Thermal design rules for electronic components on conducting boards in passively cooled enclosures

A better understanding is needed of the approaches and limitations for rejecting heat dissipated from VLSI components mounted on multi-layer printed circuit boards housed in small enclosures, as for example those encountered in small consumer electronics, and, in notebook, laptop, or hand-held personal computers. This paper derives new first order thermal design formulae for determining the peak temperatures of sources on conducting substrates, and for determining the thermal “zone of influence” or “footprint” associated with a component on a board. A one-dimensional thin board radial fin approach is used with inclusion of a circular source to represent the heat dissipating component. Exact solutions are presented for sources at the center, edge, and corner of a rectangular board. The results are compared with with both 2-d and 3-d calculations for rectangular sources on a board using the finite element method. Excellent agreement is found in predicting the maximum temperature, with maximum differences of order 10%. Simple algebraic design formulae, useful for rapid estimation, are derived from the complete solutions by taking advantage of the asymptotic behavior at small and large values of the board parameter, m. An unambiguous thermal footprint radius is defined in terms of the tangent line at the inflection point of the temperature profile. Parametric studies show that the radius corresponds to the point at which the board temperature drops to roughly 18% of its peak temperature, for all variations of board thickness and conductivity of practical interest. The simple analytical model is used to predict the temperatures on a populated board, using a linear superposition principle, and it is found to be in good agreement with experimental results for boards with multiple heat sources