Science in China Series F: Information Sciences

, Volume 45, Issue 6, pp 453–461 | Cite as

A minimum-order boundary element method to extract the 3-D inductance and resistance of the interconnects in VLSI



The high frequency resistance and inductance of the 3-D complex interconnect structures can be calculated by solving an eddy current electromagnetic problem. In this paper, a model for charactering such a 3-D eddy current problem is proposed, in which the electromagnetic fields in both the conducting and non-conducting regions are described in terms of the magnetic vector potential, and a set of the indirect boundary integral equations (IBIE) is obtained. The IBIEs can be solved by boundary element method, so this method avoids discretizing the domain of the conductors. As an indirect boundary element method, it is of minimum order. It does not restrict the direction of the current in conductors, and hence it can consider the mutual impedance between two perpendicular conductors. The numerical results can well meet the analytical solution of a 2-D problem. The mutual impedance of two perpendicular conductors is also shown under the different gaps between conductors and different frequencies.


VLSI circuits interconnects parasitic inductance and resistance indirect boundary integral equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kamon, M., Tsuk, M. J., White, J., FastHenry: A multiple accelerated 3-D inductance extraction program, IEEE Trans. on MTT, 1994, 42(9): 1750–1758.CrossRefGoogle Scholar
  2. 2.
    Beattie, M. W., Pileggi, L. T., Electromagnetic parasitic extraction via a multipole method with hierarchical refinement, International Conference on Computer-Aided Design, Digest of Technical Papers, San Jose: IEEE/ACM, 1999, 437–444.Google Scholar
  3. 3.
    Wang Junfeng, Tausch, J., White, J., A wide frequency range surface formulation for distributed RLC extraction, International Conference on Computer Aided Design, Digest of Technical Papers, San Jose: IEEE/ACM, 1999, 453–457.Google Scholar
  4. 4.
    Davis, J. A., Meindl, J. D., Compact distributed RLC models for multilevel interconnect networks, IEEE Transactions on Electron Devices, 1999, 47(11): 2078–2087.CrossRefGoogle Scholar
  5. 5.
    Ismail, Y. I., Friedman, E. G., Effects of inductance on the propagation delay and repeater insertion in VLSI circuits, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 2000, 8(2): 195–206CrossRefGoogle Scholar
  6. 6.
    Sinha, A., Gupta, S. K., Breuer, M. A., Validation and test generation for oscillatory noise in VLSI interconnects, International Conference on Computer-Aided Design, Digest of Technical Papers, San Jose: IEEE/ACM, 1999, 289–296.Google Scholar
  7. 7.
    Ruehli, E., Inductance calculations in a complex integrated circuit environment, IBM J. Res. Develop., 1972, 16: 470–481.CrossRefGoogle Scholar
  8. 8.
    Cao, Y., Li, Z. F., Mao, J. F. et al., A PEEC with a new capacitance model for circuit simulation of interconnects and packaging structures, IEEE Trans. on MTT, 2000, 48(2): 281–287.CrossRefGoogle Scholar
  9. 9.
    Yuan, J. S., Kost, A., A three-component boundary element algorithm for three-dimensional eddy current calculation, IEEE Trans on Magnetics, 1994, 30(5): 3028–3031.CrossRefGoogle Scholar
  10. 10.
    Konrad, Integrodifferential finite element formulation of two-dimensional steady-state skin effect problems, IEEE Trans. on Magnetics, 1982, 18(1): 284–292.CrossRefGoogle Scholar
  11. 11.
    Lean, M., Dual simple-layer source formulation for two-dimensional eddy current and skin effect problems, J. Appl. Phys., 1985, 57(8): 3844.CrossRefGoogle Scholar
  12. 12.
    Cao, M., Biringer, P. P., BIE formulation for skin and proximity effect problems of parallel conductors, IEEE Trans. on Magnetics, 1990, 26(5): 2768–2770.CrossRefGoogle Scholar
  13. 13.
    Silvester, P., Modern Electromagnetic Fields, N.J.: Prentice-Hall, INC., 1968, 86–88.Google Scholar
  14. 14.
    Mayergoyz, D., Boundary integral equations of minimum order for the calculation of three-dimensional eddy current problems, IEEE Trans. on Magnetics, 1982, 18(2): 536–539CrossRefMathSciNetGoogle Scholar
  15. 15.
    Morse, P. M., Feshbach, H., Methods of Theoretical Physics, New York: McGraw-Hill, 1953, 1762–1767.MATHGoogle Scholar
  16. 16.
    Smythe, W. R., Static and Dynamic Electricity, New York: McGraw-Hill, 1968, 284–287.Google Scholar
  17. 17.
    Hammond, P. M., Use of potentials in calculation of electromagnetic fields, IEE PROC., 1982, Vol. 129, Pt A, No. 2: 106–112.Google Scholar
  18. 18.
    Kriezis, E. E., Xypteras, I. E., Eddy current distribution and loss in a semi-infinite conducting space due to a vertical current loop, ETZ Archiv., 1979, 201–207.Google Scholar
  19. 19.
    Morisue, T., Magnetic vector potential and electric scalar potential in three-dimensional eddy current problem, IEEE Trans. on Magnetics, 1982, 18(2): 531–535.CrossRefGoogle Scholar
  20. 20.
    Morisue, T., Fukumi, M., 3-D eddy current calculation using the magnetic vector potential, IEEE Trans. on Magnetics, 1988, 24(1): 106–109.CrossRefGoogle Scholar
  21. 21.
    Morisue, T., A new formulation of the magnetic vector potential method in 3-D multiply connected regions, IEEE Trans. on Magnetics, 1988, 24(1): 110–113.CrossRefGoogle Scholar
  22. 22.
    Morisue, T., Electric charges induced in an eddy current field, IEEE Trans. on Magnetics, 1995, 31(3): 1313–1318.CrossRefGoogle Scholar
  23. 23.
    Tsuk, M. J., Kong, J. A., A hybrid method for the calculation of the resistance and inductance of transmission lines with arbitrary cross sections, IEEE Trans. on MTT., 1991, 39(8): 1338–1347.CrossRefGoogle Scholar
  24. 24.
    Li, Z. F., Fang Xing, Calculation of the resistance matrix of the interconnects in high-speed large scale circuit system by the boundary integral equations, Journal of Microwave (in Chinese), 1993, 35(4): 22–28.MathSciNetGoogle Scholar

Copyright information

© Science in China Press 2002

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijingChina

Personalised recommendations