ABSTRACT
Multiconductor transmission lines (MTL), or multiconductor buses as they are also often called, are found in almost every electrical packaging technology and on every technology level from digital chips, over MMICs (monolithic microwave integrated circuits) to MCMs (multichip modules), boards, and backplanes. MTL are electrical conducting structures with a constant cross section (the x-, y-plane) which propagates signals in the direction perpendicular to that cross section (the z-axis) (see also Figure 16.1). Being restricted to a constant cross section we are in fact dealing with the so-called uniform MTL. The more general case using a nonuniform cross section is much more difficult to handle and constitutes a fully three-dimensional problem. It is not the purpose of this chapter to give a detailed account of the physical properties and the use of
the different types of MTL. The literature on this subject is abundant and any particular reference is bound to be both subjective and totally incomplete. Hence, we put forward only Refs. [1-3] as references here, as they contain a wealth of information and additional references. In the frequency domain, i.e., for harmonic signals, solution of Maxwell’s sourceless equations yields a
number of (evanescent and propagating) modes characterized by modal propagation factors exp (j bz) and by a modal field distribution, which depends only upon the (x, y) coordinates of the cross section. In the presence of losses and for evanescent modes b can take complex values, and jb is then replaced by g¼aþ jb (see Equation 16.3). In the propagation direction, the modal field amplitudes essentially behave as voltage and current along a transmission line. This immediately suggests that MTL should be represented on the circuit level by a set of coupled circuit transmission lines. The relationship between the typical circuit quantities, such as voltages, currents, coupling impedances, and signal velocities, on the one hand, and the original field quantities (modal fields and modal propagation factors) is not straightforward [4]. In general, the circuit quantities will be frequency dependent. The frequency domain circuit model parameters can be used as the starting point for time domain analysis of
networks, including multiconductor lines. This is, again, a vast research topic with important technical applications. Section 16.2 describes the circuit modeling in the frequency domain of uniform MTL based on the
Telegrapher’s equations. The meaning of the voltages and currents in these equations is explained both at lower frequencies in which the quasi transverse electromagnetic (TEM) approach is valid as well as in the so-called full-wave regime valid for any frequency. The notions TEM, quasi-TEM, and full wave are elucidated. We introduce the capacitance, inductance, resistance, and conductance matrices together with the characteristic impedance matrix of the coupled transmission line model. Finally, for some simple MTL configurations analytical formulas are presented expressing the previous quantities and the propagation factors as a function of the geometric and electrical parameters of these configurations. It would be a formidable task to give a comprehensive overview of all the methods that are actually
used for the time domain analysis of MTL. In the remaining part of this paragraph a very short overview (both for uniform and nonuniform structures) is presented along with some references. In the case of linear loads and drivers, frequency domain methods in combination with (fast) Fourier transform techniques are certainly most effective [5-7]. In the presence of nonlinear loads and drivers other approaches must be used. Simulations based on harmonic balance techniques [8,9] are, again, mainly frequency domain methods. All signals are approximated by a finite sum of harmonics and the nonlinear loads and drivers are taken into account by converting their time domain behavior to the frequency domain. Kirchhoff laws are then imposed for each harmonic in an iterative way. Harmonic balance techniques are not very well suited for transient analysis or in the presence of strong nonlinearities, but are excellent for mixers, amplifiers, filters, etc. Many recent efforts were directed toward the development of time domain simulation methods (for both uniform and nonuniform interconnection structures) based on advanced convolution-type approaches. It is, of course, impossible to picture all the ramifications in this research field. We refer the reader to a recent special issue of IEEE Circuits and Systems Transactions [10], to the ‘‘Simulation techniques for passive devices and structures’’ section of a special issue of IEEE Microwave Theory and Techniques Transactions [11], and to a 1994 special issue of the Analog Integrated Circuits and Signal Processing Journal [12] and to the wealth of references therein. Both frequency and time domain experimental characterization techniques for uniform and nonuni-
form multiconductor structures can be found in Chapter 17.
