**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 =0, in YZ-plane =90°.

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 169

elements are selected so that the input resistance was equal to *Rinp* 50 Ohm, *Xinp* 0. The results have been obtained for the case *Zs* =0. Fig. 8 shows (as an example) a directional pattern in the E-plane at N=10. Fig. 9 shows the dependences of gain and «Front to Bback» ratio (F/B) on the reflector sizes in the Н-plane at a different distance between conductors of the reflector: *Dh* =0,05 λ = 50 mm and *Dh* =0,1 λ = 100 mm; fig. 10 shows the dependence of

It follows from the cited and other results of simulation that the antenna gain and input resistance depend little on reflector sizes. If *Re* changes in the interval (0,2 … 1)λ , the active input resistance varies in the interval (48,5 … 50) Ohm, and the reactive input resistance varies in the interval (0 …-1,7) Ohm. If the distance between the conductors of the reflector is *Dh* <0,1 λ, the antenna parametres hardly depend on *Dh* . The maximum of the F/B

0 5 10 15 20

N

G and F/B on the reflector size in the E-plane.

parametre corresponds to *R R e h* (0,6 … 0,7) λ.

dB

 G F/B

**Figure 7.** Gain and Front to Back ratio

**Figure 8.** Radiation pattern

**Figure 6.** Versions of reflectors

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 ( *D* ) of the relations F/B on the number of directors ( *N* ). The antenna reflector is linear. For each N value, sizes of the antenna elements are selected so that the input resistance was equal to *Rinp* 50 Ohm, *Xinp* 0. The results have been obtained for the case *Zs* =0. Fig. 8 shows (as an example) a directional pattern in the E-plane at N=10. Fig. 9 shows the dependences of gain and «Front to Bback» ratio (F/B) on the reflector sizes in the Н-plane at a different distance between conductors of the reflector: *Dh* =0,05 λ = 50 mm and *Dh* =0,1 λ = 100 mm; fig. 10 shows the dependence of G and F/B on the reflector size in the E-plane.

It follows from the cited and other results of simulation that the antenna gain and input resistance depend little on reflector sizes. If *Re* changes in the interval (0,2 … 1)λ , the active input resistance varies in the interval (48,5 … 50) Ohm, and the reactive input resistance varies in the interval (0 …-1,7) Ohm. If the distance between the conductors of the reflector is *Dh* <0,1 λ, the antenna parametres hardly depend on *Dh* . The maximum of the F/B parametre corresponds to *R R e h* (0,6 … 0,7) λ.

**Figure 7.** Gain and Front to Back ratio

168 Numerical Simulation – From Theory to Industry

=90°.

1

=0, in YZ-plane

**Figure 5.** Yagi-Uda antenna

**Figure 6.** Versions of reflectors

*3.1.1. Radiation characteristics* 

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

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

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

2 3

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

Fig. 7 shows the dependence of a directivity factor ( *D* ) of the relations F/B on the number of directors ( *N* ). The antenna reflector is linear. For each N value, sizes of the antenna

antenna elements are not shown because of the lack of space.

**Figure 8.** Radiation pattern

Segment namber 4020 1201008060 180160140 200

Segment number (b) *θ<sup>i</sup>*

=45°

Direction of wave

(b) *θ<sup>i</sup>* =45°

80 100 120 140 160 180 <sup>10</sup>

Betta, deg.

m

A

1

**Figure 11.** Dependence of G and F/B on angle β

Segment namber 4020 1201008060 180160140 200

Segment number (a) *θi*=0

15

20

dB

25

30

35

 G F/B

**Figure 12.** Current distribution

m

A

1

**Figure 13.** Scattering pattern of dipole

(a) *θ<sup>i</sup>* =0

Direction of wave

**Figure 9.** Dependence of Gain (G) and Front to Back ratio (F/B) on reflector size in H-plane

The graph in fig. 11 illustrates the influence of shape of a reflector on the antenna parametres. The dependences of *G* and F/B on an angle are shown for the reflector sizes *Re* = *Rh* = 0,6 λ = 600 mm, *Dh* =0,1 λ and N=10. It follows from fig. 11 that when an angle decreases, the F/B parametre worsens, and that the gain hardly depend on . When increases, the input resistance increases.

#### *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 <sup>2</sup> *RCSN RCS* / in the direction of a maximum of the scattering pattern.

**Figure 10.** Dependence of G and F/B on reflector size in E-plane

**Figure 11.** Dependence of G and F/B on angle β

Dh=100 mm

12.5 Dh=50 mm

**Figure 9.** Dependence of Gain (G) and Front to Back ratio (F/B) on reflector size in H-plane

decreases, the F/B parametre worsens, and that the gain hardly depend on

parametres. The dependences of *G* and F/B on an angle

200 300 400 500 600 700 800 11.7

Rh [mm[

The figures give the values of <sup>2</sup> *RCSN RCS* /

**Figure 10.** Dependence of G and F/B on reflector size in E-plane

dB

increases, the input resistance increases.

*3.1.2. Scattering characteristics* 

scattering pattern.

11.8 11.9 12.0 12.1 12.2 12.3 12.4

G [dB]

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

(a) Gain (b)Front to Back ratio

F/B [dB]

*Re* = *Rh* = 0,6 λ = 600 mm, *Dh* =0,1 λ and N=10. It follows from fig. 11 that when an angle

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.

 G F/B

500 600 700 800 900 1000 <sup>10</sup>

Re [mm]

are shown for the reflector sizes

200 300 400 500 600 700 800

Rh [mm]

 Dh=50 mm Dh=100 mm

in the direction of a maximum of the

. When **Figure 12.** Current distribution

**Figure 13.** Scattering pattern of dipole

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.

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 173

, the reactance *Xs* turns out to be equal to [Yurtsev, Runov, Kazarin

2

<sup>4</sup> 480 Ohm . *o o*

The more the ratio / *Ao o H* , the more *Xs* and the less the antenna resonance frequency. Fig. 17 shows the dependence *Rinp* and *Xinp* on frequency at two values of *Ao* for a dipole tuned

In case of decrease of *Ho* and *Ao const* , the resonance frequency and a frequency band according to a matching criterion decreases. Fig. 18 shows the dependence of a resonance

tuned to a frequency of 300 MHz. In case of decrease of *Ho* , the resonance frequency *<sup>o</sup>*

decreases, the band properties worsen, the F/B parametre decreases, the antenna gain G decreases. Fig. 19 shows the dependence of the percentage bandwidth / *<sup>o</sup> df f* , in which

*H*

*o*

*f* for a Yagi-Uda antenna with a linear reflector and one director, depending on

mm. The antenna of a smooth conductor with a radius *Ao* =3 mm was

 and

(37)

*f*

application of, instead of a smooth conductor, a conductor rolled into a helix with a radius

0 5 10 15 20

N

*Ao* , fig. 16. The surface resistance is the ratio of the tangential components of vectors *E*

of the helix field on a cylinder of the radius *Ao* , namely / *Z R iX E H ss stt* .

35 Rl=Rinp; Xl=-Xinp Rl=0; Xl=0

**Figure 15.** Dependence of RCSN on number of directors

RCSN

in resonance at a frequency of 300 MHz at *Xs* =0.

*s*

*A A <sup>X</sup>*

*H* 

**Figure 16.** Нelix

frequency *<sup>o</sup>*

*Ho* at 0,003 3 *Ao*

1974]:

For the case *Ao*

**Figure 14.** Radiation pattern (a) and scattering pattern (b, c, d)

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 \* *Z Z l inp* and for the short-circuit load ( 0 *Zl* ). It follows from fig. 15 that a structural component of RCS is much more bigger, than its antenna component.

#### *3.1.3. Dipole antennas with reduced dimensions*

In case of increase of a surface inductive reactance *Xs* , resonance frequency fo 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 *Ao* , fig. 16. The surface resistance is the ratio of the tangential components of vectors *E* and *H* of the helix field on a cylinder of the radius *Ao* , namely / *Z R iX E H ss stt* .

**Figure 15.** Dependence of RCSN on number of directors

#### **Figure 16.** Нelix

172 Numerical Simulation – From Theory to Industry

antenna with a linear reflector, frequency f=300 MHz, N=10.

**Figure 14.** Radiation pattern (a) and scattering pattern (b, c, d)

(c) Scattering pattern: *θ<sup>i</sup>*

*3.1.3. Dipole antennas with reduced dimensions* 

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 \* *Z Z l inp* and for the short-circuit load ( 0 *Zl* ). It follows from fig. 15 that a

Direction of wave Direction of wave

=45° (d) Scattering pattern: *θ<sup>i</sup>*

(a) Radiation pattern (b) Scattering pattern: *θ<sup>i</sup>*

=0

Direction of wave

=180°

In case of increase of a surface inductive reactance *Xs* , resonance frequency fo of an antenna decreases. The most practical version of implementation of the surface resistance is the

structural component of RCS is much more bigger, than its antenna component.

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

> For the case *Ao* , the reactance *Xs* turns out to be equal to [Yurtsev, Runov, Kazarin 1974]:

$$X\_s \approx 480\pi^4 \left(\frac{A\_o}{\mathcal{X}}\right) \left(\frac{A\_o}{H\_o}\right)^2 \left[\text{Ohm}\right].\tag{37}$$

The more the ratio / *Ao o H* , the more *Xs* and the less the antenna resonance frequency. Fig. 17 shows the dependence *Rinp* and *Xinp* on frequency at two values of *Ao* for a dipole tuned in resonance at a frequency of 300 MHz at *Xs* =0.

In case of decrease of *Ho* and *Ao const* , the resonance frequency and a frequency band according to a matching criterion decreases. Fig. 18 shows the dependence of a resonance frequency *<sup>o</sup> f* for a Yagi-Uda antenna with a linear reflector and one director, depending on *Ho* at 0,003 3 *Ao* mm. The antenna of a smooth conductor with a radius *Ao* =3 mm was tuned to a frequency of 300 MHz. In case of decrease of *Ho* , the resonance frequency *<sup>o</sup> f* decreases, the band properties worsen, the F/B parametre decreases, the antenna gain G decreases. Fig. 19 shows the dependence of the percentage bandwidth / *<sup>o</sup> df f* , in which

VSWR <2 (the antenna input resistance at a medium frequency *<sup>o</sup> f* is equal to *Rinp* =50 Ohm, *Xinp* =0, the wave resistance of a feeder is =50 Ohm).

1 2 *S II* 12 ( / ) , where 1*I* is the amplitude of an input current of an

, the interaction of the antennas is negligible. It means that it is sufficient to

. It ensures the absence of

, where *R* is

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

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

Fig. 20 shows directional patterns of the antenna located at the centre of a fragment of a plane array with the number of emitters *Nx* on the X axis, and *Ny* on the Y axis. The

diffraction maximums in the directional pattern at the phase scanning in a sector of 40°. All the emitters of the fragment, except for the central one, are loaded with matched loads. The numerical simulation shows that, with the increase of the number of the emitters in the fragment, the shape of the directional pattern in some angular sector tends to become rightangled. The degree of interaction of two emitters between themselves can be evaluated by

emitter in the radiation mode, 2*I* is the amplitude of an input current of a passive emitter

the distance between emitters. The calculations have been made for a Yagi-Uda antenna at the number of directors N=1 and N=5 with the location of antennas in the E-plane and the H-plane. It follows from the simulation results that 12 *S* depends little on the number of directors. If the antennas are located in the H-plane, the interaction is stronger. If the

take into consideration the interaction of a Yagi-Uda antenna as a part of a plane array only

(a) Nx = Ny = 1 (b) Nx=3, Ny=1 (c) Nx=5, Ny=1

loaded with a matched load. Fig. 22 shows the dependence 12 *S* on *R R* ' /

communication nodes.

Ohm at a medium frequency of 300 MHz.

the isolation ratio <sup>2</sup>

with an environment of two rings.

**Figure 20.** Radiation pattern in E-plane

distance is *R*

distances between the emitters on the X, Y axes are equal to 0,55

**Figure 17.** Dependence of input resistance on frequency at 0,01 *Ho* 

**Figure 18.** Resonance frequency

**Figure 19.** Percentage bandwidth

In case of change of *Ho* from 10 mm to 2.5 mm, G decreases from 7 dB to 5 dB, F/B decreases from 20 dB to 14 dB.

#### **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 *Nx* on the X axis, and *Ny* on the Y axis. The distances between the emitters on the X, Y axes are equal to 0,55 . It ensures the absence of diffraction maximums in the directional pattern at the phase scanning in a sector of 40°. All the emitters of the fragment, except for the central one, are loaded with matched loads. The numerical simulation shows that, with the increase of the number of the emitters in the fragment, the shape of the directional pattern in some angular sector tends to become rightangled. The degree of interaction of two emitters between themselves can be evaluated by the isolation ratio <sup>2</sup> 1 2 *S II* 12 ( / ) , where 1*I* is the amplitude of an input current of an emitter in the radiation mode, 2*I* is the amplitude of an input current of a passive emitter loaded with a matched load. Fig. 22 shows the dependence 12 *S* on *R R* ' / , where *R* is the distance between emitters. The calculations have been made for a Yagi-Uda antenna at the number of directors N=1 and N=5 with the location of antennas in the E-plane and the H-plane. It follows from the simulation results that 12 *S* depends little on the number of directors. If the antennas are located in the H-plane, the interaction is stronger. If the distance is *R* , the interaction of the antennas is negligible. It means that it is sufficient to take into consideration the interaction of a Yagi-Uda antenna as a part of a plane array only with an environment of two rings.

**Figure 20.** Radiation pattern in E-plane

174 Numerical Simulation – From Theory to Industry

**Figure 18.** Resonance frequency

**Figure 19.** Percentage bandwidth

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

from 20 dB to 14 dB.

*Xinp* =0, the wave resistance of a feeder is

VSWR <2 (the antenna input resistance at a medium frequency *<sup>o</sup>*

**Figure 17.** Dependence of input resistance on frequency at 0,01 *Ho*

9.5 10.0 10.5 11.0 11.5 12.0 12.5

df/fo [%]

fo [MHz]

=50 Ohm).

(a) A0=0 (b) A0=0 ,01*<sup>λ</sup>*

In case of change of *Ho* from 10 mm to 2.5 mm, G decreases from 7 dB to 5 dB, F/B decreases

2 4 6 8 10 9.0

Ho [mm]

2 4 6 8 10 <sup>200</sup>

Ho [mm]

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

*f* is equal to *Rinp* =50 Ohm,

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 177

. The angle between the axes of the neighboring antennas is equal

= 0,7 and two values of the number of directors N=1 and N=5.

, when an angle β increases, the isolation ratio decreases at first, then

0 50 100 150 200

Betta, deg.

 D'=0,7 D'=0,5

, when an angle β increases, the isolation ratio increases at first, then

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; P1 is projection of point Р on XY plane; R, θ,

(a)N=1 (b) N=5

S12, dB

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

to β =22°. In both cases, the middle antenna is excited, the remaining ones are loaded with

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

between emitters D=0,55

= 0,5 and D' =D/

D'=0,5

45 D'=0,7

1. If *D* (0,5 0,6)

2. If *D* (0,65 0,7)

matched loads.

D' =D/ 

S12, dB

**Figure 24.** Dependence of isolation ratio on angle between axes of antennas

It follows from the results of numerical simulation that

0 50 100 150 200 <sup>5</sup>

Betta, deg.

it decreases and reaches a minimum at β 90°.

it increases and reaches a maximum at β =110°-120°.

φ are the spherical coordinates of point P.

**3.3. Linear Antenna Array with Series Excitation** 

**Figure 21.** Radiation pattern in H-plane

**Figure 22.** Isolation ratio between two antennas

**Figure 23.** Directional pattern of director antenna as part of arc array

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 . The angle between the axes of the neighboring antennas is equal to β =22°. In both cases, the middle antenna is excited, the remaining ones are loaded with matched loads.

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/ = 0,5 and D' =D/ = 0,7 and two values of the number of directors N=1 and N=5.

**Figure 24.** Dependence of isolation ratio on angle between axes of antennas

It follows from the results of numerical simulation that

176 Numerical Simulation – From Theory to Industry

**Figure 21.** Radiation pattern in H-plane

S12 [dB]

(a) Nx=Ny=1 (b) Nx=1, Ny=3 (c) Nx=1, Ny=5

(a) N=1 (b) N=5

S12 [dB]

0.5 1.0 1.5 2.0 2.5 3.0 <sup>0</sup>

E-plane H-plane

R'

 E-plane H-plane

 E-plane H-plane

**Figure 22.** Isolation ratio between two antennas

0.5 1.0 1.5 2.0 2.5 3.0 <sup>0</sup>

R'

**Figure 23.** Directional pattern of director antenna as part of arc array

(a) N=3 (b) N=5


### **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; P1 is projection of point Р on XY plane; R, θ, φ are the spherical coordinates of point P.

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).

**Figure 26.** Dependences of input resistance on frequency

**Figure 28.** Input resistance and VSWR in bandwidth

800 1 000

f, MHz

f, MHz

800 1 000

VSWR

4

3

2

**Figure 27.** Antenna

Ohm

**Figure 25.** Fig. 25. Franklin antennas

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 *<sup>o</sup>* ; H=0,25 *<sup>o</sup>* ; d=0,1 *<sup>o</sup>* , where *<sup>o</sup>* is the medium wavelength of the given bandwidth. The calculations have been made for *<sup>o</sup>* =300 mm (medium frequency *<sup>o</sup> f* =1000 MHz) and two cases with the number of dipoles N=3 and N=5. It is evident that at N=3, there is a frequency, at which Х=0. At N 5, there is no such a frequency (Х <0). It complicates the antenna matching with a transmission line.

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).

**Figure 26.** Dependences of input resistance on frequency

178 Numerical Simulation – From Theory to Industry

B

**Figure 25.** Fig. 25. Franklin antennas

C

D

pattern in the E-plane.

; H=0,25 *<sup>o</sup>*

symbols of R and X are the same as in fig. 26.

bandwidth. The calculations have been made for *<sup>o</sup>*

; d=0,1 *<sup>o</sup>*

complicates the antenna matching with a transmission line.

, where *<sup>o</sup>*

sizes: L=0,5 *<sup>o</sup>*

The polarization of an antenna radiation field is linear: the E-plane is the plane φ=const, the

<sup>X</sup> <sup>Y</sup>

φ

(a) (b)

A Z H

θ

R

P

L

d

P1

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

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

MHz) and two cases with the number of dipoles N=3 and N=5. It is evident that at N=3, there is a frequency, at which Х=0. At N 5, there is no such a frequency (Х <0). It

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

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

is the medium wavelength of the given

*f* =1000

=300 mm (medium frequency *<sup>o</sup>*

H-plane is the plane XY. In the H-plane, the directional pattern is quasi-isotropic.

**Figure 28.** Input resistance and VSWR in bandwidth

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.

Numerical Simulations of Radiation and Scattering Characteristics of Dipole and LOOP Antennas 181

**Figure 30.** Nonsymmetric excitation of antenna

**Figure 32.** Dependence of VSWR on frequency K1 – N=2; K2 – N=4; K3 – N=7; K4 – N=13

VSWR

5

4

3

2

1

**Figure 31.** Directional patterns within bandwidth: N=13, the lower stub is excited.

К1 К2 К3 К4

Frequency, MHz 900 1 000 1 100

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.

**Figure 29.** Directional patterns in Е and Н-planes

**Figure 30.** Nonsymmetric excitation of antenna

make the number of dipoles N> 5.

**Figure 29.** Directional patterns in Е and Н-planes

VSVR(f) at different values of the number of dipoles.

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

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

**Figure 31.** Directional patterns within bandwidth: N=13, the lower stub is excited.

**Figure 32.** Dependence of VSWR on frequency K1 – N=2; K2 – N=4; K3 – N=7; K4 – N=13

**Figure 34.** Franklin antenna with linear reflectors and directors

Z

θ

1 2 1 2

φ

X

P

Le

Dh

Lh

Y

(a) (b) (c)

**Figure 35.** Loop antennas

β

d

L

**Figure 36.** Loop antennas with directors

**Figure 33.** Dependence of Gain on number of dipoles

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.
