## Classical electromagnetism

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Underground cables such as pipe-type cables are widely used in urban power industry. In this study, an advanced thin-wire model of the pipe-type cables is 3D FDTD simulations. In this model, the multi-conductor cables are represented with two-level transmission line equations. A stabilising technique with a 1D spatial low-pass filter is proposed to maintain computational stability. Frequency-dependent losses are fully considered by using a vector-fitting technique. The proposed thin-wire model is validated with the multi-conductor transmission line theory analytically and the traditional FDTD method numerically. Good agreements are observed. It is found that the simulation maintains stability for 360,000-time steps. Compared to the traditional FDTD method, the memory space and computation time of the proposed model can be reduced by 73% and 98%, respectively. Induced lightning currents in a cable connection station are analysed. It is found that, without considering soil ionisation and soil stratification, the peak current in the metallic armour is 1.54 times as much as the one with considering these non-linear effects. It can be reduced by 9.04% and 18.6% if the cable is buried at depths of 1 m and 1.5 m, compared with the case of a 0.5 m buried depth.

We have identified the sources of the different problems plaguing the EFIE at low frequencies in both the frequency and the TD, as well as their traditional cures. Despite their apparent effectiveness, these techniques have been shown to have a limited applicability because they introduce their own set of problems which include the high computational burden of the LS decomposition and its effect on the high-refinement conditioning of the FD-EFIE and the numerical instabilities introduced by the treatment of the TD-EFIE. Techniques leveraging qH projectors, immune from the aforementioned side-effects, have been introduced to address the different aspects of the low-frequency breakdown of the FD formulation and of the large time step breakdown of its TD counterpart. In case of the FD, using projectors allows the same re-scaling of the solenoidal and non-solenoidal parts of the RWG space as traditional LS, but it has the added benefits of not requiring identification of the global loops of the structure as well as not introducing any further high-refinement ill-conditioning. In the TD case, the projectors are still used to separate the loop and star parts of the discretized space, but this separation is used to apply the correct derivative and integrative terms to the different parts of the operators. Coupled with an adequate mixed time-discretization scheme, this technique fully addresses the low-frequency limitations of the TD-EFIE. Along with presenting these purely theoretical concepts, we have provided implemen-tation related hints, allowing the techniques presented in this chapter to be reliably and readily implemented into existing solvers. Finally, while we have addressed their low-frequency breakdown, both EFIE formulations still suffer from a high-refinement breakdown. While in standard low-frequency scenarios, a curing of low-frequency issues may suffice, for more pathological cases techniques addressing both break-downs may be required. Strategies based on qH projectors and Calderon identities have recently been introduced for the frequency and TD formulations [23,40] and should be used in this case.

In this chapter, we rigorously examine the differences in the far-field patterns (9.1) for an E-polarised scatterer with a single corner. The scattering of a plane wave by such a scatterer is formulated in Section 9.1, and an appropriate integral equation for the surface distribution on the obstacle is given. As a motivation for the analysis to follow, a brief discussion of numerical results is given in Section 9.2. In Section 9.3, the lemniscate (having a right-angled corner) and its rounded counter-part is used as a test case to establish analytic bounds for the maximum difference in the far field. An integral equation is obtained for the difference in the surface distributions on each obstacle; its approximate solution is shown to be O((kp)^{2/3}), as kp -> 0 (Theorem 9.1). It then follows that the non-dimensionalised far-field difference √κ∥μ_{ο}-μ_{ρ}∥_{∞ }is O((kp)^{4/3}), as kp -> 0 (Theorem 9.2).

This chapter describes the principal ideas and more advanced techniques of the analytical regularization method (ARM) and the semi-inversion procedure (SIP) particularly as they apply to the integral and integro-differential equations that arise in scattering and diffraction problems in electromagnetics. The purpose of this chapter is to present a clear and unified treatment of the ARM and SIP approaches that draws on some recent developments using the language of pseudo-differential operators, while clarifying its rigorous application to boundary value problems (BVP) of electromagnetics.

Green's theorem and Green's functions are applied to general dynamical systems on discrete structures in discrete time. The discrete structures may arise from spatial discretisations of continuum fields described by PDEs but are not restricted to this source and may be purely graph-theoretical in origin. The case of Maxwell's equations, spatially discretised by finite-differences on a simplicial cell complex, is introduced as an example leading to coupled discrete potentials that each satisfies a second-order discrete dynamical system, one on vertices and the other on edges. The main result is a generic expression (4.95) for the second-order discrete-time Green operator G as a polynomial in the system operator F whose order m is equal to the time step. The discrete-time matrix elements G_{tf} (m) in this expression are exactly computable in a finite number of arithmetic steps when the operator F has finite adjacency measure. The typical candidate for F is the discrete Laplacian L, which has adjacency 1. A fully discrete form of Huygens' principle is obtained for this system, which predicts the field in an exterior region D from its values at earlier times in the cut-set S consisting of pairs of elements, one in D and one in D, using explicit time-stepping. The representation of Huygens' principle in this form is a superposition of expanding wavelets each radiated from an element of the cutset S. While the general setting for the description of the dynamical system is a cell complex with Hodge dual, deletion of the metric Hodge data from the Laplacian weights leaves behind a topological dynamical system, that is essentially exploring the connectivity of a graph in discrete time steps.

An imperfectly conducting surface may support surface waves provided appropriate impedance boundary conditions (Leontovich conditions) are satisfied. Electromagnetic surface waves propagate along an impedance surface and interact with its singular points such as edges or conical vertices giving rise to the reflection and transmission of such surface waves as well as to those diffracted into the space surrounding the canonical body. In this work, we discuss a mathematical approach describing some physical processes dealing with the diffraction of surface waves by canonical singularities like wedges and cones. We develop a mathematically justified theory of such processes with the attention centred on diffraction of a skew incident surface wave at the edge of an impedance wedge. Questions of excitation of the electromagnetic surface waves by a Hertzian dipole are also addressed as well as the Geometrical Optics laws of reflection and transmission of a surface wave across the edge of an impedance wedge.

The chapter deals with a spherical-multipole expansion of scalar or electromagnetic fields in the presence of a semi-infinite elliptic cone and a uniform complex-source beam (CSB) as the incident field. The analysis is performed in sphero-conal coordinates that can be understood as generalized spherical coordinates. The corresponding coordinate surfaces include the elliptic cone, the plane angular sector, and the wedge. As the uniform CSB paraxially approximates a Gaussian beam, its waist represents a localized inhomogeneous plane wave. Hence, it is possible to exclusively illuminate a desired area of the structure-for instance the tip of the cone-by a localized plane wave. Differently from using a homogeneous plane wave as the incident field in that case the resulting spherical-multipole series converges even if the scattered far field is evaluated. Therefore, the proposed technique should allow to extract diffraction and scattering characteristics of any desired part of the scattering objects without applying series transformations as it has been found to be necessary in the case of a homogeneous plane wave.

Integral equations have proven their popularity for the electromagnetic analysis of radiation and scattering problems. The workhorse equations are the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) [1,2]. The development of these equations will be reviewed in the following for perfectly conducting targets and homogeneous dielectric targets. When applied to certain closed surfaces, the original equations exhibit uniqueness difficulties at frequencies where the target surface coincides with a resonant cavity [3]. In addition, the original EFIE and MFIE also fail under certain circumstances for electrically small bodies. Alternate integral equations were proposed to remedy those situations, and these will also be summarized in the following section. In addition, we describe the numerical solution of these equations and report the progress made in recent years associated with the use of hierarchical vector basis functions, and the recent use of singular basis functions.

The Wiener-Hopf (WH) technique is a very powerful tool in the spectral domain to solve field problems in the presence of discontinuities. This chapter introduces a generalization of this technique that allows to study geometries where coupled planar and angular region are present. Since exact solutions with closed form factorizations are available only in few cases, most of the problems require an alternative approximate technique as the Fredholm factorization. The Fredholm factorization reduces the solution of WH equations to a system of Fredholm integral equations (FIE) of second kind amenable of very efficient numerical solution. The deduction of the FIE is presented in this chapter for a relatively simple novel problem. The numerical solution of FIEs provides an analytical element of the spectra, which in general is not sufficient to evaluate the different components of the diffracted field. To obtain the whole spectrum of the unknowns, analytical continuations and recursive equations deduced by the WH equations are presented. The work ends with a short description of the numerical simulations for the novel scattering problem.

This book covers recent achievements in the area of advanced analytical and associated numerical methods as applied to various problems arising in all branches of electromagnetics. The unifying theme is the application of advanced or novel mathematical techniques to produce analytical solutions or effective analytical-numerical methods for computational electromagnetics addressing more general problems. Each chapter contains an outline of its topic, discusses its scientific context and importance, describes approaches to date, gives an exposition of the author's approach to the problem tackled in the chapter, describes the results, and concludes with a discussion of the range or class of problems where the techniques described work most appropriately and effectively. Intended primarily for researchers in the fields of electrical engineering, mathematics, physics and related disciplines, the book offers systematic and thorough coverage of this complex topic. It is hoped that the book will help to stimulate further investigation and discussion of the important problems in electromagnetics within this research community.

An efficient analytical method based on the Cohn model and mirror procedure is proposed for the prediction of the shielding effectiveness (SE) of a rectangular enclosure with electrically large apertures. Firstly, the coupled electromagnetic fields are represented by the equivalent dipole moments located at the centre of the aperture according to the Cohn model, where the dipole moments can be calculated based on the mirror procedure. Then, the vector potentials can be obtained through the relation between the dipole moments and the current distribution. Finally, the total electric fields are obtained by using the vector potentials, thus the SE and most of the resonant modes can be predicted accurately and fast. The SE results which have been verified by the CST over a wide frequency band of 0.01–3.0 GHz show that the proposed method is effective in handling the enclosure with electrically large apertures, including the multiple apertures arranged in different ways and hybrid apertures in different shapes and dimensions.

This communication investigates the convergence behaviour and accuracy of a near-field corrected iterative physical optics scheme for scatterings by imperfectly conducting and dielectric objects. The scatterings by large and complex impedance surface bodies such as fixed wing aircraft, rotatory wing aircraft, tank, vessel, and boat are computed and compared with those of a full-wave method. Several RF characteristics of a Von-Karman radome are also analysed and compared as a wave penetration application. Based on these comparisons, the near-field correction effect is examined on the accuracy of the iterative physical optics scheme.

Diffraction at a strip with one face soft (electric) and the other hard (magnetic) is studied. New results obtained by the method of moments (MoM) are compared with the asymptotic theory of edge diffraction (TED) for the totally soft and hard strips. Attention is given to diffraction of oblique incident waves including the grazing diffraction. Novel analytic estimations for the forward and backward grazing scattering are established via the advanced physical theory of diffraction (PTD) free of grazing singularities. These estimations are demonstrated for the infinitely long strips and finite size rectangular plates.

The revised and updated second edition of this textbook teaches students to create modeling codes used to analyze, design, and optimize structures and systems used in wireless communications, microwave circuits, and other applications of electromagnetic fields and waves. Worked code examples are provided for key algorithms using the MATLAB technical computing language. The book begins by introducing the field of numerical analysis and providing an overview of the fundamentals of electromagnetic field theory. Further chapters cover basic numerical tasks, finite difference methods, numerical integration, integral equations and the method of moments, solving linear systems, the finite element method, optimization methods, and inverse problems. Developing and using numerical methods helps students to learn the theory of wave propagation in a concrete, visual, and hands-on way. This book fills the missing space of current textbooks that either lack depth on key topics or treat the topic at a level that is too advanced for undergraduates or first-year graduate students. Presenting the topic with clear explanations, relevant examples, and problem sets that move from simple algorithms to complex codes with real-world capabilities, this book helps its readers develop the skills required for taking a mathematical prescription for a numerical method and translating it into a working, validated software code, providing a valuable resource for understanding the finite difference method, the method of moments, the finite element method, and other tools used in the RF and wireless industry.

As the surface conditions play a significant role on corona discharge and its related effects of the conductors, the influence of fine particulate matter on positive-polarity, direct-current conductors was studied experimentally in this study. The surface morphologies of the conductor could be discovered from the experiments. The typical morphologies are the parallel chains of particles. To evaluate the surface condition quantitively, the surface roughness of the conductors is measured. It is found that the applied voltage and testing time have a great influence on the surface condition. After that, the corona characteristics of conductors are tested. It reveals that the total ground level electric field and ion flow density increases with the surface roughness growing.

In this study, first, the problem of time reversal imaging of an embedded dielectric or metallic cylinder inside another dielectric cylinder is studied. The direct problem is carried out numerically using the finite-element method. The background dyadic Green function (DGF) is computed analytically by computing radiated fields of an infinitesimal electric dipole near an infinitely long dielectric cylinder. The time reversal technique is then applied using this background DGF to image the embedded cylinder. Results demonstrate that while using a free-space DGF for imaging yields erroneous results, by employing a proper background DGF, the scatterer inside the cylinder is correctly localised. Next, the authors extract permittivities in two different scenarios with two different techniques. In the first case, the permittivity of a background cylinder that contains a metallic cylinder inside it is extracted by calculating the image entropy. In the second case, the permittivity of an embedded cylinder inside a priori known background dielectric cylinder is estimated by a new method based on an optimisation and evaluation of the largest eigenvalues of the multistatic data matrices. The proposed techniques are then verified using simulated and measured data. For measurements, a newly developed combined tapered sectorial antenna is designed, fabricated, and characterised.

In this study, a dual-broadband reflective polarisation converter with high efficiency for both linear-to-linear and linear-to-circular polarisations based on a metasurface is proposed. Owing to the characteristics of strong anisotropy and multi-order plasmon resonances, the proposed polarisation converter can rotate an *y*/*x* polarised electromagnetic wave to its cross-polarised (*x*/*y*) direction in the lower frequency band of 7.74–14.44 GHz (a fractional bandwidth of 60.4%) with over 0.9 polarisation conversion ratio. Besides, the proposed structure can also convert a linearly polarised incident wave to a circularly polarised one after reflection in the higher frequency band of 14.95–17.35 GHz (a fractional bandwidth of 14.9%). The performance in the two bands can be controlled separately by altering the proper parameters of the structure. Numerical analysis is used to predict the polarisation states of the proposed polarisation converter. Moreover, the physical mechanism of multiple resonances is discussed based on surface current distributions. A prototype of the polarisation converter is fabricated and measured. A reasonable agreement between the experiments and simulations is obtained. The design has a simple and scalable geometry, and is a good candidate for polarisation control devices in microwave, terahertz and optical frequency regions.

This Letter presents an artificial magnetic conductor (AMC) with self-complementary unit cells (SCUCs). The use of SCUCs results in very high angular stability of the structure; up to an incident angle of 88° for both transverse electric and transverse magnetic polarised incident waves with no shift in resonant frequency *f* _{0}. No other structure in the literature has zero frequency shifts throughout the whole range of incident angle variations. Another interesting feature of this structure is the gradual increase of the ±90° reflection phase bandwidth with an increase in E-field polarisation angle from *φ* = 0° to *φ* = 90° while maintaining the same *f* _{0}. These two features make the structure unique and useful in applications where angular stability and polarisation independent *f* _{0} are the primary concerns.

In recent years, the characteristic basis function method has been developed as an efficient approach for the solution of large electromagnetic radiation or scattering problems. According to this technique, the currents over the scenario under analysis are defined using a set of pre-computed characteristic basis functions, associated with a number of blocks into which the geometry is partitioned. This involves some computational advantages due to the reduction of the number of unknowns compared to conventional approaches. However, additional pre-processing time is introduced due to the computation of the CBFs and the reduced coupling matrix. A novel strategy is presented in this study in order to accelerate the generation of the reduced matrix, based on the application of the multilevel fast multipole algorithm.

The small slope approximation (SSA) is a widely employed way to analyse the electromagnetic (EM) scattering problems from rough surfaces. However, it requires a surface sampling interval less than one-eighth of the wavelength, which makes it suffer from the problems of large memory consumption and low calculation efficiency for large rough surfaces, especially for higher microwave bands. In this study, a new way to generate the electrically large rough surfaces is proposed and its application to the calculation of EM scattering field is introduced on the base of first-order SSA. The calculation accuracy and consumed computer memory of the proposed method analysed by numerical comparisons show good performance.