5. Fractal array antennas

An array antenna is one of the best solutions for the long-range communication systems rather than aperture antennas. The multiplicity of antenna elements allows more particular control of the radiation pattern, thus resulting in lower side lobe level and high directive scanned beams. Due to these fundamental properties, array antennas play a vital role in military, defence and other space applications. Owing to novel insights into array antenna parameters like low side lobe level with narrow beams and wider side lobe level angles, ultrawideband, multibeams, feasible and simple design methodologies and algorithms of fractal array antennas, usage of these arrays increases quite commonly in the antenna literature from the past two to three decades. Due to these properties, fractal array antennas find applications in celestial and other advanced communication systems.

Random and deterministic fractal array antennas are the two basic types of fractal arrays based on their geometric construction. Again, deterministic fractal array antennas are also divided into three types based on their geometric patterns [33–36]:


Like Sierpinski fractal gasket antennas, Sierpinski fractal carpet structures have also been used in the designing of antenna elements [19]. One of the serious setbacks with a small loop antenna is that the input resistance is very small, building it hard to couple power to the antenna. By using a fractal loop, the input impedance of the antenna increases. Koch Island, Minkowski and hexagonal geometry with triangular loop are the best examples for the fractal

Fractal geometric technology was also introduced in microstrip patch antennas instead of conventional rectangular, circular and square geometries, and this leads to improved gain of those antennas with multiband and ultrawideband behaviour [25]. The analogous insight of raising the electrical length of a radiator can be applied to a patch antenna [26]. The patch antenna can be analysed as a 'microstrip transmission line'. So, if the current will be forced to pass through the convoluted path of a fractal structure rather than a conventional Euclidean pathway, the area needed to engage the resonant transmission line will be reduced. This method has been applied to patch antennas in a range of forms [27–29]. Recently, novel patterns of fractal antennas are projected for miniaturization applications, and miniaturized

An array antenna is one of the best solutions for the long-range communication systems rather than aperture antennas. The multiplicity of antenna elements allows more particular control of the radiation pattern, thus resulting in lower side lobe level and high directive scanned beams. Due to these fundamental properties, array antennas play a vital role in military, defence and other space applications. Owing to novel insights into array antenna parameters like low side lobe level with narrow beams and wider side lobe level angles, ultrawideband, multibeams,

Giuseppe Peano microstrip patch is shown Figure 9 [30–32].

Figure 9. Miniaturized Giuseppe Peano microstrip patch antenna [32].

20 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

loop antennas [20–24].

5. Fractal array antennas

3. Conformal (3D) fractal array antennas

This chapter focused on the design methodology of linear and planar deterministic fractal array antennas using concentric elliptical ring sub-array design methodology. In this process of design, the behaviour of fractal nature should apply to the regular concentric elliptical antenna array. This recursive process will produce self-similar concentric elliptical geometry as depicted in Figure 10. It is clear from the definition of fractal that the same shape repeats again and again; in this manner geometric structure considered here also repeats again and again. The array antennas proposed in this work can be defined as arrays of arrays, which means that the original counterpart of the array antennas repeats again and again. The general array factor of fractal nature is defined in Eq. (1), which is based on the definition of self-similar nature. The equation for the fractal array factor is the product of generating sub-array factor [37–38]:

$$A.F\_p(A.F(\theta,\varphi)) = \prod\_{p=1}^p \mathcal{GSA}\left(\mathcal{S}^{p-1}(A.F(\theta,\varphi))\right) \tag{1}$$

where GSA and A.F stand for generating sub-array and array factors, respectively. The array factor of concentric elliptical ring sub-array geometric generator for the design of linear and planar deterministic fractal array antennas is given in Eq. (2):

$$A.F\_P(\boldsymbol{\Theta}, \boldsymbol{\varphi}) = \prod\_{p=1}^{P} \left[ \sum\_{m=1}^{M} \sum\_{n=1}^{N} I\_{mn} e^{\boldsymbol{k} \cdot \boldsymbol{S}^{p-1} \boldsymbol{\Psi}\_{mn}} \right] \tag{2}$$

$$\begin{aligned} \psi\_{mn} &= \begin{pmatrix} a\cos\varphi\_{mn}\cos\varphi + b\sin\varphi\_{mn}\sin\varphi \end{pmatrix} \sin\theta\\ &- \begin{pmatrix} a\cos\varphi\_{mn}\cos\varphi\_0 + b\sin\varphi\_{mn}\sin\varphi\_0 \end{pmatrix} \sin\theta\_0 \end{aligned} \tag{3}$$

$$
\varphi\_{mn} = \frac{2\pi}{N}(mn - 1)\tag{4}
$$

where S is the scaling factor and two is the scaling factor of the considered sub-array; P is the iterations and four successive iterations have considered in this work, and it can be extended up to infinite iterations; M is the number of concentric rings and here only one concentric has been considered; N is the number of antenna elements and a number of antenna elements are depending on the iterations; k is the wave equation, Imn, uniform current amplitudes; φmn is the position of the antenna element in x-y region; and θ<sup>0</sup> and φ<sup>0</sup> are steering angles. This geometric technique replicates the concentric circular ring sub-array design methodology, but in this case, the circular generator is replaced with the elliptical generator as depicted in Figure 10. The proposed methodology permits choice in broadening the shape of a radiation beam or for designing multiple beams for any deterministic 1D and 2D fractal array antennas without entailing any amplitude variation and with less power constrain. The triangular fractal array antenna of expansion factor of 2 and four successive iterations have been designed by concentric elliptical ring sub-array design methodology which is observed in Figure 11, and

the corresponding design equation is described in Eq. (4). The basic triangular array starts with three elements and expands up to four iterations in this chapter. Same distance (d = λ/2) between the antenna elements maintained for both expansion factors (S) of 1 and 2. Nearly one third of the antenna elements can be thinned in the first iterations due to recursive nature of the proposed methodology. Figure 12 depicts array factor of proposed triangular fractal array antenna. Wide side lobe level angles of 53.1�, 55.4�, 54.1� and 55.8� are observed in four successive iterations with a proper balance between the remaining array factor properties:

Figure 12. Array factors of triangular fractal planar array antennas generated by concentric elliptical sub-array method-

<sup>A</sup>:FPð Þ¼ <sup>θ</sup>; <sup>φ</sup> <sup>Y</sup>

ology for S = 1 and up to four iterations [37].

Figure 11. The first four iterations of linear fractal array antenna for an expansion factor of 1 [37].

4

X 1

X 3

Imne

jkSP�1Ψmn " # (5)

Fractal Array Antennas and Applications http://dx.doi.org/10.5772/intechopen.74729 23

n¼1

m¼1

p¼1

Figure 10. Concentric elliptical sub-array geometric generators for (a) stage 1 and (b) stage 2 [38].

Figure 11. The first four iterations of linear fractal array antenna for an expansion factor of 1 [37].

depending on the iterations; k is the wave equation, Imn, uniform current amplitudes; φmn is the position of the antenna element in x-y region; and θ<sup>0</sup> and φ<sup>0</sup> are steering angles. This geometric technique replicates the concentric circular ring sub-array design methodology, but in this case, the circular generator is replaced with the elliptical generator as depicted in Figure 10. The proposed methodology permits choice in broadening the shape of a radiation beam or for designing multiple beams for any deterministic 1D and 2D fractal array antennas without entailing any amplitude variation and with less power constrain. The triangular fractal array antenna of expansion factor of 2 and four successive iterations have been designed by concentric elliptical ring sub-array design methodology which is observed in Figure 11, and

22 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

Figure 10. Concentric elliptical sub-array geometric generators for (a) stage 1 and (b) stage 2 [38].

Figure 12. Array factors of triangular fractal planar array antennas generated by concentric elliptical sub-array methodology for S = 1 and up to four iterations [37].

the corresponding design equation is described in Eq. (4). The basic triangular array starts with three elements and expands up to four iterations in this chapter. Same distance (d = λ/2) between the antenna elements maintained for both expansion factors (S) of 1 and 2. Nearly one third of the antenna elements can be thinned in the first iterations due to recursive nature of the proposed methodology. Figure 12 depicts array factor of proposed triangular fractal array antenna. Wide side lobe level angles of 53.1�, 55.4�, 54.1� and 55.8� are observed in four successive iterations with a proper balance between the remaining array factor properties:

$$A.F\_P(\theta,\varphi) = \prod\_{p=1}^4 \left[ \sum\_{m=1}^3 \sum\_{n=1}^3 I\_{mn} e^{i\mathbf{k}S^{P-1}\Psi\_{mn}} \right] \tag{5}$$

### 6. Conclusions

Fractals are self-similar structures. Various fields of science and technology have inspired by these self-similar structures to develop easy and reliable systems. Application of fractal concepts to the antenna engineering leads to new insights into the antenna parameters. Any polygon-shaped fractal array can be constructed using concentric elliptical ring sub-array design methodology. This design methodology and fractal array antennas generated by this methodology can be helpful for the generation of multiple beams with different array factor properties using a single fractal array antenna without any hardware complexity.

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Fractal Array Antennas and Applications http://dx.doi.org/10.5772/intechopen.74729 25

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### Author details

V. A. Sankar Ponnapalli<sup>1</sup> \* and P. V. Y. Jayasree<sup>2</sup>

\*Address all correspondence to: vadityasankar3@gmail.com

1 Department of Electronics and Communication Engineering, Sreyas Institute of Engineering and Technology, Hyderabad, India

2 Department of Electronics and Communication Engineering, GITAM (Deemed to be University), Visakhapatnam, India

#### References


6. Conclusions

Author details

References

V. A. Sankar Ponnapalli<sup>1</sup>

and Technology, Hyderabad, India

University), Visakhapatnam, India

York: Springer-Verlag; 1992

ogy and Economics; 2013

Wiley; 1990

Fractals are self-similar structures. Various fields of science and technology have inspired by these self-similar structures to develop easy and reliable systems. Application of fractal concepts to the antenna engineering leads to new insights into the antenna parameters. Any polygon-shaped fractal array can be constructed using concentric elliptical ring sub-array design methodology. This design methodology and fractal array antennas generated by this methodology can be helpful for the generation of multiple beams with different array factor

1 Department of Electronics and Communication Engineering, Sreyas Institute of Engineering

2 Department of Electronics and Communication Engineering, GITAM (Deemed to be

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properties using a single fractal array antenna without any hardware complexity.

\* and P. V. Y. Jayasree<sup>2</sup>

24 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

\*Address all correspondence to: vadityasankar3@gmail.com


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**Chapter 3**

Provisional chapter

**Resonant Systems for Measurement of**

Kuzmichev Igor K. and Popkov Aleksey Yu.

Additional information is available at the end of the chapter

Kuzmichev Igor K. and Popkov Aleksey Yu.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73643

waveguide, oversized waveguide

**Frequencies**

Abstract

1. Introduction

**Electromagnetic Properties of Substances at V-Band**

Properties of Substances at V-Band Frequencies

Resonant Systems for Measurement of Electromagnetic

DOI: 10.5772/intechopen.73643

Hemispherical open resonator (OR) with the segment of the oversized circular waveguide is considered. The cavity is formed by cylindrical, conical and spherical surfaces, and only axial-symmetric modes are excited. The power and spectral characteristics of such cavity with a dielectric bead and a rod have been studied. The quasi-periodic behavior of those dependencies was found out. Their qualitative agreement with similar dependencies for a cylindrical cavity is shown. It was found that physical processes in the cavity and in the hemispherical ОR with the segment of the oversized circular waveguide are identical. Dielectric permittivity and loss-angle tangent measurements have been carried out in millimeter wavelength range for the Teflon and Plexiglas samples having the shape of the bead as well as for the fused quartz and glass samples of the rod shape. It was found out that such a resonant system allows measuring samples with high losses, that is, especially important for quality control of food stuffs and analysis of biological liquids. Energy analysis of the ОR with the segment of the oversized rectangular waveguide has been performed. Basic possibility to apply such a resonant system for measurement of the dielectric permittivity of cylindrical samples with high losses has been shown as well.

Keywords: permittivity, cavity, open resonator, circular waveguide, rectangular

Measurement of electromagnetic properties of existing and novel materials in new frequency ranges is an important issue of the day. Recently emerged new class of artificial materials composite materials (composites) are characterized by the negative refraction coefficient. Investigations showed that devices based on such materials can possess entirely unique

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.


## **Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies** Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

DOI: 10.5772/intechopen.73643

Kuzmichev Igor K. and Popkov Aleksey Yu. Kuzmichev Igor K. and Popkov Aleksey Yu. Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73643

#### Abstract

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Hemispherical open resonator (OR) with the segment of the oversized circular waveguide is considered. The cavity is formed by cylindrical, conical and spherical surfaces, and only axial-symmetric modes are excited. The power and spectral characteristics of such cavity with a dielectric bead and a rod have been studied. The quasi-periodic behavior of those dependencies was found out. Their qualitative agreement with similar dependencies for a cylindrical cavity is shown. It was found that physical processes in the cavity and in the hemispherical ОR with the segment of the oversized circular waveguide are identical. Dielectric permittivity and loss-angle tangent measurements have been carried out in millimeter wavelength range for the Teflon and Plexiglas samples having the shape of the bead as well as for the fused quartz and glass samples of the rod shape. It was found out that such a resonant system allows measuring samples with high losses, that is, especially important for quality control of food stuffs and analysis of biological liquids. Energy analysis of the ОR with the segment of the oversized rectangular waveguide has been performed. Basic possibility to apply such a resonant system for measurement of the dielectric permittivity of cylindrical samples with high losses has been shown as well.

Keywords: permittivity, cavity, open resonator, circular waveguide, rectangular waveguide, oversized waveguide
