Advanced Time Domain Modeling for Electrical Engineering
Time domain modeling is a fascinating world which brings together several complex phenomena and methods of essential interest to engineers. This book is a reference guide which discusses the most advanced timedomain modeling methods and applications in electromagnetics and electrical engineering.
The book starts by clearly explaining why timedomain modeling may be worth doing; then, it provides guidelines about why some choices must be made among the principal modeling approaches and next guides the reader through the state of the art in time domain modeling, concerning either numerical and analytical methods, and applications. Finally, it highlights areas for future timedomain modeling research.
The book is a collection of chapters written by leading research groups in the fields, following a logical development set out by the editor.
Topics covered include finite element methods in time domain with applications to lowfrequency problems; transient analysis of scattering from composite objects using latetime stable TDIEs; the transmissionline modeling method, partial element equivalent circuit method in timedomain; unconditionally stable timedomain methods; timedomain linear macromodeling, analytical techniques for transient analysis; the application of the finitedifference timedomain (FDTD) technique to lightning studies; modeling of lightning and its interaction with overhead conductors; transient behaviour of grounding systems; and statistics of electromagnetic reverberation chambers and their simulation through time domain modeling.
 Book DOI: 10.1049/SBEW550E
 Chapter DOI: 10.1049/SBEW550E
 ISBN: 9781839531538
 eISBN: 9781839531545
 Page count: 596
 Format: PDF

Front Matter
 + Show details  Hide details

p.
(1)

1 Introduction to timedomain electromagnetics
 + Show details  Hide details

p.
1
–41
(41)
A complete description of the macroscopic electromagnetism is provided by Maxwell's equations whose validity is taken as a postulate. Maxwell's equations can be used either in a differential (local) form or in an integral (global) form, and there has been a long debate over which is the best representation. When stationary media are considered, the main difference between the two representations consists in how they account for discontinuities of materials and/or sources. Basically, if one adopts the differential form, some boundary conditions at surface discontinuities must be postulated; on the other hand, if the integral forms are chosen, one must postulate their validity across such discontinuities.

2 Finitedifference timedomain method
 + Show details  Hide details

p.
43
–71
(29)
The finitedifference timedomain (FDTD) method has been used to solve grand challenge problems in computational electromagnetics for the past five decades. This simple, yet, elegant method has O(N) computational efficiency and readily exploits modern computer architectures. The focus of this chapter is to provide a review of the FDTD method and several significant advances of the method, including the convolutional perfectly matched layer (CPML) mesh truncation algorithm, subcell algorithms, and more. Emphasis is given on the efficient implementation of the FDTD method on modern day multithreaded and multiprocessor computers, and the importance of establishing a common framework in the implementation. It is shown that several advanced concepts such as subcell algorithms can be reduced to effective local anisotropic materials and maintaining a highly efficient implementation.

3 Finiteelement methods in time domain with applications to lowfrequency problems
 + Show details  Hide details

p.
73
–105
(33)
In this chapter, several lowfrequency formulations were presented and discretized with the FEM in the appropriate function spaces. We considered two settings to calculate the steady state of the spacediscrete system: the initial value and the TP problems, whose solutions are commonly calculated using implicit time stepping methods. Since the classical sequential solution often leads to prohibitively long computation times, PinT integration methods were proposed. Additionally, for TP problems a multiharmonic (MH) diagonalization was used to simplify the solution of large block systems. Incorporation of the MH approach into the PinT framework allowed to introduce parallelism on the coarse level. Finally, the considered parallelization approaches were applied to an induction machine model and to a coaxial cable models which showed a considerably smaller computational effort compared to the sequential timestepping approach.

4 Transient analysis of scattering from composite objects using latetime stable TDIEs
 + Show details  Hide details

p.
107
–136
(30)
In this chapter, we have presented an extension of the method of separable expansion to the analysis of composite objects, i.e., objects comprising a mixture of piecewise homogeneous dielectric regions as well a perfectly conducting surfaces. This method is used to allow for accurate computation of impedance matrix elements using numerical quadrature, which is a necessary condition for obtaining stable solutions to TDIEs. The method, being based purely on numerical integration, is easily extended to higher order geometric surface discretizations. The stability of the method on these types of scatterers has been demonstrated for a variety of scatterers using quasiexact integration solutions as a benchmark. In addition to these results, we have developed an analysis procedure for the errors incurred in the approximation. Altogether, this body of work indicates that using a separable approach to evaluating TDIE matrices is a viable approach. The slowly growing lowfrequency instability we see in some of the results (for both the separable expansion and the quasiexact method) has a wellknown remedy and can be readily integrated into the system of equations presented here.

5 Transmissionline modeling method  TLM
 + Show details  Hide details

p.
137
–180
(44)
We have presented the TLM method as it applies to solving electrical problems of some complexity. The basic techniques have been presented in some detail. Emphasis was placed in dealing with complexity by describing embedding techniques whereby advantage is taken of local solutions in the vicinity of complex objects to interface these solutions with surrounding fields obtained from the TLM method. This approach permits a largely uniform and structured mesh throughout the problem space thus enhancing efficiency and at the same time dealing with problematic features through the embedding of special nodes. This appears to be the optimum approach, however, each problem has its own characteristics and modelers must be prepared to adapt to particular requirements. In this respect, TLM, through its stability, efficiency, versatility, and physical feel, is an ideal tool at the hands of modelers confronted by increasingly complex practical problems.

6 Partial element equivalent circuit method in timedomain
 + Show details  Hide details

p.
181
–216
(36)
The partial element equivalent method is born in the 1970s and continues to evolve with various improvements and major breakthroughs. The request for more accurate modeling in electrical design has pushed forward the development of PEEC technique. This chapter gives an overview of the timedomain PEEC formulations. Among them, the quasistatic formulation is revised in which the propagation delays are totally neglected and, as such, is accurate for electrically small problems. Further, the delayed PEEC formulation is also revised which takes the centertocenter delays into account. Several temporal basis functions can be chosen to expand the circuit quantities. Since PEEC models are typically large, model order reduction techniques have been developed over the years for both quasistatic and delayed PEEC models and they are briefly presented in the chapter. The literature on timedomain analysis of PEEC models is quite large and the interested reader can refer to the references listed.

7 Unconditionally stable timedomain methods
 + Show details  Hide details

p.
217
–252
(36)
In this chapter, the fundamental aspects and several applications of unconditional stable timedomain methods for electromagnetic wave analysis are described. The update equations of alternatingdirectionimplicit (ADI) and locallyonedimensional (LOD) finitedifference timedomain (FDTD) algorithms and their numerical errors are explained. The Perfectly Matched Layer (PML) implementation to truncate the computational domain is discussed. The iterative implementation to the ADI and LODFDTD algorithms is presented to reduce their numerical errors. In addition, the implementation of unconditionally stable FDTD algorithms in complex media is discussed. Finally, variant unconditionally stable FDTD methods are briefly discussed.

8 Timedomain linear macromodeling
 + Show details  Hide details

p.
253
–288
(36)
In this chapter, we have presented both basic and advanced material on linear macromodeling, with emphasis on scalable parameterized models. We have discussed the various aspects that are relevant for a successful macromodel extraction, including choice of model structure, identification algorithms with stability and passivity constraints, and finally model realization in terms of differential equations or equivalent circuits. Although concise, the material in this chapter is reasonably selfcontained and can be used as an introduction to the topic. We refer the reader to the list of references, in particular [1] for a complete treatment.

9 A charged particle traveling along the axis of a circular waveguide
 + Show details  Hide details

p.
289
–307
(19)
The aim of this chapter is to demonstrate that a satisfactory choice of the reference system and the use of Lorentz transformations can help determine the solution of electromagnetic problems, if moving sources are present in the system. This is a typical situation that occurs in particle accelerators, in particular, in linear accelerators, a functional scheme is shown in Figure 9.1. A linear particle accelerator, often shortened to linac, is a type of particle accelerator that accelerates charged subatomic particles or ions to a high speed by subjecting them to a series of oscillating electric potentials along a linear beamline. The design of a linac depends on the type of particle that is being accelerated: electrons, protons, or ions [1]. However, the length of the machine must be quite large in order to give sufficient thrust to accelerated particles, while in a circular machine, the packets can pass many times before being collided. The most powerful linear accelerator is located in Stanford, California at the Stanford linear accelerator centre (SLAC) laboratory and is about 3 km long. Electrons and positrons can be accelerated simultaneously and then separated and brought to collide. The maximum energy of each beam reached almost 50 GeV when, in the 1990s of the last century, the machine was used to study collisions e^{+}e^{−} at the Z _{0} resonance, in competition with the LEP, circular accelerator, which in those years, was operating at CERN in Geneva. For the future, there are plans to build even longer and more powerful linear accelerators for fundamental research, such as International Linear Collider (ILC) for collisions e^{+}e^{−} at about 5001,000 GeV in the center of mass or even Compact LInear Collider (CLIC) for energies up to 2,0004,000 GeV, achieving such energies with circular machines would be practically impossible.

10 Analytical techniques for transient analysis
 + Show details  Hide details

p.
309
–334
(26)
In this chapter, a variety of analytical techniques will be presented for the transient electromagnetic analysis of planar stratified structures made of linear, isotropic media. The field excited by simple canonical sources such as vertical or horizontal electric or magnetic dipoles will be considered, whence the solution of the electromagnetic problem for general extended sources can then be obtained by linear superposition.
The classical Cagniardde Hoop (CDH) method will be addressed first. Originally developed for the transient analysis of a dipole source in the presence of a single interface between two semiinfinite nondispersive media, it has been adapted over the years to considerably more general configurations. It will be exposed here in connection with the problem of a pulsed vertical electric dipole (VED) radiating in the presence of a thin metal screen. A modified version of the method will also be presented, considering a canonical configuration of importance for plasmonic applications, namely the excitation of a freestanding graphene sheet by a VED.
Alternative approaches will then be addressed that, in contrast with the CDH method, have a modal character, i.e., express the transient response of the multilayer structure in terms related to the relevant modal spectrum. These include the doubledeformation method of Tsang and Kong, the Haddon leakymode method, and the FelsenNiu unconventional spectral synthesis procedure. The first will be exposed for the case of a freestanding graphene sheet, whereas the latter two will be applied to a grounded dielectric slab.
Finally, an introduction to the timedomain version of the exact image theory developed by Nikoskinen and Lindell will conclude the chapter.

11 The application of the finitedifference timedomain (FDTD) technique to lightning studies
 + Show details  Hide details

p.
335
–370
(36)
We have seen in this chapter that the FDTD method for solving Maxwell's equations is accurate and versatile in a very wide variety of applications related to lightning. One can analyze the lightning electromagnetic field propagation over distances ranging from a few meters to hundreds of kilometers by just adjusting the smallest wavelength governed by the highest frequency of the band being simulated with the FDTD method. One can also take into account complex media in the EIWG by adding the media parameters through the simple discretization process. Moreover, a number of powerful software packages exist for FDTD modeling, including free, opensource packages and commercial program suites. One can also modify the basic codes to adapt them to any problem of interest. With the continued growth of computing resources, FDTD method will continue to serve as a helpful tool in research and engineering applications requiring lightning electromagnetic pulse (LEMP) simulations.

12 Modeling of lightning and its interaction with overhead conductors
 + Show details  Hide details

p.
371
–399
(29)
Lightning is one of the most powerful and dangerous natural phenomena on the Earth. It has had a profound impact on human society. We will focus here on the socalled "cloudtoground" lightning discharges. These discharges can cause damage not only when they strike the structure directly but also when they hit ground nearby. In the case of overhead conductors, whether they are power or telecommunication lines, due to overvoltages produced, these discharges can cause outages, disturbances on the network, or failure of electronic components and/or electrical equipment. In order to mitigate lightning effects via effective technological solutions and protective measures, we need to know lightning parameters and have models of both lightning itself and its interaction with the strike object or system. We will focus on the socalled indirect or nearby lightning and its effects. The following topics will be covered: (1) specification of channelbase current, (2) returnstroke models, (3) calculation of lightning electromagnetic field, and (4) electromagnetic coupling between the lightning channel and overhead conductors. Both analytical and distributedcircuit model (based on generalized telegrapher's equations) approaches will be considered.

13 Transient behaviour of grounding systems
 + Show details  Hide details

p.
401
–433
(33)
Grounding is a hot topic in electrical engineering, especially when it is necessary to deal with its transient behaviour under a lightning surge where inductive and conductive couplings among elements and the soil ionisation are not negligible. The chapter will introduce to the nonlinear and dynamic phenomenon caused by soil ionisation and the frequency variation of soil parameters first, and then the numerical methods applied to simulate the transient behaviour of grounding grids.

14 Statistics of electromagnetic reverberation chambers and their simulation through time domain modeling
 + Show details  Hide details

p.
435
–465
(31)
This chapter discusses the fundamental properties of electromagnetic reverberation chambers (RCs) and the simulation of their behavior through a simplified timedomain numerical approach. Electromagnetic RCs are used as a test environment with growing range of applications. Their unique properties allow for powerful radiated immunity and radiated emissivity tests for electromagnetic compatibility assessment of various pieces of electronic equipment in industry. Moreover, their application is also spread to antenna characterization, performance assessment of wireless communication systems, exposure of animals to electromagnetic fields, and radar crosssection estimations. This list may be further extended and will be in the future, researchers around the world being very active in this matter.

15 Analysis of a class of dynamical systems with applications to power conversion circuits
 + Show details  Hide details

p.
467
–508
(42)
In the past, remarkable developments in the field of nonlinear smooth dynamical systems and bifurcation theory have taken place [13]. Recently, nonlinear dynamics and bifurcation behavior of nonsmooth switched systems have attracted the interest of many research groups all over the world. Outstanding works have been published in different magazines and presented in specialized conferences [411]. During the last years, there has been also published several textbooks on this subject [1215].

16 Pure time domain multiconductor transmission line formalism
 + Show details  Hide details

p.
509
–537
(29)
An important electrical engineering subject, the classical theme of multiconductor transmission lines (MTL) has been receiving continued attention for quite a long time  for almost a century indeed [13]. Eclectic, the theme is of interest for a wide variety of areas, ranging from power systems engineering ("slow" timevarying phenomena) to microwave engineering ("rapid" timevarying phenomena). Overhead lines, power cables, crosstalk, electromagnetic compatibility, printedcircuit boards, ribbon cables, highspeed interconnects, very largescale integration, monolithic microwave integrated circuits, coplanar waveguides, couplers, filters, etc., are a few items in the list of topics where MTL is involved.
Considering the various electrical engineering subfields concerned with MTL problems, considering the timedomain and frequencydomain approaches utilized, considering the analytical and numerical methods developed, the counting of significant contributions published on MTL theory and applications could easily reach a few thousand. This work is not the place for that; in any case, key contributions by Wedepohl and coworkers, mainly published in IEE Proceedings, related to matrix algebra procedures in MTL analysis, are worthy of recognition [49]. Also, a few books published before 2000 deserve mentioning, namely those by Frankel [10], Djordjevic et al. [11], Faria [12], and Paul [13]. The last book is specially recommended not only because of its quality and coverage, but also because of the amount of bibliographic references there offered.
This chapter builds around the issue if MTL theory can or cannot be formulated and solved purely in the timedomain when losses are considered, excluding any detours or shortcuts into the frequency domain. Closely related to the above question a pure timedomain approach aimed to include conductors' skin effect in MTL equations is offered. Also, based directly on Maxwell's equations, timedomain conservation laws regarding multiconductor line voltages and currents are presented.

17 Shielding in time domain
 + Show details  Hide details

p.
539
–566
(28)
Analysis and design of shielding enclosures [14] are often complex tasks with several conflicting constraints coming from the need of assuring an adequate protection level at reasonable cost and weight and with the maximum possible accessibility, ventilation, and so forth [5]. The protection level must be guaranteed in the presence of either intentional or unintentional electromagnetic (EM) sources [6], and the emissions from the enclosure must be kept below the thresholds fixed by standards.

Back Matter
 + Show details  Hide details

p.
(1)