Efficient Fourier Series Solutions to Nonlinear Steady-State Heat Conduction Problems in Microwave Circuits

This paper presents a method of obtaining rapidly convergent Fourier series solutions to heat conduction problems in media containing temperature-dependent and orthotropic thermal conductivities subject to applied surface flux. Such problems occur frequently in the analysis of thermal heating of integrated circuits, with the heat sources consisting primarily of transistors, resistors and other conductors on the top surface of the die. The key development in obtaining improved convergence behavior, relative to similar existing series solutions, is the use of smoothly varying Gaussian distributions for the applied loads versus the more conventional, discontinuous, uniformly applied (isoflux) heat loads. It is shown that the smoothness of a Gaussian distribution effectively suppresses the higher order series expansion coefficients normally introduced by the uniformly applied loads and reduces the number of terms required for convergence by up to 98%. The rate of decay of the series expansion coefficients for three load distributions, with varying degrees of smoothness, and the corresponding rate of convergence of the respective doubly infinite series, are examined in detail.