Towards a Sparse Partial Inductance Matrix A.J. Dammers and N.P.van der Meijs Delft University of Technology Delft Institute of Microelectronics and Submicron Technology Faculty of Information Technology and Systems (ITS) Dept. of Electrical Engineering, Circuits and Systems Section P.O. Box 5031, 2600 GA Delft, the Netherlands Phone: +31 (0)15 - 278 1442 Fax: +31 (0)15 - 278 6190 fdammers@cas.et.tudelft.nl nick@cas.et.tudelft.nl Abstract— Currently available techniques for extraction and modelling of interconnect related inductance effects are either too restrictive on the geometries to be handled or computationally too demanding to be useful for large scale layout-to-circuit extraction. We present the results of a crit- ical evaluation of the partial inductances which form the building blocks in the PEEC method [1]. The physical origin of the well known passivity problem related to truncation of the partial inductance matrix is identified. This leads in a natural way to a preprocessing procedure of the partial in- ductances, which yields a modified matrix less sensitive to truncation. The proposed method disturbs the symmetry of the matrix, but for the important class of orthogonal systems of conductors symmetry is preserved. Therefore it will be of use for a majority of the circuit components where induc- tance effects are considered to be relevant in the first place (long lines: data, clock, V SS , V DD ). Our results quantify the existence of a tubular shell with return currents, as recently suggested by He et al. [2] in a qualitative way. Keywords—Inductance, PEEC method, Passivity,Layout- to-circuit extraction I. I NTRODUCTION In deep submicron technology feature sizes decrease, clock speeds get higher and chips get larger and more complex. Therefore, parasitic inductive effects in on-chip interconnects are playing an increasingly important role. Well known phenomena include clock skew, cross-talk, LdI/dT noise and ground bounce. The high degree of complexity of the circuits renders the modelling of such phenomena a non-trivial task. Current paths may be rather erratic, depending on the details of the signals present. This implies that description of inductive effects in terms of well-defined simple geometric structures is ruled out be- forehand. Layout-to-circuit extraction aims at reducing the physi- cal description of an integrated circuit to a highly reduced electrical lumped circuit representation, which may subse- quently be handled by circuit simulators such as SPICE. This objective is at first sight incompatible with the dis- tributed character of inductive phenomena, as self and mu- tual inductances are defined for closed current loops only, and these may span a major part of a circuit. The partial inductance concept [1] provides a handle to this problem. Unclosed conductor segments can formally be treated as magnetically coupled lumped circuit elements. The prac- tical problem, however, is the fact that this results in net- works with a high degree of connectivity. This implies that the associated partial inductance matrix has a high degree of fill-in, which strongly diminishes the efficiency of nu- merical methods which have proven to be useful for RC - problems. The most efficient method currently available for reduction of a system of partial inductances to a re- duced (smaller) one is GMRES with multipole accelera- tion, as implemented in FastHenry [3]. This method is, however, in its present form far too slow to be useful for full circuits, which may comprise up to one million tran- sistors or even more. Attempts to sparsify the inductance matrix by merely discarding relatively small elements of- ten lead to unphysical behaviour. Some eigenvalues of the truncated partial inductance matrix may become neg- ative, whereas the original inductance matrix is positive semidefinite (all eigenvalues are non-negative). As a re- sult, the overall electrical circuit may exhibit non-passive behaviour. We will analyse this problem from a physical point of view. This will provide guidelines for the intro- duction of a modified partial inductance matrix, which is expected to behave better under truncation. II. FLUX RELATED ISSUES OF THE PEEC APPROACH A. Interpretation of partial inductance elements Ruehli [1] relates the partial inductance which charac- terizes interaction between unclosed conductor segments to the magnetic flux enclosed by a loop which extends from a conductor segment to infinity. We summarize this interpretation, as it forms the basis for our introduction of ISBN: 90-73461-15-4 93 c STW, 1998 10 26-01:018