A Simple Closed Form Solution to Single Layer Heat Spreading Angle Appropriate for Microwave Hybrid Modules

Abstract

A simple equation for heat spreading angle is derived which is useful for cases with a single layer thermal spreader. The derivation starts with Fourier’s heat transfer law. Heat spreading in two dimensions is then introduced which results in a quadratic equation relative to spreading angle. The result is a closed form equation for heat spreading angle. Calculations using the equation are compared to 3D finite element simulations which show agreement acceptable for most practical applications and over a wide range of physical dimensions and thermal conductivities. A norma-lized dimensional parameter is defined which is used to generate a curve fit equation of the spreading angle. A three step procedure is then presented which allows the calculation of the spreading angle and temperature rise in the thermal spreader. The result has application for initial calculations of temperature rise in microwave hybrid modules and electronic packages such as heat sinks for high power amplifiers. This is because it is common for these types of modules and packages to use a single layer heat spreader in copper-tungsten (CuW) or copper-molybde-num (CuMo) connected to a cold plate. An important benefit of this method is that it allows microwave hybrid designers and high power amplifier packaging engineers a method to quickly perform trade studies to determine the maximum mounting temperature for integrated circuits.