Effective Impedance over Ordered Fields

TL;DR

This paper investigates the properties of effective impedance in finite and infinite electrical ladder networks over ordered fields, highlighting convergence behaviors in the Levi-Civita field.

Contribution

It introduces the analysis of effective impedance in finite and infinite ladder networks over ordered fields, including convergence properties in the Levi-Civita field.

Findings

Finite LC-network impedances converge in Levi-Civita field topology.

Finite CL-network impedances do not converge in the same topology.

Study extends impedance analysis to networks over ordered fields.

Abstract

In this paper, we study properties of effective impedance of finite electrical networks and calculate the effective impedance of a finite ladder network over an ordered field. Moreover, we consider two particular examples of infinite ladder networks (Feynman's network or LC-network and CL-network, both with zero on infinity) as networks over the ordered Levi-Civita field. We show, that effective impedances of finite LC-networks converge to the limit in order topology of Levi-Civita field, but the effective impedances of finite CL-networks do not converge in the same topology.

Equations

$$\Gamma =(V,\rho ,a_0,B),$$

$$\rho (x)=\sum _y{\rho }_{xy},$$

$$\{\begin{array}{l}{\Delta }_{\rho }v(x)=0\text{mbox}onV\setminus B_0,\\ v(x)=0\text{mbox}onB,\\ v(a_0)=1,\end{array}$$

$${\rho }_{xy}=\frac{\lambda }{L_{xy}{\lambda }^2+R_{xy}\lambda +D_{xy}},\text{mbox}{L}_{xy},R_{xy},D_{xy}\ge 0\text{mbox}and{L}_{xy}+R_{xy}+D_{xy}\ne 0,$$

$$\{\begin{array}{l}A\hat{v}=b,\\ v(a_0)=1,\\ v(a_i)=0,i=\overline{1,k},\end{array}$$

$$A=\left[\begin{array}{cccc}{\sum }_{x\sim x_1}{\rho }_{x{x}_1}& -{\rho }_{x_1{x}_2}& \dots & -{\rho }_{x_1{x}_{n-k-1}}\\ -{\rho }_{x_1{x}_2}& {\sum }_{x\sim x_2}{\rho }_{x{x}_2}& \dots & -{\rho }_{x_2{x}_{n-k-1}}\\ \dots & & & \\ -{\rho }_{x_1{x}_{n-k-1}}& -{\rho }_{x_2{x}_{n-k-1}}& \dots & {\sum }_{x\sim x_{n-k-1}}{\rho }_{x{x}_{n-k-1}}\end{array}\right],$$

$$b=({\rho }_{a_0{x}_1},{\rho }_{a_0{x}_2},\dots ,{\rho }_{a_0{x}_{n-k-1}}{)}^T,$$

$$\hat{v}=(v(x_1),v(x_2),\dots ,v(x_{n-k-1}){)}^T.$$

$$Z_{eff}(\Gamma )=\frac{1}{{\sum }_{x:x\sim a_0}(1-v(x)){\rho }_{x{a}_0}},$$

$${\mathcal{P}}_{eff}(\Gamma )=\sum _{x:x\sim a_0}(1-v(x)){\rho }_{x{a}_0}.$$

$${\mathcal{P}}_{eff}(\Gamma )=\sum _{i=1}^{|B|}\sum _{x:x\sim a_i}v(x){\rho }_{x{a}_i}=\sum _{i=1}^{|B|}{\Delta }_{\rho }v(a_i)=-{\Delta }_{\rho }v(a_0)=\frac{1}{2}\sum _{\begin{array}{c}x\sim y\\ x,y\in V\end{array}}({\nabla }_{xy}v{)}^2{\rho }_{xy},$$

$${\rho }_{x_i{x}_j}^{\prime }={\rho }_{x_i{x}_j}+\frac{{\rho }_{x_1{x}_i}{\rho }_{x_1{x}_j}}{\rho (x_1)},$$

$$A=\left[\begin{array}{ccccccc}\rho (x_1)& -{\rho }_{x_1{x}_2}& \dots & -{\rho }_{x_1{x}_l}& 0& \dots & 0\\ -{\rho }_{x_1{x}_2}& \rho (x_2)& \dots & -{\rho }_{x_2{x}_l}& -{\rho }_{x_2{x}_{m+1}}& \dots & -{\rho }_{x_2{x}_{n-k-1}}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ -{\rho }_{x_1{x}_l}& -{\rho }_{x_2{x}_l}& \dots & \rho (x_l)& -{\rho }_{x_l{x}_{m+1}}& \dots & -{\rho }_{x_l{x}_{n-k-1}}\\ 0& -{\rho }_{x_2{x}_{m+1}}& \dots & -{\rho }_{x_l{x}_{m+1}}& \rho (x_{m+1})& \dots & -{\rho }_{x_{m+1}{x}_{n-k-1}}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& -{\rho }_{x_2{x}_{n-k-1}}& \dots & -{\rho }_{x_l{x}_{n-k-1}}& -{\rho }_{x_{m+1}{x}_{n-k-1}}& \dots & \rho (x_{n-k-1})\end{array}\right],$$

$$b=({\rho }_{a_0{x}_1},{\rho }_{a_0{x}_2},\dots ,{\rho }_{a_0{x}_l},{\rho }_{a_0{x}_{m+1}},\dots ,{\rho }_{a_0{x}_{n-k-1}}{)}^T,$$

$$\left[\begin{array}[]{ccccccc|c}1&-\frac{\rho_{x_{1}x_{2}}}{\rho(x_{1})}&\dots&-\frac{\rho_{x_{1}x_{l}}}{\rho(x_{1})}&0&\dots&0&\frac{\rho_{a_{0}x_{1}}}{\rho(x_{1})}\\ 0&\rho^{*}(x_{2})&\dots&-\rho^{\prime}_{x_{2}x_{l}}&-\rho_{x_{2}x_{m+1}}&\dots&-\rho_{x_{2}x_{n-k-1}}&\rho^{*}_{a_{0}x_{2}}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&-\rho^{\prime}_{x_{2}x_{l}}&\dots&\rho^{*}(x_{l})&-\rho_{x_{l}x_{m+1}}&\dots&-\rho_{x_{l}x_{n-k-1}}&\rho^{*}_{a_{0}x_{l}}\\ 0&-\rho_{x_{2}x_{m+1}}&\dots&-\rho_{x_{l}x_{m+1}}&\rho(x_{m+1})&\dots&-\rho_{x_{m+1}x_{n-k-1}}&\rho_{a_{0}x_{m+1}}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&-\rho_{x_{2}x_{n-k-1}}&\dots&-\rho_{x_{l}x_{n-k-1}}&-\rho_{x_{m+1}x_{n-k-1}}&\dots&\rho(x_{n-k-1})&\rho_{a_{0}x_{n-k-1}}\\ \end{array}\right]$$

$${\rho }^{\ast }(x_i)=\rho (x_i)-\frac{{\rho }_{x_1{x}_i}^2}{\rho (x_1)}\text{mbox}and{\rho }_{a_0{x}_i}^{\ast }={\rho }_{a_0{x}_i}+\frac{{\rho }_{x_1{x}_i}{\rho }_{a_0{x}_1}}{\rho (x_1)}\text{mbox}foralli=\overline{2,l}.$$

$$\begin{array}{rl}{\rho }^{\prime }(x_i)=& \rho (x_i)-{\rho }_{x_1{x}_i}-\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m{\rho }_{x_i{x}_j}+\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m{\rho }_{x_i{x}_j}^{\prime }\\ =& \rho (x_i)-{\rho }_{x_1{x}_i}-\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m{\rho }_{x_i{x}_j}+\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m\left({\rho }_{x_i{x}_j}+\frac{{\rho }_{x_1{x}_i}{\rho }_{x_1{x}_j}}{\rho (x_1)}\right)\\ =& \rho (x_i)-{\rho }_{x_1{x}_i}+\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m\frac{{\rho }_{x_1{x}_i}{\rho }_{x_1{x}_j}}{\rho (x_1)}\\ =& \rho (x_i)-{\rho }_{x_1{x}_i}+\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}\sum _{\begin{array}{c}j=2\\ j\ne i\end{array}}^m{\rho }_{x_1{x}_j}\\ =& \rho (x_i)-{\rho }_{x_1{x}_i}+\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}\sum _{j=2}^m{\rho }_{x_1{x}_j}-\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}{\rho }_{x_1{x}_i}\\ =& \rho (x_i)-{\rho }_{x_1{x}_i}+\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}\rho (x_1)-\frac{{\rho }_{x_1{x}_i}^2}{\rho (x_1)}\\ =& \rho (x_i)-\frac{{\rho }_{x_1{x}_i}^2}{\rho (x_1)}={\rho }^{\ast }(x_i)\end{array}$$

$$\begin{array}{r}{\rho }_{a_0{x}_i}^{\ast }={\rho }_{a_0{x}_i}+\frac{{\rho }_{x_1{x}_i}{\rho }_{a_0{x}_1}}{\rho (x_1)}=\{\begin{array}{ll}{\rho }_{a_0{x}_i}^{\prime },& \text{if}{a}_0\in \{x_2,\dots ,x_m\}\\ {\rho }_{a_0{x}_i},& \text{otherwise, since}{\rho }_{a_0{x}_1}=0.\end{array}\end{array}$$

$$\left[\begin{array}[]{ccccccc|c}1&-\frac{\rho_{x_{1}x_{2}}}{\rho(x_{1})}&\dots&-\frac{\rho_{x_{1}x_{l}}}{\rho(x_{1})}&0&\dots&0&\frac{\rho_{a_{0}x_{1}}}{\rho(x_{1})}\\ 0&\rho^{\prime}(x_{2})&\dots&-\rho^{\prime}_{x_{2}x_{l}}&-\rho_{x_{2}x_{m+1}}&\dots&-\rho_{x_{2}x_{n-k-1}}&\rho^{\prime}_{a_{0}x_{2}}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&-\rho^{\prime}_{x_{2}x_{l}}&\dots&\rho^{\prime}(x_{l})&-\rho_{x_{l}x_{m+1}}&\dots&-\rho_{x_{l}x_{n-k-1}}&\rho^{\prime}_{a_{0}x_{l}}\\ 0&-\rho_{x_{2}x_{m+1}}&\dots&-\rho_{x_{l}x_{m+1}}&\rho(x_{m+1})&\dots&-\rho_{x_{m+1}x_{n-k-1}}&\rho_{a_{0}x_{m+1}}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&-\rho_{x_{2}x_{n-k-1}}&\dots&-\rho_{x_{l}x_{n-k-1}}&-\rho_{x_{m+1}x_{n-k-1}}&\dots&\rho(x_{n-k-1})&\rho_{a_{0}x_{n-k-1}}\\ \end{array}\right],$$

$$x_{m-j}=a_j,j=\overline{0,|\{x_1,\dots ,x_m\}\cap B_0|},$$

$${\mathcal{P}}_{eff}(\Gamma )=$$

$$=$$

$$\rho (x_1)=\sum _{i=2}^m{\rho }_{x_1{x}_i}\text{mbox}andv(x_1)=\sum _{i=2}^{l}v(x_i)\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}+\frac{{\rho }_{x_1{a}_0}}{\rho (x_1)}=\sum _{i=2}^{m-1}v(x_i)\frac{{\rho }_{x_1{x}_i}}{\rho (x_1)}+\frac{{\rho }_{x_1{a}_0}}{\rho (x_1)}$$

$${\rho }_{ac}^{\prime }=\frac{{\rho }_{ab}{\rho }_{ac}}{{\rho }_{ab}+{\rho }_{ac}},$$

$$z_{ac}^{\prime }=z_{ab}+z_{ac},$$

$${\rho }_{ac}^{\prime }=\frac{{\rho }_{ab}{\rho }_{bc}}{{\rho }_{ab}+{\rho }_{bc}}+{\rho }_{ac},$$

$$\frac{1}{z_{ac}^{\prime }}=\frac{1}{z_{ab}+z_{bc}}+\frac{1}{z_{ac}}$$

Full text

This research was supported by IRTG 2235 Bielefeld-Seoul “Searching for the regular in the irregular: Analysis of singular and random systems”.

Keywords: weighted graphs, electrical network, ladder network, effective impedance, Laplace operator, ordered field, non-Archimedean field, Levi-Civita field.

**Mathematics Subject Classification 2010: 05C22, 34B45, 05C25, 39A12, 12J15. **

1. Introduction

It is known that electrical networks with resistances are related to weighted graphs (see e.g. [4], [10], [13]). Moreover, it is shown in [8] and [10], that effective resistance for finite networks satisfies the basic physical properties (e.g. parallel and series laws). In [4] and [8] the notion of effective resistance for infinite network is introduced. An effective resistance is tightly related to random walk and Dirichlet problem on graphs, which are described in many papers and books (e. g. [1], [7], [14],[13]). In [11] a finite electrical network with alternating current and passive elements is considered as a generalization of electrical network with resistances. It is shown there, that such a network is related to weighted graphs over non-Archimedean ordered field of rational functions $\mathbb{R}(\lambda)$. The generalization of effective resistance for this case is called effective impedance. The inverse of effective impedance is called effective admittance. The most known in physics infinite network with passive elements is Feynman’s ladder network ($LC$-network, see [6]). In [15] the effective impedances of $LC$-network and $CL$-network are considered as limits of complex-valued effective impedances of corresponding sequences of finite networks.

The present paper consists of two parts. In the first part we describe some properties of electrical network over an ordered field. The main result of this part is Theorem 4, which gives the mathematical description of well-known in physics star-mesh transform. The mathematical conceptions of parallel and series laws, as well as $\Delta-Y$ and $Y-\Delta$ transform, follow from Theorem 4 as corollaries.

In the second part of the present paper we discuss the question whether one can generalize the notion of effective resistance for infinite networks with zero on infinity (see e.g. [8]) for the case of non-Archimedean weighted graphs. The main theorem of this section is Theorem 15. It shows, that a sequence of effective admittances of finite networks, exhausted a given infinite network, decreases. Unfortunately, it does not give a convergence over non-Archimedean field. As examples, we consider $LC$-network and $CL$-network (with zero on infinity) as electrical networks over ordered Levi-Civita field $\mathcal{R}$, which contains a subfield isomorphic to $\mathbb{R}(\lambda)$ (see [2], [9]). Firstly, we present the general calculation of admittance of a finite ladder network (see Figure 7) over an ordered field. Then the closeness of the Levi-Civita field in order topology ([2], [12]) gives us an opportunity to arise the question whether effective admittance of infinite network could be defined as limit of effective admittances of corresponding finite electrical networks in this case. We show, that in case of the $LC$-network the sequence of effective admittances of finite networks converge in ordered topology of the Levi-Civita field (Theorem 20). Moreover, we show, that admittances of finite $CL$-network do not converge in the same topology (Example 23). This shows, that in general it is not possible to generalize the notion of effective resistance for infinite networks for the case of non-Archimedean weights.

2. Properties of effective impedance of the finite network over ordered field

Definition 1**.**

A network over an ordered field $(K,\succ)$ is a structure

[TABLE]

where

• •

$(V,E)$ is a locally finite connected graph ($|V|\geq 2$),

• •

$\rho:E\rightarrow K$ is a positive function called admittance,

• •

$a_{0}\in V$ is a fixed vertex,

• •

$B=\{a_{1},\dots,a_{|B|}\}\in V\setminus\{a_{0}\}$ is a fixed non-empty subset of vertices.

Let us denote by $B_{0}=B\cup\{a_{0}\}$ the set of boundary vertices and by $z=\frac{1}{\rho}$ the positive function of impedance.

The network is called finite if $|V|<\infty$. Otherwise, it is called infinite.

Note that we can consider $\rho$ as a function from $V\times V$ to $K$ by setting $\rho_{xy}=0$, if $xy$ is not an edge. Then the weight $\rho_{xy}$ gives rise to a function on vertices as follows:

[TABLE]

where the notation $\sum\limits_{y}$ means $\sum\limits_{y\in V}$. Then $\rho(x)$ is called the weight of a vertex $x$. We have $0<\rho(x)<\infty$ for any vertex $x$ of a network .

Let us consider the following Dirichlet problem on the given finite network $\Gamma=(V,\rho,a_{0},B)$:

[TABLE]

where $\Delta_{\rho}v(x)=\sum_{y}(v(y)-v(x))\rho_{xy}.$

The physical meaning of Dirichlet problem is the following: if we take $K=\mathbb{R}(\lambda)$ and admittance of each edge in the form

[TABLE]

then the real voltage at the vertex $x$ at time $t$ will be equal to $\Re(v(x)e^{i\omega t})$, assuming that we keep potential $1$ at the vertex $a_{0}$, ground all the vertices from $B$ and apply alternating current of frequency $\omega=-i\lambda$ to the network (see [11]).

Note that if $|V|=n$, then the Dirichlet problem (2) is a $n\times n$ system of linear equations over the field $K$. It can be also written in a matrix form (note, that here we already have substituted $v(a_{0})=1,v(a_{i})=0,i=\overline{1,|B|}$ in the first $(n-|B|-1)$ equations):

[TABLE]

where $k=|B|$, $A$ is a symmetric matrix ($A=A^{T}$), ${\hat{v}},{b}$ are vector-columns of length $(n-k-1)$:

[TABLE]

[TABLE]

[TABLE]

In [11] it is proved, that the Dirichlet problem (2) has a unique solution for any finite network over an ordered field.

Definition 2**.**

We define effective impedance of the finite network $\Gamma$ as

[TABLE]

where $v$ is the solution of the Dirichlet problem (2).

The effective admittance is defined by

[TABLE]

Lemma 3**.**

For the solution $v$ of the Dirichlet problem (2) we have

[TABLE]

where $\nabla_{xy}v=v(y)-v(x)$.

The proof of this result follows the same outline as the proof of the similar result in [11].

Theorem 4**.**

(Star-mesh transform)* Let $\Gamma=(V,\rho,a_{0},B)$ be a finite network, $|V|=n$, $B_{0}=B\cup\{a_{0}\}$, and $x_{1},\dots,x_{m}\in V$, $3\leq m\leq n$, are such that*

1. (1)

$x_{1}\not\in B_{0}$, 2. (2)

$y\not\sim x_{1}$* for all $y\in V\setminus\{x_{2},\dots,x_{m}\}$,*

If one removes the vertex $x_{1}$, edges $(x_{1},x_{i}),i=\overline{2,m}$ and change the admittances of the edges $(x_{i},x_{j}),i,j=\overline{2,m},i\neq j$ as follows:

[TABLE]

not changing the other admittances, then for the new network the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network at corresponding vertices.

Proof.

Let us consider the Dirichlet problem for the network $\Gamma$ in a matrix form (3). Obviously, it is enough to solve the matrix equation $A\hat{v}=b$. Without loss of generality we can assume that $x_{1},\dots,x_{l}\not\in B_{0}$, where $l=m-|\{x_{1},\dots,x_{m}\}\cap B_{0}|$. Writing equations for $x_{1},\dots,x_{l}$ as the first ones and denoting $k=|B|$, we have

[TABLE]

since $y\not\sim x_{1}$ for all $y\in V\setminus\{x_{2},\dots,x_{m}\}$, and

[TABLE]

Now it is easy to calculate, that the star-mesh transform is just applying of Gaussian elimination method for the first row. Indeed, applying Gaussian elimination method for the first row of the augmented matrix $\bar{A}=[A|b]$ we obtain:

[TABLE]

since $\rho(x_{1})\neq 0$, where

[TABLE]

Note, that for all $i=\overline{2,l}$

[TABLE]

and

[TABLE]

Hence, $\bar{A}=$

[TABLE]

Therefore, we can eliminate the variable $v(x_{1})$ from the Dirichlet problem, changing admittances as in the statement of the theorem. ∎

Corollary 5**.**

Under the star-mesh transform of the network the effective impedance and effective admittance do not change.

Proof.

In the proof we will use the notations from the proof of the Theorem 4.

The case $\{x_{1},\dots,x_{m}\}\cap B_{0}=\emptyset$ is trivial. The cases, when $\{x_{1},\dots,x_{m}\}\cap B=\emptyset$ or $\{x_{1},\dots,x_{m}\}\cap\{a_{0}\}=\emptyset$ are obvious, due to (4).

Otherwise, we can assume, without loss of generality, that

[TABLE]

in particular, $x_{m}=a_{0}$. Then, if we denote the new network by $\Gamma^{\prime}$ we have by (4)

[TABLE]

since

[TABLE]

(see the first line of $\bar{A}$ and note that $v(x_{j})=0$ for all $j=\overline{j+1,m-1}$ and $a_{0}=x_{m}$). ∎

Series law and $Y-\Delta$ transform are just particular cases of star-mesh transform. Since multigraphs are not allowed in this paper, we will use a modification of parallel law and call it parallel-series law.

Corollary 6**.**

(Series law)* Let $\Gamma=(V,\rho,a_{0},B)$ be a finite network, $B_{0}=B\cup\{a_{0}\}$. Let $a,b,c\in V$ are such, that*

1. (1)

$b\not\in B_{0}$, 2. (2)

$a\not\sim c$, $a\sim b$, $b\sim c$, 3. (3)

$b\not\sim x$* for all $x\not\in\{a,c\}$.*

If one removes the vertex $b$, edges $(a,b),(b,c)$ and add the edge $(a,c)$ with the addmittance

[TABLE]

not changing other admittances, then for the new network the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network at corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (addmittance) of the original one.

Remark 7**.**

The corresponding equation for impedances is then

[TABLE]

which corresponds to the well-known physical series law.

Proof.

Apply Theorem 4 and Corollary 5 ($x_{1}=b$) for the case $m=3$ and $\rho_{ac}=0$. ∎

Corollary 8**.**

(Parallel-series law)* Let $\Gamma=(V,\rho,a_{0},B)$ be a finite network, $B_{0}=B\cup\{a_{0}\}$.

Let $a,b,c\in V$ are such, that*

1. (1)

$b\not\in B_{0}$, 2. (2)

$a\sim b,b\sim c,a\sim c$, 3. (3)

$b\not\sim x$* for all $x\not\in\{a,c\}$.*

Then if one removes the vertex $b$, edges $(a,b),(b,c)$ and add the edge $(a,c)$ with the admittance

[TABLE]

not changing other admittances, then for the new network the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network for corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.

Remark 9**.**

The corresponding equation for impedances is then

[TABLE]

which corresponds to the application of the physical series law and then the physical parallel law.

Proof.

Apply Theorem 4 and Corollary 5 ($x_{1}=b$) for the case $m=3$. ∎

Theorem 10**.**

($Y-\Delta$ transform)* Let $\Gamma=(V,\rho,a_{0},B)$ be a finite network, $B_{0}=B\cup\{a_{0}\}$. Let $a,b,c,d\in V$ are such, that*

1. (1)

$d\not\in B_{0}$, 2. (2)

$d\sim a,d\sim b,d\sim c$, 3. (3)

$d\not\sim x$* for all $x\not\in\{a,b,c\}$.*

If one removes the vertex $d$, edges $(d,a),(d,b),(d,c)$ and set

[TABLE]

not changing other admittances, then for the new network the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.

Remark 11**.**

The corresponding equalities for the impedances are

[TABLE]

From the physical point of view, if $\rho_{ab},\rho_{bc},\rho_{ac}$ are all equal to zero, then it is just $Y-\Delta$ transform, otherwise, it is $Y-\Delta$ transform and the parallel law.

Proof.

Theorem 4 and Corollary 5 ($x_{1}=d$) for the case $m=4$. ∎

The $Y-\Delta$ transform is invertible. In general, it is not the case for star-mesh transform.

Theorem 12**.**

($\Delta-Y$ transform)* Let $\Gamma^{\prime}=(V^{\prime},\rho^{\prime},a_{0},B)$ be a finite network and let $a,b,c\in V$ are such, that $a\sim b,b\sim c$, and $a\sim c$. If one add a vertex $d$ and edges $(d,a),(d,b),(d,c)$ setting*

[TABLE]

and remove the edges $(a,b),(b,c),(a,c)$ not changing other admittances, then for the new network

[TABLE]

the solution of the Dirichlet problem (2) for any vertex in $V$ will be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertex. Moreover, the effective impedance and effective admittance do not change under this transform.

Remark 13**.**

The corresponding equalities for the impedances are

[TABLE]

Proof.

To prove the theorem it is enough to express $\rho_{da}$, $\rho_{db}$ and $\rho_{dc}$ from (7), assuming $\rho_{ab}=0$, $\rho_{bc}=0$, and $\rho_{ac}=0$. Summing up the inverses of all three equations one obtains

[TABLE]

Since both sides are strictly positive, the last equation is equivalent to

[TABLE]

Multiplying the both sides of the last equation by

[TABLE]

which follows from (7), we get

[TABLE]

Then the equation for $\rho_{da}$ follows. To obtain the equations for $\rho_{db}$ and $\rho_{dc}$ one should multiply (9) by $\frac{1}{\rho^{\prime}_{ab}\rho^{\prime}_{bc}}$ and $\frac{1}{\rho^{\prime}_{ac}\rho^{\prime}_{bc}}$ respectively.

The fact that effective impedance and effective admittance do not change follows from Theorem 10. ∎

Remark 14**.**

All described in this section transforms preserve the positivity of admittances and impedances on the edges.

3. Effective impedance of infinite networks over an ordered field

3.1. Infinite networks with zero potential on infinity

Let $\Gamma=(V,\rho,a_{0},B)$ be an infinite network over an ordered field $(K,\succ)$. Let us consider the sequence of finite graphs $(V_{n},\left.\rho\right|_{V_{n}})$, where $V_{n}=\{x\in V\mid\operatorname{dist}(a_{0},x)\leq n\}$, $n\in\mathbb{N}$.

We denote by

[TABLE]

the boundary of the graph $(V_{n},\left.\rho\right|_{V_{n}})$. Note that $V_{n+1}=\partial V_{n+1}\cup V_{n}$.

Let us denote $B_{n}=B\cap V_{n}$. Then

[TABLE]

is a sequence of finite networks exhausted the infinite network $\Gamma$.

This is an analogue to the approach to infinite networks in [8].

Let us consider the Dirichlet problem (2) on each $\Gamma_{n}$:

[TABLE]

Theorem 15**.**

[TABLE]

Proof.

By Dirichlet/Thomson’s principle [11] we have

[TABLE]

for any $f:V_{n+1}\rightarrow K$ such that $f(a_{0})=1,\left.f\right|_{\partial V_{n+1}\cup B_{n+1}}\equiv 0$.

Since $(V_{n+1}\setminus\partial V_{n+1})=V_{n}$ and $B_{n+1}\cap V_{n}=B_{n}$, the inequality (12) is true for

[TABLE]

where $v^{(n)}$ is the solution of (10) for $\Gamma_{n}$. Then

[TABLE]

The last equality, together with (12), gives us $\eqref{monotonicity}$. ∎

Remark 16**.**

Even in a Cauchy complete non-Archimedean ordered field inequalities (11) for all $n\in\mathbb{N}$ do not imply, that the sequence $\{\mathcal{P}_{eff}(\Gamma_{n})\}_{n=1}^{\infty}$ converges. Obviously, if the sequence of effective admittances of finite networks converges, then the corresponding sequence of the effective impedances also has a limit (finite or infinite).

Definition 17**.**

If for given infinite network $\Gamma$ the limit of effective admittances (impedances) of exhausted finite networks exists in $K$, we call it effective admittance (impedance) of the network $\Gamma$ with zero potential at infinity and denote it by $\mathcal{P}_{eff}(\Gamma)$ ($Z_{eff}(\Gamma)$).

3.2. Examples: ladder networks over Levi-Civita field

In this subsection we will investigate the behavior of the sequence $\{\mathcal{P}_{eff}(\Gamma^{\alpha\beta}_{n})\}_{n=1}^{\infty}$ of effective admittances of finite networks exhausted the ladder network ($\alpha\beta$-network) at the Figure 6 ($\alpha,\beta\in K$, $\alpha,\beta\succ 0$). More precisely, $\alpha\beta$-network is a network $\Gamma^{\alpha\beta}=\{V,\rho,a_{0},B\}$, where

• •

$V=\{a_{0},a_{1},a_{2},\dots,x_{1},x_{2},\dots\}$,

• •

$\rho_{a_{0}x_{1}}=\alpha$, $\rho_{x_{i}x_{i+1}}=\alpha,\rho_{a_{i}x_{i}}=\beta,i\in\mathbb{N}$, and $\rho_{xy}=0,\mbox{otherwise}$,

• •

$B=\{a_{0},a_{1},a_{2},\dots\}$.

This network is similar to Feynman’s ladder network and $CL$-network (see [6], [15]), but has zero potential at infinity. Therefore, for any ordered field $K$ the Theorem 15 is true for this network. We will show (Theorem 20 and Example 23) that whether $\{\mathcal{P}_{eff}(\Gamma_{n})\}_{n=1}^{\infty}$ converges in Cauchy completness of $K$ depends on $\alpha$ and $\beta$.

3.2.1. Finite ladder network over ordered field

Let $\Gamma_{n}$ be a sequence of finite networks exhausted an $\alpha\beta$-network (see Figure 7).

The Dirichlet problem (2) for this network is the following

[TABLE]

Using the second line in (13) we obtain the following recurrence relation for $v(x_{i}),i=\overline{2,n-1}$

[TABLE]

since $v(a_{i})=0$ for $i=\overline{2,n-1}$. The characteristic polynomial of (14) is

[TABLE]

Its roots are

[TABLE]

where

[TABLE]

Note that $\xi$ should not necessary belong to $K$. It is known, that any ordered field posses a real-closed (or maximal ordered) extension $\overline{K}$. Then in $\overline{K}$ exists exactly one positive square root of ${\frac{\beta}{\alpha}+\left(\frac{\beta}{2\alpha}\right)^{2}}$ ([3]). Therefore, we fix the extension $\overline{K}$, denote the positive square root by $\xi$, and make all the further calculations in $\overline{K}$.

The solution of the recurrence equation (14) is

[TABLE]

where $c_{1},c_{2}\in\overline{K}$ are arbitrary constants.

We use first and third equations in (13) as boundary conditions for this recurrence equation. Substituting (17) in the boundary conditions we obtain the following equations for the constants:

[TABLE]

which, by (15) is equivalent to

[TABLE]

Therefore,

[TABLE]

since $\psi_{1}\psi_{2}=1$ by (15).

Now we can calculate the effective admittance of $\Gamma_{n}$:

[TABLE]

Since $\mathcal{P}_{eff}\left(\Gamma_{n}\right)$ is an element of $K$ as the solution of the Dirichlet problem (2) over $K$, it can be written without $\xi$:

[TABLE]

where in the last line we have used binomial expansion.

3.2.2. Infinite ladder networks over Levi-Civita field $\mathcal{R}$

We will consider two examples of $\alpha\beta$-network over Levi-Civita field $\mathcal{R}$. Firstly, let us describe the Levi-Civita field $\mathcal{R}$. We take the definition of $\mathcal{R}$ and theorems about its properties from [2] and [12].

Definition 18**.**

A subset $M$ of the rational numbers $\mathbb{Q}$ is called left-finite if for every $r\in\mathbb{Q}$ there are only finitely elements of $M$ that are smaller than $r$.

Then the Levi-Civita field is the set of all real valued functions on $\mathbb{Q}$ with left-finite support with the following operations:

• •

addition is defined component-wise

[TABLE]

• •

multiplication is defined as follows

[TABLE]

It is proved in [2], that $\mathcal{R}$ is an ordered field with a set of positive elements

[TABLE]

We denote by $\tau$ the following element in $\mathcal{R}$:

[TABLE]

which plays role of infinitesimal in Levi-Civita field. Therefore, the Levi-Civita field is non-Archimedean.

By [2] we can write any $\alpha\in\mathcal{R}$ as

[TABLE]

since $\alpha_{n}=\sum_{i=1}^{n}\alpha(q_{i})\tau^{q_{i}}$ converges strongly to the limit $\alpha$ in the order topology.

The set of all polynomials over real numbers $\mathbb{R}[\tau]=\{a_{n}\tau^{n}+\dots a_{1}\tau+a_{0}\mid a_{i}\in\mathbb{R},n\in\mathbb{N}\}$ is a subring of Levi-Civita field $\mathcal{R}$ due to (21). Therefore, since $\mathcal{R}$ is a field, the field of rational functions over real numbers

[TABLE]

is isomorphic to a subfield of $\mathcal{R}$.

Example 19**.**

Let us find the element in $\mathcal{R}$ which corresponds to the rational function $\frac{1}{\tau^{2}-4\tau+3}$, i. e. we should find the sequences $\{q_{i}\}\in\mathbb{Q}$ and $\{\alpha(q_{i})\}\in\mathbb{R}$ such that

[TABLE]

Comparing the coefficients at powers of $\tau$ at right hand side and left hand side, starting from the lowest power, one obtains

[TABLE]

[TABLE]

and the recurrence relation

[TABLE]

Therefore, solving the recurrence relation for $\alpha(q_{i})$ we obtain

[TABLE]

and

[TABLE]

Note, that the corresponding order in the field $\mathbb{R}(\tau)$ is the following:

[TABLE]

Therefore, we can consider Levi-Civita field $\mathcal{R}$ as an ordered extension of the ordered field $\mathbb{R}(\tau)$ with the positiveness defined as (23). To consider $\mathcal{R}$ as an ordered extension of the ordered field $\mathbb{R}(\tau)$ with the positiveness defined in [11] we make a substitution

[TABLE]

Consequently, we can consider electrical networks over field of rational numbers [11], as networks over Levi-Civita field and investigate the behavior of the sequence of effective admittances of finite electrical networks. By [2] the Levi-Civita field is Cauchy complete in order topology and real-closed.

From the physical point of view we have the following impedances of passive elements

• •

$(Li\omega)^{-1}=L^{-1}\tau$, $L>0$ for the coil,

• •

$(Ci\omega)=C\tau^{-1}$, $C>0$ for the capacitor,

• •

$R>0$ for the resistor,

where $\omega$ is a frequency of the alternating current (see [11]).

Let us consider the Feynman’s infinite ladder $LC$-network, assuming that it has zero potential at infinity. It is an $\alpha\beta$-network with $\alpha=L^{-1}\tau$, $\beta={C}\tau^{-1}$, where $L,C>0$, $\alpha,\beta\in\mathcal{R}$.

Theorem 20**.**

For the Feynman’s ladder $LC$-network ($\alpha=L^{-1}\tau$, $\beta={C}\tau^{-1}$, where $L,C>0$) with zero potential at infinity

[TABLE]

in the order topology of Levi-Civita field $\mathcal{R}$, where $\Gamma_{n}$ is the sequence of the exhausted finite networks

Remark 21**.**

For the Feynman’s ladder $LC$-network

[TABLE]

and the motivation for this quantity was Feynman’s impedance for infinite ladder $LC$-network (see [6, 22-13]).

Proof.

Firstly, we should write $\xi$ as an element of Levi-Civita field, i. e. as power series (21).

[TABLE]

Note that here and further $o\left(\tau^{m}\right)$, where $m\in\mathbb{Z}$ means $\sum_{k=m+1}^{\infty}a_{k}\tau^{k},a_{k}\in\mathbb{R}$.

Let us calculate the difference $\mathcal{P}_{eff}\left(\Gamma_{n}\right)-\frac{\beta}{\frac{\beta}{2\alpha}+\xi}$.

[TABLE]

The nominator $A$ of the last expression is

[TABLE]

where $D=\left(1+\frac{\beta}{2\alpha}+\xi\right)^{2n-1}$.

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

Therefore,

[TABLE]

The right hand side of the last expression is, obviously, positive in $(\mathcal{R},\succ)$, therefore

[TABLE]

when $n\rightarrow\infty$. ∎

Remark 22**.**

From the proof one can see that (25) is true for $\alpha\beta$-network whenever for any $\gamma\in\mathcal{R}$ exists $N_{0}\in\mathbb{N}$ such that $n>N_{0}$ implies $\left(\frac{\beta}{\alpha}\right)^{n}\succ\gamma$.

Example 23**.**

For the $CL$-network ($\alpha={C}\tau^{-1}$, $\beta=L^{-1}\tau$, $L,C>0$) effective admittances of the exhausted finite networks do not converge in the Levi-Civita field $\mathcal{R}$.

Proof.

In this case $\xi=\frac{\tau}{\sqrt{CL}}+\left(\frac{\tau}{2\sqrt{CL}}\right)^{3}+o(\tau^{3})$. Let us prove, that $\{\mathcal{P}(\Gamma_{n})\}_{n=1}^{\infty}$ is not a Cauchy sequence in $\mathcal{R}$. Indeed

[TABLE]

Since

[TABLE]

we can rewrite

[TABLE]

Substituting

[TABLE]

and

[TABLE]

we obtain

[TABLE]

Therefore, $\{\mathcal{P}(\Gamma_{n})\}_{n=1}^{\infty}$ is not a Cauchy sequence in $\mathcal{R}$. ∎

Therefore, the following question:

Under what conditions the effective admittance of infinite network over non-Archimedean field could be defined?

remains open. Note, that Remark 22 gives some sufficient condition for $\alpha\beta$-network.

Acknowledgement

The author thanks her scientific advisor, Professor Alexander Grigor’yan, for fruitful discussions on the topic.