Cmos Single Chip Fast Frequency Hopping Synthesizers For Wireless Multi-Gigahertz Applications

CMOS SINGLE CHIP FAST FREQUENCY HOPPING SYNTHESIZERS FOR WIRELESS MULTI-GIGAHERTZ APPLICATIONS

ANALOG CIRCUITS AND SIGNAL PROCESSING SERIES Consulting Editor: Mohammed Ismail. Ohio State University Titles in Series: ANALOG CIRCUIT DESIGN TECHNIQUES AT 0.5V Chatterjee, S., Kinget, P., Tsividis, Y., Pun, K.P. ISBN-10: 0-387-69953-8 IQ CALIBRATION TECHNIQUES FOR CMOS RADIO TRANCEIVERS Chen, Sao-Jie, Hsieh, Yong-Hsiang ISBN-10: 1-4020-5082-8 LOW-FREQUENCY NOISE IN ADVANCED MOS DEVICES Haartman, Martin v., Östling, Mikael ISBN-10: 1-4020-5909-4 THE GM/ID DESIGN METHODOLOGY FOR CMOS ANALOG LOW POWER INTEGRATED CIRCUITS Jespers, Paul G.A. ISBN-10: 0-387-47100-6 PRECISION TEMPERATURE SENSORS IN CMOS TECHNOLOGY Pertijs, Michiel A.P., Huijsing, Johan H. ISBN-10: 1-4020-5257-X CMOS CURRENT-MODE CIRCUITS FOR DATA COMMUNICATIONS Yuan, Fei ISBN: 0-387-29758-8 RF POWER AMPLIFIERS FOR MOBILE COMMUNICATIONS Reynaert, Patrick, Steyaert, Michiel ISBN: 1-4020-5116-6 IQ CALIBRATION TECHNIQUES FOR CMOS RADIO TRANCEIVERS Chen, Sao-Jie, Hsieh, Yong-Hsiang ISBN: 1-4020-5082-8 ADVANCED DESIGN TECHNIQUES FOR RF POWER AMPLIFIERS Rudiakova, A.N., Krizhanovski, V. ISBN 1-4020-4638-3 CMOS CASCADE SIGMA-DELTA MODULATORS FOR SENSORS AND TELECOM del Río, R., Medeiro, F., Pérez-Verdú, B., de la Rosa, J.M., Rodríguez-Vázquez, A. ISBN 1-4020-4775-4 SIGMA DELTA A/D CONVERSION FOR SIGNAL CONDITIONING Philips, K., van Roermund, A.H.M. Vol. 874, ISBN 1-4020-4679-0 CALIBRATION TECHNIQUES IN NYQUIST A/D CONVERTERS van der Ploeg, H., Nauta, B. Vol. 873, ISBN 1-4020-4634-0 ADAPTIVE TECHNIQUES FOR MIXED SIGNAL SYSTEM ON CHIP Fayed, A., Ismail, M. Vol. 872, ISBN 0-387-32154-3 WIDE-BANDWIDTH HIGH-DYNAMIC RANGE D/A CONVERTERS Doris, Konstantinos, van Roermund, Arthur, Leenaerts, Domine Vol. 871 ISBN: 0-387-30415-0 METHODOLOGY FOR THE DIGITAL CALIBRATION OF ANALOG CIRCUITS AND SYSTEMS: WITH CASE STUDIES Pastre, Marc, Kayal, Maher Vol. 870, ISBN: 1-4020-4252-3 HIGH-SPEED PHOTODIODES IN STANDARD CMOS TECHNOLOGY Radovanovic, Sasa, Annema, Anne-Johan, Nauta, Bram Vol. 869, ISBN: 0-387-28591-1 LOW-POWER LOW-VOLTAGE SIGMA-DELTA MODULATORS IN NANOMETER CMOS Yao, Libin, Steyaert, Michiel, Sansen, Willy Vol. 868, ISBN: 1-4020-4139-X DESIGN OF VERY HIGH-FREQUENCY MULTIRATE SWITCHED-CAPACITOR CIRCUITS U, Seng Pan, Martins, Rui Paulo, Epifânio da Franca, José Vol. 867, ISBN: 0-387-26121-4 DYNAMIC CHARACTERISATION OF ANALOGUE-TO-DIGITAL CONVERTERS Dallet, Dominique; Machado da Silva, José (Eds.) Vol. 860, ISBN: 0-387-25902-3 ANALOG DESIGN ESSENTIALS Sansen, Willy Vol. 859, ISBN: 0-387-25746-2

CMOS Single Chip Fast Frequency Hopping Synthesizers for Wireless Multi-Gigahertz Applications Design Methodology, Analysis, and Implementation

By

TAOUFIK BOURDI

Beceem Communications Inc., Santa Clara, California, USA

and

IZZET KALE

Westminster University, London, UK and Eastern Mediterranean University, Famagusta, North Cyprus

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5927-2 (HB) 978-1-4020-5927-8 (HB) 1-4020-5928-5 (e-book) 978-1-4020-5928-5 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Contents

Preface Nomenclature

ix xi

1 INTRODUCTION

1

1.1 Introduction 1.2 Research Contribution 2 WIRELESS COMMUNICATION SYSTEMS 2.1 Introduction 2.2 The WLAN Standards 2.3 WLAN Transceiver Systems 2.3.1 The Transmitter 2.3.2 The Receiver 2.3.3 The Frequency Synthesizer (Local Oscillator) 3 PHASE-LOCKED LOOP FREQUENCY SYNTHESIZERS 3.1 Introduction 3.2 Phase-Locked Loop Frequency Synthesizer 3.2.1 Phase-Locked Loop Main Blocks 3.2.1.1 Phase-Frequency Detector 3.2.1.2 Charge Pump 3.2.1.3 Voltage-Controlled Oscillator 3.2.1.4 Voltage-Controlled Crystal Oscillator 3.2.1.5 Dividers 3.3 Phase-Locked Loop Parameters 3.3.1 Loop Filter Design

1 2 7 7 8 10 12 12 13 15 15 15 16 16 17 18 19 19 19 20

vi

Contents 3.4 Noise in Phase-Locked Loops 3.4.1 Component Noise Models 3.4.1.1 Reference Oscillator and VCO Phase Noise 3.4.1.2 Charge Pump Current Noise 3.4.1.3 Loop Filter Resistor Noise 3.4.1.4 Main Divider Noise 3.4.1.5 Phase-Frequency Detector Phase Noise 3.4.1.6 Overall Phase Noise Contribution 3.5 Fractional-N Synthesizers 3.5.1 ∆−Σ Modulators in Frequency Synthesizers 3.5.1.1 Fractional-N Case Study 3.6 RMS Phase Error (φrms) and Error Vector Magnitude 3.7 Conclusion

27 29 30 30 30 31 31 32 34 36 39 41 42

4 SYSTEM SIMULATION OF ∆−Σ-BASED FRACTIONAL-N SYNTHESIZERS

45

4.1 Introduction 4.2 Phase-Domain Model 4.2.1 A Constituent Blocks Behavioral Models 4.2.1.1 The Reference Oscillator 4.2.1.2 The ∆−Σ Modulator/Feedback Integer Divider 4.2.1.3 The VCO 4.2.1.4 The PFD/CP 4.2.1.5 The Loop Filter 4.2.2 Noise Modeling Summary 4.3 Synthesizer Platform Evaluation 4.3.1 Dithering Effect 4.3.2 Close-to-Integer Operation 4.3.3 Noise Folding 4.3.4 Effect of Prescaler Divider 4.4 Conclusion

45 46 50 50 50 51 52 52 52 53 57 60 60 62 65

5 MULTIMODE ∆–Σ-BASED FRACTIONAL-N FREQUENCY SYNTHESIZER

67

5.1 Introduction 5.2 An overview 5.3 A Multimode Multistandard ∆−Σ-Based PLL Synthesizer Design 5.3.1 Design Methodology 5.4 The ∆−Σ Frequency Synthesizer SubBlocks Implementation 5.4.1 The Phase-Frequency Detector 5.4.2 The Charge Pump 5.4.2.1 Dead-Zone Nonlinearity 5.4.2.2 Linear Range and Cycle Slipping

67 67 69 69 71 71 73 76 78

Contents

vii

5.4.2.3 DC Offset Current 78 5.4.2.4 PFD/CP Transient Simulation 82 85 5.4.3 3.6 GHz Voltage-Controlled Oscillator 89 5.4.4 The Multimodulus Divider 5.4.4.1 MMD Operation 90 90 5.4.5 The Fractional Noise Shaping Coder (the ∆−Σ Modulator) 90 5.4.5.1 The Digital Accumulator and the First-Order Linear Model 5.4.5.2 The 30-bit Structural MASH Coder Implementation 92 93 5.4.5.3 The 24-bit Pipelined Adder Design 95 5.4.5.4 Error Cancellation Algorithm 5.4.5.5 Design Issues: Limit Cycle Cancellation in Fractional Mode 97 5.4.5.6 Design Issue: Integer Mode and Close-to-Integer Option 99 5.5 Measured Performance of the Implemented Synthesizer 102 5.6 Summary and Conclusion 107 6 IMPROVED PERFORMANCE FRACTIONAL-N FREQUENCY SYNTHESIZER

111

6.1 Introduction 6.2 Overview 6.3 Delta–Sigma-Controlled Adaptive Charge Pump 6.3.1 PLL Gain and Phase Variations 6.3.2 Charge Pump System 6.4 Synthesizer Loop Calibration 6.5 Process Calibration I/C Slew Rate and RC Time Constant 6.6 VCO Tuning Gain Calibration 6.6.1 VCO Calibration Algorithm Description 6.6.1.1 ∆N Values 6.6.1.2 Summary of Tuning Algorithm Operation 6.7 Improved VCO Band Switching 6.8 Experimental Results 6.9 Comparison with Published Results 6.10 Conclusion

111 111 113 113 116 117 119 121 121 125 125 126 127 128 128

7 CONCLUSION AND FURTHER WORK 7.1 Conclusion 7.2 Further Work

131 131 132

APPENDIX A PHASE-FREQUENCY DETECTORS AND CHARGE PUMPS

135 135

1 Phase-Frequency Detectors 2 Charge Pump 3 PFD/CP Characteristics

135 138 140

viii

Contents

APPENDIX B CONTROLLED OSCILLATORS 1 Reference Oscillators 1.1 Voltage-Controlled Crystal Oscillator 1.2 Temperature-Compensated Crystal Oscillator 2 Voltage-Controlled Oscillators 2.1 Voltage-Controlled Oscillators: Phase Noise Analysis 2.2 VCO Design Methodology 2.2.1 VCO Design Phase Noise Optimization 2.2.2

143 143 143 143 145 146 146 149 150 153

APPENDIX C PHASE NOISE

157 157

1 2 3 4

Calculation of Global Phase Error From L(f) Phase Noise and Phase Modulation RMS Phase Error From Phase Noise Residual FM

157 159 161 163

APPENDIX D FREQUENCY DIVIDERS 1 Reference Divider 1.1 Synchronous Dividers 1.2 Asynchronous Reference Frequency Divider 2 Feedback Divider 2.1 Specification and Different Architecture Evaluation 2.1.1 Direct Division versus Prescaler Method 3 High-Speed CMOS Divider Design 3.1 Current-Mode Logic Design: An Overview 4 Implemented CML Gates

165 165 165 165 166 168 168 168 177 178 183

APPENDIX E PROGRAMS AND CODES 1 MathcadTM Program used for the Simulations of all the Mathcad Figures 2 MatlabTM Program used for the Simulations of the Fractional-N PLL Noise Spectrum

187 187

INDEX

207

187 198

Preface

Frequency synthesizers are at the heart of the each transmitter/receiver system. Almost every communications consumer product employs a frequency synthesizer often operating as a local oscillator providing the carrier frequency of interest. Mobile phones, radios, and televisions are a few among the many applications that incorporate frequency synthesizers. Recently, wireless local area network (WLAN) standards have emerged in the market. Those standards operate in various frequency ranges. To reduce component count, it is of importance to design a multimode frequency synthesizer that serves all WLAN standards including 802.11a, b, and g standards. With different specifications for those standards, designing integer- based phase-locked loop frequency synthesizers can not be achieved. Fractional-N frequency synthesizers offer the solution required for a common multimode local oscillator. Those fractional-N synthesizers are based on delta–sigma modulators which in combination with a divider yield the fractional division required for the desired frequency of interest. In this book, the authors outline detailed design methodology for fast frequency hopping synthesizers for radio frequency (RF) and wireless communications applications. Great emphasis on fractional-N delta–sigmabased phase-locked loops from specifications, system analysis, and architecture planning to circuit design and silicon implementation. The book describes an efficient design and characterization methodology that has been developed to study loop trade-offs in both open- and closeloop modeling techniques. This is based on a simulation platform that incorporates both behavioral models and measured/simulated subblocks of the chosen frequency synthesizer. The platform predicts accurately the phase noise, spurious and switching performance of the final design. Therefore,

ix

x

Preface

excellent phase noise and spurious performance can be achieved while meeting all the specified requirements. The design methodology reduces the need for silicon re-spin enabling circuit designers to directly meet cost, performance, and schedule milestones. The developed knowledge and techniques have been used in the successful design and implementation of two high-speed multimode fractional-N frequency synthesizers for the IEEE 801.11a/b/g standards. Both synthesizer designs are described in details.

NOMENCLATURE

DC DCOC DFF DGA Div DMD D-S DSC DSSS EVM F FDC FOM IC IF Inv IP

Absolute Tolerance (CadenceTM specific) Alternating Current Analog-to-digital converter Application-Specific Integrated Circuit Balanced to unbalanced Bit Error Rate Complementary CodeKeying Carry Look-Ahead Current-Mode Logic Charge Pump Digital-to-analog converter Decibels per Hertz, SSB Phase Noise PSD Relative to the Carrier Direct Current DC Offset Cancellation Data-type Flip-Flop Digitally Controlled Variable-Gain Amplifier Divider Dual-Modulus Divider Delta–Sigma Differential to Single-Ended Converter Direct-Sequence Spread Spectrum Error Vector Magnitude Noise Figure Frequency to Digital Converter Figure of Merit Initial Condition Intermediate Frequency Inverter Intellectual Property

L(f) LF LPF LTI LTV LUT MC MLF

Single-Sideband Phase Noise Loop Filter Low-Pass Filter Linear Time Invariant Linear Time Variant Lookup Table Modulus Control Micro-Lead Frame

abstol AC ADC ASIC Balun BER CCK CLA CML CP DAC dBc/Hz

xi

xii

Nomenclature

MMD NTC PA PCB PFD PLL PRBS PSD PSDMD PVT QAM QPSK Ref

Multimodulus Divider Negative Temperature Coefficient Capacitor Power Amplifier Printed Circuit Board Phase-Frequency Detector Phase-Locked Loop Pseudorandom Binary Sequence Power Spectral Density Phase Switching Dual-Modulus Divider Process Voltage Temperature Quadrature-Amplitude Modulation Quadrature Phase-Shift Keying Reference

Reltol RF RFIC Rx SS SSB SSBN SSBPSD TCXO TDC TFF TRx Tx VCO VCXO VGA Vtune WLAN

Relative Tolerance (Cadence specific) Radio Frequency Radio Frequency Integrated Circuit Receiver Steady State Single Sideband Single-Sideband Noise Single-Sideband Power Spectral Density Temperature-Compensated Crystal Oscillator Time to Digital Converter Toggle Flip-Flop Transceiver Transmitter Voltage-Controlled Oscillator Voltage-Controlled Crystal Oscillator Variable-Gain Amplifier VCO Tuning Voltage Wireless Local Area Network

TM

Chapter 1 INTRODUCTION Outline and Contributions

1.1

INTRODUCTION

Frequency synthesizers are used as local oscillators in all transceiver systems. It is paramount that those frequency synthesizers are low noise as their behavior affects the entire performance of the transmission system. The work presented in this monograph focuses on the research, study, and improved design and implementation of low-noise frequency synthesizers for multimode wireless local area network (WLAN) applications covering all IEEE802.11a, b, and g standards. Performed measurements on those synthesizers show the low noise obtained by the design presented in this work. Complete test results highlighting the superior behavior of the designed synthesizers are shown in both chapters 5 and 6. In chapter 2, a brief description of those standards is presented in terms of their relevance to radio frequency transmission. An adequate transceiver that operates in all WLAN modes is also described and its transmitter/receiver chains are detailed. Architecture for the frequency synthesizer acting as a local oscillator for the transceiver is proposed. Direct frequency synthesis is not used to avoid frequency pulling in the transceiver [1]. In chapter 3, detailed analyses of integer and fractional-N phaselocked loops (PLL) frequency synthesizers are treated. Open-loop and closed-loop transfer functions of the PLL are derived. Noise contributions of individual subblocks of the synthesizers are detailed. Loop filter design is also included. The derived equations form the basis for

1

Chapter 1

2

the optimum design and implementation of the designed and implemented frequency synthesizer chips described in this book. The behavioral modeling for the proposed fractional-N delta– Sigma (∆−Σ)-based PLL is carried out in chapter 4 to check for architectural limitations, identify dominant noise sources, automate loop filter optimization, and generate phase-frquency detector/charge pump (PFD/CP) linearity specifications. Also, a phase-domain model of the proposed architecture is constructed using The CadenceTM Verilog-A Language. The model combines the voltage-controlled oscillator (VCO), reference, and divider integrators into one resettable integrator within the PFD. The ∆−Σ modulator model is also included. The divider adds ∆−Σ noise to the frequency variable, then divides the sum by the average divide ratio. The simulation results obtained in this chapter contribute to the optimum design and implementation of fractional-N synthesizers presented in this monograph. In chapter 5, simulation results presented in chapter 4 are used for the optimum chip design of a multimode frequency synthesizer for the WLAN standards. Unconditionally stable ∆−Σ modulators of the third-order (namely MASH-1-1-1) are implemented and employed in a PLL fractional-N synthesizer providing a good average estimate for fractional-N dividers. Using a deep submicron 0.18 µm complementary metal-oxide semiconductor (CMOS) process with a supply voltage of 1.8 V, a ∆−Σ-based fractional-N synthesizer is designed, simulated, laid out, fabricated, and tested. In chapter 6, additional circuit designs are proposed and incorporated to enhance the performance of the synthesizer at the cost of increased circuit complexity. Those additions include adaptive charge pump (CP) architecture to maintain loop gain and phase transfer functions while operating in fractional mode. Another circuit proposal is that of an adaptive band switching control to maintain frequency agility while offering optimum phase noise performance in the band of interest. The conclusion in chapter 7 wraps up the research monograph by describing the achievement of the work presented and offers suggestions for future work.

1.2

RESEARCH CONTRIBUTION

The main contributions of the work presented in this book are the research, study, design, and implementation of two fractional-N

Introduction

3

frequency synthesizers. Both synthesizers are incorporated in a multimode transceiver system and operate in all WLAN standards 802.11a, b, and g. The synthesizers designed and built have a unique architecture that avoids frequency pulling1 in the transceiver and it is based on the indirect frequency synthesis of the desired frequencies of interest. The first synthesizer is based on a conventional ∆−Σbased fractional-N frequency synthesizer that has been thoroughly investigated and modeled to achieve an outstanding phase noise and spurious performance. The second synthesizer uses special circuit design ideas to enhance the phase noise performance of the first frequency synthesizer making it the most agile and the most adaptive for many applications, modes of operations, and various wireless standards. The research work the authors have endeavored to produce was culminated in the publication of several articles in international conferences and journals and patents. Those are listed below. 1. T. Bourdi and I. Kale, “On the Efficient Design and Characterization of Multi-GHz ∆−Σ Fractional-N Frequency Synthesizer,” IEEE Transaction of Circuits and Systems CAS-I, Submitted April 2006. 2. T. Bourdi, A. Borjak, and I. Kale, “A Modeling Platform for Efficient Characterization of Phase-Locked Loop ∆−Σ Frequency Synthesizers,” IEEE International Symposium on Circuits and Systems, Kos Island Greece, May 2006, pp. 3221–3224. 3. Z. Pengfei, L. Der, G. Dawei, I. Sever, T. Bourdi, C. Lam, A. Zolfaghari, J. Chen, D. Gambetta, Baohong Cheng, S. Gowder, S. Hart, L. Huynh, T. Nguyen, and B. Razavi, “A Single-Chip Dual-Band Direct-Conversion IEEE 802.11a/b/g WLAN Transceiver in 0.18-um CMOS,” IEEE Journal of Solid-State Circuits, 40 (9), Sept. 2005, pp. 1932–1939. 4. Z. Pengfei, L. Der, G. Dawei, I. Sever, T. Bourdi C. Lam, A. Zolfaghari, J. Chen, D. Gambetta, B. Cheng, S. Gower, S. Hart, L. Huynh, T. Nguyen, and B. Razavi, “A CMOS Direct-Conversion Transceiver for IEEE 802.11a/b/g WLANs,” IEEE Custom Integrated Circuits Conference, 3–6 Oct. 2004, pp. 409–412.

1

Frequency pulling of the VCO is the frequency change due to nonideal load., i.e. change in the load causes frequency change in the VCO (hence the term pulling). This is most severe when the PA/TX frequency is directly related to the VCO frequency.

4

Chapter 1

5. T. Bourdi, A. Borjak, and I. Kale, “A Novel Delta–Sigma Based RF Frequency Synthesizer Architecture For Cellular Applications,” IEEE Transactions of Instrumentation and Measurement, Under Review, Submitted July 2003. 6. Z. Pengfei, N. Thai, C. Lam, D. Gambetta, C. Soorapanth, C. Baohong, S. Hart, I. Sever, T. Bourdi, A. Tham, and B. Razavi, “A 5-GHz DirectConversion CMOS Transceiver,” IEEE Journal of Solid-State Circuits, 38 (12), Dec. 2003, pp. 2232–2238. 7. Z. Pengfei, N. Thai, C. Lam, D. Gambetta, C. Soorapanth, C. Baohong, S. Hart, I. Sever, T. Bourdi, A. Tham, and B. Razavi, “A Direct Conversion CMOS Transceiver for IEEE 802.11a WLANs,” Digest of International Solid-State Circuit Conference, 2003, 1, pp. 354–498. 8. T. Bourdi, A. Borjak, and I. Kale, “Agile Multi-Band Delta–Sigma Frequency Synthesizer Architecture,” IEEE International Symposium on Circuits and Systems, ISCAS2002, May 2002, 5, pp. 413–416. 9. T. Bourdi, A. Borjak, and I. Kale, “A Delta–Sigma Frequency Synthesizer with Enhanced Phase Noise Performance,” IEEE Instrumentation and Measurement Technology Conference, May 2002, 1, pp. 247–251. 10. A. Borjak and T. Bourdi, “Intermodulation Products in a Mixer Subjected to a Multi-Carrier Signal,’’ Microwave Journal, 45 (2) Feb. 2002, pp. 130–143. 11. M. Kozak, I. Kale, A. Borjak, T. Bourdi, “A Pipelined All-Digital Delta– Sigma Modulator for Fractional-n Frequency Synthesis,” IEEE Instrumentation and Measurement Technology Conference, May 2000, 2, pp. 1153–1157. 12. T. Bourdi, A. Borjak, and D. Gambetta, “Cell-Based Charge-Pump Architecture for Delta–Sigma Fractional-n Synthesizers,” Resonext Patent, Submitted Oct. 2001. 13. T. Bourdi and A. Borjak, “A Delta–Sigma Controlled Charge Pump Architecture for Enhanced PLL Synthesizer Performance,” Resonext Patent, Submitted Mar. 2001. 14. T. Bourdi and A. Borjak, M. Henriksson, and I. Kale, “A Novel Algorithm for Effective Control of Fractional-n Synthesizers,” Nokia Patent, submitted Jan. 2000, Filed.

Introduction

5

REFERENCES [1] J.A. Weldon, et al., “A 1.75 GHz Highly-Integrated Narrow-Band CMOS Transmitter with Harmonic-Rejection Mixers,” IEEE Solid-State Circuits Conference, 2001, Digest of Technical Papers, 2001 IEEE International 5–7 Feb. 2001, pp. 160–161, 442.

Chapter 2 WIRELESS COMMUNICATION SYSTEMS An Overview

2.1

INTRODUCTION

With the turn of the new millennium, there has been an explosion in the usage of wireless equipment. Wireless cellular devices like mobile phones are now ubiquitous. Mobile Internet employing wireless devices is now available in most coffee shops in many countries. Such devices can achieve short-and medium-range coverage. Devices with short-range coverage are based on communications standards like Bluetooth [1], while medium-range coverage devices are based on the wireless local area networks (WLAN) communications standards [2]. The latter are categorized under two bands 2.4 and 5 GHz. The lower band is mainly based on the 802.11b standard and employs the complementary code keying (CCK) modulation [2]. The higher band (5 GHz) is based on the 802.11a standard and employs the orthogonal frequency division multiplexing (OFDM) modulation. Recently, the same OFDM modulation has been ported to the lower frequency of 2.4 GHz of 802.11b. This standard was termed 802.11g. WLAN devices are now used in most indoor places from homes to shops to offices. Such devices are incorporated in appliances like televisions, laptops, computers, telephones, PDAs, printers, etc. Figure 2-1 illustrates a typical usage where coverage is also shown. The synthesizer described in this book is designed and implemented for direct incorporation in to radio frequency integrated circuit (RFIC) transceivers for all the WLAN standards (802.11a, b, and g). In this chapter, a brief overview of those standards is given. An adequate transceiver system for such standards is also described.

7

8

Chapter 2

Figure 2-1. WLAN Usage

2.2

THE WLAN STANDARDS

As mentioned above the WLAN standards fall in two frequency bands: the 2.4 GHz band (802.11b and 802.11g) and the 5 GHz band (802.11a). Here is a brief description of those standards. The 2.4 GHz 802.11 standards require a pass band of approximately 22 MHz wide for one operating network. Using direct sequence spread-spectrum modulation (DSSS), the WLAN provides an 11 Mbps data rate to the network users. Thus, the 2.4 GHz band’s 83.5 MHz supports three nonoverlapping, simultaneously operating WLAN networks (66/83.5), and roughly 33 Mbps of data rate (11 Mbps * three networks) to be shared among common users across the coverage area as illustrated in Figure 2-2. This has been proven to adequately distribute sufficient bandwidth to support the majority of applications across many environments.

Wireless Communication Systems

9

Figure 2-2. Nonoverlapping and Overlapping US Channel Selection

Figure 2-3. OFDM Physical Layer Frequency Channel Plan Implementation for US Standards

10

Chapter 2

The 802.11a standard requires a 16.6 MHz pass band for one operating network. The modulation technique allowed in 5 GHz (orthogonal frequency division multiplexing (OFDM) with 64QAM subcarriers) is more efficient than the spread spectrum techniques WLAN uses (more bits/second/hertz), and provides up to 54 Mbps of data rate to network users. For the US-based 802.11a standard, the 5 GHz unlicensed band covers 300 MHz of spectrum and therefore supports 12 nonoverlapping, simultaneously operating networks, as shown in Figure 2-3. The 802.11g occupies similar frequency bands as the 802.11b, however the single subcarrier is OFDM modulated as in the case of 802.11a. A summary of the main characteristics for all the WLAN standards is shown in Table 2-1. Table 2-1. WLAN Characteristics Parameters Frequency in GHz

802.11b 2.4–2.58

802.11g 2.4–2.58

Modulation Data rate (Mbits/s)

DSSS– CCK 1, 2, 5.5 & 11

Receiver sensitivity (dBm) Transmit power (dBm) Transmit EVM (dB)

−74 @ 11 Mbps

OFDM 6, 9, 12, 18, 36, 48 & 54 −65 @ 54 Mbps

802.11a 4.91–4.99 & 5.03– 5.24 Japanese Bands 5.15–5.25 Lower 5.25–5.35 Middle 5.725–5.825 Upper OFDM 6, 9, 12, 18, 24, 36, 48 & 54 −65 @ 54 Mbps

30

30

16, 23 & 29

−9

−25

−25

2.3

WLAN TRANSCEIVER SYSTEMS

A basic diagram of a transceiver system used in the WLAN standards is shown in Figures 2-4 and 2-5. It incorporates both base band and radio frequency parts. The radio frequency part is common to all the wireless standards (802.11a, b, and g) whereas the modem varies with those standards. It provides both 2.4 and 5 GHz frequency bands for the 802.11a, b, and g WLAN standards. The transceiver incorporates both low frequency blocks like lowpass filters (LPF), variable-gain amplifiers (VGA) and high-frequency blocks like mixers, a low-noise amplifier (LNA), a power amplifier

Wireless Communication Systems

11

Figure 2-4. Transceiver System

Figure 2-5. WLAN Transceiver Architecture

(PA), and frequency synthesizers acting as local oscillators (LO). In this book, emphasis is made on the research, design, and implementation of those frequency synthesizers. The transceiver is conventionally divided into two paths: the transmitter and the receiver both incorporating the local oscillator treated here. All those blocks are described below.

12

2.3.1

Chapter 2

The Transmitter

A detailed transmitter chain block diagram is shown in Figure 2-6. The shown LPF are preceded by digital-to-analog converters (DAC) that are implemented on the base band part of the transceiver chip. Those image quality (IQ) filters are used for signal reconstruction. The base-band signal level can be adjusted through a digitally controlled VGA. The modulated base-band signal is then up-converted to either 5 GHz or 2 GHz. The two front-end signal paths take the same circuit topology. Each cut is independently optimized for its own frequency band. The upconverted differential signal is subsequently transformed to a single-ended version by an active differential to single-ended conversion (DSC) circuit to obviate the need for an external BALUN (balanced to unbalanced transformer).

Figure 2-6. Transmitter Signal Path for the WLAN Transceiver

2.3.2

The Receiver

Figure 2-7 shows the receiver chain block diagram. The shared receiver base-band section consists of a buffer amplifier, a digitally controlled VGA, an LPF and an output buffer (A2) whose gain can be adjusted. The base-band section of the Q-channel is identical, not shown here for brevity. The channel selection LPF is designed for its rapid increase of attenuation outside the pass band, and thus a narrow transition band. A nominal cutoff frequency of 8.7 MHz is automatically calibrated to account for process variations. This satisfies the channel selection requirement for 802.11a, b, and g standards. Due to a significantly

Wireless Communication Systems

13

more severe adjacent channel rejection requirement, higher rejection ratio at lower base-band signal frequency is needed for CCK modulation in 802.11b mode. This extra filtering is implemented in the digital domain.

Figure 2-7. Receiver Chain for WLAN Transceiver

Included in the receiver is a low-flicker noise mixer [3], active-LC preselection notch filter (F1) and the look-up table (LUT)-based DC offset compensation [4]. WLAN applications require the receiver to accommodate an input signal range from a few microvolts to tens of millivolts, often demanding two or more gain stages in the (LNA). The radio frequency (RF) front-end circuits for both frequency bands use exactly the same topology with independently sized devices. The only difference is that a direct bypass across the LNA has been implemented in 2 GHz to accommodate the even higher input power requirement for 802.11b.

2.3.3

The Frequency Synthesizer (Local Oscillator)

In a direct conversion transceiver, the required local oscillation (LO) frequency coincides with the RF, which entails adverse effects such as LO–RF interaction and VCO frequency pulling [5]. The LO generation scheme used consists of a quadrature VCO operating at two-thirds of the LO frequency and a divide-by-2 circuit producing quadrature outputs at one-third of the LO frequency. Two quadrature mixers subsequently multiply the VCO signal by the divide-by-2 signal to generate the quadrature LO signals (upper band), with significant suppression of the undesired lower band at one-third of the LO frequency, alleviating the image problem in the receiver and the spurious emission problem in the transmitter.

14

Chapter 2

Figure 2-8. Local Oscillator Frequency Synthesizer For WLAN Transceiver

The quadrature LO signals for 2 GHz band are then generated by dividing the 5-GHz LO signal by two as shown in Figure 2-8. Two quadrature VCOs are used (VCO1: 3.2–3.6 GHz, VCO2: 3.5–3.9 GHz) in order to cover the required frequency range with enough margins for process, voltage, and temperature (PVT) variations.

REFERENCES [1] IEEE Bluetooth Drafts Standards, http://grouper.ieee.org/groups/Bluetooth/ [2] IEEE 802.11 Drafts Standards, http://grouper.ieee.org/groups/802/11/ [3] G. Chien et al., “A 2.4 GHz CMOS Transceiver and Baseband Processor Chipset for 802.11b Wireless LAN Application,” IEEE International Solid-State Circuits Conference Digest of Technical Papers, Feb. 2003, pp. 356–357. [4] B. Razavi and P. Zhang, “Mixer Noise Reduction Technique,” US Patent, 6,748,204, June 2004. [5] J.A. Weldon, et al., “A 1.75 GHz Highly-Integrated Narrow-Band CMOS Transmitter with Harmonic-Rejection Mixers,” IEEE Solid-State Circuits Conference, 2001. Digest of Technical Papers, 2001 IEEE International 5–7 Feb. 2001, pp. 160–161, 442.

Chapter 3 PHASE-LOCKED LOOP FREQUENCY SYNTHESIZERS Principles, Analyses, and Design

3.1

INTRODUCTION

In this chapter, detailed analyses of PLL frequency synthesizers are treated. Both integer and ∆−Σ-based fractional-N are considered. Open-loop and Closed-loop gain and phase equations are derived and phase noise theory is introduced. Gain and noise contributions of individual subblocks of the synthesizers are detailed. Loop filter design is also included. Together with simulations performed in chapter 4, the derived equations aid the optimum design and implementation of the two presented fractional-N synthesizer chips described in chapters 5 and 6.

3.2

PHASE-LOCKED LOOP FREQUENCY SYNTHESIZER

A PLL frequency synthesizer is a circuit that faithfully follows and reproduces a scaled reference signal over a wide frequency range. A typical frequency synthesizer block diagram is shown in Figure 3-1. The phase Φsamp of a divided-down reference signal, namely sampling signal, is compared to the phase Φfeed of the feedback signal obtained by dividing down the oscillator output signal. The mean value of the output signal from the PFD/CP combination is equal to

15

16

Chapter 3

the phase error between phases Φsamp and Φfeed. When passive loop filter is employed, a CP is used to convert the voltage to current. The CP phase error ΦCP then drives the loop filter. Other high-frequency components also present at the output of the CP are removed by the loop filter. The phase error at the output of the filter, Φfilt controls the input voltage of the oscillator to obtain the frequency of interest fout with a phase Φout. A brief description of the PLL subblocks is listed below. VCXO

fref Φref

f Reference samp PFD Divider Φ samp

Charge Pump

Φfeed ffdbk

Icp

VCO

ΦCP

Z(s)

Φfilt

Kvco/s

fout Φout

Divider N

Figure 3-1. Phase-Locked Loop Frequency Synthesizer

3.2.1

Phase-Locked Loop Main Blocks

The main blocks used in the PLL are briefly described below. Detailed description of those blocks is included in the appendices. 3.2.1.1

Phase-Frequency Detector

The PFD [2] compares the divided down reference signal with the divided down feedback signal to generate a signal proportional to the phase error. Several types of frequency detectors are used in PLLs [13]; however, the most commonly used is the PFD as it offers both phase and frequency comparison. A conventional PFD is shown in Figure 3-2. The timing diagram for this PFD for the case of a reference signal lagging the VCO feedback signal is shown in Figure 3-3. The up and down pulses shown control the source and sink currents that charge or discharge the loop filter capacitor as described below. A detailed description of this PFD as well as other types of PFDs is included in Appendix A.

Phase-Locked Loop Frequency Synthesizers

17

Figure 3-2. Conventional Dual-Type DFF Phase-Frequency Detector

Ref

VCOf

Dn

Up Reset

Figure 3-3. Phase-Frequency Detector Timing Diagram

3.2.1.2

Charge Pump

The CP is used to convert the PFD output signal voltage to a current signal to drive the passive loop filter. Optimum CP design yield matched source and sink currents [10]. The up and down signals control two switches to source or sink current into the loop filter

18

Chapter 3

capacitive elements. The loop filter is usually of second order. However, a third-order loop filter could be used if spurii of the PFD sampling feed-through are to be suppressed [14]. A typical illustration for the sourcing and sinking of the CP currents is shown in Figure 3-4. The gain of such a CP is normally given by ICP and its unit is in amperes. Vsup

Vsup

Vsup

Iup

Iup

Iup

Up

Up

Up Icp

Icp

Icp Dn

Dn

Dn Zs Idn

Zs

Zs Idn

Idn

Figure 3-4. Illustration of Sourcing and Sinking in Charge Pump

3.2.1.3

Voltage-Controlled Oscillator

The VCO converts a continuous input voltage to a high-frequency signal. Several performance criteria for the VCO design are of interest. Those are: power consumption, phase noise, jitter, linearity, tuning range, supply voltage, and substrate noise rejection. The frequency versus tuning voltage characteristics is usually nonlinear; however, a linear approximation is often used in the analysis of the entire PLL. The linear slope approximation for the VCO gain is called KVCO and its unit is in Hz/V.

Phase-Locked Loop Frequency Synthesizers 3.2.1.4

19

Voltage-Controlled Crystal Oscillator

The crystal oscillator is used to generate the reference signal. Crystal oscillators have high spectral purity and low phase noise performance. 3.2.1.5

Dividers

There are two types of dividers. The ones used as reference dividers; those are usually low frequency. The others are high- frequency dividers and those are used as feedback dividers. Detailed circuit topologies of synchronous and asynchronous, as well as dual-modulus prescaler dividers are found in the Appendix.

3.3

PHASE-LOCKED LOOP PARAMETERS

PLLs are of nonlinear nature. To simplify their analyses, a linear approximation is often used. Important parameters that describe the PLL are the open-loop and closed-loop transfer functions, as well as the phase noise functions. The PFD shown in Figure 3-1 is modeled as a subtractor with its gain modeled as a multiplication factor Kd. The CP gain factor ICP is the value of the current used. The oscillator transfer function is given by KVCO/s. The feedback divider transfer function is given by 1/N. The open-loop transfer function is derived from basic control theory and is given by: AOpenloop ( s ) = K d I CP Z ( s )

K VCO 1 s N

(3.1)

Kd is equal to 1. The CP current ICP is in amps. The VCO gain KVCO is given in Hz/V. The closed-loop transfer function is given by:

AClosedloop ( s ) =

AOpenLoop ( s ) Φout = Φsamp 1 + AOpenLoop ( s )

(3.2)

A typical passive loop filter is a second-order filter that yields a third-order PLL. Figure 3-5 shows a passive second-order loop filter with optional third- and fourth- order extra spurious cancellation.

20

Chapter 3

ICP

R3 R2

C1

C2

R4

C3

Vtune C4

Figure 3-5. A typical Second-Order Loop Filter with Optimal third-and-fourth order Spur Cancelation Network

The transfer function for the loop filter shown in Figure 3-5 is given by:

Z (s) =

1 + sC 2 R2 s (C1 + C 2 + sR2 C1C 2 )

(3.3)

The loop bandwidth (LBW) frequency fp (radial frequency is ωp) is defined as the frequency at which the absolute value of the open-loop gain is equal to 1 (or 0 dB):

AOpenLoop ( s )

f = fn

=1

(3.4)

The phase margin is defined in the following equation:

φ = 180 + phase( AOpenLoop ( s ))

(3.5)

Equations (3.4) and (3.5) are the main equations used in the design of the optimum loop filter.

3.3.1

Loop Filter Design

Using equations (3.4) and (3.5), values for the second-order loop filter components can be easily derived. Those are shown below after some algebraic manipulation [14]

Phase-Locked Loop Frequency Synthesizers

21

2 2 K VCO T1 1 + ωp T2 C1 = I CP K d N ωp2T2 1 + ωp2T12

(3.6)

⎛T ⎞ C 2 = C1 ⎜⎜ 2 − 1⎟⎟ ⎝ T1 ⎠

(3.7)

and R2 =

T2 C2

(3.8)

where T1 =

sec(φ ) − tan(φ )

(3.9)

ωp

and T2 =

1

(3.10)

T1ωp2

For a stable loop, a good phase margin must be between 45o and 60 . The damping ratio of the loop is also given here as a function of the phase margin o

⎡ ⎤ (tan(φ )) 4 ξ =⎢ 2 ⎥ ⎣16(1 + (tan(φ )) ) ⎦

0.25

(3.11)

For a 56o phase margin the damping ratio is equal to 0.55. The other values R3, C3, R4, and C4 can be selected to reject the PFD feed-through frequency signals. Other equations for third- and fourth- order loop filters could be found in [14].

22

Chapter 3

CASE STUDY The case presented here is for a possible usage in the WLAN standard. The specified parameters are shown in Table 3-1. Table 3-1. Phase-Locked Loop Specified Parameters Parameter Synthesized frequency Sampling frequency VCO gain Charge pump gain Loop bandwidth Phase margin

Value 1.72 GHz 40/3 MHz 100 MHz/V 2 mA 100 kHz 56o

Using the above-derived equations, the values for the second-order loop filter components are obtained. Together with other loop parameters, those are shown in Table 3-2. Table 3-2. Loop Filter Designed Parameters Parameter Capacitor C2 Resistor R2 Capacitor C1 Main divider Time constant T1 Time constant T2

Value 11.6 nF 447 Ω 1.2 nF 129 0.49 µs 5.2 µs

Figure 3-6 shows the open-loop gain and phase transfer functions for the design of the PLL, whereas Figure 3-7 shows the closed-loop gain and phase transfer functions. It can be seen from Figure 3-6 that the gain drops to 1 (0 dB) at the specified LBW frequency (100 kHz) and the phase is at its peak of −124o which corresponds to a phase margin of 56o (180o–124o). The loop filter transfer function Z(s) is also plotted in Figure 3-8. The gain of Z(s) shows the change in the 20 dB/decade slope for fp /10 (10 kHz) and 10fp (1 MHz).

Phase-Locked Loop Frequency Synthesizers

23

Open-Loop Gain Transfer Function

200

100 20⋅log ( Aol (f)

)

0

100

200 100

3

1 .10

4

1 .10

5

f

1.10

6

1.10

6

1 .10

1.10

7

Opn-Loop Phase Transfer Function

100

120 180 ⋅arg (Aol (f)) 140 π

160

180 100

3

1 .10

4

1.10

5

f

1 .10

1.10

Figure 3-6. Open-Loop Gain and Phase Transfer Functions

7

24

Chapter 3 Closed-Loop Gain Transfer Function

5

20 ⋅log ( Acl (f)

)

0

5 100

3

1.10

4

1 .10

5

f

1 .10

6

1 .10

7

1 .10

Closed-Loop Phase Transfer Function

0

50 180 ⋅arg (Acl (f)) π

100

150

100

3

1 .10

4

1 .10

5

f

1 .10

6

1 .10

Figure 3-7. Closed-Loop Gain and Phase Transfer Functions

7

1 .10

Phase-Locked Loop Frequency Synthesizers

Loop Filter Gain Transfer Function

100

20log ⋅ ( Z(f)

)

25

50

0 100

3

1 .10

4

5

1 .10

f

1 .10

6

1 .10

7

1 .10

Loop Filter Phase Transfer Function

0

180 ⋅arg(Z(f)) 50 π

100 100

3

1.10

4

1.10

5

f

1.10

Figure 3-8. Loop Filter Transfer Functions

6

1.10

7

1.10

26

Chapter 3

The time-domain function for the closed-loop transfer function can be obtained by performing an inverse Laplace transform. For the third-order PLL (second-order loop filter), this is given by [14]:

(

)

(

)

⎡ ⎤ ξ − R2C2ωn F (t ) = f 2 + ( f1 − f 2 )e −ξω n t ⎢cos ωn 1 − ξ 2 t + sin ωn 1 − ξ 2 t ⎥ 1−ξ 2 ⎣⎢ ⎦⎥ (3.12)

where ξ is the damping ratio, ωn is the natural loop radial frequency, and f1 is the new frequency after a jump from the frequency f2 at the output. The locking time can be obtained from equation (3.12) and is often approximated to be:

⎛ tol ⎞ − ln ⎜ 1− ξ 2 ⎟ ⎝ ( f 2 − f1 ) ⎠ LockTime =

ξωn

(3.13)

A classical model for the settling time for the closed-loop PLL with the second-order loop filter is shown in Figure 3-9.

Figure 3-9. Time-Domain Transfer Function for the PLL

Phase-Locked Loop Frequency Synthesizers

3.4

27

NOISE IN PHASE-LOCKED LOOPS

Each sub-block of the PLL system contributes to the overall noise of the loop. Those are: PFD/CP combination noise contribution, VCO, and VCXO phase noise and phase noise of the low- and high- frequency dividers. Extra noise is contributed by the thermal noise of the loop filter resistance values. Phase noise and amplitude noise contributions are not usually specified separately. A linearized model for the noise contributions of the subblocks in the PLL is shown in Figure 3-10. Each subblock is assumed to contribute a small-signal noise source that can be referred to the input or output of the subblock. In this monograph, the noise source is placed after the functional transfer function of the subblock.

Φr

ICP _

Σ

Φvco

Vf

Ipd 1/R

F(s)

Kv

Σ

1/s

Σ

Σ

Φout

Σ 1/N Φn Figure 3-10. Noise Contributions in the Phase-Locked Loop System Table 3-3. Gain and Noise Terms and their Units Gain terms Reference Divider R Charge pump Gain ICP Loop filter Z(s) VCO tuning gain KVCO PLL feedback Divider N

Gain units No units Amps/rad Ω rad/V No units

Noise terms Reference oscillator Phase noise Φr Charge pump Current noise Ipd Loop filter Voltage noise Vf VCO phase Noise

ΦVCO

Feedback Divider phase Noise Φn Output phase Noise

Φout

Noise units rad-rms or radrms/√Hz Amps or Amps/√Hz Volt or Volt/√Hz rad-rms or radrms/√Hz rad-rms or radrms/√Hz rad-rms or radrms/√Hz

The notations used in Figure 3-10 are listed below. Gain terms and their units, as well as phase noise and their units are included (Table 3-3). From basic control theory, it is easy to determine the transfer

28

Chapter 3

function of the individual noise contributors. Those are given in the following equations. The output to reference transfer function is given by:

I CP K VCO Z ( s ) Φ out s = Φ r 1 + I CP K VCO Z ( s ) sN

(3.14)

The output to PFD/CP transfer function is given by: K VCO Z ( s ) Φ out s = I pd 1 + I CP K VCO Z ( s ) sN

(3.15)

The output to filter transfer function is given by: Φ out Vf

K VCO s = I CP K VCO Z ( s ) 1+ sN

(3.16)

The output to VCO transfer function is given by: Φ out 1 = Φ vco 1 + I CP K VCO Z ( s ) sN

(3.17)

The output to feedback divider transfer function is given by: I K Z ( s) − CP VCO Φ out s = Φ n 1 + I CP K VCO Z ( s ) sN

(3.18)

The output noise power is the product of the input noise power and the magnitude squared of the transfer function. The total output noise power is calculated by summing the output noise power contributed from each noise source (assuming the noise sources are uncorrelated).

Phase-Locked Loop Frequency Synthesizers

3.4.1

29

Component Noise Models

The component noise models must generally account for three different types of noise power spectral density (Figure 3-11). These types are primarily distinguished, for modeling purposes, by the slope of the noise spectrum [15]: • White noise is characterized by a flat, or uniform, noise power

density in the band of interest. Thermal and shot noise sources have white noise spectra. Examples include resistors (including MOSFET channel resistance) [16] (this is called white noise 1/f 0 noise, for consistency with the remaining spectra). • 1/f noise is characterized by a noise power spectrum that decreases at a rate of 3 dB per octave, or 10 dB per decade. Flicker noise sources have 1/f noise spectra. Examples include MOSFET channel (drain) current and polysilicon resistors [17]. • 1/f 2 noise is characterized by a noise power spectrum that decreases at a rate of 6 dB per octave, or 20 dB per decade. Oscillator phase noise in the thermal noise region has a 1/f 2 noise spectrum. • 1/f 3 noise is characterized by a noise power spectrum that decreases at a rate of 9 dB per octave, or 30 dB per decade. Oscillator phase noise in the (upconverted) flicker noise region has a 1/f 3 noise spectrum.

L(f) (dB)

1/f3 1/f2 1/f1

fc32

fc21

1/f0

fc10

log(f)

Figure 3-11. Phase Noise slopes for White Noise (1/f 0 ) , Flicker Noise (1/f 1 ) , Oscillator Noise in the Thermal Region (1/f 2), and Oscillator Noise in the Upconverted Flicker Noise Region (1/f 3 ) .

30

Chapter 3

Noise power spectra for some components (such as the loop filter resistors) can be directly calculated with good accuracy. More complex noise sources, such as oscillator phase noise, must often be measured in order to provide accurate results. Measured noise spectra can be characterized by noting the actual noise power at a certain frequency, the 1/f n region in which that frequency lies, and the “corner” frequencies for the different 1/f n regions. 3.4.1.1

Reference Oscillator and VCO Phase Noise

Oscillators generally have 1/f 3, 1/f 2, and white noise (1/f 0) regions. Phase noise models and simulations (e.g. Leeson’s equation [15], impulse sensitivity functions [18], and periodic noise analysis [19]) can be used to estimate the phase noise spectra, but direct measurements are preferred. Degradations to the VCO and reference noise from the buffering and divider circuits should be included, as well. For example, it is difficult to obtain noise floors below –145 dBc/Hz without significant effort at reference frequencies around 20–30MHz. 3.4.1.2

Charge Pump Current Noise

The CP current sources have 1/f and white noise (1/f 0 ) regions, which can be estimated with reasonable accuracy from simulations (e.g. SPICE). A complication that arises in the CP PLL is the aliasing effect caused by the periodic switching of the CP current. In lock, the CP has a duty cycle determined by delays in the phase detector, leakage in the loop filter, and other systematic design choices. While periodic noise simulations should be used to accurately estimate the net noise power spectrum coupled into the loop filter, in some cases a good approximation is obtained by attenuating the CP current noise by the (average) duty cycle of the CP pulses. The noise power is attenuated by multiplying it by the average duty cycle. 3.4.1.3

Loop Filter Resistor Noise

For the loop filter of Figure 3-5 we calculate the voltage noise present at the output from resistors R2 and R3 (a third-order loop filter). In general, a noisy resistor is modeled as an ideal resistor of the same value in series with a noise voltage generator [20]. The noise voltage density is given by:

Phase-Locked Loop Frequency Synthesizers

v 2 = 4kTR

31

(3.19)

where k is Boltzmann’s constant, T is the device temperature, and R is the resistance value. At room temperature, 4kT = 1.66 × 10−20 V–C. Using circuit analysis, the transfer function can be calculated from each noise voltage generator to the filter output voltage. (These equations are not simplified, but are reduced enough for computer implementation. Note the low-pass characteristic on R2 and R3. R3 needs to be placed at much higher frequency to prevent oscillation.) Vo C1 (3.20) = (4kTR2 ) VR2 C1 ( sR3C3 + 1) + ( sR2C1 + 1)( sR3C2C3 + C2 + C3 ) Vo sR2C1C2 + C1 + C2 (3.21) = (4kTR3 ) VR3 C3 ( sR2C1 + 1) + ( sR3C3 + 1)( sR2C1C2 + C1 + C2 ) 3.4.1.4

Main Divider Noise

The divider is a periodically time-varying circuit. The fixed-ratio frequency divider gives an ideal noise figure F = 20log(N). An internal noise contribution is also given by the divider and the output noise spectral density in the case of fixed division ratio is given by [1, 14]: SΦ ,n ( f ) = 3.4.1.5

SΦ ,in_divider ( f ) 10−14.7 + + 10−16.5 2 N f

(3.22)

Phase-Frequency Detector Phase Noise

Measurements made on frequency synthesizers with a decreasing division ratio showed that there is a lowering in the phase noise plateau [3]. However, this 20log(N) improvement of phase noise is somewhat offset by the increase of sampling frequency at the PFD. This has been proven in [4] and illustrated in the single-sided power spectral density at the output of the PLL as given by:

+( f ) =

⎛ 4π 2 ∆t 2 f 02 ⎞ 1 Sφout ( f ) = 10 log10 ⎜ ⎟ [dBc/Hz] 2 fs ⎝ ⎠

(3.23)

32

Chapter 3

Rewriting equation (3.23) in a more convenient dB format yield: 2 + ( f ) = FOM{dBc / Hz } + 20 log10 f 0 − 10 log10 fs [dBc/Hz] (3.24)

Substituting the divider contribution in equation (3.24) gives [4]: 2 + ( f ) = FOM{dBc / Hz } + 20 log10 N + 10 log10 fs [dBc/Hz] (3. 25)

where FOM is a noise figure of merit of the PFD. FOM is a constant at a specific frequency. 3.4.1.6

Overall Phase Noise Contribution

The contribution of the phase noise of individual subblocks to the PLL is illustrated in Figure 3-12. As can be seen in this figure, the logic noise (including divider noise) and the reference oscillator phase noise are dominant within the PLL LBW. The VCO phase noise is dominant outside the LBW. Table 3-4. Phase Noise Parameters Parameter Reference Oscillator Plateau Reference Divider plateau PFD Normalized Plateau VCO noise Plateau Main Divider plateau

Symbol Lref

Value −143

Units dBc/Hz

Lrefdiv

−173

dBc/Hz

Lpd

−216

dBc/Hz

Lvco Ldiv

−159 −173

dBc/Hz dBc/Hz

Using equations 3.14–3.18, a Mathcad program (see Appendix F) was written to predict the closed-loop phase noise contributions of the individual subblocks and the entire PLL. The same loop parameters used in the case study discussed in this chapter are used in the phase noise calculations. Example phase noise plateaus used in this program are shown in Table 3-4.

Phase-Locked Loop Frequency Synthesizers

33

phase noise dBc/Hz

VCO phase noise

loop bandwidth

logic plateau noise

Filter roll off

VCO Phase noise skirts

20log(fout/fref) 20log(fout/fsamp)

logic noise

VCXO phase noise

1/f3 LBW

1/f2 frequency offset

Figure 3-12. Phase Noise Contributions in PLL

The MathcadTM simulation results obtained using the data from the case study and data from Table 3-4 are shown in Figure 3-13. They correlate well with the conceptual phase noise contributions of Figure 3-l2.

34

Chapter 3

Figure 3-13. Phase Noise Contributions for the Case Study

3.5

FRACTIONAL-N SYNTHESIZERS

From the study of noise in the previous section, it was shown that the noise improves if a higher sampling frequency is used [8, 9]. That results in the usage of fractional division ratio to satisfy the output VCO frequency and frequency step of interest. Frequency synthesizers employing such fractional dividers are called fractional-N frequency synthesizers. Figure 3-14 shows a conventional fractional-N frequency synthesizer. Early Implementation of the fractional dividers employs a digital accumulator [12] that controls a dual-modulus divider. The synthesizer shown in Figure 3-14 is termed first-order fractional-N frequency synthesizer. The fractional divider is composed of two parts: the integral part N and the fractional part F and is often

Phase-Locked Loop Frequency Synthesizers VCXO

fref

1/R

fsamp

35

PFD

VCO

f out

DMD N/N+1

overflow

Frac latch

Figure 3-14. Conventional Fractional-N Synthesizer

represented by N.Frac (e.g. 10.5 where N is 10 and 0.Frac is 0.5). The fractional part 0.Frac controls a digital accumulator whose overflow controls a dual-modulus prescaler N/N + 1. The size of the accumulator used depends on the frequency error as well as the sampling frequency. The output frequency is given by: K⎞ ⎛ f out = fsamp × N .Frac = fsamp × ⎜ N + ⎟ F⎠ ⎝

(3.26)

The fractional part 0.Frac is usually represented by a fraction whose integer numerator is called K and whose integer denominator is called F. Since the overflow controlling the DMD changes the value of the divider from N + 1 to N within the cycle, this resets the phase error at the output of the PFD, generating signals that modulate the VCO and appear at the output of the VCO. These signals are deterministic in nature and can be predicted, they are often termed fractional spurious signals [13]. These spurious signals appear at fractional multiples of the reference and are difficult to remove and hence are not used in commercially viable solutions. However, fractional-N frequency synthesizers that employ ∆−Σ modulators in place of the first-order

36

Chapter 3

digital accumulator have recently gained popularity as they provide excellent phase noise suppression within the LBW of the PLL used [5]. A typical frequency synthesizer employing a ∆−Σ modulator is shown in Figure 3-15. VCXO

fref

Reference Divider

fsamp

Charge Pump

PFD

ffdbk

VCO

fout

Divider

∆−Σ Modulator

N.Frac

Figure 3-15. Typical Delta–Sigma (∆−Σ)-Based Fractional-N Frequency Synthesizer

3.5.1

∆−Σ Modulators in Frequency Synthesizers

In this section, ∆−Σ modulators of third order are described as they are unconditionally stable [5–7, 11]. This type of modulators is often termed MASH-1-1-1 modulators as they incorporate three first-order modulators in parallel. Figure 3-16 shows a basic block diagram of this type of modulator. Typical time-domain output of this modulator is obtained in Matlab Simulation for the case of a fractional divisor of 0.835 and is shown in Figure 3-17. As can be seen from Figure 3-17, the ∆−Σ modulator emulates the average fractional part of the divider into instantaneous several integral levels. Those vary in a random manner between −7 and 8 for this order of modulator. In general, for Norder MASH modulator, the output would vary between –( 2 Norder − 1 ) and 2 Norder [21]. The fractional divider N.Frac would then be emulated by instantaneous dividers that vary in a random manner between (N–( 2 Norder − 1 )) and ( N + 2 Norder )

Phase-Locked Loop Frequency Synthesizers

37

such that the average value of the sequence of those divisors is equal to the desired fractional-N divider value. X(n)

E 1(n)

Σ

Σ

+

Z -1

Σ

Y(n)

Σ

-

-

-E 1(n)

Z -1 E 2 (n)

Σ

Σ

+

Z -1

-E 2 (n)

Σ

Σ

-

Z -1

E 3 (n)

Σ

Σ

+

Z -1

Σ

-

Figure 3-16. A Third- Order MASH1-1-1 ∆−Σ Modulator Block Diagram

A linearized noise equation has been derived that accurately predicts the divider output phase noise spectrum [3]: L( f ) =

(2π ) 2 12 f samp ( N .Frac )

2

[2sin(

πf fsamp

)]2( Norder −1)

[rad 2 /Hz] (3.27)

38

Chapter 3

Figure 3-17. Time-Domain Output of MASH1-1-1 ∆−Σ Modulator

However, a fast Fourier transform and some algebraic manipulation are usually performed on the sequence of instantaneous dividers to yield the single-sidedband (SSB) power spectral density of the ∆−Σ modulators. Typical SSB of this modulator is shown in Figure 3-18. As can be seen from this figure, the phase noise of this modulator is very low, close to base band (below −130 dB, below 500 kHz). This will help suppress the in-band phase noise of the ∆−Σ frequency synthesizer. Above 500 kHz, the PLL loop filter should filter out the phase noise of the MASH modulator. Therefore, the usage of the MASH modulator is a key factor in reducing the phase noise plateau in fractional-N frequency synthesizer.

Phase-Locked Loop Frequency Synthesizers

39

Figure 3-18. Typical SSB Power Spectral Density of third- Order MASH1-1-1 ∆−Σ Modulator

3.5.1.1

Fractional-N Case Study

In this section, a case study is presented for a ∆−Σ-based frequency synthesizer used in the WLAN standard. The selected output frequency was chosen to yield a fractional division ratio. The specified parameters are similar to those presented in the case of the integer PLL but with a higher sampling frequency of 40 MHz as shown in the shaded cell of Table 3-5. Similar analyses to those presented for the integer PLL case will be repeated here.

40

Chapter 3

Table 3-5. ∆−Σ PLL Parameters Used Parameter Frequency Sampling frequency VCO gain Charge pump gain Loop bandwidth Phase margin

Value 1.725 GHz 40 MHz 100 MHz/V 2 mA 100 kHz 56o

Using [14], the values for the third-order loop filter components are obtained. Those are shown in Table 3-6. Table 3-6. Loop Filter-Designed Parameters Parameter Capacitor C2 Resistor R2 Capacitor C1 Main divider Capacitor C3 Resistor R2

Value 34.8 nF 150 Ω 3.6 nF 43.125 366 pF 25 Ω

The phase noise for this fractional-N frequency synthesizer was analyzed using a MatlabTM program. The results obtained are shown in Figure 3-19. They include the phase noise contributions of all the subblocks of the PLL including the ∆−Σ modulator/divider combination (top curve). Due to the suppressed phase noise of the ∆−Σ modulator, the improvement in the overall phase noise is apparent in Figure 3-19 when compared to Figure 3-13. The ∆−Σ noise contribution is however apparent outside the loop filter BW. Detailed simulation for the ∆−Σ fractional-N frequency synthesizer using a commercial circuit/system simulator package (CadenceTM) is presented in chapter 4. The simulation in chapter 4 will help the designer make the right implementation optimizations for such a synthesizer.

Phase-Locked Loop Frequency Synthesizers

41

Figure 3-19. Phase Noise Contributions of SubBlocks in ∆−Σ-Based Fractional-N Synthesizer

3.6

RMS PHASE ERROR ( φrms ) AND ERROR VECTOR MAGNITUDE

Most wireless standards specify the noise in terms of rms phase error unit rather than phase noise at spot frequencies [14]. The phase error is the area under the phase noise mask between two spot frequencies. It usually is obtained by integrating the normalized phase noise plot between the mentioned spot frequencies. The rms phase error in degrees is shown in Figure 3-20 and is given by:

φrms =

2

π

f2

+( f )

∫ 10 f1

10

df

(3.28)

42

Chapter 3

Figure 3-20. RMS Phase Error and Error Vector Magnitude in a QPSK System

The constellation diagram of a quadrature phase-shift key (QPSK) system is shown in Figure 3-20, displaying the four possible constellation points. At the receiver, the constellation point for the symbol “11” due to anomalies in the transmission, is received with a phase and amplitude error as shown in Figure 3-20 and is located at the red point. The vector connecting the original constellation point to the new location (outside the circle) is called error vector magnitude and is usually referred to as EVM.

3.7

CONCLUSION

In this chapter, detailed analyses of PLL frequency synthesizers were presented. Both integer and ∆−Σ-based fractional-N were considered. Open-loop and closed-loop gain and phase equations were derived. Phase noise of individual PLL subblocks was introduced. White noise (1/f 0), flicker noise (1/f 1), Oscillator noise in the thermal region (1/ f 2), and oscillator noise in the upconverted flicker noise region (1/f 3) were also described. Loop filter design equations were shown and used in the case study of a frequency synthesizer potentially used in the WLAN standard.

Phase-Locked Loop Frequency Synthesizers

43

The theory presented in this chapter as well as the detailed system level simulation presented in the chapter 4 aid the design and implementation of the two fractional-N synthesizer chips described in chapters 5 and 6.

REFERENCES [1] W.F. Egan, “Modeling Phase Noise in Frequency Dividers,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 37 (4), pp. 307–315, July 1990. [2] J.A. Crawford, “The Phase Frequency Detector,” R.F. Design, Feb. 1985, pp. 46–57. [3] I. Thompson and P.V. Brennan, “Phase/Frequency Detector Phase Noise Contribution in PLL Frequency Synthesizer,” IEEE Electronics Letters, July 2001, 37 (15), pp. 939–940. [4] I. Thompson and P.V. Brennan, “Phase Noise Contribution of the Phase/Frequency Detector in a Digital PLL Frequency Synthesizer,” IEE Proceedings on Circuits, Devices and Systems, Feb. 2003, 150 (1), pp. 1–5. [5] T.A.D. Riley, M.A. Copeland, and T.A. Kwasniewski, “Delta–Sigma Modulation in Fractional-N Frequency Synthesis,” IEEE Journal SolidState Circuits, 28, pp. 553–559, May 1993. [6] B. Miller and B. Conley, “A Multiple Modulator Fractional Divider,” Proceedings of IEEE 44th Annual Symposium Frequency Control, 1990, pp. 559–567. [7] B. Miller and B. Conley, “A Multiple Modulator Fractional Divider,” IEEE Transactions of Instrumentation Measurement, 40, pp. 578–583, June 1991. [8] V. Manassewitsch, Frequency Synthesizers, Theory and Design, 3rd edn., Wiley: New York, 1987. [9] R.E. Best, Phase Locked-Loop Design Simulation and Applications, 3rd edn. MacGraw-Hill, New Jersey, 1997. [10] F.M. Gardner, “Charge-Pump Phase Lock Loops.” IEEE Transactions on Communication, COM-28:1849–1858, Nov. 1980. Description of the charge pump mechanism in a PLL. [11] B.-G. Goldberg. Digital Techniques in Frequency Synthesis, MacGrawHill, New Jersey, 1996. [12] B.-G. Goldberg, “Analog and Digital Fractional-n PLL Frequency Synthesis: A Survey and Update,” Applied microwave and wireless, June 1999. Tutorial presenting fractional-N frequency synthesis. [13] P.V. Brennan, Phase-Locked Loops, Principles and Practice, McGrawHill, New Jersey, 1996. [14] D. Banerjee, “PLL Performance, Simulation and Design,” 3rd edn., 2003, National Semiconductor, (http://www.national.com/appinfo/wireless/ files/Deansbook3.pdf) [15] D.B. Leeson, “A Simple Model of Feedback Oscillator Noise Spectrum,” Proceedings of the IEEE, MI, 54 (2), pp. 329–330, 1966.

44

Chapter 3

[16] C.-H. Chen, et al., “Direct Calculation of the MOSFET High Frequency Noise Parameters,” Proceedings of the 14th International Conference on Noise and Physical Systems and 1/f Fluctuations, pp. 488–491, July 1997. [17] W. Liu, et al., “RF MOSFET Modeling Accounting for Distributed Substrate and Channel Resistance with Emphasis on the BSIM3v3 SPICE Model,” IEDM Technical Digest, pp. 309–312, Dec. 1997. [18] A. Hajimiri and T.H. Lee, “A General Theory of Phase Noise in Electrical Oscillators.” IEEE Journal of Solid-State Circuits, 33 (2), pp. 179–194, Feb. 1998. [19] Cadence Design Systems, “Periodic S-Parameter and Noise Analysis using SpectreRF PSP/PNOISE Analyses,” Application Notes and White Papers, http://www.cadence.com/whitepapers/pspapn1.pdf [20] A. Mehrotra, “Noise in Radio Frequency Circuits: Analysis and Design Implications,” International Symposium on Quality Electronic Design, ISQED San Jose, Mar. 2001. [21] M. Kozak, I. Kale, A. Borjak, and T. Bourdi, “A pipelined All-Digital Delta–Sigma Modulator for Fractional-N Frequency Synthesis,” IEEE Instrumentation and Measurement Technology Conference (IMTC 2000), Vol. 2, pp. 1153–1157, Baltimore, MD, May 2000.

Chapter 4 SYSTEM SIMULATION OF ∆−Σ-BASED FRACTIONAL-N SYNTHESIZERS Efficient Modeling and Characterization

4.1

INTRODUCTION

The aim of the work presented in this monograph is the research, study, design, and implementation of high-speed ∆−Σ-based fractional synthesizers for WLAN standards (802.11a, b, and g). In chapter 3, detailed analyses of integer-N and ∆−Σ-based fractional-N phaselocked loops have been presented. Open-loop, closed loop, and phase noise equations have been derived. In this chapter, behavioral modeling for a proposed fractional-N ∆−Σ-based PLL is carried out to evaluate architectural limitations, identify dominant noise sources, automate loop filter optimization, and generate PFD/CP linearity specifications. Also, a phase-domain model of the proposed architecture is constructed using The CadenceTM Verilog-A Language. The model combines the VCO, reference, and divider integrators into one resettable integrator within the PFD. The ∆−Σ modulator model is also included. The divider adds ∆−Σ noise to the frequency variable then divides the sum by the average divide ratio. The simulation results obtained in this chapter and measured results of subblocks of the chip designed in chapter 5 contribute to the optimum design and implementation of fractional-N synthesizers presented in chapters 5 and 6.

45

46

Chapter 4

4.2

PHASE-DOMAIN MODEL

Figure 4-1 shows a block diagram of a ∆−Σ-based fractional-N synthesizer. This synthesizer employs voltage-controlled oscillators synthesizing 802.11a frequencies that will be described in chapter 5. However, for simplicity, Figure 4-1 shows the synthesized frequency to be two-thirds of the desired frequency. This is mixed with a divided-by-2 version of it to generate the LO for the 802.11a bands. The CadenceTM model of this synthesizer is shown in Figure 4-2. The PLL model is a phase-domain model; in steady state, the VCO model generates a ramp instead of an oscillatory voltage (voltage-domain model).

Figure 4-1. A Conventional ∆−Σ-Based Fractional-N Frequency Synthesizer

Tuning Curve

PFD/CP LUT

Φ 40 MHz = 40V

Integrator

∫

CP

Divider ∆−Σ MMD Frequency

VCO Frequency

Figure 4-2. Phase-Domain Model of the Synthesizer

VCO phase

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

47

The purpose of behavioral modeling of the fractional-N ∆−Σ-based PLL is to check for performance limitations, identify dominant noise sources, automate loop filter optimization, and generate PFD/CP linearity specifications. Phase-domain models have several advantages over voltage-domain models: 1. A phase-domain model is time-invariant. Consequently, the DC operating point analysis quickly brings the model to steady state conditions, eliminating the need to simulate long startup transients. A legitimate DC operating point also makes the AC and noise analysis available. The AC and noise analyses are very fast because they run in the frequency domain with linearized models. Small signal frequency- domain analyses are used to minimize the rms. phase noise with respect to the loop filter, subject to bandwidth and phase margin constraints. 2. PFD/CP nonlinearities are easily modeled and specified by a simple transfer curve. 3. Phase-domain models are compatible with the CadenceTM environment, which means that the architecture can be changed without having to rewrite any code, top-down design flow. 4. A phase-domain model suppresses the carrier, making time domain faster than voltage-domain models in the CadenceTM environment. Rather than simulating transient voltage, the model simulates phases/frequency of the individual blocks. The voltage at the output of the reference signal represents the frequency used in the design (in this case a 40 MHz, hence the reference is 40 V). The PFD model simulates cycle slips by combining the reference and VCO integrators into one resettable integrator within the PFD model. The PFD integrates frequency error to generate a duty cycle that in turn drives the CP model. The PFD model gives a duty cycle output that can take values in the range of −1 to +1. When the duty cycle is negative there is discharge to the loop filter and when the duty cycle is positive there is a charge to the loop filter.

48

Chapter 4

Figure 4-3. PFD/CP Characteristics Showing the Dead-Zone Region

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

49

The PFD/CP linearity curve, whether taking into account the deadzone or not, is provided by a lookup table immediately after the PFD. One case where the PFD suffers from the dead-zone [4, 5] is presented in Figure 4-3. The non-linear PFD data could be easily stored in a file and loaded within the used block. The output of this lookup table is a voltage driving the subsequent CP that has normalized up and down currents yielding the desired CP values. The output of the CP current creates a voltage at the output of the loop filter that in turn drives the VCO. The VCO gain characteristic can either be a lookup table obtained by either simulating or measuring the gain of the VCO or a polynomial generated by curve fitting the characteristic gain curve [8]. The gain curves of this multiband VCO are shown in Figure 4-4. The output of the VCO is a frequency which is represented by a voltage value in this proposed model. To obtain a phase value at the output of the VCO, it is mandatory to use an integrator as shown in Figure 4-2. What follows is a brief description of the behavioral models of each constituent blocks in the synthesizer.

Figure 4-4. Multiband VCO Tuning Characteristics Showing the Frequency (a Voltage in the Phase-Domain Model) Versus the Tuning Voltage

50

Chapter 4

4.2.1

A Constituent Blocks Behavioral Models

4.2.1.1

The Reference Oscillator

For the reference oscillator phase-domain model, the reference is represented by a fixed DC voltage representing reference frequency instead of phase. This is possible and beneficial as the reference integration has been moved to the PFD and hence DC analysis can now be used to skip long startup transients. In this work, the selected reference signal frequency Fref which is equal to 40 MHz is represented by a fixed 40 V DC source. At the oscillator sampling time Tref = Fref−1 , the oscillator phase noise data could easily be superimposed on the reference signal. The measured open-loop single-sided power spectral density (PSD) for the employed reference oscillator Lref ( f ) is −145 dBc/Hz. The noise updated reference frequency signal is given by:

Fref = Fref + 10Lref ( fO ) / 20 × f O × Tref × normal _ dist ( rand ,0,1) 1 (4.1) × + white _ noise(10Lref ( fO ) /10 × fO2 ) Tref The noise term has two parts as shown in the equation above. The first term is taken at each reference sample time. It should be noted that the noise here is frequency noise. f O is the offset frequency where the measured PSD Lref ( f O ) is read. The normal distribution and the white noise functions are Verilog-A™ built-in proprietary functions. 4.2.1.2

The ∆−Σ Modulator/ Feedback Integer Divider

The ∆−Σ modulator and the feedback divider are treated jointly. The combined model represents the model of the desired fractional divider. The third-order ∆−Σ MASH modulator model is derived by employing the sampled difference equations of each node. The modeled ∆−Σ noise; mainly quantization noise; enters the loop linearly after passing thorough a digital integrator [6]. Since integration is a linear operation and since operation is at frequencies where the digital integrator could be replaced with an analog integrator, both integrators

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

51

(divider and VCO) could be pulled into the PFD model, and hence the ∆−Σ divider as well as the VCO operate on frequency rather than phase. ∆−Σ quantization noise is straightforward to model in the time domain; it just implements the difference equations. An equivalent frequency-domain noise source for the third-order ∆−Σ MASH modulator can be found from the z-domain expression for the output noise. Consider the linearized noise transfer function and PSD of the input quantization [2]:

(

H n ( z ) = 1 − z −1

)

3

(4.2)

And the PSD = T / 12, where T is the clock period. The output PSD is: PSD. H n ( z )

2

(4.3)

Note that differentiation as first order with respect to time can be approximated as (1 − z −1 ) / T , where T is the sampling period. The continuous time equivalent of the discrete time output PSD is found by replacing (1 − z −1 ) / T with s. Thus, the equivalent continuous time 3 2 noise PSD is ((sT ) ) T /12 , which is a white noise source with PSD 7 equal to (T ) / 12 , differentiated three times. 4.2.1.3

The VCO

The noisy signal generated by the VCO is generated in a similar manner to the one described for the reference signal. However, here tuning curves of the VCO are first generated with the aid of polynomial fitting performed on the measured gain characteristics [8].

K VCO = F1 + 2 F2 × x + 3F3 × x 2 + 4 F4 × x 3

(4.4)

Where x is the tuning voltage. The noise modeled here is frequency noise and hence to get phase noise, integration of this frequency noise is required at the VCO output.

52 4.2.1.4

Chapter 4 The PFD/CP

The noise sources of the PFD and CP are added as random noise similar to the reference oscillator case [1]. The PFD/CP linearity curve could also be included as a data file, showing the duty cycle versus CP, to gauge the effect of the dead zone as shown in Figure 4-3. 4.2.1.5

The Loop Filter

The loop filter uses real components rather than a transfer function and therefore the noise due to the resistors although negligible adds up to the overall noise of the loop; thus their noise should be taken into account when phase noise frequency-domain analysis is performed.

4.2.2

Noise Modeling Summary

Placing all non-loop-filter integrations inside the PFD reduces the VCO model to a tuning curve (Figure 4-4). The VCO block generates a voltage representing the VCO frequency, not the VCO phase. To get phase noise, we must integrate the VCO frequency noise. That is why the synthesizer model has an ideal extra integration at the VCO output. Placing the VCO integration in the PFD means the reference and VCO noise sources are now white noise sources. The reference and VCO are both oscillators and their noise sources are assumed to result from Wiener processes [3]. A Wiener process integrates white noise. The PSD of an oscillator’s white noise process is chosen to align the integrated noise and measured VCO noise at one frequency. The reference and VCO models have noise sources for frequency-domain and time-domain analysis. For time-domain analysis, the VCO and reference models add a Gaussian random variable to their outputs. The random variable is updated at a user-defined rate. 40 MHz was chosen for the update rate because that was the ∆−Σ clock frequency. The standard deviation of the random variable depends on the sample rate and is automatically scaled to produce the correct PSD. The open-loop single-sided power spectral densities of all the aforementioned blocks are shown in Figure 4-5. Those results are obtained by direct Cadence™ PSD transform on the phase-domain time-domain signals. The produced phase noise plots are as expected.

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

53

Figure 4-5. Open Loop of Synthesizer Constituent Blocks Obtained by Direct PSD Transform

4.3

SYNTHESIZER PLATFORM EVALUATION

The simulation in the proposed platform can be carried out in both time domain and frequency domain. The time-domain simulation aids the monitoring of the settling in the phase-locked loop. Figure 4-6 shows the phase-domain model time-domain simulation illustrating the voltages at each individual node in the loop. The reference frequency of 40 MHz (represented here by 40 V) is shown along with the feedback frequency (divider output) illustrating its average to 40 MHz (i.e. 40 V) after 7 µs. The PFD output is showing its convergence to 0 in 7 µs (i.e. locking condition). The settling of the loop is best viewed by monitoring the tuning voltage that reaches its desired value as illustrated in the figure. The synthesized VCO frequency and its correspondent local oscillator frequency are both shown to reach their respective values within 7 µs. It can be easily seen that after 10.5 µs the ∆–Σ fully settles and the effect of the initial seed disappears completely, hence increasing the modulator activity. It should be

54

Chapter 4

mentioned here that this simulation takes a couple of seconds compared to a few days if transistor-level transient simulations were run. In frequency-domain, open- and closed-loop phase noise can be characterized. At this level, open-loop phase noise data must be included in the phase domain model mentioned above. Each of the individual blocks except the loop filter employs noise data derived from a phase noise mask that has been simulated and measured. The phase noise masks data obtained from Figure 4-6 are incorporated in the phase model. Those can be enabled when loop phase noise is needed.

Figure 4-6. Time-Domain Simulation of the Phase-Domain Model of the Platform Showing the Voltages at each Node in the Synthesizer

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

55

Closed-loop phase noise of the synthesizer is then obtained by performing a PSD transform on the integrated signal shown in Figure 4-2. The divider and the ∆−Σ modulator are combined into one unique block. The block simply divides the voltage in fractional mode and contains the phase noise mask of the ∆−Σ modulator. All the results of individual blocks have been discussed separately above and it is time now to close the synthesizer loop by implementing initial off-chip loop filter components. With the aid of this platform with fast simulation time, it is possible to optimize those filter components to yield optimum phase noise performance. Initial loop filter values were calculated using [8] and are shown in Table 4-1. Table 4-1. Frequency Synthesizer Loop filter Parameters Parameter C1 R2 C2 R3 C3 R4 C4 Charge pump, Max current Average divider ratio

Value 270p 1000 2.5n 91 120p 91 120p 1 mA 91.65

The performance in the frequency domain (i.e. phase noise) can be easily obtained by performing power-spectral density transform on the time-domain phase-domain model results of the synthesizer in locked condition. A snapshot of the time-domain phase model of this synthesizer in lock is shown in Figure 4-7a and b, unzoomed and zoomed, respectively. To show the efficacy of the developed model, the synthesizer is simulated with two different loop filter bandwidths, namely 300 kHz and 1 MHz. Figure 4-8 overlays the phase noise mask for both cases. It could be easily demonstrated that this model correlates well with the phase noise mask for the synthesizer if linear-model control loop equations were used outside this platform. Clear observations can be summarized as follows: As the loop bandwidth is increased, the closein phase noise plateau is suppressed. However, the deterministic spurious noise as well as the quantization noise due to the ∆−Σ modulator is exacerbated.

56

Chapter 4

(a)

(b) Figure 4-7. Voltages at Each Node in the Synthesizer during Lock (a) Unzoomed, (b) Zoomed

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

57

Figure 4-8 also shows the presence of several spurious signals that can be suppressed if further dithering is applied to the ∆–Σ modulator as will be discussed in the ensuing section. These spurious signals are due to close-to-integer operation.

Figure 4-8. Phase Noise Waveforms for the LO Synthesizer Obtained by PSD Transform for Two Loop Bandwidth Cases (a) 300 kHz and (b) 1 MHz

4.3.1

Dithering Effect

It is useful to describe the dithering applied to the ∆–Σ modulator before discussing the performance of the synthesizer with dithering. One of the consequences of using ∆−Σ modulators with DC inputs is the presence of limit cycles or spurs [2] that are strongly visible for inputs that are inverses of power of 2 such as 0.75, 0.5, and 0.25. This is due to the fact that the binary representation of such DC values has much less randomness. Figures 4-9 and 4-10 show the noise spectrum for 0.5 dc input with and without spurious limit cycles in linear and log scales, respectively.

58

Chapter 4

Figure 4-9. Power Spectral Density of Delta–Sigma Noise Shaper with and without Dithering

Figure 4-10. Power Spectral Density of Delta–Sigma Noise Shaper with and without Dithering, Log Scale

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

59

The effect of these limit cycles are reduced greatly as we introduce dithering or randomness. Here, the spurs have been eliminated by introducing an error (nonzero initial condition) in the least significant bit (LSB) of the input word [7]. The error is too small to affect the synthesized frequency within the permissible frequency error but is good enough to eliminate the spurs to some extent. As can be seen from Figure 4-11, there are two remnant fractional spurs seen in the region of 1 MHz. Those spurs exist despite the LSB dithering applied to the ∆–Σ modulator. Those can be further removed by increasing the efficiency of the dithering employed. The low-frequency effect below 25 kHz is a deficiency in the power-spectral density transform function built in CadenceTM. If low-frequency phase noise is of interest, it is advisable to export the data to a mathematical package for further accurate processing [3].

Figure 4-11. Phase Noise Mask for the LO Synthesizer with 300 kHz Loop Bandwidth with and without Dithering

Figure 4-11 shows the phase noise profile for the frequency synthesizer local oscillator at 5.5 GHz with a loop bandwidth of 300

60

Chapter 4

kHz. Two cases are superimposed. The first case (bottom trace) is when no dithering is employed which shows the presence of spurious fractional content whereas the top trace is the case where dithering is applied. In this case, the spurious energy is spread across the spectrum and hence the lifting of the phase noise as illustrated in the figure. This effect must be taken into account when designing synthesizers to strike a compromise between deterministic spurious noise and random phase noise.

4.3.2

Close-to-Integer Operation

One of the practical issues that are often overlooked in fractional-N PLL designs is the problem of having to synthesize frequencies that are integer multiples of the reference frequency, i.e. the divider value is an integer. This becomes a problem if not catered for in advance in the design. If the input is an integer, the ∆−Σ noise shaper input is zero. Above, we have introduced an error to remove any fractional spurs that may arise from limit cycles. That error will propagate in the modulator and cause the accumulators to overflow in a determined and cyclic manner causing spurious tones for integer frequencies. For an input dc value of 0.998, Figures 4-12 and 4-13 show the PSD of the modulator without and with dithering, respectively. It can be clearly seen in Figure 4-12 that the modulator exhibits lowfrequency spurs due to insufficient dither. This problem is rectified by effectively increasing the dither via introducing an initial seed [7] in the modulator as illustrated in Figure 4-13. Figure 4-14 illustrates the phase noise masks when 1-LSB and 5-LSB (nonzero initial condition) dithering is applied to the modulator when operating with a fractional division close to integer. With 5-LSB dithering the spurious level is reduced at the expense of lifting the phase noise level.

4.3.3

Noise Folding

One important factor that deteriorates the close-in phase noise and that is easily seen in measurement but not proven by simulation is the effect of noise folding [2]. The platform developed in this research proves this phenomenon with ease. Figure 4-15 shows the effect of noise folding due to the PFD/CP nonlinearity. The high-frequency ∆–Σ quantization is folded back to within the loop bandwidth. This is illustrated in the 10 dB deterioration of the phase noise plateau rendering the fractional-N synthesizer unattractive. Fortunately, the developed platform outlines this problem and shows how to mitigate

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

Figure 4-12. Modulator Output for DC input 0.998

Figure 4-13. Modulator Output for DC Input 0.998 with Dither Applied

61

62

Chapter 4

Figure 4-14. Effect of 1-LSB and 5-LSB Dithering for Close-to-Integer Divide Ratio

the PFD/CP nonlinearity. To solve the problem of noise folding, offset charge pump current could be added to shift the PFD/CP gain characterristics to a linear operating region. Figure 4-16 shows the effect of introducing such currents on the phase noise profile. It is easily seen how the phase noise is improved first with introducing a 5% offset current. With 10% increase in offset CP for this case, it was possible to mitigate the noise folding effect almost completely. It should be noted that the amount of offset current is crucial and hence further increases to its value might deteriorate the phase noise. This was observed in both measured and simulation in the presented platform. With more than 12% increase in the offset CP current, the phase noise got worse and hence demonstrated the presence of an optimum offset CP current to be employed.

4.3.4 Effect of Prescaler Divider The synthesized VCO frequency in the 3–4 GHz region warrants the usage of high-frequency dividers. To reduce the power consumption in those dividers, it is possible to employ a prescaler preceding the main

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

63

Figure 4-15. Effect of Noise Folding on phase Noise Profile Due to Noise Folding Nonlinearity

Figure 4-16. Introduction of Offset Leakage current to Mitigate the Effect of Noise Folding Due to Delta-Sigma Modulation

64

Chapter 4

divider that is controlled by the ∆–Σ modulator as shown in Figure 4-17. However, this comes at a hefty price in phase noise performance as will be illustrated in this section. Figure 4-18 shows the lifting of the phase noise when a divide-by-4 prescaler is used to drive the multimodulus divider (MMD). This increases the ∆–Σ quantization noise by 12 dB. Therefore, it is recommended to incorporate the prescaler within the main divider. PFD/CP LUT

Φ 40 MHz = 40V

Tuning Curve

Integrator

∫

CP

Divider ∆−Σ MMD Frequency

Div-by-4

VCO Frequency

VCO phase

Figure 4-17. Phase-Domain Model for the Synthesizer Employing a Divide-by-4 Prescaler Preceding the Main Feedback Divider

Figure 4-18. Deterioration of Phase Noise Performance due to Placement of a Divide-by-4 Prescaler before the Delta–sigma-Controlled Multi-Modulus Divider

System Simulation of ∆−Σ -Based Fractional-N Synthesizers

4.4

65

CONCLUSION

A thorough simulation-based system analysis of ∆−Σ-based fractional-N synthesizer was studied. This system was based on an implemented model platform constructed with a combination of measured raw data and behavioral Verilog-A models to speed up the simulation. It was demonstrated that by having direct division versus the use of prescaler preceding the main feedback divider controlled by the ∆−Σ modulator, the quantization noise increases by 20*log10(N ) dB, where N is the preceding divider value. It was also shown that the nonlinearities in the CP/PFD combination cause the noise to fold back to the in-band. Removing those nonlinearities shows the elimination of this phenomenon. Hence, this platform has enabled the reproduction of all witnessed behaviors in the laboratory of my first implemented synthesizer chip that showed unpredicted phenomenon at the time. The platform presented in this chapter can help predict accurately the effect of nonlinearities of the frequency synthesizer subblocks on the overall performance. The developed platform has aided the design and successful implementation of the synthesizers presented in chapters 5 and 6, respectively.

REFERENCES [1] Affirma RF Simulator (SpectreRF) User Guide, “An Introduction to the PLL Library: How the PFD Model Works”. [2] M.H. Perrott, M.D. Trott, and C.G. Sodini, “A Modeling Approach for Sigma–Delta Fractional-N Frequency Synthesizer Allowing Straightforward Noise Analysis.” IEEE Journal of Solid-State Circuits, 37 (8), Aug. 2002. [3] K. Kundert, http://www.designers-guide.com/Analysis/PLLnoise+jitter.pdf [4] J. Crawford, “Frequency Synthesizer Design Handbook.” Equation (7.81) on page 349. [5] B. De Muer and M. Steyaert. “CMOS Fractional-N Synthesizers.” [6] J. van Engelen, R. van de Plassche. “Bandpass Sigma Delta Modulators”. [7] N.M. Filiol, T.A.D. Riley, C. Plett, and M.A. Copeland “An Agile ISM Band Frequency Synthesiser with Built-In GMSK Data Modulation,” IEEE Journal of Solid-State Circuits, 33 (7), July 1998. [8] D. Banerjee, PLL Performance, Simulation and Design, 3rd edn., 2003, National Semiconductor, (http://www.national.com/appinfo/wireless/files/ Deansbook3.pdf )

Chapter 5 MULTIMODE ∆−Σ-BASED FRACTIONAL-N FREQUENCY SYNTHESIZER

5.1

INTRODUCTION

In chapter 4, we performed system-level simulation to aid the implementation of fractional-N synthesizers presented in this chapter. Effects of the different subblocks in the PLL on the entire phase noise of the closed-loop fractional-N synthesizer were monitored. In this chapter, unconditionally stable ∆−Σ modulators of the third order (namely MASH-1-1-1) are implemented and employed in a phaselocked loop fractional-N synthesizer providing a good average estimate for fractional-N dividers. Using a deep sub micron 0.18 µm CMOS process with a supply voltage of 1.8 V, a ∆−Σ-based fractional-N synthesizer is designed, simulated, laid out, fabricated, and tested. Results obtained from measurements on this synthesizer outperform all synthesizers reported to date [1–6].

5.2

AN OVERVIEW

Fractional-N frequency synthesizers employing ∆−Σ noise shapers have been developed extensively in the past decade [7–16] replacing the conventional single accumulator-based synthesizer. Figure 5-1 shows the block diagram of a fractional-N frequency synthesizer. Conventional fractional-N synthesizers employing first-order ∆−Σ modulator (single accumulator) are known to suffer greatly from fractional spurs that occur every time the accumulator cyclically

67

68

Chapter 5

overflows to control the dual-modulus divider to switch between N + 1 instead of N [14]. VCXO

fref

Reference Divider

fsamp

Charge Pump

PFD

ffdbk

VCO

fout

Divider

∆−Σ Modulator

N.K

Figure 5-1. Fractional-N PLL Frequency Synthesizer

Techniques to correct for those spurs include the use of analog compensation [15] using a digital-to-analogue converter, which injects an analog format of the error to cancel out the spurious signals. However, those analog techniques require great precision and matching, which is difficult to implement in practice rendering the counter measures insufficient to totally suppress the spurious signals within the band of interest. Digital compensation techniques using higher-order ∆−Σ modulators have been demonstrated [7-12] to work well in dithering and shaping the noise pushing it to high frequencies. The loop filter can therefore filter out the noise at those frequencies. In this chapter, the design and implementation of a complete fractional-N PLL frequency synthesizer is described in details. The synthesizer is designed using a commercial 0.18 µm CMOS process. All the subblocks including the PFD/CP, VCO, dividers, and loop filter were optimized for implementation using simulation results obtained in chapter 4.

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

5.3

69

A MULTIMODE MULTISTANDARD ∆−Σ-BASED PLL SYNTHESIZER DESIGN

The aim of this chapter is to show the results of a ∆−Σ-based PLL synthesizer operating in the 2.4/5 GHz region. A typical use of such a synthesizer is in the WLAN standard 802.11a, b, and g. The synthesizer (operates from a 1.8 V supply voltage and is simulated over PVT Process [fast, slow, and typical], voltage [1.7, 1.8, and 1.9 V] and temperature [−40o, 40o, and 125o]) corners. The obtained performance is excellent and supersedes most published results in the WLAN arena. In what follows, the detailed design and implementation of the synthesizer is described. The specification of each individual subblock in the synthesizer is derived from the set of specifications shown in Table 5-1.

5.3.1

Design Methodology

The first step is to devise the architecture for the frequency synthesizer that covers all the frequency ranges shown in the specification table (Table 5-1). A scheme based on oscillator synthesis that generates two-thirds of the desired frequency mixed with a divided-by-2 version of it is employed for the 802.11 standards. This architecture is shown below in Figure 5-2. As can be seen from Figure 5.2, the required frequencies of interest are not directly generated by the respective voltage-controlled oscillators. That was done to avoid frequency pulling in the transceiver [17]. Table 5-1. Frequency Ranges for the Proposed Synthesized Architecture Parameters Frequency B, g (GHz) for a 802.11 Current consumption (mA) Supply voltage (V) Locking time (us)

Minimum 2.4 4.8

1.7

Nominal

20 1.8 224

Maximum 2.5 5.805

1.9

70

Chapter 5 f802.11a

f802.11b,g

X

/2

40MHz VCXO

R

PFD

CP

VCO1 VCO2

LF

/2

MMD n 6

∆−Σ

30

Figure 5-2. Proposed Synthesizer Architecture for 802.11WLAN Standards

The steps taken in the design of the proposed synthesizer architecture are listed below. 1. Initial frequency planning is required for the employed oscillators derived from the required synthesized frequencies of interest for the 802.11a, b, and g WLAN standards [18]. 2. VCO design is dictated by the specified phase noise requirement. Phase noise requirement is typically derived from the standards specification and usually derived from exhaustive system-level simulation that relates the EVM and bit error rate (BER) to the phase noise of the entire transmitter/receiver chain [18]. 3. Once the frequencies of the oscillators are selected, the current consumption for the front-end divide-by-2 prescalers is estimated. 4. The MMD architecture is determined based on its input frequency (1.7–2.4 GHz for 802.11a, b, and g, respectively). A typical implementation of the MMD warrants the usage of a P/P + 1 dual-modulus

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

5.

6. 7. 8.

71

prescaler controlling two low-frequency counters, namely A and B (see Appendix D). The reference frequency of the crystal oscillator is chosen to be high (40 MHz) to yield improved settling time and phase noise performance. However, it can not be too high to worsen the system performance (see the 20 log10(N) and a 10 log10( fsamp) contributions to the phase noise plateau in Chapter 3). To cover all the channels while preserving the switching speed, a fractional divider is required and hence the use of a ∆−Σ modulator. Once the VCO frequencies are selected, the limits for the required division ratios needed are obtained. The PFD and CP have to be modeled correctly to show the effect of noise and nonlinearity on the performance of the entire PLL.

The simulation results obtained in chapter 4 aid the implementation of the proposed synthesizer architecture. The implementation and simulation of the individual subblocks of the PLL are described next.

5.4

THE ∆−Σ FREQUENCY SYNTHESIZER SUBBLOCKS IMPLEMENTATION

5.4.1 The Phase-Frequency Detector The PFD circuit used in the design presented in this chapter is based on the standard Dual type flip-flop (DFF) circuit. Figure 5-3 shows the block diagram of the PFD. Since the delay of the reset can affect the PFD behavior in the vicinity of zero phase difference (e.g. dead zone in the PFD characteristic as shown in Appendix A) [19], an adjustable two-state delay circuit is added in the reset path. This circuit allows the choice between two modes of short and long delay in the reset line of the PFD. The delay in the reset path is used when operating the synthesizer in integer mode to get rid of the dead zone. The delay is minimized when operating the synthesizer in fractional mode as the synthesizer operates at an offset of the dead zone. One of the main contributions to the phase noise from the PFD point of view is its inability to drive the CP switches at the required high sampling frequency associated with the fractional-N PLL. Figure 5-4 shows the implemented PFD circuit using hand-crafted gates instead of the flip-flops to ensure the

72

Chapter 5

Figure 5-3. PFD with Enabled Delay in the Reset Path

Up Ref

delay

Enable

Dn VCOf

Figure 5-4. PFD Schematic Used in the Synthesizer

best noise performance. The UP and DN pulses are designed in order to easily drive the required CP switches without compromising the inband phase noise due to the PFD.

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

73

This PFD is best simulated in conjunction with the CP to check for the linearity of both blocks and the effect on the overall performance of the frequency synthesizer. However, time-domain characterization of the PFD in Figure 5-4 is found in Appendix A.

5.4.2

The Charge Pump

The behavioral model of the CP was shown in Figure 3-4 and repeated here in Figure 5-5. In this section, a detailed circuit design for a CP is described. CP parameters that affect the performance of the PLL are: UP and DOWN current mismatches, unequal rise and fall times, glitches, and feed-through [20]. Several circuit simulations that characterize the CP are shown. Vdd

Iup

SwP Icp Dn

SwN Zs Idn

Figure 5-5. Behavioral Model for the Charge Pump

A single-ended CP circuit was designed using a 0.18 µm CMOS process. The schematic of this CP is shown in Figure 5-6.

74

Chapter 5 Vdd

Up

P3

Vdd

N3

SwP2

SwP1

P2

P1

N4

N5

N2 Dn

SwN1

P8

P7 Iout ip

P5 P6

∆I N9

Vdd

N1

P4

ip=∆I

Z(s)

N8 N6

N7

(a) Vdd

Ip +

Iout

Ib

Vss

(b) Figure 5-6. Circuit Schematic of the Charge Pump (CP Block) used in the ∆−Σ-based Synthesizer (a) CP, (b) Push–Pull Op-amp in the CP circuit

The position of the UP and DN switches of Figure 5-5 was swapped with the current sources to buffer the switching spikes at the loop filter. SwP1 and SwN1 are the PMOS and NMOS UP and DN

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

75

Figure 5-7. UP, DN, and Charge Pump Currents versus Tuning Voltage

switches, respectively. P2 and N5 are the UP and DN PMOS and NMOS current sources, respectively. Usually these UP and DN currents are mismatched and for higher tuning voltage at the loop filter, the UP current is smaller than the DN current and vice versa. A typical DC performance of a unit 62.5 µA CP with the mismatch cancellation technique disabled is shown in Figure 5-7. It is seen that ICP deviates from its horizontal zero net when the tuning voltage exceeds the 1.2 V value. At PLL lock, there is a net current given by:

∆I = I UP − I DN

(5.1)

The role of this proposed architecture of Figure 5-6 is to minimize the mismatch (∆I ). This is achieved by using an always-on replica UP and DN CP (P4, P7, N8, and N9) and a rail-to-rail op-amp in push–pull architecture (figure 5.7). The op-amp senses the mismatch, from ip, and feeds it back via N6, N7, P5, and P6 to the PMOS DN current as illustrated by P8 PMOS current source. This will yield a near-zero

76

Chapter 5

mismatch between the UP and DN currents. A residual mismatch helps suppress the spurious feed-through signals that modulate the VCO and appear at the output of the synthesizer. The mismatch cancellation technique shown in Figure 5-6 is useful when the frequency synthesizer is operating in integer division mode. However, as we will see later, this op-amp sensing cancellation technique is disabled when operating the synthesizer in fractional division mode. This is due to the need for an extra offset in the CP to guarantee linearity of the PFD/CP combination (refer to chapter 4 for more simulation details). 5.4.2.1 Dead-zone nonlinearity

The dead-zone issue is well known in integer PLL and is the consequence of the PFD narrow output pulses as the PLL approaches lock. Those pulses get smaller due to the finite reset pulse which is the

Figure 5-8. PFD/CP Linearity Curve (the Dead Zone Shown inside the Eclipse)

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

77

result of UP and DN pulses driving reset logic. Those pulses drive the CP switch transistors that have some time constants related to their input capacitances. Those switches cannot respond to small pulses and never fully switch ON depending on the speed of operation, hence the dead-zone effect. Figure 5-8 shows the dead-zone nonlinearity plot for an ideal PFD (with no delay in the reset path) driving a real CMOS CP. The reference frequency used is 40 MHz.

Figure 5-9. Nonideal PFD Characteristic Showing the Dead zone

Figure 5-9 shows a zoomed in view around the dead-zone nonlinearity region. The dead-zone nonlinearity causes an increase in the in-band phase noise of the synthesizer due to the PFD not being able to correct for small errors creating a state where the loop keeps going into and out of lock all the time. The size of the dead-zone is proportional to the PFD sampling clock speed and therefore becomes

78

Chapter 5

more serious at higher frequency. Correcting this problem is very simple and requires the use of a longer reset pulse by putting a delay in front of the reset logic. This ensures that the UP and DN pulses of the PFD are ON for a longer period of time allowing the CP to respond fully. 5.4.2.2 Linear Range and Cycle Slipping

Cycle slips due to PFD range limits in the PLL is a long-standing problem that has been addressed in the past [21]. This issue arises if the feedback VCO frequency and the reference frequency at the PFD inputs are too different and their comparison phase error is too large and falls outside the range of the PFD. This problem can cause the PLL to cycle slip and hence increase its settling time. For fractional-N PLL, this is of a concern only if the ∆−Σ modulator is of higher order. This can be resolved by extending the range of the PFD. 5.4.2.3 DC Offset Current

When both PFD inputs are in phase, the PFD/CP combination is subject to a dead zone in the characteristic curve. Moreover, if the positive and negative currents are not exactly the same, there would be a mismatch between the gains of the PFD/CP for positive and negative phase difference between the inputs. This can degrade the performance of the PLL when used in the fractional-N mode because the input of the PFD in this mode is never zero; instead, it is a variable number with a zero mean. This poses more stringent requirements on the PFD/CP in the fractional mode. One way to avoid the issues around the zero is shifting the operating point away by adding some offset current. This, in fact, gives a systematic phase offset which is not important in a fractional synthesizer. Figure 5-10 shows the schematic of the offset current circuit. The output current of this circuit is not an absolute value. Instead, it adds (or subtracts) a percentage of the CP current to the loop filter. Therefore, it should have the same topology as the CP with some extra logic to control the offset current. This circuit has two control signals

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

79

Figure 5-10. Offset Current for the Charge Pump

(OFFSET<1:0>) that set the offset current. As shown in the table of Figure 5.10, the offset current can have four values: 0, 10%, 20% or –10% depending on the control signals. A positive current means a current provided by the PMOS transistors and alternatively a negative current is drawn by the NMOS transistors. The table on the right-top corner of the figure summarizes the output current value based on control signals. Figure 5-11 shows the CP blocks and their corresponding offset current circuit. Each CP can have a current of 1 or 2 mA, depending on CP<0>. The offset current is proportional to the total CP current. For 1 mA CP current the offset is +100, +200, and −100 uA which corresponds to +10%, +20%, and –10% respectively. Figure 5-12 shows the schematic of the simulation setup for the offset current and Figure 5-13 shows the transient response of the offset current circuit.

80

Chapter 5

Figure 5-11. Charge Pump Block and Its Offset Current Block

Figure 5-12. Simulation Setup of the Offset Current Circuit

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

81

In this simulation, the control signals of the circuit are varied and the output current is changing accordingly from 0 to 100, 200, and −100 µA for a CP current of 1 mA (CP<0> = 0) and 200, 400, and –200 µA for a CP current of 2 mA (CP<0> = 1).

Figure 5-13. Transient Response of the Offset Current Circuit

82

Chapter 5

5.4.2.4 PFD/CP Transient Simulation

Figure 5-14 shows the simulation setup of the PFD/CP circuit. In this simulation, the output short circuit current of the CP is measured. Figures 5-15–5-17 show the transient responses of the PFD and CP outputs for three different cases in which the two inputs (a) have almost the same phase, (b) the reference signal from the crystal is leading, and (c) the output of the divider in the PLL is leading. As it can be seen, there is always a positive glitch after the output is reset. This is due to the extra delay in the signal that turns off the PMOS switch of the CP and also the PMOS switch itself.

Figure 5-14. PFD/CP Simulation Setup

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

Figure 5-15. Transient Response of the PFD/CP (Case [a])

83

84

Chapter 5

Figure 5-16. Transient of the PFD/CP (Case [b])

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

85

Figure 5-17. Transient Response of the PFD/CP (Case [c])

5.4.3

3.6 GHz Voltage-Controlled Oscillator

The main types of oscillators employed in radio frequency synthesizers are namely: the ring oscillator and the LC-tuned oscillator. The ring oscillator is simple and is typically constructed with multistage

86

Chapter 5

inverters. The ring oscillator usually has worse phase noise performance [22] and is not suitable for high-performance design as the one presented in this chapter. Figure 5-18 shows a current supplied LC VCO used in the design. This VCO provides an in-phase and quadrature-phase signals (namely I and Q) as it is used to drive an IQ image rejection mixer. The structure is fully differential as it offers better power supply rejection. The LC oscillator is constructed using all PMOS transistors as PMOS provide better noise performance since they have lower flicker noise compared to their NMOS counterparts [23] (more on that in Appendix B). Vdd X1

X4

X4

Vdd

Vdd

X4

X4

750u

Q- I+

Q+

I+

I-

Q+

I-

Q-

Vcntl Vss

Figure 5-18. Simplified Schematic of the Implemented Quadrature VCO

The oscillator shown in Figure 5-18 employs a bank of tunable varactors to cover the entire frequency range of interest for 802.11a, b, and g (see Table 5-1). The oscillation is obtained for the described VCO and it is illustrated in Figure 5-19 for the middle tuning range of 0.9V (3.663 GHz).

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

Figure 5-19. VCO Oscillations for the Mid-tuning Range

Figure 5-20. Phase Noise Profile for the Employed LC VCO

87

88

Chapter 5

Figure 5-20 shows the simulated phase noise profile VCO. The phase noise at 100 kHz offset from the carrier is −103 dBc/Hz and at 20 MHz is −156 dBc/Hz. Figure 5-21 shows the simulated VCO tuning curves covering all the VCO-synthesized frequencies for 802.11 standards. These are the synthesized frequencies at point X in Figure 5-2.

Figure 5-21. VCO Tuning Curves for all WLAN Bands

Figure 5-22. VCO Gain KVCO in MHz/V

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

89

The gain of the VCO is directly derived from the simulated data in Figure 5-21 and is shown in Figure 5-22 for VCO1 and VCO2.

5.4.4

The Multimodulus Divider

The MMD employed in the designed synthesizer incorporates a P/P +1 dual-modulus divider (in this case 8/9) designed in current-mode logic (CML) technology. The P/P + 1 divider controls two lowfrequency dividers A and B whose control bits are derived from the fractional divider noise shaper as discussed in the following section. The A and B counters are implemented in CMOS technology.

Figure 5-23. Multimodulus Divider Used in the Designed Synthesizer

The block diagram of this MMD is shown in Figure 5-23. The MMD circuit and system implementations are described in detail in Appendix D, however, its operation is shown briefly below:

90

Chapter 5

5.4.4.1 MMD Operation

1. B and A are loaded (B ≥ A) and modulus control = low, the prescaler divides by P + 1. 2. Counters decremented after rising edge of prescaler until counter A reaches 0. 3. Modulus control = high, the prescaler divides by P until the content of B is 0. 4. Counters are reset and cycle begins again. 5. Prescaler divides by P + 1 for A and by P for (B–A). The total division is BP + A.

5.4.5

The Fractional Noise Shaping Coder (the ∆−Σ Modulator)

In implementing the ∆−Σ-based fractional-N frequency synthesizer, a close look at the hardware implementation of the noise-shaping modulator is required. In the next few pages, a detailed design description of the implementation of the MASH-1-1-1 ∆−Σ modulator is given [28]. A step-by-step methodology is used from linear system model to actual hardware implementation. The basic system-level block diagram of the ∆−Σ modulator was shown in Figure 3-17 and its implementation is detailed below. 5.4.5.1 The Digital Accumulator and the First-Order Linear Model

Figure 5-24 shows the first-order linear model of the ∆−Σ modulator and its hardware accumulator-based implementation. To establish the link between the model and the hardware implementation, its time domain behavior is first analyzed. Using Figure 5-24, the time-domain equations of the ∆−Σ modulator are as follows: u[n] = X − b[n − 1]

(5.2)

v[n] = u[n] + v[n − 1]

(5.3)

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

u[n] X[n] Σ + Σ -

v[n]

Z

-1

Z

-1

E[n]

b[n]

Σ

X(n) m

91

b(n)

X

C X+Y

m

Σ

-E(n)=v(n)-b(n)

Y

m

-

-E[n] (a) Linear Model

Latch (b) Hardware Implementation

Figure 5-24. The First-Order modulator (a) Linear Model, (b) Hardware Implementation

b[n] =

+ 1 if v[n] ≥ 0 − 1 otherwise

(5.4)

The quantization error is defined as: E[n] = b[n] − v[n]

(5.5)

Substituting equation (5.3) in equation (5.2), yields v[n] = X − b[n − 1] + v[n − 1]

(5.6)

Substituting equation (5.5) in equation (5.6), yields v[n] = X − E[n − 1]

(5.7)

To illustrate the equivalence between the modulator and the accumulator, the following example is considered: • No. of input bits m = 2 • Maximum accumulator range = 3 • Input X = 2, Input Y = 2

92

Chapter 5

On the next calculation cycle, the accumulator content and the carry are shown in equations (5.8) and (5.9), respectively: E[n] = 1

(5.8)

b[n] = 1

(5.9)

Straightforward accumulation and quantization is assumed, then v[n] = X + Y = 4

(5.10)

Hence E[n] = −1 and Accumulator content = − E[ n] = 1

(5.11)

5.4.5.2 The 30-bit Structural MASH Coder Implementation

Using the above-described analogy between the digital accumulator and the first-order ∆−Σ modulator, the digital implementation of the MASH-1-1-1 ∆−Σ modulator of Figure 3-17 is shown in Figure 5-25. Y[n]

Σ X[n]

X

Σ

Σ

LATCH

C1[n]

X+Y

LATCH

C2[n] -e1[n]

C3[n]

X

Y

LATCH

Σ

X+Y

-e2[n]

X

Y

LATCH

X+Y Y

LATCH

Figure 5-25. Third-Order Noise Shaper Hardware Implementation

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

93

The third-order MASH ∆−Σ modulator of Figure 5-25 is implemented using 30 bits representing the divider value, with 6 bits for integer part, and 24 bits for the fraction part. The clock speed (sampling clock) at what the modulator can operate up to is critical. This is very important since the higher the clock of operation, the better is the noise shaping. Other requirements such as area and/or power consumption can also be important depending on the application. However, since noise performance is critical in frequency synthesizers, the speed of operation forms a major part of the modulator design. 5.4.5.3 The 24-bit Pipelined Adder Design

The implementation of high-speed accumulators is the most important part in the implementation of the modulator. Each accumulator is implemented using a 24-bit pipelined adder. The resolution of the accumulator can easily be calculated using the frequency error allowed in the IEEE standards specification [18]. Figure 5.26a shows the implementation of the adder of the first accumulator stage and Figure 5.26b the adder implemented in the subsequent accumulators. The 24-bit pipelined adder is implemented using three-stage 8-bit carry look-ahead (CLA) adders to achieve very high clocking speed. Since at each CLA stage, the output is calculated during one clock cycle, the second-stage input is delayed one clock cycle and the thirdstage CLA input is delayed two clock cycles. The clocked delays synchronize the output of the CLA adders so that the output of the 24bit adder is arrives at the same time. The implementation of the 8-bit CLA adder uses the following logic equations:

⎛ Si = ⎡( X i + Yi ) • ( X i • Yi ) ⎤ ⊕ Ci ⎞ ⎣ ⎦ ⎜ ⎟ ⎜ Ci+1 = ( X i + Yi ) • ⎡⎣( X i • Yi ) + Ci ⎤⎦ ⎟ ⎝ ⎠

(5.12)

Where Si is the ith full sum of the ith input vectors and Ci + 1 is the carry of the next operation.

94

Chapter 5 C in '0'

24-bit Pipelined Adder CLA

8

C in 8-bit C out

8

1-bit Latch

8

x(23:0)

8

8

8-bit Latch

24 s(23:0)

1-bit Latch

16-bit Latch

8-bit C CLA out

8

C in

24

8

C in

8

8-bit C CLA out C 1

clk

8

24 y(23:0)

Cout

(a)

8-bitC CLA out

1-bit Latch clk

Cin x(23:16)

clk

Cin

y(15:8)

1-bit Latch

x(15:8)

y(7:0)

x(7:0)

8-bitC CLA out

s(23:16)

8-bitC CLA out C1 y(23:16)

s(15:8)

s(7:0)

(b) Figure 5-26. Pipelined 24-bit Adder (a) for First-stage Accumulator, (b) for Subsequent Stages

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

95

5.4.5.4 Error Cancellation Algorithm

Using the accumulator-based implementation of Figure 5-25, the modulated fractional output with respect to the accumulators overflow is given by [24–26]: 3

Yout3 ( n ) = ∑ k =1

k

∑ (−1) r =1

r +1

. α r . Ck ( n − r + 1)

(5.13)

where α r are the coefficients taken from Pascal’s triangle. Equation (5.13) represents the error cancellation algorithm due to the integration mechanism of the second and third accumulators. The correction is implemented by equally weighted differentiators. Expanding equation (5.13) gives: Yout3 ( n ) = C1 ( n ) + C2 ( n ) − C2 ( n − 1) + C3 ( n ) − 2C3 ( n − 1) + C3 ( n −)2 C1 C2 C3

Logic1

3

3-bit Latch

+ C2(n-1) C3(n-1) C3(n-2)

Logic2

(5.14)

3

Yout(n)

CLK

Figure 5-27. Error Cancellation Network

Figure 5-28 shows the implementation of the error cancellation algorithm based on the 2’s complement network of Figure 5.27. The Mapping Logic1 (Map_log1) and Mapping Logic2 (Map_log2) are determined by the truth table shown in Table 5-2 and implemented using the logic of equations (5.15) and (5.16), respectively. Figure 5-29

96

Chapter 5

Table 5-2. Error Correction Network and Logic 1 and Logic 2 Truth Tables C1 0 0 0 0 1 1 1 1

C2 0 0 1 1 0 0 1 1

C3 0 1 0 1 0 1 0 1

Out1 0 1 1 2 1 2 2 3

C2(n − 1) 0 0 0 0 1 1 1 1

C3(n − 1) 0 0 1 1 0 0 1 1

C3(n − 2) 0 1 0 1 0 1 0 1

Out2 0 1 −2 −1 −1 0 −3 −2

and Table 5-3 show the special 2’s complement representation and the special handling of number 4. ⎛ ML2 = ⎡⎣C2 • ( C1 + C3 ) ⎤⎦ + (C3 • C1 ) ⎞ ⎟ Map _ Log1 ⎜ ⎜⎜ ML = ⎡C • ⎡ C • C + ( C • C ) ⎤ ⎤ + C • C • C + (C • C ) ⎟⎟ 1 3 ⎦ 2 1 3 1 3 ⎦ ⎝ 1 ⎣ 2 ⎣ 1 3 ⎠

(

)

(

)

(5.15)

(

) ) ( ) ( )

⎛ ML1 = A + B • C ⎞ ⎜ ⎟ Map _ Log 2 ⎜ ML2 = A • B • C + A + B • C + ( A • B ) ⎟ ⎜⎜ ML = B • C + B • C ⎟⎟ ⎝ 3 ⎠

( (

)

A

1 - b it Latc h

cl k

B C

1 - b it Latc h cl k

1 - b it Latc h C3

C3

1 - b it Latc h

1 - b it Latc h C2

C2 cl k

C1

1 - b it L atc h

(5.16)

ML1

M a p p ing M L 2 L og ic 2

'0 ' X Y

M L3

X

M a p p ing L og ic 1

M L1 Y M L2

X '0 ' Y

C1

Figure 5-28. Error Correction Algorithm Implementation

C o ut

A d d er

S

y3

C in C o ut

A d d er

S

y2

C in C o ut

A d d er C in

S

y1

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

97

Table 5-3. 2’s Complement Arithmetic with Specila Handling of Number “4” Yout(2) 0 0 0 0 1 1 1 1

Yout(1) 0 0 1 1 0 0 1 1

Yout(0) 0 1 0 1 0 1 0 1

Mapping 000000 000001 000010 000011 000100 111101 111110 111111

Figure 5-29. Divider Interface

5.4.5.5 Design Issues: Limit Cycle Cancellation in Fractional Mode

One of the consequences of using ∆−Σ modulators with DC inputs is the presence of limit cycles or spurs [1] that are strongly visible for inputs that are inverses of power of 2 such as 0.75, 0.5, and 0.25. This is due to the fact that the binary representation of such DC values has much less randomness. Figures 5-30 and 5-31 show the noise spectrum for 0.5 DC input with and without spurious limit cycles in linear and log scales, respectively. The effect of these limit cycles is greatly reduced as we introduce dithering (or randomness). Here, the spurs have been eliminated by introducing an error in the LSB or the input word [1].

98

Chapter 5

Figure 5-30. Power Spectral Density of Delta–Sigma Noise Shaper with and without Dithering (Linear Plot)

Figure 5-31. Power Spectral Density of Delta–Sigma Noise Shaper with and without Dithering (Log Plot)

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

99

5.4.5.6 Design Issue: Integer Mode and Close-to-Integer Option

One of the practical issues that are often overlooked in fractional-N PLL designs is the problem of having to synthesize frequencies that are integer multiples of the reference frequency, i.e. the divider value is an integer. This becomes a problem if not catered for in advance in the design. If the input is an integer, the ∆−Σ noise shaper input is zero. Previously, we introduced an error to remove any fractional spurs that may arise from limit cycles. That error will propagate in the modulator and cause the accumulators to overflow in a determined and cyclic manner causing spurious tones for integer frequencies. Figures 5-32 and 5-33 show that the modulator behaves like a secondorder ∆–Σ with small DC input that eventually produces fractional spurs. It is advisable to bypass the noise shaper when selecting to operate in integer division in the synthesizer. In other words, only use the ∆−Σ noise shaper when operating the synthesizer in fractional mode. Figures 5-34 and 5-35 show the spectral densities of the modulator when the input is a DC with values of 0.998 and 0.005, respectively. It can be clearly seen that for both cases, the modulator exhibits lowfrequency spurs due to insufficient dither. This problem is rectified by effectively increasing the dither via the introduction of an initial seed [1] in the modulator as illustrated in Figure 5-36.

Figure 5-32. Modulator Output in Integer Mode, Linear Plot

100

Chapter 5

Figure 5-33. Modulator Output in Integer Mode, Logarithmic Scale

Figure 5-34. Modulator Output for DC Input 0.998

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

101

Figure 5-35. Modulator Output for DC input 0.005 Showing Low-Frequency Spurs due to Insufficient Dither

Figure 5-36. Increasing Dither Suppresses the Spurs to Some Extent in the Fractional-N Modulator for Close-to-Integer Divider Values

102

5.5

Chapter 5

MEASURED PERFORMANCE OF THE IMPLEMENTED SYNTHESIZER

All the subblocks of the implemented synthesizer have been described. In what follows, we describe detailed performance of the implemented synthesizer and compare the obtained results with published performance of similar state-of-the-art synthesizers. Figures 5-37 and 5-38 show a detailed block diagram and a photomicrograph of the fabricated synthesizer, respectively. It must be noted that this synthesizer was incorporated in an entire transceiver. The synthesizer performance was also monitored at the transmit node to show the effect of several anomalies of the RF section on the overall synthesizer performance.

Figure 5-37. Detailed Block Diagram of the Fractional-N Synthesizer

This synthesizer was fabricated in 0.18 µm mixed-mode CMOS process and was incorporated in an entire transmitter/receiver chip whose die size is 17 mm2 [6]. The synthesizer/VCO chip area is 2 mm2. The entire chip is packaged in a 64-pin micro-lead frame (MLF) and operates from a 1.8 V power supply. The entire LO generation consumes 20 mA including the synthesizer and VCO. A 3.3 V power supply is also provided for chip I/O’s.

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

103

The measured band switching curves are shown in Figure 5-39 for the VCO1 part. They correlate pretty well with those simulated and shown in Figure 5-21.

Figure 5-38. Synthesizer Photomicrograph

104

Chapter 5

Figure 5-39. Measured Switching Curves for the Implemented VCO1 at LO Frequency

Figure 5-40. 802.11g Synthesizer-Measured Frequency Spectrum

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

105

The spectrum at the output of the local oscillator is shown in Figure 5-40. Adjacent 20 MHz-spaced channels are shown for the 802.11g case. The 2442 MHz was measured with a 5 kHz resolution bandwidth whereas the 2462 MHz was measured with a 200 Hz resolution bandwidth. No reference or fractional spurs are visible above −70 dBc for close-in or −68 dBc for far-out offset frequencies. The frequency synthesizer achieves an integrated phase error of 0.54o/1.1o for 2/5-GHz band with a loop filter bandwidth of 400 kHz [6]. This is illustrated in Figure 5-41. The LO signal is monitored and FM demodulated while the receiver and transmitter are switched on and off, respectively. It is found that the LO signal settles to 2 ppm (<5 kHz) of accuracy within 4 s, indicating a fast settling of the PLL. This is shown in Figure 5-42.

Figure 5-41. Phase Noise Profile of the Synthesizer for 802.11a, b, and g

106

Chapter 5

Figure 5-42. PLL During Receive/Transmit Switch

Figure 5-43. 64-QAM Constellation and EVM at the Output of the Transmitter

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

107

Figure 5-43 shows the measured transmit constellation and EVM for the 802.11g standard showing the minimized phase noise effect of the employed synthesizer.

5.6

SUMMARY AND CONCLUSION

In direct-conversion WLAN transceivers, the required LO frequency coincides with the RF, which entails adverse effects such as LO–RF interaction and VCO frequency pulling. Hence, an LO generation scheme that consists of a quadrature VCO operating at two-thirds of the LO frequency and a divide-by-2 circuit producing quadrature outputs at one-third of the LO frequency is employed. Two quadrature mixers subsequently multiply the VCO signal by the divide-by-2 signal to generate the quadrature LO signals (upper band), with significant suppression of the undesired lower band at one-third of the LO frequency, alleviating the image problem in the receiver and the spurious emission problem in the transmitter. The quadrature LO signals for the 2-GHz bands are then generated by dividing the 5-GHz LO signal by two. Two quadrature VCOs are used (VCO1: 3.2–3.6 GHz, VCO2: 3.5–3.9 GHz) in order to cover the required frequency range with enough margins for PVT variations. The main challenge in the frequency synthesizer design for this multistandard transceiver lies in reducing the in-band phase noise while maintaining a fast loop locking time [15]. Phase noise degrades signal EVM by introducing inter-subcarrier interferences. A narrower loop filter BW reduces out of band phase noise. On the other hand, during receive/transmit switching, the existing perturbation to the VCO produces a transient LO frequency offset. If the loop filter BW is narrow and the loop takes too long to settle, frequency estimation in the baseband modem becomes inaccurate, thus degrading the total system performance. These two seemingly conflicting requirements can be fulfilled with the described fractionalN PLL. The main advantage of a fractional-N PLL is that it breaks the traditional relationship between the channel spacing and the reference frequency in an integer-N PLL. By having a fractional divider, the reference frequency can be much higher, thus reducing the overall division ratio and in-band phase noise floor. Moreover, with a higher reference frequency, the PLL loop BW can be increased to reduce the locking/settling time.

108

Chapter 5

An automatic VCO band selection scheme is employed [12, 16]. Designed in a 0.18 µm process, the 8/9 prescaler works well above 4 GHz. The N-counter consists of an 8/9 prescaler and a 7-bit pulseswallow A/B counter. A 4-bit A counter with a 3-bit B counter covers the required frequency range. The ∆−Σ modulator employs a third-order noise-shaping (MASH) architecture for its higher-order noise shaping capability and better stability than a single loop high-order modulator [17]. The final ∆−Σ MASH modulator is programmed by a 27-bit digital word with 7-bit in the integer portion and 20-bit in the fractional portion, producing a minimum frequency step of 38.2 Hz (relative to VCO frequency), more than sufficient for any standard channel-spacing requirement. As a noisy digital circuit, the modulator may generate substantial cross talk noise. Since the VCO is an analog component, running continuously, it should be placed sufficiently far away from the ∆−Σ modulator with proper isolation guard rings. Furthermore, noise cross talk between the modulator and the PFD/CP can also produce spurs and degrade phase noise. Table 5-4. Performance Comparison with Other State-of-the-art Work Work

Process

PLL type Int-N

Freq.

Rategh 2000

0.24 um CMOS

Liu 2000

0.25 um CMOS

Int-N

5 GHz

Zargari 2002

0.25 um CMOS

Int-N

5 GHz

8 MHz

250 kHz

This work

0.18 um CMOS

FracN

5–6 GHz

40 MHz

400 kHz

4.8 – 5 GHz

PFD freq. 11 MHz

Loop BW 280 kHz

Lock time

4 uS

PN @ Offset −101 dBc/Hz @1 MHz −112 dBc/Hz @1 MHz −112 dBc/Hz @1 MHz −113 dBc/Hz @1 MHz

Power

Vsup

25 mW

3V

180 mW

2.5 V

36 mW

1.8 V

In this design, the ∆−Σ modulator and the instant divider ratio loading flip-flops (DFFA, DFFB) are negative edge-triggered. Knowing that the duty cycle of the feedback signal Nout (see Figure 5-39) is almost always less than 10%, the ∆−Σ modulator and the divider ratio loading will have enough time to settle before Nout goes

Multimode ∆−Σ-Based Fractional-N Frequency Synthesizer

109

high and the PFD/CP becomes active. In essence, noise cross talk is avoided by time-domain isolation. As a conclusion, in this chapter, the author has detailed the design and performance of a multimode fractional-N synthesizer. The synthesizer was designed as a local oscillator and constructed as part of a complete direct conversion transceiver. The measured results obtained for this synthesizer supersede most published results (see Table 5-4). The developed platform of chapter 4 has helped the designed synthesizer in achieving the best performance to date. In chapter 6, we propose a new adaptive and enhanced synthesizer architecture that offers optimum performance.

REFERENCES [1] N.M. Filiol, T.A.D. Riley, C. Plett, and M.A. Copeland “An Agile ISM Band Frequency Synthesiser with Built-In GMSK Data Modulation,” IEEE Journal of Solid-State Circuits, 33 (7), July 1998. [2] B. De Muer, M.S.J. Steyaert “A CMOS Monolithic ∆Σ-Controlled Fractional-N Frequency Synthesizer for DCS-1800,” IEEE Journal of Solid-States Circuits, 37 (7), pp. 835–844, July 2002. [3] Z. Shu, Ka Lok Lee, B.H. Leung “A 2.4-GHz Ring-Oscillator-Based CMOS Frequency Synthesizer with a Fractional Divider Dual-PLL Architecture,” IEEE Journal of Solid-States, 39 (3), pp. 452–462, Mar. 2004. [4] P. Zhang, T. Nguyen, C. Lam, D. Gambetta, T. Soorapanth, B. Cheng, S. Hart, I. Sever, T. Bourdi, A. Tham, and B. Razavi, “A Direct-Conversion CMOS Transceiver for IEEE 802.11a WLANs,” ISSCC Digest of Technical Papers, pp. 354–355, Feb. 2003. [5] P. Zhang, T. Nguyen, C. Lam, D. Gambetta, T. Soorapanth, B. Cheng, S. Hart, I. Sever, T. Bourdi, A. Tham, and B. Razavi, “A 5-GHz DirectConversion CMOS Transceiver,” IEEE Journal of Solid-State Circuits, 38 (12), pp. 2232–2238, Dec. 2003. [6] P. Zhang, L. Der, D. Guo, I. Sever, T. Bourdi, C. Lam, A. Zolfaghari, J. Chen, D. Gambetta, B. Cheng, S. Gower, S. Hart, L. Huynh, T. Nguyen, and B. Razavi, “A CMOS Direct-Conversion Transceiver for IEEE 802.11a/b/g WLANs,” IEEE Custom Integrated Circuits Conference, Digest of Technical Papers, Sect. 18, No. 4, Oct. 2004. [7] T.A.D. Riley, M.A. Copeland, and T.A. Kwasniewski, “Delta–Sigma Modulation in Fractional-N Frequency Synthesis,” IEEE Journal of Solid-State Circuits, 28, pp. 553–559, May 1993. [8] B. Miller and B. Conley, “A Multiple Modulator Fractional Divider,” Proceedings of IEEE 44th Annual Symposium Frequency Control, 1990, pp. 559–567. [9] B. Miller and B. Conley, “A Multiple Modulator Fractional Divider,” IEEE Transactions on Instrumentation Measurements, 40, pp. 578–583, June 1991.

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[10] M. Kozak, I. Kale, A. Borjak, and T. Bourdi, “A Pipelined All-Digital Delta–Sigma Modulator for Fractional-N Frequency Synthesis,” IEEE Instrumentation and Measurement Technology Conference (IMTC 2000), Vol. 2, pp. 1153–1157, Baltimore, MD, May 2000. [11] B.M. Miller, HP “Multiple-Modulator Fractional-N Divider,” US Patent No 5,038,117, 6 Aug. 1991. [12] T.A.D. Riley, Osgoode, Canada “Frequency Synthesizers Having Dividing Ratio Controlled by Sigma–Delta Modulator,” US Patent No. 4,965,531, 23 Oct. 1990. [13] C.E. Hill, Sciteq/Osicom “All Digital Fractional-N Synthesizer for High Resolution Phase Locked Loops,” Applied Microwave and Wireless, Nov./Dec. 1997. [14] Cosmo Little, “Fractional-N Synthesisers,” Electronics World, Mar. 1996. [15] B.G. Goldberg, Sciteq Electronics Inc. “The Evolution and Maturity of Fractional-N PLL Synthesis,” Microwave Journal, Sept. 1996. [16] U.L. Rohde, Synergy Microwave Corp. “Fractional-N Methods Tune Base-Station Synthesizer,” Microwaves & RF, Apr. 1998. [17] J.A. Weldon, et al., “A 1.75 GHz Highly-Integrated Narrow-Band CMOS Transmitter with Harmonic-Rejection Mixers,” IEEE SolidState Circuits Conference, 2001. Digest of Technical Papers, 2001 IEEE International 5–7 Feb. 2001, pp. 160–161, 442. [18] IEEE 802.11 Drafts Standards, http://grouper.ieee.org/groups/802/11/ [19] H.O. Johansson, “A Simple Precharged CMOS Phase Frequency Detector”, IEEE Journal of Solid-State Circuits, 33 (2), pp. 295–299, Feb. 1998. [20] B. Razavi, RF Microelectronics, Prentice Hall, Upper Saddle River, NJ, 1998. [21] P.V. Brennan, Phase-Locked Loops, Principles and Practice, McGrawHill, New Jersey 1996. [22] S. Verma, J. Xu, and T.H. Lee, “A Multiply-by-3 Coupled-Ring Oscillator for Low-Power Frequency Synthesis,’’ IEEE Journal of Solid-State Circuits, 39 (4), Apr. 2004. [23] H.-W. Chiu, Y.-C. Chen, and S.-S. Lu, “Optimization of CMOSIntegrated LC Oscillators using the Generic Algorithm,” Microwave and Optical Technology Letters, 42 (2), July 20, 2004. [24] T. Bourdi, et al., “A Novel Delta-Sigma Based RF Frequency Synthesizer Architecture For Cellular Applications,” IEEE Transactions of Instrumentation and Measurement, Under Review, Submitted June 2002. [25] T. Bourdi, et al., “Agile Multi-band Delta-Sigma Frequency Synthesizer Architecture,” IEEE International Symposium on Circuits and Systems 2002 (proc ISCAS 2002). [26] T. Bourdi, et al., “A Delta-Sigma Frequency Synthesizer with Enhanced Phase Noise Performance,” IEEE Instrumentation and Measurement Technology Conference, May 2002. [27] A. Borjak, T. Bourdi, “Intermodulation Products in a Mixer Subjected to a Multi-Carrier Signal,” Microwave Journal, Jan. 2002. [28] M. Kozak, I. Kale, A. Borjak, and T. Bourdi, “A Pipelined All-Digital Delta–Sigma Modulator for Fractional-n Frequency Synthesis,” IEEE Instrumentation and Measurement Technology Conference, May 2000.

Chapter 6 IMPROVED PERFORMANCE FRACTIONAL-N FREQUENCY SYNTHESIZER

6.1

INTRODUCTION

In chapter 5, we detailed the design, implementation, and measurement of a multimode fractional-N frequency synthesizer for WLAN standards. The synthesizer offered the best performance to date. However, additional circuit could be designed to enhance the performance of the synthesizer at the cost of increased circuit complexity. Those additions include adaptive CP architecture to maintain loop gain and phase transfer functions while operating in fractional mode, i.e. instantaneous different integer divisions. Also included is an adaptive band switching control to maintain frequency agility while offering optimum phase noise performance in the band of interest. Along with other additional techniques that improve the synthesizer performance, those additions will be described in this chapter in detail.

6.2

OVERVIEW

Figure 6-1 shows the basic fractional-N architecture that was implemented in chapter 5. Several circuit techniques could be added to this synthesizer architecture to enhance its performance.

111

112

Chapter 6

Figure 6-1. Conventional Fractional-N Synthesizer Architecture

Figure 6-2. Fractional-N Synthesizer with Enhanced Performance

Improved Performance Fractional-N Frequency Synthesizer

113

Those circuit improvements are shown in Figure 6-2 using the dashed blocks and lines and are identified as follows: 1. ∆–Σ-controlled adaptive CP architecture 2. VCO gain calibration to maintain best loop dynamics for best phase noise performance 3. Improved VCO band switching to maintain linear PLL gain operation Those improvements could be used jointly for best performance. Each one of those is described in detail in the next few pages and measured results based on all these joint techniques applied to the synthesizer are shown at the end of the chapter.

6.3

DELTA—SIGMA-CONTROLLED ADAPTIVE CHARGE PUMP

∆−Σ-based fractional-N frequency synthesizers similar to the ones described in chapter 5 have been introduced to alleviate integer-N PLL frequency resolution problems as the requirement for fast switching time and low phase noise become increasingly stringent. Until recently [1–5], the use of noise shaping has been limited to fractional divider control. The ∆−Σ modulator generates a pseudorandom sequence of bits representing random integer divider values, at the speed of the sampling clock, yielding an average fractional divisor to synthesize the required frequency whose resolution is a fraction of the sampling clock frequency. These instantaneous divider variations continuously alter the PLL loop dynamics and deteriorate phase noise performance. In what follows, we will describe a new fractional-N synthesizer architecture based on an adaptive CP that was fully implemented in an RF CMOS process and show measured phase noise results that satisfy several system requirements.

6.3.1

PLL Gain and Phase Variations

In conventional ∆−Σ fractional-N synthesizers the divider ratio is modified while the CP current is fixed. This alters the loop dynamics that impact the phase noise performance. It also has direct influence on the PLL open loop dynamics, as is proven in the following derivation.

114

Chapter 6

The open loop gain of the PLL synthesizer of Figure 6-1 at the averaged fractional divider ratio is given by:

Aolaverage = I cp F ( s )

K vco

1

s

N frac

(6.1)

Icp is the CP current, F(s) is the loop filter impedance transfer function, Kvco is the VCO gain, and Nfrac is the operating fractional division. Also, the open loop gain of the PLL synthesizer with the instantaneous divider ratio Ninst is given by [6, 7]

Aolinst = I cp F ( s )

K vco

1

s N inst

N − 2 m −1 + 1 < N inst < N + 2 m −1

(6.2)

(6.3)

N is the integral part of Nfrac and m is the order of the ∆–Σ MASH modulator. Therefore, the output of a third-order MASH modulator would vary between −3 and 4 and equation (6.3) becomes:

N − 3 < N inst < N + 4

(6.4)

The closed-loop equation for the PLL with the average case is given by: Aclaverage =

Aolaverage 1 + Aolaverage

(6.5)

and the closed loop equation for the PLL with the instantaneous case is given by:

Aclinst =

Aolinst 1 + Aolinst

(6.6)

Improved Performance Fractional-N Frequency Synthesizer

115

Notice here that since equations (6.1) and (6.2) yield different results, equations (6.5) and (6.6) also yield different results. So the higher the order of the ∆−Σ modulator, the larger the divider variations and consequently the more loop gain and phase disturbances, as shown in Figure 6-3.

Closed Loop Gain (dB)

3.5 Nmin

3 2.5

Ncenter

2 1.5 Nmax

1 0.5

5

Closed Loop Phase (rad)

10 -0.6 -0.8

Nmin

-1 Ncenter

-1.2

Nmax

-1.4 -1.6

5

10 Frequency (Hz)

Figure 6-3. Closed-Loop Gain and Phase Characteristics

The example given here illustrates a typical system and clearly shows the gain and phase variations during the synthesis of the middle frequency band. A solution to this problem has been reported in [6, 7], alleviating these issues by introducing an adaptive noise-shaped Charge Pump (NSCP) architecture (see the block diagram in Figure 6-2 in orange color), which is controlled by the same ∆−Σ modulator that is controlling the divider. Manipulating equations (6.1)–(6.5) the new proposed PLL architecture is achieved.

116

Chapter 6

The modified open-loop equation that takes into account the above-mentioned variations is then given by:

Aolinst = I cp(inst) F ( s )

K vco

1

s N inst

(6.7)

where the Icp(inst) is the instantaneous adaptive CP value that is directly controlled by the ∆−Σ modulator. Notice that since Ninst changes according to equation (6.5) for a third-order modulator, Icp(inst) would be varying accordingly to keep the value of Aolinst = Aol . Aol is the open-loop gain corresponding to the case with the PLL running at the instantaneous integral divisor N and is given by:

Aol = I cp F ( s )

K vco 1 s N

(6.8)

where I cp is the nominal CP current (corresponding to PLL case with integral division N) that around which I cp(inst) varies. For a third- order modulator I cp(inst) is then given by:

I N + 4 < I cp(inst) < I N −3

(6.9)

I N + 4 is the minimum CP value and corresponds to the case of the instantaneous divider having a maximum value of N + 4 and I N − 3 is

the maximum CP value and corresponds to the case of the instantaneous divider having a minimum value of N − 3 . The new closed-loop equation for the proposed architecture would then be given by equation (6.6) but with Aolinst = Aol .

6.3.2

Charge Pump System

Figure 6-4 shows the system overview of the control of the implemented adaptive CP architecture. The coding method employed on the b bits is thermometer coding as used in current-segmented digital-to-analog converters [8]. The outputs of the thermometer encoders control two banks of UP and DOWN CP circuitry.

Improved Performance Fractional-N Frequency Synthesizer b<5:3> Thermometer M<7:1> b<5:0>

Encoder 3to 7 bits

D ck

UP

Q Thermometer L<7:1> Encoder 3to 7 b<2:0> bits

117 Iup-

Matrix Encoder Control Logic & Unit Cells

Iup

ck IdownMatrix Encoder Control Logic & Unit Cells

DOWN

Idown

ck

Figure 6-4. Adaptive Charge Pump Architecture

6.4

SYNTHESIZER LOOP CALIBRATION

Calibrating the PLL synthesizer to maintain a constant loop gain and phase is very important, as these variations affect the phase noise performance due to the CP and loop filter elements. The loop gain affects the loop bandwidth, phase noise performance, and predistortion matching. Equation (3.1) shows that the loop gain is determined by the CP current, Icp, the VCO gain, KVCO, divider ration, N, and the loop filter capacitance. In a multimode PLL frequency synthesizer, the integrated loop filter pole and zero locations, vary greatly from frequency to frequency, as well as with process. The pole and zero locations are determined by the loop filter RC time constant. To minimize the CP current and loop filter noise contribution to the VCO, the CP current and loop filter need to be designed to meet phase noise specifications with the worst KVCO gain at nominal. This will make sure exceeding or at least meeting those specifications after the final calibration is done. Starting from the previously defined loop gain in chapter 3 as shown in equation (3.1), the actual loop gain constant may be expressed as a function of the typical loop gain and the loop element physical and process tolerances:

118

Chapter 6

Aol =

( I CP + I CP .∆ Icp ).( K VCO + K VCO .∆ Kvco ) ( N nom + N nom ∆ N ).(Cnom + Cnom ∆ C )

(6.10)

The delta terms in equation (6.10), represent the variations or tolerances from the nominal or desired values. Similarly we can derive the typical RC time constant as: RC = ( Rnom + Rnom .∆ R ).(Cnom + Cnom .∆ C )

(6.11)

From equations (6.10) and (6.11), it is apparent that in the PLL loop, there are five different element variations that need to be corrected for, N divider variations, VCO gain variations, CP current reference variations, and RC process variations. N divider variations have been taken care of in section 6.3. The VCO gain is designed to be within a well-determined window that can meet the specified performance criteria at nominal and hence will not be calibrated directly. Hence, the CP current are the only elements of the loop that will be calibrated. The slew rate, I/C, and RC can be independently calibrated against the reference clock. Therefore, we do not need to program the capacitance, C, and only need to calibrate the CP current and the resistor R to account for all five PLL loop elements variations. Rearranging equations (6.10) and (6.11), gives: Aol =

I CP .K VCO .(1 + ∆ Icp ).(1 + ∆ Kvco ) N nom .Cnom .(1 + ∆ N ).(1 + ∆C )

(6.12)

and,

RC = Rnom Cnom .(1 + ∆ R ).(1 + ∆ C )

(6.13)

The calibration can be thought of as “correction factors” that null out the deviations from the various sources of error, setting the loop gain and the RC time constant to their desired, nominal values. Hence, the final open loop gain can be expressed as:

Improved Performance Fractional-N Frequency Synthesizer

Aol =

f I .I CP .K VCO .(1 + ∆ Icp ).(1 + ∆ Kvco ) N nom .Cnom .(1 + ∆ N ).(1 + ∆ C )

=

I CP .K VCO , N nom.Cnom

(1 + ∆ N ).(1 + ∆ C ) fI = (1 + ∆ Icp ).(1 + ∆ Kvco )

119

(6.14)

and the time constant, RC = f R .Rnom Cnom .(1 + ∆ R ).(1 + ∆ C ), f R =

1 (1 + ∆ R ).(1 + ∆ C )

(6.15)

fR and fI are the correction factors. Looking at the above equations, we can see that the CP current need to adjust for three sources of errors in the loop. In addition to the divider, N, variations that have already been addressed in the adaptive part of the CP, the Icp correction factor, fI, need to compensate for the VCO gain and I/C ratio. Hence,

f I = f IN . f IKvco . f IIC

(6.16)

Since N variations have been taken care of, equation (6.16) can be written as:

1+ ∆I ) f I = (1 + ∆ Kvco ).( 1 + ∆C

(6.17)

These adjustments will be determined by calibrations for I/C and KVCO. I/C calibration can be done in conjunction of RC calibration.

6.5

PROCESS CALIBRATION I/C SLEW RATE AND RC TIME CONSTANT

The CP current is derived from a well-defined bandgap reference voltage, VBG and a reference resistor R, Figure 6-5. The reference current is derived as:

120

Chapter 6

Figure 6-5. Reference Current Generation from Bandgap

I ref =

VBG R

(6.18)

Using the same reference current and using a counter clocked by the crystal reference clock used in the PLL, we can charge a capacitor, C and note the time variations, ∆T, from nominal corresponds to the RC and I/C variations, and hence the capacitor, C, in the loop can be adjusted to compensate for those variations based on the counter word output. Hence,

VBG I ref .R I ref = = T T C

(6.19)

Therefore,

T = RC and,

(6.20)

Improved Performance Fractional-N Frequency Synthesizer

VBG I = T C

121

(6.21)

From the above equations, it is apparent that as long as the bandgap voltage reference is accurate and that all the used resistor and capacitors have a good matching and of the same type, the calibration will be accurate.

6.6

VCO TUNING GAIN CALIBRATION

The VCO tuning gain (KVCO) differentiates the sensitivity of the VCO output frequency (FVCO) to changes in its tuning voltage (Vtune). It is defined as: K VCO =

dFvco dVtune

(6.22)

The tuning gain is not constant, as illustrated in the plots of VCO frequency and tuning gain versus tuning voltage (Figure 5-21 and 5-22). It is apparent that the VCO gain changes across the tuning voltage curve. Instead of compensating the VCO directly, the loop gain of the PLL can be compensated by adjusting another gain term of the CP current to cancel out the variations in the KVCO tuning gain. The compensation requires a method of measuring the tuning gain, and hence an algorithm can be derived and applied appropriately for the KVCO tuning gain adjustment.

6.6.1

VCO Calibration Algorithm Description

The used tuning gain calibration system approximates the tuning gain by measuring the difference in tuning voltage for a predetermined difference in VCO frequency. This approximates the tuning gain with finite number differences: K VCO =

dFvco ( F2 − F1 ) ∆F ≅ = dVtune (Vtune2 − Vtune1 ) ∆Vtune

(6.23)

122

Chapter 6

The calibration algorithm works such that it locks the PLL synthesizer to a slightly different frequency close to the final required frequency and the tuning voltage is recorded. It then locks to the actual frequency of interest recording the final tuning voltage. A difference in the measured tuning voltage is then calculated and hence the ratio of the difference of the measured tuning voltage ∆Vtune to the nominal difference ∆Vtune_nom equates to the ratio of the nominal tuning KVCO called KVCO_nom to the measured tuning KVCO, as illustrated in the following equations.

K vco_nom =

∆F ∆Vtune_nom

(6.24)

and ∆F ∆Vtune

(6.25)

∆Vtune_nom K vco = K vco_nom ∆Vtune

(6.26)

K vco =

Hence,

Using an analog-to-digital converter (ADC) as illustrated in Figure 6-2, the ratio in equation (6.26) is expressed in binary word format:

∆Vtune = Vtune2 − Vtune1 ⇒ ∆ADBits = ADBits2 − ADBits1

(6.27)

where ADBits represents the binary output word of the ADC and the ∆ADBits is the binary difference of final and first ADC readings. For the nominal tuning gain, there is a nominal difference in ADC output values, ∆ ADBitsnom. The ratio of the measured to the nominal ADC value is then equal to the ratio of the nominal to the measured tuning gain.

K vco_nom K vco

=

∆Vtune ∆ADBits = ∆Vtune_nom ∆ADBitsnom

(6.28)

Improved Performance Fractional-N Frequency Synthesizer

123

Introducing the above ratio to the loop gain of equation (6.14), it can be used to adjust the CP current and hence null out the VCO gain variations with respect to the golden design value:

⎛ ∆ADBits ⎞ I cpnom ⎜ ⎟ K vco I cp K vco ⎝ ∆ADBitsnom ⎠ Aol = = NC NC

(6.29)

After calibrating the loop, the effect of the loop gain Aol on the loop should be the same as the golden nominal loop gain Aolnom. This can be verified by examining the loop gains ratio, equation (6.30): I .K Aol N .C = cp vco Aolnom N .C I cp_nom .K vco_nom ⎛ ∆ADBits ⎞ ⎜ I cp_nom . ⎟ .K vco ∆ADBitsnom ⎠ ⎝ = I cp o K v o K vco_nom ⎞ ⎛ ⎜ I cp_nom . ⎟ .K vco K vco ⎠ ⎝ = I cp_nom .K vco_nom =

(6.30)

I cp_nom .K vco_nom I cp_nom .K vco_nom

=1

The CP mirror ratio is implemented as shown in Figure 6-6. The mirror devices are binary weighted and hence can be driven directly by the ADC word, ∆ ADBits. The nominal CP current corresponding to the nominal VCO gain and hence nominal loop gain is designed into the input current mirror.

124

Chapter 6 ∆<1>

∆<0>

∆

1

2n

2 To CP switch

Icp_nom

Icpp

Figure 6-6. CP Mirror Ratios VDD ∆<0>

∆nom

∆ ∆

∆<0> Vbp

Vbp 2n

1 Icp_nom

UP

VDD

DN

Charge Pump Switches

Icp_nom Iref

∆<0>

Vbn

∆<0>

1 Vbn

∆ 2

n

∆

∆nom VSS

Figure 6-7. Charge Pump Employing Mirroring as in Figure 6-6 for VCO Gain Calibration

Figure 6-7 shows the CP circuit employing the mirror ratio. The CP comprises a typical current reference cell, with the first current branches mirrors weighted according to the expected ∆ ADBitsnom

Improved Performance Fractional-N Frequency Synthesizer

125

(denoted ∆ nom in the figure), and the output mirrors binary weighted, and switched according to the measured ∆ ADBits (denoted ∆<> for simplicity). 6.6.1.1 ∆N Values

The ∆ N values are chosen based on the nominal VCO gain KVCO_nom and the nominal ADC output, ∆ ADBitsnom. Based on the ADC size and the nominal KVCO, the nominal tuning voltage, ∆Vtune_nom is determined by:

∆Vtune_nom =

∆ADBitsnom .VDD 2n

(6.31)

VDD is the supply voltage and n is the number of bits of the ADC used in the algorithm. The nominal VCO frequency change is determined by ∆Fnom:

∆Fnom = ∆Vtune_nom .K VCO_nom

(6.32)

∆N is the change in N value which is related to the change in frequency, ∆F, and hence is: ∆N =

∆Fo Fref

(6.33)

Fref is the PLL synthesizer sampling frequency. Equation (6.33) can be rewritten as shown in equation (6.34).

∆N =

∆ADBitsnom .VDD.K vco 2n ⋅ Fref

(6.34)

6.6.1.2 Summary of Tuning Algorithm Operation

To summarize the operation of the PLL synthesizer VCO gain calibration system, the calibration algorithm is outlined as follows: • The channel frequency is set and hence determines the final divider value, N. • A ∆N value is then chosen based on the ADC size and the nominal VCO gain that is determined by design.

126

Chapter 6

• The divider is then programmed to (N−∆N) value and the synthesizer is given time Tlock which is set by allowing a counter clocked by the reference clock within the algorithm implementation to count down from it to zero indicating PLL lock. • Once the counter reaches zero after Tlock, the value of the tuning voltage (Vtune) is read by the ADC and registered in regiser-1. • The synthesizer is then programmed to lock to the final frequency, N*fref. After Tlock, Vtune is similarly read by the ADC and the value is stored in register-2. • The algorithm calculates the difference between register-2 and register-1 giving ∆ADBits which then program the CP current accordingly to null out the VCO gain variations.

Figure 6-8. Half-Band Tuning Shown in Terms of third Band from the Bottom

Improved Performance Fractional-N Frequency Synthesizer

6.7

127

IMPROVED VCO BAND SWITCHING

Here we describe a technique to improve the VCO band switching used in chapter 5. An additional bit is added to the 8-bit VCO tuning to control half-band tuning. In other words, once lock has been achieved with the coarse tuning using of eight bands, an additional half band is switched on to further enhance the phase noise performance of the synthesizer. This is illustrated in Figure 6-8. The half band in red can be switched between any two full bands by switching additional half-unit capacitance. This can be regarded as fine tuning to reduce the tuning window and hence allow for an efficacy in CP linearization used as we employ the synthesizer across bands.

6.8

EXPERIMENTAL RESULTS

The design, implementation, and measurement of the proposed multiband fractional-N PLL synthesizer using an RFCMOS technology have been undertaken. The complete synthesizer architecture has been

Figure 6-9. Phase Noise for the Enhanced Fractional-N PLL @ 2.4 GHz, Spurs are Shown Transposed on Phase Noise Curve

128

Chapter 6

built for performance and to establish structure viability. Figure 6-9 shows the measured phase noise performance for the WLAN standard at 2.4 GHz band. The measured phase noise curve has two parts: phase noise curve and spurious transposed on the phase noise curve. The result shows an improvement when the NSCP, VCO calibration, and improved VCO band switching are employed.

6.9

COMPARISON WITH PUBLISHED RESULTS

A summary of the achieved results obtained by measuring the fractional-N synthesizer is shown in the Table 6-1. Table 6-1. Comparison with other Work Ref.

Process (CMOS)

PLL Type

Freq. (GHz)

PFD Freq. (MHz)

Loop BW (KHz)

[9] [10] [11] This work

0.24 0.25 0.25 0.18

Integer

5 5 5 2.4–5

11

280

8 40

250 280

6.10

Integer Fractional

L (f) @ offset (d Bc / Hz @ 1 MHz) –101 –112 –112 –119

Power Usage (mW)

Supply Voltage (V)

25 180 40

3 2.5 1.8

CONCLUSION

We have presented a novel frequency synthesizer structure based on noise-shaped fractional-N PLL employing a ∆−Σ-controlled adaptive CP. Extensive calibration techniques were employed through out the PLL. Measurement results obtained from testing the new architecture showed enhancement in the noise performance by 10 dB. This noise improvement was translated into a reduction in the synthesizer locking time since wider loop filter BW is used. The above improvements come at a price of increased complexity, current consumption, and chip area.

REFERENCES [1] T. Riley, M. Copeland, and T. Kwasniewski, “Delta-Sigma Modulation in Fractional-N Frequency Synthesis,” IEEE Journal of Solid-State Circuits, 28, pp. 553–559, May 1993.

Improved Performance Fractional-N Frequency Synthesizer

129

[2] P.T. Kenny, T. Riley, N. Filiol, and M. Copeland, “Design and Realization of a Digital Delta–Sigma Modulator for Fractional-N Frequency Synthesis,” IEEE Transactions on Vehicular Technology, 48, pp. 510–521, Mar. 1999. [3] B. Miller and R. Conley, “A Multiple Modulator Fractional Divider,” IEEE Transactions on Instrumentation and Measurements, 40, pp. 578–582, June 1991. [4] M. Kozak, I. Kale, A. Borjak, and T. Bourdi, “A Pipelined All-digital Delta–Sigma Modulator for Fractional-N Frequency Synthesis,” IEEE Instrumentation and Measurement Technology Conference (IMTC 2000), Vol. 2, pp. 1153–1157, Baltimore, MD, May 2000. [5] W. Rhee, B. Song, and A. Ali, “A 1.1-GHz CMOS Fractional-N Frequency Synthesizer with a 3-b Third-Order DS Modulator,” IEEE Journal of Solid-State Circuits, 35 (10), Oct. 2000. [6] T. Bourdi, A. Borjak, and I. Kale, “Agile Multi-Band Delta Sigma Frequency Synthesizer Architecture,” Proceedings of ISCAS 2002, pp. 413–416. [7] T. Bourdi, A. Borjak, and I. Kale, “A Delta Sigma Frequency Synthesizer with Enhanced Phase Noise Performance”, Proceedings of IMTC 2002, pp. 247–251. [8] J. Vandenbussche, et al., Systematic Design of High-Accuracy CurrentSteering D/A Converter Macrocells for Integrated VLSI Systems, IEEE Transactions on Circuits and Systems II, 48 (3), Mar. 2001. [9] H.R. Rategh, H. Samavati, and T.H. Lee, “A CMOS Frequency Synthesizer with an Injection-Locked Frequency Divider for a 5-GHz Wireless LAN Receiver,” IEEE Journal on Solid-state Circuits, 35 (5), May 2000. [10] T.-P. Liu and E. Westerwick, “2- 5-GHz CMOS Radio Transceiver Front-End Chipset,” IEEE Journal of Solid States Circuits, 35 (12), Dec. 2000. [11] M. Zargari, D.K. Su, C.P. Yue, S. Rabii, D. Weber, B.J. Kaczynski, S. Mehta, K. Singh, S. Mendis, and B.A. Wooley, “A 5-GHz CMOS Transceiver for IEEE 802.11a Wireless LAN Systems,” ISSCC Digest of Technical Papers, Feb. 2002.

Chapter 7 CONCLUSION AND FURTHER WORK

7.1

CONCLUSION

The work presented in this monograph focused on an enhanced design and implementation of low phase noise frequency synthesizers as local oscillators in transceivers for multimode WLAN applications covering all 802.11a, b, and g standards. Brief descriptions of those standards were discussed in terms of frequency bands for the radio frequency transmission. A suitable transceiver operating in all WLAN modes was described along with its transmitter/receiver chains. Architecture for the frequency synthesizer acting as local oscillator for the transceiver was proposed and employed in this book. Direct frequency synthesis was not used to avoid frequency pulling in the transceiver [1]. Detailed analyses of integer and fractional-N phase-locked loop frequency synthesizers were treated. Open-loop and closed-loop transfer functions of the phase-locked loop were derived. Noise contributions of individual subblocks of the synthesizers were detailed. Loop filter design is also included. The thorough analyses, derived equations, and simulation results obtained from the undertaken research formed the basis for an optimum design and implementation of the implemented frequency synthesizer chips described in this book. Unconditionally stable ∆−Σ modulators of the third-order (namely MASH-1-1-1) were studied and employed in the analyzed phaselocked loop fractional-N synthesizer providing a good average estimate for fractional-N dividers. Using a deep submicron 0.18 µm CMOS process with a supply voltage of 1.8 V, a ∆−Σ-based

131

132

Chapter 7

fractional-N synthesizer was designed, simulated, laid out, fabricated, and tested. The obtained phase noise performance was superior to all WLAN synthesizers in the literature [2–4]. Novel circuit techniques and advanced design ideas were also proposed and incorporated in the design, enhancing the overall performance of the synthesizer. Those additions included adaptive CP architecture to maintain loop gain and phase transfer functions while operating in fractional mode; adaptive band switching control to maintain frequency agility while offering optimum phase noise performance in the band of interest was also incorporated and tested. One of the main demonstrations and achievement of the work inhand is the development of the behavioral synthesizer platform using the existing Cadence tools. The platform has been used in the accurate simulations and predictions of all observed anomalies associated with fractional-N ∆−Σ-based frequency synthesizers. The author did not leave anything to chance by rigorously analyzing the synthesizer performance before and after different fixes and remedies have been applied. Issues like dead-zone, close-to-integer operation, and CP mismatch nonlinearity to name but a few have been observed as they affect the synthesized VCO frequency. Various techniques have been developed to mitigate these anomalies. Techniques such as operating in well simulated and determined linear region of the PFD/CP to mitigate the quantization noise folding phenomenon has been developed. Additionally, methods of ∆−Σ dithering with the right amount to completely reduce the fractional spurs to levels acceptable by the system specifications have been proven. Last but not least, the viability to optimize the loop BW on the fly in a closed loop PLL in the presence of all noise sources and nonlinear effects made the design much more predictable. Once all the tools and methods have been developed, the results have been demonstrated in the design and implementation of an ultra low noise synthesizer system for deployment in the WLAN standards. The developed techniques can be used in any synthesizer development and design with an emphasis on fractional-N ∆−Σ-based systems.

7.2

FURTHER WORK

Although the work presented in this book formed a major contribution to the superior implementation of frequency synthesizers acting as local oscillator in WLAN transceivers, a few concept ideas could be

Conclusion and Further Work

133

identified to form major future research work for the implementation of low phase noise frequency synthesizers. These are as follows: 1. The use of the developed techniques to study the feasibility of a totally calibrated and digitally controlled frequency synthesizer that can form the basis of an all digital transmission system. Such system can make use of DSP functions to solve some of the system anomalies that are usually difficult to fix using analog techniques. 2. Direct modulation frequency synthesizer to reduce nonlinear analog component count like mixers and analog modulators. In addition to a power amplifier, such synthesizers form the basis of ultralow power transmitters. ∆−Σ-based direct modulation transmitters can take the in-phase and quadrature base-band data and modulate it using the ∆−Σ modulator then upconvert directly using the VCO to drive the power amplifier. The main difficulty in such work will be the wide range of bandwidths used by different systems. The loop BW needs to be at least as wide as the transmitted data bandwidth. Hence, issues like quantization noise will be significant unless the ∆−Σ modulator is sampled at much higher frequency than the loop BW. An all digitally controlled ∆−Σ-based frequency synthesizer capable of data modulation will have many benefits over its traditional counterparts. Since all signals are digital, the use of known signal processing techniques to alleviate most of the anomalies and issues normally present in a PLL system will be much easier. The fixes will be much more robust and predictable in the digital domain. An example of such a system is shown in Figure 7-1. Data

Data

Ref. TDC

m-bits + -

error n-bits Filter

C-bits

Synthesized Freq NCO

TDC

Time to digital Converter Figure 7-1. An all Digital Frequency Synthesizer Example

134

Chapter 7

The proposed system of Figure 7-1 can either use a frequency to digital converter (FDC), or a time to digital converter TDC [5], to convert the reference signal to a digital word that will be used for comparison with the feedback data coming from an NCO. The data representing the resulting comparison error can be processed using a ∆−Σ modulator acting as a digital filter before it is passed on to control the NCO. Data modulation can be applied to either the ∆−Σ modulator or directly to the NCO.

REFERENCES [1] J.A. Weldon, et al., “A 1.75 GHz Highly-integrated Narrow-band CMOS Transmitter with Harmonic-rejection Mixers,” IEEE Solid-State Circuits Conference, 2001. Digest of Technical Papers, 2001 IEEE International 5–7 Feb. 2001, pp. 160–161, 442. [2] H.R. Rategh, H. Samavati, and T.H. Lee, “A CMOS Frequency Synthesizer with an Injection-Locked Frequency Divider for a 5-GHz Wireless LAN Receiver,” IEEE Journal on Solid-State Circuits, 35 (5), May 2000. [3] T.-P. Liu and E. Westerwick, “2- 5-GHz CMOS Radio Transceiver Front-End Chipset,” IEEE Journal Solid States Circuits, 35 (12), Dec. 2000. [4] M. Zargari, D.K. Su, C.P. Yue, S. Rabii, D. Weber, B.J. Kaczynski, S. Mehta, K. Singh, S. Mendis, and B.A. Wooley, “A 5-GHz CMOS Transceiver for IEEE 802.11a Wireless LAN Systems,” ISSCC Digest of Technical Papers, Feb. 2002. [5] A. Mantyneemi, “An Integrated CMOS High Precision Time-To-Digital Converter Based on Stabilised Three-Stage Delay Line Interpolation,” Academic Dissertation, Faculty of Technology, University of Oulu, Dec. 3, 2004.

Appendix A PHASE-FREQUENCY DETECTORS AND CHARGE PUMPS

1

PHASE-FREQUENCY DETECTORS

The typical PFD employed in frequency synthesizers is shown in Figure A-1. IN1 and IN2 are the input signals to the PFD and they conventionally represent the reference and feedback signals in the PLL, respectively. The state diagram of the PFD of Figure A-1 is shown in Figure A-2. Three states can be distinguished. Those are denoted state 1, state 0, and state +1. The state diagram can be further illustrated with the aid of a timing diagram.

Figure A-1. Phase-Frequency Detector Block Diagram

135

136

Appendix A

Three cases can be distinguished. The first case is when IN2 is leading IN1. In this case, the UP goes high with the rising edge of IN1 and is reset to low with the rising edge of IN2. The DN goes high with the rising edge of IN2 and resets instantaneously. The second case IN2 is lagging IN1. In this case, the DN goes high with the rising edge of IN2 and is reset to low with the rising edge of IN1. The UP goes high with the rising edge of IN1 and resets instantaneously. The third case is when IN2 tracks IN1, while the phase-locked loop is locked. All the three cases are illustrated in Figures A-3–A-5.

Figure A-2. State Diagram of the PFD in Figure A-1

Phase-Frequency Detectors and Charge Pumps IN1

IN2

DN

UP RESET

Figure A-3. Timing Diagram of the PFD, Case IN2 Leads IN1

IN2

IN1

UP

DN RESET

Figure A-4. Case IN2 Lags IN1

IN2

IN1

UP

DN

RESET

Figure A-5. Case IN2 Tracks IN1

137

138

Appendix A

In practice, the UP and DN signals do not reset to low instantaneously but go to zero after a certain delay. This is illustrated in Figure A-6 for the case of Figure A-3.

IN1

IN2

UP

DN RESET Delay in Reset Path

Figure A-6. Practical Timing Diagram of the PFD for IN2 Leading IN1. The Figure is not to Scale as in Reality the RESET Pulse is a Full Pulse and not an Impulse as the Figure might Suggest

One important disadvantage of the PFD described above is the dead zone. Dead zone occurs when IN1 and IN2 are very close and there is not enough delay in the reset path. This causes the UP and DN pulses not to fully turn ON and stay ON for a short time to allow for the CP switches to respond. Too much delay in the reset path, however, can adversely cause additional increase of phase noise due to the inherently noisy current sources that are ON for a longer time.

2

CHARGE PUMP

A behavioral model for a CP is shown in Figure A-7. The CP UP current source sources current into the loop filter whereas the loop filter sinks currents into the DN CP current source. The effect of sourcing and sinking the CP current is also illustrated in Figure A-7.

Phase-Frequency Detectors and Charge Pumps Vsup

139 Vsup

Vsup

Iup

Iup

Iup

Up

Up

Up Icp

Icp

Icp Dn

Dn

Dn Zs

Zs

Zs Idn

Idn

Idn

Figure A-7. Charge Pump Sourcing and Sinking Illustration

Vdd

Vdd

UP

vp

vp

UP

V dd

V dd

Icp

vn

Vss

Icp

DN

Vss

vn

DN

(a)

(b)

Figure A-8. Single-Ended Charge Pump and its Modification

A conventional single-ended CP circuit is shown in Figure A-8a. Figure A-8b shows a modification of the CP circuit to isolate the feed-through obtained by the switches’ UP and DN pulses as the current sources act as buffers for the switches.

140

Appendix A

There are few issues that need to be considered when designing a CP. These are as follows: • • • •

Output impedance of CP Current mismatch between UP and DN parts Switching time and switch feed-forward Leakage when UP and DN are off (CP Icp = 0)

Example charge pump UP, DN, and net currents are shown in Figure A-9. It can easily be seen from the net current that there is a mismatch between the UP and DN currents for tuning voltage values less than 0.3 V and greater than 1.2 V.

Figure A-9. Example DC Current Curves of the Charge Pump

3

PFD/CP CHARACTERISTICS

The combined PFD/CP performance is of particular interest. The characteristic curve is important because it can be used in system simulation to predict the PLL behavior in presence of the PFD/CP nonlinearity. Figures A-10 and A-11 show the PFD/CP characteristics and the gain slope variations. Figure A-10 is very useful as it shows the areas where the CP gain has most of the variations for a specific PFD/CP architecture. The more variations in the gain the more susceptible the PLL loop will be to nonlinearities and noise folding. The average current here is the total charge multiplied by the reference frequency (40 MHz), i.e. the measured charge over the sampling period.

Phase-Frequency Detectors and Charge Pumps

Figure A-10. PFD/CP Characteristic

Figure A-11. PFD/CP Gain Slope Variations

141

Appendix B CONTROLLED OSCILLATORS

1

REFERENCE OSCILLATORS

Wireless frequency synthesizers have very stringent phase noise requirements. Working those requirements backwards to the various blocks of the PLL, the first block that determines the initial purity of the system is the reference oscillator. The main contributors to the phase noise plateau of the PLL are the reference oscillator, the PFD and the dividers. A low phase noise plateau warrants the usage of a clean and very stable reference oscillator.

1.1

Voltage-Controlled Crystal Oscillator

VCXOs are the most accurate, clean, and stable frequency sources that are used as frequency references in the PLL. The crystal is made of a piezoelectric resonator that is electromechanical in nature. Figure B-1 shows a basic crystal model and a configuration that forms the basis of a VCXO. The motional capacitance Cs and the motional inductance Ls determine the series resonance of the crystal equation (B.1) as their impedances cancel out. The crystal impedance at the series resonance is approximated to Rs, the motional resistance. This resonant frequency is the operation frequency.

143

144

Appendix B

Ls

Rs

Cs

Cp (a)

(b) Figure B-1. Crystal Oscillator (a) Equivalent Circuit, (b) Simplified VCXO Circuit Implementation

fs =

1 2π LsCs

(B.1)

Another crystal resonant frequency is achieved when the parallel capacitance becomes significant. This happens when the inductance

Controlled Oscillators

145

impedance becomes much larger than the capacitance impedance and dominates. The magnitude of the inductive impedance becomes equal to the magnitude of the parallel capacitive impedance, Cp, and cancels out at a parallel resonance frequency equation (B.2).

1 Cp + Cs + Δf 2π LsCpCs

fp =

(B.2)

where Δf is the shift in frequency from series to parallel resonance and is determined by the load capacitance, CL, of the VCXO circuit configuration. This shift in frequency is given by:

Δf =

1.2

Cs 2(CL + Cp )

(B.3)

Temperature-Compensated Crystal Oscillator

VCXOs have a superior phase noise characteristic and hence are very suitable for PLL synthesizers with stringent phase noise requirements. However, crystal oscillators exhibit a fundamental drift in frequency with temperature. Equation B.4 [B1] shows frequency drifts due to ambient temperature variations of an AT-cut1 crystal. Δf = [α1 (T − T0 ) + α 2 (T − T0 ) 2 + α 3 (T − T0 )3 ] f 0

(B.4)

where f 0 is the nominal resonant frequency at T0 = 25o C ambient room temperature, α1 , α 2 and α 3 are constants that depend on the physical properties of the crystal and its angle of cut, and T is the ambient temperature. For highly accurate applications, a temperature-compensated crystal oscillator (TCXO) can be built where by the crystal temperature is kept constant. Compensation is achieved using temperature-dependent circuit elements such as thermistors and negative temperature coefficient (NTC) capacitors. Figure B-2 shows a typical circuit implementation of a TCXO using a colpitts oscillator topology [B2].

146

Appendix B

VCC VDD R b1 Rg1

M1 M1

Re2

Crystal Crystal

Cc

`

RT

RT

CNPO1CNPO1CNPO2CNPO2CNPO3CNPO3

R b2 Rg2 C 1

C2

C1

C2

Cc

Output

M2 Output

R e1 RS

Figure B-2. TCXO Circuit

2

VOLTAGE-CONTROLLED OSCILLATORS

VCO are the heart of PLL frequency synthesizers. The stringent requirement on their spectral purity requires very good design practices and guidelines. Improving phase noise and jitter requires, however, the exploring of new techniques such as the ones cited in [B3–B6]. To design the VCOs with low noise performance, proper noise analysis have is required as presented in the next paragraphs.

2.1

Voltage-Controlled Oscillators: Phase Noise Analysis

Equation B5 shows the SSB power spectral density (PSD) of the total phase noise.

⎡P (ω + Δω ,1Hz ) ⎤ Ltotal {Δω} = 10 ⋅ log ⎢ sideband 0 ⎥ (dBc/Hz) Pcarrier ⎣ ⎦

(B.5)

Controlled Oscillators

147

where Psideband(ω0 + Δω, 1 Hz) represents the SSB power at a frequency offset of Δω from the carrier over 1 Hz bandwidth. For LC tank oscillators, Leeson’s equation, B6, models the phase noise based on a linear time invariant (LTI) system. ⎪⎧ 2 FkT L(Δω ) = 10.log ⎨ P ⎩⎪ sig

L(Δω) (dB)

⎡ ⎛ ω ⎞ 2 ⎤ ⎛ Δω1/ f 3 0 ⋅ ⎢1 + ⎜ ⎥ ⋅ ⎜1 + ω ⎟⎠ ⎦⎥ ⎝⎜ 2 Q Δ Δω ⎢⎣ ⎝

⎞ ⎪⎫ ⎟⎟ ⎬ ⎠ ⎭⎪

(B.6)

1/f3 1/f2 1/f1

Δω 1

f3

1/f0

Δω

Figure B-3. Typical Curve of the Phase Noise of an Oscillator Versus Offset from Carrier

F is called the device excess noise number, k is Boltzman’s constant, T is the absolute temperature, Psig is the average power dissipated in the lossy resistive part of the tank, ω 0 is the oscillation frequency, Q is the loaded quality factor of the tank, Δω is the offset from the carrier and Δω1/f 3 is the frequency of the corner between 1/f 3 and 1/f 2 region (as shown in Figure B-3). This equation, however, makes use of F and Δω1/f 3 that are fitting factors and cannot be calculated beforehand. Using LTI method [B5] treats oscillator as a feedback system and considers each source noise as an input X( jω) (Figure B-4). The phase noise at the output, Y( jω), is made of the noise contributions of various elements in the circuit and the noise shaped by the feedback, equation (B.7).

148

Appendix B

X(jω) +

H(jω)

Y(jω)

-

Figure B-4. Oscillator as a Linear System

2

Y [ j(ω0 + Δω )] = X

1 dH (Δω ) dω

2

(B.7)

2

B7 can be used to get the output noise PSD. This approach, however, is based on linear analysis and cannot be used to predict realistic phase noise of real VCOs. Linear time variant (LTV) method used in [B6], however, is more useful for predicting and optimizing phase noise in oscillators. This analysis makes use of a special function, ISF that describes how much phase shift results from applying a unit impulse at any point in time. Equation (B.8) shows the phase shift due to applying a unit impulse.

hΦ (t ,τ ) =

Γ(ω0t ) u (t − τ ) qmax

(B.8)

where Γ(ω0t) is the ISF function of the output and qmax is the maximum charge offset across the capacitor. The total excess phase due to a noise current can be described by the following equation:

θ (t ) =

+∞

∫

−∞

hΦ (t ,ι )i (ι )dι =

Γ(ω0ι ) i (ι )dι qmax −∞ t

∫

(B.9)

The phase can hence be converted to voltage to get the SSBPSD, equation (B.10).

Controlled Oscillators

⎛ in2 + ∞ 2 ⎜ cn ⎜ Δf ∑ n =0 L{Δω } = 10 ⋅ log⎜ 2 2 ⎜ 8q max Δω ⎜ ⎝

149

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(B.10)

in2 is the power spectral density of the input noise current, cn Δf is the coefficient of the Fourier transform of the ISF function, and Δω is the frequency shift from the carrier frequency.

where,

2.2

VCO Design Methodology

Figure B-5 shows the steady state parallel LC oscillator model. The tank loss is represented by gtank, and the effective negative conductance of the active devices, required to compensate for the tank losses is represented by –gacitve. Typically, LC oscillators operate in two different modes: namely current- and voltage-limited modes [B7]. In the current-limited mode, the tank amplitude, Vtank, linearly increases with the bias current according to Vtank = Ibias/gtank until the oscillator enters the voltagelimited mode where the amplitude is limited to Vlimit, which is determined by the supply voltage. Vtank can be expressed in Equation (B.11) where Ibias is the independent variable.

Figure B-5. Steady State parallel LC oscillator model

150

Appendix B

Vtank

⎧ I bias ⎪ = ⎨ g tank ⎪V ⎩ limit

( I − Limited )

(B.11)

(V − Limited )

Using the tank inductance, L, as the independent variable, we can express equation Vtank in equation B.12 as: 2 Vtank =

2 Etank = 2 Etankω02 L C

(B.12)

where Etank is the tank energy defined in equation B.13 as:

Etank =

2 CVtank 2

(B.13)

and, ω 0 = 1 / LC is the oscillation frequency. The tank amplitude grows with L for a given Etank and ω0. This is referred to as inductance-limited mode where the inductance is limited by the given size or available area. The V – limit is constrained by the supply voltage; hence Vtank can be expressed as: 2

Vtank

2.2.1

⎧⎪2E tankω02 L =⎨ 2 ⎪⎩Vlimit

(L − Limited) (V − Limited)

(B.14)

VCO Design

Figure B-6 shows the building block of the used LC VCO core. In addition to the inductor dimensions, Figure B-7, the core can be optimized by optimizing the MOS W/L dimensions, the varactor tuning range, and the bias current. Using the parallel oscillator model on the Figure B-7, the tank loss gtank, the negative conductance -gactive, the tank inductance Ltank, and the capacitance Ctank are given by:

Controlled Oscillators

151

2 g tank = g op + g v + g L

(B.15)

2 g active = g mp

(B.16)

Ltank = 2 L

(B.17)

2Ctank = CPMOS + CL + Cv + C load

(B.18)

where g L and gv are the effective parallel conductance of the inductors and varactors, respectively.

Figure B-6. VCO Core Schematic

Appendix B

152

Figure B-7. Equivalent Oscillator Model and Symmetrical Spiral Inductor Model

The primary parameter in the oscillator specifications requiring optimization is the phase noise. This is followed by the power consumption, frequency tuning range, start-up conditions, tank amplitude, and diameter of spiral inductor. Derived from the system power budget, the maximum power constraint is imposed in the form of maximum bias current Imax drawn from a given supply voltage, i.e. I bias ≤ I max

(B.19)

The tank amplitude is required to be large, Vtank,min is to provide a large enough voltage swing to drive the next stage:

Vtank =

I bias g tank,max

≥ Vtank,min

(B.20)

The tuning range of the oscillation frequency needs to be maximized for greater frequency coverage, hence: L tank Ctank, min ≤

1 2 ωmax

, Max. tuning frequency

(B.21)

Controlled Oscillators

L tank Ctank, max ≥

1 2 ωmin

153

, Min. tuning frequency

(B.22)

To guarantee a reasonable start-up condition with a small-signal loop gain of at least αmin, the worst case condition can be expressed in equation (B.23), where gtank,max guarantees start-up in the worst case, hence: g active ≥ α min g tank,max

(B.23)

The final spec. is size or area which needs to be kept small. Being the most area-consuming component in the VCO core, the inductor has its size specified so as not to exceed a certain value and hence its diameter is specified as dmax, i.e.

d ≤ d max

(B.24)

For a given chip area or specified dmax, the inductance, L, is also constrained and so is the tank amplitude as shown earlier in equation (B.14). 2.2.2

Phase Noise Optimization

Using the LTV method of analysis, phase noise is given by:

L { f off } =

⎛ in2 ⎞ 1 1 2 ⋅ ⋅ ⋅ Γ ⎜ ⎟ ∑ rms,n 2 ⎜ ⎟ 8π 2 f off2 qmax n ⎝ Δf ⎠

(B.25)

in2 Δf represents the equivalent differential noise PSD due to drain current noise, inductor noise, and varactor noise, expressed as:

where foff is the offset frequency from the carrier frequency.

i M2 ,d Δf

= 2 kTγ ( g d 0 ,n + g d 0 , p )

(B.26)

154

Appendix B 2 iind = 2kTg L Δf

(B.27)

2 ivar = 2kTg v,max Δf

(B.28)

where γ is approximately

2 3

and

5 2

for long- and short-channel

transistors, respectively. It has been proven [B8] that drain current noise is predominantly amongst the three noise sources. Taking only the drain current noise term into consideration in B.25, and replacing qmax by Vtank 2 I drain , and g d0 = for short-channel transistors, (Γ 2rms = 1/2 2 Ltankω Lchannel Esat is used for pure sinusoidal waveform), the phase noise can be expressed as: ⎧⎪ L2 g L2 / I bias L{ f off } ∝ ⎨ 2 2 ⎪⎩ L I bias / Vsupply

(L − Limited) (V − Limited)

(B.29)

B.29 states that for a given bias current, the phase noise rises with increasing L in the voltage-limited mode, hence once the V-limit is reached any excess in inductance, L, will worsen the phase noise. Additionally, for a given inductance, increasing the bias current translates to an increase of the phase noise in the voltage-limited mode, inducing power wastage. For a typical on-chip spiral inductor, the minimum effective parallel conductance gL decreases with an increasing inductance; the factor L2g2L also increases. Thus, for a given Ibias, the phase noise increases with the inductance in the inductance-limited mode and hence a smaller inductance results in a better phase noise. So the design needs to be based on finding the smallest inductor that satisfies both the tanks amplitude and start-up condition for the maximum allowable bias current allowed by the current budget.

Controlled Oscillators

155

Here is a summary of the described design methodology: • Set the bias current to Imax, and pick an initial guess for inductance value minimizing gL. • Plot C versus W of the active device for the chosen inductance • If there are more than one possible points on the graph, reduce the inductance and replot until the possible region reduces to a single point. That optimum point represents the optimum C and W, corresponding to the optimum inductance yielding the optimum phase noise. A summary of constraint and design optimization methodology is given in Figure B-8 [B9].

Figure B-8. C–W plane, Summary of Constraint and Design Optimization Methodology

REFERENCES [B1] Frerking and E. Marvin, Crystal Oscillator Design and Temperature Compensation, Van Nostrand, New York, 1978. [B2] M.A. Haney, “Design Technique for Analog Temperature Compensation of Crystal Oscillators”, Master thesis, Faculty of the Virginia Polytechnic Institute and State University, Blacksburg, VI, 2001.

156

Appendix B

[B3] T.C. Weigandt, B. Kim, and P.R. Gray, “Analysis of Timing Jitter in CMOS Ring Oscillators,” Proceedings of ISCAS, June 1994. [B4] J. McNeill, “Jitter in Ring Oscillators,” IEEE Journal of Solid-State Circuits, 32, pp. 870–879, June 1997. [B5] B. Razavi, “A Study of Phase Noise in CMOS oscillators,” IEEE Journal of Solid-State Circuit, 31, pp. 331–343, Mar. 1996. [B6] A. Hajimiri and T.H. Lee, “A General Theory of Phase Noise in Electrical Oscillators,” IEEE Journal of Solid-State Circuits, 33, pp. 179–194, Feb. 1998. [B7] D.B. Leeson, “A Simple Model of Feedback Oscillator Noises Spectrum,” Proceedings on IEEE, 54, pp. 329–330, Feb. 1966. [B8] D. Ham and A. Hajimiri, “Concepts and Methods in Optimization of Integrated LC VCOs,” IEEE Journal of Solid State Circuits, 36, pp. 896–909, June 2001. [B9] F. Fang and K. Phang, “Phase Noise Analysis of VCO and Design Approach to LC VCOs,” Term Paper University of Toronto, pp. 3–14, 2001.

Appendix C PHASE NOISE Analysis

1

CALCULATION OF GLOBAL PHASE ERROR FROM L(f )

A signal s(t) having a peak amplitude A, a radian frequency ωc, and a periodic phase modulation with peak amplitude Δφpk and radian frequency ω m is given by: s (t ) = A cos(ωct + Δφpk sin ωm t )

(C.1)

Using trigonometric identities,

s (t ) = A cos(ωct ) cos(Δφpk sin ωmt ) − A sin(ωct ) sin(Δφpk sin ωmt ) (C.2) If the phase modulation index is small, Δφpk << 1, then the second cosine term approaches unity and the second sine term approaches its argument. s (t ) ≈ A cos(ωct ) − AΔφpk sin(ωc t) sin(ωm t ) AΔφpk = A cos(ωct ) − [cos(ωc − ωm )t − cos(ωc + ωm )t ] , 2

157

(C.3)

158

Appendix C

which is a single carrier tone having two sideband tones with relative amplitude (Δφpk/2), spaced by ω m from the carrier. Note that the peak phase error is related to the amplitude and power of the phase modulating signal. Similarly, the RMS phase error is given by the RMS amplitude of the phase modulating signal, Δφrms =

Δφpk

(C.4)

2

Single-sideband phase noise (SSBN) is a measure of the power in a sideband region relative to the carrier power. The sideband noise ratio (SBNR) is defined numerically for the sideband tone as: 2

2

2 ⎛ Δφpk ⎞ Δφrms power in sideband ⎛ AΔφpk ⎞ −2 SBNR = =⎜ ⎟ ⋅ ( A) = ⎜ ⎟ = power in carrier 2 ⎝ 2 ⎠ ⎝ 2 ⎠ (C.5)

SSBN, expressed in dBc/Hz is the most common measure of phase noise, describing the power in a 1-Hz bandwidth relative to the carrier at an offset frequency f from the carrier. The script-L symbol is used. ⎛ Δφ 2 ( f ) ⎞ L( f ) = 10 log10 ⎜ rms ⎟ 2 ⎝ ⎠

(C.6)

where the variable ( f ) indicates the power is measured in a 1-Hz bandwidth at an offset f Hz from the carrier. L( f ) is a convenient term because it can be measured directly using a spectrum analyzer or similar setup in the laboratory. We can also calculate the total RMS phase error contribution for phase noise in a span of frequencies offset from the carrier. In that case the sideband noise power is integrated over the bandwidth of interest. Δφrms =

∫

f2 f1

f2

2 Δφrms ( f )df = 2 ∫ 10L(f ) /10 df f1

(C.7)

This is the relationship between total, or “global,” RMS phase error and the phase noise spectrum L( f ). The equation returns the RMS phase error in radians.

Phase Noise

159

As an example, the GSM system requires a total global RMS phase error of 5° degree symbol. To allow margin for other transmit path impairments, we can arbitrarily set a specification of less than 2° for the PLL and VCO. Using the above relationship, and assuming that the in-band phase noise is constant and will dominate the result, we find: 2 ⎛ Δφrms ⎞ L( f ) = 10 log10 ⎜ ⎟ ⎝ 2 ⋅ BW ⎠

(C.8)

where BW is the loop bandwidth. For a 150 kHz bandwidth and 2° error, L( f ) must be less than 84 dBc/Hz in-band.

2

PHASE NOISE AND PHASE MODULATION

Random noise in the vicinity of a signal causes both amplitude and phase noise modulation of that signal. A single noise sideband (at any offset from the carrier), produces both amplitude and phase modulation and can be represented vectorially (Figure C-1). B

A Resultant θ

Figure C-1. Single Sideband and Carrier

The resultant vector changes in both amplitude and phase, the amplitude varying from A − B to A + B, with a modulation depth of B/A multiplied by 100%. The phase varies by ± Φ, where Φ = B/A, in radians, provided B is small compared to A.

160

Appendix C B B

A θ

Figure C-2. Carrier and Phase Modulating Sidebands

Two sidebands (Figure C-2), with equal initial phase, each in quadrature with the carrier A, but at equal frequency offsets at either side of it can be represented as positively rotating vectors, producing only phase modulation. The peak phase variation is ±2B/A. This is representative of what occurs when we measure the noise of signals where phase noise dominates, which is the usual case (the natural limiting of the oscillator in a phase-locked loop restricts the amplitude noise, but not the phase noise). The single-sideband power level is thus 20log10 (B/A) dBc. At a particular offset frequency, the sum vector is:

e(t ) = A cos(ωct ) + B cos(ωc − ωm )t + B cos(ωc + ωm )t

(C.9)

If we assume that 2B/A << π/2, then we can arrive at a value for e(t), representative of the phase modulation of this carrier at this particular frequency offset, m: e(t ) = A cos(ωct + 2AB sin(ωm t )) where 2B/A sin(ω mt) is the phase modulation.

(C.10)

Phase Noise

3

161

RMS PHASE ERROR FROM PHASE NOISE

The output of a PLL is a sinusoid with phase noise. The ultimate goal is to calculate from this phase noise spectrum the RMS phase error of this sinusoid. First considering a pure sinusoid with white noise, such as that associated with a resistor, in communications terminology there is an associated signal-to-noise ratio, (S/N), and bandwidth. Here C will be used to represent the carrier replacing S, the sinusoid, in the commonly accepted 1 Hz bandwidth of SSB phase noise, L(f ). If we look at a 1 Hz bandwidth of this noise power at an offset frequency fm, from the carrier, and calculate the effect of this small amount of noise, the effect of all the noise in the total bandwidth can be found. The relationship between the vector and spectral representations of the carrier with noise is shown in Figure C-3;

Vc C N

fo + fm fo fo

Vn

fo + fm

Figure C-3. Single Sideband and Carrier V2

2

where C, N are the powers c R , Vn R . Calculation of the ratio of carrier to noise is as follows:

θ pk = tan −1 ( VV ) ≈ VV , in power n

n

c

c

N C

(C.11)

The RMS phase deviation is:

φ=

θ pk 2

=

N 2C

(C.12)

162

Appendix C

In the case of two sidebands at ± fm with respect to fo, the incoherent random phase noise contributions add in a sum of squares fashion (Figure C-4).

C Vc

Vn

N

N

θ fo

fo - fm

fo + fm

Figure C-4. Carrier and Phase-Modulating Sidebands

VnTotal = Vn21 + Vn22 = 2Vn2 = 2 Vn

Thus θ pk =

2 Vn Vc

=

2N C

in RMSφ =

θ pk 2

(C.13)

=

N C

rad RMS

(C.14)

Before looking at the total noise in a given bandwidth, it is important to note what kind of noise spectrum is being considered. Superimposed noise vectors, from such things as amplifiers or thermal noise sources, create both AM and FM sidebands. The diagrams above considered this case. However, if the spectrum is phase noise AM noise is insignificant (because of the self-limiting action of the VCO), requiring the above equations to be altered from noise, N, to phase noise N P. Since the noise power is split evenly into AM and PM noise, N has twice the power of NP. Thus, the equations can be altered with the relationship N = 2 NP . For white noise from amplifiers or resistors, the noise can be viewed as many single sidebands and integrated over the required bandwidth. Phase Error =

1800

π

∫

b

0

N C

df =

1800

π

∫

b

N − b 2C

df

(C.15)

For the case of a phase noise spectrum: Phase Error =

1800

π

∫

b

−b

NP ( f ) C

df Deg RMS

(C.16)

Phase Noise

163

Thus, it is possible to integrate the normal SSB phase noise spectrum, which expresses phase noise as a noise power at a particular offset from the carrier, with respect to the total carrier noise power.

4

RESIDUAL FM

Residual FM is a measure of frequency instability related to Sφ( f ) (the spectral density of phase fluctuations), expressing the total RMS frequency deviation within a specified bandwidth. Commonly used bandwidths are 50 Hz–3 kHz, 300 Hz–3 kHz and 20 Hz–15 kHz. Only the short-term frequency instability occurring at rates within the bandwidth is indicated and no information regarding relative instability rates is conveyed. The presence of large spurious signals at frequencies near the carrier frequency can greatly exaggerate the measured level of residual FM, since the spurious signals are detected as FM sidebands.

S Δf ( f ) =

2 2 Δf rms ( f ) f Sφ ( f ) = B B

b

b

a

a

res FM = ∫ S Δf ( f ) = ∫

f 2 Sφ ( f ) B

(C.17)

[Hz]

(C.18)

To convert the quantity Sφ( f ) to L( f ) (SSBN), the following can be used. L( f ) =

Power density ( one phase modulation sideband) ⎡ dBc/Hz ⎤ ⎢⎣ Hz ⎥⎦ Carrier Power

(C.19)

L( f ) =

Sφ ( f ) 2

(C.20)

L( f ) is the ratio of the power in one phase-modulated sideband per Hz, to the total signal power.

Appendix D FREQUENCY DIVIDERS

1

REFERENCE DIVIDER

The reference divider operates at low frequencies (frequencies below 500 MHz for 0.18 um CMOS technology). This type of divider is constructed as a programmable counter designed in standard CMOS standard cell logic. This divider usually divides the reference frequency down to the sampling frequency of the PFD. There are two types of counters used as reference frequency dividers. These are either synchronous or asynchronous. Both dividers will be described below.

1.1

Synchronous Dividers

Synchronous frequency dividers can be constructed as pseudorandom binary sequence (PRBS) counters. A 6-bit PRBS-based counter is shown in Figure D-1. These dividers are formed from a shift register of length N with some taps (either 2 or 4) fed to a modulo-2 adder whose output is connected to the input of the shift register. If the correct taps are used, the shift register will clock through a PRBS of length 2N−1. It also has a forbidden state of all zeros; a state if it falls into then it never leaves. The sequence of 2N−1 states does not include the all-zeros state. Therefore, the divider cannot get into the forbidden state from its normal counting mode. A problem can possibly occur at switch-on, which may yield all zeros, or if the counter is accidentally loaded with all zeros. A good way to protect the divider from this state is to fix the

165

166

Appendix D

terminal count as a code which will be zero. This makes the bit which is “1” a “don’t care” bit. With this arrangement, terminal count is detected at the defined value and for the forbidden all-zeros state. The terminal count forces a parallel load on the shift register as part of the normal divider operation, so it will load a permissible code if the forbidden state occurs. q6 q5

L<5:0>

S Q q1 D1 D0 C

S Q q2 D1 D0 C

S Q q3 D1 D0 C

S Q q5 D1 D0 C

S Q q4 D1 D0 C

S Q q6 D1 D0 C

fin

q5 q4 q3

fout

q2 q1

Figure D-1. A 6-bit PRBS Synchronous Divider

These dividers are fast, simple, and fully programmable. They are nearly as fast and simple for big division ratios as for small ones. The disadvantages are that they count in pseudorandom fashion (not binary) so a lookup table is required for division ratios versus parallel load value. They also consume more power than ripple counters because all the D-type flip-flops used are clocked at the input clock-rate. If designed carefully however, the maximum frequency of operation is not much less than that of a single divide by two using the same D-type flip-flop.

1.2

Asynchronous Reference Frequency Divider

A simple 6-bit programmable reference divider is shown in figure D-1. Figure D-2 shows an asynchronous divider comprised of six toggle flip-flop (TFF) stages. Each TFF can be designed from a simple set/reset D-type flip-flop and some logic (see Figure D-3).

Frequency Dividers

167 Rout

D

Rsyn

Y

SN

C

R<5> R<4> R<3> R<2>

Q QN

X

X Y

SN

D C

Y

R<1>

Q QN

RN

Vdd

R<5:0>

C

Q

L

S

C

QN L

Q

C

QN L

S

Q

C

QN L

S

Q QN

L

S

S

R<5>

S

R<0>

L

Q QN

R<4>

C

R<3>

Q QN

R<2>

C

R<1>

Rin

Figure D-2. A Simple Programmable Reference Divider

SN C

Q QN

L

S

D

Q

C

QN RN

L

L

S

0

0

RN SN 1

0

0

1

0

1

1

0

0

0

1

1

0

0

S

Figure D-3. A Basic Toggle Flip-Flop Implementation

The clock drives only the first TFF and each subsequent stage is driven by the previous output stage. Figure D-4 shows the timing diagram of the reference divider programmed to divide by 16. Additional flip-flops and gates are used as decision logic to detect when the count reaches final count and to reload the divider value.

168

Appendix D

Figure D-4. Reference Divider Timing Diagram

2

FEEDBACK DIVIDER

The RF feedback divider is the only high-frequency block that operates at the VCO frequency. Driven by the VCO output clock frequency, the divider scales the feedback RF signal to the PFD input clock frequency for comparison with the reference VCXO clock.

2.1

Specification and Different Architecture Evaluation

High speed, power consumption, and low noise are the main requirements of the VCO frequency divider. The divider needs to operate at the VCO frequency making it unavoidably power hungry. However, some architectures have shown to consume less power than others. A good design is one that can be low noise and consume low power. These requirements are important as most wireless communication devices are portable and battery operated. 2.1.1

Direct Division versus Prescaler Method

Ideally a MMD such as the one in Figure D-5 driven directly by the RF VCO frequency (RFin) will give minimal noise as the ratio

Frequency Dividers

169

between RFin and the clock frequency of the fractional Δ−Σ modulator is at a maximum. However, the MMD is implemented using synchronous CML and hence each stage is driven and operated at the RF VCO frequency. This implementation yields maximum power consumption and is difficult to design especially as the divider modulus is increased from 6 to 7 bits to cover frequency bands such as 802.11b, 802.11g, and BluetoothTM (2.4–2.5 GHz) or higher frequency bands such as 802.11a (4.9–5.805 GHz). Additionally, if the divider is designed to marginally meet its highest frequency of operation, this leads to worst noise performance. The following example illustrates the speed constraint of the feedback divider: taking the 802.11b band, if the RF channel frequency is: f RF = 2.46 GHz

(D.1)

and the crystal oscillator frequency is: f VCXO = 40 MHz

(D.2)

then the required divider ratio is: f RF

N=

f VCXO

=

2460 = 61.5 40

(D.3) Div Out AND

MUX1

MUX2

A

A

Y B S

div0

Y B S

MUX3

MUX4

A

Y B S

div1

MUX5

A

Y B S

div2

MUX6

A

Y B S

div3

A Y B S

div4

DFF1

DFF2

DFF3

DFF4

DFF5

DFF6

D

D

D

D

D

D

Q

C Q

Q

C Q

Q

C Q

Q

C Q

Q

C Q

div5

Q

C Q

RF in XNOR NOR

XORa

XOR

NAND

XNORa NORa

NANDa

Figure D-5. Multimodulus Divider Implementation

XNORb

170

Appendix D

Ref.

VCO

PFD/CP

fcomp

MMD

fp

RF

2/4 Prescaler

Feedback Divider Figure D-6. Divide-by-2/4 Prescaler and MMD in the Feedback Divider

Using a simple third-order Mash modulator, the divisor can have a maximum value of 65 which clearly cannot be covered by a 6-bit MMD. To overcome this constraint, a prescaler such as a divide-by-2 or a divide-by-4, as shown in Figure D-6, can be used. Having an intermediate RF prescaler in the feedback path reduces the speed constraints of the MMD, this in turn reduces the power consumption of the MMD as it operates at half the VCO frequency for the 2.4 GHz bands and a quarter of the VCO frequency for the 5–6 GHz bands. However, this prescaling in the feedback increases the noise of the PLL as the quantization noise of the Δ−Σ modulator gets amplified in the loop by a factor of: 20 log10 N p

(D.4)

where N p is the intermediate prescaler value. The additional phase noise due to this phenomenon is approximately 12 dB for a 5 GHz synthesized frequency which significantly increases the in-band phase noise of the fractional-N PLL. Other architectures that have been used in the past based on dualmodulus architectures such as the phase switching dual-modulus divider (PSDMD) are reported in [D1, D2]. The PSDMD architecture of Figure D-7 has a much lower power than the conventional direct MMD as only the first and the second divide-by-2 prescalers operate at full and at half VCO speeds, respectively. The divide-by-2 prescalers are single ended input to differential output and operate from rail-to-rail and hence do not require level shifting to interface to the low frequency CMOS phase multiplexers. The main drawbacks of this architecture are the possibility of spikes occurring as the multiplexer switch transitions between phases.

Frequency Dividers INP INN

171

/2

0 90

/2

180 270

MUX 4 to 1

/16

OUT

Control

Vdd INP

Vdd

Vdd

Vdd INN OUTP OUTN

Figure D-7. The Phase Switching Dual-Modulus Prescaler

This could send the wrong information to the modulus control and hence divide by the wrong divisor. Also the input sensitivity of the divide-by-2 is much worse than its CML counterpart [D3]. The low frequency CMOS divide-by-16 counter needs to operate at a quarter of VCO frequency which might be fine for a 2 GHz range VCO frequency. However, as we move higher in the spectrum to cover the 802.11a high band of (5.805 GHz), that might be difficult to achieve. From a noise standpoint, this architecture is similar to that of the divide-by-4 prescaler-driven MMD as it exhibits similar quantization noise amplification of 12 dB, equation (D.4). The ordinary DMD in Figure D-8 works somewhat differently from the PSDMD. A and B counters are loaded with the desired divider ratio and the prescaler divides by P + 1 as long as the modulus control is low. When the content of the B counter reaches 0, the MC goes high and the prescaler divides by P until the content of the counter A reaches 0. For proper division, counter A has to be greater

172

Appendix D STATIC CMOS

A A,B Value RFin

P/P+1

Logic

Fdiv

B

Modulus Control

Figure D-8. P/P + 1 Dual-Modulus-Based Prescaler architecture

or equal to B. Hence the prescaler divides by P + 1 for B and by P for A – B, yielding a division of: B( P + 1) + ( A − B) P = AP + B

(D.5)

where A and B values can be determined as follows: ⎛ RFin ⎞ − B⎟ ⎜ Fdiv ⎠ and B = RFin − AP A= ⎝ P Fdiv

(D.6)

The advantage of this method of feedback division is illustrated in the synchronized operation of the critical high-frequency P/P + 1 prescaler with the low-frequency A/B counters. Its feedback operation makes it behave as if it were a direct division MMD without the disadvantage of high power consumption. Consequently, this method does not amplify the quantization noise. Its major disadvantage is its continuous minimum divider value that is limited to P 2 − P . The maximum divider value however is only limited by the size of the A and B counters.

173

Frequency Dividers

Figure D-9 shows logic implementation of 2/3 DMD. Having the reset signal at logic high, the divider divides by 3 for MC signal at logic high and divides by 2 for MC signal at logic low. Figure D-10 shows a simulated operation of a 2/3 DMD. Using the designed divide-by-2/3 dual-modulus prescaler architecture of Figure D-9, we can design any higher modulus divider. Figures D-11 and D-13 show the implementation of a 4/5 and 8/9 dualmodulus prescalers, respectively. Figures D-12 and D-14 show their respective simulated behaviors.

NOR

DFF D

Output

Q

NOR

C Q R

DFF D

Control

C

Q

R

Input

Reset

Figure D-9. Divide-by-2/3 Dual-Modulus Prescaler Logic

Figure D-10. Simulated Divide-by-2/3 DMD

Q

Appendix D

174 Divide by 2/3

NOR

DFF D

Q

NOR

C Q R

DFF D

Q

C Q R

Input Reset

Output

Divide by 2 DFF D

NOR

Q

C Q R

Control

Figure D-11. Divide-by-4/5 MMD implementation

Figure D-12. Simulated 4/5 Divider

Frequency Dividers

175

Divide by 2/3

NOR

DFF D

Q

DFF

NOR

C Q R

D C

Q R

Q

Input Reset Divide by 4

Output OR

DFF DFF D

Q

C Q R

D C

Q R

Q

Control

Figure D-13. Divide-by-8/9 DMD implementation

Figure D-14. Simulated 8/9 DMD divider

176

Appendix D

In implementing the multistandard WLAN of Figure 5-2, the synthesized frequency is two-thirds of the transmitter frequency. This indirect method of transmission is used to prevent power amplifier (PA) pulling of the VCO. The minimum and the maximum synthesized frequencies are 3.2 and 3.9 GHz, respectively. Bearing in mind that the reference used is 40 MHz, the minimum divider value is:

2 2 4800. 3 −n = 3 − 3 = 77 min Re f 40

RFmin .

(D.7)

where RFmin is the minimum transmitter frequency in MHz, Ref is the reference frequency in MHz and nmin is the order of the Δ−Σ MASH modulator. The maximum divider value is: 2 2 5805. 3 +n = 3 + 4 = 100 max Re f 40

RFmax .

(D.8)

RFmax is the maximum transmitter frequency in MHz and n max is the maximum Δ−Σ MASH modulator offset value. Examining all the required divisors, the divider modulus needs to be 7 bits with the MSB set to logic high. Another observation is the fact that only the first three bits of the divisor are modulated due to the implemented third-order Δ−Σ modulator. Having this in mind, we can use a 8/9 prescaler method with the A/B swallow counter with four and three bits, respectively and the MSB of the A counter (fourth bit) set to a logic high. Figure D-15, shows the implemented divider architecture. Having the MSB of counter A tied to Vdd (logic high), the fractional divider value can be programmed as follows:

N frac = 8 ⋅ [8 + N < 5 : 3 >] + N < 2 : 0 > ]

(D.9)

where N<5:0> is the output of the Δ−Σ modulator described in chapter 5. The A and B counters are full-scale static CMOS and operate at a maximum frequency of 500 MHz. The 8/9 prescaler is the bottleneck

Frequency Dividers

177 Decision Logic SN Q C QN

D

Reset

SN Q C QN RN

D

Mod Control

RFin

8/9 Prescaler

0 SN Q C QN RN

Nout

D

Vdd

<3:0>

3

Q<3:0> Counter A Clk D<6:3>

L<3:0>

Q<2> Q<1> Q<3> Clk Counter B D<2:0>

L<2:0>

N<5:0>

N <2:0>

N <5:3>

Vdd

Figure D-15. Multimodulus Divider (MMD) Architecture using 8/9 Prescaler

of the MMD operating at high VCO frequency. It is implemented using CML. The design of the 8/9 prescaler is described and covered in the next section.

3

HIGH-SPEED CMOS DIVIDER DESIGN

Implementing the high-speed prescaler in CMOS requires careful attention. The high-frequency operation means high power consumption.

178

Appendix D

However, using the architecture outlined in Figure D-13, only the flipflops of the front divide-by-2/3 stage operate at full VCO speed. The rest of the logic operates at half speed. The flip-flops of the divide-by-4 stages operate at half and quarter speeds. All 8/9 prescaler blocks are designed in high-speed CML.

3.1

Current-Mode Logic Design: An Overview

Unlike static CMOS, CML gates can operate at high frequency in the GHz region. To help us understand the various elements that set the speed and hence the power consumption of a CML gate, let us examine a simple differential CMOS amplifier also a CML inverter. Figure D-16 shows a basic CML inverter gate. The speed of the gate is determined by the transient characteristics, the current drive capability of its load and the propagation delay.

Vdd

Vdd R

R

R

Q

Q Vdd

R

Vswing

CL A

A

CL

vin

vin

Vb

(a)

Rs (b)

Figure D-16. Basic CML Gate (a) Biased Gate (b) Simplified Gate with Capacitive Load

Frequency Dividers

179

Figure D-17. CML Gate Half-Circuit Model

Using the half circuit of Figure D-16b, we can derive the propagation delay of the CML inverter gate. Figure D-17 shows half the CML gate circuit model. We can treat the delay as four components for simplicity:

τ = τ1 + τ 2 + τ 3 + τ 4

(D.10)

Each delay component can be studied separately. For τ 1 , delay term due to gate–source capacitance, use Figure D-18, we can ignore gmb and Ro: R1 =

R + Rs vT = g iT 1 + g m Rs ⎛ Rg + Rs ⎞ ⎟ Cgs ⎝ 1 + g m Rs ⎠

τ 1 = R1Cgs = ⎜

where VT is the input voltage at the gate of the MOS gate.

(D.11)

(D.12)

180

Appendix D

Figure D-18. Simplified Model to Derive First Delay Term

To derive the second delay term due to gate–drain junction capacitance, we ignore gmb and Ro, Figure D-19 is used as follows: R2 =

vT gm = Rg + R + Rg R 1 + g m Rs iT

Figure D-19. Simplified Model to Derive Second Delay Term

(D.13)

Frequency Dividers

⎛

τ 2 = ⎜ Rg + R + ⎝

181

⎞ gm Rg R ⎟ Cgd 1 + g m Rs ⎠

(D.14)

For the third delay term, τ 3 due to source–bulk junction capacitance, the simplified circuit model of Figure D-20 is used:

Figure D-20. Simplified Model to Derive Component due to Source–bulk Capacitance

R3 =

Rs 1 + g m Rs

⎛

Rs ⎞ ⎟ Csb ⎝ 1 + g m Rs ⎠

τ 3 = R3Csb = ⎜

(D.15)

(D.16)

Similarly for the fourth delay term, τ 3 due to the capacitive load CL and drain–bulk junction capacitance, Figure D-21 shows the simplified circuit model, where we can ignore Ro:

τ 4 = R ( Cdb + CL )

(D.17)

182

Appendix D

Figure D-21. Simplified Circuit Model to Derive fourth Delay Term

The total propagation delay of the CML inverter then becomes:

τ=

Rg + Rs

⎛ ⎞ RC gm Cgs + ⎜ Rg + R + Rg R ⎟ Cgd + s sb 1 + g m Rs 1 + g m Rs 1 + g m Rs ⎝ ⎠ + ( Cdb + CL ) R

(D.18)

Rs = 0 (AC ground for differential mode) and consider Rg = 0 negligible series gate resistance. This leads to:

τ = R ( Cgd + Cdb + CL ) =

Vswing I

(WC

ov

+ WLd C jd + CL )

(D.19)

Where Cov is the oxide capacitance, Cjd the drain junction capacitor, W and L are transistor dimensions. Vswing is the required output voltage swing. Knowing the required swing and the required speed we can choose the transistor dimensions and the differential tail current required to drive the CML gate.

Frequency Dividers

4

183

IMPLEMENTED CML GATES NOR/OR alternative

NOR

Vdd R

Vdd

R Q

Q

R

R

Or B

Nor

B A

A

B

Vb2

A V b1 Vb

(a)

(b) Vdd

NAND/AND R

R Q

Q

B

B

A

A

Vb

(c)

Figure D-22. Basic CML Gates Implementation (a), (b) NOR/OR, and (c) NAND/AND

184

Appendix D MUX

XOR/XNOR

Vdd

Vdd R

R

R

Q

R

Q

Q A B

A

S

Q

B

A

S

B

A A

Vb

Vb

(a)

(b) D-Latch

Vdd R

R

Q Q

D

D

Clk

Clk

Vb

(c)

Figure D-23. CML gates (a) Multiplexer, (b) XOR/XNOR, and (c) Latch

B

185

Frequency Dividers Vdd

Vdd R

R

R

R

Nor Or

A

B

Vb2

Clk

Clk Clk Vb1

Vb1

Slave

Master

(a) Vdd

Vdd R

Vdd R

R

R

R

R

Q Q

D

S

D

Clk

Clk

S

S

Master

Clk Clk

Vb

Vb

S

Vb

MUX

Slave

(b) Figure D-24. Combination CML Logic (a) NOR/OR Mater/Slave D-Type Flip-Flop (b) MUX/D-Type Flip-Flop

REFERENCES [D1] J. Craninckx and M. Steyaert “A 1.75-GHz/3-V Dual-Modulus Divideby-128/129 Prescaler in 0.7-μm CMOS,” JSSC, 31, 7, pp. 890–897, July 1996. [D2] R. Ahola and K. Halonen, “A 4 GHz CMOS Multiple Modulus Prescaler,” Proceedings of IEEE ICECS 1998, Lisbon, Portugal, Sept. 1998, pp. 2.323–2.326. [D3] T. Seneff et al., “A Sub-1 mA 1.6 GHz Silicon Bipolar Dual Modulus Prescaler,” IEEE Journal of Solid-State Circuits, 29, pp. 1206–1211, Oct. 1994.

Appendix E PROGRAMS AND CODES

MATHCADTM PROGRAM USED FOR THE SIMULATIONS OF ALL THE MATHCAD FIGURES

1

Mathcad program used for the simulation of all Mathcad figures presented in the book. Defining the Units:f ≡ 10

− 15

M ≡ 10

6

p ≡ 10 G ≡ 10

− 12

9

n ≡ 10

T ≡ 10

12

−9

µ ≡ 10

R := 10

−6

m ≡ 10

Set Resolution:-

−3 Hz

Defining the Loop Parameters:K vco := 100⋅ M

MHz/V

VCXO_Ref_Freq := 40⋅ MHz

Kφ := 2.0⋅ m

mA

Fsamp := 40⋅ M Hz

Samples per decade

Start_Freq := 10

The Log sweep...

Stop_Freq := 10⋅ M Duty Cycle := 2⋅ π Fmax :=

F

Hz

XScale (F ) := 10

Duty Cycle

1725⋅ M Hz

Fmin := 1725⋅ M Hz

187

10

k ≡ 10

3

188

Appendix E

The Linear Sweep... F := 10⋅log ( Start_Freq ) , 10⋅log ( Start_Freq ) + If FM modulation is applied at the VCO then this value must be entered .

10 R

.. 10⋅log( Stop_Freq )

K fm := K vco K fm := 200⋅k Enter values for calculating the loop filter components:-

LBW := 100⋅k Hz φp := 56 Degrees ATTEN := 15 dB (Make this value“0”to remove the additional R3 C3 low pass filter) RF Noise Source Values:VCO Noise

VCO_9dB := 1.3⋅k VCONoise_Plateau := −159 VCO_3dB:= 8 M VCO_6dB := 8⋅ M

dBc/Hz

189

Programs and Codes Noise Sources Reference Oscillator Noise

VCO Divider Noise

Ref Noise_Plateau := −143 dBc/Hz

VCODiv_Plateau := −173

Ref_3dB := 10⋅k Ref_6dB := 1⋅ k

RF Amplifier Noise

Ref_9dB := 100

Amp_NF := 8

Ref_12dB := 10

dBc/Hz

VCODiv_Noise_3dB := 0.3⋅

dBm

Power_in := −15

dB

Hz

Reference Divider Noise

Ref Div_Plateau := −173 dBc/Hz R_Div_3dB := 0.3⋅ k Amp_Gain := 25 dB

R_Div_6dB := 10

RFAmp_noise_3dB := 100

Φn_Plateau := −216 dBc/Hz Φn 3dB := 8⋅ M Φ_3dB := 1⋅ k

Φ_6dB := 50

Φ_9dB := 20 Φ_12dB := 5

Loop Filter Value CALCULATIONS Type 2 Second-Order Filter K v := K vco 8

5

K v = 1 × 10

⎛ ⎝

T1 :=

ωp

rad

π

⎞ − tan ⎛ φp ⋅ π ⎞ ⎜ ⎜ ⎜ ⎝ 180 ⎠

180 ⎠

C1T22O :=

1

T2 :=

ωp

:=

Fsamp

N = 43.125

LBW = 1 × 10

sec ⎜ φp ⋅

Fmax⋅Fmin

N :=

ωp := 2⋅ π⋅ LBW

2

ω p ⋅ T1 2

2

2

2

1 + ( 2⋅ π⋅ LBW) ⋅ T2

−7

T1 = 4.866 × 10

1 + ( 2⋅ π⋅ LBW) ⋅ T1

K ⋅ Kφ T1⋅ vco ⋅ ω p rad N T2⋅ ( 2⋅ π⋅ LBW) −9

C1T22O = 3.591536 × 10

2

−6

T2 = 5.206 × 10 C2T22O := C1T22O ⋅ ⎛⎜

T2

⎝ T1 −8

C2T22O = 3.483242 × 10 ATTEN

Type 2 Third-Order Filter

T3 :=

10

20

−1

( 2⋅ π⋅ Fsamp) 2

− 1⎞⎜

⎠

R2T22O :=

T2 C2T22O

R2T22O = 149.451

190

Appendix E 1

T2 :=

⎡ ( T1 + T3) ⋅ tan ⎛ φp⋅ ⎞ ⎜ ⎢ ⎜⎡ ⎝ 180 ⎠ ⋅ ⎢ ( T1 + T3) ⋅ ⎢ ⎢ ( T1 + T3) 2 + T1⋅ T3 ⎢ ⎢ ⎢ ⎣ ⎣ π

2

⎡⎢ ( T1 + T3) + T1⋅ T3 ⎤⎥ 2⎥ ⎢ π 2 ⎢ ( T1 + T3) ⋅ tan ⎛⎜ φp⋅ ⎞⎜ ⎥ 180 ⎣ ⎝ ⎠ ⎦

T1 ⎞ ⎡ ⎜⋅ ⎢ ⎝ T2 ⎠ ⎢ ⎡ ⎛ π ⎞ ⎢ ⎢ ( T1 + T3) ⋅ tan ⎜ φp⋅ 180 ⎜ ⎡ ⎝ ⎠ ⋅⎢ ⎢⎢ ⎢ ⎢ ( T1 + T3) 2 + T1⋅ T3 ⎢ ⎢ ⎢⎢ ⎣⎣ ⎣

C1T23O := ⎛⎜

⎤ ⎤⎥ + 1 − 1⎥ ⎥ ⎥⎥ ⎥⎥ ⎦⎦

2

1 2

⎡⎢ ( T1 + T3) + T1⋅ T3 ⎤⎥ 2⎥ ⎢ π 2 ⎢ ( T1 + T3) ⋅ tan ⎛⎜ φp⋅ ⎞⎜ ⎥ 180 ⎣ ⎝ ⎠ ⎦

⎤ ⎤⎥ + 1 − 1⎥ ⎥ ⎥⎥ ⎥⎥ ⎦⎦

2

⎤⎥ ⋅ K vco⋅ Kφ⋅ N ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⋅

2

⎡ ( T1 + T3) ⋅ tan ⎛ φp⋅ π ⎞ ⎜ ⎢ ⎜⎡ 2 ⎝ 180 ⎠ ⋅ ⎢ T2 ⋅ ⎢ 2 ⎢ ( T1 + T3) + T1⋅ T3 ⎢ ⎢ ⎢ ⎣ ⎣

⎤ ⎤⎥ ⎡⎢ ( T1 + T3) + T1⋅ T3 ⎤⎥ ⎥ + 1−1⎥ + 1 2⎥ ⎥⎥ ⎢ π ⎞ 2 ⎛ ⎥⎥ ⎢ ( T1 + T3) ⋅ tan ⎜ φp⋅ ⎥ ⎜ ⎣ ⎝ 180 ⎠ ⎦ ⎦⎦ 2 ⎤ ⎤ ⎥ ⎤⎥ ⎡⎢ ( T1 + T3) 2 + T1⋅ T3 ⎤⎥ 2 + 1 − 1⎥ ⎥ ⋅ T1 ⎥ ⎥ 2⎥ ⎥⎥ ⎢ π 2 ⎥ ⎥⎥ ⎢ ( T1 + T3) ⋅ tan ⎛⎜ φp⋅ ⎞⎜ ⎥ ⎣ ⎝ 180 ⎠ ⎦ ⎦⎦ ⎦ 2

⎡ ⎡ ⎢ ⎢ ( T1 + T3) ⋅ tan ⎛⎜ φp⋅ π ⎞⎜ ⎡ ⎝ 180 ⎠ ⋅ ⎢ ⎢1 + ⎢ ⎢ ⎢ ( T1 + T3) 2 + T1⋅ T3 ⎢ ⎢ ⎢ ⎢ ⎣ ⎣ ⎣

⋅

1

⎡ ⎡ ⎢ ⎢ ( T1 + T3) ⋅ tan ⎛⎜ φp⋅ π ⎞⎜ ⎡ ⎝ 180 ⎠ ⋅ ⎢ ⋅⎢1 + ⎢ ⎢ ⎢ ( T1 + T3) 2 + T1⋅ T3 ⎢ ⎢ ⎢ ⎢ ⎣ ⎣ ⎣ C2T23O := C1T23O⋅ ⎡⎢⎛⎜

T2 ⎞

⎤ ⎜ − 1⎥ ⎦

⎣⎝ T1 ⎠ −9

C1T 23O = 3.665 × 10

R2 T23O :=

C1 := if ( ATTEN 0, C1T22O , C1T23O)

⎡⎢ ( T1 + T3) + T1⋅ T3 ⎥⎤ 2⎥ ⎢ π ⎢ ( T1 + T3) 2⋅ tan ⎛⎜ φp⋅ ⎞⎜ ⎥ ⎣ ⎝ 180 ⎠ ⎦

T2 C2T23O

− 10

C3 T23O = 3.665 × 10

2

C3 T23O :=

C1T23O

−8

C2 T23O = 3.635 × 10

2 ⎤ ⎤ ⎥ ⎤⎥ 2 + 1 − 1⎥ ⎥ ⋅ T3 ⎥ ⎥ ⎥⎥ ⎥ ⎥⎥ ⎦⎦ ⎦

10

R3 T23O :=

R2 T23O = 146.14

C2 := if (ATTEN 0, C2T22O , C2T23O)

T3 C3 T23O

R3 T23O = 23.342

C3 := if (ATTEN 0, 0, C3 T23O)

191

Programs and Codes R2 := if ( ATTEN 0, R2 T22O , R2 T23O)

R3 := if ( ATTEN 0, 0, R3T23O)

Giving the following values:NOISE CALCULATIONS Reference Noise Calculation

R :=

VCXO_Ref_Freq Fsamp

R =1

This line calculates the R divider logic noise floor including the 1/f corners for this R divider.

R_Div_Noise ( FrqPoint) := Ref Div_Plateau+ 10⋅log⎛⎜ 1 + j⋅

⎝

R_Div_3dB FrqPoint

⎞ + 10⋅log⎛ 1 + j⋅ R_Div_6dB ⎞ ⎜ ⎜ ⎜ FrqPoint ⎠ ⎝ ⎠

This line calculates the ideal divided down (due to the R term) of the incoming reference VCXO source. Note that this is a direct curve fit on the specification and makes no attempt to correct for the usual oscillator . 1/f corners, giving 10Log breakpoints. Ref_3dB ⎞ Ref_6dB ⎞ ⎛ Ref_Div ( FrqPoint) := Ref Noise_Plateau+ 10⋅log⎛⎜ 1 + j⋅ + ⎜+ 10⋅log⎜ 1 + j⋅ FrqPoint ⎠ FrqPoint ⎠⎜ ⎝ ⎝ + 10⋅log⎛⎜ 1 + j⋅

⎝

Ref_9dB FrqPoint

⎞⎜+ 10⋅log⎛ 1 + j⋅ Ref_12dB ⎞⎜− 20⋅log(R) ⎜ FrqPoint ⎠ ⎠ ⎝

This VXCO SSB Phase Noise Plot represents the free running VCXO which is considerably cleaner than the 16 MHz clean upused. This is because the 16 MHz cleanup loop has a loop bandwidth of the order of 40 Hz, there after the phase noise follows the logic noise and Op-Amp noise profile of the other components in that loop. These values are not modeled here, giving an optimistic account of this loop’s output. This will reflect in a better than measured value in the main synthesizer output for this part of the phase noise profile. Hence, the actual phase noise profile of the signal at the sampling frequency is:-

⎛

Ref_Div ( FrqPoint)

⎝

10

Samp_Freq_Noise( FrqPoint) := 10⋅log ⎜10

+ 10

R_Div_Noise ( FrqPoint) ⎞ 10

⎜ ⎠

192

Appendix E

The effect of the R divider logic noise plateau becomes particuraly important when considering low sampling frequencies and very large VCO frequencies and hence multiplication of this reference noise. The clear distinction has to be made between the sampling frequency phase noise and the phase frequency logic noise. VCO Noise Calculation

VCONoise ( FrqPoint) := VCONoise_Plateau + 10⋅log⎛⎜ 1 + j⋅

⎝

10⋅ log⎛⎜ 1 + j⋅

⎝

VCO_9dB FrqPoint

VCO_3dB FrqPoint

⎞ + 10⋅ log⎛ 1 + j⋅ VCO_6dB ⎞ + ⎜ ⎜ ⎜ FrqPoint ⎠ ⎠ ⎝

⎞ ⎜ ⎠

Phase Noise at different offsets:-

⎛ 40 ⎞ ⎛ 50 ⎞ ⎜ 10 ⎜ ⎜ 10 ⎜ VCONoise ⎝ 10 ⎠ = −120.937 VCONoise ⎝ 10 ⎠ = −100.902 L(800 kHz)=

⎛ 30 ⎞ ⎜ 10 ⎜ VCONoise ⎝ 10 ⎠ = −78.789

⎛ 10⋅ log( 800⋅ k) ⎞ ⎜ ⎜ 10 VCONoise ⎝ 10 ⎠ = −138.957

Phase Detector Noise Calculations This first line calculates the frequency profile of the normalized phase detector noise.

⎛ ⎝

PD_Noise_Norm( FrqPoint) := Φn_Plateau + 10⋅log ⎜ 1 +

⎛ ⎝

Φ_9dB

+ 10⋅log ⎜ 1 +

FrqPoint

Φ_3dB FrqPoint

⎞ + 10⋅log ⎛ 1 + Φ_6dB ⎞ + ⎜ ⎜ ⎜ FrqPoint ⎠ ⎝ ⎠

⎞ + 10⋅log ⎛ 1 + Φ_12dB ⎞ ⎜ ⎜ ⎜ rqPoint ⎠ ⎠ ⎝

This line denormalizes this normalized phase detector noise profile.

) + 10⋅log⎛⎜ 1 + PD_Noise ( FrqPoint) := PD_Noise_Norm( FrqPoint) − 10⋅log( Fsamp) + 20⋅log( FmaxFmin ⋅ Phase Noise at 1 MHz offset:-

⎝

⎛ ⎜

PD_Noise⎝ 10

60 ⎞ 10

⎜ ⎠ = −99.499

Fsamp Φn3dB

⎞ ⎜ ⎠

193

Programs and Codes

VCO Divider Noise Calculation VCODiv_Noise ( FrqPoint ) := VCODiv_Plateau + 10⋅ log⎛⎜ 1 + j ⋅

VCODiv_Noise_3dB

⎝

FrqPoint

⎞ ⎜ ⎠

RF Amplifier Noise Calculation Some constants...

− 23 J/K

K ≡ 1.38066210 ⋅

T ≡ 273.14

K

B≡1

Hz

Calculation of Amplifier Noise RFAmp_Plateau := Amp_NF + 10⋅ log( K⋅ T⋅ B) + 30 − 3 − Power_in giving...

dBc/Hz

RFAmp_Plateau = −154.235

Hence TOTAL RF Noise of the VCO and the RF Amplifier is calculated... RFAmp_noise_3dB RFAmp_noise ( FrqPoint ) := RFAmp_Plateau + 10⋅log⎛⎜ 1 + j⋅ FrqPoint ⎝

⎞ ⎜ ⎠

RFAmp_noise ( FrqPoint ) ⎞ ⎛ VCONoise ( FrqPoint ) ⎜ ⎜ 10 10 VCOEffective_Noise( FrqPoint ) := 10⋅log⎝ 10 + 10 ⎠

194

Appendix E LOOP GAIN PARAMETER CALCULATIONS

N m := K pd :=

Fmin Fmax ⋅ Fsamp Fsamp

N m = 43.125

Fvco := Fmax ⋅ Fmin

Kφ

− 4 A/Rad

K pd = 3.183 × 10

DutyCycle

8

K v := Kvco⋅ 2⋅ π

K v = 6.283 × 10

Rad/V

R2 = 146.14

Filter Transfer Calculation ⎡

Fs ( FrqPoint) := if ⎢ATTEN

⎢ ⎣

1 + j⋅ ( 2⋅ π⋅ FrqPoint) ⋅ R2⋅ C2

1 ⎤ , ⎥ C1⋅ C2 ⎤ ⎡⎣j⋅ ( 2⋅ π⋅ FrqPoint) ⋅ C3⎤⎦ ⎡ ( ) ( ) ⎥ ⋅ + ⋅ ⋅ ⋅ ( C1 + C2 ) 1 j ⋅ 2 ⋅ π ⋅ FrqPoint ⋅ R2 j 2 ⋅ π ⋅ FrqPoint ⎡⎣ ⎤⎦ ⎢ ⎥ C1 + C2⎦ ⎦ ⎣

0,

1 ⎡ ⎤ ⎥ 1 ⎞⎡ C1⋅ C2 ⎤ ⎤ ⎥ ⎛ ⎡ ⎥ ⋅ ⎢ 1 + j⋅ ( 2⋅ π⋅ FrqPoint) ⋅ R2⋅ C2 + ⎡⎣ 1 + j⋅ ( 2⋅ π⋅ FrqPoint) ⋅ R3⋅ C3⎤⎦ ⋅ ( C1 + C2) ⋅ ⎜ ⎜⋅ ⎢ 1 + j⋅ ( 2⋅ π⋅ FrqPoint) ⋅ R2⋅ C1 + C2⎥ ⎥ ⎥ ⎦⎦ ⎦ ⎝ C3 ⎠ ⎣ ⎣ ⎣

195

Programs and Codes Calculating the Loop Gain Terms Aol ( FrqPoint ) := Fs ( FrqPoint ) ⋅

Kv

j⋅ ( 2⋅ π⋅ FrqPoint )

⋅ K pd

FB :=

1 Nm

−4

K pd = 3.183 × 10

196

Appendix E

Final Noise Summation Calculation VCO_LoopGain_dB (FrqPoint) := 20⋅log⎛⎜

1 ⎞ ⎜ ⎝ 1 + Aol(FrqPoint)⋅ FB ⎠

Ref_in_LoopGain_dB ( FrqPoint) := 20⋅log⎛⎜

Aol( FrqPoint) ⎞ ⎜ ⎝ 1 + Aol(FrqPoint)⋅ FB ⎠

PD_LoopGain_dB ( FrqPoint) := 20⋅log⎛⎜

Aol(FrqPoint)⋅ FB ⎞ ⎜ ⎝ 1 + Aol(FrqPoint)⋅ FB ⎠ Aol( FrqPoint) ⎞ VCODiv_LoopGain_dB ( FrqPoint) := 20⋅log⎛⎜ ⎜ ⎝ 1 + Aol(FrqPoint)⋅ FB ⎠ VCO_Loop_Noise_dB (FrqPoint) := VCO_LoopGain_dB ( FrqPoint) + VCOEffective_Noise ( FrqPoint) Samp_Freq_Loop_Noise_dB (FrqPoint) := Ref_in_LoopGain_dB( FrqPoint) + Samp_Freq_Noise(FrqPoint) VCODiv_Loop_Noise_dB( FrqPoint) := VCODiv_LoopGain_dB( FrqPoint) + VCODiv_Noise (FrqPoint) PD_Loop_Noise_dB( FrqPoint) := PD_Noise ( FrqPoint) + PD_LoopGain_dB(FrqPoint)

Samp_Freq_Loop_Noise_dB ( FrqPoint) ⎞ ⎛ VCO_Loop_Noise_dB ( FrqPoint) ⎜ ⎜ 10 10 Total_Noise(FrqPoint) := 10⋅log⎝ 10 + 10 + ⎠ PD_Loop_Noise_dB( FrqPoint) VCODiv_Loop_Noise_dB( FrqPoint)⎞ ⎛ ⎜ ⎜ 10 10 + 10⋅log⎝ + 10 + 10 ⎠

⎛ 10 ⋅log ( 1.7⋅ k) ⎞ ⎜ ⎜ 10 Total_Noise ⎝ 10 ⎠ = −96 Small_Angle ( FrqPoint) := −10⋅log( FrqPoint) − 30 Marker1:=

1

log( Marker1)

Marker1:= 10

577⋅ µ

L(1.5 kHz)=

Phase Noise at:-

⎛ ⎜

10

Filter Values:-

Marker2:= 10

L(800 kHz)=

Phase Noise at:-

10⋅ log(1.5⋅ k) ⎞

Total_Noise⎝ 10

log( Marker2)

Marker2:= 135⋅k

⎛ ⎜ ⎜ dBc/Hz = − 95.647 Total_Noise ⎠ ⎝ 10

10⋅log (800⋅k) ⎞

⎜ ⎠ = −125.955

10

dBc/Hz

Loop Gain Values:-

−9

8

C1 = 3.665 × 10

F

−8

C2 = 3.635 × 10

K vco = 1 × 10

Hz/V

Kφ = 0.002

Amps

7 Hz

F

Fsamp = 4 × 10

9

Fvco = 1.725 × 10

R2 = 146.14

Ohms

R3 = 23.342

Ohms

Phase Margin:-

F

LBW = 1 × 10 φp = 56

− 10

Hz

Loop Filter Dynamic Values:-

5

C3 = 3.665 × 10

Hz Degrees

Programs and Codes

197

198

2

Appendix E

MATLABTM PROGRAM USED FOR THE SIMULATIONS OF THE FRACTIONAL-N PLL NOISE SPECTRUM % Definition of the PLL elements f_max=1898.208/90; f_min=1880.928/92; f_s=20e06; N=91; V_max=5; V_min=0; % Wanted T_L approx 2*pi/Wn <= 30 us % <=> Wn >= 2*pi/ 30 us ~ 209.439 krads T_L=30e-06; Wn=2*pi/T_L; xi=0.7; K_pfd=(V_max-V_min)/(4*pi); % gain for classic PFD with CP K_vco=2*pi*20e06/(V_max-V_min); K_filt=1; K_t=K_vco*K_pfd*K_filt; K=K_t/N; K_gain=1000; % K> 1.3614 = Wn*N/(2*xi*K_t) insure that Wz >=0 K=K*K_gain; Wp= (Wn^2)/K; % Wp/(Wn)^2 =1/K; % xi = 0.5* (Wp/Wn + Wn/Wz) % 1/Wz= 2 *xi/Wn – 1/ K if (–1/K +2 *xi/Wn) >=0 Wz= 1/(–1/K + 2 *xi/Wn); else Wz=0; sprintf('Increase the loop gain or the lock in time') end

Programs and Codes dsm_order=5; f_ref=20e06; % Noise contribution in the PLL f_min=1; % 10^fmin=10 Hz f_max=7; % 10^fmax=10 MHz N_f=100; % Number of points f=logspace(f_min,f_max,N_f); s=j*2*pi*f; % Definition of the close-loop transfer function H and 1-H Hcl_2nd_order=Wn^2*[s/Wz + 1]./[(s.^2) + 2*xi*Wn*s+ Wn^2]; % Closed-Loop Gain of 2nd order Gol_2nd_order=Hcl_2nd_order./(1-Hcl_2nd_order); % Open-Loop Gain of 2nd order Wp2=10*Wp; % Further suppression at 10 times the frequency Gol_3rd_order=Gol_2nd_order./(1+s/Wp2); % Open-Loop Gain of Loop Filter + Suppression Filter Hcl_3rd_order=Gol_3rd_order./(1+Gol_3rd_order); % Closed-Loop Gain of 3rd Order Hcl=Hcl_3rd_order; % Closed-Loop Gain of 3rd Order one_min_H_cl=1-Hcl; % Complement of Closed-Loop Gain of 3rd order %semilogx(f,10*log10(abs(Hcl))) % contribution from the VCO f_p1=50; f_z1=10^(6.5); s_phi_vco1_f=100./((f/f_p1).^3).*((f/f_z1).^3+1); % article micro and RF nov 1994 % does not qualify as the phase noise is too high at high frequency % ... and S_phi_out = S_phi_vco at high frequency !! s_phi_vco2_f= 10^-14.5+10^-1.5 ./f.^3+10^-1.5 ./f.^2; % formula p 171 poly goldberg 1998 s_phi_vco_f=s_phi_vco2_f; figure(1)

199

200

Appendix E

subplot(2,1,1),semilogx(f,10*log10(s_phi_vco1_f),'b',f,10* log10(s_phi_vco2_f),'r') title('different Phase noise characteristic for the crystal oscillator') % contribution from the crystal frequency reference % p 93 poly Goldberg, state of the art Xtal s_phi_state1(1)=10^(−10); s_phi_state1(2)=10^(−13); s_phi_state1(3)=10^(−14.3); s_phi_state1(4)=10^(−15.8); s_phi_state1(5)=10^(−16.4); s_phi_state1(6)=10^(−17); s_phi_state1(7)=10^(−17); f1=logspace(0,7,8); s_phi_in1_f=s_phi_state1(1)./(f.^3).*((f/10) + 1).* ((f/100) + 1).*((f/5e03) + 1); % s_phi_in1_f is the linear approximation from the vco p93 % Microwave and RF 1994 s_phi_in2_f=1e-04./((f).^3).*((f/10^3).^2+1).*((f/(4*10^4))+1); % (Pretty noise at low frequencies !!) % too ideal xtal below !! (from example) s_phi_in3_f=1e-011./((f).^3).*((f/10).^2 + 1).*((f/100) + 1) ; % phase noise of Hy-Q oscillator s_phi_in4_f=10^(-5.3)./(f.^4).*((f.^3/5e04) + 1).*((f/8e03) + 1); figure(1) subplot(2,1,2),semilogx(f,10*log10(s_phi_in1_f),'b',... f,10*log10(s_phi_in2_f),'r',f,10*log10(s_phi_in3_f),'g',f,10* log10(s_phi_in4_f),'m') title('different Phase noise characteristic for the crystal oscillator') s_phi_in_f=s_phi_in1_f; % s_phi_in3 is too ideal and s_phi_in2 is too bad (too much noise % with in the loop bandwidth, one should achieve −80 dBc for DECT % contribution from the loop filter s_phi_loop=10^(-11)./((f).^3).*((f/(30)).^2+1).*((f/1e02)+1); % contribution from the frequency detector

Programs and Codes % sphi= 10^{−10.6 +/–0.3}/f −22 dB s_phi_pd_f=10^(−2.2)*10^(−10.6)./f; %s_phi_pd=10^(−22/10)* tf([10^(−10.6)],[1 0]); % contribution from the frequency divider % Integer-N case s_phidn=(10^(−14.7)./f + 10^(−16.5)); s_phi_dn_int=s_phidn; %s_phi_dn_int=s_phidn + s_phidn.*abs(Hcl).^2; % plot of the phase noise source before filtering, N integer figure(2) subplot(2,1,1),semilogx(f,10*log10(s_phi_dn_int),'b',f,10* log10(s_phi_pd_f),'r',... f,10*log10(s_phi_loop),'g',... f,10*log10(s_phi_in_f),'m',f,10*log10(s_phi_vco_f),'c'); grid on set(gcf,'DefaultTextColor','k') xlabel('frequency (Hz)') ylabel('S_\Phi (dB)') title('Phase noise source before filtering N integer') set(gcf,'DefaultTextColor','c') text(1e03,-70,'S_\Phi_{ vco}') set(gcf,'DefaultTextColor','m') text(1e01,-120,'S_\Phi_{ ref}') set(gcf,'DefaultTextColor','b') text(1e06,-170,'S_\Phi_{ dn}') set(gcf,'DefaultTextColor','r') text(1e03,-150,'S_\Phi_{ pfd}') set(gcf,'DefaultTextColor','g') text(1e06,-154,'S_\Phi_{ loop filter}') set(gcf,'DefaultTextColor','k') % Fractional-N case % depends on the variable dsm_order s_phi_switch=(2*pi)^2/(12*f_ref).*abs(2*sin(f*pi/f_ref)).^(2* (dsm_order-1)); s_phi_dn_frac=s_phidn+s_phi_switch;

201

202

Appendix E

% plot of the phase noise source before filtering , N Fractional figure(2) subplot(2,1,2),semilogx(f,10*log10(s_phi_dn_frac),'b',f, 10*log10(s_phi_pd_f),'r',... f,10*log10(s_phi_loop),'g',... f,10*log10(s_phi_in_f),'m',f,10*log10(s_phi_vco_f),'c'); grid on set(gcf,'DefaultTextColor','k') xlabel('frequency (Hz)') ylabel('S_\Phi (dB)') title('Phase noise source before filtering, N frac') set(gcf,'DefaultTextColor','c') text(1e03,-70,'S_\Phi_{ vco}') set(gcf,'DefaultTextColor','m') text(1e01,-120,'S_\Phi_{ ref}') set(gcf,'DefaultTextColor','b') text(1e06,-170,'S_\Phi_{ dn}') set(gcf,'DefaultTextColor','r') text(1e03,-150,'S_\Phi_{ pfd}') set(gcf,'DefaultTextColor','g') text(1e06,-154,'S_\Phi_{ loop filter}') set(gcf,'DefaultTextColor','k') % Summation of all the contribution S_phi_inband_int=(s_phi_in_f+s_phi_dn_int+s_phi_pd_f)*N^2.* (abs(Hcl)).^2; % multiply by |H|^2 S_phi_out=(s_phi_vco_f+s_phi_loop./f.^2).*(abs(one_min_H_cl)).^2; % multiply by |1-H|^2 S_phi_tot=S_phi_inband_int+S_phi_out; figure(3) subplot(2,1,1),semilogx(f,10*log10(s_phi_dn_int*N^2. *(abs(Hcl)).^2),'b',... f,10*log10(s_phi_pd_f*N^2.*(abs(Hcl)).^2),'r',... f,10*log10(s_phi_loop./f.^2 .*(abs(one_min_H_cl)).^2),'g',... f,10*log10(s_phi_in_f*N^2.*(abs(Hcl)).^2),'m',... f,10*log10(s_phi_vco_f.*(abs(one_min_H_cl)).^2),'c', f,10*log10(S_phi_tot),'b-.'); grid on set(gcf,'DefaultTextColor','k')

Programs and Codes xlabel('frequency (Hz)') ylabel('S_\Phi (dB)') title('Phase noise source after filtering') set(gcf,'DefaultTextColor','c') text(1e06,-130,'(1-H)^2 S_\Phi_{ vco}') set(gcf,'DefaultTextColor','m') text(1e01,-70,'N^2 H^2 S_\Phi_{ ref}') set(gcf,'DefaultTextColor','b') text(1e07,-165,'N^2 H^2 S_\Phi_{ dn}') set(gcf,'DefaultTextColor','r') text(1e05,-200,'N^2 H^2 S_\Phi_{ pfd}') set(gcf,'DefaultTextColor','g') text(1e02,-250,'(1-H)^2 f^{-2} S_\Phi_{ loop filter}') set(gcf,'DefaultTextColor','k') % DECT phase noise mask N_l=4*N_f; f_l=logspace(log10(1.2e06),f_max,N_l); dect_mask(1:2)=10^-(8); dect_mask(3:4)=10^-(8.5); % -85 dBc @ 100 kHz decal=4; dect_mask(decal+1:decal+174)=10^(−9.5); f_l(1) =1.2 MHz dect_mask(decal+175:decal+257)=10^(−11.7); % f_l(174) ~3 MHz dect_mask(decal+258:decal+N_l)=10^(−13.5); % f_l(257) 4.67 MH < 4.7 MHz figure(4) subplot(2,1,1),semilogx(f,10*log10(S_phi_tot),'b',f,10* log10(s_phi_vco_f),'c',... [10,1e05-1,1e05,1.2000e+06,f_l],10*log10(dect_mask),'k') title('Overall phase noise and phase mask'); grid % fractional case S_phi_inband_frac=(s_phi_in_f+s_phi_dn_frac+s_phi_pd_f)*N^2.* (abs(Hcl)).^2; S_phi_tot_frac=S_phi_inband_frac+S_phi_out;

203

204

Appendix E

figure(3) subplot(2,1,2),semilogx(f,10*log10(s_phi_dn_frac*N^2. *(abs(Hcl)).^2),'b',... f,10*log10(s_phi_pd_f*N^2.*(abs(Hcl)).^2),'r',... f,10*log10(s_phi_loop./f.^2 .*(abs(one_min_H_cl)).^2),'g',... f,10*log10(s_phi_in_f*N^2.*(abs(Hcl)).^2),'m',... f,10*log10(s_phi_vco_f.*(abs(one_min_H_cl)).^2),'c', f,10*log10(S_phi_tot_frac),'b-.'); grid on set(gcf,'DefaultTextColor','k') xlabel('frequency (Hz)') ylabel('S_\Phi (dB)') title('Phase noise source after filtering') set(gcf,'DefaultTextColor','c') text(1e06,-130,'(1-H)^2 S_\Phi_{ vco}') set(gcf,'DefaultTextColor','m') text(1e01,-70,'N^2 H^2 S_\Phi_{ ref}') set(gcf,'DefaultTextColor','b') text(1e07,-165,'N^2 H^2 S_\Phi_{ dn}') set(gcf,'DefaultTextColor','r') text(1e05,-200,'N^2 H^2 S_\Phi_{ pfd}') set(gcf,'DefaultTextColor','g') text(1e02,-250,'(1-H)^2 f^{-2} S_\Phi_{ loop filter}') set(gcf,'DefaultTextColor','k') figure(4) subplot(2,1,2),semilogx(f,10*log10(S_phi_tot_frac),'b',f,10* log10(s_phi_vco_f),'c',... [10,1e05-1,1e05,1.2000e+06,f_l],10*log10(dect_mask),'k') title('Overall phase noise and phase mask'); grid The following M-file reads in the captured output of the simulated HDL of the delta — sigma block as applied to the MMD. clear all; Ts=25; %sampling period is 24nS fid=fopen('HDL_deltasigma_out_data70p5.dat','r'); out=fscanf(fid,'%i %i %i\n',[3 inf]); fclose(fid);

Programs and Codes

205

%M=1000044; %M=length(out(1,:)); frac=out(2,:); int=out(1,:); inst_div=out(3,:); %M=length(frac); M=2^19; y2=inst_div; y2=y2(20001:M); mean(y2) M=length(y2); fs=1/(Ts*1e-9); f=(1:M)*fs/M/1e3; win=hanning(M); win=win'; a2=abs(fft(y2.*win)); a2=(a2.*a2)/M; a2=a2(1:M/2); f=f(1:M/2); a2=10*log10(a2/max(a2)); figure(1),plot(f,a2); xlabel('FREQUENCY (KHz)'); ylabel('MAGNITUDE (dB)'); axis([min(f) max(f) -350 0]),grid; title('Output spectrum of Nemo Delta —Sigma Modulator (VHDL)'); delta_f=fs/M; place=floor(1000e3/delta_f); a3=a2(1:place); f3=f(1:place); figure(2),plot(f3,a3); xlabel('FREQUENCY (KHz)'); ylabel('POWER SPECTRUM (dB)'); axis([min(f3) max(f3) -350 0]),grid; title('Baseband output spectrum of Nemo Delta — Sigma Modulator (VHDL)'); fB=200e03; % The cutoff frequency = fpass fstop=2000e03; % The stop frequency rp=3; % pass band attenuation in dB rs=30; % stop band attenuation in dB wp=fB*2/fs; % pass normalized frequency ws=fstop*2/fs; % stop normalized freuqnecy [filt_order,wn]=buttord(wp,ws,rp,rs); [b1,a1]=butter(filt_order,wn); % calculate coefficents of butterworth filter filtered=filter(b1,a1,y2);

206

Appendix E

a4=abs(fft(filtered.*win)); a4=(a4.*a4)/M; a4=a4(1:M/2); a4=10*log10(a4/max(a4)); figure(3),plot(f,a4); xlabel('FREQUENCY (KHz)'); ylabel('MAGNITUDE (dB)'); axis([min(f) max(f) -350 0]),grid; title('Filtered spectrum of MASH output'); % to be able to print the results you have to decimate the results before presenting a5=a2(1:64:length(a2)); f5=f(1:64:length(f)); figure(4),plot(f5,a5); xlabel('FREQUENCY (KHz)'); ylabel('MAGNITUDE (dB)'); axis([min(f5) max(f5) -350 0]),grid; title('Decimated output spectrum of MASH'); a6=a3(1:64:length(a3)); f6=f3(1:64:length(f3)); figure(5),plot(f6,a6); xlabel('FREQUENCY (KHz)'); ylabel('MAGNITUDE (dB)'); axis([min(f6) max(f6) -350 0]),grid; title('Decimated baseband output spectrum of MASH'); a7=a4(1:64:length(a4)); figure(6),plot(f5,a7); xlabel('FREQUENCY (KHz)'); ylabel('MAGNITUDE (dB)'); axis([min(f5) max(f5) -350 0]),grid; title('Decimated filtered spectrum of MASH output');

Index

DC offset compensation 13 dead zone 48, 71, 76, 77, 132, 138 delta–sigma modulators ix, 2, 67 delta–sigma noise 50, 58, 98 design methodology ix, x, 155 differential to single-ended conversion 12 differentiators 95 digital accumulator 34, 35, 36, 90 digital domain 13, 133 digital-to-analog converters 12, 116 direct frequency synthesis 1, 131 direct modulation 133 direct-sequence spread spectrum 8 direct conversion 107, 109 dithering 57, 98, 132 divide-by-2 13, 107, 170 dividers 19, 36, 62, 89, 131, 143, 165 feedback 19 reference 19 dual-type flip-flop 71 dual-modulus prescaler 19, 35 duty cycle 30, 47, 52, 108

2’s complement 96 802.11 a, b and g 3, 10 accumulator-based implementation 95 adaptive band switching 2, 111, 132 adaptive charge pump 2, 111, 113, 116, 117, 128, 132 adjacent channel rejection 13 attenuation 12, 205 automatically calibrated 12 average divide ratio 2, 45 balanced to unbalanced transformer 12 band switching 103, 126, 127, 128 bandgap reference 119 bandwidth 8, 32, 47, 105, 117, 133, 147, 158, 202 base band 10, 38 behavioral modelling 2, 45, 47 Bluetooth 7, 14 calibration 117, 122, 125, 128 calibration algorithm 125 carrier frequency 149, 153, 163 carry look ahead 93 channel selection 12 charge pump 17, 71, 113, 138 closed loop 1 CML see current mode logic CMOS process 2, 67, 73, 102, 113, 131 complementary code keying 7 CP linearization 127 current mismatches 73 current-mode logic 89 cutoff frequency 12

error cancellation algorithm 95 error vector magnitude 42 extra filtering 13 feed-through 18, 73, 139 flicker noise 13, 29, 42, 86 FM demodulated 105 frequency bands 13, 131, 169 frequency domain 47, 55 frequency pulling 1 frequency synthesizers

207

Index

208 fractional-N 1, 34 integer-N 1 front-end signal paths 12 GSM system 159 guard rings 108 impulse sensitivity functions 30 in-band phase noise 38, 72, 170 indirect frequency synthesis 3 integer multiples 60, 99 inter-subcarrier interferences 107 IQ filters 12 IQ image rejection mixer 86 limit cycles 57, 59, 60, 97, 99 linear time invariant xi local oscillator 11, 53, 105, 131 local oscillators 1, 11, 131 locking time 26, 107, 128 lookup table 13 loop filter 20 loop filter design 15, 42, 131 low-noise amplifier 10 MASH-1-1-1 36, 67, 90, 131 mismatch cancellation 75, 76 mixers 5, 13, 107, 133 mobile phones 7 motional capacitance 143 motional inductance 143 negative-edge triggered 108 NMOS 74, 79, 86 noise contributions 1, 15, 27, 40, 131, 147 noise shaping 90, 93, 108, 113 noise spectrum 29, 57, 97, 161, 162, 198 nonoverlapping 10 normal distribution 50 OFDM 7, 9, 10 offset current 62, 78, 79, 81 offset frequency 50, 153, 158, 160, 161 open loop 1, 42, 45, 53 PFD/CP linearity 2, 45, 47, 49, 52 phase-domain model 2, 45, 46, 47, 50, 53 phase error 16, 35, 78, 105, 157 phase margin 21 phase noise 15, 27, 41 phase-frequency detector 16, 28, 71, 135, 65 phase-locked loop 15, 27, 67, 131 pipelined adder 93 PMOS 74, 75, 79, 82, 86

power amplifier 10, 133 power spectral density 29, 38, 50, 55, 149 process tolerances 117 pseudorandom binary sequence 165 QPSK system 42 quadrature-phase signals 86 quantization noise 50, 51, 55, 64, 65, 132, 133, 172 rms phase error 41 range coverage 7 receiver ix, 1, 10, 11, 12, 13, 42, 70, 102, 105, 107, 131 reference frequency 50, 53, 60, 71, 77, 78, 99, 107, 140, 165, 176 rejection ratio 13 RF front-end circuits 13 RFIC transceivers 7 sampling frequency 31, 34, 35, 39, 40, 71, 125, 165 short-channel transistors 154 single-sideband phase noise 158, 163 single subcarrier 10 slew rate 118 spectral densities 52, 99 spurious performance x, 3 spurious signals 35, 57, 163 substrate noise rejection 18 superimposed noise vectors 162 tank amplitude 149, 150, 152, 153 time to digital converter xii, 134 time domain 26, 36, 55, 90 transceiver 1, 7, 69, 102, 109, 131 transfer functions 1 transmitter ix, 1, 5, 12, 13, 14, 70, 102, 105, 107, 131, 133, 176 tunable varactors 86 tuning curves 51, 88 tuning voltage 18, 49, 122, 140 unlicensed band 10 upconverted differential signal 12 variable-gain amplifiers 10 VCO 18, 40, 86, 132, 162, 201 wireless cellular devices 7 wireless LAN 7, 8, 10, 11, 13, 39, 45, 69, 70, 88, 107, 111, 127, 131, 132 wireless local area networks 1, 7 ∆–Σ 36, 51, 69, 90, 113, 128, 169 ∆–Σ:modulators 50