Microwave Devices and Antennas Based on Negative-Refractive-Index Transmission-Line Metamaterials. Marc A. Antoniades

Transcription

1 Microwave Devices and Antennas Based on Negative-Refractive-Index Transmission-Line Metamaterials by Marc A. Antoniades A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 29 by Marc A. Antoniades

2 Abstract Microwave Devices and Antennas Based on Negative-Refractive-Index Transmission-Line Metamaterials Marc A. Antoniades Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 29 Several microwave devices and antennas that are based on negative-refractive-index transmission-line (NRI-TL) metamaterials are presented in this thesis, which exhibit superior performance features compared to their conventional counterparts. These are a Wilkinson balun, a 1:4 series power divider, a four-element printed dipole array, a leakywave antenna, and an electrically small folded-monopole antenna. The Wilkinson balun employs +9 and 9 NRI-TL metamaterial lines at the output branches of a Wilkinson divider, to achieve a six-fold increase in the measured differential output phase bandwidth compared to that of an analogous balun employing transmission lines, while occupying only 55% of the area. The 1:4 series power divider comprises four non-radiating NRI- TL metamaterial lines, each with a compact length of λ /8, to provide equal power split to all four output ports. Compared to a conventional series power divider employing onewavelength long transmission lines, the metamaterial divider provides a 154% increase in the measured through-power bandwidth, while occupying only 54% of the area. The metamaterial series power dividing concept is also applied to a four-element fully-printed dipole array that is designed to radiate at broadside, in order to demonstrate that the array exhibits reduced beam squinting characteristics. It is shown that the metamaterialfed array has a measured scan-angle bandwidth that is 173% greater than an array that is fed using a conventional low-pass loaded line. The reduced-beam squinting property that NRI-TL metamaterial lines offer is subsequently exploited to create a leaky-wave antenna ii

3 that radiates a near-fixed beam in the forward +45 direction, with an average measured beam squint of only.31 /MHz. This is achieved by operating the antenna in the upper right-handed band where the phase and group velocities are the closest to the speed of light. Finally, an electrically small antenna comprising four NRI-TL metamaterial unit cells is presented which supports a predominantly even-mode current, thus enabling it to be modeled as a multi-arm folded monopole. This significantly increases its radiation resistance, which allows it to be matched to 5 Ω, while maintaining a high measured efficiency of 7%. iii

4 Αρχή σοφίας φόβος Κυρίου, και βουλή αγίων σύνεσις το γάρ γνώναι νόμον, διανοίας εστίν αγαθής. Παροιμίαι Σολωμώντος, ΙΧ. 1 The fear of the Lord is the beginning of wisdom, and the counsel of the saints is understanding; for to know the law is the character of a sound mind. Proverbs 9:1 iv

5 Acknowledgements I would like to express my sincere appreciation to my supervisor Prof. George V. Eleftheriades, who has skilfully mentored me throughout my graduate studies. I am grateful for all of your guidance and support, and for teaching me discerning new ways to approach engineering problems. Perhaps more importantly, however, you have taught me the importance of taking personal responsibility and pride in one s own work and to do all things in life with ethical integrity. For this and for our close friendship I am very grateful. I would like to thank Prof. Keith G. Balmain, Prof. Costas D. Sarris, Prof. Sean V. Hum, Prof. Mo Mojahedi and Prof. Sergei Dmitrevsky of the Electromagnetics group for numerous motivating discussions which have shaped my understanding of many concepts in Electromagnetics. I would also like to thank Prof. Sarris, Prof. Hum, Prof. Mojahedi and Prof. Michal Okoniewski from the University of Calgary for being members of my Ph.D. examination committee and for providing me with valuable feedback on this thesis. I would like to acknowledge our lab managers Gerald Dubois and Tse Chan for their assistance with the many practical aspects of my work. Thanks are also due to all of my fellow graduate students in the Electromagnetics group who have greatly enriched my graduate experience at U of T through our lively technical discussions, our adventurous outings, as well as our many coffee breaks. Many of you I consider friends for life. I would also like to acknowledge the financial support that I have received from the Ontario Graduate Scholarship, the Ontario Graduate Scholarship in Science and Technology, the Edward S. Rogers Sr. Ontario Graduate Scholarship, and the University of Toronto Department of Electrical and Computer Engineering Graduate Scholarship. Last but not least, I am deeply grateful for the support, love and encouragement of my parents, my sister Margarita, my brother-in-law Vincent, and my beloved girlfriend Diana. The journey throughout my graduate studies has been long and rewarding, but one which I could not have completed without your prayers and loving support. v

6 Contents List of Acronyms List of Symbols List of Figures List of Tables x xi xiv xx 1 Introduction Motivation Metamaterial Background Overview Negative-Refractive-Index Transmission-Line Metamaterials Thesis Outline Negative-Refractive-Index Transmission-Line Theory Background Positive Refractive Index (PRI) Media Incremental Circuit Analyzed Using Telegrapher s Equations Lumped-Element Circuit Analyzed Using Periodic Analysis Negative Refractive Index (NRI) Media Incremental Circuit Analyzed Using Telegrapher s Equations Lumped-Element Circuit Analyzed Using Periodic Analysis Negative-Refractive-Index Transmission-Line (NRI-TL) Metamaterial Medium Proposed Phase Compensating Structure Propagation Characteristics of the T Unit Cell vi

7 2.4.3 Propagation Characteristics of the π Unit Cell Effective Medium Propagation Characteristics Multi-Stage NRI-TL Metamaterial Phase-Shifting Lines Loading-Element Values Non-radiating NRI-TL Metamaterial Phase Shifting Lines Analysis of MTM, NR-MTM and TL Phase-Shifting Lines Choice of the Number of Unit Cells Phase Variation Characteristics A NRI-TL Metamaterial Wilkinson Balun Introduction Principle of Operation Design Practical Implementation Simulation and Experimental Results A NRI-TL Metamaterial Series Power Divider Introduction Power Divider Architecture Design of the NR-MTM lines Design of the NR-MTM and TL Power Dividers Simulation Results Using Ideal, Lossless Components Non-Ideal Simulation and Experimental Results A NRI-TL Metamaterial Series-Fed Antenna Array Introduction Uniform Linear Arrays Employing True-Time Delay Phase Shifters Tapered Amplitude Distribution in Linear Arrays Transmission-Line Fed Series Uniform Linear Arrays Low-Pass Loaded Transmission Line Metamaterial-Fed Series Uniform Linear Arrays Transmission Line and Metamaterial Series-Fed Printed Dipole Arrays Four-Element Series-Fed Printed Dipole Array Grounded Four-Element Series-Fed Printed Dipole Array Physical Realizations of the Proposed Structures vii

8 5.6.4 Simulation and Experimental Results A NRI-TL Metamaterial Leaky-Wave Antenna Introduction Proposed Structure LWA Design Leaky Transmission Line Model Reduced Beam Squinting Principle of Operation Design of the NRI-TL Metamaterial LWA General Design Considerations Design Procedure for the NRI-TL Metamaterial Unit Cell Physical Realization in CPS Technology Determination of the Complex Propagation Constant Printed Balun Design Simulation and Experimental Results Return Loss Far-Field Radiation Patterns Beam Squinting and Gain Characteristics An Electrically Small NRI-TL Metamaterial Antenna Introduction Theory of Operation Proposed Topology Even-Odd Mode Analysis Design Physical Implementation Simulation and Experimental Results Conclusion Summary Contributions Future Directions A Equivalent Circuit for the NRI-TL MTM Medium 211 A.1 Incremental Circuit Analyzed Using Telegrapher s Equations viii

9 A.2 Lumped-Element Circuit Analyzed Using Periodic Analysis B Choice of Z in a Series Power Divider 217 C Beam Squinting Analysis 223 C.1 Derivation of the Beam Squinting Equation C.2 Derivation of the Approximate Beam Squinting Equations C.3 Group Velocity Analysis D Measured Efficiency Using the G/D Method 23 D.1 Directivity D.1.1 Numerical Integration Technique D.2 Gain and Efficiency D.3 Measured Radiation Pattern Data E Measured Efficiency Using the Wheeler-Cap Method 239 E.1 Background E.2 A Modified Wheeler Cap Method by McKinzie E.3 Practical Considerations E.4 Measured Reflection Coefficient Data References 25 ix

10 List of Acronyms ADS AF BW CPS HFSS LH LHM LPL LWA MIMO MTM NRI NRI-TL NR-MTM PRI RH RHM SMA SRF SRR TEM TL TLM TTD Advanced Design System by Agilent Technologies Array Factor Backward-Wave Coplanar-Strip High-Frequency Structure Simulator by Ansoft Corporation Left-Handed Left-Handed Medium Low-Pass Loaded Leaky-Wave Antenna Multiple-Input Multiple-Output Metamaterial Negative Refractive Index Negative-Refractive-Index Transmission-Line Non-Radiating Metamaterial Positive Refractive Index Right-Handed Right-Handed Medium Sub-Miniature version A Self-Resonant Frequency Split-Ring Resonator Transverse Electro-Magnetic Transmission Line Transmission-Line Matrix True-Time Delay x

11 List of Symbols α Attenuation constant or leakage constant β Propagation constant β tl β bw β bl β mtm γ Propagation constant of a transmission line Propagation constant of a backward-wave line Bloch propagation constant Propagation constant of a metamaterial phase-shifting line Complex propagation constant γ sc Antenna array main beam scan angle, measured from the y-axis (Ch. 5) ɛ Permittivity ɛ ɛ r ɛ eff η rad θ Permittivity of free space Relative permittivity Effective relative permittivity Radiation efficiency Electrical length of a transmission line with length d tl θ bw Electrical length of a backward-wave line with length d bw θ mtm Electrical length of a metamaterial line with length d mtm θ sc Antenna array main beam scan angle, measured from the z-axis (Ch. 6) λ λ g μ μ μ r σ Free-space wavelength Guided wavelength Permeability Permeability of free space Relative permeability Electric conductivity xi

12 τ g τ g, φ tl φ h-tl φ bw φ mtm ω ω Γ Δz Φ mtm Φ nr-mtm Φ c-tl Φ c d d tl d bw d h-tl d c-tl d mtm d tot d E d G f k n tan δ v φ v g Group delay Group delay in vacuum Phase incurred by a transmission line of length d tl Phase incurred by a transmission line of length d h-tl Phase incurred by a backward-wave line of length d bw Phase shift per unit cell of a metamaterial phase-shifting line Angular frequency Angular design frequency Reflection coefficient Incremental length along the z-direction Total phase incurred by a metamaterial phase-shifting line Total phase incurred by a non-radiating metamaterial phase-shifting line Total phase incurred by a phase-compensating section of transmission line Inter-element phase shift in a linear antenna array Speed of light in vacuum Length of a periodic unit cell (Ch. 2 and 6). Length of a meandered transmission line in a TL-fed series antenna array (Ch. 5) Length of a section of transmission line Length of a section of backward-wave line Length of the host transmission line in a NRI-TL metamaterial unit cell Length of the phase-compensating transmission line in a NR-MTM line Length of a section of metamaterial line Total length of a phase-shifting line Inter-element spacing in a linear antenna array Ground plane distance in a grounded linear antenna array Design frequency Free-space wave vector Refractive index Dielectric loss tangent Phase velocity Group velocity xii

13 C C L L I I e I o R r R l V Y Z Z bl Z Z,bw Z in Z term k B D Ē H M P S Capacitance Series loading capacitance in a metamaterial unit cell Inductance Shunt loading inductance in a metamaterial unit cell Current Even-mode current Odd-mode current Radiation resistance Loss resistance Voltage Admittance Impedance Bloch impedance Characteristic impedance of a transmission line Characteristic impedance of a backward-wave line Input impedance Termination impedance wave vector Magnetic flux density vector Electric flux density vector Electric field intensity vector Magnetic field intensity vector Magnetic polarization vector Electric polarization vector Poynting vector xiii

14 List of Figures 1.1 Two-dimensional NRI-TL metamaterial unit cell Circuit diagrams used to demonstrate the link between transmission-line based and wire/srr based NRI metamaterials Incremental equivalent circuit of a continuous PRI medium Generic incremental equivalent circuit of a continuous medium Generic lumped-element unit cell used to model a lumped-element line Dispersion diagrams for a representative low-pass PRI medium Incremental equivalent circuit of a continuous NRI medium Dispersion diagrams for a representative high-pass NRI medium Method of phase compensation using PRI and NRI lines NRI-TL metamaterial phase-compensating structure n-stage NRI-TL metamaterial phase-shifting line NRI-TL metamaterial unit cells Generic NRI-TL metamaterial T unit cell Characteristics of a NRI-TL metamaterial T unit cell where Z,bw >Z Characteristics of a NRI-TL metamaterial T unit cell where Z,bw <Z Generic NRI-TL metamaterial π unit cell Characteristics of a NRI-TL metamaterial π unit cell with Z,bw <Z Characteristics of a NRI-TL metamaterial π unit cell with Z,bw >Z Characteristics of representative NRI-TL metamaterial T and π unit cells with Z,bw = Z Phase responses of a four-stage NRI-TL metamaterial line analyzed using periodic analysis and Agilent-ADS Non-radiating NRI-TL metamaterial phase-shifting line xiv

15 2.2 Dispersion diagram indicating the non-radiating NRI backward-wave region of operation Schematic diagrams of a single-stage and an n-stage MTM line Phase responses of various MTM lines Schematic diagrams of a single-stage and an m-stage NR-MTM line Phase responses of various NR-MTM lines Schematic diagrams of an n-stage MTM line, an m-stage NR-MTM line, and a conventional TL Schematic diagrams of a 32-stage MTM line, a 16-stage NR-MTM line, and a TL Phase responses of a 32-stage MTM line, a 16-stage NR-MTM line and a TL Phase responses of various lines Insertion magnitude responses of various lines Proposed architecture of the NRI-TL metamaterial balun General balun architecture TL balun architecture MTM-TL balun architecture MTM balun architecture Schematic circuit of the MTM Wilkinson balun Photographs of the fabricated MTM and TL baluns Measured and simulated magnitude responses of the MTM balun Measured and simulated phase responses of the MTM balun Measured and simulated differential phase comparison Series-fed and corporate-fed microstrip patch arrays Schematic diagram of a TL 1:4 series power divider Schematic diagram of a NR-MTM 1:4 series power divider Insertion phase responses of a NR-MTM line and a λ g -long TL Ideal simulated magnitude responses for the NR-MTM and TL dividers Ideal simulated phase responses for the NR-MTM and TL dividers Photographs of the fabricated NR-MTM and TL power dividers Measured vs. simulated S 11 magnitude responses Measured vs. simulated S 21 magnitude responses xv

16 4.1 Measured vs. simulated S 31 magnitude responses Measured vs. simulated S 41 magnitude responses Measured vs. simulated S 51 magnitude responses N-element uniform linear array of isotropic elements Normalized array factor patterns for various amplitude factors Transmission-line-fed series uniform linear array of 4 elements Rectangular-to-polar graphical representation of the array factor Modified phase vs. frequency characteristic for a TL Unit cell of a low-pass loaded (LPL) transmission line Metamaterial-fed series uniform linear array of 4 elements Comparison of the scan angle from broadside for various arrays Geometrical arrangement of the 4-element printed dipole array Normalized ideal AF patterns for the TL-fed dipole array Normalized ideal AF patterns for the MTM-fed dipole array Normalized maximum AF for the TL-fed dipole array Normalized maximum AF for the MTM-fed dipole array Geometrical arrangement of the grounded 4-element printed dipole array Normalized ideal AF patterns for the grounded TL-fed dipole array Normalized ideal AF patterns for the grounded MTM-fed dipole array Comparison of the scan angle from broadside for various grounded arrays Normalized maximum AF for the grounded TL-fed dipole array Normalized maximum AF for the grounded MTM-fed dipole array Proposed series feed-network structures Simulated S 21 for the proposed feed-network structures Photograph of the fabricated MTM-fed printed dipole array Close-up view of the components of the non-radiating MTM line Photograph of the fabricated LPL-fed printed dipole array Close-up view of the printed components of the 36 LPL line Measured and simulated scan angles of the MTM-fed and LPL-fed arrays Measured patterns for the printed MTM-fed array around 5 GHz Measured patterns for the printed MTM-fed array around 7 GHz Measured patterns for the printed LPL-fed array around 7 GHz Measured patterns for the MTM-fed array without a ferrite balun xvi

17 5.31 Measured patterns for the MTM-fed array with a ferrite balun Gain vs. frequency for the MTM and LPL-fed arrays Measured and simulated S 11 responses for the MTM-fed array Measured and simulated S 11 responses for the LPL-fed array N-element uniform linear array of isotropic elements Dispersion diagram for a NRI-TL metamaterial unit cell Bloch impedance diagram for a NRI-TL metamaterial unit cell Bloch and approximate Bloch propagation constants, β bl and β bl,eff Scan angle versus frequency for a uniform NRI-TL metamaterial array Group velocity v g and phase velocity v φ using the values in Table Beam squinting performance of a NRI-TL metamaterial array Dispersion diagram for the proposed NRI-TL metamaterial unit cell Group velocity v g and phase velocity v φ for the proposed NRI-TL line Physical layout of the fully-printed CPS MTM unit cell Geometrical details of the printed unit cell components Phase and magnitude of S 21 for the proposed MTM unit cell Normalized leakage constant α/k and propagation constant β/k Broadband microstrip-to-cps balun transition Back-to-back microstrip-to-cps balun transition S 11 and S 21 for the two back-to-back microstrip-to-cps baluns Photograph of the NRI-TL metamaterial leaky-wave antenna Measured return-loss of the 2-element metamaterial LWA Simulated patterns for the printed metamaterial LWA Measured patterns for the printed metamaterial LWA Measured co/cross-polarization patterns for the MTM LWA at 5 GHz Scan angle vs. frequency for the 2-element metamaterial LWA Gain vs. frequency for the 2-element metamaterial LWA Electrically small NRI-TL metamaterial antenna Even-mode equivalent circuits for a single unit cell of the MTM antenna Odd-mode equivalent circuits for a single unit cell of the MTM antenna Microstrip test fixture used to determine the series capacitance C Microstrip test fixture used to determine the shunt inductance L Equivalent circuit and capacitance of a Panasonic ECD-GER capacitor. 189 xvii

18 7.7 Equivalent circuit and inductance of a Coilcraft 32CS inductor Microstrip test fixture used to determine the series capacitance C using the Panasonic equivalent circuit model shown in Figure 7.6(a) Microstrip test fixture used to determine the shunt inductance L using the Coilcraft equivalent circuit model shown in Figure 7.7(a) Total capacitance of the Panasonic ECD-GER89 capacitor, and total inductance of the Coilcraft 32CS-9N2XJL inductor Final metamaterial unit cell implemented in Ansoft HFSS S 21 phase and magnitude response of the MTM unit cell of Figure Current density on the coaxial feed cable for three ground plane sizes Current density on the four vertical vias and the exterior of the coaxial feed cable, and the E-plane linear patterns Current density on the top and bottom of the ground plane Fabricated prototype of the electrically small MTM antenna Simulated and measured return-loss for the MTM antenna Simulated and measured input impedance for the MTM antenna E-plane (xz-plane) patterns for the MTM antenna at 3.8 GHz H-plane (xy-plane) patterns for the MTM antenna at 3.8 GHz A.1 Equivalent circuit of a continuous NRI-TL metamaterial medium A.2 Dispersion diagrams for a representative band-pass NRI-TL medium B.1 Schematic diagram of a 1:2 TL series power divider B.2 Reflection coefficient of a 1:2 TL series divider as a function of θ B.3 Fractional bandwidth of a 1:2 TL series divider B.4 Fractional bandwidth of a 1:4 TL series divider C.1 Group velocity v g using the values in Table C.2 Group velocity dispersion using the values in Table D.1 DRH-118 double-ridged horn used for the antenna measurements E.1 Equivalent circuit of a small electric dipole E.2 Typical input reflection coefficient data used to illustrate the modified Wheeler cap method xviii

19 E.3 Various metallic spherical caps used to measure the radiation efficiency using the Wheeler cap method E.4 Photographs of the equipment used in the Wheeler cap method E.5 Measured input reflection coefficient data used to calculate the radiation efficiency using the modified Wheeler cap method xix

20 List of Tables 2.1 Characteristics for various MTM lines using approximate design equations Characteristics for various MTM lines using exact design equations Characteristics for various NR-MTM lines using approximate design equations Characteristics for various NR-MTM lines using exact design equations Phase variation characteristics of various MTM lines Phase variation characteristics of various NR-MTM lines Phase variation characteristics of the NR-MTM divider Phase variation characteristics of the TL divider Geometrical details of the MTM printed dipole array components Geometrical details of the LPL printed dipole array components Measured and simulated efficiency values for the MTM-fed array Measured and simulated efficiency values for the LPL-fed array NRI-TL metamaterial unit cell representative parameters Metamaterial unit cell characteristics for various unit cell sizes Geometrical details of the fully-printed MTM unit cell Geometrical details of the fully-printed microstrip-to-cps balun Comparison of LWA beam squinting characteristics xx

21 Chapter 1 Introduction 1.1 Motivation The challenges facing next-generation wireless communication systems are multi-faceted. On the one hand there is a trend towards the miniaturization of components associated with handheld mobile devices, including RF front-end subsystems. On the other hand there is a growing demand for faster data transfers, which in turn requires broadband and multi-band components. These two conflicting requirements must be met using low-cost solutions, that simultaneously maintain a high efficiency. The emergence of a new class of materials that have electromagnetic properties not typically found in nature, known as metamaterials, has generated much interest within the electrical engineering and physics communities for their potential to create new devices that exhibit superior qualities compared to their conventional counterparts. It has therefore been recognized by many researchers in the field that several important design constraints imposed by next-generation wireless systems can be addressed through the use of metamaterial technology. 1.2 Metamaterial Background Overview The recent growth in metamaterial research has also resulted in an analogous increase in the amount of publications in this area. Notable among these are four recent books that have been published on metamaterials, each with a different focus [1 4]. 1

22 Chapter 1. Introduction 2 A survey of metamaterial publications indicates that metamaterials can be generally classified into several categories. The first type of metamaterials, which will be the subject of this thesis, are ones that possess simultaneously negative values of electric permittivity ɛ and magnetic permeability μ, therefore resulting in a negative refractive index (NRI) [1]. Other types of metamaterials include electromagnetic bandgap (EBG) structures, otherwise known as photonic bandgap (PBG) materials or photonic crystals, which are artificial periodic structures which can control the propagation of electromagnetic waves within certain frequency bands [2] (Chapters 8 14). Two-dimensional EBGs are more commonly known as artificial surfaces, and have been used to implement high-impedance ground planes that form artificial magnetic conductors (AMC) [5] and frequency-selective surfaces (FSS) [6]. These artificial surfaces have been used to implement low-profile, conformal and flush-mounted antennas with improved radiation characteristics [5], and have also been used to reduce mutual coupling between array elements [7] and to suppress surface-wave modes [8] and parallel-plate waveguide modes [9]. Yet another type of metamaterial can be implemented using a hyperbolically anisotropic grid, which can be used to realize diplexers [1] or beam-steerable antennas [11] through the formation of resonance cones [12]. Negative refractive index materials were first theoretically proposed by Veselago [13], who theoretically predicted that a slab of NRI material would form a flat lens that could focus light from a source on one side of the lens to the opposite side. He also investigated some of the properties of NRI materials, and found that the group velocity would be oppositely directed to the phase velocity, and therefore these materials would support backward-waves. Separate realizations of negative permittivity and permeability were subsequently proposed by Pendry et. al. using thin metal wires [14] and splitring resonators (SRR) [15], respectively. These were combined together to produce a composite medium with simultaneously negative permittivity and permeability in [16]. Pendry also suggested that a slab of metamaterial with n = 1 would act as a perfect lens, restoring the evanescent-wave spectrum at the focal point [17], and the first experimental verification of negative refraction using a volumetric composite wire/srr medium was demonstrated by Shelby et. al. in [18]. Beyond this initial ground-breaking work, many microwave and antenna applications using the wire/srr medium have subsequently been developed, many of which are detailed in [4].

23 Chapter 1. Introduction 3 2 C 2 C 2 C 2 C L y x z d Figure 1.1: Two-dimensional NRI-TL metamaterial unit cell [21]. c 23 OSA Negative-Refractive-Index Transmission-Line Metamaterials An alternative method of synthesizing metamaterials with a negative index of refraction has been proposed by Professor G.V. Eleftheriades group at the University of Toronto, and is based on the concept of periodically loading a conventional microwave transmission line with lumped-element series capacitors and shunt inductors, to form a negativerefractive-index transmission-line (NRI-TL) metamaterial as shown in Figure 1.1 [19], [2], [21]. It can be recognized that the lumped-element loading of the line C and L has a high-pass configuration that forms a dual network representation to that of the host transmission line, which has a low-pass configuration. This duality provides the NRI-TL metamaterial structure with a unique phase-compensating ability that was inherent in Veselago s original lens and which is exploited in the work of this thesis. A detailed explanation of how negative material parameters are obtained from a loaded transmission-line structure similar to that of Figure 1.1 is provided in Chapter 3 of [1], based on simple electrodynamic considerations. The underlying mechanism that gives rise to a negative permittivity is based on the fact that the shunt inductors L introduce an electric polarization P that is antiparallel to the original electric flux density D = ɛ Ē, resulting in a total electric flux density of D = ɛ Ē + P. When the inductive loading is adjusted such that the electric polarization P becomes greater than D, the effective permittivity of the medium therefore becomes negative. In a similar manner, the series capacitors C generate a magnetic polarization P m that is antiparallel to the original magnetic flux density B = μ H, resulting in a total magnetic flux density of B = μ H + μ Pm. When the capacitive loading is adjusted such that the

24 Chapter 1. Introduction 4 i 1 2C 2C i 2 v 1 L 2 v d C (a) C C y.. r o.. z x d (b) Figure 1.2: Circuit diagrams used in [27] to demonstrate the link between transmission-line based and wire/srr based NRI metamaterials. (a) Unit cell of an ideal backward-wave line, and (b) a representation of the backward-wave line of (a) using an array of capacitively loaded coupled loop resonators, i.e. SRRs. c 27 IEEE. magnetic polarization P m becomes greater than B, the effective permeability of the medium becomes negative. Therefore, there is a direct correspondence between the values of the shunt loading inductors L and a negative permittivity and the values of the series loading capacitors C and a negative permeability, which consequently allows these types of transmission-line metamaterials to exhibit a negative refractive index. When the size of each unit cell d is much smaller than the guided wavelength, a periodic arrangement of the NRI-TL metamaterial unit cells canberegardedasahomo- geneous effective medium, and as such can be described by effective material parameters μ eff and ɛ eff.whenμ eff and ɛ eff are simultaneously negative, the metamaterial structures will exhibit a negative index of refraction, and this property has been used to experimentally demonstrate negative refraction and focusing [19 22], as well as sub-wavelength imaging beyond the diffraction limit [23], [24]. More recently, sub-diffraction imaging in free-space has also been experimentally demonstrated [25], [26]. The work on NRI-TL metamaterials has demonstrated that a large bandwidth can be obtained over which the refractive index remains negative. In addition, it has also been shown that the transmission losses through the media are lower than in the original wire/srr media [2]. An intuitive explanation for the increased bandwidth in NRI- TL media has been outlined in [27], and is based on the fact that the transmission-line

25 Chapter 1. Introduction 5 approach does not rely on loosely coupled resonators to synthesize the negative permeability, as in the case of the wire/srr medium, but rather depends on the tight coupling between adjacent resonant loops that are formed by loading the host transmission line with series capacitors and shunt inductors. Therefore, in relation to the familiar NRI backward-wave line shown in Figure 1.2(a), it has been shown analytically in [27] that an array of capacitively loaded coupled loop resonators, (i.e. SRRs) shown in Figure 1.2(b) can be used to represent such a backward-wave line. Furthermore, if the loaded resonators are closely coupled, this leads to an increased NRI bandwidth that is infinite in the ideal limiting case, which however is not causal. The analysis of the NRI-TL medium based on the coupled resonator approach therefore provides a link between the wire/srr and the transmission-line based NRI metamaterials, and offers further insight into the design and optimization of both types of metamaterials. The transmission-line approach to synthesizing NRI metamaterials has proven to be very practical, and has since been adopted by many research groups, due to the many performance advantages that it has to offer. In addition, the compact, planar nature of the NRI-TL metamaterials enables easy integration with other circuits and devices. Much work has been done to create metamaterial-based antennas and microwave devices with enhanced properties. Microwave applications of NRI-TL metamaterials include phase-shifting lines with small group delays [28], active phase-shifting lines [29 32], coupled-line couplers [33 37], branch-line couplers [38], [39], ring hybrid couplers [4], [41], resonators [42 46], filters [47 51], baluns [52 54], series power dividers [55 57], corporate power dividers [58], [59], and distributed power amplifiers [1] (pp ), [6]. Antenna applications of NRI-TL metamaterials include leaky-wave antennas [61 66], series-fed antenna arrays [67 71] and various types of compact radiators [72 78]. A useful reference that describes some of the aforementioned NRI-TL metamaterial antennas and devices, as well as many other useful antenna designs and concepts, is the recent antenna handbook by Balanis [79]. 1.3 Thesis Outline In this thesis, a number of microwave and antenna applications that employ one-dimensional NRI-TL metamaterials are presented, together with their advantages compared to conventional designs. In Chapter 2, the properties of conventional positive refractive index (PRI) and neg-

26 Chapter 1. Introduction 6 ative refractive index (NRI) materials are reviewed, followed by a method of synthesizing NRI-TL metamaterial media by combining sections of PRI and NRI media to form compact, broadband phase-compensating structures. The propagation and impedance characteristics of two types of elementary NRI-TL metamaterial unit cells are then studied, and a technique to ensure that these structures do not radiate is outlined. Finally, design equations governing the values of the loading elements, the optimal choice of the number of unit cells for a particular length of line, and the phase variation of each line with respect to frequency are presented. In Chapter 3, the NRI-TL metamaterials lines are used to implement a broadband metamaterial balun, which consists of a Wilkinson divider and two metamaterial lines. One of the output branches of the Wilkinson divider is connected to a +9 metamaterial phase-shifting line, while the other output branch is connected to a 9 metamaterial phase-shifting line. The metamaterial lines are designed to have similar phase responses with frequency, which enables the balun to exhibit an almost constant phase difference of 18 over a large bandwidth, as well as excellent return loss, isolation and through characteristics, all while maintaining a small form factor. Chapter 4 describes how non-radiating NRI-TL metamaterial lines can be used to implement a 1:4 series power divider by cascading four metamaterial lines in series, that each incur a near-zero insertion phase over a large bandwidth. This configuration provides equal power split to all four output ports over a large bandwidth, while maintaining an overall small form factor of λ /8 for each section of metamaterial line. Additionally, it is demonstrated that the power divider exhibits comparable losses compared to a conventional series power divider employing meandered transmission lines, while occupying a significantly smaller area. In Chapter 5, the metamaterial power dividing scheme is used to create a series feed network for a linear array of printed dipoles, designed to radiate in the broadside direction. Series-fed arrays employing transmission-lines, low-pass loaded lines and metamaterial feed networks are theoretically analyzed and it is shown that the metamaterial-fed arrays exhibit significantly more broadband beam scanning characteristics, which leads to a reduction in the amount of beam squinting that they experience with a change in frequency. The theoretical reduced-beam squinting characteristics of the metamaterialfed array are subsequently verified using fabricated prototypes of a low-pass loaded array and a metamaterial array. It is also shown that the metamaterial feed networks can be used to create linear arrays whose main beam remains virtually fixed at a positive angle

27 Chapter 1. Introduction 7 from broadside as the frequency is varied. In Chapter 6, a line consisting of cascaded NRI-TL metamaterial unit cells that operate in the fast-wave region are used to form a leaky-wave antenna. The beam squinting characteristics of such a metamaterial line are investigated under an open stopband condition, and it is shown that by operating the antenna in the upper right-handed band of the dispersion diagram, where the phase and group velocities of the line are closest to the speed of light, the beam squinting that the antenna experiences can be minimized. Chapter 7 presents an electrically small antenna, which consists of four NRI-TL metamaterial unit cells. An even-odd mode analysis is conducted on the circuit model of the antenna, and it is shown that the antenna supports a predominately even-mode current on the vertical vias, enabling it to be modeled as a multiply-folded monopole. This provides a substantial increase in the radiation resistance of the antenna, which together with the top-loading effect of the microstrip line on the vias, enables the antenna to be matched to 5 Ω without the use of an external matching network. It is also demonstrated that the antenna maintains a high radiation efficiency, while radiating a vertical linear electric field polarization. Finally, Chapter 8 summarizes the findings of this thesis, outlines the publications that have resulted from this work, and suggests some future research directions.

28 Chapter 2 Negative-Refractive-Index Transmission-Line Theory A method of synthesizing negative refractive index transmission-line (NRI-TL) metamaterials is presented in this chapter. To begin, the properties of conventional positive refractive index (PRI) materials are presented. This is followed by a simple method developed at the University of Toronto by which negative refractive index (NRI) materials can be synthesized by considering the dual network representation of a conventional PRI medium. The two are subsequently combined together to form a NRI-TL metamaterial medium that has the form of a host transmission line loaded with lumped-element series capacitors and shunt inductors in a high-pass configuration. Periodic analysis is applied to the loaded-tl structure and its pertinent propagation and impedance characteristics are outlined. This is subsequently followed by a method of synthesizing non-radiating NRI-TL structures. The chapter concludes by providing an in-depth analysis of various zero-degree metamaterial phase-shifting lines, including the factors affecting their loading-element values, and the phase variation of each line with respect to frequency. 2.1 Background The general properties of a medium which supports electromagnetic waves can be described by its material parameters, the electric permittivity ɛ = ɛ r ɛ and the magnetic permeability μ = μ r μ. These can be used to relate the speed of a wave in the medium 8

29 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 9 v φ to the speed of light c through the index of refraction, defined as [8] n = c v φ. (2.1) From Equation (2.1) one can infer that the index of refraction can be either positive or negative, depending on the sign of the phase velocity. Inserting the expressions for ɛ and μ into Equation (2.1), we can also arrive at an alternate expression for the index of refraction [8] n = ± ɛ r μ r. (2.2) 2.2 Positive Refractive Index (PRI) Media The first type of materials considered herein are ones that possess a positive refractive index (PRI). In these materials the electric field vector Ē, magnetic field vector H and wave vector k form a right-handed triplet, and were therefore termed right-handed media (RHM) by Veselago [13]. Consequently, the Poynting vector S = Ē H, andthewave vector k are co-directed, implying that the direction of propagation is the same as the direction of power flow. Additionally, a positive k implies that the phase fronts of the waves propagating in these media travel forward with a positive phase velocity. Therefore, from Equation (2.1) these materials have a positive index of refraction. Conventional dielectric materials operating within the microwave frequency regime, as well as free space are examples of PRI materials. If we consider a lossless homogeneous isotropic PRI medium with positive values for ɛ and μ, and assume no variation in the z direction, Maxwell s equations for onedimensional propagation can be written as [8] de x dz = jωμh y, (2.3) dh y dz = jωɛe x. (2.4) The solutions to the above equations are plane waves whose electric and magnetic fields are transverse to the direction of propagation ẑ, i.e. they form a transverse electromagnetic (TEM) mode where E z =andh z =. The TEM mode has a propagation constant of k = ω μɛ, and a wave impedance of Z w = μ/ɛ. Maxwell s equations can also be applied to any two-conductor transmission line (TL) which has an arbitrary cross-section that does not vary along the direction of propagation

30 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 1 ẑ [81] (Section 5.3). If the medium between the two conductors is homogeneous and isotropic, then the solutions to Maxwell s equations will yield propagating TEM waves. Maxwell s equations can be re-formulated from the field quantities Ē and H to the circuit quantities of the voltage between the two lines V (z) and the current I(z) along the transmission line using Faraday s and Ampere s laws, respectively. The resulting relations are known as the Telegrapher s equations and can be written as follows for the lossless case dv (z) = jωli(z), (2.5) dz di(z) = jωcv (z). (2.6) dz The solutions to the Telegrapher s equations confirm that the transmission line supports TEM waves, with a propagation constant β = ω LC and a characteristic impedance Z = L/C, wherel and C are the per unit inductance and capacitance, respectively. There is therefore a direct analogy that can be drawn between TEM plane-wave propagation in a homogeneous isotropic medium and TEM plane-wave propagation on a two-wire transmission line. By comparing Equations (2.3) and (2.4) with Equations (2.5) and (2.6), this analogy can be extended to the individual parameters describing the media, where the permeability μ is analogous to the inductance L, and the permittivity ɛ is analogous to the capacitance C, i.e. μ L & ɛ C. (2.7) The analogy between the material parameters of a medium and the circuit elements of a transmission line has also been used in the transmission-line matrix (TLM) method for solving Maxwell s equations numerically [82], [83] Incremental Circuit Analyzed Using Telegrapher s Equations The analysis of distributed electromagnetic structures can be greatly simplified if classical circuit theory, with its associated synthesis and analysis methods, can be applied to these structures. To this end, the field expressions from Maxwell s equations must be rigorously mapped to the circuit expressions obtained from Kirchhoff s laws, a process which is elegantly outlined in Chapter 4 of [8]. Using the results of [8], a distributed

31 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 11 I(z) LΔz jω 2 LΔz jω 2 I(z+Δz) + V(z) - jωcδz + V(z+Δz) - Δz Figure 2.1: Incremental equivalent circuit of a continuous positive refractive index (PRI) medium. Note that the equivalent circuit has a low-pass topology and can be used to model a uniform distributed transmission line. transmission line is analyzed in this section using circuit theory in order to reveal its general characteristics. The application of circuit theory requires the physical dimensions of an electrical network to be much smaller than the wavelength, such that the voltages and currents at each node are not functions of position. Under these conditions, Kirchhoff s voltage and current laws can be conveniently used to analyze any circuit. Conventional transmission lines, however, are uniform distributed structures where the voltage V (z) and the current I(z) are functions of position along the line, and as such circuit analysis cannot be applied directly to these. By subdividing a transmission line into smaller sections that have an incremental length Δz as shown in Figure 2.1, each section can be treated as a lumpedelement equivalent circuit and therefore Kirchhoff s laws can be applied to it. Because each Δz section is electrically small, in the limit as Δz goes to zero the voltage drop across the lumped-element circuit will also go to zero and the current passing through the circuit will be constant. In this manner, a uniform distributed transmission line can be modeled as a series of incremental lumped-element circuits that each have an infinitesimal size. If, however, the size of each repeating lumped-element circuit has a finite size, then the circuit must be treated using periodic analysis, as will be shown in Section Let us therefore apply Kirchhoff s laws in order to determine the propagation characteristics of the lossless PRI transmission line shown in Figure 2.1. To do so, we consider the more general symmetric equivalent circuit shown in Figure 2.2. By applying Kirch-

32 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 12 I(z) ZΔz 2 ZΔz 2 I(z+Δz) + V(z) - YΔz + V(z+Δz) - Δz Figure 2.2: Generic incremental equivalent circuit of a continuous medium. hoff s voltage law to the outer loop, we obtain V (z +Δz) V (z) = I(z) ZΔz 2 I(z +Δz) ZΔz 2. (2.8) By applying Kirchhoff s current law at the central node, we obtain I(z +Δz) I(z) =I(z)Y Δz ZΔz 2 Y ΔzV (z). (2.9) Dividing both Equations (2.8) and (2.9) by Δz and taking the limit as Δz, yields the general form of the Telegrapher s equations dv (z) dz = ZI(z), (2.1) di(z) = YV(z). (2.11) dz Thus, Equations (2.1) and (2.11) can be solved simultaneously to produce wave equations in V (z) andi(z) d 2 V (z) = γ 2 V (z), (2.12) dz 2 d 2 I(z) = γ 2 I(z), (2.13) dz 2 where the complex propagation constant is defined as [8] γ = α + jβ = ZY. (2.14) For the lossless case considered here α =, therefore the propagation constant can be written as β = j ZY. (2.15)

33 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 13 The corresponding characteristic impedance can be found by taking the ratio of the amplitude of the forward-traveling voltage to the forward-traveling current, and is given by [8] Z Z = Y. (2.16) Returning to the PRI transmission line of Figure 2.1, the total impedance and admittance elements for this case are given by ZΔz 2 = jω LΔz 2 & Y Δz = jωcδz (2.17) The per-unit-length impedance and admittance in terms of the per-unit-length inductance and capacitance L and C are therefore Z = jωl & Y = jωc. (2.18) From Equations (2.15) and (2.18), the propagation constant for an ideal transmission line is given by [84] β tl = ω LC. (2.19) The corresponding characteristic impedance can be found from Equation (2.16) to be Z = L C. (2.2) The electrical length of a section of transmission line with physical length d tl is θ = β tl d tl = ω LCd tl. (2.21) The corresponding phase incurred by the same section of transmission line is the negative quantity of the electrical length θ and can be written as φ tl = β tl d tl = ω LCd tl. (2.22) The general expression for phase velocity is given by [84] v φ = ω β. (2.23) For a conventional lossless transmission line the phase velocity then becomes v φ,tl = 1 LC. (2.24)

34 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 14 I n + V n - Z'/2 Y' Z'/2 I n+1 + V n+1 - d Figure 2.3: Generic lumped-element unit cell used to model a lumped-element line. It can be observed that the phase velocity of a transmission line is positive, therefore resulting in a positive refractive index from Equation (2.1). Equation (2.24) also indicates that a lossless transmission line is non-dispersive, which is advantageous for signal transmission. Furthermore, the general expression for the group velocity can be written as [84] v g = dω dβ. (2.25) Thus, for a conventional lossless transmission line the group velocity is the same as the phase velocity, and is given by v g,tl = ( dβtl dω ) 1 = 1 LC. (2.26) Note that the group velocity for a transmission line is positive, which is consistent with the fact that the Poynting vector is also positive Lumped-Element Circuit Analyzed Using Periodic Analysis A transmission line can also be modeled using discrete lumped elements arranged in a low-pass topology as was shown in Figure 2.1, but with a finite periodicity d. Acascade of the individual low-pass unit cells therefore forms a periodic structure, and can be analyzed using periodic Bloch-Floquet analysis (see Section 2.4.2). This procedure has been carried out in [85] (Section 5.2), and the results are summarized below. Figure 2.3 shows the generic lumped-element unit cell under consideration. Each of the series impedances and the shunt admittance represent individual lumped elements. Thus,inrelationtotheper-unit-length parameters Z and Y considered in the previous section, the parameters Z and Y can be considered the total impedance and admittance

35 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 15 over the length d, and are given as follows Z = Zd & Y = Yd. (2.27) According to the Floquet theorem, for a forward traveling wave the voltages and currents at the terminals of the nth unit cell are related to the voltages and currents at the terminals of the (n + 1)th unit cell by the propagation factor e jβbld. Here, β bl is the Bloch propagation constant for the periodic structure of Figure 2.3 that has a periodicity of d. Following the analysis presented in [85], the general dispersion relation for the structure of Figure 2.3 is given by ( ) sin 2 βbl d = Z Y. (2.28) For the transmission line case, the total series impedance and shunt admittance are Z = jωl & Y = jωc, (2.29) where the total lumped inductance and capacitance are given by L = Ld & C = Cd. (2.3) Substituting the expressions of Z and Y from Equation (2.29) into Equation (2.28), the dispersion relation for the lumped-element transmission-line becomes ( ) βbl d sin = ω L C. (2.31) The dispersion relations for both the incremental circuit analyzed using the Telegrapher s equations and the lumped-element circuit analyzed using periodic analysis from Equations (2.21) and (2.31) are shown in Figure 2.4 using the parameters C =1.11 pf and L =2.78 nh, which are equivalent to a transmission line with parameters Z =5Ω, d =13.61 mm and θ =4 at 2 GHz. It can be observed that the incremental transmission line model exhibits a continuous dispersion relation, while the lumped-element model, by virtue of its periodicity, exhibits a high-frequency cutoff condition at β bl d = π. Thus, the lumped-element periodic structure actually acts as a low-pass filter with a cutoff frequency f lp, above which no propagation occurs. From Figure 2.4 it can also be observed that for small phase shifts per unit cell, i.e. βd 1, the characteristics of the incremental and lumped-element circuits correspond

36 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Incremental circuit (Telegrapher s eq.) Lumped element circuit (Periodic) Frequency (GHz) 6 4 f LP 2 π π/2 π/2 π βd (rad) Figure 2.4: Dispersion diagrams for a representative low-pass forward-wave PRI medium. The incremental circuit was analyzed using the Telegrapher s equations and is described by Equation (2.21): β tl d = ω LCd, and the lumped-element circuit was analyzed using periodic ) analysis and is described by Equation (2.31): sin = 1 2 ω LCd. ( βbld 2 very closely. This can also be observed by applying the above condition to the lumpedelement dispersion relation of Equation (2.31), which is equivalent to shrinking the size of each unit cell d. Under the condition that β bl d 1, we can approximate the left-hand side of Equation (2.31) simply with the argument of the sine term, therefore resulting in (L )( ) C β bl ω = ω LC. (2.32) d d It can be recognized that Equation (2.32) for the lumped-element case in the β bl d 1 limit is identical to Equation (2.19) for the incremental case. Therefore, it can be concluded that as the size of each unit cell in the lumped-element circuit tends to zero, the characteristics of the periodic lumped-element line approach those of a uniform transmission line that comprises an infinite number of incremental unit cells, and whose cutoff frequency lies at infinity. Nonetheless, for practically realizable transmission lines and for lumped-element transmission lines with a finite size the cutoff frequency will be finite.

37 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 17 Δz Δz I(z) jω2c' jω2c' I(z+Δz) + V(z) - Δz jωl' + V(z+Δz) - Δz Figure 2.5: Incremental equivalent circuit of a continuous negative refractive index (NRI) medium. Note that the equivalent circuit has a high-pass topology and can be used to model a uniform backward-wave line. 2.3 Negative Refractive Index (NRI) Media The next type of materials considered herein are ones that possess a negative refractive index (NRI). In these materials the electric field vector Ē, magnetic field vector H and wave vector k form a left-handed triplet, and were therefore termed left-handed media (LHM) by Veselago. Consequently, the Poynting vector S, and the wave vector k are oppositely directed, implying that the direction of propagation is opposite to the direction of power flow. Additionally, a negative k implies that the phase fronts of the waves propagating in these media travel backwards with a negative phase velocity. Therefore, from Equation (2.1) these materials have a negative index of refraction. NRI materials do not exist in nature, and must therefore be synthesized artificially. The transmission-line analogy considered in the previous section provides an indication of how to obtain negative material parameters μ and ɛ, and therefore a negative refractive index. Since it was found that for the transmission-line model μ was directly related to the series inductance L, andɛ was directly related to the shunt capacitance C, asimpleway to synthesize a negative μ and ɛ is to create a negative series inductance and a negative shunt capacitance. Therefore from an impedance perspective, since a negative inductance is analogous to an equivalent capacitance, and a negative capacitance is analogous to an equivalent inductance, a NRI material can simply be obtained by exchanging the roles of the inductors and the capacitors in a conventional transmission line, resulting in the dual transmission-line model shown in Figure 2.5. This method of synthesizing NRI media by simply using series capacitors and shunt inductors was proposed by Eleftheriades et. al. in [19] and [61] and was further elabo-

38 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 18 rated upon in [2]. These landmark papers paved the way for what would subsequently be termed negative refractive index transmission-line (NRI-TL) metamaterials which were based on periodically loading a host transmission line with lumped-element series capacitors and shunt inductors. The Telegrapher s equations in terms of the times-unitlength lumped-element loading C and L for the one-dimensional dual transmission-line medium therefore become ( ) dv (z) 1 = I(z), (2.33) dz jωc ( ) di(z) 1 = V (z). (2.34) dz jωl Comparing Equations (2.33) and (2.34) with Equations (2.3) and (2.4), the analogy between the material parameters μ and ɛ and the loading-element values of the NRI medium can be written as [2] μ 1 ω 2 C & ɛ 1. (2.35) ω 2 L It can be observed from Equation (2.35) that ɛ and μ are both negative, therefore resulting in a negative index of refraction. It should mentioned here, that the dual transmission-line model shown in Figure 2.5 was not novel in and of itself, since it had been commonly known as a backward-wave line for many years and can be found in many text books [8], [81]. In addition, the dualtransmission line structure is in fact equivalent to the commonly known high-pass topology, which had also been extensively used in filter theory [86]. What was indeed novel about the work presented in [19], [2] and [61], was the realization that the backwardwave structure possessed a negative index of refraction, which was something that was not appreciated in previous work on backward-wave lines. The practical implications of this discovery subsequently led to the experimental demonstration of negative refraction and focusing using a planar NRI lens [19], [21]. Furthermore, it was demonstrated that a low-loss planar NRI lens could in fact overcome the conventional diffraction limit [23] Incremental Circuit Analyzed Using Telegrapher s Equations The propagation characteristics of the lossless NRI backward-wave line shown in Figure2.5canbedeterminedusingthesameprocedure outlined in Section Provided that in the limit as Δz goes to zero the voltage drop across the circuit also goes to zero,

39 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 19 and that the current passing through the circuit is constant, then a uniform backwardwave line can be modeled as a series of infinitesimal lumped-element circuits as the one shown in Figure 2.5. It will be shown in this section that the voltage and current conditions indeed hold, which allows us to apply the Telegrapher s equations of Equations (2.1) and (2.11) to the NRI structure. Considering the NRI backward-wave line of Figure 2.5, let us begin by introducing the times-unit-length capacitance and inductance C and L, respectively C = C Δz & L = L Δz. (2.36) Here, C and L are the lumped-element series capacitance and shunt inductance, respectively. The total impedance and admittance elements for this case are given by ZΔz 2 = Δz jω2c & Y Δz = Δz. (2.37) jωl The per-unit-length impedance and admittance in terms of the times-unit-length capacitance and inductance C and L are therefore Z = 1 jωc & Y = 1. (2.38) jωl The results derived earlier for the generic symmetric circuit of Figure 2.2 can also be used for the NRI structure. From Equations (2.15) and (2.38), the propagation constant for backward-wave line can be written as β bw = 1 ω L C = 1 ω (L Δz)(C Δz). (2.39) The corresponding characteristic impedance of the backward-wave line can be found from Equations (2.16) and (2.38) to be L L Z,bw = =. (2.4) C C The electrical length of a section of backward-wave line with physical length d bw is θ bw = β bw d bw = 1 ω d L bw = C 1 ω L C. (2.41) The corresponding phase incurred by the same section of line is the negative quantity of the electrical length, and can be written as φ bw = β bw d bw = 1 ω L C. (2.42)

40 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 2 The phase velocity of the lossless backward-wave line from Equation (2.23) is therefore v φ,bw = ω 2 L C. (2.43) Contrary to the transmission-line case, the phase velocity of a backward-wave line is negative, therefore verifying from Equation (2.1) that this line indeed exhibits a negative index of refraction. Equation (2.43) however also indicates that a lossless backward-wave line is highly dispersive, since the phase velocity is not constant but rather is proportional to the frequency squared. Finally, the group velocity of the lossless backward-wave line from Equation (2.25) is given by v g,bw = ω 2 L C. (2.44) Note that even though the phase velocity of a backward-wave line is negative, the group velocity is nevertheless positive, which is consistent with the fact that the Poynting vector is also positive. Therefore, even though the phase fronts of the waves propagating on the line travel backwards, power flow is still in the forward direction Lumped-Element Circuit Analyzed Using Periodic Analysis The backward-wave line has most commonly been analyzed using discrete lumped elements arranged in a high-pass topology rather than as a continuous structure [8], [81]. The reason for this stems from the fact that there are no known naturally occurring media that exhibit backward-wave characteristics, and as such, these have typically been synthesized using lumped-element series capacitors and shunt inductors. Herein we will consider a periodic lumped-element backward-wave line consisting of finite unit cells similar to the ones shown in Figure 2.5, but with a periodicity d. The results derived earlier for the generic lumped-element unit cell of Figure 2.3 can therefore be used for the backward-wave line as well. Recall that in Figure 2.3, the parameters Z and Y are the total impedance and admittance over the length d. Therefore, for the backward-wave line these are Z = 1 & Y = 1, (2.45) jωc jωl where the total lumped capacitance and inductance are given by C = C d & L = L d. (2.46)

41 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 21 1 Incremental circuit (Telegrapher s eq.) Lumped element circuit (Periodic) 8 Frequency (GHz) π π/2 π/2 π βd (rad) f HP Figure 2.6: Dispersion diagrams for a representative high-pass backward-wave NRI medium. The incremental circuit was analyzed using the Telegrapher s equations and is described by Equation (2.41): β bw d = 1 ω, and the lumped-element circuit was analyzed using periodic L C ( ) analysis and is described by Equation (2.47): sin βbld 1 2 = 2ω. L C The dispersion relation for the backward-wave line can be obtained by substituting the expressions of Z and Y from Equation (2.45) into Equation (2.28) ( ) βbl d 1 sin = 2 2ω. (2.47) L C The dispersion relations for both the incremental circuit analyzed using the Telegrapher s equations and the lumped-element circuit analyzed using periodic analysis from Equations (2.41) and (2.47) are shown in Figure 2.6 using the parameters C =2.19 pf and L =5.47 nh. As in the case of the incremental transmission line model shown in the previous section, the uniform backward-wave line exhibits a continuous dispersion relation, while the lumped-element model, by virtue of its periodicity, exhibits a lowfrequency cutoff condition at β bl d = π. In this case, the lumped-element backward-wave periodic structure acts as a high-pass filter with a cutoff frequency f hp,belowwhichno propagation occurs. From Figure 2.6 it can also be observed that for small phase shifts per unit cell, i.e. βd 1, the characteristics of the incremental and lumped-element circuits correspond very closely. This can also be observed by applying the above condition to the lumpedelement dispersion relation of Equation (2.47), which is equivalent to shrinking the size

42 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 22 of each unit cell d. Under the condition that β bl d 1, we can approximate the left-hand side of Equation (2.47) simply with the argument of the sine term, therefore resulting in 1 β bl ω (L d)(c d) = 1 ω. (2.48) L C It can be recognized that Equation (2.48) for the lumped-element case in the β bl d 1 limit is identical to Equation (2.39) for the incremental case. Therefore, it can be concluded that as the size of each unit cell in the lumped-element circuit tends to zero, the characteristics of the periodic lumped-element line approach those of the uniform backward-wave line. This implies that under the condition that the unit cell size tends to zero, the use of the Telegrapher s equations is justified since it produces an almost identical response to that obtained using periodic theory, which does not assume that the voltages and currents on each side of the unit cell are equal. This result is also consistent intuitively, since at high frequencies where the two responses match, the series capacitance of the equivalent circuit shown in Figure 2.5 resembles a short-circuit, and the shunt inductance resembles an open-circuit. Thus, there is very little voltage drop across the circuit and an almost constant current passing through it. These conditions for the voltage and the current are necessary in order to justify the use of the Telegrapher s equations for the backward-wave circuit of Figure 2.5. By inspection of Equation (2.42) and Figure 2.6 it can be observed that at no time does the phase of a backward-wave line pass through zero, since even for an infinitely large frequency the phase is still positive. Thus, a backward-wave line cannot be used to provide any desired phase shift. This is in contrast to a transmission line, which can achieve any effective phase shift with a simple change of its length, and by virtue of the fact that the phase is periodic every 2π. Having outlined the general characteristics of both the incremental and lumpedelement high-pass NRI structures, let us now return to a subtle, yet quite important point relating to their group velocity. Observation of Equation (2.44) indicates that as the frequency increases, the group velocity will also increase in an unbounded manner, and will eventually exceed the speed of light, c. This scenario is evidently unphysical, since it would violate causality. It can therefore be concluded that there is an element missing from the backward-wave circuit of Figure 2.5, which is required in any physically realizable form of a NRI medium that possesses a group velocity that is less than the speed of light. It will be shown that the NRI-TL medium presented in the next section elegantly resolves this apparent discrepancy, and accounts for the missing element in the

43 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 23 NRI equivalent circuit by introducing a small section of host transmission line that is loaded with the series capacitive and shunt inductive NRI loading. The host transmission line accounts for the finite size of the lumped-element components while the fundamental backward-wave nature of the loaded line is still maintained, therefore resulting in a finite and subluminal group velocity for the physical realization of the NRI medium. It is also instructive to analyze a medium that consists of both PRI and NRI components from the perspective employed in Sections 2.2 and 2.3, in order to ascertain under which conditions it can be considered a truly uniform structure. The details of this analysis are shown in Appendix A. 2.4 Negative-Refractive-Index Transmission-Line (NRI-TL) Metamaterial Medium Proposed Phase Compensating Structure Thus far, media that exhibit either a positive or negative refractive index have been discussed. In this section the two aforementioned media are combined into a single structure to produce a negative refractive index transmission-line (NRI-TL) metamaterial, which is a versatile medium that exhibits both forward and backward-wave propagation characteristics. As outlined in Section 2.2, in a PRI medium the phase lags in the direction of positive group velocity, thus incurring a negative phase. A forward-traveling wave propagating along a PRI medium, for example a conventional TL, will therefore have the form e jβtlz, since β tl is a positive quantity. On the contrary, in a NRI medium, the phase leads in the direction of positive group velocity, thus incurring a positive phase. A forward-traveling wave propagating along a NRI medium, for example a backward-wave line, will therefore have the form e +jβbwz,sinceβ bw is a negative quantity. From the above arguments, it therefore follows that phase compensation can be achieved at a given frequency by cascading a section of a PRI line with a section of a NRI line as shown graphically in Figure 2.7, thus creating a NRI-TL metamaterial line. This enables the synthesis of a positive, negative or zero transmission phase, while keeping the overall physical length of the structure short. This idea of phase compensation using two phase-complementary structures is inherent in the original flat lens proposed

44 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 24 PRI Transmission Line NRI Backward-Wave Line Φ TL Φ BW Φ MTM = Φ TL + Φ BW Figure 2.7: Method of phase compensation using a PRI transmission line and a NRI backwardwave line to create a NRI-TL metamaterial line. by Veselago [13], and has also been subsequently proposed by Engheta for implementing thin sub-wavelength cavity resonators [87]. Under certain conditions that will be discussed in Section the total phase shift across the metamaterial phase-compensating structure can be written as Φ mtm =Φ tl +Φ bw. (2.49) Thus, Equation (2.49) highlights the method by which the phase compensation can be achieved. Since Φ tl is a negative quantity and Φ bw is a positive quantity, this implies that the total phase shift Φ mtm can be synthesized to be positive, negative or even zero if the two quantities Φ tl and Φ bw are equal and opposite. A physical implementation of the NRI-TL metamaterial phase-compensating structure is shown in Figure 2.8, where the transmission line section and the backward-wave section are explicitly shown. The individual PRI and NRI sections of Figure 2.8 can be theoretically analyzed and simulated in a circuit simulator, as was done in Sections 2.2 and 2.3, however the idealized line cannot be realized in practice. This is because in any practical realization, the NRI backward-wave section of the structure will occupy a finite length of line and will therefore inherently possess a right-handed component, which is in addition to the desired left-handed component created by the lumped-element loading. A close approximation to a backward-wave line can nonetheless be obtained by loading a host TL with series capacitors and shunt inductors, while ensuring that the length of the host TL is small compared to the wavelength. The equivalent circuit of the approximate backward-wave line used in Figure 2.8 would therefore have small sections of transmission lines distributed in between the series capacitors. These small sections of host transmission lines need not

45 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 25 1 n 2C 2C 2C 2C Z L L ΦTL = ω LCd TL Φ BW = + ω n L C Figure 2.8: NRI-TL metamaterial phase-compensating structure using a PRI transmission line and an n-stage NRI backward-wave line. 1 2 n 2C 2C 2C 2C 2C Z Z Z Z Z Z 2C L L L Φ MTM = nφ MTM = n ω LCd H TL + ω 1 L C Figure 2.9: n-stage NRI-TL metamaterial phase-shifting line. alter the operation of the NRI-TL metamaterial line as a phase-compensating structure, because their right-handed effect can in fact be exploited to compensate the left-handed nature of the backward-wave line. Therefore, by simply dividing the transmission line on the left-hand side of Figure 2.8 into smaller sections and distributing them evenly between the series capacitors, a uniform structure can be created as shown in Figure 2.9, that consists of identical repeating symmetrical unit cells. Each one of these unit cells can be considered to be a small NRI-TL metamaterial phase-compensating structure in its own right, since it contains both a right-handed component (the host TL) and a left-handed component (the loading BW line). Figure 2.1 shows two possible implementations of the elementary phasecompensating NRI-TL metamaterial unit cell; the T configuration was developed in

46 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 26 2C θ/2 Z θ/2 2C Z θ/2 Z C θ/2 Z L 2L 2L d H-TL (a) T unit cell d H-TL (b) π unit cell Figure 2.1: NRI-TL metamaterial unit cells. [2], [28] and the π configuration was subsequently developed in [88]. It can be recognized that the T and π unit cells are simply related by a shift of the reference planes defining each unit cell within the larger periodic structure shown in Figure 2.9. As such, it is expected that their propagation characteristics will be identical, something which will be subsequently verified in the following section. The T unit cell has a host transmission line with characteristic impedance Z and length d h-tl that is loaded with two series capacitors 2C and a shunt inductor L, while the π unit cell has a host transmission line also with characteristic impedance Z and length d h-tl that is loaded with one series capacitor C and two shunt inductors 2L. Depending on the intended application and the technology in which the circuits are realized, the T or the π configuration could prove to be more useful. In this thesis, both the configurations are used in order to synthesize distributed NRI-TL metamaterial phase shifting lines with specific phase characteristics Propagation Characteristics of the T Unit Cell The propagation characteristics of a metamaterial line that consists of identical symmetric unit cells can be determined by conducting a periodic Bloch-Floquet analysis on the T unit cell of Figure 2.1(a). The Bloch-Floquet theorem for periodic structures can be found in Chapter 8.2 of [89], and the implementation method using transmission matrices can be found in Chapter 8 of [9] and Chapter 8 of [84]. The details of how this pertains to the analysis of the particular metamaterial T unit cell of Figure 2.1(a) can be found in Chapter 3 of [91], and a summary of the results presented therein is repeated here for completeness.

47 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 27 I n + V n - Z/2 θ/2 Z Y θ/2 Z Z/2 I n+1 + V n+1 - d = d H-TL Figure 2.11: Generic NRI-TL metamaterial T unit cell. The generic unit cell of the metamaterial T unit cell of Figure 2.1(a) that will be used in the subsequent analysis is shown in Figure Here, the lumped-element capacitors and inductors have been replaced with the following generic expressions for their impedances and admittances in order to simplify the expressions: Z = 1 jωc & Y = 1 jωl. (2.5) Referring to Figure 2.11, the voltages and currents at the terminals of the nth unit cell are related to the voltages and currents at the terminals of the adjacent (n + 1)th unit cell through the transfer function, that can be expressed as an ABCD matrix, yielding the following relationship [ Vn I n ] = [ A B C D ][ Vn+1 I n+1 ]. (2.51) The ABCD matrix of the unit cell of Figure 2.11 can be written as the product of the ABCD matrices of the individual elements that constitute the unit cell: [ ] [ ][ ][ ] A B 1 Z/2 cos(θ/2) jz sin(θ/2) 1 = C D 1 jy sin(θ/2) cos(θ/2) Y 1 [ ][ ] cos(θ/2) jz sin(θ/2) 1 Z/2. (2.52) jy sin(θ/2) cos(θ/2) 1 After expanding the above expression and performing some simplifications, the individual

48 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 28 elements of the ABCD matrix can be written as ( A = 1+ ZY ) cos(θ)+ j 4 2 (ZY + YZ )sin(θ)+ ZY 4 B = ( Z + Z2 Y 8 + YZ 2 ) cos(θ)+ j ( ) Z 2 Y + YZZ +2Z sin(θ) C = Y 2 cos(θ)+jy sin(θ)+ Y ( 2 D = 1+ ZY ) cos(θ)+ j 4 2 (ZY + YZ )sin(θ)+ ZY 4. + Z2 Y 8 YZ 2 2 (2.53a) (2.53b) (2.53c) (2.53d) Note that in the above expressions, the terms A and D are equal, which is expected for a symmetric unit cell. According to the Floquet theorem, for a forward traveling wave the voltages and currents at the terminals of the nth unit cell are related to the voltages and currents at the terminals of the (n + 1)th unit cell by the propagation factor e jβbld. Here, β bl is the Bloch propagation constant for the periodic structure with periodicity d = d h-tl, as shown in Figure Therefore Equation (2.51) can be written as [ ] [ ][ ] [ ] Vn A B Vn+1 = = e jβbld Vn+1. (2.54) I n C D I n+1 I n+1 Equation (2.54) forms a system of linear equations in terms of V n+1 and I n+1,whichcan be solved by finding the determinant of the matrix and selecting the non-trivial solutions, resulting in the following dispersion relation: cos(β bl d)= A + D = A. (2.55) 2 In Equation (2.55), the fact that A = D has been used for a symmetric unit cell. From Equation (2.53a), the dispersion relation therefore becomes ( cos(β bl d)= 1+ ZY ) cos(θ)+ j 4 2 (ZY + YZ )sin(θ)+ ZY 4. (2.56) Substituting for the values of Z and Y from Equation (2.5) we can write the final dispersion relation for the periodic structure comprising an infinite number of T metamaterial unit cells as ( ) ( ) 1 cos(β bl d)= 1 cos(θ)+ sin(θ) 4ω 2 L C 1 2ωC Z + Z 2ωL 1 4ω 2 L C. (2.57) A representative dispersion diagram obtained using Equation (2.57) is shown in Fig-

49 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Frequency (GHz) Light line RH band Frequency (GHz) LH band 2 π π/2 π/2 π β d (rad) BL (a) Dispersion diagram Re{Z } (Ω) BL (b) Bloch impedance diagram Figure 2.12: Characteristics of a representative NRI-TL metamaterial T unit cell with parameters C =5pF,L =25nH,Z =5Ω,d = d h-tl =3mmandθ =8.8 at 2 GHz. Note that in this case, Z,bw = L /C >Z. ure 2.12(a). It can be observed that the propagation characteristics of the structure exhibit alternating passbands and stopbands. The two passbands that are of interest for this work are the lower left-handed band and the upper right-handed band. The light line is also shown in Figure 2.12(a), which demarcates the transition between slow-wave and fast-wave propagation. It can therefore be verified that the metamaterial structure supports backward as well as forward waves that can be either slow or fast. This indicates that the metamaterial line can be used for either transmission-line or antenna applications, depending on the region of operation. The characteristic impedance of the periodic structure, known as the Bloch impedance, is defined at the terminals of each unit cell to be Z bl = V n I n = V n+1 I n+1. (2.58) It should be noted that the Bloch impedance of a periodic structure is not unique, and will depend on the location of the reference planes for each unit cell. Therefore it is expected that the two metamaterial unit cells of Figure 2.1 will have different Bloch impedances. Using Equation (2.54) and the relations A = D for a symmetrical network and AD BC = 1 for a reciprocal lossless network, it can be shown that the Bloch impedance can

50 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 3 be written as [9] Z ± bl = ±B A2 1 = ± B C. (2.59) The ± Bloch solutions correspond to the Bloch impedances for forward and reflected traveling waves, respectively. By substituting the appropriate expressions for B and C from Equations (2.53b) and (2.53c), the Bloch impedance for the periodic structure comprising an infinite number of T metamaterial unit cells can be written as ( ) ( Z + Z2 Y + YZ 2 cos(θ)+ j Z ) 2 Y + YZZ Z sin(θ)+ Z 2 Y YZ Z bl,t = ± Y cos(θ)+jy. 2 sin(θ)+ Y 2 (2.6) A representative Bloch impedance diagram for the metamaterial T unit cell obtained using Equation (2.6) is shown in Figure 2.12(b). It can be observed that a real Bloch impedance exists only within the passbands of the periodic structure, while within the stopbands the Bloch impedance is imaginary. It can also be observed that the Bloch impedance exhibits a large variation throughout the left-handed band, and remains around 5 Ω for a very small frequency range. As such, the periodically loaded metamaterial line will exhibit the undesirable feature of a very narrow impedance bandwidth when it is matched to a 5 Ω feed line. On the contrary, in the right-handed band the Bloch impedance converges to a constant value of 5 Ω as the frequency increases. This is a very desirable feature that enables broadband matching of the periodically loaded metamaterial line to a feed line or a terminating load. In the design of the metamaterial unit cells, it is interesting to note that depending on the values of the elements loading the host transmission line, the Bloch impedance characteristics can be significantly different. In fact, under certain conditions there is a frequency region within the left-handed band where the Bloch impedance cannot be designed for a specific value. In order to further understand the Bloch impedance behaviour, we must compare the characteristic impedance of the host transmission line, Z with the characteristic impedance of the backward-wave line that loads the host TL, Z,bw = L /C. Thus, for the representative example shown in Figure 2.12 with loading-element parameters of C =5pFandL = 25 nh, the characteristic impedance of the backward-wave line is Z,bw = 7.7 Ω, which is greater than the characteristic impedance of the host transmission line, Z = 5 Ω. Now, let us consider a case where Z,bw is less than Z. For this case we choose C =15pFandL = 25 nh, resulting in Z,bw = 4.8 Ω.

51 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Frequency (GHz) RH band Frequency (GHz) LH band 2 π π/2 π/2 π β d (rad) BL (a) Dispersion diagram Re{Z } (Ω) BL (b) Bloch impedance diagram Figure 2.13: Characteristics of a representative NRI-TL metamaterial T unit cell with parameters C =15pF,L =25nH,Z =5Ω,d = d h-tl =3mmandθ =8.8 at 2 GHz. Note that in this case, Z,bw = L /C <Z. The dispersion and Bloch impedance diagrams for this second case are shown in Figure Here, it can be observed that even though the general dispersion characteristics are very similar to the ones shown in Figure 2.12(a), the Bloch impedance characteristics have changed significantly. Within the lower left-handed band, the Bloch impedance no longer passes through the 5 Ω point, but rather attains a maximum value of only 39 Ω, indicating that it is impossible to match the metamaterial line to 5 Ω within the lower left-handed band. This is an important consideration that should be taken into account when designing metamaterial lines using T unit cells with Z,bw <Z. If, however, it is desired to match the line to 5 Ω within the upper right-handed band, then the T unit cell with Z,bw <Z is still an attractive option. It can therefore be concluded that in order to obtain a Bloch impedance diagram of the form shown in Figure 2.12(b), where the metamaterial line consisting of T unit cells can be matched to a specific impedance (in this case Z = 5 Ω) in both the lower left-handed and the upper right-handed bands, then the following condition must be satisfied: Z,bw >Z. (2.61) Even though the above analysis was carried out assuming an infinitely long periodic structure, the propagation and impedance characteristics of the line can be retained even for a finite structure. This can be achieved by simply truncating the metamaterial line

52 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 32 I n + V n - Y/2 θ/2 Z Z θ/2 Z Y/2 I n+1 + V n+1 - d = d H-TL Figure 2.14: Generic NRI-TL metamaterial π unit cell. along the terminal planes of one of its constituent unit cells and terminating the line on both ends in its Bloch impedance. This eliminates any reflections at the source and load and allows the propagating waves along the line to effectively see an infinite periodic medium. Therefore, in the limit of an increasingly short periodic structure, a single unit cell can be employed as the transmission medium, which will retain its propagation and impedance characteristics provided that it is excited and terminated in its Bloch impedance Propagation Characteristics of the π Unit Cell A similar procedure can be carried out for the analysis of the metamaterial π unit cell of Figure 2.1(b), whose generic unit cell is shown in Figure The ABCD matrix of the entire unit cell of Figure 2.14 can be written as [ ] [ ][ ][ ] A B 1 cos(θ/2) jz sin(θ/2) 1 Z = C D Y/2 1 jy sin(θ/2) cos(θ/2) 1 [ ][ ] cos(θ/2) jz sin(θ/2) 1. (2.62) jy sin(θ/2) cos(θ/2) Y/2 1 After expanding the above expression and performing some simplifications, the individual elements of the ABCD matrix can be written as

53 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 33 A = ( 1+ ZY 4 ) cos(θ)+ j 2 (ZY + YZ )sin(θ)+ ZY 4 (2.63a) B = Z 2 cos(θ)+jz sin(θ)+ Z 2 C = (Y + ZY 2 + ZY 2 ) cos(θ)+ j ( ) Y 2 Z + YZY +2Y sin(θ) D = ( 1+ ZY 4 + ZY 2 ) 8 cos(θ)+ j 2 (ZY + YZ )sin(θ)+ ZY 4. ZY 2 2 (2.63b) (2.63c) (2.63d) As expected, the terms A and D in Equations (2.63a) and (2.63d) are equal for the symmetric π unit cell of Figure 2.14, and they are also equal to the terms A and D in Equations (2.53a) and (2.53d) for the metamaterial T unit cell. As such, the dispersion characteristics of the metamaterial π unit cell are identical to those of the T unit cell, and are given by Equation (2.57). It can also be observed that the terms B and C of the T unit cell have a similar form to the terms C and B of the π unit cell respectively, with the impedance and admittance terms in each of the unit cells playing a complementary role. The Bloch impedance of the metamaterial π unit cell can be evaluated using Equations (2.63b) and (2.63c) in Equation (2.59), and can be written as Z 2 Z bl,π = ± cos(θ)+jz sin(θ)+ Z ( ) 2 ( Y + ZY 2 + ZY 2 cos(θ)+ j Y ). 2 Z + YZY Y sin(θ)+ ZY 2 ZY (2.64) Representative dispersion and Bloch impedance diagrams for the metamaterial π unit cell are shown in Figure 2.15 for the case where Z,bw <Z and in Figure 2.16 for the case where Z,bw >Z. As expected, the dispersion characteristics of the π unit cell for the case where Z,bw <Z shown in Figure 2.15(a) are identical to the dispersion characteristics of the T unit cell shown in Figure 2.13(a). The Bloch impedance of the two unit cells, also has very similar characteristics in the upper right-handed band, as can be verified from Figures 2.13(b) and 2.15(b). In the lower left-handed band however, the Bloch impedance for π unit cell with Z,bw <Z has the additional advantage that it passes through 5 Ω, thus also enabling the matching of the metamaterial line within this band. For the π unit cell operating under the condition that Z,bw >Z shown in Figure 2.16, the Bloch impedance does not pass through the 5 Ω point in the left-handed band, therefore it is not possible to match the line to 5 Ω within this band. Thus, in

54 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Frequency (GHz) RH band Frequency (GHz) LH band 2 π π/2 π/2 π β d (rad) BL (a) Dispersion diagram Re{Z } (Ω) BL (b) Bloch impedance diagram Figure 2.15: Characteristics of a representative NRI-TL metamaterial π unit cell with parameters C =15pF,L =25nH,Z =5Ω,d = d h-tl =3mmandθ =8.8 at 2 GHz. Note that in this case, Z,bw = L /C <Z. order to match the line to 5 Ω in both the left-handed and right-handed bands the following condition for the π unit cells must be satisfied: Z,bw <Z. (2.65) Furthermore, we can compare the useful cases of the T unit cell with Z,bw >Z shown in Figure 2.12 and the π unit cell with Z,bw <Z showninfigure2.15. Itcanbe observed that the dispersion characteristics of the π unit cell are similar to the dispersion characteristics of the T unit cell, however the Bloch impedances of the two unit cells exhibit complementary characteristics, as can be verified from Figures 2.12(b) and 2.15(b). These complementary characteristics can be attributed to the complementary forms of the Bloch impedances given by Equations (2.6) and (2.64). The Bloch impedance of the T unit cell, begins from zero at the bottom of the left-handed band and tends to infinity at the top of the band. On the contrary, for the π unit cell, the Bloch impedance at the bottom of the left-handed band tends to infinity, while it goes to zero at the top of the band. In the right-handed band, Z bl for the T unit cell begins from zero and tends towards 5 Ω from below as the frequency increases, while Z bl for the π unit cell begins from infinity and tends towards 5 Ω from above as the frequency increases. By inspection of both Figures 2.12(b) and 2.15(b), it can also be observed that if the stopband were closed, then a metamaterial line that consisted of either T or π unit

55 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Frequency (GHz) RH band f c2 Frequency (GHz) f c1 2 LH band π π/2 π/2 π β d (rad) BL (a) Dispersion diagram Re{Z } (Ω) BL (b) Bloch impedance diagram Figure 2.16: Characteristics of a representative NRI-TL metamaterial π unit cell with parameters C =5pF,L =25nH,Z =5Ω,d = d h-tl =3mmandθ =8.8 at 2 GHz. Note that in this case, Z,bw = L /C >Z. cells would not suffer from the rapid Bloch impedance changes around the vicinity of the stopband, and would remain close to 5 Ω over a much wider bandwidth extending down to low frequencies. The method by which the stopband can be closed will be outlined in the following section, where it will be demonstrated that indeed when the stopband is closed the Bloch impedance exhibits a significantly wider bandwidth Effective Medium Propagation Characteristics The dispersion relation of Equation (2.57) is sufficient to fully characterize the propagation characteristics of the NRI-TL metamaterial structure. It is instructive, however, to consider the case where the periodically loaded metamaterial line can be considered an effective medium, thus providing further insight into its operation. As such, the propagation constant can be expressed in an intuitive form that reveals the salient propagation characteristics of the medium. Beginning from the dispersion relation of Equation (2.57), this can be simplified to: [91] ( ) [ ][ ] sin 2 βbl d = sin(θ/2) cos(θ/2) sin(θ/2) Z cos(θ/2). (2.66) 2 2ωC Z 2ωL In order to consider a series of cascaded unit cells as an effective periodic medium, the physical length of the unit cell must be much smaller than a wavelength, i.e. θ 1. In

56 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 36 addition, a small phase shift per unit cell is required in order to avoid the large phase shifts associated with the lower Bragg cutoff frequency, i.e. β bl d 1. In order to reflect the effective nature of the periodic metamaterial medium, the Bloch propagation constant β bl will be written as β mtm henceforth. Thus, under the aforementioned assumptions that θ 1andβ bl d 1 and considering a periodicity of d = d h-tl for each unit cell, the effective propagation constant can be written as ][ ] βmtm [β 2 = 1 Z tl β tl. (2.67) ωc Z d h-tl ωl d h-tl Finally, by replacing the transmission-line parameters β tl and Z from Equations (2.19) and (2.2), the effective metamaterial propagation constant can be written as β mtm = ±ω [ ][ ] L eff C eff = ±ω 1 1 L C. (2.68) ω 2 C d h-tl ω 2 L d h-tl Equation (2.68) demonstrates that the effective propagation constant of the metamaterial medium has a similar form to the propagation constant of a conventional transmission line but with effective inductance and capacitance terms, L eff and C eff given by the expressions in the square brackets of Equation (2.68). Examining this equation, it can be observed how this medium can exhibit a positive, negative as well as a zero propagation constant, simply by changing the values of the loading parameters L and C. When the loading reactance is negligible, the host transmission line parameters L and C dominate and a forward wave is present on the structure. On the other hand, when the loading reactance is dominant and greater than the individual L and C values, L eff and C eff become negative and thus the structure supports a backward wave. Finally, when the loading reactances are equal to the host transmission line parameters L and C, then the effective propagation constant is zero, and there is no propagation along the structure. In this case, a standing wave is established along the structure that has a constant amplitude (i.e. no variation along the z-direction) that varies with time. By setting each of the effective material parameters L eff and C eff equal to zero in Equation (2.68), we can obtain expressions for the cutoff frequencies of the lower and the upper edges of the stopband, f c1 and f c2, respectively, as shown in Figure 2.16(a), { } 1 f c1 =min 2π 1, LC d h-tl 2π, (2.69) L Cd { h-tl } 1 f c2 =max 2π 1, LC d h-tl 2π. (2.7) L Cd h-tl

57 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Z BL,T Frequency (GHz) Light line RH band Frequency (GHz) Z BL,π 2 LH band 2 π π/2 π/2 π β d (rad) BL (a) Dispersion diagram Re{Z } (Ω) BL (b) Bloch impedance diagram Figure 2.17: Characteristics of representative NRI-TL metamaterial T and π unit cells with parameters C =1pF,L =25nH,Z =5Ω,d = d h-tl =3mmandθ =8.8 at 2 GHz. Note that in this case the impedance matching condition of Z = L /C is imposed. Considering the scenario where 1/2π LC d h-tl < 1/2π L Cd h-tl, the lower edge of the stopband, f c1, is given by the series resonance between the total series inductance of the transmission line, Ld h-tl, and the loading capacitor C, while the upper edge of the stopband, f c2, is given by the shunt resonance between the total shunt capacitance of the transmission line section, Cd h-tl, and the loading inductance L. By equating f c1 and f c2, the stopband between the two cutoff frequencies can be closed, and a continuous band is formed between the lower left-handed and the upper right-handed bands. The impedance matching condition for closing the stopband in a NRI-TL metamaterial structure can therefore be written as [2] L Z = C = L = Z,bw. (2.71) C Thus, according to the above impedance matching condition, for a uniform transition between the left-handed band and the right-handed band, the characteristic impedance of the PRI host transmission line must be the same as the characteristic impedance of the NRI backward-wave line components that load the line. Figure 2.17(a) shows a representative dispersion diagram for either a T or a π metamaterial unit cell when the impedance matching condition of Equation (2.71) is imposed. It can be observed that compared to the open dispersion diagram of Figure 2.16(a), there is now a single passband comprising the lower left-handed band that smoothly transi-

58 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 38 tions into the upper right-handed band at the design frequency of f =2GHz. Thus, depending on the region of operation, Figure 2.17(a) highlights the fact that the NRI-TL metamaterial structure can support forward and backward waves as well as the peculiar case of the structure supporting standing waves with a zero degree phase shift at exactly the closed-stopband point of f = 2 GHz. Additionally, around the design frequency the phase response is linear and broadband with frequency. Figure 2.17(b) shows the Bloch impedance diagrams for both the T and π unit cells under the impedance matching condition of Equation (2.71), which exhibit as expected complementary characteristics. From Figure 2.17(b), it can be observed that the Bloch impedance remains remarkably constant around 5 Ω for a much wider bandwidth compared to the open stopband cases of Figures 2.12(b) and Figures 2.15(b), which suffer from rapid changes in the Bloch impedance around the vicinity of the stopbands. The broadband nature of the Bloch impedance can be better understood if we consider a more simplified expression for the Bloch impedance, derived under the effective medium conditions. Under the first condition that θ 1, the Bloch impedance expressions of Equations (2.6) and (2.64) simplify, respectively, to Z bl,t = Z bl,π = (Y Z Y + Z2 4, (2.72) Z + Y 2 4 ) 1. (2.73) Additionally, for typical loading parameters necessary to achieve the β bl d 1 condition, (see for example the parameters used in Figure 2.17), it can be shown that the first term in each of the expressions of Equations (2.72) and (2.73) dominate. Thus, we can make the approximations Z + Z2 Z and Y + Y 2 Y, and the Bloch impedance expression Y 4 Y Z 4 Z for both cases reduces to Z Z bl = Y = L. (2.74) C Furthermore, under the closed stopband condition of Equation (2.71), the characteristic impedance of the host transmission line is equal to the characteristic impedance of the backward-wave line. Therefore, under the effective medium conditions all three impedances are equal, resulting in a perfectly matched NRI-TL metamaterial structure: Z bl = Z = Z,bw. (2.75)

59 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 39 The impedance matching condition of Equation (2.71) can subsequently be used to derive an approximate expression for the effective phase shift per unit cell of a NRI- TL metamaterial line. By inserting Equation (2.71) into Equation (2.68), the effective propagation constant of the line can be expressed as β mtm = ω 1 LC + ω (2.76) L C d h-tl β mtm = β h-tl + β bw. (2.77) This expression can be interpreted as the sum of the propagation constants of the host transmission line, β h-tl, and a uniform backward-wave line, β bw formed by the loading elements L and C. The analogous effective phase shift per unit cell can be written as φ mtm = β mtm d h-tl. (2.78) Inserting Equation (2.76) into Equation (2.78), the effective phase shift per NRI-TL metamaterial unit cell under the impedance matching condition of Equation (2.71) can be written as [28] φ mtm = ω 1 LCd h-tl + ω (2.79) L C φ mtm = φ h-tl + φ bw. (2.8) Equation (2.79) can be recognized as the sum of the phases incurred by the host transmission line, φ h-tl, and the backward-wave line, φ bw, as indicated in Equation (2.8), which was described initially in Figure 2.7. This equation succinctly describes the inherent phase compensating nature of each NRI-TL metamaterial unit cell. By adjusting the values of the loading elements L and C while maintaining the impedance matching condition, the effective phase shift across each unit cell can be tailored to produce a net positive, negative or even a phase shift at a given frequency. It should be emphasized that the necessary conditions for the phase relationship presented in Equation (2.79) to be valid are that the impedance matching condition of Equation (2.71) must be satisfied, the physical length of the unit cell must be small compared to the wavelength, i.e. φ h-tl 1, and the phase shift per unit cell must also be small, i.e. φ mtm 1. The phase velocity of the lossless NRI-TL metamaterial line can also be calculated from Equation (2.23) to be v φ,mtm = ω β mtm = ω 2 L C ω 2 LCL C 1. (2.81)

60 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 4 Therefore, depending on the sign of β mtm, the line will exhibit a positive or a negative phase velocity, which will therefore determine the sign of the refractive index. Equation (2.81) also indicates that a NRI-TL line is necessarily dispersive, since the phase velocity is not constant with frequency. Nevertheless, it will be subsequently shown that by operating the NRI-TL metamaterial line in the vicinity of the β mtm =pointand above, its dispersive nature can be minimized. The group velocity of the NRI-TL metamaterial line can be obtained from Equations (2.25) and (2.68) to be ω 2 L v g,mtm = C ω 2 LCL C +1. (2.82) Thus, the group velocity of a NRI-TL metamaterial line for any loading-element values will always be positive, which is consistent with the fact that the Poynting vector is always positive, regardless of the direction of propagation of the waves on the line. It should also be pointed out that in the high-frequency limit, the group velocity of the metamaterial line is bounded, and approaches that of the unloaded host transmission line given by Equation (2.26). Thus, the metamaterial line can be effectively used to synthesize a negative refractive index medium at low frequencies as described in Section 2.3, without the unphysical problem of a superluminal group velocity at high frequencies. A final note pertains to the method of power transfer along a NRI-TL line. Since real power is carried forward only by the transverse fields of the line, this implies that it is the underlying TEM host transmission line within the NRI-TL medium that is responsible for power transfer. The high-pass loading elements C and L act to manipulate the reactive fields locally in order to shape the overall dispersion, however they do not contribute to any real power flow along the line. This can also be argued from the point of view that the orientation of the lumped loading elements should not affect the propagation characteristics, and as such the fields carrying the real power should not change. For example, in order to implement the series loading capacitance C the use of a series plate capacitor with an electric field that is oriented in the z-direction along the line, should be no different from a metal-insulator-metal (MIM) capacitor that is oriented such that its electric field is in the y-direction, perpendicular to the NRI-TL line. The same argument holds for the shunt loading inductance L. The use of an x-directed vertical wire between the two transmission-line conductors with a rotating magnetic field in the yz-plane, should be no different from the use of a wound coil that produces an x-directed magnetic field between the two transmission-line conductors.

61 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 41 By the same reasoning it can be concluded that although the theoretical NRI medium shown in Section 2.3 is useful for our understanding of the NRI-TL medium, its physical realization is impossible, since there is no method to transfer real power along the line. As was remarked in the discussion of Equation (2.82) above, the addition of an underlying TEM transmission line not only provides a means of transfer for real power along the line, but also resolves the problem of an otherwise superluminal group velocity in the NRI medium Multi-Stage NRI-TL Metamaterial Phase-Shifting Lines The propagation characteristics of a single metamaterial unit cell presented in Sections and can also be used to design multi-stage metamaterial lines, provided that the boundary conditions at the terminals of each unit cell do not change compared to the infinite periodic case. This can be achieved by terminating the metamaterial line in its Bloch impedance, thus eliminating any reflections along the terminated periodic structure. In this manner, each unit cell within the finite metamaterial line will effectively see an infinite periodic medium and will thus retain its propagation characteristics. Consequently, the phase expressions derived assuming an infinitely periodic medium can be used in order to characterize the phase response of a finite metamaterial structure. If the source and load terminations of the metamaterial line are equal to its Bloch impedance Z bl, which in the matched case is also equal to the characteristic impedance of the host transmission line Z and the characteristic impedance of the loading backwardwave line Z bw from Equation (2.75), then from Equation (2.79) the total phase incurred by an n-stage metamaterial line akin to the one shown in Figure 2.9 can be written as the sum of the phase incurred by each constituent unit cell, ( Φ mtm = nφ mtm = n ω ) 1 LCd h-tl + ω. (2.83) L C Alternatively, the full dispersion relation of Equation (2.57) can be used for φ mtm,which results in a more accurate, yet less intuitive, expression for the phase. In order to verify the validity of the above arguments, the results obtained by analyzing a representative four-stage metamaterial line using the periodic analysis outlined in the preceding sections were compared to those obtained by analyzing the same metamaterial line in a circuit simulator. The periodic analysis proceeded as outlined in Section for the T unit cell of Figure 2.1(a). The phase response of the four cascaded unit

62 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Periodic Analysis Agilent ADS Phase of S21 ( ) Frequency (GHz) Figure 2.18: Phase responses of a four-stage NRI-TL metamaterial line analyzed using periodic analysis and the Agilent-ADS circuit simulator. Parameters used: Z term = 5 Ω, C = 1 pf, L = 25 nh, Z =5Ω,d h-tl =3mmandθ =8.8 at 2 GHz. cells was obtained by multiplying the ABCD matrix of Equation (2.52) four times, and then finding the analogous dispersion relation according to Equation (2.55). The unit cell parameters were the same as the ones used in Figure 2.17, and were C = 1 pf, L =25nH,Z =5Ω,d h-tl =3mmandθ =8.8 at 2 GHz. Subsequently, a four-stage metamaterial line of T unit cells similar to the one shown in Figure 2.9 was designed in the Agilent-ADS circuit simulator with source and load terminations of Z term =5Ω. The phase responses obtained using both methods are shown in Figure It can be observed that the phase responses obtained from the periodic analysis and from Agilent-ADS are identical within the passband of the structure. At low frequencies below.15 GHz the structure has a stopband, and therefore the periodic analysis is not able to predict the phase. Nevertheless, these results confirm that the propagation characteristics derived for the infinitely periodic case can also be used to design finitelength metamaterial phase-shifting lines, provided that the lines are terminated in their Bloch impedance. It should also be mentioned that the phase response was also calculated using Equation (2.83) and was found to be very similar to both the results shown in Figure 2.18, with only a minor deviation in the low-frequency region of the passband. Additionally, as expected, it did not exhibit a low-frequency cutoff. Equations (2.83) and (2.71) were used in [28] and [91] to design various metama-

63 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 43 terial phase-shifting lines with different physical lengths. These were then compared to a conventional transmission line that was one guided wavelength long (λ g )andaslowwave low-pass loaded line, both of which incurred a 2π rad phase shift at the design frequency. The results of the above comparison concluded that the metamaterial phase-shifting lines offer some significant advantages compared to both conventional transmission lines and slow-wave low-pass loaded lines. Specifically, since the metamaterial lines comprise a series of cascaded unit cells, their length can be arbitrarily long while incurring any desired phase shift. Consequently, the physical size of the metamaterial lines can be significantly smaller than that of a λ g transmission line, and in the limiting case can be simply a single metamaterial unit cell. The small physical size of the metamaterial lines also results in a linear and broadband phase response around the design frequency, i.e. a large group velocity. The low-pass loaded lines can also be designed to occupy the same small physical size as the metamaterial lines, however their phase response is very similar to that of the λ g transmission line, and are therefore significantly more narrowband than the analogous metamaterial lines. In summary, NRI-TL metamaterial lines can be very compact in size, they can be easily fabricated using standard etching techniques, and they exhibit a linear and broadband phase response (i.e. a large group velocity) around the design frequency. They can incur either a negative or a positive phase, as well as a phase depending on the values of the loading elements, while maintaining a small physical length. In addition, the phase incurred is independent of the length of the structure. Due to their compact, planar design, they lend themselves easily towards integration with other microwave components and devices and are well suited for broadband applications. As such, the following chapters in this thesis will focus on various practical microwave and antenna applications that harness the advantages that the NRI-TL metamaterial lines have to offer Loading-Element Values Approximate Expressions In the design of metamaterial lines, it is very useful to have closed-form expressions for the values of the loading elements L and C required to produce a desired phase shift per unit cell φ mtm at a given frequency ω, given a particular section of host transmission line with parameters Z and φ h-tl. Approximate expressions for the loading elements can

64 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 44 be obtained by substituting the impedance matching condition of Equation (2.71) into Equation (2.79) and solving for L and C Z L = ω (φ mtm φ h-tl ), (2.84) 1 C = ω Z (φ mtm φ h-tl ). (2.85) It should be noted that since the impedance matching condition of Equation (2.71) was used in the derivation of Equations (2.84) and (2.85), these are only valid when the stopband is closed and in the effective medium limit of φ mtm 1and φ h-tl 1. Nonetheless, these expressions are very simple and intuitive and clearly highlight the factors that affect the values of the loading elements. Exact Expressions The exact expressions for the loading elements can be obtained by employing the full dispersion relation of Equation (2.57) for the general case where the stopband can be either open or closed. In both cases, it is desired to calculate the loading elements for a given phase shift per unit cell and a given Bloch impedance. Considering first the case when the stopband is closed, recall from Figure 2.17 that the Bloch impedance remains very close to the value of Z over a large frequency range. Thus, a very good approximation for the Bloch impedance for a large range of frequencies is actually the value of Z. We therefore use the impedance matching condition of Equation (2.71) and re-arrange it to obtain an expression of the loading inductance L in terms of C and Z L = C Z 2. (2.86) By substituting Equation (2.86) into Equation (2.57), we obtain a quadratic equation in terms of C ( ) cos(β bl d) cos(θ) C 2 ( sin(θ) ω Z The solution to the above quadratic equation can be written as ( ) ( ) 2 sin(θ) ω Z ± sin(θ) ω Z 4 (cos(β bl d) cos(θ) ) C + 1+cos(θ) =. (2.87) 4ωZ 2 2 )( 1+cos(θ) 4ω 2 Z2 C =. (2.88) 2(cos(β bl d) cos(θ)) We seek the positive solution of the quadratic equation, therefore resulting in unique values for L and C from Equations (2.86) and (2.88). A particular case of Equation )

65 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 45 (2.87) arises when β bl d is equal to θ (or equivalently, φ mtm = φ h-tl ). In this case, the first term of Equation (2.87) is equal to zero, and the loading capacitance C can be simply expressed as C = 1+cos(θ) 4ω Z sin(θ). (2.89) The open stopband case is the most general case and proceeds in a similar, yet slightly more complicated manner. The two design parameters in this case are the phase shift per unit cell, given by the dispersion relation of Equation (2.57), and the Bloch impedance given by Equations (2.6) or (2.64). Both the dispersion and Bloch impedance relations are functions of L and C, however an attempt to find a closed-form expression for these reveals that the expression quickly becomes very complicated. A simple way to overcome this problem is to solve for L and C graphically by expressing C in terms of L through the dispersion relation of Equation (2.56). Therefore, Equation (2.56) can be written as Z = 1 = cos(β bld) cos(θ) j YZ sin(θ) 2 jωc Y cos(θ)+j. (2.9) Y 4 2 sin(θ)+ Y 4 Recall that Z and Y are related to C and L through Equation (2.5). Substituting Equation (2.9) into the Bloch impedance expressions of Equation (2.6) or (2.64) and by sweeping the value of L, a range of values for the Bloch impedance is obtained. The final value of L is obtained by selecting the value that corresponds to the desired Bloch impedance. The loading capacitance C is subsequently found by substituting the value of L into Equation (2.9). 2.5 Non-radiating NRI-TL Metamaterial Phase Shifting Lines Any artificial transmission line that supports fast waves, i.e. waves whose phase velocity is greater than the speed of light (v φ >c), will tend to radiate into free-space provided that its electrical length is sufficiently long, and the geometry of the artificial line is such that there are unbalanced currents on the line. This is the underlying operating principle of a class of radiators known as leaky-wave antennas, which are designed to support waves with a superluminal phase velocity. An example of such an antenna using NRI-TL metamaterial unit cells will be described in detail in Chapter 6. There are, nonetheless, some cases in which a structure supports a superluminal phase velocity in a certain frequency band, but this structure is not intended for antenna

66 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 46 applications, but rather for guided-wave applications. This is the case for the NRI-TL metamaterial lines presented in the previous section when they are used to create phase-shifting lines. For the case of the metamaterial lines, the phase incurred by each unit cell φ mtm, and therefore the propagation constant β mtm, are designed to equal zero at the design frequency. Since the phase velocity is defined as v φ = ω/β, this will be infinite and therefore superluminal at the design frequency. Thus the metamaterial lines will support fast waves and it appears that they will be prone to radiation. It should be noted, however, that leaky-wave radiation will only occur from metamaterial lines when these are made sufficiently long, thus establishing a fixed leakage constant, α, along the line (for more details refer to Chapter 6). Additionally, in order to establish a fixed leakage constant, the geometry of the line itself must support unbalanced currents, which constitute the main radiation mechanism. It follows therefore that if the currents on the line are balanced or if the line itself is shielded, then it will not be prone to leaky-wave radiation. Hence, a superluminal phase velocity is not in itself a sufficient condition for leaky-wave radiation, but rather must be accompanied with a suitably designed artificial transmission line that facilitates the radiation. The ability of the metamaterial lines to act as leaky-wave radiators can also be verified by examining the dispersion diagram of Figure 2.17(a). It can be observed that portions of both the left-handed and the right-handed bands lie within the light cone of the structure, which is defined as the area bounded on both sides by the light lines, and which demarcates the transition between slow and fast-wave propagation relative to free space (i.e. along these lines v φ = c). Within the light cone, the phase velocity of the propagating waves is faster than the speed of light, i.e. v φ >c. Therefore if the metamaterial lines are designed to operate anywhere within the light cone, then there is the possibility of producing leaky-wave radiation. This is precisely the case for the metamaterial lines, which are designed to operate at the transition point between the left-handed and the right-handed bands, which lies in the middle of the radiation region for the metamaterial line. Herein, a new topology is proposed for a type of metamaterial line that has a phase velocity that is slow compared to the speed of light, and is therefore not prone to radiation. The non-radiating metamaterial (NR-MTM) line is shown in Figure 2.19 and consists of two sections: a metamaterial line that is specifically designed to operate in the slow-wave regime, and a conventional transmission line that is inherently slow. Since each section of the line is slow, waves traveling on the composite slow-wave metamaterial line will be

67 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 47 Metamaterial Line 1 m 2C 2C 2C 2C Transmission Line Z Z Z L L Φ MTM = m MTM Φ TL Φ NR-MTM = Φ MTM + Φ TL Figure 2.19: Non-radiating (slow-wave) NRI-TL metamaterial phase-shifting line. guided by the structure and will therefore not radiate, regardless of its length. Considering first the section of metamaterial line, in order to ensure that this does not radiate, each unit cell of the line must be operated in the NRI backward-wave region, while simultaneously ensuring that the phase velocity of the line is less than the speed of light (v φ <c). In other words, the propagation constant of the line must exceed that of free space ( β mtm >k ). This will effectively produce a slow-wave structure with a positive insertion phase, φ mtm. The dispersion diagram for this scenario is shown in Figure 2.2. It can be observed that at the design frequency of f = 2 GHz, propagation occurs well outside of the light cone. The complete m-stage slow-wave metamaterial line is then formed by cascading each of the non-radiating metamaterial unit cells as shown in Figure 2.9, to produce a total positive phase of Φ mtm = mφ mtm. Subsequently, the m-stage metamaterial line is cascaded with a conventional (PRI) transmission line, as shown in Figure 2.19, that inherently incurs a negative insertion phase, Φ c-tl. The function of the transmission line is to compensate the positive phase incurred by the metamaterial line. This phase compensating technique allows the synthesis of a positive, negative or zero insertion phase, and is analogous to the phase compensation achieved in a single metamaterial unit cell, but with the additional advantage that the line is non-radiating. The total phase incurred by the non-radiating metamaterial phase-shifting line is therefore given by the sum of the phases of each constituent line. From Equations (2.22) and (2.83) this is given by ( Φ nr-mtm =Φ mtm +Φ c-tl = m ω ) 1 LCd h-tl + ω + ( ω ) LCd c-tl. (2.91) L C It should be noted that Equation (2.91) was derived based on the impedance matching

68 Chapter 2. Negative-Refractive-Index Transmission-Line Theory Frequency (GHz) 6 4 Non radiating NRI BW region Light line 2 π π/2 φ MTM π/2 π β BL d (rad) Figure 2.2: Dispersion diagram indicating the non-radiating NRI backward-wave region of operation, which is outside of the light cone. Metamaterial unit cell parameters: C =2pF, L =5nH,Z =5Ω,d =3mmandθ =8.8 at 2 GHz. Note that the impedance matching condition of Z = L /C is imposed. condition of Equation (2.71), which ensures that there is no stopband between the lower left-handed band and the upper right-handed band. This is a necessary condition for the use of the approximate expression of Equation (2.79) for the phase shift per metamaterial unit cell, however it is not in general necessary for the operation of the NR-MTM line around the Φ nr-mtm = point. This is because the MTM section of the NR-MTM line operates in the lower left-handed region, away from the φ mtm = operating point. As such, the metamaterial line can be designed with an open stopband, as long as the desired operating region of the NR-MTM line does not overlap with the stopband region of the MTM line. Interestingly though, the total phase shift across the NR-MTM line can still be maintained at Φ nr-mtm = even for the case of an open stopband, something that of course is not possible with a conventional metamaterial line. The main advantage of using an open stopband in a NR-MTM line is that the values of the loading elements work out to be significantly smaller than for the closed stopband case, and are therefore much easier to practically implement. For the open stopband case, the complete dispersion relation of Equation (2.57) should then be used in Equation (2.91) to calculate Φ mtm for the metamaterial line.

69 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 49 Potential applications for non-radiating metamaterial lines are in series feed networks for antenna arrays, and in electrically long phase-shifting lines. The series feed networks can have an arbitrary length, can be designed to have a broadband phase response, and have the significant advantage that they would not affect the radiation patterns of the antenna arrays. 2.6 Analysis of MTM, NR-MTM and TL Phase-Shifting Lines Choice of the Number of Unit Cells Let us consider the case where we would like to design either a MTM or a NR-MTM phase-shifting line with a total length of d tot. Some natural questions that arise in the design of the lines are, how many unit cells should be used to implement the lines, how the values of the loading elements L and C are affected by the choice of the number of unit cells, and whether there are any differences in the phase characteristics of the equidistant metamaterial lines. Metamaterial Lines To begin to answer these questions, let us first consider the two metamaterial lines shown in Figure In the first case, the phase-shifting line of length d tot is implemented using a single MTM unit cell, and in the second case the same length of line d tot is implemented using n MTM unit cells. In both cases it is assumed that the desired phase shift across the entire MTM line is Φ mtm = at the design frequency of ω, therefore each unit cell must also incur φ mtm = at ω. For the single-stage MTM line of Figure 2.21(a), where the host transmission line incurs a phase of φ h-tl1, Equations (2.84) and (2.85) therefore become L 1 = ω ( φ h-tl1 ), (2.92) 1 C 1 = ω Z ( φ h-tl1 ). (2.93) Note that the phase incurred by each host TL section, φ h-tl1, is a negative quantity, therefore this results in positive and physically realizable expressions for both L 1 and C 1. Now if we consider the n-stage MTM line of Figure 2.21(b), the desired phase shift Z

70 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 5 2C Z 1 2C Z L d TOT (a) n 2C 2C 2C 2C 2C 2C 2C 2C Z Z Z Z L L L L d TOT (b) Figure 2.21: Schematic diagrams of (a) a single-stage MTM line, (b) an n-stage MTM line, both of which have a total length of d tot. per unit cell is still φ mtm =, however the phase incurred by each host TL section is now φ h-tl2 = φ h-tl1 /n. Substituting this into Equations (2.84) and (2.85), we obtain the loading-element values required for the n-stage MTM line Z L n = n ω ( φ h-tl1 ), (2.94) 1 C n = n ω Z ( φ h-tl1 ). (2.95) Thus, Equations (2.94) and (2.95) effectively demonstrate that the loading-element values for an n-stage MTM line will be n times greater than the loading-element values required for a single-stage MTM line that has the same length. In the limit as the unit cell size becomes infinitesimally small, n will tend to infinity and so will the values of the loading elements. This has significant implications on the practical realization of the loading elements. It can therefore be deduced that in order to maintain small loadingelement values that are easier to realize, it is beneficial to use the fewest possible MTM stages. This poses no problem if the total length of the MTM phase-shifting line is small compared to the wavelength. If, however the total length of the line is electrically large, then implementing the line with a single unit cell will result in a cell than can no longer be described by effective medium parameters, which puts into question the use of Equation (2.79). In this case, the full dispersion relation of Equation (2.57) must be

71 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 51 Phase of S21 ( ) n=1 MTM line n=2 MTM line n=4 MTM line n=8 MTM line n=16 MTM line Phase of S21 ( ) n=1 MTM line n=2 MTM line n=4 MTM line n=8 MTM line n=16 MTM line Frequency (GHz) (a) Frequency (GHz) (b) Figure 2.22: Phase responses of various MTM lines which all have a length of d tot = λ g /4. The loading elements for each line were calculated using (a) the approximate Equations (2.84) and (2.85), and (b) the exact Equations (2.86) and (2.88). used to calculate the values of the loading elements, which can be found from Equations (2.86) and (2.88). In order to illustrate better the above points, it is instructive to consider a specific example where five MTM lines are implemented using 1, 2, 4, 8 and 16 stages, all of which are designed to have the same length of d tot = λ g /4. The operating frequency is chosen to be f = 2 GHz, the host TL has an effective permittivity of ɛ eff =1.2, and Z = 5 Ω, which results in a guided wavelength of λ g = mm. Thus, in this case, d tot = λ g /4 = mm. The approximate loading-element values for each MTM line were first calculated using Equations (2.84) and (2.85), and the exact values were subsequently calculated using Equations (2.86) and (2.88). Figures 2.22(a) and (b) show the phase responses of the five MTM phase-shifting lines, and Tables 2.1 and 2.2 summarize the loading-element values and pertinent characteristics of the lines. The first observation that we can make about Figure 2.22(a) is that the single-stage MTM line does not incur the desired phase shift at f = 2 GHz, but rather incurs This can be attributed to the fact that the host TL used in this case has an electrical size of φ h-tl = 9, and is therefore not small compared to the wavelength. As such, the approximate phase relation of Equation (2.79) does not hold for this case where the line cannot be characterized as an effective medium. As the number of stages increases, and therefore the electrical size φ h-tl of each unit cell decreases, the phase responses of the various MTM lines in Figure 2.22(a) converge to the same charac-

72 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 52 Table 2.1: Loading element values and performance characteristics for various MTM lines which all have a length of d tot = λ g /4. The loading elements were calculated using the approximate Equation (2.79). Design φ h-tl ( ) C (pf) L (nh) Error in Φ mtm ( ) ΔΦ/Δf ( /GHz) n= n= n= n= n= Table 2.2: Loading element values and performance characteristics for various MTM lines which all have a length of d tot = λ g /4. The loading elements were calculated using the exact Equation (2.57). Design φ h-tl ( ) C (pf) L (nh) Error in Φ mtm ( ) ΔΦ/Δf ( /GHz) n= n= n= n= n= teristic. This can also be verified from the deviation of Φ mtm from and the phase variation ΔΦ/Δf from the last two columns of Table 2.1, both of which converge to constant values of and 9 /GHz, respectively, as the number of stages increases. Referring to Figure 2.22(b), it can be observed that all five of the MTM lines that are designed using the exact expressions for the loading elements of Equations (2.86) and (2.88) exhibit the desired phase shift of at f = 2 GHz. All of the lines also exhibit very similar phase responses, except for the single-stage MTM line which has a steeper phase characteristic with frequency, especially at lower frequencies. The above observations can also be verified by the last two columns in Table 2.2, which demonstrate that there is no deviation from the desired Φ mtm = phase point and the phase variation ΔΦ/Δf converges to a value of 9 /GHz as the number of stages increases. These results also agree well with the ones presented in Table 2.1. The observations made earlier about the increase in the loading-element values as

73 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 53 the unit cell size decreases can also be verified by observing the loading-element values shown in the third and fourth columns of Tables 2.1 and 2.2. For practical realizations, especially in printed form, the values of L and C become increasingly difficult to realize as the unit cell size decreases. The phase characteristics, however, remain approximately the same for unit cell sizes that are equal to or less than φ h-tl = 22.5, i.e. for the 4, 8 and 16-stage lines, as can be observed from Figure Thus, a reasonable choice for the size of each unit cell used to realize the metamaterial line is d h-tl = λ g /16. This provides a good compromise, because it results in reasonably small loading-element values, while still allowing the insertion phase to be predicted accurately using the approximate expression of Equation (2.79), and the phase variation ΔΦ/Δf is close to that of the infinitesimally small unit cell. A unit cell size of φ h-tl 22.5 also corresponds well with the effectivemedium condition of φ h-tl 1 rad used in Section Of course, if the goal is to create MTM lines with the smallest possible phase variation, then a maximum number of stages should be used. On the other hand, if the phase variation is not of interest, but it is desired to minimize the loading-element values, then even a single-stage MTM line could be used, which should then be designed using Equations (2.86) and (2.88) in order to obtain the correct insertion phase. Non-Radiating Metamaterial Lines Let us now consider a similar scenario with the two non-radiating metamaterial lines shown in Figure In the first case, the metamaterial phase-shifting section of the line that has a length of d tot /2 is implemented using a single NR-MTM unit cell, and in the second case m unit cells are used to implement the same section of line. In both caseshalfofthetotallength,i.e.d tot /2, is allocated to the metamaterial section of the line and the other d tot /2 is allocated to the transmission-line section of the line. As will be shown in Section 4.3, this scheme ensures that the NR-MTM phase-shifting lines will be slow and therefore non-radiating. In order to obtain a net phase-shift across the entire line, the phase shift incurred by the metamaterial section of the line must therefore be equal and opposite to that incurred by the transmission-line section. From Equation (2.91) for a total phase shift across the line of Φ nr-mtm = we can write Φ mtm = Φ c-tl. (2.96) Let us now examine how the loading-element values of Equations (2.84) and (2.85) are affected by the condition of Equation (2.96). For an m-stage NR-MTM line, the

74 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 54 2C Z 1 2C Z Z L d TOT (a) 1 m 2C 2C Z 2C 2C Z Z L L d TOT (b) Figure 2.23: Schematic diagrams of (a) a single-stage NR-MTM line, (b) an m-stage NR- MTM line, both of which have a total length of d tot. metamaterial section will incur a positive phase shift of Φ mtm = mφ mtm.fromequation (2.96) the phase shift per unit cell can therefore be written as φ mtm = Φ c-tl m. (2.97) Since the total length of the line is divided equally between the metamaterial section and the transmission-line section, we can write Substituting Equation (2.98) into Equation (2.97) we obtain mφ h-tl =Φ c-tl. (2.98) φ mtm = φ h-tl. (2.99) Finally, substituting Equation (2.99) into Equations (2.84) and (2.85) we obtain the approximate loading-element values for each unit cell of a NR-MTM line L = 1 2 ω ( φ h-tl ), (2.1) C = ω Z ( φ h-tl ). (2.11) Considering now the first case shown in Figure 2.23(a) where a single-stage NR-MTM line is used to incur a positive phase shift Φ mtm = φ mtm using a host TL section that Z

75 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 55 incurs a phase of φ h-tl1, the loading-element values from Equations (2.1) and (2.11) can be written as L 1 = 1 2 ω ( φ h-tl1 ), (2.12) C 1 = ω Z ( φ h-tl1 ). (2.13) Comparing the loading-element values for the MTM line from Equations (2.92) and (2.93) to the ones for the NR-MTM line from Equations (2.12) and (2.13) it is can be observed that the NR-MTM line requires elements with half of the values compared to those of a MTM line using the same size unit cells. This is a significant advantage that the NR- MTM lines have over the MTM lines, which can greatly assist in the practical realization of these phase-shifting lines. Now let us consider the second case shown in Figure 2.23(b) where an m-stage NR- MTM line is used to incur the same positive phase shift of Φ mtm as in the first case. Each unit cell of the line is therefore required to incur a phase of φ mtm =Φ mtm /m with a unit cell size of φ h-tl2 = φ h-tl1 /m. Substituting this into Equations (2.12) and (2.13) we obtain the loading-element values required for an m-stage NR-MTM line L m = m 2 ω ( φ h-tl1 ), (2.14) C m = m 1 2 ω Z ( φ h-tl1 ). (2.15) Thus, Equations (2.14) and (2.15) demonstrate that the loading-element values for an m-stage NR-MTM line will be m times greater than the loading-element values required for a single-stage NR-MTM line of the same length, which is the same result obtained for the MTM line. Now consider a representative example where the characteristics of four NR-MTM lines are compared, each employing 1, 2, 4 and 8 stages to realize a line with a total length of d tot = λ g /4. As with the case of the MTM lines, the operating frequency is chosen to be f = 2 GHz, the host TL has an effective permittivity of ɛ eff =1.2, and Z =5Ω,whichresultsind tot = λ g /4=34.23 mm. The approximate loadingelement values for each NR-MTM line were first calculated using Equations (2.1) and (2.11), and the exact values were subsequently calculated using Equations (2.86) and (2.89). Figures 2.24(a) and (b) show the phase responses of the four NR-MTM phase-shifting lines, and Tables 2.3 and 2.4 summarize the loading-element values and pertinent characteristics of the lines. Z Z

76 Chapter 2. Negative-Refractive-Index Transmission-Line Theory m=1 NR MTM line m=2 NR MTM line m=4 NR MTM line m=8 NR MTM line 2 15 m=1 NR MTM line m=2 NR MTM line m=4 NR MTM line m=8 NR MTM line Phase of S21 ( ) 1 5 Phase of S21 ( ) Frequency (GHz) (a) Frequency (GHz) (b) Figure 2.24: Phase responses of various NR-MTM lines which all have a length of d tot = λ g /4. The loading elements for each line were calculated using (a) the approximate Equations (2.1) and (2.11), and (b) the exact Equations (2.86) and (2.89). Table 2.3: Loading element values and performance characteristics for various NR-MTM lines which all have a length of d tot = λ g /4. The loading elements were calculated using the approximate Equations (2.1) and (2.11). Design φ h-tl ( ) C (pf) L (nh) Error in Φ nr-mtm ( ) ΔΦ/Δf ( /GHz) m= m= m= m= Table 2.4: Loading element values and performance characteristics for various NR-MTM lines which all have a length of d tot = λ g /4. The loading elements were calculated using the exact Equations (2.86) and (2.89). Design φ h-tl ( ) C (pf) L (nh) Error in Φ mtm ( ) ΔΦ/Δf ( /GHz) m= m= m= m=

77 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 57 From Figures 2.24(a) and (b), it can be observed that the phase responses of the lines calculated using the approximate expressions for the loading elements match quite well with those that were calculated using the exact expressions, even for the m = 1 case where the largest unit cell size of φ h-tl = 45 was used. As per the discussion of the MTM lines, as the number of stages used in the realization of the NR-MTM line increases, the phase responses of the lines implemented using different numbers of unit cells converge to the same characteristics. Nonetheless, since the length that the metamaterial section of the NR-MTM line occupies is half the length of the corresponding MTM line, the NR-MTM line can achieve comparable performance to the MTM line by using half of the same-size unit cells. Additionally, as was pointed out in Equations (2.1) and (2.11), the values of the loading elements required to achieve this are half of those required to implement the analogous MTM line. Finally, based on the discussion in the previous section, a unit cell size of φ h-tl 22.5 is recommended for the effective design of NR-MTM lines Phase Variation Characteristics A key characteristic that can be used to compare the performance of various metamaterial lines to that of conventional transmission lines, is the phase variation of each line with respect to frequency. This an especially important consideration when a net insertion phase of is desired for in-phase power division applications. A detailed analysis of these applications will be outlined in Chapter 4, where it will be shown that the phase variation will also affect the return-loss and insertion-loss bandwidths of such devices. The three types of lines that will be considered herein are a MTM line, a NR-MTM line and a one-wavelength conventional TL shown schematically in Figure 2.25, which all have the same overall length of d tot = λ g. Beginning with an n-stage MTM line with total physical length d tot = nd h-tl as shown in Figure 2.25(a), and the assumption that the phase shift per unit cell and the length of the host transmission lines are small, the total phase shift incurred by the line under the impedance matching condition of Equation (2.71), is given by Equation (2.83). By taking the derivative of Equation (2.83) with respect to ω, weobtain dφ mtm dω ( = n LCd h-tl 1 ω 2 L C ). (2.16) It is desired that the MTM line incur a phase shift of Φ mtm =, therefore setting

78 Chapter 2. Negative-Refractive-Index Transmission-Line Theory n 2C 2C 2C 2C 2C 2C 2C 2C Z Z Z Z L L L L dφ dω ( ) MTM = 2 LC nd H TL d H-TL d TOT = λ g (a) 1 m 2C 2C Z 2C 2C Z Z dφ dω NR MTM = 2 LC ( md + d ) H TL C TL L L d H-TL d C-TL d TOT = λ g (b) Z d TOT = λ g dφ dω TL = LCd TOT (c) Figure 2.25: Schematic diagrams of (a) an n-stage MTM line, (b) an m-stage NR-MTM line, and (c) a conventional TL, all of which have a length of d tot = λ g. The phase variation with frequency for each line is shown on the right-hand side. Equation (2.83) equal to zero, we can obtain the following relation between the two terms n LCnd h-tl = ω 2. (2.17) L C Substituting Equation (2.17) into Equation (2.16) we obtain the following expression for the variation of the phase with respect to frequency for the metamaterial line at Φ mtm = : dφ mtm dω = 2 LC(nd h-tl ). (2.18) The same procedure can be carried out for the NR-MTM lines presented in Section 2.5 and shown in Figure 2.25(b). The total phase shift incurred by an m-stage NR-MTM line with total physical length d tot = md h-tl + d c-tl is given by Equation (2.91). By

79 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 59 taking the derivative of Equation (2.91) with respect to ω, weobtain ( dφ nr-mtm = m ) 1 LCd h-tl dω ω 2 LCd c-tl. (2.19) L C Setting Equation (2.91) equal to zero, we can obtain the following relation m LC(md h-tl + d c-tl )= ω 2. (2.11) L C Substituting Equation (2.11) into Equation (2.19) we obtain the following expression for the variation of the phase with respect to frequency for the NR-MTM line at Φ nr-mtm = : dφ nr-mtm = 2 LC(md h-tl + d c-tl ). (2.111) dω Now if we consider a transmission line with a total length d tot, its insertion phase as a function of frequency is given by Equation (2.22). Taking the derivative of Equation (2.22) with respect to ω, weobtain dφ tl dω = LCd tot. (2.112) Examining the results derived above, there are some pertinent observations that can be made. The first is that when comparing a MTM and a NR-MTM line that are of the same length, i.e. nd h-tl = md h-tl + d c-tl, the variation of the phase with respect to frequency at Φ mtm =, from Equations (2.18) and (2.111), are exactly the same. Thus, the use of either type of metamaterial line will result in exactly the same phase response around the Φ mtm = point, and one will not provide any advantage over the other in terms of the phase bandwidth. The second observation can be made when comparing the phase responses of metamaterial lines and conventional transmission lines that have the same length, i.e. nd h-tl = md h-tl + d c-tl = d tot. In order for the TL to incur 36 (or equivalently )at the design frequency, it is necessary to set its length (and therefore also the lengths of the metamaterial lines) equal to d tot = λ g,whereλ g is one guided wavelength. From Equations (2.18), (2.111) and (2.112), it can be observed that the slope of the metamaterial lines will be twice as steep as that of the transmission line when all the lines incur. This can also be verified by examining the phase responses of three representative lines shown in Figure 2.26, designed to incur an insertion phase of at the design frequency of 2 GHz. Note that the pertinent details for each of these lines can be found next to the corresponding schematic in Figure The phase responses of the three lines are shown

80 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 6 in Figure 2.27, and it can be clearly observed that the slope of the MTM and NR-MTM responses is approximately twice as steep as the phase response of the TL. Specifically, when the slope of the lines is measured around a 2 MHz bandwidth centred at 2 GHz, the MTM line exhibits a phase variation of ΔΦ mtm /Δf = /GHz, the NR-MTM line exhibits a phase variation of ΔΦ nr-mtm /Δf = /GHz, and the TL exhibits the expected phase variation of ΔΦ tl /Δf = 18 /GHz. Thus, one might conclude that the metamaterial lines perform worse than a conventional transmission line, and that they would in fact exhibit half the phase bandwidth that a transmission line would. This would indeed be correct if the metamaterial lines were constrained to be the same length as their TL counterparts. It has been shown in Section 2.4.4, however that the phase incurred by a metamaterial line is independent of its length. The significant advantage that the MTM lines have to offer, therefore, is that they can be made arbitrarily long (or short) while still incurring a phase shift. The only limitation on the size of a MTM line is that it cannot be shorter than a single constituent unit cell. It is therefore instructive to derive an upper limit on the allowable length of a metamaterial line, below which the metamaterial line will provide a wider phase bandwidth than a conventional TL. To this end, if we let Equation (2.18) equal to Equation (2.112), and we set d tot = λ g, we can obtain the minimum length for the MTM line nd h-tl < λ g 2. (2.113) Similarly, the minimum length for the non-radiating metamaterial line from Equations (2.111) and (2.112) is md h-tl + d c-tl < λ g 2. (2.114) Therefore, as long as the physical length of a metamaterial line is less than half of the length of the equivalent λ g -long TL, the metamaterial line will exhibit a wider phase bandwidth. In order to illustrate the above observations, Figures 2.28(a) and (b) show the insertion phase responses of various MTM and NR-MTM phase-shifting lines, compared to the response of a λ g -TL. Tables 2.5 and 2.6 summarize the phase variation characteristics of the MTM and NR-MTM lines, measured around a 2 MHz bandwidth, centred at 2 GHz. The general configuration of the metamaterial lines is as shown in Figure 2.26, however the number of stages was adjusted in order to obtain lines with lengths of λ g /2, λ g /4, λ g /8andλ g /16. Specifically, for the MTM lines, the

81 Chapter 2. Negative-Refractive-Index Transmission-Line Theory C 2C 2C 2C 2C 2C 2C 2C Z Z L L d H-TL d TOT = λ g Z Z L L n = 32 Z = 5 Ω ε eff = 1.2 d TOT =λ g = mm d H-TL = 4.28 mm C = 8.8 pf L = 2.2 nh (a) C 2C Z L d H-TL 2C 2C Z Z L d C-TL d TOT = λ g m = 16 Z = 5 Ω d TOT =λ g = mm d C-TL = mm d H-TL = 4.28 mm C = 4.4 pf L = 1.1 nh (b) Z d TOT = λ g (c) Z = 5 Ω d TOT =λ g = mm Figure 2.26: Schematic diagrams of (a) a 32-stage MTM line, (b) a 16-stage NR-MTM line and (c) a TL, all of which are λ g long and incur at the design frequency of 2 GHz. 6 4 λ g MTM line λ g NR MTM line λ g TL Phase of S21 ( ) Frequency (GHz) Figure 2.27: Phase responses of a 32-stage MTM line, a 16-stage NR-MTM line and a TL, all of which are λ g long and incur at the design frequency of 2 GHz.

82 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 62 Phase of S21 ( ) λ g /2 MTM line λ g /4 MTM line λ g /8 MTM line λ g /16 MTM line λ g TL Phase of S21 ( ) λ g /2 NR MTM line λ g /4 NR MTM line λ g /8 NR MTM line λ g /16 NR MTM line λ g TL Frequency (GHz) (a) Frequency (GHz) (b) Figure 2.28: Phase responses of (a) various MTM lines (b) various NR-MTM lines, all of which incur at 2 GHz, compared to the insertion phase response of a λ g -long TL. Magnitude of S21 (db).1 λ g /2 MTM line λ g /4 MTM line λ g /8 MTM line λ g /16 MTM line λ TL g Frequency (GHz) (a) Magnitude of S21 (db).1 λ g /2 NR MTM line λ g /4 NR MTM line λ g /8 NR MTM line λ g /16 NR MTM line λ TL g Frequency (GHz) (b) Figure 2.29: Insertion magnitude responses of (a) various MTM lines (b) various NR-MTM lines, all of which incur at 2 GHz, compared to the magnitude response of a λ g -long TL. λ g /2 line employed 16 -unit cells, the λ g /4 line employed eight -unit cells, the λ g /8 line employed four -unit cells and the λ g /16 line employed two -unit cells. For the NR-MTM lines, the λ g /2 line employed eight unit cells incurring a total phase shift of Φ nr-mtm = +9 and a compensating TL incurring Φ c-tl = 9, the λ g /4 line employed four unit cells incurring a total phase shift of Φ nr-mtm =+45 and a compensating TL incurring Φ c-tl = 45,theλ g /8 line employed two unit cells incurring a total phase shift of Φ nr-mtm = and a compensating TL

83 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 63 Table 2.5: Phase variation characteristics of various MTM lines around f =2GHz. Length of MTM line λ g λ g /2 λ g /4 λ g /8 λ g /16 λ g -TL ΔΦ/Δf ( /GHz) Table 2.6: Phase variation characteristics of various NR-MTM lines around f =2GHz. Length of NR-MTM line λ g λ g /2 λ g /4 λ g /8 λ g /16 λ g -TL ΔΦ/Δf ( /GHz) incurring Φ c-tl = 22.5 and finally, the λ g /16 line employed one unit cell and a compensating TL incurring Φ c-tl = The first observation that can be made is that the phase responses of the MTM lines shown in Figure 2.28(a) are almost identical to the phase responses of the NR-MTM lines in Figure 2.28(b), something that is verified by the phase variation characteristics of the lines in Tables 2.5 and 2.6. As such, the comments made about the MTM lines also hold for the NR-MTM lines. As was shown in Equations (2.1) and (2.11), however, the NR-MTM lines possess a significant advantage over the MTM lines in as much as the loading elements required to achieve the same phase shift on an NR-MTM line have half the values of the corresponding loading elements of the MTM line. This can also be verified by observing the loading-element values required for each line shown on the right-hand side of Figure Returning to Figure 2.28, when comparing the phase responses of the λ g /2 MTM line with the λ g TL, we can see that around the design frequency of 2 GHz, the two lines have approximately the same slope, however at lower frequencies the slope of the MTM line is steeper (less broadband), and at higher frequencies the slope is shallower (more broadband) than the slope of the TL. This is due to the non-linear phase relationship of the MTM lines, from Equations (2.83) and (2.91). The advantage that the metamaterial lines have to offer, however, begins to become evident when the total length of the lines is reduced to less than λ g /2. When the metamaterial lines are λ g /4 long, they exhibit approximately half of the phase variation compared to the λ g -long TL (see Tables 2.5 and 2.6), i.e. they have twice the phase bandwidth, when the metamaterial lines are λ g /8 long, they exhibit a quarter of the phase variation, i.e. they have four times the phase bandwidth, and finally, when the metamaterial lines are λ g /16 long, they exhibit an eighth of the phase variation, i.e. they have eight times the phase bandwidth.

84 Chapter 2. Negative-Refractive-Index Transmission-Line Theory 64 The magnitude responses of the lines shown in Figure 2.28 are shown in Figure 2.29, where it should be noted that the scale in Figure 2.29(a) is an order of magnitude smaller than the one in Figure 2.29(b). It can be observed that the magnitude responses of the MTM lines clearly exhibit the expected band-pass behaviour in the frequency range shown, while for the NR-MTM lines only the low-frequency cutoff is evident. Nevertheless, both the MTM and NR-MTM lines exhibit very low levels of insertion loss. A comprehensive analysis of MTM lines, including design criteria, and return-loss, insertion-loss and phase characteristics can be found in [91]. In order to take advantage of the broadband nature of the metamaterial lines, the lengths of the lines should therefore be kept short compared to the wavelength, which has the additional advantage that the circuits employing the metamaterial lines are naturally very compact. Thus, the metamaterial phase-shifting lines are well suited for series feed networks that can provide in-phase excitations to an arbitrary number of ports, which can be made broadband and very compact. This will be the subject of Chapter 4, where the series feed network will take on the form of an in-phase power divider.

85 Chapter 3 A NRI-TL Metamaterial Wilkinson Balun In this chapter, the first of a series of devices based on NRI-TL metamaterials is presented. The metamaterial balun presented herein converts a single-ended input to a differential output over a large bandwidth. The device also exhibits excellent return loss, isolation and through characteristics over the same frequency band. The balun comprises a Wilkinson divider, followed by a +9 NRI-TL metamaterial (MTM) phase-shifting line along the top branch, and a 9 MTM phase-shifting line along the bottom branch. Utilizing MTM lines for both the +9 and 9 branches allows the slopes of their phase responses to be matched, resulting in a broadband differential output signal. The theoretical performance of the balun is verified through circuit simulations and measurements of a fabricated prototype at 1.5 GHz. The MTM balun exhibits a measured differential output phase bandwidth (18 ± 1 ) of 1.16 GHz (77%), from 1.17 to 2.33 GHz. The measured isolation and return loss for all three ports remain below 1 db over a bandwidth in excess of 2 GHz, while the output quantities S 21 and S 31 remain above 4 db from.5 to 2.5 GHz. 3.1 Introduction A balun is a 3-port device that converts an unbalanced single-ended signal to a balanced differential one. It is useful for feeding two-wire antennas, where balanced currents on each branch are necessary to maintain symmetrical radiation patterns with a given polarization. Two-wire antennas have input ports that are closely spaced, therefore 65

86 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 66 9 TL 2Z +9 MTM line Z β 1 L 1 C 1 d H-TL1 P2 P1 Z R n 1 stages n 2 stages 2Z Z β 2 L 2 C 2 d H-TL2 9 TL 9 MTM line P3 Figure 3.1: Proposed architecture of the NRI-TL metamaterial balun. c 25 IEEE. their feeding structures should be chosen to accommodate for this requirement. This precludes the use of certain balun designs, whose output ports are spaced far apart [92]. Printed balun designs can generally be classified as distributed transmission-line (TL) or lumped-element type. Distributed-TL designs are inherently narrowband due to the frequency dependence of the transmission lines used. These can be made broadband, however they usually require transmission lines that are at least several wavelengths long and are therefore not very compact [93] (pp ), [94]. Lumped-element designs, albeit compact, can suffer from a narrowband differential output phase, resulting from the inherent mismatch between the phase response of the low-pass/high-pass output lines that they employ [95]. The balun proposed herein and seen schematically in Figure 3.1, uses a conventional microstrip Wilkinson divider in combination with two periodically loaded NRI-TL metamaterial (MTM) phase-shifting lines, each incurring +9 and 9 at the design frequency f. Thus, at f the proposed balun provides a 18 phase shift between the output ports. Compared to a distributed TL Wilkinson balun employing 27 and 9 transmission lines instead of the +9 and 9 MTM lines, the MTM balun is more compact and exhibits a significantly larger differential phase bandwidth. The large bandwidth of the MTM balun is achieved by using MTM lines for both the +9 and 9 branches, which allows the phase responses of the two lines to be matched over a large frequency range. In addition, the two output ports can be spaced arbitrarily close together, while still maintaining high isolation between the two. It is therefore well suited for feeding planar devices that require a broadband differential signal, such as printed bow-tie antennas, spiral antennas or log-periodic arrays. Since the development of NRI-TL metamaterials, many research groups have inves-

87 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 67 tigated the use of one-dimensional metamaterials to improve the performance of baluns. Most of the approaches have focused on replacing the 27 transmission line in one of the output branches of the TL balun with a +9 metamaterial line, while keeping the 9 conventional right-handed transmission line in the other branch [53], [96 98]. As will be subsequently demonstrated, this does not result in the most optimal differential phase bandwidth, where the phase difference between the two output ports remains within a certain phase margin of 18. In addition, the metamaterial baluns in [53], [96] and [97] do not employ an isolation resistor between the two output ports, therefore their isolation performance is poor. A similar architecture employing a Wilkinson power divider followed by two phaseshifting lines has also been used to implement quadrature power splitters, which exhibit a wide differential phase bandwidth and good amplitude balance [99], [1]. Quadrature power splitters find applications in balanced amplifiers, image-rejection mixers and reflection-type phase shifters. It should also be mentioned that recently a broadband via-free microstrip balun using metamaterial lines has been presented by Liu and Menzel [54], which uses a different architecture to the one proposed herein that is more akin to a 18 coupled-line coupler. It was demonstrated in [54] that a very wide phase bandwidth can be achieved with a small amplitude imbalance if a sufficient number of metamaterial unit cells is used to implement the balun. 3.2 Principle of Operation Before outlining the operation principle of the metamaterial balun, it would be useful to define certain quantities relating to the phase of the balun s output signals. The differential phase of the balun is given by the difference between the insertion phase from port 1 to port 2, and the insertion phase from port 1 to port 3, and is given by ΔΦ = S 21 S 31. (3.1) An ideal balun would maintain a differential phase of 18 for all frequencies, however for practical devices this will only be the case over a finite bandwidth. We denote the bandwidth over which the differential phase remains at 18 within a phase margin of ±1 as BW 1 = f ΔΦ=18 ±1 f + ΔΦ=18 ±1. (3.2)

88 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 68 P1 +9 Line 9 Line P2 + ΔΦ=18 P3 Wilkinson Divider Figure 3.2: General balun architecture consisting of a Wilkinson divider, followed by a +9 phase-shifting line on the top branch and a 9 phase-shifting line on the bottom branch. Thus, BW 1 effectively denotes the frequency bandwidth over which ΔΦ remains within the range of Alternatively, if the phase response has an upward concave characteristic, BW 1 denotes the frequency bandwidth over which ΔΦ remains within the range of on either side of 18, and if the phase response has a downward concave characteristic, BW 1 denotes the frequency bandwidth over which ΔΦ remains within the range of on either side of 18. The generalized architecture of the balun considered herein is shown in Figure 3.2, and consists of a Wilkinson power divider, followed by a +9 phase-shifting line on the top branch and a 9 phase-shifting line on the bottom branch. The Wilkinson divider provides an equal power split, high output port isolation and good return loss at all three ports, while the difference in the insertion phase of each of the output lines enables a differential phase of 18 at the output. The first practical implementation of the generalized balun architecture of Figure 3.2 is shown in Figure 3.3(a), where the two phase-shifting lines have been realized using transmission lines of analogous length. That is, the +9 phase-shifting line on the top branch has been realized using a 27 transmission line, and the 9 phase-shifting line on the bottom branch has been realized using a 9 transmission line. From Figure 3.3(a), it can be observed that the necessary meandering of the 27 transmission line renders the size of the TL balun quite large. The TL balun was designed to operate at f =1.5 GHz, and Figure 3.3(b) demonstrates the phase responses of the two individual TLs, as well as the differential phase ΔΦ. It can be observed that at the design frequency of f =1.5 GHz,eachofthe 27 and 9 transmission lines incur the desired insertion phases of 27 and 9 respectively, and the differential phase is 18. However, due to the linear nature of the phase response of both the transmission lines, the differential phase is not constant, but rather

89 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun TL 9 9 o TL P1 P2 + ΔΦ=18 Phase ( ) ΔΦ 27 o TL Wilkinson Divider (a) 9 TL P Frequency (GHz) (b) Figure 3.3: TL balun architecture: (a) Physical realization, and (b) phase responses of each of the output branches, together with the differential output phase, ΔΦ. is also a linear function of frequency, given by ΔΦ = Φ tl1 Φ tl2 = ω LC(d tl1 d tl2 ). (3.3) Here, Φ tl1 and d tl1 are the phase and the length of the top 27 TL, and Φ tl2 and d tl2 are the phase and the length of the bottom 9 TL, according to Equation (2.22). Thus, it is evident from Equation (3.3) that the differential phase will not remain constant at 18, but will vary linearly with frequency, thus resulting in a narrowband differentialphase bandwidth. The second practical implementation of the generalized balun architecture of Figure 3.2 is shown in Figure 3.4(a), where the +9 phase-shifting line on the top branch has been realized using a +9 MTM line with n = 2 stages and the 9 phase-shifting line on the bottom branch has been realized using a 9 TL. From Figure 3.4(a), it can be observed that the size of the MTM-TL balun has been significantly reduced compared to the TL balun of Figure 3.3(a) because the 27 meandered transmission line has been effectively eliminated. The MTM-TL balun was also designed to operate at f = 1.5 GHz, and Figure 3.4(b) shows the phase responses of the individual +9 MTM line and the 9 TL, as well as the differential phase, ΔΦ. It can be observed that at f =1.5GHzeachoftheMTM and TL phase-shifting lines exhibits the required phase shift, and the differential phase is 18. It can also be observed that ΔΦ is highly nonlinear at low frequencies, and has a constant slope at higher frequencies. This can be attributed to the difference in the

90 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun MTM line L L TL TL P1 2C C 2C 9 TL Wilkinson Divider (a) P2 + ΔΦ=18 P3 Phase ( ) ΔΦ o MTM 9 9 o TL Frequency (GHz) (b) Figure 3.4: MTM-TL balun architecture: (a) Physical realization, and (b) phase responses of each of the output branches, together with the differential output phase, ΔΦ. phase characteristics of the two individual phase-shifting lines shown in Figure 3.4(b). Analytically, the differential phase between the MTM and TL branches can be expressed from Equations (2.83) and (2.22) as ΔΦ = Φ mtm Φ tl = ω LC(d tl nd h-tl )+ n ω L C. (3.4) Here, Φ mtm and nd h-tl are the phase and total length of the +9 MTM line, and Φ tl and d tl are the phase and length of the 9 TL. Equation (3.4) confirms the results shown in Figure 3.4(b), where at low frequencies the term n/ω L C which is inversely proportional to ω dominates and produces a nonlinear ΔΦ response, while at high frequencies the term ω LC(d tl nd h-tl ), which is directly proportional to ω, dominates and produces a linear ΔΦ response. The net result is that the differential-phase bandwidth of the MTM-TL balun is larger than the TL balun, however overall it is still quite narrowband. The final realization of the generalized balun architecture of Figure 3.2 is shown in Figure 3.5(a), where the +9 phase-shifting line on the top branch has been realized using a +9 MTM line with n 1 = 2 stages and the 9 phase-shifting line on the bottom branch has been realized also using a 9 MTM line with n 2 = 2 stages. The premise of using metamaterial lines to implement both the top and bottom phase-shifting lines is that the two lines, being of the same type, exhibit similar phase responses with frequency, therefore resulting in a broadband differential-phase bandwidth. Figure 3.5(b) shows the phase responses of the individual +9 and 9 MTM lines,

91 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun MTM line ΔΦ P1 2C 1 C 1 L 1 L 1 TL TL 2C 1 TL TL 2C 2 C 2 2C 2 L 2 L 2 P2 + ΔΦ=18 P3 Phase ( ) o MTM 9 o MTM Wilkinson Divider (a) 9 MTM line Frequency (GHz) (b) Figure 3.5: MTM balun architecture: (a) Physical realization, and (b) phase responses of each of the output branches, together with the differential output phase, ΔΦ. as well as the differential phase, ΔΦ. It can be observed that both the MTM phaseshifting lines have similar phase responses, and at the design frequency of f =1.5GHz each of the MTM lines exhibits the required phase shift, resulting in a differential phase of 18. It can also be observed that ΔΦ has a nonlinear response below approximately 1 GHz, however from 1 to 2 GHz it remains virtually constant at 18. Analytically, the differential phase between the two metamaterial branches can be expressed from Equation (2.83) as ΔΦ = Φ mtm1 Φ mtm2 = ω ( ) n 1 LC(n 2 d h-tl2 n 1 d h-tl1 )+ ω n 2 L 1 C 1 ω. (3.5) L 2 C 2 Here, Φ mtm1 and n 1 d h-tl1 are the phase and total length of the top +9 MTM line, and Φ mtm2 and n 2 d h-tl2 are the phase and total length of the bottom 9 MTM line. At low frequencies, the 1/ω-dependent term in Equation (3.5) dominates and produces a nonlinear ΔΦ response, while at high frequencies the ω-dependent term dominates and produces a linear ΔΦ response. At the intermediate frequencies around f =1.5GHz, however, Equation (3.5) highlights the method by which a virtually flat ΔΦ response is obtained over a wide band of frequencies. With the appropriate choice of the parameters n 1, n 2, d h-tl1, d h-tl2, L 1, C 1, L 2 and C 2,boththeω and 1/ω-dependent terms can be designed to compensate each other and produce a differential phase of 18 over a wide bandwidth. Thus, the MTM balun exhibits a broadband differential-phase bandwidth, which is wider than both the MTM-TL and TL baluns.

92 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun Design Based on the results shown in the previous section for the various balun designs, the MTM Wilkinson balun shown in Figure 3.5(a) and schematically in Figure 3.1 was chosen as the most suitable design to achieve the widest differential-phase bandwidth. The MTM balun was implemented in microstrip technology on a Rogers RO33 substrate with ɛ r = 3 and height h =.762 mm at a design frequency of f = 1.5 GHz. Using the above parameters in Agilent-ADS, a Z = 5 Ω microstrip line was found to have a guided wavelength of λ g,5ω = mm and a width of W 5Ω =1.95 mm, while a Z =7.71 Ω microstrip line had a guided wavelength of λ g,7.71ω = mm and a width of W 7.71Ω =1.9 mm. The metamaterial unit cell of Figure 2.1(a) was used to implement both the +9 and 9 MTM phase-shifting lines. Keeping in mind that the phase shift per unit cell and the length of the host TL sections should be kept small in order for the effective medium equations presented in Section to hold, the +9 MTM line was first designed to produce a +9 phase-shift at the design frequency of f = 1.5 GHz, while maintaining a short overall length. Then, the pertinent parameters for the 9 MTM line were calculated such that the shape of the phase responses of the +9 and 9 MTM lines matched, thus maintaining a 18 phase difference over a large bandwidth. A five-stage design was chosen for the +9 MTM phase-shifting line, with an initial host TL length of d H-TL1 = 4 mm per unit cell, which corresponds to φ h-tl = 11.23, and Z = 5 Ω. Thus, each metamaterial unit cell was required to incur a phase shift of φ mtm =+18 at f = 1.5 GHz. It can be verified using the approach outlined in Section 2.5, that a 4 mm MTM line that incurs a phase of β bl d = φ mtm =+18 at f = 1.5 GHz operates sufficiently into the backward-wave region, such that it becomes a slow-wave structure that is not prone to leaky-wave radiation. This can be inferred from the fact that at 1.5 GHz, the light cone that corresponds to a line with a length of 4 mm will have a phase of k d =+7.2. Since the phase incurred by the MTM line is significantly larger than the phase of the light line, this implies that the phase velocity of the waves traveling on the MTM line is slow compared to the speed of light, and therefore the line will not radiate. The ideal loading element values for the +9 MTM line were evaluated using Equations (2.83) and (2.71). Equation (2.83) is repeated below using the specific parameters

93 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 73 that describe the +9 MTM line, Φ mtm1 = n 1 ( ω ) 1 LCd h-tl1 + ω =+ π. (3.6) L 1 C 1 2 ω Therefore, using the parameters n 1 =5,Φ mtm1 =+π/2, φ h-tl1 = 11.23, Z =5Ω and f = 1.5 GHz in Equations (3.6) and (2.71), the loading elements were determined to be C 1 =4.16pFandL 1 = 1.4 nh. A five-stage design was also chosen for the 9 MTM phase-shifting line, which also had a characteristic impedance of Z = 5 Ω. Equation (2.83) can therefore also be written for the specific parameters that describe the 9 MTM line as Φ mtm2 = n 2 ( ω ) 1 LCd h-tl2 + ω = π. (3.7) L 2 C 2 2 ω By fixing the unit cell size of each of the constituent unit cells comprising the 9 MTM line, the loading parameters C 2 and L 2 can be evaluated in order to produce the desired phase shift of Φ mtm2 = π/2 atf = 1.5 GHz. However this does not guarantee that the two metamaterial lines will have the same slope around the design frequency, which is necessary for a large differential-phase bandwidth. Therefore, an additional parameter is necessary in order to ensure that the phase responses of the two lines match at the design frequency. This parameter was chosen to be d h-tl2, the length of each of the constituent MTM unit cells in the 9 MTM line, which can be determined by enforcing that the slopes of the +9 and 9 MTM lines are equal at the design frequency, expressed as dφ mtm1 dω = dφ mtm2 ω dω. (3.8) ω Equation (3.8) can therefore be used in conjunction with Equations (3.6) and (3.7) to determine d h-tl2 as follows: By taking the derivatives of Equations (3.6) and (3.7), each of the two expressions in Equation (3.8) can be expressed as ), (3.9) dφ mtm1 dω = n 1 ( 1 LCd h-tl1 ω 2 L 1 C 1 dφ mtm2 = n 2 ( ) 1 LCd h-tl2 dω ω 2. (3.1) L 2 C 2 Next, Equations (3.6) and (3.7) can be re-arranged to produce the following two equations respectively, at the design frequency ω : 1 = ω LCd 2 h-tl1 + π 1, (3.11) L1 C 1 2 n 1 ω

94 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 74 1 = ω LCd 2 h-tl2 π 1. (3.12) L2 C 2 2 n 2 ω Substituting Equations (3.11) and (3.12) into Equations (3.9) and (3.1) respectively, the two terms of Equation (3.8) can be expressed as dφ mtm1 dω = 2 LCn 1 d h-tl1 π ω 2 1 ω, (3.13) dφ mtm2 dω = 2 LCn 2 d h-tl2 + π 1. (3.14) ω 2 ω Finally, substituting Equations (3.13) and (3.14) into Equation (3.8) and re-arranging, we can obtain an expression for the required length of each MTM unit cell in the 9 MTM line: π d h-tl2 = + n 1 d h-tl1. (3.15) 2ω LCn2 n 2 Substituting the parameters n 1 =5,n 2 =5,d h-tl1 = 4 mm, β tl = ω LC =49.1 rad/m and f = 1.5 GHz into Equation (3.15), the length of each MTM unit cell in the 9 MTM line was determined to be d h-tl2 = 1.41 mm. The loading elements for the 9 MTM line were finally determined by substituting the parameters n 2 =5,Φ mtm2 = π/2, φ h-tl2 = β tl d h-tl2 = 29.23, Z =5Ωand f = 1.5 GHz into Equations (3.7) and (2.71), resulting in loading-element values of C 2 =1.83 pf and L 2 = 27.7 nh. 3.4 Practical Implementation The microstrip Wilkinson power divider used in the MTM balun of Figure 3.1 was designed to provide equal power split between the two output branches and is used to achieve high output port isolation as well as good return loss characteristics at all three ports. The input feed line was designed with a characteristic impedance of Z =5Ω, while the two λ g /4 branches were designed with a characteristic impedance of 2Z = 7.71 Ω, resulting in an isolation resistor value of R = 1 Ω. The Wilkinson divider was designed and simulated in Agilent-ADS using non-ideal microstrip components, and its various geometrical parameters were optimized within the simulator in order to provide a good return loss at all three ports. The final dimensions of the divider were: for the 5 Ω input line, W 5Ω =1.95 mm and d 5Ω = 5 mm, and for each of the 7.71 Ω λ g /4 branches, W 7.71Ω =1.9 mm and d 7.71Ω =32.73 mm.

95 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 75 Each of the +9 and 9 MTM lines were subsequently designed separately in Agilent-ADS using non-ideal models for both the chip lumped-element components and the microstrip lines, in order to account for the parasitic effects introduced by the nonideal models. When synthesizing a MTM phase shifting line by cascading multiple MTM unit cells, the two adjacent series loading capacitors 2C in each of the MTM unit cells can be combined into one capacitor of value C. Thus, a chip lumped-element capacitor was chosen to implement C, and the two capacitors at the beginning and at the end of the line that have a value of 2C were simply implemented by placing two of the C capacitors in parallel. In addition, a.4 mm gap was cut out of each section of host transmission line in order to accommodate each of the series loading capacitors. This gap was also modeled in the ADS simulations by including a microstrip gap component that was provided in the ADS microstrip library. The loading elements C 1, L 1, C 2 and L 2 were realized using standard size 42 murata chip components, and the S-parameter files containing all the parasitic values associated with each of the components were used in the simulations. As such, the murata capacitors and inductors were selected from a set of discrete values provided from the manufacturer, in order to obtain the closest effective values to the ones theoretically calculated above. Because the chip components came only in predefined discrete values, the length of each MTM unit cell was subsequently adjusted from the ideal lengths of d h-tl1 =4mmandd h-tl2 = 1.41 mm in order to obtain exactly +9 and 9 at f = 1.5 GHz for each of the MTM lines. In order to achieve the design constraints, each unit cell of the +9 MTM phaseshifting line employed a microstrip transmission line with width W h-tl1 =1.95 mm and length d h-tl1 = mm, and chip lumped-element components with values of C 1 = 3.6 pfandl 1 = 1 nh. The murata capacitors used for C 1 were multilayer ceramic chip capacitors with part number GRM1555C1H3R6CZ1 and had a selfresonant frequency (SRF) of 3.9 GHz, while the murata inductors used for L 1 were wire-wound chip inductors with part number LQW15AN1NH and had an SRF of 6.1 GHz. Correspondingly, each unit cell of the 9 MTM phase-shifting line employed a microstrip transmission line with width W h-tl1 =1.95 mm and length d h-tl2 =1.46 mm, and murata chip lumped-element components with values of C 2 =7pF(part number GRM1555C1H7RDZ1, SRF = 2.8 GHz) and L 2 = 27 nh (part number LQW15AN27NH, SRF = 4.9 GHz). The complete MTM Wilkinson balun was designed and simulated in Agilent-ADS

96 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 76 MLIN TL1 W=1.95 mm L=5 mm Term Term1 Z=5 Ohm MCURVE Curve1 W=1.9 mm Angle=18 Radius=9.854 mm MLIN TL2 W=1.9 mm L=1.773 mm MTEE Tee1 W1=1.9 mm W2=1.9 mm W3=1.95 mm MLIN TL3 W=1.9 mm L=1.773 mm MCURVE Curve2 W=1.9 mm Angle=18 Radius=9.854 mm MTEE Tee2 W1=1.9 mm W2=1.9 mm W3=1.95 mm MLIN TL8 W=1.9 mm L=.5 mm R R R=1 Ohm MLIN TL9 W=1.9 mm L=.5 mm MTEE Tee3 W1=1.9 mm W2=1.9 mm W3=1.95 mm C murata1 C=3.6 pf MGAP Gap7 W=1.95 mm S=.4 mm C murata2 C=3.6 pf C murata15 C=7 pf MGAP Gap2 W=1.95 mm S=.4 mm C murata16 C=7 pf MLIN TL3 W=1.95 mm L=1.949 mm MLIN TL51 W=1.95 mm L=5.23 mm MLIN TL31 W=1.95 mm L=1.949 mm L muratal1 L=1 nh MLIN TL5 W=1.95 mm L=5.23 mm L muratal1 L=27 nh MGAP Gap1 W=1.95 mm S=.4 mm C murata3 C=3.6 pf MGAP Gap19 W=1.95 mm S=.4 mm C murata14 C=7 pf MLIN TL35 W=1.95 mm L=1.949 mm MLIN TL48 W=1.95 mm L=5.23 mm MLIN TL34 W=1.95 mm L=1.949 mm MGAP Gap11 W=1.95 mm S=.4 mm L muratal2 L=1 nh C murata4 C=3.6 pf MLIN TL49 W=1.95 mm L=5.23 mm MGAP Gap18 W=1.95 mm S=.4 mm L muratal9 L=27 nh C murata13 C=7 pf MLIN TL36 W=1.95 mm L=1.949 mm MLIN TL47 W=1.95 mm L=5.23 mm MLIN TL37 W=1.95 mm L=1.949 mm MGAP Gap12 W=1.95 mm S=.4 mm L muratal3 L=1 nh C murata5 C=3.6 pf MLIN TL46 W=1.95 mm L=5.23 mm MGAP Gap17 W=1.95 mm S=.4 mm L muratal8 L=27 nh C murata12 C=7 pf MLIN TL39 W=1.95 mm L=1.949 mm MLIN TL44 W=1.95 mm L=5.23 mm MLIN TL38 W=1.95 mm L=1.949 mm MGAP Gap13 W=1.95 mm S=.4 mm L muratal4 L=1 nh C murata6 C=3.6 pf MLIN TL45 W=1.95 mm L=5.23 mm MGAP Gap16 W=1.95 mm S=.4 mm L muratal7 L=27 nh C murata11 C=7 pf MLIN TL4 W=1.95 mm L=1.949 mm MLIN TL43 W=1.95 mm L=5.23 mm C murata7 C=3.6 pf MLIN TL41 W=1.95 mm L=1.949 mm MGAP Gap14 W=1.95 mm S=.4 mm L muratal5 L=1 nh C murata8 C=3.6 pf C murata1 C=7 pf MLIN TL42 W=1.95 mm L=5.23 mm MGAP Gap15 W=1.95 mm S=.4 mm L muratal6 L=27 nh C murata9 C=7 pf Term Term2 Z=5 Ohm Term Term3 Z=5 Ohm Figure 3.6: Schematic circuit of the MTM Wilkinson balun implemented in Agilent-ADS.

97 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 77 (a) Metamaterial balun (b) Transmission-line balun Figure 3.7: Photographs of the fabricated MTM and TL baluns. c 25 IEEE. using the non-ideal models for both the chip lumped-element components and the microstrip lines described above, and the schematic circuit of the entire MTM Wilkinson balun is shown in Figure 3.6. For comparison purposes, a similar TL Wilkinson balun was also designed in ADS by replacing the +9 MTM line with a 27 TL with width W tl1 =1.95 mm and length d tl1 =96.58 mm, and replacing the 9 MTM line with a 9 TL with width W tl1 =1.95 mm and length d tl1 =32.6 mm. The MTM and TL baluns were then fabricated at the University of Toronto, and photographs of the two prototypes are shown in Figure Simulation and Experimental Results Figure 3.8(a) shows the measured versus the simulated return loss magnitude response for port 1, showing good agreement between the two. The measured and simulated return loss is below 1 db for the bandwidth shown from.6 GHz to above 2.5 GHz, indicating that the device is well matched over this entire band, and especially around f =1.5 GHz with a minimum measured value of S 11 = db at 1.46 GHz. The measured and simulated return loss for port 2 is shown in Figure 3.8(b), where at the lower end of the frequency range the measured and simulated results match very well,

98 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 78 Magnitude of S11 (db) Measured Simulated Frequency (GHz) (a) S 11 magnitude response Magnitude of S22 (db) Measured Simulated Frequency (GHz) (b) S 22 magnitude response Magnitude of S33 (db) Measured Simulated Frequency (GHz) (c) S 33 magnitude response Magnitude (db) S21 S31 S23 Measured Simulated Frequency (GHz) (d) S 21, S 31 and S 23 magnitude responses Figure 3.8: Measured and simulated magnitude responses of the MTM balun. c 25 IEEE. however above 1 GHz the two somewhat deviate. Nevertheless, the general characteristics of the measured response are similar to those of the simulated response with both responses exhibiting a return loss below approximately 1 db for the entire bandwidth shownfrom.5ghztoabove2.5ghz.atf =1.5 GHz the minimum measured return loss value is S 22 = 2.7 db, therefore indicating that the MTM balun is also well matchedatport2. Figure 3.8(c) shows the measured versus the simulated return loss magnitude response for port 3, also showing good agreement between the two. The measured and simulated return loss is below 1 db for the entire bandwidth shown from.5 GHz to above 2.5 GHz, indicating that the device is well matched over this entire band. The minimum measured return loss has a value of S 33 = db at 1.73 GHz, which is slightly

99 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 79 shifted from the design frequency of f =1.5 GHz, at which point the return loss still has a reasonable value of S 33 = db. Figure 3.8(d) shows the isolation, S 23, and through-power to each of the output ports S 21 and S 31. It can be observed that the MTM balun exhibits excellent isolation, as well as equal power split between the two output ports. The measured and simulated results are in good agreement. The measured isolation remains below 1 db from less than.5 GHz to 2.3 GHz, with a minimum measured value of S 23 = 39.6 db at 1.46 GHz. Both the measured and simulated S 21 and S 31 remain above 4 dbforthe entire range from.5 to 2.5 GHz, with maximum measured values of S 21 = 3.46 db and S 31 = 3.38 db around f =1.5 GHz, resulting in insertion losses of.45 db and.37 db respectively for each of the two branches. These values for the insertion losses are quite reasonable considering the fact that each branch contains a transmission line with an minimum length of 6 mm and 13 chip lumped-element components. Figure 3.9 shows the measured versus the simulated phase responses of the two balun branches, where it can be seen that the experimental results agree very closely with the simulated results. It should be noted that the additional phase delay introduced by the 9 transmission lines in each of the branches of the Wilkinson divider has been removed for clarity. As was designed for, the phase of S 21 is equal to +9 at f =1.5 GHz, while the phase of S 31 is equal to 9 at f =1.5GHz, and it can also be observed that the phase characteristics with frequency of the two branches are very similar. Figure 3.1 shows the measured and simulated differential output phase of the MTM balun, with excellent agreement between the two. The differential output phase remains flat for a large frequency band, which follows directly from the fact that the phase characteristics of the +9 and 9 MTM lines shown in Figure 3.9 correspond very closely. The flat differential output phase has a bandwidth of BW 1 =1.16 GHz, from 1.17 to 2.33 GHz. Since the device exhibits excellent return loss, isolation and through characteristics over this frequency range, it can be concluded that the MTM balun can be used as a broadband single-ended to differential converter in the frequency range from 1.17 to 2.33 GHz. For comparison, the TL Wilkinson balun of Figure 3.7 was also simulated in Agilent- ADS, and measured in the laboratory, and the differential output phase of the TL balun is also shown in Figure 3.1. It can be observed that the phase response of the TL balun is linear with frequency, with a slope equal to the difference between the phase slopes of the 27 and 9 transmission lines, similar to the case shown in Figure 3.3. Since

100 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun Phase ( ) +9 9 S21 2 S31 Measured Simulated Frequency (GHz) Figure 3.9: Measured and simulated phase responses of S 21 (+9 MTM line) and S 31 ( 9 MTM line). c 25 IEEE. 25 Differential Output Phase ( ) 18 9 BW TL BW MTM MTM Balun TL Balun Measured Simulated Frequency (GHz) Figure 3.1: Measured and simulated differential phase comparison between the MTM balun and the TL balun. c 25 IEEE.

101 Chapter 3. A NRI-TL Metamaterial Wilkinson Balun 81 the gradient of the resulting phase characteristic is quite steep, this renders the output differential phase response of the TL balun narrowband. Thus, the TL balun exhibits a measured differential phase bandwidth of only BW 1 =.16 GHz, from 1.42 to 1.58 GHz, compared to BW 1 =1.16 GHz exhibited by the MTM balun. In relative terms, the MTM balun has a relative differential phase bandwidth of 77%, compared to a 11% bandwidth exhibited by the TL balun. In addition, the TL balun occupies an area of 33.5 cm 2 compared to 18.5 cm 2 for the MTM balun. Thus, the MTM balun is more compact, occupying only 55% of the area that the conventional TL balun occupies. Furthermore, the MTM balun exhibits more than double the bandwidth compared to a lumped-element implementation using low-pass/high-pass lines, which typically exhibits a bandwidth of 32% [95]. This can be attributed to the fact that the low-pass line has a linear phase response, while the response of the high-pass line has a varying slope with frequency, similar to the case of the MTM-TL balun shown in Figure 3.4. Thus, the shapes of the phase responses of the two lines do not match, resulting in a more narrowband differential output phase. In conclusion, a new metamaterial balun has been presented in this chapter that offers a broadband differential output signal as well as excellent return loss, isolation and through characteristics, while still maintaining a small form factor. The broadband nature of the balun arises from the fact that both the +9 and the 9 branches of the balun use metamaterial lines, and therefore the shape of their phase responses is similar over a large bandwidth. This allows the MTM balun to maintain a differential output signal over a broader bandwidth compared to both a distributed TL balun employing 27 and 9 transmission lines and a lumped-element high-pass/low-pass balun. For this reason, combined with its compact, planar design, and the fact that the two output ports can be spaced closely together, the MTM balun is well-suited for feeding planar devices that require a broadband differential input signal, for example a printed bow-tie antenna.

102 Chapter 4 A NRI-TL Metamaterial Series Power Divider A NRI-TL metamaterial 1:4 series power divider that provides equal power split to all four output ports over a large bandwidth is presented in this chapter, which can be extended to an arbitrary number of output ports. The divider comprises four non-radiating metamaterial (NR-MTM) lines in series, incurring a zero insertion phase over a large bandwidth, while simultaneously maintaining a compact length of λ /8. Compared to a series power divider employing conventional one-wavelength long meandered transmission lines to provide in-phase signals at the output ports, the metamaterial divider provides a 165% increase in the input return-loss bandwidth and a 155% and 154% increase in the through-power bandwidth to ports 3 and 4 respectively. In addition, the metamaterial divider is more compact, occupying only 54% of the area that the transmission line divider occupies, while exhibiting comparable insertion losses. 4.1 Introduction Series power dividers have been used to provide a simple, compact and low-loss feed network to an array of load elements, as shown in the example of a series-fed microstrip patch array in Figure 4.1(a). By selecting a series rather than a corporate feed topology shown in Figure 4.1(b), also known as a parallel feed topology, the binary-tree TL networks associated with corporate power dividers can be collapsed into a single feed line, reducing significantly the overall area of the structure. Series dividers are therefore more compact and exhibit lower conductor, dielectric and radiation losses compared to corpo- 82

103 Chapter 4. A NRI-TL Metamaterial Series Power Divider 83 Microstrip patch Microstrip patch Microstrip patch Microstrip patch (a) Microstrip patch Microstrip patch Microstrip patch Microstrip patch Binary-tree Feed nework Figure 4.1: (a) Series-fed microstrip patch array, (b) Corporate-fed (parallel-fed) microstrip patch array. (b) rate dividers, leading to higher overall efficiencies when used in antenna arrays [11 14]. However, one of the major disadvantages of conventional TL-based series dividers is that the ratio of power delivered to each port varies with frequency, due to the inherent frequency dependence of the feed transmission lines used. Therefore, the input return loss bandwidth of the TL dividers is quite narrow, and the bandwidth over which equal power is delivered to each output port is limited. In this chapter, a compact, broadband 1:4 series power divider is presented that employs non-radiating metamaterial (NR-MTM) phase-shifting lines in order to provide an equal power split to all four output ports over a significantly larger bandwidth compared to a conventional TL series divider. In addition, the NR-MTM divider is very compact, scalable in size and can be extended to an arbitrary number of ports, therefore it is well suited for various applications including planar antenna feed networks [15] and

104 Chapter 4. A NRI-TL Metamaterial Series Power Divider 84 λ /8 λ /8 λ /8 λ /8 Z Z Z Z P1 5 Ω 36 TL 36 TL 36 TL 36 TL P2 P3 P4 P5 2 Ω 2 Ω 2 Ω 2 Ω Figure 4.2: Schematic diagram of the transmission-line (TL) 1:4 series power divider. power-combining amplifiers [1], [6]. 4.2 Power Divider Architecture Applications that require equal, in-phase power division to a series of loads that are spaced less than a wavelength apart have traditionally used a one guided-wavelength (λ g ) TL to feed each of the loads. The reference structure considered herein is shown schematically in Figure 4.2 and consists of four series-connected λ g -long transmission lines that feed four 2 Ω loads, spaced λ /8 apart. Thus at the design frequency f,the four loads appear in parallel, and the circuit is matched to 5 Ω. In this structure, the spacing between the loads was chosen to be λ /8, which necessitates meandering each of the λ g transmission lines. As it can be observed from Figure 4.2, the resulting power divider structure is physically large, and is expected to have a narrowband performance since the variation of the phase for each of the λ g TL sections is quite large (see Figure 2.28). In addition, the TL-divider is prone to parasitic radiation caused by scattering from the bends in the meandered lines. The proposed 1:4 series power divider, shown schematically in Figure 4.3, employs NR-MTM phase-shifting lines to mitigate some of the problems encountered with conventional TL series dividers. The structure consists of four series-connected non-radiating

105 Chapter 4. A NRI-TL Metamaterial Series Power Divider 85 non-radiating metamaterial line 2C C Z 2C Z Z L L Φ MTM Φ TL λ /8 λ /8 λ /8 λ /8 5 Ω P1 NR-MTM P2 NR-MTM P3 NR-MTM P4 NR-MTM P5 2 Ω 2 Ω 2 Ω 2 Ω Figure 4.3: Schematic diagram of the non-radiating metamaterial (NR-MTM) 1:4 series power divider. c 25 IEEE. NR-MTM lines that feed four 2 Ω loads, spaced λ /8 apart. Thus, at the design frequency f, the four loads appear in parallel, and the circuit is matched to 5 Ω. A spacing of λ /8 between the loads was chosen in order to highlight the broadband nature of the metamaterial lines over the λ g -long transmission lines, as outlined at the end of Section In addition, the choice of such a small spacing between the output ports demonstrates that metamaterial-based power dividers can have a very small form factor, and are not limited by typical space constraints associated with meandered transmission lines. In order to compare the performances of the TL and NR-MTM power dividers shown in Figures 4.2 and 4.3, one must compare their relevant characteristics, namely the input return loss, through power to each output port, and the isolation between the output ports. As outlined above, the input impedance seen looking into port 1 (P1) will be 5 Ω at the design frequency f, therefore at this frequency the circuit will be perfectly matched. This, however, will not hold for all frequencies due to the frequency dependence of the λ g -long transmission line and the NR-MTM lines. Therefore, we must consider the impedance bandwidth over which the input return loss remains below a certain level in order to compare the performances of the two dividers. Similarly, the impedance bandwidth can be evaluated for the through power delivered to each output port and the

106 Chapter 4. A NRI-TL Metamaterial Series Power Divider 86 isolation of each device. The impedance bandwidth for the input return loss, however, will be the determining factor for the bandwidth of the entire device, since the transformation of four 2 Ω loads to 5 Ω through a resonant-type structure will tend to limit the bandwidth. Appendix B includes a comprehensive study of the input returnloss bandwidth of a 1:2 TL divider as a function of the characteristic impedance of the interconnecting transmission lines, which is also extended to a 1:4 TL divider. It has been shown in Appendix B that in order to achieve the maximum input returnloss bandwidth of 13.55% for a 1:4 TL divider, it should be designed with interconnecting transmission lines that have a characteristic impedance of Z =67.5 Ω. Inviewofthis, a convenient characteristic impedance of Z = 2 5 = 7.71 Ω was chosen for both the TL and the NR-MTM dividers, in order to provide a fair comparison of the performances. By choosing Z =7.71 Ω, this results in a 13.5% fractional bandwidth for the 1:4 TL divider from Figure B.4, which is quite close to the maximum obtainable value of 13.55%. 4.3 Design of the NR-MTM lines The design of the non-radiating metamaterial phase-shifting lines was based on the design procedure presented in Section 2.5, that allows the synthesis of phase shifting lines that incur at the design frequency, f. The total phase shift incurred by a non-radiating metamaterial line is given by Equation (2.91) and naturally if the desired phase shift is, this implies that the phase shift incurred by the MTM section of the line must be equal and opposite to that of the TL section. Thus, for the particular case of a phase shift, Equation (2.91) becomes Φ nr-mtm =Φ mtm +Φ c-tl =. (4.1) Equation (4.1) can be used together with the impedance matching condition of Equation (2.71) to determine the loading element values L and C and the TL parameters required to design the NR-MTM phase-shifting lines. The following design procedure outlines how to design a NR-MTM phase-shifting line of total length d tot : 1. Specify the characteristic impedance Z for the line. Assume that both the host transmission lines of the MTM section and the TL section have the same Z. 2. Based on the technology chosen to implement the printed transmission lines (e.g. microstrip), determine λ g, the guided wavelength of the line. The propagation

107 Chapter 4. A NRI-TL Metamaterial Series Power Divider 87 constant of the line will therefore be β c-tl = β h-tl = 2π λ g. 3. Specify the total line length, d tot, which is equal to the load separation for the NR-MTM series power divider. 4. Divide d tot in half and assign half to the positive phase-shifting MTM line and half to the negative phase-shifting TL. d 5. The phase incurred by the TL will be Φ c-tl = β tot c-tl, therefore the phase that 2 the MTM line must incur from Equation (4.1) is Φ mtm = Φ c-tl. 6. Select m, the number of MTM stages to be used. 7.Basedontheselectionofm, determine the phase shift per MTM unit cell as φ mtm = Φmtm. m 8. Determine the length of the host TL for each MTM unit cell as d h-tl = dtot. 2m Note: The choice of m and subsequently φ mtm and d h-tl should satisfy the criteria used to derive Equation (2.79), namely that φ mtm 1and φ h-tl Use Equations (2.1) and (2.11) and the values of Z, β c-tl, d h-tl and φ mtm from above to calculate the approximate values of the loading elements L and C. Alternatively, the exact values of L and C can be found using the above parameters in Equations (2.86) and (2.89). 1. Finally, use the above parameters in Equation (2.91) to verify that indeed the phase incurred by the NR-MTM line is. The above design procedure ensures that in fact the NR-MTM phase-shifting line will be slow, something that is verified by the following arguments. We begin from the condition that the phase incurred by the metamaterial section of the NR-MTM line is equal and opposite to that incurred by the transmission-line section of the line, namely that Φ mtm = Φ c-tl. (4.2) Each section of line has a total length of d tot /2, therefore from Equation (4.2) we can write β mtm = β c-tl. (4.3)

108 Chapter 4. A NRI-TL Metamaterial Series Power Divider 88 The guided wavelength of the transmission line λ g is related to the free-space wavelength λ through the effective permittivity ɛ eff of the medium in which the transmission line is realized, λ g = λ. (4.4) ɛeff Thus, the propagation constant of the metamaterial line can be related to the parameters of the transmission line as follows: β mtm = 2π λ g = 2π ɛeff = k ɛ eff. (4.5) λ Finally, substituting Equation (4.5) into the condition for slow-wave propagation outlined in Section 2.5, namely that β mtm >k, we can verify that the NR-MTM lines designed using the procedure presented above will indeed operate in the slow-wave regime, i.e. β mtm = k ɛ eff >k. (4.6) 4.4 Design of the NR-MTM and TL Power Dividers The metamaterial and TL dividers were implemented in microstrip technology on a Rogers RT 588 substrate with ɛ r =2.2 and height h =.787 mm at a design frequency of f =1.92 GHz, which is at the centre of the PCS 19 GSM frequency band. The design and simulations were carried out with the help of the Agilent-ADS microwave simulator. The design procedure for this particular choice of parameters proceeded as follows: 1. The characteristic impedance was chosen to be Z = 2 5 = 7.71 Ω, as outlined at the end of Section From Agilent-ADS, for a Z =7.71 Ω microstrip line the guided wavelength is λ ms = mm and the width of line is W ms =1.388 mm. 3. As indicated in Figures 4.2 and 4.3, the total distance between the loads was chosen to be equal to d tot = λ 8 = mm, where λ = mm at f =1.92 GHz. 4. The length of the positive phase-shifting MTM line was therefore equal to dtot λ =9.766 mm, and the length of the negative phase-shifting TL was also equal to 16 λ 16 =9.766 mm. 2 =

109 Chapter 4. A NRI-TL Metamaterial Series Power Divider The phase incurred by the mm long microstrip TL was equal to Φ c-tl = 3.413, therefore the phase that the MTM line had to incur was Φ mtm = The number of MTM stages was chosen to be m =2. 7. The required phase shift per MTM unit cell was therefore equal to φ mtm = Φmtm m = = The length of the host TL for each MTM unit cell was d h-tl = dtot = mm = 2m mm, corresponding to φ h-tl = The choice of m =2,resultsin φ mtm =.265 1and φ h-tl =.265 1, which satisfy the criteria used to derive Equation (2.79). 9. Substituting the values of Z =7.71 Ω, φ h-tl = and φ mtm = into Equations (2.1) and (2.11) the resulting approximate values of the loading elements are L =11.42 nh and C =2.29 pf. The exact values from Equations (2.86) and (2.89) are L =1.978 nh and C =2.196 pf, which are quite close to the approximate values calculated above. Using the exact values of L =1.978 nh and C =2.196 pf, the resulting phase response for the NR-MTM line is shown in Figure 4.4, together with the phase response of the reference λ g -long TL. It can be verified that indeed the phase incurred by the NR-MTM line is at the design frequency of f = 1.92 GHz. The NR-MTM line exhibits a phase variation of ΔΦ/Δf = /GHz and the TL has a phase variation of /GHz, indicating that the NR-MTM line has approximately three times the phase bandwidth compared to the conventional λ g -long TL. It will be subsequently shown that this increased phase bandwidth for the NR-MTM line translates into a larger input impedance bandwidth as well as a larger through-power bandwidth to each of the output ports of the NR-MTM power divider. 4.5 Simulation Results Using Ideal, Lossless Components Using the above calculated parameters for each of the NR-MTM lines and the transmission lines, the power divider architectures of Figures 4.2 and 4.3 were designed and

110 Chapter 4. A NRI-TL Metamaterial Series Power Divider NR MTM 36 TL 1 Phase of S21 ( ) Frequency (GHz) Figure 4.4: Insertion phase responses of a 2-stage NR-MTM line and a λ g -long TL at f =1.92GHz. simulated in Agilent-ADS initially using ideal lossless components. The S-parameter performances of both the NR-MTM and the TL dividers are shown in Figures 4.5 and 4.6. Figure 4.5(a) shows the simulated return loss of the NR-MTM and TL dividers. The TL divider exhibits a 1 db return loss bandwidth, BW S11,of.26GHzfrom GHz, centred at f = 1.92 GHz, while the NR-MTM divider has a BW S11 of.75 GHz from GHz, indicating that the NR-MTM divider exhibits approximately three times the return-loss bandwidth compared to TL divider. The two-fold increase in return-loss bandwidth is a direct consequence of the two-fold increase of the phase bandwidth of the NR-MTM lines compared to the transmission lines, as outlined at the end of the previous Section. Figures 4.5(b)-(e) show that there is equal power split to all four ports of both the TL and NR-MTM dividers, with 6 db through power (i.e. one quarter of the input power) delivered to each port at f = 1.92 GHz. It can be observed that the power delivered to each of the four output ports remains constant over a larger bandwidth for the NR-MTM divider compared to the TL divider. The TL divider demonstrates a 3 db through bandwidth to port 3, BW S31,of.33GHzanda 3 db through bandwidth to port 4, BW S41,of.55GHz,comparedtoaBW S31 of.94 GHz and a BW S41 of 1.58 GHz for the NR-MTM divider, which again corresponds to approximately a two-fold increase

111 Chapter 4. A NRI-TL Metamaterial Series Power Divider 91 Magnitude of S11 (db) NR MTM Divider TL Divider Frequency (GHz) (a) S 11 magnitude response Magnitude of S21 (db) Magnitude of S31 (db) NR MTM Divider TL Divider Frequency (GHz) NR MTM Divider TL Divider Frequency (GHz) (b) S 21 magnitude response (c) S 31 magnitude response Magnitude of S41 (db) Magnitude of S51 (db) NR MTM Divider TL Divider Frequency (GHz) NR MTM Divider TL Divider Frequency (GHz) (d) S 41 magnitude response (e) S 51 magnitude response Figure 4.5: Ideal simulated magnitude responses S 11 - S 51 for the NR-MTM and TL dividers.

112 Chapter 4. A NRI-TL Metamaterial Series Power Divider NR MTM Divider TL Divider 8 6 NR MTM Divider TL Divider 4 4 Phase of S21 ( ) 2 2 Phase of S31 ( ) Frequency (GHz) (a) S 21 phase response Frequency (GHz) (b) S 31 phase response 8 6 NR MTM Divider TL Divider 8 6 NR MTM Divider TL Divider 4 4 Phase of S41 ( ) 2 2 Phase of S51 ( ) Frequency (GHz) (c) S 41 phase response Frequency (GHz) (d) S 51 phase response Figure 4.6: Ideal simulated phase responses S 21 - S 51 for the NR-MTM and TL dividers. in bandwidth. Within the frequency range of 1 to 3 GHz, the signal delivered to ports 2 and 5 does not fluctuate beyond 3 db from the 6 db value at f = 1.92 GHz, therefore a 3 db through bandwidth is not applicable. Here, it should be noted that the return-loss at the output ports, i.e. S 22, S 33, S 44 and S 55, is quite poor and is equal to 2.5 db at the design frequency of f =1.92GHz for both the NR-MTM and TL dividers. In addition, the isolation between the output ports, i.e. S 23, S 24, S 25, S 34, S 35 and S 45, is also relatively poor and is equal to 12.4 db at the design frequency of f = 1.92 GHz for both the NR-MTM and TL dividers. Due to its poor return-loss at the output ports and its poor isolation, this type of power divider would not function well as a power combiner and therefore its use as a power combiner is not recommended.

113 Chapter 4. A NRI-TL Metamaterial Series Power Divider 93 Table 4.1: Phase variation characteristics around f =1.92 GHz from port 1 to each of the output ports of the NR-MTM divider. NR-MTM divider path (length) S 21 (λ g /8) S 31 (λ g /4) S 41 (3λ g /8) S 51 (λ g /2) ΔΦ/Δf ( /GHz) Table 4.2: Phase variation characteristics around f =1.92 GHz from port 1 to each of the output ports of the TL divider. TL divider path (length) S 21 (λ g /8) S 31 (λ g /4) S 41 (3λ g /8) S 51 (λ g /2) ΔΦ/Δf ( /GHz) Figures 4.6(a)-(d) show the phase responses from port 1 to each of the output ports for both the NR-MTM and the TL dividers. As expected, moving along the output ports of the power divider from port 2 up to port 5, the path lengths from port 1 become longer, therefore the phase responses exhibit steeper slopes, becoming more narrowband. The path lengths from port 1 to port 2 up to port 5 are λ g /8, λ g /4, 3λ g /8andλ g /2 respectively. Tables 4.1 and 4.2 summarize the phase variation characteristics of the NR- MTM and TL dividers, measured around a 2 MHz bandwidth, centred at 1.92 GHz. It can be observed that the phase responses of all four of the output ports are steeper for the TL divider than for the corresponding ports of the NR-MTM divider. It is interesting to note, however, that the phase variation does not increase linearly with an increase in the length of each line, as in the case of NR-MTM lines and TLs that are analyzed in isolation outside of the context of a series power divider. The phase characteristics of various NR-MTM lines of different lengths have been shown in Figure 2.28(b) and the results summarized in Table 2.6, where it was shown that the phase variation indeed increases linearly with a commensurate increase in the length of the lines. The same, of course, holds for conventional transmission lines. When the NR-MTM lines and TLs are used within the context of a series power divider, however, the rate of increase of the phase variation appears to decrease with an increase in the length of each line. It should be noted, that the phase variation at each output port is significantly higher than if the corresponding length of line were tested in isolation, as can be verified by comparing the same length lines in Tables 2.6 and 4.1. The above discussion indicates that there are other effects taking place when the lines are placed within the series power divider, that do not manifest themselves when

114 Chapter 4. A NRI-TL Metamaterial Series Power Divider 94 the lines are tested in isolation. The ripples in both the magnitude and phase responses of all the output ports indicate that when the device operates in a region outside of the intended operating point where all of the lines incur, the mismatch between adjacent sections establishes standing waves within the structure. This is a consequence of the specific architecture chosen for the series power divider, and does not depend on the type of lines used to implement the divider, something that is verified by the fact that the ripples in the responses appear both in the NR-MTM divider and the TL divider. The standing waves, therefore, act in such a way as to redistribute the power delivered to each port over frequency, while always adhering to power conservation principles. One advantageous consequence of this is that the amount of power delivered to port 5 of the divider is almost constant for all frequencies, as can be observed from Figure 4.5(e), and this despite the large phase variation from port 1 to port 5 as seen in Figure 4.6(d). 4.6 Non-Ideal Simulation and Experimental Results The NR-MTM and TL dividers were subsequently designed and simulated in Agilent- ADS using non-ideal microstrip transmission-line models and S-parameter models for the lumped-elements. In order to fully account for all of the electromagnetic effects within the power divider structures, ideally both NR-MTM and the TL dividers would be simulated in a full-wave simulator, such as the Momentum method-of-moments package within Agilent-ADS, which was available at the time. The presence, however, of lumped-element components within the NR-MTM divider did not allow its simulation in Momentum, since this simulator can only analyze planar structures. The final simulations of the NR-MTM divider were therefore carried out in a hybrid simulation mode within Agilent-ADS, called a Co-simulation. This allows the placement of both lumped-element models as well as other planar models imported from Momentum within a single schematic design. When the simulation is run, ADS calls the Momentum simulator which analyzes the planar structures within the design using the method-of-moments, and then returns the result to the schematic as a frequency-dependent S-parameter file. The ADS schematic simulator then performs a nodal analysis on all of the interconnected elements within the design and produces the final results. Thus, the NR-MTM divider was analyzed using the Cosimulation technique within ADS, while the TL divider, being a fully-planar structure, was analyzed using Momentum. The TL divider of Figure 4.2 was implemented using four meanderered TLs, each

115 Chapter 4. A NRI-TL Metamaterial Series Power Divider 95 with a characteristic impedance of Z = 7.71 Ω, width W λ tl = 1.39 mm and length λ g = mm. The NR-MTM divider of Figure 4.3 was implemented using four NR- MTM phase-shifting lines, each consisting of two positive phase-shifting MTM unit cells and a section of negative phase-shifting TL. Each MTM unit cell employed a host transmission line with Z = 7.71 Ω, width W h-tl = 1.39 mm and length d h-tl = 4.48 mm. The loading elements were obtained from MuRata Manufacturing Co., together with each component s S-parameter model file which included all the measured parasitic components associated with each lumped element. Standard size 42 components were used for all the loading elements, which had values and self-resonant frequencies (SRF) of C = 2.2pF(SRF=5GHz),2C =3.3pF(SRF=4GHz)andL =11nH(SRF=5.5GHz). Note that the lumped-element components were chosen such that their self-resonant frequencies were well above the operating frequency, in order to avoid large variations in the component values. The negative phase-shifting section of TL was implemented with Z = 7.71 Ω, W c-tl =1.39mmandd c-tl = 9.77 mm. At each of the output ports of the dividers, a two-stage quarter-wavelength impedance transformation network was added to provide a broadband impedance transformation from the 5 Ω test equipment impedance to the required load impedance of 2 Ω. The first quarter-wavelength transformer converted the 5 Ω load impedance to an intermediate impedance of 125 Ω, thus requiring a characteristic impedance of Z 1 = (5)(125) = 79.6 Ω, which was realized by using a microstrip transmission line with width W TL1 =.21 mm and length d TL1 = mm. The second quarterwavelength transformer converted the intermediate 125 Ω impedance to the required load impedance of 2 Ω at each port of the divider, therefore requiring an impedance of Z 2 = (125)(2) = Ω, which was realized by using another microstrip transmission line with width W TL2 = 1.16 mm and length d TL2 = 29.7 mm. The two power dividers were fabricated at the University of Toronto laboratories and the working prototypes are shown in Figure 4.7. The two quarter-wavelength transformers described above can be seen connected to each of the output ports of the power dividers. As evident from the photograph, the NR-MTM divider occupies significantly less space than the analogous TL divider. The phase-shifting lines of the NR-MTM divider occupy an area of 18 mm 2, which corresponds to a mere 2.6% of the 498 mm 2 area that the meandered transmission lines of the TL divider occupy. Considering the entire areas of the two dividers, including the four quarter-wavelength transformers, the NR- MTM divider occupies an area of 4717 mm 2, which corresponds to 54% of the 877 mm 2

116 Chapter 4. A NRI-TL Metamaterial Series Power Divider 96 Figure 4.7: Photographs of the fabricated NR-MTM power divider (left) and the TL power divider (right). c 25 IEEE. 1 2 Metamaterial Divider Magnitude of S11 (db) Measured Simulated Transmission Line Divider 4 Measured Simulated Frequency (GHz) Figure 4.8: Measured vs. simulated S 11 magnitude responses for the NR-MTM and TL dividers. c 25 IEEE. area that the TL divider occupies. Figure 4.8 shows the simulated and measured return loss of the NR-MTM and TL series power dividers, which show good agreement. The TL divider exhibits a simulated 1 db return loss bandwidth, BW S11, of.26 GHz from GHz, while the

117 Chapter 4. A NRI-TL Metamaterial Series Power Divider 97 NR-MTM divider has a BW S11 of.74 GHz from GHz, which corresponds to a 185% increase in the simulated bandwidth. For the measured results, the TL divider exhibits a measured 1 db return loss bandwidth, BW S11, of.26 GHz from GHz, centred at f = 1.92 GHz, while the NR-MTM divider has a BW S11 of.69 GHz from GHz, which corresponds to a 165% increase in the measured bandwidth. These results also agree quite favourably with the ideal, predicted results from Section 4.5. Figures 4.9 to 4.12 show the simulated and measured transmission characteristics to each of the output ports, where it can be observed that the results match quite well. There is approximately equal power split to all four output ports of both the NR-MTM and TL dividers at f = 1.92 GHz, and as expected, the power delivered to each of the four ports remains constant over a larger bandwidth for the NR-MTM divider compared to the TL divider. The TL divider exhibits a simulated 3 db through bandwidth to port 3, BW S31,of.34GHzandaBW S41 of.56 GHz, compared to a BW S31 of.84 GHz and a BW S41 of 1.33 GHz for the NR-MTM divider, corresponding to a 147% and a 138% increase in the simulated bandwidth respectively. For the measured results, the TL divider exhibits a measured 3 db through bandwidth to port 3, BW S31,of.31GHz and a BW S41 of.54 GHz, compared to a BW S31 of.79 GHz and a BW S41 of 1.37 GHz for the NR-MTM divider, corresponding to a 155% and a 154% increase in the measured bandwidth respectively. Within the frequency range of 1 to 3 GHz, the signal delivered to ports 2 and 5 does not fluctuate beyond ±3 dbfromthevalueatf =1.92GHz, therefore a 3 db bandwidth is not applicable. The total losses in each divider can be found by performing a power balance calculation, which reveals that the total measured losses in the TL divider are 9%, while in the NR-MTM divider they are 13%. More specifically, the total measured losses in each path from port 1 to ports 2, 3, 4 and 5 of the TL divider are.3 db,.47 db,.49 db and.38 db respectively, while for the NR-MTM divider they are.82 db,.62 db,.7 db and.63 db respectively. The average measured loss per port is.41 db for the TL divider and.69 db for the NR-MTM divider. The total losses for the two dividers are comparable, and include both the material and the reflection losses. Thus, the conductor losses in the meandered transmission lines, which are the main loss mechanism of the TL divider, are comparable to the losses in the lumped-element chip components of the NR-MTM divider. Since the losses are expected to increase from ports 2 to port 5, the results indicate that there is some power redistributed amongst the output ports. This can be attributed to imperfections in the fabrication of the circuits, leading to small

118 Chapter 4. A NRI-TL Metamaterial Series Power Divider 98 Metamaterial Divider 1 Magnitude of S21 (db) 2 Measured Simulated Transmission Line Divider 1 2 Measured Simulated Frequency (GHz) Figure 4.9: Measured vs. simulated S 21 magnitude responses for the NR-MTM and TL dividers. c 25 IEEE. Metamaterial Divider 1 Magnitude of S31 (db) 2 Measured Simulated Transmission Line Divider 1 2 Measured Simulated Frequency (GHz) Figure 4.1: Measured vs. simulated S 31 magnitude responses for the NR-MTM and TL dividers. c 25 IEEE.

119 Chapter 4. A NRI-TL Metamaterial Series Power Divider 99 Metamaterial Divider 1 Magnitude of S41 (db) 2 Measured Simulated Transmission Line Divider 1 2 Measured Simulated Frequency (GHz) Figure 4.11: Measured vs. simulated S 41 magnitude responses for the NR-MTM and TL dividers. c 25 IEEE. Metamaterial Divider 1 Magnitude of S51 (db) 2 Measured Simulated Transmission Line Divider 1 2 Measured Simulated Frequency (GHz) Figure 4.12: Measured vs. simulated S 51 magnitude responses for the NR-MTM and TL dividers. c 25 IEEE.

120 Chapter 4. A NRI-TL Metamaterial Series Power Divider 1 deviations from the ideal phase incurred by each of the TL and NR-MTM lines at the design frequency of f = 1.92 GHz. This in turn results in reflections along the line, which redistribute the power to each of the output ports. It has therefore been demonstrated that by employing NR-MTM phase-shifting lines within a 1:4 series power divider, a broadband in-phase power division can be achieved to all the output ports, while maintaining a small form factor and exhibiting comparable losses when compared to a conventional series power divider employing meandered transmission lines. The NR-MTM divider is fully planar and offers the flexibility of spacing the output ports arbitrarily apart and can moreover be scaled to an arbitrary number of ports. It is therefore suitable for use in planar antenna feed networks, power combining amplifiers, and synchronized distribution of carrier signals in digital circuits.

121 Chapter 5 A NRI-TL Metamaterial Series-Fed Antenna Array The compact and broadband metamaterial power divider presented in Chapter 4 is employed in this chapter to create a series-fed antenna array that exhibits reduced beam squinting characteristics. The chapter begins by describing the general beam scanning characteristics of uniform linear arrays fed in series by true-time delay phase-shifting lines, and then proceeds to describe the scanning characteristics when the phase-shifting lines are implemented first by using transmission lines (TL) and then by using non-radiating metamaterial (MTM) lines. It is shown that the metamaterial lines have a quasi true-time delay nature, which enables them to exhibit a broadband phase response and therefore a broadband beam squinting characteristic. Using an array factor approach, the performance of a four-element series-fed printed dipole array and its grounded counterpart are analyzed using the two types of aforementioned feedlines, as well as a low-pass loaded (LPL) transmission line feed. Subsequently, the metamaterial and low-pass loaded arrays are physically implemented in fully printed form with an inter-element spacing of d E = λ /4, and it is shown that the MTM array exhibits a measured scan-angle bandwidth that is 173% greater than that of the LPL array. 5.1 Introduction In a typical series-fed linear array designed to radiate at broadside, the antenna elements must be fed in phase, while in addition, an inter-element spacing d E of less than a half freespace wavelength (d E <λ /2) is necessary to avoid capturing grating lobes in the visible 11

122 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 12 region of the array pattern. In order to achieve these design constraints, traditional designs employing TL-based feed-networks have resorted to a meander-line approach. This allows the antenna elements to be physically separated by a distance that is less than a guided wavelength λ g, while still being fed in phase using a meandered TL that has a total length of λ g, and therefore incurs a phase of 2π rad at the design frequency f. It will be shown in Section 5.4 that even though the phase incurred by a transmission line is a linear function of frequency, when transmission lines are used within a series-fed array that is printed on a dielectric substrate, a change in the operating frequency will cause the emerging beam to squint from broadside, which is generally an undesirable phenomenon. In addition, the fact that the lines are meandered causes the radiation pattern to experience high cross-polarization levels, as a result of parasitic radiation due to scattering from the corners of the meandered lines [16]. The feed-networks proposed in this chapter employ non-radiating metamaterial phaseshifting lines, which are significantly more compact and broadband when compared to meandered-tl feed networks. Thus, when the MTM feed networks are used within a series-fed linear array, this results in a significant reduction in the amount of beam squint that the array experiences with a change in frequency. The idea of using NRI-TL metamaterials to reduce beam squinting in series-fed antenna arrays was proposed in [15] for a four-element slot antenna, and subsequently in [68] and [73] for the four-element linear dipole array discussed herein. Since then, there has been significant interest in this area, with many new series-fed antenna arrays presented, mostly using active circuits to eliminate beam squinting within a certain frequency range. Notable among these are an aperture-coupled patch array that uses an active metamaterial block to achieve a negligible beam squint over a 11% bandwidth [69], and the concept of using an active series feed network with a superluminal group delay to eliminate beam squinting [17]. Furthermore, a steerable series-fed patch array that employs tunable metamaterial phase shifters to achieve a 43 scan-angle range around 2.4 GHz can also be used to provide reduced beam squinting capabilities [7].

123 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 13 y γ d E sin( γ ) FEED N d E φ x Φ Φ Φ Figure 5.1: N-element uniform linear array of isotropic elements along the x-axis. c 28 Wiley. 5.2 Uniform Linear Arrays Employing True-Time Delay Phase Shifters True-time delay (TTD) phase shifters are ones which exhibit a constant time delay (group delay) that is independent of frequency. As such, TTD phase shifters have a phase response that is a linear function of frequency, which also passes through the origin [18]. A simple section of a TEM transmission line, whose insertion phase is a linear function of frequency given by Equation (2.22), is therefore a very straightforward implementation of a TTD phase shifter. The use of true-time delay phase shifters in linear antenna arrays is particularly desirable because it eliminates the phenomenon known as beam squinting, which is a shift of the main radiated beam with a change in frequency. Although theoretically beam squinting can be eliminated in both series-fed and corporate-fed arrays, the practical limitations of a series-fed array outlined below and in Section 5.4, dictate that series-fed arrays always experience a certain amount of beam squint. Beam squinting can therefore only be completely eliminated in corporate-fed arrays [91] (pp ). In order to further understand the importance of a linear phase characteristic and its effect on beam squinting, it is instructive to consider a uniform linear array of N isotropic radiating elements arranged along the x-direction as shown in Figure 5.1. Here, the numbers from one to N indicate the locations of the N radiating elements in the

124 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 14 array and the angle γ indicates the location of the emerging beam from broadside. It is related to the azimuthal angle φ as follows: γ = π/2 φ. The inter-element spacing is d E, the progressive phase shift between each of the elements is Φ, k is the wave vector in free space, and c is the speed of light. For generality, it is assumed that Φ can take on positive or negative values corresponding to a current phase lead or lag respectively. The normalized array factor (AF) for the N-element linear array can therefore be written as [19] ( AF = 1 ( sin Nψ )) ψ N ej(n 1) 2 2 sin ( ), (5.1) ψ 2 where ψ = k d E sin(γ)+φ. (5.2) The main beam will attain its maximum value from the array factor when ψ is equal to zeroorwhenitisequalto2mπ, wherem is an integer. For the case where ψ is equal to zero, the inter-element phase shift required to produce a beam at the scan angle γ sc is Φ = ω c d E sin(γ sc ). (5.3) The scan angle for a uniform linear array can therefore be written as ( γ sc =sin 1 Φ ) ( ) =sin 1 Φ ( k d ω ). (5.4) E de c In typical phased array systems the inter-element phasing is carried out through the use of phase shifters. If the phase shifters are independent of frequency, then a simple inspection of Equation (5.4) reveals that the scan angle of the array will be frequency dependent, since c and d E are constant. However, if Φ is a linear function of frequency, then the frequency terms in the expression for γ sc will cancel, therefore rendering the scan angle frequency independent. Thus, for a TTD phase shifter, we can write the inter-element phasing relationship as Φ = ωτ g, (5.5) where τ g is the group delay, which is a constant that is determined by the type of true-time delay phase shifter used. Inserting Equation (5.5) into Equation (5.4), the frequency-independent scan angle can be written as ( ) ( ) γ sc =sin 1 τ g d Ec =sin 1 τg. (5.6) τ g,

125 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 15 In Equation (5.6) the group delay in vacuum over a distance d E, τ g, = d E /c, has been used. Re-arranging Equation (5.6), we obtain an expression for the group delay between successive array elements τ g = τ g, sin(γ sc ). (5.7) There are some interesting observations that can be made regarding Equation (5.7). Provided that the inter-element phase shift is a linear function of frequency as in Equation (5.5), then the successive time delay between array elements is given by Equation (5.7) and is constant with frequency. Thus, the phase shifting line will have a true-time delay, resulting in a radiated beam at a fixed angle γ sc that does not experience any beam squint. However, in order to produce a radiated beam at an angle γ sc between 9 and +9, Equation (5.7) also provides an indication of the amount of group delay that a phase shifting feed-line in a series-fed array must provide in order to achieve this. Since the sine function varies between the values of 1 and 1, passing through zero, this implies that the group delay along the line, τ g, must be less than or the group delay in vacuum, τ g,. Otherwise stated, the group velocity along the line, v g = d E /τ g must be greater than the speed of light, v g c. (5.8) Evidently, this condition would violate causality for passive media. A technique to overcome this problem in practical realizations of series-fed arrays is outlined in Section 5.4, however it is demonstrated that by doing so the zero beam squinting property is eliminated. In corporate-fed arrays, the inter-element phase shift of Equation (5.5) is in fact the difference between the phase shift of the individual lines that feed the array elements, therefore the effective differential group delay can be made arbitrarily small, zero or even negative. It is this property that enables corporate-fed arrays to exhibit zero beam squinting, albeit at the expense of bulky, complicated and lossy feed networks. 5.3 Tapered Amplitude Distribution in Linear Arrays Even though an expression for the array factor of an N-element uniform linear array has been presented in Equation (5.1), it is useful to derive a more general expression for

126 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 16 the array factor that does not assume a uniform excitation of the antenna elements. We begin by considering the general normalized array factor, given by [19] AF = 1 N N I n e jkˆr rn, (5.9) n=1 where N is the total number of antenna elements, I n is the current excitation of the n-th element, ˆr is the unit vector for the chosen coordinate system and r n is the position vector from the origin of the n-th element. In general, the current excitation I n has both an amplitude, a n, and phase, Φ, associated with it and is given by I n = a n e j(n 1)Φ. (5.1) If we assume that the current amplitude a n has a tapered distribution along the array given by a n = a n 1,wherea is the amplitude factor that is less than one, the array factor can therefore be written as AF = 1 N N a n 1 e j(n 1)Φ e jkˆr rn. (5.11) n=1 If we further express the current amplitude a n as a n = a n 1 = e (n 1) ln a = e (n 1)b, (5.12) where b =lna is a constant, and assume that the array elements are arranged along the x-axis as shown in Figure 5.1, then the array factor can be written as AF = 1 N N ( ) e b+jψ n 1, (5.13) n=1 where ψ = k d E sin(γ)+φ is given by Equation (5.2). Equation (5.13) can be recognized as a geometric series with a common ratio term of r = e b+jψ. The sum of the first N terms of a geometric series has a closed-form solution given by N n=1 Thus, Equation (5.13) can be written as AF = 1 N r n 1 = 1 rn 1 r. (5.14) ( 1 e Nb+jNψ 1 e b+jψ ). (5.15)

127 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 17 After some factoring this can be re-expressed as ( Nb AF = 1 sinh b N e(n 1) 2 +j(n 1) ψ j Nψ ) 2 ( b sinh 2 + j ψ ). (5.16) 2 It can be recognized that the array factor expression of Equation (5.16) for an array with a tapered amplitude distribution has a similar form to the array factor for a uniformly-fed array given by Equation (5.1). In fact, if the amplitude taper is eliminated and the array is uniformly fed by setting the amplitude factor to a = 1 (i.e. b = from Equation (5.12)), the array factor expression of Equation (5.16) reduces to that of Equation (5.1). The location of the main beam can be obtained by finding the phase ψ at which the array factor attains its maximum value. The phase ψ canthenbemappedtoanangleγ in the xy-plane using Equation (5.2). The first step in finding the location of the main beam is to take the absolute value of the array factor squared, since the array factor is a complex quantity: ( ) ( ) Nψ Nb AF 2 = 1 sin 2 +sinh 2 N 2 e(n 1)b 2 2 ( ) ( ) ψ b. (5.17) sin 2 +sinh The values of ψ for which AF 2 attains its maximum value can then be found by taking its first derivative with respect to ψ and setting this equal to zero. The first derivative of AF 2 can be written as ( d AF 2 = 1 N sin (Nψ) sin (ψ) sin ( ) ( ) ) 2 Nψ dψ 2N 2 e(n 1)b sin ( ) ( 2 +sinh 2 Nb 2 ) 2 ψ 2 +sinh 2 b ( 2 sin ( ) ( ) ) 2 2. ψ 2 +sinh 2 b 2 (5.18) By simple inspection, it can be observed that Equation (5.18) will be equal to zero when ψ =, irrespective of the value of the amplitude tapering, determined by b =lna. Furthermore, by taking the second derivative of AF 2 and evaluating this at the extremal point of ψ =, the resulting expression is negative, indicating that at ψ = the array factor indeed attains a maximum value. Thus, it can be concluded that Equation (5.2) can be used to accurately determine the location of the main beam of both uniformly-fed arrays, as well as arrays that have a tapered amplitude distribution. The maximum value of the normalized array factor, which is a measure of the gain of the array in the direction of maximum radiation, can be evaluated by letting ψ =in

128 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array a=1 a=.8 a=.5 a=.2 a= AF γ ( ) Figure 5.2: Normalized array factor patterns as a function of the azimuthal angle γ = π/2 φ for five representative amplitude factors a using the parameters N =4,k = rad/m, d E =.15 m and Φ = 1.36 rad. Equation (5.16), which after some simplifications results in the following expression: AF max = 1 ( ) 1 a N. (5.19) N 1 a Thus Equation (5.19) can be used to evaluate the relative decrease in gain of an array that has a tapered amplitude distribution with an amplitude factor a, comparedtoa uniformly-fed array that has a normalized gain of AF max =1. The effect of enforcing an amplitude taper along an array can be easily visualized by considering the example shown in Figure 5.2, which shows the magnitude of the array factor for a four-element linear array designed to radiate a beam at an angle of γ sc =+45. Equation (5.16) was used to evaluate the array factor for five representative values of the amplitude factor, a =1,.8,.5,.2 and using the parameters shown in the caption of the figure. It can be observed that when the array is uniformly fed, i.e. a = 1, the array factor has as expected a maximum value of one at γ sc =+45.Asadecreases, the main beam remains fixed at the γ sc =+45 point, however the maximum values of the array factor decrease to the levels described by Equation (5.19). When a =, thisimpliesthat only the first element in the array is fed, therefore the array can no longer be used to control the direction of the beam, and the array factor pattern becomes omnidirectional. The results of this section verify that an arbitrary amplitude taper can be enforced

129 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 19 along the array without affecting the main beam direction. This result is particularly useful when analyzing the beam scanning properties of realistic series-fed linear arrays. In such arrays, the series feed networks that interconnect the array elements will inevitably exhibit a certain amount of loss, thus enforcing an amplitude taper along the array. Nevertheless, the direction of the main radiated beam can be accurately described with the simple knowledge of the phase response of the interconnecting feed lines. Thus, the feed lines can be designed according to a certain phase specification, while ensuring that their loss is minimized in order to maintain high gain levels. In the following sections, two types of uniform series-fed linear arrays are considered that are fed using lossless interconnecting feed lines in order to investigate their beam scanning characteristics. Subsequently, more realistic models of two types of dipole arrays are presented where various amplitude tapers are enforced along the arrays, and the findings of this section with respect to the their beam scanning and gain characteristics are verified. 5.4 Transmission-Line Fed Series Uniform Linear Arrays Figure 5.3 shows a 4-element series-fed uniform linear array, where the inter-element phase shift Φ is carried out using sections of conventional printed transmission lines that have an effective permittivity of ɛ eff. The phase shift Φ for a section of line that has a physical length of d can therefore be written in terms of the transmission line parameters using Equation (2.22) as ɛeff Φ = β tl d = ω d = ωd. (5.2) v φ c It should be noted that Φ in Equation (5.2) is now a negative quantity, to reflect the phase delay nature of transmission lines. In Equation (5.2) we have used the following expression for the phase velocity of the transmission line v φ = c ɛeff. (5.21) Using Equations (5.4) and (5.2), the scan angle for a transmission-line fed array can therefore be expressed as ( ) γ sc,tl =sin 1 ɛeff d. (5.22) d E

130 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 11 y γ sc z x FEED Φ Φ Φ d d d d E d E d E Figure 5.3: Transmission-line-fed series uniform linear array of 4 elements. c 28 Wiley. The expression for the scan angle of a transmission-line fed array is frequency independent, therefore it would appear that one could design squint-free series-fed linear arrays using conventional true-time delay transmission lines, similar to the case considered in Section 5.2. However, it will be shown below that the physical constraints of a printed series-fed array actually eliminate the true-time delay nature of the transmission lines, therefore causing the beam to squint with frequency. An informative way to visualize the direction of the scan angle γ sc is to construct a rectangular-to-polar graphical representation of the array factor, as shown in Figure 5.4 for an N = 4 element uniform array with an inter-element spacing of d E = λ /4anda progressive phase shift of Φ = π/4 rad. The radius R ofthecircleisgivenby R = k d E, (5.23) and the centre of the circle with respect to the origin is given by C =Φ, (5.24) while the visible region of the array pattern is given by [93] Φ k d E ψ Φ + k d E. (5.25)

131 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array VISIBLE REGION AF(ψ).5 2π 3π/4 π/4 2π ψ (rad) R C 9 γ Figure 5.4: Rectangular-to-polar graphical representation of the array factor for a linear array with N =4,d E = λ /4andΦ = π/4 rad. c 28 Wiley. Thus, the radius of the circle for the example shown in Figure 5.4 is R = π/2 rad and the centre of the circle is located at C = π/4 rad, i.e. it is shifted to the left by π/4 rad. The visible region for this example is therefore 3π/4 ψ π/4 rad. It can be seen that within the visible region, the radiation pattern exhibits a major lobe at a scan angle of γ sc =3, a single sidelobe at γ = 75 and a null γ = 3.AsΦ is decreased from zero to π/2 rad, the location of the main beam will move from broadside (γ sc = ) when Φ = rad, through γ sc =3 when Φ = π/4 rad, to endfire (γ sc =9 )when Φ = π/2 rad, thus covering the whole scanning range of the array. Inspection of Equation (5.22) reveals that if ɛ eff d exceeds d E, then the argument of the sin 1 ( ) function will exceed one, resulting in a physically unrealizable scan angle γ sc. On the polar plot of Figure 5.4, this corresponds to the case where the distance between the centre of the circle and the origin is physically larger than the radius of the circle, i.e. C > R. Therefore, this creates a range of inter-element phase shifts (Φ )forwhich a scan angle does not exist, because the main lobe can no longer be captured by the polar plot. Physically, this will result in a radiation pattern that contains some of the sidelobes, without capturing the peak of the main lobe. Thus, the following condition

132 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 112 should be satisfied if conventional transmission lines are to be used as true-time delay phase shifting devices in a printed TL-fed series linear array: ɛeff d d E. (5.26) The physical implication of Equation (5.26) is that the antenna elements must be spaced a distance of at least the length of the interconnecting transmission lines, multiplied by the square root of the effective permittivity, which is typically larger than one for circuits printed on a dielectric substrate. More generally for any type of phase shifter, by following the same arguments outlined above, Equation (5.4) yields the following condition that must be satisfied in order for the scanning angle γ sc to have a physical meaning: Φ k d E. (5.27) The series configuration of Figure 5.3 imposes yet another important physical limitation on the choice of d and d E. Since the sections of transmission lines must be long enough to physically connect the array elements, the length of the transmission lines cannot be less than the inter-element spacing, i.e. d d E. (5.28) This condition, however, directly contradicts the condition imposed by Equation (5.26), since as already stated, ɛ eff is typically larger than one for antennas printed on a dielectric substrate. This seemingly contradictory situation can be resolved by adding a section of transmission line that is one guided wavelength λ g long to the transmission line length, d. This is equivalent to shifting the visible region of the radiation pattern shown in Figure 5.4 to the left by 2π rad, transforming d into d : d = d + λ g. (5.29) This new value of d now satisfies Equation (5.28) for typical values of d E <λ g in order to avoid grating lobes. In addition, because d is now greater than d E, this implies that the interconnecting transmission line must be meandered in order for it to fit in between the antenna elements, as shown in Figure 5.3. The inter-element phase shift Φ remains effectively the same because of its periodicity in 2π, which has a modulo-2π property. Therefore, Equation (5.27) is also satisfied, implying that the series-fed array will produce

133 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 113 φ +2π Δ φ / Δω ω ω 2π Modified +2π Original Figure 5.5: Modified phase vs. frequency characteristic for a TL. c 28 Wiley. an emerging beam at an angle γ sc. However, because the centre of the visible region has now been shifted to 2π, Φ undergoes the following transformation and becomes Φ : Φ = 2π +Φ. (5.3) By replacing Φ with 2π+Φ, this is equivalent to selecting m = 1 in the discussion after Equation (5.2), and has the effect of eliminating the true-time delay characteristic that the delay lines would otherwise offer. This can be observed by considering the modified phase-frequency characteristic of the delay lines after the transformation of Equation (5.3). This is shown graphically in Figure 5.5 for the representative case of Φ =. The transformation has the effect of shifting the phase characteristic up by a factor of 2π, while still maintaining an effective phase shift of Φ at the design frequency of ω.thus for frequencies below ω the transmission line will effectively produce a positive phase shift and for frequencies above ω it will produce a negative phase shift. The modified phase as a function of frequency for the TL is therefore given by φ tl = Δφ ω + b. (5.31) Δω Here, Δφ/Δω is the slope of the line, and b is the φ-axis intercept point. For the particular example shown in Figure 5.5, and for all transmission lines that are augmented by a one guided wavelength λ g section, b =2π. The true-time delay characteristic of the delay lines is eliminated because the transformation of Equation (5.3) introduces a 2π constant term in the previously linear

134 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 114 relationship between Φ and ω. Therefore the expression for the scan angle γ sc from Equation (5.4) now contains an extra term that is inversely proportional to frequency, and can be written as ( ) ( ( 2πc ) 2π γ sc,tl =sin 1 +Φo 1 =sin 1 k o d E d E ω + ) ɛeff d. (5.32) Thus, it has been shown that the TL-fed series uniform linear array can be physically implemented by adding a section of transmission line that is one guided wavelength λ g long to the interconnecting transmission lines. This, however, results in a frequencydependent scan angle γ sc, where any variation from the design frequency will lead to beam squinting. It will be shown in the following section that contrary to conventional delay lines, NRI-TL metamaterial lines possess an inherent quasi-ttd characteristic that can be exploited in series-fed arrays to effectively reduce the amount of beam squint that they experience with frequency. d E Low-Pass Loaded Transmission Line Before further investigating the beam scanning properties of series-fed arrays, it is useful to introduce an equivalent implementation of a transmission line that retains its main propagation and impedance properties, but has the additional benefit of a significantly smaller size. By loading a conventional distributed transmission line using lumped-element series inductors and shunt capacitors in a low-pass manner as shown in Figure 5.6, a slow-wave structure can be obtained similar to the one presented in Section By nature of its slow-wave property, this structure has the advantage that it can achieve a large phase shift that is directly dependent on the loading-element values, while occupying a small physical size. The low-pass loaded (LPL) transmission line of Figure 5.6 can be analyzed using periodic theory as was done for the generic T unit cell of Figure The loading series impedance and shunt admittance of the LPL line are given by Z = jωl lpl & Y = jωc lpl. (5.33) Inserting these into the generic dispersion relation of Equation (2.56), results in the

135 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 115 L LPL /2 θ/2 θ/2 L LPL /2 Z Z C LPL d LPL Figure 5.6: Unit cell of a low-pass loaded (LPL) transmission line. dispersion relation for the LPL line ( cos(β bl d)= 1 ω2 L lpl C lpl 4 ) cos(θ) ( ωllpl + ωc ) lplz 2Z 2 sin(θ) ω2 L lpl C lpl. 4 (5.34) Following a similar approach to that outlined in Section 2.4.4, the periodically loaded transmission line can be considered an effective medium under the conditions that θ 1 and β bl d 1, thus resulting in the following effective propagation constant for the LPL line: [ ][ ] β lpl = ±ω L + L lpl C + C lpl. (5.35) d lpl d lpl Furthermore, under the condition that the characteristic impedance of the host transmission line is equal to that of the loading elements, namely that L Z = C = Llpl, (5.36) C lpl the effective propagation constant for the LPL line can be expressed simply as the sum of the propagationconstants of the host transmission line and the low-pass loading elements, β lpl = ω LC + ω (Llpl )( Clpl d lpl d lpl ). (5.37) 5.5 Metamaterial-Fed Series Uniform Linear Arrays The feed-networks presented in this section employ non-radiating metamaterial (MTM) phase-shifting lines within a series-fed linear array, as shown in Figure 5.7, to mitigate some of the problems encountered with conventional TL-based feed-networks. In order to investigate how the main beam of the uniformly excited MTM array shifts from the

136 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 116 y γ sc z x FEED Φ Φ Φ NR-MTM NR-MTM NR-MTM d E d E d E Figure 5.7: Metamaterial-fed series uniform linear array of 4 elements. c 28 Wiley. broadside direction as the frequency is varied, we must first consider the phase response of the MTM feed lines. The inter-element phase shift Φ then takes on the form of the phase response of a non-radiating MTM line, given by Equation (2.91) as ( Φ =Φ mtm +Φ c-tl = m ω ) 1 LCd h-tl + ω + ( ω ) LCd c-tl. (5.38) L C By substituting Equation (5.38) into Equation (5.4), the scan angle for the uniformly excited MTM array with the antenna elements equally distributed along the x-axis as shown in Figure 5.7 can therefore be written as ( γ sc,mtm =sin 1 Φ ) mtm +Φ c-tl (5.39) k d ( E ( ) ) =sin 1 cn 1 d E L C ω + c LC(mdh-tl + d 2 c-tl ). (5.4) d E Inspection of Equation (5.4) reveals that the first term in the sin 1 ( ) expression, cn/ ( d E L C ω 2), is inversely proportional to the frequency squared, and the second term c LC(md h-tl + d c-tl )/d E is a constant. Thus, if the first term that is frequency dependent can be made negligible compared to the second term, then the dependence of γ sc on ω will diminish and the scan angle will become relatively insensitive to frequency variations. Due to the inverse-frequency-squared dependence of the cn/ ( d E L C ω 2)

137 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Scan angle, γ sc ( ) Metamaterial Line Transmission Line Low Pass Loaded Line Frequency (GHz) Figure 5.8: Comparison of the scan angle performance from broadside of a series-fed linear array with d E = λ /4 using a metamaterial, a transmission-line and a low-pass-loaded line feed network. c 28 Wiley. term, this will become negligible compared to c LC(md h-tl + d c-tl )/d E as the frequency increases. Thus, at high frequencies the constant term c LC(md h-tl + d c-tl )/d E will dominate, and γ sc will remain relatively constant with frequency. Therefore, it can be concluded that even though the metamaterial lines do not inherently possess a pure TTD characteristic, by operating at high enough frequencies, a quasi-ttd characteristic can be obtained, thus rendering the scan angle relatively frequency insensitive. If we return to scan angle characteristic of the TL-fed array given by Equation (5.32), we can see that the frequency dependent term in this expression, 2πc/(d E ω), is inversely proportional to frequency, and not inversely proportional to the frequency squared as in the case for the MTM-fed array. Thus, the scan angle of the TL-fed array will inherently be more frequency dependent, even at higher frequencies where the scan angle of the MTM-fed array is relatively frequency independent. The theoretical scan angles from broadside for the TL-fed and MTM-fed linear arrays with d E = λ /4 = 15 mm were calculated at f = 5 GHz using Equations (5.32) and (5.4) respectively, and the scan angle characteristics are shown in Figure 5.8. The parameters used to calculate the phase responses of the TL feed lines were d = λ g =52.44mm, Z = 17 Ω and of the MTM feed lines, m =1,d h-tl = 3 mm, d c-tl = 12 mm,

138 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 118 Z = 17 Ω, C =.13pF,L =2.44nH. It can be observed that the scan angle for the TL-fed array exhibits its full scanning range from endfire (γ sc =+9 )tobackfire(γ sc = 9 ) within a bandwidth of 2.67 GHz, while the corresponding scanning bandwidth for the metamaterial-fed array is 1.44 GHz, which amounts to a 291% increase in scan-angle bandwidth. The larger scanning bandwidth of the MTM-fed array agrees favourably with the previous discussion relating to the scan angle expressions for the two arrays from Equations (5.32) and (5.4). It is interesting to note that as the frequency increases above 5 GHz, the gradient of the slope of γ sc versus frequency for the MTM-fed array is the smallest in the region from around 7 to 13.5 GHz, which is consistent with the discussion of Equation (5.4) that the MTM lines exhibit quasi-ttd characteristics at higher frequencies. Also shown in Figure 5.8 is the scan-angle characteristic for a low-pass loaded (LPL) TL with a total length of λ /4 = 15 mm and designed to incur 2π rad at 5 GHz. The LPL line was designed using five of the unit cells shown in Figure 5.6, each designed to incur a phase of 72. The parameters Z = 17 Ω, d lpl =3mmandθ =2.6 were used in Equations (5.34) and (5.36) to produce loading-element values of L lpl =4.53nHand C lpl =.156 pf. From Figure 5.8, it can be observed that the performance of this line is identical to that of the TL meandered feed-line. Thus, although the low-pass loaded line can eliminate the need for meander lines, it does not provide the advantage of an increased scan-angle bandwidth that the metamaterial feed lines offer. 5.6 Transmission Line and Metamaterial Series-Fed Printed Dipole Arrays Four-Element Series-Fed Printed Dipole Array In order to investigate further the radiation characteristics of a series-fed array as the frequency is varied, it is instructive to consider the total field of the array, which by the pattern multiplication principle is equal to the antenna element factor multiplied by the array factor. The series-fed linear array considered herein is a four-element printed dipole array with the z-directed dipole elements arranged symmetrically along the x-axis with an inter-element spacing of d E = λ /4, as shown in Figure 5.9. The position vectors for

139 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 119 z θ FEED I 1 I 2 I 4 I 3 γ y d E x φ Figure 5.9: Geometrical arrangement of the 4-element printed dipole array. c 28 Wiley. each of the dipoles are given by r 1 = 3 8 λ ˆx, r 2 = 1 8 λ ˆx, r 3 =+ 1 8 λ ˆx, r 4 =+ 3 8 λ ˆx, (5.41) and the unit vector in the rectangular coordinate system is given by ˆr =cosφ sin θˆx +sinφ sin θŷ +cosθẑ. (5.42) The array factor from Equation (5.9) therefore becomes AF = 1 ( ) I 1 e jk 3 8 λ cos φ sin θ + I 2 e jk 1 8 λ cos φ sin θ + I 3 e +jk 1 8 λ cos φ sin θ + I 4 e +jk 3 8 λ cos φ sin θ. 4 (5.43) If we consider half-wavelength dipoles, then the electric field expression in the far field can be written as [19] E θ = jη I e jkr 2πr ( ( ) cos π cos θ) 2. (5.44) sin θ The total field pattern for the array is given by the pattern multiplication principle as E tot = E θ AF. (5.45) The main beam of the series-fed array shown in Figure 5.9 will scan in the xy-plane, which is perpendicular to the axis of the array. This is also the H-plane of the dipoles

140 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 12 and is obtained by setting θ = π/2. Considering Equation (5.44) and setting θ = π/2 we can observe the well-known fact that the field of a dipole does not vary in its H-plane. Since any variation of the total electric field of the array will be produced solely by the array factor, henceforth only the array factor will be considered when investigating the scanning characteristics of the array. If the dipoles are oriented in another direction or a different antenna element is used in the array, its field pattern would have to be considered together with the array factor as in Equation (5.45). The array factor in the xy-plane for the TL-fed array can be written using Equation (5.11) as AF = 1 4 ( a e jk 3 8 λ cos φ + a 1 e jφ,t L e jk 1 8 λ cos φ ) + a 2 e j2φ,t L e +jk 1 8 λ cos φ + a 3 e j3φ,t L e +jk 3 8 λ cos φ, (5.46) where a is the amplitude factor per section of interconnecting TL. An amplitude taper along the array can be achieved by setting a to a value less than one. The phase response of the TL feed line is given by Equation (5.31) as Φ,T L = Δφ ω +2π. (5.47) Δω Similarly, the array factor in the xy-plane for the MTM-fed array can also be written using Equation (5.11) as AF = 1 4 ( a e jk 3 8 λ cos φ + a 1 e jφ,mt M e jk 1 8 λ cos φ ) + a 2 e j2φ,mt M e +jk 1 8 λ cos φ + a 3 e j3φ,mt M e +jk 3 8 λ cos φ, (5.48) where the phase response of the MTM feed line is given by Equation (5.38) and can be re-written as ( ) Δφh-tl Φ,MT M = m Δω ω + 1 ω + Δφ c-tl ω. (5.49) L C Δω When the absolute values of Equations (5.46) and (5.48) are plotted as a function of the azimuthal angle φ (or equivalently, the angle from broadside γ = π/2 φ), the radiation pattern of the array is obtained. By varying the frequency one can observe how the location of the main beam shifts with frequency, as shown in Figures 5.1 and 5.11 for the three frequencies of 4, 5 and 6 GHz. The parameters used to calculate the phase responses of the TL feed lines were Δφ/Δω = sandofthe MTM feed lines,

141 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array γ (a) 4GHz (b) 5GHz (c) 6GHz Figure 5.1: Normalized ideal array factor patterns in the xy-plane from Equation (5.46) for the TL-fed dipole array. c 28 Wiley γ (a) 4GHz (b) 5GHz (c) 6GHz Figure 5.11: Normalized ideal array factor patterns in the xy-plane from Equation (5.48) for the MTM-fed dipole array. c 28 Wiley. n =1,Δφ h-tl /Δω = s, Δφ c-tl /Δω = s, C =.13 pf and L =2.44 nh. From Figures 5.1 and 5.11, it can be observed that both the TL-fed and the MTM-fed arrays radiate at broadside (γ = ) at 5 GHz, however when the frequency is decreased the beam shifts to the left towards the backfire direction and when the frequency is increased the beam shifts to the right towards the endfire direction. It can also be observed that within the same frequency range of 4 to 6 GHz, the beam of the MTM-fed array shifts less to the left and right off of broadside than the beam of the TL-fed array. By performing a frequency sweep and recording the azimuthal location of the beam

142 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Normalized AF max a=1.3 a=.9 a= Frequency (GHz) Figure 5.12: Normalized maximum array factor as a function of frequency for the TL-fed dipole array for three representative amplitude factor values: a = 1, a =.9 anda =.8. c 28 Wiley. maximum, the scan angles for both the TL-fed and the MTM-fed arrays were evaluated over the entire range of γ for various values of the amplitude factor a. It was found that the scan angle characteristics of the two arrays were identical to the ones presented in Figure 5.8 for the uniformly excited linear arrays regardless of the value of the amplitude factor. This result is in accordance with the findings of Section 5.3. Subsequently, the value of the array factor was recorded at the peak of each radiation pattern, and Figure 5.12 shows the amplitude of the maximum value of the array factor for the TL-fed array as a function of frequency for three representative cases of the amplitude factor a. According to Equation (5.9), the data is normalized such that the value of the maximum array factor is equal to one when the beam is at broadside, i.e. at f = 5 GHz. By plotting the maximum value of the array factor, this is equivalent to the maximum amplitude of the main beam in the patterns shown for example in Figures 5.1 and This can be considered a measure of the gain in the direction of maximum radiation. When no amplitude taper is enforced along the array, i.e. a = 1, the maximum array factor remains constant at a value of one from 4 to 6.67 GHz, which is the full scanning range of the TL-fed array as shown in Figure 5.8. In the region from 6.67 to 7.5 GHz, the beam is directed towards the endfire direction and a variation in frequency causes

143 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Normalized AF max a=1.3 a=.9 a= Frequency (GHz) Figure 5.13: Normalized maximum array factor as a function of frequency for the MTM-fed dipole array for three representative amplitude factor values: a = 1, a =.9 anda =.8. c 28 Wiley. the array factor maximum to drop because less of the endfire main lobe is being captured within the visible region of the array. From 7.5 GHz to 8 GHz the main beam reverses direction towards the backfire direction and increases in amplitude until 8 GHz. Above 8 GHz the entire main lobe is captured again within the visible region of the array, and therefore the maximum amplitude of the array factor is restored to one. By observing the three responses shown in Figure 5.12, it can be seen that by enforcing an amplitude taper along the array, this reduces the value of the maximum array factor, however the general shape of the curves is maintained the same as for the uniformly exited array. Figure 5.13 shows the amplitude of the normalized maximum value of the array factor as a function of frequency for the MTM-fed array for the same three representative cases of the amplitude factor a. By comparing Figures 5.12 and 5.13, it can be observed that the MTM-fed array has a clear advantage over the TL-fed array in as much as the amplitude of the main beam does not drop within the whole scanning range of the array, namely from 3.65 to 14.1 GHz. This is because the peak of the main beam remains within the visible region over this entire frequency range. As in the case of the TL-fed array, enforcing an amplitude taper along the array reduces the value of the maximum array factor, but the shape of the curves is maintained.

144 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 124 z θ FEED I 5 I 6 I 1 I 8 I 7 d G d G I 3 I 2 γ y I 4 d E x φ GROUND PLANE Figure 5.14: Geometrical arrangement of the grounded 4-element printed dipole array. c 28 Wiley Grounded Four-Element Series-Fed Printed Dipole Array In many applications it is desirable to have an antenna with a unidirectional radiation pattern. The printed dipole array described in the previous section will radiate a bidirectional broadside pattern at its design frequency, as shown in Figures 5.1(b) and 5.11(b). A well known technique to obtain a unidirectional pattern from a bidirectional antenna is to add a ground plane at a distance of d G = λ /4 below the antenna, as shown in Figure Shown in Figure 5.14 is a four-element printed dipole array with +z-directed currents I 1 I 4 at a distance of d G in front of an infinite ground plane in the xz-plane. By image theory, the ground plane creates z-directed image currents I 5 I 8 at a distance of d G behind the ground plane, thus effectively forming an 8-element antenna array. Thus, for currents I 1 I 4, each current I n can be expressed using Equation (5.1), while currents I 5 I 8 can be expressed as I n. The position vectors for each of the dipoles for d E = λ /4

145 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 125 and d G = λ /4aregivenby r 1 = 3 8 λ ˆx+ 1 4 λ ŷ, r 2 = 1 8 λ ˆx+ 1 4 λ ŷ, r 3 =+ 1 8 λ ˆx+ 1 4 λ ŷ, r 4 =+ 3 8 λ ˆx+ 1 4 λ ŷ. (5.5) The position vectors for each of the image currents are given by r 5 = 3 8 λ ˆx 1 4 λ ŷ, r 6 = 1 8 λ ˆx 1 4 λ ŷ, r 7 =+ 1 8 λ ˆx 1 4 λ ŷ, r 8 =+ 3 8 λ ˆx 1 4 λ ŷ. (5.51) Using the unit vector from Equation (5.42), the array factor from Equation (5.9) therefore becomes ( AF = 1 8 I 1 e jk( 3 8 λ cos φ sin θ+ 1 4 λ sin φ sin θ) + I2 e jk( 1 8 λ cos φ sin θ+ 1 4 λ sin φ sin θ) +I 3 e jk(+ 1 8 λ cos φ sin θ+ 1 4 λ sin φ sin θ) + I4 e jk(+ 3 8 λ cos φ sin θ+ 1 4 λ sin φ sin θ) +I 5 e jk( 3 8 λ cos φ sin θ 1 4 λ sin φ sin θ) + I6 e jk( 1 8 λ cos φ sin θ 1 4 λ sin φ sin θ) ) +I 7 e jk(+ 1 8 λ cos φ sin θ 1 4 λ sin φ sin θ) + I8 e jk(+ 3 8 λ cos φ sin θ 1 4 λ sin φ sin θ). (5.52) Using the general expression for each current excitation from Equation (5.1), we can re-write the array factor in the xy-plane (θ = π/2) from Equation (5.52) as ( AF = 1 a e jk( 3 8 λ cos φ+ 1 4 λ sin φ) + a 1 e jφ e jk( 1 8 λ cos φ+ 1 4 λ sin φ) 8 +a 2 e j2φ e jk(+ 1 8 λ cos φ+ 1 4 λ sin φ) + a 3 e j3φ e jk(+ 3 8 λ cos φ+ 1 4 λ sin φ) a e jk( 3 8 λ cos φ 1 4 λ sin φ) a 1 e jφ e jk( 1 8 λ cos φ 1 4 λ sin φ) ) a 2 e j2φ e jk(+ 1 8 λ cos φ 1 4 λ sin φ) a 3 e j3φ e jk(+ 3 8 λ cos φ 1 4 λ sin φ). (5.53) Here, Φ takes on the form of Φ,T L from Equation (5.47) for the TL-fed array and Φ,MT M from Equation (5.49) for the MTM-fed array. Using the same parameters for the TL and MTM lines as in Section 5.6.1, the radiation patterns for the grounded TL-fed array and the MTM-fed array are shown in Figures 5.15 and 5.16 for the three representative frequencies of 4, 5 and 6 GHz. As in the case of the ungrounded arrays, it can be observed that the main beam of the MTM-fed array squints less with frequency from the broadside location at 5 GHz. The complete scan angle characteristics for the two grounded arrays are shown in Figure The most striking observation from Figure 5.17 is that the scanning range of both the

146 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array γ (a) 4GHz (b) 5GHz (c) 6GHz Figure 5.15: Normalized ideal array factor patterns in the xy-plane using Equation (5.47) in Equation (5.53) for the grounded TL-fed dipole array. c 28 Wiley γ (a) 4GHz (b) 5GHz (c) 6GHz Figure 5.16: Normalized ideal array factor patterns in the xy-plane using Equation (5.49) in Equation (5.53) for the grounded MTM-fed dipole array. c 28 Wiley. arrays is reduced by the addition of the ground plane. The TL-fed array exhibits a scanning range of 54 <γ sc < 65 within a bandwidth of 3.9 GHz, from 3.5 to 7.4 GHz. The MTM-fed array exhibits a scanning range of 52 <γ sc < 72 within a bandwidth of 8.5 GHz, from 3.1 to 11.6 GHz. The reduced scanning range of both the arrays can be attributed to the fact that at the edge of the scanning range the maximum of the array factor shifts to another lobe other than the main one. Thus, when one of the backlobes becomes larger than the main lobe, this determines the limit on the scanning range. Another observation is that although the scanning range of the grounded arrays is decreased compared to the ungrounded arrays, the scanning bandwidth actually increases

147 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Scan Angle, γ sc ( ) Metamaterial Line 4 Transmission Line Low Pass Loaded Line Frequency (GHz) Figure 5.17: Scan angle performance from broadside of a grounded series-fed linear array with d E = λ /4 using a metamaterial, a transmission-line and a low-pass-loaded line feed network. c 28 Wiley. by 1.23 GHz for the TL-fed array and by 1.16 GHz for the MTM-fed array. As with the case of the ungrounded arrays, the scanning characteristics of the grounded arrays remained the same regardless of the value of the amplitude factor a. Figure 5.18 shows the normalized amplitude of the maximum value of the array factor as a function of frequency for the grounded TL-fed array for three representative cases of the amplitude factor a. Around the design frequency of f = 5 GHz, although the scanning bandwidth of the grounded TL-fed array from Figure 5.17 is 3.9 GHz, Figure 5.18 indicates that the frequency range over which the peak values of the main lobe remain at a maximum value is reduced to 1.3 GHz, from 4.7 to 6 GHz. This reduces the usefulness of the grounded TL-fed array, due to the limited frequency range that it can be operated in. At higher frequencies above 6 GHz the amplitude of the maximum array factor varies rapidly with frequency, and in the range of 9.5 to 11.5 GHz it drops to very low values. In the regions where the maximum array factor drops to such low values, the power is re-distributed to other lobes in the pattern, thus reducing the amplitude of the main lobe. This is generally an undesirable condition, as a lot of the power in the array is lost to these other parasitic sidelobes or backlobes. Figure 5.19 shows the amplitude of the normalized maximum value of the array factor as a function of frequency for the grounded MTM-fed array for the same three

148 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Normalized AF max a=1.2 a=.9 a= Frequency (GHz) Figure 5.18: Normalized maximum array factor as a function of frequency for the grounded TL-fed dipole array for three representative amplitude factor values: a =1,a =.9 anda =.8. c 28 Wiley. 1.9 Normalized AF max a=1.2 a=.9 a= Frequency (GHz) Figure 5.19: Normalized maximum array factor as a function of frequency for the grounded MTM-fed dipole array for three representative amplitude factor values: a = 1,a =.9 and a =.8. c 28 Wiley.

149 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 129 representative cases of the amplitude factor a. Comparing Figures 5.18 and 5.19, it can be observed that the amplitude of the main beam of the grounded MTM-array fluctuates very little compared to the grounded TL-fed array. Thus, the grounded MTM-fed array can be effectively operated within the frequency range of 4 to 14 GHz, without large variations in the amount of power delivered to the main lobe. This is because the peak of the main beam remains mainly within the visible region over this entire frequency range. As in the case of the ungrounded arrays, by enforcing an amplitude taper along either of the arrays, this reduces the value of the maximum array factor, however the general shape of the curves is maintained the same as for the uniformly exited array Physical Realizations of the Proposed Structures With sight of a physical realization of the arrays discussed in the previous section, three types of feed lines were considered as potential candidates for feeding the antenna elements in a series-fed linear array: a non-radiating metamaterial line, a 36 meandered TL and a low-pass loaded line that also incurs 36. Each of the feed lines were designed and simulated in Agilent s Momentum planar structure simulator in co-planar strip (CPS) technology, similar to the CPS implementations presented in [88], [11]. A Rogers RT 588 substrate with ɛ r =2.2, and dielectric height h = 2 mils was used, and both dielectric and conductor losses were taken into account by specifying a loss tangent of tan δ =.9 for the dielectric substrate and a thickness of 17 μm for the copper layer with a conductivity of S/m. Using the above parameters, the feed lines were realized using CPS lines with a length of d E = 15 mm, a width of 2 mm, and a separation of.9 mm, resulting in a characteristic impedance of Z = 17 Ω. The non-radiating MTM line used to feed the MTM array is shown in Figure 5.2(a) and consisted of a single MTM π unit cell of length d h-tl = 3 mm, and characteristic impedance Z = 17 Ω, followed by a negative phase-compensating section of TL of length d c-tl = 12 mm. Each series lumped-element loading capacitance value of C =.13 pf was realized using an interdigitated capacitor, and each shunt lumped-element loading inductance value of 2L =4.88 nh was realized using a meandered inductor. The design of the loading elements for the non-radiating MTM line was governed by the geometrical constraints of the CPS line and the achievable tolerances during manufacturing. Since the length of the unit cell was chosen to be 3 mm and the width of each CPS line was 2mm, this determined the total area that the series interdigitated

150 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 13 (a) (b) (c) Figure 5.2: Proposed structures for (a) the CPS MTM feed line, (b) the CPS 36 meanderedtl,and(c)thecps 36 low-pass loaded line. c 28 Wiley. capacitor could occupy. Additionally, a minimum line width and gap width of.1 mm was selected in order to enable accurate manufacturing. Each interdigital capacitor was designed by first selecting the number of interdigital fingers in the design, based on the total width of 2 mm for each CPS line and minimum line width and gap width of.1 mm. The goal was to confine the capacitor as much as possible to the centre of the unit cell, in order to minimize any distributed effects. Using the above geometric constraints, a maximum of four fingers, each with a width of.133 mm and a separation of.1333 mm were designed to fit vertically in the CPS line. The capacitor was designed within a CPS transmission line and the simulation ports were de-embedded to the centre of the line. The Y -parameter data was then used to construct an equivalent π model of the two-port network and the capacitance was extracted from the series Y 12 parameter [84] (p. 188). Finally, the length of the fingers was swept from.1 to 3 mm, until the desired value of C =.13 pf was achieved at a finger length of mm. Each meandered inductor was also designed with the goal of confining it as much as possible in the lateral direction, while utilizing the vertical space between the two CPS lines to achieve the maximum number of lines. Given a.9 mm separation between the CPS lines, four meandered lines were designed to fit between these, each with a width and separation of.1 mm. In this case, however the lateral space to the left and right of the

151 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 131 MTM unit cell was used to house the inductors in order to reduce the coupling between the inductors and the interdigital capacitors at the centre of the unit cell. The inductor was designed within a CPS transmission line and the simulation ports were de-embedded to the centre of the line. The Z-parameter data was then used to construct an equivalent T model of the two-port network and the inductance was extracted from the shunt Z 12 parameter [84] (p. 188). Finally, the length of each line was swept from to 6mm, until the desired value of 2L =4.88 nh was achieved at a line length of mm, resulting in a total line length for both inductors of 27.8 mm. The 36 meandered TL used to feed the TL array is shown in Figure 5.2(b) and also had a characteristic impedance Z = 17 Ω. By keeping the lateral size fixed at d E = 15 mm, the vertical length was adjusted until a total phase shift of 36 was achieved at f = 5 GHz, resulting in a total length of λ g =52.44 mm. The 36 LPL feed line used to feed the LPL array is shown in Figure 5.2(c) and consisted of five unit cells of length d lpl = 3 mm, and characteristic impedance Z = 17 Ω, each incurring a phase of 72. Each series lumped-element loading inductance value of L lpl /2=2.265 nh was realized using a meandered inductor, and each shunt lumped-element loading capacitance value of C lpl =.156 pf was realized using an interdigitated capacitor. The procedure and the geometrical constraints in the design of the loading components of the LPL line were similar to the ones described for the MTM feed line, with the notable differences that the meandered inductor was realized in the series branch using the Y -parameter data and the interdigital capacitor was realized in the shunt branch using the Z-parameter data. Figure 5.21 shows the magnitude of S 21 for the three feed lines. At f =5GHz the magnitude of S 21 is.49 db for the MTM line,.36 db for the meandered TL and 1.3 db for the LPL line. It can be observed that the MTM line exhibits the expected high-pass behaviour with low insertion loss at higher frequencies, while the LPL line exhibits the expected low-pass behaviour with significantly higher insertion loss at higher frequencies. The meandered TL exhibits similar insertion loss compared to the MTM line between 5 and 6 GHz, however above 6 GHz its performance begins to degrade, exhibiting higher levels of insertion loss. Although the insertion loss performance of the meandered TL was quite constant throughout the frequency band of interest, it was determined that the geometrical constraint of spacing the antenna elements of the array a distance of λ /4 apart, prohibited the use of this type of feed line because of increased coupling between the feed line and

152 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Magnitude of S21 (db) (a) NR MTM Line 12 (b) Meandered TL (c) LPL Line Frequency (GHz) Figure 5.21: Agilent-Momentum simulated S 21 magnitude responses for the each of the feed lines shown in Figure 5.2. c 28 Wiley. the antenna elements in a uniplanar array design. This can be verified by considering the meandered TL of Figure 5.2(b), where it can be observed that because of the lateral space constraint, the meandered TL is tightly coupled to itself and it would also couple strongly to adjacent antenna elements placed on each side of the line. It was therefore decided to implement the four-element printed series-fed linear dipole array using only the physically realizable MTM and LPL feed lines. Both the four-element MTM dipole array and the four-element LPL dipole array were designed and simulated in Agilent s Momentum planar structure simulator at a centre frequency of f = 5 GHz. The lengths of the dipole elements were initially designed to be approximately λ g /2, however these were subsequently increased in order to account for the mutual coupling between the antenna elements. Table 5.1 and Table 5.2 summarize the geometrical details of the MTM and LPL antenna array components respectively. A prototype of the MTM dipole array was built and is shown in Figure Figure 5.23 is a close-up view of the non-radiating MTM line, offering a better view of the printed series capacitors and shunt inductors that are placed at the centre of the line. Figure 5.24 shows the LPL dipole array structure, with Figure 5.25 showing the details of the low-pass unit cells, namely the printed series inductors and shunt capacitors. A backing ground plane that had a size of 1 cm 5 cm was added at a distance

153 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 133 Table 5.1: Geometrical details of the MTM printed dipole array components. c 28 Wiley. length (mm) width (mm) separation (mm) CPS line Dipole Capacitor fingers Inductor Table 5.2: Geometrical details of the LPL printed dipole array components. c 28 Wiley. length (mm) width (mm) separation (mm) CPS line Dipole Capacitor fingers Inductor of λ /4 = 15 mm below both of the antennas in order to create unidirectional radiation patterns. In addition, both antennas were fed directly by a coaxial cable, whose centre conductor was connected to the top CPS conductor and the outer sheath was connected to the bottom CPS conductor, as shown in Figures 5.22 and In order to prevent parasitic currents from flowing on the outer surface of the coaxial feed line, caused by the inherent imbalance between the balanced CPS line and the unbalanced coaxial line, ferrite beads were placed around the coaxial line, acting as current chokes. The ferrite beads can also be seen in Figures 5.22 and The use of ferrite beads to suppress the unwanted currents on the exterior of the coaxial feed line is quite commonplace, and provides an effective broadband balun [111] (p. 249), [112] (p. 23), which was required in order to investigate the scanning characteristics of the arrays over a wide frequency range. Other types of baluns were also considered, however the ferrite beads were deemed to be the simplest and most effective solution to achieve the broadband balun effect. Their good performance, even up to the range of 7 GHz will be demonstrated in the following section with the help of measured radiation patterns.

154 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 134 z θ Φ = x y λ /4 Figure 5.22: Photograph of the fabricated MTM-fed printed dipole array at 5 GHz, including the coaxial feed-line surrounded by ferrite beads. c 28 Wiley. capacitor inductor Figure 5.23: Close-up view of the printed components of the non-radiating MTM line, showing the series interdigitated capacitors and shunt meandered inductors. c 28 Wiley Simulation and Experimental Results Figure 5.26 shows the measured and simulated scan angles versus frequency of the emerging beam from the broadside direction for the MTM and LPL arrays. Although both the arrays were designed to exhibit broadside radiation at 5 GHz, the main beam of the fabricated prototypes passed through broadside at 5.2 GHz and 5.1 GHz for the MTM and the LPL arrays, respectively. This shift in frequency can be attributed to imperfections in the manufacturing of the printed array components. Specifically, it was observed that there was slight over-etching of the copper strips that form the interdigitated capacitors and meandered inductors. As a result, the values of the loading elements of the MTM and LPL feed lines were slightly altered, leading to an increase in the phase incurred by the lines and thus resulting in a broadside beam at a slightly higher frequency. In order to provide an equal comparison of the phase versus frequency characteristics of the two

155 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 135 z θ Φ = 36 x y λ /4 Figure 5.24: Photograph of the fabricated LPL-fed printed dipole array at 5 GHz, including the coaxial feed-line surrounded by ferrite beads. c 28 Wiley. inductor capacitor Figure 5.25: Close-up view of the printed components of the 36 LPL line, showing the series meandered inductors and the shunt interdigitated capacitors. c 28 Wiley. arrays, the appropriate data was offset in frequency such that the beam passed through broadside at exactly 5.1 GHz for all four of the cases shown in Figure It can be observed that the measured data matches the simulated data quite well. It is of interest to note that the measured and simulated scan angle plots of Figure 5.26 follow the same general patterns as the theoretical ideal scan angle plots of Figure 5.17 for both the MTM and the LPL arrays. It can be seen that at around 5 GHz the performance of the LPL array is similar to the MTM array, however below 4.8 GHz and above 5.3 GHz their performances begin to diverge, with the MTM array exhibiting significantly more broadband scan angle characteristics at frequencies above 6 GHz. The LPL array exhibits a measured scan-angle bandwidth of2ghz,from4.33to6.33ghz,whilethe MTM array has a measured scan-angle bandwidth of 5.45 GHz, from 4.25 to 9.7 GHz. This corresponds to an increase of 173% in the scan-angle bandwidth over the LPL array. Figure 5.27 shows the measured normalized co-polarization radiation patterns in the

156 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Scan angle, γ sc ( ) MTM Line (Measured) MTM Line (Simulated) LPL Line (Measured) LPL Line (Simulated) Frequency (GHz) Figure 5.26: Measured and simulated scan angles from broadside of the MTM-fed and LPL-fed arrays. c 28 Wiley γ (a) 4.9 GHz (b) 5.1 GHz (c) 5.3 GHz Figure 5.27: Measured normalized co-polarization patterns in the xy-plane for the printed MTM-fed array around 5 GHz. c 28 Wiley. xy-plane (θ = π/2) for the three selected frequencies of 4.9, 5.1 and 5.3 GHz for the MTM array. It can be observed that at 4.9 GHz the main beam is directed off broadside at an angle of approximately 13, while at 5.1 GHz the main beam is at broadside. When the frequency is further increased to 5.3 GHz, the main beam shifts to approximately +1 from broadside. In all three cases, the main beam is clearly distinguishable. Thus, it has been demonstrated that the fabricated MTM dipole array exhibits nonlinear beam scan-

157 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 137 ning characteristics with frequency about the broadside direction, which follows directly from the fact that the phase response of the MTM feed lines have a non-linear phase characteristic given by Equation (5.38). Additionally, one can also observe that if the MTM array is operated at a higher frequency region around 7 GHz, the linearity of the scan angle versus frequency response can be exploited to obtain an off-broadside main beam at approximately +45 (see Figure 5.26), that will not experience significant beam squinting with a change in frequency. This can be deduced from the fact that the gradient of the scan angle with frequency in this region is quite small. Indeed, by inspection of Equation (5.38) it can be observed that the effect of the second term ( 1/(ω L C )), which is inversely proportional to ω, diminishes as the frequency is increased. This results in a linear phase response for the MTM line, with a constant phase offset at DC, which is characteristic of a quasi true-time delay line. This in turn results in a linear scan angle characteristic with frequency that has a small gradient, as outlined in Section 5.5. In order to highlight the fact that the MTM-fed array does not exhibit significant beam squinting at higher frequencies, Figure 5.28 shows the measured normalized copolarization radiation patterns in the xy-plane (θ = π/2) for 7.1, 7.3 and 7.5 GHz. It can be observed that in all three cases the main beam is directed off broadside at an angle of approximately +45. When the frequency is changed from 7.1 to 7.5 GHz the scan angle changes a mere 6,from43 to 49, thus verifying that within this frequency range the MTM-fed array exhibits only modest beam squinting. It should be noted that even at the highest operating frequency of 7.5 GHz, the MTM feed lines still operate in the nonradiating slow-wave region (see Section 2.5). For the particular loading element values chosen for this design, the theoretical maximum operating frequency before propagation begins to occur within the radiation cone is 7.76 GHz. Above this frequency the MTM lines could act as leaky-wave structures if their electrical length is made sufficiently long. Figure 5.29 shows the measured normalized co-polarization radiation patterns for the LPL-fed array in the xy-plane (θ = π/2) for 7.1, 7.3 and 7.5 GHz. For the LPL-fed array, the main lobe is no longer captured in the visible region of the array pattern above 6.33 GHz, therefore a scan angle does not exist above this frequency (see Figure 5.26). This is verified by the plots of Figure 5.29, where the pattern changes shape rapidly with frequency, and in some cases (e.g. at 7.1 GHz) there is no clearly distinguishable main lobe, which is generally an undesirable condition. Therefore, at high frequencies the location and in fact the existence of a main beam cannot be accurately predicted for

158 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array γ (a) 7.1 GHz (b) 7.3 GHz (c) 7.5 GHz Figure 5.28: Measured normalized co-polarization patterns in the xy-plane for the printed MTM-fed array around 7 GHz (observe the almost fixed beam direction). c 28 Wiley γ (a) 7.1 GHz (b) 7.3 GHz (c) 7.5 GHz Figure 5.29: Measured normalized co-polarization patterns in the xy-plane for the printed LPL-fed array around 7 GHz (observe the non-uniform beam direction and shape). c 28 Wiley. the LPL-array, which is a distinct disadvantage compared to the MTM-fed array. In order to demonstrate the effectiveness of the ferrite beads to act as a balun at the frequencies of interest around 5 GHz and 7 GHz, Figures 5.3 and 5.31 show the measured co- and cross-polarization patterns at 5.1 GHz and 7.3 GHz for the MTM-fed array with and without the use of the ferrite balun on the coaxial feed line. These patterns were measured before the backing ground plane was added to the antenna, however the effect of the balun can still be examined. Comparing Figures 5.3(a) and 5.31(a) at 5.1 GHz, it can be observed that the cross-polarization levels, which are indicative of currents along

159 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Measured Pattern (db) E co pol Measured Pattern (db) E co pol E x pol γ ( ) (a) No ferrite balun, 5.1 GHz E x pol γ ( ) (b) No ferrite balun, 7.3 GHz Figure 5.3: Measured co- and cross-polarization patterns in the xy plane for the MTM-fed array without a backing ground plane and without a ferrite balun on the coaxial feed line. 1 1 Measured Pattern (db) E co pol Measured Pattern (db) E co pol E x pol γ ( ) (a) With ferrite balun, 5.1 GHz E x pol γ ( ) (b) With ferrite balun, 7.3 GHz Figure 5.31: Measured co- and cross-polarization patterns in the xy plane for the MTM-fed array without a backing ground plane and with a ferrite balun on the coaxial feed line. the coaxial cable, are significantly reduced in the case of Figure 5.31(a) where the ferrite balun is used. This indicates that the unbalanced current does not flow on the exterior of the coaxial cable, and therefore the currents on the two conductors of the CPS line and the two arms of the dipoles are balanced. Supporting this argument is a corresponding increase in the maximum gain of the array from 4.3 dbi for the case of Figure 5.3(a) with no balun to a maximum gain of 4.9 dbi for the case of Figure 5.31(a) with the ferrite balun.

160 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array MTM fed array LPL fed array Measured Gain (db) Frequency (GHz) Figure 5.32: Measured gain versus frequency for the MTM and LPL-fed arrays. Examining Figures 5.3(b) and 5.31(b) at 7.3 GHz, reveals that a similar decrease in the cross-polarization levels is also observed, especially around the γ = 8 and γ = 26 directions. There is also an analogous increase in the maximum gain of the array from 2. dbi for the case of Figure 5.3(b) with no balun to a maximum gain of 2.9 dbi for the case of Figure 5.31(b) with the ferrite balun, suggesting that the balun operates effectively even at the upper frequency region. This can also be verified by comparing Figures 5.31(a) and 5.31(b), where the cross-polarization levels remain at similar levels at both 5.1 GHz and 7.3 GHz. The measured gain versus frequency results for both the MTM-fed array and the LPL-fed array are shown in Figure The MTM-fed array exhibits a maximum gain of 9 dbi, while the LPL-fed array exhibits a maximum gain of 8.2 dbi around the design frequency of 5 GHz. In the upper region of operation around 7.5 GHz, the MTM-fed array exhibits a gain of 7.5 dbi, while the LPL-fed array exhibits a gain of 4.7 dbi. The simulated and measured efficiencies of the MTM-fed and the LPL-fed arrays are summarized in Tables 5.3 and 5.4, respectively, where it can be observed that the MTMfed array exhibits higher efficiency values in both the frequency bands of interest. Figures 5.33 and 5.34 show the measured and simulated S 11 magnitude responses for the MTM-fed and LPL-fed arrays, respectively. For the MTM-fed array, the measured results match quite closely with the simulated results, with a measured 1 db S 11

161 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array Magnitude of S11 (db) Measured Simulated Frequency (GHz) Figure 5.33: Measured and simulated S 11 magnitude responses for the MTM-fed array. c 28 Wiley. 5 Magnitude of S11 (db) Measured Simulated Frequency (GHz) Figure 5.34: Measured and simulated S 11 magnitude responses for the LPL-fed array. c 28 Wiley.

162 Chapter 5. A NRI-TL Metamaterial Series-Fed Antenna Array 142 Table 5.3: Measured and simulated efficiency values for the MTM-fed array. Simulated (Momentum) Measured (G/D) Efficiency at 5.1 GHz 86.7% 76.6% Efficiency at 7.3 GHz 77.6% 54.2% Table 5.4: Measured and simulated efficiency values for the LPL-fed array. Simulated (Momentum) Measured (G/D) Efficiency at 5.1 GHz 65.2% 56.6% Efficiency at 7.3 GHz 23.5% 21.9% bandwidth of 61 MHz centred around 5 GHz and a minimum S 11 value of 23.9 db at 5.1 GHz. The MTM array also exhibits reasonably good S 11 performance around 7 GHz with a bandwidth of 2 MHz and a minimum S 11 value of 1.5 dbat7.6 GHz, indicating that the array can be optimally designed to operate around this frequency, where the scan angle characteristic is quite broadband (see Figure 5.26). The corresponding measured results for the LPL-fed array match the simulated results reasonably well around 5 GHz, however at higher frequencies the two diverge. The measured 1 db S 11 bandwidth for the LPL-fed array is 72 MHz centred around 5 GHz, with a minimum S 11 value of 22.9 db at 4.92 GHz. Finally, even though the measured S 11 performance of the TL-fed array is good around 7 GHz, with a minimum S 11 value of 29.2 db at 7.4 GHz, the bandwidth is only 12 MHz. More importantly though, as has already been shown in Figure 5.29, the main beam in the frequency range around 7 GHz has a non-uniform direction and shape which severely limits the usefulness of the LPL-array. In summary, it has been demonstrated in this chapter that NRI-TL metamaterial feed networks can be used in series-fed linear arrays to effectively replace one-wavelength long conventional meandered transmission lines, thus achieving compact, broadside radiators whose beam squints much less with frequency. In addition, the metamaterial feed networks can be used to create linear arrays whose main beam remains virtually fixed at a positive angle from broadside as the frequency is varied.

163 Chapter 6 A NRI-TL Metamaterial Leaky-Wave Antenna Following the same theme as Chapter 5, a reduced beam-squinting leaky-wave antenna (LWA) is presented in this chapter, comprising a series of cascaded NRI-TL metamaterial (MTM) unit cells. Periodic analysis is applied to a single MTM unit cell in order to extract the dispersion and Bloch impedance characteristics. Subsequently, the angular variation of the main radiated beam with frequency or beam squinting is derived, based on the expression for the Bloch propagation constant of the MTM line. It is shown that by operating the LWA in the upper right-handed band where the phase and group velocities are closest to the speed of light, the beam squinting that the antenna experiences can be minimized. The theoretical performance of the LWA is verified through full-wave simulations and measurements of a fabricated prototype designed to produce a radiated beam at an angle of θ sc =+45 at f = 5 GHz. This 2-element metamaterial LWA exhibits a measured return-loss bandwidth below 1 db of.91 GHz (18.2%), and an average beam squint of.31 /MHz. The proposed metamaterial LWA is uniplanar, differential and broadband, and therefore suitable for integration with other microwave components and devices. 6.1 Introduction Leaky-wave antennas (LWAs) form a subset of the general class of traveling-wave antennas, and are thus termed because as a wave propagates along the guiding structure a small amount of energy gradually leaks out in the form of coherent radiation [113]. Since 143

164 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 144 the rate of the energy leakage is small along the guiding structure, this does not perturb significantly the mode inside the guide, which results in a large effective aperture illumination, in turn leading to a narrow radiated beamwidth. In addition, since the structure is designed to be electrically long, by the time the wave reaches the end termination most of the energy has leaked out into free space and therefore any reflected power does not significantly affect the input match, leading to a wide input impedance bandwidth [113]. This is in contrast to series-fed resonant antennas, which inherently tend to exhibit narrowband input impedance bandwidths. LWAs have therefore been considered a desirable radiating structure because of their ability to offer sharp directional beams, which can be scanned with frequency over a large bandwidth. Specifically, printed LWAs have recently attracted considerable attention for the aforementioned reasons, in addition to their ability to be easily integrated on the same substrate with other circuits and devices [114 12]. An additional advantage is their low profile, which enables them to be flush-mounted on various structures including vehicles, cellular base stations, or mobile devices. In some applications, however, it is not desirable to scan the beam with frequency, but rather it is required that the radiated beam remain fixed at a certain location over a large frequency range. Some of these applications include wireless local area networks (WLAN), ultra-wideband (UWB) technology, satellite communications, electromagnetic compatibility (EMC) testing and radiometric field sensing [121]. Such a wideband leakywave antenna was recently presented by Neto et. al. in [122], and consisted of a tapered elliptical dielectric lens mounted on top of a slot etched in a ground plane. In the aforementioned work, it was shown that the main beam remains at a virtually fixed angle in both the E and H-planes over a frequency range from 13 to 17 GHz. An even more broadband variant of this antenna, the Pyramid antenna was presented by the same authors in [121], where it was shown that the antenna operates over a decade bandwidth, from 4 to 4 GHz. One drawback of the broadband LWAs of [121] and [122] is the fact that they are inherently three-dimensional structures consisting of large dielectric lenses mounted on top of a ground plane, and therefore do not exhibit a low profile. In this chapter, it will be shown that the one-dimensional NRI-TL metamaterial phase-shifting lines presented in Chapter 2 can be used to implement a fully-printed lowprofile LWA in co-planar strip (CPS) technology, that radiates in the forward direction and whose main beam remains virtually fixed at a positive angle from broadside as the frequency is varied. This is achieved by operating the NRI-TL metamaterial line in the

165 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 145 upper (right-handed) band, where the phase and group velocities are closest to the speed of light and where the MTM line exhibits a quasi-true time delay characteristic. It should be noted that the CPS implementation of the LWA is the dual of the CPW line used in [62] to create a backward-wave LWA, and as such their characteristics are similar. The concept of using MTM lines to reduce beam squinting in series-fed antenna arrays was presented in the previous chapter, however the MTM lines presented therein were specifically designed to operate in the non-radiating regime. The main radiating elements in that case were the printed dipole elements themselves and not the interconnecting MTM feed lines. Here, it is emphasized that even though the topology of MTM unit cells employed in this chapter is identical to the one used in Chapter 5, the unit cells employed herein are designed such that they operate in the leaky-wave regime, therefore allowing the entire MTM line to radiate as a leaky-wave antenna. Additionally, the theory presented in this chapter pertaining to the scanning characteristics of the metamaterial LWA is more general, since the restrictions of a MTM unit cell with a closed-stopband, a small electrical size and a small phase shift per unit cell are lifted. This provides additional flexibility with regards to the values of the loading elements required for each MTM unit cell, and facilitates their implementation in a fully-printed form. It should finally be mentioned that since the recent development of NRI metamaterials, numerous other metamaterial-based leaky-wave antennas have been presented, including one-dimensional radiating structures [65], [123], [124], two-dimensional frequencyscanned antennas [63], [125], and electronically controlled radiators [66], [126]. 6.2 Proposed Structure LWA Design The design of the LWA was based on simple array theory, assuming infinitesimal isotropic radiators uniformly spaced along the x-axisataregularintervalofd E, as shown in Figure 6.1 [12]. Here, the numbers from one to N indicate the locations of the N radiating elements in the array, the angle θ sc indicates the location of the main beam from the broadside direction and Φ is the progressive phase shift between each of the elements. For generality, it is assumed that Φ can take on positive or negative values corresponding to a current phase lead or lag respectively. By applying suitable far-field approximations, the array factor (AF) for the N-element uniform linear array can be

166 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 146 z θ sc d E sin( θ sc ) FEED N d E x Φ Φ Φ Figure 6.1: N-element uniform linear array of isotropic elements along the x-axis. c 28 IEEE. derived as in Equation (5.1), where the location of the main beam is now given at a scan angle θ sc by ( θ sc =sin 1 Φ ). (6.1) k d E A simple inspection of Equation (6.1) reveals that if the inter-element phase shift Φ is a linear function of frequency, then the frequency terms in the expression for θ sc will cancel, therefore rendering the scan angle frequency independent. The condition that Φ has a linear characteristic with frequency which also passes though the origin, corresponds to a true-time delay (TTD) response as discussed in Section 5.2, which is responsible for the elimination of beam squinting in series-fed linear array systems. It will be shown in this chapter that when the metamaterial lines used to implement the inter-element phase shift Φ are operated in the appropriate frequency region, then the lines exhibit a quasi true-time delay behaviour which reduces the amount of beam squinting that the antenna experiences. For leaky-wave antennas, the propagation constant along the axially symmetric structure is complex and can be written as γ = α + jβ. (6.2) Here, α is the attenuation constant or leakage constant and β is the propagation constant along the structure. If the structure is uniform longitudinally, α and β will be constant along its length. The phase matching condition along the interface between the surface

167 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 147 Table 6.1: NRI-TL metamaterial unit cell representative loading-element values and transmission-line parameters. c 28 IEEE. L (nh) C (pf) Z (Ω) d h-tl (mm) θ ( )at5ghz of the leaky-wave structure and air requires that the propagation constants be continuous along this interface. Therefore, for small values of α, the angle of leakage will be approximately equal to the angle that the maximum beam emerges from broadside, θ sc, and is given by [127] ( θ sc sin 1 β ). (6.3) k By inspection of Equations (6.1) and (6.3), the equivalence between the two can be recognized for small values of α. Thus, by accurately designing for the phase along the leaky-wave structure and ensuring that the leakage constant is small, the location of the main beam can be accurately predicted Leaky Transmission Line Model The NRI-TL metamaterial π unit-cell shown in Figure 2.1(b) was used as a building block to design the leaky-wave antenna, which was formed by cascading a series of these metamaterial unit cells to create a traveling-wave radiating structure. The dispersion characteristics of the metamaterial line were determined in Section and are given by Equation (2.56). Figure 6.2 shows the dispersion relation obtained using Equation (2.56) and the representative parameters given in Table 6.1, which were computed assuming an effective relative permittivity of ɛ eff =1.2. It can be observed that the dispersion diagram exhibits a band structure with a low frequency stopband up to 1.92 GHz, followed by a lower left-handed (LH) passband from 1.92 GHz to 3.8 GHz, another stopband from 3.8 GHz to 4.3 GHz, and an upper right-handed (RH) passband from 4.3 GHz to GHz. Here, it should be noted that for generality the closed stopband matching condition Z = L/C = L /C of Equation (2.71) has not been imposed, resulting in a stopband between the lower LH band and the upper RH band. The ability to operate this structure in the leaky-wave regime can be deduced from the fact that portions of both the LH and the RH passbands exist within the light cone. The light cone is the area enclosed on both sides by the dashed light line in Figure 6.2

168 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna f t2 Frequency (GHz) LH band f t1 RH band Light line 2 π π/2 π/2 π β BL d (rad) Figure 6.2: Dispersion diagram for the NRI-TL metamaterial unit cell of Figure 2.1(b) with representative loading-element values and transmission-line parameters given in Table 6.1. c 28 IEEE. and demarcates the transition between slow and fast-wave propagation relative to free space (i.e. along this line k = β bl ). In Figure 6.2, the transition points into and out of the light cone have been denoted as f t1 and f t2 respectively and the frequency range between the two is denoted as the radiation region. The radiating passbands for this structure are therefore from f t1 = 2.98 GHz to 3.8 GHz for the LH band, and from 4.3 GHz to f t2 = 1.83 GHz for the RH band. If the frequency of operation is within these radiating passbands, the phase velocity of the waves traveling on the line will be greater than the speed of light, which is a necessary condition for leaky-wave radiation. Consequently, by appropriately designing the loading of each metamaterial unit cell, the excitation of leaky-wave modes is possible on these structures. The Bloch impedance of the metamaterial π unit cell is given by Equation (2.64), and Figure 6.3 shows the real component of the Bloch impedance as a function of frequency for the same representative loading-element values and transmission-line parameters as Figure 6.2. It can be observed that since Z,bw <Z, the Bloch impedance characteristic is similar to that of Figure 2.15(b), where the real part of the Bloch impedance goes to zero within the stopband regions (in fact the Bloch impedance is imaginary in these regions) and varies smoothly within the passbands.

169 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna Frequency (GHz) Re{Z } (Ω) BL Figure 6.3: Bloch impedance diagram for NRI-TL metamaterial unit cell of Figure 2.1(b) using the parameters of Table 6.1. c 28 IEEE. If we consider Z bl in the region close to 1 Ω, it can be seen from Figure 6.3 that the Bloch impedance characteristic has a steeper slope with frequency in the LH band compared to the RH band. In the RH region between 6 and 9 GHz, the Bloch impedance has a linear dependence on frequency and exhibits the least variation with frequency, which is desirable when attempting to match the structure to the feedline impedance over a large frequency band. It can also be observed from Figure 6.2 that within the same region between 6 and 9 GHz, the phase shift per unit cell, β bl d, also has a linear dependence on frequency and exhibits the least variation with frequency. In addition, this region is conveniently located within the light cone, indicating that for the particular parameters chosen, the RH band from 6 to 9 GHz is most ideal for broadband operation of the leaky-wave antenna Reduced Beam Squinting Principle of Operation Propagation Constant Analysis The observations from the previous section indicate that with the appropriate design of the metamaterial unit cells, these can exhibit desirable broadband return loss and radiation characteristics. Herein, we will describe more quantitatively the approach taken for

170 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 15 the design of the LWA in order to minimize the amount of beam squinting, by investigating how the scan angle θ sc changes as a function of frequency according to Equation (6.3). We begin by examining how the propagation constant of the periodic structure changes with frequency. Re-arranging Equation (2.56), we can obtain the following expression for the periodic Bloch propagation constant β bl = 1 (( 1+ ZY ) cos(θ)+ j d h-tl 4 2 (ZY + YZ )sin(θ)+ ZY ). (6.4) 4 Equation (6.4) was derived using a non-radiating transmission line analysis, where to a first order approximation the Bloch propagation constant is assumed to be equal to the propagation constant of the line. In fact, it will be shown in Section that the Bloch propagation constant matches closely to the actual propagation constant of the periodic structure evaluated using a full-wave method-of-moments simulator, thus justifying the use of Equation (6.4). A useful approximation to the above expression has been presented in Equation (2.76) for an effective medium with a small unit cell size and a small phase shift per unit cell, and is repeated here β bl,eff = ω 1 LC ω. (6.5) L C d h-tl In the derivation of Equation (6.5), the impedance matching condition has been used, which ensures that the stopband between the lower LH band and the upper RH band is closed and therefore the propagation constant is continuous between these two regions. Both the expressions of β from Equations (6.4) and (6.5) are shown in Figure 6.4 versus frequency for the same representative parameters shown in Table 6.1. It can be observed that the two curves agree quite closely, except at the high and low frequency regions where, as expected, the effective propagation constant deviates slightly from the exact one. In addition, since β bl,eff from Equation (6.5) was derived assuming the closed stopband condition, the propagation characteristics of β bl,eff in Figure 6.4 do not indicate the location of the stopband. The exact expression for β bl, however, indicates a stopband from 3.8 GHz to 4.3 GHz, in accordance with Figure 6.2. The results of Figure 6.4 verify that the effective propagation constant β bl,eff is a reasonably good approximation of the exact propagation constant β bl over the frequency range of interest between 2 and 11 GHz. Therefore, the simple and intuitive expression of β bl,eff from Equation (6.5) will be subsequently used to further our understanding of the frequency scanning characteristics of the LWA.

171 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 151 LH band RH band Propagation Constant (rad/m) β BL β BL,eff Frequency (GHz) Figure 6.4: Bloch and approximate Bloch propagation constants, β bl and β bl,eff,foranri- TL metamaterial line consisting of unit cells employing the parameters of Table 6.1. β bl was obtained using Equation (6.4) and β bl,eff using Equation (6.5). 9 Scan angle, θ sc ( ) θ sc = sin 1 (β BL /k ) θ sc = sin 1 (β BL,eff /k ) Frequency (GHz) Figure 6.5: Scan angle versus frequency for an array that employs NRI-TL metamaterial unit cells with the parameters given in Table 6.1. The Bloch propagation constant expression, β bl, was obtained from Equation (6.4) and the approximate expression of β bl,eff was obtained from Equation (6.5). c 28 IEEE.

172 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 152 Using β bl and β bl,eff from Equations (6.4) and (6.5) respectively, the scan angle versus frequency from Equation (6.3) is shown in Figure 6.5. Since β bl and β bl,eff are in close agreement, the beam squinting performance also agree quite closely. The LWA using β bl exhibits a scanning range from backfire (θ sc = 9 ) to broadside (θ sc = )within a very narrow bandwidth of.82 GHz, from 2.98 to 3.8 GHz. Within this region, the metamaterial structure operates in the backward LH band and therefore the main beam points in the backward direction. The corresponding forward scanning range from broadside to endfire (θ sc =+9 ) spans a significantly larger bandwidth of 6.55 GHz, from 4.3 to 1.83 GHz. Within this region the metamaterial structure operates in the forward RH band, which results in the main beam of the LWA pointing in the forward direction. It can therefore be concluded that it is advantageous to operate the metamaterial structure in the upper RH band where the dependence of the resulting scan angle on frequency is significantly smaller than in the lower LH band. It should be pointed out that both the LH and the RH scanning bandwidths presented above correspond to the radiating passbands of the structure, as outlined in the discussion of Figure 6.2. Expression for the Scan Angle Variation with Frequency Analytically, we can express the dependency of the scan angle on frequency by taking the derivative of Equation (6.3) with respect to ω and letting β = β bl, obtaining the following expression for the beam squinting ( dθ sc dω = ). (6.6) k 2 βbl 2 v g v φ The details of the derivation of Equation (6.6) can be found in Appendix C.1. Examining Equation (6.6), we can identify the factors that affect the beam squinting. If we consider the first term, 1/ k 2 βbl 2, we can see that there are two singularities when β bl = k. These can be recognized as the edges of the radiation region, and have been previously identified as the frequency points f t1 =2.98GHzandf t2 = 1.83 GHz in Figure 6.2. In order to avoid the two singularities that cause large variations of the scan angle, the region of operation should be limited within the radiation region, above f t1 and below f t2. In this region, the value of β bl will always be less than k,andfrom Figure 6.2, we can observe that the difference between β bl and k is relatively constant, except around the β bl = k points. Therefore, it can be concluded that within the radiation region, the term 1/ k 2 βbl 2 is relatively constant and does not significantly affect the beam squinting.

173 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 153 Turning our attention to the second term of Equation (6.6), (1/v g 1/v φ ), we can see that in order for this term to tend to zero, the group velocity must be equal to the phase velocity. Therefore, if the line can be designed such that the phase and group velocities are equal over an appreciable bandwidth, then the beam squinting can be eliminated over this bandwidth. In the case where the phase and group velocities are not equal, the array will experience a certain unavoidable amount of beam squinting, which can be minimized by operating in the region where the difference between the group and phase velocities is the smallest. For the LWA under consideration, within the radiation region the phase velocity will always be greater than the speed of light, i.e. v φ >c, and the group velocity will always be smaller than the speed of light, i.e. v g <c, in order to satisfy causality. Therefore, the phase and group velocities will never be equal, and the array will always experience a certain amount of beam squinting. In order to minimize this amount of beam squinting, the goal therefore becomes to design the metamaterial line with phase and group velocities as close as possible to the speed of light within a wide bandwidth. The phase velocity should be greater than the speed of light such that the structure still radiates, approaching c from above, and the group velocity will be less than the speed of light, as is the case for all passive structures, approaching c from below. The group and phase velocities were calculated by substituting Equation (6.4) into Equations (2.25) and (2.23), respectively, and the graphs of v g and v φ are shown in Figure 6.6. The complete derivation of the group velocity expression can be found in Appendix C.3, where some further analysis is also presented. For the group velocity, the negative sign in front of the square root was chosen in Equation (C.22) for the LH band and the positive sign for the RH band, thus ensuring a positive group velocity in both bands, which is consistent with a positive power flow along the line (i.e. a positive Poynting vector). Within the stopband, the group velocity is purely imaginary, therefore the real part of the group velocity goes to zero. The phase velocity is negative in the LH band, which corresponds to backward-wave propagation, positive in the RH band, which corresponds to forward-wave propagation, and goes to infinity at the edges of the stopband between the LH and RH bands. The graphs of v g and v φ verify that within the RH region between 6 GHz to f t2 =1.83 GHz, the phase velocity exceeds the speed of light, and approaches c from above and the group velocity is less than c, and approaches c from below. Within this region, both velocities have values close to c and remain relatively constant, therefore the term (1/v g 1/v φ ) will be small and will remain relatively constant with frequency,

174 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna x 18 Velocity (m/s) 5 c f t2 v g Frequency (GHz) Figure 6.6: Real part of the group velocity v g and phase velocity v φ for a line consisting of NRI-TL metamaterial unit cells with loading-element values and transmission-line parameters given in Table 6.1. c 28 IEEE. v φ minimizing the amount of beam squinting. In the corresponding LH region, the group velocity remains relatively constant, while the phase velocity varies rapidly; therefore the term (1/v g 1/v φ ) does not remain constant with frequency and is not small. Within this region, the array is therefore expected to experience a significant amount of beam squinting. Within the stopband from 3.8 to 4.3 GHz, since the real part of the group velocity is equal to zero and the phase velocity is equal to infinity, the term (1/v g 1/v φ ) also goes to infinity. Combining both the terms of Equation (6.6), the graph of dθ sc /dω isshowninfigure 6.7. The two singularities caused by the term 1/ k 2 β2 bl can be clearly seen at the edges of the radiation region, i.e. at the frequency points f t1 =2.98GHzand f t2 = 1.83 GHz. In addition, the stopband between 3.8 and 4.3 GHz is revealed by the (1/v g 1/v φ ) term. It can be observed that dθ sc /dω never reaches zero, which would indicate the ideal frequency-invariant scan angle. However it does remain at a relatively constant and small value within the RH band from 5 to 1 GHz, reaching a minimum value at 8.8 GHz. Within the corresponding LH band, the value of dθ sc /dω remains at a much higher value throughout the whole region and is not constant over any appreciable

175 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna x 1 9 dθ /dω sc (dθ /dω) sc eff dθ sc /dω (s) Frequency (GHz) Figure 6.7: Scan angle variation (beam squinting) with frequency for an array that employs NRI-TL metamaterial unit cells with the parameters given in Table 6.1. c 28 IEEE. bandwidth. Thus, from a scan angle perspective, the RH region between 5 and 1 GHz is the most desirable for minimum beam squinting operation. Approximate Expression for the Scan Angle Variation with Frequency In order to gain further insight into the factors affecting the variation of the scan angle with frequency, it is instructive to also derive dθ sc /dω based on the approximate Bloch propagation constant β bl,eff. Therefore, by using the expression for β bl,eff from Equation (6.5), an approximate beam squinting expression (dθ sc /dω) eff was derived in Appendix C.2 and is shown below ( ) ( dθsc = 1+ 1 ( ) ) 2 βbl,eff 2c dω eff 2 k ω β bw. (6.7) 2 The graph of the approximate beam squinting expression (dθ sc /dω) eff from Equation (6.7) is shown together with the graph of the beam squinting expression dθ sc /dω from Equation (6.6) in Figure 6.7. The two match quite closely, especially in the region of interest between 5 and 1 GHz, thus justifying the validity of the simplified model of Equation (6.7). The absence of the stopband in the graph of (dθ sc /dω) eff in Figure 6.7 allows one to easily visualize the inverse cubic dependency of (dθ sc /dω) eff on ω, since

176 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 156 β bw = 1 ω L C d h-tl in the expression of Equation (6.7). The beam squinting will therefore be minimized when the frequency of operation is high. Here, it should be noted that the approximate beam squinting expression of Equation (6.7) is only valid under the effective medium conditions used to derive Equation (6.5), namely that the impedance matching condition Z = L/C = L /C is satisfied and that the unit cell size and phase shift per unit cell are small. In view of Equation (6.5), also note that the term L C d h-tl remains fixed for a given beam direction at the design frequency f, therefore this term does not affect the beam squinting. A simple observation of the first term (1 + 1(β 2 bl,eff/k ) 2 ), reveals that it does not largely affect (dθ sc /dω) eff within the RH band of interest. At the lower edge of the band, β bl,eff =, therefore this term equals to one. At f t2, the upper edge of the radiation band, β bl,eff = k, therefore this term equals to 1.5. Thus, the value of (1 + 1(β 2 bl,eff/k ) 2 ) varies only between 1 and 1.5, indicating that the variation of the scan angle is mostly affected by the other terms in equation Equation (6.7). From Equation (6.7), the quantity (dθ sc /dω) eff is directly proportional to β bw,therefore by striving to minimize β bw, the beam squinting can be minimized. If we let β bw = in Equation (6.5), then β bl,eff = β tl, implying that the line would consist only of a conventional transmission line with no loading. For a conventional TL, β tl = ω LC, which corresponds to a pure true-time delay line, i.e. the propagation constant is a linear function of frequency and passes through the origin. Additionally for a TL, v g = v φ =1/ LC, therefore from Equation (6.6) this corresponds to an ideal zero beam squinting array for all frequencies. However, conventional TLs are slow-wave structures where β tl <k, therefore they are not suitable for leaky-wave applications. Consequently, for the NRI- TL metamaterial structure under consideration, β bw can never be completely eliminated, but is rather a necessary component, that makes the loaded line fast as implied by Equation (6.5), and it should be minimized in order to minimize the amount of beam squinting. Thus, by creating a line that has a propagation constant close to but less than that of free space, a quasi-ttd response can be achieved and the beam squinting can be minimized. This can be achieved by designing the line such that the β tl term dominates over the β bw term in Equation (6.5), which by inspection occurs at high frequencies. This can be verified from the dispersion diagram of Figure 6.2, where β bl approaches (but is less than) k in the upper RH band. The requirement of achieving a propagation constant close to that of free space, corresponds to the requirement outlined in the discussion of the phase and group velocities,

177 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 157 that these should be as close as possible to the speed of light for the NRI-TL metamaterial structure, in order to minimize the amount of beam squinting. For printed transmission lines, in order to achieve a phase velocity close to the speed of light, the structure must be constructed in air with an effective dielectric constant ɛ eff equal to one, according to v φ = c/ɛ eff. Recognizing that this requirement is impossible for printed structures on commercially available substrates where ɛ eff is typically in the range of 1.2 to 2 for low permittivity substrates, this dictates the use of a substrate with the lowest possible relative permittivity in the design of the NRI-TL metamaterial line. β =OperatingPoint The scan angle variation takes on a different form when the propagation constant is equal to zero. From Equation (6.5), when β bl,eff =,β bw = β tl = ω LC, therefore replacing this result in Equation (6.7), the approximate beam squinting at the β = point can be written as: ( ) dθsc = 2c LC (6.8) dω ω eff β= It can be observed that at the β = operating point, the variation of the scan angle is dependent on ω, and not ω 3 as in the general case of Equation (6.7). Therefore, around broadside (θ sc = ), it is expected that there will be significantly more beam squinting than in the upper RH band. The above observation can be verified from the scan angle characteristics of Figure 6.5, as well as in the related case of the series-fed dipole array presented in Figure Design of the NRI-TL Metamaterial LWA General Design Considerations For the LWA structure, the primary design criterion was to produce a main beam at an angle of θ sc =+45 at f = 5 GHz. This angle was specifically chosen in order to produce a radiation pattern with a distinct main beam whose scanning characteristics could be easily investigated, and which is not affected by potential discontinuities in the broadside direction or edge effects in the endfire and backfire directions. Considering the above criterion, a simple algebraic manipulation of Equation (6.1) reveals the phase shift

178 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 158 that each MTM unit cell must incur: Φ = k d E sin(θ sc ). (6.9) For the NRI-TL periodic structure under consideration, the progressive phase shift between each of the radiating elements, Φ, is equal to the phase shift per unit cell β bl d h-tl, and the spacing between each of the radiating elements is equal to the length of each unit cell d E, which in this case is equal to the physical length of each unit cell, d h-tl. Therefore, for a fixed angle θ sc =+45 at f = 5 GHz, the phase shift that each unit cell must incur, Φ = β bl d h-tl, reduces to a choice of the unit cell size d h-tl. In turn, the realization of Φ is governed by the dispersion characteristics of the metamaterial line. Therefore, based on the length and characteristic impedance Z of the host TL, and the amount of phase shift Φ required, Equation (2.56) will produce the required loading element values L and C for the MTM unit cell. What remains is to determine whether the values of L and C are physically realizable in the chosen technology and to verify whether the region of operation is indeed within the light cone Design Procedure for the NRI-TL Metamaterial Unit Cell The design of the NRI-TL metamaterial unit cell was based on the structure of Figure 2.1(b) implemented in CPS technology at a design frequency of f =5GHz. The substrate chosen was a Rogers RO43C laminate with relative permittivity ɛ r = 3.38, loss tangent tan δ =.27 and height h = 32 mil. All the circuit simulations were carried out in the Agilent-Advanced Design System circuit simulator, while the full-wave simulations were carried out in the Agilent-Momentum method-of-moments simulator. The first step in the design of the MTM unit cell was to design the host TL using a fullwave analysis in Agilent-Momentum in order to have a realistic value for the characteristic impedance of the line. In the realization of the host TL in CPS technology, it was determined that the realizable characteristic impedance values were relatively high and in the range of approximately 15 Ω. This is due to the inherently low value of the shunt capacitance C between the edge coupled co-planar strips. In order to proceed with the design, a convenient geometry of the CPS TL was chosen that could be physically realized in the laboratory and that would allow easy integration of series and shunt loading elements. The geometry chosen had strips of width w = 2 mm and separation s =.9mm, which resulted in a characteristic impedance of Z = Ω and an effective relative permittivity of ɛ eff =

179 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 159 The next step in the design procedure was to calculate the values of the loading elements L and C and to determine whether these were physically realizable within the constraints of the chosen host TL, given that these were to be implemented in fullyprinted form using meandered inductors for L and interdigitated capacitors for C.The design procedure used to calculate L and C was as follows: 1. Create a table with various unit cell sizes, d h-tl. 2. For each unit cell size, determine from Equation (6.9) the required value of Φ for the main beam to emerge at an angle of θ sc =+45 at f =5GHz. 3. Let Φ = β bl d h-tl in Equation (2.56), as calculated from Equation (6.9). 4. Let Z bl = Z = Ω in Equation (2.64) in order to ensure that the periodic structure is matched and that each constituent unit cell effectively sees an infinite periodic medium. 5. Equations (2.56) and (2.64) now form two simultaneous equations with two unknowns, L and C, which can be solved numerically or graphically, as described at the end of Section Using the above procedure, MTM unit cells of various sizes were designed and the loading element values required in each case were determined. Using the calculated loading element values as well as the host TL parameters, the dispersion, group velocity, phase velocity and beam squinting diagrams similar to the ones shown in Figure 6.2, Figure 6.6 and Figure 6.7 were produced for each unit cell. The phase velocity for each unit cell at f = 5 GHz was identical and equal to v φ = m/s, since for each unit cell the main radiated beam was fixed at an angle of θ sc =+45 according to Equation (6.1), therefore fixing the value of β, and subsequently the value of v φ from Equation (2.23). Thus, the main factor affecting the beam squinting characteristics of the LWA around the operating point was the group velocity, according to the discussion in Section Table 6.2 summarizes the loading element values as well as the pertinent characteristics of each unit cell. By analyzing the presented data, various observations can be made. Beginning from the first row, it can be observed that as the unit cell size shrinks, the group velocity becomes larger at the design frequency, approaching the speed of light from below. Since the phase velocity is fixed at this point, the term (1/v g 1/v φ )of

180 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 16 Table 6.2: Metamaterial unit cell characteristics for various unit cell sizes. c 28 IEEE. Unit cell size, d h-tl (mm) Group velocity v g at 5 GHz ( 1 8 m/s) Beam squinting dθ/dω at 5 GHz ( /MHz) Width of radiating LH passband (GHz) Width of radiating RH passband (GHz) L (nh) C (pf) Equation (6.6) shrinks, therefore reducing the amount of beam squinting. This can be clearly seen in the third row of Table 6.2, where the amount of beam squinting decreases as the unit cell size decreases. Here, it should be noted that the group velocity and beam squinting data obtained using the approximate Bloch propagation constant β bl,eff of Equation (6.5) agree well with the data of Table 6.2 that were obtained using the Bloch propagation constant β bl from Equation (6.4), up to a unit cell size of d h-tl =12 mm. For larger unit cell sizes the approximate model of Equation (6.5) does not hold, therefore the approximate data is not valid. Observing the following two rows in Table 6.2, it can be seen that smaller unit cell sizes result in larger radiating passbands for both the LH and the RH bands. Here, the radiating passbands are defined as the frequency bands where the dispersion relation is within the radiation cone (see Figure 6.2). Therefore, from a radiation bandwidth perspective it is also advantageous to have a small unit cell size. However, if we have a look at the bottom two rows of Table 6.2, we can observe that the values of the loading inductance L increase monotonically as d h-tl is decreased, while the values of the loading capacitance C decrease as d h-tl is decreased to a minimum when d h-tl = 12 mm, and then increase when d h-tl is decreased further. The behaviour of C as d h-tl is varied can be attributed to the non-linear natures of both Equations (2.56) and (2.64) that are used to determine C. Since the practical realization of the loading elements was one of the major factors in designing the unit cell, it was therefore determined that a unit cell size of

181 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna Light Line Frequency (GHz) Operating Point π π/2 π/2 π β BL d (rad) Figure 6.8: Dispersion diagram for the NRI-TL metamaterial unit cell with values of L = 3.71 nh, C =.45 pf, Z = Ω, d h-tl =12mmandθ =95.4 at 5 GHz. d h-tl = 12 mm was a good compromise between the physically realizable loading elements in printed form and the performance of the NRI-TL metamaterial line. Furthermore, the length of d h-tl =12mmisequaltoλ /5atf = 5 GHz, which is relatively small compared to a free-space wavelength. Thus, the periodicity of the structure is sufficiently small, allowing the effective medium theory for periodic structures to be applied, although this is not a necessary condition for the operation of the metamaterial line. The dispersion diagram of the ideal MTM unit-cell of length d h-tl =12mmandwith loading element values of L =3.71nHandC =.45 pf is shown in Figure 6.8 and the corresponding group and phase velocity diagrams are shown in Figure 6.9. From Figure 6.8, the LH passband spans from 1.48 GHz to 2.61 GHz, while the RH passband spans from 3.99 GHz to 9.43 GHz. However, due to the higher value of the effective relative permittivity (compared to the previous case of ɛ eff = 1.2 shown in Figure 6.2), the dispersion curve becomes flatter, and therefore the radiation cone covers a smaller proportion of each band. In addition, for a given unit cell size, the shape of the dispersion curve is dictated by the specific phase shift Φ required at 5 GHz in order to produce a main beam at an angle of θ sc =+45.Inthiscase,Φ =.89 rad at 5 GHz from Equation (6.9), which can be verified in Figure 6.8. Both the aforementioned factors contribute to shrink the LH and the RH radiating passbands, thus reducing the frequency range

182 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna x 18 Velocity (m/s) 5 c f t2 v g Frequency (GHz) v φ Figure 6.9: Real part of the group velocity v g and phase velocity v φ for a line consisting of NRI-TL metamaterial unit cells with values of L =3.71nH,C =.45pF,Z = Ω, d h-tl =12mmandθ =95.4 at 5 GHz. that the structure can be used as a leaky-wave antenna. Albeit, for the specific structure under consideration and with the constraints of the substrate chosen, the RH radiating passband spans from3.99ghzto 6.58 GHz, which corresponds to an appreciable scanning bandwidth of 52%. The corresponding LH radiating passband spans from 2.34 GHz to 2.61 GHz, which is a very narrow band and therefore not worth pursuing. From Figure 6.9, the group velocity has values that are closest to the phase velocity in the upper region of the RH radiating passband, above the operating point of f =5GHz. This is the region within the radiation cone where both the phase and group velocities remain relatively constant with frequency, and have the most linear characteristic. Thus, it is expected that the scanning performance of the LWA will be better at frequencies above f compared to frequencies below f. As expected, the group velocity in the RH band is significantly higher than in the LH band, confirming that it is more advantageous to operate in the upper RH band.

183 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 163 Meandered Inductor, L Interdigitated Capacitor, C w s w y z x d = λ /5 Figure 6.1: Physical layout of the fully-printed CPS MTM unit cell. c 28 IEEE Physical Realization in CPS Technology The NRI-TL metamaterial unit cell was designed in view that many such unit cells would be cascaded in series to form the LWA. Considering the schematic of Figure 2.1(b), by placing two such unit cells in series, it can be observed that at the junction between the two unit cells there are two parallel connected 2L inductors. Hence, these two inductors were lumped together into one inductor of value L. At the two outer edges of the series connected unit cells, however, the original 2L inductors were retained in order to preserve the original circuit and therefore the propagation characteristics derived earlier. Thus, an n-element LWA would consist of n 2 identical internal unit cells each with two shunt inductors L and a series capacitor C, and two edge unit cells each with one shunt inductor 2L on one side and another shunt inductor L on the other side, together with the series capacitor C in the middle. Having previously established the size of the MTM unit cell and therefore the values of the loading elements, the next step was to realize the shunt inductors L and 2L and series capacitor C in printed form. Each of the loading elements was realized in printed form and was simulated separately in Agilent-Momentum, and an equivalent circuit for each was extracted using the simulated S-parameters at the design frequency of f = 5 GHz. The geometry of each of the meandered inductors and the interdigitated capacitors was then adjusted until the desired value of inductance and capacitance was achieved. The physical layout the final fully-printed MTM unit cell is shown in Fig-

184 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 164 Interdigitated Capacitor, C (top half) g 1 w 1 l 1 Meandered Inductor, L w 2 g 2 l 2 Meandered Inductor, 2L w 3 g 3 l 3 Figure 6.11: Geometrical details of the interdigitated capacitor C, and the meandered inductors L and 2L. c 28 IEEE. Table 6.3: Geometrical details of the fully-printed MTM unit cell. c 28 IEEE. CPS line Capacitor C Inductor L Inductor 2L d =12mm l 1 =3.673 mm l 2 =4.156 mm l 3 =7.558 mm w =2mm w 1 =.11 mm w 2 =.1 mm w 3 =.1 mm s =.9 mm g 1 =.1 mm g 2 =.1 mm g 3 =.1 mm ure 6.1, with the geometrical details of the loading elements 2L, L and C shown in Figure 6.11 and summarized in Table 6.3. In Figure 6.12 the performance of the fully-printed MTM unit cell is compared to that of the ideal circuit schematic of Figure 2.1(b). The printed unit cell used in the simulation was similar to the one shown in Figure 6.1, with the L inductors replaced

185 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 165 Phase of S21 ( ) Full wave Simulation Circuit Simulation Magnitude of S21 (db) Frequency (GHz) Figure 6.12: Phase of S 21 (top), and magnitude of S 21 (bottom) for a single MTM unit cell with parameters 2L =7.42 nh, C =.45 pf, Z = Ω,d h-tl =12mmandθ =95.4 at 5GHz. c 28 IEEE. by 2L inductors. In all the simulations the structure was excited and terminated with ports having an impedance equal to the Bloch impedance, Z bl = Ω. There is relatively good agreement between the full-wave data and the circuit simulation data, especially in the range between 4 to 6 GHz. Outside of this range, the performance of the printed components deviates from the expected design values. Specifically, between 6.5 and 7 GHz, the printed unit cell exhibits abnormal behaviour, which can be attributed to resonances formed within the interdigitated capacitors and/or the meandered inductors. Fortunately, these resonances emerge just above the useful RH radiating band and therefore do not affect the performance of the final LWA. The above observations, however, highlight the fact that the printed lumped-element components have a finite bandwidth of operation. The exact location of the stopband between the lower LH band and the upper RH band as predicted in Figure 6.8 is not evident from the phase response characteristic in Figure However, an observation of the magnitude response indicates the approximate location of the stopband by a drop in the value of S 21 between2and4ghz. A cascade of more identical unit cells would sharpen the cutoff between the passbands and the stopbands, revealing the exact locations of the transitions between the two.

186 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna α/k β/k Number of unit cells Figure 6.13: Normalized leakage constant α/k and propagation constant β/k as a function of the number of MTM unit cells in the LWA at f = 5 GHz. The results were obtained using a full-wave analysis in Agilent-Momentum. c 28 IEEE Determination of the Complex Propagation Constant As outlined in Section 6.2.1, an accurate knowledge of the complex propagation constant γ along the leaky guiding structure is crucial in the design of leaky-wave antennas. A small value of the leakage constant α, leads to a large effective aperture illumination, and a constant value of the propagation constant β, enables an accurate prediction of the location of the main beam. The analysis of a single unit cell in Agilent-Momentum, however, does not accurately predict the value of the leakage constant, due to mutual coupling between the loading elements and the host TL and edge effects due to the small size of the unit cell [119], [12]. Therefore, the leakage constant was empirically determined by simulating a number of arrays with an increasing number of unit cells until α converged to a constant value. Alternatively, α, andβ can be determined from the numerical analysis of a single unit cell using an FDTD approach [119] or a methodof-moments approach [128]. Figure 6.13 shows the normalized leakage constant α/k and propagation constant β/k as a function of the number of MTM unit cells in the LWA. It can be observed that the propagation constant remains constant at a value of approximately β/k =.75, regardless of the number of unit cells in the array. The leakage constant, however,

187 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 167 requires an array of approximately 3 unit cells long in order to converge to a value of α/k =.226. This value for the leakage constant is fixed for the antenna geometry presented, however it can be controlled by adjusting the number of shunt inductors per unit length, which act as the main radiating elements. A change in the number of shunt inductors per unit length implies a change in the elementary unit cell size, therefore the design procedure outlined in Section can be used to determine the required loading element values for the chosen unit cell size. The results presented above are consistent with the ones presented in [12], although it should be pointed out that the value of the normalized leakage constant for the metamaterial LWA is significantly lower than the value of α/k =.36 from [12]. Therefore, it is expected that the metamaterial LWA will have a larger effective aperture and therefore a sharper radiated beam for the same (but sufficiently large) number of unit cells. The value of α determined from Figure 6.13 was used to calculate the length of the LWA. To do so, the following expression for the power flow along a periodic structure with periodicity d was used [84] P n = P e 2αdn. (6.1) Here, P is the incident power at the input, and P n is the power at the n th terminal of the periodic structure. Ideally, all the input power should be radiated into free space before reaching the end termination, however, this would require an antenna that is infinitely long, which is impossible to implement practically. If the length of the antenna is chosen such that 9% of the input power leaks out into free-space before it reaches the end termination, then from Equation (6.1) and using the converged value for α/k =.226, this results in an antenna with n = 41 unit cells. Thus only 1% of the input power manifests itself as residual power that is absorbed by the end termination. Alternatively, if the end termination is removed, then the residual power that reaches the end of the antenna will be reflected by the open circuit and will result in a backward lobe in the antenna radiation pattern. However, because the residual power is only 1% of the input power, the backward lobe will be very small compared to the main forward lobe. Since the length of each unit cell was chosen to be d h-tl = 12 mm, the total length of an antenna comprising 41 unit cells would be 492 mm. However, due to limitations in the fabrication size, and in order to consistently replicate the smallest 1 μm features of the loading elements along the entire length of the structure, it was decided to implement a 2 unit cell design which was 24 mm long. For the 2 unit cell design, 68% of the input

188 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 168 Ground R θ r y TL 3 TL 2 TL 1 z x Microstrip CPS Figure 6.14: Broadband microstrip-to-cps balun transition from [129]. c 28 IEEE. power is radiated into free-space and the remaining 32% of the power is dissipated in the termination resistor. Therefore, even though the efficiency of the 2 unit cell LWA is expected to be less than that of the 41 unit cell antenna, it is still possible to investigate the beam scanning characteristics of the antenna, which was the main objective in this work Printed Balun Design In order to feed the CPS antenna, a broadband transition was required in order to transform the unbalanced signal from the coaxial feed line of the measurement equipment to the balanced CPS line. In addition, the transition had to also provide an impedance transformation from the Ω characteristic impedance of the CPS line to the 5 Ω feedline. This was achieved through a broadband microstrip-to-cps balun transition as shown in Figure 6.14, with the geometrical details of the balun shown in Table 6.4. The coaxial feedline was therefore connected to the CPS line through the intermediate microstrip transition. This topology for the balun, together with the complete design procedure was adopted from [129]. The transition of the electric field from a microstrip mode (vertical E-field) to a CPS

189 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 169 Table 6.4: Geometrical details of the fully-printed microstrip-to-cps balun. c 28 IEEE. TL 1 TL 2 TL 3 Radial Stub l =8.28 mm l =1.52 mm l =6mm R =1.64 mm w =.38 mm w =1.11 mm w =1.918 mm θ r =8 Z 1 = Ω Z 2 =69.91 Ω Z 3 =5Ω Microstrip ground plane: mm mode (horizontal E-field) was achieved through the use of a broadband radial stub with radius R and sector angle θ r. The radius was designed such that it is approximately equal to λ g /4atf = 5 GHz, where λ g is the guided wavelength of the microstrip TL. The quarter-wavelength radial stub is terminated in an open circuit, therefore it presents a virtual short circuit to the upper arm of the CPS line over a wide frequency range. Since the virtual short circuit is at the same potential as the microstrip ground plane, this allows the microstrip mode to easily transition to the CPS mode. The impedance transformation was achieved by a two-section λ g /4 impedance matching transformer comprising TL 1 and TL 2. Naturally, if more sections are added to the multi-section transformer, this will increase the impedance bandwidth of the entire balun, however a two-section transformer was deemed adequate considering the existing space constraints. TL 1 was designed to transform the Ω Bloch impedance of the CPS line to the intermediate impedance of Ω, and TL 2 was designed to transform the intermediate impedance of Ω to the feedline impedance of Z 3 =5Ω. Thus,the two λ g /4 impedance transformers took on the values of Z 1 = Ω and Z 2 =69.91 Ω respectively. In order to fully account for all the electromagnetic effects between the printed balun and the metamaterial line, two back-to-back baluns were connected through a two-stage MTM line, as shown in Figure 6.15, and the microstrip-to-cps-to-microstrip transition was then simulated in Agilent-Momentum. The reflection and transmission characteristics of the back-to-back balun are shown in Figure It can be observed that around f = 5 GHz, the balun is well matched, exhibiting a 1 db return-loss bandwidth of 2.38 GHz from 4.2 to 6.58 GHz. Within the same bandwidth, S 21 remains mostly between 2 and 3 db and drops to around 5 db at the edges of the band, exhibiting a bandpass characteristic. It should be noted that S 21 contains radiative losses, the losses for both the baluns as well as the two-stage MTM line. The finite bandwidth of

190 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 17 Figure 6.15: Back-to-back microstrip-to-cps balun transition connected through a two-stage MTM line. S11 (db) S21 (db) Frequency (GHz) Figure 6.16: Magnitude of S 11 and S 21 for the two back-to-back microstrip-to-cps baluns connected through a two-stage MTM line shown in Figure The results were obtained using a full-wave analysis in Agilent-Momentum. the balun can be attributed to the fact that the Bloch impedance of the MTM line varies as a function of frequency, and to the fact that the quarter-wavelength transformers TL 1 and TL 2 have a finite bandwidth of operation. Nevertheless, the balun still exhibits an appreciable S 11 fractional bandwidth of 47.6%.

191 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 171 Capacitor Inductor Figure 6.17: Photograph of the NRI-TL metamaterial leaky-wave antenna. The inset photograph shows a close-up view of the host CPS TL that is loaded with series interdigitated capacitors and shunt meandered inductors. c 28 IEEE. 6.4 Simulation and Experimental Results Return Loss The final 2-element metamaterial LWA consisted of 18 identical internal unit cells as shown in Figure 6.1(a) connected in series, and two edge unit cells with an outermost shunt loading inductor of value 2L. The array was terminated using a Ω resistor. The complete metamaterial LWA was fabricated at the University of Toronto laboratories and the working prototype is shown in Figure The LWA was fed using the printed balun of Figure 6.14, also seen on the left-hand-side of Figure 6.17, where a standard SMA connector was attached to the 5 Ω microstrip TL of the balun, allowing the antenna to be easily connected to the test equipment. A close-up view of the series interdigitated capacitor and the shunt meandered inductor can also be seen in the inset photograph of Figure The total size of the substrate that the antenna was printed on was 28 6 cm,

192 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna S11 (db) Frequency (GHz) Figure 6.18: Measured return-loss magnitude response of the 2-element metamaterial LWA, fed using the printed balun of Figure The LWA was backed by a finite ground plane (28 6cm)atadistanceofλ /4 = 15 mm. c 28 IEEE. and this was backed by the same size ground plane at a distance of λ /4=15mmbelow the antenna. The measured return-loss magnitude response of the metamaterial LWA is shown in Figure As shown, the antenna is well matched around the design frequency of f = 5 GHz, with a minimum S 11 value of 2 db at 4.9 GHz. It exhibits a 1 db S 11 bandwidth of.91 GHz, from 4.54 to 5.45 GHz, which corresponds to an 18.2% fractional bandwidth. When the same return-loss measurement was repeated without the ground plane, the bandwidth of the antenna increased slightly to.98 GHz or 19.6%. It was not possible to obtain a simulated return-loss response because the addition of the microstripto-cps balun to the MTM line exceeded the available computational resources Far-Field Radiation Patterns The metamaterial LWA was simulated in Agilent-Momentum using a differential port excitation, and the far-field radiation patterns were extracted. In order to achieve unidirectional patterns, an infinite ground plane was added at a distance of λ /4=15mm from the antenna for all the simulations. The printed balun was not included in the simulated design because, as mentioned above, the added complexity of the microstrip-to-cps transition exceeded the available computational resources.

193 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna (a) 4.5 GHz (b) 5. GHz (c) 5.5 GHz Figure 6.19: Simulated normalized co-polarization patterns (in db) in the xz-plane for the printed metamaterial LWA. c 28 IEEE (a) 4.5 GHz (b) 5. GHz (c) 5.5 GHz Figure 6.2: Measured normalized co-polarization patterns (in db) in the xz-plane for the printed metamaterial LWA. c 28 IEEE. The corresponding far-field patterns of the fabricated LWA prototype were measured in the anechoic chamber at the University of Toronto. Three representative simulated normalized co-polarization patterns are shown in Figure 6.19 for the selected frequencies of 4.5, 5 and 5.5 GHz, and the corresponding measured patterns are shown in Figure 6.2. The patterns were measured in the xz-plane (see the coordinate system shown in Figure 6.1), and are potted against the elevation angle, θ. The measured data agrees well with the simulated data, with excellent agreement as to the location and shape of the main beam. A small increase in the sidelobe levels of the measured patterns is observed at higher frequencies, which can be attributed to parasitic radiation from the printed balun.

194 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna Figure 6.21: Measured normalized co- and cross-polarization patterns in the xz-plane for the printed metamaterial LWA backed by a 28 6 cm ground plane at 5 GHz. Solid line: co-polarization, dashed line: cross-polarization. c 28 IEEE. Additionally, there is a small amount of radiation observed in the lower half- plane of the measured data, which is present due to the finite size of the ground plane. The polar plots shown, allow one to visualize how the beam evolves as the frequency is varied from the design frequency of f = 5 GHz. As the frequency is increased, the beam tends towards the endfire direction, while as the frequency is decreased the beam tends towards broadside. Nevertheless, within the demonstrated bandwidth of 1 GHz, the location of the main beam does not change significantly. Figure 6.21 shows both the measured co-polarization and the cross-polarization patterns in the xz-plane at the design frequency of f = 5 GHz. It can be observed that the main beam emerges at the angle of θ sc =45, which agrees exactly with the specified angle in the design procedure. This indicates that the fabricated MTM line was designed according to specification, exhibiting the appropriate propagation constant, β, and thus producing the correct scan angle according to Equation (6.3). From Figure 6.21, it can be observed that the measured cross-polarization remains below -12 db throughout most of the top-half plane, while the maximum level is at -1.6 db below the co-polarization peak. This level of cross-polarization is comparable to other printed LWAs reported in the literature [12], [13], [131], and would be useful for wireless telecommunications applications where polarization diversity is required. For the simulated LWA without the balun, the maximum cross-polarization was db below the co-polarization peak, indicating that the polarization purity of the measured LWA degrades to a large extent due

195 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna Scan angle, θ sc ( ) Theoretical Simulated Measured Frequency (GHz) 6 7 Figure 6.22: Scan angle versus frequency for the 2-element metamaterial LWA. The theoretical results were obtained by using Equation (6.3): θ sc =sin 1 ( β/k ), and the simulated results were obtained from Agilent-Momentum. c 28 IEEE. to parasitic radiation from the balun and to a smaller extent due to parasitic radiation from the series interdigitated capacitors and the shunt meandered inductors. Thus, if a different type of balun is chosen to feed the antenna the cross-polarization level could be reduced Beam Squinting and Gain Characteristics In order to analyze the beam squinting behaviour of the metamaterial LWA, the radiation patterns for both the simulated and measured LWAs were plotted in the xz-plane for a range of frequencies. This data was then analyzed, and the location and value of the peak of the main beam was recorded for each frequency step. The beam squinting results are shown in Figure 6.22, together with the theoretical scan angle versus frequency calculated using Equations (6.3) and (6.4), and the loading element values for d h-tl = 12mm from Table 6.2. It can be observed that there is excellent agreement between the theoretical, simulated and measured results. Additionally, the measured gain versus frequency results are shown in Figure 6.23, where it can be observed that the LWA exhibits a maximum gain of 12 dbi around the design frequency of 5 GHz. The simulated efficiency of the LWA at 5 GHz was 47.3% compared to a measured efficiency of 42.8%.

196 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna Measured Gain (db) Frequency (GHz) Figure 6.23: Measured gain versus frequency for the 2-element metamaterial LWA. Table 6.5: Comparison of LWA beam squinting characteristics This work [125] [126] [132] [133] Δθ sc /Δf ( /MHz) The scanning bandwidth of each array was determined by identifying the range of frequencies over which the peak of the main beam was clearly identifiable. For the simulated array, θ sc varied a total of 65 within a scanning bandwidth of 2.3 GHz, from 4 to 6.3 GHz. Since the scan angle versus frequency response is approximately linear within the scanning range of the array, the approximate variation of the scan angle with frequency can be written as Δθ sc /Δf =.28 /MHz. For the measured array, θ sc varied a total of 56 within a scanning bandwidth of 1.8 GHz, from 4.1 to 5.9 GHz. Therefore, the measured approximate variation of the scan angle with frequency is Δθ sc /Δf =.31 /MHz, which is quite close to the simulated case. The beam squinting characteristics of the metamaterial LWA can be compared to that of other printed LWAs found in the literature, as summarized in Table 6.5, where it can be verified that the metamaterial antenna consistently exhibits a smaller scan angle variation with frequency. The results of Figure 6.22 therefore confirm that a LWA with reduced beam squinting can be achieved when operating each of the constituent NRI-TL

197 Chapter 6. A NRI-TL Metamaterial Leaky-Wave Antenna 177 metamaterial unit cells in the upper RH region, where their phase response is the most linear, and most importantly, the phase and group velocities are as close as possible to the speed of light. Potential applications where near fixed-beam antennas in the 5 GHz range could be used include indoor cellular base stations, broadband wireless LAN terminals and short-distance point-to-point microwave links. The fully-planar, differential and broadband nature of the proposed metamaterial LWA therefore renders it ideal for use in the above mentioned applications.

198 Chapter 7 An Electrically Small NRI-TL Metamaterial Antenna In this chapter, an electrically small antenna is presented, which consists of four NRI-TL metamaterial unit cells. An improved model for analyzing such electrically small metamaterial antennas is proposed, that highlights the methods that enable these antennas to offer a good impedance match and a high radiation efficiency compared to previously reported designs. It is found that the antenna supports a predominately evenmode current on the vertical vias, allowing the antenna to be modeled using a multiarm folded monopole topology, which provides a substantial increase in the radiation resistance. This, together with the top-loading effect of the microstrip line on the vias, enables the antenna to be matched to 5 Ω without the use of an external matching network, while maintaining a high radiation efficiency. The validity of the proposed model is confirmed with a fabricated prototype, that consists of four microstrip metamaterial unit cells with dimensions of λ /1 λ /1 λ /2 over a.45λ.45λ ground plane. The antenna s performance is verified by full-wave simulations and experimental data obtained at 3.8 GHz, which yield a vertical linear electric field polarization, a measured 1 db return-loss bandwidth of 53 MHz and a measured efficiency of 7%. 7.1 Introduction The transmission-line approach to synthesizing negative-refractive-index metamaterials, has proven to be very attractive for the design of compact antennas that exhibit desirable characteristics, such as very small form factors, high efficiencies and multiple bands of 178

199 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 179 operation that are not harmonically related. By exploiting the fact that a zero-degree phase shift can be achieved independently of the length of the metamaterial line used, several researchers have developed small, zero-degree resonant antennas that act as very compact in-phase feed networks [73], [ ]. These designs, although compact, suffer from limited impedance bandwidths and/or low radiation efficiencies. Several researchers have also used metamaterial loading to significantly reduce the size of microstrip patch antennas [74], [76], [137], while others have used metamaterial loading to reduce the size of dipole and loop antennas, all at the expense of bandwidth and efficiency [75], [138]. Furthermore, antennas inspired by metamaterial concepts have been reported to be both electrically small and highly efficient, however these designs exhibit extremely narrow bandwidths [77], [139]. In this chapter, a compact metamaterial ring antenna is presented that consists of four microstrip metamaterial unit cells that are arranged in a ring configuration over a ground plane, as shown in Figure 7.1. A comprehensive model for analyzing this electrically small metamaterial antenna is presented, that highlights the methods by which this antenna offers a good impedance match and a high radiation efficiency compared to previously reported designs [73], [135], [137]. The model developed for the metamaterial antenna is based on the multiple monopole folding technique, which is a well established method of increasing the radiation resistance of electrically short monopoles. Implicit in the operation of the folding technique is the fact that all the monopoles must be fed in phase, and this is achieved in the current design by feeding four vertical monopoles (vias) using a compact feed network comprising four low-loss metamaterial unit cells, each designed to incur zero degrees. The antenna is fed at the base of one of the vias, and the metamaterial feed network ensures that the currents on the other three non-driven vias are in phase with the driven one. Even-odd mode analysis reveals that the antenna supports a predominately even-mode current on the vertical vias, therefore verifying that the vias are indeed fed in phase, and justifying the use of the folded monopole model. Furthermore, the microstrip implementation of the metamaterial feed network provides an effective top-loading of the four vias, which results in a constant current distribution along the vias, therefore maximizing their radiation resistance. The antenna can thus be effectively matched to 5 Ω over a reasonable bandwidth without the need for an external matching network, while maintaining a high radiation efficiency. In addition, by carefully adjusting the size of the microstrip ground plane, the use of an external balun can be avoided altogether.

200 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 18 Ground Plane Coaxial Feed (a) Shunt Inductor L Series Capacitor C 1.23mm C via via via via.4mm 4.8mm C L L L C L C MTM unit cell Coaxial Feed (b) (c) Figure 7.1: Electrically small NRI-TL metamaterial antenna: (a) 3D diagram including the ground plane and the coaxial feed cable, (b) 3D diagram showing the chip loading elements, and (c) Top view with geometrical details. c 28 IEEE. 7.2 Theory of Operation Proposed Topology The design of the small antenna was based on the symmetric NRI-TL metamaterial π unit cell shown in Figure 2.1(b). The π topology was chosen because it reveals the natural monopole folding effect that the circuit provides, which is achieved by feeding the antenna from the base of one of the inductors.

201 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 181 In order to feed each of the vertical vias of the antenna in phase, it is required that each metamaterial unit cell incurs at the design frequency, f. Therefore, under the assumption that the phase shift per unit cell and the size of each unit cell are small (which is indeed the case here), the phase shift per unit cell, φ mtm, is given by Equation (2.79). The impedance matching condition of Equation (2.71) is also imposed, which ensures that there is no stopband at the φ mtm = point between the lower left-handed band and the upper right-handed band. Thus, for a specified unit cell size, Equations (2.84) and (2.85) can be used to determine the loading element values of C and L required to produce a NRI-TL metamaterial unit cell Even-Odd Mode Analysis The proposed MTM antenna of Figure 7.1 can be analyzed by decomposing the antenna current into a superposition of an even-mode (I e ) and an odd-mode (I o ). Since the antenna is symmetrical, analysis of a single unit cell that contains two vias will reveal the general characteristics of the entire antenna, namely that the even-mode current is dominant on the structure. Even though the analysis of a single unit cell only models two folded arms (vias), nevertheless this provides a qualitative measure of the antenna performance. Furthermore, Goubau has analyzed the case of an N-element folded monopole where only one of the monopoles is excited, and has demonstrated that the results can be generalized from the two-element case [14], [141] (pp ). Beginning with the even-mode excitation, we apply two equal voltages (V/2) to each end of the metamaterial unit cell of Figure 2.1(b) as shown in Figure 7.2(a). Due to the symmetry of the unit cell, this effectively places an open circuit (O.C.) at the centre of the structure, thus producing two identical decoupled circuits. Note that in the circuit of Figure 7.2(a) the vertical vias connecting the inductors to ground have been explicitly included in order to highlight their importance as the main radiating elements in the structure. We can then replace each via with its corresponding radiation resistance R r and add its associated inductance L via to the lumped inductor value L, resulting in a total inductance of L. By replacing the short TL sections with their equivalent series inductance L = Ld and capacitance C = Cd, we obtain the circuit of Figure 7.2(b). No current will pass through the series resonator formed by L /2and2C because it is open circuited on one end, therefore these components can be removed from the circuit. At resonance

202 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 182 2L' Z, θ/2 C Z, θ/2 2L' (a) I e + via via O.C. V/2 V/2 I e + 2L L'/2 2C 2C L'/2 2L (b) I e + R r R r C'/2 C'/2 V/2 V/2 I e + (c) I e + R r V/2 V/2 R r I e + Figure 7.2: Even-mode equivalent circuits for a single unit cell of the MTM antenna: (a) TLbased MTM unit cell (b) Lumped-element equivalent circuit, and (c) Simplified lumped-element circuit at resonance. c 28 IEEE. (when φ mtm = ), the series resonator formed by 2L and C /2willbecomeashort circuit, therefore resulting in the final simplified circuit of Figure 7.2(c). Since no other impedances are present in the circuit other than R r, maximum current will be delivered to the radiation resistance, given by I e =(V/2)/R r. If we now consider the odd-mode excitation, we apply an equal and opposite voltage to each end of the metamaterial unit cell as shown in Figure 7.3(a). Due to the symmetry of the unit cell, this effectively places a short circuit (S.C.) at the centre of the structure, thus producing two identical decoupled circuits with equal and opposite excitations. If we then replace the vias with their corresponding radiation resistance R r and inductance L via and the short TL sections with their equivalent series inductance L = Ld and capacitance C = Cd, we obtain the circuit of Figure 7.3(b). At resonance, the series resonator formed by L /2and2C will become a short circuit, thus shorting out C /2and therefore resulting in the final simplified circuit of Figure 7.3(c). It can now be observed

203 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 183 2L' Z, θ/2 C Z, θ/2 2L' (a) I o + via via S.C. V/2 V/2 I o + 2L 2L (b) I o + R r L'/2 L'/2 C'/2 C'/2 V/2 2C 2C V/2 R r I o + 2L 2L (c) I o + R r V/2 V/2 R r I o + Figure 7.3: Odd-mode equivalent circuits for a single unit cell of the MTM antenna: (a) TLbased MTM unit cell (b) Lumped-element equivalent circuit, and (c) Simplified lumped-element circuit at resonance. c 28 IEEE. that the odd-mode current will be given by I o =(V/2)/(R r + jω2l ). Therefore, the 2L inductor plays a pivotal role in adjusting the odd-mode current, and for any value of the loading inductor 2L, I e will always be greater than I o. Considering some representative values, a short monopole with a uniform current distribution will have a radiation resistance of [142] R r = 16π 2 (h/λ) 2. (7.1) A close approximation to a uniform current distribution can be achieved by top loading the monopole with a metal plate, something that is inherently achieved in the microstrip design shown in Figure 7.1. Therefore for a monopole with h/λ = 1/2, the radiation resistance from Equation (7.1) will be R r = 4 Ω. Assuming that the excitation voltage is V = 1 Volt, for a typical value of L = 2 nh, at 3 GHz the magnitude of the odd-mode current is I o =.66 ma. The corresponding even-mode current is I e = 127 ma, which is approximately 19 times greater than I o. Therefore, since the majority of the current

204 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 184 flowing on the antenna lies in the even-mode, this enables the metamaterial structure to act as a good radiator. From the above discussion it can be seen that a short monopole has a radiation resistance in the range of a few ohms, which makes it hard to match to 5 Ω. A simple way to increase the input impedance of a resonant antenna is to use a multiple folding technique [143], which relies on the fact that the currents in each arm of the antenna are in phase. As has been demonstrated with the even-odd mode analysis, this is naturally achieved with the NRI-TL metamaterial unit cell. For an antenna with N folded arms, the input impedance is given by [143] R in N 2 R r. (7.2) Thus, an antenna with N = 4 folded arms that are each λ/2 long and have R r =4Ω, would have an input impedance of 64 Ω. This is sufficiently close to 5 Ω to provide a good match, therefore the height of the antenna was chosen to be h = λ/2 = 5 mm. 7.3 Design The antenna was initially designed in microstrip technology at a design frequency of f = 3 GHz. In order to ensure that the design was electrically small, Wheeler s condition for small antennas was imposed, namely that the radius of the imaginary sphere enclosing the antenna, a, islessthanλ/4π [144]. The length of the antenna was chosen to be 1 mm, resulting in a radius of a =7.7mm=λ/14.14 at 3 GHz, which satisfies Wheeler s small antenna criterion. Note that the aforementioned antenna size does not include the ground plane. The reasoning behind this is that the hemisphere enclosing a monopole element above a ground plane would have the same radius as a sphere enclosing an equivalent dipole created by the original monopole and its mirror image below the ground plane. Since the dipole configuration no longer includes the ground plane, yet is enclosed by the same size sphere as the monopole, the radius of this sphere can be used to describe the size of the antenna, and need not include the size of the ground plane [141] (p. 56), [143]. The lower limit on the quality factor, Q, for a linearly polarized antenna was then used to find the maximum fractional bandwidth that an antenna with the above size

205 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 185 could attain [145], [146]: BW Q 1 = 1 ( 1 η rad k 3 a + 1 ) 1. (7.3) 3 ka Thus, for the antenna under consideration, for the lossless case where the radiation efficiency η rad = 1, the upper limit on the achievable bandwidth is 7.3%. However, since the antenna was implemented in planar microstrip technology, it did not effectively fill the 4 3 πa3 spherical volume, and thus as will be seen in Section 7.4, the actual bandwidth was less than 7.3% Physical Implementation Each metamaterial unit cell was implemented in microstrip technology, with the top copper strips printed on a thin.38 mm layer of Rogers RT588 which was suspended in air 4.62 mm above the ground plane, resulting in a total height of 5 mm. Since the length of the antenna was chosen to be 1 mm, this also fixed the length of each unit cell to 5 mm. Furthermore, a.4 mm gap was cut out of each side of the antenna in order to place the series capacitor, therefore each copper strip of the host microstrip transmission line had a size of 4.8 mm 4.8 mm, resulting in a characteristic impedance Z = Ω and a phase incurred by the host TL of φ h-tl = Using the above parameters, together with φ mtm = in Equations (2.84) and (2.85), results in approximate ideal loading-element values of C =1.36pFandL = nh for a metamaterial unit cell. The exact values were also calculated using Equations (2.86) and (2.88) using the same parameters, and were found to be C =1.35 pf and L =21.5 nh, which are quite close to the approximate ones. With the prospect of realizing the loading-element values of C = 1.35 pf and L =21.5 nh using chip lumped-element components, these were then tested in each of their respective microstrip environments in Ansoft HFSS, in order to account for any additional parasitic effects introduced by the microstrip structure. To test the series loading capacitance C, a single 1 mm long microstrip line was designed with a.4 mm gap in it, across which an ideal lumped capacitance, C L, was placed, which is analogous to placing the lumped capacitance along one of the sides of the MTM antenna of Figure 7.1. The test fixture used to determine the total capacitance of this configuration is shown in Figure 7.4. The presence of the gap in the microstrip line introduces an additional capacitance term, C gap, which appears in parallel with the lumped capacitance C L.Thus,

206 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 186 Ground Copper C L C gap RT Port 2 Air 4.62 Port 1 Figure 7.4: Microstrip test fixture used in Ansoft HFSS to determine the total effective series capacitance C using an ideal lumped capacitance C L. Note that Port 1 and Port 2 are not drawn to scale. All the dimensions are in mm. by de-embedding to the edges of the gap in the microstrip line, the total capacitance of this arrangement is given by C = C tot = C L + C gap. (7.4) The Y-parameter data from the simulation were used to construct an equivalent π- representation of the two-port network [84] (p. 188), and the total capacitance was extracted from this data using C tot = Im( Y 12). (7.5) ω The value of C L was subsequently tuned, until the total capacitance was equal to the desired value of C =1.35 pf. Therefore, the final value of the lumped capacitance required to produce a total capacitance of C =1.35 pf at f =3GHzwasC L =.9pF. The shunt loading inductance, L, was tested in an analogous way using the test fixture shown in Figure 7.5, where an ideal lumped inductance, L L, was connected to the ground through a via with a diameter of.51 mm. In order to avoid placing the shunt lumped-component vertically, which would significantly increase the fabrication complexity, a mm hole was cut out of the microstrip line, enabling the shunt component to be mounted horizontally between the microstrip conductor and the top of the via. During the fabrication process, this would allow the chip lumped-element to be easily soldered to the surface of the antenna, while still providing a shunt connection of

207 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna Ground 4.8 L L Copper 4.8 RT Port 1 L via Port 2 Air Figure 7.5: Microstrip test fixture used in Ansoft HFSS to determine the effective shunt inductance L using an ideal lumped inductance L L. Note that Port 1 and Port 2 are not drawn to scale. All the dimensions are in mm. the component through the via to the ground. The shunt via introduces an additional inductance term, L via, that appears in series with the lumped inductance L L. In addition, there is another parasitic inductance term, L para, which arises due to the additional complexity of the circuit associated with the lumped inductance being placed on the surface of the circuit. Thus, by de-embedding to the centre of the microstrip line, the total inductance of the arrangement is given by L = L tot = L L + L via + L para. (7.6) The Z-parameter data from the simulation were used to construct an equivalent T - representation of the two-port network [84] (p. 188), and the total inductance was extracted from this data using L tot = Im(Z 12). (7.7) ω The value of L L was subsequently tuned until the total inductance was equal to the desired value of L =21.5 nh. The final value of the lumped inductance required to produce a total inductance of L =21.5 nhatf =3GHzwasL L =11.7 nh. Through further simulations, it was determined that the shunt via introduced an additional inductance of approximately L via = 2.2 nh. Therefore, the additional parasitic inductance L para, introduced by the hole in the microstrip line, contributed an extra 7.6 nh to the total inductance. This value might initially appear to be quite large, but it can be justified by taking into account the capacitance formed between the outer microstrip ring and the

208 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 188 small patch that is connected to the top of the via, which resonates with the lumped inductance L L, thus increasing the effective shunt inductance of this configuration. This self resonance will be subsequently shown in Figure 7.1(b), where a lossy model for the lumped inductance is considered. To summarize, within the microstrip environment considered, ideal lumped-element values of C L =.9pFandL L = 11.7 nh were required to produce the desired loading element values of C =1.35 pf and L =21.5 nh that when used within the predescribed metamaterial unit cell produce a phase shift. The ideal lumped-element values were subsequently realized by selecting the appropriate low-loss chip components from various manufacturers. It was decided to use 42-ECD-GE series multilayer ceramic chip capacitors from Panasonic and 32CS series wirewound chip inductors from Coilcraft. The S-parameter files containing all the parasitics for each chip component were obtained from their manufacturers, and the componentsweretestedinagilentadsinorder to extract the equivalent capacitance or inductance of each chip at the desired frequency of f = 3 GHz. Subsequently, an equivalent circuit model of each component was developed that was used in the HFSS simulations in order to account for the non-idealities of the chip components. This was necessary because HFSS does not allow S-parameter definition files to be defined within its simulations, therefore the equivalent circuit of each chip element had to be defined as a series of ideal lumped-element components. The Panasonic component ECD-GER89 with a nominal capacitance of.8 pf was selected as the best candidate to realize the lumped-element value of C L =.9pF. The equivalent circuit of the chip capacitor is shown in Figure 7.6(a), and was obtained by curve fitting the response of the equivalent circuit to that produced by the S-parameter file. For the ECD-GER89 capacitor, the extracted component values were L P =.68 nh, C P =.8 pf,andr P =.38 Ω. Figure 7.6(b) shows the effective lumped capacitance obtained from both the Panasonic S-parameter model file and the equivalent circuit model of Figure 7.6(a). The capacitance was obtained by performing a two-port S-parameter simulation on each of the two circuits as shown in Figure 7.6(a), and Equation (7.5) was then used to extract the capacitance. It can be observed that the two responses are almost identical, producing an effective lumped capacitance of.99 pf at 3 GHz. It can also be observed that the capacitor s self-resonant frequency (SRF) is around 6.78 GHz, which is well above the operating frequency of f =3GHz.

209 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna Panasonic Capacitor L P C P R P Capacitance (pf) 1 1 Port 1 5 Ω (a) Port 2 5 Ω 2 S parameter File Equivalent Circuit Frequency (GHz) (b) Figure 7.6: (a) Equivalent circuit of a Panasonic ECD-GER series chip capacitor. The extracted component values for the ECD-GER89 chip capacitor were L P =.68nH,C P =.8 pf, and R P =.38 Ω. (b) Effective lumped capacitance of a Panasonic ECD-GER89 chip capacitor extracted from both the S-parameter file and the equivalent circuit of (a). The Coilcraft component 32CS-9N2XJL with a nominal inductance of 9.2 nh was selected as the best candidate to realize the lumped-element value of L L = 11.7 nh. The equivalent circuit of the chip inductor is shown in Figure 7.7(a), and was obtained by curve fitting the response of the equivalent circuit to that produced by the S-parameter file. For the 32CS-9N2XJL inductor, the extracted component values were C C =.45 pf, R C =1.23 Ω, and L C =9. nh. Figure 7.7(b) shows the effective lumped inductance obtained from both the Coilcraft S-parameter model file and the equivalent circuit model of Figure 7.7(a). The inductance was obtained by performing a two-port S-parameter simulation on each of the two circuits as shown in Figure 7.7(a), and Equation (7.7) was then used to extract the inductance. It can be observed that the two responses are almost identical, producing an effective lumped inductance of 1.51 nh at 3 GHz. It can also be observed that the inductor s self-resonant frequency is around 7.9 GHz, which is well above the operating frequency of f =3GHz. The equivalent circuit models that were developed for each of the Panasonic and Coilcraft chip components were subsequently inserted back into their respective microstrip test fixture environments in order to verify that the desired effective loading values of C =1.35 pf and L =21.5 nh were still obtained. Figures 7.8 and 7.9 show the

210 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 19 Coilcraft Inductor C C R C L C Inductance (nh) 1 1 Port 1 5 Ω Port 2 5 Ω 2 3 S parameter File Equivalent Circuit Frequency (GHz) (a) (b) Figure 7.7: (a) Equivalent circuit of a Coilcraft 32CS series chip inductor. The extracted component values for the 32CS-9N2XJL chip inductor were C C =.45 pf, R C =1.23Ω, and L C = 9. nh. (b) Effective lumped inductance of a Coilcraft 32CS-9N2XJL chip inductor extracted from both the S-parameter file and the equivalent circuit of (a). implementations of the equivalent circuits of Figures 7.6(a) and 7.7(a) within each of the capacitor and inductor microstrip test fixtures, respectively. Using the same procedure described earlier, the total capacitance and inductance were extracted for each of the circuits, and these are shown as a function of frequency in Figures 7.1(a) and (b) respectively. The total capacitance at 3 GHz was C = 1.42 pf, while the total inductance at 3 GHz was L =21.1nH. Even though these values were not exactly equal to the values of C =1.35 pf and L =21.5 nh which are required to produce an ideal metamaterial unit cell, they were deemed to be adequate considering the availability of the chip elements only in discrete predetermined values. As can be observed from both Figures 7.1(a) and (b), with the use of the equivalent circuits within the microstrip environment, the self-resonant frequencies of both components were significantly reduced compared to the equivalent circuits tested in isolation. The SRF of the capacitor was reduced to 4.8 GHz, while the SRF of the inductor was reduced to 4.2 GHz. Nevertheless, the values of C and L still remained relatively constant around the design frequency of f = 3 GHz, therefore the frequency limitations of the lumpedcomponents were not expected to adversely affect the operation of the antenna around the design frequency. The microstrip structures of Figures 7.8 and 7.9 were then combined into a single

211 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 191 Ground Copper Port 2 RT588 Air L P C P R P Panasonic Capacitor Model 4.62 Port 1 Figure 7.8: Microstrip test fixture used in Ansoft HFSS to determine the total effective series capacitance C using the Panasonic equivalent circuit model shown in Figure 7.6(a). Note that Port 1 and Port 2 are not drawn to scale. All the dimensions are in mm. Ground 4.8 Copper RT Port 1 Via Port 2 Air L C R C C C Coilcraft Inductor Model Figure 7.9: Microstrip test fixture used in Ansoft HFSS to determine the total effective shunt inductance L using the Coilcraft equivalent circuit model shown in Figure 7.7(a). Note that Port 1 and Port 2 are not drawn to scale. All the dimensions are in mm.

212 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna Effective Capacitance, C (pf) Effective Inductance, L (nh) Frequency (GHz) Frequency (GHz) (a) (b) Figure 7.1: (a) Total capacitance of the Panasonic ECD-GER89 capacitor equivalent circuit within its microstrip test fixture shown in Figure 7.8. (b) Total inductance of the Coilcraft 32CS-9N2XJL inductor equivalent circuit within its microstrip test fixture of Figure 7.9. Panasonic Capacitor Model Coilcraft Inductor Model 5 mm Port 2 Reference Plane 2 Port 1 Reference Plane 1 Figure 7.11: Metamaterial unit cell implemented in Ansoft HFSS, which includes equivalent circuit models for the Panasonic and Coilcraft chip loading elements. Note that P1 and P2 not drawn to scale. metamaterial unit cell as shown in Figure 7.11, in order to verify that the above-calculated values for C and L would indeed result in a insertion phase. The two simulation ports were each de-embedded by 2.4 mm into the reference planes shown in Figure 7.11,

213 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna Phase of S21 ( ) Magnitude of S21 (db) Frequency (GHz) (a) Frequency (GHz) (b) Figure 7.12: (a) S 21 phase response, and (b) S 21 magnitude response of the metamaterial unit cell of Figure 7.11 with a.8 pf Panasonic ECD-GER89 capacitor and a 7.2 nh Coilcraft 32CS-7N2XJ inductor. in order to effectively simulate a single 5 mm unit cell. The complex interactions between the various components did not initially result in a insertion phase, therefore the capacitance and inductance were varied within the discrete values available from the manufacturers, until the desired insertion phase was obtained. It was found that this was effectively achieved by replacing the 9.2 nh Coilcraft inductor with a 7.2 nh Coilcraft inductor. Therefore, a Coilcraft 32CS-7N2XJL chip inductor with a nominal inductance of 7.2 nh and with equivalent-circuit parameters of C C =.419 pf, R C =1.13Ω,and L C = 7.2 nh, and an SRF of 9.16 GHz, was used within the metamaterial unit cell of Figure The phase and magnitude responses of the metamaterial unit cell of Figure 7.11 with a.8 pf Panasonic ECD-GER89 capacitor and a 7.2 nh Coilcraft 32CS-7N2XJ inductor are shown in Figures 7.12(a) and (b) respectively. It can be observed that the metamaterial unit cell indeed exhibits the desired phase characteristics around 3 GHz. Exactly at 3 GHz, the phase incurred is φ mtm =2.85, which is very close to the desired value of. The phase passes exactly through zero slightly above the design frequency, at 3.13 GHz. At this point, the insertion loss is.13 db, which is quite reasonable considering the lossy nature of the unit cell considered. As will be subsequently shown, this slight frequency shift in the zero insertion phase of the individual metamaterial unit cell also resulted in a similar shift in the response of the entire antenna, but did

214 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 194 not affect its overall performance. Also apparent in the responses of Figures 7.12 is a lossy resonance region that occurs at around 6.6 GHz, which can be attributed to the self resonances of the individual loading components within the unit cell. Fortunately, this resonance is well above the operating region of the antenna, and does not affect its operation. 7.4 Simulation and Experimental Results Having verified that each constituent unit cell of the metamaterial antenna would produce a phase shift, the complete metamaterial antenna was then designed in HFSS using four of the metamaterial unit cells of Figure 7.11 arranged in a ring configuration, as shown in Figure 7.1. Initially the antenna was designed with an infinite ground plane in order to provide a reference for the subsequent comparison of antennas on finite-sized ground planes. The ground plane size was then gradually reduced in order to investigate the effect on the input impedance of the antenna. It was found that by decreasing the ground plane size from L g W g = 1 1 mm to 3 3 mm, the input impedance at resonance decreased from 8.2 Ω to 48.4 Ω. A further reduction in the ground plane size to below 2 2 mm, resulted in an increase of the input impedance at resonance to values above 85 Ω. The change in the input impedance of the folded monopole metamaterial antenna as a function of the ground plane size is consistent with the data reported in the literature for various shapes and sizes of ground planes [112], [141] (Chapter 8), [147] (Chapter 3). It was also observed that for ground plane sizes less than 4 4 mm, parasitic currents began to appear on the outer conductor of the coaxial cable feeding the antenna, which became stronger as the ground plane size was reduced. This was an indication that the unbalanced current leaking onto the exterior of the coaxial feed cable would cause undesired radiation from the cable. Figure 7.13 shows the surface current density on the exterior of the coaxial cables feeding three metamaterial antennas with ground plane sizes of 2 2 mm, mm and 1 1 mm. In all three cases the current density scales have been set to span from.1 to 2 A/m. It can be observed that for the smallest ground plane size of 2 2 mm shown in Figure 7.13(a), the current on the exterior of the cable is relatively strong compared to the current shown in Figure 7.13(c) for the largest ground plane, which is negligible. The current on the coaxial cable feeding the antenna with a mm ground plane shown in Figure 7.13(b) is slightly stronger

215 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 195 (a) L g W g =2 2 mm (b) L g W g =45 45 mm (c) L g W g = 1 1 mm Figure 7.13: Simulated surface current density on the exterior of the coaxial feed cables of three metamaterial antennas with different representative ground plane sizes at 3.8 GHz. Note that the strength of the current is proportional to the size of the current vectors depicted. than the one shown in Figure 7.13(c), however it is still significantly weaker than the one shown in Figure 7.13(a), indicating that even for a moderately sized ground plane of mm, currents are not strongly excited on the exterior of the feed cable. This

216 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna (a) (b) 15 Figure 7.14: (a) Simulated surface current density on the four vertical vias and the exterior of the coaxial feed cable of the metamaterial antenna with a mm ground plane at 3.8 GHz. Note that the strength of the current is proportional to the size of the current vectors depicted. (b) Simulated E-plane (xz-plane) patterns at 3.8 GHz displayed on an un-normalized, linear scale. Solid line: co-polarization, dashed line: cross-polarization. can also be verified by referring to the current density scale shown in Figure 7.13, which indicates that the currents on the cable in Figure 7.13(b) are approximately two orders of magnitude less than the ones of Figure 7.13(a). In view of the above observations, it was therefore decided to truncate the ground plane at an intermediate size of mm, where minimal currents were observed on the coaxial cable and a reasonable input impedance was obtained. The electrical size of a45 45 mm ground plane is approximately λ /2 λ /2 at the design frequency of f = 3 GHz, which adheres to the rule of thumb provided on p. 56 of [141] for the size of finite rectangular ground planes of small top-loaded monopole antennas. By having a sufficiently large ground plane that prevents the undesired currents from flowing on the exterior of the cable, this also eliminates the need for an external balun placed beneath the antenna, which would otherwise act to choke these currents. The current distribution on the four vertical vias and the exterior of the coaxial feed cable of the metamaterial antenna with a mm ground plane is shown in Figure 7.14(a). The current density scale in this case spans from.1 to 1 A/m. It

217 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 197 can be observed that the currents on all four vias are in phase and they are significantly stronger than the current on the coaxial feed cable, which in relative terms appears to be negligible. Additionally, if one considers that the surface area of the coaxial cable is approximately four times larger than the surface area of each individual via, this would result in a current density on the coaxial cable that is four times smaller than on each via for the same current amplitude. Nevertheless, observation of the current density scale indicates that the current density on the coaxial cable is close to three orders of magnitude less than on the vias. The above observations confirm that the metamaterial antenna indeed operates as outlined in Section 7.2.2, where the in-phase even-mode currents on the four vias act as the main source of radiation, and the contribution of the currents on the coaxial cable are minimal. Figure 7.14(b) shows the simulated E-plane patterns in the xz-plane for the metamaterial antenna, where these have been plotted on a linear scale in order to emphasize the locations of maximum radiation. It can be observed that most of the radiation is concentrated in the upper-half plane, and the antenna exhibits an almost purely vertical polarization with very low cross-polarization levels. Due to the finite size of the ground plane, however, there is still some radiation present in the lower-half plane, which is accounted for in part by diffraction at the ground plane edges [112]. It should also be noted that the radiation pattern of Figure 7.14(b) is comparable to the pattern of a quarter-wavelength monopole on a ground plane of approximately the same size depicted in Figure 8(b) of [112]. The current distributions on the top and bottom of the finite-thickness ground plane of the metamaterial antenna with a size of mm are shown in Figure The current density scale in this case spans from.1 to 2 A/m. Note that the size of the current vectors has not been scaled according to their strength in order to facilitate the visualization of the current, which is quite weak at the edges and the bottom of the ground plane. The radial flow of the currents outward from the via sources is clearly visible in Figure 7.15(a). It is interesting to note that the current on the bottom side of the ground plane in Figure 7.15(b) is also radially directed outwards from the centre, which is a direct consequence of the size of the λ /2 λ /2 ground plane. Nevertheless, the current changes direction along the outer edges in order to satisfy the boundary conditions, which dictate that the net current is equal to zero along these edges. The currents on the bottom side of the ground plane form a partial image below the ground, thus accounting for part of the radiation in this region as indicated by Figure 7.14(b).

218 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 198 (a) Top currents (b) Bottom currents Figure 7.15: Simulated surface current density on the top and bottom of the finite-thickness ground plane ( mm) of the metamaterial antenna at 3.8 GHz. Note that in this case the strength of the current is not proportional to the size of the current vectors depicted. The image is only partially formed because the currents along the top and along the edges of the ground radiate, which results in the currents on the bottom of the ground having a much smaller magnitude than the ones on the top. This is also evident from the colour scale shown in Figure The metamaterial antenna was subsequently constructed at the University of Toronto, and was populated with four.8 pf Panasonic ECD-GER89 capacitors and four 7.2 nh Coilcraft 32CS-7N2XJ inductors. Photographs of the fabricated antenna prototype are shown in Figure As can be seen in the figures, four plastic screws were used to support the top thin layer of RT588 above the ground plane, which were used only for structural support of the antenna and did not affect its performance. Figure 7.17 shows the simulated and measured return loss for the antenna obtained from HFSS simulations and lab measurements, respectively. It can be observed that the antenna is well matched around 3.8 GHz, with a simulated return loss bandwidth below 1 db of 42 MHz, and a measured return loss bandwidth of 53 MHz, which corresponds to approximately 1.7%. As explained in the discussion of Figure 7.12, the slight shift in the operating frequency from 3 to 3.8 GHz can be attributed to the fact that each constituent unit cell of the metamaterial antenna incurs a insertion phase around 3.8 GHz, and not 3 GHz. This slight shift in operating frequency was caused because the chip lumped-element values were available only in predetermined discrete increments

219 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 199 z x y 5mm 45mm 45mm Coaxial Feed (a) (b) Figure 7.16: Fabricated prototype of the electrically small NRI-TL metamaterial antenna: (a) Perspective view, (b) Side view. c 28 IEEE. 5 Magnitude of S11 (db) Measured Simulated Frequency (GHz) Figure 7.17: Simulated and measured return-loss for the metamaterial antenna. c 28 IEEE. that did not match exactly with the ideal values that were required to produce a phase shift of exactly at 3 GHz, thus producing the phase shift at a slightly higher frequency. Figure 7.18 shows the simulated and measured input impedance of the antenna, which match quite favourably. At the new resonant frequency of 3.8 GHz, the real part of the input impedance was 58.3 Ω in simulation and almost identically 58.2 Ω in measurement, enabling it to be well matched to the 5 Ω feed line. Comparing the input impedance re-

220 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna x x Zin ( Ω).5 Zin ( Ω) Real (Zin) Imag (Zin) Frequency (GHz) Real (Zin) Imag (Zin) Frequency (GHz) (a) (b) Figure 7.18: (a) Simulated, and (b) Measured input impedance for the metamaterial antenna. sponse of the metamaterial antenna to a conventional folded dipole or monopole antenna [19] (p. 463), [143], it can be observed that the two exhibit very similar characteristics. Both are inductive at low frequencies, and as the frequency is increased they both exhibit an anti-resonance, followed by a resonance and then another anti-resonance. These characteristics are also very similar to those of a conventional loop antenna [19] (p. 227), and in fact a folded dipole antenna can be considered a version of a compressed loop antenna. Antennas with these types of characteristics are typically operated at the first resonance where the input resistance is relatively constant and close to 5 Ω and the slope of the input reactance does not change rapidly with frequency. It is interesting to note that the highly cited Goubau antenna [14], [148], also has similar input impedance characteristics to the ones shown in Figure The Goubau antenna is essentially a segmented disc-loaded folded monopole antenna, where each of the four segments that load the short monopoles are interconnected with series wire inductors. It achieves a very wide impedance bandwidth of approximately an octave by using top-loading segments of different sizes, different radii for the driven and undriven monopoles, as well as different inductance values for the various interconnecting wire inductors. However, it cannot be classified as an electrically small antenna, since it has a size of ka = 1.4 [149]. In contrast, the metamaterial antenna presented herein exhibits

221 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna 21 only a 1.7% bandwidth but is electrically small with a size of ka =.44. It is therefore envisioned that future versions of the small metamaterial antenna could incorporate some of the techniques used to increase the bandwidth of the Goubau antenna, while keeping the electrical size of the antenna small. Figures 7.19 and 7.2 show the simulated and measured E-plane and H-plane patterns for the metamaterial antenna at 3.8 GHz. It can be observed that the antenna exhibits a radiation pattern with a vertical linear electric field polarization, similar to that of a short monopole over a small ground plane. The simulated directivity and gain obtained from HFSS at 3.8 GHz were 1.31 (1.18 dbi) and.95 (.24 dbi) respectively, resulting in a simulated efficiency of 72.3%. The measured efficiency was calculated using the gain/directivity method described in Appendix D, which resulted in a measured efficiency of 69.1% at 3.8 GHz, from a measured directivity of 1.78 (2.5 dbi), and a measured gain of 1.23 (.9 dbi). In order to verify the previous measurements, the efficiency of the antenna was also measured using a modified Wheeler cap method described in Appendix E, which resulted in a measured efficiency of 69.8% at 3.8 GHz. The two measurements therefore validated the expected result from the simulations. To summarize the findings of this chapter, an improved model for analyzing electrically small NRI-TL metamaterial antennas has been presented, that highlights the methods that enable these antennas to offer a good impedance match and a high radiation efficiency. Even-odd mode analysis was applied to the antenna and it was demonstrated that the structure acts as a good radiator due to the prevalence of the even-mode current. The antenna can be easily matched to 5 Ω due to the inherent top-loading and multiple folding of the short monopoles used in the antenna. In addition, the antenna exhibits a vertical linear electric field polarization with a relatively high radiation efficiency of 7% and a measured return-loss bandwidth of 53 MHz. Its compact size and good electrical performance therefore make it well suited for emerging wireless applications.

222 Chapter 7. An Electrically Small NRI-TL Metamaterial Antenna (a) (b) Figure 7.19: E-plane (xz-plane) patterns for the metamaterial antenna at 3.8 GHz: (a) Simulated and (b) Measured. Solid line: co-polarization, dashed line: cross-polarization. c 28 IEEE (a) (b) Figure 7.2: H-plane (xy-plane) patterns for the metamaterial antenna at 3.8 GHz: (a) Simulated and (b) Measured. Solid line: co-polarization, dashed line: cross-polarization. c 28 IEEE.

223 Chapter 8 Conclusion 8.1 Summary In this thesis, the salient features of one-dimensional negative-refractive-index transmission-line (NRI-TL) metamaterials have been presented, with the goal of demonstrating their effective use in various microwave and antenna applications. The properties of conventional positive refractive index (PRI) materials were first presented, followed the properties of the newly-developed negative refractive index (NRI) materials. The PRI and NRI materials were subsequently combined together to form a NRI-TL metamaterial medium that has the form of a host transmission line loaded with lumped-element series capacitors and shunt inductors in a high-pass configuration. It was demonstrated that the NRI-TL metamaterial medium possesses an inherent phase compensating property, which enables the creation of very compact and broadband phase shifting lines. Two elementary phase-compensating NRI-TL metamaterial unit cells were subsequently presented, one in a T configuration and another in a π configuration, and their propagation and impedance characteristics were explored. This was followed by a technique to ensure that the metamaterial lines operate in the slow-wave regime, thus creating non-radiating metamaterial phase-shifting lines. An in-depth analysis of various zero-degree metamaterial phase-shifting lines was then presented, including the factors affecting the loading-element values used, the optimal choice of the number of unit cells, and the phase variation of each line with respect to frequency. The NRI-TL metamaterial lines were first used to implement a broadband metamaterial balun, that employs a +9 metamaterial phase-shifting line along one of the output 23

224 Chapter 8. Conclusion 24 branches, and a 9 metamaterial phase-shifting line along the other output branch. This allows the shape of the two phase responses to be matched, while maintaining a constant phase difference of 18 over a large bandwidth. It has been demonstrated that the metamaterial balun maintains a differential output signal over a broader bandwidth compared to both a distributed transmission-line balun employing 27 and 9 transmission lines and a lumped-element high-pass/low-pass balun. In addition, the metamaterial balun exhibits excellent return loss, isolation and through characteristics, while still maintaining a small form factor. Non-radiating NRI-TL metamaterial lines were also used to implement a 1:4 series power divider that provides equal power split to all four output ports over a large bandwidth. The divider comprises four non-radiating metamaterial lines in series, incurring a near-zero insertion phase over a large bandwidth, while simultaneously maintaining a compact length of λ /8 for each section of line. It has been demonstrated that the metamaterial divider achieves a broadband in-phase power division to all the output ports, while maintaining an overall small form factor and exhibiting comparable losses when compared to a conventional series power divider employing meandered transmission lines. Additionally, the metamaterial divider offers the flexibility of spacing the output ports arbitrarily apart and can be scaled to an arbitrary number of ports. The metamaterial series power dividing scheme was subsequently used to create a series feed network for a linear array of printed dipoles designed to radiate in the broadside direction. The performance of a four-element array employing a conventional onewavelength transmission-line feed network was compared to one that employed nonradiating metamaterial lines. It has been demonstrated that the metamaterial-fed array has a significantly more compact form factor compared to its transmission-line counterpart. More importantly though, due to the broadband phase response of the metamaterial lines, these exhibit a quasi true-time delay characteristic, which enables the metamaterialfed array to exhibit significantly less beam squint compared to the transmission-line fed array, or equivalently a low-pass loaded array. In addition, the metamaterial feed networks can be used to create linear arrays whose main beam remains virtually fixed at a positive angle from broadside as the frequency is varied. It was also recognized that by operating a NRI-TL metamaterial line in the appropriate fast-wave region, then this can in and of itself form a leaky-wave antenna. Such an antenna was developed in co-planar strip technology by cascading NRI-TL metamaterial unit cells in series, and the beam squinting characteristics of the leaky-wave antenna

225 Chapter 8. Conclusion 25 were investigated. It has been shown that by operating the antenna in the upper righthanded band of the dispersion diagram, where the phase and group velocities of the line are closest to the speed of light, the beam squinting that the antenna experiences is minimized. Moreover, the leaky-wave antenna is fully-planar, differential and has a broadband return-loss characteristic. Finally, an electrically small antenna has been presented, which consists of four NRI-TL metamaterial unit cells. A circuit model for analyzing such electrically small metamaterial antennas has been proposed, that highlights the methods that enable these antennas to offer a good impedance match and a high radiation efficiency. Through an even-odd mode analysis, it has been shown that the antenna supports a predominately even-mode current on the vertical vias, allowing the antenna to be modeled as a multiplyfolded monopole, which provides a substantial increase in the radiation resistance. This, together with the top-loading effect of the microstrip line on the vias, enables the antenna to be matched to 5 Ω without the use of an external matching network, while maintaining a high radiation efficiency. Additionally, the antenna radiates a vertical linear electric field polarization from a structure that is only λ /2 in height. 8.2 Contributions The work presented in this thesis has resulted in a number of publications, which are listed below. Book Chapters 1. G.V. Eleftheriades and M.A. Antoniades, Antenna Applications of Negative Refractive Index Transmission Line (NRI-TL) Metamaterials, in Modern Antenna Handbook, edited by C.A. Balanis, John Wiley and Sons, Inc., G.V. Eleftheriades, Antenna Applications and Subwavelength Focusing Using Negative-Refractive-Index Transmission-Line (NRI-TL) Structures, in Metamaterials: Physics and Engineering Explorations, edited by N. Engheta and R.W. Ziolkowski, John Wiley and Sons, Inc., 26. (major contributions by M.A. Antoniades)

226 Chapter 8. Conclusion G.V. Eleftheriades, Microwave Devices and Antennas Using Negative-Refractive- Index Transmission-Line (NRI-TL) Metamaterials, in Negative-Refraction Metamaterials: Fundamental Principles and Applications, editedbyg.v.eleftheriades and K.G. Balmain, John Wiley and Sons, Inc., 25. (major contributions by M.A. Antoniades) Refereed Journal Papers 1. M.A. Antoniades and G.V. Eleftheriades, A folded-monopole model for electrically small NRI-TL metamaterial antennas, IEEE Antennas and Wireless Propagation Letters, vol. 7, pp , Oct P.P. Wang, M.A. Antoniades and G.V. Eleftheriades, An investigation of printed Franklin antennas at X-band using artificial (metamaterial) phase-shifting lines, IEEE Transactions on Antennas and Propagation, vol. 56, no. 1, pp , Oct M.A. Antoniades and G.V. Eleftheriades, A CPS leaky-wave antenna with reduced beam squinting using NRI-TL metamaterials, IEEE Transactions on Antennas and Propagation, vol. 56, no.3, pp , Mar G.V. Eleftheriades and M.A. Antoniades, Antenna applications of negativerefractive-index transmission-line (NRI-TL) structures, IET Microwaves, Antennas and Propagation, Special Issue on Metamaterials, vol. 1, no. 1, pp , Feb M.A. Antoniades and G.V. Eleftheriades, A broadband series power divider using zero-degree metamaterial phase-shifting lines, IEEE Microwave and Wireless Components Letters, vol.15, no.11, pp , Nov F. Qureshi, M.A. Antoniades and G.V. Eleftheriades, A compact and low-profile metamaterial ring antenna with vertical polarization, IEEE Antennas and Wireless Propagation Letters, vol. 4, pp , M.A. Antoniades and G.V. Eleftheriades, A broadband Wilkinson balun using microstrip metamaterial lines, IEEE Antennas and Wireless Propagation Letters, vol. 4, no. 1, pp , 25.

227 Chapter 8. Conclusion 27 Refereed Conference Proceedings 1. M.A. Antoniades and G.V. Eleftheriades, Efficiency measurement of electrically small negative-refractive-index transmission-line (NRI-TL) antennas, in Proc. USNC/URSI National Radio Science Meeting, San Diego, CA, Jul M.A. Antoniades and G.V. Eleftheriades, Applications of negative-refractive-index transmission-line (NRI-TL) metamaterials in planar antennas and their feed networks, in Proc. Second European Conference on Antennas and Propagation (EuCAP27), Nov. 27, pp M.A. Antoniades and G.V. Eleftheriades, A negative-refractive-index transmission-line (NRI-TL) leaky-wave antenna with reduced beam squinting, in Proc. IEEE International Symposium on Antennas and Propagation, Honolulu, HI, Jun. 27, pp G.V. Eleftheriades, M.A. Antoniades and F. Qureshi, Some antenna applications of negative-refractive-index transmission-line (NRI-TL) metamaterials, in Proc. IEEE International Symposium on Antennas and Propagation, Albuquerque, NM, Jul. 26, pp M.A. Antoniades and G.V. Eleftheriades, A metamaterial series-fed linear dipole array with reduced beam squinting, in Proc. IEEE International Symposium on Antennas and Propagation, Albuquerque, NM, Jul. 26, pp G.V. Eleftheriades, M.A. Antoniades and F. Qureshi, Selected antenna applications of negative-refractive-index transmission-line (NRI-TL) metamaterials, in Proc. IEEE Mediterranean Electrotechnical Conference (MELECON 26), Benalmadena (Malaga), Spain, May 26, pp M.A. Antoniades, F. Qureshi and G.V. Eleftheriades, Antenna applications of negative-refractive-index transmission-line metamaterials, in Proc. IEEE International Workshop on Antenna Technology: Small Antennas and Novel Metamaterials (IWAT 26), Mar. 26, pp M.A. Antoniades and G.V. Eleftheriades, A broadband 1:4 series power divider using metamaterial phase-shifting lines, in Proc. 35th European Microwave Conference (EuMC 25), Paris, France, Oct. 25, pp

228 Chapter 8. Conclusion M.A. Antoniades and G.V. Eleftheriades, A broadband balun using metamaterial phase-shifting lines, in Proc. IEEE International Symposium on Antennas and Propagation, vol. 2B, Washington, DC, Jul. 25, pp G.V. Eleftheriades, A. Grbic and M.A. Antoniades, Negative-refractive-index transmission-line metamaterials and enabling electromagnetic applications, in Proc. IEEE International Symposium on Antennas and Propagation, vol.2,monterey, CA, Jun. 24, pp G.V. Eleftheriades, M.A. Antoniades, A. Grbic, A.K. Iyer, and R. Islam, Electromagnetic applications of negative-refractive-index transmission-line metamaterials, in Proc. 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas, Santiago de Compostela, Spain, Mar. 24, pp G.V. Eleftheriades, A. Grbic, A.K. Iyer, M.A. Antoniades, and R. Islam, Enabling RF/microwave devices using negative-refractive-index transmission-line metamaterials, in Proc. IEE Seminar on Metamaterials for Microwave and (Sub) Millimetre Wave Applications: Photonic Bandgap and Double Negative Designs, Components, and Experiments, London, UK, Nov. 23, pp. 12/ Future Directions In this thesis, the multi-faceted nature of NRI-TL metamaterial phase-shifting lines has been emphasized. It has been shown that the metamaterial lines offer great versatility in terms of their phase response that can be tailored to any specification, and their size which can be made arbitrarily small and which does not depend on the desired phase shift. In addition, it has been shown that the metamaterial lines can be used as phase-shifting lines for the distribution of signals, or they can be used as radiating elements themselves. It is therefore envisioned that any device that employs conventional transmission lines stands to benefit from the use of metamaterials, by simply replacing the transmission lines with metamaterial phase-shifting lines. The benefit of using the metamaterial phaseshifting lines is even more evident when the distribution of in-phase signals is necessary, which can be performed by metamaterial lines that occupy significantly less space, while providing a more constant phase distribution with frequency. An area that stands to benefit tremendously from the use of metamaterial technology

229 Chapter 8. Conclusion 29 is that of electrically small antennas intended for next-generation broadband wireless communication systems. The requirements for these antennas are in general a high efficiency, a compact size that can conform to the size requirements of handheld wireless devices, as well as multi-band operation within the existing wireless standards, such as IEEE 82.11a/b/g/n (WiFi). Additionally, the next generation of antennas, used for example in the IEEE 82.16e (WiMax) standard, are required to provide orthogonal polarizations in order to exploit the spatially-encoded information sent over a multipleinput, multiple-output (MIMO) channel in order to increase the overall system capacity. These requirements can be met by using antennas which are based on metamaterials or which employ metamaterial loading in order to accurately manipulate the phase of various radiating sections within the antenna. The folded monopole technique outlined in Chapter 7 can be used to provide in-phase currents that can therefore be used to increase the effective aperture size of an electrically small antenna. Furthermore, by changing the values of the loading-elements in adjacent metamaterial unit cells, multiple resonances can be achieved within the same radiating structure that are not harmonically related, and which can be merged to increase the bandwidth of the antenna or alternatively they can be used to create multi-band antennas. In addition, if the unit cells are arranged such that their main radiating elements are orthogonal to each other, then the requirement of achieving orthogonal in-band or out-of-band polarizations can be satisfied. MIMO systems rely upon orthogonal transmission and reception of signals within a multipath environment, therefore minimal mutual coupling between adjacent antenna elements is required. A natural extension of this work is to develop metamaterial antennas that can address this issue by intelligently placing the minima and maxima of the antenna radiation patterns in key locations that reduce the mutual coupling between adjacent antennas. This process can be performed adaptively by incorporating active metamaterials onto the radiating elements. Alternatively, by combining both the metamaterial circuits and the metamaterial radiating elements onto a single substrate, active integrated metamaterial antennas can be created that are efficient, compact, and broadband. Furthermore, metamaterials can be used on the ground plane interconnecting adjacent antennas in order to control the cross-talk of signals, thus reducing mutual coupling. Future work along these lines would include further explorations into the use of metamaterials for the development of complete MIMO front-end radio systems. Possible areas where metamaterials could be used to improve system performance include filters, diplex-

230 Chapter 8. Conclusion 21 ers, mixers, power amplifiers, phase shifters and further antenna optimization in terms of size and bandwidth. Another extension of the existing work is to construct such devices in integrated circuit form in a low-cost CMOS process, which can also be integrated with MEMS devices in order to realize active metamaterials.

231 Appendix A Equivalent Circuit for the NRI-TL MTM Medium In this Appendix, we establish that the Telegrapher s equations can indeed be applied to the NRI-TL structure without violating the conditions under which the differential equations hold, provided that the phase shift incurred by each unit cell is small, i.e. βd 1. We begin from the loaded transmission line model of the NRI-TL line shown in Figure 2.1(a). In order to model this as a uniform structure both the sections of distributed transmission line and the lumped-element loading 2C and L must be discretized into equivalent incremental lumped-element circuits that have an infinitesimal size. These can then be combined to form an incremental equivalent circuit for a continuous NRI-TL line as shown in Figure A.1. The Telegrapher s equations for the one-dimensional NRI-TL metamaterial medium can be written as ( dv (z) = jωl + 1 ) I(z), (A.1) dz jωc di(z) dz ( = jωc + 1 ) V (z). jωl (A.2) Here, recall that C and L are the per-unit-length shunt capacitance and series inductance, and C = C Δz and L = L Δz are the times-unit-length series capacitance and shunt inductance. Comparing Equations (A.1) and (A.2) with Equations (2.3) and (2.4), the analogy between the material parameters μ and ɛ and the loading-element values of the 211

232 Appendix A. Equivalent Circuit for the NRI-TL MTM Medium 212 I(z) ZΔz 2 ZΔz 2 I(z+Δz) + V(z) - 2C' L/ 2 L / 2 2C' C L' YΔz Δz + V(z+Δz) - Figure A.1: Incremental equivalent circuit of a continuous negative-refractive-index transmission-line (NRI-TL) metamaterial medium. Note that the equivalent circuit has a bandpass topology and can be used to model a uniform NRI-TL metamaterial line. NRI-TL medium can be written as [2] μ L 1 ω 2 C & ɛ C 1. (A.3) ω 2 L It can be observed from Equation (A.3) that ɛ and μ can both take on negative, positive, as well as zero values, depending on the values of the elements comprising the line. Consequently, these expressions indicate that it is possible to achieve a positive, negative, as well as zero index of refraction with a NRI-TL metamaterial medium. It can also be observed that the values of μ and ɛ from Equation (A.3) are identical to the effective inductance and capacitance terms L eff and C eff from Equation (2.68), which were obtained by considering the loaded-tl NRI-TL medium as an effective medium. This implies that when the unit cell size is small, the incremental equivalent circuit of Figure A.1 can be used to model a uniform NRI-TL line that is characterized by the effective material parameters μ and ɛ given by Equation (A.3). A.1 Incremental Circuit Analyzed Using Telegrapher s Equations The method used to determine the propagation characteristics of the lossless NRI-TL line follows the same procedure outlined in Section The total impedance and admittance for the series and shunt elements shown in Figure A.1 are ZΔz 2 = ( jωl + 1 ) Δz jωc 2 & Y Δz = ( jωc + 1 ) Δz. jωl (A.4)

233 Appendix A. Equivalent Circuit for the NRI-TL MTM Medium 213 The per-unit-length impedance and admittance in terms of the above parameters are therefore Z = jωl + 1 & Y = jωc + 1. (A.5) jωc jωl Using the results obtained for the generic equivalent circuit of Figure 2.2, the propagation constant of the distributed NRI-TL line can be written using Equations (2.15) and (A.5) as ( β mtm = j jωl + 1 )( jωc + 1 ). (A.6) jωc jωl Expanding the above expression and using the impedance matching condition of Equation (2.71), the propagation constant simplifies to β mtm = ω LC + 1 ω L C Δz. (A.7) This familiar result can be recognized to be equal to Equation (2.68), which was obtained when considering the periodically loaded transmission line under the effective medium conditions θ 1andβ bl d 1. The corresponding electrical length of a section of NRI-TL metamaterial line with physical length d mtm is θ mtm = β mtm d mtm = ω LCd mtm + 1 ω L C. (A.8) The characteristic impedance of the NRI-TL metamaterial line can be found from Equations (2.16) and (A.5) to be Z,mtm = jωl + 1 jωc jωc + 1. (A.9) jωl Under the impedance matching condition of Equation (2.71), Equation (A.9) simplifies to the characteristic impedance of the transmission line, which is also equal to that of the backward-wave line, Z,mtm = Z = Z,bw. (A.1) Equation (A.1) indicates that the characteristic impedance of the uniform NRI-TL line is constant for all frequencies. As was shown in Figure 2.17(b) this is not strictly true, however this approximation is valid for a large range of frequencies under the assumption that the unit cell size is small.

234 Appendix A. Equivalent Circuit for the NRI-TL MTM Medium 214 A.2 Lumped-Element Circuit Analyzed Using Periodic Analysis In this section we will consider a periodic lumped-element NRI-TL line consisting of finite unit cells similar to the ones shown in Figure A.1, but with a periodicity d. Since the unit cell of Figure A.1 has the same topology as the generic lumped-element unit cell of Figure 2.3, the results derived for the latter circuit will therefore be applied to the periodic lumped-element NRI-TL line. Recall that in Figure 2.3, the parameters Z and Y are the total impedance and admittance over the length d. Therefore, for the NRI-TL line these can be written as Z = jωl + 1 & Y = jωc + 1. (A.11) jωc jωl The dispersion relation for the NRI-TL line can be obtained by substituting the expressions of Z and Y from Equation (A.11) into Equation (2.28) ( ) sin 2 βbl d = 1 ( jωl + 1 )( jωc + 1 ). (A.12) 2 4 jωc jωl Under the impedance matching condition of Equation (2.71), the dispersion relation of Equation (A.12) simplifies to ( ) βbl d sin = 1 ( ω LCd + 1 ) 2 2 ω. (A.13) L C The dispersion relations for both the incremental and the lumped-element structures from Equations (A.8) and (A.13), as well as the loaded-tl structure from Equation (2.57) are shown in Figure A.2 using the parameters C =1.11 pf, L =2.78 nh, C =2.19 pf, L =5.47 nh, Z =5Ω,d =13.61 mm and θ =4 at 2 GHz, which were chosen in order to achieve a insertion phase at 2 GHz. It can be observed that for small values of βd, the three responses match quite well, and the response of the incremental structure of Figure A.1 matches very well with that of the loaded-tl structure of Figure 2.1(a) over almost the entire frequency range of operation. This highlights the fact that the incremental NRI-TL circuit models accurately the behaviour of the loaded-tl over a large frequency range. It can also be observed that the incremental NRI-TL circuit exhibits as expected a continuous dispersion relation, while the lumped-element and the loaded-tl periodic models both exhibit cutoff conditions at β bl d = π. The lumped-element and the loaded-tl periodic structures therefore act as band-pass filters with respective upper and lower cutoff

235 Appendix A. Equivalent Circuit for the NRI-TL MTM Medium Incremental circuit (Telegrapher s eq.) Lumped element circuit (Periodic) Loaded TL circuit (Periodic) 8 Frequency (GHz) π π/2 π/2 π βd (rad) Figure A.2: Dispersion diagrams for a representative band-pass NRI-TL metamaterial medium. The incremental circuit was analyzed using the Telegrapher s equations and is described by Equation (A.8), the lumped-element circuit was analyzed using periodic analysis and is described by Equation (A.13), and the loaded-tl circuit was also analyzed using periodic analysis and is described by Equation (2.57). frequencies. At higher frequencies the lumped-element structure reaches the β bl d = π cutoff condition at a much lower frequency than the loaded-tl structure. This can be attributed to the fact that at high frequencies a distributed transmission line that has an appreciable electrical length cannot be accurately modeled using a single lumped-element low-pass unit cell. Therefore, from a practical perspective, the approach using a loaded- TL is not only more convenient to implement, but also provides a right-handed passband that extends to much higher frequencies. From Figure A.2 it can also be observed that for small phase shifts per unit cell, i.e. βd 1, the characteristics of the incremental and lumped-element lines correspond very closely. This can also be observed by applying the above condition to the lumped-element dispersion relation of Equation (A.13), which is equivalent to shrinking the size of each unit cell d. Under the condition that β bl d 1, we can approximate the left-hand side of Equation (A.13) simply with the argument of the sine term to obtain β bl ω LC + 1 ω L C d. (A.14)

236 Appendix A. Equivalent Circuit for the NRI-TL MTM Medium 216 It can be recognized that the expression for the propagation constant of Equation (A.14) for the lumped-element case in the β bl d 1 limit is identical to Equation (A.7) for the incremental case, which is also identical to Equation (2.68) for the effective medium case. Therefore, it can be concluded that as the size of each unit cell in the lumped-element circuit tends to zero, the characteristics of the periodic lumped-element line approach those of the uniform backward-wave line. This implies that under the condition that the unit cell size tends to zero, the use of the Telegrapher s equations for the NRI-TL circuit of Figure A.1 is justified, since it produces an almost identical response to that obtained using periodic theory, which does not assume that the voltages and currents on each side of the unit cell are equal. This result is also consistent intuitively, since around the frequency where β mtm =, the series resonator formed between 2C and L/2 will be a short-circuit, and the shunt resonator formed between C and L will be an open-circuit. Thus, there will be no voltage drop across the circuit and there will be a constant current passing through it, allowing us to use Kirchhoff s laws to write the Telegrapher s equations for the NRI-TL circuit.

237 Appendix B Choice of Z in a Series Power Divider There are many factors that affect the input impedance bandwidth of a series power divider, however for the particular implementation chosen in Figures 4.2 and 4.3 where the loads and the spacing between them are fixed, the input impedance bandwidth of both the TL divider and the NR-MTM divider will be a function of the characteristic impedance Z of the transmission lines used. Herein, we will therefore consider the effect of Z on the input impedance bandwidth by using simple transmission-line analysis. In order to do so, we will first consider the input impedance as a function of frequency for a 1:2 TL power divider, and subsequently it will be shown that this can be extended to a 1:4 TL power divider. Considering the circuit shown in Figure B.1, in order for the TL divider to be matched to the 5 Ω feed line (Z F ), the value of the load resistors must be R = 1 Ω. The impedance at the input port of the device, P1, is given by Z in3 and can be calculated by starting at the right-hand side of the circuit and working towards the left. The input impedance bandwidth can then be found analytically by following a procedure analogous to the one presented in [84] (pp ). The input impedance looking into the first TL that is terminated in a load resistance R is given by Z in1, and can be found by using the general form for the impedance of a lossless TL of characteristic impedance Z that is terminated in a load Z L [84], Z in = Z Z L + jz tan θ Z + jz L tan θ. (B.1) For the TL of Figure B.1, Z L = R, therefore Z in1 can be written as 217

238 Appendix B. Choice of Z in a Series Power Divider 218 Γ Z F =5 Ω P1 Z, λ g P2 Z, λ g P3 R R Z in3 Z in2 Z in1 Figure B.1: Schematic diagram of a 1:2 TL series power divider, fed by a transmission line with characteristic impedance, Z F. R + jz tan θ Z in1 = Z Z + jrtan θ. (B.2) Here, θ is the electrical length of the TL and is given by Equation (2.21) as θ = β tl d tl = β tl λ g. Z in1 can now be used to calculate Z in2 by simply combining R and Z in1 in parallel, Z in2 = R Z in1 R 2 Z + jrz 2 = t 2RZ + jt(r 2 + Z 2). (B.3) Here we have used t =tanθ in order to simplify the expression. Z in2 can now be regarded as the load of the second TL, therefore Z in3 takes on the form Z in2 + jz t Z in3 = Z Z + jz in2 t. (B.4) Substituting Equation (B.3) into Equation (B.4) we obtain Z in3 = R2 Z (R 2 Z + Z 3)t2 + j3rz 2t 2RZ RZ t 2 + jt(2r 2 + Z). (B.5) 2 At this stage we have a simple expression for the input impedance of the 1:2 power divider. We can therefore go ahead and evaluate the reflection coefficient for this device. The reflection coefficient at the interface between a feed line with characteristic impedance Z F and a load Z L is given by [84] Γ= Z L Z F. (B.6) Z L + Z F Substituting the expression for Z L = Z in3 from Equation (B.5) into Equation (B.6), and after some algebraic manipulations we obtain the reflection coefficient at P1: Γ= Z in3 Z F Z in3 + Z F = R2 Z 2RZ Z F +(RZ Z F R 2 Z Z 3 )t2 + j(3rz 2 2R2 Z F Z 2 Z F )t R 2 Z +2RZ Z F (RZ Z F R 2 Z Z 3 )t2 + j(3rz 2 +2R2 Z F + Z 2 Z F )t. (B.7)

239 Appendix B. Choice of Z in a Series Power Divider 219 The magnitude of the reflection coefficient is given by (A + Bt Γ = 2 ) 2 +(Ct) 2 (D Et 2 ) 2 +(Ft), (B.8a) 2 A = R 2 Z 2RZ Z F B = RZ Z F R 2 Z Z 3 C =3RZ 2 2R 2 Z F ZZ 2 F D = R 2 Z +2RZ Z F E = RZ Z F R 2 Z Z 3 F =3RZ 2 +2R 2 Z F + ZZ 2 F. (B.8b) (B.8c) (B.8d) (B.8e) (B.8f) (B.8g) We can now define a maximum value of Γ that can be tolerated and this will set the upper limit on the amount of power reflected back to the source. Typically, a maximum value of Γ =Γ m = 1 db =.316 is used, such that no more than 1% of the incident power is reflected back to the source. Since the expression for Γ is symmetric about θ =2π (the electrical length of the λ g TL), this results in two values of θ, θ m1 and θ m2, for which Γ =Γ m, as can be observed for the representative example of Figure B.2. The angular bandwidth of the divider will therefore be the difference between θ m2 and θ m1, expressed as Δθ = θ m2 θ m1. (B.9) Equation (B.8) can therefore be solved for the values of θ where Γ is equal to Γ m.by substituting Γ =Γ m into Equation (B.8), we obtain a quadratic equation in x = t 2 : ax 2 + bx + c =, (B.1a) a =Γ 2 me 2 B 2 b =Γ 2 mf 2 2Γ 2 mde 2AB C 2 c =Γ 2 md 2 A 2. (B.1b) (B.1c) (B.1d) The solutions to the above quadratic equation are x 1 = b b 2 4ac, (B.11a) 2a x 2 = b + b 2 4ac. (B.11b) 2a

240 Appendix B. Choice of Z in a Series Power Divider Δθ Γ m Γ.2.1 θ m1 θ m2 π 2π 3π θ (rad) Figure B.2: Magnitude of the reflection coefficient, Γ, for a 1:2 TL series divider as a function of the electrical length, θ, of the transmission lines, calculated using the parameters R = 1 Ω, Z =5ΩandZ F = 5 Ω in Equation (B.8). Therefore, in view of the fact that x = t 2 =tan 2 θ, the four possible solutions are θ m1 = tan 1 ( x 1 )+2π, θ m2 = tan 1 (+ x 1 )+2π, θ m3 = tan 1 ( x 2 )+2π, θ m4 = tan 1 (+ x 2 )+2π. (B.12a) (B.12b) (B.12c) (B.12d) The solutions corresponding to x 2 are discarded, since x 2 is negative for all values of Z, therefore leading to imaginary values for θ m3 and θ m4.sinceθ m1 and θ m2 are now known, Equation (B.9) can be used to find the angular bandwidth of the power divider. The next step is to translate the angular bandwidth of the device into a frequency bandwidth. Since the transmission lines used have a TEM response, the electrical length of a λ g -long TL around the design frequency f can be written as θ = β tl d tl = 2π λ λ g = 2πf v φ v φ = 2πf. (B.13) f f At the lower band edge where Γ =Γ m, θ is equal to θ m1. Thus, from Equation (B.13) we can find the frequency which corresponds to the lower band edge, f = f m1 to be f m1 = θ m1f 2π. (B.14)

241 Appendix B. Choice of Z in a Series Power Divider Δ f/f (%) Characteristic Impedance, Z (Ω) Figure B.3: Fractional bandwidth of a 1:2 TL series divider as a function of the characteristic impedance Z of the transmission lines, calculated using Equation (B.17). Similarly, at the higher band edge, Γ =Γ m, θ = θ m2 and f = f m2,givenby f m2 = θ m2f 2π. (B.15) The fractional bandwidth can therefore be expressed as Δf f = f m2 f m1 f. (B.16) Substituting into Equation (B.16) the expressions for f m1 and f m2 from Equations (B.14) and (B.15), and using Equation (B.9) we obtain Δf f = θ m2 θ m1 2π = Δθ 2π. (B.17) Thus, the closed-form solutions for θ m1 and θ m2 from Equations (B.12a) and (B.12b), together with Equation (B.17) provide an analytic expression for the fractional bandwidth of the 1:2 TL series divider. This allows the characteristic impedance of the lines to be subsequently varied in order to determine the effect of Z on the fractional bandwidth of the device, and the results of the above analysis are shown in Figure B.3. It is interesting to note that the maximum bandwidth does not occur at Z =5Ω as might be expected, but rather reaches a peak value of 31% at Z =57.5 Ω. Therefore, the interconnecting transmission lines used in the 1:2 TL series power divider should be designed with the optimal characteristic impedance of Z =57.5 Ω.

242 Appendix B. Choice of Z in a Series Power Divider Δ f/f (%) Characteristic Impedance, Z (Ω) Figure B.4: Fractional bandwidth of a 1:4 TL series divider as a function of the characteristic impedance Z of the transmission lines, evaluated using Agilent-ADS simulations. Extending the above analysis to the 1:4 TL divider in order to find a closed-form expression for Γ proved to be very cumbersome, considering the complexity of the expression for Γ from Equation (B.8) for the 1:2 TL divider. However, by simulating the circuit of Figure 4.2 in Agilent-ADS and measuring the bandwidth over which S 11 < 1 db, the fractional bandwidth as a function of Z was evaluated and is shown in Figure B.4. It can be observed that the bandwidth characteristics for the 1:4 TL divider shown in Figure B.4 are very similar to the ones exhibited by the 1:2 TL divider shown in Figure B.3. The 1:4 TL divider, however, exhibits a smaller fractional bandwidth compared to the 1:2 TL divider, which is a direct consequence of the larger impedance transformation from 2 Ω to 5 Ω. As such, the 1:4 TL divider exhibits a maximum fractional bandwidth of 13.55% at a higher value for the characteristic impedance of Z =67.5 Ω. Thus, in order to obtain the maximum possible bandwidth from the 1:4 TL divider, it should be designed with interconnecting transmission lines that have a characteristic impedance of Z =67.5 Ω.

243 Appendix C Beam Squinting Analysis C.1 Derivation of the Beam Squinting Equation In order to obtain an analytic expression for the dependency of the scan angle on frequency, dθ sc /dω, we begin by re-arranging Equation (6.3) to obtain sin(θ sc )= β k. (C.1) By taking the derivative of Equation (C.1) and using the chain rule, we obtain d sin(θ sc ) = d (β/k ) = dβ 1 + d(1/k ) β. (C.2) dω dω dω k dω Using the relations v g =(dβ/dω) 1, v φ = ω/β and k = ω/c, Equation (C.2) becomes d sin(θ sc ) dω = 1 cβ v g k ω 2 = 1 1 ω ω v g k β c (C.3a) (C.3b) = 1 1 (C.3c) v g k v φ k = 1 ( 1 1 ). (C.3d) k v g v φ Now, differentiating the left-hand side of Equation (C.1) and using the chain rule, we obtain d sin(θ sc ) dω = d sin(θ sc) dθ sc dθ sc dω =cos(θ sc ) dθ sc dω = 1 sin 2 (θ sc ) dθ sc dω. (C.4a) (C.4b) (C.4c) 223

244 Appendix C. Beam Squinting Analysis 224 We can now substitute Equation (C.1) into Equation (C.4c) to obtain ( ) 2 d sin(θ sc ) β dθ sc = 1 dω dω. k (C.5) In order to eliminate the d sin(θ sc )/dω term, we set Equation (C.5) equal to Equation (C.3d): ( ) 2 β dθ sc 1 k dω = 1 ( 1 1 ). (C.6) k v g v φ The final expression for the beam squinting can be obtained by re-arranging Equation (C.6) and is given by ( dθ sc dω = ). (C.7) k 2 β 2 v g v φ C.2 Derivation of the Approximate Beam Squinting Equations Equation (C.7) together with the approximate Bloch propagation constant, β bl,eff from Equation (6.5), repeated below for convenience, can be used to derive an approximate and intuitive expression for the variation of the scan angle with frequency. β bl,eff = ω 1 LC ω (C.8) L C d h-tl Equation (C.8) can be used to evaluate the approximate group velocity and phase velocity terms of Equation (C.7) as follows: 1 = dβ bl,eff v g dω 1 = β bl,eff v φ ω Therefore, the term (1/v g 1/v φ ) becomes ( 1 1 ) = v g v φ = LC + = LC 1 ω 2 L C d h-tl, 1 ω 2 L C d h-tl. 2 ω 2 L C d h-tl. (C.9) (C.1) (C.11) Inserting Equation (C.8) and Equation (C.11) into Equation (C.7) we can obtain an approximate beam squinting expression, (dθ sc /dω) eff : ( ) dθsc 1 2c = dω ( ) 2 eff βbl,eff L C d h-tl ω. (C.12) 3 1 k

245 Appendix C. Beam Squinting Analysis 225 A further simplification of Equation (C.12) can be carried out by replacing the first term 1/ 1 (β bl,eff /k ) 2 with the first two terms of its Taylor series, whose general form can be written as 1 1+ x ( ) 2 βbl,eff ; x =. (C.13) 1 x 2 k Therefore, the first term of Equation (C.12) can be approximated as ( ) βbl,eff ( ) 1+. (C.14) 2 βbl,eff 2 k 1 k Furthermore, we can use the expression for the backward-wave propagation constant 1 β bw = ω L C d h-tl from Equation (2.39) to write the final approximate beam squinting expression, (dθ sc /dω) eff, presented in Equation (6.7): ( ) ( dθsc = 1+ 1 ( ) ) 2 βbl,eff 2c dω 2 ω β bw. (C.15) 2 C.3 Group Velocity Analysis eff Analytically, the group velocity, v g, of a metamaterial line can be calculated by taking the inverse of the derivative of β with respect to ω as shown in Equation (2.25). To this end, we begin to evaluate v g by considering the dispersion relation of Equation (2.57), and let A(ω) =cos(β bl d). (C.16) From Equation (2.57), the full expression for A(ω) in terms of the transmission line and loading parameters becomes ( ) 1 A(ω) = 1 cos(ω LCd) 4ω 2 L C ( Z ) sin(ω 1 LCd). (C.17) 2ωC Z 2ωL 4ω 2 L C In Equation (C.17) the full expression of the electrical length of the host TL has been used, i.e. θ = ω LCd, to explicitly indicate its dependence on ω. Differentiating Equation (C.17) with respect to ω, we can obtain an expression for da/dω, ( ) da ω 2 dω = d(cl + LC )+1 cos(ω LCd) 2ω 3 L C ( ) LCd(4L C ω 2 1) + 2Y L +2Z C sin(ω LCd)+ 4ω 2 L C k 1 2ω 3 L C. (C.18)

246 Appendix C. Beam Squinting Analysis 226 Concurrently, we can re-arrange Equation (C.16), to obtain an expression for the periodic Bloch propagation constant, β bl = 1 d cos 1 (A). (C.19) Using the chain rule, the derivative of the propagation constant along the line can be written as dβ bl dω = 1 ( )( ) 1 da d ±. (C.2) 1 A 2 dω The expression for the group velocity in terms of A can be obtained by inserting Equation (C.2) into Equation (2.25) to obtain ( ) 1 dβbl v g = = d 1 A 2 dω da/dω. (C.21) The final expression for the group velocity can be obtained by inserting Equations (C.17) and (C.18) into Equation (C.21) and can be written as ( ) (( ) ( ) ) ± 1 1 4ω 2 L C cos(θ)+ 2ωC Z + Z 1 2ωL sin(θ) 4ω 2 L C v g = d ( ) ( ). ω 2 d(cl +LC )+1 2ω 3 L C cos(θ) LCd(4L C ω 2 1)+2Y L +2Z C 1 4ω 2 L C sin(θ)+ 2ω 3 L C (C.22) Equation (C.22) can be tested for its validity for the two limiting cases of low frequency operation and high frequency operation. By expressing cos(θ) andsin(θ) with their equivalent Taylor series and applying suitable low-frequency approximations, it can be shown that the group velocity of the metamaterial line approaches that of a purely backward-wave line, which was presented in Section 2.3, i.e. v g ω = v g,bw = ω 2 L C d. (C.23) Thus, at low frequencies the effects of the host transmission line in Figure 2.1(b) become negligible, leaving only the loading elements arranged in a high-pass manner. On this type of a line it can be observed that the group velocity has a parabolic characteristic with frequency, which is desirable for broadband operation, because this allows for large group velocities at relatively low frequencies. However, the phase response of a backwardwave line is not a linear function of frequency that passes through the origin, as can be observed from Equation (2.42), therefore a purely backward-wave line does not exhibit the desired true-time-delay (TTD) behaviour. Consequently, operating the metamaterial line at low frequencies where it exhibits backward-wave behaviour is not suitable for implementing squint-free antenna arrays.

247 Appendix C. Beam Squinting Analysis x 18 Group velocity (m/s) v g,max1 v g,max2 2 f f 12 t1 Frequency (GHz) t2 Figure C.1: Real part of the group velocity v g for a line consisting of MTM unit cells with loading-element values and transmission-line parameters given in Table 6.1. In a similar fashion, by applying suitable high-frequency approximations to Equation (C.22), it can be shown that the group velocity of the metamaterial line approaches that of the unloaded host TL, whose characteristics were outlined in Section 2.2, i.e. v g ω = v g,tl = 1. (C.24) LC Thus, at high frequencies the effects of the loading elements diminish, leaving only the host TL, which has an inherently low-pass characteristic. This verifies the well-known fact that the group velocity on a conventional TL is constant with frequency and equal to the phase velocity. In addition, since conventional transmission lines have a linear phase response that passes through the origin, they exhibit TTD characteristics. It follows, that by operating the metamaterial line in the upper RH band, it exhibits a TL-like behaviour, and can therefore be used to mitigate beam squinting in linear antenna arrays. Moreover, if the metamaterial line is operated within the light cone in the upper RH band, then it can be used as a leaky-wave antenna that exhibits reduced beam squinting. The group velocity diagram corresponding to the dispersion diagram of Figure 6.2 can be seen in Figure 6.6 and is repeated in Figure C.1 on a larger scale. From Figure C.1 it can be observed that the group velocity exhibits two distinct peaks, one in the lower LH band at v g,max1 and one in the upper RH band at v g,max2, and both of the peaks

248 Appendix C. Beam Squinting Analysis dv g /dω (m/rad) dv g /dω= Frequency (GHz) Figure C.2: Group velocity dispersion for the MTM unit cell using the loading-element values and transmission-line parameters of Table 6.1. are within the radiation region of the LWA, between f t1 and f t2. It is evident however, that the peak of the group velocity in the RH band is significantly higher than the peak in the LH band. Specifically, the group velocity in the RH band reaches a maximum value of v g,max2 = m/s at 9.4 GHz, while the group velocity in the LH band reaches a local maximum value of v g,max1 = m/s at 3.49 GHz. Moreover, the peak in the RH band is significantly broader than the one in the LH band, indicating that the rate of change of the group velocity with frequency in the RH band is much smaller than in the RH band. This is a measure of the linearity of the phase response shown in Figure 6.2, where as indicated previously, the phase is considerably more linear in the upper RH band than in the lower LH band. To illustrate this point further, the group velocity dispersion is shown in Figure C.2. The locations where the gradient of the group velocity with respect to ω is equal to zero, i.e. dv g /dω =, correspond to the locations of the maximum group velocity in Figure C.1, i.e. at 3.49 GHz and 9.4 GHz. In the ideal case, the group velocity should be constant with frequency in order to ensure phase linearity, as shown in Equation (C.24) for the case of a conventional TL. This translates to a zero group velocity dispersion for all frequencies. By considering Figure C.2, it can be observed that the group velocity dispersion is linear and remains close to zero in the RH region between 5 and 9.5 GHz. Therefore, in this region the phase response is the most linear. For the corresponding LH

249 Appendix C. Beam Squinting Analysis 229 region, around the dv g /dω = point the group velocity is highly dispersive, indicating that the phase response in this region is highly non-linear and therefore undesirable for reduced beam squinting. The expression for the group velocity dispersion was obtained by differentiating Equation (C.21) with respect to ω, resulting in ( ( ) 2 ( ) dv g dω = d A da d 2 ± 1 A + A ( ± ) ) 1 A 2 dω dω 2. (C.25) 2 In order to evaluate Equation (C.25), the expressions of A(ω) andda/dω from Equations (C.17) and (C.18) were used respectively, together with the derivative of Equation (C.18), given by d 2 A dω = K 2 1 cos(θ) K 2 sin(θ) K 3, (C.26) where ω 2 d(cl + LC )+ω 2 ( LCd(2ω ) LCd 2 L C.5) + Y L + Z C K 1 =, (C.27) 2ω 4 L C LCd (ω 2 d(cl + LC )+2) 2Y L 2Z C K 2 =, (C.28) 2ω 3 L C 3 K 3 =. (C.29) 2ω 4 L C

250 Appendix D Measured Efficiency Using the G/D Method The radiation efficiency of an antenna is defined as the ratio of the radiated power to the input power, and in some cases can be quite challenging to evaluate. The most common technique used to evaluate the efficiency of an antenna experimentally is to use the gain/directivity (G/D) method. This involves measuring the two-dimensional radiation pattern in various plane cuts of the total three-dimensional antenna radiation pattern. Using the data from the various plane cuts, the gain can then be evaluated using the gain comparison method, and the directivity can be evaluated using the numerical integration techniques that will be described below. It should be noted that most of the reference material for this section was obtained from Sections and of [19] and Section 12.6 of [15], and was adapted to suit the specific measurements taken in the anechoic chamber at the University of Toronto. D.1 Directivity Before we describe the directivity calculation procedure, it would be useful to outline the definitions of various quantities that will be used in the subsequent calculations. We begin with the radiation intensity, U(θ, φ), which is defined as the power radiated from an antenna per unit solid angle in a given direction, and has units of (W/unit solid angle). The radiation intensity is related to the electric field of an antenna in the far-field as 23

251 Appendix D. Measured Efficiency Using the G/D Method 231 follows: U(θ, φ) U θ (θ, φ)+u φ (θ, φ) = 1 [ E 2η θ (θ, φ) 2 + Eφ(θ, φ) 2]. (D.1) Here, Eθ (θ, φ) istheθ-directed component of the electric field in the far field, E φ (θ, φ) is the φ-directed component of the electric field in the far field, and η is the intrinsic impedance of the medium. Since the two terms Eθ (θ, φ) 2 /(2η) and Eφ (θ, φ) 2 /(2η) in Equation (D.1) are proportional to the radiated power in each of the θ and φ components, the radiation patterns measured in the anechoic chamber therefore represent a measure of the radiation intensity. It will subsequently be shown in Equation (D.35) that the measured radiation patterns, given by G meas (θ, φ), are simply related to the radiation intensity by a constant scaling factor. The radiation intensity can be written as the product of a constant term, U max, which is the maximum radiation intensity that occurs at the angles (θ max,φ max ), and a pattern factor, F (θ, φ), which is normalized to a maximum value of one at (θ max,φ max ). Thus, and U max = U(θ max,φ max ), U(θ, φ) =U max F (θ, φ). (D.2) (D.3) The pattern factor, F (θ, φ), contains the angular variation of the radiation pattern in the θ and φ directions, and can be considered a normalized form of the radiation intensity, written as U(θ, φ) F (θ, φ) =. (D.4) U max By integrating the radiation intensity over the entire solid angle of 4π, the total radiated power from the antenna can be obtained π 2π P rad = U(θ, φ)dω = U(θ, φ) sin(θ) dθdφ. (D.5) Ω In Equation (D.5), dω is an element of solid angle and is defined as dω =sin(θ)dθdφ. (D.6) Since dω represents an effective area, Equation (D.6) is only valid for positive values of sin(θ), which occurs for values of θ from to π. In Equation (D.5), however, the limits of θ are from to 2π in order to reflect the fact that the pattern cuts measured in the anechoic chamber were taken in the elevation plane from θ =1 to 36. As such, in

252 Appendix D. Measured Efficiency Using the G/D Method 232 order to account for the general case where sin(θ) can be negative, the element of solid angle was modified to dω = sin(θ) dθdφ. (D.7) Now, the directivity of an antenna is equal to the ratio of the radiation intensity in a given direction to the radiation intensity of an isotropic source, which from Equation (D.5) is equal to U iso = P rad /(4π). Therefore, the directivity can be written as U(θ, φ) U(θ, φ) D(θ, φ) = =4π. (D.8) U iso P rad The radiation intensity obtains its maximum value, U max, at a single set of θ and φ angles, denoted as (θ max,φ max ). The maximum directivity therefore also occurs at (θ max,φ max ), and can be written as D =4π U max. (D.9) P rad Here it should be noted that if the direction in which the directivity is evaluated is not explicitly stated, as is the case in Equation (D.9), it is implied that the directivity referred to is the maximum value. By substituting Equation (D.5) into Equation (D.8) we can obtain an expression for the directivity in terms of the measured radiation pattern data, expressed in the form of the radiation intensity U(θ, φ) D(θ, φ) =4π π 2π U(θ, φ) sin(θ) dθdφ. (D.1) From Equations (D.5) and (D.9), the maximum directivity can be written as D =4π π U max 2π U(θ, φ) sin(θ) dθdφ. (D.11) Now if we substitute Equation (D.3) into Equation (D.1), the maximum radiation intensities from the numerator and denominator cancel, and we obtain an expression of the directivity in terms of the pattern factor, F (θ, φ) F (θ, φ) D(θ, φ) =4π π 2π F (θ, φ) sin(θ) dθdφ. (D.12) Equation (D.12) effectively demonstrates that the directivity of an antenna is solely determined by its radiation pattern and does not depend on the maximum radiation intensity, U max. In fact, the absolute power level of the radiation patterns is irrelevant to the evaluation of the directivity, which is instead dependent on the accurate measurement of the shape of the radiation pattern. As we will subsequently see, the same does not

253 Appendix D. Measured Efficiency Using the G/D Method 233 E r Model DRH GHz Broad Band Antenna Figure D.1: DRH-118 double-ridged horn used for the antenna measurements. hold when determining the maximum gain of an antenna, which is directly related to the value of U max. In most cases an antenna will not have a single polarization, but will rather radiate fields with orthogonal polarizations in both the θ and φ directions. In this case, the fields in both polarizations must be measured in order to account for all of the power radiated by the antenna. In the case that a dual-polarized measurement antenna is not available, a singly-polarized antenna can be used to measure each polarization separately. The antennas available in the University of Toronto lab were DRH-118 double-ridged horns with a singe linear polarization, collinear with the ridge of the horn, as shown in Figure D.1. Therefore, for each of the antenna plane cuts the measurement was conducted twice: once with the horn antenna mounted horizontally (vertical E-field polarization) and once with the horn antenna mounted vertically (horizontal E-field polarization), in order to measure both polarizations. The metamaterial test antenna was mounted with the axis of the short monopoles in the horizontal direction, therefore these two measurements correspond to the θ and φ components of the radiation intensity, U θ (θ, φ) and U φ (θ, φ), and they also correspond to the co-polarization and cross-polarization components of the radiated electric field, respectively. The total maximum radiation intensity can be found by summing U θ (θ, φ) andu φ (θ, φ) according to Equation (D.1) and finding the maximum value. The angles at which the radiation intensity is a maximum are denoted as θ max and φ max,andu max,θ and U max,φ are the corresponding θ and φ components of the radiation intensity evaluated at (θ max,φ max ). The corresponding partial

254 Appendix D. Measured Efficiency Using the G/D Method 234 directivities, D θ and D φ can therefore be written as U max,θ D θ =4π, P rad,θ + P rad,φ (D.13a) U max,φ D φ =4π. P rad,θ + P rad,φ (D.13b) Finally, the total directivity is given by the sum of the two partial directivities as D tot = D θ + D φ. (D.14) In Equations (D.13a) and (D.13b), the expressions for the total radiated power that is contained in the θ-directed and the φ-directed field components are P rad,θ = P rad,φ = π 2π π 2π U θ (θ, φ) sin(θ) dθdφ, U φ (θ, φ) sin(θ) dθdφ. (D.15a) (D.15b) D.1.1 Numerical Integration Technique The measured radiation patterns were obtained by sampling the electric field in both the θ and φ polarizations in two-dimensional elevation plane cuts and in discrete steps of one degree from θ =1 to 36. In order to calculate the directivity from Equation (D.14), first the radiated power must be calculated. Since the radiation pattern does not have a closed-form analytical expression, but is rather in discrete form, the integrations in Equations (D.15a) and (D.15b) cannot be carried out analytically. Instead, a double series representation is used to approximate the double integrals within the radiated power expressions, and are given by [ M N ] P rad,θ U θ (θ i,φ j ) sin(θ i ) Δθ i Δφ j, (D.16a) P rad,φ j=1 i=1 [ M N ] U φ (θ i,φ j ) sin(θ i ) Δθ i Δφ j. (D.16b) j=1 i=1 The θ interval from to 2π is divided into N uniform divisions, which in this case was N = 36, therefore the incremental elevation angle is given by ( ) ( ) 2π 2π Δθ i = =. (D.17) N 36

255 Appendix D. Measured Efficiency Using the G/D Method 235 The i-th θ value is given by ( ) ( ) 2π 2π θ i = i = i i =1, 2, 3,..., N. (D.18) N 36 Analogously, the φ interval from to π is divided into M uniform divisions, which in this case was M = 4, therefore the incremental azimuthal angle is given by ( ) ( ) π π Δφ j = =. (D.19) M 4 The j-th φ value is given by ( ) ( ) π π φ j = j = j j =1, 2, 3,..., M. (D.2) M 4 Using Equations (D.17) and (D.19) in Equations (D.16a) and (D.16b), the final expressions used to calculate the radiated power in both the θ and φ polarizations are ( )( ) M [ 2π π N ] P rad,θ U θ (θ i,φ j ) sin(θ i ), (D.21a) N M j=1 i=1 ( )( ) M [ 2π π N ] P rad,φ U φ (θ i,φ j ) sin(θ i ). (D.21b) N M j=1 Practically, the radiation patterns in the elevation plane were measured eight times, twice for each value of φ :45, 9, 135 and 18 in each of the θ and φ polarizations. Subsequently, Equations (D.21a) and (D.21b) were used to find the radiated power in each of the orthogonal polarizations. The measured data, U(θ, φ), was then used to find U max,θ and U max,φ, and finally Equations (D.13) and (D.14) were used to find the total maximum directivity. The complete, final expression for the total maximum directivity is given by D tot 4π ( 2π N )( π M i=1 U max,θ + U max,φ ] ) [ [ M N j=1 i=1 U θ(θ i,φ j ) sin(θ i ) + M j=1 [ N ] ]. i=1 U φ(θ i,φ j ) sin(θ i ) (D.22) D.2 Gain and Efficiency The absolute gain of an antenna is the ratio of the radiation intensity in a given direction to the radiation intensity that would be obtained if the power accepted by the antenna, P in, were radiated isotropically, and can be expressed as G(θ, φ) =4π U(θ, φ) P in. (D.23)

256 Appendix D. Measured Efficiency Using the G/D Method 236 The absolute gain will obtain its maximum value when the radiation intensity is a maximum, U max,at(θ max,φ max ). Thus, the maximum gain can be written as G = G max =4π U max P in. (D.24) The relative gain of an antenna is the ratio of the absolute gain of an antenna in a given direction to the absolute gain of a reference antenna in its referenced direction that has the same input power. From Equation (D.24), the relative gain can therefore be written as G rel = G Umax 4π P = in. (D.25) G ref 4π U rel P in In most cases the reference antenna used is a lossless isotropic source, which has a radiation intensity of U rel = U iso = P in /(4π). Equation (D.25) therefore simplifies to G rel = G G ref = 4π Umax P in 4π P in P in 4π =4π U max P in. (D.26) Comparing Equations (D.24) and (D.26), it can be observed that they are identical. Therefore, it can be concluded that when using an ideal isotropic source as a reference antenna, the absolute gain of an antenna is identical to its relative gain. In this work, since the reference antenna considered is always an ideal isotropic source, the absolute and relative gain will simply be referred to as the gain. Since there will inevitably be some power lost in the antenna in the form of conduction and dielectric losses, we can therefore define the radiation efficiency, η rad, which relates the power available at the input terminals of the antenna to the total power radiated by the antenna η rad = P rad. (D.27) P in Inserting Equation (D.27) into Equation (D.23), and using the expression for the directivity from Equation (D.8) we obtain [ ] U(θ, φ) G(θ, φ) =η rad 4π = η rad D(θ, φ). (D.28) P rad Equation (D.28) can be re-arranged to express the radiation efficiency as a function of (θ, φ) in terms of the gain and the directivity η rad (θ, φ) = G(θ, φ) D(θ, φ). (D.29)

257 Appendix D. Measured Efficiency Using the G/D Method 237 In a similar manner, the maximum efficiency can be expressed in terms of the maximum value of the gain and the directivity η rad = G D. (D.3) In the general case of antennas that have orthogonal radiation components in both the θ and φ directions, we can define partial gains, G θ and G φ, analogous to the directivity. From Equation (D.24), these are G θ =4π U max,θ P in, (D.31a) G φ =4π U max,φ. (D.31b) P in The total gain is given by the sum of the two partial gains as G tot = G θ + G φ. (D.32) Thus, the total radiation efficiency of the antenna can be expressed as η rad = G tot D tot. (D.33) D.3 Measured Radiation Pattern Data The antenna measurements taken in the anechoic chamber were acquired using the Orbit/FR 959 Spectrum antenna measurement system. This allows the acquisition of radiation pattern data for an entire frequency range within a single plane cut. The Orbit software uses the gain comparison method [19] (pp ), in order to measure the gain of the antenna-under-test for all values of θ or φ, depending on which pattern cut is being measured. As such, the system must first be calibrated using the two DRH- 118 standard gain horn antennas, whose relative gain profile (in dbi) as a function of frequency has been loaded into the system from the vendor s datasheet. The resulting radiation pattern data displayed by the program represents the relative gain of the antenna (in dbi) as a function of θ or φ, referenced to an ideal isotropic radiator. The measured gain can be expressed in terms of the maximum gain measured in the direction of (θ max,φ max ), and the pattern factor, F (θ, φ) G meas (θ, φ) =G max F (θ, φ). (D.34)

258 Appendix D. Measured Efficiency Using the G/D Method 238 Now, if we re-arrange Equation (D.23), we can express the radiation intensity in terms of the measured gain pattern ( ) Pin U(θ, φ) = G meas (θ, φ). (D.35) 4π Thus, it can be observed that the radiation intensity is directly related to the measured gain through a constant scaling factor of P in /(4π). Furthermore, we can express the radiation intensity in terms of the pattern factor by inserting Equation (D.34) into Equation (D.35), obtaining ( ) Pin U(θ, φ) = G max F (θ, φ) =U max F (θ, φ). (D.36) 4π The equivalence between Equations (D.36) and (D.3), which is used to calculate the directivity in Section D.1, can be recognized. Therefore, Equation (D.36), which is a function of the measured radiation pattern data in the form of the gain pattern, can be directly used to calculate the directivity. Note, however, that as indicated by Equation (D.12), only the normalized pattern factor, F (θ, φ), is ultimately used in the calculation of directivity. In conclusion, a total of eight radiation patterns in the elevation plane were measured in the anechoic chamber: two patterns were measured, one for each of the θ and φ polarizations, for each value of φ :45, 9, 135 and 18. The radiation pattern data in each of the θ and φ polarizations was combined, and the maximum gain of the four cuts was determined according to Equation (D.32) to be 1.23 (.9 dbi) at the operating frequency of f = 3.8 GHz. Subsequently, the data was used to calculate a directivity of 1.78 (2.5 dbi) according to Equation (D.22), and finally Equation (D.33) was used to calculate a total measured radiation efficiency of 69.1% for the metamaterial antenna.

259 Appendix E Measured Efficiency Using the Wheeler-Cap Method E.1 Background The radiation efficiency of an antenna is given by the ratio of the power radiated by the antenna, which is equal to the power delivered to the radiation resistance R r,tothe power available at the input of the antenna, which is equal to the power delivered to both the radiation resistance and the loss resistance, R l. Thus, the radiation efficiency can be written as [19] (p. 78) η rad = R r. (E.1) R r + R l Wheeler first introduced a simple method of evaluating the efficiency of an electrically small antenna by suggesting that the radiation resistance and loss resistance could be measured separately [151]. By measuring the antenna in free-space, the total resistance of the antenna at resonance, R r + R l, can be obtained. Wheeler suggested that the radiation resistance can be removed while retaining the loss resistance of the antenna by fully enclosing the antenna in a radiation shield. The radiation shield, subsequently termed the Wheeler cap, is a perfectly conducting sphere whose inner surface is located at the radiansphere, which is the boundary between the near-field of stored energy and the far-field of radiated power, and has a radius of λ/(2π). It is assumed that the radiation shield does not change the current distribution on the antenna, and that there is negligible loss in the shield. Thus, by conducting two measurements, one in free-space and one within the Wheeler cap, the radiation efficiency can be evaluated as follows: 239

260 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 24 Small Antenna L tune C a R l Z in R r Figure E.1: Equivalent circuit of a small electric dipole [151]. We denote the total antenna resistance measured during the free-space measurement as R FS = R r + R l, and the loss resistance measured during the Wheeler cap measurement as R WCap = R l. The radiation efficiency can therefore be expressed in terms of the two measured resistances as η rad = R r +(R l R l ) R r + R l = R r + R l R r + R l =1 R l R r + R l R l R r + R l =1 R WCap. (E.2) R FS Implicit in Wheeler s method of isolating the loss resistance from the radiation resistance is that the two appear in series within an equivalent circuit of the antenna. Thus, Wheeler s model for an electrically small electric dipole takes on the form of Figure E.1, where C a is the inherent low-frequency capacitance of the antenna and L tune is an externally added inductance. By adding the external inductor in series with the antenna capacitance, the antenna can be resonated, therefore transferring power directly to the radiation resistance. Thus, the maximum radiation efficiency will occur exactly at the resonance of the antenna, where maximum power is transferred to the radiation resistance, which then manifests itself in the form of radiation. For a series RLC resonator, the real part of the input impedance will not change, and only its reactive part will change with frequency. Therefore, when viewed on a Smith chart, the input impedance will follow a constant resistance circle, and will be inductive above resonance and capacitive below resonance. However, it is well known that the radiation resistance of a dipole will not remain constant with frequency, but is in fact

261 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 241 proportional to the frequency squared, according to R r =2π 2 (l/λ) 2 [19] (p. 145). The loss resistance, which is represented by the surface resistance, is also proportional to the square-root of the frequency, according to R s = ωμ /2σ [19] (p. 79). Therefore, when considering the input impedance of a realistic antenna, R r + R l in Figure E.1 will not remain constant with frequency, and thus the input impedance will deviate from the response of an ideal series RLC resonator. Chu, in his landmark paper [152], presented a slightly different equivalent circuit for a small electric dipole, which accounts for this change in the value of the radiation resistance with frequency. Nevertheless, Chu himself uses the exact same simple series RLC circuit of Figure E.1 to approximate the behaviour of a multi-mode (TM n ) antenna in the vicinity of its resonance. In fact, he presents closed-form expressions that relate the input impedance of an antenna, expressed in terms of n-th order spherical Bessel functions, to the individual RLC components of the equivalent series RLC circuit. Therefore, it can be concluded that the equivalent circuit of Figure E.1 adequately models the characteristics of an electric antenna in the vicinity of its resonance, where it is desired to calculate the radiation efficiency of the antenna. The implication of Chu s use of an equivalent series RLC circuit for a multi-mode antenna, is that Wheeler s efficiency calculation method is not limited only to electrically small dipoles, but can be applied to any antenna whose input impedance in the vicinity of the resonance resembles the response of a corresponding series RLC circuit. Many researchers have exploited this fact and have used the Wheeler cap method to successfully measure the efficiencies of antennas that are not classified as being electrically small [ ]. More recently, the Wheeler cap method has received a lot of attention in the measurement of the efficiency of antennas on mobile handset devices, which are typically electrically quite large [ ]. E.2 A Modified Wheeler Cap Method by McKinzie The measurement of an antenna s input impedance in free-space and within the Wheeler cap will not in general have the same form as that of a series RLC circuit assumed in the Wheeler cap method, and this has been the main criticism of the original Wheeler cap technique. In order to remedy this problem, McKinzie has proposed a very simple technique that transforms the measured reflection coefficient data into the required form of an equivalent series RLC circuit in the vicinity of its resonance [154]. His technique

262 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 242 Wheeler_Cap..S(2,2)[p] Free_Space..S(1,1)[n] Wheeler_Cap..S(2,2) Free_Space..S(1,1) Z ' FS Wheeler_Cap..S(2,2)[p] Free_Space..S(1,1)[n] Wheeler_Cap..S(2,2) Free_Space..S(1,1) R WCap R FS Z ' WCap freq (3.4GHz to 3.7GHz) freq (3.45GHz (a) to 3.75GHz) freq (3.4GHz (b) to 3.7GHz) (. to.) freq (3.45GHz to 3.75GHz) (. to.) Figure E.2: Typical input reflection coefficient data used to illustrate the modified Wheeler cap method proposed by McKinzie [154]. (a) Raw data obtained from the measurement of an antenna in free space and within the Wheeler cap. (b) Rotated data of (a) such that the response of the antenna matches that of a canonical series RLC resonator. Exterior blue curve: Wheeler-cap data, and interior red curve: free-space data. involves first determining the resonant frequency at which the S 11 of the free-space measurement is a minimum, and denoting this as f FS. The corresponding impedance at the resonant frequency f FS is denoted as Z FS. Then, an ideal lossless transmission line is added to the input port of the antenna, whose length is adjusted until the impedance Z FS lies on the real axis of the Smith chart, thus becoming R FS. At this point, the measured reflection coefficient data is tangent to a constant resistance circle, which corresponds to the response of the desired canonical series RLC resonator. The Wheeler cap reflection coefficient data is also rotated by the same electrical length as the free-space data, and the point where this intersects the real axis of the Smith chart determines R WCap.This way both the free-space data as well as the Wheeler cap data have the desired canonical form of a series RLC resonator. The above procedure can be better understood by considering the representative example shown in Figure E.2. The measured input reflection coefficient data of an antenna in free space that has a minimum in its S 11 response at f FS is shown in Figure E.2(a). The impedance at f FS is denoted as Z FS, and it can be observed that around the vicinity of this resonance, the response of the antenna is dissimilar to that of a canonical series RLC resonator, which would ideally follow a constant-resistance circle on the Smith

263 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 243 chart as the frequency is varied. However, by adding an ideal transmission line with an electrical length of approximately λ/8 atf FS, the free-space measurement data can be rotated clockwise along a constant VSWR circle to reach the purely real point of R FS in Figure E.2(b). The response of the rotated free-space data, although not perfect, now closely resembles that of a canonical RLC circuit. The same rotation is then applied to the Wheeler cap measurement data, which typically exhibits a slightly different minimum S 11 resonant frequency of f WCap. Thus, the complex impedance of Z WCap is transformed to the purely real resistance of R WCap at f WCap. Finally, Equation (E.2) can be used to determine the measured efficiency using the two determined values of R FS and R WCap. As mentioned earlier, the main criticism of the Wheeler cap method for evaluating the radiation efficiency is that the input impedance of a given antenna will not necessarily conform to that of a canonical series RLC resonator, therefore the technique s applicability is not universal [153], [156], [159]. In order to address this issue, a generalized Wheeler cap method was developed by Johnston and McRory [159], which does not require any preconceived knowledge of the antenna s equivalent circuit. The antenna is simply assumed to be a linear two-port network that resides between the input feed line of the antenna and free space. It is also assumed that the antenna operates in a single mode when radiating into free space. Johnston s technique involves placing the antenna within a closed rectangular waveguide that has a sliding short on one end. The measured input reflection coefficient of the antenna for different positions of the sliding short, i.e. different cavity sizes, will describe a circle on the Smith chart, whose centre and radius will be a function of the two-port S-parameters of the antenna. The indirectly-measured S-parameters can then be used evaluate the efficiency of the antenna. Due to the increased complexity of this method compared to the one proposed by McKinzie, as well as the fact that this method is more suitable for measuring large ground-plane-mounted antennas, where the ground plane forms one of the waveguide sidewalls, it was decided to measure the radiation efficiency of the small metamaterial antenna using the McKinzie technique, while keeping in mind that Johnston s technique could be used for other more complex antennas. As will be demonstrated shortly, this choice was well justified, since the characteristics of the metamaterial antenna match closely to those of a series RLC resonator.

264 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 244 E.3 Practical Considerations When carrying out measurements using the Wheeler cap method, there are some practical issues that should be taken into consideration. The first relates to the Wheeler cap itself, and how well it acts to eliminate the radiation resistance from the input impedance of the antenna without disturbing the near fields. Since a perfectly conducting Wheeler cap is not possible to practically achieve, a reasonable alternative is to use a metallic enclosure. Wheeler points out that In practice, the size, shape and material (of the radiation shield) are not critical. A cylinder with one or both ends open may suffice. [151]. Therefore, from the onset Wheeler determined that the properties of the cap did not affect greatly the results of the efficiency measurement, something that other researchers have subsequently verified theoretically as well as experimentally in their own investigations. Newman et. al. [16] used cubic Wheeler caps of different sizes made of copper or aluminium, and found that their shape, size and material varied the measured efficiency by only ±2%. They also found that with a larger cap, the input resistance was easier to read, therefore they used the largest of the caps that they tested. Vu et. al [158] investigated in simulation the effects of varying the size of rectangular, cylindrical and spherical caps placed around a PIFA, and found that the efficiency varied by less than 1%. They also noted that in order to accurately extract the loss resistance of the antenna from the Wheeler cap measurement, it was necessary to operate below the first resonant mode of the cavity, although many of the results that they presented were actually taken below the first resonant frequency, and matched quite well with ones taken above the first resonant frequency. Huang et. al. [161] pointed out that working below the first resonance is acceptable, as long as the operating frequency of the antenna does not coincide with one of the cavity resonances. They suggested that the cavity resonances can be tuned away from the antenna operating frequency by employing a cavity with a sliding wall. Alternatively, cavities of different fixed sizes could be used. Another property of the Wheeler caps that has been the subject of many investigations, is the loss that the metallic cavities incur. Smith [162] investigated the effect of a change in the conductivity of a spherical cap enclosing a circular loop antenna, and found that the conductivity does not have to be very high in order to produce accurate efficiency results. He did caution, however, that for electrically small caps, the error associated with the method can be quite large, since the loss resistance can be comparable

265 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 245 to the radiation resistance of the antenna. A natural conclusion from this observation is to therefore restrict the minimum radius of the spherical cap to λ/(2π), as suggested by Wheeler, in order to keep the cavity losses to a minimum and to avoid disturbing the near-fields of the antenna. A thorough analysis of the electromagnetic coupling effects between a dipole antenna and a rectangular Wheeler cap made of copper has been presented by Huang et. al. [161]. It was shown that the input resistance is actually the sum of the loss resistance of the antenna and the coupling resistance between the antenna and the cavity. It was also shown that around the cavity resonances the coupling resistance can attain values in the order of 1 3 Ω, thus resulting in unphysical negative values of the radiation efficiency. Off resonance, however, the coupling resistance is in the order of 1 2 Ω, thus verifying that when the antenna operates outside of the cavity resonances, the Wheeler cap method produces accurate efficiency results. More recently, Johnston [163] has used numerical simulations using WIPL-D to separate the losses in the antenna from the losses in the cavity, and has shown that for antennas that have efficiencies less than 95%, the cavity losses introduce minimal errors in the efficiency results. Most of the antennas found in the literature that were tested using the Wheeler cap method employed relatively large ground planes. These include microstrip patch antennas, monopoles and small loop antennas, whose ground planes were used to form one side of the Wheeler cap. For instance, in [153] the ground plane of the microstrip patch antenna formed the bottom face of the hemispherical cap, and in [159] and [161] the monopole ground plane formed one of the waveguide sidewalls. From a construction standpoint, it is therefore very important that the cap makes good electrical contact with the ground plane in order to prevent any unwanted energy from leaking out of the cavity during the loss resistance measurement. Some authors have used flexible metallic fingers similar to EMI shielding gaskets in order to easily install and remove the Wheeler caps while still maintaining a good electrical contact [154], [16], while others have used conductive strips such as copper tape to ensure good electrical contact [153]. Other options to ensure that a perfectly sealed cavity is obtained could include installing conductive foam, metalized fabric or knitted wire mesh gaskets at the junctions of the cavity. Finally, although most of the implementation details of the Wheeler cap technique pertain to the cap itself, the second measurement carried out in free space also deserves a special note. As pointed out by Newman et. al. [16], the environment in which the free-space measurement is conducted can greatly affect the value of the total resistance

266 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 246 R FS = R r + R l, especially when other conducting objects or people are in close proximity to the antenna. Therefore, this measurement should preferably be taken in an anechoic chamber in order to avoid any external influences on the value of R FS. E.4 Measured Reflection Coefficient Data Based on the practical considerations outlined in the previous section, a spherical Wheeler cap was chosen in order to evaluate the radiation efficiency of the metamaterial antenna. Since the antenna is electrically small and its ground plane measures only.45λ.45λ, it was deemed impractical to use the ground plane to form one of the Wheeler cap faces. Therefore, as Wheeler initially suggested, a spherical cap was chosen that enables the small antenna to be placed at the centre of the sphere, therefore capturing all of the electromagnetic effects from the antenna, including any effects caused by the small, finite ground plane. Three Wheeler caps of different sizes were constructed, as shown in Figure E.3. Each consisted of two almost hemispherical stainless-steel bowls, which were placed on top of each other and fastened in order to form the spherical caps. A hole was drilled at the centre of the top bowl, and a bulkhead Jack-Jack SMA connector was fastened into place. The coaxial feed cable from the Vector Network Analyzer (VNA) was attached to the exterior of the SMA connector, while a semi-rigid coaxial cable whose length was slightly less than the radius of each bowl was attached to the interior of the SMA connector. This enabled the coaxial feed line to extend into the centre of each bowl, where the antenna was mounted. It should be mentioned that the extra length of semi-rigid coaxial cable was included in the calibration of the system in order to bring the reference plane of the measurements up to the input of the antenna. The complete spectrum of the modes of a spherical metallic cavity can be calculated analytically, of which the three lowest-order transverse magnetic (TM) modes are degenerate. Therefore, the lowest-order resonant frequency for a spherical cavity can be written as [164] (pp ) f tm mnp = ζ np 2πa μɛ = f 11 tm tm tm (even) = f111 (even) = f111 (odd). (E.3) Here, ζ np represents the p zeroes of the derivative of the n-th order spherical Bessel function, and for n =1andp =1,ζ 11 = Additionally, m, n, p are integers, μ

267 Appendix E. Measured Efficiency Using the Wheeler-Cap Method cm 26 cm 13 cm Figure E.3: Various metallic spherical caps used to measure the radiation efficiency using the Wheeler cap method. and ɛ are the permeability and permittivity of the medium within the cavity, and a is the radius of the cavity. Therefore, from Equation (E.3) the lowest cutoff frequencies for spherical cavities with diameters of 38 cm, 26 cm and 13 cm, similar to each of the Wheeler caps shown in Figure E.3, are f c1 =.69GHz,f c2 =1.1GHz,andf c3 = 2.2 GHz, respectively. Although the bowls used were not perfectly spherical, the above results give a reasonable indication of the approximate location of the cutoff frequency of each cavity, below which there will be no cavity resonances. Since the operating frequency of the metamaterial antenna is around 3 GHz, the above calculations suggest that some cavity resonances will inevitably be encountered if either of the three Wheeler caps are chosen. This was indeed the case, and it was observed that the number of cavity resonances increase with the size of the Wheeler cap, as was expected. Nonetheless, in the vicinity of 3 GHz the response of the larger Wheeler cap with a diameter of 38 cm was the only one that did not exhibit a cavity resonance. Therefore, this cap was chosen in order to carry out the efficiency measurements of the metamaterial antenna. First, the free-space reflection-coefficient measurement of the metamaterial antenna was conducted within the anechoic chamber at the University of Toronto, as shown in Figure E.4(a). Note that the small semi-rigid coaxial cable of length 1.5 cm as well as the bulkhead Jack-Jack SMA connector were included in this measurement in order to be consistent with the Wheeler cap measurement. Subsequently, the antenna was

268 Appendix E. Measured Efficiency Using the Wheeler-Cap Method 248 AUT 25 cm 38 cm (a) (b) AUT (c) (d) Figure E.4: (a) Measurement of the antenna under test (AUT) in free space, (b) sealed spherical Wheeler cap indicating its dimensions, (c) measurement of the AUT within the spherical Wheeler cap, and (d) view of the AUT within the spherical Wheeler cap. mounted inside the spherical Wheeler cap as shown in Figure E.4(d), and the cap was carefully sealed around the junction of the two bowls with a strip of copper tape, followed by closely-spaced metallic clips, as shown in Figure E.4(b). Finally, the Wheeler cap measurement was taken as shown in Figure E.4(c). Figure E.5(a) shows the input reflection coefficients for both the free-space measurement and the Wheeler-cap measurement, where it can be observed that even though the responses follow a smooth circular trajectory, they do not conform to the characteristics of a series RLC resonator. The minimum S 11 frequency of the free-space measurement occurred at a frequency of f FS = 3.8 GHz, while the minimum S 11 frequency of the Wheeler-cap measurement occurred at a slightly shifted frequency of f WCap =3.9GHz. The McKinzie technique was then used to rotate the free-space data along a constant