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Y.L. Le Coz (Task Supervisor, lecozy@rpi.edu), D. Krishna (Student, PhD '04)
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute
Troy , NY 121803590
ABSTRACT
Understanding and predicting the multiGHz behavior of IC interconnects is critical to meeting objectives of the semiconductordesign industry in the present decade. Essentially, we need solve Maxwell’s equations. There are two ways in which this can be done: (i) onestep, direct solution of the underlying field equations; (ii) twostep, lumpedelement 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 circuitelement extraction from the electrical engineering of circuit solution. We will presume that twostep Maxwell solution has been selected above. It is the second step, efficient solution of complex RLCM interconnectcircuit equations, that we consider our primary motivating factor. We have discovered a new impulseresponse (IR) momentextraction algorithm for RLCM circuit networks. It employs a Feynman sumoverpaths postulate. Our approach begins with generation of sdomain nodalvoltage equations. We then perform a Taylorseries 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 sumoverpaths 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 secondorder IR moments, respectively, after only 1000 sampled pathsum terms. In addition, we observed excellent convergence to exact, analytical moment values with increasing number of samples. Our sumoverpaths postulate, in fact, implies generality for arbitraryRLCMinterconnect networks, beyond those specific examples presented in this work. We believe, in conclusion, that this type of IRmoment extraction algorithm may find useful application in a massively coupled electrical system, such as that encountered in highend digitalIC 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 CustomFunding 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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