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Finite Antenna Arrays, An Eigencurrent Approach
Summary of the Ph.D. Thesis of Dave Bekers
Radar, in full 'radio detection and ranging', is used for many different goals, such as directing air traffic, measuring vehicle speeds, and detecting and tracking ships and airplanes. The principle of radar is based on the phenomenon that metallic objects reflect radio waves. These waves are transmitted and received by the antenna of a radar system. The two main goals of the design of such an antenna are: the energy induced in the antenna as electromagnetic radiation should be radiated in a specific direction and the transfer of energy from source to electromagnetic radiation should be optimal. Examples of antenna types are wire antennas, parabolic antennas, and antenna arrays.
Figure 1: (left) a radar system with a (phased) antenna array, (right) an antenna element
Antenna arrays consist of separate antennas, called elements. The number of elements varies from about ten to many hundreds. In many applications, the elements exhibit all the same shape and are positioned in a regular geometry. The main advantage of antenna arrays over other types of antennas is the possibility of electronic beam steering by using phase shifts between the elements. The beam compares to a light beam used in drama plays, but it is invisible for the human eye. Contrary to mechanical beam steering, electronic steering is accomplished without time delay due to mechanical constraints. Therefore, electronic steering in (phased) arrays facilitates multiple functions, for example, scanning, tracking and missile guiding.
The design and development of radar systems is complex and costly. To reduce design costs and design risks, and to improve the performance of the arrays, simulations are used. Simulations should meet a number of criteria: they should be fast executable, they should show boundary effects and effects of mutual coupling, and they should determine the antenna performance parameters accurately. Simulations based on the generally applied infinite-array approach and simulations based on the the finite-element method do not satisfy these criteria. Simulations of the first type do not describe boundary effects, while simulations of the second type are computationally too expensive. Both types of simulations do not provide direct insight into the physics relevant to the design. In this thesis, we propose an approach, which satisfies the mentioned criteria and provides insight into the physics. The approach can predict the disadvantageous, and sometimes even disastrous, influence of standing waves on the performance of finite arrays. Moreover, the approach can indicate how this influence can be reduced for the entire scan range of an array. The approach of this thesis was tested on line arrays with regular geometry, where the elements are rectangular-shaped microstrips or ring-shaped microstrips. In both cases, the arrays are positioned in free space or above a conducting plane. Contrary to the length and circumference of the microstrips, their circumference is small with respect to the wavelength. Moreover, a manual is provided in which the steps of the application of the approach to arrays of arbitrary elements are described.
Figure 2: A Uniform Line Array of Strips
In the proposed approach, the behavior of a finite array is described by its eigenvibrations or eigencurrents. These eigencurrents are the eigenfunctions of the impedance operator that relates the currents on the elements to their excitation fields, which are induced by a planar wave or local sources, for example. From a physical point of view, the eigencurrents are standing waves of the array. The corresponding eigenvalues represent the characteristic impedances of the eigencurrents. The larger the characteristic impedance of an eigencurrent, the less this eigencurrent contributes to the current on the elements for a given excitation field. The concept of eigencurrent appears extremely suitable for the design of arrays, because the design characteristics are one-to-one related to the excitation of specific eigencurrents. This thesis reveals that eigencurrents and their corresponding eigenvalues are one-to-one related to scan lobes, to monopulse lobes, to grating lobes, to modulated impedance oscillations, to impedance variations contributed to surface waves, and to many other properties of the array. Besides a physical interpretation, the approach with eigencurrents leads to fast-executable simulations; for, although the performance parameters of an array vary as a function of the geometry paramaters and the frequency, the eigencurrents vary hardly. Only the eigenvalues vary, in a regular way, as a function of the geometry parameters and the frequency. Therefore, eigencurrents can be fixed for a certain choice of parameters in order to use them in simulations for other choices of parameters. The corresponding eigenvalues can be approximated by the Rayleigh-Ritz quotient.
Figure 3: (left) electric far field in the xz-plane of the eigencurrent that induces the grating lobe of a line array of 40 rings with spacing 3/5*wavelength, (right) electric far field in the xz-plane of the eigencurrents that induce the broadside beam (solid curve) and the monopulse (dashed-dotted curve).
Starting point of the approach is the determination of the eigencurrents, and the corresponding eigenvalues, of a single element. The eigenvalues and eigencurrents are computed from a `normalized' moment matrix related to chosen expansion functions for the current on the element. Subsequently, an inner product is determined with respect to which the single-element eigencurrents are orthonormal. The corresponding moment matrix in terms of these eigencurrents is a diagonal matrix with respect to the new inner product. In the second step, a reduced moment matrix for the array is computed with respect to the composition of the new element inner products, where the expansion functions are the eigencurrents per element. Only single-element eigencurrents that contribute to the mutual coupling in the array are taken into account. Since the eigencurrents are not known a priori, the number of coupling single-element eigencurrents is estimated on basis of the behavior of the single-element eigenvalues. Single-element eigencurrents with large eigenvalues will or hardly contribute to the mutual coupling. The result of the second step are the array eigencurrents, which are described as concatenations of linear combinations of coupling single-element eigencurrents. The array eigencurrents and their corresponding eigenvalues are divided into groups, where each group corresponds to one single-element eigencurrent, called the dominant single-element eigencurrent. The eigencurrents in a group are perturbations, not necessarily small, of the eigenvalue of the corresponding dominant single-element eigencurrent. Their spread appears to be a quantitative measure for the mutual coupling in the array. If the spread of a group is small, the coupling of this group with itself and with other groups needs not be taken into account. Thus, whether a sufficient number of single-element eigencurrents to describe mutual coupling was taken into account can be determined a posteriori on basis of the spreads. Numerical simulations reveal that the mutual coupling of arrays, composed of elements typically designed for the excitation of one specific eigencurrent, can be described with one or two groups of coupling eigencurrents. If mutual coupling between the other groups is neglected, the computation times are reduced by a factor 10 to 50 with respect to the conventional moment method. The spread is also a quantitative measure to determine the number of neighboring elements needed to describe mutual coupling. If coupling between neighbors is neglected, a further reduction of the computation time is obtained.
Figure 4: Color representation of the (magnitude of) the expansion coefficients of the array eigencurrents of an array of 29 rings with respect to the single-ring eigencurrents (only one group of coupling eigencurrents).
A further investigation of the eigencurrents of line arrays composed of rings and strips has led to two important observations. First, the coefficients of the dominant single-element eigencurrents in each group depend hardly on the element shape. On basis of this observation, a first-order approximation of the behavior of the behavior of line arrays with complex elements is described by the coefficient distributions of the eigencurrents of line arrays with simpler elements. Second, the coefficients of the dominant single-element eigencurrent show the same pattern as the coefficients of the eigencurrents of a single strip obtained with piecewise expansion functions. On basis of this observations, a first-order approximation of the behavior of line arrays with complex elements is described by the coefficient distributions of the eigencurrents of a single strip with piecewise expansion functions. It is to be expected that the coefficient distributions of the eigencurrents of a rectangular patch can be used as a first-order approximation for the behavior of a rectangular array. From the second observation, we conclude that the array is an entire object rather than a collection of separate elements.
It should not go unrecorded that the characteristic behavior of arrays is caused by resonant behavior and that resonant behavior is caused by the excitation of specific eigencurrents. The eigenvalues, or characteristic impedances, of these eigencurrents are small in comparison with the eigenvalue that corresponds to a scan lobe. Both modulated impedance oscillations and variations of element impedances contributed to surface waves of the array structure are caused by the excitation of eigenvalues with relatively low eigenvalues. The distribution of the group with the lowest eigenvalues predicts the loading of the system needed to reduce or avoid resonant behavior. Finally, we mention that the relevance of the concept eigencurrent is demonstrated in this thesis by a number of specific physical effects, observed in the practice of antenna design.
Eigencurrent Analysis of Resonant Behavior in Finite Antenna Arrays' in IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 6, part 2 (Special Issue on 35th European Microwave Conference), pp. 2821 - 2829, June 2006
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