Center, New York: Rensselaer -Task
Y.L. Le Coz (Task Supervisor, firstname.lastname@example.org), D. Krishna (Student, PhD '04)
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute
Troy , NY 12180-3590
Understanding and predicting the multi-GHz behavior of IC interconnects is critical to meeting objectives of the semiconductor-design industry in the present decade. Essentially, we need solve Maxwell’s equations. There are two ways in which this can be done: (i) one-step, direct solution of the underlying field equations; (ii) two-step, lumped-element parasitic extraction followed by solution of the resulting circuit equations. Both methods are viable; however, the second has a certain appeal, since it conveniently separates the physics of circuit-element extraction from the electrical engineering of circuit solution. We will presume that two-step Maxwell solution has been selected above. It is the second step, efficient solution of complex RLCM interconnect-circuit equations, that we consider our primary motivating factor. We have discovered a new impulse-response (IR) moment-extraction algorithm for RLCM circuit networks. It employs a Feynman sum-over-paths postulate. Our approach begins with generation of s-domain nodal-voltage equations. We then perform a Taylor-series expansion of the circuit transfer function. These expansions yield transition diagrams involving mathematical coupling constants, or weight factors, in integral powers of complex frequency s. Our sum-over-paths postulate supports stochastic evaluation of path sums within the circuit transition diagram to any desired order of s. The specific order of s in the sum corresponds, as well, to the order of IR moment we seek to extract. In developing the algorithm, importantly, we maintain computational efficiency and full parallelism. Initial verification studies of benchmark uncoupled and coupled RLCM lines furnished promising results: a maximum of 2% and 7% approximate error for first- and second-order IR moments, respectively, after only 1000 sampled path-sum terms. In addition, we observed excellent convergence to exact, analytical moment values with increasing number of samples. Our sum-over-paths postulate, in fact, implies generality for arbitraryRLCM-interconnect networks, beyond those specific examples presented in this work. We believe, in conclusion, that this type of IR-moment extraction algorithm may find useful application in a massively coupled electrical system, such as that encountered in high-end digital-IC interconnects.
This work has been sponsored in part by the Defense Advanced Research Projects Agency (DARPA); the New York State Office of Science, Technology, and Academic Research (NYSTAR); the Semiconductor Research Corporation (SRC) Microelectronics Advanced Research Corporation (MARCO); and the SRC Custom-Funding Program (LSI Logic Corporation). The authors thank Dr. R.D. Schinella of LSI Logic Corporation for his continuing support of this work.
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