George Fikioris

The use of the frill generator in thin-wire integral equations

For a frill-generator feed, certain fundamental mathematical properties of Hallens and Pocklingtons equations with the approximate kernel are stated and derived. For a particular moment-method procedure (Galerkins method with pulse functions applied to Hallens equation), the consequences to the numerical solutions are carefully examined. Generalizations to other numerical methods with subsectional basis functions are discussed. Many of our results come from studying the simpler problem of the infinite dipole analytically and applying the understanding thus obtained to the case of the finite dipole.

The resonant circular array of electrically short elements

Different resonant distributions of current in circular arrays with N=60, 72, and 90 elements are investigated theoretically with the method of symmetrical components and a two‐term representation of current in each element. The arrays studied have only one driven element. The remarkable results—which are displayed in a sequence of graphs—indicate that the relative currents in the elements can be distributed in triplets, triplets alternating with singlets, two independent sets of singlets, or one set of singlets. The driving‐point impedance of this last distribution is a pure resistance, the Q of the 90‐element array is over 30 000, and the half‐power beamwidth of each of the 90 sharp peaks in the power pattern is 2°.

Rapidly converging spectral-domain analysis of rectangularly shielded layered microstrip lines

A moment-method-oriented direct integral-equation technique is presented for the exact analysis of rectangularly shielded layered microstrip lines. This technique retains the simplicity of conventional moment methods while optimizing them by recasting all matrix elements into rapidly converging series. Filling up the matrix requires no numerical integration. The proposed algorithms yield highly accurate results both for the modal currents and propagation constants.

Exact solutions for shielded printed microstrip lines by the Carleman-Vekua method

Exact analytical solutions for the field of the quasi-TEM (transverse electromagnetic) mode in various cross-sectional configurations of rectangularly shielded printed microstrip lines are obtained on the basis of Carleman-type singular integral equations (SIEs). For the kernel of the SIE, strongly and uniformly convergent series expansions have recently been developed that are suitable for the exact solution of the equation by the Carleman-Vekua regularization method, which proceeds by first solving the so-called dominant equation. The procedure leads to rapidly convergent series solutions for the field of the quasi-TEM mode even when the conductors are large or very near the shield, i.e. in situations for which numerical techniques becomes inadequate. Characteristic values of the shielded microstrip lines are evaluated by summing a few terms, while field plots, requiring more terms, are shown for various configurations including the case of close proximity. >

A note on the method of analytical regularization

Many problems of electromagnetics are governed by singular integral equations of the first kind. As discussed by Nosich (1999), it is often possible to obtain a different equation describing the problem by applying the method of analytical regularization, and analytically inverting part of the original equation. The transformed equation is of the second kind. Therefore, as a rule, it is usually preferable to apply a numerical method to the transformed equation than to the original one. What appears to be an exception to that rule is discussed in the present paper: under proper conditions, and for a particular numerical method, results obtained by application to the transformed equation are shown to be identical to those obtained by application to the original equation. Some consequences of this equivalence are discussed.

On the Phenomenon of Oscillations in the Method of Auxiliary Sources

When one applies the method of auxiliary sources to scattering problems involving perfect conductors, one first seeks fictitious auxiliary currents located inside the conductor, and then determines the field from these currents. For a simple two-dimensional problem involving an infinite circular cylinder illuminated by an electric current filament, it has recently been shown analytically that it is possible to have divergent auxiliary currents (to make this statement precise, one must properly normalize the currents), together with a convergent field. It was also shown-through numerical investigations-that the aforementioned divergence appears as abnormal, rapid oscillations. In the present paper, we investigate such phenomena in more detail, with particular emphasis on oscillations. For a perfectly conducting ground plane illuminated by an electric current filament, we once again demonstrate the possibility of having divergent, oscillating currents producing a convergent field. We develop an asymptotic formula for the oscillating current values, which sheds light on the nature of the oscillations. We revisit the circular-cylinder problem to develop a similar asymptotic formula. We also discuss roundoff errors, and possible generalizations to scatterers of other shapes. The present study is to a great extent analytical, with the analytical predictions confirmed and supplemented by numerical results

On the Oscillations Appearing in Numerical Solutions of Solvable and Nonsolvable Integral Equations for Thin-Wire Antennas

Differences between certain solvable and nonsolvable ill-posed integral equations, with the same nonsingular kernel, are discussed. The main results come from constructing a solvable equation in the context of straight thin-wire antennas. The kernel of this equation is the usual approximate (also called reduced) kernel, while its exact solution is the familiar sinusoidal current. Numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual-Halle¿n and Pocklington-equations with the approximate kernel; it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large. It is found that the difficulties encountered in the former (solvable) equation are much less important compared to those of the nonsolvable ones. The same conclusion is brought out from other integral equations, arising in different contexts (thin-wire circular-loop antenna, Method of Auxiliary Sources, and straight wire antenna of infinite length). We discuss the consistency of our results with Picards theorem. The results in this paper supplement previous publications regarding the difficulties of numerically solving thin-wire integral equations with the approximate kernel.

Mellin Transform Method for Integral Evaluation

This book introduces the Mellin-transform method for the exact calculation of one-dimensional definite integrals, and illustrates the application if this method to electromagnetics problems. Once the basics have been mastered, one quickly realizes that the method is extremely powerful, often yielding closed-form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals. Yet, as opposed to other methods, the present method is very straightforward to apply; it usually requires laborious calculations, but little ingenuity. Two functions, the generalized hypergeometric function and the Meijer G-function, are very much related to the Mellin-transform method and arise frequently when the method is applied. Because these functions can be automatically handled by modern numerical routines, they are now much more useful than they were in the past. The Mellin-transform method and the two aforementioned functions are discussed first. Then the method is applied in three examples to obtain results, which, at least in the antenna/electromagnetics literature, are believed to be new. In the first example, a closed-form expression, as a generalized hypergeometric function, is obtained for the power radiated by a constant-current circular-loop antenna. The second example concerns the admittance of a 2-D slot antenna. In both these examples, the exact closed-form expressions are applied to improve upon existing formulas in standard antenna textbooks. In the third example, a very simple expression for an integral arising in recent, unpublished studies of unbounded, biaxially anisotropic media is derived. Additional examples are also briefly discussed.

An application of convergence acceleration methods

It is pointed out that certain convergence acceleration methods can be applied to sequences of driving-point conductances obtained by solving Hallens equation numerically. Five easily applied methods are considered and all seem to greatly improve convergence. Reasons for having confidence in the resulting extrapolations are discussed.

Near fields of resonant circular arrays of cylindrical dipoles

The far fields of resonant circular arrays of cylindrical dipoles with only one or two dipoles driven have been examined in a number of recent works. The discrete currents on such arrays can be thought of as a slow standing or traveling wave. In the present paper, the near fields of such arrays are studied. In order to better understand the somewhat unusual behavior of the near fields, we first discuss some simpler two-dimensional problems involving slow traveling current waves. We find that the associated fields possess certain surface-wave characteristics. We also find important differences from the case of linear arrays, which has been extensively studied in the past. Our discussions include methods to measure the efficiencies of our radiators, the extent of the near-field region, and the onset of the far-field region.