European Journal of Applied Mathematics

Table of Contents - April 2012 - Volume 23, Issue 02  

Papers

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

MARIA AGUARELESa1, JAUME HAROa2, JOSEP RIUSa3 and J. SOLÀ-MORALESa2

a1 IMA, UdG, Campus Montilivi, EPS-Ed. P4, E-17071 Girona, Spain email: maria.aguareles@udg.edu

a2 MA1, UPC, ETSEIB, Av. Diagonal 647, E-08028 Barcelona, Spain email: jaime.haro@upc.edu, jc.sola-morales@upc.edu

a3 EE, UPC, ETSEIB, Av. Diagonal 647, E-08028 Barcelona, Spain email: rius@eel.upc.edu

Abstract

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.

(Received March 17 2011)

(Revised September 21 2011)

(Accepted September 21 2011)

(Online publication October 19 2011)

Key words:

  • Asymptotic analysis of PDEs;
  • Weierstrass elliptic function;
  • Gauss constant