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), 2024. Published by Science Publishing Group |
Calculation of Resistance, Infinite Two-Dimensional Grid of Resistances, Equivalent Resistance Method
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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
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
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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Download@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}
}
TY - JOUR T1 - About Calculation the Resistance of Two-dimensional Infinite Grid Systems AU - Spivak-Lavrov Igor Y1 - 2021/12/24 PY - 2021 N1 - https://doi.org/10.11648/j.jeee.20210906.13 DO - 10.11648/j.jeee.20210906.13 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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