Introduction to Active Phased Arrays 1630818666, 9781630818661

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Introduction to Active Phased Arrays

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For a complete listing of titles in the Artech House Antennas and Propagation Library, turn to the back of this book.

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Introduction to Active Phased Arrays Thomas Sikina

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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress British Library Cataloguing in Publication Data A catalog record for this book is available from the British Library. ISBN-13: 978-1-63081-866-1 Cover design by Publishers’ Design and Production Services, Inc. © 2023 Artech House 685 Canton St. Norwood, MA All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. This document does not contain technology or technical data controlled under either the U.S. International Traffic in Arms regulations or the U.S. Export Administration regulations. 10 9 8 7 6 5 4 3 2 1

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Contents CHAPTER 1 Introduction to Phased Arrays

1

1.1 Phased Array History and Perspective 2 1.2 Fundamentals of Wave Propagation: The Wave Equation 7 1.2.1  Boundary Condition Cases 13 1.3 Array Antennas 26 1.4 Aperture State Fundamentals 28 1.4.1  General Aperture State Relationships 40 1.4.2  Radiation Integrals for Circular Apertures 47 1.5 Array Far-Field Fundamentals 50 1.6 Frequency-Time Domains 54 1.6.1  Frequency-Time Domain: Fast Fourier Transform 58 References 62 CHAPTER 2 Array Theory

63

2.1 2.2 2.3 2.4 2.5

Array Far-Field Radiation 63 Array Far-Field Fundamental Observations 63 General Array Theory 68 Two-Element Arrays 73 Linear Arrays 79 2.5.1  Linear Arrays in Sine Space 84 2.5.2  Linear Array Aperture Projection 93 2.6 Planar Arrays 104 2.6.1  Planar Arrays with No Real-Space Grating Lobes 108 2.6.2  Planar Arrays with Real-Space Grating Lobes 114 2.7 Conformal Arrays 125 126 2.7.1  Radius of Curvature Embedded Element Geometry 132 2.7.2  Conformal Array Phase Alignment 2.7.3  Eclipsed Elements in Conformal Arrays 140 References 148 CHAPTER 3 Lattice Theory

149

3.1 Introduction 3.2 Floquet’s Theorem

149 149 v

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viContents

3.2.1  Phased Array Surface Wave Condition 154 3.2.2  Phased Array Scan Volume 159 3.3 Lattice Theory 162 3.3.1  Rectangular Lattice 164 3.3.2  Equilateral Triangular Lattice 177 3.3.3  Isosceles Triangular Lattice 187 3.4 Reordered Lattice Theory 194 3.4.1  Ring Lattice Arrays 196 3.4.2  Spiral Lattice Arrays 205 3.5 Finite Array and Surface Wave Effects 208 References 213 CHAPTER 4 Array Fundamentals: Supporting Theories, Part I

215

4.1 Introduction 215 4.2 Radiating Aperture Fundamentals: Three Domains 215 4.3 Array Architecture 222 4.3.1  Case 1: Hybrid Beamformed, Single Polarization 223 4.3.2  Case 2: Analog Beamformed, Dual Simultaneous Polarization 225 4.4 Practical Limits 226 4.4.1  Theorem of Reciprocity 226 4.4.2  Conservation of Energy 228 4.4.3 Superposition 228 4.4.4  Duality Theorem 229 4.5 Near and Far Fields 229 4.5.1  The Far-Field Criterion 230 4.5.2  Array Reactive and Near Fields 234 4.6 Rotational Transforms 243 4.6.1  Coordinate Frames 243 4.6.2  Sine Space 248 4.6.3  Rotated Coordinate Frames 251 4.6.4  Inverted Rotated Coordinate Frames 257 References 260

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CHAPTER 5 Array Fundamentals: Supporting Theories, Part II

263

5.1 Introduction 5.2 Radiated Gain 5.3 Polarization Domain 5.3.1  Polarization Transforms 5.3.2  Stokes Parameters 5.3.3  Polarization Isolation 5.3.4 Cross-Polarization 5.3.5  Scan-Dependent Polarization Properties 5.3.6  Polarization Compensation 5.4 Phased Array Noise Temperature

263 264 270 270 277 278 280 284 289 292

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Contents

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5.4.1  Antenna Noise Sources 293 5.4.2  Noise Wave Theory 301 References 305 CHAPTER 6 Phased Array Radiating Elements

307

307 6.1 Introduction 6.2 Single-Element Dipole over Ground Plane Radiators 309 6.2.1  Dipole Boundary Conditions 310 6.2.2  Dipole Radiation 312 6.3 Single-Element Waveguide Radiators 317 6.3.1  Rectangular Waveguide 318 6.3.2  Circular Waveguide 326 6.3.3  Circular Waveguide Radiator 330 6.4 Single-Element Patch Radiators 334 6.4.1  Square Patch Boundary Conditions 335 6.4.2  Square Patch Design Methods 337 6.4.3  Square Patch Radiation 341 6.4.4  Circular Patch Boundary Conditions 349 6.4.5  Circular Patch Radiation 352 References 356 CHAPTER 7 Active Radiating Elements

359

359 7.1 Introduction 7.2 Mutual Coupling and Embedded Elements in Arrays 360 7.2.1  Active Impedance, Reflection Coefficient, and Embedded 365 Element Gain 7.2.2 WAIM 374 7.2.3  Real-Space Grating Lobes 375 7.2.4  Surface Impedance Effects 378 7.3 Active Radiating Element Cases 379 7.4 Active Dipole over Ground Plane Radiators 380 7.4.1  Linear Dipole Array 380 7.4.2  Vee Dipole Array 392 7.4.3  PUMA Array 405 7.5 Active Patch Radiators 429 430 7.5.1  Balanced Patch Radiator Array 7.5.2  Unbalanced Patch Radiator Array 441 7.5.3  Balanced Stacked Patch Radiator in a Rectangular Lattice Array 452 References 466

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CHAPTER 8 Far-Field Synthesis, Part I

467

8.1 Introduction 8.2 Fourier Transform Method for Linear Arrays

467 468

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viiiContents

8.3 Schelkunoff’s Form 484 8.4 Canonic Forms 493 8.5 Truncated Complex Gaussian Forms 493 8.5.1  Truncated Gaussian Magnitude Aperture Taper 493 8.5.2  Truncated Gaussian Phase Aperture Taper 499 8.6 Modified sin(x)/x Distribution 499 References 505 CHAPTER 9 Far-Field Array Synthesis, Part II

507

9.1 Introduction 507 9.2 Woodward-Lawson Method 508 9.2.1  Nonorthogonal Woodward-Lawson Method 517 9.2.2  Difference Patterns by the Woodward-Lawson Method 526 9.2.3  Woodward-Lawson Synthesis Method with Controlled Sidelobes 526 9.3 Dolph-Chebyshev Synthesis 531 9.4 Taylor Line Source Synthesis 533 9.5 Planar 2-D Array Distributions 544 9.6 Circular Aperture Distributions 546 9.6.1  Taylor Circular Array Sources 549 9.7 Iterative Synthesis Methods 555 9.8 MLE 557 References 558 CHAPTER 10 Stochastic Aperture Errors in Phased Arrays

559

559 10.1 Introduction 10.1.1  Stochastic (Random) Errors in Arrays 560 10.1.2  Average (rms) Far-Field Characteristics 567 10.1.3  Beam-Pointing Error 576 10.1.4  Peak and rms Sidelobes 581 10.1.5  Dispersion and Its Impact on Instantaneous Bandwidth 582 10.1.6  Polarization Isolation 585 10.2 Stochastic Error Budgets 585 10.3 Periodic (Correlated) Array Errors 588 10.3.1  Element-Level Phase Quantization 589 590 10.3.2  Subarray Spatial Effects 10.3.3  Subarray Frequency-Domain Effects 599 10.3.4  Aperture Blockage 604 References 606 About the Author

607

Index 609

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CHAPTER 1

Introduction to Phased Arrays

Contemporary phased array systems use an arrangement of radiating elements in a single aperture with complex (i.e., real and imaginary) vector control for each and a beamforming network that combines these element vectors. The latter performs a Fourier transform, adding microwave signals from all elements into a single beam port. Thus, we have a device that transitions from one or more single mode beam ports to a radiated electromagnetic (EM) field in free space. Included in the EM transformation are the domains of frequency, time, three-dimensional (3-D) space, and polarization space. The beam ports connect to modern digital-to-radio frequency (RF) transceivers; these convert the RF to baseband information in digital form for both transmit (Tx) and receive (Rx) waveforms. With the information in digital form, it moves to a data processor that schedules the information, waveforms, control, and timing in a contemporary space-time adaptive process (STAP), for example, in sensor systems and a modem in communications systems. The subject of this book is the phased array antenna, as it provides the degrees of freedom needed in contemporary systems and expected future systems. The phased array is a powerful tool that offers expanded sensing and communications capabilities in 3-D space, and so it is the subject of expanding study. Today, the phased array is recognized as the essential technology for a wide variety of contemporary systems that extend human awareness. These range from advanced radars used for target surveillance, tracking, and identification to sensors used for terrestrial, sea, and space applications, to precision radars used for air traffic control and precision landing, to advanced satellite communications (SATCOM) systems, and to fifth-generation (5G) communications networks and their backhaul links. By studying the basics behind advanced phased array design, we advance the technical frontier. To this end, the industry today seeks new technologies to make these arrays smaller, lighter, and less expensive, while increasing their operating frequency range from the conventional communications and radar bands to millimeter-wave frequencies. The phased array studies included in this work rely on the excellent work already accomplished in the subjects of electromagnetics, antennas, and microwave engineering and metrology. This book of phased array studies is designed for today’s engineering community, graduate students, and investigators. The studies leverage other established technical references to illustrate the parallels between the advances of the background material and the cause-and-effect relationships found in phased arrays. 1

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2

1.1

Introduction to Phased Arrays

Phased Array History and Perspective A brief introspective is useful in such a diverse subject as phased arrays since it can provide context and a grounding in today’s advanced array theory. A tremendous amount of science, engineering, and mathematics is and has been involved, and the advances in this subject are a direct result of human exploration, interaction, and recognized need. The progress, recorded as inherently human achievements, shows that our understanding of this complex subject is incremental. A history reveals both a perspective of the origins of our thinking in this subject area, and our trends towards the future. We can consider arrays as complex technology comprised of simple devices. An array is an assembly of radiating elements, amplifiers, phase shifters, combiners, and transceivers designed to work together to direct EM radiation into free space with precision. Complicating notions eventually emerge in the form of mutual coupling or the EM field interaction among the radiators, in beam formation, which collects the radiation of many radiating elements, in Floquet’s theory that equates periodic array conditions to the radiated field, and in reactive field theory that characterizes confined antenna aperture fields. This trend continues with the theory behind scan blindness, which can fundamentally limit an array; digital beamforming, which completes the Fourier transform without the use of EM hardware; and polarization diversity, which allows an array to produce all possible forms of polarized radiation, just to name a few of these. A consideration of the history behind array development can offer a guide as our consideration of the subject moves from the elementary to a more comprehensive viewpoint. Figure 1.1 shows a functional block diagram of a phased array system. This array is composed of radiating antenna elements within a common aperture, phase shifters that provide a controllable electrical delay to the array elements, and a combiner or beamformer that distributes the microwave energy from a common port to the multiple branch ports that connect to the phase shifters. This combination is commonly considered a phase-controlled aperture system. Analysis of their functions and the cause-and-effect relationships at the array level is fundamental to this study. The electric fields produced on the array aperture transform in free space to form a main beam where the radiated field reaches its maximum, a series of minor peaks (sidelobes) and nulls where the radiated fields in angle space collapse. The means by which this occurs and the associated design space are the primary subjects that we wish to consider. One way to start the thought progression is to review some of the history of this subject, and how the human family has responded to each major stage. The history of innovation in phased arrays can be viewed in the form of three waves of innovation. Each wave found differing human views of this progression as fundamental array radiation effects were discovered. The first wave introduced the explorers, the investigators without peers who were among the first to recognize consistent EM behavior. Their work bridged across theoretical divides in human thought, establishing the construct of a unified theory. •

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1670 to 1900: The first wave explorers introduce first principles: – Christaan Huygens: In 1678, he recognized that light emerging from a source is readily represented as an accumulation of equivalent spherical

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1.1  Phased Array History and Perspective3

Figure 1.1  A functional diagram of a linear phased array and a representation of the far-field, radiated electric field that it can produce.

–

–

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waves. This led to the first examples of interferometry, where distinct bright peaks and dark nulls are observed. These radiation characteristics are an intrinsic aspect of phased array radiation still studied today as grating lobes. Joseph Fourier: In 1822, he recognized that any function can be represented as a summation of simple orthogonal functions, such as sines and cosines. Accordingly, the powerful Fourier transform relationship between frequency and time domains becomes available. The transform also relates the electric fields of an array aperture to its radiated fields, so this theory is widely used today. James Clerk Maxwell: In 1865, his article, “A Dynamical Theory of the Electromagnetic Field,” showed the space-time connection between electric and magnetic fields. Well before his time, Maxwell pioneered the EM fundamentals. The EM field properties are a key element of phased array design today.

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4

Introduction to Phased Arrays

–

–

Heinrich Hertz: From 1886 to 1889, Hertz published two articles on what became known as the field of contact mechanics, illustrating some of the first examples of radiation. He confirmed Maxwell’s work, providing the first demonstrations of wireless propagation. His discoveries and experimental techniques are commonly used in array design verification today. Guglielmo Marconi: In 1896, he used Morse code signals over a distance of about 6 km to verify wireless communication as a practical mechanism. His subsequent successful transatlantic wireless link stunned the human world. Marconi woke up the human race to the possibilities of wireless propagation, an essential element of array operation.

These early efforts led to a wave of designers and developers who turned theoretical and experimental efforts into practical designs that serve human needs. In this era, we see the emergence of radio, radar, the first wireless communications systems, and a world that begins to adapt to a communications network for the first time in human history. •

1900 to 1970: The second wave brought practical designers that served human needs: – Shintaro Uda of Tohoku Imperial University, Japan, invented the Yagi-Uda antenna in 1926, along with Hidetsugu Yagi. These were the first simple but effective end-fire arrays that illustrate the coherent behavior of an array of radiating dipole radiators. – S. A. Schelkunoff published some of the first books on EM theory, antennas, and array theory from 1937 to 1952. His classical work on the Schelkunoff circle remains a basis for a family of array synthesis methods used to this day. – Samuel Silver. In 1949, Silver published Microwave Antenna Theory and Design, Volume 12 of the MIT Radiation Laboratory Series. This insightful series explores the interplay between radiation, antennas, and phased arrays from an advance mathematical perspective. – John Kraus: In 1950, he published his pioneering book [1] on the subject of antennas and revealed the fundamentals of mutual coupling, elementary array theory, and the first finite-element treatment of the dipole antenna. – Roger Harrington: In 1961, he published the book Time-Harmonic Electromagnetic Fields, which established the Method of Moments theory and early development that forms the basis of today’s full-wave 3-D computational solvers.

This exceptional foundation in the new field of EM arrays reveals the pathway towards the third era of array discovery. These designers, developers, and researchers expand the degrees of freedom and open new capabilities. Their efforts lead to today’s ongoing advancement in multichannel communications networks, sensor systems, SATCOM, 5G communications, solar system mapping, advanced radar, and information technology. •

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1970 to today: The third wave expands the degrees of freedom and system capabilities:

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1.1  Phased Array History and Perspective5

–

–

–

–

–

–

In 1972, Amitay, Galindo, and Wu published the book Theory and Analysis of Phased Array Antennas, which revealed the theory and simplicity of sine space analysis and complex mutual coupling in arrays. In 1988, Lo and Lee published Antenna Handbook, which established a sound theoretical basis for a wide variety of phased array designs and techniques, including the connection between Maxwell’s theory and practical array design. In 2002, Skolnik published Introduction to Radar Systems, a text that studies many of the tools commonly used in contemporary array design, including subarray effects (i.e., repeated subsets within an array aperture, system noise temperature, and digital beamforming). In 2005, Balanis published Antenna Theory, Analysis and Design. This work establishes the foundation for many modern array analysis tools, ranging from far-field analysis and radiating element types to array synthesis. Array synthesis allows array designers to produce an intended radiated field magnitude at precise locations within the array’s observation field of view. In 2012, Pozar published Microwave Engineering. This book forms the engineering foundation for modern transmission line and network analysis and design, with a comprehensive study of the active (i.e., amplified) microwave devices commonly used in arrays. In 2018, Mailloux published Phased Array Antenna Handbook, which establishes advanced phased array design principles that are commonly used today.

One insightful aspect of the third era is the recognition of Snell’s Law and its implications to phased array design. An intrinsic advantage of a phased array is its ability to radiate its primary beam towards a specific location in free space. So the array approximates an impulse function, thus acting as an ideal sampling device that can focus its radiated energy towards one specific spatial location while ignoring all others within its field of view. The phased array approximates this, and, in the process, it produces what is called scan loss, or a reduction in its radiated energy proportional to the scan angle relative to its boresight or its aperture surface normal. Today, we know that it is Snell’s Law that accurately predicts the impedance mismatch that serves as its primary cause. Knowing this cause-and-effect relationship is fundamental to our current approach to phased array aperture impedance matching. This serves as one example that illustrates the benefits of an in depth understanding of the fundamental behavior found in phased array systems. A grasp of the fundamental physics coupled with the powerful 3-D full-wave computational solvers is a key element to some of the most effective design practices that support today’s amazing successes in phased array technology. The brilliant efforts of many engineers, scientists, investigators, and students represent the backdrop for the continued advances in the phased array field. Examples of the work already accomplished include the Cobra Dane array, with its ability to sense over a large spatial region. There is also the IRIDIUM Main Mission Antenna (MMA), with its 16 simultaneous transmit and receive beams, as documented in Figure 1.2. These simultaneous beams allow direct communications with terrestrial transceivers, supporting multiple links in each beam. Because of this, systems such

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6

Introduction to Phased Arrays

Figure 1.2  Photographs of the Cobra Dane and IRIDIUM MMA, along with measured (solid) and predicted (dashed) simultaneous beam coverage of the latter [2].

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1.2  Fundamentals of Wave Propagation: The Wave Equation7

as IRIDIUM allow communications links anywhere on the face of the Earth, truly a milestone in contemporary technology. Our studies explore the design principles behind these assets and the associated science.

1.2

Fundamentals of Wave Propagation: The Wave Equation To prepare for an informed understanding of phased array antennas, it is important that we have a thorough and broad understanding of wave propagation fundamentals, by means of the wave equation. Derived from Maxwell’s equations, the wave equation, also known as the Helmholtz equation, governs the space and time behavior of EM waves. It is a second-order linear differential equation whose solutions form the basis functions for all known forms of wave propagation. A verification of the wave equation can be seen in a simple wave represented as a plane wave originating from an ideal isotropic voltage source located within sourceless free space. This isotropic time-varying energy source produces a uniform radiated EM field in all directions from its phase center, just as a stone tossed into a calm body of water creates a spherical wave in the water, seen as a circular wave on the surface. The radiated field decomposes to a plane wave at an electrical far distance from the source [3]. The following properties of an isotropic source and its propagating free space wave are illustrated in a Cartesian coordinate frame, as shown in Figure 1.3. This offers a limited view of a wave segment as it propagated along the z-axis, although the wave is propagating in all radial directions from its source. We may observe the following: •

The propagating electric field vector (E) is parallel to the x-axis, so the wave is linearly polarized, and the electric field is aligned with the x-axis.

Figure 1.3  An isotropic voltage source and its free space radiation plane wave is broken into its electric field (x-axis), magnetic field (y-axis), and Poynting vector (z-axis).

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8

Introduction to Phased Arrays

•

•

•

The magnetic field (H) is orthogonal to the electric field and the propagation direction and polarized parallel to the y-axis. The energy propagation direction is orthogonal to the above fields and parallel to the z-axis. The radiated power travels in the propagation direction, known as the Poynting – vector (P). It represents the power in the propagation direction.

The propagating mode is termed transverse electromagnetic (TEM), as both electric and magnetic fields are perpendicular to each other and to the Poynting vector. The magnetic field is shown as in-phase with the electric field, with the relationship, H = E/Z = E/120 π . Because the medium is free space, its impedance is 120 π or 377Ω, a real number, producing a real relationship between electric and magnetic fields. If the medium’s impedance was complex, it would introduce a phase shift between the electric and magnetic fields. We can use Maxwell’s second-order differential equations to resolve EM field properties based on observations. The wave equation tells us that this free space TEM wave is a solution and that the electric and magnetic fields are orthogonal to each other (i.e., they are at right angles to each other in all of time and space). Both are orthogonal to the Poynting vector, and the magnitudes of each follow from the differential equation and the boundary conditions of the source and the propagating medium. For the time being, we place our point of observation electrically far (10 wavelengths or more) from the source, just to observe the far-field behavior. Also, since the wave source is a simple point in space, the properties of Floquet’s theory are simplified for the time being. Floquet’s theorem relates any periodicity in the aperture to periodic effect in its radiation. The wave equation is the source of a host of solutions and expansion methods that we can use to explore the space-time characteristics produced by phased arrays. For example, the above observations about a TEM wave are consistent with the general wave equation, indicating that the spatial derivative of the electric field (E) is related to the time derivative of the magnetic field (H), and the magnetic field within the medium (B),





∇×E = − ⎡ ⎢ ∇×E = ⎢ ⎢ ⎢ ⎢⎣

∂B (1.1) ∂t

xˆ yˆ zˆ ⎤ ⎥ ∂ ∂ ∂ ⎥ ∂x ∂y ∂z ⎥ (1.2) Ex Ey Ez ⎥⎥ ⎦

∇ is the spatial differential operator. The cross-product of the spatial differential operator with the magnetic field equals the displacement current (J) and the time derivative of the electric field in the medium (D),



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∇×H = J +

∂D (1.3) ∂t

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1.2  Fundamentals of Wave Propagation: The Wave Equation9

The material normalization constants apply, where the medium’s permittivity is (𝜖) and (μ ) is its permeability. In free space, these material constants are in relatively simple form, where 𝜖 = 8.85 pF/m and μ = 400 π nH/m, so, depending on the medium used, these constants change the internal magnetic and electric fields in the medium, D = εE (1.4) B = mH



The observed relationship between a source and its fields indicates that an electric charge density (ρ ) generates an electric field. The magnetic fields arise from a source electric field, not from a physical magnetic source. This observation applies to both static or fixed in time fields and time-varying fields, where time-varying fields are often characterized as sinusoids. ∇⋅D = r (1.5) ∇⋅B =0

∇⋅A =



∂Ax ∂Ay ∂Az + + (1.6) ∂x ∂y ∂z

We can use these equations to produce the wave equation. To do so, we include the material parameters for a source-free medium, ∂H ∂t ∂E (1.7) ∇×H =ε ∂t ∇ × E = −m

Applying the curl to the first equation, ∇ × ( ∇ × E ) = − 



∂2 E ∂ ⎛ ∂E ⎞ = −mε mε (1.8) ∂t ⎜⎝ ∂t ⎟⎠ ∂t 2

Rearranging,



⎛ ∂2 ⎞ E = 0 (1.9) ∇ × ∇ + mε ⎜⎝ ∂t 2 ⎟⎠

As the fields are present because of the original source, there are no other sources to affect these, so we can use the following vector identity, ∇ × ∇ × E = ∇ ( ∇ ⋅ E ) − ∇2 E

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∇2 =

∂2 ∂2 ∂2 + + ∂x2 ∂y2 ∂z2

(1.10)

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10

Introduction to Phased Arrays

–

The result is the homogenous wave equation for E, ⎛ 2 ∂2 ⎞ ⎜⎝ ∇ − mε ∂t 2 ⎟⎠ E = 0 (1.11)



Using the separation of variables technique, we use the wave equation to calculate the following solution set, assuming that we freeze the action at time zero and introduce a propagation constant (k = 2π / λ = w mε ), ⎛ ∂2 ∂2 ∂2 ∂2 ⎞ ± j(kx x+ky y+kz z ) Ee + + − mε =0 ⎜⎝ ∂x2 ∂y2 ∂z2 ∂t 2 ⎟⎠ ⎛ ∂2 ∂2 ∂2 ∂2 ⎞ = ⎜ 2 + 2 + 2 − mε 2 ⎟ Ee ± j( k⋅r ) ∂y ∂z ∂t ⎠ ⎝ ∂x



(1.12)

The propagation constant is 3-D, independent on all three axes, producing the wave continuity equation, k = kx xˆ + ky yˆ + kz zˆ r = xxˆ + yyˆ + zzˆ k2 = w 2 mε = kx2 + ky2 + kz2

(1.13)

w = 2pf

where (f) is the frequency. So a solution is

E = Ee ± j( k⋅r ) (1.14)



The solution to a differential equation needs the appropriate boundary conditions in order to resolve the fields in a given configuration. Example 1.1

We can describe all of the characteristics of the wave produced using the electric field equation, ˆ zˆ − j k z⋅z ˆ oe− jwt e ( z ) (1.15) E = yE



From this equation, we observe the following field characteristics: • • •

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The electric field is a spatial vector. The electric field is y-axis-oriented and linearly polarized. The electric field has a peak voltage magnitude of Eo.

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1.2  Fundamentals of Wave Propagation: The Wave Equation11

• •

The field oscillates at frequency ω = 2πf, where f is the frequency. The wave propagates in the z-direction.

Example 1.2

We can show that the equation given in the previous problem is a solution to the wave equation. ⎛ 2 ∂2 ⎞ ⎜⎝ ∇ − mε ∂t 2 ⎟⎠ E = 0 (1.16)

∇2 =

∂2 ∂2 ∂2 + + ∂x2 ∂y2 ∂z2

∂2 − j k z⋅z ∂2 − j k z ˆ zˆ ∇ E = 2e ( z ) = 2e ( z ) ∂z ∂z

(1.17)

2

Since

∂ − j(kz z ) e = − jkz e− jkz z ∂z ∂2 − j(kz z ) e = kz2e− jkz z (1.18) ∂z2 2 2 ∂ ∂ mε 2 E = mε 2 e− jwt = −mεw 2e− jwt ∂t ∂t

Since

k2 = w 2 mε = kx2 + ky2 + kz2 ∂2 E = −k2e− jwt (1.19) ∂t 2 2 ⎛ 2 ∂ ⎞ 2 − jkz z − kz2e− jwt = 0 ⎜⎝ ∇ − mε ∂t 2 ⎟⎠ E = kz e −mε



So time and space exponential phase expressions are equal: e− jkz z = e− jwt (1.20)

This can only be true if

kz z = wt (1.21)

relating space and time for the TEM wave.

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12

Introduction to Phased Arrays

An interesting observation about wave equation solutions is the case where the voltage is static, that is time-invariant, such as the case for a battery or a direct current (DC) distribution circuit. The wave equation solution above indicates that, when ω = 0, k = 0, meaning that there is no propagation, other than static charge. This is the reason why all wireless communications operate with a nonzero RF. Typical radiated frequencies start at very high frequencies of 50 MHz, extending to W-band at 100 GHz because of this. Considering more general wave conditions, this example is one of many possible solutions to the wave equation. Others include waveguide modal solutions, which have geometrically differing solutions indexed by an integer modal sequence of [m, n], where each integer is limited, for example, −∞ ≤ m ≤ ∞. There are also separate field types, transverse electric (TE), transverse magnetic (TM), and TEM, each with orthogonal properties. The latter refers to the 90° orientation between electric and magnetic fields, a key aspect of phased array radiation characteristics. The TE01 mode is orthogonal to the TE10 mode, for example. These examples illustrate the concept that multiple mode types are each a solution to the wave equation, so a phased array design may generate a superposition or electric field sum of many of these modes, each with differing propagation velocity and impedance characteristics. Decomposing a complex field into its modal components becomes an important analysis technique. Additional wave types encountered in array design include surface waves, creeping waves, reactive fields, near fields, and far fields, for example. We can use the wave equation to separate and identify their properties by modal expansion. We can use the fact that the electric and magnetic field characteristics of each mode (propagation velocity, impedance, and field orientation) correlate to the geometry and dimensions of the boundary conditions used in elements of a phased array. So, while multiple wave equation solutions apply to the study of phased array systems, they also illustrate an orthogonal solution expansion, a key analytical method for the complex fields generated by arrays. The reason that we introduce the concept of modal expansion is so we can use it to develop phased array designs with solutions formed from a superposition or field summation of multiple modes. The superposition of two or more modes or waves means that, at any location in space, the EM fields (modes) add as vectors. Separating the modes involved is a key element of design, as the modes respond to the boundary conditions in the design, and it is often possible to introduce a design condition that affects one mode, but not others. An illustrative example of multiple simultaneous modes occurs on a transmission line, where an impedance discontinuity can create both forward and reflected waves. The total electric field is the sum or superposition of these two waves, producing a standing wave. When significant, the standing wave inhibits the transmission of microwave energy along the transmission line, thus limiting its intended design function. A second, more intuitive example of superimposed solutions is a Floquet mode expansion, which separates wave equation solutions according to the periodic properties of the array. This forms the basis of the Floquet theory and is a fundamental of array pattern synthesis. Synthesis methods such as this allow the phased array designer to study peak radiation and to understand the production of both the

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1.2  Fundamentals of Wave Propagation: The Wave Equation13

coherent peak radiation at specific far-field angular locations and nulls, or inherent electric field cancellation at other far-field locations. The multiple superimposed modal solutions that we consider here stem from an important property of the wave equation: It states that any one solution and any vector sum of solutions to the wave equation are also wave equation solutions. We will see evidence of this as we continue. In one case, we use the superposition of deterministic (directly calculated) and stochastic (random error) fields into a composite field, representing a practical array far-field solution. Knowing this, we can treat the deterministic response of an antenna aperture, such as the sinc or sine(x)-over-(x) function of a far-field variable (x) separately,



E( u) =

sin ( x ) (1.22) x

We recognize that the microwave circuits used to create an aperture field distribution also produce Gaussian-distributed error vectors so we can treat the latter with stochastic relationships to solve for the random error components in the far field. The final result is the sum or superposition of the deterministic or error-free and stochastic or randomly distributed error fields, each solved separately. The theory of superimposed field solutions of the wave equation also allows us to decompose the Floquet modes in a composite far field in terms of ideal sinc sample functions. This allows us to construct an ideal far-field radiation pattern in terms of its Floquet beams. This method forms the basis of Woodward-Lawson and Fourier transform synthesis methods, both of which are powerful phased array design tools. So we will find that modal expansion in general is a tremendously insightful and powerful design mechanism describing the multiple degrees of freedom available in phased array systems. 1.2.1  Boundary Condition Cases

Several specific boundary condition cases are helpful in illustrating some modal solutions to the wave equation. These are useful from the design perspective, as they can be used to separate distinctive modal properties found in a total field result. Consider the case in which a design team has worked diligently to design a phased array verification model. At many possible stages of the development, a farfield result is produced and compared with the prediction model. Often, we find that the results and predictions do not correlate well. We are left with the intricate problem of isolating cause-and-effect relationships. To address this, we need a powerful tool in this process. The most common approach is to use models that allow us to recognize specific field behavior that reflects a given boundary condition case, such as the presence of surface waves. This ability to recognize the boundary condition case effects and superimpose these into the total resulting field allows the engineering team to isolate specific boundary conditions in the design. This, in turn, is a part of predictive model development that allows designers the ability to iterate towards specific design objectives.

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Introduction to Phased Arrays

Consider the case where an array is produced with building blocks that contain subsets of the total array (i.e., subarrays). Often, we find that a measurable gap occurs between these subarrays, producing a periodic gap geometry on the array aperture. Recognizing the effects of such a periodic gap as a set of periodic sidelobes in the far field allows the design team to identify a primary cause-and-effect relationship. Sidelobes are radiated far fields having a magnitude less than the main beam and are distributed over regions of space other than the main beam. The logical next step becomes the consideration about the cause for these gaps and their possible minimization or elimination as needed to meet required sidelobe levels. As examples of modal wave superposition, we next consider three cases where each originates with wave equation solutions and a specific set of boundary conditions. These lead to effects that become common limiting factors in contemporary phased arrays: •

•

•

Standing waves reduce the transferred microwave power in an array circuit and can limit the phase states available. A plane wave dielectric interface introduces the effects of surface waves and scan blindness. Surface impedance effects are the originating cause for scan loss and can cause scan blindness reducing the radiated power depending on the scan angle.

1.2.1.1  Standing Waves

We can start our case study analysis with basic models that illustrate EM fundamentals and then consider how these become an intrinsic part of phased array design. Expanding on the free-space propagation model indicated in Figure 1.4, we introduce a material barrier, initially a conductive material that can be extended further to arbitrary materials. The incident field in this case is a linearly polarized TEM wave in free space, as shown in Figure 1.4. The Poynting vector is ( zˆ ), where the wave encounters a conductive surface occupying the x-y plane. The conductive plane causes a complete reflection in the incident wave creating an equal and opposite plane wave, now propagating in the (− zˆ ) direction. The incident and reflected waves interact, producing a standing wave. So we can write the equations for this simple case, revealing important relationships with the reflection coefficient, total fields, and the phase of the resultant. For now, we assume that the conductive plate has perfect conductivity. The analysis starts with the incident electric field, including the magnetic fields and Poynting vector. The electric field can be represented by a wave equation solution and a function of the electric field magnitude (Eo), the complex time-varying sinusoid (ωt), where (ω = 2πf), and the free-space phase propagation component (β = 2π/λ o),

ˆ oe j( wt + bz ) (1.23) Ei = xE

Similarly, the reflected electric field uses the same equation, except for its complex reflection coefficient (ρ ), and reversal in propagation direction. The reflection coefficient is defined by the reflected wave at the material boundary,

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1.2  Fundamentals of Wave Propagation: The Wave Equation15

Figure 1.4 A z-axis propagating a free-space TEM plane wave incident on a conducting boundary occupying the x-y plane.





ˆ o re j( wt − bz ) (1.24) Er = xE r=

Er (1.25) Ei –

Using the superposition principle, the total electric field (Et) for (z < 0) is equal to the sum of incident and reflected waves, all of which are solutions to the wave equation,

Et = Ei + Er (1.26)

Following the work of Kraus and Carver [1], the instantaneous electric field is either the real or imaginary part of the complex field, so choosing the latter, and removing the vector notation,

Et = Eo sin ( wt + bz ) + rEo sin ( wt − bz ) (1.27) Since sin ( A ± B) = sin Acos B ± cos Asin B ,

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Introduction to Phased Arrays



Et = Eo (1 + r ) sin ( wt ) cos ( bz ) + Eo (1 − r ) cos ( wt ) sin ( bz ) (1.28) Introducing a variable substitution, A = (1 + r ) cos ( bz )

B = (1 − r ) sin ( bz )



(1.29)

Also, since

Asin ( wt ) + Bcos ( wt ) =

(

)

A2 + B2 sin ( wt + g ) (1.30)

⎛ B⎞ g = atan ⎜ ⎟ (1.31) ⎝ A⎠



The electric field maximum is just the magnitude of (1.30), with the incident field magnitude normalized (Eo = 1),

Et =

((1 + r ) cos ( bz ))2 + ((1 − r ) sin ( bz ))2 (1.32)

The fact that there are forward and reverse waves is sufficient to produce a standing wave, where the electric field magnitude oscillates at the radial frequency (ω t), with peaks and minimums that are fixed along the propagating axis. With a conductive boundary condition, the reflected wave is equal to the incident wave’s magnitude, causing a fixed sinusoidal standing wave (Figure 1.5). Alternatively, by replacing the perfectly conductive boundary with a partially absorptive boundary condition, the reverse wave’s magnitude can be significantly reduced, leaving the forward wave as the most significant in the total electric field summation. 1. For the total reflection or perfectly conductive boundary condition case (ρ = −1): a. At the peaks (z/λ = (2n − 1)/4), the forward and reverse magnitudes accumulate to a value of 2.0, or the coherent sum of two unity valued waves, alternating in polarity (not shown in Figure 1.5). At these locations, the electric field oscillates at the radial frequency (ω t) in the time domain. b. At the nulls (z/λ = (n − 1)/2), we observe that the electric field magnitude is zero for all time, meaning that it is a fixed spatial null. c. The net result is a standing wave, fixed in space, and composed of two equal magnitude waves propagating in opposite directions. 2. For the partial absorption case (ρ = −0.01), a. Peaks and nulls exist, but the absorptive boundary attenuates the reflected wave, producing a peak electric field spatial response that is essentially

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1.2  Fundamentals of Wave Propagation: The Wave Equation17

Figure 1.5  The total free-space electric field absolute magnitude along the propagating axis for a total reflection (ρ = −1) and near total absorption (ρ = −0.01).

uniform as a function of the propagation direction. This is indicative of a forward-traveling wave because the magnitude of the reverse wave is controlled by the reflection coefficient. We can go further and consider the response to a range of reflection coefficient values, while studying the total superimposed forward and reflection mode field response in the left hemisphere of the planar interface boundary condition. Allowing reflection coefficient values over the range of (ρ = [−1.0, −0.707, −0.5, −0.1, −0.01]) reveals a continuum of total field magnitude responses (Figure 1.6). The reflection coefficient is the ratio of forward and reflected voltages. The physical locations of the peaks and nulls are the same for all cases because the reflection coefficient is constrained to real values in these cases. We observe that the voltage standing-wave ratio (VSWR) or the standing wave’s magnitude is directly correlated to the reflection coefficient; thus, the relationships: S = VSWR =



r =

(1 + r ) (1 − r ) (1.33)

( S − 1) ( S + 1) (1.34)

Because the reflection coefficient is limited to the range (0 ≤ ⎪ ρ ⎪ ≤ 1) for passive circuits and boundary conditions, the VSWR is also constrained (1 ≤ S ≤ ∞).

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Introduction to Phased Arrays

Figure 1.6  The total free-space electric field absolute magnitude along the propagating axis for a range of reflection coefficient values of (ρ = [−1.0, −0.707, −0.5, −0.1, −0.01]).

The reflection coefficient is complex, while the VSWR is a scalar (i.e., it has no complex component). Recognizing that the total reflection case (ρ = −1) produces a step function of in-phase and out-of-phase values along the propagating axis, while the near total absorption (ρ = −0.01) case produces a linear phase response, the total field phase can be determined from (1.32). Intervening states have properties of both edge conditions, as seen in Figure 1.7. It is useful to note that, under the large reflection coefficient conditions, there are only two-phase states available, due to the standing wave. By contrast, under small reflection coefficient conditions, essentially any phase state is possible, because the phase associated with the single traveling wave is (ej(β z)). At all intermediate reflection coefficient magnitudes, the available phase states become restricted, leading to potentially significant phase error in a controlled phase state system. The phase

Figure 1.7  The total free space electric field relative phase along the propagating axis for a range of reflection coefficient values of (ρ = [−1.0, −0.707, −0.5, −0.1, −0.01]).

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1.2  Fundamentals of Wave Propagation: The Wave Equation19

error comes into the picture in phased array design when a specific transmission line insertion phase is produced by fixed transmission line lengths. This design assumes that the reflection coefficient is insignificant, but when it is significant, the insertion phase deviates from the assumed case, producing phase errors. These properties are useful in phased array design, and several observations are important to consider: 1. Reflection coefficients are commonly found in microwave circuits used in a phased array, replicating the effects predicted in the above equations. Where they occur with significance, the transfer (propagation) of the EM wave is reduced, due to the reflected fields. 2. Typical microwave circuits used to form a phased array system often have multiple reflection coefficients, each arising from the cascade chains of components used. Where these occur, there is a proportional system performance degradation at the peak reflection coefficient frequencies with the system’s operating band. 3. At frequencies where significant reflection coefficient magnitudes occur, there is a corresponding phase error. The error occurs when microwave component designs use the insertion phase to produce the phased array aperture phase. Deviation from the low reflection coefficient case produces a phase error that correlates to errors produced in the radiated space. 1.2.1.2  Surface Waves and Scan Loss

Mode conversion is one of the most interesting aspects of EM field behavior, and it has a direct applicability to phased array design and development. Insightful publications yield the equations for dielectric planar interface boundary conditions, Snell’s law, and surface waves, such as the work of Kraus and Carver [1]. In the first case, surface waves represent an array condition where the total field received or transmitted by a planar array aperture is trapped or unexpectedly contained within the aperture plane, yielding no radiation to or from free space. This condition causes the array to be scan-blind at the associated critical incidence angle and essentially inoperable. Snell’s Law connects the surface impedance of a planar phased array to its incidence or scan angle and polarization, indicating the conditions for impedance matching and subsequent scanned array energy transfer to space. Consider surface wave effects with a revised coordinate system from Kraus and Carver’s original work [1]. It is useful to start with a Cartesian coordinate system between two different dielectric media (1 and 2), as shown in Figure 1.8, one representing the radiation space and the array aperture. There are two pertinent polarization cases for TEM incident waves because each represents different outcomes: •

•

•

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The incident electric field can be polarized perpendicular (TM) or parallel (transverse electric (TE)) to the plane of incidence (y-z plane in the figure). In Figure 1.8, the incident field is perpendicular to the incidence plane, making the electric field vector ( xˆ ) orthogonal to the y-z plane. The incident polarization is parallel polarized when the incident electric field vector orients parallel to the plane of incidence, with its magnetic vector orthogonal and so aligned with the x-axis.

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20

Introduction to Phased Arrays

•

•

The Poynting vector is oriented along the propagating ray, as shown in the figure for both perpendicular and parallel polarization cases. Any incident wave can be accurately characterized as a vector-weighted combination of these two polarization states.

Scan blindness occurs when a phased array’s radiated field cannot escape the array aperture because of impedance mismatch or Snell’s Law mode conversion, introducing the importance of this fundamental theory. According to Kraus and Carver [1], the electric fields can be written in terms of the directional cosines at the – – interface, considering the incident (Ei), reflected (Er), and transmitted (into medium – 2) (Et) components. We can make use of existing transform relationship formulas between the rotated material layers ([x′,y′,z′]), and the coordinate system reference,



⎡ x′ ˆ ⎤ ⎡ 1 0 0 ⎤ ⎡ xˆ ⎤ ⎥ ⎢ y′ ⎢ ˆ = 0 sin q cos q ⎥ ⎢ yˆ ⎥ (1.35) ⎥ ⎢ 0 − cos q sin q ⎥ ⎢ ⎥ ⎢ ˆ ⎦ ⎣ ⎦ ⎣ zˆ ⎦ ⎣ z′ ˆ oe jk1( y sin qi +z cos qi ) Ei = xE ˆ o r⊥ e jk1( y sin qr −z cos qr ) (1.36) Er = xE



ˆ ot⊥ e jk2 ( y sin qt +z cos qt ) Et = xE

These expressions reference conventional definitions for reflection (ρ ⊥ ) and transmission coefficients (τ ⊥ ), which are in ratios relative to the incident field,

Figure 1.8  A planar interface between media 1 and 2, with incident, reflected, and transmitted waves for perpendicular polarization.

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1.2  Fundamentals of Wave Propagation: The Wave Equation21





r⊥ =

Er (1.37) Ei

t⊥ =

Et Ei (1.38)

The tangential electric field components are equal in both media at the interface between the two, providing an additional set of equations needed to solve for the total scattered fields,

e jk1( y sin qi +z cos qi ) + r⊥ e jk1( y sin qr −z cos qr ) = t⊥ e jk2 ( y sin qt +z cos qt ) (1.39) At a voltage junction, the interface voltages must be equal, so (1 + ρ ⊥ = τ ⊥ ), k1 sin qi = k1 sin qr = k2 sin qt (1.40)



In accordance with optical observations and using the first of the above two equations, the incidence and reflected angles are equal, qi = qr (1.41)



The second equation is the basis of Snell’s law of refraction, sin qt = k1 k2 sin qi sin qt =



m1ε1 m2ε2 sin qi (1.42)

A second set of equations is needed for a complete reflection and transmission coefficient solution, and these originate from the magnetic field equations, applying the characteristic impedance of the appropriate media, jk y sin qi +z cos qi ) Hi = ( − yˆ cos qi + zˆ sin qi ) Eo Z1e 1(



Hr = ( yˆ cos qr + zˆ sin qi ) r⊥ Eo Z1e jk1( y sin qr −z cos qr ) (1.43) H = − yˆ cos q + zˆ sin q t E Z e jk2 ( y sin qt +z cos qt ) t

(

t

i

)

⊥

o

2

The tangential magnetic fields must be equivalent in both media at the interface, again considered as a voltage junction (1 + ρ ⊥ = τ ⊥ ),



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− cos qi + r⊥ cos qi = t⊥

Z1 Z Z cos qt (1.44) Z2 1 2

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Introduction to Phased Arrays

− cos qi + r⊥ cos qi = − Z1 Z2 cos qt − r⊥ Z1 Z2 cos qt r⊥ cos qi + r⊥ Z1 Z2 cos qt = cos qi − Z1 Z2 cos qt



r⊥

( cos qi − Z1 = ( cos qi + Z1

Z2 cos qt )

(1.45)

Z2 cos qt )

In ideal dielectric media, where the relative permeability is unity, the scattered field relation that we seek becomes clear,

r⊥ =

Z1 =

1 ε1

Z2 =

1 ε2

( cos q ( cos q

Using Snell’s law and (cos qt = r⊥ =

) (1.47) ε cos q )

i

− ε2 ε1 cos qt

i

+ ε2

1

t

1 − sin2 qt ),

( cos q ( cos q

i i



(1.46)

) (1.48) q)

− ε2 ε1 − sin2 qi 2

+ ε2 ε1 − sin

i

When the source wave initiates from the denser material, (𝜖1 > 𝜖2), the square root radical, can become imaginary. At a critical angle, it becomes zero, causing the reflection coefficient to become unity (1 + j0), indicating total reflection into the source material at the boundary interface. We can then define the critical angle for perpendicular polarization in the equation, sin qic =



ε2 ε1 (1.49)

When the incidence angle exceeds the critical angle (θ i > θ ic), the radical in (1.49) becomes imaginary, so (⎪ ρ ⊥ ⎪ = 1). This leads to the conclusion that (sinθ t > 1), from Snell’s law, and (cosθ t) is imaginary since sin2 q + cos2 q = 1 (1.50)



We can reconstruct the expression (cosθ t) as (jA), where A is purely real, cos qt =



1 − sin2 qt = jA (1.51)

Applying Snell’s law,

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1.2  Fundamentals of Wave Propagation: The Wave Equation23

A=



sin2 qt − 1 =

ε1 ε2 sin2 qi − 1 (1.52)

We can now rewrite (1.37) for the transmitted fields located in the medium 2, ˆ ot⊥ e jk2 ( y sin qt )e jk2 ( z cos qt ) Et = xE −k z ˆ ot⊥ e 2 ( = xE

ε1 ε2 sin2 qi −1

)e jk ( y sin q ) (1.53) 2

t

ˆ ot⊥ e−k2zAe jk2 ( y sin qt ) = xE



The transmitted field expression now has two distinct terms with differing properties at angles exceeding the critical angle: •

•

•

(e−k 2 zA) describes a wave that attenuates very rapidly in the less dense medium 2, with a real and in-phase propagation constant of (e−Cz). This wave is contained within the interface between the two materials and so cannot propagate further into the second material. (ejk 2(ysinθ t)) describes a wave that propagates along the y-axis with sinusoidal properties, (i.e., a standing wave). The surface wave propagates in the x-y plane with no significant loss.

In conclusion, these results show that the transmitted wave is trapped or contained within the denser dielectric material, as illustrated in Figure 1.9. Considering

Figure 1.9  A planar medium 1 and 2 interface, showing the incident, reflected, and transmitted waves for perpendicular polarization at incidence angles exceeding the critical angle, where the transmitted wave produces a surface wave at the interface.

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Introduction to Phased Arrays

that the primary role of dielectric materials used in phased array design is to provide electrical isolation and allow the free propagation of the source wave, this wave containment is generally counterproductive and can cause scan blindness. The above evaluation indicates one set of boundary conditions that can cause a surface wave. As we study the aperture conditions prevalent in phased arrays, other related conditions that lead to a surface wave condition will appear. The end result of a surface wave is nevertheless clear, and a consequential scan blindness condition remains as a significant potential performance-limiting factor. Consider next a planar array with its materials occupying the x-y plane, so the above analysis represents a sectional view of the same geometry, with the (z > 0) hemisphere representing the intended free-space propagation region. Under the above conditions, where the incidence angle exceeds the critical angle (θ i > θ ic), the array creates a surface wave instead of propagating a wave into its forward hemisphere. The intended transmitted field is trapped within the aperture plane, only to reflect from the array edge boundaries in a 2-D standing wave, considering that its surface wave attenuation in the x-y plane has no significant loss. As a result, the surface wave condition creates a total reflection (⎪ ρ ⊥ ⎪ = 1) to its microwave sources and so is scan-blind in the associated scan region. We can show how to examine surface wave conditions in a phased array. An example is an array composed of continuous dielectric materials or a periodic grid of radiating elements, since the wave transition phenomena is comparable for both. As it produces a scan blind condition, the array cannot meet its intended radiation characteristics under these conditions and may lead to the destruction of its transmitting microwave sources. This constitutes one of the possible and avoidable major failures of such systems. The second analysis applies to the TE case, where the incident polarization is parallel to the incidence plane. This case produces a unique condition when the incidence angle produces a zero-reflection condition, termed the Brewster angle. This result differs significantly from the TM case, pointing out the significance of the incident polarization within the phased array’s scan volume. Such a divergent polarized result can be a limiting factor in fully polarized arrays. In the TE case, the matched voltages at the interface yield 1 + r! =



cos qt cos qi t! (1.54)

With lossless and nonmagnetic dielectrics, this becomes



r! =

− ε2 ε1 cos qi − ε2 ε1 − sin2 qi ε2 ε1 cos qi + ε2 ε1 − sin2 qi

(1.55)

The TE case produces a reflection coefficient of zero at the Brewster angle, equivalent to total transmission between the dielectric media when



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sin qiB =

ε2 ε1 (1.56) 1 + ε2 ε1

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1.2  Fundamentals of Wave Propagation: The Wave Equation25

So the critical surface wave angle condition is the same for both polarizations. These critical angle conditions are generally avoided in phased array designs by careful selection of dielectric material parameters. 1.2.1.3  Scan-Dependent Surface Impedance

When we consider the benefits of array design, its ability to scan or position its far-field radiation to specific directions is fundamental. So the behavior of planar dielectric surfaces to incidence or scan angles becomes a critical subject. Thus, this highlights the importance of the work of Collin [4], where he examines the impedance relationships at the same interface, using the properties of Snell’s law. Collin’s work provided the basis for what is frequently the common cause for array losses due to the scan angle, generally termed scan loss. His analysis shows that scan loss is due to the polarization and incidence angle dependency of a planar aperture’s surface impedance. Consider the geometry of an incident plane wave at the boundary of a planar surface (Figure 1.8). We know that the scattered fields include reflected and transmitted components, as shown, but we can extend these further by considering all relations as wave fronts in both media. Collin showed that, by using transform relationships, the E- and H-plane surface impedances differ and both are scan dependent. So this bears a similarity to the Snell’s Law analysis, where the differing polarizations from TE and TM incidence waves produce divergent reflection coefficient results. In this case, we consider the surface impedance response in terms of the free-space impedance (η = μ /𝜖 = 120 π = 377Ω). The H-plane (TM) surface impedance is a function of the material dielectric constant (𝜖r) at an air-dielectric, the free-space impedance (η ), and the incidence angle (θ i), as described by



ZH =

(

h εr2 − sin2 qi εr cos qi

)

(1.57)

Meanwhile, the E-plane (TE) surface impedance is ZE =

h(εr cos qi )

εr2 − sin2 qi

(1.58)

The surface impedance diverges between TE and TM waves significantly at wide scan angles because it is dependent on the relative dielectric constant value of the dielectric material used (Figure 1.10). It changes from (η = 120 π Ω) at boresight scan (0° relative to the surface normal) to the limits of 0Ω in the E-plane, and (∞Ω) for the H-plane end-fire scan (90° relative to the surface normal). Considering that the TEM port impedance of a typical array radiator is 50Ω, the impedance matching problem is formidable. This means that, when the surface impedance values differ significantly from the port impedance, the radiator becomes difficult to impedancematch. An impedance mismatch at the aperture will, in turn, reduce the energy

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Introduction to Phased Arrays

Figure 1.10  The calculated surface impedance for TE and TM waves at an air-to-Teflon dielectric interface and their ratio.

transfer to space and be scan and polarization-dependent. With increasing scan angle, the surface impedance mismatch loss can increase significantly because of Snell’s law, introducing an intrinsic scan loss effect. While most arrays limit their maximum scan angle to angles considerably less than (θ max = 90°) or end-fire for this and other reasons, the inherent impedance divergence exacerbates the array element’s impedance mismatch problem. Because of the Bode-Fano theorem [5], the impedance divergence with respect to the scan angle also equates to a reduced operating bandwidth. A commonly used scan volume is conical, extending to (θ = 60°) including all orthogonal (ϕ ) angles. Within this scan restriction notable array performance has been achieved. For example, a single linearly polarization array scanning in its E-plane (TE) and impedance matched at boresight scan (θ = 0°) produces a surface impedance ranging from 377 to 207Ω. Using a median impedance value of 292Ω, the impedance ratio over the scan range is 1.4, with a reflection coefficient of −15.3 dB and a relatively small associated insertion loss of 0.13 dB. The scan and polarization-dependent surface impedance variation nevertheless makes the impedance-matching task a noteworthy one because of contemporary needs for wide operating bandwidth. Just as with the dielectric substrate analysis, effective dielectrics show similar behavior, making surface impedance analysis general in its principles.

1.3

Array Antennas Phased array systems have become a natural development path, originating in the microwave design of passive antennas. There are multiple examples of constrained feed systems, such as those formed from TEM and waveguide transmission lines, power dividers, and delay circuits. These designs generally use a single Tx source or receiver but have migrated to distributed systems with multiple sources and receivers. These in turn have progressed to digitally beamformed systems using multiple

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1.3  Array Antennas27

digital-to-analog converters (DACs) and analog-to-digital converters (ADCs). The next generation of devices seek more general applicability in low cost, size, weight, and power (C-SWAP) designs. Throughout these progressive steps, the intrinsic phased array value has remained consistent, offering the degrees of freedom in the domains of space, frequency time, and polarization. So, while there are many possible design adaptations to the technology, the focus of our studies into phased arrays relies on the fundamentals and limiting factors, with a few examples for illustration and practical application. Phased arrays have multiple advantages over fixed beam antennas, including: •

•

•

•

•

The ability to scan the beam to a specified region of free space with beamswitching speeds of 1 μ s, thus avoiding the inertia of a fixed beam antenna mounted on a rotating pedestal. High directivity and the resulting large signal-to-noise ratio (SNR), where directivity is defined as the ability of the antenna to focus its radiated field to a specific region of free space. While high directivity is also available in fixed beam antennas, the use of distributed power amplifiers in a phased array architecture can significantly improve the total radiated power and thus the SNR. The ability to reduce the system noise temperature. The use of distributed low noise amplifiers results in low antenna system noise temperature, thus expanding the SNR. The ability to control the polarization state and its isolation within the system scan volume. Modular building blocks that reduce the system cost.

By comparison to some of the best existing publications, Balanis [6] noted that the total far-field propagating mode produced by an array is determined by the vector sum of the fields radiated by the individual elements. We refine this observation by studying the integration of the aperture tangential fields and the embedded element field, so we can observe and predict the radiated fields produced by vector potentials available on the antenna aperture. Mailloux [7] indicated that radiated element fields of an array accumulate in the far field. This is equivalent to a condition where the multiple electric field modes present on the aperture converge to a single far-field propagating mode. We explore this and develop the theory associated with the total embedded element field that results from the mutual coupling. The factors affecting the array far-field performance include those cited by Mailloux and perhaps a few more: •

•

•

•

7060_Book.indb 27

Array edge geometry: This affects the near-in sidelobes, or those closest to the main beam. Lattice: The geometrical arrangement of phase centers affects the grating lobes and scan volume. Element complex excitation vectors: The amplitude and phase of each element affects many far-field characteristics, including error sidelobes, scan conditions, beam-pointing accuracy, and polarization characteristics. Embedded element factor: This controls a limiting function for array far fields, polarization characteristics, and the associated scan loss.

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28

1.4

Introduction to Phased Arrays

Aperture State Fundamentals Although the behavior of the EM fields produced and received by phased apertures has extraordinary complexity, there are amazing parallels to the scattering and optical projection effects predicted by Huygens’ theory. A brief consideration of the associated theory offers an intuitive basis for our study of arrays. Central to the study of aperture antennas is Huygens’ principle, summarized as: •

•

•

Each point on a primary wavefront can be considered to be a new source of a secondary spherical wave and a secondary wavefront can be constructed as the envelope of these secondary spherical waves. Christaan Huygens (1629–1695) made his fundamental discoveries on the observed properties of light and developed the basis for the theory that light was composed of both waves and particles. Later, Einstein named these particles as photons. From him and others, we conclude that wave theory applies to all EM radiation, including microwaves and the related EM arrays. Huygens theorized that each light source produced a spherical wave that we as observers see is the composite of the spherical light waves produced by all sources within our focused field of view. The fundamentals of Huygens’ theory come from his 1678 treatise [8] and can be visualized from one of the book’s illustrations (Figure 1.11). The image shows multiple light sources in a candle’s flame (A, B, C, and so forth) and the multiple spherical waves that each source produces. Each wave propagates from its source in a low loss medium, such as air, at the speed of light, until these waves arrive at the observer (i.e., our eyes as a composite wave).

Huygens described the process by which light propagates from its source using a method where light waves create a secondary sphere, produced by the summation of interior or primary waves. In Figure 1.12, the secondary sphere of light at points

Figure 1.11  An image of the spherical light waves produced by a candle, as described by Huygens [8].

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1.4  Aperture State Fundamentals29

Figure 1.12  An illustration of light waves emanating from a point source and producing successive secondary spheres of reconstituted light [8].

(d) is constructed from the waves sourced at the sphere produced by points (b). Light propagation is represented as an expansion of successive spheres as a function of time. Thus, the light energy dissipates as it is spread over larger surface areas. We recognize today that the successive spheres visualized by Huygens represent the superimposed fields produced by a single point source, whose radiated voltage is a function of the distance from the source (r) and the wave number (k = 2π / λ ),



E(r ) =

e− jkr (1.59) r

Consider several observations that accompany this principle in the form of various case studies. One of the most illustrative of these is the propagation of a plane wave. The radiation from a source has TEM characteristics, so the electric and magnetic field vectors lie in a plane orthogonal to the Poynting vector. In a plane wave, we ideally have an infinite plane of these Poynting vectors, each with the same vector orientation. In the case where the magnitudes and phase (time phasing) of the electric and magnetic vectors are the same at all locations on the plane, we have a uniform phase wavefront. Where the wavefront is aligned with a plane, we have a plane wave. Next consider the propagation of the wavefront through a low loss medium, such as air or a vacuum. The Poynting vectors at each point on the planar wavefront

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30

Introduction to Phased Arrays

indicate that the wave propagates along a constant vector in the direction of propagation. At a distance of one wavelength from the original (primary) wavefront, the plane wave creates a secondary planar wavefront, again with all EM vectors having equal magnitude and phase. The primary wavefront recreates an equal secondary wavefront, both traveling at the speed of light (c). In a lossless medium, the secondary wavefront is equal to the primary plane wave and is simply displaced in time by (t = λ /c). This forms a conceptual framework for plane wave propagation. We can offer the following general considerations: •

•

•

•

•

•

•

•

A wavefront is a complex EM field distributed over a surface of interest. Typically, the evaluation surface is a plane positioned perpendicular to the Poynting vector or propagation direction, although our equations can evaluate any surface that is immersed in a propagating EM field. An elementary example of a primary wavefront is the complex field distribution produced by an ideal isotropic point source at a far-field plane whose normal vector points towards the point source phase center. Because the source is viewed from the far field, the radiated spherical wave expands into a plane wave, its amplitude and phase distributions approximate uniform functions over a virtual plane that is oriented perpendicular to the Poynting vector. The secondary wavefront is an equivalent of the primary, but displaced in propagating space, and often formed differently than the primary. This description allows us to convert the primary form of the field distribution into the secondary. Primary and secondary fields occur sequentially as the wave propagates but can be considered to occur at the same time. Huygens’ principle allows us to decompose and reconstruct the field, transforming the primary wave into the secondary one. This field construction method offers a wave propagation explanation. In the far field of a point source, the fields determined at the sample point reradiate to form the next wavefront of the expanding field. The progression of a primary wave decomposing into its secondary wave is a way of describing a wave transformation. Huygens’ principle is reciprocal, so the primary fields and the resulting secondary fields are directly related by forward and reverse transform relationships. The Fourier transform provides the needed relationship between source and observed fields. For example, at a large (far-field) distance from a candle, it appears to be an isotropic source, radiating a uniform field magnitude at any angle. The transform converts the primary spherical wave to a plane wave in the eyes of the far-field observer.

Example 1.3

We can test the idea that a single-frequency infinite plane wave reproduces itself as it propagates in free space. Our experiment can be formed in several parts: •

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Construct a plane wave composed of isotropic source points located on the [x,y] plane, as shown in Figure 1.13.

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1.4  Aperture State Fundamentals31

Figure 1.13  An illustration of multiple point sources with the z = 0 plane and a single external sample point. •

•

•

Discretize this plane with a square lattice of the nine point sources shown using fixed separations of [dx,dy] (black dots in the figure). In this initial assessment, both dx and dy dimensions are set to of 0.5λ . Construct a second [x′,y′] plane, displaced 1 wavelength from the source plane wave and along and perpendicular to the z-axis Poynting vector, with the same periodic spacing (blue dot in the figure). Calculate the field relations between the primary and secondary (source and observer) planes.

The total electric field at a given secondary point is the superposition or sum of the fields from neighboring source points. So we calculate the following field summation: •

•

•

From the nearest source point: Because the intervening path length is (dz), the received electric field phase is delayed by (e –jkdz), where (k) is the propagation constant in the media. From the next adjacent eight source points: Because the intervening path length is (dzcosθ 1), the received electric field phase is delayed by (e –jkdzcosθ 1). From the next adjacent 16 source points: Because the intervening path length is (dzcosθ 2), the received electric field phase is delayed by (e –jkdzcosθ 2).

As the source points are taken as isotropic, the magnitude of all superimposed fields is unity or 1.0, and the total superimposed field, with all source points associated with each index value (n) included in the field summation, is

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32

Introduction to Phased Arrays

Et =



∞

∑ e− jkd cos q z

n=1

n

(1.60)

We repeat this process for all source points. Next, we change our focus to the next observer point, repeating the process indicated above, and reach the same result. Therefore, the second secondary sample point has the same complex voltage as the first one. Repeating this for all secondary wavefront points, we conclude that the source field produces an identical secondary wavefront, and a planar wavefront reproduces itself as it propagates. In conclusion, a primary plane wave produces a secondary plane wave, so a plane wave propagates unchanged in its wave properties.

One of the most important values originating from Huygens’ principle is the ability to introduce mode transformation into the antenna design process. There are many possible mode forms, for example: •

•

•

The wavefront mode can be planar, cylindrical, spherical, or arbitrary, but must have a fixed geometric construction. Each mode represents a wavefront of corresponding geometries, for example, a cylindrical wavefront has uniform amplitude and phase over a prescribed cylindrical surface. Waveguide modes can be in the form of TE, TM, TEM, and hybrid modes. A standard waveguide used as a radiator mode converts a TE01 mode to TEM in the far field, for example. Mode types can convert from one type to another, based on the boundary conditions used in an array design. For example, the reactive evanescent modes produced by a phased array aperture mode convert to a single TEM propagating mode in the far field. These evanescent modes are confined to the aperture because of surface wave conditions and attenuate rapidly with the observer’s separating distance. The TEM mode is also present in the aperture but propagates to the far field with considerably less loss, becoming the dominant far-field mode. As a result, the aperture field modes transform from evanescent to a single TEM propagating mode.

It is useful to define mode transformation: the use of boundary conditions to modify one type of propagating mode into another. Frequently in mode transformations, the propagation velocity, polarization, and impedance of the source mode differs in the newly transformed mode. In general, an antenna can be considered a mode transformer because it transforms a bound TEM source at its port into a free-space TEM propagating wave. The antenna’s aperture mode field distribution mode converts from a complex set of reactive field modes to a single TEM far-field propagating mode. An electrically small opening in a conductive plate provides an example of mode transformation (Figure 1.14). From its single-frequency point source emitter, the propagating fields encounter a conductive plane containing the electrically small opening or aperture. When the dimensions are considerably less than 1 wavelength

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1.4  Aperture State Fundamentals33

Figure 1.14  The mode transformation caused by an electrically small aperture in a large conductive plate.

within the aperture plane, the source wave enters the opening and produces a mode transform. The aperture’s set of boundary conditions produces a new spherical wave on the other side of the aperture opening. The transformation produces an equivalent source phase center in the middle of the opening in the conductive plane. As a result of the Huygens wave formation process, the plane wave at the aperture transforms into a spherical wave, of which we can observe the following: •

E=

•

•

•

7060_Book.indb 33

The source fields radiate as a function of time with the spherical expansion – function at a fixed operating frequency, propagating along a vector (r), so the – phase progression follows along a vector aligned in the same direction (k), e− jk ⋅r (1.61) r

We observe that the (1/r) electric field dependency stems from the idea that a normalized source power of 1.0W spreads over a sphere of radius (r) with a power ratio of 1/4π r 2 and the proportional voltage is the square root, removing constants, producing a (1/r) dependency. The source fields encounter the conductive plate boundary condition and so reflect, except at the aperture opening. From Huygens’ principle, these aperture-bound fields form a primary wave, leading to the secondary waves that propagate into the (z > 0) half-space. The source field produces a nearly uniform complex field distribution at the aperture as the latter is located in the far field of the left hemisphere’s source.

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34

Introduction to Phased Arrays

•

•

The aperture fields radiate into the z > 0 half-space as if there were an infinite number of source emitters within the electrically small aperture boundaries, producing a uniform amplitude and phase distribution. The electrically small aperture can therefore be approximated as a point source. The far-field response from the aperture is the result of the integral of all aperture fields. This produces a secondary spherical wavefront in the (z > 0) half-space.

The source field transforms from a point source to a spherical wave, to an aperture source, and again to a half-spherical wave as illustrated in Figure 1.14. From the perspective of wave expansion, we observe the following: •

•

•

The electrically small aperture boundary conditions at the aperture or conductive plane opening control the mode transformation in the z > 0 half-space. We will see that this holds in general. As the aperture is electrically small, it approximates a point source. The incident modes also play a role in the transformation. In this case, the source field incidents on the aperture are those of a plane wave as the aperture opening in the z < 0 half-space latter is in the source’s far field. The aperture becomes a mode transformer, converting the point source into a spherical wave on the z > 0 half-space. The point source at the aperture is the primary wave that propagates and transforms into the spherical secondary waves in the (z > 0) half-space.

Next, we introduce two apertures into the boundary conditions (Figure 1.15) at a fixed frequency, and although this may seem like a simple adaptation, it opens

Figure 1.15  The mode transformation caused by two fixed frequency electrically small apertures in a large conductive plate.

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1.4  Aperture State Fundamentals35

the door to the concepts of wave theory, interference, and superposition. The experimental conditions are straightforward: produce a single frequency source and a reflecting or absorbing plane with two electrically small openings. The separation between the apertures is initially 1.5 wavelengths, but we will treat this in general terms in later sections of this book. The wave behavior in general is the same as in the single aperture boundary condition set; a spherical wave in the (z < 0) half-space creates a spherical wave that illuminates a uniform field in the two apertures, and each generates spherical waves in the (z > 0) half-space. The path length from the point source in the z < 0 half-space to the two apertures is equal. Because of this, the fields within each electrically small aperture have the same amplitude and phase. We can make the following observations about the field distributions produced: •

•

•

•

•

•

•

•

•

• •

•

The EM fields at the two apertures are identical, represented as two equal vectors, each has a voltage of (1.0 + j0.0) or (1.0 ∠ 0°), in rectangular and polar forms. This occurs because each aperture is equidistant from the point source; thus, the path lengths are the same. For simplicity, we ignore any mutual coupling, where the fields of one source effect those of the other. Because both apertures are electrically small, the field distribution within each is uniformly distributed and represented as an independent point source. The aperture fields that radiate into the (z > 0) half-space interact and add as vectors. There are spatial locations where the two aperture fields have the same phase, causing the vector fields to add. There are other spatial locations where the phase of the two aperture fields are unequal, causing a combined field magnitude less than the in-phase case. There are also locations where the phase of the two radiated aperture fields are equal in amplitude with a phase 180° apart, causing total cancellation of the accumulated field. The interaction between the fields from the two apertures produces radiation fields with peaks and minimums. There are locations in the (z > 0) half-space where the radiated fields of the two apertures are in phase, as indicated by red and green dots. These inphase locations occur at fixed θ angles in the coordinate system centered at the (z = 0) and centered between the two apertures, as shown. There are multiple separate angular locations where in-phase fields occur. In the (z > 0) half-space, there are locations where the radiated fields of the two apertures are in an out-of-phase (180°) phase relationship, causing total cancellation. These occur at fixed angles between the angular locations of the peaks. There are multiple separate angular locations where out-of-phase fields occur.

These observations are not new and were first documented by Thomas Young in 1807, as shown in Figure 1.16 [9]. In this case, the nulls (minima) are indicated, these being locations where the two aperture sources produce a cancellation of the aperture’s fields. Both Huygens and Young made their observations using

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36

Introduction to Phased Arrays

Figure 1.16  A replica of Young’s two-source interference diagram (1807), with sources A and B, producing minima at locations C, D, E, and F [9].

light as the source, so the illustrated nulls appear as dark lines where the light waves cancel. The field accumulation effects are directly calculated using plane geometry, as shown in the following example. Example 1.4

Consider two-point source apertures as point sources, as illustrated in Figure 1.17. Determine the angular locations in the (z > 0) hemisphere where nulls form. The necessary geometry as it exists in the x-z plane has the following characteristics: • •

•

•

•

Aperture source points 1 and 2 exist on the (z = 0) axis at a separation of x1. Propagation rays (straight lines) in the (z > 0) half-space derive from each of the point sources to an observation point (P) at a significant distance from the x-axis or aperture plane and form a triangle. Based on our previous observations, nulls in the field form at locations where the propagated field vectors are equal in magnitude but aligned at a 180° phase difference and generally at a nonzero angle (θ ). An inner triangle can be drawn perpendicular to the source 1 and to point (P) line. The smallest inner angle is also angle (θ ), by similar triangles. The smallest leg of the inner triangle has a length of (x1sinθ ).

In order to form a null, the path length from source 1 to (P) must exceed that of the source 2 to (P) by an odd integer of 180°. Two free-space waves meet this condition when their path lengths differ by the odd integer multiple of a half-wavelength or 0.5λ . Accordingly, we can set up the following equation: 0.5nl = x1 sin qn=odd

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sin q = 0.5 nl x1    n=odd (1.62)

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1.4  Aperture State Fundamentals37

Figure 1.17  An illustration of two point-source apertures within the x–z plane.

In the case where (x1 = 1.5λ , (1.62) reduces to sin q = 0.5 nl 1.5l



n=odd

=

n 3

n=odd

(1.63)

Accordingly, there are two possible solutions in the z > 0 hemisphere, seen in Table 1.1. Table 1.1  Possible Solutions in the z > 0 Hemisphere for x1 = 1.5λ N

θ (deg)

1.00

19.47

3.00

90.00

If we consider the geometry associated with Figure 1.16, the point source aperture spacing appears to be eight wavelengths. Accordingly, there are more nulls in the (z > 0) hemisphere, as seen in Table 1.2. So, we have a similar effect, but more nulls produced. This indicates that the number of nulls and their locations depend on the electrical spacing between the two apertures. Wave tanks found in some museums produce a similar and very visible effect. Using the observations that we have uncovered already, we can now introduce the concept of the Nyquist sampling theorem. It states that [10]: “the sampling

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38

Introduction to Phased Arrays Table 1.2  More Nulls in the (z > 0) Hemisphere for x1 = 8λ N

θ (deg)

1.00

3.58

3.00

10.81

5.00

18.21

7.00

25.94

9.00

34.23

11.00

43.43

13.00

54.34

15.00

69.64

frequency should be at least twice the highest frequency contained in the signal” to accurately sample a wave. This statement can be converted into spatial sampling by noting that frequency (f) and wavelength (λ ) are related by the speed of light (c),



l=

c f (1.64)

The corresponding maximum spatial sample period is (0.5λ ), as 2f correlates to 0.5λ and c is constant. We can readily test this conclusion. Consider Example 1.4, where the aperture spacing is constrained to the Nyquist limit of (0.5λ ). As a result, (x1 = 0.5λ ), and the angles where nulls can form in the forward hemisphere become

sin q = 0.5nl 0.5l n=odd = nn=odd (1.65)

So the only solution becomes (n = 1), and (θ = 90°), which is orthogonal to the direction of propagation, meaning that there are no nulls in the forward hemisphere, using Nyquist spatial sampling. This observation becomes more useful when we consider the multiple apertures inherent in a phased array, as the individual apertures become radiating elements. A useful, multiple aperture experiment leads to a scanned far-field mode transformation. By adding a second aperture to the plate and adjusting the plate angle to the point source in the (z < 0) hemisphere, we introduce the conditions for a scanned far-field mode transformation (Figure 1.18). As in the previous case, the left hemispherical source projects a spherical wave towards two apertures located in an ideally conductive plate. The path length from the point source to the two apertures differs, with (l1 > l2). These modifications produce spherical waves from each of the two apertures to illuminate the (z > 0) hemisphere at differing times, aperture 1 preceding aperture 2, as aperture 1 is closer to the original source and aperture 2 is further away. As a result, the two (z > 0) spherical waves are displaced in time, producing in-phase propagation at angles scanned away from boresight (θ = 0) (i.e., the surface normal to the conductive plate). Similarly, the (z > 0) nulls and sidelobes all displace in angle space by the same amount. The (z > 0) hemisphere’s radiation is therefore scanned (i.e., displaced from θ = 0°).

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1.4  Aperture State Fundamentals39

Figure 1.18  The mode transformation caused by two displaced electrically small apertures in a large conductive plate from a single displaced source.

This example illustrates some of the fundamental conditions for scanned behavior, which can be summarized as follows: •

•

•

•

•

•

Two or more apertures on a common surface radiate independent waves (at the same frequency) into a hemisphere in free space. The two apertures are illuminated with the same magnitude excitation in this case. The phase of the two apertures is unequal. In this case, with only two apertures involved, the phase difference between the two apertures forms a linear phase gradient. The phase displacement causes the peaks and nulls in the radiation half-space to shift relative to the boresight case, where the two apertures radiate in phase. The displacement in the radiation half-space moves all radiation characteristics by a constant in sine space, with the conversion (u = sinθ ), where the sine space variable is (u). The two apertures are treated as elements of a common aperture. Logically, there can be more than two elements grouped into an array aperture that includes all elements.

This section started with Huygens theory and observations with light, progressing through several wave fundamentals, up to and including a simple scanned array. It introduced the concepts of wave propagation, primary and secondary waves, and the equi-phased wavefront. Elementary examples revealed the essentials of mode transformation and Nyquist spatial sampling. This led us to a two-aperture case, revealing the principles of superimposed waves and of field addition and cancellation. A scanned case also shows us that the radiated field produced by two apertures

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40

Introduction to Phased Arrays

can be readily translated in angle space. These fundamentals serve as a basis that allows our considerably deeper investigation. 1.4.1  General Aperture State Relationships

With a grasp of aperture effects, we can expand the aperture conditions to include more complex phenomena and to uncover useful derivations. The steps involved in generating the secondary wave from the aperture fields are detailed in the master work by Balanis [6]. Following this, we can determine the radiated fields from an assumed rectangular aperture with dimensions [a,b], found in the [x,y] plane, as a starting point (Figure 1.19). Governing assumptions are as follows, starting with rectangular apertures: •

• •

• •

The coordinates of source points within the source aperture are indicated in primed notation, for example, [x′,y′]. The magnitude of the aperture field is uniform and constant at all source points. The source points are contained within the aperture [x,y] plane and extend along dimensions along (a) the x-axis and (b) the y-axis. – We integrate the current sources together to determine the vector potential (A). We solve for the radiated fields in a spherical coordinate system, (E θ ,E ϕ ), since (Er ≈ 0), in the far field.

This process yields a deterministic form for the far-field radiation produced by a given aperture. This serves as a foundation for a similar determination for a

Figure 1.19  The geometry associated with an aperture parallel to the x-y plane and centered in Cartesian space.

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1.4  Aperture State Fundamentals41

complete phased array. The process starts by considering the equivalent magnetic fields produced at a rectangular aperture. The integrated equivalent magnetic source (M) field distribution over the aperture dimensions (a,b) for the far-field integrals can be expressed as:





Lq = cos q cos f

Lf = − sin f

b /2 a/2

∫ ∫

Mxe jk( x′ sin q cos f+ y′ sin q sin f ) dx′ dy′ (1.66)

−b /2 − a/2

b /2 a/2

∫ ∫

Mxe jk( x′ sin q cos f+ y′ sin qsin f ) dx′ dy′ (1.67)

−b /2 − a/2

For convenience, we introduce a set of sine space transform relationships,



⎡ u ⎤ = ⎡ sin q cos f ⎤ ⎢⎣ v ⎥⎦ ⎢ sin q sin f ⎥ (1.68) ⎣ ⎦ The integral equations yield a closed-form sinc function solution as

−c /2

sin ( ac 2) (ac 2) (1.69)

Eq = jEo

sin X sinY abke− jkr sin f (1.70) 2pr X Y

c /2









∫

Ef = jEo

e jaz dz =

sin X sinY abke− jkr sin q cos f (1.71) 2pr X Y ⎡ X ⎤ = ⎡ ka 2u ⎤ ⎢ ⎥ (1.72) ⎣⎢ Y ⎥⎦ ⎣ kb 2v ⎦

For a linearly polarized aperture, the E-plane (ϕ = 90° = π /2) far field reduces further,



Eq = jEo

abke− jkr ⎡ sinkb 2sin q ⎤ 2pr ⎢⎣ kb 2sin q ⎥⎦ (1.73) Er = Ef = 0 (1.74)

It is interesting to note the similarity to the dipole far-field equation, noting that a dipole is one of the simplest of the aperture radiators. From the same source

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42

Introduction to Phased Arrays

[6], the dipole radiation is a function of the dipole length (l) and the free-space impedance (η ), Eq = jIo



he− jkr ⎡ coskl 2cos q − coskl 2 ⎤ ⎥⎦ (1.75) 2pr ⎢⎣ sin q

which, at its resonance frequency, becomes Eq = jIo



he− jkr ⎡ coskl 2cos q ⎤ ⎥⎦ (1.76) 2pr ⎢⎣ sin q

So we see the similarity in the far-field equations for both the rectangular aperture and the elementary dipole, both of which use the sinc function. Note that the dipole is orientated differently and so uses the cosine function while the aperture is parallel to the x-y plane and uses the sine function instead. These two equation sets indicate that similar far-field behavior exists for both fields within an aperture, where dipole elements act as a current source. Beyond that, the most significant difference lies in the denominator term in the aperture expression, kb 2 (1.77)



which produces sidelobes of diminishing magnitude as the observer moves away from the main beam in either angle or sine space. Returning to the aperture source, for a linearly polarized aperture, the H-plane (ϕ = 0°) far field reduces further, defining the radiation as a function of the [r, θ , ϕ ] polarizations as well, Ef = jEo



abke− jkr ⎡ sin ( ka 2sin q ) ⎤ 2pr ⎢⎣ ( ka 2) sin q ⎥⎦ (1.78) Er = Eq = 0 (1.79)



We note that the E-plane spatial fields are dependent on the aperture b-dimension, while the H-plane field distribution depends on the a dimension. The far-field spatial harmonics can be explored using a simple spreadsheet. Interesting variables to consider are: •

•

•

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The E-plane (ϕ = 90° = π /2) and H-plane (ϕ = 0°) 3-dB beamwidths and null locations are quite similar for electrically large apertures and less so for small apertures. The H-plane (cosθ ) factor dominates for small apertures because the aperture distribution is nonuniform. The 3-dB beamwidth is (λ /a, λ /b) in their respective planes.

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1.4  Aperture State Fundamentals43

• • • •

• •

•

• •

The null locations occur at (mλ /a, nλ /ab) in their respective planes. The first null width is (2λ /a, 2λ /b). The sidelobe magnitudes follow (E SLL = 1/X, 1/Y) in their respective planes. The sidelobe magnitudes in the two principal planes converge to the same values for large apertures. Principal plane first sidelobe magnitudes are approximately −13.3 dB. In the diagonal planes, (ϕ = 45° = π /4), E θ and E ϕ patterns converge near the main beam for large apertures. Diagonal plane first sidelobes are −26.5 dB (i.e., the product of the principal plane magnitudes). 2 2 Diagonal plane null locations occur at (w = n ( l/a ) + ( l/b ) ), when a = b. The directivity is proportional to the aperture area D = 4π ab/ λ 2 , for a uniform amplitude distribution.

The null pattern is an inverse reflection of the edge boundary condition, where the excited aperture creates nulls spaced uniformly at the rate of (λ /a, λ /b) within the confines of real space (Figure 1.18). This also means that the null placement in the diagonal planes is formed by the relation,

(w



n

= 

( l a)2 + ( l b )2

) (1.80)

The angular locations for all null relations stems from the inverse sine of the sine space coordinates, whether they are in the principal or diagonal planes. The sidelobes fall in between these nulls and so at the locations,

[ u, v ] =

⎡⎣ l ( 2n + 1) 2a, l ( 2m + 1) 2b ⎤⎦

n=1,2,3… m=1,2,3… (1.81)

The magnitude of these sidelobes relative to the main beam is governed by the denominator in the far field expression, so for a square aperture, as the numerator is confined to values of −1 ≤ ( sinka 2) u ≤ 1 (1.82)

the sidelobe magnitude is ESLL =



1 1 , X Y

⎡ X ⎤ = ⎡ ka 2u ⎤ ⎡ p ( 2n + 1) 2 ⎤ ⎢⎣ Y ⎥⎦ ⎢ ka 2v ⎥ = ⎢ p ( 2n + 1) 2 ⎥ ⎥⎦ ⎣ ⎦ ⎢⎣

(1.83) n=1,2,3…

The first three sidelobe levels (seen in Table 1.3) in the principal planes are



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ESLL =

1

,

1

,

1

(3p 2) (5p 2) (7p 2)

…  (1.84)

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44

Introduction to Phased Arrays Table 1.3  The First Three Sidelobe Levels in the Principal Planes E (V)

E (dB)

SL 1

0.2122

–13.46

SL 2

0.1273

–17.90

SL 3

0.0909

–20.82

For large apertures, it is useful to know the total power contained in the sidelobes because of an interest in minimizing the sidelobe radiation magnitude. This is readily accessible using the root-mean-square (RMS) sidelobe calculation, which sums the power of all sidelobe peaks. Example 1.5

Consider an aperture with dimensions of 1.5m by 1.5m (a,b), excited at a frequency of 1.0 GHz. We have all of the needed information to evaluate the far field, including its peaks and nulls. Using (1.78), the null locations can be determined, as shown in Figure 1.20. Confining our calculations to the region of observable space, where (θ ≤ 90°), or by equivalence, ( u2 + v2 ≤ 1), we observe a periodic grid of nulls, spaced by [λ /a, λ /b], respectively, in the sine space planes [u,v].

Figure 1.20  A sine space distribution of the nulls produced by a 5λ square aperture.

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1.4  Aperture State Fundamentals45

We can determine planar sections of the 3-D far-field radiation field profile. For example, consider the radiation pattern in the x-z principal plane, where ϕ = 0°, as shown in Figure 1.21. In this case, there are two radiated polarizations possible, E θ and E ϕ , where only the latter dominates and the figure shows the result for both in two formats, sine space (top) and angle-space (bottom). Both are equivalent and simply expressed in terms of differing but related dependent parameters. The angle space data is frequently more useful to phased array users, while the sine space data is more practical to the design community. We note that the null locations correspond to those shown in the sine space projection shown in Figure 1.20, in the sectional u-plane where v = 0. The main beam is located at the origin in both plots. The sidelobe magnitudes correspond to the maximums of the sinc function. Because there are no stochastic (random) electric field components to the source field distribution, these observations are exact. The far-field sectional progression continues with a consideration of the radiation pattern in the y-z principal plane, where ϕ = 90°, as shown in Figure 1.22. Again,

Figure 1.21  A sine space and angular space plot of far-field radiation (phi component) for a 5λ square aperture in sine and angular space for ϕ = 0°.

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46

Introduction to Phased Arrays

we note agreement between projected and sectional far-field data in terms of null location and depth, main beam location, and sidelobe magnitudes. In order to continue the far-field sectional progression, consider the radiation pattern in the diagonal plane, where ϕ = ±45° and u = v as shown in Figure 1.23, again in both sine and angle space. In this case, the null spacing stems from the use of the Pythagorean theorem in sine space,



[ u, v ] =

(((2n + 1) a) λ ) + (((2m + 1) ab) λ ) 2

2 n=1,2,3… m=1,2,3…

(1.85)

The sidelobe magnitudes are represented as the decibel sum of the adjacent sidelobes in the principal planes. For example, the second diagonal plane sidelobe magnitude is approximately −37 dB, the sum of −19 dB and −18 dB.

Figure 1.22  A sine space and angular space plot of far-field radiation (theta component) for a 5λ square aperture in sine and angular space for ϕ = 90°.

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1.4  Aperture State Fundamentals47

Figure 1.23  A sine space and angular space plot of far-field radiation (red is theta component, black is phi coponent) for a 5λ square aperture in sine and angular space for ϕ = 45°.

1.4.2  Radiation Integrals for Circular Apertures

We can follow a similar process for circular apertures, again following the inciteful work of Balanis [6]. The far-field derivation follows this similar approach, where the aperture has a radius of (a), a circumference (C), and an aperture field maximum of (Eo), and, with the following radiated field results, noting the first-order Bessel function J1(x), ka2Eoe− jkr J ( kasin q ) ka2Eoe− jkr J ( kaw ) sin f 1 = j sin f 1 r kasin q r kaw 2 − jkr 2 − jkr ka Eoe J ( kasin q ) ka Eoe J ( kaw ) Ef = j cos q cos f 1 = j cos q cos f 1 (1.86) 2pr kasin q 2pr kaw Er = 0 Eq = j

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48

Introduction to Phased Arrays

w =



u2 + v2 (1.87)

The far-field spatial harmonics can be explored using a spreadsheet. Interesting variables and results to consider are: •

• • •

The E-plane (ϕ = 90° = π /2) and H-plane (ϕ = 0°) 3-dB beamwidths and null locations are the same for electrically large apertures, less so for small apertures. The H-plane (cosθ ) factor dominates for small apertures. The far-field patterns are circularly symmetric. The 3-dB beamwidth is (1.154λ /2a) in all constant ϕ planes.

Figure 1.24  A sine space and angular space plot of far-field radiation for a 2.935λ radius circular aperture in sine and angular space for ϕ = 45°.

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1.4  Aperture State Fundamentals49

• • •

• • •

The null locations occur at an interval of (1.154λ /2a) in their respective planes. Principal plane first sidelobe magnitudes are approximately −17.6 dB. In the diagonal planes (ϕ = 45° = π /4) E θ and E ϕ patterns converge near the main beam for electrically large apertures. Diagonal plane first sidelobes are −17.6 dB. Diagonal plane null locations occur at intervals of (1.154λ /2a). The directivity is proportional to the aperture area D = 4π π a2 / λ 2 = C 2 / λ 2 , for a uniform amplitude distribution.

Sectional plots of the far field in sine and angle space are shown in Figure 1.24. Example 1.6

Determine the angle between the main beam and the first null in the diagonal plane for the following rectangular aperture, and the associated angular locations. a. Operating frequency: 10.0 GHz b. Aperture dimensions: a = 0.15m, b = 0.3m c. Aperture geometry: Rectangular d. Aperture amplitude distribution: Uniform e. Scan coordinates: u 0 = v 0 = 0 The sine-space map of the nulls is as shown in Figure 1.25, given that the principal plane nulls occur at u, v locations of (n(λ /a, λ /b)) = n(±0.200, ±0.100). In the off-diagonal plane, the first null is located at

Figure 1.25  A sine space distribution of the nulls produced by a 5λ square aperture with a revised lattice.

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50

Introduction to Phased Arrays

(w = 



( l a)2 + ( l b )2

)=

0.2002 + 0.1002 = 0.2236 sines (1.88)

The angular spacing is q = sin−1 0.2236 = 12.9° (1.89) ⎛ 0.100 ⎞ = 26.56° f = tan−1 ⎜ ⎝ 0.200 ⎟⎠



1.5

Array Far-Field Fundamentals When we arrange a set of more than one microwave radiator in a common aperture, we create the current or voltage sources corresponding to the radiating elements and so increase the spatial degrees of freedom relative to a single radiating element. The single element is an antenna that radiates a fixed beam into space at a single operating frequency. The frequency domain is a sample space where we measure the array system response at the sample frequency, an impulse function within the domain, and then repeat the analysis for other ideal frequency samples within an operating frequency range. If we expand the radiator notion to a linear array of elements with a fixed combiner or voltage exciter network, the array of radiating elements increases the directivity over the single element case, again providing a fixed beam in far-field space. The directivity of the array (i.e., its ability to focus the radiated field to a specific region of free space) increases compared to that of a single element. Adding phase shifters at each element provides the phase control needed to orient the radiated beam at prescribed positions in the far field of the array (Figure 1.26). The phase control performs the same function as the path length difference in the displaced two-aperture case studied in Figure 1.18. In free space, we can observe how the electric fields produced by a phased array combine in free space by examining their single TEM mode propagation. This can be visualized by considering the array as a set of single-mode sources, and the far field as a point of observation electrically far (i.e., many wavelengths from these sources). Figures 1.26 and 1.27 show how the electric fields of a linear array combine, where only the forward hemispheric propagating fields are shown. The antenna is an ideal mode transformation device in that it converts a bound EM wave, usually TEM or a waveguide mode at the port, into a free-space TEM propagating wave. Several clarification points are useful to consider in this context: •

•

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The antenna has a port where the source wave is initiated, depending on the mode properties and port impedance, the latter being a characteristic impedance for a TEM port and a wave impedance for waveguides. The transmission line is a mechanism that transfers the source wave from the port to the antenna element using fixed and contained boundary conditions.

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1.5  Array Far-Field Fundamentals51

Figure 1.26  Illustrations of a single element, a fixed array, and a phased array, along with their respective far-field radiation patterns.

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52

Introduction to Phased Arrays

Figure 1.27  A time-domain illustration of the electric fields of a phased array at the boresight scan.

•

•

A combiner is a multiport device that either divides the incident wave at the port into branch ports or, because it is a reciprocal device, it accumulates the wave vectors present at the branch ports to a common port. A reciprocal device produces the same insertion response whether it is used in its forward or reverse direction. So a resistor is reciprocal, but an amplifier is nonreciprocal. A phase shifter is an ideal device that introduces a fixed insertion delay of phase (ϕ ) between its two ports at a single operating frequency. It is also assumed to be an ideal reciprocal device.

We know from our study of multiple in-phase apertures that the electric fields from each of the elements radiate in all directions within the forward hemisphere but accumulate to a maximum only at the boresight location (θ = 0), where they phase align to form a plane wave in the far field. We observe that the same field alignment occurs at a scan angle (θ o) relative to the aperture surface normal (Figure 1.27) when there is a linear phase slope across the antenna aperture. Figure 1.27 illustrates the time-domain representation of an ideal single-mode wave alignment of a linear array at its boresight far field. Each radiating element produces an in-phase source wave, oscillating at a single, fixed frequency, and produces EM fields within the array aperture. The observable field includes reactive fields among the radiating elements, fields in the near field of the array, and in the far field. At a distance sufficiently far from the array’s aperture, an observer would detect the radiation from each of the elements. If the observer is located at the boresight, the path length from each element to the observer approaches a common value, resulting in an in-phase accumulation of the vectors. The result of this vector accumulation yields a beam peak or the maximum radiation magnitude, as the observer detects all sources in phase alignment. The far-field wave properties

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1.5  Array Far-Field Fundamentals53

can be represented as a virtual plane where all oscillating fields are time-phased in alignment at the observer’s aperture plane, as shown. When the observer is located in the far field, but at an angular location other than boresight or the surface normal, the phase alignment along the virtual plane changes, producing a misalignment in phase or time, producing a reduced electric field magnitude. This wave and phase misalignment produces the sidelobe peaks and nulls that we observed in previous examples. It is also evident in the equivalent sinc function solutions for the far field for rectangular apertures. However, if the source array includes phase shifters, we can reestablish the time-phase alignment at an angular position other than boresight and produce a scanned electric field maximum or beam peak (Figure 1.28). In this case, the array is phased and can produce its maximum far-field radiation at specific angular locations at will, depending on the phase state of the aperture. The aperture phase state has a linear phase gradient and a singly periodic element spacing, as represented by the linear progression of wave delays used to produce the scanned plane wave shown in the figure. In this case, the incremental insertion phase delay results from the phase shifter settings. These are some of the ideal conditions for a phased array system. Naturally, there are multiple additional parameters and conditions that affect phased array systems, which we will investigate, one at a time. These factors range from the less-than-ideal characteristics of the radiators, as they are largely governed by the associated boundary conditions, to the periodicity of the array itself, its polarization properties, beamforming principles, the effects of nonreciprocal components, and array synthesis methods, for example. The elementary concept of phased array wave propagation should nevertheless serve as a basis for a deeper exploration of the topic in the next sections.

Figure 1.28  A time-domain illustration of the electric fields of a phased array at a 30° scan.

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54

1.6

Introduction to Phased Arrays

Frequency-Time Domains Before we dive deeply into phased array antennas, it is important that we have a solid understanding of wave propagation fundamentals, especially the wave equation. One way to understand a common wave equation solution is to explore the time-frequency relationship in terms of the Fourier transform. Important phased array far-field calculations can be quickly processed using the same transform in matrix form. The Fourier transform is defined by the complex integral [11],



F ( y) =

∞

∫ f ( x) e− j2pxy dx (1.90)

−∞

provided that f(x) is continuous for all (x). The inverse Fourier transform is also defined as



F ( x) =

∞

∫ −∞f ( y ) e j2pxydx (1.91)

Generally, the time and frequency domains substitute as the (x) and (y) variables in the above equations, noting that



w = 2pf (1.92) F ( w ) = 2p

∞

∫ f (t ) e− jwt dt (1.93)

−∞

∞



1 F (t ) = ∫ f ( w ) e jwt dw (1.94) 2p −∞

In our consideration of rectangular apertures, the distribution of the electric field within the radiating aperture was assumed to be uniform, meaning that it forms a step function in space. The electric field magnitude is zero over the aperture plane outside of the radiating aperture, abruptly changing to a normalized value of unity within the aperture dimensions. A step function in the time domain transforms to a sinc (sin(ω )/ω ) function in the frequency domain. Using substitution and observing the far field, the step function of the aperture fields also transforms to a sinc function, as shown in (1.70) and (1.71), in two orthogonal planes. The Fourier transform relationship between aperture and far fields is equivalent to the frequency-time transform, with the needed parameter substitutions. This allows us to study the aperture-to-far field transformation using frequency-time Fourier pairs.

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1.6  Frequency-Time Domains55

The Fourier transform is quite useful as it converts time-dependent functions into the associated and equivalent frequency-domain spectrum. The transform relationships between these two domains form pairs that become useful operators in antenna theory. Staying temporarily in the time-frequency domains, however, a number of these pairs illustrate the principle. A step function in the time domain transforms to a sinc (sin(ω )/ω ) function in the frequency domain, as illustrated in Figure 1.29. Several properties of the transform pairs stem from this simple example: • •

•

•

A constant multiplier in one domain translates directly to the other. The step function is discontinuous at its time-domain edges, leading to the significant frequency-domain sidelobes. Complex addition in the frequency domain produces multiplication in the time domain and vice versa. The sampling rate in the time domain must be less than or equal to the Nyquist rate to accurately reproduce the frequency-domain function. Related observations include: – The Nyquist rate in the time domain is (tNy = 1/fw), where (fw) is the singularity (spurious) free frequency-domain window, 400 kHz, in this case. – When the sampling rate is considerably less than the Nyquist rate, it becomes oversampled, with no waveform distortion, only a surplus of data points (Figure 1.30). In this case, the frequency window is 1.0 MHz.

Figure 1.29  Time and frequency-domain Fourier transform pairs for the single time step shown with a sampling rate of 201 over the pulse repetition width (2.5 μ s each).

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56

Introduction to Phased Arrays

If the time-domain sampling rate exceeds the Nyquist rate, grating lobes or periodic replicas appear in the frequency domain, as shown in Figure 1.31. In this example, the frequency window is 100 kHz, producing singularities (grating lobes) at ±50 MHz. The time-domain pulse width is inversely proportional to the spectral width in the frequency domain. The 0.5 × 10 –3 second pulse width produces a spectral width of 2.0 kHz at the 3-dB points, 4 kHz at the first nulls (Figure 1.30): – The first frequency-domain sidelobe from the fundamental peak has a magnitude of −13 dBc and is located at (±1.5/t pw), where (t pw) is the pulse width in the time domain. – Frequency domain nulls occur at (±n1.5/t pw), where (n) is all integers in the real number domain and bounded by ±∞. – The 3-dB width of the frequency-domain fundamental is (1/t pw). The transform equations show that the Fourier transform of the sinc function in the time domain produces the step function in the frequency domain (Figure 1.32). The Fourier transform of an impulse time function produces a broad or white noise spectrum in the frequency domain (Figure 1.33). –

•

•

•

The Fourier transform relationship between aperture reactive fields and the radiated far fields is powerful and simple. It will be the subject of a number of simplifications when we consider far-field analysis, and it will be even more useful when

Figure 1.30  Time and frequency-domain Fourier transform pairs for the single time step shown with a sampling rate of 1,001 over the pulse repetition width (1.0 μ s each).

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1.6  Frequency-Time Domains57

Figure 1.31  Time and frequency-domain Fourier transform pairs for the single time step shown with a sampling rate of 51 over the pulse repetition width (10 μ s each).

Figure 1.32  Time and frequency-domain Fourier transform pairs for the single (sin t/t) time function shown with a sampling rate of 201 over the pulse repetition width (2.5 μ s each).

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58

Introduction to Phased Arrays

Figure 1.33  Time and frequency-domain Fourier transform pairs for the single impulse time function shown with a sampling rate of 1.001 over the pulse repetition width (0.5 μ s each).

we consider the reverse process of array synthesis, where we solve for the aperture conditions needed to produce an intended far field. Example 1.7

What type of frequency-domain function at what spectral width is produced by a time-domain sinc function with a 3-dB width of 1.0 μ s? It produces a step function in the frequency domain, with a 3-dB width of 1 MHz. Example 1.8

What is the first null spectral width of a step time function with a pulse width of 5.0 ms? The 3-dB frequency domain width is the inverse of the time function or 200 Hz, so the null width is twice that, or 400 Hz. 1.6.1  Frequency-Time Domain: Fast Fourier Transform

Many professionals in the signal processing domain use Fourier transforms routinely. Also, the fast Fourier transform (FFT) or discrete version is calculated instead of the continuous integral version used in the previous section. Its primary advantage is computational speed, which becomes a significant issue when studying arrays with large numbers of elements.

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1.6  Frequency-Time Domains59

The FFT takes advantage of a square matrix decomposition technique, which treats the frequency and time components of the transform as binomial (2n) arrays of equal size. Because of this, many terms in the Fourier summation simplify or intrinsically become zeros, requiring no complex multiplication, the latter being responsible for the largest part of the computational time needed to perform the integration. This also means that the FFT may require modifications to the time-frequency vectors before and after processing. These can be summarized as: •

•

Scaling: The frequency and time vector units need scaling, as shown in the example that follows. Zero filling: When the time or frequency vector’s size is less than a given binomial power, additional zeros need to be added to fill the vector until its size is binomial.

The time-to-frequency transform process is illustrated in Figure 1.34. A timedomain waveform is shown having several frequency-domain components. These frequency-domain components are separated visually, showing the correlation between frequency and time domains. The result is the frequency-domain components of the original time-domain function. The process shows the equivalence in both domains and can be processed in both forward and reverse directions. The frequency and time domains represent two different ways of observing the same condition set. A parallel domain comparison exists between the aperture or near field and the radiation or far field for an antenna or array. The correlation to arrays is straightforward: while the time domain represents the array aperture distribution, the frequency-domain represents the farfield distribution. In the following MATLAB example (also see Figure 1.35), a simple pulse waveform in the time-domain transforms to its FFT result in the frequency domain. Part of the scaling involves mirror imaging the frequency-domain result about its origin, in order to produce a symmetric function, in this case, (sin(t)/t). % Fast Fourier Transform %_______________________________________________________________ clear all; % Input Parameters iPlot = 1;      % Plot toggle switch Fs = 1000;      % Sampling frequency t = -0.5:1/Fs:0.5;  % Time vector L = length(t);    % Signal length Lt = 0.5*Fs; w = 0.2;       % pulse width (msec) w1 = w*100/Fs; X = zeros(1,L); Xp = zeros(1,L); for m = 1:L   if (abs(t(m)) 5λ ), this being the fourth simplification. The normalized embedded element far-field pattern is then



fi ( q, f ) = f ( q, f )

∀i (2.9)

N ⎛ e− jkRo ⎞ E (r ) = ⎜ f q, f ( ) ∑ aie− j(k⋅ri ) (2.10) ⎝ 4pRo ⎟⎠ i=1

As a result, the total far-field response is the product of three dominant factors, the far-field propagation constants, the element far-field factor, and the array factor. This convenient separation of terms allows designers to treat each independently.

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72

Array Theory

•

•

•

The propagation constants (e–jkRo /4π Ro) determine the free-space loss involved in the propagation path and form the basis for the Friis equation. This is also the proportionality constant for spherical wave function and an indication that the phased array can be treated as a single point source in the far field, with a single phase center. The embedded element factor (f(θ , ϕ )) is the far-field radiation function for a single representative radiating element in the presence of all other elements. It therefore includes the reactive fields of the array when a single element is excited and so the mutual coupling effects. This factor is the only term that includes the polarization characteristics of the array, depending on whether linear, circular, or other polarization is used. The embedded element function represents the angular envelope and so limits the over scan performance or scan loss, as the array factor’s magnitude cannot exceed a maximum of (N). Consider the case where a null is produced within the embedded element function; there is nothing within the array factor that can overcome such a condition, often viewed as a phased array deficiency termed scan blindness. N The array factor is the summation ( a e− j( k⋅ri )) , which accounts for the

∑ i=1

i

accumulated electric field of all elements and is often termed the array factor. It is convenient to determine the array factor’s aie –jk(k·ri) dot product at this stage, without a loss in generality. Using free space as the propagation medium,

(

)

⎛ 2p ⎞ ˆ + yv ˆ + zˆ 1 − u2 − v2   ⋅ ( xx ˆ i + yy ˆ i + zz ˆ i) k ⋅ ri = ⎜ ⎟ xu ⎝ l⎠

(

⎛ 2p ⎞ = ⎜ ⎟ xi u + yi v + zi 1 − w2 ⎝ l⎠



)

(2.11)

The far-field integral equation reduces to the following:



E (r ) =

N e− jkRo − j( 2p / l )( xi u+ yiv+zi f ( q, f ) ∑ ai e 4pRo i=1

1−w2

)

(2.12)

The element excitation is phase controlled to locate the beam peak at a specified position in sine space (ko), and so can readily be converted into angle space. The complex excitation function (ai) becomes



ai = ai e− jk( ko ⋅ri ) = ai e

(

j( 2p / l ) xi uo + yivo +zi 1−wo2

)

(2.13)

where



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(

)

⎛ 2p ⎞ ˆ o + yv ˆ o + zˆ 1 − wo2 (2.14) ko = ⎜ ⎟ xu ⎝ l⎠

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2.4  Two-Element Arrays73

uo = sin qo cos fo

vo = sin qo sin fo wo =



uo2

+

vo2

(2.15) = sin qo

It is quite useful to note that when the design beam peak (ko) aligns with the observation point in the far field (k), the observer integrates a phase-aligned aperture. So (ko = k) under these conditions, and the far-field equation reduces to the following when the element excitation magnitudes are equal to unity, a uniform illumination, N ⎛ e− jkRo ⎞ j( 2p / l )( xi uo + yivo +zi E (r ) = ⎜ f ( q, f ) ∑ ai e ⎟ 4pR ⎝ o⎠ i=1

1−wo2

N ⎛ e− jkRo ⎞ j( 2p / l )( xi ( uo −u ) + yi ( vo −v ) +zi ( =⎜ e f q, f ( ) ∑ ⎟ ⎝ 4pRo ⎠ i=1



)e− j(2pp / l )( x u+ y v+z i

1− ( wo2 ) − 1−w2

i

))

i

1−w2

) (2.16)

⎛ e− jkRo ⎞ =⎜ f ( qo , fo ) N ⎝ 4pRo ⎟⎠

The result is the Friis propagation loss multiplied by the number of elements and the embedded element function sampled at the observation point. Because many phased arrays use a planar geometry, a further simplification uses (zi = 0), ∀i,



N ⎛ e− jkRo ⎞ E (r ) = ⎜ ai e− j(2p / l )( xi u+ yiv ) (2.17) f q, f ( ) ∑ ⎟ 4pR ⎝ o⎠ i=1

Thanks to a sound theoretical background, we are prepared to test it by examining the behavior of simple array systems, expanding from two-element array conditions to larger linear arrays, planar arrays, and the general conformal array.

2.4

Two-Element Arrays Probably one of the most elementary and instructive array examples is a simple twoelement array. Because of its geometry, it is relatively easy to observe basic radiation effects, such as the onset of grating lobes or spatial harmonics, array scanning, and the linear relationships of sine space [3, 4]. Initially, and for simplicity, we assume that the radiating elements each have isotropic radiation field patterns and no appreciable mutual coupling, producing a uniform electric field distribution in all directions from the point source. So, when we construct a two-element array, we need only describe the locations of these two points within a 3-D coordinate frame, as indicated in Figure 2.5. Some simplifications help to better grasp the fundamentals of the far-field radiation. For example, the free-space loss or the spherical expansion factor in (2.10) becomes unity if we consider only the normalized far-field radiation. Many arrays are arranged with a linear separation in a row-column format, so we introduce a

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74

Array Theory

Figure 2.5  A two-element array within a 3-D coordinate frame.

double summation over integers [m,n]. For convenience, the elements occupy the (z = 0) plane and we can temporarily assume a unity that is isotropic element factor (f(θ ,ϕ )) = xˆ ), so the far-field integration becomes N



M

E ( r ) = xˆ ∑ ∑ am,ne

(

− j2p / l xm ,n u+ ym ,nv

)

m=1 n=1

(2.18)

For a two-element array, the far-field summation process reduces further, especially if we also assume that the element excitations are equal in amplitude, and that the phase of the first element is the reference value 0°, while the second element’s phase is arbitrary (ϕ ). For a two-element array, the element indices are [M,N] = [1,2], and elements are separated by a constant (d x), centered at the origin of the coordinate system. a1 = a2 = 1.0

a1 = e j0 = cos (0) + j sin (0) = 1.0

(2.19)

x1,1 = 0.0 x1,2 = dx y1,1 = y1,2 = 0

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(2.20)

z1,1 = z1,2 = 0

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2.4  Two-Element Arrays75

In the far field, the observer’s position is defined in the spherical coordinates of angle space. As a convenience, we temporarily limit our observations to the array plane (y = 0), so (ϕ = 0°). The far-field integration reduces as follows:



u = sin q cos f = sin q (2.21) v = sin q sin f = 0 1

2

E ( r ) = xˆ ∑ ∑ am,ne

(

− j2p / l xm ,n u+ ym ,nv

)

m=1 n=1 1

2

= xˆ ∑ ∑ e

(

− j2p / l xm ,n sin q

) 

m=1 n=1



(

= xˆ 1 + e− j((2p / l )dx sin q+j )

 

(2.22)

)

The far-field radiation magnitude cycles between a relative maximum of 2.0 when the argument 2π / λ (xm,nsinθ + ϕ ) equals (2n π ), where (n) is a separate integer. The far-field magnitude also equals zero when the argument equals (π ± 2n π = (1 + 2n)π ) or odd integer multiples of π . The three controlling variables of the far-field equation then are the element spacing (dx), the spatial angle (θ ), and the relative phase (ϕ ). Plotting the magnitude of these far-field array responses with respect to the dependent observer angle (θ ) over a range of independent element spacings and relative phase differences yields the maps as shown in Figure 2.6, consistent with the work of Kraus [1].

Figure 2.6  Far-field patterns for a two-element array as shown in Figure 2.5, under various element phase and separation conditions.

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76

Array Theory

Several observations stand out when we study the two-element far-field spatial patterns. Their characteristics have trends that eventually lead to more general theories that apply to larger linear, planar, conformal, and infinite arrays. Several of these observations are summarized as follows: •

•

•

Spatial harmonics: Studying the column associated with an equal phase relationship between the elements (ϕ = 0°), as the spacing increases beyond dx ≥ 0.25λ , multiple peaks and nulls emerge in the far-field radiation. Similar to the frequency-domain harmonics that occur when two or more signals are combined, these spatial harmonics represent peaks, where the complex products accumulate in-phase, and nulls when the incremental phase relationship is (ϕ = π ± 2n π ). As we will see later, some of the multiple peaks represent replicas of the main beam, because the exponential argument repeatedly returns to a value of (kdxu + ϕ = 2n π ); these multiple maxima are termed grating lobes. Maxima and minima: The maximum magnitude in all cases equals the number of elements (MN), while the minimum magnitude in all cases is zero. In other words, the maximum is the result of a complex voltage addition, as is evident from the source equations, while the minimum value results from perfect cancellation. Phased scanned behavior: Consider the cases where the element spacing is (dx = 0.5λ ), where the location of the beam peak translates in angle space with respect to the differential phase (ϕ ). This is logically termed phase scanning, and the relationship between the phase and beam position in sine space follows: uo = − φ kdx



(2.23)

or more generally, with an incremental phase of (ϕ = α ), uo = − a kdx (2.24)

as verified by Table 2.1. •

Nyquist limit: The maximum element spacing that produces a single maxima in the forward hemispherical space (z ≥ 0) is (dx = 0.5λ ), similar to the Nyquist limit used in the study of frequency analysis and sampling theory. Although Table 2.1  Beam Scanning Equation Verification dx(λ ) = 0.50 Peak locations

α (deg.)

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θ (deg.)

u(sines)

0.00

0.00

0.00

0.000

45.00

14.48

0.25

44.992

90.00

30.00

0.50

90.000

135.00

48.59

0.75

134.999

180.00

90.00

1.00

180.000

kdu(deg.)

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2.4  Two-Element Arrays77

•

the single beam criterion can be met with smaller element spacing, it is often of practical importance to maximize the element spacing up to the Nyquist limit, thus minimizing the total elements needed. Grating lobes: At an element spacing of (dx ≥ λ ), multiple peaks or grating lobes are produced, as the exponential argument repeatedly returns to a value of (kdxu + ϕ = 2n π ). They have the unique property of being computational replicas of the main beam, as the element phase that produces the propagating wave represents integer multiples of (2π ). This means that the grating lobe is indistinguishable from the main beam from the observer’s position, as much as are element phase values of 0°, 360°, and 720°. This is also equivalent to saying that the grating lobes represent a spatial ambiguity, as the array cannot distinguish the angle of arrival between the main beam and a grating lobe for a receiving wave.

Example 2.1

Show that the beam-steering equation holds for an equal magnitude two-element array with a spacing of (dx). The beam peak occurs when the exponential argument in (2.22) is equal to

(kdxu + a = ±2np ) , so uo = − a ± 2np kdx (2.25)

Since the incremental phase (α ) can only exist between the values of (± π ) on the complex plane, (2π ) multiples are redundant and can be removed, leaving

uo = − a kdx (2.26)

Examining the case where the element spacing is (dx = 0.5λ ), and considering the forward hemisphere (z ≥ 0), the far-field magnitude can be considered both in angular and sine space (Figure 2.7). The point of reference shifts in this comparison, even if the far-field pattern remains the same. This will make it convenient to consider a physical analogy for the equations and observations indicated above. For example, consider the geometry of the two-element array from the perspective of its phase centers (Figure 2.7). With a far-field observer located at boresight, (θ = 0), rays drawn from the two elements to the observer located a large distance away have equal path length. Assuming each element produces a spherical propagating wave and that the two elements do not interfere with each other, the two rays arrive with the same insertion phase, as the propagation phases are (dx = 0.5λ ), equal



⎛ 2p ⎞ φ = −kl = − ⎜ ⎟ l (2.27) ⎝ l⎠

If the radiated phase of each element is equal, the vectors add, resulting in a magnitude of 2.0, which is the maximum possible for this case. The combined vectors generate a plane wave, with an equiphase front, as indicated in Figure 2.8.

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78

Array Theory

Figure 2.7  Two-element array far-field magnitude plotted in both angle (a) and sine space (b).

Minima can be explained in the same manner. Consider Figure 2.8, where the far-field incidence angle (θ ) is moved from the boresight location until the path length difference from each element is (0.5λ ). The received phase difference between the two vectors is 180°, resulting in perfect cancellation of the vectors and a total farfield magnitude of 0.0. At an element separation of (0.5λ ), the minimum occurs at (θ = 90°), consistent with the far-field magnitude calculations. For the scanned case shown in the figure, a phase difference of (α ) is balanced by the path length (kl) to produce a plane wave at an angle (θ ) relative to the aperture surface normal. The connection is as follows, indicating a linear sine space relationship between the relative element phase and the beam peak location in sine space, uo = − a dx



a = −kdx uo (2.28)

Figure 2.8  Two-element array geometry and phase centers.

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2.5  Linear Arrays79

Moreover, when the beam peak is scanned away from boresight (θ = 90°), the phase difference between the elements advances or delays the signal of one element relative to the other. The IEEE convention for path length to phase conversion is as indicated in (2.28), where a transmission line or propagation path length (l) produces a negative insertion phase. The phase of the left element in Figure 2.8 therefore has positive phase relative to the right element, compensating for its longer propagation path length. Example 2.2

For a two-element equal amplitude array with a spacing of (dx = 0.5λ ) and arbitrary phase, determine the minima locations and show whether they are related to the expression uo = − a dx (2.29)



Nulls occur when the complex argument equals (±m π |m=odd), expressed as: kdx u + a = ±mp|m=odd u = − a dx ± mp kdx



(2.30) |m=odd

So the two-element main beam and null locations are displaced from each other by a constant (±m π /kdx|m=odd) in sine space. For (dx = 0.5λ ), kdx = π : u = − a p ± m|m=odd (2.31)



For the simple case where m = ±1,

u = − a p ± 1 (2.32)

2.5 Linear Arrays Linear arrays use a single plane expansion of the two-element array, with multiple radiating elements linearly displaced along the x-axis, for example, as with the two elements shown in Figure 2.5. Depending on the element spacing, the mutual coupling between elements will often be significant and generally should not be ignored. Mutual coupling between radiating elements causes the radiated voltage of one element to couple with and so affect the radiated voltage of neighboring elements. We will temporarily use ideal element factors for now, assuming controlled mutual coupling, without losing generality. Our study of linear arrays in this section reveals a single solution to the wave equation and the far-field integral equations, small array effects that can distort the far-field radiation, and the effects of sine

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80

Array Theory

space truncation, which can cut off segments of the radiated fields that we study and quantify. Most phased arrays seek to radiate into a forward hemisphere (z ≥ 0), so the aperture surface normally aligns with the positive z-axis, (θ = 0°). Given this, a linear phased array aperture can be considered from a projected area perspective. This means that when the aperture is projected to boresight, or (z = ∞), there is no projection loss, and the full aperture is visible to the far-field observer. When projected to angles other than boresight (i.e., (θ ≠ 0)), a scan-dependent loss is encountered, as the linear aperture projects a length less than its actual size to a far-field observer. The ideal power projection loss is proportional to the scan angle, p ( q, f ) = cos q (2.33)



As (P = E2/R), an ideal set of linearly polarized radiating elements oriented along the x-axis, the embedded element voltage factor becomes f ( q, f ) = xˆ cos q (2.34)



This is an ideal element factor in that it represents an ideal voltage source on a planar aperture and can readily be used to illustrate linear array effects. Before we enter a detailed study of linear array behavior, it is useful to become more familiar with sine space illustrations of the far-field magnitudes. Returning to the two-element case studied previously, we can consider a scanned case, where (θ o = 45°) and an element spacing of (dx = 0.5λ ). The relative phase between the two elements is



a = −kdx uo =

2p (0.5l sin (45°)) = 0.7071p = 127.28° (2.35) l

The far-field magnitude pattern is as shown in Figure 2.9 in the (ϕ = 0°) plane for all (θ ≤ 90°) angles, using multiple solutions to the integral equation. When plotted in sine space, in the forward hemisphere (θ ≤ 90°) and using a cosine element power function, a similar far-field response is as shown in Figure 2.10. It is shown in the form of contour and projection far-field plots, as well as sectional plots through the principal axes of the main beam. The sectional plot in the u-plane occurs at (v = 0), or (ϕ = 0°), while the v-plane sectional plot occurs at the beam peak in the orthogonal plane. The contour and 3-D plots give a wide view of the radiated field, while the sectional plots give more precise magnitude and sine space information for the same radiation response. All plots are shown in sine space and are normalized to a maximum voltage of 1.0 or 0.0 dB. Both these representations of similar radiated fields show the effect caused by scan conditions. Because the element radiation pattern has no nulls in the forward hemisphere, clearly the scan conditions produce the null at (u = 0.2929)(θ = 17.03°), −a −0.7071p ±1= ± 1 = 0.2929 p p (2.36) q = asin (0.2929) = 17.03° u=



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2.5  Linear Arrays81

Figure 2.9  The far-field magnitude of a two-element array scanned to (θ o = 45°) plotted in the (ϕ = 0°) plane in angle space.

Figure 2.10  The far-field magnitude of a two-element array scanned to (θ o = 45°) in the (ϕ = 0°) plane in sine space, with (a) a contour plot, (b) a 3-D version, with (c) u and (d) v sectional plots.

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82

Array Theory

The null location is consistent in both cases. The beam peak is located at the intended (θ o = 45°) scan angle, using an isotropic element pattern. The cosine power element pattern shifts the beam peak location to (u o = 0.5)(θ o = 30°) because the element pattern reduces the field strength at the scan angle. This effect is attributable to the small size of the array and is a small array effect. This same effect has little impact to the −6.0-dB sidelobe magnitude as the element power is the same magnitude at both the beam peak and the sidelobe location, a unique occurrence, due to the chosen scan condition. The cosine power element pattern also produces nulls at (u = ±1.0)(θ = ±90.0°), due to the element nulls. The sine space representation also offers a comprehensive view of the effects in the forward hemisphere. By comparison, an element count increased to 4, while keeping all other parameters the same, introduces interesting small array and scanned effects. The array parameters are as shown in Table 2.2. In a four-element array, the far-field summation expands because of the number of radiating elements. We can again assume that the element excitations are equal in amplitude and that the phase of the first element is the reference value 0°, while the remaining element phase values are arbitrary (ϕ i). The element indices are [M, N] = [1, 4], and elements are separated by a constant (d x), centered at the origin of the coordinate system. a1 = a2 = a3 = a4 = e j0 = 1.0 (2.37)



x1,1 = 0.0 x1,2 = dx x1,3 = 2dx x1,4 = 3dx

(2.38)

x1,n = ( n − 1) dx y1,1 = y1,2 = y1,n = 0 z1,1 = z1,2 = z1,n = 0



As before, we temporarily limit our observations to the array plane (y = 0), so (ϕ = 0). The far-field integration becomes 1

4

E ( r ) = xˆ ∑ ∑ am,n e

(

− j2p / l xm ,n u+ ym ,n v

)

m=1 n=1 1

4

= xˆ ∑ ∑ am,n e

(

− j2p / l xm ,n u

)

m=1 n=1



7060_Book.indb 82

(

= xˆ 1 + e − j2p / l( dxu+φ1 ) + e − j2p / l(2dxu+φ2 ) + e − j2p / l(3dxu+φ3 )

(2.39)

)

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2.5  Linear Arrays83 Table 2.2  Parameters for a Four-Element Linear Array Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

4

Y-axis element count (M)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.7071

Main beam scan location (vo)

0.0000

The relative phase between any two elements for (θ o = 45°) an element spacing of (dx = 0.5λ ), and the phase values for the four elements is: 2p (0.5l sin (45°)) = 0.7071p = 127.28° l (2.40) = ( n − 1) a = [0°,127.28°, −105.44°,43.68° ]

a = −kdx uo =

ai i=1,N

Note that the phase values are constrained between the limits of (−180° ≤ α i ≤ 180°). The far-field radiation magnitude cycles between a relative maximum of 4.0 when the argument 2π / λ (xm,nsinθ + ϕ ) equals (2n π ), with the normalized far field as shown in Figure 2.10. The far-field magnitude also equals zero when the argument equals (π ± 2n π = (1 + 2n)π ). Again, we have contour and projection sine space plots, as well as sectional plots through the principal axes of the main beam. The sectional plot in the u-plane occurs at (v = 0), or (ϕ = 0°), while the v-plane sectional plot occurs at the beam peak in the orthogonal plane. The far-field behavior follows the theories of Huygens and Balanis; a uniformly illuminated aperture produces a far-field sinc function, ⎛ abke − jkr ⎞ ⎛ sin X ⎞ ⎛ sinY ⎞ sin f ⎜ Eq = jEo ⎜ ⎟ ⎝ X ⎟⎠ ⎜⎝ Y ⎟⎠ ⎝ 2pr ⎠ ⎛ abke − jkr ⎞ ⎛ sin X ⎞ ⎛ sinY ⎞ Ef = jEo ⎜ sin q cos f ⎜ (2.41) ⎟ ⎝ X ⎟⎠ ⎜⎝ Y ⎟⎠ 2pr ⎝ ⎠



⎡ X ⎤ = ⎡ ( ka 2) u ⎢⎣ Y ⎥⎦ ⎢ ( kb 2) v ⎢⎣

⎤ ⎥ ⎥⎦

where the aperture dimensions are [a, b] = [4dx, dx] = [2λ , 0.5λ ].

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84

Array Theory

For the linear array, the electric field in the unscanned plane (y-axis, ϕ = 90°) is governed by the boundary conditions of the radiating element as it is composed of a single element with no scan plane controllable phase capability. The far field in the orthogonal plane (x-axis, ϕ = 0°) has four elements, each with a controllable phase, so we observe a series of peaks and nulls, with the main beam located near the intended location (θ o = 45°, uo = −0.7071). The nulls occur when the sinc argument equals (n π ⎪ n=0,1,2…). Applying this condition to (2.41) in the scanned plane, we arrive at an integer-based series null sequence in sine space,

(ka 2) u = (2p (2l ) 2pu = np n=0,1,2… u = 0.5n n=0,1,2…



2l ) u = 2pu (2.42)

The null sequence is evident in Figure 2.10 with the sequence offset by the scan angle, so the above equation can be modified, u + uo = 0.5n n=0,1,2… (2.43)



So the null locations predicted by this theory fall at values of [u null] = [−0.207, 0.293, 0.793]. The agreement with the far-field equation is precise. It is also useful to notice that there are (N − 1) nulls within observable (real) space, as shown. Examining the beam peak, we see that the peak location approximates the intended scan angle at (θ o = 45°, uo = −0.7071), and the small array distortion is significantly reduced compared to the two-element case. The general boundary between electrically small and large arrays is an aperture size of 5 wavelengths. Apertures that exceed this criterion evidence increasingly little of the small array effect. 2.5.1  Linear Arrays in Sine Space

Next, we can consider the benefits of sine space far-field calculations in comparison to the transform pair in angle space. So, although the far-field equation can be evaluated in either spatial representation, sine space offers several advantages: •

•

•

•

7060_Book.indb 84

A clear distinction between observable or real space where (θ ≤ 90°) in angle space, or by equivalence, (w = u2 + v2 ≤ 1), and nonobservable imaginary space, where (w ≥ 1) and the associated angle is imaginary. Truncation effects, where components of the radiated far field truncate at the real-space boundary. The main beam can be truncated by scanning it to the edge of real space. This creates a class of antennas (Hansen-Woodyard) that achieve reduced beamwidth using sine space truncation methods [5]. The inherent simplicity of array design in sine space, where far-field calculations operate in a linear space, in contrast to the same in angle space, where the same calculations occur in a nonlinear space.

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2.5  Linear Arrays85

One way to observe the effects of sine space more clearly minimizes the small array effects by considering the far-field integral equation with a relatively large set of array elements arranged in a single plane. We continue to use the cosine embedded element power because of its generality and simplicity, recognizing that this simplification allows us to see general trends that apply to a general phased array parameter set. Linear arrays are an expansion of the simple two-element construct on a single axis. Continuing with the original theme, the electrically large array conditions are as follows: • • • • • •

ˆ Linearly polarized ( x) Ideal cosine element power function, (f(θ ,ϕ ) = xˆ cos q) Uniform aperture amplitude distribution, ⎪ai⎪ = 1.0, ∀i Uniform element spacing, (d x = 0.5λ = constant) Boresight scan, ([uo, vo] = [0, 0]) Electrically large linear array, ([M, N] = [1, 100])

Because of these and related simplifications, the far-field integral equation can be reduced further, combining the excitation with the spatial directional cosines, 1 100

E ( r ) = xˆ cos q ∑ ∑ am,ne

(

− j2p / l xm ,n u+ ym ,nv

)

m=1 n=1

100

= xˆ cos q ∑ am,ne am,n = e

(

n=1

j2p / l xm ,n uo + ym ,nvo 100

(

− j2p / l xm ,n u+ ym ,nv

)

)

E ( r ) = xˆ cos q ∑ am,ne

(2.44)

(

j2p / l xm ,n ( uo −u ) + ym ,n ( vo −v )

)

n=1

100



= xˆ cos q ∑ e

(

j2p / l xm ,n ( uo −u )

)

n=1

The far-field radiation function is readily calculated using contemporary mathematical tools. The 100-element array produces the result shown in Figure 2.12, using the parameters in Table 2.3. We note several interesting observations of the far-field pattern results, including the following: 1. The peak magnitude occurs precisely at boresight, ([uo, vo] = [0, 0]). 2. The beam peak response falls off sharply in the u-plane, with a very wide response in the v-plane, following the reciprocal of the aperture size in units of (λ –1) in the x and y-planes, respectively. 3. The u-axis response contains 99 or (N − 1) nulls, comparable to the number of elements located on the x-axis. There are no nulls (M − 1) along the v-axis, again comparable to the element count along the y-axis. 4. The magnitude of the peaks in locations other than the main beam follows a trend of 20log10(uo − u), or essentially a voltage function of ((uo − u) –1),

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86

Array Theory

Figure 2.11  The far-field magnitude of a two-element array scanned to (μ o = −0.7071) in the (ϕ = 0°) plane and plotted in sine space with a contour plot is shown in the upper left, and a 3-D version is shown in the upper right, with u and v section plots at the bottom of the illustration.

except at the extremes of real space, where the element function dominates. Consider that at each of the sidelobes nearest to the main beam the magnitudes of the sidelobes are −13.3 dB, −17.9 dB, −20.8 dB, and so forth, so the sidelobe level (SLL) is: E ( SLL ) =

sin ( np 2) ( np 2)

(2.45) n=odd

5. The width of the main beam at its half-power locations (3 dB) is approximately (u3 ≈ λ /Ndx), and more precisely (u3 = 0.886λ /Ndx). The width of the main beam at its first nulls is twice that of the half-power width, or (unull = 2λ /Ndx). Likewise, the distance from the main beam peak to its nearest null in the linear array plane is also (u1st null = λ /Ndx). 6. Each of the nulls in the array plane is uniformly positioned along the u-axis in sine space, a notable departure from the same nulls in nonlinear angle space. The far-field array factor pattern in the scanned plane converges to the (sin(u)/u) function,

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2.5  Linear Arrays87

Figure 2.12  The far-field response in sine space for a 100-element linear array at boresight scan, with the parameters cited. Table 2.3  Parameters for a 100-Element Linear Array Parameter Element polarization Element power function X-axis element count (N) Y-axis element count (M) X-axis element separation (dx) Y-axis element separation (dy) X-axis amplitude distribution Y-axis amplitude distribution Main beam scan location (uo) Main beam scan location (vo)



EA =

(

Value x-vector cosine (θ ) 100 1 0.5λ N/A Uniform N/A 0.0000 0.0000

)

sin Ndx p lu (2.46) N sindx p lu

7. The far-field characteristics of the u and v-planes are independent, and each is a function of the array parameters in the corresponding aperture plane.

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88

Array Theory

Next, we can explore scanned characteristics of the electrically large linear array by considering several scanned cases. These cases allow us to observe several array characteristics: • •

The linear translation effects of scanning in sine space in the scan plane. Truncation effects, as the scanned beam samples the orthogonal plane fields, some of which may fall outside of real space and so become truncated (i.e., nonradiating).

For example, consider a 100-element array scanned along its array plane to a position ([uo, vo] = [−0.500, 0.0000]) or ([θ o, ϕ o] = [30°, 0°]), with (d x = 0.5λ ) using the following interelement phasing,

a = −kdx uo = 2p l (0.5l ( −0.5)) = −0.500p = −90.0° (2.47)

It produces the far-field result shown in Figure 2.13, using the array parameters in Table 2.4. We note several interesting aspects of the far-field pattern. The main beam has experienced a linear shift or translation in sine space compared to the boresight case, so the scan produces a change in the observation window alone. This is quite

Figure 2.13  The far-field function for a 30° scanned 100-element linear array, with the parameters cited.

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2.5  Linear Arrays89 Table 2.4  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [30°, 180°]) Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

100

Y-axis element count (M)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.5000

Main beam scan location (vo)

0.0000

important; because the translation is a constant offset, the scanned behavior can be characterized by adding a constant to the far-field function in sine space, other than the effects of the fixed embedded element pattern. Because the angular version of the scanned beam moves in a nonlinear space, it distorts accordingly. A small but observable effect stems from the embedded element function, (f(θ , ϕ ) ˆ = x cos q ); it is fixed in sine space and does not scan with the array function. Instead, its amplitude modulates the array far field, producing asymmetric sidelobes around the main beam, as is barely visible in Figure 2.13. The sidelobe closest to boresight ([uo, vo] = [0, 0]) has a magnitude slightly greater than its image sidelobe on the opposite side of the main beam. This amplitude asymmetry is due to the magnitude of the embedded element function at these two different sidelobe locations. Accordingly, the general sidelobe magnitude resembles the boresight scanned case. As the main beam is scanned away from boresight, it samples the embedded element pattern at the scanned location, truncating the radiation at the edges of real space. This is apparent in the projected view in sine space and in the v-plane sectional plot, particularly when these are contrasted with the boresight case. In the orthogonal or scan plane, no such sine space transition boundary truncation is evident. The sidelobe asymmetry due to the embedded element function and sine space truncation becomes significant when the beam is scanned to the position ([uo, vo] = [−0.9660, 0.0000]) or ([θ o, ϕ o] = [75°, 0°]), as shown in Figure 2.14 and Table 2.5. In this case, the first sidelobe furthest from boresight has a significantly reduced magnitude (−17.5 dB) because it lies at the edge of visible space. The main beam magnitude is reduced because of the element function, while the first sidelobe nearest to boresight has increased in magnitude to approximately −12.0 dB compared with the boresight case. As the main beam is scanned to a region near the real-space transition boundary, the orthogonal plane beam is now predominantly truncated to (−0.259 ≤ v ≤ 0.259). The extent of real space is defined by:

7060_Book.indb 89

w =

u2 + v2 ≤ 1 (2.48)

2/13/23 10:52 AM

90

Array Theory

Figure 2.14  The far-field function for a 75° scanned 100-element linear array, with the parameters cited. Table 2.5  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [75°, 180°])

7060_Book.indb 90

Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

100

Y-axis element count (M)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.9660

Main beam scan location (vo)

0.0000

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2.5  Linear Arrays91

Linear array sine space truncation primarily affects the unscanned v-axis component of the radiation pattern. We note that although there is only a single element degree of freedom in that plane, resulting in an otherwise broad beam represented by the embedded element factor (f(θ , ϕ ) = xˆ cos q), the far-field radiation pattern is also truncated at the edge of real space, as is evident in Figure 2.14, the v-axis sectional plot. The radiation sharply drops to zero magnitude at the real-space boundary, because the entire array is not visible to the observer. The truncation point can be determined from the u-axis scan location ([uo, vo] = [−0.9660, 0.0000]) by triangulation. Because the real-space boundary occurs when (wb = 1.0), the corresponding v-axis coordinate is:

w =

u2 + v2 ≤ 1 (2.49)

vb2 = wb2 − uo2

vb2 = 1.0 − 0.9662 = ±0.259 sines

(2.50)

Example 2.3

A linear array is scanned with the parameters from Table 2.4; determine the v-axis truncation coordinates of the main beam.

vb2 = 1.0 − 0.5002 = ±0.866 (2.51)

This result is consistent with the data shown in Figure 2.13. The real-space truncation effect introduces another interesting aspect of phased array theory. Because the real radiation cannot exist outside of real space but the energy created by the array must be released in some form, a logical explanation is that the region extending beyond real space has purely imaginary impedance, representing stored energy. Indeed, there is a class of antennas where the main beam is intentionally scanned to the edge of real space to reduce the width of its main beam beyond classical beam width theory, and so to increase its gain. Hansen-Woodyard antennas do this for beams scanned near end-fire ([uo, vo] = [−1.0000, 0.0000]), for example. The greater the portion of the main beam scanned outside of real space, the greater the aperture reflection produced, with corresponding reductions in antenna bandwidth. So sine space truncation can also affect the main beam and can be used as a design parameter in terms of its beam width control, increasing the antenna gain. Continuing this theory, we can construct an array excitation that scans the main beam to the edge of real space, as indicated in the array parameters in Table 2.6, and the spatial plot shown in Figure 2.15. The main beam is scanned to the real-space boundary, producing only sidelobes and half of the main beam in real space. Half of the main beam energy moves into imaginary space, while at the opposite edge of real space, half of an imaginary main beam has now scanned into

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Array Theory

Figure 2.15  The far-field function for a 90° scanned 100-element linear array, with the parameters cited. Table 2.6  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [90°, 180°])

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

100

Y-axis element count (M)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−1.0000

Main beam scan location (vo)

0.0000

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2.5  Linear Arrays93

real space. The beam is halved, as shown in the figure inset. As imaginary space now has the preponderance of the array’s energy, the effective port impedance at each radiator can become largely imaginary, reflecting transmitted energy back to its source, whether it be an RF source that energizes the array or an incoming place wave. Evidently, only a fraction of the main beam exists within real space and on the v-axis, as shown. 2.5.2  Linear Array Aperture Projection

Before we progress further with linear arrays, it is important to recognize several properties that emerged during our studies of two-element arrays. First, we recognized that array elements could be accurately treated as complex voltage sources, so there is a clear physical analog to the way phased sources accumulate fields in space. Second, consistent with Huygens’ theory, linear arrays can generate a grating lobe series. We can again consider the physical arrangement of ideal voltage sources, this time along a line, as shown in Figure 2.16. Just as in the case of the two-element array, the phase gradient along the array axis (α ) creates a time delay across the array along with free-space propagation delays that produce a plane wave. The source elements are represented by ideal uniform amplitude voltages (ejα ). These create an equiphased phase front according to the same equation used with the two-element array and is a function of the scalar propagation constant (k), the element spacing (dx), and the main beam location is sine space (uo), uo = a kdx



a = kdx uo (2.52)

At the far-field beam peak, the linear array’s aperture source vectors align inphase. This can be viewed conceptually; if each element’s complex vector is referenced to the previous, the result is equivalent to the beam peak accumulation, as shown in Figure 2.17 for an eight-element array. The vector phase is the combination of the

Figure 2.16  A linear array of complex voltage sources and the far-field plane wave phase front produced from them.

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94

Array Theory

Figure 2.17  A complex plane illustration of the element vectors of an eight-element linear array at the far-field beam peak and the first null locations.

far-field location in sine space and the element’s phase (kdxuo + ϕ n), so at the beam peak location, these terms are equal (kdxuo = − ϕ n = −kndxuo). When the far-field angle moves to the position of the first null, as shown on the right side of the figure, the array vectors do not accumulate and instead gently rotate until the sum vector returns to the origin with a total magnitude of zero. The null location is one beamwidth from the subject array’s beam peak. When the aperture illumination is uniform, a sinc function response to the far-field integral equation is a useful solution. For example, at far-field null locations, the sine argument reaches odd multiples of (π ). The sidelobe peak magnitudes weighting function is (u –1), as the numerator equals a value of 1.0 at the sidelobe peaks. The scanned version of the sinc expression is: N

EA = xˆ cos q ∑  e

(

j( 2p p / l ) xm ,n ( uo −u )

⎛ ⎞ ) = xˆ cos q ⎜ sin (( Ndxp l ) ( uo − u )) ⎟

n=1

an = 1.0

(

)

⎜⎝ N sin dx p l ( uo − u ) ⎟⎠ (2.53)

∀n Table 2.7  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [0°, 0°]) Parameter Element polarization

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Value x-vector

Element power function

cosine (θ )

X-axis element count (N)

100

Y-axis element count (M)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

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2.5  Linear Arrays95

The sinc function is useful in large part because it is a closed form expression, although it has a singularity when the argument equals zero. For convenience, the latter can be corrected by applying a small constant offset to the argument. For example, consider the linear array parameters in Table 2.7. The far-field response, as shown in Figure 2.18, is identical to the same array derived from the summation equation (2.44), as shown in Figure 2.12. The closed-form equation also allows a scanned beam, as indicated in the following set of array parameters. The far-field results, as shown in Figure 2.19, are again consistent with the summation form, as shown in Figure 2.14 and Table 2.8. Another useful way to visualize wave formation in a phased array system is to consider the array in the time domain and as a set of ideal spherical wave sources at each phase center. The analogy stems from the spherical wave expansion functions for the radiator elements. Each ideal voltage source emits an electric field function described as described in Section 2.2, with a constant magnitude component, a spatially dependent magnitude term, and a phase expansion term, the latter consistent with a spherical wave.

Figure 2.18  The far-field function for a scanned 100-element linear array, with the parameters cited. Table 2.8  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [60°, 180°]) Parameter Element polarization Element power function X-axis element count (N) Y-axis element count (M) X-axis element separation (dx) Y-axis element separation (dy) X-axis amplitude distribution Y-axis amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

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Value x-vector cosine (θ ) 100 1 0.5λ N/A Uniform N/A −0.8660 0.0000

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96

Array Theory

Figure 2.19  The far-field function for a scanned 100-element linear array, with the parameters cited.

where

E(q) = A

cos q − jkr e (2.54) r

A = 60πIdz / λ r (constant magnitude) I = Radiated current magnitude dz = Radiator length r = Distance from the radiating source to the observer cosθ /r = Spatially dependent magnitude e –jkr = Spherical wave phase function In the time domain, each phase center is the source of radiation that expands spherically as a function of time. The time expansion can be used as a parallel to the electrical phase, so (e –jkr = e –jω t), where the TEM wave propagation in space (kr) equates to the time (phase) associated with the propagating wave. So we can construct a linear array, where a circular wave expansion (a spherical wave viewed in the x-z plane, for example) represents the spherical wave phase function. Going one step further, if each element in the linear array radiates with a linearly sequential delay, it represents a linear phase gradient, as shown in Figure 2.20. The delayed excitation proceeds from right to left in the figure (i.e., the first element excited is on the right side of the array), and the element to its immediate left is the next to radiate in sequence. The emerging wave then produces a planar wavefront as the time delays align. The wave propagates at an angle relative to the array normal in proportion to the incremental element delays, as shown. Thus, the delay produces a scanned plane wave propagation. This linear array example can be reproduced on the surface of a wave-tank or a still body of water. The elements in this case would be excitation points that

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2.5  Linear Arrays97

Figure 2.20  A planar representation of the fields emerging from a five-element linear array with a timedelayed excitation.

produce the spherical wave on the water’s surface. It is important to note that the excitation needs to be continuous to represent a similar radiated spherical wave, so a single pulse will not produce the same effects, unless the element delays are synchronized in time. We noted above that the two-element array could produce grating lobes, where the main beam and grating lobe are replicas of each other and, under these conditions, the array cannot distinguish one from the other. This condition can be produced in a linear array, just as is the case for the two-element array, although the linear array effects are pronounced. Consider, for example, the linear array with parameters listed below, where the element spacing is expanded to (dx = 0.577). The far-field radiation, determined using the closed-form expression in (2.53), is as shown in Figure 2.21 and Table 2.9. It shows two main beams within real space, at (u = ±0.866), both at the same magnitude, so the array cannot distinguish one from the other in a receiving mode, and divides its radiated field between the two when operating in transmit mode. The obvious spatial ambiguity represents a significant part of the reason why grating lobes are generally avoided in phased array designs, not to mention the impedance properties. We can predict the grating lobe locations for a linear array using existing equations. Grating lobes appear in the far field when the integral equation exponential argument equals integer multiples of phase alignment (kdx + α = n2π ) from (2.53). Substituting (2.28), we replace the phase variable with the equivalent beam peak expression,

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a = −kdx uo (2.55)

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98

Array Theory

Figure 2.21  The far-field function for a scanned 100-element linear array, with the parameters cited.

So the integer 2π multiples are a function of the sine space separation of the observation and intended scan locations, kdx u − kdx uo = n2p (2.56)

Rearranging,

u − uo = nl dx n=0,1,2,3… (2.57)



Table 2.9  Parameters for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [60°, 180°])

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

100

Y-axis element count (M)

1

X-axis element separation (dx)

0.577λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.8660

Main beam scan location (vo)

0.0000

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2.5  Linear Arrays99

This expression tells us the sine space separation between the main beam and the grating lobes is a function of the element spacing in free-space wavelengths. For example, in the above case, (λ /dx = 1.7322), so for the first two (n) integers, (n = [0,1]), the grating lobes are within real space, (−1 ≤ u ≤ 1). When (n = 0), the grating lobe is the main beam, so the separation between the two is zero. When (n = 1), the grating lobe location is as follows and as shown in Figure 2.21.



u = uo +

l = −0.8660 + 1 0.577 = −0.8660 + 1.7322 = 0.8662 (2.58) dx

Calculating (2.57) and Table 2.10 for a small range of (n) integers, there are two highlighted beams within real space and an infinite number of grating lobes in imaginary space. The real-space beams have maxima falling within the range of real space, that is, (⎪ug⎪ ≤ 1). The highlighted two real-space beams are the same as those shown in Figure 2.21, illustrating the alignment of the (ug) locations with the beam peaks. The fact that there are an infinite number of these beams, the vast majority of which are outside of observable or real space, indicates that observable space is only a small fraction of the total radiation produced. This leads to the conclusion that the beam peak and grating lobes are a part of an infinite spatial harmonic set, independent of the array size. Because the presence of grating lobes within real space is a recognized deficiency in phased array design, (2.57) can be used to guarantee no real-space grating lobes for a given linear array maximum scan angle. Because the (n = 0) case represents the main beam, the grating lobes nearest to real space correspond to (n = ±1). Considering the additional condition where the maximum scan in sine space reaches to the maximum in sine space, u ± 1 = ± l dx (2.59)



Table 2.10  Grating Lobe Locations in Sine Space for a 100-Element Linear Array Scanned to ([θ o, ϕ o] = [60°, 180°])

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dx/λ

uo

0.5773

–0.866

n

u – uo

ug

–3

–5.197

–6.063

–2

–3.464

–4.330

–1

–1.732

–2.598

0

0.000

–0.866

1

1.732

0.866

2

3.464

2.598

3

5.197

4.331

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100

Array Theory

As the observation angle in sine space is also confined to real space, the following expression defines a worst condition that prevents real-space grating lobes,



l dx (2.60)



dx = 0.5l (2.61)

2=

This is also termed the Nyquist spatial sampling periodicity and indicates the maximum linear array element spacing needed to prevent real-space grating lobes for all possible scan angles. Example 2.4

Determine the interelement phase needed to produce the linear array shown in Table 2.11 without producing real-space grating lobes. Table 2.11  Linear Array Parameters for a Scanned Condition Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

1

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.7070

Main beam scan location (vo)

0.0000

The same expression uses for the two-element array applies, so using (2.18) for Nyquist spacing,



⎛ 2p ⎞ a = −kdx uo = − ⎜ ⎟ 0.5l ( −0.707 ) = 0.707p = 127.28° (2.62) ⎝ l⎠

A time-domain representation of the linear array can also produce the grating lobe effect. By setting up the wave sources at the proper spacing and producing a similar scan angle indicated in the above array example, both main beam and

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2.5  Linear Arrays101

Figure 2.22  A planar representation of the fields emerging from a five-element linear array with a time-delayed excitation that produces a grating lobe.

grating lobes can be produced, as indicated in Figure 2.22. The grating lobe results from the wave combination stemming from the second and third 2π replicas of the wave sources, producing plane waves that are continuous. In other words, if the wave sources produce only a single sine wave cycle or pulse, neither main beam nor grating lobes will appear unless they are time-synchronized.

Example 2.5

Determine the main beam and grating lobe locations within real space for the linear array parameters shown in Table 2.12. Table 2.12  Linear Array Parameters for a RealSpace Grating Lobe Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

1

X-axis element separation (dx)

0.75λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

−0.9000

Main beam scan location (vo)

0.0000

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102

Array Theory

Using the above parameters, there are two beams within real space (Table 2.13), the main beam located at −0.900 and the grating lobe at 0.433 along the u-axis. Table 2.13  Grating Lobe Occurrences Within Real Space dx/λ

uo

0.750

–0.900

n

u–uo

ug

–3

–4.000

–4.900

–2

–2.667

–3.567

–1

–1.333

–2.233

0

0.000

–0.900

1

1.333

0.433

2

2.667

1.767

3

4.000

3.100

Example 2.6

Determine the number and location for all grating lobes within real space for the linear array parameters shown in Table 2.14. Using (2.57) and the above parameters, there are 10 grating lobes and the intended main beam within real space, as shown in Table 2.15.

Table 2.14  Linear Array Parameters for Multiple Real-Space Grating Lobe Occurrences

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

1

X-axis element separation (dx)

5.0λ

Y-axis element separation (dy)

N/A

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

N/A

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

2/13/23 10:52 AM

2.5  Linear Arrays103 Table 2.15  Multiple Grating Lobe Occurrences Within Real Space dx/λ

uo

5.000

0.000

n

u–uo

ug

–5

–1.000

–1.000

–4

–0.800

–0.800

–3

–0.600

–0.600

–2

–0.400

–0.400

–1

–0.200

–0.200

0

0.000

0.000

1

0.200

0.200

2

0.400

0.400

3

0.600

0.600

4

0.800

0.800

5

1.000

1.000

The far field can be determined from the closed-form expression and is shown in Figure 2.23.

Figure 2.23  The far-field function for a scanned 50-element linear array, with the parameters cited.

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104

Array Theory

2.6 Planar Arrays The preceding study of radiation fundamentals for both two-element and linear arrays gives us a solid body of information, sufficient to explore the more complex subject of planar arrays. On the surface, planar arrays can be considered as an extension of an additional plane to the study of linear arrays. The effects of grating lobes, real-space truncation, and scanned behavior simply expand into another orthogonal dimension. However, planar arrays introduce the effects of cardinal and intercardinal plane effects, near-field and far-field distinctions, and the array gain effects. Linear arrays allow us to see many of these effects from a single dimension, so with planar arrays, the full 2-D aspects become clear. We subdivide this subject into two parts: arrays with no real-space grating lobes, and those with real-space grating lobes. Like our study of linear arrays, the discussion will start with the general wave expansion techniques and the far-field integral equation, using examples to guide our understanding. Consider the far-field integral in the form of a finite summation over the array elements, from (2.17) [4]. As the array is planar, the elements are in the (z = 0) ∀ m, n plane. We use the convenient row-column format as a temporary convenience and conduct the calculation in sine space.



⎛ e− jkRo ⎞ M N − j2p / l( xm ,n u+ ym ,nv ) E (r ) = ⎜ am,ne ∑ ∑ ⎟ (2.63) 4pR ⎝ o ⎠ m=1 n=1

The first term represents the free-space loss incurred as a single TEM wave escapes the array aperture and propagates to a fixed distance from the array aperture in the far field. The integral equation can be normalized because of these conditions, along with the cosine element power pattern assumption, M



N

E ( r ) = xˆ cos q ∑ ∑ am,ne m=1 n=1

(

− j2p / l xm ,n u+ ym ,nv

)

(2.64)

This expression is calculated using modern computational systems. The double summation component, driven by indices (M, N), determines the computational resources and processing time needed. Many contemporary phased arrays can be determined using this method, although large arrays can take an extensive amount of processing time. Example 2.7

If the planar array aperture amplitude distribution is other than uniform, does that change the beam-pointing angle of the array compared to the uniform amplitude case? No, (2.63) indicates that only the phase component of the element (am,n) excitation voltage can affect the beam position.

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2.6  Planar Arrays105

In a related question, can the embedded element factor affect the beam peak position? Yes, because the element factor in (2.64) (f(θ ,  ϕ )) acts as an angle dependent weighting function in sine or angle space, so it can affect the beam position, particularly in small arrays with small element counts. The effect is generally small in large arrays (Ndx > 10λ ). In the small array case, the spatial dependency between the element factor and the array factor has a nearly equal effect on the total farfield result. The array factor is M

N

∑ ∑ am,ne



(

− j2p / l xm ,n u+ ym ,nv

)

m=1 n=1

(2.65)

We can calculate a few array examples to observe the far-field behavior of the planar array far-field equations. The first is a uniformly distributed boresight scanned array using a 50 by 50 element composition in a square grid or lattice configuration. The array parameters are shown in Table 2.16, and the far-field response is shown in Figure 2.24. We note that its far-field behavior is similar to an arrangement of two orthogonal linear arrays, at least in terms of the principal plane behavior (along the u and v-axes). Sectioned through these planes, the far-field response is the same as the scan plane far field determined from linear array theory. In the midway intercardinal or diagonal planes, where (⎪u⎪ = ⎪v⎪), the sidelobes have a magnitude considerably less than in the principal planes, and in the case of separable aperture distributions such as the one chosen, are the product of the principal plane response. For example, consider the following: •

Sidelobes and nulls in the two orthogonal planes in the sine space follow double sinc functions, due to the uniform aperture illumination used, also called the array factor, Table 2.16  Planar Array Conditions for a Boresight Scan Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

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106

Array Theory

Figure 2.24  The far-field function for a boresight 50 by 50 element planar array, with the planar array scanned to [uo, vo] = [0, 0].

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2.6  Planar Arrays107

⎛ sin X ⎞ ⎛ sinY ⎞ E ( u,v ) = ⎜ ⎝ X ⎟⎠ ⎜⎝ Y ⎟⎠

((

) )

) ⎞⎟ ⎛⎜ sin (( Md p l)(v

⎛ sin Nd p l ( u − u ) x o = xˆ ⎜ ⎜⎝ N sin dx p l ( uo − u )

•

•

•

(

(

)

o

The first sidelobe magnitude in the diagonal plane is the product of first sidelobe magnitudes in the principal planes. In the principal planes, the first sidelobe magnitude is −13.46 dB, or 20 log10(2/3π ) in both the u and v-planes. In the diagonal plane, the first sidelobe magnitude is their product, 20 log10(2/3π )2 = 40 log10(2/3π ) or −26.9 dB. The same relationship exists for all cardinal and diagonal plane sidelobes. The location of the first diagonal plane sidelobe is determined from plane geometry applied in sine space. In the principal planes, the first sidelobe occurs at un = vn = 1.5λ /Ndx = 0.06 sines from the main beam. In the diagonal plane, the first sidelobe is located at w n = un2 + vn2 = un 2 = 0.085 sines. Then the same relationship exists for all cardinal and diagonal plane sidelobes. The array gain is indicated as 38.951 dBi (decibels relative to an isotropic radiator). This result derives from the directivity (D), which is the integral of the far-field radiation response over the full range of real space, relative to the magnitude of the far-field beam peak, D=



)

(2.66) − v) ⎞ ⎟ ⎟⎠ ⎜ M sin dy p l ( vo − v ) ⎟ ⎝ ⎠ y

4p

(2.67) ∫∫ ( r, q, f ) sin q dq df Eo2

The directivity is defined only within the realm of real space. The gain (G) is the product of the directivity and the total efficiency (η ), and for this discussion we assume that the efficiency is perfect (η = 1), G = hD



•

•

7060_Book.indb 107

The same result is reached if we consider that the integrated directivity of a cosine power pattern is (π ) or 4.97 dBi. Because of the uniform distribution, the integrated gain of the array factor is (MN) or 33.98 dBi. The total array directivity is the product of the two, (MN π ) or 38.95 dBi. The beam peak location is reported as [uo, vo] = [0.00016, 0.00016] in sine space, indicative of a small computational error. This is consistent with the designed beam-pointing position within the accuracy of the spline routines used. The beam peak location is the midway location between the 3-dB locations in sine space. The 3-dB beamwidth is reported as [u3, v3] = [0.0356, 0.0356] in sine space, with a small computational inaccuracy. This is the result of the difference between accurate 3-dB locations. It separately comes from characteristics of the sinc function, because the beamwidth in the principal planes is

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108

Array Theory

u3 =

0.886l 0.886 = = 0.354 sines Ndx 50 (0.5)

0.886l 0.886 = = 0.354 sines v3 = Mdy 50 (0.5)

•

(2.68)

The rms sidelobe level is reported as −40.051 dB. This is the rms accumulation of all sidelobes other than the main beam. It is interpreted as the average power available in all sidelobes.

2.6.1  Planar Arrays with No Real-Space Grating Lobes

Several scan cases reveal the behavior of planar array radiation in sine space. The doubly periodic array effects also occur when the main beam scans along the principal plane axes, as shown in Figure 2.25 and according to the array parameters in Table 2.17. Like a linear array, the sine space far field linearly translates by the constant offset in the scan coordinates. The most significant sine space distortion stems from the embedded element function, which is fixed. We can make the following observations about the above scanned case when compared with the boresight case: •

•

•

•

The array factor component of the far-field equation translates by a constant [uo, vo] = [−0.5000, 0.0000] in sine space, while the embedded element factor (f(θ , ϕ ) = xˆ cos q) remains fixed. The primary far-field characteristics (beamwidth, beam position, sidelobes) in the unscanned plane are same as those of the boresight case. The gain reduces by 0.625 dB, consistent with the embedded element loss at the scan angle 10 log10(cosθ o /cos0) = 5 log10(1 − uo2) = 0.624 dB. The first sidelobe asymmetry in the scan plane occurs because of the embedded element sampling by the array factor, as seen in the linear array case. Table 2.17  Planar Array Conditions for a U-Axis Scan Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.5000

Main beam scan location (vo)

0.0000

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2.6  Planar Arrays109

Figure 2.25  The far-field function for a scanned 50 by 50-element planar array, with the planar array scanned to [uo, vo] = [−0.5, 0].

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110

Array Theory

Example 2.8

For a uniformly illuminated planar array having a square lattice with dx = dy = 0.5λ and M = N = 50, determine the aperture element phase gradients (α x, α y) need to produce a beam peak location of [uo, vo] = [−0.3333, −0.3333]. ⎛ 2p ⎞ ax = kdx uo = ⎜ ⎟ 0.5l ( −0.3333) = −0.3333p = 60° ⎝ l⎠ ⎛ 2p ⎞ a y = kdyvo = ⎜ ⎟ 0.5l ( −0.3333) = −0.3333p = 60° ⎝ l⎠



(2.69)

Repeat the above calculation for a larger array with the same array conditions but with an element count of M = N = 50,000. The phase gradients are the same since the incremental phase equation has no element count dependency. Similar observations occur when the main beam is scanned by the same magnitude as the previous case, but now on the v-axis, as shown in Figure 2.26 and according to the array parameters in Table 2.18. The sine space far field linearly translates in the v-plane by the constant offset in the scan coordinates. The most significant sine space distortion stems from the fixed embedded element function, now sampled at the same sine space point, but in the v-axis. We therefore make the following observations about the above scanned case, when compared with the boresight case: •

The array factor component of the far-field equation translates by a constant [uo, vo] = [0.0000, −0.5000] in sine space, while the embedded element factor (f(θ , ϕ ) = xˆ cos q) is fixed.

Table 2.18  Planar Array Conditions for a 30° U-Axis Scan Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

−0.5000

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2.6  Planar Arrays111

Figure 2.26  The far-field function for a scanned 50 by 50-element planar array, with the planar array scanned to [uo, vo] = [0, −0.5].

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112

Array Theory

•

•

•

•

The primary far-field characteristics (beamwidth, beam position, sidelobes) in the unscanned plane are same as those of the boresight case. The gain reduces by 0.625 dB, by approximately the embedded element loss at the scan angle 10 log10(cosθ o /cos0) = 5 log10(1 − vo2) = 0.624 dB. Because of the inherent embedded element symmetry and the array factor orthogonality, scan characteristics in either u or v-planes are the same and interchangeable. The first sidelobe asymmetry in the scan plane occurs because of the embedded element sampling, as seen in the linear array case.

In the next design case, when the main beam is scanned along the diagonal plane, again the same linear translation occurs independently in both the orthogonal u and v-planes in sine space (similar to that observed in the principal plane scan cases), as shown in Figure 2.27 and Table 2.19. Asymmetric sidelobe behavior is also evident, only in this case, as the array factor samples the embedded element function. We therefore make the following observations about the above diagonal plane scanned case, when compared with the boresight case. •

•

•

•

•

The array factor component of the far-field equation translates by a constant [uo, vo] = [−0.5000, −0.5000] in sine space, while the embedded element factor (f(θ , ϕ ) = xˆ cos q ) is fixed. The gain reduces by 1.505 dB, consistent with the embedded element loss at the diagonal plane scan angle 10 log10(cosθ o /cos0) = 5 log10(1 − uo2 − vo2) = 1.505 dB. Because of the inherent embedded element symmetry and the array factor orthogonality, scan characteristics in either u or v-planes are the same and coincident in this case. The first sidelobe asymmetry in the scan plane occurs because of the embedded element sampling, as seen in the linear array case. All radiation beyond the first eight nulls and sidelobes in the u and v-planes is truncated on one side of the main beam. Table 2.19  Planar Array Conditions for a 45° Diagonal Plane Scan Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.5000

Main beam scan location (vo)

−0.5000

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2.6  Planar Arrays113

Figure 2.27  The far-field function for a scanned 50 by 50-element planar array, with the planar array scanned to [uo, vo] = [−0.5, −0.5].

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114

Array Theory

In our next design case, when the main beam is scanned along the diagonal plane to the edge of real space, again the same linear translation occurs in the orthogonal u and v-planes in sine space, but radiation beyond the real-space limit is truncated, as shown in Figure 2.28 and Table 2.20. In this interesting case, all radiation beyond the center of the main beam in the u and v-planes is truncated at the real-space boundary. We can make the following observations about the above diagonal plane scanned case, when compared with the boresight case. •

•

•

•

•

The array factor component of the far-field equation translates by a constant [uo, vo] = [−0.7071, −0.7071] in sine space, while the embedded element factor (f(θ , ϕ ) = xˆ cos q) is fixed. The beamwidth in both scan planes reduces by a 0.2319 ratio due to the severe main beam truncation at the real-space boundary. This is an example of the Hansen-Woodyard effect, as one-half of the main beam is scanned into imaginary space, where (w > 1). Because of the inherent embedded element symmetry and the array factor orthogonality, scan characteristics in either u or v-planes are the same and coincident in this case. All radiation beyond the main beam center in the u and v-planes is truncated on one side of the main beam and at the edge of real space.

2.6.2  Planar Arrays with Real-Space Grating Lobes

Grating lobe phenomena are more complex in the planar array since the array periodicity in two orthogonal aperture planes is the source of two independent infinite grating lobe sets. There are more complex array lattice and surface wave effects not found in linear arrays, as discussed briefly in this section and expanded upon elsewhere in this book. Grating lobe effects can be effectively studied from both the fundamentals of the far-field integral equation and from specific parametric cases. The argument Table 2.20  Planar Array Conditions for a 90° Diagonal Plane Scan Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.7071

Main beam scan location (vo)

−0.7071

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2.6  Planar Arrays115

Figure 2.28  The far-field function for a scanned 50 by 50-element planar array, with the planar array scanned to [uo, vo] = [−0.7071, −0.7071].

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116

Array Theory

in the far-field summation equation (2.44) produces a grating lobe when it reaches integer multiples of 2π . M N ⎛ e− jkRo ⎞ − j( 2p / l )( xm ,n u+ ym ,nv ) E (r ) = ⎜ am,ne f q, f ( ) ∑ ∑ ⎟ 4pR ⎝ o⎠ m=1 n=1 M N ⎛ e− jkRo ⎞ − j 2p / l x u− x u + y v− y v =⎜ f q, f ( ) ∑ ∑ am,n e ( )( m,n m,n o m,n m,n o ) (2.70) ⎝ 4pRo ⎟⎠ m=1 n=1



M N ⎛ e− jkRo ⎞ − j( 2p / l )( xm ,n ( u−uo ) + ym ,n ( v−vo )) =⎜ am,n e f q, f ( ) ∑ ∑ ⎟ ⎝ 4pRo ⎠ m=1 n=1

In the following examples, we apply the same array parameters used previously, M





N

E ( r ) = xˆ cos q ∑ ∑ am,n e

(

− j( 2p / l ) xm ,n ( u−uo ) + ym ,n ( v−vo )

m=1 n=1

(

)

(2.71)

)

k xm,n ( u − uo ) + ym,n ( v − vo ) = 2np (2.72) Rearranging, and using a fixed element periodicity,



⎛ dy ⎞ ⎛ dx ⎞ ⎜⎝ l ⎟⎠ ( u − uo ) + ⎜ l ⎟ ( v − vo ) = n (2.73) ⎝ ⎠

So, grating lobes can occur from periodicity in either the x or y-aperture planes and from the scan conditions in u and v-axes of sine space. Following the example of (2.57) for a linear array, grating lobes depend on two independent integer indices, each linked to the two aperture axes, ⎛ l⎞ u − uo = n ⎜ ⎟ ⎝ dx ⎠ n=0,±1,±2,±3…



⎛ l⎞ v − vo = m ⎜ ⎟ ⎝ dy ⎠ m=0,±1,±2,±3…

(2.74)

This becomes more apparent when we use a square array lattice with equal x and y-axis spacing, as indicated in the following array parameters (Table 2.21) and the far-field plot in Figure 2.29. The large element spacing creates real-space grating lobes, even for a beam scanned to boresight. The grating lobe locations follow the above equations with real-space harmonics occurring for (⎪m⎪ ≥ 1, ⎪n⎪ ≥ 1), producing multiple spatial harmonics within real space, one of which is the main beam. The zero-order spatial harmonic is frequently termed the main beam, but this is not always the case, as illustrated in the following cases.

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2.6  Planar Arrays117

Consider, for example, the array parametric case in Table 2.21. The uniform element spacing in both x and y-planes exceeds the Nyquist criterion, introducing the spatial harmonic sets in two orthogonal planes in sine space. This further illustrates both the linearity of the sine space calculation and sine space truncation effects. In this case, the spatial harmonic series is limited to (⎪m⎪ ≤ 1, ⎪n⎪ ≤ 1) to initiate this study. Table 2.21  Planar Array Conditions for a 0° Scan Condition Producing Real-Space Grating Lobes Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

1.5λ

Y-axis element separation (dy)

1.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

Figure 2.29  The far-field function for a scanned 50 by 50-element planar array, with an intentional grating lobe series.

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118

Array Theory

We can make the multiple grating lobes that appear in the real-space farfield data. •

Grating lobes appear in both u and v-planes within real space due to the element spacing. These results are consistent with (2.74). With μ o = 0, the grating lobe locations are expected at ug = [−0.6667, 0.0000, 0.6667], as determined from: ⎛ l⎞ u − uo = n ⎜ ⎟ ⎝ dx ⎠ n=0,±1,±2,±3… (2.75)



•

•

•

An identical grating lobe series exists in the v-plane, due to the real space symmetry of the element function and the identical element spacing in the aperture’s y-plane, so vg = [−0.6667, 0.0000, 0.6667]. The grating lobes provide unity impulse sampling of the embedded element factor (f(θ , ϕ ) = xˆ cos q) at the grating lobe locations. So grating lobe peak magnitudes that fall in the u or v-planes are Eg = [−1.27 dB, 0.00 dB, −1.27 dB]. The magnitude and location of sidelobes near each of the grating lobes are consistent with the sinc function. The grating lobes in the diagonal planes are located at [w = −0.9428, 0.0000, 0.9428], in accordance with wg = u2g + v2g .

The behavior attributed to planar array grating lobes becomes more evident if we consider another example, this time where the main beam is again steered to boresight, but the element spacing is increased farther than the above case. The array parameters in Table 2.22 and far-field pattern plot (Figure 2.30) show the effect of a square lattice having 3.0λ element spacing. The grating lobe locations follow (2.74), with real-space harmonics occurring for (⎪m⎪ ≤ 2, ⎪n⎪ ≤ 2), with a spacing of 0.333 along u and v-axes in sine space. This produces 25 spatial harmonics within real space, one of which is the main beam. Table 2.22  Planar Array Conditions for a 0° Scan Condition Producing Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

3.0λ

Y-axis element separation (dy)

3.0λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

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2.6  Planar Arrays119

Figure 2.30  The far-field function for a scanned 50 by 50-element planar array, with the parameters cited.

Just as was the case with linear arrays, a scanned main beam produces the same linear shift in all grating lobes. By scanning the main beam with the sine-space coordinates [uo, vo] = [−0.1667, −0.1667], all visible grating lobes move by the same amount in sine space. For the parameters in Table 2.23, the far-field radiation pattern is as shown in Figure 2.31. Table 2.23  Planar Array Conditions for a 13.66° Diagonal Plane Scan Condition Producing RealSpace Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

3.0λ

Y-axis element separation (dy)

3.0λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.1670

Main beam scan location (vo)

−0.1670

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120

Array Theory

Figure 2.31  The far-field function for a scanned 50 by 50-element planar array, with the parameters cited.

We have noted that grating lobes are a replica of the main beam and so are indistinguishable from one another, beyond the spatial dependency of the embedded element factor. This becomes evident in another example using the same lattice as indicated above, this time with a scan location of [uo, vo] = [−0.3333, −0.3333]. This produces a far-field pattern unchanged from the above boresight scanned example, other than the main beam’s displacement to the above coordinate. So, essentially, the main beam has traded places with one of the grating lobes, and all other farfield characteristics remain unchanged, as shown in Figure 2.32 and Table 2.24. The planar array can be treated as two separate linear arrays in the principal planes, and that is where the peak first sidelobes occur. Because the amplitude distribution is uniform, the sinc function will accurately predict the sidelobe levels. Considering the previous cases, each having real-space grating lobes, it may become apparent that the main beam is simply the zero-order grating lobe. All other grating lobes are replicas of it, weighted by the fixed element factor. The location of these grating lobes and their appearance in real space is governed by the element lattice geometry and element spacing. Temporarily, we have studied the effects caused by a square lattice, so in additional studies we can expand on this initial simplification. The element phase difference that was used to scan the main beam in a twoelement array becomes a linear phase gradient in a planar array. The phase gradient

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2.6  Planar Arrays121 Table 2.24  Planar Array Conditions for a 28.1° Diagonal Plane Scan Condition Producing RealSpace Grating Lobes Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

3.0λ

Y-axis element separation (dy)

3.0λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.3333

Main beam scan location (vo)

−0.3333

function for a linear array is the same function for the planar phased array, although the latter can use either or both of the two components, [uo, vo]. ax = kdx uo

ay = kdyvo (2.76)

Figure 2.32  The far-field function for a scanned 50 by 50-element planar array, with the parameters cited.

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122

Array Theory

This also means that the element excitation or distribution function can be written in exponential form, am,n = am,n e am,n = am,n e



(

jk xm ,n uo + ym ,nvo

(

jk ax +ay

)

) = a e jk(a ) m,n

(2.77)

m ,n

The element phased arrangement leads to the logical conclusion that planar phased arrays can be phased using row-column beam steering, where the phase components are those indicated in (2.75), and the phase at each element is the sum of the two linear phase gradients, am,n = a x + ay (2.78)



Returning to the far-field summation equation, we can use the above element phase representation to show that the far-field response is the integral of the product of two exponentials, one referenced to the aperture excitation and the other to the coordinates of the far-field observer, E (r ) =

M N e− jkRo − j( 2p / l )( xm ,n u+ ym ,nv ) f ( q, f ) ∑ ∑ am,ne 4pRo m=1 n=1 M



N

E ( r ) = Cf ( q, f ) ∑ ∑ am,n e

(

j( 2p / l ) xm ,n ( uo ) + ym ,n ( vo )

)e− j(2p / l)( x

m ,n

( u ) + ym ,n (v ))

(2.79)

m=1 n=1

This means that when the far-field coordinates match those of the aperture excitation, the exponential function reaches its maximum or the beam peak when uo = upeak

vo = vpeak (2.80)



Elsewhere, the integral of the exponential product produces sidelobes. The uniformly illuminated planar array can also be represented in closed form with a 2-D version of the same equation (2.63) used to represent the linear array, M

N

E ( r ) = xˆ cos q ∑ ∑ am,ne

(

− j( 2p / l ) xm ,n u+ ym ,nv

)

m=1 n=1

⎛ sin X ⎞ ⎛ sinY ⎞ E ( u,v ) = xˆ cos q ⎜ ⎝ X ⎟⎠ ⎜⎝ Y ⎟⎠

((

)

((

) )

)

⎛ sin Nd p l ( u − u ) ⎞ ⎛ sin Mdy p l ( vo − v ) ⎞ x o ⎟ ⎟⎜ = xˆ cos q ⎜ ⎜⎝ N sin dx p l ( uo − u ) ⎟⎠ ⎜ M sin dy p l ( vo − v ) ⎟ ⎝ ⎠

(

) )

(

(2.81)

am,n = 1.0, ∀m,n

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2.6  Planar Arrays123

This form, although limited to uniformly illuminated arrays, allows a rapid far-field calculation, noting that the more general form, given in (2.63), can become computationally time-consuming for especially large apertures. Example 2.9

For an array having the parameters in Table 2.25, determine the locations of the grating lobes residing within real space. From (2.74), ⎛ l⎞ u − uo = n ⎜ ⎟ = 0.2n ⎝ dx ⎠ n=0,±1,±2,±3… u = 0.2n + uo = −0.5 + 2n ⎛ l⎞ = 0.2m v − vo = m ⎜ ⎟ ⎝ dy ⎠ m=0,±1,±2,±3…

(2.82)

v = 0.2n + vo = −0.5 + 2m

The real-space harmonics are limited by w = Table 2.26.

u2 + v2 ≤ 1, as indicated in

Table 2.25  Planar Array Conditions for a 45° Diagonal Plane Scan Condition Producing RealSpace Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

5.0λ

Y-axis element separation (dy)

5.0λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.5000

Main beam scan location (vo)

−0.5000

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7060_Book.indb 124

u–uo

–1.000 –1.500

–0.800 –1.300

–0.600 –1.100

–0.400 –0.900

–0.200 –0.700

0.000 –0.500

0.200 –0.300

0.400 –0.100

0.600

0.800

1.000

1.200

1.400

1.600

1.800

n

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

7

8

9

1.300

1.100

0.900

0.700

0.500

0.300

0.100

–5.000

5.000

ug

uo

dx/λ

–5

5.000

dx/λ

Table 2.26  Sine-Space Map

–4

–5.000

vo

–3

0

2

–0.300 –0.100

1

–0.700 –0.500 –0.300 –0.100

–0.700 –0.500 –0.300 –0.100

–1

–0.300 –0.100

–0.700 –0.500 –0.300 –0.100

–0.700 –0.500 –0.300 –0.100

–0.900 –0.700 –0.500 –0.300 –0.100

–0.900 –0.700 –0.500 –0.300 –0.100

–0.900 –0.700 –0.500 –0.300 –0.100

–0.900 –0.700 –0.500 –0.300 –0.100

–2

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

0.100

3

vg

0.300

0.300

0.300

0.300

0.300

0.300

0.300

0.300

0.300

0.300

4

0.500

0.500

0.500

0.500

0.500

0.500

0.500

0.500

5

0.700

0.700

0.700

0.700

0.700

0.700

0.700

0.700

6

0.900

0.900

0.900

0.900

7

8

9

124 Array Theory

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2.7  Conformal Arrays125

Example 2.10

For an array with the parameters of Table 2.27, determine the magnitude of the first sidelobes around the main beam. Table 2.27  Planar Array Conditions for a 28.1° Diagonal Plane Scan Condition Producing RealSpace Grating Lobes Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

50

Y-axis element count (N)

50

X-axis element separation (dx)

3.0λ

Y-axis element separation (dy)

3.0λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.3330

Main beam scan location (vo)

−0.3330

The array conditions allow the use of the sinc function (Table 2.28). Table 2.28  Planar Array First Sidelobe Locations and Magnitudes

2.7

u

v

SLL(dB)

–0.323

–0.333

–13.476

–0.342

–0.333

–13.521

–0.333

–0.323

–13.521

–0.333

–0.342

–13.447

Conformal Arrays Drawing on the above methods to design and predict the radiated performance of planar arrays, we can use related tools to extend the associated spatial degrees of freedom to conformal surfaces. These take on a variety of 3-D nonplanar aperture forms, many of which are solved by canonical methods. The far-field integral equation is a key element of the approach offered in this section, made more practical using 3-D radius of curvature methods for the radiating aperture. This allows a straightforward extension to various Gaussian curvature surfaces, such as hyperboloid, cylindrical, and spherical, by simply stating two orthogonal principal curvatures [κ x, κ y] [6]. The method uses the ideal voltage sources developed for planar arrays, and uniform element spacing along the surface of curvature, consistent with Nyquist theory, following the extensive work by Mailloux [5]. The referenced work

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126

Array Theory

considers a wide range of conformal array types with important design conclusions, so the following discussion applies to the practical subset of densely spaced large radius of curvature designs in particular. Several unique properties emerge from the conformal array studies. These include embedded element effects, where the incidence angle differs for each element of the array, also producing gain degradation. The principal curvatures and the total scan volume play an important role in this behavior. The embedded element effects extend to eclipsing effects, where certain elements of the array become cut off from the intended field of view depending again on scan volume and radius of curvature. Positive and negative surfaces also contribute to the primary design variants made possible with radius of curvature array analysis. We also find projection effects produced by nonplanar conformal apertures; these effects impact the radiated far-field pattern and the gain, both of which are considered in this section. The benefits of conformal arrays come in part from a contemporary effort to extend array technology to a wide range of applications, a significant number having nonplanar surfaces. This aspiring goal often introduces a variety of manufacturing difficulties because of the extensive existing planar manufacturing base. Commonly used planar manufacturing methods exist for array radiators and a wide range of the active monolithic microwave integrated circuits (MMICs) are used today. The latter have made astounding advancements in the range of operating frequencies attainable, wide operating bandwidth, low noise figure and phase noise, high radiated power, and reduced packaging size. We have seen the advancements of printed circuit boards (PCBs), chip-scale packaging (CSP), wafer-scale packaging (WSP), and highly integrated microwave and millimeter-wave circuits in just the past few decades. These are largely planar methods, and as printed conformal electronic technology is currently in its early development stages, fully conformal arrays remain a challenge on multiple fronts. Faceted conformal arrays have been one approach that is finding some traction. Thus, the emphasis in this section on radius of curvature methods that accommodate these conformal array design variants to explore the fundamentals of a conformal aperture surface. We begin with a study of nonplanar aperture surfaces formed as a product of dual orthogonal principal curvatures with radiating elements spaced at the Nyquist spatial limit. Next, we determine the critical angles associated with the embedded radiating element in sine and angle space. This information is sufficient to produce far-field radiation results by means of the far-field integral equation. We leave important subjects such as the element impedance, polarization, and gaps between elements as subjects of study in later sections. 2.7.1  Radius of Curvature Embedded Element Geometry

Radius of curvature surfaces have the valuable property that a wide range of surface contours are possible using relatively simple mathematics. A continuous surface can be decomposed into sectional profiles where the deviations from linear are readily represented by circles of fixed radii (ρ = 1/κ ). Concave and convex surfaces are represented using negative and positive radii. So, a given conformal profile, essentially a curved line, is represented by a series of circle tangent lines, each controlled by their radii of curvature (ρ n), as shown in Figure 2.33.

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2.7  Conformal Arrays127

Figure 2.33  A curved line as represented by a series of radii of curvature.

Given this convenience, our objective in this section is to determine the Cartesian coordinates for a given (i) radiating element ([xi, yi, zi]) considered a part of a series of elements on a subject conformal surface. Given these data and a set of aperture excitation functions, we can use the far-field integral equation to determine the radiation characteristics. For each radius of curvature surface, we seek to place element phase centers on a curved surface using the following deterministic conditions: •

•

The element phase centers are separated at the Nyquist spatial rate (0.5λ ) along the aperture surface. The surface and element phase center placements are formed using orthogonal surface sections and their associated radii of curvature (ρ x, ρ y).

Radiating elements are placed on a curved line using the Nyquist spacing criterion by subdividing the radius of curvature into equiangular segments, as shown in Figure 2.34. Because the element separation along the curved line is fixed (ds), so is the angle between (α ), drawn from the center of curvature in a single plane as



ex =

dsx rx

( radians ) (2.83)

The Nyquist element spacing (ds) is an effective approach for large radius of curvature conformal arrays, where the effect of element eclipsing is limited to a small subset of the total array element count. Conformal arrays having a small radius of curvature are in a different design category where element spacing criteria apply to

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2/13/23 10:52 AM

128

Array Theory

Figure 2.34  A curved line as represented by a series of radii of curvature.

the arc length instead, and element eclipsing may affect a significant percentage of the total element count. Because the angle associated with each element along the circular curvature is the same, the angle at any element phase center is represented by the sum of the angles,

eix = iax  ( radians ) (2.84)

The 2-D coordinates for all phase centers along the curved line are in the x-z plane, xi = rx sin eix



zi = rx cos eix (2.85)

We repeat this process for the orthogonal (y-z) plane, using the following expressions with independent variables associated with that plane,



7060_Book.indb 128

dsy ry

( radians ) (2.86)

eiy = iey

( radians ) (2.87)

ey =

yi = ry sin eiy (2.88)

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2.7  Conformal Arrays129

The z-coordinate is based on the compound angle from both the x and y-axes and is expressed as the sum in sine space, zi = rx cos eix + ry cos eiy (2.89)



The compound angle from the orthogonal planes uses sine space linearity and orthogonality, revealing the projection effect, wip =



2 eix + eiy2 (2.90)

qip = sin−1 wi (2.91)



The conformal aperture’s aperture surface normal differs for all elements, unlike a planar array. This therefore violates the assumption that the embedded element can be removed from the array factor integration in the far-field integral equation. The individual element needs to be included in the far-field integration process as a result. To accomplish this, we need to know the location of the element surface normal for each element. This is best done in sine space, where the calculations can be expected to be linear equations. Accordingly, since we already have determined the (θ ) angle for each element, the orthogonal (ϕ ) angle in spherical coordinates is, taking careful note of the associated quadrant, fip = tan−1



eiy eix (2.92)

The sine space coordinates of the surface projected element normals are uip = wip cos fip



vip = wip sin fip (2.93)

Because the compound angle is drawn from a common reference at the aperture surface, it also represents the element surface normal. We can see that conformal surfaces intrinsically generate a set of surface normals for the elements that differ, depending on their location on the surface. The following example may be illustrative. Example 2.11

Generate the cylindrical surface and element Cartesian positions in three orthogonal dimensions for the set of parameters in Table 2.29. Determine the element surface normal angles [θ ip, ϕ ip] relative to the center of the aperture (see Figure 2.35).

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130

Array Theory

Figure 2.35  A cylindrically curved surface and its element surface normal. Table 2.29  Conformal Array Parameters for a Cylinder Oriented Along the Y-Axis Parameter

Value

Operating frequency (GHz)

10.0

X-axis element count (N)

10

Y-axis element count (M)

10

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis radius of curvature (ρ x)

10λ

Y-axis radius of curvature (ρ x)

105λ

Using the procedure outlined above, several intermediate values lead to the final coordinate mapping shown here: ex =



dsx = 0.05 radians ( 2.86° ) rx

(2.94) dsy −6 ey = = 5.0 ∗ 10 radians (0.0000286° ) rx xi = rx sin eix yi = ry sin eiy



zi = rx cos eix + ry cos eiy

(2.95)

The surface normals for each of the elements relative to the aperture center are (Table 2.30):

7060_Book.indb 130

wi =

2 eix + eiy2 (2.96)

qip = sin−1 wi (2.97)

2/13/23 10:52 AM

2.7  Conformal Arrays131 Table 2.30  Conformal Array Element Coordinates in Cartesian and Polar Forms Y-axis Cylindrical Contour Surface Coordinates ax= 5.00E–02rad ay= 5.00E–06rad Z(x,y) (in.)

x(in.) –2.6333 –2.055

–1.4715 –0.8844 –0.295

0.295

0.8844

1.4715

2.055

2.6333

–2.6557 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 –2.0655 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 –1.4754 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 y(in.)

–0.8852 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 –0.2951 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 0.2951 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 0.8852 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 1.4754 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 2.0655 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975 2.6557 –0.2975 –0.1803 –0.0921 –0.0332 –0.0037 –0.0037 –0.0332 –0.0921 –0.1803 –0.2975

Y-axis Cylindrical Contour Surface Normal (degrees)

y(in.)

θ p(x,y) (deg.)

x(in.) –2.6333 –2.055

–1.4715 –0.8844 –0.295

0.8844

1.4715

2.055

2.6333

–2.6557 12.892

10.027

7.162

4.297

1.432

0.295 1.432

4.297

7.162

10.027

12.892

–2.0655 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

–1.4754 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

–0.8852 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

–0.2951 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

0.2951 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

0.8852 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

1.4754 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

2.0655 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

2.6557 12.892

10.027

7.162

4.297

1.432

1.432

4.297

7.162

10.027

12.892

Y-axis Cylindrical Contour Surface Normal (degrees)

y(in.)

ϕ p(x,y) (deg.)

x(in.) 0.295

0.8844

1.4715

2.055

2.6333

–2.6557 –90.006 –90.007 –90.010 –90.017 –90.052

90.052

90.017

90.010

90.007

90.006

–2.0655 –90.005 –90.006 –90.008 –90.013 –90.040

90.040

90.013

90.008

90.006

90.005

–1.4754 –90.003 –90.004 –90.006 –90.010 –90.029

90.029

90.010

90.006

90.004

90.003

–0.8852 –90.002 –90.003 –90.003 –90.006 –90.017

90.017

90.006

90.003

90.003

90.002

–0.2951 –90.001 –90.001 –90.001 –90.002 –90.006

90.006

90.002

90.001

90.001

90.001

0.2951 –89.999 –89.999 –89.999 –89.998 –89.994

89.994

89.998

89.999

89.999

89.999

0.8852 –89.998 –89.998 –89.997 –89.994 –89.983

89.983

89.994

89.997

89.998

89.998

1.4754 –89.997 –89.996 –89.994 –89.991 –89.971

89.971

89.991

89.994

89.996

89.997

2.0655 –89.996 –89.994 –89.992 –89.987 –89.960

89.960

89.987

89.992

89.994

89.996

2.6557 –89.994 –89.993 –89.990 –89.983 –89.948

89.948

89.983

89.990

89.993

89.994

7060_Book.indb 131

–2.6333 –2.055

–1.4715 –0.8844 –0.295

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132

Array Theory

Conformal arrays in general introduce the concept of eclipsed elements. Because conformal apertures have unequal element surface normals, some of these will not align well when projected to far-field observation points within the scan volume. In some cases, elements of the aperture may be shadowed by neighboring elements or eclipsed. These may not contribute effectively to the scanned main beam, so it is better to identify and disable them, as considered in the next section. 2.7.2  Conformal Array Phase Alignment

A conformal aperture introduces a phase error due to the path length differences between the element phase centers and an ideal planar surface normal to the surface tangent of the center element. The planar sectional geometry makes this clear, as shown in Figure 2.36, where the path length error increases with elements located farthest from the central element. The conformal array projection also introduces a scan and element location-dependent unit cell projected area. This effect is generally corrected in low sidelobe applications by increasing the unit cell amplitude correspondingly. The graphical representation in the figure presents both a simple illustration of the resulting phase error as well as a correction method that is effective over a wide range of scan angles. The path length error as shown is determined by projecting the element locations onto a virtual aperture plane normal to the surface tangent of the center element, as shown in the figure. For element position coordinates in Cartesian space, the boresight scanned error path length is the difference between the z-coordinates of the two. The virtual aperture plane is positioned at the z-coordinate of the central element’s phase center (zo), so the virtual plane is represented as (zi = zo, ∀i). The differential length for each element (i) is: li = zi − zo (2.98)



Because the radiated waves for each element propagate to the far field, these are free-space TEM propagating modes, so the boresight scanned phase error is:

(

Δφi = ai = koli = 2p lo zi − zo

( uo = vo = 0)



)

(2.99)

In more general terms, for a scan angle within the array’s scan volume the aperture phase aligns with the intended scan angle when the exponentials for the embedded element and the far-field observer are equal and opposite in sign, from (2.8), E (r ) =



e− jkRo N ai fi ( q, f ) e− j( k⋅ri ) (2.100) 4pRo ∑ i=1

Aperture phase alignment occurs when all element amplitudes are equal and ai = e− j( k⋅ri )

7060_Book.indb 132

ai e

(

j( 2p / l ) xi u+ yiv+zi 1−u2 −v2

) = e− j( k⋅r ) = e j(2p / l)( x u + y v +z i

i o

i o

i

1−uo2 −vo2

) (2.101)

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2.7  Conformal Arrays133

Figure 2.36  An illustration of the path lengths between element phase centers located on a curved surface aperture and a virtual aperture plane.

So the aligned phase state for each array element is

(

)

⎛ 2p ⎞ Δai = koli = ⎜ ⎟ xi uo + yi vo + zi 1 − uo2 − vo2 (2.102) ⎝ l⎠



The above expression reduces to (2.99) under a boresight scan condition. This is equivalent to the phase mode excitation method described in some of the foremost conformal phased array literature [7]. The theory supporting both the conformal element coordinates and the phase correction is verified with a boresight array case, as shown in Figure 2.36. In this case, we choose a 10 × 10 element array, conformal in two planes, with 20 wavelength radii of curvature in x- and y-planes, a uniform illumination, scanned to boresight, as given by the array parameters in Table 2.31. The conformal array’s aperture and far-field performance follow in Figure 2.37. The results point to multiple conclusions, including with the described phase correction, but also indicate performance limitations associated with the subject of eclipsed elements. Summarizing the data, we observe the following: •

7060_Book.indb 133

The aperture coordinate plot shows the effects of a doubly conformal aperture, with equal radii of curvature in the x and y-planes, and the product of both curvatures in the diagonal plane, where (x = y).

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134

Array Theory

Figure 2.37  The projected aperture complex distribution, aperture geometry, and far-field response for a 10 by 10-element doubly conformal array, with scan coordinates of [uo, vo] = [0.0, 0.0]

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2/13/23 10:52 AM

2.7  Conformal Arrays135 Table 2.31  Conformal Array Parameters for a Dual Conformal Array with Two Equal Radii of Curvature in X and Y-Planes

•

•

•

•

Value

Operating frequency (GHz)

10.0

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

11

Y-axis element count (M)

11

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis radius of curvature (ρ x)

20λ

Y-axis radius of curvature (ρ x)

20λ

Element eclipse limit (sines)

0.0000

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

Radiating element normal locations in sine space spread out over the 11 × 11 elements with an almost uniform spatial distribution when plotted in sine space, due to the conformal aperture projection onto the x-y plane (Figure 2.38). The radiated near-field amplitude at each element in the antenna aperture is 0.0 dB, indicating a nearly uniform distribution when projected to the x-y plane. The radiated phase at each element reflects the z-axis dimensions of the aperture elements, creating a truncated circular distribution in the two orthogonal x and y-dimensions of the aperture plane consistent with (2.99). The far-field response results from (2.8), as shown below with the leading constants normalized to a value of unity, E (r ) =

•

e− jkRo N ai fi ( q, f ) e− j( k⋅ri ) (2.103) 4pRo ∑ i=1

The embedded element factor is an adjusted cosine power function, normalized to the projected element surface normal [θ ip, ϕ ip]. fi ( q, f ) =

•

7060_Book.indb 135

Parameter

(

)

cos q − qip (2.104)

The element normals spread over sine space with a maximum of [⎪uip⎪⎪vip⎪] max = [0.1247, 0.1247], so [wip]max = 0.1763, and is most significant in the diagonal aperture planes. This increases the probability for an element eclipse,

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136

Array Theory

Figure 2.38  The projected aperture complex distribution, aperture geometry and far-field function for a 11 by 11-element doubly conformal array, with scan coordinates of [uo, vo] = [−0.5, −0.5].

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2.7  Conformal Arrays137

•

•

•

where elements of the array are shadowed by nearby elements, depending on the conformal array curvature and the observation location in the far field. The far-field radiation has a main beam located at [u, v] = [0.0000003, 0.0000003] sines. The 3-dB beamwidth is [u3, v3] = [0.16142, 0.16142], just slightly larger (0.2%) than the uniform aperture beamwidth prediction for a planar array, [u3, u3] = [0.1611, 0.1611], or u3 = 0.886λ /Ndx. The radiated sidelobes in all of sine space are consistent with dual orthogonal sinc functions, ⎛ sin X ⎞ ⎛ sinY ⎞ E ( u, v ) = ⎜ ⎝ X ⎟⎠ ⎜⎝ Y ⎟⎠

((



) )

) ⎞⎟ ⎛⎜ sin (( Md p l)(v

⎛ sin Nd p l ( u − u ) x o =⎜ ⎜⎝ N sin dx p l ( uo − u )

(

)

− v) ⎞ y o ⎟ ⎟⎠ ⎜ M sin dy p l ( vo − v ) ⎟ ⎝ ⎠

(

)

(2.105)

When this moderately curved array is scanned, it verifies the aperture phasing, as illustrated in the following example, which uses the same array as above, except that it is scanned to [uo, vo] = [−0.5000, −0.5000]. Example 2.12

In this example, we determine the element phase values needed to scan a conformal array’s far-field radiation beam in sine space. As is the case for a planar array, the element phase values are a function of the 3-D coordinates of each. The example calculations in Table 2.32 provide the confirming data in both the near (aperture) fields and the far field, as shown in Table 2.33. Table 2.32  Conformal Array Parameters for Diagonal Plane Scan Condition Parameter Operating frequency (GHz) Element polarization Element power function X-axis element count (N) Y-axis element count (M) X-axis element separation (dx) Y-axis element separation (dy) X-axis radius of curvature (ρ x) Y-axis radius of curvature (ρ x) Element eclipse limit (sines) X-axis amplitude distribution Y-axis amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

7060_Book.indb 137

Value 10.0 x-vector cosine (θ ) 11 11 0.5λ 0.5λ 20λ 20λ 0.0000 Uniform Uniform −0.5000 −0.5000

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138

Array Theory Table 2.33  Conformal Array Element Coordinates and Complex Excitation Voltages El No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

7060_Book.indb 138

x(in) –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.9431 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –2.3567 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.7688 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –1.1798 –0.5901 –0.5901 –0.5901 –0.5901

y (in) –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798

Z (in) –0.3684 –0.3021 –0.2505 –0.2137 –0.1916 –0.1842 –0.1916 –0.2137 –0.2505 –0.3021 –0.3684 –0.3021 –0.2359 –0.1843 –0.1474 –0.1253 –0.1179 –0.1253 –0.1474 –0.1843 –0.2359 –0.3021 –0.2505 –0.1843 –0.1327 –0.0959 –0.0737 –0.0664 –0.0737 –0.0959 –0.1327 –0.1843 –0.2505 –0.2137 –0.1474 –0.0959 –0.0590 –0.0369 –0.0295 –0.0369 –0.0590 –0.0959 –0.1474 –0.2137 –0.1916 –0.1253 –0.0737 –0.0369

|E| (dB) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Phase(°) 65.30 –3.92 –77.84 –156.42 120.39 32.65 –59.59 –156.27 102.66 –2.72 –112.35 –3.92 –73.14 –147.06 134.36 51.17 –36.57 –128.81 134.51 33.44 –71.94 178.42 –77.84 –147.06 139.01 60.43 –22.75 –110.49 157.27 60.58 –40.48 –145.86 104.50 –156.42 134.36 60.43 –18.15 –101.33 170.93 78.69 –18.00 –119.06 135.56 25.92 120.39 51.17 –22.75 –101.33

2/13/23 10:52 AM

2.7  Conformal Arrays139 Table 2.33  (Continued) El No 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

7060_Book.indb 139

x(in) –0.5901 –0.5901 –0.5901 –0.5901 –0.5901 –0.5901 –0.5901 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 0.5901 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.1798 1.7688 1.7688 1.7688 1.7688 1.7688 1.7688 1.7688 1.7688

y (in) –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798

Z (in) –0.0148 –0.0074 –0.0148 –0.0369 –0.0737 –0.1253 –0.1916 –0.1842 –0.1179 –0.0664 –0.0295 –0.0074 0.0000 –0.0074 –0.0295 –0.0664 –0.1179 –0.1842 –0.1916 –0.1253 –0.0737 –0.0369 –0.0148 –0.0074 –0.0148 –0.0369 –0.0737 –0.1253 –0.1916 –0.2137 –0.1474 –0.0959 –0.0590 –0.0369 –0.0295 –0.0369 –0.0590 –0.0959 –0.1474 –0.2137 –0.2505 –0.1843 –0.1327 –0.0959 –0.0737 –0.0664 –0.0737 –0.0959

|E| (dB) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Phase(°) 175.48 87.74 –4.50 –101.18 157.75 52.37 –57.27 32.65 –36.57 –110.49 170.93 87.74 0.00 –92.24 171.08 70.01 –35.37 –145.01 –59.59 –128.81 157.27 78.69 –4.50 –92.24 175.52 78.84 –22.23 –127.61 122.75 –156.27 134.51 60.58 –18.00 –101.18 171.08 78.84 –17.85 –118.91 135.71 26.07 102.66 33.44 –40.48 –119.06 157.75 70.01 –22.23 –118.91

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140

Array Theory Table 2.33  (Continued) El No 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121

x(in) 1.7688 1.7688 1.7688 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.3567 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431 2.9431

y (in) 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431 –2.9431 –2.3567 –1.7688 –1.1798 –0.5901 0.0000 0.5901 1.1798 1.7688 2.3567 2.9431

Z (in) –0.1327 –0.1843 –0.2505 –0.3021 –0.2359 –0.1843 –0.1474 –0.1253 –0.1179 –0.1253 –0.1474 –0.1843 –0.2359 –0.3021 –0.3684 –0.3021 –0.2505 –0.2137 –0.1916 –0.1842 –0.1916 –0.2137 –0.2505 –0.3021 –0.3684

|E| (dB) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Phase(°) 140.03 34.64 –74.99 –2.72 –71.94 –145.86 135.56 52.37 –35.37 –127.61 135.71 34.64 –70.74 179.62 –112.35 178.42 104.50 25.92 –57.27 –145.01 122.75 26.07 –74.99 179.62 69.99

2.7.3  Eclipsed Elements in Conformal Arrays

Unlike planar arrays, conformal apertures introduce conditions where elements in a part of the aperture lie in the shadow of, and so are eclipsed by, neighboring elements. This unique condition is responsible for several limiting factor and a few advantages, as follows: •

•

7060_Book.indb 140

Eclipsed field of view: Due to the conformal aperture curvature, some aperture portions are eclipsed within the field of view. With contemporary electronics, these elements can be disabled on an individual beam basis. This capability avoids radiation occurring in unwanted sectors, while also limiting the available elements; it is generally performance-limiting. Embedded element field-of-view growth: Because some elements in a conformal aperture are eclipsed and so effectively blocked by others, some elements also have a wider field of view than the (θ ≤ 90°) sector available to elements in a planar array. The field-of-view growth does little to improve the performance of the main beam, but the embedded element field determination requires full wave 3-D simulation or the equivalent. It exceeds the limits of a traditional (cosn θ ) embedded element characterization and so can be approximated with

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2.7  Conformal Arrays141

•

•

extensions to the embedded element equation. The residual effects occur near the edges of real space, generally affecting sidelobes, but sometimes the main beam under wide scan conditions. This is also performance-limiting. 360° azimuth scanning: Using electronic beam control, the elements needed can be selected and enabled electronically for each beam created. This allows conformal arrays (such as a cylindrical array) to cover a 360° azimuth plane field of view by electronically selecting the elements that best cover specific azimuth sectors. This avoids mechanical commutation schemes and so is considered a performance advantage. Conformal aperture applications: There are ample cases today where the degrees of freedom in a phased array are needed on varying airborne, terrestrial, and space-based objects. The science and engineering of conformal arrays are making this capability possible, and so is considered a performance advantage.

Considering these eclipse-related factors, we first study the physics associated with the eclipse phenomena and the means to control it is contemporary phased array systems. Figures 2.39 and 2.40 show examples of conformal apertures having Nyquist-spaced elements. Propagating ray lines that represent TEM waves originated at several of the element traces from the element phase centers to a location orthogonal to a 60° scanned and a boresight planar wavefront. The projected wavefront requires that each ray intercept to this plane with a path length is correction in the element phase setting to produce an equiphase condition. Examining the ray intercept with the embedded element pattern reveals significant differences in the conformal element response. The embedded element far-field

Figure 2.39  An illustration of a conformal array aperture with both eclipsed elements and those within the field of view of a scanned plane wave.

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142

Array Theory

Figure 2.40  An illustration of a conformal array aperture with elements within the field of view of a boresight planewave.

radiation pattern is an assumed (cos1.25θ ), representing expected performance. We can examine several of the elements in detail for two different scan conditions, the first at 60° incidence: •

•

•

•

•

• •

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The center element located at 3:00 clock reference, has a ray that intersects the element at a 60° angle relative to its surface normal vector. This means that the amplitude weighting for that element is −3.75 dB, or (10 log10cos1.25cos60°). Considering the element located at 4:30 clock reference, the ray intercept angle is 30°, so its relative embedded element response is −0.78 dB, or (12.5 log10cos30°). The element at 6:00 clock reference ray intercept angle is 0°, so its relative embedded element response is −0.00 dB. Elements located at a 1:30 clock reference and further counterclockwise are eclipsed. The element at 1:30 clock reference has a ray intercept angle of 90°, so its relative embedded element response is inconsequential, or (12.5 log10cos90°). This element radiates its primary field in the direction of sidelobes far from the array’s main beam, so it is counterproductive to the beam formation process. Logically, it and all succeeding counterclockwise elements are disabled for the specific beam case under consideration. The element at 12:00 is blocked by all clockwise elements, and it is disabled. Elements having ray vectors within ±60° have 0° relative phase.

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2.7  Conformal Arrays143

Considering the boresight case, all elements are within the field of view, so none are eclipsed (Figure 2.39). However, in Figure 2.39 elements are eclipsed since the surface normal vectors vary over a 60° range in this case, due to the aperture’s curvature. The contributions of the embedded element differ, using the same process considered above. The individual element’s surface normal can be converted into sine space coordinates, where it is more convenient to determine the element voltage at an arbitrary real-space location [u,v] from (2.87),



⎡ uip,vip ⎤ = ⎡sin qip cos fip ,sin qip sin fip ⎤ = ⎡ wip cos fip , wip sin fip ⎤ (2.106) ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

So, the first level of eclipse control considers the range of element surface normals. Where any of these exceed a predetermined sine space radius, the element is disabled in the beam formation process. When a conformal array has either a small radius of curvature or a large number of elements the surface normal angle range can be considerable. Converting these normals into sine space coordinates and converting these into the sine space distance parameter (w), we can identify elements that exceed this value and disable them. This has the effect of creating a more circular array edge boundary condition, introducing sidelobes that resemble the rings of a circular aperture, as illustrated in the following example. Example 2.13

Table 2.34 introduces both a large element count and a small radius of curvature to illustrate both the wide element surface normal condition and a truncation method. Table 2.34  Conformal Array Eclipsed Element Parameters for Diagonal Plane Scan Condition

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Parameter

Value

Operating frequency (GHz)

10.0

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

21

Y-axis element count (M)

21

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis radius of curvature (ρ x)

20λ

Y-axis radius of curvature (ρ x)

20λ

Element eclipse limit (sines)

0.2000

X-axis amplitude distribution

Uniform, truncated

Y-axis amplitude distribution

Uniform, truncated

Main beam scan location (uo)

−0.5000

Main beam scan location (vo)

−0.5000

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144

Array Theory

Figure 2.41  The projected aperture complex distribution, aperture geometry, and far-field function for a 21 by 21-element doubly conformal array, with scan coordinates of [uo, vo] = [−0.5, −0.5].

The example illustrates several useful aspects of eclipsed conformal elements and the associated corrections, as summarized here: •

•

7060_Book.indb 144

Because of the 441 radiating elements used on a 20 × 20 wavelength radius of curvature surface element normal vectors range over [u ip,  v ip]max = [±0.2476, ±0.2476] and (wip) = 0.3499 range. When these are added to the intended scan angle of [uo, vo] = [−0.5000, −0.5000] and (wo) = 0.7071, some of the main beam falls outside of the local element normal vector range. An element surface normal restriction of (wip(max)) = 0.2000 disables elements with surface normals that fall outside of this range. This truncates a significant percentage of these elements producing a circular boundary at the aperture’s edge.

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2.7  Conformal Arrays145

•

•

The circular array boundary produced by the element surface normal truncation produces ring sidelobes around the main beam, as shown. The null depth near the main beam is less than −20 dB, indicating reasonable residual errors from the untruncated aperture elements.

Turning next to the embedded element field-of-view growth, it may be apparent that the element field of view exceeds (θ e ≤ 90°), as illustrated in Figure 2.42. This condition is determined by the radius of curvature, so it applies for all conformal apertures with finite curvature radii. Because the mutual coupling associated with planar arrays produces a predicted embedded element field only within these limits, the (cosn θ ) remains undefined outside of this region. This simplified geometry is sufficient to approximate the associated far-field effects produced. The sine space separation between a given observation position in sine space (u, v) and the projected individual element surface normal (uip, vip) is determined by the Pythagorean theorem,



( u − u ) + (v − v ) ) = ( u − u ) + (v − v ) 2

w − wip =

ip

(w − w

ip

2

ip

2

ip

2

2

ip

(

= sin2 q − qip

)

(2.107)

Figure 2.42  An illustration of a conformal array aperture with an element field of view that exceeds 90°.

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146

Array Theory

Because

(

)

(

)

cos2 q − qip = 1 − sin2 q − qip (2.108)



The embedded element function for a cosine power pattern can be determined as:



fi ( q, f ) =

(

)

(

cos q − qip = ⎛ 1 − u − uip ⎝

) ( 2

)

2 − v − vip ⎞ ⎠

0.25

(2.109)

As the total observation angle (θ − θ ip) can exceed 90°, the sine space representation above generates complex angles when

( u − u ) + (v − v ) 2

ip



2

ip

≥ 1 (2.110)

We can sense for this condition, applying a fixed embedded field value in place of the complex cosine argument. Doing so nevertheless leaves irregular far-field results at locations near the edge of real space, as is apparent in the u and v-plane far-field results from Example 2.13 in the regions where (u ≥ 0.75), and (v ≥ 0.75). Both the eclipsed element truncation and element field-of-view growth effects are illustrated in Table 2.35. As in Example 2.13, this array has the same aperture conditions but is scanned to boresight. Ring sidelobes near the main beam illustrate

Table 2.35  Conformal Array Eclipsed Element Parameters for a Boresight Scan Condition

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Parameter

Value

Operating frequency (GHz)

10.0

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

21

Y-axis element count (M)

21

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis radius of curvature (ρ x)

20λ

Y-axis radius of curvature (ρ x)

20λ

Element eclipse limit (sines)

0.2000

X-axis amplitude distribution

Uniform, truncated

Y-axis amplitude distribution

Uniform, truncated

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

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2.7  Conformal Arrays147

Figure 2.43  The projected aperture complex distribution, aperture geometry, and far-field response for a 10 by 10-element doubly conformal array, with scan coordinates of [uo, vo] = [0.0, 0.0].

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148

Array Theory

the circular aperture edge boundary produced by eclipsed element truncation (Figure 2.38). Slight deviations occur near the real-space edge, due to the embedded element field approximations due to field-of-view growth from the aperture’s curvature.

References [1] [2] [3] [4] [5] [6] [7]

7060_Book.indb 148

Kraus, J. D., Antennas, New York: McGraw-Hill, 1950. Collin, R. E., Field Theory of Guided Waves, New York: IEEE Press, 1990. Brookner, E., Practical Phased Array Antennas, Norwood, MA: Artech House, 1991. Lo, Y. T., and S. W. Lee, “Theorems and Formulas,” Ch. 2 in Antenna Handbook, New York: Van Nostrand Reinhold Co., 1988. Mailloux, R. J., Phased Array Antenna Handbook, Norwood, MA: Artech House, 2018. Kreyszig, E., Differential Geometry, New York: Dover Publications, 1991. Rudge, A. W., et al., The Handbook of Antenna Design, London, U.K.: Peter Peregrinus/ IEE, 1983.

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CHAPTER 3

Lattice Theory 3.1 Introduction The geometrical arrangement of the radiating elements in a phased array is generally termed its aperture lattice and the associated theory has been the subject of considerable study and development [1–4]. We know that lattice theory is one of the major controls of an array’s wide scan radiation in sine space. From this theory, we can determine critical array design factors, such as the maximum element phase center spacing, their geometrical arrangement, and the effects of element count, and so the array size. Our investigation into this topic starts with Floquet’s theory, which relates the periodic arrangement of the radiating elements to the propagating modes generated by the array. This theory is the foundation for the array’s radiating modes, with similarities to a waveguide’s propagating modes. It also lays the basis for the maximum spacing that the elements can have to support a given scan volume. The element spacing is a critical parameter in phased array design, as it determines the space available for the supporting electronics. The radiating element geometry also has significant control over the distribution of an array’s radiation in real and imaginary space, so we explore several planar array cases. This gives us a better understanding of some of the design techniques used to position and distribute the radiated field in the grating lobes. Our study of lattice theory examines the more commonly used forms of rectangular and triangular lattices for infinite planar arrays first. We consider case studies and examples of these using both Floquet’s theory and the finite array far-field integral equation and then consider the cases of ring and spiral lattices using the far-field integral alone. Small array effects are important, and we consider these using the finite array far-field integral equation. In retrospect, this subject area has significant scope in the phased array design. The intended result is a recognition of the lattice effects on an array’s far-field radiation.

3.2

Floquet’s Theorem One of the most powerful tools in phased array analysis is Floquet’s expansion of the wave equation in periodic form. Achille Marie Gaston Floquet (1847–1920) was a notable French mathematician who solved differential equations with periodic boundary conditions. This is especially important in phased array design, because of the generally periodic (i.e., modular) building block nature of the array aperture. 149

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150

Lattice Theory

Floquet’s theorem and the related Bloch’s theorem in solid state physics use these periodic boundary conditions to produce periodic solutions (i.e., far-field radiation in space in terms of multiple correlated peaks). The aperture boundary conditions arrange in the form of a periodic 2-D grid of phase centers, one at the center of each radiating element. Unit cells form within the boundaries between unit cells, so a rectangular lattice of phase centers produces rectangular unit cells and a rectangular arrangement of unit cell, unit cell lines, or phase walls (Figure 3.1). As a result [5], “the field at each element differs from that of adjacent cells by only a phase shift.” The theory forms a relationship between the array lattice geometry, or the arrangement of the unit cells, and the array’s farfield grating lobes, surface waves, and evanescent waves. Floquet’s theorem is based on the following array conditions [1]: • • •

•



The antenna aperture is composed of a periodic grid of identical elements. It is infinite in extent and planar (general case). Phase-wall boundary conditions exist at each unit cell. Phase walls refer to virtual unit cell edge boundary conditions that replicate the adjacent fields while introducing a phase shift associated with the wave incidence angle. The unit cell is both a physical and virtual building block representing a single radiating element. Controllable complex excitations also exist at each unit cell.

am,n = am,n e

(

− j mφx +nφy

)

(3.1)

Figure 3.1  An illustration of a planar array’s aperture geometry, including unit cells, phase centers, and phase walls.

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3.2  Floquet’s Theorem151

where the complex excitations (am,n) are indexed by the 2-D unit integers, [m, n] are a function of a peak magnitude (a 0,0), and a phase component. The latter being a periodic function of the aperture integers and linear phase constants [ϕ x, ϕ y] associated with the aperture’s x and y-axis unit cell phase center locations and dimensions (b, d) = (dx, dy).

φx = kbsin q cos f = kdx uo

φy = kd sin q sin f = kdy vo (3.2)



The aperture fields are solutions to the scalar wave equation, with (ξ ) representing either (Ez) or (Hz). The wave equation can be expressed as

(∇2 − k2 ) x ( x, y, z ) = 0 (3.3)



∇2 =



∂2 ∂2 ∂2 + + (3.4) ∂x2 ∂y2 ∂xz2

The radiated field is assumed to vary in the z-direction as a function of a complex propagation constant (Γ), x ( x, y, z ) = x ( x, y ) e− jΓz (3.5)



Because of the periodic aperture boundary conditions, the uniform voltage at each unit cell is the same other than a progressive phase slope,



(

)

x x + mdx , y+, ndy , z = x ( x, y, z ) e

(

− j mφx +nφy

)

(3.6)

So the wave equation can be expanded, using a transverse del operator,

(∇2t − k2 + Γ2 ) x ( x, y ) = 0

∇2t =

∂2 ∂2 + ∂x2 ∂y2

(3.7)

We can apply the technique of separation of variables to the expanded wave equation, ⎛ ∂2 2⎞ ⎜⎝ ∂x2 + kx ⎟⎠ f ( x ) = 0

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(3.8) ⎛ ∂2 2⎞ ⎜⎝ ∂y2 + ky ⎟⎠ g ( y ) = 0

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Lattice Theory

We expand the propagation constant into the wave continuity equation, Γ 2 = kz2 = k2 − kx2 − ky2 (3.9)



We can now structure the equations to take advantage of the periodic boundary conditions. First, the aperture field distribution is periodic with a linear phase shift,

( ) (3.10) f ( x + mdx ) = f ( x ) e − j( mφ ) f x + dx = f ( x ) e − j(φx )

x



Then it can be expressed as a periodic function, F ( x ) = f ( x ) e jx(φx / dx )

(

)

(

(3.11)

)

F x + dx = f x + dx e jx(φx / dx )e jφx = F ( x ) So F(x) can be expressed as a Fourier series,

(

)

F x + dx =

(

)

F x + dx =



∞

∑

n=−∞ ∞

an e jx(2pm / dx ) an e (

jx ( 2pm−φx ) / dx

∑

n=−∞

)

(3.12)

Each term also satisfies the wave equation, so kx = kx =

(2pm − φx ) dx

m

ky = ky =

(

2pn − φy dy

n



)

(3.13)

The propagation constants can then be expressed as Γ =

(

k2 − 2pm − φx dx

x ( x, y, z ) = e − jkm x e − jkn y e

) − (2pn − φx 2

dy

− jΓ m ,n z

)

2

(3.14)

Rearranging the propagation constants,



7060_Book.indb 152

(

Γ m,n = ±k 1 − 2pm − φx kdx

) − (2pn − φy 2

kdy

)

2

(3.15)

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3.2  Floquet’s Theorem153

which is in the same form as the waveguide propagation constant equation, Γ m,n = ± 1 − ( pm xo ) + ( pn yo ) − w 2 mε (3.16) 2



2

This indicates several observations, the foremost being that array periodicity produces a propagation constant similar to that of a waveguide with metal walls as boundary conditions. The principle can be extended further to include phase walls, which represent phase discontinuities similar to the waveguide walls. The latter is illustrated in Figure 3.2, along with the phase distribution of the unit cells in between the phase walls. The complex parts of the propagation constant reveal the type of the wave: 1. Modes propagating away from the aperture have only a positive real part: – Γ m,n > 0, real; – These are TEM waves, propagating towards the source or receiver. 2. Modes propagating towards the aperture have only a negative real part: – Γ m,n < 0, real; – These are also TEM waves, propagating away from the source or receiver. 3. Nonpropagating (evanescent) modes have only an imaginary part: – Γ m,n = ±j⎪Γ m,n⎪; – These are generalized as reactive fields, including cutoff conditions and creeping waves. 4. Surface wave modes have no z-axis propagation: – Γ m,n = 0; – These are also generalized as imaginary space grating lobes and associated surface waves.

Figure 3.2  An array aperture, unit cells, and its near-field phase condition.

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154

Lattice Theory

–

Since the z-axis propagation constant is zero, the x and y propagation terms must be finite because of the wave continuity equation, indicating that the propagation is confined to the aperture surface,

Γ 2 = k2 = kx2 + ky2 + kz2 (3.17)



Surface waves are an essential effect in phased array antennas, so we will study them in particular. They can cause: •

•

•

•

Scan blindness: The array gain drops precipitously and can defeat the system’s primary purpose at specific spatial locations. A radar system, for example, would be unable to detect intended targets in this case. Gain reduction: The gain is reduced to the extent that the system’s performance is compromised and weakened. It is important to consider that this effect generally cannot be offset by the microwave electronics or signal processing subsystems. Active impedance mismatch: The transmitted energy reflects back to its sources. This can cause damage and degradation to the array’s Tx components and has caused fires in the past. Wide scan effects: These include the size of the array, its edge boundary conditions, and the aperture surface materials and geometry.

3.2.1  Phased Array Surface Wave Condition

We can use Floquet’s equation to examine the properties of the grating lobe surface wave condition throughout sine space. This will offer a design basis from which we can develop the array lattice for the most common condition of a single grating lobe within real space. This will leverage previous studies that concluded that the main beam is the [m, n] = [0, 0] or zero-order beam, with an infinite number of grating lobes in imaginary space [⎪m⎪, ⎪n⎪] = [≠0, ≠0]. Using the sine-space notation, Floquet’s propagation solution can be simplified:



(

Γ m,n = ±k 1 − 2pm − φx kdx

) − (2pn − φy 2

kdy

)

2

(3.18)

uo = sin qo cos fo = Tx

vo = sin qo sin fo = Ty wo =



7060_Book.indb 154

k=

uo2 + vo2 = sin qo

(3.19)

2p l

(

Γ m,n = ±k 1 − ml dx − uo

) − ( nl dy − vo ) 2

2

(3.20)

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3.2  Floquet’s Theorem155

When a grating lobe moves from imaginary to real space, it enters real space at the boundary between the two, so its sine-space and angle-space positions are represented, respectively, by wo = sin2 qo = 1 qo = 90°



(3.21)

As this is also consistent with an observation angle aligned with the aperture’s surface, we treat it as a surface wave. So, to solve for the grating lobe emergence conditions, we command the propagation constant to the same location in angle and sine space. The equations map these conditions to the 2π multiples, thus generating the doubly infinite grating lobe distribution in sine space. Solving for the surface wave condition,

(

Γ m,n = ±k 1 − ml dx − uo



(

1 − ml dx − uo



) − ( nl dy − vo )

2

2

) − ( nl dy − vo )

2

2

( ml dx − uo ) − ( nl dy − uo ) 2



2

= 0 (3.22)

= 0 (3.23)

= 1 (3.24)

An example illustrates a simple case of grating lobe onset. Example 3.1

Consider the case where the lattice spacing is set to the Nyquist rate of 0.5 wavelength for both x and y-axes, so [dx, dy] = [0.5λ , 0.5λ ], indicative of a square lattice. We will locate the main beam at boresight [uo, vo]. Starting with the grating lobe solution, l l = = 2.0 dx dy

( ml dx − uo ) − ( nl dy − uo ) (2m − uo )2 + (2n − vo )2 = 1 2



2

= 1 (3.25)

The above equation represents a doubly infinite series of [m, n] circles with unit radius. These occupy the real and imaginary regions of sine space with centers at [m, n]m=0,±1,±2,…n=0,±1,±2,…. The real-space boundary defines the possible scan volume for this Nyquist spaced array, and as given by the equation (Figures 3.3 and 3.4):



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( uo )2 + (vo )2 = 1 (3.26)

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156

Lattice Theory

This is also the equation of the unit circle in sine space, as shown below. We repeat this for all grating lobe circles, as shown below. In this case, the array is scanned to [uo, vo] = [0.707, −0.707], where (wo = 1).

Figure 3.3  The grating lobe unit circle at the boundary of real and imaginary space.

Figure 3.4  The Floquet series grating lobe solution for a planar array with [dx, dy] = [0.5λ , 0.5λ ], scanned to [uo, vo] = [0.707, −0.707].

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3.2  Floquet’s Theorem157

One way to interpret this grating lobe solution is that when the main beam is scanned to any of the main beam sine-space locations indicated, grating lobes will appear at the opposite end of real space. This is readily verified. For example, when the main beam is scanned to the edge of real space at [uo, vo] = [1.0, 0.0], grating lobes appear at [ug, vg] = [−1.0, 0.0]. The element incremental phase needed to produce the example main beam is



⎛ 2p ⎞ ⎛ 2p ⎞ a = kdx uo = ⎜ ⎟ dx (1.0) = ⎜ ⎟ l 2 = p (3.27) ⎝ l⎠ ⎝ l⎠

An element incremental phase of π , or 180°, is indistinguishable from − π , or −180°. This element incremental phase produces a replica of the main beam at the opposite edge of real space,



ug = −1.0 =

kdx ⎛ 2p ⎞ ⎛ l ⎞ = −1.0 (3.28) =⎜ ⎝ −p ⎟⎠ ⎜⎝ −2 ⎟⎠ a

The integral equation far-field radiation pattern for the above example is illustrated in Table 3.1 and Figure 3.5, with the grating lobe located at ug = −1.0 and the main beam scanned to [uo, vo] = [1.0, 0.0]. Having solved the grating lobe equation for the main beam, we can see from (3.24) that there is a 2-D infinite series of these grating lobes, because the indices [m, n] can extend over all integers on the real number plane, although, in this case, we have only solved Floquet’s equation for a specific main beam scan case (i.e., where [uo, vo] = [−1.0, 0.0]). The Floquet terms produced indicate a double infinite series of solutions for all scan angles within the coverage volume. Each is represented as a double infinite series of points in sine space, and the scan edge condition is as illustrated partially above. In this case, the main beam is scanned to a different location on the grating lobe circle [uo, vo] = [0.707, −0.707], although the results

Table 3.1  Array Conditions Showing a Grating Lobe Condition

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5λ

Y-axis element separation (dy)

0.5λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

1.0000

Main beam scan location (vo)

0.0000

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Lattice Theory

Figure 3.5  The near-field and far-field functions for a scanned 40 by 40-element planar array, with a square lattice scanned to [uo, vo] = [−1.0, 0.0].

are consistent and all grating lobes displace by the same amount in sine space as the main beam or [0, 0] zeroth-order Floquet propagating mode. An important note about the grating lobe analysis thus considered involves the embedded element and the array, which is treated as ideal and infinite. In subsequent sections, we consider element and array conditions that extend the surface wave conditions. Electrically small array size is an example that does this. The Floquet mode principles indicated thus far remain unchanged, although additional modifying factors may enter the picture.

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3.2  Floquet’s Theorem159

3.2.2  Phased Array Scan Volume

Having generalized Floquet’s theorem to the extent where EM field radiation modes can be characterized in terms of a Floquet series, it is important to study the role introduced by the array geometry. One way to start this process is to consider the straightforward case of a linear array. As it can only scan along its array x-axis, the determination of a maximum scan angle is uncomplicated and can be generalized to the planar array case. So, by determining the maximum linear array scan angle, we arrive at the specifics of a phased array’s scan volume. Starting with Floquet’s solution for the surface wave condition (5.1-23), we confine the maximum scan angle to that supported by the electrical element spacing with no grating lobes within real space other than the zero-order Floquet mode,

( ml dx − uo ) + ( nl dxy − vo )

2

2



= 1 (3.29)

Because the element spacing in the x-axis is (dx), the element spacing in the orthogonal aperture plane is infinite (dy = ∞), and the array can only scan in the array axis (vo = 0), we again solve for the grating lobe condition using Floquet theory,

( ml dx − uo ) = 1 (3.30)



We are interested in the grating lobe closest to the main beam, and as only single axis scanning is involved, we solve for the first-order Floquet mode m = 1, l dx − uo = 1 l dx = 1 + uo (3.31) dx = l 1 + uo



The array can only scan in one plane, that is, (vo = 0), vo = sin qo sin fo = 0 fo = 0

uo = sin qo cos fo = sin qo



(3.32)

So, applying (3.31),



l = 1 + sin qo (3.33) dx The surface wave occurs when the main beam reaches its maximum scan angle, l = 1 + sin qmax dx dx =



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(3.34) l 1 + sin qmax

(

)

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160

Lattice Theory

This expression is quite useful but does not include surface wave factors often found in the radiating aperture nor small array effects. Both factors are deterministic, but the former generally requires a full-wave analysis and is dependent on the aperture design conditions. Because of this, a general array surface wave factor can be applied, representing a 100-millisine allowance for surface wave factors and assuming an electrically large aperture size. A conservative expression for element spacing that includes array surface wave factors is l = 1.1 + sin qmax dx

l dx = 1.1 + sin qmax

(



(3.35)

)

There are two independent aperture axes available for lattice definition in a planar phased array, represented by the element spacing constants (dx, dy). The maximum scan angle in the far-field sine-space planes associated with the two lattice dimensions yields maximum scan angles independently. So the maximum scan angle in u and v-planes can differ, producing a rectangular lattice, and square is they are the same. The maximum scan angle in the diagonal plane, where (u = v) is the Pythagorean sum of the u and v maximum scan angles from (3.19), wo =



uo2 + vo2 = sin qo (3.36)

For example, a rectangular lattice with maximum scan angles of: ⎡⎣ qmax ( f = 0° ) , qmax ( f = 90° ) ⎤⎦ = [30°,45° ] or

⎡⎣ qmax ( f = 0° ) , qmax ( f = 90° ) ⎤⎦ = ⎡⎣ umax ,vmax ⎤⎦ = [0.5,0.707 ]

(3.37)

yields a maximum scan angle of wmax ( f = 45° ) =

2 2 umax + vmax =

sin qmax ( f = 45° ) = 60°

0.52 + 0.7072 = 0.8660

(3.38)

The diagonal plane effect of a rectangular lattice is evident above, where the sine-space distance from the main beam to the nearest grating lobe is greater than on the principal (u, v) planes. Other lattice types, such as triangular, provide an improved use of the lattice geometry and the scan volume. In this same light, square lattice Nyquist sampling has been successfully used to illustrate far-field behavior in several cases because it guarantees only a single zero-order propagating Floquet mode in real space. Because of both an expanded diagonal plane scan limit and an unspecified scan volume, it is not always optimal from a unit cell area perspective. For example, at Nyquist spacing, a square lattice

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3.2  Floquet’s Theorem161

has dimensions of [dx, dy] = [0.5λ , 0.5λ ]. The lattice produces maximum scan angles extending to the real-space boundary, l = 1 + umax dx

umax = l dx − 1 = l 0.5l − 1 = 1.0 qmax = sin−1 1.0 = 90° umax = vmax



(3.39)

(3.40)

In the diagonal plane, the maximum scan angle is beyond the edge of real space, indicating that an improved lattice arrangement is possible.

wmax ( f = 45° ) =

2 2 umax + vmax =

1.02 + 1.02 = 1.414 (3.41)

The implications of lattice geometry and spacing are significant in terms of array manufacturability. Because of the importance of packaging microwave components, beamforming hardware, DC power circuits, thermal management, array control electronics, and array structure within the unit cell space, the lattice spacing needs to be as large as possible. As contemporary phased arrays use an operating frequency range that includes one or more information bandwidth segments, it is important to determine the lattice spacing sufficient to cover the operating frequency band. This is accomplished by sizing the maximum scan angle and the lattice at the highest operating frequency, because lower frequencies produce a larger scan angle for the same lattice size. Example 3.2

Consider a square array where the maximum scan angle is specified as 60° in all (ϕ ) planes, or within a conical scan volume. Determine the maximum square lattice unit cell dimensions for an operating frequency range of 12.0 to 12.5 GHz without additional array surface wave factors. Determine the maximum scan angle in the diagonal plane of the far field. Using (3.34) at the highest operating frequency, as it is the most stressing case, dx =

0.9442 l = (1 + 0.8660) 1 + sin qmax

(

)

= 0.5060 inches (12.85 mm)

(3.42)

As the scan volume is conical the maximum scan angle is the same in both u and v-planes, consistent with a square lattice, so

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dx = dy (3.43)

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Lattice Theory

In the diagonal far-field plane, uo = vo = 0.8660 wo =

uo2 + vo2 = sin qo (3.44)

= 1.2247



This shows that the array can scan to the edge of real space in the diagonal planes, meeting the maximum scan condition with margin.

3.3

Lattice Theory Floquet’s theorem indicates that the element separation in wavelengths is inversely proportional to the grating lobe separation in sine space. Equation (3.29) shows this, as the sine-space separation is set by ⎡ ml nl ⎤ ,   ⎥ (3.45) ⎢ ⎢⎣ dx dy ⎥⎦



Although this relationship holds for multiple types of periodic planar apertures its scalar properties contain no vector information, opening an opportunity to expand the relationship. By extension, Floquet’s theorem can predict the behavior between the aperture lattice geometry and the inverse relationship to the geometrical arrangement of grating lobes and surface waves in free space. This observation can be made specific by writing a vector relationship between the lattice and scan spaces. This astute observation comes from Brookner [2]. It relates the aperture lattice lines as vectors in two dimensions [dx, dy] = [d1, d2] to vectors that connect the grating lobes in sine space [b1, b2], the unit magnitude aperture surface normal (zo), and the unit cell area (Au). The lattice vectors connect the phase centers of adjacent unit cells in the aperture plane, while the grating lobe vectors connect the grating lobe centers of adjacent Floquet mode grating lobes, with conversion from Cartesian aperture to sine-space coordinates. The following equation applies to all ordered lattice types and shows the inverse vector relationship between the lattice and grating lobe vectors b1 = l Au ( d2 × zo )

b2 = l Au ( d1 × zo )

(3.46)

The coordinate conversion is



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⎡ uˆ ⎤ ⎡ xˆ ⎤ ⎢⎣ vˆ ⎥⎦ = ⎢ yˆ ⎥ (3.47) ⎣ ⎦

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3.3  Lattice Theory163

These equations indicate that the grating lobe vectors in sine space are orthogonal to the lattice vectors. Both aperture and sine-space vectors occupy separate 2-D spaces, [x, y] and [u, v], respectively. The equations are valid for all ordered periodic lattices, as will be more evident in several examples that follow. However, several observations relate the lattice vector equations to already observed behavior of phased array grating lobes. We can use a general notation for all ordered types of periodic lattices where the lattice vectors use the aperture Cartesian coordinates, ˆ x2 xˆ + dy2 yˆ ⎤⎦ (3.48) ⎡⎣ d1 , d2 ⎤⎦ = ⎡⎣dx1xˆ + dy1y,d



Using the general equation for the cross-product of two general vectors [a, b], ˆ and the scalar angle between the two vectors (α ), their normal unit vector (n), a × b =  nˆ a b sina xˆ yˆ zˆ a × b = ax ay az

(3.49)

bx by bz



The unit cell area scalar can be derived from the magnitudes of the aperture lattice line vectors, Au = d1d2 sina (3.50)



The grating lobe vector magnitudes are b1 = l Au ( d2 × zo ) = b2 = l Au ( d1 × zo ) =



ld2 l = (d1d2 sina ) (d1 sina )

(3.51) ld1 l = (d1d2 sina ) (d2 sina )

The maximum scan angle can be solved in both sine space and angle space,

wo =

uo2 + vo2 =

sin2 qo cos2 fo + sin2 qo sin2 fo2 = sin qo (3.52)

The Floquet grating lobe scalar locations (wp,q⎪ p=0,±1,±2,…q=0,±1,±2,…) can be related to the scan angle (wo) and the lattice dimensions by the Floquet integer indices (p, q), considering that there are a doubly infinite set of grating lobes for a rectangular lattice. The grating lobe vector magnitudes are w1 ( p ) = pl ( d1 sina ) + wo

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w2 ( q ) = ql ( d2 sina ) + wo (3.53)

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Lattice Theory

When the absolute value of the above equation reaches a value of 1.0, a grating lobe is at the transition to real space. For first-order grating lobes (p = ±1, q = ±1) and (α = 90°) with a rectangular lattice, this occurs when w1 ( p ) = pl ( d1 sina ) + wo = 1 p=1 l = 1 + wo (d1 sina )

l d1 = 1 + sin qmax

(



(3.54)

)

This shows the consistency between the vector form of the lattice equation and the scalar Floquet scan limit determined in (3.34). As a result, the lattice equation in vector form allows us to specify the lattice dimensions needed for a needed scan volume (i.e., in multiple lattice directions).

3.3.1  Rectangular Lattice

A rectangular lattice allows the phased array designer the latitude to define a scan volume in two orthogonal sine-space dimensions (u, v). The next example uses a square lattice and the above vector notation to define such a scan volume, so the scan volume is the same in the principal planes. Example 3.3

An example square lattice shows the practical utility of the Floquet geometrical theory. Unit cell lattice dimensions are equal for a square lattice and set to the ˆ ˆ Nyquist spatial limit of [dx, dy] = [ x0.5l, (Figure 3.6). The inner angle of y0.5l] the lattice is (α = 90°), below, and in this example let the maximum scan angle be 60° in a conical scan volume. Selecting the lattice variables,

d1 = xˆ 0.5l d 2 = yˆ 0.5l

(3.55)

z o = zˆ

Au = d1d2 sina = 0.25l2 Two additional lattice vectors can also be drawn for separate calculations, ˆ d1 = − x0.5l



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ˆ d2 = − y0.5l

(3.56)

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3.3  Lattice Theory165

Figure 3.6  An illustration of a square lattice phased array in its aperture plane and in Cartesian coordinates.

The relevant cross-products are defined as xˆ yˆ zˆ xˆ × zˆ = 1 0 0 = − yˆ 0 0 1 xˆ yˆ zˆ yˆ × zˆ = 0 1 0 = xˆ 0 0 1



(3.57)

The grating lobe vectors can now be calculated:

( ) (3.58) ˆ Au ( d1 × zo ) = xˆ × zˆ ( l0.5l 0.25l2 ) = − y2.0

ˆ b1 = l Au ( d2 × zo ) = yˆ × zˆ l0.5l 0.25l2 = x2.0

b2 = l

First-order grating lobe vectors for the remaining lattice vectors are

( ) ˆ Au ( d1 × zo ) = − xˆ × zˆ ( l0.5l 0.25l2 ) = y2.0

ˆ b1 = l Au ( d2 × zo ) = − yˆ × zˆ l0.5l 0.25l2 = − x2.0

b2 = l

(3.59)

Since the maximum scan angle is 60° in a conical scan volume, it forms a circle in sine space. This and the four nearest grating lobe vectors are as shown below.

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Lattice Theory

These repeat infinitely in the two orthogonal planes in sine space. Converting from Cartesian coordinates to sine space, ⎡ uˆ ⎤ ⎡ xˆ ⎤ ⎢⎣ vˆ ⎥⎦ = ⎢ yˆ ⎥ ⎣ ⎦



⎡ b1 ( p ) ⎤ ⎡ ±uˆ 2.0p ⎤ ⎢ ⎥= ⎢ ⎥ ⎢⎣ b2 ( q ) ⎥⎦ ⎣ ±vˆ 2.0q ⎦

(3.60) p=0,1,2,…, q=0,1,2,…

The first-order grating lobes other than the real-space main beam are those closest to real space, so they are of the most interest. These are



⎡ b1 ( p ) ⎤ ⎡ ±uˆ 2.0p ⎤ ⎢ ⎥= ⎢ ⎥ ⎢⎣ b2 ( q ) ⎥⎦ ⎣ ±vˆ 2.0q ⎦

(3.61) p=1, q=1

When the zeroth Floquet mode (main beam) is scanned to a maximum at a location of [u (0,0),  v(0,0)] = [−0.866,  0.000], the nearest grating lobe separation [u(1,0), v(1,0)] = [2.000, 0.000] is displaced accordingly: w(1,0) ( p ) = pl d2 sina + w(0,0)

w(1,0) ( p ) = [ 2.000] + [ −0.866] (3.62)

w(1,0) ( p ) = [1.134 ]

The subject grating lobe lies outside of real space, as shown below. This same procedure can be repeated for the remaining sine-space data points on the main beam circle, producing the grating lobe circles shown (Figure 3.7). Each grating lobe circle is the same scan volume produced by the main beam, but it is displaced by the main beam scan volume and the grating lobe location, each point of which is represented as wo = sin qo (3.63)



The conical scan volume described in the above example is limited by a maximum scan angle of (θ max = 60°) in a spherical coordinate system. It includes all scan angles between the maximum and boresight, including all rotational angles (0° ≤ ϕ ≤ 360°). When projected in sine space, the conical maximum scan boundary defines a circle, equivalent to the projection of the cone defined by the maximum scan angle on to the x-y plane, forming a circle defined by



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wo =

uo2 + vo2 = sin qmax (3.64)

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3.3  Lattice Theory167

Figure 3.7  A sine-space map of first-order grating lobes and the main beam for an example square lattice array with a unit cell spacing set to 0.5λ and the maximum scan angle 60°.

Because the scan angle translates all radiation in sine space by the constant [uo, vo], the grating lobes translate equally. So the projected circle of the maximum main beam scan angles produces identical circles around each of the grating lobes as shown in Figure 3.7. When the main beam scans to its extreme on the u-axis, all grating lobes do the same, making the opposite grating lobe at [p, q] = [1, 0], the nearest to the real space boundary. In general, many designs prevent grating lobes from entering real space for several reasons: •

•

•

When more than one Floquet mode (beam) exists in real space, the transmitted power divides between both real space beams, the zero-order and first-order, although other index combinations are equally possible. When a Floquet mode transitions the real-space boundary, it may produce a significant change in the active impedance of the radiators. If the aperture is impedance-matched to a single fundamental (zeroth order) Floquet beam, generally, it will mismatch during the real-space transition. The real-space transition impedance mismatch can be quite severe, with the active reflection coefficient approaching unity or total reflection. Multiple real-space grating lobes cause spatial ambiguities in receive mode.

Given the design needs for a single zero-order Floquet mode in real space (the main beam), we can use lattice theory to design accordingly and then verify the far-field results. We do this by modifying the above example to place the first-order

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168

Lattice Theory

grating lobes at or outside of the real-space transition. In this case, the maximum conical scan angle is (θ max = 60°) and the maximum operating frequency is 2.0 GHz. Using (3.34) for an ideal array with no additional surface wave allowance, the largest square lattice spacing supporting the required scan volume is dx = dy =

l l = = 0.536l = 3.1626 in. 1.866 (3.65) 1 + sin qmax

(

)

Au = d1d2 sina = 0.2873l2 (3.66)



The grating lobe vectors can now be calculated:

( ) (3.67) ˆ Au ( d1 × zo ) = xˆ × zˆ ( l0.536l 0.2873l2 ) = − y1.866

ˆ b1 = l Au ( d2 × zo ) = yˆ × zˆ l0.536l 0.2873l2 = x1.866

b2 = l

The first-order grating lobes are



⎡ b ( p ) ⎤ ⎡ ±u1.866 ˆ p ⎤ ⎢ 1 ⎥=⎢ ⎥ ˆ ⎢⎣ b2 ( q ) ⎥⎦ ⎣ ± v1.866q ⎦

(3.68) p=1,q=1

When the zeroth Floquet mode is scanned to a maximum at a location of [u(0,0), v(0,0)] = [−0.866, 0.000], the nearest grating lobe [u(1,0), v(1,0)] = [1.866, 0.000] is displaced by linear arithmetic in sine space, w(1,0) ( p ) = pl ( d2 sin α ) + w(0,0)

w(1,0) ( p ) = [1.866] + [ −0.866] (3.69)

w(1,0) ( p ) = [1.000]

The sine-space map now shows a grating lobe at the edge of real space when the main beam is scanned to its maximum (Figure 3.8). The lattice spacing positions the first-order grating lobes precisely at the real-space boundary. Because this array uses a larger unit cell spacing than the previous example, the element count is reduced by 14.9%. The results of such a revision are substantial, as the reduction applies to all parts of the array, often including radiating elements, power system, cooling, DC and logic distribution, array structure, weight, and cost. These results are confirmed with a far-field integral calculation. Table 3.2 gives the array parameters for this test case. It uses a cosine element power function, representing ideal conditions and an element count designed to yield large array far-field characteristics. The square lattice element separation relies on the above

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3.3  Lattice Theory169

Figure 3.8  A sine-space map first-order grating lobes and the main beam for a square lattice array with a unit cell spacing set to 0.536λ and the maximum scan angle 60°.

calculations and the array scans to the same [u(0,0), v(0,0)] = [−0.866, 0.000] scan coordinates as above, with far-field results as shown in Figure 3.9. The integral equation far-field results show the following characteristics: •

A main beam scanned to [u(0,0), v(0,0)] = [−0.866, 0.000] produces a grating lobe at [u(1,0), v(1,0)] = [1.000, 0.000], or at the edge of real space, consistent with lattice theory. The grating lobe’s radiated field near the edge of real space reaches a peak of approximately −6 dB and is all that remains of the grating because of an element factor null occurring near the same location. Even Table 3.2  Array Conditions for a Rectangular Lattice with No Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.536λ

Y-axis element separation (dy)

0.536λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.8660

Main beam scan location (vo)

0.0000

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Lattice Theory

Figure 3.9  The near-field and far-field functions for a scanned 40 by 40-element planar array, with a square lattice scanned to [u(0,0), v(0,0)] = [−0.866, 0.000].

•

•

•

7060_Book.indb 170

so, the effect of the surface wave impedance remains and can cause serious array problems. The radiation of the grating lobe at the grating lobe boundary is canceled by an element pattern null, because cos q = 0⎪ θ =90°. The element pattern also reduces the main beam magnitude by cos q = 0.5⎪ θ =60°. Due to the uniform aperture illumination used, sinc function sidelobes appear on both sides of the main beam and the real-space side of the grating lobe.

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3.3  Lattice Theory171

•

Those proceeding from the main beam toward the real-space boundary are significantly reduced by the element pattern function. The sectional far-field pattern in the orthogonal v-plane is unscanned and so retains the characteristics of a sinc function, other than the sine-space truncation associated with the scanned beam in the orthogonal u-plane.

Example 3.4

An example rectangular lattice further illustrates the practical utility of the Floquet geometrical theory. In this case, the lattice is a design task, and the elliptical scan volume is limited by [umax, vmax] = [0.7071, 0.500], so θ max = 45° at ϕ = 0° and θ max = 30° at ϕ = 90°. The inner angle of the planar array lattice is (α = 90°), and the highest operating frequency is 10.0 GHz. The lattice solution proceeds along the line indicated above, but now with separate calculations for two independent aperture planes. Using (3.34) for an ideal array with no additional surface wave allowance, the largest square lattice spacing supporting the required scan volume in the x-plane is dx = d2 =

l l = 1.7071 1 + sin qmax

(

)

= 0.5858l = 0.6914 in. (17.56 mm )



(3.70)

It is important to align the lattice with the scan volume. Because the maximum scan value in sine space and in the u-direction is 0.7071, it is aligned with the x-axis because, from (3.19), uo = sin qo cos fo (3.71)



Since (ϕ o = 0) satisfies this equation and ϕ is defined relative to the x-axis, the associated scan angle aligns with the x-axis of the aperture. Repeating the process for the y-axis, dy = d1 =

l l = 1.500 1 + sin qmax

(

)

= 0.6667 l = 0.7869 in.  (19.99 mm )

(3.72)

The unit cell area is

Au = d1d2 sina = 0.3905l2 (3.73) The grating lobe vectors can now be calculated,

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Lattice Theory

b1 = λ Au (d 2 × z o ) = yˆ × zˆ ( l0.5858l 0.3905l2 ) = xˆ 1.7071

b2 = λ Au (d1 × z o ) = xˆ × z ( l0.6667 l 0.3905l2 ) = − yˆ 1.5000

(3.74)

The choice of the aperture lattice vectors [d1, d2] is important. The results are readily checked by confirming that the magnitude of the grating lobe vectors [b1, b2],



b1 =

l = 1.7071 d1

b2 =

l = 1.5000 d2

(3.75)

The first-order grating lobes are at the following vector locations relative to a boresight main beam,



⎡ b1 ( p ) ⎤ ⎡ ±uˆ1.7071p ⎤ ⎢ ⎥= ⎢ ⎥ ⎢⎣ b2 ( q ) ⎥⎦ ⎣ ±vˆ 1.5000q ⎦

(3.76) p=1,q=1

We can test the scan volume in two orthogonal planes. When the zeroth Floquet mode is scanned to a maximum at a location of [u(0,0), v(0,0)] = [−0.7071, 0.0000], the nearest grating lobe separation in the scan plane [u(1,0), v(1,0)] = [1.7071, 0.000], is displaced by linear arithmetic in sine space, again in the scan plane, w(1,0) ( p ) = pl ( d2 sin α ) + w(0,0)

w(1,0) ( p ) = [1.7071] + [ −0.7071] (3.77)

w(1,0) ( p ) = [1.000]

Repeating the test for the v-axis in sine space, when the zeroth Floquet mode is scanned to a maximum at a location of [u(0,0), v(0,0)] = [0.0000, −0.5000], the nearest grating lobe separation in the scan plane [u(0,1), v(0,1)] = [0.0000, 1.5000], is displaced by linear arithmetic in sine space, again in the scan plane, w(0,1) ( q ) = ql ( d1 sina ) + w(0,0)

w(0,1) ( q ) = [1.5000] + [ −0.5000] (3.78)

w(0,1) ( q ) = [1.000]

The sine-space map now shows a grating lobe at the edge of real space when the main beam is scanned to its maximum (Figure 3.10). The lattice spacing positions the first-order grating lobes precisely at the real-space boundary. The elliptical scan volume is bound by maximum scan coordinates in the principal sine-space planes, as shown.

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3.3  Lattice Theory173

Figure 3.10  A sine-space map first-order grating lobes and the main beam for a rectangular lattice array with an elliptical scan volume, a unit cell spacing set to [dx, dy] = [ xˆ 0.5858λ ,  yˆ 0.6667λ ], and the maximum scan angle 45°.

We can verify these results at two critical scan locations on the u and v-axes in sine space, respectively. In the first test, the main beam scans to [u(0,0), v(0,0)] = [−0.7071, 0.0000], its extreme along the u-axis (Table 3.3). This tests whether the nearest grating lobe originating at [u(1,0), v(1,0)] = [1.7071, 0.0000], emerges into real space at coordinates [u(1,0), v(1,0)] = [1.0000, 0.0000]. The far-field integral solution Table 3.3  Array Conditions for a Rectangular Lattice with No Real-Space Grating Lobes

Table 3.4  Array Conditions for a Rectangular Lattice with No Real-Space Grating Lobes

Parameter

Value

Parameter

Value

Element polarization

x-vector

Element polarization

x-vector

Element power function

cosine (θ )

Element power function

cosine (θ )

X-axis element count (N)

40

X-axis element count (N)

40

Y-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5858λ

X-axis element separation (dx)

0.5858λ

Y-axis element separation (dy)

0.6667λ

Y-axis element separation (dy)

0.6667λ

X-axis amplitude distribution

Uniform

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.7071

Main beam scan location (uo)

0.0000

0.0000

Main beam scan location (vo)

−0.5000

Main beam scan location (vo)

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Lattice Theory

confirms this as shown in Figure 3.11, with the grating lobe suppressed by the element factor null at the same location. In the second test, the main beam scans to [u(0,0), v(0,0)] = [0.0000, −0.5000], its extreme along the v-axis (Table 3.4). This tests whether the nearest grating lobe originating at [u(0,1), v(0,1)] = [0.0000, 0.5000] emerges into real space at coordinates [u(0,1), v(0,1)] = [0.0000, 1.0000]. The far-field integral solution again confirms this as shown in Figure 3.12, with the grating lobe suppressed by the element factor null at the same location.

Figure 3.11  The near-field and far-field functions for a scanned 40 by 40-element planar array, with a square lattice scanned to [u (0,0), v (0,0)] = [−0.7071, 0.0000].

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3.3  Lattice Theory175

Figure 3.12  The near-field and far-field functions for a scanned 40 by 40-element planar array, with a square lattice scanned to [u(0,0), v(0,0)] = [0.0000, −0.5000].

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Next, it is useful to consider the sine-space off-cardinal plane characteristics of a scanned array. Using the same aperture lattice as in Example 3.4, consider a rectangular scan volume. The maximum scan angles in the u and v-planes remain the same, as do the rectangular lattice dimensions. The difference lies in the fact that the rectangular scan volume includes the case where both u and v-plane maximum scan angles occur simultaneously, producing an off-cardinal maximum scan angle of



⎡ u(0,0) ,v(0,0) ⎤ = [ −0.7071, −0.5000] (3.79) ⎣ ⎦ w(0,0) =



2 2 u(0,0) + u(0,0) = 0.8659

q(0,0) = sin−1 w(0,0) = 59.99°

(3.80)

This indicates that the off-cardinal plane scan angle exceeds that of the cardinal planes, as illustrated in Figure 3.13. Moreover, the off-cardinal scan angle introduces two grating lobes nearest to real space, neither of which enters real space. The u and v-plane grating locations are the result of expanding (3.53) to more general [u(p,q), v(p,q)] coordinates,

Figure 3.13  A sine-space map first-order grating lobes and the main beam for a rectangular lattice array with a rectangular scan volume, a unit cell spacing set to [dx, dy] = [ xˆ 0.5858λ ,  yˆ 0.6667λ ], and the maximum scan angle 60°.

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3.3  Lattice Theory177

⎡ u(1,0) ,v(1,0) ⎤ = pl ( d2 sin α ) + ⎡ u(0,0) ,v(0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(1,0) ,v(1,0) ⎤ = [1.7071,0.0000] + [ −0.7071, −0.5000] (3.81) ⎣ ⎦

⎡ u(1,0) ,v(1,0) ⎤ = [1.0000, −0.5000] ⎣ ⎦ The test for the v-axis in sine space is simultaneous in this case, ⎡ u(1,0) ,v(1,0) ⎤ = ql ( d1 sin α ) + ⎡ u(0,0) ,v(0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(1,0) ,v(1,0) ⎤ = [0.0000,1.5000] + [ −0.7071, −0.5000] (3.82) ⎣ ⎦



⎡ u(1,0) ,v(1,0) ⎤ = [ −0.7071,1.0000] ⎣ ⎦

We can test the scan volume in two orthogonal planes simultaneously (Table 3.5). When the zeroth Floquet mode is scanned to a maximum at a location of [u(0,0), v(0,0)] = [−0.7071, −0.5000], the nearest grating lobes in both u and v-planes remain outside of real space, as shown in Figure 3.14, substantiating the lattice theory. 3.3.2  Equilateral Triangular Lattice

The Floquet lattice vector equation also applies to lattice geometries other than rectangular, although the level of complexity increases. A rectangular lattice is a straightforward extension from the square lattice, where the [d1, d2] vectors have unequal magnitudes and an orthogonal orientation, but otherwise retain the same vector relationships as the square lattice. A triangular lattice differs from the square in several ways, including in the geometry of both the aperture and sine-space maps. For simplicity, we will consider

Table 3.5  Array Conditions for a Rectangular Lattice with No Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5858λ

Y-axis element separation (dy)

0.6667λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.7071

Main beam scan location (vo)

−0.5000

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Figure 3.14  The near-field and far-field functions for a scanned 40 by 40-element planar array, with a square lattice scanned to [u(0,0), v(0,0)] = [−0.7071, −0.5000].

the equiangular triangle with dimensions similar to the square lattice, although the principle applies equally for other triangles, including isosceles. For the equiangular triangle, the inner angles are equal and 60°, and all legs of the triangle are equal, so

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3.3  Lattice Theory179



d1 = d2 (3.83)

There are both stable triangular lattices, having a y-axis base, and unstable, with an x-axis base. Although these can be readily visualized by rotating the planar aperture by 90° about its surface normal, these two geometries place grating lobes on the u or v-axes, respectively, in sine space. This can have important effects in terms of system coverage and the mechanical arrangement of a practical phased array system. The phase centers are arranged in a triangle, in this case, an unstable triangle because the base is aligned with the y-axis, as shown in Figure 3.15. There are three nonorthogonal lattice vectors for Nyquist element spacing. The lattice vectors as well as the unit cell area can be written from aperture plane geometry as ⎛ ⎛ 1 ⎞⎞ d1 = 0.5l ⎜ xˆ + yˆ ⎜ ⎝ 3 ⎟⎠ ⎟⎠ ⎝ ⎛ ⎛ 2 ⎞⎞ d2 = 0.5l ⎜ yˆ ⎜ ⎝ ⎝ 3 ⎟⎠ ⎟⎠

(3.84)

⎛ ⎛ 1 ⎞⎞ d3 = 0.5l ⎜ xˆ − yˆ ⎜ ⎝ 3 ⎟⎠ ⎟⎠ ⎝

Figure 3.15  An unstable equiangular triangular phased array lattice in its aperture plane and Cartesian coordinates.

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Lattice Theory

Figure 3.16  A stable equiangular triangular phased array lattice in its aperture plane and Cartesian coordinates.

The lattice vectors originate at the phase center of each element. zo = zˆ Au = d1d2 sina

1⎞ ⎛ l 2 ⎞ 3 l2 2 2 3 l2 ⎛l 1+ ⎟⎜ = = = 0.2887 l2 Au = ⎜ ⎟ 4 3 3 2 3⎠ ⎝ 2 3⎠ 2 2 3 ⎝2

(3.85)

A stable triangular lattice is just as useful, as illustrated in Figure 3.16. It has essentially the same as the unstable geometry, but rotated 90°; however, this configuration also rotates the grating lobe vectors in sine space. Depending on the phased array application, one of these may be more useful. For example, an unstable triangular lattice locates grating lobes on the u-axis, which may be a disadvantage for a terrestrial-based system that operates near the horizon. Both unstable and stable triangular latices use both the [d1, d2] lattice convention and the [dx, dy] notation. The latter indicates whether an unstable or stable lattice applies, as generally dx > dy indicates an unstable triangular lattice, and dx < dy indicates a stable version. Example 3.5

We can proceed to increase the element spacing above the Nyquist limit for a limited, but practical, scan volume. As a result, the above case is revised to place the first-order grating lobes at the real-space transition. Using (3.34) for an ideal array

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3.3  Lattice Theory181

with a 60° conical scan volume, the largest square lattice spacing supporting the required scan volume is dx =

l l = = 0.5359l (3.86) (1.866) 1 + sin qmax

(

)

The three nonorthogonal lattice vectors and the unit cell area can be written from the expanded plane geometry as ⎛ ⎛ 1 ⎞⎞ d1 = 0.5359l ⎜ xˆ + yˆ ⎜ ⎝ 3 ⎟⎠ ⎟⎠ ⎝ ⎛ ⎛ 2 ⎞⎞ d2 = 0.5359l ⎜ yˆ ⎜ ⎝ ⎝ 3 ⎟⎠ ⎟⎠

(3.87)

⎛ ⎛ 1 ⎞⎞ d3 = 0.5359l ⎜ xˆ − yˆ ⎜ ⎝ 3 ⎟⎠ ⎟⎠ ⎝



Accordingly, the unit cell area is Au = d1d2 sina 2 ⎞ 3 1⎞ ⎛ 2 2 3 ⎛ Au = ⎜ 0.5359l 1 + ⎟ ⎜ 0.5359l = 0.2872l2 ⎟ 3⎠ ⎝ 3⎠ 2 3 3 2 (3.88) ⎝

=

0.5744l2 = 0.3316l2 3

The [dx, dy] parameters are as follows, and for an unstable (also referred to as unbalanced) equilateral triangle have the relationship dy ⎛ 1 ⎞ = dx ⎜⎝ 3 ⎟⎠ Equilateral dy =

d2 ⎛ 2 ⎞ = 0.3094l = 0.2680l ⎜ ⎝ 3 ⎟⎠ 2

⎛ ⎛ 1⎞ ⎞ 3 dx = d1 sin60° = 0.5359l ⎜ 1 + ⎜ ⎟ ⎟ ⎝ 3⎠ ⎠ 2 ⎝

(3.89)

⎛ 2 ⎞ ⎛ 3⎞ = 0.5359l = 0.5359l ⎜ ⎝ 3 ⎟⎠ ⎜⎝ 2 ⎟⎠

The grating lobe vectors are as indicated below and as illustrated in Figure 3.17. Note that, when scanned, the grating lobes now move to the real-space edge.

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Lattice Theory

⎛ ⎛ ⎞⎞ 2 b1 = λ Au (d 2 × z o ) = xˆ ⎜ λ 0.5359l ⎜ ⎟ ⎟ = xˆ 1.8660 2 ⎜⎝ ⎝ (0.3316l 3 ) ⎠ ⎟⎠ ⎛ ⎛ ⎞⎞ 1 b2 = λ Au (d1 × z o ) = xˆ ⎜ λ 0.5359l ⎜ ⎟⎟ 2 ⎜⎝ ⎝ (0.3316l 3 ) ⎠ ⎟⎠ − y!ˆ λ 0.5359l (0.3316l2 ) = 0.933 ( xˆ − yˆ 3 )

b3 = λ Au (d 3 × z o )

(

⎛ ⎛ 1 = − xˆ ⎜ λ 0.5359l ⎜ 2 ⎜⎝ ⎝ 0.3316l 3

(

)

)

(3.90)

⎞⎞ ⎟ ⎟⎟ ⎠⎠

− y!ˆ λ 0.5359l 0.3316l2 = 0.933 ( − xˆ − yˆ 3 )



⎡ b ( p) ⎢ 1 ⎢ b2 ( q ) ⎢ ⎢⎣ b3 ( r )

⎤ ⎤ ⎡ uˆ 2.0p ⎥ ⎥ ⎢ ⎥ = 0.933 ⎢ uˆ q − vˆ q 3 ⎥ ⎥ ⎢ uˆ r + vˆ r 3 ⎥ ⎦ ⎣ ⎥⎦

p=0,±1,±2,…,q=0,±1,±2,…,r =0,±1,±2,…

The first-order grating lobe series is



⎡ b ( p) ⎤ ⎤ ⎡ uˆ 2.0p ⎢ 1 ⎥ ⎥ ⎢ ⎢ b2 ( q ) ⎥ = 0.933 ⎢ uˆ q − vˆ q 3 ⎥ ⎢ b (r ) ⎥ ⎢ uˆ r + vˆ r 3 ⎥ ⎦ ⎣ ⎣ 3 ⎦

(3.91) p= ±1,q= ±1,r = ±1

When the zeroth Floquet mode is scanned to its maximum at a location of [u(0,0,0), v(0,0,0)] = [−0.8660, 0.0000], one of the nearest grating lobes [u(1,0,0), v(1,0,0)] = [1.8660, 0.0000] is displaced accordingly, as highlighted in the figure. ⎡ u(1,0,0) ,v(1,0,0) ⎤ = b1 + ⎡ u(0,0,0) ,v(0,0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(1,0,0) ,v(1,0,0) ⎤ = [1.8660,0.0000] + [ −0.8660,0.0000] ⎣ ⎦ ⎡ u(1,0,0) ,v(1,0,0) ⎤ = [1.0000,0.0000] ⎣ ⎦

w(0,1,0) =

(3.92)

u(20,1,0) + v(20,1,0) = 1.000

The grating lobe lies at the real-space boundary, as shown in Table 3.6 and Figure 3.17, and in the far-field integral equation solution shown in Figure 3.18. Sidelobe lines from two other grating lobes appear near the (u = 0) plane, confirming their presence, even if these sidelobes have a magnitude of approximately −40 dB, relative to the beam peak.

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3.3  Lattice Theory183

Figure 3.17  A sine-space map of first-order grating lobes and the main beam for an unstable equiangular triangular lattice array with a unit cell spacing comparable to 0.5359λ and a maximum scan angle 60°.

Table 3.6  Array Conditions for a Triangular Lattice with No Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5359λ

Y-axis element separation (dy)

0.3094λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.8660

Main beam scan location (vo)

−0.0000

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Lattice Theory

Figure 3.18  The near-field and far-field functions for a scanned 40 by 40-element planar array, with an unstable triangular lattice scanned to [u (0,1,0), v(0,1,0)] = [−0.8660, 0.0000].

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3.3  Lattice Theory185

Similar results occur for additional points on the main beam maximum conical scan volume circle, and this can be verified by direct substitution or by visual observation of the sine-space maps. For example (Table 3.7), ⎡ u(0,1,0) ,v(0,1,0) ⎤ = b2 + ⎡ u(0,0,0) ,v(0,0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(0,1,0) ,v(0,1,0) ⎤ = [0.9330, −1.6160] + [ −0.4330,0.7500] ⎣ ⎦ ⎡ u(0,1,0) ,v(0,1,0) ⎤ = [0.5000, −0.8660] ⎣ ⎦

w(0,1,0) =

(3.93)

u(20,1,0) + v(20,1,0) = 1.000

The grating lobe lies at the edge of real space, as (w(0,1,0) = 1.0), as shown in the integral equation solution in Figure 3.19. The area ratios of equivalent scan volumes for square and equiangular triangular lattices show a striking comparison. In the above examples, a 60° conical scan volume, sized to place grating lobes at the real space boundary, produces a square lattice unit cell area of Au = d1d2 sina = 0.2873l2

dx = dy = 0.536l

(3.94)

The equiangular triangular lattice under otherwise the same conditions produces a larger unit cell area of Au = d1d2 sina Au = 0.3316l2 ⎛ ⎛ 1 ⎞⎞ d1 = 0.5359l ⎜ xˆ + yˆ ⎜ ⎝ 3 ⎟⎠ ⎟⎠ (3.95) ⎝

⎛ ⎛ 1 ⎞⎞ d2 = 0.5359l ⎜ yˆ ⎜ ⎝ ⎝ 3 ⎟⎠ ⎟⎠ Table 3.7  Array Conditions for a Triangular Lattice with No Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5359λ

Y-axis element separation (dy)

0.3094λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.4330

Main beam scan location (vo)

−0.7500

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Lattice Theory

Figure 3.19  The near-field and far-field functions for a scanned 40 by 40-element planar array, with an unstable triangular lattice scanned to [u (0,0,0), v(0,0,0)] = [−0.4330, −0.7500].

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3.3  Lattice Theory187

The area ratio of the two is



Au ( triangular ) 0.3316l2 ⎛ 2 ⎞ = 1.155 (3.96) = = Au ( square) 0.2873l2 ⎜⎝ 3 ⎟⎠

This means that an equiangular triangular lattice array will produce the same scan volume coverage with 15.5% fewer elements. It is important to note that this observation is valid for electrically large apertures (25 square wavelengths or greater) and can be degraded under the electrically small aperture condition. The equiangular triangle effect is valid independent of scan volume, including the unscanned case, where qmax = 0 (3.97)



Confirmation of this statement is offered as an interesting calculation. 3.3.3  Isosceles Triangular Lattice

An equilateral triangular lattice has the advantage that all first-order grating lobes are equally spaced from the main beam. There are applications where an elliptical scan volume and a triangular lattice are both beneficial. This set of design goals leads to an isosceles triangular lattice, where two of the three lattice vectors have equal magnitude, but are unequal to the third. This is an important class of planar array lattices, and so is worthy of our study of the lattice fundamentals. We start with the recognition that the isosceles triangle has one lattice vector length unequal to its two cousins. The unequal lattice line therefore lies parallel to the y-axis for an unstable triangle or is parallel to the x-axis for a stable triangle. We can conveniently relate the isosceles triangle to the elliptical scan volume by considering the lattice variables (dx, dy), instead of the variables (d1, d2). As a result, the equilateral relationship between (dx, dy) no longer holds: dy ≠ dx



1 3 Isosceles

(3.98)

We can control the maximum scan angle in the two orthogonal principal planes by modulating the (dx, dy) parameters of an unbalanced equilateral triangle, as indicated here: dy =

l

(2(1 + sin q )) max



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dx =

⎛ 2 ⎞ ⎜⎝ 3 ⎟⎠

⎛ l ⎛ 3⎞⎞ 1+1 3⎜ ⎜ ⎝ 2 ⎟⎠ ⎟⎠ 1 + sin qmax ⎝

(

(3.99)

)

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Lattice Theory

For example, consider the case where the unbalanced isosceles triangular lattice studied previously is revised to produce an elliptical scan volume of 75° × 60° (u-x-v-planes). Accordingly, the (dx, dy) parameters are dy =

l

(2(1 + sin q )) max

dx = =

l ⎛ 2 ⎞ ⎛ 2 ⎞ = 0.3094l ⎜⎝ 3 ⎟⎠ = 2 1 + 60° )) ⎜⎝ 3 ⎟⎠ ( (

⎛ l ⎛ 3⎞⎞ 1+1 3⎜ ⎜ ⎝ 2 ⎟⎠ ⎟⎠ 1 + sin qmax ⎝

(

)

(3.100)

⎛⎛ 2 ⎞ ⎛ 3⎞⎞ l = 0.5087 l 1 + 75° ( ) ⎜⎝ ⎜⎝ 3 ⎟⎠ ⎜⎝ 2 ⎟⎠ ⎟⎠

Given these relationships, we can calculate the aperture lattice and grating lobe vectors. We start with the lattice of unit cell radiators, as shown in Figure 3.20. For the example case, the angle α < 60°, because dx has been reduced due to the larger scan volume in the associated u-plane in sine space. It is determined from plane geometry as

a tan−1 dx dy = 58.69° (3.101)

We need the unit cell area to determine the grating lobe vectors, and there are two equivalent means of accomplishing it. First, defining the lattice vectors, written from the plane geometry shown in Figure 3.20,

Figure 3.20  An unstable isosceles triangular phased array lattice in its aperture plane and Cartesian coordinates.

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3.3  Lattice Theory189

ˆ x + yd ˆ y = x0.5087 ˆ ˆ d1 = xd l + y0.3094l ˆ y = y0.6188l ˆ d2 = y2d

(3.102)

ˆ x − yd ˆ y = x0.5087 ˆ ˆ l − y0.3094l d3 = xd



The unit cell area is d1 =

dx2 + dy2 = 0.5954l

Au = d1d2 sina = (0.5954l ) 0.6188l sin58.69° = 0.3148l2  

(3.103)

The unit cell area is also determined from the two triangles that form the unit cell,

Au = 2dxdy = 2 (0.5087 l ) 0.3094l = 0.3148l2 (3.104) The grating lobe vectors are ⎛ ⎛ ⎞⎞ 2 = xˆ 1.9657 b1 = l Au (d 2 × z o ) = xˆ ⎜ λ 0.6188l ⎜ 2 ⎟⎟ ⎜⎝ ⎝ (0.3148l ) ⎠ ⎟⎠ ⎛ ⎛ ⎞ ⎞ y!ˆ l0.5087 l 1 − b2 = l Au (d1 × z o ) = xˆ ⎜ l0.3094l ⎜ (3.105) 2 ⎟⎟ 2 ⎜⎝ ⎝ (0.3148l ) ⎠ ⎟⎠ (0.3148l ) = xˆ 0.9828 − yˆ 1.6159



⎛ ⎛ ⎞ ⎞ y!ˆ l0.5087 l 1 b3 = l Au (d 3 × z o ) = − xˆ ⎜ l0.3094l ⎜ ⎟⎟ − 2 2 ⎜⎝ ⎝ (0.3148l ) ⎠ ⎟⎠ (0.3148l ) = − xˆ 0.9828 − yˆ 1.6159

⎡ b ( p) ⎤ ⎡ uˆ1.9657p ⎢ 1 ⎥ ⎢ = b q ⎢ 2 ( ) ⎥ ⎢ uˆ 0.9828q − vˆ 1.6159q ⎢ b ( r ) ⎥ ⎢ uˆ 0.9828r − vˆ 1.6159r ⎣ 3 ⎦ ⎣

⎤ ⎥ ⎥ ⎥⎦

(3.106) p=0,±1,±2,…,q=0,±1,±2,…,r =0,±1,±2,…

When mapping these vectors in sine space, the elliptical scan volume edge is also important. The maximum scan angle follows the equation of an ellipse in sine space, uo2



uo2( max ) uo( max ) =



7060_Book.indb 189

vo( max ) =

+

vo2

vo2( max )

= 1 (3.107)

l = sin75° = 0.9659 dx

l = sin60° = 0.866 2dy

(3.108)

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Lattice Theory

From the sine-space geometry shown in Figure 3.21, we see that the angle between the upper right first-order grating lobe and the u-axis is the same as the inner angle of the triangular lattice,

a = tan−1 yˆ ⋅ b2 xˆ ⋅ b2 = tan−1 1.6159 0.9828 = 58.69° (3.109)

This relationship offers another equation, which, when substituted into the elliptical scan edge, produces uo2

uo2( max )



+

vo2 tan2 a = 1 (3.110) vo2( max )

Solving for uo,



uo =

1 = 0.4624 (3.111) 1 0.9659 + tan2 a 0.86602 2

Figure 3.21  A sine-space map of first-order grating lobes and the main beam for an unstable isosceles triangular lattice array with an elliptical scan volume of 75° × 60° (u-v planes).

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3.3  Lattice Theory191

Using (3.107) for the (vo) coordinate yields the main beam position, producing a first-order grating lobe at the edge of real space, vo = vo( max ) 1 − uo2 uo2( max ) = 0.7603 ⎡⎣ uo ,vo ⎤⎦ = [0.4624,0.7603]



(3.112)

Equivalently, we have ⎡⎣ uo ,vo ⎤⎦ = [ −0.4624, −0.7603] (3.113)



These conclusions can be tested using the far-field integral simulation to confirm the aperture and sine-space calculations. Several scan conditions are illustrative, for example, ⎡ u(0,0,0) ,v(0,0,0) ⎤ = [ −0.4624, −0.7603] (3.114) ⎣ ⎦



With the main beam is scanned to its maximum at a location of [u(0,0,0), v(0,0,0)] = [−0.9659,  0.0000], the nearest grating lobe to real space, [u (1,0,0),  v (1,0,0)] = [1.9659, 0.0000] is displaced accordingly as highlighted in the Figure 3.17, ⎡ u(1,0,0) ,v(1,0,0) ⎤ = b1 + ⎡ u(0,0,0) ,v(0,0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(1,0,0) ,v(1,0,0) ⎤ = [1.9659,0.0000] + [ −0.9659,0.0000] ⎣ ⎦ ⎡ u(1,0,0) ,v(1,0,0) ⎤ = [1.0000,0.0000] ⎣ ⎦

w(0,1,0) =

(3.115)

2 2 u(0,1,0) + v(0,1,0) = 1.000

The same scan condition and grating lobe onset are confirmed by the far-field integral simulation shown in Table 3.8 and Figure 3.22. As we have seen previously, the grating lobe is compressed at its center by an element null. Table 3.8  Array Conditions for a Triangular Lattice with No Real-Space Grating Lobes Parameter Element polarization Element power function X-axis element count (N) Y-axis element count (N) X-axis element separation (dx) Y-axis element separation (dy) X-axis amplitude distribution Y-axis amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

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Value x-vector cosine (θ ) 40 40 0.5087λ 0.3094λ Uniform Uniform −0.9659 0.0000

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Lattice Theory

Figure 3.22  The near-field and far-field functions for a scanned 40 by 40-element planar array, with an unstable isosceles triangular lattice scanned to [u (0,0,0), v(0,0,0)] = [−0.9659, 0.0000].

An off-principal plane scan is also illustrative, even if somewhat more complex to calculate. With the main beam is scanned to its maximum at a location of [u(0,0,0), v(0,0,0)] = [−0.4624,  −0.7603], the nearest grating lobe to real space, [u (0,1,0),  v(0,1,0)] = [0.9828, 1.6159] is displaced accordingly as highlighted in the Figure 3.23,

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3.3  Lattice Theory193

Figure 3.23  The near-field and far-field functions for a scanned 40 by 40-element planar array, with an unstable isosceles triangular lattice scanned to [u (0,0,0), v(0,0,0)] = [−0.4624, −0.7603].

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194

Lattice Theory Table 3.9  Array Conditions for a Triangular Lattice with No Real-Space Grating Lobes Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

X-axis element count (N)

40

Y-axis element count (N)

40

X-axis element separation (dx)

0.5359λ

Y-axis element separation (dy)

0.3094λ

X-axis amplitude distribution

Uniform

Y-axis amplitude distribution

Uniform

Main beam scan location (uo)

−0.4624

Main beam scan location (vo)

−0.7603

⎡ u(0,1,0) ,v(0,1,0) ⎤ = b2 + ⎡ u(0,0,0) ,v(0,0,0) ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ u(0,1,0) ,v(0,1,0) ⎤ = [ −0.4624, −0.7603] + [0.9828,1.6159] ⎣ ⎦ ⎡ u(0,1,0) ,v(0,1,0) ⎤ = [0.5564,0.8566] ⎣ ⎦

w(0,1,0) =

(3.116)

2 2 u(0,1,0) + v(0,1,0) = 1.020

The grating lobe lies at the edge of real space, allowing for an accumulated computational inaccuracy, because (w(0,1,0) = 1.0), as shown in the integral equation solution in Table 3.9 and Figure 3.23. There are no grating lobes in the principal plane sectional patterns as these do not intersect the grating lobe. The grating lobe is evident in the projected and 3-D calculations and at the predicted location.

3.4

Reordered Lattice Theory The array aperture lattices studied thus far are ordered in the sense that two or three unique lattice vectors repeat throughout the aperture and are sufficient to characterize the positions of all elements of the array. There are good reasons for such lattice structures, principally that they set up a Cartesian ordered system that yields modular physical building blocks, used to build an array. The building blocks, generally termed subarrays, include a subset of the array components in a replicated manufacturing unit. An assembly of multiple subarrays becomes the full array, following the ordered aperture lattice vector set. The Cartesian ordering of the element lattice also leads to identical embedded element functions other than array edge effects, and this attribute generally produces deterministic element boundary value problems. As we have seen, the correlated lattice vectors also produce distinct grating lobes in sine space. Nevertheless, if we are to understand the scope of lattice theory, reordered lattices that describe the element positions of the array using non-Cartesian methods

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3.4  Reordered Lattice Theory195

with many uncorrelated lattice vectors also need consideration, as studied in [6]. A simple example of a reordered lattice such as a ring type locates elements in concentric rings, as illustrated in Figure 3.24. As lattice vectors connect unit cell phase centers, the concentric ring lattice has repeated rings, along with a large set of unique or uncorrelated radial lattice lines. As a result, it violates the first premise of Floquet’s theory, in that the radiating elements, as defined by their phase wall geometrical features, are not identical. Instead, the lattice lines associated with each element differ considerably near the center of the aperture, converging to a more uniform element geometry near the periphery of an electrically large ring array. There are geometrically radiated field results produced by the observed divergence in element phase wall conditions. Some of these may be stated from observation, such as: •

•

The reordered lattice vector variety can be expected to decorrelate the grating lobe magnitudes and locations in sine space, because of the lack of correlated lattice vector alignment. Using the Floquet lattice theory, a reordered lattice producing such a wide variety of lattice vectors would logically produce a similar variety of grating lobe vectors. Due to the range of these grating lobe vectors, the grating lobe fields would spread over a region of sine space rather than correlate, as is found with the ordered lattices (e.g., rectangular and triangular). The beam peak magnitude should be minimally affected since the element phases can be aligned to produce vector alignment, just as in the case for ordered lattices.

Figure 3.24  Concentric ring array lattice of element phase centers located in a Cartesian coordinate system.

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Lattice Theory

•

•

•

Although the magnitude and location of grating lobes may be decorrelated by the multiple lattice lines, the average element spacing can be expected to predict the general location of the peak grating lobe effects. So, in some respects, the reordered lattices can be treated as a stochastic variable, having mean and standard deviation properties. The field concentration in sine space would logically exhibit similar stochastic properties. The radiating element dissimilarities logically lead to nonidentical embedded element radiation functions. In calculations of the far-field behavior, it is useful to consider an average embedded element function as a representative sample of the distribution present in the ring array. Due to the ring lattice geometry, the edge boundary of the array is logically circular, leading to the presence of ring sidelobes, with reduced sidelobes near the main beam, as has been studied previously.

We explore these hypothetical results using the far-field integral equation. Even though reordered lattices do not meet the leading premises of Floquet’s theory, the latter provides general, but not specific guidance. The integral equation serves as the deterministic guide to the far-field behavior of these reordered lattice types. There are two reordered lattices that serve as a representative sample of similar lattice types. These are electrically large ring and equilateral spiral lattices. A consideration of the aperture lattice geometry and the associated aperture conditions are applied to a far-field integral simulation. 3.4.1  Ring Lattice Arrays

The ring lattice is constructed from a set of concentric circles with a uniform spacing. At the center of each ring, the element phase centers locate at a uniform spacing along the circumference of the ring, where the circumferential spacing is equal to or less than the ring spacing. These guidelines are sufficient to define the element phase center locations in a cylindrical coordinate system. Conversion to a Cartesian frame allows a convenient fit to the far-field integral equation, which is also structured in a Cartesian framework. For convenience, we start with the first element at the center of the aperture coordinate frame and in the (zi = 0), ∀i, although we can readily apply an offset in the x-y plane as needed,

⎡⎣ x1 , y1 , z1 ⎤⎦ = [0,0,0] (3.117)

The ring circumferences derive from the common element spacing (do) for both the ring radius and element spacing within the ring, for each ring (i), up to the specified number of rings (I),

Ci = 2pri = 2pido (3.118)

The number of elements (ni) contained in each ring is the truncated division of the circumference and the element spacing within the ring, rounded down to the nearest integer,

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3.4  Reordered Lattice Theory197



ni =

Ci = fix ( 2pi ) (3.119) do

The total number of elements is the sum of the ring element counts plus the central element,



nt = 1 +

I

∑ ni (3.120) i=1

The polar angle associated with each element (j) within the ring is



qi,j =

j2p ni (3.121)

This produces polar coordinates for each element [ni, θ i,j]. These readily convert to Cartesian coordinates with

⎡ xi,j , yi,j ⎤ = ⎡ ni do cos qi,j , ni do sin qi,j ⎤ (3.122) ⎣ ⎦ ⎣ ⎦

For example, calculating the element positions for a 10.0-GHz array with (do = 1.5λ ) and three rings generates 37 elements. The element position list and an illustration in Cartesian space are shown in Figure 3.25.

Figure 3.25  A concentric ring array lattice of element phase centers with 3 rings, (do = 1.5λ ) at 10.0 GHz and located in a Cartesian coordinate system.

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Lattice Theory

A useful way to study the far-field effects caused by the concentric ring array is with a limited case study. Several important considerations help to form the parameters for the first case, as summarized here: •

•

•

•

•

The element separation is intentionally large enough to bring grating lobe fields into real space, making their spatial behavior visible. An element separation of (do = 1.5λ ) should produce grating lobes located λ /do = 0.6667 from the main beam location. The array should initially be scanned to boresight to reveal the first-order grating lobe fields within real or visible space. The element factor is initially set to a cosine power function for simplicity. This means that significant gaps exist between the elements and that the element gain is constrained to that produced by a unit cell area of 0.25λ 2 . This also avoids possible complexities introduced by the array mutual coupling that can generally be resolved with complex full-wave finite element analysis. The aperture illumination function is uniform, as in previous lattice case studies, to simplify the far-field analysis. This restriction is removed in later chapters of this book. The far-field simulation employs the integral equation in Cartesian form, determined in sine space.

Far-field radiation results of the subject ring array analysis are shown in Figure 3.26. The calculations use the above aperture lattice relationships and the far-field integral equation in the following form:



E (r ) =

N e− jkRo − j2p / l( xi u+ yiv+zi f ( q, f ) ∑ ai e 4pRo i=1

1−w2

)

(3.123)

The first term is normalized to a value of unity at the peak far-field response and so is essentially removed from the equation. The element function (f(θ , ϕ )) uses a cosine power pattern, so (f(θ , ϕ )) = xˆ cos q = xˆ 1 − u2 − v2 . Because the aperture Table 3.10  Array Conditions for a Ring Lattice with Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

Lattice type

Concentric ring

Element ring count (I)

10

Total element count (N)

341

Element separation (do)

1.5λ

Amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

0.0000

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3.4  Reordered Lattice Theory199

illumination is uniform and aligned for boresight scan, (ai = 1.0⎪ ∀i). The far-field results are sampled throughout the extent of real space, so [⎪u⎪, ⎪v⎪] ≤ [1, 1]. The most significant introduction is the locations of the elements in Cartesian space. We consider these far-field data (Table 3.10 and Figure 3.26) in terms of the postulated hypothetical results: •

Decorrelated grating lobe magnitudes: The far-field data shows the presence of grating lobe fields in the vicinity of the expected grating lobes. The peak

Figure 3.26  The near-field and far-field functions for a 341-element planar array, with a 10 concentric ring lattice, with (do = 1.5λ ), scanned to [u (0,0,0), v(0,0,0)] = [0.0000, 0.0000].

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200

Lattice Theory

•

•

•

•

grating lobe field is approximately −10 dB relative to the beam peak, marking a substantial reduction compared to ordered lattices. The beam peak magnitude: The main beam magnitude appears to be minimally affected by the aperture lattice. However, should gaps exist between the elements, the reduced aperture area will reduce the main beam gain, even though this result is not indicated. Grating lobe location: The raised electric field magnitude associated with the grating lobe occurs at a distance of λ /do = 0.6667 from the main beam at boresight. The ring lattice translates into a ring of grating lobes with six major localized peak areas. The grating lobe fields scatter into peaks and nulls attributed to the multiple aperture lattice vectors in the vicinity of these expected maxima. Although patterned behavior is evident in the grating lobe regions, the general effect also appears increasingly stochastic. Average embedded element function: Although a deterministic element function is used, the associated deep nulls at the edge of real space, where (w = 1) evidence increased field magnitude. This general behavior is also characteristic of a random lattice, where the element positions have a purely stochastic characteristic. Ring sidelobes: Ring sidelobes are evident in the region between the main beam and the grating lobe region. These are attributed to the circular aperture edge boundary condition.

In summary, we find that all postulated far-field effects are confirmed and, in many cases, quantified. To better understand the characteristics of ring lattice arrays, several additional cases are illustrative. Consider, for example, the effects produced by doubling the array spacing while maintaining the remaining array characteristics. The far-field analysis process uses the same integral equation, with the processed results shown in Table 3.11 and Figure 3.27. We would expect that the sine-space separation between grating lobe regions would be reduced by a factor of 2 in all directions. Our observations of the far-field results are as follows: •

Decorrelated grating lobe magnitudes: The far-field data again shows the presence of grating lobe fields in the vicinity of the additional real-space grating Table 3.11  Array Conditions for a Ring Lattice with Real-Space Grating Lobes Parameter Element polarization Element power function Lattice type Element ring count (I) Total element count (N) Element separation (do) Amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

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Value x-vector cosine (θ ) Concentric ring 10 341 3.0λ Uniform 0.0000 0.0000

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3.4  Reordered Lattice Theory201

Figure 3.27  The near-field and far-field functions for a 341-element planar array, with a 10 concentric ring lattice, with (do = 3.0λ ), scanned to [u (0,0,0), v(0,0,0)] = [0.0000, 0.0000].

•

•

7060_Book.indb 201

lobes. The peak grating lobe field compared to the smaller element spacing is unchanged at approximately −10 dB relative to the beam peak, indicating that the grating lobe field magnitude is not directly affected by the element spacing. The beam peak magnitude: Compared to the smaller lattice (do = 1.5λ ), the main beam magnitude produces a similar result. Grating lobe location: The raised electric field magnitude associated with the grating lobes occurs at distances of integer multiples of (λ /do = 0.3333) from the main beam at boresight. The ring lattice translates into a ring of grating lobes with 6 near-in and 12 additional major localized peak areas.

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Lattice Theory

•

•

There also appears to be increased stochastic behavior compared with the smaller ring lattice. Average embedded element function: The edge of real space far-field behavior, where (w = 1), shows increased field magnitude in comparison to the (do = 1.5λ ) case. Ring sidelobes: Ring sidelobes are again evident but in a smaller region between the main beam and the grating lobe region, due to the increased element spacing.

We observe that the number of regions of raised grating lobe far-field magnitude increases in proportion to the Floquet mode periodicity and the average element separation. The peak decorrelated grating lobe field magnitude remains the same as the previous case where the element spacing is twice as large, just as is the case for ordered lattices. So the dominant effect of the ring lattice is a spatial decorrelation of the grating lobe fields. The decorrelation occurs in the same region of sine-space regions as the grating lobes; near the main beam region, there is minimal effect. Based on these observations, we could conclude that the best use of a ring lattice would be at an element spacing near the Nyquist rate. In the previous two cases, scan effects were not introduced, so the next case tests both factors. In this case, the average element spacing is (do = 0.75λ ) so the first-order grating lobes are (λ /do = 1.3333). The main beam is deliberately scanned to [u(0,0,0), v(0,0,0)] = [0.0000, −0.7500], to bring a single grating lobe region into real space. In order to maintain the peak grating lobe field magnitude, the ring and element counts remain the same as the previous example (Table 3.12). We see from the integral equation far-field simulation in Figure 3.28 that the farfield behavior follows the estimates indicated previously. With the average element spacing near the Nyquist limit, the sidelobes for much of sine space and surrounding the main beam are largely unaffected by the lattice type. In the region around the single first-order grating lobe within real space, we see the decorrelated grating lobe fields with peaks and nulls spread out near the grating lobe maximum location at approximately [u(0,1,0), v(0,1,0)] = [0.0000, 0.95]. In this general region, the radiated fields show a mix of stochastic and reordered maxima and minima. This example Table 3.12  Array Conditions for a Ring Lattice with Real-Space Grating Lobes

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Parameter

Value

Element polarization

x-vector

Element power function

cosine (θ )

Lattice type

Concentric ring

Element ring count (I)

10

Total element count (N)

341

Element separation (do)

0.75λ

Amplitude distribution

Uniform

Main beam scan location (uo)

0.0000

Main beam scan location (vo)

−0.7500

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3.4  Reordered Lattice Theory203

Figure 3.28  The near-field and far-field functions for a 341-element planar array, with a 10 concentric ring lattice, with (do = 0.75λ ), scanned to [u (0,0,0), v(0,0,0)] = [0.0000, −0.7500].

probably best illustrates the value of this reordered lattice; it can allow a greater element spacing than ordered lattices for the same scan volume, while spreading and decorrelating any grating lobes that enter real space. Finally, we consider the role of the size of the ring lattice array. By increasing the ring count to 20 and the element count to 1,310, we see that many of the far-field characteristics follow the trends described above, as shown in Table 3.13 and Figure 3.29. The exception is the magnitude of the peak grating lobe field, which decreases

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204

Lattice Theory Table 3.13  Array Conditions for a Ring Lattice with Real-Space Grating Lobes Parameter Element polarization Element power function Lattice type Element ring count (I) Total element count (N) Element separation (do) Amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

Value x-vector cosine (θ ) Concentric ring 20 1,310 0.75λ Uniform 0.0000 −0.7500

Figure 3.29  The near-field and far-field functions for a 1,310-element planar array, with a 20 concentric ring lattice, with (do = 0.75λ ), scanned to [u (0,0,0), v(0,0,0)] = [0.0000, −0.7500].

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3.4  Reordered Lattice Theory205

from approximately −12 to −18 dB, consistent with the ratio of the element count 10log10(341/1,310) = −5.85 dB. As a result, electrically small arrays having 100 elements or less can be expected to have reduced grating lobe decorrelation effects. 3.4.2  Spiral Lattice Arrays

The spiral lattice is based on the Archimedes spiral and is a reordered lattice type. Its element phase centers lie on a spiral curve and have far-field characteristics related to the ring lattice. The spiral arms converge towards the outer edge of the aperture into a ring geometry, and this effect increases with the number of array elements. Because of this analogy to the ring lattice, our case study can be a relatively simple one. The geometry of an Archimedes spiral is based on a simple cylindrical coordinate equation that can be readily cast into Cartesian coordinates. We constrain the element spacing in both radial and angular dimensions to a fixed value, to avoid introducing grating lobe fields that could spread towards the main beam. Because of this, several mathematical extrapolations become useful. The Archimedes spiral equation relates the radius (r) to the angular location (θ ), using a constant (a) and a radial weighting coefficient (b) that controls the separation between spiral arms or rings,

r = a + bq (3.124)

The resulting spiral line has (i) discrete points that readily convert to Cartesian coordinates, xi = ri cos q

yi = ri sin q

(3.125)

It is useful to normalize the separation between element phase centers both in the radial and angular dimensions to unity in order to allow a direct scaling by the intended element separation (do). We can produce such a spiral curve with a sufficient point density to define it with precision by zeroing the constant (a = 0) and setting the angular slope to (b = 1/2π ), as shown in Figure 3.30. One way to maintain nearly equal point density is to set the step size between element points to



⎛ 2 ⎞ q2 = ⎜ ⎝ pq1 ⎟⎠

0.8

(3.126)

Because of the importance of maintaining the same spacing between elements along the spiral curve, multiple spline functions are useful, and can generate the intended result, with an exception near the center (Figure 3.30). For a useful comparison with the ring lattice, we use the same array parameters using 480 elements (Table 3.14 and Figure 3.30), with the Archimedes spiral lattice substituted for the ring lattice (Figures 3.31 and 3.32).

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Lattice Theory

Figure 3.30  An illustration of an Archimedes spiral with unit arm spacing.

Figure 3.31  An illustration of an Archimedes spiral with uniform element spacing. Table 3.14  Array Conditions for a Spiral Lattice with Real-Space Grating Lobes Parameter Element polarization Element power function Lattice type Element ring count (I) Total element count (N) Element separation (do) Amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

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Value x-vector cosine (θ ) Archimedes spiral N/A 480 0.75λ Uniform 0.0000 −0.7500

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3.4  Reordered Lattice Theory207

Figure 3.32  The near-field and far-field functions for a 1,310-element planar array, with a 20 concentric Archimedes spiral lattice, with (do = 0.75λ ), scanned to [u (0,0,0), v(0,0,0)] = [0.0000, −0.7500].

As with previous case studies, our observations of the far-field results are as follows: •

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Decorrelated grating lobe magnitudes: The far-field data show many similarities to the ring lattice array. Grating lobe fields in the vicinity of the expected real-space grating lobes are spread further towards the main beam and further decorrelated than the ring lattice. The peak grating lobe field is reduced to approximately −12.5 dB relative to the beam peak, indicating that the grating lobe field magnitude is indirectly affected by the element spacing.

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Lattice Theory

•

•

•

•

The beam peak magnitude: The main beam magnitude shows limited decorrelation effects other than the reduced gain associated with gaps between the elements (and so the effective element area). Grating lobe location: The raised electric field magnitude associated with the grating lobes occurs at distances of integer multiples of (λ /do = 1.3333) from the main beam at boresight. The spiral lattice translates these into a ring of decorrelated grating lobes with multiple decorrelated peaks in an almost random distribution in sine space. There is increased stochastic behavior in comparison with the ring lattice and considerably more compared with ordered lattices. Average embedded element function: The edge of real-space far-field behavior, where (w = 1) shows increased field magnitude but only in the grating ring area. Elsewhere, the fields at the real-space boundary are comparable to intercardinal plane behavior seen in a rectangular lattice. Ring sidelobes: Ring sidelobes are evident in a smaller region between the main beam and the grating lobe region compared to the ring lattice, due to the additional decorrelation in the spiral lattice. The sidelobe magnitudes in regions outside of the grating lobe region evidence ring sidelobes, typical of a circular aperture edge, albeit with decorrelated effects arising from the spiral lattice.

It is reasonable to conclude that the spiral lattice is a notable entry in the reordered lattice class because of its grating lobe decorrelation effects. The ring lattice appears to offer better overall performance characteristics in this aspect, but a merit determination may include factors such as manufacturing and specific application features not considered in this study.

3.5

Finite Array and Surface Wave Effects The study of array lattice effects would be incomplete without considering finite array and surface wave effects on the far-field array performance. The small array effects are quantitative and can be applied to ordered lattice theory directly. The surface wave effects are more complex, and although these can be estimated from theory, they are dependent on specific aperture boundary conditions, materials, and geometry, so an estimate is used at this stage of our study. A more detailed surface wave consideration is included in Chapter 7, which studies surface waves produced by the radiating aperture. When we consider that Floquet theory assumes an infinite aperture size, the grating lobes become impulse functions having zero cross-sectional area in sine space. A finite array produces finite main beam and grating lobe beamwidths instead. So, as we generally seek array designs with no real-space grating lobes, the finite beamwidth becomes a pertinent factor. The deterministic aspects of finite array effects on the scan volume are surprisingly straightforward. The calculations follow a few basic principles: •

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The most common method to ensure that no grating lobes enter real space within a defined scan volume is to locate the grating lobes nearest to the realspace boundary at least 2.0 beamwidths from the same.

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3.5  Finite Array and Surface Wave Effects209

•

•

•

•

•

•

The beamwidth is defined in the two cardinal aperture planes for a rectangular or elliptical aperture and in any sectional plane through the boresight beam for a circular aperture. The beamwidth is 0.886λ /d for a rectangular aperture with uniform illumination, where (d) is the aperture dimension. The two-beamwidth criterion ensures that the first two nulls nearest the grating lobe are positioned outside of real space. The finite array effect reduces the element spacing needed to ensure gratinglobe free real-space scan within the scan volume. The lattice calculation is determined at the highest operating frequency, thus representing the worst-case condition. The finite array effects should be determined under the worst-case aperture illumination condition, as reduced sidelobe conditions expand the beamwidth of the main beam and all grating lobes.

In addition to the finite array beamwidth effects, array surface waves also impose a scan restriction. As, at maximum scan, array grating lobes can appear at the real space boundary, where (w = 1), or where (θ = 90°), this is the same spatial location of a surface wave. Aperture material and boundary conditions can and, in most cases, will reduce the surface wave velocity from a maximum of the freespace velocity. This, in turn, brings the effective grating lobe fields further into real space. A deterministic calculation of this effect requires accurate knowledge of the aperture boundary conditions, materials, lattice geometry, and radome conditions. This grating lobe onset calculation involves the asymptotic effects of the grating lobe singularity (Floquet mode impulse) occurring extremely close to a root (embedded element null). Modern commercial full-wave 3-D solvers can solve such a problem, but it often requires a large computational capacity. So, for convenience, we can use an approximation of 100 millisines, in lieu of this detailed calculation. We combine both the finite aperture and a surface wave approximation to calculate the array lattice dimensions. In this case, the maximum conical scan angle is (θ max = 60°) and the maximum operating frequency is 2.0 GHz, as in a previous analysis. To this, we impose an array element condition having 40 elements each on both x and y aperture axes for a total element count of 1,600. The surface wave condition is a conservative 100 millisines. Using a modified (3.35) for an ideal array with an intended surface wave and finite array allowance, the largest square lattice spacing supporting the required scan volume is

dx = dy =





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=

l 1.1 + (0.5 (0.886) + 2.0) l N xdx + sin qmax

(

l = 0.4933l = 2.9354 in. 2.0271

)

(3.127)

Au = d1d2 sina = 0.2434l2 (3.128)

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Lattice Theory

The grating lobe vectors are

( ) (3.129) ˆ ˆ = xˆ × zl0.4933l (0.2434l2 ) = − y2.0271

ˆ ˆ b1 = ld2 × zo Au = yˆ × zl0.4933l 0.2434l2 = x2.0271

b2 = ld1 × zo Au

The first-order grating lobes are



⎡ b1 ( p ) ⎤ ⎡ ±uˆ 2.0271p ⎤ ⎢ ⎥= ⎢ ⎥ ⎢⎣ b2 ( q ) ⎥⎦ ⎣ ±vˆ 2.0271q ⎦

p=1,q=1

(3.130)

When the zeroth Floquet mode is scanned to a maximum at a location of [u(0,0), v(0,0)] = [−0.866, 0.000], the nearest grating lobe [u(1,0), v(1,0)] = [1.1445, 0.000] is w(1,0) ( p ) = pl ( d2 sina ) + w(0,0)

w(1,0) ( p ) = [ 2.0271] + [ −0.866] (3.131)

w(1,0) ( p ) = [1.1611]

The sine-space map now shows a modified grating lobe at the edge of real space when the main beam is scanned to its maximum (Figure 3.33). It shows both the grating lobe coverage region and that of the surface wave. The net result is that the surface wave lies at the real-space boundary [u(1,0), v(1,0)] = [1.0000, 0.000], the two beamwidth condition at the next segment [u(1,0), v(1,0)] = [1.0611, 0.000], and the actual grating lobe is at [u(1,0), v(1,0)] = [1.1611, 0.000]. These results are confirmed with a far-field integral calculation, where we can examine the finite array and surface wave allowances. Table 3.15 gives the array parameters for this test case, using a cosine element power function. The square lattice element separation relies on the above calculations, with the array scanned to [u(0,0), v(0,0)] = [−0.866, 0.000] and far-field results as shown in Figure 3.34. Table 3.15  Array Conditions for a Rectangular Lattice with No Real-Space Grating Lobes Parameter Element polarization Element power function X-axis element count (N) Y-axis element count (N) X-axis element separation (dx) Y-axis element separation (dy) X-axis amplitude distribution Y-axis amplitude distribution Main beam scan location (uo) Main beam scan location (vo)

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Value x-vector cosine (θ ) 40 40 0.4974λ 0.4974λ Uniform Uniform −0.8660 0.0000

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3.5  Finite Array and Surface Wave Effects211

Figure 3.33  A sine-space map first-order grating lobes and the main beam for a square lattice array with a unit cell spacing set to 0.4974λ including finite array beamwidth and surface wave allowances and the maximum scan angle 60°.

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Lattice Theory

Figure 3.34  The near-field and far-field functions for a scanned 40 by 40-element planar array including finite array beamwidth and surface wave allowances, with a square lattice scanned to [u (0,0), v(0,0)] = [−0.866, 0.000].

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3.5  Finite Array and Surface Wave Effects213

The integral equation far-field results show the following characteristics: •

•

•

A main beam scanned to [u(0,0), v(0,0)] = [−0.866, 0.000] produces a grating lobe outside the edge of real space, so only the third sidelobe remains within real space. Based on this observation, the grating lobe is at least 2.5 beamwidths outside of the real-space boundary, or at 50 millisines from the boundary. This is consistent with the allocations combined with lattice theory. It also provides an assurance that the surface wave and grating lobe cannot enter real space, avoiding possible surface wave impedance discontinuities. There is a remarkable contrast between the compensated array illustrated above and the same but uncompensated array shown in Figure 3.32, particularly in the grating lobe region near the real-space boundary. In the compensated case, all vestiges and near-in sidelobes associated with the grating lobe are removed from visible space. The main beam characteristics are barely affected, other than the area gain reduction associated with the smaller lattice. The main beam characteristics, element pattern effects, and sinc sidelobe distribution stemming from the uniform aperture distribution are the same as those found in the same array with no finite array or surface wave allocations. The illustrated lattice calculations indicate that the effects occur largely at the real-space edge and at maximum scan.

References [1] [2] [3] [4] [5] [6]

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Amitay, N., V. Galindo, and C. P. Wu, Theory and Analysis of Phased Array Antennas, New York: Wiley-Interscience, 1972. Brookner, E., Practical Phased Array Antennas, Norwood, MA: Artech House, 1991. Mailloux, R. J., Phased Array Antenna Handbook, 3rd ed., Norwood, MA: Artech House, 2018. Lo, Y. T., and S. W. Lee, “Theorems and Formulas,” Ch. 2 in Antenna Handbook, New York: Van Nostrand Reinhold, 1988. Rudge, A. W., et al., The Handbook of Antenna Design, London: Peter Peregrinus/IEE, 1983. Sikina, T. V., “Reordered Lattices for Phased Array Antennas,” 2010 IEEE International Symposium on Phased Array Systems and Technology, 2010, pp. 801–805.

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CHAPTER 4

Array Fundamentals: Supporting Theories, Part I 4.1 Introduction Having established the fundamentals of array far-field, scan, and lattice theories to some extent, we can now open our field of view to the domains involved and several important underlying theories. This subject will introduce a variety of critical array design subjects, including the degrees of freedom available, and the design methods in common practice today. As such, it will serve as a launch point for array design pursuits and one to which we will return often. The subjects involved in this wider view of array technology include: •

•

•

4.2

The three relevant domains: Frequency, space, and polarization domains are explored, and we describe how these are used in commercial practice, along with the reasoning behind these technology development areas. Practical limiting theories: These include reciprocity, energy conservation, Fresnel zones, and superposition, as these are often encountered in practical array design. Rotational transforms: Practical array use will require a consideration of their orientation in space and the associated scan volume in transformed coordinates.

Radiating Aperture Fundamentals: Three Domains It is important to recognize the domains controlled by phased array systems, and the degrees of freedom available. This gives us as designers a broader view of the possible design paths available and the technologies involved. We can consider a few natural examples of radiating systems in the three domains to provide context: •

The Sun, for example, is a large radiating source operating in the same three domains of a phased array: – Frequency time: The Sun is a radiating sphere that emits its radiation over a very wide frequency spectrum, much of which is in the optical frequency range (Figure 4.1), representative of a black body at 5,525K, constant over the time domain, with a cyclic variation, sun-spot activity, and solar flares. – Spatial: The radiation is generally omnidirectional, with significant spatial variation in various parts of the frequency spectrum because the source is electrically very large in terms of the wavelengths involved. 215

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Polarization: The radiation is randomly polarized (essentially completing the Poincaré sphere) and time-variant. A beacon transceiver, for example, can be a critical life saver in remote areas, because it transmits both location and vital health statistics in the event of an emergency, such as an avalanche: – Frequency time: The W-link frequency is 869.8 MHz in the general European area and 916–926 MHz in the United States and Canada, with a 1.0MHz information bandwidth. – Spatial: The radiation is a fixed generally omnidirectional main beam in the azimuth plane with high antenna gain in the elevation plane. – Polarization: A single vertical polarization is used. A phased array system, with multiple degrees of freedom: – Frequency time: It emits a restricted spectrum or operating bandwidth and modulation for a communications signal and a radar waveform for sensors. The radio frequency (RF) cycles between transmit (Tx) and receive (Rx) functions for half-duplex systems. – Spatial: The radiation is directional, scannable in 3-D space, capable of multiple radiation beams, half-duplex or full-duplex, with separate Tx and Rx beam functions. – Polarization: The Tx and Rx beams are controlled to specific Poincaré polarization states and generally a single state in the Tx mode, with multiple simultaneous states possible in Rx mode. –

•

•

There are multiple design methods and technology pathways that have been developed to provide useful capabilities in each of these domains, with definitions and differentiating factors, as illustrated in Figure 4.2. In the frequency-time domain, we have an extensive set of tools and design techniques for today’s communications and sensor systems, for example:

Figure 4.1  The solar spectrum produced by our Sun.

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4.2  Radiating Aperture Fundamentals: Three Domains217

Figure 4.2  Phased array design techniques and technologies in terms of the three domains.

•

Bandwidth: – Operating frequency bandwidth: This is the range of frequencies over which the array meets its performance requirements, often analyzed at discrete frequencies. Alternatively, some analysis tools solve in the time domain, converting to the frequency domain with the Fourier transform. – Instantaneous bandwidth: The instantaneous bandwidth refers to the array performance within a subset of operating frequencies containing the transferred information. For example, an array may have an operating bandwidth of 1.0 GHz, from 7.7 to 8.7 GHz. The operating band may be composed of 10 instantaneous bands, each 100 MHz wide. Calibration frequencies would be at frequencies 7.55, 7.65, 7.75, … GHz, centered at each instantaneous band, limited by its dispersive or (∂/∂f) effects. – Bandwidth differentiations in electronic hardware: Much of the RF hardware used to form the microwave circuits in phased array systems differentiates by the operating frequency. There are the microwave and millimeter radar and communications bands, extending through L, S, C, X, Ku, K, Ka, and W bands, or from 1 through 300 GHz. The operating bandwidth is often expressed in terms of a fractional bandwidth as the ratio of highest operating frequency (f H) to the lowest (f L), relative to center (fC), Δf =



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( fH − fL ) fC

(4.1) fH >fL

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•

•

System timing: – Continuous mode operation uses a fixed Tx or Rx mode. – Full-duplex mode operates the array in both Tx and Rx modes simultaneously and so requires an extensive isolation, as there is none in the time domain. One way to achieve full-duplex isolation involves physical separation between Tx and Rx apertures, often with barriers introduced between the two. – Half-duplex mode operates the Tx and Rx components of a single array during separate sequential time periods. A time-domain pulse sequence is often used, where a Tx pulse is followed by a short time interval and then an Rx pulse and subsequently the pulse sequence repeats. This provides time-domain isolation between Tx and Rx modes. Waveforms: – Although the operating frequency is the major determinant of the free-space propagation mechanism, its modulation introduces the information content into the waveform, generally being a fraction of the operating frequency. For example, conventional video signals have a bandwidth of 1 MHz or less. – Frequency modulation: The modulated signal (f m) is superimposed on the carrier frequency (fc) in conventional analog frequency modulation, so the ratio between the carrier center frequency is generally a small fraction. The time-domain signal is (fc), where ω c = 2π fc, and Am(τ ) is the modulation signal amplitude over a modulation frequency range of (ω Δ ) t ⎛ ⎞ y = E cos2w t + 2w A t dt ⎟ (4.2) ( t) o⎜ c Δ∫ m( ) ⎝ ⎠ 0

–

( yt ) = Eo cos ( wct + ft ) (4.3)

–

Amplitude modulation: In an amplitude modulated signal, the amplitude of the carrier frequency is modulated using the parameter (ω m),

( yt ) = Eo (1 + cos wmt ) sin wct (4.4)

•

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Phase modulation: In a phase modulated signal, the phase of the carrier frequency is modulated (ϕ t),

Time processing: – Time sidelobes: Reflection coefficients occur in the components of a phased array and the effects spread into both the frequency and time domains. Abrupt rise and fall times of the pulse produce Gibbs phenomena, with damped sinusoidal responses in the time domain, also called time sidelobes. These may be controlled by reducing the reflection coefficients in the array microwave circuits or by accurately measuring the effect and canceling it in signal processing software.

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4.2  Radiating Aperture Fundamentals: Three Domains219

Pulse compression: Pulse compression is a method that code modulates the Tx waveform in a half-duplex radar, only to process the Rx return waveform to remove all but the modulated components in the time domain [1]. Space-time adaptive processing (STAP): – This advanced processing technique uses far-field space and time-dependent waveforms. One example is the use of pulse-Doppler waveforms and known target and clutter characteristics to differentiate radar returns into objects of interest. – Pulse Doppler waveforms: These use pulse-timing techniques and Rx signal Doppler effects to determine the target velocity and angle information. They combine features of both pulse and continuous-wave radars [2]. – Target and clutter filtering: A 2-D filtering technique can use a phased array antenna with multiple spatial channels. Applying the statistics of the interference environment, a STAP weight vector is formed. This weight vector is applied to the coherent samples received by the radar [3]. –

•

The spatial domain is well suited for phased array systems because these are designed to maximize the available degrees of freedom, for example: •

Scan capability: Phased arrays are designed with the ability to point the maximum directivity towards a predetermined far-field spatial location within a given scan volume and beam-pointing accuracy: – Directivity: This is a measure of the angular resolution of the array in terms of its main beam cross-sectional area. It is expressed as the ratio of the power available in the forward hemisphere to the integrated radiated power within the main beam’s angular space (E 2(θ , ϕ )), D=



4p

∫∫ E ( q, f ) sin q dq df 2

(4.5)

Scan volume: The region of space where an array’s main beam can be scanned while maintaining all performance requirements. – Beam-pointing accuracy: The standard deviation or peak range of actual beam-pointing locations in free space relative to the commanded beam pointing location. Independent scan beam: Phased arrays are generally designed to scan their main beam into free space independent of other domains and degrees of freedom, such as frequency-time and polarization domains. Frequency scanned beam: Phased arrays can be designed to scan the main beam as a function of the operating frequency. This method can be useful when the information bandwidth is small, and it can be used to design large array apertures at a relatively low cost. Beamforming: The array’s aperture state produced by the beamforming implementation method can contain a wide range of possibilities, depending on the design goals. These vector states can also be predetermined and implemented independently for each beam produced: –

•

•

•

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Single beam: The array often produces a single focused beam, maximizing the radiated power at a given far-field beam-pointing location. – Simultaneous beams: The array can produce beams that point main beams at multiple spatial locations at a single operating frequency and time. In the Rx mode, simultaneous beams provide the full array directivity at multiple beam locations simultaneously, with significant benefits in multipoint communications and in surveillance radar applications. Simultaneous beams can also include monopulse, which, in general, has three simultaneous beams used for accurate angle accuracy applications. The beams are a sum (Σ) beam that provides high array directivity at a commanded beam position within the coverage region and orthogonal (Δ Az, Δ El) difference beams in orthogonal spatial planes at the same time and the same Rx pulse. – Synthesized beams: The far-field beam or beams produced can be shaped to predefined far-field power profiles within the limits of Floquet theory. Beamforming implementation: Multiple implementation methods can be invoked to arrange the beamforming hardware or software. These form the connection between a beam port to the radiating elements in the array aperture, performing the Fourier transform: – Analog beamforming: This is confined to a series of power dividers, which can be either terminated devices, such as Wilkinson or Geysel devices, or reactive, such as a tee divider. – Spatial beamforming: The spatial beamformer can use single or multiple point feed radiators propagating to multiple receiving elements on the back of a phased array, operating in free space as a reactive feed network. – Digital beamforming: A digital beamformer connects a single radiating element to a single element or a subarray of elements to a digital receiverexciter (DREX). The beamforming is done by software, because the DREX converts the microwave signal for RF to digital form. – Hybrid beamforming: This beamforming method uses both analog and ­d igital methods and is becoming the most common implementation used today. Modular subarrays: Subarray geometrical shapes and their arrangement with the aperture have a direct impact on the far-field performance, with several versions worthy of our study: – Geometrically defined: These subarrays can take outline geometries such as square, rectangular, circular, ellipsoidal, and spiral, affecting the distribution of the grating lobe fields at the instantaneous bandwidth edge. – Random subarray sizes: Subarray edge geometry can take a random shape, decorrelating grating lobe far fields. – Overlapped subarrays: By analog combining elements from multiple subarrays into a single beam port, we can synthesize the subarray beam to minimize the grating lobe magnitude. This method produces a complex analog beamformer with the advantage of low magnitude real-space grating lobes for a simultaneous beam set operating at the edges of their instantaneous bandwidth. Aperture distributions: Far-field peaks, sidelobes, and nulls are intrinsic to the vector field produced by time-varying EM systems. Phased arrays have –

•

•

•

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4.2  Radiating Aperture Fundamentals: Three Domains221

•

the necessary degrees of freedom to control these sidelobes, dependent on the aperture vector state, for example: – Sinc function: A uniform aperture distribution, with all array elements radiating or receiving with the same amplitude, is a commonly used Tx weighting. It produces a sinc far-field function for rectangular apertures and a Bessel function for circular and elliptical apertures. It is also the most straightforward to produce with RF hardware, because Tx power amplifiers (PAs) operate with their highest power-added efficiency (PAE) when used in full saturation. – Weighted (Taylor): There is a range of aperture weighting distributions possible, each with differing sidelobe control capabilities and aperture efficiencies. Of these, the Taylor two-parameter model is the most efficient, with deterministic forms for rectangular, circular, and elliptical apertures. – Aperture efficiency: Aperture efficiency is defined as the ratio of the total power radiated relative to a uniform aperture weighting. – Synthesized: Modern far-field synthesis methods allow various sidelobe and main beam controls, including sidelobe sector control, multiple sidelobe levels (magnitudes), shaped sidelobe regions, sector beams, and various tapered and shoulder regions. – Nulled sectors: One of the advantages of a vector-controlled array is the ability to control sidelobes within a sector. Fresnel zones: The radiated field characteristics depend on the separating distance between aperture and observer, for example: – Reactive fields: Within a region extending over the aperture surface dimensions and approximately 1 wavelength above the aperture plane reactive fields propagate freely, excited by the radiating elements. These fields dissipate and transform at locations outside of the reactive field region, converging to the near-field and far-field regions. – Near fields: These fields exist in a region extending from the reactive field region to approximately 0.25D 2 / λ , where (D) is the maximum aperture dimension. The radiated fields are generally contained within the aperture surface dimensions throughout the near-field region making this a good area for array metrology. Near-field test ranges are a valuable metrology method and have largely replaced far-field design verification techniques. – Far field: The far field is located at a minimal distance of 2D 2 / λ . At this separation distance, the near and reactive field aperture vectors correlate to form the peaks and sidelobes, associated with interferometry. The vast majority of antennas and arrays operate in the far field.

The polarization domain introduces additional degrees of freedom in terms of aligned and intentionally misaligned states, generally considered in the far field, with the following degrees of freedom: •

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Polarization state: All EM radiation is polarized, meaning that the electric field aligns with one or a combination of linear, circular, elliptical, or arbitrary orientations. This is a major design parameter in microwave systems, because it can be used to polarization-align intended communications and sensor links or to isolate them.

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Linear polarization: The electric field vector is aligned with a plane. Linearly polarized far-field vectors can be aligned in the principal planes, referred to as horizontal and vertical polarizations or as E θ , E ϕ . – Circular polarization: The electric field rotates at the rate of its oscillation in the time domain (ω c). The circularly polarized TEM field vector forms a circle whose plane is oriented perpendicular to the axis of propagation or the Poynting vector. – Elliptical polarization: The electric field rotates at the rate of its oscillation in the time domain, forming an ellipse in the electric field plane. – Diagonal linear polarization: The electric field vector is aligned with a plane oriented midway between the principal planes. Multiple states: The various polarization states can be either separated or combined in the time domain: – Single polarization: A single polarization state exists and dominates the others. – Dual orthogonal polarization: Two orthogonal polarization states exist at the same time, sometimes with separate modulations. – Polarization diversity: The diverse polarization state arises from a dual orthogonally polarized array and can produce any polarization state on the Poincaré sphere [4]. Simultaneity: Simultaneity refers to the time-domain use of a dual polarized system: – Sequential dual polarization: Dual orthogonal polarizations operate at separate times. Two separate radar pulses are needed to retrieve the full polarization state information. – Simultaneous dual polarization: The simultaneous dual polarization states produce a polarization diverse system with a single pulse. Polarization isolation: Polarization isolation refers to the cross-polarization or orthogonal polarization content relative to the copolarization or dominant polarization. For example, an elliptical polarization state with the right-hand circular component defines the dominant polarization state, with the ratio to the left-hand circular component defining the polarization isolation. A perfect RHCP wave has no LHCP content, for example: – Projection effects: When scanned in the principal and diagonal planes, the polarization isolation has significant polarization isolation effects due to vector projection. – Scan dependency: The polarization isolation is separately affected by the divergent impedance of E and H-plane polarizations at wide scan angles. – Active compensation: Polarization isolation errors introduced by a variety of sources can be corrected by superimposing the cross-polarization at 180° phase. –

•

•

•

4.3

Array Architecture An understanding of the available degrees of freedom makes the definition of possible array architectures and design alternatives a tractable problem. Our scope is

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4.3  Array Architecture223

confined to current and next generation architectures using planar and conformal active apertures, advanced lattice structures, synthesized far-field beams, STAPcapable systems, polarization diversity, and arrays that can be hybrid beamformed, several of which are highlighted for the sake of illustration. 4.3.1  Case 1: Hybrid Beamformed, Single Polarization

By parameterizing the degrees of freedom, we define the phased array key performance parameters (or requirements) and subsequently the architecture of components needed to complete the array functions. The following case of a hybrid beamformed, single polarization array (Table 4.1) represents a variety of sensor systems, and, although other values are equally possible, it serves as an illustrative example. In Table 4.1, we have an assembly of phased array performance parameters that are generally suited to a sensor, whether used for terrestrial, air, or space applications. Proceeding from right to left in the array block diagram (Figure 4.3), we have the following components: •

Radiator: The free-space performance in terms of scan volume, scan loss, polarization and its isolation, and operating bandwidth are predominantly governed by the radiator. It uses an assembly of boundary conditions to form the reactive fields and eventually the far-field embedded element response. In this case, the radiator uses a fixed single vertical polarization, so there is a single port that interfaces between the radiator and the T/R module.

Table 4.1  Case 1: Phased Array Degrees of Freedom and Key Performance Parameters Parameter

Performance

Affected Architecture

9.5–10.5 GHz

All RF components

Instantaneous bandwidth

100 MHz

All RF components

System timing

Half-duplex

Switch control

Waveforms

Frequency modulated

RF source

Time processing

Pulse compressed

Radar processor

Time sidelobes

Compensated

Radar processor

STAP

Pulse-Doppler

Radar processor

60° conical

Radiator and T/R module Power combiners, DREX

Frequency Time Operating frequency range

Real and Imaginary Space Independently scanned beam Beamforming

Single beam

Beamforming implementation

Hybrid beamformed

Power combiners, DREX

Modular subarrays

Geometrically defined

Subarray design

Aperture distributions

Sinc (Tx), Taylor weighted (Rx)

Radar processor, T/R module

Fresnel Zone

Far-field

Array

  Polarization state

Single polarization

Radiator and T/R module

  Multiple states

Linear, vertical

Radiator and T/R module

Polarization

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Array Fundamentals: Supporting Theories, Part I

•

•

•

•

•

Radiator boundary conditions: The radiator assembly is a geometrical arrangement of dielectric and conducting materials that form its boundary conditions. These transition EM fields from the single-mode source or port to the reactive fields of the aperture and the radiated fields in free space. T/R module: The transmit-receive module controls the RF in the Tx and Rx operating modes. For a half-duplex system, the switches and internal logic activate the Tx and Rx circuits in a series repetitions of Tx, gap, Rx, gap in the time domain. During the gap period between Tx and Rx actions, the T/R module accepts beam commands for the next pulse. – The T/R module supplies the Tx and Rx functions using separate components. The module design isolates functions while optimizing the electrical performance for both. – Vector control originates with the attenuator and phase shifter. Logic commands to these two components control the vector state (i.e., the amplitude and phase for each beam state and pulse). – Tx/Rx switches control the RF pathway and half-duplex timing in the T/R module. Some systems use a circulator instead of a switch between the radiator and the PA and LNA. Combiners: In simple terms, these are termed power combiners, but they also perform the Fourier transform by combining the associated unit cell vectors, generally in two orthogonal planes. Small arrays may have a single subarray, so the combiners perform the full aperture Fourier transform in that case. Hybrid beamforming: By connecting each subarray beam port to a DREX, the beamforming architecture becomes a hybrid version. Not shown: Components not shown, include the functional equipment needed for structure, DC, logic, thermal management, and environmental protection.

Figure 4.3  A single polarization hybrid beamformed phased array RF architecture.

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4.3  Array Architecture225

4.3.2  Case 2: Analog Beamformed, Dual Simultaneous Polarization

In a similar manner, we next consider a phased array system used for a polarizationdiverse communications link. In this system, the modem is the source of both the Tx and Rx communications waveforms and often employs advanced modulation schemes, including phase modulation to improve the system SNR. The array is as summarized in Table 4.2 and Figure 4.4. Table 4.2  Case 2: Phased Array Degrees of Freedom and Key Performance Parameters Parameter Frequency Time Operating frequency range Instantaneous bandwidth System timing Waveforms Time processing Time sidelobes STAP Real and Imaginary Space Independently scanned beam Beamforming Beamforming implementation Modular subarrays Aperture distributions Fresnel zone Polarization Polarization state Multiple states

Performance

Affected Architecture

9.5–10.5 GHz 100 MHz Half-duplex Phase modulated None None None

All RF components All RF components Switch control RF source Modem processor Modem processor Modem processor

60° conical Single beam Analog Geometrically defined Taylor weighted (Tx), Taylor weighted (Rx) Far-field

Radiator and T/R module Power combiners, DREX Power combiners, DREX Subarray design Radar processor, T/R module

Dual polarization Simultaneous polarization

Radiator and T/R module Radiator and T/R module

Array

Figure 4.4  A simultaneous dual-polarization analog beamformed phased array RF architecture.

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4.4

Array Fundamentals: Supporting Theories, Part I

Practical Limits When considering phased arrays, network theory becomes an integral part of the discussion, along with the important physical aspects, such as the energy conservation theorem, the near-field and far-field regions, reciprocity, duality, and the subject of superposition. We consider each of these principles from the perspective of their fundamental physics but apply them to contemporary microwave and millimeterwave EM systems. 4.4.1  Theorem of Reciprocity

Reciprocity in antennas relies on both the Lorentz reciprocity theorem [5] and network theory [1, 6]. The former can be stated in terms of circuit parameters [7]: “in any network composed of linear, bilateral lumped elements, if one places a constant current (voltage) generator between two nodes in any one branch and places a voltage (current) meter between any other two nodes (in any other branch), makes observations of the meter reading, then interchanges the locations of the source and the meter, the meter reading will be unchanged.” Starting with the fundamental theory, consider a volume (V) an enclosing surface (S) with two radiating sources, producing radiated current sources (J1, J2) and magnetic current sources (M1, M 2), both oscillating in the time domain at (ej ω t) (although we suppress this last term). It is important to recognize that the timevarying aspect of these fields is responsible for their propagation in free space. These sources produce electric and magnetic fields (E1, H1) and (E2 , H2). The propagating medium is assumed to be linear, passive, and isotropic, and the source and receiver are assumed to be in each other’s far-field regions. Balanis stated that the source and field relations become [7]: −∇ ⋅ ( E1 × H2 − E2 × H1 ) = E1 ⋅ J 2 + H2 ⋅ M1 − E2 ⋅ J1 − H1 ⋅ M2 (4.6)



Converting into a volume integral and applying the divergence theorem results in the Lorentz Reciprocity Theorem,

!∫ ( E1 × H2 − E2 × H1 ) ⋅ ds′ = ∫∫∫ ( E1 ⋅ J2 + H2 ⋅ M1 − E2 ⋅ J1 − H1 ⋅ M2 ) dv′ (4.7) S

V

In a source-free region,

J 1 = J 2 = M 1 = M 2 = 0 (4.8)

So

!∫ ( E1 × H2 − E2 × H1 ) ⋅ ds′ = 0 = ∫∫∫ ( E1 ⋅ J2 + H2 ⋅ M1 − E2 ⋅ J1 − H1 ⋅ M2 ) dv′ S

V

(4.9)

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4.4  Practical Limits227

Separating terms within the volume integral,



∫∫∫ ( E1 ⋅ J2 + H2 ⋅ M1 ) dv′ = ∫∫∫ ( E2 ⋅ J1 − H1 ⋅ M2 ) dv′ (4.10) V

V

So the reaction or coupling between source and receiver is equivalent independent of whether the source is 1 or 2 and the receiver 2 or 1. The coupling between the source and receiver is reciprocal, in terms of their bra-ket notation [8], 1|2 =

∫∫∫ ( E1 ⋅ J2 + H2 ⋅ M1 ) dv′ V

2|1 =

∫∫∫ ( E2 ⋅ J1 − H1 ⋅ M2 ) dv′ (4.11) V



1|2 = 2|1

This indicates that the source and receiver can be swapped with no change in results. Using standard network notation, the reciprocity theorem can be converted to a standard two-port network, where ports 1 and 2 represent antennas in each other’s far field [9]. So the relationship between the mutual coupling (V21, V12) and the source currents (I1, I2) becomes: V21 V12 = I1 I2 (4.12)



Therefore, the mutual impedances must be equal:



Z12 = −

V12 V = − 21 = Z21 (4.13) I2 I1

The trivial case illustrating this effect includes two antennas, with a free-space medium between them and no other sources present in that space. If we generate a current in one antenna and measure the open circuit voltage in the other and then reverse the current source and meter, the voltage measurement is unchanged. This is also true if we excite one antenna with a given voltage and then measure the current in the other antenna and then reverse the source and meter locations. This means that a uniform aperture distribution on an array produces a sinc far-field response independent of whether the array operates in the Tx or Rx mode. Also, an array aperture is reciprocal unless it contains nonreciprocal components. For example, a Tx/Rx switch often occurs between the radiators and the PA and LNA amplifier chains. If a circulator is used instead, it isolates the aperture impedance from that of the amplifiers, so the network becomes nonreciprocal. When a switch is used instead, it can result in amplifier pulling effects that can cause significant array performance degradation. For instance, the PA pulling effect can change the aperture distribution and reduce the effective isotropic radiated power (EIRP).

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A network is reciprocal if each and every one of its constituent components is reciprocal. Similarly, a network is nonreciprocal if one or more of its components is nonreciprocal. 4.4.2  Conservation of Energy

From circuit theory, Kirchhoff’s second law, states that for any closed system the sum of the energy (power) at all input ports equals the total power on all output ports, including equivalent ports representing the conversion of microwave power to heat. Total input power equals total output power. An important interpretation of this law comes from Parseval’s theorem, which states that the Fourier transform is unitary. This can be interpreted as the relationship between a time-dependent signal x(t) and its frequency-domain equivalent X(ω ), ∞

∫



x2 ( t ) dt =

−∞

∞

1 ∫ X 2 ( w ) dw (4.14) 2p −∞

From our study of aperture fields, a similar expression applies to the total power available on an array aperture and its far field, M

N

∑ ∑ a m,n

m=1 n=1



2

p p /2

=  ∫

∫

E2 ( r ) sin q dq df (4.15)

−p 0

4.4.3 Superposition

According to [10, 11], “The superposition principle … states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.” In terms of electromagnetics, for a time-varying source field x1(t) producing a resultant field X1(t) and one or more additional source fields xn(t) with resultants Xn(t), the total response is:

F ( x1 ( t ) + xn ( t )) = X1 ( t ) + Xn ( t ) n=2,3,… (4.16)

There are many natural examples of the superposition principle. One is the superposition of both a spherical wave on a water body with other planar and stochastic waves (Figure 4.5). Each individual water molecule simply responds to the total force exerted on it, independent of the wave’s source or direction. So its response is to the arithmetic sum of these forces. An orchestra is another example, where the sound produced by any one instrument is generally independent of the sounds produced by others in the orchestra. They superimpose in the time domain to produce what we visualize as a multidimensional combination, complete with harmonics produced in the frequency domain. When we consider a phased array, one way to apply the superposition principle is to superimpose two aperture phase distributions onto a common aperture at the

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4.5  Near and Far Fields229

Figure 4.5  A water pattern set showing the superposition of both spherical and stochastic waves [12].

same frequency. If these individual aperture distributions scan the main beam to two different sine-space locations [uo(1), uo(2)], the vector addition of the two onto the aperture produces two main beams at the intended locations at the same time. This is a fundamental element of a far-field radiation pattern synthesis, a powerful array tool that we will study in subsequent sections. 4.4.4  Duality Theorem

Balanis introduced the subject of the duality theorem in clear and simple terms. It is a theory that is so routinely used in the EM industry that it often is simply assumed to be generally understood by all involved. This makes its precise statement all the more important, so to quote [7], consider that “when two equations that describe the behavior of two different variables are of the same mathematical form, their solutions will also be identical. The variables in the two equations that occupy identical positions are known as dual quantities and a solution of one can be formed by a systematic interchange of symbols for the other. This concept is known as the duality theorem.” Although there are many examples, as adequately illustrated in [7], a common one underlies the fact that many array analysis outcomes solve for the electric field, recognizing that the magnetic field is readily obtained from it. Even though there are no natural magnetic sources, there is a dual equivalence in the form of the wave equation, relating the vector potential (A) to the source currents (J) and the dual equation relating the magnetic vector potential (F) to the virtual magnetic source currents (M) ∇2 A + k2 A = −mJ

4.5

∇2 F + k2 F = −εM

(4.17)

Near and Far Fields The field behavior in an array’s near-field and far-field regions is considerably different, an observation that is also true of antennas and radiating devices in general. In

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the far-field or Fraunhofer region, we know that the radiated fields spread in angle space as a single TEM propagating mode, producing peaks and nulls related to the aperture fields by the Fourier transform. This applies to all array types, whether planar or conformal. In the near field, by comparison, the radiated fields are both more amorphous and more confined and produce electric field variation generally mapped in Cartesian, rather than angle space. On the aperture itself, the fields are reactive and so composed of many (thousands) of equivalent cavity modes, dependent on the aperture’s electrical size and radiator boundary conditions. An example of the near-field behavior can be seen in a typical spotlight used for performances, where the light is confined to the aperture dimension (usually circular), gradually spreading as it progresses into the far field. The wavelength at the center of the visible light spectrum is approximately 600 nm, so the far-field distance for a 12-inch aperture diameter spotlight yields a standard far-field distance (R) of 300 km:



R=

2D2 (4.18) l

As a result, at a representative stage distance of 50m or less, the light is contained in a cylindrical tube confined by the near-field of the spotlight, with a separation of



R=

0.0003D2 (4.19) l

By contrast, in the far field, an array’s radiation appears to project from a single point source, located at its equivalent phase center, projecting a spherical TEM wave to the far field. The spherical wave produces a spherical phase and amplitude projection onto a virtual plane at the observer’s location, representing a planar sampling aperture in the far field. We can now explore the aperture field regions using this representation of the physics. 4.5.1  The Far-Field Criterion

When the observation point is separated from a planar antenna or array aperture by a finite distance, a phase error is introduced. From separation distances less than those associated with the far field, the observer records the near-field or Fresnel properties of the antenna under test. The far-field or Fraunhofer region is nevertheless approximated by a separation distance that is proportional to the maximum aperture dimension (D),



R≥

2D2 (4.20) l

This far-field criterion is predicated on a maximum induced phase error of 22.5° or λ /16 and is considered the standard far-field definition [13]. The derivation shows that the resulting phase error distribution has a quadratic form.

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4.5  Near and Far Fields231

If the test array has ultralow sidelobes, accurate determination of the far-field performance requires a reduced phase error and a greater source separation, as the 22.5° error becomes a significant error source relative to the sidelobe level. For example, an equivalent error of 2.25° may be needed instead, in which case,



R≥

10D2 l

(low sidelobes ) (4.21)

Consider a planar antenna aperture aligned parallel to the x-z plane, with its center element at the origin of a Cartesian coordinate system (Figure 4.6) and the

Figure 4.6  Far-field antenna geometry showing source and far-field phase centers (above) and simplified planar geometry (below).

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far field interpreted as a planar arrangement of phase centers at the observer’s location. We examine the far-field projection to a plane at a finite distance from the antenna, satisfying the condition that (y = r). Consider a far-field observer at a distance (R) from the aperture, where the aperture fields have resolved into single far-field propagating mode. In the receiving aperture, the observer records a spherical wave that projects rays of path lengths (R) from the center of the array aperture and (R + ε ) to the aperture edge. By the Pythagorean theorem at the edge of the observer’s receiving aperture, we have

( R + e )2



= R2 +

D2 4 (4.22)

Expanding, R2 + 2Re + e 2 = R2 +



D2 (4.23) 4

Combining terms, R=



D2 e − 8e 2 (4.24)

Substituting ε = λ /16, equivalent to the 22.5° error, and removing the insignificant term, R=



2D2 (4.25) l

This is the standard far-field separation distance for any antenna, including array. For a specific far-field distance, the (R) term can be generalized, R=



nD2 l (4.26)

Back-substituting, the phase difference of the antenna aperture projection to the observation point is: R−r =

( z′ )2



2r

l ( z′ ) l = 2 (4.27) 2nD 2nD ( z′ ) 2

=

The incremental phase error for z′ = D/2 is: Δf =

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l

2nD ( z′ )

2

=

pD2 p  ( radians ) (4.28) 2 = 4n 4nD

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4.5  Near and Far Fields233

Figure 4.7  Two examples of the phase error produced by finite far-field separation from a source antenna aperture.

For (n = 2), the phase error becomes the standard of 22.5°. We also recognize that (4.27) has the same form as a parabolic (y = az 2). Plots of the phase error for two conditions of n illustrate this (Figure 4.7). The parabolic phase distribution represents the projection of a spherical wave emitted by the source antenna, projected onto the y = r plane. At greater separations from the source antenna, the phase front shows less phase variation, converging to a plane wave when (r = ∞). When an antenna operates over a range of frequencies, it is important to evaluate the far-field separation condition at the highest frequency, as this represents the most stressing condition. Example 4.1

For an array whose maximum aperture dimension is 20.0m, operating at a frequency of 10 GHz, the far-field phase error must be no greater than 2°. Determine the farfield separation needed. Relative to our Earth system, where would you think this array would be located? Δf =





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n=

p (4.29) 4n

180° p = = 22.5 (4.30) 4Δf 4 ∗ 2°

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l=

c 2.99793 ∗ 108 = = 0.0299 m (4.31) f 2.99793 ∗ 108

r =

2D2 2 * 202 = = 301.1 km (4.32) l 0.0299

The 300-km separation needed to be in this array’s far field is comparable to the altitude of the International Space Station (ISS), at 350 km. The array could be in the lower region of low Earth orbit (LEO) or further from the Earth’s surface.

4.5.2  Array Reactive and Near Fields

The near-field radiation conditions are considerably more complex than the singlemode, far-field analysis and can be solved using Geometric Theory of Diffraction (GTD). Consider as a starting point having both source and observation arrays represented as planar arrangements of point sources within a common coordinate system (Figure 4.8). Both regions can be accurately represented as arrays of source and receiving elements, consistent with Huygens’ and Nyquist’s theories. The total field received by any element in the receiving (observer) array is the net sum of the fields produced by all visible source elements. In this case where both source and receiving apertures are planar, all elements of each array are visible to each other. Several other factors are involved in the field summation, summarized as:

Figure 4.8  The geometry of a source and observer apertures in a 3-D Cartesian coordinate system.

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4.5  Near and Far Fields235

•

The radiated field produced by each source element propagates in the form of the ideal free-space Green’s function, where (r) is the separation distance between source and receiving element phase centers, E=

•

The separation between element phase centers is a function of the Cartesian coordinates of each, r =

• •

( xs − xr )2 + ( ys − yr )2 + ( zs − zr )2 (4.34)

The source and receiving elements are assumed to be polarization-matched. The source elements produce an embedded element radiation pattern proportional to the transmission angle (θ ) and an (xs − xr)2 + (ys − yr)2 exponential index (n), Ee =

cosn q = cos0.5n q

cos q = n

1 − sin q = 2

•

•

e− jkr r (4.33)

2 2 xs − xr ) − ( ys − yr ) ( 1− ( xs − xr )2 + ( ys − yr )2 + ( zs − zr )2

(4.35)

The receiving array is assumed to be an ideal Huygens’ sampling array, so the embedded element factor is isotropic. Single bounce (reflection) analysis is generally sufficient for initial trends.

The total field at any observation point (i) is the sum of the source fields, weighted by the source excitation coefficients (an), Ei =



N

∑ anEe ( n)

n=1

e− jkrn rn (4.36)

It is useful to consider that the source elements generally fall into an ordered 2-D array, so that the summation occurs over these two dimensions. The observer array similarly is 2-D, bringing the total summation over an observer array to four dimensions in this simple illustration. When the source and observer arrays are 3-D, the total summation takes on six dimensions. When polarization and frequency domains are added, the domain space expands with two additional dimensions. It is not uncommon to process nine dimensions or more, with additional degrees of freedom. Several useful cases are: • • •

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A single source element; A small rectangular array; A large rectangular array.

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In the case of the single element, the far-field region is a relatively small distance away because the ideal element has a size of 0.5λ , 2 (0.5l ) 2D2 = = 0.5l (4.37) l l 2

R=



Also, as the separation distance from the element increases beyond the near-field region, the electric field magnitude decreases in proportion to the Friis equation, E =

Eo l 4pr

2 (4.38) ⎛P ⎞⎛ l ⎞ P = C⎜ o⎟ ⎜ ⎝ 2 ⎠ ⎝ 4pr ⎟⎠



As, according to Poynting’s theorem, P=



PoE2 2h (4.39)

where η is the free space impedance of 120 π Ω. This means that the electric field strength magnitude is reduced by a factor of 4 (−6 dB) for each doubling of the separation distance. So a single Nyquistsized element (A = 0.25λ 2) has an electrically small near-field region, the far field extending from a region starting at a 0.5λ separation from the element phase center and extending to infinity. The far fields spread in a spherical wave and attenuate according to the Friis equation as all that remains of the source fields is a single propagating TEM wave. The near field is comparatively small, and the reactive field region is smaller yet. Near-field conditions change considerably when we next consider a source array composed of 100 elements arranged in a 10 × 10 wavelength square grid at a spacing of λ , scanned to boresight. The far-field distance becomes 2D (10l ) 2D2 = = 200l (4.40) l l 2

R=



The radiated fields derived from (4.36) now show a considerable difference in both near-field and far-field regions (Figure 4.9). Observations can be summarized as: •

•

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In the far field, observed distinct peak and null magnitudes occur, and their arrangement is doubly periodic, with symmetry in the two cardinal planes [xr, yr]. The far field has a peak radiation region and secondary (sidelobe) radiation regions, where the latter have a maximum magnitude of −13 dB relative to the peak, and so is consistent with a double sinc function in the observer’s cardinal planes.

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4.5  Near and Far Fields237

Figure 4.9  Radiated fields for a 10 × 10 element source array, up to the far-field region (2 D 2/λ ).

•

•

•

•

•

The main beam radiated phase at the beam maximum stabilizes and has a parabolic distribution in two orthogonal planes, consistent with the phase distributions predicted for a far-field source radiating a double sinc function. The radiation magnitude follows the Friis equation as a function of separation distance from the source aperture and in the far-field region (2 D 2 / λ ), but begins to deviate from this relationship in the near field (