A Fixed-Angle Heat Spreading Model for Dynamic Thermal Characterization of Rear-Cooled Substrates

During a period of almost 40 years already, various fixed-angle heat spreading models have been developed in the literature. These models can be 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 t_s) under a fixed angle Phi. An analytical solution for the complex thermal impedance Z_th(jomega) in phasor notation is derived. The obtained expression, in which Phi is used as a fitting parameter, is compared with exact results. A very good agreement is observed (average relative error less than 6%) for a wide range of the normalized thickness lambda = t_s/a. A compact expression for the optimal heat spreading angle as a function of lambda is given. Finally the temperature response to a heat power step is investigated. A simple formula for the thermal rise time is provided, allowing a thermal designer to make quick yet accurate estimations about the dynamic behavior of the device.