## 1. Introduction

Dipole and loop antennas are used in communication lines, in radar and navigation systems, TV and other fields of radio-engineering. Structures of antennas depend on their bandwidth. In VHF, UHF, L and S ranges, these antennas are made, as a rule, of conductors with a radius

In communication nodes and antenna arrays, dipole and loop antennas function in the conditions of strong coupling. Therefore, one of the problems of numerical simulation is the research into the influence of interaction on the electrical characteristics of antennas.

Scattering characteristics of antennas are a strong factor that facilitates radar reconnaissance. These characteristics of loop and dipole antennas have hardly been studied in the literature.

In the literature, there is no enough information or there is no information at all about a number of questions related to research into and the versions of construction of dipole and loop antenna. The following is related to these problems:

optimization of dipole and loop antennas of complex shape according to various criteria (input resistance, directional pattern, bandness); Yagi-Uda multielement antennas, discone and bicone antennas, loop antennas with a quasi-isotropic directional pattern in the E-plane, a combination of loop and dipole antennas, all these antennas are attributed to complex antennas;

arrays of dipole and loop antennas with series excitation;

interaction of dipole and loop antennas in the near-field region;

scattering characteristics of dipole and loop antennas.

The above specified questions are the contents of the present chapter.

## 2. Method of analysis and mathematical model of wire antenna

### 2.1. Integral and matrix equations to determine current in arbitrary thin conductor

Wire antennas are simulated numerically by the method of integral equations. In the present chapter, we have used Pockington’s integral equation [Mittra, 1973]. The integral equation is solved by Galerkin method (method of the moments) [Harrington, 1968; Mittra, 1973; Fletcher, 1984]. To integrate the integral equation to a system of linear algebraic equations, impulse functions have been used further as basis and weight functions. In this case, the basic part of each matrix coefficient of this system is defined by an exact analytical expression. That ensures the enhancement of accuracy of solution of the integral equation and reduces this problem time.

According to [Mittra, 1973], Pockington’s integral equation looks like:

where

The kernel of the integral equation

The right side depends on distribution of the extraneous field that excites the conductor:

where

Green’s function is defined by the expression:

The remaining quantities included in (1) – (4) are shown in fig. 1.

When solving an integral equation by the method of moments, a conductor of arbitrary shape is divided into M rectilinear segments of length

The use of impulse basis and weight functions leads to the following system of linear algebraic equations:

where

M is the number of segments, into which the whole conductor of length L is divided;

m is the number of a segment, in which point Q (source point) is located (fig. 1),

n is the number of a segment, in which point P (point of observation) is located (fig. 1),

The matrix coefficients are determined by the expression:

where

The right side of the system of equations (5) according to (3) turns out to be equal to:

The weight function is distinct from zero and is equal to unity only within a segment with number n. Therefore, it follows from (15) that:

where

The further transformation into (16) is different for radiation and scattering problems.

### 2.2. Radiation problem

When solving a radiation problem, the extraneous field is considered to be distinct from zero only in the space of excitation and constant as for its amplitude and phase within the space. In this case, it follows from (16) that:

where

U is the excitation voltage;

As a result of numerical solution of the system of linear algebraic equations (5), the current in each segment

where

If there is one segment in the space of excitation,

According to the found current distribution in the conductor, the geometry of the conductor, it is easy to determine numerically the conductor field in space in any zone, and further, the basic characteristics and parametres of a wire antenna: its directional pattern, phase diagram, polarization pattern, directivity factor. These questions compose the contents of an exterior problem of the theory, it will be studied in more detail below.

### 2.3. Scattering problem

The mathematical model of a wire antenna in the scattering mode includes the above relationships that determine current in scattering conductors, and the expression that defines the right side of the matrix equation (3). We will study (3) for the case of excitation of a conductor by a plane wave falling from the given direction. The problem is explicated by fig. 3.

The figure shows a scattering conductor in the X, Y, Z coordinate system, vectors _{i} that is the angle between the Z axis and vector

where

where

It follows from (3), (20), (21) that:

If the antenna is irradiated by a plane electromagnetic wave, the amplitude of the electric field is the same at all the points of the antenna and equal to

where

The phase

where

### 2.4. Radiation (scattered) field of thin conductor

The field at an arbitrary point of space P is determined according to the found current distribution

where

When calculating the current

where

Vectors

The cross components

### 2.5. Directional pattern, directivity factor and gain

The directional pattern

where

The shape of

The directivity factor (

The antenna gain (

### 2.6. Radar Cross Section

The radar cross section (RCS) of antennas can exceed considerably the effective area of an object, on which the antennas are installed. That is why it is necessary to take into consideration the scattering properties of the antenna, when developping countermeasures to radar reconnaissance. In accordance with [Crispin and Maffett, 1965; Scolnic, 1970;]

where

S is the radar cross section;

The scattering pattern and RCS depend on the antenna structure, frequency, polarization, and direction of falling of an electromagnetic wave on the antenna. RCS is described in general by the polarization scattering matrix:

where

The scattered field and RCS of the antenna can be represented as the sum of two components: antenna component

The scattering pattern differs from the directional pattern (DP) of an antenna in the transmission (reception) mode. It is related to the fact that a structural component predominates over an antenna component in a scattered field. The structural component of the scattered field is the result of radiation of the currents created by an incident EMW in the antenna. Distribution of these currents can differ essentially from current distribution in the antenna in the transmission mode.

### 2.7. Interaction of antennas located in near-field region

An integral equation and the subsequent part of a mathematical model of a wire antenna allow modelling wire antennas with an arbitrary geometry, located in the near-field region. If several antennas are located in the near-field region, the length

The described mathematical model has been used in the program, with the help of which the results below have been obtained.

## 3. Modelling of radiation and scattering characteristics of dipole antennas

### 3.1. Dipole antennas and Yagi-Uda antenna

Fig.5 shows a Yagi-Uda antenna. Antenna elements: 1 is a linea reflector; 2 is a dipole; 3 are directors. Plane and corner reflectors are used too( see fig. 6). The polarization of the antenna is linear. Plane XZ is Е-plane; plane YZ is Н-plane. According to fig. 4, in XZ-plane

The antenna is optimised as for its input resistance, gain, backscattered radiation level («Front to back» parameter: F/B). Its directional pattern, input resistance, directivity factor, gain depend on the antenna geometrical parametres, frequency, the surface resistance of conductors and the connected lumped resistances. Further, all the regularities are illustrated by the example of antennas intended for a medium frequency f=300 MHz. Size values of the antenna elements are not shown because of the lack of space.

#### 3.1.1. Radiation characteristics

Fig. 7 shows the dependence of a directivity factor (

It follows from the cited and other results of simulation that the antenna gain and input resistance depend little on reflector sizes. If

The graph in fig. 11 illustrates the influence of shape of a reflector on the antenna parametres. The dependences of

#### 3.1.2. Scattering characteristics

By the example of a dipole, it is convenient to show the dependence of current distribution in the scattering mode on the direction of radiation. Fig. 12 shows current distribution in a dipole with nonresonant length L=1,25λ. The dipole is irradiated from the direction that is perpendicular to the dipole axis (along the Z axis, see fig. 5), and at an angle of 45° to the axis. Fig. 13 shows the corresponding scattering patterns in the E-plane. The figures give the values of

Fig. 14 illustrates the difference of a directional pattern and a scattering pattern and the dependence of the scattering pattern on the direction of propagation of an irradiating wave in relation to the antenna axis (Z axis). The calculations have been made for a Yagi-Uda antenna with a linear reflector, frequency f=300 MHz, N=10.

Fig. 15 shows the dependence of RCSN in the direction of a maximum of the scattering pattern on the number of directors N. The calculation has been made for the matched load of the antenna

#### 3.1.3. Dipole antennas with reduced dimensions

In case of increase of a surface inductive reactance_{o} of an antenna decreases. The most practical version of implementation of the surface resistance is the application of, instead of a smooth conductor, a conductor rolled into a helix with a radius

For the case

The more the ratio

In case of decrease of

In case of change of

### 3.2. Flat and convex dipole and Yagi arrays

One of the main questions of the numerical analysis of antenna arrays and antennas as a part of a group in the near-field region is the question of interaction of antennas. The coupling effect of antennas, located in a near-field region, results in changes of all the characteristics of the antennas. Further, the differences in interaction between emitters in plane and convex arrays have been studied by the example of comparison of linear and arc arrays. The interaction of two Yagi-Uda antennas has also been studied, depending on a distance between them and their cross orientation. It is important for the organisation of communication nodes.

The received regularities are illustrated by the example of a Yagi-Uda antenna with the number of directors N=1. The antenna is tuned so that the input resistance is equal to 50 Ohm at a medium frequency of 300 MHz.

Fig. 20 shows directional patterns of the antenna located at the centre of a fragment of a plane array with the number of emitters

The influence of interaction on the directional pattern of a Yagi-Uda antenna is different in plane and convex arrays. Fig. 23 shows the directional patterns in the E-plane of a director antenna with N=1 in the arc array for the number of emitters М=3 and М=5 at a distance between emitters D=0,55

It follows from the comparison of fig. 20b, c and
fig. 23b, c that, at the same linear distance between the neighboring emitters (D), the interaction between director antennas as a part of an arc array results in more considerable changes, than if they are a part of a linear array. The dependence of an isolation ratio between two director antennas on an angle between their axes of antennas (β) is shown in fig. 24 for two distances between the antennas entries D’ =D/

It follows from the results of numerical simulation that

### 3.3. Linear Antenna Array with Series Excitation

The dipole arrays of half-wave dipoles with series excitation are decribed briefly in the book [Rothammels, 1995] with the reference to the works [Brown, Liwis and Epstein, 1937; Bruckmann, 1938). Such antennas are gone by the name of Franklin antennas (Franckin C.S. – British patent № 242342, 1924). Versions of such antennas are shown in fig. 25. The drawing symbols are the following: A is half-wave dipoles: B is closed quarter-wave stubs; C is double-wire line exciting line; D is stub bonding; L is dipole length; H is stub length; d is stub width; P is point of observation in space; P_{1} is projection of point Р on XY plane; R, θ, φ are the spherical coordinates of point P.

The polarization of an antenna radiation field is linear: the E-plane is the plane φ=const, the H-plane is the plane XY. In the H-plane, the directional pattern is quasi-isotropic.

Antenna characteristics depend on geometrical sizes of L, H, d, the number of dipoles in the antenna, an angle between the neighbouring stubs, frequency f. In the literature, these regularities are not studied, and they are the subject of the numerical simulation in the present chapter. Besides, in the chapter, we study the modifications of a Franklin antenna that are more wideband as for the matching criterion and have a sector-shaped directional pattern in the E-plane.

The version shown in fig. 25a at a resonance frequency, which corresponds to the equality to zero of a reactive part of an input resistance (Х=0), has a high active input resistance (R). For illustration, in fig. 26, the dependences R(f) X(f) are shown for an antenna with geometrical sizes: L=0,5

The more convenient version for matching is the one, in which the exciting voltage is introduced into a bonding rupture. Such a version is shown conditionally in fig. 27. The place of the antenna excitation is marked by a big point. The dependences R(f) and X(f) are shown in fig. 28 a. In such an antenna, the choice of values L, H, d can be X=0, and R=50 Ohm or 75 Ohm. The graphs of fig. 28 correspond to the antenna tuning to a frequency of 1000 MHz in order to obtain R=50 Ohm, X=0 (L=142 mm; H=71,5 mm; d=8 mm). The symbols of R and X are the same as in fig. 26.

The numerical simulation shows that with increase of the number of dipoles in the antenna (N), the antenna band properties worsen. The VSWR minimum does not match to the most acceptable directional pattern in the E-plane. Fig. 29 shows the directional patterns at three frequencies of the band, in which VSWR<2. The frequency values, gain (G), R, X, VSWR are also given. The graphs of directional patterns in the H-plane give the values of nonuniformity of the directional pattern (Nu). The nonuniformity Nu decreases, if to direct the neighbouring stubs in the contrary directions (fig. 25b).

Antenna characteristics depend on a number of an excited stub. In fig. 27, the central stub is excited. If to displace an excitation point from the central stub, the maximum of a directional pattern in the E-plane declines from a normal to the Z axis. Fig. 29 illustrates these properties by the example of the antenna with N=6. When the frequency increases, the angle of deflection of the maximum of a directional pattern decreases. Fig. 30 shows the DP of the antenna with N=13 within a bandwidth, in which VSWR<2. The lower stub is excited, as in fig. 29. The band of matching frequencies depends little on the number of dipoles in the antenna, if the extreme stub is excited. For illustration, fig. 31 shows the dependence of VSVR(f) at different values of the number of dipoles.

When the number of dipoles increases, the misphasing between the stub that is excited and the extreme stubs increases. Therefore, when the number of dipoles increases, the antenna gain at first increases promptly, then slowly. The dependence of gain (G) on the number of dipoles is shown in fig. 32. At N=1, the antenna represents a half-wave dipole that has G=2 dB. The lower stub is excited in the antenna. It follows from fig. 32 that there is no sense to make the number of dipoles N> 5.

K1 – N=2; K2 – N=4; K3 – N=7; K4 – N=13

If to apply a system of reflectors or (and) directors to an antenna, it is possible to make a directional pattern sector-shaped in the H-plane. Fig. 33 shows such an antenna and its directional pattern.

A Franklin antenna and its modifications can be used as transmitting antennas in radio links.

## 4. Modelling of radiation and scattering characteristics of frame antennas

### 4.1. Loop antenna of closed and open arrays

Various types of loop antennas are used in radio communication, radiolocation, TV. Further in this chapter, two types of loop antennas are studied: on the basis of closed arrays with a lateral length

The polarization of the antenna is linear. The E-plane is the XY plane, the H-plane is the plane φ=const. The antenna characteristics depend on geometrical sizes of L, β, d, Le, Lh, Dh, H. The antenna is excited by voltage between points 1 and 2. In version (a), the resistance between points (input resistance of the antenna) can be made equal to 50 Ohm, in version (b) – an active part of the input resistance – 200-300 Ohm. Gain in the version (b) of the antenna is 3 dB bigger, than in the version (a) of the antenna. In versions (a) and (b), directors can be used to increase the gain, fig. 35.

In the literature, the methods of antenna excitation were studied, their key properties were studied briefly [Rothammels K, 1995], but there is no information about the main regularities. Further, these regularities are examined for version 34а only, because this version is generally used.

Fig. 36a shows the dependence of an active part of an input resistance (R) on an angle β, if a reactive part of the input resistance is equal to zero (

It follows from fig. 36а that, by selecting the values of L and β at preset Н, it is possible to make input resistance R preset and to meet the condition

The antenna band properties are illustrated by fig. 37, 38. The calculation has been made for the antennas calculated so that at a frequency of 300 MHz the input resistance was equal to

It follows from the results of the numerical simulation, that at a level of VSWR<2, the percentage bandwidth is

### 4.2. Interaction of loop antennas as part of antenna array

Interaction of loop of antennas as a part of an array results in changing of their characteristics. It shall be taken into consideration, when designing antenna arrays. Fig. 39 shows directional patterns in the Е and Н-planes of a loop antenna with an input resistance of 50 Ohm: isolated one (a); when taking into account its interaction with two neighbouring emitters – one on the left and one on the right (b); and taking into account its interaction with four neighbouring emitters – two on the left and two on the right. The values of input resistance, gain and parametre F/B are also shown in the figures. The calculations have been made for a linear array, in which emitters are located in the E-plane, at a frequency of 300 MHz (as an example). The distance between the neighbouring emitters is equal to 0,65λ. It follows from the results of simulation that the interaction influences an input resistance and F/B parametre insignificantly. The directional pattern and gain influence the interaction greatly, but differences in influence of various number of interacting loop antennas are insignificant (fig.39b and 39c). It means that when simulating multiple-unit arrays of loop antennas, it is possible to take into consideration interaction of each emitter only with its nearest emitters, and further to use the theorem of multiplication of directional patterns.

Fig. 40 illustrates difference in influence of interaction in a linear array and in an annular array. The figure shows directional patterns of a loop antenna with the same geometrical sizes without taking interaction into consideration (a), when taking interaction into consideration with two neighbouring emitters (b), and with four neighbouring emitters (c). The linear distance between the neighbouring emitters is equal to 0.65λ, the angle between axes of antennas is equal to 12°.

It follows from the comparison of fig. 39 and fig. 40 that interaction between loop antennas in an annular array is less than interaction in a linear array.

### 4.3. Linear antenna arrays with coherent excitation

To increase gain, it is possible to use linear arrays of loop antennas with series excitation. Fig. 41 shows the schema of connections of arrays among themselves at a lateral length of the array

It follows from the results of simulation that it is not reasonable to make Mр>4 in the antenna. For each M value, it is necessary to optimise the sizes of elements of arrays in order to obtain the preset input resistance.

In a linear array consisting of arrays with a lateral length

Fig. 43а shows the version of a linear array with a quasi-isotropic directional pattern in the E-plane. Such a pattern can be obtained, if to bend all the arrays along the Z axis at an angle

## 5. Conclusion

The simulation results, given in this chapter, complete the information about dipole and loop antennas that is available in the literature, and can be used to choice a type of an antenna according to specified requirements and after estimating its main characteristics. Numerical simulation of an antenna can be done with the use of the formulas given in the description of a mathematical model. When using a program developed on the basis of these formulas, it is necessary to carry out research into the convergence of the results of calculation. The most sensitive parametre of the antenna in relation to the number of segments of division of conductors (M) is the input resistance. When using impulse functions as the basis and weight ones, it is possible to be oriented towards the following recommendations: the ratio of a segment length to a conductor diameter

In some cases, the results given in the graphs can be used directly with the use of electrodynamic scaling.

We express aur gratitude to D.Moskalev and V.Kizimenko for the useful discussion of the results described in this chapter.