As it is known [1], some freedom in choosing the integral representation of electromagnetic fields in diffraction problems makes it possible to get various integral equations. The essence of the proposed method comes down to determining the intermediate characteristic — the charge distribution over the metal plate and then by means of integration — the electric current. The vectors of the electromagnetic field are determined via the vector potential of electric currents using the known equations:

(1) |

where .

The radiator is regarded as part of an infinite periodic antenna array, which makes it possible to proceed to the analysis of the field within one period. Due to this, it is reasonable to use a representation in the form of expansion in plane waves for the vector potential [2]:

(2) |

where

and is the vector of the electric current volume density, (*x*,*y*,*z*), (*x'*,*y'*,*z'*) are the coordinates of the point of observation and the point of integration respectively.

Fig.1

Presuming that the metal radiators are ideal conductors and they are also infinitely thin, the vector of the electric current volume density can be written down in the following way:

(3) |

and it is reasonable to replace the effect of the screen with a mirror view of the radiator:

(4) |

where is the vector of the electric charge surface density, is the Dirac delta function.

By integrating expression (2) using the axial coordinate *z* and taking into account (3) and (4), it is possible to get the values of the vector potential harmonic in two characteristic areas of the structure:

(5) |

The wave of the potential is presented in the form of the superposition of waves related to *Е*- and *Н*-waves of the spatial waveguide:

(6) |

The specified waves are determined by the following expressions:

(7) |

where .

The above relations are obtained for the case of homogeneous space above the array (*z*>0). Now taking into account the influence of obstacles in the form of the border between dielectric layers, it is possible to transform expression (6) in the following way for the area *z _{0}*<

*z*<

*z*, where

_{1}*z*and

_{0}*z*are distances from the screen to the metal plate and to the upper dielectric layer respectively:

_{1}(8) |

In this relation the quantities:

(9) |

are the functions of the factors of the reflection of *Е*- and *H*-waves from the dielectric interface *z*=*z _{1}*. The reflector factors can be derived from a system of equations corresponding to the boundary conditions on the dielectric interface:

(10) |

where

and , are the conductivities of the (*m*,*n*th harmonic of the *E*- and *H*-wave respectively.

The component of the vector quantities in (8) can be obtained by substituting (5) into (7):

(11) |

where

(12) |

and the symbol means the *X* or *Y* coordinate.

Integrating the last expression by parts and taking into account the boundary conditions for the normal components of the current on the radiator edges and the continuity equation:

(13) |

we will get

(14) |

where are the components of the electric charge surface density related to the corresponding components of the current density.

Thus, the field vectors (1) expressed via electrodynamic potentials are completely determined by the so far unknown distribution of the electric charge over the radiator surface. The required charge can be found with the help of the integral equation method. To do it, we use a boundary condition for the complete electric field on the surface of an ideally conducting radiator:

(15) |

where

is an extraneous electric field and *E _{0}* is the known value of the electric field intensity in the gap between vibrator arms. By integrating the first equation from (1) using transverse coordinates, we can get:

(16) |

where *С* is the integration constant.

By using the expressions found above in (16) for electric potentials, it is possible to get a system of two integral equations of the first type:

(17) |

The presence of two components of the charge actually requires the solution of the system of equations (17). In the particular case when the radiators are narrow vibrators with one nonzero component of the current whose transverse distribution is known, the system of equations (17) is reduced to one-dimensional integral equation:

(18) |

where

(19) |

(20) |

(21) |

The kernel (19) of the integral equation (18) is presented as a sum of two summands (20). In case the arguments coincide, the summand has an integrable singularity and, if we isolate it [3], we can reduce the integral equation of the first type (18) to the equation of the second type:

(22) |

where

the numerical solution of which is a well-defined problem [4]. In the last expression, the summand of the kernel:

(23) |

is a smooth function of position. Where *R* is the radius of a circle with its center in the singular point *x*=*x'*, and

The orthogonal function series expansion coefficients for the isolated singularity of the type can be found in the following form:

(24) |

where , , are zero-order and first-order Bessel functions, , are zero-order and first-order Struve functions respectively.

The function in the expression (22) is a results of isolating the singularity:

(25) |

Thus, we get the integral equation (22) and the algorithms of its numerical solution are well known. In this problem, it is convenient to use the Krylov-Bogolyubov interpolation method according to which the integral equation is reduced to a system of linear algebraic equations in case of the piecewise constant approximation of the unknown function. In order to do it, the vibrator is divided into *N* finite elements that go along only axis *X* because of the above mentioned small vibrator width. If we consider the unknown function is constant within each finite element and coordinate the solution in their middle points, we can come to the system of equations:

(26) |

where

Solving this system of equations, we can find the electric charge distribution over the metallic vibrator that is used to find all integral characteristics of a radiator in the array: the partial radiation pattern, the input impedance and others. In case the finite elements are not very small, the matrix of the system will be well-conditioned because the isolation of the singularity of the integral equation kernel leads to the absolute domination of diagonal elements over the rest of the elements in the matrix of the system.

It should be noted that the characteristics of a radiator are determined in this method via the charge distribution over the vibrator surface, instead of current as it is done in many related problems. The main advantage of the method comes down to the faster convergence of the necessary radiator characteristics in case of the specified calculation accuracy. That is due to the fact that the polynomial approximant selected for describing the charge distribution is equivalent to the polynomial for the current whose power is greater by one. It is known that the integral characteristics of radiators have better convergence as compared to the current distribution in case of the numerical solution of boundary value problems. In this case, the current distribution itself is the integral characteristic of the charge distribution, which determines the advantages of the method.

The developed methods of analyzing antenna arrays with printed radiators serve as the basis for creating the program for calculating the characteristics of a printed-circuit vibrator-type PAA. In fig.2 you can see the results of the numerical solution of the test problem for determining the electric charge on the vibrator of an array radiator with the following parameters: *d _{x}*= 0,6,

*d*= 0,25,

_{y}*a*= 0,5,

*b*= 0,03,

*c*= 0,01,

*z*= 0,15,

_{0}*ε*= 4,

_{1}*ε*=

_{2}*ε*= 1, φ= 90

_{3}^{0}, θ= 5

^{0}.

Fig.2