RF Antenna Beam Forming: Focusing and Steering in Near and Far Field 3031217640, 9783031217647

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Shun-Ping Chen · Heinz Schmiedel

RF Antenna Beam Forming Focusing and Steering in Near and Far Field

RF Antenna Beam Forming

Shun-Ping Chen · Heinz Schmiedel

RF Antenna Beam Forming Focusing and Steering in Near and Far Field

Shun-Ping Chen Hochschule Darmstadt University of Applied Science Darmstadt, Germany

Heinz Schmiedel Hochschule Darmstadt University of Applied Sciences Darmstadt, Germany

ISBN 978-3-031-21764-7 ISBN 978-3-031-21765-4  (eBook) https://doi.org/10.1007/978-3-031-21765-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Reinhard Dapper This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Michael, Anna, Katja and Angela

Preface

This book has been written for students in Electrical and Electronics Engineering and Information Technology. It is also written for radio frequency (RF) engineers to give a technology oriented overview of phased array antennas, beam steering and beam forming. This may allow engineers and also project managers, involved in RF design, to develop visions and opportunities with their own applications. Phased array antennas have been developed since the 1960s and have seen many applications. Some of these use fixed arrays to increase the aperture of an antenna system to obtain a desired gain and performance. Others use controlled phases and amplitudes to allow beam steering and beam forming. The antenna system performance depends on the antenna elements and on the proper phase and amplitude control. With phase control for the individual antenna elements an antenna beam can be steered over a wide angle range. By additionally controlling the amplitudes for all antenna elements individually the beam can be formed to given specification, e.g. side lobes suppression. Well known, successful applications of steerable antenna arrays are e.g. part of the PATRIOT air defense system. Also IRIDIUM relies on phased array panels with steerable beams for worldwide telephone and data communication. Beam steering is also essential for existing and future 5G mobile communication systems. A whole chapter in this book has been dedicated to these important MIMO-antenna systems. Most readers will be familiar with Maxwell’s equations and electromagnetic waves. Nevertheless a first chapter is a brush up for Maxwell’s equations and electromagnetic wave theory, as far as needed as foundation for the antennas described later in the book. It may help the reader to get started into the antenna topic. In the following chapter basic antenna elements are described. All these may be used in phased array antenna systems. Well known are dipole arrays for TV reception, helical antenna arrays on the GPS satellites, patch antennas on the IRIDIUM satellites, and many more. Next is a comprehensive chapter on linear antenna arrays. It explains a complete set of arrangements for defined steering and beam forming. Comprehensive simulation results, along with measurement results, are discussed in detail. Examples are shown for defined beam steering angles. With an appropriate distribution of amplitude weights the shape of the beam can be designed. Equal weight, Chebyshev and binomial, that VII

VIII

Preface

is similar to a Gaussian weight, are investigated, simulated and measured for different beam steering angles. The performance of these linear antenna arrays is investigated for far field and near field also. The results for these linear arrays are put into a two-dimensional context in the next chapter. Planar arrays, where the antenna elements are placed in a two-dimensional plane, are investigated. Again beam steering is simulated and beamforming is applied to obtain desired beam shapes. So far all descriptions were related to antenna arrays oriented in one plane only. In a logical next step so called conformal arrays will be considered. It begins with a modification of the linear array which will have a defined curvature now. All antenna elements are placed on a contour which is a part of a circle’s circumference. There are two options, one is the concave and the other the convex configuration. The concave arrangement allows, as a special case, to irradiate a target in a focus point with equal distance and equal phase to all antenna elements. The convex arrangement on the other side is interesting to consider for an antenna tower and 360° coverage. Here the antenna elements can be controlled in phase, and amplitude, so that the antenna beam has full 360° coverage. In our results a fraction of the whole circle is investigated in detail, inclusive beam steering and beam forming. After describing these specific antenna designs, applications are considered in the next chapters. After an introduction the specific application of state-of-the art MIMOantenna systems is covered comprehensively. In MIMO applications antenna beams need to be switched fast between different users. Also interference cancellation is an issue, when many users are present. The application part is complemented by considering so called thinned arrays for deep space communication applications. The authors would like to interest the readers into the exciting world, visions and opportunities of phased array antenna systems with all the options of beam steering and forming. Darmstadt October, 2022

Shun-Ping Chen Heinz Schmiedel

Acknowledgements

The authors would like to thank all colleagues of the Institute of Communication Technologies, Darmstadt University of Applied Sciences h-da, for enabling and supporting the research activities, and for many helpful discussions. Shun-Ping Chen would like to thank Prof. Arne Jacob, former head of the Institute of High Frequency Engineering at Technical University Hamburg, for involving him in the investigation projects of near field beam forming and beam focusing in August 2011– March 2012 during his research sabbatical. Since then these became one of his favorite research topics. The authors also appreciate the strong support of the Phase and Amplitude Control Matrix supplier Mitron Inc. Especially they like to thank its president and system designer, Mr. Wei Li, and his developer colleagues, for rendering every assistance and several video conferences during the first period after the implementation of the system in our laboratory, enabling us to successfully start with the experiments.

IX

Contents

1 Introduction to Electromagnetic Wave Propagation. . . . . . . . . . . . . . . . . . . . 1 1.1 Maxwell’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vector Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Wave Propagation in Different Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Basic Antenna Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Isotropic Radiator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Dipole Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Patch Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Horn Antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Linear Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Introduction into Phased Array Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Phase Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Measurement Setup and Verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Far Field Characteristics of a Linear Patch Antenna Array. . . . . . . . . . . . . 31 3.5 Amplitude Weights and Side Lobes of the Radiation Pattern. . . . . . . . . . . 36 3.6 Near Field Characteristics at a Distance of 10  . . . . . . . . . . . . . . . . . . . . . 37 3.7 Near Field Characteristics at a Distance of 5.6 . . . . . . . . . . . . . . . . . . . . . 44 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Planar Antenna Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Beam Forming, Focusing and Steering of Planar Arrays. . . . . . . . . . . . . . . 50 4.2 Side Lobe Suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Simulation Results for Planar Antenna Arrays . . . . . . . . . . . . . . . . . . . . . . 56 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Conformal Antenna Array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Beam Forming of One-Dimensional Conformal Concave Arrays. . . . . . . . 63 5.2 Beam Forming of One-Dimensional Conformal Convex Arrays. . . . . . . . . 77 XI

XII

Contents

5.3 Beam Forming of 2-Dimensional Conformal Arrays . . . . . . . . . . . . . . . . . 92 5.4 Comparison of Convex, Concave and Planar Profiles. . . . . . . . . . . . . . . . . 94 5.5 Simulation Results for Conformal Antenna Arrays. . . . . . . . . . . . . . . . . . . 97 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6 MIMO Antenna Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7 Thinned Antenna Array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8 Applications of Phased Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.1 Radar, Broadcasting and Positioning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2 Smart Antenna Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.3 5G Massive MIMO Basestation for Multi-User Coverage . . . . . . . . . . . . . 124 8.4 Deep Space Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5 Radio Astronomy Antenna Array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 9 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Radio Frequency Bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

1

Introduction to Electromagnetic Wave Propagation

1.1

Maxwell’s Equations

Maxwell’s equations are the governing equations to analyze all electromagnetic wave propagation problems, from RF radio waves used in cellular mobile communications 1–2 GHz, mm waves (frequency range of 30–100 GHz) up to optical wave propagation in the frequency range of 200 THz (see also [1–5]). Depending on the applications, different carrier frequencies are used from 1 GHz to mm-waves or even to the optical spectrum. These frequencies have different abbreviations, standardized by different organizations like IEEE, ITU, EU and NATO, and are listed in the appendix A Tab 1.1 shows electromagnetic field parameters and units. ∂B (1.1) ∇ ×E=− , ∂t ∂D + J, (1.2) ∇ ×H= ∂t ∇ · D = ρ, (1.3) ∇ · B = 0.

(1.4)

Table 1.1 Electromagnetic field parameters and units Field Vector

Symbol

Unit

Electric field

E

V/m

Dielectric displacement/density

D

As/m2

Magnetic field

H

A/m

Magnetic flux density/induction

B

T = Vs/m2

Current density

J

A/m2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-P. Chen and H. Schmiedel, RF Antenna Beam Forming, https://doi.org/10.1007/978-3-031-21765-4_1

1

2

1 Introduction to Electromagnetic Wave Propagation

1.2

Vector Algebra

By using vector algebra, the above Maxwell’s equations can be derived in different coordinate systems depending on the problems to be investigated. For example, for rectangular metallic waveguides, the Cartesian coordinate system (x, y, z, t) will be meaningful, whereas for problems of cylindrical metallic or quartz glass optical waveguides, the cylindrical coordinate system (r , φ, z, t) will be used. For the antenna problems the polar coordinate system (r , θ, φ, t) will be helpful. In the following we will briefly discuss the vector algebra and the derivation of Maxwell’s equations into the Helmholtz equations as a good approximation [2, 5]. Generally a dynamic field is a physical quantity with different magnitudes and orientations at different positions and different points in time. Vector fields that vary with different frequencies f have always characteristic values or magnitudes, and point into certain directions at certain positions. For example: Electric field E(x, y, z, t) and dielectric displacement D(x, y, z, t); Magnetic field H(x, y, z, t) and magnetic induction B(x, y, z, t); Vector potential A(x, y, z, t); Mechanical force F(x, y, z, t); Velocity of solid body or fluid v(x, y, z, t). In comparison with vector fields, scalar fields have only a value or a magnitude, and are direction-independent. For example: Temperature field T(x, y, z, t); Potential Φ(x, y, z, t); Electric charges q(x, y, z, t) or charge density ρ(x, y, z, t). Vector algebra definitions are the following   ∂ ∂ ∂ , , , ∇= ∂ x ∂ y ∂z  ∇E =

∂ Ex ∂ E y ∂ Ez , , ∂ x ∂ y ∂z

(1.5)

 = grad E,

(1.6)

∇ · E = ∇x E x + ∇ y E y + ∇z E z = div E,

(1.7)

∇ × E = rot E = curl E,

(1.8)

( ∇ × E) z = ∇x E y − ∇ y E x ,

(1.9)

1.3 Wave Propagation in Different Media

3

( ∇ × E) x = ∇ y E z − ∇z E y ,

(1.10)

( ∇ × E) y = ∇z E x − ∇x E z .

(1.11)

Useful rules of vector algebra are

1.3

∇ · ( ∇T ) = ∇ 2 T ,

(1.12)

∇ × ( ∇T ) = 0,

(1.13)

∇( ∇ · E) = a vector,

(1.14)

∇ · ( ∇ × E) = 0,

(1.15)

∇ × ( ∇ × E) = ∇( ∇ · E) − ∇ 2 E,

(1.16)

( ∇ · ∇)E = ∇ 2 E.

(1.17)

Wave Propagation in Different Media

Forces, acting on a charge q, due to electric and magnetic fields F = q(E + v × B).

(1.18)

Principle of superposition of fields in linear media E = E1 + E2 .

(1.19)

From Eqs. (1.1) and (1.2) we see that electric and magnetic fields are always related to each other and form the electromagnetic waves transporting energy in a certain direction, as defined by the Poynting vector S = E × H.

(1.20)

4

1 Introduction to Electromagnetic Wave Propagation

The total power, that is transported through the area S  , will be  P= E × H · ds. S

(1.21)

For the investigation of wave propagation or antenna problems, the waves are generated by the current density J of a source, with a certain carrier frequency f , modulated with the information data in manifold modulation schemes. These waves will be matched to the antenna (for example dipole antenna, patch antenna, horn antenna, helix antenna or parabolic antenna) through the so-called reactive near field, then radiated to the near field and far field free space which is vacuum in the simplest case. The reactive near field (r < R1 ), radiating near field (Fresnel region, R1 < r < R2 ) and far field (Fraunhofer region, r > R2 ) are characterized by R1 and R2 [3] where D is the diameter of the antenna or largest dimension of an antenna array  R1 = 0.62

D3 , λ

(1.22)

2 D2 . (1.23) λ The description of electromagnetic waves in a generally inhomogeneous dielectric isotropic medium requires generally numerical methods, in order to solve the differential equations. In some cases, some simplified assumptions help to solve these differential equations analytically, as will be discussed later. By considering the generally frequencydependent characteristics of the electromagnetic waves in a generally inhomogeneous dielectric medium at the frequency f or angular frequency ω = 2π f , we have  ∞ E(r, t)e− jωt · dt. (1.24) E(r, ω) = R2 =

−∞

This equation is the Fourier transform of the electric field. It allows us to calculate the frequency spectrum from a given function versus time. Interpreting this equation, we see that a signal with a high data rate, and where the signal amplitude changes rapidly with time, has a wide frequency spectrum or needs a large bandwidth. With this complex notation, Eqs. (1.1) and (1.2) give ∇ × H = jωD + J,

(1.25)

∇ × E = − jωB.

(1.26)

In generally inhomogeneous media with complex dielectric permittivity coefficients and complex magnetic permeability coefficients, the relations between the electric field, the

1.3 Wave Propagation in Different Media

5

magnetic field, the dielectric displacement and the magnetic flux density can be described as follows 







D(r, ω) = (ε (ω) − jε (ω))E(r, ω), B(r, ω) = (μ (ω) − jμ (ω))H(r, ω),

(1.27) (1.28)

whereas the real parts correspond to the so-called dispersion, the imaginary parts correspond to the losses caused by absorptions. Dispersion describes the frequency-dependent characteristics, i.e. different wave propagation velocities at the different wavelengths. These relations are described by the so-called Debye Dispersion Model which describes the delayed reaction of the molecular dipoles to the applied electromagnetic waves. See for example [4], with the relaxation time τ which describes the retardation of the molecular dipole in response to the excitation field. 

εr (0) − εr (∞) ε (ω) = εr (ω) = εr (∞) + , ε0 1 + (ωτ )2

(1.29)



ε (ω) εr (0) − εr (∞) = ωτ . ε0 1 + (ωτ )2

(1.30)

Additionally the phase shift between D and E is caused by the absorption loss. By considering the absorption loss of the dielectric medium and the Ohm’s loss caused by the limited conductivity κ, the total power generated by the source will be 1 − 2



1 E · J · dV = 2 ∗

V

+



E × H∗ · dA 

A

1 2

κ | E |2 · dV − V

1 2



(E · D∗ − B · H∗ ) · dV .

(1.31)

V

The left hand term represents the generated power at the transmitter source, for example by a dipole antenna with the excitation current of a certain carrier frequency f, whereas the first term of the right hand expressions is the Poynting vector, or the radiated power from this antenna through the medium, e.g. free space. The real part of the second and third term represent the loss caused by the absorption, polarization loss and the limited conductivity  of the medium that is directly related to ε or tanδ . 



ε = ε − jε − j

κ  = ε (1 − j tan δ), ω

(1.32)



κ + ωε tan δ = .  ωε

(1.33)

6

1 Introduction to Electromagnetic Wave Propagation

For homogeneous, time-invariant media (grad ε = 0), i.e. free space in the most relevant cases of antenna propagation problems, we get the simplified Helmholtz equations by using the complex dielectric permittivity ∇ 2 E + ω2 μεE = 0,

(1.34)

∇ 2 H + ω2 μεH = 0,

(1.35)



κ + ωε ε =ε 1+  jωε



 .

(1.36)

With the definition of the so-called wave number k and wave propagation velocity v, where v will be the free space velocity of light, if the medium in the most simple case is vacuum (μ = μ0 , ε = ε0 ). In many special situations the inhomogeneous media have to be taken into account, for example electromagnetic waves in optical frequency range suffer amplitude scintillation due to the dynamic refractive index scintillation caused by atmospheric turbulences, or radio frequency amplitude and phase scintillation during the propagation through plasma. In the first case the laser optical beam propagation will be disturbed [6] leading to beam scattering and beam wander, whereas in the second case the radio frequency waves will suffer phase shift [7]. It should be pointed out that these special applications are not being discussed in this book. Instead we would like to treat the most general cases of free space antenna propagation scenarios ω √ (1.37) k = ω με = . v This simplest case is also most important for wireless radio frequency communications in vacuum (μr = 1, εr = 1, v = c), which is considered in the following chapters in this book. For homogeneous anisotropic dielectric media the independent permittivity coefficients become dependent on the orientation in certain media, for example crystals, semiconductors, etc. In this case the dielectric permittivity coefficients must be described by using a tensor matrix. A similar tensor matrix is valid for the anisotropic magnetic medium. For inhomogeneous dielectric, isotropic, non-magnetic and lossless media, the field components of the electromagnetic waves travelling in an inhomogeneous, dielectric, isotropic, lossless medium, which is an ideal assumption of course, could be described by the differential equations generally depending on permittivity and permeability parameters. These differential equations can be solved by using numerical methods. In some cases the inhomogeneous medium can be separated into several homogenous media. By doing so, the differential equations can be solved in each region by using analytical methods, for example in case of step-index dielectric optical fiber consisting of core and cladding [2].

1.3 Wave Propagation in Different Media

7

Considering a homogeneous medium (ε = constant or ∇ε = 0), for example free space which is the case for antenna problems, the differential equations can be simplified to the so-called Helmholtz equations with the amplitude E of the electric field E and unit vector of the polarization plane u with u = x · ux + y · uy + z · uz , ∇2 E =

∂2 E ∂2 E ∂2 E ∂2 E + + = με ∂x2 ∂ y2 ∂z 2 ∂t 2

(1.38)

(1.39)

in Cartesian coordinate systems (x, y, z). Correspondingly the differential equations can be given in cylindrical coordinate systems (ρ, φ, z) as ∇2 E =

1 ∂ ρ ∂ρ

  ∂E 1 ∂2 E ∂2 E ∂2 E ρ + 2 + = με ∂ρ ρ ∂φ 2 ∂z 2 ∂t 2

(1.40)

or in spherical coordinate systems (r , θ, φ), depending of the antenna problems under investigation as 1 ∂ ∇ E= 2 r ∂r 2

    ∂2 E 1 ∂ ∂ 1 ∂2 E 2∂E = με . r + 2 sin θ + 2 2 ∂r r sin φ ∂θ ∂θ ∂t 2 r sin θ ∂φ 2

(1.41)

Correspondingly we can derive the Helmholtz equations for the magnetic fields for homogeneous free space in a Cartesian coordinate system, a cylindrical coordinate system and a spherical coordinate system ∂2 H ∂2 H ∂2 H ∂2 H + + = με 2 , 2 2 2 ∂x ∂y ∂z ∂t   ∂H 1 ∂2 H 1 ∂ ∂2 H ∂2 H ρ + 2 ∇2 H = + = με 2 , 2 2 ρ ∂ρ ∂ρ ρ ∂φ ∂z ∂t ∇2 H =

1 ∂ ∇ H= 2 r ∂r 2

(1.42)

(1.43)

    ∂2 H 1 ∂ ∂ 1 ∂2 H 2∂H = με . (1.44) r + 2 sin θ + 2 2 ∂r r sin φ ∂θ ∂θ ∂t 2 r sin θ ∂φ 2

One very useful transformation between a spherical coordinate system and a Cartesian coordinate system can help to convert the vector potential, electric or magnetic field components into another coordinate system, in order to enable a very efficient analysis process ⎡

⎤ ⎡ ⎤ ⎤⎡ Ar sin θ cos φ sin θ sin φ cos θ Ax ⎣ Aθ ⎦ = ⎣ cos θ cos φ cos θ sin φ − sin θ ⎦ ⎣ A y ⎦ . Aφ Az − sin φ cos φ 0

(1.45)

8

1 Introduction to Electromagnetic Wave Propagation

Similarly, for the extremely small variation of the relative dielectric permittivity less than 1%, for example in case of a dielectric waveguide like quartz optical fiber, the so-called “weekly guidance condition” is valid, where the relative gradient of the permittivity vanishes to zero, so that the Helmholtz equation is also valid. The wave number vector is pointing into the propagation direction of the electromagnetic waves and c is the vacuum speed of the light. The corresponding solution can be found similarly. The simplest solution for Maxwell’s equations or the Helmholtz equation could be the so-called transverse electromagnetic (TEM) waves. For a TEM wave polarized in x-direction, the power density, represented by the Poynting vector S, will be transported in z-direction E(r, t) = E(r, t) · ux ,

(1.46)

H(r, t) = H (r, t) · uy ,

(1.47)

S(r, t) =| E(r, t) × H(r, t) | ·uz .

(1.48)

where r is a vector pointing to the observation position in space. The ratio between the perpendicular electrical and magnetic field 120 π Ω or 377 Ω is a constant, and is defined as the characteristic impedance of the medium which could be air or a homogeneous dielectric medium. Helmholtz equations are valid for the far field. The governing Maxwell’s equations can be used to explain almost all the wave propagation mechanisms, both for free space wave propagation, antenna problems and optical waveguides.

References 1. 2. 3. 4.

R. E. Collin, F. J. Zucker: Antenna Theory, Part 1. McGraw-Hill Book Company (1969). H.-G. Unger: Planar Optical Waveguides and Fibres. Oxford Clarendon Press (1977). C. A. Balanis: Antenna Theory. John Wiley & Sons, Inc. Fourth Edition (2016). K. W. Kark: Antenna Antennas and Radiation Fields (in German: Antennen und Strahlungsfelder). Springer Vieweg (2018). 5. S.-P. Chen: Fundamentals of Information and Communication Technologies. Cambridge Scholars Publishing (2020). 6. S.-P. Chen: Investigations of free space and deep space optical communication scenarios. CEAS Space Journal, Springer Nature (2021). 7. S.-P. Chen, J. Villalvilla: Comparison of Modified Woo’s Solar Phase Scintillation Model with ESA’s BepiColombo Superior Solar Conjunction Measurement Data for X-Band and Ka-Band. CEAS Space Journal, Springer Nature (2022).

2

Basic Antenna Elements

Antenna problems, free space radio frequency and optical wave propagation, multipath propagation or scattering problems can be analyzed by using Maxwell’s equations described in the last chapter. Generally the transmit antenna is excited by the current density J, generating and emitting a wave which propagates into free space in a certain defined direction, thus transmitting information to a receiver. By considering the current density distribution of different radiators like dipoles, patches and horns, the vector potential A(r, t) is used to derive the magnetic field H and electric field E at an observation point at a distance r from the source r  A(r) =

S

J(r )



e−jk|r−r | dS  , 4π | r − r |

H = ∇ × A, E=

1 1 ∇ ×H= (grad divA + k 2 A). jω0 jω0

(2.1) (2.2) (2.3)

Typical and often-used antenna types (Fig. 2.1) with special radiation characteristics are: • • • • • • • •

Theoretical, hypothetical isotropic radiator Dipole antenna Patch antenna Horn antenna Helix antenna Parabolic antenna Cassegrain antenna Slot antenna, etc.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-P. Chen and H. Schmiedel, RF Antenna Beam Forming, https://doi.org/10.1007/978-3-031-21765-4_2

9

10

2 Basic Antenna Elements

The performance and characteristics of the different types of antennas are characterized by the antenna gain G which describes the increase of the power density in the wanted direction of the antenna radiation beam in comparison with the isotropic radiator (dBi). The isotropic antenna radiates into all directions equal power density. It is a very useful theoretical hypothetical model. The higher the gain, the narrower the antenna radiation beam will be. The increase in power density into the wanted direction is related to the decrease of radiated power density into other, unwanted directions. The total radiated power stays constant. The hypothetical isotropic antenna is considered as a reference. The antenna radiation characteristics can be described by the so-called directivity D(r ) =

S(r , θ, φ)max . Sr ,θ,φ

(2.4)

Since the radiation field distributions in the near field and far field are not exactly the same, D(r ) varies slightly depending on the distance between the antenna source and the observation point. For the far field where r  R2 , the directivity will be the same for r [4] D=

S(θ, φ)max , < Sθ,φ >

(2.5)

Pt (2.6) 4πr 2 with Pt as transmit power of the source or generator and Pr as radiated power from the antenna, S(θ, φ)max as the maximum power density, and < Sθ,φ > as power density of the isotropic radiator. Therefore the directivity of the antenna will be < Sθ,φ >=

D = 4πr 2

S(θ, φ)max . Pr

(2.7)

The above relations do not consider a) reflexion caused by possible non-perfect matching of the source, tapered waveguides and antenna elements, b) conductor and c) dielectric losses. These effects can be generally taken into account by an efficiency coefficient η representing the ratio of the effectively radiated power Pr and the source generator power Pt including the power losses Pl , in order to calculate the antenna gain G η=

Pr Pr = , Pt Pr + Pl

G = η D = 4πr 2

S(θ, φ)max . Pt

(2.8) (2.9)

The antenna gain is normally expressed in dBi with respect to the isotropic radiator gdBi = 10 · log10 (G).

(2.10)

2.1

Isotropic Radiator

11

Fig. 2.1 Often used antennas Table 2.1 Typical antenna properties Type

Polarization

Isotropic antenna

Gain 0 dBi

Dipole antenna

Linear

1.5–2.3 dBi

Patch antenna

Linear, circular

7–10 dBi

Horn antenna

Linear

20 dBi

Helix antenna

Circular

10–18 dBi

Parabolic antenna or Cassegrain antenna

Linear, circular

20–70 dBi

In Fig. 2.1 and Table 2.1 the radiation characteristics of typical or often-used antennas, which will be utilized in this book for the discussion of antenna array beam forming, are summarized.

2.1

Isotropic Radiator

An isotropic radiator is a theoretical or hypothetical antenna which radiates homogeneously in all directions. The magnitude of the Poynting vector or power density S at a certain distance r is constant    S(r , θ, φ)   = 1.  (2.11)   S max

12

2 Basic Antenna Elements

The isotropic radiator is used as a reference to compare the directivities of different other antennas. The enhanced directivity in a certain direction, in comparison with the isotropic radiator, is also called antenna gain G > 1 (obviously the gain of the isotropic radiator is then G = 1 or g = 0 dBi). In terms of logarithmic dB values, the gain of a generally directive antennas will be g > 0 dBi, whereas the gain of the isotropic antenna will be g = 0 dBi.

2.2

Dipole Antenna

A perfect dipole antenna is an infinitesimally thin wire (diameter d  λ) along the z-axis, positioned at the arbitrary point (r  , θ, φ) of the spherical coordinate system, for example at the center of the coordinate system (Fig. 2.2). The surface integral of the vector potential (2.1) becomes a line integral with a source current I = I0 · uz (see for example [3, 4]) 

 l/2   e−jk|r−r | e−jk|r−r |   = μI(r ) dl dl  4π | r − r | 4π | r − r | −l/2 l  l/2  e−jk|r−r | μI0 l  = uz μI0 dl  = uz e−jk|r−r | ,   4π | r − r | −l/2 4π | r − r |

A(r) =

μI(r )

Ar = A z cos θ =

(2.12)

μI0 l  e−jk|r−r | cos θ, 4π | r − r |

(2.13)

μI0 l  e−jk|r−r | sin θ,  4π | r − r |

(2.14)

Aφ = 0.

(2.15)

Aθ = −A z sin θ =

By using the Eqs. (2.1)–(2.3) the electric field and magnetic field components can be generally calculated Hr = Hθ = 0,   k I0 l sin θ 1  · e−jk|r−r | , · 1 + 4π | r − r | jk | r − r |   η I0 l cos θ 1  · e−jk|r−r | , Er = · 1 +   4π | r − r | jk | r − r |   k I0 l sin θ 1 1  · e−jk|r−r | , =j · 1+ − 2    2 4π | r − r | jk | r − r | k | r − r | Hφ =

Eθ

E φ = 0.

(2.16)

(2.17)

(2.18)

(2.19) (2.20)

2.2

Dipole Antenna

13

Fig. 2.2 Dipole antenna coordinate system

Depending on the distances between the antenna and the observation point d = | r − r | and depending on the ratio k · d, the electric field and magnetic field components behave differently in reactive near field k · d  1, Fresnel region or radiating near field k · d > 1, Fraunhofer region or far field region k · d  1, since the second and third term of the equations will either dominate or vanish. In Fig. 2.3 the radiation characteristic is shown, with the dipole oriented in z-direction. Figures 2.4 and 2.5 show the so-called E-plane and H-plane characteristics. E-plane is the plane where the magnetic field component H is zero, whereas H-plane is the plane where the electric field E is zero. Fig. 2.3 Dipole antenna radiation characteristics

14 Fig. 2.4 Dipole antenna E-plane Eθ (θ) at φ = 0◦

Fig. 2.5 Dipole antenna H-plane Eθ (φ) at θ = 90◦ or elevation angle 0◦

2 Basic Antenna Elements

2.3

2.3

Patch Antenna

15

Patch Antenna

For a microstrip patch antenna with the effective length L e , width w, height or distance h between the microstrip patch and the groundplane, the radiated fields can be calculated in E-plane by E φ (φ) = j

     sin( k02 h cos φ) k0 L e k0 w h E 0 e−jk0 |r−r | · cos · sin φ k0 h π | r − r | 2 cos φ

(2.21)

2

and in H-plane by E φ (θ ) = j

 e−jk0 |r−r |

k0 w h E 0 π | r − r |

·

 sin



k0 h 2 sin θ k0 h 2 sin θ



sin

k0 w 2 cos θ k0 w 2 cos θ



 · sin(θ ).

(2.22)

Since the coordinate system has to be defined exactly before deriving the Maxwell’s equations, the coordinate systems for the microstrip patch antenna are defined in Fig. 2.6, similarly like in [3] (Figs. 2.7, 2.8 and 2.9).

Fig. 2.6 Patch antenna coordinate system

16

Fig. 2.7 Microstrip Patch antenna radiation characteristic

Fig. 2.8 Microstrip patch antenna E-plane Eθ (φ) at θ = 90◦

2 Basic Antenna Elements

2.4

Horn Antenna

17

Fig. 2.9 Microstrip patch antenna H-plane Eθ (θ) at θ = 0◦

2.4

Horn Antenna

Even though the dipole antenna elements and microstrip patch antenna elements are mainly used for the array discussions in this book, we would like to mention a horn antenna as another important antenna type which can be used directly as a radiator or as an antenna feed in combination with a reflector antenna or a Cassegrain antenna (see for example [3]). Satellite TV receivers typically use parabolic reflector antennas. Also reflector antennas are applied for space telecommunications. Especially by using a large parabolic aperture as a reflector, an extremely large antenna gain up to 70 dBi can be achieved, improving the horn antenna gain of 20 dBi by about 50 dB. This is very important for deep space telecommunication applications, where the radio frequency signal is strongly attenuated during the transmission over some astronomic units. 1 AU (astronomic unit) is about 150,000,000 km. Radiation characteristic, antenna directivity or the antenna gain depend on the geometrical dimensions of the horn antenna. A typical radiation pattern is shown in Fig. 2.10 calculated by using the 3-dimensional numerical simulation system CST Studio Suite [6] based on the so-called FIT (Finite Integration Technique) developed by Prof. Thomas Weiland in Darmstadt. In Fig. 2.11 the results can be calculated by using a Matlab algorithm [7] showing high antenna gain in boresight direction. In this example, the diameter of the parabolic reflector

18

Fig. 2.10 Horn antenna radiation characteristic

Fig. 2.11 Reflector antenna radiation characteristic

2 Basic Antenna Elements

References

19

is designed to be 1.2 m, approximately 10 λ. By using a larger diameter for the parabolic reflector, up to 35 m or even larger, used in space telecommunications, the antenna gain can be further increased. This is also indispensable due to the large distance between the spacecraft and the Earth groundstation and the large attenuation.

References 1. 2. 3. 4.

R. E. Collin, F. J. Zucker: Antenna Theory, Part 1. McGraw-Hill Book Company (1969). H.-G. Unger: Planar Optical Waveguides and Fibres. Oxford Clarendon Press (1977). C. A. Balanis: Antenna Theory. John Wiley & Sons, Inc. Fourth Edition (2016). K. W. Kark: Antenna Antennas and Radiation Fields (in German: Antennen und Strahlungsfelder). Springer Vieweg (2018). 5. S.-P. Chen: Fundamentals of Information and Communication Technologies. Cambridge Scholars Publishing (2020). 6. CST (Computer Simulation Technology): https://www.3ds.com/de/produkte-und-services/ simulia/produkte/cst-studio-suite/student-edition/. 7. Matlab Antenna Toolbox. https://www.mathworks.com/products/matlab.html.

3

Linear Antenna Array

This chapter starts with an introduction into phased array antennas, discusses required phase shifters and then presents simulation and measurement results of one-dimensional, linear antenna array characteristics. The measurement results are compared with simulation results. The simulation and measurement scenario consists of a linear 1 × 8 patch antenna array as transmitter and an identical, but single patch antenna as receiver. The idea is to validate the simulation results obtained by an analytical method both for the far field region and the near field. For this purpose, the amplitude weights and the phase shifts of the individual patches are adjusted by a control matrix, taking into account the calibrated antenna coaxial cables. Different beam steering angles have been studied, i.e. ±0◦ , ±15◦ , and ±30◦ with respect to the forward direction, with homogeneous, binomial, or Chebyshev weights. The measurements validate and confirm the simulation results of the analytical method. In this chapter we would like to present both simulation [1, 2] and measurement results of near field and far field characteristics of a linear microstrip patch antenna array designed for the ISM frequency of 5.8 GHz with the free space wavelength λ = 0.052 m. Antenna arrays are widely used in various applications such as mobile communications, synthetic aperture radar, medicine, sensing, imaging, or radio astronomy [3–13] to enable fast and precise beam forming. In some cases the systems operate in the near field region. For investigating the far field radiation characteristics and beam forming, far field approximation is accurate enough for antenna arrays [3–8]. The coupling effect between the array elements can be neglected as the distances between the array elements are λ/2 and radiation is low into the direction of the adjacent antennas. Far field and near field are commonly distinguished by the far field distance definition 2D 2 , (3.1) dR = λ also called Rayleigh distance d R , with the largest dimension of the antenna array D and the wavelength λ. For the investigations of generally conformal antenna arrays, the linear planar array will be the best starting point. The licensefree ISM frequency f = 5.8 GHz is chosen © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-P. Chen and H. Schmiedel, RF Antenna Beam Forming, https://doi.org/10.1007/978-3-031-21765-4_3

21

22

3 Linear Antenna Array

both for the simulations and the measurements. 8 microstrip patch antenna elements with vertical polarization are used as a linear array for the measurements at the transmit side. On the other side one patch antenna element is used as receiver antenna. The amplitude weights and phase shifts for the single patch antenna elements are adjusted by using a control matrix [14]. With the number of the antenna array elements M = 8 and the distance between the patch antenna elements dx = λ/2, the near field range (distance between the center point of the array and the observation point) remains shorter than the Rayleigh distance 24.5 λ or 1.274 m. At the distance of 1.8 m or 35 λ we are in far field distance larger than the Rayleigh distance, whereas at 5.6 λ or 10 λ we will be in the near field.

3.1

Introduction into Phased Array Antennas

Antenna arrays consist of many basic antenna elements, as discussed in Chap. 2. To achieve beam steering and beam forming, these antenna elements are controlled in phase and amplitude. To explain beam steering a simple arrangement is shown for far field in Fig. 3.1 and for near field in Fig. 3.2.

Fig. 3.1 Simple phased array antenna in far field

3.1

Introduction into Phased Array Antennas

23

We define an x–y coordinate system, as shown in Fig. 3.1. The different antenna elements 1–8 are on the x axis, at locations x1 − x8 respectively. These elements are to be controlled in phase, so that a main beam consisting of parallel beams is generated. The direction of the main beam to be generated is into the φ0 angle, measured from the y axis. The goal is, that the electric fields from all antenna elements in a far field point, will have identical phase, so that the superposition of all electric fields will be maximum, to achieve high gain. From Fig. 3.1 we see, that the signal from source 2, at x2 , obviously is earlier by Δ/v = (Δd1 − Δd2 )/v compared to source 1, at x1 , where v is the wave velocity. Δ is the path difference between the neighboring antenna elements. For example, the signal from source 5, at x5 , is earlier by 4Δ/v with respect to source 1 respectively. These differences can be expressed in phase Δφ also (one time period T equals 360◦ in degrees or 2π in radian). To achieve identical phases in the far field at the far field focal point, the time differences 0, Δ/v …7Δ/v need to be compensated for, if the source 1 is taken as reference. With increasing index i the sources are more close to the point in the far field, so that their signals must be delayed respectively. This can either be done by adjusting proper individual time delays, or, for one frequency or a relatively small frequency band, by adjusting the phases of the sources, where delay is represented by a negative phase. Trigonometry tells us, that (3.2) sin(−φ0 ) = Δ/dx , where dx is the spacing between the center points of the antenna elements. Typically dx is chosen to be dx = λ0 /2 (Remark: For dx > λ0 /2 the Shannon theorem of the Fourier Transform is violated, cf. Sect. 3.3.), thus Δ = λ0 /2 · sin(−φ0 ).

(3.3)

Generally the reference phase could be chosen arbitrarily. The most deciding factor is to guarantee the phase differences between the antenna elements. Let us assume, that reference phase φr e f = 0◦ at the center point of the 1 × 8 antenna array. The phases for the other elements, with the index i, can then be calculated as Δφi = (i − 9/2) · or

Δdi · 360◦ . λ0

Δφi = (i − 4.5) · sin(−φ0 ) · 180◦ .

(3.4)

(3.5)

Let us consider a simple example, where the wanted steering angle be φ0 = 30◦ , and the spacings between the centers of the antenna elements be λ0 /2. The required phases for the antenna elements are shown in Table 3.1. For an opposite steering angle φ0 , all phases will have the opposite sign respectively. In case of the focus point in the near field, generally known as near field focusing (Fig. 3.2), with a beam steering angle φ0 , the above-mentioned far field approximation will not be applicable, so that the distances Δdi (i = 1, 2, 3, ..., 7, 8) or phase shifts Δφi must be

24

3 Linear Antenna Array

Tab. 3.1 Required phase shifts for a beam steering angle of 30◦ with the focal point in far field Antenna element

Phase shift in radian

Phase shift

1

5.512

315◦

2

3.934

225◦

3

2.359

135◦

4

0.786

45◦

5

−0.785

−45◦

6

−2.354

−135◦

7

−3.920

−225◦

8

−5.483

−315◦

Fig. 3.2 Simple phased array antenna in near field

calculated individually and considered for the phase differences which define the beam steering (Table 3.2). Correspondingly the phases of the antenna elements would be chosen, in order to focus the beam at the desired near field focal point, for example at 10 λ with a beam steering angle of 30◦ . The required phase shifts are shown in Table 3.2.

3.2

Phase Control

25

Tab. 3.2 Required phase shifts for a beam steering angle of 30◦ with the focal point in near field at d = 10 λ Antenna element

Phase shift in radian

Phase shift

1

6.158

353◦

2

4.273

245◦

3

2.484

142◦

4

0.800

46◦

5

−0.770

−44◦

6

−2.219

−127◦

7

−3.526

−203◦

8

−4.712

−270◦

3.2

Phase Control

For beam steering, as mentioned in previous sections, it is important to properly control the phases of all antenna elements involved. For beam forming, additionally, all amplitudes need to be controlled. Phase control means to adjust the phases of all antenna elements such that the electromagnetic waves emanating from all antenna elements will arrive coherently at a defined focal point in space. The same discussion is true for the reverse consideration, that is to receive a signal which origins in that point in space. Since reprocity is true for transmit and receive antennas, the same, individual phase shifts for all antenna elements will be needed for transmit and receive applications. In most applications in communications there is a transmitter and a receiver. For both modes all phases need to be controlled. This can either be done in the radio frequency (RF) range with analog RF phase shifters and attenuators to also control individually all amplitudes, or on the other hand this can be achieved at an intermediate frequency (IF) with preferably digital technologies. We will come back to the second option at IF after discussing problems in the RF range. Typically RF-power amplifiers are needed to transmit RF signals. If there are many antenna elements, each element may have its own medium or low power amplifier, depending on the number of antenna elements. Also passive power dividers, phase shifters and attenuators may be used. For receiving, also amplifiers will be needed. In this case these will be low noise amplifiers. All these amplifiers inherently have phase and gain variations. Many of the critical parameters are temperature dependent and may change over long term operation. Thus, the antenna elements, amplifiers, phase shifters, attenuators, and the RF distribution networks, need to be carefully calibrated for phased array applications. Typically all RF communications signals are down converted anyway, either to a lower intermediate frequency (IF) first, or right down to the baseband. The inverse process is of

26

3 Linear Antenna Array

Fig. 3.3 Mixer with RF, LO and IF signals

course required for up converting signals from baseband to the RF range. Down converting a signal from the RF range to the IF range is accomplished by a so called mixer and a stable local oscillator (LO) signal. Ideally this mixer can be considered as a multiplication device which multiplies the RF signal with a LO signal. Figure 3.3 shows such a mixer. On the left hand input side, we feed a RF signal into the mixer. This RF signal has a certain amplitude A R F and a certain phase φ R F . Its angular frequency is ω R F . This is of course the description for a single frequency. For a complete frequency band, or a spectrum, this spectrum would be the superposition of many of those frequency lines. R F = A R F · cos(ω R F t + φ R F ).

(3.6)

This signal, called RF, is then multiplied with a LO signal. The mixer is assumed as an ideal multiplication stage. The LO signal is L O = 1 · cos(ω L O t + φ L O )

(3.7)

where the amplitude is normalized to 1. The amplitude is unimportant, as a mixer typically is a switched-type mixer, where the LO switches diodes or transistors on and off, with low amplitude noise. Also the LO phase φ L O can be set to 0. It is however important, to understand, that all antenna elements, in the RF range, have mixers which down convert the individual RF signals into individual IF signals. Of course all mixers require a coherent, identical LO signal in phase. Thus we obtain, for a single antenna element, the IF signal as I F = R F · L O, that is I F = R F · L O = A R F · cos(ω R F t + φ R F ) · cos(ω L O t).

(3.8)

or I F = 0.5 · A R F · [cos((ω R F + ω L O )t + φ R F ) + · cos((ω R F − ω L O )t + φ R F )]. After low-pass filtering, we remove the higher spectral components

(3.9)

3.3

Measurement Setup and Verification

cos((ω R F + ω L O )t + φ R F )

27

(3.10)

and obtain the down converted IF signal as I F = 0.5 · A R F · cos((ω R F − ω L O )t + φ R F ).

(3.11)

We see, that the original amplitude A R F and the original phase φ R F have been conserved. This is also true for a complete spectrum, as has been mentioned. That important feature means, that we can do the phase and amplitude control in the IF range. The IF signal is an identical replica of the RF signal. This offers a lot of flexibility, since phase and amplitude control, can now be done digitally. Typical digital systems are less expensive than their analog counterparts and offer unlimited long term stability. Still the mixers and all amplifiers, if required, will have to be calibrated and checked for long-term effects. Above, we considered the receive case, where a RF signal is down converted to a lower IF, where phase and amplitude control are accomplished. However a mixer can also be operated to up convert an IF signal to a RF signal at a higher frequency. In this case all phase and amplitude control can be accomplished also in the IF range. Again each transmit antenna element will require its own mixer and the coherent LO signal for up converting. It should be mentioned at this point, that all the practical measurements in this book have been done with phase and amplitude control in the RF range with the help of a special RF matrix “MiCable Control Matrix” [14]. This calibrated phase and amplitude matrix can be controlled by the help of an external computer.

3.3

Measurement Setup and Verification

Ideally theory, simulations and practical measurements of a specific antenna type should give identical results. “In theory there is no difference between theory and practice—in practice there is.” (Richard P. Feynman used this sentence). Theory and simulations can only be as good as the models that are being used. Often a model does not consider all the effects existing in reality or practice. For example in our case, the coupling effects between adjacent antenna elements is not considered in the model being used for the simulation. Also in general, sometimes there are unforeseen or undetected flaws in the theory calculation process or in the simulations. This is not to blame theory or simulations, but mistakes occur… Anyway, the ultimate benchmark is practical tests and measurements. These, of course, have their own host of problems and side effects, as we may see below. To verify our simulation results of phased-array antenna radiation patterns we did our practical measurements in the microwave student lab of the Department of Electrical Engineering of University of Applied Sciences, Hochschule Darmstadt, Darmstadt, Germany. This relatively simple equipment setup is sufficient to verify all our simulations and findings for various phased-array antenna configurations. This should encourage colleagues, engineers and students to perform their own simple, inexpensive radiation pattern

28

3 Linear Antenna Array

experiments, even as they may not have access to a sophisticated, professional anechoic antenna measurement chamber. Next follows a more detailed description of the experimental setup for the measurement of a radiation pattern of a phased-array antenna system (Fig. 3.6). A medium power signal generator SMP22 (Rohde & Schwarz) is tuned to 5.8 GHz and the output power set up to +20 dBm. The signal generator is pulsed in “Pulse Mode” (more, see below). The output of the signal generator is then fed into the Phase-Amplitude Matrix (MiCable Control Matrix NT-VPAM-1x8-5.8) [14] which has 8 output channels. Each of these channels can be adjusted individually in phase and amplitude by an external computer. Because of the passive power division and the losses of the internal phase shifters and attenuators, the basic loss of the matrix is some 25 dB. These 8 outputs are then connected by 8 coaxial cables with “equal length” (more on equal length below) to the 8 patch antennas of the phased-array antenna setup. This is our transmitter setup. Different antenna setups are being used, linear, convex or concave oriented patch antennas. The spacing between adjacent patch antenna elements is λ0 /2. The individual antenna elements are linearly, vertically polarized patch antennas. The reflection (S11) of a single antenna element is better than −15 dB. The coupling (S21) between two adjacent patch antenna elements is 35λ), a parabolic reflector is used additionally

3.3

Measurement Setup and Verification

29

to increase sensitivity. At the output of the amplifier is a detector diode which has a dynamic range from +3 dBm to some −50 dBm. The DC video output goes to the Leybold system and is chopper amplified internally, then processed and displayed by the Cassy system versus rotation angle of the turntable. The measurement of low side lobes, in a radiation pattern measurement, is always critical because of unwanted reflections from near-by metallic devices, as are shelves, test equipment and so on. To reduce these effects, anti-reflection absorber mats are being used effectively. Also the parabolic reflector on the receive side, during the far-field measurements, helps to reduce incoming unwanted, arbitrarily reflected signals from the sides. All measurements were performed and plotted with this simple experimental setup with high repeatability. These measurements, which have to cope with non-ideal conditions, however nicely confirm the simulations that assume ideal conditions. A linear patch antenna array is shown in Figs. 3.4 and 3.5, where the patch antenna elements are allocated along the x-axis and the polarization is in z direction. For the measurements the center of the transmitter patch antenna array is the coordinate origin (0, 0, 0). The patch antenna array, connected to the control matrix, is mounted on top of a rotating table which is controlled by the antenna measurement software of Leybold (Fig. 3.6). The coordinate origin and the focal point F(x F , y F ) define the wanted azimuth beam steering angle φ0 . The receiver patch antenna is positioned exactly in this focal point F(x F , y F ) for the steering angle φ0 . The vector dxi is pointing to the i-th array element from the coordinate origin (0, 0, 0) which is the reference point of the antenna array, whereas d is the vector pointing to the

Fig. 3.4 Patch array with characteristic parameters

30

3 Linear Antenna Array

Fig. 3.5 5.8 GHz 1 × 8 patch antenna array transmitter and the single patch antenna receiver

Fig. 3.6 Schematic diagram of the measurement setup

focal point from the coordinate origin. Then the vector di = d − dxi can be defined between each array element and any arbitrary observation point, or focal point, at for instance d = 10 λ (Fig. 3.4). To be more precise, the beam steering angle is the angle with respect to the main lobe in y-direction. Neglecting the coupling effect between the array elements, the radiated electric fields of single patch antenna array elements can be superimposed [1, 2]:

3.4

Far Field Characteristics of a Linear Patch Antenna Array

E(d, φ) =

M 

Ai · Ei (d − dxi , φ) · exp(jαi )

31

(3.12)

i=1

with Ai as the amplitude and αi as the properly chosen phase shift of the i-th patch antenna element. The total phase shifts αi must include all the phase shifts φi configured in the control matrix and possible phase variations δi of each corresponding connecting cable. For this purpose the phase variations of all connecting cables must be measured precisely in advance. Ei is the azimuth radiation characteristic of the single patch antenna. Characteristics of all simulation results achieved by using our own Matlab codes and measurement results in this book are shown with identical scales of −30 dB to 0 dB. Measurement plots are genuine plots (with tiny number fonts, but identical scales).

3.4

Far Field Characteristics of a Linear Patch Antenna Array

In Figs. 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14 and 3.15 the results of the analytical methods [1, 2] are compared pairwise with the measurement results. The logarithmic scales for both graphs are identical. It can be clearly seen that the relatively high side lobes in case of homogeneous amplitude weights can be significantly reduced by using binomial and Chebyshev amplitude weights. In case of binomial amplitude weights, the beam width is also remarkably broadened. On the other hand, narrower beams can be achieved by using a larger number of antenna elements.

Fig. 3.7 Far field characteristics of a patch antenna array, d = 35 λ, homogeneous amplitude weights, beam steering 0◦ . a Simulation; b measurement

32

3 Linear Antenna Array

Fig. 3.8 Far field characteristics of a patch antenna array, d = 35 λ, homogeneous amplitude weights, beam steering 15◦ . a Simulation; b measurement

Fig. 3.9 Far field radiation characteristics of a patch antenna array, d = 35 λ, homogeneous amplitude weights, beam steering 30◦ . a Simulation; b measurement

3.4

Far Field Characteristics of a Linear Patch Antenna Array

33

Fig. 3.10 Far field characteristics of a patch antenna array, d = 35 λ, binomial amplitude weights, beam steering 0◦ . a Simulation; b measurement

Fig. 3.11 Far field characteristics of a patch antenna array, d = 35 λ, binomial amplitude weights, beam steering 15◦ . a Simulation; b measurement

34

3 Linear Antenna Array

Fig. 3.12 Far field radiation characteristics of a patch antenna array, d = 35 λ, binomial amplitude weights, beam steering 30◦ . a Simulation; b measurement

Fig. 3.13 Far field characteristics of a patch antenna array, d = 35 λ, Chebyshev amplitude weights, beam steering 0◦ . a Simulation; b measurement

3.4

Far Field Characteristics of a Linear Patch Antenna Array

35

Fig. 3.14 Far field characteristics of a patch antenna array, d = 35 λ, Chebyshev amplitude weights, beam steering 15◦ . a Simulation; b measurement

Fig. 3.15 Far field radiation characteristics of a patch antenna array, d = 35 λ, Chebyshev amplitude weights, beam steering 30◦ . a Simulation; b measurement

36

3 Linear Antenna Array

We see, that there is a nice agreement between simulation and measurement results, qualitatively and in quantity. That are shapes of the main beam, half-power beam widths, steering angles, shape of the side lobes and nulls. So all simulations are verified by the measurements. Looking into the details, we will see, that the half-power beam widths of the measurements seem to be a little larger than those of the simulations. This may be attributed to phase errors in the experiments. These errors are estimated to be less than ±10◦ . (Remark: For typical beam widths an increase in half-power beam width of 1◦ is related to a gain reduction of some 0.1 dB.) As results, we see that the linear patch array antenna has a half-power beam width of 13◦ –15◦ for steering angles 0◦ –30◦ . With increasing steering angle the half-power beam width increases. This is expected, as the array broadside, as seen from the far point at the receive antenna, gets a little smaller for a higher steering angle. The side lobes for the linear patch antenna array, with homogeneous amplitudes, are relatively high which is expected from Fourier Transform considerations, that is (sin x)/x, cf. Sect. 3.5. For a steering angle of 0◦ the first side lobe is −14 dB compared to the main lobe. For higher steering angles, the side lobes increase a little, i.e. −12 dB for 30◦ steering angle. To effectively reduce these side lobes, amplitude weights will be introduced in the next chapter.

3.5

Amplitude Weights and Side Lobes of the Radiation Pattern

By choosing proper amplitude weights and phases for the antenna elements beamforming can be achieved. Considering Eq. (2.12) for dipole source current I and the amplitude weights Ai in case of discrete antenna array, we see that the antenna radiation pattern relates to the current density space distribution by the Fourier Transform or Discrete Fourier Transform [15–17]. In other words, if we know the current density space distribution, we will obtain the radiation pattern in the far field by Fourier Transform. Vice versa we will get the current distribution, both in amplitude and phase, by Inverse Fourier Transform from a given radiation pattern. This simple relationship allows us some helpful insights into antenna radiation patterns, by using simple, known Fourier Transform relationships, i.e. • Uniform current density space distribution, or uniform illumination, is transformed into a radiation pattern with sin(x)/x shape, resulting in strong antenna sidelobes. (Remark: in our simulated and measured examples, all antenna elements are driven with the same amplitude (Figs. 3.7–3.9). Observe high sidelobes and also high gain). • Gaussian illumination will be Fourier transformed into a Gaussian shaped beam pattern with low sidelobes as is typically preferred. (Remark: in our simulated and measured examples all the amplitudes of the different antenna elements are not Gaussian weighted, but binomial weighted or Chebyshev weighted).

3.6

Near Field Characteristics at a Distance of 10 λ

37

• A small antenna aperture will generate a wide antenna pattern with relatively low gain, whereas a large antenna aperture will generate a narrow antenna beam pattern with relatively high gain. • When an antenna beam is steered from a linear antenna array, as in our examples, it is easy to see, that the beamwidth will broaden, when the beam is steered from the main direction to a side. This is because the antenna array aperture seen from a point in the far field is smaller compared to the aperture seen in the main direction. Consequently a broader beam width is related to a smaller antenna gain. The similar effects can be observed for all antenna elements or antenna arrays, like linear antenna arrays, planar antenna arrays or conformal antenna arrays which will be discussed in the following chapters. By using Inverse Fourier Transform, or Inverse Fast Fourier Transform with windowing [16], wanted, also arbitrary, antenna beam patterns can be synthesized. For more specific details, see [16, 17]. Two different amplitude weighting methods have been examined, that is binomial amplitude distribution, which is close to a pseudo-Gaussian distribution, and on the other hand a Chebyshev distribution. The phases were not changed, compared to the homogeneous amplitude distribution from Sect. 3.2. A binomial amplitude weighting reduces the side lobes significantly to be less than −30 dB compared to the main lobe, cf. Figs. 3.10–3.12. As a tradeoff the half-power beam width increases from some 14◦ , for homogeneous amplitudes, to some 23◦ for binomial amplitudes. Consequently the gain will be reduced. (Remark: The gain reduction may be aggravated, if the total feeding power for a transmit antenna is reduced because of lossy RF attenuators that may be used to adjust the amplitudes for antenna array elements at the sides of the array. This is another reason for amplitude adjustment in IF stages.) Nevertheless a binomial amplitude weighting may be of interest, if extremely low side lobes are the design goal. Next Chebyshev amplitude weights are examined, see Figs. 3.13–3.15. We see, that the side lobes are also down by −30 dB. The half-power beam widths are between 17◦ for a steering angle of 0◦ , and 19◦ for a steering angle of 30◦ respectively. So obviously Chebyshev amplitude weights are a nice compromise with small half-power beam width and at the same time low side lobes. (Remark: to fully exploit the relatively high gain, the amplitudes are to be set with no RF attenuators. That is either low-loss power dividers or amplitude control in the IF stages.)

3.6

Near Field Characteristics at a Distance of 10 λ

In Figs. 3.16, 3.17, 3.18, 3.19, 3.20, 3.21, 3.22, 3.23 and 3.24 the near field characteristics of the analytical methods [1, 2] are compared pairwise with the measurement results for a very short focal point distance of 10 λ. Also for the near field it can be clearly seen that the

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3 Linear Antenna Array

Fig. 3.16 Near field characteristics of a patch antenna array, d = 10 λ, homogeneous amplitude weights, beam steering 0◦ . a Simulation; b measurement

Fig. 3.17 Near field characteristics of a patch antenna array, d = 10 λ, homogeneous amplitude weights, beam steering −15◦ . a Simulation; b measurement

3.6

Near Field Characteristics at a Distance of 10 λ

39

Fig. 3.18 Near field radiation characteristics of a patch antenna array, d = 10 λ, homogeneous amplitude weights, beam steering −30◦ . a Simulation; b measurement

Fig. 3.19 Near field characteristics of a patch antenna array, d = 10 λ, binomial amplitude weights, beam steering 0◦ . a Simulation; b measurement

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3 Linear Antenna Array

Fig. 3.20 Near field characteristics of a patch antenna array, d = 10 λ, binomial amplitude weights, beam steering −15◦ . a Simulation; b measurement

Fig. 3.21 Near field radiation characteristics of a patch antenna array, d = 10 λ, binomial amplitude weights, beam steering −30◦ . a Simulation; b measurement

3.6

Near Field Characteristics at a Distance of 10 λ

41

Fig. 3.22 Near field characteristics of a patch antenna array, d = 10 λ, Chebyshev amplitude weights, beam steering 0◦ . a Simulation; b measurement

Fig. 3.23 Near field characteristics of a patch antenna array, d = 10 λ, Chebyshev amplitude weights, beam steering −15◦ . a Simulation; b measurement

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3 Linear Antenna Array

Fig. 3.24 Near field radiation characteristics of a patch antenna array, d = 10 λ, Chebyshev amplitude weights, beam steering −30◦ . a Simulation; b measurement

relatively high side lobes, in case of homogeneous amplitude weights, can be significantly reduced by using binomial amplitude weights and Chebyshev amplitude weights. Narrower beams can be also achieved by using a larger number of antenna elements.

Fig. 3.25 Near field characteristics of a patch antenna array, d = 5.6 λ, homogeneous amplitude weights, beam steering 0◦ . a Simulation; b measurement

3.6

Near Field Characteristics at a Distance of 10 λ

43

Fig. 3.26 Near field characteristics of a patch antenna array, d = 5.6 λ, homogeneous amplitude weights, beam steering 15◦ . a Simulation; b measurement

Fig. 3.27 Near field radiation characteristics of a patch antenna array, d = 5.6 λ, homogeneous amplitude weights, beam steering 30◦ . a Simulation; b measurement

44

3.7

3 Linear Antenna Array

Near Field Characteristics at a Distance of 5.6 λ

In Figs. 3.25, 3.26, 3.27, 3.28, 3.29, 3.30, 3.31, 3.32 and 3.33 the near field characteristics of the analytical methods [1, 2] are compared pairwise with the measurement results for a focal point distance of 5.6 λ. Also for the near field it can be clearly seen that the relatively high side lobes in case of homogeneous amplitude weights can be significantly reduced by using binomial amplitude weights and Chebyshev amplitude weights. Narrower beams can be achieved by using a larger number of antenna elements. Looking at all the Figs. 3.25–3.33 we see again, that the measurements validate the simulation results nicely. Slightly higher side lobes in the measurements, compared to the simulations may be the result of unwanted reflections within the experimental setup. For these near field measurements no parabolic reflector was used, in order to reduce the receive antenna aperture for more accurate measurements. Distances of 5.6 λ and 10 λ were chosen. With proper phases, corrected for near field, and the same amplitude weights as used for far field, we obtain very similar patterns as for the far field case. Beam widths and side lobe distributions are almost identical. So near field applications can be designed easily with wanted beam shapes and steering angles. In this chapter far field and near field beam steering characteristics of one-dimensional, linear microstrip patch antenna arrays were investigated. Simulation results by using an analytical method were compared with measurements by using an antenna measurement setup for the ISM frequency of 5.8 GHz. Amplitude tapering with binomial amplitude weights and

Fig. 3.28 Near field characteristics of a patch antenna array, d = 5.6 λ, binomial amplitude weights, beam steering 0◦ . a Simulation; b measurement

3.7

Near Field Characteristics at a Distance of 5.6 λ

45

Fig. 3.29 Near field characteristics of a patch antenna array, d = 5.6 λ, binomial amplitude weights, beam steering 15◦ . a Simulation; b measurement

Fig. 3.30 Near field radiation characteristics of a patch antenna array, d = 5.6 λ, binomial amplitude weights, beam steering 30◦ . a Simulation; b measurement

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3 Linear Antenna Array

Fig. 3.31 Near field characteristics of a patch antenna array, d = 5.6 λ, Chebyshev amplitude weights, beam steering 0◦ . a Simulation; b measurement

Fig. 3.32 Near field characteristics of a patch antenna array, d = 5.6 λ, Chebyshev amplitude weights, beam steering 15◦ . a Simulation; b measurement

References

47

Fig. 3.33 Near field radiation characteristics of a patch antenna array, d = 5.6 λ, Chebyshev amplitude weights, beam steering 30◦ . a Simulation; b measurement

Chebyshev amplitude weights efficiently reduces the side lobes over a larger beam steering angle for the far field and near field radiation characteristics. The beamwidth increases with binomial and Chebyshev amplitude weights and also increases with an increased beam steering angle. Depending on the beam forming requirement, in case of a wanted narrow beam, this method can be combined with a larger number of patch antenna array elements. The analytical and measurement methods presented in this chapter can be used to provide the optimum parameters for dynamic beam forming problems for adaptive planar or conformal antenna arrays.

References 1. S.-P. Chen: Improved Near Field Focusing of Antenna Arrays with Novel Weighting Coefficients. IEEE WiVeC 2014, 6th International Symposium on Wireless Vehicular Communications (2014). 2. S.-P. Chen: An Efficient Method for Investigating Near Field Characteristics of Planar Antenna Arrays. Wireless Personal Communications 95 (2), pp. 223–232. Springer Nature (2017). 3. R. E. Collin, F. J. Zucker: Antenna Theory, Part 1. McGraw-Hill Book Company (1969). 4. K. F. Lee: Principles of Antenna Theory. John Wiley & Sons Ltd. (1984). 5. B. D. Steinberg, H. M. Subbaram: Microwave Imaging Techniques. John Wiley & Sons, Inc. (1991). 6. R. J. Mailloux: Phased Array Antenna Handbook. Artech House Inc. (1994). 7. L. V. Blake, M. W. Long: Antennas: Fundamentals, Design, Measurement. 3rd Edition. Scitech Publishing, Inc. (2009). 8. D. G. Fang: Antenna Theory and Microstrip Antennas. CRC Press (2010).

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9. W. H. Carter: On Refocusing a Radio Telescope to Image Sources in the Near Field of the Antenna Array. IEEE Transactions on Antennas and Propagation, Vol. 37, pp. 314–319 (1999). 10. A. Badawi, A. Sebak, L. Shafai: Array Near Field Focusing. WESCNEX’97 Proceedings of Conference on Communications, Power and Computing, pp. 242–245 (1997). 11. M. Bogosanovic, A. G. Williamsoni: Antenna Array with Beam Focused in Near Field Zone. Electronics Letters, vol. 39, pp. 704–705 (2005). 12. J. Grubert: A Measurement Technique for Characterization of Vehicles in Wireless Communications. PhD Thesis (in German) of Technical University Hamburg-Harburg, Cuvillier Verlag Goettingen (2006). 13. S. Ebadi, R. V. Gatti, L. Marcaccioli, R. Sorrentinoi: Near Field Focusing in Large Reflector Array Antennas Using 1-bit Digital Phase Shifters. Proceedings of the 39th European Microwave Conference, pp. 1029–1032 (2009). 14. MiCable Inc.: http://en.micable.cn. 15. C. Balanis, Antenna Theory Analysis and Design, John Wiley, 1997. 16. F. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, Proc. of IEEE, vol. 66, No. 1, pp. 51–83, 1978. 17. X. Wang, Y. Zhong, and Y. Wang, “An Improved Antenna Array Pattern Synthesis Method Using Fast Fourier Transform”, International Journal of Antennas and Propagation, 2015, 1–9, 2015.

4

Planar Antenna Arrays

Near field focusing of 2-dimensional planar antenna arrays is analyzed by using efficient analytical methods, which were validated by comparing the simulation results with the measurement results of one-dimensional linear patch antenna array in the last chapter, instead of time consuming numerical methods. This is especially important for the design and optimization process for dynamic beam forming and other near field applications. Antenna arrays show increased side lobes at larger beam steering angles in the near field. Conventional techniques such as inhomogeneous but symmetrical amplitude illuminations, to assure a certain side lobe suppression level, do not reduce the side lobes completely or sufficiently for the near field case, especially in the back fire direction. In this chapter, asymmetrical amplitude weighting coefficients in combination with Dolph Chebyshev are used to further improve the side lobe suppression over a larger angular range. Antenna arrays are widely used in various applications such as mobile communications, synthetic aperture radar, medicine, sensing, imaging or radio astronomy [1–11] to enable fast and precise beam forming. Generally a high resolution is required for beam forming which leads to relatively large antenna arrays and complex signal processing systems. In some cases the systems should operate in the near field region. The far field radiation characteristics and beam forming methods for antenna arrays have been thoroughly investigated and discussed [1–6]. The coupling effect between the array elements can often be neglected for the distances between the array elements larger than or equal to λ/2, if radiation from an antenna element is mimimum into the direction of the neighoring antenna elements. Far field and near field are commonly distinguished by the far field distance definition rmin = 2D 2 /λ with the largest dimension of the antenna array D and the wavelength λ. For the investigations in this chapter, i.e. for the number of the antenna array elements M, N ≤ 10 and dx , dz = λ/2 (Fig. 4.1–4.3), the near field range (distance between the center point of the array and the observation point) remains shorter than 32λ, even though the principles are valid for arbitrary antenna arrays with large numbers M and N [12]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S.-P. Chen and H. Schmiedel, RF Antenna Beam Forming, https://doi.org/10.1007/978-3-031-21765-4_4

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50

4.1

4 Planar Antenna Arrays

Beam Forming, Focusing and Steering of Planar Arrays

A planar antenna array is illustrated schematically in Fig. 4.1, where xm and z n are the xand z-coordinates of the single array elements. The center of the array and the focal point F(x, y, z) form the beam steering angle θ0 or φ0 . r − rmn is the vector between the array element (xm , ymn , z n ) and the focal point F(x, y, z). rmn is the vector pointing to the array element from the coordinate origin (0, 0, 0) as reference point of the antenna array. To be more precise, the beam steering angle is the angle with respect to the main lobe in y-direction. Neglecting the coupling effect between the array elements, the planar array can be further investigated by superimposing two orthogonal linear arrays parallel to x axis and z axis (see for example Fig. 4.2 and 4.3). For simplicity the planar antenna arrays are analyzed first, i.e. ymn = 0, even though the proposed method is generally valid for conformal antenna arrays. The near field characteristics of this type of plannar antenna arrays or modified conformal antenna arrays can be analyzed either by using numerical methods or by using the analytical method by superimposing all the radiation fields resulting from all the array elements located along the x − z plane. The total radiation pattern of the arbitrary array at the focal point or point of observation F can be obtained by E(r,θ .φ) =

N M   m

Fig. 4.1 Planar antenna M×N array in the x-z plane with the F(x, y. z)

n

amn · Emn (r − rmn , θ, φ) · exp(jα mn )

(4.1)

4.1

Beam Forming, Focusing and Steering of Planar Arrays

51

Fig. 4.2 Linear array along x-axis, side by side, with the focal point F(x,y,z), projection on the x-y plane for z-oriented dipoles

Fig. 4.3 Linear array along z-axis, side by side, with the focal point F(x,y,z) projection on the y-z plane for z-oriented dipoles

with amn as amplitude, and αmn as properly chosen phase shift of the corresponding antenna element located at (xm , 0, z n ) to control the desired beam steering. The elements can be Hertzian dipoles, patches or other radiators. For focusing the radiation of the antenna array, the side lobes should be reduced to a minimum. The desired narrow beam width can be achieved by using a large number of array elements M × N . For simplicity, and in case of a planar antenna array located in the x − z plane, ymn = 0 is valid. Emn (r − rmn , θ, φ) is the basic function describing the field radiation characteristics

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of each antenna element. r is the vector pointing at the focus F from the coordinate system origin (0, 0, 0). For the far field radiation characteristics investigations some simplifications are possible, i.e. the E-fields are reciprocally proportional to the distance r between the source and the observation point or focal point in the first approximation, but for the near field case either the simplified isotropic radiator or the exact near field pattern of each single radiator (dipoles or patches) located at (xm , 0, z n ) are to be considered. For realistic cases, especially by investigating the planar antenna arrays, Hertzian dipoles or patch antenna elements along the z-axis are investigated (cf. [6]). For planar arrays the beam can be formed and steered by properly choosing the amplitudes amn and phase shifts αmn in a one-dimensional manner. The 2-dimensional problem of beam forming can be simplified, if we neglect the coupling effect between the array elements and assume some amplitude and phase distributions which allow the separation of the double sum into the product of simple sums equations (4.2) – (4.4): E(r, θ. φ) = EM (r − rm , θ, φ) · EN (r − rn , θ, φ),

EM (r, θ .φ) =

M 

am · Em (r − rm , θ, φ) · exp(jαx (m)),

(4.2)

(4.3)

m

EN (r, θ. φ) =

N 

an · En (r − rn , θ, φ) · exp(jαz (n)).

(4.4)

n

For this case the planar array can be simply replaced by two othogonal linear arrays parallel to x− and z−axis separately. The analyses become much simpler, especially if the focus is on the near field focusing. By doing so, the beam forming tasks to achieve φ0 can also be done for instance by properly choosing the amplitude paramenters am and phase shifts αm first in x-y plane (Fig. 4.2) and then in x-z plane to achieve θ0 by properly choosing the amplitude coefficients an and phase shifts αn (Fig. 4.3). To steer the beam to a desired direction in the near-field, i.e. if the focus F is near the sources, say in a distance < 30λ, the connecting lines between the focal point and the array elements are no more parallel like in far field case, so that the easy far field model leads to wrong estimation of the phase shifts of the antenna elements. The phases from the array elements to the focal point F must be calculated by considering the exact distance between the array element and the focal point, not only the steering angle like in the far field case [10]:  2 + (md )2 − 2R · m · d · cos(φ ), αx (m) = k0 Rm x m x 0  αz (n) = k0 Rn2 + (ndz )2 − 2Rn · n · dz · sin(θ0 ).

(4.5) (4.6)

4.2

Side Lobe Suppression

53

With the distances Rm , Rn and the steering angles θ0 , φ0 , the phase differences between the elements, especially the difference with respect to the origin, can be simply calculated as Δα = αm − k0 · r . Fig. 4.2 illustrates the relationship. In all simulations in this chapter, the phase shifts will be based on this exact near field assumption. In order to simulate the near field focusing problems quickly, an analytical method is prefered. To ensure this, the analytical method was compared with commonly known sophiscated, but time-consuming numerical simulation techniques like CST and HFSS in [12]. The relatively small differences between these solutions confirm the low coupling effect for the parameter settings also used in this chapter, and justify the usage of the proposed analytical method to investigate the near field focusing problems as a good approximation, especially for patch antennas. In [10, 12] it could be shown that the side lobe level increases significantly in near field, when the beam steering angle becomes larger, if homogeneous amplitude weights are applied.

4.2

Side Lobe Suppression

Firstly a 10 dipole array is analyzed for different beam steering angles 15◦ – 45◦ with the focal point designed to be at a distance of 10λ to show the typical radiation characteristics with and without additional asymmetrical amplitude weights. There are many well-known steering techniques for beam forming and side lobe suppression, such as binomial and Chebyshev illumination coefficients, defined as weighting function W (m) and W (n) (e. g. [4, 5]). By using the binomial distribution the side lobe level can be reduced without ripple, but the beam width is also increased at the same time. Chebyshev amplitude weights lead to better results both in terms of narrow beam width and reduced side lobe level which are eminently important for near field focusing. The side lobe suppression is improved by using Chebyshev illumination coefficients, but the major side lobe increases for the beam steering angles of 15◦ to 45◦ [10, 12], especially for near field cases. The significant increase of the side lobes is disadvantageous for signal transmission, sensing or imaging applications. These could lead to errors and interferences due to multipath propagation effects. In [10] a simple but effective way was proposed to reduce these side lobes further by using an additional asymmetrical amplitude illumination, which was also further improved in [12] for θ0 and the array parallel for x-axis (see definitions in equations (4.7)–(4.10), the same equations are valid for φ0 and the array parallel to z-axis) ns =

(180 − θ0 ) · M, 180

aasym (i) = 1 + s · (i − 1); for 1