When internal and external problems of electrodynamics are solved, the successive reflections (successive diffractions) method is used [1, 2] that ensures enough efficiency for the numerical algorithms of determining the electrodynamic characteristics of complex objects.

If modified, the specified method can be used to define the characteristics of finite antenna arrays taking into account edge effects and the interaction of the radiators.

Let us assume that a separate radiator in the array is an aggregate of metal constructive elements located above a plane or circular cylindrical screen and apertures in the screen connecting the radiator with the attenuator or feeder line to the reflective phase shifter. The structure of the radiating curtain may include dielectric substratum and a multilayer dielectric covering in the form of a sheet of the corresponding shape located parallel to the screen.

Within the induced EMF method, the currents in the radiators are defined by the following equation:

when are the matrices of internal and mutual impedances, conductivities, and transmission coefficients defined relative to basis functions in the expansions of electric and equivalent magnetic currents of the radiators

where р, t is the number of the radiator and the constructive elements in the radiator respectively; are basis functions in the expansions of electric and magnetic currents; is the coefficient vector in the abovementioned expansions; is the vector of EMF and MMF defined relative to the corresponding basis functions in expansions (2) expressed via the electric and magnetic field of the reflective array feed or the waves of the attenuator of a transmissive array. The dimension of basis functions in (2) can be chosen in such a way that the coefficients of the vector will have the dimensions of the electric current and voltage. So these coefficients will be called the currents and voltages of the radiators. Let us assume that the finite array is composed of exactly the same equally spaced radiators.

The matrix of internal and mutual conductivities is a sum of two summands

where is the matrix of the external, internal and mutual conductivities; is the matrix if internal conductivities characterizing the radiation of coupling apertures in the transmission line that connect radiators with reflective phase shifters or the attenuator. The coefficients of the matrix represent a sum of partial conductivities

where is the wave admittance of the q-th internal wave of the transmission line; is the reflection coefficient of this wave regarding the load in the plane of the coupling aperture; is the coefficient taking into account the "interaction" between the basis functions of the t-th aperture via the transmission line over the q-th natural wave.

The solution of equation (1) using the methods we know becomes difficult when there is a lot of radiators and there are dielectric coverings and substrata. We can make it considerably easier if we manage to express the solution of equation (1) via the solution of the problem regading the excitation of the corresponding regular structure. To do it, we expand the finite array to an infinite one and select loads for additional radiators so that we exclude the influence of these radiators on current distribution in the finite array while modelling. After that we use the successive reflections method. According to this method, currents in the elements of an infinite array are represented as a sum of currents excited during the successive reflection of waves between the apertures of the radiators and loads (the phase shifters of reflective arrays or the attenuator of transmissive arrays, as well as the loads of radiators expanding an finite array to an infinite one). We need to solve the well studied problem regarding the excitation of a regular radiating structure for each reflection.

As you will see later, the models of loads in the radiators of a finite antenna array and additional radiators are different from regular models of physical loads. That is why the method below can be called the generalized method of successive reflections.

Let us present the matrix of equations (1) as a formal sum

where matrix is defined by expression (4). Matrix is diagonal; let us define the coefficients of this matrix using the following expression

where

in fact we have

However, now every distribution of the electric current corresponding to the basis functions in expansions (2) is considered as a radiator connected with the load of the transmission line having wave impedance . The coefficients of the reflection of loads are defined by expression (7). Obviously, short circuits will be such loads, since the length of transmission lines connecting these loads with radiators must equal zero.

The distributions of magnetic current in (2) can also be considered as radiators loaded to a multiport transducer with the matrix of conductivities . Let us assume that radiators and the multiport transducer are connected between each other with zeo-length lines with wave conductivities .

The distributions of electric and magnetic current specified above will be called electric and magnetic partial radiators or simply partial radiators further on.

Let us replace the initial matrix in equation (1) with the matrix of an infinite equally spaced array. Taking into account (5), equation (1) can be presented as

The coordinates of the vector in the right part (9) corresponding to radiators expanding the finite array to an infinite one equal zero.

Equations (1) and (9) are equivalent if we assign for additional radiators in (4), (6) the following

It follows from (9) that the currents and voltages of additional radiators equal zero if conditions (10) are met, therefore these radiators are in fact not there.

Equation (9) describes the excitation of an infinite array, but this array is irregular since the loads of its radiators are not the same. To switch to the equation describing the excitation of a regular structure, let us picture the excitation of the array as excitation by generators with EMF and MMF and waves reflected from the loads of partial radiators.

Let us assume that

Then the partial radiators with transmission lines form a regular structure.

Let us introduce these designations

where are the vectors of voltages and currents of loads of electric and magnetic partial radiators respectively. Let us present (12) as a sum of voltages and currents of falling and reflected waves in the corresponding transmission lines

where the upper +, - indices indicate falling and reflected from loads waves respectively; are the diagonal matrices of the wave impedances and conductivities of transmission lines connecting loads with electric and magnetic partial radiators.

Using obvious correlations

let us present expression (13) as

As a result equation (9) starts looking like this

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

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

As a result equation (16) starts looking like this

where

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

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

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

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

where is the operator of the inverse discrete Fourier transform.

The equation of the problem regarding currents in a single unit of a regular structure with partial excitation is (23). Thus, correlations (23), (24) make it possible to express the solution of problem regarding the excitation of the finite reflective array by solving the problem regarding the excitation of a regular structure.

In case of numerical implementation of this algorithm, the discrete Fourier transforms of the functions in (23), (24) are found with the help of the fast Fourier transform. The necessary number of iterations depends on the required accuracy of determining currents. The practice of calculations shows that in most cases error in determining the characteristics of an antenna array does not get higher than fractions of or several percent if we use the simplified algorithm. This simplification comes down to the fact that we "connect" regular loads to radiators that expand the finite array to an infinite one, in case of which there are practically no currents excited in these radiators. For instance, if we model a dipole antenna, it is enough to "add" infinitely large impedances to the gap of additional dipoles. In this case, it is possible to use existing software for modelling regular radiating structures with minimum tweaks to determine the characteristics of transmissive and reflective antenna arrays both with a small number of radiators and with the number of radiators up to 10⁴ and more.

Fig.1

As an example, in Fig. 1 you can see H-plane diagrams of the radiation pattern of a dipole antenna array with a square radiating curtain containing 256 radiators located above the flat screens in the nodes of a rectangular grid with spacing = 0,8, = 0,6. The amplitude distribution of the waves exciting the grid corresponded to the cosine-squared-on-a-pedestal function ensuring the level of side lobes not higher than -40 dB. To calculate the radiation pattern with the accuracy enough to reproduce side lobes whose level reaches -60 dB, we had to take into account 10 reflections and 32×32 DFT zero points.

With the specified DFT parameters, the problem of regular structure periodic excitation with one period containing 32×32 elements is solved at each stage of the iteration procedure.

The current in radiators expanding the finite array to an infinite one did not get higher than -60 dB relative to the maximum value of current in the radiators of the array under examination.

The computations time with the specified number of reflections was not more than 20 s on an ES-1045 computer. It took 24 min of computational time to calculate the characteristics of a regular array for the specified DFT parameter with the approximation of current distribution on dipoles by three basis functions.

Using the same array of characteristics for a regular dipole structure, it is evidently possible to determine currents both in a degenerate array consisting of one radiator and in an array with any shape of its radiating curtain that fits in some part of a regular structure containing 32×32 elements, including a square array with 1024 elements. Any option will require the same 20 s of computational time on an ES-1045 computer.

If the size of the array with the characteristics of a regular radiating structure does not change, the accuracy of determining the currents of the finite array reduces if the number of radiators increases. However, it should be noted that, in case of amplitude distributions that decrease closer to the edge of the radiating curtain, error in determining the characteristics of finite antenna arrays meeting the ratio specified above between the array of DFT zero points and the area of the radiating curtain barely depends on the number of radiators and the shape of the radiating curtain.

Calculating the inverse discrete Fourier transform (24) has no difficulties because function does not have peculiarities, such as poles, connected with surface waves. There are no peculiarities like these because value corresponds to a regular structure composed of partial radiators with loads in the form of match-terminated transmission lines or multiport transducers with losses. There are no surface waves in structures like these.

The appropriate choice of basis in (2) and the loads of partial radiators makes it possible to study the characteristics of arrays composed of different radiators as well as unequally spaced antenna arrays where distances between elements are divisible by arbitrary constant.