Multiconductor Transmission-line Theory in the TEM Approximation

Starting with Maxwell's equations, the transmission line equations are derived for a system consisting of an arbitrary number of conductors. The derivation is rigorous for long lossless conductors embedded in a uniform perfect dielectric. The presentation is essentially tutorial, most of the results being well known, at least for twoand three-conductor systems. The novelty lies in the point of view adopted in obtaining a systematic generalization to the case of an arbitrary number of conductors. Explicit expressions are obtained for the electric and magnetic fields in the dielectric surrounding the conductors, and a rigorous formulation is given for the problem of calculating the coefficients of capacitance and inductance. Introduction Interest in the theory of mutually coupled, multiple conductor, parallel transmission lines extends over the past forty years. With the exception of early works by Levin [ 11 and Pipes [2,3], and more recent investigations by Kuznetsov and Stratonovich [4], by Amemiya [5], and by Matsumoto [lo], most of the published work has focused on the theory of two parallel, mutually coupled transmission lines with considerable attention given to the subject of directional coupling [6,9]. Papers treating arbitrary numbers of conductors fall into two groups. Papers of the first group [2,5,10] take the generalized transmission line equations as a starting point and assume the inductance and capacitance matrices given. Papers of the second group [ 1,3,4] take Maxwell's equations for the electromagnetic field as a starting point. This paper belongs to the second group. The purpose of the present paper is essentially tutorial. Most of the results arrived at are well-known, at least for twoand three-conductor systems, and can be found in the papers cited above as well as in the widely known texts of King [ 11 ] and Collin [ 121. It is hoped, however, that some novelty will be found in the point of view adopted here to obtain a systematic generalization of familiar results to systems of arbitrary numbers of conductors. There are two motivations for such an investigation. The first of these has to do with the generalization of the familiar two-conductor result, LC = l/d2, to multicon604 ductor systems. In the multiconductor case L and C are matrices of inductances and capacitances per unit length, respectively; u is the velocity of propagation, and the unity element is a unit matrix of the same dimensions as L and C. This generalization has been applied explicitly to multiconductor TEM propagations by several authors [5,10,13] and is implicit in the work of several others [6-91. In the usual justification, stated quite nicely by Amemiya [5], it is shown that the solution of the generalized transmission line equations for an N + 1 conductor system consists of the superposition of N independent modes whose propagation velocities are the square roots of the reciprocals of the eigenvalues of the product of the L and C matrices. Then an ad hoc assumption is made that for TEM systems, all these propagation velocities must be equal. It follows that the product of the L and C matrices must be l/dL times a unit matrix. I t will be shown that to assume constant propagation velocities is unnecessary; it can instead be derived rigorously from Maxwell's equations. Similarly, it will be shown to be a strict consequence of Maxwell's equations that the product of the L and C matrices is l/u2 times a unit matrix for TEM systems. The main results are contained in Eqs. (8) and (69) (7 1 ). The second motivation for the investigation is to find an equation for the electric and magnetic fields produced by a given distribution of currents and voltages on a multiconductor TEM transmission line system. The result is embodied in Eqs. ( 5 ) and (6). A typical application of this equation would be to estimate the stray W. T. WEEKS IBM J. RES. DEVELOP magnetic field at an unselected bit position in a thin magnetic film memory when several adjacent lines are pulsed. Granted that a memory plane is not a TEM system, the TEM solution nevertheless serves as a useful starting point for the computation. The subsequent discussion is limited to propagation on lossless systems in a uniform dielectric; in short, to TEM systems. By making such a restriction it is possible to derive results which apply to a wide variety of conductor geometries. To include losses or multilayer dielectrics would soon force the analysis to a discussion of a single system with fixed geometry. By way of justification it might be added that practical analysis of real systems is based on TEM assumptions. Also a thorough understanding of the properties of lossless systems certainly is a prerequisite for the understanding of the more complicated systems encountered in practice. TEM solutions of Maxwell’s equations The specific configuration to be considered in this paper consists of N + 1 parallel, infinitely long, lossless conductors. The cross sections of these conductors may vary from conductor to conductor, but the cross section of any given conductor must be uniform over its entire length. It is supposed that the space surrounding the conductors is filled with a uniform perfect insulator of constant permittivity E and permeability p. The system is to be operated so that the sum of the currents in the N + 1 conductors is zero at all times. In this case, any one of the conductors can be selected as the common reference conductor for the system. In particular, let the conductor chosen as the reference conductor be called the N + 1st conductor and the remaining conductors be numbered from 1 to N , in any order. Throughout this paper, the common reference conductor will be referred to as the N + 1st conductor. All voltages will be measured with respect to this conductor. Since the space outside the conductors is assumed to be a perfect insulator, there can be no flow of charge into or out of this region. If this space is uncharged initially, it remains uncharged. Hence, it can be assumed that both the conduction current density and charge density vanish outside the conductors. Thus, in the space outside the conductors, the electric and magnetic fields satisfy Maxwell’s equations: V X H = E ; aE at The boundary conditions at the surfaces of the conductors are that the tangential component of E and the normal component of B = pH must be continuous across the conductor surfaces. Since the conductors are assumed lossless, both E and H vanish in their interiors. Thus, the boundary conditions reduce to the requirement that the tangential component of E and the normal component of H vanish at all conductor surfaces. The N + 1 conductor configuration just described is capable of sustaining a TEM field, that is, a field in which the E and H vectors lie in planes perpendicular to the direction of propagation. It is the TEM solutions of Maxwell’s equations that are of particular interest in transmission line theory. If one chooses a rectangular coordinate system, with the z axis parallel to the conductors and the x and y axes perpendicular to the conductors, and looks for solutions of Maxwell’s equations representing wave propagation in a direction parallel to the conductors, then the requirement for a TEM solution is that the z component of E and H vanish everywhere in the space outside the conductors. The primary objective of this paper is to demonstrate that, for the N + 1 conductor transmission line system described, such a TEM solution of Maxwell’s equations exists in the form E =-x Vi(z , t )V$Ji(x ,y) ; ( 5 ) H =-x x Lijfj( , ) z t ) k X V$Ji(x,y) ; ( 6 ) where k is a unit vector parallel to and directed along the positive z axis, the z axis being parallel to the conductors (Fig. I ) . The N functions Vi(z,t) appearing in Eq. (5) are assumed to be linearly independent, at least once differentiable, functions of z and t as are the N functions I , ( z , t ) appearing in Eq. ( 6 ) . Both the V , ( z , t ) and I , ( z , t ) are assumed to be independent of the variables x and y . The N functions I#J~(x,Y) appearing in Eqs. ( 5 ) and (6) are assumed to be at least twice differentiable functions of x and y but independent of the variables z and t. The N 2 quantities L i j in Eq. (6) are constants. The following five requirements are to be placed on the fields given by Eqs. ( 5 ) and (6). I . The tangential component of E and normal component 2. The function Vi(z,t) of Eq. ( 5 ) is the voltage of the ith conductor, with respect to the N + I st (reference) conductor. 3. The function I i ( z , t ) of Eq. ( 6 ) is the current carried by the ith conductor. 4. The sum of the currents carried by all N + 1 conductors must vanish for all times t and positions z , and v