Reduced RC Models for IC Interconnections with Coupling Capacitances

Reduced RC Models for IC Interconnections with Coupling Capacitances

A.J.van Genderen,N.P.van der Meijs

Delft University of Technology

Department of Electrical Engineering

Mekelweg4,2628CD Delft,The Netherlands

ABSTRACT

The transmission behavior of interconnections in integrated circuits is often determined by their distributed RC effects.In this paper,we present a modeling technique,for incorporation in a layout-to-circuit extraction program,that accurately represents these effects.The method consists offirst replacing IC interconnections by a complex RC network and then transforming this complex RC network into a simple RC network.It extends previous work in that(1)besides preserving the Elmore time constants between the terminals of the interconnections,the method also preserves the total resistances between the terminals and the total capacitances,and(2)the method handles ground capacitances as well as coupling capacitances.

1.Introduction

When neglecting the inductive effects,the transmission behavior of an IC interconnection is determined by distributed resistance and distributed capacitance.Experiments show that these distributed RC effects are accurately modeled(see e.g.[1])by subdividing the interconnections into pieces and replacing each piece by a lumpedπ-section.For each piece,its total capacitance is thereby symmetrically divided over the terminals of the lumped resistance model for the piece.In order to obtain sufficiently accurate results,the interconnection should be subdivided into a sufficient number of pieces,such that the distributed properties of the interconnection are accurately reflected in the RC ladder network.

In practice,for arbitrary IC interconnections,a few problems can arise with this method:(1)It may be hard tofind interconnection pieces that are accurately represented by aπsection because of irregular interconnection polygons and non-uniformly distributed resistive and capacitive effects and(2)the number of sections that are required to obtain an accurate RC model for the interconnection may be too many to allow an efficient verification of the behavior of the circuit afterwards.To solve these problems,some extensions to the RC modeling method have been described[2,3,4,5].All of these methods use a technique in which,first,a complex RC network is constructed that models the distributed resistive and capacitive effects of the interconnection in detail.Then a network reduction technique is applied to obtain afinal network that has a much lower number of nodes and elements but that displays approximately the same transmission behavior.To guarantee a close resemblance between the transmission behavior of the initial network and thefinal network,all methods preserve the value of the Elmore time constant[6].

In this paper,we present an RC modeling technique that is based on a similar technique.It transforms the initial RC network into thefinal RC network such that(1)the Elmore time constants between the terminal pairs are preserved,(2)the total resistances between the terminal pairs are preserved,and(3)the total capacitances of the conductors are preserved.Further,the node reduction technique that is described here is suited for RC networks containing resistance loops and it models ground capacitances as well as coupling capacitances.It delivers "weighted"lumpedπ-models that accurately reflect the total resistances and capacitances of the interconnections as well as their distributed effects.

2.RC mesh construction

The initial RC mesh that is constructed for an interconnection provides a discretization of the distributed RC effects of the interconnection by means of simple,uniform,lumped RC sections.To construct the initial RC mesh,the interconnect polygons arefirst decomposed into boxes and triangles as for example shown in Figure1(a).Each of the boxes and triangles is then replaced by a lumped RC section as described below.

Whenσis the conductivity of the material,and A is the area of the element,the conductance G ij(=1/R ij)between two nodes i,j of a triangle which has nodes i,j,k,is given by

G ij=σ

4A

(x k−x i)(x k−x j)+(y k−y i)(y k−y j)

.(1) For a box,the conductance between two nodes i,j is given by

G ij=

σ

2A

(y i−y j)2

σ

2A

(x i−x j)2

otherwise

if x i=x j

if y i=y j

(2) which is equivalent to using the length-width ratio of the box.