In this paper, we study the properties of effective impedances of finite electrical networks, considering them as weighted graphs over an ordered field. We prove that a star-mesh transform of finite network does not change its effective impedance. Moreover, we consider two particular examples of infinite ladder networks (Feynman’s network or LC-network and CL-network, both with zero at infinity) as networks over the ordered Levi–Cività field R. We show that the sequence of effective impedances of finite LC-networks converges to the limit in the order topology of R, but the sequence of effective impedances of finite CL-networks does not converge in the same topology. We calculate an effective impedance of a finite ladder network as an auxiliary result.

It is known that electrical networks with resistances are related to weighted graphs, where weights of edges are positive real numbers (see, e.g., Refs. 9, 19, and 22). Moreover, it is known that an effective resistance of the finite network does not change under some transforms of a network [star-mesh transform, parallel and series laws, etc. (see, e.g., Refs. 14 and 19)]. In Refs. 9 and 14, the notion of effective resistance for the infinite network is stated. An effective resistance is tightly related to the random walk and Dirichlet problem on a graph, which are described in many papers and books (e.g., Refs. 2, 13, 22, and 26). In Ref. 20, a finite electrical network with alternating current and passive elements is considered as a generalization of the electrical network with resistances. It is shown there that such a network is related to weighted graphs over non-Archimedean ordered fields of rational functions R(λ). The generalization of effective resistance for this case is called effective impedance. The inverse of effective impedance is called effective admittance. The most known physics infinite network with passive elements is Feynman’s ladder network (LC-network, see Ref. 12). In Ref. 27, the effective impedances of the LC-network and CL-network are considered as limits of complex-valued effective impedances of exhausted sequences of finite networks. Moreover, in Refs. 1 and 7, electrical networks on fractals are considered. In the present paper, we bring together the theory of random walks on graphs (where weights are always positive, see, e.g., Ref. 9) and electrical engineering, where electrical networks with impedances are widely considered. Moreover, we present a new application of the Levi–Cività fields whose other applications to mathematical physics are described in Refs. 3 and 4.

This paper is organized as follows: In Sec. II, we describe some properties of an electrical network over an ordered field. We consider an electrical network with passive elements as a weighted graph whose weights are positive elements of any ordered field. This is a natural generalization of weighted graphs over R. Moreover, in the previous paper by the same author,20 some statements about physical voltage and physical effective impedance can be shown using a theory of networks over ordered fields. The main result of Sec. II is given in Theorem 4, which gives the mathematical description of well-known in electrical engineering star-mesh transform (see, e.g., Ref. 16). The mathematical conceptions of parallel and series laws, as well as Δ − Y and Y − Δ transform, follow from Theorem 4 as corollaries.

In Sec. III, we discuss the question whether one can generalize the notion of effective resistance for infinite networks with zero on infinity (see, e.g., Refs. 14 and 22) 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 a non-Archimedean field. First, we present the general calculation of admittance of a finite ladder network (see Fig. 7) over an ordered field. Then, we consider the LC-network and CL-network (with zero on infinity) as electrical networks over ordered Levi–Cività field R, which contains a subfield isomorphic to R(λ) (see Refs. 3, 4, and 15). The ordered Levi–Cività field (which is non-Archimedean) is widely used in different branches of mathematics, mathematical physics, and programming. This field R has “infinitely small” elements, which qualitatively differ it from the field of real numbers R, where all elements are finite. The concept of “infinitely small” element allows more precise computations, particularly in computational differentiation (see Refs. 3, 4, and 17, p. 373–383). In Ref. 4, how techniques of the Levi–Cività field can be used for the solving of ordinary differential equations (ODE) and partial differential equations (PDE) over real numbers is described. All the above show the importance of the R as an extension of R. Indeed, considering networks with weights from R, we show more interesting and smaller results for infinite networks in the case of real-weights. A Cauchy completeness of the Levi–Cività field in order topology (Refs. 3 and 21) gives us an opportunity to arise the question whether effective admittance of the infinite network could be defined as the limit of effective admittances of corresponding finite electrical networks in this case. We show that in the case of the LC-network, the sequence of effective admittances of finite networks converges in ordered topology of the Levi–Cività field (Theorem 20). The result is similar to the classical Feynman’s result, which was stated in Ref. 12, but is not carefully explained there. Other explanations for this result are given in Refs. 10, 18, 23, 24, and 27. Moreover, we show that admittances of the finite CL-network do not converge in the same topology (example 23). The latest example 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, or, more precisely, in the case of R-weights, there is some other possibility for a network, apart from being transient or recurrent. This again shows that computations over the Levi–Cività field are more precise than over the real numbers. Moreover, we give a sufficient condition on an infinite ladder network for convergence of the sequence of admittances (Remark 22).

We consider locally finite connected graphs whose weights are positive elements of some fixed ordered field. Moreover, at each graph, we fix a vertex with potential 1 and a set of vertices with potential 0. Such a structure we call a network. Let us give a precise definition.

Definition 1.
A network over an ordered field (K, ≻) is a structure,
Γ=((V,E),ρ,a0,B),
where
  • (V, E) is a locally finite connected graph without loops (V is a set of vertices, |V| ≥ 2, E is a set of edges),

  • ρ: V × VK is a non-negative function (ρxy ⪰ 0) such that ρxy = 0 if and only if xyE,

  • a0V is a fixed vertex, and

  • B = {a1, …, a|B|} ∈ V \{a0} is a fixed non-empty subset of vertices.

The function ρ is called admittance. We denote by B0 = B ∪ {a0} the set of boundary vertices and by z=1ρ a positive function of impedance, defined on E. Since the function ρ uniquely determines the set of edges, we will omit edges in the notation of a network and write Γ = (V, ρ, a0, B).

For any x, yV, we write xy if xyE, and x, yV otherwise.

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

Since the graph is locally finite, the weight ρxy gives rise to a function on vertices as follows:

ρ(x)=yρxy,
(1)

where here and further in notations y means yV. Then, ρ(x) is called the weight of a vertexx. We have 0 < ρ(x) ∈ K for any vertex x of a network.

Let us consider the following system of linear equations on a given finite network Γ = (V, ρ, a0, B):

Δρv(x)=0 on V\B0,v(x)=0 on B,v(a0)=1,
(2)

where Δρv(x)=y(v(y)v(x))ρxy.

We consider (2) as a discrete boundary value Dirichlet problem. This Dirichlet problem arises from electrical engineering. Indeed, by Ohm’s complex law (see, e.g., Refs. 8 and 11), an impedance of any edge xy, containing passive elements (resistor, coil, and capacitor), can be written in the following form:

zxy=Rxy+Lxyiω+Dxyiω,

where Rxy > 0 is a resistance, Lxy > 0 is an inductance, Dxy > 0 is the inverse of a capacitance (Dxy=1Cxy), ω is a frequency of the alternating voltage, and i is the imaginary unit. Then, denoting λ = (this idea goes back to Ref. 6), we have for an admittance of an edge,

ρxy=λLxyλ2+Rxyλ+Dxy,Lxy,Rxy,Dxy0, and Lxy+Rxy+Dxy0,

where Lxy = 0, Rxy = 0, or Dxy = 0 means the lack of the corresponding element on the edge.

Hence, defined ρxy is a positive element in the ordered field of rational functions with real coefficients R(λ) with the order, defined as follows [e.g., Ref. 5 (p. A.VI.21) and Ref. 25 (p. 234)]: for any rational function,

f(λ)=bkλk++b1λ+b0dmλm++d1λ+d0R(λ),

with bk ≠ 0, dm ≠ 0; write

f(λ)0  if bkdm>0
(3)

and

f(λ)g(λ),  if f(λ)g(λ)0.

Afterward, we assume that we keep potential (voltage) 1 at the vertex a0 and ground the set B (i.e., keep potential zero). Then, all the other vertices should be considered as inner nodes, and by Kirchhoff’s law, Δρv(x) = 0 for them (a sum of incoming flows is equal to the sum of outgoing flows, see, e.g., Refs. 8, 11, and 16). Therefore, the Dirichlet problem (2) appears.

The Dirichlet problem (2) is a n × n system of linear equations over the field K, where n = |V|. It can also be written in a matrix form [here, we already have substituted v(a0)=1,v(ai)=0,i=1,|B|̄ in the first (n − |B| − 1) equations],

Av̂=b,v(a0)=1,v(ai)=0,i=1,k̄,
(4)

where k = |B|, A is a symmetric matrix (A = AT), and v̂,b are vector-columns of length (nk − 1),

A=xx1ρxx1ρx1x2ρx1xnk1ρx1x2xx2ρxx2ρx2xnk1ρx1xnk1ρx2xnk1xxnk1ρxxnk1,
b=(ρa0x1,ρa0x2,,ρa0xnk1)T,
v̂=(v(x1),v(x2),,v(xnk1))T.

In Ref. 20, it is proved that the Dirichlet problem (2) has a unique solution for any finite network over an ordered field in the case of |B| = 1. Moreover, exactly this fact, applied to the ordered field of rational functions, allows for the strict definition of the physical effective impedance (as a complex number), which shows a deep relation of the present theory to mathematical physics (see Ref. 20). In the case of |B| > 1, the proof follows exactly the same outline; therefore, we do not present it here.

FIG. 1.

Star-mesh transform over an ordered field for m = 7.

FIG. 1.

Star-mesh transform over an ordered field for m = 7.

Close modal
FIG. 2.

Series law.

FIG. 3.

Parallel-series law.

FIG. 3.

Parallel-series law.

Close modal
FIG. 4.

Y − Δ transform.

FIG. 4.

Y − Δ transform.

Close modal
FIG. 5.

Δ − Y transform.

FIG. 5.

Δ − Y transform.

Close modal

Definition 2.
We define the effective admittance of a finite network Γ as
Peff(Γ)=x:xa0(1v(x))ρxa0,
where v is the solution of the Dirichlet problem (2).
Then, the effective impedance is defined by
Zeff(Γ)=1Peff(Γ)=1x:xa0(1v(x))ρxa0,
i.e., Zeff(Γ) ∈ K ∪ {}.

Lemma 3.
For the solutionvof the Dirichlet problem (2), we have
Peff(Γ)=i=1|B|x:xaiv(x)ρxai=i=1|B|Δρv(ai)=Δρv(a0)=12xyx,yV(xyv)2ρxy,
(5)
wherexyv = v(y) − v(x).

The proof of this result follows the same outline as the proof of the similar result in Ref. 20.

There are transforms of electrical networks, which are widely used in electrical engineering for calculation of effective impedances. The most known transforms are star-mesh transform, Y − Δ and Δ − Y transform, and parallel and series laws (Figs. 15). A mathematical justification of these transforms in the case of weighted graphs over R is known (see, e.g., Ref. 19). We justify these transforms in the case of arbitrary ordered field. Therefore, we can also apply our theorems to R in order to get classical results about these transforms. The main statement is Theorem 4 (star-mesh transform), since other transform happens to be its corollaries. Multigraphs are not allowed in this paper; therefore, we use a modification of the parallel law and refer to it as a parallel-series law.

Theorem 4

(star-mesh transform). Let Γ = (V, ρ, a0, B) be a finite network, |V| = n,B0 = B ∪ {a0}, andx1, …, xmV, 3 ≤ mn, are such that

  • x1B0and

  • yx1for allyV \{x2, …, xm}.

If one removes the vertexx1, edgesx1xi,i=2,m̄and changes the admittances of the edgesxixj,i,j=2,m̄,ijas follows:
ρxixj=ρxixj+ρx1xiρx1xjρ(x1),
(6)
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 the corresponding vertices.

Proof.
Let us consider the Dirichlet problem for the network Γ in a matrix form (4). Obviously, it is enough to compare solutions of the corresponding matrix equations. Let Av̂=b be a matrix equation for the original network [as in (4)]. Without loss of generality, we can assume that x1, …, xlB0, where l = m − |{x1, …, xm} ∩ B0|. Writing equations for x1, …, xl as the first ones and denoting k = |B|, we have
A=ρ(x1)ρx1x2ρx1xl00ρx1x2ρ(x2)ρx2xlρx2xm+1ρx2xnk1ρx1xlρx2xlρ(xl)ρxlxm+1ρxlxnk10ρx2xm+1ρxlxm+1ρ(xm+1)ρxm+1xnk10ρx2xnk1ρxlxnk1ρxm+1xnk1ρ(xnk1),
since yx1 for all yV \{x2, …, xm}, and
b=(ρa0x1,ρa0x2,,ρa0xl,ρa0xm+1,,ρa0xnk1)T.
We will show that the star-mesh transform is an application of the Gaussian elimination method to the first row. Indeed, applying the Gaussian elimination method for the first row of the augmented matrix Ā=[A|b], we obtain the matrix
formula
since ρ(x1) ≠ 0, where
ρ*(xi)=ρ(xi)ρx1xi2ρ(x1) and ρa0xi*=ρa0xi+ρx1xiρa0x1ρ(x1) for all i=2,l̄.
Note that for all i=2,l̄,
ρ(xi)=ρ(xi)ρx1xij=2jimρxixj+j=2jimρxixj=ρ(xi)ρx1xij=2jimρxixj+j=2jimρxixj+ρx1xiρx1xjρ(x1)=ρ(xi)ρx1xi+j=2jimρx1xiρx1xjρ(x1)=ρ(xi)ρx1xi+ρx1xiρ(x1)j=2jimρx1xj=ρ(xi)ρx1xi+ρx1xiρ(x1)j=2mρx1xjρx1xiρ(x1)ρx1xi=ρ(xi)ρx1xi+ρx1xiρ(x1)ρ(x1)ρx1xi2ρ(x1)=ρ(xi)ρx1xi2ρ(x1)=ρ*(xi)
and
ρa0xi*=ρa0xi+ρx1xiρa0x1ρ(x1)=ρa0xiifa0{x2,,xm}ρa0xiotherwise, since thenρa0x1=0.
Hence,
formula
Therefore, we can eliminate the variable v(x1) from the Dirichlet problem, changing admittances as in the statement of the theorem.□

Corollary 5.

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

Proof.

In the proof, we are using notations from the proof of Theorem 4.

The case {x1, …, xm} ∩ B0 = ∅ is trivial. The cases, when {x1, …, xm} ∩ B = ∅ or {x1, …, xm} ∩ {a0} = ∅ are obvious due to (5).

Otherwise, we can assume, without loss of generality, that
xmj=aj,j=0,|{x1,,xm}B0|̄,
(7)
in particular, xm = a0. Then, if we denote the new network by Γ′ we have, by (5),
Peff(Γ)=xa0(1v(x))ρxa0=(1v(x1))ρx1a0+i=2m1(1v(xi))ρxia0+x{x1,,xm}(1v(x))ρxa0=Peff(Γ)i=2m1(1v(xi))ρxia0+(1v(x1))ρx1a0+i=2m1(1v(xi))ρxia0=Peff(Γ)i=2m1(1v(xi))ρx1a0ρx1xiρ(x1)+(1v(x1))ρx1a0=Peff(Γ)ρx1a0i=2m1(1v(xi))ρx1xiρ(x1)+1i=2m1v(xi)ρx1xiρ(x1)ρx1a0ρ(x1)ρx1a0=Peff(Γ)ρx1a0i=2m1ρx1xiρ(x1)i=2m1v(xi)ρx1xiρ(x1)+1i=2m1v(xi)ρx1xiρ(x1)ρx1a0ρ(x1)ρx1a0=Peff(Γ)ρx1a0ρ(x1)ρ(x1)ρx1xmρ(x1)i=2m1v(xi)ρx1xiρ(x1)+1i=2m1v(xi)ρx1xiρ(x1)ρx1a0ρ(x1)ρx1a0=Peff(Γ),
since
ρ(x1)=i=2mρx1xi andv(x1)=i=2lv(xi)ρx1xiρ(x1)+ρx1a0ρ(x1)=i=2m1v(xi)ρx1xiρ(x1)+ρx1a0ρ(x1)
[see the first line of à and note that v(xj) = 0 for all j=j+1,m1̄ and a0 = xm].□

Corollary 6

(series law). Let Γ = (V, ρ, a0, B) be a finite network,B0 = B ∪ {a0}. Leta, b, cVare such that

  • bB0,

  • ac,ab,bc, and

  • bxfor allx ∉ {a, c}.

If one removes the vertexband edgesab, bcand adds the edgeacwith the admittance,
ρac=ρabρacρab+ρac,
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 the corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.

Remark 7.
The corresponding equation for impedances is then
zac=zab+zac,
which corresponds to the well-known physical series law.

Proof.

Apply Theorem 4 and Corollary 5 (x1 = b) for the case m = 3 and ρac = 0.□

Corollary 8

(parallel-series law). Let Γ = (V, ρ, a0, B) be a finite network,B0 = B ∪ {a0}.

Leta, b, cVare such that

  • bB0,

  • ab, bc, ac, and

  • bxfor allx ∉ {a, c}.

If one removes the vertexband edgesab, bcand adds the edgeacwith the admittance,
ρac=ρabρbcρab+ρbc+ρac,
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 9.
The corresponding equation for impedances is then
1zac=1zab+zbc+1zac,
which corresponds to an application of the physical series law and then an application of the physical parallel law.

Proof.

Apply Theorem 4 and Corollary 5 (x1 = b) for the case m = 3.□

Theorem 10

(Y − Δ transform). Let Γ = (V, ρ, a0, B) be a finite network,B0 = B ∪ {a0}. Leta, b, c, dVare such that

  • dB0,

  • da, db, dc, and

  • dxfor allx ∉ {a, b, c}.

If one removes the vertexdand edgesda, db, dcand sets
ρab=ρdaρdbρda+ρdb+ρdc+ρab,ρbc=ρdbρdcρda+ρdb+ρdc+ρbc,ρac=ρdaρdcρda+ρdb+ρdc+ρac,
(8)
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
zab=zdczdazdb+zdbzdc+zdazdc+1zab,zbc=zdazdazdb+zdbzdc+zdazdc+1zbc,zac=zdbzdazdb+zdbzdc+zdazdc+1zac.
(9)
From the physical point of view, if ρab, ρbc, ρac are all equal to zero, then it is an Y − Δ transform; otherwise, it is an Y − Δ transform and a parallel law.

Proof.

Theorem 4 and Corollary 5 (x1 = d) for the case m = 4.□

The Y − Δ transform is invertible. In general, it is not the case for a star-mesh transform.

Theorem 12
(Δ − Y transform). Let Γ′ = (V′, ρ′, a0, B) be a finite network, and leta, b, cVare such thatab, bc, andac. If one adds a vertexdand edgesda, db, dcsetting
ρda=ρacρbc+ρacρab+ρabρbcρbc,ρdb=ρacρbc+ρacρab+ρabρbcρac,ρdc=ρacρbc+ρacρab+ρabρbcρab
and removes the edgesab, bc, ac, not changing other admittances, then for the new network,
Γ=(V{d},ρ,a0,B),
the solution of the Dirichlet problem (2) for any vertex inVwill be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertex. Moreover, the effective impedance and the effective admittance do not change under this transform.

Remark 13.
The corresponding equalities for the impedances are
zda=zabzaczab+zbc+zac,zdb=zabzbczab+zbc+zac,zdc=zbczaczab+zbc+zac.

Proof.
To prove the theorem, it is enough to express ρda, ρdb, and ρdc from (8), assuming ρab = 0, ρbc = 0, and ρac = 0. Summing up the inverses of all three equations, one obtains
1ρab+1ρbc+1ρac=(ρad+ρbd+ρcd)2ρdaρdbρdc.
Since both sides are strictly positive, the last equation is equivalent to
ρabρbcρacρabρbc+ρbcρac+ρabρac=ρdaρdbρdc(ρad+ρbd+ρcd)2.
(10)
Multiplying both sides of the last equation by
1ρabρac=(ρad+ρbd+ρcd)2ρda2ρdbρdc,
which follows from (8), we get
ρbcρabρbc+ρbcρac+ρabρac=1ρda.
Then, the equation for ρda follows. To obtain the equations for ρdb and ρdc, one should multiply (10) by 1ρabρbc and 1ρacρbc, respectively.

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

Remark 14.

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

In this section, we introduce a notion of finite approximations for any infinite network over an ordered field and discuss the convergence of the sequence of effective admittances of finite approximations in the order topology. We assume that any infinite network has a potential zero at infinity. We remind that a subset MK is open in the order topology of an ordered field (K, ≻) if and only if for any k0M, there exists ɛ ≻ 0, ɛK such that an ɛ-ball

O(ko,ε)={kK|kk0|ε}

is a subset of M.

As an important example, we calculate an effective impedance of an infinite LC-network over the Levi–Cività field.

Let Γ = (V, ρ, a0, B) be an infinite network over an ordered field (K, ≻). Let us consider the sequence of finite graphs (Vn,ρVn×Vn), where Vn = {xV∣dist(a0, x) ≤ n}, nN.

We denote by

Vn={xVdist(a0,x)=n},

the boundary of the graph (Vn,ρVn×Vn). Note that Vn+1 = ∂Vn+1Vn.

Let us denote Bn = BVn. Then,

Γn=(Vn,ρVn×Vn,a0,BnVn),nN

is a sequence of finite networks exhausted the infinite network Γ.

This is an analog to the classical approach to infinite networks with resistors (see, e.g., Refs. 9, 14, and 22).

Let us consider the Dirichlet problem (2) on each Γn,

y:yx(v(n)(y)v(n)(x))ρxy=0 on Vn\(VnBn{a0}),v(n)(x)=0 on VnBn,v(n)(a0)=1,
(11)

Theorem 15.
Peff(Γn+1)Peff(Γn).
(12)

Proof.
By Dirichlet/Thomson’s principle,20 we have
Peff(Γn+1)12x,yVn+1(xyf)2ρxy
(13)
for any f: Vn+1K such that f(a0)=1,fVn+1Bn+10.
Since (Vn+1\∂Vn+1) = Vn and Bn+1Vn = Bn, the inequality (13) is true for
f(x)=v(n)(x) if  xVn,0 if  xVn+1,
where v(n) is the solution of (11) for Γn. Then,
12x,yVn+1(xyf)2ρxy=12x,yVn(xyf)2ρxy+12x,yVn+1(xyf)2ρxy=Peff(Γn)+0.
The last equality, together with (13), gives us (12).□

Remark 16.

Even in a Cauchy complete, non-Archimedean ordered field inequalities (12) for all nN do not imply that the sequence {Peff(Γn)}n=1 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 the given infinite network Γ, the limit of effective admittances (impedances) of exhausted finite networks exists in K, we call this limit the effective admittance (impedance) of the network Γ with zero potential at infinity and denote it by Peff(Γ) [Zeff(Γ)].

Ladder networks are objects of interest of mathematical physics and electrical engineering for a very long time, since Richard Feynman (in 1966) has described an infinite LC-network in his Lectures on Physics [see Ref. 12 (p. 22-13) and Refs. 10, 18, 23, 24, and 27]. In all these sources, the effective impedance was calculated over the field of real numbers. We would like to consider admittances as elements of the Levi–Cività field R. In this section, we show that the sequence of effective impedances of finite LC-networks converges to the limit in the order topology of R, but the sequence of effective impedances of finite CL-networks does not converge in the same topology. These examples show a difference with the case of real positive weights, since there any network is either transient (effective resistance is finite) or recurrent (effective resistance is infinite) (see, e.g., Refs. 9 and 26). Let us investigate the behavior of the sequence {Peff(Γnαβ)}n=1 of effective admittances of finite networks exhausted the ladder network in Fig. 6 (α, βK, α, β ≻ 0). More precisely, let us consider a network Γαβ = (V, ρ, a0, B), where

  • V = {a0, a1, a2, …, x1, x2, … },

  • ρa0x1=α, ρxkxk+1=α,ρakxk=β,kN, and ρxy = 0, otherwise, and

  • B = {a1, a2, … }.

FIG. 6.

αβ-network.

This network is similar to Feynman’s ladder network and CL-network (see Refs. 12 and 27) but has zero potential at infinity. Therefore, for any ordered field K, Theorem 15 is true for this network. We will refer to this network as an αβ-network. We show (Theorem 20 and example 23) that whether {Peff(Γn)}n=1 converges in a Cauchy completion of K depends on α and β.

1. Finite ladder network over ordered field

Let Γn be a sequence of finite networks exhausted an αβ-network (see Fig. 7).

FIG. 7.

Finite ladder network.

FIG. 7.

Finite ladder network.

Close modal

The Dirichlet problem (2) for this network is as follows:

αv(a0)+βv(a1)+αv(x2)(2α+β)v(x1)=0,αv(xk1)+βv(ak)+αv(xk+1)(2α+β)v(xk)=0 for k=2,n1̄,v(xn)=0,v(ak)=0 for k=1,n1̄,v(a0)=1.
(14)

Using the second line in (14), we obtain the following recurrence relation for v(xk),k=2,n1̄:

v(xk+1)2+βαv(xk)+v(xk1)=0,
(15)

since v(ak) = 0 for k=2,n1̄. The characteristic polynomial of (15) is

ψ22+βαψ+1=0.
(16)

Its roots are

ψ1,2=1+β2α±ξ,

where

ξ=βα+β2α2.
(17)

Note that ξ should not necessarily belong to K. However, it is known that any ordered field possess a real-closed extension K̄. Then, in K̄ exists exactly one positive square root of βα+β2α2 (see, e.g., Ref. 5). Therefore, we fix the extension K̄, denote the positive square root by ξ, and make all the further calculations in K̄.

The solution of the recurrence equation (15) is

v(xk)=c1ψ1k+c2ψ2k,
(18)

where c1,c2K̄ are arbitrary constants.

We use first and third equations in (14) as boundary conditions for this recurrence equation. Substituting (18) in the boundary conditions, we obtain the following equations for the constants:

1+c1ψ12+c2ψ222+βα(c1ψ1+c2ψ2)=0,c1ψ1n+c2ψ2n=0,

which, by (16), is equivalent to

c1+c2=1,c1ψ1n+c2ψ2n=0.

Therefore,

c1=11ψ12n,c2=11ψ22n=ψ12n1ψ12n,

since ψ1ψ2 = 1 by (16).

Now, we can calculate the effective admittance of Γn,

PeffΓn=α1v(x1)=α1c1ψ1c2ψ2=α1+β2α+ξ2n1+1β2α+ξ1+β2α+ξ2n1.
(19)

Since PeffΓn is an element of K as a rational function of the solution of Dirichlet problem (2) over K, it can also be written without ξ,

PeffΓn=α1c1ψ1c2ψ2=α1ψ11ψ12nψ21ψ22n=α1ψ11ψ22n+ψ21ψ12n1ψ12n1ψ22n=α1ψ1+ψ2ψ22n1+ψ12n12ψ12n+ψ22n=α12+βα2k=0n12n12k1+β2α2n2k1βα+β2α2k22k=0n2n2k1+β2α2n2kβα+β2α2k,

where in the last line, we have used binomial expansion.

2. Infinite ladder networks over the Levi–Cività field

We consider two examples of the αβ-network over the Levi–Cività field R. First, let us describe the Levi–Cività field R. We take the definition of R and theorems about its properties from Refs. 3, 4, and 21.

Definition 18.

A subset M of the rational numbers Q is called left-finite, if for every rQ, there are only finitely elements of M that are smaller than r.

Then, the Levi–Cività field is the set of all real-valued functions on Q with left-finite support with the following operations:
  • addition is defined component-wise,

(α+β)(q)=α(q)+β(q),
  • multiplication is defined as follows:

(αβ)(q)=qα,qβQ,qα+qβ=qα(qα)β(qβ).

It is proved in Ref. 3 that R is an ordered field with a set of positive elements,

R+={αR|α(min{qQα(q)0})>0}.
(20)

We denote by τ the following element in R:

τ(q)=1 if q=10 otherwise,
(21)

which plays a role of infinitesimal in the Levi–Cività field. Therefore, the Levi–Cività field is a non-Archimedean ordered field.

In Ref. 3, we can write any αR as

α=i=1α(qi)τqi,
(22)

since αn=i=1nα(qi)τqi converges strongly to the limit α in the order topology.

The set of all polynomials over real numbers R[τ]={anτn+a1τ+a0aiR,nN} is a subring of Levi–Cività field R due to (22). Therefore, since R is a field, the field of rational functions over real numbers

R(τ)=i=knaiτii=lmbiτi=akτk+ak+1τk+1++anτnblτl+bl+1τl+1++bmτmai,biR,n,m,k,lN0
(23)

is isomorphic to a subfield of R.

Example 19.
Let us find the element in R, which corresponds to the rational function 134τ+τ2, i.e., we should find the sequences {qi}Q and {α(qi)}R such that
34τ+τ2qiα(qi)τqi=1.
Comparing the coefficients at powers of τ at right hand side and left hand side, starting from the lowest power, one obtains
q1=0,α(q1)=13,
q2=1,α(q2)=49,
and recurrence relations,
qi=qi1+1 and 3α(qi)4α(qi1)+α(qi2)=0
for i > 2. Therefore, solving the recurrence relation for α(qi), we obtain
α(qi)=123i+12
and
134τ+τ2=iN123i+12τi1.

Note that the corresponding order in the field R(τ) is as follows:

akτk+ak+1τk+1++anτnblτl+bl+1τl+1++bmτm0 if akbl>0.
(24)

Therefore, we can consider the Levi–Cività field R as an ordered extension of the ordered field R(τ) with the positiveness defined as (24). The isomorphism between R and the ordered extension of the ordered field R(λ) with the positiveness defined by (3) is given by a substitution,

τ=1λ.
(25)

Consequently, we can consider electrical networks over fields of rational numbers,20 as networks over the Levi–Cività field, and investigate the behavior of the sequence of effective admittances of finite electrical networks. In Ref. 3, the Levi–Cività field is Cauchy complete in the order topology and real-closed.

The most known ladder network is Feynman’s ladder (see Ref. 12). Its properties from the point of view of electrical engineering are described in Refs. 10, 18, 23, 24, and 27. The other studied ladder network is the CL-network (see, e.g., Ref. 27). We consider both these networks over the Levi–Cività field, which is a completely new approach.

Let us consider Feynman’s infinite ladder LC-network, assuming that it has zero potential at infinity (see Fig. 8). It is an αβ-network with α = L−1τ and β = −1, where L, C > 0, α,βR (substitutions λ = , τ = 1/λ).

FIG. 8.

Feynman’s ladder with zero at infinity.

FIG. 8.

Feynman’s ladder with zero at infinity.

Close modal

Theorem 20.
For Feynmans ladderLC-network (α = L−1τ,β = −1, whereL, C > 0) with zero potential at infinity,
PeffΓnββ2α+ξ as n
(26)
in the order topology of the Levi–Cività fieldR, where Γnis the sequence of the exhausted finite networks,ξis defined by (17).

Remark 21.
For Feynman’s ladder LC-network,
ββ2α+ξ=C2ττLCLτ2+CL2τ22,
and the motivation for this quantity was Feynman’s impedance for the infinite ladder LC-network (see Ref. 12).

Proof.
First, we write ξ as an element of the Levi–Cività field, i.e., as power series (22),
ξ=CLτ2+CL2τ22=CL2τ24τ2CL+1=CL2τ2k=012k4τ2CLk=CL2τ21+CL2τ2124τ2CL18CL2τ24τ2CL2+oτ2=CL2τ2+1τ2+oτ2.
Note that here and further oτm, where mZ means k=m+1akτk,akR.
Let us calculate the difference PeffΓnββ2α+ξ,
PeffΓnββ2α+ξ=αβ2α+ξ1+β2α+ξ2n1+11+β2α+ξ2n1ββ2α+ξ=αβ2α+ξ21+β2α+ξ2n1+1β1+β2α+ξ2n11+β2α+ξ2n1β2α+ξ.
The nominator A of the last expression is
A=α1+β2α+ξ2n1+1β2α+ξ2β1+β2α+ξ2n1=αD+1β2α+ξ2β1+β2α+ξD1,
where D=1+β2α+ξ2n1.
Since
β2α+ξ2=β24α2+βαξ+ξ2=β22α2+βα+βαξ=βαβ2α+1+ξ
and
ξ2=βα+β24α2,
we have
A=αD+1βαβ2α+1+ξβ1+β2α+ξD1=D0+2β+β22α+βξ=2αβα+β24α2+β2αξ=2αξ2+β2αξ=2αξξ+β2α.
Therefore,
PeffΓnββ2α+ξ=2αξ1+β2α+ξ2n1.
(27)
The right-hand side of the last expression is, obviously, positive in (R,); therefore,
PeffΓnββ2α+ξ=2αξ1+β2α+ξ2n1=2τξL1+LC2τ2+ξ2n1=LCτ1+2τ2τ3+oτ3LLCτ2+21LCτ2+oτ22n1=LCτ1+2τ2CLτ3+oτ3LLCτ22n+oτ4n=Cτ1+oτ11LC2nτ4n+oτ4n=CLC2nτ4n1+oτ4n10,
when n. Indeed, for any ɛ ≻ 0, ε=i=1, we have
C(LC)2nτ4n1+o(τ4n1)ε
for any n>q1+14+1.□

Remark 22.

From the proof, one can see that (26) is true for the αβ-network whenever for any γR exists N0N such that n > N0 implies βαnγ [see the denominator in (27)].

Example 23.

For the CL-network (α = −1, β = L−1τ, L, C > 0, see Fig. 9), effective admittances of the exhausted finite networks do not converge in the Levi–Cività field R.

FIG. 9.

CL-network.

Proof.
In this case,
ξ=τCL+τ2CL3+o(τ3).
Let us prove that {P(Γn)}n=1 is not a Cauchy sequence in R. Indeed,
P(Γn+1)P(Γn)=αβ2α+ξ1+β2α+ξ2n+1+11+β2α+ξ2n+21αβ2α+ξ1+β2α+ξ2n1+11+β2α+ξ2n1=αβ2α+ξ1+β2α+ξ2n+1+11+β2α+ξ2n+211+β2α+ξ2n1+11+β2α+ξ2n1.
Since
ψ1=1+β2α+ξ=1+τCL+oτ1,
we can rewrite
P(Γn+1)P(Γn)=αβ2α+ξψ12n+1+1ψ12n+21ψ12n1+1ψ12n1=αβ2α+ξψ12n+1+1ψ12n1ψ12n1+1ψ12n+21ψ12n+21ψ12n1=Cτ1τCL+oτ1ψ12n+1+1ψ12n1ψ12n1+1ψ12n+21ψ12n+21ψ12n1.
Substituting
ψ12n+1+1ψ12n1ψ12n1+1ψ12n+21=2+τCL(2n+1)+oτ1τCL(2n)+oτ12+τCL(2n1)+oτ1τCL(2n+2)+oτ1=4τCL+oτ1
in the nominator and
ψ12n+21ψ12n1=τCL(2n+2)+oτ1τCL(2n)+oτ1=τ2CL(4n2+4n)+oτ2
in the denominator, we obtain
P(Γn+1)P(Γn)=CCL+0(τ0)4τCL+oτ1CL4n2+4nτ2+oτ2=4C4n2+nτ1+oτ1τ for any nN and for any L,C>0.
Therefore, {P(Γn)}n=1 is not a Cauchy sequence in R.□

Therefore, the following question:

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

remains open. Note that Remark 22 gives some sufficient condition for the αβ-network.

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

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

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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