An Analytical Approach to Thermal Design and Optimization With a Temperature-Dependent Power Model

The heat equation and the electrothermal equation are widely used in chip design, and generally supposed to be equivalent. We prove this equivalence in mathematical manner, but show that it is only valid when the boundary condition is convective. Recent technologies have a much increased leakage power, which is highly temperature-dependent. The modified thermal equations which model this dependency have no closed form, and can only be solved iteratively; but these solutions are slow and provide no intuitive understanding of the relationship. We model power consumption as a curve made up of two quadratic polynomial segments, which provide analytic formula for steady-state and transient temperature. The accuracy of this approximation can be assessed mathematically, providing a level of confidence, and a basis for refinement of the power consumption model, as may be required. We show how this approach can be applied to the optimization of thermal parameters in temperature-constrained design and the prevention of thermal runaway.