**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 *I* is optimizing function. Since both the amplitude and phase of complex function *I* are optimizing parameters, the considered problem is the amplitude-phase synthesis problem.

#### *3.1.1. The integral equation approach*

Let the generatrix of the antenna has the length 2*l* and be parallel to *Oz* axis, and form of cross-section be determined by close curve *S* which be described by formula *r* = *r*(*ϕ*� ), where *ϕ*� is the angular coordinate on *S*.

In many practical applications, the antennas with currents linearly polarized along *Oz* axis are of interest. For such case, the RP has only the component *f<sup>θ</sup>* (*θ*, *ϕ*), denote it as *f*(*θ*, *ϕ*). On account of formula (18):

$$f(\theta,\varphi) = \int\limits\_{S} \int\_{-l}^{l} I(r(\varphi'), z) \sin\theta e^{il[r(\varphi')\sin\theta\cos(\varphi-\varphi') + z\cos\theta]} dS\_{\varphi'} dz. \tag{53}$$

Let the current distribution in antenna surface be determined as

$$I(r, z) = I\_1(r(\varphi')) \cdot I\_2(z),\tag{54}$$

then the spatial RP is completely determined by the RP *f*1(*θ*, *ϕ*), created by the distribution of current *I*1(*r*(*ϕ*� )) in *S*, and RP *f*2(*θ*) in the longitudinal plane, created by the current distribution in the generatrix of cylinder

$$f(\theta,\varphi) = f\_1(\theta,\varphi) \cdot f\_2(\theta). \tag{55}$$

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

8 Will-be-set-by-IN-TECH

is solved when the synthesis problem is formulated in the term of functional *σ* minimization.

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

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 *I* is optimizing function. Since both the amplitude and phase of complex function *I* are optimizing parameters, the considered problem is the amplitude-phase

Let the generatrix of the antenna has the length 2*l* and be parallel to *Oz* axis, and form of

In many practical applications, the antennas with currents linearly polarized along *Oz* axis are of interest. For such case, the RP has only the component *f<sup>θ</sup>* (*θ*, *ϕ*), denote it as *f*(*θ*, *ϕ*). On

*ik*[*r*(*ϕ*�

then the spatial RP is completely determined by the RP *f*1(*θ*, *ϕ*), created by the distribution

) sin *θ* cos(*ϕ*−*ϕ*�

)) in *S*, and RP *f*2(*θ*) in the longitudinal plane, created by the current

)+*z* cos *θ*]

*f*(*θ*, *ϕ*) = *f*1(*θ*, *ϕ*) · *f*2(*θ*). (55)

)) · *I*2(*z*), (54)

cross-section be determined by close curve *S* which be described by formula *r* = *r*(*ϕ*�

), *z*) sin *θe*

*I*(*r*, *z*) = *I*1(*r*(*ϕ*�

*t fn*+<sup>1</sup> + *AA*<sup>∗</sup> *fn*+<sup>1</sup> = *AA*∗(*Fe*−*<sup>i</sup>* arg *fn* ). (51)

*fn*+<sup>1</sup> <sup>=</sup> <sup>−</sup>1/*tAA*∗(*fn* <sup>−</sup> *Fe*−*<sup>i</sup>* arg *fn* ), (52)

), where

*dSϕ*� *dz*. (53)

If functional *σ<sup>t</sup>* is minimized then the respective iterative procedure has form

In the last case, one can use the iterative process

analysis of the obtained solutions [11].

synthesis problem.

*3.1.1. The integral equation approach*

*ϕ*� is the angular coordinate on *S*.

*f*(*θ*, *ϕ*) =

distribution in the generatrix of cylinder

 *l*

*I*(*r*(*ϕ*�

Let the current distribution in antenna surface be determined as

−*l*

*S*

account of formula (18):

of current *I*1(*r*(*ϕ*�

**3. Application to specific antennas**

but the convergence domain of this process is limited.

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

$$f\_1(\theta, \varphi) = \int\_S I\_1 r(\varphi')\_\prime z \dot{\varphi} \sin \theta e^{i\vec{k}[r(\varphi') \sin \theta \cos(\varphi - \varphi')]} dS\_{\varphi'} \tag{56}$$

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 and plane curvilinear antennas.

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 *θ* determines effective electrical scale of antenna. Therefore one can suppose *θ* = *π*/2. Omitting the indices "1"' in the distribution of current and RP, we represent the RP (56) in form

$$f(\boldsymbol{\varphi}) = A\boldsymbol{I} := \int\_0^{2\pi} \boldsymbol{I}(\boldsymbol{\varphi}') e^{i\boldsymbol{k}[\boldsymbol{r}(\boldsymbol{\varphi}')\cos(\boldsymbol{\varphi}-\boldsymbol{q}')]} \sqrt{r^2 + (d\boldsymbol{r}/d\boldsymbol{q}')^2} d\boldsymbol{q}',\tag{57}$$

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

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*� ), that 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 currents are defined as

$$(f\_1, f\_2)\_f = \int\_0^{2\pi} p(\varphi) f\_1(\varphi) f\_2^\*(\varphi) d\varphi. \tag{58}$$

$$(I\_1, I\_2)\_I = \int\_0^{2\pi} I\_1(\phi') I\_2^\*(\phi') \sqrt{r^2 + (dr/d\phi')^2} d\phi'. \tag{59}$$

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

$$tf(\varphi) + \int\_0^{2\pi} p(\varphi\_1) K(\varphi, \varphi\_1) f(\varphi\_1) d\varphi\_1 = Bf\_\prime \tag{60}$$

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

$$K(\varphi, \varphi\_1) = \int\_0^{2\pi} e^{ikr(\varphi')[\cos(\varphi - \varphi') - \cos(\varphi\_1 - \varphi')]} \sqrt{r^2 + (dr/d\varphi')^2} d\varphi',\tag{61}$$

and nonlinear operator *B* is determined as

$$Bf = \int\_0^{2\pi} p(\varphi\_1) K(\varphi, \varphi\_1) F(\varphi\_1) e^{i \arg f(\varphi\_1)} d\varphi\_1. \tag{62}$$

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

$$f\_{n+1} = 1/t(B - AA^\*)f\_n \tag{63}$$

For method (70), the problem of minimization of *σ<sup>t</sup>* is reduced to the solution of nonlinear

approximations substituting in its right hand side the function arg *fn* from previous iteration. Such iterative process is converging and similarly to iterative process (51) yields the

The numerical results are shown for the prescribed amplitude RPs *F*(*ϕ*) = sin2(*ϕ*/2) and *F*(*ϕ*) = sin128(*ϕ*/2) (Fig. 1a and Fig. 1b respectively). The influence of the parameter *t* in the functional (34) on the quality of synthesis is investigated. One can see that decrease of *t* improves the proximity of given *F* and synthesized | *f* | RPs. But the norm ||*I*|| of currents grows if *t* decreases. In the case of narrow *F* it is necessary to diminish *t* in order to decrease the mean-square deviation of the RPs. The detailed information about the synthesis quality is

*F*(*ϕ*) = sin2(*ϕ*/2) *F*(*ϕ*) = sin128(*ϕ*/2)

(a) *F*(*ϕ*) = sin2(*ϕ*/2) (b) *F*(*ϕ*) = sin128(*ϕ*/2)

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

**Figure 1.** Dependence of synthesis quality on the parameter *t* in the functional *σ<sup>t</sup>*

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

*t σ κ t σ κ* 0.1 0.0024 0.3419 0.01 0.0411 0.3450 1.0 0.1437 0.4367 0.10 0.0612 0.3581 10.0 1.2744 1.1065 1.0 0.0977 0.4272

(*m*) *<sup>n</sup>* = (*Fe<sup>i</sup>* arg *fn*<sup>+</sup><sup>1</sup> , *Ar*(*j*)

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 201

(*m*) *<sup>n</sup>* . This system is solved effectively by the method of successive

*<sup>n</sup>* )*<sup>f</sup>* ,(*j* = 1, ..., *M*) (71)

*<sup>n</sup>* )*<sup>f</sup>* ]*δ*

*<sup>n</sup>* )*<sup>I</sup>* + (*Ar*(*m*) *<sup>n</sup>* , *Ar*(*j*)

converging sequence of *σtn* which satisfies the condition (65).

**Table 1.** The values of *σ* and *κ* for two desired RPs *F*(*ϕ*)

algebraic system

with unknown *δ*

shown in Table 1.

*M* ∑ *m*=1 [*t*(*r* (*m*) *<sup>n</sup>* ,*<sup>r</sup>* (*j*)

*3.1.3. The numerical results*

has the limited region of convergence determined by formula

$$t > 2\sqrt{2\pi}l(\int\_0^{2\pi} p^2(\varphi)d\varphi)^{1/2},\tag{64}$$

where *l* is the length of contour *S*. Once the function *f* is found, the optimal current *I* is determined by formula (48).

The iterative process (51) is more preferable, it yields the converging sequence of functional *σ<sup>t</sup>* which satisfies the condition

$$
\sigma\_t(f\_{n+1}) < \sigma\_t(f\_n) \tag{65}
$$

for arbitrary *t*.

#### *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 *σ<sup>t</sup>* functional by the gradient methods can be also applied for solving the synthesis problem. The simplest gradient method is defined by the formula

$$I\_{n+1} = I\_n + \delta\_n z\_{n\prime} \tag{66}$$

where *δ<sup>n</sup>* is an optimizing multiplier, *zn* is a gradient of functional *σ<sup>t</sup>* (34) on the function *In*:

$$z\_n = A^\* \left( f\_n - F e^{i \arg f\_n} \right) + t I\_{\text{th}} \tag{67}$$

and

$$\delta\_{\mathfrak{N}} = ||z\_{\mathfrak{n}}|| / a\_{\mathfrak{n}\_{\mathfrak{I}}} \tag{68}$$

where *an* is a number determined by known values in the *n* + 1-th iteration [4].

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]

$$I\_{n+1} = I\_n + \delta\_n \hbar\_{n\prime} \tag{69}$$

and proposed in [3] generalized gradient method

$$I\_{n+1} = \sum\_{m=1}^{M} \delta\_n^{(m)} r\_n^{(m)} \,. \tag{70}$$

do not have such disadvantage. Here *hn* is a combination of *zn* from previous iterations, and *r* (1) *<sup>n</sup>* = *In*,*r* (2) *<sup>n</sup>* = *zn*,*r* (3) *<sup>n</sup>* ...,*r* (*M*) *<sup>n</sup>* is a set of some orthogonal functions, *<sup>δ</sup><sup>n</sup>* and *<sup>δ</sup>* (*m*) *<sup>n</sup>* are the coefficients subject to determination.

For method (70), the problem of minimization of *σ<sup>t</sup>* is reduced to the solution of nonlinear algebraic system

$$\sum\_{m=1}^{M} \left[ t(r\_n^{(m)}, r\_n^{(j)})\_I + (Ar\_n^{(m)}, Ar\_n^{(j)})\_f \right] \delta\_n^{(m)} = (Fe^{i \arg f\_{n+1}}, Ar\_n^{(j)})\_{f^\*} (j = 1, \dots, M) \tag{71}$$

with unknown *δ* (*m*) *<sup>n</sup>* . This system is solved effectively by the method of successive approximations substituting in its right hand side the function arg *fn* from previous iteration. Such iterative process is converging and similarly to iterative process (51) yields the converging sequence of *σtn* which satisfies the condition (65).

#### *3.1.3. The numerical results*

10 Will-be-set-by-IN-TECH

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

0

where *l* is the length of contour *S*. Once the function *f* is found, the optimal current *I* is

The iterative process (51) is more preferable, it yields the converging sequence of functional *σ<sup>t</sup>*

Above we mentioned the methods of successive approximations for solving the arising nonlinear equations. The direct optimization of *σ<sup>t</sup>* functional by the gradient methods can be also applied for solving the synthesis problem. The simplest gradient method is defined by

where *δ<sup>n</sup>* is an optimizing multiplier, *zn* is a gradient of functional *σ<sup>t</sup>* (34) on the function *In*:

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

> *M* ∑ *m*=1 *δ* (*m*) *<sup>n</sup> <sup>r</sup>*

do not have such disadvantage. Here *hn* is a combination of *zn* from previous iterations, and

(*M*) *<sup>n</sup>* is a set of some orthogonal functions, *<sup>δ</sup><sup>n</sup>* and *<sup>δ</sup>*

where *an* is a number determined by known values in the *n* + 1-th iteration [4].

*In*+<sup>1</sup> =

has the limited region of convergence determined by formula

*t >* 2 √ 2*πl*( 2*π*

*fn*+<sup>1</sup> = 1/*t*(*B* − *AA*∗)*fn* (63)

*σt*(*fn*+1) *< σt*(*fn*) (65)

*In*+<sup>1</sup> = *In* + *δnzn*, (66)

*δ<sup>n</sup>* = ||*zn*||/*an*, (68)

*In*+<sup>1</sup> = *In* + *δnhn*, (69)

(*m*) *<sup>n</sup>* , (70)

(*m*) *<sup>n</sup>* are the

*zn* <sup>=</sup> *<sup>A</sup>*∗(*fn* <sup>−</sup> *Fe<sup>i</sup>* arg *fn* ) + *tIn*, (67)

*p*2(*ϕ*)*dϕ*)1/2, (64)

The simplest of them

determined by formula (48).

which satisfies the condition

*3.1.2. The gradient methods of optimization*

process. The method of conjugate gradients [28]

and proposed in [3] generalized gradient method

(3) *<sup>n</sup>* ...,*r*

for arbitrary *t*.

the formula

and

*r* (1) *<sup>n</sup>* = *In*,*r*

(2) *<sup>n</sup>* = *zn*,*r*

coefficients subject to determination.

The numerical results are shown for the prescribed amplitude RPs *F*(*ϕ*) = sin2(*ϕ*/2) and *F*(*ϕ*) = sin128(*ϕ*/2) (Fig. 1a and Fig. 1b respectively). The influence of the parameter *t* in the functional (34) on the quality of synthesis is investigated. One can see that decrease of *t* improves the proximity of given *F* and synthesized | *f* | RPs. But the norm ||*I*|| of currents grows if *t* decreases. In the case of narrow *F* it is necessary to diminish *t* in order to decrease the mean-square deviation of the RPs. The detailed information about the synthesis quality is shown in Table 1.


**Table 1.** The values of *σ* and *κ* for two desired RPs *F*(*ϕ*)

**Figure 1.** Dependence of synthesis quality on the parameter *t* in the functional *σ<sup>t</sup>*

#### **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

#### 12 Will-be-set-by-IN-TECH 202 Numerical Simulation – From Theory to Industry Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics <sup>13</sup>

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 *E*- polarization) is considered.

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 *f*(*ϕ*) by the known field *v*(*S*) in the outer boundary *S* of the given form. The RP created by this field can be presented similarly to (26). The operator *A* in the case of circular external boundary has form [37]

$$Av = \int\_0^{2\pi} K(\boldsymbol{\varphi}, \boldsymbol{\varphi'}) v(\boldsymbol{\varphi'}) d\boldsymbol{\varphi'},\tag{72}$$

Solving the synthesis problem, we determine the field in outer boundary *S* of the antenna and transparency of this boundary. The form of inner boundary *S*<sup>0</sup> is determined as a curve of

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

which is generalization of (34), allows to take into account these requirements. Here *Ui*(*Si*) are prescribed values of the field's amplitude in the areas of restriction, *ui*(*Si*) are the obtained values of the field. Functions *p*(*ϕ*), *pi*(*Si*) are the weight functions, allowing to adjust a degree of proximity of the given and received values of RP and field, *t* is the parameter limiting norm

In the first step of solving the synthesis problem, the field *v* on the outer boundary *S* is determined from a condition of minimum of the functional (76). Minimization of functional can be carried out by the gradient methods, or by solving the respective Lagrange-Euler's

(1) *<sup>n</sup> vn* <sup>+</sup> *<sup>δ</sup>*

is used. The gradient *z* of the functional (76) (by virtue of the requirement of real field) has

process (77) reduces *σt*. Since *σ<sup>t</sup>* is limited from below (*σ<sup>t</sup>* ≥ 0), the process (77) is converging. In the second step, the transparency *ρ* of the outer boundary *S* and the form of inner metal

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

*N* ∑ *n*=0

In the case of antenna with arbitrary outer boundary, similarly to [37], the distribution of

The numerical calculations are carried out for the resonant antenna with a given outer elliptic boundary. The prescribed amplitude RP is: *F*(*ϕ*) = sin8(*ϕ*/2). In Fig. 2, the results are presented for the antenna with parameters *kb* = 15 and different *ka*: *ka* = 12.75, *ka* = 14.25,

*ρ*(*ϕ*) = 1/[

*J*2

*∂ψ*(*S*(*ϕ*))

(2) *<sup>n</sup> zn* <sup>+</sup> *<sup>δ</sup>*

Re*B*∗

*<sup>i</sup>* are the operators adjoined to *A* and *Bi*, respectively [4]. Each step of iterative

*an* cos *nϕ*

*<sup>n</sup>*(*ka*)

*<sup>n</sup>*(*ka*) + *N*<sup>2</sup>

*M* ∑ *i*=1

*pi*(*Si*)[*Ui*(*Si*) − |*ui*(*Si*)|]

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 203

<sup>2</sup>*dSi* + *t*

 2*π*

*v*2(*ϕ*)*dϕ*, (76)

0

(3) *<sup>n</sup> hn* (77)

*<sup>i</sup>* [*Ui* exp(*i* arg *ui*) − *ui*], (78)

), (79)

*<sup>∂</sup><sup>N</sup>* ]. (80)

constant phase of the field *u*(*r*, *ϕ*) [37].

*p*(*ϕ*)[*F*(*ϕ*) − | *f*(*ϕ*)|]

<sup>2</sup>*dϕ* +

equation. In the first case, the generalized gradient method [3]

*z* = *tv* − Re*A*∗[*F* exp(*i* arg *f*) − *f* ] −

*vn*+<sup>1</sup> = *δ*

*ρ*(*ϕ*) = *πkv*(*ϕ*)/(2

where *Jn* and *Nn* are the Bessel and Neumann functions, respectively.

*M* ∑ *i*=1

*Si*

functional

*σ<sup>t</sup>* = 2*π*

of the field *v*.

form

where *A*∗ and *B*∗

in the explicit form

boundary *S*<sup>0</sup> are determined.

*3.2.2. The numerical results*

transparency is determined by the formula

0

where kernel

$$K(\boldsymbol{\varphi}, \boldsymbol{\varphi}') = \sum\_{n=0}^{\infty} \frac{i^n \cos n(\boldsymbol{\varphi} - \boldsymbol{\varphi}')}{(1 + \delta\_{0n}) H\_n^2(ka)},\tag{73}$$

*δ*0*<sup>n</sup>* is the Kronecker delta, function *H*<sup>2</sup> *<sup>n</sup>*(*ka*) is the Hankel function of second kind, *a* is the circle radius.

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 *f* by the field *v*. In this method, the field *u*(*r*, *ϕ*) outside of antenna is represented approximately by the finite sum

$$u(r, \boldsymbol{\varphi}) = \sum\_{n=1}^{N} a\_n H\_0^2(k\mathbf{R}\_n)\_\prime \tag{74}$$

where *Rn* = *r*<sup>2</sup> + *r*<sup>2</sup> *<sup>n</sup>* − 2*rrn* cos(*ϕ* − *ϕn*) is the distance between an observation point and *n*-th auxiliary source; *r*, *ϕ* and *rn*, *ϕ<sup>n</sup>* are the polar coordinates of a point of observation and *n*-th source, respectively; *an* are the unknown coefficients subject to determination in the process of solving the synthesis problem.

The RP is given by

$$f(\varphi) = \sqrt{\frac{2}{\pi}} e^{i\pi/4} \sum\_{n=1}^{N} a\_{\mathbb{II}} e^{i\text{kr}\_{\mathbb{II}} \cos \left(\varphi - \varphi\_{\text{t}}\right)}.\tag{75}$$

Solving the synthesis problem, we determine the field in outer boundary *S* of the antenna and transparency of this boundary. The form of inner boundary *S*<sup>0</sup> is determined as a curve of constant phase of the field *u*(*r*, *ϕ*) [37].

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

$$\sigma\_{l} = \int\_{0}^{2\pi} p(\varphi) \left[ F(\varphi) - |f(\varphi)| \right]^2 d\varphi + \sum\_{i=1}^{M} \int\_{S\_{l}} p\_{i}(S\_{i}) \left[ \mathcal{U}\_{l}(S\_{i}) - |u\_{l}(S\_{l})| \right]^2 dS\_{l} + t \int\_{0}^{2\pi} v^2(\varphi) d\varphi,\tag{76}$$

which is generalization of (34), allows to take into account these requirements. Here *Ui*(*Si*) are prescribed values of the field's amplitude in the areas of restriction, *ui*(*Si*) are the obtained values of the field. Functions *p*(*ϕ*), *pi*(*Si*) are the weight functions, allowing to adjust a degree of proximity of the given and received values of RP and field, *t* is the parameter limiting norm of the field *v*.

In the first step of solving the synthesis problem, the field *v* on the outer boundary *S* is determined from a condition of minimum of the functional (76). Minimization of functional can be carried out by the gradient methods, or by solving the respective Lagrange-Euler's equation. In the first case, the generalized gradient method [3]

$$
\upsilon\_{n+1} = \delta\_n^{(1)} \upsilon\_n + \delta\_n^{(2)} z\_n + \delta\_n^{(3)} h\_n \tag{77}
$$

is used. The gradient *z* of the functional (76) (by virtue of the requirement of real field) has form

$$z = tv - \operatorname{Re} A^\* \left[ F \exp(i \arg f) - f \right] - \sum\_{i=1}^{M} \operatorname{Re} B\_i^\* \left[ \mathcal{U}\_l \exp(i \arg u\_i) - u\_i \right] \tag{78}$$

where *A*∗ and *B*∗ *<sup>i</sup>* are the operators adjoined to *A* and *Bi*, respectively [4]. Each step of iterative process (77) reduces *σt*. Since *σ<sup>t</sup>* is limited from below (*σ<sup>t</sup>* ≥ 0), the process (77) is converging.

In the second step, the transparency *ρ* of the outer boundary *S* and the form of inner metal boundary *S*<sup>0</sup> are determined.

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

$$\rho(\boldsymbol{\varrho}) = \pi k \upsilon(\boldsymbol{\varrho}) / (2 \sum\_{n=0}^{N} \frac{a\_n \cos n\boldsymbol{\varrho}}{f\_n^2(ka) + \mathcal{N}\_n^2(ka)}),\tag{79}$$

where *Jn* and *Nn* are the Bessel and Neumann functions, respectively.

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

$$\rho(\varphi) = 1/[\frac{\partial \psi(\mathcal{S}(\varphi))}{\partial N}].\tag{80}$$

#### *3.2.2. The numerical results*

12 Will-be-set-by-IN-TECH

are formed by several surfaces, one of which is semitransparent. Antennas with one

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

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

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 *f*(*ϕ*) by the known field *v*(*S*) in the outer boundary *S* of the given form. The RP created by this field can be presented similarly to

(26). The operator *A* in the case of circular external boundary has form [37]

*Av* = 2*π*

*K*(*ϕ*, *ϕ* � ) = ∞ ∑ *n*=0

outside of antenna is represented approximately by the finite sum

*f*(*ϕ*) =

*u*(*r*, *ϕ*) =

 2 *π e iπ*/4 *N* ∑ *n*=1 *ane*

0

*K*(*ϕ*, *ϕ* � )*v*(*ϕ* � )*dϕ* �

*i*

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 *f* by the field *v*. In this method, the field *u*(*r*, *ϕ*)

> *N* ∑ *n*=1

*n*-th auxiliary source; *r*, *ϕ* and *rn*, *ϕ<sup>n</sup>* are the polar coordinates of a point of observation and *n*-th source, respectively; *an* are the unknown coefficients subject to determination in the

*anH*<sup>2</sup>

*<sup>n</sup>* cos *<sup>n</sup>*(*<sup>ϕ</sup>* <sup>−</sup> *<sup>ϕ</sup>*

(1 + *δ*0*n*)*H*<sup>2</sup>

� )

*<sup>n</sup>*(*ka*) is the Hankel function of second kind, *a* is the circle

*<sup>n</sup>*(*ka*)

*<sup>n</sup>* − 2*rrn* cos(*ϕ* − *ϕn*) is the distance between an observation point and

*ikrn* cos(*ϕ*−*ϕn*)

, (72)

, (73)

<sup>0</sup> (*kRn*), (74)

. (75)

semitransparent and other metal boundary are considered here.

front-to-rear factor (FRF) the most close to the prescribed ones.

*3.2.1. Generalization of variational statement*

case of *E*- polarization) is considered.

*δ*0*<sup>n</sup>* is the Kronecker delta, function *H*<sup>2</sup>

where kernel

radius.

where *Rn* =

The RP is given by

 *r*<sup>2</sup> + *r*<sup>2</sup>

process of solving the synthesis problem.

The numerical calculations are carried out for the resonant antenna with a given outer elliptic boundary. The prescribed amplitude RP is: *F*(*ϕ*) = sin8(*ϕ*/2). In Fig. 2, the results are presented for the antenna with parameters *kb* = 15 and different *ka*: *ka* = 12.75, *ka* = 14.25,

#### 14 Will-be-set-by-IN-TECH 204 Numerical Simulation – From Theory to Industry Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics <sup>15</sup>

where *b* and *a* are the big and small semiaxes of ellipse. For such antenna the level of side lobes in the synthesized amplitude RP | *f* | is smaller than –20 dB, and distribution of transparency is smoother in the area of main radiation (the continuous lines in Figs. 2a, 2b correspond to *ka* = 12.75, and the dashed ones correspond to *ka* = 14.25). The outer elliptic boundaries (dashed lines), the found form of inner metallic boundaries (continuous lines), and the inner contour *Sa* of placement of the auxiliary sources (dash-and-dot lines) are shown in Figs. 3a, 3b. The auxiliary sources are distributed uniformly on the *Sa*.

(a) *ka* = 12.75 (b) *ka* = 14.25

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 205

(a) the RPs and field in area of restriction (b) the field *v* and transparency *ρ* in outer boundary

The geometrical parameters of resonant antenna with waveguide excitation are shown in Fig. 5. In order to create RP enough narrow, the width *L* of antenna should be much larger than the wavelength *λ*. The height *d* is of the order of *λ*/2. Excitation is carried out by a metal single-mode waveguide with semitransparent grid at its end; the width *l* of waveguide is of the order *λ*/2, and both its length and the length *D* of the antenna along the *Ox* axis are of the

The direct (analysis) problem on determination of the electromagnetic field components in the semitransparent aperture is reduced to two separate problems in the planes *xOz* and *yOz* respectively. We consider here the case of *E*-polarization. The unknown function *u* is the *Ey*

*u*(*x*, *y*) exp[*ik*(*x* sin *θ* cos *ϕ*+*y* sin *θ* sin *ϕ*)]*dxdy*. (81)

**Figure 3.** Form of antenna boundaries *S* and *S*<sup>0</sup>

*3.3.1. The physical description of problem*

*<sup>f</sup>*(*θ*, *<sup>ϕ</sup>*) =

*S*

order of *L*.

The RP of antenna has form

**Figure 4.** Synthesis of resonant antenna with restriction on the field

**Figure 2.** Synthesis of resonant antenna with elliptic outer boundary

The distribution of transparency *ρ* in the area opposite to direction of main radiation has a spasmodic character. Such distribution cannot be realized by the physical reason. Therefore the values of *ρ* are averaged in this range in order to receive the smooth distribution of *ρ*. This leads to some change of field *v* on *S*, but the numerical calculations show small change of the synthesized amplitude RP | *f* |. The more smooth distribution of *ρ* in the area mentioned above can be achieved by increasing the number of auxiliary sources here.

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: *F*(*ϕ*) = sin8(*ϕ*/2); *ka* = 15. Minimization of a field was carried out in two points *ϕ* = *π*/2, 3*π*/2 on the additional circle with radius *kb* = 20. These points were allocated in the second summand of the functional (76) using the weight function *p*1(*ϕ*) = *δ*(*π*/2, 3*π*/2); *p*(*ϕ*) ≡ 1, *t* = 0.01. In Fig. 4a, the prescribed RP *F* (thick continuous line) and synthesized | *f* | (thin continuous line) amplitude RP are shown. The amplitude |*u*1| of obtained field on the circle of restrictions is marked by dashed line.

It can be seen that the field at restriction points is reduced up to level -37 dB. The synthesized field *v* (continuous line) and transparency *ρ* (dashed line) are presented in Fig. 4b. The form of the inner synthesized metallic boundary is more complicate than in the previous example.

#### **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.

**Figure 3.** Form of antenna boundaries *S* and *S*<sup>0</sup>

14 Will-be-set-by-IN-TECH

where *b* and *a* are the big and small semiaxes of ellipse. For such antenna the level of side lobes in the synthesized amplitude RP | *f* | is smaller than –20 dB, and distribution of transparency is smoother in the area of main radiation (the continuous lines in Figs. 2a, 2b correspond to *ka* = 12.75, and the dashed ones correspond to *ka* = 14.25). The outer elliptic boundaries (dashed lines), the found form of inner metallic boundaries (continuous lines), and the inner contour *Sa* of placement of the auxiliary sources (dash-and-dot lines) are shown in Figs. 3a,

(a) the prescribed *F* (thick line) and synthesized RPs | *f* | (b) the transparency *ρ* of outer boundary *S*

The distribution of transparency *ρ* in the area opposite to direction of main radiation has a spasmodic character. Such distribution cannot be realized by the physical reason. Therefore the values of *ρ* are averaged in this range in order to receive the smooth distribution of *ρ*. This leads to some change of field *v* on *S*, but the numerical calculations show small change of the synthesized amplitude RP | *f* |. The more smooth distribution of *ρ* in the area mentioned above

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: *F*(*ϕ*) = sin8(*ϕ*/2); *ka* = 15. Minimization of a field was carried out in two points *ϕ* = *π*/2, 3*π*/2 on the additional circle with radius *kb* = 20. These points were allocated in the second summand of the functional (76) using the weight function *p*1(*ϕ*) = *δ*(*π*/2, 3*π*/2); *p*(*ϕ*) ≡ 1, *t* = 0.01. In Fig. 4a, the prescribed RP *F* (thick continuous line) and synthesized | *f* | (thin continuous line) amplitude RP are shown. The amplitude |*u*1| of obtained field on the

It can be seen that the field at restriction points is reduced up to level -37 dB. The synthesized field *v* (continuous line) and transparency *ρ* (dashed line) are presented in Fig. 4b. The form of the inner synthesized metallic boundary is more complicate than in the previous example.

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.

3b. The auxiliary sources are distributed uniformly on the *Sa*.

**Figure 2.** Synthesis of resonant antenna with elliptic outer boundary

can be achieved by increasing the number of auxiliary sources here.

circle of restrictions is marked by dashed line.

**3.3. Waveguide resonant antenna**

boundary

**Figure 4.** Synthesis of resonant antenna with restriction on the field

#### *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 *L* of antenna should be much larger than the wavelength *λ*. The height *d* is of the order of *λ*/2. Excitation is carried out by a metal single-mode waveguide with semitransparent grid at its end; the width *l* of waveguide is of the order *λ*/2, and both its length and the length *D* of the antenna along the *Ox* axis are of the order of *L*.

The RP of antenna has form

$$f(\theta,\varphi) = \iint\limits\_{S} \mathfrak{u}(\mathbf{x},\mathbf{y}) \exp[ik(\mathbf{x}\sin\theta\cos\varphi + \mathbf{y}\sin\theta\sin\varphi)] dxd\mathbf{y}.\tag{81}$$

The direct (analysis) problem on determination of the electromagnetic field components in the semitransparent aperture is reduced to two separate problems in the planes *xOz* and *yOz* respectively. We consider here the case of *E*-polarization. The unknown function *u* is the *Ey*

**Figure 5.** Geometry of the resonant antenna

component of electromagnetic field. In the region over the antenna this component satisfies the Helmholtz equation and the boundary conditions

$$
\mu = 0 \tag{82}
$$

In such a way, the RP in the *yOz* plane can be written down in the form [37]


values of the FRF outside this range, that is for *k* ∈ [*k*0, *k*1) and *k* ∈ (*k*2, *k*3].

 *L*/2

*u*(*y*, 0) exp(*iky* sin *ϕ*)*dy*. (86)

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 207

∼= *u*(*y*, 0) exp(−*ikz*). (87)

*η*(*k*). (88)

2, where *R*1(*k*) is the reflection factor

*η*(*k*), (89)

−*L*/2

Numerical calculations can be essentially simplified, if one assumes that the field above the

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 [*k*1, *k*2] and the minimal

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 *d*(*y*) and *ρ*(*y*), which maximize FRF *η*<sup>1</sup> in the operating range [*k*1, *k*2]. In this case, the least value of FRF in this range is specified as a criterion of optimization, and this value is maximized by a choice of functions *d* and *ρ*. That is, the

*η*<sup>1</sup> = max min

*k*1≤*k*≤*k*<sup>2</sup>

of the main wave in exciting waveguide, with some weight multiplier can be included into functional (88). In this case, the transparency of waveguide aperture can be also used as an

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 *d* and *ρ*, maximizing the functional *η*1, and, at the same time, minimizing additionally the following

*k*0≤*k<k*1,*k*2*<k*≤*k*<sup>3</sup>

*d*<sup>0</sup> *< d*(*y*) *< dm*, *ρ*<sup>0</sup> *< ρ*(*y*) *< ρm*. (90)

that is, minimization of the maximal FRF value outside the [*k*1, *k*2] range is required. Of course, it is necessary to take into account the restrictions on the functions *d*(*x*) and *ρ*(*x*) owing the

For example, *d*0, *dm* are the boundary values of height, which provide the single-mode conditions in antenna, and *ρ*0, *ρ<sup>m</sup>* are the boundary values of transparency, which provide a good quality of antenna in the required ranges. Moreover, the received functions should

*η*<sup>2</sup> = min max

*f*(*θ*, *ϕ*) = *k* sin *θ*

antenna is represented approximately in the form [37]

The additional parameter of optimization 1 − |*R*1(*k*)|

additional parameter of optimization.

*3.3.2. The objective functionals*

functional is maximized

functional [6]

physical reason:

*u* 

on the all metallic walls;

$$
\mu^{+} = \mu^{-}, \quad \frac{\partial \mu^{-}}{\partial \mathbf{x}} - \frac{\partial \mu^{+}}{\partial \mathbf{x}} = \frac{\mu}{\rho(\mathcal{S})} \tag{83}
$$

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

$$u\_{\parallel r \to \infty} = \frac{\exp(-ikr)}{\sqrt{kr}} f(\varphi)\_{\prime} \tag{84}$$
  $x > 0$ 

and asymptotical condition in the exciting waveguide

$$u\_{\parallel \infty \to -\infty} = \cos \frac{\pi y}{2a} (\exp(-i\beta\_1 \mathbf{x}) + R\_1 \exp(i\beta\_1 \mathbf{x})).\tag{85}$$
  $y+$ 

The problem of determination of the field *u*(*x*, *y*) in the semitransparent boundary *S* is solved in three steps [37]. In the first step, the field in the irregular region of antenna is determined using the cross-section method [19]. In the second step, the field over the exciting waveguide is sought for, and the matching of field in the regular and irregular regions of antenna is fulfilled. The reflection coefficient *R*<sup>1</sup> is determined by the fulfillment of the adjoint boundary conditions (85) in the third step.

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 *d*(*y*) of the upper wall and transparency *ρ*(*y*) of the lower wall, and RP of the linear antenna in the *xOz* plane.

In such a way, the RP in the *yOz* plane can be written down in the form [37]

$$f(\theta,\varphi) = k \sin \theta \int\_{-L/2}^{L/2} u(y,0) \exp(iky \sin \varphi) dy. \tag{86}$$

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

$$u \Big|\limits\_{z} |y| < L/2 \quad \cong u(y, 0) \exp(-ikz). \tag{87}$$

#### *3.3.2. The objective functionals*

16 Will-be-set-by-IN-TECH

component of electromagnetic field. In the region over the antenna this component satisfies

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>∂</sup>u*<sup>+</sup>

<sup>=</sup> exp(−*ikr*)

on the semitransparent upper boundary, the same conditions in the aperture of the exciting

The problem of determination of the field *u*(*x*, *y*) in the semitransparent boundary *S* is solved in three steps [37]. In the first step, the field in the irregular region of antenna is determined using the cross-section method [19]. In the second step, the field over the exciting waveguide is sought for, and the matching of field in the regular and irregular regions of antenna is fulfilled. The reflection coefficient *R*<sup>1</sup> is determined by the fulfillment of the adjoint boundary

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 *d*(*y*) of the upper wall and transparency *ρ*(*y*) of the lower wall, and RP of the

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> *<sup>u</sup>*

*<sup>u</sup>*<sup>+</sup> <sup>=</sup> *<sup>u</sup>*−, *<sup>∂</sup>u*<sup>−</sup>

*r* → ∞ *x >* 0

= cos

*πy*

*u* = 0 (82)

<sup>√</sup>*kr <sup>f</sup>*(*ϕ*), (84)

<sup>2</sup>*<sup>a</sup>* (exp(−*iβ*1*x*) + *<sup>R</sup>*<sup>1</sup> exp(*iβ*1*x*)). (85)

*<sup>ρ</sup>*(*S*) (83)

**Figure 5.** Geometry of the resonant antenna

on the all metallic walls;

the Helmholtz equation and the boundary conditions

waveguide; the condition of radiation on the infinity

and asymptotical condition in the exciting waveguide

*x* → −∞ *y*+ *< a*

*u* 

conditions (85) in the third step.

linear antenna in the *xOz* plane.

*u*  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 [*k*1, *k*2] and the minimal values of the FRF outside this range, that is for *k* ∈ [*k*0, *k*1) and *k* ∈ (*k*2, *k*3].

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 *d*(*y*) and *ρ*(*y*), which maximize FRF *η*<sup>1</sup> in the operating range [*k*1, *k*2]. In this case, the least value of FRF in this range is specified as a criterion of optimization, and this value is maximized by a choice of functions *d* and *ρ*. That is, the functional is maximized

$$\eta\_1 = \max\_{k\_1 \le k \le k\_2} \min\_{k \le k\_2} \eta(k). \tag{88}$$

The additional parameter of optimization 1 − |*R*1(*k*)| 2, where *R*1(*k*) is the reflection factor of the main wave in exciting waveguide, with some weight multiplier can be included into functional (88). In this case, the transparency of waveguide aperture can be also used as an 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 *d* and *ρ*, maximizing the functional *η*1, and, at the same time, minimizing additionally the following functional [6]

$$\eta\_2 = \min\_{k\_0 \le k < k\_1, k\_2 < k \le k\_3} \eta(k),\tag{89}$$

that is, minimization of the maximal FRF value outside the [*k*1, *k*2] range is required. Of course, it is necessary to take into account the restrictions on the functions *d*(*x*) and *ρ*(*x*) owing the physical reason:

$$d\_0 < d(y) < d\_{m\prime} \quad \rho\_0 < \rho(y) < \rho\_{m\prime} \tag{90}$$

For example, *d*0, *dm* are the boundary values of height, which provide the single-mode conditions in antenna, and *ρ*0, *ρ<sup>m</sup>* are the boundary values of transparency, which provide a good quality of antenna in the required ranges. Moreover, the received functions should satisfy the additional conditions of smoothness, which can be formulated as restriction on the second derivatives

$$|\boldsymbol{d}''(\boldsymbol{y})| \le M\_{1\prime} \quad |\boldsymbol{\rho}''(\boldsymbol{y})| \le M\_2. \tag{91}$$

(a) the main range (b) the additional range

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 209

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

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

> 1 *nm* (*x*) · *I* 2

It is assumed that the radiating elements are flat apertures, for example, end of open waveguide. The RP depends on the angular coordinates *θ* and *ϕ*. From the practical point of view, the consideration of such models of the arrays is justified by the fact that they allow to receive the values of required radiation characteristics with the accuracy of 2% - 5%, while the time of solution for the respective problems of analysis (determination of the RP of array), considerably decreases. Such approach to the solution of direct electrodynamic problems is effective especially for the arrays with constant coordinate surfaces, e.g., for the plane, cylindrical and conical arrays. Proceeding from the above assumptions, we separate the synthesis problem of such arrays into two synthesis problems for the linear and circular

where *N* is the number of radiators in circular subarray (identical for all subarrays), *M* is the number of circular subarrays, *a* is radius of cylinder. The currents *Inm* are complex numbers, by means of which choice the approximation to the given amplitude RP *F*(*ϑ*, *ϕ*) is carried out.

*nm* (*y*), (92)

*f*(*θ*, *ϕ*) = *f*1(*θ*) · *f*2(*ϕ*). (93)

*Inm fnm*(*θ*, *ϕ*) exp(*ik*(*zm* cos *θ* + *a* sin *θ* cos(*ϕ* − *ϕn*))), (94)

Thus, the expressions for current *Inm*(*x*, *y*) in radiators and RP *f*(*θ*, *ϕ*) have the form [4]

*Inm*(*x*, *y*) = *I*

methods for direct optimization of functionals are used in practical applications.

**Figure 7.** The FRF values versus the iteration number *N*

*3.4.1. The RP of array*

arrays.

one of the simplifications.

The spatial RP of cylindrical array [9] is:

*M* ∑ *m*=1

*N* ∑ *n*=1

*f*(*θ*, *ϕ*) =

**3.4. Phase synthesis problem for cylindrical array**

### *3.3.3. The modeling results*

The numerical results are shown in Fig. 6. Calculations were carried out for the problem of *η* maximization in range ±5.0% in neighborhood of the central frequency (*kc* = 6.0). It is necessary to minimize the FRF outside of this range for 5.2 *< k <* 5.6 and 6.4 *< k* ≤ 6.8. Parameters of antenna are the following: the length of antenna *L* = 6.0, half-width of excited waveguide *l*/2 = 0.3*L*, number of considered reflected waves in waveguide *N*<sup>1</sup> = 5, number of waves in a background *N*<sup>2</sup> = 20.

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 *η* in the main and additional ranges.

**Figure 6.** Optimized values of FRF in the basic and additional ranges versus the frequency (*k*)

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 [6]. Therefore, the values of FRF in the basic range are slightly decreased in comparison with the FRF values for initial problem.

The optimal form *d*(*y*) of the lower boundary of antenna and transparency *ρ*(*y*) of the upper boundary are slightly different for the both cases of optimization.

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 *k* = 12.0, and curves 2 and 3 correspond to the extreme points *k* = 11.0 and *k* = 13.0 respectively. In Fig. 7b, curve 1 corresponds to the central frequency *k* = 14.0, and curves 2 and 3 correspond to the frequency values *k* = 13.01 and *k* = 15.0. One can see that the main optimization takes place in the first iterations; there is the improvement only in low-order digit in the next steps. Therefore, it is enough to make 3-5 steps for the main range and 9-11 steps for the additional range, in iterative procedure to receive the practically interesting results.

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*

18 Will-be-set-by-IN-TECH

satisfy the additional conditions of smoothness, which can be formulated as restriction on the

The numerical results are shown in Fig. 6. Calculations were carried out for the problem of *η* maximization in range ±5.0% in neighborhood of the central frequency (*kc* = 6.0). It is necessary to minimize the FRF outside of this range for 5.2 *< k <* 5.6 and 6.4 *< k* ≤ 6.8. Parameters of antenna are the following: the length of antenna *L* = 6.0, half-width of excited waveguide *l*/2 = 0.3*L*, number of considered reflected waves in waveguide *N*<sup>1</sup> = 5, number

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 *η* in the main and additional ranges.

**Figure 6.** Optimized values of FRF in the basic and additional ranges versus the frequency (*k*)

range are slightly decreased in comparison with the FRF values for initial problem.

boundary are slightly different for the both cases of optimization.

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 [6]. Therefore, the values of FRF in the basic

The optimal form *d*(*y*) of the lower boundary of antenna and transparency *ρ*(*y*) of the upper

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 *k* = 12.0, and curves 2 and 3 correspond to the extreme points *k* = 11.0 and *k* = 13.0 respectively. In Fig. 7b, curve 1 corresponds to the central frequency *k* = 14.0, and curves 2 and 3 correspond to the frequency values *k* = 13.01 and *k* = 15.0. One can see that the main optimization takes place in the first iterations; there is the improvement only in low-order digit in the next steps. Therefore, it is enough to make 3-5 steps for the main range and 9-11 steps for the additional range, in iterative procedure to receive the practically

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

��

(*y*)| ≤ *M*2. (91)

(*y*)| ≤ *M*1, |*ρ*


second derivatives

interesting results.

*3.3.3. The modeling results*

of waves in a background *N*<sup>2</sup> = 20.

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 *Inm*(*x*, *y*) in radiators and RP *f*(*θ*, *ϕ*) have the form [4]

$$I\_{nm}(\mathbf{x}, y) = I\_{nm}^1(\mathbf{x}) \cdot I\_{nm}^2(y),\tag{92}$$

$$f(\theta, \varphi) = f\_1(\theta) \cdot f\_2(\varphi). \tag{93}$$

It is assumed that the radiating elements are flat apertures, for example, end of open waveguide. The RP depends on the angular coordinates *θ* and *ϕ*. From the practical point of view, the consideration of such models of the arrays is justified by the fact that they allow to receive the values of required radiation characteristics with the accuracy of 2% - 5%, while the time of solution for the respective problems of analysis (determination of the RP of array), considerably decreases. Such approach to the solution of direct electrodynamic problems is effective especially for the arrays with constant coordinate surfaces, e.g., for the plane, cylindrical and conical arrays. Proceeding from the above assumptions, we separate the synthesis problem of such arrays into two synthesis problems for the linear and circular arrays.

The spatial RP of cylindrical array [9] is:

$$f(\theta,\boldsymbol{\varphi}) = \sum\_{m=1}^{M} \sum\_{n=1}^{N} I\_{nm} f\_{nm}(\theta,\boldsymbol{\varphi}) \exp(ik(z\_m \cos \theta + a \sin \theta \cos(\varphi - \varphi\_n))),\tag{94}$$

where *N* is the number of radiators in circular subarray (identical for all subarrays), *M* is the number of circular subarrays, *a* is radius of cylinder. The currents *Inm* are complex numbers, by means of which choice the approximation to the given amplitude RP *F*(*ϑ*, *ϕ*) is carried out. The functions *fnm*(*ϑ*, *ϕ*) are the RPs of separate radiators, (*zm*, *a*, *ϕn*) are the coordinates of radiators. The RP of separate radiators is identical and can be presented in the following form

$$f\_{nm}(\theta,\varphi) = f\_n(\varphi - \varphi\_n) f\_m(\theta). \tag{95}$$

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

$$f(\theta, \varphi) = f\_l(\theta) f\_\mathfrak{c}(\varphi),\tag{96}$$

*ψ*(*k*+1) = arg *A*∗(*F* exp(*i* arg *f* (*k*)

The gradient methods are more convenient to solve the minimization problem for the functional *σ*. The method of conjugated gradients [28] is the most suitable for this purpose. In this method the next approximation of phase vector *ψ* = {*ψn*, *n* = 1, ...*N*} is calculated by the

*ψ*(*k*+1) = *ψ*(*k*) + *δ*(*k*)

*h*(*k*) = *z*(*k*) + *γ*(*k*)

Parameter *δ*(*k*) is determined from minimum of *σ* being a function of this parameter.

*z*(*k*) is the gradient of *σ* with respect to the phases of currents in the "point" *ψ*(*k*). The

<sup>|</sup>) exp(*<sup>i</sup>* arg *<sup>f</sup>* (*k*)

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 Δ,

The algorithm consisting of two enclosed iterative processes is used for the solution of this problem. The value of phase *ψ<sup>n</sup>* is improved in the *n*-th step of the internal iterative process, the phases in other radiators remain fixed. At the same time, the mean-square deviation of the synthesized RP and function *F* exp(*i* arg *f* (*k*)), where arg *f*(*k*) is the phase RP, which is received in the previous step of external iterative process, is minimized. The phase RP arg *f* is

The internal cycle consists of the successive improvement of phases in the separate radiators changing their number from 1 up to *N*. The value of *σ* decreases in each step of internal cycle.

The new phase RP arg *<sup>f</sup>*(*k*+1) is calculated by the found values of {*ψ*} in external cycle. The values of *σ* corresponding to new phase RP also decrease, what provides the convergence of the whole algorithm. In view of the step-type behavior of *ψ<sup>n</sup>* values, this convergence exists not only for a sequence of *σ*, but also for the phase distributions. The iterative process is

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

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 *N* = 32, the radiators are placed in active sector *β* = 90*o*.

*h*(*k*)

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 211

)))*<sup>n</sup>* · *I*

*h*(*k*−1)

to find the phases *ψ*(*k*+1).

*z* (*k*)

improved in the external iterations.

several external cycles in the latter algorithm.

*3.4.3. The results of numerical modeling*

components of vector *z*(*k*) are the following

*<sup>n</sup>* <sup>=</sup> <sup>−</sup>Im{(*A*∗((*<sup>F</sup>* <sup>−</sup> *<sup>s</sup>*<sup>|</sup> *<sup>f</sup>* (*k*)

that is *ψ<sup>n</sup>* = *λn*Δ, the integers *λ<sup>n</sup>* are unknown in this case [5, 7].

considered completed, if there is no change of *ψ<sup>n</sup>* in the internal cycle.

formula

Here

) (104)

. (105)

, (106)

(*k*) *<sup>n</sup>* }, *<sup>n</sup>* <sup>=</sup> 1, 2, ..., *<sup>N</sup>*. (107)

where

$$f\_l(\theta) = A\_1 I := \sum\_{m=1}^{M} I\_m f\_m(\theta) \exp(ikz\_m \cos \theta) \tag{97}$$

is the RP of linear array, and

$$f\_{\mathfrak{C}}(\theta,\mathfrak{g}) = A\_2 I \equiv \sum\_{n=1}^{N} I\_{\mathfrak{n}} f\_{\mathfrak{n}}(\mathfrak{g} - \mathfrak{g}\_n) \exp(ika \sin \theta \cos(\mathfrak{g} - \mathfrak{g}\_n)) \tag{98}$$

is the RP of circular array for each *θ*. Below we consider the synthesis problem for the circular array.

#### *3.4.2. Optimization criteria*

The complex currents in the array elements are determined by their amplitudes and phases. We denote these values |*In*| and *ψn*, respectively. The amplitudes |*In*| of currents are prescribed together with the amplitude RP *F* in the problem of phase synthesis. The phases *ψ<sup>n</sup>* are the optimizing parameters. We use the functionals (29) and (33) for optimization.

The equalities

$$\chi = \arg A(|I|e^{i\psi}),\tag{99}$$

$$
\psi = \arg A^\*(Fe^{j\chi}).\tag{100}
$$

should be satisfied at the points of functional (33) maximum. This set yields the system of transcendental equations for the phases *ψ* of current and phase RP *χ*.

Using normalization of the current *I* values: ||*I*|| = 1, we write down the functional (33) in two equivalent forms

$$\kappa = (Fe^{i\chi}, A(|I|e^{i\psi}))\_f = (A^\*(Fe^{i\chi}), |I|e^{i\psi})\_{I\prime} \tag{101}$$

where *χ* = arg *f* is the phase RP. The operator *A*∗ is adjoint to *A* and determined similarly to [4].

At first, we consider the problem of *κ* maximization. Substituting (99) into (100), we receive the system of nonlinear algebraic equations for optimal phase distribution of currents

$$\psi\_{\rm nl} = \arg A^\*(F \exp(i \arg A(|I|e^{i\psi}))), \textit{n} = 1, 2, \dots, \textit{N}. \tag{102}$$

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

$$\arg f^{(k)} = \arg A(|I|\exp(i\psi^{(k)})),\tag{103}$$

210 Numerical Simulation – From Theory to Industry Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics <sup>21</sup> Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 211

$$\psi^{(k+1)} = \arg A^\*(F \exp(i \arg f^{(k)}) \tag{104}$$

to find the phases *ψ*(*k*+1).

The gradient methods are more convenient to solve the minimization problem for the functional *σ*. The method of conjugated gradients [28] is the most suitable for this purpose. In this method the next approximation of phase vector *ψ* = {*ψn*, *n* = 1, ...*N*} is calculated by the formula

$$
\psi^{(k+1)} = \psi^{(k)} + \delta^{(k)} h^{(k)}.\tag{105}
$$

Here

20 Will-be-set-by-IN-TECH

The functions *fnm*(*ϑ*, *ϕ*) are the RPs of separate radiators, (*zm*, *a*, *ϕn*) are the coordinates of radiators. The RP of separate radiators is identical and can be presented in the following form

> *M* ∑ *m*=1

is the RP of circular array for each *θ*. Below we consider the synthesis problem for the circular

The complex currents in the array elements are determined by their amplitudes and phases. We denote these values |*In*| and *ψn*, respectively. The amplitudes |*In*| of currents are prescribed together with the amplitude RP *F* in the problem of phase synthesis. The phases *ψ<sup>n</sup>* are the optimizing parameters. We use the functionals (29) and (33) for optimization.

*χ* = arg *A*(|*I*|*e*

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

Using normalization of the current *I* values: ||*I*|| = 1, we write down the functional (33) in

where *χ* = arg *f* is the phase RP. The operator *A*∗ is adjoint to *A* and determined similarly to

At first, we consider the problem of *κ* maximization. Substituting (99) into (100), we receive the system of nonlinear algebraic equations for optimal phase distribution of currents

In practice, the system (99), (100) is more convenient for the numerical solution. For this

arg *<sup>f</sup>*(*k*) <sup>=</sup> arg *<sup>A</sup>*(|*I*<sup>|</sup> exp(*iψ*(*k*)

*<sup>i</sup>ψ*))*<sup>f</sup>* = (*A*∗(*Feiχ*), <sup>|</sup>*I*|*<sup>e</sup>*

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

*fl*(*ϑ*) = *A*<sup>1</sup> *I* :=

*N* ∑ *n*=1

transcendental equations for the phases *ψ* of current and phase RP *χ*.

*<sup>κ</sup>* = (*Feiχ*, *<sup>A</sup>*(|*I*|*<sup>e</sup>*

*ψ<sup>n</sup>* = arg *A*∗(*F* exp(*i* arg *A*(|*I*|*e*

reason, we use the following iterative process

*fc*(*θ*, *ϕ*) = *A*<sup>2</sup> *I* ≡

where

array.

The equalities

two equivalent forms

[4].

is the RP of linear array, and

*3.4.2. Optimization criteria*

*fnm*(*θ*, *ϕ*) = *fn*(*ϕ* − *ϕn*)*fm*(*θ*). (95)

*f*(*θ*, *ϕ*) = *fl*(*θ*)*fc*(*ϕ*), (96)

*In fn*(*ϕ* − *ϕn*) exp(*ika* sin *θ* cos(*ϕ* − *ϕn*)) (98)

*ψ* = arg *A*∗(*Feiχ*). (100)

*Im fm*(*ϑ*) exp(*ikzm* cos *ϑ*) (97)

*<sup>i</sup>ψ*), (99)

*<sup>i</sup>ψ*))), *n* = 1, 2, ..., *N*. (102)

)), (103)

*<sup>i</sup>ψ*)*I*, (101)

$$h^{(k)} = z^{(k)} + \gamma^{(k)} h^{(k-1)} \, \prime \tag{106}$$

*z*(*k*) is the gradient of *σ* with respect to the phases of currents in the "point" *ψ*(*k*). The components of vector *z*(*k*) are the following

$$z\_n^{(k)} = -\text{Im}\{ (A^\*( (F - s | f^{(k)} |) \exp(i \arg f^{(k)}))) \} \cdot I\_n^{(k)} \}, n = 1, 2, \ldots, N. \tag{107}$$

Parameter *δ*(*k*) is determined from minimum of *σ* being a function of this 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 Δ, that is *ψ<sup>n</sup>* = *λn*Δ, the integers *λ<sup>n</sup>* are unknown in this case [5, 7].

The algorithm consisting of two enclosed iterative processes is used for the solution of this problem. The value of phase *ψ<sup>n</sup>* is improved in the *n*-th step of the internal iterative process, the phases in other radiators remain fixed. At the same time, the mean-square deviation of the synthesized RP and function *F* exp(*i* arg *f* (*k*)), where arg *f*(*k*) is the phase RP, which is received in the previous step of external iterative process, is minimized. The phase RP arg *f* is improved in the external iterations.

The internal cycle consists of the successive improvement of phases in the separate radiators changing their number from 1 up to *N*. The value of *σ* decreases in each step of internal cycle.

The new phase RP arg *<sup>f</sup>*(*k*+1) is calculated by the found values of {*ψ*} in external cycle. The values of *σ* corresponding to new phase RP also decrease, what provides the convergence of the whole algorithm. In view of the step-type behavior of *ψ<sup>n</sup>* values, this convergence exists not only for a sequence of *σ*, but also for the phase distributions. The iterative process is considered completed, if there is no change of *ψ<sup>n</sup>* in the internal cycle.

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 *N* = 32, the radiators are placed in active sector *β* = 90*o*.

#### 22 Will-be-set-by-IN-TECH 212 Numerical Simulation – From Theory to Industry Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics <sup>23</sup>

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

$$F(\varphi) = \begin{cases} |\sin 18\varrho\_{\prime}|\varphi| < 5.0^0, \\ 0, |\varphi| \ge 5.0^0. \end{cases} \tag{108}$$

investigation of its properties by the example of the nonlinear equation (46), corresponding to

where *c* = *ka* sin *α*, *a* is the radius of array, *α* is the angle in which the RP *F* is non-zero, *N* = 2*M* + 1 is a total number of array elements, *ξ* is the generalized angular coordinate. The

*M* ∑ *n*=−*M*

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 213

*vn* exp(*icnξ*), (109)

*f*(*ξ*) exp(−*icnξ*)*dξ*, (110)


 *c* 2*π*

> *π* /*c*

−*π*/*c*

*π* /*c*

−*π*/*c*

The numerical calculations are carried out for the prescribed amplitude radiation pattern *F*(*ξ*) = cos(*πξ*/2) and are shown in Figs. 9 and 10. In Fig. 9a, the values of *κ* and *σ* are shown for various types of initial approximation of the current's phase *ψ*(0)(*x*). For this case, the maximization problem of *κ* is equivalent in some sense to minimization problem of *σ* [4]. The solid lines correspond to values of *κ*, and the dashed lines correspond to values of *σ*. The number of array elements *N* = 11, parameter *c* changes from *c* = 0 up to *c* = 2. For the values of *Nc* which do not exceed *Nc* = 5 all types of solutions give the same values of *κ* and *σ*. At *Nc* ≈ 2*π* the branching of solutions appears, and optimal value for *κ* and *σ* functionals gives the solution with even phase *ψ*. The prescribed amplitude radiation pattern *F* and synthesized | *f* | are shown in Fig. 9b. The amplitudes | *f* | in the considerable extent

The optimal values of sought phase distributions *ψ* are shown in Fig. 10a, the given current amplitude distribution is |*I*| ≡ 1. The optimal phase distributions *ψ* keep the parity properties of corresponding initial approximations *ψ*0. The optimal values of *κ* and *σ* provide the

The quality of approximation to prescribed amplitude pattern *F* too much depends on the parameter *c* (see Fig. 10b). At *c* = 3.14 the *κ* and *σ* values are noticeably smaller than for

The mutual coupling of the separate elements of array is taken into account in the process of

Let the operator *A* describe the RP (array factor) of linear array [4].

*f*(*ξ*) = *Av* :=

*A*∗ *f* =

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

 *c* 2*π*

*κ*(*ψ*) =

differ from the amplitude *F* because of small value of *c* parameter (*c* = 1.6).

solution with phase distribution *ψ*0(*x*) = cos(*x*) (see Fig. 9).

*c* = 1.6, although the value of *N* is larger in case of Fig. 9.

solution of a direct problem (analysis problem) [7, 14].

**3.6. Synthesis of waveguide array**

functional *κ* (33).

operator *A*∗acts in the following way

*3.5.1. The numerical results*

The thick line corresponds to the prescribed amplitude RP *F*, and thin line corresponds to the synthesized RP | *f* |. The prescribed *F* and synthesized | *f* | RPs coincide in the main lobe up to level –20 dB, the level of side lobes does not exceed -20 dB (see Fig. 8a).

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 {*ψn*} only; the amplitudes {|*In*|} of current remain constant. In fact, the problem of phase synthesis is solved for each scanning angle *ϕ<sup>s</sup>* separately.

The difference between the results of continuous and discrete synthesis depends on the value of phase discrete Δ. The smaller the difference, the smaller this value. In Fig. 8b the values of *σ* for two types of synthesis are shown for the process of scanning, *ϕ<sup>s</sup>* is changed in range from 10 up to 90. Solid line corresponds to the case of continuous synthesis, and dashed line correspond to the case of discrete synthesis, the value of Δ = 22.50. This value of Δ gives not big difference for two types of synthesis. So, the difference between *F* and | *f* | in main lobe of RPs does not exceed 1dB, and this difference does not exceed 10dB in the side lobes. Such difference is satisfactory for the engineering practice. Of course, the above mentioned difference grows if the scanning angle *ϕ<sup>s</sup>* approaches to the left or right border of the active sector *β*.

(a) the prescribed and synthesized RPs for the angle of scanning *ϕ<sup>s</sup>* = 0<sup>0</sup> (b) the normalized values of *σ* for the continuous and discrete phase synthesis

**Figure 8.** Synthesis of bi-directional RP *F*

#### **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 *κ* (33).

Let the operator *A* describe the RP (array factor) of linear array [4].

$$f(\xi) = Av := \sqrt{\frac{c}{2\pi}} \sum\_{n=-M}^{M} v\_n \exp(icn\xi),\tag{109}$$

where *c* = *ka* sin *α*, *a* is the radius of array, *α* is the angle in which the RP *F* is non-zero, *N* = 2*M* + 1 is a total number of array elements, *ξ* is the generalized angular coordinate. The operator *A*∗acts in the following way

$$A^\*f = \sqrt{\frac{c}{2\pi}} \int\_{-\pi/c}^{\pi/c} f(\xi) \exp(-icn\xi) d\xi \,\tag{110}$$

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

$$\kappa(\psi) = \int\_{-\pi/c}^{\pi/c} |f(\xi)| F(\xi) d\xi. \tag{111}$$

#### *3.5.1. The numerical results*

22 Will-be-set-by-IN-TECH

The thick line corresponds to the prescribed amplitude RP *F*, and thin line corresponds to the synthesized RP | *f* |. The prescribed *F* and synthesized | *f* | RPs coincide in the main lobe up to

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 {*ψn*} only; the amplitudes {|*In*|} of current remain constant. In fact, the problem of phase synthesis is solved for each scanning

The difference between the results of continuous and discrete synthesis depends on the value of phase discrete Δ. The smaller the difference, the smaller this value. In Fig. 8b the values of *σ* for two types of synthesis are shown for the process of scanning, *ϕ<sup>s</sup>* is changed in range from 10 up to 90. Solid line corresponds to the case of continuous synthesis, and dashed line correspond to the case of discrete synthesis, the value of Δ = 22.50. This value of Δ gives not big difference for two types of synthesis. So, the difference between *F* and | *f* | in main lobe of RPs does not exceed 1dB, and this difference does not exceed 10dB in the side lobes. Such difference is satisfactory for the engineering practice. Of course, the above mentioned difference grows if the scanning angle *ϕ<sup>s</sup>* approaches to the left or right border of the active

The problem of the non-uniqueness of solutions for phase synthesis problem is investigated

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

<sup>|</sup> sin 18*ϕ*, <sup>|</sup>*ϕ*<sup>|</sup> *<sup>&</sup>lt;* 5.00,

0, <sup>|</sup>*ϕ*| ≥ 5.00. (108)

(b) the normalized values of *σ* for the continuous and discrete phase synthesis

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

*F*(*ϕ*) =

level –20 dB, the level of side lobes does not exceed -20 dB (see Fig. 8a).

(a) the prescribed and synthesized RPs for the

angle of scanning *ϕ<sup>s</sup>* = 0<sup>0</sup>

**3.5. Investigation of branching solution**

**Figure 8.** Synthesis of bi-directional RP *F*

on the example of linear array.

angle *ϕ<sup>s</sup>* separately.

sector *β*.

The numerical calculations are carried out for the prescribed amplitude radiation pattern *F*(*ξ*) = cos(*πξ*/2) and are shown in Figs. 9 and 10. In Fig. 9a, the values of *κ* and *σ* are shown for various types of initial approximation of the current's phase *ψ*(0)(*x*). For this case, the maximization problem of *κ* is equivalent in some sense to minimization problem of *σ* [4].

The solid lines correspond to values of *κ*, and the dashed lines correspond to values of *σ*. The number of array elements *N* = 11, parameter *c* changes from *c* = 0 up to *c* = 2. For the values of *Nc* which do not exceed *Nc* = 5 all types of solutions give the same values of *κ* and *σ*. At *Nc* ≈ 2*π* the branching of solutions appears, and optimal value for *κ* and *σ* functionals gives the solution with even phase *ψ*. The prescribed amplitude radiation pattern *F* and synthesized | *f* | are shown in Fig. 9b. The amplitudes | *f* | in the considerable extent differ from the amplitude *F* because of small value of *c* parameter (*c* = 1.6).

The optimal values of sought phase distributions *ψ* are shown in Fig. 10a, the given current amplitude distribution is |*I*| ≡ 1. The optimal phase distributions *ψ* keep the parity properties of corresponding initial approximations *ψ*0. The optimal values of *κ* and *σ* provide the solution with phase distribution *ψ*0(*x*) = cos(*x*) (see Fig. 9).

The quality of approximation to prescribed amplitude pattern *F* too much depends on the parameter *c* (see Fig. 10b). At *c* = 3.14 the *κ* and *σ* values are noticeably smaller than for *c* = 1.6, although the value of *N* is larger in case of Fig. 9.

#### **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].

(a) the *κ* and *σ* values versus the array electrical size *Nc* (b) the synthesized amplitude radiation patterns for various types of current phase, *c* = 1.6

**Figure 11.** Geometry of plane periodical waveguide array

*f*(*ξ*1, *ξ*2) =

excitation coefficient for *n*-th waveguide.

RP

where

and finally

functional *σ<sup>t</sup>* (34).

where

*3.6.2. The modeling results*

where *xn*<sup>0</sup> and *yn*<sup>0</sup> are the coordinates of central points of apertures, *an* is the complex

Introducing the generalized angular coordinates *ξ*<sup>1</sup> and *ξ*2, we receive the expression for the

*N* ∑ *n*=1

The expression (116) indicates that the calculation of array factor *f*(*ξ*1, *ξ*2) using the excitation coefficients *an* is realized by the linear operator *A*. The coefficients *an* will be the optimization parameters in the synthesis problem. Solving the synthesis problem, we minimize the

The results of numerical calculations are presented for the waveguide arrays consisting of 15 and 31 radiators; *kl* = 1.2, *kL* = 18.75, *k* is wavenumber, *l* and *L* are width and length of waveguide aperture respectively. The prescribed amplitude RP is: *F*(*θ*, *ϕ*) = *F*1(*θ*)*F*2(*ϕ*),

In Fig. 12, the dependence of the synthesis results on the value *N* of waveguides is shown. It is easy to see, that the synthesized amplitude RP has narrower main lobe if *N* increases. The level of the first sidelobe is -30.26dB and -30.71dB respectively. The low level of sidelobes and velocity of its decrease is very important characteristic of the synthesized amplitude RPs. As rule, one requires the level of first sidelobe not greater than -20dB and not very slow decreasing the next sidelobes. The above mentioned characteristic for the synthesized amplitude RP in the plane *ξ*2*Oz* are shown in Table 2. The amplitude RP | *f* | at *N* = 31 has

*an* exp[*ik*(*xn*<sup>0</sup> *ξ*<sup>1</sup> + *yn*<sup>0</sup> *ξ*2)], (114)

*an fn*(*ξ*1, *ξ*2), (116)

*ξ*<sup>1</sup> = sin *θ* cos *ϕ*, *ξ*<sup>2</sup> = sin *θ* sin *ϕ*, (115)

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 215

0, <sup>|</sup>*θ*<sup>|</sup> *<sup>&</sup>gt; <sup>π</sup>*/20 , *<sup>F</sup>*2(*ϕ*)=(cos *<sup>ϕ</sup>*)64. (117)

*N* ∑ *n*=1

*f*(*ξ*1, *ξ*2) = *A�a* :=

where*�a* = {*a*1, *a*2, ...*aN*}, *fn*(*ξ*1, *ξ*2) are the RPs of separate waveguides.

*<sup>F</sup>*1(*θ*) = 1, <sup>|</sup>*θ*| ≤ *<sup>π</sup>*/20

lower first sidelobe -30.71dB and faster decrease of the far sidelobes.

**Figure 9.** The values of optimizing functionals and synthesized RPs for various type of solutions

**Figure 10.** The optimal phase distributions at *N* = 11 (a), and synthesis results at *c* = 3.14 (b)

#### *3.6.1. Statement of synthesis problem*

The objective functional is formulated as [4]

$$\sigma = \iint\limits\_{\Omega} \left( F(\theta, \varphi) - |f(\theta, \varphi)| \right)^2 d\theta d\varphi + t \sum\_{n=1}^{N} \int |I\_n(\mathbf{x}\_n, y\_n)|^2 d\mathbf{x}\_n dy\_n \tag{112}$$

where *N* = 2*M* + 1 is a number of exciting waveguides, *F*(*θ*, *ϕ*) is the prescribed amplitude RP, | *f*(*θ*, *ϕ*)| is the amplitude of synthesized RP, *In*(*xn*, *yn*) are the currents in the waveguide apertures. Geometry of waveguide array is shown in Fig. 11.

The determination of the currents *In*(*xn*, *yn*) in the waveguide apertures (the solution of analysis problem) results in solution of the integral equation system [14].

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

$$f(\theta,\varphi) = \sum\_{n=1}^{N} a\_{n} \exp[ik(\mathbf{x}\_{\text{lv}\_{0}} \sin \theta \cos \varphi + y\_{\text{lv}\_{0}} \sin \theta \sin \varphi)],\tag{113}$$

**Figure 11.** Geometry of plane periodical waveguide array

where *xn*<sup>0</sup> and *yn*<sup>0</sup> are the coordinates of central points of apertures, *an* is the complex excitation coefficient for *n*-th waveguide.

Introducing the generalized angular coordinates *ξ*<sup>1</sup> and *ξ*2, we receive the expression for the RP

$$f(\vec{\xi}\_1, \vec{\xi}\_2) = \sum\_{n=1}^{N} a\_n \exp\left[ik(\mathbf{x}\_{n\_0}\vec{\xi}\_1 + y\_{n\_0}\vec{\xi}\_2)\right] \tag{114}$$

where

24 Will-be-set-by-IN-TECH

**Figure 9.** The values of optimizing functionals and synthesized RPs for various type of solutions

(a) the optimal phase distributions (b) the desired *F* and synthesized | *f* | RPs

where *N* = 2*M* + 1 is a number of exciting waveguides, *F*(*θ*, *ϕ*) is the prescribed amplitude RP, | *f*(*θ*, *ϕ*)| is the amplitude of synthesized RP, *In*(*xn*, *yn*) are the currents in the waveguide

The determination of the currents *In*(*xn*, *yn*) in the waveguide apertures (the solution of

*N* ∑ *n*=1


*an* exp[*ik*(*xn*<sup>0</sup> sin *θ* cos *ϕ*+*yn*<sup>0</sup> sin *θ* sin *ϕ*)], (113)

<sup>2</sup>*dxndyn*, (112)

*Sn*

**Figure 10.** The optimal phase distributions at *N* = 11 (a), and synthesis results at *c* = 3.14 (b)

(*F*(*θ*, *<sup>ϕ</sup>*) − | *<sup>f</sup>*(*θ*, *<sup>ϕ</sup>*)|)2*dθd<sup>ϕ</sup>* <sup>+</sup> *<sup>t</sup>*

analysis problem) results in solution of the integral equation system [14].

apertures. Geometry of waveguide array is shown in Fig. 11.

*N* ∑ *n*=1 (b) the synthesized amplitude radiation patterns for various types of current phase, *c* = 1.6

(a) the *κ* and *σ* values versus the array electrical

size *Nc*

*3.6.1. Statement of synthesis problem*

*σ* = 

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

*f*(*θ*, *ϕ*) =

The objective functional is formulated as [4]

Ω

$$\not\subset \mathfrak{z}\_1 = \sin \theta \cos \varphi, \quad \not\subset \mathfrak{z}\_2 = \sin \theta \sin \varphi,\tag{115}$$

and finally

$$f(\mathfrak{F}\_1, \mathfrak{F}\_2) = A\vec{a} := \sum\_{n=1}^{N} a\_n f\_n(\mathfrak{F}\_1, \mathfrak{F}\_2) \, , \tag{116}$$

where*�a* = {*a*1, *a*2, ...*aN*}, *fn*(*ξ*1, *ξ*2) are the RPs of separate waveguides.

The expression (116) indicates that the calculation of array factor *f*(*ξ*1, *ξ*2) using the excitation coefficients *an* is realized by the linear operator *A*. The coefficients *an* will be the optimization parameters in the synthesis problem. Solving the synthesis problem, we minimize the functional *σ<sup>t</sup>* (34).

#### *3.6.2. The modeling results*

The results of numerical calculations are presented for the waveguide arrays consisting of 15 and 31 radiators; *kl* = 1.2, *kL* = 18.75, *k* is wavenumber, *l* and *L* are width and length of waveguide aperture respectively. The prescribed amplitude RP is: *F*(*θ*, *ϕ*) = *F*1(*θ*)*F*2(*ϕ*), where

$$F\_1(\theta) = \begin{cases} 1, |\theta| \le \pi/20 \\ 0, |\theta| > \pi/20 \end{cases}, F\_2(\rho) = (\cos \rho)^{64}. \tag{117}$$

In Fig. 12, the dependence of the synthesis results on the value *N* of waveguides is shown. It is easy to see, that the synthesized amplitude RP has narrower main lobe if *N* increases. The level of the first sidelobe is -30.26dB and -30.71dB respectively. The low level of sidelobes and velocity of its decrease is very important characteristic of the synthesized amplitude RPs. As rule, one requires the level of first sidelobe not greater than -20dB and not very slow decreasing the next sidelobes. The above mentioned characteristic for the synthesized amplitude RP in the plane *ξ*2*Oz* are shown in Table 2. The amplitude RP | *f* | at *N* = 31 has lower first sidelobe -30.71dB and faster decrease of the far sidelobes.



that one can receive the solutions with various properties starting the iterative process with

Synthesis of Antenna Systems According to the Desired Amplitude Radiation Characteristics 217

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 *σ* of the prescribed and synthesized amplitude RPs, and to optimize simultaneously the restrictions on the phase or amplitude characteristics of the excited fields.

*Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, NASU, Ukraine*

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[3] Andriychuk, M. I., Voitovich, N. N. (1985). Synthesis of a Closed Planar Antenna with Given Amplitude Pattern. *Radio Eng. and Electron. Phys*., Vol. 30, No 5, pp. 35-40. [4] Andriychuk, M. I., Voitovich, N. N., Savenko, P. A., Tkachuk, V. P. (1993). *The Antenna Synthesis According to Amplitude Radiation Pattern: Numerical Methods and Algorithms*, (in

[5] Andriychuk, M. I. (1998). The Analytical-numerical Solution Method of the Nonlinear Problems of the Antenna Phase Synthesis. *Proc. of IIIrd Intern. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-98), Tbilisi,*

[6] Andriychuk, M. I., Zamorska, O. F. (2004). The Antenna Systems Synthesis According to the Amplitude Characteristics under Condition of Electromagnetic Compatibility. *2004 Second Intern. Workshop on Ultrawideband and Ultrashort Impulse Signals, Sevastopol,*

[7] Andriychuk, M. I., Klakovych, L. M., Savenko, P. O., Tkach, M. D. (2005). Numerical Solution of Nonlinear Synthesis Problems of Antenna Arrays with Regard for Coupling of Radiators. *Proc. of Vth International Conference on Antenna Theory and Techniques, 24-27*

[8] Andriychuk, M., Zamorska, O. (2006). Waveguide Antenna Synthesis According to the Amplitude Radiation Characteristics in the Frequency Band. *Proc. of I*st *European*

[10] Bityutskov, V. I. (2001). Bunyakovskii Inequality, in M. Hazewinkel, *Encyclopedia of*

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(in Russian). *Uspekhi Mat. Nauk*, vol. 22, No 2, pp. 59-107.

different initial approximations.

**Author details**

**5. References**

Mykhaylo Andriychuk

*Functions*, (in Russian). Moscow, Nauka.

Russian). Kiev, Nauk. Dumka.

*Ukraine. Sept. 2004*, pp. 135-137.

*May 2005, Kyiv, Ukraine*, pp. 213-216.

*Georgia*, pp. 86-89.

Wiley.

*Mathematics*. Springer.

**Table 2.** The level of sidelobes (in dB) corresponding to array with various *N*

**Figure 12.** The synthesized amplitude RPs for various *N*
