Armen H. Zemanian

Three-dimensional capacitance computations for VLSI/ULSI interconnections

Three-dimensional simulations of metallization wires of VLSI/ULSI interconnections that are plagued with unreasonably large memory requirements and execution times are discussed. A strategy is presented for overcoming these problems. A principal feature is the use of a domain contraction technique, which accounts for the fringing electric field throughout the infinite domain above and below the levels where the wires appear and provides a major reduction in the number of nodal points for a finite-difference computation. Moreover, an iterative method (successive over-relaxation) is used to alleviate memory requirements, a nonuniformly distributed nodal array is used to reduce the number of nodal points still further, and parallel processing is used to reduce execution time. It is argued that rounded edges and corners for the simulation of the wires are the only appropriate configurations at current levels of miniaturization. This avoids the problem of electric-field singularities at sharp edges and corners and results in significantly reduced capacitance coefficients. >

The limb analysis of countably infinite electrical networks

Abstract A method, called “limb analysis,” for analyzing certain countably infinite electrical networks was developed in a prior publication. It consisted of two parts, a graph-theoretic decomposition of the network and an analysis wherein Kirchhoffs node and loop laws were applied. The present paper is primarily devoted to proving that every countable network possesses the required graphtheoretic decomposition. In addition, it also establishes a sufficient set of conditions under which the analytic portion of limb analysis has a solution.

Operating points in infinite nonlinear networks approximated by finite networks

Given a nonlinear infinite resistive network, an operating point can be determined by approximating the network by finite networks obtained by shorting together various infinite sets of nodes, and then taking a limit of the nodal potential functions of the finite networks. Initially, by taking a completion of the node set of the infinite network under a metric given by the resistances, limit points are obtained that represent generalized ends, which we call “terminals,” of the infinite network. These terminals can be shorted together to obtain a generalized kind of node, a special case of a 1-node. An operating point will involve Kirchhoff’s current law holding at 1-nodes, and so the flow of current into these terminals is studied. We give existence and bounds for an operating point that also has a nodal potential function, which is continuous at the 1-nodes. The existence is derived from the said approximations.

A Generalized Weierstrass Transformation

maps a suitably restricted function f( r) into an analytic function on a strip a, < Re s < a2 in the complex s-plane. Other names for it are the Gauss transformation [2], the Gauss-Weierstrass transformation [3, p. 578], and the Hille transformation [4]. The objective of this work is to extend the Weierstrass transformation and a certain inversion formula [1, p. 191] to allow f(Qr) to be a certain type of generalized function. This work is a sequel to a previous one [5] in which the convolution transformation of Hirschman and Widder [1],

Pristine transfinite graphs and permissive electrical networks

1 Introduction.- 1.1 Notations and Terminology.- 1.2 Transfinite Nodes and Graphs.- 1.3 A Need for Transfiniteness.- 1.4 Pristine Graphs.- 2 Pristine Transfinite Graphs.- 2.1 0-Graphs and 1-Graphs.- 2.2 ?-Graphs and (? + 1)-Graphs.- 2.3

Nonstandard versions of conventionally infinite networks

\mathop{\omega }\limits^{ \to }

Hyperreal Transients in Transfinite RLC Networks

-Graphs and ?-Graphs.- 2.4 Transfinite Graphs of Higher Ranks.- 3 Some Transfinite Graph Theory.- 3.1 Nondisconnectable Tips and Connectedness.- 3.2 Sections.- 3.3 Transfinite Versions of Konigs Lemma.- 3.4 Countable Graphs.- 3.5 Locally Finite Graphs.- 3.6 Transfinite Ends.- 4 Permissive Transfinite Networks.- 4.1 Linear Electrical Networks.- 4.2 Permissive 1-Networks.- 4.3 The 1-Metric.- 4.4 The Recursive Assumptions.- 4.5 Permissive (? + l)-Networks.- 4.6 Permissive Networks of Ranks