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.     

Stable and Effective Full-Wave PEEC Models by Full-Spectrum Convolution Macromodeling

A novel technique for time domain partial element equivalent circuits (PEECs) modeling is presented. The PEEC method is a well-known numerical method for creating full-wave models of interconnection structures in the frequency and time domains, which are being used for modeling electromagnetic compatibility (EMC) problems. The time domain solutions by PEEC can show the so-called late-time instabilities. Several attempts to overcome this problem have been made in the literature. The cause for instability has been revealed, and a stable time domain model has been given, however, with a reduced computational efficiency. A stable full-wave PEEC model based on a convolution macromodeling with a faster computation time is developed and tested in this paper     

Stability of discretized partial element equivalent EFIE circuitmodels

The instabilities associated with integral equation techniques for-the solution of electromagnetic problems in the time domain are well known. Instabilities may be due to either the numerical technique used for the time integration, or problems created by the discrete representation for the numerical solution of the problem. In this paper, we concentrate on the discretization issue. The stability problem occurs for various discretizations and formulations. Here, we use the partial element equivalent circuit (PEEC) formulation of the electric field integral equation (EFIE) in the circuit domain. This leads to a better understanding of the issues at hand. We show why the discretized model can be unstable and we suggest a circuit motivated technique to stabilize the solution     

Nonorthogonal PEEC formulation for time- and frequency-domain EM and circuit modeling

Electromagnetic solvers based on the partial element equivalent circuit (PEEC) approach have proven to be well suited for the solution of combined circuit and EM problems. The inclusion of all types of Spice circuit elements is possible. Due to this, the approach has been used in many different tools. Most of these solvers have been based on a rectangular or Manhattan representation of the geometries. In this paper, we systematically extend the PEEC formulation to nonorthogonal geometries since many practical EM problems require a more general formulation. Importantly, the model given in this paper is consistent with the classical PEEC model for rectangular geometries. Some examples illustrating the application of the approach are given for both the time and frequency domain.     

Accuracy and stability improvements of integral equation modelsusing the partial element equivalent circuit (PEEC) approach

The partial element equivalent circuit (PEEC) technique is a formulation which transforms an electric field integral equation (EFIE) into a full-wave equivalent circuit solution. In this paper, improvements are made to the PEEC model through the development of a refined method of computing both the partial inductances as well as the coefficients of potential. The method does not increase the number of unknowns. In addition, damping is added to the PEEC model in order to further reduce nonphysical resonances which may occur above the useful frequency range, The observations and solutions presented in this paper are especially important for time domain solvers. The effectiveness of the method is illustrated with several examples     

Fast multipole and multifunction PEEC methods

A key use of the partial element equivalent circuit (PEEC) method is the solution of combined electromagnetic and circuit problems as they occur in many situations such as today's integrated circuit (VLSI) systems and as components in mobile devices. The method, which has been applied to a multitude of electrical interconnect and package problems, is very flexible since it is easy to add new features to the approach. However, faster solutions are of interest since the problems to be solved are continuously increasing size. A class of fast methods is evolving based on the faster evaluation of the matrix elements and the use of iterative or other matrix solvers of the resultant system for the frequency domain. Fast circuit matrix solvers are easier to obtain in the time domain than the frequency domain since the delay or retardation can be utilized to sparsify the circuit matrix. In this paper, we concentrate on techniques for the fast evaluation of the PEEC circuit element for both the frequency and time domain where possible since both are important for the solution of specific problems.     

Solving Low-Frequency EM-CKT Problems Using the PEEC Method

The partial element equivalent circuit (PEEC) formulation is an integral equation based approach for the solution of combined electromagnetic and circuit (EM-CKT) problems. In this paper, the low-frequency behavior of the PEEC matrix is investigated. Traditional EM solution methods, like the method of moments, suffer from singularity of the system matrix due to the decoupling of the charge and currents at low frequencies. Remedial techniques for this problem, like loop-star decomposition, require detection of loops and therefore present a complicated problem with nonlinear time scaling for practical geometries with holes and handles. Furthermore, for an adaptive mesh of an electrically large structure, the low-frequency problem may still occur at certain finely meshed regions. A widespread application of loop-star basis functions for the entire mesh is counterproductive to the matrix conditioning. Therefore, it is necessary to preidentify regions of low-frequency ill conditioning, which in itself represents a complex problem. In contrast, the charge and current basis functions are separated in the PEEC formulation and the system matrix is formulated accordingly. The incorporation of the resistive loss (R) for conductors and dielectric loss (G) for the surrounding medium leads to better system matrix conditioning throughout the entire frequency spectrum, and it also leads to a clean dc solution. We demonstrate that the system matrix is well behaved from a full-wave solution at high frequencies to a pure resistive circuit solution at dc, thereby enabling dc-to-daylight simulations. Finally, these techniques are applied to remedy the low-frequency conditioning of the electric field integral equation matrix     

Speeding Up PEEC Partial Inductance Computations Using a QR-Based Algorithm

The partial element equivalent circuit (PEEC) approach has been used in different forms for the computation of equivalent circuit elements for quasi-static and full-wave electromagnetic models. In this paper, we focus on the topic of large scale inductance computations. For many problems as part of PEEC modeling, partial inductances need to be computed to model interactions between a large numbers of objects. These computations can be very time and memory consuming. To date, several techniques have been devised to reduce the memory and time required to compute the partial inductance entities, as well as the time required to use them in a circuit analysis compute step. Some of the existing methods use hierarchical compression while some others are based on issues like properties of the inverse of the partial inductance matrix. However, because of inherent limitations, most of these methods are less suitable for PEEC applications. In this paper, we present an approach which is based on the compression of the partial inductance matrix utilizing the QR decomposition of the far coefficients submatrices. The QR-decomposed form is represented as a compressed SPICE-compatible circuit. This yields an efficient and mathematically consistent approach for reducing the storage and time requirements     

Circuit models for three-dimensional geometries includingdielectrics

The partial element equivalent circuit (PEEC) approach has proved useful for modeling many different electromagnetic problems. The technique can be viewed as an approach for the electrical circuit modeling for arbitrary 3-D geometries. Recently, the authors extended the method to include retardation with the rPEEC models. So far the dielectrics have been taken into account only in an approximate way. In this work, they generalize the technique to include arbitrary homogeneous dielectric regions. The new circuit models are applied in the frequency as well as the time domain. The time solution allows the modeling of VLSI systems which involve interconnects as well as nonlinear transistor circuits     

Modelling of Microstrip Circuit Using a Hybrid PEEC/FDTD Approach

Currently available electromagnetic analysis methods are showing their limitations when large complex circuit structures, such as modern printed circuit boards and their environments, need to be analyzed. Two methods which may be used for the purpose are the partial element equivalent circuit (PEEC) approach and the finite-difference time-domain (FDTD) method. Each of these has their own advantages and limitations. In this paper, a hybrid method is described which overcomes many of the limitations of PEEC and FDTD while retaining their strengths. Results using the method for low-pass and bandpass microstrip filters are presented.     

Comprehensive Broad-Band Electromagnetic Modeling of On-Chip Interconnects With a Surface Discretization-Based Generalized PEEC Model

A surface integral equation formalism is proposed for broad-band electromagnetic modeling of on-chip signal and power distribution networks. The discrete model is developed in the spirit of the partial element equivalent circuit (PEEC) model, which is extended with several attributes that lead to enhanced modeling versatility, modeling accuracy, and numerical solution robustness from dc to multigigahertz frequencies. Instead of the volumetric discretization model, which has dominated the PEEC-based schemes for handling the tall and slim cross sections of the on-chip wiring, the proposed model relies on a computationally more efficient conductor surface discretization. Key to the effectiveness and accuracy of the proposed surface discretization is the definition of a frequency- and position-dependent impedance quantity on the conductor surface. Its numerical computation over the frequency bandwidth of interest is expedited through the implementation of a complex frequency-hopping algorithm. The resulting effective surface impedance is combined with a mixed triangular/rectangular meshing of the conducting surfaces for the approximation of the surface electric current and charge densities. A systematic strategy for the identification of loops in the resulting discrete model is used to ensure a numerically stable mesh analysis-based PEEC formulation for on-chip signal and power distribution modeling with electromagnetic accuracy from dc to multigigahertz frequencies.     

A PEEC with a new capacitance model for circuit simulation ofinterconnects and packaging structures

In this paper, a modified partial-element equivalent-circuit (PEEC) model, i.e., (L_p, A&oarr;, R, ϵf)PEEC, is introduced. In such a model, no equivalent circuit, but a set of state equations for the variables representing the function of circuit, are given to model a three-dimensional structure. Unlike the original (L_p, P, R, ϵf) PEEC model, the definition of vector potential A&oarr; with integral form and the Lorentz gauge are used in expanding the basic integral equation instead of the definition of the scalar potential φ with integral form. This can directly lead to the state equations, and the capacitance extraction can be replaced by the calculation of the divergence of A&oarr;, which is analytical. For analysis of most interconnect and packaging problems, generally containing complex dielectric structures, the new model can save a large part of computing time. The validity of the new model is verified by the analysis in time and frequency domain with several examples of typical interconnect and packaging structures, and the results with this new method agree well with those of other papers     

Fast multipole method for time domain PEEC analysis

High-speed electronic circuits are becoming more and more important in modern communication systems, thus leading to an increasing interest in printed circuit boards, interconnect, and packaging. Nowadays, full-wave numerical methods are widely used in order to investigate both signal integrity and electromagnetic compatibility issues arising in PCBs design. When broadband information is desired and transient effects dominate, it is more efficient using time domain numerical techniques, which may scale better than corresponding frequency-domain methods. This paper presents the derivation of the time domain partial element equivalent circuit (PEEC) method enhanced by the three-dimensional (3D) fast multipole method (FMM). It is shown that combining the full-wave time domain PEEC method with the FMM allows performing the analysis of electrically large electronic systems, which reduces both memory and CPU-time requirements. Several examples are presented confirming the capability of the proposed approach to provide a significant reduction of the computational complexity associated with the transient analysis of large systems.     

Impact of partial element accuracy on PEEC model stability

This paper details the impact of partial element accuracy on quasi-static partial element equivalent circuit (PEEC) model stability in the time domain. The potential sources of inaccurate partial element values are found to be poor geometrical meshing and the use of unsuitable partial element calculation routines. The impact on PEEC model stability of erroneous partial element values, and the coefficients of potential and partial inductances, are shown as theoretical constraints and practical results. Projection meshing, which is a discretization strategy suitable for the PEEC method, is shown to improve calculated partial element values for the same number of unknowns, thus improving model stability.     

Extrapolation of time-domain responses from three-dimensionalconducting objects utilizing the matrix pencil technique

In this paper, we use the matrix pencil approach to extrapolate time-domain responses from three-dimensional (3-D) conducting objects that arise in the numerical solution of electromagnetic field problems. By modeling the time functions as a sum of complex exponentials, we can eliminate some of the instabilities that arise in late times for the electric-field integral equation in the time domain. However, this method can also be utilized for extending the responses obtained using a finite-difference time-domain (FDTD) formulation     

加速SPICE进行三维电磁分析

互连结构的电磁分析越来越受到人们的重视.针对三维互连,A.E.Ruehli提出了部分元等效电路法.但该法生成的等效电路具有紧耦合性,用SPICE进行分析时稀疏矩阵技术已失去原先的优越性.本文采用广义残量法作为大型紧耦合线性方程组的求解工具以取代SPICE中的LU分解法,并辅以初值预估.实际计算表明,本文的方法提高了运算速度.广义残量法也可用于矩量法的方程求解中.     

高速双面共面结构印刷电路板电特性仿真

为了适应高速双面共面印刷电路板的不规则布线结构,本文采用三维电磁参数提取的部分元等效电路方法对任意形状接地/馈电板进行自动分割及单元建模,然后对包括I/O缓冲器在内的非线性电路进行时域信号响应分析.高速双面共面结构印刷电路板电特性的仿真与实际测试结果吻合较好,表明了方法的有效性.     

RF-MEMS电感三维衬底耦合的扩展PEEC法分析

本文扩展了部分元等效电路(PEEC)法来分析RF-MEMS电感的三维结构衬底耦合.将电边界条件引入PEEC法中,从而得到包含了衬底耦合的电容矩阵;通过电荷源等效使得在部分元的计算中能应用均匀介质空间的格林函数;应用合适的网格单元尺寸而引入电磁场滞后效应,使得在较大尺寸器件和较高频段时仍能保证PEEC法的高精度.这些改进降低了分析的复杂度和计算量,并保持了高精度.根据该方法编制的程序的数值结果与已发表的实验数据吻合得很好.     

基于LTCC的小型化中频带通滤波器设计

采用LTCC技术实现滤波器时,大电感值会引起加工复杂、寄生参数多等问题。文中通过部分电路等效法(PEEC)、三角形和星形联结等效变换法,将二阶直接耦合滤波器中的三角形连接电感等效为两个并联的耦合电感,减少了电感数量及感值。使得滤波器的设计复杂程度降低,标准化程度及一致性更高,封装更小。最终实现了一个中心频率70 MHz、相对带宽14.28%、插入损耗1.83 dB、回波损耗＜-15 dB的小型化中频滤波器。     

Frequency response characteristics of reference plane effectiveinductance and resistance [IC packaging]

A method which uses the partial element equivalent circuit (PEEC) method and electrical network theory to solve for the effective impedance matrix of reference planes is presented. The convergence and accuracy of the method are checked. The frequency responses of the effective inductance (L_eff(f)) and resistance (R_eff(f)) of reference plane are discussed. The effects of current redistribution and the skin effect on L_eff(f) and R _eff(f) are discussed. The effect of number of sinks and sources is examined