 ElectroDynamic Systems Software ScientificTM Antenna array

Complex directional antenna consists of separate near-omnidirectional antennas (radiating elements) positioned in the space and driven by high-frequency …

(from 'Glossary' of our web–site)  Vladimir Sergeevich Filippov, Professor of the Chair of Radio-physics, Antennas and Microwave Devices of MAI (Moscow), Doctor of Science in technics. Dmitry Vitalievich Tatarnikov, Professor of the Chair of Radio-physics, Antennas and Microwave Devices of MAI (Moscow), Doctor of Science in technics.
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Boundary waves and characteristics of slot array

Published: 02/05/2006
Original: Proceedings of institutes of MV and SSO USSR. Radioelectronics. (Moscow), 1986, №2, p.p.41...46

One of the most efficient methods of numeric research of characteristics of finite antenna arrays is the method suggested in . This method is based on the transformation of the mutual resistance matrix into a circulant matrix and iterative procedure of solution of the system of linear algebraic equations with the transformed matrix. Also efficient is another method based on the concept of boundary waves , which states that the distribution of current in a finite antenna array with uniform amplitude and phase distribution of waves propagating in the feeder lines, can be represented as a sum of the regular part and the boundary wave propagating from the border of the radiating matter deep into the array. The regular part is the distribution of the current in an infinite array when the excitation is partial, a part of which is the excitation of the finite array. The boundary effects are caused by the interference of the regular part of the current and the boundary wave current.

According to the method given in , the equation for the regular part of the current and the boundary wave can be derived from the original equation of antenna array (1)

where Z — matrix of the system of linear algebraic equations; — vector of unknown coefficients in factorizations of radiator currents by basis functions; — vector of the excitation of the finite array. To do this, the Z matrix is complemented to the matrix of an infinite antenna array: (2)

and the vector is represented as a sum of the regular part of the current and the current of the boundary wave (3)

By substituting (2), (3) to (1) we get the equivalent system of equations: (4)-(5)

where — partial excitation of the infinite array, which is the “continuous” continuation of to the radiator, which complements the finite array to the infinite one. Arbitrary excitation of the finite array can be represented as a superposition of partial feeds using the discrete Fourier transform. The equation (4) defines the regular part of the current, and (5) defines the boundary wave.

To find the regular part of the current we use the known method of solving periodical structure excitation problems. The boundary wave equation is the equation of the finite array excitation problem. To solve it, the iterative procedure  is used, which transforms the equation (5) like the original equation (1) to the equivalent system of equations: (6)-(7)

where vectors , related to by the relation (8)

The vector is the first approximation of the boundary wave, is the correction of . Equation (6) corresponds to the problem of the excitation of a finite array amounting to the infinite one, radiators of which are loaded to the matched load, and is solved similar to equation (4). To solve equation (7), the next step of the iterative procedure is performed and so on. At the n-th step we have (9)-(10)

The boundary wave is defined as the sum of solutions , received at each step of the iterative procedure (11)

The numerical experiment shows that in most cases, the error is reduced to less than 1% after the third step of the iterative procedure. The convergence of the iterative procedure is ensued from the asymptotic approximations of the right part of equation (9), which state that the coordinates of vector converge to zero when as the terms of the decreasing progression. The characteristic feature of the algorithm (6)…(11) is the fact that each step of the iterative procedure solves the problem of the excitation of a finite array amounting to the infinite one. This allows using of the existing programs for infinite antenna arrays with minor changes. The principal possibility of using this algorithm does not depend on the step and number of radiators in the array, existence or absence of dielectric coating and surface waves, which it causes.

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