Mathematical models for dispersive electromagnetic waves: an overview

In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.Comment: 46 pages, 16 figure

Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented

A new breakthrough in linear-system theory: Kharitonov's result

Given a real coefficient polynomial D(s), there exist several procedures for testing whether it is strictly Hurwitz (i.e., whether it has all its zeros in the open left-half plane). If the coefficients of D(s) are uncertain and belong to a known interval, such testing becomes more complicated because there is an infinitely large family of polynomials to which D(s) now belongs. It was shown by Kharitonov that in this case it is necessary and sufficient to test only four polynomials in order to know whether every polynomial in the family is strictly Hurwitz. An interpretation of this result in terms of reactance functions (i.e., LC impedances) was recently proposed. These results were also extended recently for the testing of positive real property of rational transfer functions with uncertain denominators. In this paper we review these results along with detailed proofs and discuss extensions to the discrete-time case

Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations

The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system's matrix. A constructive proof of the existence of directional wave-field decomposition with respect to the normal of the boundary is presented. In the process of defining the wave-field decomposition (wave-splitting), the resolvent set of the time-Laplace representation of the system's matrix is analyzed. This set is shown to contain a strip around the imaginary axis. We construct a splitting matrix as a Dunford-Taylor type integral over the resolvent of the unbounded operator defined by the electromagnetic system's matrix. The splitting matrix commutes with the system's matrix and the decomposition is obtained via a generalized eigenvalue-eigenvector procedure. The decomposition is expressed in terms of components of the splitting matrix. The constructive solution to the question on the existence of a decomposition also generates an impedance mapping solution to an algebraic Riccati operator equation. This solution is the electromagnetic generalization in an anisotropic media of a Dirichlet-to-Neumann map.Comment: 45 pages, 2 figure

Modeling and Simulation of Nonlinearly Loaded Electromagnetic Systems via Reduced Order Models - A Case Study: Energy Selective Surfaces

L'abstract è presente nell'allegato / the abstract is in the attachmen

Dispersion Relations in Scattering and Antenna Problems

This dissertation deals with physical bounds on scattering and absorption of acoustic and electromagnetic waves. A general dispersion relation or sum rule for the extinction cross section of such waves is derived from the holomorphic properties of the scattering amplitude in the forward direction. The derivation is based on the forward scattering theorem via certain Herglotz functions and their asymptotic expansions in the low-frequency and high-frequency regimes. The result states that, for a given interacting target, there is only a limited amount of scattering and absorption available in the entire frequency range. The forward dispersion relation is shown to be valuable for a broad range of frequency domain problems involving acoustic and electromagnetic interaction with matter on a macroscopic scale. In the modeling of a metamaterial, i.e., an engineered composite material that gains its properties by its structure rather than its composition, it is demonstrated that for a narrow frequency band, such a material may possess extraordinary characteristics, but that tradeoffs are necessary to increase its usefulness over a larger bandwidth. The dispersion relation for electromagnetic waves is also applied to a large class of causal and reciprocal antennas to establish a priori estimates on the input impedance, partial realized gain, and bandwidth of electrically small and wideband antennas. The results are compared to the classical antenna bounds based on eigenfunction expansions, and it is demonstrated that the estimates presented in this dissertation offer sharper inequalities, and, more importantly, a new understanding of antenna dynamics in terms of low-frequency considerations. The dissertation consists of 11 scientific papers of which several have been published in peer-reviewed international journals. Both experimental results and numerical illustrations are included. The General Introduction addresses closely related subjects in theoretical physics and classical dispersion theory, e.g., the origin of the Kramers-Kronig relations, the mathematical foundations of Herglotz functions, the extinction paradox for scattering of waves and particles, and non-forward dispersion relations with application to the prediction of bistatic radar cross sections

On the Synthesis of Passive Networks without Transformers

This thesis is concerned with the synthesis of passive networks, motivated by the recent invention of a new mechanical component, the inerter, which establishes a direct analogy between mechanical and electrical networks. We investigate the minimum numbers of inductors, capacitors and resistors required to synthesise a given impedance, with a particular focus on transformerless network synthesis. The conclusions of this thesis are relevant to the design of compact and cost-effective mechanical and electrical networks for a broad range of applications. In Part 1, we unify the Laplace-domain and phasor approach to the analysis of transformerless networks, using the framework of the behavioural approach. We show that the autonomous part of any driving-point trajectory of a transformerless network decays to zero as time passes. We then consider the trajectories of a transformerless network, which describe the permissible currents and voltages in the elements and at the driving-point terminals. We show that the autonomous part of any trajectory of a transformerless network is bounded into the future, but need not decay to zero. We then show that the value of the network's impedance at a particular point in the closed right half plane can be determined by finding a special type of network trajectory. In Part 2, we establish lower bounds on the numbers of inductors and capacitors required to realise a given impedance. These lower bounds are expressed in terms of the extended Cauchy index for the impedance, a property defined in that part. Explicit algebraic conditions are also stated in terms of a Sylvester and a Bezoutian matrix. The lower bounds are generalised to multi-port networks. Also, a connection is established with continued fraction expansions, with implications for network synthesis. In Part 3, we first present four procedures for the realisation of a general impedance with a transformerless network. These include two known procedures, the Bott-Duffin procedure and the Reza-Pantell-Fialkow-Gerst simplification, and two new procedures. We then show that the networks produced by the Bott-Duffin procedure, and one of our new alternatives, contain the least possible number of reactive elements (inductors and capacitors) and resistors, for the realisation of a certain type of impedance (called a biquadratic minimum function), among all series-parallel networks. Moreover, we show that these procedures produce the only series-parallel networks which contain exactly six reactive elements and two resistors and realise a biquadratic minimum function. We further show that the networks produced by the Reza-Pantell-Fialkow-Gerst simplification, and the second of our new alternatives, contain the least possible number of reactive elements and resistors for the realisation of almost all biquadratic minimum functions among the class of transformerless networks. We group the networks obtained by these two procedures into two quartets, and we show that these are the only quartets of transformerless networks which contain exactly five reactive elements and two resistors and realise all of the biquadratic minimum functions. Finally, we investigate the minimum number of reactive elements required to realise certain impedances, of greater complexity than the biquadratic minimum function, with series-parallel networks.Funded in part by the EPSRC Programme Grant on Control For Energy and Sustainabilit

Guided modes and resonant transmission in periodic structures

We analyze resonant scattering phenomena of scalar fields in periodic slab and pillar structures that are related to the interaction between guided modes of the structure and plane waves emanating from the exterior. The mechanism for the resonance is the nonrobust nature of the guided modes with respect to perturbations of the wavenumber, which reflects the fact that the frequency of the mode is embedded in the continuous spectrum of the pseudo-periodic Helmholtz equation. We extend previous complex perturbation analysis of transmission anomalies to structures whose coefficients are only required to be measurable and bounded from above and below, and we establish sufficient conditions involving structural symmetry that guarantee that the transmission coefficient reach 0% and 100% at nearby frequencies close to those of the guided modes. Our analysis demonstrates a few more patterns of anomalies in nongeneric cases, including anomalies of two peaks and one dip on the transmission graph with total background transmission, anomalies of one peak and two dips with total background reflection, and multiple anomalies, and we also prove sufficient conditions for these transmission coefficients to reach 0% and 100%. For pillar structures, we establish a fundamental framework using Bessel functions for the analysis of guided modes, and prove the existence and nonexistence in structures in analogy to results for slabs. We provide a new existence result of nontrivial embedded guided modes, which are stable with respect to the wavenumber and nonrobust under perturbations of the structural geometry, in periodic pillars with smaller periodic cells