Theory of impedance networks: The two-point impedance and LC resonances

Transcription

1 arxiv:math-ph/060048v6 4 Jul 2006 Theory of impedance networks: The two-point impedance and LC resonances W. J. Tzeng Department of Physics Tamkang University, Taipei, Taiwan and F. Y. Wu Department of Physics ortheastern University, Boston, Massachusetts 025, U.S.A. Abstract We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation Lu α = λ α u α with orthonormal vectors u a, the effective impedance between nodes p and q of the network is Z pq = α (u α p u α q ) 2 /λ α where the summation is over all λ α not identically equal to zero and u α p is the p-th component of u α. For networks consisting of inductances L and capacitances C, the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of λ α. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples. Key words: Impedance network, complex matrices, resonances.

2 Introduction A classic problem in electric circuit theory that has attracted attention from Kirchhoff s time [] to the present is the consideration of network resistances and impedances. While the evaluation of resistances and impedances can in principle be carried out for any given network using traditional, but often tedious, analysis such as the Kirchhoff s laws, there has been no conceptually simple solution. Indeed, the problem of computing the effective resistance between two arbitrary nodes in a resistor network has been studied by numerous authors (for a list of relevant references on resistor networks up to 2000 see, e.g., [2]). Particularly, an elementary exposition of the material can be found in Doyle and Snell [3]. However, past efforts prior to 2004 have been focused mainly on regular lattices and the use of the Green s function technique, for which the analysis is most conveniently carried out when the network size is infinite [2, 4]. Little attention has been paid to finite networks, even though the latter are those occurring in applications. Furthermore, there has been very few studies on impedance networks. To be sure, studies have been carried out on electric and optical properties of random impedance networks in binary composite media (for a review see [5]) and in dielectric resonances occurring in clusters embedded in a regular lattice [6]. But these are mostly approximate treatments on random media. More recently Asad et al. [7] evaluated the two-point capacitance in an infinite network of identical capacitances. When all impedances in a network are identical, however, the Green s function technique used and the results are essentially the same as those of identical resistors. In 2004 one of us proposed[8] a new formulation of resistor networks which leads to an expression of the effective resistance between any two nodes in a network in terms of the eigenvalues and eigenvectors of the Laplacian matrix. Using this formulation one computes the effective resistance between two arbitrary nodes in any network which can be either finite or infinite [8]. This is a fundamentally new formulation. But the analysis presented in [8] makes use of the fact that, for resistors the Laplacian matrix has real matrix elements. Consequently the method does not extend to impedances whose Laplacian matrix elements are generally complex (see e.g., [9]). In this paper we resolve this difficulty and extend the formulation of [8] to impedance networks. 2

3 Consider an impedance network L consisting of nodes numbered α =,2,...,. Let the impedance connecting nodes α and β be z αβ = z βα = r αβ + ix αβ () where r αβ = r βα 0 is the resistive part and x αβ = x βα the reactive part, which is positive for inductances and negative for capacitances. Here, i = often denoted by j = in alternating current (AC) circuit theory [9]. In this paper we shall use i and j interchangeably. The admittance y connecting two nodes is the reciprocal of the impedance. For example, y αβ = y βα = /z αβ. Denote the electric potential at node α by V α and the net current flowing into the network (from the outside world) at node α by I α. Both V α and I α are generally complex in the phasor notation used in AC circuit theory [9]. Since there is neither source nor sink of currents, one has the conservation rule The Kirchhoff equation for the network reads I α = 0. (2) α= L V = I (3) where with L = y y 2... y y 2 y 2... y , (4) y y 2... y y α y αβ, (5) β=(β α) is the Laplacian matrix associated with the network L. In (3), V and I are -vectors whose components are respectively V α and I α. Here, we need to solve (3) for V for a given current configuration I. The effective impedance between nodes p and q, the quantity we wish to compute, is by definition the ratio Z pq = V p V q I (6) 3

4 where V p and V q are solved from (3) with I α = I(δ α p δ α q ). (7) The crux of matter is to solve the Kirchhoff equation (3) for I given by (7). The difficulty lies in the fact that, since the matrix L is singular, the equation (3) cannot be formally inverted. To circumvent this difficulty we proceed as in [8] to consider instead the equation L(ǫ) V (ǫ) = I (8) where L(ǫ) = L + ǫi (9) and I is the identity matrix. The matrix L(ǫ) now has an inverse and we can proceed by applying the arsenal of linear algebra. We take the ǫ 0 limit at the end and do not expect any problem since we know there is a physical solution. The crucial step is the computation of the inverse matrix L (ǫ). For this purpose it is useful to first recall the approach for resistor networks. In the case of resistor networks the matrix L(ǫ) is real symmetric and hence it has orthonormal eigenvectors ψ α (ǫ) with eigenvalues λ α (ǫ) = λ α + ǫ determined from the eigenvalue equation L(ǫ)ψ α (ǫ) = λ α (ǫ)ψ α (ǫ), i =,2,...,. (0) ow a real hermitian matrix L(ǫ) is diagonalized by the unitary transformation U (ǫ)l(ǫ)u(ǫ) = Λ(ǫ), where U(ǫ) is a unitary matrix whose columns are the orthonormal eigenvectors ψ α (ǫ) and Λ(ǫ) is a diagonal matrix with diagonal elements λ α (ǫ) = λ α + ǫ. The inverse of this relation leads to L (ǫ) = U(ǫ)Λ (ǫ)u (ǫ). In this way we find the effective resistance between nodes p and q to be [8] R pq = α=2 λ α ψ α p ψ α q 2 () where the summation is over all nonzero eigenvalues, and ψ α p is the pth component of ψ α (0). Here the α = term in the summation with The equivalent of the method we use in obtaining () is known in mathematics literature as the pseudo-inverse method (see, e.g., [0, ]) 4

5 λ (ǫ) = ǫ and ψ p (ǫ) = / drops out (before taking the ǫ 0 limit) due to the conservation rule (2). It can be shown that there is no other zero eigenvalue if the network is singly-connected. The relation () is the main result of [8]. 2 Impedance networks For impedance networks the Laplacian matrix L is symmetric and generally complex and thus L = L L where denotes the complex conjugation and denotes the hermitian conjugation. Therefore L is not hermitian and cannot be diagonalized as described in the preceding section. However, the matrix L L is always hermitian and has nonnegative eigenvalues. Write the eigenvalue equation as L Lψ α = σ α ψ α, σ α 0, α =,2,...,. (2) One verifies that one eigenvalue is σ = 0 with ψ = {,,...,} T /, where the superscript T denotes the transpose. For complex L there can exist other zero eigenvalues (see below). To facilitate considerations, we again introduce L(ǫ) as in (9) and rewrite (2) as L (ǫ)l(ǫ)ψ α (ǫ) = σ α (ǫ)ψ α (ǫ), σ α (ǫ) 0, α =,2,...,, (3) where ǫ is small. ow one eigenvalue is σ (ǫ) = ǫ 2 with ψ (ǫ) = {,,...,} T /. For other eigenvectors we make use of the theorem established in the next section (see also [2]) that there exist orthonormal vectors u α (ǫ) satisfying the equation where L(ǫ)u α (ǫ) = λ α (ǫ)u α (ǫ), a =,2,..., (4) λ α (ǫ) = Particularly, we can take λ (ǫ) = σ α (ǫ)e iθα(ǫ), θ α (ǫ) = real. (5) σ (ǫ) = ǫ, θ (ǫ) = 0. (6) 5

6 Equation (4) plays the role of the eigenvalue equation (0) for resistors. We next construct a unitary matrix U(ǫ) whose columns are u α (ǫ). Using (4) and the fact that L(ǫ) is symmetric, one verifies that L(ǫ) is diagonalized by the transformation U T (ǫ)l(ǫ)u(ǫ) = (ǫ) where (ǫ) is a diagonal matrix with diagonal elements λ α (ǫ). The inverse of this relation leads to L (ǫ) = U(ǫ) (ǫ)u T (ǫ). (7) where (ǫ) is a diagonal matrix with diagonal elements /λ α (ǫ). We can now use (7) to solve (8) to obtain,after using (6), Z pq = lim ǫ 0 α= ( 2, u α p (ǫ) u α q (ǫ)) (8) λ α (ǫ) where u α p is the pth component of the orthonormal vector u α (ǫ). ow the term α = in the summation drops out before taking the limit just like in the case of resistors[8] since λ (ǫ) = ǫ and u p (ǫ) = u q (ǫ) = constant. If there exist other eigenvalues λ α (ǫ) = ǫ with u α p (ǫ) constant, a situation which can happen when there are pure reactances L and C, the corresponding terms in (8) diverge in the ǫ 0 limit at specific frequencies ω in an AC circuit. Then one obtains the effective impedance Z pq = α=2 ( ) 2, u α p u α q if λα 0, α 2 λ α =, if there exists λ α = 0, α 2. (9) Here u α p = u α p (0). The physical interpretation of Z = is the occurrence of a resonance in an AC circuit at frequencies where l α = 0, meaning it requires essentially a zero input current I to maintain potential differences at these frequencies. The expression (9) is our main result for impedance networks. In the case of pure resistors, the Laplacian L(ǫ) and the eigenvalues l α (ǫ) in (0) are real, so without the loss of generality we can take ψ α (ǫ) to be real (see Example 3 in Sec. 5 below), and use u α (ǫ) = ψ α (ǫ) in (4) with θ α (ǫ) = 0. Then u α p (ǫ) in (8) is real and (9) coincides with () for resistors. There is no l α = 0 other than l = 0, and there is no resonance. 6

7 3 Complex symmetric matrix For completeness in this section we give a proof of the theorem which asserts (4) and determines u α for a complex symmetric matrix. Our proof parallels that in [2]. Theorem: Let L be an n n symmetric matrix with generally complex elements. Write the eigenvalue equation of L L as L Lψ α = σ α ψ α, σ α 0, α =,2,...,n. (20) Then, there exist n orthonormal vectors u α satisfying the relation Lu α = λ α u α, α =,2,...,n (2) where denotes complex conjugation and λ α = σ α e iθα, θ α = real. For nondegenerate σ α we can take u α = ψ α ; for degenerate σ α, the u s are linear combinations of the degenerate ψ α. In either case the phase factor θ α of λ α is determined by applying (2), Remarks:. The λ α s are the eigenvalues of L if u α s are real. 2. If {u α, λ α } is a solution of (2), then {u α e iτ,λ α e 2iτ }, τ = real, is also a solution of (2). 3. While the procedure of constructing u α in the degenerate case appears to be involved, as demonstrated in examples given in section 5 the orthonormal u s can often be determined quite directly in practice. 4. If L is real, then as aforementioned it has real eigenvalues and eigenvectors and we can take these real eigenvectors to be the u α in (2) with l α real non-negative. Proof. Since L L is Hermitian its nondegenerate eigenvectors ψ α can be chosen to be orthonormal. For the eigenvector ψ α with nondegenerate eigenvalue σ α, construct a vector φ α = (Lψ α ) + c α ψ α (22) where c α is any complex number. It is readily verified that we have L Lφ α = σ α φ α, (23) 7

8 so φ α is also an eigenvector of L L with the same eigenvalue σ α. It follows that if σ α is nondegenerate then φ α and ψ α must be proportional, namely, Lψ α = λ α ψ α (24) for some λ α. The substitution of (24) into (23) with φ α given by (22) now yields λ α 2 = σ α or λ α = σ α e iθα. Thus, for nondegenerate σ α we simply choose u α = ψ α and use (2) and (20) to determine the phase factor θ α. This establishes the theorem for non-degenerate λ α. For degenerate eigenvalues of L L, say, σ = σ 2 = σ with linearly independent eigenvectors ψ and ψ 2, we construct v = (Lψ ) + σ e iθ ψ v 2 = (Lψ 2 ) + σ e iθ 2 ψ 2 (25) where the choice of the real phase factors θ,θ 2 is at our disposal. We choose θ,θ 2 to make v and v 2 linearly independent to satisfy e i(θ θ 2 ) = (v 2,v ) /(v 2,v ) (26) where (y,z) = (y T ) z is the inner product of vectors y and z. ow one has Lv = σ e iθ v, Lv 2 = σ e iθ 2 v2 L Lv = σ v, L Lv 2 = σ v 2. (27) Write u = v / v, (28) where v = (v,v) is the norm of v, and construct y = v 2 (v 2,u )u which is orthogonal to u. ext write u 2 = y/ y. (29) Then, it can be verified by using (26) that u and u 2 are orthonormal and satisfy Lu = σ e iθ u Lu 2 = σ e iθ 2 u 2. (30) In addition, both u and u 2 are eigenvectors of L L with the same eigenvalue σ, hence are orthogonal to ψ α, α 3. This establishes the theorem. 8

9 In the case of multi-degeneracy, a similar analysis can be carried out by starting from a set of v α to construct u α s by using, say, the Gram-Schmidt orthonormalization procedure. For details we refer to [3]. 4 Resonances If there exist eigenvalues λ α = 0, α 2, a situation which can occur at specific frequencies ω in an AC circuit, then the effective impedance (9) between any two nodes diverge and the network is in resonance. In an AC circuit resonances occur when the impedances are pure reactances (capacitances or inductances). The simplest example of a resonance is a circuit containing two nodes connecting an inductance L and capacitance C in parallel. It is well-known that this LC circuit is resonant with an external AC source at the frequency ω = / LC. This is most simply seen by noting that the two nodes are connected by an admittance y 2 = jωc + /jωl = j(ωc /ωl), and hence Z 2 = /y 2 diverges at ω = / LC. Alternately, using our formulation, the Laplacian matrix is ( ) L = y 2 (3) so that L L has eigenvalues σ = 0, σ 2 = 4 y 2 2 and we have λ = 0 as expected. In addition, we also have λ 2 = 0 when y 2 = 0 at the frequency ω = / LC. This is the occurrence of a resonance. An extension of this consideration to reactances in a ring is discussed in Example 2 in the next section. 5 Examples Examples of applications of the formulation (9) is given in this section. Example. A numerical example. It is instructive to work out a numerical example as an illustration. 9

10 Consider three impedances z 2 = i 3,z 23 = i 3,z 3 = connected in a ring as shown in Fig. where i = j =. We have the Laplacian i/ 3 i/ 3 L = i/ 3 0 i/ 3 i/ 3 + i/. (32) 3 Substituting L into (2) we find the following nondegenerate eigenvalues and orthonormal eigenvectors of L L, σ = 0, ψ =, σ 2 = i 3 2, ψ 2 = i 3 2 2, 2 σ 3 = i 3 2, ψ 3 = i (33) 2 Since the eigenvalues are nondegenerate according to the theorem we take u i = ψ i,i =,2,3. Using these expressions we obtain from (2) σ2 = 2, e iθ 2 = 7[ i 3(2 ] 2 + ) σ3 = 2 +, e iθ 3 = 7[ i 3(2 ] 2 ). (34) ow (9) reads Z pq = e iθ 2 σ2 ( u 2p u 2q ) 2 + e iθ 3 σ3 (u 3p u 3q ) 2, (35) using which one obtains the impedances Z 2 = 3 + i 3, Z 23 = 3 i 3, Z 3 = 0. (36) These values agree with results of direct calculation using the Ohm s law. Example 2. Resonance in a one-dimensional ring of reactances. Consider reactances jx,jx 2,...,jx connected in a ring as shown in Fig. 2, where x = ωl for inductance L and x = /ωc 0

11 for capacitance C at AC frequency ω. The Laplacian assumes the form y + y y y L = y y + y 2 y j ,(37) y y + y y y y y + y where y i = /x i. The Laplacian L has one zero eigenvalue λ = 0 as aforementioned. The product of the other eigenvalues λ α of L is known from graph theory [4, 5] to be equal to times its spanning tree generating function with edge weights y,y 2,...,y. ow the spanning trees are easily written down and as a result we obtain i=2 ( λ α = ( j) ) y y 2 y y y 2 y = ( j) (x + x x )/x x 2 x. (38) It follows that there exists another zero eigenvalue, and hence a resonance, if x + x x = 0. This determines the resonance frequency ω. Example 3. A one-dimensional ring of equal impedances. In this example we consider equal impedances z connected in a ring. We have L = y T per, L = y T per, L L = y 2 (T per )2 (39) where y = /z and T per = (40) Thus L and L L all have the same eigenvectors. The eigenvalues and orthonormal eigenvectors of T per are µ n = 2[ cos(2nπ/)] = 4cos 2 (nπ/)

12 ψ n = ω n ω 2n. ω ( )n where ω = e i2π/. The eigenvalues of L L are Since, n = 0,,..., (4) σ n = y 2 µ 2 n. (42) σ n = σ n, (43) the corresponding eigenvectors are degenerate and we need to construct vectors u n and u n2 for 0 < n < /2. For = even, however, the eigenvalue σ /2 is non-degenerate and needs to be considered separately. For 0 < n < /2 the degenerate eigenvectors ψ n and ψ n = ψ n (44) are not orthonormal. Then we construct linear combinations u n = ψ n + ψn cos 2nπ 2 = cos 4nπ 2,. u n2 = ψ n ψn 2 = 2 i cos 2( )nπ 0 sin 2nπ sin 4nπ. sin 2( )nπ [, n =,2,, 2 ], (45) which are orthonormal, where [x] = the integral part of x. The u s are eigenvectors of L L with the same eigenvalue σ n = y 2 µ 2 n. For = even we have an additional non-degenerate eigenvector u /2 = 2. (46).

13 We next use (2) to determine the phase factors θ n and θ n2. Comparing the eigenvalue equation we obtain Lu n = (yµ n )u n with Lu n = ( y µ n )e iθ n u n, Lu n2 = (yµ n )u n2 with Lu n2 = ( y µ n )e iθ n2 u n2, and Lu /2 = 4(y)u /2 with Lu /2 = 4 y e iθ /2 u /2 (47) where θ is given by y = y e iθ. θ n = θ n2 = θ /2 = θ, (48) We now use (9) to compute the impedance between nodes p and q to obtain Z pq = 2 y [ 2 ] n= [ ( cos 2npπ ) 2nqπ 2 ( cos i 2 sin 2npπ sin 2nqπ ) ] 2 µ n + E (49) where [x] denotes the integral part of x and [ ] 2 E = ( ) p ( ) q, = even 2y = 0, = odd. (50) After some manipulation it is reduced to Z pq = z n= e i2npπ/ e i2nqπ/ 2 2[ cos(2nπ/)] This expression has been evaluated in [8] with the result [ Z pq = z p q. (5) ] p q, (52) which is the expected impedance of two impedances p q z and ( p q )z connected in parallel as in a ring. This completes the evaluation of Z pq. Example 4. etworks of inductances and capacitances. As an example of networks of inductances and capacitances, we consider an M array of nodes forming a rectangular net with free 3

14 boundaries as shown in Fig. 3. The nodes are connected by capacitances C in the M directions and inductances L in the direction. The Laplacian of the network is L = (jωc)t free M ( j ) I I M T free (53) ωl where T free M is the M M matrix T free M = , (54) and I is the identity matrix. This gives L L = (ωc) 2 ( C ) ( ) U free M I 2 T free M T free 2 + IM U free (55) L ωl where U free M U free M = is the M M matrix (56) ow T free M has eigenvalues λ m = 2( cos θ m ) = 4sin 2 (θ m /2), θ m = mπ M (57) and eigenvector ψ (M) m ψ mx (M) = whose components are M, m = 0, for all x, ) θ m, m =,2,...,M, for all x. 2 M cos ( x + 2 (58) 4

15 It follows that L L has eigenvectors and eigenvalues ψ(m,n);(x,y) free = ψ(m) mx ψ ny () (59) ( σ mn = 6 ωc sin 2 θ m 2 φ ) ωl sin2 n 2 (60) 2 where θ m = mπ/m, φ n = nπ/. This gives l mn = 4j [ωc sin 2 (θ m /2) ] ωl sin2 (φ n /2) = σ mn e iθmn, θ mn = ± π/2. (6) Since the vectors ψ(m,n);(x,y) free are orthonormal and non-degenerate, according to the Theorem we can use these vectors in (9) to obtain the impedance between nodes (x,y ) and (x 2,y 2 ). This gives M Z(x free,y );(x 2,y 2 ) = M m= n= m=0 n=0(m,n) (0,0) = j ωc x x 2 + jωl M y y 2 + 2j M [ ( ) ( cos x + 2 θ m cos y + 2 ( ) 2 ψ(m,n);(x free,y ) ψfree (m,n);(x 2,y 2 ) l mn ) φ n cos ωc( cos θ m ) + ωl ( cos φ n) ( ) ( ] 2 x θ m cos y )φ n As discussed in section 4, resonances occur at AC frequencies determined from l mn = 0. Thus, there are (M )( ) distinct resonance frequencies given by sin(nπ/2) ω mn =, m =,...,M ; n =,...,. sin(mπ/2m) LC (63) A similar result can be found for an M net with toroidal boundary conditions. However, due to the degeneracy of eigenvalues, in that case there are [(M + )/2][( + )/2] distinct resonance frequencies, where [x] is the integral part of x. It is of pertinent interest to note that a network can become resonant at a spectrum of distinct frequencies, and these resonances occur in the effective impedances between any two nodes. 5. (62)

16 In the limit of M,, (63) becomes continuous indicating that the network is resonant at all frequencies. This is verified by replacing the summations by integrals in (62) to yield the effective impedance between two nodes (x,y ) and (x 2,y 2 ) Z(x,y );(x 2,y 2 ) 2π 2π = j 4π 2 0 dθ which diverges logarithmically. 2 6 Summary 0 [ ] cos[(x x 2 )θ] cos[(y y 2 )φ] dφ ωc( cos θ) +,(64) ωl ( cos φ) We have presented a formulation of impedance networks which permits the evaluation of the effective impedance between arbitrary two nodes. The resulting expression is (9) where u α and l a are those given in (4). In the case of reactance networks, our analysis indicates that resonances occur at AC frequencies ω determined by the vanishing of l a. This curious result suggests the possibility of practical applications of our formulation to resonant circuits. Acknowledgment This work was initiated while both authors were at the ational Center of Theoretical Sciences (CTS) in Taipei. The support of the CTS is gratefully acknowledged. Work of WJT has been supported in part by ational Science Council grant SC M We are grateful to J. M. Luck for calling our attention to [5, 6] and M. L. Glasser for pointing us to [7]. 2 Detailed steps leading to (62) and (64) can be found in Eqs. (37) and (40) of [8]. 6

17 References [] G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird, Ann. Phys. und Chemie 72 (847) [2] Cserti J 2002 Application of the lattice Green s function for calculating the resistance of infinite networks of resistors Am. J. Phys (Preprint cond-mat/990920) [3] P. G. Doyle and J. L. Snell, Random walks and electric networks, The Carus Mathematical Monograph, Series 22 (The Mathematical Association of America, USA, 984), pp ; Also in arxiv:math.pr/ [4] Cserti J, Dávid G and Piróth A 2002 Perturbation of infinite networks of resistors Am. J. Phys (Preprint cond-mat/007362) [5] Clerc J-P, Giraud G, Laugier J-M, and Luck J M 990 The AC electrical conductivity of binary disordered systems, percolation clusters, fractals and related models Adv. Phys [6] Clerc J-P, Giraud G, Luck J M and Robin T 996 Dielectric resonances of lattice animals and other fractal clusters J. Phys. A: Math. Gen (Preprint cond-mat/ ) [7] Asad J H, Hijjawi R S, Sakaji A J and Khalifeh J M 2005 Infinite network of identical capacitors by Green s function, Int. J. Mod. Phys. B [8] Wu F Y 2004 Theory of resistor networks: The two-point resistance, J. Phys. A: Math. Gen (Preprint math-ph/ ) [9] Alexander C and Sadiku M 2003 Fundamentals of Electric Circuits, 2nd Ed. (McGraw Hill, ew York). [0] Ben-Israel A and Greville T E 2003 Generalized Inverses: Theory and Applications 2nd Ed. (Springer-Verlag, ew York) [] Friedberg S H, Insel A J and Spence L E 2002 Linear Algebra 4th Ed. (Prentice Hall, Upper Saddle River, ew Jersey) Sec. 6.7 [2] Horn R A and Johnson C R 985 Matrix Analysis (Cambridge University Press, Cambridge) Section 4.4 [3] See, for example, pp. 5-6 of [2]. 7

18 [4] Biggs L 993 Algebraic Graph Theory 2nd Ed. (Cambridge University Press, Cambridge) [5] Tzeng W J and Wu F Y 2000 Spanning trees on hypercubic lattices and nonorientable surfaces Appl. Math. Lett. 3 (6) 9-25 (Preprint cond-mat/000408) 8

19 z 2 z z 23 Fig. An example of three impedances in a ring. 9

20 x x x - x 2 x 3 Fig. 2 A ring of reactances. 20

21 C C C C C L L L Fig. 3 A 6 4 network of capacitances C and inductances L. 2