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.     

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     

Loop star basis functions and a robust preconditioner for EFIE scattering problems

An electric field integral equation (EFIE) formulation using the loop-star basis functions has been developed for modeling plane wave scattering from perfect conducting objects. A stability analysis at the DC limit shows that the use of the Rao-Wilton-Glisson (RWG) basis functions results in a singular matrix operator. However, the use of the loop-star basis functions results in a well-conditioned matrix. Moreover, a preconditioner constructed from a two-step process, based on near interactions and an incomplete factorization with a heuristic drop strategy, has been proposed in conjunction with the conjugate gradient method to solve the resulting matrix equation. The approach is shown to be effective for resolving both the low frequency instability and the bad conditioning of the EFIE method. The computational complexity of the proposed approach is shown to be O(N/sup 2/).     

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     

Integral equation solution of Maxwell's equations from zerofrequency to microwave frequencies

We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies     

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.     

Computer methods of network analysis

In this tutorial paper, the influence of the computer not only on the modus operandi of circuit design, but also on network theory itself, is discussed. The topological properties of linear graphs are reviewed and a matrix-topological formulation of the network problem is described. In addition to the classical mesh, node, and cutset methods, a mixed method of analysis is described which is applicable to dc, ac, and transient problems. Numerical methods of solving linear and nonlinear dc network problems are discussed and a new approach to ac analysis, using the mixed method and a numerical solution of the matrix eigenvalue problem, is described. The extension of this method to the transient analysis of linear networks is also explained. Finally, the problem of instability in the numerical integration of differential equations is discussed and several means of solving the problem are outlined.     

Generalized Partial-Element Equivalent-Circuit Analysis for Planar Circuits With Slotted Ground

A generalized partial-element equivalent-circuit (PEEC) method is proposed for modeling a planar circuit with a thin narrow slot on the ground. The approach is based on the coupled mixed potential integral equations for a problem with mixed electric and magnetic currents. The coupled integral equations are converted into a lumped-element circuit network using Kirchhoff's voltage law and Kirchhoff's current law of the circuit theory. The full-wave Green's functions for a grounded dielectric substrate problem are used. The interactions between electric current on a microstrip line and magnetic current on a slot are taken into account by introducing two kinds of controlled sources. This generalized PEEC model will be very useful in signal-integrity analysis for multilayered circuits. To validate the generalized model, three numerical examples consisting of microstrip lines and slots on the ground are presented. The results obtained by the proposed generalized PEEC model are compared with those obtained by commercial electromagnetic simulation software and published experimental results. Good agreement is obtained.      

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.     

互连封装结构电特性分析中的改进PEEC三维建模

本文提出了一种改进的PEEC模型,为便于在大规模互连封装结构分析中利用规模缩减技术,它以描述系统的状态方程代替了具体的等效电路.为此它以矢量磁位的积分表达式和洛仑兹规范代替了矢量磁位和标量电位的积分表达式,对积分方程进行展开.这样做可以避免复杂介质结构中的电容矩阵提取,大大节省了计算时间.这一模型可方便地嵌入更大的系统进行分层次的综合分析和利用PVL等规模缩减技术.数值计算的结果与其他文献吻合较好,表明该方法有较高的可靠性.     

Two-dimensional SPICE-linked multiresolution impedance method for low-frequency electromagnetic interactions

A multiresolution impedance method for the solution of low-frequency electromagnetic interaction problems typically encountered in bioelectromagnetics is presented. While the impedance method in its original form is based on the discretization of the scattering objects into equal-sized cells, our formulation decreases the number of unknowns by using an automatic mesh generation method that does not yield equal-sized cells in the modeling space. Results indicate that our multiresolution mesh generation scheme can provide a 50%-80% reduction in cell count, providing new opportunities for the solution of low-frequency bioelectromagnetic problems that require a high level of detail only in specific regions of the modeling space. Furthermore, linking the mesh generator to a circuit simulator such as SPICE permits the addition of arbitrarily complex passive and active circuit elements to the generated impedance network, opening the door to significant advances in the modeling of bioelectromagnetic phenomena.     

Generating compact, guaranteed passive reduced-order models of 3-D RLC interconnects

As very large scale integration (VLSI) circuit speeds and density continue to increase, the need to accurately model the effects of three-dimensional (3-D) interconnects has become essential for reliable chip and system design and verification. Since such models are commonly used inside standard circuit simulators for time or frequency domain computations, it is imperative that they be kept compact without compromising accuracy, and also retain relevant physical properties of the original system, such as passivity. In this paper, we describe an approach to generate accurate, compact, and guaranteed passive models of RLC interconnects and packaging structures. The procedure is based on a partial element equivalent circuit (PEEC)-like approach to modeling the impedance of interconnect structures accounting for both the charge accumulation on the surface of conductors and the current traveling in their interior. The resulting formulation, based on nodal or mixed nodal and mesh analysis, enables the application of existing model order reduction techniques. Compactness and passivity of the model are then ensured with a two-step reduction procedure where Krylov-subspace moment-matching methods are followed by a recently proposed, nearly optimal, passive truncated balanced realization-like algorithm. The proposed approach was used for extracting passive models for several industrial examples, whose accuracy was validated both in the frequency domain as well as against measured time-domain data.     

A hybrid method for the calculation of the resistance andinductance of transmission lines with arbitrary cross sections

The frequency-dependent resistance and inductance of uniform transmission lines are calculated with a hybrid technique that combines a cross-section coupled circuit method with a surface integral equation approach. The coupled circuit approach is most applicable for low-frequency calculations, while the integral equation approach is best for high frequencies. The low-frequency method consists in subdividing the cross section of each conductor into triangular filaments, each with an assumed uniform current distribution. The high-frequency method expresses the resistance and inductance of each conductor in terms of the current normal to the surface. An interpolation between the results of these two methods gives very good results over the entire frequency range, even when few basis functions are used. Results for a variety of configurations are shown and are compared with experimental data and other numerical techniques     

Evaluation of Green's function integrals in conducting media

This work presents an accurate integration method for computing Green's function operators related to lossy conducting media. The presented approach is ultrawideband, i.e., the integration schemes cover the entire range of frequency behavior, from high frequencies where skin current is prevalent to low frequencies where volume current flow dominates. The scheme is a step toward permitting exact ultrawide-band frequency domain surface-only-based integral-equation simulation of arbitrarily-shaped three-dimensional conductors, and toward obviating the need for volume-based explicit frequency-dependent skin effect modeling. This work deals specifically with the computation of Green's functions and not with the unrelated but important low-frequency conditioning issue associated with the standard electric field integral equation.     

PEEC Formulation Based on Dyadic Green's Functions for Layered Media in the Time and Frequency Domains

This paper presents a novel time- and frequency-domain concept of modeling with the partial element equivalent circuit (PEEC) method, which applies the mixed potential integral equation (MPIE) with dyadic Green's functions for layered media (DGFLM-PEEC). On the one hand, it represents an exact full-wave semianalytical solution for an arbitrary configuration of traces and via holes in multilayered printed circuit boards. On the other hand, the DGFLM-PEEC model is represented in a circuit form, and thus, may be included in general-purpose circuit simulators. The paper derives a general DGFLM-PEEC formulation, which may be applied to all types of the MPIE with dyadic Green's functions. Using this concept, a particular type of layered media, namely a lossy dielectric between two grounds (stripline region), is thoroughly investigated and used to set up a particular DGFLM-PEEC model. The closed-form expressions for partial inductances and potential coefficients have been derived for this case. The time- and frequency-domain DGFLM-PEEC models for the stripline region have been validated using the measurements and the simulation by the method of moments.      

Broadband Macromodels for Retarded Partial Element Equivalent Circuit (rPEEC) Method

The partial element equivalent circuit (PEEC) method is, nowadays, widely used in electromagnetic compatibility and signal integrity problems in both the time and frequency domains. Similar to other integral-equation-based techniques, its time domain implementation may suffer from late time instabilities, especially when considering delays [(Lp,P,R,tau)PEEC] (rPEEC). The cause of the instabilities may be either the numerical technique used for the time integration or problems created by the discrete representation of the electromagnetic continuous problem. In this paper, we concentrate on the latter and show that frequency dispersion plays an important role and must be taken into account in order to preserve accuracy and mitigate instabilities issues. An enhanced formulation of the PEEC method is presented that is based on a more accurate computation of partial elements describing the electric and magnetic field couplings; broadband macromodels are generated incorporating the frequency dependence of such elements, thus, allowing us to obtain better stability properties of the resulting (Lp,P,R,tau)PEEC model. The proposed equivalent circuits resemble those of the standard PEEC formulation but are able to capture the dispersion that, when neglected, might contribute to inaccuracies and late time instabilities     

Low-Frequency Shielding Effectiveness of Nonuniform Enclosures

This paper presents a quasi-static approximate solution to the magnetic shielding of several nonuniform enclosures using the integral form of Maxwell's equations and insight gained from other approaches. The solution is called quasi-static as the assumptions made are from physical arguments based on low-frequency cases where the enclosure size is much less than a wavelength. The integral form of Maxwell's equations is used to obtain a first order correction to the static solution to obtain induced currents in the time-varying case. A cylindrical shell immersed in an axial magnetic field is used to illustrate the method, which is then extended to derive a formula for a similarly excited rectangular enclosure. These shields are seen to behave like a low-pass filter. Although the enclosure dimensions are small compared to the wavelength, the skin depth effects in the walls cannot be neglected even for relatively thin material as usually encountered in an enclosure. These skin effects are included in the analysis and experimental checks performed on a variety of enclosure sizes and materials, excited by a Helmholtz coil show agreement within two decibels over the 4-octave frequency range examined. No one can say whether this method offers a better solution to the shielding problem, as all solutions are approximate, but the author attempts to present an alternative formulation that aids in understanding the physical processes involved in the shielding effectiveness of an enclosure and fills some of the gaps between the plane-wave analysis and circuit approaches presently used.     

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.     

Hybrid numerical method for harmonic 3-D Maxwell equations:scattering by a mixed conducting and inhomogeneous anisotropicdielectric medium

This article deals with a hybrid numerical method for solving harmonic Maxwell equations in the classical electrodynamic context. This formulation can be used with any body of arbitrary three-dimensional geometry, of perfectly conducting material or dielectric, with locally inhomogeneous and anisotropic behavior laws, and with or without dielectric losses. The mathematical formulation is presented along with applications validating it. The exterior problem is treated by the integral equation method while local equations are used for the dielectric parts of the body. A global variational formulation of the coupled problem is developed for use in discretization by the finite element method. Boundary finite elements are used for integral operators connected with the exterior problem. Localized finite elements are used for the interior problem. Difficulties of irregular frequencies, also called resonant frequencies in the perfectly conducting case, arising from the integral formulation are analyzed in detail and an efficient solution is developed     

A simplified broad-band large-signal nonquasi-static table-basedFET model

In this paper, a simplified nonquasi-static table-based approach is developed for high-frequency broad-band large-signal field-effect-transistor modeling. As well as low-frequency dispersion, the quadratic frequency dependency of the γ-parameters at high frequencies is taken into account through the use of linear delays. This model is suitable for applications related to nonlinear microwave computer-aided design and can be both easily extracted from dc and S-parameter measurements and implemented in commercially available simulation tools. Model formulation, small-signal, and large-signal validation will be described in this paper. Excellent results are obtained from dc up to the device f_T frequencies, even when f _T is as high as 100 GHz