• analytical solution
• central difference approximation
• current density analysis
• finite volume method
• modeling methodology
• PCB tracks
• thermal analysis of PCB structure

DC analysis of PCB tracks could be useful for evaluating both the IR drop and the influence of electromigration in tracks, as well as estimating the Joule heating rate for further thermal analysis. One methodology based on the Cartesian-grid Finite Volume Method is proposed to realize the DC analysis. A vertex-centered method is introduced for approximately reconstructing the track boundaries. The central difference approximation method has been used for calculating the current densities. For typical tracks with 45° inclined boundaries, the control volumes at the sloping boundaries and the bending corners are discussed. For general boundaries of arbitrary shape, a piecewise linearization method is proposed for segmenting the actual boundary into a number of linear pieces, and 16 types of right triangles can be used for delineating the boundary. Moreover, the necessary and sufficient condition for solving the electric distributions in multi-terminal tracks is discussed, described and verified through both the analysis of the equivalent resistor network in a multi-terminal track and the mathematical analysis of a matrix equation, which relates all the terminal currents and terminal electric potentials. A test solver has been programmed in the MATLAB in order to test the accuracy and the efficiency of the methodology, as well as its improvement compared to the traditional Cartesian-grid method with fully square mesh cells. The layout maps of tracks can be analyzed in the solver and used for generating the mesh grid. Four track layouts with linear or curvilinear boundaries have been tested and the corresponding CAD models were also built in the COMSOL for providing reference results in the comparisons.
PCBs, as the carriers for electronic circuits, also play the role of heat sinks or heat spreaders for ICs and components. The other modeling methodology based on integrating both the FVM-based numerical solution and the Fourier-series-based analytical solution of temperature is proposed for thermal analysis of the PCB structure. Particularly, the heat spreading through tracks and the vertical heat transfer through vias are taken into account in the numerical way and regarded as additional thermal boundary conditions of insulating layers, which can be assumed homogeneous. Moreover, the methods for analyzing the multilayer structure, the non-uniform HTC distributions and the influence of the IC package are also discussed. A thermal solver has been developed in the MATLAB based on the methodology, and the test solver for current density analysis has been embedded within the thermal solver for further realizing the electrical-thermal coupled analysis of the Joule heating in tracks. Several layouts were modeled in the solver and in the COMSOL for testing the validity and investigating the influence factors of the analytical solution, the coupled analytical-numerical solution and the coupled electrical-thermal solution. Based on the analysis and the comparisons, the mesh density and the number of eigenvalues are the main influence factors. Furthermore, the vertical and horizontal heat transfer contributions of vias have been investigated through modeling the footprint layout of a power mosfet in order to test the modeling assumptions. The consistencies between the modeling results and the reference results can be found, which further validate the proposed methodologies and the algorithms of the solvers. Both the advantages and the defects of the methodologies have been discussed throughout the analysis and the comparisons.