Transients on lossy transmission lines with arbitrary boundary conditions

In this work the problem of transients on a lossy transmission line terminated by an arbitrary, including nonlinear, load is formulated. The tranmission line parameters are the constantsR, L, G, andC. The exact relation between the input and output voltages and currents in the form of two coupled integral equations is derived by the Laplace transform method. It is shown that the kernels of the integral equations may be represented in terms of either Lommel functions or integrals involving zeroth order modified Bessel functions. Simultaneous (numerical) solutions of these integral equations fulfilling the boundary conditions at the input and output of the line yields the input and output voltages and currents on the line. Finally the exact analytical relations in time domain of the voltage and current at an arbitrary point on the line (and the voltages and currents at the input and output terminals) are derived. In all parts, the problem has been formulated in such a way as to impose the causality condition explicitly.     

Optimum Setting for Proportional Controller

On the basis of four common criteria?integral square error (ISE), integral absolute error (IAE), time integral-square-error (ITSE), and time integral absolute error (ITAE)?optimal lines for typical sampled data servo drives for computer numerical control (CNC) are derived and approximated by simple equations. The objective is to provide the control engineer with a simple equation for the selection of the control parameters. A line of critical damping is also derived and simply approximated.     

Synthesis of nonuniform transmission lines with terminaldiscontinuities

An integral equation was previously derived by the authors (ibid., vol.AP-34, p.546-53, Apr. 1986) for the inverse problem associated with finite-length nonuniform lines having known impedance discontinuities at the input and output. A procedure for determining the characteristic impedance profile of the line from the solution of the integral equation was also presented. Here, the numerical aspects of this inverse problem are treated. This equation is solved using general expansions of both the kernel and unknown functions, thereby reducing the integral equation to a system of linear algebraic equations. This makes possible a numerical implementation of the theory by which the impedance profile of a nonuniform line may be reconstructed from given spectral data. The realizability aspects of the problem are treated and several examples of computer-assisted profile inversion are presented     

Scattering from narrow rectangular filled grooves

The solution of the integral equation for a small width rectangular groove is considered. It is shown that by retaining the dominant mode supported by the rectangular groove, the resulting quasi-static integral equations are comparable to those associated with the perfectly conducting narrow strip. They are, therefore, amenable to analytic solution yielding the exact field distribution or equivalent currents across the groove's aperture. The derived currents exhibit the same edge behavior as that associated with the currents of a perfectly conducting half-plane. The corresponding current behavior based on a (numerical) impedance simulation of the groove is quite different. However the resulting echowidths are comparable. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated     

Electromagnetic field coupling to a line of finite length: theoryand fast iterative solutions in frequency and time domains

A system of integral-differential equations for evaluating currents and voltages induced by external electromagnetic fields on a finite-length horizontal wire above a perfectly conducting ground is derived under the thin wire approximation. Based on perturbation theory, an iterative procedure is proposed to solve the derived coupling equations, where the zeroth iteration term is determined by using the transmission line (TL) approximation. The method can be applied both in the frequency and in the time domains. The proposed iterative procedure converges rapidly to the exact analytical solution for the case of an infinite line, and to the NEC solution for a line of finite length. Moreover, with only one iteration, an excellent approximation to the exact solution can be obtained. The method is applied to assess the validity of the TL approximation for plane wave coupling to an overhead line of finite length. It is shown that the resulting errors for the early-time response are generally higher than those corresponding to infinite lines     

Theory of microstrip lines on artificial periodic substrates

This paper presents the theory of a microstrip line on artificial periodic substrates. A two-stage moment method in conjunction with an array-scanning scheme is proposed for the microstrip characterization. The analytic and numerical methods dealing with the interaction of microstrip components (continuous plane-wave spectrum) with artificial periodic materials (discrete plane-wave spectrum, Floquet modes) are discussed. The method of solution involves two stages of vector integral equations and moment methods. The first integral-equation formulation is to find the Green's function for a planar periodic structure. A spectral-domain moment method is applied to the second vector integral equation to determine the fields or currents on the circuit components and the associated parameters of interest. Guided-wave characteristics of a microstrip line on artificial periodic substrates, including the propagation constant and the characteristic impedance, are investigated. Propagation bandgap of a microstrip line due to periodic elements is characterized. Experiment on a three-layer microstrip-line structure with a periodic mid-layer is conducted to validate the theory     

Numerical solution of integral equations for dielectric objects ofprismatic shapes

A numerical method to investigate scattering from dielectric geometries of prismatic shapes has been developed. The surface integral equations are formulated by Schelkunoff's equivalence principle in terms of equivalent surface electric and magnetic currents. To solve these integral equations for the unknown currents, the object's cross-section is mapped onto a circle. In the transformed space, Fourier type entire-domain basis functions are used in the cross section and triangular subdomain basis functions are selected along the generating curve to represent the currents. A moment method is then used to reduce the integral equations to a matrix equation to compute the current coefficients. It is found that the transformation of the object's surface to a circular shape improves the convergence of the current mode in the cross-section. However, the current modes are coupled on the surface and the matrix equation includes all the modes     

Electromagnetic scattering from electrically large coated flat and curved strips: Entire domain Galerkin formulation

Efficient numerical solutions are presented for electromagnetic scattering for classes of electrically large, coated, perfectly conducting strips which are flat or curved. The formulation is based on the solution of a coupled system of electric- and magnetic-field integral equations using the method of moments (MM). Entire domain Galerkin representations for the currents are used on the surface of the coating and at the coating-conductor interface. The resulting symmetric matrix equation is well conditioned and admits rapid, accurate solutions. Numerical results are presented for various coating thicknesses, strip widths, and curvatures for the transverse electric (TE) and transverse magnetic (TM) cases. The convergence of the Galerkin solution is examined as a function of these parameters. The effect of the edge approximation on the choice of expansion functions is discussed. The numerical results are compared with experimental measurements.     

A Method to Measure Radar Cross Section Parameters of Antennas

A new method to measure radar cross section (RCS) parameters of antennas is presented in this paper. This method relies on an equation derived for the received signal power which is represented as the superposition of structural-mode scattered, antenna-mode scattered, and leakage signals. The method also measures relative phase and provides the effect of the load connected to the antenna. The received signal power equation is solved using minimum mean square error estimation, and the solution is used to obtain structural-mode RCS, antenna-mode RCS, and relative phase of an antenna by applying short, open, and match load cases. A measurement example of RCS parameters of a microstrip patch antenna using a simple experimental setup is presented. The structural-mode RCS, antenna-mode RCS, and relative phase factor $cos(phi_{0})$ of the antenna, obtained using minimum mean square error estimation solution, are ${-}18.307$ dBsm, ${-}20.386$ dBsm, and 0.970, respectively. The present method requires less lengthy experimental measurements, while providing results that are more informative and accurate as obtained from previous methods.      

Electromagnetic analysis of an antenna embedded in a compositeenvironment

This paper addresses the problem of an antenna embedded in a hole dug in the ground. The composite medium configuration consists of a half-space dielectric (representing the Earth-air interface) containing a cylindrical hole filled with a different dielectric medium. The wire antenna resides within this hole, on the axis. The solution strategy is based on decomposing the problem into simpler subproblems, which are treated sequentially. First we calculate a numerical dyadic Green's function for the composite medium by solving an integral equation formulated over a background consisting of the unperturbed dielectric half space (for which the Green's functions are known in a spectral integral form). This integral equation is solved via the fictitious currents method, which is a special case of the method of moments. We then solve the integral equation for the antenna currents using this numerical Green's function and determine the input impedance and radiation pattern     

Integral equation based analysis of scattering from 3-D inhomogeneous anisotropic bodies

This paper presents an integral equation based scheme to analyze scattering from inhomogeneous bodies with anisotropic electromagnetic properties. Both the permittivity and permeability are assumed to be generalized tensors. Requisite integral equations are derived using volume equivalence theorem with the electric and magnetic flux densities being the unknown quantities. Matrix equations are derived by discretizing these unknowns using three dimensional Rao-Wilton-Glisson basis functions. Reduction of the integral equation to a corresponding matrix equation is considerably more involved due to the presence of anisotropy and the use of vector basis function; methods for evaluation of the integrals involved in the construction of this matrix is elucidated in detail. The method of moments technique is augmented with the fast multipole method and a compression scheme. The latter two enable large scale analysis. Finally, several numerical results are presented and compared against analytical solutions to validate the proposed scheme. An appendix provides analytical derivations for the formulae that are used to validate numerical method, and the necessary formulae that extends the approach presented herein to the analysis of scattering bianisotropic bodies.     

Full-wave analysis of microwave scattering from short vegetation: an investigation on the effect of multiple scattering

A full-wave solution for polarimetric scattering from a cluster of randomly oriented three-dimensional lossy dielectric structures above an impedance surface is presented to investigate the importance of multiple scattering. The problem is formulated using an integral equation in conjunction with the exact image representation of dyadic Green's function for the half-space problem. Then, the integral equation is solved for the induced equivalent polarization currents using the method of moments. The accuracy of the numerical code is verified using other existing numerical results and experimental observations. The model is then used to examine the effect of multiple scattering among a cluster of relatively short stems and is shown that multiple scattering significantly affects the cross-polarized backscatter whereas it has a moderate effect on the copolarized backscattering depending on the stem density.     

Numerical solution of time domain integral equations for arbitrarily shaped conductor/dielectric composite bodies

In this work, we present a numerical solution of the coupled time domain integral equations to obtain induced currents and scattered far fields on a three-dimensional, arbitrary shaped conducting/dielectric composite body illuminated by a Gaussian electromagnetic plane wave pulse. The coupled integral equations are derived utilizing the equivalence principle. The solution method is based on the method of moments and involves the triangular patch modeling of the composite body, in conjunction with the patch basis functions. Detailed mathematical steps along with several numerical results are presented to illustrate the efficacy of this approach.     

Dynamic analysis of a microstrip line over a perforated groundplane

A full wave analysis is presented to compute the characteristic impedance and propagation constant of a microstrip line over a perforated ground plane. The perforations in the ground plane are modeled by equivalent magnetic currents. The method of moments is applied to solve the coupled integral equations for the unknown electric current on the microstrip line and the unknown magnetic currents in the apertures. The fields are formulated using the space domain Sommerfeld type Green's functions. The matrix pencil technique is used to obtain the amplitude and the propagation constant of the fundamental modes for both current and the voltage on the microstrip line. Typical numerical results are given     

Root-mean-square measure of nonlinear effects to the transient response of thin wires

A root-mean-square (rms) measure of effect of nonlinear loading on the transient response of thin wires is proposed. The transient behavior of nonlinearly loaded wires is analyzed directly in the time domain. The problem is formulated via the space-time Hallen integral equation. The equation is solved by the space-time Galerkin Bubnov boundary element procedure. Numerical results for the transient response of a thin wire computed by a time domain code based on this method are compared with results obtained from a frequency domain code. Some illustrative numerical results for the spatial distribution of the rms values of time varying currents are also presented.     

MoM solutions to building blockage of mobile satellite communications

This article presents a full-wave propagation model for arbitrary profile of building blockage in mobile satellite communications, by solving the electric field integral equation for induced surface currents using the method of moments. Asymptotic expressions are used to simplify the integrals. Scattered fields are then found by the radiation equations derived from Maxwell equations. The total received fields around different profiles of buildings are calculated as a function of space, elevation angle and frequency. The results agree well with measurements and other published data. Various useful parameters for designing robust and reliable communication systems like frequency response, average fade duration and coherence bandwidth are found. Performance of mobile satellite system is evaluated in terms of bit error rate of mobile satellite system in frequency non-selective, slowly fading channel.     

PCB上微带线的电磁耦合时域建模分析方法

基于时域有限差分方法和传输线方程,结合高效网格建模技术,文中提出了一种高效的时域建模算 法,它能有效解决微带线的电磁耦合建模问题,实现空间电磁场与微带线瞬态响应的同步计算。首先,结合经验公 式,计算得到微带线的单位长度分布参数,构建适用于微带线电磁耦合分析的传输线方程。然后,采用时域有限差 分(Finite-Difference Time-Domain, FDTD)方法,结合非均匀网格技术和自动网格生成技术,仿真得到微带线激励场, 并在每个时间步进上引入传输线方程获得等效分布源项。最后,对传输线方程使用FDTD 的中心差分格式进行离 散,实现微带线及其端接电路上瞬态响应的迭代求解。为了验证时域建模算法的正确性和高效性,通过自由空间和 屏蔽腔内PCB 上微带线电磁耦合的数值模拟,从计算精度和耗时两方面与传统FDTD 方法的计算结果进行了对比。     

Surface currents and RCS of a spherical shell with a circular aperture

The electromagnetic scattering behavior of a perfectly conducting, infinitesimally thin, spherical shell with a circular aperture is studied. A time-harmonic plane wave is symmetrically incident upon the aperture. The problem is formulated in terms of theE-field integral equation. This produces two coupled integral equations for the tangential components of the currents on the scatterer surface. The equations are cast into matrix form by application of the method of moments, and expressions for the matrix elements are derived. Calculated values of the surface currents and radar cross sections, not previously available in the open literature, are presented and discussed for several cases of interest.     

Space domain approach for the analysis of printed circuits

A numerical approach to the solution of printed circuit structures of arbitrary shapes, embedded in a single or multilayer dielectric medium is presented. The electromagnetic fields are described in terms of the classical Sommerfeld integrals. The method of moments has been used to solve the derived integral equations for the surface electric and magnetic currents flowing on the conductors and/or the electric field distribution across the apertures. The matrix pencil technique is employed to decompose the current or the voltage waves along the line into their components like the fundamental modes, higher order modes, etc. The finite structures including discontinuities like bends, T junctions, crossovers, etc. are solved for their scattering parameters utilizing this method. The main advantage of this method is the generality which allows a large variety of problems to be covered     

Theoretical and experimental characterization of transientelectromagnetic fields radiated by electrostatic discharge (ESD)currents

The problem of the evaluation of the electromagnetic (EM) field radiated by an electrostatic discharge (ESD) current is examined, and an efficient numerical code for the evaluation of the radiated field is developed. The considered radiating structures, a monopole and a loop, are analyzed in the time domain using a modified electric field integral equation (EFIE). A modification of the integral equation was introduced in order to take into account the reflection of the incident pulse at the input terminals of the antenna, when fed by a transmission line. The reflected wave is very significant and its evaluation is fundamental for the comparison of theoretical and experimental data. The ESD current flowing along the wire is determined using the method of moments in time domain (MoMTD). From the knowledge of the transient current, the radiated EM field is evaluated by a standard technique. The developed model is the first stage of a project for the characterization of the measurement environment during an ESD test. All numerical results are validated by measurements and good agreement is shown