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Analysis and design of systems of coupled microwave oscillators

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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
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This dissertation o f Jonathan J. Lynch is approved
Committee Chairperson
March 1995
i
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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
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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
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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.
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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.
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Table of Contents
Historical Development
Overview
1. Analysis Techniques for Periodic and Almost Periodic Frequency
Locking
1
5
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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
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2. Synchronous Arrays
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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
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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
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3. Mode Locked Arrays
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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
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4. The Design of Microwave Phase Locked Loops
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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
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Areas for Continuing Study
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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[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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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£ = 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
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/(/)= / ( 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
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’■®(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
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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
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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
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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
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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
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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
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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
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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
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[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
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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
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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
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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
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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
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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
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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
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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 $)
0
1
cos(A^2)
V=
0
c o s(A ^ J
(15)
The frequency pulling equations (10) become
A/?= £(cos(0)y4w+sin(d>)5v)
(16)
and the perturbational system is
A S = -C A S
(17)
45
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with stability matrix
C = fi(cos(<I>)j4F'-sin(<I>)2?£/)
(18)
Equations (16) and (17) completely describe the array frequency, phases, and
stability, to the first order o f approximation. The simple form o f the equations
allows us to gather further insight into the array characteristics. For example, the
condition for broadside radiation, A ^ = 0 , gives u = (0,—,0 )r and v = (l,—,1)T.
For a coupling phase o f <J>= 0 equation (16) shows that the broadside radiation
condition is A 0 - 0 which implies identical element tunings across the array. In
addition, we will show in the next section that for 0 = 0 the stability matrix
C = sA V has positive eigenvalues when each phase satisfies -*A< A$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
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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
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^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
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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
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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
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' 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.
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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
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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
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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.
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'.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
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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
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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
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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
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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
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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
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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
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<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)
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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
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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
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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.
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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
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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
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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
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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.
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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,
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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
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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
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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
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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
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(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
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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
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>
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
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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
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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
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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
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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
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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
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[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
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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
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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
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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
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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.
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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
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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 :
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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 $)+2an(2O )cos(A A 0 1)-sin(2O )] (17)
which, with the help o f a trig identity, can be written
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(18)
Allowing AA^ to span all o f its possible values creates the existence region on the
AA/?, line. The length o f the region is
- M A L = 2 ^ |? V 5 - 4 co<2«»)
(19)
which is maximized for <X>= %, and 77= 34. These are optimum values o f coupling
phase and nonlinearity. The stability matrix C has only one element
“ 4 cos(2 0 ) cos AA$ + 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
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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
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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
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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
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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
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~ 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
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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
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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
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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
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-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)
^
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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)
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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(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
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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
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[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
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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.
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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
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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
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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
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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
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(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
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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
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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
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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
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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 )
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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[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
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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
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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
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