搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

基于递推-变换方法计算圆柱面网络的等效电阻及复阻抗

谭志中 张庆华

引用本文:
Citation:

基于递推-变换方法计算圆柱面网络的等效电阻及复阻抗

谭志中, 张庆华

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

Tan Zhi-Zhong, Zhang Qing-Hua
PDF
导出引用
  • 获得任意电阻网络等效电阻的解析解一直是科学和数学上的难题.本文采用递推-变换方法研究了一类任意mn阶圆柱面网络的等效电阻及复阻抗问题.首先采用网络分析建立递推矩阵方程模型;其次构造对角化矩阵变换方法以便获得矩阵的特征值和特征向量,从而获得矩阵方程的通解;再次采用网络分析建立边界条件约束方程模型,进而获得矩阵方程的特解;最后利用矩阵逆变换给出支路电流的解析解,从而获得任意mn阶圆柱面网络轴线上等效电阻的解析解,所得结果由特征根构成及单求和表达.作为公式的应用,给出了任意半无限和无限情形时的数个新的等效电阻公式,在与其他文献结论的对比研究中得到了一个有趣的新的三角函数恒等式.研究了圆柱面RLC网络的等效复阻抗问题,给出了精确的等效复阻抗公式.
    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.
      通信作者: 谭志中, tanz@ntu.edu.cn;tanzzh@163.com
    • 基金项目: 江苏省基础研究计划(自然科学基金)面上项目(批准号:BK20161278)资助的课题.
      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

  • [1] 赵超樱, 谭维翰. 三能级钾原子气体三维傅里叶变换频谱的解析解. 物理学报, 2020, 69(2): 020201. doi: 10.7498/aps.69.20190964
    [2] 韩忠明, 陈炎, 李梦琪, 刘雯, 杨伟杰. 一种有效的基于三角结构的复杂网络节点影响力度量模型. 物理学报, 2016, 65(16): 168901. doi: 10.7498/aps.65.168901
    [3] 蒋燕华, 陈佳民, 施娟, 周锦阳, 李华兵. 三角波脉动流通栓的晶格玻尔兹曼方法模型. 物理学报, 2016, 65(7): 074701. doi: 10.7498/aps.65.074701
    [4] 张志东, 高思敏, 王辉, 王红艳. 三角缺口正三角形纳米结构的共振模式. 物理学报, 2014, 63(12): 127301. doi: 10.7498/aps.63.127301
    [5] 王亮, 曹英晖, 贾峰, 刘震宇. 超椭圆柱面梯度线圈设计. 物理学报, 2014, 63(23): 238301. doi: 10.7498/aps.63.238301
    [6] 杨晓勇, 薛海斌, 梁九卿. 自旋相干态变换和自旋-玻色模型的基于变分法的基态解析解. 物理学报, 2013, 62(11): 114205. doi: 10.7498/aps.62.114205
    [7] 王飞, 魏兵. 电各向异性色散介质电磁散射时域有限差分分析的半解析递推卷积方法. 物理学报, 2013, 62(4): 044101. doi: 10.7498/aps.62.044101
    [8] 宋端, 刘畅, 郭永新. 高阶非完整约束系统嵌入变分恒等式的积分变分原理. 物理学报, 2013, 62(9): 094501. doi: 10.7498/aps.62.094501
    [9] 张希, 包伯成, 王金平, 马正华, 许建平. 固定关断时间控制Buck变换器输出电容等效串联电阻的稳定性分析. 物理学报, 2012, 61(16): 160503. doi: 10.7498/aps.61.160503
    [10] 范洪义, 展德会, 于文健, 周军. 厄米多项式算符的新恒等式及其在量子压缩中的应用. 物理学报, 2012, 61(11): 110302. doi: 10.7498/aps.61.110302
    [11] 周南润, 龚黎华, 贾芳. 基于双模相干-纠缠态表象的算符恒等式构造法. 物理学报, 2009, 58(4): 2179-2183. doi: 10.7498/aps.58.2179
    [12] 杨利霞, 葛德彪, 魏 兵. 电各向异性色散介质电磁散射的三维递推卷积-时域有限差分方法分析. 物理学报, 2007, 56(8): 4509-4514. doi: 10.7498/aps.56.4509
    [13] 贺 锋, 郭启波, 刘 辽. 用三角函数法获得非线性Boussinesq方程的广义孤子解. 物理学报, 2007, 56(8): 4326-4330. doi: 10.7498/aps.56.4326
    [14] 杨鹏飞. 一类带限定变换的二阶耦合线性微分方程组的解析解. 物理学报, 2006, 55(11): 5579-5584. doi: 10.7498/aps.55.5579
    [15] 蔡长英, 任中洲, 鞠国兴. 指数型变化有效质量的三维Schr?dinger方程的解析解. 物理学报, 2005, 54(6): 2528-2533. doi: 10.7498/aps.54.2528
    [16] 隆正文, 李子平. 高阶微商系统中正则Ward恒等式和Abel规范理论中动力学质量的产生. 物理学报, 2004, 53(7): 2100-2105. doi: 10.7498/aps.53.2100
    [17] 李孝申, 彭跃南. 相干三能级J—C模型的解析解. 物理学报, 1986, 35(1): 115-118. doi: 10.7498/aps.35.115
    [18] 李子平. 广义Noether恒等式及其应用. 物理学报, 1986, 35(4): 553-555. doi: 10.7498/aps.35.553
    [19] 姚希贤. 关于电阻分路约瑟夫森结的解析解. 物理学报, 1978, 27(5): 559-568. doi: 10.7498/aps.27.559
    [20] 赵保恒, 范洪义. 规范条件和Slavnov-Taylor恒等式的破坏. 物理学报, 1977, 26(6): 531-534. doi: 10.7498/aps.26.531
计量
  • 文章访问数:  5186
  • PDF下载量:  230
  • 被引次数: 0
出版历程
  • 收稿日期:  2016-10-06
  • 修回日期:  2017-01-12
  • 刊出日期:  2017-04-05

/

返回文章
返回