The values of

## 1. Introduction

The problems of antenna synthesis, which use the amplitude RP as input information, are often used in the process of antenna design for many practical applications [4, 16, 27, 31]. In spite of the fact that the respective mathematical problems are ill-posed [32] and they have the branching solutions [29], the antenna synthesis according to the desired amplitude characteristics is very useful and perspective.

As a rule, the branching of solutions depends on the properties of prescribed amplitude RP, geometrical and physical parameters of the considered antenna. The methods of nonlinear functional analysis [23] allowing to localize the branching solutions are applied for investigation of solutions and determination of their number and qualitative characteristics. Such approach too much simplifies determination of the optimal solutions by the numerical methods. The iterative processes for the numerical solving of the corresponding non-linear equations were elaborated in [3, 8, 11].

The Chapter is organized as follows.

In Section 2 we derive the main formulas for RP of antennas and introduce the objective functionals for the synthesis problem. Also in Section 2 we consider the variational statement of problems and derive the fundamental nonlinear equations of the synthesis.

Section 3 contains the application of the proposed approach to several types of antennas. Depending on the restrictions which are imposed on the sought distribution of current or field in the antenna elements and type of antenna, the problems of amplitude-phase, amplitude, and phase synthesis are considered for the specific antennas. The methods of successive approximations are applied for solving the derived non-linear integral equations; the convergence of the elaborated methods is discussed. The direct optimization of the proposed functionals by the gradient methods is performed and successfully applied to solving the amplitude and phase synthesis problems.

In Section 4 conclusions are formulated.

## 2. The theoretical background of the synthesis problems

In this Section, we present the necessary information about the properties of the electromagnetic (EM) field in far zone, introduce the variational approach to the antenna synthesis problems, as well as discuss the arising nonlinear integral and matrix equations.

### 2.1. RP of electromagnetic field

The EM field in the non-limited homogeneous medium satisfies the Maxwell equations [25]

where

The scalar and vector potentials are used for solving the equations (1)-(4). Introducing the vector potential

we satisfy the equation (4). Substituting (5) into (2), we receive the equation

which testifies that the vector field in the parenthesis of (6) is potential. This yields the equation

where

Using Lorentz lemma [15]

we receive the inhomogeneous Helmholtz equation for the vector potential

where

The vector potential

where

Using formulas (5), (7), and (11), we receive the solution to system (1)-(4) in the term of electric vector potential

In the process of solving the synthesis problem, the representation of field in far zone is of interest. Using the approximate representation of distance

where

Substituting (14) into (13) and neglecting by terms of

Consequently, the formulas for

Using formulas (15), (16) and relation

we receive the formulas for non-zero

Function

The functions (20) and (21) characterize the angular distribution of

In such a way, the vectors

The functions

is the power RP and it characterizes the angular distribution of power intensity radiation. The functions

#### 2.2.Variational. statement of the synthesis problems

Abstracting of the specific type of antenna, we present the functions

acting from some complex Hilbertian space

for operator expression of the RP

The specific form of the operator

where

In the previous subsection, we consider the direct external problem of electrodynamics consisting of determination of the asymptotic (RP) of EM field in far zone. The inverse problem, namely determination of such a current

The RPs are due to satisfy a series of requirements for the main lobe and sidelobes. In one case, the RP with narrow main beam is required, another time, this beam should have the specific wide (for example, cosecant) form; the sidelobes be as low as possible.

The angular distribution of the radiation power is characterized by the amplitude of RP, but not the whole RP. Therefore, only amplitude

In this way, in the process of synthesis we prescribe not the whole complex function

Let us introduce the Hilbertian spaces of radiation patterns

The functional

is used in most cases as optimization criterion. Dependence of

Let us introduce the normalization

where

The generalized functional, allowing to diminish the mean-square deviation of RPs and relative norm of current, has the form

Choosing the weight multiplier

### 2.3. The fundamental equations of synthesis

Let us demonstrate how one can receive the respective Lagrange-Euler’s equation [17] in the process of minimization of functional

In order to determine the derivative

and to extract the linear term

Substituting this expression into (35), we receive

or

where

Using the Cauchy-Bunyakovsky-Schwarz inequality [10] and maximizing

which is used usually for the numerical minimization of

The equation (42) contains

If operator

One can receive the similar expression for

Equating the right part of (44) to zero, we receive the explicit expression for

Equations (26) and (45) yield the system of nonlinear integral equations for the optimal current

System of nonlinear equations (26), (45) can be reduced to one nonlinear equation

Since right hand side of (46) is a result of acting by the operator

Taking into account the above considerations, we receive the following nonlinear Hammerstein equation of the second kind for the RP

in the case of functional

The integral equations (43), (46), and (47) are the fundamental equations for the synthesis problems according to the prescribed amplitude RP

### 2.4. The numerical solution of the integral equations

The equations (43), (46), and (47) are solved numerically by the method of successive approximations. The new approximation is determined by explicit formula

in the process of

is solved when the synthesis problem is formulated in the term of functional

In the last case, one can use the iterative process

but the convergence domain of this process is limited.

In accordance with the procedure used in Subsection 2.3, the solutions to the nonlinear equations (43), (46), and (47) are the stationary points of the respective functionals. Since the used functionals are nonconvex, the several solutions can appear, what corresponds to existence of several local minima or saddle points. The number of solutions can vary depending on the physical parameters of the problem what requires the special careful analysis of the obtained solutions [11].

## 3. Application to specific antennas

### 3.1. Amplitude-phase synthesis of cylindrical antenna

In this Subsection, we consider the application of variational approach to the synthesis problem of cylindrical curvilinear antenna. Functional (34) is used as criterion of optimization, complex function

#### 3.1.1. The integral equation approach

Let the generatrix of the antenna has the length

In many practical applications, the antennas with currents linearly polarized along

Let the current distribution in antenna surface be determined as

then the spatial RP is completely determined by the RP

Function

is the RP of a plane curvilinear antenna with form

There is a many literature sources on the synthesis problem of linear antennas (see, e.g. [9] and references there). Therefore we consider here the synthesis problem for plane curvilinear antenna.

It is easily seen from formula (53) that the angle

that is, RP is determined due to action of linear bounded operator

The amplitude-phase synthesis problem for closed plane curvilinear antenna according to desired amplitude RP

Using (47), (57), and definitions (58) and (59), we receive the nonlinear equation with respect to RP

where

and nonlinear operator

The methods of successive approximations are applied for solving the nonlinear equation (60). The simplest of them

has the limited region of convergence determined by formula

where

The iterative process (51) is more preferable, it yields the converging sequence of functional

for arbitrary

#### 3.1.2. The gradient methods of optimization

Above we mentioned the methods of successive approximations for solving the arising nonlinear equations. The direct optimization of

where

where

The disadvantage of method (66) is that only the information about optimizing function from previous iteration is used, in addition it has the slow convergence at the end of iterative process. The method of conjugate gradients [28]

and proposed in [3] generalized gradient method

do not have such disadvantage. Here

For method (70), the problem of minimization of

with unknown

#### 3.1.3. The numerical results

The numerical results are shown for the prescribed amplitude RPs

### 3.2. The problem of amplitude synthesis for resonant antennas

Resonant antennas are a new type of antennas [24], which allow to form the radiation characteristics satisfying a wide spectrum of practical requirements. Such antennas are formed by several surfaces, one of which is semitransparent. Antennas with one semitransparent and other metal boundary are considered here.

The synthesis problem consists of determination of such parameters of antenna (the geometry of inner boundary and transparency of the outer boundary), which form the amplitude RP or front-to-rear factor (FRF) the most close to the prescribed ones.

#### 3.2.1. Generalization of variational statement

The generalized method of eigen oscillations [1] is the mathematical basis for solving the analysis (direct) problem of resonant antennas. The two-dimensional model of antennas (the case of

The main constructive parameter of resonant antennas is the cophased field in the outer surface. This field can be considered quite real (i.e., only its amplitude can be considered) since the constant phase shift of field does not change the amplitude RP. In this connection, the synthesis problem for resonant antennas is formulated as the amplitude synthesis problem.

The direct problem consists of determination of the RP

where kernel

In the case of resonant antenna with arbitrary outer boundary, the method of auxiliary sources [2, 12, 18] is used for determination of the RP

where

The RP is given by

Solving the synthesis problem, we determine the field in outer boundary

Additionally, the restrictions on a field in some areas of a near zone can be prescribed. The functional

which is generalization of (34), allows to take into account these requirements. Here

In the first step of solving the synthesis problem, the field

is used. The gradient

where

In the second step, the transparency

For the antenna with circular outer boundary, the transparency distribution can be presented in the explicit form

where

In the case of antenna with arbitrary outer boundary, similarly to [37], the distribution of transparency is determined by the formula

#### 3.2.2. The numerical results

The numerical calculations are carried out for the resonant antenna with a given outer elliptic boundary. The prescribed amplitude RP is:

The distribution of transparency

The numerical results for solution of the synthesis problem with restrictions on the field in a near zone are given for the antenna with circular outer boundary. The prescribed amplitude RP is:

It can be seen that the field at restriction points is reduced up to level -37 dB. The synthesized field

### 3.3. Waveguide resonant antenna

Synthesis of resonant antenna with waveguide excitation is carried out according to the FRF. The optimizing functional enables to take into account a various requirements to the FRF of antenna in the operating frequency range, as well as outside this range.

#### 3.3.1. The physical description of problem

The geometrical parameters of resonant antenna with waveguide excitation are shown in Fig. 5. In order to create RP enough narrow, the width

The RP of antenna has form

The direct (analysis) problem on determination of the electromagnetic field components in the semitransparent aperture is reduced to two separate problems in the planes

on the all metallic walls;

on the semitransparent upper boundary, the same conditions in the aperture of the exciting waveguide; the condition of radiation on the infinity

and asymptotical condition in the exciting waveguide

The problem of determination of the field

Under condition of the linear polarization of field in the aperture of exciting waveguide, the RP (81) can be represented as product of two functions, namely the RP of plane antenna with variable height

In such a way, the RP in the

Numerical calculations can be essentially simplified, if one assumes that the field above the antenna is represented approximately in the form [37]

#### 3.3.2. The objective functionals

In the process of statement of the synthesis problem one requires to provide the best approximation to the prescribed FRF in the operating frequency range

The variational approach for solving this problem was developed in [37]. Modification of variational statement of the synthesis problem is proposed. The problem consists of determination of functions

The additional parameter of optimization

Minimization of FRF outside the operating frequency range is one of requirements of electromagnetic compatibility for radiating systems [36]. Thus, the value of FRF should remain the largest in the main frequency range. Under these requirements, the following generalization of the variational statement of problem is considered: to find functions

that is, minimization of the maximal FRF value outside the

For example,

#### 3.3.3. The modeling results

The numerical results are shown in Fig. 6. Calculations were carried out for the problem of

The optimized values of FRF are marked by solid line in the basic and additional ranges; the dashed line corresponds to not optimized values of

Process of additional optimization is carried out on the simplified procedure, that is the control of decrease of the FRF in the basic range is omitted [Andriychuk Zamorska, 2004]. Therefore, the values of FRF in the basic range are slightly decreased in comparison with the FRF values for initial problem.

The optimal form

In Fig. 7, the change of the FRF values at three points of main and additional ranges of frequency (two extreme points and middle one) is shown. The width of main range is equal to 8.33%, and width of additional range is equal to 10.0%. In Fig. 7a, curve 1 corresponds to the central value of frequency

The total number of iterations also depends on the width of the considered frequency ranges.

### 3.4. Phase synthesis problem for cylindrical array

The phase distributions of excitation currents in the array’s elements are the optimizing parameters in the problem of phase synthesis. The optimization of considered functionals is reduced to the solution of the corresponding system of nonlinear equations. The gradient methods for direct optimization of functionals are used in practical applications.

#### 3.4.1. The RP of array

A series of simplifications in process of synthesis of the cylindrical array is used in order to reduce the computing time [9, 22]. Separation of variables onto the vertical and horizontal components for the distributions of currents in the array elements, as well as for the RPs, is one of the simplifications.

Thus, the expressions for current

It is assumed that the radiating elements are flat apertures, for example, end of open waveguide. The RP depends on the angular coordinates

The spatial RP of cylindrical array [9] is:

where

The functions

Following the above assumptions, the RP (94) can be written as

where

is the RP of linear array, and

is the RP of circular array for each

#### 3.4.2. Optimization criteria

The complex currents in the array elements are determined by their amplitudes and phases. We denote these values

The equalities

should be satisfied at the points of functional (33) maximum. This set yields the system of transcendental equations for the phases

Using normalization of the current

where

At first, we consider the problem of

In practice, the system (99), (100) is more convenient for the numerical solution. For this reason, we use the following iterative process

to find the phases

The gradient methods are more convenient to solve the minimization problem for the functional

Here

Parameter

In practice, it is necessary to solve the problem of discrete phase synthesis [9, 22], because the arbitrary phase distributions cannot be realized in the array radiators. These distributions are prescribed as a set of discrete values, which are multiple to the given phase discrete value

The algorithm consisting of two enclosed iterative processes is used for the solution of this problem. The value of phase

The internal cycle consists of the successive improvement of phases in the separate radiators changing their number from

The new phase RP

The problem of discrete phase synthesis is solved in two steps. In the first step, the synthesis (with small accuracy) without the account of phase discrete values is carried out. After that, the found phases are approximated up to the nearest discrete values. In the second step, the described above algorithm of discrete synthesis is used. As a rule, one is enough to make several external cycles in the latter algorithm.

#### 3.4.3. The results of numerical modeling

The numerical results are given for the sector array. The RPs of separate radiators have a cosine form, and mutual coupling of separate radiators is not taken into account [35]. The number of radiators

The synthesis results are shown in Fig. 8 for the bi-directional RP

The thick line corresponds to the prescribed amplitude RP

In the practical applications, the problem of phase scanning [35] is considered for arrays. The array alongside with creating the amplitude RP which is more close to the desired one should provide the moving this RP along the angular coordinate in the scanning process. This moving is carried out by the change of the phase distribution

The difference between the results of continuous and discrete synthesis depends on the value of phase discrete

### 3.5. Investigation of branching solution

The problem of the non-uniqueness of solutions for phase synthesis problem is investigated on the example of linear array.

The various modifications of the Newton method have been developed for solving the nonlinear equations in [13], and have been detailed for the synthesis problems in [11]. We consider here the above approach for determination of the number of solutions and investigation of its properties by the example of the nonlinear equation (46), corresponding to functional

Let the operator

where

In this case, the functional (33) can be presented as

#### 3.5.1. The numerical results

The numerical calculations are carried out for the prescribed amplitude radiation pattern

The solid lines correspond to values of

The optimal values of sought phase distributions

The quality of approximation to prescribed amplitude pattern

### 3.6. Synthesis of waveguide array

The mutual coupling of the separate elements of array is taken into account in the process of solution of a direct problem (analysis problem) [7, 14].

#### 3.6.1. Statement of synthesis problem

The objective functional is formulated as [4]

where

The determination of the currents

The RP (array factor) [9] of array is:

where

Introducing the generalized angular coordinates

where

and finally

where

The expression (116) indicates that the calculation of array factor

#### 3.6.2. The modeling results

The results of numerical calculations are presented for the waveguide arrays consisting of 15 and 31 radiators;

In Fig. 12, the dependence of the synthesis results on the value

Sidelobe: | 1st | 2nd | 3rd | 4th | 5th |

## 4. Conclusions

In the process of solving the amplitude-phase synthesis problem, the influence of weight multiplier

The proposed approach for solving the synthesis problems of resonant antennas is universal, and it provides the possibility to synthesize antennas with the arbitrary form of external boundary. The calculation time of the RP of antenna is small enough what it is very important in the process of solution of the synthesis problem. The used variational statement of the synthesis problem also allows to take into account restrictions on the field at the given points (areas) of a near zone.

The synthesis of the resonant antenna with waveguide excitation gives the possibility to take into account the various requirements to the FRF in the operating frequency range. The developed algorithms enable to optimize the values of FRF in the single range, as well as in the several frequency ranges. The values of the objective parameters

The variational approach for solving the phase synthesis problem can be applied effectively for the plane, cylindrical and conical arrays. It allows to decrease the computational time, at the same time the accuracy of determination of the array characteristics is sufficient for practice. The branching solutions are investigated for the case of linear array. It is shown that one can receive the solutions with various properties starting the iterative process with different initial approximations.

The considered optimization problems of waveguide array give the possibility to take into account the requirements to the amplitude RP and amplitude-phase distribution of field in the aperture of exciting waveguides. The developed algorithms enable to achieve the minimal mean-square deviation

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