The Exact Analytical Solution for Large Circular Loops Radiating Around a Dielectric Coated Conducting Sphere

In this paper, the electromagnetic field of electrically large circular loop antennas, with different radii and different nonuniform current values, around a dielectric coated conducting sphere is considered. One or more loop antennas are located on the outer surface of a spherical dielectric shell covering a conducting sphere. Eigenfunction series solutions for the field are assumed in two regions. The current distribution on the wire loop, driven by a voltage source, is determined by Fourier series expansion and all necessary harmonics are taken into account. Exact analytical field expressions in closed forms are derived and field patterns are plotted. The antenna model and formulation presented in this paper offer exact analytical solutions to several loop antenna problems.      

Spherical harmonics and Earth-rotation synthesis in radio astronomy

That spherical harmonic functionsY_{nm}(theta,phi)can be used to advantage in the Earth-rotation synthesis of radio astronomy maps is shown in this paper. As Earth rotates the baseline of a radio interferometer generates a cone whose angletheta, measured from theN-Spolar axis, can be varied by changing the baseline's azimuthal direction on the surface of Earth. A series of Earth-rotation measurements, at different cone angles but with baselines of equal lengthb, can be regarded as being made on a baseline sphere of radiusb, the analog of theuvplane in Fourier-type synthesis. The measured output distribution can be expanded as a spherical harmonic series on the baseline sphere. The coefficients of the series are related to the coefficients of the spherical harmonic series expansion of the source distribution on the celestial sphere by a matrix transformation. The matrix [B] is a function only of the baseline configuration (it does not vary with source declination). Inversion of the matrix leads to the solution for the source coefficients, from which a spherical harmonic map is formed of the source distribution.     

Integrating the dyadic Green's function near sources

Formulas are derived which allow the dyadic Green's function to be integrated for well-behaved currents in the source region. The result is that the electric field due to a current distribution local to an observer can be expressed as a function of the current and its spatial derivatives at the point of observation plus a nonsingular integral over a surface containing the local currents. Although a spherical principal volume is used to derive the theory, the field due to this principal volume is exactly canceled by other terms. The exact form for pulse currents is derived. The theory is extended to nonpulse currents in an appendix     

Time-domain near-field analysis of short-pulse antennas .I.Spherical wave (multipole) expansion

The radiation from a time-dependent source distribution in free-space is analyzed using time-domain (TD) spherical wave (multipole) expansion. The multipole moment functions are calculated from the time-dependent source distribution. The series convergence rate in the near and far zone and the bounds on the near-zone reactive field are determined as functions of the source support and of the pulse length. The formulation involves a spherical transmission line representation that can be extended to more general spherical configurations. This formulation also describes the field and energy transmission mechanisms in a physically transparent fashion that will be used in a companion paper to define and explore fundamental concepts such as TD reactive energy and Q and to derive bounds on the antenna properties. Finally, the concepts discussed above are demonstrated numerically for pulsed radiation by a circular current disk     

Resonance in spherical-circular microstrip structures

The resonance problem of a circular microstrip disk mounted on a spherical surface is studied theoretically. The radiator is replaced by a surface current distribution. The effects of the dielectric substrate as well as the curvature effect are taken into account by the Green's function formulation in the spectral domain. A vector Legendre series is defined. Cavity model current distribution is used as the current basis. Galerkin's procedure is used to solve for the complex resonant frequencies. Numerical results show that the effect of the curved surface on the resonant frequency and quality factor of the microstrip patch may be significant. In general, when the radius of the sphere is decreased, the effective radius of the patch will be reduced and the radiation loss is increased     

Quasi-static image theory for the bi-isotropic sphere

Image theory for the static point charge and the conducting sphere, produced by Kelvin's inversion theory, is extended to the bi-isotropic sphere, including the chiral sphere as a special case. Image expressions for the bi-isotropic sphere can be derived in a manner similar to that of the dielectric sphere except that the quasi-static problem now involves both electric and magnetic scalar potentials, coupled through the interface conditions at the spherical surface. The image is a combination of electric and magnetic line charges along the axis connecting the point charge and the center of the sphere, and their expressions are obtained through what can be labeled as finite Mellin transformation. The expressions derived can find application in more complete quasi-static analyses of interactions of bi-isotropic spherical particles in artificial bi-isotropic media     

A fast method to compute the potential in the multisphere model[EEG application]

A series expansion is derived for the potential distribution, caused by a dipole source in a multilayered sphere with piecewise constant conductivity. When the radial coordinate of the source approaches the radial coordinate of the field point the spherical harmonics expansion converges only very slowly. It is shown how the convergence can be improved by first calculating an asymptotic approximation of the potential and using the so-called addition-subtraction method. Since the asymptotic solution is an approximation of the true solution, it gives some insight on the dependence of the potential on the conductivities. The formulas are given in Cartesian coordinates, so that difficulties with coordinate transformations are avoided. Attention is paid to the (fast) computation of the partial derivatives of the potential, which is useful for inverse algorithms     

Exact circuit analysis of spherical waves

A circuit representation for the impedance of spherical waves radiating from a spherical boundary was derived using a partial fraction expansion by Chu in order to establish gain bandwidth limitations for antennas. These circuits are derived directly from the recurrence relations for spherical Bessel functions. They provide an exact analog of the field solution both inside and outside of a spherical surface for any values of permittivity (epsilon) and permeability (mu). Since the circuits have the form of high-pass filters, they provide significant physical insight into scattering and radiation problems, suggest suitable asymptotic or approximate forms and allow circuit concepts and theorems to be brought to bear in order to solve specific configurations or to set general performance bounds. As an example of their application, compact computer programs for the radar cross section of conducting and dielectric spheres, the minimumQ's for antennas, theQ's and resonant frequencies of dielectric spheres and the induced current at the specular and shadow points of a conducting sphere are given. Other illustrative results are the short pulse responses of a solid dielectric and a dielectric-coated sphere computed using the circuit surge impedance and multiple reflection terms; and the response of a hypothetical scatterer in which the TE and TM modes are coupled.     

Electric Potential in a Dielectric Sphere Head Produced by a Time-Harmonic Equivalent Current Dipole

A time-harmonic equivalent current dipole model is proposed to simulate EEG source which deals with the problem concerning the capacitance effect. The expressions of potentials in both homogeneous infinite dielectric medium and dielectric sphere on the electroquasistatic condition are presented. The potential in a 3-layer inhomogeneous spherical head is computed by using this model. The influences on potential produced by time-harmonic character and permittivity are discussed. The results show that potentials in dielectric sphere are affected by frequency and permittivity.     

Studies of the Focal Region of a Spherical Reflector: Polarization Effects

The polarization of the reflection of a uniform plane wave incident on a spherical reflector is analyzed using the current distribution method for scattered fields. The current distribution on the reflector is derived. For reflectors subtending about60degor less, the radiation scattered in the direction of the circle of least confusion has essentially the same polarization as that reflected specularly from the tangent plane. The effective current, the component of surface current density radiating toward the focal region, is derived in several representations. Assuminghat{i}incident polarization, contour plots are provided forhat{i},hat{j}, andhat{k}components in spherical coordinates. Next, general formulas are derived for thehat{i},hat{j},hat{k}components of the reflected fields, in terms of the direction cosines of the normal to the reflecting surface. These are displayed in terms of projections, and apply directly to the spherical reflector.     

Asymptotically dense spherical codes. I. Wrapped spherical codes

A new class of spherical codes called wrapped spherical codes is constructed by “wrapping” any sphere packing Λ in Euclidean space onto a finite subset of the unit sphere in one higher dimension. The mapping preserves much of the structure of Λ, and unlike previously proposed maps, the density of the wrapped spherical codes approaches the density of Λ as the minimum distance approaches zero. We show that this implies that the asymptotically maximum spherical coding density is achieved by wrapped spherical codes whenever Λ is the densest possible sphere packing     

Spherical space-time codes (SSTC)

We present the extension of the trellis space-time code (STC) concept to include signal mapping drawn from an N-dimensional sphere. The signal points were designed as such to increase the minimum squared distance between points in the constellation without increasing the average transmit energy. The mapping of the N-dimensional spherical constellations was performed in accordance to set partitioning rules for STC developed by AT&T but optimized for these codes. Performance evaluation for these spherical space-time codes (SSTC) are illustrated in an OFDM framework, which is a natural choice for multidimensional signaling     

A Multistage Process for Computing Virtual Dipolar Sources of EEG Discharges from Surface Information

A mathematical model consisting of a homogeneous spherical conductive medium was used to simulate the human head. A current dipole situated in the interior of the sphere produces, on the surface, a theoretical distribution of potential similar to certain EEG activities recorded on the human scalp. We propose a multi-stage method for computing the six parameters of a such virtual dipolar source by using only measurements on the surface of the scalp. Epileptic discharges observed on a young patient were thus analyzed. The computed dipole depends on the placement of pairs of electrodes used for the record. We compare the results of four ``assemblies' using different pairings of electrodes. The center of the dipole varies within a volume of 2 cm3 but its direction is stable. We discuss the importance and the influence of the approximations introduced by the model and the method. Although these approximations cannot be neglected, they do not change the signification of our results. The question is now raised: Has the virtual source a physical interpretation? Only intracerebral investigations can bring about a clear answer. However, whatever the answer may be, it appears that the method presented here can be helpful in the analysis of EEG surface situations.     

Potential and current density distributions of cranialelectrotherapy stimulation (CES) in a four-concentric-spheres model

Cranial electrotherapy stimulation (CES) has been successfully used for treatment of many psychiatric diseases. Its noninvasive nature is its major advantage over other forms of treatments such as drugs. It is postulated that the low electric current of CES causes the release of neurotransmitters. However, the current pathways have not been extensively investigated. In this paper, analytical and numerical methods are used to determine the distribution of potential and current density in a four zone concentric spheres model of the human head when excited by two electrodes diametrically opposite to each other. Because of the azimuthal symmetry, which is assumed in this study, a two-dimensional (2-D) finite difference approximation is derived in the spherical grid. The current density distribution is projected around the center of the model, where the thalamus is modeled as a concentric sphere. All dimensions and electrical properties of the model are adapted from clinical data. Results of this simulation indicate that, in contrast to previous beliefs, a small fraction of the CES current does reach the thalamic area and may facilitate the release of neurotransmitters     

An Accurate Equivalent Behavioral Model of Antenna Radiation Using a Mode-Matching Technique Based on Spherical Near Field Measurements

A new and simple method for modeling an antenna under test (AUT) from spherical near-field (NF) measurements is presented. This method utilizes NF data to determine an equivalent behavioral model composed of magnetic and electric dipoles placed over a fictitious sphere surrounding the AUT. A spherical wave expansion (SWE) of the measured NF is developed to derive a linear relation between the transmission coefficients of the AUT and the transmission coefficients of each dipole. Dipole transmission coefficients are determined using the translational and rotational addition theorems. Finally, a least square method is employed to compute the excitation of each current source. Once the equivalent model is obtained, it can be used to study the behavior of the original AUT in different environments. Computations with electromagnetic simulation data illustrate the accuracy of the proposed method and the reliability of the derived model.     

Electromagnetic radiation of antennas in the presence of anarbitrarily shaped dielectric object: Green dyadics and theirapplications

Although numerical solutions to the electromagnetic scattering by an arbitrarily shaped object have been obtained using Waterman's (1971) T-matrix method (TMM), the general electromagnetic radiation due to an antenna of a three-dimensional (3-D) current distribution in the presence of an arbitrarily shaped object has not been well considered. In this paper, the technique of surface integral equations has been employed; and as a result, a terse and analytical representation of the dyadic Green's functions (DGFs) in the presence of an arbitrarily shaped dielectric object is obtained for the antenna radiation. In a form similar to that associated with the electromagnetic radiation in the presence of a dielectric sphere, the DGFs inside and outside of the object of arbitrary shape are expanded in terms of spherical vector wave functions. However, their coefficients are no longer decoupled due to the arbitrary surface of a 3-D object. The coupled coefficients are then determined using the surface integral equation approach, in a fashion similar to that in the T-matrix method. To confirm the applicability and correctness of the approach in this paper a dielectric sphere, as a special case, is utilized as an illustration. It is found that exactly the same expressions as in the rigorous analysis for the inner and outer spherical regions of the object are obtained using the different approaches. As applications of the approach in this paper, radiation problems of an electric dipole in the presence of superspheroids and rotational parabolic bodies are solved     

Frequency series expansion of an explicit solution for a dipoleinside a conducting sphere at low frequency

The electromagnetic field generated by a current dipole situated at an arbitrary position inside a conducting sphere is derived using the expansions of the spherical vector wave functions. The first few terms in a series expansion of this field with respect to the frequency are given for the normal magnetic field (used in magnetoencephalogram) and the tangential electric field (used in electroencephalogram) outside the conducting sphere at low frequency. It is shown that the first correction term to the static solution is linear in the frequency ω (the second correction term is proportional to ω^3/2 ) and, thus, the static solution can be used as a good approximation for the solution at a very low frequency     

An efficient calculational approach to evaluation of microwave specific attenuation

An efficient calculational approach using the scattering-radiation conversion is developed in this paper to evaluate the microwave attenuation by arbitrarily distorted raindrops. For this modified first-order approach, the perturbation technique and the spherical vector eigenfunction expansion method are employed. A method of obtaining the volumetric current distribution of the assumed source that generates the plane waves is developed in the paper and the current distribution of such a source is derived. The electromagnetic fields outside the distorted raindrop scatterers are formulated in terms of integrals consisting of a volumetric current distribution located at infinity and the dyadic Green's functions. To illustrate the validity of this approach, the spheroidal raindrop and the Pruppacher and Pitter (1971) raindrop model of varying shapes are specifically investigated. Numerical results of the extinction cross sections and the specific attenuation due to the two models are obtained. While the former agrees well with the published results, the latter is in good agreement with the experimental specific attenuation data collected at 21.225 GHz in Singapore.     

Quality factor of an electrically small antenna radiating close toa conducting plane

Expressions are derived for the smallest achievable radiation quality factor (Q) of an electrically small antenna in front of a conducting plane. Applying the low-frequency approximation to the source region involving an electric or a magnetic point dipole plus their images behind the plane, an expression is formed for the field in the radiation zone. The contribution of non-propagating energy in the near field is obtained explicitly using a spherical harmonics decomposition. The radiation Q is found to depend on the radius (relative to wavelength) of the smallest sphere that encloses the antenna and its image, the ratio of the vertical and horizontal dipole moments, as well as the positions of the dipoles relative to each other and to the plane. A number of simple wire structures are analysed with NEC (based on the method of moments), and the approximate Q obtained from their fractional bandwidth are compared to the corresponding theoretical minima