Ahmet Tekin

K21542

High Frequency Communication and Sensing: Traveling-Wave Techniques introduces novel traveling wave circuit techniques to boost the performance of high-speed circuits in standard low-cost production technologies, like complementary metal oxide semiconductor (CMOS). A valuable resource for experienced analog/radio frequency (RF) circuit designers as well as undergraduate-level microelectronics researchers, this book:

Explains the basics of high-speed signaling, such as transmission lines, distributed signaling, impedance matching, and other common practical RF background material

Promotes a dual-loop coupled traveling wave oscillator topology, the trigger mode distributed wave oscillator, as a high-frequency multiphase signal source

Introduces a force-based starter mechanism for dual-loop, even-symmetry, multiphase traveling wave oscillators, presenting a single-loop version as a force mode distributed wave antenna (FMDWA)

Describes higher-frequency, passive inductive, and quarter-wave-length-based pumped distributed wave oscillators (PDWOs)

Examines phased-array transceiver architectures and front-end circuits in detail, along with distributed oscillator topologies

Devotes a chapter to THz sensing, illustrating a unique method of traveling wave frequency multiplication and power combining

Discusses various data converter topologies, such as digital-to-analog converters (DACs), analog-to-digital converters (ADCs), and GHz-band-width sigma-delta modulators

Covers critical circuits including phase rotators and interpolators, phase shifters, phase-locked loops (PLLs), delay-locked loops (DLLs), and more

It is a signicantly challenging task to generate and distribute high-speed clocks. Multiphase low-speed clocks with sharp transition are proposed to be a better option to accommodate the desired timing resolution. High Frequency Communication and Sensing: Traveling-Wave Techniques provides new horizons in the quest for greater speed and performance.

Engineering Electrical

High FrequencyCommunicationand Sensing

Ahmet TekinAhmed Emira

TRAVELING-WAVE TECHNIQUES

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TEKIN

EMIRA

Devices, Circuits, and Systems

Series Editor Krzysztof Iniewski

CMOS Emerging Technologies Research Inc., Vancouver, British Columbia, Canada

PUBLISHED TITLES:

Atomic Nanoscale Technology in the Nuclear IndustryTaeho Woo

Biological and Medical Sensor TechnologiesKrzysztof Iniewski

Building Sensor Networks: From Design to ApplicationsIoanis Nikolaidis and Krzysztof Iniewski

Circuits at the Nanoscale: Communications, Imaging, and SensingKrzysztof Iniewski

Design of 3D Integrated Circuits and SystemsRohit Sharma

Electrical Solitons: Theory, Design, and Applications David Ricketts and Donhee Ham

Electronics for Radiation Detection Krzysztof Iniewski

Embedded and Networking Systems: Design, Software, and Implementation

Gul N. Khan and Krzysztof Iniewski

Energy Harvesting with Functional Materials and MicrosystemsMadhu Bhaskaran, Sharath Sriram, and Krzysztof Iniewski

Graphene, Carbon Nanotubes, and Nanostuctures: Techniques and Applications

James E. Morris and Krzysztof Iniewski

High-Speed Devices and Circuits with THz ApplicationsJung Han Choi

High-Speed Photonics InterconnectsLukas Chrostowski and Krzysztof Iniewski

High Frequency Communication and Sensing: Traveling-Wave Techniques

Ahmet Tekin and Ahmed Emira

Integrated Microsystems: Electronics, Photonics, and Biotechnology Krzysztof Iniewski

Integrated Power Devices and TCAD SimulationYue Fu, Zhanming Li, Wai Tung Ng, and Johnny K.O. Sin

Internet Networks: Wired, Wireless, and Optical Technologies Krzysztof Iniewski

Labs-on-Chip: Physics, Design and TechnologyEugenio Iannone

Low Power Emerging Wireless TechnologiesReza Mahmoudi and Krzysztof Iniewski

Medical Imaging: Technology and ApplicationsTroy Farncombe and Krzysztof Iniewski

Metallic Spintronic DevicesXiaobin Wang

MEMS: Fundamental Technology and ApplicationsVikas Choudhary and Krzysztof Iniewski

Micro and Nanoelectronics: Emerging Device Challenges and SolutionsTomasz Brozek

Microfluidics and Nanotechnology: Biosensing to the Single Molecule LimitEric Lagally

MIMO Power Line Communications: Narrow and Broadband Standards, EMC, and Advanced Processing

Lars Torsten Berger, Andreas Schwager, Pascal Pagani, and Daniel Schneider

Mobile Point-of-Care Monitors and Diagnostic Device DesignWalter Karlen

Nano-Semiconductors: Devices and TechnologyKrzysztof Iniewski

Nanoelectronic Device Applications HandbookJames E. Morris and Krzysztof Iniewski

Nanopatterning and Nanoscale Devices for Biological Applicationseila Selimovic

Nanoplasmonics: Advanced Device Applications James W. M. Chon and Krzysztof Iniewski

Nanoscale Semiconductor Memories: Technology and Applications Santosh K. Kurinec and Krzysztof Iniewski

Novel Advances in Microsystems Technologies and Their ApplicationsLaurent A. Francis and Krzysztof Iniewski

Optical, Acoustic, Magnetic, and Mechanical Sensor TechnologiesKrzysztof Iniewski

Radiation Effects in Semiconductors Krzysztof Iniewski

PUBLISHED TITLES:

Semiconductor Radiation Detection Systems Krzysztof Iniewski

Smart Grids: Clouds, Communications, Open Source, and AutomationDavid Bakken

Smart Sensors for Industrial Applications Krzysztof Iniewski

Technologies for Smart Sensors and Sensor FusionKevin Yallup and Krzysztof Iniewski

Telecommunication NetworksEugenio Iannone

Testing for Small-Delay Defects in Nanoscale CMOS Integrated CircuitsSandeep K. Goel and Krishnendu Chakrabarty

VLSI: Circuits for Emerging ApplicationsTomasz Wojcicki

Wireless Technologies: Circuits, Systems, and DevicesKrzysztof Iniewski

FORTHCOMING TITLES:

Analog Electronics for Radiation DetectionRenato Turchetta

Cell and Material Interface: Advances in Tissue Engineering, Biosensor, Implant, and Imaging Technologies

Nihal Engin Vrana

Circuits and Systems for Security and PrivacyFarhana Sheikh and Leonel Sousa

CMOS: Front-End Electronics for Radiation SensorsAngelo Rivetti

CMOS Time-Mode Circuits and Systems: Fundamentals and ApplicationsFei Yuan

Electrostatic Discharge Protection of Semiconductor Devices and Integrated Circuits

Juin J. Liou

Gallium Nitride (GaN): Physics, Devices, and TechnologyFarid Medjdoub and Krzysztof Iniewski

Implantable Wireless Medical Devices: Design and ApplicationsPietro Salvo

Laser-Based Optical Detection of ExplosivesPaul M. Pellegrino, Ellen L. Holthoff, and Mikella E. Farrell

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Mixed-Signal CircuitsThomas Noulis and Mani Soma

MRI: Physics, Image Reconstruction, and AnalysisAngshul Majumdar and Rabab Ward

Multisensor Data Fusion: From Algorithm and Architecture Design to ApplicationsHassen Fourati

Nanoelectronics: Devices, Circuits, and Systems Nikos Konofaos

Nanomaterials: A Guide to Fabrication and ApplicationsGordon Harling, Krzysztof Iniewski, and Sivashankar Krishnamoorthy

Optical Fibre Sensors: Advanced Techniques and Applications Ginu Rajan

Organic Solar Cells: Materials, Devices, Interfaces, and ModelingQiquan Qiao and Krzysztof Iniewski

Power Management Integrated Circuits and Technologies Mona M. Hella and Patrick Mercier

Radiation Detectors for Medical Imaging Jan S. Iwanczyk and Polad M. Shikhaliev

Radio Frequency Integrated Circuit Design Sebastian Magierowski

Reconfigurable Logic: Architecture, Tools, and Applications Pierre-Emmanuel Gaillardon

Soft Errors: From Particles to CircuitsJean-Luc Autran and Daniela Munteanu

Solid-State Radiation Detectors: Technology and ApplicationsSalah Awadalla

Wireless Transceiver Circuits: System Perspectives and Design AspectsWoogeun Rhee and Krzysztof Iniewski

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Boca Raton London New York


Ahmet TekinWaveworks , Inc .Sunnyva le , Ca l i fo rn ia , USA

Ahmed EmiraCai ro Un ivers i tyG iza , Egypt

TRAVELING-WAVE TECHNIQUES

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAbout the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1. What This Book Is About . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Lumped vs. Distributed Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Infinite Transmission Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2 Dispersionless Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Lossless Transmission Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.4 Shorted Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Voltage Standing Wave Ratio (VSWR) . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Trigger Mode Distributed Wave Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Commonly Used Wave Oscillator Topologies. . . . . . . . . . . . . . . . . . .263.2 Rotary Traveling Wave Oscillator (RTWO) . . . . . . . . . . . . . . . . . . . . . 273.3 Standing Wave Oscillator (SWO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 TMDWO Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Phase Noise in Traveling Wave Oscillators . . . . . . . . . . . . . . . . . . . . . 38

3.5.1 Phase Noise in SWO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5.1.1 Noise of Transmission Line . . . . . . . . . . . . . . . . . . . 413.5.1.2 Tail Transistor Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5.1.3 Cross-Coupled Pair Noise . . . . . . . . . . . . . . . . . . . . 43

3.5.2 Phase Noise in RTWO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5.3 Phase Noise in Differential Wave Oscillator (DWO) . . . . . 51

3.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4. Force Mode Distributed Wave Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1 Force Mode Distributed Wave Oscillation Mechanisms . . . . . . . . . 574.2 Single-Ended Force Mode Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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5. Wave-Based RF Circuit Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Pumped Distributed Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Traveling Wave Phased-Array Transceiver . . . . . . . . . . . . . . . . . . . . . 745.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6. THz Signal Generation and Sensing Techniques . . . . . . . . . . . . . . . . . . . . 836.1 Frequency Multiplication Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2 Traveling Wave Reflectometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .886.3 Wafer-Level THz Sensing Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .956.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7. Traveling Wave-Based High-Speed Data Conversion Circuits . . . . . . . 997.1 Traveling-Wave Noise Shaping Modulator . . . . . . . . . . . . . . . . . . . . 1077.2 A High-Speed Phase Interleaving Topology . . . . . . . . . . . . . . . . . . . 1097.3 A Traveling Wave Multiphase DAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4 Traveling Wave Phased-Array DAC Transmitter . . . . . . . . . . . . . . 1177.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8. Traveling Wave High-Speed Serial Link Design for Fiberand Backplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.1 Traveling Wave-Based Multiphase Rx-Tx Front End . . . . . . . . . . . 1268.2 A Full-Rate Phase-Interpolating Topology . . . . . . . . . . . . . . . . . . . . 1288.3 An ADC-Based DSP Link Front End . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Preface

Telecommunication and medical applications have become two main driv-ing forces for emerging semiconductor technologies. Computational powerslowly shifting into the cloud servers has made communication speed a limi-tation in personal computing. Hence, demand for speed from backhaul serialoptical wireline networks has experienced a surge. The existing optical net-works with 10 Gb/s per lane link rates has been targeted to boost the systemwith a 40 Gb/s and even 100 Gb/s per lane data rate over the existing fiberoptics infrastructure. This, however, has pushed the burden to the interface in-tegrated circuits, demanding more speed with less noise margins. The spreadof many WiFi and cellular small cell standards on the wireless side has alsoboosted the effort for more wireless channel capacity per available spectrum.One remedy in the search for more throughputs on the wireless side is toallocate some new, very high frequency bands, mainly around 60 GHz range,as an unlicensed free spectrum. Although a relatively large chunk of spec-trum has become available for local wireless networks, design of high-speedintegrated circuits at such frequencies remains a challenge. This book intro-duces some new traveling wave circuit techniques to boost the performanceof high-speed circuits in standard low-cost production technologies like thecomplementary metal oxide semiconductor (CMOS). While the book servesas a valuable resource for experienced analog/radio frequency (RF) circuitdesigners, it also provides much information for undergraduate-level micro-electronics researchers.

The main theme of this book involves utilizing multiple phases of travelingwaves as a fine high-speed timing reference rather than carrying high-speedsignals across a microchip. It is a significantly challenging task to generateand distribute high-speed clocks. Multiphase low-speed clocks with sharptransition are proposed to be a better option to accommodate the desired tim-ing resolution. The authors believe that the proposed techniques will providenew horizons in the quest for more speed and performance.

Chapter 2 describes high-speed signaling basics, such as transmissionlines, distributed signaling, impedance matching, and other common prac-tical RF background material. This chapter introduces entry-level readers tothe chapters that follow.

In Chapter 3, a dual-loop coupled traveling wave oscillator topology,the trigger mode distributed wave oscillator, is described in detail asa high-frequency multiphase signal source. Thanks to the proposedtrigger-mechanisms, two independent transmission line oscillators can be

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xii Preface

cross-coupled to form a single differential oscillator. Multiple oscillationphases become readily available along these symmetric oscillation tracks.These clock phases are to be used in much of the work presented in the fol-lowing chapters. The advantages and disadvantages of such a traveling waveoscillator are described in detail.

Chapter 4 introduces a force-based starter mechanism for dual-loop, even-symmetry, multiphase traveling wave oscillators. A single-loop versionof this oscillator is presented as a force mode distributed wave antenna(FMDWA). Moreover, coupling this single-ended structure with secondarypickup coils results in various integrated microwave transformer configu-rations. A phased-array transmitter system is also presented as one of theapplications of the mentioned traveling wave structure.

In Chapter 5, even higher-frequency, passive inductive, or quarter-wave-length-based pumped distributed wave oscillators (PDWOs) are described indetail. As in the case of other wave oscillators, these techniques also providemultiple high-frequency oscillation phases. Phased-array transceiver archi-tectures and front-end circuits are illustrated in detail along with the dis-tributed oscillator topologies.

Chapter 6 is devoted to THz sensing. A unique method of traveling wavefrequency multiplication and power combining is illustrated. The mentionedTHz signal source is then employed in a reflection-based integrated sensingtransceiver device that forms a unit pixel element for a low-cost, nonradioac-tive imaging device. Creating a large array of these elements on a silicon waferwill result in a complete THz image sensor.

Chapter 7 discusses various data converter topologies such as digital-to-analog converters (DACs), analog-to-digital converters (ADCs), and GHz-bandwidth sigma-delta modulators using the integrated multiphase clocksources described in previous chapters. Low-noise clock phases this timeprovide opportunity for high-accuracy interleaved time intervals for the men-tioned sampling systems.

Another set of serial link sampling systems is discussed in Chapter 8. Oncemore, many symmetric phases of the traveling wave oscillators find a specialplace in circuit blocks employed in these systems. Phase rotators and inter-polators, phase shifters, phase-locked loops (PLLs), and delay-locked loops(DLLs) are some of the critical circuits that are discussed.

About theAuthors

Ahmet Tekin received his EE PhD degree from the University of CaliforniaSanta Cruz, California, EE MS degree from North Carolina A&T State Univer-sity, Greensboro, North Carolina, and EE BS degree from Bogazici University,Istanbul, Turkey, in 2008, 2004, and 2002, respectively. During his MS stud-ies at NCA&T State University, he worked on a NASA transceiver projectdesigning very low power radiation-hard SOI complementary metal oxidesemiconductor (CMOS) circuits. While obtaining his PhD, he designed a verylow noise analog radio baseband with noise-shaping circuit techniques. Hehas worked for many innovative companies: Multigig, Inc. on the technicalstaff; Newport Media as a senior analog design engineer; Aydeekay LLC as asenior mixed-signal design engineer; Broadcom Corporation as a senior staffscientist, Semtech Corporation as a principle design engineer; and NuvotonTechnology Corporation as an analog design manager. He was co-founderof Waveworks, Inc., focusing on novel traveling wave-based high-frequencycommunication circuits and biochips.

Ahmed Emira received his BSc and MSc degrees in electronics and commu-nications from Cairo University, Giza, Egypt, in 1997 and 1999, respectively.In December 2003, he received his PhD degree in electrical engineering atTexas A&M University, College Station, Texas. From 2001 to 2002 he was withMotorola, Austin, Texas, where he worked as a radio frequency integratedcircuit (RFIC) design engineer. Following his PhD, he worked as an RFIC de-sign engineer in the wireless division of Silicon Laboratories, Austin, Texas,from 2003 to 2006. Then he worked as a senior RFIC design engineer and aleader for the power management team in Newport Media, Inc., Lake Forest,California, from 2006 to 2008. He joined Cairo University as an assistant pro-fessor in the electronics and communications department in 2008. Dr. Emirais currently the manager of the RFIC group at Atmel Inc., Egypt Design Cen-ter, and is also an associate professor at Cairo University. He has more than40 journal and conference publications, 5 U.S. patents, and several pending.His current interests include low-power mixed-signal circuits for portabledevices/energy harvesting systems, mm-wave and RF circuits, MEMS inter-face electronics, and wireless communication system architectures.

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1What This Book Is About

Communications, one form or other, has been one of the most valuable toolsfor man. The speed of developments in the last two decades, though, wasunmatched. The most cited inventions of the century have been wireless cel-lular communications and the Internet. It could only be science fiction forthis living generation 30 years ago that one would be able to experience alive conversation with a family or friend on the other side of the world in thenear future. Although we can enjoy an amazing amount of communicationexperience inexpensively through a small handheld device today, the costfor these technologies to reach this point was not low. Generations of engi-neers have worked years to come upon every incremental improvement tomake this experience smoother and more enjoyable. From transatlantic under-ocean links and satellites to cellular and wireless local area network (LAN)technologies, the incredible amount of infrastructure web around the worlddid not stop humans from desiring more. The increasing demand for wirelesscommunication has made the frequency spectrum one of the most valuablecommodities in recent human history. Hence, the technologies that can helpin better utilization of this valuable resource deserve valuable recognition aswell. Figure 1.1 shows the crowding in todays wireless spectrum. Most ofthe spectrum is allocated with very fine spacing between various standards,and the channel widths are all limited. Heavy traffic around the ultra-highfrequencies (UHFs) does not allow much possibility for wider channels. Eventhe communication in the existing channels can be hijacked by nearby inter-ferers in adjacent frequency bands. The propagation characteristics at thesefrequencies have proven to be optimal for long-range communication; hence,most 2G, 3G, and 4G cellular networks are centered at around these frequen-cies. However, as the demand for more bandwidth grows, the microwavefrequency bands come into play as resources to be tapped despite the hard-ware design challenges and cost limitations. The 23 and 38 GHz microwavebands are two strong candidates for the emerging 5G femtocell or picocell typeof cellular networks that are under review across the globe. Moreover, the 7 to10 GHz industrial, scientific, and medical (ISM) band around 60 GHz has longbeen targeted by many technology companies since its release by the FederalCommunications Commission (FCC) and its counterparts around the world;however, no widespread deployment of hardware is recorded yet due to thedesign challenges at such high frequencies. High oxygen absorption and poorradiation characteristics are only a few to mention.

1

2 High Frequency Communication and Sensing: Traveling-Wave Techniques

VLF LF MF HF VHF UHF SHF EHF

3KHz 300KHz 3MHz 30MHz 300MHz 3GHz 30GHz

MobileTV

AnalogTV

HDTVWiMax

30KHz

Long-waveRadio

Medium-waveRadio

FMRadio

DABDigitalRadio

DigitalTV

GSM

WiFiPMSE

60GHz ISMSatellite

2G-3G-4GMobile

Short-waveInternationalBroadcasting

MicrowaveRadio

Backhaul

FIGURE 1.1Todays wireless communication spectrum.

Wi-Fi LAN devices constitute a significant portion of wireless devicesaround us. The standard is called 802.11 after the name of the group formedto oversee its development. First, 802.11 only supported a maximum networkbandwidth of 2 Mbps. For this reason, the initial version of 802.11 wire-less products are no longer available. The Institute of Electrical and Elec-tronics Engineers (IEEE) expanded on the original 802.11 standard in July1999, creating the 802.11b specification. 802.11b supports bandwidths up to11 Mbps. 802.11b uses the same ISM band at 2.4 GHz as the original 802.11standard. 802.11b devices can incur interference from microwave ovens,cordless phones, Bluetooth, and other appliances using the same 2.4 GHzrange. While 802.11b was in development, IEEE created a second extensionto the original 802.11 standard called 802.11a. 802.11b gained much morepopularity than 802.11a due to its lower cost. 802.11a was used on businessnetworks, whereas 802.11b better served the home market. 802.11a supportsbandwidths up to 54 Mbps and signals in a regulated frequency spectrumof around 5 GHz. This higher frequency compared to 802.11b shortens therange of 802.11a networks. The higher frequency also means 802.11a sig-nals have more difficulty penetrating walls and other obstructions. Because802.11a and 802.11b utilize different frequencies, the two technologies are in-compatible with each other. In 2003, a new wireless (WLAN) standard, called802.11g, hit the market. 802.11g combines the best of both 802.11a and 802.11b.802.11g supports bandwidths up to 54 Mbps, similar to 802.11a, but it uses the2.4 GHz frequency for long-range transmission. 802.11g is backwards compat-ible with 802.11b, meaning that 802.11g access points will work with 802.11bwireless network adapters.

802.11n came along to improve on 802.11g in the amount of bandwidthsupported by utilizing multiple-input multiple-output (MIMO) technology.Standards groups ratified 802.11n in 2009 with specifications providing forup to 300 Mbps of network bandwidth. 802.11n also offers somewhat betterrange over earlier Wi-Fi standards due to its increased signal intensity, and itis backwards compatible with 802.11b/g. One of the latest standards, 802.11acutilizes dual-band wireless technology, supporting simultaneous connectionson both the 2.4 and 5 GHz Wi-Fi bands. 802.11ac offers backward compatibility

What This Book Is About 3

to 802.11b/g/n and bandwidths rated up to 1300 Mbps on the 5 GHz band,plus up to 450 Mbps on 2.4 GHz. In addition to these four Wi-Fi standards,other related wireless network technologies have to share the same 2.4 GHzISM band. Bluetooth, for example, uses 2.4 GHz. Bluetooth supports a veryshort range of up to 10 m and relatively low bandwidth of 1 Mbps. It is used inlow-power network devices such as medical devices, wireless keyboards andmouses, and handhelds. Other than 5 and 2.4 GHz ISM bands, many otherbands are used for machine-to-machine wireless connectivity solutions. Someof the most used ISM bands around the world are 315 and 915 MHz in theUnited States; 187, 230, 433, and 868 MHz in Europe; 426, 429, 449, 950, and1200 MHz in Japan; and 223, 230, 315, and 433 MHz in China. These devicesare generally deployed in cost-sensitive consumer products such as garagedoor openers, remote keyless entry, toys, and remote controllers. Mobile cel-lular networks also consume a significant amount of spectral resources duemainly to the number of active users at a given time. Initial forms of cellularnetworks were used to deliver voice-only data through analog modulation;hence, the sound quality was poor and the speed of transfer was only at9.6 Kbps. With the advance of 2G networks, the transmission quality had im-proved by introducing the concept of digital modulation. 2.5G is a transitionbetween 2G and 3G. In 2.5G, the most popular services, like Short MessageService (SMS), General Packet Radio Service (GPRS), Enhanced Data for GSM(Global System for Mobile Communication) Evolution (EDGE), and more,had been introduced. Shortly after the introduction of 2.5G systems, the 3Ggeneration of mobile telecommunication standards had emerged. It allowssimultaneous use of speech and data services and offers data rates of up to2 Mbps, which provide services like video calls, mobile TV, mobile Internet,and downloading. There are many technologies that fall under 3G, like Wide-band Code Division Multiple Access (WCDMA) and High Speed PackageAccess (HSPA). In telecommunications, 4G is the fourth generation of cellularwireless standards. It is a successor to the 3G and 2G families of standards. In2008, the ITU-R (International Telecommunication Union Radiocommunica-tion Sector) organization specified the IMT-Advanced (International MobileTelecommunications Advanced) requirements for 4G standards, setting datatransfer requirements for 4G service at 100 Mbit/s for high-mobility commu-nication, such as from trains and cars, and 1 Gbit/s for low-mobility commu-nication stationary users. A 4G system is expected to provide a comprehensiveand secure all-(Internet Protocol) IP-based mobile broadband solution to lap-top computers, wireless modems, smartphones, and other mobile devices.New services such as ultra-broadband Internet access, IP telephony, gamingservices, and streamed multimedia may be provided to users. Early 4G tech-nologies such as mobile WiMAX and Long-term Evolution (LTE) have beenaround for a while. The current versions of these technologies did not fulfill theoriginal ITU-R requirements of data rates approximately up to 1 Gbit/s for 4Gsystems. Another wireless communication field is GPS. All satellites broad-cast at the same two frequencies, 1.57542 GHz (L1 signal) and 1.2276 GHz


(L2 signal). The satellite network uses a code division multiple access (CDMA)spread-spectrum technique where the low-bitrate message data are encodedwith a high-rate pseudorandom number (PRN) sequence that is different foreach satellite. The receiver must be aware of the PRN codes for each satelliteto reconstruct the actual message data. The L1 carrier is modulated by boththe C/A and P codes, while the L2 carrier is only modulated by the P code.The P code can be encrypted as a so-called P(Y) code that is only availableto military equipment with a proper decryption key. Both the C/A and P(Y)codes impart the precise time of day to the user. The L3 signal at a frequencyof 1.38105 GHz is used to transmit data from the satellites to ground sta-tions. These data are used by the U.S. Nuclear Detonation Detection System(USNDS) to detect, locate, and report nuclear detonations (NUDETs) in theearths atmosphere and near space. One usage is the enforcement of nucleartest ban treaties. The L4 band at 1.379913 GHz is being studied for additionalionospheric correction. The L5 frequency band at 1.17645 GHz was added inthe process of GPS modernization. This frequency falls into an internationallyprotected range for aeronautical navigation, promising little or no interferenceunder all circumstances. The L5 consists of two carrier components that arein phase quadrature with each other. Each carrier component is bi-phase shiftkey (BPSK) modulated by a separate bit train.

High operating frequencies, and hence the required hardware speeds, havebecome a challenge not only for wireless networks but also for wireline net-works. Ethernet and optical communication networks constantly seek higherand higher speeds. It should be noted that in addition to the wireline In-ternet traffic, the mentioned wireless cellular traffic has to eventually flowthrough these networks at the backhauls. Newly emerging connectivity so-lutions such as cloud servers, Internet-based TV, social media servers, videoconferencing technologies, and online gaming have put even more pressureon these wireline Ethernet/optical networks. Recently, for example, the IEEE802.3 400Gb/s Ethernet Study Group was formed to explore development ofa 400Gb/s Ethernet standard to address the demand for more bandwidth.Although the technology scaling has helped to track the demand well in re-cent decades, Moores law has started to break apart from its original trenddue to the physical limitations of lithography. Hence, more is needed at thedesign side to catch up with the projected demand for the mentioned commu-nication technologies. This book introduces many unique, high-performancetraveling wave-based circuits, such as multi phase oscillators, transceivers,interleaved analog-to-digital converters (ADCs), and interleaved digital-to-analog converters (DACs) for future communication electronics. Special typesof traveling wave oscillators presented in the chapters ahead can provide veryhigh frequency signals in a low-cost standard complementary metal oxidesemiconductor (CMOS) process.

In addition to communications, high-frequency signals have found a newrole in the emerging field of THz sensing. Figure 1.2 shows the electromagneticspectrum. There is an interesting electromagnetic band in the range from 100

What This Book Is About 5

400 nm

0.1

1 0.1 nm

1 nm

10 nm

100 nm

1000 nm1 m

10 m

100 m

1000 m1 m

m

1 cm

10 cm

1 m10 m

100 m

1000 mAM

Long-waves

VHF

VHF

UHF

THz g

ap

FM

10 6

10 7

10 8

10 9

10 10

10 111000 M

Hz

500 MH

z

100 MH

z

50 MH

z

10 12

10 13

10 14

10 15

10 16

10 17

10 18

10 19

Radio, TV

Microwaves

Infra-red

Visible

Ultraviolet

X-rays

Gam

ma-rays

Radar

Far IR

Therm

al IR

Near IR

Wavelength

Frequency500 nm

600 nm

700 nm

FIGURE 1.2Electromagnetic spectrum.

up to 10 GHz that contains unique frequencies serving as specific markersfor many medical and scientific applications. This band is nowadays calledTHz gap since neither electronics nor optics can handle it well. High-speedlow-cost electronics still struggle at sub-THz speeds, whereas optics mostcommonly requires multiple-source frequency translation to generate THzsignals. Significant research effort is underway to close the THz gap withvarious techniques, but no low-cost efficient way is available yet.

Some microwave sensing circuits were introduced as well to be the basisfor future sensing applications such as THz cancer detection and medicalimaging, chemical and explosive detection, space research, metal detectors,and security systems. The enabling element in these systems is the THz signalsource. Traveling wave-based THz signal generation and utilization circuitswere introduced.

2Lumped vs. Distributed Elements

Lumped circuit models, such as resistors, capacitors, and inductors, are usedto evaluate the response of a circuit relative to low frequency. At sufficientlyhigh frequencies, the lumped model of an element is no longer an accurate rep-resentation of its behavior. More specifically, this occurs when the operationfrequency is comparable with the traveling speed of the voltage and currentwaves across the element boundaries. In other words, when the wavelengthof the traveling wave becomes comparable with the element dimensions, wecan no longer assume that the voltage or current waveforms are constantacross the element dimension. In this case, we must consider modeling theelement using distributed circuit model. The distinction between lumped anddistributed elements can be best understood by studying the traveling wavein a transmission line [Collin (1966), G. D. Vendelin and Pavio (1990), andPozar (1997)].

2.1 Infinite Transmission Line

Suppose we have an infinitely long transmission line, which is driven with asinusoidal voltage source as shown in Figure 2.1. The transmission line canbe modeled as an infinite number of cascaded sections of infinitesimal lengthdx. Each section can be modeled with a series impedance Zsdx and a shuntadmittance Ypdx, as illustrated in Figure 2.2, where Zs and Yp are the seriesimpedance and shunt admittance per unit length. The input impedance ofthis section can be written as

Zin(x) = 1Ypdx + 1/(Zin(x + dx) + Zsdx) (2.1)

Z0Ypdx + Z0Z0 + Zsdx = 1 (2.2)

Using the fact that Zsdx Z0, the second term in the left-hand side ofEquation (2.2) is expanded with the first two terms of the Taylor series as

Z0Ypdx + 1 ZsdxZ0 = 1 (2.3)

7


ZL

x

ZL(x)

0-

Transmission Line

Source

FIGURE 2.1Transmission line with infinite length.

which can be simplified to

Z0 =

ZsYp

(2.4)

This is called the characteristic impedance of the transmission line. It isinteresting to note that the input impedance of an infinitely long transmissionline is actually a finite value. It is also important to calculate the output voltageat distance x + dx with respect to the voltage at x. By performing Kirchhoffscurrent laws (KCL) and Kirchhoffs voltage laws (KVL) in Figure 2.2, weobtain

V(x) = V(x + dx) + I (x + dx)Zsdx (2.5)I (x) = I (x + dx) + V(x)Ypdx (2.6)

From Equations (2.5) and (2.6), we obtain the following first-order differentialequations:

dV(x)dx

= Zs I (x + dx) Zs I (x) (2.7)d I (x)

dx= YpV(x) (2.8)

Therefore, by combining Equations (2.7) and (2.8), we obtain the followingsecond-order differential equation:

d2V(x)dx2

= ZsYpV(x) = 2V(x) (2.9)

= ZsYp (2.10)

Ypdx

Zsdx

V(x) V(x + dx)++

I(x) I(x+dx)Zin(x+dx)Zin(x)

FIGURE 2.2Circuit model of an infinitesimal section of the transmission line.

Lumped vs. Distributed Elements 9

where is the propagation constant that represents the change in the waveamplitude and phase per unit length. The general solution for the abovesecond-order differential equation is

V(x) = Vf e x + Vr e x (2.11)

By substituting from Equation (2.11) into Equation (2.7), we get the generalcurrent equation

I (x) = 1Zs

dV(x)dx

= Zs

(Vf e x Vr e x

) = 1Z0

(Vf e x Vr e x

)(2.12)

The above solution is composed of two traveling waves; the first term rep-resents a forward wave traveling toward the load (in the x-direction), whilethe second term represents the reverse wave traveling back to the source (op-posite to the x-direction). The propagation constant is generally a complexterm:

= + j (2.13)where is the attenuation constant that denotes the amplitude attenuationper unit length in the direction of the traveling wave. Specifically, the reversewave amplitude drops as it travels in the x-direction, while the forward waveamplitude drops in the x-direction. is the phase constant that denotes thephase change per unit length of the transmission line. For a lossless transmis-sion line, with = 0, the wave undergoes only a phase change as it travelsthrough the transmission line. For the general case where Zs = jL + R andYp = jC + G, the propagation constant is expressed as

=

( jL + R) ( jC + G) (2.14)

And the characteristic impedance is expressed as

Z0 =

jL + RjC + G (2.15)

2.2 Dispersionless Transmission Line

At sufficiently higher frequencies for a weakly lossy transmission line, we canassume that R L and G C , and therefore we can approximate thepropagation constant as

j

LC(

1 + Rj2L

) (1 + G

j2C

) j

LC+ R

2

CL

+ G2

LC

= + j(2.16)


Therefore, we can write

= R2

CL

+ G2

LC

(2.17)

=

LC (2.18)

For a forward wave:

V(x) = V(0)e jtexe jx = V(0)e j (tx)ex = V(0)e j (x,t)ex (2.19)where (x, t) is total phase. To determine the wave propagation velocity alongthe transmission line, we can track a constant phase point and check how fastit travels.

ddt

= dxdt

= 0 (2.20)This constant phase point propagates in the x-direction at a velocity vp, whichis expressed as

vp = dxdt =

1

LC(2.21)

Since the propagation velocity and the attenuation constant are indepen-dent of frequency, this transmission line is dispersionless. The general condi-tion for dispersionless transmission can be obtained by forcing to be in thefollowing form:

= j vp

+ =

( jL + R)( jC + G) (2.22)

where vp and in the above equation are independent of frequency. By squar-ing both sides of the equation:(

j

vp+

)2= ( jL + R)( jC + G) (2.23)

2 j

vp+ 2

(

vp

)2= j(LG + C R) + RG 2LC (2.24)

2vp

= LG + C R (2.25)

2 (

vp

)2= RG 2LC (2.26)

Therefore, from equalizing the coefficients of 2 and the free term, we get

=

RG (2.27)

vp = 1LC

(2.28)


Then we substitute in the equation of the imaginary term

2

RLGC = LG + C R (2.29)

2 =

LGC R

+

C RLG

(2.30)

The only solution for the above equation is when

LGC R

= 1 (2.31)

which can be rewritten asGC

= RL

(2.32)

The above equation presents the general condition for a dispersionlesstransmission line. So even for a lossy transmission line, it can be made dis-persionless if we ensure the above condition is met.

2.3 Lossless Transmission Line

For a lossless transmission line (R = 0 and G = 0), = j and the character-istic impedance is real and is expressed as (from Equation (2.15))

Z0 =

LC

(2.33)

For a lossless transmission line of length that is terminated with a loadimpedance ZL at x = 0, we can write the following equations to obtain theinput impedance:

Zin() = V()I () =Vf e j + Vr e j

Vf e j Vr e j Z0 (2.34)

ZL = V(0)I (0) =Vf + VrVf Vr Z0 (2.35)

Lets define the ratio of reflected to forward wave amplitudes (Vf /Vr ) atx = 0 as the load reflection coefficient.

L = VrVf (2.36)


From Equations (2.35) and (2.36), we can write

ZL = 1 + L1 L Z0 (2.37)

which can also be rewritten as

L = ZL Z0ZL + Z0 (2.38)

It should be noted that the reflection coefficient L = 0 when ZL = Z0. Inthis condition, we say that the transmission line is terminated with a matchedload, and hence no reflection occurs at the load end. In other words, theimpedance Zin(0) = Z0 and the transmission line has similar impedance asthe infinitely long transmission line with no reflections. The deviation of ZLfrom Z0 determines the amount of reflected wave from the load. Now we canwrite Zin() in terms of ZL and Z0 as follows:

Zin() = 1 + Le j2

1 Le j2 Z0 (2.39)

Zin() =ZL

(1 + e j2) + Z0 (1 e j2)

Z0(1 + e j2) + ZL (1 e j2) Z0 (2.40)

which leads to the famous equation

Zin() = ZL + j Z0 tan ()Z0 + j ZL tan () Z0 (2.41)

The above equation is quite interesting and will prove very useful in laterchapters. The average power flow in the transmission line in the forwarddirection is expressed as

Pav = 12 Re(V(x) I (x)

)(2.42)

Substituting from (2.11) and (2.12):

Pav(x) = 12Z0 Re((

Vf e x + Vr e x)(

Vf e x Vr e

x)) (2.43)= 1

2Z0Re

(|Vf |2e(+ )x |Vr |2e (+ )x+ Vf Vr e (

)x Vf Vr e()x) (2.44)

= 12Z0

Re(|Vf |2e2x |Vr |2e2x + Vf Vr e2 jx Vf Vr e2 jx) (2.45)

= 12Z0

(|Vr |2 e2x |Vr |2e2x) (2.46)= |Vf |

2e2x

2Z0

(1 |L |2e4x

)(2.47)


For a lossless transmission line, this average power flow is reduced to

Pav(x) = |Vf |2

2Z0

(1 |L |2

)(2.48)

which is independent of x since there is no power loss through the transmis-sion line. From the above equation, we can calculate the input power at thesource at x = and the power delivered to the load at x = 0:

PL = Pav(0) = |Vf |2

2Z0

(1 |L |2

)(2.49)

Pin = Pav() = |Vf |2e2

2Z0

(1 |L |2e4

)(2.50)

Therefore, we can express the power transfer efficiency of the transmissionline as the ratio of the power delivered to the load to the input power,

= PLPin

= e2 1 |L |2

1 |L |2e4 (2.51)

which is dependent on the load reflection coefficient, the transmission linelength, and the attenuation constant. Zero efficiency occurs when |L | = 1,where no power is delivered to the load. The maximum power transfer ef-ficiency for a given transmission line length occurs for a matched load withL = 0 and is expressed as

max = e2 (2.52)

2.4 Shorted Transmission Line

For the case of shorted transmission line (ZL = 0), Equation (2.41) can besimplified to

Zin() = j Z0 tan () (2.53)At = 2 and its odd multiples, the input impedance of a shorted transmis-sion line is infinity. Since = 2

, where is the wavelength of the traveling

wave, this infinite input impedance condition occurs at = 4 and its oddmultiples as illustrated in Figure 2.3. This observation is particularly usefulin bias circuits since the transmission line will be acting as a short circuit at DCwhile it presents an open circuit at the desired frequency. At even multiplesof = 4 , the input impedance is zero. It is also important to note the inputimpedance of the shorted transmission line is inductive for 0 < < 4 , andit becomes capacitive for 4 < 0 ) and see what happens when approaches zero. For a lossy shortedtransmission line, the input impedance can be written as

Zin () = tanh ( ) Z0 = tanh ( + j) Z0 = Rin + j Xin (2.54)The real and imaginary components of the above expression are plotted

in Figure 2.4 for several values of . It is noted that as increases from zerotoward 4 , both input resistance and reactance increase. As approaches

4 ,

the input resistance continues to increase, but the input reactance peaks atsome value of , and then drops to zero at = 4 . At this value of , theinput resistance is at its maximum. As decreases toward zero, the point ofmaximum reactance approaches (, Xin) = ( 4 , ), while the maximum inputresistance decreases until it becomes zero for a lossless transmission line. It isimportant to note that for a finite nonzero value of , the input impedance ofthe transmission line is always finite, regardless of its length .

2.5 Voltage Standing Wave Ratio (VSWR)

For a lossless transmission line, the magnitude of the voltage waveform atany point x can be expressed as

|V(x)| = |Vf ||1 + Le2 jx| = |Vf ||1 + |L |e j (+2x)| (2.55)


Alpha = 0Alpha = 0.1Alpha = 0.2

Imaginary Part of Zin (Xin)

Real Part of Zin (Rin)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

010

5

0

5

10

0

5

10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FIGURE 2.4Input resistance and reactance of a lossy transmission line terminated with a short circuit.

where is the phase of the load reflection coefficient. From the above expres-sion, the maximum and minimum voltage magnitudes along the transmissionline can be calculated as

Vmax = |Vf |(1 + |L |) (2.56)Vmin = |Vf |(1 |L |) (2.57)

and the ratio of maximum to minimum voltage magnitudes is called thevoltage standing wave ratio (VSWR) and is expressed as

VSWR = 1 + |L |1 |L | (2.58)

The importance of the VSWR comes from the ease of measurement since it isa voltage ratio. The load impedance can be indirectly calculated by measuringthe maximum and minimum voltage magnitudes along the transmission lineand the location of the voltage minimum. The VSWR can be used to calculate|L |, while the location of the voltage minimum (min) is used to calculate from the following equations:

|L | = VSWR 1VSWR + 1 (2.59)

= + 2min (2.60)


2.6 Smith Chart

The Smith chart is a graphical tool to calculate the impedance as a function ofthe reflection coefficient. Equation (2.6) can be generalized for the impedanceand reflection coefficient at any point x on the transmission line:

Z(x) = 1 + (x)1 (x) Z0 (2.61)

To simplify our equations, we normalize the impedance as Zn(x) =Z(x)/Z0.

Zn(x) = 1 + (x)1 (x) (2.62)

Lets define Zn(x) = Rn + j Xn and (x) = a + jb; then

Rn + j Xn = 1 + a + jb1 a jb =(1 + a + jb) (1 a + jb)

(1 a )2 + b2 (2.63)

Therefore, we can write Rn and Xn in terms of a and b:

Rn = 1 a2 b2

(1 a )2 + b2 (2.64)

Xn = 2b(1 a )2 + b2 (2.65)

We can rewrite Equation (2.64) as follows:

(a Rn

1 + Rn

)2+ b2 = 1

(1 + Rn)2(2.66)

which is an equation of a circle in the -plane, centered at ( Rn1+Rn , 0) with aradius of 11+Rn . We can also rewrite Equation (2.65) as follows:

(a 1)2 +(

b 1Xn

)2= 1

X2n(2.67)

which is an equation of another circle in the -plane, with a center at (1, 1Xn )and a radius of 1Xn . The transformation from horizontal lines in the z-planeinto circles in the -plane is illustrated in Figure 2.6. The graphical represen-tations of Equations (2.66) and (2.67) are illustrated in Figures 2.5 and 2.6. InFigure 2.5, the vertical Rn lines in the z-plane are transformed into circles in the-plane with centers along the a -axis. All Rn circles are tangent to the a = 1line at the point (a, b) = (1, 0), where |Zn| = according to Equation (2.62). Itis important to recognize the special circles in the -plane. The circle Rn = 0 is


Rn

Xn

0 1Rn = 2 a

bz-plane

Rn = 1Rn = 0.5

2

Rn = 0

(0,0)

-plane

(0,1)

a(1,0)(1,0)

(0,1)

0.5

FIGURE 2.5Transformation from vertical lines in the z-plane into circles in the -plane.

equivalent to || = 1, which is called the unit circle. The unit circle respresentsthe boundary between negative values of Rn outside the unit circle (|| > 1where reflected power from the load is larger than incident power) and pos-itive values of Rn inside the unit circle (|| < 1 where reflected power is lessthan incident power) . For Rn = 0, the circle in the -plane passes throughthe origin, which represents a matched termination (|| = 0). As Rn ,

Xn

Rn0

0.5

1

2

0.5

2

1

Xn = 2

Xn = 2

Xn = 1

Xn = 1

Xn = 0.5

Xn = 0.5

Xn = 0 a

bz-plane

(0,0)(1,0)

(1,0)

(0,1)

(0,1)

-plane

FIGURE 2.6Transformation from horizontal lines in the z-plane into circles in the -plane.


Rn = 2

Xn = 0

R n = 0 X

n = 0.5R n

= 0.

5

X n = 0

.5

Xn =

2

Xn = 1

R n = 1

X n =

2

aopen

b

(0,1)

Inductive

Capacitivematched

short

FIGURE 2.7Basic Smith chart showing Xn and Rn lines.

the Rn circles shrink toward the point (a, b) = (1, 0). For passive elements,0 < Rn < , and we always operate inside the unit circle (|| < 1). Therefore,Smith charts only show the -plane inside the unit circle.

On the other hand, the horizontal Xn lines in the z-plane are transformedinto circles in the -plane with centers along the vertical line a = 1. For passiveelements, only the part of Xn circles inside the unit circle of the -plane isshown. All Xn circles are tangent to the a -axis at the point (a, b) = (1, 0).For Xn = 0, the circle has an infinite radius according to Equation (2.67) andreduces to a point on the a -axis. For inductive impedances where Xn > 0, thecircles reside in the top half of the -plane. On the other hand, for capacitiveimpedances where Xn < 0, circles reside in the bottom half of the -plane.

The above observations from Figures 2.5 and 2.6 are summarized in Figure2.7, which is considered the basic Smith chart. Both Rn and Xn circles are shownon the same chart so that we can read Zn and values instantaneously. It isimportant to note that the short-circuit (Zn = 0) and open-circuit (|Zn| = )terminations are located at leftmost and rightmost points on the unity circleperiphery. The matched point (Zn = 1 and = 0) is located at the center ofthe unit circle.

As we move toward the generator, the reflection coefficient (x) can becalculated as follows:

(x) = Vr e x

Vf e x= Le2 x (2.68)


j0.5

j0.4

j0.3

j0.2

j0.1

j0.1

j0.2

j0.3

j0.4

j0.5

j1

j2

j3

j4j5

j10j20

j20j10

j5j4

j3

j2

j1

inf0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 10 20

FIGURE 2.8 vs. normalized length for a lossless transmission line.

For a lossless transmission line, (x) = Le2 jx. The magnitude of is in-dependent of x and phase is 4 x

= 4xn, where xn = x is the normalized

distance from the load. Hence, moving toward the generator (in the x direc-tion) on the transmission line is equivalent to moving counter-clockwise ona circle with its center at the origin of the Smith chart. We spin 360 aroundthe origin of the Smith chart for every 2 of the transmission line. This is il-lustrated in Figure 2.8 for the case of a normalized load impedance Zn = 0.5.The reflection coefficient at the load L = (0) = 13 .

For the case of a lossy transmission line, (x) = Le2xe2 jx, and the mag-nitude of drops exponentially as we move toward the generator (awayfrom the load). This is illustrated in Figure 2.9 for the case of Zn = 0.5. For aninfinitely long transmssion line, (x) drops to zero as x approaches . How-ever, this does not mean that higher power is delivered to the load for longertransmission line. It simply means that higher power is transmitted from thegenerator into the transmission line and higher losses in the transmission lineitself. This is evident from Equation (2.51).


j0.5

j0.4

j0.3

j0.2

j0.1

j0.1

j0.2

j0.3

j0.4

j0.5

j1

j2

j3

j4j5

j10j20

j20j10

j5j4

j3

j2

j1

inf0 0.1 0.2 0.3 0.4 1 2 3 4 5 10 200.5

FIGURE 2.9 vs. normalized length for a lossy transmission line.

The Smith chart that we have discussed so far is also called the impedanceSmith chart. Alternatively, one can derive the admittance Smith chart bywriting

Yn(x) = Zn(x) = 1 (x)1 + (x) (2.69)

Figure 2.10 illustrates the admittance Smith chart where constant-conductance and constant-susceptance circles are drawn. The constant-conductance circles are all tangent to the a = 1 line on the -plane at thepoint (a, b) = (1, 0). The constant-susceptance circles are all tangent to the a -axis at the same point. Note that the locations of the short circuit (Yn = ) andopen circuit (Yn = 0) in the -plane are the same as they are in the impedanceSmith chart. Note also that negative susceptance cirles are at the top half ofthe admittance chart, while positive susceptance circles are at the bottom half.Therefore, the top half still represents inductive elements, while the bottomhalf represents capacitive elements, as in the impedance chart.


j0.5

j0.4

j0.3

j0.2

j0.1

j0.1

j0.2

j0.3

j0.4

j0.5

j1

j2

j3

j4j5

j10j20

j20j10

j5j4

j3

j2

j1

inf 00.10.20.30.40.5123451020

FIGURE 2.10Admittance Smith chart.

Bibliography

1. R. E. Collin. (1966). Foundations for Microwave Engineering. New York:McGraw-Hill, 1966.

2. U. L. Rohde, G. D. Vendelin, and A. M. Pavio. (1990). Microwave CircuitDesign Using Linear and Nonlinear Techniques. New York: John Wiley, 1990.

3. D. M. Pozar. (1997). Microwave Egineering. 2nd Edition. New York: JohnWiley, 1997.

3Trigger Mode Distributed Wave Oscillator

Many of todays electronic systems use an oscillator circuit as a high frequencysignal source. In the last decade or two, many transmission line-based newtraveling or standing wave oscillator techniques have been introduced as avery high frequency signal source with the availability of oscillation phases atthe tap points along the line. Availability of such GHz-range, high-resolutionoscillation phases is one of the most significant advantages of these oscil-lators compared to their LC-based lumped counterparts. LC-based lumpedones use two reactive components, an inductor and a capacitor, to create aresonant circuit, in an ideal case indefinitely transferring the energy from oneto the other. However, in reality, the loss mechanisms associated with thesereactive devices (can be modeled as resistance (R) or transconductance (G)elements) require active amplifying circuitry to compensate for these losses.The well-known classical implementation for such an active compensationcircuit is a negative resistance circuit formed by cross-coupled active devices.The metal oxide semiconductor field effect transistor (MOSFET) implemen-tation of this configuration is shown in Figure 3.1. The resultant oscillationfrequency depends on the inductance and the capacitance values and can bewritten as

fosc = 12LtankCtank(3.1)

Similarly, distributed counterparts can be constructed using transmissionlines. A transmission line is, in general, parallel running conductors sepa-rated by a dielectric material. Microstrip line (Figure 3.2), coplanar waveguide(Figure 3.3), coplanar strip line (Figure 3.4), and differential coplanar wave-guide (Figure 3.5) are some of the most common transmission line structures.Although any of these structures can be used to construct an oscillator, thedifferentially symmetric ones are more favorable since the opposite phasesof a signal are already available (coplanar strip line and differential coplanarwaveguide).

Since these transmission lines effectively represent a distributed LC struc-ture, an oscillator similar to a lumped LC tank oscillator can be formed asshown in Figure 3.6. In this figure, L0, C0, R0, and G0 represent inductanceper unit length, capacitance per unit length, resistance per unit length, andconductance per unit length for a differential transmission line stretching inthe z-direction. The inductance per unit length and capacitance per unit lengthdetermine the phase velocity of the wave propagating. The phase velocity of

23


Ltank Rloss

Ctank

Gloss

Ltank Rloss

Ctank

Gloss

IOSC

VDD

FIGURE 3.1Lumped LC tank oscillator.

a wave is as follows:

v = 1L0C0

(3.2)

where L0 and C0 are inductance per unit length and capacitance per unitlength, respectively. Then, for a given total length of transmission line, the

Cross-section

Top view

GroundDielectric

Signal

FIGURE 3.2Microstrip line.

Trigger Mode Distributed Wave Oscillator 25

Cross-sectionTop view

Signal

Ground

FIGURE 3.3Coplanar waveguide.

Cross-section

Top view

Signal+ Signal

FIGURE 3.4Coplanar strip line.

Cross-section Top view

Signal+

Ground

Signal

FIGURE 3.5Differential coplanar waveguide.

L0dx R0dx

L0dx R0dx

C0dxG0dx

x

FIGURE 3.6Distributed oscillator structure using transmission lines.


oscillation frequency can be calculated as

fosc = 1LtotCtot(3.3)

where Ltot and Ctot are the total inductance and total capacitance along thetransmission line. Again, cross-coupled active amplifiers are used to compen-sate for the conductor and substrate losses. Thanks to the distributed natureof these transmission line oscillators, multiple phases of an oscillation areavailable along the transmission line, whereas only two 180 opposite phasesare available in the case of lumped LC tank oscillators. Distributed wave os-cillators, rotary traveling wave oscillators, and standing wave oscillators aredifferent classes of existing transmission line-based oscillators all utilizing thedistributed LC nature of a transmission line structure. These existing topolo-gies will be touched upon briefly in Sections 3.1 to 3.3. Section 3.4 introducesa new topology, trigger mode distributed wave oscillator (TMDWO) [7], anddiscusses its advantages and disadvantages compared to existing topologies.Section 3.6 presents a test structure and the related measurement results ofthe proposed technique, and Section 3.7 is the conclusion.

3.1 Commonly Used Wave Oscillator Topologies

One of the earliest traveling wave oscillator inventions was distributed waveoscillator [24,9,10,16]. Figure 3.7 shows a simplified distributed oscillatorstructure. The actual shape can be in any closing geometric form bringingpoint A to the vicinity of point B so that a dashed AC coupled connection can

VDD

VBIAS

Rmatch

Cbyp

Transmissionlines with

characteristicimpedance Z0

Rmatch

A

B

FIGURE 3.7Basic distributed oscillator structure.


be obtained using a capacitor Cbyp. The reflections resulting from the mismatchof the biasing resistor, Rmatch, to the line impedance, Z0, can be a significantsource of disturbance in the steady-state oscillation waveforms. This effect,together with an additional nonideality due to the bypass capacitor Cbyp, isthe main drawback of the technique.

3.2 Rotary Traveling Wave Oscillator (RTWO)

Another transmission line oscillator technique, rotary traveling wave oscil-lator technique, avoids this disadvantage by direct cross-coupling of the endpoints with an additional cost of odd symmetry introduced by this cross-ing of the transmission lines (Figure 3.8) [8,15]. The single-wire closed-loopstructure of an RTWO limits the disturbances to one crossover, which can stillbe significant, especially at high frequencies. Once enough gain is provided,there is no latch-up danger for the technique, since it utilizes a single-lineDC-coupled closed-loop structure. A practical circular layout of the RTWO isshown in Figure 3.9, where the end points are brought together to implementthe crossover connection. Mutlimode oscillation may occur in RTWO withfrequencies that satisfy the periodic boundary condition in the continuouslimit [1]. The mode frequencies may not be exact multiples of the fundamen-tal frequency due to dispersion (see Section 2.2). Even modes, where the twolines are excited with the same signal, are suppressed with the cross-coupledinverters along the RTWO. Therefore, only odd modes may exist, where thetwo lines are excited with opposite phases. Odd modes are supported bythe cross-coupled inverters along the RTWO. So, the differential coupled linecan be simplified with a single-ended model. To analyze the odd modes ofoscillation in the RTWO, lets first consider the poles of the short-circuitedlossless transmission line of length . The driving point admittance of such a

FIGURE 3.8Rotary traveling wave oscillator.


Gsec

Csec

Lsec

Rsec

FIGURE 3.9Rotary traveling wave oscillator: circular layout.

transmission line can be obtained from (2.53) as

Yin = 1Zin() = jZ0

cot () (3.4)

The poles of the short-circuit driving point admittance occur at frequencieswhere Yin = or = n , where n is the pole number. Using Equation (2.18),we obtain the pole locations as follows:

pn = j n

LC= j n

LsecCsec(3.5)

where Lsec = L and Csec = C are the total inductance and capacitance of thetransmission line section, respectively. Equation (3.5) gives an infinite numberof possible oscillation modes. Realistically, however, the maximum oscillationfrequency is determined by the transmission line losses or the periodic loadingof the transmission line.

The RTWO is loaded periodically with the cross-coupled inverters as shownin Figure 3.9, where Csec represents the lumped capacitance of the transmis-sion line, the parasitic capacitances of the cross-coupled inverters, and any


other capacitaces needed for biasing or tuning the RTWO. If the RTWO isapproximated as a chain of transmission line sections as shown in Figure 3.9,the poles in the case of lossless network (assuming Rsec = 0 and Gsec = 0)are [1]:

pn = 2 jLsecCsecsin

( n2N

)(3.6)

where N is the number of transmission line sections in the RTWO. For largevalues of N, Equation (3.6) reduces to Equation (3.5), due to the fact thatRTWO is more distributed in this case. The highest-order mode occurs whenn = N when the pole frequency is

pn = 2 jLsecCsec= C (3.7)

This mode frequency represents the cutoff frequency of the resonator, whichlimits the maximum possible rise time, and hence limits the minimum oscil-lator phase noise that can be obtained.

It is neccessary to calculate the quality factor (Q) of the resonator in order tocalculate the phase noise and determine the size of the cross-coupled invertersto guarantee oscillation. By definition, the quality factor is determined by

Q = 2 energy storedEnergy dissipated in one cycle

= 2 f0 energy storedaverage power dissipation (3.8)

where the numerator is the energy stored in the L0 and C0 distributed compo-nents, while the denominator is the average power dissipated in the R0 andG0 components. Therefore, we can obtain an expression of the transmissionquality factor as

Q = 012 L0 I

2p + 12 C0V2p

12 R0 I

2p + 12 G0V2p

(3.9)

where Ip and Vp are the peaks of current and voltage waveforms, respectively.Substituting Vp = Z0 Ip, we get

Q = L0 + C0 Z20

R0 + G Z20= L0/Z0 + C0 Z0

R0/Z0 + G Z0 (3.10)

For a low loss transmission line, we can substitute Z0 =

L0C0

; therefore, weobtain

Q = 2

L0C0R0/Z0 + G Z0 =

3db(3.11)


100 101 102 1030

2

4

6

8

10

12

14

16

18

20

Frequency (GHz)

QT

FIGURE 3.10QT of periodically loaded transmission line with 3db = 10 GHz and c = 500 GHz.

where

3db = R0/Z0 + G Z02L0C0(3.12)

The quality factor equation (3.11) is valid only for distributed transmissionline. For periodically loaded transmission line, Equation (3.11), the qualityfactor can be modified as

QT = Q(

1 (

c

)2)(3.13)

= 3db

(1

(

c

)2)(3.14)

Therefore, the above equation is very important in determining the qualityfactor at different modes of oscillation, and hence the possibility of oscillation.Equation (3.14) is plotted in Figure 3.10 for 3db = 10 GHz and c = 500 GHz.The maximum QT can be obtained by differentiating Equation (3.14) withrespect to and equalizing it to zero:

d QTd

= 0 = 13db

33db

(

c

)2(3.15)


Therefore, the frequency of the maximum QT is

QTmax =c

3(3.16)

which is substituted in Equation (3.14) to obtain the maximum QT :

QTmax = 23

3

c

3db(3.17)

3.3 Standing Wave Oscillator (SWO)

Standing wave oscillators are another group of transmission line oscillatorsthat would utilize transmission line structures [5,6,12,13]. Standing wavesare formed by superimposing the forward and backward traveling waves onthe same transmission medium simultaneously. The two basic standing waveoscillator topologies, /4 SWO and /2 SWO, are shown in Figures 3.11 and3.12, respectively [12]. A /2 SWO is basically a combination of two /4 SWOsaround a center symmetry point, with the fundamental operating principlestaying the same. In this type of oscillator, the differential transmission linestructure is driven by a cross-coupled amplifier pair at one end, whereas theother end is shorted. The waves created at the amplifier end are reflected backat the short end, causing a reverse propagating wave along the transmissionline. In the steady state, the forward and reverse waves coexist, creating astanding wave along the line. This would imply amplitude variations in theoscillation phases along the line, gradually diminishing and eventually reach-ing zero at the short end. To derive an expression for the standing wave alongthe transmission line in Figure 3.11, we recall Equation (2.11):

V(x) = Vf e x + Vr e x (3.18)

Then we substitute = j for a lossless transmission line and use Equation(2.36):

V(x) = Vf(e jx + Le jx

)(3.19)

For a shorted transmission line, L = 1 according to Equation (2.38); there-fore:

V(x) = Vf(e jx e jx) = 2 j Vf sin(x) (3.20)

The above equation indicates a standing wave that has zero amplitude at theshorted load (x = 0) and maximum amplitude at the cross-coupled amplifierpair (x = 4 ). The phase of the voltage waveform is independent of x. In otherwords, the sum of forward traveling wave (where phase increases with x) andreverse traveling wave (where phase decreases with x) results in a standing


/4

Short

x

FIGURE 3.11/4 standing wave oscillator.

wave where phase is constant along the transmission line. We can write theexpression of the current along the transmission line from Equation (2.12):

I (x) = VfZ0

(e jx + e jx) = 2Vf

Z0cos(x) (3.21)

which indicates maximum current amplitude at the shorted load and zeroampliude at the cross-coupled amplifier pair.

Circular standing wave oscillator (CSWO) is another standing wave tech-nique that would not require any reflection mechanism, but rather a circularsymmetry to create reverse propagating waves along the transmission linemedium [5,6]. As shown in Figure 3.13, the energy is injected into a closed-loop transmission line structure equally and travels symmetrically along thering in clockwise and counterclockwise directions. These counter-travelingwaves create standing waves with an amplitude profile as shown in Figure3.14. The energy is injected at two opposite points (A and B) with additionaldashed connections to force the main mode. Additionally, at least one of thequiet ports has to be shorted to prevent any latch-up problems. This alsoreduces this structure to a single-line structure.

All of these standing wave oscillator structures have a critical drawback ofamplitude variations, which would limit their usage to a very limited set of ap-plications. The oscillation phases corresponding to the quiet ports would noteven exist, compromising the main advantage of transmission line oscillators.

ShortShort

/4 /4

FIGURE 3.12/2 standing wave oscillator.


VDDShort

A

B

FIGURE 3.13CSWO structure.

VDD

Magnitudeof standing

wave

FIGURE 3.14Amplitude profile along the CSWO structure.


3.4 TMDWO Topology

The trigger mode distributed wave oscillator described in this section pro-poses an alternative approach for a wave oscillator that can provide uniqueadvantages compared to existing prior techniques described in the precedingsection. Figure 3.15 shows the diagram of the proposed traveling wave tech-nique. In this technique, an oscillation is triggered in two independent trans-mission lines, each carrying opposite phases of the oscillation at any particularlocation along the parallel running differential transmission lines. The wavestraveling on these two independent lines would possess the same propagationcharacteristic since the lines and the loading corresponding to each line areidentical and symmetric. This means that once successfully created, the oppo-site phases of an oscillation signal can propagate indefinitely together with thehelp of cross-coupled amplifiers that are finely distributed along the lines com-pensating for the losses. These cross-coupled amplifier unit cells should bedistributed in maximally symmetric fashion, resulting in a smooth travelingwave without any amplitude or phase distortion. These inverting amplifiersuse the signal in one of the lines as booster for the opposing phase travelingin the other line. However, the difficulty arises with the initial existence ofthese opposite oscillation phases in the independent identical lines. Since theconstituent transmission lines are two independent conductors, there is nomechanism to guarantee traveling wave buildup. Thus, during the power-up, the system would latchup even before any oscillation buildup (i.e., oneof the lines would be pulled up to VDD and the other one to GND due to thecross-coupled amplifiers). This condition is overcome by using another aux-iliary oscillator that injects multiple opposite phases into the correspondingphase locations along the loop during the power-up. This would also ensurean oscillation only in the fundamental mode.

The closed shape of these parallel-running differentially triggered transmis-sion lines, which is shown to be circular, can take any symmetrical geometricform ending up with symmetrical injection points corresponding to the avail-able triggering phases. Figure 3.16 shows the direction of these resulting op-posite traveling waves, while Figure 3.17 shows the time domain waveformsfor some of the oscillation phases corresponding to the various locations alongthe transmission line. Another unique feature of the technique that relaxes therouting requirements of the available phases in the actual physical layout isthat it can provide all of the oscillation phases in both of the two independentconductors, the same phase being available at the two opposite sides withrespect to the axis of symmetry. Hence, the structure can be viewed as trig-gering two identical independent transmission line loops in opposite phasesso that each one of them uses the other as a sustainer.

Figure 3.15 illustrates a four-phase trigger mode traveling wave oscillator.Any well-known quadrature oscillator type, such as a four-stage differentialring voltage-controlled oscillator (VCO) or a quadrature LC tank VCO, can be


AuxiliaryTriggerOscillator

ph90

ph270

ph90

ph270

ph180

ph0

ph18

0

ph0

Oscillationdetectoren

Freq_ctr

Vtune

Vtune

Vtune

Vtune

A

B

VDD

A B

FIGURE 3.15TMDWO circuit diagram.

used as a triggering auxiliary oscillator. The schematic of the four-stagedifferential ring VCO, including an individual delay element, is shown inFigure 3.18.

Every delay cell in this VCO is connected to a voltage-controlled currentsource to provide frequency tuning. An oscillator with 8, 16, or more numberof phases can also be used for this purpose, routing the available phases totheir corresponding locations along the transmission lines. The quadrature


180

0 135

315

90 270 270 90

225

45

180

0

315

135

45

225

FIGURE 3.16Opposite wave propagation in TMDWO conductors.

phases for this particular case are denoted as ph0, ph90, ph180, and ph270. Animportant requirement for this triggering oscillator is that its tuning rangeshould cover the estimated frequency range of the traveling wave oscilla-tor. This triggering oscillator is powered up first, injecting the quadraturesignals into the corresponding quadrature locations on the transmission linering. As this close-by oscillation is injected into the ring, the supply for thecross-coupled amplifiers, VDD_ring, starts to ramp up. As the ring supplyramps up, the weak injection oscillation is amplified by these cross-coupledamplifiers, tracking the supply and effectively preventing the lines to latchup. Finally, after a successful oscillation buildup, the detector circuit detectsthe oscillation and powers down the triggering auxiliary oscillator to savepower, and thereafter the resultant traveling wave would sustain itself unlessa long-lasting power glitch causes it to latch up. In the case of such an occasionresulting in latch-up, the detector circuit reinitiates the start-up sequence torebuild the oscillation. However, in order for the defined start-up sequenceto be successful, the triggering oscillation frequency should be in the close


0

45

90

135

180

225

270

315

Time

Ampl

itude

FIGURE 3.17Time domain representation of some of the oscillation phases on the lines.

vicinity of the actual traveling wave frequency that would end up in the lines.Since the parameters governing the oscillation frequencies of both oscillatorsare process dependent, they can vary considerably. In order to guarantee asuccessful triggering in varying environments and process conditions, thementioned power-up sequence is repeated for a wide range of frequencies asthe triggering oscillator frequency is swept across a wide frequency range. Inevery sweep step, the oscillation detector looks for an oscillation. In case nooscillation is detected, the detector circuit updates the triggering frequencyto the next step by a frequency control word, freq-ctr, and restarts the supplyramp sequence. Once a successful oscillation is detected, the sweep process

PH0

PH180

PH45

PH225

PH90

PH270 PH315

PH135

Vtune

FIGURE 3.18A four-stage trimmable differential VCO as a trigger oscillator.


is terminated and the triggering oscillator is powered down. Although thetechnique might sound inferior considering the additional triggering circuitryrequired, the even symmetry proposed by the technique can provide a veryhigh phase accuracy that is very difficult to obtain at very high frequencieswith the classical wave oscillator techniques. This is mainly due to the fact thatthere is no source of asymmetry, such as odd number of line crossings or realtermination impedances that may not match the line impedance accurately.Moreover, since each of these independent transmission lines corresponds toa full lap of traveling wave rather than half, using the same total conduc-tor length, one can obtain double-oscillation frequency compared to existingdistributed wave oscillators. This implies that much higher frequencies canbe obtained using this technique. Regarding the triggering circuitry that maybe considered to be a disadvantage of this circuit, the overhead is insignif-icant. The area cost of an extra triggering circuitry in an integrated circuitimplementation is minimal, since the area of a ring oscillator is an order ofmagnitude smaller than the area of the main oscillator. Power is also not aconcern since the triggering circuitry is powered down after triggering thesequence, consuming no power during normal operation. It should also benoted that since the triggering is applied in a symmetric distributed fashion,and the amplifiers are also finely distributed in a very symmetric fashion,there is source of reflection or any distinctive preferred way for the energy tosplit. Thus, the wave is forced into traveling mode rather than standing mode.Standing wave oscillation generally is not a preferred mode of operation dueto severe amplitude variations in the oscillation phases. The distributed var-actors shown in Figure 3.15 constitute a fraction of the total capacitance on theline providing a tuning range for the TMTWO, and hence enabling its use inphase-locked loops. The control voltage, Vtune, changes the total amount ofcapacitor on the line, thus determining the frequency of the traveling wave.

3.5 Phase Noise in Traveling Wave Oscillators

The main advantage of traveling wave oscillators over the traditional CMOSring oscillators is that phases are generated with transmission line delaysrather than transistor delays. Therefore, we expect to have better phase noiseperformance in traveling wave-based oscillators. In this section, we aim toderive the phase noise of the SWO and RTWO [11]. We will start with thephase noise in SWO, from which we can derive the phase noise in RTWO.

3.5.1 Phase Noise in SWO

The /2 SWO discussed in Section 3.3 can be decomposed into two parallel/4 SWOs, as shown in Figure 3.11. To simplify the analysis, we start with /4


L0dx R0dx

L0dx R0dx

C0dxG0dx

-Gm

Zin(0 + )= Rp//(jXp)

FIGURE 3.19Model of the /4 SWO.

SWO phase noise calculation, and then we can derive the phase noise of the/2 SWO. The /4 SWO is modeled as shown in Figure 3.19.

The input impedance of the /4 SWO, at an offset frequency from theoscillation frequency 0, can be calculated from (2.53) by setting = /4.Therefore:

Zin(0 + ) = j Z0 tan (/4) (3.22)where is the propagation constant and is a function of the input signalfrequency. Therefore, in this equation, we cannot write = 2/ becausewe are interested in finding the value at 0 + rather than 0. We recallEquation (2.18), which is written as

(0 + ) = (0 + )

LC (3.23)

Hence, we obtain the expression of Zin as a function of and 0:

Zin(0 + ) = j Z0 tan((

1 + 0

)0

LC/4

)(3.24)

= j Z0 tan((

1 + 0

)/2

) j Z0 20

(3.25)

The frequency of oscillation is related to the propagation velocity and thewavelength as

0 = 2vp

= vp2q

(3.26)

where q is the length of the quarter-wave shorted transmission line. Substi-tuting from Equation (2.28), we obtain

0 = 2q

LC

= 2

LsecCsec(3.27)

where Lsec and Csec are the total inductance and capacitance of the quarter-wave transmission line, including the input capacitances of the cross-coupled


inverters. Substituting from Equation (3.27) into Equation (3.25), we get themagnitude of the input impedance as

|Zin(0 + )| 420 Lsec

2(3.28)

To obtain the quality factor of the resonator, the resonator loss is representedwith a shunt resistance Rp at the transmission line. To obtain the value of Rpfor a distributed shorted transmission line, we recall Equation (2.54) for alossy transmission line:

Zin (0 + ) = tanh ( ) Z0 = tanh ( + j) Z0

= 1 e22 j

1 + e22 j Z0

= 1 e2 j (1+

0)

1 + e2 j (1+ 0 )Z0

= 1 + e2 j

0

1 e2 j 0 Z0

2Z0+2 + j 0

= Z0

4 + j 0 2= 11

Rp+ jXp

(3.29)

Therefore, we can obtain the values of Rp and Xp as follows:

Rp = Z0/(

4

)(3.30)

Xp = Z0 20

(3.31)

Note that the expression for Xp agrees with Equation (3.25). By substitutingthe value of from Equation (2.17), we obtain the general expression for Rp:

Rp = 8Z0( RZ0 + G Z0)

(3.32)

Now we can find the 3d B bandwidth of the input impedance Zin by writing Zin(0 + 3d B)Zin(0) = 12 (3.33)

It is easier to write it in terms of input admittance: Yin(0)Yin(0 + 3d B) = 12 (3.34) 1/Rp1/Rp + j/Xp (0 + 3d B) = 12 (3.35)


Therefore, at the 3dB frequency we have

Xp(0 + 3d B) = Rp = Z0 20

(3.36)

where Equation (3.31) is used. Now we can write the expression for the 3dBbandwidth as

3d B = Z0 20 Rp

(3.37)

Finally, we obtain the quality factor of the resonator as

Q = 023d B

= 02Z0 20 Rp

= Rp4Z0

(3.38)

By substituting Rp from Equation (3.30), we get

Q =

(3.39)

To study the phase noise of /4 SWO, assume the NMOS cross-coupledpair shown in Figure 3.21 is used. The cross-coupled pair is biased throughthe short-circuit side of the /4 transmission line. Assuming current-limitedoperation, the differential current flowing into the transmission line is ideallya square wave of amplitude IB . The fundamental component of this currentwaveform is 4

IB . Therefore, the amplitude of oscillation at the fundamental

frequency is expressed as

Vp = 4

IB Rp (3.40)

In this oscillator structure, there are three sources of noise: the transmissionline loss represented by Rp, the tail transistor noise, and the commutatingpair noise. Next, we will study the phase noise due to these individual noises

Ahmet Tekin

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