Mutualinfluence of two dipole antennas

Accounting for the mutual influence of emitters is one of the important problems in antenna theory. Due to the fact that the results of numerical calculations are poorly presented in the scientific literature, it is very important to develop a theory and perform calculations. In this paper, we obtain a one-dimensional integral equation with a dedicated main operator for two identical emitters. An efficient numerical method is constructed, a program is developed, and numerical calculations are performed over a wide range of input parameters.


Introduction
Electrodynamic analysis of dipole antennas is based on solving integral equations for surface currents. Knowledge of surface currents allows finding the main characteristics of antennas: input resistance, field in the far zone, field in the near zone, active and reactive parts of the radiation power. The kernels of integral equations have singularities; therefore, the numerical solution of integral equations presents certain difficulties. To overcome them, the problem operator is represented as a sum of two operators: the main one and the compact one. The principal operator matrix can be found analytically. The matrix elements of a compact operator are efficiently calculated on a computer using standard programs. In this work, the method of separating the main operator is supposed to be applied to calculate the input resistance of a system of two vibrators.

Formulation of the problem. Integral equation
Let us consider two identical tubular linear vibrators of length 2l and radius a . The distance between the axes of the vibrators is equal to d . It is required to determine the currents flowing through the vibrators and other characteristics through them. We compose the integral equation to determine the currents from the condition of equality to zero of the tangential component of the electric field [1][2][3][4].
x -are variables of integration,  -is an observation variable,   It -is an unknown current function. Let us note that the second term on the left-hand side of (1) is due to the presence of the second vibrator,

Allocation of the main positive definite operator
Using the asymptotics of the modified Bessel functions     00 1 2 the equation (1) can be reduced to the form with a distinguished main operator [4] where The method for isolating the principal operator based on asymptotics (2) is widely used in scientific literature and is one of the main mathematical techniques. The main operator A in this problem is the same as in the problem of analyzing a dipole antenna in free space. As follows from formula (2), the effectiveness of this approach depends on the radius of the vibrator a , and is not very effective for very small asymptotics a (2) The operator

Numerical method for solving an integral equation An operator
A is a positive definite operator in space   An important property of the system of functions is its orthonormality Thus, the matrix elements of the operator  In conclusion, we describe a method for solving system (11). The first N of unknowns are found from the solution of the truncated system and the rest -analytically by the formula

Results of numerical calculations
In all calculations, the results of which are presented below, the primary field was specified as  Analysis of the calculated dependencies allows us to make an unambiguous conclusion about the mutual influence of vibrators, and in an oscillating form. The closer the antennas are to each other, the stronger the effect is. As the antennas move away, the input impedance values tend to those for free space. The decrease in mutual influence occurs rather slowly, the effect is clearly visible at a distance 10 d   .

Conclusion
Thus, a theory has been developed that makes it possible to calculate the parameters of antennas taking into account their mutual influence. This is necessary for practical purposes when setting up and matching antennas with feed feeders. The results can also be useful in the field of electromagnetic compatibility.