An Accurate Calculation of Spreading Resistance

There was an attempt to find an analytic expression3, however that paper contains an error in the application of the boundary conditions. More recently, Gelmont and Shur4, found an analytic expression for very thin substrates and presented the result of a more general numerical solution. They showed that the formula above was inaccurate for the range of the ratio of substrate thickness to contact radius 0.5< b/a < 3. The numerical results were reported as a line on a graph and the details of the calculation were not given. The thermal spreading resistance of a round heat conducting contact has the same mathematical form as the electrical problem. In a review of that work5, the results are also limited to approximate analytic formula and numerical calculations. In this paper, a formal, analytic solution of the spreading resistance problem will be given. Two methods of obtaining numerical results will be presented for the case where the substrate thickness is greater than the contact radius. A finite element method will be used to find the spreading resistance when the thickness is less than the contact radius. In each case, the results are calculated to an accuracy better than 0.1%. These results will be compared with previously published calculations. From these solutions, polynomials are found which can be used to find the spreading resistance for any ratio of substrate thickness to contact radius.