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Analysis and design of coupled-oscillator arrays for microwave systems

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ANALYSIS AND DESIGN OF COUPLED-OSCILLATOR
ARRAYS FOR MICROWAVE SYSTEMS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of the Ohio State University
By
Renaud Moussounda, BSEE,MSEE
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2013
Dissertation Committee:
Prof. Roberto Rojas, Advisor
Prof. Patrick Roblin
Prof. Fernando Teixeira
UMI Number: 3710325
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UMI 3710325
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�
Renaud Moussounda
2013
ABSTRACT
The concept of synchronized nonlinear coupled oscillators is applied to microwave
and antenna engineering for the analysis and design of wireless communication and
sensing systems operating at the microwave and/or millimeter (mm)-wave frequencies. The significance of such approach is justified from the potential gain in efficiency,
weight, cost and functionality although technical challenges stand in the way. Unlike
typical phased array systems, which are currently used to construct such systems,
coupled-oscillator systems present additional challenges that mainly arise from maintaining stability and synchronization as the the coupled nonlinear system is operated.
Linear systems do not present such stability issues and are consequently faster since
they do not rely on any gradual synchronization mechanism in order to function. However, at significantly higher frequencies in the quasi-optical domain, coupled-oscillator
systems can make up for the speed difference and present significant efficiency advantages over typical phased array architectures. In addition, coupled nonlinear systems
possess inherent analog properties that can be used for a multitude of functions.
This dissertation advances the topic of coupled-oscillator arrays by 1) developing
an alternative set of techniques for designing the oscillating unit cells called active
integrated antennas (AIAs) at microwave or mm-wave frequencies, 2) developing a
more accurate description of the dynamics of the array, 3) developing and implementing a new topology for a coupling network that is able to extend stability, 4)
ii
implementing a fully non-reciprocally coupled array able to produce large scan angle
without loss of stability, 5) proposing an architecture based on a single phase-locked
loop (PLL) and containing a self-calibration mechanism, and finally 6) implementing a phase-boosting mechanism using simple circuits to amplify the phase difference
between adjacent radiating antennas in order to increase scanning range.
iii
This dissertation is dedicated to my family, especially to my mother and father who
I will always love.
iv
ACKNOWLEDGMENTS
First, I would like to thank Professor Roberto Rojas for his guidance and insight
for the last five years. I am grateful to him for believing in my abilities to complete
this program. Second, I would like to thank all the members of my dissertation
committee for taking the time and effort to read and evaluate this work. Thank you
Professor Roblin and Professor Teixeira for your efforts. Lastly, I would like thank
all my former and present colleagues at the ElectroScience Laboratory (ESL) for the
insightful and stimulating discussions on various topics. Specifically, I would like
to acknowledge Bryan, Khaled, Id, Keum-Su, Ezdeen, Aseim, Ethan and Professor
Ismael Martinez for the memorable moments and the collaborative research work.
You all had a positive impact on the fulfillment of this dissertation.
v
VITA
2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BAC, LTNOB (High School), Gabon
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Montgomery College, MD (62 credits)
2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BSEE, Georgia Institute of Technology
2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MSEE, The Ohio State University
2009-Present . . . . . . . . . . . . . . . . . . . . . . . . . .
Graduate Research/Teaching Associate, The
Ohio State University
PUBLICATIONS
Journal Articles
R. Moussounda and R.G. Rojas, “Analysis of array of coupled nonlinear microwave
oscillators for active integrated antennas,” (submitted for publication) Physical Review E.
R. Moussounda and R.G. Rojas, “Parametric Design Techniques for Self-Oscillating
Active Integrated Antennas”, (submitted for publication) IEEE Microwave Theory
and Techniques, July 2013.
A. M. Elfrgani, R. Moussounda, R.G. Rojas and K.-S. Song, “Stability Analysis of
Non-Foster Circuits Based on Time-Domain Method,” (Submitted for publication),
IEEE Microwave Theory and Techniques, July 2013.
J.I. Martinez-Lopez, R. Moussounda and R.G. Rojas, “Non-reciprocal Coupling Network for Beam Steering Coupled Oscillator Arrays,” IET Microwave Antennas Propag.
Vol.5, Is. 8, pg. 940-947, 2011.
vi
Conferences Presentations and Papers
R. Moussounda, E. A. Elghannai and R.G. Rojas, “Harmonically Loaded Active Integrated Antenna Using Characteristic Mode Theory,” to be presented at URSI/USNC
Meeting, Boulder, CO, Jan. 2014.
A. M. Elfrgani, R. Moussounda and R.G. Rojas, “Time-domain Stability Analysis/Design of Negative Impedance Inverters and Converters,” Presented at Int. Microw. Symp, Seattle, WA, July 2-7, 2013.
C.-C. Liu, R. Moussounda and R.G. Rojas, “ A 60-GHz Active Integrated Antenna
oscillator,” presented at USNC-URSI/USNC Meeting, Boulder, CO, Jan. 2013.
R. Moussounda and R.G. Rojas, “Phase-Amplifying Architecture for Coupled-Oscillator
Arrays”, Presented at IEEE AP-S/URSI-USNC Conference, Chicago, IL, 2012.
R. Moussounda and R.G. Rojas, “Parametric Design and Synthesis Technique to
Minimize Phase Noise and Improve Power Efficiency for Active Integrated Antennas”,
Presented at IEEE AP-S Int. Symp. & USNC/URSI NRS Meeting, Spokane, WA,
July 3-8, 2011.
J.I. Martinez Lopez, R. Moussounda and R.G. Rojas, “Nonreciprocal Coupling Network for Beam Steering Coupled-Oscillator Arrays,” Presented at the IEEE Int.
Symp. Phased Array Systems Tech., Boston, 2010.
FIELDS OF STUDY
Major Field: Electrical Engineering
Specialization: Electromagnetics
vii
TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
CHAPTER
1
2
PAGE
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Key Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . .
1
5
6
Coupled Nonlinear Oscillators: Theory and Applications . . . . . . . .
9
2.1 Basic Oscillating Unit . . . . . . . . . . . . . . . . . . . . .
2.2 Self-regulation, Frequency Entrainment and Chaos . . . . .
2.3 Review of Nonlinear Oscillators in Phased Array Antennas
2.3.1 Active Integrated Antennas . . . . . . . . . . . . .
2.3.2 Coupled-Oscillator Arrays . . . . . . . . . . . . . .
3
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9
11
14
15
18
Analysis and Design of Active Integrated Antennas . . . . . . . . . . .
27
3.1 Introduction . . . . . . . . . . . . . . . . .
3.2 Parametric Modeling and Design Approach
3.2.1 Convexity of Antenna Immittance .
3.2.2 Feedback Termination Modeling . .
3.3 Application of Concepts . . . . . . . . . .
3.3.1 Design with Existing Antenna . . .
3.3.2 Design with Synthesized Antenna .
3.4 Fabrication and Measurements . . . . . . .
27
29
29
33
36
36
42
48
viii
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4
5
6
Dynamics of Coupled Nonlinear Oscillator Arrays . . . . . . . . . . . .
52
4.1
4.2
4.3
4.4
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74
Design Consideration for Coupled-Oscillator Arrays . . . . . . . . . . .
79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Coupling Considerations . . . . . . . . . . . . . . . . . . . .
5.2.1 Transmission-Line based Coupling Networks . . . .
5.2.2 Practical Implementation . . . . . . . . . . . . . . .
5.3 Design Method and Considerations . . . . . . . . . . . . . .
5.3.1 Counteracting Coupling Effects . . . . . . . . . . .
5.3.2 Non-reciprocal Coupling Networks . . . . . . . . . .
5.4 Implementation of a Fully Nonreciprocal Coupled-Oscillator
79
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Array
Coupled-Oscillator based Applications and Array Architectures . . . . . 106
6.1
6.2
6.3
6.4
7
Introduction . . . . . . . . . . . . . . .
Amplitude and Phase Dynamics . . . .
Transient and Stability Analysis . . . .
Numerical Simulations and Verification
4.4.1 Frequency Deviation . . . . . .
4.4.2 Synchronization . . . . . . . . .
4.4.3 Comparison to Previous Models
Introduction . . . . . . . . . . . . . . .
Array Architecture . . . . . . . . . . . .
Array Calibration and Pattern Shaping
Phase-Amplifying Arrays . . . . . . . .
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106
107
112
114
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1 Summary . . . . . . . . . . . . . . . . . . . . .
7.2 Future Work . . . . . . . . . . . . . . . . . . .
7.2.1 Harmonically Loaded AIA and COAs .
7.2.2 Non-Foster Loads for Phase-Amplifying
CHAPTER
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Arrays
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127
129
129
130
PAGE
A
Coupling Properties of Resistive Lines
. . . . . . . . . . . . . . . . . . 131
B
Matlab/FEKO Co-Simulation Commands . . . . . . . . . . . . . . . . 134
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
ix
LIST OF FIGURES
FIGURE
PAGE
1.1
Architecture of a typical phased array. . . . . . . . . . . . . . . . . .
3
1.2
Correspoding architecture of coupled-oscillator array. . . . . . . . . .
5
2.1
Basic model of oscillator.
. . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
Self-regulation depicted by a limit cycle with �=0.5 and τ =20 . . . .
12
2.3
Van der Pol oscillator with injection current source. . . . . . . . . . .
13
2.4
Diagram (bifurcation) showing behavior of forced van der Pol oscillator
for ωin /ω0 = 1.5 and F =1.2. . . . . . . . . . . . . . . . . . . . . . . .
14
2.5
Instance of a self-oscillating AIA. . . . . . . . . . . . . . . . . . . . .
17
2.6
Inter-injection phased array architecture . . . . . . . . . . . . . . . .
20
2.7
York beam-steering mechanism for weakly coupled systems . . . . . .
22
2.8
PLL-based architecture . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1
Plots of antenna locus and device lines . . . . . . . . . . . . . . . . .
32
3.2
Simple oscillator model with i(t) = A cos(ωt + φ). . . . . . . . . . . .
34
3.3
Parametrization of the input impedance of the drain terminal of a
FET transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Real and imaginary parts of the input admittance of a patch antenna
and its locus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Mapping of the convex region described by antenna locus into the
transistor design space. . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Time series of current signal and corresponding spectrum waveform. .
41
3.4
3.5
3.6
x
3.7
Schematic of circuit for obtaining behavior of circuit . . . . . . . . .
42
3.8
Plot of low-variation points (|∆Z|) . . . . . . . . . . . . . . . . . . .
44
3.9
Frequency derivative of total nonlinear core impedance at 5 GHz and
7 dbm RF level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Antenna impedance as a function of pin location and corresponding
gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
(a) Synthesized patch antenna input impedance around 5 GHz for f
= 4.58 mm. (b) Corresponding impedance locus. . . . . . . . . . . .
47
3.12
Performance of AIA for 7 dBm output power. . . . . . . . . . . . . .
48
3.13
CAD generated layout implementing design of AIA at 5 GHz. . . . .
50
3.14
Spectrum measurement of the radiated power. The oscillation frequency is 5.018 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.1
Circuit representation of N linearly coupled nonlinear oscillators . . .
54
4.2
Approximation of the normalized van der Pol oscillation frequency
(dashed) with exact (solid). The deviation increases as the nonlinear
factor increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.10
3.11
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Chain of RL-coupled van der Pol oscillators with nonlinearity described by i = −av + bv 3 . . . . . . . . . . . . . . . . . . . . . . . . .
67
Loss and gain of synchronization between oscillators as the tank capacitance of first oscillator is varied . . . . . . . . . . . . . . . . . . .
69
Phase difference φp − φk between adjacent oscillators showing a loss
of synchronicity around C1 = 0.935 pF . . . . . . . . . . . . . . . . .
70
Phase difference between oscillators 10 and 11 from the chain of oscillators.The coupling strength varies from 0.0285 to 0.0292 across the
array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Left: phase difference between oscillator 10 and 11. Right: amplitude
comparison at oscillator 11 . . . . . . . . . . . . . . . . . . . . . . .
72
Left: phase difference between oscillator 10 and 11. Right: Amplitude
comparison at oscillator 11.The coupling strength varies from 1.2739
to 1.4171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Amplitude and phase difference between oscillator 1 and 2. . . . . . .
75
xi
4.10
Comparison of phase differences in degrees between oscillator 1 and 2
using York’s equations, derived equations and exact dynamics . . . .
78
A pair of oscillators coupled through a transmission line-based circuit
(model #1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
A pair of oscillators coupled through a transmission line-based circuit
(model #2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Comparison of quality factors for various values of the electrical length
of the coupling transmission lines. . . . . . . . . . . . . . . . . . . . .
85
Comparison of coupling coefficients for various values of the electrical
length of the coupling transmission lines. . . . . . . . . . . . . . . . .
86
Two models of folded transmission lines for linking oscillators. (a)
Modeled after a sine function. (b) A rectangular shape. . . . . . . . .
88
Top and bottom views of array of patch antennas Antenna Dimension:
W=20.5mm, L=19.5 mm. Coupling line: Z0 = 100Ω. . . . . . . . . .
90
5.7
Flowchart describing optimization process by co-simulation
. . . . .
91
5.8
Progression of the algorithm as the array converges to a uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
5.9
Plot and comparison of array factors for 5-element patch array. . . .
94
5.10
Final dimensions of the array design . . . . . . . . . . . . . . . . . .
95
5.11
Optimization of scan angle in the presence of coupling. . . . . . . . .
96
5.12
Architecture of half of the nonreciprocal coupling network . . . . . .
98
5.13
Architecture of a fully nonreciprocal array. . . . . . . . . . . . . . . . 100
5.14
Layout and fabricated array showing active components and bias lines .102
5.15
Measured power spectrum of array . . . . . . . . . . . . . . . . . . . 103
5.16
Spectrum of phase fluctuations of the array . . . . . . . . . . . . . . 104
5.17
Comparison of measured radiation pattern to expected simulated results.105
6.1
COA phase-locked loop based architecture. . . . . . . . . . . . . . . . 107
6.2
Performance comparison between typical and PLL-based COA . . . . 110
6.3
Phase dynamics of five-element array for typical and PLL-based COA 111
5.1
5.2
5.3
5.4
5.5
5.6
xii
6.4
COA architecture with self-calibration mechanism. . . . . . . . . . . 113
6.5
Tracking of array factor during calibration process. . . . . . . . . . . 115
6.6
Mechanical and electric models of Omnia Ochrace
6.7
Generic representation of coupling loads between oscillators. . . . . . 117
6.8
Phase difference and amplification factor of currents I2 and I1 . . . . . 118
6.9
Plots of AF and DF generated by sweeping reactive loads . . . . . . 119
6.10
Two microstrip patch antennas coupled through a phase-boosting network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.11
Simulation results for two coupled microstrip patch antennas. . . . . 121
6.12
Total array gain with and without the coupling circuit. . . . . . . . . 122
6.13
Layout of two-element patch array for phase amplification . . . . . . 123
6.14
Performance of phase-boosting circuit from two coupled patch antennas124
6.15
Phase-amplifying architecture of coupled-oscillator arrays. . . . . . . 125
6.16
Comparison of array factors of a 10-element array (uniform and phaseamplified). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.1
Coupling circuit # 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2
Coupling circuit # 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.1
Plot of reflection coefficient of antenna during optimization. . . . . . 137
xiii
. . . . . . . . . . 116
CHAPTER 1
INTRODUCTION
1.1
Background
As civilization progresses, there is a corresponding and increasing need to exchange information at faster rates while shrinking the physical sizes of communication systems.
This continual shrinking of communication devices, demand for higher speed and efficiency, coupled with the requirement for higher power, integration and linearity, are
becoming the dominating aspects of today’s engineering designs. These challenges
are global and are encountered in handset devices, base stations, radars and satellite
transmitters. Meeting such demands necessitates producing more bandwidth, reducing antenna size, volume or footprint, increasing multi-functionality, using faster
switching circuits and coming up with superior information-coding schemes.
In addition, given the fact that the vast majority of personal communication systems operate at the lower microwave bands and that more mobile users and channels
have sprung up over time, overcrowding and interference at these lower frequency
bands are slowly becoming pressing issues. One potential solution to some of the
concerns mentioned earlier, is to operate at higher frequency bands. That would
mean engineering a new class of devices and protocols using higher microwave and
mm-wave frequencies. If that is the case, not only would the rate of information
transfer be superior from the gain in instantaneous bandwidth, but there would also
1
be a substantial and corresponding reduction in circuits and antennas dimensions.
The sizeable bandwidth expansion would not only be coveted by mobile operators,
but also by anyone interested in building a variety of commercial and military applications/gadgets going from high-resolution mapping/imaging systems, weapon detection, navigation systems, long and short-range communication to automotive collision
avoidance [1].
Over the last few decades, there has been an increasing push to mount highly sophisticated systems on moving platforms for surveillance, tracking, satellite imaging,
remote sensing, automobile safety and so on. These systems can be seen on drones,
airplane radars, high-resolution imaging systems and other various radar systems.
Such systems require the ability to scan around or have a beam-scanning mechanism
for fast and agile coverage of the scrutinized area. Typically such systems are built
on phased array technology, which have the ability to create one or multiple radiation
beams and an arbitrary large amount of power. These systems are customarily used
at microwave and/or mm-wave frequencies to due the requirements for small size and
weight
However, as advantageous as the operation of systems at higher frequencies is, it
is not without significant drawbacks. First, it is well known that the performance of
solid-state devices is degraded as the frequency of operation is increased. For modern
microwave transistors, for instance, a large drop in power gain and an increase in noise
figure are widely documented aspects of poorer performance with frequency increase.
Consequently, creating enough power efficiently at these frequencies using solid-state
devices is a challenge. Secondly, the ensuing size reduction will generally limit the
amount of the power that can be pushed through the system. This is mainly due to the
thermal properties of the composing materials: heat dissipation or sinking becomes a
major performance factor. Thirdly, there is a considerable increase in production costs
2
due to the need to use special low-loss material and advanced fabrication, design and
modeling techniques. Finally, besides the higher material (ohmic) and radiative loss,
other undesired effects affecting efficiency include an increase of mutual coupling,
possible excitation of surface waves, and unwanted radiation from the rest of the
system that might become comparable to that of the antennas. All these mentioned
issues become more pronounced as the frequency gets higher.
Historically, the generation of large amount of power has been done using phased
antenna antennas. For typical phased array systems at lower frequencies, a single
source, usually a klystron, generates a large amount of power that is split numerous
times, individually phased, and fed to each antenna as shown in Figure 1.1. Yet,
using the same approach at higher frequencies will be inadequate because the feeding
network and phase shifters (or phasers) are simply too lossy and bulky. The loss creates a highly inefficient system and the added bulkiness does indeed defeat the initial
purpose of reducing overall size and weight. Fortunately, the concepts of quasi-optical
Figure 1.1: Architecture of a typical phased array.
3
spatial power combining and active integrated antenna (AIA) have been introduced
to overcome these fundamental limitations on output power and efficiency of semiconductors circuits at the mm-wave frequencies [2]. Figure 1.2 shows the modified
phased array architecture using coupled oscillators. The main difference is that phase
shifters and power distribution networks can be completely eliminated and power directly delivered to each antenna, thus making the system compact and highly efficient
even if a small amount of power is used for coupling and synchronization purposes.
In general, the concept of AIA particularly deals with treating the antenna, the active device, and all embedded linear elements as part of the same monolithic entity.
The antenna not only serves as a load, radiator and a part of the resonator, but
also as a harmonic filter and tuning element without necessarily needing any additional matching network. This leads to a compact design with lower losses. Spatial
power combining refers to the fact that by arraying many AIAs, power is combined
in space and not within the circuit, thus completely eliminating loss due to feeding
networks and transmission lines. These methods not only allow power to combine
coherently and efficiently, but also reduce size by integrating and removing feeding
waveguides/transmission loss [1, 3, 4]. Besides the benefits in efficiency and size, this
approach also increases reliability and manufacturability due to the integration. The
coupled-oscillator solution requires arraying multiple AIAs. Each AIA is a source
contributing to the overall output power of the array. Producing an independent
source requires building a standalone nonlinear oscillator. The fact that they are autonomous means that the AIA elements will produce diverse oscillation frequencies.
In order to make sure that the overall radiated signal is coherent, all the oscillators
will need to be synchronized. Synchronization is accomplished by either establishing
a proper coupling mechanism or by providing a common driving or reference signal
to the array. Consequently, an understanding of the dynamics of nonlinear coupled
4
Figure 1.2: Correspoding architecture of coupled-oscillator array.
oscillators is crucial in providing an elegant solution to high-frequency phased array
antennas. Apart from considerable nonlinear dynamics, designing coupled-oscillator
arrays encompasses many areas of knowledge since it requires the use of microwave engineering, solid-state devices, nonlinear dynamics, circuits and antenna engineering.
In other words, it is a highly interdisciplinary endeavor.
1.2
Key Accomplishments
The work presented in this dissertation advances our current understanding of coupled
nonlinear oscillators as microwave systems by providing:
• An alternative parametric design approach for active an integrated antenna that
uses the convex-like properties of the input immittance of resonant narrowband
antennas.
• A derivation of a set of expandable differential equations that help address
the inaccuracy issues of other popular equations of coupled-oscillator systems,
especially when nearing bifurcation points. A clear advantage of these equations
5
is that all quantities are expressed in terms of directly measurable network
parameters.
• The implementation of a nonreciprocal coupling network for resolving the scanning limitation of coupled-oscillator systems.
• An implementation of a three-element oscillator array that is fully non-reciprocally
coupled for the purpose of generating the largest achieved phase difference without any loss of stability.
• A proposed architecture based on a single phase-locked loop (PLL) that not
only helps stabilize the array but also contains a self-calibration mechanism.
• The use of simple circuits between adjacent oscillators for the purpose of boosting a small phase difference into larger phase gradients for transmitting coupledoscillator systems.
1.3
Organization of Dissertation
This dissertation is organized in the following manner. First, Chapter 2 introduces
the reader to the concept of nonlinear oscillators and their various uses in modeling
various phenomena in multiple fields of study. The discussion is naturally steered
towards electric oscillators where a simple analysis is carried. The concept of frequency entrainment, synchronization, noise and chaos are introduced with forced or
coupled oscillators. Later, the topics of active integrated antennas (AIA) and coupledoscillator arrays (COA) are reviewed, and various past research are presented.
Chapter 3 is concerned with the analysis and design of active integrated antennas,
which form the building blocks of many coupled-oscillator systems. The analysis
method is highly parametric and applicable to a wide number of oscillatory systems
6
that present similar characteristics. Due to the limited degrees of freedom available
to the design of highly integrated AIAs, the presented techniques focus on some
particular properties of the integrated antenna that directly affect the output power,
phase noise and power efficiency of the system, making these techniques efficient
for quickly achieving a working first-cut design on which to carry an optimization
process. A prototype active antenna is built and its performance is measured to show
agreement with the developed analysis.
Chapter 4 is theoretical and numerical by nature and focuses on deriving more
accurate equations for the study of coupled nonlinear microwave oscillators. Using
averaging methods, the voltage amplitude and phase dynamics of an arbitrarily coupled array is derived with all quantities expressed strictly in measurable network
parameters, in a form familiar with antenna and microwave engineers. Numerous
simulations are carried out to verify the accuracy of the equations.
Chapter 5 deals with the analysis and design of the coupling system of an oscillator
array. Coupling properties of various transmission-line based networks are addressed
and other types of coupling mechanisms are discussed. A non-reciprocal coupling
structure is presented along with its implementation. An array design process using
various software tools is presented along with the implementation of a fully nonreciprocally coupled array at 5 GHz.
Chapter 6 mainly addresses the various types of proposed architectures for coupledoscillator systems. In order to have a robust system, an architecture based on PLL is
presented along with a phase calibration mechanism. Another architecture based on
the hearing system of a fly is presented that amplifies small voltage phase differences
into large current phase gradients that are radiated. This type of phase-boosting
feature is beneficial to the stability of the array in general.
7
Finally, Chapter 7 summarizes the contributions of the dissertation and suggests
future area of research related to this work.
8
CHAPTER 2
COUPLED NONLINEAR OSCILLATORS: THEORY AND
APPLICATIONS
2.1
Basic Oscillating Unit
Oscillators are a ubiquitous part of our existence as they can effortlessly be observed in
nature, in many fields of study, as well as in our technological devices. In nature, the
migration of birds, seasonal flora and fauna and other cyclical spectacles can be easily
noticed. A simple pendulum in mechanics, firing neurons and sleep-wake cycles in
biology, charge-density waves and Josephson junctions in condensed-matter physics,
just to name a few, are all example of systems exhibiting oscillatory properties. A
clock or the signals produced by many electronic devices also display oscillatory states.
In electrical engineering and related fields, oscillators are used internally as power
sources, and in the simplest case, they can be modeled as a basic RLC circuits with
additional nonlinear devices as shown in Figure 2.1. The active device needs to have
an operating range where negative differential resistance1 is observed. In general,
negative resistance is a property observable in many semiconductor devices, and it is
used to make high-frequency amplifiers and oscillators. The Gunn effect, for instance,
1
Negative differential resistance corresponds to a region where an increase in applied voltage leads
to a decrease in current flow (r ≡ dV /dI < 0). This is different from the static resistance
(R ≡ V /I), which is always positive. For ohmic devices both quantities are indistinguishable.
9
is a well-known occurrence of negative differential resistance that first puzzled its
discoverer because of the unexpected observed oscillations [5].
Figure 2.1: Basic model of oscillator.
A study of the circuit above concludes that it is fully described by the following
differential equation:
d2 v
1
−
2
dt
C
�
din
+G
dv
�
dv
v
+
= 0,
dt LC
(2.1)
where G = 1/R.
For many active devices, a cubic or van der Pol nonlinearity is common and
therefore the electric current entering such devices can be written as
in = av + bv 3 ,
(2.2)
where a and b are positive constants. The initial differential equation can now be
written in the following generalized form:
�
�
ẍ − � 1 − x2 ẋ + x = 0,
10
(2.3)
with the related quantities given by

dx


ẋ =


dτ




 τ = ω0 t
�
3b


x=v


a−G



a
−
G

 �=
ω0 C
(2.4)
√
Here, ω0 = 1/ LC, and the oscillator is clearly nonlinear since its properties are
function of the amplitude of oscillation and characterized by a limit cycle.
2.2
Self-regulation, Frequency Entrainment and Chaos
Nonlinear oscillators have the property to self-regulate and create what is called a
limit cycle. Figure 2.2 shows a limit cycle in the phase space based on equation
(2.3) for � = 0.5 and τ = 20. The x-axis represents x, while the y-axis is ẋ. It is
observed that regardless of the initial conditions, each trajectory ends up describing
the same path after a sufficient amount of time. This is the basic property of stable
limit cycles. In other words, trajectories near a limit cycle will be attracted to it and
consequently small disturbances from the limit cycle will slowly decay over time. The
limit cycle describes a control mechanism for the amplitude of oscillation such that
it is maintained over time.
However, free-running (uncoupled) oscillators are autonomous systems, meaning
that any time-shifted version of the oscillating signal is also a valid solution. Unlike
the amplitude, there is no controlling or restoring mechanism for the phase of the
oscillatory signal. This means that disturbances in the phase are not corrected leading
to oscillators having generally more phase noise than amplitude noise.
Nonlinear oscillators also display a richer behavior when driven or coupled to
11
Figure 2.2: Self-regulation depicted by a limit cycle with �=0.5 and τ =20. Abscissa
x, ordinate ẋ
other oscillators. An interesting property of nonlinear oscillators is their ability to
synchronize to an external signal. This property is also known as frequency entrainment or injection locking. Besides synchronization, other phenomena ranging from
period doubling to deterministic chaos can be induced even with a simple circuit. To
demonstrate those properties, the basic oscillator shown previously is now forced to
operate with an independent source as depicted in Figure 2.3. Historically, van der
Pol studied similar circuits using triodes [6].
As in Figure 2.1, this second circuit can be described by the following generic
equation
ẍ − �(1 − x)ẋ + x = F cos
where the factor F is given by
12
�
ωin
τ
ω0
�
,
(2.5)
Figure 2.3: Van der Pol oscillator with injection current source.
ωin
F = I0 2
ω0 C
�
3b
.
a−G
(2.6)
For an arbitrarily frequency ratio ωin /ω0 = 1.5, it is seen that the oscillator
perfectly locks to the injection frequency up to about � = 3, as shown in Figure 2.4.
Deterministic chaos then occurs, followed by alternating small regions of periodicity
and chaotic behavior. For � = 5.28 to 5.75, there is a period doubling state where
sub-harmonic (ωin /2) of the injection signal dominates. For � = 4.49 to 4.63, smaller
sub-harmonics occur, including ωin /5. In general, the oscillation frequency locks to
ωin × p/q, where p and q are integers. In 1927, van der Pol and van der Mark called
this phenomenon “frequency de-multiplication” [7] in the regions where sub-harmonic
frequency locking occurs. They did not recognize the nature of the irregular noise they
heard between locking regions. Today, it is safe to assume that the noise was from
the chaotic properties of the circuit. The occurrence of chaos is common for forced
nonlinear oscillators and such effect can be easily reproduced in simple mechanical
systems, as well, using a driven damped pendulum, for instance [8].
When multiple oscillators interact, all of the above phenomena are observed in
13
Figure 2.4: Diagram (bifurcation) showing behavior of forced van der Pol oscillator
for ωin /ω0 = 1.5 and F =1.2.
addition to states of full and partial synchronization. Except for special applications
in cryptology and information security, chaotic systems are rarely used [9]. However,
non-chaotic frequency entrained systems are widely used for various applications [10].
In phased array systems, coupled-oscillators present an attractive solution for microwave and millimeter-wave power combining applications as it will be discussed
next.
2.3
Review of Nonlinear Oscillators in Phased Array Antennas
In order to reproduce and take advantage of the useful properties of coupled nonlinear
oscillators in phased arrays and related technologies, it is necessary to come up with
14
a set of independent oscillatory units. For monolithic and high-frequency systems,
these units are known as self-oscillating active integrated antennas (AIAs).
2.3.1
Active Integrated Antennas
Active integrated antennas refer to circuits in which antenna elements and nonlinear devices are integrated to form a monolithic transmitting and/or receiving system [1, 3, 4, 11]. The integration is of paramount importance because it is well known
that assembling pre-fabricated circuits to construct a single system creates discontinuities that lead to detrimental parasitic effects. This issue is mitigated by having
a monolithic design approach where all circuits are fabricated on a single layout.
This ensures that no discontinuities are created, thus resulting in a more predictable
high-frequency performance. However, designing and fabricating such circuits require
hybrid modeling software [12] and precise fabrication techniques.
In general, AIAs can be classified by their RF functions, which are: (1) amplification, (2) signal generation (oscillators) and (3) frequency conversion. For the first
type, the antenna is located at the output of an amplifier. In this case, the antenna
also plays the role of the matching network and therefore there is no need to match
to the standard 50 Ω, as the matching is done directly. The second type is the selfoscillating active antenna. Here, the required signals are synthesized locally by the
active device and the antenna. For highly integrated systems, the antenna not only
acts as a radiator but may also act as part of the resonator (tank), matching network
or as a filter that implementing harmonic loads for improving DC-to-RF conversion
efficiency [13]. Since antennas radiate, they cannot have a very high quality factor
(Q). However, oscillators require high-Q resonators; therefore, the antenna can be
part of the tank circuit, but other components should be present to help increase the
overall Q of the oscillator. Besides power generation, self-oscillating AIAs circuits can
15
also be used for other functions such as polarization-agile switching, beam-switching
using injection-locking for radar, and omni-directional coverage [4]. Finally, frequency
conversion types are mainly used in receivers where emphasis is on minimizing conversion loss.
The bulk of this work is on the second type of AIAs since they are autonomous
nonlinear oscillators capable of generating all nonlinear phenomena mentioned earlier.
Except for the high integration, additional roles of the antenna and its properties,
designing a self-oscillating AIA is similar to dealing with a typical microwave oscillator. Various working prototypes of active antennas have been developed [12, 14–16]
and Figure 2.5 shows one typical topology. Here a patch antenna is integrated with
a transistor, forming a monolithic self-oscillatory system. Many linear and nonlinear
techniques have been applied to design microwave oscillators [17–19]. For nonlinear
techniques, the device-line technique [20] is used to measure the large-signal properties
of a negative resistance mono-port in a non-oscillating mode of operation. In essence,
it consists of measuring the reflection coefficient of the mono-port as a function of
the amplitude and phase of an injected signal. Other measurement-based processes
use active-load techniques that provide more flexibility and speed over the device-line
method. Using these newer techniques, the active device can be fully characterized
in its large-signal regime [21, 22]. These techniques can produce wide-ranging measurement data that is in turn used to design an oscillator. These techniques bypass
the need to have an accurate nonlinear model of the active device. For designs that
require an accurate nonlinear model, some techniques based on auxiliary generators
are used to control the output voltage waveforms [23]. Finally all of these nonlinear
techniques have shown the relevance of using harmonic loading to improve either the
output power or the power efficiency for nonlinear circuits such as oscillators and
power amplifiers [16, 24–27].
16
Figure 2.5: Instance of a self-oscillating AIA.
AIAs are systems within systems and they require many design considerations.
These types of active antennas are particularly useful in quasi-optical power combining applications where their arraying and integration help reduce physical size, eliminate the need for high-power solid-state sources, as well as increase output power and
system efficiency. Current AIAs are based on solid-state devices such as transistors
and diodes as opposed to vacuum or gas-discharge tube devices. Diodes reach higher
operating frequencies with higher RF output power, but transistors have higher DCto-RF conversion efficiency, operate at lower DC input power, and provide many
different functions—switching, tuning and amplifying—that would otherwise require
several types of diodes. Choosing an active device depends on the end application
and many other factors that range from cost, output power, operating frequency, efficiency, noise to stability and biasing requirements. Selecting the antenna’s geometry
17
rests on many factors that mainly include ease of integration with the rest of the
circuit, bandwidth, size, array topology, impedance and radiation pattern profiles.
Furthermore, antennas on thick substrates will tend to excite surface wave modes
that can lead to poor radiation efficiency. Scan-blindness and mutual coupling are
other aspects of the design to take into account in order to avoid undesirable effects
in an array environment [28]. All these issues have to be considered when designing
a single or many AIAs.
Finally, in high-frequency oscillator design, noise plays a particularly critical role
as all active devices produce noise with a significant time-varying power level. Noise is
a limiting factor of low-power, high-performance electronic circuits. A microwave oscillator can also be viewed as a dedicated noise generator followed by a resonator/filter
to funnel a finite amount of power into a small frequency band. Noise can be separated into AM and PM components near the carrier frequency. Oscillators are mainly
concerned with phase noise as amplitude fluctuations are relatively small, but become
significant away from the carrier [29]. Phase noise near the carrier frequency is due
to a wide variety of correlated phenomena caused not only by the active device, but
also by the rest of the embedded circuit and biasing sources [30, 31]. A fair amount
of nonlinear dynamics is crucial in simplified analysis where bifurcation points can be
monitored in order to track stability transitions. In actual nonlinear systems, chaotic
behavior may ensue, and it is important to differentiate it from typical electronic
noise [10].
2.3.2
Coupled-Oscillator Arrays
Coupling many AIAs into a single system creates a coupled-oscillator array (COA).
Such systems have interesting properties [32] and in general, coupled nonlinear oscillators have been used to model many phenomena in physics, engineering, biology
18
and finance [10, 33]. For example, the cardiac pacemaker, the flashing of fireflies [34],
menstrual cycles in groups of women [35, 36], arrays of laser diodes [37], Josephson
junction arrays [38,39], phase-locked loops, heart cell synchronization [40], neural networks, economies, immune systems and ecosystems, and so on, are all able to display
a synchronous behavior through mutual interaction. Synchronization is a property
that is only observed in interacting or coupled nonlinear systems. Such systems selfregulate so that deviations from a common or intrinsic behavior are eliminated. In
electrical engineering and related circles, the terms frequency entrainment, injection
locking and phase-locked loops (PLL) are all describing the synchronous property of
a nonlinear system that is forced to operate in a slightly different regime from its
normal state. Practically, it is done to stabilize the oscillation frequency through the
use of external reference signals so that information can be encoded/decoded and
noise level reduced [41].
COAs are fundamentally different from passive antenna arrays due to their nonlinear nature. They provide an analog computing layer capable of performing signal
processing such as determining the direction of arrival of an incident signal. The
same task on a typical phased array would require a signal processing chip or a computer. Historically, the study of COAs was preceded by that of grid oscillators [42].
Grid oscillators were periodic arrays embedded with active solid-state devices. The
grid resided in a Fabry-Perot resonator in order to provide positive feedback necessary for oscillation. However, work on grid oscillators was mostly abandoned because
they were intrinsically low-Q, and thus could not produce a tone with phase noise
low enough for RF communication or radar applications [43]. When injection locking was found to help improve grid oscillator noise performance, most researchers
invested their efforts in studying COAs with injection-locking as a noise reduction
mechanism [44, 45].
19
As the research went on, a few properties of injection-locked oscillators were already known. In 1946, Adler [46] theoretically showed that a weakly driven oscillator had a locking range—a region where the oscillator and the impressed signal
synchronized without producing a beat frequency. Later, additional theoretical and
experimental works were added, especially by Kurokawa, who extended Alder’s work
for large-signal injection locking [47], multiple resonant circuits [48] noise [49] and
stability analysis [47, 48].
All this work provided the foundation for researchers such as Mink [50], Stephan
[51,52] and Dinger [53] to go from a proposal on quasi-optical power combining to low
and high-frequency array prototypes. By bilaterally injecting a phased signal at both
ends of an array, Stephan and Morgan [51] demonstrated that a phase difference
(evenly divided between oscillators) can be generated and used for beam steering
purposes. A clear advantage of this technique over traditional phased arrays was
that beam steering was accomplished with only one phase shifter. However, since
the total available phase was 3600 , the major limitation of this technique was that
the more the oscillators, the smaller the scan angle and thus this technique was not
scalable. Figure 2.6 shows the structure of such system.
Figure 2.6: Inter-injection phased array architecture [51].
20
Since then, it has been known that coupled-oscillator systems are crippled with a
limited scanning range issue. Additional inherent issues are instabilities, inadequate
modulation schemes, and slow data rates. After additional years of research, York
and colleagues generalized many aspects of coupled oscillators by providing a set
of amplitude and phase equations describing the dynamics of an array of coupled
nonlinear oscillators [54, 55]. The amplitude and phase dynamics of each oscillator
was provided by the following equations:
N
�
dAk
µωk � 2
ωk �
=
αk − A2k Ak +
�kp Ap cos (θp − θk + Φkp )
dt
2Qk
2Qk p=1
N
dθk
ωk �
= ωk +
�kp Ap sin (θp − θk + Φkp ) ,
dt
2Qk p=1
(2.7)
(2.8)
where Ak is the amplitude and θk is the relative phase of the signal at the k-th oscillator. The parameters ωk and Qk are the free-running frequency and the tank quality
factor of that oscillator, respectively. The parameter �kp is the coupling strength between oscillator k and p. Due to the relative simplistic nature of the model above, the
equations serve to only give a qualitative description of the first-order dynamics of an
actual oscillator system. More elaborate models include additional observed effects
such as frequency dependence and asymmetric properties of the nonlinear cores [56].
One of the most useful properties derived from York’s work was the discovery that,
for a weakly coupled system, beam-steering can be scalable and accomplished without
the need of any phase shifters, simply by detuning the free-running frequencies of the
edge oscillators in equal but opposite directions [57] as shown in Figure 2.7. For
a planar array, controls would only be needed at the edge oscillators. One main
advantage of this beam-steering technique over the ones relying on injection-locking
is that the total phase shift is not a function of the number of oscillators and therefore
not restricted to 3600 (2π). However, there is still a limitation of π/2 in terms of the
21
maximum phase difference between two adjacent oscillators. For a λ/2-spaced linear
array, that would correspond to a maximum scan angle of π/6. Explicitly, to generate
a phase gradient of ∆φ, the end oscillators’ natural frequencies have to be offset by
�ω0
δω =
, with �kp ≈ �, and Qk ≈ Q.
2Q
Figure 2.7: York beam-steering mechanism (weakly coupled systems) [57].
Besides York’s technique, additional beam-steering techniques were introduced,
mostly dealing with the control of the coupling angles between adjacent oscillators.
Hwang and Myung [58] showed that a uniform phase gradient can be generated across
an array by having the edge coupling phases be opposite to one another. Unlike
York’s work, there was a linear relationship between the coupling phase produced
and the corresponding phase gradient of the array, which were equal. One of the
drawbacks of Hwang and Myung’s technique is that the two end oscillators are not
allowed to radiate because their phase relationships are different from the rest of
the array. This drawback, however, was solved by Lee and Rojas [59] by using nonreciprocal coupling devices to connect these edge oscillators to the rest of the array.
The maximum oscillator phase difference possible was also π/2 for both cases.
Another seeding work on beam-steering for coupled-oscillator array was done by
Heath [60], in which the author generalized beam-steering through manipulation of
22
coupling phases. Mainly, the author proposed two ways of performing beam-steering
from the nonreciprocal manipulation of coupling phases. One technique involved introducing a pair of nonreciprocal coupling phases at the two edges of a linear array
(same as in [59]), while the second technique had a nonreciprocal coupling circuit
between every oscillator. The latter technique was shown to bypass the π/2 limitation of the other techniques, thus providing the most stable solution to coupled
oscillators systems. The work also included a linear stability analysis for each technique. Additional beam-steering techniques for extending scanning range made use of
frequency multipliers [61] or radiation at frequency harmonics [62]. The rest of beamsteering techniques in literature involved injection locking at either fundamental [63]
or harmonic frequencies [64].
Although beam-steering is crucial in many applications, the overall shape of the
radiation pattern is of similar importance. The natural frequency and coupling phase
of each oscillator and coupling network, respectively, can be individually controlled
in order to change the phase distribution across the array, which in turn changes the
radiation pattern. This property has been used to create adaptive radiation patterns
with simultaneous beams and nulls. For example, a difference pattern for scanning
applications can be accomplished by modifying the coupling between the central
elements in a linear array [65]. Other pattern shaping techniques that produced
simultaneous beams and arbitrarily placed nulls were proposed by Heath [66] and
Georgiadis [67] with an accompanying stability analysis.
In terms of coming with additional ways to analyze and design COAs, many
other authors have made significant advances. Pogorzelski and colleagues proposed
a simplified continuum model of coupled oscillators that treats the relative phases
of the oscillators as a continuous function over the array satisfying a diffusion-type
23
partial differential equation [68, 69]. It was noticed that the phase dynamics equations were equivalent to the discretized Poison’s equations as seen in electrostatics.
Their findings showed that transient performance of large arrays could be computed
rather quickly using their techniques. For designing coupled-oscillator systems, a few
more authors have proposed a set of numerical techniques based on harmonic balance analysis where the determination of the array’s properties are carried out using
nonlinear models. The stability analysis is studied through a pole-zero identification
and bifurcation-detection techniques [70, 71].
As coupled oscillators are built, electronic noise and instability are prevalent due
to the intrinsic nature of oscillators. To increase mode stability and reduce phase
noise, array architectures using PLL or mixers have been used [72, 73]. Figure 2.8
shows an example of such architectures. Although this architecture can theoretically
achieve higher stability margins and lower phase noise, it is, however, complex due to
the number active components required, and the additional cost might be prohibitive
when the array is scaled because of the larger number of devices that is necessary.
Stability is also dictated by the physical and electrical properties of the coupling
network used as shown in references [74, 75]. Specifically on phase noise, some works
have used sub-harmonic injection locking as a phase noise reduction technique [76].
Furthermore, it was shown that the phase noise and the locking range of a COA
improved when inter-element coupling was made stronger [77]. Finally, it was found
that phase noise in a reciprocal N -element COA could be reduced to 1/N of that of
a single oscillator [78].
In terms of the performance of a COA as a basic communication system, some
works [79, 80] investigated the performance of an injection-locked COA in terms of
data-rate limitations, modulation effects on radiation pattern and the bit-error rate
for one and two-dimensional arrays under BPSK modulation. As expected, it was
24
Figure 2.8: PLL-based architecture [72].
found that linearly coupled COAs have lower maximum data rates than traditional
phased arrays at the same frequency. Moreover, the array transient or settling time
increased as the number of elements increased. However, it was noted that the overall
performance could be improved by having a stronger injection signal, more injection
points and stronger coupling between oscillators, with all these additional solutions
having their own tradeoffs.
Finally, in addition to the fully synchronized mode where all oscillators are synchronized to the same frequency, coupled oscillators can also operate in a mode-locked
state [81], where the array oscillates with a number of equally spaced spectral modes,
with comparable amplitude and locked phases. This mode of operation is extensively
used in lasers to produce short and high-power optical pulses. When applied to COA,
high-power RF pulses can be generated. In summary, despite the inherent challenges
emanating from operating COAs, they remain attractive because of their simplified
control architectures, higher efficiency, lower weight as supposed to traditional phased
25
arrays, and the potential of integrating all required circuitry in a chip, apart from
the antennas.
26
CHAPTER 3
ANALYSIS AND DESIGN OF ACTIVE INTEGRATED
ANTENNAS
3.1
Introduction
Since the advent of high-performance solid-state transistors and diodes, designing
microwave oscillators has been widely studied and accomplished using a variety of
linear and nonlinear methods [17]. For nonlinear approaches, the device-line measurement technique has been used for some designs [21, 82], whereas more recently,
active-load techniques have proven to be more reliable and efficient for designing
microwave oscillators [13, 83].
A nonlinear device can be fully characterized either by an accurate nonlinear
model or by extensive measurements in the small and large-signal regimes. However,
even with a complete characterization of a nonlinear device, designing a practical
oscillator is still not straightforward due the possibility of multiple operating points
which requires additional synthesis, optimization or tuning steps. Other additional
steps might be needed in terms of selecting the loads, implementing the feedback
terminations, choosing the bias voltages and fixing other embedded loads for optimal
power, efficiency or noise criteria. Furthermore, in the case of active integrated antennas (AIAs), only a few authors have made use of the properties of the specific antenna
when designing AIAs [12, 16, 84–86]. It should be noted that in the case of AIAs, the
27
nonlinear core is highly integrated with the antenna so that only assuming the usual
quasi-TEM mode is erroneous as the antenna generates other modes that are not captured by a simple circuit simulator. This requires the simultaneous use of a full-wave
electromagnetic (EM) solver along with a typical circuit solver. Consequently, generic
oscillator design concepts, techniques and equations are indiscriminately applied to
any AIA design, thus resulting in potentially suboptimal systems since the antenna
is treated as a standalone load/or resonator, while ignoring potential key properties
that can simplify and speed up the design process.
The purpose of this chapter is to provide an alternative scheme to quickly get to an
AIA’s operating points before the optimization or tuning steps—as always required
for any practical design at microwave or higher frequencies. The analysis and design
method that is presented is efficient, multi-purposed, and amenable to any harmonic
balance (HB) circuit simulator. It has been shown that harmonically loading an
oscillator can improve the output power and power efficiency at the fundamental
tone [24, 87]; however, this is the first time, to the best of the author’s knowledge,
that the local properties (convexity) of the antenna’s immittance locus and related
concepts are used to analyze and design a self-oscillating active integrated antenna
system. It will be shown that regardless of the active device used, the appended
antenna will vastly dictate the design performance. This work will show how the
antenna effectively draws up the boundaries of the possible design space and related
restrictions.
This work is split into two parts. The first segment deals with designing a selfoscillating active antenna given a previously available passive antenna. In this case,
the input impedance or admittance of the antenna is already determined and cannot
be modified. This can be valuable in situations where an antenna or an antenna array
system needs to be retrofitted with newer front-end components. The second part of
28
this chapter is concerned with the simultaneous design, from the ground up, of both
the antenna and the active core, thus presenting a general case of AIA design.
3.2
Parametric Modeling and Design Approach
An oscillator can be regarded as a dedicated noise generator with a filter or resonator
appended to the active device’s output in order to confine oscillations to a desired
and usually narrow frequency band. Self-oscillating active antennas are usually built
such that the antenna element is a functional part of the tank. In this case, the
antenna plays a dual role, namely, a tank circuit as well as a transmitter (or receiver)
of electromagnetic energy. This dual role of the antenna element can be problematic
since a good antenna is expected to have a low Q; however, a superior oscillator must
have a very high-Q tank circuit. Consequently, to reduce phase noise in AIAs, it is
imperative to have high-Q embedded linear elements (terminal loads) in the oscillator
in addition to the antenna in order to increase the overall or loaded Q. In other words,
the antenna alone will not be good a good resonator so other linear elements around
the transistors are needed to achieve a higher Q. In highly integrated systems—where
there is no dedicated high-Q resonator—the antenna and the embedded elements
provide the only degrees of freedom available to the designer apart from the voltage
bias and frequency of operation which are usually pre-determined.
3.2.1
Convexity of Antenna Immittance
This section will only consider resonant and narrowband antennas whose input immittance can be modeled, within a constant, as a series or parallel RLC circuit
within a particular bandwidth. We’re interested in only narrowband antennas because they have to be part of the distributed resonator in order to help with the
overall Q. Let’s define a bandwidth ∆ωC , characterized as the smallest frequency
29
interval ∆ωC = |ωb − ωa | such that ZL (ωa ) = ZL (ωb ), where ZL (ω) is the input
impedance of the antenna in use. This specific definition has to do with the convexity of the antenna input immittance in the complex plane, and the justification for
such definition will be provided shortly.
In general, the input admittance of a passive antenna can be written as follows:
YL (ω) = G(ω) + jB(ω).
(3.1)
If the antenna load can be approximately modeled as a parallel RLC circuit within a
specific bandwidth, it follows that
G(ω) ≈ G0 ,
(3.2)
and
B(ω) = ωC −
1
.
ωL
(3.3)
The impedance of the antenna is therefore computed as
ZL (ω) =
1
G0
−B(ω)
+j 2
,
≈ 2
2
YL (ω)
G0 + B (ω)
G0 + B 2 (ω)
(3.4)
and the locus of ZL (ω) describes a circle in the complex plane. To show that it is the
case, a transformation is applied by introducing an additional variable ξ such that
cos ξ =
G20 − B 2 (ω)
G20 + B 2 (ω)
(3.5)
sin ξ =
−2G0 B(ω)
.
G20 + B 2 (ω)
(3.6)
and also
30
Thus, it turns out that ZL (ω) =
1
1
(1 + cos ξ) + j
sin ξ, or simply
2G0
2G0
ZL (ξ) =
�
1 �
1 + ejξ .
2G0
(3.7)
This latest form of the antenna impedance clearly shows that the locus depicts a
displaced circle with a center at (1/2G0 , 0) and a diameter of 1/G0 in the complex
plane. The same result is obtained if we started out by modeling the antenna input
impedance as a series RLC circuit; and in that case, the input admittance will describe
a circle with similar characteristics.
The above step showed that if an antenna immittance can be modeled as an
RLC circuit—which generally occurs for narrowband antennas near resonance—the
actual antenna input immittance describes a circle-like or ellipsoidal locus within a
specified bandwidth ∆ωC . The fact that antennas subscribing to the above criteria
form approximate convex sets [88] is a property that can be readily and amply used
to quickly get the AIA’s operating points, as it is shown now.
From the analysis performed by Kurokawa [48] on negative-resistance oscillators,
it was graphically shown that stable oscillations in the nonlinear regime occur when
the device line and load locus intersect in a particular manner. To illustrate those
concepts, some antenna locus and various device lines are drawn in Figure 3.1. The
arrows indicate the directions of increasing frequency and RF current level the for
the antenna locus (input impedance looking into antenna terminal) and device lines
(negative of input impedance looking into terminal of active device), respectively. The
three loci intersect at three points A, B and C. In [48], the stability was graphically
characterized by looking at the angles of intersection—counterclockwise from the
device line to the antenna locus—being less than 1800 . Consequently, point A is
unstable while points B and C are stable. The antenna locus intersects itself at point
D. This latest point characterizes the previously defined bandwidth ∆ωC . This is
31
Figure 3.1: Plots of antenna locus (solid line) and various device lines (dashed lines)
for increasing frequency and RF current amplitude, respectively, as indicated by direction of arrows. Point A is unstable while points B and C
are stable. Point D defines the bandwidth ∆ωC .
the minimum bandwidth necessary for the antenna input immittance to return to an
earlier value around resonance. However, from the illustration in Figure 3.1, a generic
stability notion can be established as follows: it can be observed that a stable point
is always created whenever the device line exits the convex set (gray area). Inversely,
an unstable point is formed when the device line enters the convex region. Therefore,
if the device line originates from anywhere within the closed manifold or grayed area
described by the antenna locus, then the first intersection point of the two loci will
always corresponds to a stable oscillation point regardless of the path taken by the
device line. Mathematically, this outcome is always satisfied if the grayed area is
closed or encircling the device line. As a consequence, this stability notion hints
that during the initial small-signal design steps, it is desirable that the device line is
located within the closed region or at the right of it, and made to enter this region
as the RF level increases.
32
It should be noted that this is a simplistic two-dimensional description of a higherdimensional system as the device line is also frequency dependent and that harmonic
components are part of the same dynamics. Moreover, the antenna might also display
very complex modes of operation with multiple resonances; however, the goal of this
analysis is to quickly find oscillation points and during computation/simulation, all
these effects are accounted for.
3.2.2
Feedback Termination Modeling
In its most simplified form, a transmitting active integrated antenna can be regarded
as a negative-resistance oscillator. As a result, a traditional analysis is performed by
splitting the oscillator into a linear and nonlinear part as shown in Figure 3.2. The
linear part mostly consists of the antenna with an impedance of ZL (ω). The nonlinear
part comprises the active devices. However, for convenience, some linear elements or
parts can be grouped with the nonlinear core since terminal feedback/loads and bias
lines can be efficiently used to shape the negative resistance function. The nonlinear
impedance ZN (A, ω) is both a function of the RF current magnitude A and frequency
ω. For convenience, the admittance YL (ω) and YN (A, ω) of the antenna and nonlinear
device should be used if more appropriate. In this case, A would be the amplitude of
the RF voltage.
The oscillator design process commonly starts with a small-signal analysis because
it is imperative that the DC solution is unstable in order to initiate oscillations when
the system is switched on. The sufficient small-signal conditions for the onset of
oscillations, for Z = ZN + ZL , are ( see Chapter 4 for derivation):
33
Figure 3.2: Simple oscillator model with i(t) = A cos(ωt+φ).The nonlinear core comprises the active device and any other linear elements such as the feedback
terminations that shape the unstability/negative resistance. The linear
part mostly consists of the antenna.



Zr < 0





r
i


 s = ∂Z Z i − ∂Z Z r > 0
∂ω
∂ω
(3.8)
The symbols i and r denote the imaginary and real part of a quantity, respectively.
Since AIAs find many applications in quasi-optical power combining, minimizing
non-radiating loss is of paramount importance. In order to improve the loaded Q,
the feedback terminations of the active device are designed to be purely reactive. For
such case, the reflection coefficient looking into such terminal loads can be simply
written as Γ = ejθ , as in the Smith chart, thus allowing a fast sweep of all possible
reactance values. Practically, the amplitude of the reflection coefficient of any passive
termination is less than unity due to various loss mechanisms at microwave frequencies; however, these effects are accounted for during the design process by the use
of a full-wave EM solver. Consequently, the entire design space can be plotted out
34
concisely. For convenience, the following feedback impedance ZF model is used:
ZF (ω, θ) =
tan
�
jZ0
�θ
ω
ω0
2
−π
�� .
(3.9)
This specific feedback termination can be implemented by a simple open transmission
ω
θ
(π − ). Here, ω0 is an
line with a corresponding electrical length of EF (ω, θ) =
ω0
2
arbitrary reference frequency that is practically selected to be the expected oscillation
frequency. Z0 is a reference impedance, which is assumed to be 50 Ω throughout this
work. Equation (3.9) is chosen only for convenience as it was desired that the phase
θ be swept from zero to 2π; however any arbitrarily equation can be used as long as
it provides a way to sweep all reactive values from a single parameter. The reflection
coefficient of the above feedback load is ejβ , where β can be quickly calculated to be
(ω/ω0 )(θ − 2π).
For planar designs on substrates, the feedback terminations can be implemented
by using either shorted or open transmission lines (OTL). However, open microstrip
lines are easier to build and model than shorted lines with full-wave EM simulators
since open lines do not require via-holes, pins and solder that are all susceptible to
modeling errors. One disadvantage of OTLs is the presence of parasitic capacitance
(fringing fields) and possible radiation losses that all might contribute to higher EM
coupling.
For designs with an injected signal, at least one of the parametric loads must be
designed to have a real part. As a result, an additional real and positive parameter
γ should be added to model the resistive part of the input impedance of the terminal
load such that ΓF = γejθ .
To test the usefulness of the equations above, two feedback loads are placed at
the gate and source terminals of a FET transistor (NEC’s HJ-FET NE3210SO1)
35
at 10 GHz. The transistor is biased to VDS = 2 V and IDS = 10 mA. Figure 3.3
shows how the impedance looking into the drain terminal of such device evolves over
all possible reactance values at the other two terminals. This plot summarizes the
behavior of the FET under all reactive loading conditions in a concise manner by
highlighting regions of negative differential resistance. It is expected that the actual
curves summarizing the behavior of the transistor will be slightly different due to
the small resistance (loss) of the feedback terminations; however, it is not of critical
importance as the goal of this analysis is to provide a global behavior of the system.
The small deviations generated by some unavoidable loss will be accounted for at a
later stage of the design process.
3.3
Application of Concepts
The concepts of antenna locus convexity and parametric loads introduced in the
previous section will be used to design two types of AIAs. First, a specific design
scheme will be shown in order to realize an AIA from a previously fabricated or
existing antenna element. The second example will demonstrate the design of an
AIA where the antenna element and the nonlinear part are designed simultaneously.
3.3.1
Design with Existing Antenna
The concepts introduced earlier can be readily applied to the design of an active
antenna. To accomplish such task, a typical Ku-band patch antenna is provided [89].
It is built on a 31-mil RT/Duroid 5880 substrate. The resonant length is 6.97 mm and
its width is 9.12 mm. Its feeding pin is located 2 mm from the non-radiating edge.
It is desired to operate this patch at its first parallel resonance; therefore, its input
admittance will be used as plotted in Figure 3.4(a). Common rectangular patches
are narrowband antennas by nature, and from the previous analysis, it is expected
36
Figure 3.3: Parametrization of the input impedance of the drain terminal of a FET
transistor. Bias: VDS = 2 V and IDS =10 mA. The solid lines represent
the resistance , while the dashed lines is the rectance of the transistor in
its small-signal regime at 10 GHz. All contour values in Ohms.
37
(a)
(b)
Figure 3.4: (a) Real and imaginary parts of the input admittance of a patch antenna. (b) Locus of patch antenna in complex plane with ∆ωC =4.26
GHz. ωa =11.23 GHz and ωb =15.49 GHz.
that the locus of a narrowband antenna near resonance resembles a circle. Figure
3.4(b) shows that it is the case. This antenna satisfies the narrowband specifications
highlighted in Section 3.2.1, thus creating the convexity needed to apply the previous
concepts. Assuming this antenna is given and no changes can be made, the design
process moves on to the nonlinear core.
The next phase in the design process is to project the antenna locus area (gray
area) onto the parametric design space of Figure 3.3, however at 13.5 GHz. This step
is crucial because it will map the closed locus region described by the antenna onto
the transistor design space. And since it was shown earlier (Section 3.2.1) that any
device line starting inside such regions is bound to intersect the antenna locus at a
stable point, the nonlinear device synthesis step is made efficient and predictable.
Once the real and imaginary parts of the closed region are individually mapped
onto the design space, the final projection is found by juxtaposing them on top of
38
each other and tracing out their intersection. Technically, the mapping is done by
merging areas corresponding to small rectangular areas of the shaded area of Figure
3.4(b). Figure 3.5 shows the corresponding mapping. It can be seen that the convex
region has been mapped into two distinct shaded regions (shaded areas) attesting to
the highly nonlinear nature of this transformation. Figure 3.5 also shows the effect of
enforcing the small-signal startup condition s > 0, where s is expressed in Equation
(3.8). It is clearly visible that the design space has been further reduced by enforcing
such condition.
Figure 3.5: Mapping of the convex region described by antenna locus into the transistor design space. The contour lines (r, i, s = 0) represent the real,
imaginary and startup conditions delimiting the projection of the antenna impedance. Enforcing the startup condition further reduces the
considered design space.
At this stage, the designer is free to pick her or his favorite nonlinear analysis
tools and perform an oscillation analysis starting from any point within the resulting
and smaller mapped region. Generally, the end result is achieved when some design
39
performance or specifications are met. However, it is well known that maximum
power, higher power efficiency and minimum phase noise are not achieved at the
same point. This is a familiar tradeoff every designer has to deal with. At the end,
it is up to the designer to select the best point based on the initial specifications in
terms of output power, power efficiency and/or phase noise characteristics.
Before any optimization step can take place, the key pieces of information to
get out of this section are the following: First, the design space is consisely plotted
out by the use of parametric feedback loads. Second, the mapping of the antenna
convex region and startup conditions reduce the realizable design space into a smaller
region. Third, picking a point within those mapped regions will likely generate a
stable steady-state oscillation point. Without performing any optimization, choosing
a random point in the upper lobe from Figure 3.5, as the starting point, it is quickly
found that a stable oscillation is generated. The parametric point picked is (θG , θS ) =
(1600 , 2710 ), which corresponds to a reactance of j8.82 Ω at the gate and j50.9 Ω at the
source. A transient simulation produces the two plots in Figure 3.6 that describe the
AIA performance. For verification, an ensuing harmonic balance simulation concludes
that the fundamental oscillation frequency is 13.76 GHz. The output power is 6.41
dBm. The DC-to-RF conversion efficiency is 17.5% [24] and the phase noise is -122.8
dBc/Hz at an offset of 1 MHz. Many other points were picked in the upper and lower
lobes, with most of them showing displaying stable oscillation at different power level
and slightly different frequencies between 13 and 14 GHz.
In summary, the above section presented a fast and efficient way of quickly coming
up with a set of AIA’s operating points prior to any optimization and synthesis
process. The analysis and design steps are reduced since the design space is condensed
to a relatively small collection of points, thus letting the designer focus on getting the
requested specifications without an extended trial-and-error process that notoriously
40
Figure 3.6: (a) Timeseries of current at antenna port. (b) Corresponding normalized
power spectrum of current waveform with peak power (fundamental tone)
at 13.76 GHz.
41
lengthens the traditional design process. Concerning the types of terminal loads
used to parameterize the active device, it is clear that other loads can be utilized
depending on the nature of the design and implementation technology. In addition,
this technique can help quantify the limitations of the nonlinear core and its ability
to construct an oscillator at a specific frequency range based on the projected area.
3.3.2
Design with Synthesized Antenna
In the general case where the active core and the antenna have to be designed simultaneously, there are additional degrees of freedom because the antenna element
can be synthesized to enhance a particular characteristic such as phase noise, output
power, conversion efficiency and/or tuning range. The type of antenna used largely
depends on the architecture and implementation technology. Here, a pin-fed rectangular patch antenna will be used. The following circuit schematic (Figure 3.7) is used
to get a global behavior of the the system using Agilent ADS [90].
Figure 3.7: Simple schematic of AIA circuit showing transistor feedback loads and
antenna.
42
To optimize output power or conversion efficiency, for instance, a harmonic loadpull analysis or measurements can be performed and the antenna can be subsequently
modified to display specific loads at the frequencies of interest. This approach is not
applied here. On the other hand, if the focus is on reducing phase noise, the transistor terminations, antenna, and RF level must be selected such that the frequency
derivative of the total immittance is maximized at the oscillation frequency since the
phase dynamics is inversely proportional to this gradient (refer to Chapter 4). This
has the effect of increasing the overall Q of the system. Additionally, the total immittance should also be made least sensitive to other key variables, if possible. The
nonlinear device (FET in this case) has a high dependency on many variables and its
drain impedance can be written as follows for a common-source configuration
ZN = f (A, ω, θG , θS , Vbias , ...)
(3.10)
where G and S stand for gate and source, respectively. In the case where the designer
has control over the antenna’s geometry, the active device impedance’s gradient can
be optimized separately; otherwise, it is advisable to work the gradient of the total
immittance. From the design specifications, the frequency is generally fixed and other
parameters are allowed to vary. To limit the complexity of the upcoming analysis
without loss of generality, only the AIA’s conversion efficiency and phase noise are
considered and other design parameters are fixed to VDS = 2 V, IDS = 10 mA, RF
level to 7 dBm and the oscillation frequency to 5 GHz. Consequently, only the
immittance of the drain terminal will partially depend on the two terminal loads.
The oscillator operating point can therefore be selected around the point where the
input immittance is least sensitive to the oscillation amplitude and transistor loads’
lengths. Numerically, these low-variation points can be plotted out by bounding the
differential of the total immittance, such that
43
�
�
� ∂Z
�
∂Z
∂Z
|∆Z| = ��
dA +
dθG +
dθS + ...�� ≤ �
∂A
∂θG
∂θS
(3.11)
at the operating frequency and RF power level. Here, � is an arbitrarily small number
that is related to the maximum amount of variation the designer is willing to allow.
Figure 3.8 depicts a large-signal plot showing regions where the drain immittance
variation is affected based on gate and source loads at 5 GHz. Note that these types
Source Feeback, deg
350
1
300
10
6
2
150
3
1
250
200
6 10
2
3
20
20
0.3
10
100
50
20
20
50
100
150
200
250
300
350
Gate Feedback, deg
Figure 3.8: Plot of low-variation points (|∆Z|) where the transistor device input
impedance (drain) is least sensitive to amplitude variation, gate and
source loading. The lowest- variation point is observed at approximately
(251,180) at 5 GHz.
of plots should be also be computed at the small-signal level (∂(.)/∂A = 0), since they
provide valuable information on the initial transient behavior. Figure 3.9 depicts the
frequency gradient of the nonlinear core as a function of gate and source loading. It
44
can be observed that the maximum occurs near the edges and not at the same point
as the lowest-variation point of Figure 3.8.
350
0.5
Source Feedback, deg
10
300
0.3
4
2
250
0.1
1
200
150
0.1
0.3
0.5
100
1
4
2
50
10
50
100
150
200
250
300
350
Gate Feedback, deg
Figure 3.9: Frequency derivative of total nonlinear core impedance at 5 GHz and 7
dbm RF level.
When a resonant patch antenna is used, the location of the feeding pin should
be selected such that the input immittance frequency gradient is maximized. Figure
3.10 shows how the input impedance of a particular patch antenna is affected as
the feeding pin moves away from the radiating edge at 5 GHz. The impedance
differential |∆ZL (ω, f )|, parameterized here as a function of the feed-pin location
f and frequency ω, is highest around 4 mm. Note that at each pin location, a few
neighboring frequency points are also simulated. As expected the resistance decreases
as the pin moves deeper into the patch, reaching a minimum at half the length of the
45
patch. The excess inductance is due to both the pin and non-propagating higher-order
TM modes [91].
Figure 3.10: Antenna impedance plotted as a function of feeding pin location on left
at 5 GHz. Plot of normalized gradient in log scale showing maximum
frequency variation. Maximum impedance variation occurs around 4
mm. Patch dimensions: L=19.5 mm, W=20.5 mm.
After joining the antenna and the nonlinear core together, the location f = 4.58
� r�
� ∂Z �
mm was picked for two reasons. First, the quantity �� L �� had its maximum value
∂ω
and second, HB simulations showed that a higher conversion efficiency was obtained
there and not at the overall maximum location f = 4.5 mm. The corresponding
antenna impedance is plotted in Figure 3.11.
46
Figure 3.11: (a) Synthesized patch antenna input impedance around 5 GHz for f =
4.58 mm. (b) Corresponding impedance locus.
As noticed from Figures 3.8 and 3.9, the lowest-variation point and maximum
frequency derivative points do not coincide. Having these two points overlap and
operating the oscillator there would in theory minimize phase noise. The separation
between these two points and various operating points of the AIA is showed in Figure
3.12. The operating points consist of the constant output-power points (7 dBm) using
the synthesized patch as the gate and source loading is changed.
After a few simulation trials within the antenna mapped area, the point (θG , θS )
= (1200 , 2030 ) was picked as the final design point based on its compromise in terms
of producing a relatively acceptable phase noise (-115 dBc/Hz, at 1 MHz offset), a
DC-to-RF conversion efficiency of 24.7%, and an output power of 7.08 dBm. The
oscillation frequency was 5.002 GHz. The corresponding gate and source loads can
be implemented by two ideal 50-Ω OTLs with electrical lengths of 1200 and 78.50 ,
respectively.
47
Source Feedback, deg
350
300
250
200
150
100
7 dBm
50
50
100
150
200
250
300
350
Gate Feedback, deg
Figure 3.12: Performance of AIA for 7 dBm output power (square). Plots of lowvariation (solid line) and high frequency-derivative points (dashed line)
of drain impedance juxtaposed on background to show separation.
3.4
Fabrication and Measurements
The design presented in section 3.3.2 is implemented on a planar substrate with
microstrip lines. To avoid unnecessary coupling and additional cross polarization
radiation, two substrate pieces were joined together to a common ground plane. The
active device and its embedded linear network were built on one side (top) of one
substrate, only leaving the radiating patch on the other side (bottom) of the second
substrate. The substrate material used was RT/Duroid 5880 with a 31-mil thickness,
a dielectric constant of 2.2 and loss tangent of 0.001. Following the design process
in the previous section, a 5-GHz patch antenna was milled. The patch’s width and
length were 20.5 mm and 19.5 mm, respectively. The antenna and transistor were
connected by a 62-mil-long pin with a diameter of 0.4 mm located 4.58 mm inside
48
the patch. The ground plane’s dimensions were 61.5 mm and 58.5 mm. The terminal
loads were implemented with 2-mm-wide microstrip traces and the remaining biasing
traces were 20-mil thick at most. Due to the finite ground plane, coupling effects,
and the need for DC biasing lines, the transmission lines lengths were continuously
adjusted/optimized to present the transistor with loads close to the simulation values
obtained in Section 3.B. The common-source transistor HJ-FET NE3210S01 was DC
biased to VDS = 2V and IDS = 10 mA using a biasing chip (BCR400W) and various
discrete components. Figure 3.13 shows the final CAD layout, as well as the top and
bottom views of the fabricated design.
The measurement setup consisted of placing the active antenna and a receiving
horn about 30 free-space wavelengths apart. The horn was connected to an Agilent
E4407B spectrum analyzer with a resolution bandwidth of 1 MHz, 0-dB attenuation, 4-ms sweep, and 0-dBm reference power. Figure 3.14 is the actual measured
spectrum. The measured oscillation frequency was 5.018 GHz, which is very close
to the simulation in the previous section. The estimated radiated power, using the
Friis transmission equation, was 5.662 mW or 7.53 dBm; therefore the EIRP is 14.54
dBm. The DC-to-RF efficiency of the AIA was 21.8%.
49
(a)
(b)
Figure 3.13: (a) CAD layout implementing design steps of section 3.3.2. (b) Fabricated bottom view showing transistor, bias circuit and embedded linear
elements. On top, there is a patch antenna connected to the drain terminal.
50
Measured Spectrum, dBm
-20
-30
-40
-50
-60
4.98
5
5.02
5.04
5.06
Freq, GHz
Figure 3.14: Spectrum measurement of the radiated power. The oscillation frequency
is 5.018 GHz.
51
CHAPTER 4
DYNAMICS OF COUPLED NONLINEAR OSCILLATOR
ARRAYS
4.1
Introduction
As mentioned in Chapter 2, coupled nonlinear oscillators have been used to effectively
describe a wide variety of naturally observable phenomena in various fields of study
[92, 93]. Their applications in engineering have been primarily associated with lasers,
arrays of Josephson junctions, chaotic circuits and phase-locked loops [94–98].
Various sets of slowly-varying amplitude and phase equations have been derived for
microwave coupled oscillators [4, 46, 47, 60]. However, the simplifications were based
either on weak nonlinearity, broadband coupling, identical natural frequencies, and
nearest-neighbor or global topologies, and so on, leaving out many unavoidable and
significant effects that can highly affect the end performance of actual phased arrays.
Since most equations in coupled oscillators applications are special or simplified cases
of generic models developed for weakly [99–101] and strongly coupled systems [102–
107], it is clear that deviations from simulations is in part due to the difficulty of
modeling all the interactions and expressing all parameters in terms of measurable
network data or well-known figures of merit.
This chapter derives a system of expandable ordinary differential equations that
describe more accurately the dynamics of a group of oscillators coupled through an
52
arbitrarily structured linear network. The equations will be valid for weakly and
fairly strong coupled nonlinear systems with the major limiting restriction being on
the accuracy of the model for each regime and devices in use. The equations are
expressed in a form familiar with physical or electrical metrics needed to actually
analyze, design and build such systems.
4.2
Amplitude and Phase Dynamics
A simplified circuit diagram representing an array of N oscillators coupled through a
linear network is shown in Figure 4.1. Here the system is explicitly split into a linear
and nonlinear part. The linear part includes passive elements such as the antennas
and coupling circuits, which can all be systematically characterized by a N × N
immittance matrix Ykp , equal to
Ykp = |Ykp |e
jϕkp
�
IL,k ��
=
,
Vp �Vq =0 for q�=k
(4.1)
where (k, p) ∈ N2+ . The nonlinear part consists of all active and nonlinear devices.
In most prototypes, the dependent current sources in Figure 4.1 are implemented by
one or multiple transistors, while the independent sources can be used to model noise
or injection signals. For most practical applications, the desired mode of operation
consists of having all the oscillators synchronized to the same frequency ωs or phaselocked to one another. This mode of operation is critical for power-combining systems, coherent communications, transceiver sensitivity and continuous beam-steering
applications.
For common nonlinearities, the voltage vk (t) at each node or port k can be written
as a linear combination of harmonic phasors such that
53
Figure 4.1: Circuit representation of N linearly coupled nonlinear oscillators. The linear network Y includes all linear elements such the antennas and coupling
networks while the rest of the circuit is mostly made of active devices and
injection/control sources.
54
vk (t) =
M
�
�
�
� ṽk,m (t)ejmωs t ,
(4.2)
m=1
where M is the total number of harmonic terms or modes being considered and
ṽk,m (t) = Ak,m (t)ejφk,m (t) . The current going into the linear network at port k is
given by the inverse Fourier transform of its spectral counterpart
iL,k
where Vp (jω) =
�
∞
1
=
2π
�
∞
M �
N
�
Ykp (mωs )Vp (jmωs )ejωt dω,
(4.3)
−∞ m=1 p=1
vp (t)e−jωt dt. Due to the linear nature of Fourier transformations,
−∞
each harmonic term can be treated separately. Using Euler’s formula, Vp (jω) can be
written as
Vp (jω) =
�
∞
vp (t)e−jωt dt =
−∞
�
1�
Ṽp (ω − ωs ) + Ṽp∗ (ω + ωs ) .
2
(4.4)
Here, Ṽp is the Fourier transform of ṽp . Putting Eq. (4.3) and (4.4) together, one
obtains:
1
iL,k (t) =
2π
�
∞
N
�
1
−∞ p=1
2
�
Ykp Ṽp (ω − ωs ) +
Ṽp∗ (ω
�
+ ωs ) ejωt dω.
(4.5)
Using the properties of the integral above, Eq. (4.5) can also be written as follows:
�
��
� ∞
N �
�
1
iL,k (t) = �
Ykp (ω + ωs )Ṽp (ω)ejωt dωejωs t
.
2π
−∞
k=
(4.6)
The Taylor expansion of the immittance Ykp is given as:
∞
�
1 dn Ykp (ωs ) n
Ykp (ω + ωs ) =
ω .
n! dω n
n=0
Merging Eq. (4.6) and (4.7), one obtains
55
(4.7)
�
�
� � ∞
�
N �
∞
�
1 dn Ykp (ωs ) 1
1
iL,k (t) = �
(jω)n Ṽp (ω)ejωt dω ejωs t ,
n
n
n!
dω
j
2π
−∞
p=1 n=0
(4.8)
which is equivalent to the following equation:
�
�
�
�
N �
∞
�
1 dn Ykp (ωs ) 1 dn ṽp (t) jωs t
iL,k (t) = �
e
.
n! dω n j n
dtn
p=1 n=0
(4.9)
Due to the linearity of the coupling or immittance matrix, the total current at port
k is simply the linear combination of all considered M terms or harmonic components.
Since in general, the n-derivative of the phasor ṽp can be written as
n � �
dn ṽp (t) � n (k) � jφp �(n−k)
=
Ap e
dtn
k
k=0
(4.10)
and assuming that the current can be described well enough by a few dominant terms
(n = 0, 1), one obtains
�
iL,k (t) = � e
jωs t
N �
�
p=1
Ykp Ap ejφp +
1 dYkp (ωs ) dAp jφp
e
j
dω
dt
dYkp (ωs ) dφp
+
Ap ejφp + O
dω
dt
�
d2 Ykp (ωs )
dω 2
���
(4.11)
From Fig. 4.1, the Kirchhoff’s current relations at node k give
iD,k (t) + iL,k (t) = iinj,k (t).
(4.12)
In terms of averaging equations, which translate into the determination of the input
56
admittance, the following equation
2
Ak Ts
�
t
(iD,k (t) + iL,k (t))e−j(ωs t+φk ) dt
t−Ts
2
=
Ak Ts
�
t
iinj,k (t)e−j(ωs t+φk ) dt (4.13)
t−Ts
is also equivalent to:
� ωs ) + YL,k (ωs ) = Yinj,k (ωs , A,
� φ,
� ρk ),
YD,k (A,
(4.14)
assuming that the amplitudes and phases of the voltage signals are slowly-varying
1 dAk
dφk
functions of time (
� ωs and
� ωs ). Here Ts is the period of the
Ak dt
dt
� = [φ1 , φ2 , ..., φN ] are the voltage
� = [A1 , A2 , ..., AN ] and φ
fundamental mode. A
amplitude and phase vectors, respectively. ρk is the average amplitude of the injection
of noise signal at port k. To find the dynamics of the voltage amplitude and phase
at each port, the self-admittance vector, which is made of the diagonal entries of
the Y matrix, needs to be isolated from the mutual or coupling effects (remaining
matrix entries). Using Eq. (4.11) and (4.13) and assuming that d2 Ykp (ωs )/dω 2 is
comparatively small in magnitude to the first-order terms, one obtains:
�
�
�
1 dAk �� dYkk �� jϕ�kk
jϕkk
�
YD,k (A, ωs ) + |Ykk |e
+
e
jAk dt � dω �
�
�
�
dφk �� dYkk �� jϕ�kk
� ρk ), (4.15)
+
e
+ αkp = Yinj,k (ωs , A,
dt � dω �
�
where ϕkk = arg{Ykk (ωs )} and ϕkk = arg{dYkk (ωs )/dω}. The other complex quantity
αkp is given by
αkp
N �
�
Ap
1 dAp
=
|Ykp |ejθ1 +
Ak
jAk dt
p=1,�=k
�
�
� dYkp � jθ
�
� 2 Ap dφp
� dω � e + Ak dt
57
�
�
�
� dYkp � jθ
�
�e 2 ,
� dω �
(4.16)
�
with θ1 = φp − φk + ϕkk and θ2 = φp − φk + ϕkk . Splitting Eq. (4.15) into its real
and imaginary parts, one gets
r
YT,k
�
�
�
�
1 dAk �� dYkk ��
dφk �� dYkk ��
r
r
+
sin θ2 +
cos θ2 + αkp
= Yinj,k
Ak dt � dω �
dt � dω �
1 dAk
i
YT,k
−
Ak dt
�
�
�
�
� dYkk �
� dYkk �
dφ
k
i
i
�
�
�
�
� dω � cos θ2 + dt � dω � sin θ2 + αkp = Yinj,k
(4.17)
(4.18)
� ωs ) + Ykk (ωs ) and the superscripts r and i represent the real
� φ,
where YT,k = YD,k (A,
and imaginary part of a quantity, respectively. Multiplying Eq. (4.17) by sin θ2 and
subtracting the result by Eq. (4.18) times cos θ2 , one obtains
�
�
�
�
i
r
� dYkk � r
dAk
Ak
r dYkk
i dYkk
i
�
�
= −�
− YT
+�
(α sin θ2 − αkp cos θ2 )
� Y
dt
dω
dω � kp
� dYkk �2 T dω
dω
�
�
� dYkk � r
i
� (Y
+ ��
sin θ2 − Yinj,k
cos θ2 ) (4.19)
dω � inj,k
Similarly, multiplying Eq. (4.17) by cos θ2 and adding the result to Eq. (4.18) times
sin θ2 , one gets
�
�
�
�
r
i
� dYkk � r
dφk
1
r dYkk
i dYkk
i
�
�
= −�
− YT
+�
(α cos θ2 + αkp sin θ2 )
� Y
dt
dω
dω � kp
� dYkk �2 T dω
dω
�
�
� dYkk � r
i
� (Y
+ ��
cos θ2 + Yinj,k
sin θ2 ) (4.20)
dω � inj,k
However, the coupling terms emanating from αkp in Eq. (4.19) and (4.20) can be
further simplified such that
�
�
N
�
� dYkk � r
Ap
i
�
� (αkp sin θ2 − αkp
cos θ2 ) =
Hkp cos(φp − φk + Φkp )
� dω �
A
k
p=1,�=k
58
(4.21)
and
where
�
�
N
�
� dYkk � r
Ap
i
�
� (αkp cos θ2 + αkp
sin
θ
)
=
Hkp sin(φp − φk + Φkp ),
2
� dω �
A
k
p=1,�=k
Hkp
�
��
�
���
�2
��1/2
� dYkk �
� dYkp �
dφp
�
2
�
�
�
�
=�
|Ykp | + �
+ 2|Ykp | cos(ϕkp − ϕkp )
dω �
dω �
dt
(4.22)
(4.23)
In the equation above, it is assumed that the amplitude dynamics reach their steady
�
state quickly (dA/dt
→ 0). Using Eq. (4.19) and (4.20), the coupling phase Φkp
between oscillator k and p can be written as:
tan Φkp
�
�
�
� dY � p
|Ykp | cos(ϕkk − ϕkp ) + � dωkp � dφ
cos(ϕkk − ϕkp )
dt
�
�
=
.
� dY � p
�
|Ykp | sin(ϕkk − ϕkp ) + � dωkp � dφ
sin(ϕ
−
ϕ
)
kk
kp
dt
(4.24)
�
��
�
� dYkp � � dφp �
��
� is small in magnitude compared to |Ykp |, then Hkp
If the quantity ��
dt � � dt �
and Φkp can be further simplified to:
Hkp
and
�
�
� dYkk �
� |Ykp |,
= ��
dω �
tan Φkp = cot(ϕkk − ϕkp ).
(4.25)
(4.26)
dYkk
dY i
≈ j kk as in practical microwave oscillators in which an adedω
dω �
�
� r�
i �
� dYkk
� dY �
�
�
quate resonant circuit is used (�
� �� kk ��), one gets Φkp ≈ ϕkp = arg{Ykp } =
�
dω
dω
∠Ykp .
Furthermore, if
Strictly in terms of real and imaginary parts, Eq. (4.24) can be written as:
59
Φkp (ω) =
r
∂YT,k
∂ω
∠
i
r ∂YT,k
Ykp
∂ω
r
Ykp
i
∂YT,k
∂ω
r
i ∂YT,k
Ykp
∂ω
i
+ Ykp
−
.
(4.27)
The injection terms, which can be sources of noise or control signals, can also be
simplified. In the case where the injection signal at node k is a coherent signal of the
form ρinj,k cos(ωs t + Θinj,k ), one finds the following two equations
�
�
� dYkk � � r
�
i
�
� Yinj,k sin θ2 − Yinj,k
cos
θ
= ηA,k
2
� dω �
ρinj,k
=
Ak
and
�
�
� dYkk � � r
�
i
�
� Yinj,k cos θ2 + Yinj,k
sin
θ
= ηφ,k
2
� dω �
ρinj,k
=
Ak
�
�
� dYkk �
�
�
�
� dω � cos(Φinj,k − φk − ϕkk ) (4.28)
�
�
� dYkk �
�
�
�
� dω � sin(Φinj,k − φk − ϕkk ) (4.29)
Equations (4.19) and (4.20) can therefore be written as follows
�
i
r
dAk
Ak
r dYkk
i dYkk
= −�
− YT
� Y
dt
dω
� dYkk �2 T dω
dω
�
N
�
Ap
+
Hkp cos(φp − φk + Φkp ) + ηA,k , (4.30)
Ak
p=1,�=k
60
and
�
r
i
dφk
1
r dYkk
i dYkk
= −�
Y
+
Y
�
T
dt
dω
� dYkk �2 T dω
dω
�
N
�
Ap
+
Hkp sin(φp − φk + Φkp ) + ηφ,k . (4.31)
A
k
p=1,�=k
In practical microwave oscillators, the frequency dependence of the input immittance of the active device is many orders of magnitude smaller compared to that
of the resonator, and in turn, the equations above are sufficiently accurate. In the
case where this assumption is not valid, equations (4.30)-(4.31) can be adjusted by
replacing the index kk by T, k to account for the device’s high frequency variation.
Finally, for general analysis, the equations can be rendered dimensionless with a slow
time τ = ωs t such that for the amplitude dynamics
�
�
N
�
dAk
Ap
�
�
= −Ak Gk (A, φ, ωs ) +
Λkp cos(φp − φk + Φkp ) + σA,k
dτ
Ak
p=1,�=k
(4.32)
and for the phase dynamics
�
�
N
�
dφk
A
p
� ωs ) +
� φ,
= − Ωk (A,
Λkp sin(φp − φk + Φkp ) + σφ,k ,
dτ
A
k
p=1,�=k
(4.33)
where the quantities Gk and Ωk in the above equations are:
� ωs ) =
� φ,
Gk (A,
� ωs ) =
� φ,
Ωk (A,
1
�
�2
ωs � dYdωkk �
1
�
�2
ωs � dYdωkk �
61
�
�
dY i
YTr kk
dω
dY r
YTr kk
dω
−
+
dY r
YTi kk
�
dY i
YTi kk
�
dω
dω
,
(4.34)
.
(4.35)
The terms of the coupling coefficient [Λ] are given by
Λkp =
1
� dY �2 Hkp ,
ωs � dωkk �
(4.36)
and the injection terms terms are σA,k = ωs ηA,k and σφ,k = ωs ηφ,k .
4.3
Transient and Stability Analysis
Maintaining the stability of the phase-locking mechanism of the array as some control
parameters are changed is of paramount importance in practical applications. To
study the stability properties of the above equations, the start-up transient stability
will be considered. In this regime, the voltage amplitudes are at the noise level and
� ≈ �0). Neglecting higher-order derivative and injection terms, the
thus very small (A
quantity ΓA,k is given by:
ΓA,k
∂
≡
∂Ak
�
dAk
dt
�
1
≈�
�
� dYkk �2
dω
�
dY r
YTi kk
dω
−
−
dY i
YTr kk
dω
�
N
�
Ap
Hkp cos(φp − φk + Φkp ). (4.37)
A
k
p=1,�=k
The oscillation amplitudes at the noise level will grow only if there is instability which
requires that ΓA,k > 0. For a single oscillator (uncoupled, [H]=0) , that implies
dY r
dY i
YTi kk − YTr kk > 0. It should be noted that at startup, YTr is always negative for
dω
dω
dY i
a well-designed oscillatory system, therefore a sufficiently large and positive kk will
dω
make the array oscillate. The array reaches its steady state when the power provided
by the active devices is equally consumed by the rest of the array. This results in a
cancellation of the total reactance/susceptance and resistance/conductance. Another
consequence of reaching the steady-state regime is that ΓA,k becomes negative. At
62
this regime, ΓA,k is given by
1
ΓA,k = �
�
� dYkk �2
dω
�
r
i
dYTi dYkk
dY r dYkk
− T
dAk dω
dAk dω
�
N
�
Ap
−
Hkp cos (φp − φk + Φkp )
Ak
p=1,�=k
��
�
��
�
��
� dYT,k �
∂
1 �� dYT,k ��
�
�
×
−
(4.38)
∂Ak � dω �
Ak � dω �
For an uncoupled oscillator, it is clear that ΓA,k < 0 when
i
r
r
i
dYT,k
dYT,k
dYT,k
dYT,k
−
<
dAk dω
dAk dω
0 as also shown in [48].
The phase dynamics at the noise or small-signal regime is irrelevant for an oscillator. However, at the steady-state regime, maintaining phase stability or phase
locking is crucial. For that to happen, the following quantity Γφ,k must be negative.
Γφ,k
∂
≡
∂φk
�
dφk
dt
�
−ρinj,k
≈
Ak
�
�
� dYkk �
�
�
�
� dω � cos(Θinj,k − φk − ϕkk )
� N
�
� Ap
1
+�
Hkp cos(φp − φk + Φkp )
(4.39)
�
� dYkk �2 p=1,�=k Ak
dω
Here, it is clear that an uncoupled oscillator (without injection) will have Γφ = 0.
This result shows that phase perturbations of a single oscillator will neither decay nor
increase since there is no phase restoring mechanism, unlike its amplitude dynamics.
However, when the oscillators are coupled or injection-locked, a partial or full phase
restoring mechanism is present and the resulting phase disturbances or noise can be
minimized by making sure that Γφ,k is sufficiently negative. It is also important to
point the importance of the quality factor Qk of each oscillator and its effect on phase
noise. The quality factor can be approximately derived from the frequency derivative
from the self-immittance terms as follows:
63
�
�
� dYkk �
2Qk r
�
�
≈
Y .
� dω �
ωs kk
ω=ωs
(4.40)
Thus the phase dynamics from Eq. (4.31) is now equivalent to
dφk
=−
dt
�
ωs
r
2Qk Ykk
�2
dY r
(YTr kk
dω
�
�
N
�
Ap ωs |Ykp |
sin(φp − φk + Φkp )
r
dω
Ak 2Qk Ykk
p=1,�=k
�
�
r
ρinj,k 2Qk Ykk
�
+
sin(Θinj,k − φk − ϕkk ). (4.41)
Ak
ωs
dY i
+ YTi kk ) −
From the equation above, it is clear that a high quality factor limits the phase disturbances coming from the oscillator itself and its coupling mechanism.
4.4
4.4.1
Numerical Simulations and Verification
Frequency Deviation
To illustrate the validity of the above equations, a system of coupled van der Pol
oscillators will be studied as depicted in Fig. 4.3. The van der Pol nonlinearity can
be readily implemented using a set of cross-coupled transistors and such architecture
is common in integrated circuits. Before performing an in-depth verification of the
derived equations, it is also necessary to accurately determine the oscillation frequency
of the array before fabrication. To help with this endeavor, let’s assume that the
voltage at one uncoupled RLC oscillator can be written as follows:
v = A1 cos(ωt) + A3 cos(3ωt + φ).
64
(4.42)
Only the fundamental and its third harmonic terms are considered since we are dealing with a van der Pol oscillator. It is also assumed that A1 � A3 . The fundamental
and third harmonic input admittance looking into the nonlinear device are quickly
calculated to give
3
3
3
YN (ω) = −a + b( A21 + A23 ) + bA1 A3 ejφ
4
2
4
(4.43)
3
3
b A31 −jφ
YN (3ω) = −a + b( A23 + A21 ) +
e ,
4
2
4 A3
(4.44)
assuming that the nonlinearity is defined as i = −av + bv 3 . The oscillator will settle
at a frequency corresponding to a cancellation of the imaginary part of the total immittance, thus
3
1
bσA41 sin φ + ωC −
= 0,
4
ωL
(4.45)
with A3 = σA31 . This makes the oscillation frequency equal to
ω=−
3
bσA41
4
sin φ
+
2C
�
�3
4
bσA41 sin φ
2C
�2
+ ω02 .
(4.46)
Here, ω0 is (LC)−1/2 . By defining the nonlinear factor as � = Lω0 (a − 1/R), the
oscillation frequency given by Eq. (4.46) can be normalized and expressed as
�
ω
= 1 + γ 2 �4 − γ�2 ,
ω0
65
(4.47)
with
γ=
3
bσA41
4
sin φ
2Cω0 �2
(4.48)
A plot of this normalized synchronization frequency as a function of the nonlinearity
factor � is shown in Fig. 4.2. As the nonlinearity factor increases, the initial approxi-
Normalized Frequency
1
Exact
Approx
0.95
0.9
0.85
0.8
0
0.5
1
1.5
2
Nonlinearity Factor, ε
Figure 4.2: Approximation of the normalized van der Pol oscillation frequency
(dashed) with exact (solid). The deviation increases as the nonlinear
factor increases.
mation gradually deviates from the true frequency. However, for common microwave
oscillators, the nonlinearity factor is kept well below unity to avoid relaxation oscillations and other undesired effects. Regions where � is larger than 0.5 are consequently
avoided. For larger values of �, a relaxation oscillator is gradually produced and its
frequency characteristics are based on the Airy function [108].
66
4.4.2
Synchronization
As previously mentioned, in many applications, frequency synchronization has to
be maintained to correctly operate coupled-oscillator arrays. Typically, the instantaneous frequencies of all oscillators need to be equal for efficient communication
coding, signal processing and signal coherence. Let’s investigate the properties of a
set of van der Pol oscillators coupled through a simple RL circuit is shown in Fig.
4.3. The array loading and coupling will be broadly changed from weak to strong in
order to illustrate its synchronization and stability properties.
Figure 4.3: Chain of RL-coupled van der Pol oscillators. The nominal values for the
coupling resistors and inductors are RC = 500 Ω and LC = 5 nH. The
RLC tank circuits have nominal values of Ri = 50 Ω, Ci = 1 pF and Li =
0.2533 nH, i=1,2,..,N for a 10 GHz oscillation frequency. The nonlinearity
is described by i(v) = −av + bv 3 , where a = 1/30 and b = 1/90.
The dynamics of the above array is exactly described by the following system of
67
second-order differential equations:
� N
�
�
1 �
v̇p − v̇k vp − Vk
v̈k =
M (k, p)
+
Ck p=1
Rkp
Lkp
+ v̇k
�
1
−a+
+ 3bvk2
Rk
�
�
vk
−
, (4.49)
Rk
for k = 1 to N and where M (k, p) is a topology or interconnection matrix. In other
words, M (k, p) is unity if the oscillator k and p are connected, otherwise the entry
is null. The configuration in Fig. 4.3 is a nearest-neighbor topology, which means
that M (k, p) is unity only when |k − p| = 1. For N = 11, this array’s dynamics are
simulated to track the loss of synchronicity as the capacitance of the first tank circuit
(Fig. 4.3) is gradually changed. Fig. 4.4 shows how the synchronization frequency of
all eleven oscillators is gradually lost as the natural frequency of the edge oscillator
deviates from the group frequency. There is a region (C = [0.935, 1.07] pF) where all
oscillators hold the same frequency. Outside this region, frequency synchronization
is gradually lost.
To illustrate the accuracy of the derived equations from Section 4.2 versus the
exact equations (Eq. 4.49), the 11-element array of Fig. 4.3 will be analyzed. The
linear part of the array can be described by an admittance matrix [Y ], whose diagonal
entries are given by:
�
�
N
�
1
1
1
1
Ykk (ω) =
+
+ jωCk +
M (k, p)
+
,
Rk jωLk
R
jωL
kp
kp
p=1
68
(4.50)
11
Oscillation Frequency, GHz
10.8
10.8
10.6
10.6
10.4
10.2
10.4
10
0.89
0.9
0.91
0.92
0.93
10.2
10
9.8
9.6
0.85
0.9
0.95
1
1.05
1.1
1.15
Capacitance, pF
Figure 4.4: Loss and gain of synchronization between oscillators as the tank capacitance of first oscillator is varied. These results are from Eq. (4.49). From
approximately 0.935 to 1.07 pF, all oscillators are synchronized. Outside these values, partial synchronization occurs where multiple groups
of identical frequencies form. Chaotic behavior is noticed below C=0.935
pF (dotted box).
69
and the remaining terms are expressed as follows
Ykp (ω) = −
�
1
1
+
Rkp jωLkp
�
.
(4.51)
The loss of frequency synchronization can also be accurately predicted by the derived
first-order equations (Eq. 4.30-4.31). It can be seen from Fig. 4.5 that the phase
difference between multiple adjacent oscillators quickly diverges or becomes invalid
(as the frequencies differ) around C1 = 0.935 pF, thus confirming that the oscillators
have lost their synchronicity. These results are in agreement with the earlier numerical simulations involving the exact second-order differential equations. To further
Phase Difference, deg
140
120
100
80
60
0.934
0.935
0.936
0.937
0.938
0.939
0.94
Capacitance, pF
Figure 4.5: Ten curves showing the phase difference φp − φk between adjacent oscillators showing a loss of synchronicity around C1 = 0.935 pF. Another
loss of synchronicity also appears around C1 = 1.07 pF (not shown here).
verify the accuracy of the derived equations, the comparisons will be carried out at
70
areas where the most differences are noticed, mostly in the phase dynamics. The
array depicted in Fig. 4.3 can be assumed to be relatively weakly coupled since the
coupling strength as defined by Eq. (4.36) is around 0.029. The phase and amplitude
dynamics comparisons are shown in Fig. 4.6. All comparison plots are chosen from
where the most discrepancy is observed, and as expected, for Fig 4.6, the highest disagreement occurs at the other edge of the chain between oscillator 10 and 11 when the
80
1.3
60
1.25
Approx. Phase
Exact Phase
Approx. Amp
Exact Amp
40
20
1.2
Amplitude, V
Phase Difference, deg
capacitance of the first oscillator is changed. When the coupling strength is increased
1.15
0
0.94
0.96
0.98
1
1.02
1.04
1.1
1.06
Capacitance, pF
Figure 4.6: Phase difference between oscillators 10 and 11 from the chain of oscillators
depicted in Fig. 4.3. (solid line, green) is computed from the first-order
derived equations while the marked line represents results from the exact
second-order differential equation. Similarly, the amplitude comparison
at oscillator 10. (solid line, red) is computed from the first-order derived
equation while the marked line represents results from the exact secondorder differential equation.The coupling strength varies from 0.0285 to
0.0292 across the array.
71
by setting RC = 100 Ω and LC = 1 nH, the highest disagreement in amplitude and
phase dynamics are again observed between oscillators 10 and 11. Fig. 4.7 shows
results obtained with the exact and approximate equations. Again, the results are
close even for a greater range of capacitance values. Furthermore, a stronger coupling
1.4
20
1.35
15
1.3
10
5
0
0.4
Approx. Phase
Exact Phase
Approx. Amp
Exact Amp
0.6
0.8
Amplitude, V
Phase Difference, deg
25
1.25
1
1.2
1.2
Capacitance, pF
Figure 4.7: Left: phase difference between oscillator 10 and 11. (solid line, green)
is computed from the first-order derived equations while the marked line
represents results from the exact equations. Right: amplitude comparison
at oscillator 11. (solid line, red) is computed from the first-order derived
equation while the marked line represents results from the exact secondorder differential equation. The coupling strength varies from 0.1193 to
0.1328.
arrangement is introduced by setting RC = 5 Ω and LC = 1 nH. This results in a
strongly coupled system with the results shown in Fig. 4.8. It is noticed that the
stronger the coupling, the more accurate the amplitude dynamics becomes. Moreover,
72
1.27
0.8
1.26
0.6
0.4
1.25
0.2
Approx. Phase
Exact Phase
Approx. Amp
Exact Amp
0
-0.2
-0.4
0.4
0.6
0.8
Amplitude, V
Phase Difference, deg
1
1.24
1
1.23
1.2
Capacitance, pF
Figure 4.8: Left: phase difference between oscillator 10 and 11. (solid line, green)
is computed from the first-order derived equations while the marked line
represents results from the exact equations. Right: Amplitude comparison at oscillator 11. (solid line, green) is computed from the first-order
derived equation while the marked line represents results from the exact second-order differential equation. The coupling strength varies from
1.2739 to 1.4171.
73
throughout the increase in coupling strength, the first-order equations maintained a
satisfactory level of accuracy.
Finally, the response of the array from a change of nonlinearity is undertaken.
Here the nonlinearity factor is solely dependent on the loading of each active device
and it is re-defined here as:
�k =
2
2 (a − Ykk
)
� dY � .
ωs � dωkk �
(4.52)
When all the tank resistances are gradually changed from 35 to 500 Ω and the
capacitance of the first tank is increased to 1.05 pF (to generate an asymmetric phase
gradient), the nonlinear factor goes from about 0.08 to 0.5. The rest of the circuits
elements keep their nominal values as given in Fig. 4.3. Fig. 4.9 shows how the
amplitude and phase dynamics are affected from varying the nonlinearity. It is clear
that the increase of nonlinearity up to 0.5 does not greatly affect the accuracy of the
derived equations. Moreover, the amplitudes derived from the approximate equations
are nearly indiscernible from the exact response.
4.4.3
Comparison to Previous Models
Similar equations modeling arrays of coupled microwave oscillators have been presented with various degree of accuracy based on the severity of the nonlinearity or
the strength of the coupling. One of the most commonly used models was developed
by York [4, 55]. The equations are provided here for convenience for a van der Pol
74
5
58
4
56
3
Approx. Phase
Exact Phase
Approx. Amp
Exact Amp
54
2
52
50
Amplitude, V
Phase Difference, deg
60
1
0.1
0.2
0.3
0.4
0
0.5
Nonlinear Factor
Figure 4.9: Left: phase difference between oscillator 1 and 2 as the nonlinear factor
is varied. A difference of about 2 degrees is observed as the maximum
difference. Right: amplitude comparison at oscillator 1. The difference is
minimal as the two curves are indiscernible justifying the accuracy of the
amplitude dynamics (RC = 500 Ω).
75
array.
Fk =
1+
1
2Q
�N
1
Ykp Vp
∂
p=1 j∂ω ( GL ) Vk
×
�
�
�
�
ωk
A2k
1+µ 1− 2
2Q
αk
−
�
�
Iinj,k
+
+ j(ωk − ω)
(4.53)
GL V k GL V k
N
�
Ykp Vp
p=1
with the amplitude and phase dynamics being Ȧk = Ak �{Fk } and φ̇k = �{Fk },
respectively. αk and ωk are the free-running or natural amplitudes and oscillation
frequencies, respectively. GL is the load of each oscillator. Compared to equations
(4.30) and (4.31), the York equations produce similar results when the coupling is
weak and nonlinearity not severe. The disparity at other regimes is mostly caused
by the fact that the York’s equations neglect the input susceptance of the active
device and its frequency variation. This results in the absence of an additional term
in the phase dynamics that affect the accuracy of the inter-element phasing due to
their sensitivity to small parameters. Furthermore, using free-running frequencies
and amplitudes is problematic and impractical because the oscillators are always
coupled or loaded by other oscillators’ signals and circuitry and their values are not
accurately known in advance. Unlike the amplitude inaccuracies that are highly
forgiving in phased arrays dynamics, the phase dynamics on the other hand is highly
sensitive to small parameters and predictions might not be accurate if some terms
are dropped.
To illustrate some accuracy difference, an array of five coupled oscillators with
a topology similar to Fig. 4.3 is analyzed using equations (4.53), (4.30,4.31) and
(4.49). The coupling resistance is 200 Ω and the array is asymmetrically loaded by
only changing the capacitance of the first tank until the array becomes unlocked. It
is found that the amplitude dynamics is similar between all models, but the phase
76
dynamics on the other hand shows some discrepancies. The phase difference between
oscillator 1 and 2 are plotted in Fig. 4.10. It is observed that the exact differential
equations and the approximate equations provided here predict that the array will
loose its frequency synchronicity when C ≈ 1.095 and C ≈ 1.09 pF, respectively.
However the York equations predicts a lost of synchronization at C ≈ 1.1275 pF.
In terms of phase differences, the York equations become more inaccurate as the
array is pushed near the edge of stability. it is also clear that the York equations, in
this case, overestimate the controlling capabilities of the array, by predicting a wider
capacitance range of each oscillator. Although the difference seems to be small when
looking at the radiation pattern, for larger arrays with many more elements and a
sharp pencil radiation beam, small differences in oscillator phasing could introduce
significant errors on the exact location of the main beam.
In practical engineering designs, a simple circuit for an array as the one studied
earlier is not common since more complex coupling electromagnetic (EM) phenomena
are involved. In this case, it is convenient to deal with the network parameters of
the array, which are systematically obtained through an EM software suite. The
equations in this work (section 4.3) were purposefully derived to make sure that
network parameters of any kind and other related figures of merit are used by scientists
and engineers tasked with designing such systems.
In conclusion, a set of equations, for more accurately computing the dynamics
of an array of nonlinear coupled oscillators through a linear network, has been presented with focus on measurable microwave network quantities. Various degrees of
accuracy can be reached by involving additional modes of operations. While the targeted application is active and coupled integrated antennas arrays, the equations are
useful in any field dealing with coupled nonlinear oscillators arranged in an arbitrary
topology.
77
-20
Approx.
York
Exact
Phase Difference, deg
-30
-40
-50
-60
-70
-80
-90
1.06
1.08
1.1
1.12
1.14
Capacitance, pF
Figure 4.10: Comparison of phase differences in degrees between oscillator 1 and 2
using the York model (4.53), the derived equations (4.30,4.31), and exact differential dynamics (4.49). The derived equations show a higher
accuracy in reproducing the true dynamics of the array.
78
CHAPTER 5
DESIGN CONSIDERATION FOR
COUPLED-OSCILLATOR ARRAYS
5.1
Introduction
The goal of the previous chapter was to express the dynamics of coupled-oscillators
systems with readily obtainable network parameters. The motivation for producing
these equations was to deal with the drastic difference observed at times between
the design predictions or expectations to the actual performance of coupled-oscillator
array. As many simplifications are introduced to reduce the complexity of such systems, potential interesting effects may be overlooked and dismissed for the sake of
a quick design. This chapter will look at the effects of the coupling networks and
how their properties can be used to control the overall performance of the array. A
multi-simulation tool will be developed to make use of the coupling properties and
an three-element array will be designed and built to test the validity of the equations
presented in the previous chapter.
79
5.2
Coupling Considerations
Designing a system based on coupled oscillators necessitates a well-designed coupling
mechanism in order to regulate the phase and amplitude dynamics that will be generated once the system is operational. Linking two or more oscillators by a transmission
line is one of the simplest and most affordable solutions. Two simple cases will be
discussed shortly.
5.2.1
Transmission-Line based Coupling Networks
In order to illustrate even how simple coupling networks present interesting properties,
let’s look at two van der Pol oscillators coupled through a transmission line network
as shown in Figure 5.1. The black boxes represent the active devices. Let’s call this
model #1. It is a piece of uniform transmission line of characteristic impedance Z0 ,
electrical length θ, and terminated with two resistors RC at each end.
Figure 5.1: A pair of oscillators coupled through a transmission line-based circuit
(model #1).
The Y-parameter matrix of this coupling circuit between the two oscillators is
80
given by (see Appendix A for derivation):

RC
−1
 cos θ + j Z0 sin θ

j sin θ
��
�
[Y ] =

�2

RC
Z0 cos θ + j Z0 sin θ − 1 
RC
−1
cos θ + j
sin θ
Z0




.


(5.1)
The Y parameters of the entire circuit, as seen by the two active devices, are given by:

�
�

1


Ykk = Gk + j ωCk −
+


ωLk






cos θ + j RZC0 sin θ
��
�
�2
RC
Z0
cos θ + j Z0 sin θ − 1
j sin θ




−j sin θ


��
�
Y
=
Y
=

12
21
�2


RC

Z0 cos θ + j Z0 sin θ − 1

,
(5.2)
for k = 1, 2. In the particular case where RC = Z0 , the equations (5.2) simplify to
�
�

1
1


Ykk = Gk + j ωCk −
+


ωLk
2Z0






−e−jθ

 Y12 = Y21 =
2Z0
(5.3)
From equations (5.3), it can be noticed that the admittance of the oscillator is independent of the electrical length of the coupling transmission line. This behavior leads
to a wideband coupling network that is relatively simple to implement. By defining
the loaded quality factor1 Qk
1
see equation (4.40)
81
Qk (ω) = ω
� ∂Y �
� kk �
∂ω
r
2Ykk
Ck +
=ω
1
ω 2 Lk
2Gk + Y0
≈
2ωCk
,
2Gk + Y0
(5.4)
the coupling strength exerted by each oscillator, as defined in equation (4.25), is
Λkp (ω) =
|Ykp |
1
Y0
Y0
=
·
≈
,
r
2Qi Ykk
2Qk (ω) (2Gk + Y0 )
4ωCk
(5.5)
where, Y0 = 1/Z0 . Moreover, the coupling phase Φ(ω) is equal to θ(ω) since |∂Ykk /∂ω =
0| and assuming that the frequency derivative of the real part of the active device is
negligible. To find the actual coupling phase of this circuit, equation (4.27) is used.
It follows that the coupling phase is:
Φkp (ω) =
r
∂YT,k
∂ω
∠
i
r ∂YT,k
Ykp ∂ω
r
Ykp
i
∂YT,k
∂ω
r
i ∂YT,k
Ykp ∂ω
i
+ Ykp
−
≈ θ(ω),
(5.6)
r
r
∂YT,k
∂Ykk
since
≈
= 0.
∂ω
∂ω
Another coupling configuration is shown in Figure 5.2. Here, two identical sections
of transmission lines are connected by a resistive load of value 2RC . Each transmission
line section has a characteristic impedance of Z0 and an electrical length of θ/2. Let’s
call this configuration model # 2. At first, the two coupling mechanisms can be argued
to produce the same coupling properties as they are both made of the same amount
of resistance and transmission line length; however, it will be shown that these two
coupling circuits behave quite differently.
Going through a similar analysis (Appendix A), as in the first circuit, the Y matrix of the coupling circuit alone is:
82
Figure 5.2: A pair of oscillators coupled through a transmission line-based circuit
(model #2).

RC
−1
 cos θ + j Z0 sin θ

1

[Y ] =

RC (1 + cos θ) + jZ0 sin θ 

RC
−1
cos θ + j
sin θ
Z0







(5.7)
The Y matrix of the entire circuit from the active devices’ perspective is:

�
�
cos θ + j RZC0 sin θ
1



Y
=
G
+
j
ωC
−
+
kk
k
k


ωLk
RC (1 + cos θ) + jZ0 sin θ






 Y12 = Y21 =
−1
RC (1 + cos θ) + jZ0 sin θ
Equation (5.8) reduces to the following equation when RC = Z0 .
83
(5.8)
�
�

tan 2θ
1
1


Y
=
G
+
j
ωC
−
+
+
j

kk
k
k

ωLk
2Z0
2Z0








Y12 = Y21 =
(5.9)
tan 2θ
1
−j
2Z0
2Z0
Unlike the first circuit, the impedance here is highly dependent of the electrical length
of the transmission line. However, the coupling phase remains identical to that of the
r
∂Ykk
first model, that is Φ(ω) = θ(ω), since
= 0. The quality factor is given by
∂ω
Qk (ω) = ω
� ∂Y �
� kk �
∂ω
r
2Ykk
=ω
Ck +
1
ω 2 Lk
+
Y0 τg
2(1+cosθ)
2Gk + Y0
≈ω
2Ck +
Y 0 τg
2(1+cosθ)
2Gk + Y0
,
(5.10)
and the coupling strength is
Λkp (ω) =
|Ykp |
1
Y0
Y0
�
�
=
·
≈
.
r
θ
Y 0 τg
2Qi Ykk
2Qk (ω) (2Gk + Y0 )|2 cos 2 |
2ω 2Ck + 2(1+cosθ)
| cos 2θ |
(5.11)
Here, τg is the group delay of the transmission line (τg = ∂θ/∂ω). From all the
above equations, it is clear that the properties of model #2 are highly affected by
the electrical length of the coupling transmission lines. The following set of plots
compare the quality factors (Figure 5.3) and coupling strengths/coefficients (Figure
5.4) of circuit models #1 and #2. The constant ω0 is the reference frequency at
which the initial electrical lengths of all transmission lines are specified.
84
(a) θ = π/2
(b) θ = π
(c) θ = 2π
(d) θ = 4π
Figure 5.3: Comparison of quality factors for various values of the electrical length of
the coupling transmission lines.
85
(a) θ = π/2
(b) θ = π
(c) θ = 2π
(d) θ = 4π
Figure 5.4: Comparison of coupling coefficients for various values of the electrical
length of the coupling transmission lines.
86
The quality factors are normalized to Q0 = ω0
2Ck
and the coupling coef2Gk + Y0
Y0
. From the precedent plots, it is clear that
4ω0 Ck
the coupling properties of the first model are not affected by the change in electrical
ficients are normalized to Λ0 =
length of the coupling structure. This presents a uniform and easily predictable behavior of the array when expanded to multiple elements. The second model, however,
has a high dependence on the coupling angle. For case (b) in Figure 5.4, it is clear
that no coupling occurs at ω = ω0 . This explains why even two identical oscillators
coupled through an out-of-phase (θ = π) line will not synchronize at their natural
frequencies, since the coupling coefficient is null. Another observation from model #
2 is that the coupling strength can be made null at some specific frequencies. For case
(c), it is shown that coupling is zero at 0.5ω0 and 1.5ω0 . This can be useful in either
suppressing potential oscillations at those frequencies when dealing with the design
of coupled oscillating circuits. It can also be useful in applications where harmonics
or sub-harmonics oscillations are needed to be explicitly suppressed. In general, the
nulls observed in the coupling strength occurs at
ω
π
=
+n
ω0
θ0
�
2π
θ0
�
, n = 0, 1, 2, ...,
(5.12)
where θ0 is the initial electrical length of the transmission line at ω0 .
5.2.2
Practical Implementation
During the design of a phased array, a few factors need to be considered. To avoid
grating lobes, the antenna elements are placed closer to one another. Typically,
a distance d = λ0 /2 (or smaller), where λ0 = c0 /f is the free-space wavelength.
87
(a)
(b)
Figure 5.5: Two models of folded transmission lines for linking oscillators. (a) Modeled after a sine function. (b) A rectangular shape.
However, to generate a coupling phase of 0[2π], the transmission lines need to be
√
about λg = c0 /(f �e ) in length, where �e is the effective dielectric constant of the
substrate. As expected, λg will most likely will not be equal to λ0 /2. This means
that coupling transmission lines and circuits will need to be folded or made longer to
fit the fixed length between the antennas. Figure 5.5 shows two proposed layouts to
accomplish such task.
For the layout in Figure 5.5 (a), a smooth function is chosen to guide coupling
power from one oscillator to the next. The curves described by f : y = f (x) and
�
�
�
f : y = f (x ) are based on a sine function. Any convenient smooth function can be
picked for the purpose of offering a gradual transition. The reason for it is to make
the coupling circuit less prone to radiation leakage and susceptible to the excitation of
evanescent waves. However, these types of circuits can be difficult to implement, and
an alternative layout is shown in Figure 5.5 (b). This is a rectangular structure made
with straight lines that is simpler to fabricate. Depending on the type of coupling
used, a gap can be easily introduced to accommodate a resistive load (model #2).
88
5.3
5.3.1
Design Method and Considerations
Counteracting Coupling Effects
In practical designs, even when a transmission line coupling is used, the coupling
between the oscillators has additional components caused by other electromagnetic
phenomena. Typically in layered media or substrates, any source will excite one or
more surface-wave modes a well as a radiation field. In typical applications with
patch antennas, for instance, that require a substrate material, the TM0 surface wave
mode will usually propagate radially away from the patch, decaying proportionally
√
to 1/ ρ. Away from the air dielectric interface, the surface wave become negligible
and the radiation field or space wave dominates, which decay at a rate proportional
to 1/r. Along the air-dielectric interface, there is a lateral-wave field that propagates
with the same wavenumber as the space wave, but it however decays proportionally
to 1/ρ2 . Consequently, in typical layered media, surface wave and lateral-wave fields
contribute to additional coupling, with the surface-wave field dominant at larger
distances [109].
To reduce the additional coupling between the antennas and the coupling transmission lines, the antennas will be built on one side of the substrate while the coupling
circuits will be built on the other side for better isolation and greater control of the
coupling phase. For typical substrates, this results in the antennas being milled or
etched on one substrate and the rest of the system built on another substrate. The
two ground planes are soldered together and top and bottom layers are connected via
drilled holes (where pins/wires are inserted). A high-level design topology is shown
in Figure 5.6 with five patch antennas.
Since the antennas in an array are inherently coupled to one another and to the rest
of the system by surface and lateral wave fields, the phasing between the oscillators
89
Figure 5.6: Top and bottom views of array of patch antennas Antenna Dimension:
W=20.5mm, L=19.5 mm. Coupling line: Z0 = 100Ω.
is difficult to accurately predict based on the coupling transmission lines alone. In
order to maintain a desired phase distribution, it is necessary to come up with a set
of transmission lines or other circuits that force the array to adopt a predetermined
phase distribution. Unlike all designs presented in the literature review, this array
will be allowed to get the optimum performance by using transmission lines of various
length or widths so that the desired performance criterion is reached. In this example,
it is desired to obtain a uniform broadside array that is also equipped with a large
scanning range.
To solve this technical design challenge, the equations in Chapter 4, together with
a co-simulation approach will be used. In other words, a full-wave EM simulator will
control the parameterized dimensions of the array and output the necessary network
parameters. The N-port network parameters (S or Y matrix) are then exported into a
time-domain simulator which calculates the steady state of the amplitude and phase
at each port based on equations developed in the previous chapter. An objective
function, that uses the amplitude and phase dynamics, is created pointing to the
desired performance criteria. This optimization loop is executed until satisfactory
results are found. Figure 5.7 shows a flowchart describing the optimization process. It
is clear that a wide variety of optimization routines can be used to achieve the desired
90
Figure 5.7: Flowchart describing the optimization process of the array. Many algorithms can be used, especially genetic, particle swarm optimization,
conjugate gradient, etc..
91
results. In this task, the co-simulating full-wave EM software will be FEKO [110]
and the software controlling and implementing the various algorithms and the array
dynamics will be MATLAB [111].
The optimization process can be lengthy if the size of the array is substantial;
therefore for quick convergence, a fast-converging control algorithm and objective
functions must be used. Here, a hybrid and multi-objective discrete and continuous
particle swarm optimization (PSO) was developed by combining available algorithms
in MATLAB. The objective function was based on the radiation pattern or the computed array factor from the array dynamics.
The five-element patch antenna array of Figure 5.6 is simulated with the performance criteria being the best possible gain at broadside. This condition is picked
because it also implies that the phase difference between the antenna signal has to
be 0[2π]. The operational frequency is 5 GHz and the initial length of all connecting
transmission lines is 4.0 mm. Each transmission line is loaded with a 200-Ω resistor
since the characteristic impedance of the lines is 100 Ω. As the length of each coupling line is adjusted, Figure 5.8 shows the progression of the optimization process
as the array dynamics converges to adopt the characteristics of a uniform array. It is
noticed that the amplitude dynamics are not uniform at first when all the coupling
transmission lines have the same length of 4.0 mm. As the algorithmic process iterates, by adjusting the length of each coupling line, the array slowly converges to a
state where all amplitudes become equal since the initial objective was to obtain a
uniform array radiation pattern.
The phase dynamics is tracked as well, and it tends to converge to a uniform
distribution; however, both uniform amplitude and zero phasing distributions are not
fully reached because the algorithm was not able to perfectly cancel the effect of the
surface and lateral wave coupling. Nevertheless, their effects are minimized and only
92
Figure 5.8: Normal progression of the algorithm as the array converges to a uniform
phased array. The length of each coupling line is iteratively adjusted as
the optimization progresses.
93
observed in the array sidelobes as shown in Figure 5.9. The comparison is made to
an ideal uniform array with equal signal amplitudes and zero phase gradient. The
final design for the transmission lines are shown in Figure 5.10. As it can be seen,
the coupling transmission lines have different lengths of 4.3023 and 3.5971 mm. This
implementation of coupled oscillators is very different from what has been presented in
related literature as the oscillators are coupled with identical transmission lines, thus
yielding sub-optimal results. The variation of transmission lines’ lengths is needed
to improve the stability of the array because each antenna senses a different loading
environment.
Figure 5.9: Plot of the array factor of the 5-element array (red) compared to the ideal
uniform array (blue). A small difference is noticed in the sidelobes while
the main beam remained unaffected.
A more involved optimization could have also changed the characteristic impedance
of each transmission line, but for most cases, the coupling phases or line lengths are
the most important and sensible variables to be tuned. Using the same optimization process, the array is made to scan off broadside to its maximum angle before
94
becoming unstable. By unstable, it is meant to point to the fact that either the array de-synchronizes (which can be tracked by a sudden change in phasing) or adopt
another mode (different phase distribution). For both cases, the array operates at 5
GHz and keeps the same loading characteristics. The time-domain simulation runs
for at least 200 ns. The initial conditions for the amplitudes are small compared to
their steady-state counterparts and also randomly generated.
Figure 5.10: Final dimensions of the array design (bottom coupling lines not to scale).
The array is symmetric with respect to the third antenna. All resistors
are 200 Ω.
Since the theoretical maximum phasing is π/2, which corresponds to a scan angle
of π/3 for an inter-element separation of λ/2, the objective function for this setup
is written so that the algorithm drives the array dynamics to produce a peak of the
95
array factor at the expected scan angle of π/3. The result of the process is shown in
Figure 5.11.
Figure 5.11: Optimization of scan angle in the presence of coupling.
The realized scan for this simulation is 25.730 , which is lower than the 300 predicted by the theoretical equations. This result tells us that even though the natural
coupling of the array can be minimized, it still plays a limiting factor in the performance of coupled-oscillator arrays. It should also be noted that the closer the phase
dynamics reaches the theoretical limit, the slower the array becomes as reaching the
steady-state takes longer.
The presented optimization process is generic and can be used with any design
where a network parameter or a radiation characteristic is desired with the added
benefit of a time-domain simulation to solve the required differential equations. Appendix B provides the basic Matlab script to accomplish such task. This process will
be used as part of the design and implementation of any subsequent array design in
this chapter.
96
5.3.2
Non-reciprocal Coupling Networks
Maintaining the mode of operation necessary to operate a coupled-oscillator system
as a phased array requires that both signal amplitude and phase of each element is not
vastly different. This makes COA systems not very attractive for applications that require wide scanning range. As stated multiple times in this dissertation, typical COA
systems have a fundamental phase gradient limitation of π/2 under either injectionlocking or mutual/reciprocal coupling. To put this in perspective, a half-wavelength
spaced linear array can only generate a maximum scan angle of approximately ±300 .
This theoretical limit is generous since it is was shown earlier that the optimized scan
angle is closer to 25.730 . In most published works, it is also assumed that in a weakly
coupled/injected system, the amplitude dynamics can be ignored. In practice, noise
and other unaccounted coupling effects will substantially reduce that theoretical 300
figure.
Overcoming the scanning range limitation is an area of active research. Heath [60]
provided a description of a fully nonreciprocal coupling scheme necessary for removing the scanning limitation. This beam-steering method is based on a bidirectional
network that uses two explementary coupling phases. The set of coupled phases for
an N element 1-D array are defined as:


 Φp,p+1 = θ

 Φp+1,p = −θ
,
(5.13)
for p = 1, 2, . . . , N −1. Simply put, this describes a coupling network where the phase
shift going from oscillator p to p + 1 is the opposite of the phase shift from p + 1 to
p. The coupling magnitudes, on the other hand, are assumed to be the same. It was
97
shown that this phase coupling scheme is stable for all phase gradients. Furthermore,
it was shown that the array stability is no longer a function of the scanning angle.
This property allows us to obtain a more robust system in the presence of noise and
amplitude variations. For typical coupling, the arrays stability gradually decreases
as a function of the scan angle. In terms of performance, a less stable array means
that the dynamics becomes slower and that steady state takes longer to be reached.
Implementing working nonreciprocal coupling networks is a challenge. Since most
of the practical implementations of nonreciprocal networks are based on bulky ferrite
devices that are incompatible with smaller microwave monolithic circuits, Martinez
and colleagues [112] proposed a coupling network based on microwave transistor amplifiers that are fully capable of being integrated and compatible with various array
architectures at the mm-wave range. Half of the proposed circuit is shown in Figure
5.12. The gain block is needed to generate the non-reciprocity. The variable transmission lines (VTL) are used to dial the needed phase gradient to be applied to the
array.
Figure 5.12: Architecture of half of the nonreciprocal coupling network [112].
98
The major issue with the nonreciprocal circuit in Figure 5.12 is its low efficiency
and large size. In order to make the coupling circuit stable such that no oscillations
can be generated on their own, a set of resistors are put at the end of the gain
block. For more efficiency, the resistors can be removed as long as the following
two concepts are implemented. First, the resistors’ main function is to reduce the
gain of the transistor such that no oscillation is produced by the coupling circuit
itself; therefore, they are needed when the coupling signal from the gain block is too
strong. However, when an extremely small amount of power is allowed to couple
to the nonreciprocal circuit, the transistors can be fully used as amplifiers in order
to get the coupling signal back to an arbitrary level that does not make the circuit
potentially unstable. In that case, the resistors are not needed. Secondly, the nonlinearity of the the coupling circuit can be embraced so that the instability can be
used to complement the generation of the oscillation signal at the desired frequency
or for coupling reasons at one of its harmonic frequency. It is even possible to already
imagine a case where the oscillation is only due to the presence of the nonreciprocal
coupling circuit alone. In this case, a few more design steps are required in order
to make sure that the oscillator, as an entire circuit, is still working even with the
change of phase shift that might be implemented for beam-steering purposes.
5.4
Implementation of a Fully Nonreciprocal Coupled-Oscillator
Array
As it was stated earlier, a fully non-reciprocal coupling circuit, inserted between
adjacent oscillators in a linear array, removes the beam-scanning limitation. Reference [112] was the first work in COA-related literature that successfully implemented
99
a non-reciprocal coupling network using solid-state devices that can be used to accomplish such task. To show that such circuits can improve the scanning range of a
coupled-oscillator system, the following three-element array in Figure 5.13 is implemented.
Figure 5.13: Architecture of a fully nonreciprocal array.
To show that a large scan angle can be achieved, a phase gradient of 1500 is chosen
for the value of θ. Note that the theoretical maximum for a reciprocally coupled array
is 900 . Due to a lack of low-loss phase shifters available, the coupling network has
no ability to change its phasing. Due to the large number of transistors needed
to implement this array, it was decided to pick the minimum number of elements
necessary for such array. In order to replicate any nonlinear behavior observed in
coupled-oscillator systems, the minimum number of elements needed is three, thus
requiring at least seven2 transistors.
To summarize the design steps, the two coupling circuits are initially designed together making sure that they do not generate any stable oscillation at the frequency
of operation (5 GHz) or anywhere else based on the nonlinear model of the transistor
2
In general, for a fully nonreciprocal linear array, the minimum number of transistors is 3N − 2,
where N is the number of antenna elements
100
(NE3201SO1). The transistors are basically operated in the small-signal regime since
most of the power is designed to be directed to the antennas. Since there are only
three elements, the two coupling networks are required to have the same physical
dimensions due to symmetry. Once the coupling circuits display the desired properties, the three antennas are added and the power ratio is selected by increasing or
decreasing the input impedance of the antenna (moving pin location). For this design, it was desired that that between 10 and 15% of the generated power is coupled
while the rest is sent to the antennas. Once the pin location is set, based on the
power distribution ratio, the oscillators are designed as standalone oscillators (with
no coupling) using the parametrization techniques introduced in Chapter 2. Once
oscillations are achieved and stable at the desired frequency of operation, coupling
is introduced and the system is verified to oscillation. This verification process is
repeatedly done until a stable oscillation is found with the appropriate phasing. The
design of the coupling circuit is done using the co-simulation optimization process
shown earlier with FEKO/MATLAB, whereas the design of the main oscillators is
done using a harmonic balance simulator and the stability verification is carried out
using a transient or time-domain methods in ADS [90].
Figure 5.14 shows the final layout and the fabricated array. As done previously,
the antennas and the rest of the nonlinear circuits are built on two separate substrates
(with common ground planes) in order to help with isolation and cross-polarization
of the radiation pattern.
To verify the proper operation of the array, a few measurements were carried
out. The first measurement dealt with recording the spectrum of the radiated signal.
Figure 5.15 shows the recorded spectrum with a horn antenna placed about 2 m
away from the array without any smoothing applied. The spectrum analyzer used
was Agilent E4407B with a resolution bandwidth of 0.5 MHz and 0-dB attenuation.
101
(a) Top View: Patch antennas
(b) Fabricated antennas
(c) Bottom View: Layout of front-end circuits
(d) Actual front-end circuits
Figure 5.14: Layout and fabricated array showing active components and bias lines .
102
Each FET transistor was DC-biased to VDC =2 V and IDC =10 mA. The measured
peak power occurs at an oscillation frequency of 4.96 GHz. The spectrum seems to
contain a significant amount of RF power in the sidebands, indicating a high phase
noise.
Figure 5.15: Measured power spectrum of the array, 30λ0 away from antennas.
To gather more performance on noise, the phase noise was also measured with
the array put closer to the horn for more input power. A setup using a signal source
analyzer (E5052B) with microwave downconverter (E5053A) provided a phase noise
measurement of -96 dBc/Hz at an offset of 10 MHz as shown in Figure 5.16. At 1
MHz, the phase noise is about -85 dBc/Hz. As expected, the phase noise is somewhat
high for this array to be used as a current communication system as phase noise values
below -110 dBc/Hz are expected at this offset [113, 114]. For comparison, the phase
noise obtained by York and colleagues [4] (pp. 170) for a five-element array was -68
103
Figure 5.16: Spectrum of phase fluctuations of the array
dBc/Hz at 100 KHz, while for this design the phase noise is -64 dBc/Hz. The phase
noise in general is expected to decrease as the number of array elements is increased.
One common way to lower phase noise is to use a PLL or a dedicated resonator/cavity.
In this case, other authors have reported phase noise lower than -100 dBc/Hz at 1
MHz offset [115, 116]. Potential sources of added noise could have been caused by
the large number of transistors. Moreover, the array was optimized to generate the
required phasing and not to display a higher Q.
Finally, the radiation pattern was measured rudimentary (by hand) due to the
difficulty of using the circuit with all its required DC voltage bias sources on the
setup of the anechoic chamber. Figure 5.17 shows the measured points with the
simulated expectations. As noticed, the array is radiating off broadside (angle=0),
and the results are satisfactory. The location of the maximum radiation is at 37.40 .
This measurement shows that the inter-element phasing is approximately 1410 , thus
104
Figure 5.17: Comparison of measured radiation pattern to expected simulated results.
realizing the highest phasing reported to date for coupled-oscillator arrays. A work
by Liao and York [117] achieved a scan angle of 600 which corresponded to a phasing
of about 400 since the spacing was 0.26λ0 .
The estimated radiated power is 11.78 dBm, resulting in a DC-to-RF efficiency of
10.13%. The computer EIRP is 16.34 dBm. It is clear that the efficiency of the array
is decreased compared to the efficiency of a single oscillator because of the addition of
the four coupling transistors. The initial goal of generating a phase gradient of 1500
was partially achieved with a measured phasing of 1410 , thus only missing the target
by a 6% error; however, more efforts could have been undertaken to also reduce the
phase noise.
105
CHAPTER 6
COUPLED-OSCILLATOR BASED APPLICATIONS AND
ARRAY ARCHITECTURES
6.1
Introduction
Coupled-oscillator systems derive their specific properties from the mutual interactions of independent oscillators. For that reason, each oscillator has its own characteristics even though at the design level, every effort is taken to create practically
identical oscillators. However intrinsic differences exist between oscillators due to various causes such as fabrication process variation of passive and active devices, human
errors and asymmetric coupling from topological placement. Even if process variation
and human inputs are effectively accounted for, the coupling architecture/topology
might create an imbalance in loading. For instance, in linear arrays, the end elements
are subject to a different type of coupling environment than the other elements. This
change in loading environment actively creates a difference in natural frequencies for
each oscillator in an array. To mitigate all of the mentioned problems, fabricated
arrays have been manually tuned to correct this imbalance.
106
6.2
Array Architecture
To partially resolve the issues mentioned earlier, an architecture containing a phaselocked loop is proposed. Unlike various architectures found in available literature, an
alternative architecture is proposed as depicted in Figure 6.1. The most attractive
benefit of this architecture is its simplicity. In other words, N oscillators are only
controlled by one phase-locked loop (PLL) instead of many as seen in other works
[72, 73]. The reduction in the number of loops and other controller elements is due
to a need to reduce cost, size, weight and complexity. This array is also expected
Figure 6.1: COA phase-locked loop based architecture.
to be simpler to analyze and operate without losing any performance metric. The
properties of the control loop of this proposed architecture can be analyzed by first
considering that the system is noiseless. Assuming that the reference oscillator has a
signal of the form
107
1
a(t) = Aa sin(ωa t + φa ),
2
(6.1)
and that the signal coming from the output of the frequency divider is
b(t) = B cos
�ω
s
N
�
t + φb ,
(6.2)
the output of the mixer is therefore given by
c(t) = Aa BKm
�
�
�
�
��
ωs
ωs
sin (ωa − )t + φa − φb + sin (ωa + )t + φa + φb
N
N
(6.3)
where Km is the mixer gain. The output of the mixer is filtered out by the low-pass
filter (LPF) resulting in the high-frequency term to be dropped, meaning that
d(t) ≈ Aa BKm
�
�
��
ωs
sin (ωa − )t + φa − φb .
N
(6.4)
In typical PLL, the signal d(t) is used to control the oscillation frequency of the
oscillator, usually through a varactor diode, resulting in voltage-controlled oscillator
(VCO). The natural frequency of the controlled oscillator is therefore given by
ω(t) = ω0 + Kv d(t),
(6.5)
where Kv is the VCO gain [118, 119], and ω0 is the free-running frequency of the
oscillator.
This PLL feedback loop forces the controlled oscillator to change its oscillation
frequency to ωs = N ωa . At this stage, d(t) becomes a quasi-DC signal of value
Aa BKm sin(φa − φb ).
When dealing with oscillators with identical natural frequencies, the benefits of
this architecture can be noticeable. The following simulations show a comparison
108
between a typical COA and the current PLL-based COA, both having five oscillators coupled similarly, with the only difference being the addition of the PLL. The
oscillators are considered to be weakly coupled, therefore the array’s properties can
be assumed to be uniform. The coupling strength is �=0.1, the quality factor Q=10,
the coupling phase Φ is zero, and the natural frequencies are all equal to 10 GHz.
The PLL parameters for this simulation are Aa =1, B=0.1, N=1, φa = φb =0, Km =5,
Kv =0.05.
Figure 6.2 shows the difference in settling time—amount of time taken by the
array dynamics to reach its steady state—between a typical 5-element COA and the
PLL-based COA from the previous figure. In this case, the controller module sends
the same correcting signal to all oscillators since the natural frequencies are identical.
As noticed, the PLL-based COA settles to its steady state faster (10 ns) than the
typical COA with the same initial conditions (20 ns). Inserting the PLL circuit has
shown to double the speed of the typical array, showing the possible improvement.
However, when slightly different oscillators are used, as practically expected, the
same approach cannot be used because the oscillators have intrinsically different
natural frequencies. Feeding the control signal coming from a single loop to the
entire array will create a conflict between the oscillators and the final steady state
might be unpredictable since each oscillator is controlled by a signal that might not
match its current state.
To help mitigate this issue, a controller stage, with more controlling capabilities,
is used. The new controller introduces the possibility of independently managing the
free-running frequency of each oscillator based on the signal d(t). However, resolving
all the required signals that have to be fed to each oscillator requires a seeding or
calibration mechanism.
109
(a)
(b)
Figure 6.2: Performance comparison between (a) typical COA and (b) PLL-based
COA.
To show how this architecture treats the presence of multiple free-running frequencies on the array, the following simulation is carried out with the initial controller.
Let’s assume that the five-element array used earlier has the following natural frequency distribution [f0 ] in GHz :[f0 ]=[10.0262, 9.9956, 9.9519, 10.0321, 9.9945]. The
distribution is centered around a mean of 10 GHz. The PLL parameters remain the
same as before. Figure 6.3 shows a comparison of the dynamics of a typical fiveelement array to that of the PLL-based COA with the same initial conditions. As
expected, the PLL-based array converges faster than the typical COA; however, the
same steady state is reached. This shows that the speed, and therefore the stability
of the array is improved, but it does not change the stability range of the array itself
as a fully nonreciprocal coupling circuit does. In other words, if a typical COA is
unstable because of a large difference in natural frequencies or large coupling phases,
110
(a)
(b)
Figure 6.3: Phase dynamics of five-element array. (a) Typical COA. (b) PLL-based
COA.
having this PLL structure will not drastically change this stability/instability state.
This proposed architecture only increases the array stability margin, which results
in higher speeds and higher noise cancelling properties, but the stability space is
unchanged.
In terms of general stability, it has been already shown by Heath [60] that for
reciprocally coupled linear array, the maximum Lyapunov exponent is
�
π�
λm = −2� cos θ 1 − cos
,
N
(6.6)
where � is the coupling strength, θ is the uniform phase gradient along the array, and
N the number of oscillator. The Lyapunov exponent is a measure of the stability of a
dynamical system. The more negative its value, the more stable the system becomes.
111
The addition of the single PLL loop does not change the intrinsic stability of the
array; however, it results in an effective increase of the coupling strength, making the
exponent more negative, thus the array faster.
However, giving this array topology the ability to control each oscillator independently results in the array being to expand its stability domain. In other words,
the array is able to move from different modes of operation by dialing the natural
frequencies of each oscillator.
6.3
Array Calibration and Pattern Shaping
As mentioned earlier, building independent oscillators with truly identical free-running
frequencies is inherently unfeasible because of the inaccuracies in fabrication techniques and fluctuations in electronic devices. For high-frequency oscillators, these
small errors translate in various free-running frequencies, which in turn create a distorted radiation pattern even when all oscillators synchronize to the same frequency.
The distortion is caused by the phase gradient not being constant across the array.
Moreover, as demonstrated earlier, feeding all oscillators with the same PLL feedback loop is problematic because of the nonuniform phase gradient create at steady
state. To address the issues emanating from the architecture of Figure 6.1, the following changes are added as depicted in Figure 6.4. This change is necessary so that the
array seeds or calibrates itself to the small differences in natural frequencies of each
oscillator. This is another example of how nonlinear systems differ from linear ones.
As a mean to accomplish this calibration, the array pattern is used as controlling
mechanism.
An antenna is placed in the far-field of the array at the position where the maximum radiation is desired to be, typically at broadside. This error correction scheme
is mainly concerned about changing the natural frequency of each oscillator through
112
Figure 6.4: COA architecture with self-calibration mechanism.
113
a gradient optimization process or any other implemented algorithm. Put differently,
at each step, the radiated power is measured and fed to the controller/optimizer,
which in turn slightly adjusts each oscillator’s natural frequency as to maximize the
power observed at the testing antenna location. This is a simple setup that can be
expanded to include more antennas and implement more accurate calibrations. This
calibration setup can also be used to shape the radiation pattern of the array if one
or more antennas are added. For instance, a radiation null can be requested at the
locations of the test antennas if desired.
Simulations show that this calibration technique is viable for COAs as shown in
Figure 6.5. It can be seen that the array slowly calibrates itself to an ideal broadside
radiation after a few iterations for a 10-element array. Using this process, the calibration is completed when no changes in voltages are recorded based on the achievement
of the optimization goals. These individual voltages are maintained throughout the
subsequent operation of the array. A major benefit of this system is that this calibration is done by the array itself without any human intervention.
6.4
Phase-Amplifying Arrays
As mentioned in previous chapters, microwave coupled oscillators with reciprocal coupling networks have been plagued with limited scanning ranges. Using nonreciprocal
coupling networks has shown that this limitation can be lifted. However, nonreciprocal coupling circuits add complexity, additional cost and further study and analysis.
Recently, it was shown that the hearing system of a fly (Omnia Ochracea) [120, 121]
can serve as a model to amplify the phase difference between two closely spaced small
antennas [122]. In this work, a pair of passively coupled electrically small antennas
are used to amplify the phase difference between the received voltage signals. The
author suggests that this process can be used to build phase sensitive receiving array
114
(a)
(b)
(c)
Figure 6.5: Array factor of 10 isotropic antennas. Solid line is COA array factor and
dashed line is ideal broadside array factor. a) Initial random frequency
distribution. b) After a few iterations. c) After many more iterations,
the COA array factor approaches its ideal state, thus accomplishing calibration in the process.
115
(a)
(b)
Figure 6.6: (a) Mechanical model of hearing system of Omnia Ochracea. (b) Corresponding electric model [122].
for direction-finding applications. Figure 6.6 shows the mechanical and corresponding
electric model.
In this dissertation, this phase-boosting process is reversed to cater to transmitting
systems and expanded to more than two antennas. Moreover, this process is applied
to electrically large antennas and was used to ease the phase gradient requirement
for coupled-oscillator arrays. To understand this phase-amplifying mechanism, this
simple study is undertaken. The impressed or receiving voltages are replaced by the
voltages generated by the oscillators in the transmitting case. The input is the phase
difference between the oscillator voltages V1 and V2 while the output is the phase
difference between the currents I1 and I2 as shown in Figure 6.6. To show the desired
effect, the lumped elements are replaced by generic reactive loads as in Figure 6.7.
This model can be used to represent a two-element array using single mode antennas1 .
In this case, the phasing of the port currents corresponds to the individual radiating
phasing as well.
The reactive loads are used so that no loss of radiative power is consumed by
1
small dipoles, thin slots, etc
116
Figure 6.7: Generic representation of coupling loads between oscillators.
the coupling circuits. These loads are modeled here as Xi = Z0 tan θi , with θi ∈
[−π/2, π/2]. The output is the phase difference between the two branch currents
φI = ∠I2 − ∠I1 ,
(6.7)
meanwhile, the input is the initial phase difference between the oscillator voltages:
φV = ∠V2 − ∠V1 = δφ.
(6.8)
To get a better sense of the properties of such circuits, let’s define two quantities,
namely the current amplification factor AF
�
�
� dφI �
�
�,
AF = �
dφV �
(6.9)
and the current magnitude difference factor DF
� �
� I1 �
DF = 20 log �� ��
I2
117
(6.10)
By sweeping the loads, the following current-based plots are generated for an
initial voltage phase difference of 100 (Figure 6.8) for Z0 = R1 = R2 = 50 Ω and
θ1 = 610 and θ2 = 110 .
Figure 6.8: Phase difference and amplification factor of currents I2 and I1 .
Figure 6.9 shows that it is possible to generate any arbitrary current phase gradient
by choosing the appropriate set of reactive loads. However, it is clearly visible that
tradeoffs exit: the more phase amplification, the more the signal amplitude difference
in various cases, but an optimal point can be picked for a desired design. The other
half on the plot is not shown because there is symmetry.
To illustrate the effect of such coupling circuits on actual coupled oscillators, two
microstrip patch antennas will be used. Although the equivalent electric circuits for
118
Figure 6.9: Plots of AF (dotted) and DF (solid) generated by sweeping the reactive
loads as shown in Figure 6.7.
119
coupled patches are more complex than the depiction in Figure 6.7, it is nevertheless noticed that some phase-boosting effects are observed. Figure 6.10 shows two
microstrip patch antennas replacing the resistors of Figure 6.7. A reactive coupling
circuit is inserted between the two coupling ports at the top while a small and constant phase difference is maintained between feeding ports #1 and #2. After trying a
Figure 6.10: Two microstrip patch antennas coupled through a phase-boosting network.
few candidate circuits with φV =100 , and monitoring the output phase, the following
T-network was selected to be inserted between the two patches due to the observed
phase amplification. As shown in Figure 6.11, the coupling circuit comprises two
capacitors of value 0.2 pF and a shunt inductor of 1.96 nH. Figure 6.11 also shows
the overall phase-boosting performance when the voltage phasing is fully swept. It
is seen that the maximum amplification occurs at φV ≈ 150 , and that the amplitude
difference is highest around φV = 200 . These values can be tuned to obtain a higher
amplification factor, but as mentioned earlier, a significant difference in current amplitudes occurs if AF is made too high.
120
Figure 6.11: Simulation results for two coupled microstrip patch antennas.
In terms of the direction of the maximum radiation of the antenna array, it is
observed that without the coupling circuit, about 1500 of phase difference is needed
to steer the main beam to 300 off broadside. With the phase-boosting circuit present,
only 900 are needed to accomplish approximately the same result with the added
benefit of smaller sidelobes. Figure 6.12 shows the effect of placing the coupling
circuit between the antennas. It is clear that an improvement in scanning angle is
gained when using such circuits.
To show a more complete performance, Figure 6.13 shows a practical implementation of a two-element array with the patch antennas on one side of the substrate
and the remaining functional elements on the other side. This arrangement is used
to isolate the antennas from the rest of the circuit, thus barely affecting the radiation
pattern and simplifying the initial analysis.
The system shown in Figure 6.13 was designed to work at 10 GHz on a 31mil substrate with a dielectric constant of 2.2. The FET transistors used to make
121
Figure 6.12: Total array gain with and without the coupling circuit.
the individual active antenna are CEL’s NE3201SO1. The design methodology to
design each AIA follows the techniques developed in Chapter 2. By monitoring the
current phase at each feeding port of the antennas, Figure 6.14 shows that a phaseboosting mechanism is generated, as expected. In Figure 6.14(a), three curves are
shown in order to compare various performance. In summary, a 100 phase-boosted
voltage phase shift nearly generates the same scan angle as a non-coupled array
with a 650 phase shift. In Figure 6.14(b), the locations of the main beam from the
array factor (solid line) and the non-coupled array (marked line) are plotted for an
increasingly large voltage phase shift between the two oscillators. By inserting the
phase-amplifying circuit, it is possible to generate a much larger scan angle than the
one permitted by the array factor when the phase shift is small.
As noticed earlier, increasing the amplification factor AF although helps with
small phase gradients, there is however a tradeoff in terms of amplitude differences
122
(a) Top view
(b) Bottom view
Figure 6.13: Layout of two-element array. (a) Top view made of two patch antennas.
(b) Bottom view showing necessary circuitry. All dimensions in mm.
123
(a)
(b)
Figure 6.14: (a) Comparison of phase-boosting effect to normal array. (b) Snapshot
of the scan angle of phase-amplifying array (red mark) compared to scan
angle of array factor (solid line) and scan angle of normal patch array
(marked line).
124
between the oscillators. Another drawback is that a high AF would mean that small
changes in input phasing could result in substantial changes in output phase, creating potential issues with system modulation and eventually beam squinting. Another
disadvantage is that this transmitting system works well with two antennas at a time.
Attempts to expand the transmitting type to multiple antennas have proven to be
problematic because each antenna pair would need to be phase matched to its adjacent pairs. Coupled-oscillators array inherently solve this problem because each
oscillator is phase-locked to one other. Unlike a uniform phased array, this type of
phase-booting system will have varying current amplitudes when scanning since each
pair will inherently have a non-zero DF . Figure 6.15 shows the resulting architecture necessary to accomplish such characteristics. As expected an even number of
oscillators are needed.
Figure 6.15: Phase-amplifying architecture of coupled-oscillator arrays.
Following a simulation, Figure 6.16 shows the array factor of a 10-element array for
a uniform array and the corresponding array factor its phase-amplified counterpart.
125
As observed, the phase-boosted array present larger sidelobe level, but the main beam
remains unaffected.
Figure 6.16: Comparison of array factors of a 10-element array (uniform and phaseamplified).
Even though this technique does not solve the phase limitation of reciprocally coupled oscillators, it helps the array operation by accomplishing the same scan angle
by just producing a smaller amount of phase, which ultimately helps the array stability by not being pushed to the brink of instability when beam-scanning is needed.
In other applications, the phase-boosting mechanism can also be used to produce a
“phased-reducing” system in which AF is made smaller than unity. This process can
be used to control highly varying phase differences in a system.
126
CHAPTER 7
CONCLUSION
7.1
Summary
This dissertation discussed the promising technology of coupled microwave oscillators
made from independent self-oscillating active integrated antennas. From the problem
statement, it was urged that developing functional COAs would be cost-effective and
highly valuable for high-frequency phased array systems, mainly due to their higher
efficiency, smaller size, their simplified control structure and high integrability.
In the first chapter, a new set of techniques for analyzing and designing active
integrated antennas was presented. The techniques relied on using the convex-like
properties of resonant antennas’ input immittance in order to localize stable oscillation design parameters at small and large-signal steady-state operation. The potential
of the overall method or idea was not limited to active antennas as it could be applicable to the analysis and design of generic negative-resistance microwave oscillators
displaying similar characteristics. The analysis and design method was illustrated by
a simulation of an AIA at 13 GHz, and an experimental agreement with simulation
was conducted at 5 GHz.
Secondly, a mathematical description for accurately computing the dynamics of
an array of nonlinear coupled oscillators through a linear network was presented with
focus on solely using measurable network parameters such the impedance/admittance
127
matrix. The derived equations are shown to be expandable and written for each harmonic components although the focus of Chapter 3 was on presenting the equations
at the fundamental tone. While the targeted application for this dissertation is active integrated antennas based phased arrays, the equations can be useful in many
fields dealing with coupled nonlinear oscillators arranged in any topology. Numerical
simulations showed that a higher degree of accuracy is achieved when compared to
other equations provided for coupled microwave oscillators in related literature.
Using the newly generated equations from Chapter 3, a set of design considerations were examined closely. At first, the properties of most common types of coupling
circuits were studied with an emphasis on their coupling strength and phasing properties. It was found that common transmission-line based coupling networks display
broadband and extremely narrowband behavior depending on the number of line sections and location of the resistive loads. It was also shown that the coupling phase
can be a complex function of many variables that may only reduce to the expected
electrical length of the transmission line in some particular cases. Secondly, due to
the required analysis breadth of COA systems, a generic optimization method was
developed to sequentially link multiple analysis tools to perform full-wave EM calculations (FEKO) and differential equations calculations under the control of a single
algorithm (MATLAB). This provided an efficient optimization process for the design
of COAs. Thirdly, the implementation of a 10-GHz transistor-based non-reciprocal
coupling circuit was presented. The circuit’s role was to generate a set of explementary phases between two adjacent oscillators—this is a needed condition to remove
the scanning range limitation in a linear array. Lastly, a three-element array using a
compacted version of the previous non-reciprocal coupling circuit was designed and
fabricated at 5 GHz. The array produced a phase gradient of about 1410 , without
128
loss of stability, thus realizing the highest achieved phasing in current literature. Although the phase noise of the array could have been improved by the design method
involved, the design goal of the array was to primarily display the correct phase
differences between the oscillators.
Finally, a few array architectures were presented for the purpose of increasing
stability, speed, and usability in a communication or tracking system. The first
architecture made use of a single PLL for an arbitrarily large array. It was shown
that for this architecture to be viable, it required the independent control of each
oscillator through a controller. The seeding of such controller was subsequently done
by a calibration mechanism that could be applied by the array itself. The second
mechanism used to improve a typical COA’s scanning range was the implementation
of phase-amplifying technique based on the auditory system of an insect. In essence,
by using a well-designed coupling circuit, small phase differences at the oscillators
ports (voltages) are amplified into larger current phase differences at the antennas.
Even though this technique does not solve the phase limitation of reciprocally coupled
oscillators, it helps the array operation by accomplishing the same scan angle by
producing a smaller amount of phase, which ultimately helps the array stability by
not being pushed to the brink of instability when significant beam-steering is desired.
7.2
7.2.1
Future Work
Harmonically Loaded AIA and COAs
It’s been known that harmonically loading an oscillator can have positive effect in
increasing its output power level or the DC-to-RF power efficiency [24]. However,
this approach was not applied in the techniques presented in Chapter 3. It would be
129
worthwhile to include the effects of harmonic loading into the design and optimization
process of a single AIA.
In addition to that, the array equations presented in Chapter 4 could be used to
generalize the techniques of Chapter 3 into a more involved set of techniques that are
applied to the design of a COA. At this moment, each AIA is optimized individually,
thus potentially resulting in a sub-optimal performance. In general, a design process
that would harmonically load to the array to either optimize output power or power
efficiency would be highly beneficial. In summary, the parametric design techniques
presented in Chapter 3, the equations of Chapter 4, coupled with harmonic loading
can all be applied to generate a generic design method for coupled-oscillator systems.
7.2.2
Non-Foster Loads for Phase-Amplifying Arrays
The use of non-Foster circuits primarily stems from their applications in impedance
matching, loading of antennas, transmission lines and waveguides since they are not
restricted by the well-known gain-bandwidth product [123]. Consequently, it is of
importance to study the effect of using non-Foster loads in the phase-amplifying
circuits located between the oscillators. It is expected that some performance criteria
could be improved. Preliminary investigation has suggested that broadband operation
of the circuit could be achieved, but more importantly, there is evidence that similar
amplification factor (AF ) could be achieved with smaller values of reactance while
keeping DF constant or smaller.
130
Appendix A
COUPLING PROPERTIES OF RESISTIVE LINES
The network parameters of a couple of commonly used coupling circuits based on one
or two sections of transmission line are examined. Figure A.1 depicts the first circuit.
Figure A.1: Coupling circuit # 1.
The ABCD transmission matrix of the circuit above is given by [124]:






jZ0 sin θ   1 RC 
 A B   1 RC   cos θ

=



C D
0 1
jY0 sin θ
cos θ
0 1
(A.1)
which corresponds to




RC + jZ0 sin θ 
 A B   cos θ + jY0 RC sin θ

=

C D
jY0 sin θ
cos θ + jY0 RC sin θ
131
(A.2)
However, since the circuit is reciprocal, it leads to

A


Y
11 = Y22 =


B

(A.3)




 Y21 = Y12 = −1
B
Computing the coupling admittance, we have
[Y ] =
Z0
j sin θ
��
1
c
cos θ + j R
Z0


Rc
−1
cos θ + j Z0 sin θ





�
 (A.4)
�2



sin θ − 1 
Rc
−1
cos θ + j
sin θ
Z0
For for RC = Z0 , Equation A.4 reduces to
Y0
[Y ] =
2






1
−e−jθ
−e
−jθ
1






(A.5)
Applying a similar analysis, the second circuit showed in Figure A.2, the ABCD
matrix is






θ
θ
θ
θ
jZ0 sin   1 2RC   cos
jZ0 sin 
 A B   cos 2
2 
2
2

=


θ
θ
θ
θ  (A.6)
C D
jY0 sin
cos
0
1
jY0 sin
cos
2
2
2
2
This matrix can be simplified to:
132
Figure A.2: Coupling circuit # 2.




cos θ + jY0 RC sin θ
B
 A B  


=

C D
Y02 RC (cos θ − 1) + jY0 sin θ cos θ + jY0 RC sin θ
(A.7)
The coupling admittance is given by


Rc
−1

cos θ + j Z0 sin θ


1


[Y ] =




RC (1 + cos θ) + jZ0 sin θ


Rc
−1
cos θ + j
sin θ
Z0
(A.8)
For for RC = Z0 , Equation A.8 reduces to
[Y ] =

jθ
1
e

jθ
Z0 (1 + e ) 1


j θ2
−j θ2

1
e 
1
e
=

θ 
jθ
−j θ2
j θ2
2Z
cos
0
2
e
e
e
133
(A.9)
Appendix B
MATLAB/FEKO CO-SIMULATION COMMANDS
This appendix provides an example MATLAB program that controls a FEKO design
file to optimize the reflection coefficient of a microstrip antenna. A few options will
need to be modified to fit the installations on your machine, mainly, the location or
path of the programs and you will also have to provide your own .cfx file. This demo
was run with FEKO 5.5 although versions 6.0 and 6.1 were also tested to work.
Filename: runOptimzation.m
global TRACKING hAx
TRACKING=[];
figure;
hAx=axes;
%initial values of variables (note the order)
x0=[3;18];
%Matlab search function. This can be manually programmed. With the optimisation
toolbox, you might use [fmincon] for constrained multivariable minimization
instead of [fminsearch]
[X,FVAL,EXITFLAG] = fminsearch(@runYourFekoCommands,x0);
Filename: runYourFekoCommands.m
function T = runYourFekoCommands(X)
%This function runs FEKO in batch mode.
%%CHANGE/ADD PROPERTIES BELOW
filename=’patch1’;
numberOfPorts=1;
fekopath=’"\Program Files\FEKO\5.5\bin\’; %change this path if needed
134
%Assign your variables as in the design itself
f=X(1);
l=X(2);
variableArray=[’ -#f=’ num2str(f) ’ -#l=’ num2str(l)] ; %notice the white
space
global TRACKING hAx
FEKO DISPLAY=true; %or false
PLOTTING=true;
%%STOP MAKING CHANGES HERE
%Update variables
path3=[fekoPath ’cadfeko batch" ’];
dispFeko([’Running Cadfeko batch: updating Variables:’ variableArray ],
FEKO DISPLAY)
[a,b]=system(sprintf(’%s’,path3, filename, variableArray));
path=[fekoPath ’prefeko" ’];
argument=’ > output.txt’;
%Running Prefeko
path2=[fekoPath ’runfeko" ’];
argument2=’ > FekoOutput.txt’;
dispFeko(’running PREFEKO: meshing’,FEKO DISPLAY)
system(sprintf(’%s’,path, filename, argument));
pause(1)
%Running FEKO
dispFeko(’running FEKO: field calculation’,FEKO DISPLAY);
system(sprintf(’%s’,path2, filename, argument2));
pause(0.5); %wait for feko mkl to finish or write a wait loop
%Reading SNP File
dispFeko(’Reading output file’,FEKO DISPLAY);
% Code your own S parameter reading function for (’patch1.s1p’)
SM=readSNPDataFromFile([filename ’.s’ num2str(numberOfPorts) ’p’]);
%Generate your own cost function here based on the S-parameters
T=abs(SM.matrixData);
%plotting
TRACKING=[TRACKING T];
135
if PLOTTING
plot(hAx,TRACKING);
xlabel(hAx, ’Number of Iterations’);
ylabel(hAx,’Goal’)
title(hAx,variableArray);
grid(hAx,’on’);
end
end
Filename: dispFeko.m
function dispFeko(message,yn)
if yn
disp(message);
end
end
Either the S-parameter file is read or the explicitly requested output.txt file is
read in order to define a goal function that will need to be minimized. When running
this code, the following plot should be obtained after a few iterations.
136
Figure B.1: Plot of reflection coefficient of antenna during optimization.
137
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