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

Stored energies in electric and magnetic current densities for small antennas

Electric and magnetic currents are essential to describe electromagnetic stored energy, as well as the associated quantities of antenna Q and the partial directivity to antenna Q-ratio, D/Q, for general structures. The upper bound of previous D/Q-results for antennas modeled by electric currents is accurate enough to be predictive, this motivates us here to extend the analysis to include magnetic currents. In the present paper we investigate antenna Q bounds and D/Q-bounds for the combination of electric- and magnetic-currents, in the limit of electrically small antennas. This investigation is both analytical and numerical, and we illustrate how the bounds depend on the shape of the antenna. We show that the antenna Q can be associated with the largest eigenvalue of certain combinations of the electric and magnetic polarizability tensors. The results are a fully compatible extension of the electric only currents, which come as a special case. The here proposed method for antenna Q provides the minimum Q-value, and it also yields families of minimizers for optimal electric and magnetic currents that can lend insight into the antenna design.Comment: 27 pages 7 figure

Stored energies for electric and magnetic current densities

Electric and magnetic current densities are an essential part of electromagnetic theory. The goal of the present paper is to define and investigate stored energies that are valid for structures that can support both electric and magnetic current densities. Stored energies normalized with the dissipated power give us the Q factor, or antenna Q, for the structure. Lower bounds of the Q factor provide information about the available bandwidth for passive antennas that can be realized in the structure. The definition that we propose is valid beyond the leading order small antenna limit. Our starting point is the energy density with subtracted far-field form which we obtain an explicit and numerically attractive current density representation. This representation gives us the insight to propose a coordinate independent stored energy. Furthermore, we find here that lower bounds on antenna Q for structures with e.g. electric dipole radiation can be formulated as convex optimization problems. We determine lower bounds on both open and closed surfaces that support electric and magnetic current densities. The here derived representation of stored energies has in its electrical small limit an associated Q factor that agrees with known small antenna bounds. These stored energies have similarities to earlier efforts to define stored energies. However, one of the advantages with this method is the above mentioned formulation as convex optimization problems, which makes it easy to predict lower bounds for antennas of arbitrary shapes. The present formulation also gives us insight into the components that contribute to Chu's lower bound for spherical shapes. We utilize scalar and vector potentials to obtain a compact direct derivation of these stored energies. Examples and comparisons end the paper.Comment: Minor updates to figures and tex

Stored Electromagnetic Energy and Antenna Q

Decomposition of the electromagnetic energy into its stored and radiated parts is instrumental in the evaluation of antenna Q and the corresponding fundamental limitations on antennas. This decomposition is not unique and there are several proposals in the literature. Here, it is shown that stored energy defined from the difference between the energy density and the far field energy equals the new energy expressions proposed by Vandenbosch for many cases. This also explains the observed cases with negative stored energy and suggests a possible remedy to them. The results are compared with the classical explicit expressions for spherical regions where the results only differ by ka that is interpreted as the far-field energy in the interior of the sphere. Numerical results of the Q-factors for dipole, loop, and inverted L-antennas are also compared with estimates from circuit models and differentiation of the impedance. The results indicate that the stored energy in the field agrees with the stored energy in the Brune synthesized circuit models whereas the differentiated impedance gives a lower value for some cases. The corresponding results for the bandwidth suggest that the inverse proportionality between bandwidth and Q depends on the relative bandwidth or equivalent the threshold of the reflection coefficient. The Q from the differentiated impedance and stored energy are most useful for relative narrow and wide bandwidths, respectively

On the Accuracy of Equivalent Antenna Representations

The accuracy of two equivalent antenna representations, near-field sources and far-field sources, are evaluated for an antenna installed on a simplified platform in a series of case studies using different configurations of equivalent antenna representations. The accuracy is evaluated in terms of installed far-fields and surface currents on the platform. The results show large variations between configurations. The root-mean-square installed far-field error is 4.4% for the most accurate equivalent representation. When using far-field sources, the design parameters have a large influence of the achieved accuracy. There is also a varying accuracy depending on the type of numerical method used. Based on the results, some recommendations on the choice of sub-domain for the equivalent antenna representation are given. In industrial antenna applications, the accuracy in determining e.g. installed far-fields and antenna isolation on large platforms are critical. Equivalent representations can reduce the fine-detail complexity of antennas and thus give an efficient numerical descriptions to be used in large-scale simulations. The results in this paper can be used as a guideline by antenna designers or system engineers when using equivalent sources

Long time motion of NLS solitary waves in a confining potential

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schroedinger equations with a confining, slowly varying external potential, $V(x)$. A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval. We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential $V(x)$ over a long time interval.Comment: 42 pages, 2 figure

On methods to determine bounds on the Q-factor for a given directivity

This paper revisit and extend the interesting case of bounds on the Q-factor for a given directivity for a small antenna of arbitrary shape. A higher directivity in a small antenna is closely connected with a narrow impedance bandwidth. The relation between bandwidth and a desired directivity is still not fully understood, not even for small antennas. Initial investigations in this direction has related the radius of a circumscribing sphere to the directivity, and bounds on the Q-factor has also been derived for a partial directivity in a given direction. In this paper we derive lower bounds on the Q-factor for a total desired directivity for an arbitrarily shaped antenna in a given direction as a convex problem using semi-definite relaxation techniques (SDR). We also show that the relaxed solution is also a solution of the original problem of determining the lower Q-factor bound for a total desired directivity. SDR can also be used to relax a class of other interesting non-convex constraints in antenna optimization such as tuning, losses, front-to-back ratio. We compare two different new methods to determine the lowest Q-factor for arbitrary shaped antennas for a given total directivity. We also compare our results with full EM-simulations of a parasitic element antenna with high directivity.Comment: Correct some minor typos in the previous versio