Xiaoyan LIN,Zhizhong TAN

Department of Physics,Nantong University,Nantong 226019,China

Abstract: In this paper,two different n-order topological circuit networks are connected by diodes to establish a unified network model,which is a previously unexplored problem.The network model includes not only five resistive elements but also diode devices,so the network contains many different network types.This problem can be solved through three main steps: First,the network is simplified into two different equivalent circuit models.Second,the nonlinear difference equation model is established by applying Kirchhoff’s law.Finally,the two equations with similar structures are processed uniformly,and the general solutions of the nonlinear difference equations are obtained by using the transformation technique.As an example,several interesting specific results are deduced.Our study on the network model has significant value,as it can be applied to relevant interdisciplinary research.

Key words: Complex networks;Equivalent transform;Nonlinear difference equation;Equivalent resistance

1 Introduction

In 1845,Kirchhoff established the node current law and loop voltage law as the first thorough math‐ematical description of electrical circuits,and laid the theoretical basis for research on large-scale circuits(Kirchhoff,1847).Since then,more and more scholars have studied the resistor network and investigated its various forms,such as square,triangular,honeycomb,and hypercube (Aitchison,1964;Venezian,1994;Atkinson and van Steenwijk,1999;Cserti,2000).Cur‐rently,the research on resistor network models is no longer limited to the field of circuits,but has been extended to many other areas.The calculation of resis‐tance in electric circuit theory can be used to solve many abstract and complex scientific and engineering problems,such as random walks (Doyle and Snell,1984),first-passage processes (Redner,2001),graphene properties (Kimouche et al.,2015),electronic plants(Stavrinidou et al.,2015),metagratings (Xu et al.,2021),reflectarray antennas (Hum and Du,2017),and topological properties (Albert et al.,2015).Therefore,the construction and application of resis‐tor network models have gradually become the basic solution methods for a series of scientific prob‐lems,playing an important role in the natural and engineering fields.

In the past 170 years,much progress has been made in the study of resistor network models,provid‐ing solutions for a series of resistor network problems(Brayton and Moser,1964a,1964b;Desoer and Wu,1974;Bianco et al.,2000;Bianco and Giordano,2003).Some circuits including diodes have also been dis‐cussed;for example,electrical characterization of ran‐dom networks and mixtures was investigated (Bianco et al., 2000;Bianco and Giordano,2003).General small-scale circuit problems can be summarized into solving difference equations governed by Ohm’s and Kirchhoff’s laws.However,when the circuit becomes a complex large-scale network structure,it is not enough to use Kirchhoff’s law alone.Therefore,re‐searchers have proposed several new methods.Cserti(2000) proposed the Green function technique,which opened the door to the study of infinite resistor net‐works.He calculated the equivalent resistance be‐tween any two points of several infinite lattice resis‐tance structures using this method (Cserti et al.,2002).He also applied this method to the problem of a per‐turbed lattice,in which one of the bonds is missing(Cserti et al.,2011).Asad used this method in several networks and found it to be a useful tool for study‐ing capacitor networks (Asad et al.,2013;Asad,2013a,2013b).Hijjawi et al.(2008) discussed the Green function of anisotropic diamond lattice,and the ana‐lytical properties of the Green function in all dimen‐sions were investigated by Guttmann (2010).Gior‐dano (2007) applied Green function technique to the study of two-dimensional anisotropic random lattices(a class of anisotropic infinite networks),which is a new research progress.Green function technique has been widely applied in research on infinite net‐works.As is well known,an infinite network is a kind of ideal model,yet the finite network is a prob‐lem occurring in real life.Therefore,Green function technique is not suitable for studying finite resistor networks.Wu (2004) developed a theory called the Laplace matrix method,which can calculate the re‐sistance between arbitrary nodes for a finite lattice of resistors.Since then,this method has been further de‐veloped and applied to impedance networks (Tzeng and Wu,2006;Izmailian et al.,2014;Essam et al.,2014,2015).For example,corner-to-corner resistance and its asymptotic expansion for free boundary condi‐tions were obtained by Essam and Wu (2009).Izmail‐ian and Huang (2010) also calculated the resistance for other boundary conditions.However,the Laplace matrix method has some limitations on the finite net‐works with complex boundary conditions;therefore,Tan ZZ (2011,2015a,2015b,2015c,2015d,2017) set up a new method called the recursion-transform (RT)method,and the original theory has been further im‐proved (Tan ZZ and Tan,2020a,2020b,2020c).This approach is based on recursive techniques and vari‐able transformation techniques.He has applied this method to solve the resistor network problems of vari‐ous resistor network models,such as various basic and applied research of this theory in the literature(Tan ZZ and Zhang,2015;Tan ZZ,2015a,2015b,2016;Zhou et al.,2017;Tan Z and Tan,2018;Tan Z et al.,2018a,2018b).Nowadays,the RT method is still applied to studyn-order networks in different sit‐uations,and a large number of complex networks have been solved.For example,Tan ZZ et al.(2017)studied the multi-purposen-order resistor network model and a series of other complexn-order circuit network models (Chen et al.,2019,2020;Chen and Tan,2020;Chen and Yang,2020;Fang and Tan,2022;Tan ZZ,2022).The above-mentioned studies indicate that the RT method can be applied to many different types of resistor network models,and a series of inter‐esting conclusions have been drawn.

Although many breakthroughs and much prog‐ress have been made in resistor network research,there are still several complicated resistor network models that have not been solved.In this work,two differentn-order topological circuit networks (Fig.1) are con‐sidered.On one hand,the resistor network model can be used to simulate the properties of some mate‐rials to carry out theoretical innovation research.On the other hand,the two different resistor networks above can be unified into a single network model due to the unidirectional conductivity of diodes.Al‐though the electrical properties of the two resistor networks are different,they can be described in a unified form,which is an interesting and novel as‐pect of the relevant research.Even the analysis and derivation of these unified networks can be applied to LC networks.The unified network model con‐structed in this paper contains not only five different resistor elements but also diode devices (Fig.2).Our study belongs to theoretical research;therefore,its main purpose is to provide a theoretical basis for fu‐ture related research in different fields,such as phy?sics,mathematics,biology,and engineering.

Fig.1 Two circuit network models with different topologies

The RT method is used to study the new net‐work model.It can be divided into four steps: step 1 is to build an equivalent circuit model,step 2 is to create a nonlinear difference equation model using Kirchhoff’s laws,step 3 is to construct the method of equivalent transformation to obtain the general solution of the nonlinear difference equation,and step 4 is to discuss special situations.

Fig.1 presents two circuit network models with two different topologies and different resistance ele‐ments distributed on the branches.The two circuits may look very similar,but they have different inter‐nal structures.In this paper,the two models in Fig.1 are unified into the model depicted in Fig.2.

Fig.2 A multi-functional n-order resistor network with X circuits and diodes,which contains five different resistor elements (including R0) and ideal diodes in the upper boundary

Fig.2 presents a type ofn-order resistor net‐work model with ideal diodes and X circuits.The characteristics of such a network are as follows: the number of network elements isn,the upper bound‐ary element is an ideal diode,the element of lower boundary resistance isr,the element of resistance in the direction of vertical section isr0,the resistors on the X-type cross line arer1andr2,and the load resistor on the right end isR0.The network is com‐posed of five different resistance elements and a diode,which means that it is a multi-functional net‐work.This paper focuses on the analytical expres‐sion of the resistance betweenAnandBnin the net‐work in Fig.1.Two equivalent resistance values are given,as well as their derivation and proof processes.

2 Total equivalent resistance formulae

In Fig.1,letAkbe thektharbitrary node betweenA0andAnof the upper axis (counted from the right end,the first node on the right end isA0).Bkis thektharbitrary node betweenB0andBnof the lower axis.The ideal diode D (meaning its absolute unidirectional conductivity) is connected to the upper boundary and X circuits in the network,and all parameters are shown in Fig.2.The two main results are as follows:

Result 1When the currentIis input fromAnand output fromBnas shown in Fig.2,the equivalent re‐sistance can be written as

Result 2When the currentIis input fromBnand output fromAnas shown in Fig.2,the equivalent re‐sistance can be written as

Eqs.(1) and (5) are the first of their kind,and show an innovation in theory and methodology.These two conclusions are derived from the RT theory,which involves building equivalent models and recursive equations.

3 Equivalent models and recursive equations

According to the structural features of Fig.2,as‐suming that the equivalent resistance between the two nodes ofAnandBnat the left end of the network isRn,the equivalent resistance between the two nodes at the left end ofn-1 network will beRn-1.Since the ideal diode D has zero forward resistance (short cir‐cuit) and infinite reverse resistance (open circuit),when the current is input fromAnand output fromBn,we can simplify Fig.2 into a simple model shown in Fig.3.

Fig.3 Equivalent model of a two-terminal circuit network with triangular structure

Next,we establish the relationship betweenRnandRn-1using Kirchhoff’s law.We suppose that the constant currentIis input at nodeAnand output at nodeBn.Meanwhile,other branch currents are de‐fined in Fig.3.

From Fig.3,according to Kirchhoff’s law,the circuit current equations can be written as

The node current equations can be written as

Then,we substitute Eq.(10) into Eq.(9),eliminateI3,I4,I5,and simplify the equations as

By solving Eq.(11),a current relationship is obtained:

Then,Eq.(12) is simplified,and can be written as

wherea1andb1are given by Eq.(4).SinceRn=U/I=I0r0/I,using Eq.(13),one can obtain

Eq.(14) is a simple recurrence formula that we are looking for.

The second condition is assuming that the cur‐rent is input at nodeBnand output at nodeAn;in this case,the diode D has zero resistance (short circuit).Similar to the analysis above,we establish an equiva‐lent model in Fig.4,where other branch currents are defined.The relationship betweenRnandRn-1is es‐tablished using Kirchhoff’s law.

Fig.4 Equivalent model of a two-terminal circuit network with Y circuits

From Fig.4,according to Kirchhoff’s law,the circuit current equations can be written as

From Fig.4,the node current equations can be writ‐ten as

We substitute Eq.(16) into Eq.(15) and eliminateI3,I4,I5,to simplify the equations as

By solving Eq.(17),a current relationship can be obtained:

wherea2andb2are given by Eq.(8).AsRn=U/I=I0r0/I,substituting Eq.(19) into this equation yields

In this way,we have deduced Eq.(20),a simple recurrence formula that is classed as a nonlinear difference equation.It is interesting to note that Eq.(14) is similar to Eq.(20) in structure,so we can deal with them together.The work that follows is to seek the general and special solution to Eqs.(14)and (20).

4 Transformation and derivation

According to the structural similarity between Eqs.(14) and (20),we rewrite the two recursive equa‐tions as

In this way,we need only to study the solution to Eq.(21) to prove Eqs.(1) and (5).Here,we use the variable substitution method described in Tan ZZ(2011);that is,supposing that there is a sequence {xn},we use the following transformation relationship:

The initial term can be specified asx0=1.On using Eq.(22),one has

By substituting Eq.(22) and its recurrence formulaRn-1into Eq.(21) and simplifying it,we can obtain

Assuming thatαandβare two roots of the characteristic equation of Eq.(24),and solving the eigenvalue of Eq.(24),we can obtain the values of Eqs.(3) and (7).Therefore,from Eq.(24),we can obtain

According to the method offered in Tan ZZ (2011)to solve Eq.(25),we can deduce

Substituting the initial term Eq.(23) into Eq.(26)yields

To further simplify Eq.(27),from Eqs.(24) and (25),we can obtain

By substituting Eq.(28) into Eq.(27),we can obtain

Substituting Eq.(29) and its recurrencexninto Eq.(22)gives

By further simplifying it with functionFn=(αn-βn)/(α-β),we can obtain

Eq.(31) is a general equivalent resistance for‐mula,containing two different conclusions to be proved.For example,whena=a1,b=b1,andλ=λ1,Eq.(1) can be obtained from Eq.(31);whena=a2,b=b2,andλ=λ2,Eq.(5) can be obtained from Eq.(31).

Obviously,the above unified derivation is a the‐oretical and methodological innovation;it is mean‐ingful because it has solved a profound equivalent re‐sistance problem.Since all calculation processes are precise and rigorous,and all calculation equations are self-consistent,the conclusions drawn are necessarily correct.

5 Special cases

The network model in Fig.2 contains five dif‐ferent resistor elements andnideal diodes.Since the resistor elements are arbitrary (the resistor value can be zero or infinite),this multi-parameter network has many special cases.Several of these special cases are given below.

Case 1: In the network of Fig.2,whenr1=∞,from Eqs.(2) and (4),we can obtain

Then,using Eq.(1),the equivalent resistance is obtained:

Whenr1=∞,from Eqs.(6) and (8),we can obtain

Then,the equivalent resistance Eq.(5) is simlified to

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Case 2: In the network of Fig.2,whenr1=0,from Eqs.(2) and (4),we can obtain

Substituting them into Eq.(3) yields

This is an interesting result: the equivalent resistance is independent ofn,which is completely consistent with the actual situation,since the circuit network in this case is as shown in Fig.5.

Fig.5 An n-order resistor network with diodes,which contains three different resistor elements and an ideal diode at the upper boundary

Whenr1=0,from Eqs.(6) and (8),we can obtain

Substituting them into Eq.(7) yields

Becauseα2=β2,if taking the limit,we can obtain

Substituting Eq.(44) into Eq.(5) gives

Then,simplifying Eq.(45) yields

whereα2=r0r2/(r2+2r0) is given by Eq.(43).Eq.(46) is an interesting concise result,in full agree‐ment with the actual situation,since the circuit net‐work in this case is as shown in Fig.5.

Case 3: Whenn=0,one can verify the correctness of Eqs.(1) and (5) in this simple case;for example,

Eqs.(49) and (50) show that the general results,i.e.,Eqs.(1) and (5),are applicable to the case ofn=0,and the actual circuit shows thatR0(A0,B0)=R0.

Case 4: Whenn=1,one can verify the correct‐ness of Eqs.(1) and (5) in this simple case;for example,

wherea1andb1are given by Eq.(4),anda2andb2are given by Eq.(8).

It is found that,ifn=1 is substituted into Eqs.(14) and (20),Eqs.(54) and (55) can also be derived.Obviously,Eqs.(54) and (55) are completely consistent with the results obtained by the actual cir‐cuit calculation,which proves that the conclusion is correct whenn=1.

6 Discussion and summary

In this paper,ann-order resistor network model with X circuits and diodes (Fig.2) is proposed which has not been studied previously.The recursiontransform (RT) method is used to evaluate the equiv‐alent resistance of this new resistor network.The equivalent models given by Eqs.(14) and (20) are es‐tablished for the forward and reverse resistance of the circuit network (due to the unidirectional con‐ductivity of the diode),respectively.Then,a unified difference model given by Eq.(21) is established by the similarity of the structure.The general solution of the unified difference equation is given by establishing a variable substitution method given by Eq.(31).The general formula of equivalent resistance is expressed as a functionFn=(αn-βn)/(α-β),and a highly con‐cise result is obtained.Since the general equivalent resistance is given in this paper,based on its formula,a series of special equivalent resistance conclusions are derived.If the variable substitutionri=a+jbis used for resistance elements,it can be seen that the re‐search methods and conclusions of this paper are also applicable to the study of complex impedance net‐works in Fig.1.

The innovative research in this paper has ob‐tained two new equivalent resistance formulae,which establish a new theoretical tool for future research on the resistor network model.The research methods and innovative ideas in this paper have theoretical and practical significance for future research-based teach‐ing and scientific exploration.

Contributors

Xiaoyan LIN designed the research.Xiaoyan LIN and Zhizhong TAN processed the data.Xiaoyan LIN drafted the paper.Zhizhong TAN helped organize the paper.Xiaoyan LIN and Zhizhong TAN revised and finalized the paper.

Compliance with ethics guidelines

Xiaoyan LIN and Zhizhong TAN declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.