1. Introduction
Resistor network models are important in the field of physics and engineering since the issues of various disciplines can be studied by simulating resistor network, such as percolation and conduction [1], Nonlinear localized modes in two-dimensional electrical lattices [2], Electric circuit networks equivalent to chaotic quantum billiards [3], photonic crystal circuits [4], Manifesting the evolution of eigenstates from quantum billiards [5], topological properties of linear circuit lattices [6], three-dimensional printed meshes [7], topological insulator surface [8], the mean field theory [9, 10], lattice Green’s functions [11–14], resistance distance [15], a recursion formula for resistance distances [16], and so on. In particular, two important equations of Poisson equation and Laplace equation [17, 18] can be simulated by resistor network model [19]. In addition, a real plane network of graphene exists in the real nature.
It is well known that calculating the equivalent resistance between two arbitrary lattice sites in a resistor network is always an important but difficult problem since it requires not only the circuit theory but also the innovative algebra. For example, when the boundary of resistor network is arbitrary, it is usually very difficult to obtain the exact potential and resistance of the complex networks with arbitrary boundaries. In fact, the boundary is like a wall or trap, which affects the solution of the problem. Therefore, the reality requires us to create new theories to accurately calculate the electrical characteristics (voltage and resistance) of the complex circuit network.
Let’s review the research history of resistor networks. In 1845 Kirchhoff established the basic circuit theory (the node current law and the circuit voltage law). After 150 years, Cserti [20] calculated the two-point resistance of the infinite network by Green’s function technique, which is mainly focused on infinite lattices, and some applications of Green’s function technique were obtained in later literature [21, 22]. In 2004 Wu [23] formulated a different approach (call the Laplacian matrix method) and derived the explicit resistance in arbitrary finite and infinite lattices with normative boundary (such as free, periodic boundary etc) in terms of the eigenvalues and eigenvectors of the Laplacian matrix, which relies on two matrices along two vertical directions. Later, the Laplacian matrix analysis has also been applied to impedance networks [24], after some improvements, several new resistor network problems have been resolved [25–27]. However, the Laplacian approach cannot apply to the network with arbitrary boundary since it is impossible to give the explicit eigenvalues for the arbitrary matrix elements (associating arbitrary boundaries). But the boundary condition is important since it is real case occurring in real life.
In 2011 Tan pioneered a new technique for studying complex resistor networks [28], which now is called recursion-transform (RT) theory of resistor networks [19]. Tan’s RT method depends on one matrix containing one directions, which is obviously different from the Laplacian method which depends on two matrices along two directions. With the development of the RT technique, many problems of non-regular network with zero resistor edges have been resolved [29–38]. In addition, the advantage of the RT method is that all resistance results are in a single summation differs from the Laplacian approach gave resistance results are in the form of a double summation. Recently, the RT method has been subdivided into two forms: one form is the matrix equation expressed by current parameters [31–37], which is simply called the RT-I method; another form is the matrix equation expressed by potential parameters [19, 38], which is simply called the RT-V method. Summarizing the previous applications of the RT (including RT-I and RT-V) method, it is not hard to see that the previous studies have not solved all the resistor networks, but only solved some personalized problems that depend on zero resistor boundary conditions, such as the globe network [29, 37] belongs to cylindrical network with two zero resistor boundaries, the cobweb network [19, 33] belongs to cylindrical network with one zero resistor boundary, the fan network [30, 38] belongs to nonregular rectangular network with one zero resistor boundary, and the hammock network [27, 36] belongs to nonregular rectangular network with two zero resistor boundaries. Obviously, how to study the complex network without zero resistor boundary by the RT method is a question.
This paper developed the RT theory to allow us to study arbitrary resistor networks without relying on zero resistor boundary, which can derive the electrical properties (potential and resistance) of the arbitrary m × n complex networks with complex boundaries. Here we build two new theorems lead to large problems to be resolved. Our study shows the universal RT method is very interesting and useful to solve the complex network. We focus on researching the electrical properties (potential and resistance) of figures 1 and 2 on two complex m × n resistor networks with two arbitrary boundaries by the advanced RT-V method, which have not been resolved before. It is worth emphasizing that the non-regular complex networks with two arbitrary boundaries are the multi-purpose network model because it can produce various geometrical structure as shown in figures 4 and 5. Thus a large number of problems of resistor networks will be resolved by this paper.
Figure 1. An arbitrary m × n resistor network with two arbitrary boundary resistors, where n and m are the maximum coordinate values of (n, m). Bonds in the horizontal and vertical directions are resistors r and r0 except for two arbitrary boundary resistors of r1 and r2. |
Figure 2. A nonregular cylindrical m × n resistor network, where n and m are the maximum coordinate value of (n, m), with the resistors r and r0 in the respective horizontal and vertical (loop) directions except for two arbitrary boundary resistors of r1 and r2. |
Case-1. Consider figure 1 with r1 = r2 = r0 is a regular m × n rectangle network, where n and m are the maximum coordinate values of (n, m), resistors r and r0 are bonded respectively in the horizontal and vertical directions. The resistance formula for figure 1 is
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{R}_{m\times n}^{}({d}_{1},{d}_{2})=\displaystyle \frac{r}{m+1}\left|{x}_{1}-{x}_{2}\right|+\displaystyle \frac{{r}_{0}}{n+1}\left|{y}_{1}-{y}_{2}\right|\\ +\,\displaystyle \frac{2}{(m+1)(n+1)}\\ \times \,\displaystyle \displaystyle \sum _{i=1}^{m}\displaystyle \displaystyle \sum _{j=1}^{n}\displaystyle \frac{{\left[{C}_{{x}_{1},j}\cos \left({y}_{1}+\tfrac{1}{2}\right){\theta }_{i}-{C}_{{x}_{2},j}\cos \left({y}_{2}+\tfrac{1}{2}\right){\theta }_{i}\right]}^{2}}{{r}^{-1}(1-\cos {\phi }_{j})+{r}_{0}^{-1}(1-\cos {\theta }_{i})},\end{array}\end{array}\end{eqnarray}$
where ${C}_{{x}_{k},j}=\cos \left({x}_{k}+\tfrac{1}{2}\right){\phi }_{j}$, ${\theta }_{{\rm{i}}}={i}\pi /(m+1)$, ${\phi }_{j}=j\pi /(n+1)$ and ${d}_{1}({x}_{1},{y}_{1})$ and ${d}_{2}({x}_{2},{y}_{2})$ are arbitrary two nodes in the network.Case-2. Consider figure 2 with r1 = r2 = r0 is a cylindrical m × n resistor network, where n and m are the maximum coordinate values of (n, m), resistors r and r0 are bonded respectively in the horizontal and vertical directions. The resistance formula for figure 2 is
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{{r}_{0}}{n+1}\left(\left|{y}_{1}-{y}_{2}\right|-\displaystyle \frac{{\left({y}_{1}-{y}_{2}\right)}^{2}}{m+1}\right)\\ +\,\displaystyle \frac{r}{m+1}\left|{x}_{1}-{x}_{2}\right|+\displaystyle \frac{1}{(m+1)(n+1)}\\ \times \,\displaystyle \displaystyle \sum _{i=1}^{m}\displaystyle \displaystyle \sum _{j=1}^{n}\displaystyle \frac{{C}_{{x}_{1},j}^{2}+{C}_{{x}_{2},j}^{2}-2{C}_{{x}_{1},j}{C}_{{x}_{2},j}\cos 2({y}_{2}-{y}_{1}){\theta }_{i}}{{r}_{0}^{-1}(1-\cos 2{\theta }_{i})+{r}^{-1}(1-\cos {\phi }_{j})},\end{array}\end{array}\end{eqnarray}$
where ${C}_{{x}_{k},j}=\cos \left({x}_{k}+\tfrac{1}{2}\right){\phi }_{j}$, ${\theta }_{i}={i}\pi /(m+1)$, ${\phi }_{j}\,=j\pi /(n+1)$.The above results were found for the first time by Wu. Later [25–27] improved the Laplacian matrix method to make it applicable to regular cobweb and hammock networks. However, the improved Wu method still cannot resolve the resistor network with arbitrary boundary, such as the networks with two arbitrary boundaries of figures 1 and 2. In addition, the equivalent resistances in equations (1 ) and (2 ) are in the double summation but not in a single sum.
2. RT-V theory and Poisson equation
Consider two kinds of complex m × n resistor networks of figures 1 and 2, where n and m are the maximum coordinate values of (n, m). Assume A0 (0, 0) is the origin of the rectangular coordinate system, and denoting nodes of the network by coordinate (x, y). Assume the electric current J goes from the input d1 (x1, y1) to the output d2 (x2, y2). Denote the nodal potential of the sub-network is shown in figure 3, and expressing the nodal potential at d (x, y) by ${U}_{m\times n}(x,y)={V}_{x}^{(y)}$. We will study the complex resistor networks in four steps.
Figure 3. The resistor sub-network with the potential and resistor parameters. |
The first step, setting up discrete Poisson equation based on the sub-network of figure 3. By Kirchhoff law ($\sum {r}_{i}^{-1}{V}_{k}=0$) to set up the nodal potential equations along the vertical direction, we achieve a discrete static field equation (or call Poisson equation) for any network3 ) reduces to the discrete Laplace equation $({{\rm{\Delta }}}_{x}^{2}+h{{\rm{\Delta }}}_{y}^{2}){V}_{x}^{(y)}\,=0$. For the arbitrary network together with the upper and lower boundary conditions, by equation (3 ) we are led to
$\begin{eqnarray}({{\rm{\Delta }}}_{x}^{2}+h{{\rm{\Delta }}}_{y}^{2}){V}_{x}^{(y)}=-{{rI}}_{x}^{(y)}{\delta }_{x,{x}_{k}},\end{eqnarray}$
where $h=r/{r}_{0}$, and ${I}_{x}^{(y)}=J({\delta }_{y,{y}_{1}}-{\delta }_{y,{y}_{2}})$ contains the input and output conditions of the current, ${{\rm{\Delta }}}_{x}^{2}{V}_{x}^{(y)}={V}_{x\,+\,1}^{(y)}\,-2{V}_{x}^{(y)}+{V}_{x-1}^{(y)}$ and ${{\rm{\Delta }}}_{y}^{2}{V}_{x}^{(y)}={V}_{x}^{(y+1)}-2{V}_{x}^{(y)}+{V}_{x}^{(y-1)}$ denote second order discrete equation, and when ${x}_{k}\ne x$, equation ( $\begin{eqnarray}{{\boldsymbol{V}}}_{k+1}={{\boldsymbol{A}}}_{m+1}{{\boldsymbol{V}}}_{k}-{{\boldsymbol{V}}}_{k-1}-r{{\boldsymbol{I}}}_{k}{\delta }_{k,x},\end{eqnarray}$
where ${{\boldsymbol{V}}}_{k}$ and ${{\boldsymbol{I}}}_{x}$ are respectively two column matrixes, and reads $\begin{eqnarray}{{\boldsymbol{V}}}_{k}={\left[{V}_{k}^{(0)},{V}_{k}^{(1)},{V}_{k}^{(2)},\cdots ,{V}_{k}^{\left(m\right)}\right]}^{{\rm{T}}},\end{eqnarray}$
$\begin{eqnarray}{{\boldsymbol{I}}}_{k}={\left[{I}_{k}^{(0)},{I}_{k}^{(1)},{I}_{k}^{(2)},\cdots ,{I}_{k}^{(m)}\right]}^{{\rm{T}}},\end{eqnarray}$
and ${{\boldsymbol{A}}}_{m+1}$ is the matrix built along the vertical direction. For figures 1 and 2, the ${{\boldsymbol{A}}}_{m+1}$ is $\begin{eqnarray}\begin{array}{l}{{\boldsymbol{A}}}_{m+1}\\ =\,\left(\begin{array}{ccccc}2+h+{bh} & -h & 0 & \cdots & -{bh}\\ -h & 2+2h & -h & \cdots & \cdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & \cdots & -h & \,\,2+2h & -h\\ -{bh} & \cdots & 0 & -h & 2+h+{bh}\end{array}\right),\end{array}\end{eqnarray}$
where $b={r}_{0}/{r}_{3}$, and r3 is the resistor between (x, 0) and (x, m) in figure 2, when b = 0 $({r}_{3}=\infty )$, the ${{\boldsymbol{A}}}_{m+1}$ belongs to figure 1; when b = 1 (r3 = r0), the ${{\boldsymbol{A}}}_{m+1}$ belongs to figure 2. The purpose of introducing r3 is to express two different resistor networks uniformly.The second step, consider the boundary conditions of the left and right edges in the network of figures 1 and 2. Applying Kirchhoff’s law ($\sum {r}_{i}^{-1}{V}_{k}=0$) to each of the left and right boundaries, we obtain two matrix equations of boundary conditions7 ).
$\begin{eqnarray}{h}_{1}{{\boldsymbol{V}}}_{1}=[{{\boldsymbol{A}}}_{m+1}-(2-{h}_{1}){\boldsymbol{E}}]{{\boldsymbol{V}}}_{0},\end{eqnarray}$
$\begin{eqnarray}{h}_{2}{{\boldsymbol{V}}}_{n-1}=[{{\boldsymbol{A}}}_{m+1}-(2-{h}_{2}){\boldsymbol{E}}]{{\boldsymbol{V}}}_{n},\end{eqnarray}$
where ${h}_{1}={r}_{1}/{r}_{0},{h}_{2}={r}_{2}/{r}_{0}$, ${\boldsymbol{E}}$ is the $(m+1)\times (m+1)$ identity matrix, matrix ${{\boldsymbol{A}}}_{m+1}$ is given by equation (Equations (4 )–(9 ) are all the equations we need to compute the node potential. However, it is impossible for us to get the solution of the above equations directly. Thanks to the RT theory of Tan that gave the matrix transform method [19, 31–33] and we create the new technique here. In the following we are going to give the transformation technology based on RT-V theory.
The third step, creating matrix transform. Firstly, we work out the eigenvalue ${t}_{i}$ of matrix ${{\boldsymbol{A}}}_{m+1}$, which is given by solving determinantal equation of $\det \left|{{\boldsymbol{A}}}_{m+1}-t{\boldsymbol{E}}\right|=0$ (just b = 0 and b = 1), yields4 )–(9 ) by the following approaches4 ) from the left-hand side by ${{\boldsymbol{P}}}_{m+1}$, we get11 ) and (12 ) are used.
$\begin{eqnarray}{t}_{i}=2(1+h)-2h\cos {\theta }_{i},\quad (i=0,1,\cdots ,\ m),\end{eqnarray}$
where ${\theta }_{i}=(1+b){i}\pi /(m+1)$, and b = 0 for figure 1, b = 1 for figure 2. Next to transform equations ( $\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}{{\boldsymbol{A}}}_{m+1}=\mathrm{diag}\{{t}_{0},{t}_{1},\cdots ,\ {t}_{m}\}{{\boldsymbol{P}}}_{m+1},\end{eqnarray}$
$\begin{eqnarray}{{\boldsymbol{X}}}_{k}={{\boldsymbol{P}}}_{m+1}{{\boldsymbol{V}}}_{k}\,\mathrm{or}\,{{\boldsymbol{V}}}_{k}={\left({{\boldsymbol{P}}}_{m+1}\right)}^{-1}{{\boldsymbol{X}}}_{k},\end{eqnarray}$
where ${{\boldsymbol{X}}}_{k}={\left[{X}_{k}^{(0)},{X}_{k}^{(1)},\cdots ,{X}_{k}^{(m)}\right]}^{{\rm{T}}}$. Assuming Pi is the row vectors of matrix ${{\boldsymbol{P}}}_{m+1}$, such as $\begin{eqnarray}{P}_{i}=\left[{\zeta }_{0,i},{\zeta }_{1,i},{\zeta }_{2,i},\cdots ,\ {\zeta }_{m,i}\right].\end{eqnarray}$
Thus, we multiply equation ( $\begin{eqnarray}{X}_{k+1}^{(i)}={t}_{i}{X}_{k}^{(i)}-{X}_{k-1}^{(i)}-{rJ}({\delta }_{{x}_{1},k}{\zeta }_{{y}_{1},i}-{\delta }_{{x}_{2},k}{\zeta }_{{y}_{2},i}),\end{eqnarray}$
where equations (Similarly, applying ${{\boldsymbol{P}}}_{m+1}$ to equations (8 ) and (9 ), we are led to10 )–(16 ) are all essential equations for evaluating the node potential.
$\begin{eqnarray}{h}_{1}{X}_{1}^{(i)}=({t}_{i}+{h}_{1}-2){X}_{0}^{(i)},\end{eqnarray}$
$\begin{eqnarray}{h}_{2}{X}_{n-1}^{(i)}=({t}_{i}+{h}_{2}-2){X}_{n}^{(i)}.\end{eqnarray}$
The above equations (The fourth step, solving the matrix equations (13 )–(16 ) Selecting ${\sum }_{i=0}^{m}{V}_{0}^{(i)}=\tfrac{1}{2}({x}_{2}-{x}_{1}){rJ}$ as the reference potential (notice: the potential is a relative reference value that you can artificially assume), by equations (14 )–(16 ) we obtain after some algebra and reduction the solution28 ) below, and have19 )–(25 ) below.
$\begin{eqnarray}{X}_{k}^{(0)}=\displaystyle \frac{1}{\sqrt{2-b}}\left(\displaystyle \frac{{x}_{1}+{x}_{2}}{2}-{x}_{\tau }\right){rJ},\end{eqnarray}$
where b = 0 for figure 1, b = 1 for figure 2, and xτ is defined in equation ( $\begin{eqnarray}{X}_{k}^{(i)}=\displaystyle \frac{{\beta }_{k\vee {x}_{1}}^{(i)}{\zeta }_{{y}_{1},i}-{\beta }_{k\vee {x}_{2}}^{(i)}{\zeta }_{{y}_{2},i}}{({t}_{i}-2){G}_{n}^{(i)}}{rJ},(i\geqslant 1),\end{eqnarray}$
where ${C}_{k,i}$, ${\beta }_{k,s}^{(i)}$, ${G}_{k}^{(i)}$ are, respectively, defined in equations (The RT-V theory. The above method of establishing recursive matrix equations with voltage parameters, implementing matrix transform and obtaining the solutions of matrix equations is called RT-V theory. The detailed content of the RT-V theory (recursion-transform theory with potential parameters) can be found by the above four steps in equation (3 )–(18 ).
3. Two theorems of resistor networks
3.1. Several definitions
In order to facilitate and simplify the expression of the solutions of matrix equations, we define several variables of ${C}_{k,i}$ and ${\lambda }_{i},{\bar{\lambda }}_{i}$ for later uses
$\begin{eqnarray}{C}_{{y}_{k},i}=\cos \left({y}_{k}+\displaystyle \frac{1}{2}\right){\theta }_{i},\,{C}_{{y}_{k}-y}=\cos ({y}_{k}-y){\theta }_{i},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{i} & = & 1+h-h\cos {\theta }_{i}+\sqrt{{(1+h-h\cos {\theta }_{i})}^{2}-1},\\ {\bar{\lambda }}_{i} & = & 1+h-h\cos {\theta }_{i}-\sqrt{{(1+h-h\cos {\theta }_{i})}^{2}-1}.\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{c}{\theta }_{i}=(1+b){i}\pi /(m+1),\left\{\begin{array}{l}b=0\,{\rm{f}}{\rm{o}}{\rm{r}}\,{\rm{f}}{\rm{i}}{\rm{g}}{\rm{u}}{\rm{r}}{\rm{e}}\,1,\\ b=1\,{\rm{f}}{\rm{o}}{\rm{r}}\,{\rm{f}}{\rm{i}}{\rm{g}}{\rm{u}}{\rm{r}}{\rm{e}}\,2\end{array}\right..\end{array}\end{eqnarray}$
And define variables ${F}_{k}^{(i)}$ , ${\alpha }_{s,x}^{(i)}$, ${\beta }_{k,s}^{(i)}$ and ${G}_{k}^{(i)}$ for later uses by $\begin{eqnarray}{F}_{k}^{(i)}=({\lambda }_{i}^{k}-{\bar{\lambda }}_{i}^{k})/({\lambda }_{i}-{\bar{\lambda }}_{i}),\,{\rm{\Delta }}{F}_{k}^{(i)}={F}_{k+1}^{(i)}-{F}_{k}^{(i)}.\end{eqnarray}$
$\begin{eqnarray}{\alpha }_{s,x}^{(i)}={\rm{\Delta }}{F}_{x}^{(i)}+({h}_{s}-1){\rm{\Delta }}{F}_{x-1}^{(i)},\,{h}_{s}={r}_{s}/{r}_{0}.\end{eqnarray}$
$\begin{eqnarray}{\beta }_{x\vee {x}_{s}}^{(i)}=\left\{\begin{array}{l}{\beta }_{x,{x}_{s}}^{(i)}={\alpha }_{1,x}^{(i)}{\alpha }_{2,n-{x}_{s}}^{(i)},\,\mathrm{if}\ \ \,x\leqslant {x}_{s}\\ {\beta }_{{x}_{s},x}^{(i)}={\alpha }_{1,{x}_{s}}^{(i)}{\alpha }_{2,n-x}^{(i)},\,\mathrm{if}\ \ \,x\geqslant {x}_{s}\end{array}\right.,\end{eqnarray}$
$\begin{eqnarray}{G}_{n}^{(i)}={F}_{n+1}^{(i)}+({h}_{1}+{h}_{2}-2){F}_{n}^{(i)}+({h}_{2}-1)({h}_{1}-1){F}_{n-1}^{(i)}.\end{eqnarray}$
The above definitions are applicable to the entire article. All of these definitions are meant to illustrate the following two fundamental theorems, and we always assume that the electric current J goes from the input ${d}_{1}({x}_{1},{y}_{1})$ to the output ${d}_{2}({x}_{2},{y}_{2})$ in our entire paper.
3.2. Two fundamental theorems
Consider the arbitrary $m\times n$ resistor networks of figures 1 and 2 whose maximum coordinate value is $(n,m)$. Then the potential of node $d(x,y)$ in the $m\times n$ resistor network can be written as
$\begin{eqnarray}{V}_{m\times n}(x,y)=\displaystyle \frac{1}{({\zeta }_{k,i},{\bar{\zeta }}_{k,i})}\displaystyle \sum _{i=0}^{m}{X}_{x}^{(i)}{\bar{\zeta }}_{y,i},\end{eqnarray}$
where $({\zeta }_{k,i},{\bar{\zeta }}_{k,i})={\sum }_{k=0}^{m}{\zeta }_{k,i}{\bar{\zeta }}_{k,i}$, and ${\zeta }_{y,i}$ is defined in equation (In particular, when selecting ${\sum }_{i=0}^{m}{V}_{0}^{(i)}=\tfrac{1}{2}({x}_{2}-{x}_{1}){rJ}$ as the reference potential, the potential of node d(x, y) in the m × n resistor networks can be written as18 ) which is the solution of equations (13 )–(16 ).
$\begin{eqnarray}{V}_{m\times n}(x,y)=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}{rJ}+\displaystyle \frac{2-b}{m+1}\displaystyle \sum _{i=1}^{m}{X}_{x}^{(i)}{\bar{\zeta }}_{y,i},\end{eqnarray}$
where $\bar{x}=({x}_{1}+{x}_{2})/2$, ${\bar{\zeta }}_{y,i}$ is the conjugate complex of ${\zeta }_{y,i}$ (there be ${\bar{\zeta }}_{y,i}={\zeta }_{y,i}$ if ${\zeta }_{y,i}$ is just a real number), and xτ is a piecewise function $\begin{eqnarray}\begin{array}{l}{x}_{\tau }=\left\{{x}_{1},0\leqslant x\leqslant {x}_{1}\right\}\cup \left\{x,{x}_{1}\leqslant x\leqslant {x}_{2}\right\}\\ \quad \cup \,\left\{{x}_{2},{x}_{2}\leqslant x\leqslant n\right\}\end{array}\end{eqnarray}$
and ${X}_{k}^{(i)}$ is given by (Consider the arbitrary $m\times n$ resistor networks of figures 1 and 2 whose maximum coordinate value is $(n,m)$. Then the resistance between any two nodes ${d}_{1}({x}_{1},{y}_{1})$ and ${d}_{2}({x}_{2},{y}_{2})$ in the network is given by
$\begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{1}{({\zeta }_{y,i},{\bar{\zeta }}_{y,i})}\displaystyle \sum _{i=0}^{m}\displaystyle \frac{{X}_{{x}_{1}}^{(i)}{\bar{\zeta }}_{{y}_{1},i}-{X}_{{x}_{2}}^{(i)}{\bar{\zeta }}_{{y}_{2},i}}{J},\end{eqnarray}$
where ${X}_{k}^{(i)}$ is the solution of the matrix equation (In particular, for the networks of figures 1 and 2, the resistance between two nodes ${d}_{1}({x}_{1},{y}_{1})$ and d2 (x2, y2) can be written as18 ) which is the solution of equations (13 )–(16 ).
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{1}-{x}_{2}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{2-b}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{X}_{{x}_{1}}^{(i)}{\bar{\zeta }}_{{y}_{1},i}-{X}_{{x}_{2}}^{(i)}{\bar{\zeta }}_{{y}_{2},i}}{J},\end{array}\end{eqnarray}$
where b = 0 is the case of figure 1, and b = 1 is the case of figure 2, and ${X}_{k}^{(i)}$ is given by (The above two new theorems contain a wide variety of geometric structure of the network model, which can produce many new results of potential and resistance, we are going to prove the correctness of two theorems.
3.3. Proof of theorems
Consider the m × n resistor network with two arbitrary boundaries shown in figures 1 and 2, in the introduction, we have built the key equations (4 )–(9 ) by the RT-V theory, and converted the equations to equations (14 )–(16 ) and derived equations (17 ) and (18 ). Now we will work out the exact eigenvalues of matrix ${{\boldsymbol{A}}}_{m+1}$ in equation (7 ). Equation (10 ) can be derived by solving equation $\det \left|{{\boldsymbol{A}}}_{m+1}-t{\boldsymbol{E}}\right|=0$, and then we need to consider two cases below.
One is for figure 1, substituting equation (10 ) into (11 ) with b = 0 in ${{\boldsymbol{A}}}_{m+1}$, we get the eigenvectors
$\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}=\left(\begin{array}{cccc}1/\sqrt{2} & 1/\sqrt{2} & \cdots & 1/\sqrt{2}\\ \cos ({v}_{0}{\theta }_{1}) & \cos ({v}_{1}{\theta }_{1}) & \cdots & \cos ({v}_{m}{\theta }_{1})\\ \vdots & \vdots & \ddots & \vdots \\ \cos ({v}_{0}{\theta }_{m}) & \cos ({v}_{1}{\theta }_{m}) & \cdots & \cos ({v}_{m}{\theta }_{m})\end{array}\right),\end{eqnarray}$
where ${v}_{k}=k+1/2$, and ${\theta }_{i}={i}\pi /(m+1)$. By careful calculation, the inverse matrix can be easily obtained $\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}^{-1}=\displaystyle \frac{2}{m+1}{[{{\boldsymbol{P}}}_{m+1}]}^{{\rm{T}}},\end{eqnarray}$
where ${[]}^{{\rm{T}}}$ denote matrix transpose.Thus, the term ${\zeta }_{k,i}$ appearing in equation (13 ) can be specifically rewritten as
$\begin{eqnarray}{\bar{\zeta }}_{y,0}={\zeta }_{y,0}=1/\sqrt{2},\,(0\leqslant y\leqslant m),\end{eqnarray}$
$\begin{eqnarray}{\bar{\zeta }}_{y,i}={\zeta }_{y,i}=\cos \left(y+\displaystyle \frac{1}{2}\right){\theta }_{i},\,(i\geqslant 1).\end{eqnarray}$
Another is for figure 2, substituting equation (10 ) into (11 ) with b = 1 in ${{\boldsymbol{A}}}_{m+1}$, the eigenvector is obtained after some algebra and derivation
$\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}^{}=\left(\begin{array}{cc}\begin{array}{cc}1 & 1\qquad \,\,1\\ 1 & \exp ({\rm{i}}{\theta }_{1})\,\exp ({\rm{i}}2{\theta }_{1})\end{array} & \begin{array}{cc}\cdots & 1\\ \cdots & \exp ({\rm{i}}m{\theta }_{1})\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ 1 & \exp ({\rm{i}}{\theta }_{m})\,\exp ({\rm{i}}2{\theta }_{m})\end{array} & \begin{array}{cc}\ddots & \vdots \\ \cdots & \exp ({\rm{i}}m{\theta }_{m})\end{array}\end{array}\right),\end{eqnarray}$
where ${{\rm{i}}}^{2}=-1$, ${\theta }_{i}=2{i}\pi /(m+1)\,(i=0,1,2,\cdots ,m)$. According to strict calculations, the inverse matrix reads $\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}^{-1}=\displaystyle \frac{1}{m+1}\left(\begin{array}{cc}\begin{array}{cc}1 & 1\\ 1 & \exp (-{\rm{i}}{\theta }_{1})\end{array} & \begin{array}{cc}\cdots & 1\\ \cdots & \exp (-{\rm{i}}{\theta }_{m})\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ 1 & \exp (-{\rm{i}}m{\theta }_{1})\end{array} & \begin{array}{cc}\ddots & \vdots \\ \cdots & \exp (-{\rm{i}}m{\theta }_{m})\end{array}\end{array}\right).\end{eqnarray}$
Thus, the term ${\zeta }_{k,i}$ appearing in equation (13 ) can be specifically rewritten as
$\begin{eqnarray}{\zeta }_{y,0}={\bar{\zeta }}_{y,0}=1,\,(0\leqslant y\leqslant m),\end{eqnarray}$
$\begin{eqnarray}{\zeta }_{y,i}=\exp ({\rm{i}}y{\theta }_{i}),\,{\bar{\zeta }}_{y,i}=\exp (-{\rm{i}}y{\theta }_{i}).\end{eqnarray}$
We find that equations (32 ) and (36 ) can be rewritten as a unified form below39 ) is an important innovation which is the key to our unified study of resistor networks.
$\begin{eqnarray}{{\boldsymbol{P}}}_{m+1}^{-1}=\displaystyle \frac{1}{({\zeta }_{k,i},{\bar{\zeta }}_{k,i})}\left(\begin{array}{cc}\begin{array}{cc}{\bar{\zeta }}_{\mathrm{0,0}} & {\bar{\zeta }}_{\mathrm{0,1}}\\ {\bar{\zeta }}_{\mathrm{1,0}} & {\bar{\zeta }}_{\mathrm{1,1}}\end{array} & \begin{array}{cc}\cdots & {\bar{\zeta }}_{0,m}\\ \cdots & {\bar{\zeta }}_{1,m}\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ {\bar{\zeta }}_{m,0} & {\bar{\zeta }}_{m,1}\end{array} & \begin{array}{cc}\ddots & \vdots \\ \cdots & {\bar{\zeta }}_{m,m}\end{array}\end{array}\right),\end{eqnarray}$
where $({\zeta }_{k,i},{\bar{\zeta }}_{k,i})={\sum }_{k=0}^{m}{\zeta }_{k,i}{\bar{\zeta }}_{k,i}$, and ${\bar{\zeta }}_{y,i}$ is the conjugate complex of ${\zeta }_{y,i}$. Equation (Using equation (12 ), we have ${{\boldsymbol{V}}}_{k}={({{\boldsymbol{P}}}_{m+1})}^{-1}{{\boldsymbol{X}}}_{k}$, expanding this matrix equation, then we get40 ) agrees with the formula (26 ) that we need to verify.
$\begin{eqnarray}{V}_{k}^{(y)}=\displaystyle \frac{1}{({\zeta }_{k,i},{\bar{\zeta }}_{k,i})}\left({X}_{k}^{(0)}{\bar{\zeta }}_{y,0}+\displaystyle \sum _{i=1}^{m}{X}_{k}^{(i)}{\bar{\zeta }}_{y,i}\right),\end{eqnarray}$
Equation (Further, we get $({\zeta }_{k,i},{\bar{\zeta }}_{k,i})=(m+1)/(2-b)$ by comparing equation (39 ) with equations (32 ) and (36 ). And when selecting ${\sum }_{i=0}^{m}{V}_{0}^{(i)}=\tfrac{1}{2}({x}_{2}-{x}_{1}){rJ}$, we have equation (17 ). Substituting equations (17 ), (33 ) and (37 ) into equation (40 ), then equation (27 ) can be verified immediately.
Next, we verify equations (29 ) and (30 ) by Ohm’s Law, we have26 ) with $x=\{{x}_{1},{x}_{2}\}$ and $y=\{{y}_{1},{y}_{2}\}$ into equation (41 ), we therefore obtain equation (29 ). Substituting equation (27 ) with $x=\{{x}_{1},{x}_{2}\}$ and $y=\{{y}_{1},{y}_{2}\}$ into equation (41 ), we immediately obtain equation (30 ). Thus, two theorems are verified.
$\begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{1}{J}[V({x}_{1},{y}_{1})-V({x}_{2},{y}_{2})].\end{eqnarray}$
Substituting equation (In subsequent sections we consider applications of theorems to arbitrary lattices. In all applications, we stipulate all parameters in equations (18 )–(39 ) apply to all resistor networks, and denote the resistors along the two principal directions by r and r0 except for resistors on the left-right boundaries, and the input and output nodes of current are respectively at ${d}_{1}({x}_{1},{y}_{1})$ and ${d}_{2}({x}_{2},{y}_{2})$.
4. Electrical properties of complex rectangular network
4.1. Nodal potential of complex rectangular network
Consider the non-regular m × n resistor network shown in figure 1, where the maximum coordinate is (n, m), selecting ${\sum }_{i=0}^{m}{V}_{0}^{(i)}=\tfrac{1}{2}({x}_{2}-{x}_{1}){rJ}$ as the reference potential, the potential of any node d(x, y) in the finite and sem-infinite networks can be written as19 )–(25 ). For equation (43 ), there be $n\to \infty $, ${x}_{1},{x}_{2}\to \infty $ with finite ${x}_{k}-x$.
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r+\displaystyle \frac{{r}_{0}}{m+1}\\ \,\times \,\displaystyle \displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1},i}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2},i}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}{C}_{y,i},\end{array}\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}\displaystyle \frac{{U}_{m\times \infty }(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r+\displaystyle \frac{r}{m+1}\\ \,\times \,\displaystyle \displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\bar{\lambda }}_{i}^{\left|{x}_{1}-x\right|}{C}_{{y}_{1},i}-{\bar{\lambda }}_{i}^{\left|{x}_{2}-x\right|}{C}_{{y}_{2},i}}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}}{C}_{y,i},\end{array}\end{array}\end{eqnarray}$
where ${\theta }_{i}={i}\pi /(m+1)$, and ${C}_{k,i}$, ${\beta }_{k,s}^{(i)}$ , ${G}_{k}^{(i)}$ are, respectively, defined in equations (In particular, when x2 = x1 (means the input and output nodes of currents are at the same vertical axis), formulae (42 ) and (43 ) reduce to
$\begin{eqnarray}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{({C}_{{y}_{1},i}-{C}_{{y}_{2},i}){C}_{y,i}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}{\beta }_{{x}_{1}\vee x}^{(i)},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{U}_{m\times \infty }(x,y)}{J}=\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\bar{\lambda }}_{i}^{\left|{x}_{1}-x\right|}({C}_{{y}_{1},i}-{C}_{{y}_{2},i}){C}_{y,i}}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}}.\end{eqnarray}$
For figure 1, substituting equation (
$\begin{eqnarray}{X}_{k}^{(i)}=\displaystyle \frac{{\beta }_{k\vee {x}_{1}}^{(i)}{C}_{{y}_{1},i}-{\beta }_{k\vee {x}_{2}}^{(i)}{C}_{{y}_{2},i}}{({t}_{i}-2){G}_{n}^{(i)}}{rJ},(1\leqslant k\leqslant n).\end{eqnarray}$
Substituting equation (For proving equation (43 ), when $n\to \infty $, ${x}_{1},{x}_{2}\to \infty $ with finite ${x}_{k}-x$, it can be got a limit by using equations (20 )–(25 )47 ) into (42 ) with $n\to \infty $, we therefore verified equation (43 ).
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{\displaystyle \genfrac{}{}{0em}{}{n\to \infty }{x\to \infty }}\displaystyle \frac{{\beta }_{{x}_{k}\vee x}^{(i)}}{{G}_{n}^{(i)}}=({t}_{i}-2)\displaystyle \frac{{\bar{\lambda }}_{k}^{\left|{x}_{k}-x\right|}}{{\lambda }_{i}-{\bar{\lambda }}_{i}}.\end{eqnarray}$
So, substituting equation (Formula (42 ) is a meaningful result because the network of figure 1 is very complex and has not been resolved before, which contains a lot of different network models since the different boundary resistors can produce different geometric structures. Here several special applications of formula (42 ) are given below.
When ${r}_{1}={r}_{2}={r}_{0}$, figure 1 degrades into a regular $m\times n$ rectangular network, the potential of a node $d(x,y)$ in the network is
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{U}_{(x,y)}}{J} & = & \displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ & & +\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{x\vee {x}_{1}}^{(i)}{C}_{{y}_{1},i}-{\beta }_{x\vee {x}_{2}}^{(i)}{C}_{{y}_{2},i}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}{C}_{y,i},\end{array}\end{eqnarray}$
where ${\beta }_{x,{x}_{s}}^{(i)}$ reduces to ${\beta }_{x,{x}_{s}}^{(i)}={\rm{\Delta }}{F}_{x}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$.In particular, when ${x}_{2}={x}_{1}$, potential formula (48 ) reduces further to
$\begin{eqnarray}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{({C}_{{y}_{1},i}-{C}_{{y}_{2},i}){C}_{y,i}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}{\beta }_{{x}_{1}\vee x}^{(i)}.\end{eqnarray}$
When ${h}_{1}=0\,({r}_{1}=0)$, figure 1 degrades into a Fan network as shown in figure 4(a), where $r$ and ${r}_{0}$ are the respective resistors along longitude (radius) and latitude (arc) directions, and the resistor element on the outer arc is ${r}_{2}$ (an arbitrary boundary resistor). The potential of a node $d(x,y)$ in the $m\times n$ Fan network can be written as
$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}\displaystyle \frac{U(x,y)}{J} & = & \displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ & & +\displaystyle \frac{2r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{x\vee {x}_{1}}^{(i)}{C}_{{y}_{1},i}-{\beta }_{x\vee {x}_{2}}^{(i)}{C}_{{y}_{2},i}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{(i)}}{C}_{y,i},\end{array}\end{array}\end{eqnarray}$
where we redefine ${\beta }_{x\vee {x}_{s}}^{(i)}={F}_{x}^{(i)}{\alpha }_{2,n-{x}_{s}}^{(i)}(\mathrm{if}\,x\leqslant {x}_{s})$ and ${\beta }_{x\vee {x}_{s}}^{(i)}\,={F}_{{x}_{s}}^{(i)}{\alpha }_{2,n-x}^{(i)}(\mathrm{if}\ \ \,x\geqslant {x}_{s})$.
Figure 4. Two resistor network models. (a) is a Fan network with an arbitrary boundary resistor r2; (b) is an arbitrary hammock network. |
Please note that a non-regular Fan network (the outer arc resistor r2 is arbitrary) is a scientific conundrum, which has not been solved before. Reference [19] has researched just the regular Fan network (the outer arc resistor is ${r}_{2}={r}_{0}$), but our formula (50 ) with ${r}_{2}={r}_{0}$ is different from the result in [19] because two results depends on the different matrices along the orthogonal direction.
When ${r}_{1}={r}_{2}=0$, figure 1 degrades into a hammock network as shown in figure 4(b), the potential of a node $d(x,y)$ in the $m\times n$ hammock network can be written as
$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}\displaystyle \frac{U(x,y)}{J} & = & \displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ & & +\displaystyle \frac{2r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{x\vee {x}_{1}}^{(i)}{C}_{{y}_{1},i}-{\beta }_{x\vee {x}_{2}}^{(i)}{C}_{{y}_{2},i}}{{F}_{n}^{(i)}}{C}_{y,i},\end{array}\end{array}\end{eqnarray}$
where we redefine ${\beta }_{x\vee {x}_{s}}^{(i)}={F}_{x}^{(i)}{F}_{n-{x}_{s}}^{(i)}(\mathrm{if}\ \ \,x\leqslant {x}_{s})$ and ${\beta }_{x\vee {x}_{s}}^{(i)}\,={F}_{{x}_{s}}^{(i)}{F}_{n-x}^{(i)}(\mathrm{if}\ \ \,x\geqslant {x}_{s})$.In particular, when ${d}_{1}(0,{y}_{1})$ and ${d}_{2}(n,{y}_{2})$ are respectively at the left and right poles, the potential of equation (51 ) reduces to
$\begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r.\end{eqnarray}$
Assume figure 1 is a semi-infinite $\infty \times n$ network, and $m\to \infty $ but n, x and y are finite. Consider ${d}_{1}(0,{y}_{1})$ is on the left edge, and ${d}_{2}(n,{y}_{2})$ is on the right edge, when ${r}_{1}={r}_{2}={r}_{0}$, the potential of a node $d(x,y)$ in the semi-infinite $\infty \times n$ rectangular network is
$\begin{eqnarray}\displaystyle \frac{{U}_{\infty \times n}(x,y)}{J}=\displaystyle \frac{{r}_{0}}{\pi }{\int }_{0}^{\pi }\displaystyle \frac{({\rm{\Delta }}{F}_{n-x}{C}_{{y}_{1}}-{\rm{\Delta }}{F}_{x}{C}_{{y}_{2}}){C}_{y}}{(1-\cos \theta ){F}_{n+1}}{\rm{d}}\theta ,\end{eqnarray}$
where ${C}_{{y}_{k}}=\cos \left({y}_{k}+\tfrac{1}{2}\right)\theta $, ${F}_{k}^{}=({\lambda }^{k}-{\bar{\lambda }}^{k})/(\lambda -\bar{\lambda })$ with $\lambda =1+h-h\cos \theta +\sqrt{{(1+h-h\cos \theta )}^{2}-1}$. Please note that these definitions apply to all such issues as appear below.4.2. Resistance of complex rectangular network
Consider an m × n rectangular network with two arbitrary boundaries shown in figure 1, where the maximum coordinate is $(n,m)$. Defining ${\beta }_{k,s}^{(i)}={\beta }_{{x}_{k},{x}_{s}}^{(i)}$, the resistance between two nodes ${d}_{1}({x}_{1},{y}_{1})$ and ${d}_{2}({x}_{2},{y}_{2})$ in the finite and semi-infinite networks are respectively19 )–(25 ) For equation (55 ), there be $n\to \infty $, ${x}_{1},{x}_{2}\to \infty $ with finite ${x}_{1}-{x}_{2}$. Equation (55 ) can be derived by taking the limit $n\to \infty $ in equation (54 ).
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \,+\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}{C}_{{y}_{1},i}^{2}-2{\beta }_{1,2}^{(i)}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{\beta }_{2,2}^{(i)}{C}_{{y}_{2},i}^{2}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{R}_{m\times \infty }({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \,+\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{C}_{{y}_{1},i}^{2}+{C}_{{y}_{2},i}^{2}-2{\bar{\lambda }}_{i}^{\left|{x}_{2}-{x}_{1}\right|}{C}_{{y}_{1},i}{C}_{{y}_{2},i}}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}},\end{array}\end{array}\end{eqnarray}$
where ${\theta }_{i}={i}\pi /(m+1)$, and ${C}_{k,i}$, ${G}_{k}^{(i)}$ are, respectively, defined in equations ( For figure 1, substituting equation (
Equation (54 ) is an exact expression which still contains a variety of resistance results with all kinds of boundary conditions because the left and right boundaries are the arbitrary resistors. For clearly understanding formula (54 ), we set h1 and h2, m or n as special values, and give several special cases to understand its application and meaning.
Case 1. When ${h}_{1}=1$, the network of figure 1 degrades into a rectangular m × n network with an arbitrary right boundary, then formula (54 ) reduces to
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \,+\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}{C}_{{y}_{1},i}^{2}-2{\beta }_{1,2}^{(i)}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{\beta }_{2,2}^{(i)}{C}_{{y}_{2},i}^{2}}{(1-\cos {\theta }_{i})[{F}_{n+1}^{(i)}+({h}_{2}-1){F}_{n}^{(i)}]},\end{array}\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ reduces to ${\beta }_{k,s}^{(i)}={\rm{\Delta }}{F}_{{x}_{k}}^{(i)}[{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-{x}_{s}-1}^{(i)}]$.Case 2. When ${h}_{2}={h}_{1}=1$, the network of figure 1 degrades into a normal m × n rectangular network, then formula (54 ) reduces to1 ) with a double sums. Clearly, our result (57 ) is different from equation (1 ). This also shows that the equivalent resistance can be expressed in different forms.
$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \,+\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}{C}_{{y}_{1},i}^{2}-2{\beta }_{1,2}^{(i)}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{\beta }_{2,2}^{(i)}{C}_{{y}_{2},i}^{2}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{array}\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ reduces to ${\beta }_{k,s}^{(i)}={\rm{\Delta }}{F}_{{x}_{k}}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$. This problem has been researched in [23], and gave equation (Case 3. When ${h}_{1}=0$, the network of figure 1 degrades into a non-regular m × n Fan network with an arbitrary boundary as shown in figure 4(a), by equation (54 ), we obtain the resistance of a Fan network
$\begin{eqnarray}\begin{array}{c}\begin{array}{r}\begin{array}{c}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \,+\,\displaystyle \frac{2r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{\left(i\right)}{C}_{{y}_{1},i}^{2}-2{\beta }_{1,2}^{\left(i\right)}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{\beta }_{2,2}^{\left(i\right)}{C}_{{y}_{2},i}^{2}}{{\rm{\Delta }}{F}_{n}^{\left(i\right)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{\left(i\right)}},\end{array}\end{array}\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ is re-defined as ${\beta }_{k,s}^{(i)}={F}_{{x}_{k}}^{(i)}[{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}\,+({h}_{2}-1){\rm{\Delta }}{F}_{n-{x}_{s}-1}^{(i)}]$.In particular, when ${h}_{1}=0,\,{h}_{2}=1$, the network of figure 1 degrades into a normal m × n Fan network, then formula (58 ) reduces to59 ), however they are equivalent to each other. The reason is that they choice the different matrix along different direction. This also shows that the equivalent resistance can be expressed in different forms.
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{2r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}{C}_{{y}_{1},i}^{2}-2{\beta }_{1,2}^{(i)}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{\beta }_{2,2}^{(i)}{C}_{{y}_{2},i}^{2}}{{\rm{\Delta }}{F}_{n}^{(i)}},\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ is re-defined as ${\beta }_{k,s}^{(i)}={F}_{{x}_{k}}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$. This case has been researched in [30], but the result is different from equation (Case 4. When ${d}_{1}(0,{y}_{1})$ is on the left edge and ${d}_{2}(n,{y}_{2})$ is on the right edge, formula (54 ) reduces to
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,{y}_{1}\},\{n,{y}_{2}\})=\displaystyle \frac{{nr}}{m+1}+\displaystyle \frac{{r}_{0}}{m+1}\\ \quad \times \,\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(i)}{C}_{{y}_{1},i}^{2}-2{h}_{1}{h}_{2}{C}_{{y}_{1},i}^{}{C}_{{y}_{2},i}^{}+{h}_{2}{\alpha }_{1,n}^{(i)}{C}_{{y}_{2},i}^{2}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{array}\end{eqnarray}$
where ${\alpha }_{k,n}^{(i)}={\rm{\Delta }}{F}_{n}^{(i)}+({h}_{k}-1){\rm{\Delta }}{F}_{n-1}^{(i)}$.In particular, when ${h}_{1}={h}_{2}=1$, equation (60 ) reduces to
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,{y}_{1}\},\{n,{y}_{2}\})=\displaystyle \frac{{nr}}{m+1}\\ \quad +\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{({C}_{{y}_{1},i}^{2}+{C}_{{y}_{2},i}^{2}){\rm{\Delta }}{F}_{n}^{(i)}-2{C}_{{y}_{1},i}{C}_{{y}_{2},i}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}.\end{array}\end{eqnarray}$
Case 5. When ${d}_{1}({x}_{1},0)$ is at the bottom edge and ${d}_{2}({x}_{2},m)$ is on the top edge, then formula (54 ) reduces to24 ) and (25 ).
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{{x}_{1},0\},\{{x}_{2},m\})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\left(-1\right)}^{i}{\beta }_{1,2}^{(i)}+{\beta }_{2,2}^{(i)}}{2{G}_{n}^{(i)}}{\cot }^{2}\left(\displaystyle \frac{1}{2}{\theta }_{i}\right),\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ and ${G}_{n}^{(i)}$ are defined in (Case 6. When ${d}_{1}(0,0)$ and ${d}_{2}(n,m)$ are two diagonal nodes, by (54 ) we have the resistance between two maximally separated nodes
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,0\},\{n,m\})=\displaystyle \frac{{nr}}{m+1}\\ \quad +\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(i)}-2{\left(-1\right)}^{i}{h}_{1}{h}_{2}+{h}_{2}{\alpha }_{1,n}^{(i)}}{2{G}_{n}^{(i)}}{\cot }^{2}\left(\displaystyle \frac{1}{2}{\theta }_{i}\right),\end{array}\end{eqnarray}$
where ${\alpha }_{k,n}^{(i)}={\rm{\Delta }}{F}_{n}^{(i)}+({h}_{k}-1){\rm{\Delta }}{F}_{n-1}^{(i)}$, ${\theta }_{i}={i}\pi /(m+1)$.In particular, when ${r}_{1}={r}_{2}={r}_{0}$, equation (63 ) reduces to64 ) is a desired equivalent resistance between two maximum separated nodes in an arbitrary m × n resistor network. This is an interesting result because it is simple and easy to research the asymptotic expansion for the maximum resistance. References [39, 40] studied the asymptotic expansion by making use of the result (1 ). Obviously, the concise equation (64 ) is more conducive to the study of the asymptotic expression of the maximum resistance.
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,0\},\{n,m\})=\displaystyle \frac{n}{m+1}r\\ \quad +\ \displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\left(\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(i)}-{\left(-1\right)}^{i}}{{F}_{n+1}^{(i)}}\right){\cot }^{2}\left(\displaystyle \frac{1}{2}{\theta }_{i}\right).\end{array}\end{eqnarray}$
Please note that equation (From the above derivation, we find that formula (54 ) is a generalized result, which is applicable to many network problems and summarized a variety of complex network models since it contains six arbitrary elements $({r}_{0},r,{r}_{1},{r}_{2},n,m)$.
5. Electrical properties of complex cylindrical network
5.1. Nodal potential of complex cylindrical network
Consider the non-regular m × n cylindrical network shown in figure 2, where the maximum coordinate is $(n,m)$, selecting ${\sum }_{i=0}^{m}{V}_{0}^{(i)}=\tfrac{1}{2}({x}_{2}-{x}_{1}){rJ}$ as the reference potential, defining ${\theta }_{i}=2i\pi /(m+1)$, and ${C}_{{y}_{k}-y}=\cos ({y}_{k}-y){\theta }_{i}$, the potential of any node $d(x,y)$ in the finite and sem-infinite networks can be written as22 )–(25 ). For equation (66 ), there be $n\to \infty $, ${x}_{1},{x}_{2}\to \infty $ with finite ${x}_{k}-x$. Equation (66 ) can be derived by taking the limit $n\to \infty $ in equation (65 ).
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{U}_{m\times \infty }(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ \quad +\,\displaystyle \frac{r}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\bar{\lambda }}_{i}^{\left|{x}_{1}-x\right|}{C}_{{y}_{1}-y}-{\bar{\lambda }}_{i}^{\left|{x}_{2}-x\right|}{C}_{{y}_{2}-y}}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}},\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ , ${G}_{k}^{(i)}$ are, respectively, defined in equations (In particular, when ${x}_{2}={x}_{1}$ (means the input and output nodes of currents are at the same vertical axis), formulae (65 ) and (66 ) reduce to
$\begin{eqnarray}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{C}_{{y}_{1}-y}-{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}{\beta }_{{x}_{1}\vee x}^{(i)},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{U}_{m\times \infty }(x,y)}{J}\\ \quad =\,\displaystyle \frac{r}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{C}_{{y}_{1}-y}-{C}_{{y}_{2}-y}}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}}{\bar{\lambda }}_{i}^{\left|{x}_{1}-x\right|}.\end{array}\end{eqnarray}$
For figure 2, substituting equation (
$\begin{eqnarray}{X}_{k}^{(i)}=\displaystyle \frac{{\beta }_{k\vee {x}_{1}}^{(i)}\exp ({\rm{i}}{y}_{1}{\theta }_{i})-{\beta }_{k\vee {x}_{2}}^{(i)}\exp ({\rm{i}}{y}_{2}{\theta }_{i})}{({t}_{i}-2){G}_{n}^{(i)}}{rJ}.\end{eqnarray}$
The substitution of ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r+\displaystyle \frac{{r}_{0}}{m+1}\\ \times \,\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}+{\rm{i}}\displaystyle \frac{{r}_{0}}{m+1}\\ \times \,\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}\sin [({y}_{1}-y){\theta }_{i}]-{\beta }_{{x}_{2}\vee x}^{(i)}\sin [({y}_{2}-y){\theta }_{i}]}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{array}\end{eqnarray}$
Because the elements rk in the network is real number, the potential $U(x,y)$ must be real number. Thus, extracting the real part of equation (Formula (65 ) is a meaningful result because the network of figure 2 is very complex and has not been resolved before, contains a lot of resistor network models, where each of the different boundary resistor represents a different network structure. So Formula (65 ) can create many interesting results. In the following applications we always assume that the current J goes from ${d}_{1}({x}_{1},{y}_{1})$ to ${d}_{2}({x}_{2},{y}_{2})$ except for special instructions.
Consider an arbitrary $m\times n$ cylindrical network of figure 2 with ${r}_{1}={r}_{2}={r}_{0}$ , by (
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{array}\end{eqnarray}$
where ${\beta }_{x,{x}_{s}}^{(i)}$ reduces to ${\beta }_{x,{x}_{s}}^{(i)}={\rm{\Delta }}{F}_{x}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$.In particular, when ${x}_{2}={x}_{1}$, potential formula (71 ) reduces further to
$\begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{C}_{{y}_{1}-y}-{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}{\beta }_{{x}_{1}\vee x}^{(i)}.\end{eqnarray}$
Consider an $m\times n$ cylindrical network of figure 2. When ${r}_{1}=0$, figure 2 degrades into an $m\times n$ cobweb network as shown in figure 5(a), by (
$\begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r+\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{(i)}},\end{eqnarray}$
where ${\beta }_{x\vee {x}_{s}}^{(i)}$ is redefined as ${\beta }_{x\vee {x}_{s}}^{(i)}={F}_{x}^{(i)}{\alpha }_{2,n-{x}_{s}}^{(i)}(\mathrm{if}\,x\leqslant {x}_{s})$ and ${\beta }_{x\vee {x}_{s}}^{(i)}={F}_{{x}_{s}}^{(i)}{\alpha }_{2,n-x}^{(i)}(\mathrm{if}\,x\geqslant {x}_{s})$.
Figure 5. Two resistor network models. (a) is a cobweb network with an arbitrary boundary resistor r2; (b) is an arbitrary globe network. |
In particular, when ${d}_{1}(0,{y}_{1})$ is at left edge, and ${d}_{2}(n,{y}_{2})$ is at right edge. Equation (73 ) reduces to74 ) is an original result.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r\\ \quad -\,\displaystyle \frac{{r}_{2}h}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{F}_{x}^{(i)}\cos ({y}_{2}-y){\theta }_{i}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{(i)}}.\end{array}\end{eqnarray}$
Please note that the cobweb network with an arbitrary boundary has not been resolved before, the previous work only studied the normal cobweb network (the boundary resistor is ${r}_{2}={r}_{0}$) [19], equation ( Consider an arbitrary $m\times n$ globe network shown in figure 5(b). That is to say that figure 2 degrades into a globe network when ${r}_{2}={r}_{1}=0$, from (
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{\bar{x}-{x}_{\tau }}{m+1}r\\ \quad +\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}}{{F}_{n}^{(i)}},\end{array}\end{eqnarray}$
where we redefine ${\beta }_{x\vee {x}_{s}}^{(i)}={F}_{x}^{(i)}{F}_{n-{x}_{s}}^{(i)}(\mathrm{if}\,x\leqslant {x}_{s})$ and ${\beta }_{x\vee {x}_{s}}^{(i)}\,={F}_{{x}_{s}}^{(i)}{F}_{n-x}^{(i)}(\mathrm{if}\,x\geqslant {x}_{s})$.In particular, when ${d}_{1}(0,{y}_{1})$ is at left pole, and ${d}_{2}(n,{y}_{2})$ is at right pole, equation (75 ) reduces to
$\begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r.\end{eqnarray}$
Formula (76 ) is very simple and very interesting because the potential distribution is only related to the x and has nothing to do with y, which shows the nodal potential is equal in the same latitude.
Consider a non-regular $m\times n$ cylindrical network of figure 2. Assume ${d}_{1}(0,{y}_{1})$ is on the left edge, and ${d}_{2}(n,{y}_{2})$ is on the right edge. By (
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(i)}{C}_{{y}_{1}-y}-{h}_{2}{\alpha }_{1,x}^{(i)}{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{array}\end{eqnarray}$
where ${\alpha }_{k,x}^{(i)}={\rm{\Delta }}{F}_{x}^{(i)}+({h}_{k}-1){\rm{\Delta }}{F}_{x-1}^{(i)}$ is defined in equation (In particular, when ${h}_{1}={h}_{2}=1$ , equation (77 ) reduces to77 ) reduces to
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\rm{\Delta }}{F}_{n-x}^{(i)}{C}_{{y}_{1}-y}-{\rm{\Delta }}{F}_{x}^{(i)}{C}_{{y}_{2}-y}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{array}\end{eqnarray}$
when ${h}_{1}={h}_{2}=1$, ${y}_{2}={y}_{1}$, equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-2x}{2(m+1)}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\rm{\Delta }}{F}_{n-x}^{(i)}-{\rm{\Delta }}{F}_{x}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}{C}_{{y}_{1}-y}.\end{array}\end{eqnarray}$
One know the potential function have important application value for solving the Laplace equation. In this paper, the analytical solutions of node potential functions under various conditions are given, which provides a new theory for practical application.
5.2. Resistance of complex $m\times n$ cylindrical network
Consider an m × n cylindrical network with two arbitrary boundaries shown in figure 2, where the maximum coordinate is $(n,m)$. Defining ${\beta }_{k,s}^{(i)}={\beta }_{{x}_{k},{x}_{s}}^{(i)}$, the resistance between two nodes ${d}_{1}({x}_{1},{y}_{1})$ and ${d}_{2}({x}_{2},{y}_{2})$ in the finite and semi-infinite networks are respectively24 )–(25 ). For equation (81 ), there be $n\to \infty $, ${x}_{1},{x}_{2}\to \infty $, with finite m. Equation (81 ) can be derived by taking the limit $n\to \infty $ in equation (80 ).
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{R}_{m\times \infty }({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{1-{\bar{\lambda }}_{i}^{\left|{x}_{2}-{x}_{1}\right|}\cos (y{\theta }_{i})}{\sqrt{{\left(1+h-h\cos {\theta }_{i}\right)}^{2}-1}},\end{array}\end{eqnarray}$
where ${\theta }_{i}=2{i}\pi /(m+1)$, $y={y}_{2}-{y}_{1}$, ${\beta }_{k,s}^{(i)}$ and ${G}_{k}^{(i)}$ are, respectively, defined in equations ( For figure 2, substituting equation (
Formula (80 ) is an exact and exciting result because the network of figure 2 is very complex and has not been resolved before, and contains a lot of resistor network models, where each of the different boundary resistor represents a different network structure. In particular, when taking some specific value for r1 and r2, equation (80 ) gives rise to a series of special cases below.
Case 1. Consider a non-regular m × n cylindrical network of figure 2. When ${r}_{1}={r}_{0}$, the resistance of equation (80 ) reduces to
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i})[{F}_{n+1}^{(i)}+({h}_{2}-1){F}_{n}^{(i)}]},\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ reduces to ${\beta }_{k,s}^{(i)}={\rm{\Delta }}{F}_{{x}_{k}}^{(i)}[{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-{x}_{s}-1}^{(i)}]$.Case 2. Consider a normal m × n cylindrical network of figure 2 with ${r}_{2}={r}_{1}={r}_{0}$, the resistance of equation (80 ) reduces to2 ) with a double sums. Clearly, our result (83 ) is different from equation (2 ). This also shows that the equivalent resistance can be expressed in different forms.
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{array}\end{eqnarray}$
where $y={y}_{2}-{y}_{1}$, ${\beta }_{k,s}^{(i)}$ reduces to ${\beta }_{k,s}^{(i)}={\rm{\Delta }}{F}_{{x}_{k}}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$. This problem has been researched in [23], and gave equation (Case 3. Consider a non-regular m × n cylindrical network of figure 2, when ${h}_{1}=0$, the left boundary collapses to a pole, the network of figure 2 degrades into a cobweb network with an arbitrary boundary resistor r2 as shown in figure 5(a), we have the equivalent resistance
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{(i)}},\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ is re-defined as ${\beta }_{k,s}^{(i)}={F}_{{x}_{k}}^{(i)}[{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-{x}_{s}-1}^{(i)}]$.In particular, when ${h}_{1}=0,\,{h}_{2}=1$, the network of figure 5(a) degrades into a regular cobweb network, the resistance of equation (84 ) reduces to
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}}\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ is re-defined as ${\beta }_{k,s}^{(i)}={F}_{{x}_{k}}^{(i)}{\rm{\Delta }}{F}_{n-{x}_{s}}^{(i)}$.Please note that Case 2 has been researched in [30], but the result is different from equation (85 ), however they are equivalent to each other. The reason is that they choice the different matrix along different direction, where [30] set up matrix along the longitude, but this paper set up matrix along the latitude.
Case 4. When ${h}_{1}={h}_{2}=0$, the left and right boundary collapse respectively to two poles, the network of figure 2 degrades into an m × n globe network as shown in figure 5(b), we have
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{r}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{{F}_{n}^{(i)}},\end{array}\end{eqnarray}$
where ${\beta }_{k,s}^{(i)}$ is re-defined as ${\beta }_{k,s}^{(i)}={F}_{{x}_{k}}^{(i)}{F}_{n-{x}_{s}}^{(i)}$.Please note that Case 4 has been researched in [29], but the result is different from equation (86 ), however they are equivalent to each other. The reason is that they choice the different matrix along different axes. This also shows that the equivalent resistance can be expressed in different forms.
Case 5. Consider a non-regular m × n cylindrical network of figure 2, when both ${d}_{1}({x}_{1},0)$ and ${d}_{2}({x}_{2},0)$ are on the same horizontal axis, we have
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{{x}_{1},0\},\{{x}_{2},0\})=\displaystyle \frac{\left|{x}_{2}-{x}_{1}\right|}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{array}\end{eqnarray}$
Case 6. Consider a non-regular m × n cylindrical network of figure 2, when ${d}_{1}(0,0)$ is on the left edge and ${d}_{2}(n,y)$ is on the right edge, the resistance between two edges is88 ) reduces to88 ) reduces to
$\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{2(m+1)}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(i)}+{h}_{2}{\alpha }_{1,n}^{(i)}-2{h}_{1}{h}_{2}\cos (y{\theta }_{i})}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{array}\end{eqnarray}$
In particular, when ${h}_{1}={h}_{2}=1$, equation ( $\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m+1}r\\ \quad +\,\displaystyle \frac{{r}_{0}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(i)}-\cos (y{\theta }_{i})}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}.\end{array}\end{eqnarray}$
And when ${h}_{1}=0$, equation ( $\begin{eqnarray}\begin{array}{l}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m+1}r\\ \quad +\,\displaystyle \frac{{{rh}}_{2}}{m+1}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{F}_{n}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{2}-1){\rm{\Delta }}{F}_{n-1}^{(i)}}.\end{array}\end{eqnarray}$
From the above results we know formula (80 ) is a general results which, contains many results in a variety of lattice structures, can produce many new resistance formulae.
6. Conclusion and comment
This paper developed the RT-V theory (RT theory with potential parameters) and reveals the basic principle of electrical characteristics of complex resistor networks for the first time, such as two theorems of theorems 1 and 2 are proposed, and the explicit electrical characteristics (potential and resistance) formulae of the complex networks are given, which contains the results of finite and infinite networks. As applications of two theorems, the analytical solutions of the electrical characteristics (potential function and equivalent resistance) in the complex m × n resistor networks with arbitrary boundaries are given, and many interesting results of the various types of resistor networks are produced.
It must be emphasized that the previous theories (Mainly refers Green’s function technique and Laplacian matrix method) cannot solve resistor networks with complex boundaries, because the Green’s function technique is usually used to solve infinite network problems, and the Laplacian matrix method depends on the solution of two eigenvalues which relies on two matrices along two orthogonal directions. Using Tan’s RT-V method to study resistor networks just relies on one matrix along one vertical directions, which avoids the confusion of another matrix with arbitrary elements that cannot be solved explicitly, and also gives concise results in a single summation, such as the all equations given by this paper.
In addition, resistance formulae (54 ), (55 ), (80 ) and (81 ) et al can be extended to impedance networks since the grid elements rk can be either resistors or impedances in figures 1 and 2. For example, assume $r={Z}_{L}=R+j\omega L,\,{r}_{0}={Z}_{C}=-j/\omega C$, then we can therefore study the arbitrary m × n RLC network if we do a plural analysis [24, 34] to the resistance results obtained in this paper.