**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.

©2012 Andriychuk, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **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]

$$\text{rot}H = \text{ik}\varepsilon \mathbf{E} + \frac{4\pi}{c}I,\tag{1}$$

where *RPQ* <sup>=</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*�)<sup>2</sup> + (*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*�)<sup>2</sup> + (*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*�)<sup>2</sup> is the distance between the observation point

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

*H* = rot*A<sup>e</sup>*

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

where *α* is the angle between the vectors, directed into observation and integration point, *r*�

*ωε* graddiv*A<sup>e</sup>*

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 193

<sup>∞</sup>*ϕ*, *<sup>H</sup><sup>ϕ</sup>* <sup>=</sup> <sup>−</sup>*ikA<sup>e</sup>*

<sup>∞</sup>*<sup>θ</sup>* , *<sup>E</sup><sup>ϕ</sup>* <sup>=</sup> <sup>−</sup>*iωμA<sup>e</sup>*

*e*−*ikr*

*e*−*ikr*

<sup>4</sup>*π<sup>r</sup>* is a spherical wave and it depends on *r* only. The second terms in (18) and (19)

) sin *θ*]*eik*(*x*� sin *<sup>θ</sup>* cos *<sup>ϕ</sup>*+*y*� sin *<sup>θ</sup>* sin *<sup>ϕ</sup>*+*z*� cos *<sup>θ</sup>*)*dV*,

) cos *ϕ* − *Ix*(*x*�

, *y*� , *z*�

, *y*� , *z*�

are the functions of angular coordinates of the observation point and are determined by the

*eik*(*x*� sin *<sup>θ</sup>* cos *<sup>ϕ</sup>*+*y*� sin *<sup>θ</sup>* sin *<sup>ϕ</sup>*+*z*� cos *<sup>θ</sup>*)*dV*.

) cos *θ* cos *ϕ* + *Iy*(*x*�

*ikr*� cos *<sup>α</sup>dV* <sup>+</sup> *<sup>O</sup>*( <sup>1</sup>

) is the density of current in the domain

. (13)

, (12)

*<sup>r</sup>*<sup>2</sup> ), (14)

<sup>∞</sup>. These formulas in the

<sup>∞</sup>*<sup>θ</sup>* . (15)

<sup>∞</sup>*ϕ*. (16)

rot*H*, (17)

<sup>4</sup>*π<sup>r</sup> <sup>f</sup><sup>θ</sup>* (*θ*, *<sup>ϕ</sup>*), (18)

<sup>4</sup>*π<sup>r</sup> <sup>f</sup>ϕ*(*θ*, *<sup>ϕ</sup>*). (19)

) cos *θ* sin *ϕ*−

) sin *ϕ*]×

(20)

(21)

*<sup>r</sup>*<sup>2</sup> ) order, we receive the formulas

, *y*� , *z*� ), *I*(*x*� , *y*� *z*�

*<sup>E</sup>* <sup>=</sup> <sup>−</sup>*iωμA<sup>e</sup>* <sup>−</sup> <sup>1</sup>

*Q*(*x*, *y*, *z*) and integration point *P*(*x*�

formula for vector potential in this region

spherical coordinates have form

Using formulas (15), (16) and relation

Function *<sup>e</sup>*−*ikr*

current *I*(*x*�

, *y*� , *z*� )

*<sup>f</sup><sup>θ</sup>* (*θ*, *<sup>ϕ</sup>*) =

*Iz*(*x*� , *y*� , *z*�

*V*

*<sup>f</sup>ϕ*(*θ*, *<sup>ϕ</sup>*) =

[*Ix*(*x*� , *y*� , *z*�

*V*

[*Iy*(*x*� , *y*� , *z*�

*Ae*

Consequently, the formulas for *E* are the following

<sup>∞</sup>(*Q*) = <sup>−</sup>*eikr*

and *r* are the radius-vector of the points *P* and *Q*, respectively. Substituting (14) into (13) and neglecting by terms of *O*( <sup>1</sup>

for components of magnetic field *H* in term of vector potential *A<sup>e</sup>*

*Hr* = 0, *<sup>H</sup><sup>θ</sup>* = *ikA<sup>e</sup>*

*Er* <sup>=</sup> 0, *<sup>E</sup><sup>θ</sup>* <sup>=</sup> <sup>−</sup>*iωμA<sup>e</sup>*

we receive the formulas for non-zero *E* components of EM field

*<sup>E</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *ωε*

*E<sup>θ</sup>* (*r*, *θ*, *ϕ*) = −*iωμ*

*Eϕ*(*r*, *θ*, *ϕ*) = −*iωμ*

4*πr*

*I*(*x*� , *y*� , *z*� )*e*

*V*

*V*.

vector potential

$$\text{rotE} = -ik\mu H\_\prime \tag{2}$$

$$\mathbf{div}\mathbf{E} = \rho/\varepsilon,\tag{3}$$

$$\text{div}H = 0,\tag{4}$$

where *E* is a component of electric field, *H* is a component of magnetic field, *I* is the extrinsic current density, *ρ* is the volume density of electric charge.

The scalar and vector potentials are used for solving the equations (1)-(4). Introducing the vector potential *A<sup>e</sup>* as [34]

$$H = \text{rot}A^{\ell},\tag{5}$$

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

$$\text{rot}(E + i\omega\mu A^{\ell}) = 0,\tag{6}$$

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

$$E = -i\omega\mu A^{\varepsilon} - \text{grad}\varphi^{\varepsilon},\tag{7}$$

where *ϕ<sup>e</sup>* is the scalar potential. Substituting (5) and (7) into (1), we receive the following equation for *I*:

$$
\nabla^2 A^\varepsilon + \omega^2 \varepsilon \mu A^\varepsilon - \text{grad}(\text{div} A^\varepsilon + i\omega \varphi^\varepsilon) = -I. \tag{8}
$$

Using Lorentz lemma [15]

$$i\operatorname{div} A^{\varepsilon} + i\omega \varepsilon \varphi^{\varepsilon} = 0,\tag{9}$$

we receive the inhomogeneous Helmholtz equation for the vector potential

$$
\nabla^2 A^\varepsilon + k^2 A^\varepsilon = -I,\tag{10}
$$

where *k* = *ω*√*εμ* is a propagation coefficient. For a free space, the values *ε* and *μ* are real and related with the light velocity as *c* = 1/√*εμ*, and coefficient *k* = *ω*/*c* = 2*π*/*λ* is the wavenumber, *λ* is the length of wave.

The vector potential *A<sup>e</sup>* in the arbitrary observation point *Q*(*x*, *y*, *z*) is determined by formula [26]

$$A^{\varepsilon}(\mathbf{x}, y, z) = \frac{1}{4\pi} \int\_{V} I(\mathbf{x}', y', z') \frac{-e^{i\mathbf{k}\cdot\mathbf{R}\_{PQ}}}{R\_{PQ}} dV\_{\prime} \tag{11}$$

where *RPQ* <sup>=</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*�)<sup>2</sup> + (*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*�)<sup>2</sup> + (*<sup>z</sup>* <sup>−</sup> *<sup>z</sup>*�)<sup>2</sup> is the distance between the observation point *Q*(*x*, *y*, *z*) and integration point *P*(*x*� , *y*� , *z*� ), *I*(*x*� , *y*� *z*� ) is the density of current in the domain *V*.

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

$$E = -i\omega\mu A^{\varepsilon} - \frac{1}{\omega\varepsilon} \text{grad}\text{div}A^{\varepsilon} \tag{12}$$

$$H = \text{rot}A^{\ell}.\tag{13}$$

In the process of solving the synthesis problem, the representation of field in far zone is of interest. Using the approximate representation of distance *RPQ* in far zone, we receive the formula for vector potential in this region

$$A^\ell\_\infty(Q) = \frac{-e^{ikr}}{4\pi r} \int\_V I(\mathbf{x'}, y', z') e^{ikr'\cos\alpha} dV + O(\frac{1}{r^2}),\tag{14}$$

where *α* is the angle between the vectors, directed into observation and integration point, *r*� and *r* are the radius-vector of the points *P* and *Q*, respectively.

Substituting (14) into (13) and neglecting by terms of *O*( <sup>1</sup> *<sup>r</sup>*<sup>2</sup> ) order, we receive the formulas for components of magnetic field *H* in term of vector potential *A<sup>e</sup>* <sup>∞</sup>. These formulas in the spherical coordinates have form

$$H\_{\mathbb{T}} = 0, \quad H\_{\theta} = ikA^{\ell}\_{\infty\varphi\prime} \quad H\_{\varphi} = -ikA^{\ell}\_{\infty\theta}.\tag{15}$$

Consequently, the formulas for *E* are the following

$$E\_{\prime} = 0, \quad E\_{\theta} = -i\omega\mu A^{\varepsilon}\_{\infty\theta\prime} \quad E\_{\theta} = -i\omega\mu A^{\varepsilon}\_{\infty\theta\prime}.\tag{16}$$

Using formulas (15), (16) and relation

2 Will-be-set-by-IN-TECH

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.

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

where *E* is a component of electric field, *H* is a component of magnetic field, *I* is the extrinsic

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

*H* = rot*A<sup>e</sup>*

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

*<sup>E</sup>* <sup>=</sup> <sup>−</sup>*iωμA<sup>e</sup>* <sup>−</sup> *gradϕ<sup>e</sup>*

where *ϕ<sup>e</sup>* is the scalar potential. Substituting (5) and (7) into (1), we receive the following

where *k* = *ω*√*εμ* is a propagation coefficient. For a free space, the values *ε* and *μ* are real and related with the light velocity as *c* = 1/√*εμ*, and coefficient *k* = *ω*/*c* = 2*π*/*λ* is the

The vector potential *A<sup>e</sup>* in the arbitrary observation point *Q*(*x*, *y*, *z*) is determined by formula

*I*(*x*� , *y*� , *z*� ) <sup>−</sup>*eikRPQ RPQ*

rot(*E* + *iωμA<sup>e</sup>*

<sup>∇</sup>2*A<sup>e</sup>* <sup>+</sup> *<sup>ω</sup>*2*εμA<sup>e</sup>* <sup>−</sup> grad(div*A<sup>e</sup>* <sup>+</sup> *<sup>i</sup>ωϕ<sup>e</sup>*

we receive the inhomogeneous Helmholtz equation for the vector potential

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

4*π c*

*I*, (1)

, (5)

) = 0, (6)

div*A<sup>e</sup>* + *iωεϕ<sup>e</sup>* = 0, (9)

<sup>∇</sup>2*A<sup>e</sup>* <sup>+</sup> *<sup>k</sup>*2*A<sup>e</sup>* <sup>=</sup> <sup>−</sup>*I*, (10)

, (7)

) = −*I*. (8)

*dV*, (11)

rot*E* = −*ikμH*, (2) div*E* = *ρ*/*ε*, (3) div*H* = 0, (4)

rot*H* = *ikεE* +

**2. The theoretical background of the synthesis problems**

current density, *ρ* is the volume density of electric charge.

**2.1. RP of electromagnetic field**

vector potential *A<sup>e</sup>* as [34]

equation

[26]

equation for *I*:

Using Lorentz lemma [15]

wavenumber, *λ* is the length of wave.

*Ae*

(*x*, *<sup>y</sup>*, *<sup>z</sup>*) = <sup>1</sup>

4*π* 

*V*

$$E = -\frac{1}{\omega\varepsilon} \text{rot}H,\tag{17}$$

we receive the formulas for non-zero *E* components of EM field

$$E\_{\theta}(r,\theta,\varphi) = -i\omega\mu \frac{e^{-ikr}}{4\pi r} f\_{\theta}(\theta,\varphi),\tag{18}$$

$$E\_{\varphi}(r,\theta,\varphi) = -i\omega\mu \frac{e^{-ikr}}{4\pi r} f\_{\varphi}(\theta,\varphi). \tag{19}$$

Function *<sup>e</sup>*−*ikr* <sup>4</sup>*π<sup>r</sup>* is a spherical wave and it depends on *r* only. The second terms in (18) and (19) are the functions of angular coordinates of the observation point and are determined by the current *I*(*x*� , *y*� , *z*� )

$$\begin{aligned} f\_{\theta}(\theta,\varphi) &= \int \left[ I\_{\mathbf{x}}(\mathbf{x}',\mathbf{y}',z') \cos\theta \cos\varphi + I\_{\mathbf{y}}(\mathbf{x}',\mathbf{y}',z') \cos\theta \sin\varphi - \int \mathbf{I}\_{\mathbf{y}}(\mathbf{x}',\mathbf{y}',z') \cos\theta \sin\varphi \right] d\mathbf{x}' d\mathbf{y} \\\ I\_{\mathbf{z}}(\mathbf{x}',\mathbf{y}',z') &\sin\theta \left[ e^{i\mathbf{k}(\mathbf{x}'\sin\theta\cos\varphi + \mathbf{y}'\sin\theta\sin\varphi + z'\cos\theta)} dV\_{\mathbf{z}} \right] \end{aligned} \tag{20}$$

$$\begin{cases} f\_{\boldsymbol{\theta}}(\boldsymbol{\theta},\boldsymbol{\varphi}) = \int \left[ I\_{\boldsymbol{y}}(\mathbf{x}',\mathbf{y}',\mathbf{z}') \cos \boldsymbol{\varphi} - I\_{\mathbf{x}}(\mathbf{x}',\mathbf{y}',\mathbf{z}') \sin \boldsymbol{\varphi} \right] \times \\\ \begin{array}{l} V \\ \operatorname{\bf{e}}^{\text{ik}(\mathbf{x}' \sin \boldsymbol{\theta} \cos \boldsymbol{\varphi} + \mathbf{y}' \sin \boldsymbol{\theta} \sin \boldsymbol{\varphi} + \mathbf{z}' \cos \boldsymbol{\theta}) \, dV \, \end{array} \tag{21}$$

The functions (20) and (21) characterize the angular distribution of *E* components in far zone. In such a way, the vectors *E* and *H* in far zone are expressed by formulas [4]

$$E(r,\theta,\varphi) = -i\omega\mu \frac{e^{-ikr}}{4\pi r} \{0, f\_{\theta}\left(\theta,\varphi\right), f\_{\varphi}\left(\theta,\varphi\right)\},\tag{22}$$

fixed or be subject to determination in process of solving this problem. In such interpretation, the inverse problem is defined as the synthesis problem, namely the problem of determination

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 195

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

The angular distribution of the radiation power is characterized by the amplitude of RP, but not the whole RP. Therefore, only amplitude | *f* | of function *f* is interesting in the process of statement and solving the synthesis problem. In this case, the freedom of choice of the phase arg *f* of function *f* is used for better approximation to the amplitude RP. In the synthesis

In this way, in the process of synthesis we prescribe not the whole complex function *f* , but only its amplitude. We denote this function as *F*, the created (synthesized) by antenna amplitude RP is denoted by | *f* |. The both functions are real and positive. Of course, these functions can not coincide in any real-world situation. This fact yields to use the variational statement of the synthesis problem. In such statement, one requires not the whole coincidence of the functions | *f* | and *F*, but the better approximation of function | *f* | to *F* in a certain sense only.

Let us introduce the Hilbertian spaces of radiation patterns *Hf* and currents *HI*. Let (·, ·)*<sup>f</sup>* be

is used in most cases as optimization criterion. Dependence of *σ* on the sought distribution of the current *I* in antenna is determined by formula (26). The additional multiplier *s* in (29) can be either prescribed (for example, *s* = 1) or determined from the condition *∂σ*/*∂s* = 0. In the

and ||*I*|| is the norm of current, determined by inner product (·, ·)*<sup>I</sup>* in the Hilbertian space *HI* of currents: ||*I*||<sup>2</sup> = (*I*, *<sup>I</sup>*)*I*. The factor *<sup>Q</sup>* characterizes the goodness (reactivity) of antenna,

The generalized functional, allowing to diminish the mean-square deviation of RPs and

and *κ* determines its power efficiency, namely the part of power radiated by the RP.


*σ* = (*F* − *s*| *f* |, *F* − *s*| *f* |)*<sup>f</sup>* (29)

*s* = (*F*, | *f* |)*<sup>f</sup>* /(*f* , *f*)*<sup>f</sup>* . (30)

*<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>Q</sup>κ*2, (31)

*<sup>Q</sup>* <sup>=</sup> ||*I*||2/|| *<sup>f</sup>* ||2, (32) *κ* = (*F*, | *f* |)*<sup>f</sup>* /||*I*||, (33)

*<sup>σ</sup><sup>t</sup>* = (*<sup>F</sup>* − | *<sup>f</sup>* <sup>|</sup>, *<sup>F</sup>* − | *<sup>f</sup>* <sup>|</sup>)*<sup>f</sup>* <sup>+</sup> *<sup>t</sup>*||*I*||2. (34)

specific wide (for example, cosecant) form; the sidelobes be as low as possible.

The mean-square deviation of both RPs is used as the criterion of optimization.

Let us introduce the normalization (*F*, *F*)*<sup>f</sup>* = 1, then (29) can be written as

theory, function | *f* | is amplitude RP, and function arg *f* is phase RP.

an inner product in *Hf* , and norm of *f* is defined as [4]

The functional

latter case

where

relative norm of current, has the form

of the current according to the desired RP.

$$H(r,\theta,\varphi) = \text{ik}\frac{e^{-ikr}}{4\pi r} \{0, f\_{\theta}\left(\theta,\varphi\right), -f\_{\theta}\left(\theta,\varphi\right)\}.\tag{23}$$

The functions *f<sup>θ</sup>* (*θ*, *ϕ*) and *fϕ*(*θ*, *ϕ*) are defined as the RPs. Function

$$N(\theta,\varphi) = |f\_{\theta}\left(\theta,\varphi\right)|^{2} + |f\_{\theta}\left(\theta,\varphi\right)|^{2} \tag{24}$$

is the power RP and it characterizes the angular distribution of power intensity radiation. The functions *f<sup>θ</sup>* (*θ*, *ϕ*), *f<sup>ϕ</sup>* (*θ*, *ϕ*), and *N*(*θ*, *ϕ*) are used in the process of formulating and solving the synthesis problems for various antennas.

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

Abstracting of the specific type of antenna, we present the functions *f<sup>θ</sup>* and *f<sup>ϕ</sup>* in the formulas (22)-(24) by the linear operator **A** = {*A<sup>θ</sup>* , *Aϕ*}:

$$\mathbf{f} = \mathbf{A}I, \quad (f\_{\vee} = A\_{\vee}I, \quad \boldsymbol{\nu} = \boldsymbol{\theta}, \boldsymbol{\varrho}), \tag{25}$$

acting from some complex Hilbertian space *HI*, to which the distribution functions of current or field belong, into the complex space *C*<sup>2</sup> *<sup>f</sup>* = *C*[Ω] ⊕ *C*[Ω] of vector-valued continuous functions on the compact <sup>Ω</sup>¯ <sup>∈</sup> <sup>R</sup><sup>2</sup> (or <sup>Ω</sup>¯ <sup>∈</sup> <sup>R</sup>1) [29]. The form and properties of the operators *A<sup>ν</sup>* depend on a type and geometry of antenna. In many practical applications, one can reduce the synthesis problem to separate consideration of the *f<sup>θ</sup>* and *f<sup>ϕ</sup>* components. This allows to reduce the synthesis problem to the scalar one. In that way, we will use more simple formula

$$f = AI \tag{26}$$

for operator expression of the RP *f* .

The specific form of the operator *A* depends on the antenna type. This operator is integral for the continuous antennas. As an example, for a cylindrical antenna with curvilinear generatrix and with current polarized along the cylinder axis, the RP in the transversal plane has form [21]

$$f = AI := \int\_{\mathcal{S}} I(\mathcal{S}) e^{ikr(\boldsymbol{\varphi}') \cos(\boldsymbol{\varphi} - \boldsymbol{\varphi}')} dS\_{\boldsymbol{\varphi}'} \tag{27}$$

where *f* and *I* are nonzero components *f<sup>θ</sup>* and *Iz* respectively; *ϕ* is the angular coordinate of the point in far zone, *ϕ*� is the angular coordinate of the point in antenna, *r* = *r*(*ϕ*� ) describes the generatrix in polar coordinates, *dSϕ*� <sup>=</sup> *<sup>r</sup>*<sup>2</sup> + (*dr*/*dϕ*�)<sup>2</sup> is an element of arc. In the case of array, the operator *A* is described by a finite sum.

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 *I* that create EM field with the desired RP *f* , is of specific interest in the antenna design. The characteristic parameters of antenna can be fixed or be subject to determination in process of solving this problem. In such interpretation, the inverse problem is defined as the synthesis problem, namely the problem of determination of the current according to the desired RP.

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 | *f* | of function *f* is interesting in the process of statement and solving the synthesis problem. In this case, the freedom of choice of the phase arg *f* of function *f* is used for better approximation to the amplitude RP. In the synthesis theory, function | *f* | is amplitude RP, and function arg *f* is phase RP.

In this way, in the process of synthesis we prescribe not the whole complex function *f* , but only its amplitude. We denote this function as *F*, the created (synthesized) by antenna amplitude RP is denoted by | *f* |. The both functions are real and positive. Of course, these functions can not coincide in any real-world situation. This fact yields to use the variational statement of the synthesis problem. In such statement, one requires not the whole coincidence of the functions | *f* | and *F*, but the better approximation of function | *f* | to *F* in a certain sense only. The mean-square deviation of both RPs is used as the criterion of optimization.

Let us introduce the Hilbertian spaces of radiation patterns *Hf* and currents *HI*. Let (·, ·)*<sup>f</sup>* be an inner product in *Hf* , and norm of *f* is defined as [4]

$$\left|\left|f\right|\right|^{2} = (f\_{\prime}f)\_{f}.\tag{28}$$

The functional

4 Will-be-set-by-IN-TECH

The functions (20) and (21) characterize the angular distribution of *E* components in far zone.

is the power RP and it characterizes the angular distribution of power intensity radiation. The functions *f<sup>θ</sup>* (*θ*, *ϕ*), *f<sup>ϕ</sup>* (*θ*, *ϕ*), and *N*(*θ*, *ϕ*) are used in the process of formulating and solving

Abstracting of the specific type of antenna, we present the functions *f<sup>θ</sup>* and *f<sup>ϕ</sup>* in the formulas

acting from some complex Hilbertian space *HI*, to which the distribution functions of current

functions on the compact <sup>Ω</sup>¯ <sup>∈</sup> <sup>R</sup><sup>2</sup> (or <sup>Ω</sup>¯ <sup>∈</sup> <sup>R</sup>1) [29]. The form and properties of the operators *A<sup>ν</sup>* depend on a type and geometry of antenna. In many practical applications, one can reduce the synthesis problem to separate consideration of the *f<sup>θ</sup>* and *f<sup>ϕ</sup>* components. This allows to reduce the synthesis problem to the scalar one. In that way, we will use more simple formula

The specific form of the operator *A* depends on the antenna type. This operator is integral for the continuous antennas. As an example, for a cylindrical antenna with curvilinear generatrix and with current polarized along the cylinder axis, the RP in the transversal plane has form

*ikr*(*ϕ*�

where *f* and *I* are nonzero components *f<sup>θ</sup>* and *Iz* respectively; *ϕ* is the angular coordinate of

the generatrix in polar coordinates, *dSϕ*� <sup>=</sup> *<sup>r</sup>*<sup>2</sup> + (*dr*/*dϕ*�)<sup>2</sup> is an element of arc. In the case

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 *I* that create EM field with the desired RP *f* , is of specific interest in the antenna design. The characteristic parameters of antenna can be

) cos(*ϕ*−*ϕ*� )

<sup>4</sup>*π<sup>r</sup>* { 0, *<sup>f</sup><sup>θ</sup>* (*θ*, *<sup>ϕ</sup>*), *<sup>f</sup><sup>ϕ</sup>* (*θ*, *<sup>ϕ</sup>*)}, (22)

<sup>4</sup>*π<sup>r</sup>* {0, *<sup>f</sup><sup>ϕ</sup>* (*θ*, *<sup>ϕ</sup>*), <sup>−</sup>*f<sup>θ</sup>* (*θ*, *<sup>ϕ</sup>*)}. (23)

<sup>2</sup> (24)

<sup>2</sup> <sup>+</sup> <sup>|</sup> *<sup>f</sup><sup>θ</sup>* (*θ*, *<sup>ϕ</sup>*)<sup>|</sup>

**f** = **A***I*, (*f<sup>ν</sup>* = *A<sup>ν</sup> I*, *ν* = *θ*, *ϕ*), (25)

*<sup>f</sup>* = *C*[Ω] ⊕ *C*[Ω] of vector-valued continuous

*f* = *AI* (26)

*dSϕ*� , (27)

) describes

*e*−*ikr*

In such a way, the vectors *E* and *H* in far zone are expressed by formulas [4]

*E*(*r*, *θ*, *ϕ*) = −*iωμ*

*<sup>H</sup>*(*r*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) = *ik <sup>e</sup>*−*ikr*

The functions *f<sup>θ</sup>* (*θ*, *ϕ*) and *fϕ*(*θ*, *ϕ*) are defined as the RPs. Function

the synthesis problems for various antennas.

(22)-(24) by the linear operator **A** = {*A<sup>θ</sup>* , *Aϕ*}:

or field belong, into the complex space *C*<sup>2</sup>

for operator expression of the RP *f* .

[21]

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

*f* = *AI* :=

of array, the operator *A* is described by a finite sum.

*I*(*S*)*e*

*S*

the point in far zone, *ϕ*� is the angular coordinate of the point in antenna, *r* = *r*(*ϕ*�

*N*(*θ*, *ϕ*) = | *f<sup>θ</sup>* (*θ*, *ϕ*)|

$$
\sigma = \left( F - s|f| \, \_\prime F - s|f| \right)\_f \tag{29}
$$

is used in most cases as optimization criterion. Dependence of *σ* on the sought distribution of the current *I* in antenna is determined by formula (26). The additional multiplier *s* in (29) can be either prescribed (for example, *s* = 1) or determined from the condition *∂σ*/*∂s* = 0. In the latter case

$$s = (F\_{\prime}|f|)\_{f} / (f\_{\prime}f)\_{f}. \tag{30}$$

Let us introduce the normalization (*F*, *F*)*<sup>f</sup>* = 1, then (29) can be written as

$$
\sigma = 1 - Q\kappa^2,\tag{31}
$$

where

$$Q = ||I||^2 / ||f||^2,\tag{32}$$

$$\mathfrak{k} = (F\_{\prime}|f|)\_{f} / ||I||\_{\prime} \tag{33}$$

and ||*I*|| is the norm of current, determined by inner product (·, ·)*<sup>I</sup>* in the Hilbertian space *HI* of currents: ||*I*||<sup>2</sup> = (*I*, *<sup>I</sup>*)*I*. The factor *<sup>Q</sup>* characterizes the goodness (reactivity) of antenna, and *κ* determines its power efficiency, namely the part of power radiated by the RP.

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

$$
\sigma\_l = (F - |f|\_\prime F - |f|)\_f + t||I||^2. \tag{34}
$$

Choosing the weight multiplier *t*, one can regularize the value of mean-square deviation of the RPs and norm of current. Similar functional appears when the method of Lagrange multipliers [16] is used to solving the conditional extremum problems. Simultaneously, one can use some weight function *p* ≥ 0 in the definition (28) and to improve the approximation to prescribed amplitude RP *F* in the appointed angular range.

#### **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 *σ*. Let *s* = 1. It is known [30] that the equality to zero of functional's gradient is requirement of its extremum. This yields determination of the maximum of the following value

$$\frac{\partial \sigma}{\partial z} = \frac{1}{||z||} \lim\_{\varepsilon \to 0} \frac{\sigma(I + \varepsilon z) - \sigma(I)}{\varepsilon}. \tag{35}$$

substitute formula (26) into (42) instead of *f* . In this way, we receive the integral equation for

If operator *A* is integral (consequently, *A*∗ is also integral) then (43) is the nonlinear

*z* = *A*∗(*Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

*I* = *A*∗(*Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

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

*f* = *AA*∗(*Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

Since right hand side of (46) is a result of acting by the operator *A* on some function, any solution to this equation represents the realizable RP. Once this equation is solved, the optimal distribution of current is determined by formula (45). The equation (46) is more simple than system (26), (45) because determination of its solution does not require additional operation

Taking into account the above considerations, we receive the following nonlinear

*t f* + *AA*<sup>∗</sup> *f* = *AA*∗(*Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

in the case of functional *σ<sup>t</sup>* minimization. As in the case of (46), the solution to this equation (at *t* �= 0) represents the realizable RP. Having the solution to (47), we determine the optimal

*<sup>I</sup>* <sup>=</sup> <sup>−</sup>1/*tA*∗(*<sup>f</sup>* <sup>−</sup> *Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

The integral equations (43), (46), and (47) are the fundamental equations for the synthesis

The equations (43), (46), and (47) are solved numerically by the method of successive

One can receive the similar expression for *z* in the process of *κ* maximization:

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

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

Hammerstein equation of the second kind for the RP *f*

problems according to the prescribed amplitude RP *F*.

in the process of *κ* maximization. The equation

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

approximations. The new approximation is determined by explicit formula

*A*∗*AI* = *A*∗*A*(*Fe*−*<sup>i</sup>* arg *AI*). (43)

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 197

) − *I*. (44)

). (45)

). (46)

) (47)

). (48)

*fn*+<sup>1</sup> = *AA*∗(*Fe*−*<sup>i</sup>* arg *fn* ) (49)

*A*∗*AIn*+<sup>1</sup> = *A*∗*A*(*Fe*−*<sup>i</sup>* arg *AIn* ) (50)

the optimal current distribution

Hammerstein equation [38].

*I* and RP *f* created by it.

current distribution by formula

of *A* and *A*∗.

In order to determine the derivative *∂σ <sup>∂</sup><sup>z</sup>* , one requires to factorize *σ*(*I* + *εz*) in series relatively to *ε*

$$
\sigma(I + \varepsilon z) = \sigma(I) + \varepsilon \delta \sigma(I, z) + O(\varepsilon^2) \tag{36}
$$

and to extract the linear term *δσ*. Evidently, *∂σ <sup>∂</sup><sup>z</sup>* <sup>=</sup> *δσ* ||*z*|| . The increment of amplitude RP <sup>|</sup> *<sup>f</sup>* <sup>|</sup> should be known in order to calculate *δσ*. This increment has the form [4]

$$|A(I + \varepsilon z)| = |f| + \varepsilon \text{Re}[A(z)e^{-i\arg f}] + O(\varepsilon^2). \tag{37}$$

Substituting this expression into (35), we receive

$$\frac{\partial \sigma}{\partial z} = -2(F, \text{Re}[A(z)e^{-i \text{arg}\,f}])\_f + 2\text{Re}(f, Az)\_{f'} \tag{38}$$

or

$$\frac{\partial \sigma}{\partial z} = -2\text{Re}[(A^\*(f - Fe^{-i\arg f})\_\prime z)\_\text{I}]\_\prime \tag{39}$$

where *A*∗ is an operator adjoint to *A* in the following sense:

$$(A1\_1f\_2)\_f = (I\_1A^\*f\_2)\_I. \tag{40}$$

If *A* is an integral operator then *A*∗ is the same with the complex conjugate kernel and integration with respect to the second argument.

Using the Cauchy-Bunyakovsky-Schwarz inequality [10] and maximizing *∂σ <sup>∂</sup><sup>z</sup>* , we receive the expression for gradient of *σ*:

$$z = A^\*(f - Fe^{-i\arg f}),\tag{41}$$

which is used usually for the numerical minimization of *σ*. In order to receive the Lagrange-Euler's equation one should equate to zero the function *z* (condition of *σ* minimum)

$$A^\*(f - Fe^{-i\arg f}) = 0.\tag{42}$$

The equation (42) contains *f* as an unknown function. One can turn out that its solutions represent the unreliazable patterns [20]. In order to avoid such solutions, it is necessary to substitute formula (26) into (42) instead of *f* . In this way, we receive the integral equation for the optimal current distribution

$$A^\* A I = A^\* A (F e^{-i \arg A I}).\tag{43}$$

If operator *A* is integral (consequently, *A*∗ is also integral) then (43) is the nonlinear Hammerstein equation [38].

One can receive the similar expression for *z* in the process of *κ* maximization:

6 Will-be-set-by-IN-TECH

Choosing the weight multiplier *t*, one can regularize the value of mean-square deviation of the RPs and norm of current. Similar functional appears when the method of Lagrange multipliers [16] is used to solving the conditional extremum problems. Simultaneously, one can use some weight function *p* ≥ 0 in the definition (28) and to improve the approximation to prescribed

Let us demonstrate how one can receive the respective Lagrange-Euler's equation [17] in the process of minimization of functional *σ*. Let *s* = 1. It is known [30] that the equality to zero of functional's gradient is requirement of its extremum. This yields determination of the

*σ*(*I* + *εz*) = *σ*(*I*) + *εδσ*(*I*, *z*) + *O*(*ε*

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> *δσ*

−*i* arg *f*

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>−</sup>2Re[(*A*∗(*<sup>f</sup>* <sup>−</sup> *Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

If *A* is an integral operator then *A*∗ is the same with the complex conjugate kernel and

*<sup>z</sup>* <sup>=</sup> *<sup>A</sup>*∗(*<sup>f</sup>* <sup>−</sup> *Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

*<sup>A</sup>*∗(*<sup>f</sup>* <sup>−</sup> *Fe*−*<sup>i</sup>* arg *<sup>f</sup>*

which is used usually for the numerical minimization of *σ*. In order to receive the Lagrange-Euler's equation one should equate to zero the function *z* (condition of *σ* minimum)

The equation (42) contains *f* as an unknown function. One can turn out that its solutions represent the unreliazable patterns [20]. In order to avoid such solutions, it is necessary to

−*i* arg *f*

*σ*(*I* + *εz*) − *σ*(*I*)

*<sup>ε</sup>* . (35)


])*<sup>f</sup>* + 2Re(*f* , *Az*)*<sup>f</sup>* , (38)

), *z*)*I*], (39)

), (41)

) = 0. (42)

<sup>2</sup>) (36)

<sup>2</sup>). (37)

*<sup>∂</sup><sup>z</sup>* , we receive the

*<sup>∂</sup><sup>z</sup>* , one requires to factorize *σ*(*I* + *εz*) in series relatively

] + *O*(*ε*

(*AI*1, *f*2)*<sup>f</sup>* = (*I*1, *A*<sup>∗</sup> *f*2)*I*. (40)


should be known in order to calculate *δσ*. This increment has the form [4]

*<sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>−</sup>2(*F*, Re[*A*(*z*)*<sup>e</sup>*

*∂σ*

where *A*∗ is an operator adjoint to *A* in the following sense:


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

amplitude RP *F* in the appointed angular range.

**2.3. The fundamental equations of synthesis**

*∂σ <sup>∂</sup><sup>z</sup>* <sup>=</sup> <sup>1</sup>

maximum of the following value

to *ε*

or

In order to determine the derivative *∂σ*

and to extract the linear term *δσ*. Evidently, *∂σ*

Substituting this expression into (35), we receive

*∂σ*

integration with respect to the second argument.

expression for gradient of *σ*:

$$z = A^\*(Fe^{-i\arg f}) - I.\tag{44}$$

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

$$I = A^\*(Fe^{-r\arg f}).\tag{45}$$

Equations (26) and (45) yield the system of nonlinear integral equations for the optimal current *I* and RP *f* created by it.

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

$$f = AA^\*(Fe^{-i\arg f}).\tag{46}$$

Since right hand side of (46) is a result of acting by the operator *A* on some function, any solution to this equation represents the realizable RP. Once this equation is solved, the optimal distribution of current is determined by formula (45). The equation (46) is more simple than system (26), (45) because determination of its solution does not require additional operation of *A* and *A*∗.

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

$$tf + AA^\*f = AA^\*(Fe^{-i\arg f})\tag{47}$$

in the case of functional *σ<sup>t</sup>* minimization. As in the case of (46), the solution to this equation (at *t* �= 0) represents the realizable RP. Having the solution to (47), we determine the optimal current distribution by formula

$$I = -1/tA^\*(f - Fe^{-i\arg f}).\tag{48}$$

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

#### **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

$$f\_{n+1} = AA^\*(Fe^{-i\arg f\_n})\tag{49}$$

in the process of *κ* maximization. The equation

$$A^\* A I\_{n+1} = A^\* A (F e^{-i \arg A I\_n}) \tag{50}$$

is solved when the synthesis problem is formulated in the term of functional *σ* minimization. If functional *σ<sup>t</sup>* is minimized then the respective iterative procedure has form

$$tf\_{n+1} + AA^\*f\_{n+1} = AA^\*(Fe^{-i\arg f\_n}).\tag{51}$$

Function *f*2(*θ*) is the RP of a linear antenna with length 2*l*. Function

*I*1*r*(*ϕ*�

), *z*) sin *θe*

is the RP of a plane curvilinear antenna with form *S*. Consequently, one can reduce the synthesis problem of cylindrical antenna to two independent problems for synthesis of linear

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

It is easily seen from formula (53) that the angle *θ* determines effective electrical scale of antenna. Therefore one can suppose *θ* = *π*/2. Omitting the indices "1"' in the distribution

The amplitude-phase synthesis problem for closed plane curvilinear antenna according to desired amplitude RP *F* consists of determination of such distribution of the current *I*(*ϕ*�

the amplitude RP | *f*(*ϕ*)| created (synthesized) by it, is the most close to *F*(*ϕ*). The functional (34) is used as the criterion of optimization. The inner products in the spaces of the RPs and

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

)−cos(*ϕ*1−*ϕ*�

)]

*i* arg *f*(*ϕ*1)

*p*(*ϕ*)*f*1(*ϕ*)*f* <sup>∗</sup>

*r*<sup>2</sup> + (*dr*/*dϕ*�)2*dϕ*�

*p*(*ϕ*1)*K*(*ϕ*, *ϕ*1)*f*(*ϕ*1)*dϕ*<sup>1</sup> = *B f* , (60)

*r*<sup>2</sup> + (*dr*/*dϕ*�)2*dϕ*�

 2*π*

0

*I*1(*ϕ*� )*I* ∗ <sup>2</sup> (*ϕ*� ) 

)[cos(*ϕ*−*ϕ*�

*p*(*ϕ*1)*K*(*ϕ*, *ϕ*1)*F*(*ϕ*1)*e*

) cos(*ϕ*−*ϕ*�

)]

*r*<sup>2</sup> + (*dr*/*dϕ*�)2*dϕ*�

<sup>2</sup> (*ϕ*)*dϕ*, (58)

. (59)

, (61)

*dϕ*1. (62)

*ik*[*r*(*ϕ*�

) sin *θ* cos(*ϕ*−*ϕ*�

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 199

)]*dSϕ*� (56)

, (57)

), that

*S*

 2*π*

*I*(*ϕ*� )*e ik*[*r*(*ϕ*�

0

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

(*f*1, *f*2)*<sup>f</sup>* =

 2*π*

0

2*π*

0

(*I*1, *I*2)*<sup>I</sup>* =

*t f*(*ϕ*) +

2*π*

0 *e ikr*(*ϕ*�

*B f* = 2*π*

0

where *K*(*ϕ*, *ϕ*1) is the kernel of operator *AA*<sup>∗</sup>

*<sup>K</sup>*(*ϕ*, *<sup>ϕ</sup>*1) =

and nonlinear operator *B* is determined as

*<sup>f</sup>*1(*θ*, *<sup>ϕ</sup>*) =

of current and RP, we represent the RP (56) in form

*f*(*ϕ*) = *AI* :=

and plane curvilinear antennas.

currents are defined as

to RP *f*

antenna.

In the last case, one can use the iterative process

$$f\_{n+1} = -1/tAA^\*(f\_n - Fe^{-i\arg f\_n}),\tag{52}$$

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].
