University of California - Microwave Electronics Laboratory at UCSB

University of California
Santa Barbara
Analysis and Design of
Systems of Coupled Microwave Oscillators
A Dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
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
This dissertation of Jonathan J. Lynch is approved
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Committee Chairperson
March 1995
i
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 of free thought;
My mother, who always reminded me
what is truly important in life.
ii
Acknowledgements
Although the distinguishing characteristic of a dissertation is independence
of 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 his 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 Mark 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 of the members of 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 failed to
provide intelligent and informative suggestions. Finally, and most importantly, I
thank all of my friends and family. You have provided that important element in
life that has given me the strength to achieve whatever I desire.
iii
VITA
Jonathan James Lynch was born August 27, 1965, Winchester, MA.
1987 Bachelor of Science, University of California, Santa Barbara
1993 Master of Science, University of California, Santa Barbara
1995 Doctorate of Philosophy, University of California, Santa Barbara
Industry Experience
1986-1995
1995-
Electrical Engineer, Delco Systems Operations, Santa Barbara, CA.
Electrical Engineer, Hughes Research Labs, Malibu, CA.
Publications
J. Lynch, R. York, "Stability of 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.
iv
H. Tsai, P. Liao, J. Lynch, A. Alexanian, R. York, "Active Antenna Arrays for
Millimeter wave Power Combining," 1994 International Conference on Millimeter
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 of
systems of 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 of practical importance. The author presents
techniques to improve locking characteristics of mode locked arrays, using
automatic level control oscillators, and synchronized arrays using coupled phase
locked loops. Detailed design and measurements of 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
1.1 Microwave Measurements
1.2 Systems of Coupled Oscillators
1.2.1 Derivation of the Nonlinear Dynamic Equations
1.2.1.1 Definitions of 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 of complex transfer functions
1.2.2 Solution of Periodic and Almost Periodic States--Locking
Diagrams
1.2.2.1 Existence of locked states
1.2.2.2 Stability of locked states
Appendix 1.1 Narrowband response of nonlinear circuit elements
Appendix 1.2 Response of networks to narrowband signals
Appendix 1.3 Approximate stability of locked states
References
2. Synchronous Arrays
2.1 Synchronization of 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 of a Resonant Coupling Network on the
Synchronization of Two Oscillators
2.2.1 Dynamic Equations
2.2.2 Synchronized States
2.2.3 Stability of States
2.2.4 Cases of 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 of a Stability Matrix
Appendix 2.2--Amplitudes, ∆ω c Near Line of Equally Spaced
Frequencies
References
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3. Mode Locked Arrays
3.1 The Time Domain Mode Locked Waveform
3.2 Linear Arrays of Van der Pol Oscillators
3.2.1 Three Element Array
3.2.2 Four Element Array
3.3 Mode 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--Normalized Form of Mode Locking Equations
Appendix 3.2--Frequency Pulling Equations for Mode Locked Arrays
Appendix 3.3--Dynamic Equations for ALC Oscillator
References
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4. The Design of Microwave Phase Locked Loops
4.1 Arrays of 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 Detector
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
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. 17
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.
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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 (N=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 ∆∆β 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.
Figure 9--Locking diagram for a four element ALC array.
Figure 10--Locking region size, maximum and minimum eigenvalues as functions of
the number of array elements.
Figure 11--Pulse enhancement using the time varying amplitudes of the ALC oscillators.
Figure 12--Locking region size, L, for two values of coupling phase, as a function of the
number of array elements.
Figure 13--Maximum sensitivity as a function of the number of elements.
Figure 14--Schematic of a single ALC oscillator.
Figure 15--Response of the magnitude of the amplitude variations to an injected signal.
Figure 16--The measured and theoretical mode locked waveforms.
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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 circuitry 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 of 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 27--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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Analysis and Design of
Systems of Coupled Microwave Oscillators
Recent research in quasi-optical power combining of microwave and millimeter
wave oscillators has resulted in novel methods for producing solid state high power and
electronically steerable arrays while maintaining low circuit complexity. Such systems
rely on frequency locking of individual oscillating elements to ensure coherent operation.
Proper phasing of the elements is essential in beam steering systems and this requires a
solid understanding of the dependence of oscillator phases on various circuit quantities
such as inter-element coupling strength, phase delay, and element tunings.
Professor York's research group has explored, both experimentally and
analytically, various approaches to beam steerable and pulsed, or mode locked, arrays.
The analysis of single frequency (synchronous) oscillator systems has led to simple array
designs that enable electronic beam steering without the use of phase shifters. This
present work develops the analysis further to graphically depict synchronization and
understand array characteristics through "characteristic tunings." Such characteristics
include the ability of the oscillators to synchronize, the sensitivity of the phases (array
"robustness"), and the transient phase response. Synchronization is enhanced through the
use of phase locked loops in place of conventional microwave oscillators.
Another type of phase coherent state is called mode locking, in which the
oscillator frequencies are different but evenly spaced by some small interval. Summing
the oscillator outputs produces periodic pulses of microwave energy, useful for pulsed
radar systems. The analysis of mode locked states is more difficult than the synchronous
case. This present work develops approximate methods that results in a mathematical
framework analogous to the synchronous case. Thus, all of the information derived for
the synchronous case can be carried over to the mode locked case. The tendency to mode
lock can be enhanced by using oscillators with underdamped amplitude responses.
Jonathan J. Lynch received the BS ('87), MS ('92), and Ph.D. ('95) degrees in electrical
engineering from the university of California at Santa Barbara. He was employed at
Delco Systems Operations in Santa Barbara from 1986 to 1995 as an automotive
electronics circuit designer and radar systems engineer. In February of 1995 he
transferred to Hughes Research Labs in Malibu, CA where he is currently involved in the
design and fabrication of quasi-optical arrays and millimeter wave systems for various
commercial applications.
Historical Development
As technology advances, the maximum operating frequency of electronic
systems continues to climb. The millimeter wave frequency range, typically
assumed to lie above 40 GHz, offers exceptional resolution for radar imaging
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 of system performance. Unfortunately a fundamental trade off 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 many 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
of high power systems over conventional power combining methods.
Most quasi-optical millimeter wave systems exploit engineering
techniques typically utilized at optical frequencies. A good example is the power
combining array originally proposed by Mink [1](note: references are located at
the end of the Overview section), and shown in figure 1.
Oscillator Grid
Output
Partially
Reflecting
Mirror
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 of microwave negative resistance devices is placed in an
electromagnetic cavity and excites a sustained resonant mode of oscillation.
Power from individual oscillators combines coherently within the cavity and a
partially reflecting mirror serves as the output port. The system essentially
mimics an optical laser in which the power source is a distributed gain medium.
1
Over the past few years quasi-optical power combining systems have
separated into two groups: grid systems and discrete oscillator systems. The
former group utilizes grids of 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.
Oscillating
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 of success and simplifying
performance optimization. In addition, arrays of coupled oscillators can provide
electronic beam steering and pulsed transmitting functions without the use of
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 of such systems requires a detailed understanding of the
behavior of 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 of ideal systems of
coupled nonlinear oscillators and from measurements of fabricated systems.
Appropriate analysis techniques have been developed by applying
classical analytical methods to our particular class of problems. The study of
synchronization of electrical oscillators began with B. Van der Pol at the
2
beginning of this century.[5] He utilized a method of 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 of oscillating systems.[6] The methods
presented in this dissertation are based on these averaging methods which are the
first terms of a power series expansion of the solution with respect to a small
parameter. Thus all results are approximate, but quite accurate and useful for
many microwave systems. More recently, the development of the mixed
potential theory for electrical networks,[7] and the subsequent development of the
averaged potential for oscillatory networks,[8] provides a slightly different
analysis technique that has also proved useful for understanding the behavior of
coupled microwave oscillators.[9]
The analysis of microwave oscillators was given a more physical basis by
Robert Adler who derived the dynamic equation for oscillator phase under the
influence of 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 of
microwave oscillators. These methods were generalized by Robert York to
include any number of 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 of
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 of 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 of 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 of frequency
locking is called "mode locking" in the laser community [13] and can result in
pulses of microwave energy.[14] The dynamic equations for mode locked systems
can be derived using the same techniques as for synchronous arrays, but the
solution of the equations for the stable states becomes much more complicated,
and requires different analytical methods.
This dissertation advances the analysis methods of systems of coupled
oscillators to include narrowband coupling networks, and generalizes Kurokawa's
method to give more accurate results. In addition, the analysis of synchronous
3
arrays is generalized to include mode locked arrays, and stable frequency locked
states of either type can be deduced in a fairly straightforward manner. For the
important class of linear arrays with nearest neighbor broadband coupling we
introduce the concept of characteristic tunings to relate the size of 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 of mode locked arrays using automatic level control oscillators, and
of synchronous arrays using phase locked loops. The detailed design of a
microwave phase locked loop is presented in chapter 4.
4
Overview
The purpose of this work is to provide analysis techniques for a class of
nonlinear oscillatory circuits and to apply these techniques to practical
microwave systems. The current research area of quasi-optical power combining
utilizes networks of coupled microwave or millimeter wave oscillators to
generate high power sources by combining the power of many small devices.
Such power combination must be coherent to avoid destructive interference
between oscillating elements, and coherence is obtained through frequency
locking of the elements. We will study two types of locking. The more common
we 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 of time, and therefore the amplitude and phase of the oscillation are
constant in the steady state. When a synchronized state of 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 of a
physically realized system are stable. The analysis of this type of system is
relatively straightforward and well documented in the literature.[15] We will
analyze models of systems of synchronous microwave oscillators and derive
some important characteristics that influence the design of such systems.
The less familiar type of locking occurs when many frequency
components exist in close proximity, and the components are separated by an
integer multiple of some small frequency separation. An example is the mode
locked laser that gives rise to a comb spectrum of evenly spaced components.
The resulting time domain waveform consists of a carrier with periodic amplitude
and phase modulation. Usually the periods of 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 of 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 of each type of locking phenomenon,
but none, to this author's knowledge, develop a method of analysis applicable to
both. This present study unifies the treatment of the two systems, within the
constraints of the approximations. It provides a general, and straightforward,
technique for determining the (asymptotic) stability of 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
although R. York extended their use to microwave oscillator systems.[4] Laser
physicists that concern themselves with the stability of 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 of 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 of an alternative method.
The properties of even simple nonlinear systems are myriad and highly
complex. The understanding of the dynamics of a forced Van der Pol oscillator
is, to this day, not complete despite over 60 years of vigorous study. The intent
of the following effort is to concentrate on a few important aspects of the
properties of a specific class of coupled nonlinear oscillators that are of principal
importance to the microwave systems engineer. Narrowing the class of
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 of arrays, the same principal information, namely, how we
maximize the ability of the oscillators to lock and the combination of 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 of attack and all of
the associated approximations, as well as our notation. The main contribution
here is the generalization of the well known analysis of synchronized systems to
that of 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 of the oscillators and coupling
networks. This method is an extension of Kurokawa'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 diagram, that graphically depicts the frequency locking ability
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of the arrays, and can indicate the tuning that gives rise to a desired phase
distribution.
The second chapter is a series of applications of the methods of chapter 1
to synchronous systems of practical importance, namely, beam steering arrays.
The first section contains an analysis of a simple model: a linear array with
frequency independent, nearest neighbor coupling. Much of this work was
developed elsewhere, but we take advantage of the simplicity to exemplify the
analysis methods and to gain a deeper understanding of this important class of
arrays. The methods of 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 of characteristic tunings for the array, a
concept we are led to quite naturally by the mathematics. We then apply our
methods to the synchronization of two 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 of oscillators in a resonant cavity.
The third chapter is devoted to analyses of mode locked arrays. In the
first section we analyze a simple type: a linear array of 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 of oscillator tunings. In the next section we 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 of the stability region. This type of array has nearly identical
properties to the synchronous array analyzed in the first section. A particular
choice of coupling phase enhances the pulsed power significantly.
The fourth chapter is a detailed description of the analysis and design of 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 of 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 of a single PLL, and measurements
are presented for a fabricated circuit. A block diagram of a beam steering array
of PLL's is presented and the analysis is identical to the first section of chapter 2.
7
[
1] J. W. Mink, "Quasi-Optical Power Combining of Solid-State Millimeter-Wave Sources,"
IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986.
[2] A. B. Popovic, R. M. Weikle II, M. Kim, and D. B. Rutledge, "A 100 MESFET planar grid
oscillator," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 193-200, Feb. 1991.
[3] P. Liao, R. A. York, "A New Phase-Shifterless Beam-Scanning Technique using Arrays of
Coupled Oscillators," IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1810-1815.
[4] R. York, R. Compton, "Mode Locked Oscillator Arrays," IEEE Microwave and Guided Wave
Letters, vol. 1, No. 8, Aug 1991, pp. 1810-1815.
[5] B. Van der Pol, "A Theory of the Amplitude of 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 of Nonlinear
Oscillations, Hindustan Pub. Corp., 1961.
[7] R. K. Brayton, J. K. Moser, "A Theory of Nonlinear Networks," Quarterly of Applied
Mathematics, Vol. XXII, No. 1, April 1964.
[8] M. Kuramitsu, 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. Nogi, "Mode Analytical Study of Cylindrical Cavity Power Combiners, IEEE
Trans. Microwave Theory Tech., vol. MTT-34, pp. 943-951.
[10] R. Adler, "A Study of Locking Phenomena in Oscillators," Proc. IRE, vol. 34, pp. 351-357,
June 1946.
[11] K. Kurokawa, "Injection Locking of Solid State Microwave Oscillators," Proc. IEEE, vol. 61,
pp. 1386-1409, Oct. 1973.
[12] R. A. York, P. Liao, J. J. Lynch, "Oscillator Array Dynamics with Broadband N-port
Coupling Networks," IEEE Trans. Microwave Theory Tech., vol. MTT-42, 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 of Automatic
Level Control Oscillators." IEEE MTT-S, International Symposium Digest, 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
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 of 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 of 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 harmonics. Next we determine
the differential equations that describe the amplitudes and phases of the voltages
and currents. After solving the differential equations for the desired locked states,
we test the stability of the states. Completing these steps gives the conditions that
must be satisfied by stable frequency locked states, periodic or almost periodic,
and shows how various circuit quantities influence these states. The increased
understanding allows us to optimize the design, and therefore the performance, of
practical frequency locked systems.
Subsequent analyses contain two essential approximations: the circuit
voltages and currents have relatively narrow bandwidths about a carrier, and the
amplitudes of 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 of such systems. Given a system that meets these
criteria we must find a suitable method of "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 = Go (1 − ε v 2 ) v = G( v ) v .
9
(1)
If the voltage across this element is sinusoidal, v = A cos(ωt ) , then the component
of current generated at frequency ω , found by direct substitution, is
i(1) ( t ) = G(1 − 3 4 ε A2 ) A cos(ωt ) = 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 1.1 for a particular class of
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
of 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 we will assume, the
comb spectrum can be represented as a single carrier frequency containing slowly
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 of 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 of 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 of amplitudes and phases to zero and solve the algebraic
system (or, equivalently, the frequency domain equations). Stability is then
10
determined by perturbing the steady state and observing the growth or decay of
the perturbations. This last step results in a linear system of differential equations
with constant coefficients, a system that is easily analyzed using techniques of
linear algebra. For almost periodic states, however, the 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 of averaging to the perturbed system to
approximate the stability of the state. It may seem like the approximations are
stacking up fast but they are satisfied by a fairly large class of almost periodic
systems. And if we extend our methods beyond the realm of accuracy many of 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 of microwave measurements will not be treated
in depth here, some comments are necessary. We are often faced with the
problem of characterizing a network through microwave measurement in order to
design a system or verify a model. Such measurements are typically of the large
signal scattering (S) parameters, in which the S parameters of a network are
measured over a particular frequency band for a range of input power levels.[1]
The result is a set of S parameters of the form S( A, ω ) . The question immediately
arises as to whether these parameters completely describe the network over the
range of measurements. The answer is generally "no." S parameters are measured
by energizing one port while terminating all others. Because the principle of
superposition does not apply to nonlinear networks one cannot assume the
response to combined stimuli will be the sum of the partial stimuli.
For some circuits, however, this type of characterization may be
approximately correct, and therefore useful. An example is the common source
FET. The parameters S11, S12 , and S22 often do not vary appreciably with
amplitude; only the magnitude of S21 varies significantly, and the reverse gain S12
is small.[2] Thus the significant nonlinear parameter is S21 and this 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
oscillators, is generally not amenable to this type of characterization. This
configuration often has large values of both S12 and S21 so that the input and
output ports are strongly coupled. In addition, we can no longer assume that S11
and S22 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 single 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 of the parameters respond instantaneously to
amplitude changes.[3] This assumption greatly simplifies the analysis of such
circuits and seems to be accurate in many cases of practical interest, and we will
use this assumption throughout this work. Relaxing this assumption requires
additional measurements to be taken for adequate network characterization. Also,
microwave measurements are made with particular terminating impedances. If 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 of 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 of 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 of interest, periodic and almost periodic
frequency locking. The equations can be used to determine the existence and
stability of such states.
A block diagram of a system of coupled oscillators is shown in figure 1.
The following analysis is based on an admittance description of 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 of circuits. For the sake of brevity, however, only the admittance
description will be presented here.
12
ic1
+
Osc I
v1
ic2
+
Osc II
N Port
Linear
Coupling
Network
v1
.
.
.
icN
+
vN
Osc N
Figure 1--block diagram of N oscillators coupled through a linear network.
The voltage at the terminals of the n th oscillator are related to the other circuit
voltages through the admittance matrix of the coupling network:
N
osc
n
Y
( A , ω )V + ∑ Y
n
n
m=1
coup
nm
N
(ω )Vm ≡ ∑ Ynm ( An , ω )Vm = 0 (3)
m=1
This results from applying Kirchoff's current law at the n th oscillator terminals
and must hold for each of the N ports. We have defined a "total" admittance
osc
coup
Ynm = Ynm
δ nm + Ynm
, where δ nm is the Kronecker delta function. The voltages Vn
are sinusoidal signals with slowly varying amplitudes and phases, which,
expressed in phasor form, are
( )
Vn ( t ) = An ( t ) e j (ωr t +φn t )
(4)
The frequency ω r may not represent a true frequency component in the system but
is simply a convenient reference. If the true frequency of the n th oscillator is not
ω r the difference will be provided by the time dependent phase φn ( t ) . This is
explained in more detail in section 1.2.1.1 below. Appendix 1.2 shows how the
dynamic equations for the amplitudes and phases can be derived from the
13
frequency domain equations. The method was first used by Kurokawa and is
known as Kurokawa's substitution.[3] We replace the frequency ω with the
A!
"instantaneous" frequency ωr + φ! − j , and expand the frequency dependent
A
admittances in a Taylor series about ω r , retaining only the constant and linear
terms. The result is a set of N differential equations:
N

∑ Y ( A , ω ) +
m=1

nm
m
r
∂ Ynm ( Am , ωr )  !
A!  
 φm − j m   Ame jφ = 0
∂ω
Am  

m
(5)
A!
In most cases the above equations can be manipulated so that each φ!m − j m is
Am
expressed in the standard form:
φ!n − j
A! n
= Fn ( A1 , A2 ,", φ1 , φ2 ,", ωr )
An
(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 of equations (6) we have 2N differential equations
that describe the dynamics of the system of coupled oscillators.
1.2.1.1 Definitions of Oscillator Frequency and Phase
Since we used an arbitrary reference frequency ω r in the derivation, the
phases in equations (6) may have linearly increasing or decreasing parts that
account for the difference between the true frequency of an oscillator and the
reference frequency. Usually it is advantageous to replace the above phases with
the "true" phases, defined below. With the notation above the "instantaneous"
phase of the n th oscillator is θn ( t ) = ωr t + φn ( t ) . We define the "true" frequency of
this oscillator as
1
T →∞ T
ωn = lim
∫
T
0
θ!n ( t )dt
(7)
and the "true" phase as φn′( t ) = φn ( t ) − ωr t . Replacing all of the phases in equation
(6) in this manner gives a more convenient form for the dynamic system.
14
Through the remainder of this work the words "oscillator frequency and phase"
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 fact which we will use to analyze these states.
One may wonder about the order of the resulting system of differential
equations. We would expect the system order to depend on the total number of
energy storage elements and not on the number of ports on the coupling
network.[5] Equations (6) are only approximately correct, and the heart of the
approximation lies in the Taylor series expansion of 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 of two
parallel resonant circuits containing nonlinear negative resistance devices and
coupled through a series resonant circuit, as shown in figure 2.
Cc
Lc
Rc
+
+
ic(t)
v1(t)
-
-G(A1) L1
Oscillator I
v2(t)
C
C
Y1
Yc (v2=0)
Y2
L 2 -G(A 2)
-
Oscillator II
Figure 2--Two self sustained oscillators coupled through a resonant network.
The oscillators are identical except for their resonant frequencies, or 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
and phases of the circuit variables, and withhold solution of the equations for the
next chapter. The frequency domain equations can be written by inspection
Ic = Y1 V1 , Ic = − Y2 V2 , Ic = Yc (V2 − V1 )
(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 of equations by one, as outlined in the general
analysis above, the order of the resulting system of 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 of figure 2 meets these criteria if G( A) is a decreasing function of
amplitude. We now approximate the frequency dependent parts of the admittance
functions with a linear frequency dependence, as demanded by Kurokawa's
substitution. The oscillator admittance function for oscillator I is
Y1 = − Go f ( A1 ) +

ω −ω 
C
ωo21 − ω12 ) ≅ − Go  f ( A1 ) + j o1 1  (9)
(
ωa 
jω1

1
is the tank resonant frequency, Go is the nonlinear device
L1C
conductance at zero voltage, f ( A) is the saturation function for the device
G
conductance, and 2ωa = o is the oscillator "bandwidth." The frequency ω1 is
C
an arbitrary Taylor expansion frequency and the best choice is the steady state (or
"true") frequency of oscillator I. If the frequency of oscillator I remains close to
its "free running" or uncoupled value ω o1 then the linear approximation is
extremely accurate, as illustrated in figure 3.
The admittance function for oscillator II is identical except that ω o2 , ω 2 replace
ω o1 , ω1 . Using the first and second of equations (8) and Kurokawa's substitution
we can write the dynamic equations for the two oscillators in terms of the
coupling current. The transfer function and its derivative at frequency ω1 are
where ωo1 =
16
ω a=1, ω oc =10
3π/2
5
Exact
Exact
Approx
Approx
Phase
Magnitude
4
3
2
1
π/2
0
6
8
10
12
14
16
6
Frequency
8
10
12
14
16
Frequency
Figure 3--The exact and approximate oscillator admittance magnitude and phase.
Agreement is excellent over a broad range of frequencies.

G
ω − ω1 
dY1(ω1 )
Y1 (ω1 ) = − Go  f ( A1 ) + j o1
= j o
 and
ωa 
dω1
ωa

(10)
After applying Kurokawa's substitution, and repeating the procedure for oscillator
II, we find the oscillator equations are
A!1 = ωa f ( A1 ) A1 + ωa Ic cos(θ1 − θc )
θ!1 = ωo1 − ωa
Ic
sin (θ1 − θc )
A1
A! 2 = ωa f ( A2 ) A2 − ωa Ic cos(θ2 − θc )
θ!2 = ωo 2 − ωa
(11)
Ic
sin(θ2 − θc )
A2
where we have used the instantaneous phase θi ( t ) = ωi t + φi ( t ), i = 1,2, c, to
simplify the notation. Note that we have expressed the coupling current in terms
of its slowly varying amplitude and phase as ic ( t ) = Ic ( t ) cos(ωc t + φc ( t )) . The
17
current expansion frequency ω c is arbitrary and the equations will take on
different forms depending on the choice of ω c .
We now consider the (possibly) narrowband coupling circuit.
admittance function for the coupling network is
Yc (ω ) =
1
Rc
1
The
(12)
ω2 −ω2
1 − j oc
2ωωac
If we were to use the broadband assumption and expand the admittance function
Yc in a Taylor series about ω c , as in reference [4], we would have the following
result:


ωc ωc2 + ωoc2

j
( ω − ωc ) 
1 
1
ωac 2ωc2

Yc (ω ) ≅
2
2 −
2


2
2
ω − ωc
Rc

ωoc − ωc 
1 − j oc

1 − j

2ωcωac
2ωcωac 



(13)
Figure 4 shows a plot of the magnitude and phase of the approximate and exact
transfer functions.
ω ac=1, ωoc =10, ω c =10
π/2
5
Approx
3
Phase
Magnitude
4
2
1
0
Exact
Approx
Exact
6
8
10
12
14
−π/2
16
6
Frequency
18
8
10 12 14
Frequency
16
Figure 4--Exact and approximate coupling circuit admittance magnitude and phase using
linear approximation for entire transfer function. The phase is quite close, but the
magnitude response is a very poor approximation.
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 of polynomial
N (ω )
functions Yc (ω ) = c
Dc (ω ) and write the relation between oscillator voltages
and coupling currents in (8) as
Dc (ω ) Ic (ω ) = N c (ω )(V2 (ω ) − V2 (ω ))
(14)
The transfer functions Dc and Nc operate on the current and voltage separately
and we may apply Kurokawa's method to each. This has the effect of linearizing
the numerator and denominator of the admittance function separately and leads to
a highly accurate approximation:
Yc (ω ) ≅
1
Rc
1
ω −ω
ω − ωc
1− j
+j
2ωcωac
ωac
2
oc
2
c
≅
1
Rc
1
ω −ω
1 − j oc
ωac
(15)
The magnitude and phase response of (15) are compared to the exact response
(12) in figure 5.
19
ω ac=1, ω oc=10
1
π/2
Exact
0.6
Approx
Approx
Phase
Magnitude
0.8
0.4
Exact
0.2
−π/2
0
6
8
10 12
Frequency
14
6
16
8
10 12
Frequency
14
16
Figure 5--More accurate approximation of coupling circuit admittance using separate
linear approximations of numerator and denominator.
Applying Kurokawa's substitution to (14) we have
Dc (ωc ) Ic e jθc +
dDc (ωc )  !
I!  1
A2e jθ2 − A1e jθ1 )
 φc − j c  =
(
dω 
Ic  Rc

1 !
1
ω − ωc
I!  
→ 1 − j oc
+j
A2e jθ2 − A1e jθ1 )
 φc − j c   Ic e jθc =
(
ωac
ωac 
Ic  
Rc

(16)
Rearranging terms gives the dynamic equations for the amplitude and phase of the
coupling current
(
ω
I!c = −ωac Ic + ac V2 cos(θ2 − θc ) − V1 cos(θ1 − θc )
Rc
θ!c = ωoc +
ωac
Rc Ic
(
(V sin(θ − θ ) − (V sin(θ − θ )))
2
c
2
1
1
))
(17)
c
Equations (11) and (17) together represent the dynamic equations for the
amplitudes and phases of the oscillators and the coupling current. The order of
the system matches the order of the exact system and due to the high accuracy of
the approximations, we expect the dynamics of the approximate system to give
good agreement with the exact system.
1.2.1.2.1 Approximation of Complex Transfer Functions
20
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
N
In = YnoscVn , I n = ∑ YnpcoupV p , n = 1,2,", N
(18)
p =1
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 ith and jth terms in
the sum above have strong frequency dependence. The network equations
become
In = YnoscVn , I n =
N
∑Y
p =1,
p≠i , j
V p + I ni + I nj ,
coup
np
(19)
Dnicoup I ni = N nicoupVi , Dnjcoup Inj = N njcoupV j
The narrowband admittances produce additional pairs of differential equations for
the associated coupling currents which produces an approximate system of nearly
the same order as the original (depending on the number of such terms that exist).
One may find that an admittance function cannot be adequately
represented by a linear approximation of the numerator and denominator. For
example, if in our circuit of figure 2 the coupling network was composed of two
second order resonant networks, as shown in figure 6,
ω oc2, ω ac
+
+
ω oc1, ω ac
v1(t)
v2(t)
-
-
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.
21
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 of second:


 ωoc2 1 + ωoc2 2

−ω2



2

1 − j


2ωωac






2 

Yc (ω ) =

Rc 
 ωoc2 1 + ωoc2 2

2

2
2
−ω  

2
1 − (ωoc1 − ω )(ωoc22 − ω ) − j 2


2
ωω


ac
( 2ωωac )


 
(20)
Using a partial fraction expansion expresses the admittance as the sum of two
second order functions. For this contrived example this step is easy:




1
1
1

 = Y + Y (21)
Yc =
+
ωoc2 1 − ω 2
ωoc2 2 − ω 2  c1 c 2
Rc 
1− j
1 − j 2ωω
2ωωac 
ac

and, as before, we define two coupling currents, one due to each admittance
function. Once again we are increasing the order of the system to achieve more
accurate results.
1.2.2 Solution of Periodic and Almost Periodic States--Locking
Diagrams
We now have a method of deriving accurate dynamic amplitude and phase
equations for a system of coupled oscillators. From this point we could analyze
any number of the myriad properties of such a system. Details of the dynamics
near synchronization boundaries, determination of stability in the large, response
for various values of initial conditions, are all subjects of the large field of
nonlinear oscillations.[7] For the sake of 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 of stable frequency locking.
Another is the dependence of the relative phases of the oscillators on the element
tunings. These are primary considerations for beam steering or pulsed arrays, and
throughout the remainder of this work these will be our primary focus.
22
Locked states occur only for values of 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 or 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 ωo1, ωo2 , and ωo3 can be
tuned independently, shown schematically in figure 7. Notice that the output
frequencies are the true frequencies ω1 , ω2 , ω3 . If we leave the tuning of one of the
oscillators fixed, say ωo2 , we can plot the values of the other tunings that result in
frequency locking. Such a plot might resemble figure 8. Note that the origin is
not zero frequency, but is the fixed frequency ωo2 . In the vicinity of the origin
where ωo1 ≈ ωo2 ≈ ωo3 we find the region of synchronization. By definition, if we
tune the oscillators to values within this region the frequencies can lock together
to a common value ω1 = ω2 = ω3 .
Tuning Ports
ω o1
ω o2
ω o3
ω1
ω2
ω3
Outputs
Figure 7--Three coupled oscillators. Tuning ports control the "free running frequencies,"
which are the oscillation frequencies in the absence of coupling.
23
ω o3
Region of
Periodic Locking
(ω 1=ω 2=ω ) 3
ω o1
ω o2
Region of QuasiPeriodic Locking
(ω 2 −ω 1=ω −ω
3 ) 2
Figure 8--The graphical representation of 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 ω o 2 is the origin of the graph, and the latter when
the three tunings are almost evenly spaced.
24
This plot does not indicate what this frequency will be, or of any other quantity
(e.g. the phases), only that the oscillators can lock. A almost periodic region
extends along the line ωo 2 − ωo1 ≈ ωo3 − ωo2 where the tunings are nearly evenly
spaced. Within this region the steady state frequencies of the locked state will be
exactly evenly spaced, that is, ω2 − ω1 = ω3 − ω2 . This type of 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, of any analysis that shows whether the regions
overlap. If they do overlap then the type of synchronization that will ensue upon
application of 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
of locked states, periodic or almost periodic, from the differential equations that
describe the amplitude and phase dynamics of 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
φ! = ωo − ω + ε f (φ , A, t , ωb ) = β + ε f (φ , A, t , ωb )
A! = g( φ , A, t , ωb )
(22)
where each variable φ , A, ωo , ω , and β are N element vectors, f and g are vector
functions, and ε and t are coupling and time parameters, respectively. The
parameter ω b is the beat frequency parameter which is smallest frequency
separation for almost periodic locking, and is zero for synchronized systems. The
vector β contains the amount of frequency pulling of 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 of which will be considered in the next section. The most important
assumption in equations (22) is that the steady state frequencies ω n do not appear
within the functions f and g. If they do appear, we must introduce new variables
appropriately to increase the order of the system using the methods of section
1.2.2.1.
25
For the case of synchronization the frequencies and phases are constant, so
the existence of states can be determined by setting the time derivatives in (22)
equal to zero and solving the algebraic system:
β + ε f (φ , A) = 0
(23)
g(φ , A) = 0
(Note: the time dependence in the functions f and g vanish for synchronized
systems). We assume, but do not prove, that values of frequency, phase, and
amplitude that satisfy the above system indicate the existence of a synchronized
state.
Finding states of almost periodic locking is much more difficult since
amplitudes and phases are not constant, but we do know that they are periodic
functions of 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 ε:[8]
φ ( t ) = φ ( 0 ) ( t ) + ε φ (1) ( t ) + ε 2 φ ( 2 ) ( t ) +"
A( t ) = A( 0 ) ( t ) + ε A(1) ( t ) + ε 2 A( 2 ) ( t ) +"
β=β
(0)
+εβ
(1)
(24)
+ ε β +"
2
(2)
Substituting the above variables into equations (22) and equating like powers of ε
gives a sequence of differential equations that can be solved recursively by
enforcing the periodicity of φ and A. This procedure often becomes prohibitively
complicated after the second order, so the results will be accurate only for
relatively small values of ε.
Performing the above substitution, for the zero order ( ε = 0 ) we have
φ!( 0 ) = β ( 0 )
(25)
A! ( 0 ) = g ( A( 0 ) , φ ( 0 ) , t )
Enforcing the periodicity of φ ( 0 ) and A( 0 ) gives β ( 0 ) = 0, φ ( 0 ) ≡ φo = const , and
g ( A( 0 ) , φo , t ) = 0 where the brackets denote time average over one period. The
vector of phases φ o represents, approximately, the time average value of the phase
φ ( t ) . We will see later that the dependence of the frequency pullings β on the
26
time average phases φ o can indicate stability of the state. The higher order
amplitude and phase corrections cannot be evaluated explicitly for this general
case, but we can use the periodicity of the phases to express the frequency pulling
vector as
β = −ε f (φ , A, t ) = −ε F ( φo )
(26)
Thus the frequency pullings are functions of the time average phase variables,
among other things. Given a set of tunings ω o this perturbation technique allows
us to determine, at least theoretically, the time dependent amplitudes and phases
and the frequency pullings to any desired degree of accuracy. Practically, we can
determine the solution to the first or second order, but this order of approximation
usually provides a great deal of information and insight to the conditions for the
existence of almost periodic states. The next step is to determine the stability of
states.
1.2.2.2 Stability of Locked States [9]
Once a locked state is found by solving equations (26) the stability of the
state must be tested. The following analysis shows that equations (26) contain
information about the stability, at least approximately in the case of almost
periodic locking. Stability can be tested by applying small perturbations to the
phase variables and finding the stability of 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 of an
approximate "averaged" system for the phase perturbations. This will allow us to
investigate the stability of periodic solutions using the well known techniques of
linear algebra applied to systems with constant coefficients.
The assumptions are contained in the derivation of appendix 1.3, 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 of
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 of the transient amplitude response and consider only the
"algebraic" influence of the amplitudes (see appendix 1.3 for details). We can
then reduce the 2N order system (23) to a N order system. It is important to
27
understand that we are not neglecting the influence of the amplitudes entirely. In
fact, 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 of appendix 1.3 gives an approximate perturbational system
for equations (22) that has constant, instead of periodic, coefficients. If we
perturb the phases from their steady state values
φ(t ) = φ p (t ) + δ (t )
(27)
where φ p ( t ) is the periodic steady state solution and δ ( t ) is the perturbation, then
the perturbations behave according to the linear system
~
δ!( t ) = − C ( t ) δ ( t )
(28)
~
where the matrix C ( t ) is a periodic function of 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 of the actual
perturbation δ ( t ) . The constant "stability" matrix C, obtained from (26), is
Cnm =
∂βn
∂φom
(30)
where the phase φ om is the m th element of the time average phase vector φ o .
Stability of a state is ensured when the real parts of the eigenvalues of C are
positive. Thus, both existence and stability of states is supplied by the frequency
pulling equations (26).
To illustrate the above concepts figure 9 shows a comparison of 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 of a four element array of Van der Pol oscillators. Instead of the
phase variables themselves, the second differences of 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
28
parameter ε was chosen somewhat large to show that good agreement is obtained
in this case.
ε=0.25
η=0.5
1.5
∆∆δ2
∆∆d2
1
0.5
τ
0
∆∆d1
-0.5
∆∆δ1
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
Conductance
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 of 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 i = f ( 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 of
the "instantaneous" type. The applied voltage is a narrowband signal at carrier
frequency ω with slowly varying amplitude and phase:
v ( t ) = A( t )cos(ω t + φ ( t )) ,
29
A! !
, φ << ω
A
(31)
The nonlinear device gives rise to a spectrum of frequencies
(
)
i( t ) = f A cos(ω t + φ ) = ∑ ( an cos( nωt ) + bn sin( nωt ) )
(32)
n
We can find the amplitudes of the fundamental components using the
orthogonality of the circular functions:
1
a1 =
π∫
b1 =
1
2π
0
π
∫
2π
0
i( t ) cos(ω t ) d (ωt )
i( t ) sin(ω t ) d (ωt )
(33)
The device curve, and therefore the current, can be expressed as a Taylor series
i( t ) = ∑ cn v n = ∑ cn An cos n (ωt + φ )
n
(34)
n
which gives the fundamental components in terms of the input voltage
2 T n
A ( t ) cos n (ωt + φ ( t )) cos(ωt ) dt
∫
0
T
n
2 T
b1 = ∑ cn ∫ An ( t )cos n (ωt + φ ( t )) sin (ωt ) dt
T 0
n
a1 = ∑ cn
(35)
If the amplitude and phase vary negligibly over a cycle we can simplify equations
(35). The amplitude can be pulled outside the integral and the remaining integral
can be simplified by changing variables:
2 T n
2 T +φ
A ( t ) cos n (ωt + φ ( t )) cos(ωt ) dt ≅ An ∫ cos n (ωt ) cos(ωt − φ )dt
∫
T 0
T φ
2 T
2 T


≅ An cos(φ ) ∫ cosn +1(ωt ) dt + sin(φ ) ∫ cos n (ωt ) sin(ωt ) dt 
0
0
T
T


(36)
The second integral vanishes because of symmetry of the integrand. A similar
analysis can be carried out for the second integral of equation (35). The
fundamental current components are then
30
a1 = ∑ cn An ( t )cos(φ ( t ))
2 T
cos n+1(ωt ) dt
∫
0
T
n
2 T
b1 = − ∑ cn An ( t )sin(φ ( t )) ∫ cos n +1(ωt ) dt
T 0
n
(37)
The current at the fundamental is
i(1) ( t ) = a1 cos(ωt ) + b1 sin(ωt )


2 T

= ∑ cn  ∫ cos n+1(ωt ) dt  An ( t )  cos(ωt + φ ( t ) )

 n T 0

(38)
= G( A( t )) A( t )cos(ωt + φ ( t ) )
where G( A) is the nonlinear conductance we sought that gives rise to the correct
fundamental component of current, and is given explicitly by
2 T

G( A) = ∑ cn  ∫ cos n+1 (ωt ) dt  An−1
0


T
n
(39)
Note that harmonics are not generated by this amplitude dependent conductance.
This method also shows how to compute G( A) from the Taylor series expansion
of the device I-V curve. Using the previous example of nonlinear conductance
i = Gv (1 − ε v 2 ) the above equation gives G( A) = G(1 − 3 4 ε A2 ) , the result found
previously using direct substitution. The important result of this analysis is that,
to the first approximation, the fundamental component resulting from the
narrowband excitation of an "instantaneous" nonlinear device is in phase with the
exciting signal, but with altered amplitude.
Capacitors 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 chargevoltage (Q-V) curve
q = f (v)
(40)
As before, we consider a narrowband voltage v ( t ) = A( t )cos(ω t + φ ( t )) and we
immediately see that this problem is identical to the previous. The fundamental
component of charge is therefore given by
31


2 T

q(1) ( t ) = ∑ cn  ∫ cos n+1(ωt ) dt  An ( t )  cos(ωt + φ ( t ) )


 n T 0
= F ( A)cos(ωt + φ )
(41)
Differentiating this with respect to time gives the relation between current and
voltage
dF !
i(1) ( t ) = −(ω + φ!) F ( A)sin (ωt + φ ) +
A cos(ωt + φ )
dA
(42)
Defining an amplitude dependent capacitance with F ( A) = C ( A) A gives the
fundamental component of current versus voltage,
dC !
i(1) ( t ) = −(ω + φ!) C( A) A sin (ωt + φ ) + CA! cos(ωt + φ ) +
AA cos(ωt + φ ) (43)
dA
and implies the following element admittance at constant amplitude and phase:
Ycap (ω ) = jω C( A)
(44)
This admittance would be the result of a large signal microwave measurement.
Applying Kurokawa's substitution here, we find
i(1) ( t ) = −(ω + φ!) C( A) A sin(ωt + φ ) + CA! cos(ωt + φ )
(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 of Kurokawa's substitution. The missing term creates
additional phase shift due to amplitude modulation and becomes small for weakly
C( A )
dC( A)
nonlinear capacitors where
>>
. Similar results hold for inductors
A
dA
with a nonlinear flux linkage curve ψ = f (i ).
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
32
conductance that responds instantaneously to signal amplitude at the oscillator
terminals. If the frequency response of such a network is known for a fixed
excitation amplitude, over a wide range of amplitudes, then we can determine the
network response to a signal that contains amplitude and/or frequency modulation.
In the case of narrowband signals the result collapses to Kurokawa's substitution.
Consider
a
time
invariant
network
whose
input
is
V (ω )
v1( t ) = A( t )cos(ωot + φ ( t )) and with frequency response H ( A, ω ) = 2
V1 (ω ) .
The frequency response depends on the input amplitude because of the possible
presence of nonlinear conductances but is defined by maintaining 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, we can account for the amplitude dependence of the above
transfer function by simply substituting in A(t ) for the constant value A. The
frequency dependence of H, however, gives rise to modulation dependent terms,
as this analysis will show. Since all of 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 of the output.
The input signal we will consider is
v1( t ) = A( t )e jφ ( t )e jωot
(46)
and the spectrum of the modulation is
∞
S (ω ) = ∫ Ae jφ e− jωt dt
(47)
−∞
∞
This spectrum is related to the input signal spectrum V1(ω ) = ∫ Ae j (ωot +φ )e− jωt dt
−∞
through the modulation property: [10]
V1(ω ) = S (ω − ωo )
(48)
The time domain filter output can be expressed in terms of the modulation
spectrum via the inverse Fourier transform
33
∞
v2 ( t ) = ∫ H ( A, ω )V1 (ω )e jωt
−∞
∞
dω
2π
= ∫ H ( A, ω )S (ω − ωo )e jωt
−∞
∞
dω
2π
j ω ω t
= ∫ H (ω ′ + ωo )S (ω ′ )e ( ′+ o )
−∞
(49)
dω ′
2π
The third integral is obtained through the substitution of variables ω = ω ′ + ω o .
Next, expand the filter transfer function in a Taylor series about the carrier ω o and
insert into (49)
∞
∞ 
1 d n H ( A, ωo )
dω ′
n
(
v 2 ( t ) = ∫ ∑
ω ′ )  S (ω ′ )e j (ω ′+ωo )t
n
−∞
dω
2π
 n=0 n !

(50)
Swapping the order of integration and summation we have
1 d n H ( A, ωo ) jωot ∞
dω ′
n
e ∫ ( jω ′ ) S (ω ′ )e jω ′t
n
−∞
2π
d ( jω )
n=0 n !
∞
v2 ( t ) = ∑
(51)
but the integral is simply the nth time derivative of the modulation, so the
complex output voltage is
n
jφ
∞
1 d n H ( A, ωo ) d ( Ae ) jωot
(52)
v2 ( t ) = ∑
e
n
dt n
d ( jω )
n=0 n !
The true time domain output voltage is obtained by taking the real part of (52).
Since we assumed that the input and output signals have slowly varying
amplitudes and phases the higher order time derivatives of 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 of the above series

∂ H ( A, ω )  !
A!  
 φ − j   Ae j (ωot +φ )
v2 ( t ) ≅  H ( A, ωo ) +
∂ω
A  


ωo
34
(53)
The above expression can be derived by substituting the "instantaneous"
A!
frequency ω ( t ) = ωo + φ! − j into the transfer function and linearizing about the
A
carrier frequency. This substitution was first used by Kurokawa to derive the
amplitude and phase dynamics for oscillators.[3]
35
Appendix 1.3 Approximate Stability of Locked States
The phase variables of equation (22) can either be perturbed directly,
leaving the β unchanged, or by perturbing the free running frequencies. We will
show how the relation between these two types of perturbations can be used to
determine the coefficients of the averaged linear system. Treating the first type of
perturbation, expand the φ's and the A's about a periodic solution:
φ! p + δ! = β p + ε f (φ p + δ , A p + α , t )
A! p + α! = g (φ p + δ , A p + α , t )
(54)
where δ and α are the perturbations and the superscript 'p' denotes the periodic
solution. The dynamic equations of the perturbation, or the variational equations,
result from a first order expansion about the periodic solution:
 ∂f
dδn
∂f
= ε ∑  n δm + n
dt
∂Am
m  ∂φm φ p
m

 ∂g
dα n
∂g
= ∑  n δm + n
dt
∂Am
m  ∂φm φ p
m


αm 
Amp

αm 

Amp

(55)
Note that the coefficients multiplying the perturbations are periodic functions of
time.
When ascertaining the stability of most systems of coupled oscillators it is
sufficient to study the system response to initial values of the phase perturbation
variables only, maintaining the initial amplitude perturbations at zero.[8] This is
possible as long as the transient responses of the amplitude variables decay
quickly. The second equation in (55) is an inhomogeneous system of linear
differential equations for α where the forcing functions are superpositions of the
variables δ. The solution, assuming it exists, can be written as the sum of a
transient homogeneous solution and the particular solution. In addition, the
particular solution can always be written as a linear combination of the phase
variables. Thus the general solution has the form
αn ( t ) = hnhomo ( t ) + ∑ qnm ( t ) δm ( t )
m
36
(56)
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 of equations (55) eliminates the amplitude perturbation
variables. The dynamic equations for the phase perturbations become
 ∂f

dδn
∂f
= ε ∑  n δm + n ∑ qmlδl 
dt
∂Am l
m  ∂φm

 ∂f

∂f
= ε ∑  n + ∑ n qlm  δm
m  ∂φm
l ∂Al

(57)
The partial derivatives are still evaluated at the periodic solution but this is not
explicitly shown for notational convenience. The derivatives of the phase
perturbations are proportional to the small parameter ε 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 of Krylov and Bogoliubov.[11] Using brackets to
represent an averaging operation, the slowly varying "average" value of the n th
phase perturbation is defined as
dn ( t ) = δn ( t )
(58)
and the equivalent averaged system with constant coefficients is:
 ∂f

∂f n
d
dn = ε ∑  n + ∑
qlm  d m
∂Al
dt
m  ∂φm
l

= −ε ∑ Cnm d m
(59)
m
For a complete justification of this method see references [11], [12], [13].
Fortunately, the coefficients Cnm need not be evaluated directly but can be
calculated from the frequency pulling variables β, as we will now show.
Consider an infinitesimal change dβ to the frequency pulling of 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
 ∂f

∂f
d
dφnp ) = dβn + ε ∑  n dφmp + n dAmp 
(
∂Am
dt
m  ∂φm

37
(60)
The partials are evaluated at the original periodic values, that is, before β 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 of the derivative of
the phase perturbation must vanish:
T
1 d

( dφ p ) dt = 0
∫

T 0  dt

(61)
so,
dβn = −ε ∑
m
∂f n
∂f
dφmp + n dAmp
∂φm
∂Am
(62)
Using the expression in (56) for the amplitudes we have

 ∂f
∂f
dβn = −ε ∑  n + ∑ n qlm  dφmp
m
l ∂Al

 ∂φm
(63)
The phase perturbation dφ mp is a time dependent quantity that satisfies equation
(60). Since its derivative is proportional to the small quantity ε it must vary
slowly and we can therefore approximate the average value of the product of the
bracketed terms with dφ mp in equation (63) as the product of the average values
(note that the differential change in β in equation (60) can be neglected when
determining the smallness of dφ mp because the average value of the summation
must cancel dβ in order to maintain a bounded dφmp ). The approximate
expression for dβ becomes
 ∂f

∂f n
dβn = −ε ∑  n + ∑
qlm  dφmavg
∂Al
m  ∂φm
l

(64)
where
T
dφmavg =
1
dφmp ]dt
[
∫
T 0
(65)
and T is one period. Thus the coefficients in equation (59) can be identified as
Cnm =
∂βn
∂φmavg
(66)
38
This simple relation allows us to evaluate the coefficients of the averaged system
by simply differentiating the expressions for the frequency pulling of each
oscillator. The stability of a periodic state can then be determined by computing
the eigenvalues of the matrix of coefficients. Note that this result is exact for the
case of periodic synchronization. It is satisfying to see that the result represents
the "average" values for the almost periodic system under the stated conditions.
39
[
1] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, Prentice-Hall, NJ, 1984.
[2] K. Johnson, "Large Signal GaAs MESFET Oscillator Design," IEEE Trans. Microwave
Theory Tech., vol. MTT-27, pp. 217-227, Mar. 1979.
[3] K. Kurokawa, "Some Basic Characteristics of 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 of
Vector Fields, 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 of Mode Locked States of 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 of 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.
40
Chapter 2
Synchronous Arrays
In this chapter we apply the theory developed in chapter 1 to specific
problems relevant to practical synchronous oscillator arrays. The chapter begins
with the analysis of 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 simple
case of linear arrays with nearest neighbor coupling due to the current interest
and practical importance in beam steerable transmitting arrays. The particular
choice of zero degrees coupling phase represents an optimum value and greatly
simplifies the equations. The simple form allows us to graphically depict the
synchronization diagram and to understand the relationship between the size of
the synchronization region, the phase sensitivity, and the transient response of the
phases, for arrays of any size. These attributes are tied together through the
"characteristic tunings."
The second half of the chapter is devoted to the synchronization of
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 of weak and strong
coupling, and narrow and broad coupling bandwidths. This analysis is the first
step toward an analysis of many oscillators coupled through a resonant cavity, a
case of practical importance.[5]
2.1 Synchronization of Oscillators through Broadband Coupling
Networks
Simple models of 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 of many microwave
oscillator systems. Of 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 of coupled
microwave oscillators and determine the characteristics of the system in the
synchronized state. We assume nearly sinusoidal oscillations, slowly varying
41
1 !
amplitudes and phases ( φ! n and
An << ω ), nonlinear oscillators of the type in
An
section 1.2.2, and 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 of a state by computing the eigenvalues of
the stability matrix, as described in the previous chapter. Since one of the phases
is arbitrary (due to the arbitrary time reference) one of the eigenvalues of the
stability matrix is zero. We will show how to remove the zero eigenvalue,
thereby reducing the order of the system by one, while preserving all of the
nonzero eigenvalues.
After applying the analysis methods of section 1.2.1 the equations for the
amplitudes and phases have the form, for n = 1, 2, " , N (see [6] for details of the
derivation),
N
A! n = ηS n ( An ) An + ε ∑ λnm Am cos( φn − φm + Φ nm )
m=1
1
φ!n = βn − ε
An
N
∑λ
m=1
nm
Am sin (φn − φm + Φ nm )
(1)
where β n = ω on − ω is the frequency pulling, η is a nonlinearity parameter that
controls the amount of nonlinearity present in the oscillators, ε is a small
coupling parameter, λ nm are variations in the coupling strengths, and Φ nm is the
phase delay from the n th to the m th element. Note that the free running
frequency, or tuning, of the n th oscillator is denoted ω on and the true frequency of
the synchronized system is denoted ω . The function Sn ( An ) is the conductance
saturation function for the n th 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, two
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. We
set the time derivatives in equation (1) equal to zero and solve the remaining
algebraic system for the N amplitudes, N-1 linearly independent phases (or
combinations of phases), and the frequency ω . If a solution exists for a given set
of tunings, we conclude (although we will not prove) that a synchronized state
exists. To find the stability of the state we perturb the amplitude and phase
42
variables, derive the linear dynamic system for the perturbations, and determine
its stability. In practice, however, the solution of the algebraic equations is
difficult and we usually resort to approximations. A tremendous simplification
occurs when the coupling between elements, represented by the parameter ε, is
relatively weak. We then find an approximate solution using a power series
expansion. This is the same procedure outlined in section 1.2.2 for the
approximate solution of mode locked states.
Assuming inter-element coupling is weak, we expand all unknown
variables in power series of ε,
φn = φn( 0 ) + ε φn(1) + ε 2 φn( 2 ) +"
An = An( 0 ) + ε An(1) + ε 2 An( 2 ) +"
βn = β
( 0)
n
+εβ
(1)
n
(2)
+ ε β +"
2
(2)
n
insert these into equations (1), and equate like powers of ε. For the zeroth order,
A! n( 0 ) = S n ( An( 0 ) ) An( 0 ) = 0
φ!( 0 ) = β ( 0 ) = 0
n
(3)
n
which gives the zeroth order steady state amplitudes (by the zeros of Sn ( An ) ) and
shows that the frequency pulling is at least a first order quantity. The phases are
all constant, and will be denoted φ (n0) ≡ φ on as before, but they cannot be
determined by the above equations. The first order phase equations are
N
φ!n(1) = βn(1) − ∑
m=1
Am( 0 )
λ sin( φon − φom + Φ nm ) = 0
An( 0 ) nm
(4)
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 of the oscillator phases are
N
βn = ε ∑
m=1
Am( 0 )
λ sin( φon − φom + Φ nm )
An( 0 ) nm
(5)
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
43
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 of tunings ω on we can
solve the above system for the frequency and phase differences. We can then test
the stability by applying the result of section 1.2.2.2. The dynamic system for the
phase perturbations is
N
δ!n = − ∑ Cnmδm
(6)
n =1
A state is stable when the real parts of the nonzero eigenvalues of the matrix
whose elements are
Cnm =
∂βn
A( 0 )
= −ε m( 0 ) λnm cos( φon − φom + Φ nm )
∂φom
An
(7)
are positive.
The frequency pulling variables of equations (5) depend only on the phase
differences and not the phases themselves, so one of the phases is arbitrary. Thus
the above stability matrix (7) has one zero eigenvalue. We can reduce the order
of 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 of N-1 phases through a
linear transformation φ ′ = Aφ where A is a real N − 1 × N matrix, and then
applying the same transformation to the frequency pulling variables to form a
new set β′ = Aβ . The "reduced" N − 1 × N − 1 stability matrix, derived in the
same manner as equation (7),
Cnm
′ =
∂βn′
∂φom
′
(8)
contains the same eigenvalues as the original matrix C with the zero eigenvalue
removed. This is proved in appendix 2.1.
Given a set of tunings ω o we can, in principal, determine the existence of
a synchronous state of coupled oscillators using equations (5) and test the
stability of the state by finding the stability of the linear perturbational system (6).
These two sets of equations are connected through the stability matrix of equation
44
(7) and, therefore, the tendency for the elements to synchronize, the dependence
of 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 of the equations is relatively simple. However,
in the following section we will consider a coupling scheme for a useful practical
case that considerably simplifies the analysis, and allows us to understand the
interdependence of 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 of 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 elements
that we can achieve, but also on the physical spacing of the radiating elements.
We will only consider the former here since this problem is treated in detail
elsewhere.[9
9]
Tuning Ports
ω o1
ω o2
ε,Φ
ω
ω o3
ε,Φ
ω
ω o3
ε,Φ
ω
ω
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.
45
The diagram of figure 1 shows a four element linear array of oscillators
coupled only between nearest neighbors. The coupling constraint simplifies the
design of 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 of nearest neighbor coupling is not overly restrictive. To
further simplify the analysis we will assume that the coupling magnitude and
phase 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
βn = ε
n +1
∑ sin(φ
on
m= n −1,
m≠ n
[
− φom + Φ) = ε sin( ∆φn−1 + Φ) − sin( ∆φn − Φ)
]
(9)
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 of N
equations in N unknowns: N-1 phase differences and the true frequency ω.
Because of the coupling scheme only the N-1 phase differences between adjacent
pairs appear so these represent a natural choice of independent phase variables
for the problem. Reducing the order of the system in the manner outlined in the
previous section, we form the difference between adjacent frequency pulling
variables
[
∆βn ≡ βn+1 − βn = ε − sin ( ∆φn−1 + Φ) + 2 cos( Φ) sin ( ∆φn ) − sin ( ∆φn+1 − Φ)
]
(10)
Notice that this eliminates the true frequency ω from the system and therefore
equations (10) represent N-1 equations in N-1 unknowns. The system of
differential equations governing phase perturbations for the reduced system is
N −1
∆δ!n = − ∑ Cnm ∆δm
(11)
m=1
where the ∆δ n are actually the difference between adjacent phase perturbations,
defined by ∆δ n = δ n +1 − δ n for n = 1, 2, ", N − 1. The stability matrix for the
reduced system is (we will drop the prime for simplicity)
Cnm =
∂∆βn
∂∆φm
(12)
46
When the real parts of the eigenvalues of 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 − 1 × N − 1 matrices
 2 −1

0 1





−1 2 #

−1 0 #

A=
, B = 

# # − 1
# # 1






−1 2 
−1 0


0
0
0
(13)
0
the N-1 element sine and cosine vectors
 cos( ∆φ1 ) 
 sin ( ∆φ1 ) 




 cos( ∆φ2 ) 
 sin ( ∆φ2 ) 
u=

, v = 
$
$




 cos( ∆φ )
 sin ( ∆φ )


N −1 
N −1 
(14)
and the N − 1 × N − 1 diagonal sine and cosine matrices

 sin( ∆φ )
1




sin( ∆φ2 )
U =

#



sin ( ∆φN −1 )

0
0
(15)
 cos( ∆φ )

1




cos( ∆φ2 )
V =

#



cos( ∆φN −1 )

0
0
The frequency pulling equations (10) become
∆β = ε (cos( Φ) Au + sin( Φ) Bv )
(16)
and the perturbational system is
∆δ! = −C∆δ
(17)
47
with stability matrix
C = ε ( cos( Φ) AV − sin ( Φ) BU )
(18)
Equations (16) and (17) completely describe the array frequency, phases, and
stability, to the first order of approximation. The simple form of the equations
allows us to gather further insight into the array characteristics. For example, the
T
T
condition for broadside radiation, ∆φ = 0, gives u = (0,",0) and v = (1,",1) .
For a coupling phase of Φ = 0 equation (16) shows that the broadside radiation
condition is ∆β = 0 which implies identical element tunings across the array. In
addition, we will show in the next section that for Φ = 0 the stability matrix
C = εAV has positive eigenvalues when each phase satisfies − π 2 < ∆φ n < π 2 .
Thus this desired phase distribution represents a stable state of the system.
However, for Φ = 90 degrees the matrix B is singular for even values of
N. This implies that the pulling equations are not linearly independent and we
cannot determine the phases from a given set of tunings. For odd N the matrix B
is nonsingular, so an inverse exists, but the stability matrix C = − ε BU always
has zero trace. Since the trace of a matrix is the sum of its eigenvalues, the real
parts of the eigenvalues cannot all be positive and still give a zero result, hence
no stable states exist. When Φ is some other value the stability matrix is a
weighted combination of AV and BU (equation (18)) and we can probably infer
that the stability region is maximized for Φ = 0 , although we have not proved
this. It is easy to show that the tendency for synchronization is maximized at
Φ = 0 for N=2 elements, but this is difficult to show for larger N. In summary,
Φ = 0 represents an attractive value of 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
Diagrams
Degrees
Coupling
Phase--Synchronization
When we assume Φ = 0 the frequency pulling equations and the
perturbational equations take on very simple forms
and
∆β = ε Au
(19)
∆δ! = − C ∆δ = − ε AV ∆δ
(20)
48
To gain more insight into the array behavior we utilize synchronization diagrams
to graphically depict the array element tunings that can result in stable
synchronization. Such a diagram might appear as in figure 2 for a three element
array.
ω o3
ω o1
Figure 2--Example of a synchronization region for a three element array. The second
oscillator tuning remains fixed. At the origin ω o1 = ω o 2 = ω o3
Since there are only N-1 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 exist. 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
Φ = 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 of 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 [-π,π] is sufficient) then the elements of u will span [-1,1].
This is depicted for N=3 in figure 3.
49
∆φ2
π
u2
1
Nonlinear
−π
π
-1
∆φ1
−π
1 u1
-1
∆β2
Linear
∆β1
Figure 3--The square region in ∆φ space is mapped nonlinearly to the square region in u
space which is then mapped to the rectangular parallelepiped in ∆β space. The inscribed
circle maps to an inscribed ellipse whose length and direction of major and minor axes
are given by the eigenvalues and eigenvectors of 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-1 dimensions onto a rectangular
parallelepiped of N-1 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 ∆β n , 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, ω o ,
we can compute the N-1 element vector of frequency pullings using a linear
50
transformation. We will choose the first element tuning as fixed, although this
choice is arbitrary. The frequency pullings are then written as
  ωo2   ωo1
 ∆β1   1


 
  
∆
β

  ωo 3   0 
1
1
−
2

=
 $  −  $ 
 $  
#
#


 
  

 ∆βN −1 
− 1 1   ωoN   0 
0
0
or
∆β = Dωoc − ωoe
(21)
Thus, to map the existence region in the space of frequency pullings to the space
of element tunings we use the transformation ωoc = D −1( ∆β + ωoe ) which changes
the size and eccentricity of the parallelepiped but retains its basic shape. This is
illustrated in figure 4.
ω o3
ω o2
Figure 4--Synchronization region in the plane of 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
51
stability matrix, C = ε AV . As mentioned before, when the real parts of the
eigenvalues are positive the state is stable. However, in the present case
determination of stability is simple due to the nature of the matrices A and V.
The matrix A defined by equation (13) is positive definite and the matrix V
defined by equation (15) is also positive definite when each of the phases lies in
the range [− π 2 , π 2 ]. Since the product of two positive definite matrices is also
positive definite, the eigenvalues of the stability matrix are all real and positive
when the phases lie in the above range. This range of phases is sufficient to
cause the vector u in equation (19) to span all of its possible values, which proves
that the stability region fills the entire existence region. Furthermore, over this
range of phases the sine functions within u are one-to-one, so the transformation
from ∆φ 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 Φ = 0 is an optimum choice in another respect. Although the
above analysis was illustrated graphically for the case N=3, all of the results hold
for arrays of 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.
2.1.1.1.3 Phase Sensitivity
Another attractive characteristic of zero degrees coupling phase is that the
center of 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 of loss of synchronization due to random tuning errors will be minimized.
Also, the sensitivity of 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 of phase
sensitivity in more detail.
A convenient measure of phase sensitivity is the change in length of the
phase vector for a given change in the length of 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
pullings through the linear transformation of 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:
d∆β = ε AV d∆φ = C d∆φ
52
(22)
The sensitivity defined above is then given by
S ∆∆βφ
(
)
1
 d∆β T (C −1 )T C −1 d∆β  2
 d∆φ T d∆φ 


≡
 =
T
T

d∆β d∆β
 d∆β d∆β 


1
2
(23)
If the change in tuning is an eigenvector of the matrix C −1 with eigenvalue λ−1 ,
then the sensitivity is simply S∆∆φβ = λ−1 . Note that the eigenvectors of C −1 and C
are identical and the eigenvalues are reciprocals of one another. One can show
that the eigenvectors of C are the directions for which the sensitivity is either a
local maximum or minimum. Thus a tuning change in the direction of an
eigenvector of the stability matrix C, called a "characteristic tuning" for the array,
causes an extremum of the phase sensitivity, which is given by the reciprocal of
the corresponding eigenvalue of C. This result, together with the results of
section 2.1.1.1.1, shows that the size of the existence region is directly related to
the phase sensitivity. Indeed, we expect to lose synchronization more quickly if
we tune in the direction of high phase sensitivity. The characteristic tunings also
play an important role in the transient phase response of 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 of 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 of
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 of
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 of the synchronization region, but
53
the computed settling times give approximate results for large tuning variations,
and the physical insight developed is useful in understanding the dynamic phase
behavior.
Tuning any of the array elements perturbs the oscillator phases and a
certain amount of time is required before the phases reach a new steady state.
Although a general analysis of the effects of arbitrary time dependent tuning
changes is feasible, in the following we will consider only step changes of
element tunings. This type of 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, or tunings, with infinite
rapidity. In practice, we will not have such control, but the following results will
hold approximately if the element tunings respond faster than the fastest time
constant of the array. If this is not the case the time constants of 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]
∆δ ( t ) = e − C t ∆δ ( 0)
(24)
where the matrix exponential is defined through the usual power series expansion
e− C t = I − C t + 12 ( C t ) + "
2
(25)
Thus the behavior of 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, or by
operation near the edge of 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 of 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 of linear system theory tells us that the eigenvalues of the
stability matrix in equation (20) are the reciprocals of the time constants that
govern the exponential decay of 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
tunings. As mentioned previously, the eigenvalues and eigenvectors of the
54
matrix C are directly related to the size, shape, and orientation of the locking
region.
The phase change due to an arbitrary, but infinitesimal, tuning change is
given by
d∆β = C d∆φ
(26)
This shows that if the phase change is an eigenvector of the stability matrix, then
the corresponding tuning change is proportional to the same eigenvector and the
constant of proportionality is the eigenvalue, that is,
C d∆φ = λd∆φ = d∆β
(27)
Thus, the characteristic tuning changes are found by computing the eigenvectors
of the stability matrix C.
Practical systems will be designed to operate near the center of the
locking region to avoid excessive phase sensitivity to tuning changes which
occurs near the region edge. Near the region center we can make the
approximation in equation (19) that u ≅ ∆φ , which gives a linear relation between
the phase differences and oscillator tunings differences
∆β ≅ ε A ∆φ
(28)
In addition, we can simplify the linear system (20)
∆δ! ≅ −εA ∆δ
(29)
because the matrix V in 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 of the locking region. The
tuning difference vector prior to the step change is ∆β init = ( ∆β1init " ∆βNinit−1 )
with a corresponding phase difference vector ∆φ init . After the change we have
T
∆β final = ( ∆β1 final " ∆βNfinal
and the steady state phase vector is ∆φ final . Just
−1 )
T
after the step change the phase vector is time dependent, ∆φ ( t ) , and the
difference, or perturbation, ∆δ ( t ) = ∆φ ( t ) − ∆φ final satisfies the linear system
55
(29) with the initial value ∆φ init − ∆φ final . The initial and final phase vectors are
directly related to the oscillator tunings through (29):
∆φ init =
∆φ
final
1
ε
=
A−1∆β init
1
ε
(30)
A ∆β
−1
final
so the phase perturbation at t = 0 is
∆δ ( 0) =
1
ε
(
)
A−1 ∆β init − ∆β final .
(31)
and the system dynamics will evolve according to
∆δ ( t ) =
A ( ∆β
ε[
1
−1
init
)]
− ∆β final e −εAt
(32)
The general behavior of (32) is quite complicated and probably best understood
by decomposing the response into a superposition of characteristic responses.
The next step, therefore, is to find the eigenvalues and eigenvectors of the matrix
A.
Simple closed form expressions for the eigenvalues and eigenvectors of A
can be found by noting the similarity between the matrix A and the second
d2
derivative operator 2 . The k th eigenvalue is given by
dx
 kπ 
 , k = 1,2,", N − 1
 2N 
λk = 4 sin 2 
(33)
and the corresponding eigenvector is
(e% )
k
n
=
2
 nπ 
sin  k
 , k = 1,2,", N − 1, n = 1,2,", N − 1
 N
N
(34)
where n indexes the individual elements of the vector. The set of 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.
56
λ1 = 0.382 , e(1)
 .372


.602

=
.602


 .372
λ3 = 2.618 , e( 3)
 −.602


.372 

=
.372 


 −.602
.
, e( 2 )
λ2 = 1382
 .602 


.372 

=
−.372


 −.602
λ4 = 3.618 , e( 4 )
 .372 


−.602

=
.602 


 −.372
(35)
Figure 5--Eigenvalues and eigenvectors for a five element array (N=5).
Using some basic results of linear system theory [3] one can show that if
the tuning change is proportional to one of the eigenvectors, that is, if
∆β init − ∆β final = Ω e% k
(36)
where Ω sets the step size, then the phase perturbations will decay exponentially
1
1
according to
with a unique time constant τ k =
=
ελk ε sin 2 ( 2kπN )
∆δ ( t ) = Ω τ k e
−
t
τk
e% k
(37)
This shows that the effect of the tuning change on the phases increases as the
eigenvalue decreases. This is consistent with the statement earlier concerning the
sensitivity of phase with respect to frequency variations, and is directly related to
the size of the locking region in this particular tuning direction. Referring to the
expressions for the eigenvalues (33) we see that the longest time constant is τ1
and, if this mode is excited, represents the array settling time. The eigenvectors
form a complete orthonormal set, as expected from the symmetry of A, and can
be used to represent an arbitrary tuning change. Once the change is decomposed
into a linear combination of the characteristic tuning changes the phase dynamics
can be analyzed by superposing the individual responses, each of which has a
characteristic decay time directly related to its eigenvalue. Certain tuning
variations may excite some modes to such a small degree that the phase
perturbations can be neglected.
57
2.1.1.1.4.1 Example: Settling Time for Beam Steering
Using our example of electronic beam steering discussed above, we will
treat the case of a step change away from broadside for a five element array.
Initially, broadside radiation implies that the elements are identically tuned so
that ∆β init = 0 . To steer a transmitting beam the end elements must be tuned in
equal but opposite directions, thus the final vector of tuning differences is
T
∆β final = (Ω,0,",0, Ω) where Ω is the amount of the change. Expressing the
applied tuning as a linear combination of characteristic tunings, we have
N −1
∆β = ∑ ak e% k
(38)
k =1
and taking the dot product of both sides with e% l and using the eigenvector
orthogonality relation e% l ⋅ e% k = δ k ,l , gives the weighting factors
al = ∆β ⋅ e% l
(39)
For our example we set N = 5 and find that the weighting factors are
a = ( 0.743 0 1203
.
0)
T
(40)
with corresponding time constants
τ=
( 2.618
ε
1
0.724 0.382 0.276) .
T
(41)
The zero entries are due to orthogonal symmetries between the tuning variation
and two of the characteristic tunings. We can see that the applied tuning couples
quite efficiently into the dominant time constant, which is the first element of
each vector. However, for the type of tuning considered here, where only the end
elements are varied, the lowest order mode is coupled less efficiently as the
number of oscillator elements N increases.
58
2.2 Effect of a Resonant Coupling Network on the Synchronization of
Two Oscillators [4]
Oscillator Grid
Output
Partially
Reflecting
Mirror
Figure 6--Quasi-optical power combining as originally proposed by Mink. Oscillator
coupling occurs through the resonant cavity. Operation is similar to optical lasers.
Many quasi-optical microwave systems involve arrays of 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 of the first examples of quasi-optical power
combining, where an array of 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 of 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 of 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 two Van der Pol oscillators coupled through a series
resonant coupling network, shown in figure 7. In particular the authors study the
ability of the oscillators to synchronize for wide ranges of coupling strengths and
coupling bandwidths. Simple approximate equations for the dimensions of the
frequency locking region are then compared with numerically computed results
and show good agreement for all coupling strengths and bandwidths.
59
Cc
Lc
Rc
+
+
ic(t)
v1(t)
-
-G(A1) L1
Oscillator I
v2(t)
C
C
Y1
Yc (v2=0)
Y2
L 2 -G(A2)
-
Oscillator II
Figure 7--Two oscillators coupled through resonant network. The negative conductances
depend on the amplitudes of the RF voltages across them.
2.2.1 Dynamic Equations
The linear coupling circuit can model a wide range of frequency
dependent circuits, and in particular cavity resonators. The starting point for the
following analysis is the system of differential equations, derived in section
1.2.1.2, that relates the amplitudes and phases of the oscillators and the dynamic
variables of the coupling circuit. Assuming the oscillators are locked to a
common frequency ω , the oscillator voltages can be represented in terms of their
slowly varying amplitudes and phases as
v1( t ) = A1 ( t ) cos(ωt + φ1( t ) )
(42)
v2 ( t ) = A2 ( t ) cos(ωt + φ2 ( 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 of slowly varying quantities:
(
ic ( t ) = Go Acx ( t )cos(ωt ) + Acy ( t )sin(ωt )
)
(43)
where the conductance Go = G( 0) is the magnitude of the nonlinear conductance
G( A) in figure 7 evaluated at A = 0, and is added for dimensional equality. The
"rectangular" form of 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
60
numerical evaluation. For Van der Pol oscillators, the dynamic equations relating
the slowly varying quantities of equations (42) and (43), derived in section 1.2.2,
are
[
A!1 = ωa (1 − A12 ) A1 + ωa Acx cos(φ1 ) + Acy sin (φ1 )
φ!1 = ωo1 − ω − ωa
[
1
A sin (φ1 ) − Acy cos( φ1 )
A1 cx
]
[
]
A! 2 = ωa (1 − A22 ) A2 − ωa Acx cos( φ2 ) + Acy sin (φ2 )
[
1
φ!2 = ωo 2 − ω + ωa
A sin( φ2 ) − Acy cos( φ2 )
A2 cx
]
]
(44)
[
]
λ [ A sin(φ ) − A sin( φ ) ]
A! cx = −ωac Acx + (ω − ωoc ) Acy + ωac λo A2 cos( φ2 ) − A1 cos(φ1 )
A! cy = −(ω − ωoc ) Acx − ωac Acy + ωac
o
2
where the oscillator bandwidths are 2ω a =
bandwidth is 2ω ac =
tunings, are ω o1 =
1
L1C
Rc
Lc
2
Go
C
1
1
, the unloaded coupling circuit
, the oscillator uncoupled resonant frequencies, or
and ω o2 =
1
L2 C
, and the coupling constant is λ o =
1
Go Rc
.
These five parameters directly affect the ability of the oscillators to lock and our
task is to understand the effects of 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 ∆φ = φ 2 − φ1 , and the
frequency ω . The two coupling variables Acx and Acy can be eliminated so that
the resulting system consists of four equations in four unknowns. Once a locked
state is found, stability of the state must be tested by perturbing the variables of
equation (44) and observing whether the perturbations increase or decrease in
time. Perturbing the variables produces a linear system of differential equations
with constant coefficients, and the real parts of the eigenvalues of this system
indicate stability.[8]
2.2.2 Synchronized States
Different characteristics of the system analyzed here will be important in
different situations. For example, the variation of the phase difference is
important in the design of beam scanning systems,[9][10][11] and the frequency
modulation bandwidth and array settling time are important in wideband
communication systems, and these characteristic may be examined using
equations (44). The focus of this paper, however, is on understanding how the
frequency locking ability of the oscillators depends on the coupling strength,
61
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.
∆ω o2=ω −ω
o2
oc
direction moved
to perturb
∆ ω c and A 1
2 W
line of
equal tunings
∆φ=0
∆ω o1=ω −ω
o1
2H
oc
line of equally
spaced tunings
∆ ω c=0
frequency locking
region
Figure 8--Region of frequency locking in the plane of oscillator tunings with respect to
the coupling circuit resonant frequency. The lines of symmetry are the lines of equal
tunings, ω o1 = ω o2 , and equally spaced tunings,
1
2
(ω
o1
+ ωo2 ) = ωoc . The width, W,
is the total span of ∆ω o1 + ∆ω o 2 at half the maximum value of ∆ω o 2 − ∆ω o1 . The small
arrow shows the direction of 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 oscillator tunings and the frequency ω to the coupling circuit resonant
frequency using the substitutions
∆ω o1 = ω o1 − ω oc , ∆ω o2 = ω o2 − ω oc , ∆ω c = ω − ω oc
62
(45)
Setting the derivatives equal to zero gives the algebraic equations describing the
locked states that, after eliminating the coupling variables Acx and Acy , can be
written as
(1 − λ ε
2
o
− A12 ) A1 = − λoε A2 cos( ∆φ − Φ)

ω 
A
∆ωo1 − 1 − λoε 2 a  ∆ωc = − λoε ωa 2 sin( ∆φ − Φ)
A1
ωac 

(1 − λ ε
o
2
− A22 ) A2 = −λoε A1 cos( ∆φ + Φ)
(46)

A
ω 
∆ωo2 − 1 − λoε 2 a  ∆ωc = λoε ωa 1 sin( ∆φ + Φ)
A2
ωac 

where ε =
1
2
 ∆ωc 
and Φ = tan −1 
 are, respectively, the coupling
 ωac 
 ∆ωc 
1+ 

 ωac 
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: Φ < 90 & . The form of 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
of the equations contain terms not present in the analysis of reference [12] that
account for the loading effects of the coupling circuit on the oscillators.
The concept of coupling magnitude and phase is useful in understanding
the effect of the coupling network on the ability of the oscillators to lock and was
used extensively in section 2.1. We found the tendency to lock increases with
increasing coupling strength and is maximum for zero or 180 degrees coupling
phase. In fact, for ±90 degrees of coupling phase the ability to lock ceases
entirely, at least to the first order of approximation. For the present case we can
identify a frequency dependent coupling magnitude λoε ( ∆ωc ) and coupling phase
Φ( ∆ωc ) and can immediately see that these quantities depend on the location of
the steady state frequency ω relative to the coupling circuit passband, as we
should expect. If the frequency ω lies at coupling circuit resonance, i.e.
ω = ω oc → ∆ω c = 0, the coupling strength and phase are both optimized, and the
locking tendency is strongest. As ∆ω c ω ac becomes large, coupling becomes weak
and the coupling phase approaches ±90 degrees, quickly causing loss of
synchronization. Thus frequency locking depends critically on the proximity of
63
the steady state frequency to the coupling circuit passband, but the frequency is a
complicated function of the circuit parameters that we must solve for using
equations (46). In the next section we will apply approximate methods to
estimate ∆ω c and use this result to determine how the locking region depends on
the circuit parameters.
Solutions to equations (46) indicate the existence of frequency locked
states, but we will briefly pause to consider these steady states from the
viewpoint of 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 of replacing the steady state frequency ω
A!
with the dynamic quantity ω + φ! − j for each transfer function (see section 1.2).
A
In the steady state the amplitudes An and phases φ n of 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) of 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 theory.[13]
2.2.3 Stability of States
Instead of deriving the frequency pulling variables and using these to find
the existence and stability of 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. Th
he steady state
values are perturbed by substituting the following into equations (44))
Ai → Ai + αi , φi → φi + δi ,
Acx → Acx + αcx ,
Acy → Acy + αcy ,
(47)
and retaining only first order terms where Ai , φ i , etc. are the steady state values
for the mode locked state in question and α i , δ i , etc. are the infinitesimal
perturbations. The resulting dynamic system for the perturbations is
p! = M p
(48)
64
where the vector of perturbations is
 α1 
 
 δ1 
 α2 
p= 
 δ2 
α 
 cx 
 αcy 
(49)
and the matrix of coefficients is
 ωa (1 − 3 A12 )

 ∆ωo1 − ∆ωc

A1

0
M =

0


 − λoωac

0

− A1 ( ∆ωo1 − ∆ωc )
0
0
ωa
ωa (1 − A12 )
0
0
0
0
ωa (1 − 3 A )
∆ωo 2 − ∆ωc
0
0
− λoωac A1
2
2
A2
λoωac cos( ∆φ )
λoωac sin( ∆φ )
− A2 ( ∆ωo2 − ∆ωc ) − ωa cos( ∆φ )
sin ( ∆φ )
ωa (1 − A22 )
ωa
A2
− λoωac A2 sin ( ∆φ )
− ωac
λoωac A2 cos( ∆φ )
− ∆ωc
(50)
All expressions appearing within the matrix are the time independent values of
the frequency locked state. As mentioned before one steady state oscillator phase
is arbitrary so we set φ1 = 0 and ∆φ = φ 2 . This implies that the above system has
only five degrees of freedom and therefore one of the eigenvalues is zero. It is
possible to reduce the set of equations, but the coefficients of 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 of all of 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 of locked states are
sufficiently complicated to require computer evaluation for exact solutions.
However, for many cases approximations can be made to reduce the complexity.
In the following section we will derive simple expressions for the values of
65
0
ωa
A1
− ωa sin ( ∆φ )
cos( ∆φ )
− ωa
A2
∆ωc
− ωac
oscillator tunings that result in stable frequency locking for various values of
coupling strength and coupling bandwidth.
2.2.4 Cases of Practical Interest
In order to simplify the analysis we will consider cases of weak, strong,
wideband, and narrowband coupling separately and make the appropriate
approximations for each case. Taking all of these results together gives us a
broad understanding of the system for a wide range of 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 of
oscillation. In general there may be three stable modes for the circuit considered
here, one whose frequency is located near the resonance of the coupling network
and the other two whose frequencies are located near each oscillator tuning. The
former is the mode of 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 of tunings for which the mode of interest is relatively
easy to analyze. For equal tunings, ∆ ω o1 = ∆ ω o2 , 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 ∆φ 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 of coupling. However, one can see in the figure that the locking
region becomes very 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, ∆ω o1 = − ∆ω o2 , which corresponds
to the diagonal line through the second and fourth quadrants, one can show that
∆ω c = 0 , which implies maximum coupling strength and optimum coupling
phase, and equal amplitudes. We will refer to this type of tuning as "equally
spaced" since all three frequencies are equally spaced. It is not obvious from
equations (46)) but the locking region is symmetric about the diagonal lines of
equal and equally spaced tunings. Once we determine the locking region
66
boundary in one quadrant the entire region is determined. In the analysis that
follows we will consider quantities above the line of equal tunings since the
phase difference is always positive in this region and this simplifies the
mathematics.
Moving along the line of equally spaced tunings the quantity
∆ω o = ∆ω o2 − ∆ω o1 increases and the total change in ∆ω o as we traverse the
entire locking region we will call the "height" and denote it H (the factor of 2
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 of even tuning and vary the quantity ∆ω o = 12 ( ∆ω o1 + ∆ω o2 ) , which
is the "average" oscillator tunings away from the coupling circuit resonance, and
eventually meet the locking region edge. Twice the total change in ∆ω o at half
the maximum value of ∆ω o we will refer to as the width W, indicated in figure 8..
Since ∆ω c = 0 along the line of equally spaced tunings the value of H is
relatively easy to determine. But determining W requires knowledge of the ∆ω c
variation as we move away from this line since the ratio ∆ω c ω ac has direct bearing
on W..
The functional form of the phase difference for equally spaced tunings is
derived from equations (46)) by subtracting the second and fourth equations and
setting ∆ω c = 0 . The result is
∆ωo = ∆ωo 2 − ∆ωo1 = 2λoωa sin ( ∆φ )
(51)
and we can immediately see that solutions cannot exist for ∆ω o > 2 λ o ω a .
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 of 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 ∆φ increases until the locking region boundary is
encountered.
Along the line of equally spaced tunings the amplitudes, which are equal
in this case, are found from equations (46) (see appendix 2.2):
67
2

 ∆ωo  

A = 1 − λo 1 − 1 − 


2
λ
ω

 
o
a

2
(52)
The amplitude variation across the locking region increases with increasing
coupling strength λ o but remains close to unity for weak coupling.
The functional dependence of ∆ω c can also be found by adding the
second and fourth of equations (46) (see appendix 2.2):
A
A
∆ωo + 12 λoωa ε 2  2 − 1  sin( ∆φ )
 A1 A2 
∆ωc =

A
A
ω 
1 − λoε 2 a 1 − 12  2 + 1  cos( ∆φ )
ωac 
 A1 A2 

(53)
The amplitudes and the phase difference depend on the oscillator tunings through
equations (46) and ∆ω c also appears implicitly in ε . This complexity forces us
to approximate ∆ω c for specific cases. Since the width of the locking region
depends on how fast ∆ω c changes as we move away from the line of equally
spaced tunings we will derive the change in ∆ω c for a small change in ∆ω o for a
fixed value of ∆ω o . Referring to figure 8 we will move perpendicularly away
from the diagonal as indicated. After considerable algebra (see appendix 2.2) the
approximate value for ∆ω c valid near the line of equally spaced tunings is
∆ωc ≅
∆ωo
1
ωa
∆ωo2
1 − (1 − A )
+
ωac 2 A2 − 1 + λo 4ωaωac
(54)
2
where the amplitude A is given by equation (52). This relation is simple enough
to allow us to determine the approximate locking region width for cases of
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 of ∆ω c as the coupling circuit is tuned relative to the
oscillators. This type of tuning is equivalent, in our analysis, to tuning in the
direction perpendicular to the line of equally spaced tunings, where the spacing
between the oscillators is maintained but both are tuned relative to the coupling
68
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 ∆ωc ≈ ∆ωo → ω ≈ 12 (ωo1 + ωo 2 ) . Whereas for narrowband
coupling we would expect the frequency to follow the coupling circuit resonance,
or ∆ω c ≈ 0 → ω ≈ ω oc . These two conditions give us criteria to identify the
coupling type as broad or narrow. Equation (54) tells us that for sufficiently
small ∆ω o , that is, as ∆ω o1 is tuned sufficiently close to ∆ω o2 , that the broadband
condition is satisfied even for small ω ac , which seems to contradict our usual
notion of narrowband coupling. If 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 of broadband and narrowband in the following sections are somewhat
arbitrary since the location of the steady state frequency changes in different parts
of the locking region.
The division of the coupling strength and bandwidth into regions of
weak/strong and narrow/broad coupling are expressed in graphical form in figure
9.
69
ω ac
ωa
Broadband
1/2
Narrowband
1/2
Weak
Coupling
Strong
Coupling
λo
Figure 9--Parameter diagram showing four regions of interest. Coupling strength
depends only on λ o whereas coupling bandwidth depends on ω a and λ o .
The boundaries separating the various regions will come directly out of the
analysis that follows.
2.2.4.1 Weak Coupling: λ o << 1 2
If the resistance Rc in the coupling network of figure 7 becomes large then
λ o = 1 Rc Go << 1 2 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 of ∆ω o from equation (51) we
have, using equation (52),
∆ω c
H
=
∆ω o
1 − λo
ωa
λ2o ω a
+
ω ac 1 − λ o ω ac
(55)
and, since λ o is small, the third term in the denominator is always much less than
the second and therefore can be neglected. At the edge of the locking region,
when the second term is much less than unity the coupling circuit is broadband
and as unity is approached ∆ω c blows up. This behavior is not what we would
70
expect for narrowband coupling circuits, as discussed in the previous section, and
causes loss of lock fairly close to the line of equally spaced tunings. At any rate,
the boundary for narrowband vs. broadband coupling can be taken as ω ac = λ oω a .
2.2.4.1.1 Broadband Case: ω ac >> λ oω a
Along the line of equally spaced tunings one can show that stable
solutions exist for all ∆φ < π 2 , although proof of this will be omitted here. The
height H, found from equation (51) and shown in figure 10(a), is
H = ∆ω o
max
= 2 λ oω a
(56)
Thus the height of the region is proportional to the coupling strength and the
oscillator bandwidths (assumed equal).
From equation (54) broadband coupling implies ∆ω c ≈ ∆ω o which means
that the steady state frequency is as far away from the coupling circuit resonance
as the average values of the oscillator tunings are from the coupling circuit
resonance. Or, in other words, the steady state frequency is exactly between the
71
Weak Coupling, λ o< < 1/2
Broadband, ω ac > > λ oωa
2
ω o2− ω oc
H = 2λoωa

3
λ ω
W ≅ 2ωac − 21 −
2  o a

W
2
ω o1− ω oc
H
Weak Coupling, λ o< < 1/2
Narrowband, ω ac < < λ oω a
2
2
ω o2− ω oc
H ≅ 2 2λoωaωac 1 −
ωac
2λoωa
3 

W ≅ 21 −
ω
 8 2  ac
W
ω o1− ω oc
H
Additional
Locking
Region
Figure 10--Dimensions of the locking region for weakly coupled oscillators. (a) The case of
broadband coupling has a fairly wide locking region that is bounded by the phase requirement
∆φ < π / 2 . (b) The narrowband case is quite thin and is bounded by loss of stability due to
72
high sensitivity of the steady state frequency with respect to tuning variations. The additional
locking regions appear for values of λ o near, but below, unity.
73
oscillator tunings and is independent of the coupling circuit resonant frequency.
This situation can be taken as the defining characteristic of 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
∆ωo ≅ 2λoωa
ωac2
ωac2 + ∆ωo
2
sin( ∆φ )
(57)
Thus the locking region boundary consists of the values of ∆ω o where ∆φ = ±90 &
and is plotted in figure 10(a). The width of the region when ∆ω o is half of its
maximum value occurs when ∆ω c ≅ ∆ω o = ω ac , as seen from equation (57).
Including the second term in the denominator of equation (54) for ∆ω o gives a
more accurate result for the width:
(
W = 2∆ωo ∆ω = 1 H ≅ 2ωac − 2 1 −
o
2
3
2
)λ ω
o
(58)
a
The case of 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 of equal oscillator tunings,
as we would expect from physical considerations.
2.2.4.1.2 Narrowband Case: ω ac << λ oω a
We now consider small coupling circuit bandwidths. The quantity ∆ω c
near the line of equally spaced tunings is found from equation (54) to be
∆ωc ≅
∆ωo
2
 ∆ωo  
λoωa 
1−
1− 1− 

ωac 
 2λoωa  
(59)
Near the center of the locking region, that is for small ∆ω o , ∆ω c ≈ ∆ω o and we
see the same behavior as in the previous case. But as ∆ω o increases the
denominator in (59) decreases and ∆ω c becomes much more sensitive to tuning
variations. Computer simulations show that the value of ∆ω o that causes the
denominator to vanish is (approximately) a stability boundary and for values of
∆ω o for which ∆ω c is negative the system is not stable. The stability boundary,
74
and therefore the height of the locking region, is found by setting the
denominator of (59) equal to zero:
ω
H = ∆ωo max ≅ 2 2λoωaωac 1 − 2λoacωa
(60)
For coupling bandwidths above 2 λ o ω a this stability boundary does not exist.
Below this threshold we must also meet the general existence criterion that
∆ω o < 2 λ o ω a . Using these two criteria together we find that this new stability
boundary exists only for ω ac < λ oω a and above this value of coupling bandwidth
the general existence criterion applies. Assuming the former condition applies,
the stability region is found, at least approximately, using
∆ωo ≅ 2λoωa
ωac2
ωac2 + ∆ωc 2
sin( ∆φ )
(61)
with ∆ω c given in equation (59). To find the width we find 2∆ωo ∆ω = 1 H which,
o
2
as for the previous case, occurs at ∆ω c = ω ac . Solving for the width gives
(
)
W = 2∆ωo ∆ω = 1 H ≅ 2 1 − 8 3 2 ωac
o
2
(62)
Figure 10(b) shows the approximate shape of the locking region for weak
coupling and narrow coupling bandwidth. The region is much thinner near the
edge of the odd tuning boundary due to the increased sensitivity of ∆ω c to
changes in ∆ω o near this boundary.
If we include the third term in the denominator of equation (54) we find
that for values of λ o close to but less than unity that the denominator becomes
zero a second time and for values of ∆ω o 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.
2.2.4.2 Strong Coupling: λ o >> 1 2
As the coupling strength λ o is increased the amplitudes decrease
considerably as we traverse the locking region in the direction of equally spaced
75
tunings. The physical reason for this is that as the coupling resistor Rc is reduced
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 either of the amplitudes drops too far below
unity the system becomes unstable and locking is lost. It is difficult to determine
exactly when this occurs, but we can find the approximate amplitude boundary
from the perturbational system (5
50).
The perturbational system consists of three second order subsystems, the
three diagonal blocks, and are coupled through the off diagonal blocks. If no
coupling existed then stability of the system would be insured if each of 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 of
the diagonal elements is negative. Applying this criterion to each of the oscillator
subcircuits in the variational system gives conditions for stability in the
uncoupled case but which we assume hold approximately in the general case:
1
(63)
2
This means that if either amplitude drops below 1 2 the system will become
unstable. This approximate stability condition is surprisingly accurate for most
values of coupling strength and bandwidth, and becomes inaccurate only when
these parameters both become quite large. Even in this case, however, the
dimensions of the locking region given below are fairly accurate.
A1 and A2 >
For strongly coupled oscillators the boundaries of the locking region can
be approximated as the locus of points where either oscillator amplitude is 1 2 .
Along the line of equally spaced tunings the value of ∆ω o that causes the
amplitudes to assume this value can easily be found from equation (52) and is
H = ∆ω o max = 2 λ o ω a 1 − 4λ1 for λ o > 12
o
(64)
One important consequence is that for a coupling strength λ o = 12 the
locking region height is maximized while still allowing the phase difference ∆φ
to vary 180 degrees over the locking region. This is important for beam scanning
76
systems where the designer wishes to maximize the total phase variation and the
locking range simultaneously.
If the coupling strength is sufficiently strong the width of the region will
also be determined by the amplitude criterion of equation (63). To estimate the
rate of decrease of the amplitude away from the line of equally spaced tunings we
will again resort to a perturbation analysis which is contained in the appendix 2.2.
The results show that if we move from this line in the direction of increasing
∆ω o that the amplitude of oscillator I will diminish according to


∆ωo ∆ωc

A1 ≅ A1 −
2 
 4ωaωac ( λo − 1 + 2 A ) 
(65)
where A is given by (52). We will assume strong coupling, λ o >> 2 A2 − 1, and
simplify the denominator. To find the width of the locking region we will
evaluate the amplitude at ∆ω o = 12 ∆ω o max and find the value of ∆ω o that gives
A = 12 . First, however, we must determine ∆ω c .
Using equation (52) for the amplitude for odd tunings and noting that for
large coupling strengths we can expand the square root, ∆ω c from (54) is
approximately
∆ω c ≅
1+
∆ω o
∆ω 2o
(66)
8λ o ω a ω ac
This shows that for ∆ω o << 2 2 λ o ω a ω ac the steady state frequency remains
halfway between the two oscillators, as in the case of weak broadband coupling,
but for ∆ω o >> 2 2 λ o ω a ω ac the steady state frequency follows the resonant
frequency of the coupling network. Using the above result at the maximum value
of ∆ω o given by equation (64), and assuming λ o >> 1, we can say that the
boundary for weak vs. narrowband coupling is at ω ac ≅ 12 ω a .
2.2.4.2.1 Broadband Case: ω ac >> 12 ω a
In this section we will assume that ω ac >> 12 ω a so that ∆ω c ≈ ∆ω o .
Using this result in equation (65) for the amplitude of oscillator I and setting the
amplitude to 1 2 , we find that the width of the locking region is
77
Strong Coupling, λ o> > 1 /2
Broadband, ω ac > > λ oωa
2
2
ω o2− ω oc
H = 2 λo ωa 1 −
1
4λo
1 

W ≅ 81 −
 λω

2  o ac
W
H
ω o1− ω oc
(a)
Strong Coupling, λ o> > 1 /2
Narrowband, ω ac < < λ oω a
2
ω o2− ω oc
H ≅ 2 λo ωa 1 −
1
4λo
1 

W ≅ 1 −
 λω

2 o a
W
2
ω o1− ω oc
H
(b)
Figure 11--Dimensions of the locking region for strongly coupled oscillators. (a) For
broadband coupling the region is large but increases as
λo .
(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.
78
(
W ≅ 8 1−
1
2
)
λo ωac
(67)
The locking region for this case, shown in figure 11(a), looks similar to the case
of weak and broadband coupling, but the height grows more slowly with
increasing coupling strength λ o and the width is no longer constant with λ o
2.2.4.2.2 Narrowband Case: ω ac << 12 ω a
8λ oω a ω ac
∆ω o and using this result in equation (65)
∆ω o2
and setting the amplitude to 1 2 , the width of the locking region is
We now have ∆ω c ≈
(
W ≅ 1−
1
2
)
λo ωa
(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 of 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 of the above expressions for the height and width
of the locking region, MathCAD was used to obtain solutions to equations (46)
and to compute the eigenvalues of the variational system (5
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 of parameters were correct. The coupling circuit resonant
frequency and oscillator bandwidths were kept constant at ω oc = 10 and ω a = 0.1.
The height H and width W were computed for three different values of coupling
bandwidth, ω ac = 0. 005, 0. 05, 0 . 5 as functions of the coupling strength λ o , and
the simulation results and the results calculated from the approximate expressions
are shown in figures 12, 13, and 14.
79
Locking Region Height, H
ω oc=10
ω a=0.1
ωac=0.005
0.7
0.6
0.5
Weak
Coupling
0.4
Strong
Coupling
0.3
Approximate
0.2
Simulation
0.1
0
0.1
0.01
1
10
Locking Region Width, W
λ o , log scale
0.1
0.08
0.06
ω oc=10
ω a=0.1
ωac=0.005
Weak
Coupling
Strong
Coupling
0.04
Approximate
Simulation
0.02
0
0.01
0.1
1
10
λ o , log scale
Figure 12--Comparison of approximate formulas to computer simulations for "high" Q
coupling circuit.
80
Locking Region Height, H
0.7
0.6
0.5
0.4
ω oc=10
ω a=0.1
ωac=0.05
Weak
Coupling
Strong
Coupling
0.3
0.2
Simulation
Approximate
0.1
0
0.1
0.01
1
10
Locking Region Width, W
λ o , log scale
0.5
0.4
0.3
ω oc=10
ω a=0.1
ωac=0.05
Simulation
Strong
Coupling
Weak
Coupling
0.2
Approximate
0.1
0
0.01
1
0.1
10
λ o , log scale
Figure 13--Comparison of approximate formulas to computer simulations for "moderate"
Q coupling circuit.
81
Locking Region Height, H
ω oc=10
ω a=0.1
ωac=0.5
0.8
Simulation
0.6
Approximate
0.4
0.2
Weak
Coupling
Strong
Coupling
0
0.01
1
0.1
10
Locking Region Width, W
λ o , log scale
5
4
ω oc=10
ω a=0.1
ωac=0.5
Simulation
3
Approximate
2
1
0
0.01
Strong
Coupling
Weak
Coupling
0.1
1
10
λ o , log scale
Figure 14--Comparison of approximate formulas to computer simulations for "low" Q
coupling circuit.
82
Appendix 2.1--Reducing the Order of a Stability Matrix
The following analysis shows that an N × N real stability matrix C of
rank N-1 can be reduced to an N − 1 × N − 1 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 of a set
of N-1 independent "basis" phase quantities, for example the N-1 phase
differences ∆φ n −1 ≡ φ o,n +1 − φ on . The basis phases φ′ are related to the phases φ
through a linear transformation φ ′ = Aφ , where A is a real N − 1 × N matrix
(written A ∈ℜ N −1× N ) of rank N-1. Next, express the derivatives in (7) in terms of
the new phases:
Cnm =
∂β n ∂β n ∂φ1′ ∂β n ∂φ ′2
∂β n φ ′N −1
=
+
+" +
∂φ m ∂φ1′ ∂φ m ∂φ′2 ∂φ m
∂φ ′N −1 ∂φ m
The derivatives of the phase variables are given by
(69)
∂φ ′k
= Akm , so the stability
∂φ m
matrix can be represented using matrix notation as
C = DA, D ∈ℜ N × N −1 , A ∈ℜ N −1× N
(70)
where the matrix D is
Dnm =
∂β n
∂φ ′m
(71)
Denoting an eigenvalue of C as λ and the corresponding eigenvector as x, the
eigenvalue equation for the stability matrix C is
Cx = DAx = λx
(72)
If we transform the eigenvector using the same transformation matrix, y = Ax ,
we have
DAx = Dy = λx
(73)
→ ADy = λAx = λy
Thus y is an eigenvector of the matrix
C ′ = AD, C ′ ∈ℜ N −1× N −1
83
(74)
with the same eigenvalue λ. This must hold for all of the eigenvalues and
corresponding eigenvectors of C, including λ = 0. For this latter case, however,
we must have Ax = 0 for the corresponding eigenvector. Thus, the new stability
matrix C ′ is of dimension N-1 and has the same eigenvalues with the zero
eigenvalue removed. Moreover, its eigenvectors are the eigenvectors of C
transformed by the matrix A.
Appendix 2.2--Amplitudes and ∆ω c Near Line of Equally Spaced
Frequencies
In this section we derive approximate expressions for ∆ω c and the
amplitudes A1 and A2 that are valid near the line of equally spaced tunings as we
tune perpendicularly away from that line, as indicated by the small arrow in
figure 8. The first and third of equations (46) relate the oscillator amplitudes to
various quantities and are repeated here with the coupling phase expanded:
(1 − λ ε
o
2


∆ωc
− A12 ) A1 = − λoε 2 A2  cos( ∆φ ) +
sin ( ∆φ )
ωac




(1 − λoε 2 − A22 ) A2 = −λoε A1  cos(∆φ ) − ∆ωωc sin(∆φ )
ac
(75)
Subtracting the second fourth of equations (46) we arrive at an expression for the
difference between oscillator tunings
 A

A
∆ωc  A2 A1 
∆ωo = ∆ωo 2 − ∆ωo1 = λoε 2ωa  2 + 1  sin ( ∆φ ) −
 −  cos( ∆φ ) (76)
ωac  A1 A2 
 A1 A2 

and adding them gives an expression for ∆ω c
 ∆ω 1  A

A
1 ∆ωc  A2 A1 
∆ωc = ∆ωo + λoε 2ωa  c +  2 − 1  sin( ∆φ ) −
 +  cos( ∆φ )
2 ωac  A1 A2 
 ωac 2  A1 A2 

(77)
As we vary ∆ω o an infinitesimal amount d ∆ω o away from zero, many of the
quantities in the above equations will change. For example, ∆ω c is nominally
zero, but after this perturbation it will have a nonzero value. The quantity ε ,
84
however, depends on the square of ∆ω c (see equations (46)) so, to the first order,
ε will remain unity.
The equations are perturbed by implicit differentiation and any
unperturbed terms will be evaluated on the line ∆ω o = 0 and, as a result, some
may vanish. Along this line equations (75), (76), and (77) take on particularly
simple forms, and result in the amplitude expression of equation (52) and the
following relations for the phase difference:
sin ( ∆φ ) =
∆ωo
2λoωa
cos( ∆φ ) = 1 −
1
λo
(78)
(1 − A )
2
These relations help simplify the form of the mathematics that follows.
Implicitly differentiating the amplitude equations (75) gives
(1 − λ
o



d∆ωc 
− 3 A2 ) dA1 = − λo  dA2 cos( ∆φ ) − A d∆φ −
 sin( ∆φ )
ωac 







(1 − λo − 3 A2 ) dA2 = −λo dA1 cos( ∆φ ) − A d∆φ + dω∆ωc  sin(∆φ )
ac


(79)
where the unperturbed quantities have been evaluated along ∆ω o = 0. Adding
and subtracting these equations and using the relations (78) gives
∆ω o
d∆φ
2A
∆ω o d∆ω c
A
dA2 − dA1 = − 2
2 A + λ o − 1 2 ω a ω ac
dA2 + dA1 = −
(80)
Applying the same analysis technique to equation (76), which we maintain at
zero, gives
 A

A
d∆ωo = 0 = λoωa d  2 + 1  sin( ∆φ ) + 2 cos( ∆φ )d∆φ 
  A1 A2 


2
= λoωa  ( dA1 + dA2 ) sin( ∆φ ) + 2 cos( ∆φ )d∆φ 

A
85
(81)
Using the first of equations (80) the above expression becomes
2

2A
cos( ∆φ ) ( dA1 + dA2 ) = 0
 sin( ∆φ ) −
∆ωo
A

(82)
The first factor is not generally zero, so we must have dA1 + dA2 = 0 and, from
(80), at follows that d∆φ = 0. Using these results with the second of equations
(80), we can find the approximate expression for the amplitude of oscillator II
(which is the lesser), valid for small ∆ω c , stated in equation (65):

1
∆ωo ∆ωc 
A2 ≅ A1 − 2

 2 A + λo − 1 4ωaωac 
(83)
The remaining task is to find an approximate expression for ∆ω c .
Implicitly differentiating equation (77) and using the relations (78) and those
resulting from (82), we find

ω
d∆ωc 1 − λo a 1 − cos( ∆φ )
ωac

(

) = d ∆ω
o

A
1
A
+ λoωa d  2 − 1  sin ( ∆φ )
2
 A1 A2 
= d ∆ωo − λoωa sin( ∆φ )
= d ∆ωo −
1
∆ωo d∆ωc
2 A + λo − 1 2ωaωac
2
1
∆ωo2
d∆ωc
2 A2 + λo − 1 4ωaωac
(84)
which, after rearrangement, gives equation (54) valid for small ∆ω o and ∆ω c :
∆ωc ≅
∆ωo
ωa
∆ωo2
1
1 − (1 − A )
+
ωac 2 A2 − 1 + λo 4ωaωac
(85)
2
The above perturbation technique is not only useful for finding approximate
expressions but it also helps us understand the manner in which various
quantities are affected by tuning variations.
86
[
1] C. Balanis, Antenna Theory Analysis and Design, Wiley & Sons, NY, 1982.
[2] J. J. Lynch, R. A. York, "An Analysis of Mode Locked Arrays of Automatic Level Control
Oscillators," IEEE Trans. on Circuits and 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 of Microwave Oscillators
Coupled through Resonant Circuits," Submitted to IEEE Trans. Microwave Theory Tech.
[5] J. W. Mink, "Quasi-Optical Power Combining of Solid-State Millimeter-Wave Sources,"
IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986.
[6] R. A. York, "Nonlinear Analysis of 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 New Phase-Shifterless Beam-Scanning Technique using Arrays of
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," IEEE MTT-S, International Symposium Digest, San Diego, June 1994, pp.
1231-1234
[11] P. S. Hall, I. L. Morrow, P. M. Haskins, J. S. Dahele, "Phase Control in Injection Locked
Microstrip Active Antennas," IEEE MTT-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," IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 2040-2045..
[13] M. Kuramitsu, F. Takase, "Analytical Method for Multimode Oscillators Using the Averaged
Potential," Elect. and Comm. in Japan, vol. 66-A, No. 4, 1983.
87
Chapter 3
Mode Locked Arrays
Carrier Envelope
Mode locking is commonly utilized in laser systems for producing pulses
of electromagnetic energy,[1] and the same phenomenon has been demonstrated
in systems of coupled microwave oscillators.[2] The basic idea is to synthesize a
frequency spectrum that is a periodic pulse train in the time domain. Such a
waveform produced by a four element array is depicted in figure 1.
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 of a high frequency microwave carrier, and is
assumed to have a period many times longer than that of the carrier. The
magnitude spectrum of 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. If the phase distribution
is suboptimum the peak pulse power will be diminished and the sideband power
88
will increase. When the proper conditions are met the output signal, obtained by
summing together the oscillator outputs, will appear as in figure 1.
A(ω )
Frequenc
Pulling
ω1 ω2 ω3 ω4
ω
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
ω o1
ω o2
ε,Φ
ω
ω o3
ε,Φ
ω
ω o3
ε,Φ
ω
ω
Oscillator Outputs
Figure 3--Four element linear array with nearest neighbor frequency independent
coupling. Adjacent output frequencies differ by the beat frequency ω b .
It is identical to the synchronous array of section 2.1.1, but for this case we tune
the oscillators outside the region of synchronization and maintain nearly equal
spacing between free running frequencies. We will apply the analysis methods of
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 of 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.
89
3.1 The Time Domain Mode Locked Waveform
The time domain output of 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
N
N
n =1
n =1
v( t ) = ∑ An ( t ) cos(ωot + ( n − 1) ωb t + φn ( t ) ) = ∑ An ( t ) cos(θn ( t ) )
(1)
The frequency of the first oscillator ω o is defined arbitrarily as the carrier and the
frequency separation, or beat frequency, is denoted ω b . The element frequencies
are assumed to increase from one end of the array to the other; that is, referring to
figure 3 we have ω 1 < ω 2 < ! < ω N . The waveform in equation (1) can be
written as a high frequency carrier with slowly varying amplitude and phase
modulation:
v( t ) = Ve ( t ) cos(ωot + Θ( t ) )
(2)
where the envelope, which is the mode locked waveform shown in figure 1, is
Ve ( t ) =
N
∑ A ( t ) e ((
n =1
j n −1) ωb t +φn ( t ) )
n
(3)
The time dependent phase modulation in (2) is not considered here since it does
not effect the envelope waveform.
The peak amplitude of the envelope will be maximized when the terms in
the sum of (3) add coherently, and this occurs when the phases in equation (1)
satisfy
∆∆θn ( t ) ≡ ∆θn+1 − ∆θn = (θn+ 2 − θn +1 ) − (θn+1 − θn ) = 0 (4)
for n = 1, 2, ! , N − 2 . The second difference of 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 of the phases
become ∆∆θn ( t ) = ∆∆φn ( t ) . Thus condition (4) requires a constant phase
difference ∆φ n = φ n +1 − φ n to exist between adjacent elements (for a discussion of
these phase definitions see section 1.2.1.1). Since the frequencies shown in (1)
90
are the true steady state frequencies the second differences of the phases are
periodic functions of time. For weakly coupled arrays the periodic variation will
be small and the ∆∆φ n 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 of the phases.
Under certain conditions the time dependent amplitudes can enhance the
peak of the envelope and reduce the sidelobes. If, 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:
N

sin  (ωbt + ∆φ )
2

( )
( )
Ve ( t ) = A( t ) ∑ e j ( n−1 ωbt +φn t ) = A( t )
1

n =1
sin  (ωbt + ∆φ )
2

N
(5)
The factor multiplying the amplitude A( t ) attains a maximum value of NA every
T = 2 π ω b seconds, and it is this function that is plotted in figure 1 for N=4. If the
time dependent amplitude A( t ) is maximum during these times then the peak
value of the envelope will be enhanced.[3] We will show later that this is indeed
the case for a certain class of 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 of 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 3.1 for
the derivation):
A" n = η(1 − An 2 ) An + ε [ An −1 cos(τ + φn − φn−1 + Φ ) + An+1 cos(τ + φn+1 − φn − Φ )]
 An−1

A
sin(τ + φn − φn−1 + Φ ) + n+1 sin(τ + φn+1 − φn − Φ )]
An
 An

φ"n = βn − ε 
91
(6)
where η and ε are the normalized nonlinearity and coupling parameters,
βn = (ωon − ωn ) ωb is the normalized frequency pulling, τ = ωbt is the
normalized time parameter, and the dot denotes differentiation with respect to τ.
With the dynamic equations (6) in a relatively simple form, we can derive
the existence of stable mode locked states using the methods of chapter 1. The
frequency pullings in terms of the time average phases are derived in appendix 2.
The results, in matrix form are
β = ε2
2η
1 + ( 2η)
2
( Fu + Gv ) + ε 2 k
(7)
where F ∈ℜ N × N − 2 and G ∈ℜ N × N − 2 are defined as
 12



η




 cos( 2Φ) #

 − sin( 2Φ) #





1
F =  12
η
#
#
2
 , G =  −η




(
)
(
)
# cos 2Φ
# − sin 2Φ 




1
−η 



2
0
0
0
(8)
0
and the N element vector k is defined as
1
η


2
2
 − cos ( Φ) + 2 sin ( Φ) + 2 sin( 2Φ)
Γ
Γ


2
η


(
)
sin
2
Φ
Γ2


k =
$

η


sin ( 2Φ)


Γ2


1
η
 cos2 ( Φ) − 2 sin 2 ( Φ) + 2 sin ( 2Φ) 


Γ
Γ
where Γ 2 = 1 + ( 2η) .
synchronous case:
2
(9)
The vectors u and v are defined analogously to the
92
 cos( ∆∆φ1 ) 
 sin ( ∆∆φ1 ) 




 cos( ∆∆φ2 ) 
 sin ( ∆∆φ2 ) 
u=
(10)

, v=
$
$




 cos( ∆∆φ )
 sin ( ∆∆φ )


N −2 
N −2 
where ∆∆φn ≡ φn +2 − 2φn+1 + φn . Equation (7) comprises a system of N equations
in N unknowns, the unknowns being the N-2 phases ∆∆φn , the beat frequency,
and one of the steady state frequencies. These variables completely describe all
of the frequencies and phases of the periodic state, except for two arbitrary phases
that arise from the periodicity of both the carrier and the envelope. Unlike the
synchronous case, for mode locked arrays the frequency pullings depend on the
coupling parameter ε squared. Thus we expect a significantly smaller locking
region.
Since only the second differences of 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 of section 2.1 to reduce the order of the
system. This will remove two zero eigenvalues from the stability matrix, while
preserving the remaining eigenvalues. The second difference of the phases are
related to the phases themselves through the linear transformation ∆∆φ = Dφ ,
defined by
 ∆∆φ1  

 1
 ∆∆φ2  = 
 $  


 ∆∆φN −2  
0
−2
# # #
1
0  φφ 
1
1
−2
2
 

 $
1  
 φN 
(11)
Applying the second difference transformation D ∈ℜ N −2× N to the pulling
variables in equation (7) gives the reduced system
∆∆β ≡ Dβ = ε 2
2η
(
) 2
2 Au + Bv + ε Dk
Γ
(12)
where A = DF ∈ℜ N −2× N −2 and B = DG ∈ℜ N −2× N −2 . Note that the second
difference of frequency pullings ∆∆β contains no steady state frequencies nor the
beat frequency; it depends only on the oscillator tunings. Thus equation (12) is a
system of N-2 equations in N-2 unknowns. The system is quite similar to the
corresponding system of section 2.1.1 for the synchronous case, with two
significant differences. The first is the additional factor 2η Γ 2 that depends on the
93
nonlinearity parameter η. This term tends to zero as η tends to zero and infinity,
and is maximum for η=1/2. Thus we expect the locking region to maximized for
this value of nonlinearity. Second, there is a phase independent term ε 2 Dk that
shifts the center of the locking region away from ∆∆β=0. This implies that the
tunings that place us in the center of the existence region are not quite equally
spaced.
The perturbational system for the array is derived immediately from (12)
using the methods of chapter 1:
∆∆δ" = −C ∆∆δ
(13)
The second differences of the phase perturbations δ are derived using the second
difference transformation ∆∆δ=Dδ, and the stability matrix is
C = ε2
2η
( AV − BU )
Γ2
(14)
The diagonal matrices U and V are defined as in section 2.1.1:
 sin( ∆∆φ )

1




sin( ∆∆φ2 )
U =

#




sin
∆∆
φ
(
)

N −2 
(15)
 cos( ∆∆φ )

1




cos( ∆∆φ2 )
V =

#



cos( ∆∆φN − 2 )

(16)
0
0
0
0
Unlike the synchronous case there is no value of coupling phase that
eliminates the matrix B and simplifies the equations. Deriving the locking
diagram is quite complicated and we will immediately consider simple cases of
three and four element arrays.
3.2.1 Three Element Array
94
For the three element array the one dimensional matrices A and B are
2η
A = 1 − 2 cos( 2Φ) and B = 2 sin( 2Φ) , and the "vector" Dk is Dk = − 2 sin ( 2Φ) .
Γ
The single pulling equation is, therefore,
∆∆β1 = ε 2
2η
(1 − 2 cos( 2Φ)) sin( ∆∆φ1 ) + 2 sin( 2Φ) cos( ∆∆φ1 ) − sin( 2Φ)]
Γ2
[
(17)
which, with the help of a trig identity, can be written
∆∆β1 = ε 2


2η 
2 sin( 2Φ)  
−1 
  − sin ( 2Φ)  (18)
2  5 − 4 cos( 2Φ ) sin  ∆∆φ1 + tan 
 1 − 2 cos( 2Φ)  
Γ 


Allowing ∆∆φ1 to span all of its possible values creates the existence region on
the ∆∆β1 line. The length of the region is
∆∆β1 max − ∆∆β1 min = 2ε 2
2η
5 − 4 cos( 2Φ )
Γ2
(19)
which is maximized for Φ = π 2 and η = 1 2 . These are optimum values of
coupling phase and nonlinearity. The stability matrix C has only one element
C11 = ε 2

2η
 2 sin( 2Φ)  
5 − 4 cos( 2Φ) cos ∆∆φ1 + tan −1 

2
 1 − 2 cos( 2Φ)  
Γ

(20)
and is positive when the argument of the cosine lies between -π/2 and π/2. This
condition ensures that a single stable mode exists for a given set of free running
frequencies since over the range of phases that satisfy (19) the sine function in
equation (18) is single valued.
3.2.2 Four Element Array
Since the results of the last section showed that the locking region is
largest when Φ = π 2 we will assume this value to simplify the problem. The
pulling equations are
95
 ∆∆β1 
2η  3 − 2  sin ( ∆∆φ1 )   0 − 2η  cos( ∆∆φ1 )  
 +


 = ε 2 2 


0   cos( ∆∆φ2 ) 
Γ  − 2 3   sin ( ∆∆φ2 )  2η
 ∆∆β2 
1
η


− cos2 ( Φ) + 2 sin 2 ( Φ) − 2 sin ( 2Φ)

Γ
Γ
+ ε 2

1
η
 cos2 ( Φ) − 2 sin 2 ( Φ) − 2 sin( 2Φ) 


Γ
Γ
(21)
Changing variables puts the equations in a more symmetric form. Defining the
"sum" and "difference" phase and frequency variables according to
∆∆β2 + ∆∆β1
∆∆β2 − ∆∆β1
, βd =
,
2
2ε
2ε 2
∆∆φ2 + ∆∆φ1
∆∆φ2 − ∆∆φ1
φs =
, φd =
+ θd
2
2
βs =
(22)
the equations become
βs = As sin(φs ) cos(φd − θs )
βd = −
1
+ Ad cos(φs )sin(φd + θs )
Γ2
(23)
where
2η
2η
, Ad = 2 25 + ( 2η )2 ,
Γ
Γ
1
1
 2η  
 2η  
θs =  tan −1 ( 2η ) + tan −1  2   , θd =  tan −1( 2η ) − tan −1  2  
 Γ 
 Γ 
2
2
As =
(24)
With the pulling equations in this form one can see by (thoughtful) inspection that
the minimum and maximum values of the pulling variables occur on the β s , β d
axes.
The coefficients of the stability matrix can still be found by differentiating
β s and β d with respect to φ s and φ d , since we have simply applied an additional
linear transformation to the phase and pulling variables, an operation that leaves
the eigenvalues of the stability matrix unchanged. The (modified) stability matrix
is
96
 As cos( φs ) cos(φd − θs ) − As sin( φs ) sin( φd − θs )

C = 
 − Ad sin ( φs ) sin( φd + θs ) Ad cos( φs ) cos( φd + θs ) 
(25)
and the conditions for stability can be determined by the Routh-Hurwitz criteria,
which, for the 2 × 2 case, require the trace and the determinant of C to be
positive. After some manipulation these criteria reduce to
cos(φs )cos( φd + θd ) > 0
cos( 2φd ) + cos( 2θs ) cos( 2φs ) > 0
(26)
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 π as shown by the dotted lines in figure 4.
97
φs
φs
π
π
η = 0.1
−π
π
η = 0.5
φd
π
−π
−π
−π
(a)
(b)
φs
π
φs
π
η=1
−π
φd
π
η=4
−π
φd
−π
π
φd
−π
(c)
(d)
Figure 4--Phase plane regions of stable solutions (shaded areas) for four values of
nonlinearity parameter η. Mapping these regions onto the frequency pulling planes
gives the regions of stable mode locking.
The second condition creates curved lines that further divide the phase plane.
Putting the two together gives the regions of stable solutions in the phase plane
and are shown in figures 4(a) through (d) for four different values of η. Figure 5
shows the corresponding boundaries in the plane of frequency pullings
98
∆∆β2
∆∆β2
∆∆β1
∆∆β1
η = 0 .5
η = 0 .1
(b)
(a)
∆∆β2
∆∆β2
∆∆β1
∆∆β1
η=4
η=1
Region of multiple
stable states
(c)
(d)
Figure 5--Region of stable mode locked states in the ∆∆β plane for four values of η. In
plot (d) a subregion exists where two stable states exist for each set of tunings.
For η < 5 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 η > 5 2 the inner curve
shown in figure 5(d) bounds the region where two stable modes exist. These
99
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 of 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 of central free running frequencies and ∆∆β are related through the linear
transformation
ωb ∆∆β = H ωoc + ωoe
(27)
where the N − 2 × N − 2 matrix H and the N-2 element vectors are defined as
 ωo1 
− 2 1

 ωo 2 




0 



 1 −2 #

 ωo 3 


H =
 ωoc =  $  , ωoe =  $ 
1
#
#




 0 


 ωo, N −1
1 − 2



 ωoN 
0
(28)
0
In what follows we will work with the vector ∆∆β. The central free running
frequencies can be obtained using (28). Figure 6 shows the stability region in the
plane of central free running frequencies (keeping the end elements fixed). The
origin contains the values of ω o2 and ω o3 that result in evenly spaced free running
frequencies. It is interesting to note that, unlike the N=3 case, as η → ∞ the
existence region approaches a constant area. However, near this limit the real
parts of the eigenvalues of the averaged system are almost zero, indicating that
the system is highly underdamped.
100
∆ ∆ φ 1,2= 0
ω o3
ω o3
ω o2
ω o2
η = 0.5
η = 0.1
ω o3
Region of multiple
stable states
ω o3
ω o2
ω o2
η=1
η=4
Figure 6--Locking regions in the plane of 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 of 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 ω o 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
101
Resonator
Resonator
vs(t)
Σ
H(ω )
v(t)
vs(t)
Σ
Gain
H(ω )
v(t)
Gain
G(v)
G(g)
Nonlinear
Gain
Control
Filter
(a)
g(t)
Hf(ω )
(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 of the high frequency carrier. The dynamic
equations for the variables shown in figure 7(b) are
v"" + (ωo 2 + 2ωa g" ) v + 2ωa (1 − Go + g ) v" = 2ωa v" s
(29)
g" + ω g g = ω g f ( v )
where ω a = ω o / 2Q is the half bandwidth of the filter, Go is the maximum loop
gain (at g=0), ω g is the bandwidth of the gain control lowpass filter, and f ( v ) is
the nonlinear envelope detection function. Assuming the resonator has a fairly
high quality factor (>20) 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 ( t ) = A( t )cos [ω t + φ ( t )]
vs ( t ) = As ( t )cos [ωs t + φs ( t )]
(30)
Both signals are assumed to be almost periodic and the frequencies are chosen so
that the phase φ( t ) is a periodic function of time. After some manipulating and
applying appropriate approximations (shown in appendix 3.3) the dynamic
equations for the amplitude, gain control, and phase are
102
A" = ωa [(1 − g ) A + As cos [(ωs − ω )t + φs − φ ]]
g" = −ω g g + ω g A
φ" = ωo − ω +
(31)
ωa
A
g" + ωa s sin[(ωs − ω )t + φs − φ ]
ω
A
To simplify (31) the maximum gain was chosen to be Go = 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 show that the ALC oscillator can have an underdamped
amplitude response. In the absence of an injected signal the fixed point of
equations (31) is g = A = 1, ω = ω o and the amplitude and gain control form a
second order system. Linearizing about the fixed point gives the perturbational
system
d α   0
  =
dt  γ   ω g
− ωa   α 
 
− ωg   γ 
(32)
where α and γ are the perturbations to the amplitude and gain control
respectively. The system has eigenvalues λ = − 1 2 ω g ± j ωaω g −
(
1
2
)
2
ω g , thus
the amplitude response to perturbations can be made underdamped and we can
control the damping by adjusting the bandwidth of the gain control filter.
Amplitude
ALC
Van der Pol
Time
Figure 8--Amplitude (carrier envelope) response at turn-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
103
To illustrate the above analysis the envelope of 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 of the amplitude oscillation to the spectral
spacing between elements the locking region will be proportional to the size of
the resonant amplitude variations, and can be made fairly large by adjusting the
coupling strength and gain control bandwidth. Using this method the 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 of ALC oscillators. The analysis is
similar to that for mode locked arrays of Van der Pol oscillators in the previous
section, so many of 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 of 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" n = η (1 − gn ) An + ε [ An−1 cos(τ + φn − φn−1 + Φ ) + An+1 cos(τ + φn+1 − φn − Φ )]
g" n = −
1
η
gn +
1
η
An
 An−1

A
sin(τ + φn − φn−1 + Φ ) − n+1 sin(τ + φn +1 − φn − Φ )
An
 An

(33)
φ"n = βn − ε 
where n = 1, 2, ! , N , 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
η
η

F + 4 cos 2 ( Φ )G ]u − η sin( 2Φ )Gv + cos( 2Φ )k1 + sin( 2Φ )k 2 (34)
[
2
2

β = ε2
The matrices F and G and the N element vectors k1 and k2 are defined as
104


 1
0
 − 1
 1




 
 


− 2 #
1 #
 0
 0
F =  1 # 1  , G =  0 # 0  , k1 =  $  , k 2 =  $ 
 
 


# − 2
# 1
 0
 0
 
 




 1
 1
0
1


0
0
0
(35)
0
and Γ 2 = 1 + ( 2η) . The vectors u and v are defined exactly as in equation (10).
We then reduce the order of the system using the transformation matrix D of
equation (11) (which happens to be F T ) to form the second differences of the
pulling equations. The result is
2
η
η

∆∆β = Dβ = ε 2  [ DF + 4 cos2 ( Φ ) DG ]u − η sin( 2Φ ) DGv + cos( 2Φ ) D k1 + sin( 2Φ ) D k 2
2
2

(36)
which is a system containing N-2 equations in N-2 unknowns.
3.3.1.1 The Locking Region
Equation (36) has a form similar to equation (12):
∆∆β = ε 2
η
2
( A u + B v + k)
(37)
which, as explained in section 3.2, defines a mapping from the N-2 dimensional
space of phases (∆∆φ) to the N-2 dimensional space of frequency pullings (∆∆β).
If each phase is allowed to span all of its possible values (it is sufficient to
consider the interval -π to π since the nonlinear transformation involves periodic
functions) the resulting frequency pullings span the region of existence of mode
locked states. Any vector ∆∆β lying in this region has a corresponding phase
vector ∆∆φ .
We will consider the size and location of the existence region in more
detail. The center of the region is found by averaging the pulling equations over
all possible values of ∆∆φ . Denoting the center by ∆∆β c we have
∆∆βc =
∫ ∆∆β ( ∆∆φ ) d∆∆φ = ε
Vol
105
2
η
2
k
(38)
where the integral extends over the N-2 dimensional volume of the phase space.
Thus the constant vector in equation (37) is the center of the existence region.
The shape of the region is generally quite complicated, as we saw for the
case of Van der Pol arrays, but we can estimate the size by computing the mean
length of the vector ∆∆β , measured with respect to the center of the region, over
all possible values of phase. Thus, using properties of circular functions we have,
1/ 2
∆∆βRMS


T
1
d
∆∆
β
∆∆
β
∆∆
β
∆∆
β
∆∆
φ
=
−
−
(
)
(
)

N −2 ∫
c
c
 ( N − 2)( 2π ) Vol

1/ 2


1
=
Tr( AT A) + Tr( B T B) 
 2( N − 2)

(
)
(39)
Carrying out the algebra gives, for N ≥ 7 ,
∆∆βRMS
η
N − 48 / 19
N −5/ 2

cos( 2Φ )
− 16
= ε 19
N −2
N −2
2

2
1/ 2
(40)
For N < 7 expressions must be derived for each value of N separately.
Comparing this to the simple case N=3 for which
∆∆βRMS = ε 2
η
[20 − 16 cos(2Φ )]
2
1/ 2
(41)
we can see that increasing the size of the array does not significantly affect the
overall size of 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 Φ=π/2). Equation (40) shows that the region is maximized for
a coupling phase of π/2 or 3π/2, similar to the Van der Pol case. However, unlike
the Van der Pol case the region above depends linearly on η, which allows us to
enhance the mode locking region by increasing this parameter.
Since Φ=π/2 maximizes the locking region, we will assume this value of
coupling phase to simplify the mathematics. Equation (37) becomes
∆∆β = ε 2
η
2
( Au + k )
(42)
106
which is the simple form we obtained for the synchronous array of 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.
∆∆β
2
∆∆β
1
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 of the eigenvectors with lengths proportional to the
eigenvalues of 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 of A, which is the volume of the locking
region in frequency pulling space, and raise this to the 1/(N-2) power:
L = [ det( A) ]
1
N −2
(43)
The resulting number L is the length of 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.
107
λ max
Amplitude Variations
4
3
2
L
1
λ min
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Number of Elements
Figure 10--Locking region size (L), maximum and minimum eigenvalues (scaled by
ε 2η )
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 tuning errors.
One can see that as N increases the size of 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 of the eigenvector corresponding to this eigenvalue cause large phase
variations and possibly loss of lock. This is also true for the synchronous arrays
of 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 of 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 we can
achieve these conditions we must look closely at the amplitude variations.
Appendix 3.3 contains the first order approximation of the time dependent
amplitudes and phases:
[
An ( t ) ≅ 1 + ε 1 + η 2 sin (τ + ∆φn − Φ + tan −1 η) + sin (τ + ∆φn−1 + Φ + tan −1 η)
[
φn ( t ) ≅ φon − ε cos(τ + ∆φn − Φ) − cos(τ + ∆φn−1 + Φ)
108
]
] (44)
for n = 2, 3, ! , N − 1 . The end element terms are slightly different and will affect
the results of 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 ∆φn ≡ φo,n+1 − φon = ∆φ for all n. Assuming this condition is met
the amplitudes and phases will be the same for each element. In addition, for
ninety degrees of 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 Φ = 0 so
that the phase condition ∆∆φ = 0 represents a stable state.
Assuming Φ = 0 the amplitude and phase equations become
An ( t ) ≅ 1 + 2ε 1 + η 2 sin (τ + ∆φ + tan −1 η)
φn ( t ) ≅ φon = n∆φ
(45)
Since the phase of the carrier envelope is ( n − 1)(ωbt + ∆φ ) (see section 3.1)
coherent phase addition, and hence peak power, occurs at (normalized) time
τ = −∆φ . At this time the amplitudes are An = 1 + 2εη , so the peak of the mode
locked pulse will be increased proportionally to the nonlinearity parameter. The
peak amplitude does not occur precisely at the time of coherent phase addition,
but quite close for large values of η. Figure 11 shows plots of mode locked
waveforms for unity amplitude oscillators and for enhanced ALC oscillators.
The main drawback to designing with Φ = 0 is that the locking region is
smaller than the corresponding region for Φ = π 2 . However, the Φ = 0 region for
large arrays is less eccentric. The analysis for this case will not be presented here,
but figure 12 shows a comparison of the size of the locking region, using the
equation (43). Although the region for Φ = π 2 is significantly larger for N=3, it
decreases fairly rapidly with N while for Φ = 0 the region remains essentially
constant; for large N the regions are comparable. Figure 13 shows the maximum
phase sensitivity for the two values of coupling phase. Since this sensitivity is
directly related to the region size in a particular direction, it also represents the
eccentricity of the region. For Φ = π 2 the sensitivity increases with N indicating,
as mentioned earlier, that the region becomes highly eccentric. However, for
Φ = 0 the sensitivity remains fairly constant. Thus, Φ = 0 represents a viable
choice for mode locked ALC arrays.
109
Ve(τ)
12
10
8
6
4
2
0
Time dependent
amplitudes
Unity
amplitudes
2π τ
π
0
Locking region size, L
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 ε=0.2, η=2, Φ=0.
3
2.5
Φ=90
2
1.5
1
Φ=0
0.5
0
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
Φ=90
Smax
8
6
4
Φ=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.
110
3.3.1.3 Experimental Verification
A three element mode locked array of ALC oscillators was designed,
built, and tested to verify the basic results of the above analysis. The center
frequency of the system was chosen to be 50 KHz and the resonators were second
order state variable filters with oscillator quality factors of 25. The voltage
controlled amplifier necessary for level control was realized using the RCA
CA3080 operational transconductance amplifier. The peak detector circuit
consists of a full wave rectifier, a diode, and RC filter. The complete circuit is
shown in figure 14.
First, we verified the amplitude response of a single ALC oscillator to an
injected signal. The magnitude of the amplitude variations is a function of 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 15. The discrepancy is caused by
component tolerances.
Next, three oscillators were coupled together through 90 degrees of
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 of 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 of section IIIc we calculated the
locking region size for the center oscillator free running frequency of 54 Hz, and
the measured value was 55-75 Hz. Excessive thermal drift and high phase
sensitivity near the edge of the locking region prevented precise measurements of
the relative phase vs. free running frequencies.
111
Resonator
Coupling
Circuit
20k
1000 pF
20k
1000 pF
470k
220k
_
Vin
20k
+
_
10k
+
_
1000 pF
1k
_
+
+
22k
470k
10k
Vout
CA3080 _
+
Voltage Controlled
Amplifier
.01
22k
.022
560k
33k
10k
+
_
10k
100
+
_
33k
33k
1k
10k
+
_
Envelope
Detector
Op-amps: LM347
Diodes: 1N4148
100
Figure 14--Schematic of a single ALC oscillator. The resonator is a state variable
bandpass filter, the voltage controlled amplifier uses an RC3080 transconductance
amplifier, and the envelope detector is a full wave rectifier and filter. The input coupling
circuit provides 90 degrees of phase shift.
112
Amplitude Variations (V)
0.2
Theoretical
0.15
0.1
Measured
0.05
0
0
1
2
3
4
Beat Frequency (KHz)
Figure 15--Response of the magnitude of the amplitude variations to an injected signal. A beat
frequency of 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 ε and η 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.
2
∆ ∆ φ =0.8 rad
Volts
Theoretical
Measured
2
1
1
0
0
-1
-1
-2
1
2
3
5 mS
4
∆ ∆ φ =0 rad
Volts
-2
1
2
(a)
2
Theoretical
Measured
3
4
5 mS
(b)
∆∆φ =-1.27 rad
Volts
Theoretical
Measured
1
0
-1
-2
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.
113
Appendix 3.1--Normalized Form of Mode Locking Equations
The dynamic equations of section 2.1 can be used, with the parameters slightly
altered:
A" n = µ S n ( An ) An + λ
1
φ"n = ωon − ω − λ
An
n +1
∑A
m
m= n −1
m≠ n
n +1
∑A
m
m= n −1
m≠ n
cos(φn − φm + Φ)
sin (φn − φm + Φ)
(46)
The parameters µ and λ are the nonlinearity and coupling parameters; we reserve
η and ε for the normalized versions. The equations above are written with the
phases defined relative to a common frequency ω. For a mode locked array,
however, the steady state frequencies will all be different. Thus, to maintain
periodic phases φ n we must redefine the phases with respect to the true
frequencies ω n . The instantaneous oscillator phases are defined above with
respect to a common frequency ω, that is, θ n = ω t + φ n , so we redefine them in
terms of the true frequencies:
θn = ωn t + φn′ → φn′ = (ω − ωn ) t + φn
(47)
Substituting the new phases φ′n in equations (46) we have the dynamic equations
in the desired form (dropping the primes for notational simplicity):
A" n = µ S n ( An ) An + λ
1
φ"n = ωon − ωn − λ
An
n +1
∑A
m= n −1
m≠ n
m
n +1
∑A
m= n −1
m≠ n
m
cos(ωb t + φn − φm + Φ)
sin(ωb t + φn − φm + Φ)
(48)
where the beat frequency is ω b = ω n +1 − ω n for n = 1, 2, ! , N − 1. For the Van der
Pol oscillator the amplitude damping function is Sn ( An ) = 1 − An2 , and in
synchronized arrays the explicit form of this function was irrelevant for first order
approximations. For mode locked arrays the function directly affects the locking
characteristics. In fact, setting the amplitudes to a constant value eliminates all
114
stable mode locked states. Thus, amplitude dynamics are necessary for mode
locking.
We will simplify the notation further by normalizing the parameters to a
beat period. Defining normalized time as τ ≡ ω b t , we have
A" n = η(1 − An 2 ) An + ε [ An −1 cos(τ + φn − φn−1 + Φ ) + An+1 cos(τ + φn+1 − φn − Φ )]
 An−1

A
sin(τ + φn − φn−1 + Φ ) + n+1 sin(τ + φn+1 − φn − Φ )]
An
 An

(49)
φ"n = βn − ε 
where η = µ ωb , ε = λ ωb , βn = (ωon − ωn ) ωb are the normalized nonlinear
parameter, coupling strength, and frequency pulling, respectively, and the dot
denotes differentiation with respect to τ.
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 of Van der Pol oscillators. Beginning with
the dynamic equations for the amplitudes and phases
A" n = η(1 − An 2 ) An + ε [ An −1 cos(τ + φn − φn−1 + Φ ) + An+1 cos(τ + φn+1 − φn − Φ )]
 An−1

A
sin(τ + φn − φn−1 + Φ ) + n+1 sin(τ + φn+1 − φn − Φ )]
An
 An

φ"n = βn − ε 
(50)
we expand the unknown variables in power series of the coupling parameter ε:
A( t ) = A( 0 ) ( t ) + ε A(1) ( t ) + ε 2 A( 2 ) ( t ) +!
φ ( t ) = φ ( 0 ) ( t ) + ε φ (1) ( t ) + ε 2 φ ( 2 ) ( t ) +!
(51)
β = β ( 0 ) + ε β (1) + ε 2 β ( 2 ) +!
Inserting these into equations (50) and equating like powers of ε gives a series of
equations that can be solved recursively. We find the zeroth order steady state
quantities are
2
A" n( 0 ) = η(1 − ( An( 0 ) ) ) An( 0 ) = 0 → An( 0 ) = 1
φ"n( 0 ) = βn( 0 ) = 0 → φn( 0 ) ≡ φon = const
115
(52)
The first order equations are
A" n(1) = −2η An(1) + [cos(τ + ∆φn−1 + Φ ) + cos(τ + ∆φn − Φ )]
φ"n(1) = βn(1) −[sin(τ + ∆φn−1 + Φ ) + sin(τ + ∆φn − Φ )]]
(53)
where the time average phase differences are defined as ∆φn ≡ φo,n+1 − φon .
Enforcing the periodicity of the amplitudes and phases enables us to determine
their steady state values:
An(1) =
1
1 + ( 2η)
2
[sin(τ + ∆φ
n −1
]
+ Φ + tan −1( 2η) ) + sin(τ + ∆φn − Φ + tan −1 ( 2η) )
φn(1) =[cos(τ + ∆φn−1 + Φ ) − cos(τ + ∆φn − Φ )]]
βn(1) = 0
(54)
The second order phase correction is
[
φ"n( 2 ) = βn( 2 ) − φn(1)−1 ⋅ cos(τ + ∆φn −1 + Φ) + ( An(1)−1 − An(1) ) ⋅ sin(τ + ∆φn−1 + Φ)
− φn(1) ⋅ cos(τ + ∆φn − Φ) − ( An(1)+1 − An(1) ) ⋅ sin (τ + ∆φn − − Φ)
]
(55)
Substituting the first order corrections and enforcing the periodicity of the second
order phase correction gives the frequency pullings as functions of the time
average phases:
βn = ε 2
2η  1
sin( ∆∆φn ) + η cos( ∆∆φn ) + sin( 2Φ ) + sin( ∆∆φn −1 − 2Φ )
Γ 2  2
1

+ sin( ∆∆φn −2 ) − η cos( ∆∆φn−2 ) , n = 2,3, !, N − 1
2

(56)
where ∆∆φ n ≡ ∆φ n +1 − ∆φ n is the second difference of the time average phases.
The frequency pullings of the end elements have additional terms:
116

1
2η 2
η
η

2
2
β1 = ε − cos ( Φ ) + 2 sin ( Φ ) + 2 sin( 2Φ )+ 2 sin( ∆∆φ1 ) + 2 cos( ∆∆φ1 )
Γ
Γ
Γ
Γ


2


βN = ε 2 cos2 ( Φ ) −

1
2η 2
η
η
2
+
+
−
2
sin
(
Φ
)
sin(
Φ
)
sin(
∆∆
φ
)
N −2
2
2
2
2 cos( ∆∆φN − 2 )
Γ
Γ
Γ
Γ

(57)
The subscripts for the variables ∆∆φ n run from 1 to N-2 and ∆φ n from 1 to N-1.
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 t is a slowly varying function of time, so that 2ω a g" << ω o 2 and the nonlinear
damping term in the first equation can be treated quasi-statically. Thus, the
frequency domain equations can be written
!
"
(ω
2
o
)
− ω 2 + j 2ωa ω (1 − Go + g ) V (ω ) = j 2ωa ω Vs (ω )
g" ( t ) + ω g g( t ) = ω g A( t )
(58)
Here we have assumed that the envelope detector has unity gain so that the output
is the amplitude of the ALC oscillator output. To simplify the equations further
we will assume the gain Go is two, which cause the steady state gain control
variable to be unity. The equations become
(ω
2
o
)
− ω 2 + j 2ωa ω ( g − 1) V (ω ) = j 2ωa ω Vs (ω )
g" ( t ) + ω g g( t ) = ω g A( t )
(59)
We 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( t ) = A( t ) cos(ωt + φ ( t ) ) and
vs ( t ) = As ( t ) cos(ωt + φs ( t ) ) and applying the methods of section 1.2.1 to the first
of equations (59) we have
117


A"   jφ
2ω (ωo − ω ) + j 2ωa ω ( g − 1) + j 2ωa ( g − 1) − 2ωo  φ" − j   Ae
A  


(60)
" 


A
=  j 2ωa ω + j 2ωa  φ"s + j s   Ase jφs
As  


(
)
A"
Dividing through by j 2ω a ω and assuming ω a << ω and φ" s − j s << ω we have
As

ωo − ω
1 
A"   A
+ j  φ" − j   = s e j ( φs −φ )
g −1 − j
ωa
ωa 
A   A

(61)
Equating real and imaginary parts gives
A" = ωa (1 − g ) A + ωa As cos(φs − φ )
φ" = ωo − ω + ωa
As
sin (φs − φ )
A
(62)
If the injected signal frequency is different than the ALC oscillator frequency, we
simply redefine the injected signal phase as φs′ = (ω − ωs ) t + φs . 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 λ
and phase delay Φ gives
[
A" n = ωa (1 − g ) An + λωa An+1 cos(ωbt + φn+1 − φn − Φ) + An cos(ωbt + φn −1 − φn − Φ)
]
 An+1

A
sin(ωbt + φn+1 − φn − Φ) + n−1 sin(ωbt + φn−1 − φn − Φ) 
An
 An

φ"n = ωon − ωn + λωa 
(63)
We now normalize the time parameter to the beat frequency, τ = ω b t , and
substitute into the above equation. The result, including the gain control variable,
is the normalized dynamic equations for the ALC oscillator:
118
[
A" n = η(1 − g ) An + ε An+1 cos(τ + φn+1 − φn − Φ) + An cos( τ + φn−1 − φn − Φ)
g" ( t ) +
1
η
g( t ) =
1
η
A( t )
]
(64)
 An+1

A
sin(τ + φn+1 − φn − Φ) + n−1 sin(τ + φn −1 − φn − Φ) 
An
 An

φ"n = βn + ε 
where η = ωa ωb is the normalized nonlinearity parameter, ε = λωa ωb is the
normalized coupling parameter, βn = ( on n ) ωb is the normalized frequency
pulling, and the dot now denotes differentiation with respect to τ. The gain
2
control filter bandwidth is related to the other parameters as ω g = ωb ωa so that the
ω −ω
amplitude resonance is the same as the beat frequency.
119
[
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 of Mode Locking in
Coupled Oscillator Arrays," J. Appl. Phys., vol. 71, no. 6. pp 2959-2965, March 15, 1992.
[3] 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.
[4] J. J. Lynch, R. A. York, "Mode Locked Arrays of Microwave Oscillators," 1993 Symposium
on Nonlinear Theory and Applications, vol. 2, p. 605.
[5] J. J. Lynch, R. A. York, "An Analysis of Mode Locked Arrays of Automatic Level Control
Oscillators," IEEE Trans. on Circuits and Systems, vol. 41, Nol 12, pp. 859-865, Dec. 1994.
120
Chapter 4
The Design of Microwave Phase Locked Loops
Significant progress has recently been made in the design and fabrication
of quasi-optical transmitting arrays using coupled microwave oscillators.[1][2]
However, conventional oscillator arrays, such as those discussed in the bulk of
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 of 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 of
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 of the array pattern.
To overcome these difficulties phase locked loops (PLL's) can be used in
place of 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 of 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 of a single
microwave phase locked loop for use in beam steering systems. The design is
kept simple to aid the future fabrication of arrays of PLL circuits. A frequency of
9 GHz was chosen to allow the use of readily available microwave HEMT's and
varactors. A detailed description of the analysis, design, and measured
performance of the PLL will be presented, but first we will briefly consider the
use of coupled PLL's in beam steering systems.
121
4.1 Arrays of Phase Locked Loops for Beam Steering Systems
In this section we present an array of coupled phase locked loops that
allows electronic steering of 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 fact, 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.
tuning
port
ωo1
Σ
...
...
+
ωo2
-
Σ
+
ω o,N-1
ω oN
...
output
φ0
φ1
φN-1
φN
antenna
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 of this idealized system will not be presented in this section but the
resulting phase equations are
(
)
φ!n = ωon − ω − ε sin( φn+1 − φn ) − sin( φn − φn−1 ) for n = 1,2,", N
(1)
(note: terms containing variables with subscripts zero or N+1 must be ignored).
Previous analyses have shown that the frequency and phases of the array are
controlled by the first and last elements. If 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 of array the reader should consult
reference [4].
122
4.1.1 Ideal PLL Operation
An ideal phase locked loop is shown in figure 2.
Pwr
VCO
Output
Div
Tuning
Port
Phase Detector
Input
Figure 2--Block diagram of PLL.
The PLL consists of 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 of 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 remains 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 of 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 of
locked and unlocked loops are shown in figures 3.
Oscillator Frequency
A(ω)
Oscillator Frequency
A(ω )
Injected Signal Frequency
Injected Signal Frequency
Sidebands
ω ο ωs
ωο=ωs
ω
(a)
ω
(b)
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.
123
A central concern of PLL analysis is determining the conditions required
to maintain synchronization. In addition, the assumed use of PLL's is for beam
steering systems, so the phase difference between input and output across the
locking range is of 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 of nonlinear dynamics are treated in many texts, for example [5]. As with
the analyses of previous chapters, the following will be limited to what I believe
to be the most significant aspects of PLL performance for use in beam steering
systems.
The circuit equation for a single PLL can be written as
ω + φ! = ωo + S vc
(linear VCO)
vc = −Vo cos( (ω − ωi )t + φ ( t ) − φi ) (ideal phase detector)
(2)
where ω + φ! is the instantaneous VCO output frequency, ω i + φ! i is the
instantaneous input frequency, vc is the phase detector output voltage, S is the
VCO tuning sensitivity, ω o is the VCO frequency with zero tuning voltage, and
Vo 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 of time. The differential equation
(2) reduces to an algebraic equation that relates the steady state frequency to the
phase difference between input and output:
ω = ωo − S Vo cos( ∆φ )
(3)
where ∆φ = φ − φ i . The above relation shows that minimum and maximum
values of ω exist since the sine function is bounded, and as the input frequency is
varied the phase difference ∆φ also varies. Each choice of ω within the range of
124
existence of solutions to (3) gives rise to two possible values of phase difference.
It turns out that one of the values corresponds to a stable solution and the other to
an unstable solution, much like the stable and unstable equilibria of a rigid
pendulum.
To find the stability condition formally we would perturb equations (2)
from their equilibria and observe whether the perturbations grow or decay. Since
we have already done this in section 1.2 we will simply apply those results. First
we express the frequency pulling as a function of the VCO phase
β ≡ ωo − ω = S Vo cos(φ − φi ) = S Vo cos( ∆φ )
(4)
The stability condition is
dβ
= − S Vo sin ( ∆φ ) > 0 → − π < ∆φ < 0
dφ
(5)
This limits the phase difference to a range of values for which the cosine function
of equation (4) is monotonic. Thus, the stability condition ensures that a unique
stable state exists for a given input frequency ω . For this simple case it is
instructive to establish the stability condition by inspection of the PLL block
diagram. Since we have assumed the phase of the carrier at the VCO output is
increasing in time, that is v(t ) ∝ e jωt , increasing the frequency also increases the
phase of the signal. For stable frequency locking of the closed loop system an
increase of 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
dφ
<0 →
dω
dω
<0
dφ
(6)
which is the same as (5).
Figure 4 shows a graphical representation of the synchronization region
and phase difference between PLL and injected signal for this idealized example.
Near the center of 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.
125
It is important to remember that the above synchronization range is
defined as the values of input frequency where frequency locking is possible.
Synchronization may be possible only for a narrow range of initial conditions of
ω
ωs
π /2
ωs
ωo
∆φ
ωs
ωo
− π /2
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 PLL 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 of 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 of 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 of a simple negative resistance oscillator is shown in
figure 5.
b1
b2
a1
a2
II
I
Device
Term
and
Network
Load
SII
SI
126
Figure 5--Block diagram of a negative resistance oscillator.
The circuit consists of a device, with load included, and a termination network
chosen to create oscillations at the desired frequency. For voltage control of the
oscillation frequency this block will contain a varactor diode. The incident and
reflected waves of each block are related by the circuit's S parameters
b1 = S I a1 , b2 = S II a2 , and to each other a1 = b2 , a2 = b1 .[6] Combining these
equations gives steady state conditions for oscillation in terms of the wave
incident on the device:
S I S II a2 ≡ Sa2 = a2
(7)
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 S II would be measured. Next, we express the wave a2 in
terms of its slowly varying amplitude and phase
a2 =
1
( )
A( t ) e j (ωot +φ t )
Zo
(8)
where the frequency ω o is the steady state frequency of oscillation. Thus the
condition for oscillation can be written
S ( Ao , ωo ) = S I (ωo ) S II ( Ao , ωo ) = 1
(9)
where the subscript 'o' denotes steady state quantities. Utilizing Kurokawa's
method (section 1.2.1) we derive the dynamic equations for the amplitude and
phase of the incident wave. The result is
1− S 
A! = − A Im 

 S′ 
(10)
1− S 
φ! = Re

 S′ 
where S ′ ≡
∂S
∂ω
. One can see that the oscillation condition of equation (9)
Ao ,ω o
gives fixed points, or time independent values, for the amplitude and phase. The
127
amplitude equation in (10) is stable when the factor on the right hand side is
negative. Thus the stability condition is
∂ S 
 ∂ 1− S  
∂A  < 0
Im  
  > 0 → Im 
∂
S




∂
A
S
′

 S =1
 ∂ω  S =1
(11)
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.
II
I
S
S'A
S'ω
θA−θ
ω
Figure 6--Smith chart representation of stable oscillation. Point I is the point of steady
state oscillation and point II is the value of S at power on (small signal S parameters).
The steady state is stable when the angle θ A − θ ω between the amplitude and frequency
perturbation vectors is less than 180 degrees.
The point S=1 is the location of 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, S (0, ω ) , shown at
point II. As the amplitude and phase change according to the dynamic equations
(10), the "instantaneous" value of S traces out a trajectory and eventually settles
out to S=1. The stability condition on the right side of equations (11) has a
simple graphical interpretation. If we represent S as a vector on the Smith chart
then the changes in S due to a change in A, denoted SA′ , and a change in ω,
128
denoted Sω′ , are also vectors, as shown in figure 6. Expressing these changes in
polar form
∂S
∂S
= S A′ e jθ and
= Sω′ e jθω , θA,ω < π
∂A
∂ω
A
(12)
the stability condition becomes
∂ S 
∂A  = S A′ sin(θ − θ ) < 0
Im 
ω
A
∂ S

Sω′
 ∂ω  S =1
(13)
Assuming SA′ and Sω′ are defined as positive values in equation (12), the steady
state at S=1 will be stable when − π < θ A − θ ω < 0. This condition holds when the
∂S
∂S
vector to the
vector, in the
positive angle measured from the
∂A
∂ω
counterclockwise direction, is less than 180 degrees. If 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 of the requirements for
small signal S parameters to insure stable operation. If the small signal
magnitude of 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 of power. To better understand
the mechanisms that govern oscillator power generation we consider a simple
example. Figure 7 shows an oscillator model consisting of 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 of the oscillator). At its
terminals, the device will have some I-V characteristic that may resemble typical
129
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. If the resistance is high, as in line II, we have the converse situation.
Line III represents an optimum point since both voltage and current are
maximized. In addition, the voltage and current saturation due to pinch off and
L
-R(A)
I
III
30
V
C
RL
I (mA)
+
-
20
II
10
V=Acos(ω ot)
0 0
1
2
V (Volts)
3
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.
Amplitude
cut off vary as the load resistance varies. Thus we find that the steady state
amplitude of oscillation depends on the load resistance. A hypothetical
dependence is illustrated in figure 8.
RL
Figure 8--An illustration of the dependence of oscillation amplitude on the load
resistance. The dotted lines indicate optimum load resistance and amplitude.
130
The power delivered to the load is given by
PL =
A2 ( RL )
RL
(14)
where A is the amplitude of the voltage waveform across the load. Assuming the
simple amplitude curve of 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 of (14) equal to zero. The resulting condition for optimum power
dissipation is
dA 1 A
=
dRL 2 RL
(15)
This result represents the trade off between voltage amplitude and load resistance.
The optimum point is shown in figure 8.
The results of this section are difficult to utilize directly since this would
require knowledge of the functional dependence of the voltage amplitude on the
load resistor. Characterization of microwave devices is generally difficult since
we are usually constrained to a measurement system of a particular impedance,
e.g. 50Ω, 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 of the NEC FET transistor NE32184A. No large signal
measurements were performed so we will be forced to make certain assumptions
about the amplitude dependence of the device S parameters. The Libra software
package from EEsof was used to simulate the RF circuit performance, but the
level of complexity of 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 of
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
131
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.
5V
120Ω
5V
Via
Holes
120Ω
.4V
S
D
2.5V
20Ω
G
20Ω
S
G
S
D
NE32184A
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 50Ω 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.
132
θ1
S
G
Z1
D
θ2
Zo
C
Load
Zo
SII
Via
S
G
S
D
7 GHz
To 50 Ω
Load
11 GHz
S
S
II
II
Figure 10--Schematic, 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).
As with most packaged microwave transistors, the NE32184A has two source
leads and we will connect to both of them in a symmetric fashion. In the
Touchstone simulation, however, only one source lead is available, so in the
simulation we will connect the two 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 II shown on a Smith chart. A capacitive reactance was
produced to cancel the inductive reactance of 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 I S II which may or may not contain a loop.
After the appropriate varactor network is connected, oscillations will
build up and the parameter S II at the operating point will move toward the unit
circle (or so we assume). The steady state will be reached when the phase of S II
is equal and opposite to the phase of the varactor circuit S I , and the magnitudes
are inverses.
4.1.2.2.3 Varactor Circuit Design
133
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.
.15pF
.4nH
Cv
2Ω
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 of the diode circuit
and the input reflection coefficient are shown in figure 12. The inductive
impedance is due to the effects of copper strips that connect varactor to shunt
capacitor (see figure 13).
11 GHz
θ
Varactor
I
S
Cs
Z
7 GHz
SI
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 of the microstrip trace
that represents the junction of 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
134
Radial
Stub
Cu Strip
Varactor
Varactor
Cu Strip
20Ω
S
G
S
Source
line
D
Radial
Stub (Cs)
Substrate
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).
Capacitance (pF)
The varactor capacitance depends on the varactor reverse bias voltage according
to the curve shown in figure 14 (computed from catalog parameters):
2
1
0
0
1
2
3
4
Reverse Bias (V)
5
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 of the
source transmission lines to give the desired frequency of operation. This is
possible since changing these lengths changes the phase of S II . The oscillation
135
condition is best displayed, as described in section 4.1.2.1, by displaying the
quantity S = S I S II 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 of varactor capacitance.
S
S
7 GHz
7 GHz
7 GHz
S
11 GHz
11 GHz
(b)
(a)
11 GHz
(c)
Figure 15--Total scattering parameter S = S I S II vs. frequency for three different values
of varactor capacitance: (a) Cv = 0. 5 , (b) Cv = 0. 7 , (c) Cv = 1. 0 .
Frequency (GHz)
To estimate the oscillation frequency we assume that the above curve moves
radially inward as the amplitude increases. Thus the frequency of oscillation is
the frequency for which S has zero phase. The VCO tuning curves computed
from the simulation and measured from a fabricated circuit are shown in figures
16:
9.4
Simulated
9.2
9
8.8
Measure
8.6
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 tuning curves. The large discrepancy in tuning
sensitivity is probably due to inadequate high frequency modeling of the varactor diode
and associated circuitry.
136
The discrepancy between the simulated and measured curves is probably due to
inadequate modeling of 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 of 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 of the phase detector is to produce an output voltage
proportional to the sine or cosine of the phase difference between two RF input
signals. Standard mixers are often used as phase detectors, but one of the inputs is
a high power pump (LO) and the other is a much lower level signal (RF).[8] For
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
of a Wilkinson power combiner followed by an active RF power detector.
DC Outpu
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 of the power combiner is the sum of the two input voltages.
Defining the inputs as
v1( t ) = A1 cos(ωt + φ1 )
(16)
v2 ( t ) = A2 cos(ωt + φ2 )
the output voltage is
137
v( t ) =

 A − A2
 φ − φ  
A12 + A22 + 2 A1 A2 cos( φ1 − φ2 ) cos ωt + tan −1  1
tan  1 2   
 2  
 A1 + A2

1
1
v1 + v2 ) =
(
2
2
(17)
This signal is then sent to an RF level detector whose output is proportional to the
RF power:
1
vd = Gd A12 + A22 + 2 A1 A2 cos( φ1 − φ2 )
(18)
2
(
)
where Gd is the detector gain (in dimensions of V −1 ). If both input amplitudes
(
)
are equal the above simplifies to vd = Gd A2 1 + cos( φ1 − φ2 ) . Thus the detector
output varies as the cosine of the phase difference, as desired.
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.
5V
510Ω
NE32184A
DC Outpu
RF Input
Matching
Network
AC Short
1KΩ
-1.1V
Figure 18--Detector is a common source amplifier biased near pinch off.
The transistor is biased near pinch off 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 off, 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
138
35
Drain Current (mA)
30
25
20
15
10
5
0
-1.2 -1
-0.8 -0.6 -0.4 -0.2
Gate Voltage (V)
0
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)
output voltage increases as the square of the gate voltage. Thus the circuit
operates as a power detector.
To better understand the detector operation we will consider the circuit
equations. The DC output is given by
v = VDD − Rd id
(19)
where the brackets denote the time average value. The gate voltage is a DC bias
and an RF signal
v g = VGG + A cos(ωt + θ ) = VGG + vrf
(20)
( )
We assume that the drain current is a function of the gate voltage as id = f v g .
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 f v g in a Taylor series about the gate bias
voltage that will be most accurate for small RF voltages. The result is
139
( )
id v g ≅ id (VGG ) +
did
dv g
= io + gmvrf +
⋅ vrf +
VGG
1 d 2id
2 dv g2
⋅ vrf2
VGG
(21)
1
g′ v 2
2 m rf
where io is the drain bias current and gm is the transconductance with no RF
input. Taking the time average value gives
1
RD gm′ vrf2
2
1
= VDD − RD io − RD gm′ A2
4
v = VDD − RD io −
(22)
The first two terms give the output voltage when no RF signal is applied. As
mentioned above, the transistor is biased near pinch off so that io is small and
therefore the output voltage is nominally close to the power supply. Upon
application of 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.
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 in gm . 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.
1KΩ
RF Input
DC Output
G
NE32184A
510Ω
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
transconductance with gate voltage. (Layout is approximate).
140
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 21.
Cg=.3pF Lg=.6nH
12 GHz
Rg=4Ω
FET Input
Γ in
Zin
5 GHz
Figure 21--Equivalent circuit and input impedance near 9 GHz.
Details of 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 of the input
impedance of the completed detector.
0
-1
Γin (dB)
-2
-3
-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 Pinc (RMS) the power dissipated in the
(
gate is Pg = 1 − Γin
2
)P
inc
, where Γin is the input reflection coefficient plotted
141
above. The RMS current through the resistor is I g =
Pg
Rg
, so the RMS voltage
developed across the gate terminals is Vg = Ig Rg2 + Xg2 . At 9 GHz the FET input
impedance is primarily capacitive, Zin 9GHz = 4Ω − j 24Ω . Using the above
expression with an input power of 6 dBm (4 mW RMS) produces a gate voltage
of .77 V peak. With a gate bias of -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 of 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 of 50% duty cycle. The measured curve falls below this level,
however. This is probably due to unequal rise and fall times of the output
voltage. If the FET turns on faster than its turns off then, under large signal
conditions, the average value of the output voltage will be lower than 2.5 V. This
behavior enhances the voltage range of 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.
Output Voltage (V)
5
Simulated
4
Measured
3
2
1
-20
-10
0
Input Power (dBm)
10
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.
142
Output Voltage (V)
5
4.5
4
3.5
3
0
0.5 1
1.5 2
2.5
Phase Difference (rad)
3
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.
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.
143
5V
120Ω
S
G
S
20Ω
Branchline
Coupler
PLL outpu
(10dBm)
D
NE32184A
1KΩ
50Ω
-1.1V
1KΩ
100Ω
PLL Inpu
(3 dBm)
G
NE32184A
Wilkinson
Combiner
510Ω
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 back to the
VCO tuning port. (Layout is approximate).
A branchline hybrid couples about 3 dBm of VCO output power back to the
phase detector input. With a VCO output power of 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.
144
5
10.4
10.2
4
10
3
9.8
2
9.6
8.5
8.6
8.7
8.8
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 the 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 of the PLL is for beam steering systems, the total
phase change between the PLL output and the input is an important figure of
merit. This measurement is somewhat complicated since a calibration of the
measurement apparatus is required to eliminate errors. The measurement system,
shown in figure 27, consists of a pair of power dividers, a reference cable, and a
phase detector. The PLL output biases the LO port of 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
145
50Ω
out
Pwr
Div
PLL
in
3 dBm
Calibration
Reference
Planes
RF
LO
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).
10 dBm
Pwr
Div
3 dBm
Reference
Cable
50Ω
Calibration
Reference
Planes
RF
LO
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 of the measurement system and subtract this from the PLL
measurement.
This procedure will become clearer by considering the
measurements in detail.
146
The first step of the calibration is a measurement of the mixer output with
an LO bias, but with the RF mixer input terminated, for a frequency range greater
than the locking band of the PLL. The result is plotted in figure 29.
4.4
IF Output (mV)
4.2
4
3.8
3.6
3.4
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 of figure 29 is subtracted, is shown in figure
30.
100
IF Output (mV)
80
60
40
20
0
8.5
8.6
8.7
8.8
Frequency (GHz)
Figure 30--IF output with configuration of figure 28.
147
8.9
9
We assume the mixer output is proportional to the sine of the phase difference
between the two inputs, and can be written
vcal = V sin(ωτcal + θ (ω ) ) ,
(23)
where V is a constant amplitude, τ cal is the time delay between the reference
planes shown in figure 28, and θ (ω ) 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 of ωτ cal and is shown in figure 31.
3
2
1
0
-1
-2
-3
8.5
8.6
8.7
8.8
Frequency (GHz)
8.9
9
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 θ (ω ) ,
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 of the PLL, as shown in figure 34.
148
2
1.5
θ (ω )
1
0.5
0
-0.5
-1
-1.5
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.
IF signal with PLL (mV)
100
50
0
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.
149
PLL Input-Output Phase Shift (deg)
80
100
120
140
160
180
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 fact that the phase difference between the
PLL input and output is a decreasing function of frequency, which was proved in
section 4.1.1.
The above phase change is only about 90 degrees, which is half of 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 of synchronization near the edges of the locking region
where loop stability is reduced. This further diminishes the phase range.
150
[
1] P. Liao, R. A. York, "A New Phase-Shifterless Beam-Scanning Technique using Arrays of
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," IEEE MTT-S, International Symposium Digest, San Diego, June 1994, pp.
1231-1234.
[3] S. Nogi, J. Lin, T. Itoh, "Mode Analysis of Stabilization of 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 of Phase Relationships in Quasi-Optical Oscillator Arrays,"
IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1799-1809.
[5] E. A. Jackson, Perspectives of Nonlinear Dynamics, Cambridge Unversity Press, Cambridge,
1989.
[6] G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, Prentice-Hall, 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.
151
Areas for Continuing Study
The research presented in this dissertation could proceed in many possible
directions. This work was not simply a "linear" continuation of a previous effort, but tied
together our understanding of two distinct types of quasi-optical sources, synchronous
and almost periodic. Much of this progress was due to the analytical techniques
presented here. These methods were an extension of mature classical perturbation theory
applied to nonlinear oscillatory systems, and they supplied the most important
information that we required for the design of such systems. The mathematical study of
such systems has seen some significant advances over the past thirty years.[1] 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 of microwave engineering.
Another area of study is the use of more complex coupling schemes for
synchronous and mode locked arrays. For all practical cases considered here the arrays
have been linear (i.e. arranged in a row) with nearest neighbor coupling. Two
dimensional arrays provide the possibility of beam steering in two 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 of 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 of 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 of
amplitude response. The results of 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 of interest. However, applying the
methods developed in chapter 1 may increase the accuracy of the 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
152
possible that matching the intermode spacing to the amplitude resonant frequency will
have the desired effects listed in section 3.3.
The treatment of 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 of
characteristic tunings provides a lot of insight into array behavior. Further development
of these concepts is necessary for practical systems. The effects of 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 of 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. With
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 of 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 of Solid-State Millimeter-Wave Sources," IEEE Trans.
Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb 1986.
153