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University of California Santa Barbara Analysis and Design of Systems of Coupled Microwave Oscillators A Dissertation submitted in partial satisfaction o f the requirements for the degree o f D octor ofPhilosophy in Electrical and Computer Engineering by Jonathan James Lynch Committee in charge: Professor Robert York, Chairperson Professor John Bowers Professor Petar Kokotovic Professor Umesh Mishra Professor Mark Rodwell May 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9602124 UMI Microform 9602124 Copyright 1995, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation o f Jonathan J. Lynch is approved Committee Chairperson March 1995 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedicated to: My wife, who worked at least as hard as I and gave this effort a special significance; My father, who by example taught me the power o f free thought; My mother, who always reminded me what is truly important in life. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements Although the distinguishing characteristic o f a dissertation is independence o f the conducted research, I am obviously indebted to many collegues and friends whose contributions lie unseen throughout this work. Foremost is my father, David Lynch, whose profound influence on my life reaches well beyond the usual fatherly sphere. He not only helped me develop a solid technical foundation, but also introduced me to many other intellectual pursuits that have proved at least as exciting and gratifying as my technical work. Professor Robert York deserves the highest praise for his advising skills. He attaches great value to nurturing independence and creativity in bis student's efforts, a difficult goal considering their diverse strengths and needs. In this respect I am convinced I could not have chosen a superior advisor. My Delco supervisor, Dave Fayram, provided vigorous support for my ambitions. My wife and I thank him and Dave Zubas for creating Delco's Ph.D. work/study program that enabled us to not only live comfortably, but to continue to improve our lives. I thank my committee, John Bowers, Petar Kokotovic, Umesh Mishra, and M ark Rodwell for somehow finding the time to review my work. Many other people influenced my work significantly, though less directly. I thank Nguyen Nguyen for many enlightening technical and philosophical discussions. We must continue our "coffee talks." Jeff Yen helped me temporarily forget the graduate school pressures over occasional billiards and darts, and lightened the school laboratory atmosphere through his comic relief All o f the members o f Professor York's and Mishra's groups created a relaxed, friendly atmosphere that was enjoyable to work in. My brother, Chris Lynch, was always available for advice on any subject, technical or not, and never M ed to provide intelligent and informative suggestions. Finally, and most importantly, I thank all o f my friends and family. You have provided that important element in life that has given me the strength to achieve whatever I desire. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Jonathan James Lynch was bom August 27,1965, Winchester, M A 1987 Bachelor o f Science, University o f California, Santa Barbara 1993 M aster o f Science, University o f California, Santa Barbara 1995 Doctorate ofPhilosophy, University o f California, Santa Barbara Industry Experience 1986-1995 1995- Electrical Engineer, Delco Systems Operations, Santa Barbara, C A Electrical Engineer, Hughes Research Labs, Malibu, C A Publications J. Lynch, R. York, "Stability o f Mode Locked States of Coupled Oscillators." To be published in IEEE Trans. Circuits and Systems. J. Lynch, R. York, "An Analysis of Mode Locked Arrays of Automatic Level Control Oscillators." IEEE Trans. Circuits and Systems, vol. 41, Nol 12, pp. 859-865, Dec. 1994. J. Lynch, R. York, "Oscillator Dynamics with Frequency Dependent Coupling Networks." Submitted to IEEE Trans, on Microwave Theory and Techniques Sept. 1994. J. Lynch, R. York, "Synchronization of Microwave Oscillators Coupled through Resonant Networks." Submitted to IEEE Trans, on Microwave Theory and Techniques Sept 1994. J. Lynch, R. York, "A Mode Locked Array of Coupled Phase Locked Loops." Submitted to IEEE Microwave and Guided Wave Letters Nov. 1994. J. Lynch, R. York, "Mode Locked Arrays of Microwave Oscillators." 1993 Symposium on Nonlinear Theory and Applications, vol. 2, p. 605. J. Lynch, R. York, "Pulse Power Enhancement using Mode Locked Arrays of Automatic Level Control Oscillators." IEEE MTT-S, International Symposium Digest, San Diego, June 1994, Vol. 2, pp. 969. R. York, P. Liao, J. Lynch, "Oscillator Array Dynamics with Broadband N-port Coupling Networks." IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 2040-2045. H. Tsai, P. Liao, J. Lynch, A Alexanian, R. York, "Active Antenna Arrays for Millimeter wave Power Combining," 1994 International Conference on M illimeter Waves and Far Infrared Science and Technology (Guangzhou, China), pp. 371-374, Sept 1994. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Analysis and Design of Systems of Coupled Microwave Oscillators by Jonathan J. Lynch The following work advances the analysis techniques and understanding o f systems o f coupled microwave oscillators utilized in quasi-optical beam steering and pulse power transmitting arrays. The analysis methods for periodic frequency locked systems are generalized to include almost periodic systems, and these techniques are applied to arrays o f practical importance. The author presents techniques to improve locking characteristics o f mode locked arrays, using automatic level control oscillators, and synchronized arrays using coupled phase locked loops. Detailed design and measurements o f a microwave phase locked loop are presented. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Historical Development Overview 1. Analysis Techniques for Periodic and Almost Periodic Frequency Locking 1 5 9 1.1 Microwave Measurements 1.2 Systems o f Coupled Oscillators 1.2.1 Derivation o f the Nonlinear Dynamic Equations 1.2.1.1 Definitions o f oscillator frequency and phase 1.2.1.2 Example: Dynamics Equations for Two Oscillators Coupled through a Resonant Circuit 1.2.1.2.1 Approximation o f complex transfer functions 1.2.2 Solution o f Periodic and Almost Periodic States—Locking Diagrams 1.2.2.1 Existence o f locked states 1.2.2.2 Stability o f locked states Appendix 1.1 Narrowband response o f nonlinear circuit elem ents Appendix 1.2 Response o f networks to narrowband signals Appendix 1.3 Approxim ate stability o flo cked states References 22 24 26 28 31 34 38 2. Synchronous Arrays 39 2.1 Synchronization o f coupled oscillator systems through broadband networks 2.1.1 Linear Arrays with Nearest Neighbor Coupling—Beam Steering 2.1.1.1 Zero Degrees Coupling Phase-Synchronization Diagrams 2.1.1.1.1 Existence Region 2.1.1.1.2 Stability Region 2.1.1.1.3 Phase sensitivity 2.1.1.1.4 Transient Response to Tuning Variations 2.1.1.1.4.1 Example: settling time for beam steering 2.2 The Effect o f a Resonant Coupling Network on the Synchronization o f Two Oscillators 2.2.1 Dynamic Equations 2.2.2 Synchronized States 2.2.3 Stability o f States 2.2.4 Cases o f Practical Interest 2.2.4.1 Weak Coupling 2.2.4.1.1 Broadband Case 11 12 12 14 IS 20 39 43 46 47 49 50 51 56 57 58 59 62 64 68 68 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.4.1.2 Narrowband Case 2.2.4.2 Strong Coupling 2.2.4.2.1 Broadband Case 2.2.4.2.2 Narrowband Case 2.2.5 Computer Simulations Appendix 2.1—Reducing the Order o f a Stability M atrix Appendix 2.2—Am plitudes, Aa>c N ear Line o f Equally Spaced Frequencies References 70 71 73 75 75 79 80 83 3. Mode Locked Arrays 84 3.1 The Time Domain M ode Locked Waveform 3.2 Linear Arrays o f Van der Pol Oscillators 3.2.1 Three Element Array 3.2.2 Four Element Array 3.3 M ode Locked Arrays using Automatic Level Control Oscillators 3.3.1 Linear Arrays with Nearest Neighbor Coupling 3.3.1.1 The Locking Region 3.3.1.2 Pulse Power Enhancement 3.3.1.3 Experimental Verification Appendix 3.1—N orm alized Form o f M ode Locking Equations Appendix 3.2—Frequency P ulling E quationsfor M ode Locked Arrays Appendix 3.3—Dynamic Equationsfo r ALC O scillator References 86 87 90 91 96 99 100 103 106 109 110 112 115 4. The Design of Microwave Phase Locked Loops 116 4.1 Arrays o f Phase Locked Loops for Beam Steering Systems 4.1.1 Ideal PLL Operation 4.1.2 PLL Design 4.1.2.1 Oscillator Analysis 4.1.2.2 VCO Circuit Design 4.1.2.2.1 Device Bias 4.1.2.2.2 FET Circuit Design 4.1.2.2.3 Varactor Circuit Design 4.1.2.2.4 Complete VCO Design and Measurements 4.1.2.3 Phase D etector 4.1.2.3.1 Ideal Operation 4.1.2.3.2 FET Detector Design 4.1.3 PLL Measurements 4.1.3.1 Phase Measurement References 117 118 121 121 126 126 127 128 130 131 131 132 138 139 145 Areas for Continuing Study 146 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Historical Development Figure 1—Quasi-optical power combining as originally proposed by J. Mink Figure 2-Coupled oscillator array. 1 2 Chapter 1 Figure 1—Block diagram of N oscillators coupled through a linear network. Figure 2—Two self sustained oscillators coupled through a resonant network. Figure 3—The exact and approximate oscillator admittance magnitude and phase. Figure 4—Exact and approximate coupling circuit admittance magnitude and phase using linear approximation for entire transfer function. Figure 5—More accurate approximation of coupling circuit admittance. Figure 6—Two oscillators coupled through a fourth order coupling network. Figure 7—Three coupled oscillators. Figure 8—The graphical representation of synchronization and mode locking for three oscillators. Figure 9—Phase response to perturbation for a four element mode locked array. 13 IS 17 IS 19 21 23 23 28 Chapter 2 Figure 1—Four element linear array with frequency independent nearest neighbor coupling. Figure 2—Example of a synchronization region for a three element array. Figure 3—Linear and nonlinear transformations. Figure 4—Synchronization region in the plane of free running frequencies. Figure 5—Eigenvalues and eigenvectors for a five element array (1V=5). Figure 6—Quasi-optical power combining as originally proposed by Mink Figure 7—Two oscillators coupled through resonant network. Figure 8—Region of frequency locking in the plane of oscillator tunings Figure 9—Parameter diagram showing four regions of interest Figure 10—Dimensions of the locking region for weakly coupled oscillators. Figure 11-Dimensions of the locking region for strongly coupled oscillators. Figure 12—Comparison of approximate formulas to computer simulations for "high" Q coupling circuit Figure 13—Comparison of approximate formulas to computer simulations for "moderate" Q coupling circuit Figure 14—Comparison of approximate formulas to computer simulations for "low" Q coupling circuit 43 47 48 49 55 57 58 60 67 69 74 76 77 78 Chapter 3 Figure 1—The ideal time domain mode locked waveform. Figure 2—Magnitude spectrum of a four element mode locked array. Figure 3-Four element linear array with nearest neighbor coupling. Figure 4—Phase plane regions of stable solutions. Figure 5-Region of stable mode locked states in the AAfi plane. Figure 6—Locking regions in the plane of oscillator tunings. Figure 7—Conventional and ALC oscillators. 84 85 85 93 94 96 97 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 8—Amplitude response at turn-on of a Van der Pol and ALC oscillators. 98 Figure 9—Locking diagram for a four element ALC array. 102 Figure 10—Locking region size, maximum and minimum eigenvalues as functions of the number of array elements. 103 Figure 11-Pulse enhancement using the time varying amplitudes of the ALC oscillators. 105 Figure 12—Locking region size, L, for two values of coupling phase, as a function of the number of array elements. 105 Figure 13—Maximum sensitivity as a function of the number of elements. 105 Figure 14—Schematic of a single ALC oscillator. 107 Figure 15-Response of the magnitude of the amplitude variations to an injected signal. 108 Figure 16—The measured and theoretical mode locked waveforms. 108 Chapter 4 Figure 1—Array of coupled phase locked loops for electronic beam steering. Figure 2—Block diagram of PLL. Figure 3-Magnitude spectrum of locked and unlocked PLL. Figure 4—PLL synchronization region and phase difference. Figure 5—Block diagram of a negative resistance oscillator. Figure 6—Smith chart representation of stable oscillation. Figure 7-Circuit model of a simple oscillator and possible AC I-V curves. Figure 8—Illustration of the dependence of oscillation amplitude on the load resistance. Figure 9—Schematic diagram and microstrip layout of bias drcuitiy for FET. Figure 10—Schematic, layout, and input reflection coefficient plot for the active device. Figure 11—Equivalent circuit for the varactor diode. Figure 12-Schematic diagram and input S parameter of varactor circuit. Figure 13—Physical layout of varactor circuit Figure 14—Varactor capacitance vs. reverse bias. Figure 15—Total scattering parameter vs. frequency. Figure 16—Simulated and measured VCO tuning curves. Figure 17—Phase detector block diagram. Figure 18-Detector is a common source amplifier biased near pinch off. Figure 19-FET drain current vs. gate voltage. Figure 20—Microstrip circuit layout o f detector. Figure 21—Equivalent circuit and input impedance near 9 GHz. Figure 22-Magnitude of the input reflection coefficient of the detector. Figure 23—Simulated and Measured detector output voltage vs. input power. Figure 24—Simulated phase detector output voltage vs. input phase difference. Figure 25—Complete PLL circuit Figure 26-Phase detector output voltage and PLL output power. Figure 2 7 -Measurement of PLL phase shift between input and output Figure 28-Calibration of phase measurement system. Figure 29-IF output with RF input terminated. Figure 30—IF output with configuration of figure 28. Figure 31—Phase delay between reference planes from PLL output to input Figure 32-Phase shift of measurement system. Figure 33—IF signal with PLL connected as in figure 27. Figure 34-Measured phase shift of PLL between reference planes. 117 118 118 121 121 123 125 125 127 127 128 129 129 130 130 131 132 133 133 135 135 136 137 137 138 139 140 140 141 141 142 143 143 144 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Historical Development As technology advances, the maximum operating frequency o f electronic systems continues to climb. The millime te r -wave frequency range, typically assumed to lie above 40 G H z, offers exceptional resolution for radar im aging systems, highly accurate telemetry systems, and a less crowded spectrum for broadband communication systems. Solid state device designers have continually improved high frequency device performance, and such devices often dictate the limits o f system performance. Unfortunately a fundamental trade o ff between frequency response and power handling capability causes the available output power to diminish as the operating frequency increases, but system requirements for output power generally remain the same. A possible solution is to combine the outputs from m aty low power devices to create a high powered source. These outputs can be combined electronically using circuits, or spatially using so called "quasi-optical" techniques. This latter method can increase the efficiency o f high power systems over conventional power combining methods. M ost quad-optical m illim e te r wave systems exploit engineering techniques typically utilized at optical frequencies. A good example is the power com bining array originally proposed by Mink [l](note: references are located at the end o f the Overview section), and shown in figure 1. Oscillator Grid Output Partially Reflecting M irror Figure 1—Quasi-optical power combining as originally proposed by J. Mink Oscillator coupling occurs through the resonant cavity. Operation is similar to optical lasers. An array o f microwave negative resistance devices is placed in an electromagnetic cavity and excites a sustained resonant mode o f oscillation. Power from individual oscillators combines coherently within the cavity and a partially reflecting m irro r serves as the output port. The system essentially mimics an optical laser in which the power source is a distributed gain medium. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Over the past few years quad-optical power combining systems have separated into two groups: grid systems and discrete oscillator systems. The former group utilizes grids o f strongly coupled electrically small devices, typically much smaller than a wavelength, to form a high power system. When placed within a cavity the grid acts as a "distributed" gain medium. Such systems have successfully produced high power transmitters. [2] Discrete oscillator systems contain complete microwave sources that are coupled together externally and are generally not placed within a cavity, as shown in figure 2. OstiDatmg Elements Coherent Power Combining Figure 2-Coupled oscillator array. Each element radiates power that combines coherently the others to form a directed beam. The oscillating elements can be designed and tested before they are inserted into an array, thereby increasing the likelihood o f success and simplifying performance optimization. In addition, arrays o f coupled oscillators can provide electronic beam steering and pulsed transmitting functions without the use o f phase shifting elements or high speed switches.[3][4] This dissertation deals exclusively with discrete coupled oscillator systems, such as in figure 2, as used in beam steering and pulsed power arrays. The design o f such systems requires a detailed understanding o f the behavior o f coupled microwave oscillators. In particular, one must ensure that the correct phase relationship between the oscillating elements can be reliably produced and accurately controlled. Furthermore, one must understand how the individual oscillator output amplitudes vary as the elements are tuned since this can adversely effect the output radiation pattern. Finally, and most importantly, the designer must ensure that the oscillators settle to a robust frequency locked state to maintain a coherent output signal. Understanding microwave beam steering and pulsed power arrays is achieved through analyses o f ideal systems o f coupled nonlinear oscillators and from measurements o f fabricated systems. Appropriate analysis techniques have been developed by applying classical analytical methods to our particular class o f problems. The study o f synchronization o f electrical oscillators began with B. Van der Pol at the beginning 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f this centuiy.[5] He utilized a method o f averaging to obtain approximate solutions for nearly sinusoidal systems. This method was extended by Krylov, Bogoliubov, and Mitropolsky (KBM) in the 1930's, and was successfully applied to many types o f oscillating systems. [6] The methods presented in this dissertation are based on these averaging methods which are the first terms o f a power series expansion o f the solution with respect to a small parameter. Thus all results are approximate, but quite accurate and useful for many microwave systems. M ore recently, the development o f the mixed potential theory for electrical networks,[7] and the subsequent development o f the averaged potential for oscillatory networks,[8] provides a slightly different analysis technique that has also proved useful for understanding the behavior o f coupled microwave oscillators. [9] The analysis o f microwave oscillators was given a more physical basis by Robert Adler who derived the dynamic equation for oscillator phase under the influence o f an injected signal. [10] This was taken further by Kurokawa who derived the dynamic equations for both amplitude and phase from the amplitude dependent Z parameters,[11] and thus provided a pragmatic understanding o f microwave oscillators. These methods were generalized by Robert York to include any number o f oscillators coupled through a broadband coupling network. [12] Thus a mathematical framework was established, based on parameters obtained from microwave measurements, that applied to systems o f coupled oscillators and gave excellent agreement with many fabricated arrays. However, the techniques could not be applied to narrowband systems, such as Mink's system (figure 1) where the oscillators are coupled through a resonant cavity, or to almost periodic systems such as the so-called mode locked oscillator arrays. Nearly all systems o f coupled microwave oscillators studied in the literature operate in a synchronous mode, that is, all elements are synchronized to a common frequency. However, an array o f coupled oscillators, when appropriately tuned, will lock to a state where the frequencies are exactly evenly spaced (an example is shown in chapter 3, figure 2). This type o f frequency locking is called "mode locking" in the laser community [13] and can result in pulses o f microwave energy. [14] The dynamic equations for mode locked systems can be derived using the same techniques as for synchronous arrays, but the solution o f the equations for the stable states becomes much more complicated, and requires different analytical methods. This dissertation advances the analysis methods o f systems o f coupled oscillators to include narrowband coupling networks, and generalizes Kurokawa's method to give more accurate results. In addition, the analysis o f synchronous arrays is generalized to include mode locked arrays, and stable frequency locked 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. states o f either type can be deduced in a fairly straightforward manne r For the important class o f linear arrays with nearest neighbor broadband coupling we introduce the concept o f characteristic tunings to relate the size o f the locking region, phase sensitivity, and phase transient response, represent the frequency locking ability graphically on a locking diagram. Analysis results lead to the enhancement o f mode locked arrays using automatic level control oscillators, and o f synchronous arrays using phase locked loops. The detailed design o f a microwave phase locked loop is presented in chapter 4. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Overview The purpose o f this work is to provide analysis techniques for a class o f nonlinear oscillatory circuits and to apply these techniques to practical microwave systems. The current research area o f quasi-optical power combining utilizes networks o f coupled microwave or millimeter wave oscillators to generate high power sources by combining the pow er o f many small devices. Such power combination must be coherent to avoid destructive interference between oscillating elements, and coherence is obtained through frequency locking o f the elements. We will study two types o f locking. The more common w e call synchronization because all oscillators are synchronized to a common fundamental frequency, although higher harmonics are always present due to inherent nonlinearities. Any voltage or current within such a system is a periodic function o f time, and therefore the amplitude and phase o f the oscillation are constant in the steady state. When a synchronized state o f a physical system is perturbed slightly, the amplitudes and phases in the circuit will vary in time but eventually decay back to constant values. Thus all observed locked states o f a physically realized system are stable. The analysis o f this type o f system is relatively straightforward and well documented in the literature. [15] W e will analyze models o f systems o f synchronous microwave oscillators and derive some important characteristics that influence the design o f such systems. The less familiar type o f locking occurs when many frequency components exist in close proximity, and the components are separated by an integer multiple o f some small frequency separation. An example is the mode locked laser that gives rise to a comb spectrum o f evenly spaced components. The resulting time domain waveform consists o f a carrier with periodic amplitude and phase modulation. Usually the periods o f the carrier and the modulation are not related by an integer multiple, that is, they are noncommensurate. Therefore, the time domain waveform is not strictly periodic, but is called almost periodic, as defined in [16], All observed almost periodic states o f physically realized systems are also stable, since any perturbation will decay in time and the amplitudes and phases will return to their periodic states. The literature contains many studies o f each type o f locking phenomenon, but none, to this author's knowledge, develop a method o f analysis applicable to both. This present study unifies the treatment o f the two systems, within the constraints o f the approximations. It provides a general, and straightforward, technique for determining the (asymptotic) stability o f almost periodic systems for which the dynamic equations are non-autonomous, that is, containing explicit time dependence. Almost periodic systems are common in the laser community, 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. although R. York extended their use to microwave oscillator systems. [4] Laser physicists that concern themselves with the stability o f mode locked states commonly analyze the states by discarding terms in the dynamic equations that contain explicit time dependence, thereby rendering the system autonomous. Stability is then ascertained by linearizing about fixed points and determining the eigenvalues o f a constant matrix—precisely the method used for synchronous arrays. However, for the mode locked microwave oscillators considered here, all coupling terms contain explicit time dependence so such a treatment is not possible. Thus, necessity inspired the development o f an alternative method. The properties o f even simple nonlinear systems are myriad and highly complex. The understanding o f the dynamics o f a forced Van der Pol oscillator is, to this day, not complete despite over 60 years o f vigorous study. The intent o f the following effort is to concentrate on a few important aspects o f the properties o f a specific class o f coupled nonlinear oscillators that are o f principal importance to the microwave systems engineer. Narrowing the class o f oscillating systems and limiting the properties under study serves dual purposes. First, restricting the oscillating systems to those with nearly sinusoidal outputs (i.e. low harmonics) allows us to significantly simplify the analysis and often obtain closed form expressions and considerable understanding. Second, limiting the properties we choose to study focuses the analysis and reduces the overall size to an acceptable level. To maintain our focus we will sacrifice some generality and assume particular applications throughout the succeeding analyses. For synchronized systems the application will be electronically steerable transmitting arrays, and for mode locked systems it will be pulsed power arrays. We require, for both types o f arrays, the same principal information, namely, how we maximize the ability o f the oscillators to lock and the combination o f tunings that produces the desired phase distribution. The analyses will provide much additional information, but we will never venture far from these two concerns. The following study is organized into three chapters. The first develops the general analytical methods discussed above. This chapter is the most general, and therefore the most abstract, but here we define our method o f attack and all o f the associated approximations, as well as our notation. The main contribution here is the generalization o f the well known analysis o f synchronized systems to that o f almost periodic, and in particular mode locked, systems. We show how the nonlinear equations relating the amplitudes and phases can be derived directly from the frequency domain representation o f the oscillators and coupling networks. This method is an extension ofKurokawa's method to arbitrarily complex systems, and usually requires less effort than the Krylov/Bogoliubov method. Analytical results are presented on a particular bifurcation diagram, that we call a locking 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. diagram, that graphically depicts the frequency locking ability o f the arrays, and can indicate the tuning that gives rise to a desired phase distribution. The second chapter is a series o f applications o f the methods o f chapter 1 to synchronous systems o f practical importance, namely, beam steering arrays. The first section contains an analysis o f a simple model: a linear array with frequency independent, nearest neighbor coupling. M ich o f this work was developed elsewhere, but we take advantage o f the simplicity to exemplify the analysis methods and to gain a deeper understanding o f this important class o f arrays. The methods o f chapter 1 give considerable insight into array synchronization, the phase sensitivity to tuning variations (i.e. array robustness), and the transient phase response. These characteristics are intimately related and best understood using the concept o f characteristic tunings for the array, a concept we are led to quite naturally by the mathematics. We then apply our methods to the synchronization o f tw o oscillators coupled through a resonant network. The results, depicted on synchronization diagrams, show how the ability to lock depends on the coupling circuit resonance, loss and bandwidth. This analysis is the first step to understanding complex coupling structures such as synchronization o f oscillators in a resonant cavity. The third chapter is devoted to analyses o f mode locked arrays. In the first section we analyze a simple type: a linear array o f Van der Pol oscillators with nearest neighbor coupling. The results reveal optimum values for coupling phase and nonlinearity, and show that, for high nonlinearity, multiple stable states can exist for a given set o f oscillator tunings. In the next section w e introduce the automatic level control (ALC) oscillator as an array element. When designed correctly the underdamped amplitude response enhances the locking ability and maximizes the size o f the stability region. This type o f array has nearly identical properties to the synchronous array analyzed in the first section. A particular choice o f coupling phase enhances the pulsed power significantly. The fourth chapter is a detailed description o f the analysis and design o f a microwave phase locked loop. The phase locked loop is similar to conventional oscillators in that it has an input and an output and synchronizes to a suitable injected frequency. However, it can be designed to have a larger locking bandwidth than conventional oscillators, and, if designed adequately its amplitude will not vary under the influence o f an injected signal. This latter effect occurs in conventional oscillator arrays that are strongly coupled and can unfavorably alter the array radiation pattern as the beam is steered. This chapter, however, is concerned only with the design o f a single PLL, and measurements are presented for a fabricated circuit. A block diagram o f a beam steering array o f PLL's is presented and the analysis is identical to the first section o f chapter 2. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [1] J. W. Mink, "Quasi-Optical Pow er Combining o f Solid-State Millimeter-Wave Sources," IEEE Trans. M icrowave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986. [2] A. B. Popovic, R. M. Weikle n , M. Kim, and D. B. Rutledge, "A 100 MESFET planar grid oscillator," IE EE Trans. M icrowave Theory Tech., vol. 39, pp. 193-200, Feb. 1991. [3] P. Liao, R. A York, "A New Phase-Shifteriess Beam-Scanning Technique using Arrays o f Coupled Oscillators," IE EE Trans. M icrowave Theory Tech., vol. MTT-41, pp. 1810-1815. [4] R. York, R. Compton, "Mode Locked Oscillator Arrays," IE EE M icrowave and G uided Wave Letters, voL 1, No. 8, Aug 1991, pp. 1810-1815. [5] B. Van der Pol, "A Theory o f the Amplitude o f Free and Forced Triode Vibrations," Radio Review, VoL 1, pp. 701-754,1920. [6] N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory o f Nonlinear Oscillations. Hindustan Pub. Corp., 1961. [7] R. K. Brayton, J. K. Moser, "A Theory o f Nonlinear Networks," Q uarterly o f A pplied M athem atics, VoL XXII, No. 1, April 1964. [8] M. Kuramrtsu, F. Takase, "An Analytical Method for Multimode Oscillators using the Averaged Potential," Trans. IECEJ, vol. J66-A, pp. 336-343, April 1983 (in Japanese). [9] K. Fukui, S. Nogj, "Mode Analytical Study o f Cylindrical Cavity Power Combiners, IEEE Trans. M icrowave Theory Tech., vol. MTT-34, pp. 943-951. [10] R. Adler, "A Study o f Locking Phenomena in Oscillators," Proc. IRE, vol. 34, pp. 351-357, June 1946. [11] K Kurokawa, "Injection Locking o f Solid State Microwave Oscillators," Proc. IEEE, vol. 61, pp. 1386-1409, O ct 1973. [12] R. A York, P. Liao, J. J. Lynch, "Oscillator Array Dynamics with Broadband N-port Coupling Networks," IEEE Trans. M icrowave Theory Tech., vol. MTT42, pp. 2040-2045. [13] M. Sargent, M. Scully, W. Lamb, Laser Physics. Addison-Wesley Pub. Co., 1974. [14] J. Lynch, R. York, "Pulse Power Enhancement using Mode Locked Arrays o f Automatic Level Control Oscillators." IEEEM TT-S, International Symposium D igest, San Diego, June 1994, VoL 2, pp. 969. [15] N. Minorsky, Nonlinear Oscillations. Princeton University, Princeton, NJ, 1962. [16] J. K. Hale, Oscillations in Nonlinear Systems. Dover, 1963. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Analysis Techniques for Periodic and Almost periodic Frequency Locking In this chapter we develop a general analysis method for finding periodic and almost periodic frequency locked states o f coupled oscillator systems. Physical microwave systems will be modeled using idealized circuits that contain linear and nonlinear elements. Model complexity is kept as low as possible while still retaining the most important phenomena exhibited in the physical system. Although modeling is an important aspect o f system analysis, we will not consider this subject in great detail. The analysis methods we will apply are approximate and accurate only for narrowband systems, although many practical microwave systems satisfy this constraint. Once a suitable circuit model is developed, the analysis proceeds in a stepwise fashion. First, we devise a second "equivalent" circuit that has the same voltages and currents at the fundamental, or carrier, frequency as our circuit model but has no higher hannonics. Next w e determine the differential equations that describe the amplitudes and phases o f the voltages and currents. After solving the differential equations for the desired locked states, we test the stability o f the states. Completing these steps gives the conditions that must be satisfied by stable frequency locked states, periodic o r almost periodic, and shows how various circuit quantities influence these states. The increased understanding allows us to optimize the design, and therefore the performance, o f practical frequency locked systems. Subsequent analyses contain tw o essential approximations: the circuit voltages and currents have relatively narrow bandwidths about a carrier, and the amplitudes o f the higher harmonics are small enough to be neglected. This is equivalent to requiring the voltages and currents to be nearly sinusoidal with slowly varying amplitudes and phases. These assumptions are approximately satisfied by many oscillatory microwave systems and they greatly simplify the analysis and understanding o f such systems. Given a system that meets these criteria we must find a suitable method o f "neglecting" the harmonics. As mentioned above, this is accomplished by replacing all nonlinear circuit elements with "equivalent" elements that give nearly the same response at the fundamental but generate no harmonics under sinusoidal excitation. For example, consider a lossy element with a cubic I-V curve: i = G0(l-e v 2)v = G(v)v. (1) 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I f the voltage across this element is sinusoidal, v = A cos(a t) , then the component o f current generated at frequency co, found by direct substitution, is /^ (O = G { l- % s A 2)A cos(a t) = G '(A )v. (2) Thus if we replace the original conductance G(v) with this new one G '(A ) the resulting current due to sinusoidal excitation will be purely sinusoidal. Replacing all nonlinear circuit elements in an analogous manner eliminates the harmonics. The relation between the actual nonlinear elements and their approximate "fundamental" counterparts is derived in appendix 1.0 for a particular class o f nonlinear elements. In the studies that follow we will generally begin with an appropriate circuit that generates no harmonics, that is, one that has already had its nonlinear elements replaced with approximate "fundamental" equivalents. Many microwave measurements (e.g. large signal S parameters) involve only the fundamental frequency component and neglect higher harmonics. Thus, this representation is a convenient starting point. One may think that for synchronized systems, neglecting all harmonics permits an algebraic frequency domain analysis, as in linear systems, since the oscillations are purely sinusoidal. The frequency domain equations will give all o f the possible synchronized states, but admits unstable as well as stable states. To determine stability the amplitudes and phases must be perturbed from their steady state values, and here the nonlinearities become important. We must be able to model the circuit dynamics when the amplitudes and phases vary in time. For almost periodic states the frequency domain equations are clearly not sufficient since many frequency components are present simultaneously and interact through the nonlinear devices. However, when the total system bandwidth is small compared to the carrier frequency, as w e will assume, the comb spectrum can be represented as a single carrier frequency containing siowly varying amplitude and phase modulation. Thus, if we can derive dynamic equations that accurately represent the oscillating system when the amplitudes and phases vary in time, then we can determine both synchronized and mode locked states, and test stability o f both by perturbing the states. We will show how to derive these dynamic equations directly from the frequency domain circuit equations. This method is a generalization o f Kurokawa's method, and a rigorous derivation is presented in appendix 1.2. Determining existence and stability is straightforward for synchronized oscillators since the steady state amplitude and phase variables are constant. We set the time derivatives o f amplitudes and phases to zero and solve the algebraic system (or, equivalently, the frequency domain equations). Stability is then 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. determined by perturbing the steady state and observing the growth o r decay o f the perturbations. This last step results in a linear system o f differential equations with constant coefficients, a system that is easily analyzed using techniques o f linear algebra. For almost periodic states, however, die steady states have periodic amplitudes and phases. Generally, these states must be found by some approximate method, and perturbing the steady state produces a linear system with periodic coefficients whose stability cannot be easily ascertained. To solve this problem we will use a perturbation method to approximate the almost periodic steady states, and then apply a method o f averaging to the perturbed system to approximate the stability o f the state. It may seem like the approximations are stacking up fast but they are satisfied by a fairly large class o f almost periodic systems. And if we extend our methods beyond the realm o f accuracy many o f the insights gained will often be useful in understanding these complex systems. The approximate stability analysis we will develop for almost periodic systems gives exact stability information when applied to synchronous systems. Thus the theory developed in this chapter will be used for both types. 1.1 Microwave Measurements Although the broad topic o f microwave measurements will not be treated in depth here, some comments are necessary. We are often faced with the problem o f characterizing a network through microwave measurement in order to design a system or verify a model. Such measurements are typically o f the large signal scattering (S) parameters, in which the S parameters o f a network are measured over a particular frequency band for a range o f input power levels.[l] The result is a set o f S parameters o f the form S(A, a>). The question immediately arises as to whether these parameters completely describe the network over the range o f measurements. The answer is generally "no." S parameters are measured by energizing one port while terminating all others. Because the principle o f superposition does not apply to nonlinear networks one cannot assume the response to combined stimuli will be the sum o f the partial stimuli For some circuits, however, this type o f characterization may be approximately correct, and therefore useful. An example is the common source FET. The parameters ^ 2, and often do not vary appreciably with amplitude; only the magnitude o f 5^ varies significantly, and the reverse gain is small.[2] Thus the significant nonlinear parameter is and tins depends only on the input amplitude even when a signal at the output port is present. Large signal S parameter characterization should be adequate for this circuit since the ports do not interact through nonlinear parameters. On the other hand, a common base FET with a destabilizing gate inductor, as is commonly used for FET 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oscillators, is generally not amenable to this type o f characterization. This configuration often has large values o f both <S^2 and so that the input and output ports are strongly coupled. In addition, we can no longer assume that and S2 2 are only weakly dependent on input and output amplitudes. For this case large signal characterization may not provide enough information about the circuit nonlinearities. Even for tingle port circuits large signal characterization does not provide complete information. Since measurements are made with constant input signal amplitude they do not contain information about the circuit response to modulated input signals. This is equivalent to saying that the S parameters may depend not only on the input amplitude, but also on its derivatives. Engineers often assume that the amplitude dependent part o f the parameters respond instantaneously to amplitude changes. [3] This assumption greatly simplifies the analysis o f such circuits and seems to be accurate in many cases o f practical interest, and we will use this assumption throughout tins work. Relaxing this assumption requires additional measurements to be taken for adequate network characterization. Also, microwave measurements are made with particular te rm inating impedances. I f a circuit is terminated with a different impedance than was used for the measurement, the device AC operating point, and therefore the S parameters, will change. Thus one must be acutely aware o f the approximations made in nonlinear analyses and must insure that the physical systems meet the requirements, at least approximately. 1.2 Systems of Coupled Oscillators 1.2.1 Derivation of the Nonlinear Dynamic Equations We now consider systems o f coupled microwave oscillators and derive the nonlinear differential equations that describe the amplitude and phase dynamics. The resulting mathematical model contains many properties observed in physical systems, including the properties o f interest, periodic and almost periodic frequency locking. The equations can be used to determine the existence and stability o f such states. A block diagram o f a system o f coupled oscillators is shown in figure 1. The following analysis is based on an admittance description o f the oscillator and coupling circuit networks, and closely follows reference [4]. An impedance description is equally valid and may result in more accurate results for certain types o f circuits. For the sake o f brevity, however, only the admittance description will be presented here. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Osc I NPort Linear Coupling Network OscN Figure 1-block diagram of N oscillators coupled through a linear network. The voltage at the terminals o f the oscillator are related to the other circuit voltages through the admittance matrix o f the coupling network: r r U , v ) K + f , r % * ’M K , = f x U , , < » K = o m=l (3) m=l This results from applying Kirchoffs current law at the nA oscillator terminals and must hold for each o f the N ports. We have defined a "total" admittance Y^ = +Y£?p, where is the Kronecker delta function. The voltages Vn are sinusoidal signals with slowly varying amplitudes and phases, which, expressed in phasor form, are r . M = 4 , « > * WAW) (4) The frequency cor may not represent a true frequency component in the system but is simply a convenient reference. If the true frequency o f the n th oscillator is not 6)r the difference will be provided by the time dependent phase This is explained in more detail in section 1.2.1.1 below. Appendix 1.1 shows how the dynamic equations for the amplitudes and phases can be derived from the 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency domain equations. The method was first used by Kurokawa and is known as Kurokawa's substitution. [3] We replace the frequency © with the "instantaneous" frequency cor + A and expand the frequency dependent A admittances in a Taylor series about cor, retaining only the constant and linear terms. The result is a set o fN differential equations: N z m=l A -) <*> (5) 4 ./. i In most cases the above equations can be manipulated so that each <f>m - j - 22- is expressed in the standard form: (6) This is accomplished by expressing equations (5) using matrix notation and solving for the derivative terms by matrix inversion. The resulting expressions for the functions Fn are complicated and will not be explicitly shown here. By taking the real and imaginary parts o f equations (6) we have 2N differential equations that describe the dynamics o f the system o f coupled oscillators. 1.2.1.1 Definitions of O scillator Frequency and Phase Since we used an arbitrary reference frequency cor in the derivation, the phases in equations (6) may have linearly increasing or decreasing parts that account for the difference between the true frequency o f an oscillator and the reference frequency. Usually it is advantageous to replace the above phases with the "true" phases, defined below. W ith the notation above the "instantaneous" phase o f the n th oscillator is 0n(t) = cort+ We define the "true" frequency of tins oscillator as and the "true" phase as $n{t) = <j>n(i)-c o rt. Replacing all o f the phases in equation (6) in this manner gives a more convenient form for the dynamic system. Through the remainder o f this work the words "oscillator frequency and phase" 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. will denote the true values defined above. Using this convention, direct synchronization results in constant amplitudes and phases, hence the time derivatives vanish. Almost periodic synchronization gives rise to periodic amplitudes and phases, a feet which we will use to analyze these states. One may wonder about the order o f the resulting system o f differential equations. We would expect the system order to depend on the total number o f energy storage elements and not on the number o f ports on the coupling network.[5] Equations (6) are only approximately correct, and the heart o f the approximation lies in the Taylor series expansion o f the admittance transfer functions. The linear approximation is necessary to obtain first order differential equations, but by considering the individual admittance functions in more detail, and possibly introducing additional variables, more accurate approximations may be obtained. At this point we must leave the general treatment and exemplify the above statements by considering a particular case. 1.2.1.2 Example: Dynamic Equations for Two Oscillators Coupled through a Resonant Circuit [6] We apply the above analysis technique to a circuit composed o f two parallel resonant circuits containing nonlinear negative resistance devices and coupled through a series resonant circuit, as shown in figure 2. Cc Lc Rc + Vj(t) Oscillator II Oscillator I Figure 2—Two self sustained oscillators coupled through a resonant network. The oscillators are identical except for their resonant frequencies, o r tunings All three resonant frequencies (including the coupling network) are considered arbitrary. Our task at this point is to derive the dynamic equations for amplitudes 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and phases o f the circuit variables, and withhold solution o f the equations for the next chapter. The frequency domain equations can be written by inspection I ' = Yx Vl9 Ie = -Y 2 V2 , Ie = Ye(y2 - V x) (8) and explicitly show how the coupling current Ic is related to the oscillator voltages through admittance transfer functions. If we eliminate the coupling currents and reduce the number o f equations by one, as outlined in the general analysis above, the order o f the resulting system o f differential equations will be lower. Thus, we expect that including the coupling current will lead to more accurate results. The oscillator transfer functions are necessarily nonlinear since a practical microwave oscillator requires a stable steady state amplitude, and we will assume the nonlinearity is sufficiently weak so that the outputs are nearly sinusoidal. A simple model is a linear resonant tank circuit containing a negative resistance or conductance whose magnitude saturates with increasing voltage amplitude. Our circuit o f figure 2 meets these criteria if G(A) is a decreasing function o f amplitude. We now approximate the frequency dependent parts o f the admittance functions with a linear frequency dependence, as demanded by Kurokawa's substitution. The oscillator admittance function for oscillator I is r, = - G . - ffl,2) = -G . JQ\ where = <0~ (9) is the tank resonant frequency, Ga is the nonlinear device conductance at zero voltage, f ( A ) is the saturation function for the device G conductance, and 2 a a = —— is the oscillator "bandwidth." The frequency a l is an arbitrary Taylor expansion frequency and the best choice is the steady state (or "true") frequency o f oscillator I. I f the frequency o f oscillator I remains close to its "free running" or uncoupled value a ol then the linear approximation is extremely accurate, as illustrated in figure 3. The admittance function for oscillator II is identical except that a o2, a>2 replace a ol, a x. Using the first and second o f equations (8) and Kurokawa's substitution we can write the dynamic equations for the two oscillators in terms o f the coupling current. The transfer function and its derivative at frequency a l are 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 =1, o K=10 5 Approx. Exact Exact 4 <3> 3 so ce a a© a. 2 1 0 it12 6 8 10 12 14 16 6 Frequency 8 10 12 14 16 Frequency Figure 3—The exact and approximate oscillator admittance magnitnde and phase. Agreement is excellent over a broad range of frequencies. and r ,(n ,)= -G , I ©« J ( 10 ) d (O x COa After applying Kurokawa's substitution, and repeating the procedure for oscillator n , we find the oscillator equations are A =<oJ{Ax)A + °>Jc - 0e) ex =© 0l- o a ^ - s i n ( ^ - ^ ) A A. = <»J(A)A - °>Jccos {d2- 9C) K = ©02 - ©o A 01) - d c) where we have used the instantaneous phase Oi(t) = <x>it+<f>i(t), / = 1, 2 , 0 , to amplify the notatioa N ote that we have expressed the coupling current in terms o f its slowly varying amplitude and phase as ic(/) = Ic(/) cos( (/)). The current expansion frequency <oc is arbitrary and the equations will take on diffftn»nt forms depending on the choice o f a>c. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We now consider the (possibly) narrowband coupling circuit. admittance function for the coupling network is 1 The ( 12) K l j °> 2axo„ I f we were to use the broadband assumption and expand the admittance function Ye in a Taylor series about <oc, as in reference [4], we would have the following result: . a>e o l+ c o lc , G>ac i- y W 2 <t>c«>oc \ 2 a c______________ (13) -G>lc-C& 2 G>cG>cc I Figure 4 shows a plot o f the magnitude and phase o f the approximate and exact transfer functions. ®«=1* ©c=10 5 TO2 Exact 4 3 a 60 a. 2 1 Exact 0 16 ■> 6 8 10 12 14 16 Figure 4—Exact and approximate coupling circuit admittance magnitude anrf phase using linear approximation for entire transfer function. The phase is quite close, but the magnitude response is a very poor approximation. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although the phase response is accurate the magnitude is a poor approximation We would expect good agreement only very close to the expansion frequency, or if the coupling network is extremely broadband. This is the "broadband" approximation used in reference [4] and it is this approximation we must improve to extend the analysis to more narrowband coupling networks. The first step is to express the admittance function as a ratio o f polynomial functions Yc{a>)=^c^eo^/jy ^ and write the relation between oscillator voltages and coupling currents in (8) as (14) The transfer functions Dc and N c operate on the current and voltage separately and we may apply Kurokawa's method to each. This has the effect o f linearizing the numerator and denominator o f the admittance function separately and leads to a highly accurate approximation: Ye{a>)= 1 * w 1 1 _ q - - A +y- ” z & . 2 (0 ^ <D„ * .i- j 1 oc (15) a>„ The magnitude and phase response o f (IS) are compared to the exact response (12) in figure 5. ©*=1, ©,c=10 Exact 0.8 .-§ 0.6 Approx CL, s 0.4 < r Exact 6 8 10 12 14 16 Frequency Figure 5—More accurate approximation of coupling circuit admittance using separate linear approximations of numerator and denominator 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Applying Kurokawa's substitution to (14) we have Rearranging terms gives the dynamic equations for the amplitude and phase o f the coupling current i , = -a > J ' + ^ ( k cos(d>2 K = ^ + f f ( v 2sm (e2- e c) A K « J)) - 0,))) Equations (11) and (17) together represent the dynamic equations for the amplitudes and phases o f the oscillators and the coupling current. The order o f the system matches the order o f the exact system and due to the high accuracy o f the approximations, we expect the dynamics o f the approximate system to give good agreement with the exact system. 1.2.1.2.1 Approximation of Complex Transfer Functions The procedure outlined in the previous section can be extended to higher order systems. For N oscillators coupled through an N-port network, as shown in figure 1, the frequency domain equations can be written (18) Any coupling admittances with strong frequency dependence that require the denominator expansion used in the previous section should be removed from the sum and handled separately. For example, suppose that the i^1 and jth terms in the sum above have strong frequency dependence. The network equations become p=1. p * ij (19) D T % = ^ T pVj 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The narrowband admittances produce additional pairs o f differential equations for the associated coupling currents which produces an approximate system o f nearly the same order as the original (depending on the number o f such terms that exist). One may find that an admittance function cannot be adequately represented by a linear approximation o f the numerator and denominator. For example, if in our circuit o f figure 2 the coupling network was composed o f tw o second order resonant networks, as shown in figure 6, CO0(2 , ( 0 ae Oscillator II Oscillator I Figure 6—Fourth order coupling network. The overall admittance transfer function can be divided into sums of simpler functions using the partial fraction expansion technique. This method is essentially one of approximating the poles and zeros of the coupling network admittance function. the coupling admittance transfer function would be fourth order instead o f second: 2 f . 2 ~\ G > o c l+ ® o c 2 a 2 2_________ w 2 (dCD„ (20) * ( <»lcx+oLi •ri 1- 2 <0 (0 „ (2<b< 0 2 y. Using a partial fraction expansion expresses the admittance as the sum o f tw o second order functions. For this contrived example this step is easy: 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and, as before, we define tw o coupling currents, one due to each admittance function. Once again we are increasing the order o f the system to achieve more accurate results. 1.2.2 Solution of Periodic and Almost Periodic States—Locking Diagrams We now have a method o f deriving accurate dynamic amplitude and phase equations for a system o f coupled oscillators. From this point we could analyze any number o f the myriad properties o f such a system. Details o f the dynamics near synchronization boundaries, determination o f stability in the large, response for various values o f initial conditions, are all subjects o f the large field o f nonlinear oscillations. [7] For the sake o f brevity and focus we will confine our attention to a few pragmatic details. One important practical task is to find the oscillator tunings that give rise to a particular type o f stable frequency locking. Another is the dependence o f the relative phases o f the oscillators on the element tunings. These are primary considerations for beam steering or pulsed arrays, and throughout the remainder o f this work these will be our primary focus. lo ck ed states occur only for values o f oscillator tunings that lie within specific and rather narrow ranges. For example, synchronization can occur when two or more coupled oscillators are tuned relatively close to one another, and almost periodic locking can occur when three o r more are tuned with nearly even spacing but far enough apart to avoid synchronization. These regions can be illustrated graphically using "locking diagrams." As a simple example, consider three coupled oscillators whose free running frequencies a>ol, eoo2, and a o3 can be tuned independently, shown schematically in figure 7. Notice that the output frequencies are the true frequencies eo2, eo3. If we leave the tuning of one o f the oscillators fixed, say a>o2, we can plot the values o f the other tunings that result in frequency locking. Such a plot might resemble figure 8. N ote that the origin is not zero frequency, but is the fixed frequency eoo2. In the vicinity o f the origin where coo l« a>o2 « oao3 we find the region o f synchronization. By definition, if we tune the oscillators to values within this region the frequencies can lock together to a common value g>1 = cd2 = co3. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tuning Ports <°1 “j 1 1 i > Outputs Figure 7—Three coupled oscillators. Tuning ports control the "free running frequencies." which are the oscillation frequencies in the absence o f coupling. Region of Periodic Locking ) 999999^ Region of QuasiPeriodic Locking (CO,—^ = ( 0 ,—cq.) Figure 8—The graphical representation o f synchronization and mode locking for three oscillators. The former condition exists when the outer two oscillators are tuned near the center, whose free running frequency co„2 is the origin o f the graph, and the latter when the three tunings are alm ost evenly spaced. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This plot does not indicate what this frequency will be, or o f any other quantity (e.g. the phases), only that the oscillators can lock. A almost periodic region extends along the line a>o2 —a>o l« <oo2 - eoo2 where the tunings are nearly evenly spaced. Within this region the steady state frequencies o f the locked state will be exactly evenly spaced, that is, co2 —<ox = - <a2. This type o f almost periodic locking is referred to as mode locking in the laser community. The region extends away from the origin since mode locking depends mainly on the even spectral spacing and less on the mutual proximity. The almost periodic and periodic regions are nonoverlapping in the illustration, but this may not be the case. I have not performed, nor am I aware, o f any analysis that shows whether the regions overlap. I f they do overlap then the type o f synchronization that will ensue upon application o f power depends on the initial conditions within the network. Analysis is difficult near this boundary region because perturbation techniques become inaccurate. 1.2.2.1 Existence of Locked States We will now outline an approximate method for determining the existence o f locked states, periodic or almost periodic, from the differential equations that describe the amplitude and phase dynamics o f the system. The dynamic system is assumed to have a particular form that, for the most part, results from equations (6) with the actual frequencies and phases substituted in. Using vector notation they are <P=a>0~ a>b) =fi+ sf[^,A,t, ab) Ai = ^ A , t t <ab)\ (22) where each variable 4>,A, a>0, co, and are N element vectors, / and g are vector functions, and s and t are coupling and time parameters, respectively. The parameter <ob is the beat frequency parameter which is smallest frequency separation for almost periodic locking, and is zero for synchronized systems. The vector 3 contains the amount o f frequency pulling o f each oscillator and is introduced for notational simplicity. We will not attempt to show that this form always follows from equations (6), but suggest that it will for many practical cases, some o f which will be considered in the next section. The most important assumption in equations (22) is that the steady state frequencies co„ do not appear within the functions / and g. If they do appear, we must introduce new variables appropriately to increase the order o f the system using the methods o f section 1.2 .2 . 1. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the case o f synchronization the frequencies and phases are constant, so the existence o f states can be determined by setting the time derivatives in (22) equal to zero and solving the algebraic system: P + eM A ) =0 g(<M )=o (Note: the time dependence in the functions / and g vanish for synchronized systems). We assume, but do not prove, that values o f frequency, phase, and amplitude that satisfy the above system indicate the existence o f a synchronized state. Finding states o f almost periodic locking is much more difficult since amplitudes and phases are not constant, but we do know that they are periodic functions o f time. Thus, we use a perturbation method in which we expand all unknown variables (including the frequency pullings) in a power series in the (assumed) small coupling parameter s:[S] ft) = +^ A(t) = A {0\ t ) + eA m (t) + S2A ™ (t)+ - (24) Substituting the above variables into equations (22) and equating like powers o f e gives a sequence o f differential equations that can be solved recursively by enforcing the periodicity o f <f>and A. This procedure often becomes prohibitively complicated after the second order, so the results will be accurate only for relatively small values o f 8. Performing the above substitution, for the zero order (e= 0 ) we have £°> = 0 O) A ^ = g {A ^ J ° \t) Enforcing the periodicity o f (^A ^°\ (25) and -4(0) gives 0 O) = 0, = <f>0 - const, and - 0 where the brackets denote time average over one period. The vector o f phases $Q represents, approximately, the time average value o f the phase <f(t). We will see later that the dependence o f the frequency pullings /?on the 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. time average phases can indicate stability o f the state. The higher order amplitude and phase corrections cannot be evaluated explicitly for this general case, but we can use the periodicity o f the phases to express the frequency pulling vector as P= - s { M A t ) ) = - e F f a ) (26) Thus the frequency pullings are functions o f the time average phase variables, among other things. Given a set o f tunings co0 this perturbation technique allows us to determine, at least theoretically, the time dependent amplitudes and phases and the frequency pullings to any desired degree o f accuracy. Practically, we can determine the solution to the first or second order, but this order o f approximation usually provides a great deal o f information and insight to the conditions for the existence o f almost periodic states. The next step is to determine the stability o f states. 1.2.2.2 Stability of Locked States [9] Once a locked state is found by solving equations (26) the stability o f the state must be tested. The following analysis shows that equations (26) contain information about the stability, at least approximately in the case o f almost periodic locking. Stability can be tested by applying small perturbations to the phase variables and finding the stability o f the resulting linear system. For almost periodic states this leads to a linear system with periodic coefficients, whose stability is difficult to determine. Under certain conditions, which are often satisfied in practical systems, we can find the constant coefficients o f an approximate "averaged" system for the phase perturbations. This will allow us to investigate the stability o f periodic solutions using the well known techniques o f linear algebra applied to systems with constant coefficients. The assumptions are contained in the derivation o f appendix 1.2, but the two most significant are that the inter-element coupling is weak and that the amplitude perturbations decay quickly compared to the phase perturbations. The former condition forces the changes in amplitude and phase modulation due to perturbations to vary slowly in time. This allows us to use the method o f averaging to approximate the linear system with periodic coefficients with a linear system with constant coefficients. The second assumption above allows us to neglect the effect o f the transient amplitude response and consider only the "algebraic" influence o f the amplitudes (see appendix 1.2 for details). We can then reduce the I N order system (23) to a N order system. It is important to understand that we are not neglecting the influence o f the amplitudes entirely. In 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. feet, in the systems we will study in chapter 3 stable mode locked states do not exist when the amplitudes are fixed at constant values. The derivation o f appendix 1.2 gives an approximate perturbational system for equations (22) that has constant, instead o f periodic, coefficients. I f we perturb the phases from their steady state values * 0 = ^ (0 + * * ) (27) where ^ ( / ) is the periodic steady state solution and <5(/) is the perturbation, then the perturbations behave according to the linear system %t) = - C { t ) m (28) where the matrix C(t) is a periodic function o f time. The approximate linear system with constant coefficients is denoted d(t) = -C d { t) (29) where the phase perturbation d(t) follows the "average" value o f the actual perturbation 8(r). The constant "stability" matrix C, obtained from (26), is C = -^ - (30) Bm 1 ; where the phase ^om is the m0' element o f the time average phase vector <f>0. Stability o f a state is ensured when the real parts o f the eigenvalues o f C are positive. Thus, both existence and stability o f states is supplied by the frequency pulling equations (26). To illustrate the above concepts figure 9 shows a comparison o f the approximate "averaged” system given by equation (29) to the actual perturbational system given by equation (28). The curves where generated by numerically integrating the exact and approximate perturbational systems for a stable mode locked state o f a four element array o f Van der Pol oscillators. Instead o f the phase variables themselves, the second differences o f the phase variables are plotted (see section 3.2 for definitions), but the important point is that the smooth approximate response closely follows the actual response. The "small" coupling parameter s was chosen somewhat large to show that good agreement is obtained in this case. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. £ = 0 .2 5 11= 0.5 AAdz 0 .5 ■■ AAdi -0 .5 A A 8i Figure 9—Phase response to perturbation for a four element mode locked array. The bumpy responses are due to the influence of the periodic coefficients. The smooth responses are the "averaged” approximation. There is good agreement even though the "small" coupling parameter is rather large. Appendix 1.1 Narrowband Response of Nonlinear Circuit Elements C onductance In this section we will show that a nonlinear conductance responding to a narrowband signal gives rise to a current waveform whose fundamental component is in phase with the applied voltage but with altered amplitude. Thus, as far as the fundamental is concerned, a nonlinear conductance can be represented as a conductance whose value depends on the amplitude o f the applied voltage, but not its phase. This model for a nonlinear conductance is used widely in the engineering literature. We assume that the nonlinear device has a well behaved I-V curve given by /'= /( v ) and that this relation holds at and above the carrier frequency. Time lags between the voltage and current in a physical device can be modeled using constant resistors, capacitors or inductors, but the nonlinearity assumed here is o f the "instantaneous" type. The applied voltage is a narrowband signal at carrier frequency (o with slowly varying amplitude and phase: v(0 = A (t) cos{<ot+ f i t ) ) , — '<f>«co A (31) The nonlinear device gives rise to a spectrum o f frequencies 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /(/)= / ( a cos(a>t + $ ) = 2 (a„ cos(neot)+ b„ sin(»Gtf)) (32) n We can find the amplitudes o f the fundamental components using the orthogonality o f the circular functions: ax = — f !ri(t)co4.cot)d(cot) ir J 0 f h = -f*i(f)suL(a> t)d(a*) (33) X JO The device curve, and therefore the current, can be expressed as a Taylor series /(/) = £ c „ v n = Y*cnAn cosn(aX+0) n (34) n which gives the fundamental components in terms o f the input voltage ai = S c» y j or ^"(0cos"(<af+ #/))cos(fflO<# ” 1 (3 5 ) t h = 2 X - J or ^"(O cosn( ^ + ^ ) ) s i n ( ffl/ ) ^ n I f the amplitude and phase vary negligibly over a cycle we can simplify equations (35). The amplitude can be pulled outside the integral and the rem aining integral can be simplified by changing variables: \ \TA" (t)cosn(eat+ 4(t))cos(<ot)dt = A n — \T+*oos"((ot)cos(.ajt-^)dt jo T r 2 rr 2 eT 1 = A n\ c o s($ —Jo cos'H'1( ^ ) <* + s in (^ )—J0 cosn{cot)sai(G)t)dt I ^ ^ The second integral vanishes because o f symmetry o f the integrand. A similar analysis can be carried out for the second integral o f equation (35). The fundamental current components are then <h = 'E cnA ”(0 cos( (jit)) y cosn+1{<ot)dt (37) h = - Z c^ ”(0 s in (^ (0 )|: cosn* \a t)d t T' The current at the fundamental is 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ’■®(0 = ax covert)+6t a n {cot) = |^ S c« ( f £* cosn+l((o t)d jA n(t) cos(eot+4it)) (38) = G (A(t))A(t) cos( <ot+ where G(A) is the nonlinear conductance we sought that gives rise to the correct fundamental component o f current, and is given explicitly by G(A) = cos"+1( ^ ) ^ " " 1 (39) N ote that harmonics are not generated by this amplitude dependent conductance. This method also shows how to compute G(A) from the Taylor series expansion o f the device I-V curve. Using the previous example o f nonlinear conductance i = G v ( l- e v 2) the above equation gives G(A) = G(l - % s A 2) , the result found previously using direct substitution. The important result o f this analysis is that, to the first approximation, the fundamental component resulting from the narrowband excitation o f an "instantaneous" nonlinear device is in phase with the exciting signal, but with altered amplitude. C apacitors and Inductors Although in the subsequent analyses we will assume that only constant valued capacitors and inductors are used, it is instructive to apply the above results to the nonlinear capacitor. Such a capacitor is described by a charge-voltage (QV) curve q =f(v ) (40) As before, we consider a narrowband voltage v(t) = A(t)cos(a)/+ $(/)) and we immediately see that this problem is identical to the previous. The fimdamental component o f charge is therefore given by cos(eot+d(t)) 7(1)(0 = (41) = F(A)cos{cot+4>) 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Differentiating this with respect to time gives the relation between current and voltage = -(co+ 0)F(A)sm(cot+ f t + — A cos(<ar+ f t dA (42) Defining an amplitude dependent capacitance with F (A ) = C(A)A gives the fundamental component o f current versus voltage, r® (0 = -{& + $ C (A )A sin(c* + ft+ C A cos{cot+ f t + ^ - A A cos(cat+ f t dA (43) and implies the following element admittance at constant amplitude and phase: Ycap(o')=jo>C(A) (44) This admittance would be the result o f a large signal microwave measurement. Applying Kurokawa's substitution here, we find z<1)( 0 = -{& + $ C (A )A sin (cot+ft+C A cos (g# + f t (45) This is nearly identical to (43) but it is missing the last term. Thus if we use the representation (43) for a nonlinear capacitor we must include the missing term upon the application o f Kurokawa's substitution. The missing term creates additional phase shift due to amplitude modulation and becomes small for weakly nonlinear capacitors where A dA with a nonlinear flux linkage curve y /= f( i). Similar results hold for inductors Appendix 1.2 Response of Networks to Narrowband Signals The networks that interconnect nonlinear oscillators are usually constructed from linear circuit elements, and the oscillators themselves can often be adequately modeled as a simple resonant tank circuit with a nonlinear conductance that responds instantaneously to signal amplitude at the oscillator terminals. I f the frequency response o f such a network is known for a fixed excitation amplitude, over a wide range o f amplitudes, then we can determine the network response to a signal that contains amplitude and/or frequency modulation. In the case o f narrowband signals the result collapses to Kurokawa's substitution. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Consider a time invariant network whose input is ^ (/) = A (t) cos(o>0t + ) ) and with frequency response H (A, eo) = ^2 • The frequency response depends on the input amplitude because o f the possible presence o f nonlinear conductances but is defined by m aintaining constant input amplitude and phase. This is precisely how large signal microwave measurements are made and yield amplitude and frequency dependent transfer functions in the above form. Since the nonlinear devices are assumed to respond instantaneously to the applied voltage, w e can account for the amplitude dependence o f the above transfer function by simply substituting in A (t) for the constant value A. The frequency dependence o f H, however, gives rise to modulation dependent terms, as this analysis will show. Since all o f the succeeding operations are linear, we will simplify the mathematics by expressing the input as a complex exponential and, in the end, taking the real part o f the output. The input signal we will consider is Vj(/) = A(t)eJ^ e ituJ (46) and the spectrum o f the modulation is (47) S(,a) = £ j e i*e'JO*dt This spectrum is related to the input signal spectrum Vi(eo)=J°°A ^ ° ^ e ~ la*dt through the modulation property: [10] (48) V1(o)) = S(eo-CD0) The time domain filter output can be expressed in terms o f the modulation spectrum via the inverse Fourier transform v2(l)=r l7C J-°° = £ tf ( 4 a > ) S ( o - (49) = £ # ( * / + a o)S(a>,)ej{e0'+‘0°)‘ 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The third integral is obtained through the substitution o f variables g > = 6 ) ' + g>0 . Next, expand the filter transfer function in a Taylor series about the carrier a>0 and insert into (49) d nH(A,<o0) da? v2(0 StfieOe***** — “ l ie (50) Swapping the order o f integration and summation we have v2(r) = y J_ d nH{A, a> ) ^ 2W d{ja>)n p ^ J~~U y dcd_ (51) but the integral is simply the rfi1time derivative o f the modulation, so the complex output voltage is (52) t?0n[ d(j<o)” d tn The true time domain output voltage is obtained by taking the real part o f (52). Since we assumed that the input and output signals have slowly varying amplitudes and phases the higher order time derivatives o f the input modulation diminish quickly. In addition, derivatives with respect to frequency also diminish quickly due to the (assumed) high carrier frequency. Thus, for many practical microwave systems the output signal can be represented adequately by the first two terms o f the above series „ (A v 3H(A,e>) v2(r) = H(A, <o0) + — y - ! - 9(0 .A ) - < (53) A) The above expression can be derived by substituting the "instantaneous" frequency A co(t) = a>0 + <f>-j— into the transfer function and linearizing about the carrier A frequency. This substitution was first used by Kurokawa to derive the amplitude and phase dynamics for oscillators.[3] 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix 1 3 Approximate Stability of Locked States The phase variables o f equation (22) can other be perturbed directly, leaving the fi unchanged, or by perturbing the free running frequencies. We will show how the relation between these two types o f perturbations can be used to determine the coefficients o f the averaged linear system. Treating the first type o f perturbation, expand the ^ s and them 's about a periodic solution: iff + 8 = f ? + s f(< ff+8, A p + a ,t) (54) A p + a = g (0 p + S,A p + a ,t) where 8 and a are the perturbations and the superscript 'p' denotes the periodic solution. The dynamic equations o f the perturbation, or the variational equations, result from a first order expansion about the periodic solution: ' * 8gn r a „m « m dan * S fn . *■ 45 8m+ ^ n &f>m * IB 45 . Note that the coefficients multiplying the perturbations are periodic functions o f time. When ascertaining the stability o f most systems o f coupled oscillators it is sufficient to study the system response to initial values o f the phase perturbation variables only, maintaining the initial amplitude perturbations at zero. [8] This is possible as long as the transient responses o f the amplitude variables decay quickly. The second equation in (SS) is an inhomogeneous system o f linear differential equations for a where the forcing functions are superpositions o f the variables 8. The solution, assuming it exists, can be written as the sum o f a transient homogeneous solution and the particular solution. In addition, the particular solution can always be written as a linear combination o f the phase variables. Thus the general solution has the form (56) m 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We shall assume the transient part dies out quickly and therefore does not appreciably affect the phase dynamics. Neglecting the homogeneous portion, inserting (56) into the first o f equations (55) eliminates the amplitude perturbation variables. The dynamic equations for the phase perturbations become dS, dt (57) m / <34/ The partial derivatives are still evaluated at the periodic solution but this is not explicitly shown for notational convenience. The derivatives o f the phase perturbations are proportional to the small parameter £ so the phases must vary slowly and we can "average" the equations over time in such a way that any fast variations are averaged out but the slow variations are retained. Formally, we are applying the averaging method o f Krylov and Bogoliubov.[l 1] Using brackets to represent an averaging operation, the slowly varying "average" value o f the nth phase perturbation is defined as (58) dn(t) = ( S M ) and the equivalent averaged system with constant coefficients is: — dn = s T dt " ~ * (59) For a complete justification o f this method see references [11], [12], [13]. Fortunately, the coefficients need not be evaluated directly but can be calculated from the frequency pulling variables ft, as we will now show. Consider an infinitesimal change d p to the frequency pulling o f a periodic state and allow the system to settle to a new periodic state. The difference between the new and old phases is determined by £(<<«)=<%+*z — d f m+ - ^ - d A pm (60) m 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The partials are evaluated at the original periodic values, that is, before p was perturbed. Since the new solution to (60) is also periodic, its time average over one period T must be constant and, therefore, the time average o f the derivative o f the phase perturbation must vanish: (61) so, (62) Using the expression in (56) for the amplitudes we have (63) The phase perturbation d ^m is a time dependent quantity that satisfies equation (60). Since its derivative is proportional to the small quantity s i t must vary slowly and we can therefore approximate the average value o f the product o f the bracketed terms with d<j?m in equation (63) as the product o f the average values (note that the differential change in p in equation (60) can be neglected when determining the smallness o f d<ffm because the average value o f the summation must cancel d p in order to maintain a bounded d ^ ) . The approximate expression for d p becomes (64) where (65) o and T is one period. Thus the coefficients in equation (59) can be identified as (66) 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This simple relation allows us to evaluate the coefficients o f the averaged system by amply differentiating the expressions for the frequency pulling o f each oscillator. The stability o f a periodic state can then be determined by computing the eigenvalues o f the matrix o f coefficients. Note that this result is exact for the case o f periodic synchronization. It is satisfying to see that the result represents the "average" values for the almost periodic system under the stated conditions. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [1] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, PrenticeHall, NJ, 1984. [2] K. Johnson, "Large Signal GaAs MESFET Oscillator Design," TEEF. Trans. Microwave Theory Tech., vol. MTT-27, pp. 217-227, Mar. 1979. [3] K. Kurokawa, "Some Basic Characteristics o f Broadband Negative Resistance Oscillator Circuits," Bell System Technical Journal, Aug. 1969. [4] R. A. York, P. Liao, J. J. Lynch, "Oscillator Array Dynamics with Broadband N -port Coupling Networks," to appear in IEEE Trans. Microwave Theory Tech [5] M. E. Van Valkenburg, Network Analysis. Prentice-Hall, NJ, 1974. [6] J. J. Lynch, R. A. York, "Oscillator Dynamics with Frequency Dependent Coupling Networks," IEEE Trans. Microwave Theory Tech, vol. MTT-42, pp. 2040-2045. [7] J. Guckenheimer, P. Holmes, Nonlinear Oscillations. Dynamical Systems, and Bifurcations o f Vector Reids. Springer-Verlag, NY, 1983 [8] R. Bellman. Perturbation Techniques in Mathematics. Physics, and Engineering. Holt, Rinehart, and Winston, Inc., 1964. [9] J. J. Lynch, R A. York, "Stability o f M ode Locked States o f Coupled Oscillator Arrays," Submitted to IEEE Trans, on Circuits and Systems. [10] A. V. Oppenheim, A. S. Willsky, Signals and Systems. Prentice-Hall, Inc., New Jersey, 1983. [11] N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory o f Nonlinear Oscillations. Hindustan Pub. Corp., 1961. [12] N. Minorsky, Nonlinear Oscillations. Princeton University, Princeton, NJ, 1962. [13] J. K. Hale, Oscillations in Nonliear Systems. Dover, 1963. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Synchronous Arrays In this chapter we apply the theory developed in chapter 1 to specific problems relevant to practical synchronous oscillator arrays, Tbe chapter begins with the analysis o f simple oscillators coupled in an arbitrary manner through a broadband network. This general case can be applied to specific coupling schemes, for example two dimensional arrays. However we choose the am ple case o f linear arrays with nearest neighbor coupling due to the current interest and practical importance in beam steerable transmitting arrays. The particular choice o f zero degrees coupling phase represents an optimum value and greatly simplifies the equations. The ample form allows us to graphically depict the synchronization diagram and to understand the relationship between the size o f the synchronization region, the phase sensitivity, and the transient response o f the phases, for arrays o f any size. These attributes are tied together through the "characteristic timings." The second half o f the chapter is devoted to the synchronization o f oscillators coupled through a possibly narrowband resonant circuit. This analysis is sufficiently complicated to prohibit a general treatment so we immediately consider two oscillators coupled through a simple resonant circuit. The synchronization diagrams are derived for combinations o f weak and strong coupling, and narrow and broad coupling bandwidths. This analysis is the first step toward an analysis o f many oscillators coupled through a resonant cavity, a case o f practical importance. [5] 2.1 Synchronization of Oscillators through Broadband Coupling Networks Simple models o f coupled oscillator systems have performed in good agreement with fabricated microwave systems. Parallel negative resistance oscillator circuits, as in figure 1.2, coupled through a frequency independent network have demonstrated the first order characteristics o f many microwave oscillator systems. O f course one must design the actual system with fairly broadband coupling and relatively high Q elements to meet the assumptions used in the model. In this section we will analyze this simple model o f coupled microwave oscillators and determine the characteristics o f the system in the synchronized state. We assume nearly sinusoidal oscillations, slowly varying amplitudes and • J • phases (<f>n and — A « ®)> nonlinear oscillators o f the type in section 1.2.2, and A 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency independent coupling magnitude and phase delay. After finding the first order algebraic equations for the oscillator frequency and phases, we will determine the stability o f a state by computing the eigenvalues o f the stability matrix, as described in the previous chapter. Since one o f the phases is arbitrary (due to the arbitrary time reference) one o f the eigenvalues o f the stability matrix is zero. We will show how to remove the zero eigenvalue, thereby reducing the order o f the system by one, while preserving all o f the nonzero eigenvalues. After applying the analysis methods o f section 1.2.1 the equations for the amplitudes and phases have the form, for n = \ 2 , — , N (see [6] for details o f the derivation), K = rjSn(A„)An + e £ AmAmcos(& - &, + $ J 1 - ^ fa = A - £— Ann=i (1) * * ( &- 4>m + 0 where fin = o>on-ct> is the frequency pulling, q is a nonlinearity parameter that controls the amount o f nonlinearity present in the oscillators, s is a small coupling parameter, are variations in the coupling strengths, and Q>m is the phase delay from the n* to them * element. N ote that the free running frequency, or t uning, o f the n A oscillator is denoted co^ and the true frequency o f the synchronized system is denoted eo. The function S„(An) is the conductance saturation function for the nA oscillator and is left arbitrary since it vanishes in the first approximation. The above system allows for coupling between any oscillators in the circuit and can therefore be used for linear arrays, tw o dimensional arrays, or general arrays with complex coupling structures. Finding the conditions required for stable synchronization is, in principle, straightforward because the steady state amplitudes and phases are constant. W e set the time derivatives in equation (1) equal to zero and solve the remaining algebraic system for the N amplitudes, N -l linearly independent phases (or combinations o f phases), and the frequency a>. I f a solution exists for a given set o f tunings, we conclude (although we will not prove) that a synchronized state exists. To find the stability o f the state we perturb the amplitude and phase variables, derive the linear dynamic system for the perturbations, and determine its stability. In practice, however, the solution o f the algebraic equations is difficult and we usually resort to approximations. A tremendous simplification occurs when the coupling between elements, represented by the parameter e, is relatively weak. We then find an approximate solution using a power series 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. expansion. This is the same procedure outlined in section 1.2.2 for the approximate solution o f mode locked states. Assuming inter-element coupling is weak, we expand all unknown variables in power series o f e, h = ft? An = A?> + e A ? + s?A?>+~ (2) insert these into equations (1), and equate like powers o f 8. For the zeroth order, 4S»=s.(4»>)4»>=o W = /f?= o which gives the zeroth order steady state amplitudes (by the zeros o f S„(A„)) and shows that the frequency pulling is at least a first order quantity. The phases are all constant, and will be denoted ft°* = <j>m as before, but they cannot be determined by the above equations. The first order phase equations are N 0) (4) m=l n The first order phase, and all subsequent orders, can be taken as zero if we assume that the zeroth order phases are the true oscillator phases. The first order frequency pullings in terms o f the oscillator phases are n Pn = sin(&„ -t^ om+ O (5) m =\A n Note that this depends only on zeroth order quantities. The above expression shows that a difference in oscillator amplitudes is equivalent to a variation in the inter-element coupling. Unfortunately this equivalent coupling change is not reciprocal, that is, the effective coupling from element m to n is not the same as from element n to m. A synchronous state exists when, for a given set o f tunings (O^ we can solve the above system for the frequency and phase differences. We can then test 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the stability by applying the result o f section 1.2.2.2. The dynamic system for the phase perturbations is K = - f .C „ S „ (6) n=l A state is stable when the real parts o f the nonzero eigenvalues o f the matrix whose elements are C _ = - J ^ = - * C - 4 _ « s ( A .- A .+ ® J (7) n are positive. The frequency pulling variables o f equations (5) depend only on the phase differences and not the phases themselves, so one o f the phases is arbitrary. Thus the above stability matrix (7) has one zero eigenvalue. We can reduce the order o f the system by eliminating the zero eigenvalue, but we must retain the nonzero eigenvalues to preserve the stability information. This is accomplished by transforming the N phases to a linearly independent set o f N -l phases through a linear transformation <jf = A<f>where A is a real N - I x N matrix, and then applying the same transformation to the frequency pulling variables to form a new set P = Af$. The "reduced" N - l x N - l stability matrix, derived in the same manner as equation (7), contains the same eigenvalues as the original matrix C with the zero eigenvalue removed. This is proved in appendix 2.0. Given a set o f tunings co0 we can, in principal, determine the existence o f a synchronous state o f coupled oscillators using equations (5) and test the stability o f the state by finding the stability o f the linear perturbational system (6). These two sets o f equations are connected through the stability matrix o f equation (7) and, therefore, the tendency for the elements to synchronize, the dependence o f the phases on element tunings, and the transient phase response to perturbations are all interdependent. For complex coupling schemes large arrays are not easily understood, even though the form o f the equations is relatively simple. However, in the following section we will consider a coupling scheme for a useful practical 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. case that considerably amplifies the analysis, and allows us to understand the interdependence o f the characteristics mentioned above. 2.1.1 Linear Arrays with Nearest Neighbor Coupling—Beam Steering The basic idea behind a beam steerable array, such as the one shown in figure 1, is that the radiation pattern depends on the phase shift between adjacent elements which, in turn, can be controlled by the element tunings. A broadside radiation pattern occurs when there is zero phase shift between elements [1] and this condition occurs for a particular combination o f element tunings. By varying the tuning in a certain manner we are able to produce equal phase shifts between elements and therefore steer the radiation pattern away from broadside. The total steering angle depends not only on the maximum phase shift between dements that we can achieve, but also on the physical spacing o f the radiating elements. We will only consider the former here since this problem is treated in detail elsewhere. [9] Tuning Ports 0 y © © © >r >f ^ Oscillator Outputs Figure 1—Four element linear array with frequency independent nearest neighbor coupling. Since the oscillators are assumed to be synchronized their output frequencies are identical, but their free running, or uncoupled, frequencies are generally different The diagram o f figure 1 shows a four element linear array o f oscillators coupled only between nearest neighbors. The coupling constraint simplifies the design o f the coupling network since circuit traces need not cross one another. If the coupling is radiative, elements in closest proximity will couple strongest. Thus the assumption o f nearest neighbor coupling is not overly restrictive. To further simplify the analysis we will assume that the coupling magnitude and phase 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between elements are identical across the array, and that the oscillators themselves are identical, except for their tunings. With the above assumptions, the frequency pulling equations (5) become n+I =£ ~ to n , +<&) = ^ ( ^ - I + <&) “ Sm(A& - $ )] (9) m*n where the last term on the right holds for n = 1,2,— , N - 1 if we set to zero any terms containing subscripts zero or N. Equations (9) represent a system o f N equations in N unknowns: N -l phase differences and the true frequency ©. Because o f the coupling scheme only the N -l phase differences between adjacent pairs appear so these represent a natural choice o f independent phase variables for the problem. Reducing the order o f the system in the manner outlined in the previous section, we form the difference between adjacent frequency pulling variables & Pn =Pn+x ~ P n = 4 -s in (A^»-i + $ ) + 2 cos(0)sin(A & ) - s i^ A ^ , -<&)] (10) Notice that this eliminates the true frequency 0 from the system and therefore equations (10) represent N -l equations in N -l unknowns. The system o f differential equations governing phase perturbations for the reduced system is M ,= - Z c la ^ , m=1 ( 11 ) where the ASn are actually the difference between adjacent phase perturbations, defined by A8n = Sn+1 - S n forn = \,2 ,- - - ,N - \. The stability matrix for the reduced system is (we will drop the prime for simplicity) When the real parts o f the eigenvalues o f the stability matrix for a particular state are positive the state is stable. Equations (10) through (12) can be expressed more compactly in matrix form. Define the constant N —lx iV -1 matrices 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O'] 2 -1 A = -1 2 r0 1 -1 0 , B= -1 (T ■*. ■*. *•- (13) 1 o o t— i 1 1 C-l o the N -l element sine and cosine vectors u= ' sin(A ^) " sin(A^2) r cos(A ^) ^ , v= cos(A^2) cosCa ^ (14) .J ^ and the N - lx iV -1 diagonal sine and cosine matrices 0 a n (A ^ ) sin(A^2) U= 0 sinCA^.J cos(A $) 0 1 cos(A^2) V= 0 c o s(A ^ J (15) The frequency pulling equations (10) become A/?= £(cos(0)y4w+sin(d>)5v) (16) and the perturbational system is A S = -C A S (17) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with stability matrix C = fi(cos(<I>)j4F'-sin(<I>)2?£/) (18) Equations (16) and (17) completely describe the array frequency, phases, and stability, to the first order o f approximation. The simple form o f the equations allows us to gather further insight into the array characteristics. For example, the condition for broadside radiation, A ^ = 0 , gives u = (0,—,0 )r and v = (l,—,1)T. For a coupling phase o f <J>= 0 equation (16) shows that the broadside radiation condition is A 0 - 0 which implies identical element tunings across the array. In addition, we will show in the next section that for 0 = 0 the stability matrix C = sA V has positive eigenvalues when each phase satisfies -*A< A$n < X A - Thus this desired phase distribution represents a stable state o f the system. However, for O = 90 degrees the matrix B is angular for even values o f N. This implies that the pulling equations are not linearly independent and we cannot determine the phases from a given set o f tunings. For odd N the matrix B is nonsingular, so an inverse exists, but the stability matrix C = - e B U always has zero trace. Since the trace of a matrix is the sum o f its eigenvalues, the real parts o f the eigenvalues cannot all be positive and still give a zero result, hence no stable states exist. When $ is some other value the stability matrix is a weighted combination o f A V and B U (equation (18)) and we can probably infer that the stability region is maximized for <J>= 0 , although we have not proved this. It is easy to show that the tendency for synchronization is maximized at O = 0 for N=2, elements, but this is difficult to show for larger N. In summary, <b = 0 represents an attractive value o f coupling phase since a simple tuning configuration gives the broadside phase condition and it seems to maximize the tendency to synchronize. In the following we will assume this value and explore array characteristics in more detail. 2.1.1.1 Zero Degrees Coupling Phase-Synchronization Diagrams When we assume $ = 0 the frequency pulling equations and the perturbational equations take on very simple forms and 6.0= eAu A 5 = -C A S = -sA V A S (19) (20) To gain more insight into the array behavior we utilize synchronization diagrams to graphically depict the array element tunings that can result in stable 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. synchronization. Such a diagram might appear as in figure 2 for a three element array. A C0o3 — > COol v Figure 2-E xam ple o f a synchronization region for a three element array. T he second oscillator tuning remains fixed. A t the origin <moI = coo2 = coo3 Since there are only N -l independent phases, we may leave one of the element tunings fixed, for example the second in the diagram above. The shaded region indicates the values of the remaining element tunings for which stable synchronized states ex ist Thus we can "see" what combinations of tunings allow for synchronization. These diagrams can be constructed for arrays with arbitrary coupling, but they are generally difficult, if not impossible, to construct analytically. However, the simple form of the equations for the special case of 0 = 0 allows us to easily construct the diagrams for any size array. 2.1.1.1.1 Existence Region We begin by considering the frequency pulling equations (19) and finding the region o f existence of synchronized states. The vector u depends on the element phases (specifically the phase differences) through the sine functions, as shown in equation (14). If we let the phase differences span all of their possible values (the interval [-7t,7t] is sufficient) then the elements of u will span [-1,1]. This is depicted for N=3 in figure 3. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^A<j>2 AU2 Nonlinear Ul !—Jl £s Linear Figure 3 - T h e square region in AO space is mapped nonlineariy to the square region in u space which is then mapped to the rectangular parallelepiped in AfJ space. The inscribed circle m aps to an inscribed ellipse whose length and direction o f m ajor and m inor axes are given by the eigenvalues and eigenvectors o f the transformation matrix. This region is then mapped through the linear transformation in equation (19) onto a region in the frequency pulling "space,” also shown in figure 3. The linear transformation maps the square region of N -l dimensions onto a rectangular parallelepiped of N -l dimensions. For every frequency pulling value in this region a synchronized state exists, thus it is called the existence region. Although its size and shape are completely described by the linear transformation A in (19), we can also relate them to the eigenvalues and eigenvectors of A. If we inscribe a circle in the square region in u space, as shown in figure 3, and then transform the circle onto the frequency pulling plane using equation (19), the result is an ellipsoid inscribing the parallelepiped. The eigenvalues of A are proportional to the lengths of the major and minor axes and the eigenvectors give the directions of the axes. Thus tuning the array in the direction of an eigenvector gives an indication of the size of the existence region. We will see in sections 2.1.1.1.3 and 2.1.1.1.4 that these "characteristic" tunings are directly related to the sensitivity and transient response of the phases. In the above diagrams we have represented the element tunings in terms of the frequency pulling variables A(3„, which are the differences between adjacent element tunings. It is often preferable in practice to give the results for the actual element tunings. Given an N element vector of oscillator tunings, co„, we can compute the N- 1 element vector of frequency pullings using a linear 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transformation. We will choose the first element tuning as fixed, although this choice is arbitrary. The frequency pullings are then written as f A3, ) Ap, .AP.V-, ; r i = -1 .0 O' 1 0 ®«3 -i i , ^«N J . 0 J or (21) Thus, to map the existence region in the space of frequency pullings to the space o f element tunings we use the transformation co^. = ZT'fA p+co^) which changes the size and eccentricity o f the parallelepiped but retains its basic shape. This is illustrated in figure 4. COo2 Figure 4—Synchronization region in the plane o f free running frequencies. In the following we will express the results in the frequency pulling space and not in the free running frequency space for the sake of simplicity. 2.1.1.1.2 Stability Region Once we have found the existence region, the next step is to determine the subregion of stable synchronized states since these are the only states observed in physical systems. Each point within the existence region has at least one corresponding phase vector that is found by solving equation (19). In general, for each phase vector we must compute the eigenvalues of the corresponding stability matrix, C = e A V . As mentioned before, when the real parts of the eigenvalues 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are positive the state is stable. However, in the present case determination o f stability is am ple due to the nature o f the matrices A and V. The matrix A defined by equation (13) is positive definite and the matrix V defined by equation (IS) is also positive definite when each o f the phases lies in the range [ %]. Since the product o f two positive definite matrices is also positive definite, the eigenvalues o f the stability matrix are all real and positive when the phases lie in the above range. This range o f phases is sufficient to cause the vector u in equation (19) to span all o f its possible values, which proves that the stability region fills the entire existence region. Furthermore, over this range o f phases the sine functions within u are one-to-one, so the transformation from A<j>to u has a unique inverse. Thus for each frequency pulling vector within the stability region there is a unique phase vector which implies that a unique stable synchronized state exists for a given tuning. This is an important result and shows that 0 = 0 is an optimum choice in another respect. Although the above analysis was illustrated graphically for the case N=3, all o f the results hold for arrays o f any size. Many other characteristics deduced from equations (19) and (20) are discussed in reference [2] and in section 3.3 for mode locked states with similar locking regions. 2.1.1.13 Phase Sensitivity Another attractive characteristic o f zero degrees coupling phase is that the center o f the stability region gives the phase distribution for a broadside radiation pattern. The region center is a nice "target" for the element tunings since the danger o f loss o f synchronization due to random tuning errors will be minimized Also, the sensitivity o f phase with respect to tuning variations increases as the region boundary is approached, and lower phase sensitivity gives more robust array characteristics. In the following we will consider the topic o f phase sensitivity in more detail. A convenient measure o f phase sensitivity is the change in length o f the phase vector for a given change in the length o f the frequency pulling vector in the synchronization diagram. In practice we would probably want the change in phase for a change in element tunings, but the tunings are simply related to the piillings through the linear transformation o f equation (21). Again, we will consider the frequency pullings for simplicity. Using equation (19) a differential change in phase is related to a change in frequency pulling through the stability matrix: dAfi= sAVdA<j)= Cd&<f> (22) The sensitivity defined above is then given by 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ' d L fi{ ( c r ') TC r ') d L fft * d A fd A p \d A p d A 0 ) (23) I f the change in tuning is an eigenvector o f the matrix C-1 with eigenvalue X 1, then the sensitivity is amply = |/T1|. N ote that the eigenvectors o f C-1 and C are identical and the eigenvalues are reciprocals o f one another. One can show that the eigenvectors o f C are the directions for which the sensitivity is either a local maximum or minimum. Thus a t uning change in the direction o f an eigenvector o f the stability matrix C, called a "characteristic tuning” for the array, causes an extremum o f the phase sensitivity, which is given by the reciprocal o f the corresponding eigenvalue o f C. This result, together with the results o f section 2.1.1.1.1, shows that the size o f the existence region is directly related to the phase sensitivity. Indeed, we expect to lose synchronization more quickly if we tune in the direction o f high phase sensitivity. The characteristic tunings also play an important role in the transient phase response o f the array, a topic we will consider next. 2.1.1.1.4 Transient Response to Tuning Variations Many papers in the engineering literature have discussed the design of microwave oscillator arrays to optimize various steady state parameters, but none, to this author's knowledge, has addressed dynamic problems associated with the modulation o f the carrier frequency, such as modulation bandwidth and array settling time. However, these are important concerns for system designers. For beam steering systems there is a certain array settling time required for the oscillator phases to achieve a new steady state after the beam has suffered a step change. Similarly, for communication systems relying on phase modulation this settling time implies a finite modulation bandwidth. The settling time not only depends on the coupling strength, oscillator bandwidth, and the number o f elements, but also depends on the manner that the array is modulated. The following analysis shows that the complex phase behavior due to an arbitrary change in the array element tunings can be decomposed into a combination o f responses to characteristic tunings, each giving rise to a characteristic phase perturbation that possesses a unique exponential decay time. The analysis assumes small tuning changes near the center o f the synchronization region, but the computed settling times give approximate results for large tuning variations, and the physical insight developed is useful in understanding the dynamic phase behavior. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Timing any o f the array elements perturbs the oscillator phases and a certain amount o f time is required before die phases reach a new steady state. Although a general analysis o f the effects o f arbitrary time dependent tuning changes is feasible, in the following we will consider only step changes o f element tunings. This type o f tuning could occur in beam steering systems where a step change in the transmitting beam direction might occur. We assume that we can control the oscillator free running frequencies, o r tunings, with infinite rapidity. In practice, we will not have such control, but the following results will hold approximately if the element tunings respond foster than the fastest time constant o f the array. I f this is not the case the time constants o f the tunings must be included in the analysis. For small phase changes the phase response is governed by the system (20), which has a solution [3] Ad(t) = (24) where the matrix exponential is defined through the usual power series expansion e-Cf= / - C / + i ( C / ) 2 + - (25) Thus the behavior o f the phases for a stable frequency locked state always involves exponential decay, at least for small phase changes. Large phase changes may be brought about by beam switching across large angles, o r by operation near the edge o f the frequency locking region. Such phase changes may not initially behave according to equation (20), but as the phases approach the steady state the exponential decay o f equation (24) will prevail. The phase behavior that we will describe approximates the behavior for large phase changes and should provide a good estimate. A general result o f linear system theory tells us that the eigenvalues o f the stability matrix in equation (20) are the reciprocals o f the time constants that govern the exponential decay o f the phases into the steady state for particular phase distributions, the eigenvectors. These characteristic phase perturbations can be caused by particular tuning changes that we will call characteristic t unings As mentioned previously, the eigenvalues and eigenvectors o f the matrix C are directly related to the size, shape, and orientation o f the locking region. The phase change due to an arbitrary, but infinitesimal^ tuning change is given by 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dA(3=CdA<f> (26) This shows that if the phase change is an eigenvector o f the stability matrix, then the corresponding tuning change is proportional to the same eigenvector and the constant o f proportionality is the eigenvalue, that is, C dA$= M A$= dAp (27) Thus, the characteristic tuning changes are found by computing the eigenvectors o f the stability matrix C. Practical systems will be designed to operate near the center o f the locking region to avoid excessive phase sensitivity to tuning changes which occurs near the region edge. N ear the region center we can make the approximation in equation (19) that u = A<f>, which gives a linear relation between the phase differences and oscillator tunings differences Ap=sAA<t> (28) In addition, we can simplify the linear system (20) A 5 = -s A A S (29) because the matrix Via equation (20) is nearly the identity matrix. We now consider the dynamic phase response to an arbitrary step change (at t = 0) to the oscillator tunings near the center o f the locking region. The tuning difference vector prior to the step change is Aft*1= { A 0 f‘ ••• A)5j with a corresponding phase difference vector A<jT‘. Ajjfi™1 - (AySf"®7 After the change we have A f t w ) T and the steady state phase vector is A ^® 7. Just after the step change the phase vector is time dependent, A $ t), and the difference, or perturbation, A<5(/) = A<0t) —Atff™1 satisfies the linear system (29) with the initial value A^1”' - A< 0^. The initial and final phase vectors are directly related to the oscillator tunings through (29): A tf* = —A~'A0nit \ A ^ = A A''Afi*™1 e (30) 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. so the phase perturbation at / = 0 is A<5(0) = - AyS^). s (31) and the system dynamics will evolve according to A = 2-[a -' (A 0 * - A ^ ) ] e“^ r (32) The general behavior o f (32) is quite complicated and probably best understood by decomposing the response into a superposition o f characteristic responses. The next step, therefore, is to find the eigenvalues and eigenvectors o f the m atrix^. Simple closed form expressions for the eigenvalues and eigenvectors o f A can be found by noting the similarity between the matrix A and the second derivative operator The ^ At = 4sin2^ , eigenvalue is given by k = \2 ,—, N - \ (33) and the corresponding eigenvector is ^ Wl = ^ sin( ^ ) ’ * = n = l,2,—,JV-1 (34) where n indexes the individual elements o f the vector. The set o f eigenvalues and eigenvectors, along with the characteristic tunings, for a five element array is shown in figure 13. Fixing the center element tuning rather than the first preserves symmetry in the characteristic tunings. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. '.372^ = 0 .3 8 2 , e0) = .602 .602 f .602 ^ = 1.382, eP) = 1372 .372 -.3 7 2 ^ -.6 0 2 J f-6 0 2 ^ ^ = 2 .6 1 8 , e(3) = .372 .372 ' .372 ' A4 = 3.618, e(4) = v-.6 0 2 , -.6 0 2 .602 -.3 7 2 ; (35) Figure 5—Eigenvalues and eigenvectors for a five element array (N=5). Using some base results o f linear system theory [3] one can show that if the tuning change is proportional to one o f the eigenvectors, that is, if A J ? * -A p final = n e (k) (36) where £2 sets the step size, then the phase perturbations will decay exponentially with a unique time constant zk - —— = ----- \ according to sXk ^sin ( |0 f AS(i) = Q zke r* e {k) (37) This shows that the effect o f the tuningchange on the phasesincreases as the eigenvaluedecreases. This is consistent with the statement earlier concerning the sensitivity o f phase with respect to frequency variations, and is directly related to the size o f the locking region in tins particular tuning direction. Referring to the expressions for the eigenvalues (33) we see that the longest time constant is and, if this mode is excited, represents the array settling time. The eigenvectors form a complete orthonormal set, as expected from the symmetry o f A, and can be used to represent an arbitrary tuning change. Once the change is decomposed into a linear combination o f the characteristic tuning changes the phase dynamics can be analyzed by superposing the individual responses, each o f which has a characteristic decay time directly related to its eigenvalue. Certain tuning 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variations may excite some modes to such a small degree that the phase perturbations can be neglected. 2.1.1.1.4.1 Example: Settling Time for Beam Steering Using our example o f electronic beam steering discussed above, we will treat the case o f a step change away from broadside for a five element array. Initially, broadside radiation implies that the elements are identically tuned so that A/7"“ = 0 . To steer a transmitting beam the end elements must be tuned in equal but opposite directions, thus the final vector o f timing differences is Aft*”* = (Q ,0 ,—,0 ,Q )r where Q is the amount o f the change.Expressing the applied tuning as a linear combination o f characteristic tunings, we have (38) *=i and taking the dot product o f both sides with e® and using the eigenvector orthogonality relation e , gives the weighting factors at = A p-e{1) (39) For our example we set N = 5 and find that the weighting factors are a = (0.743 0 1.203 0f (40) with corresponding time constants r = -(2 .6 1 8 e 0.724 0.382 0.276)r . (41) The zero entries are due to orthogonal symmetries between the tuning variation and two o f the characteristic tunings. We can see that the applied tuning couples quite efficiently into the dominant time constant, which is the first element o f each vector. However, for the type o f tuning considered here, where only the end elements are varied, the lowest order mode is coupled less efficiently as the number o f oscillator elements N increases. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2J2 Effect of a Resonant Coupling Network on the Synchronization of Two Oscillators [4] Oscillator Grid Partially Reflecting M irror Figure 6—Quasi-optical power combining as originally proposed by M ink Oscillator coupling occurs through the resonant cavity. Operation is sim ilar to optical lasers. Many quasi-optica! microwave systems involve arrays o f oscillators that are locked to a common frequency through mutual coupling. In some systems the coupling network is a high Q resonant structure that may, or may not, force the oscillators to lock to the coupling resonance. An example is the structure originally proposed by Mink [5], one o f the first examples o f quasi-optical power combining, where an array o f oscillators is placed in a resonant cavity. Assuming frequency locking occurs and the correct phase distribution is achieved, power from the array elements adds constructively to produce a high output power signal, as illustrated in figure 6. The designer o f such systems must understand the oscillator tuning accuracy required to maintain frequency locking and obtain the desired phase distribution, in addition to the functional dependence o f the oscillator phases on the oscillator tunings. Past papers have shown that the simple Van der Pol model leads to analytical results that agree quite well with measurements.[6] In this section we model two microwave oscillators coupled through a resonant (though possibly lossy) coupling network as tw o Van der Pol oscillators coupled through a series resonant coupling network, shown in figure 7. In particular the authors study the ability o f the oscillators to synchronize for wide ranges o f coupling strengths and coupling bandwidths. Simple approximate equations for the dimensions o f the frequency locking region are then compared with numerically computed results and show good agreement for all coupling strengths and bandwidths. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cc Lc Rc v,(t) V2(t) Oscillator I Oscillator II Figure 7—Two oscillators coupled through resonant network. conductances depend on the amplitudes of the RF voltages across them. The negative 2.2.1 Dynamic Equations The linear coupling circuit can model a wide range o f frequency dependent circuits, and in particular cavity resonators. The starting point for the following analysis is the system o f differential equations, derived in section 1.2.1.2, that relates the amplitudes and phases o f the oscillators and the dynamic variables o f the coupling circuit. Assuming the oscillators are locked to a common frequency a , the oscillator voltages can be represented in terms o f their slowly varying amplitudes and phases as v, (/) = 4 (t)co s(c* + $ (/) ) (42) v2(t) = A2(t)cos(cDt+4>2(t)) In the steady state the amplitudes and phases will be constant in time but if we perturb the state, as we will when considering stability, the amplitudes and phases become time dependent. The current flowing through the coupling network is related to the oscillator amplitudes and phases through differential, rather than algebraic, equations since the coupling network is frequency dependent. [7] The coupling current can also be expressed in terms o f slowly varying quantities: /c(t) = G0( j a (/)cos(cDt)+Av (t) sm(eot)) (43) where the conductance Ga = G( 0) is the magnitude o f the nonlinear conductance G(A) in figure 7 evaluated at A = 0, and is added for dimensional equality. The "rectangular" form o f equation (43) was chosen over the "polar" form because the current amplitude can drop to zero. This occurrence does not present formal mathematical difficulties but the does complicate subsequent analysis and numerical evaluation. For Van der Pol oscillators, the dynamic equations relating 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the slowly varying quantities o f equations (42) and (43), derived in section 1.2.2, are A = &a(l-A2)A + a>a[4c cosfo)+4* sm($,)] A = a>0x ~ cos(^,)] A A= 6Ja ( l - ^ 22) 4 - ^ [ ^ c r CO s ( ^ ) + ^ S in ( ^ ) ] & = coo2 - co+<oa-j-[A a sin (A ) ~ Av cos(& )] A Aa = -G ijia , + { ( 0 - e o jA ^ + a>JL0[ A cos(A)~A co s(^ )] K = -{c o - c o jA ^ - + ( o J . \ A sin($J2) - 4 sin (^ )] (44) where the oscillator bandwidths are 2coa = %-, the imloaded coupling circuit bandwidth is the oscillator uncoupled resonant frequencies, or tunings, are <a0, = and coo2 = and the coupling constant is Xa = yG^ . These five parameters directly affect the ability o f the oscillators to lock and our task is to understand the effects o f each. The steady frequency locked states are found by setting the derivatives in equation (44) equal to zero and solving the algebraic system for the amplitudes, phase difference A<f>= 4>2-<pu and the frequency co. The tw o coupling variables Aa and .4^ can be eliminated so that the resulting system consists o f four equations in four unknowns. Once a locked state is found, stability o f the state must be tested by perturbing the variables o f equation (44) and observing whether the perturbations increase or decrease in time. Perturbing the variables produces a linear system o f differential equations with constant coefficients, and the real parts o f the eigenvalues o f this system indicate stability.[8] 2.2.2 Synchronized States Different characteristics o f the system analyzed here will be important in different situations. For example, the variation o f the phase difference is important in the design o f beam scanning systems, [9] [10] [11] and the frequency modulation bandwidth and array settling time are important in wideband 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. communication systems, and these characteristic may be examined using equations (44). The focus o f this paper, however, is on understanding how the frequency locking ability of the oscillators depends on the coupling strength, bandwidth, and oscillator tunings for all possible combinations of each. The oscillator and coupling circuit tunings that result in frequency locking are expressed graphically in figure 8 where the axes are the oscillator tunings refered to the unloaded coupling circuit resonant frequency. AO)02—(0o2—C00 \ V2 w / direction moved / ' to perturb x Acoc and A 1/ ^ne ° f e(lual tunings Acf)=0 fs / A cD ^or-co* V2 h \ line of equally spaced tunings Acg.=0 frequency locking region Figure 8—Region o f frequency locking in the plane o f oscillator tunings with respect to the coupling circuit resonant frequency. T h e lines o f symmetry are the lines o f equal tunings. <0ol = co„2, and equally spaced tunings. - t ( c o 0, + a>oZ) = c o ^ . The width. W , is the total span o f A a ol + Ao)o2 at half the maximum value o f A(0o2 - A©ol. The small arrow show s the direction o f the perturbation used in appendix 2.2. The shaded region is where frequency locking occurs; that is, if the oscillator tunings lie within this region the oscillators will synchronize. Our task is to determine the size and shape of this region for various values of coupling strength, coupling bandwidth, and oscillator bandwidth. In equations (44) we refer the 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oscillator timings and the frequency co to the coupling circuit resonant frequency using the substitutions A&oi = a oi ~ &OC > Afflo2 = fi)o2- 0 a ) Affle = 0 - 0 (!C (45) Setting the derivatives equal to zero gives the algebraic equations describing the locked states that, after eliminating the coupling variables Aa zn d A ^, can be written as ( l - A 0€*~ Aj2)Al = -AoSAz cos(A ^- <I>) (46) (i - A < ,£?-A% ) a 2 = - A 0eA1cos(A0+ d>) an(A^+d>) where s ■j are, respectively, the coupling strength scale factor and coupling phase that result from frequency dependent attenuation and phase delay through the coupling circuit, and the coupling phase is confined to: |0 | < 90°. The form o f equations (46) is nearly identical to the form given in reference [12] describing frequency independent coupling networks except that here the coupling parameters are frequency dependent. The left sides o f the equations contain terms not present in the analysis o f reference [12] that account for the loading effects o f the coupling circuit on the oscillators. The concept o f coupling magnitude and phase is useful in understanding the effect o f the coupling network on the ability o f the oscillators to lock and was used extensively in section 2.1. W e found the tendency to lock increases with increasing coupling strength and is maximum for zero or 180 degrees coupling phase. In feet, for ±90 degrees o f coupling phase the ability to lock ceases entirely, at least to the first order o f approximation. For the present case we can identify a frequency dependent coupling magnitude A0s(Acoc) and coupling phase <D(A«uc) and can immediately see that these quantities depend on the location o f the steady state frequency co relative to the coupling circuit passband, as we should expect. If the frequency co lies at coupling circuit resonance, i.e. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <o=a>oe-> A a c = 0, the coupling strength and phase are both optimized, and the locking tendency is strongest As A'% ac becomes large, coupling becomes weak and the coupling phase approaches ±90 degrees, quickly causing loss o f synchronization. Thus frequency locking depends critically on the proximity o f the steady state frequency to the coupling circuit passband, but the frequency is a complicated function o f the circuit parameters that w e must solve for using equations (46). In the next section we will apply approximate methods to estimate Acoe and use this result to determine how the locking region depends on the circuit parameters. Solutions to equations (46) indicate the existence o f frequency locked states, but we will briefly pause to consider these steady states from the viewpoint o f linear circuit theory. As described in chapter 1 the amplitudes and phases must satisfy the frequency domain equations with Kurokawa's substitution, which essentially consists o f replacing the steady state frequency a> w ith the dynamic A quantity <o+<f>-j — for each transfer function (see section 1.2). In the steady A state the amplitudes An and phases <f>„ o f the oscillators are constant so the time derivatives vanish and the steady state system satisfies the frequency domain transfer functions. Since the amplitudes are constant we can replace the amplitude dependent conductances with constant ones, but with the same conductance values, without perturbing the steady state solution. Recalling that the locked state contains only one frequency component we can identify the state as a mode (i.e. an eigenstate) o f the linear system. This modal viewpoint can be helpful in systems with very small nonlinear conductances that can, to the first approximation, be ignored. This leads to orthonormal modes and such systems are elegantly analyzed using the average potential theoiy.[13] 2.2.3 Stability of States Instead o f deriving the frequency pulling variables and using these to find the existence and stability o f the states, we will consider the present problem more carefully. Our general method is accurate when the amplitude perturbations decay rapidly compared to the phase perturbations. We would like to consider more general cases where this condition may not be satisfied. The steady state values are perturbed by substituting the following into equations (44) Ai ^ A i + a l, Aex-+ A a' + a ae, A ^ - t A ^ + a ^ , (47) 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and retaining only first order terms where Ai, <f>{, etc. are the steady state values for the mode locked state in question and a it 8t, etc. are the infinitesimal perturbations. The resulting dynamic system for the perturbations is p-M p (48) where the vector o f perturbations is $ (49) P= a„ \ a<y j and the matrix o f coefficients is \ ( ®a ( l - 3 4 2) tua>01 - Aa>c M= &a>c) o a {\ - A ? ) A 0 0 0 0 0 0 <°a n 0 0 - A 2(Aeoo 2 -&a>c ) - o a c o s(A ^ ) r ®a ( l - 3 4 ) b a o2- b a e —X qCDqc 0 2 ZjtDg,. c o s ( A $ 0 -XofOggA^ ^ a ^ s in ^ ) toa{ \ - A l ) a \ C0S(A*) 0 <*>a A -a> a sin (A # ) sin (A ^ ) °a A wa 2 co s(A ^ ) ■A -<°ac 6a>e -b w c -<°ac , (50) All expressions appearing within the matrix are the time independent values o f the frequency locked state. As mentioned before one steady state oscillator phase is arbitrary so we set $ = 0 and A ^= <f>2. This implies that the above system has only five degrees o f freedom and therefore one o f the eigenvalues is zero. It is possible to reduce the set o f equations, but the coefficients o f the remaining system are considerably more complicated and the simple coupling structure is obscured. Since the above matrix has constant coefficients the system is stable when the real parts o f all o f the nonzero eigenvalues are negative. We are now prepared to determine the region in the tuning plane within which stable frequency locking occurs. Equations (46) that determine the existence and equations (50) that determine stability o f locked states are sufficiently complicated to require computer evaluation for exact solutions. However, for many cases approximations can be made to reduce the complexity. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the following section we will derive simple expressions for the values o f oscillator tunings that result in stable frequency locking for various values o f coupling strength and coupling bandwidth. 2.2.4 Cases of Practical Interest In order to simplify the analysis we will consider cases o f weak, strong, wideband, and narrowband coupling separately and make the appropriate approximations for each case. Taking all o f these results together gives us a broad understanding o f the system for a wide range o f parameters. In the end we will compare our approximate expressions for the locking region dimensions to exact solutions obtained by computer simulation and will find good agreement in all cases. The first difficulty we encounter is that there may be more than one solution to equations (46), each solution corresponding to a different mode o f oscillation. In general there may be three stable modes for the circuit considered here, one whose frequency is located near the resonance o f the coupling network and the other tw o whose frequencies are located near each oscillator timing The former is the mode o f practical interest and only this mode will be studied in this section. It has the largest locking region since its frequency is closest to the coupling circuit passband, and very often it is the only mode excited. The other two modes are possible only when the oscillators are tuned well within each other’s and the coupling circuit’s passbands. There are two types o f tunings for which the mode o f interest is relatively easy to analyze. For equal tunings, A<oA = A cuo2, which corresponds to the diagonal line through the first and third quadrants in figure 8, one can show using equations (46) that the phase difference equals zero and the oscillators will always lock no matter how far away from the origin we tune. This occurs because in-phase oscillation eliminates current flow through the coupling network and since the oscillators are identically tuned they will remain in phase in the absense o f coupling. However, one can see in the figure that the locking region becomes vexy small as we tune far away from the origin so that, practically speaking, locking becomes precarious. When the coupling circuit resonance is located exactly between the oscillator tunings, A<ho1 = -A o o2, which corresponds to the diagonal line through the second and fourth quadrants, one can show that Aa>c = 0, which implies maximum coupling strength and optimum coupling phase, and equal amplitudes. We will refer to this type o f tuning as "equally spaced" since all three frequencies are equally spaced. It is not obvious from 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equations (46) but the locking region is symmetric about the diagonal lines o f equal and equally spaced tunings. Once we determine the locking region boundary in one quadrant the entire region is determined. In the analysis that follows we will consider quantities above the line o f equal tunings since the phase difference is always positive in this region and this simplifies the mathematics. Moving along the line o f equally spaced tunings the quantity Ao)0 = Acao2 - Aco0] increases and the total change in Aco0 as we traverse the entire locking region we will call the "height" and denote it H (the factor o f -Jl in figure 8 is required since the measure indicated is the diagonal length). As we move away from this line perpendicularly within the locking region we move in the direction o f even tuning and vary the quantity Aco0 = ^{Aeool +Aeoo2), which is the "average" oscillator tunings away from the coupling circuit resonance, and eventually meet the locking region edge. Twice the total change in Acoa at half the maximum value o f A<oQ we will refer to as the width W , indicated in figure 8. Since Acoc = 0 along the line o f equally spaced tunings the value o f H is relatively easy to determine. But determining W requires knowledge o f the Aa e variation as we move away from this line since the ratio tL°‘/o x has direct bearing on W. The functional form o f the phase difference for equally spaced tunings is derived from equations (46) by subtracting the second and fourth equations and setting A<oc = 0 . The result is Aa>o = Aa>o2 - Ao>ol = 2 X0eoa sin(A^) (51) and we can immediately see that solutions cannot exist for Aco0 >2A0coa. Although we cannot easily prove it for the general case, extensive computer simulations suggest that a necessary condition for stability is that the phase difference lie between -90 and +90 degrees for any value o f coupling strength or bandwidth, and we will assume that this is true. Thus, as the oscillator tunings are moved apart and the coupling circuit resonance is maintained exactly halfway between, the phase difference A<j> increases until the locking region boundary is encountered. Along the line o f equally spaced tunings the amplitudes, which are equal in this case, are found from equations (46) (see appendix 2.1): A2= \-X . 1 - J l- A<o0 (52) 2h°>a. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The amplitude variation across the locking region increases with increasing coupling strength Xa but remains close to unity for weak coupling. The functional dependence o f A<oc can also be found by adding the second and fourth o f equations (46) (see appendix 2.1): AG>0 + =■ — ' -------------------------T - A 4 -ls in (A $ \ X 0o>a t ? (53) The amplitudes and the phase difference depend on the oscillator tunings through equations (46) and A<oe also appears implicitly in s . This complexity forces us to approximate A<oe for specific cases. Since the width o f the locking region depends on how fast Aeoc changes as we move away from the line o f equally spaced tunings we will derive the change in Aojc for a small change in Aco0 for a fixed value o f A<o0. Referring to figure 8 we will move perpendicularly away from the diagonal as indicated. After considerable algebra (see appendix 2.1) the approximate value for Ao)c valid near the line o f equally spaced tunings is A<a = --------------------- ^ 2 —---------- — 2 - (54) 1----------- (Doc 2A2-1 + X 0 4a>aa 0e where the amplitude A is given by equation (52). This relation is ample enough to allow us to determine the approximate locking region width for cases o f practical interest, but the approximations turn out to be quite accurate, as we will show by comparing them to computer simulations. Whether we classify a coupling network as "narrowband" or "broadband" depends on the behavior o f Aeoc as the coupling circuit is tuned relative to the oscillators. This type o f tuning is equivalent, in our analysis, to t uning in the direction perpendicular to the line o f equally spaced tunings where the spacing between the oscillators is maintained but both are tuned relative to the coupling circuit resonance. For broadband coupling we would expect the steady state frequency to be determined by the oscillator tunings and not by the coupling circuit, which implies Acoc * A<o0 - » <y~ |(< aol +eoo2). Whereas for narrowband 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coupling we would expect the frequency to follow the coupling circuit resonance, or Affle * 0 - > o « ja w. These tw o conditions give us criteria to identify the coupling type as broad or narrow. Equation (54) tells us that for sufficiently small A< d0, that is, as A< oo1 is tuned sufficiently dose to Aa>o2, that the broadband condition is satisfied even for small co^., which seems to contradict our usual notion o f narrowband coupling. I f the oscillators are both tuned within the unloaded coupling circuit passband, however, the steady state frequency will always remain within this band so this is essentially a "broadband" condition. Furthermore, the effective coupling depends not on the unloaded coupling bandwidth, but on the loaded bandwidth which involves the coupling strength and the oscillator bandwidths. When the oscillator bandwidths overlap the coupling circuit bandwidth the coupling circuit is more heavily loaded by the oscillators and hence the loaded Q is reduced. We must keep in mind that the definitions o f broadband and narrowband in the following sections are somewhat arbitrary since the location o f the steady state frequency changes in different parts o f the locking region. The division o f the coupling strength and bandwidth into regions o f weak/strong and narrow/broad coupling are expressed in graphical form in figure 9. Qac CDa Broadband llI I llI L- V2 ''I II I i Narjrowband Weak Coupling W Strong Coupling Figure 9—Parameter diagram showing four regions of interest Coupling strength depends only on XQwhereas coupling bandwidth depends on <oa and X 0. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The boundaries separating the various regions will come directly out o f the analysis that follows. 2.2.4.1 Weak Coupling: XQ« X A I f the resistance in the coupling network o f figure 7 becomes large then = Yrjs. « Yl and the oscillator amplitudes remain close to unity. The question immediately arises as to which terms, if any, in equation (54) we can neglect and under what conditions. Using the maximum value o f A<o0 from equation (51) we have, using equation (52), Aa>0 (55) and, since XQis small, the third term in the denominator is always much less than the second and therefore can be neglected. At the edge o f the locking region, when the second term is much less than unity the coupling circuit is broadband and as unity is approached Agjc blows up. This behavior is not what we would expect for narrowband coupling circuits, as discussed in the previous section, and causes loss o f lock fairly close to the line o f equally spaced tunings. At any rate, the boundary for narrowband vs. broadband coupling can be taken as cox = X0<oa. 2.2.4.1.1 Broadband Case: co^ » Xocoa Along the line o f equally spaced tunings one can show that stable solutions o ast for all |A^| < %, although proof o f this will be omitted here. The height H, found from equation (51) and shown in figure 10(a), is (56) Thus the height o f the region is proportional to the coupling strength and the oscillator bandwidths (assumed equal). From equation (54) broadband coupling implies Acoe « Aco0 which means that the steady state frequency is as far away from the coupling circuit resonance as the average values o f the oscillator tunings are from the coupling circuit resonance. Or, in other words, the steady state frequency is exactly between the 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Weak Coupling, \ « l / 2 Broadband, ©K»X,Cd, W = 2oo„. - 2 1- -> K<&* Weak Coupling, \ « \ ! 2 Narrowband, coac« l 00)a f:t , « W ati-oic Xlii* \ Additional Locking Figure 10—Dimensions o f the locking region for weakly coupled oscillators, (a) The case o f broadband coupling has a fairly wide locking region that is bounded by the phase requirem ent |A()>| < 7C/ 2 . (b) The narrowband case is quite thin and is bounded by loss of stability due to high sensitivity of the steady state frequency with respect to tuning variations. The additional locking regions appear for values o f near, but below, unity. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oscillator tunings and is independent o f the coupling circuit resonant frequency. This situation can be taken as the defining characteristic o f broadband coupling. Since the amplitudes are both nearly equal to unity throughout the region the relation between the phase difference and the oscillator tunings can be approximated from equations (46) as Aa>o = 'lX o(0a sin(A^) (57) Thus the locking region boundary consists o f the values o f Aco0 where A ^= ±90° and is plotted in figure 10(a). The width o f the region when Aa>0 is half o f its maximum value occurs when Acdc = Ato0 = a>x , as seen from equation (57). Including the second term in the denominator o f equation (54) for Aeo0 gives a more accurate result for the width: (58) The case o f a resistive coupling circuit can be found by letting the coupling circuit bandwidth approach infinity in equations (56) and (58). The result is an infinite locking region that follows the line o f equal oscillator tunings, as we would expect from physical considerations. 2.2.4.1.2 Narrowband Case: cox « Aocoa We now consider small coupling circuit bandwidths. The quantity Aa e near the line o f equally spaced tunings is found from equation (54) to be Acoc = Aa>0 (59) Near the center o f the locking region, that is for small Acoa, Acac « Aa>0 and we see the same behavior as in the previous case. But as Aa>0 increases the denominator in (59) decreases and Acoc becomes much more sensitive to tuning variations. Computer simulations show that the value o f Aa>0 that causes the denominator to vanish is (approximately) a stability boundary and for values o f Aco0 for which A<oe is negative the system is not stable. The stability boundary, 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and therefore the height o f the locking region, is found by setting the denominator o f (59) equal to zero: B = - = w For coupling bandwidths above 2Xa(oa this stability boundary does not exist Below this threshold we must also meet the general existence criterion that Acoa <2X0<oa. Using these two criteria together we find that this new stability boundary exists only for < X0coa and above this value o f coupling bandwidth the general existence criterion applies. Assuming the former condition applies, the stability region is found, at least approximately, using Aa>0 = 2Xo(0a - sin(A^) cC + A (0e2 (61) with Aeoc given in equation (59). To find the width we find 2 A a \ which, as IAd for the previous case, occurs at Aa>c - (ox . Solving for the width gives <6 2 > Figure 10(b) shows the approximate shape o f the locking region for weak coupling and narrow coupling bandwidth. The region is much thinner near the edge o f the odd tuning boundary due to the increased sensitivity o f Acoc to changes in Aco0 near this boundary. If we include the third term in the denominator o f equation (54) we find that for values o f X0 close to but less than unity that the denominator becomes zero a second time and for values o f A<o0 greater than this critical value the locked states are stable once again. Thus two new locking regions appear and are disconnected from the main region; they are shown as dotted regions in figure 10(b). In this analysis however we will limit ourselves to small coupling parameters for which case these additional stability regions do not exist. 2 .2 A .2 Strong Coupling: Xa» y2 As the coupling strength XQ is increased the amplitudes decrease considerably as we traverse the locking region in the direction o f equally spaced tunings. The physical reason for this is that as the coupling resistor Rc is reduced 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the power dissipated in it increases. The oscillator conductances must make up this power loss by becoming more negative, which is achieved by amplitude reduction. However, power dissipation in the coupling network requires a phase difference to exist between the oscillators, and this phase difference increases as we traverse the locking region. If other o f the amplitudes drops too far below unity the system becomes unstable and locking is lo st It is difficult to determine exactly when this occurs, but we can find the approximate amplitude boundary from the perturbational system (50). The perturbational system consists o f three second order subsystems, the three diagonal blocks, and are coupled through the o ff diagonal blocks. If no coupling existed then stability o f the system would be insured if each o f the three subsystems were stable. The coupling circuit is always stable since it contains some nonzero positive resistance, but the subcircuits representing the oscillators will become unstable if either amplitude drops excessively since low amplitudes imply net negative resistance. A second order system is stable when the sum o f the diagonal elements is negative. Applying this criterion to each o f the oscillator subcircuits in the variational system gives conditions for stability in the uncoupled case but which w e assume hold approximately in the general case: A (63) a n d A 2 > -j= r This means that if either amplitude drops below the system will become unstable. This approximate stability condition is surprisingly accurate for most values o f coupling strength and bandwidth, and becomes inaccurate only when these parameters both become quite large. Even in this case, however, the dimensions o f the locking region given below are fairly accurate. For strongly coupled oscillators the boundaries o f the locking region can be approximated as the locus o f points where either oscillator amplitude is jfe . Along the line o f equally spaced tunings the value o f Am0 that causes the amplitudes to assume this value can easily be found from equation (52) and is H = A6?o L = 2V ^ W 1 - ^ : for JLo > i (64) One important consequence is that for a coupling strength /t0 = y the locking region height is maximized while still allowing the phase difference A<f>to vary 180 degrees over the locking region. This is important for beam scanning systems where the designer wishes to maximize the total phase variation and the locking range simultaneously. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If the coupling strength is sufficiently strong the width o f the region will also be determined by the amplitude criterion o f equation (63). To estimate the rate o f decrease o f the amplitude away from the line o f equally spaced tunings we will again resort to a perturbation analysis which is contained in the appendix 2.1. The results show that if we move from this line in the direction o f increasing Aeoa that the amplitude o f oscillator I will diminish according to AX= A 2 _____ A<o A(o 4e>aa 0C(Ao- l+ 2 A 2) (65) where A is given by (52). We will assume strong coupling, » 2 A Z—1, and simplify the denominator. To find the width o f the locking region w e will evaluate the amplitude at Aa>0 = jA & 0\mK and find the value o f A a 0 that gives A = -fe. First, however, we must determine A<oe. Using equation (52) for the amplitude for odd tunings and noting that for large coupling strengths we can expand the square root, Acoc from (54) is approximately 1+ (66> % ----------- 2— 8wA**ocoawac <o This shows that for A(o0 « 2 ^ 2 X 0eoacoac the steady state frequency remains halfway between the tw o oscillators, as in the case o f weak broadband coupling, but for A<o0 » 2*j2A0coa6)ac the steady state frequency follows the resonant frequency o f the coupling network. Using the above result at the maximum value o f A(0 o given by equation (64), and assuming Aa » l , we can say that the boundary for weak vs. narrowband coupling is at co^ = \(o a. 2.2.4.2.1 Broadband Case: co^ » \co a In this section we will assume that co^ » j a g so that Acoc « A<o0 . Using this result in equation (65) for the amplitude o f oscillator I and setting the amplitude to y% , we find that the width o f the locking region is 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Strong Coupling, X0» l / 2 Broadband, coic»X.0coi V Strong Coupling, \ » l / 2 ~ Narrowband, ©ac«A,#a)4 (b) Figure 11-D im ensions o f the locking region for strongly coupled oscillators, (a) For broadband coupling the region is large but increases as -y /X j. (b) The narrowband case shows large region width as oscillator tunings are moved apart, but remains narrow when oscillators are tuned within the coupling circuit passband. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (67) The lockmg region for this case, shown in figure 11(a), looks similar to the case o f weak and broadband coupling, but the height grows more slowly with increasing coupling strength Xa and the width is no longer constant with Xa 2 .2 A .2 .2 Narrowband Case: (O^ « \ c o a We now have A ac « Acoc a <o0 and using this result in equation (65) Aa>0 and setting the amplitude to y ^ , the width o f the locking region is (68) The locking region for this case is shown in figure 11(b) where we can see that the region gets slightly wider as we move along the line o f equally spaced tunings. The reason for this behavior is that as the oscillators are tuned far apart they influence the steady state frequency less. Thus the frequency can follow the coupling circuit bandwidth and strong coupling is maintained over a wide range. For very low loss coupling networks the locking region can extend quite far out. 2.2.5 Computer Simulations To verify the accuracy o f the above expressions for the height and width o f the locking region, MathCAD was used to obtain solutions to equations (46) and to compute the eigenvalues o f the variational system (50) for various circuit parameters. In addition, the nonlinear differential equations (44) were also numerically integrated to verify that the steady states and eigenvalues for a particular set o f parameters were correct. The coupling circuit resonant frequency and oscillator bandwidths were kept constant at co^ =10 and coa = 0.1. The height H and width W were computed for three different values o f coupling bandwidth, eo^. = 0.005,0.05,0.5 as functions o f the coupling strength Xa, and the simulation results and the results calculated from the approximate expressions are shown in figures 12,13, and 14. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X -p A bo •r-l © X a o •1 bo © bo a •p* © 0 I 0.7 “ ©«=10 ©a=0.1 0.6 - ©ar=0-005 _ 0.5 0.4 _ Weak Coupling Strong / Coupling j f 9/ 0.3 - / * *§ 9 9* /9 #9 Approximate 0.2 - s* 0.1 II -n Simulation —™T__ 0.01 —r 1 1 i 10 ^ K , log scale Weak Coupling Strong Coupling Approximate Simulation K , log scale Figure 12-Comparison of approximate formulas to computer simulations for "high" Q coupling circuit 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. > w -p 0.7 A bo 0.6 © oc= 10 GJfO.l ©apO.05 0 0.5 cs o »H bo © « bo •sN © o Weak Coupling 0.4 Strong Coupling / / 0.3 , / r\ Approximate Simulation 0.2 “ 0.1 A , 10 0.01 K , log scale 0.5 o«=10 fflj=0.1 0.4 o 0.3 0) « 0.2 Simulation ©*=0.05 Strong Coupling / . Weak Coupling Approximate ES •iH 0.1 M o o J 0 0.01 1 0.1 10 log scale Figure 13—Comparison of approximate formulas to computer simulations for "moderate" Q coupling circuit 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H eight, H Region L ocking 0. 8 - 0 . 6 -- Approximate 0.4-■ 0.2-- Weak Coupling 0.01 Locking R egion W idth, W K , log scale Approximate Weak Coupling K , log scale Figure 14-Comparison of approximate formulas to computer simulations for "low" Q coupling circuit 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix 2.1—Reducing the Order o f a Stability Matrix The following analysis shows that an N x N real stability matrix C o f rank N -l can be reduced to an N - l x N - l matrix C' that has the same eigenvalues as C but with the zero eigenvalue removed (as long as the eigenvalues are distinct). We begin by expressing the N phases in terms o f a set o f N -l independent "baas" phase quantities, for example the N -l phase differences s 0on+1 - ^gn. The baas phases <ff are related to the phases ^ through a linear transformation <ff = A<j>, where A is a real N - l x N matrix (written A e f t ”' 1*”) o f rank N -l. Next, express the derivatives in (7) in terms o f the new phases: C = d&« = d$n W ap2af>m $n-\ (69) d#N_xaf>m OJJ The derivatives o f the phase variables are given by —*-*- = Aba, so the stability matrix can be represented using matrix notation as C = DA, Z )eK M - ', A e f t ”-1*" (70) where the matrix D is D- = l r (71) Denoting an eigenvalue o f C as X and the corresponding eigenvector as x, the eigenvalue equation for the stability matrix C is Cx = DAx = t e (72) I f we transform the eigenvector using the same transformation matrix, y = A x, we have D Ax = D y = 2x (73) —» A D y= AAx = Ay Thus y is an eigenvector o f the matrix C ' = AD, C 'e f t ”-1*”-1 (74) 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the same eigenvalue X. This must hold for all o f the eigenvalues and corresponding eigenvectors o f C, including 2 = 0. For this latter case, however, we must have A x = 0 for the corresponding eigenvector. Thus, the new stability matrix C is o f dimension N -l and has the same eigenvalues with the zero eigenvalue removed. Moreover, its eigenvectors are the eigenvectors o f C transformed by the m atrix^. Appendix 2.2—Amplitudes and Acoc Near Line of Equally Spaced Frequencies In this section we derive approximate expressions for Aa>c and the amplitudes Al and A2 that are valid near the line o f equally spaced tunings as we tune perpendicularly away from that line, as indicated by the small arrow in figure 8. The first and third o f equations (46) relate the oscillator amplitudes to various quantities and are repeated here with the coupling phase expanded: -A i)A l = A^ co s(A $ + -^ £-sin(A$l CO (75) ( l - X 0£ -A%)A2 = - A 0sA 1 cos(A $ —^-S-shn s in (A $ j Subtracting the second fourth o f equations (46) we arrive at an expression for the difference between oscillator tunings Aco0 = A coo2 - A eool = T ' + T ' l sin(A^ ^ - - ^ f ^ - ^ l c o s C A ^ ) A2) ^ a c V -dl 2) s.^1 (76) and adding them gives an expression for Aeo( A a c = A(0o +Aae l eoa A6>c — - + (0nr. 1 — 2 A A . A A, (77) As we vary A(o0 an infinitesimal amount dA<o0 away from zero, many o f the quantities in the above equations will change. For example, Acoc is nominally zero, but after this perturbation it will have a nonzero value. The quantity s , 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. however, depends on the square o f Acoc (see equations (46)) so, to the first order, s will remain unity. The equations are perturbed by implicit differentiation and any unperturbed terms will be evaluated on the line Aa>0 = 0 and, as a result, some may vanish Along this line equations (75), (76), and (77) take on particularly am ple forms, and result in the amplitude expression o f equation (52) and the following relations for the phase difference: 2 Pi0(oa (78) cos(A0) = 1 — —(l - A 2) These relations help amplify the form o f the mathematics that follows. Implicitly differentiating the amplitude equations (75) gives ( 1 - ^ - 3 A2)dAx = - X c c& 2 cos(A <j)-A dA(f>- dAcoc sin(A^) (79) ( l - A 0 ~ 3 A 2)d42 = - A < dAx c o s (A $ -X a A ^ + — sin(A^) ac j where the unperturbed quantities have been evaluated along Aeo0 = 0. Adding and subtracting these equations and using the relations (78) gives d A z+ d A ^ -^ ^ d A # 2A A<0odAa)c cU L-dA, = ------= - 2 1 2A - I 2a>„<o. (80) Applying the same analysis technique to equation (76), which we m aintain at zero, gives dAto0 = 0 = Ao<0 ajd ^ + ^ |k jsin (A $ + 2 c o s (A $ d A ^ J (81) = Ao0)aj^ -(d 4 1+dA2) sin(A0) + 2 cos(A^)£/’A ^l 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using the first o f equations (80) the above expression becomes —sin(A ^)— cos( A A<o0 ) +dA2) = 0 (82) The first factor is not generally zero, so w e must have dA1JrdA2 = 0 and, from (80), at follows that dA<p-0. Using these results with the second o f equations (80), we can find the approximate expression for the amplitude o f oscillator II (which is the lesser), valid for small Aa>c, stated in equation (65): A a 0Amc 1 A ^ = A 1— 2A*+ A0 - 1 4 CDaCD". (83) The remaining task is to find an approximate expression for Acoc. Implicitly differentiating equation (77) and using the relations (78) and those resulting from (82), we find dAco, l - ^ 0 — (l-c o s (A $ ) = dAa)0 + -2 .0{oad co 1 A(o0dA(oc = dAo)0 - A 0coa sm (A $ -— I - - 2A + X „ - \ 2 a„co. 1 = dAcon =■ Ao)2° dAco, ° 2 A 2 + Z - \4 c o nco. (84) which, after rearrangement, gives equation (54) valid for small Aa>0 and A a c : A Aco„ =■ Acol cox (85) 2A - 1 + A C4 The above perturbation technique is not only useful for finding approximate expressions but it also helps us understand the manne r in which various quantities are affected by tuning variations. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [1] C. Balanis, Antenna Theory Analysis and Design. Wiley & Sons, NY, 1982. [2] J. J. Lynch, R. A. York, "An Analysis o f M ode Locked Arrays o f Automatic Level Control Oscillators," IE E E Trans, on Circuits en d Systems, vol. 41, Nol 12, pp. 859-865, Dec. 1994. [3] G. Strang, Linear Algebra and its Applications. Academic Press, 1980. [4] J. J. Lynch, R. A. York, "Synchronization o f Microwave Oscillators Coupled through Resonant Circuits," Submitted to TFF.F. Trans. Microwave Theory Tech. [5] J. W. Mink, "Quasi-Optical Power Combining o f Solid-State Millimeter-Wave Sources," IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986. [6] R. A. York, "Nonlinear Analysis o f Phase Relationships in Quasi-Optical Oscillator Arrays," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1799-1809. [7] J. J. Lynch, R A. York, "Oscillator Array Dynamics with Frequency Dependent Coupling Networks," submitted to IEEE Trans. Microwave Theory Tech. [8] G. Strang, Linear Algebra and its Applications. Academic Press, 1980. [9] P. Liao, R. A. York, "A N ew Phase-Shifterless Beam-Scanning Technique using Arrays o f Coupled Oscillators," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1810-1815. [10] J. Lin, S. T. Chew, T. Itoh, "A Unilateral Injection-locking Type Active Phased Array for Beam Scanning," IEEEM TT-S, International Symposium Digest, San Diego, June 1994, pp. 1231-1234 [11] P. S. Hall, I. L. M orrow, P. M Haskins, J. S. Dahele, "Phase Control in Injection Locked Microstrip Active Antennas," IEEEM TT-S, International Symposium Digest, San Diego, June 1994, pp. 1227-1230. [12] R. A. York, P. Liao, J. J. Lynch, "Oscillator Array Dynamics with Broadband N -port Coupling Networks," IE E E Trans. Microwave Theory Tech., vol. MTT42, pp. 2040-2045.. [13] M Kuramhsu, F. Takase, "Analytical Method for Multimode Oscillators Using the Averaged Potential," Elect, and Comm, in Japan, vol. 66-A, No. 4, 1983. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Mode Locked Arrays Mode locking is commonly utilized in laser systems for producing pulses o f electromagnetic energy,[1] and the same phenomenon has been demonstrated in systems o f coupled microwave oscillators. [2] The bade idea is to synthesize a frequency spectrum that is a periodic pulse train in the time dom ain Such a waveform produced by a four element array is depicted in figure 1. OS Time Figure 1—The ideal time domain mode locked waveform corresponding to the spectrum above. The high frequency carrier is not shown. If the phase distribution across the array is suboptimum the above pulse shape becomes distorted. The waveform is the envelope o f a high frequency microwave carrier, and is assumed to have a period many times longer than that o f the carrier. The magnitude spectrum o f this signal is shown in figure 2. To synthesize this spectrum the fundamental requirement is that the spectral components be evenly spaced. If the components drift even slightly, the sharp pulses shown in figure 1 become distorted. This is accomplished by a frequency locking condition analogous to synchronization. If the oscillators are tuned so that their free running (uncoupled) frequencies are nearly evenly spaced, under the proper conditions the coupled system will lock to a state where the steady state frequencies are exactly evenly spaced. However, even spectral spacing is not sufficient to provide sharp pulses. The relative phases between elements must also satisfy certain conditions that we will derive below. I f the phase distribution is suboptimum the peak pulse power will be diminished and the sideband power will increase. When the proper 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conditions are met the output signal, obtained by summing together the oscillator outputs, will appear as in figure 1. M o) Frequency Pulling ®1 <°2 °3 ®4 Figure 2—Magnitude spectrum of a four element mode locked array. The solid lines are the steady state frequencies and the dotted lines are the free running (uncoupled) frequencies. The two additional sidebands are generated through oscillator nonlinearities. Figure 3 shows a four element linear oscillator array with nearest neighbor coupling. Tuning Ports Qoi e>o2 COo3 8,0 A . 8,0 — CO { ^ > ' ©o3 8,0 — CD >f CD a Oscillator Outputs Figure 3—Four element linear array with nearest neighbor frequency independent coupling. Adjacent output frequencies differ by the beat frequency eob. It is identical to the synchronous array of section 2.1.1, but for this case we tune the oscillators outside the region o f synchronization and m aintain nearly equal spacing between free running frequencies. We will apply the analysis methods o f chapter 1 to determine the tuning required for mode locked states (i.e. locking diagram), and the tuning that gives us the desired phase distribution. In addition, we would like to understand the effects o f coupling magnitude and phase on the synchronization diagram. Before embarking we need to consider in more detail the time domain waveform produced by a mode locked array. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 The Time Domain Mode Locked Waveform The time domain output o f an N element mode locked array is obtained by summing the individual oscillator outputs. Each spectral component will, in general, have a slowly varying amplitude and phase. Thus the time domain waveform can be expressed as vto = 2 X W g o * to + {n -\)c o bt+<f>„{t)) = £ 4 ,(0 cos(0„(f)) n=I (1) n=I The frequency o f the first oscillator <o0 is defined arbitrarily as the carrier and the frequency separation, o r beat frequency, is denoted a>b. The element frequencies are assumed to increase from one end o f the array to the other, that is, referring to figure 3 we have cox < a>2 < ---< a>N. The waveform in equation (1) can be written as a high frequency earner with slowly varying amplitude and phase modulation: v(0 = ^(/)cos(<yo/+0(r)) (2) where the envelope, which is the mode locked waveform shown in figure 1, is N (3) K(<) = n=l The time dependent phase modulation in (2) is not considered here since it does not effect the envelope waveform. The peak amplitude o f the envelope will be maximized when the terms in the sum o f (3) add coherently, and this occurs when the phases in equation (1) satisfy M 0 „(/) ^ - A0n = (0n+2 - 0n+x) - ( 0 n+l -0 „ ) = O (4) for n = \ 2 , — , N - 2 . The second difference o f the phases, as defined above, will prove to be an important quantity for mode locked arrays. Using the values for the instantaneous phases from equation (1) the second differences o f the phases become AAOn(t) = AA<f>n{t). Thus condition (4) requires a constant phase difference A(j>n = ^n+1 - <j>n to exist between adjacent elements (for a discussion o f these phase definitions see section 1.2.1.1). Since the frequencies shown in (1) are the true steady state frequencies the second differences o f the phases are periodic functions o f time. For weakly coupled arrays the periodic variation will 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be small and the AA^„ will be nearly constant As the periodic variation grows the pulse shape will change, but the dominant contribution for practical arrays will be the time average value o f the phases. Under certain conditions the time dependent amplitudes can enhance the peak o f the envelope and reduce the sidelobes. I£ for example, the amplitudes and phase differences between adjacent elements are all identical, A(t) can be pulled outside the summation in equation (3) and the envelope can be summed in closed form: K The factor multiplying the amplitude A{i) attains a maximum value o f NA every T = 2/ 0t seconds, and it is this function that is plotted in figure 1 for N=4. I f the time dependent amplitude A(t) is maximum during these times then the peak value o f the envelope will be enhanced.[3] W e will show later that this is indeed the case for a certain class o f mode locked oscillators using Automatic Level Control (ALC) oscillators. 3.2 Linear Arrays of Van der Pol Oscillators [4] In this section we will analyze a linear array o f N identical (except for tunings) nearly sinusoidal Van der Pol oscillators, each coupled to its nearest neighbors through frequency independent coupling parameters. This array is identical to that analyzed in section 2.1.1 for the synchronized case; however, we will now assume nearly evenly spaced tunings and look for mode locked states. The dynamic equations for the amplitudes and phases are (see appendix 3.0 for the derivation): K = T0 ~ A n2)A n + e [ A ^ co s(r+ <f>n - +<D)+An+Xco s(r+ & +1-<f>n- <&)] s i n ( r + ^ - ^ 1+<D)+ s in ( r + ^ ,- ^ - < I > ) ] (6) where t| and s are the normalized nonlinearity and coupling parameters, Pn = ~ ^ normalized frequency pulling, t= <obt is the normalized time parameter, and the dot denotes differentiation with respect to x. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. With the dynamic equations (6) in a relatively am ple form, we can derive the existence o f stable mode locked states using the methods o f chapter 1. The frequency pullings in terms o f the time average phases are derived in appendix 1. The results, in matrix form are /?= fi2 — t - 3 (F u + Gv)+ e 'k l+ (2 7 ) where F ^ NxN~2 and G 0 a are defined as ' 1 cos(2 d>) F= y2 (7) 0 > rj -s in ( 2 d>) **. T\ -?7 (8) -s in ( 2 d)) j I K , G= O ,o y cos(2 <l>) and the N element vector k is defined as -c o s 2( 0 ) + ^ s i n 2(<I>)+^-sin(2<I>) p s m ( 24>) k= (9) p-sin(24>) cos’ C®) - p - s i n J(C>) + p -sin (2 ® ) where r 2 = l+ (277)\ The vectors u and v are defined analogously to the synchronous case: f sin(AA^,) ' f cos(AA$) > u- sin(AA02) cos(AA^2) , v= ( 10) ^cosCm ^ ^ where AA^„ s - 2 ^ m.1+ <f>n. Equation (7) comprises a system o f N equations in N unknowns, the unknowns bang the N -2 phases AA0n, the beat frequency, and one o f the steady state frequencies. These variables completely describe all o f the frequencies and phases o f the periodic state, except for two arbitrary phases that 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. arise from the periodicity o f both the carrier and the envelope. Unlike the synchronous case, for mode locked arrays the frequency pullings depend on the coupling parameter s squared. Thus we expect a significantly smaller locking region. Since only the second differences o f the phases appear in the equations, these N-2 linearly independent quantities represent a natural choice for the phase variables. Thus we will use the method o f section 2.1 to reduce the order o f the system. This will remove two zero eigenvalues from the stability matrix, while preserving the remaining eigenvalues. The second difference o f the phases are related to the phases themselves through the linear transformation AA(f>=D<f>y defined by f AA£ ' f AA& 1 -2 'A ' 1 A — , a a ^ _ 2> 0 1 -2 (ii) 1 V A ; Applying the second difference transformation D e 9 ?w-2xW to the pulling variables in equation (7) gives the reduced system AAfi= D fi= gL— (A u + B v )+ s LD k (12) where A = D F e $ llf~2*N~2 and B = D G ^ - 2xN~2. N ote that the second difference o f frequency pullings A A fi contains no steady state frequencies nor the beat frequency; it depends only on the oscillator tunings. Thus equation (12) is a system o f N-2 equations in N-2 unknowns. The system is quite sim ilar to the corresponding system o f section 2 . 1.1 for the synchronous case, with tw o significant differences. The first is the additional factor 2%z that depends on the nonlinearity parameter q. This term tends to zero as q tends to zero and infinity, and is maximum for q= l/2. Thus we expect the locking region to maximized for this value o f nonlinearity. Second, there is a phase independent term £~Dk that shifts the center o f the locking region away from AA^=0. This implies that the tunings that place us in the center o f the existence region are not quite equally spaced. The perturbational system for the array is derived immediately from ( 12 ) using the methods o f chapter 1 : 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AAS= -CAA<5 (13) The second differences o f the phase perturbations 5 are derived using the second difference transformation AAS=DS, and the stability matrix is C = ^{A V -B U ) (14) The diagonal matrices U and V are defined as in section 2.1.1: u= (15) \ 0 v= (16) v 0 c o s(M ^ _ 2)y Unlike the synchronous case there is no value o f coupling phase that eliminates the matrix B and simplifies the equations. Deriving the locking diagram is quite complicated and w e will immediately consider simple cases o f three and four element arrays. 3.2.1 Three Element Array For the three element array the one dimensional matrices A and B are A = 1—2 cos(2<t>) and 5 = 2 sin(2 0 ), and the "vector" D k is D k = —^ ~ sin(20). The angle pulling equation is, therefore, AA/?, = £2 Y r[(l-2cos(2<t>))sin(A A $)+2an(2O )cos(A A 0 1)-sin(2O )] (17) which, with the help o f a trig identity, can be written 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (18) Allowing AA^ to span all o f its possible values creates the existence region on the AA/?, line. The length o f the region is - M A L = 2 ^ |? V 5 - 4 co<2«») (19) which is maximized for <X>= %, and 77= 34. These are optimum values o f coupling phase and nonlinearity. The stability matrix C has only one element “ 4 cos(2 0 ) cos AA$ + tan_! Qj = e 2 ['_ 2 s in (2 ® L 'j'| l - 2 cos(2 <I>)JJ (> and is positive when the argument o f the cosine lies between -n/2 and iz/2. This condition ensures that a single stable mode exists for a given set o f free running frequencies since over the range o f phases that satisfy (19) the sine function in equation (18) is single valued. 3.2.2 Four Element Array Since the results o f the last section showed that the locking region is largest when d> = we will assume this value to simplify the problem. The pulling equations are fA A /n in f 3 U a /? J r2 -2 ' - 2 \ f sin(A A ^) W 3 ^sin(AA^2)J 0 2*7 -2rf cos(AA^1) > cosCAA^) 1 ' -c o s 2 (<I>)+ -^ 2 sin 2(C>)—^ -s in ( 2 0 ) (21) cos2 ( 0 >) - - 4 -sin 2 (<I») —-§-sin( 2 <J>) * A < Changing variables puts the equations in a more symmetric form. Defining the "sum" and "difference" phase and frequency variables according to 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the equations become Pt = At si n ( ^ ) c o s ( ^ - ^ ) Pd = - f i + A d cos(& )sin(& + 9,) (23) where As = j l , A d = ^ 2 5 H 2 v ) \ (24) ^ =l ( tan_I(277)+tan f P ) } ^ =^[tan"1(27^"tan"1(p )] With the pulling equations in this form one can see by (thoughtful) inspection that the minimum and maximum values o f the pulling variables occur on the P ,,P d axes. The coefficients o f the stability matrix can still be found by differentiating P, and f$d with respect to and <f>d, since we have amply applied an additional linear transformation to the phase and pulling variables, an operation that leaves the eigenvalues o f the stability matrix unchanged. The (modified) stability matrix is C = { ^ cos( ^ ) cos( ^ “ ff.) sin(& ) sin(& + 9,) - 4 s m ( & ) s in ( & - 0 j Ad c o s(^ ) cos(& + 0 j (25) and the conditions for stability can be determined by the Routh-Hurwitz criteria, winch, for the 2 x 2 case, require the trace and the determinant o f C to be positive. After some manipulation these criteria reduce to cos(& )cos(& + 0d) >O cos(2 ^<f)+ co s( 2 ^t)cos( 2 ^ ) > 0 (26) 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We now apply these conditions to find the regions in the phase plane where stable states exist. The first condition divides the phase plane into square regions of dimension it as shown by the dotted lines in figure 4. (c) (cD Figure 4—Phase plane regions o f stable solutions (shaded areas) for four values of nonlinearity param eter q . M apping these regions onto the frequency pulling planes gives the regions o f stable m ode locking. The second condition creates curved lines that further divide the phase plane. Putting the two together gives the regions o f stable solutions in the phase plane and are shown in figures 4(a) through (d) for four different values of T |. Figure 5 shows the corresponding boundaries in the plane of frequency pullings 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a AAp2 AAp2 AA(3i AA(3i TJ=0.1 I (b) (a) AA(3i AA|3i ti= 4 V Region of multiple stable states (c) V (d) Figure 5-R eg io n o f stable mode locked states in the AAj3 plane for four values o f T|. In plot (d) a subregion exists where two stable states exist for each set o f tunings. For T| < V5 / 2 the mapping from the stability region in the phase plane to the frequency plane is one to one so that only one stable mode exists for each set of free running frequencies that lie in this region. For T[>4 s /2 the inner curve shown in figure 5(d) bounds the region where two stable modes exist. These 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. results were obtained graphically by observing the mapping from the phase to frequency planes and noting the conditions when mapped lines cross one another. The locking region in the plane o f free running frequencies can be obtained using a linear transformation, as in section 2.1.1.1.1. When tuning the array we will leave the end elements fixed and vary only the center elements so that the beat frequency and spectral location remain essentially constant The vector o f central free running frequencies and AA/? are related through the linear transformation <DbM .0= H < ooe + a>ot (27) where the N - 2 x N - 2 matrix H and the N-2 element vectors are defined as 1 1 -2 « • COG WO 0*3 — • • ,0 (T • • 1 1 /----- H= -2 a* c r * 0 -2 , <a cNJ In what follows we will work with the vector AA/?. The central free running frequencies can be obtained using (28). Figure 6 shows the stability region in the plane o f central free running frequencies (keeping the end elements fixed). The origin contains the values o f eoo2 and a>o3 that result in evenly spaced free running frequencies. It is interesting to note that, unlike the N=3 case, as 77—> 00 the existence region approaches a constant area. However, near this limit the real parts o f the eigenvalues o f the averaged system are almost zero, indicating that the system is highly underdamped. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ~ COo3 * COo3 A A < j)i^ = 0 C0o2 C0o2 r|=0.5 T|=0.1 C0o3 C0o3 Region o f multiple stable states ----- . COo2 T l= l COo2 rj=4 Figure 6—Locking regions in the plane o f oscillator tunings. The origin corresponds to evenly spaced free running frequencies. In each plot the existence region is oval and the irregularities (points, flat spots, etc.) are due to stability conditions. As the nonlinearity parameter gets large multiple stable states can exist for a given set o f tunings. 3.3 Mode Locked Arrays using Automatic Level Control Oscillators [5] Block diagrams of conventional and automatic level control (ALC) feedback oscillators are shown in figure 7. The resonator in both cases is a second order bandpass filter with center frequency co„ and quality factor Q. In the conventional oscillator the gain saturates as the signal v(t) increases. In the ALC oscillator the gain also depends on v(t), but in a different way. The gain control block is an envelope detector followed by a first order filter. Thus the gain 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Resonator v«(t) - * ( ? } - * Resonator » v(t) V«(t) —> ( T ) - > Gain H(«) » v (t) Gain 6(g) G(v) Nonlinear Gain Control Filter (a) Hi(o) (b) Figure 7—(a) A conventional instantaneous gain response oscillator and (b) an Automatic Level Control (ALC) delayed gain response oscillator. The nonlinear gain control filter is an envelope detector followed by a linear lowpass filter; the bandwidth adjusts the amount of damping in the amplitude perturbation response. depends on the filtered envelope o f the high frequency carrier. equations for the variables shown in figure 7(b) are v +(°>o +2a>ai ) v + 2 o a(l - Ga +g)v = 2 a avs £+<ogg=<ogf ( v ) The dynamic (29) where coa — a>0l 2 Q is the half bandwidth o f the filter, Ga is the maximum loop gain (at g=0 ), a>g is the bandwidth o f the gain control lowpass filter, and / (v) is the nonlinear envelope detection function. Assuming the resonator has a fairly high quality factor (>2 0 ) and the input signal is small, which will be true for weakly coupled microwave array elements, the dynamic variables will be nearly sinusoidal with slowly varying amplitudes and phases. Thus we can represent the output and input signals as v(/) = A (t) cos[ry/ + fct)] v,(0 =4(0«>s[®, * +&(03 (30) Both signals are assumed to be almost periodic and the frequencies are chosen so that the phase $(t) is a periodic function o f time. After some manipulating and 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applying appropriate approximations (shown in appendix 3.2) the dynamic equations for the amplitude, gain control, and phase are A = o)a[ ( l - g ) A + As cos[(<os - eo)t + 1] g = - (o gg+a>gA (31) * co A &= a>0 - m +— g + <oa -f-sm [(m , - <o)t + & - f t CO A To simplify (31) the maximum gain was chosen to be Ga = 2 and the proportionality constant between the amplitude and the gain control variable was set to unity, this form can always be achieved by proper scaling. We will now amplitude response. equations (31) is g = second order system. system show that the ALC oscillator can have an underdamped In the absence o f an injected signal the fixed point o f A = 1, co=co0 and the amplitude and gain control form a Linearizing about the fixed point gives the pertuibational where a and y are the perturbations to the amplitude and gain control respectively. The system has eigenvalues X = ~yi cos ±j^j(Oa<og - ( k ® , ) 2 , thus the amplitude response to perturbations can be made underdamped and we can control the damping by adjusting the bandwidth o f the gain control filter. ALC Van derPol Time Figure 8—Amplitude (carrier envelope) response at tum-on of a Van der Pol oscillator and ALC oscillator. For use in an array the natural frequency of the ALC response is set equal to the frequency of the mode locked pulse train 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To illustrate the above analysis the envelope o f the output waveform was simulated and plotted in figure 8 for both the Van der Pol oscillator, which has a damped amplitude response, and the ALC oscillator. The ringing in the ALC response is advantageous in mode locked arrays. As the following analysis will show, by tuning the natural frequency o f the amplitude oscillation to the spectral spacing between elements the locking region will be proportional to the size o f the resonant amplitude variations, and can be made fairly large by adjusting the coupling strength and gain control bandwidth. Using this method tire ALC locking region can be made considerably larger than the Van der Pol region. 3.3.1 Linear Arrays with Nearest Neighbor Coupling We now consider mode locked arrays o f ALC oscillators. The analysis is similar to that for mode locked arrays o f Van der Pol oscillators in the previous section, so many o f the details included there will be omitted from this section. The elements are arranged exactly as they were in the previous section for Van der Pol arrays, and all o f the assumptions made there apply to this problem. After normalizing the parameters to the beat frequency (as in the last section), the dynamic equations are A = *7(1 - & )A + *[A-i ° ° < T+ A - 4>n-1 + <*)) + A +i c o s(r+ <D)] & =--& + -A 7 7 (33) 4>n=Pn~£ ^ s i n (r + <f>n - <f>n_x + 4 > ) - ^ - s in ( t + &+I -& -< & ) .A A where n - 1,2,---,JV, and expression containing variables with other subscripts should be set to zero. The gain control bandwidth was chosen to make the amplitude resonant frequency and the array beat frequency identical. Applying our analysis methods, the frequency pulling equations are P = a2 j-^ [F + 4 cos2 ( 0 )G]« - 7 sin(2 <D)Gv+cos(2 <I>)£l + -js in (2 0 )£ 2 j (34) The matrices F and G and the N element vectors k l and k2 are defined as 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -2 **. 1 **. O 0 0] ( 1 1 , G= 0 1 0 •*. -2 r- r rr 0 0 , kl = , k2 = 0 1 0 0 1 0 ,0 , and T 2 = 1+ (2 J7)2. The vectors u and v are defined exactly as in equation (10). W e then reduce the order o f the system using the transformation matrix D o f equation (11) (which happens to be F T) to form the second differences o f the pulling equations. The result is AA/?= D/J= £ ? |—[D F + 4 co s 2(0 )D G ]« - 77sin(2 <&)DGv+cos(2 0 )D k l 1 . + ^sin (2 ® )D k2 I (36) winch is a system containing N-2 equations in N-2 unknowns. 33.1.1 The Locking Region Equation (35) has a form similar to equation (12): M 0 = i? 2 (A u + B v + k ) (37) which, as explained in section 3.2, defines a mapping from the N-2 dimensional space o f phases (AA$ to the N-2 dimensional space o f frequency pullings (AA/?) I f each phase is allowed to span all o f its possible values (it is sufficient to consider the interval -7t to tc since the nonlinear transformation involves periodic functions) the resulting frequency pullings span the region o f existence o f mode locked states. Any vector AA/? lying in this region has a corresponding phase vector AA^. We will consider the size and location o f the existence region in more detail. The center o f the region is found by averaging the pulling equations over all possible values o f AA^. Denoting the center by AA/?Cwe have AA/?C= J AA#AA 0 d&A<f>= f? ^ -k Vol (38) ^ 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the integral extends over the N-2 dimensional volume o f the phase space. Thus the constant vector in equation (37) is the center o f the existence region. The shape o f the region is generally quite complicated, as we saw for the case o f Van der Pol arrays, but we can estimate the size by computing the mean length o f the vector AA/?, measured with respect to the center o f the region, over all possible values o f phase. Thus, using properties o f circular functions we have, 1/2 A A /W = f( M l- A A A )T(A A ff-A A & )dAA# x™ 2 X 2 x ) VoI (39) Carrying out the algebra gives, for N > 7, iV—4 8/1 9 N -2 16# ~ 5 /2 cos(2 <I>) N -2 (40) For N <7 expressions must be derived for each value o fN separately. Comparing this to the simple case N=3 for which M A ms = ** f [20 - 16cos(2<D)]1/2 (41) we can see that increasing the size o f the array does not significantly affect the overall size o f the existence region, at least when measured this way. However, we will see later that the region becomes highly eccentric as N increases (we will show this only for <C>=7c/2 ). Equation (40) shows that the region is maximized for a coupling phase o f nI2 or 3x/2, similar to the Van der Pol case. However, unlike the Van der Pol case the region above depends linearly on tj, which allows us to enhance the mode locking region by increasing this parameter. Since &=tz/2 maximizes the locking region, we will assume this value o f coupling phase to simplify the mathematics. Equation (37) becomes AA/?= fi2 + k) (42) 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which is the simple form we obtained for the synchronous array o f oscillators in section 2.1.1.1. Thus the locking region is similar to the synchronization region of that section, and all of the subsequent characteristics apply to this case. Figure 9 shows a locking diagram for a four element mode array. AApi Figure 9—Locking diagram for a four element ALC array. The region has the same shape as the three element synchronous array analyzed in section 2.1.1. but is more eccentric. The arrows show the directions o f the eigenvectors with lengths proportional to the eigenvalues o f the stability matrix. As before, the region of stable mode locked states fills the entire existence region and, in this respect, the ALC oscillator array represents an optimum design. The ability of an array to mode lock depends on the number of elements, N. In section 3.3.1 we estimated the size of the existence region by computing the RMS value of the frequency pulling vector length over all of its possible values. A different, and possibly more direct approach, is to compute the total volume contained within the region. To compare sizes in different dimensional spaces, we will compute the determinant o f A, which is the volume of the locking region in frequency pulling space, and raise this to the l/(N-2) pow er L = [d e t(,4 )f -2 (43) The resulting number L is the length o f a cube in N-2 dimensional space that has the same volume as the stability region and provides a convenient figure of merit for comparison of region sizes of different dimensions. Other important parameters are the minimum and maximum eigenvalues since these show the eccentricity of the region. These three parameters are plotted in figure 10 as functions of the number of elements. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of Elements Figure 10—Locking region size (L), m axim um and m inim um eigenvalues (scaled by fi2 rf) as functions of the number of array elements. As N gets large the smallest eigenvalue diminishes quickly. This implies that large mode locked arrays will be difficult to tune since there is a large phase sensitivity for certain tim ing errors. One can see that as N increases the size o f the region slowly decreases and, more importantly, the smallest eigenvalue quickly approaches zero. This implies that large mode locked arrays will be difficult to tune since small tuning errors in the direction o f the eigenvector corresponding to this eigenvalue cause large phase variations and possibly loss o f lock. This is also true for the synchronous arrays o f section 2 . 1 . 1 . 1, but the mode locked regions here are more eccentric. 3.3.1.2 Pulse Power Enhancement In this section we will show how to enhance the peak amplitude o f the mode locked waveform by utilizing the periodic amplitude variations. Section 3.1 describes the requirements for such enhancement, namely, that the amplitude and phase variations between adjacent elements must be identical, and at the time the peak occurs the amplitudes must be maximum. To determine how w e can achieve these conditions we must look closely at the amplitude variations. Appendix 3.3 contains the first order approximation o f the time dependent amplitudes and phases: 4 , (/) = 1 + s^/l + T72 [sin( r+ - <J>+ tan "1 77)+ sin (r+ A ^ n_, +<J>+tan_ 1 77)] (44) = 0on~ t+ ® )'“ r + W n- 1 + ®)] 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for n = 2,3,—, N - 1 . The end element terms are slightly different and will affect the results o f the following analysis, but for moderate sized arrays this effect will be small and will be neglected here for simplicity. The ideal phase condition for mode locking is = ^on+1 - # on = A $ for all n. Assuming this condition is met the amplitudes and phases will be the same for each element. In addition, for ninety degrees o f coupling phase the amplitude perturbations are zero and the phase perturbations are maximum. Zero or 180 degrees coupling phase maximizes the amplitude perturbations and eliminates the phase perturbations. Peak power enhancement requires the latter condition, a we must use O = 0 so that the phase condition AA^= 0 represents a stable state. Assuming <X>= 0 the amplitude and phase equations become A„(r) = l+ 2 e J l+ 772 sin(r+ A 0+ tan- 1 77) (45) Since the phase o f the carrier envelope is {n-1)(<jobt + A $ (see section 3.1) coherent phase addition, and hence peak power, occurs at (normalized) time r = -A<f>. At this time the amplitudes are A n = 1+ 2 stj, so the peak o f the mode locked pulse will be increased proportionally to the nonlinearity parameter. The peak amplitude does not occur precisely at the time o f coherent phase addition, but quite close for large values o f 7 . Figure 11 shows plots o f mode locked waveforms for unity amplitude oscillators and for enhanced ALC oscillators. The main drawback to designing with 0 = 0 is that the locking region is smaller than the corresponding region for <t>= However, the 0 = 0 region for large arrays is less eccentric. The analysis for this case will not be presented here, but figure 12 shows a comparison o f the size o f the locking region, using the equation (43). Although the region for O = ?£ is significantly larger for N=3, it decreases fairly rapidly with N while for 0 = 0 the region rem ains essentially constant; for large N the regions are comparable. Figure 13 shows the maximum phase sensitivity for the two values o f coupling phase. Since this sensitivity is directly related to the region size in a particular direction, it also represents the eccentricity o f the region. For O = %. the sensitivity increases with N indicating, as mentioned earlier, that the region becomes highly eccentric. However, for O = 0 the sensitivity remains fairly constant. Thus, 0 = 0 represents a viable choice for mode locked ALC arrays. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z 0 2z z Figure 11—Pulse enhancement using the time varying amplitudes of the ALC oscillators. Waveforms from a six element array are shown for unity amplitudes and for amplitudes given by equation (45). The parameters are e=0.2, r|=2, <X>=0. j <D 3 .2 25 CD s 2 0=0 o 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of elements, N Figure 12-Locking region size, L, for two values of coupling phase, as a function of the number of array elements. 12 10 8 a cos 6 4 0=0 2 0 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of elements, N Figure 13—Maximum sensitivity as a function of the number of elements for two values of coupling phase. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.1.3 Experimental Verification A three dem ent mode locked array o f ALC oscillators was designed, built, and tested to verify the basic results o f the above analysis. The center frequency o f the system was chosen to be SO KHz and the resonators were second order state variable filters with oscillator quality factors o f 25. The voltage controlled amplifier necessary for level control was realized using the RCA CA3080 operational transconductance amplifier. The peak detector circuit consists o f a full wave rectifier, a diode, and RC filter. The complete circuit is shown in figure 14. First, we verified the amplitude response o f a single ALC oscillator to an injected signal. The magnitude o f the amplitude variations is a function o f the beat frequency and can be calculated using the variational system, at least for low level injection. The measured response and the response calculated from the circuit parameters are plotted in figure IS. The discrepancy is caused by component tolerances. Next, three oscillators were coupled together through 90 degrees o f coupling phase and tuned to make the beat frequency equal to the amplitude resonant frequency. The outputs were summed together through a resistor network (with an attenuation factor o f four) to form the mode locked waveform. Figures 16a-c show the measured time domain waveform and the calculated waveform using the first order amplitude and phase approximations for three different relative phases. Using the results o f section m e we calculated the locking region size for the center oscillator free running frequency o f 54 Hz, and the measured value was 55-75 Hz. Excessive thermal drift and high phase sensitivity near the edge o f the locking region prevented precise measurements o f the relative phase vs. free running frequencies. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Resonator Coupling Circuit 20k 1000 pF 220k 20k 470k 1000 pF 20k 1000 pF Vin 22k 470k 10k CA3080 Voltage Controlled Amplifier V out 100 22k 56% 33k 10k 33k 10k Envelope Detector Op-amps: LM347 Diodes: 1N4148 Figure 14—Schematic of a single ALC oscillator. The resonator is a state variable bandpass filter, the voltage controlled amplifier uses an RC308O transcnndncfancp. amplifier, and the envelope detector is a full wave rectifier and filter. The input coupling circuit provides 90 degrees of phase shift 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theoretical S (US > Measured 0 1 : 3 4 Beat Frequency (KHz) Figure 15—Response of the magnitude of the amplitude variations to as injected signal A beat frequency o f zero Hz corresponds to the edge of the fundamental locking region. The beat frequency of the mode locked array will be set equal to the resonant peak of this response. This response can be used to determine the parameters e and t) experimentally. The theoretical plot is based on calculations using the element values shown in figure 14, and the discrepancy is mainly due to component tolerances. Theoretical Theoretical Measured ^ Measured AAij) — 121rad 1 2 3 4 5 mS (c) Figure 16a-c—The measured and theoretical mode locked waveforms resulting from summing (and dividing by four) the three oscillator outputs for three different phase distributions. The theoretical curve shows only the carrier envelope whereas the data shows random samplings of the carrier. The main discrepancy is the peak amplitude which is accounted for by component tolerances. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix 3.1—Normalized Form of Mode Locking Equations The dynamic equations o f section 2.1 can be used, with the parameters slightly altered: A ' r t U k + - * 2 X <w (a « = n-l mxn t. +<i>) „ w 4 » ib = b -1 The parameters p and A. are the nonlinearity and coupling parameters; we reserve r) and e for the normalized versions. The equations above are written with the phases defined relative to a common frequency co. For a mode locked array, however, the steady state frequencies will all be different Thus, to maintain periodic phases <j>„ we must redefine the phases with respect to the true frequencies con. The instantaneous oscillator phases are defined above with respect to a common frequency to, that is, 9r = cot+<f>n, so w e redefine them in terms o f the true frequencies: 3 , = *>»*+& -> # = ( » - < » .) /+ & (47) Substituting the new phases $n in equations (46) we have the dynamic equations in the desired form (dropping the primes for notational simplicity): 4 , = mS„(A,)a„+* 1 2 X c o s ta /+ & - £ , + < & ) m*n i k = - (48) I X s in ta '+ & - <t>m+<*>) 4 ,* —i where the beat frequency is cob = con+l - con for n = 1,2,-• ■-,N - 1. For the Van der Pol oscillator the amplitude damping function is Sn(An) = l - A ^ , and in synchronized arrays the explicit form o f this function was irrelevant for first order approximations. For mode locked arrays the function directly affects the locking characteristics. In fret, setting the amplitudes to a constant value eliminates all 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stable mode locked states. Thus, amplitude dynamics are necessary for mode locking. W e will amplify the notation further by normalizing the parameters to a beat period. Defining normalized time as t= cobt , we have A = ¥ J ~ A„2)A„ + e lA -x cos( t + <f>n ^=J-sin( r+ <pn - +<J>)+4 , +1 cos( r+ <j>^ - f a - <&)] (49) + <D) + ^ - s i n ( r + < f > ^ <D)] where 77= [xfcob, s= ljcab, ftn = (co^ - <jo„)/(t>b are the normalized nonlinear parameter, coupling strength, and frequency pulling, respectively, and the dot denotes differentiation with respect to z. Appendix 3.2—Frequency Pulling Equations for Mode Locked Arrays The following derivation follows the general method described in section 1.2.1 applied to a mode locked array o f Van der Pol oscillators. Beginning with the dynamic equations for the amplitudes and phases A = ¥ 1~ A 2)A„ + s [ A ^ cos(r+ A - + <D)+ An+1cos(z+ <f>n+l-<pn - <D)] sin (r+ fa - <t>n_x + <D) + ^ - s i n ( r + <D)] ^ = A - ^L4A =L (50) we expand the unknown variables in power series o f the coupling parameter e: A{f) = A m (t) + s A 0)(t)+ s 2Ac- \t) + (51) p = 0 O) + 8 0 ^ + e 20 2)+-Inserting these into equations (50) and equating like powers o f e gives a series o f equations that can be solved recursively. We find the zeroth order steady state quantities are ^ = tfi-(4 °0 % 4 ;O)= o ^ 0 )= ^ , 0 ) = 0 -> = A° > = 1 (52) const 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first order equations are 4® = - 2 7 4 ” +[cos(r+A&_, +<D)+cos(r+A& -<D)] X 3= 0 ? - [ s in ( r + A ^ , +<J>)+sin(r+A& -<D)]] where the time average phase differences are defined as A<j>n = ^ +I - ^on. Enforcing the periodicity o f the amplitudes and phases enables us to determine their steady state values: A!** = . * |sin (r+ A ^ + <I>+ tan " 1(2 77))+ sin ( r+ A # - ® + tan ' 1(277))] V 1+(2 t7)2 =[cos( r+ A^_t + <D)- cos( r+ A^„ - O)]] ^ =0 (54) The second order phase correction is =>s») - [ ^ - i * co s(r+ A ^., + < b ) + ( ^ —^ ))-a n (r+ A ^ JI_, + O ) (55) -cos( t + A<f>„- o ) - (4 ® -4 ® )• sin (r+ A&_ - 3>)] Substituting the first order corrections and enforcing the periodicity o f the second order phase correction gives the frequency pullings as functions o f the time average phases: Pn = ^ - p d -sin (A A ^ J+ 7 cos(AA$i) + sin (2 0 ) + sin(AA^_, -2<J>) (56) +^sin(AA&_2) - 77cos(AA^_ 2 )J, n = 2 , 3 , - , N - l where AA<j>n s A ^ ., - A^„ is the second difference o f the time average phases. The frequency pullings o f the end elements have additional terms: 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A = £2|^-cos2( 0 ) + p - s in 2(< D )+ ^|-sin(2< I> )+ ^-sin(A A ^)+ ^-cos(A A ^)j 0h = ^ « « * (® )“ p-sin2(®)+^-sin(2®)+^-sin(M ^r^ )-^ -co s(A A ^ r^ )j (57) The subscripts for the variables AA^n run from 1 to N-2 and from 1 to N -l. For any subscripts occurring outside this range the associated term should be set to zero. Appendix 3.3—Dynamic Equations for ALC Oscillator Here we derive the dynamic equations for the amplitude, gain control, and phase for the ALC oscillator. Equations (29) can be simplified by observing that g it) is a slowly varying function o f time, so that 2coag « co2 and the nonlinear damping term in the first equation can be treated quasi-statically. Thus, the frequency domain equations can be written {a l ~ <*? +y'2©fl G>{l-Ga +g))v(co) =j2coa coVs(co) g(t)+cogg(t) = cogA(t) Here we have assumed that the envelope detector has unity gain so that the output is the amplitude o f the ALC oscillator output. To simplify the equations further w e will assume the gain Ga is two, which cause the steady stale gain control variable to be unity. The equations become (C0l ~ *>2 +J2 “>a & (g~ 1))V(&) = j 2 &a&K(<o) g(t)+<ogg(t) = cogA(t) W e now assume the oscillator input and output are sinusoidal signals with slowly varying amplitudes and phases, and apply Kurokawa's method to derive the amplitude and phase dynamics. Letting v(/) = A(t)cos(<ot+ a n d v,(/) = .4 (*)cos(* * + # ,(/)) and applying the methods o f section 1.2.1 to the first o f equations (59) we have 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (60) j 2 o a o + j 2 o a\ <f>3 ^ 4 ). A je* Dividing through by j 2 o a co and assuming coa « co and <f>s - j A— « o we have A. , .(0 - 0 . 1 S ~ 1 J — + J— i (o„ o„ -A (61) Equating real and imaginary parts gives A = o J 1- g ) A + o aAs cos(& - $ (62) #= o a - o + o a^ - s m A - $ A I f the injected signal frequency is different than the ALC oscillator frequency, we simply redefine the injected signal phase as <ffs = ( o - o j t + 0 s. In a mode locked array with nearest neighbor coupling there are two injected signals for each oscillator, each with a different frequency. Defining each oscillator phase with respect to the true oscillator frequency, and including a coupling attenuation X and phase delay <t>gives A„ = <oa(l - g K + te>a[An+1 cos(<V + &+i - f a - ® ) + A n cos{o bt + <j>n_x-<j>„-O)] k = <ocn- o n + A oi ^ s i n ( o bt + &+1-$„-<&) + ^ - s m { o bt + . An - <f>„- <D) n (63) We now normalize the time parameter to the beat frequency, r= a bt, and substitute into the above equation. The result, including the gain control variable, is the normalized dynamic equations for the ALC oscillator 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K = 4 \- g ) A „ + 4 c , cos( T+ C , -4> n-® )+ An cos( 2-+ C , £ W + -g W = - ^ ( 0 7] fa=P« + <&)] (64) 7] z+ C , -< P r~^) + i j Lsin( r+ C , - & - <D)j where 77= is the normalized nonlinearity parameter, €=*atf>t is the normalized coupling parameter, /?„ = is the normalized frequency pulling, and the dot now denotes differentiation with respect to x. The gain control filter bandwidth is related to the other parameters as <og = °]4,a so that the amplitude resonance is the same as the beat frequency. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [1] M. Sargent, M. Scully, W. Lamb, Laser Physics. Addison-Wesley Pub. Co., 1974. [2] R. A. York, R. C. Compton, "Experimental Observation and Simulation o f Mode Locking in Coupled Oscillator Arrays," J. Appl. Phys., v o l 71, no. 6 . pp 2959-2965, March 15, 1992. [3] J. Lynch, R. York, "Pulse Power Enhancement using M ode Locked Arrays o f Automatic Level Control Oscillators." IEEEMTT-S, International Symposium D igest, San Diego, June 1994, Vol. 2, pp. 969. [4] J. J. Lynch, R A. York, "Mode Locked Arrays o f Microwave Oscillators," 1993 Symposium on Nonlinear Theory and Applications, vol. 2, p. 605. [5] J. J. Lynch, R. A. York, "An Analysis o f M ode Locked Arrays o f Automatic Level Control Oscillators," IEEE Trans, on C ircuits and System s, vol. 41, Nol 12 , pp. 859-865, Dec. 1994. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 The Design of Microwave Phase Locked Loops Significant progress has recently been made in the design and fabrication o f quasi-optical transmitting arrays using coupled microwave oscillators. [ 1] [2 ] However, conventional oscillator arrays, such as those discussed in the bulk o f this present work, suffer some drawbacks that limit practicality and can adversely affect array performance. In practice, an array must be designed so that the elements can synchronize over a relatively large bandwidth to avoid excessive performance sensitivity to component tolerances and to allow for reasonable modulation bandwidths. The analyses o f the previous chapters show that the ability to synchronize is proportional to the coupling strength between the elements. Unfortunately, large coupling strengths can cause the excitation o f unwanted modes that must be suppressed by appropriate coupling network design. [3] Strong coupling also causes a significant change in oscillator output power as the beam is steered and can degrade the quality o f the array pattern. To overcome these difficulties phase locked loops (PLL's) can be used in place o f conventional oscillators in beam steering arrays. Although the overall circuit complexity is higher, phase locked loops can achieve larger synchronization bandwidths and do not suffer strong amplitude dependence. Traditional coupled oscillator arrays require significant development time to interconnect the elements since the necessarily strong coupling adversely affects the element characteristics. In PLL circuits the oscillator is isolated from the injected RF signal so that the output remains relatively constant over the locking range. High gain PLL's require little input power to synchronize so that most oscillator power is available to the array output. The lack o f undesired interaction between oscillating elements in a coupled PLL system simplifies their interconnection. Once a suitable PLL element is completed an array can be developed rather easily. This chapter is devoted to the design and fabrication o f a angle microwave phase locked loop for use in beam steering systems. The design is kept am ple to aid the future fabrication o f arrays o f PLL circuits. A frequency o f 9 GHz was chosen to allow the use o f readily available microwave HEM Ts and varactors. A detailed description o f the analysis, design, and measured performance o f the PLL will be presented, but first w e will briefly consider the use o f coupled PLL's in beam steering systems. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Arrays of Phase Locked Loops for Beam Steering Systems In this section we present an array o f coupled phase locked loops that allows electronic steering o f the transmitted beam. There are many ways to interconnect PLL's to form such arrays. The method outlined here is closely related to the linear arrays with nearest neighbor coupling considered in the previous chapter. In feet, the following dynamic equations for the PLL phases are identical to the phase equations in section 2 . 1 . 1 ; however the amplitude equations are absent. As before, two dimensional arrays may be advantageous for some applications, and the analytical methods used here can be applied, but we choose the linear array for its simplicity. port Ool <Do2 loN output Figure 1—Array of coupled phase locked loops that allows electronic steering of transmitted beam by tuning the end element VCO's. Figure 1 shows the block diagram for the beam steering system. The analysis o f this idealized system will not be presented in this section but the resulting phase equations are k = " ® - «(sin(^n+i - & ) - sin(& - £_ ,)) for n = 1, 2 ,—, AT ( 1) (note: terms containing variables with subscripts zero or iV+1 must be ignored). Previous analyses have shown that the frequency and phases o f the array are controlled by the first and last elements. I f both tunings are changed by equal amounts in the same direction the frequency will change but the phase distribution will remain fixed. If both are changed by equal but opposite amounts, the frequency will remain fixed and the linear phase progression will change. For details concerning this type o f array the reader should consult reference [4], 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.1 Ideal PLL Operation An ideal phase locked loop is shown in figure 2. Output T iming r* © ------------ Port Phase Detector .kT~ Input Figure 2—Block diagram of PLL. The PLL consists o f a voltage controlled oscillator (VCO), whose instantaneous output frequency is proportional to a tuning voltage, a power divider to deliver output power to a load, and a phase detector whose output depends on the phase difference between the VCO output and an external input. The phase detector output is fed back to the tuning port o f the VCO and, under conditions we will derive, locks the VCO frequency to the input frequency. In the locked, or synchronized, state the VCO frequency rem ains identical to the injected signal frequency, even when the injected frequency is changed or the oscillator tuned by a small amount. This situation will persist as the injected frequency is varied over a particular range, called the locking range. When the boundary o f the locking range is passed, the loop "unlocks" and the VCO will oscillate at a frequency different from the injected signal. Typical spectra for the cases o f locked and unlocked loops are shown in figures 3. ^sA(a>) S id e b a n d s f —-O s c illa to r F re q u e n c y * * In je c te d S ig n a l F re q u e n c y COoOOfe Oscillator Frequency --pA(a>) ^Injected Signal Frequency COo=CD* -i -i <b) (a) Figure 3—(a) magnitude spectrum of unlocked PLL. The sidebands are generated through PLL nonlinearities, and the one-sidedness is typical of such spectra, (b) Spectrum of locked PLL. Oscillator and injected frequencies are identical 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A central concern o f PLL analysis is determining the conditions required to maintain synchronization. In addition, the assumed use o f PLL's is for beam steering systems, so the phase difference between input and output across the locking range is o f primary importance. In practical systems, however, there are other characteristics that may be important. For example, the "capture" range is the range'of injected frequencies that causes synchronization from the unlocked state. If this region is exceedingly small it will be difficult to initially achieve frequency locking, although once achieved the condition may be quite robust Finding the capture range is difficult analytically since it involves solving a nonlinear differential equation near a bifurcation point. Another important consideration may be the phase or frequency response to a modulation input This also involves solving a nonlinear differential equation but for frequencies away from the locking range edges, simplifying assumptions can be made. These aspects o f nonlinear dynamics are treated in many texts, for example [5]. As with the analyses o f previous chapters, the following will be limited to what I believe to be the most significant aspects o f PLL performance for use in beam steering systems. The circuit equation for a single PLL can be written as e>+fi=a>0 +Svc (linear VCO) vc = -V a cos((© - (ot)t + fa) (ideal phase detector) where co+fa is the instantaneous VCO output frequency, (Di+fa is the instantaneous input frequency, vc is the phase detector output voltage, S is the VCO tuning sensitivity, co0 is the VCO frequency with zero tuning voltage, and VQ is the phase detector gain. The frequencies above are always defined so that the corresponding phases are bounded in time, as discussed in section 1 .2 . 1 . 1 . In the locked state the VCO frequency is identical to the injected frequency and the VCO phase is independent o f time. The differential equation (2) reduces to an algebraic equation that relates the steady state frequency to the phase difference between input and output: <o=6>0 - S V 0 cos(A $ (3) where A ^= <j>- fa. The above relation shows that minimum and m aximum values o f (o exist since the sine function is bounded, and as the input frequency is varied the phase difference also varies. Each choice o f co within the range o f existence o f solutions to (3) gives rise to two possible values o f phase difference. It turns out that one o f the values corresponds to a stable solution and the other to 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. an unstable solution, much like the stable and unstable equilibria o f a rigid pendulum. To find the stability condition formally w e would perturb equations (2) from their equilibria and observe whether the perturbations grow o r decay. Since we have already done this in section 1.2 we will amply apply those results. First we express the frequency pulling as a function o f the VCO phase p=a>0 -c o = SVQc o s(0 - ti) = SV0 cos(A0) (4) The stability condition is ^ = - S V 0 &n ( A $ > 0 dtp -» -;r<A 0<O (5) This limits the phase difference to a range o f values for which the cosine function o f equation (4) is monotonic. Thus, the stability condition ensures that a unique stable state exists for a given input frequency co. For this simple case it is instructive to establish the stability condition by inspection o f the PLL block diagram. Since we have assumed the phase o f the carrier at the VCO output is increasing in time, that is v(/) oc ejax, increasing the frequency also increases the phase o f the signal. For stable frequency locking o f the closed loop system an increase o f VCO frequency must cause the loop to produce a decrease in VCO phase in order to return the frequency to its steady state value. Thus the stability condition is dtp S n <0 dco n -*■ T * < 0 (s) which is the same as (5). Figure 4 shows a graphical representation o f the synchronization region and phase difference between PLL and injected signal for this idealized example. Near the center o f the region the phase change is fairly linear, but near the edges the phase sensitivity becomes quite large. In a practical system one must avoid the locking region edges to maintain robustness. It is important to remember that the above synchronization range is defined as the values o f input frequency where frequency locking is possible. Synchronization may be possible only for a narrow range o f initial conditions o f 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (Do (Do -%n Figure 4—Synchronization region and phase difference as functions of the PLL free running frequency. circuit components. When this is the case synchronization will generally not occur when the PUL is powered on from zero initial conditions. One must tune the input frequency well within the locking range to initiate locking and then tune back to the input frequency o f interest Thus, as mentioned previously, the capture range may be significantly smaller than the locking range. 4.1.2 PLL Design 4.1.2.1 Oscillator Analysis Before beginning the oscillator design we will develop some general oscillator theory to better understand the circuit operation. We utilize is the S parameter representation o f the oscillator circuit since this is most familiar to microwave circuit designers and is conducive to microwave measurements. Kurokawa's method is applied to the frequency domain equations to determine the stability requirements. The block diagram o f a simple negative resistance oscillator is shown in figure 5. b i-> < -b 2 ai <— —> a 2 Network Figure 5—Block diagram of a negative resistance oscillator. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The circuit consists o f a device, with load included, and a termination network chosen to create oscillations at the desired frequency. For voltage control o f the oscillation frequency this block will contain a varactor diode. The incident and reflected waves o f each block are related by the circuit's S parameters = S Ia1, b2 = Sna2, and to each other ezl =b2, a , = £ ,.[ 6 ] Combining these equations gives steady state conditions for oscillation in terms o f the wave incident on the device: , (7) S & 2 = cl We could have expressed the other waves similarly, but we chose the wave incident on the device because this is the manner that the amplitude dependent scattering parameter Sn would be measured. Next, we express the wave in terms o f its slowly varying amplitude and phase a, = -jL (8) where the frequency <oa is the steady state frequency o f oscillation. Thus the condition for oscillation can be written = =I (9) where the subscript 'o' denotes steady state quantities. Utilising Kurokawa's method (section 1 .2 . 1) we derive the dynamic equations for the amplitude and phase o f the incident wave. The result is (10) n s\ where S' = — da> One can see that the oscillation condition o f equation (9) gives fixed points, or time independent values, for the amplitude and phase. The amplitude equation in ( 10 ) is stable when the factor on the right hand side is negative. Thus the stability condition is 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Oscillator stability depends on how the scattering parameters change with both amplitude and frequency. The conditions for stable oscillation can be displayed graphically on a Smith chart, as in Figure 6 . Figure 6—Smith chart representation of stable oscillation. Point I is the point of steady state oscillation and point II is the value of 5 at power on (small signal S parameters). The steady state is stable when the angle 0A - 60 between die amplitude and frequency perturbation vectors is less than 180 degrees. The point 5= 1 is the location o f steady state oscillation, shown at point I. When oscillation amplitude is low, for example when the oscillator is initially powered on, the scattering parameter is equal to its small signal value, 5(0,® ), shown at point n . As the amplitude and phase change according to the dynamic equations ( 10 ), the "instantaneous’1 value o f 5 traces out a trajectory and eventually settles out to 5=1. The stability condition on the right side o f equations (11) has a simple graphical interpretation. If we represent 5 as a vector on the Smith chart then the changes in 5 due to a change in A , denoted 5 ', and a change in co, denoted 5 ', are also vectors, as shown in figure 6 . Expressing these changes in polar form ( 12) 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the stability condition becomes Im \d S / / aA as/ (13) . /d o . Assuming SA and S'm are defined as positive values in equation (12), the steady state at 5=1 will be stable when - j t < 9A —60 < 0. This condition holds when the dS dS positive angle measured from the — vector to the — vector, in the 2A do counterclockwise direction, is less than ISO degrees. I f the two vectors meet at right angles the amplitude decay time is minimized and represents an optimally stable oscillator. These results give us a good indication o f the requirements for small signal S parameters to insure stable operation. If the small signal magnitude o f S decreases with amplitude then it must also move in the clockwise direction with frequency. The above analysis allows us to find the conditions for stable oscillation but tell us nothing about how to optimize the output power. Probably the best method for maximizing power in practice is to vary the oscillator load impedance until the maximum is achieved. We will briefly consider how the oscillator power depends on the load resistance to better understand the oscillator characteristics. In linear circuits power transfer is maximized by impedance matching sources and loads. Matching also plays a role in nonlinear oscillators, but is not the usual matching to optimize power transfer. The load circuit must present the impedance that dissipates the maximum amount o f power. To better understand the mechanisms that govern oscillator power generation we consider a simple example. Figure 7 shows an oscillator model consisting o f a series resonant circuit with an amplitude dependent negative resistance, connected to a load resistance (we could have included a load reactance but this simply shifts the resonant frequency and we can consider this part o f the oscillator). A t its terminals, the device will have some I-V characteristic that may resemble typical DC curves, as shown in figure 7. At high frequencies the I-V curves cannot be fully represented on a two dimensional diagram since the gate and drain voltages and currents are not necessarily in phase. However, the important point is that the chosen bias and AC load line determine the voltage and current swings. If the load resistance is low, as in line I, the voltage swing will be small and the current swing large. I f the resistance is high, as in line n, we have the converse situation. Line m represents an optimum point since both voltage and current are maximized In addition, the voltage and current saturation due to pinch off and 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L -R(A) 30 -\SlSlSU---R ■c 20 G l 10 V=Acos( Oot) 0 V (Volts) Figure 7-Circuit model of a simple oscillator and possible AC I-V curves. The negative resistance depends on the amplitude of the oscillation, but this dependence also depends on the AC load line and hence on the value of the load resistance. cut o ff vary as the load resistance varies. Thus we find that the steady state amplitude o f oscillation depends on the load resistance. A hypothetical dependence is illustrated in figure 8 . ex S ■< Rl Figure 8—An illustration of the dependence of oscillation amplihidp on the load resistance. The dotted lines indicate optimum load resistant and amplitude The power delivered to the load is given by a2{rl ) rl PL= (1 4 ) 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A is the amplitude o f the voltage waveform across the load. Assuming the simple amplitude curve o f figure 8 the power tends to zero as the load resistance approaches zero and infinity. Thus a maximum exists, and is found by setting the derivative o f (14) equal to zero. The resulting condition for optimum power dissipation is — = -— dRL 2 Rl (15) J K } This result represents the trade off between voltage amplitude and load resistance. The optimum point is shown in figure 8 . The results o f this section are difficult to utilize directly since this would require knowledge o f the functional dependence o f the voltage amplitude on the load resistor. Characterization o f microwave devices is generally difficult since we are usually constrained to a measurement system o f a particular impedance, e.g. 50Q, and this value determines the AC load line. However, the results are useful in helping us understand why an optimum load impedance exists. 4.1.2.2 VCO Circuit Design The following voltage controlled oscillator design is based on the small signal parameters o f the NEC FET transistor NE32184A. N o large signal measurements were performed so we •wall be forced to make certain assumptions about the amplitude dependence o f the device S parameters. The Libra software package from EEsof was used to simulate the RF circuit performance, but the level o f complexity o f the simulated circuit was kept low. The simulations indicated general circuit performance, but many adjustments were made on the fabricated circuit. Thus, the simulation provided only a rough estimate o f performance. 4.1.2.2.1 Device Bias The common gate topology allows single supply biasing and is quite common in the literature.[7] The biasing circuit diagram and layout are shown in figure 9. The power supply is 5 V and the DC bias point is chosen to give Vds = 2.5 V and ID = 20 mA. The DC bias circuits are isolated from the RF circuits with open circuited stubs that present high impedances where bias is supplied. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5V W20Q Via Holes 120 Q 2.5V 20 Q Figure 9—Schematic diagram and microstrip layout of bias circuitry for FET. The cross structures near the resistors are quarter wavelength open circuited stubs that create a high RF impedance at the bias connection point (Layout is approximate). 4.1.2.2.2 RF Circuit Design The first step in the oscillator design is to add some gate inductance (line length) to decrease device stability and to design a network between the device drain and 50Q load to present a negative resistance at the source. The source termination network (varactor diode) is series resonant so we will design the overall oscillator, as viewed from the source, as a low impedance (approximately) series resonant circuit. The RF network (determined by trial and error) shown in figure 10 provides the desired impedance looking toward the source leads. Load 7 GHz To 50 Q Load 11 GHz Sn Figure 10-Schemalic, layout, and input reflection coefficient plot for the active device. The small loop on the Smith chart plot will not cause frequency jumping since stability depends on both the device and varactor S parameters. (Layout is approximate). 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As with most packaged microwave transistors, the NE32184A has two source leads and we will connect to both o f them in a symmetric fashion. In the Touchstone simulation, however, only one source lead is available, so in the simulation we will connect the tw o source circuits to the single point As mentioned above, the simulations are not intended to accurately predict circuit performance; only to provide a rough guide. To connect the varactor to both source leads we must extend the leads away from the device using transmission lines. This causes the impedance curve above to rotate clockwise around the Smith chart. The circuit schematic and layout appear as in figure 10, with the device input S parameter S ° shown on a Smith chart. A capacitive reactance was produced to cancel the inductive reactance o f the varactor circuit, which we will consider in the next section. One might think that the small loop in the plot above would cause instability, or a frequency jump, because the stability condition may not be met there. This is possible, but we must apply the stability condition to the entire S parameter S = S*Sa which may or may not contain a loop. After the appropriate varactor network is connected, oscillations will build up and the parameter S11 at the operating point will move toward the unit circle (or so we assume). The steady state will be reached when the phase o f Sn is equal and opposite to the phase o f the varactor circuit S1, and the magnitudes are inverses. 4.1.2.2.3 Varactor Circuit Design The varactor diode provides electronic frequency tuning via the bias dependent junction capacitance. The device used here is a GaAs hyperabrupt diode made by Alpha Industries (part number DVG5464-70). The equivalent circuit, obtained from the part catalog, is shown in figure 11 , and is a high Q series resonant circuit. .l5pF .. .4nH C* 2Cl Figure 11—Equivalent circuit for the varactor diode is a high Q series resonant circuit whose series capacitance can be adjusted by varying the diode reverse bias. We need to isolate the varactor bias from the FET circuit bias, so we will place the varactor in series with a grounded capacitor. A schematic o f the diode circuit and the input reflection coefficient are shown in figure 12. The inductive 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. impedance is due to the effects o f copper strips that connect varactor to shunt capacitor (see figure 13). 11 GHz e C. =F Varactor — iw - Z 7 GHz Figure 12—Schematic diagram and input S parameter of varactor circuit Bias is applied at capacitor Cs. To realize the above circuit we mount the varactor on top o f the microstrip trace that represents the junction o f the two source transmission lines and use copper strips to connect the varactor radial stubs that represent the grounded capacitance. The physical layout is shown in figure 13 Radial Stub Cu Strip Varactor Varactor Cu Strip Source line m 20Q Radial Substrate Stub(G) Ground plane Varactor bias (tuning port) Side View Figure 13—Physical layout of varactor circuit The diode is mounted on the top of the substrate and connected to the radial stubs via copper strips. (Layout is approximate). The varactor capacitance depends on the varactor reverse bias voltage according to the curve shown in figure 14 (computed from catalog parameters): 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. oo ce eo O Reverse Bias (V) Figure 14—Varactor capacitance vs. reverse bias. The sensitivity is greatest for low bias voltage (flattening of curve below 0.5 V not shown). Thus by varying the varactor bias we can tune the oscillator frequency. 4.1.2.2.4 Complete VCO Design and Measurements Once the basic circuitry is designed we can adjust the line lengths o f the source transmission lines to give the desired frequency o f operation. This is possible since changing these lengths changes the phase o f Sa. The oscillation condition is best displayed, as described in section 4.1.2.1, by displaying the quantity S = SJSn on the Smith chart. This is easily accomplished using Touchstone, and the results, after adjusting the source lines to give nominal 9 GHz operation, are shown in figure 15 for three values o f varactor capacitance. 7 GHz' 7 GHz 7 GHz 11 GHz 11 GHz 11 GHz (a) (b) Figure 15—Total scattering parameter S=S*SU vs. frequency for three different values of varactor capacitance: (a) Cv = 0.5, (b) Cv = 0.7, (c) Cv = L 0. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To estimate the oscillation frequency we assume that the above curve moves radially inward as the amplitude increases. Thus the frequency o f oscillation is the frequency for which S has zero phase. The VCO timing curves computed from the simulation and measured from a fabricated circuit are shown in figures 16: 9.4 Simulated 9.2 N SC 9 o 8.8 u>» Measured c<x> 3 8.6 O * oj 8.4 8.2 8 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Tuning Voltage (V) Figure 16—Simulated and measured VCO tim ing curves. The large discrepancy in tuning sensitivity is probably due to inadequate high frequency modeling of the varactor diode and associated circuitry. The discrepancy between the simulated and measured curves is probably due to inadequate modeling o f both the varactor diode characteristics and its physical mounting. The manufacturer measures the diode characteristics and parasitics at low frequencies, typically at 1 GHz, and we then extrapolate. In addition, we cannot easily model the packaged diode placed on top o f the microstrip trace, or the copper strips used to connect the diode to the radial stubs. The tuning curve slope is very sensitive to the reactance presented by the strips. Nevertheless, the basic design procedure is confirmed fairly well be the above measurements and we are ready to proceed to the next PLL block. 4.1.2.3 Phase Detector 4.1.2.3.1 Ideal Operation The purpose o f the phase detector is to produce an output voltage proportional to the sine or cosine o f the phase difference between tw o RF input signals. Standard mixers are often used as phase detectors, but one o f the inputs is a high power pump (LO) and the other is a much lower level signal (RF).[8 ] For 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the present case we would like to design the circuit so that both input levels are approximately equal and relatively small, while maintaining high output sensitivity. This is accomplished using the circuit shown in figure 17, consisting o f a Wilkinson power combiner followed by an active RF power detector. -o DC Output RF Inputs Power Combiner Power Detector Figure 17—Phase detector consists of a Wilkinson power combiner followed by an RF power detector. The output voltage o f the power combiner is the sum o f the two input voltages. Defining the inputs as v, (/) = Axcos{ a t+& ) (16) v2(t) = A2cos{eot + <f>2) the output voltage is + 4 + 2A A cos(^ ~ &) v(/) = - ^ ( v ,+ v2) = •cos a t + tan -1 (17) 4-4* tJ iiz £ V4+A I 2 . This signal is then sent to an RF level detector whose output is proportional to the RF pow er vd = G * + £ +2AXA2 cos(^ - <f>2)) (18) where Gd is the detector gain (in dimensions o f V~'). If both input amplitudes are equal the above simplifies to vd = GdA2( l + cos( - $ ,) ) . output varies as the cosine o f the phase difference, as desired. Thus the detector 4.1.2.3.2 FET Detector Design In order to provide gain, the detector was implemented using a HEMT FET, the same device used for the VCO. The detector circuit is shown in figure 18. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NE32184A * DC Output Matching Network RF Input —r—AC Short lK fl -1.1V Figure 18—Detector is a common source amplifier biased near pinch off. The transistor is biased near pinch o ff so that the drain voltage is nominally near the upper power supply when no RF input exists. An RF signal applied to the gate alternately turns the device on and o ff creating an average DC level at the drain. The drain current vs. gate voltage curve shown in figure 19 follows, approximately, a square law so that the 35 30 ■< 25 *3 20 S3 15 o .5 *3 10 5 0 -L2 -1 -0.8 -0.6 -0.4 -02 0 Gate Voltage (V) Figure 19—The drain current vs. gate voltage is approximately a quadratic function. Thus the detector output is proportional to the input power. (Above curve from nonlinear model) 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. output voltage increases as the square o f the gate voltage. operates as a power detector. Thus the circuit To better understand the detector operation we will consider the circuit equations. The DC output is given by v = r „ - .R ,( ; ,) O 9) where the brackets denote the time average value. The gate voltage is a DC bias and an RF signal vz = vg g + a c o s(n * + 6) = Vc a +vif (20) We assume that the drain current is a function o f the gate voltage as id = / ( v«)At microwave frequencies the relation becomes more complicated, but the above low frequency relation will approximately hold. We can obtain an approximation for the drain current by expanding / ( v j in a Taylor series about the gate bias voltage that will be most accurate for small RF voltages. The result is — (X^'gg) "t." dv„ 1 ' V* ' 2 dvt -v: (21) 1 = h +SmVrf+~^S,mVrf where ia is the drain bias current and gm is the transconductance w ith no RF input. Taking the time average value gives v = VDD- R J 0 = ^DD ~ R tfo ~ \ (22) 4 The first two terms give the output voltage when no RF signal is applied. As mentioned above, the transistor is biased near pinch o ff so that iQ is small and therefore the output voltage is nominally close to the power supply. Upon application o f an RF signal, the change in output voltage is proportional to the RF power, and the sensitivity is proportional to the change in transconductance with gate voltage. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From this last result we wish to maximize the change in transconductance with gate voltage. Looking back at the transistor I-V characteristics in figure 7, a vertical AC load line will produce the largest change i n . . Thus we require an RF short circuit at the FET drain and we realize the short using two quarter wavelength open circuited stubs. The complete detector circuit is shown in figure 20. lK f lg EF DC Output 510® 5V Figure 20—Microstrip circuit layout of detector, including input matching network. The two quarter wavelength open circuited stubs on the FET drain create a large change in transcondnctance with gate voltage. (Layout is approximate). To maximize the gate voltage we must design a matching network to deliver maximum power to the FET input. The equivalent circuit for the FET input, with the above output circuit, is shown in figure 2 1 . C i = .3pF |) L{= .6nH <S&SLs Z in Figure 21—Equivalent circuit and input impedance near 9 GHz. Details o f the matching network design will not be presented here, but the procedure was taken directly from reference [9], and the completed circuit is shown in figure 20 above. Figure 22 shows a Libra simulation o f the input impedance o f the completed detector. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 -1 S ' *2 b -3 Jh 4 -5 5 6 7 8 9 10 11 12 Frequency (GHz) Figure 22—Magnitude of the input reflection coefficient of the detector. The return loss above reduces the voltage that is developed across the gate terminals. For given incident input power (RMS) the power dissipated in the gate is Pg =(l-|rin|2)pine, where Tm is the input reflection coefficient plotted above. The RMS current through the resistor is Ig = so the RMS voltage developed across the gate terminals is Vg = Igyj l ^ +X% . At 9 GHz the FET input impedance is primarily capacitive, Zir\9GH. = 4£2-_/24Q . Using the above expression with an input power o f 6 dBm (4 mW RMS) produces a gate voltage o f .77 V peak. With a gate bias o f -1.1 V (experimentally determined) the voltage is adequate to completely turn on the device. The simulated and measured detector performances are shown in figure 23. The simulated curve was generated using a nonlinear model o f the DC I-V FET characteristics. The curve approaches one half the power supply voltage as the input power increases because at high power levels the transistor output is a square wave o f 50% duty cycle. The measured curve fells below this level, however. This is probably due to unequal rise and fell times o f the output voltage. I f the FET turns on fester than its turns off then, under large signal conditions, the average value o f the output voltage will be lower than 2.5 V. This behavior enhances the voltage range o f the detector. Combining the above voltage vs. power curve with the input power vs. phase difference curves calculated from equation (18) gives the phase detector output voltage vs. input phase difference shown in figure 24. 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Simulated 4 Measured 3 2 i -20 •10 10 0 Input Power (dBm) Figure 23—Simulated and Measured detector output voltage vs. input power. The simulated curve flattens out to 2.5 V as the input power is increased indefinitely. The simulated curve is based on the nonlinear DC device I-V characteristics. 5 5bo OS 4 0 04 0 o 3.5 3 0 0.5 1 1.5 2 2.5 3 Phase Difference (rad) Figure 24—Simulated phase detector output voltage vs. input phase difference for equal (3 dBm) input powers. The curve is not quite sinusoidal, but fairly close. This figure was generated assuming equal 3 dBm inputs. The phase detector response was not directly measured, although it is included implicitly in the full PLL response, presented in the following section. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.3 PLL Measurements With the VCO and the phase detector designs completed we connect the two together to form the PLL circuit. The complete circuit is shown in figure 25. Branchline Coupler ELL output (lOdBm) E i -i.iv gign 50 n 100 Q ELL Input (3 dBm) NE32184A WilkinRnn ~l§5io n Combiner 5V Figure 25—Complete PLL circuit The branchline coupler diverts some of the output power (3 dBm) to the phase detector input The phase detector output is fed bade to the VCO tuning port (Layout is approximate). A branchline hybrid couples about 3 dBm o f VCO output power back to the phase detector input. With a VCO output power o f 11 dBm, the PLL output power is 10 dBm. The phase detector output is fed back to the VCO tuning port through a resistor (the resistor provides a convenient place to open the loop for testing and troubleshooting). The DC biases are isolated by coupled line filters acting as blocking capacitors on both the VCO output and the detector input. The locking range was measured by injecting an external signal at 3 dBm and observing the input frequency range over which the PLL locks. The phase detector output voltage vs. input frequency over the locking range is plotted in figure 26. The locking range is 300 MHz and over this range the PLL output power was 10.04+ 06 dBm. The ripples are probably due to multiple reflections along measurement cables. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 S 10.2 j£ o. 2 8.5 8.6 8.7 8.8 9.6 8.9 8.6 Input Frequency (GHz) 8.7 8.8 8.9 Input Frequency (GHz) Figure 26—Phase detector output voltage and PLL output power over die 300 MHz PLL locking range. The ripples in the former are most likely due to multiple reflections along measurement cables. 4.1.3.1 Phase Measurement Since the intended use o f the PLL is for beam steering systems, the total phase change between the PLL output and the input is an important figure o f merit. This measurement is somewhat complicated since a calibration o f the measurement apparatus is required to eliminate errors. The measurement system, shown in figure 27, consists o f a pair o f power dividers, a reference cable, and a phase detector. The PLL output biases the LO port o f the mixer and an external signal is injected into the PLL and provides the mixer RF signal. As the input frequency is adjusted the phase difference between PLL input and output changes, and the IF output signal varies. To determine the PLL phase difference we must determine the correspondence between this phase shift and the IF output. The following calibration provides this information. The measurement system is calibrated by applying a 10 dBm RF signal at the input port and terminating the output port, as shown in figure 28. The reference cable attenuation is chosen to provide 3 dBm at the RF output port, which matches the required 3 dBm PLL input power. As the input frequency is changed the mixer IF output voltage changes in direct relation to the change in phase between the input and output reference planes (see figure 28). These reference planes are calibrated on a network analyzer so that we know the precise phase delay through the measurement network. Thus we can calculate the 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. son out Pwr PLL 3 dBm Calibration Reference Planes IF Output Figure 27—Measurement of PLL phase shift between input and output The injected signal power is adjusted to match the calibrated power at the same port (3 dBm). 3 dBm sow: Reference Cable AO- Calibration Reference Planes IF Output Figure 28-Calibration of phase measurement system. A reference cable is attached between power dividers and a signal is injected as shown. Recording the IF output voltage vs. frequency allows the determination of the inherent measurement system phase shift inherent phase shift o f the measurement system and subtract this from the PLL measurement. This procedure will become clearer by considering the measurements in detail. 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first step o f the calibration is a measurement o f the mixer output with an LO bias, but with the RF mixer input terminated, for a frequency range greater than the locking band o f the PLL. The result is plotted in figure 29. 4.4 3.2 8.5 8.6 8.7 8.8 Frequency (GHz) 8.9 9 Figure 29—IF output with RF input terminated. This voltage will be subtracted from future measurements. Next, the reference line is connected, as in figure 28, and the IF output voltage is recorded. The result, after the offset o f figure 29 is subtracted, is shown in figure 30. 100 0 8.5 8.6 8.7 8.8 8.9 9 Frequency (GHz) Figure 30—IF output with configuration of figure 28. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We assume the mixer output is proportional to the sine o f the phase difference between the two inputs, and can be written = Fsm(aTeal + 6(a>)), (23) where V is a constant amplitude, is the time delay between the reference planes shown in figure 28, and G(co) is a frequency dependent phase that we wish to calibrate out. Measuring the phase delay between the reference planes on a network analyzer gives the value o f a rt^ and is shown in figure 31. 3 -3 8.5 8 .6 8.7 8 .8 8.9 9 Frequency (GHz) Figure 31—Phase delay between reference planes from PLL output to input measured on a network analyzer. Using equation (23) for the calibration waveform, we compute the phase 6(e>), which is plotted in figure 32. This completes the calibration. To measure the PLL phase change, we remove the reference cable and attach the PLL and an RF source as shown in figure 27. The RF source power is adjusted to give the same power into the PLL as was measured during the calibration procedure, that is, 3 dBm. We then record the mixer IF output over the PLL locking range. The results are shown in figure 33. From this curve we calculate the phase change (using an arcsine function) and subtract the calibrated phase change from figure 32. The result is the phase difference between the reference planes at the input and output o f the PLL, as shown in figure 34. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 1.5 1 0.5 0 -0.5 -1 -15 8.5 8.6 8.7 8.8 8.9 9 Frequency (GHz) Figure 32—Phase shift of measurement system computed from the phase derived from figure 30 with the phase delay of figure 31 subtracted. 100 >■ a ► j CL, .X3 -50 -100 8.65 8.7 8.75 8.8 8.85 8.9 Frequency (GHz) Figure 33-IF signal with PLL connected as in figure 27. The ripples are due to multiple cable reflections and PLL phase noise. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bo o no CO S C8 -120 %140 O, 0 160 1 180 _3 Bu *200 8.65 8.7 8.75 8.8 8.85 8.9 Frequency (GHz) Figure 34—Measured phase shift of PLL between reference planes. In computing these results we used the feet that the phase difference between the PLL input and output is a decreasing function o f frequency, which was proved in section 4.1.1. The above phase change is only about 90 degrees, which is half o f the ideal 180 degrees computed earlier. This discrepancy may be due to many factors. Frequency dependent phase delays within the PLL directly reduce the total phase change. For example, if the phase through the branchline coupler varies 30 degrees over the locking frequency range, then the total phase range will be (ideally) only 150 degrees. Thus physically small and broadband components are necessary in the loop design. In addition, excessive VCO phase noise will cause loss o f synchronization near the edges o f the locking region where loop stability is reduced. This further diminishes the phase range. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [1] P. Liao, R. A. York, "A New Phase-Shifterless Beam-Scanning Technique using Arrays o f Coupled Oscillators," IEEE Trans. Microwave Theory Tech., voL MTT-41, pp. 1810-1815. [2] J. Lin, S. T. Chew, T. Itoh, "A Unilateral Injection-locking Type Active Phased Array for Beam Scanning," IEEEM TT-S, International Symposium Digest, San Diego, June 1994, pp. 1231-1234. [3] S. Nogi, J. Lin, T. Itoh, "Mode Analysis o f Stabilization o f a Spatial Power Combining Array with Strongly Coupled Oscillators," IEEE Trans. Microwave Theory Tech, vol. MTT-41, pp. 1827-1837. [4] R. A. York, "Nonlinear Analysis o f Phase Relationships in Quasi-Optical Oscillator Arrays," IEEE Trans. Microwave Theory Tech, vol. MTT-41, pp. 1799-1809. [5] E. A. Jackson, Perspectives o f Nonlinear Dynamics, Cambridge Unvershy Press, Cambridge, 1989. [ 6 ] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design PrenticeHall, NJ, 1984. [7] G. D. Vendelin, A. M Pavio, U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques. Wiley & Sons, NY, 1990. [ 8 ] S. A. Maas, Nonlinear Microwave Circuits. Artech House, Norwood, MA, 1988. [9] G. L. Matthaei, L. Young, E. M T. Jones, Microwave Filters. Impedance Matching Networks, and Coupling Structures. Artech House, Dedham, MA, 1980. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Areas for Continuing Study The research presented in this dissertation could proceed in many possible directions. This work was not simply a "linear" continuation o f a previous effort, but tied together our understanding o f two distinct types o f quasi-optical sources, synchronous and almost periodic. Much o f this progress was due to the analytical techniques presented here. These methods were an extension o f mature classical perturbation theory applied to nonlinear oscillatory systems, and they supplied the most important information that we required for die design o f such systems. The mathematical study o f such systems has seen some significant advances over the past thirty years.fl] A possible research direction is to apply these techniques to microwave systems. This would undoubtedly increase our understanding and would provide a broader and more solid foundation for this aspect o f microwave engineering. Another area o f study is the use o f more complex coupling schemes for synchronous and mode locked arrays. For all practical cases considered here the arrays have been linear (Le. arranged in a row) with nearest neighbor coupling. Two dimensional arrays provide the possibility o f beam steering in tw o directions, or beam steering in one direction and continuous scanning in the other. [2] The nonlinear dynamic system from section 1 .2 .1 still describes such arrays, but it is much more complicated. Concepts developed for the linear arrays, such as characteristic tunings, must be generalized, if possible. Nearest neighbor coupling greatly simplifies the mathematics and allows for a simple beam steering implementation, but increasing the coupling between non-adjacent elements can increase the locking region size and probably wider modulation bandwidths since the settling time o f the array can be reduced. Once again, the dynamic equations become quite complicated so existing methods must be generalized. The analytical techniques developed here can be applied to more complex synchronized systems such as the oscillator grid mounted in a resonant cavity, as shown in the "Historical Development" section, figure 1. The analysis o f such a system can proceed from a linear frequency domain analysis such as in [3]. Depending on the coupling circuit bandwidth and quality factor one may have to include the effects o f amplitude response. The results o f section 2.2 should provide useful information. Mode locking is prevalent in pulsed laser systems, but the analyses I have seen, which are admittedly few, use rather course simplifying assumptions. These assumptions may give adequate accuracy for practical cases o f interest. However, applying the methods developed in chapter 1 may increase the accuracy o f the 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analyses and demonstrate effects that went previously unnoticed or unexplained. In addition, some conversations with laser engineers have pointed out that some (many?) mode locked lasers already contain a resonant amplitude response, resembling the ALC oscillator. It is possible that matching the intennode spacing to the amplitude resonant frequency will have the desired effects listed in section 3.3. The treatment o f modulation bandwidth and settling time in synchronous arrays is important and deserves careful attention. The section devoted to this study (2.1.1.1.4) is highly simplified and requires experimental verification. Nonetheless, the concepts o f characteristic tunings provides a lot o f insight into array behavior. Further development o f these concepts is necessary for practical systems. The effects o f noise in oscillator arrays was not treated at all in this dissertation, but is an important subject Such analyses should be relatively straightforward for the simple arrays considered in here. The use o f mode locked arrays in radar systems presents an interesting alternative to conventional systems. A mode locked array transmits a comb spectrum in the frequency domain and a continuously sweeping beam in the time/space domain. If the array elements were configured to down convert the received signals, the baseband signals would contain information about range, range rate, and angular position. W ith one baseband signal per radiating element redundant information is available and may result in more robust parameter estimates. [1] J. Guckenheimer, P. Holmes, Nonlinear Oscillations. Dynamical Systems, and Bifurcations o f Vector Fields. Springer-Verlag, NY, 1983. [2] R. York, R. Compton, "Automatic Beam Scanning in Mode Locked Oscillator Arrays," IEEE Antennas Prop. Symp. Digest (Chicago), July, 1992. [3] J. W. Mink, "Quasi-Optical Power Combining o f Solid-State Millimeter-Wave Sources," IE EE Trans. Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986. 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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