A Fixed-Angle Dynamic Heat Spreading Model for (An)Isotropic Rear-Cooled Substrates

Abstract

During a period of almost 40 years already, various fixed-angle heat spreading models have been developed in the literature. These models are commonly used by thermal engineers as approximations for the thermal steady-state resistance of a heat source on a rear-cooled substrate. In this paper, an extension of these models to dynamic (time-dependent) phenomena is proposed. The heat dissipated by a square source (side a) is assumed to spread out into the substrate (thickness b) under an angle ϕ. An analytical solution for the complex thermal impedance Zth(jω) in phasor notation is derived. The obtained expression, in which ϕ is used as a fitting parameter, is compared with accurate analytical results. A very good agreement is observed (average relative error less than 6%) for a wide range of the normalized thickness λ=b/a. A compact expression for the optimal heat spreading angle as a function of λ is given. Also the temperature response to a heat power step is investigated, and a simple formula for the thermal rise time is provided. Finally, the model can be easily extended to anisotropic media, which often appear in electronic packaging applications. Overall the proposed model allows a thermal designer to make quick yet accurate estimations about the dynamic behavior of the device.