Partial Element Equivalent Circuit Method Research Articles

Lightning Response Transient Characteristics of Substation Grounding Grid in Multilayer Horizontal Stratified Earth Model Considering Soil Ionization Effect

This article proposes a new time-domain partial element equivalent circuit method mathematical model. The mutual impedance matrix between the potential on the discrete node and the leakage current on the conductor connected around the discrete node is adopted. To reduce the amount of calculation, the matrix equation that minimizes the unknown quantity is derived and adopted. The model can accurately calculate the lightning current distribution and lightning impulse response in a horizontal multilayered soil model considering the soil ionization effect. To improve computational efficiency, the quasi-static complex image method and its closed form of Green's function in the time domain are introduced into the model.

Investigation of Induced Overvoltages on DC Cables of PV System Subjected to Lightning Strikes Using FDTD Method

Due to installation of photovoltaic (PV) panels in outdoor areas, they are subjected to lightning strikes which may cause degradation or complete damage, resulting in service interruption and added cost. Hence, it is crucial to provide an efficient and effective lightning protection system which requires some considerations to increase its efficiency and reduce its cost. Any damage to the PV system inverter causes service interruption, so it is important to analyze the transient voltages induced on the dc cables connected to the inverter during lightning strikes. In this article, the induced voltages on dc cables are calculated using the finite difference time domain (FDTD) method, which introduces high accuracy in transient studies. Important factors affecting the transient voltages on dc cables are studied, including the effect of lightning strike polarity and changing the separation distance between the PV panel and the isolated protection system. Moreover, the effect of modifying dc cable arrangement is studied, in addition to the impact of concrete foundation presence in the grounding system. Then, some recommendations are proposed which succeeded in reducing transient voltages on dc cables. In addition, the induced voltages calculated by FDTD are compared with those obtained by partial element equivalent circuit method.

Optimized Analytical Computation of Partial Elements Using a Retarded Taylor Series Expansion

The aim of this article is to efficiently and accurately calculate the integrals of the full-wave (FW) partial element equivalent circuit (PEEC) method. The accuracy of the analytical formulas calculated by the standard precision can be compromised when using nonuniform mesh to properly model the high-frequency effects. The numerical errors can be avoided by using a high-precision arithmetic, i.e., higher number of digits, however, at the expense of significantly higher computation time. This article presents an analytical approach for calculating the FW-PEEC interaction integrals of two elementary volumes/surfaces based on the Taylor expansion, which allows a high computational speed preserving the accuracy with a relative error of less than 0.1%. The proposed solution is verified compared to the high-precision arithmetic and the standard Gaussian integration for two examples of strip lines. Moreover, it is shown that the accuracy of FW-PEEC integrals can affect the convergence of an iterative PEEC matrix solver.

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.

Full-wave electromagnetic analysis of lightning strikes to wind farm connected to medium-voltage distribution lines

This paper studies the voltages developed on a wind turbine (WT) and a medium-voltage distribution line (MVDL) connected to a wind farm subjected to lightning strikes and located on frequency-dependent (FD) soils. The ground potential rise (GPR), voltages at the blade tip and on phase conductors of the MVDL are calculated using the full-wave electromagnetic software XGSLab® employing the rigorous Partial Element Equivalent Circuit (PEEC) method. The wind farm comprises four wind turbines with interconnected grounding systems using cables buried in resistivity soils of 1,000 and 5,000 Ω m. The voltages are computed for the first positive impulse (FPI) of 100-kA 10/350μs and for the subsequent negative impulse (SNI) of 50-kA, 1/200μs. Results have shown that voltage peaks increase notably as the soil resistivity increases. When the WTs are assumed, oscillations in the GPR waveforms for the SNI occur due to the multiple reflections between the blade and the turbine’s base. However, the voltages for the FPI present smooth time-domain responses. Furthermore, the overvoltages developed at the MVDL are significantly dependent on the soil resistivity and lightning current waveform.

Precise Modeling and Design of Self-Resonant for High-Efficiency Mid-Range Wireless Power Transfer System

Transfer power and efficiency of wireless power transfer (WPT) systems are limited by the quality factor and coupling coefficient of the transmitting (Tx) coil and receiving (Rx) coil. Getting rid of compensation circuits, self-resonant coils have simple structure, high system reliability, high quality factor and high power level, making it a promising candidate for WPT applications. In modeling and computation of self-resonant coils, the results derived from the transmission line theory have obvious errors, and the finite element simulation takes a long computing time. For obtaining accurate results and achieving fast computation, a partial-element equivalent circuit (PEEC) method is applied to calculate the self-resonant frequency (SRF) and the current distribution of helical coils. To validate the accuracy of the proposed model, some prototypes were simulated and measured. The theoretical calculation predications are in great agreements with the simulation and measurement results. The magnetic coupling model between Tx and Rx is also established using PEEC method and a novel design guideline of self-resonant helical coils is proposed. Furthermore, a WPT system with 6.78MHz selfresonant helical coils was designed, fabricated and tested for a high-efficiency mid-range energy transfer. It was used to drive a 5W load wirelessly as a demonstration to validate the feasibility of the proposed system in practical applications.

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.

The design of YBCO binary current leads and its electromagnetic-thermal coupling analysis based on PEEC and finite volume method

A superconducting magnet system is developed for the application of high-frequency gyrotrons at Wuhan National High Magnetic Field Center, China. The operating cost for the magnet is dominated by refrigeration power. To reduce the heat load for cryogenic systems, a pair of YBCO binary current leads is designed by consideration of the industrial concentration of type II high-temperature superconducting YBCO material. In this paper, a simulation model is proposed to perform the electromagnetic-thermal coupling analysis of these YBCO binary leads under different operating conditions by combining a partial element equivalent circuit method and finite volume method. The simulation code can improve computational efficiency and enable high-accuracy coupling between the electromagnetic field and heat transfer process. In the steady state, the heat leakage at the 4.2 K cold end of the YBCO binary leads depends mainly on the geometric parameters of the YBCO tape, especially the cross-sectional area of the copper layer in the YBCO tape. The cooling mode and liquid helium level also have a significant impact on the heat leakage level. The optimal outer diameter of the normal copper section is identified, and the optimum value is largely influenced by the effective cooling power imposed on the cold end of the normal copper section. In the transient state, simulations for the charging process and the loss of cooling accident are performed, along with a detailed analysis of the electromagnetic-thermal response features of the leads under these conditions. The results indicate that the YBCO binary current leads possess high thermal stability and an ample time margin for the magnet system to be demagnetized.

A Spline-Based Partial Element Equivalent Circuit Method for Electrostatics

This contribution investigates the connection between isogeometric analysis (IgA) and the partial element equivalent circuit (PEEC) method for electrostatic problems. We demonstrate that using the spline-based geometry concepts from IgA allows for extracting circuit elements without an explicit meshing step. Moreover, the proposed IgA–PEEC method converges for complex geometries up to three times faster than the conventional PEEC approach and, in turn, it requires a significantly lower number of degrees of freedom to solve a problem with comparable accuracy. The resulting method is closely related to the isogeometric boundary element method. However, it uses lowest-order basis functions to allow for straightforward physical and circuit interpretations. The findings are validated by an analytical example with complex geometry, that is, significant curvature, and by a realistic model of a surge arrester.

Efficient PEEC Iterative Solver for Power Electronic Applications

A new formulation of the partial element equivalent circuit (PEEC) is proposed that is well suited to be used along with fast Fourier transform (FFT)-based acceleration techniques. FFT-based techniques are used in conjunction with iterative solvers to accelerate matrix-vector products. Despite the significant memory saving and computational time reduction, these techniques may suffer at high frequencies due to the fine voxelization that they require. In this work, a new surface impedance-based formulation is presented and a novel preconditioner is proposed. Both contributions allow the frequency range of applicability of the FFT-accelerated PEEC method to be extended up to the gigahertz range. The accuracy and efficiency of the proposed solver are demonstrated by applying the proposed modeling strategy to a toy-problem and real-world structures comprising a printed circuit board and a transistor package, involving millions of unknowns.

Fast and Accurate Transient Analysis of Thin-Wire Structure in Half-Space With PEEC Method

This article presents an efficient full-wave partial element equivalent circuit (PEEC) method of thin-wire structure in half-space for lightning transient analysis. In the framework of the PEEC formulation for thin-wire conductors, the partial elements are computed by double-folded line integrals. In layered media, the interaction integrals involve dyadic Green's functions, which contain a closed-form part and a Sommerfeld-integral part. The direct numerical computation of these integrals is very challenging and time-consuming, which hinders the practical application of the full-wave method. However, the efficient and accurate evaluation of partial elements has not been investigated in a comprehensive way so far. Accordingly, this article presents an efficient method for the evaluation of the partial elements between elementary segments in half space. The main novelty is that the interaction integrals are computed through a combination of the Taylor series expansion of the full-wave free space Green's function and the Sommerfeld-integral adaptive interpolation technique, which significantly reduces the computation time. The new method is systematically validated on examples from finite-difference time-domain (FDTD) method and method of moments and experimental results. The proposed method is validated with the NEC code in the frequency domain and the FDTD in the time domain.

Integrated Model for Lightning Strikes on a Tall Structure: Application to a Wind Turbine System

This article presents an integrated model of the lightning channel and tall structure on the ground for lightning analysis, by considering their interaction. An extended distributed circuit is proposed for modeling the lightning channel, by including the dynamic behavior of the corona sheath. The nonlinear, nonuniform, and time evolution characteristics of the lightning channel are taken into account. Meanwhile, the wire structure on the ground is modeled with the partial element equivalent circuit method. A lumped voltage source is introduced to ensure an “undisturbed” current waveform at the attachment point. A procedure is provided to determine this voltage source in a nonlinear circuit. The proposed model is verified with the experimental data of lightning-striking tall towers. This model then is applied to simulate lightning surges on a wind turbine system. A comparison with the simple current source model is made. Significant oscillation is found with the simple current source model, which is not in correspondence with the observation result under natural lightning.

Partial-inductance retarded partial coefficients: Their exact computation based on the Cagniard–DeHoop technique

The Partial Element Equivalent Circuit (PEEC) method is a well recognized integral-equation (IE) technique to solve Maxwell’s equations. Similarly to the method of moments (MoM), the electromagnetic (EM) interactions between currents and between charges are described in terms of integrals. In contrast to the standard MoM, the PEEC method keeps the electric and magnetic coupling phenomena separate, which leads to different interaction integrals to be computed. These integrals admit simplified solutions for the case of the static free-space Green’s function and orthogonal geometries but their applicability is limited to electrically small problems only. When the full-wave free-space Green’s function is considered, the integrals are typically computed in the frequency domain (FD) by resorting to Gaussian quadrature schemes. The accuracy and efficiency of such schemes is a delicate issue. Therefore, recent works have investigated the possibility of applying the Cagniard–DeHoop (CdH) technique to calculate the interaction integrals for zero-thickness elementary domains. In this paper, we close the loop and shall apply the CdH technique to calculate the partial-inductance between two elementary bricks as prescribed by the PEEC technique exactly in the time domain (TD). The analytical approach is demonstrated on the interaction between two bricks as it occurs in the modeling of the magnetic field coupling between volumetric currents. The accuracy of the proposed approach is (successfully) tested for two representative cases.

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.

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.

Analysis and test of a 1m / 275 mT large-scale high-intensity magnetic field immunity test coil system for ITER

The magnetic field immunity test is a qualification procedure for all types of electrical and electronic equipment located in ITER Tokamak areas where the static magnetic induction is higher than 5 mT. To perform the magnetic field immunity tests, a large-scale high-intensity magnetic field coil system required by ITER organization was developed and tested. The designed test coil system is capable of generating a homogeneous magnetic field with a maximum magnetic flux density of 275 mT in a cubic test zone with a side-length of 1 m. Since the scale and intensity of the test field are high, it is important to ensure the stability of the test coil system. In this paper, we report the engineering analysis and test results of the ITER test coil system. First, the water cooling analysis and design based on the finite element method (FEM) were performed to ensure the thermal stability. Then, the equivalent stress and deformation of the coil system under rated condition and seismic event were analyzed to verify the structural stability. Further, the inductance frequency characteristic of the coil system was analyzed through partial element equivalent circuit (PEEC) method to support the control strategy design of the power supply. In addition, a speeding-up inductance matrix calculation method was brought forward, which avoids redundant computation and significantly improves computational efficiency. At last, resistance, inductance, and temperature rise tests were completed to verify the design of the test coil.

Enhancing Litz Wire Power Loss Calculations by Combining a Sparse Strand Element Equivalent Circuit Method With a Voronoi-Based Geometry Model

High-frequency (HF) litz wires are increasingly used to increase the power density of power electronic systems. The wires installed in such applications consist of several hundreds or even thousands of strands sophistically twisted on multiple geometry levels. The novelty of the presented impedance extraction approach lies in a significant reduction of the calculation effort and the resolution of the litz wires with high copper fill factors (up to more than 60%) by a new multilevel Voronoi approach. The speedup is realized by a geometry-based restriction of the calculation of the inductive coupling of strand section pairs. The calculation of large coupling matrices can, therefore, be bypassed by a direct evaluation of sparse inductance matrices of lower dimension. Combined with an efficient implementation, the new sparse strand element equivalent circuit method requires less than 10% of RAM and only 0.1% of CPU time for the impedance extraction compared to the established partial element equivalent circuit method for high-frequency litz wires. Benchmarks with this established method demonstrate less than 1% deviation for the frequency-dependent impedances up to 1 MHz.

Efficient Partial Elements Computation for the Non-Orthogonal PEEC Method Including Conductive, Dielectrics, and Magnetic Materials

The partial-element equivalent circuit method is a well-known numerical technique that is used to solve Maxwell’s equations in their integral equation form. The application of the partial-element equivalent circuit (PEEC) method to modeling domains with non-orthogonal three-dimensional geometries requires the computation of the interaction integrals to be performed numerically, thus slowing down the overall computation. This work presents a new technique that allows improving the computation of the interaction integrals of the PEEC method for non-orthogonal geometries under the quasi-static hypothesis. To this purpose, a voxelization approach that automatically decomposes non-orthogonal volumes in elementary parallelepipeds is used, allowing the implementation of closed-form formulas for the interaction integrals and completely avoiding numerical integration. The proposed approach is applied to three example problems exhibiting very good accuracy and excellent speed-up compared with the standard one using the numerical integration.

Simulation Based Poynting Vector Description of the Field Regions for Simple Radiating Structures

Estimating the coupling properties between a radiating structure and other conductive elements, the field behavior of the radiating source is essential to know. One well-known classification of the field behavior are the field-regions around antennas, namely, the far-field, the radiating near-field, and the reactive near-field. The different kinds of near-fields are distinguished by the reactive and radiating parts of the electromagnetic field, whereas in the far-field region the field behaves as a plane wave in the direction of propagation. One way to describe these field characteristics is to use the complex Poynting vector, which defines the electromagnetic power flow. This work presents a Poynting-vector-based approach to classify and visualize the field behavior around simple radiators using numerical simulations. First, the approach is applied to simple antenna structures such as dipoles and loop antennas. Later, the introduced field regions are utilized to predict the coupling behavior of practical applications, the coupling between single elements of a linear antenna array, and the coupling behavior of an electrically large loop antenna. It could be shown that the introduced approach, defining a surface description of the boundary between the near-field regions, enables the possibility of predicting the coupling behavior between radiating structures. The introduced error estimator for the far-field also delivers knowledge about the far-field quality in different angular directions and distances. All simulations have been executed applying a one-dimensional partial element equivalent circuit method.

Modeling Large-Size Round Steel Plates With Surface PEEC Method for Lightning Transient Analysis

The round steel plate is the basic component of the external floating roof tank structure. Modeling large-size round steel plates is essential for analyzing the lightning response of external floating roof tanks. This article builds time-domain equivalent circuit models for round steel plates excited by the lightning current. First, a meshing scheme is presented to generate sector and cylindrical surface cells, which make use of the distribution characteristics of the currents and voltages on the round steel plates. Then, the surface partial element equivalent circuit (PEEC) approach is applied to model the plate. A method of calculating the equivalent surface impedance is proposed, which converts the frequency dependence of the skin effect into impedance based on the time domain. Finally, the time-domain surface PEEC approach is validated by the indirect time-domain solution based on the Fourier transform and the commercial software using the finite-difference time-domain technique, respectively. The comparisons show that the calculation results of different methods are consistent with each other. However, the proposed method in this article can greatly enhance the computational efficiency while meeting the accuracy requirements. It has great significance to the study of lightning protection for external floating roof tank.