ANALYSIS AND DESIGN OF COUPLED-OSCILLATOR ARRAYS FOR MICROWAVE SYSTEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Renaud Moussounda, BSEE,MSEE Graduate Program in Electrical and Computer Engineering The Ohio State University 2013 Dissertation Committee: Prof. Roberto Rojas, Advisor Prof. Patrick Roblin Prof. Fernando Teixeira UMI Number: 3710325 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3710325 Published by ProQuest LLC (2015). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 c Copyright by � Renaud Moussounda 2013 ABSTRACT The concept of synchronized nonlinear coupled oscillators is applied to microwave and antenna engineering for the analysis and design of wireless communication and sensing systems operating at the microwave and/or millimeter (mm)-wave frequencies. The significance of such approach is justified from the potential gain in eﬃciency, weight, cost and functionality although technical challenges stand in the way. Unlike typical phased array systems, which are currently used to construct such systems, coupled-oscillator systems present additional challenges that mainly arise from maintaining stability and synchronization as the the coupled nonlinear system is operated. Linear systems do not present such stability issues and are consequently faster since they do not rely on any gradual synchronization mechanism in order to function. However, at significantly higher frequencies in the quasi-optical domain, coupled-oscillator systems can make up for the speed diﬀerence and present significant eﬃciency advantages over typical phased array architectures. In addition, coupled nonlinear systems possess inherent analog properties that can be used for a multitude of functions. This dissertation advances the topic of coupled-oscillator arrays by 1) developing an alternative set of techniques for designing the oscillating unit cells called active integrated antennas (AIAs) at microwave or mm-wave frequencies, 2) developing a more accurate description of the dynamics of the array, 3) developing and implementing a new topology for a coupling network that is able to extend stability, 4) ii implementing a fully non-reciprocally coupled array able to produce large scan angle without loss of stability, 5) proposing an architecture based on a single phase-locked loop (PLL) and containing a self-calibration mechanism, and finally 6) implementing a phase-boosting mechanism using simple circuits to amplify the phase diﬀerence between adjacent radiating antennas in order to increase scanning range. iii This dissertation is dedicated to my family, especially to my mother and father who I will always love. iv ACKNOWLEDGMENTS First, I would like to thank Professor Roberto Rojas for his guidance and insight for the last five years. I am grateful to him for believing in my abilities to complete this program. Second, I would like to thank all the members of my dissertation committee for taking the time and eﬀort to read and evaluate this work. Thank you Professor Roblin and Professor Teixeira for your eﬀorts. Lastly, I would like thank all my former and present colleagues at the ElectroScience Laboratory (ESL) for the insightful and stimulating discussions on various topics. Specifically, I would like to acknowledge Bryan, Khaled, Id, Keum-Su, Ezdeen, Aseim, Ethan and Professor Ismael Martinez for the memorable moments and the collaborative research work. You all had a positive impact on the fulfillment of this dissertation. v VITA 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BAC, LTNOB (High School), Gabon 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Montgomery College, MD (62 credits) 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BSEE, Georgia Institute of Technology 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MSEE, The Ohio State University 2009-Present . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research/Teaching Associate, The Ohio State University PUBLICATIONS Journal Articles R. Moussounda and R.G. Rojas, “Analysis of array of coupled nonlinear microwave oscillators for active integrated antennas,” (submitted for publication) Physical Review E. R. Moussounda and R.G. Rojas, “Parametric Design Techniques for Self-Oscillating Active Integrated Antennas”, (submitted for publication) IEEE Microwave Theory and Techniques, July 2013. A. M. Elfrgani, R. Moussounda, R.G. Rojas and K.-S. Song, “Stability Analysis of Non-Foster Circuits Based on Time-Domain Method,” (Submitted for publication), IEEE Microwave Theory and Techniques, July 2013. J.I. Martinez-Lopez, R. Moussounda and R.G. Rojas, “Non-reciprocal Coupling Network for Beam Steering Coupled Oscillator Arrays,” IET Microwave Antennas Propag. Vol.5, Is. 8, pg. 940-947, 2011. vi Conferences Presentations and Papers R. Moussounda, E. A. Elghannai and R.G. Rojas, “Harmonically Loaded Active Integrated Antenna Using Characteristic Mode Theory,” to be presented at URSI/USNC Meeting, Boulder, CO, Jan. 2014. A. M. Elfrgani, R. Moussounda and R.G. Rojas, “Time-domain Stability Analysis/Design of Negative Impedance Inverters and Converters,” Presented at Int. Microw. Symp, Seattle, WA, July 2-7, 2013. C.-C. Liu, R. Moussounda and R.G. Rojas, “ A 60-GHz Active Integrated Antenna oscillator,” presented at USNC-URSI/USNC Meeting, Boulder, CO, Jan. 2013. R. Moussounda and R.G. Rojas, “Phase-Amplifying Architecture for Coupled-Oscillator Arrays”, Presented at IEEE AP-S/URSI-USNC Conference, Chicago, IL, 2012. R. Moussounda and R.G. Rojas, “Parametric Design and Synthesis Technique to Minimize Phase Noise and Improve Power Eﬃciency for Active Integrated Antennas”, Presented at IEEE AP-S Int. Symp. & USNC/URSI NRS Meeting, Spokane, WA, July 3-8, 2011. J.I. Martinez Lopez, R. Moussounda and R.G. Rojas, “Nonreciprocal Coupling Network for Beam Steering Coupled-Oscillator Arrays,” Presented at the IEEE Int. Symp. Phased Array Systems Tech., Boston, 2010. FIELDS OF STUDY Major Field: Electrical Engineering Specialization: Electromagnetics vii TABLE OF CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x CHAPTER 1 2 PAGE Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Key Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . 1 5 6 Coupled Nonlinear Oscillators: Theory and Applications . . . . . . . . 9 2.1 Basic Oscillating Unit . . . . . . . . . . . . . . . . . . . . . 2.2 Self-regulation, Frequency Entrainment and Chaos . . . . . 2.3 Review of Nonlinear Oscillators in Phased Array Antennas 2.3.1 Active Integrated Antennas . . . . . . . . . . . . . 2.3.2 Coupled-Oscillator Arrays . . . . . . . . . . . . . . 3 . . . . . 9 11 14 15 18 Analysis and Design of Active Integrated Antennas . . . . . . . . . . . 27 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Parametric Modeling and Design Approach 3.2.1 Convexity of Antenna Immittance . 3.2.2 Feedback Termination Modeling . . 3.3 Application of Concepts . . . . . . . . . . 3.3.1 Design with Existing Antenna . . . 3.3.2 Design with Synthesized Antenna . 3.4 Fabrication and Measurements . . . . . . . 27 29 29 33 36 36 42 48 viii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 6 Dynamics of Coupled Nonlinear Oscillator Arrays . . . . . . . . . . . . 52 4.1 4.2 4.3 4.4 . . . . . . . 52 53 62 64 64 67 74 Design Consideration for Coupled-Oscillator Arrays . . . . . . . . . . . 79 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Coupling Considerations . . . . . . . . . . . . . . . . . . . . 5.2.1 Transmission-Line based Coupling Networks . . . . 5.2.2 Practical Implementation . . . . . . . . . . . . . . . 5.3 Design Method and Considerations . . . . . . . . . . . . . . 5.3.1 Counteracting Coupling Eﬀects . . . . . . . . . . . 5.3.2 Non-reciprocal Coupling Networks . . . . . . . . . . 5.4 Implementation of a Fully Nonreciprocal Coupled-Oscillator 79 80 80 87 89 89 97 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Array Coupled-Oscillator based Applications and Array Architectures . . . . . 106 6.1 6.2 6.3 6.4 7 Introduction . . . . . . . . . . . . . . . Amplitude and Phase Dynamics . . . . Transient and Stability Analysis . . . . Numerical Simulations and Verification 4.4.1 Frequency Deviation . . . . . . 4.4.2 Synchronization . . . . . . . . . 4.4.3 Comparison to Previous Models Introduction . . . . . . . . . . . . . . . Array Architecture . . . . . . . . . . . . Array Calibration and Pattern Shaping Phase-Amplifying Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 107 112 114 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.1 Summary . . . . . . . . . . . . . . . . . . . . . 7.2 Future Work . . . . . . . . . . . . . . . . . . . 7.2.1 Harmonically Loaded AIA and COAs . 7.2.2 Non-Foster Loads for Phase-Amplifying CHAPTER . . . . . . . . . . . . Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 129 129 130 PAGE A Coupling Properties of Resistive Lines . . . . . . . . . . . . . . . . . . 131 B Matlab/FEKO Co-Simulation Commands . . . . . . . . . . . . . . . . 134 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 ix LIST OF FIGURES FIGURE PAGE 1.1 Architecture of a typical phased array. . . . . . . . . . . . . . . . . . 3 1.2 Correspoding architecture of coupled-oscillator array. . . . . . . . . . 5 2.1 Basic model of oscillator. . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Self-regulation depicted by a limit cycle with �=0.5 and τ =20 . . . . 12 2.3 Van der Pol oscillator with injection current source. . . . . . . . . . . 13 2.4 Diagram (bifurcation) showing behavior of forced van der Pol oscillator for ωin /ω0 = 1.5 and F =1.2. . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Instance of a self-oscillating AIA. . . . . . . . . . . . . . . . . . . . . 17 2.6 Inter-injection phased array architecture . . . . . . . . . . . . . . . . 20 2.7 York beam-steering mechanism for weakly coupled systems . . . . . . 22 2.8 PLL-based architecture . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Plots of antenna locus and device lines . . . . . . . . . . . . . . . . . 32 3.2 Simple oscillator model with i(t) = A cos(ωt + φ). . . . . . . . . . . . 34 3.3 Parametrization of the input impedance of the drain terminal of a FET transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Real and imaginary parts of the input admittance of a patch antenna and its locus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Mapping of the convex region described by antenna locus into the transistor design space. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Time series of current signal and corresponding spectrum waveform. . 41 3.4 3.5 3.6 x 3.7 Schematic of circuit for obtaining behavior of circuit . . . . . . . . . 42 3.8 Plot of low-variation points (|∆Z|) . . . . . . . . . . . . . . . . . . . 44 3.9 Frequency derivative of total nonlinear core impedance at 5 GHz and 7 dbm RF level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Antenna impedance as a function of pin location and corresponding gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (a) Synthesized patch antenna input impedance around 5 GHz for f = 4.58 mm. (b) Corresponding impedance locus. . . . . . . . . . . . 47 3.12 Performance of AIA for 7 dBm output power. . . . . . . . . . . . . . 48 3.13 CAD generated layout implementing design of AIA at 5 GHz. . . . . 50 3.14 Spectrum measurement of the radiated power. The oscillation frequency is 5.018 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Circuit representation of N linearly coupled nonlinear oscillators . . . 54 4.2 Approximation of the normalized van der Pol oscillation frequency (dashed) with exact (solid). The deviation increases as the nonlinear factor increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.10 3.11 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Chain of RL-coupled van der Pol oscillators with nonlinearity described by i = −av + bv 3 . . . . . . . . . . . . . . . . . . . . . . . . . 67 Loss and gain of synchronization between oscillators as the tank capacitance of first oscillator is varied . . . . . . . . . . . . . . . . . . . 69 Phase diﬀerence φp − φk between adjacent oscillators showing a loss of synchronicity around C1 = 0.935 pF . . . . . . . . . . . . . . . . . 70 Phase diﬀerence between oscillators 10 and 11 from the chain of oscillators.The coupling strength varies from 0.0285 to 0.0292 across the array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Left: phase diﬀerence between oscillator 10 and 11. Right: amplitude comparison at oscillator 11 . . . . . . . . . . . . . . . . . . . . . . . 72 Left: phase diﬀerence between oscillator 10 and 11. Right: Amplitude comparison at oscillator 11.The coupling strength varies from 1.2739 to 1.4171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Amplitude and phase diﬀerence between oscillator 1 and 2. . . . . . . 75 xi 4.10 Comparison of phase diﬀerences in degrees between oscillator 1 and 2 using York’s equations, derived equations and exact dynamics . . . . 78 A pair of oscillators coupled through a transmission line-based circuit (model #1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A pair of oscillators coupled through a transmission line-based circuit (model #2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Comparison of quality factors for various values of the electrical length of the coupling transmission lines. . . . . . . . . . . . . . . . . . . . . 85 Comparison of coupling coeﬃcients for various values of the electrical length of the coupling transmission lines. . . . . . . . . . . . . . . . . 86 Two models of folded transmission lines for linking oscillators. (a) Modeled after a sine function. (b) A rectangular shape. . . . . . . . . 88 Top and bottom views of array of patch antennas Antenna Dimension: W=20.5mm, L=19.5 mm. Coupling line: Z0 = 100Ω. . . . . . . . . . 90 5.7 Flowchart describing optimization process by co-simulation . . . . . 91 5.8 Progression of the algorithm as the array converges to a uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.9 Plot and comparison of array factors for 5-element patch array. . . . 94 5.10 Final dimensions of the array design . . . . . . . . . . . . . . . . . . 95 5.11 Optimization of scan angle in the presence of coupling. . . . . . . . . 96 5.12 Architecture of half of the nonreciprocal coupling network . . . . . . 98 5.13 Architecture of a fully nonreciprocal array. . . . . . . . . . . . . . . . 100 5.14 Layout and fabricated array showing active components and bias lines .102 5.15 Measured power spectrum of array . . . . . . . . . . . . . . . . . . . 103 5.16 Spectrum of phase fluctuations of the array . . . . . . . . . . . . . . 104 5.17 Comparison of measured radiation pattern to expected simulated results.105 6.1 COA phase-locked loop based architecture. . . . . . . . . . . . . . . . 107 6.2 Performance comparison between typical and PLL-based COA . . . . 110 6.3 Phase dynamics of five-element array for typical and PLL-based COA 111 5.1 5.2 5.3 5.4 5.5 5.6 xii 6.4 COA architecture with self-calibration mechanism. . . . . . . . . . . 113 6.5 Tracking of array factor during calibration process. . . . . . . . . . . 115 6.6 Mechanical and electric models of Omnia Ochrace 6.7 Generic representation of coupling loads between oscillators. . . . . . 117 6.8 Phase diﬀerence and amplification factor of currents I2 and I1 . . . . . 118 6.9 Plots of AF and DF generated by sweeping reactive loads . . . . . . 119 6.10 Two microstrip patch antennas coupled through a phase-boosting network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.11 Simulation results for two coupled microstrip patch antennas. . . . . 121 6.12 Total array gain with and without the coupling circuit. . . . . . . . . 122 6.13 Layout of two-element patch array for phase amplification . . . . . . 123 6.14 Performance of phase-boosting circuit from two coupled patch antennas124 6.15 Phase-amplifying architecture of coupled-oscillator arrays. . . . . . . 125 6.16 Comparison of array factors of a 10-element array (uniform and phaseamplified). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.1 Coupling circuit # 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.2 Coupling circuit # 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.1 Plot of reflection coeﬃcient of antenna during optimization. . . . . . 137 xiii . . . . . . . . . . 116 CHAPTER 1 INTRODUCTION 1.1 Background As civilization progresses, there is a corresponding and increasing need to exchange information at faster rates while shrinking the physical sizes of communication systems. This continual shrinking of communication devices, demand for higher speed and efficiency, coupled with the requirement for higher power, integration and linearity, are becoming the dominating aspects of today’s engineering designs. These challenges are global and are encountered in handset devices, base stations, radars and satellite transmitters. Meeting such demands necessitates producing more bandwidth, reducing antenna size, volume or footprint, increasing multi-functionality, using faster switching circuits and coming up with superior information-coding schemes. In addition, given the fact that the vast majority of personal communication systems operate at the lower microwave bands and that more mobile users and channels have sprung up over time, overcrowding and interference at these lower frequency bands are slowly becoming pressing issues. One potential solution to some of the concerns mentioned earlier, is to operate at higher frequency bands. That would mean engineering a new class of devices and protocols using higher microwave and mm-wave frequencies. If that is the case, not only would the rate of information transfer be superior from the gain in instantaneous bandwidth, but there would also 1 be a substantial and corresponding reduction in circuits and antennas dimensions. The sizeable bandwidth expansion would not only be coveted by mobile operators, but also by anyone interested in building a variety of commercial and military applications/gadgets going from high-resolution mapping/imaging systems, weapon detection, navigation systems, long and short-range communication to automotive collision avoidance [1]. Over the last few decades, there has been an increasing push to mount highly sophisticated systems on moving platforms for surveillance, tracking, satellite imaging, remote sensing, automobile safety and so on. These systems can be seen on drones, airplane radars, high-resolution imaging systems and other various radar systems. Such systems require the ability to scan around or have a beam-scanning mechanism for fast and agile coverage of the scrutinized area. Typically such systems are built on phased array technology, which have the ability to create one or multiple radiation beams and an arbitrary large amount of power. These systems are customarily used at microwave and/or mm-wave frequencies to due the requirements for small size and weight However, as advantageous as the operation of systems at higher frequencies is, it is not without significant drawbacks. First, it is well known that the performance of solid-state devices is degraded as the frequency of operation is increased. For modern microwave transistors, for instance, a large drop in power gain and an increase in noise figure are widely documented aspects of poorer performance with frequency increase. Consequently, creating enough power eﬃciently at these frequencies using solid-state devices is a challenge. Secondly, the ensuing size reduction will generally limit the amount of the power that can be pushed through the system. This is mainly due to the thermal properties of the composing materials: heat dissipation or sinking becomes a major performance factor. Thirdly, there is a considerable increase in production costs 2 due to the need to use special low-loss material and advanced fabrication, design and modeling techniques. Finally, besides the higher material (ohmic) and radiative loss, other undesired eﬀects aﬀecting eﬃciency include an increase of mutual coupling, possible excitation of surface waves, and unwanted radiation from the rest of the system that might become comparable to that of the antennas. All these mentioned issues become more pronounced as the frequency gets higher. Historically, the generation of large amount of power has been done using phased antenna antennas. For typical phased array systems at lower frequencies, a single source, usually a klystron, generates a large amount of power that is split numerous times, individually phased, and fed to each antenna as shown in Figure 1.1. Yet, using the same approach at higher frequencies will be inadequate because the feeding network and phase shifters (or phasers) are simply too lossy and bulky. The loss creates a highly ineﬃcient system and the added bulkiness does indeed defeat the initial purpose of reducing overall size and weight. Fortunately, the concepts of quasi-optical Figure 1.1: Architecture of a typical phased array. 3 spatial power combining and active integrated antenna (AIA) have been introduced to overcome these fundamental limitations on output power and eﬃciency of semiconductors circuits at the mm-wave frequencies [2]. Figure 1.2 shows the modified phased array architecture using coupled oscillators. The main diﬀerence is that phase shifters and power distribution networks can be completely eliminated and power directly delivered to each antenna, thus making the system compact and highly eﬃcient even if a small amount of power is used for coupling and synchronization purposes. In general, the concept of AIA particularly deals with treating the antenna, the active device, and all embedded linear elements as part of the same monolithic entity. The antenna not only serves as a load, radiator and a part of the resonator, but also as a harmonic filter and tuning element without necessarily needing any additional matching network. This leads to a compact design with lower losses. Spatial power combining refers to the fact that by arraying many AIAs, power is combined in space and not within the circuit, thus completely eliminating loss due to feeding networks and transmission lines. These methods not only allow power to combine coherently and eﬃciently, but also reduce size by integrating and removing feeding waveguides/transmission loss [1, 3, 4]. Besides the benefits in eﬃciency and size, this approach also increases reliability and manufacturability due to the integration. The coupled-oscillator solution requires arraying multiple AIAs. Each AIA is a source contributing to the overall output power of the array. Producing an independent source requires building a standalone nonlinear oscillator. The fact that they are autonomous means that the AIA elements will produce diverse oscillation frequencies. In order to make sure that the overall radiated signal is coherent, all the oscillators will need to be synchronized. Synchronization is accomplished by either establishing a proper coupling mechanism or by providing a common driving or reference signal to the array. Consequently, an understanding of the dynamics of nonlinear coupled 4 Figure 1.2: Correspoding architecture of coupled-oscillator array. oscillators is crucial in providing an elegant solution to high-frequency phased array antennas. Apart from considerable nonlinear dynamics, designing coupled-oscillator arrays encompasses many areas of knowledge since it requires the use of microwave engineering, solid-state devices, nonlinear dynamics, circuits and antenna engineering. In other words, it is a highly interdisciplinary endeavor. 1.2 Key Accomplishments The work presented in this dissertation advances our current understanding of coupled nonlinear oscillators as microwave systems by providing: • An alternative parametric design approach for active an integrated antenna that uses the convex-like properties of the input immittance of resonant narrowband antennas. • A derivation of a set of expandable diﬀerential equations that help address the inaccuracy issues of other popular equations of coupled-oscillator systems, especially when nearing bifurcation points. A clear advantage of these equations 5 is that all quantities are expressed in terms of directly measurable network parameters. • The implementation of a nonreciprocal coupling network for resolving the scanning limitation of coupled-oscillator systems. • An implementation of a three-element oscillator array that is fully non-reciprocally coupled for the purpose of generating the largest achieved phase diﬀerence without any loss of stability. • A proposed architecture based on a single phase-locked loop (PLL) that not only helps stabilize the array but also contains a self-calibration mechanism. • The use of simple circuits between adjacent oscillators for the purpose of boosting a small phase diﬀerence into larger phase gradients for transmitting coupledoscillator systems. 1.3 Organization of Dissertation This dissertation is organized in the following manner. First, Chapter 2 introduces the reader to the concept of nonlinear oscillators and their various uses in modeling various phenomena in multiple fields of study. The discussion is naturally steered towards electric oscillators where a simple analysis is carried. The concept of frequency entrainment, synchronization, noise and chaos are introduced with forced or coupled oscillators. Later, the topics of active integrated antennas (AIA) and coupledoscillator arrays (COA) are reviewed, and various past research are presented. Chapter 3 is concerned with the analysis and design of active integrated antennas, which form the building blocks of many coupled-oscillator systems. The analysis method is highly parametric and applicable to a wide number of oscillatory systems 6 that present similar characteristics. Due to the limited degrees of freedom available to the design of highly integrated AIAs, the presented techniques focus on some particular properties of the integrated antenna that directly aﬀect the output power, phase noise and power eﬃciency of the system, making these techniques eﬃcient for quickly achieving a working first-cut design on which to carry an optimization process. A prototype active antenna is built and its performance is measured to show agreement with the developed analysis. Chapter 4 is theoretical and numerical by nature and focuses on deriving more accurate equations for the study of coupled nonlinear microwave oscillators. Using averaging methods, the voltage amplitude and phase dynamics of an arbitrarily coupled array is derived with all quantities expressed strictly in measurable network parameters, in a form familiar with antenna and microwave engineers. Numerous simulations are carried out to verify the accuracy of the equations. Chapter 5 deals with the analysis and design of the coupling system of an oscillator array. Coupling properties of various transmission-line based networks are addressed and other types of coupling mechanisms are discussed. A non-reciprocal coupling structure is presented along with its implementation. An array design process using various software tools is presented along with the implementation of a fully nonreciprocally coupled array at 5 GHz. Chapter 6 mainly addresses the various types of proposed architectures for coupledoscillator systems. In order to have a robust system, an architecture based on PLL is presented along with a phase calibration mechanism. Another architecture based on the hearing system of a fly is presented that amplifies small voltage phase diﬀerences into large current phase gradients that are radiated. This type of phase-boosting feature is beneficial to the stability of the array in general. 7 Finally, Chapter 7 summarizes the contributions of the dissertation and suggests future area of research related to this work. 8 CHAPTER 2 COUPLED NONLINEAR OSCILLATORS: THEORY AND APPLICATIONS 2.1 Basic Oscillating Unit Oscillators are a ubiquitous part of our existence as they can eﬀortlessly be observed in nature, in many fields of study, as well as in our technological devices. In nature, the migration of birds, seasonal flora and fauna and other cyclical spectacles can be easily noticed. A simple pendulum in mechanics, firing neurons and sleep-wake cycles in biology, charge-density waves and Josephson junctions in condensed-matter physics, just to name a few, are all example of systems exhibiting oscillatory properties. A clock or the signals produced by many electronic devices also display oscillatory states. In electrical engineering and related fields, oscillators are used internally as power sources, and in the simplest case, they can be modeled as a basic RLC circuits with additional nonlinear devices as shown in Figure 2.1. The active device needs to have an operating range where negative diﬀerential resistance1 is observed. In general, negative resistance is a property observable in many semiconductor devices, and it is used to make high-frequency amplifiers and oscillators. The Gunn eﬀect, for instance, 1 Negative diﬀerential resistance corresponds to a region where an increase in applied voltage leads to a decrease in current flow (r ≡ dV /dI < 0). This is diﬀerent from the static resistance (R ≡ V /I), which is always positive. For ohmic devices both quantities are indistinguishable. 9 is a well-known occurrence of negative diﬀerential resistance that first puzzled its discoverer because of the unexpected observed oscillations [5]. Figure 2.1: Basic model of oscillator. A study of the circuit above concludes that it is fully described by the following diﬀerential equation: d2 v 1 − 2 dt C � din +G dv � dv v + = 0, dt LC (2.1) where G = 1/R. For many active devices, a cubic or van der Pol nonlinearity is common and therefore the electric current entering such devices can be written as in = av + bv 3 , (2.2) where a and b are positive constants. The initial diﬀerential equation can now be written in the following generalized form: � � ẍ − � 1 − x2 ẋ + x = 0, 10 (2.3) with the related quantities given by dx ẋ = dτ τ = ω0 t � 3b x=v a−G a − G �= ω0 C (2.4) √ Here, ω0 = 1/ LC, and the oscillator is clearly nonlinear since its properties are function of the amplitude of oscillation and characterized by a limit cycle. 2.2 Self-regulation, Frequency Entrainment and Chaos Nonlinear oscillators have the property to self-regulate and create what is called a limit cycle. Figure 2.2 shows a limit cycle in the phase space based on equation (2.3) for � = 0.5 and τ = 20. The x-axis represents x, while the y-axis is ẋ. It is observed that regardless of the initial conditions, each trajectory ends up describing the same path after a suﬃcient amount of time. This is the basic property of stable limit cycles. In other words, trajectories near a limit cycle will be attracted to it and consequently small disturbances from the limit cycle will slowly decay over time. The limit cycle describes a control mechanism for the amplitude of oscillation such that it is maintained over time. However, free-running (uncoupled) oscillators are autonomous systems, meaning that any time-shifted version of the oscillating signal is also a valid solution. Unlike the amplitude, there is no controlling or restoring mechanism for the phase of the oscillatory signal. This means that disturbances in the phase are not corrected leading to oscillators having generally more phase noise than amplitude noise. Nonlinear oscillators also display a richer behavior when driven or coupled to 11 Figure 2.2: Self-regulation depicted by a limit cycle with �=0.5 and τ =20. Abscissa x, ordinate ẋ other oscillators. An interesting property of nonlinear oscillators is their ability to synchronize to an external signal. This property is also known as frequency entrainment or injection locking. Besides synchronization, other phenomena ranging from period doubling to deterministic chaos can be induced even with a simple circuit. To demonstrate those properties, the basic oscillator shown previously is now forced to operate with an independent source as depicted in Figure 2.3. Historically, van der Pol studied similar circuits using triodes [6]. As in Figure 2.1, this second circuit can be described by the following generic equation ẍ − �(1 − x)ẋ + x = F cos where the factor F is given by 12 � ωin τ ω0 � , (2.5) Figure 2.3: Van der Pol oscillator with injection current source. ωin F = I0 2 ω0 C � 3b . a−G (2.6) For an arbitrarily frequency ratio ωin /ω0 = 1.5, it is seen that the oscillator perfectly locks to the injection frequency up to about � = 3, as shown in Figure 2.4. Deterministic chaos then occurs, followed by alternating small regions of periodicity and chaotic behavior. For � = 5.28 to 5.75, there is a period doubling state where sub-harmonic (ωin /2) of the injection signal dominates. For � = 4.49 to 4.63, smaller sub-harmonics occur, including ωin /5. In general, the oscillation frequency locks to ωin × p/q, where p and q are integers. In 1927, van der Pol and van der Mark called this phenomenon “frequency de-multiplication” [7] in the regions where sub-harmonic frequency locking occurs. They did not recognize the nature of the irregular noise they heard between locking regions. Today, it is safe to assume that the noise was from the chaotic properties of the circuit. The occurrence of chaos is common for forced nonlinear oscillators and such eﬀect can be easily reproduced in simple mechanical systems, as well, using a driven damped pendulum, for instance [8]. When multiple oscillators interact, all of the above phenomena are observed in 13 Figure 2.4: Diagram (bifurcation) showing behavior of forced van der Pol oscillator for ωin /ω0 = 1.5 and F =1.2. addition to states of full and partial synchronization. Except for special applications in cryptology and information security, chaotic systems are rarely used [9]. However, non-chaotic frequency entrained systems are widely used for various applications [10]. In phased array systems, coupled-oscillators present an attractive solution for microwave and millimeter-wave power combining applications as it will be discussed next. 2.3 Review of Nonlinear Oscillators in Phased Array Antennas In order to reproduce and take advantage of the useful properties of coupled nonlinear oscillators in phased arrays and related technologies, it is necessary to come up with 14 a set of independent oscillatory units. For monolithic and high-frequency systems, these units are known as self-oscillating active integrated antennas (AIAs). 2.3.1 Active Integrated Antennas Active integrated antennas refer to circuits in which antenna elements and nonlinear devices are integrated to form a monolithic transmitting and/or receiving system [1, 3, 4, 11]. The integration is of paramount importance because it is well known that assembling pre-fabricated circuits to construct a single system creates discontinuities that lead to detrimental parasitic eﬀects. This issue is mitigated by having a monolithic design approach where all circuits are fabricated on a single layout. This ensures that no discontinuities are created, thus resulting in a more predictable high-frequency performance. However, designing and fabricating such circuits require hybrid modeling software [12] and precise fabrication techniques. In general, AIAs can be classified by their RF functions, which are: (1) amplification, (2) signal generation (oscillators) and (3) frequency conversion. For the first type, the antenna is located at the output of an amplifier. In this case, the antenna also plays the role of the matching network and therefore there is no need to match to the standard 50 Ω, as the matching is done directly. The second type is the selfoscillating active antenna. Here, the required signals are synthesized locally by the active device and the antenna. For highly integrated systems, the antenna not only acts as a radiator but may also act as part of the resonator (tank), matching network or as a filter that implementing harmonic loads for improving DC-to-RF conversion eﬃciency [13]. Since antennas radiate, they cannot have a very high quality factor (Q). However, oscillators require high-Q resonators; therefore, the antenna can be part of the tank circuit, but other components should be present to help increase the overall Q of the oscillator. Besides power generation, self-oscillating AIAs circuits can 15 also be used for other functions such as polarization-agile switching, beam-switching using injection-locking for radar, and omni-directional coverage [4]. Finally, frequency conversion types are mainly used in receivers where emphasis is on minimizing conversion loss. The bulk of this work is on the second type of AIAs since they are autonomous nonlinear oscillators capable of generating all nonlinear phenomena mentioned earlier. Except for the high integration, additional roles of the antenna and its properties, designing a self-oscillating AIA is similar to dealing with a typical microwave oscillator. Various working prototypes of active antennas have been developed [12, 14–16] and Figure 2.5 shows one typical topology. Here a patch antenna is integrated with a transistor, forming a monolithic self-oscillatory system. Many linear and nonlinear techniques have been applied to design microwave oscillators [17–19]. For nonlinear techniques, the device-line technique [20] is used to measure the large-signal properties of a negative resistance mono-port in a non-oscillating mode of operation. In essence, it consists of measuring the reflection coeﬃcient of the mono-port as a function of the amplitude and phase of an injected signal. Other measurement-based processes use active-load techniques that provide more flexibility and speed over the device-line method. Using these newer techniques, the active device can be fully characterized in its large-signal regime [21, 22]. These techniques can produce wide-ranging measurement data that is in turn used to design an oscillator. These techniques bypass the need to have an accurate nonlinear model of the active device. For designs that require an accurate nonlinear model, some techniques based on auxiliary generators are used to control the output voltage waveforms [23]. Finally all of these nonlinear techniques have shown the relevance of using harmonic loading to improve either the output power or the power eﬃciency for nonlinear circuits such as oscillators and power amplifiers [16, 24–27]. 16 Figure 2.5: Instance of a self-oscillating AIA. AIAs are systems within systems and they require many design considerations. These types of active antennas are particularly useful in quasi-optical power combining applications where their arraying and integration help reduce physical size, eliminate the need for high-power solid-state sources, as well as increase output power and system eﬃciency. Current AIAs are based on solid-state devices such as transistors and diodes as opposed to vacuum or gas-discharge tube devices. Diodes reach higher operating frequencies with higher RF output power, but transistors have higher DCto-RF conversion eﬃciency, operate at lower DC input power, and provide many diﬀerent functions—switching, tuning and amplifying—that would otherwise require several types of diodes. Choosing an active device depends on the end application and many other factors that range from cost, output power, operating frequency, eﬃciency, noise to stability and biasing requirements. Selecting the antenna’s geometry 17 rests on many factors that mainly include ease of integration with the rest of the circuit, bandwidth, size, array topology, impedance and radiation pattern profiles. Furthermore, antennas on thick substrates will tend to excite surface wave modes that can lead to poor radiation eﬃciency. Scan-blindness and mutual coupling are other aspects of the design to take into account in order to avoid undesirable eﬀects in an array environment [28]. All these issues have to be considered when designing a single or many AIAs. Finally, in high-frequency oscillator design, noise plays a particularly critical role as all active devices produce noise with a significant time-varying power level. Noise is a limiting factor of low-power, high-performance electronic circuits. A microwave oscillator can also be viewed as a dedicated noise generator followed by a resonator/filter to funnel a finite amount of power into a small frequency band. Noise can be separated into AM and PM components near the carrier frequency. Oscillators are mainly concerned with phase noise as amplitude fluctuations are relatively small, but become significant away from the carrier [29]. Phase noise near the carrier frequency is due to a wide variety of correlated phenomena caused not only by the active device, but also by the rest of the embedded circuit and biasing sources [30, 31]. A fair amount of nonlinear dynamics is crucial in simplified analysis where bifurcation points can be monitored in order to track stability transitions. In actual nonlinear systems, chaotic behavior may ensue, and it is important to diﬀerentiate it from typical electronic noise [10]. 2.3.2 Coupled-Oscillator Arrays Coupling many AIAs into a single system creates a coupled-oscillator array (COA). Such systems have interesting properties [32] and in general, coupled nonlinear oscillators have been used to model many phenomena in physics, engineering, biology 18 and finance [10, 33]. For example, the cardiac pacemaker, the flashing of fireflies [34], menstrual cycles in groups of women [35, 36], arrays of laser diodes [37], Josephson junction arrays [38,39], phase-locked loops, heart cell synchronization [40], neural networks, economies, immune systems and ecosystems, and so on, are all able to display a synchronous behavior through mutual interaction. Synchronization is a property that is only observed in interacting or coupled nonlinear systems. Such systems selfregulate so that deviations from a common or intrinsic behavior are eliminated. In electrical engineering and related circles, the terms frequency entrainment, injection locking and phase-locked loops (PLL) are all describing the synchronous property of a nonlinear system that is forced to operate in a slightly diﬀerent regime from its normal state. Practically, it is done to stabilize the oscillation frequency through the use of external reference signals so that information can be encoded/decoded and noise level reduced [41]. COAs are fundamentally diﬀerent from passive antenna arrays due to their nonlinear nature. They provide an analog computing layer capable of performing signal processing such as determining the direction of arrival of an incident signal. The same task on a typical phased array would require a signal processing chip or a computer. Historically, the study of COAs was preceded by that of grid oscillators [42]. Grid oscillators were periodic arrays embedded with active solid-state devices. The grid resided in a Fabry-Perot resonator in order to provide positive feedback necessary for oscillation. However, work on grid oscillators was mostly abandoned because they were intrinsically low-Q, and thus could not produce a tone with phase noise low enough for RF communication or radar applications [43]. When injection locking was found to help improve grid oscillator noise performance, most researchers invested their eﬀorts in studying COAs with injection-locking as a noise reduction mechanism [44, 45]. 19 As the research went on, a few properties of injection-locked oscillators were already known. In 1946, Adler [46] theoretically showed that a weakly driven oscillator had a locking range—a region where the oscillator and the impressed signal synchronized without producing a beat frequency. Later, additional theoretical and experimental works were added, especially by Kurokawa, who extended Alder’s work for large-signal injection locking [47], multiple resonant circuits [48] noise [49] and stability analysis [47, 48]. All this work provided the foundation for researchers such as Mink [50], Stephan [51,52] and Dinger [53] to go from a proposal on quasi-optical power combining to low and high-frequency array prototypes. By bilaterally injecting a phased signal at both ends of an array, Stephan and Morgan [51] demonstrated that a phase diﬀerence (evenly divided between oscillators) can be generated and used for beam steering purposes. A clear advantage of this technique over traditional phased arrays was that beam steering was accomplished with only one phase shifter. However, since the total available phase was 3600 , the major limitation of this technique was that the more the oscillators, the smaller the scan angle and thus this technique was not scalable. Figure 2.6 shows the structure of such system. Figure 2.6: Inter-injection phased array architecture [51]. 20 Since then, it has been known that coupled-oscillator systems are crippled with a limited scanning range issue. Additional inherent issues are instabilities, inadequate modulation schemes, and slow data rates. After additional years of research, York and colleagues generalized many aspects of coupled oscillators by providing a set of amplitude and phase equations describing the dynamics of an array of coupled nonlinear oscillators [54, 55]. The amplitude and phase dynamics of each oscillator was provided by the following equations: N � dAk µωk � 2 ωk � = αk − A2k Ak + �kp Ap cos (θp − θk + Φkp ) dt 2Qk 2Qk p=1 N dθk ωk � = ωk + �kp Ap sin (θp − θk + Φkp ) , dt 2Qk p=1 (2.7) (2.8) where Ak is the amplitude and θk is the relative phase of the signal at the k-th oscillator. The parameters ωk and Qk are the free-running frequency and the tank quality factor of that oscillator, respectively. The parameter �kp is the coupling strength between oscillator k and p. Due to the relative simplistic nature of the model above, the equations serve to only give a qualitative description of the first-order dynamics of an actual oscillator system. More elaborate models include additional observed eﬀects such as frequency dependence and asymmetric properties of the nonlinear cores [56]. One of the most useful properties derived from York’s work was the discovery that, for a weakly coupled system, beam-steering can be scalable and accomplished without the need of any phase shifters, simply by detuning the free-running frequencies of the edge oscillators in equal but opposite directions [57] as shown in Figure 2.7. For a planar array, controls would only be needed at the edge oscillators. One main advantage of this beam-steering technique over the ones relying on injection-locking is that the total phase shift is not a function of the number of oscillators and therefore not restricted to 3600 (2π). However, there is still a limitation of π/2 in terms of the 21 maximum phase diﬀerence between two adjacent oscillators. For a λ/2-spaced linear array, that would correspond to a maximum scan angle of π/6. Explicitly, to generate a phase gradient of ∆φ, the end oscillators’ natural frequencies have to be oﬀset by �ω0 δω = , with �kp ≈ �, and Qk ≈ Q. 2Q Figure 2.7: York beam-steering mechanism (weakly coupled systems) [57]. Besides York’s technique, additional beam-steering techniques were introduced, mostly dealing with the control of the coupling angles between adjacent oscillators. Hwang and Myung [58] showed that a uniform phase gradient can be generated across an array by having the edge coupling phases be opposite to one another. Unlike York’s work, there was a linear relationship between the coupling phase produced and the corresponding phase gradient of the array, which were equal. One of the drawbacks of Hwang and Myung’s technique is that the two end oscillators are not allowed to radiate because their phase relationships are diﬀerent from the rest of the array. This drawback, however, was solved by Lee and Rojas [59] by using nonreciprocal coupling devices to connect these edge oscillators to the rest of the array. The maximum oscillator phase diﬀerence possible was also π/2 for both cases. Another seeding work on beam-steering for coupled-oscillator array was done by Heath [60], in which the author generalized beam-steering through manipulation of 22 coupling phases. Mainly, the author proposed two ways of performing beam-steering from the nonreciprocal manipulation of coupling phases. One technique involved introducing a pair of nonreciprocal coupling phases at the two edges of a linear array (same as in [59]), while the second technique had a nonreciprocal coupling circuit between every oscillator. The latter technique was shown to bypass the π/2 limitation of the other techniques, thus providing the most stable solution to coupled oscillators systems. The work also included a linear stability analysis for each technique. Additional beam-steering techniques for extending scanning range made use of frequency multipliers [61] or radiation at frequency harmonics [62]. The rest of beamsteering techniques in literature involved injection locking at either fundamental [63] or harmonic frequencies [64]. Although beam-steering is crucial in many applications, the overall shape of the radiation pattern is of similar importance. The natural frequency and coupling phase of each oscillator and coupling network, respectively, can be individually controlled in order to change the phase distribution across the array, which in turn changes the radiation pattern. This property has been used to create adaptive radiation patterns with simultaneous beams and nulls. For example, a diﬀerence pattern for scanning applications can be accomplished by modifying the coupling between the central elements in a linear array [65]. Other pattern shaping techniques that produced simultaneous beams and arbitrarily placed nulls were proposed by Heath [66] and Georgiadis [67] with an accompanying stability analysis. In terms of coming with additional ways to analyze and design COAs, many other authors have made significant advances. Pogorzelski and colleagues proposed a simplified continuum model of coupled oscillators that treats the relative phases of the oscillators as a continuous function over the array satisfying a diﬀusion-type 23 partial diﬀerential equation [68, 69]. It was noticed that the phase dynamics equations were equivalent to the discretized Poison’s equations as seen in electrostatics. Their findings showed that transient performance of large arrays could be computed rather quickly using their techniques. For designing coupled-oscillator systems, a few more authors have proposed a set of numerical techniques based on harmonic balance analysis where the determination of the array’s properties are carried out using nonlinear models. The stability analysis is studied through a pole-zero identification and bifurcation-detection techniques [70, 71]. As coupled oscillators are built, electronic noise and instability are prevalent due to the intrinsic nature of oscillators. To increase mode stability and reduce phase noise, array architectures using PLL or mixers have been used [72, 73]. Figure 2.8 shows an example of such architectures. Although this architecture can theoretically achieve higher stability margins and lower phase noise, it is, however, complex due to the number active components required, and the additional cost might be prohibitive when the array is scaled because of the larger number of devices that is necessary. Stability is also dictated by the physical and electrical properties of the coupling network used as shown in references [74, 75]. Specifically on phase noise, some works have used sub-harmonic injection locking as a phase noise reduction technique [76]. Furthermore, it was shown that the phase noise and the locking range of a COA improved when inter-element coupling was made stronger [77]. Finally, it was found that phase noise in a reciprocal N -element COA could be reduced to 1/N of that of a single oscillator [78]. In terms of the performance of a COA as a basic communication system, some works [79, 80] investigated the performance of an injection-locked COA in terms of data-rate limitations, modulation eﬀects on radiation pattern and the bit-error rate for one and two-dimensional arrays under BPSK modulation. As expected, it was 24 Figure 2.8: PLL-based architecture [72]. found that linearly coupled COAs have lower maximum data rates than traditional phased arrays at the same frequency. Moreover, the array transient or settling time increased as the number of elements increased. However, it was noted that the overall performance could be improved by having a stronger injection signal, more injection points and stronger coupling between oscillators, with all these additional solutions having their own tradeoﬀs. Finally, in addition to the fully synchronized mode where all oscillators are synchronized to the same frequency, coupled oscillators can also operate in a mode-locked state [81], where the array oscillates with a number of equally spaced spectral modes, with comparable amplitude and locked phases. This mode of operation is extensively used in lasers to produce short and high-power optical pulses. When applied to COA, high-power RF pulses can be generated. In summary, despite the inherent challenges emanating from operating COAs, they remain attractive because of their simplified control architectures, higher eﬃciency, lower weight as supposed to traditional phased 25 arrays, and the potential of integrating all required circuitry in a chip, apart from the antennas. 26 CHAPTER 3 ANALYSIS AND DESIGN OF ACTIVE INTEGRATED ANTENNAS 3.1 Introduction Since the advent of high-performance solid-state transistors and diodes, designing microwave oscillators has been widely studied and accomplished using a variety of linear and nonlinear methods [17]. For nonlinear approaches, the device-line measurement technique has been used for some designs [21, 82], whereas more recently, active-load techniques have proven to be more reliable and eﬃcient for designing microwave oscillators [13, 83]. A nonlinear device can be fully characterized either by an accurate nonlinear model or by extensive measurements in the small and large-signal regimes. However, even with a complete characterization of a nonlinear device, designing a practical oscillator is still not straightforward due the possibility of multiple operating points which requires additional synthesis, optimization or tuning steps. Other additional steps might be needed in terms of selecting the loads, implementing the feedback terminations, choosing the bias voltages and fixing other embedded loads for optimal power, eﬃciency or noise criteria. Furthermore, in the case of active integrated antennas (AIAs), only a few authors have made use of the properties of the specific antenna when designing AIAs [12, 16, 84–86]. It should be noted that in the case of AIAs, the 27 nonlinear core is highly integrated with the antenna so that only assuming the usual quasi-TEM mode is erroneous as the antenna generates other modes that are not captured by a simple circuit simulator. This requires the simultaneous use of a full-wave electromagnetic (EM) solver along with a typical circuit solver. Consequently, generic oscillator design concepts, techniques and equations are indiscriminately applied to any AIA design, thus resulting in potentially suboptimal systems since the antenna is treated as a standalone load/or resonator, while ignoring potential key properties that can simplify and speed up the design process. The purpose of this chapter is to provide an alternative scheme to quickly get to an AIA’s operating points before the optimization or tuning steps—as always required for any practical design at microwave or higher frequencies. The analysis and design method that is presented is eﬃcient, multi-purposed, and amenable to any harmonic balance (HB) circuit simulator. It has been shown that harmonically loading an oscillator can improve the output power and power eﬃciency at the fundamental tone [24, 87]; however, this is the first time, to the best of the author’s knowledge, that the local properties (convexity) of the antenna’s immittance locus and related concepts are used to analyze and design a self-oscillating active integrated antenna system. It will be shown that regardless of the active device used, the appended antenna will vastly dictate the design performance. This work will show how the antenna eﬀectively draws up the boundaries of the possible design space and related restrictions. This work is split into two parts. The first segment deals with designing a selfoscillating active antenna given a previously available passive antenna. In this case, the input impedance or admittance of the antenna is already determined and cannot be modified. This can be valuable in situations where an antenna or an antenna array system needs to be retrofitted with newer front-end components. The second part of 28 this chapter is concerned with the simultaneous design, from the ground up, of both the antenna and the active core, thus presenting a general case of AIA design. 3.2 Parametric Modeling and Design Approach An oscillator can be regarded as a dedicated noise generator with a filter or resonator appended to the active device’s output in order to confine oscillations to a desired and usually narrow frequency band. Self-oscillating active antennas are usually built such that the antenna element is a functional part of the tank. In this case, the antenna plays a dual role, namely, a tank circuit as well as a transmitter (or receiver) of electromagnetic energy. This dual role of the antenna element can be problematic since a good antenna is expected to have a low Q; however, a superior oscillator must have a very high-Q tank circuit. Consequently, to reduce phase noise in AIAs, it is imperative to have high-Q embedded linear elements (terminal loads) in the oscillator in addition to the antenna in order to increase the overall or loaded Q. In other words, the antenna alone will not be good a good resonator so other linear elements around the transistors are needed to achieve a higher Q. In highly integrated systems—where there is no dedicated high-Q resonator—the antenna and the embedded elements provide the only degrees of freedom available to the designer apart from the voltage bias and frequency of operation which are usually pre-determined. 3.2.1 Convexity of Antenna Immittance This section will only consider resonant and narrowband antennas whose input immittance can be modeled, within a constant, as a series or parallel RLC circuit within a particular bandwidth. We’re interested in only narrowband antennas because they have to be part of the distributed resonator in order to help with the overall Q. Let’s define a bandwidth ∆ωC , characterized as the smallest frequency 29 interval ∆ωC = |ωb − ωa | such that ZL (ωa ) = ZL (ωb ), where ZL (ω) is the input impedance of the antenna in use. This specific definition has to do with the convexity of the antenna input immittance in the complex plane, and the justification for such definition will be provided shortly. In general, the input admittance of a passive antenna can be written as follows: YL (ω) = G(ω) + jB(ω). (3.1) If the antenna load can be approximately modeled as a parallel RLC circuit within a specific bandwidth, it follows that G(ω) ≈ G0 , (3.2) and B(ω) = ωC − 1 . ωL (3.3) The impedance of the antenna is therefore computed as ZL (ω) = 1 G0 −B(ω) +j 2 , ≈ 2 2 YL (ω) G0 + B (ω) G0 + B 2 (ω) (3.4) and the locus of ZL (ω) describes a circle in the complex plane. To show that it is the case, a transformation is applied by introducing an additional variable ξ such that cos ξ = G20 − B 2 (ω) G20 + B 2 (ω) (3.5) sin ξ = −2G0 B(ω) . G20 + B 2 (ω) (3.6) and also 30 Thus, it turns out that ZL (ω) = 1 1 (1 + cos ξ) + j sin ξ, or simply 2G0 2G0 ZL (ξ) = � 1 � 1 + ejξ . 2G0 (3.7) This latest form of the antenna impedance clearly shows that the locus depicts a displaced circle with a center at (1/2G0 , 0) and a diameter of 1/G0 in the complex plane. The same result is obtained if we started out by modeling the antenna input impedance as a series RLC circuit; and in that case, the input admittance will describe a circle with similar characteristics. The above step showed that if an antenna immittance can be modeled as an RLC circuit—which generally occurs for narrowband antennas near resonance—the actual antenna input immittance describes a circle-like or ellipsoidal locus within a specified bandwidth ∆ωC . The fact that antennas subscribing to the above criteria form approximate convex sets [88] is a property that can be readily and amply used to quickly get the AIA’s operating points, as it is shown now. From the analysis performed by Kurokawa [48] on negative-resistance oscillators, it was graphically shown that stable oscillations in the nonlinear regime occur when the device line and load locus intersect in a particular manner. To illustrate those concepts, some antenna locus and various device lines are drawn in Figure 3.1. The arrows indicate the directions of increasing frequency and RF current level the for the antenna locus (input impedance looking into antenna terminal) and device lines (negative of input impedance looking into terminal of active device), respectively. The three loci intersect at three points A, B and C. In [48], the stability was graphically characterized by looking at the angles of intersection—counterclockwise from the device line to the antenna locus—being less than 1800 . Consequently, point A is unstable while points B and C are stable. The antenna locus intersects itself at point D. This latest point characterizes the previously defined bandwidth ∆ωC . This is 31 Figure 3.1: Plots of antenna locus (solid line) and various device lines (dashed lines) for increasing frequency and RF current amplitude, respectively, as indicated by direction of arrows. Point A is unstable while points B and C are stable. Point D defines the bandwidth ∆ωC . the minimum bandwidth necessary for the antenna input immittance to return to an earlier value around resonance. However, from the illustration in Figure 3.1, a generic stability notion can be established as follows: it can be observed that a stable point is always created whenever the device line exits the convex set (gray area). Inversely, an unstable point is formed when the device line enters the convex region. Therefore, if the device line originates from anywhere within the closed manifold or grayed area described by the antenna locus, then the first intersection point of the two loci will always corresponds to a stable oscillation point regardless of the path taken by the device line. Mathematically, this outcome is always satisfied if the grayed area is closed or encircling the device line. As a consequence, this stability notion hints that during the initial small-signal design steps, it is desirable that the device line is located within the closed region or at the right of it, and made to enter this region as the RF level increases. 32 It should be noted that this is a simplistic two-dimensional description of a higherdimensional system as the device line is also frequency dependent and that harmonic components are part of the same dynamics. Moreover, the antenna might also display very complex modes of operation with multiple resonances; however, the goal of this analysis is to quickly find oscillation points and during computation/simulation, all these eﬀects are accounted for. 3.2.2 Feedback Termination Modeling In its most simplified form, a transmitting active integrated antenna can be regarded as a negative-resistance oscillator. As a result, a traditional analysis is performed by splitting the oscillator into a linear and nonlinear part as shown in Figure 3.2. The linear part mostly consists of the antenna with an impedance of ZL (ω). The nonlinear part comprises the active devices. However, for convenience, some linear elements or parts can be grouped with the nonlinear core since terminal feedback/loads and bias lines can be eﬃciently used to shape the negative resistance function. The nonlinear impedance ZN (A, ω) is both a function of the RF current magnitude A and frequency ω. For convenience, the admittance YL (ω) and YN (A, ω) of the antenna and nonlinear device should be used if more appropriate. In this case, A would be the amplitude of the RF voltage. The oscillator design process commonly starts with a small-signal analysis because it is imperative that the DC solution is unstable in order to initiate oscillations when the system is switched on. The suﬃcient small-signal conditions for the onset of oscillations, for Z = ZN + ZL , are ( see Chapter 4 for derivation): 33 Figure 3.2: Simple oscillator model with i(t) = A cos(ωt+φ).The nonlinear core comprises the active device and any other linear elements such as the feedback terminations that shape the unstability/negative resistance. The linear part mostly consists of the antenna. Zr < 0 r i s = ∂Z Z i − ∂Z Z r > 0 ∂ω ∂ω (3.8) The symbols i and r denote the imaginary and real part of a quantity, respectively. Since AIAs find many applications in quasi-optical power combining, minimizing non-radiating loss is of paramount importance. In order to improve the loaded Q, the feedback terminations of the active device are designed to be purely reactive. For such case, the reflection coeﬃcient looking into such terminal loads can be simply written as Γ = ejθ , as in the Smith chart, thus allowing a fast sweep of all possible reactance values. Practically, the amplitude of the reflection coeﬃcient of any passive termination is less than unity due to various loss mechanisms at microwave frequencies; however, these eﬀects are accounted for during the design process by the use of a full-wave EM solver. Consequently, the entire design space can be plotted out 34 concisely. For convenience, the following feedback impedance ZF model is used: ZF (ω, θ) = tan � jZ0 �θ ω ω0 2 −π �� . (3.9) This specific feedback termination can be implemented by a simple open transmission ω θ (π − ). Here, ω0 is an line with a corresponding electrical length of EF (ω, θ) = ω0 2 arbitrary reference frequency that is practically selected to be the expected oscillation frequency. Z0 is a reference impedance, which is assumed to be 50 Ω throughout this work. Equation (3.9) is chosen only for convenience as it was desired that the phase θ be swept from zero to 2π; however any arbitrarily equation can be used as long as it provides a way to sweep all reactive values from a single parameter. The reflection coeﬃcient of the above feedback load is ejβ , where β can be quickly calculated to be (ω/ω0 )(θ − 2π). For planar designs on substrates, the feedback terminations can be implemented by using either shorted or open transmission lines (OTL). However, open microstrip lines are easier to build and model than shorted lines with full-wave EM simulators since open lines do not require via-holes, pins and solder that are all susceptible to modeling errors. One disadvantage of OTLs is the presence of parasitic capacitance (fringing fields) and possible radiation losses that all might contribute to higher EM coupling. For designs with an injected signal, at least one of the parametric loads must be designed to have a real part. As a result, an additional real and positive parameter γ should be added to model the resistive part of the input impedance of the terminal load such that ΓF = γejθ . To test the usefulness of the equations above, two feedback loads are placed at the gate and source terminals of a FET transistor (NEC’s HJ-FET NE3210SO1) 35 at 10 GHz. The transistor is biased to VDS = 2 V and IDS = 10 mA. Figure 3.3 shows how the impedance looking into the drain terminal of such device evolves over all possible reactance values at the other two terminals. This plot summarizes the behavior of the FET under all reactive loading conditions in a concise manner by highlighting regions of negative diﬀerential resistance. It is expected that the actual curves summarizing the behavior of the transistor will be slightly diﬀerent due to the small resistance (loss) of the feedback terminations; however, it is not of critical importance as the goal of this analysis is to provide a global behavior of the system. The small deviations generated by some unavoidable loss will be accounted for at a later stage of the design process. 3.3 Application of Concepts The concepts of antenna locus convexity and parametric loads introduced in the previous section will be used to design two types of AIAs. First, a specific design scheme will be shown in order to realize an AIA from a previously fabricated or existing antenna element. The second example will demonstrate the design of an AIA where the antenna element and the nonlinear part are designed simultaneously. 3.3.1 Design with Existing Antenna The concepts introduced earlier can be readily applied to the design of an active antenna. To accomplish such task, a typical Ku-band patch antenna is provided [89]. It is built on a 31-mil RT/Duroid 5880 substrate. The resonant length is 6.97 mm and its width is 9.12 mm. Its feeding pin is located 2 mm from the non-radiating edge. It is desired to operate this patch at its first parallel resonance; therefore, its input admittance will be used as plotted in Figure 3.4(a). Common rectangular patches are narrowband antennas by nature, and from the previous analysis, it is expected 36 Figure 3.3: Parametrization of the input impedance of the drain terminal of a FET transistor. Bias: VDS = 2 V and IDS =10 mA. The solid lines represent the resistance , while the dashed lines is the rectance of the transistor in its small-signal regime at 10 GHz. All contour values in Ohms. 37 (a) (b) Figure 3.4: (a) Real and imaginary parts of the input admittance of a patch antenna. (b) Locus of patch antenna in complex plane with ∆ωC =4.26 GHz. ωa =11.23 GHz and ωb =15.49 GHz. that the locus of a narrowband antenna near resonance resembles a circle. Figure 3.4(b) shows that it is the case. This antenna satisfies the narrowband specifications highlighted in Section 3.2.1, thus creating the convexity needed to apply the previous concepts. Assuming this antenna is given and no changes can be made, the design process moves on to the nonlinear core. The next phase in the design process is to project the antenna locus area (gray area) onto the parametric design space of Figure 3.3, however at 13.5 GHz. This step is crucial because it will map the closed locus region described by the antenna onto the transistor design space. And since it was shown earlier (Section 3.2.1) that any device line starting inside such regions is bound to intersect the antenna locus at a stable point, the nonlinear device synthesis step is made eﬃcient and predictable. Once the real and imaginary parts of the closed region are individually mapped onto the design space, the final projection is found by juxtaposing them on top of 38 each other and tracing out their intersection. Technically, the mapping is done by merging areas corresponding to small rectangular areas of the shaded area of Figure 3.4(b). Figure 3.5 shows the corresponding mapping. It can be seen that the convex region has been mapped into two distinct shaded regions (shaded areas) attesting to the highly nonlinear nature of this transformation. Figure 3.5 also shows the eﬀect of enforcing the small-signal startup condition s > 0, where s is expressed in Equation (3.8). It is clearly visible that the design space has been further reduced by enforcing such condition. Figure 3.5: Mapping of the convex region described by antenna locus into the transistor design space. The contour lines (r, i, s = 0) represent the real, imaginary and startup conditions delimiting the projection of the antenna impedance. Enforcing the startup condition further reduces the considered design space. At this stage, the designer is free to pick her or his favorite nonlinear analysis tools and perform an oscillation analysis starting from any point within the resulting and smaller mapped region. Generally, the end result is achieved when some design 39 performance or specifications are met. However, it is well known that maximum power, higher power eﬃciency and minimum phase noise are not achieved at the same point. This is a familiar tradeoﬀ every designer has to deal with. At the end, it is up to the designer to select the best point based on the initial specifications in terms of output power, power eﬃciency and/or phase noise characteristics. Before any optimization step can take place, the key pieces of information to get out of this section are the following: First, the design space is consisely plotted out by the use of parametric feedback loads. Second, the mapping of the antenna convex region and startup conditions reduce the realizable design space into a smaller region. Third, picking a point within those mapped regions will likely generate a stable steady-state oscillation point. Without performing any optimization, choosing a random point in the upper lobe from Figure 3.5, as the starting point, it is quickly found that a stable oscillation is generated. The parametric point picked is (θG , θS ) = (1600 , 2710 ), which corresponds to a reactance of j8.82 Ω at the gate and j50.9 Ω at the source. A transient simulation produces the two plots in Figure 3.6 that describe the AIA performance. For verification, an ensuing harmonic balance simulation concludes that the fundamental oscillation frequency is 13.76 GHz. The output power is 6.41 dBm. The DC-to-RF conversion eﬃciency is 17.5% [24] and the phase noise is -122.8 dBc/Hz at an oﬀset of 1 MHz. Many other points were picked in the upper and lower lobes, with most of them showing displaying stable oscillation at diﬀerent power level and slightly diﬀerent frequencies between 13 and 14 GHz. In summary, the above section presented a fast and eﬃcient way of quickly coming up with a set of AIA’s operating points prior to any optimization and synthesis process. The analysis and design steps are reduced since the design space is condensed to a relatively small collection of points, thus letting the designer focus on getting the requested specifications without an extended trial-and-error process that notoriously 40 Figure 3.6: (a) Timeseries of current at antenna port. (b) Corresponding normalized power spectrum of current waveform with peak power (fundamental tone) at 13.76 GHz. 41 lengthens the traditional design process. Concerning the types of terminal loads used to parameterize the active device, it is clear that other loads can be utilized depending on the nature of the design and implementation technology. In addition, this technique can help quantify the limitations of the nonlinear core and its ability to construct an oscillator at a specific frequency range based on the projected area. 3.3.2 Design with Synthesized Antenna In the general case where the active core and the antenna have to be designed simultaneously, there are additional degrees of freedom because the antenna element can be synthesized to enhance a particular characteristic such as phase noise, output power, conversion eﬃciency and/or tuning range. The type of antenna used largely depends on the architecture and implementation technology. Here, a pin-fed rectangular patch antenna will be used. The following circuit schematic (Figure 3.7) is used to get a global behavior of the the system using Agilent ADS [90]. Figure 3.7: Simple schematic of AIA circuit showing transistor feedback loads and antenna. 42 To optimize output power or conversion eﬃciency, for instance, a harmonic loadpull analysis or measurements can be performed and the antenna can be subsequently modified to display specific loads at the frequencies of interest. This approach is not applied here. On the other hand, if the focus is on reducing phase noise, the transistor terminations, antenna, and RF level must be selected such that the frequency derivative of the total immittance is maximized at the oscillation frequency since the phase dynamics is inversely proportional to this gradient (refer to Chapter 4). This has the eﬀect of increasing the overall Q of the system. Additionally, the total immittance should also be made least sensitive to other key variables, if possible. The nonlinear device (FET in this case) has a high dependency on many variables and its drain impedance can be written as follows for a common-source configuration ZN = f (A, ω, θG , θS , Vbias , ...) (3.10) where G and S stand for gate and source, respectively. In the case where the designer has control over the antenna’s geometry, the active device impedance’s gradient can be optimized separately; otherwise, it is advisable to work the gradient of the total immittance. From the design specifications, the frequency is generally fixed and other parameters are allowed to vary. To limit the complexity of the upcoming analysis without loss of generality, only the AIA’s conversion eﬃciency and phase noise are considered and other design parameters are fixed to VDS = 2 V, IDS = 10 mA, RF level to 7 dBm and the oscillation frequency to 5 GHz. Consequently, only the immittance of the drain terminal will partially depend on the two terminal loads. The oscillator operating point can therefore be selected around the point where the input immittance is least sensitive to the oscillation amplitude and transistor loads’ lengths. Numerically, these low-variation points can be plotted out by bounding the diﬀerential of the total immittance, such that 43 � � � ∂Z � ∂Z ∂Z |∆Z| = �� dA + dθG + dθS + ...�� ≤ � ∂A ∂θG ∂θS (3.11) at the operating frequency and RF power level. Here, � is an arbitrarily small number that is related to the maximum amount of variation the designer is willing to allow. Figure 3.8 depicts a large-signal plot showing regions where the drain immittance variation is aﬀected based on gate and source loads at 5 GHz. Note that these types Source Feeback, deg 350 1 300 10 6 2 150 3 1 250 200 6 10 2 3 20 20 0.3 10 100 50 20 20 50 100 150 200 250 300 350 Gate Feedback, deg Figure 3.8: Plot of low-variation points (|∆Z|) where the transistor device input impedance (drain) is least sensitive to amplitude variation, gate and source loading. The lowest- variation point is observed at approximately (251,180) at 5 GHz. of plots should be also be computed at the small-signal level (∂(.)/∂A = 0), since they provide valuable information on the initial transient behavior. Figure 3.9 depicts the frequency gradient of the nonlinear core as a function of gate and source loading. It 44 can be observed that the maximum occurs near the edges and not at the same point as the lowest-variation point of Figure 3.8. 350 0.5 Source Feedback, deg 10 300 0.3 4 2 250 0.1 1 200 150 0.1 0.3 0.5 100 1 4 2 50 10 50 100 150 200 250 300 350 Gate Feedback, deg Figure 3.9: Frequency derivative of total nonlinear core impedance at 5 GHz and 7 dbm RF level. When a resonant patch antenna is used, the location of the feeding pin should be selected such that the input immittance frequency gradient is maximized. Figure 3.10 shows how the input impedance of a particular patch antenna is aﬀected as the feeding pin moves away from the radiating edge at 5 GHz. The impedance diﬀerential |∆ZL (ω, f )|, parameterized here as a function of the feed-pin location f and frequency ω, is highest around 4 mm. Note that at each pin location, a few neighboring frequency points are also simulated. As expected the resistance decreases as the pin moves deeper into the patch, reaching a minimum at half the length of the 45 patch. The excess inductance is due to both the pin and non-propagating higher-order TM modes [91]. Figure 3.10: Antenna impedance plotted as a function of feeding pin location on left at 5 GHz. Plot of normalized gradient in log scale showing maximum frequency variation. Maximum impedance variation occurs around 4 mm. Patch dimensions: L=19.5 mm, W=20.5 mm. After joining the antenna and the nonlinear core together, the location f = 4.58 � r� � ∂Z � mm was picked for two reasons. First, the quantity �� L �� had its maximum value ∂ω and second, HB simulations showed that a higher conversion eﬃciency was obtained there and not at the overall maximum location f = 4.5 mm. The corresponding antenna impedance is plotted in Figure 3.11. 46 Figure 3.11: (a) Synthesized patch antenna input impedance around 5 GHz for f = 4.58 mm. (b) Corresponding impedance locus. As noticed from Figures 3.8 and 3.9, the lowest-variation point and maximum frequency derivative points do not coincide. Having these two points overlap and operating the oscillator there would in theory minimize phase noise. The separation between these two points and various operating points of the AIA is showed in Figure 3.12. The operating points consist of the constant output-power points (7 dBm) using the synthesized patch as the gate and source loading is changed. After a few simulation trials within the antenna mapped area, the point (θG , θS ) = (1200 , 2030 ) was picked as the final design point based on its compromise in terms of producing a relatively acceptable phase noise (-115 dBc/Hz, at 1 MHz oﬀset), a DC-to-RF conversion eﬃciency of 24.7%, and an output power of 7.08 dBm. The oscillation frequency was 5.002 GHz. The corresponding gate and source loads can be implemented by two ideal 50-Ω OTLs with electrical lengths of 1200 and 78.50 , respectively. 47 Source Feedback, deg 350 300 250 200 150 100 7 dBm 50 50 100 150 200 250 300 350 Gate Feedback, deg Figure 3.12: Performance of AIA for 7 dBm output power (square). Plots of lowvariation (solid line) and high frequency-derivative points (dashed line) of drain impedance juxtaposed on background to show separation. 3.4 Fabrication and Measurements The design presented in section 3.3.2 is implemented on a planar substrate with microstrip lines. To avoid unnecessary coupling and additional cross polarization radiation, two substrate pieces were joined together to a common ground plane. The active device and its embedded linear network were built on one side (top) of one substrate, only leaving the radiating patch on the other side (bottom) of the second substrate. The substrate material used was RT/Duroid 5880 with a 31-mil thickness, a dielectric constant of 2.2 and loss tangent of 0.001. Following the design process in the previous section, a 5-GHz patch antenna was milled. The patch’s width and length were 20.5 mm and 19.5 mm, respectively. The antenna and transistor were connected by a 62-mil-long pin with a diameter of 0.4 mm located 4.58 mm inside 48 the patch. The ground plane’s dimensions were 61.5 mm and 58.5 mm. The terminal loads were implemented with 2-mm-wide microstrip traces and the remaining biasing traces were 20-mil thick at most. Due to the finite ground plane, coupling eﬀects, and the need for DC biasing lines, the transmission lines lengths were continuously adjusted/optimized to present the transistor with loads close to the simulation values obtained in Section 3.B. The common-source transistor HJ-FET NE3210S01 was DC biased to VDS = 2V and IDS = 10 mA using a biasing chip (BCR400W) and various discrete components. Figure 3.13 shows the final CAD layout, as well as the top and bottom views of the fabricated design. The measurement setup consisted of placing the active antenna and a receiving horn about 30 free-space wavelengths apart. The horn was connected to an Agilent E4407B spectrum analyzer with a resolution bandwidth of 1 MHz, 0-dB attenuation, 4-ms sweep, and 0-dBm reference power. Figure 3.14 is the actual measured spectrum. The measured oscillation frequency was 5.018 GHz, which is very close to the simulation in the previous section. The estimated radiated power, using the Friis transmission equation, was 5.662 mW or 7.53 dBm; therefore the EIRP is 14.54 dBm. The DC-to-RF eﬃciency of the AIA was 21.8%. 49 (a) (b) Figure 3.13: (a) CAD layout implementing design steps of section 3.3.2. (b) Fabricated bottom view showing transistor, bias circuit and embedded linear elements. On top, there is a patch antenna connected to the drain terminal. 50 Measured Spectrum, dBm -20 -30 -40 -50 -60 4.98 5 5.02 5.04 5.06 Freq, GHz Figure 3.14: Spectrum measurement of the radiated power. The oscillation frequency is 5.018 GHz. 51 CHAPTER 4 DYNAMICS OF COUPLED NONLINEAR OSCILLATOR ARRAYS 4.1 Introduction As mentioned in Chapter 2, coupled nonlinear oscillators have been used to eﬀectively describe a wide variety of naturally observable phenomena in various fields of study [92, 93]. Their applications in engineering have been primarily associated with lasers, arrays of Josephson junctions, chaotic circuits and phase-locked loops [94–98]. Various sets of slowly-varying amplitude and phase equations have been derived for microwave coupled oscillators [4, 46, 47, 60]. However, the simplifications were based either on weak nonlinearity, broadband coupling, identical natural frequencies, and nearest-neighbor or global topologies, and so on, leaving out many unavoidable and significant eﬀects that can highly aﬀect the end performance of actual phased arrays. Since most equations in coupled oscillators applications are special or simplified cases of generic models developed for weakly [99–101] and strongly coupled systems [102– 107], it is clear that deviations from simulations is in part due to the diﬃculty of modeling all the interactions and expressing all parameters in terms of measurable network data or well-known figures of merit. This chapter derives a system of expandable ordinary diﬀerential equations that describe more accurately the dynamics of a group of oscillators coupled through an 52 arbitrarily structured linear network. The equations will be valid for weakly and fairly strong coupled nonlinear systems with the major limiting restriction being on the accuracy of the model for each regime and devices in use. The equations are expressed in a form familiar with physical or electrical metrics needed to actually analyze, design and build such systems. 4.2 Amplitude and Phase Dynamics A simplified circuit diagram representing an array of N oscillators coupled through a linear network is shown in Figure 4.1. Here the system is explicitly split into a linear and nonlinear part. The linear part includes passive elements such as the antennas and coupling circuits, which can all be systematically characterized by a N × N immittance matrix Ykp , equal to Ykp = |Ykp |e jϕkp � IL,k �� = , Vp �Vq =0 for q�=k (4.1) where (k, p) ∈ N2+ . The nonlinear part consists of all active and nonlinear devices. In most prototypes, the dependent current sources in Figure 4.1 are implemented by one or multiple transistors, while the independent sources can be used to model noise or injection signals. For most practical applications, the desired mode of operation consists of having all the oscillators synchronized to the same frequency ωs or phaselocked to one another. This mode of operation is critical for power-combining systems, coherent communications, transceiver sensitivity and continuous beam-steering applications. For common nonlinearities, the voltage vk (t) at each node or port k can be written as a linear combination of harmonic phasors such that 53 Figure 4.1: Circuit representation of N linearly coupled nonlinear oscillators. The linear network Y includes all linear elements such the antennas and coupling networks while the rest of the circuit is mostly made of active devices and injection/control sources. 54 vk (t) = M � � � � ṽk,m (t)ejmωs t , (4.2) m=1 where M is the total number of harmonic terms or modes being considered and ṽk,m (t) = Ak,m (t)ejφk,m (t) . The current going into the linear network at port k is given by the inverse Fourier transform of its spectral counterpart iL,k where Vp (jω) = � ∞ 1 = 2π � ∞ M � N � Ykp (mωs )Vp (jmωs )ejωt dω, (4.3) −∞ m=1 p=1 vp (t)e−jωt dt. Due to the linear nature of Fourier transformations, −∞ each harmonic term can be treated separately. Using Euler’s formula, Vp (jω) can be written as Vp (jω) = � ∞ vp (t)e−jωt dt = −∞ � 1� Ṽp (ω − ωs ) + Ṽp∗ (ω + ωs ) . 2 (4.4) Here, Ṽp is the Fourier transform of ṽp . Putting Eq. (4.3) and (4.4) together, one obtains: 1 iL,k (t) = 2π � ∞ N � 1 −∞ p=1 2 � Ykp Ṽp (ω − ωs ) + Ṽp∗ (ω � + ωs ) ejωt dω. (4.5) Using the properties of the integral above, Eq. (4.5) can also be written as follows: � �� � ∞ N � � 1 iL,k (t) = � Ykp (ω + ωs )Ṽp (ω)ejωt dωejωs t . 2π −∞ k= (4.6) The Taylor expansion of the immittance Ykp is given as: ∞ � 1 dn Ykp (ωs ) n Ykp (ω + ωs ) = ω . n! dω n n=0 Merging Eq. (4.6) and (4.7), one obtains 55 (4.7) � � � � ∞ � N � ∞ � 1 dn Ykp (ωs ) 1 1 iL,k (t) = � (jω)n Ṽp (ω)ejωt dω ejωs t , n n n! dω j 2π −∞ p=1 n=0 (4.8) which is equivalent to the following equation: � � � � N � ∞ � 1 dn Ykp (ωs ) 1 dn ṽp (t) jωs t iL,k (t) = � e . n! dω n j n dtn p=1 n=0 (4.9) Due to the linearity of the coupling or immittance matrix, the total current at port k is simply the linear combination of all considered M terms or harmonic components. Since in general, the n-derivative of the phasor ṽp can be written as n � � dn ṽp (t) � n (k) � jφp �(n−k) = Ap e dtn k k=0 (4.10) and assuming that the current can be described well enough by a few dominant terms (n = 0, 1), one obtains � iL,k (t) = � e jωs t N � � p=1 Ykp Ap ejφp + 1 dYkp (ωs ) dAp jφp e j dω dt dYkp (ωs ) dφp + Ap ejφp + O dω dt � d2 Ykp (ωs ) dω 2 ��� (4.11) From Fig. 4.1, the Kirchhoﬀ’s current relations at node k give iD,k (t) + iL,k (t) = iinj,k (t). (4.12) In terms of averaging equations, which translate into the determination of the input 56 admittance, the following equation 2 Ak Ts � t (iD,k (t) + iL,k (t))e−j(ωs t+φk ) dt t−Ts 2 = Ak Ts � t iinj,k (t)e−j(ωs t+φk ) dt (4.13) t−Ts is also equivalent to: � ωs ) + YL,k (ωs ) = Yinj,k (ωs , A, � φ, � ρk ), YD,k (A, (4.14) assuming that the amplitudes and phases of the voltage signals are slowly-varying 1 dAk dφk functions of time ( � ωs and � ωs ). Here Ts is the period of the Ak dt dt � = [φ1 , φ2 , ..., φN ] are the voltage � = [A1 , A2 , ..., AN ] and φ fundamental mode. A amplitude and phase vectors, respectively. ρk is the average amplitude of the injection of noise signal at port k. To find the dynamics of the voltage amplitude and phase at each port, the self-admittance vector, which is made of the diagonal entries of the Y matrix, needs to be isolated from the mutual or coupling eﬀects (remaining matrix entries). Using Eq. (4.11) and (4.13) and assuming that d2 Ykp (ωs )/dω 2 is comparatively small in magnitude to the first-order terms, one obtains: � � � 1 dAk �� dYkk �� jϕ�kk jϕkk � YD,k (A, ωs ) + |Ykk |e + e jAk dt � dω � � � � dφk �� dYkk �� jϕ�kk � ρk ), (4.15) + e + αkp = Yinj,k (ωs , A, dt � dω � � where ϕkk = arg{Ykk (ωs )} and ϕkk = arg{dYkk (ωs )/dω}. The other complex quantity αkp is given by αkp N � � Ap 1 dAp = |Ykp |ejθ1 + Ak jAk dt p=1,�=k � � � dYkp � jθ � � 2 Ap dφp � dω � e + Ak dt 57 � � � � dYkp � jθ � �e 2 , � dω � (4.16) � with θ1 = φp − φk + ϕkk and θ2 = φp − φk + ϕkk . Splitting Eq. (4.15) into its real and imaginary parts, one gets r YT,k � � � � 1 dAk �� dYkk �� dφk �� dYkk �� r r + sin θ2 + cos θ2 + αkp = Yinj,k Ak dt � dω � dt � dω � 1 dAk i YT,k − Ak dt � � � � � dYkk � � dYkk � dφ k i i � � � � � dω � cos θ2 + dt � dω � sin θ2 + αkp = Yinj,k (4.17) (4.18) � ωs ) + Ykk (ωs ) and the superscripts r and i represent the real � φ, where YT,k = YD,k (A, and imaginary part of a quantity, respectively. Multiplying Eq. (4.17) by sin θ2 and subtracting the result by Eq. (4.18) times cos θ2 , one obtains � � � � i r � dYkk � r dAk Ak r dYkk i dYkk i � � = −� − YT +� (α sin θ2 − αkp cos θ2 ) � Y dt dω dω � kp � dYkk �2 T dω dω � � � dYkk � r i � (Y + �� sin θ2 − Yinj,k cos θ2 ) (4.19) dω � inj,k Similarly, multiplying Eq. (4.17) by cos θ2 and adding the result to Eq. (4.18) times sin θ2 , one gets � � � � r i � dYkk � r dφk 1 r dYkk i dYkk i � � = −� − YT +� (α cos θ2 + αkp sin θ2 ) � Y dt dω dω � kp � dYkk �2 T dω dω � � � dYkk � r i � (Y + �� cos θ2 + Yinj,k sin θ2 ) (4.20) dω � inj,k However, the coupling terms emanating from αkp in Eq. (4.19) and (4.20) can be further simplified such that � � N � � dYkk � r Ap i � � (αkp sin θ2 − αkp cos θ2 ) = Hkp cos(φp − φk + Φkp ) � dω � A k p=1,�=k 58 (4.21) and where � � N � � dYkk � r Ap i � � (αkp cos θ2 + αkp sin θ ) = Hkp sin(φp − φk + Φkp ), 2 � dω � A k p=1,�=k Hkp � �� � ��� �2 ��1/2 � dYkk � � dYkp � dφp � 2 � � � � =� |Ykp | + � + 2|Ykp | cos(ϕkp − ϕkp ) dω � dω � dt (4.22) (4.23) In the equation above, it is assumed that the amplitude dynamics reach their steady � state quickly (dA/dt → 0). Using Eq. (4.19) and (4.20), the coupling phase Φkp between oscillator k and p can be written as: tan Φkp � � � � dY � p |Ykp | cos(ϕkk − ϕkp ) + � dωkp � dφ cos(ϕkk − ϕkp ) dt � � = . � dY � p � |Ykp | sin(ϕkk − ϕkp ) + � dωkp � dφ sin(ϕ − ϕ ) kk kp dt (4.24) � �� � � dYkp � � dφp � �� � is small in magnitude compared to |Ykp |, then Hkp If the quantity �� dt � � dt � and Φkp can be further simplified to: Hkp and � � � dYkk � � |Ykp |, = �� dω � tan Φkp = cot(ϕkk − ϕkp ). (4.25) (4.26) dYkk dY i ≈ j kk as in practical microwave oscillators in which an adedω dω � � � r� i � � dYkk � dY � � � quate resonant circuit is used (� � �� kk ��), one gets Φkp ≈ ϕkp = arg{Ykp } = � dω dω ∠Ykp . Furthermore, if Strictly in terms of real and imaginary parts, Eq. (4.24) can be written as: 59 Φkp (ω) = r ∂YT,k ∂ω ∠ i r ∂YT,k Ykp ∂ω r Ykp i ∂YT,k ∂ω r i ∂YT,k Ykp ∂ω i + Ykp − . (4.27) The injection terms, which can be sources of noise or control signals, can also be simplified. In the case where the injection signal at node k is a coherent signal of the form ρinj,k cos(ωs t + Θinj,k ), one finds the following two equations � � � dYkk � � r � i � � Yinj,k sin θ2 − Yinj,k cos θ = ηA,k 2 � dω � ρinj,k = Ak and � � � dYkk � � r � i � � Yinj,k cos θ2 + Yinj,k sin θ = ηφ,k 2 � dω � ρinj,k = Ak � � � dYkk � � � � � dω � cos(Φinj,k − φk − ϕkk ) (4.28) � � � dYkk � � � � � dω � sin(Φinj,k − φk − ϕkk ) (4.29) Equations (4.19) and (4.20) can therefore be written as follows � i r dAk Ak r dYkk i dYkk = −� − YT � Y dt dω � dYkk �2 T dω dω � N � Ap + Hkp cos(φp − φk + Φkp ) + ηA,k , (4.30) Ak p=1,�=k 60 and � r i dφk 1 r dYkk i dYkk = −� Y + Y � T dt dω � dYkk �2 T dω dω � N � Ap + Hkp sin(φp − φk + Φkp ) + ηφ,k . (4.31) A k p=1,�=k In practical microwave oscillators, the frequency dependence of the input immittance of the active device is many orders of magnitude smaller compared to that of the resonator, and in turn, the equations above are suﬃciently accurate. In the case where this assumption is not valid, equations (4.30)-(4.31) can be adjusted by replacing the index kk by T, k to account for the device’s high frequency variation. Finally, for general analysis, the equations can be rendered dimensionless with a slow time τ = ωs t such that for the amplitude dynamics � � N � dAk Ap � � = −Ak Gk (A, φ, ωs ) + Λkp cos(φp − φk + Φkp ) + σA,k dτ Ak p=1,�=k (4.32) and for the phase dynamics � � N � dφk A p � ωs ) + � φ, = − Ωk (A, Λkp sin(φp − φk + Φkp ) + σφ,k , dτ A k p=1,�=k (4.33) where the quantities Gk and Ωk in the above equations are: � ωs ) = � φ, Gk (A, � ωs ) = � φ, Ωk (A, 1 � �2 ωs � dYdωkk � 1 � �2 ωs � dYdωkk � 61 � � dY i YTr kk dω dY r YTr kk dω − + dY r YTi kk � dY i YTi kk � dω dω , (4.34) . (4.35) The terms of the coupling coeﬃcient [Λ] are given by Λkp = 1 � dY �2 Hkp , ωs � dωkk � (4.36) and the injection terms terms are σA,k = ωs ηA,k and σφ,k = ωs ηφ,k . 4.3 Transient and Stability Analysis Maintaining the stability of the phase-locking mechanism of the array as some control parameters are changed is of paramount importance in practical applications. To study the stability properties of the above equations, the start-up transient stability will be considered. In this regime, the voltage amplitudes are at the noise level and � ≈ �0). Neglecting higher-order derivative and injection terms, the thus very small (A quantity ΓA,k is given by: ΓA,k ∂ ≡ ∂Ak � dAk dt � 1 ≈� � � dYkk �2 dω � dY r YTi kk dω − − dY i YTr kk dω � N � Ap Hkp cos(φp − φk + Φkp ). (4.37) A k p=1,�=k The oscillation amplitudes at the noise level will grow only if there is instability which requires that ΓA,k > 0. For a single oscillator (uncoupled, [H]=0) , that implies dY r dY i YTi kk − YTr kk > 0. It should be noted that at startup, YTr is always negative for dω dω dY i a well-designed oscillatory system, therefore a suﬃciently large and positive kk will dω make the array oscillate. The array reaches its steady state when the power provided by the active devices is equally consumed by the rest of the array. This results in a cancellation of the total reactance/susceptance and resistance/conductance. Another consequence of reaching the steady-state regime is that ΓA,k becomes negative. At 62 this regime, ΓA,k is given by 1 ΓA,k = � � � dYkk �2 dω � r i dYTi dYkk dY r dYkk − T dAk dω dAk dω � N � Ap − Hkp cos (φp − φk + Φkp ) Ak p=1,�=k �� � �� � �� � dYT,k � ∂ 1 �� dYT,k �� � � × − (4.38) ∂Ak � dω � Ak � dω � For an uncoupled oscillator, it is clear that ΓA,k < 0 when i r r i dYT,k dYT,k dYT,k dYT,k − < dAk dω dAk dω 0 as also shown in [48]. The phase dynamics at the noise or small-signal regime is irrelevant for an oscillator. However, at the steady-state regime, maintaining phase stability or phase locking is crucial. For that to happen, the following quantity Γφ,k must be negative. Γφ,k ∂ ≡ ∂φk � dφk dt � −ρinj,k ≈ Ak � � � dYkk � � � � � dω � cos(Θinj,k − φk − ϕkk ) � N � � Ap 1 +� Hkp cos(φp − φk + Φkp ) (4.39) � � dYkk �2 p=1,�=k Ak dω Here, it is clear that an uncoupled oscillator (without injection) will have Γφ = 0. This result shows that phase perturbations of a single oscillator will neither decay nor increase since there is no phase restoring mechanism, unlike its amplitude dynamics. However, when the oscillators are coupled or injection-locked, a partial or full phase restoring mechanism is present and the resulting phase disturbances or noise can be minimized by making sure that Γφ,k is suﬃciently negative. It is also important to point the importance of the quality factor Qk of each oscillator and its eﬀect on phase noise. The quality factor can be approximately derived from the frequency derivative from the self-immittance terms as follows: 63 � � � dYkk � 2Qk r � � ≈ Y . � dω � ωs kk ω=ωs (4.40) Thus the phase dynamics from Eq. (4.31) is now equivalent to dφk =− dt � ωs r 2Qk Ykk �2 dY r (YTr kk dω � � N � Ap ωs |Ykp | sin(φp − φk + Φkp ) r dω Ak 2Qk Ykk p=1,�=k � � r ρinj,k 2Qk Ykk � + sin(Θinj,k − φk − ϕkk ). (4.41) Ak ωs dY i + YTi kk ) − From the equation above, it is clear that a high quality factor limits the phase disturbances coming from the oscillator itself and its coupling mechanism. 4.4 4.4.1 Numerical Simulations and Verification Frequency Deviation To illustrate the validity of the above equations, a system of coupled van der Pol oscillators will be studied as depicted in Fig. 4.3. The van der Pol nonlinearity can be readily implemented using a set of cross-coupled transistors and such architecture is common in integrated circuits. Before performing an in-depth verification of the derived equations, it is also necessary to accurately determine the oscillation frequency of the array before fabrication. To help with this endeavor, let’s assume that the voltage at one uncoupled RLC oscillator can be written as follows: v = A1 cos(ωt) + A3 cos(3ωt + φ). 64 (4.42) Only the fundamental and its third harmonic terms are considered since we are dealing with a van der Pol oscillator. It is also assumed that A1 � A3 . The fundamental and third harmonic input admittance looking into the nonlinear device are quickly calculated to give 3 3 3 YN (ω) = −a + b( A21 + A23 ) + bA1 A3 ejφ 4 2 4 (4.43) 3 3 b A31 −jφ YN (3ω) = −a + b( A23 + A21 ) + e , 4 2 4 A3 (4.44) assuming that the nonlinearity is defined as i = −av + bv 3 . The oscillator will settle at a frequency corresponding to a cancellation of the imaginary part of the total immittance, thus 3 1 bσA41 sin φ + ωC − = 0, 4 ωL (4.45) with A3 = σA31 . This makes the oscillation frequency equal to ω=− 3 bσA41 4 sin φ + 2C � �3 4 bσA41 sin φ 2C �2 + ω02 . (4.46) Here, ω0 is (LC)−1/2 . By defining the nonlinear factor as � = Lω0 (a − 1/R), the oscillation frequency given by Eq. (4.46) can be normalized and expressed as � ω = 1 + γ 2 �4 − γ�2 , ω0 65 (4.47) with γ= 3 bσA41 4 sin φ 2Cω0 �2 (4.48) A plot of this normalized synchronization frequency as a function of the nonlinearity factor � is shown in Fig. 4.2. As the nonlinearity factor increases, the initial approxi- Normalized Frequency 1 Exact Approx 0.95 0.9 0.85 0.8 0 0.5 1 1.5 2 Nonlinearity Factor, ε Figure 4.2: Approximation of the normalized van der Pol oscillation frequency (dashed) with exact (solid). The deviation increases as the nonlinear factor increases. mation gradually deviates from the true frequency. However, for common microwave oscillators, the nonlinearity factor is kept well below unity to avoid relaxation oscillations and other undesired eﬀects. Regions where � is larger than 0.5 are consequently avoided. For larger values of �, a relaxation oscillator is gradually produced and its frequency characteristics are based on the Airy function [108]. 66 4.4.2 Synchronization As previously mentioned, in many applications, frequency synchronization has to be maintained to correctly operate coupled-oscillator arrays. Typically, the instantaneous frequencies of all oscillators need to be equal for eﬃcient communication coding, signal processing and signal coherence. Let’s investigate the properties of a set of van der Pol oscillators coupled through a simple RL circuit is shown in Fig. 4.3. The array loading and coupling will be broadly changed from weak to strong in order to illustrate its synchronization and stability properties. Figure 4.3: Chain of RL-coupled van der Pol oscillators. The nominal values for the coupling resistors and inductors are RC = 500 Ω and LC = 5 nH. The RLC tank circuits have nominal values of Ri = 50 Ω, Ci = 1 pF and Li = 0.2533 nH, i=1,2,..,N for a 10 GHz oscillation frequency. The nonlinearity is described by i(v) = −av + bv 3 , where a = 1/30 and b = 1/90. The dynamics of the above array is exactly described by the following system of 67 second-order diﬀerential equations: � N � � 1 � v̇p − v̇k vp − Vk v̈k = M (k, p) + Ck p=1 Rkp Lkp + v̇k � 1 −a+ + 3bvk2 Rk � � vk − , (4.49) Rk for k = 1 to N and where M (k, p) is a topology or interconnection matrix. In other words, M (k, p) is unity if the oscillator k and p are connected, otherwise the entry is null. The configuration in Fig. 4.3 is a nearest-neighbor topology, which means that M (k, p) is unity only when |k − p| = 1. For N = 11, this array’s dynamics are simulated to track the loss of synchronicity as the capacitance of the first tank circuit (Fig. 4.3) is gradually changed. Fig. 4.4 shows how the synchronization frequency of all eleven oscillators is gradually lost as the natural frequency of the edge oscillator deviates from the group frequency. There is a region (C = [0.935, 1.07] pF) where all oscillators hold the same frequency. Outside this region, frequency synchronization is gradually lost. To illustrate the accuracy of the derived equations from Section 4.2 versus the exact equations (Eq. 4.49), the 11-element array of Fig. 4.3 will be analyzed. The linear part of the array can be described by an admittance matrix [Y ], whose diagonal entries are given by: � � N � 1 1 1 1 Ykk (ω) = + + jωCk + M (k, p) + , Rk jωLk R jωL kp kp p=1 68 (4.50) 11 Oscillation Frequency, GHz 10.8 10.8 10.6 10.6 10.4 10.2 10.4 10 0.89 0.9 0.91 0.92 0.93 10.2 10 9.8 9.6 0.85 0.9 0.95 1 1.05 1.1 1.15 Capacitance, pF Figure 4.4: Loss and gain of synchronization between oscillators as the tank capacitance of first oscillator is varied. These results are from Eq. (4.49). From approximately 0.935 to 1.07 pF, all oscillators are synchronized. Outside these values, partial synchronization occurs where multiple groups of identical frequencies form. Chaotic behavior is noticed below C=0.935 pF (dotted box). 69 and the remaining terms are expressed as follows Ykp (ω) = − � 1 1 + Rkp jωLkp � . (4.51) The loss of frequency synchronization can also be accurately predicted by the derived first-order equations (Eq. 4.30-4.31). It can be seen from Fig. 4.5 that the phase diﬀerence between multiple adjacent oscillators quickly diverges or becomes invalid (as the frequencies diﬀer) around C1 = 0.935 pF, thus confirming that the oscillators have lost their synchronicity. These results are in agreement with the earlier numerical simulations involving the exact second-order diﬀerential equations. To further Phase Difference, deg 140 120 100 80 60 0.934 0.935 0.936 0.937 0.938 0.939 0.94 Capacitance, pF Figure 4.5: Ten curves showing the phase diﬀerence φp − φk between adjacent oscillators showing a loss of synchronicity around C1 = 0.935 pF. Another loss of synchronicity also appears around C1 = 1.07 pF (not shown here). verify the accuracy of the derived equations, the comparisons will be carried out at 70 areas where the most diﬀerences are noticed, mostly in the phase dynamics. The array depicted in Fig. 4.3 can be assumed to be relatively weakly coupled since the coupling strength as defined by Eq. (4.36) is around 0.029. The phase and amplitude dynamics comparisons are shown in Fig. 4.6. All comparison plots are chosen from where the most discrepancy is observed, and as expected, for Fig 4.6, the highest disagreement occurs at the other edge of the chain between oscillator 10 and 11 when the 80 1.3 60 1.25 Approx. Phase Exact Phase Approx. Amp Exact Amp 40 20 1.2 Amplitude, V Phase Difference, deg capacitance of the first oscillator is changed. When the coupling strength is increased 1.15 0 0.94 0.96 0.98 1 1.02 1.04 1.1 1.06 Capacitance, pF Figure 4.6: Phase diﬀerence between oscillators 10 and 11 from the chain of oscillators depicted in Fig. 4.3. (solid line, green) is computed from the first-order derived equations while the marked line represents results from the exact second-order diﬀerential equation. Similarly, the amplitude comparison at oscillator 10. (solid line, red) is computed from the first-order derived equation while the marked line represents results from the exact secondorder diﬀerential equation.The coupling strength varies from 0.0285 to 0.0292 across the array. 71 by setting RC = 100 Ω and LC = 1 nH, the highest disagreement in amplitude and phase dynamics are again observed between oscillators 10 and 11. Fig. 4.7 shows results obtained with the exact and approximate equations. Again, the results are close even for a greater range of capacitance values. Furthermore, a stronger coupling 1.4 20 1.35 15 1.3 10 5 0 0.4 Approx. Phase Exact Phase Approx. Amp Exact Amp 0.6 0.8 Amplitude, V Phase Difference, deg 25 1.25 1 1.2 1.2 Capacitance, pF Figure 4.7: Left: phase diﬀerence between oscillator 10 and 11. (solid line, green) is computed from the first-order derived equations while the marked line represents results from the exact equations. Right: amplitude comparison at oscillator 11. (solid line, red) is computed from the first-order derived equation while the marked line represents results from the exact secondorder diﬀerential equation. The coupling strength varies from 0.1193 to 0.1328. arrangement is introduced by setting RC = 5 Ω and LC = 1 nH. This results in a strongly coupled system with the results shown in Fig. 4.8. It is noticed that the stronger the coupling, the more accurate the amplitude dynamics becomes. Moreover, 72 1.27 0.8 1.26 0.6 0.4 1.25 0.2 Approx. Phase Exact Phase Approx. Amp Exact Amp 0 -0.2 -0.4 0.4 0.6 0.8 Amplitude, V Phase Difference, deg 1 1.24 1 1.23 1.2 Capacitance, pF Figure 4.8: Left: phase diﬀerence between oscillator 10 and 11. (solid line, green) is computed from the first-order derived equations while the marked line represents results from the exact equations. Right: Amplitude comparison at oscillator 11. (solid line, green) is computed from the first-order derived equation while the marked line represents results from the exact second-order diﬀerential equation. The coupling strength varies from 1.2739 to 1.4171. 73 throughout the increase in coupling strength, the first-order equations maintained a satisfactory level of accuracy. Finally, the response of the array from a change of nonlinearity is undertaken. Here the nonlinearity factor is solely dependent on the loading of each active device and it is re-defined here as: �k = 2 2 (a − Ykk ) � dY � . ωs � dωkk � (4.52) When all the tank resistances are gradually changed from 35 to 500 Ω and the capacitance of the first tank is increased to 1.05 pF (to generate an asymmetric phase gradient), the nonlinear factor goes from about 0.08 to 0.5. The rest of the circuits elements keep their nominal values as given in Fig. 4.3. Fig. 4.9 shows how the amplitude and phase dynamics are aﬀected from varying the nonlinearity. It is clear that the increase of nonlinearity up to 0.5 does not greatly aﬀect the accuracy of the derived equations. Moreover, the amplitudes derived from the approximate equations are nearly indiscernible from the exact response. 4.4.3 Comparison to Previous Models Similar equations modeling arrays of coupled microwave oscillators have been presented with various degree of accuracy based on the severity of the nonlinearity or the strength of the coupling. One of the most commonly used models was developed by York [4, 55]. The equations are provided here for convenience for a van der Pol 74 5 58 4 56 3 Approx. Phase Exact Phase Approx. Amp Exact Amp 54 2 52 50 Amplitude, V Phase Difference, deg 60 1 0.1 0.2 0.3 0.4 0 0.5 Nonlinear Factor Figure 4.9: Left: phase diﬀerence between oscillator 1 and 2 as the nonlinear factor is varied. A diﬀerence of about 2 degrees is observed as the maximum diﬀerence. Right: amplitude comparison at oscillator 1. The diﬀerence is minimal as the two curves are indiscernible justifying the accuracy of the amplitude dynamics (RC = 500 Ω). 75 array. Fk = 1+ 1 2Q �N 1 Ykp Vp ∂ p=1 j∂ω ( GL ) Vk × � � � � ωk A2k 1+µ 1− 2 2Q αk − � � Iinj,k + + j(ωk − ω) (4.53) GL V k GL V k N � Ykp Vp p=1 with the amplitude and phase dynamics being Ȧk = Ak �{Fk } and φ̇k = �{Fk }, respectively. αk and ωk are the free-running or natural amplitudes and oscillation frequencies, respectively. GL is the load of each oscillator. Compared to equations (4.30) and (4.31), the York equations produce similar results when the coupling is weak and nonlinearity not severe. The disparity at other regimes is mostly caused by the fact that the York’s equations neglect the input susceptance of the active device and its frequency variation. This results in the absence of an additional term in the phase dynamics that aﬀect the accuracy of the inter-element phasing due to their sensitivity to small parameters. Furthermore, using free-running frequencies and amplitudes is problematic and impractical because the oscillators are always coupled or loaded by other oscillators’ signals and circuitry and their values are not accurately known in advance. Unlike the amplitude inaccuracies that are highly forgiving in phased arrays dynamics, the phase dynamics on the other hand is highly sensitive to small parameters and predictions might not be accurate if some terms are dropped. To illustrate some accuracy diﬀerence, an array of five coupled oscillators with a topology similar to Fig. 4.3 is analyzed using equations (4.53), (4.30,4.31) and (4.49). The coupling resistance is 200 Ω and the array is asymmetrically loaded by only changing the capacitance of the first tank until the array becomes unlocked. It is found that the amplitude dynamics is similar between all models, but the phase 76 dynamics on the other hand shows some discrepancies. The phase diﬀerence between oscillator 1 and 2 are plotted in Fig. 4.10. It is observed that the exact diﬀerential equations and the approximate equations provided here predict that the array will loose its frequency synchronicity when C ≈ 1.095 and C ≈ 1.09 pF, respectively. However the York equations predicts a lost of synchronization at C ≈ 1.1275 pF. In terms of phase diﬀerences, the York equations become more inaccurate as the array is pushed near the edge of stability. it is also clear that the York equations, in this case, overestimate the controlling capabilities of the array, by predicting a wider capacitance range of each oscillator. Although the diﬀerence seems to be small when looking at the radiation pattern, for larger arrays with many more elements and a sharp pencil radiation beam, small diﬀerences in oscillator phasing could introduce significant errors on the exact location of the main beam. In practical engineering designs, a simple circuit for an array as the one studied earlier is not common since more complex coupling electromagnetic (EM) phenomena are involved. In this case, it is convenient to deal with the network parameters of the array, which are systematically obtained through an EM software suite. The equations in this work (section 4.3) were purposefully derived to make sure that network parameters of any kind and other related figures of merit are used by scientists and engineers tasked with designing such systems. In conclusion, a set of equations, for more accurately computing the dynamics of an array of nonlinear coupled oscillators through a linear network, has been presented with focus on measurable microwave network quantities. Various degrees of accuracy can be reached by involving additional modes of operations. While the targeted application is active and coupled integrated antennas arrays, the equations are useful in any field dealing with coupled nonlinear oscillators arranged in an arbitrary topology. 77 -20 Approx. York Exact Phase Difference, deg -30 -40 -50 -60 -70 -80 -90 1.06 1.08 1.1 1.12 1.14 Capacitance, pF Figure 4.10: Comparison of phase diﬀerences in degrees between oscillator 1 and 2 using the York model (4.53), the derived equations (4.30,4.31), and exact diﬀerential dynamics (4.49). The derived equations show a higher accuracy in reproducing the true dynamics of the array. 78 CHAPTER 5 DESIGN CONSIDERATION FOR COUPLED-OSCILLATOR ARRAYS 5.1 Introduction The goal of the previous chapter was to express the dynamics of coupled-oscillators systems with readily obtainable network parameters. The motivation for producing these equations was to deal with the drastic diﬀerence observed at times between the design predictions or expectations to the actual performance of coupled-oscillator array. As many simplifications are introduced to reduce the complexity of such systems, potential interesting eﬀects may be overlooked and dismissed for the sake of a quick design. This chapter will look at the eﬀects of the coupling networks and how their properties can be used to control the overall performance of the array. A multi-simulation tool will be developed to make use of the coupling properties and an three-element array will be designed and built to test the validity of the equations presented in the previous chapter. 79 5.2 Coupling Considerations Designing a system based on coupled oscillators necessitates a well-designed coupling mechanism in order to regulate the phase and amplitude dynamics that will be generated once the system is operational. Linking two or more oscillators by a transmission line is one of the simplest and most aﬀordable solutions. Two simple cases will be discussed shortly. 5.2.1 Transmission-Line based Coupling Networks In order to illustrate even how simple coupling networks present interesting properties, let’s look at two van der Pol oscillators coupled through a transmission line network as shown in Figure 5.1. The black boxes represent the active devices. Let’s call this model #1. It is a piece of uniform transmission line of characteristic impedance Z0 , electrical length θ, and terminated with two resistors RC at each end. Figure 5.1: A pair of oscillators coupled through a transmission line-based circuit (model #1). The Y-parameter matrix of this coupling circuit between the two oscillators is 80 given by (see Appendix A for derivation): RC −1 cos θ + j Z0 sin θ j sin θ �� � [Y ] = �2 RC Z0 cos θ + j Z0 sin θ − 1 RC −1 cos θ + j sin θ Z0 . (5.1) The Y parameters of the entire circuit, as seen by the two active devices, are given by: � � 1 Ykk = Gk + j ωCk − + ωLk cos θ + j RZC0 sin θ �� � �2 RC Z0 cos θ + j Z0 sin θ − 1 j sin θ −j sin θ �� � Y = Y = 12 21 �2 RC Z0 cos θ + j Z0 sin θ − 1 , (5.2) for k = 1, 2. In the particular case where RC = Z0 , the equations (5.2) simplify to � � 1 1 Ykk = Gk + j ωCk − + ωLk 2Z0 −e−jθ Y12 = Y21 = 2Z0 (5.3) From equations (5.3), it can be noticed that the admittance of the oscillator is independent of the electrical length of the coupling transmission line. This behavior leads to a wideband coupling network that is relatively simple to implement. By defining the loaded quality factor1 Qk 1 see equation (4.40) 81 Qk (ω) = ω � ∂Y � � kk � ∂ω r 2Ykk Ck + =ω 1 ω 2 Lk 2Gk + Y0 ≈ 2ωCk , 2Gk + Y0 (5.4) the coupling strength exerted by each oscillator, as defined in equation (4.25), is Λkp (ω) = |Ykp | 1 Y0 Y0 = · ≈ , r 2Qi Ykk 2Qk (ω) (2Gk + Y0 ) 4ωCk (5.5) where, Y0 = 1/Z0 . Moreover, the coupling phase Φ(ω) is equal to θ(ω) since |∂Ykk /∂ω = 0| and assuming that the frequency derivative of the real part of the active device is negligible. To find the actual coupling phase of this circuit, equation (4.27) is used. It follows that the coupling phase is: Φkp (ω) = r ∂YT,k ∂ω ∠ i r ∂YT,k Ykp ∂ω r Ykp i ∂YT,k ∂ω r i ∂YT,k Ykp ∂ω i + Ykp − ≈ θ(ω), (5.6) r r ∂YT,k ∂Ykk since ≈ = 0. ∂ω ∂ω Another coupling configuration is shown in Figure 5.2. Here, two identical sections of transmission lines are connected by a resistive load of value 2RC . Each transmission line section has a characteristic impedance of Z0 and an electrical length of θ/2. Let’s call this configuration model # 2. At first, the two coupling mechanisms can be argued to produce the same coupling properties as they are both made of the same amount of resistance and transmission line length; however, it will be shown that these two coupling circuits behave quite diﬀerently. Going through a similar analysis (Appendix A), as in the first circuit, the Y matrix of the coupling circuit alone is: 82 Figure 5.2: A pair of oscillators coupled through a transmission line-based circuit (model #2). RC −1 cos θ + j Z0 sin θ 1 [Y ] = RC (1 + cos θ) + jZ0 sin θ RC −1 cos θ + j sin θ Z0 (5.7) The Y matrix of the entire circuit from the active devices’ perspective is: � � cos θ + j RZC0 sin θ 1 Y = G + j ωC − + kk k k ωLk RC (1 + cos θ) + jZ0 sin θ Y12 = Y21 = −1 RC (1 + cos θ) + jZ0 sin θ Equation (5.8) reduces to the following equation when RC = Z0 . 83 (5.8) � � tan 2θ 1 1 Y = G + j ωC − + + j kk k k ωLk 2Z0 2Z0 Y12 = Y21 = (5.9) tan 2θ 1 −j 2Z0 2Z0 Unlike the first circuit, the impedance here is highly dependent of the electrical length of the transmission line. However, the coupling phase remains identical to that of the r ∂Ykk first model, that is Φ(ω) = θ(ω), since = 0. The quality factor is given by ∂ω Qk (ω) = ω � ∂Y � � kk � ∂ω r 2Ykk =ω Ck + 1 ω 2 Lk + Y0 τg 2(1+cosθ) 2Gk + Y0 ≈ω 2Ck + Y 0 τg 2(1+cosθ) 2Gk + Y0 , (5.10) and the coupling strength is Λkp (ω) = |Ykp | 1 Y0 Y0 � � = · ≈ . r θ Y 0 τg 2Qi Ykk 2Qk (ω) (2Gk + Y0 )|2 cos 2 | 2ω 2Ck + 2(1+cosθ) | cos 2θ | (5.11) Here, τg is the group delay of the transmission line (τg = ∂θ/∂ω). From all the above equations, it is clear that the properties of model #2 are highly aﬀected by the electrical length of the coupling transmission lines. The following set of plots compare the quality factors (Figure 5.3) and coupling strengths/coeﬃcients (Figure 5.4) of circuit models #1 and #2. The constant ω0 is the reference frequency at which the initial electrical lengths of all transmission lines are specified. 84 (a) θ = π/2 (b) θ = π (c) θ = 2π (d) θ = 4π Figure 5.3: Comparison of quality factors for various values of the electrical length of the coupling transmission lines. 85 (a) θ = π/2 (b) θ = π (c) θ = 2π (d) θ = 4π Figure 5.4: Comparison of coupling coeﬃcients for various values of the electrical length of the coupling transmission lines. 86 The quality factors are normalized to Q0 = ω0 2Ck and the coupling coef2Gk + Y0 Y0 . From the precedent plots, it is clear that 4ω0 Ck the coupling properties of the first model are not aﬀected by the change in electrical ficients are normalized to Λ0 = length of the coupling structure. This presents a uniform and easily predictable behavior of the array when expanded to multiple elements. The second model, however, has a high dependence on the coupling angle. For case (b) in Figure 5.4, it is clear that no coupling occurs at ω = ω0 . This explains why even two identical oscillators coupled through an out-of-phase (θ = π) line will not synchronize at their natural frequencies, since the coupling coeﬃcient is null. Another observation from model # 2 is that the coupling strength can be made null at some specific frequencies. For case (c), it is shown that coupling is zero at 0.5ω0 and 1.5ω0 . This can be useful in either suppressing potential oscillations at those frequencies when dealing with the design of coupled oscillating circuits. It can also be useful in applications where harmonics or sub-harmonics oscillations are needed to be explicitly suppressed. In general, the nulls observed in the coupling strength occurs at ω π = +n ω0 θ0 � 2π θ0 � , n = 0, 1, 2, ..., (5.12) where θ0 is the initial electrical length of the transmission line at ω0 . 5.2.2 Practical Implementation During the design of a phased array, a few factors need to be considered. To avoid grating lobes, the antenna elements are placed closer to one another. Typically, a distance d = λ0 /2 (or smaller), where λ0 = c0 /f is the free-space wavelength. 87 (a) (b) Figure 5.5: Two models of folded transmission lines for linking oscillators. (a) Modeled after a sine function. (b) A rectangular shape. However, to generate a coupling phase of 0[2π], the transmission lines need to be √ about λg = c0 /(f �e ) in length, where �e is the eﬀective dielectric constant of the substrate. As expected, λg will most likely will not be equal to λ0 /2. This means that coupling transmission lines and circuits will need to be folded or made longer to fit the fixed length between the antennas. Figure 5.5 shows two proposed layouts to accomplish such task. For the layout in Figure 5.5 (a), a smooth function is chosen to guide coupling power from one oscillator to the next. The curves described by f : y = f (x) and � � � f : y = f (x ) are based on a sine function. Any convenient smooth function can be picked for the purpose of oﬀering a gradual transition. The reason for it is to make the coupling circuit less prone to radiation leakage and susceptible to the excitation of evanescent waves. However, these types of circuits can be diﬃcult to implement, and an alternative layout is shown in Figure 5.5 (b). This is a rectangular structure made with straight lines that is simpler to fabricate. Depending on the type of coupling used, a gap can be easily introduced to accommodate a resistive load (model #2). 88 5.3 5.3.1 Design Method and Considerations Counteracting Coupling Eﬀects In practical designs, even when a transmission line coupling is used, the coupling between the oscillators has additional components caused by other electromagnetic phenomena. Typically in layered media or substrates, any source will excite one or more surface-wave modes a well as a radiation field. In typical applications with patch antennas, for instance, that require a substrate material, the TM0 surface wave mode will usually propagate radially away from the patch, decaying proportionally √ to 1/ ρ. Away from the air dielectric interface, the surface wave become negligible and the radiation field or space wave dominates, which decay at a rate proportional to 1/r. Along the air-dielectric interface, there is a lateral-wave field that propagates with the same wavenumber as the space wave, but it however decays proportionally to 1/ρ2 . Consequently, in typical layered media, surface wave and lateral-wave fields contribute to additional coupling, with the surface-wave field dominant at larger distances [109]. To reduce the additional coupling between the antennas and the coupling transmission lines, the antennas will be built on one side of the substrate while the coupling circuits will be built on the other side for better isolation and greater control of the coupling phase. For typical substrates, this results in the antennas being milled or etched on one substrate and the rest of the system built on another substrate. The two ground planes are soldered together and top and bottom layers are connected via drilled holes (where pins/wires are inserted). A high-level design topology is shown in Figure 5.6 with five patch antennas. Since the antennas in an array are inherently coupled to one another and to the rest of the system by surface and lateral wave fields, the phasing between the oscillators 89 Figure 5.6: Top and bottom views of array of patch antennas Antenna Dimension: W=20.5mm, L=19.5 mm. Coupling line: Z0 = 100Ω. is diﬃcult to accurately predict based on the coupling transmission lines alone. In order to maintain a desired phase distribution, it is necessary to come up with a set of transmission lines or other circuits that force the array to adopt a predetermined phase distribution. Unlike all designs presented in the literature review, this array will be allowed to get the optimum performance by using transmission lines of various length or widths so that the desired performance criterion is reached. In this example, it is desired to obtain a uniform broadside array that is also equipped with a large scanning range. To solve this technical design challenge, the equations in Chapter 4, together with a co-simulation approach will be used. In other words, a full-wave EM simulator will control the parameterized dimensions of the array and output the necessary network parameters. The N-port network parameters (S or Y matrix) are then exported into a time-domain simulator which calculates the steady state of the amplitude and phase at each port based on equations developed in the previous chapter. An objective function, that uses the amplitude and phase dynamics, is created pointing to the desired performance criteria. This optimization loop is executed until satisfactory results are found. Figure 5.7 shows a flowchart describing the optimization process. It is clear that a wide variety of optimization routines can be used to achieve the desired 90 Figure 5.7: Flowchart describing the optimization process of the array. Many algorithms can be used, especially genetic, particle swarm optimization, conjugate gradient, etc.. 91 results. In this task, the co-simulating full-wave EM software will be FEKO [110] and the software controlling and implementing the various algorithms and the array dynamics will be MATLAB [111]. The optimization process can be lengthy if the size of the array is substantial; therefore for quick convergence, a fast-converging control algorithm and objective functions must be used. Here, a hybrid and multi-objective discrete and continuous particle swarm optimization (PSO) was developed by combining available algorithms in MATLAB. The objective function was based on the radiation pattern or the computed array factor from the array dynamics. The five-element patch antenna array of Figure 5.6 is simulated with the performance criteria being the best possible gain at broadside. This condition is picked because it also implies that the phase diﬀerence between the antenna signal has to be 0[2π]. The operational frequency is 5 GHz and the initial length of all connecting transmission lines is 4.0 mm. Each transmission line is loaded with a 200-Ω resistor since the characteristic impedance of the lines is 100 Ω. As the length of each coupling line is adjusted, Figure 5.8 shows the progression of the optimization process as the array dynamics converges to adopt the characteristics of a uniform array. It is noticed that the amplitude dynamics are not uniform at first when all the coupling transmission lines have the same length of 4.0 mm. As the algorithmic process iterates, by adjusting the length of each coupling line, the array slowly converges to a state where all amplitudes become equal since the initial objective was to obtain a uniform array radiation pattern. The phase dynamics is tracked as well, and it tends to converge to a uniform distribution; however, both uniform amplitude and zero phasing distributions are not fully reached because the algorithm was not able to perfectly cancel the eﬀect of the surface and lateral wave coupling. Nevertheless, their eﬀects are minimized and only 92 Figure 5.8: Normal progression of the algorithm as the array converges to a uniform phased array. The length of each coupling line is iteratively adjusted as the optimization progresses. 93 observed in the array sidelobes as shown in Figure 5.9. The comparison is made to an ideal uniform array with equal signal amplitudes and zero phase gradient. The final design for the transmission lines are shown in Figure 5.10. As it can be seen, the coupling transmission lines have diﬀerent lengths of 4.3023 and 3.5971 mm. This implementation of coupled oscillators is very diﬀerent from what has been presented in related literature as the oscillators are coupled with identical transmission lines, thus yielding sub-optimal results. The variation of transmission lines’ lengths is needed to improve the stability of the array because each antenna senses a diﬀerent loading environment. Figure 5.9: Plot of the array factor of the 5-element array (red) compared to the ideal uniform array (blue). A small diﬀerence is noticed in the sidelobes while the main beam remained unaﬀected. A more involved optimization could have also changed the characteristic impedance of each transmission line, but for most cases, the coupling phases or line lengths are the most important and sensible variables to be tuned. Using the same optimization process, the array is made to scan oﬀ broadside to its maximum angle before 94 becoming unstable. By unstable, it is meant to point to the fact that either the array de-synchronizes (which can be tracked by a sudden change in phasing) or adopt another mode (diﬀerent phase distribution). For both cases, the array operates at 5 GHz and keeps the same loading characteristics. The time-domain simulation runs for at least 200 ns. The initial conditions for the amplitudes are small compared to their steady-state counterparts and also randomly generated. Figure 5.10: Final dimensions of the array design (bottom coupling lines not to scale). The array is symmetric with respect to the third antenna. All resistors are 200 Ω. Since the theoretical maximum phasing is π/2, which corresponds to a scan angle of π/3 for an inter-element separation of λ/2, the objective function for this setup is written so that the algorithm drives the array dynamics to produce a peak of the 95 array factor at the expected scan angle of π/3. The result of the process is shown in Figure 5.11. Figure 5.11: Optimization of scan angle in the presence of coupling. The realized scan for this simulation is 25.730 , which is lower than the 300 predicted by the theoretical equations. This result tells us that even though the natural coupling of the array can be minimized, it still plays a limiting factor in the performance of coupled-oscillator arrays. It should also be noted that the closer the phase dynamics reaches the theoretical limit, the slower the array becomes as reaching the steady-state takes longer. The presented optimization process is generic and can be used with any design where a network parameter or a radiation characteristic is desired with the added benefit of a time-domain simulation to solve the required diﬀerential equations. Appendix B provides the basic Matlab script to accomplish such task. This process will be used as part of the design and implementation of any subsequent array design in this chapter. 96 5.3.2 Non-reciprocal Coupling Networks Maintaining the mode of operation necessary to operate a coupled-oscillator system as a phased array requires that both signal amplitude and phase of each element is not vastly diﬀerent. This makes COA systems not very attractive for applications that require wide scanning range. As stated multiple times in this dissertation, typical COA systems have a fundamental phase gradient limitation of π/2 under either injectionlocking or mutual/reciprocal coupling. To put this in perspective, a half-wavelength spaced linear array can only generate a maximum scan angle of approximately ±300 . This theoretical limit is generous since it is was shown earlier that the optimized scan angle is closer to 25.730 . In most published works, it is also assumed that in a weakly coupled/injected system, the amplitude dynamics can be ignored. In practice, noise and other unaccounted coupling eﬀects will substantially reduce that theoretical 300 figure. Overcoming the scanning range limitation is an area of active research. Heath [60] provided a description of a fully nonreciprocal coupling scheme necessary for removing the scanning limitation. This beam-steering method is based on a bidirectional network that uses two explementary coupling phases. The set of coupled phases for an N element 1-D array are defined as: Φp,p+1 = θ Φp+1,p = −θ , (5.13) for p = 1, 2, . . . , N −1. Simply put, this describes a coupling network where the phase shift going from oscillator p to p + 1 is the opposite of the phase shift from p + 1 to p. The coupling magnitudes, on the other hand, are assumed to be the same. It was 97 shown that this phase coupling scheme is stable for all phase gradients. Furthermore, it was shown that the array stability is no longer a function of the scanning angle. This property allows us to obtain a more robust system in the presence of noise and amplitude variations. For typical coupling, the arrays stability gradually decreases as a function of the scan angle. In terms of performance, a less stable array means that the dynamics becomes slower and that steady state takes longer to be reached. Implementing working nonreciprocal coupling networks is a challenge. Since most of the practical implementations of nonreciprocal networks are based on bulky ferrite devices that are incompatible with smaller microwave monolithic circuits, Martinez and colleagues [112] proposed a coupling network based on microwave transistor amplifiers that are fully capable of being integrated and compatible with various array architectures at the mm-wave range. Half of the proposed circuit is shown in Figure 5.12. The gain block is needed to generate the non-reciprocity. The variable transmission lines (VTL) are used to dial the needed phase gradient to be applied to the array. Figure 5.12: Architecture of half of the nonreciprocal coupling network [112]. 98 The major issue with the nonreciprocal circuit in Figure 5.12 is its low eﬃciency and large size. In order to make the coupling circuit stable such that no oscillations can be generated on their own, a set of resistors are put at the end of the gain block. For more eﬃciency, the resistors can be removed as long as the following two concepts are implemented. First, the resistors’ main function is to reduce the gain of the transistor such that no oscillation is produced by the coupling circuit itself; therefore, they are needed when the coupling signal from the gain block is too strong. However, when an extremely small amount of power is allowed to couple to the nonreciprocal circuit, the transistors can be fully used as amplifiers in order to get the coupling signal back to an arbitrary level that does not make the circuit potentially unstable. In that case, the resistors are not needed. Secondly, the nonlinearity of the the coupling circuit can be embraced so that the instability can be used to complement the generation of the oscillation signal at the desired frequency or for coupling reasons at one of its harmonic frequency. It is even possible to already imagine a case where the oscillation is only due to the presence of the nonreciprocal coupling circuit alone. In this case, a few more design steps are required in order to make sure that the oscillator, as an entire circuit, is still working even with the change of phase shift that might be implemented for beam-steering purposes. 5.4 Implementation of a Fully Nonreciprocal Coupled-Oscillator Array As it was stated earlier, a fully non-reciprocal coupling circuit, inserted between adjacent oscillators in a linear array, removes the beam-scanning limitation. Reference [112] was the first work in COA-related literature that successfully implemented 99 a non-reciprocal coupling network using solid-state devices that can be used to accomplish such task. To show that such circuits can improve the scanning range of a coupled-oscillator system, the following three-element array in Figure 5.13 is implemented. Figure 5.13: Architecture of a fully nonreciprocal array. To show that a large scan angle can be achieved, a phase gradient of 1500 is chosen for the value of θ. Note that the theoretical maximum for a reciprocally coupled array is 900 . Due to a lack of low-loss phase shifters available, the coupling network has no ability to change its phasing. Due to the large number of transistors needed to implement this array, it was decided to pick the minimum number of elements necessary for such array. In order to replicate any nonlinear behavior observed in coupled-oscillator systems, the minimum number of elements needed is three, thus requiring at least seven2 transistors. To summarize the design steps, the two coupling circuits are initially designed together making sure that they do not generate any stable oscillation at the frequency of operation (5 GHz) or anywhere else based on the nonlinear model of the transistor 2 In general, for a fully nonreciprocal linear array, the minimum number of transistors is 3N − 2, where N is the number of antenna elements 100 (NE3201SO1). The transistors are basically operated in the small-signal regime since most of the power is designed to be directed to the antennas. Since there are only three elements, the two coupling networks are required to have the same physical dimensions due to symmetry. Once the coupling circuits display the desired properties, the three antennas are added and the power ratio is selected by increasing or decreasing the input impedance of the antenna (moving pin location). For this design, it was desired that that between 10 and 15% of the generated power is coupled while the rest is sent to the antennas. Once the pin location is set, based on the power distribution ratio, the oscillators are designed as standalone oscillators (with no coupling) using the parametrization techniques introduced in Chapter 2. Once oscillations are achieved and stable at the desired frequency of operation, coupling is introduced and the system is verified to oscillation. This verification process is repeatedly done until a stable oscillation is found with the appropriate phasing. The design of the coupling circuit is done using the co-simulation optimization process shown earlier with FEKO/MATLAB, whereas the design of the main oscillators is done using a harmonic balance simulator and the stability verification is carried out using a transient or time-domain methods in ADS [90]. Figure 5.14 shows the final layout and the fabricated array. As done previously, the antennas and the rest of the nonlinear circuits are built on two separate substrates (with common ground planes) in order to help with isolation and cross-polarization of the radiation pattern. To verify the proper operation of the array, a few measurements were carried out. The first measurement dealt with recording the spectrum of the radiated signal. Figure 5.15 shows the recorded spectrum with a horn antenna placed about 2 m away from the array without any smoothing applied. The spectrum analyzer used was Agilent E4407B with a resolution bandwidth of 0.5 MHz and 0-dB attenuation. 101 (a) Top View: Patch antennas (b) Fabricated antennas (c) Bottom View: Layout of front-end circuits (d) Actual front-end circuits Figure 5.14: Layout and fabricated array showing active components and bias lines . 102 Each FET transistor was DC-biased to VDC =2 V and IDC =10 mA. The measured peak power occurs at an oscillation frequency of 4.96 GHz. The spectrum seems to contain a significant amount of RF power in the sidebands, indicating a high phase noise. Figure 5.15: Measured power spectrum of the array, 30λ0 away from antennas. To gather more performance on noise, the phase noise was also measured with the array put closer to the horn for more input power. A setup using a signal source analyzer (E5052B) with microwave downconverter (E5053A) provided a phase noise measurement of -96 dBc/Hz at an oﬀset of 10 MHz as shown in Figure 5.16. At 1 MHz, the phase noise is about -85 dBc/Hz. As expected, the phase noise is somewhat high for this array to be used as a current communication system as phase noise values below -110 dBc/Hz are expected at this oﬀset [113, 114]. For comparison, the phase noise obtained by York and colleagues [4] (pp. 170) for a five-element array was -68 103 Figure 5.16: Spectrum of phase fluctuations of the array dBc/Hz at 100 KHz, while for this design the phase noise is -64 dBc/Hz. The phase noise in general is expected to decrease as the number of array elements is increased. One common way to lower phase noise is to use a PLL or a dedicated resonator/cavity. In this case, other authors have reported phase noise lower than -100 dBc/Hz at 1 MHz oﬀset [115, 116]. Potential sources of added noise could have been caused by the large number of transistors. Moreover, the array was optimized to generate the required phasing and not to display a higher Q. Finally, the radiation pattern was measured rudimentary (by hand) due to the diﬃculty of using the circuit with all its required DC voltage bias sources on the setup of the anechoic chamber. Figure 5.17 shows the measured points with the simulated expectations. As noticed, the array is radiating oﬀ broadside (angle=0), and the results are satisfactory. The location of the maximum radiation is at 37.40 . This measurement shows that the inter-element phasing is approximately 1410 , thus 104 Figure 5.17: Comparison of measured radiation pattern to expected simulated results. realizing the highest phasing reported to date for coupled-oscillator arrays. A work by Liao and York [117] achieved a scan angle of 600 which corresponded to a phasing of about 400 since the spacing was 0.26λ0 . The estimated radiated power is 11.78 dBm, resulting in a DC-to-RF eﬃciency of 10.13%. The computer EIRP is 16.34 dBm. It is clear that the eﬃciency of the array is decreased compared to the eﬃciency of a single oscillator because of the addition of the four coupling transistors. The initial goal of generating a phase gradient of 1500 was partially achieved with a measured phasing of 1410 , thus only missing the target by a 6% error; however, more eﬀorts could have been undertaken to also reduce the phase noise. 105 CHAPTER 6 COUPLED-OSCILLATOR BASED APPLICATIONS AND ARRAY ARCHITECTURES 6.1 Introduction Coupled-oscillator systems derive their specific properties from the mutual interactions of independent oscillators. For that reason, each oscillator has its own characteristics even though at the design level, every eﬀort is taken to create practically identical oscillators. However intrinsic diﬀerences exist between oscillators due to various causes such as fabrication process variation of passive and active devices, human errors and asymmetric coupling from topological placement. Even if process variation and human inputs are eﬀectively accounted for, the coupling architecture/topology might create an imbalance in loading. For instance, in linear arrays, the end elements are subject to a diﬀerent type of coupling environment than the other elements. This change in loading environment actively creates a diﬀerence in natural frequencies for each oscillator in an array. To mitigate all of the mentioned problems, fabricated arrays have been manually tuned to correct this imbalance. 106 6.2 Array Architecture To partially resolve the issues mentioned earlier, an architecture containing a phaselocked loop is proposed. Unlike various architectures found in available literature, an alternative architecture is proposed as depicted in Figure 6.1. The most attractive benefit of this architecture is its simplicity. In other words, N oscillators are only controlled by one phase-locked loop (PLL) instead of many as seen in other works [72, 73]. The reduction in the number of loops and other controller elements is due to a need to reduce cost, size, weight and complexity. This array is also expected Figure 6.1: COA phase-locked loop based architecture. to be simpler to analyze and operate without losing any performance metric. The properties of the control loop of this proposed architecture can be analyzed by first considering that the system is noiseless. Assuming that the reference oscillator has a signal of the form 107 1 a(t) = Aa sin(ωa t + φa ), 2 (6.1) and that the signal coming from the output of the frequency divider is b(t) = B cos �ω s N � t + φb , (6.2) the output of the mixer is therefore given by c(t) = Aa BKm � � � � �� ωs ωs sin (ωa − )t + φa − φb + sin (ωa + )t + φa + φb N N (6.3) where Km is the mixer gain. The output of the mixer is filtered out by the low-pass filter (LPF) resulting in the high-frequency term to be dropped, meaning that d(t) ≈ Aa BKm � � �� ωs sin (ωa − )t + φa − φb . N (6.4) In typical PLL, the signal d(t) is used to control the oscillation frequency of the oscillator, usually through a varactor diode, resulting in voltage-controlled oscillator (VCO). The natural frequency of the controlled oscillator is therefore given by ω(t) = ω0 + Kv d(t), (6.5) where Kv is the VCO gain [118, 119], and ω0 is the free-running frequency of the oscillator. This PLL feedback loop forces the controlled oscillator to change its oscillation frequency to ωs = N ωa . At this stage, d(t) becomes a quasi-DC signal of value Aa BKm sin(φa − φb ). When dealing with oscillators with identical natural frequencies, the benefits of this architecture can be noticeable. The following simulations show a comparison 108 between a typical COA and the current PLL-based COA, both having five oscillators coupled similarly, with the only diﬀerence being the addition of the PLL. The oscillators are considered to be weakly coupled, therefore the array’s properties can be assumed to be uniform. The coupling strength is �=0.1, the quality factor Q=10, the coupling phase Φ is zero, and the natural frequencies are all equal to 10 GHz. The PLL parameters for this simulation are Aa =1, B=0.1, N=1, φa = φb =0, Km =5, Kv =0.05. Figure 6.2 shows the diﬀerence in settling time—amount of time taken by the array dynamics to reach its steady state—between a typical 5-element COA and the PLL-based COA from the previous figure. In this case, the controller module sends the same correcting signal to all oscillators since the natural frequencies are identical. As noticed, the PLL-based COA settles to its steady state faster (10 ns) than the typical COA with the same initial conditions (20 ns). Inserting the PLL circuit has shown to double the speed of the typical array, showing the possible improvement. However, when slightly diﬀerent oscillators are used, as practically expected, the same approach cannot be used because the oscillators have intrinsically diﬀerent natural frequencies. Feeding the control signal coming from a single loop to the entire array will create a conflict between the oscillators and the final steady state might be unpredictable since each oscillator is controlled by a signal that might not match its current state. To help mitigate this issue, a controller stage, with more controlling capabilities, is used. The new controller introduces the possibility of independently managing the free-running frequency of each oscillator based on the signal d(t). However, resolving all the required signals that have to be fed to each oscillator requires a seeding or calibration mechanism. 109 (a) (b) Figure 6.2: Performance comparison between (a) typical COA and (b) PLL-based COA. To show how this architecture treats the presence of multiple free-running frequencies on the array, the following simulation is carried out with the initial controller. Let’s assume that the five-element array used earlier has the following natural frequency distribution [f0 ] in GHz :[f0 ]=[10.0262, 9.9956, 9.9519, 10.0321, 9.9945]. The distribution is centered around a mean of 10 GHz. The PLL parameters remain the same as before. Figure 6.3 shows a comparison of the dynamics of a typical fiveelement array to that of the PLL-based COA with the same initial conditions. As expected, the PLL-based array converges faster than the typical COA; however, the same steady state is reached. This shows that the speed, and therefore the stability of the array is improved, but it does not change the stability range of the array itself as a fully nonreciprocal coupling circuit does. In other words, if a typical COA is unstable because of a large diﬀerence in natural frequencies or large coupling phases, 110 (a) (b) Figure 6.3: Phase dynamics of five-element array. (a) Typical COA. (b) PLL-based COA. having this PLL structure will not drastically change this stability/instability state. This proposed architecture only increases the array stability margin, which results in higher speeds and higher noise cancelling properties, but the stability space is unchanged. In terms of general stability, it has been already shown by Heath [60] that for reciprocally coupled linear array, the maximum Lyapunov exponent is � π� λm = −2� cos θ 1 − cos , N (6.6) where � is the coupling strength, θ is the uniform phase gradient along the array, and N the number of oscillator. The Lyapunov exponent is a measure of the stability of a dynamical system. The more negative its value, the more stable the system becomes. 111 The addition of the single PLL loop does not change the intrinsic stability of the array; however, it results in an eﬀective increase of the coupling strength, making the exponent more negative, thus the array faster. However, giving this array topology the ability to control each oscillator independently results in the array being to expand its stability domain. In other words, the array is able to move from diﬀerent modes of operation by dialing the natural frequencies of each oscillator. 6.3 Array Calibration and Pattern Shaping As mentioned earlier, building independent oscillators with truly identical free-running frequencies is inherently unfeasible because of the inaccuracies in fabrication techniques and fluctuations in electronic devices. For high-frequency oscillators, these small errors translate in various free-running frequencies, which in turn create a distorted radiation pattern even when all oscillators synchronize to the same frequency. The distortion is caused by the phase gradient not being constant across the array. Moreover, as demonstrated earlier, feeding all oscillators with the same PLL feedback loop is problematic because of the nonuniform phase gradient create at steady state. To address the issues emanating from the architecture of Figure 6.1, the following changes are added as depicted in Figure 6.4. This change is necessary so that the array seeds or calibrates itself to the small diﬀerences in natural frequencies of each oscillator. This is another example of how nonlinear systems diﬀer from linear ones. As a mean to accomplish this calibration, the array pattern is used as controlling mechanism. An antenna is placed in the far-field of the array at the position where the maximum radiation is desired to be, typically at broadside. This error correction scheme is mainly concerned about changing the natural frequency of each oscillator through 112 Figure 6.4: COA architecture with self-calibration mechanism. 113 a gradient optimization process or any other implemented algorithm. Put diﬀerently, at each step, the radiated power is measured and fed to the controller/optimizer, which in turn slightly adjusts each oscillator’s natural frequency as to maximize the power observed at the testing antenna location. This is a simple setup that can be expanded to include more antennas and implement more accurate calibrations. This calibration setup can also be used to shape the radiation pattern of the array if one or more antennas are added. For instance, a radiation null can be requested at the locations of the test antennas if desired. Simulations show that this calibration technique is viable for COAs as shown in Figure 6.5. It can be seen that the array slowly calibrates itself to an ideal broadside radiation after a few iterations for a 10-element array. Using this process, the calibration is completed when no changes in voltages are recorded based on the achievement of the optimization goals. These individual voltages are maintained throughout the subsequent operation of the array. A major benefit of this system is that this calibration is done by the array itself without any human intervention. 6.4 Phase-Amplifying Arrays As mentioned in previous chapters, microwave coupled oscillators with reciprocal coupling networks have been plagued with limited scanning ranges. Using nonreciprocal coupling networks has shown that this limitation can be lifted. However, nonreciprocal coupling circuits add complexity, additional cost and further study and analysis. Recently, it was shown that the hearing system of a fly (Omnia Ochracea) [120, 121] can serve as a model to amplify the phase diﬀerence between two closely spaced small antennas [122]. In this work, a pair of passively coupled electrically small antennas are used to amplify the phase diﬀerence between the received voltage signals. The author suggests that this process can be used to build phase sensitive receiving array 114 (a) (b) (c) Figure 6.5: Array factor of 10 isotropic antennas. Solid line is COA array factor and dashed line is ideal broadside array factor. a) Initial random frequency distribution. b) After a few iterations. c) After many more iterations, the COA array factor approaches its ideal state, thus accomplishing calibration in the process. 115 (a) (b) Figure 6.6: (a) Mechanical model of hearing system of Omnia Ochracea. (b) Corresponding electric model [122]. for direction-finding applications. Figure 6.6 shows the mechanical and corresponding electric model. In this dissertation, this phase-boosting process is reversed to cater to transmitting systems and expanded to more than two antennas. Moreover, this process is applied to electrically large antennas and was used to ease the phase gradient requirement for coupled-oscillator arrays. To understand this phase-amplifying mechanism, this simple study is undertaken. The impressed or receiving voltages are replaced by the voltages generated by the oscillators in the transmitting case. The input is the phase diﬀerence between the oscillator voltages V1 and V2 while the output is the phase diﬀerence between the currents I1 and I2 as shown in Figure 6.6. To show the desired eﬀect, the lumped elements are replaced by generic reactive loads as in Figure 6.7. This model can be used to represent a two-element array using single mode antennas1 . In this case, the phasing of the port currents corresponds to the individual radiating phasing as well. The reactive loads are used so that no loss of radiative power is consumed by 1 small dipoles, thin slots, etc 116 Figure 6.7: Generic representation of coupling loads between oscillators. the coupling circuits. These loads are modeled here as Xi = Z0 tan θi , with θi ∈ [−π/2, π/2]. The output is the phase diﬀerence between the two branch currents φI = ∠I2 − ∠I1 , (6.7) meanwhile, the input is the initial phase diﬀerence between the oscillator voltages: φV = ∠V2 − ∠V1 = δφ. (6.8) To get a better sense of the properties of such circuits, let’s define two quantities, namely the current amplification factor AF � � � dφI � � �, AF = � dφV � (6.9) and the current magnitude diﬀerence factor DF � � � I1 � DF = 20 log �� �� I2 117 (6.10) By sweeping the loads, the following current-based plots are generated for an initial voltage phase diﬀerence of 100 (Figure 6.8) for Z0 = R1 = R2 = 50 Ω and θ1 = 610 and θ2 = 110 . Figure 6.8: Phase diﬀerence and amplification factor of currents I2 and I1 . Figure 6.9 shows that it is possible to generate any arbitrary current phase gradient by choosing the appropriate set of reactive loads. However, it is clearly visible that tradeoﬀs exit: the more phase amplification, the more the signal amplitude diﬀerence in various cases, but an optimal point can be picked for a desired design. The other half on the plot is not shown because there is symmetry. To illustrate the eﬀect of such coupling circuits on actual coupled oscillators, two microstrip patch antennas will be used. Although the equivalent electric circuits for 118 Figure 6.9: Plots of AF (dotted) and DF (solid) generated by sweeping the reactive loads as shown in Figure 6.7. 119 coupled patches are more complex than the depiction in Figure 6.7, it is nevertheless noticed that some phase-boosting eﬀects are observed. Figure 6.10 shows two microstrip patch antennas replacing the resistors of Figure 6.7. A reactive coupling circuit is inserted between the two coupling ports at the top while a small and constant phase diﬀerence is maintained between feeding ports #1 and #2. After trying a Figure 6.10: Two microstrip patch antennas coupled through a phase-boosting network. few candidate circuits with φV =100 , and monitoring the output phase, the following T-network was selected to be inserted between the two patches due to the observed phase amplification. As shown in Figure 6.11, the coupling circuit comprises two capacitors of value 0.2 pF and a shunt inductor of 1.96 nH. Figure 6.11 also shows the overall phase-boosting performance when the voltage phasing is fully swept. It is seen that the maximum amplification occurs at φV ≈ 150 , and that the amplitude diﬀerence is highest around φV = 200 . These values can be tuned to obtain a higher amplification factor, but as mentioned earlier, a significant diﬀerence in current amplitudes occurs if AF is made too high. 120 Figure 6.11: Simulation results for two coupled microstrip patch antennas. In terms of the direction of the maximum radiation of the antenna array, it is observed that without the coupling circuit, about 1500 of phase diﬀerence is needed to steer the main beam to 300 oﬀ broadside. With the phase-boosting circuit present, only 900 are needed to accomplish approximately the same result with the added benefit of smaller sidelobes. Figure 6.12 shows the eﬀect of placing the coupling circuit between the antennas. It is clear that an improvement in scanning angle is gained when using such circuits. To show a more complete performance, Figure 6.13 shows a practical implementation of a two-element array with the patch antennas on one side of the substrate and the remaining functional elements on the other side. This arrangement is used to isolate the antennas from the rest of the circuit, thus barely aﬀecting the radiation pattern and simplifying the initial analysis. The system shown in Figure 6.13 was designed to work at 10 GHz on a 31mil substrate with a dielectric constant of 2.2. The FET transistors used to make 121 Figure 6.12: Total array gain with and without the coupling circuit. the individual active antenna are CEL’s NE3201SO1. The design methodology to design each AIA follows the techniques developed in Chapter 2. By monitoring the current phase at each feeding port of the antennas, Figure 6.14 shows that a phaseboosting mechanism is generated, as expected. In Figure 6.14(a), three curves are shown in order to compare various performance. In summary, a 100 phase-boosted voltage phase shift nearly generates the same scan angle as a non-coupled array with a 650 phase shift. In Figure 6.14(b), the locations of the main beam from the array factor (solid line) and the non-coupled array (marked line) are plotted for an increasingly large voltage phase shift between the two oscillators. By inserting the phase-amplifying circuit, it is possible to generate a much larger scan angle than the one permitted by the array factor when the phase shift is small. As noticed earlier, increasing the amplification factor AF although helps with small phase gradients, there is however a tradeoﬀ in terms of amplitude diﬀerences 122 (a) Top view (b) Bottom view Figure 6.13: Layout of two-element array. (a) Top view made of two patch antennas. (b) Bottom view showing necessary circuitry. All dimensions in mm. 123 (a) (b) Figure 6.14: (a) Comparison of phase-boosting eﬀect to normal array. (b) Snapshot of the scan angle of phase-amplifying array (red mark) compared to scan angle of array factor (solid line) and scan angle of normal patch array (marked line). 124 between the oscillators. Another drawback is that a high AF would mean that small changes in input phasing could result in substantial changes in output phase, creating potential issues with system modulation and eventually beam squinting. Another disadvantage is that this transmitting system works well with two antennas at a time. Attempts to expand the transmitting type to multiple antennas have proven to be problematic because each antenna pair would need to be phase matched to its adjacent pairs. Coupled-oscillators array inherently solve this problem because each oscillator is phase-locked to one other. Unlike a uniform phased array, this type of phase-booting system will have varying current amplitudes when scanning since each pair will inherently have a non-zero DF . Figure 6.15 shows the resulting architecture necessary to accomplish such characteristics. As expected an even number of oscillators are needed. Figure 6.15: Phase-amplifying architecture of coupled-oscillator arrays. Following a simulation, Figure 6.16 shows the array factor of a 10-element array for a uniform array and the corresponding array factor its phase-amplified counterpart. 125 As observed, the phase-boosted array present larger sidelobe level, but the main beam remains unaﬀected. Figure 6.16: Comparison of array factors of a 10-element array (uniform and phaseamplified). Even though this technique does not solve the phase limitation of reciprocally coupled oscillators, it helps the array operation by accomplishing the same scan angle by just producing a smaller amount of phase, which ultimately helps the array stability by not being pushed to the brink of instability when beam-scanning is needed. In other applications, the phase-boosting mechanism can also be used to produce a “phased-reducing” system in which AF is made smaller than unity. This process can be used to control highly varying phase diﬀerences in a system. 126 CHAPTER 7 CONCLUSION 7.1 Summary This dissertation discussed the promising technology of coupled microwave oscillators made from independent self-oscillating active integrated antennas. From the problem statement, it was urged that developing functional COAs would be cost-eﬀective and highly valuable for high-frequency phased array systems, mainly due to their higher eﬃciency, smaller size, their simplified control structure and high integrability. In the first chapter, a new set of techniques for analyzing and designing active integrated antennas was presented. The techniques relied on using the convex-like properties of resonant antennas’ input immittance in order to localize stable oscillation design parameters at small and large-signal steady-state operation. The potential of the overall method or idea was not limited to active antennas as it could be applicable to the analysis and design of generic negative-resistance microwave oscillators displaying similar characteristics. The analysis and design method was illustrated by a simulation of an AIA at 13 GHz, and an experimental agreement with simulation was conducted at 5 GHz. Secondly, a mathematical description for accurately computing the dynamics of an array of nonlinear coupled oscillators through a linear network was presented with focus on solely using measurable network parameters such the impedance/admittance 127 matrix. The derived equations are shown to be expandable and written for each harmonic components although the focus of Chapter 3 was on presenting the equations at the fundamental tone. While the targeted application for this dissertation is active integrated antennas based phased arrays, the equations can be useful in many fields dealing with coupled nonlinear oscillators arranged in any topology. Numerical simulations showed that a higher degree of accuracy is achieved when compared to other equations provided for coupled microwave oscillators in related literature. Using the newly generated equations from Chapter 3, a set of design considerations were examined closely. At first, the properties of most common types of coupling circuits were studied with an emphasis on their coupling strength and phasing properties. It was found that common transmission-line based coupling networks display broadband and extremely narrowband behavior depending on the number of line sections and location of the resistive loads. It was also shown that the coupling phase can be a complex function of many variables that may only reduce to the expected electrical length of the transmission line in some particular cases. Secondly, due to the required analysis breadth of COA systems, a generic optimization method was developed to sequentially link multiple analysis tools to perform full-wave EM calculations (FEKO) and diﬀerential equations calculations under the control of a single algorithm (MATLAB). This provided an eﬃcient optimization process for the design of COAs. Thirdly, the implementation of a 10-GHz transistor-based non-reciprocal coupling circuit was presented. The circuit’s role was to generate a set of explementary phases between two adjacent oscillators—this is a needed condition to remove the scanning range limitation in a linear array. Lastly, a three-element array using a compacted version of the previous non-reciprocal coupling circuit was designed and fabricated at 5 GHz. The array produced a phase gradient of about 1410 , without 128 loss of stability, thus realizing the highest achieved phasing in current literature. Although the phase noise of the array could have been improved by the design method involved, the design goal of the array was to primarily display the correct phase diﬀerences between the oscillators. Finally, a few array architectures were presented for the purpose of increasing stability, speed, and usability in a communication or tracking system. The first architecture made use of a single PLL for an arbitrarily large array. It was shown that for this architecture to be viable, it required the independent control of each oscillator through a controller. The seeding of such controller was subsequently done by a calibration mechanism that could be applied by the array itself. The second mechanism used to improve a typical COA’s scanning range was the implementation of phase-amplifying technique based on the auditory system of an insect. In essence, by using a well-designed coupling circuit, small phase diﬀerences at the oscillators ports (voltages) are amplified into larger current phase diﬀerences at the antennas. Even though this technique does not solve the phase limitation of reciprocally coupled oscillators, it helps the array operation by accomplishing the same scan angle by producing a smaller amount of phase, which ultimately helps the array stability by not being pushed to the brink of instability when significant beam-steering is desired. 7.2 7.2.1 Future Work Harmonically Loaded AIA and COAs It’s been known that harmonically loading an oscillator can have positive eﬀect in increasing its output power level or the DC-to-RF power eﬃciency [24]. However, this approach was not applied in the techniques presented in Chapter 3. It would be 129 worthwhile to include the eﬀects of harmonic loading into the design and optimization process of a single AIA. In addition to that, the array equations presented in Chapter 4 could be used to generalize the techniques of Chapter 3 into a more involved set of techniques that are applied to the design of a COA. At this moment, each AIA is optimized individually, thus potentially resulting in a sub-optimal performance. In general, a design process that would harmonically load to the array to either optimize output power or power eﬃciency would be highly beneficial. In summary, the parametric design techniques presented in Chapter 3, the equations of Chapter 4, coupled with harmonic loading can all be applied to generate a generic design method for coupled-oscillator systems. 7.2.2 Non-Foster Loads for Phase-Amplifying Arrays The use of non-Foster circuits primarily stems from their applications in impedance matching, loading of antennas, transmission lines and waveguides since they are not restricted by the well-known gain-bandwidth product [123]. Consequently, it is of importance to study the eﬀect of using non-Foster loads in the phase-amplifying circuits located between the oscillators. It is expected that some performance criteria could be improved. Preliminary investigation has suggested that broadband operation of the circuit could be achieved, but more importantly, there is evidence that similar amplification factor (AF ) could be achieved with smaller values of reactance while keeping DF constant or smaller. 130 Appendix A COUPLING PROPERTIES OF RESISTIVE LINES The network parameters of a couple of commonly used coupling circuits based on one or two sections of transmission line are examined. Figure A.1 depicts the first circuit. Figure A.1: Coupling circuit # 1. The ABCD transmission matrix of the circuit above is given by [124]: jZ0 sin θ 1 RC A B 1 RC cos θ = C D 0 1 jY0 sin θ cos θ 0 1 (A.1) which corresponds to RC + jZ0 sin θ A B cos θ + jY0 RC sin θ = C D jY0 sin θ cos θ + jY0 RC sin θ 131 (A.2) However, since the circuit is reciprocal, it leads to A Y 11 = Y22 = B (A.3) Y21 = Y12 = −1 B Computing the coupling admittance, we have [Y ] = Z0 j sin θ �� 1 c cos θ + j R Z0 Rc −1 cos θ + j Z0 sin θ � (A.4) �2 sin θ − 1 Rc −1 cos θ + j sin θ Z0 For for RC = Z0 , Equation A.4 reduces to Y0 [Y ] = 2 1 −e−jθ −e −jθ 1 (A.5) Applying a similar analysis, the second circuit showed in Figure A.2, the ABCD matrix is θ θ θ θ jZ0 sin 1 2RC cos jZ0 sin A B cos 2 2 2 2 = θ θ θ θ (A.6) C D jY0 sin cos 0 1 jY0 sin cos 2 2 2 2 This matrix can be simplified to: 132 Figure A.2: Coupling circuit # 2. cos θ + jY0 RC sin θ B A B = C D Y02 RC (cos θ − 1) + jY0 sin θ cos θ + jY0 RC sin θ (A.7) The coupling admittance is given by Rc −1 cos θ + j Z0 sin θ 1 [Y ] = RC (1 + cos θ) + jZ0 sin θ Rc −1 cos θ + j sin θ Z0 (A.8) For for RC = Z0 , Equation A.8 reduces to [Y ] = jθ 1 e jθ Z0 (1 + e ) 1 j θ2 −j θ2 1 e 1 e = θ jθ −j θ2 j θ2 2Z cos 0 2 e e e 133 (A.9) Appendix B MATLAB/FEKO CO-SIMULATION COMMANDS This appendix provides an example MATLAB program that controls a FEKO design file to optimize the reflection coeﬃcient of a microstrip antenna. A few options will need to be modified to fit the installations on your machine, mainly, the location or path of the programs and you will also have to provide your own .cfx file. This demo was run with FEKO 5.5 although versions 6.0 and 6.1 were also tested to work. Filename: runOptimzation.m global TRACKING hAx TRACKING=[]; figure; hAx=axes; %initial values of variables (note the order) x0=[3;18]; %Matlab search function. This can be manually programmed. With the optimisation toolbox, you might use [fmincon] for constrained multivariable minimization instead of [fminsearch] [X,FVAL,EXITFLAG] = fminsearch(@runYourFekoCommands,x0); Filename: runYourFekoCommands.m function T = runYourFekoCommands(X) %This function runs FEKO in batch mode. %%CHANGE/ADD PROPERTIES BELOW filename=’patch1’; numberOfPorts=1; fekopath=’"\Program Files\FEKO\5.5\bin\’; %change this path if needed 134 %Assign your variables as in the design itself f=X(1); l=X(2); variableArray=[’ -#f=’ num2str(f) ’ -#l=’ num2str(l)] ; %notice the white space global TRACKING hAx FEKO DISPLAY=true; %or false PLOTTING=true; %%STOP MAKING CHANGES HERE %Update variables path3=[fekoPath ’cadfeko batch" ’]; dispFeko([’Running Cadfeko batch: updating Variables:’ variableArray ], FEKO DISPLAY) [a,b]=system(sprintf(’%s’,path3, filename, variableArray)); path=[fekoPath ’prefeko" ’]; argument=’ > output.txt’; %Running Prefeko path2=[fekoPath ’runfeko" ’]; argument2=’ > FekoOutput.txt’; dispFeko(’running PREFEKO: meshing’,FEKO DISPLAY) system(sprintf(’%s’,path, filename, argument)); pause(1) %Running FEKO dispFeko(’running FEKO: field calculation’,FEKO DISPLAY); system(sprintf(’%s’,path2, filename, argument2)); pause(0.5); %wait for feko mkl to finish or write a wait loop %Reading SNP File dispFeko(’Reading output file’,FEKO DISPLAY); % Code your own S parameter reading function for (’patch1.s1p’) SM=readSNPDataFromFile([filename ’.s’ num2str(numberOfPorts) ’p’]); %Generate your own cost function here based on the S-parameters T=abs(SM.matrixData); %plotting TRACKING=[TRACKING T]; 135 if PLOTTING plot(hAx,TRACKING); xlabel(hAx, ’Number of Iterations’); ylabel(hAx,’Goal’) title(hAx,variableArray); grid(hAx,’on’); end end Filename: dispFeko.m function dispFeko(message,yn) if yn disp(message); end end Either the S-parameter file is read or the explicitly requested output.txt file is read in order to define a goal function that will need to be minimized. 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