Thermal analysis of a substrate with power dissipation in the vias

The authors present an analysis of the temperature field resulting from the Joule heating in a metal wire surrounded by a non-heat-generating (electrically insulating) material. Exact analytical solutions are given for when the heat generation rate is constant and linearly dependent on the temperature. Asymptotic solutions are given for arbitrarily temperature-dependent heat generation, when the insulating material thermal conductivity is much less than the thermal conductivity of the wire. A numerical example of practical interest is considered. It is shown that neglecting the Joule heating in the wires can result in significant underestimation of the temperature. The effect of temperature on the electrical resistivity of the wire is shown to be negligible. The phenomenon of thermal runaway is also examined using stability theory and is shown to be unimportant in practical circumstances