384 IEEE TRANSACTIONS ON ELEmOMAGNETIC COMPATIBILITY, VOL. 37, NO. 3, AUGUST 1995

TEM Modes in Parallel-Line Transmission Structures Peter P. Silvester, Fellow, IEEE, and Simon V. Chamlian

Abstruct- A transmission structure of many parallel conduc- tors is capable of propagating many distinct quasi-TEM modes. A field theory of such modes is developed for a theoretically infinite number of very thin conductors. These possess infinitely many distinct propagation modes, which are the eigenfunctions of a pair of integral operators with Green’s function kernels. The modes depend strongly on the structure cross-sectional shape and size, only slightly on the number and placement of individual conductors. The theory is illustrated by a structure comprising many thin parallel wires on a dielectric substrate backed by a ground plane, whose modal functions and velocity distributions are used to solve a simple crosstalk problem.

I. INTRODUCTION

HE classical problem of a multiconductor transmission T line has been treated by two distinct approaches-one based on circuit theory, the other on field theory. The circuit- theoretic formulation originated with W. Thomson (Lord Kelvin) [l], who stated it in a form surprisingly similar to what most authors have used since the middle of the twentieth century-even though mutual inductance was hardly a well- developed concept at the time, and the notion of displacement current was yet to be published! The field-theoretic approach to the problem of the three-conductor line (two wires and ground) was first published by Mie [2], recognizably in its modern form, though without the benefit (as yet) of Gibbs’ vector notation. As electrical engineering science swung away from field theory toward circuits at mid-century, formulations based on Laplace transforms and matrix notation became popular; this viewpoint seems to have been first advanced by Pipes [3]. More recently, the general multiwire case was treated by Marx [4] with great care and detail, and key points of the published work of this school were summarized in a book by Frankel [5 ] . On the field-theoretic side, a somewhat comparable, though less well-integrated, overview is given in the monograph by Kuznetsov and Stratonovich [6]. The recent monograph by Brandao Faria [7] reviews the key papers of the past twenty or more years, particularly as applied to microwave circuits.

Multimodal propagation in transmission structures first at- tracted interest in the area now called electromagnetic com- patibility-the suppression of crosstalk and the design of transpositions in telegraph lines. Much of the field-theoretic work throughout the long history of this problem has been directed to applications in the power industry-balancing of polyphase transmission lines and their concurrent use for signaling. Because such lines are inherently dispersive, almost all of this work has been formulated in the frequency domain.

Manuscript received September 7, 1994; revised March 20, 1995. The authors are with The Department of Electrical Engineering, McGill

University, 3480 University Street, Montreal, F Q H3A 2A7 Canada. IEEE Log Number 9413151.

Fig. 1. Transmission structure composed of many parallel wires or strips, separated by small gaps.

In recent decades, there has been a resurgence of interest in time-domain formulations, as electromagnetic compatibility problems in digital circuits have moved into prominence.

The now very extensive literature on modal propagation studies is characterized by what might be termed a “bottom- up” approach-the multiconductor transmission line is seen as a generalization of the two-wire line. This paper takes the contrary, or “top-down” approach, which does not appear to have been employed in the past. It first considers the problem of a continuum containing an infinite number of wires, Le., it presupposes continuous distributions of potentials and current densities. The N-conductor problem is subsequently viewed as a specialization of that general case.

This paper develops a quasi-TEM modal theory of transmis- sion lines with infinitely many conductors. In keeping with the application areas of current interest, it takes a field-theoretic viewpoint and formulates the problem entirely in the time domain. The classical telegrapher’s equations are first derived in a novel operator form, then reduced to a pair of operator wave equations. These lead to a pair of eigenvalue problems of one-dimensional propagation, whose solutions are propagation modes described by bi-orthogonal pairs of voltage and current distributions. The paper concludes by displaying the mode spectrum for a planar multiwire structure, similar to Fig. 1, and by indicating how the modal theory can be made practically useful in crosstalk calculation.

11. TELEGRAPHER’S EQUATIONS

Although the theory presented in this paper is applicable to a broad class of problems, a specific case will be used to focus discussion and to furnish illustrative examples. The transmission structure shown in Fig. 1 comprises a multiplicity of thin parallel conductors separated from a large ground plane by a dielectric sheet or substrate. This structure is assumed infinitely long in the z direction (the direction of wave propagation), and the individual wires are taken to be separated by thin insulating spaces so that currents can flow only in the longitudinal or z direction.

The classical telegrapher’s equations of transmission line theory may be stated in an operator form that follows directly from field theory, without reference to a circuit interpretation. It will be assumed that all propagation is quasi-TEM, i.e.,

0018-9375/95$04.00 0 1995 IEEE

SILVESTER AND CHAMLIAN: TEM MODES IN PARALLEL-LINE TRANSMISSION STRUCTURES

~

385

that E and H are purely transverse to the direction of wave propagation z. The electric field E can always be written in terms of the conventional vector and scalar potentials A and V a s

aA E = - - - VV. dt

The potentials A and V can be determined from the distribu- tions of current densities and charges, respectively. Taking the magnetic vector potential first

where G M ( P ; Q ) is the Green’s function appropriate to magnetic fields. In a TEM wave, the longitudinal (z-directed) electric field must vanish. Equation (1) then requires that

(3)

If all dielectric and magnetic materials are isotropic with re- spect to the direction of propagation z (though not necessarily with respect to 2 and y), then D, vanishes along with the longitudinal electric field E,. Consequently, (3) may be written

dV - aA, az at . - -~ -

?! = - JG,(P; Q)% a J dRQ. az (4)

Note that the subscript z on J , has been omitted, for J has only a longitudinal component.

Consider the scalar potential V next. It may be derived from the electric charge distribution p ( z , y, z , t ) as

VP = / G E ( P ; Q ) P Q d f l ~ ( 5 )

where GE( P ; Q ) is the Green’s function appropriate to electric fields. At any space point, the law of conservation of electric charge may be written

dP div J = -- at

so that on differentiating (5 ) , with respect to time and sub- stituting,

at = - / G E ( P ; Q)div JdS2Q. (7)

Because the current density vector is purely longitudinal, the time derivative of potential is given by

?! = - / G E ( P ; Q ) z d J dOQ. at

For brevity in further discussion, let the linear integral operator ‘$3 be defined by

(9)

In analogy with the potential coefficients that arise in the circuit theory of transmission lines, ‘$3 will be referred to as the potential operator. In a similar way, let the inductance operator C be defined by

where G M ( P ; Q ) is the Green’s function appropriate to magnetic fields. Note that in both operator definitions, the quantity u represents a source density, which is nonzero only in the conductors themselves. The domain of integration is, therefore, the conductor cross section in the xy plane.

The potential operator !&I is semidefinite, but becomes definite once a potential reference has been selected for V . Choosing V = 0 at some fixed point establishes the reference, and makes ‘$3 positive definite, hence invertible. Its inverse will be denoted by C

(1 1)

and will be referred to as the capacitance operator. Using this operator notation, equations (4) and (8) may be rewritten as the generalized telegrapher’s equations.

CQU = ‘$3Cu = u

d J - = -E--, dV az at av = -e-. a J dz dt -

A precise way of stating the TEM assumption is to say that the Green’s functions GE(P; Q ) and G M ( P ; Q ) shall be taken as two-dimensional, i.e., as the Green’s functions appropriate to the transverse electrostatic and transverse magnetostatic problem, respectively. By this means, the suppression of longitudinal vector components is automatic. Then B, C, and C are purely transverse operators, i.e., they operate on the transverse coordinates x and y only, not on z or t.

111. WAVE EQUATIONS AND SOLUTIONS The scalar potential V and the longitudinal current density

J that satisfy the telegrapher’s equations also satisfy a pair of operator wave equations. To arrive at these, differentiate the telegrapher’s equations (12) and (13) with respect to distance, then interchange time and distance differentiations

Cross-substitution of the space derivatives, as given by the telegrapher’s equations (13) and (12), respectively, then pro- duces

a2 v - = CC- a2v

dz2 at2 d2 J a2 J

at2 dz2 = ec-. These may be regarded as wave equations in operator form. Note that there are two wave equations, and that they are distinct. There is no a priori reason to expect that the two operators C and C commute; indeed in many interesting cases they do not-CC # CC.

An important special case arises if the transmission structure contains only a single isotropic dielectric material and only a single isotropic magnetic material. In that circumstance, the Green’s functions GE and GM are identical, except for

386 IEEE TRANSACIIONS ON ELECTROMAGNEnC COMPATIBILITY, VOL. 37, NO. 3, AUGUST 1995

material constants, so the operators C and C collapse into scalars

(18) CCU = CCU = pEU.

Both operator wave equations then reduce to the conventional scalar wave equation.

To show that the general wave equations (16)-(17) do possess propagating solutions, substitute a traveling wave of current density into the wave equation (17) as a trial solution. That is, suppose

J = $(x, y)f(ct F 2 ) (19)

where the upper sign denotes propagation in the +z direction. Substituting into (17), rewriting, and collecting terms, there results

where, for brevity, s = ct F z. This equation is satisfied by any twice differentiable function f ( s ) = f ( c t 7 z), so any such function may represent an acceptable traveling wave. A similar development holds for the wave equation (16) that governs the scalar potential. Set

(21) v = $(T y)f(ct F 2) = $(IC, y)f(s) .

Then

Equations (20) and (22) represent the pair of eigenvalue problems

(23)

(24)

Solution of these two equations produces a family of pairs of eigenfunctions $k and 4 k , each pair associated with a single eigenvalue 1/c", The eigenfunctions represent modal distributions of current density and potential, respectively; the eigenvalues give the corresponding propagation velocities Ck . Although the two operators C and C are positive and self- adjoint, they do not generally commute, so their compositions CC and C2 are each other's adjoints, but not self-adjoint. The eigenfunctions {&} and {$k} are, therefore, not alike, but they form bi-orthogonal sets

1

1

CZ'd'k(X, y) = T $ k ( z , Y)

CC$k(x, y) = y$k(x , y). 'k

LW, Y ) $ j ( T Y) dQ = 0, 2 # j . (25)

The eigenfunctions may be scaled to be bi-orthonormal, i.e., scaled so that the integral (25) has unity value for i = j. In the remainder of this paper it will be assumed that the eigen- functions are bi-orthonormal, unless otherwise stated, and that sets of eigenfunctions belonging to eigenvalues of multiplicity m > 1, should any occur, have been bi-orthonormalized with respect to each other. Note, however, that such normalization does not uniquely fix the amplitudes of the eigenfunctions;

scaling +i to be bigger can always be compensated by making the amplitude of 4; smaller.

A valuable observation may be made at this point, even before any solution has been attempted. The integral operators C and are closely related to potential integrals, and may be expected to share the key properties of potential integrals, including their well-known insensitivity to local detail in their integrands. It may, therefore, be anticipated that the mode shapes and propagation velocities will depend strongly on the cross-sectional shape and size of the transmission structure in question, but only weakly on the exact number and placement of conductors in the structure. This point has been borne out by numerical experiments, but will not be explored at great length in the present paper.

Iv . POYNTINGS VECTOR AND WAVE IMPEDANCE

At any cross-sectional plane through the transmission struc- ture, the Poynting vector is popularly interpreted as a measure of power flow density. It is given by the usual expression

N ( z , y) = E x H . (26)

By the E M hypothesis, both E and H are purely transverse so the Poynting vector is everywhere purely z-directed. The electric field E is given by

E ( z , y) = -VtV (27)

where Vt is understood to represent the transverse (q) gradient operator, and

N ( z , 9) = -vtv x H . (28)

Using standard vector identities, and noting that H and D are both purely transverse, the Poynting vector may be converted into the simple form

N ( z , y) = V J . (29)

Let dS = 1, dS represent an area element of the structure cross section (here, as elsewhere in this paper, the bold numeral l,, embellished by a subscript, represents a unit vector in the direction identified by its subscript). The Poynting power flow is then

W = V J . 1 , dS = V(Z, ~ ) J ( z , y) dS. (30) J' J' The power is a suitable inner product for any function space spanned by the potential and current density functions, for it satisfies all the usual conditions that an inner product must satisfy [8]. The conventionally abbreviated notation for inner products

(4, 4 = J'$@> y)'d'(z, Y) dS (31)

will, therefore, be used freely in the following. A propagating wave comprises both potentials and currents,

and their relationship needs to be established next. Suppose that a traveling wave contains a current density distribution J ( z , y) given by just one eigenfunction +j(a, y)

(32) 4 . 1 Y, 2 , t ) = Jj'd'j(Z, Y ) f ( Z F C j t ) .

SILVESTER AND CHAMLIAN: TEM MODES IN PAWLEL-LINE TRANSMISSION STRUCTURES 387

On substituting J , as given by (32), into the first telegrapher’s equation ( 12)

- = f c j J j f ‘ ( z ~ j t ) C $ j ( ~ , y) av az (33)

because the inductance operator C acts only on the transverse coordinates. Since the eigenfunctions $k of the potential wave equation constitute a basis for all possible potential distributions, the propagating potential distribution that accom- panies the current density J ( z , y, z , t) may be written as an eigenfunction expansion

To determine the unknown functions hk(z with respect to z

Ckt), differentiate

(35)

then combine (33) with (33, so as to eliminate d V / a z . There results

To complete the argument, take inner products with $i on both sides of (37)

The two sets of eigenfunctions { & } and { $ k } are bi- orthogonal with respect to the power inner product, so that only the term for z = j = k can survive in (38). Consequently, the relationship of potentials to currents is simply

(39)

In plain words, this argument says: A current density distribu- tion describable by a single eigenfunction $k may propagate in either the +z or -z direction, provided it propagates with velocity C k , and provided it is accompanied by a potential distribution that propagates with the same velocity and has a spatial distribution describable by &. The ratio of amplitudes of the two distributions is given by (39), with the positive sign taken if propagation takes place in the positive z direction.

A similar development exactly dual to the above may be carried out beginning with the potential distribution. A

unimodal potential distribution is assumed to propagate

On substituting (40) and simplifying, the second telegrapher’s equation (1 3) becomes

The propagating current density J ( z , y, z , t) that accom- panies this potential may be written as an eigenfunction expansion in the t,bk

Differentiating longitudinally, then combining with (41) so as to eliminate 8Jld.z

(43)

Following arguments similar to the above, bi-orthogonality finally requires that

Clearly, the ratio of potential coefficient v k to current density coefficient Jk must be the same no matter whether calculated by (39) or (44). Multiplying these two equations, a stationary expression for the eigenvalue is quickly obtained.

Alternately, on dividing (39) by (44)

It is tempting to view vk/Jk as a modal characteristic impedance 2,. This viewpoint is valid, subject to a note of caution: The scaling of $k and (bk is arbitrary, so absolute impedance values have no objective meaning (though relative values may be useful). Brews [9] has discussed this point in extensive detail. On the other hand, the velocity c; is normalization-independent. If $k is rescaled to have an amplitude larger by some factor K, then 4 k must be smaller by the same factor to keep the power the same; c; therefore remains invariant. The ratio = Vk / Jk, on the other hand, grows by K 2 under such a rescaling.

It should be noted that the functions & and $k are dimen- sionless mathematical functions, but that the operator C has units of henriedmeter because it contains the permeability p. Similarly, the operator has units of faraddmeter. The ratio VklJk, therefore, has the expected dimensions of ohms.

388 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 37, NO. 3, AUGUST 1995

V. A PRACTICAL S m u m where the matrices are given by Fig. 1 depicts a situation not infrequently encountered in

electronics, both at the chip and the board level: A group of numerous parallel wires occupies a channel of predetermined

paper suggests that the natural propagation modes of such

pmn J J G . ( ~ ; <)pm(t)pn(x) d < d x (54)

width on a dielectric substrate. The theory presented in this Lmn = / ' / ' G M ( ~ ; t )pm( t )pn(x ) d t d x . (55)

a structure are substantially independent of the number and exact placement of the conductors, but depend mainly on the parameters of the transmission channel-structure width, substrate thickness, dielectric permittivity. In the balance of this paper, the modal spectrum of one such structure will be described, and a simple application to crosstalk analysis will be described briefly.

The computational problem of modal analysis is that of finding the sets of eigenfunctions that satisfy the pair of eigenvalue problems (23)-(24). It may be solved by well- established projective approximation methods [lo]. To do so, a convenient basis {pi(x) li = 1 , . . . , m} is selected for the '$k and &. In the present example, all conductors are very thin so that density variations in the y direction may be neglected, and detailed modeling in the x direction alone is sufficient. Piecewise polynomials of x were used to compute the examples shown below, though other functions could be employed. An approximation in terms of these basis functions is written

(47) m

m

Then

and

The eigenvalue problem, therefore, takes the form

or

m J

m J

If this is to hold true for all k and m, then inner products taken with any pn(x) must also yield equality. Forming inner products with every pn(x), in turn, (52) directly leads to the conventional matrix eigenvalue problem

As stated here, this is an infinite eigenvalue problem; to compute actual values it must be truncated at some convenient point, i.e., a finite-basis set {pi(x)(i = 1, . . . , N } must be used instead of {pi(x)li = 1 , . . . , m}.

The electric Green's function GE(z ; I) for a substrate of thickness h is obtained by the method of partial images [ 113, [12]. Its simplest possible form that isolates the characteristic logarithmic singularity is

k 1 - Er

k=l ( I C + 1)2 + c2

log k2 + < 2

where 5 = (x - <)/h. This form can be made very stable numerically, for reasons evident on brief physical reflection. Each term of the infinite series in (56) corresponds to a source- image pair, whose contribution to the total potential is that of a dipole, hence falls off asymptotically as the logarithm of distance. A numerical computation that calculates the value of each dipolar term log{(k + 1)2 + c 2 } / { k 2 + c2} and adds dipoles to form the Green's function, is highly stable; a computation that adds terms due to individual charges, e.g., log{(k + 1)2 + c2} to -log{k2 + S2}, is not. For large or large k , the individual logarithmic terms are very similar but oppositely signed, so addition causes a serious loss of significant figures.

The magnetic Green's function G M ( ~ ; E ) is easier to obtain, so long as the substrate is taken to be nonmagnetic, p = PO. Its form coincides with that of G.(x; E ) when E, = 1; its magnitude is cop0 times larger.

since C ~ E O ~ O = 1. An easy way of actually computing G M ( z ; <) is to use the program for G.(x; <) with E , set to unity.

The computation of the projection integrals Pmn and Lm, was carried out using the methods given by Silvester and Chamlian 1131 and will not be discussed further, here.

For a structure like that of Fig. 1, with substrate permittivity E = 960 and a width 10.0 times the substrate thickness, the first six modal function pairs ?/lk - are shown in Fig. 2. The potential distribution functions &, shown in the right column in Fig. 2, generally show the behavior one might expect-the potentials are continuous, bounded, and display the usual "k wiggles in the kth function" that characterize most families of orthogonal functions. The current distribution functions (left

SILS’ESTER AND CHAMLIAN: TEM MODES IN PARALLEL-LINE TRANSMISSION STRUCTURES 389

Fig. 2. flat transmission structure, ten substrate-thicknesses wide.

The first six current (left) and voltage (right) modal functions for a

column of Fig. 2) do likewise, in fact, they resemble the potential distribution functions, but they show a singularity at the structure edge. This certainly agrees with the well- known singular behavior of current and charge densities at sharp edges; the interesting point perhaps is that all modes show such singular behavior, not just a few dominant modes.

Table I shows the distribution of propagation velocities for the structure described, normalized to the free-space velocity of light CO. On physical grounds, all modal velocities must lie between the velocity of a wave traveling in air alone, and the velocity of a wave traveling in the substrate material alone. With E, = 9, these limits lie at co and c0/3. The first

TABLE I NORMALIZED PROPAGATION VELOCITIES AND VELOCITY DIFFERENCES

FOR THE FIRST 20 MODES OF THE STRUCTURE OF FIG. 1

k

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

~

‘k

0.35747 0.38684 0.40824 0.42318 0.43277 0.43873 0.44229 0.44439 0.44560 0.44630 0.44670 0.44692 0.44705 0.4471 2 0.44716 0.4471 8 0.44720 0.44720 0.4472 1 0.44721

‘k - ‘k-1

0.35747 0.02937 0.02140 0.01494 0.00958 0.00597 0.00356 0.00210 0.00121 0.00070 0.00040 0.00023 0.00013 0.00007 0.00004 0.00002 0.00001 0.00001 0.00000 0.00000

mode clearly carries a large proportion of its field energy in the substrate material, for its velocity is only a little higher than c0/3. Higher modes transport a larger proportion of their field energy in the air, but this proportion quickly stabilizes with increasing mode number, with velocity values after the fifteenth identical to four or more significant figures, as shown in the third column of the table. The velocity when all conductors are connected to the same source, i.e., when the structure is electrically made into a single flat strip, is 0 . 3 5 9 7 8 9 ~ . This is only a little higher than the velocity of the slowest (dominant) mode. Clearly, when the conductors are joined to make a single strip, the great majority of energy is transported by the first mode.

VI. A SIMPLE CROSSTALK PROBLEM

Use of the modal theory will now be illustrated by a simple example. Suppose that the structure of Fig. 1 is semi-infinite in the direction of propagation z , and that it comprises a large number N of similar and equi-spaced conductors, with conductor m occupying x,-~ < x < x,. Suppose the conductors are driven by N independent voltage sources, one for each conductor. The crosstalk problem may, for present purposes, be stated as follows: If all the sources are held at zero except for the source feeding conductor m that generates a short square pulse p ( t ) , what voltage appears at time t at position z on conductor n?

Let w(x, z , t) represent the potential distribution on all the conductors. At the sending end, the voltage sources force it to have a value closely approximated by

where p ( t) represents a pulse of short finite duration and unity amplitude. No reflections can exist on a semi-infinite structure,

so energy can only propagate in the +z direction. Hence, the potential distribution may be written

The unknown coefficients v k are quickly determined by form- ing inner products with ‘$j on both sides. Bi-orthogonality implies that only the terms k = j survive, so that

This integral is easily evaluated. If conductor m is very narrow, then to a very good approximation

where Z, is the midpoint of conductor m and A , is the space it occupies, A , = x,-1 < x < x,. The voltage on conductor n, i.e., the value of the series in (59) at x = C,, is then

This represents a group of right-traveling pulses p ( z - C k t ) , moving with velocity c k and having an amplitude ‘ $ k ( Z m ) $ k ( Z , ) - Fig. 3 shows a typical result in a structure that extends over -5h <_ x <_ +5h, for the choice x, = 0.80h, x, = 3.55h (h is the substrate thickness): a train of pulses of varying amplitude and slightly varying width as observed at some point z >O. The time scale in Fig. 3 is normalized to the velocity of light in air, so one time unit is the time at which a pulse traveling in free air would arrive at z ; as a result of this normalization, the actual value of z is unimportant. At time 2.3 a substantial number of small pulses are crowded together, as the fast-traveling high-order modes arrive. Subsequently, longer times elapse between pulses carried in the slower low-order modes. At time 2.8, the lowest-order mode arrives and the process is complete. Each distinct pulse represents a packet of energy traveling in a different mode, hence, at a different velocity, arriving at a different time. What happens is really quite simple: The original pulse is spatially decomposed into the orthogonal modes of the transmission structure and the individual modal components are propagated at different speeds, so they arrive at different times. It must be emphasized that none of these pulses results from reflections; the lines are infinite, so there are no reflections.

Brief examination of equation (62) shows that the con- tributions of two modes can be minimized, or even totally annihilated, by selecting m and n so that Zm and Z, coincide with zeros of +k and $ k . Driving several lines from the same voltage source can suppress an even larger number of modes. This suggests that minimum-crosstalk positions can be located in a structure even if the total number of conductors and their

-1 2.2 2.5 2.8

normalized time

Fig. 3. Voltage distribution as a function of time on conductor at I = 0.80h when conductor at I = 3.55h is excited by a short pulse at t = 0. Time units are normalized so a pulse traveling at the free-space velocity of light would arrive at t = 1 .

exact placement are unknown. Applications of this principle will be reported on in greater detail elsewhere.

VII. CONCLUSION

Electromagnetic energy propagates along any multiwire structure in a countably infinite set of normal modes, each mode having a distinct velocity of its own. Each mode com- prises a characteristic current distribution and a characteristic potential distribution; these distributions are bi-orthogonal with respect to power, i.e., no power can be shared between any two modes. The mode functions and velocities depend only on the total conductive material in the structure, not on how conductors are placed in a transmission pathway, cable, or channel. Therefore, they characterize the channel, not the individual wires in it. However, the transmission properties (such as crosstalk sensitivity) of any individual conductor are strongly dependent on the placement of the conductor, and on its relationship to the spectrum of modal functions. This point will be explored in greater detail in a future work.

REFERENCES

[ l ] W. Thomson, “On peristaltic induction by electric currents,” in Proc. Roy. SOC. Lon., vol. 8, pp. 121-132, 1856 (reprinted in: Lord Kelvin (Sir W. Thomson), Mathematical and Physical Papers, vol. 2. Cambridge: Cambridge Univ. Press, 1884, pp. 79-91).

[2] G. Mie: “Elektrische Wellen an zwei parallelen Drtihten,” Annalen der Physik, 4. Folge, vol. 2, pp. 201-249, 1900.

[3] L. A. Pipes, “Matrix theory of multiconductor transmission lines,” Philosophical Mag., vol. 24, pp. 97-1 13, 1937.

[4] K. D. Marx, “Propagation modes, equivalent circuits, and characteristic terminations for multiconductor transmission lines with inhomogeneous dielectrics,” ZEEE Trans. Microwave Theory Tech., vol. h4TT-21, pp. 450457, 1973.

[5] S. Frankel, Multiconductor Transmission Line Analysis. Norwood, MA: Artech House, 1977.

[6] P. I. Kuznetsov and R. L. Stratonovich, The Propagation of Electromag- netic Waves in Multiconductor Transmission Lines, F. F. Kelleher, trans. Oxford: Macmillan, 1964.

[7] J. A. Brandao Faria, Multiconductor Transmission-Line Structures. Modal Analysis Techniques.

[8] I. Stakgold, Boundary Value Problems of Mathematical Physics, Vol. 1 . New York Macmillan, 1967.

New York Wiley, 1994.

SILVESTER AND CHAMLIAN: TEM MODES IN PAWLEL-LINE TRANSMISSION STRUCTURES 39 1

[9] J. R. Brews, “Characteristic impedance of microstrip lines,” IEEE Trans. Microwave Theory Tech., vol. MlT-35, pp. 30-34, 1987.

[ 101 P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 2nd ed. Cambridge, UK: Cambridge Univ. Press, 1990.

[ 111 P. Silvester, “TEM wave properties of microstrip transmission lines,” in Proc. ZEE, vol. 115, no. 1, pp. 43-48, 1968.

[12] A. Bryant and J. Weiss, “Parameters of microstrip transmission lines and of coupled pairs of microstrip lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-16, pp. 1021-1027, 1968.

[I31 P. P. Silvester and S. V. Chamlian, “Symbolic generation of finite elements for skin-effect integral equations,” IEEE Trans. Magn., vol. 30, pp. 3594-3597, 1994.

Peter P. Silvester (S’60-M’64-SM’SO-F’83) re- ceived the B.S. degree in electrical engineering from the Camegie Institute of Technology, in 1956, the M.A.Sc. degree from the University of Toronto, in 1958, and the Ph.D. degree from McGill University, in 1964.

Since then, he has pursued an academic career, principally at McGill University, where he now holds the rank of professor. A Visiting Fellow Com- moner at Trinity College, Cambridge, in 198&1989, he has also held research fellowships or visiting

professorships at the General Electric Corporate Research and Development Center, Imperial College of Science and Technology, London, Institut National Polytechnique de Grenoble, Universita degli Studi di Firenze, and other institutions. He has concentrated on the numerical analysis of electromagnetic field problems and has been instrumental in introducing the finite element method to electrical engineering.

Dr. Silvester is the author, coauthor, or editor of numerous books and scientific papers on numerical electromagnetics and computer engineering. He is a Fellow of the Royal Society of Canada and the Institution of Electrical Engineers.

Simon V. Chamlian received the B.E. degree from h o l e Polytechmque de Montr6al, in 1990, followed by a M.S. degree in electrical engineering from McGill University, in 1992, and is currently working toward the Ph.D. degree at McGill University.

His research focuses on the numerical analysis of electromagnetic fields. He holds a teaching positlon as an adjunct professor at h o l e Polytechnique de Montrdal, since 1990, and has also trained profes- sionals in his field.

Mr. Chamlian is the coauthor of a teaching text- book, as well as a book on circuit simulations.