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任意矩形电路网络中的电位分布问题一直是科学研究的难题. 本研究发展了研究电阻网络的递推-变换(RT)理论使之能够用于计算任意m × n阶电路网络模型. 研究了一类含有任意边界的m × n阶矩形网络的电位分布及等效电阻, 这是一个之前一直没有解决的深刻问题, 因为之前的研究依赖于规则的边界条件或一个含有零电阻的边界条件. 其他方法如格林函数技术和拉普拉斯矩阵方法计算电位函数比较困难, 研究含有任意边界的电阻网络也是不可能的. 电位函数问题是自然科学和工程技术领域研究的一个重要内容, 如拉普拉斯方程的求解问题就是其中之一. 本文给出了含有一条任意边界的m × n矩形电阻网络的节点电位函数解析式, 并且得到了任意两节点间的等效电阻公式, 同时导出了一些特殊情形下的结果. 在对不同结果的比较研究时, 得到了一个新的数学分式恒等式.
The development of natural science raises many complex new problems and requires people to find the basic method to resolve them. It was found that many problems could be resolved by building the resistor network model. In 1845, the German scientist Kirchhoff set up the node current law and the circuit voltage law.Since then the basic theory of electric circuit has been established. At present, three general theories for studying large-scale resistor networks have been developed, for example, In 2000 Cserti [Am. J. Phys. 2000, 68 , 896] set up the Green function technique to evaluate the resistance of infinite lattices. In 2004 Wu [J. Phys. A: Math. Gen. 2014, 37 , 6653] formulated a Laplacian matrix method and calculated the resistance of arbitrary finite and infinite lattices by using the eigenvalues and eigenvectors. In 2011 Tan [Resistance Network Model (Xi’an: Xidian University Press) 2011, pp16–216] proposed the recursion-transform (RT) method which depends on the one matrix along one directions and avoids the trouble of the Laplacian method that depends on two matrices along two directions. Among them, only two theories can calculate both finite and infinite networks. One is Wu's Laplacian matrix method and the other is Tan's RT method. However, there is only one way to compute a resistor network with arbitrary boundary, that is, the Tan's RT method. Potential distribution problem in arbitrary rectangular circuit network has always been a problem of scientific research. In this paper, we develop the RT-I theory of resistor networks to calculate the arbitrary m × n circuit network model. We study the potential distribution and the equivalent resistance of a class of m × n rectangular network with an arbitrary boundary, a profound problem that has not been resolved so far, because previous research depends on the boundary conditions of rules or a zero-resistance boundary condition. Other methods, such as Green function technique and Laplacian method to calculate potential function are difficult and also impossible to study the resistor network with arbitrary boundary. Potential function problem is an important research subject in natural science and engineering technology, for example, the solution of Laplace's equation is one of research work. In this paper, we present an analytical expression of the node potential function of m × n rectangular resistor network with an arbitrary boundary, and also obtainan equivalent resistance formula between any two nodes, and the results in some special cases as well. In the comparative study of different results, a new mathematical identity and several interesting inferences are discovered. -
Keywords:
- complex networks /
- recursion-transform theory /
- matrix equation /
- potential function /
- boundary conditions /
- fraction identity
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图 1 含有一个任意边界的矩形电阻网络, 右边界上的任意电阻为r1, 其他在水平和竖直方向的电阻元素分别为r和r0
Fig. 1. An arbitrary rectangular m × n resistor network with an arbitrary boundary. Bonds in the horizontal and vertical directions are resistors r and r0 except for r1 on the right boundary.
图 2 含有电流参数和方向的部分电阻网络
Fig. 2. Segment of resistor network with current directions and parameters.
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[1] Kirkpatrick S 1973 Rev. Mod. Phys. 45 574
Google Scholar
[2] Melnikov A V, Shuba M, Lambin P 2018 Phys. Rev. E 97 043307
Google Scholar
[3] Cserti J 2000 Am. J. Phys. 68 896
Google Scholar
[4] Cserti J, David G, Piroth A 2002 Am. J. Phys. 70 153
Google Scholar
[5] Cserti J, Szechenyi G, David G 2011 J. Phys. A: Math. Theor. 44 215201
Google Scholar
[6] 梁昆淼, 刘法, 缪国庆 1998 数学物理方法 (北京: 高等教育出版社) 第459−467页
Liang K M, Liu F, Miao G Q 1998 Mathematical Physics Methods (Beijing: Higher Education Press) pp459−467 (in Chinese)
[7] Asad J H, Diab A A, Hijjawi R S, Khalifeh J M 2013 Eur. Phys. J. Plus 128 1
Google Scholar
[8] Asad J H 2013 J. Stat. Phys. 150 1177
Google Scholar
[9] Owaidat M Q, Hijjawi R S, Khalifeh J M 2014 Eur. Phys. J. Plus 129 29
Google Scholar
[10] Owaidat M Q, Asad J H, Tan Z Z 2016 Int. J. Mod. Phys. B 30 1650166
Google Scholar
[11] Owaidat M Q, Al-Badawi A A, Asad J H, Al-Twiessi M 2018 Chin. Phys. Lett. 35 020502
Google Scholar
[12] Owaidat M Q, Asad J H 2016 Eur. Phys. J. Plus 131 309
Google Scholar
[13] Wu F Y 2004 J. Phys. A: Math. Gen. 37 6653
Google Scholar
[14] Tzeng W J, Wu F Y 2006 J. Phys. A: Math. Gen. 39 8579
Google Scholar
[15] Izmailian N S, Huang M C 2010 Phys. Rev. E 82 011125
Google Scholar
[16] Essam J W, Wu F Y 2009 J. Phys. A: Math. Theor. 42 025205
Google Scholar
[17] Chair N 2012 Ann. Phys. 327 3116
Google Scholar
[18] Chair N 2014 Ann. Phys. 341 56
Google Scholar
[19] Chair N 2014 J. Stat. Phys. 154 1177
Google Scholar
[20] Izmailian N S, Kenna R, Wu F Y 2014 J. Phys. A: Math. Theor. 47 035003
Google Scholar
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Google Scholar
[24] 谭志中 2011 电阻网络模型 (西安: 西安电子科技大学出版社) 第16—216页
Tan Z Z 2011 Resistance Network Model (Xi’an: Xidian University Press) pp16–216 (in Chinese)
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Google Scholar
[26] Tan Z Z, Essam J W, Wu F Y 2014 Phys. Rev. E 90 012130
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Google Scholar
Tan Z Z, Zhang Q H 2017 Acta Phys. Sin. 66 070501
Google Scholar
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Google Scholar
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Google Scholar
[37] Tan Z Z, Asad J H, Owaidat M Q 2017 Int. J. Circ. Theor. Appl. 45 1942
Google Scholar
[38] Zhou L, Tan Z Z, Zhang Q H 2017 Front. Inf. Technol. Electron. Eng. 18 1186
Google Scholar
[39] Tan Z, Tan Z Z, Asad J H, Owaidat M Q 2019 Phys. Scripta 94 055203
Google Scholar
[40] Tan Z Z, Zhu H, Asad J H, Xu C, Tang H 2017 Front. Inf. Technol. Electron. Eng. 18 2070
Google Scholar
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Google Scholar
[42] Tan Z, Tan Z Z 2018 Sci. Rep. 8 9937
Google Scholar
[43] Tan Z Z 2017 Commun. Theor. Phys. 67 280
Google Scholar
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