基于递推-变换方法计算圆柱面网络的等效电阻及复阻抗

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

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m×n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m×n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 m × n 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 m × n 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 m × n cylindrical RLC network is also treated, where the equivalent impedance formula is also given.