Efficient and accurate approximation of infinite series summation using asymptotic approximation and fast convergent series

We present an approach for very quick and accurate approximation of infinite series summation arising in electromagnetic problems. This approach is based on using asymptotic expansions of the arguments and the use of fast convergent series to accelerate the convergence of each term. It has been validated by obtaining very accurate solution for propagation constant for shielded microstrip lines using spectral domain approach (SDA). In the spectral domain analysis of shielded microstrip lines, the elements of the Galerkin matrix are summations of infinite series of product of Bessel functions and Green\u27s function. The infinite summation is accelerated by leading term extraction using asymptotic expansions for the Bessel function and the Green\u27s function, and the summation of the leading terms is carried out using the fast convergent series

Fast full-wave analysis of multistrip transmission lines based on MPIE and complex image theory

The mixed-potential electric-field integral equation is used in conjunction with the Galerkin's method and complex image theory for analyzing a transmission line with multiple strips embedded in different layers of a multilayered uniaxially anisotropic dielectric substrate. The two-dimensional Green's functions for the scalar and vector potentials are analytically obtained in the space domain due to the approximation of its spectral-domain version with complex images, thus avoiding lengthy numerical evaluations. Double integrals involved in the computation of Galerkin's matrix entries are quasi-analytically carried out for the chosen basis functions, which are well suited to the problem

Accurate electromagnetic full-wave modeling for interconnects in semiconductor integrated circuits

Semiconductor-based integrated circuits have become the mainstream for very-large-scale integration systems such as high-speed digital circuits, radio-frequency integrated circuits, and even monolithic microwave integrated circuits. The shrinking feature size and increasing frequency promote high integration density and interconnection complexity that demand high-accuracy modeling techniques. The current design paradigm has shifted from the transistor-driven design to the interconnect-driven design. Thus the accurate electromagnetic full-wave modeling of on-chip interconnect becomes critical for the computer-aided design tools to analyze the overall system performance.;In this research, the full-wave spectral domain approach is implemented to investigate the electromagnetic properties of multilayered transmission lines with semiconductor substrates. In particular, finite thin metallization components, such as the thin metal ground layer and signal strips, are focused on. The thin metal ground layer is generally designed as a shield or a ground plane to depress the coupling and noise from neighboring components. But its fabricated thickness is often a small fraction of one micron, which may allow electromagnetic fields to penetrate through at some low frequencies. Such electromagnetic leakage phenomena play a significant role in the overall dispersive performance of transmission lines, and their consideration is inevitable.;For the spectral domain approach, the metallization layer can be rigorously modeled as a dielectric with a complex permittivity. However, due to the large conductivity of metal, the conventional transfer matrix method has potential overflow problems in obtaining the multilayered Green\u27s function. In our research, a new formulation of the cascaded matrix is developed to overcome such numerical difficulties. Based on this formulation, the complete characteristics of multilayered transmission lines with thin metallization components are studied by parameters like the propagation constant, attenuation per unit length, field distribution, characteristic impedance, transient response, and extracted resistance, inductance, capacitance, and conductance of equivalent circuits. The parallel-plate waveguide model is applied to study a metal-insulator-metal-semiconductor structure. The first- and second-order low-frequency approximations for the fundamental propagation mode are derived with corresponding equivalent circuit models. In addition, other approximate models for the thin metal ground are compared numerically to assess their validity.;Two transmission lines with the metal-insulator-metal-semiconductor and the metal-insulator-metal-insulator structures are analyzed. Numerical results indicate that the thin metallization components have significant impacts on the propagation characteristics. The thin metal layer can enhance or even excite the slow-wave mode. Thus, it is necessary to take these effects into account to achieve accurate and reliable analysis of integrated circuit interconnects from dc to millimeter-wave frequencies

Electromagnetic and acoustic propagation in strip lines and porous media

Wave propagation in two physical structures is described and analyzed in this dissertation. In the first problem, the propagation of a normally incident plane acoustic wave through a three dimensional rigid slab with periodically placed holes is modeled and analyzed. The spacing of the holes A and B, the wavelength λ and the thickness of the slab L are order one parameters compared to the characteristic size D of the holes, which is a small quantity. Scattering matrix techniques are used to derive expressions for the transmission and reflection coefficients of the lowest mode. These expressions depend only on the transmission coefficient, r0 of an infinitely long slab with the same configuration. The determination of r0 requires the solution of an infinite set of algebraic equations. These equations are approximately solved by exploiting the small parameter D/√AB. Remarkably, this structure is transparent at certain frequencies which could prove useful in narrow band filters and resonators. In the second problem, a systematic mathematical approach is given to find the solutions of microstrip transmission lines. Specifically, we employ an asymptotic method to determine an approximation to the field components and propagation constant when the wavelength is much bigger than the thickness of the substrate. It is found that the transverse electrical and magnetic fields can be expressed in terms of two potential functions which are elliptic in character and are coupled through the longitudinal electrical field boundary conditions. The solvability conditions for the longitudinal magnetic field yield an approximation to the propagation constant. Transmission line equations are also obtained for coupled microstrip transmission lines and single microstrips with smoothly changing widths by using the same techniques

The Unified-FFT Method for Fast Solution of Integral Equations as Applied to Shielded-Domain Electromagnetics

Electromagnetic (EM) solvers are widely used within computer-aided design (CAD) to improve and ensure success of circuit designs. Unfortunately, due to the complexity of Maxwell\u27s equations, they are often computationally expensive. While considerable progress has been made in the realm of speed-enhanced EM solvers, these fast solvers generally achieve their results through methods that introduce additional error components by way of geometric approximations, sparse-matrix approximations, multilevel decomposition of interactions, and more. This work introduces the new method, Unified-FFT (UFFT). A derivative of method of moments, UFFT scales as O(N log N), and achieves fast analysis by the unique combination of FFT-enhanced matrix fill operations (MFO) with FFT-enhanced matrix solve operations (MSO). In this work, two versions of UFFT are developed, UFFT-Precorrected (UFFT-P) and UFFT-Grid Totalizing (UFFT-GT). UFFT-P uses precorrected FFT for MSO and allows the use of basis functions that do not conform to a regular grid. UFFT-GT uses conjugate gradient FFT for MSO and features the capability of reducing the error of the solution down to machine precision. The main contribution of UFFT-P is a fast solver, which utilizes FFT for both MFO and MSO. It is demonstrated in this work to not only provide simulation results for large problems considerably faster than state of the art commercial tools, but also to be capable of simulating geometries which are too complex for conventional simulation. In UFFT-P these benefits come at the expense of a minor penalty to accuracy. UFFT-GT contains further contributions as it demonstrates that such a fast solver can be accurate to numerical precision as compared to a full, direct analysis. It is shown to provide even more algorithmic efficiency and faster performance than UFFT-P. UFFT-GT makes an additional contribution in that it is developed not only for planar geometries, but also for the case of multilayered dielectrics and metallization. This functionality is particularly useful for multi-layered printed circuit boards (PCBs) and integrated circuits (ICs). Finally, UFFT-GT contributes a 3D planar solver, which allows for current to be discretized in the z-direction. This allows for similar fast and accurate simulation with the inclusion of some 3D features, such as vias connecting metallization planes

The finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides

This thesis presents a new variational finite element formulation and its implementation for the analysis of microwave and optical waveguide problem with arbitrarily- shaped cross section, inhomogeneous, transverse-anisotropic, and lossy dielectrics. In this approach, the spurious, nonphysical solutions, which ordinarily appear interspersed with the correct results of earlier vectorial finite element methods and thus have been the most serious problem in finite element analysis of waveguides, are totally eliminated. In this formulation either the propagation constant or the frequency may be treated as eigenvalues of the resulting generalized eigenvalue problem. This formulation also has the capability to find complex modes of lossless waveguides. Furthermore, the numerical efficiency of the solution is maximized since this formulation uses the most economical representation of a problem, in terms of only two vector components. This is achieved without losing the sparsity of the matrices of the resultant eigenvalue equation, which only depends on the topology of mesh used. This property is very important for solving large-size problems by efficient sparse matrix algorithms. In this work, a basic vector wave equation which involves only transverse components of magnetic field is straightforwardly derived from Maxwell equations. This differential equation incorporates the divergence condition V.B = 0 and leads to a canonical form of the resultant eigenvalue equation. The Local Potential Method is used to obtain the variational formulation. When implementing the finite element method, the Rayleigh-Ritz procedure is used to find stationary values of the functional to get the resulting generalized matrix eigenvalue equation. To show the validity and applicability of the method, a series of examples of microwave and optical waveguides including inhomogeneity, anisotropy and loss are studied. These examples show good accuracy and complete absence of spurious modes, demonstrating the effectiveness of the new formulation developed