Analytical solutions for two-dimensional heat spread in ASICs using conformal mapping techniques

Abstract

The subject of this work is the calculation of the temperature distribution in ASICs, generated by rectangular sources at the top surface of the ASIC while the backside of the substrate is held at a fixed temperature. Using conformal mapping, the steady state temperature distribution generated by surface sources of variable width in ASICs of variable thickness is calculated. Conformal mapping theory, especially the Schwarz-Christoffel transformation, is used to transform the given structure into simpler topology so that the solution of the heat equation becomes much easier and allows the use of given solutions, e.g. for symmetrical structures. Additionally, methods are shown on how to obtain transformations that can be inverted analytically, so that all results appear in closed-form elementary functions containing the geometry parameters. Furthermore, some hints are given, how a multitude of thermal problems can be solved using the conformal mapping method. Finally, the results are compared with ANSYS simulations for the verification of the two approaches.