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.
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Generalized method of successive reflections in the theory of finite antenna arrays

Published: 02/21/2023
Original: Proceedings of institutes of MV and SSO USSR. Radioelectronics (Moscow), 1991, №2, p.p.26...32

Using obvious correlations

 (14)

let us present expression (13) as

 (15)

As a result equation (9) starts looking like this

 (16)

According to (16), the currents and voltages of the radiators are excited by EMF and MMF generators, as well as by the waves of transmission lines reflected from the loads of the partial radiators. Let us present the amplitude of the waves reflected from the loads using the current and voltage of falling waves

 (17)

where the coefficients of diagonal matrix are defined by expressions (7), (10),

As a result equation (16) starts looking like this

 (18)

where

 (19)

Equation (18) is solved with the help of the iteration method. At the initial stage of the iteration procedure, the solution we are looking for is represented as a sum of two summands that are the solutions of the following equations

(20а) (20б)

and equation (20а) is solved. At the K-th stage, the correction for the approximate solution obtained at the previous stage is presented as sum , the summands of which are the solution of the following equations

 (21а)
 (21б)

and equation (21а) is solved. The solution of the initial equation is a sum of the following series

 (22)

The first part of equation (21а) is the vector of doubled amplitudes of waves that reflected from loads of partial radiators at the previous stage of the interaction procedure. Thus, algorithm (20)...(22) actually describes the process of successive reflections of waves between radiator apertures and loads.

Iteration procedure (20)...(22) converges because the modules of the reflection coefficients at the apertures of the partial radiators are less than one due to the radiation and losses in constructive elements.

If we implement the iteration procedure numerically, equations (20а), (21а) are solved with the help of the Fourier transform method. After we apply discrete Fourier transform (DFT) to (20а), (21а), we get

 (23)

where the tilde indicates the discrete Fourier transforms (DFTs) of the corresponding values. Switching to solving equations (20а), (21а) is made by using inverse discrete Fourier transform

 (24)

where is the operator of the inverse discrete Fourier transform.

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References

1. Theory of Diffraction - H. Hönl, A. W. Maue, R. Westpfahl – M.: Mir, – 1964. – p. 425
2. Analysis of a Reflective Module PAA - V.P. Kudin, I.I. Lukhanin, Yu.S. Ushakov //Radioelectronics. – 1989, – No. 2, – pp. 78–80. (Higher Education Institution News).

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