Multiconductor Transmission Line Model Research Articles

A light Darwin implementation of Maxwell’s equations to quantify resistive, inductive, and capacitive couplings in windings

High operating frequency is an enabler of high key performance indicators, such as increased power density, in electrical machines. The latter enhances the cross-coupling of resistive–inductive–capacitive phenomena in windings, which may lead to significant loss in performance and reliability. The full-wave Maxwell’s equations can be employed to characterize this coupling. To address the frequency instability that arises as a result, a simplification known as the Darwin formulation can be employed, where the wave propagation effects are neglected. Still, this modification is prone to ill-conditioned systems that necessitate intricate pre-conditioning and gauging steps. To overcome these limitations, a fast 2D formulation is derived, which preserves the current continuity conservation along the model depth. This implementation is validated experimentally on a laboratory-scale medium-frequency transformer. The computed impedances for the open- and short-circuit modes of the transformer are validated using measurements and compared with the multi-conductor transmission line model that is widely adopted for the analysis mentioned above. The developed formulation demonstrates a high accuracy and outstanding frequency stability in a wide frequency range, becoming an efficient and computationally light method to investigate the interconnected resistive, inductive, and capacitive effects in windings.

Multiconductor transmission lines irradiated by plane wave: An EM‐reciprocity analysis methodology

AbstractThis paper extends previous work on electromagnetic (EM) reciprocity theorem analysis from the single‐conductor to the more general multiconductor transmission line (MTL) case. The focus is on the EM coupling problem of an ideal MTL above a perfectly electrically conducting ground when excited by a plane wave. The proposed methodology addresses mutual coupling effects and mismatched loads in the MTL. The expression for the current along the MTL in the test (radiation) state is derived, considering these factors. A novel calculation method for the induced voltage at the load is then introduced, assuming consistent coupling between different lines in both test and receiving states. Numerical examples validate the proposed EM‐reciprocity method, offering an alternative approach for analyzing the field‐to‐line coupling problem in MTL model.

Time domain characteristics of lossy multiconductor transmission lines randomly excited by electromagnetic pulse

This paper provides a time domain method for solving the induced currents (and voltages) of frequency-dependent lossy multiconductor transmission lines (MTLs) model randomly excited by electromagnetic pulse (EMP) plane wave. The model is processed based on the generalized polynomial chaos expansion (gPCE) and improved finite difference time domain (FDTD). The improved FDTD method overcomes the shortcomings of the conventional method that oversimplifies the expression of conductor impedance. In addition, when quantifying the influence of uncertainty on induced values, general formula of the global sensitivity analysis is provided by defining the gPCE basis function sequence to solve the complicated search work. The verification shows that the improved FDTD method alone or combined with gPCE to solve the lossy MTLs model excited by EMP is superior to the conventional methods in solving time and accuracy. Finally, compared with the Monte Carlo method, the effectiveness of the sensitivity method is also verified.

Unified multiconductor transmission-line model of multiple-shields multiconductor cables: evaluation of shield connections performances

This paper presents a transmission-line model of a multiple-shields multiconductor cable. This unified model includes at the same time the propagation and cross-coupling characteristics of the electrical wires and of the cable-shields. It also includes the electromagnetic characteristics of the shields (in terms of transfer impedance and transfer admittance). It is derived in compliance with the multiconductor-transmission-line theory and it is valid whatever the connection configurations at the shield ends are. Therefore, it makes it possible the modelling of realistic connection problems ranging from ideal 360° shield connections to simple bonding wires. In addition, it is suitable for both electromagnetic susceptibility and emission problems. The paper proposes a physical explanation of the derived per-unit-length matrices. This unified model is also used to define the required conditions for being able to calculate the response of a shielded-cable in a two-steps model in which the shield problem and the inner shield problem are solved in sequence. Finally, the paper illustrates an application of the model in order to evaluate performances of a shielded-cable-link on crosstalk configurations with respect to various electrical bonding techniques of the shield.

Inter-Turn Breakdown Fault Analysis and Winding Structure Optimisation of Winding of Dry-Type Transformers in Wind Farms

To address the problem of winding turn-to-turn breakdown faults in 35 kV dry-type transformers in wind farms under overvoltage conditions, this paper establishes a simulation model based on the structural dimensions and material parameters of the transformer windings. The winding distribution parameters were calculated using the finite element method. The transient processes inside the high-voltage coil were calculated by constructing a multi-conductor transmission line model (MTL) that took into account the influence of the secondary winding. The voltage distribution of the winding was analysed for both lightning shock and extra-fast transient overvoltage conditions. The simulation results show that the maximum overvoltage between turns of the transformer winding under lightning shock is 5.282 kV; the maximum overvoltage between turns of the winding under very fast transient overvoltage is 11.6 kV, which occurs between the first 2–3 layers of the section, close to the insulation breakdown margin. On this basis, the transformer winding structure was optimised and the maximum inter-turn overvoltage after optimisation was 9.104 kV, reducing the likelihood of insulation breakdown by 24.1%. Finally, the accuracy of the winding structure optimisation simulation study was verified by testing the transformer’s impulse voltage before and after optimisation, providing a reference for the stable operation of 35 kV dry-type transformers in wind farm practical applications.

Implementation of an Alternative Frequency-Dependent Three-Phase Transmission Line Model Based on the Folded Line Equivalent Model in MatLab-Simulink

This paper proposes an alternative multiconductor transmission line model that combines the folded line equivalent with the modal transformation. The folded line equivalent decomposes the nodal admittance matrix of a transmission line into its open-circuit and short-circuit contributions. These contributions are fitted to rational functions, which are associated with Norton equivalent circuits based on their state space models. The proposed model uses an orthogonal matrix to transform voltages and currents from the phase domain to the folded line equivalent domain and vice versa. Because the transformation matrix is orthogonal, we represent it using ideal transformers in simulation software. First, we use a circuit representation of Clarke’s matrix to decompose a transmission line into its modes. Then, each mode is decomposed into its open-circuit and short-circuit contributions using a circuit implementation of the proposed matrix. The proposed approach can accurately represent short lines in simulations with time steps equal to or greater than the propagation time of the transmission line. We compare the results obtained with the proposed approach to those obtained with power systems computer-aided design/electromagnetic transients including the DC universal line model.

Precise Evaluation of Repetitive Transient Overvoltages in Motor Windings in Wide-Bandgap Drive Systems

The increasing interest in employing wide-bandgap (WBG) drive systems has brought about very high power, high-frequency inverters enjoying switching frequencies up to hundreds of kilohertz. However, voltage surges with steep fronts, caused by turning semiconductor switches on/off in inverters, travel through the cable and are reflected at interfaces due to impedance mismatches, giving rise to overvoltages at motor terminals and in motor windings. The phenomena typically associated with these repetitive overvoltages are partial discharges and heating in the insulation system, both of which contribute to insulation system degradation and may lead to premature failures. In this article, taking the mentioned challenges into account, the repetitive transient overvoltage phenomenon in WBG drive systems is evaluated at motor terminals and in motor windings by implementing a precise multiconductor transmission line (MCTL) model in the time domain considering skin and proximity effects. In this regard, first, a finite element method (FEM) analysis is conducted in COMSOL Multiphysics to calculate parasitic elements of the motor; next, the vector fitting approach is employed to properly account for the frequency dependency of calculated elements, and, finally, the model is developed in EMTP-RV to assess the transient overvoltages at motor terminals and in motor windings. As shown, the harshest situation occurs in turns closer to motor terminals and/or turns closer to the neutral point depending on whether the neutral point is grounded or floating, how different phases are connected, and how motor phases are excited by pulse width modulation (PWM) voltages.

Analysis of Transient Response of Jointless Track Circuit Based on Multiconductor Transmission Lines Theory

In order to ensure the train operation safety, it is necessary to analysis the transient response of track circuit in complex electromagnetic environment. According to the distribution characteristics of transmission lines in traction electric network and the structure of ZPW‐2000 jointless track circuit, the multiconductor transmission lines model of the traction electric network combined with the track circuit is established. The model cannot be calculated directly due to the presence of parallel or terminated nonlinear lumped elements in the track circuit. For this problem, the new analytical method is proposed, which is based on finite difference time domain (FDTD) method to solve the transmission line part, Thevenin's theorem with state variable method to solve the equations of the terminated nonlinear track circuit. The solution equation of multiconductor transmission line system is obtained by combining the two parts. The relevant parameters of the model can be calculated by finite element method. In ATP‐EMTP, the simulation results verify the correctness of the expression. In addition, through the transient response experiment of track circuit, it is confirmed that the transmission characteristics of transient signal on rail are consistent with the calculation results in this paper. Finally, the influence of signal frequency, ballast resistance on track surface voltage, and the influence of interline coupling voltage by the conductivity (σ) and the permittivity (ε) are studied. It provides theoretical reference for the anti‐interference analysis of track circuits. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Electromagnetic Interference in a Buried Multiconductor Cable Due to a Dynamic Wireless Power Transfer System

The aim of this study is to predict the electromagnetic interference (EMI) effect produced by a dynamic wireless power transfer (DWPT) system on a buried multiconductor signal cable. The short-track DWPT system architecture is here considered with an operating frequency of 85 kHz and maximum power transferred to an EV equal to 10 kW. The EMI source is the DWPT transmitting coil which is activated when a vehicle passes over it. The electric and magnetic fields in the earth produced by the DWPT coil currents are calculated numerically using the finite elements method (FEM). These fields are then used to derive the voltage and current sources that appear in the field-excited multiconductor transmission line (MTL) model, used for the buried shielded cable. The MTL is analyzed considering the first ten harmonics of the current. The currents and voltages at the terminal ends are calculated considering the wireless charging of a single electric vehicle (EV) first, and then the simultaneous charging of 10 EVs which absorb a total power of 100 kW. The preliminary results reveal possible EMI problems in underground cables.

Performance Evaluation of Inductive and Capacitive Couplers for Pulsed Current Injection

Pulsed current injection (PCI), as an important conducted susceptibility testing method, has been introduced in several standards. However, the selection of couplers for PCI tests has not been clearly specified, which leads to the fact that the injected electrical quantities by different couplers might be rather different and it is inadequate to inject the proper disturbance at the testing ports. So, it is of significance to evaluate their performance quantitatively and determine the usage scenarios for couplers. This article focuses on the quantitative performance evaluation of inductive and capacitive couplers including inductive ferrite-core clamps and capacitive coupling clamps. A distributed-parameter multiconductor transmission line (MTL) model has been proposed for couplers clamped on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> wires. The validation test confirms the accuracy of the model in the bandwidth of PCI tests. The performances of an inductive coupler and a capacitive coupler are analyzed and compared based on MTL models. The selection criterion of the couplers is concluded as that a proper coupler should lead to higher time-waveform norms, which indicate possible nonlinear effects of the tested objects, such as the overcurrent, the turn-to-turn breakdown, and the component burnout.

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.

Antiresonance Phenomenon and Peak Voltage Stress Within PWM Inverter Fed Stator Winding

The high-frequency behavior of the stator winding is synthesized herein using a multiconductor transmission line model to study the potential location of excessive voltage stress, being the main cause of insulation failure within the winding. The comprehensive modeling approach proclaims that the voltage distribution within the winding is a result of the antiresonance phenomenon, which can be characterized by two oscillatory responses. One of the oscillatory response may lead to excessive voltage stress at the terminals of the machine, which have been reported extensively. However, it is the other oscillatory response, which engenders maximum voltage stress at the neutral end to the winding and prevails due to the superposition of traveling voltage waves. The latter is a sole characteristic of the stator winding and may be more detrimental to the winding insulation. Therefore, the stator winding remains under stress even with short or no cable. Furthermore, the article illustrates that increasing the cable up to a certain length can be potentially treacherous. The simulation results validate the identified phenomenon through the high-frequency model of the cable and the stator winding developed in MATLAB/Simulink environment. Further, the experimental results verify the aforementioned phenomenon on an automotive-grade 60 kW permanent magnet synchronous machine.

Holistic Modeling of High-Frequency Behavior of Inverter-Fed Machine Winding, Considering Mutual Couplings in Time Domain

Prediction of the voltage distribution within the winding due to pulsewidth modulated voltage impinging at the machine terminals is prevalent in the literature. For this purpose, the state-of-art work employs a high-frequency model of the stator winding but fails to represent the mutual inductive and resistive couplings between the turns and the coil. Herein, this article proposes a multiconductor transmission line model for representing the high-frequency behavior of the stator winding. A distinctive feature of the proposed model is that it incorporates the mutual inductive and resistive coupling between the turns and the coils using a current-dependent voltage source with due account of frequency-dependent parameters in the form of a ladder network. It enables the modeling of the mutual coupling of both single and multilayer winding in the time domain without the explicit use of modal transformation, convolution integral, or inverse Fourier transform. The close agreement between the simulation and the experimental results validates the proposed model of an automotive-grade 60-kW permanent magnet synchronous machine adapted in the Toyota Prius vehicle. In addition, a more intuitive analytical model is proposed to explain the voltage distribution and to characterize the peak voltage stress within the winding. The analytical model can be used to predict voltage stress as an alternative to the high-frequency model.

Multi-Conductor Transmission Line Model of Split-Winding Transformer for Frequency Response and Disk-to-Disk Fault Analysis

For transformer designers, split-winding transformer modeling in different frequency ranges is of great importance. In this paper, for the first time, a multi-conductor transmission line model is proposed for high-frequency modeling of the split-winding transformer. In this model, all the turns in layers and disks have been considered and the model's parameters have been calculated using the finite element method. In order to validate the proposed model, the results are compared with the result of a model, which is based on finite element and coupled field-circuit. It is shown that the introduced model has good accuracy and it can be employed for split-winding transformer modeling in different frequency ranges. In addition, using the validated multi-conductor transmission line model, the frequency response of the split-winding transformer and disk-to-disk short circuit fault are analyzed.

Voltage Distribution in the Windings of Medium-Frequency Transformers Operated With Wide Bandgap Devices

The short rise time observed in the PWM voltages generated by ultrafast wide bandgap devices increases the amplitude of voltage harmonics at higher frequencies. These harmonics can excite the resonances of medium-frequency transformers (MFTs), resulting in overvoltages inside the windings during continuous operation. Without further measures, these overvoltages can lead to unexpectedly high electric fields in the insulation material, which can result in partial discharge, accelerated aging, and possible failure of the MFT. To avoid these effects, the mechanism causing the overvoltages has to be understood and quantified during the design process. Based on this, the MFT can be designed in a way that the overvoltages vanish or are tolerable. Therefore, the voltage distribution inside the MFT windings is analyzed by a fully coupled multiconductor transmission line model, which includes the damping effect of electromagnetic losses in the litz wire and the core. This method is verified by measuring the transfer functions of the voltage to ground of individual turns and their voltage waveforms during continuous operation. The waveforms indicate repeating overvoltages inside the windings. A guideline for the design verification and a simplified approach to speed-up the modeling process are presented.

Radiated Susceptibility Analysis of Multiconductor Transmission Lines Based on Polynomial Chaos

To address the uncertainties of the radiated susceptibility of multiconductor transmission lines (MTLs), a surrogate model of the MTLs radiated susceptibility is established based on generalized polynomial chaos (gPC), and the gPC is made sparser by combining the adaptive hyperbolic truncation (AHT) scheme and the least angle regression (LAR) method. The uncertainties of the radiated susceptibility of transmission lines are calculated using the adaptive-sparse polynomial chaos (AS-PC) scheme. The parameters related to the incident field, such as elevation angle theta, azimuth angle psi, polarization angle eta, and field amplitude E, are inevitably random. Therefore, these four variables are taken as random input variables, and each of them is subject to different variable distributions. The MTLs model with infinite ground as the reference conductor is adopted, different impedances are used and the AS-PC scheme is combined with transmission line theory to calculate the average, standard deviation and probability distribution of the radiated susceptibility of MTLs. Sobol global sensitivity analysis based on variance decomposition is adopted to calculate the influence of random input variables on the MTLs radiated susceptibility model. The calculation results are compared with the results of the Monte Carlo (MC) method, proving that the proposed method is correct and feasible.

Approach for CM/DM Current Extraction and Crosstalk Analysis of Twisted-Wire Pairs with Random Non-uniform Twisting

In this paper, a twisted-wire pairs (TWP) with random non-uniform twisting is established. It is divided into a complete pitch segment and a noncomplete pitch segment by the ratio between the pitch and the length. The randomness of the actual TWP cable is accurately simulated by the following methods: 1) random combination of complete pitch segments; 2) random combination of non-complete pitch segments; 3) random combination between 1) and 2). Based on the TWP model, an equivalent multi-conductor transmission lines (MTLs) model can be obtained. The neural network algorithm is introduced to describe the complex relationship between the arbitrary position of the TWP and the per-unit-length (p.u.l) parameter matrix. In addition, the crosstalk and the common-mode (CM) and differential-mode (DM) noise under field-to-wire coupling are predicted. The numerical results show that crosstalk and CM/DM noise in TWP cable are susceptible to the twisted pitch at high frequencies. Compared with full-wave simulation, the accuracy of the proposed method is proved.

A Node-to-Node Admittance Functions Implementation of an Improved Frequency Dependent Multiconductor Transmission Line Model

The electromagnetic interference (EMI) performances of the interconnects and cables can be predicted via a standard multi-conductor transmission line (MTL) model, while the latter is not valid for the evaluation of power rail collapse and ground bounce responses. To circumvent the limitations, a more general and feasible improved MTL representation is presented in this paper. It physically incorporates the partial resistance and partial inductance parameters of all signal and reference conductors. To consider the frequency dependent behavior of the per-unit-length (PUL) electrical parameters in time domain simulations, a terminal description for this improved MTL model with any desired length is demonstrated. Subsequently, an equivalent node-to-node admittance functions (NAFs) implementation for this terminal representation is carried out. The correctness and effectiveness of the NAFs circuit model in time domain is then numerically validated by analyzing two dedicated examples.

Comparing the New Improved RLC and CMTL Models for Measuring Partial Discharge in Transformer Winding

Nowadays, one of the most well-known methods for accurate simulation of the fast-transient phenomenon as well as the study and location of partial discharge (PD) in transformer windings is the method based on winding modeling. In this regard, the circular multi-conductor transmission line (CMTL) model, which is an improved model of the MTL model, has been proposed as the finest solution since it is acceptable in an extensive frequency range and it can investigate the effect of winding rotation by considering the radius and angle variables. However, one of the fundamental problems of using the CMTL model is that a set of large equations, known as complete admittance matrices, must be solved for all winding turns, which typically contain thousands of turns. To overcome this problem, we presented an improved RLC model, which has the same accuracy as the CMTL model in terms of frequency validity range and less computational complexity. In this model, the variables of length and curvature of the turns are also considered in the formulations. Although this method is theoretically accurate, its reliability method must also be determined in practice and how the PD signal propagation must be carefully considered due to the problems in high-frequency modeling of the winding. Numerical analyses and laboratory measurements resulting from the use of PD pulses on a 20 kV winding of a distribution transformer, as well as a 63 kV, 30 MV winding, revealed that a valid frequency range is extendable up to 5 MHz for the proposed improved model without a significant increase in computational complexity.

Feasibility Study of Power Line Communications for Backhauling Outdoor Small Cells

Small cells are low-power and low-range radio access nodes that can be used to increase network capacity in reduced areas with high traffic demand and to provide coverage to small isolated zones. As their number is expected to be very high, connecting them to the aggregation point in a cost-efficient manner is a key challenge, and there is consensus that none of the current technologies is valid for all scenarios. Since lamp-posts are one of the most convenient locations for outdoor small cells, this work analyzes the use of power line communications (PLC) over outdoor public lighting networks (OPLN) as a backhaul technology for outdoor small cell deployments. To this end, the characteristics of the PLC channels established in OPLN are firstly assessed. The analysis combines noise measurements performed in existing OPLN and channel responses obtained from a multiconductor transmission line (MTL) model. The attenuation, delay spread and spatial correlation of the multiple-input multiple-output (MIMO) channels are investigated and their influence in the physical layer parameters of PLC systems is discussed. Afterward, the performance achieved by state-of-the-art PLC systems is assessed. To this end, estimations obtained by means of simulations and measurements taken in actual networks are included. Data throughput achieved by PLC systems based on the ITU-T G.hn standard, as well as the expected improvements obtained by $3\times 3$ MIMO systems, are given. Results indicate that PLC can be an interesting technology for coverage-driven small cell deployments with backhaul length shorter than 150 m, offering throughput values similar to existing Sub-6 GHz wireless solutions in non-line-of-sight (NLOS) conditions and wired ones like G.fast.