Modeling of Plane Arrays Using a Variational Approach

The variational statement of synthesis problem is generalized in order to account the additional requirements to the synthesized radiation pattern (RP) and field distribution in the specified points of near zone. For this aim, the minimizing functional is supplemented by term providing the possibility to minimize the values of field in these points; creating the deep zeros in the RP for the certain angular coordinates is realized too. The approach foresees reduction of an explicit formula for field values in a near zone. The results of computational modeling testify the possibility to create zeros in the given RP and to minimize the values of field in a near zone of plane arrays in a great extent.


Introduction
A lot of modern antenna systems are functioning by condition of a different series of various requirements to their radiation characteristics. One of such requirements is forming the deep minimums or zeros in the given points of near or far zones of antenna. This requirement corresponds to issues of the electromagnetic compatibility (EMC). Of course, all antenna systems are working at the Some general recommendations how to improve EMC issues for radiating systems are set out in [6]. Different technical solutions and measurements for improving the EMC requirements in the neighboring environment are discussed in papers [7] [8] [9]. Application of numerical techniques in order to provide the series of restrictions on the radiation characteristics of antennas on the whole [10] [11], to RP only [12] [13] or to characteristics radiation in a near zone [14] gives possible to elaborate the general vision to EMC problem solving. However, a comprehensive solution to the EMC problem for radiating systems and antennas is far from complete. The generalized variational approach for solving the array's synthesis problems, proposed in this paper, is one of the means that allows to solve this problem at a more fundamental level. The mathematical statement of synthesis problem including the EMC specifications for plane arrays and results of modeling are discussed in the presented paper.
The paper is organized as follows.
The generalization of usual synthesis problem for plane array allowing to prescribe the restrictions on the radiation characteristics in a near zone is proposed and explanations to terms of formulated functional are given in Section 2. The formula for RP of array (array factor) is discussed with introducing the generalized angular coordinates. In Section 3, the procedure of obtaining the nonlinear integral equation for optimal distribution of currents in array elements, corresponding to minimizing functional, is described. Reduction of explicit formula for components of electromagnetic (EM) field in a near zone of array is made in Section 4. Section 5 and Section 6 are devoted to methods of creating zeros (deep gaps) in RP of array and values of EM field in a near zone. The computational results of modeling for rectangular and hexagonal arrays are presented and discussed in Sections 7; Section 8 contains concluding remarks and proposal for the future investigations.

Variational Statement of Synthesis Problem
The functional aimed to take into account a series of requirements to both the RP and field distribution in a near zone is formulated as sum of several terms [15] Open Journal of Antennas and Propagation [16] [17], each of them ensures the best approximation to the array's radiation characteristics [18] ( ) ( ) ( ) The last term in (1) is applied to restrict the amplitude of currents in the array's elements. This provides to solve the problem to make away with the effect of super-directivity [19] [20] [21] for array. The additional optimizing values nm t are applied to restrict the amplitude of currents in the array's radiators.
We use for RP (array factor) of rectangular array the next formula [18] ( ) ( ) ( ) the RP of hexagonal array is determined similarly [22]. nm I are the currents in array's radiators, , nm f s s are the RPs of separate radiators.
We will use below the generalized angular coordinates 1 2 , s s instead of usual , θ ϕ . This provides ability to present the dependence of RP of array on the one generalized parameter that includes the geometrical size of array and length of wave or frequency simultaneously.

Numerical Minimization of Functional
In order to find the minimum of (1), we apply the approach of the variation calculus that yields in receiving the formula for the gradient of functional. Using such formula, we pass to non-linear integral equation [18] with respect to unknown currents in the array's elements.
The notations can be simplified in a great extent if to use the operator form as ( ) Here, operator B presents explicit relation for the field's components in a near zone. It is determined similarly to A, the difference is that the additional multiplier, depending on the distance r to array is presented. Equating gradient to zero, we receive the non-linear vector-matrix equation for the optimal currents in array's elements In practice, we do not deal with solving (4) because in the iterative procedure of its solving there is necessary to calculate the actions of A, B, * A and * B operators. If one to act on (4) by operator A, we have the nonlinear integral equation of Hammerstein's type with respect to RP f [18], because the ( )

Determination of Field Components in Near Zone
In order to carry out with explicit relation for the field components in a near zone using the currents nm I in the radiators of array, formula for the electric vector potential [23] , , e d 4π is applied. Here ( ) that is we will restrict ourselves to the value of order  Following the above assumptions, we will accept the same relations for the E and H field components in a near zone. As result, we have In such a way, we have obtained relations (9), (11) or (9)

Creation of Zeros in the RP
We rewrite (2) that is important for consideration below.
Formulas (13)-(15) are used for presentation of RP using the Kotelnikov's formula [25] and forming zeros in the given points ( ) We use the greed Kotelnikov's formula for (17) is where ( ) Substituting (18) into (13), we receive ( ) Finally, the Kotelnikov's formula for plane array is ( ) where j is iteration number.

Creation of Zeros in a Near Zone
Let us , E ϕ θ ϕ ′ ′ too, and we have optimal currents * nm I [27]. The components of electric vector potential (9) are determined by known * nm I . After this, we determine the rest of components of EM field. The radiation power density (RPD) is determined as [22] ( ) ( )

Creating Zeros in RP
Firstly, we show the computational results for the hexagonal array that has 127 elements. By this we assume known RPs Results are shown for frequency equal to 11.99 GHz, distance between radiators in the linear subarrays is equal to 0.0078 m, and distance between the linear subarrays is equal to 0.0087 m. The prescribed amplitude RP 1 F ≡ in the given range 1 F ≡ (Figure 1). In Figure 2, Figure 3    The level of the first sidelobe was reduced to −52.6 dB ( Figure 3). The obtained results testify that such a way to create zeros has some limitation. Use of Kotelnikov's procedure (25) or (26) is more complicate, but it can provide much small zeros.

Creating Zeros in a Near Zone
Secondly, the results for antenna with rectangular placement of radiator are considered. Array that consists of 121 radiators is examined, and distance d between them is equal to 0.01 m, that is square of array is equal to 0.01 m 2 ; the work frequency is equal to 5.976 GHz, by this the condition 2 d λ < is fulfilled that allows to form the main lobe of RP without several maximums (see [22]). In Fig Such modification of initial data allows to decrease the RPD from quantity of 0.1831 W/m 2 to quantity of 0.0373 W/m 2 that is 21.7% of its initial value. In the rest range of θ ′ and ϕ′ values, the RPD increased not more than on 10.2% (see Figure 5). The obtained results testify that the field values and its structure outside the considered θ ′ and ϕ′ for such r is close to constant value.
In Figure 6, the results related to minimization of the RPD in one point are shown. Similarly to the previous example, the RPR values diminish significantly.
If distance r to array's plane increases the values of the RPD decrease quickly.
The respective results are shown in Figure 7 for the distance that is equal to 10.0 m r = to array.
The presented results testify that the RPD diminishes in a great extent for such parameters of array; by this its values are more two order lower than for values corresponding to 0.5 m r = . For example, the maximum of RPD for 6.0 m r = reaches     One more requirement to increase the array's performance consists of the ability to take into account the additional limitations on the amplitude and phase distribution of the currents nm I  in the array elements. This is can be carry out by the prescribing the weight multipliers nm t instead of fixed t [28]. For goal to decrease nm I  , the values nm t should be increased. The numerical calculations demonstrate that such a way is suitable to avoid the superdirectivity of array's elements to a large degree. The weight multipliers nm t are prescribed so that the amplitude of currents nm I  is close to constant distribution.

Conclusions
The variational approach for solving the antenna synthesis problem by the given amplitude RP is modified for the goal to take into account the additional restric-