Optimization of Impedance Plane Reducing Coupling between Antennas ()
1. Introduction
In recent years, there has been growing interest in artificial electromagnetic materials, such as electromagnetic band-gap (EBG) structures. An EBG material is a periodic structure in which electromagnetic states are not allowed at certain frequency bands (bandgaps) [1]. The EBG structure has also been applied in antenna design to suppress surface waves and to improve the radiation performance of the antenna [2-9]. The majority of EBG structures used in the microstrip patch antenna application are of a single period [2].
In practice, 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 widespread ways of reducing coupling between antennas is the use of a periodic structure [10-17]. The main point of this method consists in the fact that under certain conditions such a structure “pushes away” the field from its surface, in this way reducing the amount of energy which enters the receiving antenna.
Well-known papers from different journals [10-17] are usually devoted to the research of decoupling efficiency between antennas located on different surfaces, by using different electrodynamic structures (interference coverings and corrugated structures). In these papers, the influence on the level of antenna decoupling of different combinations of structural parameters is considered; i.e., only the task of analysis is solved. However, effective design of decoupling structures requires the solution of the design optimization problem. Probably the only example of known research where an effort is made to solve the design optimization problem of decoupling devices is in Reference [18]. In this book, the law of distribution of purely reactive impedance was obtained, providing faster reduction of the field along a geodesic line which connects the antennas, compared with an ideal conducting surface. However, the authors of the book limited their research to only a specific case of purely reactive impedance. Furthermore, the results in Reference [18] do not present the distribution formulas of the synthesized impedance.
In this paper, we propose a solution to the design optimization problem of impedance surfaces with the goal of creating effective decoupling structures. In particular, we investigate the degree of electromagnetic field (EMF) attenuation along the structure, the degree of reduction of the complete field level across the impedance part of the structure, and the decoupling level between the antennas. In addition, we determine the degree of influence of the resistive part of the impedance on the rate of the field attenuation along the impedance structure and the influence of the initial and final parts of the impedance structure on the level of the complete field.
The paper is organized as follows: In Section 2, we consider a solution to the design optimization problem of complex passive surface impedance using the law of electromagnetic field distribution. A solution to the problem using the pointwise synthesis method is given in Section 3, and numerical results are discussed in Section 4.
2. Optimizing the Impedance Plane
2.1. Statement of the Design Optimization Problem
In this section, we consider a solution to the design optimization of a complex passive 
surface impedance 
for a given EMF distribution, and we study the achieved spatial decoupling of antenna devices located on the same plane. In practice, the problem of design optimization is solved in the absence of the second antenna, meaning the electromagnetic field decreases as a function of increasing distance from the source [18]. Furthermore, a receiving antenna is placed at the points with minimum intensity of the “interfering” field. This is the reason why we only solve the design optimization problem for a single transmitting antenna. The optimization problem can also be explained by considering that in order to provide electromagnetic compatibility between antennas by means of a surface impedance
, it is necessary to minimize the functional
, where 
and 
are the powers of the signals at the exit of the receiving and the transmitting antennas, respectively. The decoupling coefficient, 
, is inversely related to the value of
, and is defined as 
[19]. Minimization of the functional is usually performed by means of non-linear programming and requires the specification of an initial choice for the variation of the impedance distribution,
. This choice is of great significance because it will influence the number of necessary iteration steps for the minimization of 
in order to reach global and local minima. The closer the initial choice is to the optimized design, the fewer the number of iterations steps. However, to widen the applicability of the results, we chose an initial solution which is nearly arbitrary (given the constraints imposed on the antenna), and work from there towards the optimized design.
To begin the process of design optimization, we first consider a solution to the two-dimensional design problem for the arrangement shown in Figure 1. On the plane, 
, at the point with coordinates 
let an antenna be located which is in the shape of an infinite thread of in-phase magnetic current, directed along 
axis. The opening of a narrow parallel-plate waveguide (Figure 1) can serve as a physical model of such a radiator. This kind of source creates an electromagnetic field in the upper space with a magnetic field vector of intensity:
, (1)
where 
is the wave number; 
is the wavelength; 
is the zeroth-order Hankel function of the 2nd kind; 
is the current amplitude; and 
is the characteristic resistance of free space. On the surface
, the boundary impedance conditions of Shukin-Leontovich are fulfilled:
. (2)
It is necessary to determine the dependence of the passive impedance 
for a given variation of the magnetic field, 
on the surface
. Once 
is obtained, the complete field in the upper space is found, and then the degree of decoupling between antennas can be obtained.
2.2. Solution of the Design Optimization Problem
To solve the problem, we use the Lorentz lemma for the upper space in Figure 1. As a result, we obtain the Fredholm integral equation of the second kind relative to the complete field 
on the surface
:
(3)
where 
and
is the field of an antenna radiation.
Figure 1. a narrow parallel-plate waveguide with impedance.
Now, we consider the complete magnetic field on the section 
for the entire upper space. From Equation. (3), relative to the complete field, 
, we obtain a Fredholm integral equation of the first kind:
(4)
The solution of Equation (4), a complete magnetic field, 
, on the final interval 
relative to
, can be obtained numerically. For example, through the method of Krylov-Bogolyubov [18], the required dependence of the impedance distribution can be found from the boundary conditions of Equation (2),
. In this case, the feasibility of the required impedance is not imposed by any limitations and therefore, can be checked in the process of calculation. In this way, we define a category of the passive impedance decoupling structures.
3. Pointwise Synthesis
The problem of the optimization of the design of the complex passive surface impedance is solved in the previous section. The resulting structure gives the complex dependence of the impedance, where its real part is large and positive. The realization of such an impedance with a large real (resistive) part is a complicated task on a planar surface. However, there is another way of achieving a large value of impedance: this can in practice be realized by a corrugated structure with the depth of corrugations divisible approximately by
, 
, even though such a structure is narrowbanded.
In this section, we consider the variation of antenna decoupling with the help of a purely reactive structure, for which the real part of the synthesized dependence of the complex impedance is simply taken as equal to zero. We assume a weak dependence of tangential components 
and 
on the impedance in Equation (1) and express the boundary impedance conditions, Equation (2), in a complex form:
(5)
Then, the solution of the system of equations
(6)
can be obtained with the help of the method of linear programming [20-23]. Here, if the real part of the impedance in Equation (6) is set to equal zero (i.e., the resistive impedance
, we obtain the minimum deviation of the solution of the system of equations in Equation (6) at each point of the surface of the decoupling structure (such a synthesis is called “pointwise” [20]). This minimum deviation point can be obtained using the exchange method of Stiefel [24]:
(7)
In order to look at the behavior of the overall impedance, which is the essence of the exchange method, the relationship between the imaginary and real parts of the impedance is explicitly plotted in Figure 2. The solution of Equation (7) is a point which is equidistant from where the straight lines, given by Equations. (7), where they cross the ordinate axis
.
4. Results and Discussion
· We state the complete magnetic field on the impedance part, specifying the shape of the field as:
, (8)
where 
is the coefficient of attenuation
.
The synthesized impedance must provide sharper EMF attenuation along the structure, as compared with an ideal conducting plane. The attenuation factor is defined by the value of the coefficient
. As an example, Figure 3 shows the dependence of the synthesized impedance
Figure 2. Illustration for the exchange method.
[
and
] for two different coefficients of attenuation, where the length of the impedance structure is
. It is seen that the real parts of the impedance are positive [
for 
(solid) and 
(dashed)] and two imaginary parts are negative (capacitive) [
for 
(dotted) and 
(dash-dotted)]. We note here that the impedance has a monotonic increase and the magnitude of the impedance is larger when
.
Next, we study the degree of the influence of this impedance on the coupling of antennas. Figure 4(a) shows the dependence of 
on the synthesized impedance (Figure 3), normalized relative to the field 
above an ideal conducting plane, for 
(solid curve) and 
(dashed curve). As we can see, a greater attenuation of the field is accompanied by a steeper slope of the impedance alteration, which coincides with the results in the Ref. [18]. The main difference consists in the fact that the impedance obtained in this paper not only has a reactive component but also a resistive component,
. In order to measure the degree of influence of the resistive part 
of the impedance, 
, on the field attenuation along the impedance structure, we show in Figure 4(b) the De-
pendence of 
for the synthesized impedance (
), normalized relative to the field 
above an ideal conducting plane, with the active component, 
(solid curve) and without it, 
, (dashed curve). The calculations show that the presence of the resistive part of the impedance not only doesn’t worsen the level of decoupling between antennas, as stated in Ref. [13], but, in fact, increases it by about an additional 5 dB. These results are probably caused by the different dependence of the impedance obtained in this paper compared with what was analyzed in Ref. [13] (this dependence is called uniform). The results of the design optimization show that greater attenuation of the field is reached with a higher rate of impedance growth (generally of its capacitive part, Figure 3). However, the rate of impedance growth (slope of the curves in Figure 4(b)) cannot be arbitrarily large because of practical limitations on the precision of the production of the structures. Therefore, one way to increase the rate of impedance alteration (increase the decoupling) substitutes the monotonically growing impedance with a periodic variation.
Figure 5 shows the dependence of the initial synthesized impedance with
, for the structure with length 
(
, red solid curve and
, blue dotted curve) and compressed by a factor of three (
, black dashed curve and
, purple dash-dotted curve), i.e., with the rate of impedance alteration three times greater. Figure 6 shows the dependence of the field, 
, normalized relative to the field 
above an ideal conducting plane, for the initial impedance (synthesized in Figure 5, solid and dotted curves) and periodic impedance (see Figure 5, dashed and dash-dotted curves). Here, the dash-dotted curve shows the case for homogeneous purely reactive