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Calculation of the equivalent resistance and impedance of the cylindrical network based on recursion-transform method

Tan Zhi-Zhong Zhang Qing-Hua

Calculation of the equivalent resistance and impedance of the cylindrical network based on recursion-transform method

Tan Zhi-Zhong, Zhang Qing-Hua
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  • A classic problem in circuit theory first studied by German physicist Kirchhoff more than 170 years ago is the computation of resistances in resistor networks. Nowadays, resistor network has been an important model in the fields of natural science and engineering technology, but it is very difficult to calculate the equivalent resistance between two arbitrary nodes in an arbitrary resistor network. In 2004, Wu F Y formulated a Laplacian matrix method and derived expressions for the two-point resistance in arbitrary finite and infinite lattices in terms of the eigenvalues and eigenvectors of the Laplacian matrix, and the resistance results obtained by Laplacian matrix method is composed of double sums. The weakness of the Laplacian matrix approach is that it depends on the two matrices along two orthogonal directions. In 2011, Tan Z Z created the recursion-transform (RT) method, which can resolve the resistor network with arbitrary boundary. Using the RT method to compute the equivalent resistance relies on just one matrix along one direction, and the resistance is expressed by single summation. In the present paper, we investigate the equivalent resistance and complex impedance of an arbitrary mn cylindrical network by the RT method. Firstly, based on the network analysis, a recursion relation between the current distributions on three successive vertical lines is established through a matrix equation. In order to obtain the eigenvalues and eigenvectors of the matrix, and the general solution of the matrix equation, we then perform a diagonalizing transformation on the driving matrix.Secondly, we derive a recursion relation between the current distributions on the boundary, and construct some particular solutions of the matrix equation. Finally by using the matrix equation of inverse transformation, we obtain the analytical solution of the branch current, and gain the equivalent resistance formula along the axis of the arbitrary mn cylindrical network, which consists of the characteristic root and expressed by only single summation. As applications, several new formulae of equivalent resistances in the semi-infinite and infinite cases are given. These formulae are compared with those in other literature, meanwhile an interesting new identity of trigonometric function is discovered. At the end of the article, the equivalent impedance of the mn cylindrical RLC network is also treated, where the equivalent impedance formula is also given.
      Corresponding author: Tan Zhi-Zhong, tanz@ntu.edu.cn;tanzzh@163.com
    • Funds: Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20161278).
    [1]

    Kirchhoff G 1847 Ann. Phys. Chem. 148 497

    [2]

    Kirkpatrick S 1973 Rev. Mod. Phys. 45 574

    [3]

    Klein D J, Randi M 1993 J. Math. Chem. 12 81

    [4]

    Jafarizadeh S, Sufiani R, Jafarizadeh M A 2010 J. Stat. Phys. 139 177

    [5]

    Jzsef C 2000 Am. J. Phys. 68 896

    [6]

    Giordano S 2005 Int. J. Circ. Theor. Appl. 33 519

    [7]

    Asad J H 2013 J. Stat. Phys. 150 1177

    [8]

    Asad J H 2013 Mod. Phys. Lett. B 27 1350112

    [9]

    Wu F Y 2004 J. Phys. A:Math. Gen. 37 6653

    [10]

    Tzeng W J, Wu F Y 2006 J. Phys. A:Math. Gen. 39 8579

    [11]

    Izmailian N Sh, Kenna R, Wu F Y 2014 J. Phys. A:Math. Theor. 47 035003

    [12]

    Essam J W, Izmailian N S, Kenna R, Tan Z Z 2015 Royal Society Open Science 2 140420

    [13]

    Izmailian N S, Kenna R 2014 J. Stat. Mech. 09 P09016

    [14]

    Izmailian N S, Kenna R 2015 Chin. J. Phys. 53 040703

    [15]

    Tan Z Z 2011 Resistance Network Model (Xi'an:Xidian University Press) pp16-216(in Chinese)[谭志中2011电阻网络模型(西安:西安电子科技大学出版社)第16216页]

    [16]

    Tan Z Z, Zhou L, Yang J H 2013 J. Phys. A:Math. Theor. 46 195202

    [17]

    Tan Z Z, Zhou L, Luo D F 2015 Int. J. Circ. Theor. Appl. 43 329

    [18]

    Tan Z Z 2015 Int. J. Circ. Theor. Appl. 43 1687

    [19]

    Tan Z Z, Fang J H 2015 Commun. Theor. Phys. 63 36

    [20]

    Tan Z Z, Essam J W, Wu F Y 2014 Phys. Rev. E 90 012130

    [21]

    Essam J W, Tan Z Z, Wu F Y 2014 Phys. Rev. E 90 032130

    [22]

    Tan Z Z 2015 Chin. Phys. B 24 020503

    [23]

    Tan Z Z 2015 Phys. Rev. E 91 052122

    [24]

    Tan Z Z 2015 Sci. Reports 5 11266

    [25]

    Tan Z Z, Zhang Q H 2015 Int. J. Circ. Theor. Appl. 43 944

    [26]

    Tan Z Z 2016 Chin. Phys. B 25 050504

    [27]

    Zhuang J, Yu G R, Nakayama K 2014 Sci. Reports 4 06720

    [28]

    Jia L P, Jasmina T, Duan W S 2015 Chin. Phys. Lett. 32 040501

    [29]

    Wang Y, Yang X R 2015 Chin. Phys. B 24 118902

    [30]

    Wang B, Huang H L, Sun Z Y, Kou S P 2012 Chin. Phys. Lett. 29 120301

  • [1]

    Kirchhoff G 1847 Ann. Phys. Chem. 148 497

    [2]

    Kirkpatrick S 1973 Rev. Mod. Phys. 45 574

    [3]

    Klein D J, Randi M 1993 J. Math. Chem. 12 81

    [4]

    Jafarizadeh S, Sufiani R, Jafarizadeh M A 2010 J. Stat. Phys. 139 177

    [5]

    Jzsef C 2000 Am. J. Phys. 68 896

    [6]

    Giordano S 2005 Int. J. Circ. Theor. Appl. 33 519

    [7]

    Asad J H 2013 J. Stat. Phys. 150 1177

    [8]

    Asad J H 2013 Mod. Phys. Lett. B 27 1350112

    [9]

    Wu F Y 2004 J. Phys. A:Math. Gen. 37 6653

    [10]

    Tzeng W J, Wu F Y 2006 J. Phys. A:Math. Gen. 39 8579

    [11]

    Izmailian N Sh, Kenna R, Wu F Y 2014 J. Phys. A:Math. Theor. 47 035003

    [12]

    Essam J W, Izmailian N S, Kenna R, Tan Z Z 2015 Royal Society Open Science 2 140420

    [13]

    Izmailian N S, Kenna R 2014 J. Stat. Mech. 09 P09016

    [14]

    Izmailian N S, Kenna R 2015 Chin. J. Phys. 53 040703

    [15]

    Tan Z Z 2011 Resistance Network Model (Xi'an:Xidian University Press) pp16-216(in Chinese)[谭志中2011电阻网络模型(西安:西安电子科技大学出版社)第16216页]

    [16]

    Tan Z Z, Zhou L, Yang J H 2013 J. Phys. A:Math. Theor. 46 195202

    [17]

    Tan Z Z, Zhou L, Luo D F 2015 Int. J. Circ. Theor. Appl. 43 329

    [18]

    Tan Z Z 2015 Int. J. Circ. Theor. Appl. 43 1687

    [19]

    Tan Z Z, Fang J H 2015 Commun. Theor. Phys. 63 36

    [20]

    Tan Z Z, Essam J W, Wu F Y 2014 Phys. Rev. E 90 012130

    [21]

    Essam J W, Tan Z Z, Wu F Y 2014 Phys. Rev. E 90 032130

    [22]

    Tan Z Z 2015 Chin. Phys. B 24 020503

    [23]

    Tan Z Z 2015 Phys. Rev. E 91 052122

    [24]

    Tan Z Z 2015 Sci. Reports 5 11266

    [25]

    Tan Z Z, Zhang Q H 2015 Int. J. Circ. Theor. Appl. 43 944

    [26]

    Tan Z Z 2016 Chin. Phys. B 25 050504

    [27]

    Zhuang J, Yu G R, Nakayama K 2014 Sci. Reports 4 06720

    [28]

    Jia L P, Jasmina T, Duan W S 2015 Chin. Phys. Lett. 32 040501

    [29]

    Wang Y, Yang X R 2015 Chin. Phys. B 24 118902

    [30]

    Wang B, Huang H L, Sun Z Y, Kou S P 2012 Chin. Phys. Lett. 29 120301

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Publishing process
  • Received Date:  06 October 2016
  • Accepted Date:  12 January 2017
  • Published Online:  05 April 2017

Calculation of the equivalent resistance and impedance of the cylindrical network based on recursion-transform method

    Corresponding author: Tan Zhi-Zhong, tanz@ntu.edu.cn;tanzzh@163.com
  • 1. Department of Physics, Nantong University, Nantong 226019, China;
  • 2. Department of Mathematics, Nantong University, Nantong 226019, China
Fund Project:  Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20161278).

Abstract: A classic problem in circuit theory first studied by German physicist Kirchhoff more than 170 years ago is the computation of resistances in resistor networks. Nowadays, resistor network has been an important model in the fields of natural science and engineering technology, but it is very difficult to calculate the equivalent resistance between two arbitrary nodes in an arbitrary resistor network. In 2004, Wu F Y formulated a Laplacian matrix method and derived expressions for the two-point resistance in arbitrary finite and infinite lattices in terms of the eigenvalues and eigenvectors of the Laplacian matrix, and the resistance results obtained by Laplacian matrix method is composed of double sums. The weakness of the Laplacian matrix approach is that it depends on the two matrices along two orthogonal directions. In 2011, Tan Z Z created the recursion-transform (RT) method, which can resolve the resistor network with arbitrary boundary. Using the RT method to compute the equivalent resistance relies on just one matrix along one direction, and the resistance is expressed by single summation. In the present paper, we investigate the equivalent resistance and complex impedance of an arbitrary mn cylindrical network by the RT method. Firstly, based on the network analysis, a recursion relation between the current distributions on three successive vertical lines is established through a matrix equation. In order to obtain the eigenvalues and eigenvectors of the matrix, and the general solution of the matrix equation, we then perform a diagonalizing transformation on the driving matrix.Secondly, we derive a recursion relation between the current distributions on the boundary, and construct some particular solutions of the matrix equation. Finally by using the matrix equation of inverse transformation, we obtain the analytical solution of the branch current, and gain the equivalent resistance formula along the axis of the arbitrary mn cylindrical network, which consists of the characteristic root and expressed by only single summation. As applications, several new formulae of equivalent resistances in the semi-infinite and infinite cases are given. These formulae are compared with those in other literature, meanwhile an interesting new identity of trigonometric function is discovered. At the end of the article, the equivalent impedance of the mn cylindrical RLC network is also treated, where the equivalent impedance formula is also given.

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