Abstract
The electric copper lines in printed circuit boards (PCBs) introduce anisotropy in the various layers in multilayered PCBs. This anisotropy in combination with a general lack of symmetry with respect to the PCB midplane is a cause for thermally induced warpage. An analytical model which can predict the thermomechanical properties and warpage of PCBs with various electric artworks was developed. The model is based on Kirchhoff plate theory. An effective medium assumption was employed for the layers consisting of copper lines and composite side by side. The properties of the effective medium were obtained by a combination of Voigt (constant strain) and Reuss (constant stress) predictions, in conjunction with three-dimensional finite element unit cell calculations to determine a few constants. The prediction of the model conform well to some recent experiments. The model was finally used to “tune” the electric artwork of a PCB to minimize thermally induce warpage by changing trace widths and spacings in a six layered board. The calculations predict that isothermal warpage could be reduced by more than 90% for this board.
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