Partial Element Equivalent Circuit Formulation Research Articles

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.

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.

Error Estimation and Adaptive Mesh Refinement for the Unstructured Inductive PEEC Formulation

An adaptive mesh refinement procedure is presented in order to address efficiently low-frequency electromagnetic problems with the unstructured partial element equivalent circuit (PEEC) method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A posteriori</i> error estimator based on an equilibrated energy criterion is proposed, which means that two dual solutions are necessary. In fact, knowing that there is no dual formulation for the PEEC method, an auxiliary method is proposed in order to calculate a second admissible solution. This method is based on a projection of the PEEC solution from facets to edges function space in order to ensure the continuity of the electric field in the mesh. Two test cases have been treated in order to show the efficiency of the procedure. First, an inductive coupling test case is the TEAM Workshop problem no°7. Second, a circuit coupling test case is a laminated busbar, this last one being treated for a low-frequency case and also for the dc case. The adaptive mesh refinement procedure has shown great potential and makes the unstructured PEEC method much more functional and attractive for modeling complex devices.

Efficient Computation of Partial Elements in the Full-Wave Surface-PEEC Method

The partial element equivalent circuit (PEEC) method provides an electromagnetic model of interconnections and packaging structures in terms of standard circuit elements. The surface-based PEEC (S-PEEC) formulation can reduce the number of unknowns compared to the standard volume-based PEEC (V-PEEC) method. This reduction is of particular use in the case of high-speed circuits and high-switching power electronics, where the bandwidth extends from low frequencies to the GHz range. In this article, the S-PEEC formulation is revised and cast in a matrix form. The main novelty is that the interaction integrals involving the curl of the magnetic and electric vector potentials are computed through the Taylor series expansion of the full-wave Green's function, leading to analytical forms that are rigorously derived. Therefore, the numerical integration is avoided, with a consequent reduction of the computation time. The proposed formulas are studied in terms of the frequency, size of the mesh, and distance between the basis function domains. Three examples are presented, confirming the accuracy of the proposed method compared to the V-PEEC method and surface-based numerical methods from literature.

Accurate Calculation of Partial Inductances for the Orthogonal PEEC Formulation

The Partial Element Equivalent Circuit (PEEC) method is promising numerical technique for three-dimension electromagnetic modeling across various application fields. In the framework of the PEEC method, the partial elements modeling the magnetic and electric field coupling between elementary volumes and surfaces are computed by double-folded volume and surface integrals. Assuming the quasi-static hypothesis and an orthogonal mesh, the integrals have been computed by the analytical formulas derived in literature, which significantly reduces the computational time in comparison to the numerical integration. However, the existing analytical formulas are affected by significant numerical errors for certain PEEC structural mesh necessary to model the skin and proximity effects with a higher accuracy. To utilize the full potential of the PEEC method, the calculation of partial elements has to be carefully addressed, which has not been investigated in a comprehensive way so far. Accordingly, this paper presents a systematic accuracy analysis of the existing closed-form analytical formulas and methods for calculating the self and mutual inductances between two rectangular conductors. Additionally, a new strategy to select a proper analytical formula depending on the dimensions and positions of two conductors is proposed, which allows the mutual inductance extraction with a relative error of less than 0.1%. The new method is systematically validated on examples of 3-D dense PEEC systems using the quadruple precision arithmetic as reference.

Modeling Stripline With Slotted Ground Plane in Multilayered Circuit Board Using PEEC Formulation

In this study, the structure that a stripline crosses with a narrow slot on the adjacent ground plane was rigorously studied in both a signal integrity and an electromagnetic interference point of view. A partial-element equivalent-circuit (PEEC) based circuit model was proposed to analyze the local region close to the slot, and it can be connected to the plane model and the transmission line model later. Based on an equivalent principal, the problem under a study can be converted to a mixed coupling problem between electrical surface current at the stripline and magnetic surface current on the slot. Therefore, the structure under the study can be treated as a complete cavity surrounded by the perfect electrical conductor at the top and bottom surfaces and a perfect magnetic conductor at four sides, so that the cavity Green's functions can be used in the PEEC formulation. The results obtained by the proposed method matched well with those obtained by a finite-element-based commercial tool.

A Volume PEEC Formulation Based on the Cell Method for Electromagnetic Problems From Low to High Frequency

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.

Partial Element Equivalent Circuit Formulation for Moving Objects

This paper presents the partial element equivalent circuit formulation of electromagnetic systems with moving objects. The motion of portion of the system generates two effects: additional specific electromotive forces related to the speed and magnetic flux density and time-varying partial elements. The proposed formulation is presented in a general context including conductors, dielectrics, and magnetic materials. Exploiting Galerkin's testing procedure, additional contributions to the electric field integral equation that are proportional to electrical currents and magnetization appear. Three numerical examples are presented to support the theoretical analysis.

Time-Domain PEEC Transient Analysis for a Wire Structure Above the Perfectly Conducting Ground With the Incident Field From a Distant Lightning Channel

This paper presents the time-domain partial element equivalent circuit (PEEC) analysis of induced transients in a wire structure during an indirect lightning strike. The full-wave PEEC formulation with an external incident field is employed. The incident field is calculated with either the Uman's or the Jefimenko's formula by using the engineering model of a lightning channel. An instability problem is, however, observed. It becomes a hurdle in the application of PEEC for evaluating induced transient currents in closed wire loops. The problem is investigated, and is primarily caused by discretization of the electric field in numerical evaluation of external voltage sources. An algorithm is provided to solve this instability problem, and is tested for both near- and far-field cases. It is validated by the finite-difference time-domain method. The issue of time step and segment length is addressed. It is also found that the Uman's formula is generally incapable of producing stable induced currents in a wire loop structure. Finally, the proposed PEEC procedure is applied to evaluate induced lightning transients in a photovoltaic panel during an indirect lightning strike.

Fast and Versatile Inverse Hysteresis Model in Quasi-Static Regime: Derivation and Implementation in Simulink and PEEC Frameworks

In this paper, a fast H(M) magnetostatic hysteresis model is presented, tested, and used in Simulink and partial element equivalent circuit (PEEC) environments. The model is primarily designed to achieve full compatibility with the current 3-D PEEC formalism for nonlinear isotropic magnetic materials, compatibility not met by any direct model. The capabilities of the proposed model are shown in stand-alone configuration and as a core of circuit components within the Simulink environment. Its effectiveness in the 3-D PEEC formulation is then demonstrated.

A 3-D PEEC Formulation Based on the Cell Method for Full-Wave Analyses With Conductive, Dielectric, and Magnetic Media

A novel partial element equivalent circuit (PEEC) formulation for solving full-Maxwell’s equations, with piecewise homogeneous and linear conductive, dielectric, and magnetic media, is presented. It is based on the cell method which by using integral variables as problem unknowns is naturally suited for developing circuit-like approaches such as PEEC. Volume meshing allows complex 3-D geometries, with electric and magnetic materials, to be discretized. Electromagnetic couplings in the air domain are modeled by integral equations.

Quasi-static PEEC planar solver using a weighted combination of 2D and 3D analytical Green's functions and a predictive meshing generator

In this work, a quasi-static implementation of the partial element equivalent circuit (PEEC) method for the analysis of planar radiofrequency (RF) and microwave (uW) components is proposed. The procedure is divided in three parts. First, an alternative PEEC formulation based on energy concepts is described. Second, a smart mesh generator is developed in order to provide an accurate solution at minimum computational costs, taking into account both geometry and device physics as metrics for the correct sizing of mesh elements. And third, a weighted combination of the 2D and 3D quasi-static Green's functions (GF) is proposed for extending the valid frequency range of the quasi-static approximation. It is shown that the 3D-GF is very accurate at low frequency, whereas the 2D-GF is more suitable at higher frequencies. Numerical examples are compared to experimental data for different passive components and technologies in a wide frequency range.

Quasi‐static partial element equivalent circuit models of magneto‐dielectric materials

This study aims to present the formulation of the partial element equivalent circuit (PEEC) for dispersive and lossy linear magneto-dielectric materials under the quasi-static hypothesis. The magnetic permeability and dielectric permittivity are usually available as tabulated data at different frequency samples. They are incorporated in the time-domain PEEC formulation by resorting to a rational approximation of their frequency-dependent constitutive relations. Thus, a set of ordinary differential equations is obtained besides those obtained by enforcing Kirchoff voltage and current laws to the PEEC equivalent circuit. The resulting augmented model preserves the circuit structure of the usual PEEC model and, hence, can be run in Spice-like circuit solvers. The robustness and accuracy of the proposed time-domain approach is tested by comparison with its frequency-domain counterpart despite the larger number of unknowns.

Time-Domain Partial Element Equivalent Circuit Solver Including Non-Linear Magnetic Materials

This paper presents a novel partial element equivalent circuit formulation to handle 3-D non-linear problems. Since the displacement currents are taken into account, the proposed formulation is not restricted to low-frequency applications, although the propagation delays are neglected. The non-linear problem is solved by means of the Newton–Raphson algorithm. Special care is taken in order to analytically compute the Jacobian of the problem. Furthermore, an adaptive setting of the time step is adopted in the transient analysis allowing to maintain the size of the time step as large as possible while preserving the accuracy. Computed results for a closed magnetic circuit and the testing electromagnetic analysis methods workshop problem 10 are reported.

A Surface PEEC Formulation for High-Fidelity Analysis of the Current Return Networks in Composite Aircrafts

This paper describes the approach developed to model the current return networks installed aboard aircrafts having parts made in composite materials. The surface partial element equivalent circuit (PEEC) method is adopted for its high-fidelity modeling capabilities, and its accuracy in the low-frequency region, which is of interest for the characterization of the return networks. State of the art of PEEC modeling is implemented in order to allow real-life aircrafts to be modeled. A special complex mock-up has been realized and measured. The numerical results are compared with measurements to assess their adequacy.

Modeling Local Via Structures Using Innovative PEEC Formulation Based on Cavity Green's Functions With Wave Port Excitation

Complex via structures such as noncircular antipads are rigorously studied, in the local region close to vias, using an innovative partial-element equivalent-circuit (PEEC) formulation based on the cavity Green's functions. The problem under study is an essential building block that can be used to model complex multilayer structures. With the usage of the cavity Green's functions, the number of unknowns is significantly reduced. Further, nonorthogonal quadrilateral mesh is used to better accommodate the complex shape of the antipads in practical designs. A novel wave port excitation method is proposed, which is new for the PEEC formulations. Singularity extraction in self element calculations involving the nonorthogonal cell and the cavity Green's functions is studied. The impedance parameters of several via structures are obtained using the proposed method, and the results are validated with the finite-element solution from a commercial tool.

Circuit Parameters of Vertical Wires Above a Lossy Ground in PEEC Models

This paper presents an investigation into circuit parameters of vertical wires over a lossy ground in partial element equivalent circuit (PEEC) models. The full-wave PEEC formulation of the vertical wires is presented, and formulas of circuit parameters for the vertical wires are derived using the Sommerfeld integral. Frequency responses of circuit parameters are discussed for both poor and good grounds, respectively. The results are validated numerically using the numerical electromagnetic code (NEC2). It is found that the lossy ground can be replaced with the weighted images in the electromagnetic analysis for the vertical wires. The lossy ground can be even replaced with the perfect conducting ground for the closely spaced vertical wires. The difference of the current and potential on the wires might not be significant if the imaginary part of ground dielectric constant is much greater than its real part. Using the perfect conducting ground can reduce computing resources significantly in the electromagnetic analysis. The complex plane method could lead to a large error if it is applied for vertical wires.

PERFORMANCE ANALYSIS OF PARALLEL NON-ORTHOGONAL PEEC-BASED SOLVER FOR EMC APPLICATIONS

A parallel implementation of a quasi-static Partial Element Equivalent Circuit (PEEC)-based solver that can handle electromagnetic problems with non-orthogonal structures is presented in this paper. The solver has been written in C++ and employs GMM++ and ScaLAPACK computational libraries to make the solver fast, e-cient, and adaptable to current parallel computer systems. The parallel PEEC-based solver has been tested and studied on high performance computing clusters and the correctness of the solver has been verifled by doing comparisons between results from orthogonal routines and also another type of electromagnetic solver, namely FEKO. Two non-orthogonal numerical test cases have been analysed in the time and frequency domain. The results are given for solution time and memory consumption while bottlenecks are pointed out and discussed. The benchmarks show a good speedup which gets improved as the problem size is increased. With the capability of the presented solver, the non-orthogonal PEEC formulation is a viable tool for modelling geometrically complex problems.

Multipoint Full-Wave Model Order Reduction for Delayed PEEC Models With Large Delays

The increase of operating frequencies requires 3-D electromagnetic (EM) methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D EM methods and model order reduction (MOR) techniques are used to reduce such a high complexity. When signal waveform rise times decrease and the corresponding frequency content increases, or the geometric dimensions become electrically large, time delays must be included in the modeling. A PEEC formulation, which include delay elements called τ PEEC method, becomes necessary and leads to systems of neutral delayed differential equations (NDDE). The reduction of large NDDE is still a very challenging research topic, especially for electrically large structures, where delays among coupled elements cannot be neglected or easily approximated by rational basis functions. We propose a novel model order technique for τ PEEC models that is able to accurately reduce electrically large systems with large delays. It is based on an adaptive multipoint expansion and MOR of equivalent first-order systems. The neutral delayed differential formulation is preserved in the reduced model. Pertinent numerical examples based on τ PEEC models validate the proposed MOR approach.

Waveform Relaxation Time Domain Solver for Subsystem Arrays

In this paper we present a waveform relaxation approach for the transient analysis of 3-D electromagnetic problems using the partial element equivalent circuit (PEEC) method. Relying on weaker couplings among separated systems, a waveform relaxation scheme is proposed to accelerate the transient analysis of large electromagnetic problems. The results are compared with those obtained using a conventional PEEC formulation. They exhibit a significant speed-up while preserving the solution accuracy.