Low Wavenumber Extraction Research Articles

Broadband Green's Function (BBGFL) Method With Imaginary Wavenumber Extractions for Simulations of Radiated Emissions From Irregular Shaped Printed Circuit Board

In this article, we apply the broadband Green's function for fast full-wave simulations of the radiated emissions from the printed circuit boards (PCBs) of irregular shape. Changes were made on the previous version that improve the computational efficiency and accuracies of the solutions. The improvements include imaginary wavenumber extractions for the acceleration of convergence of modal expansions without the need of high-order quadrature, and the fast-exact normalization of modes. Results of the radiated fields are illustrated over a broad frequency range up to 10 GHz and for irregular shaped PCB up to 4 in in sizes. The results are in good agreement with the method of moment for both the modal solutions and the radiated emissions. The results are shown to be more accurate than the previous version of broadband Green's functions with low wavenumber extractions. The significant computational efficiency makes this technique useful for electronic design automation to electromagnetic interference/electromagnetic compatibility applications.

BROADBAND GREEN’S FUNCTION WITH HIGHER ORDER LOW WAVENUMBER EXTRACTIONS FOR AN INHOMOGENEOUS WAVEGUIDE WITH IRREGULAR SHAPE

The method of broadband Green's functions with low wavenumber extractions (BBGFL) is used to calculate Green's function for inhomogeneous waveguides filled with different dielectrics and with irregular boundaries. To construct the BBGFL modal solutions, we derive governing equations of the linear eigen-matrix problem and orthonormalization condition. In BBGFL, the Green's function is represented in modal expansions with convergence accelerated by higher order low wavenumber extractions. To obtain a linear eigenvalue problem for the modes, we use two BBGFLs of rectangular waveguides with two dielectric wavenumbers. The orthonormalized mode functions are used to construct the Green's function. Current wavenumber derivatives and Green's function wavenumber derivatives are computed by a single low wavenumber MoM impedance matrix. The wavenumber derivatives are used to accelerate the convergence of modal summations to 6th order. Numerical results are illustrated and compared with the direct MoM method of using free space Green's function. Results show accuracies and computation efficiencies for broadband simulations of Green's functions.

Modeling of Scattering in Arbitrary-Shape Waveguide Using Broadband Green's Function With Higher Order Low Wavenumber Extractions

The broadband Green's function with low wavenumber extractions (BBGFL) consists of low wavenumber method of moments (MoM) solutions combined with modal expansions. BBGFL is suitable for broadband simulations because the modal functions are independent of frequencies. The modal expansion in this paper has sixth-order convergence. Using wavenumber derivatives of the surface current, it is shown that the higher order extraction requires only a single MoM impedance matrix on the left-hand side. For the same number of modes, the sixth-order convergence has significant improvement in accuracy over the ${\text{second}}$ - and ${\text{fourth}}$ -order convergence. Scattering in an arbitrary-shape waveguide is modeled by using a surface integral equation formulated with the sixth-order BBGFL. Using BBGFL, the unknowns are only on the surface of the scatterer. Results of simulations are illustrated and the results are in good agreement with those of direct MoM.

Scattering of waves by a half-space of periodic scatterers using broadband Green's function.

An efficient scatterer-free full-wave solution for plane wave scattering from a half-space of two-dimensional (2D) periodic scatterers is derived using broadband Green's function. The Green's function is constructed using band solutions of the infinite periodic structure, and it satisfies boundary conditions on all the scatterers. A low wavenumber extraction technique is applied to the Green's function to accelerate the convergence of the modal expansion. This facilitates the Green's function with low wavenumber extraction (BBGFL) to be evaluated over a broadband as the modal solutions are independent of wavenumber. Coupled surface integral equations (SIE) are constructed using the BBGFL and the free-space Green's function respectively for the two half-spaces with unknowns only on the interface. The method is distinct from the effective medium approach which represents the periodic scatters with an effective medium. This new approach provides accurate near-field solutions around the interface with localized field patterns useful for surface plasmon polaritons and topological edge states examinations.

Fast Broadband Modeling of Traces Connecting Vias in Printed Circuit Boards Using Broadband Green’s Function Method

In this paper, the broadband Green’s function with low wavenumber extraction (BBGFL) is applied to modeling of traces connecting vias in arbitrarily shaped power/ground planes of printed circuit boards (PCBs). The proposed modeling method is a hybrid technique based on mode decomposition: BBGFL is used to fast compute the vias in power/ground planes, MoM is used to calculate the impedance matrix of the traces, and a physical circuit model is then applied to combine the vias and traces in the power/ground planes. The present method is compared to the method of moments (MoM) and the commercial tool HFSS. Results show that the results of the present technique are in good agreement with MoM and HFSS on the radiated emissions from PCBs. BBGFL is several hundred times faster than HFSS in CPU time for broadband simulations. The technique provides a fast broadband solution for system-level modeling of high-speed interconnects with vias and traces in electronic design automation design and applications.

Green’s functions, including scatterers, for photonic crystals and metamaterials

The Green’s functions are physical responses due to a single point source in a periodic lattice. The single point source can also correspond to an impurity or a defect. In this paper, the Green’s functions, including the scatterers, for periodic structures such as in photonic crystals and metamaterials are calculated. The Green’s functions are in terms of the multiband solutions of the periodic structures. The Green’s functions are broadband solutions so that the frequency or wavelength dependences of the physical responses can be calculated readily. They are obtained by integrating the periodic Green’s function including the scatterers in the Brillouin zone. Low wavenumber extraction methods are used to accelerate the convergence rate of the multiband expansions. The low wavenumber component represents the reactive near field. The band solutions of the periodic structure are obtained from a surface integral equation solution, which is effectively converted to a linear eigenvalue problem, giving multiple band solutions simultaneously. Numerical results are illustrated for the band modal fields, the periodic Green’s functions, and the single point source Green’s functions for two-dimensional (2D) perfect-electric-conductor (PEC) scatterers in a 2D lattice.

Fast Electromagnetic Analysis of Emissions From Printed Circuit Board Using Broadband Green's Function Method

In this paper, we apply the broadband Green's function with low wavenumber extraction (BBGFL) to fast modeling of the radiated emissions from printed circuit boards (PCBs). We study power bus structures for electromagnetic interference/compatibility (EMI/EMC). We use the BBGFL to compute the Green's function over broadband frequencies along the boundary walls of arbitrarily shaped power/ground planes. The broadband Green's function is calculated by using modal expansions and low wavenumber extraction. Because the modal expressions are independent of frequencies, they are computed once and used for all frequencies. The radiated fields are derived from the equivalent magnetic surface currents on boundary walls using Huygen's equivalent principle. Results are illustrated up to 10 GHz and for arbitrary shaped PCB up to 5 inches in size. The accuracy and computational efficiency of the present method are compared with the method of moments (MoM) and the commercial tool HFSS. The BBGFL results are in good agreement with MoM and HFSS. The BBGFL is several hundred times faster than MoM and HFSS for broadband simulations. The significant improvement in computational efficiency makes this technique useful for electronic design automation to EMI/EMC applications.

Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scatterers using broadband Green's function with low wavenumber extraction (BBGFL).

The broadband Green's function with low wavenumber extraction (BBGFL) is applied to the calculations of band diagrams of two-dimensional (2D) periodic structures with dielectric scatterers. Periodic Green's functions of both the background and the scatterers are used to formulate the dual surface integral equations by approaching the surface of the scatterer from outside and inside the scatterer. The BBGFL are applied to both periodic Green's functions. By subtracting a low wavenumber component of the periodic Green's functions, the broadband part of the Green's functions converge with a small number of Bloch waves. The method of Moment (MoM) is applied to convert the surface integral equations to a matrix eigenvalue problem. Using the BBGFL, a linear eigenvalue problem is obtained with all the eigenmodes computed simultaneously giving the multiband results at a point in the Brillouin zone Numerical results are illustrated for the honeycomb structure. The results of the band diagrams are in good agreement with the planewave method and the Korringa Kohn Rostoker (KKR) method. By using the lowest band around the Γ point, the low frequency dispersion relations are calculated which also give the effective propagation constants and the effective permittivity in the low frequency limit.

BROADBAND GREEN'S FUNCTION WITH LOW WAVENUMBER EXTRACTION FOR ARBITRARY SHAPED WAVEGUIDE AND APPLICATIONS TO MODELING OF VIAS IN FINITE POWER/GROUND PLANE

In this paper we developed the method of broadband Green's function with low wavenumber extraction (BBGFL) for arbitrary shaped waveguide. The case of Neumann boundary condition is treated. The BBGFL has the advantage that when using it to solve boundary value problems in a waveguide, the boundary conditions have been satisfied already. The broadband Green's function is expressed in modal expansion of modes that are frequency independent. To accelerate the convergence of the Green's function, a low wavenumber extraction is performed. The singularity of the Green's function is also extracted by such low wavenumber extraction. Numerical results show that BBGLF and direct MoM are in good agreement. We next illustrate the application of BBGFL for broadband simulations of vias in printed circuit boards (PCB) by combining with the method of Foldy-Lax multiple scattering equation. The results show that BBGFL are in good agreement with MoM and HFSS. It is also shown that BBGFL is many times faster than direct MoM and HFSS. The computational efficiency in broadband simulations makes this technique useful for fast computer-aided design (CAD). The effects of waveguide or cavity structures are critical for the electrical performance of electronic devices and components in signal integrity (SI), power integrity (PI), electromagnetic interference (EMI), and electromagnetic compatibility (EMC). Harmful electromagnetic signal noises or interferences are often generated and amplified at the resonant frequencies of the waveguide or cavity structures. The issues deteriorate when the electronic devices or computer systems operate at higher frequency or faster speed. In printed circuited boards (PCBs), two adjacent power/ground planes form a waveguide/cavity structure. The propagating modes satisfy the PMC (Neumann boundary conditions) at the edges of PCB power/ground plane structures. The power/ground plane structures are the key root causes in SI/PI and EMI/EMC problems. Vias are used for vertical interconnects for multilayer PCBs. At frequencies near the resonant frequencies, the propagating electromagnetic waves excite resonant modes, that result in strong edge radiations. These cause EMI/EMC problems. The switching noises induced by voltage regulator module (VRM) generate voltage fluctuations and lead to PI problems. The high frequency power noise can also couple into signal vias and cause SI/PI coupling issues. Therefore, the modeling of PCB cavity with vias is critical in practical designs and applications of high speed PCBs and packages. Fast and accurate modeling technique is desired for broadband simulations in electronic design and application. The finiteness of the parallel power/ground planes make them waveguide/cavity structures. The power/ground planes are also of arbitrary shape. Commercial tools such as HFSS provide solutions for the analysis of the via-cavity coupling problem. The tools require large CPU and memory and are not suited for broadband analysis. The physical problem is that of TM modes in a cavity with PMC boundary conditions on the side walls. Various methods have been used for waveguide

BROADBAND CALCULATIONS OF BAND DIAGRAMS IN PERIODIC STRUCTURES USING THE BROADBAND GREEN'S FUNCTION WITH LOW WAVENUMBER EXTRACTION (BBGFL)

We apply the method of the Broadband Green's Functions with Low wavenumber extraction (BBGFL) to calculate band diagrams in periodic structures. We consider 2D impenetrable objects placed in a 2D periodic lattice. The low wavenumber extraction is applied to the 2D periodic Green's function for the lattice which is used to formulate the surface integral equation. The low wavenumber extraction accelerates the convergence of the Floquet modes expansion. Using the BBGFL to the surface integral equation and the Method of Moments gives a linear eigenvalue equation that gives the broadband (multi-band) solutions simultaneously for a given point in the first Brillouin zone. The method only requires the calculation of the periodic Green's function at a single low wavenumber. Numerical results are illustrated for the 2D hexagonal lattice to show the computational efficiency and accuracy of the method. Because of the acceleration of convergence, an eigenvalue problem with dimensions 49 plane wave Floquet modes are sufficient to give the multi-band solutions that are in excellent agreement with results of the Korringa Kohn Rostoker (KKR) method. The multiband solutions for the band problem and the complementary band problem are also discussed.