Application of a hybrid electromagnetic circuit method to lightning surge analysis

SummaryThe high level of integration has made the analysis and design of integrated circuits and packages increasingly challenging. Hence, there exists an urgent need to reduce the computational complexity of existing numerical methods. The integral equation‐based method known as the partial element equivalent circuit (PEEC) method naturally generates an equivalent circuit that can be analyzed in both the time and frequency domains. The enforcement of Kirchhoff laws to the equivalent circuit may easily result into a very large set of equations whose solution can be extremely time‐consuming.In this paper, we propose a vectorized version, over the frequency sweep, of the adaptive cross approximation algorithm. Furthermore, the multiscale block decomposition is applied to the PEEC method, powered by a vectorization strategy and an efficient management of the random access memory. It is found that the proposed use of vectorization and compression techniques in the framework of the multiscale block decomposition results in a significant computational speedup of the frequency‐domain analysis of PEEC models. The efficiency and accuracy of the proposed method are demonstrated through its application to two pertinent problems. Copyright © 2014 John Wiley & Sons, Ltd.

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.

PV systems are considered one of the most important and popular forms of renewable energy sources worldwide. Due to the installation of PV modules on rooftops and outdoor areas, they are frequently subjected to lightning strikes, leading to a deterioration in PV systems. To provide an efficient lightning protection system, the accurate modeling of PV systems during lightning strikes is required. The main objective of this paper is to assess the transient behavior of PV systems when struck by lightning, where the PV system is modeled using the partial element equivalent circuit (PEEC) and finite difference time-domain (FDTD) methods. The PEEC method is one of the accurate methods used to represent the PV system by its equivalent RLC circuit to study lightning transients. Moreover, the FDTD method provides high accuracy when studying lightning transients by analyzing the electric and magnetic fields inside the PV system. The PEEC and FDTD methods are used to calculate the transient overvoltages at different points of the PV system. Then, a small-scale PV system is tested in the laboratory using an impulse generator representing the lightning strike to verify the correctness of the results obtained by simulation. Most of the existing research works introduce low accuracy in PV system modeling during lightning transients by neglecting the effect of the mounting system, metal frame, and mutual coupling, which are considered in this work. Moreover, the high-frequency nature of lightning current is considered when modeling the elements of the grounding system. Verifying the results by laboratory experiments can guarantee the application of the simulation model on large-scale PV systems.

Broadband electromagnetic (EM) modeling increases in importance for virtual prototyping of advanced power electronics systems (PES), enabling a more accurate prediction of fast switching converter operation and its impact on energy conversion efficiency and EM interference. With the aim to predict and reduce an adverse impact of parasitics on the dynamic performance of fast switching power semiconductor devices, the circuit-oriented EM modeling based on the extraction of equivalent lumped R-L-C-G circuits is frequently selected over the Finite Element Method (FEM)-based EM modeling, mainly due to its lower computational complexity. With requirements for more accurate virtual prototyping of fast-switching PES, the modeling accuracy of the equivalent-RLCG-circuit-based EM modeling has to be re-evaluated. In the literature, the equivalent-RLCG-circuit-based EM techniques are frequently misinterpreted as the quasi-static (QS) 3-D Partial Element Equivalent Circuit (PEEC) method, and the observed inaccuracies of modeling HF effects are attributed to the QS field assumption. This paper presents a comprehensive analysis on the differences between the QS 3-D PEEC-based and the equivalent-RLCG-circuit-based EM modeling for simulating the dynamics of fast switching power devices. Using two modeling examples of fast switching power MOSFETs, a 3-D PEEC solver developed in-house and the well-known equivalent-RLCG-circuit-based EM modeling tool, ANSYS Q3D, are compared to the full-wave 3-D FEM-based EM tool, ANSYS HFSS. It is shown that the QS 3-D PEEC method can model the fast switching transients more accurately than Q3D. Accordingly, the accuracy of equivalent-RLCG-circuit-based modeling approaches in the HF range is rather related to the approximations made on modeling electric-field induced effects than to the QS field assumption.

In this paper, the analytical formula for the magnetic-to-electric field coupling of time-varying magnetization is presented in the framework of the partial element equivalent circuit (PEEC) method with a Manhattan-type tessellation. In the virtue of the results presented in a previous work, the proposed formula allows obtaining a quasi-magnetostatic PEEC solver in which all the matrix entries can be computed analytically. This ensures a significant speedup of the resulting PEEC solver if compared to the one where the evaluation of the integral was performed numerically. The numerical results confirm the accuracy and robustness of the proposed formula and the expected significant speedups. Detailed pseudocodes are provided to make the results easily reproducible.

In order to introduce thin electromagnetic shell into quasi-static partial element equivalent circuit (PEEC) method, a coupling with an integro-differential method (IDM) is proposed. This coupling allows us to benefit from both methods. The complex conductor geometry is modelled with the PEEC method while thin shell is taken into account, thanks to the IDM.

Memristors have great application potential in neuromorphic chips due to their integrated functionality of storage and computing. Research on the signal integrity of memristor crossbars under the excitation of bionic spiking signals is in urgent need to promote the rapid development of neuromorphic chip industry. This paper applies the Partial Element Equivalent Circuit (PEEC) method to analyze the influence of parasitic effects on the spike-timing-dependent plasticity (STDP) learning behavior of a memristor under the excitation of bionic spiking signal pairs. Study shows appreciable differences in the STDP behaviors under extremely narrow spiking signal excitations.

This paper presents hybridization approaches of the method of moments (MoMs) and the partial-element equivalent-circuit (PEEC) method in order to achieve a significant reduction of numerical complexity. The MoMs is applied as a full-wave method-only where it is necessary-to 3-D metallic scatterers such as antennas. Furthermore, the PEEC method is applied to single-layer printed circuit boards combined with an effective dielectric constant assuming an infinite dielectric in order to avoid discretization of the substrate. In the hybrid approaches of the MoMs and the PEEC method, both techniques are applied consecutively compared to a full-wave MoMs model. The validity of the hybrid approaches and the gain in numerical computation time are demonstrated by three numerical examples, which are related to electromagnetic compatibility. The results are verified by full-wave MoMs reference calculations and by a measurement.

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.

A new and general partial element equivalent circuit (PEEC) formulation based on electric and magnetic vector potentials and the cell method is presented. The purpose of this paper is manifold. First, a comprehensive review of well-established PEEC method is given and the state-of-the-art of the method with focus on the inclusion of magnetic media in the formulation is discussed. Then, a new and general formulation of the PEEC method based on integral equations is proposed. The formulation is valid for conductive, dielectric, and magnetic media and its advantages with respect to the state-of-the-art PEEC formulations are underlined. Unlike the existing PEEC methods, this novel formulation, thanks to the reinterpretation of PEEC in the context of integral equation methods, allows for maintaining a circuit interpretation for both electric and magnetic media. Thus, a standard Spice-like solver can in principle be adopted for the solution of electromagnetic (EM) problems also when magnetic media are involved. The imposition of the divergence condition for the current density vectors is guaranteed and also the case of inhomogeneous and anisotropic media is considered. A thorough discussion on the suitability of the method for the study of both low- and high-frequency EM problems is also presented. Finally, the case study of a near-field communication antenna is considered as an example of the applicability of the formulation to problems of industrial interest. Comparisons with commercial softwares, measurements, and the state-of-the-art PEEC method are given in order to demonstrate the accuracy and the advantages of the proposed PEEC formulation.

It is well known that the common mode (CM) current accounts for the main reasons of the far-field radiated electromagnetic noise of power cables. Therefore, this paper studies the far-field CM radiation prediction from a V-shaped power cables, which is based on the retarded partial element equivalent circuit (τPEEC) method. In this paper, the circuit matrix model and radiation formulas of τPEEC method are firstly introduced, then the planar V-shaped flat cables are modeled using this method. The radiation results predicted by this τPEEC method are compared with the simulated results and the measured results in anechoic chamber. These results verified the effectiveness of the proposed method.

This chapter discusses important issues related to the quality of the solutions obtained with a partial element equivalent circuit (PEEC) solver. Issues are considered such as the passivity in both the time and frequency domain responses and their stability in the time domain. Time domain stability has impacted differential equation???based electromagnetic (EM) techniques such as the finite difference time domain (FDTD) or the finite integration technique (FIT) models, as well as FEM methods. The chapter also considers techniques for the improvement of stability as well as passivity for the solution obtained with (EM) PEEC methods. It further summarizes the different relevant assessment, enhancement, and enforcement techniques, including solver stability enhancement, solver passivity enhancement, solution passivity assessment, and solution passivity enforcement. Finally, the chapter considers passivity of quasistatic (QS) PEEC models and general properties of all PEEC models.

The new stochastic partial element equivalent circuit (PEEC) method is proposed for uncertainty quantification in electromagnetic problems when material parameters are considered as random variables. The proposed formulation is derived using polynomial chaos expansion (PCE) and Galerkin projection. For the first time, the well-known advantages of PEEC are combined with those of PCE techniques, which can tackle also large variations in the random parameters. Volume conductive media are first considered in the formulation, which is then extended to dielectric, magnetic, and surface conductive media.

The partial element equivalent circuit (PEEC) method is a well-established technique for obtaining a circuit equivalent for an electromagnetic problem. The time domain solution of such models is usually performed using nodal voltages and branch currents, or sometimes charge and currents. The present paper describes a possible alternative approach, which can be obtained expressing and solving the problem in the waves domain. The digital wave theory is used to find an equivalent representation of the PEEC circuit in the wave domain. Through a pertinent continuous to discrete time transformation, the constitutive relations for partial inductances, capacitances, and resistances are translated in an explicit form. The combination of such equations with Kirchhoff laws allows to achieve a semiexplicit resolution scheme. Three different physical configurations are analyzed and their extracted digital wave PEEC models are simulated at growing sizes using the general-purpose digital wave simulator. The results are compared to those obtained by using standard SPICE simulators in both linear and nonlinear cases. When the size of the model is manageable by SPICE, an excellent accuracy and a speed-up factor of up to three orders of magnitude are observed with much lower memory requirements. A comparative analysis of results including the effect of parameters like the simulation time step choice is also presented.

Partial element equivalent circuit (PEEC) is a convenient method for modeling transient behavior of high temperature superconductors (HTS) magnets made of metal-as insulation windings. In HTS coils, especially those made of tapes, screening currents are known to lower the magnetic field, to generate losses, and to contribute to the mechanical stress. Including screening current calculation in PEEC method can be achieved by subdividing the superconducting tape along its width, leading to larger models and much longer computation time. Herein, we propose a finite element (FE) model for screening currents calculation, which can be combined with PEEC model for transport current addition. The FE model is defined as thin shell using formulation with normal component of electric vector potential T as nodal variable, calculated by an iterative procedure. Source term in the FE model is obtained from variation of radial flux density. Critical current densities calculated in the FE model are fed back to the PEEC model at each step of transport current calculation. This work presents in detail the FE model and its coupling with PEEC model. The model is validated by comparison with other 3D FE models and models with axial symmetry. The effect of reducing the size of FE model, by averaging screening currents in adjacent turns, is also presented in this study. Finally, we present simulation results for current ramps and a quench of a double-pancake magnet, obtained by PEEC with and without adding FE screening current model.