Abstract
The voltage-current regime of an infinite resistive network, whose current sources are distributed throughout the network and also connected to the network at infinity, is shown to be the limit in a certain Hilbert space of the regimes of an expanding sequence of subnetworks. This result is then applied to an infinite gridlike structure. A decomposition of that structure into ∞-ports yields an equivalent ladder network of operators. A procedure is established for determining the exact solution of the infinite gridlike structure by solving the finite truncations of the operator ladder and then passing to the limit appropriately. This closes a lacuna appearing in a number of prior finite-difference analyses of several exterior problems based upon infinite network theory.
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References
[AZ] H. K. An and A. H. Zemanian, Computationally efficient determination of threshold voltages in narrow-channel MOSFETs including fringing and inversion effects.IEEE Trans. Electron Devices,36 (1989), 534–541.
[C] G. A. Campbell, Physical theory of the electric wave-filter,Bell System Tech. J.,1 (1922), 1–32.
[CFL] R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengliechungen der mathematischen Physik,Math. Ann.,100 (1928), 32–74.
[DS1] P. G. Doyle and J. L. Snell,Random Walks and Electric Networks, Mathematical Association of America, Washington, DC, 1984.
[D1] R. J. Duffin, Discrete potential theory,Duke Math. J.,20 (1953), 233–251.
[D2] R. J. Duffin, Basic properties of discrete analytic functions,Duke Math. J.,23 (1956), 335–364.
[DS2] R. J. Duffin and D. H. Shaffer, Asymptotic expansion of double Fourier transforms,Duke Math. J.,27 (1960), 581–596.
[DS3] R. J. Duffin and E. P. Shelly, Difference equations of polyharmonic type,Duke Math. J.,25 (1958), 209–238.
[F1] H. Flanders, Infinite networks: I—Resistive networks,IEEE Trans Circuit Theory,18 (1971), 326–331.
[F2] H. Flanders, Infinite networks: II—Resistance in an infinite grid,J. Math. Anal. Appl.,40 (1972), 30–35.
[F3] R. M. Foster, The average impedance of an electric network, inContributions to Applied Mathematics, Reisner Anniversary Volume, pp. 333–340, Edwards, Ann Arbor, MI, 1949.
[H] H. A. Heilbron, On discrete harmonic functions,Math. Proc. Cambridge Philos. Soc.,45 (1949), 194–206.
[K] A. E. Kennelly,Artificial Lines and Nets, Wiley, New York, 1912.
[L] L. Lavatelli, The resistive net and finite-difference equations,Amer. J. Phys.,40 (1972), 1246–1257.
[NS] A. W. Naylor and G. R. Sell,Linear Operator Theory, Holt, Rinehart, and Winston, New York, 1971.
[P1] B. Peikari,Fundamentals of Network Analysis and Synthesis, Prentice-Hall, Englewood Cliffs, NJ, 1974.
[P2] B. van der Pol, The finite difference analogy of the periodic wave equation and the potential equation, Appendix IV, inProbability and Related Topics in Physical Sciences (M. Kac, ed.), pp. 237–257, Interscience, London, 1959.
[S1] A. Stöhr, Über einige lineare partielle Differenzengleichungen mit konstanten Koeffizienten,Math. Nach.,3 (1949/50), 208–242, 295–315, 330–357.
[S2] J. L. Synge, The fundamental theorem of electrical networks,Quart. Appl. Math.,9 (1951), 113–127.
[S3] J. L. Synge, Addendum to the fundamental theorem of electrical networks,Quart. Appl. Math.,11 (1953), 215.
[W1] K. W. Wagner, Die Theorie der Kettenleiten nebst Anwendungen,Arch. Elektrotech.,3 (1915), 315–332.
[W2] L. Weinberg,Network Analysis and Synthesis, McGraw-Hill, New York, 1962.
[Z1] A. H. Zemanian,Realizability Theory for Continuous Linear Systems, Academic Press, New York, 1972.
[Z2] A. H. Zemanian, Countably infinite networks that need not be locally finite,IEEE Trans. Circuits and Systems,21 (1974), 274–277.
[Z3] A. H. Zemanian, Connections at infinity of a countable resistive network,Circuit Theory Appl.,3 (1975), 333–337.
[Z4] A. H. Zemanian, Nonuniform semi-infinite grounded grids,SIAM J. Math. Anal.,13 (1982), 770–788.
[Z5] A. H. Zemanian, Operator-valued transmission lines in the analysis of two-dimensional anomalies imbedded in a horizontally layered earth under transient polarized electromagnetic excitation,SIAM J. Appl. Math.,45 (1985), 591–620.
[Z6] A. H. Zemanian, Infinite electrical networks with finite sources at infinity,IEEE Trans. Circuits and Systems,34 (1987), 1518–1534.
[Z7] A. H. Zemanian, A finite-difference procedure for the exterior problem inherent incapacitance computations of VLSI interconnections,IEEE Trans. Electron Devices 35 (1988), 985–992.
[ZA1] A. H. Zemanian and H. K. An, Finite-difference analysis of borehole flows involving domain contractions around three-dimensional anomalies,Appl. Math. Comput. 26 (1988), 45–75.
[ZA2] A. H. Zemanian and B. Anderson, Models of borehole resistivity measurements using infinite electrical grids,Geophysics,52 (1987), 1525–1534.
[ZS] A. H. Zemanian and P. Subramaniam, A theory for ungrounded electrical grids and its application to the geophysical exploration of layered strate,Studia Math.,LXXXVII (1983), 163–181.
[ZZ] A. H. Zemanian and T. S. Zemanian, Domain contractions around three-dimensional anomalies in spherical finite-difference computations of Poisson's equation,SIAM J. Appl. Math.,46 (1986), 1126–1149.
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This work was supported by National Science Foundation Grants DMS-8521824 and MIP-8822774.
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Zemanian, A.H. An electrical gridlike structure excited at infinity. Math. Control Signal Systems 4, 217–231 (1991). https://doi.org/10.1007/BF02551268
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DOI: https://doi.org/10.1007/BF02551268