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## Charge method in mathematical modeling of printed radiators

Published: 08/18/2010
Original: Computer-aided design of microwave devices and systems // Interuniversity digest of scientific papers. (Moscow, MIREA), 1982, p.p.138…148
© 1982, V. S. Filippov, A. A. Sapozhnikov
© 2010, EDS–Soft,  http://www.eldys.org,   E-mail: publications@eldys.org

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 z0<z<z1, where z0 and z1 are distances from the screen to the metal plate and to the upper dielectric layer respectively:

 (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=z1. 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,nth 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 E0 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)

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References

1. E.N. Vasiliev Algorithmization of Diffraction Problems Based on Integral Equations. - In Applied Electrodynamics Moscow: Higher School, 1977, #1.
2. G.T. Markov Antennas. – Moscow/Leningrad: Gosenergoizdat, 1960.
3. A.S. Ilyinsky, A.G. Sveshnikov Numerical Methods in Diffraction Theory. Moscow: MGU, 1975.
4. A.N. Tikhonov, V.Ya. Arsenin Solution Methods for Ill-Defined Problems. Moscow: Science, 1979.