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We study the coupling problem of two waveguide antennas using the design of a two-dimensional inhomogeneous impedance structure with a fixed reflected field. Since this structure enables electromagnetic compatibility between antennas located on a plane, the behaviors of the electromagnetic field along the impedance structure are investigated. The method of moments is used to solve the integral equations and the numerical results are presented and analyzed. To reduce coupling between antennas, we need to take into account both the amplitude distribution of the field along the structure and in the openings of the antennas. In addition, while designing the structure, it is necessary to control the coefficient of decoupling.

During last decade, the process of development of radio electronics, radio location, radio navigation, and radio communication worldwide was characterized by the following basic tendencies: technical realization of enhanced physical effects and technical solutions, aspiration to accomplish transmission and information processing in real time with the broad use of computers, and the expansion of the applications solved by technology. As a consequence, despite micro-miniaturization of radio electronics facilities (REF), the volume occupied by such equipment on mobile and stationary objects is increasing [1,2]. The progression of these modern trends is vitally necessary, but it aggravates even more the serious problem of the provision in radio engineering complexes (REC) of electromagnetic compatibility, which is understood as the ability of REF and REC to function together with limited degradation of their own essential parameters and features.

Practically, it is often required to provide significant decoupling between the receiving and transmitting antennas, located on a common surface at a small distance from each other. One of the most well-known ways to reduce coupling between antennas is the application of electromagnetic bandgap (EBG) structures [3-8]. The EBG structures have received increased attention in recent years [

In this paper, we re-visit the bandgap structure and present another interesting mathematical model for designing the structure and suppressing surface waves on metals. In addition, we examine the possibility of reducing the coupling between antennas located on the plane, using an inhomogeneous synthesized impedance. In particular, we investigate the design problem of the impedance surface when an infinite thread of in-phase magnetic current is located above the plane at a certain height, and also the case with its location right on the impedance surface. Finally, the behaviors of the complete field on the impedance surface and the decoupling level between antennas are also investigated.

The paper is organized as follows: in Section 2, we consider a solution to the problem of synthesis of an inhomogeneous impedance plane by a fixed reflected field.

A solution to the problem of coupling of antennas on an impedance plane is given in Section 3, and numerical results are discussed in Section 4. Finally, Section 5 is devoted to conclusions.

First, we consider a solution to the two-dimensional design problem for the arrangement shown in

where is the unit normal to the plane, is the surface impedance, is the electric field, and is the magnetic field.

It is necessary to determine the dependence of the passive impedance on the surface S. Once Z(x) is obtained, the complete field in the upper space is found, and then the degree of decoupling between antennas can be obtained.

where and are reflected fields, and. Here, is the ze-

roth-order Hankel function of the second kind, is the wave number, is the wavelength, is the imaginary unit, is the characteristic resistance of free space, , and is the first order Hankel function of the second kind.

The reflected field can be written as a sum of the reflected field in the fixed horizontal direction and the mirror-image field with unknown amplitude [

andwhere and are field vector components of an imaginary mirror source, and are field vector components of the given reflected fields. The solution to the design problem given in this paper differs from the solution in [

We now consider the analytical presentation of the distributed field on the plane. As long as the amplitude of the plane wave does not vary along the direction of its distribution, then for the reflected field in the direction, it is possible to write:

where is the distribution of the scattered field on the surface. We represent the mirrorimage field on the impedance plane as the following way:

where is the constant amplitude. Then, the summative magnetic field on the surface can be written

where. From the first Maxwell equation, neglecting the derivative multiplier and, we obtain for normalized on:

where. As a result, for the required impedance normalized on, we also have:

where and. In the general case, the resulting correlation gives the dependence of the passive impedance which gives a real part that can acquire positive as well as negative values.

Next, let us consider the design problem of the purely reactive impedance. Presenting the correlation of Equation (6) as a real and imaginary part, we can obtain the condition of feasibility of purely reactive impedance:

An additional degree of freedom in the form of a mirror-image field gives an opportunity to realize the impedance structure with [

where and is an angle of reflection. When the source of the field in Equation (6) is located right on the impedance surface, which provides a completely normal (at the angle) reflection of the incident wave (without a mirror-image,), the required impedance can be expressed:

where and. From the condition of purely reactive impedance feasibility

it is not difficult to find the variation of the wave reflected from the inhomogeneous impedance plane,:

where. In this case, the impedance can also be found from a straightforward expression:

where,

and are the zeroth and first-order Bessel functions, respectively, and and are the zeroth and firstorder Neumann functions, respectively.

The fact that variation of the surface impedance causes radiation of energy can be used to increase the decoupling between antennas, as well as to reduce the backscattering of the antennas. An example of a similar application of the surface impedance appears in

The general system studied in this section has two aperture antennas in the shape of the open ends of parallel-plate waveguides (transmitting and receiving ones) with opening sizes of a and b, which are located on the y = 0 plane at a distance L from each other. On the y = 0 plane, several boundary conditions of Shukin-Leontovich [Equation (1)] are fulfilled. To solve the problem of analysis, we use the Lorentz lemma in the integral form for each of the three areas:, , and, shown in

(in;

in):

where the subsidiary magnetic fields, , and are solutions of the nonuniform Helmholtz equations for complex amplitudes of the vector potentials for regions, , and, respectively. In this way, the fields in the opening of the antennas and on the impedance part of the flange can be found. From this, the minimum level of coupling between the two antennas can then be determined.

It is necessary to note that development of an algorithm for the mathematical model under consideration is based on the specifics of the electric field at the edges and on the numerical solution of a system of integral equations through the KrylovBogolyubov method [

We next study the behavior of the complete field on the impedance surface as a function of its dimensions and the parameters, and. In

Figures 4(a) and (b) show the dependence of, normalized relative to the field above an ideal conducting plane for fixed and various angles:

(solid line), (dashed line) and (dotted line); and for the fixed angle with various values of the parameter, (solid line), (dashed line) and (dotted line), respectively. The length of the impedance structure is equal to for both cases. The results of calculations show that the best data (greatest decoupling) are obtained with the parameters and when the impedance acquires the greatest capacitive value near the source of radiation. The greatest decoupling level is obtained with and. The synthesized impedance which gives appropriate results for increased decoupling should be taken into account, because it has a large negative value of the reactive part in close proximity to the antenna. This leads to the fact that the impedance practically creates an anti-phase field relative to the ideal conducting surface. As a result, all the energy of the electromagnetic field transfers into the energy stored around the antenna. The structure turns into a resonator without losses (for the reactive impedance), including radiation. As an example,