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On an asymptotic formula for the maximum voltage drop in a on-chip power distribution network


We present a new asymptotic formula for the maximum static voltage in a simplified model for on-chip power distribution networks of array bonded integrated circuits. In this model the voltage is the solution of the Poisson's equation in an infinite planar domain whose boundary is an array of circular pads of radius ϵ, and we deal with the singular limit ϵ → 0 case. In comparison with approximations that appear in the electronics engineering literature, our formula is more complete, since we have obtained terms up to order ϵ15. A procedure will be presented to compute all the successive terms, which can be interpreted by using multipole solutions of equations involving spatial derivatives of δ-functions. To deduce the formula, we use the method of matched asymptotic expansions. Our results are completely analytical and we make an extensive use of special functions and the Gauss constant G.

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[4] C. M. Bender & S. A. Orszag (1999) Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, New York.

[7] R. Finch (2003) Mathematical Constants, Cambridge University Press, Cambridge, UK.

[9] E. J. Hinch (1991) Perturbation Methods, Cambridge University Press, Cambridge, UK.

[11] N. Koblitz (1984) Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York.

[14] M. Popovich , A. V. Mezhiba & E. G. Friedman (2008) Power Distribution Networks with On-Chip Decoupling Capacitors, Springer-Verlag, New York.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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