Contribution to the study of synchronized differential

THÈSE

Pour l'obtention du grade deDOCTEUR DE L'UNIVERSITÉ DE POITIERS

École nationale supérieure d'ingénieurs (Poitiers)Laboratoire d'informatique et d'automatique pour les systèmes - LIAS

(Diplôme National - Arrêté du 7 août 2006)

École doctorale : Sciences et ingénierie pour l'information, mathématiques - S2IMSecteur de recherche : Electronique, microélectronique et nanoélectronique

Cotutelle : Universitatea politehnica (Bucarest)

Présentée par :Mihaela-Izabela Ionita

Contribution to the study of synchronized differentialoscillators used to controm antenna arrays

Directeur(s) de Thèse :Jean-Marie Paillot, David Cordeau, Mihai Iordache

Soutenue le 18 octobre 2012 devant le jury

Jury :

Président Marina Topa Domnule Profesor, Universitatii tehnice din Cluj Napoca, Romania

Rapporteur Farid Temcamani Professeur des Universités, ENSEA de Cergy-Pontoise

Rapporteur Lucian Mandache Domnule Profesor, Universitatea din Craiova, Romania

Membre Jean-Marie Paillot Professeur des Universités, Université de Poitiers

Membre David Cordeau Maître de conférences, Université de Poitiers

Membre Mihai Iordache Domnule Profesor, Universitatii Politehnice din Bucuresti, Romania

Pour citer cette thèse :Mihaela-Izabela Ionita. Contribution to the study of synchronized differential oscillators used to controm antennaarrays [En ligne]. Thèse Electronique, microélectronique et nanoélectronique. Poitiers : Université de Poitiers,2012. Disponible sur Internet <http://theses.univ-poitiers.fr>

false

THESE en cotutelle

Pour l’obtention du Grade de

DOCTEUR DE L’UNIVERSITE DE POITIERS (ECOLE SUPERIEURE d’INGENIEURS de POITIERS)

(Diplôme National - Arrêté du 7 août 2006)

Ecole Doctorale : Sciences et Ingénierie pour l’Information

et de

DOCTOR AL UNIVERSITATEA POLITEHNICA DIN BUCURE ŞTI Facultatea INGINERIE ELECTRICĂ

Catedra de ELECTROTEHNICĂ

Secteur de Recherche : ELECTRONIQUE, MICROELECTRONIQUE ET NANOELECTRONIQUE

Présentée par :

MIHAELA-IZABELA IONITA

************************

CONTRIBUTION TO THE STUDY OF SYNCHRONIZED DIFFERENT IAL

OSCILLATORS USED TO CONTROL ANTENNA ARRAYS

************************

Directeurs de Thèse : Jean-Marie PAILLOT, Mihai IORDACHE Co-directeur de Thèse : David CORDEAU

************************

Soutenue le 18 Octobre 2012

devant la Commission d’Examen

************************

JURY

Rapporteurs : Farid TEMCAMANI Professeur à l’ENSEA de Cergy-Pontoise Lucian MANDACHE Professeur à l’Université de Craiova Examinateurs : Marina TOPA Professeur à l’Université technique de Cluj Napoca Jean-Marie PAILLOT Professeur à l’Université de Poitiers Mihai IORDACHE Professeur à l’Université Polytechnique de Bucarest David CORDEAU Maître de conférences à l’Université de Poitiers

iii

ACKNOWLEDGEMENTS

First and foremost I want to thank my advisors Mr. Jean-Marie Paillot, Mr. David

Cordeau, professors at University of Poitiers, France and Mr. Mihai Iordache, professor at

University Politehnica of Bucharest. It has been an honor to work with you as a Ph.D.

student. I appreciate all your contributions of time, ideas, and funding to make my Ph.D.

experience productive and stimulating. I have learned so much from you, from figuring out

what research is, to choosing a research agenda, to learning how to present my work. Your

constructive criticism and collaboration have been tremendous assets throughout my Ph.D.

I would also like to thank the rest of my thesis committee for their support. Mr. Farid

Temcamani, professor at University of Cergy-Pontoise, Mr. Lucian Mandache, professor at

University of Craiova and Mrs. Marina Topa, professor at Technical University of Cluj-

Napoca, which provided me invaluable advice and comments on both my research and my

future research career plans. I would also like to thank to Mrs. Lucia Dumitriu, for her

support and encouragements.

I’ve been very lucky throughout most of my life as a Ph.D student, in that I’ve been

able to concentrate mostly on my research. This is due in a large part to the gracious support

of the team from LAII (Laboratoire d’Automatique et d’Informatique Industrielle) of

University of Poitiers, France. The team has been a source of friendships as well as good

advice and collaboration. I am especially grateful to Mr. Smail Bachir and Mr. Claude

Duvanaud, professors at University of Poitiers.

I would also like to express my thanks to the head of the LAII, Mr. Gerard

Champenois, for accepting me as Ph. D student in his laboratory and for the funding sources

that made my Ph.D. work possible in France.

I gratefully acknowledge the funding sources from University Politehnica of

Bucharest and I would also like to express my thanks to prof.dr.ing. Ecaterina Andronescu.

I’ve been fortunate to have a great group of friends that became a part of my life. Not

only are you the people I can discuss my research with and goof off with, but also you are

confidants who I can discuss my troubles with and who stand by me through thick and thin.

This, I believe, is the key to getting through a Ph.D. program – having good friends to have

fun with and complain to.

iv

Finally, I would like to dedicate this work to my family: my parents and my sister.

Without your unending support and love from childhood to now, I never would have made it

through this process or any of the tough times in my life. Thank you.

v

TABLE OF CONTENT

Introduction ………………………………………………………….. 1

Chapter I – Coupled-Oscillator Arrays – Application.………….. 4

1.1. Introduction...…………………………………………………... 5

1.2. Oscillator principle……...……………………………………... 6

1.2.1. The sinusoidal oscillator………………………….. 9

1.2.1.1. The RC oscillator…………………………... 10

1.2.1.2. The LC oscillator…………………………... 12

1.2.1.3. Crystal oscillator……………………........... 13

1.2.1.4. The Armstrong oscillator………………….. 14

1.2.1.5. The Hartley oscillator……………………… 15

1.2.1.6. The Colpitts oscillator……………………... 16

1.2.1.7. The Pierce oscillator……………………….. 16

1.2.1.8. The differential oscillator………………….. 17

1.2.1.9. The Van der Pol oscillator…....................... 19

1.3. State of the art of coupled oscillators theory………………….. 20

1.4. Applications: Beamsteering of antenna arrays………………... 22

1.4.1. Antenna arrays……………………………………. 23

1.4.1.1. Uniform linear network……………............ 24

1.4.1.2. Controlling the shape of the radiation pattern………………………………............ 26

1.4.1.2.1. Phase synthesis……………….. 27

1.4.1.2.2. Amplitude and phase synthesis 31

1.4.2. The control of the radiation pattern using coupled oscillators…………………………………………. 33

1.5. Conclusion……………………………………………………… 35

vi

Chapter II – Theoretical Analysis of Coupled Oscillators Applied to an Antenna Array……………………….. 36

2.1. Introduction…………………………………………………….. 37

2.2. Dynamics of two Van der Pol oscillators coupled through a resonant circuit…………………………………………………. 38

2.3. Dynamics of two Van der Pol oscillators coupled through a broad-band circuit……………………………………………… 48

2.4. New formulation of the equations describing the locked states of two Van der Pol coupled oscillators allowing a more accurate prediction of the amplitudes……………………….... 51

2.4.1. Two Van der Pol oscillators coupled through a resonant network……………………………………………. 51

2.4.2. Resistive coupling case…………………………… 60

2.5. New formulation of the equations describing the locked states of two Van der Pol oscillators allowing an easier numerical solving method…………………………………………………. 61

2.6. CAD tool “ASVAL”………………………………………….... 68

2.6.1. The objective of “ASVAL”………………………. 68

2.6.2. Variables estimation technique…………………... 71

2.6.3. Stability of synchronized states………………….. 72

2.7. The cartography of the synchronization area…………………. 74

2.8. Conclusion…………………………….……………………….. 77

Chapter III – Study and Analysis of an Array of Differential Oscillators and VCOs Coupled Through a Resistive Network………………………………………………… 79

3.1. Introduction…………………………………………………….. 80

3.2. Analysis and design of two differential oscillators coupled through a resistive network……………………………………. 81

3.2.1. RLC differential oscillator schematic…………. 81

3.2.2. The modeling of the differential oscillator as a 84

vii

Van der Pol oscillator

3.2.2.1. The modeling of the passive part………….. 84

3.2.2.2. The modeling of the active part…………… 85

3.2.2.3. Simulations of the Van der Pol oscillator…. 87

3.2.3. Two coupled differential Van der Pol oscillators.. 88

3.2.4. Two coupled differential oscillators……………... 93

3.2.5. Comparison between the theory, the Van der Pol model and the differential structure……………… 96

3.2.6. Study and analysis of the two coupled differential oscillators in the weak coupling case…………….. 100

3.2.7. Study and analysis of the two coupled differential oscillators in the strong coupling case…………… 103

3.3. Analysis and design of two VCOs coupled through a resistive network…...……………………………………………………..

106

3.3.1. Introduction……………………………………….. 106

3.3.2. The LC VCO architecture………………………… 109

3.3.2.1. The design of the passive part……………... 110

3.3.2.2. The design of the active part………………. 111

3.3.2.3. VCO simulation results……………………. 112

3.3.3. The modeling of a differential VCO as a differential Van der Pol oscillator………………... 115

3.3.3.1. The modeling of the passive part………….. 115

3.3.3.2. The modeling of the active part…………… 117

3.3.4. Two coupled differential VCOs………………….. 120

3.3.4.1. Study and analysis of two coupled differential VCOs of an optimal coupling case………………. 120

3.3.4.1.1. Study and analysis of two coupled differential VCOs using the state equation approach………………………….. 125

3.3.4.2. Study and analysis of two coupled differential 131

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VCOs in the weak coupling case………………….

3.3.4.3. Study and analysis of two coupled differential VCOs in the strong coupling case……………… 133

3.3.4.4. Study and analysis of the variation of the phase shift ∆φ versus the coupling resistor Rc 135

3.3.4.5. The effect of a mismatch between the two Rc on

the phase shift ∆φ…………………………………. 138

3.3.5. Four coupled differential VCOs………………….. 140

3.4. Conclusion.……………………………………………………... 142

Final conclusion.….………………………………………………….. 144

List of publications………………………………………………….. 150

Appendix A…………………………………………………………… 151

Appendix B…………………………………………………………… 156

References……………………………………………………………. 166

Abstract………………………………………………………………. 173

ix

x

LIST OF FIGURES

No. fig. Title of the figure Page

Figure 1.1 Block diagram of an array of N coupled oscillators…. 6

Figure 1.2 Linear model of an oscillator….…………………....... 7

Figure 1.3 Model of a one-port oscillator..………………………. 8

Figure 1.4 The block diagram of a typical feedback amplifier...... 10

Figure 1.5 The RC oscillator…………………………………..….. 10

Figure 1.6 The LC oscillator…………………………………..….. 12

Figure 1.7 Feedback signal coupling for the LC oscillators……... 13

Figure 1.8 The symbol and the equivalent circuit of a quartz crystal………………………………………………...... 13

Figure 1.9 (a) Series-fed Armstrong oscillator; (b) Shunt fed Armstrong oscillator………………………………….. 14

Figure 1.10 The Hartley oscillator…………................................... 15

Figure 1.11 The Colpitts oscillator………………………………… 16

Figure 1.12 A Pierce oscillator with PNP transistor in the common-base configuration…………………………... 17

Figure 1.13 Electronic structure of a differential oscillator………. 18

Figure 1.14 Electronic structure of a Van der Pol oscillator……… 20

Figure 1.15 Linear network scheme………………………..……… 25

Figure 1.16 The imposed sizes of the level and opening for the main and side lobes…………………………………… 27

Figure 1.17 The control on RF path………………….……………. 30

Figure 1.18 The architecture with polyphase oscillators represented here with four antennas………………….. 30

Figure 1.19 The architecture with four coupled VCOs…………… 31

Figure 1.20 Vector modulator architecture used on LO path……... 32

xi

Figure 1.21 Vector modulator architecture used on RF path.……. 32

Figure 1.22 Block diagram of an antenna array using coupled oscillators…………………………………………….... 34

Figure 2.1 Two Van der Pol oscillators coupled through a RLC circuit……………………………………………….…. 39

Figure 2.2 Van der Pol oscillator model……………………...….. 52

Figure 2.3 Two Van der Pol oscillators coupled through a series RLC circuit………………………...………………….. 53

Figure 2.4 The graphical representation of a synchronization area……...……………………………………………... 70

Figure 2.5 Two coupled differential Van der Pol oscillators……. 75

Figure 2.6 The cartographies of the phase shift ∆φ, the amplitudes A1 and A2 and the synchronization frequency fS of the coupled oscillators……………… 76

Figure 2.7 The cartography of the phase shift: example ∆φ = 42° 77

Figure 3.1 The schematic of the RLC differential oscillator…….. 81

Figure 3.2 a) The waveforms of the output voltage of the differential oscillator; b) The output spectrum………. 83

Figure 3.3 The identification of the Van der Pol passive part parameters……………………………………………... 85

Figure 3.4 The Van der Pol characteristic………………………... 86

Figure 3.5 The differential Van der Pol oscillator model with i = -av +bv3……………………………………………….. 87

Figure 3.6 Comparison between the output voltages of the differential oscillator and the differential Van der Pol oscillator model………………………………………..

88

Figure 3.7 Two coupled Van der Pol oscillators…………………. 88

Figure 3.8.a Two single-ended oscillators coupled through a resistor…………………………………………………. 89

Figure 3.8.b Two differential oscillators coupled through a resistor 90

xii

Figure 3.9 a) The waveforms of the output voltages of the coupled differential Van der Pol oscillators for

∆=7,4%; b) The output spectrum……………………... 92

Figure 3.10 Two differential oscillators coupled through a resistor 93

Figure 3.11 a) The waveforms of the output voltages of the coupled differential oscillators for ∆=7.4%; b) The output spectrum……………………………………….. 95

Figure 3.12 Cartography of the oscillators’ locked states provided by the CAD tool……………………………………….. 97

Figure 3.13 Waveforms of the output voltages of the two coupled differential NMOS oscillators, when ∆φ = 31.23° and A = 2.72 V 98

Figure 3.14 a) Comparison of the phase shift; b) Comparison of the amplitude………………………………………….. 99

Figure 3.15 The weak coupling case – a) Comparison of the phase shift; b) Comparison of the amplitude 101

Figure 3.16 The cartographies of the phase shift ∆φ, the amplitudes A1 and A2 and the synchronization frequency fs of the coupled oscillators for the weak coupling case………………………………………….. 103

Figure 3.17 The strong coupling case – a) Comparison of the phase shift; b) Comparison of the amplitude………… 104

Figure 3.18 The cartographies of the phase shift ∆φ, the amplitudes A1 and A2 and the synchronization frequency fs of the coupled oscillators for the strong coupling case………………………………………….. 105

Figure 3.19 The RLC NMOS differential oscillator: a)Comparison of ∆φ while changing L and C; b) Comparison of the amplitude while changing L and C…………………… 107

Figure 3.20 The Van der Pol oscillator: a) Comparison of ∆φ while changing L and C; b) Comparison of the amplitude while changing L and C…………………… 108

xiii

Figure 3.21 The LC VCO schematic………………………………. 109

Figure 3.22 a) Variation of C versus Vtune; b) Variation of Q versus Vtune…………………………………………… 111

Figure 3.23 The VCO oscillation frequency versus Vtune………... 112

Figure 3.24 The output power of the VCO………………………… 113

Figure 3.25 The output voltages of the VCO……………………… 113

Figure 3.26 Simulated phase noise of the VCO for a tuning voltage of 0.62 V……………………………………… 114

Figure 3.27 Simulated phase noise at 1 MHz versus tuning voltage…………………………………………………. 114

Figure 3.28 The identification of the parameters of the Van der Pol resonator…………………………………………... 116

Figure 3.29 The real and imaginary part of the two impedances Z11 and Z22……………………………………………... 117

Figure 3.30 The Van der Pol characteristic obtained for a VCO…. 118

Figure 3.31 The differential output voltage of a VCO at 5.89 GHz 119

Figure 3.32 The differential Van der Pol oscillator………………... 119

Figure 3.33 Two coupled differential VCOs………………………. 120

Figure 3.34 Two differential Van der Pol coupled oscillators……. 120

Figure 3.35 Cartography of the VCOs’ locked states provided by the CAD tool…………………………………………... 122

Figure 3.36 Waveforms of the output voltages of the two differential NMOS VCOs for ∆φ = 65.6° and A≈1.5 V 123


Figure 3.38 Two differential Van der Pol oscillators coupled through a resistor with GNL = -a + buC1

2(t)…………... 126

Figure 3.39 The output voltages of two coupled differential Van der Pol oscillators obtained with Matlab for ∆φ = 75.6°, A = 1.35 V and fS = 5.89 GHz………………… 129

xiv


Figure 3.41 The weak coupling case – a) Comparison of the phase shift; b) Comparison of the amplitude………………... 132

Figure 3.42 The cartographies of the phase shift ∆φ, the amplitudes A1 and A2 and the synchronization frequency fs of the coupled VCOs for the weak coupling case………………………………………….. 133

Figure 3.43 The strong coupling case – a) Comparison of the phase shift; b) Comparison of the amplitude………… 134

Figure 3.44 The cartographies of the phase shift ∆φ, the amplitudes A1 and A2 and the synchronization frequency fs of the coupled VCOs for the strong coupling case………………………………………….. 135

Figure 3.45 The variation of the phase shift versus Rc when f0 = 100 MHz………………………………………………. 137

Figure 3.46 The variation of the phase shift versus Rc when f0 = 200 MHz………………………………………………. 137

Figure 3.47 a) The variation of the phase shift for 5% mismatch; b) The variation of the amplitude for 5% mismatch…. 139

Figure 3.48 a) The variation of the phase shift for 7% mismatch; b) The variation of the amplitude for 7% mismatch…. 140

Figure 3.49 Schematic of four coupled VCOs…………………….. 141

Figure 3.50 Waveforms of the output voltages of the four

differential NMOS VCOs for ∆φ ≈ -37°…………….. 141

xv

LIST OF TABLES

No. table Title of the table Page

Table 1 The synchronization frequency, phase shift and amplitude obtained for two coupled differential Van der Pol oscillators……………………………………... 91

Table 2 The synchronization frequency, phase shift and amplitude obtained for two coupled differential oscillators……………………………………………… 94

Table 3 The varactor diode’s performances…………………... 111

1

INTRODUCTION

Introduction

2

Arrays of coupled oscillators offer a potentially useful technique for producing

higher powers at millimeter-wave frequencies with better efficiency than is possible with

conventional power-combining techniques. Another application is the beam steering of

antenna arrays. In this case, the radiation pattern of a phased antenna array is steered in a

particular direction by establishing a constant phase progression in the oscillator chain

which is obtained by detuning the free-running frequencies of the outermost oscillators in

the array. Moreover, it is shown that the resulting inter-stage phase shift is independent of

the number of oscillators in the array. Furthermore, synchronization phenomena in arrays

of coupled oscillators are very important models to describe various higher-dimensional

nonlinear phenomena in the field of natural science.

Many techniques have been used to analyse the behaviour of coupled oscillators for

many years such as time domain approaches or frequency domain approaches. Concerning

the last ones, R. York & al. made use of simple Van der Pol oscillators to model

microwave oscillators coupled through either a resistive network or a broad-band network.

Since these works are limited to cases where the coupling network bandwidth is much

greater than the oscillators’ bandwidth, he used more accurate approximations based on a

generalization of Kurokawa’s method to extend the study to the case of a narrow-band

circuit. This theory allows the equations for the amplitude and phase dynamics of two

oscillators coupled through many types of circuits to be derived. As a consequence, it

provides a full analytical formulation allowing to predict the performances of microwave

oscillator arrays. Unfortunately, it is shown that the theoretical limit of the phase shift that

can be obtained by slightly detuning the end elements of the array by equal amounts but in

opposite directions is only ±90°. Thus, it seems interesting to study and analyze the

behavior of an array of coupled differential oscillators or Voltage Controlled Oscillators

(VCOs) since, in this case, the theoretical limit of the phase shift is within 360° due to the

differential operation of the array, leading to an efficient beam-scanning architecture for

example. Furthermore, differential oscillators are widely used in high-frequency circuit

design due to their relatively good phase noise performances and ease of integration.

Due to these considerations, the aim of this work is to study and analyze the

behavior of coupled differential oscillators and VCOs used to control antenna arrays.

Introduction

3

In the first chapter, we will first remind the principle of an oscillator, including the

Van der Pol oscillator. Then, a state of the art of coupled-oscillator theory will be provided

followed by a brief presentation of antenna arrays theory and their applications in the

communication systems. Finally, few technical solutions to control the radiation pattern

will be presented, including the coupled oscillators approach.

In chapter 2, an overview over R. York’s theory giving the dynamics for two Van

der Pol oscillators coupled through a resonant network will be presented. Then, the case of

a broadband coupling circuit will be showed. Since the Van der Pol model used is too

simple and doesn’t allow an accurate prediction of the amplitudes, a new formulation of the

equations describing the locked states of these two coupled oscillators using an accurate

model allowing a good prediction of the amplitudes will be then described. Finally,

mathematical manipulations will be applied to the dynamic equations describing the locked

states of the coupled Van der Pol oscillators. A reduced system of equations with no

trigonometric aspects will be obtained, leading to the elaboration of a CAD tool that

provides, in a considerably short simulation time, the frequency locking region of two

differential oscillators coupled through a resistive network, in terms of the amplitudes of

their output signals and the phase shift between them.

The last chapter will be dedicated to the study and the analysis of an array of

differential oscillators and VCOs coupled through a broadband network. Hence, in the first

part of this chapter dealing with the analysis of two coupled differential oscillators, a

modeling procedure of the differential oscillator as a differential Van der Pol oscillator will

be presented. Then, the proposed CAD tool will be used in order to obtain the cartography

of the oscillators’ locked-states. The validation of the results provided by our CAD tool

will be showed by comparing them to the simulation results of the two coupled differential

oscillators obtained with Agilent’s ADS software for different cases of coupling strength.

Then, the same study will be performed for the case of two coupled differential Voltage

Controlled Oscillators (VCOs). Furthermore, the study of the variation of the phase shift

versus the coupling resistor will also be investigated as well as the effect of a mismatch

between the two coupling resistors on the phase shift. Finally, the behavior of four coupled

differential VCOs will be presented.

4

CHAPTER I

Coupled-Oscillator Arrays - Application

Chapter I – Coupled-Oscillator Arrays - Application

5

1.1. INTRODUCTION

During the past decade, arrays of coupled oscillators are the subject of increasing

research activity due to successful modeling of many diverse biological and physical

phenomena. Biological examples include swarms of synchronously flashing fireflies, the

coordinated firing of cardiac pacemaker cells, rhythmic spinal locomotion in vertebrates

and the synchronized activity of nerve cells in response to external stimuli. In physical

sciences, examples include oscillations in certain nonlinear chemical reactions, the

collective behavior of Josephson junction arrays and laser diode arrays [1]. Almost any

system of discrete or distinguishable behavior can be modeled by a system of coupled

oscillators.

In electronics, in particular, the synchronization behavior of oscillators has been

exploited in many relevant applications. Frequency locking effects have been used to

realize low-cost high-performance quadrature oscillators [2] or to reduce the effect of noise

[3]. Injection locking is the principle, which is at the basis of phase-locked loops (PLL)

circuits, furthermore it is employed to realize low-power consumption frequency dividers

for high-frequency applications [4]. Moreover, arrays of coupled oscillators offer a

potentially useful technique for producing higher powers at millimeter-wave frequencies

with better efficiency than is possible with conventional power-combining techniques [5,

6]. Another application is the beam steering of antenna arrays [7]. In this case, the radiation

pattern of a phased antenna array is steered in a particular direction by establishing a

constant phase progression in the oscillator chain. For a linear array presented in Figure

1.1, a phase shift ∆φ between adjacent elements results in steering the beam to an angle θ

off broadside, which is given by:

φ∆πλ

=θd

arcsin2

0 , (1.1)

where d is the distance separating two antennas and λ0 is the free-space wavelength

[8]. The required inter-stage phase shift can be obtained by detuning the free-running

frequencies of the outermost oscillators in the array [9, 10]. Moreover, it is shown that the


6

resulting inter-stage phase shift is independent of the number of oscillators in the array [1,

11, 12].

Figure 1.1 – Block diagram of an array of N coupled oscillators

In this chapter we will first remind the principle of an oscillator, where different

types of sinusoidal oscillators will be presented including the differential and Van der Pol

oscillators. Then, a state of art of coupled-oscillator theory will be provided followed by a

brief presentation of antenna arrays theory and their applications in the communication

system. Finally, few technical solutions to control the radiation pattern will be presented,

including the coupled oscillators approach.

1.2. OSCILLATOR PRINCIPLE

An oscillator is usually represented as a closed loop system. In Figure 1.2, we can

see a linear model of an oscillator, where A(jω) represents the transfer function of the

active element of the oscillator, and B(jω) represents the transfer function of the passive

part of the reaction, which gives the selection and stability of the oscillation frequency.

This passive part is represented by the resonator.


7

Figure 1.2 - Linear model of an oscillator.

The transfer function of this system can be written as follows:

ω) ω1

ω

ω

ω

) B ( j - A ( j

)A ( j

) ( jV

) ( j V

e

s = . (1.2)

Considering this expression, there is a pulsation, ω0, for which: Vs(jω0) ≠ 0 whereas

Ve(jω0) = 0; this pulsation must fulfill the relation A(jω0) .B(jω0) = 1.

Therefore, the oscillation conditions, known as Barkhausen, are written as follows:

==

π2ωω

1ωω

00

00

k ) ) .B ( j j Arg ( A (

) ) .B ( j A ( j (1.3)

where k ∈ N.

Therefore, in steady state, the module of the opened loop gain must be equal to one

and the total phase shift must be zero or 2kπ.

An oscillator circuit can also be represented by a nonlinear impedance, ZNL(I, ω),

which represents the active part of the oscillator, in parallel with the equivalent impedance

of the resonator, ZR(ω), as shown in Figure 1.3.


8

Figure 1.3 – Model of a one-port oscillator.

In these conditions, Kirchoff laws give:

(ZNL ( I,ω0) + ZR (ω0)) .I = 0, (1.4)

In order to obtain an oscillation phenomena with i(t) = I cos(ω0t), the oscillation

condition becomes:

ZT (I, ω0) = ZNL (I, ω0) + ZR (ω0) = 0. (1.5)

This condition can be expressed in terms of real parts, RT, and imaginary part, XT, of

the total impedance ZT, so that:

RT ( I, ω0) = 0, (1.6)

XT ( I, ω0) = 0. (1.7)

These two equations give the condition for the sustaining of the oscillations and the

oscillation frequency of the circuit. As the real part of the equivalent impedance of the

passive part is positive at the pulsation ω0, the condition for sustaining the oscillation,

defined by equation (1.6) can be fulfilled only if the real part of the non-linear impedance

is negative at the pulsation ω0, which can be obtained by an active element.


9

Oscillators are classified in accordance with the waveforms they produce and the

circuitry required for producing the desired oscillations, as presented in [13]:

The sinusoidal oscillator - the output voltage is sinusoidal;

The non-sinusoidal oscillator - the output voltage is non-sinusoidal but it has

triangular, square and saw tooth waveforms.

1.2.1. The Sinusoidal Oscillator

A sinusoidal oscillator is an oscillator that produces a sine-wave output signal. An

ideal oscillator should produce an output signal with constant amplitude with no variation

in frequency. A practical oscillator cannot have these criteria, the degree to which the ideal

is approached depends on the class of amplifier operation, amplifier characteristics,

frequency stability, and amplitude stability. Sinusoidal oscillators generate signals ranging

from low audio frequencies to ultrahigh radio and microwave frequencies.

The sinusoidal oscillators are classified as follows:

• RC oscillators;

• LC oscillators;

• the crystal-controlled oscillator.

Many low-frequency oscillators use resistors and capacitors to form their frequency-

determining networks and are referred to as RC oscillators. These are used in the audio-

frequency range.

The LC oscillators are commonly used for the higher radio frequencies. They are not

suitable for use as extremely low-frequency oscillators because the inductors and

capacitors would be large in size, heavy, and costly to manufacture.

The third category of sinusoidal oscillator is the crystal-controlled oscillator. The

crystal-controlled oscillator provides excellent frequency stability and is used from the

middle of the audio range through the radio frequency range.


10

An oscillator must provide amplification where the amplification of signal power

occurs from the input to the output of the oscillator and a portion of the output is feedback

to the input to sustain a constant input.

Figure 1.4 represents the block diagram of a typical feedback amplifier, where:

• A – is the open-loop gain of the amplifier and Vout = AV;

• β – is the feedback factor and Vf = β Vout .

Figure 1.4 – The block diagram of a typical feedback amplifier.

If V = Vin – Vf, the feedback is negative and the amplifier is provided with negative

feedback. If V = Vin + Vf, the feedback is positive and the amplifier is provided with

positive feedback. As a consequence, the general expression for a feedback loop is as

follows:

A

A

V

VA

in

outf

β1±==

, (1.8)

where βA is the loop gain, and 1±βA is the amount of feedback.

Practically, all amplifiers use negative feedback, but the sinusoidal oscillators use

positive feedback.

1.2.1.1. The RC Oscillator

In Figure 1.5 an RC oscillator is represented.


11

Figure 1.5 – The RC oscillator.

This oscillator is composed of a RC network and an amplifier. This oscillator is also

well-known as phase-shift oscillator. Theoretically, to satisfy the Barkhausen criterion and

to sustain oscillations, both gain and phase conditions must be fulfilled. The bipolar

transistor provides the amplification to achieve an open loop gain greater than one required

for the start-up of the oscillation phenomena:

|β(jω0)A(jω0)| ≥ 1.

This structure is in a common-emitter configuration and then the phase shift is close to

180°. In these conditions, the passive part must present a phase shift greater than 180° at

the oscillation frequency ω0. To obtain such a phase shift with RC components, a transfer

function presenting three poles is required. This implies the three RC network. The output

of the oscillator contains only a single sinusoidal frequency. When the oscillator is

powered on, the loop gain βA is greater than unity and the amplitude of the oscillations will

increase. A level is reached when the gain of the amplifier decreases, and the value of the

loop gain decreases to unity and constant amplitude oscillations are sustained. The

frequency of oscillations is determined by the values of resistance and capacitance in the

three sections. Variable resistors and capacitors are usually used to provide tuning in the

feedback network for variations in phase shift.

Let us consider that the resistors R and Rb are greater than the input bipolar

transistor impedance h11. In these conditions, for the RC oscillator of Figure 1.5, the

oscillation frequency is given by the following formula:


12

R

RRC

oL4

6

1

+=ω

(1.9)

In general, for an RC phase-shift oscillator, the frequency of oscillation (resonant

frequency) can be approximated with the following relation:

nRC 2

1ω0 = , (1.10)

where n is the number of RC sections.

1.2.1.2. The LC Oscillator

The LC oscillators use resonant circuits. A resonant circuit stores energy alternately

in the inductor and capacitor. However, every circuit contains some resistance and this

resistance causes reduction in the amplitude of the oscillations. Figure 1.6 shows a typical

diagram of an LC oscillator.

Figure 1.6 – The LC oscillator.

In an LC oscillator the sinusoidal signal is generated by the action of an inductor

and a capacitor. The feedback signal is coupled from the LC tank of the oscillator circuit by

using a coil tickler or a coil pair as shown in Figure 1.7(a) and 1.7(b) or by using a


13

capacitor pair in the tank circuit as shown in Figure 1.7(c). A tickler coil is an inductor that

is inductively coupled to the inductor of the LC tank circuit.

Figure 1.7 - Feedback signal coupling for the LC oscillators.

1.2.1.3. Crystal Oscillator

Crystal oscillators are oscillators where the primary frequency determining element

is a quartz crystal. Because of the inherent characteristics of the quartz crystal the crystal

oscillator may be held to extreme accuracy of frequency stability. The frequency depends

almost entirely on the thickness where the thinner the thickness, the higher the frequency

of oscillation.

The symbol and the equivalent circuit of a quartz crystal are shown in Figure 1.8,

where capacitor C1 represents the electrostatic capacitance between the electrodes of the

crystal and in general C1»C2.

Figure 1.8 - The symbol and the equivalent circuit of a quartz crystal.


14

The available power is limited by the heat the crystal will withstand without

fracturing. The amount of heating is dependent upon the amount of current that can safely

pass through a crystal and this current may be in the order of 50 to 200 milliamperes.

Accordingly, temperature compensation must be applied to crystal oscillators to improve

thermal stability of the crystal oscillator.

Crystal oscillators are used in applications where frequency accuracy and stability

are of outmost importance such as broadcast transmitters and radar. The frequency stability

of crystal-controlled oscillators depends on the quality factor Q. The Q of a crystal may

vary from 10,000 to 100,000.

1.2.1.4. The Armstrong Oscillator

In an Armstrong oscillator, the feedback is provided through a tickler coil. There are

two types of Armstrong oscillator:

• series-fed tuned-collector Armstrong oscillator

• shunt-fed tuned-collector Armstrong oscillator

Figure 1.9 shows the two circuits known as Armstrong oscillators.

(a) (b)

Figure 1.9 – (a) Series-fed Armstrong oscillator; (b) Shunt fed Armstrong oscillator.


15

The series-fed tuned-collector Armstrong oscillator is so-called because the power

supply voltage Vcc supplied to the transistor is through the tank circuit. The shunt-fed

tuned-collector Armstrong oscillator is so-called because the power supply voltage Vcc

supplied to the transistor is through a path parallel to the tank circuit.

For both types of Armstrong oscillators, the power through Vcc is supplied to the

transistor and the tank circuit begins to oscillate.

The transistor conducts for a short period of time and returns sufficient energy to the

tank circuit to ensure a constant amplitude output signal.

1.2.1.5. The Hartley Oscillator

The Hartley oscillator is an improvement over the Armstrong oscillator. Although

its frequency stability is not the best possible of all the oscillators, the Hartley oscillator

can generate a wide range of frequencies and is very easy to tune. Such a type of oscillator

is shown in Figure 1.10.

Figure 1.10 – The Hartley oscillator.

The main difference between the Armstrong and the Hartley oscillators lies in the

design of the feedback (tickler) coil. A separate coil is not used. Instead, in the Hartley

oscillator, the coil in the tank circuit is a split inductor. Current flow through one section

induces a voltage in the other section to develop a feedback signal.


16

1.2.1.6. The Colpitts Oscillator

Figure 1.11 shows a simplified version of the Colpitts oscillator.

Figure 1.11 – The Colpitts oscillator.

In a Colpitts oscillator the feedback is provided through a capacitor pair. The

Colpitts oscillator provides better frequency stability than the Armstrong and Hartley

oscillators. Moreover, the Colpitts oscillator is easier to tune and thus can be used for a

wide range of frequencies.

For the oscillator circuit of Figure 1.11 the frequency of oscillation is as follows:

21

210ω CLC

CC += , (1.11)

1.2.1.7. The Pierce Oscillator

Figure 1.12 shows a Pierce oscillator with a PNP transistor as an amplifier in the

common-base configuration.


17

Figure 1.12 – A Pierce oscillator with PNP transistor in the common-base

configuration.

The Pierce oscillator is a modified Colpitts oscillator that uses a crystal as a

parallel-resonant circuit and for this reason is often referred to as crystal-controlled Pierce

oscillator.

In the oscillator circuit of Figure 1.12, feedback is provided from the collector to the

emitter of the transistor through capacitor C1 and resistors R1, RB, and RC are used to

establish the proper bias conditions. Besides the crystal, the frequency of oscillation is also

determined by the settings of the variable capacitors CE and CB.

1.2.1.8. The differential oscillator

Like the previous architectures, this type of oscillator presents a structure that

resonates at a frequency for which the losses are compensated by the active part, [14].

Thus, this resistance Rp is in parallel with a structure that can present a negative resistance.

In steady state, to sustain the oscillation, the negative resistance must be equal to the losses

resistance of the resonator. Usually, the negative resistance is represented by an element or

an active circuit (a structure based on transistors), as shown in Figure 1.13.

The output voltage between the points vx and vy is as follows:

Chapter I – C

The small-signal ana

conductance between points

a single transistor.

Figure 1.13 –

Thus, to ensure the

provided by the pair of trans

so that:

In steady state, the tr

This implies that the losses r

Coupled-Oscillator Arrays - Appli

18

voutput = vx – vy .

analysis of the pair of transistors shows

ts vx and vy is equal to -2mg

where gm is the

Electronic structure of a differential osc

he start-up condition of the oscillator, th

ansistors, 2mg

, must be greater than the losse

pm R

g2> .

transistor’s operating point varies from blo

s resistance Rp is canceled by gm :

plication

(1.12)

s that the equivalent

the transconductance of

scillator.

the transconductance

sses conductance pR

1 ,

(1.13)

blocking to saturation.


19

pm R

g1−= . (1.14)

The problem of oscillation start-up is a critical problem in oscillators design.

Equations (1.13) and (1.14) indicate that the transistors must be oversized. Moreover,

because we can’t know precisely Rp, generally, the transistors have their transconductance

three or four times greater than that required in steady state.

The amplitude of the output voltage for this type of oscillator can be calculated

according to the limits of the transistor’s operating point. If one transistor of the

differential pair is off and the other is saturated, then, all the current IDC is flowing in this

latter. In this case we have:

=−

=

DCp

yDD

DDx

IR

v V

V v

2

. (1.15)

And the amplitude is:

DCp

yx IR

v v A 2

=−= .

(1.16)

1.2.1.9. The Van der Pol oscillator

The Van der Pol oscillator is an oscillator with nonlinear damping governed by the

second-order differential equation:

0 1 ε 2 =+−− xx)x(x , (1.17)

where:

• ε - is a positive constant, which measures the degree of linearity of the system;


20

• x- is the dynamical variable.

The Van der Pol model commonly used is made of a nonlinear conductance and a

resonator, as shown in Figure 1.14. The general expression of the nonlinear conductance of

the Van der Pol model is written as follows:

2γα V - GNL += , (1.18)

where:

• α− - is the negative conductance necessary to start the oscillation;

• 2 γV - is the nonlinear conductance which modelises the saturation phenomenon.

Figure 1.14 – Electronic structure of a Van der Pol oscillator.

1.3. STATE OF THE ART OF COUPLED OSCILLATORS THEORY

Many techniques have been used to analyze the behavior of coupled oscillators for

many years such as time domain approaches and frequency domain approaches.

Concerning the frequency domain approaches, B. Van der Pol [15] started to study the

oscillators’ synchronization phenomenon using an "averaging" method to obtain

approximate solutions for quasi-sinusoidal systems. Then, R. Adler gave to the microwave


21

oscillator analysis a more physical basis defining the phase dynamic equation of an

oscillator under the influence of an injected signal [16]. This was sustained by K.

Kurokawa who derived the dynamic equations for both the amplitude and phase [17],

providing a pragmatic understanding of coupled microwave oscillators. In R. York’s

previous works, coupled microwave oscillators have been modeled as simple single-ended

Van der Pol oscillators coupled through either a resistive network or a broad-band network

[1, 11, 18]. Unfortunately, this works are limited to the cases when the coupling network

bandwidth is much greater than the oscillators’ bandwidth. In these conditions, a

generalization of Kurokawa’s method [17] was used to extend the study to a narrow-band

circuit allowing the equations for the amplitude and phase dynamics of two oscillators

coupled through many types of circuits to be derived [19]. Since these works, only few

papers present new techniques for the analysis of coupled-oscillator arrays in the frequency

domain [20, 21, 22]. In [22], a semi-analytical formulation is presented for the design of

coupled oscillator systems, avoiding the computational expensiveness of a full harmonic

balance synthesis presented in [20] and [21]. In [23], a simplified closed-form of the semi-

analytical formulation proposed in [22] for the optimized design of coupled-oscillator

systems is presented. Nevertheless, even if this new semi-analytical formulation allows a

good prediction of the coupled oscillator solution, it is only valid for the weak coupling

case.

Now, concerning the time domain approaches [24-29], among other, D. Aoun and

D.A. Linkens studied in [24] nonlinear oscillators used in bioelectronics applications, in

particular, towards the electrical activity of the mammalian gastro-intestinal tract. This

activity known as “slow-waves” has been extensively modeled using nonlinear oscillatory

dynamics. Therefore, they applied a matrix extension of the Krylov-Bogolioubov

linearization technique to a wide range of structures comprising chains, arrays, rings and

tubes. Unfortunately, it has been demonstrated that this technique produces complicated

stability criteria for the existence of stable limit cycles. Hence, they decided to engineer a

CAD package which solves these criteria for the structures mentioned above with an

arbitrary number of either third or fifth power conductance Van der Pol oscillators coupled

mutually. Later on, Chai Wah Wu and Leon O. Chua demonstrated in [25] how an array of

resistively coupled identical oscillators can be synchronized if the coupling conductances


22

are large enough. To do so, they used algebraic graph theory to derive sufficient conditions

for an array of resistively coupled nonlinear oscillators to synchronize. These conditions

are derived, in fact, from the connectivity graph, which describes how the oscillators are

connected. Moreover, they showed that the upper bound on the coupling conductance

required for synchronization for arbitrary graphs, is in order of n2, where n is the number of

oscillators. In [26], P. Maffezoni studies the synchronization phenomenon of the weakly

coupled oscillators using a phase-domain macromodel based on perturbation projection

vector that describes the linear periodically time-varying behavior of an oscillator in the

neighbor of its stable limit cycle. Using this method, the mutual locking range and the

common locking frequency of the locked oscillators could be predicted with great

precision. The reliability and accuracy of the method have been demonstrated also when

the mutual coupling results in an anomalous synchronization frequency. Furthermore, in

this way Maffezoni could estimate, by means of closed-form expressions, the mutual

pulling effects that arise between two self-sustained oscillators in the presence of weak

interactions.

Nevertheless, even if the time domain approaches offer a good prediction of the

coupled oscillator solution, the frequency domain approaches are less complex and more

preferred in the design of RF and microwave coupled oscillators. For instance, R. York’s

theory [19] provides a full analytical formulation allowing to predict the performances of

microwave oscillator arrays for both weak and strong coupling.

1.4. APPLICATIONS: BEAMSTEERING OF ANTENNA ARRAYS

Mobile communications are the subject of increasing research activity covering

many technical areas. Worldwide activities in this growing industry are perhaps an

indication of its importance.

An application of antenna arrays has been suggested in the recent years for mobile

communication systems in order to solve the problem of limited bandwidth of the channel,

and therefore, to satisfy the increasing demands for a large number of mobile

communication channels [30, 31]. Antenna arrays have many other advantages, in fact,


23

they contribute to the improving of the communication system performances while

increasing the channel capacity, providing a wider band of coverage, and minimizing the

multipath fading and also the interference between channels. Beside the advantages

mentioned in this paragraph, the two properties below also describe a few assets of antenna

arrays:

− The decrease of the electromagnetic pollution: the shape of the radiation pattern

can be optimized in order to reduce the side lobes. Similarly, the radiation pattern

can be orientated in the desired direction, and therefore any radiations in useless

directions are minimized.

− A better quality of the transmission / reception: the emitted power can be focused

in the desired direction and therefore the wasted power in useless directions is

reduced. One of the advantages resulting from this decrease is also the distribution

of the power on all power amplifiers constituting the transmission network.

Similarly, for the reception, the noise provided by the interfering signals is

minimized, leading to a reduced Bit Error Rate (BER), which is the priority of any

transmission architecture.

Thus, antenna arrays are essential to increase the efficiency of mobile

communications systems. For the transport area, these devices are installed on vehicles,

boats, planes, satellites and base stations in order to fulfill the channel requirements for this

service.

1.4.1. Antenna Arrays

An antenna array is defined as a set of N radiating elements distributed in space.

The amplitude and/or phase of the signal injected into each of these antennas can be

commanded so that it can control the shape of the radiation pattern of the network as well

as its orientation. These commands can be chosen so that several lobes can be created

simultaneously or a single lobe in the direction of the incident signal and zero in the

direction of the interference wave.


24

The main characteristics of an antenna array are determined by the number and type

of antennas constituting the network and also their geometric arrangement.

For reasons of simplicity and implementing, identical elements are chosen.

An antenna array has the following possible configurations:

• Linear - antennas are aligned in a straight line;

• Circular - the antennas are arranged in a circle;

• Planar - the antennas are arranged on a plane;

• Surface - the antennas are arranged on a surface with a curvature, such as a sphere

or a cylinder;

• Volume - the antennas are distributed in a volume.

1.4.1.1. Uniform linear network

The radiation pattern of an antenna array is based on basic physical structure of

antennas and network geometry, but also on their command signals [32]. When the basic

sources of a linear network are excited with the same amplitude, the linear network is

considered to be uniform, and therefore is called equi-amplitude.

We consider a uniform network of N elements identical and equidistant with a

distance d between them along an axis x, commanded by N sources with a phase gradient

φ∆ , one against another. This network is illustrated in Figure 1.15, where r is the maximum

distance between reference antenna and the observation plane, and θ is the angle of the

main lobe direction.


25

Figure 1.15 - Linear network scheme.

In these conditions, in far field, the total electrical field radiated by an antenna array

is:

)(AEEtot ψ×= 0 , (1.19)

where:

• E0 - is the electrical field radiated by an elementary antenna. It's called “element

factor” and depends only on the physical characteristics of the elementary antenna;

• A(ψ) – is the “array factor” and depends on the geometry of the network and on the

size of the amplitude and phase applied to the network elements. This array factor

has the following formula:

) ( sin

)N

( sin) / ) ( j ( N -exp

N )A (

2

2 2 ψ11

ψ

ψ

=ψ

,

(1.20)

where φ∆θλπ=ψ -) (

d 2 sin , and λ is the wavelength.


26

The total radiation pattern of the antenna array is then defined by the product of the

module of the electric field radiated by an elementary antenna by the array factor, which is

given by the following expression:

) ( sin

)N

( sin

N )A (

2

2

1

ψ

ψ

=ψ

.

(1.21)

Under these conditions, it is important to observe that when the number N of

elementary antennas is increased, the total radiation pattern of the network depends more

on the array factor and less on the radiation pattern of each elementary antenna.

1.4.1.2. Controlling the shape of the radiation pattern

The main purpose of an array of N radiating elements is to control the distribution of

the energy radiated or received in the space. This control is based on the establishment of

an appropriate control law. This is possible by applying the optimal values of the

amplitudes and / or phases to the signals injected into each elementary antenna of the array.

Thus, the synthesis of the radiation pattern of an antenna array consists in determining the

set of parameters of the control law able to produce a radiation pattern which meets all the

required elements.

The characteristics of the radiation pattern of an array are determined by the level

and / or the opening of the side lobes compared to those of the main lobe, as shown in

Figure 1.16. Thus, more the opening of the latter decreases, the level of the side lobes

increases, and therefore, a significant part of the power is “wasted”. In these conditions, it

is necessary to model the radiation pattern in order to obtain the lowest level of the side

lobes so that the required opening of the main lobe is obtained and the losses induced in

other directions are minimized.


27

Figure 1.16 – The imposed sizes of the level and opening for the main and side

lobes.

Due to these considerations, three synthesis solutions can be envisaged. The first is

to synthesize only the amplitude. This technique yields the direction lobes with the

possibility of controlling the level of the side lobes. The coefficients of the array can be

calculated using analytical techniques developed among others by Fourier or Chebyshev.

However, applications of this type of synthesis are limited. Thus, this solution is not

studied in detail in this manuscript. The other two solutions are presented in the following.

1.4.1.2.1. Phase synthesis

This solution allows to perform beam-steering of the antenna array. To do so,

several solutions with phase shifters or phase shifterless can be used as described below:

• A first approach illustrated in Figure 1.17 is relatively well known. The intermediate

frequency IF of the signal is converted in a RF frequency thanks to the oscillator

LO. The resulting signal is then divided using power dividers. Each of the obtained

signals is then amplified and phase shifted before being sent to the antenna elements

of the array. The concept of this approach is simple, but these devices can introduce

significant losses and a significant production cost. Moreover, the phase shifters are


28

passive devices difficult to integrate, and therefore their use in hybrid circuits is

limited.

• Furthermore, in the area of passive beam distributors, two classes coexist [33]:

1- The quasi-optical types: they imply a hybrid arrangement meaning either a

reflector or a lens objective with an antenna array. The quasi-optical type which

is the best known is the Rotman lens [34] which is a time delay device having

the synthesis procedure based on the optical geometry principles. The input or

output ports drive the cavity of a lens whose periphery is well defined. The

excitation of an input port produces a uniform amplitude distribution and a linear

phase gradient (constant) at the output ports. The Rotman lens is interesting

because it allows a certain freedom of design with many parameters to adjust,

one can obtain many beams with a stable frequency system. However, its

disadvantages are not negligible, especially regarding the complexity of the

design due to the number of variables to adjust and the mutual coupling between

its ports.

2- The circuits’ types using the microstrip technology, striplines or waveguides

such as the Butler matrix [35] which is a symmetrical passive mutual circuit with

N input ports and N output ports. This circuit drives N radiating elements

producing N different orthogonal beams. This is a parallel system made of

junctions which connect the input ports to the output ports through equal length

transmission lines. Thus, an input signal is many times divided without losses

until the output ports. Despite its simple and lossless architecture, the Butler

matrix has many disadvantages particularly regarding the number of components

which increases considerably with the number of desired beams.

• Another approach illustrated in Figure 1.18, consists in using a polyphase voltage-

controlled oscillator, which is able to generate the local oscillator frequency with N

phases. In this architecture, all the N phases are conveyed to each antenna via a

distribution network. Phase selectors ensure the required phase to each element [36,

37]. In this case, the phase variations are discrete, which does not constitute a

problem as long as the discrimination steps are adequate to the application.


29

Nevertheless, the distribution network of N phases constitutes a real issue here since

each path must be forwarded to the phase selector in a symmetrical way [38].

• An alternative approach consists in using an array of coupled voltage controlled

oscillators as shown in Figure 1.19. When working, all the VCOs are synchronized

at a common frequency and have different phases according to their free-running

frequencies. In these conditions, the same phase difference is obtained between each

pair of neighboring oscillators. Therefore, the control of the oscillation frequencies

can impose the required phase shift and thus the desired radiation direction. This

solution presents the advantage of providing a continuous variation of the phase

shift, at the expense of the complexity of implementation. With this technique, it is

difficult to control precisely the synchronization frequency, but this approach is

very interesting especially for radar applications where it is important to control the

orientation of the main lobe of the radiation pattern of the antenna array [11].

Nevertheless, uncertainties on the received frequency can be tolerated. Several

methods can be applied to obtain the desired phase shift:

• By coupling the oscillators through delay lines as P. Liao and R. York

proposed in [9].

• By changing the free running frequency of all the VCOs of the array as T.

Heath demonstrated in [39]. Indeed, changing the free running frequencies of all the

VCOs of the array, leads to a phase difference between the output voltages of the

VCOs.

• By changing the free running frequencies of the two outermost VCOs of the

array [40]. In these conditions, the same phase difference is obtained between each

pair of neighboring oscillators. This third approach is discussed in detail later in this

chapter.


30

Figure 1.17 – The control on RF path.

Figure 1.18 – The architecture with polyphase oscillators represented here with four

antennas.


31

Figure 1.19 – The architecture with four coupled VCOs.

1.4.1.2.2. Amplitude and phase synthesis

This solution allows to perform beamforming of the linear antenna array. This type

of synthesis is efficient for applications of adaptive arrays but its implementation requires a

control architecture in amplitude and phase, which involves a costly and complicated

technique.

To do so, an active solution using vector modulators on LO path (Figure 1.20) or on

RF path (Figure 1.21) can be used. The main advantage of this architecture is on one hand

the simplicity of implementation (no distribution path of the complex phases) plus the

stability of the frequency obtained by a frequency synthesizer, and on the other hand the

control of the amplitude [38].


32

Figure 1.20 – Vector modulator architecture used on LO path.

Figure 1.21 – Vector modulator architecture used on RF path.


33

1.4.2. The control of the radiation pattern using coupled oscillators

As mentioned in subsection 1.4.1.2.1, R. York & al. introduced a new approach

based on using the synchronization propriety of coupled oscillators to steer the radiation

pattern of an antenna array. This approach is illustrated in Figure 1.22, where:

• N – is the number of the array elements;

• θ – the main lobe steering angle - is the angle between z axis and the vector that

links the origin of the coordinate system with an arbitrary point chosen in far field,

and r is the distance between them;

• d – is the distance between two adjacent elements of the array;

• φ∆ – is the phase gradient between the output voltages of two adjacent oscillators.

Regardless of the topology, all oscillator arrays must satisfy two key requirements.

First, the oscillators must synchronize to a common frequency. Second, they must maintain

a desired phase relationship in the steady-state. The synchronization requirement can be

satisfied by coupling the oscillators or to an external source (injection-locking phenomena

[16, 17]). In practice and in all the cases, the most delicate task is to ensure the proper

phase relationship between the elements and also to control the phase shift. This requires a

good understanding of the influence of various circuit parameters, as the coupling strength

and the difference between the oscillators’ free running frequencies, on the values of

phases of the output voltages of each oscillator. When these free running frequencies are

within a particular locking range, the oscillators are synchronized and the inter-stage phase

shift φ∆ is related to the original distribution of the free running frequencies [1, 9, 39].

Moreover, a constant phase gradient between adjacent elements can be obtained by

controlling the free running frequencies of the outermost oscillators [9]. Compared with the

method that uses phase shifters, this approach based on coupled oscillators has the

advantage of limiting the number of commands. Indeed, each phase shifter must be

controlled by a number of commands more or less important, depending on the complexity

of the phase shifter, whereas in the coupled oscillators approach only the outermost

oscillators will be controlled. Let us note that for both techniques, the amplitudes of the

output voltages cannot be controlled.


34

Figure 1.22 – Block diagram of an antenna array using coupled oscillators.

Nevertheless, it is shown in [1] that the theoretical limit of the phase shift that can

be obtained by slightly detuning the oscillators of the array by equal amounts but in

opposite directions is only ±90o. Therefore, it seems to be interesting to analyze the

behavior of an array of coupled differential VCOs since, in this case the theoretical limit of

the phase shift is within 360o due to the differential nature of the array. In these conditions,

a continuously controlled 360o phase shifting range can be obtained leading to a more

efficient beam-scanning architecture. Furthermore, differential oscillators or VCOs are

widely used in high-frequency circuit design due to their relatively good phase noise

performances and ease of integration. In this context, R. York’s theory is used to analyze

the behavior of an array of coupled differential VCOs. This theoretical study has then been

used to develop a CAD tool, allowing to obtain, in a very short simulation time, the

graphical representation of the locked states of the coupled differential oscillators. Thus,

from this cartography it will be possible to extract the values of the free running

frequencies required to generate the desired phase shift. The results obtained with the CAD

tool will then be compared with simulations using Agilent’s ADS software.


35

1.5. CONCLUSION

In this chapter a review of coupled-oscillator arrays as well as their applications was

presented. Thus, after a classification of oscillators in accordance with the waveform they

produce and the circuitry required for producing the desired oscillations.

A state of the art of coupled-oscillators theory was presented followed by a few

applications of antenna arrays. Due to their different geometric configuration, antenna

arrays can have an important role in controlling the radiation angle of the pattern.

Therefore, and also for simplicity reasons, a linear configuration was presented in this

chapter. Controlling the antenna array consists in generating the amplitudes and/or the

phases necessary for orientating the radiation pattern in the desired direction. As a

consequence, various technical solutions were proposed, including the coupled oscillators

approach.

36

CHAPTER II

Theoretical Analysis of Coupled Oscillators Applied

to an Antenna Array

Chapter II– Theoretical Analysis of Coupled Oscillators Applied to an Antenna Array

37

2.1. INTRODUCTION

Arrays of coupled oscillators are the subject of increasing research activity in the

communications systems due to their use in new applications such as power-combining

techniques and beam steering of antenna arrays.

In this context, let us remind that, the theoretical limit of the phase shift obtained

for an array of single-ended coupled oscillators is within the range of ±90° [1]. Thus, it

seems interesting to analyze the behavior of an array of differential VCOs. Indeed, in

this case, the theoretical limit of the phase shift is within 360° due to the differential

nature of the array, leading to a more efficient beam-scanning architecture for instance.

Furthermore, differential oscillators are widely used in high-frequency circuit design

due to their relatively good phase noise performances and ease of integration.

Moreover, the use of a broadband coupling network, i.e. a resistor, instead of a resonant

one, can lead to a substantial save in chip area in the case of RF integrated circuit

design.

Due to these considerations, in this chapter we will first remind R. York’s theory

giving the dynamics for two Van der Pol oscillators coupled through a resonant

network. Then, the dynamics of two Van der Pol oscillators coupled through a

broadband circuit will be presented. A new formulation of the equations describing the

locked states of two coupled Van der Pol oscillators using an accurate model allowing a

good prediction of the amplitudes will be then described. Finally, mathematical

manipulations will be applied to the dynamic equations describing the locked states of

the coupled Van der Pol oscillators. A reduced system of equations with no

trigonometric aspects will be obtained, leading to the elaboration of a CAD tool that

provides, in a considerably short simulation time, the frequency locking region of two

differential oscillators coupled through a resistive network, in terms of the amplitudes

of their output signals and the phase shift between them.


38

2.2. DYNAMICS OF TWO VAN DER POL OSCILLATORS

COUPLED THROUGH A RESONANT CIRCUIT

Let us remember that the primary objective of this work is to orientate the

radiation pattern of an antenna array by controlling the phase gradient existing between

the signals applied to adjacent elements of the array. Thus, this chapter considers the

case of a system of two oscillators coupled first through a RLC circuit which provides

the synchronization frequency [19]. Indeed, the radiation pattern of a phased antenna

array is steered in a particular direction through a constant phase progression in the

oscillator chain which is obtained by detuning the free running frequencies of the

outermost oscillators in the array. Furthermore, it is shown that the resulting inter-stage

phase shift is independent on the number of oscillators in the array [9].

First, the oscillators are modeled by two Van der Pol circuits. The choice of this

model is justified by its simplicity regarding the analytical calculations, as presented in

[1]. This model is made of a RLC resonator and a nonlinear conductance, which has the

following expression:

231 VaaGNL +−= , (2.1)

where:

• -a1 is the negative conductance necessary to start the oscillation;

• a3V 2 is the nonlinear conductance which modelizes the saturation phenomenon.

In Figure 2.1, this nonlinear conductance is represented by the term G(A), given

by the following expression:

)A(fG)A(G 0−= , (2.2)

where:

• G0 is the negative conductance for a null control voltage;

• )A()A(f 21−= is the saturation function of the nonlinear conductance.


39

Figure 2.1 – Two Van der Pol oscillators coupled through a RLC circuit.

Therefore, each oscillator is modeled by the negative conductance G(A),

associated to a parallel resonant circuit. These two oscillators are coupled through a

series resonant circuit, as shown in Figure 2.1. Let us note that both oscillators have the

same characteristics, except for their free running frequencies.

The main objective is to determine the phase shift φ∆ between the output

voltages of both oscillators, depending on their free-running frequencies referred to the

resonant frequency of the coupling circuit, when the system is synchronized.

The coupling current Ic can be expressed in terms of the admittance of the

oscillator 1, Y1(A1, ω1), of the admittance of the oscillator 2, Y2(A2, ω2), or depending on

the admittance of the coupling circuit, Yc(ωc), so that:

11111111 V)](Y)A(G[V),A(YI Lc ω+=ω=

22222222 V)](Y)A(G[V),A(YI Lc ω+−=ω−=

)VV)((YI ccc 21 −ω−= ,

(2.3)

where A1, ω1 represents the amplitude and pulsation of oscillators 1, A2, ω2

represents the amplitude and pulsation of oscillators 2, and V1 and V2 are the output

voltages of oscillator 1 and oscillator 2 respectively.

The admittance of oscillator 1, Y1(A1,ω1), can be expressed as follows:


40

)(j

C)A(fGCj

Lj)A(fG),A(Y 2

1201

1101

1110111

1 ω−ωω

+−=ω+ω

+−=ω , (2.4)

where CL1

011=ω is the resonance pulsation of the resonator of oscillator 1.

Now, if 011 ω≅ω then 1101 2ω≅ω+ω and if C

Ga

02 =ω is the frequency bandwidth

of the two oscillators, then Y1(A1,ω1) can be written as is equation (2.5):

ωω−ω

+−=ωa

j)A(fG),A(Y 10110111 . (2.5)

Similarly, the admittance of oscillator 2, Y2(A2,ω2), is calculated and expressed in

the following equation:

ωω−ω

+−=ωa

j)A(fG),A(Y 20220222 , (2.6)

where CL2

021=ω is the resonance pulsation of the resonator of oscillator 2.

Knowing the expressions (2.5) and (2.6) of the admittances, it is important to

determine the expression of the current through the coupling circuit. To do so, the

complex output voltages of oscillators 1 and 2, are expressed as in equation (2.7):

111

θ= jeAV

222

θ= jeAV ,

(2.7)

where 111 φ+ω=θ t and 222 φ+ω=θ t . In this expression, the amplitudes A1 and A2,

and the phases 1φ and 2φ , are considered constants. In the case of oscillators, the values

of the amplitudes and phases are slowly varying [41], so that:


41

)t(j)t(j eA)jj(eAdt

Vd1111

11111 φ+ωφ+ω φ+ω+=

)t(j)t(j eA)jj(eAdt

Vd2222

22222 φ+ωφ+ω φ+ω+= .

(2.8)

Hence,

)t(j

.

e)A

Aj(jA

dt

Vd11

1

1111

1 φ+ω−φ+ω=

)t(j

.

e)A

Aj(jA

dt

Vd22ω

2

2222

2ω

φ+−φ+=

.

(2.9)

Comparing equation (2.9) with the result from Fourier theory, Kurokawa

concluded that the expression in the brackets must be the time-domain representation of

the instantaneous frequency [17]. Using this expression for the pulsation ω, so that

ω = ω1,2 + δω with δω=−φ

.

A

Aj

1

11

or δω=−φ

.

A

Aj

2

22

, allows (2.5) and (2.6) to be expanded

in a Taylor series about the pulsations ω1,2 in order to study the behavior around the

oscillation pulsation ω1,2. Thus, for δω ‹‹ ω1,2, we have:

1

1

11

11

11111

θ

ω

φ+ω

−φω

ω+ω= j

.

)t(jc eA)

A

Aj(

d

),A(dY),A(YeI cc

2

2

22

22

22222

θ

ω

φ+ω

−φω

ω+ω−= j

.

)t(jc eA)

A

Aj(

d

),A(dY),A(YeI cc

.

(2.10)

If we divide the expressions (2.10) by 1θje and respectively 2θje , and replace

the derivate of the admittance depending on the pulsation ω, we obtain:


42

11

11

0101010

1 A)A

Aj(

GjjG)A(fGeI

.

aa

)(jc

c

−φω

+ω

ω−ω−−=θ−θ

22

22

0202020

2 A)A

Aj(

GjjG)A(fGeI

.

aa

)(jc

c

−φω

−ω

ω−ω+=θ−θ

.

(2.11)

By separating the real part from the imaginary part, we obtain the following

expressions:

acc

AGA)A(fG)cos(I

ω+−=θ−θ 10

1101

acc

AGA)A(fG)cos(I

ω−=θ−θ 20

2202

110101

101 φω

+ω

ω−ω−=θ−θ A

GAG)sin(I

aacc

220202

202 φω

−ω

ω−ω=θ−θ A

GAG)sin(I

aacc .

(2.12)

Thus, from the equations above we can deduce 1A , 2A , 1θ and 2θ , as follows:

)cos(G

IA)A(fA c

caa θ−θω+ω= 1

0111

)sin(AG

Ic

ca θ−θω

−ω=θ 110

011

)cos(G

IA)A(fA c

caa θ−θω−ω= 2

0222

(2.13)


43

)sin(AG

Ic

ca θ−θω+ω=θ 220

022 .

In (2.13), the dynamic equations for the amplitude and the phase of two coupled

oscillators are expressed [42]. Therefore, the next step is to find the dynamic equations

for the amplitude and the phase of the coupling current, Ic. The admittance of the

coupling circuit, Yc has the following expression:

ωωω−ω

−

=

ω+ω+

=ω

acc

ccc

ccccc

cc

jR

CjLjR

)(Y

21

111

122

0

, (2.14)

where ccCLc

10 =ω is the resonance pulsation of the coupling circuit, and

c

cac L

R=ω2 is the bandwidth of the coupling circuit. If we suppose that the

synchronization frequency of the coupling circuit ωc is close to the resonance pulsation

of the coupling circuit ω0c, then (2.14) can be written as follows:

)(D

)(N

jR

)(Ycc

cc

ac

ccccc ω

ω=

ωω−ω

−

≅ω01

11

. (2.15)

Thus, the coupling current can be written as follows:

)](V)(V[)(D

)(N)](V)(V)[(Y)(I

c

ccc ω−ω×

ωω

=ω−ωω=ω 1212 , (2.16)

and

)](V)(V[)(N)(I)(D ccc ω−ω×ω=ω×ω 12 . (2.17)

By applying a Taylor series development about ωc, to Dc(ω), we obtain:


44

( )1212

1 θθθ

ω−=

ω−ω

ω+ω jj

c

jcc

ccc eAeA

ReI)(

d

dD)(D c

c

. (2.18)

Hence,

( ) ( )[ ])sin(j)cos(A)sin(j)cos(AR

II

Ij

jj cccc

cc

c

cc

acac

cc θ−θ+θ−θ−θ−θ+θ−θ=

−φ

ω+

ωω−ω− 111222

0 11

.

(2.19)

By separating the real part from the imaginary part, we can deduce the dynamic

equations of the amplitude and the phase of the coupling current, so that:

[ ])cos(A)cos(AR

II ccc

accacc θ−θ−θ−θ

ω+ω−= 1122

[ ].

cccc

accc )sin(A)sin(A

IRθ−θ−θ−θω+ω=θ 11220

.

(2.20)

Let us remind that the purpose of this work is to steer the beam of an antenna

array in a required direction. This direction leads to a phase gradient that exists between

the signals applied to adjacent elements of the array. This phase gradient is controlled

by the free-running frequencies of the oscillators which are located at the extremes of

the array. However, an important condition must be satisfied in order to achieve this

scope: the oscillators of the array must be synchronized at a common frequency. Thus,

we will present here a system of equations which defines the synchronization states of

the oscillators, using the dynamic equations written above.

Let us first define the notations Acx and Acy, from the following expression of the

coupling current:

)sin(jI)cos(IeI ccccj

cc θ+θ=θ

. (2.21)

So that,


45

)cos(G

IA c

ccx φ=

0

)sin(G

IA c

ccy φ=

0,

(2.22)

where

)t(t)t( cc φ+ω=θ , )t(t)t( 11 φ+ω=θ , )t(t)t( 22 φ+ω=θ . (2.23)

Thus, using the notations above, and for a Van der Pol oscillator so that

)A()A(f 21−= , the equations (2.13) can be written as follows:

[ ])sin(A)cos(AA)A(A cycxaa 111211 1 φ+φω+−ω=

[ ])cos(A)sin(AA cycx

a11

1011 φ−φ

ω−ω−ω=φ

[ ])sin(A)cos(AA)A(A cycxaa 222222 1 φ+φω−−ω=

[ ])cos(A)sin(AA cycx

a22

2022 φ−φ

ω+ω−ω=φ

)sin(G

I)cos(

G

IA cc

cc

ccx φφ−φ=

00

)cos(G

I)sin(

G

IA cc

cc

ccy φφ+φ=

00,

(2.24)

where ω−θ=φ 11 , ω−θ=φ 22

and ω−θ=φ cc .

Replacing cI and cφ from the equations (2.20), we obtain the following

expressions for cxA and cyA :


46

[ ])cos(A)cos(AA)(AA accyccxaccx 112200 φ−φλω+ω−ω+ω−=

[ ])sin(A)sin(AA)(AA accxccyaccy 112200 φ−φλω+ω−ω−ω−= ,

(2.25)

with the coupling constant 0

01

GRc

=λ .

The system is synchronized if the variations in time of amplitudes and phases are

null. Thus, the synchronization states can be determined by canceling the equations

(2.24). Under these conditions, cxA = 0 and cyA = 0, leading to:

))cos(A)cos(A(A)(A accyccxac 112200 φ−φλω=ω−ω−ω

))sin(A)sin(A(A)(A accxccyac 112200 φ−φλω=ω−ω+ω ,

(2.26)

We can see that the equations above can be put in matrix form, and Cramer's law

allows finding the expressions for Acx and Acy as shown below:

)]sin(A)sin(A[)]cos(A)cos(A[Aac

ccx 1122

201122

20 φ−φελ+φ−φελ= ω

ω∆

)]cos(A)cos(A[)]sin(A)sin(A[Aac

ccy 1122

201122

20 φ−φελ−φ−φελ= ω

ω∆ ,

(2.27)

where 2

1

1

ωω∆

+

=ε

ac

c

is the coupling strength scale factor and )Yarctan( 1=φ∆

is difference between the synchronization pulsation and the resonance pulsation of the

coupling circuit.

Now setting the other derivatives equal zero, so that 01 =A , 01 =φ , 02 =A and

02 =φ , gives the algebraic equations describing the locked states that, after eliminating

the coupling variables Acx and Acy, can be written as:


47

)cos(AA)A( Φ−φ∆ελ−=ελ−− 2012

0211

)sin(A

Aac

ac

a Φ−φ∆εωλ−=ω∆

ωω

ελ−−ω∆1

20

2001 1

)cos(AA)A( Φ+φ∆ελ−=ελ−− 1022

0221

)sin(A

Aac

ac

a Φ+φ∆εωλ=ω∆

ωω

ελ−−ω∆2

10

2002 1

,

(2.28)

where

ωω∆

=Φac

carctan is the coupling phase, and the parameters ω01 and ω02,

are used to express the free running frequencies of the oscillators, ω01 and ω02, and the

resonance pulsation of the coupling circuit ω0c , as shown below:

c00101 ω−ω=ω∆

c00202 ω−ω=ω∆ .

Thus, a solution to (2.28) indicates the existence of a frequency-locked state and

provides the user with the amplitudes A1 and A2 of the two oscillators as well as the

inter-stage phase shift φ∆ and the synchronization pulsation ω for a combination

( )0201 ω∆ω∆ , .

Furthermore, let us note that the ability of the oscillators to synchronize

increases with the coupling strength and is maximum when the coupling phase is 0° or

180°. This can be demonstrated by expressing the scale factor of the coupling strength

as depending of the coupling phase Φ: 21

1

)][tan(Φ+=ε . Thus, for Φ = 0° or Φ = 180°,

ε = 1, the scale factor of the coupling strength is maximum. However, for a coupling

phase Φ = ±90°, ε = 0 and the ability of the oscillators to synchronize is then minimal.

Therefore, we can note that the ability of the oscillators to synchronize depends

critically on the phase of the coupling and thus on the proximity between the


48

synchronization frequency and the resonant frequency of the coupling circuit compared

to the bandwidth of the coupling circuit.

Nevertheless, to simplify the analysis, J. Lynch and R. York made use of cases of

practical interest according to the tunings of oscillators 1 and 2 [19]. Indeed, for equal

tunings, i.e. for 0201 ω∆=ω∆ , and using the second and fourth equations of (2.28), one

can show that the phase difference φ∆ equals zero and the oscillators will always lock

no matter how far away from the origin we tune with equal amplitudes ( 21 AA = ).

On the other hand, when the coupling circuit resonance ( ocω ) is located exactly

between the two oscillators free-running frequencies, i.e. for 0201 ω∆−=ω∆ , one can

show, again using the second and fourth equations of (2.28), that 0 =ω∆ c , which

implies 0 =Φ and 21 AA = . In these conditions, the frequency difference between the

two free-running frequencies of oscillators 1 and 2 can be expressed, by subtracting the

fourth and second equations, as follows:

( )φ∆ωλ=ω∆ sina00 2 , (2.29)

with 01020 ω−ω=ω∆ as mentioned previously.

Furthermore, in the same conditions, the amplitudes are equal and found from

(2.28), by combining the first and third equation, as follows:

. 2

111 2

0

00

2

ωλω∆

−−λ−=a

A (2.30)

2.3. DYNAMICS OF TWO VAN DER POL OSCILLATORS

COUPLED THROUGH A BROAD-BAND CIRCUIT

Let us remember that, R. York described in [19] the oscillators’ dynamic

equations, as well as those for the amplitude and phase of the coupling current. Then, by

setting the derivatives equal to zero, the algebraic equations describing the oscillators’


49

frequency locked states, for two Van der Pol oscillators coupled through a RLC circuit,

were obtained as detailed in the previous section. This system of four equations with

four unknowns is reminded as follows:

)cos(AA)A( Φ−φ∆ελ−=ελ−− 2012

0211

)sin(A

Aac

ac


ωω

ελ−−ω∆1

20

2001 1

)cos(AA)A( Φ+φ∆ελ−=ελ−− 1022

0221

)sin(A

Aac

ac


ωω

ελ−−ω∆2

10

2002 1

,

The ability of these coupled oscillators to synchronize to a common frequency is

affected by the following parameters [10]:

• 2

1

1

ωω∆

+

=ε

ac

c

: coupling strength scale factor;

•

ωω∆=Φ −

ac

ctan 1 : coupling phase;

• cRG0

01=λ : coupling constant, where 0G is the first-order term of Van der Pol

nonlinear conductance;

• C

Ga

02 =ω : the oscillators’ bandwidth;

• c

cac L

R=ω2 : bandwidth of the unloaded coupling circuit;

• ω01, ω02 : free-running frequencies or tunings of oscillators 1 and 2,

respectively;

• ω0c : resonant frequency of the coupling circuit;


50

• c00101 ω−ω=ω∆ c00202 ω−ω=ω∆ cc 0ω−ω=ω∆ .

In these conditions, the Broad-band case is found by letting the coupling circuit

bandwidth approach infinity so that ωac →∞, which means that the coupling circuit is

made of one resistor for instance [18]. Thus, if ωac →∞, this implies ε = 1 and Φ = 0. In

these conditions, the new equations describing the locked states can be written as:

)cos(AA)A( φ∆λ−=λ−− 2010211

)sin(A

Aac φ∆ωλ−=ω∆−ω∆

1

2001

)cos(AA)A( φ∆λ−=λ−− 1020221

)sin(A

Aac φ∆ωλ=ω∆−ω∆

2

1002

.

(2.31)

Since ω−ω=ω∆−ω∆ 0101 c and ω−ω=ω∆−ω∆ 0202 c ,

Thus, we have:

)cos(AA)A( φ∆λ−=λ−− 2010211

)sin(A

Aa φ∆ωλ−ω=ω

1

2001

)cos(AA)A( φ∆λ−=λ−− 1020221

)sin(A

Aa φ∆ωλ+ω=ω

2

1002

.

(2.32)


51

2.4. NEW FORMULATION OF THE EQUATIONS DESCRIBING

THE LOCKED STATES OF TWO VAN DER POL COUPLED

OSCILLATORS ALLOWING A MORE ACCURATE

PREDICTION OF THE AMPLITUDES

2.4.1. Two Van der Pol oscillators coupled through a resonant network

The purpose of this section is to propose a new formulation of the equations

describing the locked states of two Van der Pol oscillators coupled through a resonant

network using an accurate model allowing a good prediction of the oscillators’

amplitudes.

Indeed, let us remember that the Van der Pol model used by R. York in [19] is

described by the following equation:

200 AGG)A(G +−= ,

(2.33)

where G0 is the device conductance at zero voltage.

Nevertheless, the free-running amplitude of such a Van der Pol oscillator is

shown to be equal to 1 irrespective of the value of G0. As a consequence, this model

doesn’t allow an accurate prediction of the amplitudes of the two coupled oscillators.

To illustrate this affirmation, let us explain the assumptions and approximations

made by J. Lynch and R. York in [19] for the modelling of a Van der Pol oscillator.

Indeed, let us first consider the schematic of a Van der Pol oscillator as shown in Figure

2.2 for which the current in the nonlinear conductance GNL is equal to ( ) ( ) ( )tbvtavti 31 +−=

with –a the negative conductance necessary to start the oscillations and b a parameter

used to model the saturation phenomenon. In this case, according to the Kirchhoff's

current law, the current i(t) can be written as:

)t(v.b)t(v.Ga)t(i 3 )( ++−= . (2.34)


52

Figure 2.2. Van der Pol oscillator model

Now, using the assumption of a perfectly sinusoidal oscillation so that

( ) ( )tcos.Atv 0 ω= , from (2.34) we have:

)tcos(A

btcos.A.bAGa)t(i 0

3

02 3

4 )(

4

3 ω+ω

++−= . (2.35)

Furthermore, let us recall that such an oscillator topology can be modelled by a

quasi-linear representation allowing a very simple analytical calculation [43]. In this

case, the expression in the first bracket in (2.35) represents the negative conductance

presented by the active part. Nevertheless, in York’s model [19], this negative

conductance is equal to (2.33).

In these conditions, the expression in the first bracket in (2.35) and (2.33) leads

to:

bGaG4

3 0 =−= . (2.36)

This result clearly shows the limitations of the Van der Pol model used by Lynch

and York in [19] since this model is valid only for Van der Pol parameters values a and

b for which (2.36) is fulfilled. Furthermore, one can show that, in this case, the free-

running amplitude of the Van der Pol oscillator is always equal to unity. Indeed, the

condition of free-running oscillation of one of the oscillators of Figure 2.1 at a

frequency f0 can be written as:

G

v(t)

GNL L C

i(t)

i1(t) i2(t)


53

0 )( )( 00 =ω+ω LNL Y,AY . (2.37)

where the subscripts “NL” and “L” denote the nonlinear and linear portions of

the circuit and with )A(G)A(G,AYNL2

00 1 )( −−==ω and

ω−ω=

LCjYL

1 .

Thus, according to (2.37), the frequency of oscillations is found to be

LCf

π=

2

10 and the amplitude is 1 =A .

As a consequence, a new formulation of equations (2.28) using a more accurate

Van der Pol model will be presented in the following. The case of a resistive coupling

circuit will then be deduced from the obtained system of equations.

Hence, let us now consider the circuit of Figure 2.3 made of two Van der Pol

oscillators coupled through a RLC circuit. These oscillators are considered identical,

except of their free-running frequencies.

Figure 2.3 – Two Van der Pol oscillators coupled through a series RLC circuit

Therefore, each oscillator is modeled by the negative conductance -Gd(Ai),

associated to a parallel resonant circuit. These two oscillators are coupled through a

series resonant circuit, as shown in Figure 2.3.

In these conditions, the coupling current Ic can be expressed in terms of the

admittance of the oscillator 1, Y1(A1,ω1), of the admittance of the oscillator 2, Y2(A2,ω2),

or depending of the admittance of the coupling circuit, Yc(ωc):


54

10111

1111 21 V)(jCG

)A(GGV),A(YI

L

dLc

ω−ω+

−≅ω=

20221

2222 21 V)(jCG

)A(GGV),A(YI

L

dLc

ω−ω+

−−≅ω−=

)VV)((YI ccc 21 −ω−= ,

(2.38)

where 2

4

3iid bAa)A(G +−=− is the negative conductance presented by the active

part as explained in (2.35) and V1 and V2 are the output voltages of oscillator 1 and

respectively, oscillator 2.

Now using the notation below:

⇒−1L

id

G

)A(G)A(Si

Ai

i

i µ=

α−µ

2

2

1 ,

where L

L

G

Ga −=µ and

b

)Ga(

A)A(Si

L

ii

3

41

2

−−= , with a and b the parameters of the

Van der Pol nonlinearity.

The admittance of oscillator 1, Y1(A1,ω1), can be expressed as follows:

ωω−ω

+µ−=ωa

L)(

j)A(SiG),A(Y 1011111 , (2.39)

where CL1

011=ω is the resonance pulsation of the resonator of oscillator 1,

C

GLa =ω2 is the resonator bandwidth and 1101011 2ω≅ω+ω⇒ω≅ω .

Similarly, the admittance of oscillator 2, Y2(A2,ω2), is expressed in the following

equation:


55

ωω−ω

+µ−=ωa

L)(

j)A(SiG),A(Y 2022222 , (2.40)

where 2202022 2ω≅ω+ω⇒ω≅ω andCL2

021=ω is the resonance pulsation of

the resonator of oscillator 2.

Knowing the expressions (2.39) and (2.40) of the admittances, the same steps as

in section 2.2 are followed. Hence, to determine the expression of the current through

the coupling circuit, the output voltages of oscillator 1 and 2, are expressed as in

equation (2.41):

)t(je)t(AV 111

θ=

)t(je)t(AV 222

θ= ,

(2.41)

where )t(t)t( 111 φ+ω=θ and )t(t)t( 222 φ+ω=θ .

Then,

)t(j

.

e)A

Aj(jA

dt

Vd11

1

1111

1 φ+ω−φ+ω=

)t(j

.

e)A

Aj(jA

dt

Vd22

2

2222

2 φ+ω−φ+ω=

.

(2.42)

Thus, using the expression in brackets for the pulsation ω, so that ω = ω1,2 + δω with

δω=−φ

.

A

Aj

1

11

or δω=−φ

.

A

Aj

2

22

allows (2.39) and (2.40) to be expanded in a Taylor series

about the pulsations ω1,2 in order to study the behavior around the oscillation pulsation

ω1,2. Thus, for δω ‹‹ ω1,2, we have:


56

1

1

11

11

11111

θ

ω

θ

−φ

ωω

+ω= jjc eA

A

Aj

d

),A(dY),A(YeI c

2

2

22

22

22222

θ

ω

θ

−φ

ωω

+ω−= jjc eA

A

Aj

d

),A(dY),A(YeI c

.

(2.43)

If we divide the expressions (2.43) by 1θje and respectively 2θje , and replace the

derivate of the admittance depending on the pulsation ω, we obtain:

aL

aL

aLL

)(jc

AGAjGAjGA)A(SiGeI c

ω+

ωφ+

ωω−ω

−µ−=θ−θ 11

11

10111

1

aL

aL

aLL

)(jc

AGAjGAjGA)A(SiGeI c

ω−

ωφ−

ωω−ω

+µ=θ−θ 22

22

20222

2

.

(2.44)

By separating the real part from the imaginary part, we obtain the following

expressions:

aLLcc

AGA)A(SiG)cos(I

ω+µ−=θ−θ 1

111

aLLcc

AGA)A(SiG)cos(I

ω−µ=θ−θ 2

222

11

1101

1 AGAG)sin(Ia

La

Lcc ωφ+

ωω−ω

−=θ−θ

22

2202

2 AGAG)sin(Ia

La

Lcc ωφ−

ωω−ω

=θ−θ

,

(2.45)

Thus, from the equations above we can deduce 1A , 2A , 1θ and 2θ , as follows:

)cos(G

IA)A(SiA c

L

aca θ−θω+ωµ= 1111

(2.46)


57

)sin(AG

Ic

L

ac θ−θω

−ω=θ 11

011

)cos(G

IA)A(SiA c

L

aca θ−θω−ωµ= 2222

)sin(AG

Ic

L

ac θ−θω

+ω=θ 22

022

.

Since the coupling circuit is the same, the dynamic equations for the amplitude

and the phase of the coupling current remain the same, so that:

[ ])cos(A)cos(AR

II ccc

accacc θ−θ−θ−θω+ω−= 1122

[ ].

cccc

accc )sin(A)sin(A

IRθ−θ−θ−θ

ω+ω=θ 11220

.

(2.47)

Now, with

)cos(G

IA c

L

ccx φ=

)sin(G

IA c

L

ccy φ= ,

(2.48)

and

)t(t)t( cc φ+ω=θ )t(t)t( 11 φ+ω=θ )t(t)t( 22 φ+ω=θ , (2.49)

the equations (2.46) can be written as follows:

)]sin(A)cos(A[A)A(SiA cycxaa 11111 φ+φω+ωµ=

(2.50)


58

)]cos(A)sin(A[A cycx

a11

1011 φ−φ

ω−ω−ω=φ

)]sin(A)cos(A[A)A(SiA cycxaa 22222 φ+φω−ωµ=

)]cos(A)sin(A[A cycx

a22

2022 φ−φ

ω+ω−ω=φ

)sin(G

I)cos(

G

IA cc

L

cc

L

ccx φφ−φ=

)cos(G

I)sin(

G

IA cc

L

cc

L

ccy φφ+φ=

.

where ω−θ=φ 11 , ω−θ=φ 22

and ω−θ=φ cc .

Replacing cI and cφ from the equations (2.47), we obtain the following

expressions for cxA and cyA :

[ ])cos(A)cos(AA)(AA accyccxaccx 112200 φ−φλω+ω−ω+ω−=

[ ])sin(A)sin(AA)(AA accxccyaccy 112200 φ−φλω+ω−ω−ω−= ,

(2.51)

with the coupling constant LcGR

10 =λ .

The system is synchronized if the variations in time of the amplitudes and the

phases are null. Thus, the synchronization states can be determined by canceling the

equations (2.51). Under these conditions, cxA = 0 and cyA = 0, leading to:

))cos(A)cos(A(A)(A accyccxac 112200 φ−φλω=ω−ω−ω

))sin(A)sin(A(A)(A accxccyac 112200 φ−φλω=ω−ω+ω .

(2.52)

We can see that the equations above can be put in matrix form, and Cramer's law

allows finding the expressions for Acx and Acy as shown below:


59

)]sin(A)sin(A[)]cos(A)cos(A[Aac

ccx 1122

201122

20 φ−φελ+φ−φελ= ω

ω∆

)]cos(A)cos(A[)]sin(A)sin(A[Aac

ccy 1122

201122

20 φ−φελ−φ−φελ= ω

ω∆ ,

(2.53)

where 2

1

1

ωω∆

+

=ε

ac

c

is the coupling strength scale factor and cc 0ω−ω=ω∆ is

difference between the synchronization pulsation and the resonance pulsation of the

coupling circuit.

Now setting the derivatives equal zero, so that01 =A , 01 =φ , 02 =A and 02 =φ ,

gives the algebraic equations describing the locked states that, after eliminating the

coupling variables Acx and Acy, can be written as:

)cos(AA])A(Si[ Φ−φ∆ελ−=ελ−µ 2012

01

)sin(A

Aac

ac


ωω

ελ−−ω∆1

20

2001 1

)cos(AA])A(Si[ Φ+φ∆ελ−=ελ−µ 1022

02

)sin(A

Aac

ac


ωωελ−−ω∆

2

10

2002 1

,

(2.54)

where

ωω∆

=Φac

carctan is the coupling phase, and c00101 ω−ω=ω∆ ,

c00202 ω−ω=ω∆ .

Thus, for the specific case of 0201 ω∆−=ω∆ , for which the coupling circuit

resonance is located exactly between the two oscillators free-running, one can show,

using (2.54), that 00 =ω−ω=ω∆ cc , which implies 1=ε , 0=Φ and A1 = A2 = A. As a


60

consequence, the frequency difference between the two free-running frequencies of

oscillators 1 and 2 can also be expressed using (2.29) but with LcGR

10 =λ and

C

GLa =ω2 .

Nevertheless, since A1=A2= A then, from the system of equations above, we can

obtain the following expression:

)cos()A(Si φ∆λ−=λ−µ 00 . (2.55)

Since L

L

G

Ga −=µ and

b

)Ga(A

)A(SiL

3

41

2

−−= , replacing in equation (2.55) these

values, the new expression for the amplitude of both oscillators is:

)])(cos([Gab

A L 113

40 −φ∆λ−−= .

(2.56)

One can notice that this expression is different from the one obtained in equation

(2.30) and allows the prediction of the amplitudes of the two coupled Van der Pol

oscillators for any values of parameters a and b of the Van der Pol non linearity under a

sinusoidal assumption.

2.4.2. Resistive coupling case

As already mentioned in section 2.3, the case of a resistive coupling circuit is

found by letting the coupling circuit bandwidth approach infinity so that ∞→ωac

leading to 1 =ε and 0 =Φ . In these conditions, the new equations describing the locked

states can be deduced from equations (2.54) as follows :

)cos(AA))A(S( i φ∆λ−=λ−µ 20101

)sin(A

Aa φ∆ωλ−ω=ω

1

2001

)cos(AA))A(S( i φ∆λ−=λ−µ 10202

)sin(A

Aa φ∆ωλ+ω=ω

2

1002 .

(2.57)


61

2.5. NEW FORMULATION OF THE EQUATIONS DESCRIBING

THE LOCKED STATES OF TWO VAN DER POL COUPLED

OSCILLATORS ALLOWING AN EASIER NUMERICAL

SOLVING METHOD

Once more, a solution to (2.54) indicates the existence of a frequency-locked

state for the two van der Pol coupled oscillators. However, in this case, one can obtain

the oscillators’ amplitudes A1 and A2 for any values of parameters a and b of the Van

der Pol non linearity under the assumption of a sinusoidal behavior as well as the inter-

stage phase shift φ∆ and the synchronization pulsation ω for a combination ( 0201 ω∆ω∆ ,

).

Nevertheless, due to the trigonometric and non-linear aspect of (2.54), the

solutions of this system are very difficult or impossible to obtain. In these conditions,

mathematical manipulations are applied to the equations in order to obtain a new system

of three equations with three unknowns A1, A2 and ω, easier to solve [10]. These

equations will allow then to determine the phase shift φ∆ according to the pulsations

ω01 and ω02. First we will write the equations (2.54) in another form, as shown below:

20

12

01

A

A])A(Si[)cos(

ελελ−µ−=Φ−φ∆

1

20

2001 1

A

A)sin(

a

cac

a

εωλ

ω∆

ωωελ−−ω∆

−=Φ−φ∆

10

22

02

A

A])A(Si[)cos(

ελελ−µ−=Φ+φ∆

2

10

2002 1

A

A)sin(

a

cac

a

εωλ

ω∆

ωω

ελ−−ω∆=Φ+φ∆ ,

(2.58)


62

where 14

3 2

−−

=µLG

bAa)A(Si . Replacing Φ and ε by their expressions leads to the

following system of equations:

22

0

120

1

1

1

A

A)A(Si

arctancos

ac

c

ac

c

ac

c

ωω∆

+

λ

ωω∆

+

λ−µ

−=

ωω∆

−φ∆

1

2

2

0

20

01

1

1

1

A

Aarctansin

ac

c

a

cac

a

ac

c

ac

c

ωω∆

+

ωλ

ω∆

ωω

ωω∆

+

λ−−ω∆

−=

ωω∆

−φ∆

12

0

220

2

1

1

A

A)A(Si

arctancos

ac

c

ac

c

ac

c

ωω∆

+

λ

ωω∆

+

λ−µ

−=

ωω∆

+φ∆

2

1

2

0

20

02

1

1

1

A

Aarctansin

ac

c

a

cac

a

ac

c

ac

c

ωω∆

+

ωλ

ω∆

ωω

ωω∆

+

λ−−ω∆

=

ωω∆

+φ∆ .

(2.59)


63

A first simplification of the equations is performed by using the following

notations:

20

1

1

ωω∆

+

λ=

ac

c

X

2

02

1

ωω∆

+

λ=

ac

c

X .

Thus, the equations (2.59) can be written in the form below:

[ ]22

111

AX

AX)A(Siarctancos

ac

c −µ−=

ωω∆−φ∆ (2.60.a)

1

22

101 1

A

AX

X

arctansin

a

cac

a

ac

c

ω

ω∆

ωω

−−ω∆−=

ωω∆

−φ∆

(2.60.b)

[ ]12

212

AX

AX)A(Siarctancos

ac

c −µ−=

ωω∆+φ∆

(2.60.c)

2

12

102 1

A

AX

X

arctansin

a

cac

a

ac

c

ω

ω∆

ωω

−−ω∆=

ωω∆

+φ∆ .

(2.60.d)

Starting from this form, using combinations between the equations, we can

obtain a system of dynamic equations reduced in size and easier to solve. Thus, as the

sum of the squares of the two equations (2.60.a) and (2.60.b) is equal to unity, we can

write:


64

⇒=

ωω∆

−φ∆+

ωω∆

−φ∆ 122

ac

c

ac

c arctansinarctancos

[ ]1

1

2

1

2222

2

101

22

22

21

211 =

ω

ω∆

ωω−−ω∆

+−µ

A

AX

X

AX

AX)A(Si

a

cac

a

.

(2.61)

The same operation is used for the last two equations (2.60.c) and (2.60.d):

⇒=

ωω∆

+φ∆+

ωω∆

+φ∆ 122

ac

c

ac

c arctansinarctancos

[ ]1

1

2

2

1222

2

102

21

22

22

212 =

ω

ω∆

ωω−−ω∆

+−µ

A

AX

X

AX

AX)A(Si

a

cac

a

.

(2.62)

Now, equation (2.60.b) is divided by equation (2.60.a):

⇒

ωω∆

−φ∆=

ωω∆

−φ∆

ωω∆

−φ∆

ac

c

ac

c

ac

c

arctantan

arctancos

arctansin

a

cac

a

ac

c

ac

c

]X)A(Si[

X

)tan(

)tan(

ω−µ

ω∆

ωω−−ω∆

=

ωω∆φ∆+

ωω∆−φ∆

11

101 1

1 .

(2.63)

In the same way equation (2.60.d) is divided by equation (2.60.c):


65

⇒

ωω∆

+φ∆=

ωω∆

+φ∆

ωω∆+φ∆

ac

c

ac

c

ac

c

arctantan

arctancos

arctansin

a

cac

a

ac

c

ac

c

]X)A(Si[

X

)tan(

)tan(

ω−µ

ω∆

ωω−−ω∆

−=

ωω∆φ∆−

ωω∆+φ∆

12

102 1

1 .

(2.64)

At this step, another simplification of the equations is performed by using the

following notations:

a

cac

a

]X)A(Si[

X

Yω−µ

ω∆

ωω

−−ω∆=

11

101

1

1

a

cac

a

]X)A(Si[

X

Yω−µ

ω∆

ωω−−ω∆

=12

102

2

1

.

Therefore, equations (2.61) and (2.62) can be written as shown below:

22

22

21

21

211 1 AX)Y(A]X)A(Si[ =+−µ

21

22

22

22

212 1 AX)Y(A]X)A(Si[ =+−µ .

(2.65)

In the same way, using the notations above, equations (2.63) and (2.64) can be

written as follows:

ac

c

ac

c

Y

Y

)tan(

ωω∆

−

ωω∆

+=φ∆

1

1

1 ,

(2.66)


66

ac

c

ac

c

Y

Y

)tan(

ωω∆

−

ωω∆

+−=φ∆

2

2

1 .

(2.67)

The ratio of these two expressions is equal to unity, and therefore we can write:

2121

2

21 12 YY)YY()YY(ac

c

ac

c +=

ωω∆

−+

ωω∆

+ .

After this mathematical re-formulation, a new system of three equations with

three unknowns, A1, A2 and ωc, is presented below:

22

22

21

211

21 1 AX)Y(]X)A(Si[A =+−µ

21

22

22

212

22 1 AX)Y(]X)A(Si[A =+−µ

2121

2

21 12 YY)YY()YY(ac

c

ac

c +=

ωω∆

−+

ωω∆

+ .

(2.68)

In this new system of equations there is no trigonometric aspect, thus, it can be

much easier to solve. Once, this system of equations is solved, the solutions are

replaced in the expressions (2.66) or (2.67) to obtain the phase shift between two

adjacent elements of the array:

ωω∆

−

ωω∆

+=φ∆

ac

c

ac

c

Y

Y

arctan

1

1

1 ,

(2.69)


67

ωω∆

−

ωω∆

+−=φ∆

ac

c

ac

c

Y

Y

arctan

2

2

1 .

(2.70)

The Broad-band case

Let us remember that the Broad-band case implies ωac →∞, which means that the

coupling circuit is made of one resistor for instance.

In this case, replacing ωac→∞ in simplifications above, we obtain new

expressions for X1, X2, Y1 and Y2 as shown below:

01 λ=X

02 λ=X

a])A(Si[Y

ωλ−µω−ω

=01

011

a])A(Si[Y

ωλ−µω−ω=

02

022 .

Thus, the broad-band case implies a new system of three equations with three

unknowns, A1, A2 and ω, presented below:

22

20

21

201

21 1 A)Y(])A(Si[A λ=+λ−µ

21

20

22

202

22 1 A)Y(])A(Si[A λ=+λ−µ

021 =+YY .

(2.71)

Let us note that, this new system of equations doesn’t present a trigonometric

aspect as well, thus, it can be much easier to solve with calculation tools such as

MATLAB. Once this system of equations is solved, the solutions are replaced in the


68

expressions (2.66) or (2.67) to obtain the phase shift between two adjacent elements of

the array:

)Yarctan( 1=φ∆ , (2.72)

)Yarctan( 2−=φ∆ . (2.73)

2.6. CAD TOOL “ASVAL”

The purpose of this section is to present our CAD tool named ASVAL ("Analysis

of Synchronized VCOs by Angoulême Laboratory "), developed in MATLAB and

allowing to plot the cartography of the synchronization area of two coupled oscillators.

Thus, with this cartography and with the precise free-running frequencies, it is possible

to extract the value of the phase shift between the output voltages of the oscillators as

well as their amplitudes. Therefore, this tool can determine the free-running frequencies

required to achieve the desired phase shift between the output voltages of the two

coupled oscillators. In these conditions, with this cartography, it will be relatively

simple to control the radiation pattern of an antenna array, when applying the control

voltages to the outermost VCOs of the array, via a microcontroller, a DSP or a FPGA.

2.6.1. The objective of “ASVAL”

The primary objective of this work is to develop graphics or cartographies

illustrating the behavior of two coupled oscillators or VCOs. Then it is possible to

extract the values of the command voltages necessary to obtain the required phase shift

between the signals applied to each antenna element of a linear array. In these

conditions, one can control the orientation of the radiation pattern in the required

direction:

φ∆πλ=θd

arcsin2

,


69

where λ is the wavelength and d is the distance between two adjacent antennas.

Let us note that, it is difficult to obtain this cartography with the existing

simulation methods. Indeed, a transient analysis provides the waveforms but before any

use of the results, it is important to ensure that the steady state is obtained and then to

determine the phase difference of the coupling circuit using Fourier transforms. This

analysis should be performed for a large number of frequency values and therefore for a

large number of points. On the other hand, the results are very accurate because there is

no approximation of the model in the calculation, and this temporal analysis will

validate certain points from the cartography.

Another type of analysis is the harmonic balance method, using Agilent’s ADS

software for example. But one more time, a full harmonic balance synthesis of coupled-

oscillator system is computationally expensive as explained in section 1.3 of chapter 1.

The synchronization states of the coupled oscillators occur only if the values of

the free-running frequencies are within a specific locking-range. The synchronization

states can be illustrated graphically by obtaining the cartography of the synchronization

area.

To do so, the system of equations (2.71) has been implemented in MATLAB, and

using the Marquardt algorithm, explained in detail in Appendix A, the equations were

solved using nonlinear programming technique explained in detail in the following

subsection 2.6.2. Once the equations are solved, the tool, very quickly, provides the

curves describing the variations of the amplitudes A1 and A2 of the voltages V1 and V2,

at the output of the two oscillators, the phase shift φ∆ between these voltages and the

synchronization frequency fS, according to the free-running frequencies, f01 and f02.

Such a plot might look like Figure 2.4, where the origin represents the point where the

free-running pulsations of the oscillators 1 and 2 are equal to the synchronization

pulsation: ω01 = ω02 = ω.


70

Figure 2.4 – The graphical representation of a synchronization area.

The hatched region represents the synchronization area of the coupled

oscillators. In other words, outside of this region, the oscillators are unable to synchronize.

Thus, if we impose to the outermost VCOs of the array free-running pulsations values

inside the hatched area, then, these oscillators are synchronized at a common pulsation ω.

However, let us remember that the purpose is to impose a specific value for the

phase shift between the coupled oscillators. Solving the new system of equations (2.71),

we will be able to deduce, for each pair of pulsation (ω01, ω02), the following:

• the synchronization frequency, ω;

• the amplitude A1 of the voltage V1 of the oscillator 1;

• the amplitude A2 of the voltage V2 of the oscillator 2;

• the phase shift φ∆ between V1 and V2.

Thus, the cartography of the phase shift is very useful in the context of

controlling the orientation of the radiation pattern of an antenna array, because it allows

the determination of the free-running frequencies, (f01, f02), able to provide the desired

phase shift value.


71

2.6.2. Variables estimation technique

In this part, the problem of identifying the variables (or the roots) of the system

of equations (2.71) is discussed. The basic formulation of the general problem is the

following: given a nth dimension system of nonlinear coupled functions )(fk θ , we seek

the value of the variables’ vector θ for which ( n1,k 0 ==θ ,)(fk ).

In the case of the two Van der Pol coupled oscillators model given previously, a

new root finding procedure based on parameter estimation technique is developed. Let

us consider the previous mathematical model given by equations (2.71). The vector of

variables to be estimated is:

T

a

AA

ωω=θ 2

221 , (2.74)

Where [ .] T denotes a transposition operation.

Furthermore, the nonlinear coupled functions )(fk θ are found from (2.71) as:

( ) 222

20

2012

2

021

211 1

4

3 L

aaLL GAGGbAaA)(f λ−

ωω−

ωω

+

λ+−−=θ , (2.75)

( ) 221

20

2022

2

022

222 1

4

3 L

aaLL GAGGbAaA)(f λ−

ωω−

ωω

+

λ+−−=θ , (2.76)

( ) ( )

λ+−−

ωω−

ωω

+

λ+−−

ωω−

ωω

=θ 021

020

22

013 1

4

3 1

4

3 L

aaL

aa

GbAaGbAa)(f . (2.77)

As a general rule, estimation with Output Error technique [44, 45] is based on

minimization of a quadratic multivariable criterion defined as:


72

∑=

θ=3

1

2

kk )(fJ . (2.78)

Unlike linear equations, most of the nonlinear equations cannot be solved in a

finite number of steps. Iterative methods are being used to solve nonlinear equations.

For criterion minimization, the variables values θ can be driven iteratively to the

optimum by Non Linear Programming techniques. Practically, we are using Marquardt’s

algorithm [46] for off-line estimation:

[ ] i

ˆii J.I.Jˆˆθ=θθ

−θθ+ ′λ+′′−θ=θ

1

1 , (2.79)

where

θ is an estimation of the system solutions θ ,

∑=

θθ σ−=θ∂

∂=′3

1

2k

,kk .f.J

J is the gradient of criterion,

∑=

θθθθ σσ≈θ∂

∂=′′3

12

2

2k

T,k,k ..

JJ is the hessian,

λ is the monitoring parameter,

and θ∂∂=σ θ

k,k

f is the output sensitivity function or derivative of )(fk θ

according to θ .

It is important to note that the obtained θ is an approximated solution if J ≈ 0,

i.e. 1,3 0 =≈θ k,)(fk . This final value of the criterion corresponds to “small residual”.

2.6.3. Stability of synchronized states

A solution to (2.71) indicates that a synchronized state exists, but the stability of

the state must be ascertained. To do so, it is important to consider an initial differential


73

system describing the amplitudes ( )21 A,A and phases ( )21 φφ , dynamics and the

coupling current in Cartesian format ( )cycx A,A . Thus, dynamic equations (2.50) and

(2.51) have been used and give the following differential relations:

( )0201 ωωω= ,,,xfx a , (2.80)

where

[ ]Tcycx AAAAx 2211 φφ=

and

( )

( ) ( )( )

( ) ( )[ ]

( ) ( )( )

( ) ( )[ ]( ) ( ) ( )[ ]( ) ( ) ( )[ ]

φ−φλω+ω−ωω−

φ−φλω+ω−ω+ω−

φ−φω

−ω−ω

φ+φω−

−−ω

φ−φω

−ω−ω

φ+φω+

−−ω

=ωωω

112200

112200

222

02

22

22

2

111

01

11

21

1

0201

-

4

3

4

3

sinAsinAAA

cosAcosAAA

cosAsinAA

sinAcosAG

GbAaA

cosAsinAA

sinAcosAG

GbAaA

,,,xf

accxccyac

accyccxac

cycxa

cycxaL

L

a

cycxa

cycxaL

L

a

a

with

( )cL

ccx cos

G

IA φ= , ( )c

L

ccy sin

G

IA φ= and cφ is the phase of the coupling current.

These differential equations are nonlinear in states x , and unfortunately, the

stability theory developed for the linear problem does not apply directly to this system.

In practice, we typically linearize this system around a synchronized solution noted 0x

and consider the eigenvalues of the Jacobian matrix [ ]A so that:


74

[ ] x.Ax.x

fx

xx

δ=δ

∂∂

=δ=

⋅

0

, (2.81)

where 0 xxx −=δ are the small variations of the synchronized states x and

[ ]0

66

xxx

fA

=

×

∂∂

= is

[ ]

( ) ( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )[ ] ( ) ( )

( ) ( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )[ ] ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

−−−−

−−−−

−+−−

−−+−−

−−

−+−−

+−

−−

=

accacacacac

cacacacacac

aacycx

acycx

a

aacycxaLL

a

aacycx

acycx

a

aacycxaLL

a

AA

AA

AAAA

AAA

A

AAbAGaG

AAAA

AAA

A

AAbAGaG

A

ωωωφλωφλωφλωφλω

ωωωφλωφλωφλωφλω

φω

φω

φφω

φφω

φωφωφφωω

φω

φω

φφω

φφω

φωφωφφωω

02202011010

02202011010

22

22

222

2222

222222

11

11

111

1121

111121

cossincossin

sincossincos

cos sinsin coscos sin 00

sincoscos sin49 00

cossin 00sin cos cos sin

sincos00cos sin49

Synchronized states are asymptotically stable if and only if all the eigenvalues of

the Jacobian matrix [ ]A have negative real parts. Thus, after each estimation of

variables values θ , the resulting Jacobian matrix is evaluated and these six eigenvalues,

noted iµ , are computed. Hence, the obtained solution is asymptotically stable if all

eigenvalues iµ of the Jacobian matrix [ ]A satisfy ( )ieal µℜ < 0 whereas the solution

point is unstable if at least one eigenvalue iµ satisfies ( ) 0 ≥µℜ ieal .

Let us note that in the developed CAD tool, the eigenvalues are obtained using

the MATLAB software function eig(⋅).

2.7. THE CARTOGRAPHY OF THE SYNCHRONIZATION AREA

As mentioned before, the scope of using our CAD tool is to have a preliminary

idea of the synchronization area of the coupled oscillators. The input data necessary to

launch this tool are the circuit parameters of the two Van der Pol coupled oscillators.


75

These are in fact the parameters of the equations (2.75), (2.76) and (2.77) i.e. λ0, ωa, a

and b.

In this tool, the free running frequencies are the only parameters of the system

which can be changed. Thus, their values are changed while keeping the bandwidth of

the resonators (ωa) the same. The nonlinear programming technique detailed in 2.6.2 is

used to solve the system of equations (2.71). The solutions of this system of equations

are the synchronization pulsation ω, the amplitudes A1 and A2 of the output voltages of

the two oscillators and the phase shift φ∆ between them (using equations (2.72) and

(2.73)), for each pair of free-running frequencies (f01, f02).

We chose an example to clarify the use of the cartographies. For this, we

consider two coupled differential Van der Pol oscillators, as shown in Figure 2.5.

Figure 2.5 – Two coupled differential Van der Pol oscillators.

The coupling circuit is made of one resistor Rc/2 of 200 , the bandwidth of the

resonators, ωa = 5.64·109 rad/s and the coupling constant λ0 = 0.5. The parameters a and

b of the negative conductance presented by the active part of each oscillator are

respectively equal to 0.00755 and 0.0004.

In these conditions, the cartographies of the phase shift φ∆ between the output

voltages of the two oscillators, and those of the amplitudes A1 and A2, of the differential

voltages (vosc1-vosc2 and vosc3-vosc4) are shown in Figure 2.6. These cartographies

are plotted based on the free running frequencies f01 and f02. Outside the limits of these

cartographies, there is no synchronization.


76

Figure 2.6 – The cartographies of the phase shift ∆∆∆∆φφφφ, the amplitudes A1 and

A2 and the synchronization frequency fS of the coupled oscillators.

In the center of these cartographies, the free running frequencies of the

oscillators are identical and equal to the synchronization frequency of the system: f01 =

f02 = fS = 5.970 GHz. In this point considered as the origin, the phase shift between the

waveforms of the output voltages of the oscillators is zero.

Consider now, the cartography of the phase shift, like in Figure 2.7.

For instance, if the desired phase shift for the targeted application is 42°, the

free-running frequencies of the two oscillators are f01 = 5.67 GHz and f02 = 6.27 GHz

according to the cartography of Figure 2.7. The synchronization frequency fS = 5.97

GHz and the amplitudes A1 = A2 = 2.521 V.


77

Figure 2.7 – The cartography of the phase shift: example ∆∆∆∆φφφφ = 42°.

2.8. CONCLUSION

In the first part of this chapter, a complete review of R. York’s theory was

presented. R. York theory refers on the dynamics of two Van der Pol oscillators coupled

through a series resonant circuit. Then, starting from the admittance transfer functions

binding the coupling current to the oscillators voltage and relying on Kurokawa’s

substitution, York and Lynch described the dynamic equations of the two Van der Pol

oscillators coupled through a resonant circuit. The case of a resistive coupling circuit

was then deduced from this system of equations.

In the second part of this chapter, a new expression of the dynamics of two Van

der Pol coupled oscillators allowing a more accurate prediction of the amplitudes was

developed. To do so, the limitation of R. York’s theory regarding the prediction of the

amplitudes of the two coupled oscillators was first showed. After that, this theory was

adapted to the case of our more accurate Van der Pol model in order to obtain a new

system of four equations with four unknowns describing the locked states of two Van

der Pol oscillators coupled through a resonant network. This has led to a new expression

allowing the prediction of the amplitudes of the two coupled Van der Pol oscillators for

any values of parameters a and b of the Van Der Pol non linearity under a sinusoidal

assumption.


78

In the last part of this chapter, mathematical manipulations were applied to the

system of nonlinear equations describing the locked states of two Van der Pol

oscillators coupled through a resonant network as well as through a resistive one. A

simpler system of three equations with three unknowns was obtained allowing then to

implement a CAD tool that permits to extract the locking region of the coupled

oscillators, in an extremely short simulation time. This new system of equations was

solved on Matlab using nonlinear programming technique and the stability of the

synchronized states was also analyzed. Finally, an example of 3D graphic of the

synchronization area for two differential Van der Pol oscillators coupled through a

resistive network was showed. Thus, with this cartography and with the precise free-

running frequencies, it was possible to extract the value of the phase shift between the

output voltages of the oscillators as well as their amplitudes.

79

CHAPTER III

Study and Analysis of an Array of Differential

Oscillators and VCOs Coupled Through a Resistive

Network

Chapter III– Study and Analysis of an Array of Differential Oscillators and VCOs Coupled through a Resistive Network

80

3.1. INTRODUCTION

In the previous chapter, a new CAD tool which provides, in a considerably short

analysis time a cartography giving the phase shifts, synchronization frequencies and

amplitudes of the differential output voltages of two differential oscillators coupled

through a broadband network was presented. Indeed, as mentioned previously, it seems

to be interesting to analyze the behavior of an array of coupled differential oscillators

and VCOs since, in this case, the theoretical limit of the phase shift is within 360o due

to the differential nature of the array. Furthermore, the use of a broadband coupling

network, i.e. a resistor, instead of a resonant one, can lead to a substantial save in chip

area in the case of RF integrated circuit design.

Hence, in the first part of this chapter we will consider the case of the analysis of

two coupled differential oscillators. Since the theory implemented in our CAD tool uses

Van der Pol oscillators to model microwave coupled oscillators, a modeling procedure

of the differential oscillator as a differential Van der Pol oscillator will be presented.

Then, the proposed CAD tool was used in order to obtain the cartography of the

oscillators’ locked-states. This cartography can help the designer to rapidly find the free-

running frequencies of the two outermost VCOs of the array required to achieve the desired

phase shift. After that, in order to validate the results provided by our CAD tool, we

compared them to the simulation results of the two coupled differential oscillators

obtained with Agilent’s ADS software for different cases of coupling strength.

In the second part of this chapter, the same study was performed for the case of

two coupled differential Voltage Controlled Oscillators (VCOs. Furthermore, a new

analysis based on state equations method will be presented. The study of the variation

of the phase shift versus the coupling resistor will also be investigated as well as the

effect of a mismatch between the two coupling resistors on the phase shift. Finally, the

behavior of four coupled differential VCOs will be presented.


81

3.2. ANALYSIS AND DESIGN OF TWO DIFFERENTIAL

OSCILLATORS COUPLED THROUGH A RESISTIVE NETWORK

3.2.1. RLC differential oscillator schematic

Figure 3.1 shows the VCO schematic used in simulations with Agilent’s ADS

software and which will be part of the array made of two identical differential

oscillators coupled through a resistive network. It is based on the well-known cross-

coupled differential NMOS topology using a 0.35 µm BiCMOS SiGe process.

Figure 3.1 – The schematic of the RLC differential oscillator.

The cross connected NMOS differential pair provides the negative resistance to

compensate for the tank losses. The tail current source is a simple NMOS current mirror

and draws 28 mA with a ratio of 14 in order to reduce the power consumption. The

power supply voltage Vcc is 2.5 V. The frequency of oscillation is determined by the LC

tank at the drains, so that the oscillation frequency is close to 6 GHz to meet ETSI


82

(European Telecommunication Standard Institute) standard for ITS (Intelligent

Transport System requirements) for instance Appendix B.

In these conditions, the inductance value, L, is close to 0.8 nH and the capacitor

value, C, is close to 0.88 pF. The resistor value, R, is equal to 100 , so that the quality

factor of the tank is:

00

ωω

L

R R C Q == . (3.1)

Replacing the values for R, C, 0ω , the quality factor is equal to 3.3.

A tail capacitor, X1 of 20 pF, is used to attenuate both the high frequency noise

components of the tail-current and the voltage variation on the tail node. This latter

effect results in more symmetric waveforms and smaller harmonic distortion in LC-

VCO outputs [47].

To ensure proper start-up of the oscillator, the following condition needs to be

satisfied:

R gm1> .

(3.2)

In these conditions, the NMOS transistors sizes are identical and chosen to be:

m350

m70

.

L

w =

.

Figure 3.2 a shows the output voltages, vd1 and vd2, of the differential oscillator.

We can notice that they both have almost the same peak to peak amplitude, close to 3 V.

In these conditions, the frequency of oscillation is, 5.970 GHz, with a capacitor value of

0.88 pF, as shown in Figure 3.2 b.


83

a)

b)

Figure 3.2 – a) The waveforms of the output voltage of the differential

oscillator;

b) The output spectrum.

794.90

794.95

795.00

795.05

795.10

795.15

795.20

795.25

795.30

794.85

795.35

-2

-1

0

1

2

-3

3

time, nsec

vd1-

vd2


84

3.2.2. The modeling of the differential oscillator as a Van der Pol

oscillator

Since the theory implemented in our CAD tool “ASVAL” uses Van Der Pol

oscillators to model microwave coupled oscillators as explained in chapter 2, we

performed the modeling of the differential oscillator of Figure 3.1 as a differential Van

der Pol oscillator.

The modeling step is divided in two parts:

The modeling of the passive part which is represented by the resonator giving the

oscillation frequency;

The modeling of the active part, which includes the transistor that compensate

the losses.

3.2.2.1. The modeling of the passive part

The passive part corresponding to the resonator must be modeled by a parallel

RLC circuit. They must have the same behavior i.e. the same resonance frequency and

the same quality factor, within the frequency band of operation of the oscillator.

We set the parameters of the Van der Pol oscillator in order to ensure a

resonance frequency of the RLC circuit equal to the resonance frequency of the

differential oscillator’s resonator, like in Figure 3.3. In these conditions, the inductance

value, Lvp, is close to 1.6 nH, the capacitor value, Cvp, is close to 0.44 pF and the

resistor value, Rvp, is equal to 200 .


85

Figure 3.3 – The identification of the Van der Pol passive part parameters.

3.2.2.2. The modeling of the active part

For the modeling of the active part, the typical I = f(Vdiff) = f(Vd1-Vd2)

characteristic of the differential oscillator of Figure 3.1 had been plotted at 5.97 GHz, as

shown in Figure 3.4.


86

Figure 3.4 – The Van der Pol characteristic.

Furthermore, let us remind that the general expression of the current of a Van

der Pol oscillator is written as follows:

3diffdiff b v a vi +−= . (3.3)

The a parameter is given by the slope of the Van der Pol characteristic given in

Figure 3.4 so that:

a = 0.00755. (3.4)

Now according to the theory of a Van der Pol oscillator [15], the amplitude of

the oscillation is equal to b

Ga

32

− with

vpRG

1= .

Since the amplitude of the differential voltage of one differential oscillator is

equal to 2.885 V as shown in Figure 3.2.a, we can deduce the value of parameter b so

that:


87

b

Ga

3

− = 1.4425 ⇒ b = 0.0004. (3.5)

3.2.2.3. Simulations of the Van der Pol oscillator

According to the modeling steps described previously, the differential oscillator

of Figure 3.1 can be reduced into a differential Van der Pol oscillator synthesized on

Agilent’s ADS software, as shown in Figure 3.5.

Figure 3.5 – The differential Van der Pol oscillator model with i = -av +bv3

In order to check the validity of this model, Figure 3.6 shows the waveforms of

the differential output voltages for the RLC NMOS differential oscillator of Figure 3.1

as well as those of the previous differential Van der Pol oscillator model at a frequency

of 5.97 GHz. As can be seen in this figure, a very good agreement is found between the

behavior of the model and the circuit.


88

Figure 3.6 – Comparison between the output voltages of the differential oscillator

and the differential Van der Pol oscillator model

3.2.3. Two coupled differential Van der Pol oscillators

This section is dedicated to the analysis and the study of the two differential Van

der Pol oscillators presented previously coupled with only one resistor, as in Figure 3.7.

Figure 3.7 – Two coupled Van der Pol oscillators.


89

In this case, the value of the coupling resistor on each path must be equal to Rc/2

with Rc the value of the coupling resistor used in the theory elaborated in chapter two

and based on the use of two single-ended Van der Pol coupled oscillators.

Indeed, let us consider first the case of two single-ended oscillators coupled

through one resistor RSE as shown in Figure 3.8.a. In this case we can write:

021 =−+ VVV c ,

and cSEc IRV = ,

then cSEIRVV −=− 21 . (3.6)

Ic RSE

V1 Vc V2

Figure 3.8.a – Two single-ended oscillators coupled through a resistor.

Now if we consider the case of two differential oscillators coupled through a

resistor RD on each path, as in Figure 3.8.b, we have:

021 =+−+ cc VVVV ,

and cDc IRV = ,

then cD IRVV 221 −=− . (3.7)

Osc1 Osc2


90

RD Ic

V1 Ic V2

Figure 3.8.b – Two differential oscillators coupled through a resistor.

In order to obtain the same behavior of the two previous circuits, the following

condition must be satisfied:

2SE

DR

R =. (3.8)

The circuit shown in Figure 3.7 is then simulated using Agilent’s ADS software

in transient analysis. Hence, in order to find the range of frequencies over which these

two coupled oscillators can lock, we changed the free-running frequencies of the two

oscillators, f01 and f02, so that the synchronization frequency will remain the same.

Furthermore, the free-running frequencies of oscillator 1 (the one which is in the left)

and oscillator 2 (the one which is in the right) have been defined as follows:

) 1001 ( f f += , (3.9)

) - ( f f 1002 = , (3.10)

where:

f0 – is the desired synchronization frequency or the center frequency;

RD

Vc

Vc

Osc1 Osc2


91

∆ - is a percentage.

The simulation results are summarized in Table 1 and show the phase shift ∆φ

between the two oscillators as well as the synchronization frequency and the amplitude

of the differential output voltage. Let us note that the maximum value of the phase shift

is reached for ∆ = 7.4 % since, above this value, the two oscillators are not able to

synchronize anymore.

Table 1 – The synchronization frequency, phase shift and amplitude obtained for

two coupled differential Van der Pol oscillators

∆ f1 f2 fs φ∆ A

0% 5.970 GHz 5.970 GHz 5.970 GHz 0° 2.916 V

1% 6.030 GHz 5.910 GHz 5.940 GHz 7.2° 2.905 V

2% 6.090 GHz 5.850 GHz 5.960 GHz 15.8° 2.861 V

3% 6.150 GHz 5.790 GHz 5.960 GHz 24.7° 2.781V

4% 6.210 GHz 5.730 GHz 5.960 GHz 31.4° 2.697 V

5% 6.270 GHz 5.670 GHz 5.960 GHz 41.6° 2.528 V

6% 6.330 GHz 5.610 GHz 5.960 GHz 53.8° 2.254 V

7% 6.390 GHz 5.550 GHz 5.960 GHz 69.4° 1.759 V

7.1% 6.400 GHz 5.553 GHz 5.980 GHz 69.9° 1.737 V

7.2% 6.408 GHz 5.540 GHz 5.960 GHz 74.6° 1.538 V

7.3% 6.410 GHz 5.540 GHz 5.980 GHz 75.7° 1.493 V

7.4% 6.420 GHz 5.530 GHz 5.980 GHz 82.2° 1.142 V

Figure 3.9 shows the output voltages, vosc1, vosc2, vosc3 and vosc4, of both

oscillators, at the maximum value of ∆, for which the oscillators are able to

synchronize. In this figure we can also see the synchronization frequency.


92

a)

b)

Figure 3.9 – a) The waveforms of the output voltages of the coupled

differential Van der Pol oscillators for ∆∆∆∆=7,4%;

b) The output spectrum.

Chapter III– SOscillators and VC

3.2.4. Two coupled diffe

In this section and un

two differential Van der Po

Figure 3.1 were coupled thro

Figure 3.10 – Tw

The simulation result

Table 2. Let us note that the

7,4 % since, above this value

Study and Analysis of an Array of D VCOs Coupled through a Resistive

93

ifferential oscillators

under the same simulation conditions used p

Pol oscillators, the two identical different

rough a resistor of 200 Ω as shown in Figure

Two differential oscillators coupled throug

ults of the two coupled oscillators are the

the maximum value of the phase shift is als

lue, the two oscillators are not able to synchr

of Differential ive Network

d previously for the

ential oscillators of

ure 3.10.

ugh a resistor.

then summarized in

also reached for ∆=

hronize anymore.


94

Table 2 - The synchronization frequency, phase shift and amplitude obtained for

two coupled differential oscillators

∆ f1 f2 fs φ∆ A

0% 5.970GHz 5.970 GHz 5.960 GHz 0˚ 2.891 V

1% 6.030 GHz 5.910 GHz 5.960 GHz 7.18˚ 2.882 V

2% 6.090 GHz 5.850 GHz 5.980 GHz 15.79˚ 2.846 V

3% 6.150 GHz 5.790 GHz 5.980 GHz 24.63˚ 2.784 V

4% 6.210 GHz 5.730 GHz 5.980 GHz 31.23˚ 2.723 V

5% 6.270 GHz 5.670 GHz 5.980 GHz 41.38˚ 2.605 V

6% 6.330 GHz 5.610 GHz 5.990 GHz 53.40˚ 2.434 V

7% 6.390 GHz 5.550 GHz 5.990 GHz 68.71˚ 2.163 V

7.1% 6.400 GHz 5.553 GHz 5.990 GHz 69.28˚ 2.151 V

7.2% 6.408 GHz 5.540 GHz 5.990 GHz 73.90˚ 2.050 V

7.3% 6.410 GHz 5.540 GHz 5.990 GHz 74.81˚ 2.027 V

7.4% 6.420 GHz 5.530 GHz 5.990 GHz 81.01˚ 1.855 V

Furthermore, Figure 3.11 shows the output voltages, vd1, vd2, vd3 and vd4, of

both oscillators, at the maximum value of ∆ = 7.4%, for which the oscillators are able to

synchronize. In this figure we can also see the synchronization frequency.


95

a)

b)

Figure 3.11 – a) The waveforms of the output voltages of the coupled

differential oscillators for ∆∆∆∆=7.4%;

b) The output spectrum;


96

3.2.5. Comparison between the theory, the Van der Pol model and the

differential structure

In this section, our CAD tool is used in order to show its usefulness and its

reliability. To do so, the results provided will be compared with the simulation results

presented in sub-sections 3.4 and 3.5.

Hence, knowing the parameters λ0, ωa , a and b, the proposed CAD tool provides the

cartography of the locked states of the two differential coupled oscillators as explained in

sub-section 2.7. Hence, using the parameter of the circuit of Figure 3.7 and for a

synchronization frequency of 5.97 GHz, we can find ωa = 5.64 109 rad/s and a coupling

constant λ0 = 0.5. Furthermore, let us remind that the parameters a and b of the negative

conductance presented by the active part of each oscillator are respectively equal to

0.00755 and 0.0004 as mentioned in sub-section 3.3.2.

Thus, the cartography of the oscillators’ locked states provided by the CAD tool can

be plotted and is presented in Figure 3.12. This figure presents the variations of the phase

shift, ∆φ, the oscillators’ amplitudes A1 and A2 of the differential voltages, and the

synchronization frequency fs in function of the free-running frequencies f01 and f02.


97

Figure 3.12 - Cartography of the oscillators’ locked states provided by the CAD tool.

In order to validate the results provided by our CAD tool, we compared them to the

simulation results of the two coupled differential oscillators of Figure 3.10, obtained with

Agilent’s ADS software. Let us note that with ADS, only a transient analysis of one point at a

time of synchronization region allows to verify the synchronization results obtained with the

CAD tool. For instance, let us now consider Figure 3.12 where the point marked with an

arrow in the four subplots represents a free-running frequency f01 = 5.73 GHz for oscillator 1

and f02 = 6.21 GHz for oscillator 2. The marked points lead to a phase shift of 32.39°, a

synchronization frequency of the coupled oscillators of 5.97 GHz and an amplitude of 2.68 V

at the output of each of the coupled oscillators.


98

In the same conditions, the two differential NMOS coupled oscillators of Figure 3.10

simulated with ADS have lead to two sinusoidal waves at a synchronization frequency of

5.98 GHz, a phase shift of 31.23° and an amplitude of 2.72 V at the output of each oscillator,

as presented in Figure 3.13 and in Table 2.

Figure 3.13 – Waveforms of the output voltages of the two coupled differential NMOS

oscillators, when ∆∆∆∆φφφφ = 31.23° and A = 2.72 V

Furthermore, Figure 3.14 shows a comparison of the phase shift and amplitude

obtained for the coupled NMOS differential oscillators, the coupled differential Van der

Pol oscillators and by using “ASVAL”, for a synchronization frequency of 5.97 GHz, as

a function of 01020 fff −=∆ .


99

a)

b)

Figure 3.14 – a) Comparison of the phase shift;

b) Comparison of the amplitude.

As can be seen on this figure, a good agreement is found between the results provided

by “ASVAL” and the simulation results showing the reliability of our CAD tool.

Nevertheless, as we approach the locking-region boundary, we can observe that the

difference between the theoretical and simulated results is increasing especially for the

amplitude. This is mainly due to the fact that the modeling of each NMOS differential

-100-80-60-40-20

020406080

100

-1000 -500 0 500 1000

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)

ASVAL Simulation VDP Simulation Diff NMOS Simulation

0

0,5

1

1,5

2

2,5

3

3,5

-1000 -500 0 500 1000

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



100

oscillator as a differential Van der Pol oscillator has been performed only at one frequency,

i.e. the desired synchronization frequency. The Figure 3.14.a shows that, as the oscillator

tunings are moved apart, but the synchronization frequency is halfway between, the phase

shift increases until the locking-region boundary is encountered. Concerning the prediction of

the amplitudes, we can say that the amplitude is at a minimum at the locking-region

boundaries (i.e. when the phase shift is at the extreme values) and equal to the oscillators’

free-running amplitude for equal tunings (i.e. when the phase shift is equal to zero).

Furthermore, let us remind that these simulations where performed for a coupling

constant λ0 = 0.5 for which the value of the coupling resistor Rc must be chosen equal to 400

Ω since the value of the parallel resistor R of the resonator is equal to 200 Ω. For this typical

value, York and Lynch proved in [19] that the locking-region is maximized while still

allowing the phase difference ∆φ to vary 180° over the locking range. Hence, this constitutes

an optimal value especially for beam-scanning systems where the designer wishes to

maximize the total phase variation and the locking-range simultaneously. Furthermore, let us

remind that, in the case studied here of two differential coupled oscillators, it is possible to

obtain a continuously controlled 360° phase shifting range due to the differential nature of the

system leading to a much more efficient beam-scanning architecture.

Nevertheless, in order to verify this affirmation, it seems interesting to study the

behavior of the system in both weak and strong coupling cases i.e. for values of λ0

respectively less and greater than 0.5.

3.2.6. Study and analysis of the two coupled differential oscillators in

the weak coupling case

In the weak coupling case, the circuits were analyzed for a value of the coupling

constant λ0 = 0.25 leading to Rc/2 = 400 . Thus, Figure 3.15 presents a comparison of

the phase shift and amplitude obtained for the two coupled Van der Pol oscillators, the

two coupled RLC NMOS differential oscillators and using “ASVAL” also for a

synchronization frequency of 5.97 GHz, as a function of 01020 fff −=∆ .


101

a)

b)

Figure 3.15 – The weak coupling case – a) Comparison of the phase shift;


As can be seen on this figure, the locking range as well as the maximum phase shift

which can be obtained are clearly reduced compared to the optimal case of a coupling

-80

-60

-40

-20

0

20

40

60

80

-400 -300 -200 -100 0 100 200 300 400

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)


2,55

2,6

2,65

2,7

2,75

2,8

2,85

2,9

2,95

-400 -300 -200 -100 0 100 200 300 400

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



102

strength λ0 = 0.5. This result can be justified using the approximate formula (2.29). Indeed,

according to (2.29) we have:

aωλ<ω∆ 00 2 .

Hence, the locking range can be defined as:

amaxωλ=ω∆ 00 2 .

As a consequence, one can easily understand that the weak coupling case leads to a

smaller locking range of the system.

Furthermore, according to Figure 3.15.b, we can notice that the amplitude remains

close to its free-running value almost over the entire locking range. One more time, this

behavior can be justified using the approximate formula elaborated in chapter 2, i.e. (2.56).

Indeed, for small values of the coupling strength λ0, the expression (2.56) can be

approximated by ( )LGab

A −≅3

4 which represents the free-running amplitude of the Van

der Pol oscillator modelizing the behavior of the differential NMOS LC oscillator.

In Figure 3.16, the cartographies, obtained with “ASVAL”, of the phase shift ∆φ

between the output voltages of the two oscillators, and those of their amplitudes A1 and

A2, in the case of weak coupling are shown. Let us remind here that this cartography allows

to obtain the entire frequency locked states of the array of coupled oscillators.


Figure 3.16 - The car

A2 and the synchronizati

3.2.7. Study and analys

the strong couplin

Let us now consider th

value of λ0 = 1.25 which imp

comparison of the obtained p

oscillators, two coupled RLC

a synchronization frequency


103

cartographies of the phase shift ∆∆∆∆φφφφ, the am

ation frequency fs of the coupled oscillator

coupling case

lysis of the two coupled differential

ling case

r the strong coupling case where the circuits w

mplies Rc/2 = 80 . In these conditions, Figu

d phase shift and amplitude for the two cou

NMOS differential oscillators and using “AS

cy of 5.97 GHz, as a function of 020 fff −=∆


amplitudes A1 and

tors for the weak

ial oscillators in

were analyzed for a

igure 3.17 presents a

oupled Van der Pol

ASVAL” always for

01f .


104

a)

b)

Figure 3.17 – The strong coupling case – a) Comparison of the phase shift;


As can be seen on Figure 3.17.a, the locking range increased compared to the optimal

case of a coupling strength λ0 = 0.5. This result can be justified again using the approximate

formula (2.29). Indeed, according to (2.29), one can say that, for a given phase shift ∆φ, the

frequency difference between the two free-running frequencies of oscillators, f0, is increased

-60

-40

-20

0

20

40

60

-2000 -1500 -1000 -500 0 500 1000 1500 2000

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)


0

0,5

1

1,5

2

2,5

3

3,5

-2000 -1500 -1000 -500 0 500 1000 1500 2000

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



105

compared to the case of an optimal coupling strength λ0 = 0.5. Furthermore, we understand

intuitively that a strong coupling, i.e. a small value of Rc, leads to a wider locking range of

the system.

Furthermore, according to Figure 3.15.b, we notice that the amplitude decreases

considerably as we traverse the locking region. As explained in [19], the physical reason is

that, as the coupling resistor Rc is reduced, the power dissipated in it increases. Hence, the

oscillator conductances must make up this power loss by becoming more negative, which is

achieved by amplitude reduction.

In the case of strong coupling, the cartographies of the locked states of the two

coupled oscillators, obtained with “ASVAL”, are presented in Figure 3.18.

Figure 3.18 - The cartographies of the phase shift ∆∆∆∆φφφφ, the amplitudes A1 and

A2 and the synchronization frequency fs of the coupled oscillators for the strong

coupling case.


106

3.3. ANALYSIS AND DESIGN OF TWO VCOs COUPLED THROUGH

A RESISTIVE NETWORK

3.3.1. Introduction

The accuracy and the reliability of the equations used for the elaboration of our

CAD tool were demonstrated in the previous chapter, where an array of two coupled

differential oscillators was simulated using Agilent’s ADS software.

Let us now remind the expression of the bandwidth of the resonator of one

oscillator:

C

GLa 2

=ω . (3.11)

In the case of the two coupled oscillators, ωa was fixed since the frequency was

obtained by changing the inductance value, L, according to R. York’s theory. Now, in

the case of a VCO, the value of C is changing in order to tune the VCO leading to the

variation of ωa. In these conditions, some simulations are computed in order to see the

effect of the variation of ωa on the results. Thus, Figure 3.19 presents a comparison of

the phase shift and amplitude when ωa is varying and when it is constant, in the case of

two coupled RLC NMOS differential oscillators of Figure 3.10.


107

a)

b)

Figure 3.19 – The RLC NMOS differential oscillator:

a) Comparison of ∆∆∆∆φφφφ while changing L and C;

b) Comparison of the amplitude while changing L and C.

-100

-80

-60

-40

-20

0

20

40

60

80

100

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)

L changed C changed

0

0,5

1

1,5

2

2,5

3

3,5

-1000 -500 0 500 1000

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)

L changed C changedRLC NMOS diff. osc.

RLC NMOS diff. osc.


108

In the same way, Figure 3.20 presents a comparison of the phase shift and amplitude

when ωa is varying and when it is constant, but in the case of two coupled Van der Pol

differential oscillators of Figure 3.7.

a)

b)

Figure 3.20 – The Van der Pol oscillator:

a) Comparison of ∆∆∆∆φφφφ while changing L and C;

b) Comparison of the amplitude while changing L and C.

-100

-80

-60

-40

-20

0

20

40

60

80

100

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)

L changed C changed

0

0,5

1

1,5

2

2,5

3

3,5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)

L changed C changed

VAN der POL osc.

VAN der POL osc.


109

Since the variation of the resonator’s bandwidth doesn’t affect significantly the

results, ωa was kept constant for the following study and modeling of an array of two

differential LC VCOs. Hence, in our CAD tool, which can plot the cartography of the

synchronization area of two coupled VCOs, the value of ωa which will be used will

correspond to the capacitor value of the resonator leading to 0 0 =∆f .

3.3.2. The LC VCO architecture

The VCO’s schematic used in simulations is illustrated in Figure 3.21. The VCO

structure is also based on a crossed-coupled NMOS differential topology using a 0.35 m

BiCMOS SiGe process. Let us remind that, this VCO architecture is widely used in high-

frequency circuit design due to its relatively good phase noise performances and ease of

integration.

Figure 3.21 – The LC VCO schematic.


110

3.3.2.1. The design of the passive part

The structure of the passive part is made of the inductor, L, and the P+/N varactor

diode associated to AC coupling capacitors as shown in Figure 3.21.

In these conditions, the frequency of oscillation is chosen to be close to 6 GHz and is

determined by the LC tank at the drains, leading to the inductance value, L, close to 0.8 nH

with a quality factor, Q, equal to 15 at 6 GHz.

The tuning range depends on the global capacitance C variation and thus on the

Cmax/Cmin ratio of the varactor diodes and on the AC coupling capacitor.

In order to obtain the desired bandwidth of the VCO, the varactor diode’s parameters

are adjusted, so that:

- the length, is 8 m;

- the width is 2 m;

- the number of fingers is 8;

- the number of varactor diodes in parallel is 4.

In these conditions, Figure 3.22 presents the characteristic of the capacitor and the

quality factor versus the control voltage, Vtune.

0.5 1.0 1.5 2.0 2.5 0.0 3.0

6.0E-13

8.0E-13

1.0E-12

4.0E-13

1.2E-12

Vtune [V]

C [

F]

a)


111

0.5 1.0 1.5 2.0 2.5 0.0 3.0

16

18

20

22

24

14

26

Vtune [V]

Q@

6 G

Hz

b)

Figure 3.22 – a) Variation of C versus Vtune;

b) Variation of Q versus Vtune.

Thus, Table 3 shows the varactor diode’s performances obtained using Agilent’s ADS

software.

Table 3 – The varactor diode’s performances

C @ Vtune = 1.35V [pF] 0.68

Cmax/Cmin 2.48

Q @ 6 GHz ( Vtune = 1.35V ) 20.21

3.3.2.2. The design of the active part

As mentioned previously, the active part is based on the classical cross-coupled

NMOS differential topology. The cross connected NMOS differential pair provides the

negative resistance to compensate for the tank losses. The tail current source is a simple

NMOS current mirror and draws 12 mA with a ratio of 14 in order to reduce the power


112

consumption. A tail capacitor CT is used to attenuate both the high-frequency noise

component of the tail current and the voltage variations on the tail node [48].

Let us remember that, to ensure a proper start-up of the VCO, the following condition

needs to be satisfied:

Rgm

1 ⟩ , (3.12)

where:

- gm is the transconductance of the NMOS transistor;

- R is the resistive part of the resonator.

In these conditions, the sizes of NMOS transistors T1 and T2 are identical and chosen

to be m35.0

m70

LW =

.

3.3.2.3. VCO simulation results

In this section, simulation results regarding the VCO’s performances using Agilent’s

ADS software are presented. Hence, Figure 3.23 shows the tuning characteristic of the VCO.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.0 2.6

5.5

6.0

6.5

7.0

5.0

7.5

Vtune [V]

Fre

qu

ency

[G

Hz]

Figure 3.23 – The VCO oscillation frequency versus Vtune.


113

As mentioned in this figure, the VCO is tuned from 5.3 GHz to 7 GHz with a tuning

voltage varying from 0 to 2.5 V.

Then, Figure 3.24 illustrates the output power of the VCO on 50 load over the

frequency bandwidth.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.0 2.6

9.5

10.0

10.5

9.0

11.0

Vtune [V]

Po

wer

[d

Bm

]

Figure 3.24 – The output power of the VCO.

In Figure 3.25 the two output voltages, vd1 and vd2, of the VCO are presented for a

tuning voltage Vtune equal to 1.35V.

Figure 3.25 – The output voltages of the VCO.

779.20 779.25 779.30 779.35 779.40779.15 779.45

1.5

2.0

2.5

3.0

3.5

1.0

4.0

time, nsec

vd1,

Vvd

2, V


114

Finally, Figures 3.26 and 3.27 present the phase noise simulation results of the VCO.

Thus, Figure 3.26 shows the phase noise plot for Vtune = 0.62 V leading to an oscillation

frequency of f0 = 5.8 GHz and a current consumption Ibias = 12 mA. As can be seen in this

figure, the VCO features a phase noise of -84 dBc/Hz and -111 dBc/Hz at 100 kHz and 1

MHz frequency offset respectively. Furthermore, Figure 3.27 illustrates the phase noise at 1

MHz frequency offset versus Vtune for Ibias = 12 mA. As we can notice, the worst case

phase noise is –108 dBc/Hz at 1 MHz frequency offset.

1E6 1E5 1E7

-130

-120

-110

-100

-90

-140

-80

Frequency offset [Hz]

Pha

se n

oise

[dB

c/H

z]

Figure 3.26 – Simulated phase noise of the VCO for a tuning voltage of 0.62 V.

0.5 1.0 1.5 2.0 0.0 2.5

-111

-110

-109

-112

-108

Vtune [V]

Pha

se n

oise

[dB

c/H

z]

Figure 3.27 – Simulated phase noise at 1 MHz versus tuning voltage.


115

3.3.3. The modeling of a differential VCO as a differential Van der Pol

oscillator

As for the case of the design of two differential coupled oscillators detailed in section

3.2, we performed the modeling of the differential LC VCO of Figure 3.21 as a differential

Van der Pol oscillator at the required synchronization frequency at 5.89 GHz.

Thus, the modeling step is divided in two parts:

• The modeling of the passive part which is represented by the inductor and the varicap

diode giving the oscillation frequency;

• The modeling of the active part, which includes the transistor that compensate the

losses.

3.3.3.1. The modeling of the passive part

The passive part corresponding to the resonator of the VCO is modeled by a parallel

RLC circuit. They must have the same behavior, i.e. the same resonance frequency and the

same quality factor, within the frequency band of operation of the VCO. This modeling was

realized using a S-parameters simulation with ADS software, as shown in Figure 3.28.

Therefore, the resonator parameters of the Van der Pol oscillator were set so that the

impedances presented by the VCO’s resonator, Z11, and the Van der Pol resonator, Z22, will

be equal.


116

Figure 3.28 – The identification of the parameters of the Van der Pol resonator.

Hence, since the inductor value on each side is equal to 0.8 nH, the differential

inductor of the Van der Pol resonator will be Lvp = 1.6 nH. The resistor value, Rvp,

corresponds to the real part of Z11 at the resonance frequency and was found to be equal to

260 as shown in Figure 3.29. Then, the capacitor value Cvp was tuned so that Z11= Z22.

Hence, for Cvp = 0.456 pF, the real and imaginary part of the two impedances, Z11 and Z22

match well as illustrated in Figure 3.29.


117

5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 5.0 7.0

100

150

200

250

50

300

freq, GHz

Rea

l(Z11

)

Rea

l(Z22

) Real(Z11)=260 @ F0 = 5.87 GHz

VCO passive part

The equivalent RLC

5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 5.0 7.0

-100

-50

0

50

100

-150

150

freq, GHz

Imag

(Z11

)

Imag

(Z22

)

Imag(Z11) = 0 @ F0 = 5.87 GHz

The equivalent RLC

VCO passive part

Figure 3.29– The real and imaginary part of the two impedances Z11 and Z22.

3.3.3.2. The modeling of the active part

For the modeling of the active part, in order to obtain the Van der Pol characteristic,

the typical I = f(Vdiff) = f(Vd1- Vd2) characteristic of the Van der Pol oscillator has been

plotted at a required synchronization frequency, as shows in Figure 3.30.


118

Figure 3.30 – The Van der Pol characteristic obtained for a VCO.

Let us now remind the general expression of the current of a Van der Pol

oscillator:

3diffdiff b v a vi +−= . (3.13)

The a parameter is given by the slope of the Van der Pol characteristic when

Vdiff = 0, so that:

006620.V

Ia

diff

=∆

∆= . (3.14)

Now according to the theory of a Van der Pol oscillator [15], the amplitude of

the oscillation is equal to b

Ga

32

−, with

vpRG

1= .

Since the amplitude of the differential voltage of one VCO at the required

synchronization frequency is equal to 1.941V as shown in Figure 3.31, we can deduce

the value of parameter b so that:

b

Ga

3

− = 1.941 ⇒ b = 0.000976.

(3.15)


119

In these conditions, we can do the modeling of the VCO as a differential Van der Pol

oscillator, like in Figure 3.32.

m1

m1 time= ts(vd1)-ts(vd2)=1.941 Max

83.67psec

Figure 3.31 – The differential output voltage of a VCO at 5.89 GHz.

Figure 3.32 – The differential Van der Pol oscillator.


120

3.3.4. Two coupled differential VCOs

3.3.4.1. Study and analysis of two coupled differential VCOs for an

optimal coupling case

As presented in the previous section, using ADS simulations for one VCO at the

required synchronization frequency, it was possible to perform the modeling of this

structure as a differential Van der Pol oscillator. As a consequence, the two coupled

differential VCOs of Figure 3.33 can be reduced into two differential Van der Pol

coupled oscillators as shown in Figure 3.34. In this case, let us note that the value of the

coupling resistor on each path Rc/2 is equal to 260 leading to λ0 = 0.5, since for this

coupling strength the locking-region is maximized as mentioned previously.

Figure 3.33 – Two coupled differential VCOs.

Figure 3.34 - Two differential Van der Pol coupled oscillators.


121

Once more, the results provided by our CAD tool “ASVAL” has been compared

with simulation results of the two coupled VCOs performed with Agilent’s ADS

software. Hence, knowing the parameters λ0, ωa , a and b, of the Van der Pol model,

“ASVAL” provides the cartography of the locked states of the two differential coupled

VCOs of Figure 3.33. Thus, for ωa = 4.21 109 rad/s, λ0 = 0.5, a = 6.62 10-3 and b =

9.76 10-4, the cartography of the oscillators’ locked states provided by the CAD tool is

presented in Figure 3.35. Thus, this figure presents the variations of the phase shift, ∆φ,

the VCOs’ amplitudes A1 and A2, and the synchronization frequency fs in function of f01

and f02.

Now, we compared the results provided by “ASVAL” to the simulation results of

the two coupled differential VCOs of Figure 3.33, obtained with Agilent’s ADS

software. For instance, let us consider Figure 3.35 where the point marked with an

arrow in the four subplots represents a free-running frequency f01 = 5.59 GHz for VCO

1 and f02 = 6.18 GHz for VCO 2. The marked points lead to a phase shift of 61.59°, a

synchronization frequency of the coupled VCOs of 5.89 GHz and an amplitude of 1.55

V at the output of each of the coupled VCOs.


122

Figure 3.35 - Cartography of the VCOs’ locked states provided by the CAD tool.

Let us now compare the previous results with those obtained with the two

identical differential coupled VCOs of Figure 3.33 under the same simulation

conditions. Hence, the tuning voltages Vtune1 and Vtune2 have been adjusted in order to

obtain the same free-running frequencies. In these conditions, the two differential

NMOS coupled VCOs of Figure 3.33 simulated with ADS has lead to two sinusoidal

waves at a synchronization frequency of 5.86 GHz, a phase shift of 65.6° and an

amplitude close to 1.5 V at the output of each VCO, as presented in Figure 3.36.


123

Figure 3.36 – Waveforms of the output voltages of the two differential NMOS VCOs for

∆∆∆∆φφφφ = 65.6° and A ≈ 1.5 V

Furthermore, Figure 3.37 shows a comparison between the phase shift and the

amplitudes obtained in simulations with ADS for the two coupled differential VCOs,

the two coupled differential Van der Pol oscillators and by using our CAD tool, for a

synchronization frequency of 5.98 GHz as a function of 01020 fff −=∆ .

Hence, a good agreement was found between the results presented in Figure 3.37

showing the reliability and the accuracy of the presented CAD tool.

685.35 685.40 685.45 685.50 685.55 685.60 685.65 685.70 685.75685.30 685.80

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-2.0

2.0

time (nsec)

Ou

tput

Vo

ltag

es

(V)

vd1-vd2

vd3-vd4


124

a)

b)



As can be seen in this figure, the two coupled differential VCOs have the same

behavior as the two coupled differential oscillators presented in section 3.2.5. Indeed,

-100,00

-80,00

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

80,00

100,00

-800 -600 -400 -200 0 200 400 600 800

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)

VDP Simulation VCO Simulation ASVAL Simulation

0

0,5

1

1,5

2

2,5

-800 -600 -400 -200 0 200 400 600 800

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



125

one can notice that, as the oscillator tunings are moved apart, but the synchronization

frequency is halfway between, the phase shift increases until the locking-region

boundary is encountered. Furthermore, as shown in Figure 3.37.b, the amplitude is also

at a minimum at the locking-region boundaries and equal to the oscillators’ free-running

amplitude for equal tunings.

The main advantage of this CAD tool is that, in an extremely short simulation time,

all the phase shifts, synchronization frequencies and amplitudes of the differential coupled

system are computed, in function of the free-running frequencies f01 and f02 of the two

differential VCOs. Hence, since the inter-stage phase shift is independent of the number of

oscillators in the array, the proposed tool can also help the designer to find rapidly the free-

running frequencies of the two outermost VCOs of the array required to achieve the desired

phase shift. Indeed, without such a tool, a transient analysis of the VCO array for different

couples of VCOs’ free-running frequencies must be performed in order to find the phase shift

required for the targeted application.

As mentioned above, the previous simulations were performed for a coupling

constant λ0 = 0.5 for which the value of the coupling resistor Rc must be chosen equal to

520 Ω since the value of the parallel resistor Rvp of the resonator is equal to 260 Ω.

Hence, as for the case of two coupled differential oscillators presented in section 3.2.6,

it seems interesting to study again the behavior of the system in both weak and strong

coupling cases i.e. for values of λ0 respectively less and greater than 0.5. Furthermore,

a new analysis based on state equations method is presented in the next section.

3.3.4.1.1. Study and analysis of two coupled differential VCOs using the

state equation approach

In this section, we will study the behavior of the two coupled differential VCOs

presented previously, using the state equations method implemented in Matlab software.

In order to formulate the state equations in symbolic normal-form for nonlinear

time- invariant analog circuits, a computer program called SYSEG – Symbolic State

Equation Generation, developed by the Electrical Engineering Department of the

University Politehnica of Bucharest was used [50]. The program allows the analysis of

circuits containing both linear and nonlinear resistors, inductors and capacitors,


independent voltage and c

formulates the symbolic stat

by simplification of the ex

minimum number of state v

form, our method is based

elements, and it uses the faci

Using the state equati

circuit design and an improv

straightforward analysis, the

computed, in order to detect

Let us now consider

were coupled through a resis

Figure 3.38 – Two differe

The general state spac

following:

Furthermore, the re

representation (3.16), obtaine


126

current sources and linear controlled so

tate equations without any inverse of a sym

expressions, it obtains a symbolic comp

variables. Starting from the circuit descrip

d on Kirchhoff’s laws, constitutive equatio

acilities of symbolic simulator Maple.

ations in symbolic form, we obtain an impor

roving of the accuracy in numerical calcula

he coefficients of the characteristic polynom

ct the eigenvalues at the origin.

er Figure 3.38, where two differential Van d

sistor.

rential Van der Pol oscillators coupled thr

with GNL1,2 = -a + buC1,22(t).

pace representation, written in matrix form fo

⋅=+⋅=

)t(xC)t(y

)t(B)t(xA)t(x,

resulting equations corresponding to

ined with SYSEG program are the following


sources. This tool,

ymbolic matrix and,

mpact form with a

ription in the netlist

tions of the circuit

portant efficiency in

ulations. Also, by a

mial can be exactly

n der Pol oscillators

through a resistor

for this case, is the

(3.16)

the state space

ng:


127

)t(uLdt

)t(di

)t(uLdt

)t(di

iC

uC

bu

RCu

RRa

Cdt

)t(du

iC

uC

bu

RCu

RRa

Cdt

)t(du

CL

CL

LCCc

Cc

C

LCCc

Cc

C

!

2

2

1

1

2212

2

121

1

2

1

2

3

2222

1

3

1111

1

1

11111

11111

=

=

−−+

−−=

−−+

−−=

, (3.17)

These equations describes the system of the coupled oscillators as a set of inputs,

outputs and state variables related by first order differential equations. This modeling

method, using simple matrix operations can be applied for systems with multiple inputs

and multiple outputs (MIMO) and includes the effect of initial conditions [51].

Nevertheless, the most important advantage of this modeling form is that the

representation consists in simple first order differential equations and provides directly

a time domain solution, without adding unnecessary complexity and, also, it is more

adapted to computations and can incorporate nonlinear effects.

The state vector, noted x, composed of state variables, is a minimum set of

variables that are fully describing the system and its response to any given set of inputs.

These variables are related to energy storage elements in a circuit. For two coupled

differential Van der Pol oscillators, the state vector includes four variables, the currents

through the inductors )t(i L1 and )t(iL2

and the voltages across the capacitors )t(uC1 and

)t(uC2, as follows:

=

)t(i

)t(i

)t(u

)t(u

)t(x

L

L

C

C

2

1

2

1

, (3.18)

The state space matrices that describe the evolution of this system and

completely characterized its dynamic behavior are the following:


128

−⋅

−−

−⋅

−−

=

001

0

0001

10

1111

011111

1

1

1122

1111

L

L

CCRRa

CR

CCRCRRa

A cc

cc

,

⋅−

⋅−

=

0

0

3

2

3

1

2

1

)t(uC

b

)t(uC

b

)t(B C

C

(3.19)

and [ ]0011=C .

Now, in order to analyze the system of two coupled oscillators, the state

equations are implemented on Matlab Mathworks software and solved using the 4th

order Runge-Kutta method. This numerical technique is used to solve ordinary

differential equations. The implicit 4th order Runge-Kutta method has the following

form:

443322111 kwkwkwkwyy ii ++++=+ , (3.20)

where w1,… are the weights, and k1,... are h times various approximations to the

slopes at points in the step, and are given by:

),kbkbkby,hax(hfk

),kbkby,hax(hfk

),kby,hax(hfk

),y,x(hfk

ii

ii

ii

ii

36251434

231223

1112

1

++++=+++=

++==

(3.21)

where the a1,… and b1,… are constants to be determined.

Nevertheless, using this numerical method, the global error is of the same order

as the local error and the derivatives of the function are useless in the computation. One

of the most important advantage is that the “Automatic Error Control” can be made

easily.

Now, after solving the state equations system, in Figure 3.39 are presented the

resulting waveforms of the output voltages obtained for Rc/2 = 260 , a = 6.62 10-3 and


129

b = 9.76 10-4, R1 = R2 = 260 , L1 = L2 = 1.6 10-9 H, C1 = 5.11 10-13 F, C2 = 4.1 10-13

F. Let us note that, in this case the obtained values of the phase shift, ∆φ, is 75.6°,

amplitude, A, is 1.35 V and synchronization frequency, fS, is 5.89 GHz.

Figure 3.39 – The output voltages of two coupled differential Van der Pol

oscillators obtained with Matlab for ∆∆∆∆φφφφ = 75.6°, A = 1.35 V and fS = 5.89 GHz.

In order to validate the results provided by this method for two differential

coupled Van der Pol oscillators of Figure 3.34, we compared them with the simulation

results obtained with ASVAL and ADS. Thus, Figure 3.40 illustrates a comparison of

the phase shift and amplitude resulting from simulations with Matlab, ASVAL and

ADS.


130

a)

b)



-100,00-80,00-60,00-40,00-20,00

0,0020,0040,0060,0080,00

100,00

-800 -600 -400 -200 0 200 400 600 800

∆φ∆φ ∆φ∆φ(°

)

∆∆∆∆f0 ((((MHz)

ADS Simulation ASVAL Simulation MATLAB Simulation

0

0,5

1

1,5

2

2,5

-800 -600 -400 -200 0 200 400 600 800

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)

ADS Simulation ASVAL Simulation MATLAB Simulation


131

3.3.4.2. Study and analysis of two coupled differential VCOs in the weak

coupling case

In the case of weak coupling, the circuits were analyzed again for a value of λ0 = 0.25

leading to Rc/2 = 520 . Thus, Figure 3.41 presents a comparison of the obtained phase shift

and amplitude for the two coupled Van der Pol oscillators, the two coupled differential VCOs

and using “ASVAL” also for a synchronization frequency of 5.89 GHz as a function of

01020 fff −=∆ .

a)

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

-300 -200 -100 0 100 200 300

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)



132

b)

Figure 3.41 – The weak coupling case – a) Comparison of the phase shift;


One more time, as can be seen in this figure, the locking range as well as the

maximum phase shift which can be obtained are clearly reduced compared to the optimal

case of a coupling strength λ0 = 0.5. This result can of course also be justified using the

approximate formula (2.29) as explained in section 3.2.6.

Furthermore, according to Figure 3.41.b, the amplitude also remains close to its free-

running value almost over the entire locking range which can be justified using the

approximate formula elaborated in chapter 2, i.e. (2.56).

In Figure 3.42 the cartographies, obtained with “ASVAL”, of the phase shift ∆φ

between the output voltages of the two VCOs, and those of their amplitudes A1 and A2,

in the case of weak coupling are shown.

1,80

1,82

1,84

1,86

1,88

1,90

1,92

1,94

1,96

-300 -200 -100 0 100 200 300

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



133

Figure 3.42 - The cartographies of the phase shift ∆∆∆∆φφφφ, the amplitudes A1 and A2 and

the synchronization frequency fs of the coupled VCOs for the weak coupling case.

3.3.4.3. Study and analysis of two coupled differential VCOs in the strong

coupling case

Let us now consider the strong coupling case where the circuits were analyzed for a

value of λ0 = 1.25. In these conditions, Figure 3.43 presents a comparison of the obtained

phase shift and amplitude for the two coupled Van der Pol oscillators, the two coupled

differential VCOs and using “ASVAL” also for a synchronization frequency of 5.89 GHz as

a function of 01020 fff −=∆ .


134

a)

b)

Figure 3.43 – The strong coupling case – a) Comparison of the phase shift;


As expected, the locking range increased compared to the optimal case of a coupling

strength λ0 = 0.5. This result can be justified again using the approximate formula (2.29) as

-100,00

-80,00

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

80,00

100,00

-2000 -1500 -1000 -500 0 500 1000 1500 2000

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)


0,00

0,50

1,00

1,50

2,00

2,50

-2000 -1500 -1000 -500 0 500 1000 1500 2000

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



135

explained in detail in section 3.2.7. Furthermore, we notice that, as for the case of two

coupled differential oscillators, the amplitude decreases considerably as we traverse the

locking region.

In Figure 3.44 the cartographies, obtained with “ASVAL” in the case of strong

coupling are presented.

Figure 3.44 - The cartographies of the phase shift ∆∆∆∆φφφφ, the amplitudes A1 and A2

and the synchronization frequency fs of the coupled VCOs for the strong coupling case.

3.3.4.4. Study and analysis of the variation of the phase shift ∆∆∆∆φφφφ versus the

coupling resistor Rc

In the case of coupled VCOs, the control of the phase shift is an important aspect

especially for the beam-scanning architecture. Thus, according to (2.29), this phase shift


136

can be adjusted either by detuning the two outermost VCOs of the array or by changing

the value of the coupling resistor Rc. Furthermore, in integrated circuits field, this

resistor can be replaced by a MOS transistor in the triode region of operation in order to

synthesize a voltage controlled resistor which can lead to a substantial save in chip area.

Thus, Figure 3.45 and Figure 3.46 present the variation of the phase shift versus Rc,

when f0 = 100 MHz and f0 = 200 MHz, respectively. Let us note that in both cases λ0

is decreasing from strong to weak coupling since the value of Rc is increasing.

We notice that in both cases, the value of the phase shift is increasing with the

value of Rc. This can be justified using (2.29) which can be also expressed as follows:

ωω∆

=φ∆a

cL RG

2arcsin 0 . (3.22)

Indeed, according to (3.22), the increasing of Rc leads to a greater value of the

phase shift for a given value of ω0.

Furthermore, comparing Figure 3.45 and Figure 3.46 and according to (3.22), for

a given value of Rc, the phase shift ∆φ is increasing with the difference between the two

free-running frequencies of the VCOs, ω0, which constitute an already known

behavior as detailed in the previous sections.

Nevertheless, these results show that it is possible to adjust the phase shift by

changing only the value of the coupling resistor Rc (or the gate voltage of a MOS

transistor in triode region) over a wide range. Indeed, as shown in Figure 3.45 and 3.46,

the theoretical limit of 90° can almost be reached using this technique.


137

Figure 3.45 – The variation of the phase shift versus Rc when f0 = 100 MHz.

Figure 3.46 – The variation of the phase shift versus Rc when f0 = 200 MHz;

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000

∆φ∆φ ∆φ∆φ( °° °°

)

Rc/2 (Ohm)


0

10

20

30

40

50

60

70

80

90

0 100 200 300 400 500 600 700 800 900

∆φ∆φ ∆φ∆φ( °° °°

)

Rc/2 (Ohm)



138

3.3.4.5. The effect of a mismatch between the two Rc on the phase shift ∆∆∆∆φφφφ

Let us remind that in an integrated circuit, a gradient in the silicon process can

lead to a difference between the two coupling resistors. Therefore, the phase shift

behavior in the case of such a mismatch between the two Rc can be an important issue.

Thus, Figure 3.47 and Figure 3.48 illustrate the phase shift variation versus f0, when

the mismatch is 5% and 7%, respectively. We can note that the circuit made by two

coupled VCOs is robust and, thus, a mismatch between the two Rc is not affecting either

the phase shift or the amplitude behavior.

a)

-100,00

-80,00

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

80,00

100,00

-800 -600 -400 -200 0 200 400 600 800

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)

0% Mismatch 5% Mismatch


139

b)

Figure 3.47 – a) The variation of the phase shift for 5% mismatch;

b) The variation of the amplitude for 5% mismatch.

a)

0

0,5

1

1,5

2

2,5

-800 -600 -400 -200 0 200 400 600 800

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)


-100,00

-80,00

-60,00

-40,00

-20,00

0,00

20,00

40,00

60,00

80,00

100,00

-800 -600 -400 -200 0 200 400 600 800

∆φ∆φ ∆φ∆φ( °° °°

)

∆∆∆∆f0 ((((MHz)



140

b)

Figure 3.48 – a) The variation of the phase shift for 7% mismatch;

b) The variation of the amplitude for 7% mismatch.

3.3.5. Four coupled differential VCOs

As mentioned previously and according to York & al., the inter-stage phase shift

is independent of the number of oscillators in the array [1, 11, 12]. Thus, in order to

verify this affirmation, four identical VCOs were coupled through a resistor Rc/2 of 260

Ω. In this case, a free-running frequency of 5.89 GHz is imposed to the VCOs 2 and 3

via the tuning voltage Vtune whereas the tuning voltages of the two outermost VCOs

can be adjusted in order to obtain the desired phase shift as shown in Figure 3.49.

0

0,5

1

1,5

2

2,5

-800 -600 -400 -200 0 200 400 600 800

Am

plit

ud

e (V

)

∆∆∆∆f0 ((((MHz)



141

Figure 3.49 – Schematic of four coupled VCOs.

Now let us consider the case for which the free-running frequencies of the two

outermost VCOs are equal to f01 = 6.07 GHz for VCO 1 and f04 = 5.71 GHz for VCO 4

leading to f0= -360 MHz. In these conditions, the four differential NMOS coupled

VCOs of Figure 3.33 simulated with ADS has lead to four sinusoidal waves at a

synchronization frequency of 5.87 GHz and a phase shift of -37.7°, as shown in Figure

3.50.

Figure 3.50 – Waveforms of the output voltages of the four differential NMOS VCOs

for ∆∆∆∆φφφφ ≈≈≈≈ -37°

Rc/2 Rc/2

Rc/2 Rc/2

VCO1 VCO2

Rc/2

Rc/2

VCO3 VCO4

Vtune1 Vtune Vtune Vtune4


142

This result is close to the one obtained for two coupled VCOs as shown in Figure

3.37.a for which the phase shift is equal to -34° under the same conditions. This

confirms the theory elaborated by York & al. and shows that arrays of coupled

differential VCOs can be used to control the radiation pattern of antenna arrays with a

higher number of array elements.

3.4. CONCLUSION

In this chapter, we started with the analysis of two differential oscillators

coupled through a resistive network. The oscillator schematic used is based on the well-

known cross-coupled NMOS differential topology. In order to represent this oscillator

by a negative resistance in parallel with an RLC resonator, Agilent’s ADS software was

used. From ADS simulation results for the differential oscillator at the required

synchronization frequency, it was possible to extract the parallel RLC circuit that

models the resonator, as well as the parameters a and b of the Van der Pol equation

capable of reproducing the behaviour of the oscillator’s active part.

After that, our CAD tool “ASVAL” was used to provide, in a considerably short

analysis time a cartography giving the phase shifts, synchronization frequencies and

amplitudes of the differential output voltages of the two differential oscillators coupled

through a resistor.

Then, a comparison of the phase shift, synchronization frequency and amplitudes

obtained with ADS simulations and on the other hand obtained with “ASVAL” is

presented. A good agreement was found between the results provided by “ASVAL” and the

simulation results showing the reliability of our CAD tool for different cases of coupling

strength.

In the second part of the chapter, the study and the analysis of two coupled

VCOs was presented. Following the same steps like for the modeling of a differential

RLC NMOS oscillator, the differential VCO was modeled like a Van der Pol oscillator.

Then two differential VCOs were coupled through a resistor. The comparison of the

phase shift, synchronization frequency and amplitudes obtained with ADS simulations,

with “ASVAL” and using a state equation method also showed a good agreement


143

between the results showing the usefulness of our CAD tool for the design of an array of

differential VCOs coupled through a broadband network. Furthermore, the variation of the

phase shift versus the coupling resistor was also investigated showing that it is possible

to adjust the phase shift by changing only the value of the coupling resistor Rc over a

wide range. The robustness of the circuit regarding the mismatch between the two

coupling resistors was also presented.

Finally, four VCOs were coupled through a resistor showing that arrays of

coupled differential VCOs can be used to control the radiation pattern of antenna arrays

with a higher number of array elements.

144

FINAL CONCLUSION

Final Conclusion

145

Arrays of coupled oscillators are receiving increasing interest in both military and

commercial applications. They are used to achieve high-power RF sources through

coherent power combining. Another application is the beam steering of antenna arrays. The

radiation pattern of a phased antenna array is steered in a particular direction by

establishing a constant phase progression throughout the oscillator’s chain.

Nevertheless, it is shown that the theoretical limit of the phase shift that can be

obtained by slightly detuning the end elements of the array by equal amounts but in

opposite directions is only ±90°. In this context, it seemed interesting to study and analyze

the behavior of an array of coupled differential oscillators or Voltage Controlled

Oscillators (VCOs) since, in this case, the theoretical limit of the phase shift is within 360°

due to the differential operation of the array leading to a more efficient beam-scanning

architecture for example. As a consequence, the aim of this work was to study and analyze

the behavior of coupled differential oscillators and VCOs used to control antenna arrays.

This work was organized in 3 chapters, followed by references and appendixes.

Chapter I, Coupled-Oscillator Arrays - Application, reminds briefly the oscillators’

principle as well as their classifications and applications in the communication system. The

principle of an oscillator can be represented by a nonlinear impedance, which is the active

part of the oscillator, in parallel with the equivalent impedance of the resonator. The

condition for sustaining the oscillations can be fulfilled only if the real part of the non-

linear impedance is negative, which can be obtained by an active element. Oscillators are

classified in accordance with the wave shapes they produce and the circuitry required for

producing the desired oscillations.

In the second part of the chapter, a state of the art of coupled-oscillators theory was

presented followed by a few applications of antenna arrays. Due to their different

geometric configuration, antenna arrays can have an important role in controlling the

radiation angle of the pattern. Therefore, and also for simplicity reasons, a linear

configuration was presented in this section. The most important advantage of controlling

the antenna array consists in generating the amplitudes and/or the phases necessary for

orientating the radiation pattern in the desired direction. As a consequence, various

technical solutions were proposed, including the coupled oscillators approach.

Final Conclusion

146

Chapter II, Theoretical Analysis of Coupled Oscillators Applied to an Antenna

Array, presents in the first part a complete review of R. York’s theory, giving the dynamics

of two Van der Pol oscillators coupled through a series resonant network. Then, the case of

a resistive coupling circuit was deduced from this system of equations.

Then, showing the limitation of this approach regarding the prediction of the

oscillators’ amplitudes, a new formulation of the nonlinear equations describing the

oscillators’ locked states was proposed.

Nevertheless, due to the trigonometric and strongly non-linear aspect of this system

of nonlinear equations describing the locked states of two coupled Van der Pol oscillators,

mathematical manipulations were applied in order to obtain a new system easier to solve

numerically. Thus, a simpler system of three equations with three unknowns was obtained

allowing then to implement a CAD tool that permits to extract, in an extremely short

simulation time, a cartography giving the frequency locking region of two differential

oscillators coupled through a resistive network. This cartography can help the designer to

find rapidly the free-running frequencies of the two outermost oscillators or VCOs of the

array required to achieve the desired phase shift.

Chapter III, Study and Analysis of an Array of Differential Oscillators and VCOs

Coupled through a Resistive Network, begins with the analysis of two RLC NMOS

differential oscillators coupled through a resistive network. The oscillator schematic used

is based on the well-known cross-coupled NMOS differential topology, which provides the

negative resistance to compensate for the tank losses of the resonator. Since the theory

implemented in our CAD tool “ASVAL” uses Van der Pol oscillators to model microwave

coupled oscillators, a modeling procedure of the differential oscillator as a differential Van

der Pol oscillator was performed. To do so, Agilent’s ADS software was used. From ADS

simulation results for the differential oscillator at the required synchronization frequency, it

was possible to extract the parallel RLC circuit that models the resonator, as well as the

parameters a and b of the Van der Pol equation capable of reproducing the behaviour of the

oscillator’s active part.

After that, the proposed CAD tool was used to provide the cartography giving the

phase shifts, synchronization frequencies and amplitudes of the differential output voltages

Final Conclusion

147

of the two differential oscillators coupled through a resistor. Then, in order to validate

these results, a comparison to the simulation results of the two coupled differential

oscillators obtained with Agilent’s ADS software was presented. A good agreement was

found between the results provided by our CAD tool and the simulation results showing the

reliability of the presented tool for different cases of coupling strength

In the last part of the chapter, the same study was performed for the case of two

differential Voltage Controlled Oscillators (VCOs) coupled through a resistor. The

comparison of the phase shift, synchronization frequencies and amplitudes obtained with

ADS simulations, with “ASVAL” and using a state equation method also showed good

agreements between the results proving the usefulness of our CAD tool for the design of an

array of differential VCOs coupled through a broadband network. Furthermore, the study of the

variation of the phase shift versus the coupling resistor was also investigated as well as the

robustness of the circuit regarding the mismatch between the two coupling resistors.

Finally, four VCOs were coupled through a resistor showing that arrays of coupled

differential VCOs can be used to control the radiation pattern of antenna arrays with a

higher number of array elements.

PERSONAL CONTRIBUTIONS

• The study of generality on antenna arrays, uniform linear network and the

controlling of radiation angle of the pattern.

• The state of the art on generality and classification of oscillators.

• The study and the analysis of the dynamic equations of two oscillators coupled

through a RLC circuit.

• A nonlinear system of equations describing the oscillators’ locked states for the

specific case of a resistive coupling network was deduced.

• Showing the limitation of R. York approach regarding the prediction of the oscillators’

amplitudes, a new formulation of the nonlinear equations describing the oscillators’

locked states was developed.

Final Conclusion

148

• The specific dynamic equations of two oscillators coupled through a resistive network were

deduced.

• Because of the trigonometric and strongly non-linear aspect of the dynamic equations

describing the oscillators’ locked states, mathematical manipulations were applied in

order to obtain a new system easier to solve numerically.

• The elaboration of a CAD tool which provides, in a considerably short simulation time,

the frequency locking region of two coupled Van der Pol oscillators.

• The cartography of the synchronization area, phase shift and amplitudes of two

coupled oscillators using the CAD tool was generated.

• Simulations using Agilent Advanced Design System were performed for the analysis of

two coupled NMOS differential oscillators and two coupled differential Van der Pol

oscillators.

• The modeling procedure of two coupled NMOS differential oscillators as two coupled

differential Van der Pol oscillators, with a resistive coupling network was performed.

Good agreements between the simulations of the circuit, the model and the theoretical

results from our CAD tool were found.

• The analysis of the behavior of two coupled NMOS differential oscillators in the case

of strong and weak coupling was realized.

• The modeling procedure of two coupled differential VCOs as two coupled differential

Van der Pol oscillators, with a resistive coupling network was performed.

• The cartography of the synchronization area, phase shift and amplitudes of two

coupled differential VCOs using the CAD tool was generated.

• The study and analysis of an array of two differential VCOs coupled through a resistive

network was realized.

• The analysis and study of four differential VCOs coupled through a resistive network was

performed.

Final Conclusion

149

• Simulations using Matlab Mathworks software of an array of two differential Van der Pol

oscillators coupled through a resistor using the state space representation. Comparison of

the obtained results using the theory and Agilent Advanced Design System simulations.

FUTURE PROSPECTS

• The design of an array of coupled VCOs using a 0,25 µm BiCMOS SiGe process.

• The elaboration of a modeling procedure of two single-ended oscillators, starting from

the state equation representation leading to the parameters identification of two coupled

Van der Pol oscillators. This approach is useful in the design of an array of coupled

oscillators allowing to determine the outermost free-running frequencies of the array.

List of Publications

150

Publications

Iulia Dumitrescu, Mihaela Ionita , Jean-Marie Paillot, Mihai Iordache, Couples theory des oscillateurs couples appliquée aux antennes réseaux, The Scientific Bulletin of Electrical Engineering Faculty, year 9, No 2, pp. 27-35, 2009; Iulia Dumitrescu, Mihai Iordache, Mihaela Ionita , Analysis of Coupled Oscillators through a Series RLC Network, Proceedings of the 10th International Conference on Development and Application Systems, Suceava, Romania, pp. 65, May 2010; Iulia Dumitrescu, Ileana Calomfirescu, Mihaela Ionita, Analiza Pspice a oscilatoarelor de tip Van der Pol cuplate, Simpozionul National de Electrotehnica Teoretica, Bucuresti, Romania, November 2009; Mihai Iordache, Jean-Marie Paillot, Iulia Dumitrescu, Mihaela Ionita , Analysis of coupled Oscillators Applied to Antenna Arrays, 10th International Conference on Applied and Theoretical Electricity, Craiova, Romania, pp. 25-29, October 2010; Mihaela Ionita , Iulia Dumitrescu, Mihai Iordache, Analysis of Coupled Oscillators by Semi-State Variable Method, U.P.B. Scientific Bulletin Series C – Electrical Engineering, no. 3, August 2012, (in press).

Publications in international data base

Mihai Iordache, Lucia Dumitriu, Iulia Dumitrescu, Mihaela Ionita , Analysis of Coupled Oscillators Applied to 1D Antenna Array, 5th European Conference on Circuits and Systems for Communications, Belgrade Serbia, pp. 256-260, November 2010; Mihaela Ionita , David Cordeau, Jean-Marie Paillot, Mihai Iordache, Analysis and Design of an Array of Two Differential Oscillators Coupled Through a Resistive Network, 20th European Conference on Circuit Theory and Design, Linkoping, Sweden, pp. 73-76, August 2011; Mihaela Ionita , David Cordeau, Jean-Marie Paillot, Smail Bachir, Mihai Iordache, A CAD Tool for an Array of Differential Oscillators Coupled Through a Broadband Network, International Journal of RF and Microwave Computer-aided Engineering, (in press); Mihaela Ionita , Mihai Iordache, Lucia Dumitriu, David Cordeau, Jean-Marie Paillot, Generation of the Coupling Circuit Parameters for the Coupled Oscillators Used in Antenna Arrays, International Conference on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design, (accepted for publication).

151

APPENDIX A

Appendix A – Marquardt Algorithm

152

Marquardt algorithm

In mathematics and computing, the Levenberg–Marquardt algorithm (LMA)

provides a numerical solution to the problem of minimizing a function, generally nonlinear,

over a space of parameters of the function. These minimization problems arise especially in

least squares curve fitting and nonlinear programming [46].

The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method

of gradient descent. The LMA is more robust than the GNA, which means that in many

cases it finds a solution even if it starts very far off the final minimum. For well-behaved

functions and reasonable starting parameters, the LMA tends to be a bit slower than the

GNA. LMA can also be viewed as Gauss–Newton using a trust region approach.

The LMA is a very popular curve-fitting algorithm used in many software

applications for solving generic curve-fitting problems. However, the LMA finds only a

local minimum, not a global minimum.

Caveat Emptor

One important limitation that is very often over-looked is that it only optimizes for

residual errors in the dependant variable (y). It thereby implicitly assumes that any errors in

the independent variable are zero or at least ratio of the two is so small as to be negligible.

This is not a defect, it is intentional, but it must be taken into account when deciding

whether to use this technique to do a fit. While this may be suitable in context of a

controlled experiment there are many situations where this assumption cannot be made. In

such situations either non-least squares methods should be used or the least-squares fit

should be done in proportion to the relative errors in the two variables, not simply the

vertical "y" error. Failing to recognize this can lead to a fit which is significantly incorrect

and fundamentally wrong. It will usually underestimate the slope. This may or may not be

obvious to the eye.

Microsoft Excel's chart offers a trend fit that has this limitation that is

undocumented. Users often fall into this trap assuming the fit is correctly calculated for all

situations. OpenOffice spreadsheet copied this feature and presents the same problem.


153

The problem

The primary application of the Levenberg–Marquardt algorithm is in the least squares curve

fitting problem: given a set of m empirical datum pairs of independent and dependent variables, (xi,

yi), optimize the parameters β of the model curve f(x,β) so that the sum of the squares of the

deviations:

∑=

β−=βm

iii )],x(fy[)(S

1

2,

becomes minimal.

The solution

Like other numeric minimization algorithms, the Levenberg–Marquardt algorithm is

an iterative procedure. To start a minimization, the user has to provide an initial guess for

the parameter vector, β. In cases with only one minimum, an uninformed standard guess

like βT=(1,1,...,1) will work fine; in cases with multiple minima, the algorithm converges

only if the initial guess is already somewhat close to the final solution.

In each iteration step, the parameter vector, β, is replaced by a new estimate, β + δ.

To determine δ, the functions ),x(f i δ+β are approximated by their linearization:

δ+β≈δ+β iii J),x(f),x(f ,

where

β∂β∂

=),x(f

J ii ,

is the gradient (row-vector in this case) of f with respect to β.

At its minimum, the sum of squares, S(β), the gradient of S with respect to δ will be

zero. The above first-order approximation of ),x(f i δ+β gives:


154

∑=

δ−β−≈δ+βm

iiii )J),x(fy()(S

1

2.

Or in vector notation,

2δ−β−≈δ+β J)(fy)(S .

Taking the derivative with respect to δ and setting the result to zero gives:

)](fy[J)JJ( TT β−=δ ,

where J is the Jacobian matrix whose i th row equals Ji, and where f and y are vectors

with i th component ),x(f i β and yi, respectively. This is a set of linear equations which can

be solved for δ.

Levenberg's contribution is to replace this equation by a "damped version",

)](fy[J)IJJ( TT β−=δλ+ ,

where I is the identity matrix, giving as the increment, δ, to the estimated parameter

vector, β.

The (non-negative) damping factor, λ, is adjusted at each iteration. If reduction of S

is rapid, a smaller value can be used, bringing the algorithm closer to the Gauss–Newton

algorithm, whereas if an iteration gives insufficient reduction in the residual, λ can be

increased, giving a step closer to the gradient descent direction. Note that the gradient of S

with respect to β equals 22 )])(fy[J( T β−− . Therefore, for large values of λ, the step will

be taken approximately in the direction of the gradient. If either the length of the calculated

step, δ, or the reduction of sum of squares from the latest parameter vector, β + δ, fall

below predefined limits, iteration stops and the last parameter vector, β, is considered to be

the solution.


155

Levenberg's algorithm has the disadvantage that if the value of damping factor, λ, is

large, inverting JTJ + λI is not used at all. Marquardt provided the insight that we can scale

each component of the gradient according to the curvature so that there is larger movement

along the directions where the gradient is smaller. This avoids slow convergence in the

direction of small gradient. Therefore, Marquardt replaced the identity matrix, I, with the

diagonal matrix consisting of the diagonal elements of JTJ, resulting in the Levenberg–

Marquardt algorithm:

)](fy[J))JJ(diagJJ( TTT β−=δλ+ .

A similar damping factor appears in Tikhonov regularization, which is used to solve

linear ill-posed problems, as well as in ridge regression, an estimation technique in

statistics.

Choice of damping parameter

Various more-or-less heuristic arguments have been put forward for the best choice

for the damping parameter λ. Theoretical arguments exist showing why some of these

choices guaranteed local convergence of the algorithm; however these choices can make

the global convergence of the algorithm suffer from the undesirable properties of steepest-

descent, in particular very slow convergence close to the optimum.

The absolute values of any choice depend on how well-scaled the initial problem is.

Marquardt recommended starting with a value λ0 and a factor ν>1. Initially setting λ=λ0 and

computing the residual sum of squares S(β) after one step from the starting point with the

damping factor of λ=λ0 and secondly with λ0/ν. If both of these are worse than the initial

point then the damping is increased by successive multiplication by ν until a better point is

found with a new damping factor of λ0νk for some k.

If use of the damping factor λ/ν results in a reduction in squared residual then this is

taken as the new value of λ (and the new optimum location is taken as that obtained with

this damping factor) and the process continues; if using λ/ν resulted in a worse residual, but

using λ resulted in a better residual then λ is left unchanged and the new optimum is taken

as the value obtained with λ as damping factor.

156

APPENDIX B

Appendix B – Intelligent Transport Systems

157

Intelligent Transport Systems

The term Intelligent Transport Systems (ITS) refers to information and

communication technology (applied to transport infrastructure and vehicles) that improve

transport outcomes such as transport safety, transport productivity, travel reliability,

informed travel choices, social equity, environmental performance and network operation

resilience [49].

Background

Interest in ITS comes from the problems caused by traffic congestion and a synergy

of new information technology for simulation, real-time control, and communications

networks. Traffic congestion has been increasing worldwide as a result of increased

motorization, urbanization, population growth, and changes in population density.

Congestion reduces efficiency of transportation infrastructure and increases travel time, air

pollution, and fuel consumption.

The United States, for example, saw large increases in both motorization and

urbanization starting in the 1920s that led to migration of the population from the sparsely

populated rural areas and the densely packed urban areas into suburbs. The industrial

economy replaced the agricultural economy, leading the population to move from rural

locations into urban centers. At the same time, motorization was causing cities to expand

because motorized transportation could not support the population density that the existing

mass transit systems could. Suburbs provided a reasonable compromise between population

density and access to a wide variety of employment, goods, and services that were

available in the more densely populated urban centers. Further, suburban infrastructure

could be built quickly, supporting a rapid transition from a rural/agricultural economy to an

industrial/urban economy.

Recent governmental activity in the area of ITS – specifically in the United States –

is further motivated by an increasing focus on homeland security. Many of the proposed

ITS systems also involve surveillance of the roadways, which is a priority of homeland

security. Funding of many systems comes either directly through homeland security


158

organizations or with their approval. Further, ITS can play a role in the rapid mass

evacuation of people in urban centers after large casualty events such as a result of a

natural disaster or threat. Much of the infrastructure and planning involved with ITS

parallels the need for homeland security systems.

In the developing world, the migration of people from rural to urbanized habitats

has progressed differently. Many areas of the developing world have urbanized without

significant motorization and the formation of suburbs. In areas like Santiago, Chile, a high

population density is supported by a multimodal system of walking, bicycle transportation,

motorcycles, buses, and trains. A small portion of the population can afford automobiles,

but the automobiles greatly increase the congestion in these multimodal transportation

systems. They also produce a considerable amount of air pollution, pose a significant safety

risk, and exacerbate feelings of inequities in the society.

Other parts of the developing world, such as China, remain largely rural but are

rapidly urbanizing and industrializing. In these areas a motorized infrastructure is being

developed alongside motorization of the population. Great disparity of wealth means that

only a fraction of the population can motorize, and therefore the highly dense multimodal

transportation system for the poor is cross-cut by the highly motorized transportation

system for the rich. The urban infrastructure is being rapidly developed, providing an

opportunity to build new systems that incorporate ITS at early stages.

Intelligent transport technologies

Intelligent transport systems vary in technologies applied, from basic management

systems such as car navigation; traffic signal control systems; container management

systems; variable message signs; automatic number plate recognition or speed cameras to

monitor applications, such as security CCTV systems; and to more advanced applications

that integrate live data and feedback from a number of other sources, such as parking

guidance and information systems; weather information; bridge deicing systems; and the

like. Additionally, predictive techniques are being developed to allow advanced modeling

and comparison with historical baseline data. Some of the constituent technologies

typically implemented in ITS are described in the following sections.


159

Wireless communications

Various forms of wireless communications technologies have been proposed for

intelligent transportation systems.

Radio modem communication on UHF and VHF frequencies are widely used for

short and long range communication within ITS.

Short-range communications (less than 500 yards) can be accomplished using IEEE

802.11 protocols, specifically WAVE or the Dedicated Short Range Communications

standard being promoted by the Intelligent Transportation Society of America and the

United States Department of Transportation. Theoretically, the range of these protocols can

be extended using Mobile ad-hoc networks or Mesh networking.

Longer range communications have been proposed using infrastructure networks

such as WiMAX (IEEE 802.16), Global System for Mobile Communications (GSM), or

3G. Long-range communications using these methods are well established, but, unlike the

short-range protocols, these methods require extensive and very expensive infrastructure

deployment. There is lack of consensus as to what business model should support this

infrastructure.

Computational technologies

Recent advances in vehicle electronics have led to a move toward fewer, more

capable computer processors on a vehicle. A typical vehicle in the early 2000s would have

between 20 and 100 individual networked microcontroller/Programmable logic controller

modules with non-real-time operating systems. The current trend is toward fewer, more

costly microprocessor modules with hardware memory management and Real-Time

Operating Systems. The new embedded system platforms allow for more sophisticated

software applications to be implemented, including model-based process control, artificial

intelligence, and ubiquitous computing. Perhaps the most important of these for Intelligent

Transportation Systems is artificial intelligence.


160

Floating car data/floating cellular data

"Floating car" or "probe" data collection is a set of relatively low-cost methods for

obtaining travel time and speed data for vehicles traveling along streets, highways,

freeways, and other transportation routes. Broadly speaking, three methods have been used

to obtain the raw data:

• Triangulation Method. In developed countries a high proportion of cars contain one or

more mobile phones. The phones periodically transmit their presence information to the

mobile phone network, even when no voice connection is established. In the mid 2000s,

attempts were made to use mobile phones as anonymous traffic probes. As a car moves, so

does the signal of any mobile phones that are inside the vehicle. By measuring and

analyzing network data using triangulation, pattern matching or cell-sector statistics (in an

anonymous format), the data was converted into traffic flow information. With more

congestion, there are more cars, more phones, and thus, more probes. In metropolitan areas,

the distance between antennas is shorter and in theory accuracy increases. An advantage of

this method is that no infrastructure needs to be built along the road; only the mobile phone

network is leveraged. But in practice the triangulation method can be complicated,

especially in areas where the same mobile phone towers serve two or more parallel routes

(such as a freeway with a frontage road, a freeway and a commuter rail line, two or more

parallel streets, or a street that is also a bus line). By the early 2010s, the popularity of the

triangulation method was declining.

• Vehicle Re-Identification. Vehicle re-identification methods require sets of detectors

mounted along the road. In this technique, a unique serial number for a device in the

vehicle is detected at one location and then detected again (re-identified) further down the

road. Travel times and speed are calculated by comparing the time at which a specific

device is detected by pairs of sensors. This can be done using the MAC (Machine Access

Control) addresses from Bluetooth devices, or using the RFID serial numbers from

Electronic Toll Collection (ETC) transponders (also called "toll tags").

• GPS Based Methods. An increasing number of vehicles are equipped with in-vehicle GPS

(satellite navigation) systems that have two-way communication with a traffic data

provider. Position readings from these vehicles are used to compute vehicle speeds.


161

Floating car data technology provides advantages over other methods of traffic

measurement:

• Less expensive than sensors or cameras

• More coverage (potentially including all locations and streets)

• Faster to set up and less maintenance

• Works in all weather conditions, including heavy rain

Sensing technologies

Technological advances in telecommunications and information technology, coupled

with state-of-the-art microchip, RFID (Radio Frequency Identification), and inexpensive

intelligent beacon sensing technologies, have enhanced the technical capabilities that will

facilitate motorist safety benefits for intelligent transportation systems globally. Sensing

systems for ITS are vehicle- and infrastructure-based networked systems, i.e., Intelligent

vehicle technologies. Infrastructure sensors are indestructible (such as in-road reflectors)

devices that are installed or embedded in the road or surrounding the road (e.g., on

buildings, posts, and signs), as required, and may be manually disseminated during

preventive road construction maintenance or by sensor injection machinery for rapid

deployment. Vehicle-sensing systems include deployment of infrastructure-to-vehicle and

vehicle-to-infrastructure electronic beacons for identification communications and may

also employ video automatic number plate recognition or vehicle magnetic signature

detection technologies at desired intervals to increase sustained monitoring of vehicles

operating in critical zones.

Inductive loop detection

Inductive loops can be placed in a roadbed to detect vehicles as they pass through

the loop's magnetic field. The simplest detectors simply count the number of vehicles

during a unit of time (typically 60 seconds in the United States) that pass over the loop,

while more sophisticated sensors estimate the speed, length, and weight of vehicles and the

distance between them. Loops can be placed in a single lane or across multiple lanes, and

they work with very slow or stopped vehicles as well as vehicles moving at high-speed.


162

Video vehicle detection

Traffic flow measurement and automatic incident detection using video cameras is

another form of vehicle detection. Since video detection systems such as those used in

automatic number plate recognition do not involve installing any components directly into

the road surface or roadbed, this type of system is known as a "non-intrusive" method of

traffic detection. Video from black-and-white or color cameras is fed into processors that

analyze the changing characteristics of the video image as vehicles pass. The cameras are

typically mounted on poles or structures above or adjacent to the roadway. Most video

detection systems require some initial configuration to "teach" the processor the baseline

background image. This usually involves inputting known measurements such as the

distance between lane lines or the height of the camera above the roadway. A single video

detection processor can detect traffic simultaneously from one to eight cameras, depending

on the brand and model. The typical output from a video detection system is lane-by-lane

vehicle speeds, counts, and lane occupancy readings. Some systems provide additional

outputs including gap, headway, stopped-vehicle detection, and wrong-way vehicle alarms.

Intelligent transport applications

Emergency vehicle notification systems

The in-vehicle eCall is an emergency call generated either manually by the vehicle

occupants or automatically via activation of in-vehicle sensors after an accident. When

activated, the in-vehicle eCall device will establish an emergency call carrying both voice

and data directly to the nearest emergency point (normally the nearest E1-1-2 Public-safety

answering point, PSAP). The voice call enables the vehicle occupant to communicate with

the trained eCall operator. At the same time, a minimum set of data will be sent to the eCall

operator receiving the voice call.

The minimum set of data contains information about the incident, including time,

precise location, the direction the vehicle was traveling, and vehicle identification. The

pan-European eCall aims to be operative for all new type-approved vehicles as a standard

option. Depending on the manufacturer of the eCall system, it could be mobile phone based

(Bluetooth connection to an in-vehicle interface), an integrated eCall device, or a


163

functionality of a broader system like navigation, Telematics device, or tolling device.

eCall is expected to be offered, at earliest, by the end of 2010, pending standardization by

the European Telecommunications Standards Institute and commitment from large EU

member states such as France and the United Kingdom.

The EC funded project SafeTRIP is developing an open ITS system that will

improve road safety and provide a resilient communication through the use of S-band

satellite communication. Such platform will allow for greater coverage of the Emergency

Call Service within the EU.

Automatic road enforcement

A traffic enforcement camera system, consisting of a camera and a vehicle-

monitoring device, is used to detect and identify vehicles disobeying a speed limit or some

other road legal requirement and automatically ticket offenders based on the license plate

number. Traffic tickets are sent by mail. Applications include:

• Speed cameras that identify vehicles traveling over the legal speed limit. Many such

devices use radar to detect a vehicle's speed or electromagnetic loops buried in each lane of

the road.

• Red light cameras that detect vehicles that cross a stop line or designated stopping place

while a red traffic light is showing.

• Bus lane cameras that identify vehicles traveling in lanes reserved for buses. In some

jurisdictions, bus lanes can also be used by taxis or vehicles engaged in car pooling.

• Level crossing cameras that identify vehicles crossing railways at grade illegally.

• Double white line cameras that identify vehicles crossing these lines.

• High-occupancy vehicle lane cameras for that identify vehicles violating HOV

requirements.

• Turn cameras at intersections where specific turns are prohibited on red. This type of

camera is mostly used in cities or heavy populated areas.

Variable speed limits

Recently some jurisdictions have begun experimenting with variable speed limits

that change with road congestion and other factors. Typically such speed limits only

change to decline during poor conditions, rather than being improved in good ones. One


164

example is on Britain's M25 motorway, which circumnavigates London. On the most

heavily traveled 14-mile (23 km) section (junction 10 to 16) of the M25 variable speed

limits combined with automated enforcement have been in force since 1995. Initial results

indicated savings in journey times, smoother-flowing traffic, and a fall in the number of

accidents, so the implementation was made permanent in 1997. Further trials on the M25

have been thus far proved inconclusive.

Collision avoidance systems

Japan has installed sensors on its highways to notify motorists that a car is stalled

ahead.

Dynamic Traffic Light Sequence

Intelligent RFID traffic control has been developed for dynamic traffic light

sequence. It circumvents or avoids problems that usually arise with systems that use image

processing and beam interruption techniques. RFID technology with appropriate algorithm

and database were applied to a multi vehicle, multi lane and multi road junction area to

provide an efficient time management scheme. A dynamic time schedule was worked out

for the passage of each column. The simulation has shown that, the dynamic sequence

algorithm has the ability to intelligently adjust itself even with the presence of some

extreme cases. The real time operation of the system able to emulate the judgment of a

traffic police officer on duty, by considering the number of vehicles in each column and the

routing proprieties.

Cooperative systems on the road

Communication cooperation on the road includes car-to-car, car-to-infrastructure,

and vice versa. Data available from vehicles are acquired and transmitted to a server for

central fusion and processing. These data can be used to detect events such as rain (wiper

activity) and congestion (frequent braking activities). The server processes a driving

recommendation dedicated to a single or a specific group of drivers and transmits it

wirelessly to vehicles. The goal of cooperative systems is to use and plan communication

and sensor infrastructure to increase road safety. The definition of cooperative systems in

road traffic is according to the European Commission:


165

"Road operators, infrastructure, vehicles, their drivers and other road users will cooperate

to deliver the most efficient, safe, secure and comfortable journey. The vehicle-vehicle and

vehicle-infrastructure co-operative systems will contribute to these objectives beyond the

improvements achievable with stand-alone systems."

166

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Abstract

The work presented in this thesis deals with the study of coupled differential oscillators and Voltage Controlled Oscillators (VCO) used to control antenna arrays. After reminding the concept of antenna arrays and oscillators, an overview of R. York’s theory giving the dynamics for two Van der Pol oscillators coupled through a resonant network was presented. Then, showing the limitation of this approach regarding the prediction of the oscillators’ amplitudes, a new formulation of the nonlinear equations describing the oscillators’ locked states was proposed. Nevertheless, due to the trigonometric and strongly non-linear aspect of these equations, mathematical manipulations were applied in order to obtain a new system easier to solve numerically. This has allowed to the elaboration of a Computer Aided Design (CAD) tool, which provides a cartography giving the frequency locking region of two coupled differential Van der Pol oscillators. This cartography can help the designer to rapidly find the free-running frequencies of the two outermost differential oscillators or VCOs of the array required to achieve the desired phase shift. To do so, a modeling procedure of two coupled differential oscillators and VCOs as two coupled differential Van der Pol oscillators, with a resistive coupling network was performed. Then, in order to validate the results provided by our CAD tool, we compared them to the simulation results of two coupled differential oscillators and VCOs obtained with Agilent’s ADS software. Good agreements between the simulations of the circuits, the models and the theoretical results from our CAD tool were found.

Keywords: Antenna arrays, coupled oscillators, design automation, van der Pol differential oscillators, synchronization, Voltage Controlled Oscillator (VCO).

Résumé

Le travail présenté dans ce mémoire traite de l’étude d’oscillateurs et d’Oscillateurs Contrôlés en Tension (OCT) différentiels couplés appliqués à la commande d’un réseau d’antennes linéaire. Après avoir rappelé les concepts d’antennes réseaux et d’oscillateurs, une synthèse de la théorie élaborée par R. York et donnant les équations dynamiques modélisant deux oscillateurs de Van der Pol couplés par un circuit résonnant a été présentée. Après avoir montré la limitation de cette approche concernant la prédiction de l’amplitude des oscillateurs, une nouvelle formulation des équations non linéaires décrivant les états de synchronisation a été proposée. Néanmoins, compte tenu du caractère trigonométrique et fortement non linéaire de ces équations, une nouvelle écriture facilitant la résolution numérique a été proposée. Ceci a permis l’élaboration d’un outil de Conception Assistée par Ordinateur (CAO) fournissant une cartographie de la zone de synchronisation de deux oscillateurs de Van der Pol couplés. Celle-ci permet de déterminer rapidement les fréquences d’oscillation libres nécessaires à l’obtention du déphasage souhaité. Pour ce faire, une procédure de modélisation de deux oscillateurs et OCTs différentiels couplés, par deux oscillateurs de Van der Pol couplés par une résistance a été élaborée. Les résultats fournis par l’outil de CAO proposé ont ensuite été comparés avec les résultats de simulations de deux oscillateurs et OCTs différentiels couplés obtenus avec le logiciel ADS d’Agilent. Une très bonne concordance des résultats a alors été obtenue montrant ainsi l’utilité et la précision de l’outil présenté.

Mots-clés : Réseau d’antennes, oscillateurs couplés, conception assistée par ordinateur, oscillateurs de van der Pol différentiels, synchronisation, oscillateur commandé en tension (OCT).

Documents

Contribution to the study of synchronized differential