A compact representation is given of the electric- and magnetic-type dyadic Green's functions for plane-stratified, multilayered, uniaxial media based on the transmission-line network analog along the aids normal to the stratification. Furthermore, mixed-potential integral equations are derived within the framework of this transmission-line formalism for arbitrarily shaped, conducting or penetrable objects embedded in the multilayered medium. The development emphasizes laterally unbounded environments, but an extension to the case of a medium enclosed by a rectangular shield is also included.

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Multilayered media Green's functions in integral equation formulations

Silicon integrated circuit spiral inductors and transformers are analyzed using electromagnetic analysis. With appropriate approximations, the calculations are reduced to electrostatic and magnetostatic calculations. The important effects of substrate loss are included in the analysis. Classic circuit analysis and network analysis techniques are used to derive two-port parameters from the circuits. From two-port measurements, low-order, frequency-independent lumped circuits are used to model the physical behavior over a broad-frequency range. The analysis is applied to traditional square and polygon inductors and transformer structures as well as to multilayer metal structures and coupled inductors. A custom computer-aided-design tool called ASITIC is described, which is used for the analysis, design, and optimization of these structures. Measurements taken over a frequency range from 100 MHz to 5 GHz show good agreement with theory.

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Analysis, design, and optimization of spiral inductors and transformers for Si RF ICs

Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods.

Integral Equation Methods for Electromagnetic and Elastic Waves

Spatial-domain Green's functions for multilayer, planar geometries are cast into closed forms with two-level approximation of the spectral-domain representation of the Green's functions. This approach is very robust and much faster compared to the original one-level approximation. Moreover, it does not require the investigation of the spectral-domain behavior of the Green's functions in advance to decide on the parameters of the approximation technique, and it can be applied to any component of the dyadic Green's function with the same ease.

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A robust approach for the derivation of closed-form Green's functions

The closed-form Green's functions of the vector and scalar potentials in the spatial domain are presented for the sources of horizontal electric, magnetic, and vertical electric, magnetic dipoles embedded in general, multilayer, planar media. First, the spectral domain Green's functions in an arbitrary layer are derived analytically from the Green's functions in the source layer by using a recursive algorithm. Then, the spatial domain Green's functions are obtained by adding the contributions of the direct terms, surface waves, and complex images approximated by the Generalized Pencil of Functions Method (GPOF). In the derivations, the main emphasis is to put these closed-form representations in a suitable form for the solution of the mixed potential integral equation (MPIE) by the method of moments in a general three-dimensional geometry. The contributions of this paper are: 1) providing the complete set of closed-form Green's functions in spectral and spatial domains for general stratified media; 2) using the GPOF method, which is more robust and less noise sensitive, in the derivation of the closed-form spatial domain Green's functions; and 3) casting the closed-form Green's functions in a form to provide efficient applications of the method of moments. >

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Closed-form Green's functions for general sources and stratified media

The spatial Green's function for the open microstrip structure, especially with a thick substrate, is generally represented in the time-consuming Sommerfield integrals. Through the Sommerfield identity, a closed-form spatial Green's function of a few terms is found from the quasi-dynamic images, the complex images, and the surface waves. With the numerical integration of the Sommerfeld integrals thus avoided, this closed-form Green's function is computationally very efficient. Numerical examples show that the closed-form Green's function gives less than 1% error for all substrates and source-to-field distances. >

A closed-form spatial Green's function for the thick microstrip substrate

A rapidly convergent algorithm to find the spatial simulated images of a point charge in multilayered media is presented. The simulated images turn out to be complex; i.e. they have complex amplitudes and are located at complex positions. Surprisingly, these complex images give the static field in multilayered media very accurately (errors approximately 0.1%). The examples of two- and three-layered media are examined, together with the available exact image solutions of singly or doubly infinite series. >

Complex images for electrostatic field computation in multilayered media

Simple closed-form expressions are derived for the vector and scalar potentials of a horizontal electric dipole in homogeneous and layered dielectrics between two ground planes. For the homogeneous dielectric case, an infinite number of dipole images due to the top and bottom ground planes are replaced by a few complex dipole images. For the layered dielectric case, the dipole images due to both the dielectric interfaces and the ground planes are replaced by a few complex dipole images. In addition, the waveguide modes of LSE and LSM types trapped by the two ground planes, and the surface wave modes of LSE and LSM types trapped by the dielectric slabs, both excited by the dipole, are included in the closed-form expressions. The accuracy of the closed-form expressions is confirmed by the numerical integration of spectral integrals. >

Complex images of an electric dipole in homogeneous and layered dielectrics between two ground planes

An integral equation for solving thin conducting strip problems always involves three singularities, namely, two charge singularities at the strip edges and the Green's function singularity for close proximity of source and field points This work overcomes the singularity convergence problem using Gauss-Chebyshev quadrature for the edge charges, but more importantly by a multipipe model for the Green's function singularity This model applies equally well to both two-dimensional (2-D) and three-dimensional (3-D) problems of metallic strips embedded in multilayer dielectric substrates To reduce the scope, however, this work analyzes only the quasi-TEM (transverse electromagnetic) cases of 2-D thin-strip transmission lines in multilayer dielectric substrates >

A multipipe model of general strip transmission lines for rapid convergence of integral equation singularities

A multiconductor transmission line in a multilayer dielectric medium is complicated and therefore is frequently analyzed by the simple quasi-TEM (transverse electromagnetic) approach. Unlike the full wave eigenvalue approach, the quasi-TEM approach does not give the propagation dispersion characteristics of the line. This work overcomes this problem by first obtaining the solution of each multiconductor mode from the quasi-TEM approach. Then these solutions are used as both basis and testing functions in a variational formulation of the full wave eigenvalue analysis. The result is high accuracy in the dispersive propagation constants of the multiconductor modes. By solving only for high frequency eigenvalues, not for the high frequency eigenvectors, the method is simpler and faster than the conventional full wave dispersion analyses. >

J.J. Yang

Papers

A closed-form spatial Green's function for the thick microstrip substrate

Complex images for electrostatic field computation in multilayered media

Complex images of an electric dipole in homogeneous and layered dielectrics between two ground planes

A multipipe model of general strip transmission lines for rapid convergence of integral equation singularities

A simple technique for calculating the propagation dispersion of multiconductor transmission lines in multilayer dielectric media