(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).