitting techniques [38]. In this thesis, formulae are developed for L based on the modal summation expression. As for the impedance case, the approach developed in the thesis that removes the singular behaviour, results in computational efficient expressions. Preliminary results on the simple LC model were presented by the author in [42, 43] The results reported in the thesis extend this work in the light of the new impedance model proposed. The simple LC characterisation enables the development of new low frequency expression for the input and transfer impedance for ports on planes loaded with many decoupling capacitors. The expressions are based on a one off frequency independent decomposition of the inductance matrix associated with the placement of the capacitors. The eigen-mode decomposition eliminates the need for matrix inversion at each frequency point and leads to an efficient computational procedure for calculating the impedance of loaded planes. Furthermore, the interaction between the capacitors and planes is clearly seen in the analytical expressions. This has led to new insights regarding the interaction of multiple capacitors with supply planes in terms of location, resonance mechanisms, pole locations and damping. These insights are beneficial to the understanding and optimisation of printed circuit board power distribution systems.

The objective of this dissertation is to develop electrical modeling and co-simulation methodologies for signal and power integrity of package and board applications. The dissertation includes 1) the application of the finite element method to the optimization for decoupling capacitor selection and placement on a power delivery network (PDN), 2) the development of a PDN modeling method effective for multidimensional and multilayer geometries, 3) the analysis and modeling of return path discontinuities (RPDs), and 4) the implementation of the absorbing boundary condition for PDN modeling. The optimization technique for selection and placement of decoupling capacitors uses a genetic algorithm (GA) and the multilayer finite element method (MFEM), a PDN modeling method using FEM. The GA is customized for the decoupling problem to enhance the convergence speed of the optimization. The mathematical modifications necessary for the incorporation of the capacitor model into MFEM is also presented. The main contribution of this dissertation is the development of a new modeling method, the multilayer triangular element method (MTEM), for power/ground planes of a PDN. MTEM creates a surface mesh on each plane-pair using dual graphs; a non-uniform triangular mesh (Delaunay triangulation) and its orthogonal counterpart (Voronoi diagram), to which electromagnetic and equivalent circuit concepts are applied. The non-uniform triangulation is especially efficient for discretizing multidimensional and irregular geometries which are common in package and board PDNs. Moreover, MTEM generates a sparse, banded, and symmetric system matrix, which enables efficient computations. For a given plane-pair, MTEM extracts an equivalent circuit that is consistent with the physics-based planar-circuit model of a plane-pair. Thus, the values of the lumped elements can be simply calculated from the physical parameters, such as material properties and mesh geometries of each unit-cell. Consequently, the modeling of MTEM is flexible and easy to modify for further extensions, such as the incorporation of external circuits, e.g. decoupling capacitors and vertical interconnects. Power and ground planes provide paths for the return current of signal traces. Typically, planes have discontinuities such as via holes, plane cutouts, and split planes that disturb flow of signal return currents. At the discontinuity, return currents have to detour or switch to different layers, causing signal and power integrity problems. Therefore, a separate analysis of signal interconnects will neglect the significant coupling with a PDN, and the result will not be reliable. In this dissertation, the co-simulation of the signal and power integrity is presented focusing on the modeling of RPDs created by split planes, apertures, and vias. Plane resonance is one of the main sources of power integrity problems in package and board PDNs. A number of techniques have been developed and published in literature to reduce or prevent the resonance of a plane-pair. One of the techniques is to surround plane-pair edges with absorbing material that effectively damps the outgoing parallel-plate wave and minimizes the reflection. To model this behavior, the boundary condition of MTEM needs to be changed from its original form, the open-circuit boundary condition. In this dissertation, the application of the 1st order absorbing boundary condition to MTEM is presented.

Mixed signal system-on-package (SoP) technology is a key enabler for increasing functional integration, especially in mobile and wireless systems. Due to the presence of multiple dissimilar modules, each having unique power supply requirements, the design of the power distribution network (PDN) becomes critical. Typically, this PDN is designed as alternating layers of power and ground planes with signal interconnects routed in between or on top of the planes. The goal for the simulation of multi-layer power/ground planes, is the following: Given a stack-up and other geometrical information, it is required to find the network parameters (S/Y/Z) between port locations. Commercial packages have extremely complicated stack-ups, and the trend to increasing integration at the package level only points to increasing complexity. It is computationally intractable to solve these problems using these existing methods. The approach proposed in this thesis for obtaining the response of the PDN is the multi-layer finite difference method (M-FDM). A surface mesh / finite difference based approach is developed, which leads to a system matrix that is sparse and banded, and can be solved efficiently. The contributions of this research are the following: 1. The development of a PDN modeler for multi-layer packages and boards called the the multi-layer finite difference method. 2. The enhancement of M-FDM using multi-port connection networks to include the effect of fringe fields and gap coupling. 3. An adaptive triangular mesh based scheme called the multi-layer finite element method (MFEM) to address the limitations of M-FDM 4. The use of modal decomposition for the co-simulation of signal nets with the PDN. 5. The use of a robust GA-based optimizer for the selection and placement of decoupling capacitors in multi-layer geometries. 6. Implementation of these methods in a tool called MSDT 1.