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About Calculation the Resistance of Two-dimensional Infinite Grid Systems

Received: 14 November 2021     Accepted: 7 December 2021     Published: 24 December 2021
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Abstract

The paper considers the problem of calculating the resistance between nodes of infinite grid resistor systems with square and triangular cells. There has long been a question about the resistance between the nearest nodes of an infinite grid of resistances with square cells with the same resistance r. Here, earlier, by the method of symmetry and superposition, a result was obtained r/2 that is striking in its simplicity. However, this result is only approximate, although many physicists consider this result to be accurate. New examples are presented proving what the results obtained earlier by the superposition and symmetry method is only approximate. The result r/2 gives only the lower limit of the correct resistance value. In our work, the correctness of using the equivalent resistance method to calculate the resistance between nearest nodes of infinite grid systems is proved. Using this method, for the resistance between the nearest nodes of an infinite grid of resistances with square cells, a result is obtained about 0.5216 r that only slightly differs from r/2. The results differ from the previously obtained values by about 10%. The resistance between the diagonal points of an infinite grid of identical resistors r with square cells is calculated. For the value of this resistance, a value founded about 0.7071 r that differs from the value 2r/π obtained previously by the superposition and symmetry method.

Published in Journal of Electrical and Electronic Engineering (Volume 9, Issue 6)
DOI 10.11648/j.jeee.20210906.13
Page(s) 194-199
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Calculation of Resistance, Infinite Two-Dimensional Grid of Resistances, Equivalent Resistance Method

References
[1] Bessonov L. A. (2002) Theoretical bases of electrical engineering. Electric circuits. – Moscow: Gardariki, 2002. – P. 638.
[2] Venezian G. On the resistance between two points on a grid // Am. J. Phys. 62, 1000–1004 (1994).
[3] Van Steenwijk F. J. Equivalent resistors of polyhedral resistive structures // Am. J. Phys. 66, 90–91 (1998).
[4] Atkinson D. and F. J. van Steenwijk. Infinite resistive lattices // Am. J. Phys. 67 (6), 486–492 (1999).
[5] Q. Meng, J. He, F. P. Dawalibi and J. Ma, “A new method to decrease ground resistances of substationgrounding systems in high resisistivity regions”, IEEE Trans. On New Power Delivery, Vol. 14, pp. 911-916, 1999.
[6] H. S. Lee, J. H. Kim, F. P. Dawalibi and J. Ma, “Efficient ground grid designs in layered soils”, IEEE Trans. On Power Delivery, Vol. 13, No. 3, pp. 745-751, July 1998.
[7] Bairamkulov R., Friedman E. G. Effective resistance of finite two-dimensional grids based on infinity mirror technique // IEEE Transactions on Circuits and Systems I: Regular Papers. (2020/04/24) – DOI. 10.1109/TCSI.2020.2985652.
[8] Irodov I. E. Exercises in General Physics. Tutorial. 14 th ed. – S.-Pt. – Msk. – Krs.: Izd. "Lan", 2016. – 416 p.
[9] Spivak-Lavrov I. F, Kurmanbai M. S., Mazhit А. N. About one method of calculation of resistance of two-dimensional infinite grid systems. – Vestnik ARSU. – No. 1 (51), Aktobe, 2018. – P. 43-51.
[10] Spivak-Lavrov IF, Kurmanbai MS, Mazhit АN (2018) About One Method of Calculation of Resistance of Two-dimensional Infinite Grid Systems. Educ Res Appl: ERCA-157. DOI: 10.29011/2575-7032/100057/.
[11] Katsneison, M. I. Carbon in Two Dimensions. – New York: Cambridge University Press, 2012. – 366 p.
[12] Davydov S. A Chain Model of a Zigzag Contact of Lateral Graphene-Like Hetero-structures // Technical Physics Letters (2018).
[13] Bhattarai, S. P. Construction of Sheet Resistance Measurement Setup for Tin Dioxide Film Using Four Probe Method. – American Journal of Physics and Applications; 2017, 5 (5): 60-65.
[14] Dandekar R. and Deepak D. Proportionate growth in patterns formed in the rotor-router model // Journal of Statistical Mechanics: Theory and Experiment, Volume 2014 (2014) P. 11–30.
[15] Owaidat M. Q. Determining the resistance of a full-infinite ladder network using lattice Green's function // Advanced Studies in Theoretical Physics, Vol. 9, 2015, no. 2, 77-83. doi.org/10.12988/ astp.2015.412159.
[16] Owaidat M. Q. and Asad J. H. Resistance calculation of three-dimensional triangular and hexagonal prism lattices, The European Physical Journal Plus, 131 (2016), no. 9, 309.
Cite This Article
  • APA Style

    Spivak-Lavrov Igor. (2021). About Calculation the Resistance of Two-dimensional Infinite Grid Systems. Journal of Electrical and Electronic Engineering, 9(6), 194-199. https://doi.org/10.11648/j.jeee.20210906.13

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    ACS Style

    Spivak-Lavrov Igor. About Calculation the Resistance of Two-dimensional Infinite Grid Systems. J. Electr. Electron. Eng. 2021, 9(6), 194-199. doi: 10.11648/j.jeee.20210906.13

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    AMA Style

    Spivak-Lavrov Igor. About Calculation the Resistance of Two-dimensional Infinite Grid Systems. J Electr Electron Eng. 2021;9(6):194-199. doi: 10.11648/j.jeee.20210906.13

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  • @article{10.11648/j.jeee.20210906.13,
      author = {Spivak-Lavrov Igor},
      title = {About Calculation the Resistance of Two-dimensional Infinite Grid Systems},
      journal = {Journal of Electrical and Electronic Engineering},
      volume = {9},
      number = {6},
      pages = {194-199},
      doi = {10.11648/j.jeee.20210906.13},
      url = {https://doi.org/10.11648/j.jeee.20210906.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.jeee.20210906.13},
      abstract = {The paper considers the problem of calculating the resistance between nodes of infinite grid resistor systems with square and triangular cells. There has long been a question about the resistance between the nearest nodes of an infinite grid of resistances with square cells with the same resistance r. Here, earlier, by the method of symmetry and superposition, a result was obtained r/2 that is striking in its simplicity. However, this result is only approximate, although many physicists consider this result to be accurate. New examples are presented proving what the results obtained earlier by the superposition and symmetry method is only approximate. The result r/2 gives only the lower limit of the correct resistance value. In our work, the correctness of using the equivalent resistance method to calculate the resistance between nearest nodes of infinite grid systems is proved. Using this method, for the resistance between the nearest nodes of an infinite grid of resistances with square cells, a result is obtained about 0.5216 r that only slightly differs from r/2. The results differ from the previously obtained values by about 10%. The resistance between the diagonal points of an infinite grid of identical resistors r with square cells is calculated. For the value of this resistance, a value founded about 0.7071 r that differs from the value 2r/π obtained previously by the superposition and symmetry method.},
     year = {2021}
    }
    

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    AU  - Spivak-Lavrov Igor
    Y1  - 2021/12/24
    PY  - 2021
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    T2  - Journal of Electrical and Electronic Engineering
    JF  - Journal of Electrical and Electronic Engineering
    JO  - Journal of Electrical and Electronic Engineering
    SP  - 194
    EP  - 199
    PB  - Science Publishing Group
    SN  - 2329-1605
    UR  - https://doi.org/10.11648/j.jeee.20210906.13
    AB  - The paper considers the problem of calculating the resistance between nodes of infinite grid resistor systems with square and triangular cells. There has long been a question about the resistance between the nearest nodes of an infinite grid of resistances with square cells with the same resistance r. Here, earlier, by the method of symmetry and superposition, a result was obtained r/2 that is striking in its simplicity. However, this result is only approximate, although many physicists consider this result to be accurate. New examples are presented proving what the results obtained earlier by the superposition and symmetry method is only approximate. The result r/2 gives only the lower limit of the correct resistance value. In our work, the correctness of using the equivalent resistance method to calculate the resistance between nearest nodes of infinite grid systems is proved. Using this method, for the resistance between the nearest nodes of an infinite grid of resistances with square cells, a result is obtained about 0.5216 r that only slightly differs from r/2. The results differ from the previously obtained values by about 10%. The resistance between the diagonal points of an infinite grid of identical resistors r with square cells is calculated. For the value of this resistance, a value founded about 0.7071 r that differs from the value 2r/π obtained previously by the superposition and symmetry method.
    VL  - 9
    IS  - 6
    ER  - 

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Author Information
  • Department Physics, K. Zhubanov Aktobe Regional University, Аktobe, Kazakhstan

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