Efficient Heat Transfer Approximation for the Chip-on-Substrate Problem

An analytically based approximate solution is presented for the thermal resistance of an axisymmetric heat source mounted on a conductive substrate with bottom- and top-side convective cooling of the substrate. The approximation closely matches an exact solution for bottom-side convective cooling and reference finite element solutions for top-side and both-side cooling over broad ranges of substrate thickness (10−4 ≤ t* ≤ 104 and 10−2 ≤ t* ≤ 102), substrate outer radius (1 ≤ b* ≤ 100) and convective Blot numbers (10-4 to 102). With bottom-side cooling, a minimum in the thermal resistance can occur over a wide range of substrate thicknesses. The approximate solution possesses simplicity and ease of computation as compared to exact or computational solutions for many microelectronic applications.