Full-wave Partial Element Equivalent Circuit 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.

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.

Time-Domain Analysis of Retarded Partial Element Equivalent Circuit Models Using Numerical Inversion of Laplace Transform

Full-wave time-domain computational electromagnetic (CEM) solvers, which are integral equation (IE)-based, may suffer from what is called “late-time instability” problems. This unstable behavior occurs for CEM solvers for very fast rise-time input signals. A multitude of techniques has been devised by researchers over the years to solve the problem. In this article, we pursue an approach for the stable solution of full-wave partial element equivalent circuit (PEEC) models for fast-rising input waveforms. In particular, step and impulse response will be considered that are the most challenging from a stability point of view. For the solver part, a conventional full-wave PEEC code is used that requires one to use retarded partial elements. Unfortunately, a PEEC, as well as impedance Z-(method of moment) solvers using suitable numerical time-stepping methods have stability problems, especially for fast rising impulse or step inputs. An important step forward is achieved in this work by providing a larger class of stable solutions well above the stability achieved for time-stepping methods in the last 50 years. The time-domain stability is achieved by replacing the stepping integration methods with a numerical inversion of Laplace transform (NILT) technique. The NILT transform starts out by applying it to a frequency-domain PEEC solution. The surprising result is that the NILT-based method has a variable time-dependent bandwidth that is advantageous for the full-wave IE solution stability. In this article, we give several examples that show that a PEEC-NILT solution provides accurate and stable results for impulse, step- and piece-wise linear input waveforms.

Poles and Stability of Full-Wave PEEC

Time-domain simulation of full-wave partial element equivalent circuit (PEEC) often encounters stability problems. In the past, various approaches have been proposed to determine the stability of a given model. Most of these methods, however, have low efficiency and scalability. In this article, we propose a highly efficient method to obtain a particular group of the poles of the model; such information then serves as a necessary-condition test for stability. Furthermore, the method is extended to include the effects of damping resistors. From the root locus plots of the poles, we clearly see the unstable poles moving left as damping is added in, and by the time they move into the left-half-plane, we can quantitatively determine the minimum damping level required to achieve a stable model. Simulation results are provided to verify the proposed method.

A Full-Wave PEEC Model of Thin-Wire Structures Above the Lossy Ground

This article addresses a full-wave partial element equivalent circuit (PEEC) model of wire structures above a lossy ground (LSG) for lightning transient analysis. The PEEC model is formulated with dyadic Green's functions. An equivalent circuit is derived for the first time by including correction terms arising from the LSG. Circuit parameters are expressed using Sommerfeld integrals, which can be evaluated numerically and be presented with lookup tables. The low-frequency model of the LSG is derived, and is depicted using the mirror image of source elements, similar to the case of a perfect ground. The comparison of circuit parameters calculated with this model and the Sommerfeld integrals is made. The proposed method is validated numerically with the numerical computation code in the frequency domain, and the finite-difference time-domain (FDTD) method in the time domain. Good agreements are observed. The proposed method is then applied to analyze lightning transients in a wire structure over the LSG. It is concluded that the LSG can be substituted with the low-frequency model for transient analysis. The computational burden in the time-domain simulation can be significantly relieved.

Generalized PEEC Model for Conductor–Dielectric Problems With Radiation Effect

In this article, a passive full-wave partial element equivalent circuit (PEEC) model is formulated for modeling electromagnetic (EM) problems with finite dielectrics. The proposed PEEC model is a substantial extension of the generalized PEEC model for problems of multiple conductors in free space. The proposed generalized PEEC model reflects the physical radiation mechanism of an EM problem with finite dielectrics. The radiation effects, both from conductors and dielectrics, are incorporated in real-valued frequency-dependent inductances and resistances, while capacitances remain to be frequency invariant. Consequently, the resultant circuit model of the proposed PEEC model is numerically verified to be more passive than the existing PEEC models. The minor passivity violation can be amended without altering the circuit configuration. To model a dielectric with the irregular-shaped surface, a triangular meshing scheme with Rao–Wilton–Glisson (RWG) basis is used. The generalized PEEC model provides a passive circuit model for both the frequency- and time-domain modeling of an EM problem with finite dielectrics. The passivity and accuracy of the generalized PEEC model are validated by an low temperature co-fired ceramic (LTCC) multilayer filter module, two dielectric resonator antennas, and a patch antenna.

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.

Analytical Evaluation of Partial Elements Using a Retarded Taylor Series Expansion of the Green’s Function

The computation of the fundamental magnetic and electric field coupling terms for the full-wave partial element equivalent circuit method is time consuming for large problems. Extensive numerical integration is required for the computation of four to six fold integration of the partial element coefficients. In this paper, we present new full-wave analytical formulas for their computation, which are based on several different Taylor series expansion approaches for the case of rectangular elementary volumes and surfaces.

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.

Efficient Low-Frequency Breakdown Free Full-Wave PEEC Modeling Based on Geometrical Optics DCIM

The partial element equivalent circuit (PEEC) modeling based on geometrical optics discrete complex image method (GODCIM) is proposed. In GODCIM, the spectral domain dyadic Green's functions for layered media (DGFLM) are expanded into a geometrical optics series. The coefficient of each term is related to the Green's function for a half-space problem and then is expanded into an exponential series with frequency-independent parameters, which avoids the repeated computations of the complex images at each frequency brought by the conventional discrete complex image method (DCIM) and facilitates its application to the PEEC method to solve the layer medium problems. To address the low-frequency breakdown issue in PEEC modeling, a capacitor-opened (CO) model is introduced, which leads to fewer unknowns comparing with modified nodal analysis (MNA) model. The calculation errors between CO and MNA model are verified to be negligible. Easy switching between CO model and modified loop analysis (MLA) model is realized by a switching criterion based on Kalman filter which leads to a full spectrum simulation from dc to high frequencies. Numerical results ranged from Hz to 30 GHz are given to validate the proposed method.

Stability of Full-Wave PEEC Models: Reason for Instabilities and Approach for Correction

The partial element equivalent circuit (PEEC) method is a widespread numerical method for creating full-wave models of interconnection structures in the frequency and time domains for use in modeling EMC problems. The possible instabilities of time domain solutions-so-called late time instabilities-can complicate the use of the method. Several attempts to improve the stability of time domain solutions have been made in the literature. A new mathematically correct approach for analyzing the stability of PEEC circuits is presented in this paper. The reason for instabilities is discovered, and a method for stability improvement is developed and tested.

Full-wave PEEC time-domain method for the modeling of on-chip interconnects

With the advances in the speed of high-performance chips, inductance effects in some on-chip interconnects have become significant. Specific networks such as clock distributions and other highly optimized circuits are especially impacted by inductance. Several difficult aspects have to be overcome to obtain valid waveforms for problems where inductances contribute significantly. Mainly, the geometries are very complex and the interactions between the capacitive and inductive currents have to be taken into account simultaneously. In this paper, we show that a full-wave partial element equivalent circuit method, which includes the delays among the partial elements, leads to an efficient solver enabling the analysis of large meaningful problems. Applying this method to several examples leads to helpful insights for realistic very large scale integration wiring problems. It is shown in this paper that the impact overshoot, reflections, and inductive coupling are critical for the design of critical on-chip interconnects.

A method for reduced-order modeling and simulation of large interconnect circuits and its application to PEEC models with retardation

The continuous improvement in the performance and the increases in the sizes of VLSI systems make electrical interconnect and package (EIP) design and modeling increasingly more important. Special software tools must be used for the design of high-performance VLSI systems. Furthermore, larger and faster systems require larger and more accurate circuit models. The partial element equivalent circuit (PEEC) technique is used for modeling such systems with three-dimensional full wave models. In this paper, we present a practical, readily parallelizable procedure for generating reduced-order frequency-domain models from general full wave PEEC systems. We use multiple expansion points, and piecemeal construction of pole-residue approximations to transfer functions of the PEEC systems, as was used in the complex frequency hopping algorithms. We consider general, multiple-input/multiple-output PEEC systems. Our block procedure consists of an outer loop of local approximations to the PEEC system, coupled with an inner loop where an iterative model-reduction method is applied to the local approximations. We systematically divide the complex frequency region of interest into small regions and construct local approximations to the PEEC system in each subregion. The local approximations are constructed so that the matrix factorizations associated with each of them are the size of the original system and independent of the order of the approximation. Results of computations on these local systems are combined to obtain a reduced-order model for the original PEEC system. We demonstrate the usefulness of our approach with three interesting examples.

Analysis of 3-D interconnect structures with PEEC using SPICE

This paper discusses the possibilities of using the circuit simulation program, simulation program with integrated circuit emphasis (SPICE) for the simulation of partial element equivalent circuit (PEEC) models. After an introduction into the PEEC method, the simulation of quasi-stationary models is considered. An enhancement of SPICE is described, allowing the simulation of retarded PEEC models. This enables the computation of electric fields radiated from an interconnection structure. With the modified SPICE simulator it is possible to use existing SPICE models and combine them with full wave PEEC models.