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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

Northeastern University, Boston, Massachusetts 02115, U.S.A.

Abstract

We present a formulation of the determination of the impedancebetween any two nodes in an impedance network. An impedancenetwork is described by its Laplacian matrix L which has generallycomplex matrix elements. We show that by solving the equationLu = u

with orthonormal vectors ua, the effective impedance

between nodes p and q of the network is Zpq =

(up u q)2/where the summation is over all not identically equal to zero andup is the p-th component of u. For networks consisting of induc-tances L and capacitances C, the formulation leads to the occurrenceof resonances at frequencies associated with the vanishing of . Thiscurious result suggests the possibility of practical applications to res-onant circuits. Our formulation is illustrated by explicit examples.

Key words: Impedance network, complex matrices, resonances.

1

1 Introduction

A classic problem in electric circuit theory that has attracted attentionfrom Kirchhoffs time [1] to the present is the consideration of networkresistances and impedances. While the evaluation of resistances andimpedances can in principle be carried out for any given network usingtraditional, but often tedious, analysis such as the Kirchhoffs laws,there has been no conceptually simple solution. Indeed, the problemof computing the effective resistance between two arbitrary nodes ina resistor network has been studied by numerous authors (for a listof relevant references on resistor networks up to 2000 see, e.g., [2]).Particularly, an elementary exposition of the material can be found inDoyle and Snell [3].

However, past efforts prior to 2004 have been focused mainly onregular lattices and the use of the Greens function technique, forwhich the analysis is most conveniently carried out when the networksize is infinite [2, 4]. Little attention has been paid to finite networks,even though the latter are those occurring in applications. Further-more, there has been very few studies on impedance networks. To besure, studies have been carried out on electric and optical properties ofrandom impedance networks in binary composite media (for a reviewsee [5]) and in dielectric resonances occurring in clusters embedded ina regular lattice [6]. But these are mostly approximate treatments onrandom media. More recently Asad et al. [7] evaluated the two-pointcapacitance in an infinite network of identical capacitances. When allimpedances in a network are identical, however, the Greens functiontechnique used and the results are essentially the same as those ofidentical resistors.

In 2004 one of us proposed[8] a new formulation of resistor net-works which leads to an expression of the effective resistance betweenany two nodes in a network in terms of the eigenvalues and eigenvec-tors of the Laplacian matrix. Using this formulation one computes theeffective resistance between two arbitrary nodes in any network whichcan be either finite or infinite [8]. This is a fundamentally new formu-lation. But the analysis presented in [8] makes use of the fact that,for resistors the Laplacian matrix has real matrix elements. Conse-quently the method does not extend to impedances whose Laplacianmatrix elements are generally complex (see e.g., [9]). In this paper weresolve this difficulty and extend the formulation of [8] to impedancenetworks.

2

Consider an impedance network L consisting ofN nodes numbered = 1, 2, ...,N . Let the impedance connecting nodes and be

z = z = r + i x (1)

where r = r 0 is the resistive part and x = x the reactivepart, which is positive for inductances and negative for capacitances.Here, i =

1 often denoted by j = 1 in alternating current (AC)circuit theory [9]. In this paper we shall use i and j interchange-ably. The admittance y connecting two nodes is the reciprocal of theimpedance. For example, y = y = 1/z .

Denote the electric potential at node by V and the net currentflowing into the network (from the outside world) at node by I .Both V and I are generally complex in the phasor notation used inAC circuit theory [9]. Since there is neither source nor sink of currents,one has the conservation rule

N=1

I = 0 . (2)

The Kirchhoff equation for the network reads

L ~V = ~I (3)

where

L =

y1 y12 . . . y1Ny21 y2 . . . y2N...

.... . .

...yN1 yN2 . . . yN

, (4)

with

y N

=1 ( 6=)y, (5)

is the Laplacian matrix associated with the network L. In (3), ~V and~I are N -vectors whose components are respectively V and I.

Here, we need to solve (3) for ~V for a given current configuration~I. The effective impedance between nodes p and q, the quantity wewish to compute, is by definition the ratio

Zpq =Vp Vq

I(6)

3

where Vp and Vq are solved from (3) with

I = I( p q). (7)

The crux of matter is to solve the Kirchhoff equation (3) for ~Igiven by (7). The difficulty lies in the fact that, since the matrix L issingular, the equation (3) cannot be formally inverted.

To circumvent this difficulty we proceed as in [8] to consider insteadthe equation

L() ~V () = ~I (8)

whereL() = L+ I (9)

and I is the identity matrix. The matrix L() now has an inverse andwe can proceed by applying the arsenal of linear algebra. We take the 0 limit at the end and do not expect any problem since we knowthere is a physical solution.

The crucial step is the computation of the inverse matrix L1().For this purpose it is useful to first recall the approach for resistornetworks.

In the case of resistor networks the matrix L() is real symmet-ric and hence it has orthonormal eigenvectors () with eigenvalues() = + determined from the eigenvalue equation

L()() = ()(), i = 1, 2, ...,N . (10)Now a real hermitian matrix L() is diagonalized by the unitary trans-formation U()L()U() = (), where U() is a unitary matrixwhose columns are the orthonormal eigenvectors () and () is adiagonal matrix with diagonal elements () = + . The inverseof this relation leads to L1() = U()1()U(). 1 In this way wefind the effective resistance between nodes p and q to be [8]

Rpq =N=2

1

|p q|2 (11)

where the summation is over all nonzero eigenvalues, and p is thepth component of (0). Here the = 1 term in the summation with

1The equivalent of the method we use in obtaining (11) is known in mathematicsliterature as the pseudo-inverse method (see, e.g., [10, 11])

4

1() = and 1 p() = 1/N drops out (before taking the 0

limit) due to the conservation rule (2). It can be shown that thereis no other zero eigenvalue if the network is singly-connected. Therelation (11) is the main result of [8].

2 Impedance networks

For impedance networks the Laplacian matrix L is symmetric andgenerally complex and thus

L = L 6= L

where denotes the complex conjugation and denotes the hermitianconjugation. Therefore L is not hermitian and cannot be diagonalizedas described in the preceding section.

However, the matrix LL is always hermitian and has nonnegativeeigenvalues. Write the eigenvalue equation as

L L = , 0, = 1, 2, ...,N . (12)

One verifies that one eigenvalue is 1 = 0 with 1 = {1, 1, ..., 1}T /N ,

where the superscript T denotes the transpose. For complex L therecan exist other zero eigenvalues (see below).

To facilitate considerations, we again introduce L() as in (9) andrewrite (12) as

L()L()() = ()() , () 0, = 1, 2, ...,N , (13)

where is small. Now one eigenvalue is 1() = 2 with 1() =

{1, 1, ..., 1}T /N . For other eigenvectors we make use of the theo-rem established in the next section (see also [12]) that there exist Northonormal vectors u() satisfying the equation

L()u() = ()u(), a = 1, 2, ...,N (14)

where() =

() e

i(), () = real. (15)

Particularly, we can take

1() =1() = , 1() = 0. (16)

5

Equation (14) plays the role of the eigenvalue equation (10) for resis-tors.

We next construct a unitary matrix U() whose columns are u().Using (14) and the fact that L() is symmetric, one verifies that L()is diagonalized by the transformation

UT ()L()U() = ()

where () is a diagonal matrix with diagonal elements (). Theinverse of this relation leads to

L1() = U()1()UT (). (17)

where 1() is a diagonal matrix with diagonal elements 1/().We can now use (17) to solve (8) to obtain,after using (6),

Zpq = lim0

N=1

1

()

(up() u q()

)2, (18)

where up is the pth component of the orthonormal vector u().

Now the term = 1 in the summation drops out before taking thelimit just like in the case of resistors[8] since 1() = and u1 p() =u1 q() = constant. If there exist other eigenvalues () = withup() 6= constant, a situation which can happen when there are purereactances L and C, the corresponding terms in (18) diverge in the 0 limit at specific frequencies in an AC circuit. Then oneobtains the effective impedance

Zpq =N=2

1

(up u q

)2, if 6= 0, 2

= , if there exists = 0, 2. (19)Here up = u p(0). The physical interpretation of Z = is theoccurrence of a resonance in an AC circuit at frequencies where l =0, meaning it requires essentially a zero input current I to maintainpotential differences at these frequencies.

The expression (19) is our main result for impedance networks.

In the case of pure resistors, the Laplacian L() and the eigenvaluesl() in (10) are real, so without the loss of generality we can take ()to be real (see Example 3 in Sec. 5 below), and use u() = () in(14) with () = 0. Then up() in (18) is real and (19) coincideswith (11) for resistors. There is no l = 0 other than l1 = 0, and thereis no resonance.

6

3 Complex symmetric matrix

For completeness in this section we give a proof of the theorem whichasserts (14) and determines u for a complex symmetric matrix. Ourproof parallels that in [12].

Theorem:

Let L be an n n symmetric matrix with generally complex ele-ments. Write the eigenvalue equation of LL as

L L = , 0, = 1, 2, ..., n. (20)Then, there exist n orthonormal vectors u satisfying the relation

Lu = u, = 1, 2, ..., n (21)

where denotes complex conjugation and = ei, = real.For nondegenerate we can take u = ; for degenerate , the

us are linear combinations of the degenerate . In either case thephase factor of is determined by applying (21),

Remarks:

1. The s are the eigenvalues of L if us are real.

2. If {u, } is a solution of (21), then {u ei , e2i}, = real,is also a solution of (21).

3. While the procedure of constructing u in the degenerate caseappears to be involved, as demonstrated in examples given in section 5the orthonormal us can often be determined quite directly in practice.

4. If L is real, then as aforementioned it has real eigenvalues andeigenvectors and we can take these real eigenvectors to be the u in(21) with l real non-negative.

Proof.

Since LL is Hermitian its nondegenerate eigenvectors can bechosen to be orthonormal. For the eigenvector with nondegenerateeigenvalue , construct a vector

= (L) + c (22)

where c is any complex number. It is readily verified that we have

L L = , (23)

7

so is also an eigenvector of LL with the same eigenvalue . It fol-

lows that if is nondegenerate then and must be proportional,namely,

L = (24)

for some . The substitution of (24) into (23) with given by (22)now yields ||2 = or = ei . Thus, for nondegenerate we simply choose u = and use (21) and (20) to determine thephase factor . This establishes the theorem for non-degenerate .

For degenerate eigenvalues of L L, say, 1 = 2 = with linearlyindependent eigenvectors 1 and 2, we construct

v1 = (L1) +

ei1 1

v2 = (L2) +

ei2 2 (25)

where the choice of the real phase factors 1, 2 is at our disposal. Wechoose 1, 2 to make v1 and v2 linearly independent to satisfy

ei(12) = (v2, v1)/(v2, v1) (26)

where (y, z) = (yT )z is the inner product of vectors y and z.

Now one has

L v1 = ei1 v1 , L v2 =

ei2 v2

L L v1 = v1, LL v2 = v2 . (27)

Writeu1 = v1/|v1| , (28)

where |v| = (v, v) is the norm of v, and construct y = v2 (v2, u1)u1which is orthogonal to u1. Next write

u2 = y/|y|. (29)Then, it can be verified by using (26) that u1 and u2 are orthonormaland satisfy

Lu1 = ei1 u1

Lu2 = ei2 u2 . (30)

In addition, both u1 and u2 are eigenvectors of LL with the same

eigenvalue , hence are orthogonal to , 3. This establishes thetheorem.

8

In the case of multi-degeneracy, a similar analysis can be carriedout by starting from a set of v to construct us by using, say, theGram-Schmidt orthonormalization procedure. For details we refer to[13].

4 Resonances

If there exist eigenvalues = 0, 2, a situation which can occurat specific frequencies in an AC circuit, then the effective impedance(19) between any two nodes diverge and the network is in resonance.

In an AC circuit resonances occur when the impedances are purereactances (capacitances or inductances). The simplest example of aresonance is a circuit containing two nodes connecting an inductanceL and capacitance C in parallel. It is well-known that this LC circuitis resonant with an external AC source at the frequency = 1/

LC.

This is most simply seen by noting that the two nodes are connectedby an admittance y12 = jC + 1/jL = j(C 1/L), and henceZ12 = 1/y12 diverges at = 1/

LC.

Alternately, using our formulation, the Laplacian matrix is

L = y12

(1 11 1

)(31)

so that LL has eigenvalues 1 = 0, 2 = 4|y12|2 and we have 1 = 0as expected. In addition, we also have 2 = 0 when y12 = 0 at thefrequency = 1/

LC. This is the occurrence of a resonance.

An extension of this consideration to N reactances in a ring isdiscussed in Example 2 in the next section.

5 Examples

Examples of applications of the formulation (19) is given in this sec-tion.

Example 1. A numerical example.

It is instructive to work out a numerical example as an illustration.

9

Consider three impedances z12 = i3, z23 = i

3, z31 = 1 con-

nected in a ring as shown in Fig. 1 where i = j =1. We have the

Laplacian

L =

1 i/

3 i/

3 1

i/3 0 i/3

1 i/3 1 + i/3

. (32)

Substituting L into (12) we find the following nondegenerate eigenval-ues and orthonormal eigenvectors of L L,

1 = 0, 1 =

111

,

2 = 3 22, 2 =

124 62

2

2 + i

3

2 1 i322 1

,

3 = 3 + 22, 3 =

124 + 6

2

2 +

2 + i

3

2 1 i322 1

. (33)

Since the eigenvalues are nondegenerate according to the theorem wetake ui = i, i = 1, 2, 3. Using these expressions we obtain from (21)

2 =

2 1, ei2 = 1

7

[32 2 + i

3(2

2 + 1)

]3 =

2 + 1, ei3 =

1

7

[32 + 2 + i

3(2

2 1)

]. (34)

Now (19) reads

Zpq =ei22

(u2p u2q

)2+ei33

(u3p u3q

)2, (35)

using which one obtains the impedances

Z12 = 3 + i3, Z23 = 3 i

3, Z31 = 0. (36)

These values agree with results of direct calculation using the Ohmslaw.

Example 2. Resonance in a one-dimensional ring of N reactances.

Consider N reactances jx1, jx2, ..., jxN connected in a ring asshown in Fig. 2, where x = L for inductance L and x = 1/C

10

for capacitance C at AC frequency . The Laplacian assumes theform

L =1

j

y1 + yN y1 0 0 0 yNy1 y1 + y2 y2 0 0 0...

......

. . ....

......

0 0 0 yN1 yN1 + yN yNyN 0 0 0 y1 yN + y1

,(37)

where yi = 1/xi. The Laplacian L has one zero eigenvalue 1 = 0 asaforementioned. The product of the other N1 eigenvalues of L isknown from graph theory [14, 15] to be equal to N times its spanningtree generating function with edge weights y1, y2, ..., yN . Now the Nspanning trees are easily written down and as a result we obtain

Ni=2

= N(j)N1(1

y1+

1

y2+ + 1

yN

)y1y2 yN

= N(j)N1(x1 + x2 + + xN )/x1x2 xN . (38)

It follows that there exists another zero eigenvalue, and hence a res-onance, if x1 + x2 + + xN = 0. This determines the resonancefrequency .

Example 3. A one-dimensional ring of N equal impedances.

In this example we consider N equal impedances z connected in aring. We have

L = yTperN , L = yTperN , L

L = |y|2 (TperN )2 (39)

where y = 1/z and

TperN =

2 1 0 0 0 11 2 1 0 0 0...

......

. . ....

......

0 0 0 1 2 11 0 0 0 1 2

. (40)

Thus L and LL all have the same eigenvectors. The eigenvalues andorthonormal eigenvectors of TperN are

n = 2[1 cos(2n/N)] = 4 cos2(n/N)

11

n =1N

1n

2n...

(N1)n

, n = 0, 1, ..., N 1 (41)

where = ei2/N . The eigenvalues of L L are

n = |y|22n. (42)Since

Nn = n , (43)

the corresponding eigenvectors are degenerate and we need to con-struct vectors un1 and un2 for 0 < n < N/2. For N = even, however,the eigenvalue N/2 is non-degenerate and needs to be considered sep-arately.

For 0 < n < N/2 the degenerate eigenvectors

n and Nn = n (44)

are not orthonormal. Then we construct linear combinations

un1 =n +

n

2=

2

N

1cos 2nNcos 4nN

...cos 2(N1)nN

,

un2 =n n

2 i=

2

N

0sin 2nNsin 4nN

...sin 2(N1)nN

, n = 1, 2, ,

[N 12

],

(45)

which are orthonormal, where [x] = the integral part of x. The us areeigenvectors of L L with the same eigenvalue n = |y|22n. For N =even we have an additional non-degenerate eigenvector

uN/2 =1N

1111...1

. (46)

12

We next use (21) to determine the phase factors n1 and n2. Com-paring the eigenvalue equation

Lun1 = (yn)un1 with Lun1 = (|y|n)ein1 un1,Lun2 = (yn)un2 with Lun2 = (|y|n)ein2 un2,

and LuN/2 = 4(y)uN/2 with LuN/2 = 4|y|eiN/2uN/2 (47)we obtain

n1 = n2 = N/2 = , (48)

where is given by y = |y| ei.We now use (19) to compute the impedance between nodes p and

q to obtain

Zpq =2

Ny

[N12

]n=1

1

n

[(cos

2np

N cos 2nq

N

)2 i2

(sin

2np

N sin 2nq

N

)2]

+E (49)

where [x] denotes the integral part of x and

E =1

2Ny

[(1)p (1)q

]2, N = even

= 0, N = odd. (50)

After some manipulation it is reduced to

Zpq =z

N

N1n=1

ei2np/N ei2nq/N 22[1 cos(2n/N)] . (51)

This expression has been evaluated in [8] with the result

Zpq = z |p q|[1 |p q|

N

], (52)

which is the expected impedance of two impedances |p q|z and(N |p q|)z connected in parallel as in a ring. This completesthe evaluation of Zpq.

Example 4. Networks of inductances and capacitances.

As an example of networks of inductances and capacitances, weconsider an M N array of nodes forming a rectangular net with free

13

boundaries as shown in Fig. 3. The nodes are connected by capaci-tances C in the M directions and inductances L in the N direction.

The Laplacian of the network is

L = (jC)TfreeM IN ( jL

)IM TfreeN (53)

where TfreeM is the M M matrix

T freeM =

1 1 0 0 0 01 2 1 0 0 0...

......

. . ....

......

0 0 0 1 2 10 0 0 0 1 1

, (54)

and IN is the N N identity matrix. This gives

LL = (C)2 UfreeM IN2(CL

)TfreeM TfreeN +

( 1L

)2IMUfreeN (55)

where UfreeM is the M M matrix

UfreeM =

2 3 1 0 0 0 0 03 6 4 1 0 0 0 01 4 6 4 1 0 0 00 1 4 6 4 0 0 00 0 1 4 6 0 0 0...

......

......

. . ....

......

0 0 0 0 0 4 6 30 0 0 0 0 1 3 2

. (56)

Now TfreeM has eigenvalues

m = 2(1 cos m) = 4 sin2(m/2), m = mM

(57)

and eigenvector (M)m whose components are

(M)mx =

1M, m = 0, for all x,

2M cos

(x+ 12

)m, m = 1, 2, . . . ,M 1, for all x.

(58)

14

It follows that LL has eigenvectors

free(m,n);(x,y) = (M)mx

(N)ny (59)

and eigenvalues

mn = 16(C sin2

m2 1

Lsin2

n2

)2(60)

where m = m/M, n = n/N . This gives

lmn = 4j[C sin2(m/2) 1

Lsin2(n/2)

]=

mn e

imn , mn = /2 . (61)Since the vectors free(m,n);(x,y) are orthonormal and non-degenerate, ac-

cording to the Theorem we can use these vectors in (19) to obtain theimpedance between nodes (x1, y1) and (x2, y2). This gives

Z free(x1,y1);(x2,y2) =M1m=0

N1n=0(m,n)6=(0,0)

(free(m,n);(x1,y1) free(m,n);(x2,y2)

)2lmn

=jNC

|x1 x2|+ jLM

|y1 y2|+ 2jMN

M1m=1

N1n=1

[cos

(x1 +

12

)m cos

(y1 +

12

)n cos

(x2 +

12

)m cos

(y2 +

12

)n]2

C(1 cos m) + 1L (1 cosn).

(62)

As discussed in section 4, resonances occur at AC frequencies de-termined from lmn = 0. Thus, there are (M 1)(N 1) distinctresonance frequencies given by

mn =

sin(n/2N)sin(m/2M) 1LC , m = 1, ...,M 1; n = 1, ..., N 1.

(63)A similar result can be found for anMN net with toroidal boundaryconditions. However, due to the degeneracy of eigenvalues, in that casethere are [(M +1)/2][(N +1)/2] distinct resonance frequencies, where[x] is the integral part of x. It is of pertinent interest to note that anetwork can become resonant at a spectrum of distinct frequencies,and these resonances occur in the effective impedances between anytwo nodes.

15

In the limit of M,N , (63) becomes continuous indicatingthat the network is resonant at all frequencies. This is verified byreplacing the summations by integrals in (62) to yield the effectiveimpedance between two nodes (x1, y1) and (x2, y2)

Z(x1,y1);(x2,y2)

=j

42

20

d

20

d

[1 cos[(x1 x2)] cos[(y1 y2)]C(1 cos ) + 1L(1 cos)

],(64)

which diverges logarithmically.2

6 Summary

We have presented a formulation of impedance networks which permitsthe evaluation of the effective impedance between arbitrary two nodes.The resulting expression is (19) where u and la are those given in(14). In the case of reactance networks, our analysis indicates thatresonances occur at AC frequencies determined by the vanishing ofla. This curious result suggests the possibility of practical applicationsof our formulation to resonant circuits.

Acknowledgment

This work was initiated while both authors were at the National Cen-ter of Theoretical Sciences (NCTS) in Taipei. The support of theNCTS is gratefully acknowledged. Work of WJT has been supportedin part by National Science Council grant NSC 94-2112-M-032-008.We are grateful to J. M. Luck for calling our attention to [5, 6] andM. L. Glasser for pointing us to [7].

2Detailed steps leading to (62) and (64) can be found in Eqs. (37) and (40) of [8].

16

References

[1] G. Kirchhoff, Uber die Auflosung der Gleichungen, auf welche manbei der Untersuchung der linearen Verteilung galvanischer Stromegefuhrt wird, Ann. Phys. und Chemie 72 (1847) 497-508.

[2] Cserti J 2002 Application of the lattice Greens function for calcu-lating the resistance of infinite networks of resistors Am. J. Phys68 896-906 (Preprint cond-mat/9909120)

[3] P. G. Doyle and J. L. Snell, Random walks and electric networks,The Carus Mathematical Monograph, Series 22 (The Mathemat-ical Association of America, USA, 1984), pp. 83-149; Also inarXiv:math.PR/0001057.

[4] Cserti J, David G and Piroth A 2002 Perturbation of infi-nite networks of resistors Am. J. Phys. 70 153-9 (Preprintcond-mat/0107362)

[5] Clerc J-P, Giraud G, Laugier J-M, and Luck J M 1990 The ACelectrical conductivity of binary disordered systems, percolationclusters, fractals and related models Adv. Phys. 39 191-309

[6] Clerc J-P, Giraud G, Luck J M and Robin T 1996 Dielectric res-onances of lattice animals and other fractal clusters J. Phys. A:Math. Gen. 29 4781-4801 (Preprint cond-mat/9608079)

[7] Asad J H, Hijjawi R S, Sakaji A J and Khalifeh J M 2005 Infinitenetwork of identical capacitors by Greens function, Int. J. Mod.Phys. B 19 3713-21

[8] Wu F Y 2004 Theory of resistor networks: The two-pointresistance, J. Phys. A: Math. Gen. 37 6653-6673 (Preprintmath-ph/0402038)

[9] Alexander C and Sadiku M 2003 Fundamentals of Electric Cir-cuits, 2nd Ed. (McGraw Hill, New York).

[10] Ben-Israel A and Greville T N E 2003 Generalized Inverses: The-ory and Applications 2nd Ed. (Springer-Verlag, New York)

[11] Friedberg S H, Insel A J and Spence L E 2002 Linear Algebra 4thEd. (Prentice Hall, Upper Saddle River, New Jersey) Sec. 6.7

[12] Horn R A and Johnson C R 1985 Matrix Analysis (CambridgeUniversity Press, Cambridge) Section 4.4

[13] See, for example, pp. 15-16 of [12].

17

[14] Biggs N L 1993 Algebraic Graph Theory 2nd Ed. (CambridgeUniversity Press, Cambridge)

[15] Tzeng W J and Wu F Y 2000 Spanning trees on hypercubic lat-tices and nonorientable surfaces Appl. Math. Lett. 13 (6) 19-25(Preprint cond-mat/0001408)

18

12 3

z12 z31

z23

Fig. 1 An example of three impedances in a ring.

19

x1

x2

x3

xN

xN-1

Fig. 2 A ring of N reactances.

20

LC

L

L

C C C C

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

21

IntroductionImpedance networksComplex symmetric matrixResonancesExamplesSummary