Design, Modeling and Noise Measurement of Oscillators Using a Large Signal Network Analyzer

DESIGN, MODELING AND NOISE MEASUREMENT OF OSCILLATORS USING A LARGE SIGNAL NETWORK ANALYZER

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Inwon Suh, M.S.E.E.

Graduate Program in Electrical and Computer Engineering

The Ohio State University

2011

Dissertation Committee:

Prof. Patrick Roblin, Adviser Prof. Steven B. Bibyk Prof. Roberto Rojas-Teran

Prof. James Peck °c Copyright by

Inwon Suh

2011 ABSTRACT

Oscillators play a crucial role in wireless communication systems since they are used together with mixers for frequency translation. To design discrete oscillators in a more efﬁcient way in terms of output power, a multi-harmonic real-time open-loop active load- pull technique was developed and used to ﬁnd the optimal fundamental and harmonic load impedances. Multi-harmonic loaded load circuits were then designed and implemented to approach the optimal multi-harmonic load impedances and realize a stand-alone oscillator.

Then, a new behavioral modeling technique based on power dependent Volterra series was developed to model negative-resistance oscillators. The derived behavioral model predicts the harmonic load-pull behaviors and the output power characteristics of oscillators and assists with the oscillator design.

The noise characteristic of oscillator is also of great importance in communication circuits. A new generalized 1/f Kurokawa noise analysis applicable to both low and high

Q oscillators is proposed for 1/f phase and amplitude noise. A theoretical correspondence between the new generalized Kurokawa theory and the impulse sensitivity function and the perturbation projection vector analyses is also derived. The proposed generalized

Kurokawa theory is then applied to a Van Der Pol oscillator, a BJT Colpitts oscillator and a

CMOS ring-oscillator, and a pHEMT oscillator and is veriﬁed to yield comparable results to those obtained from the matrix conversion method and the experimental result.

ii Also, an 1/f additive phase noise analysis for the injection-locked oscillator is presented.

An additive phase noise measurement system integrated with an LSNA is also developed to effectively acquire the model parameters needed. The validity of the analytic solution is

ﬁnally veriﬁed to yield reasonable agreement to the experimental result.

The additive phase noise measurement system integrated with a LSNA and a tunable monochromatic light source is further applied to characterize the additive phase noise performance of the both passivated and unpassivated AlGaN/GaN HEMT at 2 GHz under various operating conditions. Illumination with different photon energies, different drain voltages, and different load impedances are used to probe the dependence of the additive phase noise on the trap and 2DEG population.

iii This is dedicated to my family.

iv ACKNOWLEDGMENTS

First of all, I would like to express my deep appreciation to my respected advisor Pro- fessor Patrick Roblin for his teaching and support. His outstanding insights and creative ideas have made various impossible researches possible whenever we have faced difﬁcul- ties. Most of all, he has been an excellent role model for me as a researcher and even as a human being. All of the knowledge and experiences I have learned from him will inspire me for the rest of my life.

I also would like to thank Professor Roberto Rojas and Professor Steven Bibyk for their valuable time and guidance throughout my graduate studies as excellent committee members. Without their supports and advices, it would not have been possible for me to complete my doctorate studies.

It is also my great pleasure to thank visiting scholars: Professor Hyo Dal Park, Professor

Young Gi Kim, and Dr. Dominique Chaillot. They always gave me fruitful advices and encouragements for my research whenever needed.

I would also like to give my gratitude to Jeff Strahler and Christian Bean. They were great mentors during my internship at Andrew Wireless Solutions.

I also would like to express my gratitude for all of the colleagues in our Non-Linear

RF Lab. Especially, I thank Professor Seok Joo Doo who greatly helped me for doing many experiments in the lab. Also Dr. Sukkeun Myoung, Dr. Sunyoung Lee, Dr. Jongsoo

Lee, and Dr. Xian Cui greatly helped me understanding many difﬁcult concepts and ideas

v related to the research as senior students. Also I spent many enjoyable days with Venkatesh

Balasubramanian discussing various research areas.

I also would like to thank Haedong Jang, Jiwoo Kim, and Youngseo Ko for providing great research discussions as well as their time for entertainments. Also Chie-Kai Yang, Xi

Yang, Chunjoo Yang, Andres Zarate-de-Landa, and Dounia Baiya helped me a lot understanding various different research areas.

I also owe a lot of appreciation to my parents and brother in South Korea for their continuous encouragement and support. Without their love and support, it would not have been possible to complete this work.

Finally I would like to express my deepest gratitude to my wife Junghoon Lee, son

Jonathan Suh, and daughter Jenny Suh for their endless love and support during my graduate studies. Especially Junghoon has sacriﬁced so much time for me and it will never be forgettable. I love you and let’s enjoy this memorable time of our life.

vi VITA

January, 1979 ...... Born - Seoul, Republic of Korea

August, 2005 ...... B. S. Electrical Engineering, Korea University, Seoul, Republic of Korea June, 2007 ...... M. S. Electrical & Computer Engineering, The Ohio State University. June, 2007 - present ...... Ph. D. Student, Electrical and Computer Engineering, The Ohio State University. June 2007 - September 2007 ...... Graduate Research Associate, Electrical and Computer Engineering, The Ohio State University. September 2007 - June 2008 ...... Graduate Teaching Associate, Electrical and Computer Engineering, The Ohio State University. June 2008 - September 2008 ...... Intern, Andrew Wireless Solutions, Westerville, OH. September 2008 - March 2009 ...... Graduate Teaching Associate, Electrical and Computer Engineering, The Ohio State University. March 2009 - September 2009 ...... Graduate Research Associate, Electrical and Computer Engineering, The Ohio State University. September 2009 - June 2010 ...... Graduate Teaching Associate, Electrical and Computer Engineering, The Ohio State University.

vii June 2010 - September 2010 ...... Graduate Research Associate, Electrical and Computer Engineering, The Ohio State University. September 2010 - present ...... Graduate Teaching Associate, Electrical and Computer Engineering, The Ohio State University.

PUBLICATIONS

Research Publications

P. Roblin, Y. Ko, C. K. Yang, I. Suh, and S. J. Doo, “NVNA Techniques for Pulsed RF Measurements”. IEEE Microwave Magazine, vol. 12, no. 2, pp. 65-76, Apr. 2011.

I. Suh, P. Roblin, S. J. Doo, X. Cui, J. Strahler, and R. G. Rojas, “A Measurement-Based Methodology to Design Harmonic Loaded Oscillators Using Real-Time Active Load-Pull”. IET Microwaves, Antennas and Propagation, vol. 5, no. 1, pp. 77-83, Jan. 2011.

I. Suh, P. Roblin, Y. Ko, C. K. Yang, A. Malonis, A. Arehart, S. Ringel, C. Poblenz, Y. Pei, J. Speck, and U. Mishra, “Additive Phase Noise Measurements of AlGaN/GaN HEMTs Using an Large Signal Network Analyzer and a Tunable Monochromatic Light Source”. ARFTG 74th Conf., Broomﬁeld/Boulder, CO, Dec. 2009.

J. Mukherjee, Y. G. Kim, I. Suh, P. Roblin, W. R. Liou, Y. C. Lin and M. Shojaei Baghini, “Microstrip Equivalent Parasitic Modeling of RFIC Interconnects”. 50th Midwest Symp. on Circuits and Systems, Montreal, Canada, Aug. 2007.

I. Suh, S. J. Doo, P. Roblin, X. Cui, Y. G. Kim, J. Strahler, M. V. Bossche, R. Rojas, and H. D. Park, “Negative input resistance and real-time active load-pull measurements of a 2.5GHz oscillator using a LSNA”. ARFTG 69th Conf., Honolulu, HI, June 2007.

FIELDS OF STUDY

Major Field: Electrical and Computer Engineering

Studies in Microwave and Microelectronics: Prof. Patrick Roblin

viii TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xii

List of Figures ...... xiii

Chapters:

1. Introduction ...... 1

1.1 Measurement-based oscillator design ...... 1 1.2 Oscillator 1/f phase noise models ...... 4 1.2.1 1/f phase noise model for oscillator ...... 4 1.2.2 1/f phase noise model for injection-locked oscillator ...... 8 1.3 Additive phase noise measurement system ...... 9 1.4 Research outline and new contributions ...... 10

2. Measurement-based methodology to design harmonic loaded oscillators using real-time active load pull ...... 13

2.1 Description of the Measurement System ...... 15 2.1.1 Negative Resistance Oscillator Design ...... 15 2.1.2 Real-Time Active Load-Pull Measurement System ...... 16 2.2 Experimental Results ...... 18 2.2.1 Negative Input Resistance ...... 18

ix 2.2.2 Harmonic Tuning ...... 18 2.2.3 Device Line Measurement ...... 23 2.3 Stand Alone Oscillator ...... 26 2.3.1 Design of Harmonic Load Circuits ...... 26 2.3.2 Experimental Results ...... 26 2.4 Conclusion ...... 28

3. Behavioral modeling of oscillators using real-time active load pull ...... 30

3.1 Power-Dependent Volterra Series Modeling ...... 31 3.1.1 Volterra Algorithm ...... 31 3.1.2 Model Extraction ...... 33 3.2 Model Validation ...... 35 3.2.1 Error Evaluation ...... 38 3.3 Conclusion ...... 42

4. Model comparison for 1/f noise in oscillators with and without AM to PM noise conversion ...... 43

4.1 Kurokawa 1/f Noise Models for an Oscillator ...... 47 4.1.1 Derivation of Sa,1/f (∆ω) ...... 53 4.1.2 Derivation of Sφ,1/f (∆ω) ...... 55 4.1.3 Comparison of Kurokawa theory with ISF theory for 1/f noise . 57 4.1.4 Comparison of Kurokawa theory with PPV theory for 1/f noise . 59 4.2 Modiﬁed Van der Pol Oscillator ...... 62 4.2.1 Kurokawa Coefﬁcients Extraction ...... 64 4.2.2 Amplitude and Frequency of Oscillation and Phase Noise . . . . 67 4.2.3 PPV Analytic Solution ...... 68 4.3 Model Comparison for 1/f noise in Van der Pol Oscillator ...... 69 4.3.1 Circuit Implementation in Simulators ...... 69 4.3.2 Model Comparison for Uncorrelated Case ...... 71 4.3.3 Model Comparison for Correlated Case ...... 73 4.4 Model Comparison for Other Oscillator Circuits ...... 76 4.4.1 BJT Colpitts Oscillator ...... 76 4.4.2 CMOS Ring Oscillator ...... 78 4.4.3 pHEMT Oscillator ...... 81 4.5 Conclusion ...... 82

5. 1/f additive phase noise analysis for injection-locked oscillators ...... 87

5.1 1/f Additive Phase Noise Model for Injection-Locked Oscillator . . . . 88 5.2 Measurement System Description ...... 91

x 5.3 Experimental Results ...... 92 5.4 Conclusion ...... 94

6. Additive phase noise measurements of AlGaN/GaN HEMTs using a large signal network analyzer and a tunable monochromatic light source ...... 96

6.1 Measurement Set-up Description ...... 97 6.1.1 Additive Phase Noise Measurement System ...... 97 6.1.2 LSNA and Tunable Monochromatic Light Source ...... 97 6.2 Experimental Results and Discussions ...... 99 6.2.1 Illuminations with Different Photon Energies ...... 99 6.2.2 Supportive Analysis ...... 101 6.2.3 Phase Noise Dependence on Load Impedances ...... 104 6.3 Conclusion ...... 106

7. Conclusion and future work ...... 107

Appendices:

A. Volterra series expansion of four-tone excitation ...... 112

A.1 Volterra series expansion of four-tone excitation ...... 112

B. PPV derivation ...... 116

C. Abbreviation ...... 119

Bibliography ...... 121

xi LIST OF TABLES

Table Page

2.1 Summary of the output power optimization procedure using multi-harmonic RT-ALP technique and constant-phase active load-pull (ALP) LSNA measurements ...... 24

2.2 Comparison of predicted and measured frequency of oscillation and output power with both load-lines ...... 28

4.1 Harmonics of output voltage and current using an harmonic short. (The same circuit parameters as in Figure 4.6 are used in this result) ...... 64

4.2 Comparison of the phase noise and sensitivity parameters varying NMOS gate width between ADS and Kurokawa method. Common parameters used: C=2 pF, λ =0.001, λ =0.02, Wp = 250µm , and R =R =0...... 84 N P Lp 0.24µm d s

6.1 Temperature variation during the measurements ...... 105

xii LIST OF FIGURES

Figure Page

1.1 Research outline for Ph.D studies...... 10

2.1 Block diagram of the multi-harmonic RT-ALP measurement system. . . . . 16

2.2 Measured magnitude of ΓIN (0, ω) = 7.5 for the oscillator...... 19

2.3 Comparison of output power sweeps. The maximum output powers of 35.7 mW (red dashed dotted line), 38.3 mW (blue dashed line), and 39.3 mW (black solid line) are obtained respectively. The two operating points A and B are used in the next section for realizing stand alone oscillators...... 19

2.4 Loci of ΓL(2ω0, t) obtained from the 2ω0 RT-ALP measurement with the LSNA. Note that the ΓL(2ω0, t) loci even extend outside of the Smith chart. A frequency offset (∆ω) of about 200 kHz is used...... 20

2.5 Output power contour plots in ΓL(2ω0) plane. The dashed line contour plot shows the ﬁrst 2ω0 RT-ALP result while the solid line contour indicates the second 2ω0 RT-ALP. The black dot on Smith chart indicates the ◦ ΓL,OP T 1(2ω0) = 16 143.1 which provides the maximum output power of 38.5 mW with the ﬁrst 2ω0 RT-ALP. The best performance is obtained with ◦ the ΓL,OP T 2(2ω0) = 16 150.9 (blue rectangle) which provides the maximum output power of 39.4 mW with the second 2ω0 RT-ALP. The black ◦ circle indicates the ΓL(2ω0) = 0.766 133 obtained with the implemented load circuit 1 (see section 4)...... 21

2.6 Loci of ΓL(3ω0, t) obtained from the 3ω0 RT-ALP measurement with the LSNA. A frequency offset (∆ω) of about 200 kHz is used...... 23

xiii 2.7 Output power contour plot in ΓL(3ω0) plane. The black dot on Smith chart ◦ indicates the optimal ΓL,OP T 1(3ω0) = 0.956 − 18.4 which provides the maximum output power of 39.2 mW. The black circle indicates the ΓL(3ω0) = 0.386 112 ◦ obtained with the implemented load circuit 1 later. Consid- ering the output power difference between black dot and black circle, this resulting error is negligible...... 24

2.8 Measurement result of the load-line1 ΓL(ω0) (red dashed line) and load- −1 line2 ΓL(ω0) (blue dotted line) with a device line ΓIN (|a1|, ω0) (black solid line). The operating point A for the load-line 1 corresponds to the maximum output power while B is the operating point for the load-line 2. Their respective output powers are given in Fig. 2.3...... 25

2.9 Layout of the harmonic load circuit designed for self-sustaining oscillation measurement...... 27

2.10 Test bed for self-sustaining oscillation measurement...... 27

2.11 Phase noise measurement of a stand alone oscillator with load-line 1 (black dashed line) and load-line 2 (blue solid line)...... 29

3.1 Comparison of contour plots in the ΓL(2ω0) plane between a Volterramodel which uses ﬁrst-order perturbation at 2ω0 and 3ω0 (a) and the proposed Volterra model with higher order nonlinearities for 2ω and 3ω (b). The blue dashed lines stand for the modeling result...... 36

3.2 Comparison of contour plots in the ΓL(3ω0) plane between a Volerra model which uses ﬁrst-order perturbation at 2ω0 and 3ω0 (a) and the proposed Volterra model with higher order 2ω and 3ω (b). The blue dashed lines stand for the modeling results...... 37

3.3 Comparison of the output power characteristics between LSNA measured data, MATLAB modeling results, and ADS simulation results for the (a) order 4 for ω and linear 2ω and 3ω and (b) order 4 for ω and 3, 2 2ω and 3 ω Volterra models...... 39

xiv 3.4 Graphical evaluation of the error in frequency and amplitude of oscillation using the model implemented in ADS. The black arrow on each load-line indicates the direction of the frequency increment while the black arrow on the device-line depicts the direction of the amplitude increment. The black squares predicted by the ADS simulation are at the same power level as the nearby blue square for the measured power level...... 41

4.1 An admittance model of an oscillator with low-frequency modulation of the nonlinear device impedance at the fundamental frequency...... 47

4.2 An nonlinear feedback model of an oscillator with low-frequency modulation of the nonlinear device circuits by 1/f noise and other cyclostationary noise processes...... 51

4.3 Modiﬁed Van der Pol oscillator model including AM and PM 1/f noise and AM to PM nonlinear noise correlation...... 63

4.4 Output voltage (top) and current (bottom) waveform in time domain using an harmonic short. (The same circuit parameters as in Figure 4.6 are used in this result.) ...... 64

4.5 The Kurokawa and PPV analytic model results for uncorrelated 1/f amplitude noise are compared to the ADS simulation results for three different −14 2 amplitudes of oscillation. a1 = 3, c = 1, L=1 nH, CL=1 pF, S=1e V , R=1/g0=30 Ω, and RL=1/GL=40 Ω (red), 100 Ω (blue), and 400 Ω (black) are used respectively for the comparison...... 71

4.6 The Kurokawa and PPV analytic model results for uncorrelated 1/f phase noise are compared to the ADS simulation results for three different resonator Q factors. a1 = 3, c = 1, R=1/g0=30 Ω, and RL=1/GL=40 Ω, −14 2 S=1e V , CIN1 =1 pF/V, CL=1 pF (Q=1.26), 10 pF (Q=12.6), and 100 pF (Q=126) are used respectively for the comparison while maintaining the same frequency of oscillation...... 73

4.7 Phase noise comparison result for the three different 1/f noise strength (S). a1 = 3, c = 1, R=1/g0=30 Ω, and RL=1/GL=40 Ω, CIN1 =1 nF/V, and CL=1 pF are used respectively for this comparison...... 74

xv 4.8 Total phase noise and its components at 1 Hz offset frequency versus correlation strength 10 log[αB/(βA)] for the correlated case (k1 6= k3). For this −18 2 comparison, R=1/g0=30 Ω, RL=1/GL=40 Ω, S=1e V , L=1 nH, CL=1 pF, and CIN1=0.1 pF/V are used with while sweeping CIN2 to variate the correlation strength...... 75

4.9 Comparison result of the phase noise at 1 Hz offset frequency, for four different 1/f phase noise strengths CIN1 for the correlated case. The same circuit parameters as in Figure 4.8 are used in this comparison...... 76

4.10 Comparison result of the phase noise at 100 KHz offset frequency, for four different 1/f phase noise strengths CIN1 for the correlated case. The same circuit parameters as in Figure 4.8 are used in this comparison...... 77

4.11 Schematic of the 1 GHz BJT-based Colpitts oscillator...... 78

4.12 Phase noise model comparison of a BJT-based Colpitts oscillator...... 79

4.13 Topology of a three stage ring oscillator implemented with CMOS inverters. The detailed circuit parameters used are in Table 4.2...... 80

4.14 Comparison of the calculated (top) amplitude and (bottom) frequency sensitivities (dashed lines) with those obtained from harmonic balance simulations for the ring oscillator with gate width of 30 µm...... 81

4.15 Voltage (top) and current (bottom) waveforms obtained from harmonic balance simulations for the ring oscillator with gate width of 30 µm...... 82

4.16 Comparison of phase noise versus NMOS gate width of a ring oscillator shown in Fig. 4.13 at 1 KHz offset frequency between ADS simulation and Kurokawa uncorrelated and correlated analytic model...... 83

4.17 Schematic of the 2.5 GHz pHEMT-based negative-resistance oscillator. . . 85

4.18 Phase noise model comparison of a pHEMT oscillator...... 86

5.1 Admittance model for one port injection-locked oscillator ...... 88

5.2 Additive phase noise measurement system integrated with an LSNA for an injection-locked oscillator. The N-C stands for the negative-conductance circuit...... 91

xvi 5.3 Phase noise comparison of a pHEMT oscillator with a load circuit 1 operating at 2.4828 GHz...... 94

5.4 Phase noise comparison of a pHEMT oscillator with a load circuit 2 operating at 2.485 GHz...... 95

6.1 Block diagram of the additive phase noise measurement set-up integrated with an LSNA and a tunable monochromatic light source...... 98

6.2 Comparison of additive phase noise on various illumination energies between different DC biasing conditions VDS = 10 V (solid line) and VDS = 15 V (dashed line)...... 100

6.3 RF gate and drain load-line of unpassivated GaN HEMT. In (b), DC-IV and pulsed-IVs pulsed from three different operating points are compared with drain load-line. All IVs shown are for VGS = -1 V...... 102

6.4 RF drain load-lines of the unpassivated GaN HEMT with DC-IVs (black solid line), pulsed-IVs (blue dashed line) from VGS = -3 V, VDS = 10 V and contour plot of β. For both IV characteristics, VGS is swept from -5 V (bottom) to 0 V (top) in steps of 1 V...... 103

6.5 RF drain load-lines of the passivated GaN HEMT with DC-IVs (black solid line), pulsed-IVs (blue dashed line) from VGS = -3 V, VDS = 10 V and a contour plot of β. For both IV characteristics, VGS is swept from -5 V (bottom) to 0 V (top) in steps of 1 V...... 103

6.6 Comparison of phase noise dependence between passivated and unpassivated device on load impedances...... 105

xvii CHAPTER 1

INTRODUCTION

1.1 Measurement-based oscillator design

In wireless communication systems, oscillators are one of the most critical components since they are used in conjunction with mixers for frequency translation [1]. The conventional approach for designing oscillators is to rely on circuit simulations. Alterna- tively Volterra series can be used for analyzing and designing oscillators [2]- [4]. Provided that accurate nonlinear device models are available, both methods allow for the design of oscillators with optimal performance. However circuits relying on positive feedback are very sensitive to the device nonlinearity and packaging parasitics and there is no guarantee in practice that the optimal performance is obtained for the design of individual discrete oscillators given the device process ﬂuctuation.

Alternatively with a design methodology relying on the experimental characterization of the device line, the synthesis of the load circuit providing optimal output power, power efﬁciency or alternatively optimal phase noise performance can be achieved [5]- [9].

The optimal load impedance found at the fundamental frequency can be tuned to provide the best possible output power of the oscillator [9]- [11]. The output power and the power efﬁciency can be further maximized by determining the optimal second harmonic

1 load impedance which can be searched by simulation tools [12] or active load-pull technique and the device line characterization [13]. Active load-pull measurement is, however, time consuming and might not be practical for rapidly optimizing a circuit.

The recently reported real-time active load-pull (RT-ALP) system using an LSNA can greatly reduce the acquisition time [14]. For an 100 Hz bandwidth resolution it takes 10 ms per continuous phase sweep measurements and 0.2 s of acquisition time (20 discrete power measurements) to cover the entire Smith chart. Another advantage of using RT-ALP system is that the load impedance can go on the edge of the Smith chart unlike for passive load- pull system. It also allows for a quasi-interactive optimization of the output power and the efﬁciency of power ampliﬁers [14]. RT-ALP technique with the LSNA can also be applied to the characterization of negative resistances in order to optimize in particular the effect of the second harmonic load impedance on the output power [15]. However, due to memory effects during the RT-ALP measurement, the performance predicted might not be achieved when building an actual stand alone oscillator. Actually to validate the design approach, it is necessary to perform a complete design cycle and built a stand alone oscillator to verify the entire design methodology.

In chapter 2, we shall use the multi-harmonic RT-ALP measurement technique to complete the design of a stand-alone oscillator [16]. Furthermore, the optimal multi-harmonic load impedances obtained from the LSNA measurement will be implemented with an actual load circuit to realize a stand alone multi-harmonic loaded oscillator. The measured operating frequency and the maximum output power of the oscillator will be compared with those of the LSNA predicted results to evaluate the entire design methodology. The proposed design approach differs from conventional design techniques in that large-signal measurements are used to guide the design process of the harmonic load circuit. Further

2 unlike conventional iterative design procedures based on circuit simulations, both our measurement and design simulations assume that (1) the frequency of operation is not to be searched but to be set to the targeted value through out the analysis and (2) that it is up to the load circuit to be designed to establish the desired conditions of operation in terms of frequency of oscillation and amplitude of oscillation. Note that the amplitude of oscillation could be set by the designer to achieve maximum output power or minimum phase noise [6]. The design of the load circuit involves then the synthesis of a load circuit with the proper impedance termination at each harmonic to achieve the selected operating point.

In this work, we have elected to maximize the output power of the oscillator. One of the motivation is that according to Leeson’s formula [7] which can be re-derived from the

Kurokawa analysis [6], the oscillator phase noise for white noise is inversely proportional to the output power of the oscillator at the fundamental frequency. Selecting an operating point for the oscillator which maximizes the fundamental output power is therefore of critical importance for reducing the phase noise. Note that the realization of a low phase-noise oscillator calls also for the design of a high Q termination circuit but this remains independent from the selection of the oscillator operating point. Also the condition for which the maximum output power is maximized at the fundamental frequency is usually associated with a reduction in harmonic generation.

In today’s integrated circuit (IC) industry, due to the high volume of circuit components and the complexity of IC environment, it is very challenging to accurately simulate a complete system in an efﬁcient way [17]. Behavioral modeling, also called a ”black-box” modeling, is often used in the frequency-domain to characterize such system efﬁciently and accurately simply using a mathematical relation between the input and output signals [18].

3 The Poly-Harmonic Distortion (PHD) modeling [19] provides a powerful method to model the non-linear characteristics of power ampliﬁers. The PHD model assumes that the dominant non-linear response is controlled by the fundamental tone and that the harmonic signals can be dealt with using ﬁrst-order perturbations. This elegant formalism is found to be applicable to a large class of circuits. However, the accuracy will naturally degrade when the incident power for the harmonics is high and the linear response for the harmonics is no longer applicable.

In chapter 3, to assist with the oscillator design, a power-dependent Volterra series model for the oscillator will be developed to ﬁt the multi-harmonic load-pull data and enable the harmonic balance analysis of the oscillator in a circuit simulator. The proposed modeling method aims to achieve improved performance compared to other methods by taking into account optimal higher-order perturbations for the harmonics as needed.

1.2 Oscillator 1/f phase noise models

1.2.1 1/f phase noise model for oscillator

The noise characteristic of an oscillator is of great importance in communication systems. For example, the selectivity and sensitivity of mixers as well as the system bit error rate can be affected by the phase noise of the oscillator. Also the oscillator phase noise determines the minimum detectable target signal in Doppler radar systems [20].

Dominant types of noise in RF circuits are thermal, shot and 1/f noise. Low-frequency

1/f noise generates a noise sideband with 1/∆f 3 dependence when it is up-converted to the RF carrier signal in oscillators [21]. A considerable body work is available on the analysis of oscillator phase noise resulting from 1/f noise sources.

4 The Leeson model [7] which relies on an empirical expression for the up-conversion of the 1/f noise is often used to ﬁt the up-converted 1/∆f 3 noise but is of limited applicability for the prediction of the 1/f noise performance directly from circuit analysis.

An extension of the Kurokawa noise analysis [22] to 1/f noise up-conversion was reported by Rohdin et al. [23]. More recent detailed derivations were then given in Ref. [6].

The model reported in Ref. [21] provides a phase noise analysis for 1/f noise for a nonlinear intrinsic FET device model. However, this analytic model is difﬁcult to apply to more complex oscillators.

Current commercial harmonic balanced circuit simulators rely instead on the method of conversion matrix [24,25]. In the conversion matrix method, a small-signal perturbative analysis is superposed upon large-signal harmonic balance simulations of an autonomous oscillator circuit in order to account for the noise-induced frequency modulation of the carrier at low frequency offsets. The model can account for both the up-conversion of low-frequency noise (1/f) and the down conversion of cyclostationary noise if such noise processes are implemented. Note that a cyclostationary noise can be decomposed as the product of a stationary noise source and a deterministic periodic RF oscillation. Additional cyclostationary effects such as the impact of large-signal RF oscillations on noise sources can also be accounted for by using time averages as was demonstrated in Ref. [26] the

Lorentzian (pop-corn) and 1/f noises generated by traps.

A Linear Time Varying (LTV) model based on the Impulse Sensitivity Function (ISF) which also accounts for both the up-conversion of 1/f noise and down conversion of harmonic noise was reported in Ref. [27, 28]. This model provides great insights on the phase noise mechanisms and is therefore conceptually useful to circuit designers. However the numerical calculation of the ISF in the time domain is more involved.

5 A general differential equation for the phase error in oscillators was obtained using a perturbation technique in [29]. A Perturbation Projection Vector (PPV) method for calculating the phase noise due to 1/f noise sources was then introduced in Ref. [30]. The PPV method projects the noise on the oscillator’s orbit in order to separate its tangential and transversal components and decouple the AM and PM noise.

A comparison of the ISF and PPV method was reported in Ref. [31]. This study pointed out some of the similarities between the two time-domain methods. However it was argued in Ref. [32] that depending on the ISF deﬁnition (tangential perturbation ISF) used, the ISF theory could neglect the amplitude to phase noise conversion which is accounted for by the PPV, Kurokawa and conversion methods. A correspondence between the PPV method and the matrix conversion method was also theoretically established in Ref. [32] and was supported by the good agreement in noise simulation results. The PPV model is not usually amenable to analytic solutions. Limiting results for low and high frequency offset were however reported [33] [34]. An analytic solution for white noise was however reported in Ref. [35] for a simple negative resistance oscillator. Since the oscillator studied did not feature any AM to PM noise conversion, the results reported were identical to the ISF results.

Recently, a useful analytic expression for the autocorrelation RV,1/f (∆ω) has been reported for the PPV method [36] for colored noises. The single side band voltage spectral density normalized by the oscillator power SV,1/f,ssb(∆ω)/Posc (old IEEE phase noise) derived from RV,1/f (∆ω) was found to reach a ceiling value at low offset frequencies in

6 output power. Similar analytic results for the autocorrelation RV,1/f (∆ω) were also previously reported in Ref. [37] for the PPV method and in Ref. [6] for the generalized Kurokawa theory. But the resulting SV,1/f (∆ω) were not compared and this will be pursued in this paper.

The recent ISF and PPV time-domain theories have respectively brought great insights and rigor to the noise calculation. Nevertheless the calculation of the ISF or the PPV involves numerical computations requiring access to the circuit Jacobian. In this work, as an alternate approach, we shall consider a new generalized 1/f Kurokawa noise model which is circuit-based and therefore more accessible to circuit designers. Indeed the analytic expressions for the amplitude and phase noise derived from the generalized Kurokawa theory [6] are all expressed in terms of the derivatives of immittances which can be easily evaluated either analytically (for simple circuits) or by circuit simulations or measurements. In the generalized Kurokawa model, the noise sources can not only be stationary

(white noise) but also up-converted 1/f noise, or down converted cyclostationary noise.

Finally the new generalized Kurokawa domain theory is also applicable to low-Q circuit such as ring oscillators as shall be demonstrated in this paper.

Given the circuit-based representation of its results, the question arises on how the phase noise results obtained with the 1/f Kurokawa model compares with those of the matrix conversion, ISF, and PPV methods respectively for common RF oscillator circuits.

Also of particular interest is the evaluation of the contribution of the amplitude and phase correlation and the incremental phase noise resulting from AM to PM noise conversion in these various techniques.

7 In chapter 4, to evaluate the different 1/f noise theories we shall consider four different oscillator circuits. The ﬁrst circuit will be a generalized Van der Pol oscillator with an harmonic short which features a new fourth-order nonlinear capacitance exhibiting both AM

1/f and PM 1/f noise up-conversion as well as AM to PM 1/f noise conversion. A fully analytic solution will be obtained within both the Kurokawa and PPV formalism for this oscillator which will facilitate the model evaluation and comparison with simulations and other noise theories. The second and third circuits will respectively be a BJT-based Colpitts oscillator and a CMOS-based ring oscillator implemented in a circuit simulator. The fourth circuit will be a HEMT-based transmission line oscillator fabricated and characterized.

1.2.2 1/f phase noise model for injection-locked oscillator

Low frequency 1/f noise is signiﬁcantly important in oscillator design since it generates a noise sideband with 1/∆f 3 dependence when it is up-converted to the RF carrier signal [21]. Among many approaches for reducing this detrimental near carrier noise, the injection-locking method has been effectively used for minimizing the phase noise of the oscillator. A considerable body of work [38]– [41] is already available on the analysis of injection-locked oscillator phase noise. Previous analyses show that the near carrier phase noise spectrum of the injection-locked oscillator follows that of the injecting signal [38]–

[41].

Recently, an additive noise measurement system has been used by many researchers due to its ability to measure the intrinsic noise of a device under test (DUT) by ideally canceling the noise of the external RF source [42]– [46]. This system can also be used to obtain the noise characteristic of the DUT under a large signal operation [46].

8 In chapter 5, the source of the additive noise measurement system is used to synchro- nize the free-running oscillator operating at 2.4828 GHz and 2.485 GHz respectively while measuring the additive phase noise of the injection-locked oscillator. An analytic expression for the 1/f additive phase noise of the injection-locked oscillator is derived using

Kurokawa approach. A reasonable agreement of both the phase noise level and the corner frequency is obtained between the measured 1/f additive phase noise spectrum and the obtained analytic solution for both oscillators, which conﬁrms the validity of the model.

1.3 Additive phase noise measurement system

As described in chapter 5, additive phase noise measurements are useful for accurately characterizing the phase noise contributed by a DUT in a system. An additive phase noise measurement system differs from a conventional absolute phase noise measurement system in that it ideally cancels the noise of the external RF source in the system to achieve a lower noise ﬂoor [45]. This system has been widely used to analyze and model the noise characteristics of various devices and microwave components such as GaAs heterojunction bipolar transistor (HBT), AlGaN/GaN HEMT, Silicon (Si) bipolar junction transistor (BJT), GaAs

ﬁeld effect transistor (FET), frequency dividers and multipliers [42], [43], [45], [47].

The low-frequency noise of both unpassivated and passivated AlGaN/GaN HEMT was reported in [47]. A reduced phase noise was observed at low-frequency in passivated devices compared to unpassivated devices.

In chapter 6, an additive phase noise measurement system integrated with (1) an LSNA and (2) a tunable monochromatic light source is presented to investigate the noise characteristic of both unpassivated and passivated AlGaN/GaN HEMTs under various operating

9 conditions. A physical analysis on the origin of 1/f noise in AlGaN/GaN HEMTs is also presented in support of the various measurement results.

1.4 Research outline and new contributions

Figure 1.1 summarizes the research topics covered in this dissertation. First of all, a multi-harmonic loaded oscillator is designed using a measurement-based approach relying on the RT-ALP technique. Then the behavioral model for the circuit simulation is obtained using the Volterra formalism. Also the 1/f noise of ampliﬁers is characterized under various operating conditions appropriate for trap analysis. Then the 1/f phase noise of the free-running oscillators is analyzed for different oscillators and compared with 1/f noise measurement. Finally, the 1/f additive phase noise of the injection-locked oscillators is measured and compared with a new phase noise model developed for injection-locked oscillators.

1/f Additive Phase Noise Phase Noise Analysis Measurement & Modeling of Injection−Locked Oscillators

1/f Noise Modeling Phase Noise Analysis and Model Comparison of Free−Running Oscillators

Behavioral 1/f Noise Analysis Modeling Model for Simulation of Amplifiers and Analysis Oscillator Measurement−based Oscillator For Trap Analysis Design Design using RT−ALP

Figure 1.1: Research outline for Ph.D studies.

10 In Chapter 2, the new contributions are as follows:

1. A multi-harmonic loaded oscillator is designed for the ﬁrst time using the RT-ALP design approach.

2. This is the ﬁrst experimental active load-pull to consider third-harmonic tuning in oscillator design.

3. A stable negative-resistance and a device line are measured for the ﬁrst time using the

LSNA.

In Chapter 3, the new contributions are as follows:

1. The proposed power-dependent Volterra model provides better ﬁtting performance compared to the PHD model (X-parameter).

2. The proposed Volterra formalism can account for non-linearities of inﬁnite orders using an functional expansion in terms of a ﬁnite of number of harmonic excitations.

In Chapter 4, the new contributions are as follows:

1. The traditional 1/f Kurokawa noise analysis is extended to cover both low and high Q oscillators.

2. For the ﬁrst time, a theoretical correspondence between the new generalized Kurokawa theory and the ISF and the PPV is derived in the absence and presence of AM to PM noise conversion.

3. The proposed generalized 1/f Kurokawa theory is shown to to predict the same ceiling of the voltage noise spectral density at low offset frequencies as the PPV theory and experimental results.

11 4. For the ﬁrst time, an analytic solution for the 1/f amplitude noise Sa,1/f (∆ω) is obtained for both the Kurokawa and the PPV theory for a generalized Van Der Pol oscillator featuring AM to PM noise conversion.

In Chapter 5, the new contribution is as follow:

1. An analytic expression is derived for the ﬁrst time which gives the 1/f additive phase noise of the injection-locked oscillator. This theory is validated experimentally on a transmission line pHEMT oscillator.

In Chapter 6, the new contributions are as follows:

1. For the ﬁrst time, an additive phase noise measurement system is integrated with an

LSNA and a tunable monochromatic light source to analyze the noise characteristics of an unpassivated and passivated AlGaN/GaN HEMT at 2 GHz under various operating and illumination conditions.

2. The cyclostationary noise effect is demonstrated experimentally for the ﬁrst time in a

GaN HEMT from the dependence of the measured additive phase noise upon the large- signal dynamic load-lines for different load resistances.

12 CHAPTER 2

MEASUREMENT-BASED METHODOLOGY TO DESIGN HARMONIC LOADED OSCILLATORS USING REAL-TIME ACTIVE LOAD PULL

13 load impedance which can be searched by simulation tools [12] or active load-pull technique and the device line characterization [13]. Active load-pull measurement is, however, time consuming and might not be practical for rapidly optimizing a circuit.

The recently reported real-time active load-pull (RT-ALP) system using an LSNA can greatly reduce the acquisition time [14]. For an 100 Hz bandwidth resolution it takes 10 ms per continuous phase sweep measurements and 0.2 s of acquisition time (20 discrete power measurements) to cover the entire Smith chart. Another advantage of using RT-ALP system is that the load impedance can go on the edge of the Smith chart unlike for passive load-pull system. It also allows for a quasi-interactive optimization of the output power and the efﬁciency of power ampliﬁers [14]. RT-ALP technique with the LSNA can also be applied to the characterization of negative resistances in order to optimize in particular the effect of the second harmonic load impedance on the output power [15]. However, due to memory effects during the RT-ALP measurement, the performance predicted might not be achieved when building an actual stand alone oscillator. Ultimately to validate the design approach, it is necessary to perform a complete design cycle and built a stand alone oscillator to verify the entire design methodology.

In this chapter, we shall use the multi-harmonic RT-ALP measurement technique to complete the design of a stand-alone oscillator [16]. Also, the optimal multi-harmonic load impedances obtained from the LSNA measurement will be implemented with an actual load circuit to realize a stand alone multi-harmonic loaded oscillator.

The measured operating frequency and the maximum output power of the oscillator will be compared with those of the LSNA predicted results to evaluate the entire design methodology. The proposed design approach differs from conventional design techniques in that large-signal measurements are used to guide the design process of the harmonic

14 load circuit. Further unlike conventional iterative design procedures based on circuit simulations, both our measurement and design simulations assume that (1) the frequency of operation is not to be searched but to be set to the targeted value through out the analysis and (2) that it is up to the load circuit to be designed to establish the desired conditions of operation in terms of frequency of oscillation and amplitude of oscillation. Note that the amplitude of oscillation could be set by the designer to achieve maximum output power or minimum phase noise [6]. The design of the load circuit involves then the synthesis of a load circuit with the proper impedance termination at each harmonic to achieve the selected operating point.

In this work we have elected to maximize the output power of the oscillator. One of the motivations is that according to Leeson’s formula [7] which can be re-derived from the

Kurokawa analysis [6], the oscillator phase noise is inversely proportional to the output power of the oscillator at the fundamental frequency. Selecting an operating point for the oscillator which maximizes the fundamental output power is therefore of critical importance for reducing the phase noise. Note that the realization of a low phase-noise oscillator calls also for the design of a high Q termination circuit but this remains independent from the selection of the oscillator operating point. Also the condition for which the maximum output power is maximized at the fundamental frequency is usually associated with a reduction in harmonic generation.

2.1 Description of the Measurement System

2.1.1 Negative Resistance Oscillator Design

A basic oscillator circuit can be divided into a frequency and amplitude sensitive nonlinear (device) part and a frequency sensitive linear (load) part (see Fig. 2.10). A non-linear

15 part of the oscillator circuit providing a stable negative resistance was designed at 2.5 GHz with an Avago ATF54143 HEMT for performing large signal characterization. To prevent any possible self-oscillations during the measurements, the Nyquist stability condition must

−1 be enforced. That is, the load-line ΓL(ω0) should not encircle the device line ΓIN (|a1|, ω0). This condition can be achieved if the load network consists of a broadband 50 Ω impedance termination. Series feedback with a tunable capacitor was used to set the peak of the input reﬂection coefﬁcient in the targeted frequency band. VDS of 2.0 V and VGS of 0.55 V are used for DC biasing, yielding a drain current IDS of 30 mA.

2.1.2 Real-Time Active Load-Pull Measurement System

Large Signal Network Analyzer

a1 Port1 Oscillator Bias Port2 b1 Supply Diplexer HPF 50Ω Power RF source Combiner 2ω + ∆ω 90° Couplers 0

Spectrum RF source RF source 3ω + ∆ω ω Analyzer 0 0

Figure 2.1: Block diagram of the multi-harmonic RT-ALP measurement system.

Fig. 2.1 shows a block diagram of the multi-harmonic RT-ALP measurement system.

The incident wave a1 is injected from the RF source ω0 to the non-linear part of the oscillator and the reﬂected wave b1 is then measured with an LSNA. Note that compared to

16 the previous work [15], two RF sources at 2ω0 and 3ω0 are used in this work for synthe- sizing the 2nd and 3rd harmonic active-loads in order to investigate the impact of both the

2nd and 3rd harmonic effects on the output power of the oscillator. A diplexer is used to both control the signal path of the signal sources and provide the required broadband 50 Ω impedance termination at ω0. A frequency offset (∆ω) of about 200 kHz is used and 49 single-side band tones are used for the LSNA measurements. Also a spectrum analyzer is used to conﬁrm the locking of the oscillation frequency during the measurements. When

th an incident wave of a1(nω0 + ∆ω) for n harmonic RT-ALP is injected to the non-linear part of the oscillator, the phase of the nth harmonic load reﬂection coefﬁcient represented in (2.1) is effectively continuously swept in a closed quasi-circular path within a single

LSNA measurement (10 ms). Then by varying the incident power levels in (2.2), the radius of the reflection coefficient trajectory can be controlled to cover the entire Smith chart. For example in Fig. 2.4, 19 different input power levels are used to map the second harmonic load reflection coefficient in the Smith Chart.

PSSB jp∆ωt p=−SSB a1(nω0 + p∆ω)e ΓL(nω0, t) = P (2.1) SSB jp∆ωt p=−SSB b1(nω0 + p∆ω)e

1 P (nω + ∆ω) = |a (nω + ∆ω)|2. (2.2) in 0 2 1 0

Also, the total RF output power including all the harmonics can be expressed as:

XN Pout,total = Pout(nω0, t), (2.3) n=1 where Pout(nω0, t) is given by:

17 ½ 1 SSBX SSBX P (nω , t) = − v (nω + p∆ω) out 0 2 1 0 p=−SSB q=−SSB ¾ ∗ j(p−q)∆ωt × i1(nω0 + q∆ω)e . (2.4)

Using (2.3) and (2.4), the total RF output power can then be calculated as a function of the extracted nth harmonic load reﬂection coefﬁcients as shown in Fig. 2.5 and Fig. 2.7.

2.2 Experimental Results

2.2.1 Negative Input Resistance

The incident wave a1 and reflected wave b1 are measured with the LSNA while maintaining the Nyquist stability criteria. Then the input reflection coefficient can be calculated using:

b1(ω) ΓIN (|a1|, ω) = . (2.5) a1(ω) Fig. 2.2 shows the stable input reﬂection coefﬁcient of the oscillator obtained from the

LSNA measurement using (2.5).

2.2.2 Harmonic Tuning

Multi-harmonic RT-ALP technique is used to determine the optimal harmonic load impedances yielding the maximum output power for the oscillator. Implemented with an

LSNA, the RT-ALP measurement can be performed in a short time (20 measurements of

10 ms each).

The output power delivered to the load is then expressed by: ½ ¾ 1 P (ω ) = |b (ω )|2 − |a (ω )|2 . (2.6) L 0 2 1 0 1 0

18 8

5 |

IN 4 Γ |

0 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Frequency (GHz)

Figure 2.2: Measured magnitude of ΓIN (0, ω) = 7.5 for the oscillator.

45 No harmonics applied A B 40 With Γ (2ω ) L,OPT1 0 With Γ (2ω )+Γ (3ω ) 35 L,OPT2 0 L,OPT1 0

15 Output power (mW)

0 −30 −25 −20 −15 −10 −5 0 5 10 a (dBm) 1

Figure 2.3: Comparison of output power sweeps. The maximum output powers of 35.7 mW (red dashed dotted line), 38.3 mW (blue dashed line), and 39.3 mW (black solid line) are obtained respectively. The two operating points A and B are used in the next section for realizing stand alone oscillators.

19 Fig. 2.3 summarizes the harmonic tuning method obtained with the RT-ALP technique using (2.6). The incident wave a1 is swept from -30 dBm to 8 dBm at fundamental frequency. First, the red dashed dotted line is obtained by setting the 2nd and 3rd harmonic load impedances to 50 Ω.

1.0 1 0.5 2.0 0.8 0.6

0.4 0.2 0.2 0 0.0 0.2 0.5 1.0 2.0 Inf −0.2 −0.4 −0.2 −0.6 −0.8 −0.5 −2.0 −1 −1.0 −1 −0.5 0 0.5 1

Figure 2.4: Loci of ΓL(2ω0, t) obtained from the 2ω0 RT-ALP measurement with the LSNA. Note that the ΓL(2ω0, t) loci even extend outside of the Smith chart. A frequency offset (∆ω) of about 200 kHz is used.

Fig. 2.4 shows the loci of ΓL(2ω0, t) obtained from the 2ω0 RT-ALP measurement for 19 different amplitudes of |a1(2ω0 + ∆ω)|. Unlike for passive load-pull measurements,

ΓL(2ω0, t) in RT-ALP typically extends outside of the Smith chart. The dashed line contour in Fig. 2.5 shows the output power contour plot in ΓL(2ω0) plane obtained using (2.3)-(2.4).

As shown, the variation of the output power is from around 28 mW to 39 mW on the Smith chart depending on second harmonic load impedance.

1.0 0.044 1 0.5 2.0 0.042 0.8 0.6 0.04

0.4 0.2 0.038 0.2 0.036 0 0.0 0.2 0.5 1.0 2.0 Inf −0.2 0.034

−0.4 −0.2 0.032 −0.6 0.03 −0.8 −0.5 −2.0 −1 0.028 −1.0

−1 −0.5 0 0.5 1

Figure 2.5: Output power contour plots in ΓL(2ω0) plane. The dashed line contour plot shows the ﬁrst 2ω0 RT-ALP result while the solid line contour indicates the second 2ω0 RT- ◦ ALP. The black dot on Smith chart indicates the ΓL,OP T 1(2ω0) = 16 143.1 which provides the maximum output power of 38.5 mW with the ﬁrst 2ω0 RT-ALP. The best performance is ◦ obtained with the ΓL,OP T 2(2ω0) = 16 150.9 (blue rectangle) which provides the maximum output power of 39.4 mW with the second 2ω0 RT-ALP. The black circle indicates the ◦ ΓL(2ω0) = 0.766 133 obtained with the implemented load circuit 1 (see section 4).

21 The black dot in Fig. 2.5 indicates the optimal ΓL,OP T 1(2ω0) which provides the maximum output power of 38.5 mW. Since this output power can be affected by memory effects by the phase sweeping during the RT-ALP measurement, the accuracy of the RT-ALP results was veriﬁed to be accurate using constant phase measurements. Applying the resulting

ΓL,OP T 1(2ω0) veriﬁed by constant phase measurements, the output power is swept again to

ﬁnd out a new optimal ΓL,OP T 2(ω0). The result is represented in Fig. 2.3 by a blue dashed line.

With this new optimal ΓL,OP T 2(ω0), the optimal ΓL,OP T 1(2ω0) is further optimized re- cursively using both RT-ALP technique and the constant phase measurements respectively

◦ in ΓL(2ω0) plane. As a result, a new optimal ΓL,OP T 2(2ω0) of 16 150.9 which provides the maximum output power of 38.8 mW is obtained. The solid line contour plot and the blue rectangle in Fig. 2.5 depict the second 2ω0 RT-ALP result and the optimal ΓL,OP T 2(2ω0) respectively.

rd After both optimal ΓL,OP T 2(ω0) and ΓL,OP T 2(2ω0) are found, a 3 harmonic RT-ALP is performed to further increase the maximum output power. Fig. 2.6 and Fig. 2.7 shows the loci of ΓL(3ω0, t) and the output power contour plot obtained respectively. As shown, the variation of the output power is from around 35 mW to 39 mW on the Smith chart depending on third harmonic load impedance. The black dot depicts the optimal ΓL,OP T 1(3ω0) which provides the maximum output power of 39.2 mW. Similarly, the constant phase measurement is performed to account for the memory effect.

Finally, the black solid line in Fig. 2.3 is obtained by using both of the optimal terminations ΓL,OP T 2(2ω0) and ΓL,OP T 1(3ω0). A summary of the output power obtained at each step of the optimization procedure is given in Table 2.1. Note that the results of the optimization with RT-ALP of higher harmonics does not lead always to a monotonous power

22 increase as the remaining reﬂection coefﬁcients cannot be exactly held constant during the

RT-ALP phase sweep. However the validity of the overall iterative optimization scheme is conﬁrmed by the fact that the sequential optimization of the 2nd and 3rd harmonic load impedances is veriﬁed in Fig. 2.3 to successively improves the output power of the oscillator. The modest improvement achieved in the third harmonic optimization step is indicative of the rate of convergence of the multi-harmonic optimization for the particular negative resistance under consideration. It is further to be noticed that the output power is maximized for harmonic loads located on the lossless edge on the Smith chart.

1.0 1 0.5 2.0 0.8 0.6

0.4 0.2 0.2 0 0.0 0.2 0.5 1.0 2.0 Inf −0.2 −0.4 −0.2 −0.6 −0.8 −0.5 −2.0 −1 −1.0 −1 −0.5 0 0.5 1

Figure 2.6: Loci of ΓL(3ω0, t) obtained from the 3ω0 RT-ALP measurement with the LSNA. A frequency offset (∆ω) of about 200 kHz is used.

2.2.3 Device Line Measurement

−1 The device line ΓIN (|a1|, ω0) obtained by sweeping the incident wave a1 from -1.5 dBm to 8 dBm while using both the optimal ΓL,OP T 2(2ω0) and ΓL,OP T 1(3ω0) determined

1.0 0.039 1 0.5 2.0 0.0385 0.8 0.038 0.6

0.4 0.2 0.0375

0.2 0.037 0 0.0 0.2 0.5 1.0 2.0 Inf 0.0365 −0.2 0.036 −0.4 −0.2 0.0355 −0.6 −0.8 0.035 −0.5 −2.0 −1 0.0345 −1.0

−1 −0.5 0 0.5 1

Figure 2.7: Output power contour plot in ΓL(3ω0) plane. The black dot on Smith chart ◦ indicates the optimal ΓL,OP T 1(3ω0) = 0.956 − 18.4 which provides the maximum output ◦ power of 39.2 mW. The black circle indicates the ΓL(3ω0) = 0.386 112 obtained with the implemented load circuit 1 later. Considering the output power difference between black dot and black circle, this resulting error is negligible.

Method Output Power (mW) Γ (ω ) Γ (2ω ) Γ (3ω ) L 0 L 0 L 0 RT-ALP ALP Γ (ω ) L,OP T 1 0 n.a. 35.7 0.261 6 -143.2 0 6 0 0 6 0 (Fig. 2.3) Γ (2ω ) L,OP T 1 0 38.5 38.1 0.261 6 -143.2 1 6 143.1 0 6 0 Γ (ω ) L,OP T 2 0 n.a 38.3 0.282 6 -146.2 1 6 143.1 0 6 0 (Fig. 2.3) Γ (2ω ) L,OP T 2 0 39.4 38.8 0.282 6 -146.2 1 6 150.9 0 6 0 Γ (3ω ) L,OP T 1 0 39.2 39.6 0.282 6 -146.2 1 6 150.9 0.95 6 -18.4 Γ (ω ) L,OP T 3 0 n.a 39.3 0.279 6 -153.0 1 6 150.9 0.95 6 -18.4 (Fig. 2.3)

Table 2.1: Summary of the output power optimization procedure using multi-harmonic RT-ALP technique and constant-phase active load-pull (ALP) LSNA measurements

24 from the RT-ALP measurements can now be used for designing the oscillator. This optimal devices line is represented by a black solid line in the Smith Chart shown in Fig. 2.8. The operating points A and B associated with the maximum or near-maximum output-power on Fig. 2.3, are indicated using black dots labeled A and B on the parametric plot of the optimal device line shown Fig. 2.8.

2ω 0 2ω 0.6 0 3ω 3ω 0 0 0.4

0.2

0 0.2 0.5 A 1.0 2.0

B −0.2 Device line Load−line1

−0.4 Load−line2

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

Figure 2.8: Measurement result of the load-line1 ΓL(ω0) (red dashed line) and load-line2 −1 ΓL(ω0) (blue dotted line) with a device line ΓIN (|a1|, ω0) (black solid line). The operating point A for the load-line 1 corresponds to the maximum output power while B is the operating point for the load-line 2. Their respective output powers are given in Fig. 2.3.

25 2.3 Stand Alone Oscillator

2.3.1 Design of Harmonic Load Circuits

Two different load circuits were designed according to the Kurokawa theory yielding, respectively, the operating point A and B in Fig. 2.3. The layout of the load circuit for operating point A is shown in Fig. 2.9. To position the optimal ΓL,OP T 2(2ω0) and ΓL,OP T 1(3ω0) to the desired location found on Smith chart, a λ/4 high impedance line and a λ/2 50 Ω line are used in the load circuit design. As shown in Fig. 2.8, both of the load-lines ΓL(ω) measured versus frequency with a network analyzer achieve reﬂections coefﬁcients ΓL(2ω0) at the second harmonic which are reasonably close to the desired optimal ΓL,OP T 2(2ω0) in

Fig. 2.5 determined from the LSNA measurements. On the other hand the ΓL(3ω0) obtained for both load-lines are relatively far from the desired optimal values ΓL,OP T 1(3ω0) shown in Fig.2.7. However, the ΓL(3ω0) of both load-lines still deliver a similar output power range as indicated in the contour plot of Fig. 2.7.

Fig. 2.10 shows the test bed used for the self-sustaining oscillation measurement. In order to create a stand alone oscillator while maintaining well deﬁned reference planes, a male SMA connector was used in the load circuit and a female SMA connector in the drain port of the oscillator. The other port of the load circuit is connected to the spectrum analyzer for the output power and the phase noise measurements.

2.3.2 Experimental Results

After combining the non-linear part of the oscillator with the load circuit 1 and 2 designed, the output power of the stand alone multi-harmonic loaded oscillator were measured

26 GND

GND DC bias tee λ/4 high impedance line

Oscillator port Drain port

λ/2 50 ohm line

Figure 2.9: Layout of the harmonic load circuit designed for self-sustaining oscillation measurement.

S e r Γ Γ i IN L e s Spectrum G D Load Termination Circuit Analyzer 1dB S F e cable loss e d b a c k

Figure 2.10: Test bed for self-sustaining oscillation measurement.

27 with the spectrum analyzer. As is summarized in Table 2.2, a reasonable agreement is observed between the LSNA predicted and the oscillator measured results. Also as expected, the output power with load-line 1 is slightly higher than that with load-line 2.

Method Load circuit Frequency Output power LSNA Load-line1 2.500 GHz 15.94 dBm predicted Load-line2 2.500 GHz 15.83 dBm Oscillator Load-line1 2.494 GHz 14.73 dBm measured Load-line2 2.491 GHz 14.43 dBm

Table 2.2: Comparison of predicted and measured frequency of oscillation and output power with both load-lines

Having built the stand alone oscillator, its phase noise characteristics can then be acquired. Fig. 2.11 shows the measured phase noise of the stand alone oscillator with load- line 1 and load-line 2. As shown, a relatively low noise of -102 dBc/Hz is observed with both load-lines at 100 kHz offset frequency owing to the optimization of the oscillator output power.

2.4 Conclusion

A large-signal measurement-based methodology for the design of negative-resistance oscillators with multi-harmonic loading has been presented. Harmonic-tuning was realized with a phase swept RT-ALP testbed implemented with an LSNA for each harmonic. Owing to the fact that in RT-ALP the swept-phase data are acquired in a single 10 ms LSNA measurement for each power level, the multi-harmonic optimal load impedances which optimize the output power of the oscillator can be determined in a short time (0.2 s LSNA acquisition time for 20 different power measurements). A stand alone oscillator was then

28 −60 load circuit1 load circuit2 −70

−80

−90

−100

Phase noise (dBc/Hz) −110

−120

−130 2 3 4 5 6 10 10 10 10 10 Frequency(Hz)

Figure 2.11: Phase noise measurement of a stand alone oscillator with load-line 1 (black dashed line) and load-line 2 (blue solid line).

fabricated and a reasonable agreement between the expected and measured performances was obtained in accordance with the Kurokawa design methodology. The phase noise of the stand alone oscillator was measured as well and consistent performances were observed for the two load circuits used. This work demonstrates the applicability of the phase-swept RT-

ALP technique to the design of negative-resistance oscillators. To our knowledge, this is the

ﬁrst harmonic loaded oscillator designed and implemented using the multi-harmonic RT-

ALP technique as well as the ﬁrst experimental active-load-pull to consider third harmonic tuning in oscillator design. This works further supports the notion that multi-harmonic real- time active load-pull provides the designer with a time-efﬁcient technique for the quasi- interactive design of negative-resistance oscillators.

29 CHAPTER 3

BEHAVIORAL MODELING OF OSCILLATORS USING REAL-TIME ACTIVE LOAD PULL

In wireless communication systems, due to the high volume of circuit components and the complexity of integrated-circuit (IC) environment, it is very challenging to accurately simulate a complete system in an efﬁcient way [17]. Behavioral modeling, also called a

”black-box” modeling, is often used in the frequency-domain to characterize such system efﬁciently and accurately by simply using a mathematical relation between the input and output signals [18].

The Poly-Harmonic Distortion (PHD) modeling [19] provides a powerful method to model the non-linear characteristics of power ampliﬁers. The PHD model assumes that the dominant non-linear response is controlled by the fundamental tone and that the harmonic signals can be dealt with using ﬁrst-order perturbations. This elegant formalism is found to be applicable to a large class of circuits. However, the accuracy will naturally degrade when the incident power for the harmonics is high and the linear response for the harmonics is no longer applicable.

The main purpose of this chapter is to develop a power-dependent Volterra series model for the oscillator to ﬁt the multi-harmonic load-pull data and enable the harmonic balance analysis of the oscillator in a circuit simulator. The proposed modeling method aims at

30 achieving improved performance compared to other methods by taking into account optimal higher-order perturbations for the harmonics as needed.

3.1 Power-Dependent Volterra Series Modeling

3.1.1 Volterra Algorithm

The Volterra series is a well-known approach to represent non-linear circuits in the time and frequency domain. In the frequency domain we shall see that it admits a simple representation for a ﬁnite number of harmonic tones which is valid up to inﬁnite order.

Let us consider injecting a signal consisting of three incident harmonics waves a1(ω0), a1(2ω0), and a1(3ω0) on the nonlinear negative resistance circuit of an oscillator of infinite order. All other incident harmonics a1(nω0) with n > 3 are assumed to be negligible and are set to zero. The system being of infinite order we expect an infinite number of harmonics will be generated for the reflected waves. Let us consider initially only the first three reflected waves b1(ω0), b1(2ω0), and b1(3ω0). It can be verified that they can then be

31 represented using the following Volterra series expansion:

∗ b1(ω0) = a1(ω0) · f1(x3, ω0) + a1(ω0)a1(2ω0) · f2(x3, ω0)

∗ + a1(2ω0)a1(3ω0) · f3(x3, ω0)

2∗ + a1 (ω0)a1(3ω0) · f4(x3, ω0)

∗ 2 + a1(3ω0)a1(2ω0) · f5(x3, ω0), (3.1)

2 b1(2ω0) = a1(2ω0) · f1(x3, 2ω0) + a1(ω0) · f2(x3, 2ω0)

∗ + a1(2ω0)a1(ω0)a1(3ω0) · f3(x3, 2ω0)

∗ + a1(ω0)a1(3ω0) · f4(x3, 2ω0)

2∗ 2 + a1 (2ω0)a1(3ω0) · f5(x3, 2ω0), (3.2)

3 b1(3ω0) = a1(3ω0) · f1(x3, 3ω0) + a1(ω0) · f2(x3, 3ω0)

+ a1(ω0)a1(2ω0) · f3(x3, 3ω0)

∗ 2 + a1(ω0)a1(2ω0) · f4(x3, 3ω0)

∗ 3 + a1(3ω0)a1(2ω0) · f5(x3, 3ω0). (3.3)

The 5 functions fi(x3, nω0) in (5.2) - (3.3) are found to be functionally dependent on 13

DC terms xi,3:

fi(x3, nω0) = fi(x1,3, x2,3, ..., x13,3, nω0) (3.4) where these DC terms are given by:

∗ ∗ x1,3 = a1(ω0)a1(ω0), x2,3 = a1(2ω0)a1(2ω0), ∗ ∗ x3,3 = a1(3ω0)a1(3ω0), x4,3 = a1(3ω0)a1(ω0)a1(2ω0), ∗ 2 ∗ ∗ 2 x5,3 = a1(2ω0)a1(ω0), x6,3 = a1(3ω0)a1(ω0)a1(2ω0), ∗ 3 2∗ 3 x7,3 = a1(3ω0)a1(ω0), x8,3 = a1 (3ω0)a1(2ω0) ∗ ∗ x9,3 = x4,3, x10,3 = x5,3, ∗ ∗ x11,3 = x6,3, x12,3 = x7,3, ∗ x13,3 = x8,3.

32 This expansion for three harmonic excitations holds up to inﬁnite order in non-linearity but requires obviously that the fi(x3, niω0) functions be identiﬁed. The non-linear response of the oscillator is then modeled using (5.2) - (5.3). The Volterra series expansion for four tones is presented in Appendix A.

3.1.2 Model Extraction

All the fundamental and DC terms in (5.2) and (5.3) for three harmonic case and four harmonic case in Appendix A are obtained after an exhaustive search up to order 40 and random search up to order 5000 using a simple MATLAB algorithm. Let us first briefly explain the guiding principle for the algorithm used to find the DC terms in (5.3).

For an inﬁnite order nonlinear system excited by n incident harmonic tones, a matrix

Dn(k, i) can be used to record the order (exponent) of each tone of harmonic frequency k

∗ in each DC term i such that we have using a1(−k) = a1(k):

Yn Dn(n+1+k,i) xi,n = [a1(kω0)] k=−n

The frequency of each tone k is given itself by the companion vector Vn(k). The total order P for a DC term xi,n is then given by the sum of the exponent of its components: k Dn(k, i).

For example for n = 3 the following matrix D3 is obtained,

  0011011200000   0100100010203   1000010012030   D3 = 0000000000000 (3.5)   1001203000100 0101020301000 0010000010112

In the matrix D3, each column represents a DC term and each row represents a harmonic frequency from −3ω0 (top) to +3ω0 (bottom), with the center row associated with the DC

33 component. Note that all the coefﬁcients in the matrix D3, are positive or null integers.

Negative integers would be conceptually associated with dividers and subharmonic generation which are outside the class of the systems considered here.

Let V3 be a harmonic frequency vector: £ ¤ V3 = −3 −2 −1 0 1 2 3

The xi,3 columns retained in the matrix D3 are those which generate DC contributions by verifying the condition £ ¤ V3 × D3 = 0 0 0 0 0 0 0 0 and which cannot be generated from the superposition of other lower order xi,3 exponent columns. Indeed an inﬁnite number of other xi,3 featuring higher orders are possible but can be factorized in terms of the previously extracted lower order DC terms xi,3. That is for a higher order term, say x9,3 we have     p−3 m   1 p−2  m     2  p−1  m     3  [x9,3] =  p0  = D3  .     .   p1    m12  p2  m13 p3 with all pk and mi positive integer (natural) numbers.

Using a similar methodology, a matrix F3 which shows the order of 5 fundamental terms with positive frequency can also be obtained.

  0 0 0 1 0   0 1 0 0 0   0 0 1 0 2   F3 = 0 0 0 0 0 (3.6)   1 0 0 0 0 0 0 1 2 0 0 1 0 0 1

34 The yi,3 columns retained in the matrix F3 are those which generate fundamental frequency contributions by verifying the condition

£ ¤ V3F3 = 1 1 1 1 1 1 1 1

and which cannot be reduced further by the factorization of DC terms xi,3 from the matrix

D3.

The derivation summarized above for the case of three tones can be easily implemented for a system with an arbitrary number n of incident harmonic tones.

3.2 Model Validation

To verify the applicability of the proposed behavioral model, the RT-ALP contour plots at the second and third-harmonics in Fig. 2.5 and 2.7 respectively are ﬁtted using the

Volterra model presented in Section 3.1. Any representation could be used for the functions fi(x, nω0). For convenience a polynomial expansion with contribution limited, respectively, to order 7, 3 and 3 for the tones ω0, 2ω0 and 3ω0 was used. Further the terms with negligible contribution were also pruned. For the sake of comparison, a Volterra model similar to the PHD model [19] which only uses a first-order perturbation for the second and third harmonics is also used to fit the RT-ALP contour plot. Figures 3.1 (a) and (b) compare the performance of the RT-ALP fitting in the ΓL(2ω0) plane using both models.

As expected, the Volterra model which relies on higher orders for 2ω and 3ω exhibits a superior ﬁtting performance than the linear 2ω and 3ω model. An even improved performance is observed for ΓL(3ω0) as is depicted in Fig. 3.2. These results indicate that for the non-linear circuit application considered here, higher order perturbations at harmonic frequencies have a noticeably impact on the modeling results.

1.0 1 0.04 0.5 2.0 0.8 0.039

0.6 0.038

0.4 0.2 0.037 0.2 0.036 0 0.0 0.2 0.5 1.0 2.0 Inf 0.035 −0.2 0.034 −0.4 −0.2 0.033 −0.6 −0.8 0.032 −0.5 −2.0 −1 0.031 −1.0

−1 −0.5 0 0.5 1

(a) Linear 2ω and 3ω

1.0 0.04 1 0.5 2.0 0.8 0.039 0.6 0.038

0.4 0.2 0.037

0.2 0.036 0 0.0 0.2 0.5 1.0 2.0 Inf 0.035 −0.2 0.034 −0.4 −0.2 0.033 −0.6 0.032 −0.8 −0.5 −2.0 −1 0.031 −1.0 0.03 −1 −0.5 0 0.5 1

(b) Higher order 2ω and 3ω

Figure 3.1: Comparison of contour plots in the ΓL(2ω0) plane between a Volterra model which uses ﬁrst-order perturbation at 2ω0 and 3ω0 (a) and the proposed Volterra model with higher order nonlinearities for 2ω and 3ω (b). The blue dashed lines stand for the modeling result.

1.0 1 0.039 0.5 2.0 0.8 0.0385

0.6 0.038

0.4 0.2 0.0375 0.2 0.037 0 0.0 0.2 0.5 1.0 2.0 Inf 0.0365 −0.2 0.036 −0.4 −0.2 0.0355 −0.6 0.035 −0.8 −0.5 −2.0 −1 0.0345 −1.0 0.034 −1 −0.5 0 0.5 1

(a) Linear 2ω and 3ω

1.0 1 0.0385 0.5 2.0 0.8 0.038 0.6 0.0375 0.4 0.2 0.2 0.037

0 0.0 0.2 0.5 1.0 2.0 Inf 0.0365 −0.2 0.036 −0.4 −0.2 0.0355 −0.6 −0.8 0.035 −0.5 −2.0 −1 −1.0 0.0345

−1 −0.5 0 0.5 1

(b) Higher order 2ω and 3ω

Figure 3.2: Comparison of contour plots in the ΓL(3ω0) plane between a Volerra model which uses ﬁrst-order perturbation at 2ω0 and 3ω0 (a) and the proposed Volterra model with higher order 2ω and 3ω (b). The blue dashed lines stand for the modeling results.

37 To use this model for circuit simulations, the Volterra model was implemented in Ag- ilent Advanced Design System (ADS) using the built-in 2-port Frequency Domain De- vice (FDD) component. Owing to the automated MATLAB extraction developed, all the

Volterra coefficients and associated theoretical expression required for the non-linear mixing terms are easily embedded into the simulator. The number of non-linear coefficients were reduced in the ADS simulations to prevent any convergence problems. This was made possible by reducing the fitting area of in the RT-ALP contour plots in the ΓL(2ω0) and ΓL(3ω0) planes to a region with a radius of 1.2.

Fig. 3.3 compares the output power characteristics of the higher order 2ω and 3ω model obtained with optimal harmonic loads with the linear 2ω and 3ω model in ADS and MAT-

LAB. As shown, the higher order 2ω and 3ω model exhibits better agreements to the LSNA measured results than the linear 2ω and 3ω model in both MATLAB and ADS modeling results.

3.2.1 Error Evaluation

The behavioral model can also be used to evaluate the error in frequency and amplitude of oscillation. These errors are difﬁcult to avoid given the synthesized and fabricated load circuit will depart from the desired multi-harmonic load terminations obtained from

RTALP. Fig. 3.4 depicts the error of frequency and amplitude of oscillation due to the slight disagreement of the load-line 1. Note that two load circuits were generated to test the self-oscillation. As shown, the measured load-line (red circle-plain line) slightly deviates from the desired load-line (blue-plain line). Owing to the accurate modeling, the modeled device-line (black dashed line) in ADS shows an excellent agreement with the measured device-line (blue square). The circled A indicates the desired operating point providing

38 45 Measured by LSNA 40 Modeled in MATLAB Modeled in ADS 35 45 30 40 25 35 20 30 Measured by LSNA Modeled in MATLAB 15 Output power (mW) Modeled in ADS

Output power (mW) 25 0 2 4 6 a (dBm) 10 1

0 −30 −20 −10 0 10 a (dBm) 1 (a) Linear 2ω and 3ω

45 Measured by LSNA 40 Modeled in MATLAB Modeled in ADS 35 45 30 40 25 35 20 30 Measured by LSNA Modeled in MATLAB 15 Output power (mW) Modeled in ADS 25

Output power (mW) 0 2 4 6 a (dBm) 10 1

0 −30 −20 −10 0 10 a (dBm) 1 (b) Higher order 2ω and 3ω

Figure 3.3: Comparison of the output power characteristics between LSNA measured data, MATLAB modeling results, and ADS simulation results for the (a) order 4 for ω and linear 2ω and 3ω and (b) order 4 for ω and 3, 2 2ω and 3 ω Volterra models.

39 maximum output power. The circled B indicates the predicted operating point obtained from ADS simulations using the actual load line from the circuit fabricated. The ADS simulations relied on the behavioral model and the actual impedance measured from the load circuit fabricated. The circled C shows the actual operating point obtained from the self-oscillation measurement. The error in predicted frequency and amplitude of oscillation can be calculated as follows:

∆ωK,D = |ω0,K − ω0,D| = |2.498 GHz − 2.5 GHz| = 2 MHz. (3.7)

∆AK,D = |A0,K − A0,D| = |5.4 dBm − 5.2 dBm| = 0.2 dBm. (3.8)

The self-sustained oscillation was observed at 2.494 GHz with an output power of 14.73 dBm. Moreover, the amplitude of oscillation can be obtained by comparing the output power measured with the power sweep plot in Fig. 2.3. Thus, the frequency and amplitude error between measured and desired operating points can be calculated as follows:

∆ωM,D = |ω0,M − ω0,D| = |2.494 GHz − 2.5 GHz| = 6 MHz. (3.9)

∆AM,D = |A0,M − A0,D| = |1.3 dBm − 5.2 dBm| = 3.9 dBm. (3.10)

The difference between (3.7) and (3.9) is of 4 MHz and the difference between (3.8) and (3.10) is of 3.7 dB. The predicted error in frequency and amplitude of oscillation from the ADS model shows a reasonable agreement with the measured error considering the sensitivity of oscillator circuits and experimental measurement errors. More accurate prediction may require accounting from the frequency dependence in the behavioral model at the cost of more extensive measurements and extraction.

40 −0.12

−0.122

−0.124 (ω , A ) 0,D 0,D −0.126 A

−0.128

−0.13

Im(Z) B −0.132 (ω , A ) 0,K 0,K

−0.134 Measured device−line −0.136 Modeled device−line in ADS ADS predicted device−line for measured load−line −0.138 Desired load−line (ω , A ) Measured load−line 0,M 0,M C −0.14 Curve fitted measured load−line

−0.27 −0.265 −0.26 −0.255 −0.25 −0.245 Re(Z)

Figure 3.4: Graphical evaluation of the error in frequency and amplitude of oscillation using the model implemented in ADS. The black arrow on each load-line indicates the direction of the frequency increment while the black arrow on the device-line depicts the direction of the amplitude increment. The black squares predicted by the ADS simulation are at the same power level as the nearby blue square for the measured power level.

41 3.3 Conclusion

A novel power-dependent Volterra series model and a large-signal measurement-based methodology to design oscillator have been presented. The new modeling approach holds up to infinite order for a finite of number of harmonic excitations. The behavioral model yielded a good fit of the load-pull measurement data obtained using an approximated third- order expansion of the power-dependent functions involved. This model which could be also applied to amplifiers was implemented in a circuit simulator to facilitate the harmonic balance simulation of an oscillator. Finally a methodology to estimate the error in frequency and amplitude of oscillation due to real-world non optimal loading conditions, has been presented using the behavioral model presented.

42 CHAPTER 4

MODEL COMPARISON FOR 1/F NOISE IN OSCILLATORS WITH AND WITHOUT AM TO PM NOISE CONVERSION

Dominant types of noise in RF circuits are thermal, shot and 1/f noise. Low-frequency

The Leeson model [7] which relies on an empirical expression for the up-conversion of the 1/f noise is often used to ﬁt the up-converted 1/∆f 3 noise but is of limited applicability for the prediction of the 1/f noise performance directly from circuit analysis.

An extension of the Kurokawa noise analysis [22] to 1/f noise up-conversion was reported by Rohdin et al. [23]. More recent detailed derivations were then given in Ref. [6].

43 Current commercial harmonic balanced circuit simulators rely instead on the method of conversion matrix [24], [25]. In the conversion matrix method, a small-signal perturbative analysis is superposed upon large-signal harmonic balance simulations of an autonomous oscillator circuit in order to account for the noise-induced frequency modulation of the carrier at low frequency offsets. The model can account for both the up-conversion of low-frequency noise (1/f) and the down conversion of cyclostationary noise if such noise processes are implemented. Note that a cyclostationary noise can be decomposed as the product of a stationary noise source and a deterministic periodic RF oscillation. Additional cyclostationary effects such as the impact of large-signal RF oscillations on noise sources can also be accounted for by using time averages as was demonstrated in Ref. [26] the

Lorentzian (pop-corn) and 1/f noises generated by traps.

A Linear Time Varying (LTV) model based on the Impulse Sensitivity Function (ISF) which also accounts for both the up-conversion of 1/f noise and down conversion of harmonic noise was reported in Ref. [27], [28]. This model provides great insights on the phase noise mechanisms and is therefore conceptually useful to circuit designers. However the numerical calculation of the ISF in the time domain is more involved.

A general differential equation for the phase error in oscillators was obtained using a perturbation technique in [29]. A Perturbation Projection Vector (PPV) method for calculating the phase noise due to 1/f noise sources was then introduced in Ref. [30]. The PPV method projects the noise on the oscillator’s orbit in order to separate its tangential and transversal components and decouple the AM and PM noise.

44 theory could neglect the amplitude to phase noise conversion which is accounted for by the PPV, Kurokawa and conversion methods. A correspondence between the PPV method and the matrix conversion method was also theoretically established in Ref. [32] and was supported by the good agreement in noise simulation results. The PPV model is not usually amenable to analytic solutions. Limiting results for low and high frequency offset were however reported [33] [34]. An analytic solution for white noise was however reported in Ref. [35] for a simple negative resistance oscillator. Since the oscillator studied did not feature any AM to PM noise conversion, the results reported were identical to the ISF results.

1 deviation from the phase spectral density 2 Sφ,1/f,ssb(∆ω) (new IEEE phase noise) which diverges for vanishing offset frequencies. The ceiling value results from power conservation since the integration of the voltage spectral density SV,1/f (∆ω) gives the total oscillator output power. Similar analytic results for the autocorrelation RV,1/f (∆ω) were also previously reported in Ref. [37] for the PPV method and in Ref. [6] for the generalized Kurokawa theory. But the resulting SV,1/f (∆ω) were not compared and this will be pursued in this paper.

45 which is circuit-based and therefore more accessible to circuit designers. Indeed the analytic expressions for the amplitude and phase noise derived from the generalized Kurokawa theory [6] are all expressed in terms of the derivatives of immittances which can be easily evaluated either analytically (for simple circuits) or by circuit simulations or measurements. In the generalized Kurokawa model, the noise sources can not only be stationary

(white noise) but also up-converted 1/f noise, or down converted cyclostationary noise.

Finally the new generalized Kurokawa domain theory is also applicable to low-Q circuit such as ring oscillators as shall be demonstrated in this paper.

To evaluate the different 1/f noise theories we shall consider four different oscillator circuits. The ﬁrst circuit will be a generalized Van der Pol oscillator with an harmonic short which features a new fourth-order nonlinear capacitance exhibiting both AM 1/f and

PM 1/f noise up-conversion as well as AM to PM 1/f noise conversion. A fully analytic solution will be obtained within both the Kurokawa and PPV formalism for this oscillator which will facilitate the model evaluation and comparison with simulations and other noise theories. The second and third circuits will respectively be a BJT-based Colpitts oscillator and a CMOS-based ring oscillator implemented in a circuit simulator. The fourth circuit will be a HEMT-based transmission line oscillator fabricated and characterized.

46 4.1 Kurokawa 1/f Noise Models for an Oscillator

It is customary for the Kurokawa analysis of an oscillator to divide the oscillator circuit into two independent parts. For shunt resonance, (1) a frequency and amplitude sensitive nonlinear active circuit with admittance YIN (A, ω) and (2) a frequency sensitive linear passive circuit YL(ω) are used as shown in Figure 5.1. Further in the original Kurokawa analysis the linear circuit is also assumed to a be a shunt or series resonator. It is therefore often concluded that the Kurokawa analysis is not applicable to low Q oscillator such as ring oscillators which do not feature a resonator.

i L iIN

v (t) YL YIN V N,(1/f)

Figure 4.1: An admittance model of an oscillator with low-frequency modulation of the nonlinear device impedance at the fundamental frequency.

To generalize the Kurokawa theory to arbitrary oscillator conﬁgurations we shall instead represent the oscillator by a one-port network excited by a signal at the fundamental frequency with the harmonics left free running. As we shall see this approximation is equivalent to using YL(ω) = 0 and just retaining the oscillator describing function YIN (A, ω).

Many approaches (ﬁlters) are possible for implementing the targeted circuit. In this work a one-port circuit will be realized by breaking a given pair of nodes (typically a port of

47 interest) into two ports (port 1 and 2) which are then connected together via a through such that we have v1(t) = v2(t) = v(t). With this topology we can now handle both negative resistance and feedback oscillators since no load circuit is required. The condition for sustained oscillations (in steady state or not) is obtained when the circuit current veriﬁes i1(t) + i2(t) = 0. To be able to characterize such a one-port circuit using harmonic balance one can then introduce a voltage source v3 = A cos(ωt) at the frequency ω with amplitude

A which is only connected to the circuit at the fundamental frequency of oscillation. The sustaining voltage can then be injected in the oscillator via a 3-port circuit which veriﬁes at the fundamental frequency:

V1(ω) = V2(ω) = V3(ω)

I1(ω) + I2(ω) + I3(ω) = 0

where Vk(ω) and Ik(ω) are the Fourier coefﬁcients at the fundamental frequency of the voltage vk(t) and current ik(t) respectively. The resulting Sthru matrix at the fundamental

ω is then:   −1 2 2 1 S (ω) =  2 −1 2  . thru 3 2 2 −1 For the higher harmonics no signal is injected and we have instead the close loop conditions:

V1(kω) = V2(kω)

I1(kω) + I2(kω) = 0.

The resulting Sthru matrix at the k-th harmonic kω is then:   0 1 0 Sthru(kω) =  1 0 0  . 0 0 0

48 Under the circuit condition described, the fundamental and harmonics of the oscillator current iIN1 and iIN2 are all dependent variables on the applied voltage v3. The oscillator current iIN1 and iIN2 in steady state is thus uniquely dependent on the frequency ω and the amplitude A of the excitation applied at port 3. Speciﬁcally at the fundamental frequency we have for the generator current i3(t) = iIN (t):

IIN (A, ω) = IIN1(A, ω) + IIN2(A, ω).

Note that all the harmonic are self-consistently accounted for by the close-loop circuit.

The circuit will be in its steady-state operating point (A0, ω0) when we have i3(t) = 0 for an excitation v3(t) = A0 cos ω0t. At the fundamental frequency we thus have the following identity:

IIN (A0, ω0) = IIN1(A0, ω0) + IIN2(A0, ω0) = 0.

Note that no assumption was made on the size of the harmonics of the voltage or current at port 1 and 2. Using this approach the Kurokawa analysis is now applicable to low Q oscillator circuits exhibiting large harmonics for both the voltage and currents. This will be veriﬁed in Section 4.4 with a ring oscillator example.

A four-port coupler (OSCPORT in ADS) similar to the three-port through is usually implemented in harmonic balance simulators. However that four-port circuit is asymmetric between port 1 and 2 and therefore the same voltage is not applied to port 1 and 2, except when in steady state. On the other hand, the symmetric 3-port circuit Sthru which applies the same voltage v1(t) = v2(t) = v(t) at port 1 and 2 is amenable to a simpler single-port analysis. In the remaining of the noise analysis we indeed mostly refer to the single current

IIN and further introduce the single admittance YIN deﬁned as:

I (A, ω) Y (A, ω) = IN . IN A 49 Using the usual Kurokawa modeling technique, we can next investigate any departure from the steady-state periodic solution by characterizing the variation of the frequency and amplitude of oscillation when a small perturbation is applied somewhere else in the oscillator circuit.

It is a well known fact that for small enough perturbation the operation of an oscillator can be linearized around a stable operating point: the amplitude of oscillation (A0) and the frequency of oscillation (f0) [5]. Based on this fact the Kurokawa analysis can be extended to account for 1/f noise by accounting for the variation of any DC biasing voltage

[VN (t) = VN0 + δVN (t)] or any DC biasing current [IN (t) = IN0 + δIN (t)]. A single DC bias dependence is initially assumed in this work for the sake of simplicity but the results are easily generalized to an arbitrary number of independent DC bias voltage or current sources. A voltage DC source will be used for the derivation but the results obtained are applicable to current biasing sources as well as as will be demonstrated in the Colpitts and ring oscillators in Section 4.4. Using the above assumption, the admittance of the nonlinear part YIN (A, ω, VN ) can be represented as [6]: [deﬁning YIN,0 = YIN (A0, ω0,VN0)]

∂Y Y (A, ω, V ) = Y (A , ω ,V ) + IN,0 δA IN N IN 0 0 N0 ∂A ∂YIN,0 ∂YIN,0 + δω + δVN ∂ω ∂VN where the operating points (A, ω, VN ) and (A0, ω0, VN0) represent the steady state operating point with and without noise, amplitude and frequency perturbations respectively:

A(t) = A0 + δA(t)

ω = ω0 + δω(t)

VN = VN0 + δVN .

50 Note that for self-sustained oscillations, the operating point veriﬁes: YIN (A0, ω0,VN0) = 0 and it results that for steady-state perturbations we have:

δω(∞) B δA(∞) C = and = − (4.1) δVN β δVN β where the constants β, B and C are deﬁned below in terms of the derivatives of YIN . The above sensitivity relations establish two constrains that these derivatives must verify for the linear perturbation to be accurate as shall be investigated in Section 4.4 for the sensitive case of the ring oscillator.

Oscillator Nonlinear Loop Circuit ( ω) V N k iIN2 i2 i1 iIN1 V N v1 =v2 =v3 v (t) v (t) 2 2 Sthru 1 1 Cyclo− 1/f stationary noise noise 3 i3=iIN v3 (t) YIN,2 YIN,1

(ω ) v3 (t)=A cos t

Figure 4.2: An nonlinear feedback model of an oscillator with low-frequency modulation of the nonlinear device circuits by 1/f noise and other cyclostationary noise processes.

The voltage v(t) = v1(t) = v2(t) = v3(t) across the port of observation in Figure 4.2 and the total nonlinear current ﬂowing in the device iIN (t) = iIN1(t) + iIN2(t) can be written

51 as:

£ ¤ v(t) = Re A(t)ej[ω0t+φ(t)] + harmonics £ ¤ j[ω0t+φ(t)] iIN (t) = Re IIN (t)e + harmonics £ ¤ j[ω0t+φ(t)] = Re YIN A(t)e + harmonics.

Then following the standard Kurokawa analysis we can next deﬁne the frequency deviation associated with the amplitude and phase evolution:

dφ(t) 1 dA(t) δω(t) = − j . dt A(t) dt

It results that the time evolution of the amplitude δA(t) and phase φ(t) deviation can be represented by the following differential equations [deﬁning YIN,0 = YIN (A0, ω0,VN0)]: ¯ ¯ ¯ ¯2 ¯∂YIN,0 ¯ 1 dδA(t) ¯ ¯ + β δA(t) = B δVN (t) (4.2) ∂ω A0 dt ¯ ¯ ¯∂Y ¯2 dφ(t) ¯ IN,0 ¯ + α δA(t) = −A δV (t) (4.3) ¯ ∂ω ¯ dt N

where the constants A, B, and C are deﬁned as:

∂G ∂G ∂B ∂B ∂Y ∂Y A = IN,0 IN,0 + IN,0 IN,0 ≡ IN,0 · IN,0 ∂ω ∂VN ∂ω ∂VN ∂ω ∂VN ∂G ∂B ∂B ∂G ∂Y ∂Y B = IN,0 IN,0 − IN,0 IN,0 ≡ IN,0 × IN,0 ∂ω ∂VN ∂ω ∂VN ∂ω ∂VN ∂G ∂B ∂B ∂G ∂Y ∂Y C = IN,0 IN,0 − IN,0 IN,0 ≡ IN,0 × IN,0 ∂A ∂VN ∂A ∂VN ∂A ∂VN

with YIN = GIN + j BIN expressed in term of the conductance GIN and susceptance BIN respectively. Also the correlation factor α and the stability factor β are given by:

∂G ∂G ∂B ∂B ∂Y ∂Y α = IN,0 IN,0 + IN,0 IN,0 ≡ IN,0 · IN,0 ∂ω ∂A ∂ω ∂A ∂A ∂ω 52 ∂B ∂G ∂G ∂B ∂Y ∂Y β = IN,0 IN,0 − IN,0 IN,0 ≡ IN,0 × IN,0 . ∂ω ∂A ∂ω ∂A ∂A ∂ω

The derivative terms ∂YIN,0 , ∂YIN,0 , and ∂YIN,0 represent the variation of the admittances ∂A ∂VN ∂ω with the various perturbations. The correlation factor α accounts for the correlation between the amplitude and phase deviations. Note that the deﬁnition of A and B parameters are reversed compared to those erroneously given in Ref. [6]. δVN (t) is the low frequency

1/f noise voltage which will be deﬁned by its power spectral density SδVN ,1/f (∆ω) =

S/|∆ω| in the following subsection.

4.1.1 Derivation of Sa,1/f (∆ω)

Assuming the autocorrelation function is WSS and Ornstein-Uhlenbeck process [48], the autocorrelation function of the δVN (t) for a single trap can be represented by:

−λ|τ| RδVN ,1trap(τ) = ke . (4.4)

Then the noise spectral density can be calculated by taking the Fourier transform of RδVN (τ) to be given by: 2λk S (∆ω) = . (4.5) δVN ,1trap ∆ω2 + λ2

As is seen, this noise spectral density has a Lorentzian distribution. Thus, a noise process with a 1/f distribution can be obtained by a superposition of many of these Lorentzian distributions with time constants τtrap(y) = 1/λ = τS exp(ρy) which are spatially varying

53 with position y in the oxide (MOS) or wide-bandgap region (HFET).

Z dmax

SδVN ,1/f (∆ω)= SδVN ,1trap(∆ω)dy 0Z 1 λ1 (−dλ) = S (∆ω) ρ δVN ,1trap λ λ0 · µ ¶ µ ¶¸ 2k λ λ = tan−1 0 − tan−1 1 ρ∆ω ∆ω ∆ω kπ S ' = for λ < ∆ω < λ ρ∆ω ∆ω 1 0

where λ0 = 1/τS ' ∞, λ1 = 1/τtrap(dmax) ' 0 and S = kπ/ρ. Since each of the traps capture electrons independently, this superposition is valid.

By taking the Fourier transform of (4.2) and (4.3), the following frequency domain representations are obtained: ¯ ¯ ¯ ¯2 ¯∂YIN,0¯ 1 ¯ ¯ j∆ω δA(∆ω)+ β δA(∆ω) = BδVN (∆ω) (4.6) ∂ω A0 ¯ ¯ ¯∂Y ¯2 ¯ IN,0 ¯ j∆ω φ(∆ω) + α δA(∆ω)= −AδV (∆ω). (4.7) ¯ ∂ω ¯ N

Then the expression of the amplitude noise spectral density for a single trap can be obtained by solving (4.6) using (4.5):

2B2λk S (∆ω) = µ ¶ (4.8) δA,1trap ¯ ¯4 2 2 2 ¯ ∂YIN,0 ¯ ∆ω2 (∆ω + λ ) β + ¯ ¯ 2 ∂ω A0 where 0 < ∆ω < ω0. By taking a summation over all traps in the limit of λ0 = ∞ and

λ1 = 0, SδA,1/f (∆ω) is derived to be: Z µ ¶ 1 λ1 −dλ S (∆ω) = S (∆ω) δA,1/f ρ δA,1trap λ λ0 · µ ¶ µ ¶¸ KδA 1 1 −1 λ0 −1 λ1 = 2 tan − tan ρλ (∆ω + k3) ∆ω ∆ω ∆ω 2 2 SB A0 1 1 ' ¯ ¯4 2 for λ1 < ∆ω < λ0 (4.9) ¯ ∂YIN,0 ¯ ∆ω (∆ω + k3) ¯ ∂ω ¯

54 2 2 4 2 2 4 where KδA = (2B λkA0) / |∂YIN,0/∂ω| , and k3 = (A0β ) / |∂YIN,0/∂ω| . Then

2 SδA,1/f (∆ω) is normalized by A0 for obtaining the ﬁnal analytic expression of the 1/f amplitude noise as: SB2 1 1 Sa,1/f (∆ω) = ¯ ¯4 2 . (4.10) ¯ ∂YIN,0 ¯ ∆ω (∆ω + k3) ¯ ∂ω ¯ This equation which was not derived in Ref. [6] will be needed for our study of the amplitude to phase noise conversion.

4.1.2 Derivation of Sφ,1/f (∆ω)

To derive an expression for Sφ,1trap(∆ω), (4.7) needs to be solved in terms of φ(ω) which results in:

AδVN (∆ω) + αδA(∆ω) φ(∆ω) = − ¯ ¯2 . (4.11) ¯ ∂YIN,0 ¯ j∆ω ¯ ∂ω ¯ Then, by squaring the above (4.11), the 1/f phase noise spectral density for a single trap can be calculated to be:

2λk Sφ,1trap(∆ω) = ¯ ¯4 2 ¯ ∂YIN,0 ¯ 2 2 ∆ω ¯ ∂ω ¯ (∆ω + λ ) ¯ ¯4  ¯ ∂YIN,0 ¯ ∆ω2A2 2 ¯ ¯ 2 + (βA + αB)  ∂ω A0  ×  ¯ ¯4 . (4.12) ¯ ∂YIN,0 ¯ ∆ω2 2 ¯ ¯ 2 + β ∂ω A0

55 Applying a similar summation method used in (4.9), the ﬁnal analytic expression for 1/f phase noise spectral density Sφ,1/f (∆ω) can be obtained: Z 1 λ1 (−dλ) Sφ,1/f (∆ω) = Sφ,1trap(∆ω) ρ λ0 λ 2 2 · µ ¶ µ ¶¸ 2kκ (∆ω + k1) 1 −1 λ0 −1 λ1 = 2 2 tan − tan ρ∆ω (∆ω + k3) ∆ω ∆ω ∆ω 2 2 Sκ (∆ω + k1) ' 3 2 for λ1 < ∆ω < λ0 (4.13) ∆ω (∆ω + k3)

2 2 2 4 2 2 4 where k1 = [A0(βA + αB) ] /(A |∂YIN,0/∂ω| ), k3 = (A0β ) /|∂YIN,0/∂ω| , and κ =

2 A/ |∂YIN,0/∂ω| . This ﬁnal result was reported in Ref. [6] with an incorrect ∆ω power.

Usually for a wide range of offset frequencies, the constants k1 and k3 term are much larger than ∆ω2 term, and Equation (4.13) can be simpliﬁed as:

2 2 µ ¶2 Sκ k1 Sκ αB Sφ,1/f,corr(∆ω) ' 3 = 3 1 + (4.14) ∆ω k3 ∆ω βA √ √ for λ1 < ∆ω < min{ k1, k3, λ0} .

The k1/k3 ratio depends on the correlation factor α in (4.3). As the correlation factor α increases, the k1/k3 ratio and therefore the 1/f noise strength increases. The correlation (AM to PM) effect will be investigated in more details in Section 4.3 using an analytic model and harmonic balance simulations and in Section 4.4 using a Colpitts and ring oscillator.

Note that α = 0 condition (uncorrelated case) yields a k1/k3 ratio equal to 1. When neglecting the correlation (AM to PM), the equation (4.13) can then be simpliﬁed as:

Sκ2 S (∆ω) ' for λ < ∆ω < λ . (4.15) φ,1/f,uncorr ∆ω3 1 0

Comparison of the generalized Kurokawa theory with ISF and PPV theories neglecting AM to PM effects will be pursued in the next subsections and in Sections 4.3 and 4.4.

56 4.1.3 Comparison of Kurokawa theory with ISF theory for 1/f noise

The ISF differential equation describing the phase evolution with time when a small signal (voltage or current) noise perturbation e(t) is applied to a speciﬁc location in the circuit, can be deﬁned as [49]:

dφ(t) = ΓISF[ω t + φ(t)] e(t) (4.16) dt 0 where Γ(φ) is the impulse sensitivity function which describes the response of the oscillation phase to a perturbation. The impulse sensitivity is periodic in φ with period 2π and can be expended in a Fourier series: " # X∞ jkθ Γ(θ) = Re Γke . k=0

The ISF differential equation described in (4.16) reduces to the generalized Kurokawa equation in (4.3) if we can assume that the correlation effect between amplitude (δA) and phase (φ) deviation is negligible. When the noise is injected at the measurement point, the coefﬁcient Γ1 which is responsible for white noise is identiﬁed using the Kurokawa theory to be given by:

ISF Kur 2 |Γ1 | ≡ |Γ1 (α = 0)| = ¯ ¯. ¯ ∂YIN,0 ¯ A0 ¯ ∂ω ¯

In general the coefﬁcient Γk with k equal or larger than one for noise perturbation ek(t) =

Re[δV(t) exp(ikω0t)] at kω0 located anywhere inside the circuit can be derived using a perturbative Volterra expansion which linearizes the dependence of the device current iIN

∗ at the fundamental frequency ω0 relative to the perturbating noise voltage δVk and δVk at

57 the harmonic frequency kω0:

IIN (A, ω, VN , δVk) ' YIN (A, ω, VN , 0) A X∞ h (1) + YIN,k(A0, ω0,VN0, 0) δVk(t) k=1 i (2) ∗ +YIN,k(A0, ω0,VN0, 0) δVk (t) .

This expansion is similar to the poly-harmonic distortion expansion (PHD) for nonlinear

Kur RF circuits [17]. The resulting noise sensitivity coefﬁcient |Γk | is:

ISF 2 Kur 2 4 |Γk | ≡ |Γk (α = 0)| = ¯ ¯4 × 2 ¯ ∂YIN,0 ¯ A0 ¯ ∂ω ¯ (· ¸ · ¸ ) ∂Y ³ ´ 2 ∂Y ³ ´ 2 IN,0 · Y (1) +Y (2) + IN,0× Y (2) −Y (1) . ∂ω IN,k IN,k ∂ω IN,k IN,k

(2) Note that in accordance with the PHD theory we have YIN,1 = 0 for k = 1 since no down conversion is involved for noise at ω0. To our knowledge this is the ﬁrst time an explicit expression is derived which permits to calculate the ISF coefﬁcient Γk directly from the derivative of the circuit admittances. The associated double sided phase noise spectral density Sφ(∆ω) (phase noise L(∆ω)) for multiple harmonics k up to K and multiple noise sources ei is then: XK X |Γ |2S (∆ω) L(∆ω) = S (∆ω) = k,j ej ,k φ 2∆ω2 k=1 j with Sej ,k(∆ω) the double sided spectral density of the noise ej(t) at the harmonic frequency kω0.

We place now our focus on the coefﬁcient Γ0 which is responsible for the up-converted

1/f noise. By directly equating the equation (4.16) with the equation (4.3), the correspond- ing coefﬁcient Γ in the generalized Kurokawa theory is: 0 ¯ ¯ ¯ ∂Y ∂Y ¯ ¯ ¯ ¯ ¯ ¯ IN,0 · IN,0 ¯ ¯ ISF¯ ¯ Kur ¯ ∂ω ∂VN Γ0 ≡ Γ0 (α=0) = |κ| = ¯ ¯2 (4.17) ¯ ∂YIN,0 ¯ ¯ ∂ω ¯

58 Using the Γ0 expression derived in (4.17), one can easily verify that the 1/f phase noise analytic expression of the ISF theory agrees with the generalized Kurokawa analytic expression in (4.15); |Γ |2S S (∆ω) = 0 . φ,1/f ∆ω3 One can conclude from this equivalence that the generalized Kurokawa theory provides an alternate mean to calculate Γ0 in the ISF theory directly from the derivative of the device admittance YIN . Note that following Ref. [32], the ISF theory was assumed to neglect the impact of the AM to PM noise conversion upon the phase noise.

4.1.4 Comparison of Kurokawa theory with PPV theory for 1/f noise

It is also of great interest to compare the Kurokawa model for 1/f noise with the PPV method [30], [33], [34]. The PPV analysis starts from the oscillator state equations:

d x = f(x, e(t)) ' f(x, 0) + J (x) e(t), dt e

∂f(x,0) with x(t) the state variables, e(t) the 1/f noise, and J e(x) = ∂e the noise Jacobian.

This oscillator admits the steady solution xS(t): d x = f(x , 0), dt S S in the absence of noise (e(t) = 0).

The ﬁnal results for the PPV theory takes a similar form to that of ISF (4.16) [31], [49]:

dφ(t) = ΓPPV[ω t + φ(t)] e(t). dt 0

However the oscillator sensitivity function ΓPPV(φ) is now calculated by the PPV theory using a longitudinal projection of the noise on the oscillator’s orbit which accounts for the dependence of the phase upon the transversal amplitude variation of the orbit:

PPV > Γ [ω0t] = ω0 v1(t) · J e(xS(t)),

59 where v1(t) is the Floquet vector associated with the unity Floquet exponential multiplier.

The Floquet vector v1(t) is obtained from the adjoint equation:

dv > − 1 = J (x ) v , dt S 1

> ∂f(x,0) where J(x) = ∂x is the Jacobian of the oscillator. The Floquet vector is submitted to

> dxS the bi-normality constrain v1 · dt = 1. Note that the correlation between the oscillator phase φ(t) and the amplitude process

δx⊥ (deviation of the state variables from the limit cycle) was explicitly shown in an alternate projection reported in Ref. [29]. The Floquet vector projection has the advantage of automatically accounting for the correlation of the phase noise to the amplitude noise. The

kur equivalent Γ0 obtained from the correlated analytical Kurokawa results in (4.14) is: ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ PPV¯ ¯ Kur¯ ¯ k1 ¯ |A| ¯ αB ¯ Γ0 ≡ Γ0 = ¯κ ¯ = ¯ ¯2 ¯1 + ¯ . (4.18) k3 ¯ ∂YIN,0 ¯ βA ¯ ∂ω ¯

The robustness of this equivalence will be investigated in the next sections for several oscillators.

In a recent report [36], an analytic expression for the autocorrelation function RV,1/f (∆ω) was also derived for the PPV method using Keshner’s 1/f model. The autocorrelation function RV,1/f (∆ω) at the fundamental frequency can be represented as: ½ · ¸ ¾ 3 £ ¤ R (τ)=exp −γ log 4t + τ 2 exp −γ log |τ|τ 2 V,1/f a 2 |ΓPPV|2S with γ = 0 (4.19) 2π

PPV 2 where using this paper’s notation |Γ0 | is the DC sensitivity coefﬁcient, S the 1/f noise strength, and ta the measurement starting time. Note that a more general form accounting for fractional 1/f δ noise at different harmonics and for multiple noise sources is given in

60 Ref. [36]. For simplicity of presentation, a single ﬂicker noise source is assumed in the equation presented. For all theories the extension to multiple sources is given by:

X |Γ |2S X |Γ |2S S (∆ω) = 0,i i , and γ = 0,i i . φ,1/f ∆ω3 2π i i

By numerically calculating the Fourier transform of (4.19), the voltage noise spectral density SV,1/f (∆ω) predicted by the PPV method can then be obtained.

In the Kurokawa model, Rφ,1trap(τ) can be obtained by taking an inverse Fourier transform of (4.12). Then by taking a summation over all traps the following analytic Rφ,1/f (∆ω) can be obtained for the uncorrelated case (k1 = k3): · Sκ2 e−λ0|τ| e−λ1|τ| R (τ) = − φ,1/f π 2λ2 2λ2 µ 0 1 ¶ 1 e−λ0|τ| 1 e−λ1|τ| + − − + |τ| λ 2λ λ 2λ 0 0 1 1 ¸ ¡ ¢ |τ|2 + Ei(−λ |τ|) − Ei(−λ |τ|) (4.20) 1 0 2 with Ei(x) the exponential integral. Substituting the analytic expression for Rφ,1/f (τ) in

(4.20) with Equ. 7 in Ref. [6] while optionally neglecting RδA,1/f (τ) yields a closed form expression for RV,1/f (τ):

1 £ ¤ R (τ) = A2 cos(ω τ) exp R (τ) − R (0) . (4.21) V,1/f 2 0 0 φ,1/f φ,1/f

Note that an analytic expression for the Rφ,1/f (τ) holding for the general case (k1 6= k3) can be found in [6]. The voltage noise density SV,1/f (∆ω) can then be obtained by calculating numerically the Fourier transform of RV,1/f (τ). Unlike the IEEE deﬁnition of the phase

1 noise L(∆ω) = Sφ,1/f (∆ω) = 2 Sφ,1/f,ssb(∆ω) [50] which diverges at vanishing ∆ω, the

1 2 old phase noise L(∆ω)(old) = SV,1/f,ssb(∆ω)/( 2 A0) deﬁned from the voltage spectral density (observable with a spectrum analyzer but not calculated by current commercial

61 harmonic balance simulators) is found to saturate at low offset frequencies. This saturation is a requirement of power conservation, since the integration of the voltage spectral density

1 2 SV,1/f,ssb(ω − ω0) over positive frequencies yields the oscillator output power 2 A0.

Using numerical calculations (for large enough time ta), we will verify for the four different oscillators studied that the autocorrelation function RV,1/f (∆ω) in (4.19) for the

PPV method yields a similar voltage spectral density SV,1/f (∆ω) to that obtained from the Kurokawa Rφ,1/f (∆ω) in (4.20) (see section 4.3 and 4.4 for the comparison plots).

Furthermore, no nonlinear dependence on the 1/f noise strength S, for the 1/∆ω3 offset frequency dependence of SV,1/f (∆ω) is observed at high offset frequencies (away from saturation) for the PPV method in agreement with the linear Kurokawa theory. Note that this close agreement in SV,1/f (∆ω) for the two theories is obtained despite the fact that the autocorrelation RV,1/f (τ) in (4.19) is derived using the Keshner model [51] for 1/f noise whereas a sum of physical Ornstein-Uhlenbeck (trap) processes is used for modeling the 1/f noise in the generalized Kurokawa derivation. The advantage of the generalized

Kurokawa treatment is that it provides a simple circuit-based expression (4.18) to calculate the ISF DC sensitivity coefﬁcient Γ0 which accounts for the correlation between amplitude and phase noise (AM to PM) via the factor k1/k3. The impact of the k1/k3 factor upon the

1/f phase noise will be analyzed in Sections 4.3 and 4.4.

4.2 Modiﬁed Van der Pol Oscillator

A modiﬁed Van der Pol oscillator circuit is shown in Figure 4.3. This circuit can be divided into a passive load circuit consisting of a parallel RLC circuit selecting the resonant frequency and four current sources: (1) a current source iIN1,NC generating the negative conductance (NC) sustaining the oscillations, (2) a current source i1/f,AM generating the

62 1/f amplitude noise, (3) a current source i1/f,P M generating the 1/f phase noise, and (4) a current source i1/f,corr which is responsible for the correlation between the amplitude and phase deviations. We shall now use these four current sources to calculate all the Kurokawa parameters appearing in the analytic expressions for 1/f amplitude and phase noise derived in the previous section. The LC resonator provides an efﬁcient but not perfect short for the

DC and harmonics voltage off resonance. To simplify the obtention of an exact analytic solution for this oscillator, the RF voltage harmonics and DC voltage across the tank are further shorted using an harmonic short identiﬁed in Figure 4.3. This is not a limitation of the theory and no such harmonic short will be used for the Colpitts, ring and transmission- line oscillators in Section 4.4. The frequency and time domain behavior of the harmonic is shown in Table 4.1 and Figure 4.4 respectively. As shown, all the RF voltage harmonics and DC voltage are shorted while the RF odd current harmonics are maintained.

+ v VN c _ i1/f,AM

iIN2 iIN1 + i1/f,AM i1/f,corr Harm. RL L CL Short v _ iIN1,NC i1/f,PM

CIN1 CIN2 + + 3 v V N v _ i1/f,PM _ i1/f,corr

Figure 4.3: Modiﬁed Van der Pol oscillator model including AM and PM 1/f noise and AM to PM nonlinear noise correlation.

63 Harmonics DC f0 2f0 3f0 4f0 5f0 Voltage (dBm) −∞ 5.2 −∞ −∞ −∞ −∞ Current (dBm) −∞ -26.8 −∞ -45.9 −∞ -294.6

Table 4.1: Harmonics of output voltage and current using an harmonic short. (The same circuit parameters as in Figure 4.6 are used in this result)

0.5

−0.5 Output voltage (V) −1 0 50 100 150 200 250 300 350 400 Time (ps) 0.02

0.01

−0.01 Output current (A) −0.02 0 50 100 150 200 250 300 350 400 Time (ps)

Figure 4.4: Output voltage (top) and current (bottom) waveform in time domain using an harmonic short. (The same circuit parameters as in Figure 4.6 are used in this result.)

4.2.1 Kurokawa Coefﬁcients Extraction

The nonlinear current iIN1(t) in a Van der Pol oscillator circuit shown in Figure 4.3 can be divided into a conduction current iIN1,cond(t) and a displacement current iIN1,displ(t) as follow:

iIN1(t) = iIN1,cond(t) + iIN1,displ(t) (4.22)

64 where h a i i (t) = −g [1 + c V ] v(t) − 1 v3(t) IN1,cond 0 N 3 d Q [v(t),V ] i (t) = IN N IN1,displ dt with

3 QIN [v(t),VN ] = CIN1 v(t) VN + CIN2 v (t) where g0 = 1/R is the conductance of the resistive load and the constant c represents a nonlinearity factor which controls the strength of the 1/f amplitude noise.

Using these deﬁnitions the following conduction current is obtained:

iIN1,cond(t) = iIN1,NC (t) + i1/f,AM (t) h a i i (t) = −g v(t) − 1 v3(t) (4.23) IN1,NC 0 3 h a i i (t) = −g c V v(t) − 1 v3(t) . (4.24) 1/f,AM 0 N 3 Equations (4.23) and (4.24) are the equations which were implemented in the simulators for the conduction currents. Since we are not interested in the harmonic of the voltage for

1/f noise, an harmonic short was placed across the tank to assist it in perfectly suppressing them. The oscillator voltage in (4.23) and (4.24) is therefore purely harmonic: v(t) =

A cos(ωt + φ). The conduction current at the fundamental frequency is then derived to be: · ¸ a A2 i (t)= −g A 1 − 1 cos(ωt + φ) [1 + c V ] IN1,cond 0 4 N + large harmonics. (4.25)

In addition, the displacement current can be rewritten as:

iIN1,displ(t) = i1/f,P M (t) + i1/f,corr(t) dv(t) i (t) = C V (4.26) 1/f,P M IN1 N dt dv3(t) i (t) = C . (4.27) 1/f,corr IN2 dt 65 Equations (4.26) and (4.27) are the equations which were actually implemented in the circuit simulators for the displacement currents. In the absence of noise the displacement current at the fundamental frequency is then derived to be:

iIN1,displ(t)= −CIN1VN A ω sin(ωt + φ) 3 − A3 C ω sin(ωt + φ) + large harmonics. 4 IN2 (4.28)

The nonlinear capacitance parameters CIN1 and CIN2 are the factors determining the direct 1/f phase noise (PM) strength and the correlation strength (AM to PM) respectively.

The total nonlinear current iIN1(t) in the oscillator is then obtained by summing both the conduction (4.25) and displacement (4.28) currents: µ ¶ a A2 i (t)= −g A 1 − 1 cos(ωt + φ) [1 + c V ] IN1 0 4 N 3 −C V A ω sin(ωt + φ) − A3C ω sin(ωt + φ) IN1 N 4 IN2 + large harmonics. (4.29)

The current ﬂowing in the nonlinear part of the oscillator can also be expressed as:

£ jωt ¤ iIN1(t) = Re A(t)e YIN,1 (A, ωi,VN ) + large harmonics. (4.30)

By equating (4.29) and (4.30) at the fundamental frequency, the nonlinear device admittance YIN,1(A, ω, VN ) can be calculated [5]. Adding YIN,2(A, ω, VN ) = 1/RL + j(ωCL −

1/(ωL) to the YIN,1(A, ω, VN ) obtained, the total nonlinear admittance YIN (A, ω, VN ) is then obtained to be: µ ¶ 1 1 Y (A, ω, V )= + j ωC − IN N R L ωL µ ¶ L· ¸ a A2 3 −g [1 + c V ] 1 − 1 + j C V ω + A2C ω . 0 N 4 IN1 N 4 IN2

66 From the nonlinear admittance YIN (A, ω, VN ) obtained, the Kurokawa coefﬁcients are then calculated to be: ¯ ∂Y ¯ a g A 3 IN ¯ = 1 0 0 + j A C ω (4.31) ∂A ¯ 2 2 0 IN2 0 ¯A0,ω0,VN0=0 µ ¶ ∂Y ¯ 3 IN ¯ = j 2 C + A2C (4.32) ∂ω ¯ L 4 0 IN2 ¯A0,ω0,VN0=0 µ ¶ ∂Y ¯ a A2 IN ¯ = g c 1 0 − 1 + jC ω . (4.33) ∂V ¯ 0 4 IN1 0 N A0,ω0,VN0=0

To simplify the model the DC bias voltage was set to zero VN0 = 0.

4.2.2 Amplitude and Frequency of Oscillation and Phase Noise

At the oscillation operating point, the following two conditions have to be satisﬁed:

GIN (A0, ω0,VN0) = 0 (4.34)

BIN (A0, ω0,VN0) = 0. (4.35)

Using (4.34), the expression for the amplitude of oscillation can be calculated: s 2 GL A0 = √ 1 − . (4.36) a1 g0

Also using (4.35), the expression for the frequency of oscillation can be obtained:

1 f0 = q ¡ ¢. (4.37) 3 2 2π L CL + 4 A0CIN2

Notice that the nonlinear capacitance CIN2 which introduces the amplitude to phase correlation, also induces a shift of the frequency of oscillation.

67 Substituting the Kurokawa parameters in (4.18), the phase-noise sensitivity factor is then: µ ¶ Kur CIN1ω0 3cCIN2GL Γ0 = − 1 + . (4.38) 2CT a1CIN1g0

4.2.3 PPV Analytic Solution

An frequency-domain analytic PPV solution can also be obtained for the modiﬁed Van der Pol oscillator. In the presence of the harmonic short the state equations for the oscillator are: · ¸ dv dv C + F 3C v2 L dt IN2 dt · ¸ dv = −F i (v, V ) + C V , IN1,cond N IN1 N dt di L L = v. (4.39) dt

In the above system F[·] indicates the ﬁltering action of the DC and harmonic short which only let the fundamental component of the device current iin2 reach the tank. This ideal harmonic short which facilitates the obtention of an analytic solution forces us to solve the state equations in the frequency domain. The solution is carried out in Appendix I by using a frequency domain analysis similar to that introduced in Ref. [35]. It results from this analysis that the oscillator state variables are: · ¸ · ¸ C v(t) −ω A C sin(ω t + φ(t)) x(t) = T = 0 0 T 0 , (4.40) Li(t) A0 cos(ω0t + φ(t))

3 2 where CT = CL + δ and δ = 4 CIN2A0. Using the methodology described in Appendix I, the band limited Floquet vector is then derived from the adjoint of the oscillator Jacobian

68 equation to be:

v (t) = "1 q # δ 1 L − cos(ω0t + φ(t)) − sin(ω0t + φ(t)) γ2CT A0 A0 CT , 1 δ cos(ω0t + φ(t)) − √ sin(ω0t + φ(t)) A0 γ2 LCT A0 (4.41)

2 A0 with γ2 = g0a1 4 . As required the band limited Floquet vector veriﬁes the bi-normality

> dx(t) PPV relation: v1 (t)· dt = 1. The phase noise sensitivity factor Γ0 is then given by the usual PPV equation: Z ω T ΓPPV = 0 v>(τ) · J (t)dτ 0 T 1 e 0 µ ¶ C ω 3cC G = − IN1 0 1 + IN2 L (4.42) 2CT a1CIN1g0 where the noise Jacobian vector J e(t) is: " h i h i # YIN,0 YIN,0 −Re dV A0 cos ω0t + Im dV A0 sin ω0t J e(t) = N N . 0

PPV The coefﬁcient Γ0 derived from the PPV theory is seen to be the same as that derived using the generalized Kurokawa theory in (4.38). Note that to our knowledge this is the

ﬁrst analytic solution obtained for the PPV noise theories in the presence of AM to PM noise conversion. The ISF approximation is recovered by setting CIN2 = 0, yielding the uncorrelated solution given in Ref. [27] and [35].

4.3 Model Comparison for 1/f noise in Van der Pol Oscillator

4.3.1 Circuit Implementation in Simulators

In order to compare the obtained Kurokawa and PPV analytic equations for Sa,1/f (∆ω) in (4.10) and Sφ,1/f (∆ω) in (4.13) with the conversion matrix method, the modiﬁed Van

69 der Pol oscillator shown in Figure 4.3 is implemented in two popular harmonic balance circuit simulators: Agilent ADS and AWR Microwave Ofﬁce (MWO). For example, the current source iIN1,NC in (4.23) was realized with a nonlinear voltage-controlled current source. Also, the current source i1/f,AM in (4.24) was implemented by multiplying the current iIN1,NC with the 1/f noise voltage source and the constant c using a voltage multiplier and an ideal voltage-controlled current source respectively. The current source i1/f,P M in (4.26) was developed by a voltage multiplier which multiplies the 1/f noise voltage source with the oscillator voltage v(t) in series with the nonlinear capacitance parameter CIN1 and an ideal current-controlled current source. The current source i1/f,corr in

(4.27) was implemented with a nonlinear voltage-controlled voltage source in series with a nonlinear capacitance parameter CIN2 and an ideal current-controlled current source.

Lastly, the harmonic short was developed with an equation-based 1 port S-parameter component. In ADS simulation, anmx (AM) and pnmx (PM) are used while amplitude noise spectral density (AM NOISE F, AM) and SSB noise to carrier ratio (L USB F, PM) are used in MWO simulation to calculate the 1/f amplitude and 1/f phase noise respectively.

Note that the obtained 1/f amplitude noise and phase noise from both simulators represent

Sa,1/f,ssb(∆f)/2 and Sφ,1/f,ssb(∆f)/2 respectively. However, only the simulation results for the ADS will be shown in the comparison plots in this paper since the two circuit simulators essentially gave the same results. The ADS relies on a small-signal noise mixing analysis to compute both AM and PM noise. A bias-dependent noise voltage source is used as a 1/f noise source in ADS whereas a ﬂicker noise current source with an 1 Ω resistor in shunt is used in MWO since no ﬂicker noise voltage source is supported in MWO.

70 4.3.2 Model Comparison for Uncorrelated Case

Figure 4.5 compares the obtained Kurokawa and PPV analytic model for the uncorrelated (k1 = k3) 1/f amplitude noise (AM) with the ADS simulation. For this comparison, both CIN1 and CIN2 which are responsible for the generation of the 1/f phase noise (PM) and the AM to PM noise conversion are set to 0 respectively. Three different amplitudes of oscillation are tested by varying the strength of the nonlinearity.

A =0.577V (analytic) 0 −100 A =0.577V (simulator) 0 A =0.966V (analytic) 0 −120 A =0.966V (simulator) 0 A =1.111V (analytic) −140 0 A =1.111V (simulator) 0 −160

−180 Amplitude noise (dBc/Hz) −200

−220

−2 0 2 4 6 10 10 10 10 10 Offset frequency (Hz)

Figure 4.5: The Kurokawa and PPV analytic model results for uncorrelated 1/f amplitude noise are compared to the ADS simulation results for three different amplitudes of oscilla- −14 2 tion. a1 = 3, c = 1, L=1 nH, CL=1 pF, S=1e V , R=1/g0=30 Ω, and RL=1/GL=40 Ω (red), 100 Ω (blue), and 400 Ω (black) are used respectively for the comparison.

As shown in Figure 4.5, an excellent agreement is observed for all cases tested in the entire offset frequency range. A clear 1/∆f slope is observed since the constant k3 is much larger

2 than ∆ω term in the analytic equation for Sa,1/f (∆ω) in (4.10). Furthermore, the predicted

71 amplitude and frequency of oscillation from (4.36), (4.37) are in excellent agreement with those obtained from the ADS.

To verify the Kurokawa analytic model for the uncorrelated 1/f phase noise, a 1/f phase noise generation factor CIN1 of 1 pF/V is used keeping CIN2 equal to 0. Also, three different load capacitances are used to investigate the resonator Quality (Q) factor dependence on the 1/f phase noise performance. As shown in Figure 4.6, the Kurokawa and PPV analytic model is in excellent agreement with the circuit simulation result. A clear 1/∆f 3 slope is observed due to the 1/f phase noise up-conversion process. Also a consistent 20 dB decrease in phase noise is observed as the Q factor increases by a factor of 10, which agrees with Leeson’s model [7]. The positive 1/f phase noise observed at the low offset frequency in Figure 4.6 originates from the fact that the phase of the oscillator drifts without bound as ∆ω goes 0. Note that the voltage spectral density SV,1/f (∆ω) which is directly observable on a spectrum analyzer has a Lorentzian spectra as ∆ω goes 0 to maintain the total power ﬁnite. The amplitude and the frequency of oscillation predicted in (4.36), (4.37) and the ADS are in excellent agreement.

Fig. 4.7 shows a phase noise comparison result for the three different 1/f noise strength cases between the Kurokawa and PPV analytic models and circuit simulator for comparing Sφ,1/f (∆f) and PPV numerical model and Kurokawa numerical model for comparing

SV,1/f (∆f). As shown, the Kurokawa analytic model Sφ,1/f (∆f) accurately predicts the phase noise calculated from the circuit simulator. Also both PPV and Kurokawa numerical model predict a similar ﬂattening of the phase noise spectrum for all cases of 1/f noise strength (S) tested.

100 Q=1.26 (analytic) Q=1.26 (simulator) S (∆ f) Q=12.6 (analytic) φ, 1/f Q=12.6 (simulator) 50 Q=126 (analytic) Q=126 (simulator) 0

−50 Phase noise (dBc/Hz) −100

−150

−2 0 2 4 6 10 10 10 10 10 Offset frequency (Hz)

Figure 4.6: The Kurokawa and PPV analytic model results for uncorrelated 1/f phase noise are compared to the ADS simulation results for three different resonator Q factors. −14 2 a1 = 3, c = 1, R=1/g0=30 Ω, and RL=1/GL=40 Ω, S=1e V , CIN1 =1 pF/V, CL=1 pF (Q=1.26), 10 pF (Q=12.6), and 100 pF (Q=126) are used respectively for the comparison while maintaining the same frequency of oscillation.

4.3.3 Model Comparison for Correlated Case

The analytic 1/f phase noise model is now compared with the harmonic balance circuit simulators for the correlated case (k1 6= k3). The correlation is controlled by the nonlinear capacitor parameter CIN2. Note that CIN2 will affect the Kurokawa parameters deﬁned in (4.31) and (4.32) as well as the frequency of oscillation in (4.37). Figure 4.8 presents a result for the correlated case obtained from the Kurokawa analytic model and the ADS simulation. Since the up-converted 1/f phase noise can be divided into both direct 1/f

PM noise up-conversion and AM to PM noise up-conversion, these results are also plotted and compared separately with those obtained from the ADS simulation. The direct PM noise is obtained by setting the nonlinearity factor for 1/f amplitude noise c equal to 0.

73 −20 S (∆ f) φ, 1/f −40

S=1e−16 V2 −60 S (∆ f) V, 1/f

−80

S=1e−18 V2 −100 Phase noise (dBc/Hz)

Kurokawa & PPV analytic −120 ADS simulation PPV method numerical 2 Kurokawa numerical S=1e−20 V −140 2 3 4 5 6 10 10 10 10 10 Offset Frequency (Hz)

Figure 4.7: Phase noise comparison result for the three different 1/f noise strength (S). a1 = 3, c = 1, R=1/g0=30 Ω, and RL=1/GL=40 Ω, CIN1 =1 nF/V, and CL=1 pF are used respectively for this comparison.

Also by deactivating the capacitance CIN1 responsible for the 1/f phase noise generation, the AM to PM 1/f noise is obtained. The x-axis represents the strength of the normalized correlation factor 10 log[αB/(βA)] from (4.12). As shown in Figure 4.8, the Kurokawa analytic 1/f phase noise model (black solid line) exhibits a quasi-perfect agreement with the simulator (black circle) in the full correlation region.

Moreover, four different 1/f phase noise strengths (CIN1= 10, 1, 0.1 and 0.01 pF/V) are compared at 1 Hz offset frequency in Figure 4.9 and 100 KHz offset frequency in

Figure 4.10 where the nonlinear capacitance parameter CIN2 is swept to vary the correlation strength. In the x-axis of Figure 4.9, CIN2 is normalized using CIN2/CIN2,−4dB where

CIN,−4dB is deﬁned from the shift of the frequency of oscillation f0 in (4.37) such that we

74 10

0 PM (C =0.1 pF/V) −10 IN1

−20

−30

−40

−50 Analytic PM Phase noise (dBc/Hz) Simulator PM −60 Analytic direct PM ∆ f = 1 Hz Simulator direct PM −70 Analytic AM to PM Simulator AM to PM −80 −30 −20 −10 0 10 Normalized correlation strength (log scale)

Figure 4.8: Total phase noise and its components at 1 Hz offset frequency versus correlation strength 10 log[αB/(βA)] for the correlated case (k1 6= k3). For this comparison, −18 2 R=1/g0=30 Ω, RL=1/GL=40 Ω, S=1e V , L=1 nH, CL=1 pF, and CIN1=0.1 pF/V are used with while sweeping CIN2 to variate the correlation strength.

have: 3 2 A2C = C . 4 0 IN2,−4dB 5 L

To get a measure of the relative high correlation strength involved in Figure 4.9, let us note that the frequency of oscillation in (4.37) is found to vary from 5.03 GHz to 2.69 GHz when

CIN2 is swept from its minimum to its maximum value. Overall, a quasi-perfect agreement is observed in both Figure 4.9 and Figure 4.10 between the Kurokawa and PPV analytic models (black solid line) and the ADS simulation (black circle) in all cases. Moreover, regardless of the correlation strength, the predicted frequency and amplitude of oscillation in the correlated case, exhibit, an excellent agreement with those calculated from the ADS simulator.

75 40 PM (C =10 pF/V) IN1 20 PM (C =1 pF/V) IN1 0 PM (C =0.1 pF/V) IN1 −20 PM (C =0.01 pF/V) IN1 −40

Analytic PM

Phase noise (dBc/Hz) −60 Simulator PM Analytic direct PM −80 Simulator direct PM ∆ f = 1 Hz Analytic AM to PM −100 Simulator AM to PM

−30 −20 −10 0 10 20 Normalized correlation strength [10log (C /C )] 10 IN2 IN2,−4dB

Figure 4.9: Comparison result of the phase noise at 1 Hz offset frequency, for four different 1/f phase noise strengths CIN1 for the correlated case. The same circuit parameters as in Figure 4.8 are used in this comparison.

4.4 Model Comparison for Other Oscillator Circuits

4.4.1 BJT Colpitts Oscillator

Three additional oscillator circuits are evaluated for a comparison of phase noise models. The first oscillator circuit tested is an 1 GHz BJT-based Colpitts oscillator. The schematic of the oscillator is depicted in Fig. 4.11. The noise source consists in a 1/f noise current source placed in shunt between the base and emitter. A commercial harmonic- balance circuit simulator is used to obtain the Kurokawa coefficients (4.31)–(4.33). By sweeping either A, ω, or VN about the free-running operating point while keeping the two other parameters constant, ∂YIN (A0,ω0,VN0) , ∂YIN (A0,ω0,VN0) , and ∂YIN (A0,ω0,VN0) can be cal- ∂A ∂ω ∂VN culated. Note that a least square fit is used to remove any residual numerical noise. For the

PM (C =10 pF/V) −120 IN1

PM (C =1 pF/V) −140 IN1

PM (C =0.1 pF/V) −160 IN1

−180

PM (C =0.01 pF/V) −200 IN1 Analytic PM Phase noise (dBc/Hz) Simulator PM −220 Analytic direct PM ∆ f = 100 KHz Simulator direct PM −240 Analytic AM to PM Simulator AM to PM

−15 −10 −5 0 5 10 15 20 Normalized correlation strength [10log (C /C )] 10 IN2 IN2,−4dB

Figure 4.10: Comparison result of the phase noise at 100 KHz offset frequency, for four different 1/f phase noise strengths CIN1 for the correlated case. The same circuit parameters as in Figure 4.8 are used in this comparison.

calculation of ∂YIN (A0,ω0,VN0) the 1/f current source is temporarily replaced by a small DC ∂VN current source.

The comparison of the results for the different phase noise models is presented in Fig.

4.12. Since no access to the Jacobian is provided in the harmonic balance simulation, a direct PPV result is not used in the comparison. As shown in Fig. 4.12, the 1/f phase noise spectral density of the circuit simulator are in good agreement with the Kurokawa correlation model. Also the numerical method in [36] and the Kurokawa numerical correlation model predict a similar ﬂattening for the voltage spectral density SV,1/f (∆ω) at low offset

PPV frequencies. Note that the PPV sensitivity coefﬁcient Γ0 in (4.19) was extracted from the ADS phase noise at high offset frequencies following the methodology given Ref. [36].

When neglecting the amplitude-phase correlation effect, a phase noise difference of around

77 3 KΩ 6.8 KΩ

+ 5 V − RFC

Γ Γ IN2 IN1

3.3 pF

0.01 fF 10 nH 3.9 pF 510 Ω

Figure 4.11: Schematic of the 1 GHz BJT-based Colpitts oscillator.

3 dB is observed between the Kurokawa uncorrelated results and the simulator results. The correlation factor k1/k3 in (4.14) obtained is around 2.2 yielding around 3.2 dB phase noise difference at large offset frequencies in 1/∆ω3 region between Kurokawa correlated and uncorrelated results. These simulation results demonstrate that the correlation can play an important role for the accurate prediction of the 1/f phase noise.

4.4.2 CMOS Ring Oscillator

The original Kurokawa theory was limited to high-Q oscillators, whereas the new generalized Kurokawa can now be applied to low-Q circuits such as the CMOS ring oscillator shown in Fig. 4.13. In this circuit a 1/f noise current is injected in shunt between the inverter output and the ground. The three derivatives of YIN needed by the noise model are extracted using the circuit topology shown in Fig. 4.2. In order to achieve a high level of accuracy for calculating derivatives, 31 harmonics with 14 digits of accuracy are considered in harmonic balance simulation. To further verify the quality of the derivatives extracted,

78 −10 S (∆ f) φ, 1/f −20

−30 S (∆ f) V, 1/f −40

−50

−60

−70 ADS simulation Phase noise (dBc/Hz) ISF method −80 PPV method numerical Kurokawa analytic (correlated) −90 Kurokawa analytic (uncorrelated) Kurokawa numerical (correlated) −100 0 1 2 3 4 10 10 10 10 10 Offset Frequency (Hz)

Figure 4.12: Phase noise model comparison of a BJT-based Colpitts oscillator.

the amplitude sensitivity factor δA/δIN and the frequency sensitivity factor δω/δIN calculated using (4.1) (with VN replaced by IN ) are compared to those directly obtained from harmonic balance simulations for various steady-state DC perturbations δIN . The amplitude and frequency sensitivities obtained from harmonic balance simulations are veriﬁed in Fig. 4.14 to reasonably converge for small enough δIN perturbations to the sensitivities calculated using (4.1). A δIN of 0.1 µA is used for the noise analysis.

To make a meaningful comparison of the phase noise, the NMOS gate width (Wn) is variated from 30 µm to 100 µm with a step of 10 µm while maintaining other device parameters as indicated in Table 4.2. The resulting voltage and current waveforms for the ring oscillator with gate width of 30 µm are shown in Fig. 4.15. Both the voltage and current waveforms are veriﬁed to be non sinusoidal. The phase noise predicted by the generalized Kurokawa theory with (red circle) and without (blue triangle) AM to PM

79 Γ Γ IN1 IN2

C C C

3 V

PMOS

NMOS IN

Figure 4.13: Topology of a three stage ring oscillator implemented with CMOS inverters. The detailed circuit parameters used are in Table 4.2.

correlation are compared in Fig. 4.16 versus NMOS gate width to the results obtained using the conversion matrix method in harmonic balance simulations. Since the phase noise predicted by the ADS simulator showed a slight variation with respect to the direction of the

OSCPORT component used (left or right), the average phase noise (solid line) between both results is also plotted in Fig. 4.16. Note that the ﬁlled color symbol represents the phase noise result obtained with placing the OSCPORT towards the input of the inverter (→) while the unﬁlled color symbol shows that obtained with the reverse (←). The phase noise results for both directions of the OSCPORT ADS simulation are also indicated in Table 4.2 for three selected gate width. As shown in Fig. 4.16, a reasonably good agreement (within

0.5 dB) in phase noise is observed for the correlated case compared to the ADS simulation

(the uncorrelated results are about 2 dB off). In Table 4.2 , it is observed that the improved phase noise at larger gate widths is associated with reduced a amplitude sensitivity factor

δA/δIN and reduced frequency sensitivity factor δω/δIN . These results demonstrate for

80 8.6 δ A / δ I (ADS) N δ δ

(V/A) A / I (calculated) N N

I 8.5 δ A / δ 8.4 −8 −7 −6 −5 −4 10 10 10 10 10 δ I (A) N

−0.2 δ f / δ I (ADS)

(1e19 Hz/A) N N

I δ f / δ I (calculated) δ −0.4 N

f / δ −8 −7 −6 −5 −4 10 10 10 10 10 δ I (A) N

Figure 4.14: Comparison of the calculated (top) amplitude and (bottom) frequency sensitivities (dashed lines) with those obtained from harmonic balance simulations for the ring oscillator with gate width of 30 µm.

the ﬁrst time, the capability of the generalized Kurokawa theory for predicting the phase noise in low-Q oscillator circuits.

4.4.3 pHEMT Oscillator

The fourth oscillator circuit tested is a 2.5 GHz Avago ATF54143 pHEMT negative- resistance oscillator fabricated on RT/Duroid 5880 substrate (εr = 2.2 and h = 45 mil). For the DC biasing, VDS of 1.2 V and VGS of 0.6 V are used yielding a drain current IDS of 20 mA. A simpliﬁed schematic of the pHEMT oscillator is described in Fig. 4.17. The phase noise spectrum was consecutively measured 8 times and the average value is shown in Fig.

4.18 (green solid curve).

Following the methodology in [36], the Kurokawa model coefﬁcient Sκ2 in (4.15) was extract from the measured spectrum at high offset frequencies. Applying the obtained

81 3

1 Outout voltage (V) 0 0 100 200 300 400 500 600 700 800 Time (pS)

0.1

0.05

−0.05 Outout current (mA) 0 100 200 300 400 500 600 700 800 Time (pS)

Figure 4.15: Voltage (top) and current (bottom) waveforms obtained from harmonic balance simulations for the ring oscillator with gate width of 30 µm.

Kurokawa coefﬁcient to the expression for Rφ,1.f (τ) in (4.20), the phase noise spectrum

SV,1/f (∆ω) can be calculated using numerical calculations. As shown in Fig. 4.18, the phase noise spectrum predicted by the Kurokawa numerical model (black circle) and the

PPV method in [36] (blue solid line) are in close agreement and predict fairly well the measured experimental phase noise spectrum (green solid line).

4.5 Conclusion

In this paper we have veriﬁed using alternatively analytic, simulation and experimental results in four different oscillator circuits, that the generalized Kurokawa theory yields similar results to the conversion matrix in harmonic balance simulations and the PPV and

ISF approaches for both the uncorrelated and correlated (AM to PM noise conversion) cases.

−4

−6

−8

−10

−12 ADS Oscport avg f = 1 KHz (dBc/Hz) ∆ −14 Kuro. Corr. avg Kuro. Uncorr. avg −16 ADS Oscport → Kuro. Corr. → −18 Kuro. Uncorr. →

Phase noise at ← −20 ADS Oscport Kuro. Corr. ← −22 Kuro. Uncorr. ←

30 40 50 60 70 80 90 100 NMOS gate width W (µm) n

Figure 4.16: Comparison of phase noise versus NMOS gate width of a ring oscillator shown in Fig. 4.13 at 1 KHz offset frequency between ADS simulation and Kurokawa uncorrelated and correlated analytic model.

In particular the generalized Kurokawa theory for 1/f noise was found to provide a circuit base methodology for the calculation of the coefﬁcient Γ0 (and other Γk) which is responsible in the ISF theory for the up-conversion of 1/f noise to 1/f 3 noise at RF. The generalized Kurokawa model has further the advantage to account for AM to PM noise conversion which was demonstrated to provide additional correction to the 1/f 3 phase noise.

To rigorously analyze the impact of the AM to PM noise conversion upon the phase

(PM) noise in a simple example, a modiﬁed Van der Pol oscillator featuring two new nonlinear capacitances was introduced. A fully analytic solution valid for both the generalized

Kurokawa and PPV analysis in the presence of an harmonic short across the tank, was then

83 CASE 1 CASE 2 CASE 3 Gate width Wn (µm) 30 70 100 A0 (V) 1.445 1.553 1.573 f0 (GHz) 2.572 4.145 4.936 |δω/δIN | deviation 0.02 % 0.004 % 0.015 % |δA/δIN | deviation 0.015 % 0.005 % 0.001 % |δω/δIN | 7.988e9 3.135e9 1.163e9 |δA/δIN | 8.437 2.087 0.566 2 Qeﬀ (|∂YIN0/∂ω| ) 3.516e-23 4.188e-23 4.463e-23 Angle [A in Eq. (4.3)] 128.25 ◦ 131.28 ◦ 135.50 ◦ Digits of accuracy 14 14 14 L (1 KHz) OSCPORT direction (← / →) dBc/Hz ADS simulation -5.30/-5.77 -13.74/-14.01 -22.08/-22.04 Kurokawa correlated -5.33/-4.96 -13.37/-13.08 -21.63/-21.70 Kurokawa uncorrelated -4.11/-3.90 -12.07/-11.83 -20.49/-20.56

Table 4.2: Comparison of the phase noise and sensitivity parameters varying NMOS gate width between ADS and Kurokawa method. Common parameters used: C=2 pF, λ =0.001, λ =0.02, Wp = 250µm , and R =R =0. N P Lp 0.24µm d s

derived for the modified Van der Pol oscillator. It was then verified for both the uncorrelated and correlated (AM to PM noise conversion) cases that this analytic solution yielded the same numerical results as harmonic balance based simulations using the conversion method. To our knowledge this is the first analytic solution for both Kurokawa and PPV obtained in the presence of AM to PM noise conversion.

Finally the model comparison was extended to other oscillator conﬁgurations (Colpitts, ring, and transmission line oscillators), and a good agreement with simulation and experimental results was demonstrated when AM to PM noise conversion was accounted for.

The generalized Kurokawa theory and the PPV theory were also found to predict similar

1 2 voltage spectral noise densities SV,1/f,ssb(∆ω)/( 2 A0) (old IEEE phase noise) at both low and large offset-frequency ∆ω for large enough measurement times.

84 Series feedback

Termination network Load network

50 Ω Tunable capacitor Series feedback

Figure 4.17: Schematic of the 2.5 GHz pHEMT-based negative-resistance oscillator.

Both the generalized Kurokawa and PPV theories were also found to match measured experimental data of the voltage spectral noise density SV,1/f (∆ω) while using the phase noise strength extracted at high offset frequencies. Note that this extraction approach [36] automatically accounts for amplitude-phase correlation (AM to PM noise conversion).

While numerical phase noise computations in harmonic balance simulators using the conversion matrix method [24], [25] or PPV method [52] remains presently the most con- venient approach for RF circuit designers to numerically calculate the phase noise, analytic models can provide greater insights in the noise processes and facilitate the investigation on new noise effects such as cyclostationary effects [26]. The generalized Kurokawa theory being a circuit oriented theory, its model parameters are admittance parameters which can be derived analytically or extracted from circuit simulations or experimental measurements as was demonstrated for the Van der Pol, Colpitts, ring and transmission line oscillators.

85 −20

−40

−60

−80

−100 Phase noise (dBc/Hz)

Measured spectrum −120 Kurokawa analytic (uncorrelated) PPV method numerical Kurokawa numerical (uncorrelated) −140 1 2 3 4 5 6 10 10 10 10 10 10 Offset Frequency (Hz)

Figure 4.18: Phase noise model comparison of a pHEMT oscillator.

The model can also be readily generalized to multiple 1/f noise sources and fractional

1/f δ noise sources.

The very good agreement obtained in the four oscillators studied between the generalized Kurokawa analysis and the well established conversion matrix, ISF and PPV noise analyses attest to the robustness of the Kurokawa describing function methodology for modeling the phase noise dynamics. The correspondences established between the

Kurokawa analysis and the ISF and PPV theories should provide circuit designers with greater insights in the noise processes taking place in oscillators and the circuit-oriented expressions derived for the phase noise sensitivity functions should facilitate phase noise optimization.

86 CHAPTER 5

1/F ADDITIVE PHASE NOISE ANALYSIS FOR INJECTION-LOCKED OSCILLATORS

[41].

In this chapter, the source of the additive noise measurement system is used to syn- chronize the free-running oscillator operating at 2.4828 GHz and 2.485 GHz respectively while measuring the additive phase noise of the injection-locked oscillator. An analytic expression for the 1/f additive phase noise of the injection-locked oscillator is derived using

87 Kurokawa approach. A reasonable agreement of both the phase noise level and the corner frequency is obtained between the measured 1/f additive phase noise spectrum and the obtained analytic solution for both oscillators, which conﬁrms the validity of the model. To the knowledge of the authors, this is the ﬁrst report to analyze the 1/f additive phase noise of an injection-locked oscillator.

5.1 1/f Additive Phase Noise Model for Injection-Locked Oscillator

iL iIN

v(t) ω YL YIN IS cos S t

IN,(1/f)

Figure 5.1: Admittance model for one port injection-locked oscillator .

An admittance model for an injection-locked oscillator is shown in Fig. 5.1. This model can be divided into both frequency and amplitude sensitive nonlinear active part

YIN (A, ω), a frequency sensitive linear passive part YL(ω), an 1/f noise current source, and an injection locking current source with amplitude |IS| and frequency ωS. Note that

ωS is selected to be close to the self oscillation frequency ω0 of the free running oscillator.

The voltage across the tank in Fig. 5.1 can be written as: £ ¤ v(t) = Re A(t) ej(ωt+φ(t)) + small harmonics

with A(t) = AS + δA(t) dφ(t) 1 dA(t) ω = ω + δω(t) = ω + − j . S S dt A(t) dt

88 Performing a ﬁrst-order Taylor series expansion of the total admittance (YT = YIN + YL) about the steady state injection-locked condition AS, ωS, and IN0, we obtain:

|IS| cos ωSt © ª j(ωS t+φ(t)) = Re AS(t)e [YL(ωi) + YIN (AS, ωi,IN )]

+ large harmonics © £ j(ωS t+φ(t)) 0 = Re AS(t)e YL,S + YIN,S + YT,S δω

0 0 ¤ª + YIN,S δA + YIN,I,S δIN + large harmonics (5.1)

where deﬁning YIN,S = YIN (AS, ωS,IN0) , YL,S = YL(ωS) and the 0 symbol stands for the derivative. The above equation (5.1) can be divided into real and imaginary parts given by:

0 0 0 dδφ(t) BT,S dAS(t) GIN,SδA(t) + GT,S + dt AS dt

0 |IS| +GIN,I,SδIN (t) = cos(φ(t)) (5.2) AS 0 0 0 dδφ(t) GT,S dAS(t) BIN,SδA(t) + BT,S − dt AS dt

0 |IS| +BIN,I,SδIN (t) = − sin(φ(t)). (5.3) AS

In frequency domain, the above equations can be rewritten as:

0 |IS| 0 GIN,S δA(Ω) + sin(φS) δφ(Ω) + jΩ GT,S δφ(Ω) AS 0 BT,S 0 +jΩ δA(Ω) = −GIN,I,S δIN (Ω) (5.4) AS

0 |IS| 0 BIN,S δA(Ω) + cos(φS) δφ(Ω) + jΩ BT,S δφ(Ω) AS 0 GT,S 0 −jΩ δA(Ω) = −BIN,I,S δIN (Ω). (5.5) AS

The above two equations can be rewritten in a matrix format:

89 · ¸ · ¸ 0 δA(Ω) −GIN,I,SδIN (Ω) D = 0 (5.6) δφ(Ω) −BIN,I,SδIN (Ω) where " B0 # 0 T,S |IS | 0 G + jΩ sin(φS) + jΩG IN,S AS AS T,S D = G0 . 0 T,S |IS | 0 B − jΩ cos(φS) + jΩB IN,S AS AS T,S By taking the inverse we obtain,

· ¸ · ¸ 0 δA(Ω) −1 −GIN,I,S δIN (Ω) = D 0 . (5.7) δφ(Ω) −BIN,I,S δIN (Ω) By focusing on phase noise only, an analytic expression for the 1/f additive phase noise of the injection-locked oscillator can be obtained:

¯ ¯ 2 2 2 2 ¯ ¯2 S [ASR + A δΩ ] δφ(Ω)1/f = 2 2 2 2 2 0 2 2 2 2 4 0 4 δΩ[|IS| N + δΩ {ASβ − 2|IS|N|YT | + |IS| P /AS + 2β|IS|P } + δΩ |YT | ] (5.8)

where

0 0 M = sin(φS) BIN,I,S − cos(φS) GIN,I,S

0 0 N = cos(φS) GIN,S − sin(φS) BIN,S

0 0 P = cos(φS) BT,S + sin(φS) GT,S

0 0 0 0 R = BIN,S GIN,I,S − GIN,S BIN,I,S

0 0 0 0 A = GT,S GIN,I,S + BT,S BIN,I,S

0 0 0 0 B = GT,S BIN,I,S − BT,S GIN,I,S

S = ﬂicker noise spectral density (A2).

90 5.2 Measurement System Description

Large Signal Network Analyzer

2 a2 Port 1 Port 2 b2

Load Y L Spectrum 30 dB Ref. Analyzer YIN N − C RF Oscillator LNA ADC Splitter Mixer RF=18 dBm LO

Phase shifter

Figure 5.2: Additive phase noise measurement system integrated with an LSNA for an injection-locked oscillator. The N-C stands for the negative-conductance circuit.

Fig. 5.2 shows an additive phase noise measurement system integrated with an LSNA for an injection-locked oscillator [44]. The additive phase noise testbed consists of a RF signal source, a low-noise power splitter, a 30 dB attenuator, a phase shifter, a low-noise mixer and a phase noise analyzer. The advantage of this noise measurement system is that it ideally cancels the noise of the locking RF signal source in the system, which enables us to measure the intrinsic noise of the injection-locked oscillator. An LSNA is used to provide an efficient way to measure the amplitude and the phase of the incident and the reflected waves respectively at port 2 as well as the phase (φS) of the injection-locked oscillator voltage at the fundamental frequency relative to the voltage wave. Also a circulator is used for the signal injection. An external RF signal source of 18 dBm is used to provide a sufficient power level to the LO port of the mixer. Also a 30 dB attenuator is used to set

91 the locking power level to the oscillator. A phase shifter is used to maintain a quadrature condition between RF and LO port of the mixer for measuring the additive phase noise of the injection-locked oscillator. The oscillator circuit tested is an Avago ATF54143 pHEMT negative-conductance oscillator fabricated on RF/Duroid 5880 substrate (²r = 2.2 and h =

45 mil). Note that the oscillator circuit can be divided into two separate circuit boards com- posed of the negative-conductance active circuit and the passive load circuit respectively by experimentally obtaining the required Kurokawa parameters used in (5.8). In this work, two different load circuits which provide different amplitude and frequency of oscillation are used respectively with different DC biasing to test the theoretical agreement with the measurements in two different operating conditions. Since the LSNA measures both the incident and the reﬂected waves at port 2 in Fig. 5.2, the scattering parameters of the load circuit is also accounted for shifting the measurement reference place from port 2 to the port Ref. (between the load circuit and the negative-conductance circuit) as shown in Fig.

5.2. For the DC biasing, VGS of 0.55 V and VDS of 2.00 V yielding a drain current IDS of

24 mA are used for the ﬁrst oscillator while VGS of 0.65 V, VDS of 1.18 V yielding a drain current IDS of 21 mA are used for the second oscillator respectively. A spectrum analyzer

(Agilent E4405B with phase noise option) is also used to measure the phase noise of the free-running oscillator when the locking RF signal source is turned off.

5.3 Experimental Results

The comparison of the phase noise result of the oscillator with the load circuit 1 and

2 is shown in Fig. 5.3 and Fig. 5.4 respectively. The additive phase noise spectrum of the injection-locked oscillator was consecutively measured 7 times with the measurement system described in Fig. 5.2 and the average value is plotted (red solid curve). Also the

92 experimental phase noise spectrum of the free-running oscillator is depicted as a green solid line. Note that the injection-locked additive phase noise has a slope of 1/f 0.90 and

1/f 0.68 respectively due to the rejection of the 1/f noise up-conversion by the locking mechanism while the phase noise of the free-running oscillator has a slope of 1/f 2.90 and

1/f 2.68 respectively due to the 1/f noise up-conversion.

In order to predict the 1/f additive phase noise spectrum of the injection-locked oscillator (red solid line) in Fig. 5.3 and Fig. 5.4 using (5.8), a 5 parameters ﬁtting approach is used. This ﬁtting approach is needed since the origin of the 1/f noise source is unknown

0 0 and ﬁtting is for obtaining the three 1/f noise parameters (GIN,I,S, BIN,I,S, and S) in the ﬁ-

0 0 nal 1/f phase noise expression in (5.8). The other four required parameters, GIN,S, BIN,S,

0 0 GT,S, and BT,S in (5.8), are obtained experimentally at the operating point (AS, ωS). Two

0 0 parameters GIN,S and BIN,S are extracted by measuring an admittance of the active device for several power levels near the operating point (AS) at the injection-locked frequency

0 0 (fS). Also the frequency dependent parameters GT,S and BT,S are obtained by adding

0 0 0 0 0 0 GIN,S and BIN,S with GL,S and BL,S respectively. GIN,S and BIN,S are extracted using the frequency modulation (FM) approach at the power level of AS. The network analyzer is used to measure the admittance of the load circuit near the injection-locked frequency of

0 0 0 0 fS for extracting GL,S and BL,S. The experimentally obtained GT,S and BT,S parameters are further optimized for ﬁtting 1/f additive phase noise curve (red solid line) with three

0 0 1/f noise parameters (GIN,I,S, BIN,I,S, and S) to accurately predict the corner frequency. Note that only a 0.1 dB difference in the 1/f additive phase noise level is observed for both oscillator cases only using just the three 1/f noise parameters for ﬁtting.

The black dashed line in Fig. 5.3 and Fig. 5.4 shows the ﬁnal ﬁtting result, which accurately predicts both the 1/f additive phase noise spectrum and a corner frequency

93 (black dot). To verify the validity of the optimized parameters, the phase noise slope (≈

1/f 3) of the free-running oscillator is also modeled with the same parameters using the

Kurokawa linearization approach [6]. As shown in Fig. 5.3 and Fig. 5.4, the modeled

≈ 1/f 3 curve (blue dashed dotted line) predicts fairly well the experimental phase noise spectrum of the free-running oscillator.

−20

Free−running −40

Injection−locked −60 1/f 0.90 −80 1/f 2.90 −100

Noise floor −120 Phase noise (dBC/Hz)

−140 Injection−locked additive PM Modeled 1/f additive PM −160 3 Modeled 1/f PM Corner frequency

−1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 Offset frequency (Hz)

Figure 5.3: Phase noise comparison of a pHEMT oscillator with a load circuit 1 operating at 2.4828 GHz.

5.4 Conclusion

In this chapter, the 1/f additive phase noise analysis for the injection-locked oscillator has been presented. The Kurokawa analysis has been used to obtain an analytic expression for the 1/f additive phase noise of the injection-locked oscillator. The additive

94 −30

−40 Free−running −50

−60 Injection−locked −70 1/f 0.68 −80 1/f 2.68 −90

−100

Phase noise (dBC/Hz) −110

−120 Injection−locked additive PM Corner frequency −130 Modeled 1/f additive PM Modeled 1/f3 PM −140 −1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 Offset frequency (Hz)

Figure 5.4: Phase noise comparison of a pHEMT oscillator with a load circuit 2 operating at 2.485 GHz.

phase noise measurement system integrated with an LSNA has been used to efﬁciently obtain required model parameters for predicting 1/f additive phase noise. The Kurokawa derivatives needed for the analytic expression has been experimentally obtained and further optimized to accurately predict the corner frequency. The obtained analytic solution has been veriﬁed to closely predict the experimental 1/f additive phase noise result of the pHEMT micro-strip line oscillators at 2.4828 GHz and 2.485 GHz respectively. The presented 1/f additive phase noise analytic model with the experimental agreement for the injection-locked oscillator should provide circuit designers with great insights in the noise mechanism taking place in injection-locked oscillators and should facilitate phase noise optimization.

95 CHAPTER 6

ADDITIVE PHASE NOISE MEASUREMENTS OF ALGAN/GAN HEMTS USING A LARGE SIGNAL NETWORK ANALYZER AND A TUNABLE MONOCHROMATIC LIGHT SOURCE

Additive phase noise measurements are useful for accurately characterizing the phase noise contributed by a device under test (DUT) in a system. An additive phase noise measurement system differs from a conventional absolute phase noise measurement system in that it ideally cancels the noise of the external RF source in the system to achieve a lower noise ﬂoor [45]. This system has been widely used to analyze and model the noise characteristics of various devices and microwave components such as GaAs heterojunction bipolar transistor (HBT), AlGaN/GaN HEMT, Silicon (Si) bipolar junction transistor (BJT), GaAs

ﬁeld effect transistor (FET), frequency dividers and multipliers [42], [43], [45], [47].

In this chapter, an additive phase noise measurement system integrated with (1) an

LSNA and (2) a tunable monochromatic light source is presented to investigate the noise characteristic of both unpassivated and passivated AlGaN/GaN HEMTs under various operating conditions. A physical analysis on the origin of 1/f noise in AlGaN/GaN HEMTs is also presented in support of the various measurement results.

96 6.1 Measurement Set-up Description

6.1.1 Additive Phase Noise Measurement System

An additive phase noise measurement system integrated with an LSNA and a tunable monochromatic light source is presented in Fig. 6.1. The additive phase noise system consists of a power divider, two attenuators, a phase shifter, a low-noise mixer and a phase noise analyzer. A signal source providing 16.5 dBm at 2 GHz is used to power the additive phase noise testbed. A constant power of 12 dBm is provided to the LO port of the mixer.

Two step attenuators are used to control the input and output power levels at the DUT and adjust the output power level to the mixer. The phase shifter is also used to control the relative phase of the signals in the two branches in quadrature. By maintaining a quadrature condition, the mixer can be used as a phase detector to measure the additive phase noise of the DUT [53].

6.1.2 LSNA and Tunable Monochromatic Light Source

An LSNA is used to provide an efﬁcient way to measure the input and output power levels of the DUT while monitoring the device load-line. Port 1 and port 2 are used for the gate and drain of the DUT respectively. For this study, on-wafer unpassivated and passivated AlGaN/GaN HEMTs on SiC substrate are used as a DUT. Using two external bias tees and DC power supplies, two different operating points, VGS of -3 V and VDS of 10

V and 15 V, were selected for the DC biasing, yielding a drain current IDS of 44 mA. Using a source and a load tuner, both input and output voltage swings are controlled to further investigate the noise dependence on the drain voltage swing.

97 Large Signal Network Analyzer

Port 1 Tuner DUT Tuner Port 2

Shutter Current Current Sensor Oscilloscope Sensor

Bias DC Bias Bias Tee Supply Tee

L L2 1 LNA ADC

RF=16.5 dBm 2 GHz Phase Shifter

Deep Level Monochromator Optical Spectrometer Lamp

Figure 6.1: Block diagram of the additive phase noise measurement set-up integrated with an LSNA and a tunable monochromatic light source.

98 A 1000 W Xenon lamp is used as a source of light. A monochromator allows for the tuning of the photon energy from 1.5 to 3.9 eV for the light illuminating the DUT. A shutter is used to control the duration of the illumination. Typically 30 seconds of duration is needed to reach the steady state before the RF signal is injected. The entire measurements are performed in the dark since the occupancy of the traps can be affected by the incident photons from the ambient light. Typically 3 minutes are required between different illuminations to minimize memory effects and recover quasi-equilibrium conditions [54].

6.2 Experimental Results and Discussions

6.2.1 Illuminations with Different Photon Energies

Fig. 6.2 shows a comparison result of the additive phase noise measurement between

VGS = -3 V, VDS = 10 V and VGS = -3 V, VDS = 15 V DC biasing condition. As shown, a clear additive phase noise is observed for both DC biasing conditions in the entire offset frequency range compared to the noise floor level. The noise floor is measured by using a “Through” calibration standard as a DUT. A small peak in the noise floor (black dashed dotted line) between 1 KHz and 10 KHz originates from the signal source itself. The calibration constant of the mixer is determined for each measurement to account for any difference in RF drive to the mixer. It is observed that the phase noise decreases as the illumination photon energy increases. The strongest noise decrease is observed with 3.9 eV illumination which is just below the band-gap of Al0.3Ga0.7N (4.08 eV). However, a decrease in additive phase noise is also clearly observed with increasing photon energies below the GaN band-gap (3.38 eV). This arises from the decrease of the trap population induced by photon assisted emission of electrons from the trap levels to the conduction band and photon assisted capture of electrons from the valence band by the trap levels [55].

99 −70 4 Noise floor 1/f No illumination (V =10 V) −80 DS 1.5 eV (V =10 V) 2 DS 1/f 2.3 eV (V =10 V) −90 DS 3.1 eV (V =10 V) DS 3.9 eV (V =10 V) −100 DS No illumination (V =15 V) DS 1.5 eV (V =15 V) −110 DS 2.3 eV (V =15 V) DS −120 3.1 eV (V =15 V) DS 3.9 eV (V =15 V) DS −130 1/f −140

−150 Single Sideband Phase Noise (dBc/Hz) −160

−170 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 Offset Frequency (Hz)

Figure 6.2: Comparison of additive phase noise on various illumination energies between different DC biasing conditions VDS = 10 V (solid line) and VDS = 15 V (dashed line).

The decrease in additive phase noise is observed despite the increase of the RF power gain under illumination [54]. Furthermore, the phase noise exhibits a slope close to 1/f (=

1/f 1.18) for offset frequencies ranging from 1 to 10 KHz which is usually indicative of the presence of uniformly distributed traps. Note that the 1/f noise at RF can originate from the up-conversion process of the low-frequency noise due to non-linear operation [56], or gain modulation [57] like in the present case.

Also as is shown in Fig. 6.2, a degraded phase noise performance is observed with

VDS = 15 V compared to VDS = 10 V. This increase in additive phase noise originates from the development in unpassivated AlGaN/GaN HEMTs of a highly resistive region in the

2DEG drain access region near the gate resulting from an increase in AlGaN donor trap occupation [58], [59]. It can be veriﬁed that the larger the drain resistance the stronger the

100 impact (up-conversion) of the trap occupation ﬂuctuation upon the output RF drain current noise.

6.2.2 Supportive Analysis

Using a source and a load tuner, the impact of the load impedances on the additive phase noise is investigated. Fig. 6.3 (b) shows the RF drain load-line of the unpassivated GaN

HEMT with DC-IV and pulsed-IVs pulsed from three different operating points. All IVs shown are for VGS = -1 V. Considering the gate voltage swing in Fig. 6.3 (a), the pulsed-

IV pulsed from VGS = -5 V and VDS = 11 V (green solid line) shows the closest result to the drain load-line (black solid line). This arises from the fact that the effective DC biasing point for the traps can be shifted in large signal operation due to the cyclostationary effect [26], [60] .

Both DC-IVs and pulsed-IVs from VGS = -3 V, VDS = 10 V with various RF drain load-lines of unpassivated GaN HEMT are shown in Fig. 6.4 with a contour plot of β (=

∆n/n0) which is the modulation index of the channel concentration next to the gate. Both

∆n and n0 indicate the deviation in channel concentration and the equilibrium channel concentration respectively [58]. Note that β ranges from 0 to -1. The channel concentration will be maximum for β = 0. On the other hand, the channel concentration (n0 +∆n) below the traps will decrease as β decreases. As shown in the contour plot in Fig. 6.4, β decreases as both the gate and drain voltage deviate from 0 V. In Fig. 6.4, black circles represent the

DC biasing points while black squares indicate the effective DC operating points for the traps due to the cyclostationary process. These effective DC operating points for the traps indicate an average β ranging from -0.87 to -0.89 for RL varying from 50 ohm to 170 ohm.

The gate voltage in the IVs is swept from -5 V to 0 V in steps of 1 V. It is observed that the

101 0.1

0.09

0.08

0.07

0.06

0.05 (A) DS I 0.04

0.03

0.02

0.01

−6 −5 −4 −3 −2 −1 0 V (V) GS (a) Gate load-line

DC−IV (V = −1V) 0.14 GS Pulsed−IV from V =−2V, V =9V GS DS Pulsed−IV from V =−3V, V =10V 0.12 GS DS Pulsed−IV from V =−5V, V =11V GS DS Drain load−line (maximum swing) 0.1

0.08 (A) DS I 0.06

0.04

0.02

0 2 4 6 8 10 12 14 16 18 20 V (V) DS (b) IVs and drain load-line

Figure 6.3: RF gate and drain load-line of unpassivated GaN HEMT. In (b), DC-IV and pulsed-IVs pulsed from three different operating points are compared with drain load-line. All IVs shown are for VGS = -1 V.

102 0.14 DC−IVs −0.1 Pulsed−IVs 0.12 50 ohm 80 ohm −0.2 110 ohm 0.1 140 ohm −0.3 170 ohm

0.08 −0.4

(A) −0.5 DS

I 0.06

−0.6 0.04 −0.7

0.02 −0.8

0 −0.9

0 5 10 15 20 V (V) DS

Figure 6.4: RF drain load-lines of the unpassivated GaN HEMT with DC-IVs (black solid line), pulsed-IVs (blue dashed line) from VGS = -3 V, VDS = 10 V and contour plot of β. For both IV characteristics, VGS is swept from -5 V (bottom) to 0 V (top) in steps of 1 V.

0.14 DC−IVs −0.05 Pulsed−IVs 0.12 50 ohm 80 ohm −0.1 110 ohm 0.1 140 ohm 170 ohm −0.15

0.08 −0.2 (A) DS

I 0.06 −0.25

−0.3 0.04

−0.35 0.02

−0.4 0

0 5 10 15 20 V (V) DS

Figure 6.5: RF drain load-lines of the passivated GaN HEMT with DC-IVs (black solid line), pulsed-IVs (blue dashed line) from VGS = -3 V, VDS = 10 V and a contour plot of β. For both IV characteristics, VGS is swept from -5 V (bottom) to 0 V (top) in steps of 1 V.

103 drain voltage swing is approximately proportional to the load impedance. The load-lines are seen to bend at low drain voltages due to the knee walk-out induced by the effective DC operating point of the traps.

Similar measurements with the passivated GaN HEMT are shown in Fig. 6.5. β decreases now only to a minimum value of -0.32 such that there is no knee walk-out. Thus, the load-lines are not distorted (straight lines) since the device remains operating in the saturation region. The overall range of β for passivated devices is higher compared to the that of unpassivated devices, which explains the lower noise observed for the passivated device shown in Fig. 6.6 in the next section.

6.2.3 Phase Noise Dependence on Load Impedances

The additive phase noise of unpassivated and passivated GaN HEMT is shown in Fig.

6.6 for load impedances of 20, 50, 80, and 110 ohm. The devices operate in quasi class

A for the unpassivated and in class A for the passivated. As expected, a noise increase is observed as the drain voltage swing increases in the unpassivated device, while only a slight noise increase is observed in the passivated device. Also the noise level of the unpassivated device is higher than that of the passivated device, which is in agreement with the prediction.

The temperature variation during the measurement is also calculated using:

∆T = Rth × ∆Pheat (6.1) where

Rth = 12.37C/W Z 1 T Pheat = vDS(t)iD(t) dt. T 0 104 −70 Unpassivated (20 ohm) Unpassivated (50 ohm) −80 Unpassivated (80 ohm) Unpassivated (110 ohm) −90 Passivated (20 ohm) Passivated (50 ohm) Passivated (80 ohm) −100 Passivated (110 ohm)

−110

−120

−130 1/f −140

−150 Single Sideband Phase Noise (dBc/Hz) −160

−170 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 Offset Frequency (Hz)

Figure 6.6: Comparison of phase noise dependence between passivated and unpassivated device on load impedances.

Table 6.1 shows the calculated temperature variation during the measurements. Since less than 1◦C difference is obtained, thermal effects appear to be a negligible factor for all the measurements.

Measurements ∆T (◦C) Load variation (Unpassivated) 0.55 Load variation (Passivated) 0.67

Table 6.1: Temperature variation during the measurements

105 6.3 Conclusion

An additive phase noise measurement system was integrated with a large signal network analyzer and a tunable monochromatic light source. This system was used to characterize the additive phase noise of an unpassivated AlGaN/GaN HEMT on SiC substrate.

1/f additive phase noise at 2 GHz was observed in a signiﬁcative frequency range which is indicative of the presence of uniformly traps. Illumination of the device with a monochromatic beam with different photon energies, was used to alter the trap and 2DEG population to help with the identiﬁcation of the physical origin of the additive phase noise. A decrease in additive phase noise was observed as the photon energy of illumination was increased. It was also demonstrated that the device biased with a higher drain voltage exhibited a poorer additive phase noise performance compared to the one with lower drain voltages.

To further characterize the noise performance of both passivated and unpassivated Al-

106 CHAPTER 7

CONCLUSION AND FUTURE WORK

An oscillator is one of the fundamental components in modern RF systems since it is often used with mixers for frequency translation.

A large-signal measurement-based approach using a LSNA presented can be a good candidate for designing negative-resistance harmonic-loaded oscillators. Harmonic-tuning was realized with a phase swept RT-ALP testbed implemented with an LSNA for each harmonic. Thanks to the fast acquisition time of the RT-ALP, the multi-harmonic optimal load impedances which optimize the output power of the oscillator can be determined in a short time. A stand alone oscillator was then fabricated with the optimal harmonic loads and a reasonable agreement between the expected and measured performances was obtained in accordance with the Kurokawa design methodology. This method demonstrates the applicability of the phase-swept RT-ALP technique to the design of negative-resistance oscillators. This work further supports the notion that multi-harmonic real-time active load- pull provides the designer with a time-efﬁcient technique for the quasi-interactive design of negative-resistance oscillators.

A power-dependent Volterra series model for the negative-resistance oscillator was developed to assist the oscillator design procedure. The new modeling approach holds up to inﬁnite order for a ﬁnite number of harmonic excitations. The behavioral model yielded

107 a good ﬁt of the load-pull measurement data obtained using an approximated third-order expansion of the power-dependent functions involved. This model was implemented in a circuit simulator to facilitate the harmonic balance simulation of an oscillator. As a result, a methodology to estimate the error in frequency and amplitude of oscillation due to imperfect loading conditions was presented using the behavioral model obtained.

The phase noise is one of the most signiﬁcant criteria for specifying oscillator performance since it corrupts both up-converted and down-converted signals in RF systems [1].

Among various noises, the 1/f noise which is the most critical to the overall near carrier phase noise was analyzed using analytic, simulation and experimental results in four different oscillator circuits. The generalized Kurokawa theory was veriﬁed to yield similar results to the conversion matrix in harmonic balance simulations and the PPV and ISF approaches for both the uncorrelated and correlated (AM to PM noise conversion) cases.

To rigorously analyze the impact of the AM to PM noise conversion upon the phase

(PM) noise in a simple example, a modiﬁed Van der Pol oscillator featuring two new nonlinear capacitances was introduced. A fully analytic solution valid for both the generalized

Kurokawa and PPV analysis in the presence of an harmonic short across the tank, was then derived for the modiﬁed Van der Pol oscillator. It was then veriﬁed for both the uncorrelated and correlated (AM to PM noise conversion) cases that this analytic solution yielded

108 the same numerical results as harmonic balance based simulations using the conversion method.

The generalized Kurokawa theory and the PPV theory were also found to predict similar

1 2 voltage spectral noise densities SV,1/f,ssb(∆ω)/( 2 A0) (old IEEE phase noise) at both low and large offset-frequency ∆ω for large enough measurement times.

The model can also be readily generalized to multiple 1/f noise sources and fractional

1/f δ noise sources.

109 for modeling the phase noise dynamics. The correspondences established between the

In addition to the 1/f noise analysis of the free running oscillator, the 1/f additive phase noise analysis for the injection-locked oscillator was presented. The Kurokawa analysis was used to obtain an analytic expression for the 1/f additive phase noise of the injection-locked oscillator. The additive phase noise measurement system integrated with an LSNA was used to efﬁciently obtain the required model parameters for predicting 1/f additive phase noise. The Kurokawa derivatives needed for the analytic expression was experimentally obtained and further optimized to accurately predict the corner frequency.

The obtained analytic solution was veriﬁed to closely predict the experimental 1/f additive phase noise result of the pHEMT micro-strip line oscillator at 2.485 GHz. The presented

1/f additive phase noise analytic model with the experimental agreement for the injection- locked oscillator should provide circuit designers great insights in the noise mechanism taking place in injection-locked oscillators and should facilitate the phase noise optimization.

The noise measurement system developed can also be used to characterize the noise performance of ampliﬁers. By integrating the noise measurement system with a LSNA and a tunable monochromatic light source, the additive phase noise of an AlGaN/GaN HEMT on SiC substrate was measured and analyzed at 2 GHz. Illumination of the device with a monochromatic beam with different photon energies, was used to alter the trap and 2DEG population to help with the identiﬁcation of the physical origin of the additive phase noise.

110 A decrease in additive phase noise was observed as the photon energy of illumination was increased. It was also demonstrated that the device biased with a higher drain voltage exhibited a poorer additive phase noise performance compared to the one with lower drain voltages.

To further characterize the noise performance of both passivated and unpassivated Al-

GaN/GaN HEMTs, various RF load impedances were applied to induce different drain voltage swings. The shift of the effective DC biasing point of the traps due to the cyclostationary effect was observed in large signal operation. As a result, more additive phase noise was observed with an ascending drain voltage swing. Also, less noise was observed in the passivated device compared to the unpassivated device. These results indicate that the new joint additive-phase-noise/LSNA testbed presented should prove a useful tool for the characterization of additive phase noise in transistors and ampliﬁers operating under large signal operation.

111 APPENDIX A

VOLTERRA SERIES EXPANSION OF FOUR-TONE EXCITATION

A.1 Volterra series expansion of four-tone excitation

The reﬂected waves b1(ω0), b1(2ω0), and b1(3ω0) for four incident tones can be represented using the following Volterra series expansion:

∗ b1(ω0) = a1(ω0) · f1(x4, ω0) + a1(ω0)a1(2ω0) · f2(x4, ω0)

∗ + a1(2ω0)a1(3ω0) · f3(x4, ω0)

2∗ + a1 (ω0)a1(3ω0) · f4(x4, ω0)

∗ 2 + a1(3ω0)a1(2ω0) · f5(x4, ω0)

∗ + a1(3ω0)a1(4ω0) · f6(x4, ω0)

2∗ 3 + a1 (4ω0)a1(3ω0) · f7(x4, ω0)

3∗ + a1 (ω0)a1(4ω0) · f8(x4, ω0)

∗ ∗ + a1(2ω0)a1(ω0)a1(4ω0) · f9(x4, ω0)

∗ + a1(4ω0)a1(2ω0)a1(3ω0) · f10(x4, ω0)

∗ ∗ 2 + a1(4ω0)a1(ω0)a1(3ω0) · f11(x4, ω0). (A.1)

112 2 b1(2ω0) = a1(2ω0) · f1(x4, 2ω0) + a1(ω0) · f2(x4, 2ω0)

∗ + a1(ω0)a1(3ω0) · f3(x4, 2ω0)

∗ + a1(2ω0)a1(4ω0) · f4(x4, 2ω0)

∗ 2 + a1(4ω0)a1(3ω0) · f5(x4, 2ω0)

2∗ + a1 (ω0)a1(4ω0) · f6(x4, 2ω0)

2∗ 2 + a1 (3ω0)a1(4ω0) · f7(x4, 2ω0)

2∗ 2 + a1 (2ω0)a1(3ω0) · f8(x4, 2ω0)

∗ + a1(3ω0)a1(ω0)a1(4ω0) · f9(x4, 2ω0)

∗ + a1(2ω0)a1(ω0)a1(3ω0) · f10(x4, 2ω0). (A.2)

3 b1(3ω0) = a1(3ω0) · f1(x4, 3ω0) + a1(ω0) · f2(x4, 3ω0)

+ a1(ω0)a1(2ω0) · f3(x4, 3ω0)

∗ 2 + a1(ω0)a1(2ω0) · f4(x4, 3ω0)

∗ + a1(ω0)a1(4ω0) · f5(x4, 3ω0)

∗ 3 + a1(3ω0)a1(2ω0) · f6(x4, 3ω0)

3∗ 3 + a1 (3ω0)a1(4ω0) · f7(x4, 3ω0)

∗ ∗ 2 + a1(3ω0)a1(2ω0)a1(4ω0) · f8(x4, 3ω0)

∗ 2 + a1(3ω0)a1(ω0)a1(4ω0) · f9(x4, 3ω0)

∗ + a1(3ω0)a1(2ω0)a1(4ω0) · f10(x4, 3ω0)

∗ + a1(2ω0)a1(ω0)a1(4ω0) · f11(x4, 3ω0)

2∗ 2 + a1 (3ω0)a1(ω0)a1(4ω0) · f12(x4, 3ω0). (A.3)

113 where each functions fi(x4, nω0) in (A.1) - (A.3) were found to be functionally dependent on 34 DC terms xi:

fi(x4, nω0) = fi(x1, x2, ..., x34). (A.4)

114 These DC terms are given by:

∗ x1,4 = a1(ω0)a1(ω0),

∗ x2,4 = a1(2ω0)a1(2ω0),

∗ x3,4 = a1(3ω0)a1(3ω0),

∗ x4,4 = a1(4ω0)a1(4ω0),

∗ 2 x5,4 = a1(2ω0)a1(ω0),

∗ x6,4 = a1(3ω0)a1(ω0)a1(2ω0),

∗ 3 x7,4 = a1(3ω0)a1(ω0),

∗ ∗ 2 x8,4 = a1(3ω0)a1(ω0)a1(2ω0),

2∗ 3 x9,4 = a1 (3ω0)a1(2ω0),

2∗ ∗ 3 x10,4 = a1 (4ω0)a1(ω0)a1(3ω0),

∗ x11,4 = a1(4ω0)a1(ω0)a1(3ω0),

∗ 2 x12,4 = a1(4ω0)a1(2ω0),

∗ ∗ 2 x13,4 = a1(4ω0)a1(2ω0)a1(3ω0),

∗ 4 x14,4 = a1(4ω0)a1(ω0),

∗ 2∗ x15,4 = a1(4ω0)a1 (ω0)a1(2ω0),

3∗ 4 x16,4 = a1 (4ω0)a1(3ω0),

∗ 2∗ 2 x17,4 = a1(4ω0)a1 (ω0)a1(3ω0),

2∗ 2 x18,4 = a1 (4ω0)a1(2ω)a1(3ω0),

∗ ∗ x19,4 = a1(4ω0)a1(ω0)a1(2ω0)a1(3ω0),

∗ xi,4 = xi−15,4 for 20 ≤ i ≤ 34.

115 APPENDIX B

PPV DERIVATION

In this appendix we shall seek a bandwidth limited PPV solution for the oscillator equation (4.39). Following the methodology of Ref. [35] we represent the tank voltage v(t) and inductor currents iL using it Fourier coefﬁcients Vk and Ik.

∗ −jωt jωt v(t) = V1 e + V0 + V1e ,

∗ −jωt jωt iL(t) = I1 e + I0 + I1e .

Only the fundamental Fourier coefﬁcients need to be considered due to the harmonic short.

The state equation with VN = 0 in the frequency domain are then represented by 6 vectors:

1 V ∗( − jωC ) − jωV |V |23C + I∗ + I∗ (W ), 1 R L 1 1 IN2 1 IN1 V 0 + I = 0, R L0 1 V ( + jωC ) + jωV ∗|V |23C + I + I (W ), 1 R L 1 1 IN2 1 IN1 ∗ ∗ −jωLI1 − V0 = 0, −V0 = 0, jωLI1 − V1 = 0,

> ∗ ∗ with W = [V1 ,V0,V1,I1 ,I0,I1] and IIN1(W ) = YIN,1(W )V1. This nonlinear system

1 of equations admits for solution the operating point already derived: ω = ω0, V1 = 2 A0,

V0 = 0, I1 = V1/(jωL), I0 = 0. This nonlinear system can be rewritten under the form:

jωG(W ) = H(W ) = jωDω,1 X,

116 where G and H are frequency-independent vectors of dimension 6×1. The relation to the

> ∗ ∗ ∗ state variables of the oscillator X = [CT V1 ,CT V0,CT V1 , LI1 , LI0, LI1] is also established using the derivative operator jωDω,1 deﬁned as: " # D1 D1 Dω,1 = , D1 D1 with Dk a diagonal matrix with diagonal elements [−k · · · − 1, 0, 1, . . . k].

The Jacobian equation associated with the perturbation ∆W vector can then by obtained from:

jωJ ω∆W = jωDω,1F ω,1∆W = J 0∆W, (B.1)

with the frequency and DC (6×6) Jacobian matrices J ω and J 0 respectively deﬁned by: " # ∂G F ω,11 0 J ω = = Dω,1 F ω,1 with F ω,1 = , ∂W 0 F ω,22 " # ∂H J 0,11 I J 0 = = , (B.2) ∂W −I 0

where I is a 3×3 identity matrix, 0 a 3×3 null matrix and where F ω,11, F ω,22, and J 0,11, are given by:   CL +2δ 0 δ F ω,11 =  0 C+2δ 0  , δ 0 C +2δ   L   L 0 0 γ2 0 γ2    1  F ω,22 = 0 L 0 , J 0,11 = − 0 R 0 . (B.3) 0 0 L γ2 0 γ2

To determine the adjoint state equation we note that the matrices F ω,ij are symmetric

Toeplitz matrices. It results that the Jacobian matrix equation (B.1) is the band limited

117 solution of the following Jacobian differential equation:

d d h i ∆x(t) = F (t) · ∆w(t) = J (t) · ∆w(t) dt dt ω 0 h −1 i = J 0(t) F ω (t) · ∆x(t). (B.4)

The adjoint equation is then:

d h −1 i − v>(t) = v>(t) · J (t) F (t) , dt 0 ω or equivalently > d > −F (t) v(t) = J (t) v(t). ω dt 0

In the frequency domain, the adjoint equation to solve for the band limited Floquet vector v1(t) with unity exponential multiplier is then:

B> B> −jωF ω,1 Dω,1P = J 0 P, (B.5)

B> where the notation J x is used to indicate the block transposed of the J x realized by swapping the sub-block matrices J x,12 and J x,21. Solving this adjoint linear system leads

> to P = [p−1, p0, p1, q−1, q0, q1] with p0 = q0 = 0 and the remaining pi and qi given by: r ∗ ∗ δ L 1 p1 = p−1 = jω0Lq1 = jω0Lq−1 = − + j , 2γ2CT A0 CT 2A0

> dx(t) once the bi-normality v1 (t)· dt = 1 is enforced for all time. The equivalence between the differential equation and the matrix equation was limited in this circuit to the ﬁrst harmonic due to the harmonic short. Note that the methodology used can also be extended to an arbitrary number k of harmonics. The only requirements is that the matrices F ω,ij be

Hermitian Toeplitz matrices.

118 APPENDIX C

ABBREVIATION

ADS: Advanced Design System

ALP: Active Load-Pull

AM: Amplitude Modulation

BJT: Bipolar Junction Transistor

CMOS: Complementary Metal-Oxide-Semiconductor

DUT: Device Under Test

FDD: Frequency Domain Device

FET: Field-Effect Transistor

HBT: Heterojunction Bipolar Transistor

HEMT: High Electron Mobility Transistor

IC: Integrated Circuit

ISF: Impulse Sensitivity Function

LO: Local Oscillator

LSNA: Large Signal Network Analyzer

LTV: Linear Time Varying

MWO: Microwave Ofﬁce

PHD: Poly-Harmonic Distortion

119 pHEMT: Pseudomorphic HEMT

PM: Phase Modulation

PPV: Perturbation Projection Vector

RF: Radio Frequency

RT-ALP: Real-Time Active Load-Pull

SSB: Single Side Band

2DEG: Two-Dimensional Electron Gas

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