Circularly Polarized Antenna Technology 9783110562804, 9783110561180

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Yufeng Wang Circularly Polarized Antenna Technology

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Yufeng Wang

Circularly Polarized Antenna Technology

Author Yufeng Wang Room 103 & 105, Building 1, 79 Jingshui Road, Nanhu District, Jiaxing 314000, Zhejiang Province, China Email: [email protected]

ISBN 978-3-11-056118-0 e-ISBN (PDF) 978-3-11-056280-4 e-ISBN (EPUB) 978-3-11-056156-2 Library of Congress Control Number: 2020943428 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 National Defense Industry Press and Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Driven by the science and technology revolution, technologies such as wireless communication, computer, deep-space exploration and bioengineering are developing at an unprecedented speed. Above all, the rise of wireless communication technology not only has a very profound impact on science and technology, but also accelerates a series of revolutions in the realm of politics, economics and culture. More importantly, it enables our communication to be free from space restraint, makes our vision extended to the vast universe, makes economic globalization possible and greatly boosts informatization and networking in the military domain, a key enabler for Network-Centric Warfare. Needless to say, wireless communication technology is infiltrating at an unstoppable speed into every aspect and every corner of the world. Wireless communication makes use of the characteristics of electromagnetic waves propagating in space freely to exchange information remotely. The antenna is designed as the portal of the wireless communication, whose task is converting high-frequency current to electromagnetic waves, and vise versa; thus, different electromagnetic waves can be transmitted or received by antenna. The antenna performance has a direct impact on the efficiency of the wireless communication system. Therefore，the development of antenna technology plays a critical role in the development of wireless communication. As we all know, an electromagnetic wave radiated by any antenna is an elliptical polarization wave, and its extreme case is a linear polarized wave or a circular polarized wave. However, linear polarized wave is vulnerable to many factors, such as climate, environment and carrier movement, which would cause polarization deflection loss or even failure; and although circular polarized wave suffers less polarization deflection loss, it will suffer polarization reversal when it encounters a reflector. Therefore, in the field of wireless communication, especially satellite communication, circular polarized wave is usually used to counter electromagnetic wave interference caused by many factors, such as rain, fog, ionosphere and multipath effects. Therefore, the circularly polarized antenna is a very important member in the antenna family, and it is widely used in radar, remote sensing, communication, military application and other aspects. However, circularly polarized antennae of wireless communication systems are facing more and more stringent requirements, and there are challenges due to miniaturization, high gain and their wideband nature. Fortunately, scholars worldwide are working tirelessly in the field of circularly polarized antennae and have developed a series of new circularly polarized antenna technologies, including dual-band, dual-circular polarization antenna technology, wideband circular polarization antenna technology, wide beam circular polarization antenna array technology, and so on. There are various literatures on new technologies for circularly polarized antennae. However, they are not general solutions for all kinds circularly polarized antennae; instead there are key technologies for different problems. Therefore, these new technologies need to be categorized so as to lay a solid foundation for the development of circularly polarized antenna https://doi.org/10.1515/9783110562804-202

VI

Preface

technology, and that is why this book Circularly Polarized Antenna Technology is compiled by the 36th Research Institute of China Electronic Technology Group Corporation. The authors of this book are senior engineers who have been working in the field of antennae for years, and they are very experienced in both theory and practical design. This book is very clear in context, integrating theory with practice and is easy to understand; therefore, it is ideal for engineers, college senior undergraduates, graduate students and young teachers engaged in circularly polarized antenna research. I firmly believe this book will be of great help to circular polarization antenna designers and will be welcomed and applauded by readers. Xiaoniu Yang, Academician of Chinese Academy of Engineering

Contents Preface

V

Introduction

1

Chapter 1 Parameters of antennae 4 1.1 Radiation power density and radiation intensity 1.1.1 Radiation power density 4 1.1.2 Radiation intensity 5 1.2 Directivity and gain 6 1.2.1 Directivity 6 1.2.2 Gain 6 1.3 Beam solid angle 7 1.4 Antenna field zones 7 1.5 Radiation pattern 9 1.5.1 Description of antenna pattern 10 1.5.2 HPBW 11 1.5.3 FNBW 12 1.5.4 Directivity estimation 12 1.5.5 Sidelobe level 12 1.5.6 The difference among the decibels 13 1.6 Effective length and effective area 14 1.6.1 Effective length 14 1.6.2 Effective area 15 1.7 Antenna impedance 16 1.7.1 Input impedance 16 1.7.2 Voltage standing wave ratio 17 1.8 Bandwidth 19 1.9 Polarization of the antenna 19 1.10 Equivalent isotropically radiated power 20 1.11 Friis transmission equation 21 1.12 Receiving antenna 23 1.12.1 Receiver voltage 23 1.12.2 Antenna factor 23 References 25 Chapter 2 Polarization theory 26 2.1 Polarization characteristics of the wave 26 2.1.1 Plane electromagnetic wave electric field vector

4

26

VIII

2.1.2 2.1.3 2.1.4 2.2 2.3 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 2.6 2.6.1 2.6.2 2.7 2.7.1 2.7.2 2.7.3 2.8

Contents

Linear polarization 27 Circular polarization 28 Elliptically polarization wave 29 Polarization ellipse dip angle 30 Axial ratio 31 Polarization ratio 32 Linear polarization ratio 32 Circular polarization ratio 33 Circular polarization component synthesis 34 Elliptic equation derivation 34 Synthesis of orthogonal circular polarization waves Polarization loss 37 Polarization ratio and polarization efficiency 38 Axial ratio and polarization efficiency 39 Cross-polarization 41 Cross-polarization identification 42 Cross-polarization isolation 44 Cross-polarization calculation of antenna 44 Circular polarization antenna 45 References 48

36

Chapter 3 Crossed dipole circularly polarized antenna 49 3.1 Crossed dipole antenna principle 49 3.1.1 Dipole antenna 49 3.1.2 Crossed dipole antenna 51 3.2 Balun-fed self-phased crossed dipole antenna 53 3.2.1 Balun feed structure 53 3.2.1.1 λ/4 choke sleeve 54 3.2.1.2 Split coax balun 54 3.2.1.3 Folded balun 55 3.2.1.4 Half-wavelength balun 56 3.2.1.5 Coaxial taper balun 57 3.2.1.6 Natural balun 58 3.2.2 Gain and beam width control methods 58 3.2.2.1 Crossed dipole antenna radiation characteristics 58 3.2.2.2 Effects on spacing of the dipoles and reflector on gain and beam width 59 3.2.2.3 Declinate arms for beam widening 60 3.2.3 Design example of self-phase-shifted crossed dipole antenna 63 3.2.3.1 Antenna structure 63

Contents

3.2.3.2 3.2.3.3 3.3 3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2

Effect of bending angle on antenna performance 64 Results 69 Printed self-phased crossed dipole antenna 70 Design process of compact printed crossed dipole antenna 70 Antenna parameter analysis 75 Antenna measurement and discussion 76 Broadband crossed dipole CP antenna 78 Broadband principle of CP antenna with crossed dipole 79 Design example of broadband crossed dipole CP antenna 79 References 84

Chapter 4 Circularly polarized microstrip antenna 87 4.1 Principle and solutions of microstrip CP antenna 88 4.1.1 Single probe feed 90 4.1.2 Microstrip line feed 91 4.1.3 Aperture coupling 92 4.1.4 Microstrip proximity coupling 93 4.1.5 Multifed network 94 4.1.5.1 3 dB bridge 94 4.1.5.2 Phase-shifted power divider 94 4.2 Single probe-fed circularly polarized MSA 96 4.2.1 A single probe-fed square patch with truncated corners MSA 96 4.2.2 A single probe-fed slot-loaded MSA 98 4.2.3 A single probe-fed slot-loaded dual-band MSA 102 4.2.4 A single probe-fed stub loaded MSA 106 4.3 Single-probe feed-stacked CP MSA 109 4.3.1 Dual-layer square patch CP antenna with truncated corners 111 4.3.1.1 Broadband stacked square patch CP antenna with truncated corners 111 4.3.1.2 Dual-band stacked square corner cut CP patch antenna 115 4.3.2 Stacked quasi-square patch CP antenna 118 4.3.3 Stacked circular slot CP patch antenna 120 4.4 Microstrip-fed CP MSA 125 4.4.1 Microstrip-fed CP MSA 125 4.4.2 Microstrip proximity coupled CP MSA 129 4.4.3 CPW-fed CP MSA 134 4.4.4 Aperture-coupled CP antenna 137 4.5 Multifed CP MSA 142 4.5.1 A broadband CP H-sharped aperture-coupled MSA 142 4.5.2 Multiple L-shaped probe-fed CP MSA 145

IX

X

4.5.2.1 4.5.2.2

Contents

CP MSA with different feeds 145 Engineering realization of quad-feed CP MSA References 156

Chapter 5 Helix antenna 159 5.1 Basic structure of helix antenna 159 5.2 Axial-mode helix antenna 161 5.2.1 Principle of CP 161 5.2.2 Phase difference 162 5.2.3 Radiation pattern 163 5.2.4 Antenna design 163 5.2.5 Main parameters analysis 170 5.2.5.1 The effects of the pitch angle with fixed length 5.2.5.2 The effects of helix diameter with fixed length 5.2.5.3 The effects of the number of turns 172 5.3 Modified axial-mode helix antenna 173 5.3.1 Tapered-end helix antenna 175 5.3.2 Conical helix antenna 176 5.3.3 Nonuniform helix antenna 177 5.4 Helix antenna with reflective cavity 178 5.5 Normal mode helix antenna 183 5.6 Multiarm helix antenna 187 References 191

151

170 171

Chapter 6 Quadrifilar helix antennas 194 6.1 Principle of quadrifilar helix antenna 194 6.1.1 Structure of the quadrifilar helix antenna 194 6.1.2 Radiation field of QHA 197 6.2 Parameters analysis of the QHA 198 6.2.1 The effect of axis length and number of turns on the performance of the QHA 199 6.2.1.1 The effect of axial length and number of turns on 3 dB beamwidth 199 6.2.1.2 The effect of axial length and number of turns on axial ratio 199 6.2.1.3 The effect of axial length and number of turns on front-to-back ratio 200 6.2.2 The effect of rotation angle on the performance of the QHA 201 6.2.2.1 The effect of rotation angle on impedance 202 6.2.2.2 The effect of rotation angle on gain, front-to-back ratio and beamwidth 203

XI

Contents

6.2.2.3 6.2.3 6.3 6.3.1 6.3.1.1 6.3.1.2 6.3.1.3 6.3.1.4 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.4 6.4.1 6.4.1.1 6.4.1.2 6.4.1.3 6.4.2 6.4.3

The effect of rotation angle on axial ratio 204 The effect of radius and pitch ratio on the pattern of the QHA Realization of the QHA 204 Self-phase-shifting QHA 205 Self-phase-shifting structure 206 Balanced feed structure 206 Self-phase-shifting QHA with infinite barron feed 207 Self-phase-shifting QHA with slotted balun feed 208 QHA with phase-shifting network feed 211 Phase-shifting network 211 Sense of rotation of QHA circular polarization 214 Design example 214 Research direction of QHA 216 Miniaturization of QHA 216 Meandering method 217 Folding method 219 Dielectric loading method 221 Wideband QHA 223 Dual-band QHA 224 References 227

204

Chapter 7 Circularly polarized frequency-independent antenna 229 7.1 The principle of frequency-independent antenna 230 7.2 Plane Archimedes spiral antenna 231 7.2.1 Antenna structure 231 7.2.2 Electrical characteristics 233 7.2.3 Design examples 236 7.2.3.1 A broadband low-profile Archimedes plane spiral antenna 236 7.2.3.2 Miniature planar helix antenna loading curved sawtooth 241 7.2.3.3 Ultra-wideband low-profile hybrid back-cavity plane spiral antenna 244 7.3 Planar equiangular spiral antenna 247 7.3.1 Antenna structure 247 7.3.2 Electrical properties 249 7.3.3 Design examples 251 7.3.3.1 Annular equiangular spiral filled with edge-absorbing material 251 7.3.3.2 A planar equiangular helix antenna with high power capacity 254 7.4 Conical logistic-spiral antenna 255 7.4.1 Antenna structure 255 7.4.2 Electrical properties 256

XII

7.4.2.1 7.4.2.2 7.4.2.3 7.4.2.4 7.4.3 7.5 7.5.1 7.5.1.1 7.5.1.2 7.5.2 7.5.2.1 7.5.2.2

Contents

Impedance 256 Working bandwidth 257 Polarization 257 Directivity 258 Design examples 259 Other types of frequency-independent circular polarization antenna 263 Log-periodic CP antenna 264 Basic structure of log-periodic antenna 264 Example of log-periodic CP antenna 266 Sinusoidal circularly polarized antenna 268 Basic structure of sinusoidal circularly polarized antenna 268 Examples of sinusoidal circularly polarized antennas 271 References 273

Chapter 8 Circularly polarized horn antennas 275 8.1 Circular polarizer types 275 8.1.1 Screw circular polarizer 276 8.1.2 Diaphragm circular polarizer 276 8.1.3 Baffle circular polarizer 277 8.1.4 Dielectric plate circular polarizer 279 8.1.5 Resonant cavity circular polarizer 280 8.2 Common horn antenna 281 8.2.1 Dual-mode conical horn antenna 281 8.2.2 Multimode conical horn antenna 282 8.2.3 Corrugated horn antenna 283 8.2.3.1 Small flare angle horn 283 8.2.3.2 Large flare angle horn 284 8.2.3.3 Axial groove corrugated horn 285 8.3 Circularly polarized horn antenna design 285 8.3.1 Design of baffle circular polarizer 286 8.3.2 Design of dual-mode conical horn antenna 290 8.3.3 Simulation results of the dual circularly polarized dual-mode conical horn antenna 292 References 294 Chapter 9 Omnidirectional circularly polarized antenna 296 9.1 OCP antennas using circular arrays of directional circularly polarized antenna elements 297

Contents

9.1.1 9.1.2 9.2 9.2.1 9.2.2 9.2.3 9.3 9.3.1 9.3.2

A conformal omnidirectional circularly polarized antenna using patch antennas 297 A broadband OCP antenna 298 OCP antennas using circular arrays of omnidirectional antenna elements with linear polarizations 299 OCP antenna using slant or bended dipoles 301 OCP antenna with helix antennas 305 OCP antenna utilizing bended monopoles 309 Other forms of OCP antennas 313 OCP antennas based on zero-order resonant antennas 313 OCP antennas based on omnidirectional cone antennas fed by CP waves 315 References 317

Chapter 10 Circularly polarized radial line array antenna 319 10.1 Structure and basic principle of radial antenna 319 10.1.1 Structure of radial line antenna 319 10.1.2 Radial transmission mode 320 10.1.3 Concentric ring array theory 322 10.2 Design of circularly polarized radial line array antenna 324 10.2.1 Antenna geometry 324 10.2.2 Relationship between the slow wave structure and the waveguide structure 325 10.2.3 Aperture field distribution 327 10.2.4 Derivation of position alignment of gap 328 10.2.5 Realization of dual-circular polarization radial slot array antenna 330 10.2.6 Circularly polarized radial line helical array antenna 336 References 337 Index

339

XIII

Introduction With the development and the wide application of satellite communication, remote telemetry, radar, electronic warfare and other technologies, circularly polarized antennae are increasingly used. Circular polarization antenna has many advantages over linear polarization antenna. In satellite communication, it can eliminate the polarization distortion loss caused by the Faraday rotation effect of ionosphere. It can also reduce the signal loss in communication, remote sensing and radar. It can counter interference caused by cloud and rain when used in radar. In the field of electronic warfare, circularly polarized antenna can be used to detect and jam various linear polarization and elliptical polarization signals. It can also be used to receive signal when the host platform is subject to severe swing or rolling. In brief, circularly polarized antenna is widely used in satellite communication, remote telemetry, radar, electronic warfare, and so on. In China, the earliest literature specialized on circular polarization antenna theory and technology can be traced back to 1986. It is titled The Circularly Polarized Antenna, which was coauthored by Lin Changlu and Song Ximing. In recent years, the published books on antenna also have some chapters on circularly polarized antenna but the content is brief. Since the 1980s, China has made rapid development in many fields, such as satellite communication, telemetry, radar and electronic warfare and these systems have found application in land, sea, air and space. Antenna, as the sensor of electromagnetic waves, must be mounted on the outermost surface of the platform. The antennae differ based on platforms, installation site and application scenarios. Moreover, the requirements for antenna may also vary depending on the different application fields, application requirements and scenarios. The circular polarization antenna has the advantage of being very adaptive. This feature greatly accelerates the development of circularly polarized antenna technology. The purpose of this book is to introduce the various circularly polarized antennae that the editorial team have developed in recent years or are familiar with, and they do their best to give readers a comprehensive and in-depth explanation of the modern circularly polarized antenna technology. The book is written by senior engineers who have many years of experience in the research and development of circularly polarized antenna. They have integrated their experience with the published literature in both China and abroad on the latest technology development in this field. This book systematically explains the circularly polarized antenna, starting with the basic concepts, and mainly focuses on the principle, characteristics and design methods of various circularly polarized antennae. It covers various types of circularly polarized antennae, including cross-dipole antenna, circularly polarized microstrip antenna, helical antenna, quadrifilar helix antenna, circularly polarized frequency independent antenna, circular horn antenna, omnidirectional circularly https://doi.org/10.1515/9783110562804-001

2

Introduction

polarized antenna, circular polarization radial array antenna. The contents of each chapter are as follows: Chapter 1 outlines the basic theory of antenna, the concepts associated with antennae and the relevant definitions and formulas. Chapter 2 elaborates the theory of circularly polarized antenna, and focuses on the relevant concepts of circularly polarized antenna and the commonly used formulas after introducing the concept of circularly polarized electromagnetic wave. Chapter 3 introduces the principle of crossed dipole circularly polarized antenna and gives two examples, that is, balun feed antenna and printed antenna. Finally, a wideband circularly polarized antenna with wideband ratio of 3: 1 is described according to the requirements of broadband. Chapter 4 introduces the principle of microstrip circularly polarized antenna and its realization. Since the microstrip antenna has many types of circular polarization, its classification is mainly by the ways of realization and the chapter gives design examples of different types of realization, including single feed probe, microstrip line feed and multifeed network feed. Chapter 5 presents the principle of helical antenna and focuses on the realization of axial mode helical antenna. It analyzes the impact of each parameter on antenna performance and gives a better way to improve the axial ratio. The chapter discusses spiral antenna with reflection cavity and gives an example of sidelobe suppression. The multiarm spiral antenna is also elaborated and simulation examples of adjusting parameters of dual-arm spiral antenna to realize different patterns are given. Chapter 6 elaborates the principle and implementation of the quadrifilar helical antenna. First, the structure and characteristics of the radiation field are introduced. Then the impact of main parameters on performance is analyzed. The realization method and examples of self-phase shifting and network phase-fed helical antenna are also given. In keeping with the hot topic in the field of quadrifilar helix antenna, the research status of miniaturization, wideband and dual-frequency point is discussed. Chapter 7 covers nonfrequency-varying circularly polarized antenna and mainly focuses on planar Archimedes spiral antenna, planar logarithmic spiral antenna, cone logarithmic spiral antenna, logarithmic periodic circularly polarized antenna and sinusoidal circularly polarized antenna. Chapter 8 introduces the circularly polarized horn antenna, which combines circular polarizer and horn antenna to realize circularly polarized radiation. It also covers the types and implementation of circular polarizer and a design example of double circular polarization dual mode conical horn antenna. Chapter 9 introduces the omnidirectional circularly polarized antenna and the realization of arraying by directional circularly polarized antenna elements and omnidirectional linearly polarized antenna elements, respectively. It also introduces other types of omnidirectional circularly polarized antenna.

Introduction

3

Chapter 10 introduces the circularly polarized radial line array antenna, explains its structure and principle, elaborates the realization of the double circularly polarized radial line slot array antenna and briefly outlines the radial line spiral array antenna. The team of authors of this book is as follows: Yufeng Wang, Lei Chang, Xiaoyu He, Daxin Lv, Yufeng Yu, Liang Chen, Huiying Fu, Yefang Wang. Acknowledgments: The authors thank all who offered help and encouragement during the development of this book. They also thank Dr. Chen Linglu who revised this book very carefully, and for the encouragement and direction by Jianqiang Zhang, Xingli Yao, Baoming Li, Fengqing Xu, Jingbo Cui, Shuai He and Mingfang Zhang who are engaged in the research and development of the antenna technology at the 36th Research Institute of China Electronic Technology Cooperation. This book has also cited and referred many classic books and academic materials, and hence, the authors express their greatest gratitude to all. editor August 28, 2019

Chapter 1 Parameters of antennae An antenna is an energy convertor in the radio equipment that converts bound circuit fields into propagating electromagnetic waves from the transmitter or converts propagating electromagnetic waves into bound circuit fields to the receiver. With the development of wireless technology, antennae find wider use. The performances of many wireless systems are mainly limited by antennae. The antenna requirements of a system are dependent on antenna parameters in practical engineering. This chapter mainly deals with the antenna parameters used in practical engineering.

1.1 Radiation power density and radiation intensity 1.1.1 Radiation power density Spherical waves radiated by antennas will propagate in the radial direction for a coordinate system centered on the antenna [1]–[3]. These waves can be approximated at large distances by plane waves in a small area at the antenna directional angle. Both the direction of propagation and the power density of the electromagnetic wave can be described by the Poynting vector. It is the vector cross product of the electric and magnetic fields and is denoted by S and expressed as follows:
 (1:1) S = E × H* W=m2 where E is the electric field vector measured in units of volt per meter (V/m). H is the magnetic field vector measured in units of ampere per meter (A/m). H* is the complex conjugate of the magnetic field vector. The direction of the Poynting vector is orthogonal to both electric and magnetic fields. The directions of E, H and S define a right-handed coordinate system. We can determine the intensity and direction of the energy flow at any point in the space if we know E and H by using eq. (1.1). The average power density, W, could be expressed as follows： 1 jEj2 Wðr, θ, ϕÞ = Sav = ReðE × H* Þ = 2η 2

(1:2)

where η is the intrinsic impedance of the medium (η = 376.73 Ω in free space). Power density can be expressed as a spherical energy flow. The total power P of the antenna radiation can be obtained by taking the closed-surface integration of the power density on the enclosed surface of the antenna.

https://doi.org/10.1515/9783110562804-002

1.1 Radiation power density and radiation intensity

5

ðð ð 2π ð π ð 2π ð π P =  W · gds = Wr ðr, θ, ϕÞr2 sin θdθdϕ = Wr ðr, θ, ϕÞr2 dΩ

(1:3)

0

0

0

0

where dΩ = sin θdθdϕ is the differentiation of the solid angle, shown in Fig. 1.1.

Fig. 1.1: The dΩ shown in polar coordinate system.

In the case of isotropy, eq. (1.3) can be simplified as： P = 4πr2 Wr ðrÞ

(1:4)

1.1.2 Radiation intensity The radiation intensity in a given direction is defined as the power radiated by the antenna in the unit solid angle. Consider the area of spherical with radius, r, is 4πr2 and the solid angle is 4π. The radiation intensity Uðθ, ’Þin the given direction is expressed as Uðθ, ’Þ =

Wðr, θ, ϕÞ4πr2 = r2 Wðr, θ, ϕÞ 4π

(1:5)

Since in a radiated wave, Wðr, θ, ϕÞ is proportional to 1/r2 , the radiation intensity Uðθ, ’Þ is distance independent. Uðθ, ’Þ depends only on the direction of radiation

6

Chapter 1 Parameters of antennae

and remains the same at all distances. From eq. (1.5), the calculation of the total radiation power of the antenna can be simplified as P=

ð 2π ð π 0

Wr ðr, θ, ϕÞr2 sin θdθdϕ =

ð 2π ð π

0

0

Uðθ, ϕÞdΩ

(1:6)

0

1.2 Directivity and gain 1.2.1 Directivity Directivity is the degree of concentration of the energy of the antenna radiation in the free-space. The directivity is expressed by the ratio of the power density of the antenna in a given direction to that of the isotropic source under the same radiated power as follows: Dðθ, ’Þ =

Wðr, θ, ϕÞ 4πr2 Wðr, θ, ϕÞ 4πUðθ, ϕÞ = = Pr =4πr2 Pr Pr

(1:7)

Substituting eq. (1.6) into eq. (1.7), the directivity can be further expressed as 4πUðθ, ϕÞ Dðθ, ’Þ = Ð 2π Ð π 0 0 Uðθ, ϕÞdΩ

(1:8)

From eq. (1.8), the maximum directivity D can be expressed as 4πUmax D = Ð 2π Ð π 0 0 Uðθ, ϕÞdΩ

(1:9)

1.2.2 Gain Directivity is based on the radiated power of the antenna, regardless of the antenna material loss and impedance mismatch caused by the reflected power. Gain is based on the input power of the antenna and better demonstrates the actual antenna radiation performance. Considering the input power Pi of the isotropic antenna, gain can be expressed as follows [4]–[6]: Gðθ, ’Þ =

4πUðθ, ϕÞ Pi

(1:10)

Generally, the gain of the antenna refers to the maximum radiation of the gainbiased pattern, so eq. (1.10) can be written as

1.4 Antenna field zones

G=

4πUmax Pi

7

(1:11)

According to eqs. (1.10) and (1.11), the difference between the gain and the directivity is Pi and P; thus the relation of gain and the directionality coefficient are described as follows: G=

P D = ηe D Pi

(1:12)

where ηe is antenna efficiency including the effect of loss and mismatch. The difference between gain and directionality is efficiency ηe . If ηe = 1, the gain of the antenna is equal to the directivity in ideal conditions.

1.3 Beam solid angle Beam solid angle (ΩA ) is the solid angle through which all the power is radiated with maximum value (Umax) and zero, otherwise. Beam solid angle can be derived from the eq. (1.9) as follows [7]: 4πUmax 4π 4π D = Ð 2π Ð π = = Ð 2π Ð π Uðθ, ϕÞ ΩA Uðθ, ϕÞdΩ dΩ 0 0 0 0 Umax

(1:13)

The relationship between antenna directivity and the beam solid angle is expressed in eq. (1.13), and the beam solid angle can be expressed as ΩA =

ð 2π ð π 0

Uðθ, ϕÞ sin θdθdϕ 0 Umax

(1:14)

The beam solid angle is represented by the steradian, which represents the solid angle corresponding to the region of the sphere surface, r2. The beam solid angle of the antenna is usually approximately considered as the product of θHP and ϕHP described as follows: ΩA ≈ θHP ϕHP

(1:15)

where θHP represents half-power beamwidth (HPBW) at elevation plane, and ϕHP represents (HPBW) at azimuth plane.

1.4 Antenna field zones The field around the antenna can be divided into three regions: reactive near-field, radiating near-field (Fresnel) and far-field (Fraunhofer) regions, shown in Fig. 1.2.

8

Chapter 1 Parameters of antennae

The interface between the near field and far field

The near field or reactance

The far field or Fraunhofer field R2

L

R1

The near field or Fresnel field

Fig. 1.2: Field regions of an antenna.

Reactive near-field region is defined as “that portion of the near-field region immediately surrounding the antenna wherein the reactive field predominates.” For most antennae, the outer boundary of this region exists at a distance of R1, expressed as [8] rffiffiffiffiffi L3 ðmÞ (1:16) R1 = 0.62 λ where L is the maximum dimension of the antenna, unit: m; λ is the wavelength, unit: m. Radiating near-field region is defined as “the region of the field of an antenna wherein radiation fields predominate and wherein the angular field distribution is dependent upon the distance from the antenna.” The distances from the center to the inner boundary and outer boundary are R1 and R2, respectively, where R2 is expressed as follows: R2 =

2L2 ðmÞ λ

(1:17)

The region of the field of an antenna where the angular field distribution is essentially independent of the distance from the antenna is defined as far-field region. In this region, the field components are essentially transverse, and the angular distribution is independent of the radial distance where the measurements are made. The inner boundary is taken to be the radial distance R2. The amplitude pattern of

1.5 Radiation pattern

9

an antenna changes in shape as the observation distance is varied from the reactive near field to the far field, as shown in Fig. 1.3. The direction of the antenna in the near-field of the reactance is wavy, but the amplitude changes little and is relatively flat. The direction chart of the radiation near field is smooth and an obvious lobe is formed.

Fig. 1.3: Typical changes of antenna amplitude pattern shape from reactive near field toward the far field. (From: Y. Rahmat-Samii, L. I. Williams, and R. G. Yoccarino, The UCLA bi-polar planar-nearfield antenna measurement and diagnostics range, IEEE Antennas Propag. Mag., Vol. 37, No. 6, December 1995. Copyright © 1995 IEEE.).

1.5 Radiation pattern An “antenna radiation pattern” or simply, “antenna pattern” reflects the radiation characteristics of the antenna [9]. In most cases, the radiation pattern is determined in the far-field region, also called the far-field pattern. Radiation characteristics include power flux density, radiation intensity, field strength, directivity, phase and polarization. The spatial distribution of radiated energy is the most relevant radiation property with both amplitude electric field and amplitude power pattern being their key characteristics. The amplitude electric field pattern is a trace of the received electric field at a constant radius, and the amplitude power pattern is a graph of the spatial variation of the power density along a constant radius.

10

Chapter 1 Parameters of antennae

1.5.1 Description of antenna pattern Antenna pattern is generally a three-dimensional curved surface graph, seen in Fig. 1.4. In engineering, plane patterns are used to describe the cross-sectional profile of two orthogonal principal planes. The antenna pattern can be represented in polar coordinates (Fig. 1.5) and Cartesian coordinates (Fig. 1.6). The dimension of the pattern can be electric field intensity and power, in terms of V/m and W, respectively. When the decibel scale (dB) is used in antenna pattern, the power pattern is the same as the field intensity pattern. In order to facilitate comparison and drawing, the normalized pattern is used to take the maximum value of the directional function of 1.

Fig. 1.4: Antenna radiation pattern.

The antenna pattern is generally petal-shaped, as shown in Fig. 1.5, and is also called the lobe or beam. The lobe with maximum radiation direction is known as the main lobe, and the other is known as the sidelobe. The sidelobe is divided into first sidelobe, second sidelobe, etc., and the back lobe which at the opposite direction of the main lobe. The pit between the main lobe and the first sidelobe is called the first null. Similarly, the definition of the main lobe and sidelobe is also appropriate to the pattern of the Cartesian coordinate system.

1.5 Radiation pattern

11

1.5.2 HPBW The beamwidth is the HPBW, which is defined by IEEE as: “In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the beam.” The HPBW is also expressed as2θ0.5 , shown in Figs. 1.5 and 1.6.

Main lobe

0 dB

–3 dB 2Θ0.5 HPBW FNBW

First null (FNBW) First side lobe Second side lobe

Fig. 1.5: Antenna polar coordinates of two-dimensional pattern.

+

+

HPBW First null – π

+

First side lobe

Back lobe

FNBW

– π/2

Main lobe

– 0

+ π/2

Fig. 1.6: Antenna Cartesian coordinates two-dimensional pattern.

– π

Θ

12

Chapter 1 Parameters of antennae

1.5.3 FNBW The beamwidth between the first nulls of the main lobe on both sides is referred to as the first-null beamwidth (FNBW), as shown in Figs. 1.5 and 1.6.

1.5.4 Directivity estimation For the orientation pattern as shown in Fig. 1.4, directivity can be approximately estimated by the HPBWs of the principal plane patterns expressed as [10] D=

41253 θHP ϕHP

(1:18)

where 41,253 is the number of the square degrees in the sphere, and its value is 4πð180=πÞ2 ; unit: square degrees. θHP and ϕHP is the 3 dB beamwidth of the two mutually orthogonal main radiation planes, unit: degree. The sidelobe effect is ignored in eq. (1.18), and the following approximation can be used: D=

40, 000 θHP ϕHP

(1:19)

Example 1.1：Estimate the directivity of the antenna of 3-dB beamwidths of 110° and 70° of the principal planes, respectively. D=

40, 000 = 5.2 110 × 70

ð7.16 dBÞ

(1:20)

For the omnidirectional pattern, as shown in Fig. 1.7, if the azimuthal HPBW of the antenna is known, its directivity can be expressed as D=

101 θHP − 0.0027θ2HP

(1:21)

1.5.5 Sidelobe level The sidelobe level refers to the ratio of the maximum value of the sidelobe and the maximum value of the main lobe, and is usually expressed in decibels [11]–[12]: SLLi = 20 log

jEi max j ðdBÞ jEmax j

(1:22)

1.5 Radiation pattern

13

Fig. 1.7: Omnidirectional antenna pattern.

where Ei max is the maximum field strength of the ith sidelobe and Emax is the maximum field strength of the main lobe. In this way, the sidelobe level values can be obtained for each sidelobe as SLL1 , SLL2 shown in Fig. 1.5. In engineering, the sidelobe level refers to the level of the maximum sidelobes, denoted as SLL. In general, the level of the first sidelobe close to the main lobe is the highest. In engineering, the energy in sidelobe direction is usually not required. The lower the sidelobe level of the antenna, the weaker the energy of the antenna that is radiated or received in the unwanted direction, the stronger the ability to suppress the incoming stray signals in these directions and the stronger the antiinterference capability.

1.5.6 The difference among the decibels In engineering, the pattern is usually expressed in decibels, and the decibel is the basic unit of gain. The gain with dB scaled is a relative concept, which usually compares with an isotropic pattern and a half-wave dipole pattern united with dBi and dBd, respectively. Similarly, the dimension of the gain also depends on the polarizations with different units to distinguish. The different dimensions of the antenna patterns are listed in the following table.

14

Chapter 1 Parameters of antennae

Table 1.1: The meaning of dimension. Unit Meaning

Remark

dB log(Multiple of power) dBi Based on point source dBd Based on half-wave dipole The circularly polarized component compared with point dBic source component The linear polarization component compared with point dBiL source

Purely relative value Total energy Total energy,  dBd = .dBi Part of the energy (circular polarization) Part of the energy (linear polarization)

1.6 Effective length and effective area The effective length and effective area of the antenna can be used to characterize the ability of the antenna to receive electromagnetic waves. The induced voltage on the receiving antenna is proportional to its effective length, and the power is proportional to the effective area. For a linear dipole antenna, the current is not evenly distributed, so its effective length is generally not equal to its physical length. For the aperture antenna, such as horn, reflectors, etc., the diameter of the electromagnetic field on the surface has to meet the boundary conditions and non-uniform distribution, so its effective area is generally less than its physical aperture area.

1.6.1 Effective length The effective length is defined as the length of equivalent antenna with uniform current distribution of the same field strength in the maximum radiation direction [13]–[14]. The effective length of the receiving antenna is defined as the ratio of the open-circuit voltage of the receiving antenna to the received electric field: Le =

VA Ei

(1:23)

where Le is the effective length, VA is the open-circuit induced voltage at the antenna terminal, Ei is the received electric field intensity. The effective length of the transmitting antenna is numerically equal to the effective length when the antenna is received. The effective length of the symmetrical linear antenna can be calculated using the following equation: Le =

λ kl lg π 2

(1:24)

1.6 Effective length and effective area

15

where k = 2π=λ is the free space wave number, l is the length of the single arm of the symmetrical antenna.

1.6.2 Effective area When the receiving antenna with the physical aperture of Ap (m2) is located in a uniform plane electromagnetic wave with a power density of W, the total power received by the entire physical aperture is expressed as Pr = WAp

(1:25)

In fact, the current distribution on the antenna as a converter is not uniform, and the actual effective conversion of the aperture is not Ap. The effective area Ae is less than Ap, and it is expressed as Pr = WAe

(1:26)

where Ae is the effective area of the antenna. The effective area of the antenna is demonstrated by the receiving antenna with such content. According to the reciprocity theorem of antenna, the transmitting antenna also has similar characteristics. The effective area of the antenna is usually smaller than the physical aperture area, and the ratio between them is called aperture efficiency, expressed as ηi =

Ae Ap

(1:27)

The aperture efficiency is generally 50–80% of the horn antenna and reflector antenna, and nearly 100% of the dipole or patch antenna arrays with uniform field maintained in the whole aperture, especially in the edges. The effective area of the antenna is proportional to the directivity and the operating wavelength, and is expressed as Ae =

Dλ2 4π

(1:28)

It follows that the antenna directivity can be estimated by the effective area and wavelength using the following expression: D=

4πAe λ2

(1:29)

16

Chapter 1 Parameters of antennae

Example 1.2：Suppose that a reflector antenna operates at 50 GHz with the aperture diameter of 450 mm. Calculate the antenna’s directivity with the aperture efficiency of 50%. According to (1.29) and (1.30), the directivity is calculated as
 4π × 50% × π × 0.2252 = 27, 758.3ð44.4 dBÞ D= ð0.3=50Þ2 When we assume the aperture of the antenna is a circle with the diameter of d and θHP = ϕHP , the expression of the directivity (eq. (1.29)) can be simplified as D = ηi

 2 πd λ

(1:30)

When the decibel scale (dB) is used in directivity calculation, eq. (1.30) is written as follows:   πd + 10 lg ηi ≈ − 39.6 + 20 lg d + 20 ≶ f + 10 lg ηi DðdBÞ = 20 lg λ

(1:31)

where d is the diameter of the circular aperture in meters, f is the operating frequency in MHz.

Example 1.3：Recalculate Example 1.2 using eq. (1.31). Directivity in Example 1.2 can be calculated as D ≈ − 39.6 + 20 lg 0.45 + 20 lg 50, 000 + 10 ≶ 0.5 = 44.4ðdBÞ The result is the same as that in Example 1.2.

1.7 Antenna impedance 1.7.1 Input impedance The input impedance of the antenna is the ratio of the input voltage Vi and current Ii and is expressed as follows: Zi =

Vi Ii

(1:32)

The input impedance is a pure resistance when the input voltage and current are in-phase, but they are usually out of phase, which results in the input impedance including resistance, Ri, and reactance, Xi: Zi = Ri + jXi

(1:33)

The antenna is connected to a transmitter or receiver, and its input impedance is equivalent to the load of the transmitter or receiver. In other words, the input impedance value represents the matching status of the antenna to the transmitter or receiver.

1.7 Antenna impedance

17

The resistance, Rin, of the input impedance of the antenna consists of the radiation resistance, Rr, and the loss resistance, R, and is expressed as Rin = Rr + R,

(1:34)

where the radiation resistanceRr is defined as the equivalent resistance “absorbing” the antenna radiation power when considering the radiating power to absorbing power. The loss resistance R, is defined as the equivalent resistance “absorbing” the loss of power in the antenna (including heat loss in the conductor, dielectric loss of the dielectric, ground current loss, etc.) when considering the loss power to absorbing power.

1.7.2 Voltage standing wave ratio The system obtains optimum efficiency with the antenna input impedance matching the transmitter or receiver well, whereas, it degrades due to the reflected power. According to the transmission line theory, when the antenna and the transmitter or receiver does not fully match, there are incident and reflected waves that exist together at the port. Reflection coefficient is defined to describe the degree of mismatch, and the relationship between the reflection coefficient and the antenna input impedance is expressed as follows: Z i = Zc

1+Γ 1−Γ

(1:35)

where Zc is the transmission line characteristic impedance, Γ is the reflection coefficient, Zi is a complex number. The voltage standing wave ratio (VSWR) is usually used to characterize the degree of matching between antenna and transmitter or receivers. The relationship between VSWR and Γ is given by VSWR = jΓj =

Vmax 1 + jΓj = Vmin 1 − jΓj

(1:36)

VSWR − 1 VSWR + 1

(1:37)

The transmitting power of antenna to the transmitter or the receiver can be calculated from the magnitude of VSWR or reflection coefficient. The reflected power ratio and the transmission power ratio are expressed as follows: Pref = jΓj2 Pi

(1:38)

18

Chapter 1 Parameters of antennae

Ptrans = 1 − jΓj2 Pi

(1:39)

where Pref is the reflection power, Ptrans is the transmission power, Pi is the input total power. Return loss and reflected power loss are often used in engineering. The return loss is the ratio of the reflected power and incident power, that is, the reflected power ratio. The reflected power loss is the difference between the incident power and the reflected power, that is, the transmission power ratio. The return loss and the reflected power loss are usually expressed in dB: Return loss ðdBÞ = 20 logjΓj

(1:40)

Reflected power ðdBÞ = 10 logð1 − jΓj2 Þ

(1:41)

Table 1.2 shows some relationships between VSWR, reflection coefficient, reflection power ratio, transmission power ratio, return loss and reflection power loss.

Table 1.2: VSWR, reflection coefficient, reflection and transmission power ratio, return loss, reflection power loss. VSWR

Reflection coefficient

Reflection power ratio (%)

Transmission power ratio (%)

Return loss (dB)



Reflection power loss (dB)







.

.

.

.

.

−.

−.

.

.

.

.

−.

−.

.

.

.

.

−.

−.

.

.

.

.

−.

−.

.

.





−.

−.

.

.

.

.

−.

−.

.

.

.

.

−.

−.

.

.





−.

−.

.

.





−.

−.

.

.

.

.

−.

−.

1.9 Polarization of the antenna

19

1.8 Bandwidth The antenna impedance generally changes with the frequency, and therefore the matching also changes with the frequency. The antenna operates with high efficiency when frequency is matched, otherwise it is low. In engineering, the antenna operates well within a restricted band of frequencies called frequency range in which the antenna may show good performance in one or more of following specifications: input impedance, pattern, beam pointing, polarization and gain, etc. We can define the frequency range as a frequency band with a certain performance parameter up to the standard. The frequency range can be denoted as f1 ~ f2 with the center frequency f0. The bandwidth is a relative result obtained by comparing f1 and f2, and takes presentations of the absolute bandwidth, relative bandwidth and ratio bandwidth. The absolute bandwidth is defined as the difference between the upper and lower frequencies and expressed as follows: Δf = f2 − f1

(1:42)

The relative bandwidth is defined as the ratio of the absolute bandwidth and the center frequency: Δf =

f2 − f1 × 100% f0

(1:43)

The ratio bandwidth is defined as the ratio of the upper and lower frequency that is f2: f1. The relative bandwidth is the most widely used in the three presentations. The ratio bandwidth is mainly used to describe the bandwidth of the ultra-bandwidth antennas. Bandwidth has various meanings such as impedance bandwidth, VSWR bandwidth, pattern bandwidth and gain bandwidth, etc. in different performance parameters. Impedance bandwidth is the most widely used and is generally defined as the bandwidth of VSWR less than 2 for the 50 Ω input impedance.

1.9 Polarization of the antenna The polarization of an antenna is the polarization of the radiated electromagnetic wave produced by an antenna, evaluated in the far field. The polarization of the electromagnetic wave is the property of the time varying direction and amplitude of electric field vector. The polarization of the antenna can be found through the curve traced by the end point of the vector representing the instantaneous electric field [1]. The antenna performs linear polarization if the curve is straight, circular polarization if the curve is circular and elliptical polarization if the curve is elliptical. The linear and circular polarizations are two special cases of elliptical polarization. Figure 1.8 shows the typical curves traced by the instantaneous electric field with time.

20

Chapter 1 Parameters of antennae

ωt

ωt

Linear polarization Ey

(a)

Ex

Ey (b)

Fig. 1.8: Typical trajectory curves traced by the instantaneous electric field with time.

The electric field direction of the circular polarization is rotated over time clockwise or counterclockwise along the propagating direction of the electric wave. Clockwise and counterclockwise correspond to right-handed circular polarization and left-handed circular polarization, respectively.

1.10 Equivalent isotropically radiated power Equivalent isotropically radiated power (EIRP) represents the radiated power of the transmitting system in a specified direction. In the wireless communication system, the radio frequency signal that is emitted from the transmitter travels to the transmitting antenna through the feed line. The antenna radiates the radio frequency signal to the destination through the free space, is received by the receiving antenna and travels to the receiver through the feed line. We know the more powerful the transmitter radiation, the stronger the receiver receives. Hence, it is very important to calculate the EIRP of the wireless communication system. The EIRP can be measured in watts [W] or decibels above 1 W [dBW]. The transformation of the two units is derived as follows: ½dBW = 10 × log½W ½W = 10½dBW=10 For example, the EIRP is 100 W, which is also 20 dBW. In engineering, mW and dBm are also usually used as units of power, and their relationship to W and dBW is: 1 mW = 0.001 W, 0 dBW = −30 dBm. In the wireless system, the antenna is used to convert the current wave to electromagnetic waves, and it amplifies the signal in transmitting and receiving by the

1.11 Friis transmission equation

21

antenna gain. The loss of the feed line reduces the radiated power of the transmitting system. The EIRP measured by dBW is calculated as follows: EIRP = Pt − Ls + Gt where Pt represents the output power of the transmitting device in dBw, Ls is the transmission loss between the transmitting devices in dB, Gt is the transmit antenna gain in dB. Example 1.4: The output power of the transmitting device is 100 mW, the gain of the antenna is 10dBi, the loss between the transmitter and antenna is 1 dB. The ERIP is calculated by 20 dBm −1 dB + 10 dBi = 29 dBm. It is important for engineers to understand the “dB” dimension. The followings are some corresponding relationships (approximate) between reductions or increases in dBs and proportions. −1 dB: Power value declines by about 20%. −2 dB: Power value declines by about 36%. −3 dB: Power value declines by about 50%. −6 dB: Power value declines by about 75%; + 1 dB: Power value decreases by about 25%; + 2 dB: Power value decreases by about 60%; + 3 dB: Power value decreases by about 100%; + 6 dB: Power value decreases by about 300%. For example 1, the initial EIRP of a transmitting system is 100 W or 20 dBW. If the radiation power declines by 2 dB, the EIRP changes to 18 dBW corresponding to 64 W. It is calculated by 100 W x (1–36%) = 64 W. For example 2, the initial EIRP of a transmitting system is 100 W or 20 dBW. If the radiation power declines by 7 dB, the EIRP changes to 13dBW corresponding to 20 W. As we know, 7 dB = 6 dB + 1 dB, the final EIRP can be calculated by 100 W x (1–75%) × (1–20%) = 20 W.

1.11 Friis transmission equation Harald T. Friis gives the power relationship between the transmitting and receiving antennas separated by larger than far-field distance. Suppose that the transmitting and receiving antennas are in polarization matching, the propagation link is shown in Fig. 1.9.

Fig. 1.9: Transmitting antenna and receiving antenna.

22

Chapter 1 Parameters of antennae

Substituting eqs. (1.4) and (1.28) in eq. (1.26), the following equation is obtained: Pr = Aer W =

Aer Pt Gt 4πr2

(1:44)

We can rewrite the ratio of received power Pr and transmitted power Pt as eq. (1.45), and that is the Friis formula:   Pr λ 2 = Gr Gt Pt 4πr

(1:45)

Example 1.5: There is a communication link operating at 3 GHz with 30 km space between the transmitting and receiving systems. The transmitter outputs power of 15 W with antenna gain of 15 dBi. The receiving antenna‘s gain is 18 dBi. The transmitting and receiving antennas are supposed to exactly match. Please calculate the power received by the receiving antenna. The received power can be obtained by the following expression from eq. (1.45):  Pr =

λ 4πr

2

 G r G t Pt =

0.1 4π × 30, 000

2 × 1018=10 × 1015=10 × 15 = 2.1 nW

The Friis transmission equation contains the gain of the transmitting and receiving antennas and the propagation loss in free space between them. The propagation loss (Ls) in free space is calculated by dividing eq. (1.45) by Gr and Gt, as follows:  Ls =

λ 4πr

2 (1:46)

where Ls is only related to the space between transmitting and receiving antennas (r) and operating wavelength (λ). In engineering, formula (1.46) is usually expressed in dB as follows: Ls = 32.45 + 20 log d + 20 log f ðdBÞ

(1:47)

where d is the propagation distance in km, equivalent to r in eq. (1.45), f is the operating frequency in MHz. When substituting eq. (1.45), eq. (1.47) can be written in dBs, as follows: Pr = Pt + Gt + Gr − Ls ðdBÞ

(1:48)

Example 1.6: Recalculate Example 1.5 according to eq. (1.48). Ls = 32.45 + 20 log 30 + 20 log 3, 000 = 131.54 dB Pr = 10 log 15 + 15 + 18 − 131.54 = − 86.77 dBW The result is the same as in Example 1.5.

2.1 nW

1.12 Receiving antenna

23

1.12 Receiving antenna Due to the reciprocity, the antenna has the same characteristics whether it is used as a transmitter or a receiver. It is easy to assess antenna performance. However, there are several special parameters of the receiving antenna. In this section, the received voltage and antenna coefficient which are used frequently for describing receiving antenna are explained.

1.12.1 Receiver voltage The response of the antenna while receiving the electromagnetic wave is usually expressed by the received power, that is, Pr, generally in dBm. It is also expressed by the received voltage, that is, Ur, generally in μV or dB μV. For a 50 Ω system, the conversion relation of dBm and dBμV is 0 dBm = 107 dBμV. The derivation process is as follows. According to the conversion of power and voltage Pr = Ur2 =R

(1:49)

where Pr is in W, Ur is in V. If the units of Pr and Ur change to mW and μV, eq. (1.6) can be rewritten as follows: Prm × 10 − 3 = ðUr μV × 10 − 6 Þ2 =R

(1:50)

Take the logarithm of both sides of this equation with R = 50 Ω. Equation (1.7) is rewritten as follows: 10 lg Prm = 20 lg UrμV + 30 − 120 − 10 lg 50 = 20 lg UrμV − 107

(1:51)

Prm ðdBmÞ = UrμV ðdBμVÞ − 107

(1:52)

and

1.12.2 Antenna factor According to IEEE definition, antenna factor (AF) is defined as the ratio of the electric field strength to the voltage induced across the terminals of an antenna [15]–[16]. It is defined mathematically as AF = E=V (1:53) where E is the measured electric field strength (V/m), V is the voltage (V) at the output of the antenna.

24

Chapter 1 Parameters of antennae

The unit of AF is 1/m, and the logarithmic form is in dB/m. In engineering, dB / m is used much more. The expression of AF in logarithmic form is AFðdB=mÞ = 20 lg ðE=VÞ = EðdBμV=mÞ − VðdBμVÞ

(1:54)

If E is specified, the stronger the V produces, the higher the sensitivity of the antenna, the smaller the AF. The weaker the V produces, the lower the sensitivity of the antenna, the greater the AF. The AF is typically in the range of 0 to 60 dB/m. AF and gain have a corresponding relationship, which can be derived by reciprocity method. Supposing that the transmit antenna gain Gt and the transmit power Pt, the receiving antenna gain Gr and the received power Pr, are as shown in Fig. 1.9, power density at the location of the receiving antenna is as follows: Pd =

Pt Gt 4πr2

(1:55)

where Pd is the power density in W/m2. If an incident field is part of a propagating wave, the power density of the wave is also given by Pd = EH = E2 =Zw

(1:56)

where Zw is the wave impedance of 120πΩ. From eqs. (1.55) and (1.56), it is known that Pt =

E2 4πr2 E2 4πr2 E2 r 2 = = zw G t 120πGt 30Gt

(1:57)

and E E AF = = pffiffiffiffiffiffiffiffiffi = V Pr z0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 30Pt Gt 30Pt Gt =r pffiffiffiffiffiffiffiffiffi = Pr z0 r2 Pr z0

(1:58)

Assuming that the transmitting and receiving antennas have equal gains, Gr = Gt = G, Pr is derived from eq. (1.45) as follows:  Pr =

λG 4πr

2 Pt

Substituting into eq. (1.58), the following is obtained: pffiffiffiffiffiffiffiffiffiffiffiffi 4π 30=z0 pffiffiffiffi AF = λ G

(1:59)

(1:60)

References

25

When z0 = 50 Ω, 9.73 AF = pffiffiffiffi ð1=mÞ λ G

(1:61)

The equation can be rewritten in the form of dB/m, as follows: AF = 20 ≶ ð9.73=λÞ − 10 ≶ G ðdB=mÞ

(1:62)

G = 20 ≶ ð9.73=λÞ + AFðdBÞ

(1:63)

Or

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

L. Changlu, S.X. Ming, et al. Circular polarized antenna. Bejing, Posts & Telecom Press, 1986. X. Chufang, R. Kejin. Electromagnetic Field and Electromagnetic Wave. Bejing, Higher Education Press, 1980. A.M. Thomas (Author), Modern Antenna Design (second edition), Guo Yuchun, Fang Jiayun, Zhang Guangsheng (Translator). Bejing, Publishing House of Electronics Industry, 2012. W.L. Stutzman, G.A. Thiele (Author), Antenna Theory and Design (second edition), Z. Shouzheng, A. Tongyi, et al. (Translator). Bejing, Posts & Telecom Press, 2006. K. Xingjian. Antenna Theory and Design. Bejing, Beijing Institute of Technology Press, 1993. Y. Xinyuan, Z.Z. Ma Huizhu Modern antenna technology. Bejing, Beijing Institute of Technology Press, 2009. N.A. McDonald. Approximate relationship between directivity and beamwidth for broadside collinear array. IEEE Trans. Antennas Propag. 1978 March, 26(2), 340–341. Y. Rahmat-Samii, L.I. Williams, R.G. Yoccarino. The UCLA bi-polar planar-near-field antenna measurement and diagnostics range. IEEE Antennas Propag. Mag. 1995 December, 37(6). Dennis Roddy (Author). Satellite Communications. Z. Baoyu, et al. (Translator). Bejing, China Machine Press, 2011. G. Frank (Author). Smart Antennas for Wireless Communication. H. Yejun, G. Liangqi, L. Xia (Translator). Bejing, Publishing House of Electronics Industry, 2009. J.D. Kraus, R.J. Marhefka(Author). Antenna (Third edition). Z. Wenxun (Translator). Bejing, Publishing House of Electronics Industry, 2004. L. Changlu, et al. Antenna Engineering Handbook. Bejing, Publishing House of Electronics Industry, 2002. L. li, Antenna and Radio Propagation. Science Press, 2009. C.A. Balanis. Modern Antenna Handbook. Wiley & Sons, Inc., 2008. TanKang. Correction of Site Effects on Antenna Factor Measurement. Beijing, Beijing Jiaotong University, 2009. B. Tongyun, W. Wei, C. Zhiyu. The calibration and use of antenna factor. Chinese Journal of Radio Science. 2000 Dec, 15(4).

Chapter 2 Polarization theory During the propagation of an electromagnetic wave, the direction of the electric field and magnetic field often varies with time. Polarization theory was proposed to explain the change rule of the amplitude and direction of the electric field intensity (or magnetic field intensity). The wave polarization was generally defined by the trajectories of the electric field intensity vector endpoint in space for every cycle [1]. This chapter introduces the basic concepts of electromagnetic polarization and their interrelation, as well as the concepts of antenna-relevant parameters. The methods of judging the left- and right-handed polarization characteristics of some typical circularly polarized antennas are given.

2.1 Polarization characteristics of the wave 2.1.1 Plane electromagnetic wave electric field vector An electromagnetic wave radiated by an antenna is a spherical wave that propagates in the radial direction for a coordinate system centered on the antenna [2]–[5]. If on a spherical surface with a center on the antenna and a far-field distance r as the radius, a small area adjacent to the range of the maximum direction of the antenna is taken, the electromagnetic wave on this small area can be approximated as a plane wave. In the spherical coordinate system, electric field in the far field of antenna is generally represented by Eθ and E’ , and the relationship in the coordinate system is shown in Fig. 2.1. It can also be expressed as a plane wave propagating along the z-axis, that is Ex and Ey , and the synthetic electric field can be written as E

Ey

τ Ex

Fig. 2.1: Linear polarization wave trajectory curve.

https://doi.org/10.1515/9783110562804-003

2.1 Polarization characteristics of the wave

^ Ex + y ^Ey = x ^ E0x e − jðβz + ’x Þ + y ^E0y e − jðβz + ’y Þ E=x

27

(2:1)

^ and y ^ are unit vectors, ’x and ’y are the phases of the electric field compowhere x nents Ex and Ey , respectively, E0x and E0y are the amplitudes of the electric field components Ex and Ey , respectively. Let us multiply both sides by the time factor e–jωt and take its real part, and the instantaneous synthetic electric field at z = 0 is expressed as *

^ Ex ðtÞ + y ^Ey ðtÞ E ðz, tÞjz = 0 = x

The instantaneous components are ( Ex ðtÞ = E0x cosðωt + ’x Þ Ey ðtÞ = E0y cosðωt + ’y Þ

(2:2)

(2:3)

Δ’ = ’y − ’x be the phase difference between the electric field components Ex and Ey, then eq. (2.3) can be rewritten as ( Ex ðtÞ = E0x cosðωtÞ (2:4) Ey ðtÞ = E0y cosðωt + Δ’Þ

Let

Eliminating the expression containing ωt in eq. (2.4), the equation becomes Ey2 ðtÞ Ex ðtÞEy ðtÞ Ex2 ðtÞ −2 cosðΔ’Þ + 2 = sin2 ðΔ’Þ 2 E0x E0y E0y E0x

(2:5)

where Δ’ is the phase difference between the two components. According to this equation, the time-varying trajectory of the synthetic electric field vector at z = 0 is discussed in this section. 2.1.2 Linear polarization If Δ’ is 0 or π, then eqs. (2.4) and (2.5) can be reduced to ( Ex ðtÞ = E0x cosðωtÞ Ey ðtÞ = E0y cosðωtÞ 8 E ðtÞ x >

: Ex ðtÞ = − E

(2:6)

Ey ðtÞ E 0y

(2:7) Δ’ = π

28

Chapter 2 Polarization theory

If the phases of components Ex and Ey are the same or differ by π, the synthetic electric field is a linear polarization wave. The synthetic electric field intensity is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + E2 cos ωt E = Ex2 + Ey2 = E0x (2:8) 0y The angle τ between the synthetic electric field and the x-axis is tan τ =

Ey E0y = = const Ex E0x

(2:9)

From eqs. (2.8) and (2.9), it is shown that the magnitude of the synthetic electric field varies with time, but the angle between the trajectory of the synthetic electric field and the x-axis remains constant. The synthetic electric field trajectory is shown in Fig. 2.1.

2.1.3 Circular polarization If Δ’ is ±π=2，and E0x = E0y , eqs. (2.4) and (2.5) can be reduced to ( Ex ðtÞ = E0 cosðωtÞ Ey ðtÞ = E0 cosðωt ± π=2Þ Ex2 ðtÞ Ey2 ðtÞ + 2 =1 E0 E02

(2:10)

(2:11)

It can be seen that the synthetic electric field is a circular polarized wave if the amplitude of the components Ex and Ey are equal and the phase difference is ±π=2. The synthetic electric field intensity is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi (2:12) E = Ex2 + Ey2 = E02 + E02 = 2E0 The angle τ between the synthetic electric field and the x-axis is tan τ =

Ey cosðωt ± π=2Þ ∓ sin ωt = = = ∓ tan ωt Ex cos ωt cos ωt

(2:13)

That is τ = ∓ ωt. It can be seen from eqs. (2.12) and (2.13) that the magnitude of the electric field does not change with time, but the direction does. The vector endpoint trajectory of the synthetic electric field rotates on a circle at angular velocity ω, as shown in Fig. 2.2.

2.1 Polarization characteristics of the wave

29

y ω E

Ey τ

x

Ex

Fig. 2.2: Circular polarization wave trajectory curve.

As shown in Fig. 2.2, the angular velocity ω is vector, and “±” corresponds to the sense of rotation. With the four fingers rotating in the direction of ω, the thumb points to the direction of the wave. If the right-hand screw rule is satisfied, it is the righthanded circular polarization (RHCP) wave, and conversely if the left-hand screw rule is satisfied, it is the left-handed circular polarization (LHCP) wave. If Δ’ is –π=2, as shown in Fig. 2.2, the propagation direction is perpendicular outward to the surface of the paper and it is the RHCP wave; when Δ’ is π=2, it is the LHCP wave.

2.1.4 Elliptically polarization wave Generally, the amplitudes and phases of the components Ex and Ey are not equal, and that makes it an elliptic polarization wave whose expressions are shown in eqs. (2.4) and (2.5). Equation (2.5) is a general elliptic equation. If the tip of the electric field is traced in space over time, it appears as an ellipse with the electric field rotating either clockwise (CW) or counterclockwise (CCW), as shown in Fig. 2.3. Figure 2.3 is also called the polarization ellipse, and a and b in the figure are the minor axis and the major axis of the polarization ellipse, respectively. If Δ’ < 0, the wave is RHCP, else, it is LHCP. y

Ey a b

E τ

x

Ex

Fig. 2.3: Elliptic polarization wave trajectory curve.

30

Chapter 2 Polarization theory

The sense of rotation of the circular polarization wave can also be judged by the variation characteristics of electric field in space. The sense of rotation is always determined by rotating the phase-leading component toward the phase-lagging component. Take the components Ex and Ey in eq. (2.4) as an example – the phase difference between them is Δ’. If Δ’< 0, the component Ey is the phase-lagging component, the sense of rotation is CCW along the direction of propagation and the wave is RHCP; if Δ’ > 0, the component Ey is the phase-leading component, the sense of rotation is CW along the direction of propagation and the wave is LHCP.

2.2 Polarization ellipse dip angle The polarization ellipse dip angle of the elliptic polarization wave is defined as the angle between the major axis of the polarization ellipse and the x-axis, which is parallel to the plane, expressed as σ and shown in Fig. 2.3. If we rotate the X–Y coordinate system in Fig. 2.3 with an angle of σ, make the major axis a on the x-axis and the minor axis b on the y-axis, and the dip angle of the elliptic polarization wave σ = 0°. A new coordinate system x′ − y′ is created and is shown in Fig. 2.4: y'

y

x'

P(x',y')

σ

x

Fig. 2.4: The coordinate for polarization ellipse dip angle.

(

x = x′ cos σ − y′ sin σ y = x′ sin σ + y′ cos σ

(2:14)

The synthetic field components Ex ðtÞ and Ey ðtÞ in the new coordinate system in ′ ′ Fig. 2.4 can be expressed in terms of Ex ðtÞ and Ey ðtÞ as follows: (

′

′

′

′

Ex ðtÞ = Ex ðtÞ cos σ − Ey ðtÞ sin σ Ey ðtÞ = Ex ðtÞ sin σ + Ey ðtÞ cos σ

(2:15)

2.3 Axial ratio

31

When we substitute eq. (2.15) into eq. (2.5), and then extract the terms of ′ ′ ′ ′ Ex2 ðtÞ、Ex ðtÞEy ðtÞ and Ey2 ðtÞ respectively, we obtain " # " # 2 2 sin σ sin 2σ cosðΔ’Þ cos σ sin 2σ 2 cos 2σ cosðΔ’Þ sin 2σ 2 ′ ′ E′x ðtÞ − + 2 − + 2 + Ex ðtÞEy ðtÞ 2 2 E0x E0y E0x E0y E0y E0x E0y E0x " # cos2 σ sin 2σ cosðΔ’Þ sin2 σ 2 ′ + E y ðtÞ + + 2 = sin2 ðΔ’Þ 2 E0x E0y E0y E0x

(2:16)

If the major axis and the minor axis of the polarization ellipse coincide with the new x′-axis and y′-axis, respectively, eq. (2.16) should be a standard elliptic equation expres′ ′ sion, which means that the term Ex ðtÞ, Ey ðtÞ must have a coefficient of 0. Thus, sin 2σ 2 cos 2σ cosðΔ’Þ sin 2σ − + 2 =0 2 E0x E0y E0y E0x

(2:17)

We can obtain tan 2σ =

2E0x E0y cosðΔ’Þ 2 − E2 E0x 0y

(2:18)

and 2E0x E0y cosðΔ’Þ 1 σ = tan − 1 2 − E2 2 E0x 0y

(2:19)

Equation (2.19) shows the relation between the dip angle of an elliptical polarization wave and the vectors of two linearly polarized waves. If the amplitude and phase differences between the two orthogonal linear polarization vectors are given, we can calculate the dip angle of the polarization ellipse.

2.3 Axial ratio The ratio of the major axis b of the polarization ellipse to the minor axis a (see Fig. 2.3) is referred to as the axial ratio (AR). As shown in eq. (2.16), if the coefficient ′ ′ of term Ex ðtÞEy ðtÞ is 0, it can be rewritten as " # sin2 σ sin 2σ cosðΔ’Þ cos2 σ 2 ′ E x ðtÞ 2 − + 2 E0y sin2 ðΔ’Þ E0x E0y sin2 ðΔ’Þ E0x sin2 ðΔ’Þ " # 2 2 cos σ sin 2σ cosðΔ’Þ sin σ 2 + + 2 + E′y ðtÞ 2 =1 (2:20) E0y sin2 ðΔ’Þ E0x E0y sin2 ðΔ’Þ E0x sin2 ðΔ’Þ

32

Chapter 2 Polarization theory

Equation (2.20) can be expressed as a standard elliptic equation described as follows: 2 2 E′x ðtÞ E′y ðtÞ + =1 a2 b2

(2:21)

Equation (2.21) assumes that the major axis b of the polarization ellipse coincides with x′ and the minor axis a coincides with y′. According to eqs. (2.20) and (2.21), the expression of the AR, r, can be derived: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 sin2 σ E0x cos2 σ + E0x E0y sinð2σÞ cosðΔϕÞ + E0y b u r= =t 2 (2:22) 2 2 cos2 σ a E0x sin σ − E0x E0y sinð2σÞ cosðΔϕÞ + E0y The value range of the AR, r, is 1 ≤ r < ∞, and r commonly uses decibel as an expression in engineering: AR = 20lgr

(2:23)

If AR＝1(0 dB), it is circular polarization. If AR = ∞, it is linear polarization. Thus, in the design of circularly polarized antenna, the AR is an important technical indicator in measuring the degree of circular polarization. The AR is required to be in the range of ARdB ≤ ð3⁓6ÞdB, within the main lobe of the pattern.

2.4 Polarization ratio 2.4.1 Linear polarization ratio The elliptical polarization electric field vectors in Sections 2.1–2.3 are denoted as two orthogonal linear polarization components Ex and Ey, and the ratio of the two orthogonal linear polarization components is defined as the linear polarization ratio ^ρL as follows [6]–[8]: ^L = ρ

Ey Ex

(2:24)

^L can also be expressed in terms of the amplitude ρL The linear polarization ratio ρ and the phase δL as follows: ^L = ρL ejδL ρ

(2:25)

If the dip angle of polarization ellipse σ = 0° or 180°, the amplitude ρL is equal to the AR, r; if σ = ±90°, ρL is inversely related to AR r. The synthetic electric field of elliptical polarization wave can be expressed as

2.4 Polarization ratio

^ Ex + y ^Ey = Ex ðx ^+ρ ^Þ ^L y E=x

33

(2:26)

^L is 0, eq. (2.26) represents a purely linear polarization If the imaginary part of ρ ± jπ=2 ^ , eq. (2.26) represents a purely circular polarization wave. “±” wave; if ρL is e corresponds to LHCP and RHCP, respectively (the propagation direction is outward vertical to the paper face), as shown in Fig. 2.5.

ρˆ = e L

j

π 2

−j

ρˆ = e

: LHC

L

π 2

: RHC

y

y ωt

ωt

x

x

Fig. 2.5: LHC and RHC polarization ellipses of linear polarization ratio e ± jπ=2 .

The left-handed and right-handed elliptical polarizations are judged according to the value range of δL . If 0 < δL < π, it is the left-handed elliptical polarization radiated; if − π < δL < 0, it is the right-handed elliptical polarization radiated.

2.4.2 Circular polarization ratio The electric field of elliptical polarization wave can also be represented by two orthogonal circular polarization components, and the expression is ^ + ER R ^ E = EL L

(2:27)

^ and R ^ are unit vectors of LHCP and RHCP components, respectively. Their where L representations in the Cartesian coordinate system are shown in Fig. 2.5. The lefthanded and right-handed orthogonal unit vectors defined in terms of linear components are 8 ^ = p1ffiffi ðx 0 for region 1 and t2 = εr k02 − kρ2 , ε2 = εr ε0 , n = 2, z < 0 for region 2. Equations (10.9) and (10.10) are for region 1 and region 2, respectively. The complex wave number kρ is jkρ = α + jβ, where α is the coupling factor to indicate the attenuation due to the coupling with unit m–1 and β is the phase constant. The relationship between ξ and β is shown as follows: H’n = −

ξ=

λg k0 = λ0 β

(10:11)

10.2 Design of circularly polarized radial line array antenna

327

The aperture surface impedance Z0 can be expressed as the ratio of the tangential electric field to the magnetic field: Z0 = Eρ =Hf ðz = d1 Þ

(10:12)

After calculation, the relationship is as follows: t2 sinðt2 d2 Þ½t1 cosðt1 d1 Þ + jωε0 Z0 sinðt1 d1 Þ + cosðt2 d2 Þ = 0 εr t1 ½t1 sinðt1 sinðt1 d1 Þ − jωε0 Z0 cosðt1 d1 Þ

(10:13)

kρ can be solved from the above equation, and then α and β can be obtained from the definition of kρ . The slow wave factor ξ can also be solved. If there is no slot (Z0 = 0) at the aperture, kρ becomes a real number and the attenuation in the infield will never exist.

10.2.3 Aperture field distribution In the dual-layer RLSA, when designing the slot pairs and making the radial current of the top plate excited by a constant factor, the excitation coefficient is determined by the magnetic field in the upper waveguide. If the coupling of the slot is omitted and the thickness of the upper radial waveguide d < λd/2, the symmetric mode is only TEM wave. The magnetic field is represented by the following equation: ð1Þ

Hf = H1 ðkρÞ − Γ

ð1Þ

H1 ðkρm Þ ð2Þ

H1 ðkρm Þ

ð2Þ

H1 ðkρÞ

(10:14)

Where, k = 2π/λg is the wave number of the waveguide, Γ is the reflection coefficient ð1Þ ð2Þ of the electromagnetic wave at ρ = ρm . H1 ðkρÞ and H1 ðkρÞ are the first class and the second class of first-order Hankel function. Since kρ1, Hϕ can be simplified as follows: sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi  2 jðkρ − 3πÞ 2 jfkð2ρm − ρÞ − 3πg 4 4 e e (10:15) −Γ Hf = πkρ πkρ The first item in the above equation is the radial inward (in the –ρ direction) TEM, and the second item denotes the radial outward (in the ρ direction) TEM. Since Hϕ is proportional to the radial current, the eq. (10.15) gives the excitation coefficient of the slot as well as the field distribution of the aperture. Let Γ = 0, then there is only the first item in eq. (10.15), that is, the radial inward transmission mode with pffiffiffi amplitude ρ. This results in a tilted distribution of the field, which is detrimental to antenna gain.

328

Chapter 10 Circularly polarized radial line array antenna

Assume that the radiation power of each cell area slot is proportional to the power in the waveguide and the rotational symmetry of the field is constant. Then the coupling can be considered. Using the parameter α to represent the coupling value, the aperture field distribution FðρÞ can be obtained: rffiffiffi rffiffiffi α jðk + αÞρ α jðk + αÞð2ρm − ρÞ −Γ (10:16) e e FðρÞ = ρ ρ This equation gives the excitation coefficient of the slot with equal density located at ρ. For the ideal case of Γ = 0, the initial tilt amplitude factor is corrected by factor eαρ . After calculation, the aperture distributions of different α values are shown in Fig. 10.9, where E is the field strength varying with the aperture distribution. The figure shows that when α = 5, the aperture distribution is relatively uniform. Once the uniform aperture field distribution is achieved, the aperture efficiency of the antenna is greatly improved.

Fig. 10.9: Distribution of aperture fields with different coupling factors.

10.2.4 Derivation of position alignment of gap The circular polarization of the whole aperture can be formed by a certain number of circular polarization aperture radiation cells that are distributed according to certain rules at the top plate of the upper radial waveguide. The design and placement of the slot pairs are divided into two steps: (1) the design of a single slot pair; (2) the

10.2 Design of circularly polarized radial line array antenna

329

design of slot pairs array on the whole aperture. In the design, only the wave mode of the radial inward transmission is considered and the variation of amplitude ð1Þ value is neglected. Besides, only the phase delay related to H1 ðkρÞ is considered and calculated. The slots are counted in the order of increasing sequence as shown in Fig. 10.10 (n = 1，2，3, . . ., N).

Fig. 10.10: The arrangement of the slot pair and its design parameters.

In Fig. 10.10, Pn is the distance from the origin O to the center Pn of slot Ln, β2n–1 is the angle between P2n–1O and P2nO, θ2n–1 is the dip angle of the slot to the radial direction, 2L is the length of the slot, δ is the distance between the two slots. The parameters δ and 2 L are constant and will be known as design parameters. Figure 10.10(a) represents the design parameters of the first pair slots (n = 1 and 2). As slots 1 and 2 are vertical to each other in space, the two slots have the same dip angle θ1 in the radial direction, and they are excited at the same amplitude. If the excitation phase difference between the two slots is 90°, the slot pair constitutes a circular polarization cell radiator. This condition can be expressed as follows: ð1Þ

ð1Þ

argðH1 ðkρ2 ÞÞ − argðH1 ðkρ1 ÞÞ =

π 2

(10:17)

where the arg(x) represents the phase of x and the following relationship is geometrically satisfied: ρ2 sinðθ1 Þ − ρ1 cosðθ1 Þ = L + δ

(10:18)

β1 = 2θ1 − π=2

(10:19)

When the initial value ρ1 is given, ρ2 ,β1 and θ1 can be obtained by solving the simultaneous eqs. (10.17), (10.18) and (10.19). After obtaining the geometrical parameters of one slot pair, it is necessary to find the geometric parameters of the adjacent slot pairs. To produce a circularly

330

Chapter 10 Circularly polarized radial line array antenna

polarized beam over the entire aperture surface, it is necessary to rotate the next slot pair by an angle ϕ3 to obtain an equal-phase wave front. ϕ3 satisfies the following formula: ð1Þ

ð1Þ

ϕ3 = argðH1 ðkρ3 ÞÞ − argðH1 ðkρ1 ÞÞ

(10:20)

According to eq. (10.20) and the geometric relationships in Fig. 10.10: ð1Þ

ð1Þ

ϕ3 = ρ21 + ρ23 − 2ρ1 ρ3 cosfargðH1 ðkρ3 ÞÞ − argðH1 ðkρ1 ÞÞg

(10:21)

In conclusion, ρ3 is determined by eq. (10.21), ϕ3 is determined by eq. (10.20) and the arrangement of successive slots can be obtained by iterations from eqs. (10.17)–(11.21). That is, ρ3 and θ3 are obtained from ρ1 in eqs. (10.17)–(10.20), and ρ5 and θ5 are obtained from ρ3 in eqs. (10.20)–(10.21). Thus, the changes in θn on the different slot pairs are small, so each slot is almost coupled to a radial current by a constant factor. If kρ >> 1, then eq. (10.20) can be simplified to ρ3 ðψÞ = ψ

λg + ρ1 2π

(10:22)

Equation (10.22) indicates that the trajectory change of the slot to the aperture is a helical line. Since ρðψ + 2πÞ − ρðψÞ = λg , Sρ should be approximately equal to λg , if Sϕ is set to be a constant, all slot pairs on the aperture are arranged in equal density.

10.2.5 Realization of dual-circular polarization radial slot array antenna In satellite live TV, to effectively utilize the stationary orbit and spectrum resources, the frequency-multiplexing phenomenon exists in some channel divisions. This phenomenon generates mutual interference in the effective bandwidth and so different polarized methods should be used to isolate the transmitted and received signals. Satellite live television uses a circularly polarized method to transmit signals. According to different satellite locations, the beam can be divided into LHCP or RHCP signals. Therefore, it is necessary for satellite live TV to design a high-gain antenna which can realize the LHCP and RHCP in the same frequency band. Based on the CP-RLSA antenna, this section uses a special feed structure to achieve the simultaneous radiation of LHCP and RHCP signals. For the usual single-polarization (LHCP or RHCP) RLSA antenna, its feed probe works in the lower radial waveguide region. When the feed probe excites the lower waveguide, the radial inward inner traveling wave (wave is propagated along the center of the disk) is formed in the upper radial waveguide. Its energy flow diagram is shown in Fig. 10.11(a). As previously mentioned, it is similar to the cylindrical TEM wave. When the feed probe extends into the upper waveguide region, the radial outward outer traveling wave (wave is propagated along the edge of the disk) is excited

10.2 Design of circularly polarized radial line array antenna

331

in the upper waveguide, as shown in Fig. 10.11(b). The energy flow diagram of the inner and outer waves in two different transmission directions is shown in Fig. 10.11.

(a) Inner wave

(b) Outer wave

Fig. 10.11: Radial waveguide internal energy flow.

According to Figs. 10.10 and 10.11, the phase difference between slot 1 and slot 2 is 90° when the inner traveling wave is excited in the upper waveguide of the radial line, as shown in eq. (10.17). When the outer wave is excited in the upper waveguide, the phase difference between slot 1 and the slot 2 is ð1Þ

ð1Þ

argðH1 ðkρ2 ÞÞ − argðH1 ðkρ1 ÞÞ = − π=2

(10:23)

Equation (10.23) shows that the phase difference between slot 1 and slot 2 is −90°. It can be concluded that the same radiation slot pair, when excited by the outer traveling wave and the inner travelling wave, respectively, can form circular polarization with an opposite rotation. This is the theoretical basis for the design of the dual-circularly polarized antenna in the current section. The excitation probe is located at the center of the radial waveguide. When the upper waveguide is required to generate the inner traveling wave, the probe needs to be excited in the lower waveguide. When the probe is excited in the upper waveguide, the upper waveguide transmits the outer travelling wave. According to this principle, the upper and lower excitation of the dual-layer radial line waveguide can be simultaneously achieved by using the structure of Fig. 10.12. The two-layer rectangular waveguide is used as a feed structure and the upper and lower rectangular waveguides share a common wall. A coaxial inner conductor is inserted into the lower rectangular waveguide to generate an induced current. It is transmitted along the coaxial line into the upper radial waveguide to excite the outer traveling wave. In the upper rectangular waveguide, the outer conductor of the coaxial line is regarded as its excitation probe, which is extended into the lower radial waveguide to excite the inner travelling wave in the upper radial waveguide. To improve the isolation of the two feed ports, the lower end of the outer conductor of the coaxial line is a short circuit to the common wall of the waveguide.

332

Chapter 10 Circularly polarized radial line array antenna

Fig. 10.12: Double circular polarization feed.

According to the earlier description, in Fig. 10.12, the lower rectangular waveguide of the feed structure excites the outer traveling wave in the upper radial waveguide and the upper rectangular waveguide excites the inner traveling wave in the lower radial waveguide. When the field modes of these two different transmission directions are applied to the circularly polarized slot pairs on the radial waveguide surface, it can make the RLSA antenna radiate two kinds of circularly polarized waves with opposite rotations. Figure 10.13 shows the simulation results of the feed structure. Port1 and Port2 respectively represent the upper and lower rectangular waveguides. The reflection coefficients of the ports are shown in the S11 and S22 curves. The reflection coefficient in the 11.75 GHz–12.25 GHz band is less than −10 dB. The energy transmission coefficients S13 and S23 transmitted by Port1 and Port2 to Port3 are both close to 0 dB.

Fig. 10.13: Reflection and transmission coefficient curve of each port.

10.2 Design of circularly polarized radial line array antenna

333

This section discusses the designs of a dual-CP-RLSA antenna operating at 12.0 GHz. In the upper radial waveguide, the PTFE dielectric material is filled to form a slow wave structure. The radial waveguide height d = 4 mm, the material thickness is 1 mm, the equivalent dielectric constant is 1.39, the wavelength of the waveguide at 12 GHz is 21.2 mm, the axial distance of the adjacent slot Sφ = 0.48λg and the radial distance Sρ = λg. The maximum radius of antenna ρmax = 100 mm. The design parameters are shown in Tab. 10.1.

Tab. 10.1: RLSA antenna design parameters. Maximum antenna radius ρmax Waveguide wavelength λg Upper (lower) radial waveguide height d Radial distance of slot pairs Sρ Axial distance of slot pairs Sφ Dielectric constant εr Slot length Slot width Slot pair quantity

 mm . mm  mm λg .λg .  mm  mm （pairs）

Figure 10.14 shows a model of a dual-CP-RLSA. The antenna is mainly composed of a radial waveguide. The CP slots are pairs arranged regularly on the radial waveguide surface. The slot pairs are arranged in an Archimedes spiral pattern on the waveguide plane, which is implemented in the form of a DXF file. In Fig. 10.14, when the port Port 1 is fed, in the upper radial waveguide, the outer travelling wave transmitted along the edge of the disk is excited to form the RHCP radiation wave. When the Port 2 is fed, in the lower radial waveguide, the inner traveling wave transmitted along the center of circle is excited to form the LHCP radiation wave. Figure 10.15 shows the VSWR curve of the radial waveguide excited by Port 1 and Port 2, respectively. When Port1 excites the upper radial waveguide, its VSWR is good and less than 1.3 in the entire frequency band, as shown by the red curve. When Port 2 excites the lower radial waveguide, its VSWR is large but less than 2 in the whole frequency band. The reflection characteristics of the antenna can be improved by correcting the probe structure in the lower layer radial waveguide. Figure 10.16 shows the radiation pattern at 12 GHz excited by Port 1 and Port 2, respectively. The RHCP gain is 24.3 dBic (the outer traveling wave excited by the Port 1) and the LHCP gain is 25.6 dBic (the inner traveling wave excited by the Port 1). The aperture efficiency of LHCP and RHCP at 12 GHz is about 57.5% and 42.6%, respectively. Figure 10.17 shows the axial ratio directional pattern of the antenna on the XOZ plane. The axial ratios of the two polarizations are both less than 3 dB in the range of ±4°.

334

Chapter 10 Circularly polarized radial line array antenna

z

y

x Fig. 10.14: Simulation model of dual-circular polarization antenna.

Fig. 10.15: VSWR of antenna.

Theoretically, the distribution of the field wave formed on the radial waveguide radiation plane is more uniform when the inner traveling wave excited by Port2 acts on the slot pairs and so the LHCP gain is relatively higher. When the outer traveling wave excited by the Port1 is transmitting to the waveguide edge, the wave will be attenuated during transmission. So, when the radiation slot pairs of the aperture plane are in uniform distribution, the field intensity distribution of the antenna aperture is uneven and the antenna efficiency is, consequently, reduced. To make the pattern gain of the two polarization modes consistent, the length and the arrangement

10.2 Design of circularly polarized radial line array antenna

335

27 24 21 18

RHCP _Gain

15

LHCP _Gain

12

Gain (dB)

9 6 3 0 –3 –6 –9 –12 –15 –18 –21 –180

–120

–60

0

60

120

180

Theta (deg) Fig. 10.16: A 12 GHz antenna radiation pattern.

Fig. 10.17: The AR directional pattern of the antenna at the XOZ plane at 12 GHz.

position of radiation slots on the aperture plane should be modified. In the above simulation, the LHCP gain excited by the inner traveling wave is greater than that of the RHCP excited by the outer traveling wave and the simulation results are consistent with the theoretical analysis.

336

Chapter 10 Circularly polarized radial line array antenna

10.2.6 Circularly polarized radial line helical array antenna The structure of the spiral array antenna is shown in Fig. 10.18. The radial line waveguide conducts energy transmission and distribution and the short spiral antenna radiates the internal energy to the free space. The radial line waveguide and the array cells are connected by coupling feed probes and coupling ports. By adjusting the length and position of the probes, the array cell antenna can obtain different amplitude and phase excitations.

Fig. 10.18: Circularly polarized radial line spiral array antenna.

Figure 10.19 shows a certain X-band circular polarization radial line short spiral antenna.

Fig. 10.19: An X-band circularly polarized radial short spiral antenna prototype.

References

337

References [1] [2]

H. Wang (translator), Я.Д.Ширман (author). Radio Waveguide and Cavity Resonator. Bejing, Science Press, 1962. M. Wang, S.W. Lv, R. Liu. Array antenna analysis and synthesis. Chengdu, UEST Press, 1989.

Index Antenna Factor 23 Antenna Field Zones 7 antenna radiation 4, 6, 9, 17, 97, 165, 253, 272, 300, 314 Aperture Coupling 92 Aperture Field Distribution 327 appropriate degenerate modes 120 average power density 4 axial-mode helix antenna 159, 161, 170–173, 175, 180 Axial Ratio 31, 39, 199, 204 Axis Length 199 Baffle Circular Polarizer 277, 286 balanced feed structure 194, 205 Balun-fed 53 basic sinusoidal logarithmic curve 268 Beam solid angle 7 Beam width 58, 199, 203 Beidou navigation 194 bifilar helix antenna 46, 187, 195, 197 broadband 78, 111, 142, 236 Circular Polarization 28, 33–34, 45, 263 circular polarization ratio 34, 39 Circularly Polarized Horn Antennas 275 circularly Polarized Radial Line Array antenna 3, 319 Circularly Polarized Radial Line Helical Array antenna 336 Concentric Ring Array Theory 322 Conical Helix Antenna 176 Conical Logistic-Spiral Antenna 255 coordinate system 4, 10, 26, 30, 33, 50 Corrugated Horn Antenna 283 Cross-Polarization 41 crossed dipole 2, 46, 49, 51, 53–54, 58, 60, 63, 70, 73, 76, 78–79, 194, 206 Cross-Polarization Identification 42 cross-polarization isolation 44–45 Derivation of Position Alignment of Gap 328 Diaphragm Circular Polarizer 276 Dielectric Loading Method 221 Dielectric Plate Circular Polarizer 279 direction of propagation 4, 30

https://doi.org/10.1515/9783110562804-012

directionality coefficient 7 Directivity 6, 12, 164, 258 Dual-Band 102, 115 Dual-Band Quadrifilar Helix Antenna 224 Dual-Circular Polarization Radial Slot Array Antenna 330 Dual-Layer 111 Dual-mode Conical Horn antenna 281, 292 effective area 14–15 Effective Length 14 electric and magnetic fields 4 electric field 4, 9–10, 14, 19–20, 23, 26–30, 32, 34, 36, 41, 88, 142, 198, 275–276, 279, 310 electromagnetic wave V, 4, 14–15, 19–20, 23, 26, 36, 51, 233, 275, 281 Elliptic Equation Derivation 34 Elliptically Polarization Wave 29 Equivalent isotropically radiated power (EIRP) 20 far-field (Fraunhofer) regions 7 first-null beamwidth (FNBW) 12 Folding Method 219 Frequency-independent Antenna 229 Friis Transmission Equation 21–22 Gain 6, 58 GPS 116, 164–165, 194 half-power beamwidth (HPBW) 11 Helix antenna 159 Helix antenna with reflective cavity 178 helix diameter with fixed length 171 high-efficient 275 high-gain 275, 324 H-Sharped Aperture-Coupled Microstrip Antenna 142 incident field 24 input Impedance 16 intrinsic impedance of the medium 4 Linear Polarization 27, 32, 299 linear polarization ratio 32 Log-Periodic CP Antenna 264 log-periodic dipole antenna (LPDA) 264

340

Index

magnetic field 4, 14, 26, 320, 323, 327 maritime communication 125, 194 Meandering Method 217 microstrip antenna (MSA) 87 Microstrip Line Feed 91 Microstrip Proximity Coupling 93 Modified axial-mode helix antenna 173 Multiarm Helix Antenna 187 Multifed Circularly Polarized Microstrip antenna 142 multifed network 87, 94 Multimode Conical Horn Antenna 282 Multiple L-shaped Probe-fed Circularly Polarized Microstrip Antenna 145 Nonuniform Helix Antenna 177 Normal Mode Helix Antenna 183 number of turns 160, 170, 172, 177, 187, 194–195 omnidirectional circularly polarized antenna 2, 296–297, 299, 301, 304–305, 313 Omnidirectional Circularly Polarized Antenna utilizing bended monopoles 309 Omnidirectional Circularly Polarized Antennas Based on Omnidirectional Cone Antennas fed by CP Waves 315 Orthogonal Circular Polarization Waves 36 perturbation 103–104, 111, 120, 129, 133, 137, 140 phase difference 75, 94, 99, 120, 125, 129, 131, 137, 140, 142, 162, 186, 194–195, 197–198, 204, 211, 232 Phase-Shifting Network 211 pitch angle with fixed length 170 Pitch Ratio 204 Planar Equiangular Spiral Antenna 247 Plane Archimedes Spiral Antenna 231 Plane Electromagnetic Wave Electric Field Vector 26 polarization V, VI, 237 Polarization efficiency 37–38, 40 polarization Ellipse Dip Angle 30 Polarization Loss 37 Polarization Ratio 32–33, 38 Poynting vector 4, 50 Printed Self-phased Crossed dipole Antenna 70

quadrifilar helix antenna 159, 187, 194, 217, 222 Quadrifilar Helix Antenna with Phase-shifting Network feed 211 Quasi-Square 118 Radial Antenna 319 Radial Line Array Antenna 319, 324 Radial Transmission Mode 320 radiating near-field (Fresnel) 7 radiation intensity 5, 9, 11 Radiation Pattern 9, 163 Radiation Power Density 4 reactive near-field 7 received power 22–24, 37 receiving antenna 14–15, 20–24, 37–38, 40, 44, 225, 275 Resonant Cavity Circular Polarizer 280 Return loss 18, 101 Rotation Angle 201–202, 204 Screw Circular Polarizer 276 Self-Phase-Shifting Quadrifilar Helix Antenna with Infinite Barron Feed 207 Self-Phase-Shifting Quadrifilar Helix Antenna with Slotted Balun Feed 208 Self-Phase-Shifting Structure 206 Self-phased 49, 53 Sidelobe Level 12 single probe feed 87 sinusoidal antenna 230, 268, 270–272 Sinusoidal circularly polarized antenna 268 Slot-loaded 98, 102 slow wave structure 125, 324–325 solid angle 5, 7 Spherical Waves 4 Square Patch 96, 111, 118 stacked 115, 118, 120 Stub Loaded 106 Tapered-end Helix Antenna 175 Truncated Corners 96, 111 wave impedance 24 waveguide structure 320, 325 Wideband Quadrifilar Helix Antenna 223