On the frequency-dependent line admittance of VLSI interconnect lines on silicon-based semiconductor substrates

In this paper, a method for analysis and modelling of transmission interconnect lines with zero- or non-zero thickness on Si–SiO₂ substrate is presented. The analysis is based on semi-analytical expressions for the frequency-dependent transmission line admittances. The electromagnetic concept of free charge density is applied. It allows us to obtain integral equations between electric scalar potential and charge density distributions. These equations are solved by the Galerkin procedure of the method of moments. This new model represents narrow and thick line interconnect behaviour over a wide range of frequencies up to 20 GHz. The accuracy of the developed method in this work is validated by comparing with the rigorous simulation data obtained by full-wave electromagnetic solver and CAD-oriented equivalent-circuit modelling approach. The response of the proposed model is shown to be in good agreement with the frequency-dependent capacitance and conductance characteristics of general coupled multiconductor on-chip interconnects.     

Calculation of the electrical capacitance of a truncated cone

The paper presents a method for the evaluation of the capacitance and charge distribution of a metallic truncated cone of finite length with top and bottom circular flat plates and isolated in free space, the integral equations relating the potential and unknown charge distribution on the surfaces are solved using the moment method employing pulse and delta functions as basis function and testing functions, respectively. The elements of the matrix to be inverted are found by dividing the surfaces into subsections which are combination of curvilinear rectangles and triangles. Capacitance and charge distribution are evaluated as a function of the geometrical parameters of the structure, numerical results on capacitance and charge distribution are presented     

Capacitance coefficients for VLSI multilevel metallization lines

The problem of reducing the complexity of parasitic capacitance evaluation of interconnection lines in a multilevel stratified dielectric medium (a good approximation for VLSI) is considered. We start out with a review of the Green's function method for the Si-SiO2composite and its derivation via the Fourier integral approach. Next, a piecewise linear finite-element technique is proposed to solve the integral equations that relate charges to potentials and lead to the desired capacitance matrix. We show by example that the proposed method is both less complex than methods based on piecewise constant surface charge distributions and equally accurate. This supports the accuracy and usefulness of the technique for IC design.     

Shape sensitivities of capacitances of planar conducting surfacesusing the method of moments

In this contribution, a new method is presented to obtain the sensitivities of the capacitance or the charge with respect to a geometrical parameter of planar conducting surfaces. The charge density is found by an integral equation technique. By applying the flux-transport theorem, a new integral equation for the total derivative of the charge with respect to a geometrical parameter is derived from the original electrostatic integral equation for the charge distribution. This new integral equation is solved together with the original integral equation by the method of moments using the same set of basis and test functions. The method is also applied to obtain derivatives for the inductance, impedance and effective dielectric constant. Some simple electrostatic problems are presented, which illustrate the capabilities of our approach. In these examples we also discuss the difference between the geometrical derivatives obtained in this way with geometrical derivatives which are obtained by a central finite difference estimate. Next, some examples of the calculation of geometrical derivatives of capacitance and inductance matrices of multilayer, multiconductor thin microstrip lines are discussed     

The equivalent circuit of a microstrip crossover in a dielectricsubstrate

A quasi-static analysis is carried out to examine the capacitive coupling between two nonintersecting orthogonal microstrip lines above a ground plane and in a dielectric substrate. The charge density along the width of each strip is described using a prescribed charge distribution. A pair of coupled integral equations is derived and solved by the method of moments to obtain the excess charge densities. The lumped excess capacitances are computed and compared to those obtained using wire lines with radii equal to the equivalent radii of the strips     

Capacitance of dielectric coated cylinder of finite axial lengthand truncated cone isolated in free space

The capacitance of dielectric coated metallic cylinder and truncated cone are evaluated using the method of moments based on the pulse function and point matching. The analysis is based on the boundary condition for the potential on the conductor surface and the normal component of the displacement density at the dielectric-free space interface. The total free charge on the conductor surface is found from the inversion of a matrix partitioned into submatrices. Numerical data on the capacitance and charge distribution are presented     

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     

Space-domain Green's function approach to the capacitancecalculation of multiconductor lines in multilayered dielectrics withimproved surface charge modeling

An integral equation method for the calculation of capacitance and inductance matrices is presented. The method is suitable for multiconductor transmission lines embedded in a multilayered dielectric medium on top of a ground plane. Conductors of arbitrary polygonal cross section can be handled, as well as infinitely thin conductors. The method is new in two respects. The kernel of the integral equation is the space-domain Green's function of the layered medium. The accuracy of the solution is enhanced by using basis functions that exactly model the singular behavior of the charge density in the neighborhood of a conductor edge. Numerical examples show the accuracy of the calculations and the complexity of the configurations that can be treated     

Capacitance of Parallel Rectangular Plates Separated by a Dielectric Sheet

To determine the capacitance between two rectangular parallel plates separated by a dielectric sheet, the charge distribution on the plates is formulated in terms of a Fredholm integral equation of the first kind. This equation is solved numerically by a projective method using polynomial approximants. The resulting capacitance values are given in normalized graphical form, permitting capacitance determination for any practical values of dielectric constant and geometric parameters to within a few percent.     

Equivalent Capacitances of Microstrip Open Circuits

The integral equations that describe the charge distribution near an open-circuited microstrip end are formulated and subsequently solved by a projective method. The solution hinges on development of computationally efficient techniques for dealing with the singularities that occur by special quadrature formulas. The necessary formulas are described and tabulated. These techniques are then used to find open-circuit capacitance values for microstrip. It is found that the curves of excess capacitance versus width are easily describable by empirical equations; such equations are presented, along with the curves themselves. For strip widths between 0.1 and 10 times the substrate thickness and for dielectric constants in the range 1-51, the given data are believed accurate to within a few percent.     

Capacitance computations in a multilayered dielectric medium usingclosed-form spatial Green's functions

An efficient method to compute the 2-D and 3-D capacitance matrices of multiconductor interconnects in a multilayered dielectric medium is presented. The method is based on an integral equation approach and assumes the quasi-static condition. It is applicable to conductors of arbitrary polygonal shape embedded in a multilayered dielectric medium with possible ground planes on the top or bottom of the dielectric layers. The computation time required to evaluate the space-domain Green's function for the multilayered medium, which involves an infinite summation, has been greatly reduced by obtaining a closed-form expression, which is derived by approximating the Green's function using a finite number of images in the spectral domain. Then the corresponding space-domain Green's functions are obtained using the proper closed-form integrations. In both 2-D and 3-D cases, the unknown surface charge density is represented by pulse basis functions, and the delta testing function (point matching) is used to solve the integral equation. The elements of the resulting matrix are computed using the closed-form formulation, avoiding any numerical integration. The presented method is compared with other published results and showed good agreement. Finally, the equivalent microstrip crossover capacitance is computed to illustrate the use of a combination of 2-D and 3-D Green's functions     

Transient response of nonlinearly loaded wires in a two media configuration

The transient behavior of thin wires with nonlinear resistive load in the presence of a dielectric half-space is analyzed directly in the time domain. The nonlinear wire problem is formulated by the space-time Hallen integral equation. The effect of a two-media configuration is taken into account via the space-time reflection coefficient appearing within the Green function. The resulting integral equation is handled by the space-time boundary integral equation method. The transient response for the case of a thin wire isolated in free space computed by this direct time-domain approach is compared with results obtained by another method solved using data derived from frequency-domain analysis. Results for various other configurations are also presented.     

Methodology for characterization of high-speed multi-conductor metal interconnections and evaluation of measurement errors

Analysis and design of interconnects in high speed integrated circuits and systems involves models in the form of multiconductor transmission lines. The fundamental parameters of those models are matrices of capacitance, (C), inductance, (L), resistance, (R), and conductance (G). We present a methodology for measurement of entries in capacitance matrix. The entries of capacitance matrices can be calculated using numerical solvers of electrostatic fields established under the assumption of suitable biasing of interconnect structures. Numerical calculations of complete field equations are very complex and expensive in terms of computer time, therefore several approximations are made in constructing interconnect dedicated software packages available on the market. Because of these approximations it is necessary to validate the calculations via measurements. Calculation of the off-diagonal entries of capacitance matrix from measurements of "two-terminal" capacitances is strongly corrupted by the measuring errors. The method involves direct capacitance measurement in multi-conductor structures and provides analysis of accuracy.     

A Legendre Approximation Method for the Circular Microstrip Disk Problem

The quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach. The basis functions are modified Legendre polynomials combined with a reciprocal square root to provide the correct singularity in charge density at the edge of the disk. The integrals involving the singular part of the Green's function are evaluated exactly, the remainder by using Gaussian quadrature. The method is compared in computational efficiency with recent methods based either on a Galerkin approach in the spectral domain, or the use of dual integral equations. Numerical results are given for charge distribution and capacitance; they are compared to exact results and those obtained by others, and the limitations of those methods are discussed. Closed form expressions are given for the capacitance of a disk based on two simple charge distributions.     

Two-dimensional capacitance calculation in stratified and/orarbitrary dielectric media

This paper combines the layered Green's function and bound charge technique to characterize the capacitance matrix in arbitrary two-dimensional geometries. When there exists no infinite ground plane, the authors enforce the constraint that the sum of all free charges is zero. When arbitrary dielectrics are sandwiched between two parallel ground planes, they numerically differentiate the integral of a closed-form parallel-plate Green's function. Both CPU time and storage are reduced as a result     

三维寄生电容边界元计算的半解析积分方法

在VLSI三维多介质互连寄生电容的边界元法计算中,利用互连结构特点,本文提出一种半解析积分方法,它应用原函数方法将二维面积分转化为一维线积分,再用一维高斯积分求出积分值.与二维高斯积分相比,速度更快、精度更高,并改善了解的精度.     

Decoupling between two conductor microstrip transmission line

A numerical analysis of two-conductor transmission line with a rectangular notch in the dielectric between the strips is presented. Three media integral equations are derived and solved for the charge distributions. The decoupling between such two-conductor coupled microstrip transmission lines is investigated for asymmetric conductors. It is found that the coupling between two conducting lines can be reduced significantly by removing dielectric material between the lines which has a rectangular shape. For best decoupling, the width should be as wide as possible between the conducting lines but the depth should have an optimum somewhere in the base dielectric substrate     

Analyse d’antennes diélectriques par une méthode de diffraction

In order to approach the radiation problem of dielectric antennas, the authors study the scattering of electromagnetic waves by heterogeneous dielectric obstacles situated in the near-field zone of a circular waveguide carrying the TE₁₁ mode. By making use of the polarisation current densities and the free-space Green’s function, the integral equation concerning the electric field inside the obstacle is derived and then solved by the moments method. Results on the scattering by bodies of various shapes and structures are given as a function of the distance between the waveguide aperture and the scatterer. Finally the method is tested on dielectric antennas, i.e., the structures for which this distance vanishes.     

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.     

Partial-boundary element method for analysis of striplines witharbitrary cross-sectional dielectric in multi-layered media

A new method of analysis called the partial-boundary element method (p-BEM) is proposed for the analysis of striplines with arbitrary cross-sectional dielectric in multi-layered media. By using a Green's function that satisfies the boundary conditions of a relevant structure with multi-layered media and introducing a concept of the equivalent charge density, the p-BEM formulates a potential integral and boundary integral equations only on partial-boundaries such as the surface of the arbitrary cross-sectional dielectric. The number of the equations needed to be formulated is much less than in the conventional BEM. Numerical results of analysis are presented for two kinds of striplines: 1) with a rectangular dielectric ridge and 2) with an embedded rectangular dielectric in three-layered media