Multiconductor Transmission Line Equations Research Articles

Transmission Line Model for a Shielded TWP With Apertures and Generalization to Braided–Shielded TWP Cables

An efficient multiconductor transmission line (MTL) model of a shielded twisted wire pair (TWP) with apertures on its sheath is developed for predicting common mode (CM) and differential mode (DM) currents induced in the shielded TWP cable. Its transfer impedance and admittance with one-line apertures on its sheath are derived in a closed form. Therefore, the MTL equations are formulated and solved to obtain the CM and DM currents induced in the shielded TWP. Moreover, the proposed MTL model is generalized to be applicable for the braided–shielded TWP, in which the transfer impedance and admittance are formulated analytically. Results showed that the induced DM current inside the braided–shielded TWP is suppressed dramatically due to the uniform structure of the cable. The models are validated by comparing their results with those obtained by the commercial software FEKO; good agreements are obtained.

The impact of different corona models on FD algorithms for the solution of multiconductor transmission lines equations

In this study, the implementation of different corona discharge models in a transient program for overhead multiconductor lines based on the implicit Crank-Nicolson Finite Difference Time Domain method is presented. Among different corona models available in the literature, two empirical models (Gary's and Suliciu's models) and a physics-based model (Malik's model) are discussed, which are broadly used in EMT programs, and their predictions under fast-front lightning surges and slow-front switching impulses are assessed. The authors critically review the dynamic capacitance approach in terms of numerical accuracy, when dealing with slowly varying voltages, and applicability, regarding dispersive models assuming a non-instantaneous charge-voltage relation. The alternative voltage-controlled generator approach is proposed, which is successfully included in the implicit Crank-Nicolson scheme. One of the objectives of this study is to show how different corona models may predict overvoltages with huge discrepancies, depending on the transient waveform and the propagation distance, despite the similarity of the q-v hysteresis curves. More evident discrepancies between models under fast-front voltage waves are reported, rather than for switching impulses. The consistency of these results in terms of the models' features and q-v curves is analysed, highlighting the necessity for new engineering tools for corona simulation of general predictive capability.

An efficient time-domain method for the electromagnetic transient response of multiconductor transmission lines excited by an electromagnetic field

An efficient time-domain method based on the waveform relaxation with transverse partitioning (WR-TP) techniques and the delay extraction passive compact circuit (DEPACT) macromodel is presented for evaluating the electromagnetic (EM) transient response of multiconductor transmission lines (MTLs) excited by an external EM field. Firstly, the equivalent distributed sources due to the tangential components of the electric field are introduced into the MTLs equations. Then, the TLs are divided into a cascaded of DEPACT subnetworks, where the telegrapher’s equations of lossless TLs are decoupled by applying the WR-TP techniques. And the relaxation sources over the lossy TLs are updated by using the waveform scaling algorithm, which can completely avoid performing the numerical convolution. Finally, the equivalent circuit is established and the transient responses induced on the lines can be obtained by using the circuit simulation software PSCAD.

Time-Domain Modeling of Field-to-Wire Coupling in Obliquely Oriented Multiwire Cables With Junctions Using JEMS-FDTD

Multiwire cable bundles located inside electrically large objects play a crucial role in ensuring their electromagnetic safety. The cable bundles are seldom parallel to the Cartesian coordinates and, hence, they are difficult to handle by using the finite-difference time-domain (FDTD) algorithm. Our motivation in this article is to develop an efficient time-domain field-to-wire coupling model for obliquely oriented multiwire cables with junctions, based on the multiconductor transmission line theory. The multiconductor transmission line equations are discretized by using the 1-D FDTD method. The distributed sources on the cables are obtained by adopting an interpolation technique. The junctions are handled by making use of three theorems. The per-unit-length parameters are extracted by solving an electric field integral equation using the 2-D method of moments. The proposed model has been incorporated into the 3-D full-wave in-house solver J Electromagnetic Solver-FDTD (JEMS-FDTD) to efficiently model the induced effects on the cable bundles located inside electrically large objects illuminated by an external electromagnetic field. To validate the accuracy of the algorithm, the induced currents on a five-wire cable bundle are derived and compared with those obtained from a commercial software. Additionally, the proposed methodology is applied to simulate the induced currents on a seven-wire cable bundle inside a vehicle illuminated by a plane wave.

A time-domain method for the electromagnetic transient response of multiconductor transmission lines excited by the electromagnetic pulse

An efficient field-to-line coupling method in the time-domain is developed based on the waveform relaxation (WR) techniques for evaluating the electromagnetic(EM) transient response of multiconductor transmission lines (MTLs) excited by an electromagnetic pulse(EMP). Firstly, the incident EMP electric fields on the lines are calculated as equivalent distributed sources introduced into the MTLs equations. Then, the telegraphs equations of lossless transmission lines are decoupled by applying the WR techniques and different types of equivalent current sources at the terminal of MTLs are calculated using the iterative convolution and analytical method separately. Finally, the equivalent circuit is established and the transient currents/voltages induced on the lines are obtained by using the circuit simulation software PSCAD. The proposed time-domain method has the advantages of accuracy and high efficiency compared with the method based on finite-difference time-domain (FDTD) via two numerical examples.

Research and application of the FETD method for the EM transient response of MTLs excited by HEMP

An efficient field-to-line coupling model in the time domain is developed combining the second-order decoupled multiconductor transmission lines (MTLs) equations and the finite-element time-domain (FETD) method for evaluating the electromagnetic (EM) transient response of MTLs over a lossy ground excited by a high-altitude EM pulse (HEMP). First, the incident HEMP electric fields on the lines are calculated as equivalent distributed sources introduced into the MTLs equations. Then, the discrete FETD equations for the lines against the HEMP, in which the effect of frequency-dependent parameters caused by the ground impedance is considered, are derived using the FETD method with current as the single iteration variable. Finally, the transient currents induced on the lines are obtained by assembling all the discrete FETD equations into the global FETD equation, and the transient induced voltages on the lines are calculated by discretising the coupled equations with respect to current and voltage. The proposed model is successfully compared with the finite-difference time-domain method via several examples. It is then employed to analyse the effect of various parameters on the transient currents including the polarisation and the incident angles of the HEMP, the conductivity, and the relative permittivity of the soil.

A Hybrid Perturbative-Stochastic Galerkin Method for the Variability Analysis of Nonuniform Transmission Lines

In this paper, a hybridization of the classical stochastic Galerkin method (SGM) with two perturbative solution techniques is proposed to speed up the statistical analysis of nonuniform multiconductor transmission line (MTL) structures with parameters affected by uncertainty. The first method leverages a recently developed deterministic perturbation technique (PT) to deal with nonuniformity affecting the SGM-augmented MTL equations. This approach is proven to be computationally more efficient than the traditional solution based on line subdivision into uniform cascaded sections, yet its performance is still affected by the so-called “curse of dimensionality.” To further mitigate this issue, a second method is proposed, which resorts to the solution of uncoupled MTLs having the same size as the original structure, and where the effects of both nonuniformity and stochasticity are iteratively included by means of distributed sources. The accuracy and computational efficiency of the proposed approaches are assessed based on the statistical prediction of the mixed-mode S-parameters of microstrip-line structures with different numbers of random parameters. The test cases demonstrate that the hybrid SGM-PT approach is applicable to problems with a few tens of random variables, which is an unprecedented result for state-of-the-art SGM implementations.

Accelerated frequency-dependent method of characteristics for the simulation of multiconductor transmission lines in the time domain

Simulation of transients in transmission lines can be time consuming and resource demanding when performed directly in the time domain using small time steps. This paper proposes an efficient implementation of the frequency-dependent method of characteristics. The proposed implementation exploits the circuit topology of the system to reduce the number of state equations. State-space matrices are then grouped and sparsity techniques are used to solve the system of the ordinary differential equations faster. We post-process the calculated state variables to reduce the memory usage and the number of memory accesses needed in each iteration and propose a technique to correct the error due to the time discretization of the travel time along the transmission line. To decouple the multiconductor transmission line equations, the real part of the modal transformation matrix at a fix frequency is considered. Presented results demonstrate that the proposed approach is accurate for overhead transmission lines, and significantly reduces the overall computation time and memory consumption of the model.

Artificial Material Single-Layer Method Applied to Model the Electromagnetic Field Propagation Through Anisotropic Shields

This paper deals with the extension of the artificial material single-layer (AMSL) method, recently developed to model electromagnetically a thin conductive material using the finite-element method (FEM), to the more general case of transversally anisotropic shields. The analogy between the field equations and the multiconductor transmission line (MTL) equations is here used to calculate the admittance matrix of a thin anisotropic material. This admittance matrix is then imposed to be that of an equivalent circuit with lumped parameters. Thus, it is possible to synthetize the AMSL tensors containing the specific constants to be used in commercial software tools. The adoption of the AMSL method in FEM simulations avoids a fine discretization inside the thin conductive anisotropic material required at high frequency. Simple tests are finally carried out to validate the proposed method.

Frequency-Dependent Multiconductor Transmission Line Model With Collocated Voltage and Current Propagation

This paper reviews the classical multiconductor transmission line (MTL) equations and proposes additional constraints on these equations. A fundamental physical constraint is that the voltage and current waves must be collocated and travel together with the same propagation function. Based on this condition, the Revised Multiconductor Transmission Line (RMTL) equations are proposed. As opposed to the classical MTL equations that require complex frequency-dependent transformation matrices for their diagonalization, the RMTL equations can be diagonalized very accurately using a single real constant transformation matrix. A new Frequency-Dependent Line Model (FDLM) is proposed based on the RMTL equations. FDLM is compared with the two most accepted frequency-dependent line models in the Electromagnetic Transients Program (EMTP): The JMARTI model (fdLine) that uses a constant transformation matrix as an approximation, and the phase-coordinates Universal Line Model (ULM) that fits the frequency dependence of the transformation matrices. These time-domain models are compared with a reference frequency-domain solution for a double-circuit vertical line.

A Method for Acquiring the Terminal Common Mode Impedance of Cables

A method is presented for quickly acquiring the equivalent terminal common-mode impedance of cables. Firstly, the parallel cables are treated as equivalent multi-conductor transmission line. Using vector network analyzer (VNA) and current probe to measure voltage reflection coefficients at two locations of each cable. Then a nonlinear equation about terminal equivalent impedance is constructed. By solving the nonlinear equations, broadband terminal equivalent impedance for each cable can be acquired. Compared with existing methods, the efficiency of this method is improved significantly. Furthermore, the precision is improved in that this method including the coupling effect between cables. This method helpful to the EMC test and fault diagnosis.

Effects of Undesired Asymmetries and Nonuniformities in Differential Lines

In this paper, the effects of undesired asymmetry and nonuniformity on differential line (DL) circuits are modeled as deviations from the ideal, nominal geometrical characteristics of a perfectly symmetrical and uniform DL, whose performance is taken as reference. To this end, the involved multiconductor transmission line equations are solved in the modal domain by a decoupled approach based on the perturbation of the p.u.l. parameters of the ideal DL on one hand, and on the assumption of weak coupling between the corresponding differential and common mode equivalent circuits on the other. Three illustrative DL structures are considered and analytically solved, with the objective to put in evidence the effects of the aforesaid geometrical nonidealities on the degradation of the differential mode signal propagating along the DL as well as on the generation of undesired common mode currents/voltages. A circuit interpretation of the involved phenomena is provided, and the relevant geometrical parameters are identified.

Large-signal distributed millimeter-wave multifinger pHEMT modeling using time-domain technique

The finite-difference time-domain (FDTD) method is used for the large-signal modeling of a multifinger pHEMT, which is considered as five nonlinear coupled distributed transmission lines. The developed model, which is based on the exact physical layout of multifinger pHEMT, not only accurately describes the propagation effects along the electrodes at higher frequencies but it also includes major nonlinearities of the I–V and Q–V characteristics. Using the transmission line theory, a proper nonlinear equivalent lumped circuit model is allocated for the differential length of the quintuple-line transistor and the nonlinear active multiconductor transmission line (NAMCTL) equations are derived. These nonlinear, coupled differential equations are numerically solved using the FDTD method. The proposed model is applied to a 100 nm GaAs pHEMT and the simulation results are compared with the results of conventional sliced model in Keysight ADS simulator. The developed transient nonlinear model accurately predicts both the S-parameters (1–150 GHz) and large-signal power performances especially at millimeter wave frequency range. The proposed model can be useful in design and analysis of various types of high-frequency nonlinear integrated circuits.

FDTD-Based Analysis of High Speed Data Transmission Along Interconnections Terminated with Noisy MOSFET

In this paper, we present a new comprehensive technique aimed to analyze data transmission quality along interconnections taking into account internal noise sources especially those considered within active terminations such as MOSFET transistors. The proposed analysis consists of combining a numerical method to solve the multi-conductor transmission line equations through the Finite Difference-Time Domain algorithm, with techniques of calculating BER and setting eye diagrams. As a result, the effects of several transmission parameters on signal integrity are thoroughly examined producing useful information for designers.

Multiresolution Time-Domain Analysis of Multiconductor Transmission Lines Terminated in Linear Loads

This paper derives a multiresolution time-domain (MRTD) scheme for the multiconductor transmission line (MTL) equations based on Daubechies’ scaling functions. The terminations are characterized by a state-variable formulation which allows a general description of the termination networks. For the linear load terminations, a method incorporating the terminal constraints is proposed to work out the scheme at and close to the terminations. The MRTD scheme is implemented with different basis functions for linear components including resistances, inductances, and capacitances. Numerical results show that the MRTD schemes obtain a more stable result than the conventional finite difference time-domain (FDTD) method with a coarse space step.

Multiconductor Transmission-Line Theory That Includes an Antenna Process with a Lumped-Parameter Circuit

Electromagnetic noise was studied theoretically by developing a multiconductor transmission-line (MTL) theory derived from Maxwell’s equations without any approximations. The MTL theory naturally contains the antenna process, and for the case of three lines it is written in terms of normal, common, and antenna modes. Here we develop a theoretical method to solve the MTL equations by introducing boundary conditions consisting of a lumped-parameter circuit in order to describe the performance of any electric circuit for good performance without noise. The finite-difference time-domain (FDTD) method is a powerful algorithm to solve time-dependent coupled differential equations for a combined distributed- and lumped-parameter circuit.

Accurate time-domain modeling of multi-finger pHEMT transistor based on transmission line theory

In this paper, a fully distributed modeling of double gate InGaAs pHEMT as a 5-line active coupled transmission line is described. In order to apply the transmission line theory, a proper equivalent lumped circuit model is allocated for the differential length of the double gate quintuple line transistor and the active multi-conductor transmission line equations are obtained. The passive element values are extracted through numerical analysis and conversion of the single gate active small signal model to the double gate one is performed by an introduced scaling method. These equations are solved by the finite difference time domain technique. The simulation results are compared with the results of conventional slice modeling which are in agreement at low frequencies but show differences, at high frequencies.

Delay uncertainty in MLGNR interconnects under process induced variations of width, doping, dielectric thickness and mean free path

This paper analyzes the delay performance of multi-layered graphene nanoribbon (MLGNR) interconnect under process induced variations. An equivalent multi-conductor transmission line (MTL) model driven by CMOS gate is employed for the analysis. The propagation delay is analyzed for different interconnect lengths and widths by taking into account the variations in width, dielectric thickness, dielectric constant, interlayer distance and doping concentration of MLGNR. Encouragingly, it is observed that the average deviation in delay is below 3 % for all process induced parameter variations except for the mean free path.

Extraction of VLSI Multiconductor Transmission Line Parameters by Complementarity

Solving lossy multiconductor transmission line (MTL) equations is of fundamental importance for the design and signal integrity verification of interconnections in VLSI systems. It is well established that the critical issue is the efficient and accurate electrical characterization of the MTLs through the determination of their per-unit-length parameters. In this respect, the so-called complementarity has the potential to become a fast and accurate method for the extraction of these parameters. Besides the value of the parameters, in fact, complementarity provides rigorous error bounds for them. Despite this important feature, commercial software do not use complementarity yet, due to the fact that there are unsolved theoretical issues related to the nonstandard formulation based on the electric vector potential. Some attempts to fill this gap have been already reported. The aim of this paper is to fill this gap by introducing a general formulation based on the electric vector potential highlighting the advantages of complementarity with respect to the standard first- and second-order finite element formulations.

Efficient Time-Domain Signal and Noise FET Models for Millimetre-Wave Applications

Based on the active coupled line concept, a novel approach for efficient signal and noise modeling of millimeter-wave field-effect transistors is proposed. The distributed model considers the effect of wave propagation along the device electrodes, which can significantly affect the device performance especially in the millimetre-wave range. By solving the multi-conductor transmission line equations using the Finite-Difference Time-Domain technique, the proposed procedure can accurately determine the signal and noise performance of the transistor. In order to demonstrate the proposed FET model accuracy, a distributed low-noise amplifier was designed and tested. A model selection is often a trade-off between procedure complexity and response accuracy. Using the proposed distributed model versus the circuit-based model will allow increasing the model frequency range.