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Graphs in VLSI pp 195–215Cite as

Effective resistance of finite grids

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Abstract

Conventional numerical circuit analysis tools typically scale superlinearly with the number of nodes. With the rapid increase in the number of nodes in modern VLSI systems, alternative methods are required. The effective resistance is an important characteristic of electrical systems, which is used to simplify the circuit analysis process. An infinite resistive rectangular mesh is commonly assumed in the analysis of grid structures to determine the effective resistance of a grid. The assumption of infinity provides a useful approximation when a large grid is analyzed far from the boundaries. If however the grid is analyzed in close proximity to a boundary or if the grid dimensions are small, the assumption of infinity may lead to significant error. To address this issue, the infinity mirror technique is proposed to determine the effective resistance of a two-dimensional structure, where one or both dimensions are finite. The runtime of the proposed method does not depend on the size of the grid and exhibits good agreement with nodal analysis, achieving an error below 1 Based on the infinity mirror technique, IR drops can be accurately determined at arbitrary points within a mesh without considering the entire grid. The infinity mirror technique therefore greatly accelerates the IR drop analysis process in large grids by exclusively determining the electric potential at a few nodes of primary interest. A 1,400 fold speedup is achieved in the analysis of 100 nodes within a 103 × 104 grid.

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References

  1. R. Bairamkulov, A. Roy, M. Nagarajan, V. Srinivas, and E. G. Friedman, “SPROUT - Smart Power ROUting Tool for Board-Level Exploration and Prototyping,” Proceedings of the ACM/IEEE Design Automation Conference, pp. 283–288, December 2021.

    Google Scholar 

  2. R. Bairamkulov and E. G. Friedman, “Effective Resistance of Two-Dimensional Truncated Infinite Mesh Structures,” IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 66, No. 11, pp. 4368–4376, November 2019.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Bairamkulov, A. Roy, M. Nagarajan, V. Srinivas, and E. G. Friedman, “SPROUT - smart power routing tool for board-level exploration and prototyping,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 41, No. 7, pp. 2263–2275, July 2022

    Article  Google Scholar 

  4. E. Salman and E. G. Friedman, High Performance Integrated Circuit Design, McGraw-Hill Professional, 2012.

    Google Scholar 

  5. V. Adler and E. Friedman, “Uniform Repeater Insertion in RC Trees,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 47, No. 10, pp. 1515–1523, October 2000.

    Article  Google Scholar 

  6. A. Çiprut, Grids in Very Large Scale Integration Systems, Ph.D. Thesis, University of Rochester, May 2019.

    Google Scholar 

  7. S. Köse and E. G. Friedman, “Effective Resistance of a Two Layer Mesh,” IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 58, No. 11, pp. 739–743, November 2011.

    Google Scholar 

  8. W. H. McCrea and F. J. W. Whipple, “Random Paths in Two and Three Dimensions,” Proceedings of the Royal Society of Edinburgh, Vol. 60, No. 3, pp. 281–298, January 1940.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. van der Pol and H. Bremmer, Operational Calculus based on the Two-Sided Laplace Integral, Cambridge University Press, 1950.

    MATH  Google Scholar 

  10. G. Venezian, “On the Resistance between Two Points on a Grid,” American Journal of Physics, Vol. 62, No. 11, pp. 1000–1004, November 1994.

    Article  Google Scholar 

  11. D. Atkinson and F. J. Van Steenwijk, “Infinite Resistive Lattices,” American Journal of Physics, Vol. 67, No. 6, pp. 486–492, June 1999.

    Article  Google Scholar 

  12. J. Cserti, “Application of the Lattice Green’s Function for Calculating the Resistance of an Infinite Network of Resistors,” American Journal of Physics, Vol. 68, No. 10, pp. 896–906, October 2000.

    Article  Google Scholar 

  13. S. Köse and E. G. Friedman, “Efficient Algorithms for Fast IR Drop Analysis Exploiting Locality,” Integration, the VLSI Journal, Vol. 45, No. 2, pp. 149–161, March 2012.

    Article  Google Scholar 

  14. Y. Ogasahara, M. Hashimoto, T. Kanamoto, and T. Onoye, “Measurement of Supply Noise Suppression by Substrate and Deep N-well in 90nm Process,” Proceedings of the IEEE Asian Solid-State Circuits Conference, pp. 397–400, November 2008.

    Google Scholar 

  15. R. E. Aitchison, “Resistance between Adjacent Points of Liebman Mesh,” American Journal of Physics, Vol. 32, pp. 566–566, July 1964.

    Article  Google Scholar 

  16. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Vol. 22, Mathematical Association of America, 1984.

    Google Scholar 

  17. K. Brown, “Infinite Grid of Resistors,” Available at https://www.mathpages.com/home/kmath668/kmath668.htm.

  18. M. Jeng, “Random Walks and Effective Resistances on Toroidal and Cylindrical Grids,” American Journal of Physics, Vol. 68, No. 1, pp. 37–40, January 2000.

    Article  MathSciNet  Google Scholar 

  19. “SciPy: Reference Guide,” 2014, [Online; accessed April 15, 2019].

    Google Scholar 

  20. F. Dorfler and F. Bullo, “Kron Reduction of Graphs with Applications to Electrical Networks,” IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 60, No. 1, pp. 150–163, January 2013.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Liu and E. Li, “Fast Voltage Drop Modeling of Power Grid with Application to Silicon Interposer Analysis,” Proceedings of the IEEE Electronic Components and Technology Conference, pp. 1109–1114, May 2013.

    Google Scholar 

  22. M. Popovich, M. Sotman, A. Kolodny, and E. G. Friedman, “Effective Radii of On-Chip Decoupling Capacitors,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Vol. 16, No. 7, pp. 894–907, July 2008.

    Article  Google Scholar 

  23. S. Zhao, K. Roy, and C.-K. Koh, “Decoupling Capacitance Allocation and its Application to Power-Supply Noise-Aware Floorplanning,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 21, No. 1, pp. 81–92, January 2002.

    Article  Google Scholar 

  24. S. Köse and E. G. Friedman, “Distributed On-Chip Power Delivery,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, Vol. 2, No. 4, pp. 704–713, December 2012.

    Article  Google Scholar 

  25. J. M. Kosterlitz and D. J. Thouless, “Ordering, Metastability and Phase Transitions in Two-Dimensional Systems,” Journal of Physics C: Solid-State Physics, Vol. 6, No. 7, pp. 1181, April 1973.

    Google Scholar 

  26. A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari, “The Electrical Resistance of a Graph Captures its Commute and Cover Times,” Computational Complexity, Vol. 6, No. 4, pp. 312–340, December 1996.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. H. Zemanian, “Infinite Electrical Networks: a Reprise,” IEEE Transactions on Circuits and Systems, Vol. 35, No. 11, pp. 1346–1358, November 1988.

    Article  MathSciNet  Google Scholar 

  28. A. Carmona, A. M. Encinas, and M. Mitjana, “Kirchhoff Index of Periodic Linear Chains,” Journal of Mathematical Chemistry, Vol. 53, No. 5, pp. 1195–1206, May 2015.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Zhang and Y. Yang, “Resistance Distance and Kirchhoff Index in Circulant Graphs,” International Journal of Quantum Chemistry, Vol. 107, No. 2, pp. 330–339, November 2007.

    Article  Google Scholar 

  30. F.-Y. Wu, “Theory of Resistor Networks: the Two-Point Resistance,” Journal of Physics A: Mathematical and General, Vol. 37, No. 26, pp. 6653–6673, July 2004.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Jakushokas and E. G. Friedman, “Power Network Optimization based on Link Breaking Methodology,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Vol. 21, No. 5, pp. 983–987, May 2013.

    Article  Google Scholar 

  32. C.-J. Lee, J. K. Park, S. Kim, and J.-H. Chun, “A Study on a Lattice Resistance Mesh Model of Display Cathode Electrodes for Capacitive Touch Screen Panel Sensors,” Procedia Engineering, Vol. 168, pp. 884 – 887, September 2016.

    Article  Google Scholar 

  33. W. Xu and E. G. Friedman, “Clock Feedthrough in CMOS Analog Transmission Gate Switches,” IEEE International ASIC/SOC Conference, pp. 181–185, September 2002.

    Google Scholar 

  34. R. M. Secareanu, S. Warner, S. Seabridge, C. Burke, J. Becerra, T. E. Watrobski, C. Morton, W. Staub, T. Tellier, I. S. Kourtev, and E. G. Friedman, “Substrate Coupling in Digital Circuits in Mixed-Signal Smart-Power Systems,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Vol. 12, No. 1, pp. 67–78, January 2004.

    Article  Google Scholar 

  35. J. Xie, Y. Jia, and M. Miao, “High Sensitivity Knitted Fabric Bi-Directional Pressure Sensor based on Conductive Blended Yarn,” Smart Materials and Structures, Vol. 28, No. 3, pp. 035017, February 2019.

    Google Scholar 

  36. J. Xie and H. Long, “Equivalent Resistance Calculation of Knitting Sensor under Strip Biaxial Rlongation,” Sensors and Actuators A: Physical, Vol. 220, pp. 118 – 125, December 2014.

    Article  Google Scholar 

  37. E. Bendito, A. Carmona, A. M. Encinas, and J. M. Gesto, “Characterization of Symmetric M-Matrices as Resistive Inverses,” Linear Algebra and its Applications, Vol. 430, No. 4, pp. 1336–1349, February 2009.

    Article  MathSciNet  MATH  Google Scholar 

  38. S. J. Kirkland and M. Neumann, “The M-matrix Group Generalized Inverse Problem for Weighted Trees,” SIAM Journal on Matrix Analysis and Applications, Vol. 19, No. 1, pp. 226–234, January 1998.

    Article  MathSciNet  MATH  Google Scholar 

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  1. University of Rochester, Rochester, NY, USA

    Rassul Bairamkulov & Eby Friedman

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  1. Rassul Bairamkulov
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Bairamkulov, R., Friedman, E. (2023). Effective resistance of finite grids. In: Graphs in VLSI. Springer, Cham. https://doi.org/10.1007/978-3-031-11047-4_7

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  • DOI: https://doi.org/10.1007/978-3-031-11047-4_7

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