nonlinear synthesis problem
plane antenna array
amplitude directivity pattern
nonuniqueness and branching of the solutions
V. F. Kravchenko, L. P. Protsakh, P. А. Savenko, M. D. Tkach
In the present work the variational synthesis problem of flat rectangular equidistant antenna array (AA) according to the prescribed amplitude directivity pattern (DP) is considered. At that the freedom of a choice the phase DP is used as an additional degree of freedom for improvement of approximation quality of the module synthesized DP to given. The functional minimizing the mean-square deviation of the module synthesized DP of array from given amplitude DP within the limits of the given solid angle and level of radiating capacity in sidelobes within the limits of the period multiplier of array, is used as optimization criterion. In the further the problem on finding the optimum solutions is reduced to investigation and numerical solving of the Hammerstein type two-dimensional nonlinear integral equation with singular kernel.
It is shown, that the absence of the requirements to the phase directivity pattern in conditions problem generates the uniqueness and branching of solutions dependent on two nondimensional physical parameters of array , , characterizing the electrical sizes of array and sizes of solid angle, in which is given required amplitude DP. From the practical point of view, presence of several types of various solutions creating the same or similar amplitude DP’s enables to choose solution having simple realization. From the mathematical point of view the investigation of questions of uniqueness of the solutions demands application of rigorous methods of branching theory of nonlinear equations.
The researches of the solutions properties of nonlinear synthesis problems of plane АA differ essentially from investigations of the synthesis problem of linear antenna array (LAA). In particular, unlike the existing branching points of the solutions at LAA synthesis, in the synthesis problems of plane arrays there exist the branching lines of solutions, and problem on their finding is not enough studied two-parametrical nonlinear spectral problem.
It is shown, that at any values of parameters and there exist the solution (named initial) in class of in-phase patterns, being optimum only at small values of these parameters. More effective complex solutions (in sense values of minimizing functional) branch-off from the initial solution with growth of parameters and . The properties of branching-off solutions depend on properties of the given amplitude DP and parameters of array. For finding the branching-off solutions by means numerical methods it is necessary to locate their, i.e. to determine domains of their existence and properties. The problem on finding the branching lines of the solutions is reduced to the problem on eigen values for linear homogeneous two-dimensional integral equation with nonlinear two-dimensional spectral parameter entering in kernel. The numerical method of finding the branching lines of the solutions is offered.
The iterative process for finding the optimum solutions of initial synthesis problem is constructed and justified. With this purpose the varieties of functions having the property of invariance concerning to Hammerstein integral operator describing solution of synthesis problem, are determined. It is shown, at synthesis of symmetric given amplitude DP’s the optimum solutions with odd on one argument (or on two arguments) phase DP’s branch-off in the first. At that the optimum distributions of currents in an array creating such DP’s are real but nonsymmetrical concerning to the center of array.
The numerical example of synthesis of the given two-beam amplitude DP is presented. It is shown, that finding the optimum phase DP improves considerably approximation quality of synthesized DP to given in comparison with synthesis in a class in-phase DP. In particular, it allows also decreasing the sizes of array nearly by 30 % at identical efficiency of synthesis in comparison with synthesis in a class in-phase DP.
We shall note that the investigation of non-uniqueness and branching of solutions dependent on physical parameters and is the basic difficulty at the solution of nonlinear synthesis problems. As follows from researches for special case DP , the quantity of existing solutions grows significantly with increase of parameters and . However, for synthesis problems of antenna arrays it is important obtaining the best approximation to the given amplitude DP at rather small values of parameters , what allows to be limited to investigations of several first branching points (lines). Obtaining the irrefragable answer concerning the exact quantity of existing solutions of equation when for certain values of parameters , is the subject of separate investigations.