aa r X i v : . [ m a t h . C O ] A ug ON THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITENETWORKS
ANNA MURANOVAA
BSTRACT . In this paper we deal with the notion of the effective impedanceof AC networks consisting of resistances, coils and capacitors. Mathematicallysuch a network is a locally finite graph whose edges are endowed with complex-valued weights depending on a complex parameter λ (by the physical meaning, λ = i ω , where ω is the frequency of the AC). For finite networks, we provesome estimates of the effective impedance. Using these estimates, we show that,for infinite networks, the sequence of impedances of finite graph approximationsconverges in certain domains in C to a holomorphic function of λ , which allowsus to define the effective impedance of the infinite network. C ONTENTS
1. Introduction 12. Effective impedance of finite networks 23. Estimates of the effective admittance of finite networks 73.1. An upper bound of the admittance using Re λ λ λ NTRODUCTION
Mathematically an electrical network can be represented by a connected graphwhose edges are endowed by weights that are determined by the physical propertiesof the connection between two nodes. Here we deal with the networks consistingof resistors, coils and capacitors. Assuming that the network is connected to asource of AC with the frequency ω , each edge xy between the nodes x , y receivesthe complex-valued weight (impedance) z ( λ ) xy = L xy λ + R xy + C xy λ , This research was supported by IRTG 2235 Bielefeld-Seoul “Searching for the regular in theirregular: Analysis of singular and random systems”.
Keywords : weighted graphs, Laplace operator, Kirchhoff’s equations, electrical network, effectiveimpedance, ladder network.
Mathematics Subject Classification 2010: where R xy is the resistance of this edge, L xy is the inductance, C xy is the capacitance,and λ = i ω . The goal is to determine the effective impedance of the entire network.If the network is finite then the problem amounts to a linear system of Kirch-hoff’s equations. In absence of coils and capacitors this system has always non-zero determinant, which implies that the effective impedance is well-defined (and,of course, is independent of λ ). Note that a network that consists only of resis-tances determines naturally a reversible Markov chain (see e.g. [2], [7], [9], [12]).For infinite (but locally finite) graphs, again in absence of coils and capacitors,one constructs first a sequence { Z n } of partial impedances that are the effectiveimpedances of an exhaustive sequence of finite graphs, and then defines the effec-tive impedance Z of the entire network as the limit lim Z n . This limit always existsdue to the monotonicity of the sequence { Z n } (cf. [2], [6], [7], [9], [12], [16])Although the notion of the effective impedance is widely used in physical andmathematical literature, the problem of justification of this notion in the presenceof coils and capacitors is not satisfactorily solved. In the case of finite graphs,the determinant of the system may vanish for some values of λ and the systemmay have infinitely many solutions or no solutions. Hence, the definition of theeffective impedance in this case requires substantial work that was done in theprevious paper of the author [10]. In Section 2 we present an extended version ofthese results for finite graphs.The case of infinite graphs is even more complicated because the sequence { Z n } is complex valued, depends on λ , and no monotonicity argument is available.One of the first examples of computation of effective impedance for an infinitenetwork was done by Richard Feynman [5]. As it was observed later (cf. [3],[8], [13], [14], [17]), the sequence { Z n } of partial effective impedances in thisnetwork (named Feynman’s ladder ) converges not for all values of the frequency ω , which raises the question about the validity of Feynman’s computation as wellas the problem about a careful mathematical definition of the effective impedancefor infinite networks.In this paper we make the first attempt to solve this problem. We work withadmittances ρ ( λ ) xy = z ( λ ) xy regarded as functions of λ ∈ C (similarly to [1]), and in-vestigate the problem of convergence of the sequence { P n ( λ ) } of the partial ef-fective admittances. Our main result, Theorem 22, says that P ( λ ) : = lim P n ( λ ) exists and is a holomorphic function of λ in the domain { Re λ > } as well as insome other regions. In the case of a resistance free network, Corollary 24 says that P ( λ ) is holomorphic in C \ (cid:2) − i √ S , i √ S (cid:3) where S = sup xy C xy L xy . These results about infinite networks are proved in Section 4. The proof is basedon the estimates of admittances for finite networks that are presented in Section 3.In Section 5 we give some examples, including a modified Feynman’s ladder, thatillustrate the domain of convergence of the sequence P n ( λ ) .2. E FFECTIVE IMPEDANCE OF FINITE NETWORKS
Let ( V , E ) be a finite connected graph, where V is the set of vertices, | V | ≥ E is the set of (unoriented) edges. N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 3
Assume that each edge xy ∈ E is equipped with a resistance R xy , inductance L xy ,and capacitance C xy , where R xy , L xy ∈ [ , + ∞ ) and C xy ∈ ( , + ∞ ] , which correspondto the physical resistor, inductor (coil), and capacitor (see e.g. [4]). It will beconvenient to use the inverse capacity D xy = C xy ∈ [ , + ∞ ) . We always assume that for any edge xy ∈ ER xy + L xy + D xy > . The impedance of the edge xy is defined as the following function of a complexparameter λ : z ( λ ) xy = R xy + L xy λ + D xy λ . Although the impedance has a physical meaning only for λ = i ω , where ω is thefrequency of the alternating current (that is, ω is a positive real number), it will beconvenient to allow λ to take arbitrary values in C \ { } (cf. [1]).In fact, it will be more convenient to work with the admittance ρ ( λ ) xy : ρ ( λ ) xy : = z ( λ ) xy = λ L xy λ + R xy λ + D xy . (1)Define the physical Laplacian ∆ ρ as an operator on functions f : V → C asfollows ∆ ρ f ( x ) = ∑ y ∈ V : y ∼ x ( f ( y ) − f ( x )) ρ ( λ ) xy , (2)where x ∼ y means that xy ∈ E . For convenience let us extend ρ ( λ ) xy to all pairs x , y ∈ V by setting ρ ( λ ) xy = x y . Then the summation in (2) can be extended toall y ∈ V . Let us fix a vertex a ∈ V and a non-empty subset B ∈ V such that a B . Set B = B ∪ { a } . The physical meaning of a and B is as follows: a is the source of ACwith the unit voltage, while the set B represents the ground with zero voltage. Werefer to the structure Γ = ( V , ρ , a , B ) as a finite (electrical) network .By the complex Ohm’s and Kirchhoff’s laws, the complex voltage v : V → C satisfies the following conditions: ∆ ρ v ( x ) = x ∈ V \ B , v ( x ) = x ∈ B , v ( a ) = . (3)We consider (3) as a discrete boundary value Dirichlet problem .Denote by Λ the set of all those values of λ for which ρ ( λ ) xy ∈ C \ { } for alledges xy . The complement C \ Λ consists of λ = L xy λ + R xy λ + D xy = . In particular, C \ Λ is a finite set. Clearly, for every λ ∈ C \ Λ we have Re λ ≤ Λ ⊃ { Re λ > } . Observe also that Re λ > ⇒ Re z ( λ ) xy > ⇒ Re ρ ( λ ) xy > ANNA MURANOVA and, for λ ∈ Λ , Re λ ≥ ⇒ Re z ( λ ) xy ≥ ⇒ Re ρ ( λ ) xy ≥ λ ∈ Λ . If v ( x ) is a solution of (3) then the total current through a is equal to ∑ x ∈ V ( − v ( x )) ρ ( λ ) xa , which motivates the following definition (cf. [10]). Definition 1.
For any λ ∈ Λ , the effective admittance of the network Γ is definedby P ( λ ) = ∑ x ∈ V ( − v ( x )) ρ ( λ ) xa , (6)where v is a solution of the Dirichlet problem (3). The effective impedance of Γ isdefined by Z ( λ ) = P ( λ ) = ∑ x ∈ V ( − v ( x )) ρ ( λ ) xa . If the Dirichlet problem (3) has no solution for some λ , then we set P ( λ ) = ∞ and Z ( λ ) = Z ( λ ) and P ( λ ) take values in C = C ∪ { ∞ } . We will prove belowthat in the case when (3) has multiple solution, the values of Z ( λ ) and P ( λ ) areindependent of the choice of the solution v . In the case when B is a singleton, thiswas proved in [10].Observe immediately the following symmetry properties that will be used lateron. Lemma 2. ( a ) If λ ∈ Λ then also λ ∈ Λ and P ( λ ) = P ( λ ) . (7) ( b ) Assume in addition that R xy = for all xy ∈ E. Then λ ∈ Λ implies − λ ∈ Λ and P ( − λ ) = − P ( λ ) . Proof. ( a ) If λ is a root of the equation L xy λ + R xy λ + D xy = λ is also aroot, whence the first claim follows. If v is a solution of (3) for some λ then clearly v is a solution of (3) with the parameter λ instead of λ . Substituting into (6) andusing ρ ( λ ) = ρ ( λ ) , we obtain (7). ( b ) The proof is similar to ( a ) observing that if λ is a root of L xy λ + D xy = − λ is also a root. (cid:3) The following Green’s formula was proved in [10] (for simplicity of notation,we skip the superscript in ρ ( λ ) when λ is fixed). Lemma 3 (Green’s formula) . For any λ ∈ Λ and for any two functions f , g : V → C we have ∑ x , y ∈ V ( ∇ xy f )( ∇ xy g ) ρ xy = − ∑ x ∈ V ∆ ρ f ( x ) g ( x ) = − ∑ x ∈ V ∆ ρ g ( x ) f ( x ) , (8) where ∇ xy f = f ( y ) − f ( x ) . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 5
Lemma 4.
For any solution v of ( ) we have ∑ x ∈ V ( − v ( x )) ρ xa = − ∆ ρ v ( a ) = ∑ b ∈ B ∆ ρ v ( b ) = ∑ x , y ∈ V ( ∇ xy v )( ∇ xy u ) ρ xy , (9) where u : V → C is any function such thatu ( a ) = and u (cid:12)(cid:12) B ≡ . (10) Proof.
Using v ( a ) =
1, we have ∆ ρ v ( a ) = ∑ x ∈ V ( v ( x ) − v ( a )) ρ xa = ∑ x ∈ V ( v ( x ) − ) ρ xa , which proves the first identity in (9). Since by (8) with f = v and u ≡ ∑ x ∈ V ∆ ρ v ( x ) = ∆ ρ v ( x ) = x ∈ V \ B , we obtain ∑ b ∈ B ∆ ρ v ( b ) + ∆ ρ v ( a ) = ∑ x , y ∈ V ( ∇ xy v )( ∇ xy u ) ρ xy = − ∑ x ∈ V ∆ ρ v ( x ) u ( x ) = − ∆ ρ v ( a ) , because ∆ ρ v ( x ) = x ∈ V \ B , while u (cid:12)(cid:12) B ≡ u ( a ) = (cid:3) Comparing (6) with (9), we obtain the identity P ( λ ) = ∑ x , y ∈ V ( ∇ xy v )( ∇ xy u ) ρ xy = ∑ xy ∈ E ( ∇ xy v )( ∇ xy u ) ρ xy , (11)where v is a solution of the Dirichlet problem (3) for Γ and u : V → C is anyfunction satisfying (10). Choosing here u = v we obtain also the identity P = ∑ x , y ∈ V | ∇ xy v | ρ xy (12)(conservation of the complex power). Theorem 5.
For any λ ∈ Λ , the values of Z ( λ ) and P ( λ ) do not depend on thechoice of a solution v of the Dirichlet problem ( ).Proof. The proof uses the same argument as in [10]. Let v and v be two solutionsof (3) for the same λ ∈ Λ . By (9) with v = v and u = v we have ∑ x ∈ V ( − v ( x )) ρ xa = ∑ x , y ∈ V ( ∇ xy v ) ( ∇ xy v ) ρ xy . Similarly, we have ∑ x ∈ V ( − v ( x )) ρ xa = ∑ x , y ∈ V ( ∇ xy v ) ( ∇ xy v ) ρ xy , whence the identity ∑ x ∈ V ( − v ( x )) ρ xa = ∑ x ∈ V ( − v ( x )) ρ xa follows. Hence, v and v determine the same admittance and impedance. (cid:3) ANNA MURANOVA
Theorem 6.
The Dirichlet problem ( ) has a unique solution v = v ( λ ) for all λ ∈ Λ where Λ is a subset of Λ such that Λ \ Λ is finite. Besides, Λ contains thedomains Λ ∩ { Re ρ ( λ ) xy > ∀ xy ∈ E } , (13) Λ ∩ { Im ρ ( λ ) xy > ∀ xy ∈ E } and Λ ∩ { Im ρ ( λ ) xy < ∀ xy ∈ E } . (14) Consequently, P ( λ ) is a rational C -valued function in Λ and, hence, in any ofthe domains ( ) and ( ).Proof. Let us denote the vertices V \ B by x , . . . , x n and rewrite the Dirichlet prob-lem (3) as a linear system n × n : n ∑ j = A i j X j = P i for any i = , ..., n , (15)where X j = v ( x j ) , P i = ρ x i a , A ii = ∑ y : y ∼ x i ρ x i y and A i j = − ρ x i x j for i = j . Set also D = det ( A i j ) and let D j be the determinant of the matrix obtained by replacing the column j inthe matrix { A i j } by the column { P i } . Then, by Cramer’s rule, X j = D j D provided D =
0. Of course, all these quantities are functions of λ . Since all thecoefficients A i j and P i are rational functions of λ , also D = D ( λ ) and D j = D j ( λ ) are rational functions of λ . For all λ ∈ Λ but a finite number, all functions D j ( λ ) and D ( λ ) take values in C . The existence and uniqueness of a solution of (15) isequivalent to D ( λ ) =
0. Hence, define Λ as the subset of Λ where all functions D j ( λ ) and D ( λ ) take values in C and, besides, D ( λ ) = . Since D ( λ ) is a rationalfunction on λ , it may have only finitely many zeros or vanish identically.Hence, it suffices to exclude the latter case, that is, to show that Λ = /0. For that,let us prove that Λ contains the domain (13) that in turn, by (4) and (5), contains { Re λ > } and, hence, is non-empty. In order to show that D ( λ ) = λ from (13), it suffices to verify that the homogeneous Dirichlet problem ( ∆ ρ u ( x ) = V \ B , u ( x ) = B (16)has a unique solution u ≡
0. Indeed, by Green’s formula we have ∑ xy ∈ E | ∇ xy u | ρ xy = ∑ x , y ∈ V | ∇ xy u | ρ xy = − ∑ x , y ∈ V ∆ ρ u ( x ) u ( x )= − ∑ x ∈ V \ B ∆ ρ u ( x ) u ( x ) − ∑ x ∈ B ∆ ρ u ( x ) u ( x ) = , since u is a solution of (16). Since Re ρ xy >
0, we conclude that | ∇ xy u | = u = const. Since u (cid:12)(cid:12) B ≡
0, we conclude that u ≡ Λ . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 7
Finally, by the above argument, v ( λ ) ( x ) is a rational function of λ , so that thelast claim follows from (6). (cid:3) Remark 7.
Since { Re λ > } is contained in Λ , we see that P ( λ ) is a holomor-phic function in { Re λ > } . If R xy > xy ∈ E , then also Λ ∩ { Re λ ≥ } isa subset of (13). Remark 8.
The uniqueness of the solution of the Dirichlet problem for the domain(13) follows also from [15, Lemma 4.4] and (4).
Example 9.
Let us consider the finite network as at Fig. 1, where all inductances,capacitances and resistance are equal to 1, with a = { } , B = { } . Then Λ = C \ { } . 0 λ λ λ λλ λ F IGURE
1. Example of a finite networkThe effective admittance of this network is calculated in [10, Example 28]. Wehave P ( λ ) = ∞ , λ = ± i , − , λ = − , λ + λ + λ + , otherwise , and Λ = C \ {− , , ± i } .Our next goal is to define the effective admittance of infinite networks. We willdo it in Section 4, but before that we prove some estimates for P ( λ ) on finitenetworks.3. E STIMATES OF THE EFFECTIVE ADMITTANCE OF FINITE NETWORKS
We use the same setup and notation as in Section 2. We skip λ in notations ρ ( λ ) xy and P ( λ ) when the value of λ is fixed.3.1. An upper bound of the admittance using Re λ .Theorem 10. Let λ ∈ Λ be fixed and assume that, for some ε > , inf xy ∈ E Re ρ xy | ρ xy | ≥ ε . (17) Then | P | ≤ ε ∑ x ∼ a | ρ xa | . (18) ANNA MURANOVA
The same result is true if one assumes instead of ( ) that inf xy ∈ E Im ρ xy | ρ xy | ≥ ε or inf xy ∈ E − Im ρ xy | ρ xy | ≥ ε . Proof.
Under the hypothesis (17) the Dirichlet problem (3) has by Theorem 6 aunique solution v . We have by (12) | P | ≥ Re P = ∑ xy ∈ E | ∇ xy v | Re ρ xy ≥ ε ∑ xy ∈ E | ∇ xy v | | ρ xy | . (19)Applying (11) with the function u = { a } and using the inequality 2 | ab | ≤ ε | a | + ε | b | , we obtain | P | ≤ ∑ xy ∈ E | ∇ xy v | | ∇ xy u | | ρ xy | ≤ ε ∑ xy ∈ E | ∇ xy v | | ρ xy | + ε ∑ xy ∈ E | ∇ xy u | | ρ xy | . Setting U : = ∑ xy ∈ E | ∇ xy u | | ρ xy | = ∑ x ∼ a | ρ xa | , we obtain | P | ≤ ε ∑ xy ∈ E | ∇ xy v | | ρ xy | + ε U . Combing this with (19) yields ε ∑ xy ∈ E | ∇ xy v | | ρ xy | ≤ ε U whence by (12) | P | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ xy ∈ E | ∇ xy v | ρ xy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∑ xy ∈ E | ∇ xy v | | ρ xy | ≤ ε U . The conditions with Im ρ xy are handled in the same way. (cid:3) In order to be able to verify (17), we need the following lemma.
Lemma 11.
Let L, R, D be non-negative real numbers and λ ∈ C \ { } . Ifz : = R + L λ + D λ = then Re z | z | ≥ Re λ | λ | and, for ρ = z , Re ρ | ρ | ≥ Re λ | λ | . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 9
Proof.
We haveRe z = R + L Re λ + D Re λ | λ | ≥ (cid:18) R + L | λ | + D | λ | (cid:19) Re λ | λ | and | z | ≤ R + L | λ | + D | λ | , whence Re z | z | ≥ Re λ | λ | . Finally, we have ρ = z = Re z − i Im z | z | and, hence, Re ρ | ρ | = Re z | z | | z | = Re z | z | ≥ Re λ | λ | . (cid:3) Corollary 12. If Re λ > then | P ( λ ) | ≤ | λ | ( Re λ ) ∑ x ∼ a (cid:12)(cid:12)(cid:12) ρ ( λ ) xa (cid:12)(cid:12)(cid:12) . (20) Proof.
Indeed, we have for all xy ∈ E Re ρ ( λ ) xy | ρ ( λ ) xy | ≥ Re λ | λ | = : ε . Substituting into (18) we obtain (20). (cid:3)
Corollary 13. If Re λ > then | P ( λ ) | ≤ C | λ | (cid:0) + | λ | (cid:1) ( Re λ ) , (21) where C = ∑ x ∼ a R xa + L xa + D xa . Proof.
We have for z = R + L λ + D λ | z | ≥ Re z ≥ R + L Re λ + D Re λ | λ | ≥ ( R + L + D ) min (cid:18) , Re λ , Re λ | λ | (cid:19) ≥ ( R + L + D ) min (cid:18) Re λ , Re λ | λ | (cid:19) =( R + L + D ) Re λ min (cid:0) , | λ | − (cid:1) ≥ ( R + L + D ) Re λ + | λ | , since ( Re λ ) | λ | ≤ λ ≤ Re λ | λ | ≤
1. It follows that, for ρ = z , | ρ | ≤ R + L + D + | λ | Re λ . Hence, ∑ x ∼ a | ρ xa ( λ ) | ≤ + | λ | Re λ ∑ x ∼ a R xa + L xa + D xa . Substituting into (20) we obtain (21). (cid:3)
Remark 14.
In the case L > { Re λ > } the estimate | z | ≥ L Re λ which implies | ρ | ≤ L Re λ . Hence, if L xa > x ∼ a then ∑ x ∼ a (cid:12)(cid:12)(cid:12) ρ ( λ ) xa (cid:12)(cid:12)(cid:12) ≤ C ′ λ , where C ′ = ∑ x ∼ a L xa . Therefore, by (20) in this case, in the domain { Re λ > } we have | P ( λ ) | ≤ C ′ | λ | ( Re λ ) . An upper bound of the admittance using large Im λ .Lemma 15. Let R, L, D be non-negative numbers. Let L > and λ ∈ C be suchthat Im λ > , | λ | > DL . Then z : = R + L λ + D λ = and for ρ = z we have − Im ρ | ρ | ≥ − DL | λ | | λ | + DL | λ | + RL Im λ (22) and | ρ | ≤ (cid:16) L − D | λ | (cid:17) Im λ . Proof.
We have Im z = L Im λ − D Im λ | λ | = (cid:18) L − D | λ | (cid:19) Im λ > . In particular, z =
0. We have also | z | ≤ R + L | λ | + D | λ | . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 11
It follows that − Im ρ | ρ | = Im z | z | ≥ (cid:16) L − D | λ | (cid:17) Im λ R + L | λ | + D | λ | ≥ − DL | λ | | λ | + DL | λ | + RL Im λ which proves (22). Finally, we have | ρ | = | z | ≤ z = (cid:16) L − D | λ | (cid:17) Im λ . (cid:3) Theorem 16.
Assume that L xy > for all xy ∈ E. SetS D = sup xy ∈ E D xy L xy , S R = sup xy ∈ E R xy L xy , and C ′ = ∑ x ∼ a L xa . Then, in the domain, Ω = { λ ∈ C | Im λ = and | λ | > S D } (23) the function P ( λ ) is holomorphic and | P ( λ ) | ≤ C ′ ( | λ | + S R ) | λ | ( | λ | − S D ) | Im λ | . √ S D i √ S D Re λ Im λ Ω F IGURE
2. The domain Ω = { λ ∈ C | Im λ = | λ | > S D } . Proof.
By the symmetry λ → λ , it suffices to prove the both claims in the domain Ω + = { λ ∈ C | Im λ > | λ | > S D } . For λ ∈ Ω + we have by Lemma 15 that z xy =
0, whence λ ∈ Λ . By Lemma 15 wehave for all xy ∈ E and λ ∈ Ω + − Im ρ xy | ρ xy | ≥ − D xy L xy | λ | | λ | + D xy L xy | λ | + R xy L xy Im λ ≥ − S D | λ | | λ | + S D | λ | + S R Im λ ≥ − S D | λ | | λ | + S R Im λ > , (24)since S D | λ | < | λ | . By Theorem 6 we conclude that P ( λ ) is a holomorphic functionin Ω + .Using (24), we obtain by Theorem 10 that for λ ∈ Ω + | P ( λ ) | ≤ | λ | + S R (cid:16) − S D | λ | (cid:17) Im λ ∑ x ∼ a (cid:12)(cid:12)(cid:12) ρ ( λ ) xa (cid:12)(cid:12)(cid:12) . (25)Next, we have by Lemma 15 | ρ xy | ≤ (cid:16) L xy − D xy | λ | (cid:17) Im λ ≤ L xy (cid:16) − S D | λ | (cid:17) Im λ , whence ∑ x ∼ a | ρ xa | ≤ ∑ x ∼ a ( L xa ) − (cid:16) − S D | λ | (cid:17) Im λ = C ′ (cid:16) − S D | λ | (cid:17) Im λ . It follows from (25) that | P ( λ ) | ≤ | λ | + S R (cid:16) − S D | λ | (cid:17) Im λ C ′ (cid:16) − S D | λ | (cid:17) Im λ = C ′ ( | λ | + S R ) (cid:16) − S D | λ | (cid:17) ( Im λ ) which was to be proved. (cid:3) Corollary 17.
Under the hypothesis of Theorem , assume in addition that R xy = for all xy ∈ E. Then P ( λ ) is holomorphic in C \ J whereJ = h − i p S D , i p S D i . Proof.
In this case we have the symmetry P ( − λ ) = − P ( λ ) . By Theorem 6 (seeRemark 7) and Theorem 16, P ( λ ) is holomorphic in the union { Re λ = } ∪ { ( Im λ ) > S D } , that coincides with C \ J . (cid:3) An upper bound of the admittance using small Im λ .Lemma 18. Let R, L, D be non-negative numbers. Let L > and λ ∈ C be suchthat Im λ > and | λ | < DL . Then z : = R + L λ + D λ = N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 13 and, for ρ = z , we have Im ρ | ρ | ≥ DL | λ | − | λ | + DL | λ | + RL Im λ (26) and | ρ | ≤ (cid:16) D | λ | − L (cid:17) Im λ . Proof.
We have − Im z = − L Im λ + D Im λ | λ | = (cid:18) D | λ | − L (cid:19) Im λ > . In particular, z =
0. We have also | z | ≤ R + L | λ | + D | λ | . It follows thatIm ρ | ρ | = − Im z | z | ≥ (cid:16) D | λ | − L (cid:17) Im λ R + L | λ | + D | λ | ≥ DL | λ | − | λ | + DL | λ | + RL Im λ which proves (26). Finally, we have | ρ | = | z | ≤ − Im z = (cid:16) D | λ | − L (cid:17) Im λ . (cid:3) Theorem 19.
Assume that L xy > for all xy ∈ E. SetS D = sup xy ∈ E D xy L xy , S ∗ D = inf xy ∈ E D xy L xy , S R = sup xy ∈ E R xy L xy , and C ′ = ∑ x ∼ a L xa . Then in the domain Ω ∗ = { λ ∈ C | Im λ = and | λ | < S ∗ D } (27) the function P ( λ ) is holomorphic and | P ( λ ) | ≤ C ′ ( | λ | + S R | λ | + S D ) | λ | ( S ∗ D − | λ | ) | Im λ | . (28) p S ∗ D i p S ∗ D Re λ Im λΩ ∗ F IGURE
3. The domain Ω ∗ = { λ ∈ C | Im λ = | λ | < S ∗ D } . Proof. If S ∗ D = Ω ∗ = /0 and there is nothing to prove. Let S ∗ D >
0. By thesymmetry λ → λ , it suffices to prove the both claims in the domain Ω ∗ + = { λ ∈ C | Im λ > | λ | < S ∗ D } . For λ ∈ Ω ∗ + we have by Lemma 18 that z xy =
0, whence λ ∈ Λ . By Lemma 18 wehave for all xy ∈ E and λ ∈ Ω ∗ + Im ρ xy | ρ xy | ≥ D xy L xy | λ | − | λ | + D xy L xy | λ | + R xy L xy Im λ ≥ S ∗ D | λ | − | λ | + S D | λ | + S R Im λ > , (29)By Theorem 6 we conclude that P ( λ ) is a holomorphic function in Ω ∗ + .Using (29), we obtain by Theorem 10 that for all λ ∈ Ω ∗ + | P ( λ ) | ≤ | λ | + S D | λ | + S R (cid:16) S ∗ D | λ | − (cid:17) Im λ ∑ x ∼ a (cid:12)(cid:12)(cid:12) ρ ( λ ) xa (cid:12)(cid:12)(cid:12) . Next, we have by Lemma 18 | ρ xy | ≤ (cid:16) D xy | λ | − L xy (cid:17) Im λ = L xy (cid:16) D xy L xy | λ | − (cid:17) Im λ ≤ L xy (cid:16) S ∗ D | λ | − (cid:17) Im λ , whence ∑ x ∼ a | ρ xa | ≤ ∑ x ∼ a ( L xa ) − (cid:16) S ∗ D | λ | − (cid:17) Im λ = C ′ (cid:16) S ∗ D | λ | − (cid:17) Im λ . It follows that | P ( λ ) | ≤ C ′ (cid:16) | λ | + S D | λ | + S R (cid:17) (cid:16) S ∗ D | λ | − (cid:17) ( Im λ ) whence (28) follows. (cid:3) N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 15
Corollary 20.
Under the hypothesis of Theorem , assume in addition that R xy = for all xy ∈ E. Then P ( λ ) is holomorphic in the domain C \ J ∗ whereJ ∗ = h i p S ∗ D , i p S D i ∪ h − i p S D , i p S ∗ D i ∪ { } . Proof.
In this case we have the symmetry P ( − λ ) = − P ( λ ) . By Theorem 6 (seeRemark 7), Theorems 16 and 19 P ( λ ) is holomorphic in the union { Re λ = } ∪ { ( Im λ ) > S D } ∪ { < ( Im λ ) < S ∗ D } that coincides with C \ J ∗ . (cid:3) i p S ∗ D i √ S D − i p S ∗ D − i √ S D Re λ Im λ C \ J ∗ F IGURE
4. The domain C \ J ∗ .4. E FFECTIVE IMPEDANCE OF INFINITE NETWORKS
Let ( V , E ) be an infinite locally finite connected graph equipped with the weights ρ ( λ ) xy as in Section 2. Fix a vertex a ∈ V and a set of vertices B ∈ V such that a B .Note that here the set B can be empty (which physically means that the ground willbe at infinity).Then the structure Γ = ( V , ρ , a , B ) is called an infinite network .Let dist ( x , y ) be the graph distance on V , that is, the minimal value of n suchthat there exists a path { x k } nk = connecting x and y , that is, x = x ∼ x ∼ ... ∼ x n = y . Let us consider a sequence of finite graphs ( V n , E n ) , n ∈ N , where V n = { x ∈ V | dist ( a , x ) ≤ n } and E n consists of all the edges of E with the endpoints in V n . We endow the finitegraph ( V n , E n ) with the complex weight ρ n = ρ | E n . Consider the set ∂ V n = { x ∈ V | dist ( a , x ) = n } , that will be regarded as the boundary of the graph ( V n , E n ) . Note that V n = ∂ V n ∪ V n − . Let us set B n = ( B ∩ V n ) ∪ ∂ V n and consider the following sequence of finite networks Γ n = ( V n , ρ n , a , B n ) , n ∈ N . Let P n ( λ ) be the effective admittance of Γ n . Definition 21.
Define the effective admittance of Γ as P ( λ ) = lim n → ∞ P n ( λ ) for those λ ∈ C \ { } where the limit exists. Theorem 22.
The following is true for any infinite network. ( a ) The sequence { P n ( λ ) } converges as n → ∞ locally uniformly in the do-main { Re λ > } . ( b ) If L xy > for all xy ∈ E andS D : = sup xy ∈ E D xy L xy < ∞ and S R : = sup xy ∈ E R xy L xy < ∞ (30) then { P n ( λ ) } converges as n → ∞ locally uniformly in the domain Ω = { λ ∈ C | Im λ = and | λ | > S D } . (31) ( c ) If in addition to ( ) alsoS ∗ D : = inf xy ∈ E D xy L xy > then { P n ( λ ) } converges as n → ∞ locally uniformly in the domain Ω ∗ = { λ ∈ C | Im λ = and | λ | < S ∗ D } . In all the cases, the limit P ( λ ) = lim n → ∞ P n ( λ ) is a holomorphic function in the domains in question.Proof. ( a ) By Corollary 13, the sequence { P n ( λ ) } is uniformly bounded in anydomain { Re λ ≥ ε , | λ | ≤ c } with 0 < ε < c < ∞ . Hence, the sequence { P n ( λ ) } is precompact in such a domainand, hence, has a convergent subsequence. By a diagonal process, we obtain aconvergent subsequence { P n k ( λ ) } in the entire domain { Re λ > } , and the limitis a holomorphic function in this domain. On the other hand, for positive real λ also all ρ ( λ ) xy are real and positive on the edges, and in this case the sequence { P n ( λ ) } is known to be positive and decreasing (from the theory of random walkson graphs, see e.g. [7], [12]). Hence, this sequence has a limit for all positive real λ . Since every holomorphic function in { Re λ > } is uniquely determined by itsvalues on positive reals, we obtain that lim P n k ( λ ) is independent of the choice ofa subsequence. Hence, the entire sequence { P n ( λ ) } converges as n → ∞ in thedomain { Re λ > } , and the limit is a holomorphic function in this domain. N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 17 ( b ) By Theorem 16 all the functions { P n ( λ ) } are holomorphic in the domain Ω = { λ ∈ C | Im λ = | λ | > S D } and admit the estimate | P n ( λ ) | ≤ C ( | λ | + S R ) | λ | ( | λ | − S D ) | Im λ | . Hence, the sequence { P n ( λ ) } is locally uniformly bounded in Ω and, hence, isprecompact. All the limits of convergent subsequences of { P n ( λ ) } coincide by ( a ) in the domain Ω ∩ { Re λ > } which implies that they coincide also in Ω . Hence, { P n ( λ ) } converges in Ω to aholomorphic function. ( c ) By Theorem 19 all functions { P n ( λ ) } are holomorphic Ω ∗ = { λ ∈ C | Im λ = | λ | < S ∗ D } and admit the estimate | P n ( λ ) | ≤ C ′ ( | λ | + S R | λ | + S D ) | λ | ( S ∗ D − | λ | ) | Im λ | . Hence, the sequence { P n ( λ ) } is locally uniformly bounded in Ω ∗ and, hence, isprecompact. All the limits of convergent subsequences of { P n ( λ ) } coincide in thedomain Ω ∗ ∩ { Re λ > } which implies that they coincide also in Ω ∗ . Hence, { P n ( λ ) } converges in Ω ∗ toa holomorphic function. (cid:3) Corollary 23.
Assume that R xy = for all xy ∈ E . Then P ( λ ) = lim n → ∞ P n ( λ ) is well-defined and holomorphic in the domain C \ J ∗ , whereJ ∗ = h i p S ∗ D , i p S D i ∪ h − i p S D , i p S ∗ D i ∪ { } . Proof.
Note that we assume neither S D < ∞ nor S ∗ D > . By the symmetry P ( − λ ) = − P ( λ ) an by Theorem 22, the sequence { P n ( λ ) } converges locally uniformly inthe union { Re λ = } ∪ { ( Im λ ) > S D } ∪ { < ( Im λ ) < S ∗ D } that coincides with C \ J ∗ . (cid:3) The next statement is a simplified version of Corollary 23.
Corollary 24.
Assume that R xy = for all xy ∈ E and setS : = sup xy ∈ E C xy L xy . Then P ( λ ) = lim n → ∞ P n ( λ ) is well-defined and holomorphic in the domain C \ h − i √ S , i √ S i .
5. E
XAMPLES
Example 25.
Consider the infinite graph ( V , E ) , where V = { , , , . . . } and E is given by 0 ∼ ∼ ∼ · · · ∼ n ∼ ( n + ) ∼ . . . Define the impedance of the edge k ∼ ( k + ) by z k ( k + ) = L k λ + D k λ , where L k ≥ D k > R k =
0) (see Fig. 5).0 L D L D ( n − ) L n − D n − n F IGURE
5. Chain networkSet a = B = /0. Then we have V n = { , ..., n } and B n = { n } . It follows that P n ( λ ) = ∑ n − k = (cid:16) L k λ + D k λ (cid:17) = l n λ + d n λ , where l n = n − ∑ k = L k and d n = n − ∑ k = D k . Assume further that ∞ ∑ k = L k = ∞ ∑ k = D k = ∞ that is lim n → ∞ l n = lim n → ∞ d n = + ∞ . Then for any λ with Re λ = (cid:18) l n λ + d n λ (cid:19) → ∞ whence P ( λ ) = lim n → ∞ P n ( λ ) = . For λ = i ω with real ω we have P n ( i ω ) = − il n ω − d n ω . Assume in addition that ∞ ∑ k = ( L k − D k ) = : c ∈ ( , ∞ ) that is lim n → ∞ ( l n − d n ) = c ∈ ( , ∞ ) . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 19
Then for ω = P n ( i ) = − il n − d n , whence P ( i ) = lim n → ∞ P n ( i ) = − ic . It follows that also P ( − i ) = ic . For ω = ± l n ω − d n ω = ( l n − d n ) ω + d n (cid:18) ω − ω (cid:19) → ± ∞ whence it follows that P ( i ω ) = lim n → ∞ P n ( i ω ) = . Hence, for any λ ∈ C \ { } we have P ( λ ) = ( − λ c , if λ = ± i , otherwise.In particular, P ( λ ) is holomorphic in C \ {± i } but is discontinuous at λ = ± i .On the other hand we have S D = sup n D n L n and S ∗ D = inf n D n L n . Since c < ∞ then necessarily S D =
1. Since c > S ∗ D < λ = ± i (where P looses continuity) lie in the set J ∗ = h − i p S D , − i p S ∗ D i ∪ h i p S ∗ D , i p S D i ∪ { } that matches Corollary 23. For example, choose L n = D n = − ε − n , where ε >
0. Then S D = S ∗ D = − ε , so that the interval h i p S ∗ D , i √ S D i = h i √ − ε , i i , containing λ = i , can have an arbitrary small length. Example 26.
Consider the infinite graph ( V , E ) , where V = { , , , , , . . . } and E is given by ( k − ) ∼ k and ( k − ) ∼ k for k = , ∞ . Let us make thisgraph into a network as on Fig. 6. LD LD ( n − ) nLD ( n − ) LD ( n − ) F IGURE
6. Modified ladder networkThat is, let the impedance of the edges ( k − ) ∼ k be λ and impedance of theedges 2 k − ∼ k be L λ + D λ , where L > D >
0. Set also a =
0, while B = { , , . . . } . This network is an αβ -network from [11] and it is similar to Feynman’s laddernetwork (see [5]), but we add coils to the “vertical” edges and ground at infinity.Clearly, we have V n = { , , . . . , n } \ { n − } and B n = { , , . . . , n − } ∪ { n } . The Dirichlet problem (3) for the finite network Γ n is as follows: v ( k − ) + µ v ( k − ) + v ( k + ) − ( + µ ) v ( k ) = , k = , n − , v ( ) = , v ( k − ) = , k = , n − , v ( n ) = , (32)where µ = λ L λ + D .Substituting the equations from the third line of (32) to the first line and denoting v k = v ( k ) , we obtain the following recurrence relation for v k : v k + − ( + µ ) v k + v k − = . (33)The characteristic polynomial of (33) is ψ − ( + µ ) ψ + = . (34)By the definition of a network µ =
0. If µ = −
4, then the equation (34) has twodifferent complex roots ψ , ψ and its solution is v k = c ψ k + c ψ k , (35)where c , c ∈ C are arbitrary constants.We use the second and fours equations of (32) as boundary conditions for thisrecurrence equation. Substituting (35) in the boundary conditions we obtain thefollowing equations for the constants: ( c + c = , c ψ n + c ψ n = . N THE EFFECTIVE IMPEDANCE OF FINITE AND INFINITE NETWORKS 21
Therefore, c = − ψ n = − ψ n − ψ n , c = − ψ n = − ψ n − ψ n , since ψ ψ = Γ n : P n ( λ ) = λ ( − v ( )) = λ ( − c ψ − c ψ )= (cid:0) ψ n − + (cid:1) ( ψ − ) λ (cid:0) ψ n − (cid:1) = (cid:0) ψ n − + (cid:1) ( ψ − ) λ (cid:0) ψ n − (cid:1) . Without loss of generality we can assume, that | ψ | ≤ | ψ | . Then, since ψ ψ = | ψ | < < | ψ | or | ψ | = | ψ | = | ψ | < < | ψ | we obtain P ( λ ) = lim n → ∞ P n ( λ ) = − ψ λ . In the case | ψ | = | ψ | = { P n ( λ ) } has no limit.Let us now consider the case µ = −
4. Then the solution of the recurrencerelation (33) is v k = c ( − ) k + c k ( − ) k , where c , c ∈ C are arbitrary constants. And using boundary conditions, we obtain ( c = c = − n , P n ( λ ) = λ ( − v ( )) = n − λ n , and P ( λ ) = lim n → ∞ P n ( λ ) = λ . Therefore, for the infinite network we have P ( λ ) = − ψ λ , if | ψ | < < | ψ | , λ , if ψ = ψ = − , not defined otherwise . Now we will reformulate the above identity in terms of λ . Claim 27.
Let µ = − and µ = . Then the condition | ψ | = | ψ | = occurs ifand only if µ ∈ ( − , ) .Proof. “ ⇒ ” Let | ψ | =
1. Then | ψ | =
1. Since µ = µ = −
4, it follows from(34) and | ψ | = | ψ | = ψ , ψ R . Therefore, ψ = ψ , since ψ ψ = + µ = ψ + ψ ∈ R . Also ψ , ψ R means that thedeterminant of (34) ( + µ ) − = µ + µ (36)is not positive, i.e. µ ( − ∞ , − ) ∪ ( , ∞ ) . Therefore, µ ∈ R \ (( − ∞ , − ) ∪ ( , ∞ )) which was to be proved. “ ⇐ ” Let µ ∈ ( − , ) . Then the determinant of (34) is negative and | ψ , | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + µ ± i r − µ − (cid:16) µ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:16) + µ (cid:17) − µ − (cid:16) µ (cid:17) = . (cid:3) Since µ = − λ = ± i q DL + / , we have P ( λ ) = − ψ λ , if λ ∈ C \ h − i q DL + / , i q DL + / i λ , if λ = ± i q DL + / not defined, if λ ∈ (cid:16) − i q DL + / , i q DL + / (cid:17) , (37)where ψ is the root if the equation (34) with | ψ | <
1. Clearly, this function iscontinuous at the points λ = ± i q DL + / . Indeed, ψ → − , when λ → ± i q DL + / , since roots of the quadratic equation are continuous func-tions on coefficients. Hence, the continuity of P ( λ ) at given points follows.Therefore, P ( λ ) is well-defined and continuous in the domain C \ − i s DL + / , i s DL + / ! . In particular, the domain of holomorphicity of P ( λ ) is C \ " − i s DL + / , i s DL + / . (38)On the other hand we have S ∗ D = S D = DL , therefore, the Corollary 23 states,that P ( λ ) is holomorphic in the domain C \ " − i r DL , i r DL . (39)Comparison of the intervals (38) and (39) shows the sharpness of Corollary 23.A CKNOWLEDGEMENT
The author thanks her scientific advisor, Professor Alexander Grigor’yan, forhelpful comments related to this work.R
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IELEFELD , G, G