Partial Element Equivalent Circuit Model Research Articles

New exponentially weighted inequality and its application to finite-time stability of descriptor time-delay systems

The paper considers the finite-time stability of descriptor time-delay systems. Firstly, we establish a set of exponentially weighted orthogonal polynomials, and based on this, we obtain a new exponentially weighted inequality(EWI). Secondly, an augmented Lyapunov-like functional (LLF) is established driven by the presented EWI. Finally, by applying the EWI and the LLF, some sufficient conditions for ensuring the finite-time stability of the system are obtained. A small partial element equivalent circuit model and two numerical examples are used to verify the effectiveness of the method.

Analysis of Nonuniform Contact Resistivity Distribution on No-Insulation HTS Racetrack Coil

This paper presents a numerical analysis and experiment to investigate a contact resistivity distribution of a no-insulation (NI) high-temperature superconductor (HTS) racetrack coil. Since the racetrack coil consists of straight and curved sections with continuously varing curvatures, the local inhomogeneity of contact resistivity can occur. We adopt a partial element equivalent circuit (PEEC) method to investigate this inhomogeneity. A 50-turn NI HTS single pancake test coil is fabricated to figure out the discrepancy of contact resistivity, and the coil is charged up to 100 A in a liquid nitrogen bath. Voltage profiles between inserted voltage taps and the magnetic fields at the coil center are measured and compared with two simulation cases: 1) uniform contact resistivity; 2) nonuniform contact resistivity distribution. The results indicated that nonuniform distribution of contact resistivity may exist. Lastly, the current and Joule heat distribution during the ramping process is analyzed by the PEEC model.

Performance Against Direct Lightning on 10-kV Overhead Distribution Lines With Counterpoise Wires

This article investigates the lightning performance of a 10-kV overhead distribution line equipped with grounding counterpoise wires under extremely low soil conductivity conditions. A partial element equivalent circuit model with the lossy ground being considered is adopted for lightning surge investigation. The Monte Carlo method considering natural lightning randomness is used for the flashover rate evaluation. It is found that under extremely low soil conductivity conditions, the simple vertical grounding rod is not effective in lightning protection, while the performance is significantly improved with extended counterpoise wires. The single-row counterpoise wire with a length of line span performs best. The single-row counterpoise wire with a half-length of line span is a competitive one, due to a half-cost reduction and a competitive performance. If a higher performance of lightning protection is expected, further extending the length of counterpoise wires would be ineffective. Increasing the CFO level or installing surge arresters on all poles would be the possible solution.

Electromagnetic Behavior Simulation of REBCO Pancake Coils Considering REBCO Tape Rotation Under High Magnetic Field

Since the screening current induced in rare earth-barium-copper-oxide (REBCO) magnet generates an irregular magnetic field, a few screening current simulation methods have been proposed. For an insert REBCO magnet generating ultra-high magnetic field, a new screening current method has been proposed with consideration of the coil-deformation effect. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">In this paper, we have developed a new screening current method based on a partial element equivalent circuit (PEEC) model coupling with a two-dimensional stress finite element analysis to accurately the time-transient distribution of accurate screening current. In the presented method, the changes of self/mutual inductances are considered due to the coil deformation. As the simulation results, it was found that the coil deformation affected the screening current distribution. When considering the coil deformations, the screening current-induced fields to operating current are different for charging cycles. It is necessary to simulate the coil deformation to accurately estimate the screening current and the screening current-induced field.

Axial Movement Effect on Voltage and Current Behaviors of Insert No-Insulation REBCO Pancake Coil

In 2017, an insert 12-stacked no-insulation (NI) rare-earth barium copper oxide (REBCO) magnet was used in the generation of a world-record DC magnetic field of 45.5 T. The experiment demonstrated its high thermal stability; however, NI REBCO pancake coils are facing mechanical problems. In such an ultra-high field, strong electromagnetic forces work on insert REBCO coils, in particular during sequential normal-state transitions. Strong forces may cause insert REBCO coils to deform and move. Some research revealed that some deformations and movements affected the coil voltages and the screening currents. Whereas, the coil movement in an ultra-high field has not been studied yet. Voltages and currents are induced by the movement of insert coils, and there is a possibility of degrading the coil stability. In this paper, we have investigated the electromagnetic behaviors during sequential normal-state transition, considering the coil movement due to strong axial forces. The electromagnetic behaviors are simulated using a radially-subdivided partial element equivalent circuit (PEEC) model coupled with mechanical movement analysis.

Methodology for Analyzing Coupling Mechanisms in RFI Problems Based on PEEC

In this article, a method for analyzing coupling mechanisms in radio frequency interference (RFI) problems is proposed. The partial element equivalent circuit (PEEC) method is first used to derive the retarded inductances and capacitances between different mesh cells. With the introduction of a novel partitioning algorithm, the capacitive coupling and inductive coupling between arbitrary layout parts can be quantified based on the magnitude of the displacement current and induced voltage drop. The accuracy of the PEEC models is validated by comparison with different commercial tools. The proposed coupling mechanism analysis flow provides a useful prelayout tool for RFI risk analysis.

A Full-Wave PEEC Model for Thin-Wire Structure in the Air and Homogeneous Lossy Ground

This article presents an efficient full-wave partial element equivalent circuit (PEEC) formulation of thin-wire structure in the full space of air-homogeneous lossy ground for lightning transient analysis. First, the Green's functions for the case of an air-uniform ground are derived, which involve Sommerfeld-type integrals. Next, using the singularity subtraction technique, a new form of Green's functions is obtained, which has the advantage of being less singular and smooth compared with the original form. In the full-wave method, the efficient and accurate computation of partial elements is crucial for the practical application of the PEEC formulation. A three-dimensional cubic interpolation scheme is employed to accelerate the computation of Sommerfeld integrals. To improve the computational efficiency and accuracy of the interpolation, the continuity and symmetry of the Green's functions are investigated in a comprehensive way, which is important for a successful interpolation. Numerical and experimental examples from the literature are presented to validate the accuracy of the proposed model.

PEEC Modeling in 3D IC/Packaging Applications Based on Layered Green's Functions

A circuit modeling application for 3D IC/packages is proposed in this paper. The method is based on the partial element equivalent circuit (PEEC) method and layered Green's functions (LGF). The LGFs are calculated from the discrete complex image method with three terms, direct coupling, complex images, and surface wave, extracted to analyze the wave behaviors. The dominant terms for the LGFs are analyzed for four canonical stack-ups in 3D IC/packaging systems. Analytical formulas that include the contribution of the complex images calculated from the LGFs are used for the partial capacitance calculation. A fast-modeling approach is proposed by applying the LGF in PEEC using three acceleration treatments to handle the 3D IC/packaging geometry without sacrificing accuracy. An on-chip power distribution network (PDN) geometry is used to illustrate and validate the method.

An Extension of Partial Element Equivalent Circuit Method for Frequency Selective Surface Analysis

In this communication, an extension of a partial element equivalent circuit (PEEC) method for frequency selective surfaces (FSSs) modeling is presented. Based on Floquet’s theorem and dyadic periodic Green’s functions of a layered medium, the general PEEC formulations and models for periodical structures are deduced, and Poisson’s sum formula is used to improve the convergence speed. Furthermore, the extension is also performed by developing a normalized-current-based simplified PEEC (NCB simplified PEEC), which can accelerate the calculation of the scattering coefficients significantly. Also, the circuit parameters within the interested frequency band are related to the FSS geometric parameters, which is beneficial to the optimization of FSS. The scattering coefficients of both patch and slot FSSs with a dielectric substrate are calculated, which agree well with those calculated by electromagnetic simulation software Altair FEKO 2020.

Method of Characteristic Modes Analysis and Manipulation for Antenna Design by Using Generalized Partial Element Equivalent Circuit

A method of analyzing characteristic modes (CMs) of antennas with multiple lumped <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</i> loads is proposed based on the full-wave generalized partial element equivalent circuit (GPEEC) model. Different with traditional CMs analysis methods, this method can be integrated into an optimization algorithm. With this powerful engine, we can develop a systematic method to manipulate radiation CMs of a mobile terminal while preserving its aesthetic structure features of the industrial design. This method can reveal all possible radiation structure performances as traditional methods and even create new possible radiation modes to improve antenna performance. With the GPEEC model, a comprehensive analysis of antennas can be achieved simultaneously, including but not limited to simulating time-/frequency-domain responses, evaluating radiation efficiency contributed by the self- and mutual radiated power separately, and plotting current and field distribution. Three design examples are demonstrated, including analyzing the working mechanism of the self-curing decoupling technique for mobile terminals from the view of CMs, extending the bandwidth of an antenna in LTE-A low-frequency bands, and creating a pair of MIMO antennas in the low LTE-A bands. These examples are verified theoretically and experimentally, showing the high potential of this method in analyzing and designing antennas for mobile terminals.

Multi-physics quench simulation of no-insulation HTS insert coils with high background magnetic field

Compared with traditional insulated High Temperature Superconducting (HTS) coil, no-insulation (NI) HTS coil has higher thermal stability and self-protection capability, so that is considered as an important way to generate high steady-state magnetic field. This paper focuses on the experimental phenomena that mechanical damage always occurs in NI coils during quench with high background field. A multi-physics quench simulation model is established for NI insert coils, which contains a 2-d axisymmetric partial element equivalent circuit (PEEC) model for electromagnetic calculation and two axisymmetric finite element models for thermal and mechanical analysis. Based on these models, we analyzed the electromagnetic, thermal and mechanical phenomena of insert coils during quench propagation and recovery. The simulation results show that mechanical damage may occur in the outer superconducting zone near the normal zone in no-insulation coils during quench, and thermal strain caused by temperature rise during quench may exacerbate mechanical damage. Meanwhile, quench at different single-pancake coils has different effects on the mechanical performance of the entire insert coils. Based on the electromagnetic simulation results, we also analyzed the time delay of the traditional terminal voltage quench detection method and its causes. This study analyzed the multi-physics quench phenomenon of no-insulation HTS insert coils. And the results imply that hoop strain will increase to the critical strain due to high magnetic field and current redistribution caused by radial current shunt during quench, resulting in mechanical damage. Additionally, radial current shunt and induced voltage will retard quench propagation velocity, resulting in the time delay in terminal voltage quench detection.

Model Order Reduction for Delayed PEEC Models With Guaranteed Accuracy and Observed Stability

In this work, a stable and accurate model order reduction method for large-scale partial element equivalent circuit (PEEC) models with many delays is proposed. The method reduces the dimension of the original model by interpolating the original transfer function. The interpolation points are iteratively selected using a greedy algorithm. An efficient error estimator for the reduced transfer function makes the greedy algorithm very successful in both, properly selecting the interpolation points and tightly measuring the error of the reduced transfer function. The resulting reduced-order model is accurate both in the frequency and in the time domain. Numerical tests have shown that the reduced-order model successfully filters the instability behavior of the original model and exhibits stability over a large time interval.

A Full-Wave Generalized PEEC Model for Periodic Metallic Structure With Arbitrary Shape

In this work, a full-wave partial element equivalent circuit (PEEC) model for 2-D periodic metallic structures is proposed. The mixed potential integral equations (MPIEs) with periodic Green’s function (GF) are interpreted as loop equations following Kirchhoff’s voltage law (KVL) in a circuit domain. The impact of gauge freedom in the MPIE on the corresponding PEEC model is discussed. A tailored gauge function is introduced to form a generalized PEEC model for periodic structures, which demonstrates superior passivity and frequency stability. With the proposed PEEC model, lumped and active elements can be easily integrated with conducting bodies for field-circuit co-simulation. Through the distributed circuit including coupling elements, the PEEC model provides a clear physical insight into the electromagnetic behavior of the physical models for design and optimization. Three numerical examples, including frequency-selective surface (FSS), array antenna, and pixelwise meta-atom, are given to illustrate the accuracy and efficiency of the proposed PEEC model.

Numerical Evaluation of Electromagnetic Behavior of Multi-Stacked No-Insulation REBCO Coil System for Whole-Body MRIs and Medical Cyclotrons

We have been developing a no-insulation (NI) REBCO pancake coil technology for applications in medical cyclotrons and high-field whole-body MRI. The thermal and electromagnetic behaviors of multi-stacked NI-REBCO double pancake coils are numerically analyzed in this study when a local defect occurs in one coil using coil diameter and operating temperature parameters. Based on changes in voltage at both ends of the coil and the magnetic field at the center, we present a numerical study of the possibility of continuous operation after the occurrence of a local defect. In the numerical analysis, a new simplified model based on the PEEC model (Partial Element Equivalent Circuit model) is suggested and used for current distribution analysis, and the results are compared.

Transient Behavior of a REBCO No-Insulation or Metal-as-Insulation Multi-Pancake Coil Using a Partial Element Equivalent Circuit Model

REBCO No or Metal Insulation magnets open the way for very high field magnets but their transient operation must be understood and optimized. We present then the thermo-electric behavior of an HTS pancake with different values of the radial contact resistance. The model is built from a Partial Element Equivalent Circuit (PEEC) model coupled with a 2D finite difference thermal model and with a 3D magnetic model. We consider different transient cases from the simple driven charge-discharge to the local quench inside a pancake. Then we extend the model to a magnet made of several pancakes powered in series and we observe the temperature distributions, the radial and azimuthal current of each sector of the pancakes due to a source term generated on one sector in one of the pancakes. This model makes possible to investigate the influence and optimize the turn-to-turn resistance. With low contact resistance, the current flows radially from one turn to another, bypassing the local defect, but the charging time increases. With high contact resistance, the current flows through the resistive zone, causing a local thermal overheating, but the magnet charging time is reduced. A compromise must be found.

AC Loss Evaluation of NI REBCO Pancake Coils in External Low-Frequency Magnetic Field

No-insulation (NI) rare-earth barium copper oxide (REBCO) pancake coils have attracted much attention because of their high thermal stability. It is, therefore, expected to apply NI REBCO pancake coils to various applications. In recent years, there have been discussions on the possibility of applying NI REBCO pancake coils to field coils of synchronous motors for ship and aircraft propulsion. NI REBCO field coils would be exposed to an AC magnetic field of ∼1 Hz. The AC losses must be discussed because a current is induced in an NI REBCO coil under externa AC field. AC losses in NI REBCO pancake coils exposed to changing magnetic fields should be investigated in detail. In this paper, the AC Joule loss of the NI-REBCO pancake coils is investigated by simulation using a partial element equivalent circuit (PEEC) model. The magnetization loss is also investigated based on the Bean model when a low-frequency AC magnetic field is applied to the NI REBCO pancake coil. From the simulation results, we concluded a high contact resistivity can sufficiently reduce the Joule loss.

Plastic Deformation Simulation of REBCO Tapes Using Particle Methods

Ultra-high field superconducting magnets are facing a mechanical degradation problem. On 2017, a world-record high DC magnetic field of 45.5 T was generated by insert no-insulation (NI) Rare-Earth Barium Copper Oxide (REBCO) pancake coils (14.4 T), called “LBC3”, and an outsert resistive magnet (31.1 T). Although the experiment showed a high potential of the NI REBCO magnets for ultra-high field generation, the insert NI REBCO pancake coils and the REBCO tapes were plastically deformed. The critical current properties of damaged REBCO tapes were degraded although the Hastelloy substrate of REBCO tapes is stiff and it has a high yield point of approximately 1 GPa. However, the mechanisms of the damaging REBCO tapes are not clarified. Therefore, it is necessary to clarify the mechanical phenomenon of REBCO tapes. In this paper, we model a REBCO tape with a smoothed particle hydrodynamics (SPH) method, which can represent plastic deformation. The SPH simulation is coupled with electromagnetic analysis with a partial element equivalent circuit (PEEC) model which we have previously developed. In the simulation, an external field, which linearly increases to 5 T and then decreases to 0 T, is applied to a short-length REBCO tape. The simulation results show that the REBCO tape largely deforms during the external field excitation and deexcitation.

Research on Calculation of Rail Self-Impedance of Track Circuit Based on Partial Element Equivalent Circuit

The self-impedance of the steel rail, as the signal transmission medium in the electrified railway track circuit, has a direct impact on the track circuit’s transmission performance. The finite element method is the most common method for calculating rail impedance, although it has a number of drawbacks, including a complicated model, a long computation time, and poor accuracy. The partial element equivalent circuit (PEEC) approach is used in this paper to provide a method for determining rail self-impedance. Firstly, the rail equivalent method is determined by analyzing the physical model of the rail. Considering the skin effect, the PEEC model of the rail is established. The internal impedance of the rail can be obtained by solving the equivalent circuit. The external impedance considering the influence of the earth is calculated by the Carson impedance calculation formula, which is processed by the segmented linear approximation method. The rail’s self-impedance is determined using the two procedures together. Finally, the actual measurement data verified the PEEC method to calculate the rail impedance. Compared with the finite element method (FEM), the calculation accuracy of the PEEC method is higher. The current frequency, the height of the rail from the ground, and the earth’s conductivity impact on the rail’s self-impedance are analyzed. The results show that the PEEC technique can be used to calculate the rail’s self-impedance and that the impact of current frequency, rail height above the ground, and ground conductivity on the rail’s self-impedance may be accurately represented. The self-impedance computation of the track circuit provides a theoretical basis.

A newly developed screening current simulation method for REBCO pancake coils based on extension of PEEC model

Since the screening current (SC) in rare earth-barium-copper-oxide (REBCO) coated conductor (CC) generates an undesired magnetic field, it must be accurately estimated, especially for magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR). Moreover, in recent years, it was pointed out that the screening current enhanced the stress/strain in REBCO CC, when an REBCO magnet was operated as an insert under an ultrahigh magnetic field. The previously reported SC simulation methods may be roughly categorized into finite element method (FEM) and equivalent circuit method. The FEM-based method often adopted an axisymmetric model or a thin film approximation model, while the circuit-based are the simple equivalent circuit model and the network equivalent circuit model, so-called the partial element equivalent circuit (PEEC) model. The latter is newly developed in this paper. Features of those SC simulation models are briefly compared to each other in this paper. Each SC simulation models have pros & cons. We have to adequately chose an SC simulation model depending on a purpose. We extended the original PEEC model to simulate SC. The extended model is named the advanced partial element equivalent circuit (A-PEEC) model. It is also extendable to an SC simulation of no-insulation REBCO pancake coils. To simulate the SC of a simple coil model and the LBC3 magnet, we investigated the screening current distribution maps, and the simulated screening current-induced fields were compared with the measurements. We have confirmed the validity of the newly developed A-PEEC model.

Evaluation of Green's Functions for PEEC Models in the Air and Lossy-Ground Space

This article presents the numerical evaluation of Green's functions for the generalized partial element equivalent circuit (PEEC) formulation in the air and lossy-ground space. Both the source and field points can be located in any layer of the air–ground space. Green's functions are expressed using Sommerfeld integrals (SIs), which have highly oscillating and slowly decaying integrands. As a robust and easy-implemented approach, the direct real-axis integration is presented with the techniques for removing the singularity and accelerating the convergence. Closed-form expressions are derived for the quasi-static and asymptotic approximation of SIs. The results are compared with numerical electromagnetics code NEC2, and good agreements were observed. An image model is developed for quasi-static SIs. Greens’ functions can then be contributed by the source and/or its image. This model is used to derive the frequency-invariant PEEC model for fast transient simulation in the time domain. The proposed numerical procedure can be applied to generate lookup tables of SIs, which are indispensable in the evaluation of PEEC circuit parameters for massive wire segments.