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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

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: dx= 0,6, dy= 0,25, a= 0,5, b= 0,03, c= 0,01, z0= 0,15, ε1= 4, ε2=ε3= 1, φ= 900, θ= 50.

Fig.2

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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.