(1)

where ; .

In the above formulas, , — angles in spherical coordinate system that determine phase direction, (, ) — coordinates of radiator centers in Cartesian system where the OX axis is parallel to the longitudinal axis of the vibrators and the OZ axis is perpendicular to the array plane.

Excitation (1) allows considering of array element radiation as excitation of electromagnetic field in space waveguide – Floquet channel [1]. To analyze the field in space waveguide, the following well-known ratios are used:

(2)

where , u — vector and scalar potentials.

Integral equation of the problem is received from the boundary condition for the tangent component of the electric field on the surface of an ideally conducting radiator

(3)

where — side electric field equal to zero everywhere except the area of the gap between the vibrator arms where it possesses the known value , equal to the ratio of the exciting voltage to the gap width. By integrating the boundary condition (3) taking into account (2), we get

(4)

where С — integration constant, and t = x, y.

Electrodynamical potentials are represented as a factorization by plane waves [2], and consideration of boundary values for the normal component of the current on the vibrator edge and continuity equation allows making a step from the surface density of the current to the surface density of the charge. By regarding the potential wave as a superposition of waves related to H- and E-waves of the space waveguide, boundaries between dielectric layers are taken into account, and expression (4) is reduced to an integral equation of the following kind

which is received assuming the longitudinal component of the , is directed along y axis. As numerical experiments have shown, consideration of this component of the charge on the edge of the narrow vibrator does not significantly affect the results.

To solve the received equation, the method of moments was used, which transforms the integral equation into a system of linear algebraic equations. To do this, the original function is represented as

where — basis function; — unknown coefficient. As basis functions, the eigenfunctions of the area taken by the metallic plate are used (for a narrow rectangular vibrator – set of trigonometric functions of one coordinate).

Using basis functions as weight functions, we get the following system of linear algebraic equations:

where

By solving this system of equations, we can find distribution of charge on the metallic vibrator, which can be used to find all integral characteristics of the radiator in the array: partial direction diagram, input resistance and others.

Numerical experiments were conducted using the following parameters of the array and radiator (sizes are normalized to the wave length): array step = = 0,6; vibrator width b = 0,035; = 0,175; = 2,5; = = 0°. A dielectric coating is assumed, with thickness and dielectric permettivity .

Fig 1,a shows relation between the active and reactive components of the input resistance of the printed-circuit vibrator in an infinite array, and its length. The parameter is dielectric permittivity of the substrate , equal to 1,5 (curve 1); 2,0 (curve 2) and 2,5 (curve 3). When is increased, resonance size of the vibrator is decreased, however the working bank is also decreased. When the width of the printed-circuit b is decreased, the active component of the input resistance in the area of second resonance is increased, and the working bank is significantly decreased. The following values are shown at fig 1,b b: 0,035 (curve 1); 0,055 (curve 2) and 0,075 (curve 3). It can also be mentioned that resonance length is somewhat shifted in the direction of large values of vibrator length. The same behavior of the resonance length is observed when the height of vibrator above the screen is changed. At fig 1,c, has the following values: 0,150 (curve 1); 0,175 (curve 2) and 0,2 (curve 3).

Input resistance of the vibrator radiator in the array significantly depends on the step of the periodic structure, which is caused by interaction of radiators in the phased array. Figures 2,a and b show correlation of input resistance of the printed-circuit radiator with width b = 0,075 when the array step and is changed in E- and H-planes respectively. Curves at fig 2,a correspond to the following values: 1 — = 0,7; 2 — = 0,8 and 3 — = 0,9, at fig. 2, b: = 0,6 (curve 1); 0,65 (curve 2) and 0,7 (curve 3).

The most significant changes of the input resistance occur when the array step is changed in H-plane. For example, increasing of step size in H-plane from 0,6 to 0,7 doubles the active part of the input resistance and reduces the resonance length by 15% (fig 2,b). The same changes of array step in E-plane (fig. 2,a) do not significantly affect the input resistance. Note that these relations, caused by interaction of radiators in the array are opposite to the corresponding effects in the array of lamellar radiators, input resistance of which significantly depends on the step in E-plane [3].

Fig 2,c shows correlation between the input resistance of the strip vibrator with length 2l = 0,375, located on substrate with dielectric permittivity = 1,5, and the deviation angle of the array beam in primary planes (curves 1 correspond to the H plane, curves 2 — to E plane). Drastic change in input resistance of the radiator is caused by appearance of surface waves of H- and E-types. The charts show that the radiator in questions can be nicely matched by a 50-Ohm line in the sector of angles ±30°. For comparison, fig. 3,a presents the same curves when there is no dielectric ( = 1) for half-wave vibrator situated above the screen at a distance of a quarter of the wave length.

Fig. 3,c shows relation between the resonance length of a half-wave printed-circuit vibrator and dielectric permittivity of the substrate (curve 1). Changes of resonance length is different from decreasing proportional to (dashed-line curve), and difference grows as increases. This is caused by interaction of radiators in the array. Curves 2 and 3 correspond to change in resonance length and in E-plane (curve 2, = 0,5) it is more significant than change in resonance length when array step is changed in H-plane (curve 3, = 0,5). The latter experiment has been conducted with a radiator located on a dielectric substrate with = 1,5.

Fig 3,c shows cut set of the direction diagram of a printed-circuit vibrator radiator contained in an infinite array (1 — plane H, 2 — plane E) by primary planes. Block curves correspond to the vibrator with length 2l = 0,375, situated on substrate with = 1,5, dashed-line curves — to the radiator with 2l = 0,5, = 1 and = 0,25. Angle position of dips in the give diagram corresponds to the moment of appearance of surface waves in dielectric substrate.

Mathematical model of an infinite vibrator array and algorithms and programs that implement it on a computer, can be used to conduct research of characteristics of finite arrays with vibrator printed-circuit radiators. As an example below there are results of calculation of direction diagram of vibrator radiators in a square array, located in the center, the middle of the lateral side and the angle point of the radiating curtain.

To determine direction diagram of a vibrator radiator in a finite array, radiating elements of which are loaded to matched loads, results were used that were received in [4] using boundary waves method. The direction diagram of the radiator in a finite array is formed by the radiation of the regular part of the current inducted in radiators of the infinite array by the excited vibrator and the current of the boundary wave. The boundary wave rather slightly affects the direction diagram of the edge radiator in the dip area, related to excitation of surface wave in dielectric substrate. This affection quickly decreases as the excited vibrator is moved off the edge of the radiating curtain. In the rest, the shape of the direction diagram of radiators is determined by the regular part of the current.

The regular part of the current is a divergent wave, amplitude of which decreases the same as the coefficients of interconnection when moving away from the exciting element. Strength of the inducted wave depends on the direction in the plane of radiating curtain. The inducted wave creates radiation field, maximum of which is directed in the direction of dips in the direction diagram of the radiation in an infinite array. Interference of the direct space wave and the radiation field of currents inducted in surrounding elements is the cause of resonance dips in the direction diagram of the radiator in an infinite array.

Fig 4, a shows direction diagrams of the central element in square vibrator arrays with different number of radiators, situated in lattice points of the square grid with step = = 0,6 on a dielectric substrate with thickness = 0,175 and = 1,5. In the infinite array, as has been shown above, the direction diagram has two symmetric zero dips in E plane. In the finite array with comparatively small number of radiators, there are no dips since the currents inducted in the radiators of the finite array, surrounding the exciting elements, have limited space extent and the maximum radiation level of these currents in the direction of dips of the direction diagram of the radiation in the infinite array does not compensate the radiation of a single vibrator situated above the screen on a dielectric substrate. As the number of radiators and the array size is increased, the space extent of the inducted current and the level of maximum radiation are increased. As a result, dips appear in the direction diagram, depth of which grows as the array size is increased. In the square array with the number of radiators = 1681 dip width corresponds to 60% of the maximum value of the direction diagram, which describes the direction diagram of the radiation in an infinite array outside the dips quite good.

Fig. 4,b shows direction diagrams in the E plane of the edge radiator located in the middle of the lateral side of square vibrator arrays with different number of radiating elements. Currents, inducted by this radiator in a finite array, elements of which are loaded with matched load, are similar to those inducted by the central element of the square array on one of the two halves of the radiating curtain. Therefore, the direction diagram of the edge element is nonsymmetrical. In arrays with a small number of radiators, the direction diagram of the edge element is characterized by lack of both resonance dips, but, as the number of radiators and the array size increase, one of the dips appears, directed away from the array, because in this direction, the maximum radiation of currents is formed, inducted by the excited radiator in the finite array. As this is the case with the central radiator, increase in number of vibrators and array size leads to deepening of the dip of the direction diagram of the edge vibrator, which is equal to 40% of the maximum value when = 1681. The second dip oriented to the array does not exist, since in the infinite array it is formed by currents, inducted in radiators located on the opposite side of the excited vibrator and in the finite array these radiators do not exist. Therefore, the direction diagram of the edge radiator in a finite array always has one dip.

Fig 4,c shows direction diagrams of the angle radiator in a square vibrator array containing = 1681 radiating elements. Direction diagrams are calculated for E-plane (curve 1), as well as the field component ( = 135°, curve 2).

As the charts show, in E-plane, the direction diagram of the angle vibrator is practically not different from the direction diagram of the edge radiator located in the middle of the lateral side of the radiating curtain, except for the dip depth. The radiation level of the edge vibrator in the direction of the dip has an increase of about 18%, which is obviously caused by decrease in the number of radiators that form the dip in the direction diagram of the edge radiator. In the diagonal plane, the direction diagram of the edge radiator does not have dips, since the radiation maximum of the currents inducted by the edge radiator in the finite array lay outside the plane.

Thus, the direction diagram of the central vibrator radiator in the square array is symmetrical and as the size of the array and the number of radiators is increased, it converges to the direction diagram of the radiator in the infinite array. Direction diagrams of the edge element situated in the middle of the lateral side of the radiating curtain is nonsymmetrical and has one dip, the depth of which is increased as the size of the array and the number of radiators is increased. The direction diagram of the angle radiator coincides in shape with the direction diagram of the edge radiator only in primary planes.