Edge effects in finite arrays of uniform slits on a ground plane

The results obtained by modeling a linear array as an infinite periodic structure can be used for the analysis of finite arrays as the zero-order approximation of a perturbation technique. This idea is utilized to investigate the edge effects in two arrays of uniform slits fed by parallel-plate waveguides terminated on a ground plane. It is shown that the realized gain pattern of an element depends substantially upon its position in the array. This is true particularly for the deep resonance notches in the patterns which are present for certain element spacings. When the array is excited with uniform magnitude and linear phase, the aperture voltages are the superposition of a term, corresponding to the infinite array model, plus another correction term (a "spatial transient") representing the edge effect. The influence of this term is particularly relevant when the array is scanned at endfire. In such a case, the method introduced here allows the prediction of the element terminal admittances and the array pattern, while according to the infinite array model no radiation would be permitted.     

On the size requirement for finite phased-array models

The question considered is how large an array model must be in order to capture approximately the characteristics of both the interior and the edge elements of a large multi-octave phased array. Arrays with tapered slot elements and with top-loaded dipoles are analyzed at element spacings as small as 0.1λ and it is concluded that at any frequency, a finite array model with this type of element should be at least 5λ×5λ in size. This suggests the generalization of the 10×10 element model often used as an engineering "rule of thumb" in the normal narrow-band case with 0.5λ element spacing. An array model with a 5:1 bandwidth thus needs about 25 times more elements than a narrow-band model. The array feed impedance is considered and it is found that the array active reflection coefficient in finite arrays but not in infinite arrays is dependent on the matching condition at the feed. The finite-difference time-domain (FDTD) technique is used to analyze arrays up to 49×49 elements, demonstrating that computer power now makes feasible the full wave solution for large phased arrays with complex geometry     

Performance of a protruding-dielectric waveguide element in a phased array

A solution for the active array reflection coefficient in a two-dimensional array of parallel-plate-guide fed protruding-dielectric elements is presented. Analysis for this class of elements is not available in the literature. Element pattern nulls are found as in the case of dielectric-slab-covered arrays. The resonance mechanisms are also shown to be similar. Bandwidth properties of this array are briefly considered. A good agreement with experiments on two small arrays is found. Departures from infinite array theory, which take the form of a ripple on the flat portion of the element patterns and dips beyond the grating lobe angle, are shown to result from the finiteness of the arrays. These edge effects may be interpreted as arising from the interference between the "space wave," i.e., the infinite array element pattern and the fields radiated by the sources induced at the array edges by a wave traveling along the array face.     

Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays

Some experiences from finite-differences time-domain (FDTD) analysis of infinite and finite multi-octave phased arrays are presented. First, a more unified derivation of equations suitable for unit-cell analysis of phased arrays or other types of periodic structures using FDTD is presented. Second, results from FDTD calculations of small to very large multi-octave finite arrays are summarized in order to answer the question of how large an array model must be in order to capture the characteristics of both the interior and the edge elements of a large multi-octave phased array. It is found that a considerably larger number of elements is required in the broad-band case than in the normal narrow-band case, and it is also found that FDTD is well suited for such calculations. Third, simple methods to save computer memory using locally fine grids in an otherwise coarse FDTD grid to model finite-phased arrays are explored. The two local grid methods tested were found in our application to suffer from numerical instabilities.     

Experimental Study of Planar Finite Grid Oscillator Arrays

Results of an experimental study on finite grid oscillator arrays and the effects of the edge element loading stubs in such arrays are presented. Three finite grid oscillator arrays, based on the same unit cell, with different number of unit were fabricated on RT/Duroid 5870 substrate and tested in terms of the oscillation frequencies, radiated power and radiation patterns. It is observed that the oscillation frequency of a finite grid array differs from the theoretically prediction based on the infinite array assumption and is strongly affected by the edge element loading stubs. The measurement also indicates that mode-jumping and multi-frequency (spurious) oscillation can exist in grid oscillator arrays.     

Finite phased array of microstrip patch antennas: the infinitearray approach

An efficient method of analysis of large infinite arrays based on a convolution technique that allows one to obtain the finite array characteristics from the infinite array results is presented. The edge effects are taken into account by convoluting the infinite array results with the proper current amplitude window on the array. The method is based on the use of Poisson's sum formula in the case of finite arrays applied here to microstrip antennas. It is an approximate technique that can be assimilated into a perturbation method     

Properties of a phased array of rectangular waveguides with thin walls

The radiation impedance of an infinite array of open rectangular waveguides has been calculated by a function-theoretic method forHplane and quasi-Eplane beam scanning directions. The mutual coupling between columns has also been obtained. The amplitudes of the coupling coefficients decay asymptotically asr^{-3/2}while the phase difference between successive coupling coefficients approaches that to be expected from free space wave propagation. This asymptotic behavior is independent of waveguide dimensions for both planes of scan. It is similar to the asymptotic behavior of a line-source-excited wave propagating over a lossy surface. This suggests that the interface between an array and free space may in general be treated as such a surface. The coupling coefficients are used to determine the properties of an array, which has a finite number of active elements surrounded by an infinite passive array. Also, the edge effect due to the finiteness of an array is evaluated.     

Analysis of finite phase arrays of microstrip patches

A method for the analysis of large phased arrays of microstrip patches is presented. It is based on an infinite array approach where the edge effects are taken into account through the convolution with a proper window function. In the first step, a rigorous Green's function corresponding to a finite array of elementary sources is derived. This Green's function is then used to analyze the finite phased array of microstrip patches. Results are shown for the active impedance and element patterns of several arrays, and compared with measurements or, in the case of small arrays, with results obtained by a rigorous element-by-element approach. It is shown that the method, even if developed for the analysis of large arrays, is able to handle small arrays. Indeed, the results obtained are good even for single patches. Although the method has been developed for the microstrip phased array case, the results are general and are valid for any phased array with a rectangular grid     

Large finite array analysis using infinite array data

The prediction of edge element behavior is a common problem during the design of large array antennas. The performance of the center elements can be approximated by an infinite array model, but the edge element patterns and active reflection coefficients cannot. The full element-by-element analysis of a large finite array is either excessively time consuming or impossible due to the computer power required. A study has recently been carried out to develop and test methods of fully predicting large array performance using infinite array data. The methods devised are presented, together with comparisons of predicted performance and measured data from a 163-element WG-16 array     

Generalized moment method analysis of planar reactively loadedrectangular waveguide arrays

A combined modal expansion and moment method is applied for analyzing finite planar arrays of rectangular waveguides in an infinite ground plane including reactively loaded waveguide elements. The analysis, which is based on the moment method for solving the aperture problem, in combination with the modal analysis for the reactively loaded waveguide cavities or feeding waveguides, extends the network formulation of single apertures to multiple waveguide arrays including reactively loaded elements. Two orthogonally polarized sets of basic functions are utilized to model the expansion into the complete set of eigenmodes is used to formulate the waveguide cavity problem. The theory is applied to a finite planar array of arbitrarily spaced rectangular waveguides, where reactive loads are realized by placing sliding electrical conductors at specified distances in each of the unfed waveguides. Numerical design examples show that beamforming, beamsteering, and the improvement of the crosspolarization behavior is possible by a suitable choice of the geometrical parameters. The theory is verified by results available from the literature as well as newly performed experimental measurements     

A generalized approach to the analysis of infinite planar array antennas

In this paper a generalized expression for the complex power radiated by an element in an infinite planar array antenna is derived. Since this power formula applies to a large class of phased array antennas where the aperture field distribution can be completely specified (in normal mode form), it proves to be a powerful, unifying principle. The utility of this approach is illustrated by the simplicity with which previously known results can be derived; e.g., an infinite array of slots in a ground plane and an infinite array of flat dipoles with or without a ground plane. Further demonstrations of the usefulness of the power formula are provided by the systematic and straightforward solutions of the less-well-known problems of infinite arrays of crossed-dipole pairs and infinite arrays of open-ended rectangular waveguides. The waveguide array solution is particularly interesting because it reduces to a set of equations which are identical to those one would use to characterize an N-port network on an admittance basis (N is the number of waveguide modes). Since the power formula is derived for a parallelogram element Lattice, the resultant solution for a specific type of element is in its most general form. Expressions for the scan-dependent, dominant mode radiation admittance and the element gain function for a multimode rectangular waveguide radiator are also derived. In addition, various aspects of the waveguide array solution are investigated in the light of previous studies of infinite arrays.     

Mutual coupling analysis of a finite planar waveguide array

This paper deals with mutual coupling in a finite planar array antenna, composed of open-ended circular waveguides in a ground plane. The element-by-element approach is used and the problem is formulated as an integral equation, by requiring the transverse electric and magnetic fields to be continuous across the apertures. The equation is then solved by the method of moments and the mutual coupling in a 127-element array is computed. Excellent agreement with measurements and with the active reflection coefficient for the corresponding infinite array is found. The presented method of coupling analysis can be considered as a supplement to the established periodic-structure approaches for infinite arrays and may be useful for the analysis of small or nonperiodic arrays.     

Ray analysis of conformal antenna arrays

The aperture design of conformal arrays is predicated on the knowledge of the element patterns and coupling coefficients in a mutually coupled environment. For equispaced identical slits on a perfectly conducting cylinder, a previous analysis has utilized the modal theory of periodic structures to simplify the calculations. Modal procedures are rather difficult to apply when the array surface has a more general, though rotationally symmetric and separable, shape; they become practically inapplicable when the surface is nonseparable. These difficulties may be overcome by recourse to the geometrical theory of diffraction and utilization of surface rays whose properties are determined from an appropriately defined local environment on the array surface. Depending on the problem under consideration, the local environment may involve either the unperforated array surface or a surface with periodic loading. It is shown how the surface ray technique can be applied to the analysis of mutual coupling in full ring arrays and finite arrays on a circular cylinder, and in nonperiodic or almost periodic arrays on surfaces of variable curvature. For finite arrays, a theoretical model leading to a representation of finiteness effects in terms of multiple scattering of surface rays of the periodic array structure between the edge discontinuities is confirmed by independently calculated numerical results. Although the demonstrations in this paper are confined to two-dimensional geometries, the procedure is applicable also to three-dimensional configurations.     

Analysis of general lossy inhomogeneous and anisotropic waveguidesby the finite-element method (FEM) using edge elements

Several finite element formulations based on edge elements have been developed in recent years, avoiding the appearance of spurious modes in waveguides. However, no formulation of this kind dealing with general lossy inhomogeneous and anisotropic waveguides has been found in the literature. In this paper, a new finite element scheme for the most general linear waveguides has been derived from vector wave equations via a Galerkin procedure. In this formulation, triangular and quadrilateral edge elements have been used in order to avoid the spurious solutions. Furthermore, the final eigensystem involves only very sparse matrices, thus allowing important savings in time and memory     

Admittance analysis of nonuniformly spaced phased arrays of waveguide apertures in a ground plane

A general formulation for the analysis of a phased array of waveguide apertures in a common ground plane has been given for finite number of elements and nonuniform spacings based on network representation of the system. The analysis yields the radiation pattern, reflection coefficient, and aperture field in each waveguide. The pattern of an array of physically identical elements is expressed as superposition of patterns of infinite number of arrays. The formulation, when applied to single and two aperture cases, confirms the known results. It is then applied to investigate the properties of the element position modulated phased array of 15 rectangular waveguide apertures excited by uniform incident waves. Uniformly spaced arrays are also analyzed for comparison. The dominant mode and one higher order evanescent mode are included in the computations. The results show that the overall power reflection coefficient of the nonuniform array does not undergo any peaks over a wide scan range. It is concluded that the advantages of nonuniformly spaced arrays in suppressing grating lobes and eliminating blind spots are physically realizable.     

Mutual Coupling in Infinite Planar Arrays of Rectangular Waveguide Horns

The radiation admittance and effective power transmission coefficient are derived by a field matching method for arbitrary scan of an infinite array of open rectangular waveguides. Higher order waveguide modes, as well as higher order modes in the free space cell above the array, are included in the field matching problem. The convergence of the solution was studied, and the number of external free space and internal waveguide modes necessary for an adequate answer are given. Numerical results are obtained and compared with previous theory and with experiment for rectangular grids. Comparison is also made with experiment on a triangular grid which exhibited an anomalous notch in the element pattern. The experimental results in both cases seem to substantiate the theory, at least to the extent that could be expected from the small test arrays that were used for the measurements. The comparison indicates that the method of analysis includes all of the necessary features required for the study of mutual coupling effects in infinite arrays of rectangular waveguides having thick walls.     

General relations for a phased array of printed antennas derived from infinite current sheets

Simple and general relations characterizing the behavior of infinite phased arrays of printed antenna elements are derived from a model based on infinite current sheets. The Green's function of an electric current source on a grounded dielectric slab is used in various limiting forms to treat arrays in free space, arrays above a ground plane, arrays on a semi-infinite substrate, and arrays on a grounded dielectric slab. Current sheets are selected, using the orthogonality properties of the Floquet modes of the infinite array Green's function, to excite only a few specific low-order Floquet modes. Results from this idealized model, in the form of reflection coefficient magnitudes and input resistance, are compared with rigorous moment method solutions for specific elements (dipoles and microstrip patches). It is shown how the dominant scanning characteristics of a printed phased array, such as reflection coefficient and input resistance trends, scan blindnesses, and grating lobe effects, are dictated more by factors such as element spacing and substrate parameters than by the particular element type itself.     

MoM simulation of signal-to-noise patterns in infinite and finite receiving antenna arrays

An approach based on the method of moments is presented for the computation of the sensitivity of infinite and finite receiving phased arrays with active beamforming networks. The sensitivity is characterized in terms of signal-to-noise element patterns. Coupling of noise through the array is included in the analysis, as well as noise resulting from losses in the antennas. Simulation results are shown for arrays consisting of tapered-slot antennas made of metallic plates. For finite arrays, the average signal-to-noise ratio per element is not necessarily smaller than in the infinite-arraycase. For an 8/spl times/8 array, the average signal-to-noise element pattern is somewhat more narrow than for the infinite array. At broadside, the sensitivity of relatively small arrays (4/spl times/4 to 8/spl times/8) is described within order 1 dB by the infinite-array solution.     

Finite periodic structure approach to large scanning array problems

There are two conventional techniques dealing with mutual coupling problems for antenna arrays. The "element-by-element" method is useful for small to moderate size arrays. The "infinite periodic structure" method deals with one cell of infinite periodic structures, including all the mutual coupling effects. It cannot, however, include edge effects, current tapers, and nonuniform spacings. A new technique called the "finite periodic structure" method, is presented and applied to represent the active impedance of an array, it involves two operations. The first is to convert the discrete array problem into a series of continuous aperture problems by the use of Poisson's sum formula. The second is to use spatial Fourier transforms to represent the impedance in a form similar to the infinite periodic structure approach. The active impedance is then given by a convolution integral involving the infinite periodic structure solution and the Fourier transform of the equivalent aperture distribution of the current over the entire area of the array. The formulation is particularly useful for large finite arrays, and edge effects, current tapers, and nonuniform spacings can also be included in the general formulation. Although the general formulation is valid for both the free and forced modes of excitation, the forced excitation problem is discussed to illustrate the method.