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** Description:** Analyzing the basic methods and features of determining the directional characteristics of axis-symmetric antenna arrays, which include convex antenna arrays, in the near, far and intermediate zones, distinguish a number of their differences from the directional characteristics of flat antenna arrays. These differences are mainly due to the fact that in determining the directional characteristics, axisymmetric antenna arrays should be considered as a system of extraneous non-collinear electric (magnetic) currents on the convex surface of the antenna. The application of this approach requires the use of the basics of electrodynamics in solving problems of determining the directional characteristics of such antenna systems. However, the different orientation of the partial radiation patterns of the emitters does not allow the application of the diagram multiplication theorem and the inverse methods of determining the radiation characteristics. In addition, in the known works are not fully represented mathematical models of the fields of elementary sources in the far, intermediate and near areas with respect to convex antenna arrays. The mathematical model of the electric and magnetic components of the radiation field of elements of a convex antenna array is developed. The obtained expressions allow us to construct algorithms for numerical simulation of the radiation field of a convex antenna array in the far, intermediate and near regions. During the development of the mathematical model, it was considered that an internal problem for each of the dipoles entering the convex antenna array was solved and the distribution of currents in them was found. Moreover, the dependence of the complex amplitude of the current on time is considered harmonious. The mathematical expressions given in the article can be the basis for the method of analysis of the directivity characteristics of three-dimensional models of a convex antenna array in the far, intermediate and near regions.

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Keywords:
** convex antenna array, elementary source, Hertz dipole, electric and magnetic field, near zone, far zone, intermediate zone

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