Full-Periphery Surface Impedance for Skin-Effect Approximation in Electric Field Integral Equation

A new surface impedance model for RL-extraction in lossy two-dimensional (2-D) interconnects of rectangular cross section is presented. The model is derived directly from the volumetric electric field integral equation under the approximation of the unknown volumetric current density as a product of the exponential factor describing the skin-effect and the unknown surface current density on the conductor's periphery. By proper accounting for the coupling between the boundary elements situated on the top and bottom surfaces of conductor with the elements located on the side-walls, the model maintains accuracy from dc to multi-GHz frequencies as well as for conductors with both large and small thickness/width ratios.     

Authors' reply to "Comments on On deembedding of port discontinuities in full-wave CAD models of multiport circuits"

The authors reply to the comments made by Rautio (IEEE Trans. Microwave Theory Tech., vol.52, no.10, p.2448-9, Oct. 2004) on the original paper by Okhmatovski et al. (IEEE Trans. Microwave Theory Tech., vol.51, p.2355-65, Dec. 2003).     

Evaluation of layered media Green's functions via rational function fitting

An efficient rational function fitting methodology, called VECTFIT, is utilized toward the closed-form evaluation of the Sommerfeld integrals associated with electromagnetic Green's functions in planar layered media. VECTFIT approximates the component of the spectrum of the Green's function that remains after the extraction of the primary source contribution and the quasistatic part with a rational function, thus enabling a robust and expedient closed-form evaluation of the Sommerfeld integral for electromagnetic potentials and associated field quantities.     

On deembedding of port discontinuities in full-wave CAD models of multiport circuits

A systematic numerical methodology is proposed for the accurate deembedding of multiport discontinuities in full-wave numerical models of multiconductor microwave and millimeter-wave passive structures and high-speed digital interconnects. The discussed methodology is based on the earlier-proposed short-open calibration (SOC) procedure. The latter being a numerical analog of the experimental transmission-thru-reflection technique provides a consistent removal of the feed networks of the device-under-test over a wide range of frequencies. The treatment of multiport topologies is achieved through the continuation of the original scalar SOC method into the vector space. The new vector SOC method is easily combined with integral-equation-based method-of-moments electromagnetic-field solvers and allows for substantial flexibility in the choice of excitation mechanisms. Such commonly used method-of-moments driving schemes as ports locally backed up by a vertical conducting wall, ungrounded-internal differential ports, and via-mounted ports can be accurately deembedded within the framework of the vector SOC. Several numerical experiments are provided to validate the proposed multiport deembedding methodology and demonstrate its application.     

Efficient calculation of the electromagnetic dyadic Green's function in spherical layered media

A numerical methodology is proposed for the evaluation of the electromagnetic fields of an electric or magnetic dipole of arbitrary orientation in spherical stratified media. The proposed methodology is based on the numerical solution of the differential equation in the radial coordinate that is satisfied by each coefficient in the spherical harmonics expansion of the governing Helmholtz equation. More specifically, the finite-difference solution of this equation may be cast in a pole-residue form that allows the analytic evaluation of the spherical harmonics series by means of the Watson transformation. Thus, a closed-form expression is obtained for the electromagnetic fields in terms of a short series of associated Legendre functions P/sup m//sub /spl nu//(cos(/spl theta/)) of integer order m and complex degree /spl nu/. The number of terms in the series is strongly dependent on the angle /spl theta/, and decreases very fast when the point of observation moves away from the source. The method allows for arbitrary variation in the permittivity and permeability profiles in the radial directions.     

Novel closed-form Green's function in shielded planar layered media

A new method is proposed for the construction of closed-form Green's function in planar, stratified media between two conducting planes. The new approach does not require the a priori extraction of the guided-wave poles and the quasi-static part from the Green function spectrum. The proposed methodology can be easily applied to arbitrary planar media without any restriction on the number of layers and their thickness. Based on the discrete solution of one-dimensional ordinary differential equations for the spectral-domain expressions of the appropriate vector potential components, the proposed method leads to the simultaneous extraction of all Green's function values associated with a given set of source and observation points. Krylov subspace model order reduction is used to express the generated closed-form Green's function representation in terms of a finite sum involving a small number of Hankel functions. The validity of the proposed methodology and the accuracy of the generated closed-form Green's functions are demonstrated through a series of numerical experiments involving both vertical and horizontal dipoles     

A new technique for the derivation of closed-form electromagnetic Green's functions for unbounded planar layered media

A closed-form electromagnetic Green's function for unbounded, planar, layered media is derived in terms of a finite sum of Hankel functions. The derivation is based on the direct inverse Hankel transform of a pole-residue representation of the spectral-domain form of the Green's function. Such a pole-residue form is obtained through the solution of the spectral-domain form of the governing Green's function equation numerically, through a finite-difference approximation, rather than analytically. The proposed methodology can handle any number of layers, including the general case where the planar media exhibit arbitrary variation in their electrical properties in the vertical direction. The numerical implementation of the proposed methodology is straightforward and robust, and does not require any preprocessing of the spectrum of the Green's function for the extraction of surface-wave poles or its quasi-static part. The number of terms in the derived closed-form expression is chosen adaptively with the distance between source and observation point as parameter. The development of the closed-form Green's function is presented for both vertical and horizontal dipoles. Its accuracy is verified through a series of numerical examples and comparisons with results from other established methods.     

Finite-thickness conductor models for full-wave analysis of interconnects with a fast Integral equation method

In this paper, a fast electromagnetic integral equation solution methodology is proposed for the frequency-domain modeling of lossy, interconnect structures. The proposed method utilizes a two-layer model for thick conductors where the unknown current density inside the rectangular wire strips is approximated in terms of equivalent surface currents placed at the top and bottom sides of the strip which, for the purposes of this paper, are assumed to be substantially larger than the side strips. A matrix impedance relationship, which depends on the conductor thickness and its material properties, is established between the two surface current densities to account for the skin effect behavior of the field within the conductor. The selected assignment of the unknown current densities on the planes associated with the top and bottom sides of the metallization, combined with the planarity of multilayered interconnect structures, makes possible the application of the conjugate gradient fast Fourier transform (FFT) algorithm for the computationally efficient prediction of the electromagnetic response of multiport interconnect structures. Applications of the resulting fast integral equation solver to the electromagnetic modeling of interconnect circuits with thick lossy conductors from near dc to multigigahertz frequencies are used to demonstrate the validity of the method and quantify its computational efficiency.     

Loop-tree implementation of the adaptive integral method (AIM) for numerically-stable, broadband, fast electromagnetic modeling

The adaptive integral method (AIM) is implemented in conjunction with the loop-tree (LT) decomposition of the electric current density in the method of moments approximation of the electric field integral equation. The representation of the unknown currents in terms of its solenoidal and irrotational components allows for accurate, broadband electromagnetic (EM) simulation without low-frequency numerical instability problems, while scaling of computational complexity and memory storage with the size of the problem are of the same order as in the conventional AIM algorithm. The proposed algorithm is built as an extension to the conventional AIM formulation that utilizes roof-top expansion functions, thus providing direct and easy way for the development of the new stable formulation when the roof-top based AIM is available. A new preconditioning strategy utilizing near interactions in the system which are typically available in the implementation of fast solvers is proposed and tested. The discussed preconditioner can be used with both roof-top and LT formulations of AIM and other fast algorithms. The resulting AIM implementation is validated through its application to the broadband, EM analysis of large microstrip antennas and planar interconnect structures.