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) less than or equal to t* less than or equal to 10(4) and 10(-2) less than or equal to t* less than or equal to 10(2)), substrate outer radius (1 less than or equal to b* less than or equal to 100) and convective Blot numbers (10(-4) to 10(2)). With bottom-side cooling a minimum in the thermal resistance cart 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.