Solution Of Transmission-line Equations Research Articles

Crosstalk and Mode Conversion in Adjacent Differential Lines

In this paper, the concepts of weak coupling and weak imbalance are exploited to investigate crosstalk and mode conversion occurring in perfectly balanced differential lines (DLs) due to proximity with other DLs printed on the same PCB substrate. Approximate modal circuits are derived, where the aforementioned phenomena are accounted for by the use of distributed voltage and current sources. This allows evidencing the relevant parameters at the basis of these undesired phenomena, which may significantly degrade DL performance in terms of emission and immunity properties as well as signal integrity. Accuracy of the proposed model, preliminary validated versus exact solution of transmission line equations, is eventually assessed through vector network analyzer measurements carried out on a PCB sample.

An improved modeling of TDR signal propagation for measuring complex dielectric permittivity

Time domain reflectometry (TDR) is a measurement technique based upon transmission line theory. The solutions of transmission line equations are reformulated in terms of independent physical properties, instead of coupled per-unit-length circuit parameters. The complete TDR response is effectively modeled by a non-uniform transmission line using the non-recursive ABCD matrix approach. Approaches to calibrate line parameters and perform TDR measurements based upon such model are introduced with an example on dielectric spectroscopy. TDR modeling in terms of decoupled physical parameters and non-recursive algorithm allows more convenient calibration of line parameters and facilitates interpretation of TDR measurements.

Modeling of Crosstalk Effects in Multiwall Carbon Nanotube Interconnects

Crosstalk effects in multiwall carbon nanotube (MWCNT) interconnects for future ICs are investigated by virtue of the equivalent single conductor model and the finite-difference time-domain solution of transmission line equations. The worst case time delay and the peak crosstalk voltage on victim wire of multiwire MWCNT interconnect configurations are derived and compared to those of the copper (Cu) wire counterparts for the intermediate and global interconnects at the 22- and 14-nm technology nodes. The numerical results illustrate that the crosstalk-induced time delays in the MWCNT interconnects are much smaller than those in the Cu interconnects, in particular, for longer wires. Nevertheless, the MWCNT interconnects exhibit little improvement on the crosstalk-induced noises in comparison with their Cu counterparts.

A total-variation-diminishing finite-difference scheme for the transient response of a lossless transmission line

Total-variation-diminishing (TVD) finite-difference schemes have been used in computational fluid dynamics for accurate solutions of fluid problems involving shock phenomena. This paper investigates the possibility of their application in transient electromagnetic-wave problems. A lossless transmission line with a resistive load is considered to illustrate the application. A TVD Lax-Wendroff finite-difference scheme is presented for the numerical solution of transmission-line equations in time domain. Numerical results show that the TVD scheme can approximate the discontinuous waveforms with remarkable accuracy.

Analysis of nonuniform, frequency-dependent high-speed interconnects using numerical inversion of Laplace transform

Interconnects in high-speed VLSI circuits and systems exhibit transmission line effects. Due to the complex geometries of interconnections, coupling between various layers, and inhomogeneous insulating materials, these high-speed interconnects need to be modelled as nonuniform frequency dependent transmission lines. In this paper, we describe a method of simulating the transient response of nonuniform high-speed interconnects in its most general form, i.e. nonuniformly distributed, lossy, coupled, multiple lines with frequency-dependent parameters with linear and nonlinear terminations. Transmission line equations are formulated in the frequency domain as an initial value problem and solved using numerical integration. A new algorithm is proposed for overcoming the inherent initial value instability encountered in the solution of transmission line equations. The time domain response is obtained by Numerical Inversion of Laplace Transform (NILT). Nonlinear networks containing nonuniform high-speed interconnects are analyzed using the Piecewise Decomposition Technique. The accuracy and efficiency of the proposed method is illustrated by appropriate examples and comparisons with published results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computer-based methods for induced-voltage calculations in AC railways

Calculations of induced voltage in metallic lineside cables have traditionally been made using simple methods based upon the explicit solution of transmission line equations. These methods were developed about 40 years ago with an emphasis on lead-sheathed cables which were in continuous contact with the ground. The paper reviews the limitations of the classical approach and concludes that in-secton induced voltages are significantly overstated by this approach. An alternative computer-based approach is presented which is more appropriate for modern cables, with insulated outer sheaths, which are normally earthed by discrete earth-rod connections along the cable. This new approach, which is illustrated with reference to CAD techniques developed for autotransformer and booster-supplied AC railways, should permit a more realistic and economic approach to specifying screening requirements for copper cables. Examples of the use of the developed software for assessing induced voltages in the recently electrified 2 × 25 kV North Island Main Trunk in New Zealand are also presented.

Wave Propagation in Polyphase Transmission Lines a General Solution to Include Cases Where Ordinary Modal Theory Fails

The solution of transmission line equations is usually written as a superposition of so called natural modes of exponential type. These are obtained through the use of a suitable transformation that decouples the original sets or N simultaneous 2nd order wave equations for voltages and currents into N independent equations. For such a transformation to exist the fundamental product matrix ZY must be diagonalizable. In a previous paper it has been shown that physically realizable transmission lines are possible for which ZY is not diagonalizable and to which ordinary modal theory does not apply. In the present paper a new generalized modal theory is developed for the purpose of including non-diagonalizing cases.

Numerical Representation of Multiconductor Transmission-Line Equations by Integration along Characteristics

It is often necessary to obtain the solution of the transmission-line equations in the time domain, for example, for lines excited by transient, nonuniform electromagnetic fields (fields due to nuclear detonations). This paper presents the solution of the multiconductor transmission-line equations by representing them by integration along their characteristic directions. The multiconductor transmission-line equations are first decoupled to a set of n single-line equations and then the method of characteristics is applied to solve these equations. It is shown in this paper that the multiconductor transmission-line equations can always be decoupled into a set of n single-line equations. The transmission-line equations are stable along their characteristics in the presence of high air conductivity, whereas the finite-difference solutions become unstable.

Steady State Solutions of Transmission Line Equations

Methods of obtaining the steady state voltages and currents in a uniform transmission line consisting of several parallel wires are described in Part I. This line may or may not be acted upon by an externally impressed field distributed along its length. A square matrix Γ, which is a generalization of the propagation constant γ for a single circuit, is introduced. Matrix expressions obtained for the voltages and currents involve Γ in much the same way as the corresponding single circuit expressions involve γ. In Part II similar methods are described for obtaining the voltages and currents in a transmission line composed of a number of multi-terminal symmetrical sections connected in tandem. Expressions for the voltages and currents in a line composed of unsymmetrical sections are also given. These sections may or may not contain generators.